LMFDB - Embedded newform 78.3.c.a.53.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [78,3,Mod(53,78)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(78, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("78.53");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 78 = 2 \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	78.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.12534606201$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.16845963264.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 6x^{6} + 15x^{4} + 8x^{2} + 9 $ x^8 - 6x^6 + 15x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			53.7
Root			$-1.95007 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	78.53
Dual form			78.3.c.a.53.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} +(1.18087 - 2.75782i) q^{3} -2.00000 q^{4} -5.25985i q^{5} +(3.90014 + 1.67000i) q^{6} +8.86743 q^{7} -2.82843i q^{8} +(-6.21110 - 6.51323i) q^{9} +O(q^{10})$	\(q+1.41421i q^{2} +(1.18087 - 2.75782i) q^{3} -2.00000 q^{4} -5.25985i q^{5} +(3.90014 + 1.67000i) q^{6} +8.86743 q^{7} -2.82843i q^{8} +(-6.21110 - 6.51323i) q^{9} +7.43855 q^{10} +6.35241i q^{11} +(-2.36174 + 5.51563i) q^{12} +3.60555 q^{13} +12.5404i q^{14} +(-14.5057 - 6.21118i) q^{15} +4.00000 q^{16} +22.6898i q^{17} +(9.21110 - 8.78383i) q^{18} -9.32232 q^{19} +10.5197i q^{20} +(10.4713 - 24.4547i) q^{21} -8.98366 q^{22} -19.1973i q^{23} +(-7.80028 - 3.34000i) q^{24} -2.66599 q^{25} +5.09902i q^{26} +(-25.2968 + 9.43781i) q^{27} -17.7349 q^{28} +43.0325i q^{29} +(8.78394 - 20.5141i) q^{30} -53.0245 q^{31} +5.65685i q^{32} +(17.5188 + 7.50135i) q^{33} -32.0882 q^{34} -46.6413i q^{35} +(12.4222 + 13.0265i) q^{36} +30.3400 q^{37} -13.1837i q^{38} +(4.25768 - 9.94345i) q^{39} -14.8771 q^{40} -9.56003i q^{41} +(34.5842 + 14.8086i) q^{42} +67.2373 q^{43} -12.7048i q^{44} +(-34.2586 + 32.6694i) q^{45} +27.1491 q^{46} +50.3135i q^{47} +(4.72347 - 11.0313i) q^{48} +29.6313 q^{49} -3.77028i q^{50} +(62.5743 + 26.7936i) q^{51} -7.21110 q^{52} +41.6953i q^{53} +(-13.3471 - 35.7751i) q^{54} +33.4127 q^{55} -25.0809i q^{56} +(-11.0084 + 25.7092i) q^{57} -60.8572 q^{58} -43.3630i q^{59} +(29.0114 + 12.4224i) q^{60} +106.536 q^{61} -74.9880i q^{62} +(-55.0765 - 57.7556i) q^{63} -8.00000 q^{64} -18.9646i q^{65} +(-10.6085 + 24.7753i) q^{66} -45.4233 q^{67} -45.3796i q^{68} +(-52.9426 - 22.6695i) q^{69} +65.9608 q^{70} -48.0466i q^{71} +(-18.4222 + 17.5677i) q^{72} -123.970 q^{73} +42.9072i q^{74} +(-3.14818 + 7.35232i) q^{75} +18.6446 q^{76} +56.3295i q^{77} +(14.0622 + 6.02127i) q^{78} -94.8590 q^{79} -21.0394i q^{80} +(-3.84441 + 80.9087i) q^{81} +13.5199 q^{82} -108.062i q^{83} +(-20.9425 + 48.9095i) q^{84} +119.345 q^{85} +95.0878i q^{86} +(118.676 + 50.8157i) q^{87} +17.9673 q^{88} -42.2586i q^{89} +(-46.2016 - 48.4490i) q^{90} +31.9720 q^{91} +38.3946i q^{92} +(-62.6149 + 146.232i) q^{93} -71.1541 q^{94} +49.0340i q^{95} +(15.6006 + 6.68000i) q^{96} -88.5132 q^{97} +41.9050i q^{98} +(41.3747 - 39.4554i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} - 8 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q - 16 * q^4 - 8 * q^7 + 8 * q^9	$ 8 q - 16 q^{4} - 8 q^{7} + 8 q^{9} + 16 q^{10} - 56 q^{15} + 32 q^{16} + 16 q^{18} - 24 q^{19} + 88 q^{21} + 8 q^{25} + 16 q^{28} - 64 q^{30} - 72 q^{31} - 56 q^{33} - 112 q^{34} - 16 q^{36} + 96 q^{37} - 32 q^{40} + 80 q^{42} + 208 q^{43} - 16 q^{45} + 32 q^{46} - 24 q^{49} + 112 q^{51} - 176 q^{55} + 40 q^{57} + 176 q^{58} + 112 q^{60} - 272 q^{61} - 216 q^{63} - 64 q^{64} - 64 q^{66} + 264 q^{67} - 240 q^{69} - 32 q^{70} - 32 q^{72} - 416 q^{73} + 128 q^{75} + 48 q^{76} + 160 q^{79} + 200 q^{81} + 176 q^{82} - 176 q^{84} + 464 q^{85} + 384 q^{87} + 16 q^{90} + 104 q^{91} - 264 q^{93} - 384 q^{94} - 16 q^{97} + 416 q^{99}+O(q^{100}) $ 8 * q - 16 * q^4 - 8 * q^7 + 8 * q^9 + 16 * q^10 - 56 * q^15 + 32 * q^16 + 16 * q^18 - 24 * q^19 + 88 * q^21 + 8 * q^25 + 16 * q^28 - 64 * q^30 - 72 * q^31 - 56 * q^33 - 112 * q^34 - 16 * q^36 + 96 * q^37 - 32 * q^40 + 80 * q^42 + 208 * q^43 - 16 * q^45 + 32 * q^46 - 24 * q^49 + 112 * q^51 - 176 * q^55 + 40 * q^57 + 176 * q^58 + 112 * q^60 - 272 * q^61 - 216 * q^63 - 64 * q^64 - 64 * q^66 + 264 * q^67 - 240 * q^69 - 32 * q^70 - 32 * q^72 - 416 * q^73 + 128 * q^75 + 48 * q^76 + 160 * q^79 + 200 * q^81 + 176 * q^82 - 176 * q^84 + 464 * q^85 + 384 * q^87 + 16 * q^90 + 104 * q^91 - 264 * q^93 - 384 * q^94 - 16 * q^97 + 416 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/78\mathbb{Z}\right)^\times$.

$n$	$53$	$67$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.41421i			0.707107i
$2$			1.41421i			0.707107i
$3$	1.18087	−	2.75782i	0.393623	−	0.919272i
$3$	1.18087	−	2.75782i	0.393623	−	0.919272i
$4$	−2.00000			−0.500000
$5$		−	5.25985i		−	1.05197i	−0.850494	−	0.525985i	$-0.823697\pi$
$5$		−	5.25985i		−	1.05197i	0.850494	−	0.525985i	$-0.176303\pi$
$6$	3.90014	+	1.67000i	0.650024	+	0.278333i
$7$	8.86743			1.26678			0.633388	−	0.773835i	$-0.281664\pi$
$7$	8.86743			1.26678			0.633388	+	0.773835i	$0.281664\pi$
$8$		−	2.82843i		−	0.353553i
$9$	−6.21110	−	6.51323i	−0.690123	−	0.723693i
$10$	7.43855			0.743855
$11$			6.35241i			0.577491i	0.957406	+	0.288746i	$0.0932382\pi$
$11$			6.35241i			0.577491i	−0.957406	+	0.288746i	$0.906762\pi$
$12$	−2.36174	+	5.51563i	−0.196811	+	0.459636i
$13$	3.60555			0.277350
$13$	3.60555			0.277350
$14$			12.5404i			0.895746i
$15$	−14.5057	−	6.21118i	−0.967046	−	0.414079i
$16$	4.00000			0.250000
$17$			22.6898i			1.33469i	0.744747	+	0.667347i	$0.232570\pi$
$17$			22.6898i			1.33469i	−0.744747	+	0.667347i	$0.767430\pi$
$18$	9.21110	−	8.78383i	0.511728	−	0.487990i
$19$	−9.32232			−0.490648			−0.245324	−	0.969441i	$-0.578894\pi$
$19$	−9.32232			−0.490648			−0.245324	+	0.969441i	$0.578894\pi$
$20$			10.5197i			0.525985i
$21$	10.4713	−	24.4547i	0.498631	−	1.16451i
$22$	−8.98366			−0.408348
$23$		−	19.1973i		−	0.834664i	−0.908754	−	0.417332i	$-0.862965\pi$
$23$		−	19.1973i		−	0.834664i	0.908754	−	0.417332i	$-0.137035\pi$
$24$	−7.80028	−	3.34000i	−0.325012	−	0.139167i
$25$	−2.66599			−0.106640
$26$			5.09902i			0.196116i
$27$	−25.2968	+	9.43781i	−0.936918	+	0.349549i
$28$	−17.7349			−0.633388
$29$			43.0325i			1.48388i	0.670466	+	0.741940i	$0.266094\pi$
$29$			43.0325i			1.48388i	−0.670466	+	0.741940i	$0.733906\pi$
$30$	8.78394	−	20.5141i	0.292798	−	0.683805i
$31$	−53.0245			−1.71047			−0.855234	−	0.518243i	$-0.826586\pi$
$31$	−53.0245			−1.71047			−0.855234	+	0.518243i	$0.826586\pi$
$32$			5.65685i			0.176777i
$33$	17.5188	+	7.50135i	0.530872	+	0.227314i
$34$	−32.0882			−0.943770
$35$		−	46.6413i		−	1.33261i
$36$	12.4222	+	13.0265i	0.345061	+	0.361846i
$37$	30.3400			0.820000			0.410000	−	0.912086i	$-0.365529\pi$
$37$	30.3400			0.820000			0.410000	+	0.912086i	$0.365529\pi$
$38$		−	13.1837i		−	0.346941i
$39$	4.25768	−	9.94345i	0.109171	−	0.254960i
$40$	−14.8771			−0.371927
$41$		−	9.56003i		−	0.233171i	−0.993181	−	0.116586i	$-0.962805\pi$
$41$		−	9.56003i		−	0.233171i	0.993181	−	0.116586i	$-0.0371950\pi$
$42$	34.5842	+	14.8086i	0.823434	+	0.352586i
$43$	67.2373			1.56366			0.781829	−	0.623494i	$-0.214287\pi$
$43$	67.2373			1.56366			0.781829	+	0.623494i	$0.214287\pi$
$44$		−	12.7048i		−	0.288746i
$45$	−34.2586	+	32.6694i	−0.761302	+	0.725988i
$46$	27.1491			0.590197
$47$			50.3135i			1.07050i	0.844694	+	0.535250i	$0.179783\pi$
$47$			50.3135i			1.07050i	−0.844694	+	0.535250i	$0.820217\pi$
$48$	4.72347	−	11.0313i	0.0984056	−	0.229818i
$49$	29.6313			0.604720
$50$		−	3.77028i		−	0.0754056i
$51$	62.5743	+	26.7936i	1.22695	+	0.525365i
$52$	−7.21110			−0.138675
$53$			41.6953i			0.786705i	0.919388	+	0.393352i	$0.128685\pi$
$53$			41.6953i			0.786705i	−0.919388	+	0.393352i	$0.871315\pi$
$54$	−13.3471	−	35.7751i	−0.247168	−	0.662501i
$55$	33.4127			0.607503
$56$		−	25.0809i		−	0.447873i
$57$	−11.0084	+	25.7092i	−0.193130	+	0.451039i
$58$	−60.8572			−1.04926
$59$		−	43.3630i		−	0.734965i	−0.930030	−	0.367483i	$-0.880220\pi$
$59$		−	43.3630i		−	0.734965i	0.930030	−	0.367483i	$-0.119780\pi$
$60$	29.0114	+	12.4224i	0.483523	+	0.207039i
$61$	106.536			1.74649			0.873246	−	0.487280i	$-0.162011\pi$
$61$	106.536			1.74649			0.873246	+	0.487280i	$0.162011\pi$
$62$		−	74.9880i		−	1.20948i
$63$	−55.0765	−	57.7556i	−0.874230	−	0.916756i
$64$	−8.00000			−0.125000
$65$		−	18.9646i		−	0.291764i
$66$	−10.6085	+	24.7753i	−0.160735	+	0.375383i
$67$	−45.4233			−0.677960			−0.338980	−	0.940794i	$-0.610082\pi$
$67$	−45.4233			−0.677960			−0.338980	+	0.940794i	$0.610082\pi$
$68$		−	45.3796i		−	0.667347i
$69$	−52.9426	−	22.6695i	−0.767284	−	0.328543i
$70$	65.9608			0.942297
$71$		−	48.0466i		−	0.676712i	−0.941018	−	0.338356i	$-0.890129\pi$
$71$		−	48.0466i		−	0.676712i	0.941018	−	0.338356i	$-0.109871\pi$
$72$	−18.4222	+	17.5677i	−0.255864	+	0.243995i
$73$	−123.970			−1.69822			−0.849110	−	0.528216i	$-0.822861\pi$
$73$	−123.970			−1.69822			−0.849110	+	0.528216i	$0.822861\pi$
$74$			42.9072i			0.579827i
$75$	−3.14818	+	7.35232i	−0.0419758	+	0.0980309i
$76$	18.6446			0.245324
$77$			56.3295i			0.731552i
$78$	14.0622	+	6.02127i	0.180284	+	0.0771957i
$79$	−94.8590			−1.20075			−0.600373	−	0.799720i	$-0.704981\pi$
$79$	−94.8590			−1.20075			−0.600373	+	0.799720i	$0.704981\pi$
$80$		−	21.0394i		−	0.262992i
$81$	−3.84441	+	80.9087i	−0.0474619	+	0.998873i
$82$	13.5199			0.164877
$83$		−	108.062i		−	1.30195i	−0.759098	−	0.650976i	$-0.774360\pi$
$83$		−	108.062i		−	1.30195i	0.759098	−	0.650976i	$-0.225640\pi$
$84$	−20.9425	+	48.9095i	−0.249316	+	0.582256i
$85$	119.345			1.40406
$86$			95.0878i			1.10567i
$87$	118.676	+	50.8157i	1.36409	+	0.584089i
$88$	17.9673			0.204174
$89$		−	42.2586i		−	0.474816i	−0.971410	−	0.237408i	$-0.923702\pi$
$89$		−	42.2586i		−	0.474816i	0.971410	−	0.237408i	$-0.0762978\pi$
$90$	−46.2016	−	48.4490i	−0.513351	−	0.538322i
$91$	31.9720			0.351340
$92$			38.3946i			0.417332i
$93$	−62.6149	+	146.232i	−0.673279	+	1.57239i
$94$	−71.1541			−0.756958
$95$			49.0340i			0.516147i
$96$	15.6006	+	6.68000i	0.162506	+	0.0695833i
$97$	−88.5132			−0.912507			−0.456253	−	0.889850i	$-0.650809\pi$
$97$	−88.5132			−0.912507			−0.456253	+	0.889850i	$0.650809\pi$
$98$			41.9050i			0.427602i
$99$	41.3747	−	39.4554i	0.417926	−	0.398540i
$100$	5.33198			0.0533198
$101$			2.53411i			0.0250902i	0.999921	+	0.0125451i	$0.00399334\pi$
$101$			2.53411i			0.0250902i	−0.999921	+	0.0125451i	$0.996007\pi$
$102$	−37.8919	+	88.4934i	−0.371489	+	0.867582i
$103$	−104.661			−1.01612			−0.508061	−	0.861321i	$-0.669638\pi$
$103$	−104.661			−1.01612			−0.508061	+	0.861321i	$0.669638\pi$
$104$		−	10.1980i		−	0.0980581i
$105$	−128.628	−	55.0772i	−1.22503	−	0.524545i
$106$	−58.9661			−0.556284
$107$		−	179.855i		−	1.68089i	−0.541896	−	0.840445i	$-0.682293\pi$
$107$		−	179.855i		−	1.68089i	0.541896	−	0.840445i	$-0.317707\pi$
$108$	50.5936	−	18.8756i	0.468459	−	0.174774i
$109$	104.396			0.957757			0.478878	−	0.877881i	$-0.341043\pi$
$109$	104.396			0.957757			0.478878	+	0.877881i	$0.341043\pi$
$110$			47.2527i			0.429570i
$111$	35.8275	−	83.6721i	0.322770	−	0.753803i
$112$	35.4697			0.316694
$113$		−	32.0712i		−	0.283816i	−0.989880	−	0.141908i	$-0.954676\pi$
$113$		−	32.0712i		−	0.283816i	0.989880	−	0.141908i	$-0.0453237\pi$
$114$	−36.3584	−	15.5683i	−0.318933	−	0.136564i
$115$	−100.975			−0.878041
$116$		−	86.0651i		−	0.741940i
$117$	−22.3944	−	23.4838i	−0.191406	−	0.200716i
$118$	61.3245			0.519699
$119$			201.200i			1.69076i
$120$	−17.5679	+	41.0283i	−0.146399	+	0.341902i
$121$	80.6469			0.666504
$122$			150.665i			1.23496i
$123$	−26.3648	−	11.2891i	−0.214348	−	0.0917816i
$124$	106.049			0.855234
$125$		−	117.473i		−	0.939788i
$126$	81.6788	−	77.8899i	0.648244	−	0.618174i
$127$	−10.3189			−0.0812508			−0.0406254	−	0.999174i	$-0.512935\pi$
$127$	−10.3189			−0.0812508			−0.0406254	+	0.999174i	$0.512935\pi$
$128$		−	11.3137i		−	0.0883883i
$129$	79.3983	−	185.428i	0.615491	−	1.43743i
$130$	26.8201			0.206308
$131$			47.4112i			0.361918i	0.983491	+	0.180959i	$0.0579201\pi$
$131$			47.4112i			0.361918i	−0.983491	+	0.180959i	$0.942080\pi$
$132$	−35.0375	−	15.0027i	−0.265436	−	0.113657i
$133$	−82.6650			−0.621541
$134$		−	64.2383i		−	0.479390i
$135$	49.6415	+	133.057i	0.367714	+	0.985609i
$136$	64.1764			0.471885
$137$		−	89.9553i		−	0.656608i	−0.944572	−	0.328304i	$-0.893523\pi$
$137$		−	89.9553i		−	0.656608i	0.944572	−	0.328304i	$-0.106477\pi$
$138$	32.0594	−	74.8721i	0.232315	−	0.542552i
$139$	−37.1716			−0.267422			−0.133711	−	0.991020i	$-0.542689\pi$
$139$	−37.1716			−0.267422			−0.133711	+	0.991020i	$0.542689\pi$
$140$			93.2826i			0.666305i
$141$	138.755	+	59.4136i	0.984081	+	0.421373i
$142$	67.9481			0.478508
$143$			22.9039i			0.160167i
$144$	−24.8444	−	26.0529i	−0.172531	−	0.180923i
$145$	226.345			1.56100
$146$		−	175.320i		−	1.20082i
$147$	34.9906	−	81.7176i	0.238031	−	0.555902i
$148$	−60.6800			−0.410000
$149$			115.353i			0.774182i	0.922042	+	0.387091i	$0.126520\pi$
$149$			115.353i			0.774182i	−0.922042	+	0.387091i	$0.873480\pi$
$150$	−10.3977	−	4.45220i	−0.0693183	−	0.0296814i
$151$	−62.1381			−0.411511			−0.205755	−	0.978603i	$-0.565965\pi$
$151$	−62.1381			−0.411511			−0.205755	+	0.978603i	$0.565965\pi$
$152$			26.3675i			0.173470i
$153$	147.784	−	140.929i	0.965907	−	0.921102i
$154$	−79.6619			−0.517285
$155$			278.901i			1.79936i
$156$	−8.51536	+	19.8869i	−0.0545856	+	0.127480i
$157$	−278.183			−1.77187			−0.885933	−	0.463813i	$-0.846481\pi$
$157$	−278.183			−1.77187			−0.885933	+	0.463813i	$0.846481\pi$
$158$		−	134.151i		−	0.849056i
$159$	114.988	+	49.2367i	0.723196	+	0.309665i
$160$	29.7542			0.185964
$161$		−	170.231i		−	1.05733i
$162$	−114.422	−	5.43682i	−0.706310	−	0.0335606i
$163$	−7.95933			−0.0488302			−0.0244151	−	0.999702i	$-0.507772\pi$
$163$	−7.95933			−0.0488302			−0.0244151	+	0.999702i	$0.507772\pi$
$164$			19.1201i			0.116586i
$165$	39.4560	−	92.1460i	0.239127	−	0.558461i
$166$	152.823			0.920619
$167$			108.575i			0.650151i	0.945688	+	0.325075i	$0.105390\pi$
$167$			108.575i			0.650151i	−0.945688	+	0.325075i	$0.894610\pi$
$168$	−69.1684	−	29.6172i	−0.411717	−	0.176293i
$169$	13.0000			0.0769231
$170$			168.779i			0.992818i
$171$	57.9019	+	60.7184i	0.338607	+	0.355079i
$172$	−134.475			−0.781829
$173$			247.079i			1.42820i	0.700042	+	0.714102i	$0.253165\pi$
$173$			247.079i			1.42820i	−0.700042	+	0.714102i	$0.746835\pi$
$174$	−71.8643	+	167.833i	−0.413013	+	0.964558i
$175$	−23.6405			−0.135089
$176$			25.4096i			0.144373i
$177$	−119.587	−	51.2059i	−0.675633	−	0.289299i
$178$	59.7627			0.335746
$179$			153.345i			0.856673i	0.903619	+	0.428337i	$0.140900\pi$
$179$			153.345i			0.856673i	−0.903619	+	0.428337i	$0.859100\pi$
$180$	68.5172	−	65.3389i	0.380651	−	0.362994i
$181$	67.3943			0.372344			0.186172	−	0.982517i	$-0.440392\pi$
$181$	67.3943			0.372344			0.186172	+	0.982517i	$0.440392\pi$
$182$			45.2152i			0.248435i
$183$	125.805	−	293.807i	0.687459	−	1.60550i
$184$	−54.2981			−0.295098
$185$		−	159.584i		−	0.862615i
$186$	−206.803	−	88.5509i	−1.11184	−	0.476080i
$187$	−144.135			−0.770774
$188$		−	100.627i		−	0.535250i
$189$	−224.317	+	83.6891i	−1.18687	+	0.442800i
$190$	−69.3445			−0.364971
$191$			122.592i			0.641845i	0.947105	+	0.320922i	$0.103993\pi$
$191$			122.592i			0.641845i	−0.947105	+	0.320922i	$0.896007\pi$
$192$	−9.44694	+	22.0625i	−0.0492028	+	0.114909i
$193$	142.335			0.737487			0.368744	−	0.929531i	$-0.379788\pi$
$193$	142.335			0.737487			0.368744	+	0.929531i	$0.379788\pi$
$194$		−	125.177i		−	0.645240i
$195$	−52.3010	−	22.3947i	−0.268210	−	0.114845i
$196$	−59.2626			−0.302360
$197$		−	23.8909i		−	0.121274i	−0.998160	−	0.0606368i	$-0.980687\pi$
$197$		−	23.8909i		−	0.121274i	0.998160	−	0.0606368i	$-0.0193131\pi$
$198$	55.7984	+	58.5127i	0.281810	+	0.295518i
$199$	121.893			0.612526			0.306263	−	0.951947i	$-0.400921\pi$
$199$	121.893			0.612526			0.306263	+	0.951947i	$0.400921\pi$
$200$			7.54056i			0.0377028i
$201$	−53.6389	+	125.269i	−0.266860	+	0.623230i
$202$	−3.58378			−0.0177415
$203$			381.588i			1.87974i
$204$	−125.149	−	53.5873i	−0.613473	−	0.262683i
$205$	−50.2843			−0.245289
$206$		−	148.013i		−	0.718507i
$207$	−125.036	+	119.236i	−0.604040	+	0.576021i
$208$	14.4222			0.0693375
$209$		−	59.2191i		−	0.283345i
$210$	77.8910	−	181.908i	0.370909	−	0.866227i
$211$	200.038			0.948046			0.474023	−	0.880513i	$-0.342801\pi$
$211$	200.038			0.948046			0.474023	+	0.880513i	$0.342801\pi$
$212$		−	83.3907i		−	0.393352i
$213$	−132.504	−	56.7366i	−0.622083	−	0.266369i
$214$	254.354			1.18857
$215$		−	353.658i		−	1.64492i
$216$	26.6942	+	71.5501i	0.123584	+	0.331251i
$217$	−470.191			−2.16678
$218$			147.638i			0.677236i
$219$	−146.392	+	341.887i	−0.668458	+	1.56113i
$220$	−66.8254			−0.303752
$221$			81.8092i			0.370177i
$222$	118.330	+	50.6678i	0.533019	+	0.228233i
$223$	−136.870			−0.613766			−0.306883	−	0.951747i	$-0.599286\pi$
$223$	−136.870			−0.613766			−0.306883	+	0.951747i	$0.599286\pi$
$224$			50.1617i			0.223936i
$225$	16.5587	+	17.3642i	0.0735944	+	0.0771743i
$226$	45.3555			0.200688
$227$		−	279.375i		−	1.23073i	−0.788244	−	0.615363i	$-0.789009\pi$
$227$		−	279.375i		−	1.23073i	0.788244	−	0.615363i	$-0.210991\pi$
$228$	22.0168	−	51.4185i	0.0965651	−	0.225520i
$229$	279.058			1.21859			0.609296	−	0.792943i	$-0.291452\pi$
$229$	279.058			1.21859			0.609296	+	0.792943i	$0.291452\pi$
$230$		−	142.800i		−	0.620869i
$231$	155.346	+	66.5177i	0.672495	+	0.287955i
$232$	121.714			0.524631
$233$			207.796i			0.891829i	0.895075	+	0.445915i	$0.147122\pi$
$233$			207.796i			0.891829i	−0.895075	+	0.445915i	$0.852878\pi$
$234$	33.2111	−	31.6705i	0.141928	−	0.135344i
$235$	264.641			1.12613
$236$			86.7259i			0.367483i
$237$	−112.016	+	261.604i	−0.472641	+	1.10381i
$238$	−284.540			−1.19555
$239$		−	378.965i		−	1.58563i	−0.609465	−	0.792813i	$-0.708616\pi$
$239$		−	378.965i		−	1.58563i	0.609465	−	0.792813i	$-0.291384\pi$
$240$	−58.0228	−	24.8447i	−0.241762	−	0.103520i
$241$	−157.277			−0.652604			−0.326302	−	0.945266i	$-0.605803\pi$
$241$	−157.277			−0.652604			−0.326302	+	0.945266i	$0.605803\pi$
$242$			114.052i			0.471289i
$243$	218.592	+	106.145i	0.899554	+	0.436809i
$244$	−213.072			−0.873246
$245$		−	155.856i		−	0.636147i
$246$	15.9652	−	37.2855i	0.0648994	−	0.151567i
$247$	−33.6121			−0.136081
$248$			149.976i			0.604742i
$249$	−298.015	−	127.607i	−1.19685	−	0.512478i
$250$	166.133			0.664530
$251$		−	237.146i		−	0.944805i	−0.881383	−	0.472403i	$-0.843387\pi$
$251$		−	237.146i		−	0.944805i	0.881383	−	0.472403i	$-0.156613\pi$
$252$	110.153	+	115.511i	0.437115	+	0.458378i
$253$	121.949			0.482012
$254$		−	14.5931i		−	0.0574530i
$255$	140.930	−	329.131i	0.552668	−	1.29071i
$256$	16.0000			0.0625000
$257$		−	257.298i		−	1.00116i	−0.865691	−	0.500579i	$-0.833120\pi$
$257$		−	257.298i		−	1.00116i	0.865691	−	0.500579i	$-0.166880\pi$
$258$	262.235	+	112.286i	1.01641	+	0.435218i
$259$	269.038			1.03876
$260$			37.9293i			0.145882i
$261$	280.281	−	267.280i	1.07387	−	1.02406i
$262$	−67.0496			−0.255915
$263$			262.304i			0.997352i	0.866788	+	0.498676i	$0.166180\pi$
$263$			262.304i			0.997352i	−0.866788	+	0.498676i	$0.833820\pi$
$264$	21.2170	−	49.5506i	0.0803675	−	0.187692i
$265$	219.311			0.827589
$266$		−	116.906i		−	0.439496i
$267$	−116.541	−	49.9018i	−0.436485	−	0.186898i
$268$	90.8466			0.338980
$269$			119.235i			0.443251i	0.975132	+	0.221626i	$0.0711363\pi$
$269$			119.235i			0.443251i	−0.975132	+	0.221626i	$0.928864\pi$
$270$	−188.171	+	70.2036i	−0.696931	+	0.260013i
$271$	89.0798			0.328708			0.164354	−	0.986401i	$-0.447446\pi$
$271$	89.0798			0.328708			0.164354	+	0.986401i	$0.447446\pi$
$272$			90.7591i			0.333673i
$273$	37.7547	−	88.1728i	0.138295	−	0.322977i
$274$	127.216			0.464292
$275$		−	16.9355i		−	0.0615835i
$276$	105.885	+	45.3389i	0.383642	+	0.164271i
$277$	−187.542			−0.677047			−0.338523	−	0.940958i	$-0.609927\pi$
$277$	−187.542			−0.677047			−0.338523	+	0.940958i	$0.609927\pi$
$278$		−	52.5686i		−	0.189096i
$279$	329.341	+	345.361i	1.18043	+	1.23785i
$280$	−131.922			−0.471148
$281$		−	287.799i		−	1.02420i	−0.858927	−	0.512098i	$-0.828868\pi$
$281$		−	287.799i		−	1.02420i	0.858927	−	0.512098i	$-0.171132\pi$
$282$	−84.0236	+	196.230i	−0.297956	+	0.695851i
$283$	−279.458			−0.987484			−0.493742	−	0.869609i	$-0.664371\pi$
$283$	−279.458			−0.987484			−0.493742	+	0.869609i	$0.664371\pi$
$284$			96.0931i			0.338356i
$285$	135.227	+	57.9026i	0.474480	+	0.203167i
$286$	−32.3910			−0.113255
$287$		−	84.7729i		−	0.295376i
$288$	36.8444	−	35.1353i	0.127932	−	0.121998i
$289$	−225.826			−0.781405
$290$			320.100i			1.10379i
$291$	−104.522	+	244.103i	−0.359183	+	0.838842i
$292$	247.940			0.849110
$293$			493.507i			1.68432i	0.539226	+	0.842161i	$0.318717\pi$
$293$			493.507i			1.68432i	−0.539226	+	0.842161i	$0.681283\pi$
$294$	115.566	+	49.4842i	0.393082	+	0.168314i
$295$	−228.083			−0.773161
$296$		−	85.8144i		−	0.289914i
$297$	−59.9528	−	160.695i	−0.201861	−	0.541062i
$298$	−163.134			−0.547429
$299$		−	69.2168i		−	0.231494i
$300$	6.29637	−	14.7046i	0.0209879	−	0.0490154i
$301$	596.222			1.98080
$302$		−	87.8765i		−	0.290982i
$303$	6.98861	+	2.99245i	0.0230647	+	0.00987608i
$304$	−37.2893			−0.122662
$305$		−	560.363i		−	1.83726i
$306$	199.303	+	208.998i	0.651317	+	0.683000i
$307$	−81.6460			−0.265948			−0.132974	−	0.991120i	$-0.542453\pi$
$307$	−81.6460			−0.265948			−0.132974	+	0.991120i	$0.542453\pi$
$308$		−	112.659i		−	0.365776i
$309$	−123.590	+	288.635i	−0.399969	+	0.934093i
$310$	−394.425			−1.27234
$311$		−	274.543i		−	0.882773i	−0.897317	−	0.441387i	$-0.854487\pi$
$311$		−	274.543i		−	0.882773i	0.897317	−	0.441387i	$-0.145513\pi$
$312$	−28.1243	−	12.0425i	−0.0901420	−	0.0385979i
$313$	470.031			1.50170			0.750848	−	0.660475i	$-0.229645\pi$
$313$	470.031			1.50170			0.750848	+	0.660475i	$0.229645\pi$
$314$		−	393.410i		−	1.25290i
$315$	−303.786	+	289.694i	−0.964399	+	0.919663i
$316$	189.718			0.600373
$317$		−	376.816i		−	1.18869i	−0.804209	−	0.594347i	$-0.797411\pi$
$317$		−	376.816i		−	1.18869i	0.804209	−	0.594347i	$-0.202589\pi$
$318$	−69.6312	+	162.618i	−0.218966	+	0.511377i
$319$	−273.360			−0.856928
$320$			42.0788i			0.131496i
$321$	−496.008	−	212.385i	−1.54520	−	0.661637i
$322$	240.742			0.747647
$323$		−	211.521i		−	0.654865i
$324$	7.68882	−	161.817i	0.0237309	−	0.499437i
$325$	−9.61237			−0.0295765
$326$		−	11.2562i		−	0.0345282i
$327$	123.277	−	287.904i	0.376995	−	0.880439i
$328$	−27.0399			−0.0824386
$329$			446.152i			1.35608i
$330$	130.314	+	55.7992i	0.394891	+	0.169088i
$331$	−305.739			−0.923681			−0.461841	−	0.886963i	$-0.652811\pi$
$331$	−305.739			−0.923681			−0.461841	+	0.886963i	$0.652811\pi$
$332$			216.124i			0.650976i
$333$	−188.445	−	197.611i	−0.565900	−	0.593428i
$334$	−153.548			−0.459726
$335$			238.920i			0.713193i
$336$	41.8850	−	97.8190i	0.124658	−	0.291128i
$337$	−13.7977			−0.0409429			−0.0204714	−	0.999790i	$-0.506517\pi$
$337$	−13.7977			−0.0409429			−0.0204714	+	0.999790i	$0.506517\pi$
$338$			18.3848i			0.0543928i
$339$	−88.4464	−	37.8718i	−0.260904	−	0.111716i
$340$	−238.690			−0.702028
$341$		−	336.833i		−	0.987780i
$342$	−85.8688	+	81.8856i	−0.251078	+	0.239432i
$343$	−171.751			−0.500731
$344$		−	190.176i		−	0.552836i
$345$	−119.238	+	278.470i	−0.345617	+	0.807159i
$346$	−349.423			−1.00989
$347$			170.629i			0.491726i	0.969305	+	0.245863i	$0.0790714\pi$
$347$			170.629i			0.491726i	−0.969305	+	0.245863i	$0.920929\pi$
$348$	−237.352	−	101.631i	−0.682045	−	0.292045i
$349$	−309.707			−0.887414			−0.443707	−	0.896172i	$-0.646337\pi$
$349$	−309.707			−0.887414			−0.443707	+	0.896172i	$0.646337\pi$
$350$		−	33.4327i		−	0.0955220i
$351$	−91.2089	+	34.0285i	−0.259854	+	0.0969474i
$352$	−35.9346			−0.102087
$353$			570.419i			1.61592i	0.589238	+	0.807959i	$0.299428\pi$
$353$			570.419i			1.61592i	−0.589238	+	0.807959i	$0.700572\pi$
$354$	72.4161	−	169.122i	0.204565	−	0.477745i
$355$	−252.718			−0.711881
$356$			84.5172i			0.237408i
$357$	554.873	+	237.591i	1.55427	+	0.665520i
$358$	−216.862			−0.605760
$359$			211.973i			0.590454i	0.955427	+	0.295227i	$0.0953953\pi$
$359$			211.973i			0.590454i	−0.955427	+	0.295227i	$0.904605\pi$
$360$	92.4032	+	96.8980i	0.256675	+	0.269161i
$361$	−274.094			−0.759264
$362$			95.3100i			0.263287i
$363$	95.2334	−	222.409i	0.262351	−	0.612698i
$364$	−63.9439			−0.175670
$365$			652.064i			1.78648i
$366$	415.505	+	177.915i	1.13526	+	0.486107i
$367$	502.884			1.37026			0.685128	−	0.728422i	$-0.259746\pi$
$367$	502.884			1.37026			0.685128	+	0.728422i	$0.259746\pi$
$368$		−	76.7891i		−	0.208666i
$369$	−62.2667	+	59.3783i	−0.168744	+	0.160917i
$370$	225.685			0.609961
$371$			369.731i			0.996578i
$372$	125.230	−	292.464i	0.336639	−	0.786193i
$373$	430.042			1.15293			0.576464	−	0.817123i	$-0.304432\pi$
$373$	430.042			1.15293			0.576464	+	0.817123i	$0.304432\pi$
$374$		−	203.837i		−	0.545019i
$375$	−323.970	−	138.721i	−0.863921	−	0.369922i
$376$	142.308			0.378479
$377$			155.156i			0.411555i
$378$	−118.354	−	317.233i	−0.313107	−	0.839240i
$379$	495.114			1.30637			0.653185	−	0.757199i	$-0.273433\pi$
$379$	495.114			1.30637			0.653185	+	0.757199i	$0.273433\pi$
$380$		−	98.0679i		−	0.258073i
$381$	−12.1852	+	28.4575i	−0.0319822	+	0.0746916i
$382$	−173.372			−0.453853
$383$		−	89.4468i		−	0.233543i	−0.993159	−	0.116771i	$-0.962746\pi$
$383$		−	89.4468i		−	0.233543i	0.993159	−	0.116771i	$-0.0372544\pi$
$384$	−31.2011	−	13.3600i	−0.0812529	−	0.0347917i
$385$	296.285			0.769570
$386$			201.292i			0.521482i
$387$	−417.617	−	437.932i	−1.07911	−	1.13161i
$388$	177.026			0.456253
$389$			228.216i			0.586674i	0.956009	+	0.293337i	$0.0947659\pi$
$389$			228.216i			0.586674i	−0.956009	+	0.293337i	$0.905234\pi$
$390$	31.6709	−	73.9648i	0.0812076	−	0.189653i
$391$	435.582			1.11402
$392$		−	83.8099i		−	0.213801i
$393$	130.752	+	55.9864i	0.332701	+	0.142459i
$394$	33.7868			0.0857533
$395$			498.944i			1.26315i
$396$	−82.7494	+	78.9109i	−0.208963	+	0.199270i
$397$	211.552			0.532877			0.266439	−	0.963852i	$-0.414153\pi$
$397$	211.552			0.532877			0.266439	+	0.963852i	$0.414153\pi$
$398$			172.382i			0.433121i
$399$	−97.6164	+	227.975i	−0.244653	+	0.571366i
$400$	−10.6640			−0.0266599
$401$		−	386.430i		−	0.963667i	−0.876263	−	0.481834i	$-0.839971\pi$
$401$		−	386.430i		−	0.963667i	0.876263	−	0.481834i	$-0.160029\pi$
$402$	−177.157	−	75.8569i	−0.440690	−	0.188699i
$403$	−191.183			−0.474398
$404$		−	5.06822i		−	0.0125451i
$405$	425.567	+	20.2210i	1.05078	+	0.0499284i
$406$	−539.647			−1.32918
$407$			192.732i			0.473543i
$408$	75.7838	−	176.987i	0.185745	−	0.433791i
$409$	83.1800			0.203374			0.101687	−	0.994816i	$-0.467576\pi$
$409$	83.1800			0.203374			0.101687	+	0.994816i	$0.467576\pi$
$410$		−	71.1127i		−	0.173446i
$411$	−248.080	−	106.225i	−0.603601	−	0.258456i
$412$	209.321			0.508061
$413$		−	384.518i		−	0.931036i
$414$	−168.626	−	176.828i	−0.407308	−	0.427121i
$415$	−568.390			−1.36961
$416$			20.3961i			0.0490290i
$417$	−43.8947	+	102.512i	−0.105263	+	0.245833i
$418$	83.7485			0.200355
$419$		−	468.033i		−	1.11702i	−0.829497	−	0.558511i	$-0.811373\pi$
$419$		−	468.033i		−	1.11702i	0.829497	−	0.558511i	$-0.188627\pi$
$420$	257.256	+	110.154i	0.612515	+	0.262273i
$421$	−96.5185			−0.229260			−0.114630	−	0.993408i	$-0.536568\pi$
$421$	−96.5185			−0.229260			−0.114630	+	0.993408i	$0.536568\pi$
$422$			282.896i			0.670369i
$423$	327.704	−	312.502i	0.774713	−	0.738777i
$424$	117.932			0.278142
$425$		−	60.4908i		−	0.142331i
$426$	80.2377	−	187.388i	0.188351	−	0.439879i
$427$	944.700			2.21241
$428$			359.711i			0.840445i
$429$	63.1648	+	27.0465i	0.147237	+	0.0630455i
$430$	500.147			1.16313
$431$		−	268.755i		−	0.623560i	−0.950154	−	0.311780i	$-0.899075\pi$
$431$		−	268.755i		−	0.623560i	0.950154	−	0.311780i	$-0.100925\pi$
$432$	−101.187	+	37.7513i	−0.234230	+	0.0873872i
$433$	117.618			0.271635			0.135817	−	0.990734i	$-0.456634\pi$
$433$	117.618			0.271635			0.135817	+	0.990734i	$0.456634\pi$
$434$		−	664.950i		−	1.53214i
$435$	267.283	−	624.217i	0.614444	−	1.43498i
$436$	−208.791			−0.478878
$437$			178.963i			0.409527i
$438$	−483.501	−	207.030i	−1.10388	−	0.472671i
$439$	−275.024			−0.626478			−0.313239	−	0.949674i	$-0.601414\pi$
$439$	−275.024			−0.626478			−0.313239	+	0.949674i	$0.601414\pi$
$440$		−	94.5053i		−	0.214785i
$441$	−184.043	−	192.995i	−0.417331	−	0.437631i
$442$	−115.696			−0.261755
$443$		−	536.850i		−	1.21185i	−0.795521	−	0.605926i	$-0.792803\pi$
$443$		−	536.850i		−	1.21185i	0.795521	−	0.605926i	$-0.207197\pi$
$444$	−71.6550	+	167.344i	−0.161385	+	0.376901i
$445$	−222.274			−0.499492
$446$		−	193.563i		−	0.433998i
$447$	318.123	+	136.217i	0.711684	+	0.304735i
$448$	−70.9394			−0.158347
$449$			367.318i			0.818081i	0.912516	+	0.409040i	$0.134136\pi$
$449$			367.318i			0.818081i	−0.912516	+	0.409040i	$0.865864\pi$
$450$	−24.5567	+	23.4176i	−0.0545705	+	0.0520391i
$451$	60.7292			0.134655
$452$			64.1423i			0.141908i
$453$	−73.3769	+	171.365i	−0.161980	+	0.378290i
$454$	395.096			0.870255
$455$		−	168.168i		−	0.369599i
$456$	72.7167	+	31.1365i	0.159466	+	0.0682819i
$457$	−337.847			−0.739271			−0.369635	−	0.929177i	$-0.620517\pi$
$457$	−337.847			−0.739271			−0.369635	+	0.929177i	$0.620517\pi$
$458$			394.647i			0.861675i
$459$	−214.142	−	573.979i	−0.466540	−	1.25050i
$460$	201.950			0.439021
$461$			196.400i			0.426030i	0.977049	+	0.213015i	$0.0683283\pi$
$461$			196.400i			0.426030i	−0.977049	+	0.213015i	$0.931672\pi$
$462$	−94.0702	+	219.693i	−0.203615	+	0.475526i
$463$	673.240			1.45408			0.727041	−	0.686594i	$-0.240895\pi$
$463$	673.240			1.45408			0.727041	+	0.686594i	$0.240895\pi$
$464$			172.130i			0.370970i
$465$	769.157	+	329.345i	1.65410	+	0.708269i
$466$	−293.868			−0.630619
$467$			59.6076i			0.127639i	0.997961	+	0.0638197i	$0.0203282\pi$
$467$			59.6076i			0.127639i	−0.997961	+	0.0638197i	$0.979672\pi$
$468$	44.7889	+	46.9676i	0.0957028	+	0.100358i
$469$	−402.788			−0.858823
$470$			374.260i			0.796297i
$471$	−328.497	+	767.177i	−0.697446	+	1.62883i
$472$	−122.649			−0.259850
$473$			427.118i			0.902999i
$474$	−369.964	−	158.414i	−0.780514	−	0.334208i
$475$	24.8532			0.0523226
$476$		−	402.400i		−	0.845378i
$477$	271.572	−	258.974i	0.569332	−	0.542923i
$478$	535.937			1.12121
$479$		−	227.538i		−	0.475028i	−0.971384	−	0.237514i	$-0.923667\pi$
$479$		−	227.538i		−	0.475028i	0.971384	−	0.237514i	$-0.0763325\pi$
$480$	35.1358	−	82.0566i	0.0731995	−	0.170951i
$481$	109.392			0.227427
$482$		−	222.424i		−	0.461460i
$483$	−469.465	−	201.020i	−0.971976	−	0.416190i
$484$	−161.294			−0.333252
$485$			465.566i			0.959929i
$486$	−150.111	+	309.135i	−0.308871	+	0.636081i
$487$	−144.608			−0.296937			−0.148469	−	0.988917i	$-0.547434\pi$
$487$	−144.608			−0.296937			−0.148469	+	0.988917i	$0.547434\pi$
$488$		−	301.329i		−	0.617478i
$489$	−9.39891	+	21.9504i	−0.0192207	+	0.0448883i
$490$	220.414			0.449824
$491$			470.330i			0.957901i	0.877842	+	0.478951i	$0.158983\pi$
$491$			470.330i			0.957901i	−0.877842	+	0.478951i	$0.841017\pi$
$492$	52.7296	+	22.5783i	0.107174	+	0.0458908i
$493$	−976.399			−1.98053
$494$		−	47.5347i		−	0.0962240i
$495$	−207.530	−	217.625i	−0.419252	−	0.439646i
$496$	−212.098			−0.427617
$497$		−	426.050i		−	0.857242i
$498$	180.464	−	421.457i	0.362377	−	0.846300i
$499$	62.4043			0.125059			0.0625293	−	0.998043i	$-0.480083\pi$
$499$	62.4043			0.125059			0.0625293	+	0.998043i	$0.480083\pi$
$500$			234.947i			0.469894i
$501$	299.430	+	128.213i	0.597665	+	0.255914i
$502$	335.375			0.668078
$503$			584.530i			1.16209i	0.813872	+	0.581044i	$0.197356\pi$
$503$			584.530i			1.16209i	−0.813872	+	0.581044i	$0.802644\pi$
$504$	−163.358	+	155.780i	−0.324122	+	0.309087i
$505$	13.3290			0.0263941
$506$			172.462i			0.340834i
$507$	15.3513	−	35.8516i	0.0302787	−	0.0707132i
$508$	20.6377			0.0406254
$509$		−	675.889i		−	1.32788i	−0.747788	−	0.663938i	$-0.768884\pi$
$509$		−	675.889i		−	1.32788i	0.747788	−	0.663938i	$-0.231116\pi$
$510$	465.462	+	199.306i	0.912670	+	0.390795i
$511$	−1099.30			−2.15126
$512$			22.6274i			0.0441942i
$513$	235.825	−	87.9823i	0.459697	−	0.171505i
$514$	363.874			0.707926
$515$			550.499i			1.06893i
$516$	−158.797	+	370.856i	−0.307745	+	0.718713i
$517$	−319.612			−0.618205
$518$			380.477i			0.734511i
$519$	681.399	+	291.768i	1.31291	+	0.562173i
$520$	−53.6401			−0.103154
$521$			383.819i			0.736696i	0.929688	+	0.368348i	$0.120077\pi$
$521$			383.819i			0.736696i	−0.929688	+	0.368348i	$0.879923\pi$
$522$	377.990	+	396.377i	0.724119	+	0.759343i
$523$	523.963			1.00184			0.500920	−	0.865493i	$-0.332995\pi$
$523$	523.963			1.00184			0.500920	+	0.865493i	$0.332995\pi$
$524$		−	94.8225i		−	0.180959i
$525$	−27.9163	+	65.1961i	−0.0531739	+	0.124183i
$526$	−370.953			−0.705235
$527$		−	1203.11i		−	2.28295i
$528$	70.0751	+	30.0054i	0.132718	+	0.0568284i
$529$	160.464			0.303335
$530$			310.153i			0.585194i
$531$	−282.433	+	269.332i	−0.531889	+	0.507216i
$532$	165.330			0.310771
$533$		−	34.4692i		−	0.0646701i
$534$	70.5718	−	164.815i	0.132157	−	0.308641i
$535$	−946.011			−1.76825
$536$			128.477i			0.239695i
$537$	422.896	+	181.080i	0.787516	+	0.337206i
$538$	−168.623			−0.313426
$539$			188.230i			0.349221i
$540$	−99.2829	−	266.115i	−0.183857	−	0.492805i
$541$	−559.364			−1.03395			−0.516973	−	0.856002i	$-0.672941\pi$
$541$	−559.364			−1.03395			−0.516973	+	0.856002i	$0.672941\pi$
$542$			125.978i			0.232432i
$543$	79.5838	−	185.861i	0.146563	−	0.342286i
$544$	−128.353			−0.235943
$545$		−	549.104i		−	1.00753i
$546$	124.695	+	53.3932i	0.228379	+	0.0977897i
$547$	−398.163			−0.727903			−0.363952	−	0.931418i	$-0.618573\pi$
$547$	−398.163			−0.727903			−0.363952	+	0.931418i	$0.618573\pi$
$548$			179.911i			0.328304i
$549$	−661.706	−	693.894i	−1.20529	−	1.26392i
$550$	23.9504			0.0435461
$551$		−	401.163i		−	0.728064i
$552$	−64.1189	+	149.744i	−0.116157	+	0.271276i
$553$	−841.155			−1.52108
$554$		−	265.224i		−	0.478744i
$555$	−440.103	−	188.447i	−0.792978	−	0.339545i
$556$	74.3432			0.133711
$557$			834.510i			1.49822i	0.662444	+	0.749112i	$0.269519\pi$
$557$			834.510i			1.49822i	−0.662444	+	0.749112i	$0.730481\pi$
$558$	−488.414	+	465.758i	−0.875294	+	0.834692i
$559$	242.427			0.433680
$560$		−	186.565i		−	0.333152i
$561$	−170.204	+	397.497i	−0.303394	+	0.708551i
$562$	407.010			0.724216
$563$		−	799.431i		−	1.41995i	−0.704228	−	0.709974i	$-0.748707\pi$
$563$		−	799.431i		−	1.41995i	0.704228	−	0.709974i	$-0.251293\pi$
$564$	−277.511	−	118.827i	−0.492041	−	0.210687i
$565$	−168.689			−0.298565
$566$		−	395.213i		−	0.698256i
$567$	−34.0900	+	717.452i	−0.0601235	+	1.26535i
$568$	−135.896			−0.239254
$569$			28.8146i			0.0506408i	0.999679	+	0.0253204i	$0.00806060\pi$
$569$			28.8146i			0.0506408i	−0.999679	+	0.0253204i	$0.991939\pi$
$570$	−81.8867	+	191.239i	−0.143661	+	0.335508i
$571$	722.322			1.26501			0.632506	−	0.774556i	$-0.282026\pi$
$571$	722.322			1.26501			0.632506	+	0.774556i	$0.282026\pi$
$572$		−	45.8078i		−	0.0800836i
$573$	338.087	+	144.765i	0.590030	+	0.252645i
$574$	119.887			0.208862
$575$			51.1798i			0.0890083i
$576$	49.6888	+	52.1059i	0.0862653	+	0.0904616i
$577$	−17.5710			−0.0304524			−0.0152262	−	0.999884i	$-0.504847\pi$
$577$	−17.5710			−0.0304524			−0.0152262	+	0.999884i	$0.504847\pi$
$578$		−	319.366i		−	0.552537i
$579$	168.079	−	392.534i	0.290292	−	0.677951i
$580$	−452.689			−0.780499
$581$		−	958.233i		−	1.64928i
$582$	−345.214	−	147.817i	−0.593151	−	0.253981i
$583$	−264.866			−0.454315
$584$			350.640i			0.600412i
$585$	−123.521	+	117.791i	−0.211147	+	0.201353i
$586$	−697.924			−1.19100
$587$		−	1074.50i		−	1.83050i	−0.402886	−	0.915250i	$-0.631993\pi$
$587$		−	1074.50i		−	1.83050i	0.402886	−	0.915250i	$-0.368007\pi$
$588$	−69.9813	+	163.435i	−0.119016	+	0.277951i
$589$	494.311			0.839238
$590$		−	322.557i		−	0.546707i
$591$	−65.8867	−	28.2120i	−0.111483	−	0.0477360i
$592$	121.360			0.205000
$593$		−	396.177i		−	0.668089i	−0.942557	−	0.334044i	$-0.891586\pi$
$593$		−	396.177i		−	0.668089i	0.942557	−	0.334044i	$-0.108414\pi$
$594$	227.258	−	84.7861i	0.382589	−	0.142738i
$595$	1058.28			1.77862
$596$		−	230.706i		−	0.387091i
$597$	143.939	−	336.157i	0.241104	−	0.563078i
$598$	97.8873			0.163691
$599$			741.342i			1.23763i	0.785536	+	0.618816i	$0.212387\pi$
$599$			741.342i			1.23763i	−0.785536	+	0.618816i	$0.787613\pi$
$600$	20.7955	+	8.90441i	0.0346591	+	0.0148407i
$601$	1093.82			1.82001			0.910003	−	0.414601i	$-0.136079\pi$
$601$	1093.82			1.82001			0.910003	+	0.414601i	$0.136079\pi$
$602$			843.185i			1.40064i
$603$	282.129	+	295.853i	0.467875	+	0.490635i
$604$	124.276			0.205755
$605$		−	424.191i		−	0.701141i
$606$	−4.23196	+	9.88339i	−0.00698344	+	0.0163092i
$607$	392.407			0.646469			0.323235	−	0.946319i	$-0.395230\pi$
$607$	392.407			0.646469			0.323235	+	0.946319i	$0.395230\pi$
$608$		−	52.7350i		−	0.0867352i
$609$	1052.35	+	450.605i	1.72800	+	0.739910i
$610$	792.473			1.29914
$611$			181.408i			0.296903i
$612$	−295.568	+	281.857i	−0.482954	+	0.460551i
$613$	238.314			0.388767			0.194383	−	0.980926i	$-0.437729\pi$
$613$	238.314			0.388767			0.194383	+	0.980926i	$0.437729\pi$
$614$		−	115.465i		−	0.188054i
$615$	−59.3791	+	138.675i	−0.0965514	+	0.225488i
$616$	159.324			0.258643
$617$			562.652i			0.911916i	0.890001	+	0.455958i	$0.150703\pi$
$617$			562.652i			0.911916i	−0.890001	+	0.455958i	$0.849297\pi$
$618$	−408.191	−	174.783i	−0.660504	−	0.282821i
$619$	443.139			0.715894			0.357947	−	0.933742i	$-0.383477\pi$
$619$	443.139			0.715894			0.357947	+	0.933742i	$0.383477\pi$
$620$		−	557.801i		−	0.899680i
$621$	181.180	+	485.630i	0.291756	+	0.782012i
$622$	388.262			0.624215
$623$		−	374.725i		−	0.601485i
$624$	17.0307	−	39.7738i	0.0272928	−	0.0637401i
$625$	−684.542			−1.09527
$626$			664.724i			1.06186i
$627$	−163.316	−	69.9300i	−0.260471	−	0.111531i
$628$	556.366			0.885933
$629$			688.408i			1.09445i
$630$	−409.689	−	429.618i	−0.650300	−	0.681933i
$631$	284.737			0.451247			0.225624	−	0.974215i	$-0.427558\pi$
$631$	284.737			0.451247			0.225624	+	0.974215i	$0.427558\pi$
$632$			268.302i			0.424528i
$633$	236.218	−	551.667i	0.373172	−	0.871512i
$634$	532.898			0.840533
$635$			54.2756i			0.0854734i
$636$	−229.976	−	98.4734i	−0.361598	−	0.154832i
$637$	106.837			0.167719
$638$		−	386.590i		−	0.605940i
$639$	−312.939	+	298.422i	−0.489732	+	0.467014i
$640$	−59.5084			−0.0929818
$641$		−	674.486i		−	1.05224i	−0.850410	−	0.526120i	$-0.823646\pi$
$641$		−	674.486i		−	1.05224i	0.850410	−	0.526120i	$-0.176354\pi$
$642$	300.358	−	701.461i	0.467848	−	1.09262i
$643$	−720.691			−1.12083			−0.560413	−	0.828213i	$-0.689358\pi$
$643$	−720.691			−1.12083			−0.560413	+	0.828213i	$0.689358\pi$
$644$			340.461i			0.528666i
$645$	−975.323	−	417.623i	−1.51213	−	0.647477i
$646$	299.136			0.463059
$647$			81.3261i			0.125697i	0.998023	+	0.0628486i	$0.0200185\pi$
$647$			81.3261i			0.125697i	−0.998023	+	0.0628486i	$0.979981\pi$
$648$	228.844	+	10.8736i	0.353155	+	0.0167803i
$649$	275.459			0.424436
$650$		−	13.5939i		−	0.0209138i
$651$	−555.233	+	1296.70i	−0.852893	+	1.99186i
$652$	15.9187			0.0244151
$653$		−	333.779i		−	0.511147i	−0.966790	−	0.255574i	$-0.917736\pi$
$653$		−	333.779i		−	0.511147i	0.966790	−	0.255574i	$-0.0822643\pi$
$654$	407.157	+	174.340i	0.622565	+	0.266576i
$655$	249.376			0.380727
$656$		−	38.2401i		−	0.0582929i
$657$	769.991	+	807.446i	1.17198	+	1.22899i
$658$	−630.954			−0.958896
$659$		−	213.628i		−	0.324170i	−0.986777	−	0.162085i	$-0.948178\pi$
$659$		−	213.628i		−	0.324170i	0.986777	−	0.162085i	$-0.0518218\pi$
$660$	−78.9119	+	184.292i	−0.119564	+	0.279230i
$661$	−389.924			−0.589899			−0.294950	−	0.955513i	$-0.595303\pi$
$661$	−389.924			−0.589899			−0.294950	+	0.955513i	$0.595303\pi$
$662$		−	432.380i		−	0.653141i
$663$	225.615	+	96.6058i	0.340294	+	0.145710i
$664$	−305.646			−0.460310
$665$			434.805i			0.653842i
$666$	279.465	−	266.501i	0.419617	−	0.400152i
$667$	826.108			1.23854
$668$		−	217.150i		−	0.325075i
$669$	−161.625	+	377.462i	−0.241592	+	0.564218i
$670$	−337.883			−0.504304
$671$			676.760i			1.00858i
$672$	138.337	+	59.2344i	0.205858	+	0.0881464i
$673$	−612.798			−0.910546			−0.455273	−	0.890352i	$-0.650458\pi$
$673$	−612.798			−0.910546			−0.455273	+	0.890352i	$0.650458\pi$
$674$		−	19.5130i		−	0.0289510i
$675$	67.4410	−	25.1611i	0.0999126	−	0.0372758i
$676$	−26.0000			−0.0384615
$677$		−	823.880i		−	1.21696i	−0.793570	−	0.608478i	$-0.791780\pi$
$677$		−	823.880i		−	1.21696i	0.793570	−	0.608478i	$-0.208220\pi$
$678$	53.5588	−	125.082i	0.0789953	−	0.184487i
$679$	−784.884			−1.15594
$680$		−	337.558i		−	0.496409i
$681$	−770.465	−	329.905i	−1.13137	−	0.484442i
$682$	476.354			0.698466
$683$			883.354i			1.29334i	0.762768	+	0.646672i	$0.223840\pi$
$683$			883.354i			1.29334i	−0.762768	+	0.646672i	$0.776160\pi$
$684$	−115.804	−	121.437i	−0.169304	−	0.177539i
$685$	−473.151			−0.690731
$686$		−	242.892i		−	0.354070i
$687$	329.530	−	769.589i	0.479665	−	1.12022i
$688$	268.949			0.390914
$689$			150.335i			0.218193i
$690$	−393.816	−	168.628i	−0.570748	−	0.244388i
$691$	137.792			0.199409			0.0997045	−	0.995017i	$-0.468210\pi$
$691$	137.792			0.199409			0.0997045	+	0.995017i	$0.468210\pi$
$692$		−	494.159i		−	0.714102i
$693$	366.887	−	349.868i	0.529419	−	0.504860i
$694$	−241.306			−0.347703
$695$			195.517i			0.281319i
$696$	143.729	−	335.666i	0.206507	−	0.482279i
$697$	216.915			0.311212
$698$		−	437.992i		−	0.627496i
$699$	573.064	+	245.380i	0.819834	+	0.351044i
$700$	47.2810			0.0675443
$701$		−	53.7470i		−	0.0766719i	−0.999265	−	0.0383359i	$-0.987794\pi$
$701$		−	53.7470i		−	0.0766719i	0.999265	−	0.0383359i	$-0.0122057\pi$
$702$	−48.1236	−	128.989i	−0.0685521	−	0.183745i
$703$	−282.839			−0.402331
$704$		−	50.8192i		−	0.0721864i
$705$	312.507	−	729.833i	0.443272	−	1.03522i
$706$	−806.695			−1.14263
$707$			22.4711i			0.0317837i
$708$	239.174	+	102.412i	0.337817	+	0.144649i
$709$	1053.20			1.48548			0.742740	−	0.669580i	$-0.233526\pi$
$709$	1053.20			1.48548			0.742740	+	0.669580i	$0.233526\pi$
$710$		−	357.397i		−	0.503376i
$711$	589.179	+	617.839i	0.828663	+	0.868972i
$712$	−119.525			−0.167873
$713$			1017.93i			1.42767i
$714$	−336.004	+	784.708i	−0.470594	+	1.09903i
$715$	120.471			0.168491
$716$		−	306.689i		−	0.428337i
$717$	−1045.11	−	447.507i	−1.45762	−	0.624138i
$718$	−299.775			−0.417514
$719$		−	269.797i		−	0.375240i	−0.982242	−	0.187620i	$-0.939923\pi$
$719$		−	269.797i		−	0.375240i	0.982242	−	0.187620i	$-0.0600773\pi$
$720$	−137.034	+	130.678i	−0.190326	+	0.181497i
$721$	−928.071			−1.28720
$722$		−	387.628i		−	0.536881i
$723$	−185.724	+	433.742i	−0.256880	+	0.599920i
$724$	−134.789			−0.186172
$725$		−	114.724i		−	0.158241i
$726$	314.534	+	134.680i	0.433243	+	0.185510i
$727$	−237.727			−0.326997			−0.163498	−	0.986544i	$-0.552278\pi$
$727$	−237.727			−0.326997			−0.163498	+	0.986544i	$0.552278\pi$
$728$		−	90.4304i		−	0.124218i
$729$	550.855	−	477.493i	0.755631	−	0.654997i
$730$	−922.157			−1.26323
$731$			1525.60i			2.08700i
$732$	−251.610	+	587.613i	−0.343729	+	0.802751i
$733$	340.835			0.464986			0.232493	−	0.972598i	$-0.425312\pi$
$733$	340.835			0.464986			0.232493	+	0.972598i	$0.425312\pi$
$734$			711.186i			0.968918i
$735$	−429.822	−	184.045i	−0.584792	−	0.250402i
$736$	108.596			0.147549
$737$		−	288.547i		−	0.391516i
$738$	−83.9736	−	88.0584i	−0.113785	−	0.119320i
$739$	−117.744			−0.159329			−0.0796647	−	0.996822i	$-0.525385\pi$
$739$	−117.744			−0.159329			−0.0796647	+	0.996822i	$0.525385\pi$
$740$			319.167i			0.431307i
$741$	−39.6914	+	92.6960i	−0.0535647	+	0.125096i
$742$	−522.878			−0.704687
$743$			1183.02i			1.59222i	0.605153	+	0.796109i	$0.293112\pi$
$743$			1183.02i			1.59222i	−0.605153	+	0.796109i	$0.706888\pi$
$744$	413.606	+	177.102i	0.555922	+	0.238040i
$745$	606.740			0.814416
$746$			608.171i			0.815243i
$747$	−703.833	+	671.185i	−0.942213	+	0.898507i
$748$	288.269			0.385387
$749$		−	1594.85i		−	2.12931i
$750$	196.181	−	458.163i	0.261574	−	0.610884i
$751$	270.244			0.359845			0.179922	−	0.983681i	$-0.442415\pi$
$751$	270.244			0.359845			0.179922	+	0.983681i	$0.442415\pi$
$752$			201.254i			0.267625i
$753$	−654.006	−	280.038i	−0.868533	−	0.371897i
$754$	−219.424			−0.291013
$755$			326.837i			0.432897i
$756$	448.635	−	167.378i	0.593433	−	0.221400i
$757$	−503.174			−0.664695			−0.332347	−	0.943157i	$-0.607841\pi$
$757$	−503.174			−0.664695			−0.332347	+	0.943157i	$0.607841\pi$
$758$			700.197i			0.923743i
$759$	144.006	−	336.313i	0.189731	−	0.443100i
$760$	138.689			0.182486
$761$		−	48.5675i		−	0.0638207i	−0.999491	−	0.0319103i	$-0.989841\pi$
$761$		−	48.5675i		−	0.0638207i	0.999491	−	0.0319103i	$-0.0101591\pi$
$762$	−40.2450	−	17.2325i	−0.0528149	−	0.0226148i
$763$	925.720			1.21326
$764$		−	245.185i		−	0.320922i
$765$	−741.263	−	777.320i	−0.968971	−	1.01611i
$766$	126.497			0.165140
$767$		−	156.347i		−	0.203843i
$768$	18.8939	−	44.1251i	0.0246014	−	0.0574545i
$769$	−840.813			−1.09339			−0.546693	−	0.837333i	$-0.684113\pi$
$769$	−840.813			−1.09339			−0.546693	+	0.837333i	$0.684113\pi$
$770$			419.010i			0.544168i
$771$	−709.580	−	303.835i	−0.920337	−	0.394079i
$772$	−284.670			−0.368744
$773$			432.392i			0.559369i	0.960092	+	0.279684i	$0.0902298\pi$
$773$			432.392i			0.559369i	−0.960092	+	0.279684i	$0.909770\pi$
$774$	619.329	−	590.600i	0.800167	−	0.763049i
$775$	141.363			0.182404
$776$			250.353i			0.322620i
$777$	317.698	−	741.956i	0.408878	−	0.954899i
$778$	−322.747			−0.414841
$779$			89.1216i			0.114405i
$780$	104.602	+	44.7895i	0.134105	+	0.0574224i
$781$	305.211			0.390796
$782$			616.006i			0.787732i
$783$	−406.133	−	1088.59i	−0.518689	−	1.39028i
$784$	118.525			0.151180
$785$			1463.20i			1.86395i
$786$	−79.1767	+	184.911i	−0.100734	+	0.235255i
$787$	−56.7283			−0.0720817			−0.0360408	−	0.999350i	$-0.511475\pi$
$787$	−56.7283			−0.0720817			−0.0360408	+	0.999350i	$0.511475\pi$
$788$			47.7818i			0.0606368i
$789$	723.385	+	309.746i	0.916838	+	0.392580i
$790$	−705.613			−0.893181
$791$		−	284.389i		−	0.359531i
$792$	−111.597	−	117.025i	−0.140905	−	0.147759i
$793$	384.121			0.484390
$794$			299.180i			0.376801i
$795$	258.977	−	604.820i	0.325758	−	0.760780i
$796$	−243.785			−0.306263
$797$		−	918.907i		−	1.15296i	−0.817112	−	0.576479i	$-0.804426\pi$
$797$		−	918.907i		−	1.15296i	0.817112	−	0.576479i	$-0.195574\pi$
$798$	−322.405	−	138.050i	−0.404016	−	0.172996i
$799$	−1141.60			−1.42879
$800$		−	15.0811i		−	0.0188514i
$801$	−275.240	+	262.473i	−0.343621	+	0.327681i
$802$	546.495			0.681416
$803$		−	787.508i		−	0.980708i
$804$	107.278	−	250.538i	0.133430	−	0.311615i
$805$	−895.387			−1.11228
$806$		−	270.373i		−	0.335450i
$807$	328.827	+	140.800i	0.407469	+	0.174474i
$808$	7.16755			0.00887073
$809$			691.788i			0.855115i	0.903988	+	0.427557i	$0.140626\pi$
$809$			691.788i			0.855115i	−0.903988	+	0.427557i	$0.859374\pi$
$810$	−28.5968	+	601.843i	−0.0353047	+	0.743016i
$811$	613.882			0.756945			0.378472	−	0.925613i	$-0.376449\pi$
$811$	613.882			0.756945			0.378472	+	0.925613i	$0.376449\pi$
$812$		−	763.176i		−	0.939872i
$813$	105.191	−	245.666i	0.129387	−	0.302172i
$814$	−272.564			−0.334845
$815$			41.8648i			0.0513679i
$816$	250.297	+	107.175i	0.306737	+	0.131341i
$817$	−626.807			−0.767206
$818$			117.634i			0.143807i
$819$	−198.581	−	208.241i	−0.242468	−	0.254262i
$820$	100.569			0.122645
$821$			199.766i			0.243320i	0.992572	+	0.121660i	$0.0388218\pi$
$821$			199.766i			0.243320i	−0.992572	+	0.121660i	$0.961178\pi$
$822$	150.225	−	350.838i	0.182756	−	0.426811i
$823$	479.789			0.582976			0.291488	−	0.956575i	$-0.405850\pi$
$823$	479.789			0.582976			0.291488	+	0.956575i	$0.405850\pi$
$824$			296.025i			0.359254i
$825$	−46.7049	−	19.9985i	−0.0566120	−	0.0242407i
$826$	543.790			0.658342
$827$			954.357i			1.15400i	0.816744	+	0.577000i	$0.195777\pi$
$827$			954.357i			1.15400i	−0.816744	+	0.577000i	$0.804223\pi$
$828$	250.073	−	238.473i	0.302020	−	0.288010i
$829$	539.188			0.650408			0.325204	−	0.945644i	$-0.394567\pi$
$829$	539.188			0.650408			0.325204	+	0.945644i	$0.394567\pi$
$830$		−	803.825i		−	0.968464i
$831$	−221.462	+	517.206i	−0.266501	+	0.622390i
$832$	−28.8444			−0.0346688
$833$			672.327i			0.807116i
$834$	−144.974	−	62.0765i	−0.173830	−	0.0744323i
$835$	571.089			0.683939
$836$			118.438i			0.141673i
$837$	1341.35	−	500.435i	1.60257	−	0.597892i
$838$	661.898			0.789854
$839$			823.873i			0.981971i	0.871168	+	0.490985i	$0.163363\pi$
$839$			823.873i			0.981971i	−0.871168	+	0.490985i	$0.836637\pi$
$840$	−155.782	+	363.815i	−0.185455	+	0.433114i
$841$	−1010.80			−1.20190
$842$		−	136.498i		−	0.162111i
$843$	−793.697	−	339.853i	−0.941515	−	0.403147i
$844$	−400.075			−0.474023
$845$		−	68.3780i		−	0.0809207i
$846$	441.945	+	463.443i	0.522394	+	0.547805i
$847$	715.131			0.844310
$848$			166.781i			0.196676i
$849$	−330.003	+	770.694i	−0.388696	+	0.907766i
$850$	85.5469			0.100643
$851$		−	582.445i		−	0.684425i
$852$	265.007	+	113.473i	0.311041	+	0.133185i
$853$	566.441			0.664057			0.332028	−	0.943269i	$-0.392267\pi$
$853$	566.441			0.664057			0.332028	+	0.943269i	$0.392267\pi$
$854$			1336.01i			1.56441i
$855$	319.370	−	304.555i	0.373532	−	0.356205i
$856$	−508.708			−0.594285
$857$		−	1597.01i		−	1.86349i	−0.363119	−	0.931743i	$-0.618288\pi$
$857$		−	1597.01i		−	1.86349i	0.363119	−	0.931743i	$-0.381712\pi$
$858$	−38.2495	+	89.3285i	−0.0445799	+	0.104113i
$859$	−519.003			−0.604194			−0.302097	−	0.953277i	$-0.597687\pi$
$859$	−519.003			−0.604194			−0.302097	+	0.953277i	$0.597687\pi$
$860$			707.315i			0.822460i
$861$	−233.788	−	100.106i	−0.271531	−	0.116267i
$862$	380.076			0.440924
$863$		−	552.630i		−	0.640359i	−0.947357	−	0.320180i	$-0.896257\pi$
$863$		−	552.630i		−	0.640359i	0.947357	−	0.320180i	$-0.103743\pi$
$864$	−53.3883	−	143.100i	−0.0617921	−	0.165625i
$865$	1299.60			1.50243
$866$			166.337i			0.192075i
$867$	−266.671	+	622.787i	−0.307579	+	0.718324i
$868$	940.382			1.08339
$869$		−	602.583i		−	0.693421i
$870$	882.776	+	377.995i	1.01469	+	0.434477i
$871$	−163.776			−0.188032
$872$		−	295.275i		−	0.338618i
$873$	549.764	+	576.507i	0.629742	+	0.660374i
$874$	−253.092			−0.289579
$875$		−	1041.69i		−	1.19050i
$876$	292.785	−	683.773i	0.334229	−	0.780563i
$877$	879.577			1.00294			0.501469	−	0.865176i	$-0.332793\pi$
$877$	879.577			1.00294			0.501469	+	0.865176i	$0.332793\pi$
$878$		−	388.943i		−	0.442987i
$879$	1361.00	+	582.766i	1.54835	+	0.662988i
$880$	133.651			0.151876
$881$		−	391.641i		−	0.444541i	−0.974985	−	0.222270i	$-0.928653\pi$
$881$		−	391.641i		−	0.444541i	0.974985	−	0.222270i	$-0.0713468\pi$
$882$	272.937	−	260.276i	0.309452	−	0.295098i
$883$	−535.770			−0.606761			−0.303380	−	0.952870i	$-0.598115\pi$
$883$	−535.770			−0.606761			−0.303380	+	0.952870i	$0.598115\pi$
$884$		−	163.618i		−	0.185089i
$885$	−269.335	+	629.010i	−0.304334	+	0.710745i
$886$	759.221			0.856909
$887$			1093.47i			1.23278i	0.787443	+	0.616388i	$0.211405\pi$
$887$			1093.47i			1.23278i	−0.787443	+	0.616388i	$0.788595\pi$
$888$	−236.660	−	101.336i	−0.266510	−	0.114117i
$889$	−91.5017			−0.102927
$890$		−	314.343i		−	0.353194i
$891$	−513.965	−	24.4213i	−0.576841	−	0.0274088i
$892$	273.739			0.306883
$893$		−	469.039i		−	0.525239i
$894$	−192.640	+	449.893i	−0.215481	+	0.503236i
$895$	806.569			0.901194
$896$		−	100.323i		−	0.111968i
$897$	−190.887	−	81.7359i	−0.212806	−	0.0911214i
$898$	−519.466			−0.578470
$899$		−	2281.78i		−	2.53813i
$900$	−33.1175	−	34.7285i	−0.0367972	−	0.0385872i
$901$	−946.058			−1.05001
$902$			85.8840i			0.0952151i
$903$	704.059	−	1644.27i	0.779689	−	1.82090i
$904$	−90.7110			−0.100344
$905$		−	354.484i		−	0.391695i
$906$	−242.347	−	103.771i	−0.267492	−	0.114537i
$907$	−911.197			−1.00463			−0.502314	−	0.864685i	$-0.667518\pi$
$907$	−911.197			−1.00463			−0.502314	+	0.864685i	$0.667518\pi$
$908$			558.750i			0.615363i
$909$	16.5053	−	15.7396i	0.0181576	−	0.0173153i
$910$	237.825			0.261346
$911$		−	148.918i		−	0.163466i	−0.996654	−	0.0817331i	$-0.973954\pi$
$911$		−	148.918i		−	0.163466i	0.996654	−	0.0817331i	$-0.0260455\pi$
$912$	−44.0337	+	102.837i	−0.0482826	+	0.112760i
$913$	686.454			0.751866
$914$		−	477.787i		−	0.522743i
$915$	−1545.38	−	661.715i	−1.68894	−	0.723185i
$916$	−558.115			−0.609296
$917$			420.416i			0.458469i
$918$	811.728	−	302.842i	0.884236	−	0.329894i
$919$	1073.76			1.16840			0.584201	−	0.811609i	$-0.301408\pi$
$919$	1073.76			1.16840			0.584201	+	0.811609i	$0.301408\pi$
$920$			285.600i			0.310435i
$921$	−96.4131	+	225.165i	−0.104683	+	0.244478i
$922$	−277.751			−0.301249
$923$		−	173.234i		−	0.187686i
$924$	−310.693	−	133.035i	−0.336248	−	0.143978i
$925$	−80.8862			−0.0874445
$926$			952.105i			1.02819i
$927$	650.058	+	681.679i	0.701249	+	0.735360i
$928$	−243.429			−0.262316
$929$			1276.49i			1.37405i	0.726634	+	0.687025i	$0.241084\pi$
$929$			1276.49i			1.37405i	−0.726634	+	0.687025i	$0.758916\pi$
$930$	−465.764	+	1087.75i	−0.500821	+	1.16963i
$931$	−276.232			−0.296705
$932$		−	415.592i		−	0.445915i
$933$	−757.138	−	324.198i	−0.811509	−	0.347480i
$934$	−84.2978			−0.0902546
$935$			758.126i			0.810830i
$936$	−66.4222	+	63.3411i	−0.0709639	+	0.0676721i
$937$	−335.148			−0.357682			−0.178841	−	0.983878i	$-0.557235\pi$
$937$	−335.148			−0.357682			−0.178841	+	0.983878i	$0.557235\pi$
$938$		−	569.628i		−	0.607280i
$939$	555.044	−	1296.26i	0.591102	−	1.38047i
$940$	−529.283			−0.563067
$941$		−	645.432i		−	0.685900i	−0.939354	−	0.342950i	$-0.888574\pi$
$941$		−	645.432i		−	0.685900i	0.939354	−	0.342950i	$-0.111426\pi$
$942$	−1084.95	−	464.565i	−1.15175	−	0.493169i
$943$	−183.527			−0.194620
$944$		−	173.452i		−	0.183741i
$945$	440.192	+	1179.88i	0.465812	+	1.24855i
$946$	−604.036			−0.638516
$947$			89.2133i			0.0942062i	0.998890	+	0.0471031i	$0.0149989\pi$
$947$			89.2133i			0.0942062i	−0.998890	+	0.0471031i	$0.985001\pi$
$948$	224.032	−	523.207i	0.236321	−	0.551907i
$949$	−446.980			−0.471002
$950$			35.1478i			0.0369976i
$951$	−1039.19	−	444.970i	−1.09273	−	0.467896i
$952$	569.080			0.597773
$953$			835.022i			0.876203i	0.898925	+	0.438102i	$0.144349\pi$
$953$			835.022i			0.876203i	−0.898925	+	0.438102i	$0.855651\pi$
$954$	366.245	+	384.060i	0.383904	+	0.402579i
$955$	644.817			0.675201
$956$			757.929i			0.792813i
$957$	−322.802	+	753.877i	−0.337306	+	0.787750i
$958$	321.788			0.335895
$959$		−	797.672i		−	0.831775i
$960$	116.046	+	49.6895i	0.120881	+	0.0517599i
$961$	1850.60			1.92570
$962$			154.704i			0.160815i
$963$	−1171.44	+	1117.10i	−1.21645	+	1.16002i
$964$	314.555			0.326302
$965$		−	748.660i		−	0.775814i
$966$	284.285	−	663.923i	0.294291	−	0.687291i
$967$	414.770			0.428925			0.214462	−	0.976732i	$-0.431200\pi$
$967$	414.770			0.428925			0.214462	+	0.976732i	$0.431200\pi$
$968$		−	228.104i		−	0.235645i
$969$	−583.337	−	249.779i	−0.601999	−	0.257770i
$970$	−658.409			−0.678773
$971$			1520.43i			1.56584i	0.622122	+	0.782920i	$0.286271\pi$
$971$			1520.43i			1.56584i	−0.622122	+	0.782920i	$0.713729\pi$
$972$	−437.183	−	212.289i	−0.449777	−	0.218405i
$973$	−329.616			−0.338763
$974$		−	204.507i		−	0.209966i
$975$	−11.3509	+	26.5092i	−0.0116420	+	0.0271889i
$976$	426.144			0.436623
$977$			1010.30i			1.03409i	0.855960	+	0.517043i	$0.172967\pi$
$977$			1010.30i			1.03409i	−0.855960	+	0.517043i	$0.827033\pi$
$978$	−31.0425	−	13.2921i	−0.0317408	−	0.0135911i
$979$	268.444			0.274202
$980$			311.712i			0.318074i
$981$	−648.411	−	679.952i	−0.660970	−	0.693122i
$982$	−665.146			−0.677339
$983$		−	337.991i		−	0.343836i	−0.985111	−	0.171918i	$-0.945004\pi$
$983$		−	337.991i		−	0.343836i	0.985111	−	0.171918i	$-0.0549964\pi$
$984$	−31.9305	+	74.5709i	−0.0324497	+	0.0757835i
$985$	−125.662			−0.127576
$986$		−	1380.84i		−	1.40044i
$987$	1230.40	+	526.846i	1.24661	+	0.533785i
$988$	67.2242			0.0680407
$989$		−	1290.77i		−	1.30513i
$990$	307.768	−	293.491i	0.310876	−	0.296456i
$991$	−939.188			−0.947717			−0.473859	−	0.880601i	$-0.657139\pi$
$991$	−939.188			−0.947717			−0.473859	+	0.880601i	$0.657139\pi$
$992$		−	299.952i		−	0.302371i
$993$	−361.037	+	843.171i	−0.363582	+	0.849115i
$994$	602.525			0.606162
$995$		−	641.136i		−	0.644358i
$996$	596.031	+	255.214i	0.598424	+	0.256239i
$997$	−1882.58			−1.88825			−0.944123	−	0.329593i	$-0.893089\pi$
$997$	−1882.58			−1.88825			−0.944123	+	0.329593i	$0.893089\pi$
$998$			88.2530i			0.0884298i
$999$	−767.504	+	286.343i	−0.768273	+	0.286630i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	78.3.c.a.53.7	yes	8
3.2	odd	2	inner	78.3.c.a.53.3	✓	8
4.3	odd	2		624.3.f.b.209.4		8
12.11	even	2		624.3.f.b.209.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.3.c.a.53.3	✓	8	3.2	odd	2	inner
78.3.c.a.53.7	yes	8	1.1	even	1	trivial
624.3.f.b.209.3		8	12.11	even	2
624.3.f.b.209.4		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 78 = 2 \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	78.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(1.18087 - 2.75782i) q^{3} -2.00000 q^{4} -5.25985i q^{5} +(3.90014 + 1.67000i) q^{6} +8.86743 q^{7} -2.82843i q^{8} +(-6.21110 - 6.51323i) q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} +(1.18087 - 2.75782i) q^{3} -2.00000 q^{4} -5.25985i q^{5} +(3.90014 + 1.67000i) q^{6} +8.86743 q^{7} -2.82843i q^{8} +(-6.21110 - 6.51323i) q^{9} +7.43855 q^{10} +6.35241i q^{11} +(-2.36174 + 5.51563i) q^{12} +3.60555 q^{13} +12.5404i q^{14} +(-14.5057 - 6.21118i) q^{15} +4.00000 q^{16} +22.6898i q^{17} +(9.21110 - 8.78383i) q^{18} -9.32232 q^{19} +10.5197i q^{20} +(10.4713 - 24.4547i) q^{21} -8.98366 q^{22} -19.1973i q^{23} +(-7.80028 - 3.34000i) q^{24} -2.66599 q^{25} +5.09902i q^{26} +(-25.2968 + 9.43781i) q^{27} -17.7349 q^{28} +43.0325i q^{29} +(8.78394 - 20.5141i) q^{30} -53.0245 q^{31} +5.65685i q^{32} +(17.5188 + 7.50135i) q^{33} -32.0882 q^{34} -46.6413i q^{35} +(12.4222 + 13.0265i) q^{36} +30.3400 q^{37} -13.1837i q^{38} +(4.25768 - 9.94345i) q^{39} -14.8771 q^{40} -9.56003i q^{41} +(34.5842 + 14.8086i) q^{42} +67.2373 q^{43} -12.7048i q^{44} +(-34.2586 + 32.6694i) q^{45} +27.1491 q^{46} +50.3135i q^{47} +(4.72347 - 11.0313i) q^{48} +29.6313 q^{49} -3.77028i q^{50} +(62.5743 + 26.7936i) q^{51} -7.21110 q^{52} +41.6953i q^{53} +(-13.3471 - 35.7751i) q^{54} +33.4127 q^{55} -25.0809i q^{56} +(-11.0084 + 25.7092i) q^{57} -60.8572 q^{58} -43.3630i q^{59} +(29.0114 + 12.4224i) q^{60} +106.536 q^{61} -74.9880i q^{62} +(-55.0765 - 57.7556i) q^{63} -8.00000 q^{64} -18.9646i q^{65} +(-10.6085 + 24.7753i) q^{66} -45.4233 q^{67} -45.3796i q^{68} +(-52.9426 - 22.6695i) q^{69} +65.9608 q^{70} -48.0466i q^{71} +(-18.4222 + 17.5677i) q^{72} -123.970 q^{73} +42.9072i q^{74} +(-3.14818 + 7.35232i) q^{75} +18.6446 q^{76} +56.3295i q^{77} +(14.0622 + 6.02127i) q^{78} -94.8590 q^{79} -21.0394i q^{80} +(-3.84441 + 80.9087i) q^{81} +13.5199 q^{82} -108.062i q^{83} +(-20.9425 + 48.9095i) q^{84} +119.345 q^{85} +95.0878i q^{86} +(118.676 + 50.8157i) q^{87} +17.9673 q^{88} -42.2586i q^{89} +(-46.2016 - 48.4490i) q^{90} +31.9720 q^{91} +38.3946i q^{92} +(-62.6149 + 146.232i) q^{93} -71.1541 q^{94} +49.0340i q^{95} +(15.6006 + 6.68000i) q^{96} -88.5132 q^{97} +41.9050i q^{98} +(41.3747 - 39.4554i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} - 8 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q - 16 * q^4 - 8 * q^7 + 8 * q^9	\( 8 q - 16 q^{4} - 8 q^{7} + 8 q^{9} + 16 q^{10} - 56 q^{15} + 32 q^{16} + 16 q^{18} - 24 q^{19} + 88 q^{21} + 8 q^{25} + 16 q^{28} - 64 q^{30} - 72 q^{31} - 56 q^{33} - 112 q^{34} - 16 q^{36} + 96 q^{37} - 32 q^{40} + 80 q^{42} + 208 q^{43} - 16 q^{45} + 32 q^{46} - 24 q^{49} + 112 q^{51} - 176 q^{55} + 40 q^{57} + 176 q^{58} + 112 q^{60} - 272 q^{61} - 216 q^{63} - 64 q^{64} - 64 q^{66} + 264 q^{67} - 240 q^{69} - 32 q^{70} - 32 q^{72} - 416 q^{73} + 128 q^{75} + 48 q^{76} + 160 q^{79} + 200 q^{81} + 176 q^{82} - 176 q^{84} + 464 q^{85} + 384 q^{87} + 16 q^{90} + 104 q^{91} - 264 q^{93} - 384 q^{94} - 16 q^{97} + 416 q^{99}+O(q^{100}) \) 8 * q - 16 * q^4 - 8 * q^7 + 8 * q^9 + 16 * q^10 - 56 * q^15 + 32 * q^16 + 16 * q^18 - 24 * q^19 + 88 * q^21 + 8 * q^25 + 16 * q^28 - 64 * q^30 - 72 * q^31 - 56 * q^33 - 112 * q^34 - 16 * q^36 + 96 * q^37 - 32 * q^40 + 80 * q^42 + 208 * q^43 - 16 * q^45 + 32 * q^46 - 24 * q^49 + 112 * q^51 - 176 * q^55 + 40 * q^57 + 176 * q^58 + 112 * q^60 - 272 * q^61 - 216 * q^63 - 64 * q^64 - 64 * q^66 + 264 * q^67 - 240 * q^69 - 32 * q^70 - 32 * q^72 - 416 * q^73 + 128 * q^75 + 48 * q^76 + 160 * q^79 + 200 * q^81 + 176 * q^82 - 176 * q^84 + 464 * q^85 + 384 * q^87 + 16 * q^90 + 104 * q^91 - 264 * q^93 - 384 * q^94 - 16 * q^97 + 416 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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