LMFDB - Embedded newform 342.4.g.d.163.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [342,4,Mod(163,342)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("342.163"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(342, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	342.g (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,2,0,-4,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$20.1786532220$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			163.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	342.163
Dual form			342.4.g.d.235.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(1.50000 - 2.59808i) q^{5} -32.0000 q^{7} -8.00000 q^{8} +(-3.00000 - 5.19615i) q^{10} -4.00000 q^{11} +(34.5000 + 59.7558i) q^{13} +(-32.0000 + 55.4256i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(9.50000 - 16.4545i) q^{17} +(76.0000 + 32.9090i) q^{19} -12.0000 q^{20} +(-4.00000 + 6.92820i) q^{22} +(33.5000 + 58.0237i) q^{23} +(58.0000 + 100.459i) q^{25} +138.000 q^{26} +(64.0000 + 110.851i) q^{28} +(25.5000 + 44.1673i) q^{29} -132.000 q^{31} +(16.0000 + 27.7128i) q^{32} +(-19.0000 - 32.9090i) q^{34} +(-48.0000 + 83.1384i) q^{35} -14.0000 q^{37} +(133.000 - 98.7269i) q^{38} +(-12.0000 + 20.7846i) q^{40} +(-206.500 + 357.668i) q^{41} +(-64.5000 + 111.717i) q^{43} +(8.00000 + 13.8564i) q^{44} +134.000 q^{46} +(-308.500 - 534.338i) q^{47} +681.000 q^{49} +232.000 q^{50} +(138.000 - 239.023i) q^{52} +(191.500 + 331.688i) q^{53} +(-6.00000 + 10.3923i) q^{55} +256.000 q^{56} +102.000 q^{58} +(-299.500 + 518.749i) q^{59} +(108.500 + 187.928i) q^{61} +(-132.000 + 228.631i) q^{62} +64.0000 q^{64} +207.000 q^{65} +(112.500 + 194.856i) q^{67} -76.0000 q^{68} +(96.0000 + 166.277i) q^{70} +(350.500 - 607.084i) q^{71} +(-507.500 + 879.016i) q^{73} +(-14.0000 + 24.2487i) q^{74} +(-38.0000 - 329.090i) q^{76} +128.000 q^{77} +(-174.500 + 302.243i) q^{79} +(24.0000 + 41.5692i) q^{80} +(413.000 + 715.337i) q^{82} +592.000 q^{83} +(-28.5000 - 49.3634i) q^{85} +(129.000 + 223.435i) q^{86} +32.0000 q^{88} +(-674.500 - 1168.27i) q^{89} +(-1104.00 - 1912.18i) q^{91} +(134.000 - 232.095i) q^{92} -1234.00 q^{94} +(199.500 - 148.090i) q^{95} +(306.500 - 530.874i) q^{97} +(681.000 - 1179.53i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{4} + 3 q^{5} - 64 q^{7} - 16 q^{8} - 6 q^{10} - 8 q^{11} + 69 q^{13} - 64 q^{14} - 16 q^{16} + 19 q^{17} + 152 q^{19} - 24 q^{20} - 8 q^{22} + 67 q^{23} + 116 q^{25} + 276 q^{26} + 128 q^{28}+ \cdots + 1362 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^4 + 3 * q^5 - 64 * q^7 - 16 * q^8 - 6 * q^10 - 8 * q^11 + 69 * q^13 - 64 * q^14 - 16 * q^16 + 19 * q^17 + 152 * q^19 - 24 * q^20 - 8 * q^22 + 67 * q^23 + 116 * q^25 + 276 * q^26 + 128 * q^28 + 51 * q^29 - 264 * q^31 + 32 * q^32 - 38 * q^34 - 96 * q^35 - 28 * q^37 + 266 * q^38 - 24 * q^40 - 413 * q^41 - 129 * q^43 + 16 * q^44 + 268 * q^46 - 617 * q^47 + 1362 * q^49 + 464 * q^50 + 276 * q^52 + 383 * q^53 - 12 * q^55 + 512 * q^56 + 204 * q^58 - 599 * q^59 + 217 * q^61 - 264 * q^62 + 128 * q^64 + 414 * q^65 + 225 * q^67 - 152 * q^68 + 192 * q^70 + 701 * q^71 - 1015 * q^73 - 28 * q^74 - 76 * q^76 + 256 * q^77 - 349 * q^79 + 48 * q^80 + 826 * q^82 + 1184 * q^83 - 57 * q^85 + 258 * q^86 + 64 * q^88 - 1349 * q^89 - 2208 * q^91 + 268 * q^92 - 2468 * q^94 + 399 * q^95 + 613 * q^97 + 1362 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	1.00000	−	1.73205i	0.353553	−	0.612372i
$2$	1.00000	−	1.73205i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.250000	−	0.433013i
$5$	1.50000	−	2.59808i	0.134164	−	0.232379i	−0.791114	−	0.611669i	$-0.790498\pi$
$5$	1.50000	−	2.59808i	0.134164	−	0.232379i	0.925278	+	0.379290i	$0.123832\pi$
$6$		0			0
$7$	−32.0000			−1.72784			−0.863919	−	0.503631i	$-0.831997\pi$
$7$	−32.0000			−1.72784			−0.863919	+	0.503631i	$0.831997\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$	−3.00000	−	5.19615i	−0.0948683	−	0.164317i
$11$	−4.00000			−0.109640			−0.0548202	−	0.998496i	$-0.517459\pi$
$11$	−4.00000			−0.109640			−0.0548202	+	0.998496i	$0.517459\pi$
$12$		0			0
$13$	34.5000	+	59.7558i	0.736044	+	1.27487i	0.954264	+	0.298967i	$0.0966419\pi$
$13$	34.5000	+	59.7558i	0.736044	+	1.27487i	−0.218219	+	0.975900i	$0.570025\pi$
$14$	−32.0000	+	55.4256i	−0.610883	+	1.05808i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	9.50000	−	16.4545i	0.135535	−	0.234753i	−0.790267	−	0.612763i	$-0.790058\pi$
$17$	9.50000	−	16.4545i	0.135535	−	0.234753i	0.925802	+	0.378010i	$0.123391\pi$
$18$		0			0
$19$	76.0000	+	32.9090i	0.917663	+	0.397360i
$19$	76.0000	+	32.9090i	0.917663	+	0.397360i
$20$	−12.0000			−0.134164
$21$		0			0
$22$	−4.00000	+	6.92820i	−0.0387638	+	0.0671408i
$23$	33.5000	+	58.0237i	0.303706	+	0.526034i	0.976972	−	0.213366i	$-0.0684427\pi$
$23$	33.5000	+	58.0237i	0.303706	+	0.526034i	−0.673267	+	0.739400i	$0.735109\pi$
$24$		0			0
$25$	58.0000	+	100.459i	0.464000	+	0.803672i
$26$	138.000			1.04092
$27$		0			0
$28$	64.0000	+	110.851i	0.431959	+	0.748176i
$29$	25.5000	+	44.1673i	0.163284	+	0.282816i	0.936045	−	0.351882i	$-0.114458\pi$
$29$	25.5000	+	44.1673i	0.163284	+	0.282816i	−0.772761	+	0.634698i	$0.781125\pi$
$30$		0			0
$31$	−132.000			−0.764771			−0.382385	−	0.924003i	$-0.624897\pi$
$31$	−132.000			−0.764771			−0.382385	+	0.924003i	$0.624897\pi$
$32$	16.0000	+	27.7128i	0.0883883	+	0.153093i
$33$		0			0
$34$	−19.0000	−	32.9090i	−0.0958374	−	0.165995i
$35$	−48.0000	+	83.1384i	−0.231814	+	0.401513i
$36$		0			0
$37$	−14.0000			−0.0622050			−0.0311025	−	0.999516i	$-0.509902\pi$
$37$	−14.0000			−0.0622050			−0.0311025	+	0.999516i	$0.509902\pi$
$38$	133.000	−	98.7269i	0.567775	−	0.421464i
$39$		0			0
$40$	−12.0000	+	20.7846i	−0.0474342	+	0.0821584i
$41$	−206.500	+	357.668i	−0.786582	+	1.36240i	0.141467	+	0.989943i	$0.454818\pi$
$41$	−206.500	+	357.668i	−0.786582	+	1.36240i	−0.928049	+	0.372458i	$0.878515\pi$
$42$		0			0
$43$	−64.5000	+	111.717i	−0.228748	+	0.396203i	−0.957437	−	0.288641i	$-0.906796\pi$
$43$	−64.5000	+	111.717i	−0.228748	+	0.396203i	0.728689	+	0.684844i	$0.240130\pi$
$44$	8.00000	+	13.8564i	0.0274101	+	0.0474757i
$45$		0			0
$46$	134.000			0.429505
$47$	−308.500	−	534.338i	−0.957433	−	1.65832i	−0.728700	−	0.684833i	$-0.759875\pi$
$47$	−308.500	−	534.338i	−0.957433	−	1.65832i	−0.228733	−	0.973489i	$-0.573458\pi$
$48$		0			0
$49$	681.000			1.98542
$50$	232.000			0.656195
$51$		0			0
$52$	138.000	−	239.023i	0.368022	−	0.637433i
$53$	191.500	+	331.688i	0.496312	+	0.859638i	0.999991	−	0.00425305i	$-0.00135379\pi$
$53$	191.500	+	331.688i	0.496312	+	0.859638i	−0.503679	+	0.863891i	$0.668020\pi$
$54$		0			0
$55$	−6.00000	+	10.3923i	−0.0147098	+	0.0254781i
$56$	256.000			0.610883
$57$		0			0
$58$	102.000			0.230918
$59$	−299.500	+	518.749i	−0.660874	+	1.14467i	0.319512	+	0.947582i	$0.396481\pi$
$59$	−299.500	+	518.749i	−0.660874	+	1.14467i	−0.980386	+	0.197086i	$0.936852\pi$
$60$		0			0
$61$	108.500	+	187.928i	0.227738	+	0.394453i	0.957137	−	0.289635i	$-0.0935338\pi$
$61$	108.500	+	187.928i	0.227738	+	0.394453i	−0.729400	+	0.684088i	$0.760200\pi$
$62$	−132.000	+	228.631i	−0.270387	+	0.468325i
$63$		0			0
$64$	64.0000			0.125000
$65$	207.000			0.395003
$66$		0			0
$67$	112.500	+	194.856i	0.205135	+	0.355305i	0.950176	−	0.311714i	$-0.100903\pi$
$67$	112.500	+	194.856i	0.205135	+	0.355305i	−0.745041	+	0.667019i	$0.767570\pi$
$68$	−76.0000			−0.135535
$69$		0			0
$70$	96.0000	+	166.277i	0.163917	+	0.283913i
$71$	350.500	−	607.084i	0.585869	−	1.01475i	−0.408898	−	0.912580i	$-0.634087\pi$
$71$	350.500	−	607.084i	0.585869	−	1.01475i	0.994766	−	0.102175i	$-0.0325800\pi$
$72$		0			0
$73$	−507.500	+	879.016i	−0.813676	+	1.40933i	0.0965979	+	0.995323i	$0.469204\pi$
$73$	−507.500	+	879.016i	−0.813676	+	1.40933i	−0.910274	+	0.414005i	$0.864129\pi$
$74$	−14.0000	+	24.2487i	−0.0219928	+	0.0380926i
$75$		0			0
$76$	−38.0000	−	329.090i	−0.0573539	−	0.496700i
$77$	128.000			0.189441
$78$		0			0
$79$	−174.500	+	302.243i	−0.248516	+	0.430443i	−0.963114	−	0.269092i	$-0.913276\pi$
$79$	−174.500	+	302.243i	−0.248516	+	0.430443i	0.714598	+	0.699535i	$0.246610\pi$
$80$	24.0000	+	41.5692i	0.0335410	+	0.0580948i
$81$		0			0
$82$	413.000	+	715.337i	0.556198	+	0.963363i
$83$	592.000			0.782897			0.391448	−	0.920200i	$-0.371974\pi$
$83$	592.000			0.782897			0.391448	+	0.920200i	$0.371974\pi$
$84$		0			0
$85$	−28.5000	−	49.3634i	−0.0363678	−	0.0629908i
$86$	129.000	+	223.435i	0.161749	+	0.280158i
$87$		0			0
$88$	32.0000			0.0387638
$89$	−674.500	−	1168.27i	−0.803335	−	1.39142i	−0.917409	−	0.397946i	$-0.869723\pi$
$89$	−674.500	−	1168.27i	−0.803335	−	1.39142i	0.114074	−	0.993472i	$-0.463610\pi$
$90$		0			0
$91$	−1104.00	−	1912.18i	−1.27177	−	2.20276i
$92$	134.000	−	232.095i	0.151853	−	0.263017i
$93$		0			0
$94$	−1234.00			−1.35401
$95$	199.500	−	148.090i	0.215455	−	0.159934i
$96$		0			0
$97$	306.500	−	530.874i	0.320828	−	0.555691i	−0.659831	−	0.751414i	$-0.729372\pi$
$97$	306.500	−	530.874i	0.320828	−	0.555691i	0.980659	+	0.195723i	$0.0627054\pi$
$98$	681.000	−	1179.53i	0.701953	−	1.21582i
$99$		0			0
$100$	232.000	−	401.836i	0.232000	−	0.401836i
$101$	617.500	+	1069.54i	0.608352	+	1.05370i	0.991512	+	0.130015i	$0.0415024\pi$
$101$	617.500	+	1069.54i	0.608352	+	1.05370i	−0.383160	+	0.923682i	$0.625164\pi$
$102$		0			0
$103$	1632.00			1.56122			0.780610	−	0.625018i	$-0.214908\pi$
$103$	1632.00			1.56122			0.780610	+	0.625018i	$0.214908\pi$
$104$	−276.000	−	478.046i	−0.260231	−	0.450733i
$105$		0			0
$106$	766.000			0.701891
$107$	−2060.00			−1.86119			−0.930597	−	0.366046i	$-0.880711\pi$
$107$	−2060.00			−1.86119			−0.930597	+	0.366046i	$0.880711\pi$
$108$		0			0
$109$	−527.500	+	913.657i	−0.463535	+	0.802867i	−0.999134	−	0.0416060i	$-0.986753\pi$
$109$	−527.500	+	913.657i	−0.463535	+	0.802867i	0.535599	+	0.844473i	$0.320086\pi$
$110$	12.0000	+	20.7846i	0.0104014	+	0.0180158i
$111$		0			0
$112$	256.000	−	443.405i	0.215980	−	0.374088i
$113$	−1006.00			−0.837491			−0.418746	−	0.908104i	$-0.637530\pi$
$113$	−1006.00			−0.837491			−0.418746	+	0.908104i	$0.637530\pi$
$114$		0			0
$115$	201.000			0.162986
$116$	102.000	−	176.669i	0.0816419	−	0.141408i
$117$		0			0
$118$	599.000	+	1037.50i	0.467309	+	0.809402i
$119$	−304.000	+	526.543i	−0.234182	+	0.405615i
$120$		0			0
$121$	−1315.00			−0.987979
$122$	434.000			0.322070
$123$		0			0
$124$	264.000	+	457.261i	0.191193	+	0.331156i
$125$	723.000			0.517337
$126$		0			0
$127$	−993.500	−	1720.79i	−0.694164	−	1.20233i	−0.970462	−	0.241256i	$-0.922441\pi$
$127$	−993.500	−	1720.79i	−0.694164	−	1.20233i	0.276297	−	0.961072i	$-0.410893\pi$
$128$	64.0000	−	110.851i	0.0441942	−	0.0765466i
$129$		0			0
$130$	207.000	−	358.535i	0.139655	−	0.241889i
$131$	−901.500	+	1561.44i	−0.601255	+	1.04140i	0.391376	+	0.920231i	$0.371999\pi$
$131$	−901.500	+	1561.44i	−0.601255	+	1.04140i	−0.992631	+	0.121174i	$0.961334\pi$
$132$		0			0
$133$	−2432.00	−	1053.09i	−1.58557	−	0.686573i
$134$	450.000			0.290105
$135$		0			0
$136$	−76.0000	+	131.636i	−0.0479187	+	0.0829977i
$137$	−334.500	−	579.371i	−0.208600	−	0.361307i	0.742673	−	0.669654i	$-0.233557\pi$
$137$	−334.500	−	579.371i	−0.208600	−	0.361307i	−0.951274	+	0.308347i	$0.900224\pi$
$138$		0			0
$139$	1362.50	+	2359.92i	0.831408	+	1.44004i	0.896922	+	0.442190i	$0.145798\pi$
$139$	1362.50	+	2359.92i	0.831408	+	1.44004i	−0.0655134	+	0.997852i	$0.520869\pi$
$140$	384.000			0.231814
$141$		0			0
$142$	−701.000	−	1214.17i	−0.414272	−	0.717540i
$143$	−138.000	−	239.023i	−0.0807003	−	0.139777i
$144$		0			0
$145$	153.000			0.0876273
$146$	1015.00	+	1758.03i	0.575356	+	0.996546i
$147$		0			0
$148$	28.0000	+	48.4974i	0.0155513	+	0.0269356i
$149$	35.5000	−	61.4878i	0.0195186	−	0.0338072i	−0.856101	−	0.516808i	$-0.827120\pi$
$149$	35.5000	−	61.4878i	0.0195186	−	0.0338072i	0.875620	+	0.483001i	$0.160453\pi$
$150$		0			0
$151$	−656.000			−0.353540			−0.176770	−	0.984252i	$-0.556565\pi$
$151$	−656.000			−0.353540			−0.176770	+	0.984252i	$0.556565\pi$
$152$	−608.000	−	263.272i	−0.324443	−	0.140488i
$153$		0			0
$154$	128.000	−	221.703i	0.0669775	−	0.116008i
$155$	−198.000	+	342.946i	−0.102605	+	0.177717i
$156$		0			0
$157$	526.500	−	911.925i	0.267639	−	0.463564i	−0.700613	−	0.713542i	$-0.747090\pi$
$157$	526.500	−	911.925i	0.267639	−	0.463564i	0.968252	+	0.249978i	$0.0804233\pi$
$158$	349.000	+	604.486i	0.175728	+	0.304369i
$159$		0			0
$160$	96.0000			0.0474342
$161$	−1072.00	−	1856.76i	−0.524754	−	0.908901i
$162$		0			0
$163$	68.0000			0.0326759			0.0163379	−	0.999867i	$-0.494799\pi$
$163$	68.0000			0.0326759			0.0163379	+	0.999867i	$0.494799\pi$
$164$	1652.00			0.786582
$165$		0			0
$166$	592.000	−	1025.37i	0.276796	−	0.479424i
$167$	−460.500	−	797.609i	−0.213381	−	0.369586i	0.739390	−	0.673278i	$-0.235114\pi$
$167$	−460.500	−	797.609i	−0.213381	−	0.369586i	−0.952770	+	0.303692i	$0.901781\pi$
$168$		0			0
$169$	−1282.00	+	2220.49i	−0.583523	+	1.01069i
$170$	−114.000			−0.0514318
$171$		0			0
$172$	516.000			0.228748
$173$	−946.500	+	1639.39i	−0.415960	+	0.720464i	−0.995529	−	0.0944596i	$-0.969888\pi$
$173$	−946.500	+	1639.39i	−0.415960	+	0.720464i	0.579569	+	0.814923i	$0.303221\pi$
$174$		0			0
$175$	−1856.00	−	3214.69i	−0.801717	−	1.38861i
$176$	32.0000	−	55.4256i	0.0137051	−	0.0237379i
$177$		0			0
$178$	−2698.00			−1.13609
$179$	20.0000			0.00835123			0.00417562	−	0.999991i	$-0.498671\pi$
$179$	20.0000			0.00835123			0.00417562	+	0.999991i	$0.498671\pi$
$180$		0			0
$181$	−919.500	−	1592.62i	−0.377602	−	0.654025i	0.613111	−	0.789997i	$-0.289918\pi$
$181$	−919.500	−	1592.62i	−0.377602	−	0.654025i	−0.990713	+	0.135971i	$0.956584\pi$
$182$	−4416.00			−1.79855
$183$		0			0
$184$	−268.000	−	464.190i	−0.107376	−	0.185981i
$185$	−21.0000	+	36.3731i	−0.00834568	+	0.0144551i
$186$		0			0
$187$	−38.0000	+	65.8179i	−0.0148601	+	0.0257384i
$188$	−1234.00	+	2137.35i	−0.478716	+	0.829161i
$189$		0			0
$190$	−57.0000	−	493.634i	−0.0217643	−	0.188484i
$191$	−2992.00			−1.13347			−0.566737	−	0.823899i	$-0.691794\pi$
$191$	−2992.00			−1.13347			−0.566737	+	0.823899i	$0.691794\pi$
$192$		0			0
$193$	696.500	−	1206.37i	0.259768	−	0.449931i	−0.706412	−	0.707801i	$-0.749687\pi$
$193$	696.500	−	1206.37i	0.259768	−	0.449931i	0.966180	+	0.257870i	$0.0830207\pi$
$194$	−613.000	−	1061.75i	−0.226860	−	0.392933i
$195$		0			0
$196$	−1362.00	−	2359.05i	−0.496356	−	0.859713i
$197$	4214.00			1.52404			0.762018	−	0.647556i	$-0.224209\pi$
$197$	4214.00			1.52404			0.762018	+	0.647556i	$0.224209\pi$
$198$		0			0
$199$	−383.500	−	664.241i	−0.136611	−	0.236617i	0.789601	−	0.613621i	$-0.210288\pi$
$199$	−383.500	−	664.241i	−0.136611	−	0.236617i	−0.926212	+	0.377004i	$0.876954\pi$
$200$	−464.000	−	803.672i	−0.164049	−	0.284141i
$201$		0			0
$202$	2470.00			0.860340
$203$	−816.000	−	1413.35i	−0.282128	−	0.488660i
$204$		0			0
$205$	619.500	+	1073.01i	0.211062	+	0.365571i
$206$	1632.00	−	2826.71i	0.551975	−	0.956049i
$207$		0			0
$208$	−1104.00			−0.368022
$209$	−304.000	−	131.636i	−0.100613	−	0.0435667i
$210$		0			0
$211$	−1182.50	+	2048.15i	−0.385814	+	0.668249i	−0.991882	−	0.127164i	$-0.959413\pi$
$211$	−1182.50	+	2048.15i	−0.385814	+	0.668249i	0.606068	+	0.795413i	$0.292746\pi$
$212$	766.000	−	1326.75i	0.248156	−	0.429819i
$213$		0			0
$214$	−2060.00	+	3568.02i	−0.658031	+	1.13974i
$215$	193.500	+	335.152i	0.0613795	+	0.106312i
$216$		0			0
$217$	4224.00			1.32140
$218$	1055.00	+	1827.31i	0.327769	+	0.567712i
$219$		0			0
$220$	48.0000			0.0147098
$221$	1311.00			0.399038
$222$		0			0
$223$	1219.50	−	2112.24i	0.366205	−	0.634286i	−0.622764	−	0.782410i	$-0.713990\pi$
$223$	1219.50	−	2112.24i	0.366205	−	0.634286i	0.988969	+	0.148124i	$0.0473235\pi$
$224$	−512.000	−	886.810i	−0.152721	−	0.264520i
$225$		0			0
$226$	−1006.00	+	1742.44i	−0.296098	+	0.512857i
$227$	1708.00			0.499401			0.249700	−	0.968323i	$-0.419668\pi$
$227$	1708.00			0.499401			0.249700	+	0.968323i	$0.419668\pi$
$228$		0			0
$229$	4618.00			1.33260			0.666301	−	0.745683i	$-0.267876\pi$
$229$	4618.00			1.33260			0.666301	+	0.745683i	$0.267876\pi$
$230$	201.000	−	348.142i	0.0576241	−	0.0998079i
$231$		0			0
$232$	−204.000	−	353.338i	−0.0577296	−	0.0999905i
$233$	1609.50	−	2787.74i	0.452540	−	0.783823i	−0.546003	−	0.837783i	$-0.683851\pi$
$233$	1609.50	−	2787.74i	0.452540	−	0.783823i	0.998543	+	0.0539608i	$0.0171846\pi$
$234$		0			0
$235$	−1851.00			−0.513812
$236$	2396.00			0.660874
$237$		0			0
$238$	608.000	+	1053.09i	0.165592	+	0.286813i
$239$	4236.00			1.14646			0.573230	−	0.819394i	$-0.305690\pi$
$239$	4236.00			1.14646			0.573230	+	0.819394i	$0.305690\pi$
$240$		0			0
$241$	1214.50	+	2103.58i	0.324618	+	0.562254i	0.981435	−	0.191796i	$-0.0614311\pi$
$241$	1214.50	+	2103.58i	0.324618	+	0.562254i	−0.656817	+	0.754050i	$0.728098\pi$
$242$	−1315.00	+	2277.65i	−0.349303	+	0.605011i
$243$		0			0
$244$	434.000	−	751.710i	0.113869	−	0.197227i
$245$	1021.50	−	1769.29i	0.266372	−	0.461371i
$246$		0			0
$247$	655.500	+	5676.80i	0.168860	+	1.46237i
$248$	1056.00			0.270387
$249$		0			0
$250$	723.000	−	1252.27i	0.182906	−	0.316803i
$251$	121.500	+	210.444i	0.0305538	+	0.0529208i	0.880898	−	0.473306i	$-0.156940\pi$
$251$	121.500	+	210.444i	0.0305538	+	0.0529208i	−0.850344	+	0.526227i	$0.823606\pi$
$252$		0			0
$253$	−134.000	−	232.095i	−0.0332984	−	0.0576746i
$254$	−3974.00			−0.981697
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	523.500	+	906.729i	0.127062	+	0.220079i	0.922537	−	0.385908i	$-0.126112\pi$
$257$	523.500	+	906.729i	0.127062	+	0.220079i	−0.795475	+	0.605987i	$0.792778\pi$
$258$		0			0
$259$	448.000			0.107480
$260$	−414.000	−	717.069i	−0.0987507	−	0.171041i
$261$		0			0
$262$	1803.00	+	3122.89i	0.425152	+	0.736384i
$263$	952.500	−	1649.78i	0.223322	−	0.386805i	−0.732493	−	0.680775i	$-0.761643\pi$
$263$	952.500	−	1649.78i	0.223322	−	0.386805i	0.955815	+	0.293970i	$0.0949766\pi$
$264$		0			0
$265$	1149.00			0.266349
$266$	−4256.00	+	3159.26i	−0.981023	+	0.728221i
$267$		0			0
$268$	450.000	−	779.423i	0.102568	−	0.177652i
$269$	−2916.50	+	5051.53i	−0.661049	+	1.14497i	0.319292	+	0.947657i	$0.396555\pi$
$269$	−2916.50	+	5051.53i	−0.661049	+	1.14497i	−0.980340	+	0.197314i	$0.936778\pi$
$270$		0			0
$271$	1863.50	−	3227.68i	0.417711	−	0.723496i	−0.577998	−	0.816038i	$-0.696166\pi$
$271$	1863.50	−	3227.68i	0.417711	−	0.723496i	0.995709	+	0.0925421i	$0.0294993\pi$
$272$	152.000	+	263.272i	0.0338837	+	0.0586882i
$273$		0			0
$274$	−1338.00			−0.295006
$275$	−232.000	−	401.836i	−0.0508732	−	0.0881149i
$276$		0			0
$277$	−5294.00			−1.14832			−0.574162	−	0.818742i	$-0.694672\pi$
$277$	−5294.00			−1.14832			−0.574162	+	0.818742i	$0.694672\pi$
$278$	5450.00			1.17579
$279$		0			0
$280$	384.000	−	665.108i	0.0819585	−	0.141956i
$281$	687.500	+	1190.78i	0.145953	+	0.252798i	0.929728	−	0.368247i	$-0.120042\pi$
$281$	687.500	+	1190.78i	0.145953	+	0.252798i	−0.783775	+	0.621045i	$0.786708\pi$
$282$		0			0
$283$	653.500	−	1131.90i	0.137267	−	0.237753i	−0.789194	−	0.614144i	$-0.789502\pi$
$283$	653.500	−	1131.90i	0.137267	−	0.237753i	0.926461	+	0.376390i	$0.122835\pi$
$284$	−2804.00			−0.585869
$285$		0			0
$286$	−552.000			−0.114127
$287$	6608.00	−	11445.4i	1.35909	−	2.35401i
$288$		0			0
$289$	2276.00	+	3942.15i	0.463261	+	0.802391i
$290$	153.000	−	265.004i	0.0309809	−	0.0536605i
$291$		0			0
$292$	4060.00			0.813676
$293$	−3818.00			−0.761263			−0.380631	−	0.924727i	$-0.624293\pi$
$293$	−3818.00			−0.761263			−0.380631	+	0.924727i	$0.624293\pi$
$294$		0			0
$295$	898.500	+	1556.25i	0.177331	+	0.307147i
$296$	112.000			0.0219928
$297$		0			0
$298$	−71.0000	−	122.976i	−0.0138017	−	0.0239053i
$299$	−2311.50	+	4003.64i	−0.447082	+	0.774369i
$300$		0			0
$301$	2064.00	−	3574.95i	0.395239	−	0.684574i
$302$	−656.000	+	1136.23i	−0.124995	+	0.216498i
$303$		0			0
$304$	−1064.00	+	789.815i	−0.200739	+	0.149010i
$305$	651.000			0.122217
$306$		0			0
$307$	−436.500	+	756.040i	−0.0811478	+	0.140552i	−0.903743	−	0.428075i	$-0.859192\pi$
$307$	−436.500	+	756.040i	−0.0811478	+	0.140552i	0.822595	+	0.568627i	$0.192525\pi$
$308$	−256.000	−	443.405i	−0.0473602	−	0.0820303i
$309$		0			0
$310$	396.000	+	685.892i	0.0725525	+	0.125665i
$311$	−4180.00			−0.762142			−0.381071	−	0.924546i	$-0.624445\pi$
$311$	−4180.00			−0.762142			−0.381071	+	0.924546i	$0.624445\pi$
$312$		0			0
$313$	1582.50	+	2740.97i	0.285777	+	0.494980i	0.972797	−	0.231658i	$-0.0744150\pi$
$313$	1582.50	+	2740.97i	0.285777	+	0.494980i	−0.687020	+	0.726638i	$0.741082\pi$
$314$	−1053.00	−	1823.85i	−0.189249	−	0.327789i
$315$		0			0
$316$	1396.00			0.248516
$317$	2269.50	+	3930.89i	0.402107	+	0.696469i	0.993980	−	0.109562i	$-0.0349448\pi$
$317$	2269.50	+	3930.89i	0.402107	+	0.696469i	−0.591873	+	0.806031i	$0.701611\pi$
$318$		0			0
$319$	−102.000	−	176.669i	−0.0179025	−	0.0310081i
$320$	96.0000	−	166.277i	0.0167705	−	0.0290474i
$321$		0			0
$322$	−4288.00			−0.742115
$323$	1263.50	−	937.906i	0.217656	−	0.161568i
$324$		0			0
$325$	−4002.00	+	6931.67i	−0.683049	+	1.18308i
$326$	68.0000	−	117.779i	0.0115527	−	0.0200098i
$327$		0			0
$328$	1652.00	−	2861.35i	0.278099	−	0.481681i
$329$	9872.00	+	17098.8i	1.65429	+	2.86531i
$330$		0			0
$331$	3660.00			0.607770			0.303885	−	0.952709i	$-0.401716\pi$
$331$	3660.00			0.607770			0.303885	+	0.952709i	$0.401716\pi$
$332$	−1184.00	−	2050.75i	−0.195724	−	0.339004i
$333$		0			0
$334$	−1842.00			−0.301766
$335$	675.000			0.110087
$336$		0			0
$337$	2586.50	−	4479.95i	0.418088	−	0.724150i	−0.577659	−	0.816278i	$-0.696034\pi$
$337$	2586.50	−	4479.95i	0.418088	−	0.724150i	0.995747	+	0.0921285i	$0.0293671\pi$
$338$	2564.00	+	4440.98i	0.412613	+	0.714667i
$339$		0			0
$340$	−114.000	+	197.454i	−0.0181839	+	0.0314954i
$341$	528.000			0.0838499
$342$		0			0
$343$	−10816.0			−1.70265
$344$	516.000	−	893.738i	0.0808746	−	0.140079i
$345$		0			0
$346$	1893.00	+	3278.77i	0.294128	+	0.509445i
$347$	−4295.50	+	7440.02i	−0.664538	+	1.15101i	0.314873	+	0.949134i	$0.398038\pi$
$347$	−4295.50	+	7440.02i	−0.664538	+	1.15101i	−0.979410	+	0.201879i	$0.935295\pi$
$348$		0			0
$349$	6946.00			1.06536			0.532680	−	0.846317i	$-0.321185\pi$
$349$	6946.00			1.06536			0.532680	+	0.846317i	$0.321185\pi$
$350$	−7424.00			−1.13380
$351$		0			0
$352$	−64.0000	−	110.851i	−0.00969094	−	0.0167852i
$353$	8226.00			1.24030			0.620150	−	0.784483i	$-0.287072\pi$
$353$	8226.00			1.24030			0.620150	+	0.784483i	$0.287072\pi$
$354$		0			0
$355$	−1051.50	−	1821.25i	−0.157205	−	0.272287i
$356$	−2698.00	+	4673.07i	−0.401668	+	0.695709i
$357$		0			0
$358$	20.0000	−	34.6410i	0.00295261	−	0.00511406i
$359$	5692.50	−	9859.70i	0.836876	−	1.44951i	−0.0556169	−	0.998452i	$-0.517713\pi$
$359$	5692.50	−	9859.70i	0.836876	−	1.44951i	0.892493	−	0.451060i	$-0.148954\pi$
$360$		0			0
$361$	4693.00	+	5002.16i	0.684211	+	0.729285i
$362$	−3678.00			−0.534009
$363$		0			0
$364$	−4416.00	+	7648.74i	−0.635883	+	1.10138i
$365$	1522.50	+	2637.05i	0.218332	+	0.378163i
$366$		0			0
$367$	1476.50	+	2557.37i	0.210007	+	0.363743i	0.951717	−	0.306978i	$-0.0993179\pi$
$367$	1476.50	+	2557.37i	0.210007	+	0.363743i	−0.741709	+	0.670722i	$0.765985\pi$
$368$	−1072.00			−0.151853
$369$		0			0
$370$	42.0000	+	72.7461i	0.00590129	+	0.0102213i
$371$	−6128.00	−	10614.0i	−0.857547	−	1.48531i
$372$		0			0
$373$	5006.00			0.694908			0.347454	−	0.937697i	$-0.387046\pi$
$373$	5006.00			0.694908			0.347454	+	0.937697i	$0.387046\pi$
$374$	76.0000	+	131.636i	0.0105077	+	0.0181998i
$375$		0			0
$376$	2468.00	+	4274.70i	0.338504	+	0.586306i
$377$	−1759.50	+	3047.54i	−0.240368	+	0.416330i
$378$		0			0
$379$	6764.00			0.916737			0.458369	−	0.888762i	$-0.348434\pi$
$379$	6764.00			0.916737			0.458369	+	0.888762i	$0.348434\pi$
$380$	−912.000	−	394.908i	−0.123117	−	0.0533114i
$381$		0			0
$382$	−2992.00	+	5182.30i	−0.400744	+	0.694108i
$383$	−2481.50	+	4298.08i	−0.331067	+	0.573425i	−0.982721	−	0.185091i	$-0.940742\pi$
$383$	−2481.50	+	4298.08i	−0.331067	+	0.573425i	0.651654	+	0.758516i	$0.274075\pi$
$384$		0			0
$385$	192.000	−	332.554i	0.0254162	−	0.0440221i
$386$	−1393.00	−	2412.75i	−0.183684	−	0.318149i
$387$		0			0
$388$	−2452.00			−0.320828
$389$	1811.50	+	3137.61i	0.236110	+	0.408954i	0.959595	−	0.281386i	$-0.0907942\pi$
$389$	1811.50	+	3137.61i	0.236110	+	0.408954i	−0.723485	+	0.690340i	$0.757461\pi$
$390$		0			0
$391$	1273.00			0.164651
$392$	−5448.00			−0.701953
$393$		0			0
$394$	4214.00	−	7298.86i	0.538828	−	0.933278i
$395$	523.500	+	906.729i	0.0666839	+	0.115500i
$396$		0			0
$397$	3814.50	−	6606.91i	0.482227	−	0.835242i	−0.517564	−	0.855644i	$-0.673161\pi$
$397$	3814.50	−	6606.91i	0.482227	−	0.835242i	0.999792	+	0.0204019i	$0.00649457\pi$
$398$	−1534.00			−0.193197
$399$		0			0
$400$	−1856.00			−0.232000
$401$	−4558.50	+	7895.55i	−0.567682	+	0.983255i	0.429112	+	0.903251i	$0.358826\pi$
$401$	−4558.50	+	7895.55i	−0.567682	+	0.983255i	−0.996795	+	0.0800035i	$0.974507\pi$
$402$		0			0
$403$	−4554.00	−	7887.76i	−0.562905	−	0.974981i
$404$	2470.00	−	4278.17i	0.304176	−	0.526848i
$405$		0			0
$406$	−3264.00			−0.398989
$407$	56.0000			0.00682019
$408$		0			0
$409$	−2467.50	−	4273.84i	−0.298313	−	0.516693i	0.677437	−	0.735581i	$-0.263091\pi$
$409$	−2467.50	−	4273.84i	−0.298313	−	0.516693i	−0.975750	+	0.218887i	$0.929757\pi$
$410$	2478.00			0.298487
$411$		0			0
$412$	−3264.00	−	5653.41i	−0.390305	−	0.676028i
$413$	9584.00	−	16600.0i	1.14188	−	1.97780i
$414$		0			0
$415$	888.000	−	1538.06i	0.105037	−	0.181929i
$416$	−1104.00	+	1912.18i	−0.130116	+	0.225367i
$417$		0			0
$418$	−532.000	+	394.908i	−0.0622511	+	0.0462095i
$419$	−6516.00			−0.759731			−0.379866	−	0.925042i	$-0.624030\pi$
$419$	−6516.00			−0.759731			−0.379866	+	0.925042i	$0.624030\pi$
$420$		0			0
$421$	−5457.50	+	9452.67i	−0.631787	+	1.09429i	0.355399	+	0.934715i	$0.384345\pi$
$421$	−5457.50	+	9452.67i	−0.631787	+	1.09429i	−0.987186	+	0.159572i	$0.948988\pi$
$422$	2365.00	+	4096.30i	0.272811	+	0.472523i
$423$		0			0
$424$	−1532.00	−	2653.50i	−0.175473	−	0.303928i
$425$	2204.00			0.251552
$426$		0			0
$427$	−3472.00	−	6013.68i	−0.393494	−	0.681551i
$428$	4120.00	+	7136.05i	0.465298	+	0.805920i
$429$		0			0
$430$	774.000			0.0868037
$431$	−2754.50	−	4770.93i	−0.307841	−	0.533197i	0.670049	−	0.742317i	$-0.266273\pi$
$431$	−2754.50	−	4770.93i	−0.307841	−	0.533197i	−0.977890	+	0.209120i	$0.932940\pi$
$432$		0			0
$433$	1876.50	+	3250.19i	0.208265	+	0.360726i	0.951168	−	0.308673i	$-0.0998850\pi$
$433$	1876.50	+	3250.19i	0.208265	+	0.360726i	−0.742903	+	0.669399i	$0.766552\pi$
$434$	4224.00	−	7316.18i	0.467185	−	0.809189i
$435$		0			0
$436$	4220.00			0.463535
$437$	636.500	+	5512.25i	0.0696749	+	0.603402i
$438$		0			0
$439$	−3906.50	+	6766.26i	−0.424709	+	0.735617i	−0.996393	−	0.0848566i	$-0.972957\pi$
$439$	−3906.50	+	6766.26i	−0.424709	+	0.735617i	0.571685	+	0.820473i	$0.306290\pi$
$440$	48.0000	−	83.1384i	0.00520071	−	0.00900789i
$441$		0			0
$442$	1311.00	−	2270.72i	0.141081	−	0.244360i
$443$	−3058.50	−	5297.48i	−0.328022	−	0.568151i	0.654097	−	0.756410i	$-0.273049\pi$
$443$	−3058.50	−	5297.48i	−0.328022	−	0.568151i	−0.982119	+	0.188260i	$0.939715\pi$
$444$		0			0
$445$	−4047.00			−0.431115
$446$	−2439.00	−	4224.47i	−0.258946	−	0.448508i
$447$		0			0
$448$	−2048.00			−0.215980
$449$	7146.00			0.751093			0.375546	−	0.926804i	$-0.377455\pi$
$449$	7146.00			0.751093			0.375546	+	0.926804i	$0.377455\pi$
$450$		0			0
$451$	826.000	−	1430.67i	0.0862413	−	0.149374i
$452$	2012.00	+	3484.89i	0.209373	+	0.362644i
$453$		0			0
$454$	1708.00	−	2958.34i	0.176565	−	0.305819i
$455$	−6624.00			−0.682501
$456$		0			0
$457$	−18118.0			−1.85454			−0.927269	−	0.374395i	$-0.877851\pi$
$457$	−18118.0			−1.85454			−0.927269	+	0.374395i	$0.877851\pi$
$458$	4618.00	−	7998.61i	0.471146	−	0.816049i
$459$		0			0
$460$	−402.000	−	696.284i	−0.0407464	−	0.0705748i
$461$	8467.50	−	14666.1i	0.855468	−	1.48171i	−0.0207420	−	0.999785i	$-0.506603\pi$
$461$	8467.50	−	14666.1i	0.855468	−	1.48171i	0.876210	−	0.481929i	$-0.160064\pi$
$462$		0			0
$463$	−7788.00			−0.781726			−0.390863	−	0.920449i	$-0.627823\pi$
$463$	−7788.00			−0.781726			−0.390863	+	0.920449i	$0.627823\pi$
$464$	−816.000			−0.0816419
$465$		0			0
$466$	−3219.00	−	5575.47i	−0.319994	−	0.554246i
$467$	−12500.0			−1.23861			−0.619305	−	0.785150i	$-0.712586\pi$
$467$	−12500.0			−1.23861			−0.619305	+	0.785150i	$0.712586\pi$
$468$		0			0
$469$	−3600.00	−	6235.38i	−0.354440	−	0.613909i
$470$	−1851.00	+	3206.03i	−0.181660	+	0.314645i
$471$		0			0
$472$	2396.00	−	4149.99i	0.233654	−	0.404701i
$473$	258.000	−	446.869i	0.0250800	−	0.0434399i
$474$		0			0
$475$	1102.00	+	9543.60i	0.106449	+	0.921875i
$476$	2432.00			0.234182
$477$		0			0
$478$	4236.00	−	7336.97i	0.405335	−	0.702061i
$479$	−6052.50	−	10483.2i	−0.577340	−	0.999982i	−0.995783	−	0.0917390i	$-0.970757\pi$
$479$	−6052.50	−	10483.2i	−0.577340	−	0.999982i	0.418443	−	0.908243i	$-0.362576\pi$
$480$		0			0
$481$	−483.000	−	836.581i	−0.0457857	−	0.0793031i
$482$	4858.00			0.459078
$483$		0			0
$484$	2630.00	+	4555.29i	0.246995	+	0.427807i
$485$	−919.500	−	1592.62i	−0.0860873	−	0.149108i
$486$		0			0
$487$	296.000			0.0275422			0.0137711	−	0.999905i	$-0.495616\pi$
$487$	296.000			0.0275422			0.0137711	+	0.999905i	$0.495616\pi$
$488$	−868.000	−	1503.42i	−0.0805174	−	0.139460i
$489$		0			0
$490$	−2043.00	−	3538.58i	−0.188354	−	0.326238i
$491$	1606.50	−	2782.54i	0.147659	−	0.255752i	−0.782703	−	0.622395i	$-0.786160\pi$
$491$	1606.50	−	2782.54i	0.147659	−	0.255752i	0.930362	+	0.366643i	$0.119493\pi$
$492$		0			0
$493$	969.000			0.0885224
$494$	10488.0	+	4541.44i	0.955217	+	0.413621i
$495$		0			0
$496$	1056.00	−	1829.05i	0.0955964	−	0.165578i
$497$	−11216.0	+	19426.7i	−1.01229	+	1.75333i
$498$		0			0
$499$	−1508.50	+	2612.80i	−0.135330	+	0.234399i	−0.925724	−	0.378201i	$-0.876543\pi$
$499$	−1508.50	+	2612.80i	−0.135330	+	0.234399i	0.790393	+	0.612600i	$0.209876\pi$
$500$	−1446.00	−	2504.55i	−0.129334	−	0.224013i
$501$		0			0
$502$	486.000			0.0432096
$503$	−3638.50	−	6302.07i	−0.322530	−	0.558639i	0.658479	−	0.752599i	$-0.271200\pi$
$503$	−3638.50	−	6302.07i	−0.322530	−	0.558639i	−0.981009	+	0.193960i	$0.937867\pi$
$504$		0			0
$505$	3705.00			0.326476
$506$	−536.000			−0.0470911
$507$		0			0
$508$	−3974.00	+	6883.17i	−0.347082	+	0.601164i
$509$	−5346.50	−	9260.41i	−0.465578	−	0.806406i	0.533649	−	0.845706i	$-0.320820\pi$
$509$	−5346.50	−	9260.41i	−0.465578	−	0.806406i	−0.999227	+	0.0393005i	$0.987487\pi$
$510$		0			0
$511$	16240.0	−	28128.5i	1.40590	−	2.43509i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	2094.00			0.179693
$515$	2448.00	−	4240.06i	0.209460	−	0.362795i
$516$		0			0
$517$	1234.00	+	2137.35i	0.104973	+	0.181819i
$518$	448.000	−	775.959i	0.0380000	−	0.0658179i
$519$		0			0
$520$	−1656.00			−0.139655
$521$	7190.00			0.604606			0.302303	−	0.953212i	$-0.402245\pi$
$521$	7190.00			0.604606			0.302303	+	0.953212i	$0.402245\pi$
$522$		0			0
$523$	−4323.50	−	7488.52i	−0.361479	−	0.626100i	0.626726	−	0.779240i	$-0.284395\pi$
$523$	−4323.50	−	7488.52i	−0.361479	−	0.626100i	−0.988204	+	0.153140i	$0.951061\pi$
$524$	7212.00			0.601255
$525$		0			0
$526$	−1905.00	−	3299.56i	−0.157912	−	0.273512i
$527$	−1254.00	+	2171.99i	−0.103653	+	0.179532i
$528$		0			0
$529$	3839.00	−	6649.34i	0.315526	−	0.546506i
$530$	1149.00	−	1990.13i	0.0941686	−	0.163105i
$531$		0			0
$532$	1216.00	+	10530.9i	0.0990983	+	0.858216i
$533$	−28497.0			−2.31584
$534$		0			0
$535$	−3090.00	+	5352.04i	−0.249705	+	0.432502i
$536$	−900.000	−	1558.85i	−0.0725263	−	0.125619i
$537$		0			0
$538$	5833.00	+	10103.1i	0.467432	+	0.809616i
$539$	−2724.00			−0.217683
$540$		0			0
$541$	−2021.50	−	3501.34i	−0.160649	−	0.278252i	0.774453	−	0.632632i	$-0.218025\pi$
$541$	−2021.50	−	3501.34i	−0.160649	−	0.278252i	−0.935102	+	0.354380i	$0.884692\pi$
$542$	−3727.00	−	6455.35i	−0.295366	−	0.511589i
$543$		0			0
$544$	608.000			0.0479187
$545$	1582.50	+	2740.97i	0.124380	+	0.215432i
$546$		0			0
$547$	2694.50	+	4667.01i	0.210619	+	0.364803i	0.951908	−	0.306383i	$-0.0991188\pi$
$547$	2694.50	+	4667.01i	0.210619	+	0.364803i	−0.741290	+	0.671185i	$0.765785\pi$
$548$	−1338.00	+	2317.48i	−0.104300	+	0.180653i
$549$		0			0
$550$	−928.000			−0.0719456
$551$	484.500	+	4195.89i	0.0374599	+	0.324412i
$552$		0			0
$553$	5584.00	−	9671.77i	0.429396	−	0.743735i
$554$	−5294.00	+	9169.48i	−0.405994	+	0.703202i
$555$		0			0
$556$	5450.00	−	9439.68i	0.415704	−	0.720021i
$557$	3363.50	+	5825.75i	0.255864	+	0.443169i	0.965130	−	0.261772i	$-0.0843068\pi$
$557$	3363.50	+	5825.75i	0.255864	+	0.443169i	−0.709266	+	0.704941i	$0.750974\pi$
$558$		0			0
$559$	−8901.00			−0.673474
$560$	−768.000	−	1330.22i	−0.0579534	−	0.100378i
$561$		0			0
$562$	2750.00			0.206409
$563$	19908.0			1.49027			0.745135	−	0.666914i	$-0.232385\pi$
$563$	19908.0			1.49027			0.745135	+	0.666914i	$0.232385\pi$
$564$		0			0
$565$	−1509.00	+	2613.66i	−0.112361	+	0.194615i
$566$	−1307.00	−	2263.79i	−0.0970624	−	0.168117i
$567$		0			0
$568$	−2804.00	+	4856.67i	−0.207136	+	0.358770i
$569$	8730.00			0.643200			0.321600	−	0.946876i	$-0.395779\pi$
$569$	8730.00			0.643200			0.321600	+	0.946876i	$0.395779\pi$
$570$		0			0
$571$	−4732.00			−0.346809			−0.173405	−	0.984851i	$-0.555477\pi$
$571$	−4732.00			−0.346809			−0.173405	+	0.984851i	$0.555477\pi$
$572$	−552.000	+	956.092i	−0.0403501	+	0.0698885i
$573$		0			0
$574$	−13216.0	−	22890.8i	−0.961019	−	1.66453i
$575$	−3886.00	+	6730.75i	−0.281839	+	0.488159i
$576$		0			0
$577$	−23882.0			−1.72309			−0.861543	−	0.507685i	$-0.830502\pi$
$577$	−23882.0			−1.72309			−0.861543	+	0.507685i	$0.830502\pi$
$578$	9104.00			0.655150
$579$		0			0
$580$	−306.000	−	530.008i	−0.0219068	−	0.0379437i
$581$	−18944.0			−1.35272
$582$		0			0
$583$	−766.000	−	1326.75i	−0.0544159	−	0.0942511i
$584$	4060.00	−	7032.13i	0.287678	−	0.498273i
$585$		0			0
$586$	−3818.00	+	6612.97i	−0.269147	+	0.466176i
$587$	−8273.50	+	14330.1i	−0.581744	+	1.00761i	0.413528	+	0.910491i	$0.364296\pi$
$587$	−8273.50	+	14330.1i	−0.581744	+	1.00761i	−0.995273	+	0.0971194i	$0.969037\pi$
$588$		0			0
$589$	−10032.0	−	4343.98i	−0.701802	−	0.303889i
$590$	3594.00			0.250784
$591$		0			0
$592$	112.000	−	193.990i	0.00777563	−	0.0134678i
$593$	−7514.50	−	13015.5i	−0.520377	−	0.901319i	−0.999719	−	0.0236913i	$-0.992458\pi$
$593$	−7514.50	−	13015.5i	−0.520377	−	0.901319i	0.479342	−	0.877628i	$-0.340875\pi$
$594$		0			0
$595$	912.000	+	1579.63i	0.0628376	+	0.108838i
$596$	−284.000			−0.0195186
$597$		0			0
$598$	4623.00	+	8007.27i	0.316135	+	0.547561i
$599$	6893.50	+	11939.9i	0.470218	+	0.814442i	0.999420	−	0.0340541i	$-0.0108418\pi$
$599$	6893.50	+	11939.9i	0.470218	+	0.814442i	−0.529202	+	0.848496i	$0.677509\pi$
$600$		0			0
$601$	11382.0			0.772515			0.386257	−	0.922391i	$-0.373768\pi$
$601$	11382.0			0.772515			0.386257	+	0.922391i	$0.373768\pi$
$602$	−4128.00	−	7149.91i	−0.279476	−	0.484067i
$603$		0			0
$604$	1312.00	+	2272.45i	0.0883850	+	0.153087i
$605$	−1972.50	+	3416.47i	−0.132551	+	0.229586i
$606$		0			0
$607$	−25312.0			−1.69256			−0.846279	−	0.532740i	$-0.821162\pi$
$607$	−25312.0			−1.69256			−0.846279	+	0.532740i	$0.821162\pi$
$608$	304.000	+	2632.72i	0.0202777	+	0.175610i
$609$		0			0
$610$	651.000	−	1127.57i	0.0432102	−	0.0748423i
$611$	21286.5	−	36869.3i	1.40943	−	2.44120i
$612$		0			0
$613$	11748.5	−	20349.0i	0.774090	−	1.34076i	−0.161214	−	0.986920i	$-0.551541\pi$
$613$	11748.5	−	20349.0i	0.774090	−	1.34076i	0.935304	−	0.353844i	$-0.115126\pi$
$614$	873.000	+	1512.08i	0.0573802	+	0.0993853i
$615$		0			0
$616$	−1024.00			−0.0669775
$617$	−6610.50	−	11449.7i	−0.431327	−	0.747080i	0.565661	−	0.824638i	$-0.308621\pi$
$617$	−6610.50	−	11449.7i	−0.431327	−	0.747080i	−0.996988	+	0.0775578i	$0.975288\pi$
$618$		0			0
$619$	15316.0			0.994511			0.497255	−	0.867604i	$-0.334341\pi$
$619$	15316.0			0.994511			0.497255	+	0.867604i	$0.334341\pi$
$620$	1584.00			0.102605
$621$		0			0
$622$	−4180.00	+	7239.97i	−0.269458	+	0.466715i
$623$	21584.0	+	37384.6i	1.38803	+	2.40414i
$624$		0			0
$625$	−6165.50	+	10679.0i	−0.394592	+	0.683453i
$626$	6330.00			0.404150
$627$		0			0
$628$	−4212.00			−0.267639
$629$	−133.000	+	230.363i	−0.00843093	+	0.0146028i
$630$		0			0
$631$	3024.50	+	5238.59i	0.190814	+	0.330499i	0.945520	−	0.325564i	$-0.105554\pi$
$631$	3024.50	+	5238.59i	0.190814	+	0.330499i	−0.754706	+	0.656063i	$0.772221\pi$
$632$	1396.00	−	2417.94i	0.0878638	−	0.152185i
$633$		0			0
$634$	9078.00			0.568665
$635$	−5961.00			−0.372528
$636$		0			0
$637$	23494.5	+	40693.7i	1.46136	+	2.53115i
$638$	−408.000			−0.0253180
$639$		0			0
$640$	−192.000	−	332.554i	−0.0118585	−	0.0205396i
$641$	−12044.5	+	20861.7i	−0.742167	+	1.28547i	0.209339	+	0.977843i	$0.432869\pi$
$641$	−12044.5	+	20861.7i	−0.742167	+	1.28547i	−0.951507	+	0.307628i	$0.900465\pi$
$642$		0			0
$643$	−816.500	+	1414.22i	−0.0500772	+	0.0867362i	−0.889977	−	0.456005i	$-0.849280\pi$
$643$	−816.500	+	1414.22i	−0.0500772	+	0.0867362i	0.839900	+	0.542741i	$0.182613\pi$
$644$	−4288.00	+	7427.03i	−0.262377	+	0.454451i
$645$		0			0
$646$	−361.000	−	3126.35i	−0.0219866	−	0.190410i
$647$	−14592.0			−0.886663			−0.443331	−	0.896358i	$-0.646203\pi$
$647$	−14592.0			−0.886663			−0.443331	+	0.896358i	$0.646203\pi$
$648$		0			0
$649$	1198.00	−	2075.00i	0.0724586	−	0.125502i
$650$	8004.00	+	13863.3i	0.482989	+	0.836561i
$651$		0			0
$652$	−136.000	−	235.559i	−0.00816897	−	0.0141491i
$653$	23430.0			1.40411			0.702057	−	0.712121i	$-0.252265\pi$
$653$	23430.0			1.40411			0.702057	+	0.712121i	$0.252265\pi$
$654$		0			0
$655$	2704.50	+	4684.33i	0.161334	+	0.279438i
$656$	−3304.00	−	5722.70i	−0.196646	−	0.340600i
$657$		0			0
$658$	39488.0			2.33952
$659$	2067.50	+	3581.02i	0.122213	+	0.211679i	0.920640	−	0.390412i	$-0.127668\pi$
$659$	2067.50	+	3581.02i	0.122213	+	0.211679i	−0.798427	+	0.602092i	$0.794334\pi$
$660$		0			0
$661$	4142.50	+	7175.02i	0.243759	+	0.422203i	0.961782	−	0.273817i	$-0.0882861\pi$
$661$	4142.50	+	7175.02i	0.243759	+	0.422203i	−0.718023	+	0.696019i	$0.754953\pi$
$662$	3660.00	−	6339.31i	0.214879	−	0.372181i
$663$		0			0
$664$	−4736.00			−0.276796
$665$	−6384.00	+	4738.89i	−0.372272	+	0.276340i
$666$		0			0
$667$	−1708.50	+	2959.21i	−0.0991805	+	0.171786i
$668$	−1842.00	+	3190.44i	−0.106690	+	0.184793i
$669$		0			0
$670$	675.000	−	1169.13i	0.0389217	−	0.0674143i
$671$	−434.000	−	751.710i	−0.0249693	−	0.0432481i
$672$		0			0
$673$	23990.0			1.37407			0.687033	−	0.726626i	$-0.258913\pi$
$673$	23990.0			1.37407			0.687033	+	0.726626i	$0.258913\pi$
$674$	−5173.00	−	8959.90i	−0.295633	−	0.512051i
$675$		0			0
$676$	10256.0			0.583523
$677$	−690.000			−0.0391711			−0.0195856	−	0.999808i	$-0.506235\pi$
$677$	−690.000			−0.0391711			−0.0195856	+	0.999808i	$0.506235\pi$
$678$		0			0
$679$	−9808.00	+	16988.0i	−0.554339	+	0.960144i
$680$	228.000	+	394.908i	0.0128579	+	0.0222706i
$681$		0			0
$682$	528.000	−	914.523i	0.0296454	−	0.0513473i
$683$	−5760.00			−0.322694			−0.161347	−	0.986898i	$-0.551584\pi$
$683$	−5760.00			−0.322694			−0.161347	+	0.986898i	$0.551584\pi$
$684$		0			0
$685$	−2007.00			−0.111947
$686$	−10816.0	+	18733.9i	−0.601978	+	1.04266i
$687$		0			0
$688$	−1032.00	−	1787.48i	−0.0571870	−	0.0990507i
$689$	−13213.5	+	22886.5i	−0.730616	+	1.26546i
$690$		0			0
$691$	4348.00			0.239372			0.119686	−	0.992812i	$-0.461811\pi$
$691$	4348.00			0.239372			0.119686	+	0.992812i	$0.461811\pi$
$692$	7572.00			0.415960
$693$		0			0
$694$	8591.00	+	14880.0i	0.469899	+	0.813889i
$695$	8175.00			0.446180
$696$		0			0
$697$	3923.50	+	6795.70i	0.213218	+	0.369305i
$698$	6946.00	−	12030.8i	0.376662	−	0.652397i
$699$		0			0
$700$	−7424.00	+	12858.7i	−0.400858	+	0.694307i
$701$	−5144.50	+	8910.54i	−0.277183	+	0.480095i	−0.970683	−	0.240361i	$-0.922734\pi$
$701$	−5144.50	+	8910.54i	−0.277183	+	0.480095i	0.693501	+	0.720456i	$0.256067\pi$
$702$		0			0
$703$	−1064.00	−	460.726i	−0.0570832	−	0.0247178i
$704$	−256.000			−0.0137051
$705$		0			0
$706$	8226.00	−	14247.8i	0.438512	−	0.759525i
$707$	−19760.0	−	34225.3i	−1.05113	−	1.82062i
$708$		0			0
$709$	15372.5	+	26626.0i	0.814283	+	1.41038i	0.909842	+	0.414955i	$0.136203\pi$
$709$	15372.5	+	26626.0i	0.814283	+	1.41038i	−0.0955593	+	0.995424i	$0.530464\pi$
$710$	−4206.00			−0.222322
$711$		0			0
$712$	5396.00	+	9346.15i	0.284022	+	0.491940i
$713$	−4422.00	−	7659.13i	−0.232265	−	0.402295i
$714$		0			0
$715$	−828.000			−0.0433083
$716$	−40.0000	−	69.2820i	−0.00208781	−	0.00361619i
$717$		0			0
$718$	−11385.0	−	19719.4i	−0.591761	−	1.02496i
$719$	420.500	−	728.327i	0.0218109	−	0.0377775i	−0.854914	−	0.518770i	$-0.826390\pi$
$719$	420.500	−	728.327i	0.0218109	−	0.0377775i	0.876725	+	0.480992i	$0.159724\pi$
$720$		0			0
$721$	−52224.0			−2.69754
$722$	13357.0	−	3126.35i	0.688499	−	0.161151i
$723$		0			0
$724$	−3678.00	+	6370.48i	−0.188801	+	0.327013i
$725$	−2958.00	+	5123.41i	−0.151527	+	0.262453i
$726$		0			0
$727$	135.500	−	234.693i	0.00691254	−	0.0119729i	−0.862548	−	0.505974i	$-0.831133\pi$
$727$	135.500	−	234.693i	0.00691254	−	0.0119729i	0.869461	+	0.494002i	$0.164466\pi$
$728$	8832.00	+	15297.5i	0.449637	+	0.778794i
$729$		0			0
$730$	6090.00			0.308769
$731$	1225.50	+	2122.63i	0.0620065	+	0.107398i
$732$		0			0
$733$	−8102.00			−0.408259			−0.204130	−	0.978944i	$-0.565436\pi$
$733$	−8102.00			−0.408259			−0.204130	+	0.978944i	$0.565436\pi$
$734$	5906.00			0.296995
$735$		0			0
$736$	−1072.00	+	1856.76i	−0.0536881	+	0.0929905i
$737$	−450.000	−	779.423i	−0.0224911	−	0.0389558i
$738$		0			0
$739$	−7824.50	+	13552.4i	−0.389484	+	0.674607i	−0.992380	−	0.123213i	$-0.960680\pi$
$739$	−7824.50	+	13552.4i	−0.389484	+	0.674607i	0.602896	+	0.797820i	$0.294013\pi$
$740$	168.000			0.00834568
$741$		0			0
$742$	−24512.0			−1.21275
$743$	−5167.50	+	8950.37i	−0.255151	+	0.441934i	−0.964937	−	0.262483i	$-0.915459\pi$
$743$	−5167.50	+	8950.37i	−0.255151	+	0.441934i	0.709786	+	0.704418i	$0.248792\pi$
$744$		0			0
$745$	−106.500	−	184.463i	−0.00523739	−	0.00907143i
$746$	5006.00	−	8670.65i	0.245687	−	0.425543i
$747$		0			0
$748$	304.000			0.0148601
$749$	65920.0			3.21584
$750$		0			0
$751$	7334.50	+	12703.7i	0.356378	+	0.617264i	0.987353	−	0.158539i	$-0.0506783\pi$
$751$	7334.50	+	12703.7i	0.356378	+	0.617264i	−0.630975	+	0.775803i	$0.717345\pi$
$752$	9872.00			0.478716
$753$		0			0
$754$	3519.00	+	6095.09i	0.169966	+	0.294390i
$755$	−984.000	+	1704.34i	−0.0474324	+	0.0821552i
$756$		0			0
$757$	20718.5	−	35885.5i	0.994751	−	1.72296i	0.408761	−	0.912641i	$-0.365961\pi$
$757$	20718.5	−	35885.5i	0.994751	−	1.72296i	0.585990	−	0.810318i	$-0.300706\pi$
$758$	6764.00	−	11715.6i	0.324115	−	0.561385i
$759$		0			0
$760$	−1596.00	+	1184.72i	−0.0761750	+	0.0565453i
$761$	34258.0			1.63187			0.815934	−	0.578145i	$-0.196223\pi$
$761$	34258.0			1.63187			0.815934	+	0.578145i	$0.196223\pi$
$762$		0			0
$763$	16880.0	−	29237.0i	0.800914	−	1.38722i
$764$	5984.00	+	10364.6i	0.283368	+	0.490809i
$765$		0			0
$766$	4963.00	+	8596.17i	0.234100	+	0.405473i
$767$	−41331.0			−1.94573
$768$		0			0
$769$	−3427.50	−	5936.60i	−0.160727	−	0.278387i	0.774403	−	0.632693i	$-0.218050\pi$
$769$	−3427.50	−	5936.60i	−0.160727	−	0.278387i	−0.935130	+	0.354306i	$0.884717\pi$
$770$	−384.000	−	665.108i	−0.0179719	−	0.0311283i
$771$		0			0
$772$	−5572.00			−0.259768
$773$	−14728.5	−	25510.5i	−0.685313	−	1.18700i	−0.973338	−	0.229375i	$-0.926332\pi$
$773$	−14728.5	−	25510.5i	−0.685313	−	1.18700i	0.288025	−	0.957623i	$-0.407001\pi$
$774$		0			0
$775$	−7656.00	−	13260.6i	−0.354854	−	0.614625i
$776$	−2452.00	+	4246.99i	−0.113430	+	0.196467i
$777$		0			0
$778$	7246.00			0.333910
$779$	−27464.5	+	20387.1i	−1.26318	+	0.937669i
$780$		0			0
$781$	−1402.00	+	2428.34i	−0.0642350	+	0.111258i
$782$	1273.00	−	2204.90i	0.0582128	−	0.100827i
$783$		0			0
$784$	−5448.00	+	9436.21i	−0.248178	+	0.429857i
$785$	−1579.50	−	2735.77i	−0.0718150	−	0.124387i
$786$		0			0
$787$	20716.0			0.938305			0.469152	−	0.883117i	$-0.344560\pi$
$787$	20716.0			0.938305			0.469152	+	0.883117i	$0.344560\pi$
$788$	−8428.00	−	14597.7i	−0.381009	−	0.659927i
$789$		0			0
$790$	2094.00			0.0943053
$791$	32192.0			1.44705
$792$		0			0
$793$	−7486.50	+	12967.0i	−0.335250	+	0.580670i
$794$	−7629.00	−	13213.8i	−0.340986	−	0.590606i
$795$		0			0
$796$	−1534.00	+	2656.97i	−0.0683055	+	0.118309i
$797$	−3722.00			−0.165420			−0.0827102	−	0.996574i	$-0.526358\pi$
$797$	−3722.00			−0.165420			−0.0827102	+	0.996574i	$0.526358\pi$
$798$		0			0
$799$	−11723.0			−0.519061
$800$	−1856.00	+	3214.69i	−0.0820244	+	0.142070i
$801$		0			0
$802$	9117.00	+	15791.1i	0.401412	+	0.695266i
$803$	2030.00	−	3516.06i	0.0892119	−	0.154520i
$804$		0			0
$805$	−6432.00			−0.281613
$806$	−18216.0			−0.796069
$807$		0			0
$808$	−4940.00	−	8556.33i	−0.215085	−	0.372538i
$809$	−11358.0			−0.493604			−0.246802	−	0.969066i	$-0.579380\pi$
$809$	−11358.0			−0.493604			−0.246802	+	0.969066i	$0.579380\pi$
$810$		0			0
$811$	−4239.50	−	7343.03i	−0.183562	−	0.317939i	0.759529	−	0.650474i	$-0.225430\pi$
$811$	−4239.50	−	7343.03i	−0.183562	−	0.317939i	−0.943091	+	0.332534i	$0.892096\pi$
$812$	−3264.00	+	5653.41i	−0.141064	+	0.244330i
$813$		0			0
$814$	56.0000	−	96.9948i	0.00241130	−	0.00417650i
$815$	102.000	−	176.669i	0.00438393	−	0.00759319i
$816$		0			0
$817$	−8578.50	+	6367.88i	−0.367348	+	0.272686i
$818$	−9870.00			−0.421878
$819$		0			0
$820$	2478.00	−	4292.02i	0.105531	−	0.182785i
$821$	6685.50	+	11579.6i	0.284197	+	0.492243i	0.972414	−	0.233261i	$-0.0749398\pi$
$821$	6685.50	+	11579.6i	0.284197	+	0.492243i	−0.688217	+	0.725505i	$0.741606\pi$
$822$		0			0
$823$	13292.5	+	23023.3i	0.562998	+	0.975141i	0.997233	+	0.0743415i	$0.0236855\pi$
$823$	13292.5	+	23023.3i	0.562998	+	0.975141i	−0.434235	+	0.900800i	$0.642981\pi$
$824$	−13056.0			−0.551975
$825$		0			0
$826$	−19168.0	−	33199.9i	−0.807433	−	1.39852i
$827$	18301.5	+	31699.1i	0.769535	+	1.33287i	0.937815	+	0.347135i	$0.112845\pi$
$827$	18301.5	+	31699.1i	0.769535	+	1.33287i	−0.168280	+	0.985739i	$0.553821\pi$
$828$		0			0
$829$	31238.0			1.30873			0.654367	−	0.756177i	$-0.272935\pi$
$829$	31238.0			1.30873			0.654367	+	0.756177i	$0.272935\pi$
$830$	−1776.00	−	3076.12i	−0.0742721	−	0.128643i
$831$		0			0
$832$	2208.00	+	3824.37i	0.0920056	+	0.159358i
$833$	6469.50	−	11205.5i	0.269094	−	0.466084i
$834$		0			0
$835$	−2763.00			−0.114512
$836$	152.000	+	1316.36i	0.00628831	+	0.0544584i
$837$		0			0
$838$	−6516.00	+	11286.0i	−0.268606	+	0.465239i
$839$	15328.5	−	26549.7i	0.630749	−	1.09249i	−0.356650	−	0.934238i	$-0.616081\pi$
$839$	15328.5	−	26549.7i	0.630749	−	1.09249i	0.987399	−	0.158251i	$-0.0505857\pi$
$840$		0			0
$841$	10894.0	−	18869.0i	0.446677	−	0.773667i
$842$	10915.0	+	18905.3i	0.446741	+	0.773778i
$843$		0			0
$844$	9460.00			0.385814
$845$	3846.00	+	6661.47i	0.156576	+	0.271197i
$846$		0			0
$847$	42080.0			1.70707
$848$	−6128.00			−0.248156
$849$		0			0
$850$	2204.00	−	3817.44i	0.0889371	−	0.154044i
$851$	−469.000	−	812.332i	−0.0188920	−	0.0327219i
$852$		0			0
$853$	−2221.50	+	3847.75i	−0.0891708	+	0.154448i	−0.907161	−	0.420784i	$-0.861755\pi$
$853$	−2221.50	+	3847.75i	−0.0891708	+	0.154448i	0.817990	+	0.575232i	$0.195088\pi$
$854$	−13888.0			−0.556484
$855$		0			0
$856$	16480.0			0.658031
$857$	−3484.50	+	6035.33i	−0.138889	+	0.240564i	−0.927077	−	0.374872i	$-0.877687\pi$
$857$	−3484.50	+	6035.33i	−0.138889	+	0.240564i	0.788187	+	0.615436i	$0.211020\pi$
$858$		0			0
$859$	−19659.5	−	34051.3i	−0.780877	−	1.35252i	−0.931431	−	0.363917i	$-0.881439\pi$
$859$	−19659.5	−	34051.3i	−0.780877	−	1.35252i	0.150554	−	0.988602i	$-0.451894\pi$
$860$	774.000	−	1340.61i	0.0306897	−	0.0531562i
$861$		0			0
$862$	−11018.0			−0.435353
$863$	−9380.00			−0.369987			−0.184994	−	0.982740i	$-0.559226\pi$
$863$	−9380.00			−0.369987			−0.184994	+	0.982740i	$0.559226\pi$
$864$		0			0
$865$	2839.50	+	4918.16i	0.111614	+	0.193321i
$866$	7506.00			0.294531
$867$		0			0
$868$	−8448.00	−	14632.4i	−0.330350	−	0.572183i
$869$	698.000	−	1208.97i	0.0272474	−	0.0471940i
$870$		0			0
$871$	−7762.50	+	13445.0i	−0.301977	+	0.523040i
$872$	4220.00	−	7309.25i	0.163884	−	0.283856i
$873$		0			0
$874$	10184.0	+	4409.80i	0.394141	+	0.170668i
$875$	−23136.0			−0.893874
$876$		0			0
$877$	5124.50	−	8875.89i	0.197311	−	0.341753i	−0.750344	−	0.661047i	$-0.770112\pi$
$877$	5124.50	−	8875.89i	0.197311	−	0.341753i	0.947656	+	0.319294i	$0.103446\pi$
$878$	7813.00	+	13532.5i	0.300314	+	0.520160i
$879$		0			0
$880$	−96.0000	−	166.277i	−0.00367745	−	0.00636954i
$881$	28698.0			1.09746			0.548729	−	0.836000i	$-0.315112\pi$
$881$	28698.0			1.09746			0.548729	+	0.836000i	$0.315112\pi$
$882$		0			0
$883$	−15889.5	−	27521.4i	−0.605577	−	1.04889i	−0.991960	−	0.126551i	$-0.959609\pi$
$883$	−15889.5	−	27521.4i	−0.605577	−	1.04889i	0.386383	−	0.922338i	$-0.373724\pi$
$884$	−2622.00	−	4541.44i	−0.0997595	−	0.172789i
$885$		0			0
$886$	−12234.0			−0.463893
$887$	1783.50	+	3089.11i	0.0675130	+	0.116936i	0.897806	−	0.440391i	$-0.145160\pi$
$887$	1783.50	+	3089.11i	0.0675130	+	0.116936i	−0.830293	+	0.557327i	$0.811827\pi$
$888$		0			0
$889$	31792.0	+	55065.4i	1.19940	+	2.07743i
$890$	−4047.00	+	7009.61i	−0.152422	+	0.264003i
$891$		0			0
$892$	−9756.00			−0.366205
$893$	−5861.50	−	50762.1i	−0.219650	−	1.90223i
$894$		0			0
$895$	30.0000	−	51.9615i	0.00112044	−	0.00194065i
$896$	−2048.00	+	3547.24i	−0.0763604	+	0.132260i
$897$		0			0
$898$	7146.00	−	12377.2i	0.265551	−	0.459948i
$899$	−3366.00	−	5830.08i	−0.124875	−	0.216289i
$900$		0			0
$901$	7277.00			0.269070
$902$	−1652.00	−	2861.35i	−0.0609818	−	0.105624i
$903$		0			0
$904$	8048.00			0.296098
$905$	−5517.00			−0.202642
$906$		0			0
$907$	15539.5	−	26915.2i	0.568887	−	0.985341i	−0.427789	−	0.903879i	$-0.640707\pi$
$907$	15539.5	−	26915.2i	0.568887	−	0.985341i	0.996676	−	0.0814629i	$-0.0259592\pi$
$908$	−3416.00	−	5916.69i	−0.124850	−	0.216247i
$909$		0			0
$910$	−6624.00	+	11473.1i	−0.241301	+	0.417945i
$911$	−32856.0			−1.19492			−0.597458	−	0.801900i	$-0.703822\pi$
$911$	−32856.0			−1.19492			−0.597458	+	0.801900i	$0.703822\pi$
$912$		0			0
$913$	−2368.00			−0.0858372
$914$	−18118.0	+	31381.3i	−0.655679	+	1.13567i
$915$		0			0
$916$	−9236.00	−	15997.2i	−0.333151	−	0.577034i
$917$	28848.0	−	49966.2i	1.03887	−	1.79938i
$918$		0			0
$919$	−7736.00			−0.277679			−0.138840	−	0.990315i	$-0.544337\pi$
$919$	−7736.00			−0.277679			−0.138840	+	0.990315i	$0.544337\pi$
$920$	−1608.00			−0.0576241
$921$		0			0
$922$	−16935.0	−	29332.3i	−0.604907	−	1.04773i
$923$	48369.0			1.72490
$924$		0			0
$925$	−812.000	−	1406.43i	−0.0288631	−	0.0499924i
$926$	−7788.00	+	13489.2i	−0.276382	+	0.478707i
$927$		0			0
$928$	−816.000	+	1413.35i	−0.0288648	+	0.0499953i
$929$	15259.5	−	26430.2i	0.538911	−	0.933421i	−0.460052	−	0.887892i	$-0.652169\pi$
$929$	15259.5	−	26430.2i	0.538911	−	0.933421i	0.998963	−	0.0455288i	$-0.0144973\pi$
$930$		0			0
$931$	51756.0	+	22411.0i	1.82195	+	0.788927i
$932$	−12876.0			−0.452540
$933$		0			0
$934$	−12500.0	+	21650.6i	−0.437915	+	0.758491i
$935$	114.000	+	197.454i	0.00398738	+	0.00690634i
$936$		0			0
$937$	−4703.50	−	8146.70i	−0.163988	−	0.284035i	0.772308	−	0.635249i	$-0.219102\pi$
$937$	−4703.50	−	8146.70i	−0.163988	−	0.284035i	−0.936295	+	0.351214i	$0.885769\pi$
$938$	−14400.0			−0.501254
$939$		0			0
$940$	3702.00	+	6412.05i	0.128453	+	0.222487i
$941$	13407.5	+	23222.5i	0.464476	+	0.804496i	0.999178	−	0.0405447i	$-0.0129093\pi$
$941$	13407.5	+	23222.5i	0.464476	+	0.804496i	−0.534702	+	0.845041i	$0.679576\pi$
$942$		0			0
$943$	−27671.0			−0.955559
$944$	−4792.00	−	8299.99i	−0.165219	−	0.286167i
$945$		0			0
$946$	−516.000	−	893.738i	−0.0177343	−	0.0307166i
$947$	8724.50	−	15111.3i	0.299375	−	0.518533i	−0.676618	−	0.736334i	$-0.736555\pi$
$947$	8724.50	−	15111.3i	0.299375	−	0.518533i	0.975993	+	0.217801i	$0.0698885\pi$
$948$		0			0
$949$	−70035.0			−2.39561
$950$	17632.0	+	7634.88i	0.602166	+	0.260745i
$951$		0			0
$952$	2432.00	−	4212.35i	0.0827958	−	0.143406i
$953$	−19936.5	+	34531.0i	−0.677656	+	1.17374i	0.298028	+	0.954557i	$0.403671\pi$
$953$	−19936.5	+	34531.0i	−0.677656	+	1.17374i	−0.975685	+	0.219178i	$0.929662\pi$
$954$		0			0
$955$	−4488.00	+	7773.44i	−0.152071	+	0.263396i
$956$	−8472.00	−	14673.9i	−0.286615	−	0.496432i
$957$		0			0
$958$	−24210.0			−0.816482
$959$	10704.0	+	18539.9i	0.360428	+	0.624279i
$960$		0			0
$961$	−12367.0			−0.415125
$962$	−1932.00			−0.0647507
$963$		0			0
$964$	4858.00	−	8414.30i	0.162309	−	0.281127i
$965$	−2089.50	−	3619.12i	−0.0697030	−	0.120729i
$966$		0			0
$967$	5989.50	−	10374.1i	0.199182	−	0.344994i	−0.749081	−	0.662478i	$-0.769505\pi$
$967$	5989.50	−	10374.1i	0.199182	−	0.344994i	0.948264	+	0.317484i	$0.102838\pi$
$968$	10520.0			0.349303
$969$		0			0
$970$	−3678.00			−0.121746
$971$	9100.50	−	15762.5i	0.300771	−	0.520951i	−0.675540	−	0.737324i	$-0.736089\pi$
$971$	9100.50	−	15762.5i	0.300771	−	0.520951i	0.976311	+	0.216373i	$0.0694226\pi$
$972$		0			0
$973$	−43600.0	−	75517.4i	−1.43654	−	2.48816i
$974$	296.000	−	512.687i	0.00973763	−	0.0168661i
$975$		0			0
$976$	−3472.00			−0.113869
$977$	−37398.0			−1.22463			−0.612317	−	0.790612i	$-0.709762\pi$
$977$	−37398.0			−1.22463			−0.612317	+	0.790612i	$0.709762\pi$
$978$		0			0
$979$	2698.00	+	4673.07i	0.0880781	+	0.152556i
$980$	−8172.00			−0.266372
$981$		0			0
$982$	−3213.00	−	5565.08i	−0.104410	−	0.180844i
$983$	22018.5	−	38137.2i	0.714426	−	1.23742i	−0.248754	−	0.968567i	$-0.580021\pi$
$983$	22018.5	−	38137.2i	0.714426	−	1.23742i	0.963180	−	0.268856i	$-0.0866456\pi$
$984$		0			0
$985$	6321.00	−	10948.3i	0.204471	−	0.354154i
$986$	969.000	−	1678.36i	0.0312974	−	0.0542087i
$987$		0			0
$988$	18354.0	−	13624.3i	0.591011	−	0.438712i
$989$	−8643.00			−0.277888
$990$		0			0
$991$	−4914.50	+	8512.16i	−0.157532	+	0.272853i	−0.933978	−	0.357330i	$-0.883687\pi$
$991$	−4914.50	+	8512.16i	−0.157532	+	0.272853i	0.776446	+	0.630184i	$0.217020\pi$
$992$	−2112.00	−	3658.09i	−0.0675968	−	0.117081i
$993$		0			0
$994$	22432.0	+	38853.4i	0.715795	+	1.23979i
$995$	−2301.00			−0.0733132
$996$		0			0
$997$	−19991.5	−	34626.3i	−0.635042	−	1.09993i	−0.986506	−	0.163724i	$-0.947649\pi$
$997$	−19991.5	−	34626.3i	−0.635042	−	1.09993i	0.351464	−	0.936201i	$-0.385684\pi$
$998$	3017.00	+	5225.60i	0.0956929	+	0.165745i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.4.g.d.163.1		2
3.2	odd	2		38.4.c.a.11.1	yes	2
12.11	even	2		304.4.i.b.49.1		2
19.7	even	3	inner	342.4.g.d.235.1		2
57.8	even	6		722.4.a.a.1.1		1
57.11	odd	6		722.4.a.e.1.1		1
57.26	odd	6		38.4.c.a.7.1	✓	2
228.83	even	6		304.4.i.b.273.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.c.a.7.1	✓	2	57.26	odd	6
38.4.c.a.11.1	yes	2	3.2	odd	2
304.4.i.b.49.1		2	12.11	even	2
304.4.i.b.273.1		2	228.83	even	6
342.4.g.d.163.1		2	1.1	even	1	trivial
342.4.g.d.235.1		2	19.7	even	3	inner
722.4.a.a.1.1		1	57.8	even	6
722.4.a.e.1.1		1	57.11	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	342.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(1.50000 - 2.59808i) q^{5} -32.0000 q^{7} -8.00000 q^{8} +(-3.00000 - 5.19615i) q^{10} -4.00000 q^{11} +(34.5000 + 59.7558i) q^{13} +(-32.0000 + 55.4256i) q^{14} +(-8.00000 + 13.8564i) q^{16} +(9.50000 - 16.4545i) q^{17} +(76.0000 + 32.9090i) q^{19} -12.0000 q^{20} +(-4.00000 + 6.92820i) q^{22} +(33.5000 + 58.0237i) q^{23} +(58.0000 + 100.459i) q^{25} +138.000 q^{26} +(64.0000 + 110.851i) q^{28} +(25.5000 + 44.1673i) q^{29} -132.000 q^{31} +(16.0000 + 27.7128i) q^{32} +(-19.0000 - 32.9090i) q^{34} +(-48.0000 + 83.1384i) q^{35} -14.0000 q^{37} +(133.000 - 98.7269i) q^{38} +(-12.0000 + 20.7846i) q^{40} +(-206.500 + 357.668i) q^{41} +(-64.5000 + 111.717i) q^{43} +(8.00000 + 13.8564i) q^{44} +134.000 q^{46} +(-308.500 - 534.338i) q^{47} +681.000 q^{49} +232.000 q^{50} +(138.000 - 239.023i) q^{52} +(191.500 + 331.688i) q^{53} +(-6.00000 + 10.3923i) q^{55} +256.000 q^{56} +102.000 q^{58} +(-299.500 + 518.749i) q^{59} +(108.500 + 187.928i) q^{61} +(-132.000 + 228.631i) q^{62} +64.0000 q^{64} +207.000 q^{65} +(112.500 + 194.856i) q^{67} -76.0000 q^{68} +(96.0000 + 166.277i) q^{70} +(350.500 - 607.084i) q^{71} +(-507.500 + 879.016i) q^{73} +(-14.0000 + 24.2487i) q^{74} +(-38.0000 - 329.090i) q^{76} +128.000 q^{77} +(-174.500 + 302.243i) q^{79} +(24.0000 + 41.5692i) q^{80} +(413.000 + 715.337i) q^{82} +592.000 q^{83} +(-28.5000 - 49.3634i) q^{85} +(129.000 + 223.435i) q^{86} +32.0000 q^{88} +(-674.500 - 1168.27i) q^{89} +(-1104.00 - 1912.18i) q^{91} +(134.000 - 232.095i) q^{92} -1234.00 q^{94} +(199.500 - 148.090i) q^{95} +(306.500 - 530.874i) q^{97} +(681.000 - 1179.53i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{4} + 3 q^{5} - 64 q^{7} - 16 q^{8} - 6 q^{10} - 8 q^{11} + 69 q^{13} - 64 q^{14} - 16 q^{16} + 19 q^{17} + 152 q^{19} - 24 q^{20} - 8 q^{22} + 67 q^{23} + 116 q^{25} + 276 q^{26} + 128 q^{28}+ \cdots + 1362 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^4 + 3 * q^5 - 64 * q^7 - 16 * q^8 - 6 * q^10 - 8 * q^11 + 69 * q^13 - 64 * q^14 - 16 * q^16 + 19 * q^17 + 152 * q^19 - 24 * q^20 - 8 * q^22 + 67 * q^23 + 116 * q^25 + 276 * q^26 + 128 * q^28 + 51 * q^29 - 264 * q^31 + 32 * q^32 - 38 * q^34 - 96 * q^35 - 28 * q^37 + 266 * q^38 - 24 * q^40 - 413 * q^41 - 129 * q^43 + 16 * q^44 + 268 * q^46 - 617 * q^47 + 1362 * q^49 + 464 * q^50 + 276 * q^52 + 383 * q^53 - 12 * q^55 + 512 * q^56 + 204 * q^58 - 599 * q^59 + 217 * q^61 - 264 * q^62 + 128 * q^64 + 414 * q^65 + 225 * q^67 - 152 * q^68 + 192 * q^70 + 701 * q^71 - 1015 * q^73 - 28 * q^74 - 76 * q^76 + 256 * q^77 - 349 * q^79 + 48 * q^80 + 826 * q^82 + 1184 * q^83 - 57 * q^85 + 258 * q^86 + 64 * q^88 - 1349 * q^89 - 2208 * q^91 + 268 * q^92 - 2468 * q^94 + 399 * q^95 + 613 * q^97 + 1362 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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