LMFDB - Embedded newform 108.6.b.c.107.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,6,Mod(107,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("108.107");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	108.b (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:	$17.3214525398$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 94 x^{18} + 5872 x^{16} - 207192 x^{14} + 5271952 x^{12} - 76648960 x^{10} + 792478720 x^{8} + \cdots + 41943040000 $ x^20 - 94x^18 + 5872x^16 - 207192x^14 + 5271952x^12 - 76648960x^10 + 792478720x^8 - 4371873792x^6 + 17152147456x^4 - 32033996800*x^2 + 41943040000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{50}\cdot 3^{40} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			107.16
Root			$1.31722 + 0.760496i$ of defining polynomial
Character	$\chi$	$=$	108.107
Dual form			108.6.b.c.107.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.10571 + 3.89142i) q^{2} +(1.71366 + 31.9541i) q^{4} +98.6190i q^{5} -67.7266i q^{7} +(-117.311 + 137.863i) q^{8} +O(q^{10})$	\(q+(4.10571 + 3.89142i) q^{2} +(1.71366 + 31.9541i) q^{4} +98.6190i q^{5} -67.7266i q^{7} +(-117.311 + 137.863i) q^{8} +(-383.768 + 404.901i) q^{10} -403.224 q^{11} +525.857 q^{13} +(263.553 - 278.066i) q^{14} +(-1018.13 + 109.517i) q^{16} +462.383i q^{17} -1958.86i q^{19} +(-3151.28 + 169.000i) q^{20} +(-1655.52 - 1569.11i) q^{22} +696.614 q^{23} -6600.72 q^{25} +(2159.02 + 2046.33i) q^{26} +(2164.14 - 116.060i) q^{28} +4664.43i q^{29} +8849.55i q^{31} +(-4606.31 - 3512.32i) q^{32} +(-1799.33 + 1898.41i) q^{34} +6679.13 q^{35} -14878.3 q^{37} +(7622.77 - 8042.52i) q^{38} +(-13595.9 - 11569.1i) q^{40} -5373.92i q^{41} -1880.14i q^{43} +(-690.989 - 12884.6i) q^{44} +(2860.09 + 2710.82i) q^{46} +25776.3 q^{47} +12220.1 q^{49} +(-27100.6 - 25686.2i) q^{50} +(901.142 + 16803.3i) q^{52} +31909.4i q^{53} -39765.5i q^{55} +(9336.97 + 7945.08i) q^{56} +(-18151.3 + 19150.8i) q^{58} -5011.77 q^{59} +21451.5 q^{61} +(-34437.3 + 36333.7i) q^{62} +(-5244.24 - 32345.6i) q^{64} +51859.6i q^{65} +18429.1i q^{67} +(-14775.0 + 792.368i) q^{68} +(27422.6 + 25991.3i) q^{70} +67791.2 q^{71} +52806.9 q^{73} +(-61085.9 - 57897.7i) q^{74} +(62593.7 - 3356.83i) q^{76} +27309.0i q^{77} +94935.6i q^{79} +(-10800.5 - 100407. i) q^{80} +(20912.2 - 22063.7i) q^{82} -82255.2 q^{83} -45599.8 q^{85} +(7316.42 - 7719.31i) q^{86} +(47302.6 - 55589.5i) q^{88} +20696.7i q^{89} -35614.5i q^{91} +(1193.76 + 22259.7i) q^{92} +(105830. + 100307. i) q^{94} +193181. q^{95} +19858.5 q^{97} +(50172.2 + 47553.6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{4}+O(q^{10}) $ 20 * q + 20 * q^4	$ 20 q + 20 q^{4} + 184 q^{10} - 116 q^{13} - 4168 q^{16} + 696 q^{22} - 15228 q^{25} - 4764 q^{28} - 16520 q^{34} - 6452 q^{37} + 1504 q^{40} - 9336 q^{46} - 44464 q^{49} + 8236 q^{52} - 58736 q^{58} + 84604 q^{61} - 6496 q^{64} + 138696 q^{70} + 85420 q^{73} + 89172 q^{76} + 221200 q^{82} + 180320 q^{85} - 85824 q^{88} - 60936 q^{94} - 219908 q^{97}+O(q^{100}) $ 20 * q + 20 * q^4 + 184 * q^10 - 116 * q^13 - 4168 * q^16 + 696 * q^22 - 15228 * q^25 - 4764 * q^28 - 16520 * q^34 - 6452 * q^37 + 1504 * q^40 - 9336 * q^46 - 44464 * q^49 + 8236 * q^52 - 58736 * q^58 + 84604 * q^61 - 6496 * q^64 + 138696 * q^70 + 85420 * q^73 + 89172 * q^76 + 221200 * q^82 + 180320 * q^85 - 85824 * q^88 - 60936 * q^94 - 219908 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$-1$	$-1$

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 $

$2$	4.10571	+	3.89142i	0.725793	+	0.687913i
$2$	4.10571	+	3.89142i	0.725793	+	0.687913i
$3$		0			0
$3$		0			0
$4$	1.71366	+	31.9541i	0.0535519	+	0.998565i
$5$			98.6190i			1.76415i	0.471108	+	0.882076i	$0.343854\pi$
$5$			98.6190i			1.76415i	−0.471108	+	0.882076i	$0.656146\pi$
$6$		0			0
$7$		−	67.7266i		−	0.522413i	−0.965283	−	0.261207i	$-0.915880\pi$
$7$		−	67.7266i		−	0.522413i	0.965283	−	0.261207i	$-0.0841204\pi$
$8$	−117.311	+	137.863i	−0.648058	+	0.761591i
$9$		0			0
$10$	−383.768	+	404.901i	−1.21358	+	1.28041i
$11$	−403.224			−1.00476			−0.502382	−	0.864646i	$-0.667543\pi$
$11$	−403.224			−1.00476			−0.502382	+	0.864646i	$0.667543\pi$
$12$		0			0
$13$	525.857			0.862998			0.431499	−	0.902114i	$-0.357985\pi$
$13$	525.857			0.862998			0.431499	+	0.902114i	$0.357985\pi$
$14$	263.553	−	278.066i	0.359375	−	0.379164i
$15$		0			0
$16$	−1018.13	+	109.517i	−0.994264	+	0.106950i
$17$			462.383i			0.388043i	0.980997	+	0.194021i	$0.0621531\pi$
$17$			462.383i			0.388043i	−0.980997	+	0.194021i	$0.937847\pi$
$18$		0			0
$19$		−	1958.86i		−	1.24486i	−0.782676	−	0.622430i	$-0.786146\pi$
$19$		−	1958.86i		−	1.24486i	0.782676	−	0.622430i	$-0.213854\pi$
$20$	−3151.28	+	169.000i	−1.76162	+	0.0944737i
$21$		0			0
$22$	−1655.52	−	1569.11i	−0.729252	−	0.691191i
$23$	696.614			0.274582			0.137291	−	0.990531i	$-0.456160\pi$
$23$	696.614			0.274582			0.137291	+	0.990531i	$0.456160\pi$
$24$		0			0
$25$	−6600.72			−2.11223
$26$	2159.02	+	2046.33i	0.626358	+	0.593667i
$27$		0			0
$28$	2164.14	−	116.060i	0.521664	−	0.0279762i
$29$			4664.43i			1.02992i	0.857214	+	0.514960i	$0.172193\pi$
$29$			4664.43i			1.02992i	−0.857214	+	0.514960i	$0.827807\pi$
$30$		0			0
$31$			8849.55i			1.65393i	0.562254	+	0.826965i	$0.309934\pi$
$31$			8849.55i			1.65393i	−0.562254	+	0.826965i	$0.690066\pi$
$32$	−4606.31	−	3512.32i	−0.795203	−	0.606343i
$33$		0			0
$34$	−1799.33	+	1898.41i	−0.266939	+	0.281639i
$35$	6679.13			0.921616
$36$		0			0
$37$	−14878.3			−1.78669			−0.893343	−	0.449375i	$-0.851647\pi$
$37$	−14878.3			−1.78669			−0.893343	+	0.449375i	$0.851647\pi$
$38$	7622.77	−	8042.52i	0.856355	−	0.903511i
$39$		0			0
$40$	−13595.9	−	11569.1i	−1.34356	−	1.14327i
$41$		−	5373.92i		−	0.499265i	−0.968341	−	0.249633i	$-0.919690\pi$
$41$		−	5373.92i		−	0.499265i	0.968341	−	0.249633i	$-0.0803098\pi$
$42$		0			0
$43$		−	1880.14i		−	0.155067i	−0.996990	−	0.0775335i	$-0.975296\pi$
$43$		−	1880.14i		−	0.155067i	0.996990	−	0.0775335i	$-0.0247045\pi$
$44$	−690.989	−	12884.6i	−0.0538071	−	1.00332i
$45$		0			0
$46$	2860.09	+	2710.82i	0.199290	+	0.188889i
$47$	25776.3			1.70207			0.851034	−	0.525111i	$-0.175976\pi$
$47$	25776.3			1.70207			0.851034	+	0.525111i	$0.175976\pi$
$48$		0			0
$49$	12220.1			0.727084
$50$	−27100.6	−	25686.2i	−1.53304	−	1.45303i
$51$		0			0
$52$	901.142	+	16803.3i	0.0462152	+	0.861759i
$53$			31909.4i			1.56037i	0.625547	+	0.780186i	$0.284876\pi$
$53$			31909.4i			1.56037i	−0.625547	+	0.780186i	$0.715124\pi$
$54$		0			0
$55$		−	39765.5i		−	1.77256i
$56$	9336.97	+	7945.08i	0.397865	+	0.338554i
$57$		0			0
$58$	−18151.3	+	19150.8i	−0.708495	+	0.747509i
$59$	−5011.77			−0.187439			−0.0937197	−	0.995599i	$-0.529876\pi$
$59$	−5011.77			−0.187439			−0.0937197	+	0.995599i	$0.529876\pi$
$60$		0			0
$61$	21451.5			0.738132			0.369066	−	0.929403i	$-0.379678\pi$
$61$	21451.5			0.738132			0.369066	+	0.929403i	$0.379678\pi$
$62$	−34437.3	+	36333.7i	−1.13776	+	1.20041i
$63$		0			0
$64$	−5244.24	−	32345.6i	−0.160042	−	0.987110i
$65$			51859.6i			1.52246i
$66$		0			0
$67$			18429.1i			0.501553i	0.968045	+	0.250776i	$0.0806859\pi$
$67$			18429.1i			0.501553i	−0.968045	+	0.250776i	$0.919314\pi$
$68$	−14775.0	+	792.368i	−0.387486	+	0.0207804i
$69$		0			0
$70$	27422.6	+	25991.3i	0.668903	+	0.633991i
$71$	67791.2			1.59598			0.797990	−	0.602671i	$-0.205897\pi$
$71$	67791.2			1.59598			0.797990	+	0.602671i	$0.205897\pi$
$72$		0			0
$73$	52806.9			1.15980			0.579901	−	0.814687i	$-0.303091\pi$
$73$	52806.9			1.15980			0.579901	+	0.814687i	$0.303091\pi$
$74$	−61085.9	−	57897.7i	−1.29677	−	1.22908i
$75$		0			0
$76$	62593.7	−	3356.83i	1.24307	−	0.0666646i
$77$			27309.0i			0.524902i
$78$		0			0
$79$			94935.6i			1.71144i	0.517440	+	0.855720i	$0.326885\pi$
$79$			94935.6i			1.71144i	−0.517440	+	0.855720i	$0.673115\pi$
$80$	−10800.5	−	100407.i	−0.188676	−	1.75403i
$81$		0			0
$82$	20912.2	−	22063.7i	0.343451	−	0.362363i
$83$	−82255.2			−1.31059			−0.655297	−	0.755371i	$-0.727457\pi$
$83$	−82255.2			−1.31059			−0.655297	+	0.755371i	$0.727457\pi$
$84$		0			0
$85$	−45599.8			−0.684566
$86$	7316.42	−	7719.31i	0.106673	−	0.112547i
$87$		0			0
$88$	47302.6	−	55589.5i	0.651146	−	0.765220i
$89$			20696.7i			0.276966i	0.990365	+	0.138483i	$0.0442226\pi$
$89$			20696.7i			0.276966i	−0.990365	+	0.138483i	$0.955777\pi$
$90$		0			0
$91$		−	35614.5i		−	0.450841i
$92$	1193.76	+	22259.7i	0.0147044	+	0.274188i
$93$		0			0
$94$	105830.	+	100307.i	1.23535	+	1.17087i
$95$	193181.			2.19612
$96$		0			0
$97$	19858.5			0.214298			0.107149	−	0.994243i	$-0.465828\pi$
$97$	19858.5			0.214298			0.107149	+	0.994243i	$0.465828\pi$
$98$	50172.2	+	47553.6i	0.527713	+	0.500171i
$99$		0			0
$100$	−11311.4	−	210920.i	−0.113114	−	2.10920i
$101$			42840.9i			0.417884i	0.977928	+	0.208942i	$0.0670020\pi$
$101$			42840.9i			0.417884i	−0.977928	+	0.208942i	$0.932998\pi$
$102$		0			0
$103$		−	92957.3i		−	0.863357i	−0.902027	−	0.431679i	$-0.857921\pi$
$103$		−	92957.3i		−	0.863357i	0.902027	−	0.431679i	$-0.142079\pi$
$104$	−61688.9	+	72496.1i	−0.559273	+	0.657251i
$105$		0			0
$106$	−124173.	+	131010.i	−1.07340	+	1.13251i
$107$	−48825.7			−0.412277			−0.206139	−	0.978523i	$-0.566090\pi$
$107$	−48825.7			−0.412277			−0.206139	+	0.978523i	$0.566090\pi$
$108$		0			0
$109$	104798.			0.844863			0.422432	−	0.906395i	$-0.361177\pi$
$109$	104798.			0.844863			0.422432	+	0.906395i	$0.361177\pi$
$110$	154745.	−	163266.i	1.21936	−	1.28651i
$111$		0			0
$112$	7417.21	+	68954.3i	0.0558722	+	0.519417i
$113$		−	57875.8i		−	0.426384i	−0.977010	−	0.213192i	$-0.931614\pi$
$113$		−	57875.8i		−	0.426384i	0.977010	−	0.213192i	$-0.0683860\pi$
$114$		0			0
$115$			68699.4i			0.484405i
$116$	−149048.	+	7993.26i	−1.02844	+	0.0551542i
$117$		0			0
$118$	−20576.8	−	19502.9i	−0.136042	−	0.128942i
$119$	31315.6			0.202719
$120$		0			0
$121$	1538.38			0.00955215
$122$	88073.8	+	83477.0i	0.535731	+	0.507770i
$123$		0			0
$124$	−282779.	+	15165.1i	−1.65156	+	0.0885711i
$125$		−	342772.i		−	1.96214i
$126$		0			0
$127$		−	204929.i		−	1.12744i	−0.825965	−	0.563721i	$-0.809369\pi$
$127$		−	204929.i		−	1.12744i	0.825965	−	0.563721i	$-0.190631\pi$
$128$	104339.	−	153209.i	0.562889	−	0.826533i
$129$		0			0
$130$	−201807.	+	212920.i	−1.04732	+	1.10499i
$131$	−106839.			−0.543938			−0.271969	−	0.962306i	$-0.587675\pi$
$131$	−106839.			−0.543938			−0.271969	+	0.962306i	$0.587675\pi$
$132$		0			0
$133$	−132667.			−0.650331
$134$	−71715.3	+	75664.4i	−0.345025	+	0.364024i
$135$		0			0
$136$	−63745.3	−	54242.6i	−0.295530	−	0.251474i
$137$			141277.i			0.643089i	0.946895	+	0.321544i	$0.104202\pi$
$137$			141277.i			0.643089i	−0.946895	+	0.321544i	$0.895798\pi$
$138$		0			0
$139$			202692.i			0.889816i	0.895576	+	0.444908i	$0.146764\pi$
$139$			202692.i			0.889816i	−0.895576	+	0.444908i	$0.853236\pi$
$140$	11445.8	+	213426.i	0.0493543	+	0.920293i
$141$		0			0
$142$	278331.	+	263804.i	1.15835	+	1.09789i
$143$	−212038.			−0.867110
$144$		0			0
$145$	−460002.			−1.81693
$146$	216810.	+	205494.i	0.841776	+	0.797843i
$147$		0			0
$148$	−25496.3	−	475422.i	−0.0956805	−	1.78412i
$149$		−	27975.3i		−	0.103231i	−0.998667	−	0.0516153i	$-0.983563\pi$
$149$		−	27975.3i		−	0.103231i	0.998667	−	0.0516153i	$-0.0164370\pi$
$150$		0			0
$151$			26904.4i			0.0960242i	0.998847	+	0.0480121i	$0.0152886\pi$
$151$			26904.4i			0.0960242i	−0.998847	+	0.0480121i	$0.984711\pi$
$152$	270054.	+	229796.i	0.948074	+	0.806741i
$153$		0			0
$154$	−106271.	+	112123.i	−0.361087	+	0.380971i
$155$	−872734.			−2.91778
$156$		0			0
$157$	−169696.			−0.549442			−0.274721	−	0.961524i	$-0.588585\pi$
$157$	−169696.			−0.549442			−0.274721	+	0.961524i	$0.588585\pi$
$158$	−369435.	+	389778.i	−1.17732	+	1.24215i
$159$		0			0
$160$	346381.	−	454270.i	1.06968	−	1.40286i
$161$		−	47179.3i		−	0.143445i
$162$		0			0
$163$		−	120390.i		−	0.354912i	−0.984129	−	0.177456i	$-0.943213\pi$
$163$		−	120390.i		−	0.354912i	0.984129	−	0.177456i	$-0.0567868\pi$
$164$	171719.	−	9209.08i	0.498549	−	0.0267366i
$165$		0			0
$166$	−337716.	−	320090.i	−0.951220	−	0.901574i
$167$	252246.			0.699896			0.349948	−	0.936769i	$-0.386199\pi$
$167$	252246.			0.699896			0.349948	+	0.936769i	$0.386199\pi$
$168$		0			0
$169$	−94767.0			−0.255235
$170$	−187219.	−	177448.i	−0.496853	−	0.470922i
$171$		0			0
$172$	60078.2	−	3221.93i	0.154844	−	0.00830414i
$173$			71644.3i			0.181998i	0.995851	+	0.0909989i	$0.0290060\pi$
$173$			71644.3i			0.181998i	−0.995851	+	0.0909989i	$0.970994\pi$
$174$		0			0
$175$			447044.i			1.10346i
$176$	410533.	−	44159.9i	0.999002	−	0.107460i
$177$		0			0
$178$	−80539.7	+	84974.7i	−0.190528	+	0.201020i
$179$	472794.			1.10291			0.551455	−	0.834205i	$-0.314073\pi$
$179$	472794.			1.10291			0.551455	+	0.834205i	$0.314073\pi$
$180$		0			0
$181$	142613.			0.323566			0.161783	−	0.986826i	$-0.448275\pi$
$181$	142613.			0.323566			0.161783	+	0.986826i	$0.448275\pi$
$182$	138591.	−	146223.i	0.310139	−	0.327218i
$183$		0			0
$184$	−81720.5	+	96037.1i	−0.177945	+	0.209119i
$185$		−	1.46728e6i		−	3.15199i
$186$		0			0
$187$		−	186444.i		−	0.389892i
$188$	44171.9	+	823659.i	0.0911490	+	1.69962i
$189$		0			0
$190$	793146.	+	751750.i	1.59393	+	1.51074i
$191$	−762070.			−1.51151			−0.755755	−	0.654854i	$-0.772730\pi$
$191$	−762070.			−1.51151			−0.755755	+	0.654854i	$0.772730\pi$
$192$		0			0
$193$	766564.			1.48134			0.740670	−	0.671869i	$-0.234508\pi$
$193$	766564.			1.48134			0.740670	+	0.671869i	$0.234508\pi$
$194$	81533.3	+	77277.9i	0.155536	+	0.147418i
$195$		0			0
$196$	20941.1	+	390482.i	0.0389368	+	0.726041i
$197$		−	279378.i		−	0.512893i	−0.966558	−	0.256446i	$-0.917448\pi$
$197$		−	279378.i		−	0.512893i	0.966558	−	0.256446i	$-0.0825517\pi$
$198$		0			0
$199$			50473.8i			0.0903510i	0.998979	+	0.0451755i	$0.0143847\pi$
$199$			50473.8i			0.0903510i	−0.998979	+	0.0451755i	$0.985615\pi$
$200$	774337.	−	909993.i	1.36885	−	1.60865i
$201$		0			0
$202$	−166712.	+	175892.i	−0.287468	+	0.303297i
$203$	315906.			0.538044
$204$		0			0
$205$	529970.			0.880779
$206$	361736.	−	381656.i	0.593915	−	0.626619i
$207$		0			0
$208$	−535389.	+	57590.3i	−0.858048	+	0.0922978i
$209$			789861.i			1.25079i
$210$		0			0
$211$		−	762679.i		−	1.17933i	−0.807647	−	0.589666i	$-0.799260\pi$
$211$		−	762679.i		−	1.17933i	0.807647	−	0.589666i	$-0.200740\pi$
$212$	−1.01963e6	+	54681.8i	−1.55813	+	0.0835610i
$213$		0			0
$214$	−200464.	−	190001.i	−0.299228	−	0.283611i
$215$	185418.			0.273562
$216$		0			0
$217$	599350.			0.864034
$218$	430269.	+	407813.i	0.613196	+	0.581192i
$219$		0			0
$220$	1.27067e6	−	68144.7i	1.77001	−	0.0949239i
$221$			243147.i			0.334880i
$222$		0			0
$223$		−	1.12395e6i		−	1.51351i	−0.653697	−	0.756756i	$-0.726783\pi$
$223$		−	1.12395e6i		−	1.51351i	0.653697	−	0.756756i	$-0.273217\pi$
$224$	−237877.	+	311969.i	−0.316762	+	0.415424i
$225$		0			0
$226$	225219.	−	237621.i	0.293315	−	0.309467i
$227$	630849.			0.812570			0.406285	−	0.913746i	$-0.366824\pi$
$227$	630849.			0.812570			0.406285	+	0.913746i	$0.366824\pi$
$228$		0			0
$229$	551782.			0.695311			0.347655	−	0.937622i	$-0.386978\pi$
$229$	551782.			0.695311			0.347655	+	0.937622i	$0.386978\pi$
$230$	−267338.	+	282060.i	−0.333228	+	0.351578i
$231$		0			0
$232$	−643051.	−	547189.i	−0.784378	−	0.667448i
$233$			506640.i			0.611377i	0.952132	+	0.305689i	$0.0988867\pi$
$233$			506640.i			0.611377i	−0.952132	+	0.305689i	$0.901113\pi$
$234$		0			0
$235$			2.54204e6i			3.00270i
$236$	−8588.47	−	160146.i	−0.0100377	−	0.187170i
$237$		0			0
$238$	128573.	+	121862.i	0.147132	+	0.139453i
$239$	−843448.			−0.955132			−0.477566	−	0.878596i	$-0.658481\pi$
$239$	−843448.			−0.955132			−0.477566	+	0.878596i	$0.658481\pi$
$240$		0			0
$241$	−160398.			−0.177892			−0.0889460	−	0.996036i	$-0.528350\pi$
$241$	−160398.			−0.177892			−0.0889460	+	0.996036i	$0.528350\pi$
$242$	6316.15	+	5986.50i	0.00693289	+	0.00657105i
$243$		0			0
$244$	36760.7	+	685464.i	0.0395284	+	0.737073i
$245$			1.20514e6i			1.28269i
$246$		0			0
$247$		−	1.03008e6i		−	1.07431i
$248$	−1.22002e6	−	1.03815e6i	−1.25962	−	1.07184i
$249$		0			0
$250$	1.33387e6	−	1.40732e6i	1.34978	−	1.42411i
$251$	793734.			0.795227			0.397613	−	0.917553i	$-0.369839\pi$
$251$	793734.			0.795227			0.397613	+	0.917553i	$0.369839\pi$
$252$		0			0
$253$	−280891.			−0.275891
$254$	797465.	−	841379.i	0.775582	−	0.818290i
$255$		0			0
$256$	1.02459e6	−	223004.i	0.977123	−	0.212674i
$257$		−	410870.i		−	0.388036i	−0.980998	−	0.194018i	$-0.937848\pi$
$257$		−	410870.i		−	0.388036i	0.980998	−	0.194018i	$-0.0621520\pi$
$258$		0			0
$259$			1.00766e6i			0.933389i
$260$	−1.65712e6	+	88869.7i	−1.52027	+	0.0815306i
$261$		0			0
$262$	−438648.	−	415754.i	−0.394787	−	0.374182i
$263$	−808668.			−0.720910			−0.360455	−	0.932777i	$-0.617379\pi$
$263$	−808668.			−0.720910			−0.360455	+	0.932777i	$0.617379\pi$
$264$		0			0
$265$	−3.14687e6			−2.75273
$266$	−544693.	−	516264.i	−0.472006	−	0.447371i
$267$		0			0
$268$	−588884.	+	31581.2i	−0.500833	+	0.0268591i
$269$		−	693759.i		−	0.584558i	−0.956333	−	0.292279i	$-0.905586\pi$
$269$		−	693759.i		−	0.584558i	0.956333	−	0.292279i	$-0.0944136\pi$
$270$		0			0
$271$			34964.4i			0.0289203i	0.999895	+	0.0144602i	$0.00460297\pi$
$271$			34964.4i			0.0289203i	−0.999895	+	0.0144602i	$0.995397\pi$
$272$	−50638.8	−	470764.i	−0.0415012	−	0.385817i
$273$		0			0
$274$	−549769.	+	580043.i	−0.442389	+	0.466749i
$275$	2.66157e6			2.12229
$276$		0			0
$277$	−348977.			−0.273274			−0.136637	−	0.990621i	$-0.543629\pi$
$277$	−348977.			−0.273274			−0.136637	+	0.990621i	$0.543629\pi$
$278$	−788761.	+	832195.i	−0.612116	+	0.645822i
$279$		0			0
$280$	−783536.	+	920803.i	−0.597261	+	0.701894i
$281$			1.85655e6i			1.40263i	0.712854	+	0.701313i	$0.247402\pi$
$281$			1.85655e6i			1.40263i	−0.712854	+	0.701313i	$0.752598\pi$
$282$		0			0
$283$			70998.4i			0.0526966i	0.999653	+	0.0263483i	$0.00838789\pi$
$283$			70998.4i			0.0526966i	−0.999653	+	0.0263483i	$0.991612\pi$
$284$	116171.	+	2.16620e6i	0.0854678	+	1.59369i
$285$		0			0
$286$	−870567.	−	825130.i	−0.629342	−	0.596496i
$287$	−363957.			−0.260823
$288$		0			0
$289$	1.20606e6			0.849423
$290$	−1.88863e6	−	1.79006e6i	−1.31872	−	1.24989i
$291$		0			0
$292$	90493.2	+	1.68740e6i	0.0621096	+	1.15814i
$293$			2.36601e6i			1.61008i	0.593221	+	0.805039i	$0.297856\pi$
$293$			2.36601e6i			1.61008i	−0.593221	+	0.805039i	$0.702144\pi$
$294$		0			0
$295$		−	494256.i		−	0.330671i
$296$	1.74539e6	−	2.05116e6i	1.15788	−	1.36072i
$297$		0			0
$298$	108864.	−	114858.i	0.0710137	−	0.0749241i
$299$	366320.			0.236964
$300$		0			0
$301$	−127336.			−0.0810090
$302$	−104696.	+	110461.i	−0.0660562	+	0.0696937i
$303$		0			0
$304$	214529.	+	1.99437e6i	0.133138	+	1.23772i
$305$			2.11553e6i			1.30218i
$306$		0			0
$307$		−	19907.0i		−	0.0120548i	−0.999982	−	0.00602741i	$-0.998081\pi$
$307$		−	19907.0i		−	0.0120548i	0.999982	−	0.00602741i	$-0.00191860\pi$
$308$	−872633.	+	46798.3i	−0.524149	+	0.0281095i
$309$		0			0
$310$	−3.58319e6	−	3.39618e6i	−2.11771	−	2.00718i
$311$	2.54109e6			1.48977			0.744885	−	0.667193i	$-0.232505\pi$
$311$	2.54109e6			1.48977			0.744885	+	0.667193i	$0.232505\pi$
$312$		0			0
$313$	964745.			0.556611			0.278306	−	0.960493i	$-0.410227\pi$
$313$	964745.			0.556611			0.278306	+	0.960493i	$0.410227\pi$
$314$	−696721.	−	660358.i	−0.398781	−	0.377968i
$315$		0			0
$316$	−3.03358e6	+	162688.i	−1.70898	+	0.0916509i
$317$		−	1.96724e6i		−	1.09954i	−0.835317	−	0.549769i	$-0.814716\pi$
$317$		−	1.96724e6i		−	1.09954i	0.835317	−	0.549769i	$-0.185284\pi$
$318$		0			0
$319$		−	1.88081e6i		−	1.03483i
$320$	3.18990e6	−	517182.i	1.74141	−	0.282337i
$321$		0			0
$322$	183595.	−	193704.i	0.0986779	−	0.104112i
$323$	905745.			0.483059
$324$		0			0
$325$	−3.47104e6			−1.82285
$326$	468488.	−	494286.i	0.244149	−	0.257593i
$327$		0			0
$328$	740863.	+	630420.i	0.380236	+	0.323553i
$329$		−	1.74574e6i		−	0.889182i
$330$		0			0
$331$		−	2.34318e6i		−	1.17554i	−0.809029	−	0.587769i	$-0.800007\pi$
$331$		−	2.34318e6i		−	1.17554i	0.809029	−	0.587769i	$-0.199993\pi$
$332$	−140958.	−	2.62839e6i	−0.0701849	−	1.30871i
$333$		0			0
$334$	1.03565e6	+	981597.i	0.507980	+	0.481467i
$335$	−1.81746e6			−0.884815
$336$		0			0
$337$	3.39138e6			1.62668			0.813338	−	0.581791i	$-0.197648\pi$
$337$	3.39138e6			1.62668			0.813338	+	0.581791i	$0.197648\pi$
$338$	−389086.	−	368779.i	−0.185248	−	0.175580i
$339$		0			0
$340$	−78142.6	−	1.45710e6i	−0.0366598	−	0.683584i
$341$		−	3.56835e6i		−	1.66181i
$342$		0			0
$343$		−	1.96591e6i		−	0.902252i
$344$	259201.	+	220561.i	0.118098	+	0.100492i
$345$		0			0
$346$	−278798.	+	294150.i	−0.125199	+	0.132093i
$347$	−952624.			−0.424715			−0.212358	−	0.977192i	$-0.568114\pi$
$347$	−952624.			−0.424715			−0.212358	+	0.977192i	$0.568114\pi$
$348$		0			0
$349$	−1.24091e6			−0.545350			−0.272675	−	0.962106i	$-0.587908\pi$
$349$	−1.24091e6			−0.545350			−0.272675	+	0.962106i	$0.587908\pi$
$350$	−1.73964e6	+	1.83543e6i	−0.759082	+	0.800881i
$351$		0			0
$352$	1.85737e6	+	1.41625e6i	0.798992	+	0.609233i
$353$		−	322727.i		−	0.137847i	−0.997622	−	0.0689236i	$-0.978044\pi$
$353$		−	322727.i		−	0.137847i	0.997622	−	0.0689236i	$-0.0219565\pi$
$354$		0			0
$355$			6.68550e6i			2.81555i
$356$	−661345.	+	35467.2i	−0.276569	+	0.0148321i
$357$		0			0
$358$	1.94116e6	+	1.83984e6i	0.800484	+	0.758705i
$359$	2.01638e6			0.825725			0.412862	−	0.910793i	$-0.364529\pi$
$359$	2.01638e6			0.825725			0.412862	+	0.910793i	$0.364529\pi$
$360$		0			0
$361$	−1.36105e6			−0.549675
$362$	585528.	+	554968.i	0.234842	+	0.222585i
$363$		0			0
$364$	1.13803e6	−	61031.3i	0.450194	−	0.0241434i
$365$			5.20777e6i			2.04607i
$366$		0			0
$367$			2.80423e6i			1.08680i	0.839475	+	0.543398i	$0.182862\pi$
$367$			2.80423e6i			1.08680i	−0.839475	+	0.543398i	$0.817138\pi$
$368$	−709241.	+	76291.1i	−0.273007	+	0.0293666i
$369$		0			0
$370$	5.70981e6	−	6.02423e6i	2.16829	−	2.28769i
$371$	2.16111e6			0.815159
$372$		0			0
$373$	793282.			0.295227			0.147613	−	0.989045i	$-0.452841\pi$
$373$	793282.			0.295227			0.147613	+	0.989045i	$0.452841\pi$
$374$	725531.	−	765483.i	0.268211	−	0.282981i
$375$		0			0
$376$	−3.02385e6	+	3.55360e6i	−1.10304	+	1.29628i
$377$			2.45282e6i			0.888819i
$378$		0			0
$379$		−	875935.i		−	0.313237i	−0.987659	−	0.156619i	$-0.949941\pi$
$379$		−	875935.i		−	0.313237i	0.987659	−	0.156619i	$-0.0500594\pi$
$380$	331048.	+	6.17293e6i	0.117607	+	2.19297i
$381$		0			0
$382$	−3.12883e6	−	2.96554e6i	−1.09704	−	1.03979i
$383$	−5.68554e6			−1.98050			−0.990250	−	0.139305i	$-0.955513\pi$
$383$	−5.68554e6			−1.98050			−0.990250	+	0.139305i	$0.955513\pi$
$384$		0			0
$385$	−2.69318e6			−0.926007
$386$	3.14729e6	+	2.98302e6i	1.07515	+	1.01903i
$387$		0			0
$388$	34030.8	+	634561.i	0.0114761	+	0.213990i
$389$		−	2.56686e6i		−	0.860059i	−0.902815	−	0.430029i	$-0.858503\pi$
$389$		−	2.56686e6i		−	0.860059i	0.902815	−	0.430029i	$-0.141497\pi$
$390$		0			0
$391$			322102.i			0.106550i
$392$	−1.43355e6	+	1.68470e6i	−0.471193	+	0.553741i
$393$		0			0
$394$	1.08718e6	−	1.14704e6i	0.352825	−	0.372254i
$395$	−9.36246e6			−3.01924
$396$		0			0
$397$	6.13240e6			1.95278			0.976392	−	0.216004i	$-0.0693025\pi$
$397$	6.13240e6			1.95278			0.976392	+	0.216004i	$0.0693025\pi$
$398$	−196415.	+	207231.i	−0.0621536	+	0.0655762i
$399$		0			0
$400$	6.72037e6	−	722891.i	2.10011	−	0.225903i
$401$		−	5.63051e6i		−	1.74858i	−0.485401	−	0.874292i	$-0.661326\pi$
$401$		−	5.63051e6i		−	1.74858i	0.485401	−	0.874292i	$-0.338674\pi$
$402$		0			0
$403$			4.65360e6i			1.42734i
$404$	−1.36894e6	+	73414.9i	−0.417284	+	0.0223785i
$405$		0			0
$406$	1.29702e6	+	1.22932e6i	0.390509	+	0.370127i
$407$	5.99928e6			1.79520
$408$		0			0
$409$	2.86885e6			0.848008			0.424004	−	0.905660i	$-0.360624\pi$
$409$	2.86885e6			0.848008			0.424004	+	0.905660i	$0.360624\pi$
$410$	2.17590e6	+	2.06234e6i	0.639264	+	0.605899i
$411$		0			0
$412$	2.97037e6	−	159297.i	0.862118	−	0.0462345i
$413$			339430.i			0.0979208i
$414$		0			0
$415$		−	8.11193e6i		−	2.31209i
$416$	−2.42226e6	−	1.84698e6i	−0.686258	−	0.523273i
$417$		0			0
$418$	−3.07368e6	+	3.24294e6i	−0.860435	+	0.907816i
$419$	−5.25000e6			−1.46091			−0.730456	−	0.682960i	$-0.760692\pi$
$419$	−5.25000e6			−1.46091			−0.730456	+	0.682960i	$0.760692\pi$
$420$		0			0
$421$	3.82887e6			1.05285			0.526423	−	0.850223i	$-0.323533\pi$
$421$	3.82887e6			1.05285			0.526423	+	0.850223i	$0.323533\pi$
$422$	2.96791e6	−	3.13134e6i	0.811277	−	0.855951i
$423$		0			0
$424$	−4.39911e6	−	3.74332e6i	−1.18837	−	1.01121i
$425$		−	3.05206e6i		−	0.819635i
$426$		0			0
$427$		−	1.45284e6i		−	0.385610i
$428$	−83670.8	−	1.56018e6i	−0.0220782	−	0.411685i
$429$		0			0
$430$	761271.	+	721539.i	0.198549	+	0.188186i
$431$	763450.			0.197965			0.0989823	−	0.995089i	$-0.468441\pi$
$431$	763450.			0.197965			0.0989823	+	0.995089i	$0.468441\pi$
$432$		0			0
$433$	−126626.			−0.0324566			−0.0162283	−	0.999868i	$-0.505166\pi$
$433$	−126626.			−0.0324566			−0.0162283	+	0.999868i	$0.505166\pi$
$434$	2.46075e6	+	2.33232e6i	0.627110	+	0.594380i
$435$		0			0
$436$	179588.	+	3.34872e6i	0.0452441	+	0.843651i
$437$		−	1.36457e6i		−	0.341816i
$438$		0			0
$439$		−	6.36482e6i		−	1.57625i	−0.615516	−	0.788124i	$-0.711052\pi$
$439$		−	6.36482e6i		−	1.57625i	0.615516	−	0.788124i	$-0.288948\pi$
$440$	5.48218e6	+	4.66494e6i	1.34996	+	1.14872i
$441$		0			0
$442$	−946189.	+	998292.i	−0.230368	+	0.243054i
$443$	−5.00344e6			−1.21132			−0.605661	−	0.795723i	$-0.707091\pi$
$443$	−5.00344e6			−1.21132			−0.605661	+	0.795723i	$0.707091\pi$
$444$		0			0
$445$	−2.04109e6			−0.488610
$446$	4.37378e6	−	4.61462e6i	1.04116	−	1.09850i
$447$		0			0
$448$	−2.19066e6	+	355175.i	−0.515679	+	0.0836078i
$449$		−	2.33255e6i		−	0.546028i	−0.962010	−	0.273014i	$-0.911979\pi$
$449$		−	2.33255e6i		−	0.546028i	0.962010	−	0.273014i	$-0.0880205\pi$
$450$		0			0
$451$			2.16689e6i			0.501644i
$452$	1.84937e6	−	99179.6i	0.425772	−	0.0228337i
$453$		0			0
$454$	2.59008e6	+	2.45490e6i	0.589758	+	0.558978i
$455$	3.51227e6			0.795352
$456$		0			0
$457$	3.50222e6			0.784429			0.392214	−	0.919874i	$-0.371709\pi$
$457$	3.50222e6			0.784429			0.392214	+	0.919874i	$0.371709\pi$
$458$	2.26546e6	+	2.14722e6i	0.504652	+	0.478313i
$459$		0			0
$460$	−2.19523e6	+	117728.i	−0.483710	+	0.0259408i
$461$		−	4.55640e6i		−	0.998550i	−0.866444	−	0.499275i	$-0.833600\pi$
$461$		−	4.55640e6i		−	0.998550i	0.866444	−	0.499275i	$-0.166400\pi$
$462$		0			0
$463$		−	3.28727e6i		−	0.712660i	−0.934360	−	0.356330i	$-0.884028\pi$
$463$		−	3.28727e6i		−	0.712660i	0.934360	−	0.356330i	$-0.115972\pi$
$464$	−510834.	−	4.74898e6i	−0.110150	−	1.02401i
$465$		0			0
$466$	−1.97155e6	+	2.08011e6i	−0.420574	+	0.443734i
$467$	4.27154e6			0.906343			0.453171	−	0.891423i	$-0.350293\pi$
$467$	4.27154e6			0.906343			0.453171	+	0.891423i	$0.350293\pi$
$468$		0			0
$469$	1.24814e6			0.262018
$470$	−9.89215e6	+	1.04369e7i	−2.06560	+	2.17934i
$471$		0			0
$472$	587935.	−	690935.i	0.121472	−	0.142752i
$473$			758117.i			0.155806i
$474$		0			0
$475$			1.29299e7i			2.62943i
$476$	53664.4	+	1.00066e6i	0.0108560	+	0.202428i
$477$		0			0
$478$	−3.46295e6	−	3.28221e6i	−0.693228	−	0.657048i
$479$	2.08964e6			0.416134			0.208067	−	0.978115i	$-0.433283\pi$
$479$	2.08964e6			0.416134			0.208067	+	0.978115i	$0.433283\pi$
$480$		0			0
$481$	−7.82385e6			−1.54191
$482$	−658547.	−	624176.i	−0.129113	−	0.122374i
$483$		0			0
$484$	2636.27	+	49157.6i	0.000511536	+	0.00953845i
$485$			1.95843e6i			0.378054i
$486$		0			0
$487$			2.02802e6i			0.387480i	0.981053	+	0.193740i	$0.0620618\pi$
$487$			2.02802e6i			0.387480i	−0.981053	+	0.193740i	$0.937938\pi$
$488$	−2.51650e6	+	2.95737e6i	−0.478352	+	0.562155i
$489$		0			0
$490$	−4.68969e6	+	4.94793e6i	−0.882377	+	0.930966i
$491$	−7.39231e6			−1.38381			−0.691905	−	0.721988i	$-0.743228\pi$
$491$	−7.39231e6			−1.38381			−0.691905	+	0.721988i	$0.743228\pi$
$492$		0			0
$493$	−2.15675e6			−0.399653
$494$	4.00849e6	−	4.22922e6i	0.739032	−	0.779728i
$495$		0			0
$496$	−969176.	−	9.00996e6i	−0.176888	−	1.64444i
$497$		−	4.59127e6i		−	0.833761i
$498$		0			0
$499$		−	579614.i		−	0.104205i	−0.998642	−	0.0521024i	$-0.983408\pi$
$499$		−	579614.i		−	0.104205i	0.998642	−	0.0521024i	$-0.0165922\pi$
$500$	1.09530e7	−	587395.i	1.95933	−	0.105076i
$501$		0			0
$502$	3.25884e6	+	3.08876e6i	0.577170	+	0.547047i
$503$	−1.39449e6			−0.245750			−0.122875	−	0.992422i	$-0.539211\pi$
$503$	−1.39449e6			−0.245750			−0.122875	+	0.992422i	$0.539211\pi$
$504$		0			0
$505$	−4.22493e6			−0.737210
$506$	−1.15326e6	−	1.09307e6i	−0.200240	−	0.189789i
$507$		0			0
$508$	6.54832e6	−	351179.i	1.12582	−	0.0603767i
$509$			2.31186e6i			0.395519i	0.980251	+	0.197759i	$0.0633665\pi$
$509$			2.31186e6i			0.395519i	−0.980251	+	0.197759i	$0.936634\pi$
$510$		0			0
$511$		−	3.57643e6i		−	0.605896i
$512$	5.07446e6	+	3.07151e6i	0.855490	+	0.517819i
$513$		0			0
$514$	1.59887e6	−	1.68691e6i	0.266935	−	0.281634i
$515$	9.16736e6			1.52309
$516$		0			0
$517$	−1.03936e7			−1.71018
$518$	−3.92121e6	+	4.13714e6i	−0.642090	+	0.677447i
$519$		0			0
$520$	−7.14950e6	−	6.08370e6i	−1.15949	−	0.986641i
$521$			2.65886e6i			0.429143i	0.976708	+	0.214571i	$0.0688355\pi$
$521$			2.65886e6i			0.429143i	−0.976708	+	0.214571i	$0.931165\pi$
$522$		0			0
$523$			3.85731e6i			0.616638i	0.951283	+	0.308319i	$0.0997664\pi$
$523$			3.85731e6i			0.616638i	−0.951283	+	0.308319i	$0.900234\pi$
$524$	−183085.	−	3.41393e6i	−0.0291290	−	0.543158i
$525$		0			0
$526$	−3.32016e6	−	3.14687e6i	−0.523232	−	0.495923i
$527$	−4.09188e6			−0.641795
$528$		0			0
$529$	−5.95107e6			−0.924605
$530$	−1.29201e7	−	1.22458e7i	−1.99792	−	1.89364i
$531$		0			0
$532$	−227347.	−	4.23926e6i	−0.0348265	−	0.649398i
$533$		−	2.82591e6i		−	0.430865i
$534$		0			0
$535$		−	4.81515e6i		−	0.727319i
$536$	−2.54068e6	−	2.16193e6i	−0.381978	−	0.325035i
$537$		0			0
$538$	2.69971e6	−	2.84837e6i	0.402125	−	0.424268i
$539$	−4.92744e6			−0.730549
$540$		0			0
$541$	−7.07371e6			−1.03909			−0.519546	−	0.854442i	$-0.673899\pi$
$541$	−7.07371e6			−1.03909			−0.519546	+	0.854442i	$0.673899\pi$
$542$	−136061.	+	143554.i	−0.0198947	+	0.0209902i
$543$		0			0
$544$	1.62403e6	−	2.12988e6i	0.235287	−	0.308573i
$545$			1.03351e7i			1.49047i
$546$		0			0
$547$		−	2.47235e6i		−	0.353298i	−0.984274	−	0.176649i	$-0.943474\pi$
$547$		−	2.47235e6i		−	0.353298i	0.984274	−	0.176649i	$-0.0565258\pi$
$548$	−4.51438e6	+	242101.i	−0.642166	+	0.0344386i
$549$		0			0
$550$	1.09276e7	+	1.03573e7i	1.54035	+	1.45995i
$551$	9.13699e6			1.28211
$552$		0			0
$553$	6.42967e6			0.894079
$554$	−1.43280e6	−	1.35802e6i	−0.198340	−	0.187988i
$555$		0			0
$556$	−6.47684e6	+	347346.i	−0.888539	+	0.0476514i
$557$			5.84907e6i			0.798820i	0.916772	+	0.399410i	$0.130785\pi$
$557$			5.84907e6i			0.798820i	−0.916772	+	0.399410i	$0.869215\pi$
$558$		0			0
$559$		−	988686.i		−	0.133822i
$560$	−6.80020e6	+	731478.i	−0.916330	+	0.0985670i
$561$		0			0
$562$	−7.22464e6	+	7.62247e6i	−0.964884	+	1.01802i
$563$	−310076.			−0.0412284			−0.0206142	−	0.999788i	$-0.506562\pi$
$563$	−310076.			−0.0412284			−0.0206142	+	0.999788i	$0.506562\pi$
$564$		0			0
$565$	5.70766e6			0.752206
$566$	−276285.	+	291498.i	−0.0362506	+	0.0382468i
$567$		0			0
$568$	−7.95265e6	+	9.34587e6i	−1.03429	+	1.21548i
$569$			4.87432e6i			0.631151i	0.948900	+	0.315575i	$0.102198\pi$
$569$			4.87432e6i			0.631151i	−0.948900	+	0.315575i	$0.897802\pi$
$570$		0			0
$571$			1.04563e7i			1.34211i	0.741409	+	0.671054i	$0.234158\pi$
$571$			1.04563e7i			1.34211i	−0.741409	+	0.671054i	$0.765842\pi$
$572$	−363362.	−	6.77549e6i	−0.0464354	−	0.865865i
$573$		0			0
$574$	−1.49430e6	−	1.41631e6i	−0.189303	−	0.179423i
$575$	−4.59815e6			−0.579981
$576$		0			0
$577$	9.62808e6			1.20393			0.601963	−	0.798524i	$-0.294385\pi$
$577$	9.62808e6			1.20393			0.601963	+	0.798524i	$0.294385\pi$
$578$	4.95173e6	+	4.69329e6i	0.616506	+	0.584329i
$579$		0			0
$580$	−788287.	−	1.46989e7i	−0.0973004	−	1.81433i
$581$			5.57086e6i			0.684672i
$582$		0			0
$583$		−	1.28666e7i		−	1.56781i
$584$	−6.19484e6	+	7.28011e6i	−0.751619	+	0.883295i
$585$		0			0
$586$	−9.20714e6	+	9.71414e6i	−1.10759	+	1.16858i
$587$	584872.			0.0700593			0.0350296	−	0.999386i	$-0.488847\pi$
$587$	584872.			0.0700593			0.0350296	+	0.999386i	$0.488847\pi$
$588$		0			0
$589$	1.73351e7			2.05891
$590$	1.92336e6	−	2.02927e6i	0.227473	−	0.239999i
$591$		0			0
$592$	1.51480e7	−	1.62942e6i	1.77644	−	0.191086i
$593$			6.31733e6i			0.737728i	0.929483	+	0.368864i	$0.120253\pi$
$593$			6.31733e6i			0.737728i	−0.929483	+	0.368864i	$0.879747\pi$
$594$		0			0
$595$			3.08832e6i			0.357626i
$596$	893924.	−	47940.2i	0.103083	−	0.00552820i
$597$		0			0
$598$	1.50400e6	+	1.42550e6i	0.171987	+	0.163010i
$599$	9.48535e6			1.08016			0.540078	−	0.841615i	$-0.318395\pi$
$599$	9.48535e6			1.08016			0.540078	+	0.841615i	$0.318395\pi$
$600$		0			0
$601$	−1.10737e7			−1.25057			−0.625284	−	0.780397i	$-0.715017\pi$
$601$	−1.10737e7			−1.25057			−0.625284	+	0.780397i	$0.715017\pi$
$602$	−522802.	−	495516.i	−0.0587958	−	0.0557271i
$603$		0			0
$604$	−859705.	+	46105.0i	−0.0958864	+	0.00514228i
$605$			151714.i			0.0168514i
$606$		0			0
$607$		−	7.47030e6i		−	0.822936i	−0.911424	−	0.411468i	$-0.865016\pi$
$607$		−	7.47030e6i		−	0.822936i	0.911424	−	0.411468i	$-0.134984\pi$
$608$	−6.88015e6	+	9.02313e6i	−0.754812	+	0.989916i
$609$		0			0
$610$	−8.23242e6	+	8.68575e6i	−0.895784	+	0.945111i
$611$	1.35547e7			1.46888
$612$		0			0
$613$	527227.			0.0566692			0.0283346	−	0.999598i	$-0.490980\pi$
$613$	527227.			0.0566692			0.0283346	+	0.999598i	$0.490980\pi$
$614$	77466.7	−	81732.5i	0.00829266	−	0.00874931i
$615$		0			0
$616$	−3.76489e6	−	3.20364e6i	−0.399761	−	0.340167i
$617$			9.29470e6i			0.982929i	0.870897	+	0.491465i	$0.163538\pi$
$617$			9.29470e6i			0.982929i	−0.870897	+	0.491465i	$0.836462\pi$
$618$		0			0
$619$			3.38720e6i			0.355316i	0.984092	+	0.177658i	$0.0568520\pi$
$619$			3.38720e6i			0.355316i	−0.984092	+	0.177658i	$0.943148\pi$
$620$	−1.49557e6	−	2.78874e7i	−0.156253	−	2.91359i
$621$		0			0
$622$	1.04330e7	+	9.88845e6i	1.08126	+	1.02483i
$623$	1.40172e6			0.144691
$624$		0			0
$625$	1.31766e7			1.34928
$626$	3.96096e6	+	3.75423e6i	0.403985	+	0.382900i
$627$		0			0
$628$	−290801.	−	5.42247e6i	−0.0294237	−	0.548653i
$629$		−	6.87946e6i		−	0.693311i
$630$		0			0
$631$		−	1.26464e6i		−	0.126443i	−0.998000	−	0.0632216i	$-0.979863\pi$
$631$		−	1.26464e6i		−	0.126443i	0.998000	−	0.0632216i	$-0.0201375\pi$
$632$	−1.30881e7	−	1.11370e7i	−1.30342	−	1.10911i
$633$		0			0
$634$	7.65538e6	−	8.07693e6i	0.756386	−	0.798037i
$635$	2.02099e7			1.98898
$636$		0			0
$637$	6.42603e6			0.627472
$638$	7.31902e6	−	7.72205e6i	0.711871	−	0.751071i
$639$		0			0
$640$	1.51093e7	+	1.02898e7i	1.45813	+	0.993021i
$641$		−	1.06313e7i		−	1.02197i	−0.859588	−	0.510987i	$-0.829280\pi$
$641$		−	1.06313e7i		−	1.02197i	0.859588	−	0.510987i	$-0.170720\pi$
$642$		0			0
$643$			1.49352e7i			1.42457i	0.701893	+	0.712283i	$0.252339\pi$
$643$			1.49352e7i			1.42457i	−0.701893	+	0.712283i	$0.747661\pi$
$644$	1.50757e6	−	80849.3i	0.143240	−	0.00768178i
$645$		0			0
$646$	3.71873e6	+	3.52464e6i	0.350601	+	0.332302i
$647$	−4.38900e6			−0.412197			−0.206099	−	0.978531i	$-0.566077\pi$
$647$	−4.38900e6			−0.412197			−0.206099	+	0.978531i	$0.566077\pi$
$648$		0			0
$649$	2.02086e6			0.188332
$650$	−1.42511e7	−	1.35073e7i	−1.32301	−	1.25396i
$651$		0			0
$652$	3.84695e6	−	206308.i	0.354403	−	0.0190062i
$653$		−	7.73540e6i		−	0.709905i	−0.934884	−	0.354952i	$-0.884497\pi$
$653$		−	7.73540e6i		−	0.709905i	0.934884	−	0.354952i	$-0.115503\pi$
$654$		0			0
$655$		−	1.05363e7i		−	0.959589i
$656$	588535.	+	5.47133e6i	0.0533965	+	0.496402i
$657$		0			0
$658$	6.79343e6	−	7.16751e6i	0.611680	−	0.645363i
$659$	1.45433e6			0.130452			0.0652258	−	0.997871i	$-0.479223\pi$
$659$	1.45433e6			0.130452			0.0652258	+	0.997871i	$0.479223\pi$
$660$		0			0
$661$	−1.13463e7			−1.01007			−0.505034	−	0.863100i	$-0.668520\pi$
$661$	−1.13463e7			−1.01007			−0.505034	+	0.863100i	$0.668520\pi$
$662$	9.11832e6	−	9.62043e6i	0.808667	−	0.853197i
$663$		0			0
$664$	9.64944e6	−	1.13399e7i	0.849341	−	0.998137i
$665$		−	1.30835e7i		−	1.14728i
$666$		0			0
$667$			3.24931e6i			0.282798i
$668$	432265.	+	8.06030e6i	0.0374808	+	0.698892i
$669$		0			0
$670$	−7.46195e6	−	7.07250e6i	−0.642193	−	0.608675i
$671$	−8.64977e6			−0.741649
$672$		0			0
$673$	2.63098e6			0.223913			0.111957	−	0.993713i	$-0.464288\pi$
$673$	2.63098e6			0.223913			0.111957	+	0.993713i	$0.464288\pi$
$674$	1.39240e7	+	1.31973e7i	1.18063	+	1.11901i
$675$		0			0
$676$	−162399.	−	3.02819e6i	−0.0136683	−	0.254869i
$677$			2.00948e7i			1.68504i	0.538661	+	0.842522i	$0.318930\pi$
$677$			2.00948e7i			1.68504i	−0.538661	+	0.842522i	$0.681070\pi$
$678$		0			0
$679$		−	1.34495e6i		−	0.111952i
$680$	5.34935e6	−	6.28651e6i	0.443638	−	0.521359i
$681$		0			0
$682$	1.38859e7	−	1.46506e7i	1.14318	−	1.20613i
$683$	9.59174e6			0.786766			0.393383	−	0.919375i	$-0.371305\pi$
$683$	9.59174e6			0.786766			0.393383	+	0.919375i	$0.371305\pi$
$684$		0			0
$685$	−1.39326e7			−1.13451
$686$	7.65018e6	−	8.07144e6i	0.620670	−	0.654848i
$687$		0			0
$688$	205907.	+	1.91422e6i	0.0165844	+	0.154178i
$689$			1.67798e7i			1.34660i
$690$		0			0
$691$		−	537489.i		−	0.0428227i	−0.999771	−	0.0214114i	$-0.993184\pi$
$691$		−	537489.i		−	0.0428227i	0.999771	−	0.0214114i	$-0.00681597\pi$
$692$	−2.28933e6	+	122774.i	−0.181737	+	0.00974634i
$693$		0			0
$694$	−3.91120e6	−	3.70706e6i	−0.308256	−	0.292167i
$695$	−1.99893e7			−1.56977
$696$		0			0
$697$	2.48481e6			0.193736
$698$	−5.09480e6	−	4.82889e6i	−0.395811	−	0.375153i
$699$		0			0
$700$	−1.42849e7	+	766082.i	−1.10187	+	0.0590922i
$701$			1.34046e7i			1.03029i	0.857104	+	0.515143i	$0.172261\pi$
$701$			1.34046e7i			1.03029i	−0.857104	+	0.515143i	$0.827739\pi$
$702$		0			0
$703$			2.91445e7i			2.22417i
$704$	2.11460e6	+	1.30425e7i	0.160804	+	0.991814i
$705$		0			0
$706$	1.25587e6	−	1.32502e6i	0.0948269	−	0.100049i
$707$	2.90147e6			0.218308
$708$		0			0
$709$	1.75400e7			1.31043			0.655217	−	0.755441i	$-0.272577\pi$
$709$	1.75400e7			1.31043			0.655217	+	0.755441i	$0.272577\pi$
$710$	−2.60161e7	+	2.74487e7i	−1.93685	+	2.04351i
$711$		0			0
$712$	−2.85331e6	−	2.42795e6i	−0.210935	−	0.179490i
$713$			6.16472e6i			0.454140i
$714$		0			0
$715$		−	2.09110e7i		−	1.52971i
$716$	810210.	+	1.51077e7i	0.0590629	+	1.10133i
$717$		0			0
$718$	8.27865e6	+	7.84657e6i	0.599305	+	0.568026i
$719$	−9.18586e6			−0.662670			−0.331335	−	0.943513i	$-0.607499\pi$
$719$	−9.18586e6			−0.662670			−0.331335	+	0.943513i	$0.607499\pi$
$720$		0			0
$721$	−6.29568e6			−0.451029
$722$	−5.58808e6	−	5.29642e6i	−0.398951	−	0.378129i
$723$		0			0
$724$	244391.	+	4.55707e6i	0.0173276	+	0.323102i
$725$		−	3.07886e7i		−	2.17543i
$726$		0			0
$727$			4.08483e6i			0.286641i	0.989676	+	0.143321i	$0.0457780\pi$
$727$			4.08483e6i			0.286641i	−0.989676	+	0.143321i	$0.954222\pi$
$728$	4.90991e6	+	4.17798e6i	0.343357	+	0.292171i
$729$		0			0
$730$	−2.02656e7	+	2.13816e7i	−1.40751	+	1.48502i
$731$	869345.			0.0601726
$732$		0			0
$733$	−1.95584e7			−1.34454			−0.672269	−	0.740307i	$-0.734680\pi$
$733$	−1.95584e7			−1.34454			−0.672269	+	0.740307i	$0.734680\pi$
$734$	−1.09124e7	+	1.15133e7i	−0.747621	+	0.788789i
$735$		0			0
$736$	−3.20882e6	−	2.44673e6i	−0.218349	−	0.166491i
$737$		−	7.43104e6i		−	0.503942i
$738$		0			0
$739$			1.93503e7i			1.30340i	0.758477	+	0.651699i	$0.225944\pi$
$739$			1.93503e7i			1.30340i	−0.758477	+	0.651699i	$0.774056\pi$
$740$	4.68856e7	−	2.51443e6i	3.14746	−	0.168795i
$741$		0			0
$742$	8.87289e6	+	8.40980e6i	0.591637	+	0.560758i
$743$	−1.80071e7			−1.19666			−0.598330	−	0.801249i	$-0.704169\pi$
$743$	−1.80071e7			−1.19666			−0.598330	+	0.801249i	$0.704169\pi$
$744$		0			0
$745$	2.75890e6			0.182115
$746$	3.25699e6	+	3.08700e6i	0.214274	+	0.203090i
$747$		0			0
$748$	5.95764e6	−	319502.i	0.389332	−	0.0208794i
$749$			3.30680e6i			0.215379i
$750$		0			0
$751$		−	1.64226e7i		−	1.06253i	−0.847205	−	0.531267i	$-0.821716\pi$
$751$		−	1.64226e7i		−	1.06253i	0.847205	−	0.531267i	$-0.178284\pi$
$752$	−2.62436e7	+	2.82295e6i	−1.69230	+	0.182036i
$753$		0			0
$754$	−9.54498e6	+	1.00706e7i	−0.611430	+	0.645099i
$755$	−2.65328e6			−0.169401
$756$		0			0
$757$	2.25867e7			1.43256			0.716279	−	0.697814i	$-0.245844\pi$
$757$	2.25867e7			1.43256			0.716279	+	0.697814i	$0.245844\pi$
$758$	3.40863e6	−	3.59633e6i	0.215480	−	0.227346i
$759$		0			0
$760$	−2.26623e7	+	2.66325e7i	−1.42321	+	1.67255i
$761$		−	2.24952e7i		−	1.40808i	−0.710159	−	0.704041i	$-0.751377\pi$
$761$		−	2.24952e7i		−	1.40808i	0.710159	−	0.704041i	$-0.248623\pi$
$762$		0			0
$763$		−	7.09760e6i		−	0.441368i
$764$	−1.30593e6	−	2.43512e7i	−0.0809443	−	1.50934i
$765$		0			0
$766$	−2.33432e7	−	2.21248e7i	−1.43743	−	1.36241i
$767$	−2.63547e6			−0.161760
$768$		0			0
$769$	−1.86118e7			−1.13494			−0.567468	−	0.823395i	$-0.692077\pi$
$769$	−1.86118e7			−1.13494			−0.567468	+	0.823395i	$0.692077\pi$
$770$	−1.10574e7	−	1.04803e7i	−0.672090	−	0.637012i
$771$		0			0
$772$	1.31363e6	+	2.44948e7i	0.0793287	+	1.47921i
$773$			1.63770e7i			0.985793i	0.870088	+	0.492896i	$0.164062\pi$
$773$			1.63770e7i			0.985793i	−0.870088	+	0.492896i	$0.835938\pi$
$774$		0			0
$775$		−	5.84134e7i		−	3.49348i
$776$	−2.32962e6	+	2.73775e6i	−0.138877	+	0.163207i
$777$		0			0
$778$	9.98873e6	−	1.05388e7i	0.591645	−	0.624225i
$779$	−1.05268e7			−0.621515
$780$		0			0
$781$	−2.73350e7			−1.60358
$782$	−1.25344e6	+	1.32246e6i	−0.0732969	+	0.0773330i
$783$		0			0
$784$	−1.24416e7	+	1.33831e6i	−0.722914	+	0.0777618i
$785$		−	1.67352e7i		−	0.969298i
$786$		0			0
$787$			1.08535e7i			0.624643i	0.949976	+	0.312322i	$0.101107\pi$
$787$			1.08535e7i			0.624643i	−0.949976	+	0.312322i	$0.898893\pi$
$788$	8.92726e6	−	478759.i	0.512157	−	0.0274664i
$789$		0			0
$790$	−3.84395e7	−	3.64333e7i	−2.19134	−	2.07697i
$791$	−3.91973e6			−0.222749
$792$		0			0
$793$	1.12805e7			0.637006
$794$	2.51779e7	+	2.38638e7i	1.41732	+	1.34335i
$795$		0			0
$796$	−1.61284e6	+	86495.0i	−0.0902214	+	0.00483847i
$797$		−	2.34260e7i		−	1.30633i	−0.757215	−	0.653165i	$-0.773441\pi$
$797$		−	2.34260e7i		−	1.30633i	0.757215	−	0.653165i	$-0.226559\pi$
$798$		0			0
$799$			1.19185e7i			0.660475i
$800$	3.04049e7	+	2.31838e7i	1.67965	+	1.28074i
$801$		0			0
$802$	2.19107e7	−	2.31172e7i	1.20287	−	1.26911i
$803$	−2.12930e7			−1.16533
$804$		0			0
$805$	4.65278e6			0.253059
$806$	−1.81091e7	+	1.91063e7i	−0.981883	+	1.03595i
$807$		0			0
$808$	−5.90617e6	−	5.02572e6i	−0.318257	−	0.270813i
$809$		−	3.41203e7i		−	1.83291i	−0.400133	−	0.916457i	$-0.631036\pi$
$809$		−	3.41203e7i		−	1.83291i	0.400133	−	0.916457i	$-0.368964\pi$
$810$		0			0
$811$			1.96027e7i			1.04656i	0.852160	+	0.523281i	$0.175292\pi$
$811$			1.96027e7i			1.04656i	−0.852160	+	0.523281i	$0.824708\pi$
$812$	541356.	+	1.00945e7i	0.0288133	+	0.537272i
$813$		0			0
$814$	2.46313e7	+	2.33457e7i	1.30294	+	1.23494i
$815$	1.18727e7			0.626119
$816$		0			0
$817$	−3.68294e6			−0.193037
$818$	1.17787e7	+	1.11639e7i	0.615478	+	0.583355i
$819$		0			0
$820$	908190.	+	1.69347e7i	0.0471674	+	0.879515i
$821$			2.53758e7i			1.31390i	0.753934	+	0.656950i	$0.228154\pi$
$821$			2.53758e7i			1.31390i	−0.753934	+	0.656950i	$0.771846\pi$
$822$		0			0
$823$		−	1.26091e7i		−	0.648908i	−0.945901	−	0.324454i	$-0.894819\pi$
$823$		−	1.26091e7i		−	0.648908i	0.945901	−	0.324454i	$-0.105181\pi$
$824$	1.28153e7	+	1.09049e7i	0.657525	+	0.559506i
$825$		0			0
$826$	−1.32086e6	+	1.39360e6i	−0.0673609	+	0.0710702i
$827$	1.62032e7			0.823831			0.411916	−	0.911222i	$-0.364860\pi$
$827$	1.62032e7			0.823831			0.411916	+	0.911222i	$0.364860\pi$
$828$		0			0
$829$	−1.31970e7			−0.666942			−0.333471	−	0.942760i	$-0.608220\pi$
$829$	−1.31970e7			−0.666942			−0.333471	+	0.942760i	$0.608220\pi$
$830$	3.15669e7	−	3.33052e7i	1.59051	−	1.67810i
$831$		0			0
$832$	−2.75772e6	−	1.70092e7i	−0.138115	−	0.851874i
$833$			5.65037e6i			0.282140i
$834$		0			0
$835$			2.48763e7i			1.23472i
$836$	−2.52393e7	+	1.35355e6i	−1.24900	+	0.0669823i
$837$		0			0
$838$	−2.15550e7	−	2.04300e7i	−1.06032	−	1.00498i
$839$	−2.51468e7			−1.23333			−0.616663	−	0.787227i	$-0.711516\pi$
$839$	−2.51468e7			−1.23333			−0.616663	+	0.787227i	$0.711516\pi$
$840$		0			0
$841$	−1.24576e6			−0.0607356
$842$	1.57202e7	+	1.48997e7i	0.764149	+	0.724267i
$843$		0			0
$844$	2.43707e7	−	1.30697e6i	1.17764	−	0.0631555i
$845$		−	9.34583e6i		−	0.450273i
$846$		0			0
$847$		−	104189.i		−	0.00499017i
$848$	−3.49462e6	−	3.24878e7i	−0.166882	−	1.55142i
$849$		0			0
$850$	1.18768e7	−	1.25309e7i	0.563837	−	0.594886i
$851$	−1.03644e7			−0.490593
$852$		0			0
$853$	−1.63408e7			−0.768956			−0.384478	−	0.923134i	$-0.625618\pi$
$853$	−1.63408e7			−0.768956			−0.384478	+	0.923134i	$0.625618\pi$
$854$	5.65361e6	−	5.96493e6i	0.265266	−	0.279873i
$855$		0			0
$856$	5.72780e6	−	6.73124e6i	0.267179	−	0.313986i
$857$			3.65159e7i			1.69836i	0.528101	+	0.849181i	$0.322904\pi$
$857$			3.65159e7i			1.69836i	−0.528101	+	0.849181i	$0.677096\pi$
$858$		0			0
$859$			2.09455e7i			0.968516i	0.874925	+	0.484258i	$0.160910\pi$
$859$			2.09455e7i			0.968516i	−0.874925	+	0.484258i	$0.839090\pi$
$860$	317743.	+	5.92485e6i	0.0146497	+	0.273169i
$861$		0			0
$862$	3.13450e6	+	2.97091e6i	0.143681	+	0.136182i
$863$	2.99156e7			1.36732			0.683661	−	0.729799i	$-0.260387\pi$
$863$	2.99156e7			1.36732			0.683661	+	0.729799i	$0.260387\pi$
$864$		0			0
$865$	−7.06549e6			−0.321072
$866$	−519888.	−	492754.i	−0.0235568	−	0.0223273i
$867$		0			0
$868$	1.02708e6	+	1.91517e7i	0.0462707	+	0.862794i
$869$		−	3.82803e7i		−	1.71959i
$870$		0			0
$871$			9.69107e6i			0.432839i
$872$	−1.22939e7	+	1.44477e7i	−0.547520	+	0.643440i
$873$		0			0
$874$	5.31013e6	−	5.60253e6i	0.235140	−	0.248088i
$875$	−2.32148e7			−1.02505
$876$		0			0
$877$	6.01027e6			0.263873			0.131937	−	0.991258i	$-0.457880\pi$
$877$	6.01027e6			0.263873			0.131937	+	0.991258i	$0.457880\pi$
$878$	2.47682e7	−	2.61321e7i	1.08432	−	1.14403i
$879$		0			0
$880$	4.35500e6	+	4.04864e7i	0.189575	+	1.76239i
$881$		−	3.80007e6i		−	0.164950i	−0.996593	−	0.0824750i	$-0.973718\pi$
$881$		−	3.80007e6i		−	0.164950i	0.996593	−	0.0824750i	$-0.0262825\pi$
$882$		0			0
$883$		−	3.26490e7i		−	1.40918i	−0.709613	−	0.704592i	$-0.751130\pi$
$883$		−	3.26490e7i		−	1.40918i	0.709613	−	0.704592i	$-0.248870\pi$
$884$	−7.76955e6	+	416673.i	−0.334399	+	0.0179335i
$885$		0			0
$886$	−2.05427e7	−	1.94705e7i	−0.879169	−	0.833283i
$887$	3.86440e7			1.64920			0.824600	−	0.565716i	$-0.191400\pi$
$887$	3.86440e7			1.64920			0.824600	+	0.565716i	$0.191400\pi$
$888$		0			0
$889$	−1.38791e7			−0.588991
$890$	−8.38012e6	−	7.94275e6i	−0.354630	−	0.336121i
$891$		0			0
$892$	3.59149e7	−	1.92608e6i	1.51134	−	0.0810515i
$893$		−	5.04924e7i		−	2.11883i
$894$		0			0
$895$			4.66265e7i			1.94570i
$896$	−1.03763e7	−	7.06654e6i	−0.431792	−	0.294060i
$897$		0			0
$898$	9.07693e6	−	9.57676e6i	0.375620	−	0.396304i
$899$	−4.12781e7			−1.70341
$900$		0			0
$901$	−1.47543e7			−0.605491
$902$	−8.43229e6	+	8.89662e6i	−0.345087	+	0.364090i
$903$		0			0
$904$	7.97892e6	+	6.78947e6i	0.324730	+	0.276322i
$905$			1.40644e7i			0.570820i
$906$		0			0
$907$		−	1.21928e7i		−	0.492138i	−0.969252	−	0.246069i	$-0.920861\pi$
$907$		−	1.21928e7i		−	0.492138i	0.969252	−	0.246069i	$-0.0791389\pi$
$908$	1.08106e6	+	2.01582e7i	0.0435147	+	0.811404i
$909$		0			0
$910$	1.44204e7	+	1.36677e7i	0.577261	+	0.547133i
$911$	−8.17620e6			−0.326404			−0.163202	−	0.986593i	$-0.552182\pi$
$911$	−8.17620e6			−0.326404			−0.163202	+	0.986593i	$0.552182\pi$
$912$		0			0
$913$	3.31673e7			1.31684
$914$	1.43791e7	+	1.36286e7i	0.569333	+	0.539619i
$915$		0			0
$916$	945568.	+	1.76317e7i	0.0372352	+	0.694313i
$917$			7.23581e6i			0.284161i
$918$		0			0
$919$			1.31066e6i			0.0511920i	0.999672	+	0.0255960i	$0.00814835\pi$
$919$			1.31066e6i			0.0511920i	−0.999672	+	0.0255960i	$0.991852\pi$
$920$	−9.47108e6	−	8.05920e6i	−0.368918	−	0.313922i
$921$		0			0
$922$	1.77309e7	−	1.87073e7i	0.686915	−	0.724741i
$923$	3.56485e7			1.37733
$924$		0			0
$925$	9.82073e7			3.77389
$926$	1.27921e7	−	1.34966e7i	0.490248	−	0.517244i
$927$		0			0
$928$	1.63830e7	−	2.14858e7i	0.624485	−	0.818995i
$929$			7.54802e6i			0.286942i	0.989655	+	0.143471i	$0.0458263\pi$
$929$			7.54802e6i			0.286942i	−0.989655	+	0.143471i	$0.954174\pi$
$930$		0			0
$931$		−	2.39375e7i		−	0.905118i
$932$	−1.61892e7	+	868209.i	−0.610500	+	0.0327404i
$933$		0			0
$934$	1.75377e7	+	1.66224e7i	0.657818	+	0.623485i
$935$	1.83869e7			0.687828
$936$		0			0
$937$	1.52871e7			0.568822			0.284411	−	0.958702i	$-0.408202\pi$
$937$	1.52871e7			0.568822			0.284411	+	0.958702i	$0.408202\pi$
$938$	5.12449e6	+	4.85703e6i	0.190171	+	0.180245i
$939$		0			0
$940$	−8.12285e7	+	4.35620e6i	−2.99839	+	0.160801i
$941$			3.53718e7i			1.30221i	0.758986	+	0.651107i	$0.225695\pi$
$941$			3.53718e7i			1.30221i	−0.758986	+	0.651107i	$0.774305\pi$
$942$		0			0
$943$		−	3.74354e6i		−	0.137089i
$944$	5.10261e6	−	548873.i	0.186364	−	0.0200467i
$945$		0			0
$946$	−2.95015e6	+	3.11261e6i	−0.107181	+	0.113083i
$947$	−1.28282e7			−0.464826			−0.232413	−	0.972617i	$-0.574662\pi$
$947$	−1.28282e7			−0.464826			−0.232413	+	0.972617i	$0.574662\pi$
$948$		0			0
$949$	2.77689e7			1.00091
$950$	−5.03157e7	+	5.30864e7i	−1.80882	+	1.90842i
$951$		0			0
$952$	−3.67367e6	+	4.31725e6i	−0.131373	+	0.154389i
$953$		−	1.90229e7i		−	0.678490i	−0.940698	−	0.339245i	$-0.889828\pi$
$953$		−	1.90229e7i		−	0.678490i	0.940698	−	0.339245i	$-0.110172\pi$
$954$		0			0
$955$		−	7.51546e7i		−	2.66653i
$956$	−1.44538e6	−	2.69516e7i	−0.0511492	−	0.953761i
$957$		0			0
$958$	8.57947e6	+	8.13169e6i	0.302027	+	0.286264i
$959$	9.56823e6			0.335958
$960$		0			0
$961$	−4.96854e7			−1.73548
$962$	−3.21224e7	−	3.04459e7i	−1.11911	−	1.06070i
$963$		0			0
$964$	−274868.	−	5.12537e6i	−0.00952646	−	0.177637i
$965$			7.55978e7i			2.61331i
$966$		0			0
$967$			3.63296e7i			1.24938i	0.780873	+	0.624690i	$0.214774\pi$
$967$			3.63296e7i			1.24938i	−0.780873	+	0.624690i	$0.785226\pi$
$968$	−180469.	+	212086.i	−0.00619035	+	0.00727483i
$969$		0			0
$970$	−7.62107e6	+	8.04073e6i	−0.260068	+	0.274389i
$971$	−4.25414e7			−1.44798			−0.723991	−	0.689809i	$-0.757694\pi$
$971$	−4.25414e7			−1.44798			−0.723991	+	0.689809i	$0.757694\pi$
$972$		0			0
$973$	1.37277e7			0.464851
$974$	−7.89186e6	+	8.32644e6i	−0.266552	+	0.281230i
$975$		0			0
$976$	−2.18404e7	+	2.34931e6i	−0.733898	+	0.0789433i
$977$			5.19393e7i			1.74084i	0.492308	+	0.870421i	$0.336153\pi$
$977$			5.19393e7i			1.74084i	−0.492308	+	0.870421i	$0.663847\pi$
$978$		0			0
$979$		−	8.34541e6i		−	0.278286i
$980$	−3.85090e7	+	2.06520e6i	−1.28085	+	0.0686904i
$981$		0			0
$982$	−3.03507e7	−	2.87666e7i	−1.00436	−	0.951941i
$983$	−7.20689e6			−0.237884			−0.118942	−	0.992901i	$-0.537950\pi$
$983$	−7.20689e6			−0.237884			−0.118942	+	0.992901i	$0.537950\pi$
$984$		0			0
$985$	2.75520e7			0.904820
$986$	−8.85499e6	−	8.39283e6i	−0.290065	−	0.274926i
$987$		0			0
$988$	3.29154e7	−	1.76521e6i	1.07277	−	0.0575314i
$989$		−	1.30973e6i		−	0.0425786i
$990$		0			0
$991$		−	1.43917e7i		−	0.465510i	−0.972535	−	0.232755i	$-0.925226\pi$
$991$		−	1.43917e7i		−	0.465510i	0.972535	−	0.232755i	$-0.0747740\pi$
$992$	3.10824e7	−	4.07637e7i	1.00285	−	1.31521i
$993$		0			0
$994$	1.78666e7	−	1.88504e7i	0.573555	−	0.605138i
$995$	−4.97768e6			−0.159393
$996$		0			0
$997$	3.43826e7			1.09547			0.547736	−	0.836652i	$-0.315490\pi$
$997$	3.43826e7			1.09547			0.547736	+	0.836652i	$0.315490\pi$
$998$	2.25552e6	−	2.37973e6i	0.0716838	−	0.0756311i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.6.b.c.107.16	yes	20
3.2	odd	2	inner	108.6.b.c.107.5	✓	20
4.3	odd	2	inner	108.6.b.c.107.6	yes	20
12.11	even	2	inner	108.6.b.c.107.15	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.b.c.107.5	✓	20	3.2	odd	2	inner
108.6.b.c.107.6	yes	20	4.3	odd	2	inner
108.6.b.c.107.15	yes	20	12.11	even	2	inner
108.6.b.c.107.16	yes	20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	108.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(4.10571 + 3.89142i) q^{2} +(1.71366 + 31.9541i) q^{4} +98.6190i q^{5} -67.7266i q^{7} +(-117.311 + 137.863i) q^{8} +O(q^{10})\)	\(q+(4.10571 + 3.89142i) q^{2} +(1.71366 + 31.9541i) q^{4} +98.6190i q^{5} -67.7266i q^{7} +(-117.311 + 137.863i) q^{8} +(-383.768 + 404.901i) q^{10} -403.224 q^{11} +525.857 q^{13} +(263.553 - 278.066i) q^{14} +(-1018.13 + 109.517i) q^{16} +462.383i q^{17} -1958.86i q^{19} +(-3151.28 + 169.000i) q^{20} +(-1655.52 - 1569.11i) q^{22} +696.614 q^{23} -6600.72 q^{25} +(2159.02 + 2046.33i) q^{26} +(2164.14 - 116.060i) q^{28} +4664.43i q^{29} +8849.55i q^{31} +(-4606.31 - 3512.32i) q^{32} +(-1799.33 + 1898.41i) q^{34} +6679.13 q^{35} -14878.3 q^{37} +(7622.77 - 8042.52i) q^{38} +(-13595.9 - 11569.1i) q^{40} -5373.92i q^{41} -1880.14i q^{43} +(-690.989 - 12884.6i) q^{44} +(2860.09 + 2710.82i) q^{46} +25776.3 q^{47} +12220.1 q^{49} +(-27100.6 - 25686.2i) q^{50} +(901.142 + 16803.3i) q^{52} +31909.4i q^{53} -39765.5i q^{55} +(9336.97 + 7945.08i) q^{56} +(-18151.3 + 19150.8i) q^{58} -5011.77 q^{59} +21451.5 q^{61} +(-34437.3 + 36333.7i) q^{62} +(-5244.24 - 32345.6i) q^{64} +51859.6i q^{65} +18429.1i q^{67} +(-14775.0 + 792.368i) q^{68} +(27422.6 + 25991.3i) q^{70} +67791.2 q^{71} +52806.9 q^{73} +(-61085.9 - 57897.7i) q^{74} +(62593.7 - 3356.83i) q^{76} +27309.0i q^{77} +94935.6i q^{79} +(-10800.5 - 100407. i) q^{80} +(20912.2 - 22063.7i) q^{82} -82255.2 q^{83} -45599.8 q^{85} +(7316.42 - 7719.31i) q^{86} +(47302.6 - 55589.5i) q^{88} +20696.7i q^{89} -35614.5i q^{91} +(1193.76 + 22259.7i) q^{92} +(105830. + 100307. i) q^{94} +193181. q^{95} +19858.5 q^{97} +(50172.2 + 47553.6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{4}+O(q^{10}) \) 20 * q + 20 * q^4	\( 20 q + 20 q^{4} + 184 q^{10} - 116 q^{13} - 4168 q^{16} + 696 q^{22} - 15228 q^{25} - 4764 q^{28} - 16520 q^{34} - 6452 q^{37} + 1504 q^{40} - 9336 q^{46} - 44464 q^{49} + 8236 q^{52} - 58736 q^{58} + 84604 q^{61} - 6496 q^{64} + 138696 q^{70} + 85420 q^{73} + 89172 q^{76} + 221200 q^{82} + 180320 q^{85} - 85824 q^{88} - 60936 q^{94} - 219908 q^{97}+O(q^{100}) \) 20 * q + 20 * q^4 + 184 * q^10 - 116 * q^13 - 4168 * q^16 + 696 * q^22 - 15228 * q^25 - 4764 * q^28 - 16520 * q^34 - 6452 * q^37 + 1504 * q^40 - 9336 * q^46 - 44464 * q^49 + 8236 * q^52 - 58736 * q^58 + 84604 * q^61 - 6496 * q^64 + 138696 * q^70 + 85420 * q^73 + 89172 * q^76 + 221200 * q^82 + 180320 * q^85 - 85824 * q^88 - 60936 * q^94 - 219908 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more