LMFDB - Embedded newform 608.6.b.b.303.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,6,Mod(303,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("608.303");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	608.b (of order $2$, degree $1$, not 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:	$97.5133624463$
Analytic rank:	$0$
Dimension:	$96$
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			303.7
Character	$\chi$	$=$	608.303
Dual form			608.6.b.b.303.90

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-26.7852i q^{3} -46.5237i q^{5} -208.957i q^{7} -474.446 q^{9} +O(q^{10})$	$q-26.7852i q^{3} -46.5237i q^{5} -208.957i q^{7} -474.446 q^{9} -35.8554 q^{11} -87.1027 q^{13} -1246.15 q^{15} -955.073 q^{17} +(1568.49 - 126.201i) q^{19} -5596.96 q^{21} +3396.26i q^{23} +960.543 q^{25} +6199.33i q^{27} -4924.87 q^{29} +668.108 q^{31} +960.395i q^{33} -9721.47 q^{35} +4068.94 q^{37} +2333.06i q^{39} +11858.4i q^{41} -9400.87 q^{43} +22073.0i q^{45} +8042.65i q^{47} -26856.1 q^{49} +25581.8i q^{51} +15890.1 q^{53} +1668.13i q^{55} +(-3380.32 - 42012.4i) q^{57} +35837.9i q^{59} -33919.4i q^{61} +99139.0i q^{63} +4052.34i q^{65} -69017.0i q^{67} +90969.5 q^{69} +38298.3 q^{71} -63378.5 q^{73} -25728.3i q^{75} +7492.26i q^{77} -86670.7 q^{79} +50759.7 q^{81} -90612.1 q^{83} +44433.5i q^{85} +131914. i q^{87} -26070.4i q^{89} +18200.7i q^{91} -17895.4i q^{93} +(-5871.35 - 72972.2i) q^{95} -135407. i q^{97} +17011.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q - 6168 q^{9}+O(q^{10}) $ 96 * q - 6168 * q^9	$ 96 q - 6168 q^{9} - 944 q^{11} - 3832 q^{17} - 5240 q^{19} - 62504 q^{25} - 7720 q^{35} - 45096 q^{43} - 210840 q^{49} - 36336 q^{57} - 4336 q^{73} - 20624 q^{81} - 52152 q^{83} + 752768 q^{99}+O(q^{100}) $ 96 * q - 6168 * q^9 - 944 * q^11 - 3832 * q^17 - 5240 * q^19 - 62504 * q^25 - 7720 * q^35 - 45096 * q^43 - 210840 * q^49 - 36336 * q^57 - 4336 * q^73 - 20624 * q^81 - 52152 * q^83 + 752768 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/608\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		−	26.7852i		−	1.71827i	−0.511749	−	0.859135i	$-0.671002\pi$
$3$		−	26.7852i		−	1.71827i	0.511749	−	0.859135i	$-0.328998\pi$
$4$		0			0
$5$		−	46.5237i		−	0.832242i	−0.909309	−	0.416121i	$-0.863389\pi$
$5$		−	46.5237i		−	0.832242i	0.909309	−	0.416121i	$-0.136611\pi$
$6$		0			0
$7$		−	208.957i		−	1.61180i	−0.592049	−	0.805902i	$-0.701681\pi$
$7$		−	208.957i		−	1.61180i	0.592049	−	0.805902i	$-0.298319\pi$
$8$		0			0
$9$	−474.446			−1.95245
$10$		0			0
$11$	−35.8554			−0.0893457			−0.0446728	−	0.999002i	$-0.514225\pi$
$11$	−35.8554			−0.0893457			−0.0446728	+	0.999002i	$0.514225\pi$
$12$		0			0
$13$	−87.1027			−0.142946			−0.0714732	−	0.997443i	$-0.522770\pi$
$13$	−87.1027			−0.142946			−0.0714732	+	0.997443i	$0.522770\pi$
$14$		0			0
$15$	−1246.15			−1.43002
$16$		0			0
$17$	−955.073			−0.801520			−0.400760	−	0.916183i	$-0.631254\pi$
$17$	−955.073			−0.801520			−0.400760	+	0.916183i	$0.631254\pi$
$18$		0			0
$19$	1568.49	−	126.201i	0.996779	−	0.0802010i
$19$	1568.49	−	126.201i	0.996779	−	0.0802010i
$20$		0			0
$21$	−5596.96			−2.76952
$22$		0			0
$23$			3396.26i			1.33869i	0.742950	+	0.669347i	$0.233426\pi$
$23$			3396.26i			1.33869i	−0.742950	+	0.669347i	$0.766574\pi$
$24$		0			0
$25$	960.543			0.307374
$26$		0			0
$27$			6199.33i			1.63657i
$28$		0			0
$29$	−4924.87			−1.08743			−0.543713	−	0.839271i	$-0.682982\pi$
$29$	−4924.87			−1.08743			−0.543713	+	0.839271i	$0.682982\pi$
$30$		0			0
$31$	668.108			0.124866			0.0624328	−	0.998049i	$-0.480114\pi$
$31$	668.108			0.124866			0.0624328	+	0.998049i	$0.480114\pi$
$32$		0			0
$33$			960.395i			0.153520i
$34$		0			0
$35$	−9721.47			−1.34141
$36$		0			0
$37$	4068.94			0.488627			0.244313	−	0.969696i	$-0.421437\pi$
$37$	4068.94			0.488627			0.244313	+	0.969696i	$0.421437\pi$
$38$		0			0
$39$			2333.06i			0.245620i
$40$		0			0
$41$			11858.4i			1.10171i	0.834602	+	0.550854i	$0.185698\pi$
$41$			11858.4i			1.10171i	−0.834602	+	0.550854i	$0.814302\pi$
$42$		0			0
$43$	−9400.87			−0.775348			−0.387674	−	0.921796i	$-0.626721\pi$
$43$	−9400.87			−0.775348			−0.387674	+	0.921796i	$0.626721\pi$
$44$		0			0
$45$			22073.0i			1.62491i
$46$		0			0
$47$			8042.65i			0.531074i	0.964101	+	0.265537i	$0.0855492\pi$
$47$			8042.65i			0.531074i	−0.964101	+	0.265537i	$0.914451\pi$
$48$		0			0
$49$	−26856.1			−1.59791
$50$		0			0
$51$			25581.8i			1.37723i
$52$		0			0
$53$	15890.1			0.777027			0.388514	−	0.921443i	$-0.372989\pi$
$53$	15890.1			0.777027			0.388514	+	0.921443i	$0.372989\pi$
$54$		0			0
$55$			1668.13i			0.0743572i
$56$		0			0
$57$	−3380.32	−	42012.4i	−0.137807	−	1.71274i
$58$		0			0
$59$			35837.9i			1.34033i	0.742211	+	0.670167i	$0.233777\pi$
$59$			35837.9i			1.34033i	−0.742211	+	0.670167i	$0.766223\pi$
$60$		0			0
$61$		−	33919.4i		−	1.16714i	−0.812062	−	0.583571i	$-0.801655\pi$
$61$		−	33919.4i		−	1.16714i	0.812062	−	0.583571i	$-0.198345\pi$
$62$		0			0
$63$			99139.0i			3.14697i
$64$		0			0
$65$			4052.34i			0.118966i
$66$		0			0
$67$		−	69017.0i		−	1.87832i	−0.343483	−	0.939159i	$-0.611607\pi$
$67$		−	69017.0i		−	1.87832i	0.343483	−	0.939159i	$-0.388393\pi$
$68$		0			0
$69$	90969.5			2.30024
$70$		0			0
$71$	38298.3			0.901641			0.450820	−	0.892615i	$-0.351132\pi$
$71$	38298.3			0.901641			0.450820	+	0.892615i	$0.351132\pi$
$72$		0			0
$73$	−63378.5			−1.39199			−0.695993	−	0.718049i	$-0.745035\pi$
$73$	−63378.5			−1.39199			−0.695993	+	0.718049i	$0.745035\pi$
$74$		0			0
$75$		−	25728.3i		−	0.528151i
$76$		0			0
$77$			7492.26i			0.144008i
$78$		0			0
$79$	−86670.7			−1.56244			−0.781222	−	0.624253i	$-0.785403\pi$
$79$	−86670.7			−1.56244			−0.781222	+	0.624253i	$0.785403\pi$
$80$		0			0
$81$	50759.7			0.859621
$82$		0			0
$83$	−90612.1			−1.44375			−0.721874	−	0.692025i	$-0.756719\pi$
$83$	−90612.1			−1.44375			−0.721874	+	0.692025i	$0.756719\pi$
$84$		0			0
$85$			44433.5i			0.667058i
$86$		0			0
$87$			131914.i			1.86849i
$88$		0			0
$89$		−	26070.4i		−	0.348877i	−0.984668	−	0.174439i	$-0.944189\pi$
$89$		−	26070.4i		−	0.348877i	0.984668	−	0.174439i	$-0.0558111\pi$
$90$		0			0
$91$			18200.7i			0.230402i
$92$		0			0
$93$		−	17895.4i		−	0.214553i
$94$		0			0
$95$	−5871.35	−	72972.2i	−0.0667466	−	0.829561i
$96$		0			0
$97$		−	135407.i		−	1.46121i	−0.682801	−	0.730605i	$-0.739238\pi$
$97$		−	135407.i		−	1.46121i	0.682801	−	0.730605i	$-0.260762\pi$
$98$		0			0
$99$	17011.5			0.174443
$100$		0			0
$101$		−	56704.1i		−	0.553110i	−0.960998	−	0.276555i	$-0.910807\pi$
$101$		−	56704.1i		−	0.553110i	0.960998	−	0.276555i	$-0.0891928\pi$
$102$		0			0
$103$	89254.9			0.828970			0.414485	−	0.910056i	$-0.363962\pi$
$103$	89254.9			0.828970			0.414485	+	0.910056i	$0.363962\pi$
$104$		0			0
$105$			260391.i			2.30491i
$106$		0			0
$107$			38727.5i			0.327010i	0.986543	+	0.163505i	$0.0522799\pi$
$107$			38727.5i			0.327010i	−0.986543	+	0.163505i	$0.947720\pi$
$108$		0			0
$109$	−95570.2			−0.770471			−0.385236	−	0.922818i	$-0.625880\pi$
$109$	−95570.2			−0.770471			−0.385236	+	0.922818i	$0.625880\pi$
$110$		0			0
$111$		−	108987.i		−	0.839593i
$112$		0			0
$113$			175456.i			1.29262i	0.763073	+	0.646312i	$0.223690\pi$
$113$			175456.i			1.29262i	−0.763073	+	0.646312i	$0.776310\pi$
$114$		0			0
$115$	158007.			1.11412
$116$		0			0
$117$	41325.5			0.279096
$118$		0			0
$119$			199569.i			1.29189i
$120$		0			0
$121$	−159765.			−0.992017
$122$		0			0
$123$	317630.			1.89303
$124$		0			0
$125$		−	190075.i		−	1.08805i
$126$		0			0
$127$	321898.			1.77096			0.885480	−	0.464677i	$-0.153830\pi$
$127$	321898.			1.77096			0.885480	+	0.464677i	$0.153830\pi$
$128$		0			0
$129$			251804.i			1.33226i
$130$		0			0
$131$	−263947.			−1.34381			−0.671907	−	0.740635i	$-0.734525\pi$
$131$	−263947.			−1.34381			−0.671907	+	0.740635i	$0.734525\pi$
$132$		0			0
$133$	−26370.7	−	327748.i	−0.129268	−	1.60661i
$134$		0			0
$135$	288416.			1.36202
$136$		0			0
$137$	−135367.			−0.616186			−0.308093	−	0.951356i	$-0.599691\pi$
$137$	−135367.			−0.616186			−0.308093	+	0.951356i	$0.599691\pi$
$138$		0			0
$139$	40557.9			0.178048			0.0890242	−	0.996029i	$-0.471625\pi$
$139$	40557.9			0.178048			0.0890242	+	0.996029i	$0.471625\pi$
$140$		0			0
$141$	215424.			0.912528
$142$		0			0
$143$	3123.10			0.0127716
$144$		0			0
$145$			229123.i			0.905002i
$146$		0			0
$147$			719346.i			2.74565i
$148$		0			0
$149$		−	303380.i		−	1.11949i	−0.828664	−	0.559746i	$-0.810899\pi$
$149$		−	303380.i		−	1.11949i	0.828664	−	0.559746i	$-0.189101\pi$
$150$		0			0
$151$	244773.			0.873618			0.436809	−	0.899554i	$-0.356109\pi$
$151$	244773.			0.873618			0.436809	+	0.899554i	$0.356109\pi$
$152$		0			0
$153$	453131.			1.56493
$154$		0			0
$155$		−	31082.9i		−	0.103918i
$156$		0			0
$157$			357645.i			1.15798i	0.815333	+	0.578992i	$0.196554\pi$
$157$			357645.i			1.15798i	−0.815333	+	0.578992i	$0.803446\pi$
$158$		0			0
$159$		−	425619.i		−	1.33514i
$160$		0			0
$161$	709673.			2.15771
$162$		0			0
$163$	584184.			1.72219			0.861094	−	0.508446i	$-0.169780\pi$
$163$	584184.			1.72219			0.861094	+	0.508446i	$0.169780\pi$
$164$		0			0
$165$	44681.1			0.127766
$166$		0			0
$167$	226657.			0.628896			0.314448	−	0.949275i	$-0.398181\pi$
$167$	226657.			0.628896			0.314448	+	0.949275i	$0.398181\pi$
$168$		0			0
$169$	−363706.			−0.979566
$170$		0			0
$171$	−744166.	+	59875.7i	−1.94616	+	0.156589i
$172$		0			0
$173$	191412.			0.486244			0.243122	−	0.969996i	$-0.421828\pi$
$173$	191412.			0.486244			0.243122	+	0.969996i	$0.421828\pi$
$174$		0			0
$175$		−	200712.i		−	0.495426i
$176$		0			0
$177$	959925.			2.30306
$178$		0			0
$179$		−	669795.i		−	1.56246i	−0.624242	−	0.781231i	$-0.714592\pi$
$179$		−	669795.i		−	1.56246i	0.624242	−	0.781231i	$-0.285408\pi$
$180$		0			0
$181$	−156815.			−0.355788			−0.177894	−	0.984050i	$-0.556928\pi$
$181$	−156815.			−0.355788			−0.177894	+	0.984050i	$0.556928\pi$
$182$		0			0
$183$	−908538.			−2.00547
$184$		0			0
$185$		−	189302.i		−	0.406655i
$186$		0			0
$187$	34244.6			0.0716123
$188$		0			0
$189$	1.29539e6			2.63783
$190$		0			0
$191$		−	269519.i		−	0.534572i	−0.963617	−	0.267286i	$-0.913873\pi$
$191$		−	269519.i		−	0.534572i	0.963617	−	0.267286i	$-0.0861269\pi$
$192$		0			0
$193$		−	376841.i		−	0.728224i	−0.931355	−	0.364112i	$-0.881372\pi$
$193$		−	376841.i		−	0.728224i	0.931355	−	0.364112i	$-0.118628\pi$
$194$		0			0
$195$	108543.			0.204416
$196$		0			0
$197$			761356.i			1.39773i	0.715256	+	0.698863i	$0.246310\pi$
$197$			761356.i			1.39773i	−0.715256	+	0.698863i	$0.753690\pi$
$198$		0			0
$199$			478068.i			0.855770i	0.903833	+	0.427885i	$0.140741\pi$
$199$			478068.i			0.855770i	−0.903833	+	0.427885i	$0.859259\pi$
$200$		0			0
$201$	−1.84863e6			−3.22746
$202$		0			0
$203$			1.02909e6i			1.75272i
$204$		0			0
$205$	551697.			0.916887
$206$		0			0
$207$		−	1.61134e6i		−	2.61374i
$208$		0			0
$209$	−56239.0	+	4525.00i	−0.0890578	+	0.00716561i
$210$		0			0
$211$		−	206673.i		−	0.319578i	−0.987151	−	0.159789i	$-0.948919\pi$
$211$		−	206673.i		−	0.319578i	0.987151	−	0.159789i	$-0.0510814\pi$
$212$		0			0
$213$		−	1.02583e6i		−	1.54926i
$214$		0			0
$215$			437363.i			0.645277i
$216$		0			0
$217$		−	139606.i		−	0.201259i
$218$		0			0
$219$			1.69760e6i			2.39181i
$220$		0			0
$221$	83189.4			0.114574
$222$		0			0
$223$	961700.			1.29502			0.647512	−	0.762056i	$-0.275810\pi$
$223$	961700.			1.29502			0.647512	+	0.762056i	$0.275810\pi$
$224$		0			0
$225$	−455726.			−0.600133
$226$		0			0
$227$			89750.5i			0.115604i	0.998328	+	0.0578019i	$0.0184092\pi$
$227$			89750.5i			0.115604i	−0.998328	+	0.0578019i	$0.981591\pi$
$228$		0			0
$229$			890166.i			1.12171i	0.827912	+	0.560857i	$0.189528\pi$
$229$			890166.i			1.12171i	−0.827912	+	0.560857i	$0.810472\pi$
$230$		0			0
$231$	200681.			0.247444
$232$		0			0
$233$	378174.			0.456354			0.228177	−	0.973620i	$-0.426723\pi$
$233$	378174.			0.456354			0.228177	+	0.973620i	$0.426723\pi$
$234$		0			0
$235$	374174.			0.441982
$236$		0			0
$237$			2.32149e6i			2.68470i
$238$		0			0
$239$			538846.i			0.610196i	0.952321	+	0.305098i	$0.0986893\pi$
$239$			538846.i			0.610196i	−0.952321	+	0.305098i	$0.901311\pi$
$240$		0			0
$241$			1.09661e6i			1.21621i	0.793855	+	0.608107i	$0.208071\pi$
$241$			1.09661e6i			1.21621i	−0.793855	+	0.608107i	$0.791929\pi$
$242$		0			0
$243$			146828.i			0.159512i
$244$		0			0
$245$			1.24945e6i			1.32985i
$246$		0			0
$247$	−136620.	+	10992.5i	−0.142486	+	0.0114644i
$248$		0			0
$249$			2.42706e6i			2.48075i
$250$		0			0
$251$	−249160.			−0.249628			−0.124814	−	0.992180i	$-0.539833\pi$
$251$	−249160.			−0.249628			−0.124814	+	0.992180i	$0.539833\pi$
$252$		0			0
$253$		−	121774.i		−	0.119607i
$254$		0			0
$255$	1.19016e6			1.14619
$256$		0			0
$257$		−	489707.i		−	0.462491i	−0.972895	−	0.231246i	$-0.925720\pi$
$257$		−	489707.i		−	0.462491i	0.972895	−	0.231246i	$-0.0742801\pi$
$258$		0			0
$259$		−	850235.i		−	0.787570i
$260$		0			0
$261$	2.33659e6			2.12315
$262$		0			0
$263$			832999.i			0.742601i	0.928513	+	0.371300i	$0.121088\pi$
$263$			832999.i			0.742601i	−0.928513	+	0.371300i	$0.878912\pi$
$264$		0			0
$265$		−	739266.i		−	0.646675i
$266$		0			0
$267$	−698301.			−0.599466
$268$		0			0
$269$	−2.02119e6			−1.70304			−0.851521	−	0.524320i	$-0.824320\pi$
$269$	−2.02119e6			−1.70304			−0.851521	+	0.524320i	$0.824320\pi$
$270$		0			0
$271$			1.20460e6i			0.996370i	0.867071	+	0.498185i	$0.166000\pi$
$271$			1.20460e6i			0.996370i	−0.867071	+	0.498185i	$0.834000\pi$
$272$		0			0
$273$	487510.			0.395892
$274$		0			0
$275$	−34440.7			−0.0274625
$276$		0			0
$277$		−	994518.i		−	0.778777i	−0.921074	−	0.389389i	$-0.872686\pi$
$277$		−	994518.i		−	0.778777i	0.921074	−	0.389389i	$-0.127314\pi$
$278$		0			0
$279$	−316981.			−0.243794
$280$		0			0
$281$			71283.1i			0.0538543i	0.999637	+	0.0269272i	$0.00857222\pi$
$281$			71283.1i			0.0538543i	−0.999637	+	0.0269272i	$0.991428\pi$
$282$		0			0
$283$	−2.51083e6			−1.86359			−0.931796	−	0.362982i	$-0.881759\pi$
$283$	−2.51083e6			−1.86359			−0.931796	+	0.362982i	$0.881759\pi$
$284$		0			0
$285$	−1.95457e6	+	157265.i	−1.42541	+	0.114689i
$286$		0			0
$287$	2.47790e6			1.77574
$288$		0			0
$289$	−507693.			−0.357566
$290$		0			0
$291$	−3.62691e6			−2.51075
$292$		0			0
$293$	−564118.			−0.383885			−0.191942	−	0.981406i	$-0.561479\pi$
$293$	−564118.			−0.383885			−0.191942	+	0.981406i	$0.561479\pi$
$294$		0			0
$295$	1.66731e6			1.11548
$296$		0			0
$297$		−	222280.i		−	0.146221i
$298$		0			0
$299$		−	295823.i		−	0.191361i
$300$		0			0
$301$			1.96438e6i			1.24971i
$302$		0			0
$303$	−1.51883e6			−0.950392
$304$		0			0
$305$	−1.57806e6			−0.971345
$306$		0			0
$307$		−	3.21582e6i		−	1.94736i	−0.227927	−	0.973678i	$-0.573195\pi$
$307$		−	3.21582e6i		−	1.94736i	0.227927	−	0.973678i	$-0.426805\pi$
$308$		0			0
$309$		−	2.39071e6i		−	1.42439i
$310$		0			0
$311$		−	3.09147e6i		−	1.81244i	−0.422805	−	0.906220i	$-0.638955\pi$
$311$		−	3.09147e6i		−	1.81244i	0.422805	−	0.906220i	$-0.361045\pi$
$312$		0			0
$313$	−379670.			−0.219051			−0.109525	−	0.993984i	$-0.534933\pi$
$313$	−379670.			−0.219051			−0.109525	+	0.993984i	$0.534933\pi$
$314$		0			0
$315$	4.61231e6			2.61904
$316$		0			0
$317$	−158325.			−0.0884912			−0.0442456	−	0.999021i	$-0.514088\pi$
$317$	−158325.			−0.0884912			−0.0442456	+	0.999021i	$0.514088\pi$
$318$		0			0
$319$	176584.			0.0971569
$320$		0			0
$321$	1.03732e6			0.561891
$322$		0			0
$323$	−1.49803e6	+	120531.i	−0.798938	+	0.0642826i
$324$		0			0
$325$	−83665.9			−0.0439380
$326$		0			0
$327$			2.55987e6i			1.32388i
$328$		0			0
$329$	1.68057e6			0.855987
$330$		0			0
$331$			30802.8i			0.0154533i	0.999970	+	0.00772664i	$0.00245949\pi$
$331$			30802.8i			0.0154533i	−0.999970	+	0.00772664i	$0.997541\pi$
$332$		0			0
$333$	−1.93049e6			−0.954021
$334$		0			0
$335$	−3.21093e6			−1.56321
$336$		0			0
$337$			3.91249e6i			1.87663i	0.345782	+	0.938315i	$0.387614\pi$
$337$			3.91249e6i			1.87663i	−0.345782	+	0.938315i	$0.612386\pi$
$338$		0			0
$339$	4.69962e6			2.22108
$340$		0			0
$341$	−23955.3			−0.0111562
$342$		0			0
$343$			2.09984e6i			0.963719i
$344$		0			0
$345$		−	4.23224e6i		−	1.91436i
$346$		0			0
$347$	915682.			0.408245			0.204123	−	0.978945i	$-0.434566\pi$
$347$	915682.			0.408245			0.204123	+	0.978945i	$0.434566\pi$
$348$		0			0
$349$			694461.i			0.305200i	0.988288	+	0.152600i	$0.0487646\pi$
$349$			694461.i			0.305200i	−0.988288	+	0.152600i	$0.951235\pi$
$350$		0			0
$351$		−	539978.i		−	0.233942i
$352$		0			0
$353$	−3.90432e6			−1.66767			−0.833833	−	0.552017i	$-0.813858\pi$
$353$	−3.90432e6			−1.66767			−0.833833	+	0.552017i	$0.813858\pi$
$354$		0			0
$355$		−	1.78178e6i		−	0.750383i
$356$		0			0
$357$	5.34550e6			2.21982
$358$		0			0
$359$			1.92295e6i			0.787464i	0.919225	+	0.393732i	$0.128816\pi$
$359$			1.92295e6i			0.787464i	−0.919225	+	0.393732i	$0.871184\pi$
$360$		0			0
$361$	2.44425e6	−	395892.i	0.987136	−	0.159885i
$362$		0			0
$363$			4.27935e6i			1.70455i
$364$		0			0
$365$			2.94860e6i			1.15847i
$366$		0			0
$367$		−	3.12502e6i		−	1.21112i	−0.795799	−	0.605561i	$-0.792949\pi$
$367$		−	3.12502e6i		−	1.21112i	0.795799	−	0.605561i	$-0.207051\pi$
$368$		0			0
$369$		−	5.62617e6i		−	2.15103i
$370$		0			0
$371$		−	3.32035e6i		−	1.25242i
$372$		0			0
$373$	−3.94141e6			−1.46683			−0.733415	−	0.679781i	$-0.762075\pi$
$373$	−3.94141e6			−1.46683			−0.733415	+	0.679781i	$0.762075\pi$
$374$		0			0
$375$	−5.09119e6			−1.86957
$376$		0			0
$377$	428969.			0.155444
$378$		0			0
$379$		−	73519.4i		−	0.0262908i	−0.999914	−	0.0131454i	$-0.995816\pi$
$379$		−	73519.4i		−	0.0262908i	0.999914	−	0.0131454i	$-0.00418443\pi$
$380$		0			0
$381$		−	8.62209e6i		−	3.04299i
$382$		0			0
$383$	1.71447e6			0.597220			0.298610	−	0.954375i	$-0.403477\pi$
$383$	1.71447e6			0.597220			0.298610	+	0.954375i	$0.403477\pi$
$384$		0			0
$385$	348568.			0.119849
$386$		0			0
$387$	4.46021e6			1.51383
$388$		0			0
$389$		−	4.88524e6i		−	1.63686i	−0.574604	−	0.818431i	$-0.694844\pi$
$389$		−	4.88524e6i		−	1.63686i	0.574604	−	0.818431i	$-0.305156\pi$
$390$		0			0
$391$		−	3.24368e6i		−	1.07299i
$392$		0			0
$393$			7.06988e6i			2.30904i
$394$		0			0
$395$			4.03224e6i			1.30033i
$396$		0			0
$397$		−	4.50286e6i		−	1.43388i	−0.697136	−	0.716939i	$-0.745542\pi$
$397$		−	4.50286e6i		−	1.43388i	0.697136	−	0.716939i	$-0.254458\pi$
$398$		0			0
$399$	−8.77879e6	+	706343.i	−2.76059	+	0.222118i
$400$		0			0
$401$		−	264814.i		−	0.0822394i	−0.999154	−	0.0411197i	$-0.986907\pi$
$401$		−	264814.i		−	0.0822394i	0.999154	−	0.0411197i	$-0.0130925\pi$
$402$		0			0
$403$	−58194.0			−0.0178491
$404$		0			0
$405$		−	2.36153e6i		−	0.715412i
$406$		0			0
$407$	−145894.			−0.0436567
$408$		0			0
$409$		−	1.93926e6i		−	0.573228i	−0.958046	−	0.286614i	$-0.907470\pi$
$409$		−	1.93926e6i		−	0.573228i	0.958046	−	0.286614i	$-0.0925297\pi$
$410$		0			0
$411$			3.62583e6i			1.05877i
$412$		0			0
$413$	7.48859e6			2.16036
$414$		0			0
$415$			4.21561e6i			1.20155i
$416$		0			0
$417$		−	1.08635e6i		−	0.305935i
$418$		0			0
$419$	−3.75075e6			−1.04372			−0.521859	−	0.853032i	$-0.674761\pi$
$419$	−3.75075e6			−1.04372			−0.521859	+	0.853032i	$0.674761\pi$
$420$		0			0
$421$	−2.84993e6			−0.783661			−0.391831	−	0.920037i	$-0.628158\pi$
$421$	−2.84993e6			−0.783661			−0.391831	+	0.920037i	$0.628158\pi$
$422$		0			0
$423$		−	3.81581e6i		−	1.03690i
$424$		0			0
$425$	−917389.			−0.246366
$426$		0			0
$427$	−7.08771e6			−1.88121
$428$		0			0
$429$		−	83652.9i		−	0.0219451i
$430$		0			0
$431$	5.18314e6			1.34400			0.672001	−	0.740550i	$-0.265435\pi$
$431$	5.18314e6			1.34400			0.672001	+	0.740550i	$0.265435\pi$
$432$		0			0
$433$		−	1.13879e6i		−	0.291893i	−0.989292	−	0.145947i	$-0.953377\pi$
$433$		−	1.13879e6i		−	0.291893i	0.989292	−	0.145947i	$-0.0466228\pi$
$434$		0			0
$435$	6.13711e6			1.55504
$436$		0			0
$437$	428612.	+	5.32701e6i	0.107365	+	1.33438i
$438$		0			0
$439$	2.28863e6			0.566779			0.283390	−	0.959005i	$-0.408541\pi$
$439$	2.28863e6			0.566779			0.283390	+	0.959005i	$0.408541\pi$
$440$		0			0
$441$	1.27418e7			3.11985
$442$		0			0
$443$	3.82921e6			0.927043			0.463521	−	0.886086i	$-0.346586\pi$
$443$	3.82921e6			0.927043			0.463521	+	0.886086i	$0.346586\pi$
$444$		0			0
$445$	−1.21289e6			−0.290350
$446$		0			0
$447$	−8.12609e6			−1.92359
$448$		0			0
$449$		−	4.20035e6i		−	0.983262i	−0.870804	−	0.491631i	$-0.836401\pi$
$449$		−	4.20035e6i		−	0.983262i	0.870804	−	0.491631i	$-0.163599\pi$
$450$		0			0
$451$		−	425188.i		−	0.0984328i
$452$		0			0
$453$		−	6.55630e6i		−	1.50111i
$454$		0			0
$455$	846766.			0.191750
$456$		0			0
$457$	−868949.			−0.194627			−0.0973136	−	0.995254i	$-0.531025\pi$
$457$	−868949.			−0.194627			−0.0973136	+	0.995254i	$0.531025\pi$
$458$		0			0
$459$		−	5.92081e6i		−	1.31174i
$460$		0			0
$461$			7.76528e6i			1.70178i	0.525340	+	0.850892i	$0.323938\pi$
$461$			7.76528e6i			1.70178i	−0.525340	+	0.850892i	$0.676062\pi$
$462$		0			0
$463$			4.35658e6i			0.944480i	0.881470	+	0.472240i	$0.156554\pi$
$463$			4.35658e6i			0.944480i	−0.881470	+	0.472240i	$0.843446\pi$
$464$		0			0
$465$	−832561.			−0.178560
$466$		0			0
$467$	−3.01885e6			−0.640545			−0.320272	−	0.947326i	$-0.603774\pi$
$467$	−3.01885e6			−0.640545			−0.320272	+	0.947326i	$0.603774\pi$
$468$		0			0
$469$	−1.44216e7			−3.02748
$470$		0			0
$471$	9.57958e6			1.98973
$472$		0			0
$473$	337072.			0.0692740
$474$		0			0
$475$	1.50661e6	−	121222.i	0.306384	−	0.0246517i
$476$		0			0
$477$	−7.53899e6			−1.51711
$478$		0			0
$479$		−	6.61819e6i		−	1.31795i	−0.752163	−	0.658977i	$-0.770989\pi$
$479$		−	6.61819e6i		−	1.31795i	0.752163	−	0.658977i	$-0.229011\pi$
$480$		0			0
$481$	−354416.			−0.0698474
$482$		0			0
$483$		−	1.90087e7i		−	3.70754i
$484$		0			0
$485$	−6.29965e6			−1.21608
$486$		0			0
$487$	−8.57297e6			−1.63798			−0.818990	−	0.573807i	$-0.805466\pi$
$487$	−8.57297e6			−1.63798			−0.818990	+	0.573807i	$0.805466\pi$
$488$		0			0
$489$		−	1.56475e7i		−	2.95918i
$490$		0			0
$491$	−287543.			−0.0538269			−0.0269135	−	0.999638i	$-0.508568\pi$
$491$	−287543.			−0.0538269			−0.0269135	+	0.999638i	$0.508568\pi$
$492$		0			0
$493$	4.70361e6			0.871594
$494$		0			0
$495$		−	791437.i		−	0.145179i
$496$		0			0
$497$		−	8.00270e6i		−	1.45327i
$498$		0			0
$499$	41932.3			0.00753871			0.00376936	−	0.999993i	$-0.498800\pi$
$499$	41932.3			0.00753871			0.00376936	+	0.999993i	$0.498800\pi$
$500$		0			0
$501$		−	6.07106e6i		−	1.08061i
$502$		0			0
$503$			1.80449e6i			0.318005i	0.987278	+	0.159003i	$0.0508278\pi$
$503$			1.80449e6i			0.318005i	−0.987278	+	0.159003i	$0.949172\pi$
$504$		0			0
$505$	−2.63809e6			−0.460321
$506$		0			0
$507$			9.74194e6i			1.68316i
$508$		0			0
$509$	7.01256e6			1.19973			0.599863	−	0.800103i	$-0.295222\pi$
$509$	7.01256e6			1.19973			0.599863	+	0.800103i	$0.295222\pi$
$510$		0			0
$511$			1.32434e7i			2.24361i
$512$		0			0
$513$	782363.	+	9.72361e6i	0.131255	+	1.63130i
$514$		0			0
$515$		−	4.15247e6i		−	0.689904i
$516$		0			0
$517$		−	288373.i		−	0.0474491i
$518$		0			0
$519$		−	5.12701e6i		−	0.835499i
$520$		0			0
$521$		−	4.29098e6i		−	0.692568i	−0.938130	−	0.346284i	$-0.887443\pi$
$521$		−	4.29098e6i		−	0.692568i	0.938130	−	0.346284i	$-0.112557\pi$
$522$		0			0
$523$			6.59641e6i			1.05452i	0.849705	+	0.527258i	$0.176780\pi$
$523$			6.59641e6i			1.05452i	−0.849705	+	0.527258i	$0.823220\pi$
$524$		0			0
$525$	−5.37612e6			−0.851277
$526$		0			0
$527$	−638092.			−0.100082
$528$		0			0
$529$	−5.09825e6			−0.792103
$530$		0			0
$531$		−	1.70032e7i		−	2.61694i
$532$		0			0
$533$		−	1.03290e6i		−	0.157485i
$534$		0			0
$535$	1.80175e6			0.272151
$536$		0			0
$537$	−1.79406e7			−2.68473
$538$		0			0
$539$	962938.			0.142767
$540$		0			0
$541$			4.11246e6i			0.604100i	0.953292	+	0.302050i	$0.0976709\pi$
$541$			4.11246e6i			0.604100i	−0.953292	+	0.302050i	$0.902329\pi$
$542$		0			0
$543$			4.20032e6i			0.611341i
$544$		0			0
$545$			4.44628e6i			0.641218i
$546$		0			0
$547$		−	1.39299e6i		−	0.199058i	−0.995035	−	0.0995288i	$-0.968266\pi$
$547$		−	1.39299e6i		−	0.199058i	0.995035	−	0.0995288i	$-0.0317335\pi$
$548$		0			0
$549$			1.60929e7i			2.27879i
$550$		0			0
$551$	−7.72463e6	+	621525.i	−1.08392	+	0.0872127i
$552$		0			0
$553$			1.81105e7i			2.51835i
$554$		0			0
$555$	−5.07050e6			−0.698744
$556$		0			0
$557$			1.13242e6i			0.154658i	0.997006	+	0.0773288i	$0.0246391\pi$
$557$			1.13242e6i			0.154658i	−0.997006	+	0.0773288i	$0.975361\pi$
$558$		0			0
$559$	818841.			0.110833
$560$		0			0
$561$		−	917247.i		−	0.123049i
$562$		0			0
$563$			1.29877e6i			0.172688i	0.996265	+	0.0863439i	$0.0275184\pi$
$563$			1.29877e6i			0.172688i	−0.996265	+	0.0863439i	$0.972482\pi$
$564$		0			0
$565$	8.16287e6			1.07578
$566$		0			0
$567$		−	1.06066e7i		−	1.38554i
$568$		0			0
$569$		−	2.85081e6i		−	0.369137i	−0.982820	−	0.184568i	$-0.940911\pi$
$569$		−	2.85081e6i		−	0.369137i	0.982820	−	0.184568i	$-0.0590887\pi$
$570$		0			0
$571$	2.47617e6			0.317827			0.158913	−	0.987293i	$-0.449201\pi$
$571$	2.47617e6			0.317827			0.158913	+	0.987293i	$0.449201\pi$
$572$		0			0
$573$	−7.21913e6			−0.918540
$574$		0			0
$575$			3.26226e6i			0.411480i
$576$		0			0
$577$	−5.44738e6			−0.681158			−0.340579	−	0.940216i	$-0.610623\pi$
$577$	−5.44738e6			−0.681158			−0.340579	+	0.940216i	$0.610623\pi$
$578$		0			0
$579$	−1.00938e7			−1.25129
$580$		0			0
$581$			1.89341e7i			2.32704i
$582$		0			0
$583$	−569746.			−0.0694240
$584$		0			0
$585$		−	1.92262e6i		−	0.232275i
$586$		0			0
$587$	−5.79540e6			−0.694205			−0.347103	−	0.937827i	$-0.612834\pi$
$587$	−5.79540e6			−0.694205			−0.347103	+	0.937827i	$0.612834\pi$
$588$		0			0
$589$	1.04792e6	−	84316.1i	0.124463	−	0.0100143i
$590$		0			0
$591$	2.03930e7			2.40167
$592$		0			0
$593$	6.02635e6			0.703749			0.351874	−	0.936047i	$-0.385544\pi$
$593$	6.02635e6			0.703749			0.351874	+	0.936047i	$0.385544\pi$
$594$		0			0
$595$	9.28471e6			1.07517
$596$		0			0
$597$	1.28051e7			1.47045
$598$		0			0
$599$	−3.37829e6			−0.384707			−0.192354	−	0.981326i	$-0.561612\pi$
$599$	−3.37829e6			−0.384707			−0.192354	+	0.981326i	$0.561612\pi$
$600$		0			0
$601$			6.27163e6i			0.708262i	0.935196	+	0.354131i	$0.115223\pi$
$601$			6.27163e6i			0.708262i	−0.935196	+	0.354131i	$0.884777\pi$
$602$		0			0
$603$			3.27449e7i			3.66733i
$604$		0			0
$605$			7.43288e6i			0.825598i
$606$		0			0
$607$	−1.03899e7			−1.14456			−0.572280	−	0.820058i	$-0.693941\pi$
$607$	−1.03899e7			−1.14456			−0.572280	+	0.820058i	$0.693941\pi$
$608$		0			0
$609$	2.75643e7			3.01165
$610$		0			0
$611$		−	700537.i		−	0.0759150i
$612$		0			0
$613$			6.87610e6i			0.739080i	0.929215	+	0.369540i	$0.120485\pi$
$613$			6.87610e6i			0.739080i	−0.929215	+	0.369540i	$0.879515\pi$
$614$		0			0
$615$		−	1.47773e7i		−	1.57546i
$616$		0			0
$617$	−1.33979e7			−1.41685			−0.708427	−	0.705784i	$-0.750595\pi$
$617$	−1.33979e7			−1.41685			−0.708427	+	0.705784i	$0.750595\pi$
$618$		0			0
$619$	−7.51655e6			−0.788482			−0.394241	−	0.919007i	$-0.628993\pi$
$619$	−7.51655e6			−0.788482			−0.394241	+	0.919007i	$0.628993\pi$
$620$		0			0
$621$	−2.10545e7			−2.19087
$622$		0			0
$623$	−5.44760e6			−0.562322
$624$		0			0
$625$	−5.84128e6			−0.598148
$626$		0			0
$627$	121203.	+	1.50637e6i	0.0123125	+	0.153025i
$628$		0			0
$629$	−3.88613e6			−0.391644
$630$		0			0
$631$		−	6.39890e6i		−	0.639781i	−0.947454	−	0.319891i	$-0.896354\pi$
$631$		−	6.39890e6i		−	0.639781i	0.947454	−	0.319891i	$-0.103646\pi$
$632$		0			0
$633$	−5.53577e6			−0.549121
$634$		0			0
$635$		−	1.49759e7i		−	1.47387i
$636$		0			0
$637$	2.33924e6			0.228416
$638$		0			0
$639$	−1.81705e7			−1.76041
$640$		0			0
$641$			1.08371e7i			1.04176i	0.853630	+	0.520880i	$0.174396\pi$
$641$			1.08371e7i			1.04176i	−0.853630	+	0.520880i	$0.825604\pi$
$642$		0			0
$643$	−1.48100e7			−1.41263			−0.706314	−	0.707899i	$-0.749643\pi$
$643$	−1.48100e7			−1.41263			−0.706314	+	0.707899i	$0.749643\pi$
$644$		0			0
$645$	1.17149e7			1.10876
$646$		0			0
$647$			6.43857e6i			0.604684i	0.953200	+	0.302342i	$0.0977684\pi$
$647$			6.43857e6i			0.604684i	−0.953200	+	0.302342i	$0.902232\pi$
$648$		0			0
$649$		−	1.28498e6i		−	0.119753i
$650$		0			0
$651$	−3.73937e6			−0.345817
$652$		0			0
$653$		−	4.01075e6i		−	0.368080i	−0.982919	−	0.184040i	$-0.941082\pi$
$653$		−	4.01075e6i		−	0.368080i	0.982919	−	0.184040i	$-0.0589176\pi$
$654$		0			0
$655$			1.22798e7i			1.11838i
$656$		0			0
$657$	3.00697e7			2.71779
$658$		0			0
$659$			4.28337e6i			0.384213i	0.981374	+	0.192106i	$0.0615319\pi$
$659$			4.28337e6i			0.384213i	−0.981374	+	0.192106i	$0.938468\pi$
$660$		0			0
$661$	−5.08901e6			−0.453032			−0.226516	−	0.974007i	$-0.572734\pi$
$661$	−5.08901e6			−0.453032			−0.226516	+	0.974007i	$0.572734\pi$
$662$		0			0
$663$		−	2.22824e6i		−	0.196870i
$664$		0			0
$665$	−1.52481e7	+	1.22686e6i	−1.33709	+	0.107582i
$666$		0			0
$667$		−	1.67262e7i		−	1.45573i
$668$		0			0
$669$		−	2.57593e7i		−	2.22520i
$670$		0			0
$671$			1.21620e6i			0.104279i
$672$		0			0
$673$			2.68365e6i			0.228396i	0.993458	+	0.114198i	$0.0364298\pi$
$673$			2.68365e6i			0.228396i	−0.993458	+	0.114198i	$0.963570\pi$
$674$		0			0
$675$			5.95472e6i			0.503039i
$676$		0			0
$677$	1.50512e7			1.26212			0.631059	−	0.775735i	$-0.282621\pi$
$677$	1.50512e7			1.26212			0.631059	+	0.775735i	$0.282621\pi$
$678$		0			0
$679$	−2.82943e7			−2.35518
$680$		0			0
$681$	2.40398e6			0.198639
$682$		0			0
$683$			2.12625e7i			1.74406i	0.489450	+	0.872031i	$0.337197\pi$
$683$			2.12625e7i			1.74406i	−0.489450	+	0.872031i	$0.662803\pi$
$684$		0			0
$685$			6.29778e6i			0.512815i
$686$		0			0
$687$	2.38433e7			1.92741
$688$		0			0
$689$	−1.38407e6			−0.111073
$690$		0			0
$691$	1.97533e7			1.57379			0.786893	−	0.617090i	$-0.211688\pi$
$691$	1.97533e7			1.57379			0.786893	+	0.617090i	$0.211688\pi$
$692$		0			0
$693$		−	3.55467e6i		−	0.281168i
$694$		0			0
$695$		−	1.88690e6i		−	0.148179i
$696$		0			0
$697$		−	1.13256e7i		−	0.883041i
$698$		0			0
$699$		−	1.01295e7i		−	0.784140i
$700$		0			0
$701$			3.91708e6i			0.301070i	0.988605	+	0.150535i	$0.0480996\pi$
$701$			3.91708e6i			0.301070i	−0.988605	+	0.150535i	$0.951900\pi$
$702$		0			0
$703$	6.38211e6	−	513505.i	0.487053	−	0.0391883i
$704$		0			0
$705$		−	1.00223e7i		−	0.759444i
$706$		0			0
$707$	−1.18487e7			−0.891505
$708$		0			0
$709$		−	7.05237e6i		−	0.526889i	−0.964674	−	0.263445i	$-0.915141\pi$
$709$		−	7.05237e6i		−	0.526889i	0.964674	−	0.263445i	$-0.0848587\pi$
$710$		0			0
$711$	4.11206e7			3.05060
$712$		0			0
$713$			2.26907e6i			0.167157i
$714$		0			0
$715$		−	145298.i		−	0.0106291i
$716$		0			0
$717$	1.44331e7			1.04848
$718$		0			0
$719$		−	2.51127e7i		−	1.81164i	−0.423668	−	0.905818i	$-0.639258\pi$
$719$		−	2.51127e7i		−	1.81164i	0.423668	−	0.905818i	$-0.360742\pi$
$720$		0			0
$721$		−	1.86505e7i		−	1.33614i
$722$		0			0
$723$	2.93729e7			2.08978
$724$		0			0
$725$	−4.73055e6			−0.334246
$726$		0			0
$727$			3.76567e6i			0.264245i	0.991233	+	0.132122i	$0.0421792\pi$
$727$			3.76567e6i			0.264245i	−0.991233	+	0.132122i	$0.957821\pi$
$728$		0			0
$729$	1.62674e7			1.13370
$730$		0			0
$731$	8.97851e6			0.621457
$732$		0			0
$733$		−	6.65591e6i		−	0.457559i	−0.973478	−	0.228780i	$-0.926526\pi$
$733$		−	6.65591e6i		−	0.457559i	0.973478	−	0.228780i	$-0.0734735\pi$
$734$		0			0
$735$	3.34667e7			2.28504
$736$		0			0
$737$			2.47464e6i			0.167820i
$738$		0			0
$739$	−1.17124e7			−0.788925			−0.394462	−	0.918912i	$-0.629069\pi$
$739$	−1.17124e7			−0.788925			−0.394462	+	0.918912i	$0.629069\pi$
$740$		0			0
$741$	294435.	+	3.65939e6i	0.0196990	+	0.244829i
$742$		0			0
$743$	−862163.			−0.0572951			−0.0286475	−	0.999590i	$-0.509120\pi$
$743$	−862163.			−0.0572951			−0.0286475	+	0.999590i	$0.509120\pi$
$744$		0			0
$745$	−1.41144e7			−0.931688
$746$		0			0
$747$	4.29906e7			2.81885
$748$		0			0
$749$	8.09240e6			0.527076
$750$		0			0
$751$	3.20768e6			0.207535			0.103768	−	0.994602i	$-0.466910\pi$
$751$	3.20768e6			0.207535			0.103768	+	0.994602i	$0.466910\pi$
$752$		0			0
$753$			6.67379e6i			0.428928i
$754$		0			0
$755$		−	1.13878e7i		−	0.727062i
$756$		0			0
$757$			2.56954e6i			0.162973i	0.996674	+	0.0814865i	$0.0259668\pi$
$757$			2.56954e6i			0.162973i	−0.996674	+	0.0814865i	$0.974033\pi$
$758$		0			0
$759$	−3.26175e6			−0.205516
$760$		0			0
$761$	−8.81052e6			−0.551493			−0.275746	−	0.961230i	$-0.588925\pi$
$761$	−8.81052e6			−0.551493			−0.275746	+	0.961230i	$0.588925\pi$
$762$		0			0
$763$			1.99701e7i			1.24185i
$764$		0			0
$765$		−	2.10813e7i		−	1.30240i
$766$		0			0
$767$		−	3.12158e6i		−	0.191596i
$768$		0			0
$769$	2.09606e7			1.27817			0.639083	−	0.769137i	$-0.279314\pi$
$769$	2.09606e7			1.27817			0.639083	+	0.769137i	$0.279314\pi$
$770$		0			0
$771$	−1.31169e7			−0.794685
$772$		0			0
$773$	−1.10279e7			−0.663811			−0.331906	−	0.943313i	$-0.607692\pi$
$773$	−1.10279e7			−0.663811			−0.331906	+	0.943313i	$0.607692\pi$
$774$		0			0
$775$	641747.			0.0383804
$776$		0			0
$777$	−2.27737e7			−1.35326
$778$		0			0
$779$	1.49654e6	+	1.85998e7i	0.0883580	+	1.09816i
$780$		0			0
$781$	−1.37320e6			−0.0805577
$782$		0			0
$783$		−	3.05309e7i		−	1.77965i
$784$		0			0
$785$	1.66390e7			0.963722
$786$		0			0
$787$			3.24978e6i			0.187032i	0.995618	+	0.0935162i	$0.0298107\pi$
$787$			3.24978e6i			0.187032i	−0.995618	+	0.0935162i	$0.970189\pi$
$788$		0			0
$789$	2.23120e7			1.27599
$790$		0			0
$791$	3.66628e7			2.08346
$792$		0			0
$793$			2.95447e6i			0.166839i
$794$		0			0
$795$	−1.98014e7			−1.11116
$796$		0			0
$797$	9.08440e6			0.506583			0.253292	−	0.967390i	$-0.418487\pi$
$797$	9.08440e6			0.506583			0.253292	+	0.967390i	$0.418487\pi$
$798$		0			0
$799$		−	7.68132e6i		−	0.425666i
$800$		0			0
$801$			1.23690e7i			0.681167i
$802$		0			0
$803$	2.27246e6			0.124368
$804$		0			0
$805$		−	3.30166e7i		−	1.79574i
$806$		0			0
$807$			5.41378e7i			2.92629i
$808$		0			0
$809$	3.47754e6			0.186810			0.0934051	−	0.995628i	$-0.470225\pi$
$809$	3.47754e6			0.186810			0.0934051	+	0.995628i	$0.470225\pi$
$810$		0			0
$811$			4.16309e6i			0.222261i	0.993806	+	0.111131i	$0.0354472\pi$
$811$			4.16309e6i			0.222261i	−0.993806	+	0.111131i	$0.964553\pi$
$812$		0			0
$813$	3.22655e7			1.71203
$814$		0			0
$815$		−	2.71784e7i		−	1.43328i
$816$		0			0
$817$	−1.47452e7	+	1.18640e6i	−0.772851	+	0.0621837i
$818$		0			0
$819$		−	8.63527e6i		−	0.449848i
$820$		0			0
$821$			1.51654e7i			0.785230i	0.919703	+	0.392615i	$0.128430\pi$
$821$			1.51654e7i			0.785230i	−0.919703	+	0.392615i	$0.871570\pi$
$822$		0			0
$823$			7.23765e6i			0.372476i	0.982505	+	0.186238i	$0.0596295\pi$
$823$			7.23765e6i			0.372476i	−0.982505	+	0.186238i	$0.940371\pi$
$824$		0			0
$825$			922501.i			0.0471880i
$826$		0			0
$827$			2.46106e7i			1.25129i	0.780107	+	0.625646i	$0.215165\pi$
$827$			2.46106e7i			1.25129i	−0.780107	+	0.625646i	$0.784835\pi$
$828$		0			0
$829$	1.47786e7			0.746874			0.373437	−	0.927655i	$-0.378179\pi$
$829$	1.47786e7			0.746874			0.373437	+	0.927655i	$0.378179\pi$
$830$		0			0
$831$	−2.66383e7			−1.33815
$832$		0			0
$833$	2.56496e7			1.28076
$834$		0			0
$835$		−	1.05449e7i		−	0.523393i
$836$		0			0
$837$			4.14182e6i			0.204352i
$838$		0			0
$839$	−1.49881e7			−0.735091			−0.367545	−	0.930006i	$-0.619802\pi$
$839$	−1.49881e7			−0.735091			−0.367545	+	0.930006i	$0.619802\pi$
$840$		0			0
$841$	3.74322e6			0.182497
$842$		0			0
$843$	1.90933e6			0.0925363
$844$		0			0
$845$			1.69210e7i			0.815236i
$846$		0			0
$847$			3.33841e7i			1.59894i
$848$		0			0
$849$			6.72530e7i			3.20216i
$850$		0			0
$851$			1.38192e7i			0.654122i
$852$		0			0
$853$			8.12725e6i			0.382447i	0.981547	+	0.191223i	$0.0612455\pi$
$853$			8.12725e6i			0.382447i	−0.981547	+	0.191223i	$0.938755\pi$
$854$		0			0
$855$	2.78564e6	+	3.46214e7i	0.130320	+	1.61968i
$856$		0			0
$857$			1.99678e7i			0.928704i	0.885651	+	0.464352i	$0.153713\pi$
$857$			1.99678e7i			0.928704i	−0.885651	+	0.464352i	$0.846287\pi$
$858$		0			0
$859$	2.19102e7			1.01313			0.506564	−	0.862202i	$-0.330915\pi$
$859$	2.19102e7			1.01313			0.506564	+	0.862202i	$0.330915\pi$
$860$		0			0
$861$		−	6.63710e7i		−	3.05120i
$862$		0			0
$863$	1.14632e7			0.523937			0.261969	−	0.965076i	$-0.415628\pi$
$863$	1.14632e7			0.523937			0.261969	+	0.965076i	$0.415628\pi$
$864$		0			0
$865$		−	8.90521e6i		−	0.404673i
$866$		0			0
$867$			1.35987e7i			0.614396i
$868$		0			0
$869$	3.10762e6			0.139598
$870$		0			0
$871$			6.01156e6i			0.268499i
$872$		0			0
$873$			6.42434e7i			2.85294i
$874$		0			0
$875$	−3.97175e7			−1.75373
$876$		0			0
$877$	9.70110e6			0.425914			0.212957	−	0.977062i	$-0.431691\pi$
$877$	9.70110e6			0.425914			0.212957	+	0.977062i	$0.431691\pi$
$878$		0			0
$879$			1.51100e7i			0.659618i
$880$		0			0
$881$	2.41648e7			1.04892			0.524461	−	0.851435i	$-0.324267\pi$
$881$	2.41648e7			1.04892			0.524461	+	0.851435i	$0.324267\pi$
$882$		0			0
$883$	1.21850e7			0.525923			0.262962	−	0.964806i	$-0.415301\pi$
$883$	1.21850e7			0.525923			0.262962	+	0.964806i	$0.415301\pi$
$884$		0			0
$885$		−	4.46593e7i		−	1.91670i
$886$		0			0
$887$	−4.38883e7			−1.87301			−0.936504	−	0.350657i	$-0.885958\pi$
$887$	−4.38883e7			−1.87301			−0.936504	+	0.350657i	$0.885958\pi$
$888$		0			0
$889$		−	6.72629e7i		−	2.85444i
$890$		0			0
$891$	−1.82001e6			−0.0768034
$892$		0			0
$893$	1.01499e6	+	1.26149e7i	0.0425926	+	0.529363i
$894$		0			0
$895$	−3.11614e7			−1.30035
$896$		0			0
$897$	−7.92368e6			−0.328811
$898$		0			0
$899$	−3.29035e6			−0.135782
$900$		0			0
$901$	−1.51762e7			−0.622803
$902$		0			0
$903$	5.26163e7			2.14734
$904$		0			0
$905$			7.29563e6i			0.296102i
$906$		0			0
$907$		−	2.47211e7i		−	0.997816i	−0.866655	−	0.498908i	$-0.833735\pi$
$907$		−	2.47211e7i		−	0.997816i	0.866655	−	0.498908i	$-0.166265\pi$
$908$		0			0
$909$			2.69031e7i			1.07992i
$910$		0			0
$911$	2.36658e7			0.944769			0.472384	−	0.881393i	$-0.343393\pi$
$911$	2.36658e7			0.944769			0.472384	+	0.881393i	$0.343393\pi$
$912$		0			0
$913$	3.24894e6			0.128993
$914$		0			0
$915$			4.22686e7i			1.66903i
$916$		0			0
$917$			5.51537e7i			2.16597i
$918$		0			0
$919$		−	2.69233e7i		−	1.05157i	−0.850616	−	0.525787i	$-0.823771\pi$
$919$		−	2.69233e7i		−	1.05157i	0.850616	−	0.525787i	$-0.176229\pi$
$920$		0			0
$921$	−8.61363e7			−3.34609
$922$		0			0
$923$	−3.33588e6			−0.128886
$924$		0			0
$925$	3.90839e6			0.150191
$926$		0			0
$927$	−4.23466e7			−1.61853
$928$		0			0
$929$	−2.67977e7			−1.01873			−0.509364	−	0.860551i	$-0.670119\pi$
$929$	−2.67977e7			−1.01873			−0.509364	+	0.860551i	$0.670119\pi$
$930$		0			0
$931$	−4.21237e7	+	3.38928e6i	−1.59277	+	0.128154i
$932$		0			0
$933$	−8.28056e7			−3.11426
$934$		0			0
$935$		−	1.59318e6i		−	0.0595987i
$936$		0			0
$937$	−9.87790e6			−0.367549			−0.183775	−	0.982968i	$-0.558832\pi$
$937$	−9.87790e6			−0.367549			−0.183775	+	0.982968i	$0.558832\pi$
$938$		0			0
$939$			1.01695e7i			0.376389i
$940$		0			0
$941$	−8.71643e6			−0.320896			−0.160448	−	0.987044i	$-0.551294\pi$
$941$	−8.71643e6			−0.320896			−0.160448	+	0.987044i	$0.551294\pi$
$942$		0			0
$943$	−4.02742e7			−1.47485
$944$		0			0
$945$		−	6.02666e7i		−	2.19532i
$946$		0			0
$947$	−2.76979e6			−0.100363			−0.0501813	−	0.998740i	$-0.515980\pi$
$947$	−2.76979e6			−0.100363			−0.0501813	+	0.998740i	$0.515980\pi$
$948$		0			0
$949$	5.52043e6			0.198979
$950$		0			0
$951$			4.24075e6i			0.152052i
$952$		0			0
$953$			3.15715e6i			0.112606i	0.998414	+	0.0563031i	$0.0179313\pi$
$953$			3.15715e6i			0.112606i	−0.998414	+	0.0563031i	$0.982069\pi$
$954$		0			0
$955$	−1.25390e7			−0.444893
$956$		0			0
$957$		−	4.72982e6i		−	0.166942i
$958$		0			0
$959$			2.82859e7i			0.993171i
$960$		0			0
$961$	−2.81828e7			−0.984409
$962$		0			0
$963$		−	1.83741e7i		−	0.638471i
$964$		0			0
$965$	−1.75321e7			−0.606059
$966$		0			0
$967$			1.42538e7i			0.490191i	0.969499	+	0.245096i	$0.0788194\pi$
$967$			1.42538e7i			0.490191i	−0.969499	+	0.245096i	$0.921181\pi$
$968$		0			0
$969$	3.22845e6	+	4.01249e7i	0.110455	+	1.37279i
$970$		0			0
$971$		−	4.87109e7i		−	1.65797i	−0.559267	−	0.828987i	$-0.688917\pi$
$971$		−	4.87109e7i		−	1.65797i	0.559267	−	0.828987i	$-0.311083\pi$
$972$		0			0
$973$		−	8.47486e6i		−	0.286979i
$974$		0			0
$975$			2.24101e6i			0.0754973i
$976$		0			0
$977$			5.11627e7i			1.71481i	0.514640	+	0.857406i	$0.327926\pi$
$977$			5.11627e7i			1.71481i	−0.514640	+	0.857406i	$0.672074\pi$
$978$		0			0
$979$			934766.i			0.0311707i
$980$		0			0
$981$	4.53429e7			1.50431
$982$		0			0
$983$	−6.16221e6			−0.203401			−0.101700	−	0.994815i	$-0.532428\pi$
$983$	−6.16221e6			−0.203401			−0.101700	+	0.994815i	$0.532428\pi$
$984$		0			0
$985$	3.54211e7			1.16325
$986$		0			0
$987$		−	4.50144e7i		−	1.47082i
$988$		0			0
$989$		−	3.19278e7i		−	1.03795i
$990$		0			0
$991$	5.67892e7			1.83688			0.918441	−	0.395558i	$-0.129449\pi$
$991$	5.67892e7			1.83688			0.918441	+	0.395558i	$0.129449\pi$
$992$		0			0
$993$	825060.			0.0265529
$994$		0			0
$995$	2.22415e7			0.712208
$996$		0			0
$997$			5.89936e7i			1.87961i	0.341716	+	0.939803i	$0.388992\pi$
$997$			5.89936e7i			1.87961i	−0.341716	+	0.939803i	$0.611008\pi$
$998$		0			0
$999$			2.52247e7i			0.799673i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	608.6.b.b.303.7		96
4.3	odd	2		152.6.b.b.75.20	yes	96
8.3	odd	2	inner	608.6.b.b.303.8		96
8.5	even	2		152.6.b.b.75.78	yes	96
19.18	odd	2	inner	608.6.b.b.303.89		96
76.75	even	2		152.6.b.b.75.77	yes	96
152.37	odd	2		152.6.b.b.75.19	✓	96
152.75	even	2	inner	608.6.b.b.303.90		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.6.b.b.75.19	✓	96	152.37	odd	2
152.6.b.b.75.20	yes	96	4.3	odd	2
152.6.b.b.75.77	yes	96	76.75	even	2
152.6.b.b.75.78	yes	96	8.5	even	2
608.6.b.b.303.7		96	1.1	even	1	trivial
608.6.b.b.303.8		96	8.3	odd	2	inner
608.6.b.b.303.89		96	19.18	odd	2	inner
608.6.b.b.303.90		96	152.75	even	2	inner

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:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	608.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-26.7852i q^{3} -46.5237i q^{5} -208.957i q^{7} -474.446 q^{9} +O(q^{10})\)	\(q-26.7852i q^{3} -46.5237i q^{5} -208.957i q^{7} -474.446 q^{9} -35.8554 q^{11} -87.1027 q^{13} -1246.15 q^{15} -955.073 q^{17} +(1568.49 - 126.201i) q^{19} -5596.96 q^{21} +3396.26i q^{23} +960.543 q^{25} +6199.33i q^{27} -4924.87 q^{29} +668.108 q^{31} +960.395i q^{33} -9721.47 q^{35} +4068.94 q^{37} +2333.06i q^{39} +11858.4i q^{41} -9400.87 q^{43} +22073.0i q^{45} +8042.65i q^{47} -26856.1 q^{49} +25581.8i q^{51} +15890.1 q^{53} +1668.13i q^{55} +(-3380.32 - 42012.4i) q^{57} +35837.9i q^{59} -33919.4i q^{61} +99139.0i q^{63} +4052.34i q^{65} -69017.0i q^{67} +90969.5 q^{69} +38298.3 q^{71} -63378.5 q^{73} -25728.3i q^{75} +7492.26i q^{77} -86670.7 q^{79} +50759.7 q^{81} -90612.1 q^{83} +44433.5i q^{85} +131914. i q^{87} -26070.4i q^{89} +18200.7i q^{91} -17895.4i q^{93} +(-5871.35 - 72972.2i) q^{95} -135407. i q^{97} +17011.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q - 6168 q^{9}+O(q^{10}) \) 96 * q - 6168 * q^9	\( 96 q - 6168 q^{9} - 944 q^{11} - 3832 q^{17} - 5240 q^{19} - 62504 q^{25} - 7720 q^{35} - 45096 q^{43} - 210840 q^{49} - 36336 q^{57} - 4336 q^{73} - 20624 q^{81} - 52152 q^{83} + 752768 q^{99}+O(q^{100}) \) 96 * q - 6168 * q^9 - 944 * q^11 - 3832 * q^17 - 5240 * q^19 - 62504 * q^25 - 7720 * q^35 - 45096 * q^43 - 210840 * q^49 - 36336 * q^57 - 4336 * q^73 - 20624 * q^81 - 52152 * q^83 + 752768 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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