LMFDB - Embedded newform 400.8.c.r.49.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,8,Mod(49,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.49"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-2936] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$124.954010194$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{2521})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1261x^{2} + 396900 $ x^4 + 1261*x^2 + 396900
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 100)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$25.6048i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.8.c.r.49.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-30.2096i q^{3} +161.257i q^{7} +1274.38 q^{9} +1119.43 q^{11} -5152.69i q^{13} -7125.53i q^{17} -33951.5 q^{19} +4871.51 q^{21} +24499.2i q^{23} -104567. i q^{27} +16061.1 q^{29} -114867. q^{31} -33817.5i q^{33} +492031. i q^{37} -155661. q^{39} +640276. q^{41} +69314.0i q^{43} -1.21927e6i q^{47} +797539. q^{49} -215259. q^{51} -1.38883e6i q^{53} +1.02566e6i q^{57} -2.49636e6 q^{59} +1.83520e6 q^{61} +205504. i q^{63} -1.94206e6i q^{67} +740110. q^{69} -163523. q^{71} -1.95783e6i q^{73} +180516. i q^{77} +1.31338e6 q^{79} -371844. q^{81} -4.04009e6i q^{83} -485198. i q^{87} -7.83758e6 q^{89} +830909. q^{91} +3.47009e6i q^{93} -9.70980e6i q^{97} +1.42658e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2936 q^{9} - 4560 q^{11} + 41936 q^{19} + 71704 q^{21} + 461904 q^{29} - 429344 q^{31} - 1787504 q^{39} + 898164 q^{41} + 2852748 q^{49} - 5521488 q^{51} - 1887648 q^{59} + 2532728 q^{61} + 3370152 q^{69}+ \cdots + 21498240 q^{99}+O(q^{100}) $ 4 * q - 2936 * q^9 - 4560 * q^11 + 41936 * q^19 + 71704 * q^21 + 461904 * q^29 - 429344 * q^31 - 1787504 * q^39 + 898164 * q^41 + 2852748 * q^49 - 5521488 * q^51 - 1887648 * q^59 + 2532728 * q^61 + 3370152 * q^69 - 7233552 * q^71 - 6079792 * q^79 - 7263484 * q^81 - 6478524 * q^89 + 10939424 * q^91 + 21498240 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 30.2096i	− 0.645981i	−0.946402	−	0.322991i	$-0.895312\pi$
$3$	− 30.2096i	− 0.645981i	0.946402	−	0.322991i	$-0.104688\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	161.257i	0.177695i	0.996045	+	0.0888477i	$0.0283185\pi$
$7$	161.257i	0.177695i	−0.996045	+	0.0888477i	$0.971682\pi$
$8$	0	0
$9$	1274.38	0.582708
$10$	0	0
$11$	1119.43	0.253584	0.126792	−	0.991929i	$-0.459532\pi$
$11$	1119.43	0.253584	0.126792	+	0.991929i	$0.459532\pi$
$12$	0	0
$13$	− 5152.69i	− 0.650478i	−0.945632	−	0.325239i	$-0.894555\pi$
$13$	− 5152.69i	− 0.650478i	0.945632	−	0.325239i	$-0.105445\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7125.53i	− 0.351760i	−0.984412	−	0.175880i	$-0.943723\pi$
$17$	− 7125.53i	− 0.351760i	0.984412	−	0.175880i	$-0.0562770\pi$
$18$	0	0
$19$	−33951.5	−1.13559	−0.567794	−	0.823171i	$-0.692203\pi$
$19$	−33951.5	−1.13559	−0.567794	+	0.823171i	$0.692203\pi$
$20$	0	0
$21$	4871.51	0.114788
$22$	0	0
$23$	24499.2i	0.419860i	0.977716	+	0.209930i	$0.0673236\pi$
$23$	24499.2i	0.419860i	−0.977716	+	0.209930i	$0.932676\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 104567.i	− 1.02240i
$28$	0	0
$29$	16061.1	0.122287	0.0611437	−	0.998129i	$-0.480525\pi$
$29$	16061.1	0.122287	0.0611437	+	0.998129i	$0.480525\pi$
$30$	0	0
$31$	−114867.	−0.692518	−0.346259	−	0.938139i	$-0.612548\pi$
$31$	−114867.	−0.692518	−0.346259	+	0.938139i	$0.612548\pi$
$32$	0	0
$33$	− 33817.5i	− 0.163811i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	492031.i	1.59693i	0.602041	+	0.798465i	$0.294354\pi$
$37$	492031.i	1.59693i	−0.602041	+	0.798465i	$0.705646\pi$
$38$	0	0
$39$	−155661.	−0.420196
$40$	0	0
$41$	640276.	1.45085	0.725427	−	0.688299i	$-0.241642\pi$
$41$	640276.	1.45085	0.725427	+	0.688299i	$0.241642\pi$
$42$	0	0
$43$	69314.0i	0.132948i	0.997788	+	0.0664740i	$0.0211749\pi$
$43$	69314.0i	0.132948i	−0.997788	+	0.0664740i	$0.978825\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 1.21927e6i	− 1.71299i	−0.516152	−	0.856497i	$-0.672636\pi$
$47$	− 1.21927e6i	− 1.71299i	0.516152	−	0.856497i	$-0.327364\pi$
$48$	0	0
$49$	797539.	0.968424
$50$	0	0
$51$	−215259.	−0.227230
$52$	0	0
$53$	− 1.38883e6i	− 1.28140i	−0.767791	−	0.640700i	$-0.778644\pi$
$53$	− 1.38883e6i	− 1.28140i	0.767791	−	0.640700i	$-0.221356\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.02566e6i	0.733569i
$58$	0	0
$59$	−2.49636e6	−1.58243	−0.791217	−	0.611536i	$-0.790552\pi$
$59$	−2.49636e6	−1.58243	−0.791217	+	0.611536i	$0.790552\pi$
$60$	0	0
$61$	1.83520e6	1.03521	0.517605	−	0.855620i	$-0.326824\pi$
$61$	1.83520e6	1.03521	0.517605	+	0.855620i	$0.326824\pi$
$62$	0	0
$63$	205504.i	0.103545i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.94206e6i	− 0.788861i	−0.918926	−	0.394431i	$-0.870942\pi$
$67$	− 1.94206e6i	− 0.788861i	0.918926	−	0.394431i	$-0.129058\pi$
$68$	0	0
$69$	740110.	0.271222
$70$	0	0
$71$	−163523.	−0.0542218	−0.0271109	−	0.999632i	$-0.508631\pi$
$71$	−163523.	−0.0542218	−0.0271109	+	0.999632i	$0.508631\pi$
$72$	0	0
$73$	− 1.95783e6i	− 0.589040i	−0.955645	−	0.294520i	$-0.904840\pi$
$73$	− 1.95783e6i	− 0.589040i	0.955645	−	0.294520i	$-0.0951598\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	180516.i	0.0450608i
$78$	0	0
$79$	1.31338e6	0.299706	0.149853	−	0.988708i	$-0.452120\pi$
$79$	1.31338e6	0.299706	0.149853	+	0.988708i	$0.452120\pi$
$80$	0	0
$81$	−371844.	−0.0777433
$82$	0	0
$83$	− 4.04009e6i	− 0.775564i	−0.921751	−	0.387782i	$-0.873241\pi$
$83$	− 4.04009e6i	− 0.775564i	0.921751	−	0.387782i	$-0.126759\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 485198.i	− 0.0789953i
$88$	0	0
$89$	−7.83758e6	−1.17847	−0.589233	−	0.807963i	$-0.700570\pi$
$89$	−7.83758e6	−1.17847	−0.589233	+	0.807963i	$0.700570\pi$
$90$	0	0
$91$	830909.	0.115587
$92$	0	0
$93$	3.47009e6i	0.447354i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 9.70980e6i	− 1.08021i	−0.841597	−	0.540106i	$-0.818384\pi$
$97$	− 9.70980e6i	− 1.08021i	0.841597	−	0.540106i	$-0.181616\pi$
$98$	0	0
$99$	1.42658e6	0.147766
$100$	0	0
$101$	−6.28989e6	−0.607461	−0.303731	−	0.952758i	$-0.598232\pi$
$101$	−6.28989e6	−0.607461	−0.303731	+	0.952758i	$0.598232\pi$
$102$	0	0
$103$	− 5.56437e6i	− 0.501748i	−0.968020	−	0.250874i	$-0.919282\pi$
$103$	− 5.56437e6i	− 0.501748i	0.968020	−	0.250874i	$-0.0807180\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.16208e7i	1.70619i	0.521752	+	0.853097i	$0.325279\pi$
$107$	2.16208e7i	1.70619i	−0.521752	+	0.853097i	$0.674721\pi$
$108$	0	0
$109$	−9.27249e6	−0.685809	−0.342905	−	0.939370i	$-0.611411\pi$
$109$	−9.27249e6	−0.685809	−0.342905	+	0.939370i	$0.611411\pi$
$110$	0	0
$111$	1.48640e7	1.03159
$112$	0	0
$113$	− 9.72018e6i	− 0.633724i	−0.948472	−	0.316862i	$-0.897371\pi$
$113$	− 9.72018e6i	− 0.633724i	0.948472	−	0.316862i	$-0.102629\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 6.56650e6i	− 0.379039i
$118$	0	0
$119$	1.14904e6	0.0625061
$120$	0	0
$121$	−1.82340e7	−0.935695
$122$	0	0
$123$	− 1.93425e7i	− 0.937225i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	3.14904e7i	1.36416i	0.731277	+	0.682081i	$0.238925\pi$
$127$	3.14904e7i	1.36416i	−0.731277	+	0.682081i	$0.761075\pi$
$128$	0	0
$129$	2.09395e6	0.0858819
$130$	0	0
$131$	−1.89614e7	−0.736920	−0.368460	−	0.929644i	$-0.620115\pi$
$131$	−1.89614e7	−0.736920	−0.368460	+	0.929644i	$0.620115\pi$
$132$	0	0
$133$	− 5.47492e6i	− 0.201789i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.98063e6i	0.198712i	0.995052	+	0.0993562i	$0.0316783\pi$
$137$	5.98063e6i	0.198712i	−0.995052	+	0.0993562i	$0.968322\pi$
$138$	0	0
$139$	4.86902e6	0.153776	0.0768882	−	0.997040i	$-0.475502\pi$
$139$	4.86902e6	0.153776	0.0768882	+	0.997040i	$0.475502\pi$
$140$	0	0
$141$	−3.68335e7	−1.10656
$142$	0	0
$143$	− 5.76808e6i	− 0.164951i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2.40933e7i	− 0.625584i
$148$	0	0
$149$	−4.11601e7	−1.01935	−0.509677	−	0.860366i	$-0.670235\pi$
$149$	−4.11601e7	−1.01935	−0.509677	+	0.860366i	$0.670235\pi$
$150$	0	0
$151$	−6.91055e7	−1.63340	−0.816701	−	0.577061i	$-0.804200\pi$
$151$	−6.91055e7	−1.63340	−0.816701	+	0.577061i	$0.804200\pi$
$152$	0	0
$153$	− 9.08065e6i	− 0.204973i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.91482e7i	1.21981i	0.792474	+	0.609906i	$0.208793\pi$
$157$	5.91482e7i	1.21981i	−0.792474	+	0.609906i	$0.791207\pi$
$158$	0	0
$159$	−4.19560e7	−0.827760
$160$	0	0
$161$	−3.95068e6	−0.0746073
$162$	0	0
$163$	− 1.84384e7i	− 0.333477i	−0.986001	−	0.166738i	$-0.946676\pi$
$163$	− 1.84384e7i	− 0.333477i	0.986001	−	0.166738i	$-0.0533235\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.16180e7i	− 0.525323i	−0.964888	−	0.262662i	$-0.915400\pi$
$167$	− 3.16180e7i	− 0.525323i	0.964888	−	0.262662i	$-0.0846003\pi$
$168$	0	0
$169$	3.61983e7	0.576879
$170$	0	0
$171$	−4.32671e7	−0.661716
$172$	0	0
$173$	− 8.16760e7i	− 1.19932i	−0.800257	−	0.599658i	$-0.795303\pi$
$173$	− 8.16760e7i	− 1.19932i	0.800257	−	0.599658i	$-0.204697\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	7.54140e7i	1.02222i
$178$	0	0
$179$	−8.42697e7	−1.09821	−0.549106	−	0.835753i	$-0.685032\pi$
$179$	−8.42697e7	−1.09821	−0.549106	+	0.835753i	$0.685032\pi$
$180$	0	0
$181$	3.10251e7	0.388900	0.194450	−	0.980912i	$-0.437708\pi$
$181$	3.10251e7	0.388900	0.194450	+	0.980912i	$0.437708\pi$
$182$	0	0
$183$	− 5.54406e7i	− 0.668727i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 7.97653e6i	− 0.0892008i
$188$	0	0
$189$	1.68622e7	0.181676
$190$	0	0
$191$	−6.20657e7	−0.644517	−0.322259	−	0.946652i	$-0.604442\pi$
$191$	−6.20657e7	−0.644517	−0.322259	+	0.946652i	$0.604442\pi$
$192$	0	0
$193$	− 1.73827e8i	− 1.74047i	−0.492635	−	0.870236i	$-0.663966\pi$
$193$	− 1.73827e8i	− 1.74047i	0.492635	−	0.870236i	$-0.336034\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.01949e8i	0.950064i	0.879969	+	0.475032i	$0.157563\pi$
$197$	1.01949e8i	0.950064i	−0.879969	+	0.475032i	$0.842437\pi$
$198$	0	0
$199$	−1.17985e8	−1.06130	−0.530652	−	0.847590i	$-0.678053\pi$
$199$	−1.17985e8	−1.06130	−0.530652	+	0.847590i	$0.678053\pi$
$200$	0	0
$201$	−5.86687e7	−0.509590
$202$	0	0
$203$	2.58997e6i	0.0217299i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.12214e7i	0.244656i
$208$	0	0
$209$	−3.80063e7	−0.287968
$210$	0	0
$211$	−6.99393e7	−0.512546	−0.256273	−	0.966605i	$-0.582495\pi$
$211$	−6.99393e7	−0.512546	−0.256273	+	0.966605i	$0.582495\pi$
$212$	0	0
$213$	4.93995e6i	0.0350263i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.85232e7i	− 0.123057i
$218$	0	0
$219$	−5.91452e7	−0.380509
$220$	0	0
$221$	−3.67157e7	−0.228812
$222$	0	0
$223$	− 2.95613e8i	− 1.78508i	−0.450970	−	0.892539i	$-0.648922\pi$
$223$	− 2.95613e8i	− 1.78508i	0.450970	−	0.892539i	$-0.351078\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.63799e7i	0.206429i	0.994659	+	0.103215i	$0.0329129\pi$
$227$	3.63799e7i	0.206429i	−0.994659	+	0.103215i	$0.967087\pi$
$228$	0	0
$229$	−2.88623e8	−1.58821	−0.794104	−	0.607782i	$-0.792059\pi$
$229$	−2.88623e8	−1.58821	−0.794104	+	0.607782i	$0.792059\pi$
$230$	0	0
$231$	5.45332e6	0.0291084
$232$	0	0
$233$	− 2.19155e8i	− 1.13503i	−0.823364	−	0.567514i	$-0.807905\pi$
$233$	− 2.19155e8i	− 1.13503i	0.823364	−	0.567514i	$-0.192095\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3.96766e7i	− 0.193604i
$238$	0	0
$239$	1.54502e8	0.732053	0.366027	−	0.930604i	$-0.380718\pi$
$239$	1.54502e8	0.732053	0.366027	+	0.930604i	$0.380718\pi$
$240$	0	0
$241$	−1.22833e8	−0.565267	−0.282634	−	0.959228i	$-0.591208\pi$
$241$	−1.22833e8	−0.565267	−0.282634	+	0.959228i	$0.591208\pi$
$242$	0	0
$243$	− 2.17454e8i	− 0.972179i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.74941e8i	0.738675i
$248$	0	0
$249$	−1.22049e8	−0.501000
$250$	0	0
$251$	2.00115e8	0.798771	0.399386	−	0.916783i	$-0.369223\pi$
$251$	2.00115e8	0.798771	0.399386	+	0.916783i	$0.369223\pi$
$252$	0	0
$253$	2.74252e7i	0.106470i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.46950e8i	− 1.27497i	−0.770461	−	0.637487i	$-0.779974\pi$
$257$	− 3.46950e8i	− 1.27497i	0.770461	−	0.637487i	$-0.220026\pi$
$258$	0	0
$259$	−7.93435e7	−0.283767
$260$	0	0
$261$	2.04679e7	0.0712578
$262$	0	0
$263$	− 1.07808e8i	− 0.365432i	−0.983166	−	0.182716i	$-0.941511\pi$
$263$	− 1.07808e8i	− 0.365432i	0.983166	−	0.182716i	$-0.0584888\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.36770e8i	0.761267i
$268$	0	0
$269$	−3.70830e8	−1.16156	−0.580780	−	0.814060i	$-0.697252\pi$
$269$	−3.70830e8	−1.16156	−0.580780	+	0.814060i	$0.697252\pi$
$270$	0	0
$271$	1.18015e8	0.360201	0.180100	−	0.983648i	$-0.442358\pi$
$271$	1.18015e8	0.360201	0.180100	+	0.983648i	$0.442358\pi$
$272$	0	0
$273$	− 2.51014e7i	− 0.0746670i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 2.46581e8i	− 0.697075i	−0.937295	−	0.348538i	$-0.886678\pi$
$277$	− 2.46581e8i	− 0.697075i	0.937295	−	0.348538i	$-0.113322\pi$
$278$	0	0
$279$	−1.46385e8	−0.403536
$280$	0	0
$281$	−5.56340e8	−1.49578	−0.747890	−	0.663822i	$-0.768933\pi$
$281$	−5.56340e8	−1.49578	−0.747890	+	0.663822i	$0.768933\pi$
$282$	0	0
$283$	− 2.87532e8i	− 0.754108i	−0.926191	−	0.377054i	$-0.876937\pi$
$283$	− 2.87532e8i	− 0.754108i	0.926191	−	0.377054i	$-0.123063\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.03249e8i	0.257810i
$288$	0	0
$289$	3.59565e8	0.876265
$290$	0	0
$291$	−2.93329e8	−0.697797
$292$	0	0
$293$	− 4.45120e8i	− 1.03381i	−0.856043	−	0.516905i	$-0.827084\pi$
$293$	− 4.45120e8i	− 1.03381i	0.856043	−	0.516905i	$-0.172916\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.17055e8i	− 0.259265i
$298$	0	0
$299$	1.26237e8	0.273110
$300$	0	0
$301$	−1.11774e7	−0.0236242
$302$	0	0
$303$	1.90015e8i	0.392409i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	3.80295e7i	0.0750129i	0.999296	+	0.0375065i	$0.0119415\pi$
$307$	3.80295e7i	0.0750129i	−0.999296	+	0.0375065i	$0.988059\pi$
$308$	0	0
$309$	−1.68097e8	−0.324120
$310$	0	0
$311$	1.98911e8	0.374971	0.187486	−	0.982267i	$-0.439966\pi$
$311$	1.98911e8	0.374971	0.187486	+	0.982267i	$0.439966\pi$
$312$	0	0
$313$	3.06310e8i	0.564619i	0.959323	+	0.282310i	$0.0911006\pi$
$313$	3.06310e8i	0.564619i	−0.959323	+	0.282310i	$0.908899\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.11969e9i	1.97420i	0.160117	+	0.987098i	$0.448813\pi$
$317$	1.11969e9i	1.97420i	−0.160117	+	0.987098i	$0.551187\pi$
$318$	0	0
$319$	1.79792e7	0.0310102
$320$	0	0
$321$	6.53155e8	1.10217
$322$	0	0
$323$	2.41922e8i	0.399454i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.80118e8i	0.443020i
$328$	0	0
$329$	1.96616e8	0.304391
$330$	0	0
$331$	−6.94965e8	−1.05333	−0.526665	−	0.850073i	$-0.676558\pi$
$331$	−6.94965e8	−1.05333	−0.526665	+	0.850073i	$0.676558\pi$
$332$	0	0
$333$	6.27035e8i	0.930544i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.25084e8i	0.889680i	0.895610	+	0.444840i	$0.146739\pi$
$337$	6.25084e8i	0.889680i	−0.895610	+	0.444840i	$0.853261\pi$
$338$	0	0
$339$	−2.93642e8	−0.409374
$340$	0	0
$341$	−1.28586e8	−0.175612
$342$	0	0
$343$	2.61411e8i	0.349780i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.53942e9i	− 1.97790i	−0.148259	−	0.988949i	$-0.547367\pi$
$347$	− 1.53942e9i	− 1.97790i	0.148259	−	0.988949i	$-0.452633\pi$
$348$	0	0
$349$	5.95572e8	0.749972	0.374986	−	0.927030i	$-0.377648\pi$
$349$	5.95572e8	0.749972	0.374986	+	0.927030i	$0.377648\pi$
$350$	0	0
$351$	−5.38801e8	−0.665048
$352$	0	0
$353$	− 1.00357e9i	− 1.21433i	−0.794575	−	0.607166i	$-0.792306\pi$
$353$	− 1.00357e9i	− 1.21433i	0.794575	−	0.607166i	$-0.207694\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3.47121e7i	− 0.0403778i
$358$	0	0
$359$	5.38711e8	0.614505	0.307252	−	0.951628i	$-0.400590\pi$
$359$	5.38711e8	0.614505	0.307252	+	0.951628i	$0.400590\pi$
$360$	0	0
$361$	2.58830e8	0.289561
$362$	0	0
$363$	5.50843e8i	0.604442i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 7.66487e8i	− 0.809419i	−0.914445	−	0.404710i	$-0.867373\pi$
$367$	− 7.66487e8i	− 0.809419i	0.914445	−	0.404710i	$-0.132627\pi$
$368$	0	0
$369$	8.15957e8	0.845424
$370$	0	0
$371$	2.23960e8	0.227699
$372$	0	0
$373$	1.29545e9i	1.29253i	0.763113	+	0.646265i	$0.223670\pi$
$373$	1.29545e9i	1.29253i	−0.763113	+	0.646265i	$0.776330\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.27577e7i	− 0.0795452i
$378$	0	0
$379$	−9.56773e8	−0.902758	−0.451379	−	0.892332i	$-0.649068\pi$
$379$	−9.56773e8	−0.902758	−0.451379	+	0.892332i	$0.649068\pi$
$380$	0	0
$381$	9.51313e8	0.881223
$382$	0	0
$383$	1.64891e9i	1.49969i	0.661612	+	0.749846i	$0.269873\pi$
$383$	1.64891e9i	1.49969i	−0.661612	+	0.749846i	$0.730127\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.83326e7i	0.0774698i
$388$	0	0
$389$	1.83369e8	0.157944	0.0789720	−	0.996877i	$-0.474836\pi$
$389$	1.83369e8	0.157944	0.0789720	+	0.996877i	$0.474836\pi$
$390$	0	0
$391$	1.74570e8	0.147690
$392$	0	0
$393$	5.72815e8i	0.476037i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 5.66325e8i	− 0.454254i	−0.973865	−	0.227127i	$-0.927067\pi$
$397$	− 5.66325e8i	− 0.454254i	0.973865	−	0.227127i	$-0.0729333\pi$
$398$	0	0
$399$	−1.65395e8	−0.130352
$400$	0	0
$401$	−7.51183e8	−0.581756	−0.290878	−	0.956760i	$-0.593947\pi$
$401$	−7.51183e8	−0.581756	−0.290878	+	0.956760i	$0.593947\pi$
$402$	0	0
$403$	5.91876e8i	0.450467i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.50794e8i	0.404957i
$408$	0	0
$409$	1.06838e9	0.772135	0.386067	−	0.922471i	$-0.373833\pi$
$409$	1.06838e9	0.772135	0.386067	+	0.922471i	$0.373833\pi$
$410$	0	0
$411$	1.80672e8	0.128365
$412$	0	0
$413$	− 4.02557e8i	− 0.281191i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.47091e8i	− 0.0993367i
$418$	0	0
$419$	2.27591e9	1.51149	0.755745	−	0.654866i	$-0.227275\pi$
$419$	2.27591e9	1.51149	0.755745	+	0.654866i	$0.227275\pi$
$420$	0	0
$421$	4.84009e8	0.316130	0.158065	−	0.987429i	$-0.449474\pi$
$421$	4.84009e8	0.316130	0.158065	+	0.987429i	$0.449474\pi$
$422$	0	0
$423$	− 1.55381e9i	− 0.998175i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.95939e8i	0.183952i
$428$	0	0
$429$	−1.74251e8	−0.106555
$430$	0	0
$431$	2.82597e9	1.70019	0.850095	−	0.526629i	$-0.176544\pi$
$431$	2.82597e9	1.70019	0.850095	+	0.526629i	$0.176544\pi$
$432$	0	0
$433$	2.17262e9i	1.28610i	0.765823	+	0.643051i	$0.222332\pi$
$433$	2.17262e9i	1.28610i	−0.765823	+	0.643051i	$0.777668\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 8.31784e8i	− 0.476788i
$438$	0	0
$439$	2.38136e9	1.34338	0.671692	−	0.740831i	$-0.265568\pi$
$439$	2.38136e9	1.34338	0.671692	+	0.740831i	$0.265568\pi$
$440$	0	0
$441$	1.01637e9	0.564309
$442$	0	0
$443$	2.81990e9i	1.54106i	0.637402	+	0.770532i	$0.280009\pi$
$443$	2.81990e9i	1.54106i	−0.637402	+	0.770532i	$0.719991\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.24343e9i	0.658483i
$448$	0	0
$449$	−2.04796e9	−1.06773	−0.533863	−	0.845571i	$-0.679260\pi$
$449$	−2.04796e9	−1.06773	−0.533863	+	0.845571i	$0.679260\pi$
$450$	0	0
$451$	7.16744e8	0.367914
$452$	0	0
$453$	2.08765e9i	1.05515i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 3.49506e9i	− 1.71296i	−0.516177	−	0.856482i	$-0.672645\pi$
$457$	− 3.49506e9i	− 1.71296i	0.516177	−	0.856482i	$-0.327355\pi$
$458$	0	0
$459$	−7.45094e8	−0.359639
$460$	0	0
$461$	−3.35928e8	−0.159696	−0.0798479	−	0.996807i	$-0.525443\pi$
$461$	−3.35928e8	−0.159696	−0.0798479	+	0.996807i	$0.525443\pi$
$462$	0	0
$463$	− 2.65935e9i	− 1.24521i	−0.782537	−	0.622604i	$-0.786075\pi$
$463$	− 2.65935e9i	− 1.24521i	0.782537	−	0.622604i	$-0.213925\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.93004e9i	− 0.876913i	−0.898752	−	0.438456i	$-0.855525\pi$
$467$	− 1.93004e9i	− 0.876913i	0.898752	−	0.438456i	$-0.144475\pi$
$468$	0	0
$469$	3.13171e8	0.140177
$470$	0	0
$471$	1.78684e9	0.787976
$472$	0	0
$473$	7.75922e7i	0.0337135i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.76990e9i	− 0.746682i
$478$	0	0
$479$	2.94332e9	1.22367	0.611834	−	0.790986i	$-0.290432\pi$
$479$	2.94332e9	1.22367	0.611834	+	0.790986i	$0.290432\pi$
$480$	0	0
$481$	2.53528e9	1.03877
$482$	0	0
$483$	1.19348e8i	0.0481949i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.12191e9i	0.832485i	0.909254	+	0.416243i	$0.136653\pi$
$487$	2.12191e9i	0.832485i	−0.909254	+	0.416243i	$0.863347\pi$
$488$	0	0
$489$	−5.57015e8	−0.215420
$490$	0	0
$491$	3.56167e9	1.35790	0.678952	−	0.734183i	$-0.262435\pi$
$491$	3.56167e9	1.35790	0.678952	+	0.734183i	$0.262435\pi$
$492$	0	0
$493$	− 1.14444e8i	− 0.0430157i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.63693e7i	− 0.00963497i
$498$	0	0
$499$	−8.82640e8	−0.318003	−0.159002	−	0.987278i	$-0.550828\pi$
$499$	−8.82640e8	−0.318003	−0.159002	+	0.987278i	$0.550828\pi$
$500$	0	0
$501$	−9.55166e8	−0.339349
$502$	0	0
$503$	4.72486e9i	1.65539i	0.561177	+	0.827696i	$0.310349\pi$
$503$	4.72486e9i	1.65539i	−0.561177	+	0.827696i	$0.689651\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 1.09353e9i	− 0.372653i
$508$	0	0
$509$	1.54524e9	0.519377	0.259689	−	0.965692i	$-0.416380\pi$
$509$	1.54524e9	0.519377	0.259689	+	0.965692i	$0.416380\pi$
$510$	0	0
$511$	3.15715e8	0.104670
$512$	0	0
$513$	3.55020e9i	1.16103i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.36488e9i	− 0.434389i
$518$	0	0
$519$	−2.46740e9	−0.774735
$520$	0	0
$521$	1.31300e8	0.0406754	0.0203377	−	0.999793i	$-0.493526\pi$
$521$	1.31300e8	0.0406754	0.0203377	+	0.999793i	$0.493526\pi$
$522$	0	0
$523$	1.23021e8i	0.0376031i	0.999823	+	0.0188015i	$0.00598507\pi$
$523$	1.23021e8i	0.0376031i	−0.999823	+	0.0188015i	$0.994015\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.18491e8i	0.243600i
$528$	0	0
$529$	2.80461e9	0.823717
$530$	0	0
$531$	−3.18132e9	−0.922097
$532$	0	0
$533$	− 3.29915e9i	− 0.943748i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.54575e9i	0.709424i
$538$	0	0
$539$	8.92789e8	0.245577
$540$	0	0
$541$	2.03955e9	0.553789	0.276895	−	0.960900i	$-0.410695\pi$
$541$	2.03955e9	0.553789	0.276895	+	0.960900i	$0.410695\pi$
$542$	0	0
$543$	− 9.37254e8i	− 0.251222i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 5.83612e9i	− 1.52464i	−0.647198	−	0.762322i	$-0.724059\pi$
$547$	− 5.83612e9i	− 1.52464i	0.647198	−	0.762322i	$-0.275941\pi$
$548$	0	0
$549$	2.33875e9	0.603226
$550$	0	0
$551$	−5.45297e8	−0.138868
$552$	0	0
$553$	2.11792e8i	0.0532563i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.59270e9i	0.635711i	0.948139	+	0.317855i	$0.102963\pi$
$557$	2.59270e9i	0.635711i	−0.948139	+	0.317855i	$0.897037\pi$
$558$	0	0
$559$	3.57154e8	0.0864797
$560$	0	0
$561$	−2.40968e8	−0.0576220
$562$	0	0
$563$	4.07296e9i	0.961902i	0.876748	+	0.480951i	$0.159708\pi$
$563$	4.07296e9i	0.961902i	−0.876748	+	0.480951i	$0.840292\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 5.99626e7i	− 0.0138146i
$568$	0	0
$569$	−3.79554e9	−0.863736	−0.431868	−	0.901937i	$-0.642145\pi$
$569$	−3.79554e9	−0.863736	−0.431868	+	0.901937i	$0.642145\pi$
$570$	0	0
$571$	−7.21371e9	−1.62156	−0.810778	−	0.585353i	$-0.800956\pi$
$571$	−7.21371e9	−1.62156	−0.810778	+	0.585353i	$0.800956\pi$
$572$	0	0
$573$	1.87498e9i	0.416346i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.48078e9i	0.971043i	0.874225	+	0.485522i	$0.161370\pi$
$577$	4.48078e9i	0.971043i	−0.874225	+	0.485522i	$0.838630\pi$
$578$	0	0
$579$	−5.25124e9	−1.12431
$580$	0	0
$581$	6.51494e8	0.137814
$582$	0	0
$583$	− 1.55470e9i	− 0.324943i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 6.51913e9i	− 1.33032i	−0.746700	−	0.665161i	$-0.768363\pi$
$587$	− 6.51913e9i	− 1.33032i	0.746700	−	0.665161i	$-0.231637\pi$
$588$	0	0
$589$	3.89992e9	0.786415
$590$	0	0
$591$	3.07985e9	0.613723
$592$	0	0
$593$	− 5.01545e9i	− 0.987686i	−0.869551	−	0.493843i	$-0.835592\pi$
$593$	− 5.01545e9i	− 0.987686i	0.869551	−	0.493843i	$-0.164408\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.56427e9i	0.685583i
$598$	0	0
$599$	5.07957e9	0.965680	0.482840	−	0.875708i	$-0.339605\pi$
$599$	5.07957e9	0.965680	0.482840	+	0.875708i	$0.339605\pi$
$600$	0	0
$601$	7.77131e8	0.146027	0.0730135	−	0.997331i	$-0.476738\pi$
$601$	7.77131e8	0.146027	0.0730135	+	0.997331i	$0.476738\pi$
$602$	0	0
$603$	− 2.47493e9i	− 0.459676i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.36690e9i	0.792526i	0.918137	+	0.396263i	$0.129693\pi$
$607$	4.36690e9i	0.792526i	−0.918137	+	0.396263i	$0.870307\pi$
$608$	0	0
$609$	7.82417e7	0.0140371
$610$	0	0
$611$	−6.28250e9	−1.11426
$612$	0	0
$613$	− 9.38340e8i	− 0.164531i	−0.996610	−	0.0822657i	$-0.973784\pi$
$613$	− 9.38340e8i	− 0.164531i	0.996610	−	0.0822657i	$-0.0262156\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.56965e9i	1.12602i	0.826452	+	0.563008i	$0.190356\pi$
$617$	6.56965e9i	1.12602i	−0.826452	+	0.563008i	$0.809644\pi$
$618$	0	0
$619$	−3.07773e9	−0.521570	−0.260785	−	0.965397i	$-0.583981\pi$
$619$	−3.07773e9	−0.521570	−0.260785	+	0.965397i	$0.583981\pi$
$620$	0	0
$621$	2.56181e9	0.429265
$622$	0	0
$623$	− 1.26387e9i	− 0.209408i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.14815e9i	0.186022i
$628$	0	0
$629$	3.50598e9	0.561736
$630$	0	0
$631$	5.34255e8	0.0846536	0.0423268	−	0.999104i	$-0.486523\pi$
$631$	5.34255e8	0.0846536	0.0423268	+	0.999104i	$0.486523\pi$
$632$	0	0
$633$	2.11283e9i	0.331095i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 4.10947e9i	− 0.629938i
$638$	0	0
$639$	−2.08391e8	−0.0315955
$640$	0	0
$641$	−7.42746e9	−1.11388	−0.556938	−	0.830554i	$-0.688024\pi$
$641$	−7.42746e9	−1.11388	−0.556938	+	0.830554i	$0.688024\pi$
$642$	0	0
$643$	9.71646e8i	0.144135i	0.997400	+	0.0720675i	$0.0229597\pi$
$643$	9.71646e8i	0.144135i	−0.997400	+	0.0720675i	$0.977040\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.55212e9i	1.24139i	0.784051	+	0.620696i	$0.213150\pi$
$647$	8.55212e9i	1.24139i	−0.784051	+	0.620696i	$0.786850\pi$
$648$	0	0
$649$	−2.79450e9	−0.401281
$650$	0	0
$651$	−5.59578e8	−0.0794927
$652$	0	0
$653$	− 8.05866e9i	− 1.13257i	−0.824208	−	0.566287i	$-0.808379\pi$
$653$	− 8.05866e9i	− 1.13257i	0.824208	−	0.566287i	$-0.191621\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.49502e9i	− 0.343238i
$658$	0	0
$659$	−6.24628e9	−0.850203	−0.425101	−	0.905146i	$-0.639762\pi$
$659$	−6.24628e9	−0.850203	−0.425101	+	0.905146i	$0.639762\pi$
$660$	0	0
$661$	−9.52973e9	−1.28344	−0.641720	−	0.766939i	$-0.721779\pi$
$661$	−9.52973e9	−1.28344	−0.641720	+	0.766939i	$0.721779\pi$
$662$	0	0
$663$	1.10916e9i	0.147808i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.93484e8i	0.0513436i
$668$	0	0
$669$	−8.93035e9	−1.15313
$670$	0	0
$671$	2.05438e9	0.262513
$672$	0	0
$673$	1.80205e9i	0.227884i	0.993487	+	0.113942i	$0.0363478\pi$
$673$	1.80205e9i	0.227884i	−0.993487	+	0.113942i	$0.963652\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 5.08442e9i	− 0.629768i	−0.949130	−	0.314884i	$-0.898034\pi$
$677$	− 5.08442e9i	− 0.629768i	0.949130	−	0.314884i	$-0.101966\pi$
$678$	0	0
$679$	1.56578e9	0.191949
$680$	0	0
$681$	1.09902e9	0.133350
$682$	0	0
$683$	− 7.71566e9i	− 0.926618i	−0.886197	−	0.463309i	$-0.846662\pi$
$683$	− 7.71566e9i	− 0.926618i	0.886197	−	0.463309i	$-0.153338\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.71918e9i	1.02595i
$688$	0	0
$689$	−7.15623e9	−0.833522
$690$	0	0
$691$	−6.13074e9	−0.706871	−0.353435	−	0.935459i	$-0.614987\pi$
$691$	−6.13074e9	−0.706871	−0.353435	+	0.935459i	$0.614987\pi$
$692$	0	0
$693$	2.30047e8i	0.0262573i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 4.56231e9i	− 0.510352i
$698$	0	0
$699$	−6.62059e9	−0.733207
$700$	0	0
$701$	−1.61818e10	−1.77424	−0.887121	−	0.461538i	$-0.847298\pi$
$701$	−1.61818e10	−1.77424	−0.887121	+	0.461538i	$0.847298\pi$
$702$	0	0
$703$	− 1.67052e10i	− 1.81346i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.01429e9i	− 0.107943i
$708$	0	0
$709$	−5.28706e9	−0.557124	−0.278562	−	0.960418i	$-0.589858\pi$
$709$	−5.28706e9	−0.557124	−0.278562	+	0.960418i	$0.589858\pi$
$710$	0	0
$711$	1.67375e9	0.174641
$712$	0	0
$713$	− 2.81416e9i	− 0.290761i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 4.66745e9i	− 0.472893i
$718$	0	0
$719$	1.45007e10	1.45492	0.727459	−	0.686151i	$-0.240701\pi$
$719$	1.45007e10	1.45492	0.727459	+	0.686151i	$0.240701\pi$
$720$	0	0
$721$	8.97295e8	0.0891583
$722$	0	0
$723$	3.71072e9i	0.365152i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.09696e10i	1.05882i	0.848366	+	0.529410i	$0.177587\pi$
$727$	1.09696e10i	1.05882i	−0.848366	+	0.529410i	$0.822413\pi$
$728$	0	0
$729$	−7.38243e9	−0.705753
$730$	0	0
$731$	4.93899e8	0.0467657
$732$	0	0
$733$	− 7.58134e9i	− 0.711020i	−0.934672	−	0.355510i	$-0.884307\pi$
$733$	− 7.58134e9i	− 0.711020i	0.934672	−	0.355510i	$-0.115693\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.17400e9i	− 0.200043i
$738$	0	0
$739$	−1.63393e8	−0.0148929	−0.00744643	−	0.999972i	$-0.502370\pi$
$739$	−1.63393e8	−0.0148929	−0.00744643	+	0.999972i	$0.502370\pi$
$740$	0	0
$741$	5.28490e9	0.477170
$742$	0	0
$743$	− 2.11808e10i	− 1.89444i	−0.320585	−	0.947220i	$-0.603879\pi$
$743$	− 2.11808e10i	− 1.89444i	0.320585	−	0.947220i	$-0.396121\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5.14862e9i	− 0.451927i
$748$	0	0
$749$	−3.48651e9	−0.303183
$750$	0	0
$751$	1.53997e10	1.32670	0.663350	−	0.748309i	$-0.269134\pi$
$751$	1.53997e10	1.32670	0.663350	+	0.748309i	$0.269134\pi$
$752$	0	0
$753$	− 6.04540e9i	− 0.515992i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.93493e9i	0.748610i	0.927306	+	0.374305i	$0.122119\pi$
$757$	8.93493e9i	0.748610i	−0.927306	+	0.374305i	$0.877881\pi$
$758$	0	0
$759$	8.28502e8	0.0687777
$760$	0	0
$761$	5.23885e9	0.430913	0.215456	−	0.976513i	$-0.430876\pi$
$761$	5.23885e9	0.430913	0.215456	+	0.976513i	$0.430876\pi$
$762$	0	0
$763$	− 1.49526e9i	− 0.121865i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.28630e10i	1.02934i
$768$	0	0
$769$	−9.28281e9	−0.736101	−0.368050	−	0.929806i	$-0.619975\pi$
$769$	−9.28281e9	−0.736101	−0.368050	+	0.929806i	$0.619975\pi$
$770$	0	0
$771$	−1.04812e10	−0.823610
$772$	0	0
$773$	− 5.95607e9i	− 0.463801i	−0.972740	−	0.231900i	$-0.925506\pi$
$773$	− 5.95607e9i	− 0.463801i	0.972740	−	0.231900i	$-0.0744943\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.39693e9i	0.183308i
$778$	0	0
$779$	−2.17383e10	−1.64757
$780$	0	0
$781$	−1.83052e8	−0.0137498
$782$	0	0
$783$	− 1.67946e9i	− 0.125027i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 1.83278e10i	− 1.34029i	−0.742230	−	0.670145i	$-0.766232\pi$
$787$	− 1.83278e10i	− 1.34029i	0.742230	−	0.670145i	$-0.233768\pi$
$788$	0	0
$789$	−3.25683e9	−0.236062
$790$	0	0
$791$	1.56745e9	0.112610
$792$	0	0
$793$	− 9.45621e9i	− 0.673381i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.70828e8i	0.0189492i	0.999955	+	0.00947458i	$0.00301590\pi$
$797$	2.70828e8i	0.0189492i	−0.999955	+	0.00947458i	$0.996984\pi$
$798$	0	0
$799$	−8.68791e9	−0.602562
$800$	0	0
$801$	−9.98808e9	−0.686702
$802$	0	0
$803$	− 2.19165e9i	− 0.149371i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.12026e10i	0.750346i
$808$	0	0
$809$	2.82437e10	1.87544	0.937718	−	0.347398i	$-0.112935\pi$
$809$	2.82437e10	1.87544	0.937718	+	0.347398i	$0.112935\pi$
$810$	0	0
$811$	−7.76043e9	−0.510873	−0.255436	−	0.966826i	$-0.582219\pi$
$811$	−7.76043e9	−0.510873	−0.255436	+	0.966826i	$0.582219\pi$
$812$	0	0
$813$	− 3.56518e9i	− 0.232683i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 2.35331e9i	− 0.150974i
$818$	0	0
$819$	1.05890e9	0.0673534
$820$	0	0
$821$	2.22969e10	1.40619	0.703096	−	0.711095i	$-0.251801\pi$
$821$	2.22969e10	1.40619	0.703096	+	0.711095i	$0.251801\pi$
$822$	0	0
$823$	7.95952e9i	0.497723i	0.968539	+	0.248861i	$0.0800564\pi$
$823$	7.95952e9i	0.497723i	−0.968539	+	0.248861i	$0.919944\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.86879e10i	− 1.76372i	−0.471511	−	0.881860i	$-0.656291\pi$
$827$	− 2.86879e10i	− 1.76372i	0.471511	−	0.881860i	$-0.343709\pi$
$828$	0	0
$829$	1.02676e10	0.625935	0.312968	−	0.949764i	$-0.398677\pi$
$829$	1.02676e10	0.625935	0.312968	+	0.949764i	$0.398677\pi$
$830$	0	0
$831$	−7.44909e9	−0.450298
$832$	0	0
$833$	− 5.68289e9i	− 0.340653i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.20113e10i	0.708030i
$838$	0	0
$839$	3.91270e9	0.228723	0.114361	−	0.993439i	$-0.463518\pi$
$839$	3.91270e9	0.228723	0.114361	+	0.993439i	$0.463518\pi$
$840$	0	0
$841$	−1.69919e10	−0.985046
$842$	0	0
$843$	1.68068e10i	0.966246i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.94037e9i	− 0.166269i
$848$	0	0
$849$	−8.68622e9	−0.487140
$850$	0	0
$851$	−1.20544e10	−0.670488
$852$	0	0
$853$	− 2.44066e9i	− 0.134643i	−0.997731	−	0.0673217i	$-0.978555\pi$
$853$	− 2.44066e9i	− 0.134643i	0.997731	−	0.0673217i	$-0.0214454\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.01405e10i	1.09304i	0.837446	+	0.546521i	$0.184048\pi$
$857$	2.01405e10i	1.09304i	−0.837446	+	0.546521i	$0.815952\pi$
$858$	0	0
$859$	−1.58900e10	−0.855357	−0.427678	−	0.903931i	$-0.640668\pi$
$859$	−1.58900e10	−0.855357	−0.427678	+	0.903931i	$0.640668\pi$
$860$	0	0
$861$	3.11911e9	0.166541
$862$	0	0
$863$	− 1.25359e10i	− 0.663924i	−0.943293	−	0.331962i	$-0.892289\pi$
$863$	− 1.25359e10i	− 0.663924i	0.943293	−	0.331962i	$-0.107711\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.08623e10i	− 0.566051i
$868$	0	0
$869$	1.47023e9	0.0760007
$870$	0	0
$871$	−1.00068e10	−0.513136
$872$	0	0
$873$	− 1.23740e10i	− 0.629448i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.73627e10i	0.869198i	0.900624	+	0.434599i	$0.143110\pi$
$877$	1.73627e10i	0.869198i	−0.900624	+	0.434599i	$0.856890\pi$
$878$	0	0
$879$	−1.34469e10	−0.667822
$880$	0	0
$881$	3.02287e10	1.48937	0.744687	−	0.667414i	$-0.232599\pi$
$881$	3.02287e10	1.48937	0.744687	+	0.667414i	$0.232599\pi$
$882$	0	0
$883$	1.11570e10i	0.545363i	0.962104	+	0.272682i	$0.0879105\pi$
$883$	1.11570e10i	0.545363i	−0.962104	+	0.272682i	$0.912089\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.41351e10i	1.16123i	0.814179	+	0.580613i	$0.197187\pi$
$887$	2.41351e10i	1.16123i	−0.814179	+	0.580613i	$0.802813\pi$
$888$	0	0
$889$	−5.07807e9	−0.242405
$890$	0	0
$891$	−4.16253e8	−0.0197145
$892$	0	0
$893$	4.13959e10i	1.94526i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 3.81356e9i	− 0.176424i
$898$	0	0
$899$	−1.84489e9	−0.0846862
$900$	0	0
$901$	−9.89617e9	−0.450745
$902$	0	0
$903$	3.37664e8i	0.0152608i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.87637e10i	− 1.72504i	−0.506019	−	0.862522i	$-0.668884\pi$
$907$	− 3.87637e10i	− 1.72504i	0.506019	−	0.862522i	$-0.331116\pi$
$908$	0	0
$909$	−8.01573e9	−0.353973
$910$	0	0
$911$	2.08115e10	0.911990	0.455995	−	0.889982i	$-0.349284\pi$
$911$	2.08115e10	0.911990	0.455995	+	0.889982i	$0.349284\pi$
$912$	0	0
$913$	− 4.52260e9i	− 0.196671i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3.05766e9i	− 0.130947i
$918$	0	0
$919$	4.53215e10	1.92619	0.963097	−	0.269155i	$-0.0867444\pi$
$919$	4.53215e10	1.92619	0.963097	+	0.269155i	$0.0867444\pi$
$920$	0	0
$921$	1.14885e9	0.0484570
$922$	0	0
$923$	8.42582e8i	0.0352701i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 7.09113e9i	− 0.292372i
$928$	0	0
$929$	2.95054e10	1.20739	0.603694	−	0.797216i	$-0.293695\pi$
$929$	2.95054e10	1.20739	0.603694	+	0.797216i	$0.293695\pi$
$930$	0	0
$931$	−2.70776e10	−1.09973
$932$	0	0
$933$	− 6.00902e9i	− 0.242225i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 2.26337e10i	− 0.898810i	−0.893328	−	0.449405i	$-0.851636\pi$
$937$	− 2.26337e10i	− 0.898810i	0.893328	−	0.449405i	$-0.148364\pi$
$938$	0	0
$939$	9.25349e9	0.364734
$940$	0	0
$941$	−8.98563e9	−0.351548	−0.175774	−	0.984431i	$-0.556243\pi$
$941$	−8.98563e9	−0.351548	−0.175774	+	0.984431i	$0.556243\pi$
$942$	0	0
$943$	1.56863e10i	0.609156i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.43732e10i	0.932583i	0.884631	+	0.466291i	$0.154410\pi$
$947$	2.43732e10i	0.932583i	−0.884631	+	0.466291i	$0.845590\pi$
$948$	0	0
$949$	−1.00881e10	−0.383158
$950$	0	0
$951$	3.38253e10	1.27529
$952$	0	0
$953$	3.99263e10i	1.49429i	0.664663	+	0.747143i	$0.268575\pi$
$953$	3.99263e10i	1.49429i	−0.664663	+	0.747143i	$0.731425\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 5.43145e8i	− 0.0200320i
$958$	0	0
$959$	−9.64421e8	−0.0353103
$960$	0	0
$961$	−1.43181e10	−0.520419
$962$	0	0
$963$	2.75532e10i	0.994213i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.48658e10i	1.59559i	0.602926	+	0.797797i	$0.294002\pi$
$967$	4.48658e10i	1.59559i	−0.602926	+	0.797797i	$0.705998\pi$
$968$	0	0
$969$	7.30836e9	0.258040
$970$	0	0
$971$	−2.14651e10	−0.752429	−0.376214	−	0.926533i	$-0.622774\pi$
$971$	−2.14651e10	−0.752429	−0.376214	+	0.926533i	$0.622774\pi$
$972$	0	0
$973$	7.85165e8i	0.0273254i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.17721e10i	− 0.746911i	−0.927648	−	0.373455i	$-0.878173\pi$
$977$	− 2.17721e10i	− 0.746911i	0.927648	−	0.373455i	$-0.121827\pi$
$978$	0	0
$979$	−8.77363e9	−0.298841
$980$	0	0
$981$	−1.18167e10	−0.399627
$982$	0	0
$983$	− 7.58129e9i	− 0.254569i	−0.991866	−	0.127285i	$-0.959374\pi$
$983$	− 7.58129e9i	− 0.254569i	0.991866	−	0.127285i	$-0.0406262\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 5.93967e9i	− 0.196631i
$988$	0	0
$989$	−1.69814e9	−0.0558195
$990$	0	0
$991$	5.67181e10	1.85125	0.925623	−	0.378446i	$-0.123542\pi$
$991$	5.67181e10	1.85125	0.925623	+	0.378446i	$0.123542\pi$
$992$	0	0
$993$	2.09946e10i	0.680432i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.64140e10i	− 0.844115i	−0.906569	−	0.422057i	$-0.861308\pi$
$997$	− 2.64140e10i	− 0.844115i	0.906569	−	0.422057i	$-0.138692\pi$
$998$	0	0
$999$	5.14501e10	1.63270

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.8.c.r.49.2		4
4.3	odd	2		100.8.c.c.49.3		4
5.2	odd	4		400.8.a.be.1.1		2
5.3	odd	4		400.8.a.u.1.2		2
5.4	even	2	inner	400.8.c.r.49.3		4
20.3	even	4		100.8.a.d.1.1	yes	2
20.7	even	4		100.8.a.b.1.2	✓	2
20.19	odd	2		100.8.c.c.49.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.8.a.b.1.2	✓	2	20.7	even	4
100.8.a.d.1.1	yes	2	20.3	even	4
100.8.c.c.49.2		4	20.19	odd	2
100.8.c.c.49.3		4	4.3	odd	2
400.8.a.u.1.2		2	5.3	odd	4
400.8.a.be.1.1		2	5.2	odd	4
400.8.c.r.49.2		4	1.1	even	1	trivial
400.8.c.r.49.3		4	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-30.2096i q^{3} +161.257i q^{7} +1274.38 q^{9} +1119.43 q^{11} -5152.69i q^{13} -7125.53i q^{17} -33951.5 q^{19} +4871.51 q^{21} +24499.2i q^{23} -104567. i q^{27} +16061.1 q^{29} -114867. q^{31} -33817.5i q^{33} +492031. i q^{37} -155661. q^{39} +640276. q^{41} +69314.0i q^{43} -1.21927e6i q^{47} +797539. q^{49} -215259. q^{51} -1.38883e6i q^{53} +1.02566e6i q^{57} -2.49636e6 q^{59} +1.83520e6 q^{61} +205504. i q^{63} -1.94206e6i q^{67} +740110. q^{69} -163523. q^{71} -1.95783e6i q^{73} +180516. i q^{77} +1.31338e6 q^{79} -371844. q^{81} -4.04009e6i q^{83} -485198. i q^{87} -7.83758e6 q^{89} +830909. q^{91} +3.47009e6i q^{93} -9.70980e6i q^{97} +1.42658e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2936 q^{9} - 4560 q^{11} + 41936 q^{19} + 71704 q^{21} + 461904 q^{29} - 429344 q^{31} - 1787504 q^{39} + 898164 q^{41} + 2852748 q^{49} - 5521488 q^{51} - 1887648 q^{59} + 2532728 q^{61} + 3370152 q^{69}+ \cdots + 21498240 q^{99}+O(q^{100}) \) 4 * q - 2936 * q^9 - 4560 * q^11 + 41936 * q^19 + 71704 * q^21 + 461904 * q^29 - 429344 * q^31 - 1787504 * q^39 + 898164 * q^41 + 2852748 * q^49 - 5521488 * q^51 - 1887648 * q^59 + 2532728 * q^61 + 3370152 * q^69 - 7233552 * q^71 - 6079792 * q^79 - 7263484 * q^81 - 6478524 * q^89 + 10939424 * q^91 + 21498240 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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