LMFDB - Embedded newform 400.10.c.q.49.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,10,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	400.c (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:	$206.014334466$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 1305x^{4} + 433104x^{2} + 16000000 $ x^6 + 1305x^4 + 433104x^2 + 16000000
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{8} $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.5
Root			$-22.2334i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.10.c.q.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+210.171i q^{3} +9905.49i q^{7} -24489.0 q^{9} +O(q^{10})$	$q+210.171i q^{3} +9905.49i q^{7} -24489.0 q^{9} -36453.6 q^{11} +164867. i q^{13} -82357.1i q^{17} -609617. q^{19} -2.08185e6 q^{21} -1.88578e6i q^{23} -1.01008e6i q^{27} -339235. q^{29} -547314. q^{31} -7.66150e6i q^{33} -5.25687e6i q^{37} -3.46503e7 q^{39} +2.05812e6 q^{41} +6.76158e6i q^{43} +3.15241e7i q^{47} -5.77651e7 q^{49} +1.73091e7 q^{51} -4.89593e7i q^{53} -1.28124e8i q^{57} +8.77960e7 q^{59} +3.84654e7 q^{61} -2.42575e8i q^{63} -1.36116e8i q^{67} +3.96337e8 q^{69} -3.49218e8 q^{71} +1.61345e8i q^{73} -3.61091e8i q^{77} -1.26975e8 q^{79} -2.69727e8 q^{81} +2.87494e8i q^{83} -7.12976e7i q^{87} +5.63133e8 q^{89} -1.63309e9 q^{91} -1.15030e8i q^{93} +4.71704e8i q^{97} +8.92711e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 116468 q^{9}+O(q^{10}) $ 6 * q - 116468 * q^9	$ 6 q - 116468 q^{9} + 109398 q^{11} + 1637690 q^{19} - 4750788 q^{21} - 4350960 q^{29} - 8548132 q^{31} - 100828184 q^{39} + 11852622 q^{41} + 12907858 q^{49} + 51269398 q^{51} + 11341920 q^{59} + 250613852 q^{61} + 628549884 q^{69} - 595101192 q^{71} - 620050340 q^{79} + 2797694726 q^{81} + 2207720070 q^{89} - 2366375312 q^{91} - 1784469044 q^{99}+O(q^{100}) $ 6 * q - 116468 * q^9 + 109398 * q^11 + 1637690 * q^19 - 4750788 * q^21 - 4350960 * q^29 - 8548132 * q^31 - 100828184 * q^39 + 11852622 * q^41 + 12907858 * q^49 + 51269398 * q^51 + 11341920 * q^59 + 250613852 * q^61 + 628549884 * q^69 - 595101192 * q^71 - 620050340 * q^79 + 2797694726 * q^81 + 2207720070 * q^89 - 2366375312 * q^91 - 1784469044 * q^99

Display 100 coefficients 10 coefficients

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
$3$	210.171i	1.49805i	0.662539	+	0.749027i	$0.269479\pi$
$3$	210.171i	1.49805i	−0.662539	+	0.749027i	$0.730521\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	9905.49i	1.55932i	0.626204	+	0.779659i	$0.284608\pi$
$7$	9905.49i	1.55932i	−0.626204	+	0.779659i	$0.715392\pi$
$8$	0	0
$9$	−24489.0	−1.24417
$10$	0	0
$11$	−36453.6	−0.750712	−0.375356	−	0.926881i	$-0.622480\pi$
$11$	−36453.6	−0.750712	−0.375356	+	0.926881i	$0.622480\pi$
$12$	0	0
$13$	164867.i	1.60099i	0.599341	+	0.800494i	$0.295429\pi$
$13$	164867.i	1.60099i	−0.599341	+	0.800494i	$0.704571\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 82357.1i	− 0.239156i	−0.992825	−	0.119578i	$-0.961846\pi$
$17$	− 82357.1i	− 0.239156i	0.992825	−	0.119578i	$-0.0381541\pi$
$18$	0	0
$19$	−609617.	−1.07316	−0.536582	−	0.843848i	$-0.680285\pi$
$19$	−609617.	−1.07316	−0.536582	+	0.843848i	$0.680285\pi$
$20$	0	0
$21$	−2.08185e6	−2.33594
$22$	0	0
$23$	− 1.88578e6i	− 1.40513i	−0.711620	−	0.702564i	$-0.752038\pi$
$23$	− 1.88578e6i	− 1.40513i	0.711620	−	0.702564i	$-0.247962\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.01008e6i	− 0.365778i
$28$	0	0
$29$	−339235.	−0.0890657	−0.0445328	−	0.999008i	$-0.514180\pi$
$29$	−339235.	−0.0890657	−0.0445328	+	0.999008i	$0.514180\pi$
$30$	0	0
$31$	−547314.	−0.106441	−0.0532205	−	0.998583i	$-0.516949\pi$
$31$	−547314.	−0.106441	−0.0532205	+	0.998583i	$0.516949\pi$
$32$	0	0
$33$	− 7.66150e6i	− 1.12461i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5.25687e6i	− 0.461126i	−0.973057	−	0.230563i	$-0.925943\pi$
$37$	− 5.25687e6i	− 0.461126i	0.973057	−	0.230563i	$-0.0740567\pi$
$38$	0	0
$39$	−3.46503e7	−2.39837
$40$	0	0
$41$	2.05812e6	0.113748	0.0568739	−	0.998381i	$-0.481887\pi$
$41$	2.05812e6	0.113748	0.0568739	+	0.998381i	$0.481887\pi$
$42$	0	0
$43$	6.76158e6i	0.301606i	0.988564	+	0.150803i	$0.0481859\pi$
$43$	6.76158e6i	0.301606i	−0.988564	+	0.150803i	$0.951814\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.15241e7i	0.942330i	0.882045	+	0.471165i	$0.156166\pi$
$47$	3.15241e7i	0.942330i	−0.882045	+	0.471165i	$0.843834\pi$
$48$	0	0
$49$	−5.77651e7	−1.43147
$50$	0	0
$51$	1.73091e7	0.358268
$52$	0	0
$53$	− 4.89593e7i	− 0.852303i	−0.904652	−	0.426151i	$-0.859869\pi$
$53$	− 4.89593e7i	− 0.852303i	0.904652	−	0.426151i	$-0.140131\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.28124e8i	− 1.60766i
$58$	0	0
$59$	8.77960e7	0.943281	0.471640	−	0.881791i	$-0.343662\pi$
$59$	8.77960e7	0.943281	0.471640	+	0.881791i	$0.343662\pi$
$60$	0	0
$61$	3.84654e7	0.355701	0.177851	−	0.984057i	$-0.443086\pi$
$61$	3.84654e7	0.355701	0.177851	+	0.984057i	$0.443086\pi$
$62$	0	0
$63$	− 2.42575e8i	− 1.94005i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 1.36116e8i	− 0.825227i	−0.910906	−	0.412613i	$-0.864616\pi$
$67$	− 1.36116e8i	− 0.825227i	0.910906	−	0.412613i	$-0.135384\pi$
$68$	0	0
$69$	3.96337e8	2.10496
$70$	0	0
$71$	−3.49218e8	−1.63092	−0.815462	−	0.578810i	$-0.803517\pi$
$71$	−3.49218e8	−1.63092	−0.815462	+	0.578810i	$0.803517\pi$
$72$	0	0
$73$	1.61345e8i	0.664971i	0.943108	+	0.332486i	$0.107887\pi$
$73$	1.61345e8i	0.664971i	−0.943108	+	0.332486i	$0.892113\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.61091e8i	− 1.17060i
$78$	0	0
$79$	−1.26975e8	−0.366772	−0.183386	−	0.983041i	$-0.558706\pi$
$79$	−1.26975e8	−0.366772	−0.183386	+	0.983041i	$0.558706\pi$
$80$	0	0
$81$	−2.69727e8	−0.696213
$82$	0	0
$83$	2.87494e8i	0.664931i	0.943115	+	0.332466i	$0.107881\pi$
$83$	2.87494e8i	0.664931i	−0.943115	+	0.332466i	$0.892119\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 7.12976e7i	− 0.133425i
$88$	0	0
$89$	5.63133e8	0.951385	0.475692	−	0.879612i	$-0.342198\pi$
$89$	5.63133e8	0.951385	0.475692	+	0.879612i	$0.342198\pi$
$90$	0	0
$91$	−1.63309e9	−2.49645
$92$	0	0
$93$	− 1.15030e8i	− 0.159454i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.71704e8i	0.541000i	0.962720	+	0.270500i	$0.0871890\pi$
$97$	4.71704e8i	0.541000i	−0.962720	+	0.270500i	$0.912811\pi$
$98$	0	0
$99$	8.92711e8	0.934012
$100$	0	0
$101$	−1.25673e9	−1.20170	−0.600851	−	0.799361i	$-0.705172\pi$
$101$	−1.25673e9	−1.20170	−0.600851	+	0.799361i	$0.705172\pi$
$102$	0	0
$103$	− 1.95398e9i	− 1.71062i	−0.518119	−	0.855309i	$-0.673367\pi$
$103$	− 1.95398e9i	− 1.71062i	0.518119	−	0.855309i	$-0.326633\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.12306e9i	1.56580i	0.622148	+	0.782899i	$0.286260\pi$
$107$	2.12306e9i	1.56580i	−0.622148	+	0.782899i	$0.713740\pi$
$108$	0	0
$109$	1.18863e8	0.0806544	0.0403272	−	0.999187i	$-0.487160\pi$
$109$	1.18863e8	0.0806544	0.0403272	+	0.999187i	$0.487160\pi$
$110$	0	0
$111$	1.10484e9	0.690791
$112$	0	0
$113$	− 1.85455e9i	− 1.07001i	−0.844850	−	0.535003i	$-0.820311\pi$
$113$	− 1.85455e9i	− 1.07001i	0.844850	−	0.535003i	$-0.179689\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4.03742e9i	− 1.99190i
$118$	0	0
$119$	8.15787e8	0.372920
$120$	0	0
$121$	−1.02908e9	−0.436432
$122$	0	0
$123$	4.32557e8i	0.170400i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 5.39208e9i	− 1.83924i	−0.392805	−	0.919622i	$-0.628495\pi$
$127$	− 5.39208e9i	− 1.83924i	0.392805	−	0.919622i	$-0.371505\pi$
$128$	0	0
$129$	−1.42109e9	−0.451823
$130$	0	0
$131$	−3.83690e9	−1.13831	−0.569154	−	0.822231i	$-0.692729\pi$
$131$	−3.83690e9	−1.13831	−0.569154	+	0.822231i	$0.692729\pi$
$132$	0	0
$133$	− 6.03856e9i	− 1.67340i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.55408e9i	0.619430i	0.950829	+	0.309715i	$0.100234\pi$
$137$	2.55408e9i	0.619430i	−0.950829	+	0.309715i	$0.899766\pi$
$138$	0	0
$139$	−4.09121e9	−0.929577	−0.464789	−	0.885422i	$-0.653870\pi$
$139$	−4.09121e9	−0.929577	−0.464789	+	0.885422i	$0.653870\pi$
$140$	0	0
$141$	−6.62547e9	−1.41166
$142$	0	0
$143$	− 6.00999e9i	− 1.20188i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.21406e10i	− 2.14443i
$148$	0	0
$149$	6.13643e9	1.01995	0.509974	−	0.860190i	$-0.329655\pi$
$149$	6.13643e9	1.01995	0.509974	+	0.860190i	$0.329655\pi$
$150$	0	0
$151$	5.83754e9	0.913763	0.456882	−	0.889528i	$-0.348966\pi$
$151$	5.83754e9	0.913763	0.456882	+	0.889528i	$0.348966\pi$
$152$	0	0
$153$	2.01684e9i	0.297550i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.35031e9i	0.177371i	0.996060	+	0.0886857i	$0.0282667\pi$
$157$	1.35031e9i	0.177371i	−0.996060	+	0.0886857i	$0.971733\pi$
$158$	0	0
$159$	1.02898e10	1.27680
$160$	0	0
$161$	1.86796e10	2.19104
$162$	0	0
$163$	4.50806e9i	0.500202i	0.968220	+	0.250101i	$0.0804638\pi$
$163$	4.50806e9i	0.500202i	−0.968220	+	0.250101i	$0.919536\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.81836e10i	1.80908i	0.426393	+	0.904538i	$0.359784\pi$
$167$	1.81836e10i	1.80908i	−0.426393	+	0.904538i	$0.640216\pi$
$168$	0	0
$169$	−1.65766e10	−1.56316
$170$	0	0
$171$	1.49289e10	1.33520
$172$	0	0
$173$	9.77143e9i	0.829375i	0.909964	+	0.414687i	$0.136109\pi$
$173$	9.77143e9i	0.829375i	−0.909964	+	0.414687i	$0.863891\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.84522e10i	1.41309i
$178$	0	0
$179$	6.65866e9	0.484784	0.242392	−	0.970178i	$-0.422068\pi$
$179$	6.65866e9	0.484784	0.242392	+	0.970178i	$0.422068\pi$
$180$	0	0
$181$	−7.32873e9	−0.507546	−0.253773	−	0.967264i	$-0.581672\pi$
$181$	−7.32873e9	−0.507546	−0.253773	+	0.967264i	$0.581672\pi$
$182$	0	0
$183$	8.08432e9i	0.532860i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.00221e9i	0.179537i
$188$	0	0
$189$	1.00053e10	0.570364
$190$	0	0
$191$	−8.06956e9	−0.438732	−0.219366	−	0.975643i	$-0.570399\pi$
$191$	−8.06956e9	−0.438732	−0.219366	+	0.975643i	$0.570399\pi$
$192$	0	0
$193$	3.12603e9i	0.162176i	0.996707	+	0.0810878i	$0.0258394\pi$
$193$	3.12603e9i	0.162176i	−0.996707	+	0.0810878i	$0.974161\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.40133e10i	1.13594i	0.823051	+	0.567968i	$0.192270\pi$
$197$	2.40133e10i	1.13594i	−0.823051	+	0.567968i	$0.807730\pi$
$198$	0	0
$199$	9.73605e9	0.440093	0.220046	−	0.975489i	$-0.429379\pi$
$199$	9.73605e9	0.440093	0.220046	+	0.975489i	$0.429379\pi$
$200$	0	0
$201$	2.86077e10	1.23623
$202$	0	0
$203$	− 3.36029e9i	− 0.138882i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.61808e10i	1.74822i
$208$	0	0
$209$	2.22227e10	0.805637
$210$	0	0
$211$	2.39305e10	0.831151	0.415576	−	0.909559i	$-0.363580\pi$
$211$	2.39305e10	0.831151	0.415576	+	0.909559i	$0.363580\pi$
$212$	0	0
$213$	− 7.33956e10i	− 2.44321i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5.42141e9i	− 0.165975i
$218$	0	0
$219$	−3.39101e10	−0.996163
$220$	0	0
$221$	1.35779e10	0.382885
$222$	0	0
$223$	− 2.02289e10i	− 0.547773i	−0.961762	−	0.273886i	$-0.911691\pi$
$223$	− 2.02289e10i	− 0.547773i	0.961762	−	0.273886i	$-0.0883092\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.42152e10i	1.10524i	0.833434	+	0.552618i	$0.186371\pi$
$227$	4.42152e10i	1.10524i	−0.833434	+	0.552618i	$0.813629\pi$
$228$	0	0
$229$	4.89483e10	1.17619	0.588096	−	0.808791i	$-0.299878\pi$
$229$	4.89483e10	1.17619	0.588096	+	0.808791i	$0.299878\pi$
$230$	0	0
$231$	7.58909e10	1.75362
$232$	0	0
$233$	6.68132e10i	1.48512i	0.669781	+	0.742559i	$0.266388\pi$
$233$	6.68132e10i	1.48512i	−0.669781	+	0.742559i	$0.733612\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2.66865e10i	− 0.549444i
$238$	0	0
$239$	−2.21153e10	−0.438431	−0.219216	−	0.975676i	$-0.570350\pi$
$239$	−2.21153e10	−0.438431	−0.219216	+	0.975676i	$0.570350\pi$
$240$	0	0
$241$	9.97637e10	1.90500	0.952502	−	0.304532i	$-0.0985002\pi$
$241$	9.97637e10	1.90500	0.952502	+	0.304532i	$0.0985002\pi$
$242$	0	0
$243$	− 7.65703e10i	− 1.40874i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.00506e11i	− 1.71812i
$248$	0	0
$249$	−6.04229e10	−0.996104
$250$	0	0
$251$	4.97276e10	0.790799	0.395399	−	0.918509i	$-0.370606\pi$
$251$	4.97276e10	0.790799	0.395399	+	0.918509i	$0.370606\pi$
$252$	0	0
$253$	6.87435e10i	1.05485i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 5.81308e10i	− 0.831204i	−0.909547	−	0.415602i	$-0.863571\pi$
$257$	− 5.81308e10i	− 0.831204i	0.909547	−	0.415602i	$-0.136429\pi$
$258$	0	0
$259$	5.20718e10	0.719041
$260$	0	0
$261$	8.30753e9	0.110813
$262$	0	0
$263$	3.23476e10i	0.416909i	0.978032	+	0.208454i	$0.0668433\pi$
$263$	3.23476e10i	0.416909i	−0.978032	+	0.208454i	$0.933157\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.18354e11i	1.42523i
$268$	0	0
$269$	−6.24220e9	−0.0726863	−0.0363431	−	0.999339i	$-0.511571\pi$
$269$	−6.24220e9	−0.0726863	−0.0363431	+	0.999339i	$0.511571\pi$
$270$	0	0
$271$	−8.05816e10	−0.907557	−0.453779	−	0.891114i	$-0.649924\pi$
$271$	−8.05816e10	−0.907557	−0.453779	+	0.891114i	$0.649924\pi$
$272$	0	0
$273$	− 3.43228e11i	− 3.73982i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.07976e10i	0.722536i	0.932462	+	0.361268i	$0.117656\pi$
$277$	7.07976e10i	0.722536i	−0.932462	+	0.361268i	$0.882344\pi$
$278$	0	0
$279$	1.34031e10	0.132430
$280$	0	0
$281$	8.15619e10	0.780385	0.390193	−	0.920733i	$-0.372408\pi$
$281$	8.15619e10	0.780385	0.390193	+	0.920733i	$0.372408\pi$
$282$	0	0
$283$	− 1.89301e11i	− 1.75434i	−0.480184	−	0.877168i	$-0.659430\pi$
$283$	− 1.89301e11i	− 1.75434i	0.480184	−	0.877168i	$-0.340570\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.03867e10i	0.177369i
$288$	0	0
$289$	1.11805e11	0.942805
$290$	0	0
$291$	−9.91387e10	−0.810447
$292$	0	0
$293$	− 5.39489e10i	− 0.427640i	−0.976873	−	0.213820i	$-0.931409\pi$
$293$	− 5.39489e10i	− 0.427640i	0.976873	−	0.213820i	$-0.0685906\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.68209e10i	0.274594i
$298$	0	0
$299$	3.10903e11	2.24959
$300$	0	0
$301$	−6.69768e10	−0.470300
$302$	0	0
$303$	− 2.64129e11i	− 1.80022i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 1.30037e11i	− 0.835496i	−0.908563	−	0.417748i	$-0.862819\pi$
$307$	− 1.30037e11i	− 0.835496i	0.908563	−	0.417748i	$-0.137181\pi$
$308$	0	0
$309$	4.10671e11	2.56260
$310$	0	0
$311$	−2.09580e9	−0.0127036	−0.00635182	−	0.999980i	$-0.502022\pi$
$311$	−2.09580e9	−0.0127036	−0.00635182	+	0.999980i	$0.502022\pi$
$312$	0	0
$313$	− 2.43788e11i	− 1.43570i	−0.696200	−	0.717848i	$-0.745127\pi$
$313$	− 2.43788e11i	− 1.43570i	0.696200	−	0.717848i	$-0.254873\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.46483e11i	1.92715i	0.267445	+	0.963573i	$0.413821\pi$
$317$	3.46483e11i	1.92715i	−0.267445	+	0.963573i	$0.586179\pi$
$318$	0	0
$319$	1.23664e10	0.0668626
$320$	0	0
$321$	−4.46207e11	−2.34565
$322$	0	0
$323$	5.02063e10i	0.256653i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.49816e10i	0.120825i
$328$	0	0
$329$	−3.12262e11	−1.46939
$330$	0	0
$331$	1.43303e11	0.656189	0.328094	−	0.944645i	$-0.393594\pi$
$331$	1.43303e11	0.656189	0.328094	+	0.944645i	$0.393594\pi$
$332$	0	0
$333$	1.28735e11i	0.573718i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 3.34820e11i	− 1.41409i	−0.707170	−	0.707044i	$-0.750028\pi$
$337$	− 3.34820e11i	− 1.41409i	0.707170	−	0.707044i	$-0.249972\pi$
$338$	0	0
$339$	3.89773e11	1.60293
$340$	0	0
$341$	1.99515e10	0.0799064
$342$	0	0
$343$	− 1.72469e11i	− 0.672804i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9.22101e10i	− 0.341425i	−0.985321	−	0.170713i	$-0.945393\pi$
$347$	− 9.22101e10i	− 0.341425i	0.985321	−	0.170713i	$-0.0546070\pi$
$348$	0	0
$349$	4.71738e11	1.70211	0.851053	−	0.525080i	$-0.175965\pi$
$349$	4.71738e11	1.70211	0.851053	+	0.525080i	$0.175965\pi$
$350$	0	0
$351$	1.66528e11	0.585606
$352$	0	0
$353$	− 4.01029e11i	− 1.37464i	−0.726353	−	0.687322i	$-0.758786\pi$
$353$	− 4.01029e11i	− 1.37464i	0.726353	−	0.687322i	$-0.241214\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.71455e11i	0.558654i
$358$	0	0
$359$	−2.36110e10	−0.0750220	−0.0375110	−	0.999296i	$-0.511943\pi$
$359$	−2.36110e10	−0.0750220	−0.0375110	+	0.999296i	$0.511943\pi$
$360$	0	0
$361$	4.89457e10	0.151681
$362$	0	0
$363$	− 2.16284e11i	− 0.653799i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.23035e11i	0.354022i	0.984209	+	0.177011i	$0.0566427\pi$
$367$	1.23035e11i	0.354022i	−0.984209	+	0.177011i	$0.943357\pi$
$368$	0	0
$369$	−5.04012e10	−0.141521
$370$	0	0
$371$	4.84966e11	1.32901
$372$	0	0
$373$	− 2.39248e11i	− 0.639968i	−0.947423	−	0.319984i	$-0.896322\pi$
$373$	− 2.39248e11i	− 0.639968i	0.947423	−	0.319984i	$-0.103678\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5.59287e10i	− 0.142593i
$378$	0	0
$379$	−5.90629e11	−1.47041	−0.735205	−	0.677845i	$-0.762914\pi$
$379$	−5.90629e11	−1.47041	−0.735205	+	0.677845i	$0.762914\pi$
$380$	0	0
$381$	1.13326e12	2.75529
$382$	0	0
$383$	5.30525e10i	0.125983i	0.998014	+	0.0629914i	$0.0200641\pi$
$383$	5.30525e10i	0.125983i	−0.998014	+	0.0629914i	$0.979936\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.65584e11i	− 0.375249i
$388$	0	0
$389$	−3.10395e11	−0.687293	−0.343646	−	0.939099i	$-0.611662\pi$
$389$	−3.10395e11	−0.687293	−0.343646	+	0.939099i	$0.611662\pi$
$390$	0	0
$391$	−1.55307e11	−0.336045
$392$	0	0
$393$	− 8.06407e11i	− 1.70525i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.09351e11i	0.422979i	0.977380	+	0.211489i	$0.0678314\pi$
$397$	2.09351e11i	0.422979i	−0.977380	+	0.211489i	$0.932169\pi$
$398$	0	0
$399$	1.26913e12	2.50685
$400$	0	0
$401$	4.43868e11	0.857244	0.428622	−	0.903484i	$-0.358999\pi$
$401$	4.43868e11	0.857244	0.428622	+	0.903484i	$0.358999\pi$
$402$	0	0
$403$	− 9.02338e10i	− 0.170411i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.91632e11i	0.346172i
$408$	0	0
$409$	−4.56031e11	−0.805823	−0.402912	−	0.915239i	$-0.632002\pi$
$409$	−4.56031e11	−0.805823	−0.402912	+	0.915239i	$0.632002\pi$
$410$	0	0
$411$	−5.36794e11	−0.927940
$412$	0	0
$413$	8.69663e11i	1.47087i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 8.59856e11i	− 1.39256i
$418$	0	0
$419$	−1.12392e12	−1.78144	−0.890720	−	0.454552i	$-0.849799\pi$
$419$	−1.12392e12	−1.78144	−0.890720	+	0.454552i	$0.849799\pi$
$420$	0	0
$421$	−1.02822e12	−1.59521	−0.797605	−	0.603180i	$-0.793900\pi$
$421$	−1.02822e12	−1.59521	−0.797605	+	0.603180i	$0.793900\pi$
$422$	0	0
$423$	− 7.71994e11i	− 1.17242i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.81018e11i	0.554652i
$428$	0	0
$429$	1.26313e12	1.80048
$430$	0	0
$431$	1.37714e12	1.92234	0.961171	−	0.275953i	$-0.0889934\pi$
$431$	1.37714e12	1.92234	0.961171	+	0.275953i	$0.0889934\pi$
$432$	0	0
$433$	1.94790e11i	0.266299i	0.991096	+	0.133150i	$0.0425091\pi$
$433$	1.94790e11i	0.266299i	−0.991096	+	0.133150i	$0.957491\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.14961e12i	1.50793i
$438$	0	0
$439$	−5.32533e11	−0.684315	−0.342157	−	0.939643i	$-0.611158\pi$
$439$	−5.32533e11	−0.684315	−0.342157	+	0.939643i	$0.611158\pi$
$440$	0	0
$441$	1.41461e12	1.78099
$442$	0	0
$443$	− 7.26633e11i	− 0.896393i	−0.893935	−	0.448196i	$-0.852067\pi$
$443$	− 7.26633e11i	− 0.896393i	0.893935	−	0.448196i	$-0.147933\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.28970e12i	1.52794i
$448$	0	0
$449$	−9.34830e11	−1.08549	−0.542743	−	0.839899i	$-0.682614\pi$
$449$	−9.34830e11	−1.08549	−0.542743	+	0.839899i	$0.682614\pi$
$450$	0	0
$451$	−7.50258e10	−0.0853918
$452$	0	0
$453$	1.22688e12i	1.36887i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.09190e12i	1.17101i	0.810670	+	0.585503i	$0.199103\pi$
$457$	1.09190e12i	1.17101i	−0.810670	+	0.585503i	$0.800897\pi$
$458$	0	0
$459$	−8.31870e10	−0.0874779
$460$	0	0
$461$	9.93683e11	1.02469	0.512347	−	0.858779i	$-0.328776\pi$
$461$	9.93683e11	1.02469	0.512347	+	0.858779i	$0.328776\pi$
$462$	0	0
$463$	6.95734e11i	0.703605i	0.936074	+	0.351802i	$0.114431\pi$
$463$	6.95734e11i	0.703605i	−0.936074	+	0.351802i	$0.885569\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.87957e12i	1.82866i	0.404975	+	0.914328i	$0.367280\pi$
$467$	1.87957e12i	1.82866i	−0.404975	+	0.914328i	$0.632720\pi$
$468$	0	0
$469$	1.34830e12	1.28679
$470$	0	0
$471$	−2.83795e11	−0.265712
$472$	0	0
$473$	− 2.46484e11i	− 0.226419i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.19896e12i	1.06041i
$478$	0	0
$479$	−2.29011e11	−0.198768	−0.0993838	−	0.995049i	$-0.531687\pi$
$479$	−2.29011e11	−0.198768	−0.0993838	+	0.995049i	$0.531687\pi$
$480$	0	0
$481$	8.66683e11	0.738256
$482$	0	0
$483$	3.92591e12i	3.28230i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 4.81410e11i	− 0.387824i	−0.981019	−	0.193912i	$-0.937882\pi$
$487$	− 4.81410e11i	− 0.387824i	0.981019	−	0.193912i	$-0.0621177\pi$
$488$	0	0
$489$	−9.47465e11	−0.749330
$490$	0	0
$491$	1.70320e12	1.32251	0.661257	−	0.750160i	$-0.270023\pi$
$491$	1.70320e12	1.32251	0.661257	+	0.750160i	$0.270023\pi$
$492$	0	0
$493$	2.79384e10i	0.0213006i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.45917e12i	− 2.54313i
$498$	0	0
$499$	2.06075e12	1.48789	0.743947	−	0.668238i	$-0.232951\pi$
$499$	2.06075e12	1.48789	0.743947	+	0.668238i	$0.232951\pi$
$500$	0	0
$501$	−3.82168e12	−2.71010
$502$	0	0
$503$	5.97493e11i	0.416176i	0.978110	+	0.208088i	$0.0667240\pi$
$503$	5.97493e11i	0.416176i	−0.978110	+	0.208088i	$0.933276\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 3.48392e12i	− 2.34170i
$508$	0	0
$509$	8.46938e11	0.559270	0.279635	−	0.960106i	$-0.409787\pi$
$509$	8.46938e11	0.559270	0.279635	+	0.960106i	$0.409787\pi$
$510$	0	0
$511$	−1.59820e12	−1.03690
$512$	0	0
$513$	6.15760e11i	0.392540i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.14917e12i	− 0.707418i
$518$	0	0
$519$	−2.05367e12	−1.24245
$520$	0	0
$521$	7.31013e11	0.434666	0.217333	−	0.976098i	$-0.430264\pi$
$521$	7.31013e11	0.434666	0.217333	+	0.976098i	$0.430264\pi$
$522$	0	0
$523$	− 1.04453e12i	− 0.610467i	−0.952278	−	0.305233i	$-0.901266\pi$
$523$	− 1.04453e12i	− 0.610467i	0.952278	−	0.305233i	$-0.0987345\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.50751e10i	0.0254559i
$528$	0	0
$529$	−1.75502e12	−0.974386
$530$	0	0
$531$	−2.15003e12	−1.17360
$532$	0	0
$533$	3.39315e11i	0.182109i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.39946e12i	0.726233i
$538$	0	0
$539$	2.10575e12	1.07462
$540$	0	0
$541$	−1.88034e12	−0.943733	−0.471866	−	0.881670i	$-0.656420\pi$
$541$	−1.88034e12	−0.943733	−0.471866	+	0.881670i	$0.656420\pi$
$542$	0	0
$543$	− 1.54029e12i	− 0.760331i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.08624e12i	− 0.518780i	−0.965773	−	0.259390i	$-0.916479\pi$
$547$	− 1.08624e12i	− 0.518780i	0.965773	−	0.259390i	$-0.0835215\pi$
$548$	0	0
$549$	−9.41977e11	−0.442553
$550$	0	0
$551$	2.06804e11	0.0955821
$552$	0	0
$553$	− 1.25775e12i	− 0.571914i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.11965e12i	− 0.933072i	−0.884502	−	0.466536i	$-0.845502\pi$
$557$	− 2.11965e12i	− 0.933072i	0.884502	−	0.466536i	$-0.154498\pi$
$558$	0	0
$559$	−1.11476e12	−0.482868
$560$	0	0
$561$	−6.30978e11	−0.268956
$562$	0	0
$563$	− 3.04489e12i	− 1.27727i	−0.769509	−	0.638636i	$-0.779499\pi$
$563$	− 3.04489e12i	− 1.27727i	0.769509	−	0.638636i	$-0.220501\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.67178e12i	− 1.08562i
$568$	0	0
$569$	−3.25275e12	−1.30090	−0.650452	−	0.759548i	$-0.725420\pi$
$569$	−3.25275e12	−1.30090	−0.650452	+	0.759548i	$0.725420\pi$
$570$	0	0
$571$	1.35916e12	0.535066	0.267533	−	0.963549i	$-0.413792\pi$
$571$	1.35916e12	0.535066	0.267533	+	0.963549i	$0.413792\pi$
$572$	0	0
$573$	− 1.69599e12i	− 0.657245i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.35782e12i	0.885564i	0.896629	+	0.442782i	$0.146008\pi$
$577$	2.35782e12i	0.885564i	−0.896629	+	0.442782i	$0.853992\pi$
$578$	0	0
$579$	−6.57002e11	−0.242948
$580$	0	0
$581$	−2.84777e12	−1.03684
$582$	0	0
$583$	1.78474e12i	0.639834i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.37065e12i	− 0.824130i	−0.911154	−	0.412065i	$-0.864808\pi$
$587$	− 2.37065e12i	− 0.824130i	0.911154	−	0.412065i	$-0.135192\pi$
$588$	0	0
$589$	3.33652e11	0.114229
$590$	0	0
$591$	−5.04690e12	−1.70169
$592$	0	0
$593$	4.56707e12i	1.51667i	0.651864	+	0.758336i	$0.273987\pi$
$593$	4.56707e12i	1.51667i	−0.651864	+	0.758336i	$0.726013\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.04624e12i	0.659283i
$598$	0	0
$599$	−5.30493e12	−1.68368	−0.841839	−	0.539729i	$-0.818527\pi$
$599$	−5.30493e12	−1.68368	−0.841839	+	0.539729i	$0.818527\pi$
$600$	0	0
$601$	−2.71307e12	−0.848255	−0.424128	−	0.905602i	$-0.639419\pi$
$601$	−2.71307e12	−0.848255	−0.424128	+	0.905602i	$0.639419\pi$
$602$	0	0
$603$	3.33335e12i	1.02672i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 4.00734e12i	− 1.19814i	−0.800698	−	0.599069i	$-0.795538\pi$
$607$	− 4.00734e12i	− 1.19814i	0.800698	−	0.599069i	$-0.204462\pi$
$608$	0	0
$609$	7.06237e11	0.208052
$610$	0	0
$611$	−5.19728e12	−1.50866
$612$	0	0
$613$	− 2.24954e12i	− 0.643459i	−0.946832	−	0.321730i	$-0.895736\pi$
$613$	− 2.24954e12i	− 0.643459i	0.946832	−	0.321730i	$-0.104264\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 1.19664e12i	− 0.332414i	−0.986091	−	0.166207i	$-0.946848\pi$
$617$	− 1.19664e12i	− 0.332414i	0.986091	−	0.166207i	$-0.0531519\pi$
$618$	0	0
$619$	−4.36723e12	−1.19563	−0.597817	−	0.801633i	$-0.703965\pi$
$619$	−4.36723e12	−1.19563	−0.597817	+	0.801633i	$0.703965\pi$
$620$	0	0
$621$	−1.90478e12	−0.513965
$622$	0	0
$623$	5.57811e12i	1.48351i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.67058e12i	1.20689i
$628$	0	0
$629$	−4.32940e11	−0.110281
$630$	0	0
$631$	−2.78790e12	−0.700077	−0.350039	−	0.936735i	$-0.613832\pi$
$631$	−2.78790e12	−0.700077	−0.350039	+	0.936735i	$0.613832\pi$
$632$	0	0
$633$	5.02950e12i	1.24511i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 9.52355e12i	− 2.29177i
$638$	0	0
$639$	8.55199e12	2.02915
$640$	0	0
$641$	−2.28295e12	−0.534115	−0.267058	−	0.963681i	$-0.586051\pi$
$641$	−2.28295e12	−0.534115	−0.267058	+	0.963681i	$0.586051\pi$
$642$	0	0
$643$	2.29995e12i	0.530603i	0.964166	+	0.265301i	$0.0854715\pi$
$643$	2.29995e12i	0.530603i	−0.964166	+	0.265301i	$0.914529\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.85647e12i	0.416503i	0.978075	+	0.208251i	$0.0667772\pi$
$647$	1.85647e12i	0.416503i	−0.978075	+	0.208251i	$0.933223\pi$
$648$	0	0
$649$	−3.20048e12	−0.708132
$650$	0	0
$651$	1.13942e12	0.248640
$652$	0	0
$653$	7.75397e12i	1.66884i	0.551129	+	0.834420i	$0.314197\pi$
$653$	7.75397e12i	1.66884i	−0.551129	+	0.834420i	$0.685803\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3.95117e12i	− 0.827336i
$658$	0	0
$659$	7.42166e12	1.53291	0.766455	−	0.642298i	$-0.222019\pi$
$659$	7.42166e12	1.53291	0.766455	+	0.642298i	$0.222019\pi$
$660$	0	0
$661$	5.79783e12	1.18130	0.590648	−	0.806929i	$-0.298872\pi$
$661$	5.79783e12	1.18130	0.590648	+	0.806929i	$0.298872\pi$
$662$	0	0
$663$	2.85369e12i	0.573583i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.39724e11i	0.125149i
$668$	0	0
$669$	4.25153e12	0.820593
$670$	0	0
$671$	−1.40220e12	−0.267029
$672$	0	0
$673$	− 6.32456e12i	− 1.18840i	−0.804318	−	0.594200i	$-0.797469\pi$
$673$	− 6.32456e12i	− 1.18840i	0.804318	−	0.594200i	$-0.202531\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.95066e12i	0.905762i	0.891571	+	0.452881i	$0.149604\pi$
$677$	4.95066e12i	0.905762i	−0.891571	+	0.452881i	$0.850396\pi$
$678$	0	0
$679$	−4.67246e12	−0.843591
$680$	0	0
$681$	−9.29277e12	−1.65571
$682$	0	0
$683$	− 7.32611e12i	− 1.28819i	−0.764945	−	0.644095i	$-0.777234\pi$
$683$	− 7.32611e12i	− 1.28819i	0.764945	−	0.644095i	$-0.222766\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.02875e13i	1.76200i
$688$	0	0
$689$	8.07176e12	1.36453
$690$	0	0
$691$	5.66806e12	0.945765	0.472882	−	0.881126i	$-0.343214\pi$
$691$	5.66806e12	0.945765	0.472882	+	0.881126i	$0.343214\pi$
$692$	0	0
$693$	8.84274e12i	1.45642i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.69501e11i	− 0.0272034i
$698$	0	0
$699$	−1.40422e13	−2.22479
$700$	0	0
$701$	−4.20985e12	−0.658470	−0.329235	−	0.944248i	$-0.606791\pi$
$701$	−4.20985e12	−0.658470	−0.329235	+	0.944248i	$0.606791\pi$
$702$	0	0
$703$	3.20468e12i	0.494863i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.24486e13i	− 1.87384i
$708$	0	0
$709$	−1.19654e13	−1.77835	−0.889176	−	0.457566i	$-0.848721\pi$
$709$	−1.19654e13	−1.77835	−0.889176	+	0.457566i	$0.848721\pi$
$710$	0	0
$711$	3.10949e12	0.456326
$712$	0	0
$713$	1.03211e12i	0.149563i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 4.64799e12i	− 0.656794i
$718$	0	0
$719$	−4.07118e11	−0.0568120	−0.0284060	−	0.999596i	$-0.509043\pi$
$719$	−4.07118e11	−0.0568120	−0.0284060	+	0.999596i	$0.509043\pi$
$720$	0	0
$721$	1.93551e13	2.66740
$722$	0	0
$723$	2.09675e13i	2.85380i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.44424e12i	0.855592i	0.903875	+	0.427796i	$0.140710\pi$
$727$	6.44424e12i	0.855592i	−0.903875	+	0.427796i	$0.859290\pi$
$728$	0	0
$729$	1.07838e13	1.41416
$730$	0	0
$731$	5.56864e11	0.0721308
$732$	0	0
$733$	5.46122e12i	0.698749i	0.936983	+	0.349375i	$0.113606\pi$
$733$	5.46122e12i	0.698749i	−0.936983	+	0.349375i	$0.886394\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.96192e12i	0.619507i
$738$	0	0
$739$	−1.88201e12	−0.232125	−0.116062	−	0.993242i	$-0.537027\pi$
$739$	−1.88201e12	−0.232125	−0.116062	+	0.993242i	$0.537027\pi$
$740$	0	0
$741$	2.11234e13	2.57384
$742$	0	0
$743$	− 1.36704e13i	− 1.64563i	−0.568312	−	0.822813i	$-0.692403\pi$
$743$	− 1.36704e13i	− 1.64563i	0.568312	−	0.822813i	$-0.307597\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 7.04042e12i	− 0.827287i
$748$	0	0
$749$	−2.10300e13	−2.44158
$750$	0	0
$751$	−8.52585e12	−0.978043	−0.489021	−	0.872272i	$-0.662646\pi$
$751$	−8.52585e12	−0.978043	−0.489021	+	0.872272i	$0.662646\pi$
$752$	0	0
$753$	1.04513e13i	1.18466i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7.50792e12i	0.830975i	0.909599	+	0.415487i	$0.136389\pi$
$757$	7.50792e12i	0.830975i	−0.909599	+	0.415487i	$0.863611\pi$
$758$	0	0
$759$	−1.44479e13	−1.58022
$760$	0	0
$761$	−1.61560e13	−1.74624	−0.873118	−	0.487509i	$-0.837906\pi$
$761$	−1.61560e13	−1.74624	−0.873118	+	0.487509i	$0.837906\pi$
$762$	0	0
$763$	1.17740e12i	0.125766i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.44747e13i	1.51018i
$768$	0	0
$769$	−1.11830e12	−0.115317	−0.0576583	−	0.998336i	$-0.518363\pi$
$769$	−1.11830e12	−0.115317	−0.0576583	+	0.998336i	$0.518363\pi$
$770$	0	0
$771$	1.22174e13	1.24519
$772$	0	0
$773$	− 1.73165e13i	− 1.74442i	−0.489129	−	0.872212i	$-0.662685\pi$
$773$	− 1.73165e13i	− 1.74442i	0.489129	−	0.872212i	$-0.337315\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.09440e13i	1.07716i
$778$	0	0
$779$	−1.25466e12	−0.122070
$780$	0	0
$781$	1.27302e13	1.22435
$782$	0	0
$783$	3.42654e11i	0.0325783i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 3.88515e11i	− 0.0361012i	−0.999837	−	0.0180506i	$-0.994254\pi$
$787$	− 3.88515e11i	− 0.0361012i	0.999837	−	0.0180506i	$-0.00574599\pi$
$788$	0	0
$789$	−6.79853e12	−0.624552
$790$	0	0
$791$	1.83702e13	1.66848
$792$	0	0
$793$	6.34166e12i	0.569474i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.08432e12i	0.0951910i	0.998867	+	0.0475955i	$0.0151558\pi$
$797$	1.08432e12i	0.0951910i	−0.998867	+	0.0475955i	$0.984844\pi$
$798$	0	0
$799$	2.59624e12	0.225363
$800$	0	0
$801$	−1.37906e13	−1.18368
$802$	0	0
$803$	− 5.88161e12i	− 0.499202i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.31193e12i	− 0.108888i
$808$	0	0
$809$	−4.06803e12	−0.333900	−0.166950	−	0.985965i	$-0.553392\pi$
$809$	−4.06803e12	−0.333900	−0.166950	+	0.985965i	$0.553392\pi$
$810$	0	0
$811$	−4.87468e12	−0.395688	−0.197844	−	0.980234i	$-0.563394\pi$
$811$	−4.87468e12	−0.395688	−0.197844	+	0.980234i	$0.563394\pi$
$812$	0	0
$813$	− 1.69359e13i	− 1.35957i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4.12198e12i	− 0.323673i
$818$	0	0
$819$	3.99926e13	3.10600
$820$	0	0
$821$	−1.03829e12	−0.0797582	−0.0398791	−	0.999205i	$-0.512697\pi$
$821$	−1.03829e12	−0.0797582	−0.0398791	+	0.999205i	$0.512697\pi$
$822$	0	0
$823$	− 1.10923e13i	− 0.842794i	−0.906876	−	0.421397i	$-0.861540\pi$
$823$	− 1.10923e13i	− 0.842794i	0.906876	−	0.421397i	$-0.138460\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.45526e13i	− 1.08184i	−0.841073	−	0.540922i	$-0.818075\pi$
$827$	− 1.45526e13i	− 1.08184i	0.841073	−	0.540922i	$-0.181925\pi$
$828$	0	0
$829$	−1.37985e13	−1.01470	−0.507348	−	0.861741i	$-0.669374\pi$
$829$	−1.37985e13	−1.01470	−0.507348	+	0.861741i	$0.669374\pi$
$830$	0	0
$831$	−1.48796e13	−1.08240
$832$	0	0
$833$	4.75736e12i	0.342345i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.52829e11i	0.0389337i
$838$	0	0
$839$	9.20234e11	0.0641165	0.0320582	−	0.999486i	$-0.489794\pi$
$839$	9.20234e11	0.0641165	0.0320582	+	0.999486i	$0.489794\pi$
$840$	0	0
$841$	−1.43921e13	−0.992067
$842$	0	0
$843$	1.71420e13i	1.16906i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.01936e13i	− 0.680536i
$848$	0	0
$849$	3.97855e13	2.62809
$850$	0	0
$851$	−9.91330e12	−0.647941
$852$	0	0
$853$	− 7.46964e12i	− 0.483091i	−0.970390	−	0.241545i	$-0.922346\pi$
$853$	− 7.46964e12i	− 0.483091i	0.970390	−	0.241545i	$-0.0776543\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.05568e13i	− 1.30179i	−0.759166	−	0.650897i	$-0.774393\pi$
$857$	− 2.05568e13i	− 1.30179i	0.759166	−	0.650897i	$-0.225607\pi$
$858$	0	0
$859$	−1.05923e12	−0.0663775	−0.0331888	−	0.999449i	$-0.510566\pi$
$859$	−1.05923e12	−0.0663775	−0.0331888	+	0.999449i	$0.510566\pi$
$860$	0	0
$861$	−4.28469e12	−0.265709
$862$	0	0
$863$	− 1.53023e13i	− 0.939092i	−0.882908	−	0.469546i	$-0.844418\pi$
$863$	− 1.53023e13i	− 0.939092i	0.882908	−	0.469546i	$-0.155582\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	2.34982e13i	1.41237i
$868$	0	0
$869$	4.62869e12	0.275340
$870$	0	0
$871$	2.24410e13	1.32118
$872$	0	0
$873$	− 1.15516e13i	− 0.673095i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.74558e11i	0.00996416i	0.999988	+	0.00498208i	$0.00158585\pi$
$877$	1.74558e11i	0.00996416i	−0.999988	+	0.00498208i	$0.998414\pi$
$878$	0	0
$879$	1.13385e13	0.640629
$880$	0	0
$881$	8.59039e12	0.480420	0.240210	−	0.970721i	$-0.422784\pi$
$881$	8.59039e12	0.480420	0.240210	+	0.970721i	$0.422784\pi$
$882$	0	0
$883$	1.70227e13i	0.942336i	0.882043	+	0.471168i	$0.156167\pi$
$883$	1.70227e13i	0.942336i	−0.882043	+	0.471168i	$0.843833\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.01246e13i	− 1.09162i	−0.837909	−	0.545810i	$-0.816222\pi$
$887$	− 2.01246e13i	− 1.09162i	0.837909	−	0.545810i	$-0.183778\pi$
$888$	0	0
$889$	5.34112e13	2.86797
$890$	0	0
$891$	9.83253e12	0.522655
$892$	0	0
$893$	− 1.92177e13i	− 1.01127i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.53428e13i	3.37002i
$898$	0	0
$899$	1.85668e11	0.00948023
$900$	0	0
$901$	−4.03214e12	−0.203833
$902$	0	0
$903$	− 1.40766e13i	− 0.704535i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 2.10130e13i	− 1.03099i	−0.856893	−	0.515495i	$-0.827608\pi$
$907$	− 2.10130e13i	− 1.03099i	0.856893	−	0.515495i	$-0.172392\pi$
$908$	0	0
$909$	3.07761e13	1.49512
$910$	0	0
$911$	−1.95345e12	−0.0939656	−0.0469828	−	0.998896i	$-0.514961\pi$
$911$	−1.95345e12	−0.0939656	−0.0469828	+	0.998896i	$0.514961\pi$
$912$	0	0
$913$	− 1.04802e13i	− 0.499172i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 3.80064e13i	− 1.77499i
$918$	0	0
$919$	−2.47656e12	−0.114533	−0.0572663	−	0.998359i	$-0.518238\pi$
$919$	−2.47656e12	−0.114533	−0.0572663	+	0.998359i	$0.518238\pi$
$920$	0	0
$921$	2.73301e13	1.25162
$922$	0	0
$923$	− 5.75744e13i	− 2.61109i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.78510e13i	2.12830i
$928$	0	0
$929$	−4.40844e13	−1.94184	−0.970922	−	0.239395i	$-0.923051\pi$
$929$	−4.40844e13	−1.94184	−0.970922	+	0.239395i	$0.923051\pi$
$930$	0	0
$931$	3.52146e13	1.53621
$932$	0	0
$933$	− 4.40477e11i	− 0.0190308i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 4.92250e12i	− 0.208621i	−0.994545	−	0.104310i	$-0.966736\pi$
$937$	− 4.92250e12i	− 0.208621i	0.994545	−	0.104310i	$-0.0332635\pi$
$938$	0	0
$939$	5.12372e13	2.15075
$940$	0	0
$941$	−1.54680e13	−0.643103	−0.321552	−	0.946892i	$-0.604204\pi$
$941$	−1.54680e13	−0.643103	−0.321552	+	0.946892i	$0.604204\pi$
$942$	0	0
$943$	− 3.88116e12i	− 0.159830i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.49107e12i	0.221862i	0.993828	+	0.110931i	$0.0353832\pi$
$947$	5.49107e12i	0.221862i	−0.993828	+	0.110931i	$0.964617\pi$
$948$	0	0
$949$	−2.66004e13	−1.06461
$950$	0	0
$951$	−7.28207e13	−2.88697
$952$	0	0
$953$	− 1.76932e13i	− 0.694845i	−0.937709	−	0.347422i	$-0.887057\pi$
$953$	− 1.76932e13i	− 0.694845i	0.937709	−	0.347422i	$-0.112943\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.59905e12i	0.100164i
$958$	0	0
$959$	−2.52994e13	−0.965888
$960$	0	0
$961$	−2.61401e13	−0.988670
$962$	0	0
$963$	− 5.19916e13i	− 1.94812i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	2.89231e13i	1.06372i	0.846834	+	0.531858i	$0.178506\pi$
$967$	2.89231e13i	1.06372i	−0.846834	+	0.531858i	$0.821494\pi$
$968$	0	0
$969$	−1.05519e13	−0.384481
$970$	0	0
$971$	−2.69482e13	−0.972843	−0.486422	−	0.873724i	$-0.661698\pi$
$971$	−2.69482e13	−0.972843	−0.486422	+	0.873724i	$0.661698\pi$
$972$	0	0
$973$	− 4.05255e13i	− 1.44951i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 3.90598e13i	− 1.37153i	−0.727824	−	0.685764i	$-0.759468\pi$
$977$	− 3.90598e13i	− 1.37153i	0.727824	−	0.685764i	$-0.240532\pi$
$978$	0	0
$979$	−2.05282e13	−0.714216
$980$	0	0
$981$	−2.91084e12	−0.100348
$982$	0	0
$983$	− 9.03830e12i	− 0.308742i	−0.988013	−	0.154371i	$-0.950665\pi$
$983$	− 9.03830e12i	− 0.308742i	0.988013	−	0.154371i	$-0.0493351\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6.56285e13i	− 2.20123i
$988$	0	0
$989$	1.27509e13	0.423795
$990$	0	0
$991$	−4.89096e13	−1.61088	−0.805439	−	0.592678i	$-0.798071\pi$
$991$	−4.89096e13	−1.61088	−0.805439	+	0.592678i	$0.798071\pi$
$992$	0	0
$993$	3.01181e13i	0.983007i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 4.01445e13i	− 1.28676i	−0.765547	−	0.643380i	$-0.777532\pi$
$997$	− 4.01445e13i	− 1.28676i	0.765547	−	0.643380i	$-0.222468\pi$
$998$	0	0
$999$	−5.30984e12	−0.168670

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.10.c.q.49.5		6
4.3	odd	2		25.10.b.c.24.2		6
5.2	odd	4		400.10.a.u.1.3		3
5.3	odd	4		400.10.a.y.1.1		3
5.4	even	2	inner	400.10.c.q.49.2		6
12.11	even	2		225.10.b.m.199.5		6
20.3	even	4		25.10.a.c.1.2	✓	3
20.7	even	4		25.10.a.d.1.2	yes	3
20.19	odd	2		25.10.b.c.24.5		6
60.23	odd	4		225.10.a.p.1.2		3
60.47	odd	4		225.10.a.m.1.2		3
60.59	even	2		225.10.b.m.199.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.10.a.c.1.2	✓	3	20.3	even	4
25.10.a.d.1.2	yes	3	20.7	even	4
25.10.b.c.24.2		6	4.3	odd	2
25.10.b.c.24.5		6	20.19	odd	2
225.10.a.m.1.2		3	60.47	odd	4
225.10.a.p.1.2		3	60.23	odd	4
225.10.b.m.199.2		6	60.59	even	2
225.10.b.m.199.5		6	12.11	even	2
400.10.a.u.1.3		3	5.2	odd	4
400.10.a.y.1.1		3	5.3	odd	4
400.10.c.q.49.2		6	5.4	even	2	inner
400.10.c.q.49.5		6	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+210.171i q^{3} +9905.49i q^{7} -24489.0 q^{9} +O(q^{10})\)	\(q+210.171i q^{3} +9905.49i q^{7} -24489.0 q^{9} -36453.6 q^{11} +164867. i q^{13} -82357.1i q^{17} -609617. q^{19} -2.08185e6 q^{21} -1.88578e6i q^{23} -1.01008e6i q^{27} -339235. q^{29} -547314. q^{31} -7.66150e6i q^{33} -5.25687e6i q^{37} -3.46503e7 q^{39} +2.05812e6 q^{41} +6.76158e6i q^{43} +3.15241e7i q^{47} -5.77651e7 q^{49} +1.73091e7 q^{51} -4.89593e7i q^{53} -1.28124e8i q^{57} +8.77960e7 q^{59} +3.84654e7 q^{61} -2.42575e8i q^{63} -1.36116e8i q^{67} +3.96337e8 q^{69} -3.49218e8 q^{71} +1.61345e8i q^{73} -3.61091e8i q^{77} -1.26975e8 q^{79} -2.69727e8 q^{81} +2.87494e8i q^{83} -7.12976e7i q^{87} +5.63133e8 q^{89} -1.63309e9 q^{91} -1.15030e8i q^{93} +4.71704e8i q^{97} +8.92711e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 116468 q^{9}+O(q^{10}) \) 6 * q - 116468 * q^9	\( 6 q - 116468 q^{9} + 109398 q^{11} + 1637690 q^{19} - 4750788 q^{21} - 4350960 q^{29} - 8548132 q^{31} - 100828184 q^{39} + 11852622 q^{41} + 12907858 q^{49} + 51269398 q^{51} + 11341920 q^{59} + 250613852 q^{61} + 628549884 q^{69} - 595101192 q^{71} - 620050340 q^{79} + 2797694726 q^{81} + 2207720070 q^{89} - 2366375312 q^{91} - 1784469044 q^{99}+O(q^{100}) \) 6 * q - 116468 * q^9 + 109398 * q^11 + 1637690 * q^19 - 4750788 * q^21 - 4350960 * q^29 - 8548132 * q^31 - 100828184 * q^39 + 11852622 * q^41 + 12907858 * q^49 + 51269398 * q^51 + 11341920 * q^59 + 250613852 * q^61 + 628549884 * q^69 - 595101192 * q^71 - 620050340 * q^79 + 2797694726 * q^81 + 2207720070 * q^89 - 2366375312 * q^91 - 1784469044 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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