LMFDB - Embedded newform 3150.3.c.c.449.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,3,Mod(449,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3150.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3150.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:	$85.8312832735$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.4
Root			$0.581861 + 1.28897i$ of defining polynomial
Character	$\chi$	$=$	3150.449
Dual form			3150.3.c.c.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +20.8740i q^{11} +5.64575i q^{13} -3.74166i q^{14} +4.00000 q^{16} +16.5583 q^{17} +1.77124 q^{19} -29.5203i q^{22} -23.6137 q^{23} -7.98430i q^{26} +5.29150i q^{28} +29.3593i q^{29} -17.1660 q^{31} -5.65685 q^{32} -23.4170 q^{34} +14.7712i q^{37} -2.50492 q^{38} +29.4323i q^{41} +39.1255i q^{43} +41.7480i q^{44} +33.3948 q^{46} +39.1543 q^{47} -7.00000 q^{49} +11.2915i q^{52} +19.8563 q^{53} -7.48331i q^{56} -41.5203i q^{58} +88.5630i q^{59} -85.4980 q^{61} +24.2764 q^{62} +8.00000 q^{64} -19.4797i q^{67} +33.1166 q^{68} -58.5253i q^{71} -113.601i q^{73} -20.8897i q^{74} +3.54249 q^{76} -55.2273 q^{77} +14.2693 q^{79} -41.6235i q^{82} +151.232 q^{83} -55.3318i q^{86} -59.0405i q^{88} -15.5878i q^{89} -14.9373 q^{91} -47.2273 q^{92} -55.3725 q^{94} -57.2470i q^{97} +9.89949 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4}+O(q^{10}) $ 8 * q + 16 * q^4	$ 8 q + 16 q^{4} + 32 q^{16} + 120 q^{19} + 32 q^{31} - 272 q^{34} - 8 q^{46} - 56 q^{49} - 176 q^{61} + 64 q^{64} + 240 q^{76} - 288 q^{79} - 56 q^{91} + 192 q^{94}+O(q^{100}) $ 8 * q + 16 * q^4 + 32 * q^16 + 120 * q^19 + 32 * q^31 - 272 * q^34 - 8 * q^46 - 56 * q^49 - 176 * q^61 + 64 * q^64 + 240 * q^76 - 288 * q^79 - 56 * q^91 + 192 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	−1.41421	−0.707107
$2$	−1.41421	−0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	−2.82843	−0.353553
$9$	0	0
$10$	0	0
$11$	20.8740i	1.89763i	0.315826	+	0.948817i	$0.397718\pi$
$11$	20.8740i	1.89763i	−0.315826	+	0.948817i	$0.602282\pi$
$12$	0	0
$13$	5.64575i	0.434289i	0.976140	+	0.217144i	$0.0696742\pi$
$13$	5.64575i	0.434289i	−0.976140	+	0.217144i	$0.930326\pi$
$14$	− 3.74166i	− 0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	16.5583	0.974019	0.487009	−	0.873397i	$-0.338088\pi$
$17$	16.5583	0.974019	0.487009	+	0.873397i	$0.338088\pi$
$18$	0	0
$19$	1.77124	0.0932233	0.0466117	−	0.998913i	$-0.485158\pi$
$19$	1.77124	0.0932233	0.0466117	+	0.998913i	$0.485158\pi$
$20$	0	0
$21$	0	0
$22$	− 29.5203i	− 1.34183i
$23$	−23.6137	−1.02668	−0.513341	−	0.858185i	$-0.671592\pi$
$23$	−23.6137	−1.02668	−0.513341	+	0.858185i	$0.671592\pi$
$24$	0	0
$25$	0	0
$26$	− 7.98430i	− 0.307088i
$27$	0	0
$28$	5.29150i	0.188982i
$29$	29.3593i	1.01239i	0.862420	+	0.506194i	$0.168948\pi$
$29$	29.3593i	1.01239i	−0.862420	+	0.506194i	$0.831052\pi$
$30$	0	0
$31$	−17.1660	−0.553742	−0.276871	−	0.960907i	$-0.589298\pi$
$31$	−17.1660	−0.553742	−0.276871	+	0.960907i	$0.589298\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	−23.4170	−0.688735
$35$	0	0
$36$	0	0
$37$	14.7712i	0.399223i	0.979875	+	0.199611i	$0.0639680\pi$
$37$	14.7712i	0.399223i	−0.979875	+	0.199611i	$0.936032\pi$
$38$	−2.50492	−0.0659189
$39$	0	0
$40$	0	0
$41$	29.4323i	0.717860i	0.933364	+	0.358930i	$0.116858\pi$
$41$	29.4323i	0.717860i	−0.933364	+	0.358930i	$0.883142\pi$
$42$	0	0
$43$	39.1255i	0.909895i	0.890518	+	0.454948i	$0.150342\pi$
$43$	39.1255i	0.909895i	−0.890518	+	0.454948i	$0.849658\pi$
$44$	41.7480i	0.948817i
$45$	0	0
$46$	33.3948	0.725973
$47$	39.1543	0.833070	0.416535	−	0.909120i	$-0.363244\pi$
$47$	39.1543	0.833070	0.416535	+	0.909120i	$0.363244\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	11.2915i	0.217144i
$53$	19.8563	0.374647	0.187324	−	0.982298i	$-0.440019\pi$
$53$	19.8563	0.374647	0.187324	+	0.982298i	$0.440019\pi$
$54$	0	0
$55$	0	0
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	− 41.5203i	− 0.715867i
$59$	88.5630i	1.50107i	0.660832	+	0.750534i	$0.270204\pi$
$59$	88.5630i	1.50107i	−0.660832	+	0.750534i	$0.729796\pi$
$60$	0	0
$61$	−85.4980	−1.40161	−0.700804	−	0.713354i	$-0.747175\pi$
$61$	−85.4980	−1.40161	−0.700804	+	0.713354i	$0.747175\pi$
$62$	24.2764	0.391555
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	− 19.4797i	− 0.290742i	−0.989377	−	0.145371i	$-0.953562\pi$
$67$	− 19.4797i	− 0.290742i	0.989377	−	0.145371i	$-0.0464376\pi$
$68$	33.1166	0.487009
$69$	0	0
$70$	0	0
$71$	− 58.5253i	− 0.824300i	−0.911116	−	0.412150i	$-0.864778\pi$
$71$	− 58.5253i	− 0.824300i	0.911116	−	0.412150i	$-0.135222\pi$
$72$	0	0
$73$	− 113.601i	− 1.55618i	−0.628151	−	0.778091i	$-0.716188\pi$
$73$	− 113.601i	− 1.55618i	0.628151	−	0.778091i	$-0.283812\pi$
$74$	− 20.8897i	− 0.282293i
$75$	0	0
$76$	3.54249	0.0466117
$77$	−55.2273	−0.717238
$78$	0	0
$79$	14.2693	0.180624	0.0903119	−	0.995914i	$-0.471214\pi$
$79$	14.2693	0.180624	0.0903119	+	0.995914i	$0.471214\pi$
$80$	0	0
$81$	0	0
$82$	− 41.6235i	− 0.507604i
$83$	151.232	1.82207	0.911037	−	0.412325i	$-0.135283\pi$
$83$	151.232	1.82207	0.911037	+	0.412325i	$0.135283\pi$
$84$	0	0
$85$	0	0
$86$	− 55.3318i	− 0.643393i
$87$	0	0
$88$	− 59.0405i	− 0.670915i
$89$	− 15.5878i	− 0.175144i	−0.996158	−	0.0875718i	$-0.972089\pi$
$89$	− 15.5878i	− 0.175144i	0.996158	−	0.0875718i	$-0.0279107\pi$
$90$	0	0
$91$	−14.9373	−0.164146
$92$	−47.2273	−0.513341
$93$	0	0
$94$	−55.3725	−0.589070
$95$	0	0
$96$	0	0
$97$	− 57.2470i	− 0.590176i	−0.955470	−	0.295088i	$-0.904651\pi$
$97$	− 57.2470i	− 0.590176i	0.955470	−	0.295088i	$-0.0953489\pi$
$98$	9.89949	0.101015
$99$	0	0
$100$	0	0
$101$	107.037i	1.05977i	0.848070	+	0.529884i	$0.177765\pi$
$101$	107.037i	1.05977i	−0.848070	+	0.529884i	$0.822235\pi$
$102$	0	0
$103$	− 122.103i	− 1.18547i	−0.805398	−	0.592734i	$-0.798048\pi$
$103$	− 122.103i	− 1.18547i	0.805398	−	0.592734i	$-0.201952\pi$
$104$	− 15.9686i	− 0.153544i
$105$	0	0
$106$	−28.0810	−0.264915
$107$	−46.1366	−0.431183	−0.215592	−	0.976484i	$-0.569168\pi$
$107$	−46.1366	−0.431183	−0.215592	+	0.976484i	$0.569168\pi$
$108$	0	0
$109$	12.3137	0.112970	0.0564850	−	0.998403i	$-0.482011\pi$
$109$	12.3137	0.112970	0.0564850	+	0.998403i	$0.482011\pi$
$110$	0	0
$111$	0	0
$112$	10.5830i	0.0944911i
$113$	−164.445	−1.45527	−0.727634	−	0.685965i	$-0.759380\pi$
$113$	−164.445	−1.45527	−0.727634	+	0.685965i	$0.759380\pi$
$114$	0	0
$115$	0	0
$116$	58.7185i	0.506194i
$117$	0	0
$118$	− 125.247i	− 1.06142i
$119$	43.8092i	0.368144i
$120$	0	0
$121$	−314.723	−2.60102
$122$	120.912	0.991086
$123$	0	0
$124$	−34.3320	−0.276871
$125$	0	0
$126$	0	0
$127$	87.5608i	0.689455i	0.938703	+	0.344727i	$0.112029\pi$
$127$	87.5608i	0.689455i	−0.938703	+	0.344727i	$0.887971\pi$
$128$	−11.3137	−0.0883883
$129$	0	0
$130$	0	0
$131$	− 174.240i	− 1.33008i	−0.746808	−	0.665039i	$-0.768415\pi$
$131$	− 174.240i	− 1.33008i	0.746808	−	0.665039i	$-0.231585\pi$
$132$	0	0
$133$	4.68627i	0.0352351i
$134$	27.5485i	0.205586i
$135$	0	0
$136$	−46.8340	−0.344368
$137$	55.3004	0.403652	0.201826	−	0.979421i	$-0.435312\pi$
$137$	55.3004	0.403652	0.201826	+	0.979421i	$0.435312\pi$
$138$	0	0
$139$	−138.561	−0.996840	−0.498420	−	0.866936i	$-0.666086\pi$
$139$	−138.561	−0.996840	−0.498420	+	0.866936i	$0.666086\pi$
$140$	0	0
$141$	0	0
$142$	82.7673i	0.582868i
$143$	−117.849	−0.824121
$144$	0	0
$145$	0	0
$146$	160.656i	1.10039i
$147$	0	0
$148$	29.5425i	0.199611i
$149$	149.004i	1.00002i	0.866019	+	0.500012i	$0.166671\pi$
$149$	149.004i	1.00002i	−0.866019	+	0.500012i	$0.833329\pi$
$150$	0	0
$151$	154.793	1.02512	0.512561	−	0.858651i	$-0.328697\pi$
$151$	154.793	1.02512	0.512561	+	0.858651i	$0.328697\pi$
$152$	−5.00983	−0.0329594
$153$	0	0
$154$	78.1033	0.507164
$155$	0	0
$156$	0	0
$157$	− 193.292i	− 1.23116i	−0.788076	−	0.615578i	$-0.788923\pi$
$157$	− 193.292i	− 1.23116i	0.788076	−	0.615578i	$-0.211077\pi$
$158$	−20.1798	−0.127720
$159$	0	0
$160$	0	0
$161$	− 62.4759i	− 0.388049i
$162$	0	0
$163$	− 60.4941i	− 0.371129i	−0.982632	−	0.185565i	$-0.940589\pi$
$163$	− 60.4941i	− 0.371129i	0.982632	−	0.185565i	$-0.0594114\pi$
$164$	58.8646i	0.358930i
$165$	0	0
$166$	−213.875	−1.28840
$167$	246.391	1.47540	0.737698	−	0.675131i	$-0.235913\pi$
$167$	246.391	1.47540	0.737698	+	0.675131i	$0.235913\pi$
$168$	0	0
$169$	137.125	0.811393
$170$	0	0
$171$	0	0
$172$	78.2510i	0.454948i
$173$	−190.825	−1.10303	−0.551516	−	0.834164i	$-0.685950\pi$
$173$	−190.825	−1.10303	−0.551516	+	0.834164i	$0.685950\pi$
$174$	0	0
$175$	0	0
$176$	83.4959i	0.474409i
$177$	0	0
$178$	22.0445i	0.123845i
$179$	313.303i	1.75030i	0.483856	+	0.875148i	$0.339236\pi$
$179$	313.303i	1.75030i	−0.483856	+	0.875148i	$0.660764\pi$
$180$	0	0
$181$	206.516	1.14097	0.570487	−	0.821307i	$-0.306754\pi$
$181$	206.516	1.14097	0.570487	+	0.821307i	$0.306754\pi$
$182$	21.1245	0.116069
$183$	0	0
$184$	66.7895	0.362987
$185$	0	0
$186$	0	0
$187$	345.638i	1.84833i
$188$	78.3086	0.416535
$189$	0	0
$190$	0	0
$191$	− 128.276i	− 0.671600i	−0.941933	−	0.335800i	$-0.890993\pi$
$191$	− 128.276i	− 0.671600i	0.941933	−	0.335800i	$-0.109007\pi$
$192$	0	0
$193$	− 122.431i	− 0.634359i	−0.948365	−	0.317180i	$-0.897264\pi$
$193$	− 122.431i	− 0.634359i	0.948365	−	0.317180i	$-0.102736\pi$
$194$	80.9596i	0.417317i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	23.9057	0.121349	0.0606745	−	0.998158i	$-0.480675\pi$
$197$	23.9057	0.121349	0.0606745	+	0.998158i	$0.480675\pi$
$198$	0	0
$199$	−46.7085	−0.234716	−0.117358	−	0.993090i	$-0.537443\pi$
$199$	−46.7085	−0.234716	−0.117358	+	0.993090i	$0.537443\pi$
$200$	0	0
$201$	0	0
$202$	− 151.373i	− 0.749369i
$203$	−77.6773	−0.382647
$204$	0	0
$205$	0	0
$206$	172.680i	0.838253i
$207$	0	0
$208$	22.5830i	0.108572i
$209$	36.9729i	0.176904i
$210$	0	0
$211$	−313.158	−1.48416	−0.742081	−	0.670310i	$-0.766161\pi$
$211$	−313.158	−1.48416	−0.742081	+	0.670310i	$0.766161\pi$
$212$	39.7126	0.187324
$213$	0	0
$214$	65.2470	0.304893
$215$	0	0
$216$	0	0
$217$	− 45.4170i	− 0.209295i
$218$	−17.4142	−0.0798819
$219$	0	0
$220$	0	0
$221$	93.4841i	0.423005i
$222$	0	0
$223$	− 386.553i	− 1.73342i	−0.498811	−	0.866711i	$-0.666230\pi$
$223$	− 386.553i	− 1.73342i	0.498811	−	0.866711i	$-0.333770\pi$
$224$	− 14.9666i	− 0.0668153i
$225$	0	0
$226$	232.561	1.02903
$227$	84.7326	0.373272	0.186636	−	0.982429i	$-0.440242\pi$
$227$	84.7326	0.373272	0.186636	+	0.982429i	$0.440242\pi$
$228$	0	0
$229$	54.7451	0.239061	0.119531	−	0.992831i	$-0.461861\pi$
$229$	54.7451	0.239061	0.119531	+	0.992831i	$0.461861\pi$
$230$	0	0
$231$	0	0
$232$	− 83.0405i	− 0.357933i
$233$	−386.497	−1.65879	−0.829393	−	0.558666i	$-0.811314\pi$
$233$	−386.497	−1.65879	−0.829393	+	0.558666i	$0.811314\pi$
$234$	0	0
$235$	0	0
$236$	177.126i	0.750534i
$237$	0	0
$238$	− 61.9555i	− 0.260317i
$239$	377.939i	1.58133i	0.612246	+	0.790667i	$0.290266\pi$
$239$	377.939i	1.58133i	−0.612246	+	0.790667i	$0.709734\pi$
$240$	0	0
$241$	−237.763	−0.986570	−0.493285	−	0.869868i	$-0.664204\pi$
$241$	−237.763	−0.986570	−0.493285	+	0.869868i	$0.664204\pi$
$242$	445.085	1.83920
$243$	0	0
$244$	−170.996	−0.700804
$245$	0	0
$246$	0	0
$247$	10.0000i	0.0404858i
$248$	48.5528	0.195777
$249$	0	0
$250$	0	0
$251$	356.021i	1.41841i	0.705002	+	0.709206i	$0.250946\pi$
$251$	356.021i	1.41841i	−0.705002	+	0.709206i	$0.749054\pi$
$252$	0	0
$253$	− 492.911i	− 1.94827i
$254$	− 123.830i	− 0.487518i
$255$	0	0
$256$	16.0000	0.0625000
$257$	−117.051	−0.455450	−0.227725	−	0.973725i	$-0.573129\pi$
$257$	−117.051	−0.455450	−0.227725	+	0.973725i	$0.573129\pi$
$258$	0	0
$259$	−39.0810	−0.150892
$260$	0	0
$261$	0	0
$262$	246.413i	0.940508i
$263$	−124.346	−0.472800	−0.236400	−	0.971656i	$-0.575968\pi$
$263$	−124.346	−0.472800	−0.236400	+	0.971656i	$0.575968\pi$
$264$	0	0
$265$	0	0
$266$	− 6.62739i	− 0.0249150i
$267$	0	0
$268$	− 38.9595i	− 0.145371i
$269$	− 196.132i	− 0.729115i	−0.931181	−	0.364558i	$-0.881220\pi$
$269$	− 196.132i	− 0.729115i	0.931181	−	0.364558i	$-0.118780\pi$
$270$	0	0
$271$	10.6052	0.0391337	0.0195669	−	0.999809i	$-0.493771\pi$
$271$	10.6052	0.0391337	0.0195669	+	0.999809i	$0.493771\pi$
$272$	66.2333	0.243505
$273$	0	0
$274$	−78.2065	−0.285425
$275$	0	0
$276$	0	0
$277$	4.91503i	0.0177438i	0.999961	+	0.00887189i	$0.00282405\pi$
$277$	4.91503i	0.0177438i	−0.999961	+	0.00887189i	$0.997176\pi$
$278$	195.955	0.704872
$279$	0	0
$280$	0	0
$281$	− 285.713i	− 1.01677i	−0.861130	−	0.508386i	$-0.830242\pi$
$281$	− 285.713i	− 1.01677i	0.861130	−	0.508386i	$-0.169758\pi$
$282$	0	0
$283$	248.458i	0.877942i	0.898501	+	0.438971i	$0.144657\pi$
$283$	248.458i	0.877942i	−0.898501	+	0.438971i	$0.855343\pi$
$284$	− 117.051i	− 0.412150i
$285$	0	0
$286$	166.664	0.582741
$287$	−77.8705	−0.271326
$288$	0	0
$289$	−14.8222	−0.0512878
$290$	0	0
$291$	0	0
$292$	− 227.203i	− 0.778091i
$293$	−39.5924	−0.135128	−0.0675638	−	0.997715i	$-0.521523\pi$
$293$	−39.5924	−0.135128	−0.0675638	+	0.997715i	$0.521523\pi$
$294$	0	0
$295$	0	0
$296$	− 41.7794i	− 0.141147i
$297$	0	0
$298$	− 210.723i	− 0.707124i
$299$	− 133.317i	− 0.445876i
$300$	0	0
$301$	−103.516	−0.343908
$302$	−218.911	−0.724871
$303$	0	0
$304$	7.08497	0.0233058
$305$	0	0
$306$	0	0
$307$	495.925i	1.61539i	0.589599	+	0.807696i	$0.299286\pi$
$307$	495.925i	1.61539i	−0.589599	+	0.807696i	$0.700714\pi$
$308$	−110.455	−0.358619
$309$	0	0
$310$	0	0
$311$	− 135.206i	− 0.434747i	−0.976089	−	0.217373i	$-0.930251\pi$
$311$	− 135.206i	− 0.434747i	0.976089	−	0.217373i	$-0.0697489\pi$
$312$	0	0
$313$	− 497.881i	− 1.59067i	−0.606167	−	0.795337i	$-0.707294\pi$
$313$	− 497.881i	− 1.59067i	0.606167	−	0.795337i	$-0.292706\pi$
$314$	273.355i	0.870559i
$315$	0	0
$316$	28.5385	0.0903119
$317$	60.5607	0.191043	0.0955216	−	0.995427i	$-0.469548\pi$
$317$	60.5607	0.191043	0.0955216	+	0.995427i	$0.469548\pi$
$318$	0	0
$319$	−612.844	−1.92114
$320$	0	0
$321$	0	0
$322$	88.3542i	0.274392i
$323$	29.3288	0.0908013
$324$	0	0
$325$	0	0
$326$	85.5516i	0.262428i
$327$	0	0
$328$	− 83.2470i	− 0.253802i
$329$	103.593i	0.314871i
$330$	0	0
$331$	−404.542	−1.22218	−0.611091	−	0.791560i	$-0.709269\pi$
$331$	−404.542	−1.22218	−0.611091	+	0.791560i	$0.709269\pi$
$332$	302.464	0.911037
$333$	0	0
$334$	−348.450	−1.04326
$335$	0	0
$336$	0	0
$337$	405.069i	1.20199i	0.799254	+	0.600993i	$0.205228\pi$
$337$	405.069i	1.20199i	−0.799254	+	0.600993i	$0.794772\pi$
$338$	−193.925	−0.573742
$339$	0	0
$340$	0	0
$341$	− 358.323i	− 1.05080i
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	− 110.664i	− 0.321697i
$345$	0	0
$346$	269.867	0.779961
$347$	−132.419	−0.381612	−0.190806	−	0.981628i	$-0.561110\pi$
$347$	−132.419	−0.381612	−0.190806	+	0.981628i	$0.561110\pi$
$348$	0	0
$349$	−320.310	−0.917793	−0.458897	−	0.888490i	$-0.651755\pi$
$349$	−320.310	−0.917793	−0.458897	+	0.888490i	$0.651755\pi$
$350$	0	0
$351$	0	0
$352$	− 118.081i	− 0.335457i
$353$	92.7539	0.262759	0.131380	−	0.991332i	$-0.458059\pi$
$353$	92.7539	0.262759	0.131380	+	0.991332i	$0.458059\pi$
$354$	0	0
$355$	0	0
$356$	− 31.1756i	− 0.0875718i
$357$	0	0
$358$	− 443.077i	− 1.23765i
$359$	− 209.173i	− 0.582655i	−0.956623	−	0.291328i	$-0.905903\pi$
$359$	− 209.173i	− 0.582655i	0.956623	−	0.291328i	$-0.0940970\pi$
$360$	0	0
$361$	−357.863	−0.991309
$362$	−292.058	−0.806791
$363$	0	0
$364$	−29.8745	−0.0820728
$365$	0	0
$366$	0	0
$367$	374.170i	1.01954i	0.860312	+	0.509768i	$0.170269\pi$
$367$	374.170i	1.01954i	−0.860312	+	0.509768i	$0.829731\pi$
$368$	−94.4547	−0.256670
$369$	0	0
$370$	0	0
$371$	52.5348i	0.141603i
$372$	0	0
$373$	418.741i	1.12263i	0.827602	+	0.561315i	$0.189704\pi$
$373$	418.741i	1.12263i	−0.827602	+	0.561315i	$0.810296\pi$
$374$	− 488.806i	− 1.30697i
$375$	0	0
$376$	−110.745	−0.294535
$377$	−165.755	−0.439669
$378$	0	0
$379$	−226.793	−0.598400	−0.299200	−	0.954190i	$-0.596720\pi$
$379$	−226.793	−0.598400	−0.299200	+	0.954190i	$0.596720\pi$
$380$	0	0
$381$	0	0
$382$	181.409i	0.474893i
$383$	24.0731	0.0628540	0.0314270	−	0.999506i	$-0.489995\pi$
$383$	24.0731	0.0628540	0.0314270	+	0.999506i	$0.489995\pi$
$384$	0	0
$385$	0	0
$386$	173.144i	0.448560i
$387$	0	0
$388$	− 114.494i	− 0.295088i
$389$	202.337i	0.520147i	0.965589	+	0.260073i	$0.0837467\pi$
$389$	202.337i	0.520147i	−0.965589	+	0.260073i	$0.916253\pi$
$390$	0	0
$391$	−391.003	−1.00001
$392$	19.7990	0.0505076
$393$	0	0
$394$	−33.8078	−0.0858067
$395$	0	0
$396$	0	0
$397$	457.697i	1.15289i	0.817137	+	0.576444i	$0.195560\pi$
$397$	457.697i	1.15289i	−0.817137	+	0.576444i	$0.804440\pi$
$398$	66.0558	0.165969
$399$	0	0
$400$	0	0
$401$	658.387i	1.64186i	0.571027	+	0.820931i	$0.306545\pi$
$401$	658.387i	1.64186i	−0.571027	+	0.820931i	$0.693455\pi$
$402$	0	0
$403$	− 96.9150i	− 0.240484i
$404$	214.073i	0.529884i
$405$	0	0
$406$	109.852	0.270572
$407$	−308.335	−0.757579
$408$	0	0
$409$	−152.103	−0.371891	−0.185945	−	0.982560i	$-0.559535\pi$
$409$	−152.103	−0.371891	−0.185945	+	0.982560i	$0.559535\pi$
$410$	0	0
$411$	0	0
$412$	− 244.207i	− 0.592734i
$413$	−234.316	−0.567351
$414$	0	0
$415$	0	0
$416$	− 31.9372i	− 0.0767721i
$417$	0	0
$418$	− 52.2876i	− 0.125090i
$419$	390.083i	0.930985i	0.885052	+	0.465492i	$0.154123\pi$
$419$	390.083i	0.930985i	−0.885052	+	0.465492i	$0.845877\pi$
$420$	0	0
$421$	21.2876	0.0505643	0.0252821	−	0.999680i	$-0.491952\pi$
$421$	21.2876	0.0505643	0.0252821	+	0.999680i	$0.491952\pi$
$422$	442.872	1.04946
$423$	0	0
$424$	−56.1621	−0.132458
$425$	0	0
$426$	0	0
$427$	− 226.207i	− 0.529758i
$428$	−92.2733	−0.215592
$429$	0	0
$430$	0	0
$431$	151.060i	0.350488i	0.984525	+	0.175244i	$0.0560714\pi$
$431$	151.060i	0.350488i	−0.984525	+	0.175244i	$0.943929\pi$
$432$	0	0
$433$	516.952i	1.19388i	0.802285	+	0.596942i	$0.203618\pi$
$433$	516.952i	1.19388i	−0.802285	+	0.596942i	$0.796382\pi$
$434$	64.2293i	0.147994i
$435$	0	0
$436$	24.6275	0.0564850
$437$	−41.8256	−0.0957106
$438$	0	0
$439$	547.395	1.24691	0.623456	−	0.781858i	$-0.285728\pi$
$439$	547.395	1.24691	0.623456	+	0.781858i	$0.285728\pi$
$440$	0	0
$441$	0	0
$442$	− 132.207i	− 0.299110i
$443$	−119.687	−0.270174	−0.135087	−	0.990834i	$-0.543131\pi$
$443$	−119.687	−0.270174	−0.135087	+	0.990834i	$0.543131\pi$
$444$	0	0
$445$	0	0
$446$	546.668i	1.22571i
$447$	0	0
$448$	21.1660i	0.0472456i
$449$	− 809.207i	− 1.80224i	−0.433568	−	0.901121i	$-0.642746\pi$
$449$	− 809.207i	− 1.80224i	0.433568	−	0.901121i	$-0.357254\pi$
$450$	0	0
$451$	−614.369	−1.36224
$452$	−328.891	−0.727634
$453$	0	0
$454$	−119.830	−0.263943
$455$	0	0
$456$	0	0
$457$	− 163.030i	− 0.356740i	−0.983964	−	0.178370i	$-0.942918\pi$
$457$	− 163.030i	− 0.356740i	0.983964	−	0.178370i	$-0.0570824\pi$
$458$	−77.4212	−0.169042
$459$	0	0
$460$	0	0
$461$	− 648.695i	− 1.40715i	−0.710622	−	0.703574i	$-0.751586\pi$
$461$	− 648.695i	− 1.40715i	0.710622	−	0.703574i	$-0.248414\pi$
$462$	0	0
$463$	− 367.210i	− 0.793111i	−0.918011	−	0.396556i	$-0.870205\pi$
$463$	− 367.210i	− 0.793111i	0.918011	−	0.396556i	$-0.129795\pi$
$464$	117.437i	0.253097i
$465$	0	0
$466$	546.589	1.17294
$467$	334.714	0.716732	0.358366	−	0.933581i	$-0.383334\pi$
$467$	334.714	0.716732	0.358366	+	0.933581i	$0.383334\pi$
$468$	0	0
$469$	51.5385	0.109890
$470$	0	0
$471$	0	0
$472$	− 250.494i	− 0.530708i
$473$	−816.705	−1.72665
$474$	0	0
$475$	0	0
$476$	87.6184i	0.184072i
$477$	0	0
$478$	− 534.486i	− 1.11817i
$479$	715.169i	1.49305i	0.665360	+	0.746523i	$0.268278\pi$
$479$	715.169i	1.49305i	−0.665360	+	0.746523i	$0.731722\pi$
$480$	0	0
$481$	−83.3948	−0.173378
$482$	336.248	0.697610
$483$	0	0
$484$	−629.446	−1.30051
$485$	0	0
$486$	0	0
$487$	− 528.320i	− 1.08485i	−0.840105	−	0.542423i	$-0.817507\pi$
$487$	− 528.320i	− 1.08485i	0.840105	−	0.542423i	$-0.182493\pi$
$488$	241.825	0.495543
$489$	0	0
$490$	0	0
$491$	− 205.291i	− 0.418108i	−0.977904	−	0.209054i	$-0.932962\pi$
$491$	− 205.291i	− 0.418108i	0.977904	−	0.209054i	$-0.0670385\pi$
$492$	0	0
$493$	486.140i	0.986085i
$494$	− 14.1421i	− 0.0286278i
$495$	0	0
$496$	−68.6640	−0.138436
$497$	154.843	0.311556
$498$	0	0
$499$	551.542	1.10530	0.552648	−	0.833415i	$-0.313617\pi$
$499$	551.542	1.10530	0.552648	+	0.833415i	$0.313617\pi$
$500$	0	0
$501$	0	0
$502$	− 503.490i	− 1.00297i
$503$	−968.250	−1.92495	−0.962475	−	0.271370i	$-0.912523\pi$
$503$	−968.250	−1.92495	−0.962475	+	0.271370i	$0.912523\pi$
$504$	0	0
$505$	0	0
$506$	697.082i	1.37763i
$507$	0	0
$508$	175.122i	0.344727i
$509$	− 553.218i	− 1.08687i	−0.839450	−	0.543436i	$-0.817123\pi$
$509$	− 553.218i	− 1.08687i	0.839450	−	0.543436i	$-0.182877\pi$
$510$	0	0
$511$	300.561	0.588182
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	165.535	0.322052
$515$	0	0
$516$	0	0
$517$	817.306i	1.58086i
$518$	55.2689	0.106697
$519$	0	0
$520$	0	0
$521$	344.985i	0.662159i	0.943603	+	0.331080i	$0.107413\pi$
$521$	344.985i	0.662159i	−0.943603	+	0.331080i	$0.892587\pi$
$522$	0	0
$523$	154.693i	0.295780i	0.989004	+	0.147890i	$0.0472481\pi$
$523$	154.693i	0.295780i	−0.989004	+	0.147890i	$0.952752\pi$
$524$	− 348.481i	− 0.665039i
$525$	0	0
$526$	175.852	0.334320
$527$	−284.240	−0.539355
$528$	0	0
$529$	28.6052	0.0540742
$530$	0	0
$531$	0	0
$532$	9.37254i	0.0176176i
$533$	−166.167	−0.311759
$534$	0	0
$535$	0	0
$536$	55.0970i	0.102793i
$537$	0	0
$538$	277.373i	0.515562i
$539$	− 146.118i	− 0.271091i
$540$	0	0
$541$	−158.616	−0.293190	−0.146595	−	0.989197i	$-0.546831\pi$
$541$	−158.616	−0.293190	−0.146595	+	0.989197i	$0.546831\pi$
$542$	−14.9981	−0.0276717
$543$	0	0
$544$	−93.6680	−0.172184
$545$	0	0
$546$	0	0
$547$	− 566.342i	− 1.03536i	−0.855574	−	0.517680i	$-0.826796\pi$
$547$	− 566.342i	− 1.03536i	0.855574	−	0.517680i	$-0.173204\pi$
$548$	110.601	0.201826
$549$	0	0
$550$	0	0
$551$	52.0024i	0.0943782i
$552$	0	0
$553$	37.7530i	0.0682694i
$554$	− 6.95090i	− 0.0125467i
$555$	0	0
$556$	−277.122	−0.498420
$557$	−795.517	−1.42822	−0.714109	−	0.700035i	$-0.753168\pi$
$557$	−795.517	−1.42822	−0.714109	+	0.700035i	$0.753168\pi$
$558$	0	0
$559$	−220.893	−0.395157
$560$	0	0
$561$	0	0
$562$	404.059i	0.718966i
$563$	−397.409	−0.705877	−0.352938	−	0.935647i	$-0.614818\pi$
$563$	−397.409	−0.705877	−0.352938	+	0.935647i	$0.614818\pi$
$564$	0	0
$565$	0	0
$566$	− 351.372i	− 0.620799i
$567$	0	0
$568$	165.535i	0.291434i
$569$	260.205i	0.457303i	0.973508	+	0.228651i	$0.0734316\pi$
$569$	260.205i	0.457303i	−0.973508	+	0.228651i	$0.926568\pi$
$570$	0	0
$571$	−193.383	−0.338674	−0.169337	−	0.985558i	$-0.554163\pi$
$571$	−193.383	−0.338674	−0.169337	+	0.985558i	$0.554163\pi$
$572$	−235.699	−0.412060
$573$	0	0
$574$	110.125	0.191856
$575$	0	0
$576$	0	0
$577$	373.593i	0.647476i	0.946147	+	0.323738i	$0.104940\pi$
$577$	373.593i	0.647476i	−0.946147	+	0.323738i	$0.895060\pi$
$578$	20.9617	0.0362660
$579$	0	0
$580$	0	0
$581$	400.123i	0.688679i
$582$	0	0
$583$	414.480i	0.710943i
$584$	321.313i	0.550193i
$585$	0	0
$586$	55.9921	0.0955497
$587$	−1101.38	−1.87628	−0.938142	−	0.346252i	$-0.887454\pi$
$587$	−1101.38	−1.87628	−0.938142	+	0.346252i	$0.887454\pi$
$588$	0	0
$589$	−30.4052	−0.0516217
$590$	0	0
$591$	0	0
$592$	59.0850i	0.0998057i
$593$	−98.5346	−0.166163	−0.0830814	−	0.996543i	$-0.526476\pi$
$593$	−98.5346	−0.166163	−0.0830814	+	0.996543i	$0.526476\pi$
$594$	0	0
$595$	0	0
$596$	298.007i	0.500012i
$597$	0	0
$598$	188.539i	0.315282i
$599$	597.538i	0.997560i	0.866729	+	0.498780i	$0.166218\pi$
$599$	597.538i	0.997560i	−0.866729	+	0.498780i	$0.833782\pi$
$600$	0	0
$601$	564.288	0.938914	0.469457	−	0.882955i	$-0.344450\pi$
$601$	564.288	0.938914	0.469457	+	0.882955i	$0.344450\pi$
$602$	146.394	0.243180
$603$	0	0
$604$	309.587	0.512561
$605$	0	0
$606$	0	0
$607$	− 93.8967i	− 0.154690i	−0.997004	−	0.0773449i	$-0.975356\pi$
$607$	− 93.8967i	− 0.154690i	0.997004	−	0.0773449i	$-0.0246443\pi$
$608$	−10.0197	−0.0164797
$609$	0	0
$610$	0	0
$611$	221.055i	0.361793i
$612$	0	0
$613$	1028.14i	1.67723i	0.544724	+	0.838615i	$0.316634\pi$
$613$	1028.14i	1.67723i	−0.544724	+	0.838615i	$0.683366\pi$
$614$	− 701.344i	− 1.14225i
$615$	0	0
$616$	156.207	0.253582
$617$	−365.326	−0.592101	−0.296051	−	0.955172i	$-0.595670\pi$
$617$	−365.326	−0.592101	−0.296051	+	0.955172i	$0.595670\pi$
$618$	0	0
$619$	90.7020	0.146530	0.0732650	−	0.997313i	$-0.476658\pi$
$619$	90.7020	0.146530	0.0732650	+	0.997313i	$0.476658\pi$
$620$	0	0
$621$	0	0
$622$	191.210i	0.307412i
$623$	41.2414	0.0661981
$624$	0	0
$625$	0	0
$626$	704.110i	1.12478i
$627$	0	0
$628$	− 386.583i	− 0.615578i
$629$	244.587i	0.388850i
$630$	0	0
$631$	1143.43	1.81209	0.906044	−	0.423184i	$-0.139088\pi$
$631$	1143.43	1.81209	0.906044	+	0.423184i	$0.139088\pi$
$632$	−40.3596	−0.0638601
$633$	0	0
$634$	−85.6458	−0.135088
$635$	0	0
$636$	0	0
$637$	− 39.5203i	− 0.0620412i
$638$	866.693	1.35845
$639$	0	0
$640$	0	0
$641$	− 855.704i	− 1.33495i	−0.744632	−	0.667476i	$-0.767375\pi$
$641$	− 855.704i	− 1.33495i	0.744632	−	0.667476i	$-0.232625\pi$
$642$	0	0
$643$	− 426.627i	− 0.663495i	−0.943368	−	0.331748i	$-0.892362\pi$
$643$	− 426.627i	− 0.663495i	0.943368	−	0.331748i	$-0.107638\pi$
$644$	− 124.952i	− 0.194024i
$645$	0	0
$646$	−41.4772	−0.0642062
$647$	354.252	0.547530	0.273765	−	0.961797i	$-0.411731\pi$
$647$	354.252	0.547530	0.273765	+	0.961797i	$0.411731\pi$
$648$	0	0
$649$	−1848.66	−2.84848
$650$	0	0
$651$	0	0
$652$	− 120.988i	− 0.185565i
$653$	136.177	0.208540	0.104270	−	0.994549i	$-0.466749\pi$
$653$	136.177	0.208540	0.104270	+	0.994549i	$0.466749\pi$
$654$	0	0
$655$	0	0
$656$	117.729i	0.179465i
$657$	0	0
$658$	− 146.502i	− 0.222647i
$659$	− 744.120i	− 1.12917i	−0.825376	−	0.564583i	$-0.809037\pi$
$659$	− 744.120i	− 1.12917i	0.825376	−	0.564583i	$-0.190963\pi$
$660$	0	0
$661$	−846.192	−1.28017	−0.640085	−	0.768304i	$-0.721101\pi$
$661$	−846.192	−1.28017	−0.640085	+	0.768304i	$0.721101\pi$
$662$	572.109	0.864214
$663$	0	0
$664$	−427.749	−0.644200
$665$	0	0
$666$	0	0
$667$	− 693.280i	− 1.03940i
$668$	492.782	0.737698
$669$	0	0
$670$	0	0
$671$	− 1784.68i	− 2.65974i
$672$	0	0
$673$	− 626.959i	− 0.931589i	−0.884893	−	0.465795i	$-0.845769\pi$
$673$	− 626.959i	− 0.931589i	0.884893	−	0.465795i	$-0.154231\pi$
$674$	− 572.854i	− 0.849932i
$675$	0	0
$676$	274.251	0.405697
$677$	−53.6772	−0.0792869	−0.0396435	−	0.999214i	$-0.512622\pi$
$677$	−53.6772	−0.0792869	−0.0396435	+	0.999214i	$0.512622\pi$
$678$	0	0
$679$	151.461	0.223065
$680$	0	0
$681$	0	0
$682$	506.745i	0.743028i
$683$	−333.790	−0.488712	−0.244356	−	0.969686i	$-0.578577\pi$
$683$	−333.790	−0.488712	−0.244356	+	0.969686i	$0.578577\pi$
$684$	0	0
$685$	0	0
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	156.502i	0.227474i
$689$	112.104i	0.162705i
$690$	0	0
$691$	−1157.03	−1.67443	−0.837216	−	0.546872i	$-0.815818\pi$
$691$	−1157.03	−1.67443	−0.837216	+	0.546872i	$0.815818\pi$
$692$	−381.649	−0.551516
$693$	0	0
$694$	187.269	0.269840
$695$	0	0
$696$	0	0
$697$	487.349i	0.699209i
$698$	452.986	0.648978
$699$	0	0
$700$	0	0
$701$	− 613.835i	− 0.875656i	−0.899059	−	0.437828i	$-0.855748\pi$
$701$	− 613.835i	− 0.875656i	0.899059	−	0.437828i	$-0.144252\pi$
$702$	0	0
$703$	26.1635i	0.0372169i
$704$	166.992i	0.237204i
$705$	0	0
$706$	−131.174	−0.185799
$707$	−283.192	−0.400555
$708$	0	0
$709$	−995.668	−1.40433	−0.702164	−	0.712016i	$-0.747782\pi$
$709$	−995.668	−1.40433	−0.702164	+	0.712016i	$0.747782\pi$
$710$	0	0
$711$	0	0
$712$	44.0889i	0.0619226i
$713$	405.352	0.568517
$714$	0	0
$715$	0	0
$716$	626.606i	0.875148i
$717$	0	0
$718$	295.816i	0.412000i
$719$	− 1185.52i	− 1.64884i	−0.565977	−	0.824421i	$-0.691501\pi$
$719$	− 1185.52i	− 1.64884i	0.565977	−	0.824421i	$-0.308499\pi$
$720$	0	0
$721$	323.055	0.448065
$722$	506.094	0.700962
$723$	0	0
$724$	413.033	0.570487
$725$	0	0
$726$	0	0
$727$	− 322.067i	− 0.443008i	−0.975159	−	0.221504i	$-0.928903\pi$
$727$	− 322.067i	− 0.443008i	0.975159	−	0.221504i	$-0.0710966\pi$
$728$	42.2489	0.0580343
$729$	0	0
$730$	0	0
$731$	647.852i	0.886255i
$732$	0	0
$733$	− 150.554i	− 0.205395i	−0.994713	−	0.102697i	$-0.967253\pi$
$733$	− 150.554i	− 0.205395i	0.994713	−	0.102697i	$-0.0327473\pi$
$734$	− 529.156i	− 0.720921i
$735$	0	0
$736$	133.579	0.181493
$737$	406.620	0.551723
$738$	0	0
$739$	853.906	1.15549	0.577744	−	0.816218i	$-0.303933\pi$
$739$	853.906	1.15549	0.577744	+	0.816218i	$0.303933\pi$
$740$	0	0
$741$	0	0
$742$	− 74.2954i	− 0.100129i
$743$	380.224	0.511741	0.255871	−	0.966711i	$-0.417638\pi$
$743$	380.224	0.511741	0.255871	+	0.966711i	$0.417638\pi$
$744$	0	0
$745$	0	0
$746$	− 592.189i	− 0.793820i
$747$	0	0
$748$	691.276i	0.924165i
$749$	− 122.066i	− 0.162972i
$750$	0	0
$751$	976.029	1.29964	0.649819	−	0.760089i	$-0.274845\pi$
$751$	976.029	1.29964	0.649819	+	0.760089i	$0.274845\pi$
$752$	156.617	0.208268
$753$	0	0
$754$	234.413	0.310893
$755$	0	0
$756$	0	0
$757$	− 950.889i	− 1.25613i	−0.778162	−	0.628064i	$-0.783848\pi$
$757$	− 950.889i	− 1.25613i	0.778162	−	0.628064i	$-0.216152\pi$
$758$	320.734	0.423132
$759$	0	0
$760$	0	0
$761$	− 397.357i	− 0.522151i	−0.965318	−	0.261076i	$-0.915923\pi$
$761$	− 397.357i	− 0.522151i	0.965318	−	0.261076i	$-0.0840772\pi$
$762$	0	0
$763$	32.5791i	0.0426986i
$764$	− 256.551i	− 0.335800i
$765$	0	0
$766$	−34.0445	−0.0444445
$767$	−500.005	−0.651897
$768$	0	0
$769$	683.555	0.888889	0.444444	−	0.895806i	$-0.353401\pi$
$769$	683.555	0.888889	0.444444	+	0.895806i	$0.353401\pi$
$770$	0	0
$771$	0	0
$772$	− 244.863i	− 0.317180i
$773$	266.059	0.344190	0.172095	−	0.985080i	$-0.444946\pi$
$773$	266.059	0.344190	0.172095	+	0.985080i	$0.444946\pi$
$774$	0	0
$775$	0	0
$776$	161.919i	0.208659i
$777$	0	0
$778$	− 286.148i	− 0.367799i
$779$	52.1317i	0.0669213i
$780$	0	0
$781$	1221.66	1.56422
$782$	552.961	0.707111
$783$	0	0
$784$	−28.0000	−0.0357143
$785$	0	0
$786$	0	0
$787$	1498.17i	1.90364i	0.306650	+	0.951822i	$0.400792\pi$
$787$	1498.17i	1.90364i	−0.306650	+	0.951822i	$0.599208\pi$
$788$	47.8115	0.0606745
$789$	0	0
$790$	0	0
$791$	− 435.081i	− 0.550040i
$792$	0	0
$793$	− 482.701i	− 0.608702i
$794$	− 647.281i	− 0.815215i
$795$	0	0
$796$	−93.4170	−0.117358
$797$	−459.880	−0.577014	−0.288507	−	0.957478i	$-0.593159\pi$
$797$	−459.880	−0.577014	−0.288507	+	0.957478i	$0.593159\pi$
$798$	0	0
$799$	648.329	0.811426
$800$	0	0
$801$	0	0
$802$	− 931.099i	− 1.16097i
$803$	2371.31	2.95306
$804$	0	0
$805$	0	0
$806$	137.059i	0.170048i
$807$	0	0
$808$	− 302.745i	− 0.374685i
$809$	359.649i	0.444560i	0.974983	+	0.222280i	$0.0713500\pi$
$809$	359.649i	0.444560i	−0.974983	+	0.222280i	$0.928650\pi$
$810$	0	0
$811$	−958.316	−1.18165	−0.590824	−	0.806801i	$-0.701197\pi$
$811$	−958.316	−1.18165	−0.590824	+	0.806801i	$0.701197\pi$
$812$	−155.355	−0.191323
$813$	0	0
$814$	436.051	0.535689
$815$	0	0
$816$	0	0
$817$	69.3008i	0.0848235i
$818$	215.106	0.262966
$819$	0	0
$820$	0	0
$821$	575.866i	0.701420i	0.936484	+	0.350710i	$0.114060\pi$
$821$	575.866i	0.701420i	−0.936484	+	0.350710i	$0.885940\pi$
$822$	0	0
$823$	386.852i	0.470051i	0.971989	+	0.235026i	$0.0755174\pi$
$823$	386.852i	0.470051i	−0.971989	+	0.235026i	$0.924483\pi$
$824$	345.360i	0.419126i
$825$	0	0
$826$	331.373	0.401177
$827$	1019.38	1.23262	0.616312	−	0.787502i	$-0.288626\pi$
$827$	1019.38	1.23262	0.616312	+	0.787502i	$0.288626\pi$
$828$	0	0
$829$	1074.36	1.29597	0.647987	−	0.761652i	$-0.275611\pi$
$829$	1074.36	1.29597	0.647987	+	0.761652i	$0.275611\pi$
$830$	0	0
$831$	0	0
$832$	45.1660i	0.0542861i
$833$	−115.908	−0.139146
$834$	0	0
$835$	0	0
$836$	73.9458i	0.0884519i
$837$	0	0
$838$	− 551.660i	− 0.658306i
$839$	991.362i	1.18160i	0.806818	+	0.590800i	$0.201188\pi$
$839$	991.362i	1.18160i	−0.806818	+	0.590800i	$0.798812\pi$
$840$	0	0
$841$	−20.9660	−0.0249298
$842$	−30.1052	−0.0357544
$843$	0	0
$844$	−626.316	−0.742081
$845$	0	0
$846$	0	0
$847$	− 832.678i	− 0.983091i
$848$	79.4252	0.0936618
$849$	0	0
$850$	0	0
$851$	− 348.803i	− 0.409875i
$852$	0	0
$853$	1236.18i	1.44921i	0.689165	+	0.724604i	$0.257977\pi$
$853$	1236.18i	1.44921i	−0.689165	+	0.724604i	$0.742023\pi$
$854$	319.904i	0.374595i
$855$	0	0
$856$	130.494	0.152446
$857$	−402.021	−0.469103	−0.234551	−	0.972104i	$-0.575362\pi$
$857$	−402.021	−0.469103	−0.234551	+	0.972104i	$0.575362\pi$
$858$	0	0
$859$	1315.30	1.53120	0.765602	−	0.643314i	$-0.222441\pi$
$859$	1315.30	1.53120	0.765602	+	0.643314i	$0.222441\pi$
$860$	0	0
$861$	0	0
$862$	− 213.631i	− 0.247832i
$863$	−369.887	−0.428606	−0.214303	−	0.976767i	$-0.568748\pi$
$863$	−369.887	−0.428606	−0.214303	+	0.976767i	$0.568748\pi$
$864$	0	0
$865$	0	0
$866$	− 731.080i	− 0.844203i
$867$	0	0
$868$	− 90.8340i	− 0.104647i
$869$	297.856i	0.342758i
$870$	0	0
$871$	109.978	0.126266
$872$	−34.8285	−0.0399409
$873$	0	0
$874$	59.1503	0.0676776
$875$	0	0
$876$	0	0
$877$	1185.51i	1.35177i	0.737005	+	0.675887i	$0.236239\pi$
$877$	1185.51i	1.35177i	−0.737005	+	0.675887i	$0.763761\pi$
$878$	−774.133	−0.881701
$879$	0	0
$880$	0	0
$881$	− 1202.54i	− 1.36497i	−0.730899	−	0.682486i	$-0.760899\pi$
$881$	− 1202.54i	− 1.36497i	0.730899	−	0.682486i	$-0.239101\pi$
$882$	0	0
$883$	1432.23i	1.62201i	0.585042	+	0.811003i	$0.301078\pi$
$883$	1432.23i	1.62201i	−0.585042	+	0.811003i	$0.698922\pi$
$884$	186.968i	0.211503i
$885$	0	0
$886$	169.263	0.191042
$887$	−851.053	−0.959474	−0.479737	−	0.877412i	$-0.659268\pi$
$887$	−851.053	−0.959474	−0.479737	+	0.877412i	$0.659268\pi$
$888$	0	0
$889$	−231.664	−0.260589
$890$	0	0
$891$	0	0
$892$	− 773.106i	− 0.866711i
$893$	69.3518	0.0776616
$894$	0	0
$895$	0	0
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	1144.39i	1.27438i
$899$	− 503.981i	− 0.560602i
$900$	0	0
$901$	328.787	0.364913
$902$	868.848	0.963247
$903$	0	0
$904$	465.122	0.514515
$905$	0	0
$906$	0	0
$907$	− 94.4496i	− 0.104134i	−0.998644	−	0.0520671i	$-0.983419\pi$
$907$	− 94.4496i	− 0.104134i	0.998644	−	0.0520671i	$-0.0165810\pi$
$908$	169.465	0.186636
$909$	0	0
$910$	0	0
$911$	− 1048.18i	− 1.15058i	−0.817950	−	0.575289i	$-0.804889\pi$
$911$	− 1048.18i	− 1.15058i	0.817950	−	0.575289i	$-0.195111\pi$
$912$	0	0
$913$	3156.82i	3.45763i
$914$	230.559i	0.252253i
$915$	0	0
$916$	109.490	0.119531
$917$	460.997	0.502723
$918$	0	0
$919$	1745.54	1.89939	0.949697	−	0.313169i	$-0.101391\pi$
$919$	1745.54	1.89939	0.949697	+	0.313169i	$0.101391\pi$
$920$	0	0
$921$	0	0
$922$	917.393i	0.995004i
$923$	330.419	0.357984
$924$	0	0
$925$	0	0
$926$	519.314i	0.560814i
$927$	0	0
$928$	− 166.081i	− 0.178967i
$929$	359.861i	0.387364i	0.981064	+	0.193682i	$0.0620429\pi$
$929$	359.861i	0.387364i	−0.981064	+	0.193682i	$0.937957\pi$
$930$	0	0
$931$	−12.3987	−0.0133176
$932$	−772.994	−0.829393
$933$	0	0
$934$	−473.357	−0.506806
$935$	0	0
$936$	0	0
$937$	− 510.065i	− 0.544360i	−0.962246	−	0.272180i	$-0.912255\pi$
$937$	− 510.065i	− 0.544360i	0.962246	−	0.272180i	$-0.0877446\pi$
$938$	−72.8865	−0.0777042
$939$	0	0
$940$	0	0
$941$	1007.90i	1.07110i	0.844505	+	0.535548i	$0.179895\pi$
$941$	1007.90i	1.07110i	−0.844505	+	0.535548i	$0.820105\pi$
$942$	0	0
$943$	− 695.004i	− 0.737014i
$944$	354.252i	0.375267i
$945$	0	0
$946$	1154.99	1.22092
$947$	1116.30	1.17878	0.589390	−	0.807849i	$-0.299368\pi$
$947$	1116.30	1.17878	0.589390	+	0.807849i	$0.299368\pi$
$948$	0	0
$949$	641.365	0.675832
$950$	0	0
$951$	0	0
$952$	− 123.911i	− 0.130159i
$953$	−1162.02	−1.21933	−0.609666	−	0.792658i	$-0.708697\pi$
$953$	−1162.02	−1.21933	−0.609666	+	0.792658i	$0.708697\pi$
$954$	0	0
$955$	0	0
$956$	755.878i	0.790667i
$957$	0	0
$958$	− 1011.40i	− 1.05574i
$959$	146.311i	0.152566i
$960$	0	0
$961$	−666.328	−0.693369
$962$	117.938	0.122597
$963$	0	0
$964$	−475.527	−0.493285
$965$	0	0
$966$	0	0
$967$	− 418.097i	− 0.432365i	−0.976353	−	0.216182i	$-0.930639\pi$
$967$	− 418.097i	− 0.432365i	0.976353	−	0.216182i	$-0.0693606\pi$
$968$	890.171	0.919598
$969$	0	0
$970$	0	0
$971$	− 772.290i	− 0.795355i	−0.917525	−	0.397678i	$-0.869816\pi$
$971$	− 772.290i	− 0.795355i	0.917525	−	0.397678i	$-0.130184\pi$
$972$	0	0
$973$	− 366.597i	− 0.376770i
$974$	747.158i	0.767102i
$975$	0	0
$976$	−341.992	−0.350402
$977$	405.117	0.414654	0.207327	−	0.978272i	$-0.433524\pi$
$977$	405.117	0.414654	0.207327	+	0.978272i	$0.433524\pi$
$978$	0	0
$979$	325.379	0.332359
$980$	0	0
$981$	0	0
$982$	290.326i	0.295647i
$983$	365.743	0.372068	0.186034	−	0.982543i	$-0.440436\pi$
$983$	365.743	0.372068	0.186034	+	0.982543i	$0.440436\pi$
$984$	0	0
$985$	0	0
$986$	− 687.506i	− 0.697267i
$987$	0	0
$988$	20.0000i	0.0202429i
$989$	− 923.896i	− 0.934172i
$990$	0	0
$991$	1288.88	1.30058	0.650292	−	0.759684i	$-0.274646\pi$
$991$	1288.88	1.30058	0.650292	+	0.759684i	$0.274646\pi$
$992$	97.1056	0.0978887
$993$	0	0
$994$	−218.982	−0.220304
$995$	0	0
$996$	0	0
$997$	964.982i	0.967885i	0.875100	+	0.483943i	$0.160796\pi$
$997$	964.982i	0.967885i	−0.875100	+	0.483943i	$0.839204\pi$
$998$	−779.999	−0.781562
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.3.c.c.449.4		8
3.2	odd	2	inner	3150.3.c.c.449.7		8
5.2	odd	4		3150.3.e.d.701.1	yes	4
5.3	odd	4		3150.3.e.a.701.4	yes	4
5.4	even	2	inner	3150.3.c.c.449.6		8
15.2	even	4		3150.3.e.d.701.3	yes	4
15.8	even	4		3150.3.e.a.701.2	✓	4
15.14	odd	2	inner	3150.3.c.c.449.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3150.3.c.c.449.1		8	15.14	odd	2	inner
3150.3.c.c.449.4		8	1.1	even	1	trivial
3150.3.c.c.449.6		8	5.4	even	2	inner
3150.3.c.c.449.7		8	3.2	odd	2	inner
3150.3.e.a.701.2	✓	4	15.8	even	4
3150.3.e.a.701.4	yes	4	5.3	odd	4
3150.3.e.d.701.1	yes	4	5.2	odd	4
3150.3.e.d.701.3	yes	4	15.2	even	4

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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +20.8740i q^{11} +5.64575i q^{13} -3.74166i q^{14} +4.00000 q^{16} +16.5583 q^{17} +1.77124 q^{19} -29.5203i q^{22} -23.6137 q^{23} -7.98430i q^{26} +5.29150i q^{28} +29.3593i q^{29} -17.1660 q^{31} -5.65685 q^{32} -23.4170 q^{34} +14.7712i q^{37} -2.50492 q^{38} +29.4323i q^{41} +39.1255i q^{43} +41.7480i q^{44} +33.3948 q^{46} +39.1543 q^{47} -7.00000 q^{49} +11.2915i q^{52} +19.8563 q^{53} -7.48331i q^{56} -41.5203i q^{58} +88.5630i q^{59} -85.4980 q^{61} +24.2764 q^{62} +8.00000 q^{64} -19.4797i q^{67} +33.1166 q^{68} -58.5253i q^{71} -113.601i q^{73} -20.8897i q^{74} +3.54249 q^{76} -55.2273 q^{77} +14.2693 q^{79} -41.6235i q^{82} +151.232 q^{83} -55.3318i q^{86} -59.0405i q^{88} -15.5878i q^{89} -14.9373 q^{91} -47.2273 q^{92} -55.3725 q^{94} -57.2470i q^{97} +9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \) 8 * q + 16 * q^4	\( 8 q + 16 q^{4} + 32 q^{16} + 120 q^{19} + 32 q^{31} - 272 q^{34} - 8 q^{46} - 56 q^{49} - 176 q^{61} + 64 q^{64} + 240 q^{76} - 288 q^{79} - 56 q^{91} + 192 q^{94}+O(q^{100}) \) 8 * q + 16 * q^4 + 32 * q^16 + 120 * q^19 + 32 * q^31 - 272 * q^34 - 8 * q^46 - 56 * q^49 - 176 * q^61 + 64 * q^64 + 240 * q^76 - 288 * q^79 - 56 * q^91 + 192 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more