LMFDB - Embedded newform 9900.2.c.x.5149.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9900,2,Mod(5149,9900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("9900.5149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9900.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:	$79.0518980011$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 11x^{2} + 25 $ x^4 + 11*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1100)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5149.2
Root			$2.79129i$ of defining polynomial
Character	$\chi$	$=$	9900.5149
Dual form			9900.2.c.x.5149.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.208712i q^{7} +O(q^{10})$	$q-0.208712i q^{7} +1.00000 q^{11} +1.00000i q^{13} -0.791288i q^{17} -6.58258 q^{19} -3.79129i q^{23} +6.79129 q^{29} -8.58258 q^{31} -2.58258i q^{37} +1.41742 q^{41} +10.0000i q^{43} +1.41742i q^{47} +6.95644 q^{49} +11.3739i q^{53} -10.5826 q^{59} +4.20871 q^{61} -4.00000i q^{67} +10.7477 q^{71} +7.79129i q^{73} -0.208712i q^{77} +15.5390 q^{79} -9.95644i q^{83} -0.791288 q^{89} +0.208712 q^{91} -6.20871i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 4 q^{11} - 8 q^{19} + 18 q^{29} - 16 q^{31} + 24 q^{41} - 18 q^{49} - 24 q^{59} + 26 q^{61} - 12 q^{71} - 2 q^{79} + 6 q^{89} + 10 q^{91}+O(q^{100}) $ 4 * q + 4 * q^11 - 8 * q^19 + 18 * q^29 - 16 * q^31 + 24 * q^41 - 18 * q^49 - 24 * q^59 + 26 * q^61 - 12 * q^71 - 2 * q^79 + 6 * q^89 + 10 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/9900\mathbb{Z}\right)^\times$.

$n$	$2377$	$4501$	$4951$	$5501$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.208712i	− 0.0788858i	−0.999222	−	0.0394429i	$-0.987442\pi$
$7$	− 0.208712i	− 0.0788858i	0.999222	−	0.0394429i	$-0.0125583\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 0.791288i	− 0.191915i	−0.995385	−	0.0959577i	$-0.969409\pi$
$17$	− 0.791288i	− 0.191915i	0.995385	−	0.0959577i	$-0.0305914\pi$
$18$	0	0
$19$	−6.58258	−1.51015	−0.755073	−	0.655640i	$-0.772399\pi$
$19$	−6.58258	−1.51015	−0.755073	+	0.655640i	$0.772399\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3.79129i	− 0.790538i	−0.918565	−	0.395269i	$-0.870651\pi$
$23$	− 3.79129i	− 0.790538i	0.918565	−	0.395269i	$-0.129349\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.79129	1.26111	0.630555	−	0.776144i	$-0.282827\pi$
$29$	6.79129	1.26111	0.630555	+	0.776144i	$0.282827\pi$
$30$	0	0
$31$	−8.58258	−1.54148	−0.770738	−	0.637152i	$-0.780112\pi$
$31$	−8.58258	−1.54148	−0.770738	+	0.637152i	$0.780112\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.58258i	− 0.424573i	−0.977207	−	0.212286i	$-0.931909\pi$
$37$	− 2.58258i	− 0.424573i	0.977207	−	0.212286i	$-0.0680910\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.41742	0.221364	0.110682	−	0.993856i	$-0.464696\pi$
$41$	1.41742	0.221364	0.110682	+	0.993856i	$0.464696\pi$
$42$	0	0
$43$	10.0000i	1.52499i	0.646997	+	0.762493i	$0.276025\pi$
$43$	10.0000i	1.52499i	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.41742i	0.206753i	0.994642	+	0.103376i	$0.0329646\pi$
$47$	1.41742i	0.206753i	−0.994642	+	0.103376i	$0.967035\pi$
$48$	0	0
$49$	6.95644	0.993777
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11.3739i	1.56232i	0.624331	+	0.781160i	$0.285372\pi$
$53$	11.3739i	1.56232i	−0.624331	+	0.781160i	$0.714628\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.5826	−1.37773	−0.688867	−	0.724888i	$-0.741892\pi$
$59$	−10.5826	−1.37773	−0.688867	+	0.724888i	$0.741892\pi$
$60$	0	0
$61$	4.20871	0.538870	0.269435	−	0.963019i	$-0.413163\pi$
$61$	4.20871	0.538870	0.269435	+	0.963019i	$0.413163\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.7477	1.27552	0.637760	−	0.770235i	$-0.279861\pi$
$71$	10.7477	1.27552	0.637760	+	0.770235i	$0.279861\pi$
$72$	0	0
$73$	7.79129i	0.911901i	0.890005	+	0.455951i	$0.150701\pi$
$73$	7.79129i	0.911901i	−0.890005	+	0.455951i	$0.849299\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.208712i	− 0.0237850i
$78$	0	0
$79$	15.5390	1.74828	0.874138	−	0.485678i	$-0.161427\pi$
$79$	15.5390	1.74828	0.874138	+	0.485678i	$0.161427\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 9.95644i	− 1.09286i	−0.837504	−	0.546431i	$-0.815986\pi$
$83$	− 9.95644i	− 1.09286i	0.837504	−	0.546431i	$-0.184014\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.791288	−0.0838763	−0.0419382	−	0.999120i	$-0.513353\pi$
$89$	−0.791288	−0.0838763	−0.0419382	+	0.999120i	$0.513353\pi$
$90$	0	0
$91$	0.208712	0.0218790
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.20871i	− 0.630399i	−0.949025	−	0.315200i	$-0.897929\pi$
$97$	− 6.20871i	− 0.630399i	0.949025	−	0.315200i	$-0.102071\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.3739	1.72876	0.864382	−	0.502836i	$-0.167710\pi$
$101$	17.3739	1.72876	0.864382	+	0.502836i	$0.167710\pi$
$102$	0	0
$103$	− 5.95644i	− 0.586905i	−0.955974	−	0.293453i	$-0.905196\pi$
$103$	− 5.95644i	− 0.586905i	0.955974	−	0.293453i	$-0.0948043\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 19.7477i	− 1.90908i	−0.298075	−	0.954542i	$-0.596345\pi$
$107$	− 19.7477i	− 1.90908i	0.298075	−	0.954542i	$-0.403655\pi$
$108$	0	0
$109$	4.79129	0.458922	0.229461	−	0.973318i	$-0.426304\pi$
$109$	4.79129	0.458922	0.229461	+	0.973318i	$0.426304\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.41742i	0.133340i	0.997775	+	0.0666700i	$0.0212375\pi$
$113$	1.41742i	0.133340i	−0.997775	+	0.0666700i	$0.978763\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.165151	−0.0151394
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.3739i	1.45295i	0.687195	+	0.726473i	$0.258842\pi$
$127$	16.3739i	1.45295i	−0.687195	+	0.726473i	$0.741158\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.626136	0.0547058	0.0273529	−	0.999626i	$-0.491292\pi$
$131$	0.626136	0.0547058	0.0273529	+	0.999626i	$0.491292\pi$
$132$	0	0
$133$	1.37386i	0.119129i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.5390i	0.985845i	0.870073	+	0.492922i	$0.164071\pi$
$137$	11.5390i	0.985845i	−0.870073	+	0.492922i	$0.835929\pi$
$138$	0	0
$139$	−9.74773	−0.826791	−0.413396	−	0.910551i	$-0.635657\pi$
$139$	−9.74773	−0.826791	−0.413396	+	0.910551i	$0.635657\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.00000i	0.0836242i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.16515	0.259299	0.129650	−	0.991560i	$-0.458615\pi$
$149$	3.16515	0.259299	0.129650	+	0.991560i	$0.458615\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	17.1652i	1.36993i	0.728577	+	0.684964i	$0.240182\pi$
$157$	17.1652i	1.36993i	−0.728577	+	0.684964i	$0.759818\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.791288	−0.0623622
$162$	0	0
$163$	21.3739i	1.67413i	0.547103	+	0.837065i	$0.315730\pi$
$163$	21.3739i	1.67413i	−0.547103	+	0.837065i	$0.684270\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.5826i	1.28320i	0.767040	+	0.641599i	$0.221729\pi$
$167$	16.5826i	1.28320i	−0.767040	+	0.641599i	$0.778271\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 13.7477i	− 1.04522i	−0.852572	−	0.522610i	$-0.824958\pi$
$173$	− 13.7477i	− 1.04522i	0.852572	−	0.522610i	$-0.175042\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	22.1216	1.65345	0.826723	−	0.562610i	$-0.190203\pi$
$179$	22.1216	1.65345	0.826723	+	0.562610i	$0.190203\pi$
$180$	0	0
$181$	−3.37386	−0.250777	−0.125389	−	0.992108i	$-0.540018\pi$
$181$	−3.37386	−0.250777	−0.125389	+	0.992108i	$0.540018\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.791288i	− 0.0578647i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17.2087	−1.24518	−0.622589	−	0.782549i	$-0.713919\pi$
$191$	−17.2087	−1.24518	−0.622589	+	0.782549i	$0.713919\pi$
$192$	0	0
$193$	7.16515i	0.515759i	0.966177	+	0.257879i	$0.0830237\pi$
$193$	7.16515i	0.515759i	−0.966177	+	0.257879i	$0.916976\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 26.5390i	− 1.89083i	−0.325874	−	0.945413i	$-0.605658\pi$
$197$	− 26.5390i	− 1.89083i	0.325874	−	0.945413i	$-0.394342\pi$
$198$	0	0
$199$	1.62614	0.115274	0.0576369	−	0.998338i	$-0.481643\pi$
$199$	1.62614	0.115274	0.0576369	+	0.998338i	$0.481643\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.41742i	− 0.0994837i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6.58258	−0.455326
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.79129i	0.121601i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.791288	0.0532278
$222$	0	0
$223$	20.5826i	1.37831i	0.724613	+	0.689156i	$0.242018\pi$
$223$	20.5826i	1.37831i	−0.724613	+	0.689156i	$0.757982\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 11.5390i	− 0.765871i	−0.923775	−	0.382936i	$-0.874913\pi$
$227$	− 11.5390i	− 0.765871i	0.923775	−	0.382936i	$-0.125087\pi$
$228$	0	0
$229$	7.62614	0.503949	0.251975	−	0.967734i	$-0.418920\pi$
$229$	7.62614	0.503949	0.251975	+	0.967734i	$0.418920\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.7913i	1.03452i	0.855828	+	0.517261i	$0.173048\pi$
$233$	15.7913i	1.03452i	−0.855828	+	0.517261i	$0.826952\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.20871	−0.142870	−0.0714349	−	0.997445i	$-0.522758\pi$
$239$	−2.20871	−0.142870	−0.0714349	+	0.997445i	$0.522758\pi$
$240$	0	0
$241$	−23.1216	−1.48939	−0.744696	−	0.667404i	$-0.767406\pi$
$241$	−23.1216	−1.48939	−0.744696	+	0.667404i	$0.767406\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 6.58258i	− 0.418839i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	11.5390	0.728336	0.364168	−	0.931333i	$-0.381353\pi$
$251$	11.5390	0.728336	0.364168	+	0.931333i	$0.381353\pi$
$252$	0	0
$253$	− 3.79129i	− 0.238356i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	−0.539015	−0.0334928
$260$	0	0
$261$	0	0
$262$	0	0
$263$	24.1652i	1.49009i	0.667016	+	0.745044i	$0.267571\pi$
$263$	24.1652i	1.49009i	−0.667016	+	0.745044i	$0.732429\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.7042	−1.26236	−0.631178	−	0.775638i	$-0.717428\pi$
$269$	−20.7042	−1.26236	−0.631178	+	0.775638i	$0.717428\pi$
$270$	0	0
$271$	15.7477	0.956606	0.478303	−	0.878195i	$-0.341252\pi$
$271$	15.7477	0.956606	0.478303	+	0.878195i	$0.341252\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 8.58258i	− 0.515677i	−0.966188	−	0.257838i	$-0.916990\pi$
$277$	− 8.58258i	− 0.515677i	0.966188	−	0.257838i	$-0.0830102\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	13.7477	0.820121	0.410060	−	0.912058i	$-0.365508\pi$
$281$	13.7477	0.820121	0.410060	+	0.912058i	$0.365508\pi$
$282$	0	0
$283$	30.7042i	1.82517i	0.408883	+	0.912587i	$0.365918\pi$
$283$	30.7042i	1.82517i	−0.408883	+	0.912587i	$0.634082\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 0.295834i	− 0.0174625i
$288$	0	0
$289$	16.3739	0.963168
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2.83485i	− 0.165614i	−0.996566	−	0.0828068i	$-0.973612\pi$
$293$	− 2.83485i	− 0.165614i	0.996566	−	0.0828068i	$-0.0263884\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.79129	0.219256
$300$	0	0
$301$	2.08712	0.120300
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 27.2087i	− 1.55288i	−0.630189	−	0.776442i	$-0.717023\pi$
$307$	− 27.2087i	− 1.55288i	0.630189	−	0.776442i	$-0.282977\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.83485	0.330864	0.165432	−	0.986221i	$-0.447098\pi$
$311$	5.83485	0.330864	0.165432	+	0.986221i	$0.447098\pi$
$312$	0	0
$313$	− 6.74773i	− 0.381404i	−0.981648	−	0.190702i	$-0.938924\pi$
$313$	− 6.74773i	− 0.381404i	0.981648	−	0.190702i	$-0.0610764\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.37386i	0.470323i	0.971956	+	0.235162i	$0.0755619\pi$
$317$	8.37386i	0.470323i	−0.971956	+	0.235162i	$0.924438\pi$
$318$	0	0
$319$	6.79129	0.380239
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.20871i	0.289820i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0.295834	0.0163098
$330$	0	0
$331$	17.3303	0.952560	0.476280	−	0.879294i	$-0.341985\pi$
$331$	17.3303	0.952560	0.476280	+	0.879294i	$0.341985\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 22.0000i	− 1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	− 22.0000i	− 1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8.58258	−0.464773
$342$	0	0
$343$	− 2.91288i	− 0.157281i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 12.6261i	− 0.677807i	−0.940821	−	0.338903i	$-0.889944\pi$
$347$	− 12.6261i	− 0.677807i	0.940821	−	0.338903i	$-0.110056\pi$
$348$	0	0
$349$	14.5826	0.780587	0.390294	−	0.920690i	$-0.372373\pi$
$349$	14.5826	0.780587	0.390294	+	0.920690i	$0.372373\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.1652i	0.966833i	0.875390	+	0.483417i	$0.160604\pi$
$353$	18.1652i	0.966833i	−0.875390	+	0.483417i	$0.839396\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	24.3303	1.28054
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.62614i	0.450281i	0.974326	+	0.225140i	$0.0722841\pi$
$367$	8.62614i	0.450281i	−0.974326	+	0.225140i	$0.927716\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.37386	0.123245
$372$	0	0
$373$	19.3303i	1.00089i	0.865770	+	0.500443i	$0.166829\pi$
$373$	19.3303i	1.00089i	−0.865770	+	0.500443i	$0.833171\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.79129i	0.349769i
$378$	0	0
$379$	31.0000	1.59236	0.796182	−	0.605058i	$-0.206850\pi$
$379$	31.0000	1.59236	0.796182	+	0.605058i	$0.206850\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 18.0000i	− 0.919757i	−0.887982	−	0.459879i	$-0.847893\pi$
$383$	− 18.0000i	− 0.919757i	0.887982	−	0.459879i	$-0.152107\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−3.00000	−0.151717
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.20871i	0.361795i	0.983502	+	0.180897i	$0.0579002\pi$
$397$	7.20871i	0.361795i	−0.983502	+	0.180897i	$0.942100\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.8348	0.740817	0.370408	−	0.928869i	$-0.379218\pi$
$401$	14.8348	0.740817	0.370408	+	0.928869i	$0.379218\pi$
$402$	0	0
$403$	− 8.58258i	− 0.427529i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2.58258i	− 0.128014i
$408$	0	0
$409$	−17.0000	−0.840596	−0.420298	−	0.907386i	$-0.638074\pi$
$409$	−17.0000	−0.840596	−0.420298	+	0.907386i	$0.638074\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.20871i	0.108684i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	27.0000	1.31904	0.659518	−	0.751689i	$-0.270760\pi$
$419$	27.0000	1.31904	0.659518	+	0.751689i	$0.270760\pi$
$420$	0	0
$421$	25.3739	1.23665	0.618323	−	0.785924i	$-0.287812\pi$
$421$	25.3739	1.23665	0.618323	+	0.785924i	$0.287812\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 0.878409i	− 0.0425092i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.9129	−0.525655	−0.262827	−	0.964843i	$-0.584655\pi$
$431$	−10.9129	−0.525655	−0.262827	+	0.964843i	$0.584655\pi$
$432$	0	0
$433$	13.3303i	0.640613i	0.947314	+	0.320307i	$0.103786\pi$
$433$	13.3303i	0.640613i	−0.947314	+	0.320307i	$0.896214\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24.9564i	1.19383i
$438$	0	0
$439$	−11.9564	−0.570650	−0.285325	−	0.958431i	$-0.592101\pi$
$439$	−11.9564	−0.570650	−0.285325	+	0.958431i	$0.592101\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	22.4174i	1.06508i	0.846404	+	0.532542i	$0.178763\pi$
$443$	22.4174i	1.06508i	−0.846404	+	0.532542i	$0.821237\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	26.2087	1.23687	0.618433	−	0.785838i	$-0.287768\pi$
$449$	26.2087	1.23687	0.618433	+	0.785838i	$0.287768\pi$
$450$	0	0
$451$	1.41742	0.0667439
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.3739i	0.625603i	0.949819	+	0.312801i	$0.101267\pi$
$457$	13.3739i	0.625603i	−0.949819	+	0.312801i	$0.898733\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	24.4955	1.14087	0.570434	−	0.821344i	$-0.306775\pi$
$461$	24.4955	1.14087	0.570434	+	0.821344i	$0.306775\pi$
$462$	0	0
$463$	2.25227i	0.104672i	0.998630	+	0.0523360i	$0.0166667\pi$
$463$	2.25227i	0.104672i	−0.998630	+	0.0523360i	$0.983333\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 10.5826i	− 0.489703i	−0.969561	−	0.244852i	$-0.921261\pi$
$467$	− 10.5826i	− 0.489703i	0.969561	−	0.244852i	$-0.0787392\pi$
$468$	0	0
$469$	−0.834849	−0.0385497
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10.0000i	0.459800i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0.165151	0.00754596	0.00377298	−	0.999993i	$-0.498799\pi$
$479$	0.165151	0.00754596	0.00377298	+	0.999993i	$0.498799\pi$
$480$	0	0
$481$	2.58258	0.117755
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 29.7477i	− 1.34800i	−0.738732	−	0.673999i	$-0.764575\pi$
$487$	− 29.7477i	− 1.34800i	0.738732	−	0.673999i	$-0.235425\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.41742	0.334744	0.167372	−	0.985894i	$-0.446472\pi$
$491$	7.41742	0.334744	0.167372	+	0.985894i	$0.446472\pi$
$492$	0	0
$493$	− 5.37386i	− 0.242027i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2.24318i	− 0.100620i
$498$	0	0
$499$	−14.9564	−0.669542	−0.334771	−	0.942299i	$-0.608659\pi$
$499$	−14.9564	−0.669542	−0.334771	+	0.942299i	$0.608659\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.4174i	0.732017i	0.930612	+	0.366008i	$0.119276\pi$
$503$	16.4174i	0.732017i	−0.930612	+	0.366008i	$0.880724\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.1216	0.448632	0.224316	−	0.974517i	$-0.427985\pi$
$509$	10.1216	0.448632	0.224316	+	0.974517i	$0.427985\pi$
$510$	0	0
$511$	1.62614	0.0719360
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.41742i	0.0623382i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.58258	−0.0693339	−0.0346670	−	0.999399i	$-0.511037\pi$
$521$	−1.58258	−0.0693339	−0.0346670	+	0.999399i	$0.511037\pi$
$522$	0	0
$523$	25.1652i	1.10040i	0.835034	+	0.550198i	$0.185448\pi$
$523$	25.1652i	1.10040i	−0.835034	+	0.550198i	$0.814552\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.79129i	0.295833i
$528$	0	0
$529$	8.62614	0.375049
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.41742i	0.0613955i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.95644	0.299635
$540$	0	0
$541$	−10.6261	−0.456853	−0.228427	−	0.973561i	$-0.573358\pi$
$541$	−10.6261	−0.456853	−0.228427	+	0.973561i	$0.573358\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7.37386i	0.315284i	0.987496	+	0.157642i	$0.0503891\pi$
$547$	7.37386i	0.315284i	−0.987496	+	0.157642i	$0.949611\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−44.7042	−1.90446
$552$	0	0
$553$	− 3.24318i	− 0.137914i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	45.6606i	1.93470i	0.253441	+	0.967351i	$0.418438\pi$
$557$	45.6606i	1.93470i	−0.253441	+	0.967351i	$0.581562\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 40.1216i	− 1.69092i	−0.534036	−	0.845462i	$-0.679325\pi$
$563$	− 40.1216i	− 1.69092i	0.534036	−	0.845462i	$-0.320675\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.9564	0.794695	0.397348	−	0.917668i	$-0.369931\pi$
$569$	18.9564	0.794695	0.397348	+	0.917668i	$0.369931\pi$
$570$	0	0
$571$	27.2867	1.14191	0.570957	−	0.820980i	$-0.306572\pi$
$571$	27.2867	1.14191	0.570957	+	0.820980i	$0.306572\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11.9564i	0.497753i	0.968535	+	0.248877i	$0.0800613\pi$
$577$	11.9564i	0.497753i	−0.968535	+	0.248877i	$0.919939\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.07803	−0.0862112
$582$	0	0
$583$	11.3739i	0.471057i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15.9564i	− 0.658593i	−0.944227	−	0.329296i	$-0.893188\pi$
$587$	− 15.9564i	− 0.658593i	0.944227	−	0.329296i	$-0.106812\pi$
$588$	0	0
$589$	56.4955	2.32785
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 22.5826i	− 0.927355i	−0.886004	−	0.463678i	$-0.846530\pi$
$593$	− 22.5826i	− 0.927355i	0.886004	−	0.463678i	$-0.153470\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	44.7042	1.82656	0.913281	−	0.407329i	$-0.133540\pi$
$599$	44.7042	1.82656	0.913281	+	0.407329i	$0.133540\pi$
$600$	0	0
$601$	28.5390	1.16413	0.582065	−	0.813142i	$-0.302245\pi$
$601$	28.5390	1.16413	0.582065	+	0.813142i	$0.302245\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 34.0000i	− 1.38002i	−0.723801	−	0.690009i	$-0.757607\pi$
$607$	− 34.0000i	− 1.38002i	0.723801	−	0.690009i	$-0.242393\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.41742	−0.0573428
$612$	0	0
$613$	− 33.4519i	− 1.35111i	−0.737310	−	0.675555i	$-0.763904\pi$
$613$	− 33.4519i	− 1.35111i	0.737310	−	0.675555i	$-0.236096\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	28.7477i	1.15734i	0.815562	+	0.578670i	$0.196428\pi$
$617$	28.7477i	1.15734i	−0.815562	+	0.578670i	$0.803572\pi$
$618$	0	0
$619$	−6.25227	−0.251300	−0.125650	−	0.992075i	$-0.540102\pi$
$619$	−6.25227	−0.251300	−0.125650	+	0.992075i	$0.540102\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.165151i	0.00661665i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.04356	−0.0814821
$630$	0	0
$631$	18.1216	0.721409	0.360705	−	0.932680i	$-0.382536\pi$
$631$	18.1216	0.721409	0.360705	+	0.932680i	$0.382536\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.95644i	0.275624i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23.8348	0.941420	0.470710	−	0.882288i	$-0.343998\pi$
$641$	23.8348	0.941420	0.470710	+	0.882288i	$0.343998\pi$
$642$	0	0
$643$	43.4955i	1.71529i	0.514239	+	0.857647i	$0.328074\pi$
$643$	43.4955i	1.71529i	−0.514239	+	0.857647i	$0.671926\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 21.3303i	− 0.838581i	−0.907852	−	0.419290i	$-0.862279\pi$
$647$	− 21.3303i	− 0.838581i	0.907852	−	0.419290i	$-0.137721\pi$
$648$	0	0
$649$	−10.5826	−0.415402
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.7042i	1.04501i	0.852635	+	0.522507i	$0.175003\pi$
$653$	26.7042i	1.04501i	−0.852635	+	0.522507i	$0.824997\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0.460985	0.0179574	0.00897871	−	0.999960i	$-0.497142\pi$
$659$	0.460985	0.0179574	0.00897871	+	0.999960i	$0.497142\pi$
$660$	0	0
$661$	30.5826	1.18952	0.594762	−	0.803902i	$-0.297246\pi$
$661$	30.5826	1.18952	0.594762	+	0.803902i	$0.297246\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 25.7477i	− 0.996956i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.20871	0.162476
$672$	0	0
$673$	38.9129i	1.49998i	0.661448	+	0.749991i	$0.269942\pi$
$673$	38.9129i	1.49998i	−0.661448	+	0.749991i	$0.730058\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.58258i	0.0608233i	0.999537	+	0.0304117i	$0.00968183\pi$
$677$	1.58258i	0.0608233i	−0.999537	+	0.0304117i	$0.990318\pi$
$678$	0	0
$679$	−1.29583	−0.0497295
$680$	0	0
$681$	0	0
$682$	0	0
$683$	34.9129i	1.33590i	0.744204	+	0.667952i	$0.232829\pi$
$683$	34.9129i	1.33590i	−0.744204	+	0.667952i	$0.767171\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−11.3739	−0.433310
$690$	0	0
$691$	−32.1216	−1.22196	−0.610981	−	0.791645i	$-0.709225\pi$
$691$	−32.1216	−1.22196	−0.610981	+	0.791645i	$0.709225\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.12159i	− 0.0424833i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−20.8348	−0.786921	−0.393461	−	0.919341i	$-0.628722\pi$
$701$	−20.8348	−0.786921	−0.393461	+	0.919341i	$0.628722\pi$
$702$	0	0
$703$	17.0000i	0.641167i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 3.62614i	− 0.136375i
$708$	0	0
$709$	−29.4955	−1.10773	−0.553863	−	0.832608i	$-0.686847\pi$
$709$	−29.4955	−1.10773	−0.553863	+	0.832608i	$0.686847\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	32.5390i	1.21860i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.49545	0.242240	0.121120	−	0.992638i	$-0.461351\pi$
$719$	6.49545	0.242240	0.121120	+	0.992638i	$0.461351\pi$
$720$	0	0
$721$	−1.24318	−0.0462985
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 23.1216i	− 0.857532i	−0.903416	−	0.428766i	$-0.858948\pi$
$727$	− 23.1216i	− 0.857532i	0.903416	−	0.428766i	$-0.141052\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.91288	0.292668
$732$	0	0
$733$	− 31.2432i	− 1.15399i	−0.816747	−	0.576997i	$-0.804225\pi$
$733$	− 31.2432i	− 1.15399i	0.816747	−	0.576997i	$-0.195775\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4.00000i	− 0.147342i
$738$	0	0
$739$	−42.1216	−1.54947	−0.774734	−	0.632287i	$-0.782116\pi$
$739$	−42.1216	−1.54947	−0.774734	+	0.632287i	$0.782116\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	39.7913i	1.45980i	0.683554	+	0.729900i	$0.260434\pi$
$743$	39.7913i	1.45980i	−0.683554	+	0.729900i	$0.739566\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.12159	−0.150600
$750$	0	0
$751$	13.2087	0.481993	0.240996	−	0.970526i	$-0.422526\pi$
$751$	13.2087	0.481993	0.240996	+	0.970526i	$0.422526\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 7.33030i	− 0.266424i	−0.991088	−	0.133212i	$-0.957471\pi$
$757$	− 7.33030i	− 0.266424i	0.991088	−	0.133212i	$-0.0425292\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−30.3303	−1.09947	−0.549736	−	0.835338i	$-0.685272\pi$
$761$	−30.3303	−1.09947	−0.549736	+	0.835338i	$0.685272\pi$
$762$	0	0
$763$	− 1.00000i	− 0.0362024i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 10.5826i	− 0.382115i
$768$	0	0
$769$	41.7477	1.50546	0.752731	−	0.658328i	$-0.228736\pi$
$769$	41.7477	1.50546	0.752731	+	0.658328i	$0.228736\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26.2087i	0.942662i	0.881956	+	0.471331i	$0.156226\pi$
$773$	26.2087i	0.942662i	−0.881956	+	0.471331i	$0.843774\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.33030	−0.334293
$780$	0	0
$781$	10.7477	0.384584
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 17.7477i	− 0.632638i	−0.948653	−	0.316319i	$-0.897553\pi$
$787$	− 17.7477i	− 0.632638i	0.948653	−	0.316319i	$-0.102447\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.295834	0.0105186
$792$	0	0
$793$	4.20871i	0.149456i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 34.2867i	− 1.21450i	−0.794511	−	0.607249i	$-0.792273\pi$
$797$	− 34.2867i	− 1.21450i	0.794511	−	0.607249i	$-0.207727\pi$
$798$	0	0
$799$	1.12159	0.0396790
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.79129i	0.274949i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−45.4955	−1.59953	−0.799767	−	0.600310i	$-0.795044\pi$
$809$	−45.4955	−1.59953	−0.799767	+	0.600310i	$0.795044\pi$
$810$	0	0
$811$	−55.6606	−1.95451	−0.977254	−	0.212072i	$-0.931979\pi$
$811$	−55.6606	−1.95451	−0.977254	+	0.212072i	$0.931979\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 65.8258i	− 2.30295i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−48.6606	−1.69827	−0.849133	−	0.528178i	$-0.822875\pi$
$821$	−48.6606	−1.69827	−0.849133	+	0.528178i	$0.822875\pi$
$822$	0	0
$823$	− 8.00000i	− 0.278862i	−0.990232	−	0.139431i	$-0.955473\pi$
$823$	− 8.00000i	− 0.278862i	0.990232	−	0.139431i	$-0.0445274\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	10.4174i	0.362249i	0.983460	+	0.181125i	$0.0579738\pi$
$827$	10.4174i	0.362249i	−0.983460	+	0.181125i	$0.942026\pi$
$828$	0	0
$829$	37.9564	1.31828	0.659141	−	0.752020i	$-0.270920\pi$
$829$	37.9564	1.31828	0.659141	+	0.752020i	$0.270920\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 5.50455i	− 0.190721i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23.7042	−0.818359	−0.409179	−	0.912454i	$-0.634185\pi$
$839$	−23.7042	−0.818359	−0.409179	+	0.912454i	$0.634185\pi$
$840$	0	0
$841$	17.1216	0.590400
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 0.208712i	− 0.00717143i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−9.79129	−0.335641
$852$	0	0
$853$	− 3.12159i	− 0.106881i	−0.998571	−	0.0534406i	$-0.982981\pi$
$853$	− 3.12159i	− 0.106881i	0.998571	−	0.0534406i	$-0.0170188\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.4955i	1.04170i	0.853647	+	0.520852i	$0.174386\pi$
$857$	30.4955i	1.04170i	−0.853647	+	0.520852i	$0.825614\pi$
$858$	0	0
$859$	2.08712	0.0712117	0.0356058	−	0.999366i	$-0.488664\pi$
$859$	2.08712	0.0712117	0.0356058	+	0.999366i	$0.488664\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 27.3303i	− 0.930334i	−0.885223	−	0.465167i	$-0.845994\pi$
$863$	− 27.3303i	− 0.930334i	0.885223	−	0.465167i	$-0.154006\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	15.5390	0.527125
$870$	0	0
$871$	4.00000	0.135535
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 16.6606i	− 0.562589i	−0.959622	−	0.281294i	$-0.909236\pi$
$877$	− 16.6606i	− 0.562589i	0.959622	−	0.281294i	$-0.0907637\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.87841	−0.0632852	−0.0316426	−	0.999499i	$-0.510074\pi$
$881$	−1.87841	−0.0632852	−0.0316426	+	0.999499i	$0.510074\pi$
$882$	0	0
$883$	− 26.1652i	− 0.880527i	−0.897869	−	0.440264i	$-0.854885\pi$
$883$	− 26.1652i	− 0.880527i	0.897869	−	0.440264i	$-0.145115\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	39.3303i	1.32058i	0.751010	+	0.660291i	$0.229567\pi$
$887$	39.3303i	1.32058i	−0.751010	+	0.660291i	$0.770433\pi$
$888$	0	0
$889$	3.41742	0.114617
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 9.33030i	− 0.312227i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−58.2867	−1.94397
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.6606i	1.08448i	0.840224	+	0.542239i	$0.182423\pi$
$907$	32.6606i	1.08448i	−0.840224	+	0.542239i	$0.817577\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	12.3303	0.408521	0.204261	−	0.978917i	$-0.434521\pi$
$911$	12.3303	0.408521	0.204261	+	0.978917i	$0.434521\pi$
$912$	0	0
$913$	− 9.95644i	− 0.329510i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 0.130682i	− 0.00431551i
$918$	0	0
$919$	−28.8348	−0.951174	−0.475587	−	0.879669i	$-0.657764\pi$
$919$	−28.8348	−0.951174	−0.475587	+	0.879669i	$0.657764\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10.7477i	0.353766i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	13.7477	0.451048	0.225524	−	0.974238i	$-0.427591\pi$
$929$	13.7477	0.451048	0.225524	+	0.974238i	$0.427591\pi$
$930$	0	0
$931$	−45.7913	−1.50075
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.1652i	1.14880i	0.818576	+	0.574398i	$0.194764\pi$
$937$	35.1652i	1.14880i	−0.818576	+	0.574398i	$0.805236\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	50.0780	1.63250	0.816249	−	0.577701i	$-0.196050\pi$
$941$	50.0780	1.63250	0.816249	+	0.577701i	$0.196050\pi$
$942$	0	0
$943$	− 5.37386i	− 0.174997i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.91288i	0.159647i	0.996809	+	0.0798235i	$0.0254357\pi$
$947$	4.91288i	0.159647i	−0.996809	+	0.0798235i	$0.974564\pi$
$948$	0	0
$949$	−7.79129	−0.252916
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 42.0000i	− 1.36051i	−0.732974	−	0.680257i	$-0.761868\pi$
$953$	− 42.0000i	− 1.36051i	0.732974	−	0.680257i	$-0.238132\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.40833	0.0777691
$960$	0	0
$961$	42.6606	1.37615
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	43.5390i	1.40012i	0.714084	+	0.700060i	$0.246844\pi$
$967$	43.5390i	1.40012i	−0.714084	+	0.700060i	$0.753156\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−18.9564	−0.608341	−0.304171	−	0.952618i	$-0.598379\pi$
$971$	−18.9564	−0.608341	−0.304171	+	0.952618i	$0.598379\pi$
$972$	0	0
$973$	2.03447i	0.0652221i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.7477i	0.823743i	0.911242	+	0.411871i	$0.135125\pi$
$977$	25.7477i	0.823743i	−0.911242	+	0.411871i	$0.864875\pi$
$978$	0	0
$979$	−0.791288	−0.0252897
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 48.6606i	− 1.55203i	−0.630713	−	0.776016i	$-0.717237\pi$
$983$	− 48.6606i	− 1.55203i	0.630713	−	0.776016i	$-0.282763\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	37.9129	1.20556
$990$	0	0
$991$	55.8693	1.77475	0.887374	−	0.461051i	$-0.152527\pi$
$991$	55.8693	1.77475	0.887374	+	0.461051i	$0.152527\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	40.3739i	1.27865i	0.768935	+	0.639327i	$0.220787\pi$
$997$	40.3739i	1.27865i	−0.768935	+	0.639327i	$0.779213\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9900.2.c.x.5149.2		4
3.2	odd	2		1100.2.b.d.749.4		4
5.2	odd	4		9900.2.a.bz.1.1		2
5.3	odd	4		9900.2.a.bh.1.2		2
5.4	even	2	inner	9900.2.c.x.5149.3		4
12.11	even	2		4400.2.b.s.4049.1		4
15.2	even	4		1100.2.a.h.1.2	yes	2
15.8	even	4		1100.2.a.g.1.1	✓	2
15.14	odd	2		1100.2.b.d.749.1		4
60.23	odd	4		4400.2.a.bu.1.2		2
60.47	odd	4		4400.2.a.bi.1.1		2
60.59	even	2		4400.2.b.s.4049.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.2.a.g.1.1	✓	2	15.8	even	4
1100.2.a.h.1.2	yes	2	15.2	even	4
1100.2.b.d.749.1		4	15.14	odd	2
1100.2.b.d.749.4		4	3.2	odd	2
4400.2.a.bi.1.1		2	60.47	odd	4
4400.2.a.bu.1.2		2	60.23	odd	4
4400.2.b.s.4049.1		4	12.11	even	2
4400.2.b.s.4049.4		4	60.59	even	2
9900.2.a.bh.1.2		2	5.3	odd	4
9900.2.a.bz.1.1		2	5.2	odd	4
9900.2.c.x.5149.2		4	1.1	even	1	trivial
9900.2.c.x.5149.3		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.208712i q^{7} +O(q^{10})\)	\(q-0.208712i q^{7} +1.00000 q^{11} +1.00000i q^{13} -0.791288i q^{17} -6.58258 q^{19} -3.79129i q^{23} +6.79129 q^{29} -8.58258 q^{31} -2.58258i q^{37} +1.41742 q^{41} +10.0000i q^{43} +1.41742i q^{47} +6.95644 q^{49} +11.3739i q^{53} -10.5826 q^{59} +4.20871 q^{61} -4.00000i q^{67} +10.7477 q^{71} +7.79129i q^{73} -0.208712i q^{77} +15.5390 q^{79} -9.95644i q^{83} -0.791288 q^{89} +0.208712 q^{91} -6.20871i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 4 q^{11} - 8 q^{19} + 18 q^{29} - 16 q^{31} + 24 q^{41} - 18 q^{49} - 24 q^{59} + 26 q^{61} - 12 q^{71} - 2 q^{79} + 6 q^{89} + 10 q^{91}+O(q^{100}) \) 4 * q + 4 * q^11 - 8 * q^19 + 18 * q^29 - 16 * q^31 + 24 * q^41 - 18 * q^49 - 24 * q^59 + 26 * q^61 - 12 * q^71 - 2 * q^79 + 6 * q^89 + 10 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more