LMFDB - Embedded newform 8100.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(1,8100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8100.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.a (trivial)

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:	yes
Analytic conductor:	$64.6788256372$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.456850$ of defining polynomial
Character	$\chi$	$=$	8100.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79129 q^{7} +3.10260 q^{11} -2.79129 q^{13} -4.83465 q^{17} +3.58258 q^{19} -4.83465 q^{23} +3.46410 q^{29} +6.58258 q^{31} +3.37386 q^{37} +6.56670 q^{41} -2.79129 q^{43} +8.66025 q^{47} +0.791288 q^{49} +4.83465 q^{53} -8.29875 q^{59} +10.3739 q^{61} -5.00000 q^{67} -7.93725 q^{71} -5.00000 q^{73} -8.66025 q^{77} -4.00000 q^{79} -1.00905 q^{83} -13.1334 q^{89} +7.79129 q^{91} +1.16515 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{7} - 2 q^{13} - 4 q^{19} + 8 q^{31} - 14 q^{37} - 2 q^{43} - 6 q^{49} + 14 q^{61} - 20 q^{67} - 20 q^{73} - 16 q^{79} + 22 q^{91} - 32 q^{97}+O(q^{100}) $ 4 * q - 2 * q^7 - 2 * q^13 - 4 * q^19 + 8 * q^31 - 14 * q^37 - 2 * q^43 - 6 * q^49 + 14 * q^61 - 20 * q^67 - 20 * q^73 - 16 * q^79 + 22 * q^91 - 32 * q^97

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
$7$	−2.79129	−1.05501	−0.527504	−	0.849553i	$-0.676872\pi$
$7$	−2.79129	−1.05501	−0.527504	+	0.849553i	$0.676872\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.10260	0.935470	0.467735	−	0.883869i	$-0.345070\pi$
$11$	3.10260	0.935470	0.467735	+	0.883869i	$0.345070\pi$
$12$	0	0
$13$	−2.79129	−0.774164	−0.387082	−	0.922045i	$-0.626517\pi$
$13$	−2.79129	−0.774164	−0.387082	+	0.922045i	$0.626517\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.83465	−1.17258	−0.586288	−	0.810103i	$-0.699411\pi$
$17$	−4.83465	−1.17258	−0.586288	+	0.810103i	$0.699411\pi$
$18$	0	0
$19$	3.58258	0.821899	0.410950	−	0.911658i	$-0.365197\pi$
$19$	3.58258	0.821899	0.410950	+	0.911658i	$0.365197\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.83465	−1.00809	−0.504047	−	0.863676i	$-0.668156\pi$
$23$	−4.83465	−1.00809	−0.504047	+	0.863676i	$0.668156\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	6.58258	1.18227	0.591133	−	0.806574i	$-0.298681\pi$
$31$	6.58258	1.18227	0.591133	+	0.806574i	$0.298681\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.37386	0.554660	0.277330	−	0.960775i	$-0.410551\pi$
$37$	3.37386	0.554660	0.277330	+	0.960775i	$0.410551\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.56670	1.02555	0.512773	−	0.858524i	$-0.328618\pi$
$41$	6.56670	1.02555	0.512773	+	0.858524i	$0.328618\pi$
$42$	0	0
$43$	−2.79129	−0.425667	−0.212834	−	0.977088i	$-0.568269\pi$
$43$	−2.79129	−0.425667	−0.212834	+	0.977088i	$0.568269\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.66025	1.26323	0.631614	−	0.775283i	$-0.282393\pi$
$47$	8.66025	1.26323	0.631614	+	0.775283i	$0.282393\pi$
$48$	0	0
$49$	0.791288	0.113041
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.83465	0.664091	0.332045	−	0.943263i	$-0.392261\pi$
$53$	4.83465	0.664091	0.332045	+	0.943263i	$0.392261\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.29875	−1.08041	−0.540203	−	0.841535i	$-0.681653\pi$
$59$	−8.29875	−1.08041	−0.540203	+	0.841535i	$0.681653\pi$
$60$	0	0
$61$	10.3739	1.32824	0.664119	−	0.747627i	$-0.268807\pi$
$61$	10.3739	1.32824	0.664119	+	0.747627i	$0.268807\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.93725	−0.941979	−0.470989	−	0.882139i	$-0.656103\pi$
$71$	−7.93725	−0.941979	−0.470989	+	0.882139i	$0.656103\pi$
$72$	0	0
$73$	−5.00000	−0.585206	−0.292603	−	0.956234i	$-0.594521\pi$
$73$	−5.00000	−0.585206	−0.292603	+	0.956234i	$0.594521\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.66025	−0.986928
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.00905	−0.110758	−0.0553789	−	0.998465i	$-0.517637\pi$
$83$	−1.00905	−0.110758	−0.0553789	+	0.998465i	$0.517637\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.1334	−1.39214	−0.696069	−	0.717975i	$-0.745069\pi$
$89$	−13.1334	−1.39214	−0.696069	+	0.717975i	$0.745069\pi$
$90$	0	0
$91$	7.79129	0.816749
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.16515	0.118303	0.0591516	−	0.998249i	$-0.481160\pi$
$97$	1.16515	0.118303	0.0591516	+	0.998249i	$0.481160\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.2270	−1.51514	−0.757569	−	0.652755i	$-0.773613\pi$
$101$	−15.2270	−1.51514	−0.757569	+	0.652755i	$0.773613\pi$
$102$	0	0
$103$	3.37386	0.332437	0.166218	−	0.986089i	$-0.446844\pi$
$103$	3.37386	0.332437	0.166218	+	0.986089i	$0.446844\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.66025	−0.837218	−0.418609	−	0.908166i	$-0.637482\pi$
$107$	−8.66025	−0.837218	−0.418609	+	0.908166i	$0.637482\pi$
$108$	0	0
$109$	−16.9564	−1.62413	−0.812066	−	0.583565i	$-0.801657\pi$
$109$	−16.9564	−1.62413	−0.812066	+	0.583565i	$0.801657\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.84370	0.549729	0.274865	−	0.961483i	$-0.411367\pi$
$113$	5.84370	0.549729	0.274865	+	0.961483i	$0.411367\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	13.4949	1.23708
$120$	0	0
$121$	−1.37386	−0.124897
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−21.7477	−1.92980	−0.964899	−	0.262620i	$-0.915413\pi$
$127$	−21.7477	−1.92980	−0.964899	+	0.262620i	$0.915413\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.723000	0.0631688	0.0315844	−	0.999501i	$-0.489945\pi$
$131$	0.723000	0.0631688	0.0315844	+	0.999501i	$0.489945\pi$
$132$	0	0
$133$	−10.0000	−0.867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.1552	1.89284	0.946422	−	0.322934i	$-0.104669\pi$
$137$	22.1552	1.89284	0.946422	+	0.322934i	$0.104669\pi$
$138$	0	0
$139$	−15.3739	−1.30399	−0.651997	−	0.758222i	$-0.726069\pi$
$139$	−15.3739	−1.30399	−0.651997	+	0.758222i	$0.726069\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.66025	−0.724207
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.6784	0.874805	0.437402	−	0.899266i	$-0.355899\pi$
$149$	10.6784	0.874805	0.437402	+	0.899266i	$0.355899\pi$
$150$	0	0
$151$	2.62614	0.213712	0.106856	−	0.994275i	$-0.465922\pi$
$151$	2.62614	0.213712	0.106856	+	0.994275i	$0.465922\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−9.41742	−0.751592	−0.375796	−	0.926702i	$-0.622631\pi$
$157$	−9.41742	−0.751592	−0.375796	+	0.926702i	$0.622631\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	13.4949	1.06355
$162$	0	0
$163$	−0.582576	−0.0456309	−0.0228154	−	0.999740i	$-0.507263\pi$
$163$	−0.582576	−0.0456309	−0.0228154	+	0.999740i	$0.507263\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.4859	−0.966185	−0.483092	−	0.875569i	$-0.660486\pi$
$167$	−12.4859	−0.966185	−0.483092	+	0.875569i	$0.660486\pi$
$168$	0	0
$169$	−5.20871	−0.400670
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.3386	1.47029	0.735144	−	0.677911i	$-0.237115\pi$
$173$	19.3386	1.47029	0.735144	+	0.677911i	$0.237115\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	14.1425	1.05706	0.528528	−	0.848916i	$-0.322744\pi$
$179$	14.1425	1.05706	0.528528	+	0.848916i	$0.322744\pi$
$180$	0	0
$181$	6.58258	0.489279	0.244639	−	0.969614i	$-0.421330\pi$
$181$	6.58258	0.489279	0.244639	+	0.969614i	$0.421330\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−15.0000	−1.09691
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.0616	−1.45161	−0.725804	−	0.687902i	$-0.758532\pi$
$191$	−20.0616	−1.45161	−0.725804	+	0.687902i	$0.758532\pi$
$192$	0	0
$193$	−21.7477	−1.56544	−0.782718	−	0.622377i	$-0.786167\pi$
$193$	−21.7477	−1.56544	−0.782718	+	0.622377i	$0.786167\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.5130	1.10526	0.552628	−	0.833428i	$-0.313625\pi$
$197$	15.5130	1.10526	0.552628	+	0.833428i	$0.313625\pi$
$198$	0	0
$199$	−6.20871	−0.440124	−0.220062	−	0.975486i	$-0.570626\pi$
$199$	−6.20871	−0.440124	−0.220062	+	0.975486i	$0.570626\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−9.66930	−0.678652
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	11.1153	0.768862
$210$	0	0
$211$	3.41742	0.235265	0.117633	−	0.993057i	$-0.462469\pi$
$211$	3.41742	0.235265	0.117633	+	0.993057i	$0.462469\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−18.3739	−1.24730
$218$	0	0
$219$	0	0
$220$	0	0
$221$	13.4949	0.907766
$222$	0	0
$223$	16.1652	1.08250	0.541249	−	0.840862i	$-0.317952\pi$
$223$	16.1652	1.08250	0.541249	+	0.840862i	$0.317952\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	25.9808	1.72440	0.862202	−	0.506565i	$-0.169085\pi$
$227$	25.9808	1.72440	0.862202	+	0.506565i	$0.169085\pi$
$228$	0	0
$229$	−5.41742	−0.357993	−0.178997	−	0.983850i	$-0.557285\pi$
$229$	−5.41742	−0.357993	−0.178997	+	0.983850i	$0.557285\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.6784	−0.699562	−0.349781	−	0.936831i	$-0.613744\pi$
$233$	−10.6784	−0.699562	−0.349781	+	0.936831i	$0.613744\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.9898	−1.74583	−0.872913	−	0.487876i	$-0.837772\pi$
$239$	−26.9898	−1.74583	−0.872913	+	0.487876i	$0.837772\pi$
$240$	0	0
$241$	−23.9129	−1.54036	−0.770182	−	0.637824i	$-0.779835\pi$
$241$	−23.9129	−1.54036	−0.770182	+	0.637824i	$0.779835\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.0000	−0.636285
$248$	0	0
$249$	0	0
$250$	0	0
$251$	24.1733	1.52580	0.762901	−	0.646515i	$-0.223774\pi$
$251$	24.1733	1.52580	0.762901	+	0.646515i	$0.223774\pi$
$252$	0	0
$253$	−15.0000	−0.943042
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.9898	−1.68358	−0.841789	−	0.539806i	$-0.818497\pi$
$257$	−26.9898	−1.68358	−0.841789	+	0.539806i	$0.818497\pi$
$258$	0	0
$259$	−9.41742	−0.585170
$260$	0	0
$261$	0	0
$262$	0	0
$263$	26.9898	1.66426	0.832132	−	0.554578i	$-0.187120\pi$
$263$	26.9898	1.66426	0.832132	+	0.554578i	$0.187120\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.28970	−0.444461	−0.222231	−	0.974994i	$-0.571334\pi$
$269$	−7.28970	−0.444461	−0.222231	+	0.974994i	$0.571334\pi$
$270$	0	0
$271$	0.252273	0.0153245	0.00766224	−	0.999971i	$-0.497561\pi$
$271$	0.252273	0.0153245	0.00766224	+	0.999971i	$0.497561\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−32.3303	−1.94254	−0.971270	−	0.237981i	$-0.923514\pi$
$277$	−32.3303	−1.94254	−0.971270	+	0.237981i	$0.923514\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.3115	−0.973060	−0.486530	−	0.873664i	$-0.661738\pi$
$281$	−16.3115	−0.973060	−0.486530	+	0.873664i	$0.661738\pi$
$282$	0	0
$283$	24.5390	1.45869	0.729347	−	0.684144i	$-0.239824\pi$
$283$	24.5390	1.45869	0.729347	+	0.684144i	$0.239824\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−18.3296	−1.08196
$288$	0	0
$289$	6.37386	0.374933
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−18.3296	−1.07082	−0.535412	−	0.844591i	$-0.679844\pi$
$293$	−18.3296	−1.07082	−0.535412	+	0.844591i	$0.679844\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.4949	0.780431
$300$	0	0
$301$	7.79129	0.449082
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−3.25227	−0.185617	−0.0928085	−	0.995684i	$-0.529584\pi$
$307$	−3.25227	−0.185617	−0.0928085	+	0.995684i	$0.529584\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.91915	0.335644	0.167822	−	0.985817i	$-0.446327\pi$
$311$	5.91915	0.335644	0.167822	+	0.985817i	$0.446327\pi$
$312$	0	0
$313$	−7.20871	−0.407461	−0.203730	−	0.979027i	$-0.565307\pi$
$313$	−7.20871	−0.407461	−0.203730	+	0.979027i	$0.565307\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.3115	0.916143	0.458071	−	0.888915i	$-0.348540\pi$
$317$	16.3115	0.916143	0.458071	+	0.888915i	$0.348540\pi$
$318$	0	0
$319$	10.7477	0.601757
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−17.3205	−0.963739
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−24.1733	−1.33272
$330$	0	0
$331$	30.7477	1.69005	0.845024	−	0.534728i	$-0.179586\pi$
$331$	30.7477	1.69005	0.845024	+	0.534728i	$0.179586\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−32.3303	−1.76114	−0.880572	−	0.473913i	$-0.842841\pi$
$337$	−32.3303	−1.76114	−0.880572	+	0.473913i	$0.842841\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	20.4231	1.10597
$342$	0	0
$343$	17.3303	0.935748
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.82560	−0.205369	−0.102685	−	0.994714i	$-0.532743\pi$
$347$	−3.82560	−0.205369	−0.102685	+	0.994714i	$0.532743\pi$
$348$	0	0
$349$	−25.4955	−1.36474	−0.682370	−	0.731007i	$-0.739051\pi$
$349$	−25.4955	−1.36474	−0.682370	+	0.731007i	$0.739051\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−25.1823	−1.34032	−0.670160	−	0.742217i	$-0.733774\pi$
$353$	−25.1823	−1.34032	−0.670160	+	0.742217i	$0.733774\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.37960	−0.125591	−0.0627953	−	0.998026i	$-0.520002\pi$
$359$	−2.37960	−0.125591	−0.0627953	+	0.998026i	$0.520002\pi$
$360$	0	0
$361$	−6.16515	−0.324482
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.3739	−0.698110	−0.349055	−	0.937102i	$-0.613497\pi$
$367$	−13.3739	−0.698110	−0.349055	+	0.937102i	$0.613497\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−13.4949	−0.700621
$372$	0	0
$373$	−13.8348	−0.716341	−0.358171	−	0.933656i	$-0.616599\pi$
$373$	−13.8348	−0.716341	−0.358171	+	0.933656i	$0.616599\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.66930	−0.497995
$378$	0	0
$379$	−28.7913	−1.47891	−0.739455	−	0.673206i	$-0.764917\pi$
$379$	−28.7913	−1.47891	−0.739455	+	0.673206i	$0.764917\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.84370	0.298599	0.149300	−	0.988792i	$-0.452298\pi$
$383$	5.84370	0.298599	0.149300	+	0.988792i	$0.452298\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.3115	0.827024	0.413512	−	0.910499i	$-0.364302\pi$
$389$	16.3115	0.827024	0.413512	+	0.910499i	$0.364302\pi$
$390$	0	0
$391$	23.3739	1.18207
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−13.8348	−0.694351	−0.347176	−	0.937800i	$-0.612859\pi$
$397$	−13.8348	−0.694351	−0.347176	+	0.937800i	$0.612859\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	−18.3739	−0.915267
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.4678	0.518867
$408$	0	0
$409$	−32.2867	−1.59648	−0.798238	−	0.602342i	$-0.794234\pi$
$409$	−32.2867	−1.59648	−0.798238	+	0.602342i	$0.794234\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	23.1642	1.13984
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13.4949	−0.659269	−0.329635	−	0.944109i	$-0.606926\pi$
$419$	−13.4949	−0.659269	−0.329635	+	0.944109i	$0.606926\pi$
$420$	0	0
$421$	29.4955	1.43752	0.718760	−	0.695258i	$-0.244710\pi$
$421$	29.4955	1.43752	0.718760	+	0.695258i	$0.244710\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−28.9564	−1.40130
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−15.5885	−0.750870	−0.375435	−	0.926849i	$-0.622507\pi$
$431$	−15.5885	−0.750870	−0.375435	+	0.926849i	$0.622507\pi$
$432$	0	0
$433$	5.58258	0.268281	0.134141	−	0.990962i	$-0.457173\pi$
$433$	5.58258	0.268281	0.134141	+	0.990962i	$0.457173\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−17.3205	−0.828552
$438$	0	0
$439$	12.1216	0.578532	0.289266	−	0.957249i	$-0.406589\pi$
$439$	12.1216	0.578532	0.289266	+	0.957249i	$0.406589\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.8245	1.51203	0.756013	−	0.654557i	$-0.227145\pi$
$443$	31.8245	1.51203	0.756013	+	0.654557i	$0.227145\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−25.6193	−1.20905	−0.604524	−	0.796587i	$-0.706637\pi$
$449$	−25.6193	−1.20905	−0.604524	+	0.796587i	$0.706637\pi$
$450$	0	0
$451$	20.3739	0.959368
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.25227	−0.152135	−0.0760675	−	0.997103i	$-0.524236\pi$
$457$	−3.25227	−0.152135	−0.0760675	+	0.997103i	$0.524236\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−30.0924	−1.40154	−0.700772	−	0.713386i	$-0.747161\pi$
$461$	−30.0924	−1.40154	−0.700772	+	0.713386i	$0.747161\pi$
$462$	0	0
$463$	−15.1216	−0.702760	−0.351380	−	0.936233i	$-0.614287\pi$
$463$	−15.1216	−0.702760	−0.351380	+	0.936233i	$0.614287\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.6591	−1.69638	−0.848191	−	0.529691i	$-0.822308\pi$
$467$	−36.6591	−1.69638	−0.848191	+	0.529691i	$0.822308\pi$
$468$	0	0
$469$	13.9564	0.644448
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.66025	−0.398199
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10.0308	0.458319	0.229160	−	0.973389i	$-0.426402\pi$
$479$	10.0308	0.458319	0.229160	+	0.973389i	$0.426402\pi$
$480$	0	0
$481$	−9.41742	−0.429398
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.3303	1.01188	0.505941	−	0.862568i	$-0.331145\pi$
$487$	22.3303	1.01188	0.505941	+	0.862568i	$0.331145\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	30.7400	1.38728	0.693638	−	0.720324i	$-0.256007\pi$
$491$	30.7400	1.38728	0.693638	+	0.720324i	$0.256007\pi$
$492$	0	0
$493$	−16.7477	−0.754280
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.1552	0.993795
$498$	0	0
$499$	−19.6261	−0.878587	−0.439293	−	0.898344i	$-0.644771\pi$
$499$	−19.6261	−0.878587	−0.439293	+	0.898344i	$0.644771\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12.4859	−0.556717	−0.278358	−	0.960477i	$-0.589790\pi$
$503$	−12.4859	−0.556717	−0.278358	+	0.960477i	$0.589790\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10.6784	0.473310	0.236655	−	0.971594i	$-0.423949\pi$
$509$	10.6784	0.473310	0.236655	+	0.971594i	$0.423949\pi$
$510$	0	0
$511$	13.9564	0.617397
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	26.8693	1.18171
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−21.3567	−0.935654	−0.467827	−	0.883820i	$-0.654963\pi$
$521$	−21.3567	−0.935654	−0.467827	+	0.883820i	$0.654963\pi$
$522$	0	0
$523$	−21.7477	−0.950962	−0.475481	−	0.879726i	$-0.657726\pi$
$523$	−21.7477	−0.950962	−0.475481	+	0.879726i	$0.657726\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−31.8245	−1.38630
$528$	0	0
$529$	0.373864	0.0162549
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−18.3296	−0.793941
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.45505	0.105747
$540$	0	0
$541$	11.7913	0.506947	0.253474	−	0.967342i	$-0.418427\pi$
$541$	11.7913	0.506947	0.253474	+	0.967342i	$0.418427\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	11.7477	0.502297	0.251148	−	0.967949i	$-0.419192\pi$
$547$	11.7477	0.502297	0.251148	+	0.967949i	$0.419192\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	12.4104	0.528701
$552$	0	0
$553$	11.1652	0.474791
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	7.79129	0.329536
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.66025	0.364986	0.182493	−	0.983207i	$-0.441583\pi$
$563$	8.66025	0.364986	0.182493	+	0.983207i	$0.441583\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	33.1950	1.39161	0.695804	−	0.718232i	$-0.255048\pi$
$569$	33.1950	1.39161	0.695804	+	0.718232i	$0.255048\pi$
$570$	0	0
$571$	1.66970	0.0698747	0.0349373	−	0.999390i	$-0.488877\pi$
$571$	1.66970	0.0698747	0.0349373	+	0.999390i	$0.488877\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−38.4955	−1.60259	−0.801293	−	0.598272i	$-0.795854\pi$
$577$	−38.4955	−1.60259	−0.801293	+	0.598272i	$0.795854\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.81655	0.116850
$582$	0	0
$583$	15.0000	0.621237
$584$	0	0
$585$	0	0
$586$	0	0
$587$	47.3375	1.95383	0.976913	−	0.213636i	$-0.0685307\pi$
$587$	47.3375	1.95383	0.976913	+	0.213636i	$0.0685307\pi$
$588$	0	0
$589$	23.5826	0.971703
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.8154	1.26544	0.632719	−	0.774382i	$-0.281939\pi$
$593$	30.8154	1.26544	0.632719	+	0.774382i	$0.281939\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	19.9862	0.816612	0.408306	−	0.912845i	$-0.366120\pi$
$599$	19.9862	0.816612	0.408306	+	0.912845i	$0.366120\pi$
$600$	0	0
$601$	−9.70417	−0.395841	−0.197921	−	0.980218i	$-0.563419\pi$
$601$	−9.70417	−0.395841	−0.197921	+	0.980218i	$0.563419\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	26.7477	1.08566	0.542828	−	0.839844i	$-0.317353\pi$
$607$	26.7477	1.08566	0.542828	+	0.839844i	$0.317353\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−24.1733	−0.977945
$612$	0	0
$613$	−36.7477	−1.48423	−0.742113	−	0.670274i	$-0.766176\pi$
$613$	−36.7477	−1.48423	−0.742113	+	0.670274i	$0.766176\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.3296	−0.737920	−0.368960	−	0.929445i	$-0.620286\pi$
$617$	−18.3296	−0.737920	−0.368960	+	0.929445i	$0.620286\pi$
$618$	0	0
$619$	−20.7477	−0.833922	−0.416961	−	0.908924i	$-0.636905\pi$
$619$	−20.7477	−0.833922	−0.416961	+	0.908924i	$0.636905\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	36.6591	1.46872
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.3115	−0.650380
$630$	0	0
$631$	19.7042	0.784410	0.392205	−	0.919878i	$-0.371712\pi$
$631$	19.7042	0.784410	0.392205	+	0.919878i	$0.371712\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.20871	−0.0875124
$638$	0	0
$639$	0	0
$640$	0	0
$641$	27.9234	1.10291	0.551454	−	0.834205i	$-0.314073\pi$
$641$	27.9234	1.10291	0.551454	+	0.834205i	$0.314073\pi$
$642$	0	0
$643$	−46.8693	−1.84835	−0.924173	−	0.381975i	$-0.875244\pi$
$643$	−46.8693	−1.84835	−0.924173	+	0.381975i	$0.875244\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.3477	0.799949	0.399975	−	0.916526i	$-0.369019\pi$
$647$	20.3477	0.799949	0.399975	+	0.916526i	$0.369019\pi$
$648$	0	0
$649$	−25.7477	−1.01069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−23.1642	−0.906486	−0.453243	−	0.891387i	$-0.649733\pi$
$653$	−23.1642	−0.906486	−0.453243	+	0.891387i	$0.649733\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−24.1733	−0.941657	−0.470828	−	0.882225i	$-0.656045\pi$
$659$	−24.1733	−0.941657	−0.470828	+	0.882225i	$0.656045\pi$
$660$	0	0
$661$	27.7477	1.07926	0.539631	−	0.841902i	$-0.318564\pi$
$661$	27.7477	1.07926	0.539631	+	0.841902i	$0.318564\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−16.7477	−0.648475
$668$	0	0
$669$	0	0
$670$	0	0
$671$	32.1860	1.24253
$672$	0	0
$673$	−35.0000	−1.34915	−0.674575	−	0.738206i	$-0.735673\pi$
$673$	−35.0000	−1.34915	−0.674575	+	0.738206i	$0.735673\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.3477	0.782024	0.391012	−	0.920386i	$-0.372125\pi$
$677$	20.3477	0.782024	0.391012	+	0.920386i	$0.372125\pi$
$678$	0	0
$679$	−3.25227	−0.124811
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−20.1371	−0.770523	−0.385262	−	0.922807i	$-0.625889\pi$
$683$	−20.1371	−0.770523	−0.385262	+	0.922807i	$0.625889\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−13.4949	−0.514115
$690$	0	0
$691$	−10.4955	−0.399266	−0.199633	−	0.979871i	$-0.563975\pi$
$691$	−10.4955	−0.399266	−0.199633	+	0.979871i	$0.563975\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−31.7477	−1.20253
$698$	0	0
$699$	0	0
$700$	0	0
$701$	14.5794	0.550657	0.275328	−	0.961350i	$-0.411213\pi$
$701$	14.5794	0.550657	0.275328	+	0.961350i	$0.411213\pi$
$702$	0	0
$703$	12.0871	0.455874
$704$	0	0
$705$	0	0
$706$	0	0
$707$	42.5028	1.59848
$708$	0	0
$709$	28.0780	1.05449	0.527246	−	0.849713i	$-0.323225\pi$
$709$	28.0780	1.05449	0.527246	+	0.849713i	$0.323225\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−31.8245	−1.19184
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	9.02175	0.336455	0.168227	−	0.985748i	$-0.446196\pi$
$719$	9.02175	0.336455	0.168227	+	0.985748i	$0.446196\pi$
$720$	0	0
$721$	−9.41742	−0.350723
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.37386	0.125130	0.0625648	−	0.998041i	$-0.480072\pi$
$727$	3.37386	0.125130	0.0625648	+	0.998041i	$0.480072\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	13.4949	0.499127
$732$	0	0
$733$	5.12159	0.189170	0.0945851	−	0.995517i	$-0.469848\pi$
$733$	5.12159	0.189170	0.0945851	+	0.995517i	$0.469848\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−15.5130	−0.571429
$738$	0	0
$739$	−33.8693	−1.24590	−0.622951	−	0.782260i	$-0.714067\pi$
$739$	−33.8693	−1.24590	−0.622951	+	0.782260i	$0.714067\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	24.1733	0.883272
$750$	0	0
$751$	18.1216	0.661266	0.330633	−	0.943759i	$-0.392738\pi$
$751$	18.1216	0.661266	0.330633	+	0.943759i	$0.392738\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18.3739	0.667809	0.333905	−	0.942607i	$-0.391634\pi$
$757$	18.3739	0.667809	0.333905	+	0.942607i	$0.391634\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.5656	−1.25300	−0.626500	−	0.779421i	$-0.715513\pi$
$761$	−34.5656	−1.25300	−0.626500	+	0.779421i	$0.715513\pi$
$762$	0	0
$763$	47.3303	1.71347
$764$	0	0
$765$	0	0
$766$	0	0
$767$	23.1642	0.836411
$768$	0	0
$769$	16.5390	0.596412	0.298206	−	0.954502i	$-0.403612\pi$
$769$	16.5390	0.596412	0.298206	+	0.954502i	$0.403612\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.64215	0.238902	0.119451	−	0.992840i	$-0.461887\pi$
$773$	6.64215	0.238902	0.119451	+	0.992840i	$0.461887\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23.5257	0.842896
$780$	0	0
$781$	−24.6261	−0.881192
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.37386	0.120265	0.0601326	−	0.998190i	$-0.480848\pi$
$787$	3.37386	0.120265	0.0601326	+	0.998190i	$0.480848\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16.3115	−0.579969
$792$	0	0
$793$	−28.9564	−1.02827
$794$	0	0
$795$	0	0
$796$	0	0
$797$	44.5209	1.57701	0.788506	−	0.615027i	$-0.210855\pi$
$797$	44.5209	1.57701	0.788506	+	0.615027i	$0.210855\pi$
$798$	0	0
$799$	−41.8693	−1.48123
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−15.5130	−0.547442
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.19615	−0.182687	−0.0913435	−	0.995819i	$-0.529116\pi$
$809$	−5.19615	−0.182687	−0.0913435	+	0.995819i	$0.529116\pi$
$810$	0	0
$811$	44.9564	1.57863	0.789317	−	0.613986i	$-0.210435\pi$
$811$	44.9564	1.57863	0.789317	+	0.613986i	$0.210435\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−10.0000	−0.349856
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−56.2838	−1.96432	−0.982159	−	0.188054i	$-0.939782\pi$
$821$	−56.2838	−1.96432	−0.982159	+	0.188054i	$0.939782\pi$
$822$	0	0
$823$	33.8348	1.17941	0.589704	−	0.807619i	$-0.299244\pi$
$823$	33.8348	1.17941	0.589704	+	0.807619i	$0.299244\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.86180	0.273382	0.136691	−	0.990614i	$-0.456353\pi$
$827$	7.86180	0.273382	0.136691	+	0.990614i	$0.456353\pi$
$828$	0	0
$829$	41.4955	1.44120	0.720598	−	0.693353i	$-0.243867\pi$
$829$	41.4955	1.44120	0.720598	+	0.693353i	$0.243867\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.82560	−0.132549
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−13.1334	−0.453416	−0.226708	−	0.973963i	$-0.572796\pi$
$839$	−13.1334	−0.453416	−0.226708	+	0.973963i	$0.572796\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.83485	0.131767
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−16.3115	−0.559150
$852$	0	0
$853$	−5.00000	−0.171197	−0.0855984	−	0.996330i	$-0.527280\pi$
$853$	−5.00000	−0.171197	−0.0855984	+	0.996330i	$0.527280\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−42.5028	−1.45187	−0.725934	−	0.687764i	$-0.758592\pi$
$857$	−42.5028	−1.45187	−0.725934	+	0.687764i	$0.758592\pi$
$858$	0	0
$859$	0.252273	0.00860744	0.00430372	−	0.999991i	$-0.498630\pi$
$859$	0.252273	0.00860744	0.00430372	+	0.999991i	$0.498630\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.8154	−1.04897	−0.524484	−	0.851420i	$-0.675742\pi$
$863$	−30.8154	−1.04897	−0.524484	+	0.851420i	$0.675742\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−12.4104	−0.420994
$870$	0	0
$871$	13.9564	0.472896
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.79129	−0.0942551	−0.0471275	−	0.998889i	$-0.515007\pi$
$877$	−2.79129	−0.0942551	−0.0471275	+	0.998889i	$0.515007\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	36.9452	1.24471	0.622357	−	0.782733i	$-0.286175\pi$
$881$	36.9452	1.24471	0.622357	+	0.782733i	$0.286175\pi$
$882$	0	0
$883$	34.6606	1.16642	0.583211	−	0.812321i	$-0.301796\pi$
$883$	34.6606	1.16642	0.583211	+	0.812321i	$0.301796\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−23.1642	−0.777778	−0.388889	−	0.921285i	$-0.627141\pi$
$887$	−23.1642	−0.777778	−0.388889	+	0.921285i	$0.627141\pi$
$888$	0	0
$889$	60.7042	2.03595
$890$	0	0
$891$	0	0
$892$	0	0
$893$	31.0260	1.03825
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	22.8027	0.760513
$900$	0	0
$901$	−23.3739	−0.778696
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.62614	0.0539950	0.0269975	−	0.999636i	$-0.491405\pi$
$907$	1.62614	0.0539950	0.0269975	+	0.999636i	$0.491405\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−0.286051	−0.00947728	−0.00473864	−	0.999989i	$-0.501508\pi$
$911$	−0.286051	−0.00947728	−0.00473864	+	0.999989i	$0.501508\pi$
$912$	0	0
$913$	−3.13068	−0.103610
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.01810	−0.0666436
$918$	0	0
$919$	−13.6261	−0.449485	−0.224742	−	0.974418i	$-0.572154\pi$
$919$	−13.6261	−0.449485	−0.224742	+	0.974418i	$0.572154\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	22.1552	0.729246
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	14.1425	0.463999	0.231999	−	0.972716i	$-0.425473\pi$
$929$	14.1425	0.463999	0.231999	+	0.972716i	$0.425473\pi$
$930$	0	0
$931$	2.83485	0.0929084
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−9.41742	−0.307654	−0.153827	−	0.988098i	$-0.549160\pi$
$937$	−9.41742	−0.307654	−0.153827	+	0.988098i	$0.549160\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.81655	−0.0918169	−0.0459085	−	0.998946i	$-0.514618\pi$
$941$	−2.81655	−0.0918169	−0.0459085	+	0.998946i	$0.514618\pi$
$942$	0	0
$943$	−31.7477	−1.03385
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−57.0068	−1.85247	−0.926236	−	0.376945i	$-0.876975\pi$
$947$	−57.0068	−1.85247	−0.926236	+	0.376945i	$0.876975\pi$
$948$	0	0
$949$	13.9564	0.453045
$950$	0	0
$951$	0	0
$952$	0	0
$953$	53.9796	1.74857	0.874286	−	0.485412i	$-0.161330\pi$
$953$	53.9796	1.74857	0.874286	+	0.485412i	$0.161330\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−61.8414	−1.99696
$960$	0	0
$961$	12.3303	0.397752
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.25227	0.265375	0.132688	−	0.991158i	$-0.457639\pi$
$967$	8.25227	0.265375	0.132688	+	0.991158i	$0.457639\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.55765	−0.178354	−0.0891768	−	0.996016i	$-0.528424\pi$
$971$	−5.55765	−0.178354	−0.0891768	+	0.996016i	$0.528424\pi$
$972$	0	0
$973$	42.9129	1.37572
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−35.8607	−1.14728	−0.573642	−	0.819106i	$-0.694470\pi$
$977$	−35.8607	−1.14728	−0.573642	+	0.819106i	$0.694470\pi$
$978$	0	0
$979$	−40.7477	−1.30230
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−22.3658	−0.713357	−0.356679	−	0.934227i	$-0.616091\pi$
$983$	−22.3658	−0.713357	−0.356679	+	0.934227i	$0.616091\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13.4949	0.429113
$990$	0	0
$991$	−41.1216	−1.30627	−0.653135	−	0.757241i	$-0.726547\pi$
$991$	−41.1216	−1.30627	−0.653135	+	0.757241i	$0.726547\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−6.74773	−0.213703	−0.106851	−	0.994275i	$-0.534077\pi$
$997$	−6.74773	−0.213703	−0.106851	+	0.994275i	$0.534077\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.a.w.1.2	yes	4
3.2	odd	2	inner	8100.2.a.w.1.1	✓	4
5.2	odd	4		8100.2.d.r.649.2		8
5.3	odd	4		8100.2.d.r.649.8		8
5.4	even	2		8100.2.a.bb.1.4	yes	4
15.2	even	4		8100.2.d.r.649.1		8
15.8	even	4		8100.2.d.r.649.7		8
15.14	odd	2		8100.2.a.bb.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8100.2.a.w.1.1	✓	4	3.2	odd	2	inner
8100.2.a.w.1.2	yes	4	1.1	even	1	trivial
8100.2.a.bb.1.3	yes	4	15.14	odd	2
8100.2.a.bb.1.4	yes	4	5.4	even	2
8100.2.d.r.649.1		8	15.2	even	4
8100.2.d.r.649.2		8	5.2	odd	4
8100.2.d.r.649.7		8	15.8	even	4
8100.2.d.r.649.8		8	5.3	odd	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:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79129 q^{7} +3.10260 q^{11} -2.79129 q^{13} -4.83465 q^{17} +3.58258 q^{19} -4.83465 q^{23} +3.46410 q^{29} +6.58258 q^{31} +3.37386 q^{37} +6.56670 q^{41} -2.79129 q^{43} +8.66025 q^{47} +0.791288 q^{49} +4.83465 q^{53} -8.29875 q^{59} +10.3739 q^{61} -5.00000 q^{67} -7.93725 q^{71} -5.00000 q^{73} -8.66025 q^{77} -4.00000 q^{79} -1.00905 q^{83} -13.1334 q^{89} +7.79129 q^{91} +1.16515 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{7} - 2 q^{13} - 4 q^{19} + 8 q^{31} - 14 q^{37} - 2 q^{43} - 6 q^{49} + 14 q^{61} - 20 q^{67} - 20 q^{73} - 16 q^{79} + 22 q^{91} - 32 q^{97}+O(q^{100}) \) 4 * q - 2 * q^7 - 2 * q^13 - 4 * q^19 + 8 * q^31 - 14 * q^37 - 2 * q^43 - 6 * q^49 + 14 * q^61 - 20 * q^67 - 20 * q^73 - 16 * q^79 + 22 * q^91 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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