LMFDB - Embedded newform 8100.2.a.bb.1.1

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:	$0$
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			1.1
Root			$-2.18890$ 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)

$f(q)$	$=$	$q-1.79129 q^{7} +O(q^{10})$	$q-1.79129 q^{7} -4.83465 q^{11} -1.79129 q^{13} -3.10260 q^{17} -5.58258 q^{19} -3.10260 q^{23} +3.46410 q^{29} -2.58258 q^{31} +10.3739 q^{37} -1.37055 q^{41} -1.79129 q^{43} -8.66025 q^{47} -3.79129 q^{49} +3.10260 q^{53} -0.361500 q^{59} -3.37386 q^{61} +5.00000 q^{67} +7.93725 q^{71} +5.00000 q^{73} +8.66025 q^{77} -4.00000 q^{79} -14.8655 q^{83} +2.74110 q^{89} +3.20871 q^{91} +17.1652 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{7}+O(q^{10}) $ 4 * q + 2 * q^7	$ 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

Display 100 coefficients 10 coefficients

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
$7$	−1.79129	−0.677043	−0.338522	−	0.940959i	$-0.609927\pi$
$7$	−1.79129	−0.677043	−0.338522	+	0.940959i	$0.609927\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.83465	−1.45770	−0.728851	−	0.684672i	$-0.759945\pi$
$11$	−4.83465	−1.45770	−0.728851	+	0.684672i	$0.759945\pi$
$12$	0	0
$13$	−1.79129	−0.496814	−0.248407	−	0.968656i	$-0.579907\pi$
$13$	−1.79129	−0.496814	−0.248407	+	0.968656i	$0.579907\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.10260	−0.752491	−0.376246	−	0.926520i	$-0.622785\pi$
$17$	−3.10260	−0.752491	−0.376246	+	0.926520i	$0.622785\pi$
$18$	0	0
$19$	−5.58258	−1.28073	−0.640365	−	0.768070i	$-0.721217\pi$
$19$	−5.58258	−1.28073	−0.640365	+	0.768070i	$0.721217\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.10260	−0.646937	−0.323469	−	0.946239i	$-0.604849\pi$
$23$	−3.10260	−0.646937	−0.323469	+	0.946239i	$0.604849\pi$
$24$	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
$31$	−2.58258	−0.463844	−0.231922	−	0.972734i	$-0.574501\pi$
$31$	−2.58258	−0.463844	−0.231922	+	0.972734i	$0.574501\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.3739	1.70545	0.852726	−	0.522358i	$-0.174947\pi$
$37$	10.3739	1.70545	0.852726	+	0.522358i	$0.174947\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.37055	−0.214044	−0.107022	−	0.994257i	$-0.534132\pi$
$41$	−1.37055	−0.214044	−0.107022	+	0.994257i	$0.534132\pi$
$42$	0	0
$43$	−1.79129	−0.273169	−0.136584	−	0.990628i	$-0.543612\pi$
$43$	−1.79129	−0.273169	−0.136584	+	0.990628i	$0.543612\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.66025	−1.26323	−0.631614	−	0.775283i	$-0.717607\pi$
$47$	−8.66025	−1.26323	−0.631614	+	0.775283i	$0.717607\pi$
$48$	0	0
$49$	−3.79129	−0.541613
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.10260	0.426175	0.213088	−	0.977033i	$-0.431648\pi$
$53$	3.10260	0.426175	0.213088	+	0.977033i	$0.431648\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.361500	−0.0470633	−0.0235316	−	0.999723i	$-0.507491\pi$
$59$	−0.361500	−0.0470633	−0.0235316	+	0.999723i	$0.507491\pi$
$60$	0	0
$61$	−3.37386	−0.431979	−0.215989	−	0.976396i	$-0.569298\pi$
$61$	−3.37386	−0.431979	−0.215989	+	0.976396i	$0.569298\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.93725	0.941979	0.470989	−	0.882139i	$-0.343897\pi$
$71$	7.93725	0.941979	0.470989	+	0.882139i	$0.343897\pi$
$72$	0	0
$73$	5.00000	0.585206	0.292603	−	0.956234i	$-0.405479\pi$
$73$	5.00000	0.585206	0.292603	+	0.956234i	$0.405479\pi$
$74$	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
$83$	−14.8655	−1.63170	−0.815848	−	0.578266i	$-0.803729\pi$
$83$	−14.8655	−1.63170	−0.815848	+	0.578266i	$0.803729\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.74110	0.290556	0.145278	−	0.989391i	$-0.453592\pi$
$89$	2.74110	0.290556	0.145278	+	0.989391i	$0.453592\pi$
$90$	0	0
$91$	3.20871	0.336364
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	17.1652	1.74286	0.871429	−	0.490522i	$-0.163194\pi$
$97$	17.1652	1.74286	0.871429	+	0.490522i	$0.163194\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.28970	−0.725353	−0.362676	−	0.931915i	$-0.618137\pi$
$101$	−7.28970	−0.725353	−0.362676	+	0.931915i	$0.618137\pi$
$102$	0	0
$103$	10.3739	1.02217	0.511084	−	0.859531i	$-0.329244\pi$
$103$	10.3739	1.02217	0.511084	+	0.859531i	$0.329244\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.66025	0.837218	0.418609	−	0.908166i	$-0.362518\pi$
$107$	8.66025	0.837218	0.418609	+	0.908166i	$0.362518\pi$
$108$	0	0
$109$	5.95644	0.570523	0.285262	−	0.958450i	$-0.407919\pi$
$109$	5.95644	0.570523	0.285262	+	0.958450i	$0.407919\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	17.9681	1.69029	0.845146	−	0.534535i	$-0.179513\pi$
$113$	17.9681	1.69029	0.845146	+	0.534535i	$0.179513\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.55765	0.509469
$120$	0	0
$121$	12.3739	1.12490
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−5.74773	−0.510028	−0.255014	−	0.966937i	$-0.582080\pi$
$127$	−5.74773	−0.510028	−0.255014	+	0.966937i	$0.582080\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	16.5975	1.45013	0.725066	−	0.688680i	$-0.241809\pi$
$131$	16.5975	1.45013	0.725066	+	0.688680i	$0.241809\pi$
$132$	0	0
$133$	10.0000	0.867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.2179	−1.21472	−0.607359	−	0.794428i	$-0.707771\pi$
$137$	−14.2179	−1.21472	−0.607359	+	0.794428i	$0.707771\pi$
$138$	0	0
$139$	−1.62614	−0.137927	−0.0689635	−	0.997619i	$-0.521969\pi$
$139$	−1.62614	−0.137927	−0.0689635	+	0.997619i	$0.521969\pi$
$140$	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
$149$	−21.0707	−1.72618	−0.863088	−	0.505054i	$-0.831473\pi$
$149$	−21.0707	−1.72618	−0.863088	+	0.505054i	$0.831473\pi$
$150$	0	0
$151$	16.3739	1.33249	0.666243	−	0.745735i	$-0.267901\pi$
$151$	16.3739	1.33249	0.666243	+	0.745735i	$0.267901\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.5826	1.48305	0.741525	−	0.670925i	$-0.234103\pi$
$157$	18.5826	1.48305	0.741525	+	0.670925i	$0.234103\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	5.55765	0.438004
$162$	0	0
$163$	−8.58258	−0.672239	−0.336120	−	0.941819i	$-0.609115\pi$
$163$	−8.58258	−0.672239	−0.336120	+	0.941819i	$0.609115\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.4231	1.58039	0.790194	−	0.612857i	$-0.209980\pi$
$167$	20.4231	1.58039	0.790194	+	0.612857i	$0.209980\pi$
$168$	0	0
$169$	−9.79129	−0.753176
$170$	0	0
$171$	0	0
$172$	0	0
$173$	12.4104	0.943546	0.471773	−	0.881720i	$-0.343614\pi$
$173$	12.4104	0.943546	0.471773	+	0.881720i	$0.343614\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−17.6066	−1.31598	−0.657988	−	0.753028i	$-0.728592\pi$
$179$	−17.6066	−1.31598	−0.657988	+	0.753028i	$0.728592\pi$
$180$	0	0
$181$	−2.58258	−0.191961	−0.0959807	−	0.995383i	$-0.530599\pi$
$181$	−2.58258	−0.191961	−0.0959807	+	0.995383i	$0.530599\pi$
$182$	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
$191$	−4.18710	−0.302968	−0.151484	−	0.988460i	$-0.548405\pi$
$191$	−4.18710	−0.302968	−0.151484	+	0.988460i	$0.548405\pi$
$192$	0	0
$193$	−5.74773	−0.413730	−0.206865	−	0.978369i	$-0.566326\pi$
$193$	−5.74773	−0.413730	−0.206865	+	0.978369i	$0.566326\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	24.1733	1.72227	0.861137	−	0.508373i	$-0.169753\pi$
$197$	24.1733	1.72227	0.861137	+	0.508373i	$0.169753\pi$
$198$	0	0
$199$	−10.7913	−0.764974	−0.382487	−	0.923961i	$-0.624932\pi$
$199$	−10.7913	−0.764974	−0.382487	+	0.923961i	$0.624932\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−6.20520	−0.435520
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	26.9898	1.86692
$210$	0	0
$211$	12.5826	0.866220	0.433110	−	0.901341i	$-0.357416\pi$
$211$	12.5826	0.866220	0.433110	+	0.901341i	$0.357416\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.62614	0.314043
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.55765	0.373848
$222$	0	0
$223$	2.16515	0.144989	0.0724946	−	0.997369i	$-0.476904\pi$
$223$	2.16515	0.144989	0.0724946	+	0.997369i	$0.476904\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.9808	−1.72440	−0.862202	−	0.506565i	$-0.830915\pi$
$227$	−25.9808	−1.72440	−0.862202	+	0.506565i	$0.830915\pi$
$228$	0	0
$229$	−14.5826	−0.963644	−0.481822	−	0.876269i	$-0.660025\pi$
$229$	−14.5826	−0.963644	−0.481822	+	0.876269i	$0.660025\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.0707	−1.38038	−0.690192	−	0.723626i	$-0.742474\pi$
$233$	−21.0707	−1.38038	−0.690192	+	0.723626i	$0.742474\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.1153	−0.718989	−0.359495	−	0.933147i	$-0.617051\pi$
$239$	−11.1153	−0.718989	−0.359495	+	0.933147i	$0.617051\pi$
$240$	0	0
$241$	21.9129	1.41153	0.705766	−	0.708445i	$-0.250603\pi$
$241$	21.9129	1.41153	0.705766	+	0.708445i	$0.250603\pi$
$242$	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
$251$	−15.5130	−0.979172	−0.489586	−	0.871955i	$-0.662852\pi$
$251$	−15.5130	−0.979172	−0.489586	+	0.871955i	$0.662852\pi$
$252$	0	0
$253$	15.0000	0.943042
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.1153	0.693354	0.346677	−	0.937985i	$-0.387310\pi$
$257$	11.1153	0.693354	0.346677	+	0.937985i	$0.387310\pi$
$258$	0	0
$259$	−18.5826	−1.15467
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.1153	−0.685399	−0.342700	−	0.939445i	$-0.611341\pi$
$263$	−11.1153	−0.685399	−0.342700	+	0.939445i	$0.611341\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.2270	−0.928404	−0.464202	−	0.885729i	$-0.653659\pi$
$269$	−15.2270	−0.928404	−0.464202	+	0.885729i	$0.653659\pi$
$270$	0	0
$271$	27.7477	1.68555	0.842777	−	0.538263i	$-0.180919\pi$
$271$	27.7477	1.68555	0.842777	+	0.538263i	$0.180919\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4.33030	−0.260183	−0.130091	−	0.991502i	$-0.541527\pi$
$277$	−4.33030	−0.260183	−0.130091	+	0.991502i	$0.541527\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−32.1860	−1.92005	−0.960027	−	0.279908i	$-0.909696\pi$
$281$	−32.1860	−1.92005	−0.960027	+	0.279908i	$0.909696\pi$
$282$	0	0
$283$	7.53901	0.448148	0.224074	−	0.974572i	$-0.428064\pi$
$283$	7.53901	0.448148	0.224074	+	0.974572i	$0.428064\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.45505	0.144917
$288$	0	0
$289$	−7.37386	−0.433757
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.45505	0.143426	0.0717128	−	0.997425i	$-0.477153\pi$
$293$	2.45505	0.143426	0.0717128	+	0.997425i	$0.477153\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.55765	0.321407
$300$	0	0
$301$	3.20871	0.184947
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	30.7477	1.75487	0.877433	−	0.479700i	$-0.159254\pi$
$307$	30.7477	1.75487	0.877433	+	0.479700i	$0.159254\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.7937	1.23581	0.617903	−	0.786255i	$-0.287983\pi$
$311$	21.7937	1.23581	0.617903	+	0.786255i	$0.287983\pi$
$312$	0	0
$313$	11.7913	0.666483	0.333241	−	0.942842i	$-0.391858\pi$
$313$	11.7913	0.666483	0.333241	+	0.942842i	$0.391858\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−32.1860	−1.80774	−0.903872	−	0.427803i	$-0.859288\pi$
$317$	−32.1860	−1.80774	−0.903872	+	0.427803i	$0.859288\pi$
$318$	0	0
$319$	−16.7477	−0.937693
$320$	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
$329$	15.5130	0.855260
$330$	0	0
$331$	3.25227	0.178761	0.0893805	−	0.995998i	$-0.471511\pi$
$331$	3.25227	0.178761	0.0893805	+	0.995998i	$0.471511\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−4.33030	−0.235887	−0.117943	−	0.993020i	$-0.537630\pi$
$337$	−4.33030	−0.235887	−0.117943	+	0.993020i	$0.537630\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12.4859	0.676147
$342$	0	0
$343$	19.3303	1.04374
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.7629	0.631463	0.315732	−	0.948849i	$-0.397750\pi$
$347$	11.7629	0.631463	0.315732	+	0.948849i	$0.397750\pi$
$348$	0	0
$349$	29.4955	1.57886	0.789428	−	0.613844i	$-0.210377\pi$
$349$	29.4955	1.57886	0.789428	+	0.613844i	$0.210377\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.3785	−1.61688	−0.808441	−	0.588577i	$-0.799688\pi$
$353$	−30.3785	−1.61688	−0.808441	+	0.588577i	$0.799688\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.4322	1.13115	0.565573	−	0.824698i	$-0.308655\pi$
$359$	21.4322	1.13115	0.565573	+	0.824698i	$0.308655\pi$
$360$	0	0
$361$	12.1652	0.640271
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−0.373864	−0.0195155	−0.00975776	−	0.999952i	$-0.503106\pi$
$367$	−0.373864	−0.0195155	−0.00975776	+	0.999952i	$0.503106\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.55765	−0.288539
$372$	0	0
$373$	32.1652	1.66545	0.832724	−	0.553688i	$-0.186780\pi$
$373$	32.1652	1.66545	0.832724	+	0.553688i	$0.186780\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.20520	−0.319584
$378$	0	0
$379$	−24.2087	−1.24352	−0.621759	−	0.783209i	$-0.713582\pi$
$379$	−24.2087	−1.24352	−0.621759	+	0.783209i	$0.713582\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	17.9681	0.918125	0.459062	−	0.888404i	$-0.348185\pi$
$383$	17.9681	0.918125	0.459062	+	0.888404i	$0.348185\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	32.1860	1.63189	0.815947	−	0.578127i	$-0.196216\pi$
$389$	32.1860	1.63189	0.815947	+	0.578127i	$0.196216\pi$
$390$	0	0
$391$	9.62614	0.486815
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	32.1652	1.61432	0.807161	−	0.590331i	$-0.201003\pi$
$397$	32.1652	1.61432	0.807161	+	0.590331i	$0.201003\pi$
$398$	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
$403$	4.62614	0.230444
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−50.1540	−2.48604
$408$	0	0
$409$	27.2867	1.34924	0.674621	−	0.738164i	$-0.264307\pi$
$409$	27.2867	1.34924	0.674621	+	0.738164i	$0.264307\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.647551	0.0318639
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5.55765	−0.271509	−0.135755	−	0.990743i	$-0.543346\pi$
$419$	−5.55765	−0.271509	−0.135755	+	0.990743i	$0.543346\pi$
$420$	0	0
$421$	−25.4955	−1.24257	−0.621286	−	0.783584i	$-0.713390\pi$
$421$	−25.4955	−1.24257	−0.621286	+	0.783584i	$0.713390\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.04356	0.292468
$428$	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
$433$	3.58258	0.172168	0.0860838	−	0.996288i	$-0.472565\pi$
$433$	3.58258	0.172168	0.0860838	+	0.996288i	$0.472565\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.3205	0.828552
$438$	0	0
$439$	−29.1216	−1.38990	−0.694949	−	0.719059i	$-0.744573\pi$
$439$	−29.1216	−1.38990	−0.694949	+	0.719059i	$0.744573\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8.01270	−0.380695	−0.190348	−	0.981717i	$-0.560961\pi$
$443$	−8.01270	−0.380695	−0.190348	+	0.981717i	$0.560961\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17.6820	−0.834466	−0.417233	−	0.908800i	$-0.637000\pi$
$449$	−17.6820	−0.834466	−0.417233	+	0.908800i	$0.637000\pi$
$450$	0	0
$451$	6.62614	0.312013
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.7477	1.43832	0.719159	−	0.694846i	$-0.244527\pi$
$457$	30.7477	1.43832	0.719159	+	0.694846i	$0.244527\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.28065	−0.292519	−0.146260	−	0.989246i	$-0.546723\pi$
$461$	−6.28065	−0.292519	−0.146260	+	0.989246i	$0.546723\pi$
$462$	0	0
$463$	−26.1216	−1.21397	−0.606987	−	0.794712i	$-0.707622\pi$
$463$	−26.1216	−1.21397	−0.606987	+	0.794712i	$0.707622\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.91010	0.227212	0.113606	−	0.993526i	$-0.463760\pi$
$467$	4.91010	0.227212	0.113606	+	0.993526i	$0.463760\pi$
$468$	0	0
$469$	−8.95644	−0.413570
$470$	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
$479$	2.09355	0.0956568	0.0478284	−	0.998856i	$-0.484770\pi$
$479$	2.09355	0.0956568	0.0478284	+	0.998856i	$0.484770\pi$
$480$	0	0
$481$	−18.5826	−0.847293
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.3303	0.649368	0.324684	−	0.945823i	$-0.394742\pi$
$487$	14.3303	0.649368	0.324684	+	0.945823i	$0.394742\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−16.8836	−0.761944	−0.380972	−	0.924587i	$-0.624411\pi$
$491$	−16.8836	−0.761944	−0.380972	+	0.924587i	$0.624411\pi$
$492$	0	0
$493$	−10.7477	−0.484053
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.2179	−0.637760
$498$	0	0
$499$	−33.3739	−1.49402	−0.747010	−	0.664813i	$-0.768511\pi$
$499$	−33.3739	−1.49402	−0.747010	+	0.664813i	$0.768511\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.4231	0.910621	0.455311	−	0.890333i	$-0.349528\pi$
$503$	20.4231	0.910621	0.455311	+	0.890333i	$0.349528\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−21.0707	−0.933941	−0.466970	−	0.884273i	$-0.654655\pi$
$509$	−21.0707	−0.933941	−0.466970	+	0.884273i	$0.654655\pi$
$510$	0	0
$511$	−8.95644	−0.396210
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	41.8693	1.84141
$518$	0	0
$519$	0	0
$520$	0	0
$521$	42.1413	1.84624	0.923122	−	0.384506i	$-0.125628\pi$
$521$	42.1413	1.84624	0.923122	+	0.384506i	$0.125628\pi$
$522$	0	0
$523$	−5.74773	−0.251331	−0.125665	−	0.992073i	$-0.540107\pi$
$523$	−5.74773	−0.251331	−0.125665	+	0.992073i	$0.540107\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.01270	0.349039
$528$	0	0
$529$	−13.3739	−0.581472
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.45505	0.106340
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	18.3296	0.789510
$540$	0	0
$541$	7.20871	0.309927	0.154963	−	0.987920i	$-0.450474\pi$
$541$	7.20871	0.309927	0.154963	+	0.987920i	$0.450474\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	15.7477	0.673324	0.336662	−	0.941626i	$-0.390702\pi$
$547$	15.7477	0.673324	0.336662	+	0.941626i	$0.390702\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−19.3386	−0.823852
$552$	0	0
$553$	7.16515	0.304693
$554$	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
$559$	3.20871	0.135714
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.66025	−0.364986	−0.182493	−	0.983207i	$-0.558417\pi$
$563$	−8.66025	−0.364986	−0.182493	+	0.983207i	$0.558417\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.44600	0.0606195	0.0303097	−	0.999541i	$-0.490351\pi$
$569$	1.44600	0.0606195	0.0303097	+	0.999541i	$0.490351\pi$
$570$	0	0
$571$	38.3303	1.60407	0.802037	−	0.597275i	$-0.203750\pi$
$571$	38.3303	1.60407	0.802037	+	0.597275i	$0.203750\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−16.4955	−0.686715	−0.343357	−	0.939205i	$-0.611564\pi$
$577$	−16.4955	−0.686715	−0.343357	+	0.939205i	$0.611564\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	26.6283	1.10473
$582$	0	0
$583$	−15.0000	−0.621237
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.1606	0.667018	0.333509	−	0.942747i	$-0.391767\pi$
$587$	16.1606	0.667018	0.333509	+	0.942747i	$0.391767\pi$
$588$	0	0
$589$	14.4174	0.594060
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−22.8782	−0.939493	−0.469747	−	0.882801i	$-0.655655\pi$
$593$	−22.8782	−0.939493	−0.469747	+	0.882801i	$0.655655\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−35.5746	−1.45354	−0.726770	−	0.686881i	$-0.758979\pi$
$599$	−35.5746	−1.45354	−0.726770	+	0.686881i	$0.758979\pi$
$600$	0	0
$601$	40.7042	1.66036	0.830179	−	0.557497i	$-0.188238\pi$
$601$	40.7042	1.66036	0.830179	+	0.557497i	$0.188238\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0.747727	0.0303493	0.0151747	−	0.999885i	$-0.495170\pi$
$607$	0.747727	0.0303493	0.0151747	+	0.999885i	$0.495170\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	15.5130	0.627589
$612$	0	0
$613$	9.25227	0.373696	0.186848	−	0.982389i	$-0.440173\pi$
$613$	9.25227	0.373696	0.186848	+	0.982389i	$0.440173\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.45505	0.0988366	0.0494183	−	0.998778i	$-0.484263\pi$
$617$	2.45505	0.0988366	0.0494183	+	0.998778i	$0.484263\pi$
$618$	0	0
$619$	6.74773	0.271214	0.135607	−	0.990763i	$-0.456702\pi$
$619$	6.74773	0.271214	0.135607	+	0.990763i	$0.456702\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.91010	−0.196719
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−32.1860	−1.28334
$630$	0	0
$631$	−30.7042	−1.22231	−0.611157	−	0.791510i	$-0.709296\pi$
$631$	−30.7042	−1.22231	−0.611157	+	0.791510i	$0.709296\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.79129	0.269081
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−43.5119	−1.71862	−0.859308	−	0.511459i	$-0.829105\pi$
$641$	−43.5119	−1.71862	−0.859308	+	0.511459i	$0.829105\pi$
$642$	0	0
$643$	−21.8693	−0.862442	−0.431221	−	0.902246i	$-0.641917\pi$
$643$	−21.8693	−0.862442	−0.431221	+	0.902246i	$0.641917\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.2759	1.07232	0.536162	−	0.844115i	$-0.319873\pi$
$647$	27.2759	1.07232	0.536162	+	0.844115i	$0.319873\pi$
$648$	0	0
$649$	1.74773	0.0686043
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.647551	−0.0253406	−0.0126703	−	0.999920i	$-0.504033\pi$
$653$	−0.647551	−0.0253406	−0.0126703	+	0.999920i	$0.504033\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.5130	0.604301	0.302150	−	0.953260i	$-0.402295\pi$
$659$	15.5130	0.604301	0.302150	+	0.953260i	$0.402295\pi$
$660$	0	0
$661$	0.252273	0.00981228	0.00490614	−	0.999988i	$-0.498438\pi$
$661$	0.252273	0.00981228	0.00490614	+	0.999988i	$0.498438\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−10.7477	−0.416154
$668$	0	0
$669$	0	0
$670$	0	0
$671$	16.3115	0.629697
$672$	0	0
$673$	35.0000	1.34915	0.674575	−	0.738206i	$-0.264327\pi$
$673$	35.0000	1.34915	0.674575	+	0.738206i	$0.264327\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27.2759	1.04830	0.524148	−	0.851627i	$-0.324384\pi$
$677$	27.2759	1.04830	0.524148	+	0.851627i	$0.324384\pi$
$678$	0	0
$679$	−30.7477	−1.17999
$680$	0	0
$681$	0	0
$682$	0	0
$683$	43.9488	1.68165	0.840827	−	0.541303i	$-0.182069\pi$
$683$	43.9488	1.68165	0.840827	+	0.541303i	$0.182069\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−5.55765	−0.211730
$690$	0	0
$691$	44.4955	1.69269	0.846343	−	0.532638i	$-0.178799\pi$
$691$	44.4955	1.69269	0.846343	+	0.532638i	$0.178799\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	4.25227	0.161066
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.4539	1.15023	0.575114	−	0.818073i	$-0.304958\pi$
$701$	30.4539	1.15023	0.575114	+	0.818073i	$0.304958\pi$
$702$	0	0
$703$	−57.9129	−2.18423
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.0580	0.491095
$708$	0	0
$709$	−36.0780	−1.35494	−0.677469	−	0.735551i	$-0.736923\pi$
$709$	−36.0780	−1.35494	−0.677469	+	0.735551i	$0.736923\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	8.01270	0.300078
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16.9590	0.632464	0.316232	−	0.948682i	$-0.397582\pi$
$719$	16.9590	0.632464	0.316232	+	0.948682i	$0.397582\pi$
$720$	0	0
$721$	−18.5826	−0.692051
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	10.3739	0.384745	0.192373	−	0.981322i	$-0.438382\pi$
$727$	10.3739	0.384745	0.192373	+	0.981322i	$0.438382\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.55765	0.205557
$732$	0	0
$733$	36.1216	1.33418	0.667091	−	0.744977i	$-0.267539\pi$
$733$	36.1216	1.33418	0.667091	+	0.744977i	$0.267539\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.1733	−0.890434
$738$	0	0
$739$	34.8693	1.28269	0.641344	−	0.767253i	$-0.278377\pi$
$739$	34.8693	1.28269	0.641344	+	0.767253i	$0.278377\pi$
$740$	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
$749$	−15.5130	−0.566833
$750$	0	0
$751$	−23.1216	−0.843719	−0.421859	−	0.906661i	$-0.638622\pi$
$751$	−23.1216	−0.843719	−0.421859	+	0.906661i	$0.638622\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−4.62614	−0.168140	−0.0840699	−	0.996460i	$-0.526792\pi$
$757$	−4.62614	−0.168140	−0.0840699	+	0.996460i	$0.526792\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.12070	0.185625	0.0928127	−	0.995684i	$-0.470414\pi$
$761$	5.12070	0.185625	0.0928127	+	0.995684i	$0.470414\pi$
$762$	0	0
$763$	−10.6697	−0.386269
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0.647551	0.0233817
$768$	0	0
$769$	−15.5390	−0.560351	−0.280176	−	0.959949i	$-0.590393\pi$
$769$	−15.5390	−0.560351	−0.280176	+	0.959949i	$0.590393\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−38.3912	−1.38083	−0.690417	−	0.723411i	$-0.742573\pi$
$773$	−38.3912	−1.38083	−0.690417	+	0.723411i	$0.742573\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.65120	0.274133
$780$	0	0
$781$	−38.3739	−1.37312
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	10.3739	0.369788	0.184894	−	0.982758i	$-0.440806\pi$
$787$	10.3739	0.369788	0.184894	+	0.982758i	$0.440806\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−32.1860	−1.14440
$792$	0	0
$793$	6.04356	0.214613
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.7889	1.51566	0.757830	−	0.652452i	$-0.226260\pi$
$797$	42.7889	1.51566	0.757830	+	0.652452i	$0.226260\pi$
$798$	0	0
$799$	26.8693	0.950568
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−24.1733	−0.853056
$804$	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
$811$	22.0436	0.774054	0.387027	−	0.922068i	$-0.373502\pi$
$811$	22.0436	0.774054	0.387027	+	0.922068i	$0.373502\pi$
$812$	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
$821$	38.9633	1.35983	0.679914	−	0.733292i	$-0.262017\pi$
$821$	38.9633	1.35983	0.679914	+	0.733292i	$0.262017\pi$
$822$	0	0
$823$	−52.1652	−1.81836	−0.909181	−	0.416400i	$-0.863292\pi$
$823$	−52.1652	−1.81836	−0.909181	+	0.416400i	$0.863292\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	47.6990	1.65866	0.829328	−	0.558762i	$-0.188724\pi$
$827$	47.6990	1.65866	0.829328	+	0.558762i	$0.188724\pi$
$828$	0	0
$829$	−13.4955	−0.468716	−0.234358	−	0.972150i	$-0.575299\pi$
$829$	−13.4955	−0.468716	−0.234358	+	0.972150i	$0.575299\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	11.7629	0.407559
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.74110	0.0946333	0.0473167	−	0.998880i	$-0.484933\pi$
$839$	2.74110	0.0946333	0.0473167	+	0.998880i	$0.484933\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−22.1652	−0.761604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−32.1860	−1.10332
$852$	0	0
$853$	5.00000	0.171197	0.0855984	−	0.996330i	$-0.472720\pi$
$853$	5.00000	0.171197	0.0855984	+	0.996330i	$0.472720\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13.0580	−0.446051	−0.223026	−	0.974813i	$-0.571593\pi$
$857$	−13.0580	−0.446051	−0.223026	+	0.974813i	$0.571593\pi$
$858$	0	0
$859$	27.7477	0.946740	0.473370	−	0.880864i	$-0.343037\pi$
$859$	27.7477	0.946740	0.473370	+	0.880864i	$0.343037\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22.8782	0.778782	0.389391	−	0.921073i	$-0.372686\pi$
$863$	22.8782	0.778782	0.389391	+	0.921073i	$0.372686\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	19.3386	0.656017
$870$	0	0
$871$	−8.95644	−0.303477
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.79129	−0.0604875	−0.0302437	−	0.999543i	$-0.509628\pi$
$877$	−1.79129	−0.0604875	−0.0302437	+	0.999543i	$0.509628\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−26.5529	−0.894589	−0.447294	−	0.894387i	$-0.647612\pi$
$881$	−26.5529	−0.894589	−0.447294	+	0.894387i	$0.647612\pi$
$882$	0	0
$883$	38.6606	1.30103	0.650516	−	0.759492i	$-0.274552\pi$
$883$	38.6606	1.30103	0.650516	+	0.759492i	$0.274552\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−0.647551	−0.0217426	−0.0108713	−	0.999941i	$-0.503461\pi$
$887$	−0.647551	−0.0217426	−0.0108713	+	0.999941i	$0.503461\pi$
$888$	0	0
$889$	10.2958	0.345311
$890$	0	0
$891$	0	0
$892$	0	0
$893$	48.3465	1.61785
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−8.94630	−0.298376
$900$	0	0
$901$	−9.62614	−0.320693
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−15.3739	−0.510481	−0.255240	−	0.966878i	$-0.582155\pi$
$907$	−15.3739	−0.510481	−0.255240	+	0.966878i	$0.582155\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.4630	1.04241	0.521207	−	0.853430i	$-0.325482\pi$
$911$	31.4630	1.04241	0.521207	+	0.853430i	$0.325482\pi$
$912$	0	0
$913$	71.8693	2.37853
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−29.7309	−0.981801
$918$	0	0
$919$	−27.3739	−0.902980	−0.451490	−	0.892276i	$-0.649107\pi$
$919$	−27.3739	−0.902980	−0.451490	+	0.892276i	$0.649107\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−14.2179	−0.467988
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−17.6066	−0.577652	−0.288826	−	0.957382i	$-0.593265\pi$
$929$	−17.6066	−0.577652	−0.288826	+	0.957382i	$0.593265\pi$
$930$	0	0
$931$	21.1652	0.693660
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	18.5826	0.607066	0.303533	−	0.952821i	$-0.401834\pi$
$937$	18.5826	0.607066	0.303533	+	0.952821i	$0.401834\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−26.6283	−0.868058	−0.434029	−	0.900899i	$-0.642908\pi$
$941$	−26.6283	−0.868058	−0.434029	+	0.900899i	$0.642908\pi$
$942$	0	0
$943$	4.25227	0.138473
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−22.3658	−0.726790	−0.363395	−	0.931635i	$-0.618382\pi$
$947$	−22.3658	−0.726790	−0.363395	+	0.931635i	$0.618382\pi$
$948$	0	0
$949$	−8.95644	−0.290738
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−22.2306	−0.720120	−0.360060	−	0.932929i	$-0.617244\pi$
$953$	−22.2306	−0.720120	−0.360060	+	0.932929i	$0.617244\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	25.4684	0.822416
$960$	0	0
$961$	−24.3303	−0.784848
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−35.7477	−1.14957	−0.574785	−	0.818305i	$-0.694914\pi$
$967$	−35.7477	−1.14957	−0.574785	+	0.818305i	$0.694914\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−13.4949	−0.433072	−0.216536	−	0.976275i	$-0.569476\pi$
$971$	−13.4949	−0.433072	−0.216536	+	0.976275i	$0.569476\pi$
$972$	0	0
$973$	2.91288	0.0933826
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−51.4491	−1.64600	−0.823002	−	0.568039i	$-0.807702\pi$
$977$	−51.4491	−1.64600	−0.823002	+	0.568039i	$0.807702\pi$
$978$	0	0
$979$	−13.2523	−0.423544
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−57.0068	−1.81823	−0.909117	−	0.416541i	$-0.863242\pi$
$983$	−57.0068	−1.81823	−0.909117	+	0.416541i	$0.863242\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.55765	0.176723
$990$	0	0
$991$	0.121591	0.00386245	0.00193123	−	0.999998i	$-0.499385\pi$
$991$	0.121591	0.00386245	0.00193123	+	0.999998i	$0.499385\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−20.7477	−0.657087	−0.328544	−	0.944489i	$-0.606558\pi$
$997$	−20.7477	−0.657087	−0.328544	+	0.944489i	$0.606558\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.a.bb.1.1	yes	4
3.2	odd	2	inner	8100.2.a.bb.1.2	yes	4
5.2	odd	4		8100.2.d.r.649.3		8
5.3	odd	4		8100.2.d.r.649.5		8
5.4	even	2		8100.2.a.w.1.3	✓	4
15.2	even	4		8100.2.d.r.649.4		8
15.8	even	4		8100.2.d.r.649.6		8
15.14	odd	2		8100.2.a.w.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8100.2.a.w.1.3	✓	4	5.4	even	2
8100.2.a.w.1.4	yes	4	15.14	odd	2
8100.2.a.bb.1.1	yes	4	1.1	even	1	trivial
8100.2.a.bb.1.2	yes	4	3.2	odd	2	inner
8100.2.d.r.649.3		8	5.2	odd	4
8100.2.d.r.649.4		8	15.2	even	4
8100.2.d.r.649.5		8	5.3	odd	4
8100.2.d.r.649.6		8	15.8	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:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79129 q^{7} +O(q^{10})\)	\(q-1.79129 q^{7} -4.83465 q^{11} -1.79129 q^{13} -3.10260 q^{17} -5.58258 q^{19} -3.10260 q^{23} +3.46410 q^{29} -2.58258 q^{31} +10.3739 q^{37} -1.37055 q^{41} -1.79129 q^{43} -8.66025 q^{47} -3.79129 q^{49} +3.10260 q^{53} -0.361500 q^{59} -3.37386 q^{61} +5.00000 q^{67} +7.93725 q^{71} +5.00000 q^{73} +8.66025 q^{77} -4.00000 q^{79} -14.8655 q^{83} +2.74110 q^{89} +3.20871 q^{91} +17.1652 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{7}+O(q^{10}) \) 4 * q + 2 * q^7	\( 4 q + 2 q^{7} + 2 q^{13} - 4 q^{19} + 8 q^{31} + 14 q^{37} + 2 q^{43} - 6 q^{49} + 14 q^{61} + 20 q^{67} + 20 q^{73} - 16 q^{79} + 22 q^{91} + 32 q^{97}+O(q^{100}) \) 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

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Varieties

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Representations

Groups

Database

Properties

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