LMFDB - Embedded newform 4761.2.a.bx.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4761,2,Mod(1,4761)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4761, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4761.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.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:	$38.0167764023$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 30 x^{18} + 376 x^{16} - 2566 x^{14} + 10441 x^{12} - 26158 x^{10} + 40383 x^{8} - 37458 x^{6} + \cdots + 529 $ x^20 - 30x^18 + 376x^16 - 2566x^14 + 10441x^12 - 26158x^10 + 40383x^8 - 37458x^6 + 19703x^4 - 5244*x^2 + 529
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 207)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.65128$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.65128 q^{2} +5.02931 q^{4} -1.60789 q^{5} -0.611275 q^{7} -8.03155 q^{8} +O(q^{10})$	$q-2.65128 q^{2} +5.02931 q^{4} -1.60789 q^{5} -0.611275 q^{7} -8.03155 q^{8} +4.26298 q^{10} -4.13764 q^{11} -6.29184 q^{13} +1.62066 q^{14} +11.2353 q^{16} -5.73681 q^{17} +5.12452 q^{19} -8.08659 q^{20} +10.9701 q^{22} -2.41468 q^{25} +16.6815 q^{26} -3.07429 q^{28} -3.08385 q^{29} -3.93971 q^{31} -13.7249 q^{32} +15.2099 q^{34} +0.982865 q^{35} -1.82430 q^{37} -13.5866 q^{38} +12.9139 q^{40} -1.38185 q^{41} -7.89525 q^{43} -20.8095 q^{44} -4.94390 q^{47} -6.62634 q^{49} +6.40201 q^{50} -31.6436 q^{52} -3.15049 q^{53} +6.65289 q^{55} +4.90949 q^{56} +8.17616 q^{58} +3.26551 q^{59} +7.30354 q^{61} +10.4453 q^{62} +13.9180 q^{64} +10.1166 q^{65} -11.5048 q^{67} -28.8522 q^{68} -2.60585 q^{70} -13.2606 q^{71} -3.72492 q^{73} +4.83675 q^{74} +25.7728 q^{76} +2.52924 q^{77} -0.363386 q^{79} -18.0652 q^{80} +3.66367 q^{82} +1.11340 q^{83} +9.22417 q^{85} +20.9326 q^{86} +33.2317 q^{88} -1.65263 q^{89} +3.84605 q^{91} +13.1077 q^{94} -8.23967 q^{95} +4.92979 q^{97} +17.5683 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{4} + 18 q^{7}+O(q^{10}) $ 20 * q + 20 * q^4 + 18 * q^7	$ 20 q + 20 q^{4} + 18 q^{7} + 22 q^{10} + 16 q^{16} + 40 q^{19} + 14 q^{22} + 20 q^{25} + 32 q^{28} - 22 q^{31} + 60 q^{34} + 18 q^{37} + 74 q^{40} + 32 q^{43} + 2 q^{49} - 52 q^{55} - 24 q^{58} + 70 q^{61} + 36 q^{64} + 64 q^{67} + 48 q^{70} + 40 q^{73} + 82 q^{76} + 106 q^{79} - 36 q^{82} + 2 q^{85} + 58 q^{88} + 88 q^{91} + 40 q^{94} + 24 q^{97}+O(q^{100}) $ 20 * q + 20 * q^4 + 18 * q^7 + 22 * q^10 + 16 * q^16 + 40 * q^19 + 14 * q^22 + 20 * q^25 + 32 * q^28 - 22 * q^31 + 60 * q^34 + 18 * q^37 + 74 * q^40 + 32 * q^43 + 2 * q^49 - 52 * q^55 - 24 * q^58 + 70 * q^61 + 36 * q^64 + 64 * q^67 + 48 * q^70 + 40 * q^73 + 82 * q^76 + 106 * q^79 - 36 * q^82 + 2 * q^85 + 58 * q^88 + 88 * q^91 + 40 * q^94 + 24 * 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 $

$2$	−2.65128	−1.87474	−0.937370	−	0.348334i	$-0.886748\pi$
$2$	−2.65128	−1.87474	−0.937370	+	0.348334i	$0.886748\pi$
$3$	0	0
$3$	0	0
$4$	5.02931	2.51465
$5$	−1.60789	−0.719071	−0.359536	−	0.933131i	$-0.617065\pi$
$5$	−1.60789	−0.719071	−0.359536	+	0.933131i	$0.617065\pi$
$6$	0	0
$7$	−0.611275	−0.231040	−0.115520	−	0.993305i	$-0.536853\pi$
$7$	−0.611275	−0.231040	−0.115520	+	0.993305i	$0.536853\pi$
$8$	−8.03155	−2.83958
$9$	0	0
$10$	4.26298	1.34807
$11$	−4.13764	−1.24755	−0.623773	−	0.781605i	$-0.714401\pi$
$11$	−4.13764	−1.24755	−0.623773	+	0.781605i	$0.714401\pi$
$12$	0	0
$13$	−6.29184	−1.74504	−0.872522	−	0.488576i	$-0.837517\pi$
$13$	−6.29184	−1.74504	−0.872522	+	0.488576i	$0.837517\pi$
$14$	1.62066	0.433141
$15$	0	0
$16$	11.2353	2.80883
$17$	−5.73681	−1.39138	−0.695690	−	0.718342i	$-0.744901\pi$
$17$	−5.73681	−1.39138	−0.695690	+	0.718342i	$0.744901\pi$
$18$	0	0
$19$	5.12452	1.17564	0.587822	−	0.808990i	$-0.299985\pi$
$19$	5.12452	1.17564	0.587822	+	0.808990i	$0.299985\pi$
$20$	−8.08659	−1.80822
$21$	0	0
$22$	10.9701	2.33883
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−2.41468	−0.482936
$26$	16.6815	3.27150
$27$	0	0
$28$	−3.07429	−0.580986
$29$	−3.08385	−0.572657	−0.286328	−	0.958132i	$-0.592435\pi$
$29$	−3.08385	−0.572657	−0.286328	+	0.958132i	$0.592435\pi$
$30$	0	0
$31$	−3.93971	−0.707593	−0.353797	−	0.935322i	$-0.615110\pi$
$31$	−3.93971	−0.707593	−0.353797	+	0.935322i	$0.615110\pi$
$32$	−13.7249	−2.42624
$33$	0	0
$34$	15.2099	2.60848
$35$	0.982865	0.166134
$36$	0	0
$37$	−1.82430	−0.299914	−0.149957	−	0.988693i	$-0.547913\pi$
$37$	−1.82430	−0.299914	−0.149957	+	0.988693i	$0.547913\pi$
$38$	−13.5866	−2.20403
$39$	0	0
$40$	12.9139	2.04186
$41$	−1.38185	−0.215808	−0.107904	−	0.994161i	$-0.534414\pi$
$41$	−1.38185	−0.215808	−0.107904	+	0.994161i	$0.534414\pi$
$42$	0	0
$43$	−7.89525	−1.20401	−0.602007	−	0.798491i	$-0.705632\pi$
$43$	−7.89525	−1.20401	−0.602007	+	0.798491i	$0.705632\pi$
$44$	−20.8095	−3.13715
$45$	0	0
$46$	0	0
$47$	−4.94390	−0.721142	−0.360571	−	0.932732i	$-0.617418\pi$
$47$	−4.94390	−0.721142	−0.360571	+	0.932732i	$0.617418\pi$
$48$	0	0
$49$	−6.62634	−0.946620
$50$	6.40201	0.905381
$51$	0	0
$52$	−31.6436	−4.38818
$53$	−3.15049	−0.432754	−0.216377	−	0.976310i	$-0.569424\pi$
$53$	−3.15049	−0.432754	−0.216377	+	0.976310i	$0.569424\pi$
$54$	0	0
$55$	6.65289	0.897075
$56$	4.90949	0.656058
$57$	0	0
$58$	8.17616	1.07358
$59$	3.26551	0.425133	0.212567	−	0.977147i	$-0.431818\pi$
$59$	3.26551	0.425133	0.212567	+	0.977147i	$0.431818\pi$
$60$	0	0
$61$	7.30354	0.935122	0.467561	−	0.883961i	$-0.345133\pi$
$61$	7.30354	0.935122	0.467561	+	0.883961i	$0.345133\pi$
$62$	10.4453	1.32655
$63$	0	0
$64$	13.9180	1.73975
$65$	10.1166	1.25481
$66$	0	0
$67$	−11.5048	−1.40553	−0.702765	−	0.711422i	$-0.748052\pi$
$67$	−11.5048	−1.40553	−0.702765	+	0.711422i	$0.748052\pi$
$68$	−28.8522	−3.49884
$69$	0	0
$70$	−2.60585	−0.311459
$71$	−13.2606	−1.57374	−0.786870	−	0.617118i	$-0.788300\pi$
$71$	−13.2606	−1.57374	−0.786870	+	0.617118i	$0.788300\pi$
$72$	0	0
$73$	−3.72492	−0.435969	−0.217985	−	0.975952i	$-0.569948\pi$
$73$	−3.72492	−0.435969	−0.217985	+	0.975952i	$0.569948\pi$
$74$	4.83675	0.562260
$75$	0	0
$76$	25.7728	2.95634
$77$	2.52924	0.288233
$78$	0	0
$79$	−0.363386	−0.0408842	−0.0204421	−	0.999791i	$-0.506507\pi$
$79$	−0.363386	−0.0408842	−0.0204421	+	0.999791i	$0.506507\pi$
$80$	−18.0652	−2.01975
$81$	0	0
$82$	3.66367	0.404585
$83$	1.11340	0.122212	0.0611060	−	0.998131i	$-0.480537\pi$
$83$	1.11340	0.122212	0.0611060	+	0.998131i	$0.480537\pi$
$84$	0	0
$85$	9.22417	1.00050
$86$	20.9326	2.25722
$87$	0	0
$88$	33.2317	3.54251
$89$	−1.65263	−0.175178	−0.0875891	−	0.996157i	$-0.527916\pi$
$89$	−1.65263	−0.175178	−0.0875891	+	0.996157i	$0.527916\pi$
$90$	0	0
$91$	3.84605	0.403175
$92$	0	0
$93$	0	0
$94$	13.1077	1.35195
$95$	−8.23967	−0.845373
$96$	0	0
$97$	4.92979	0.500545	0.250272	−	0.968175i	$-0.419480\pi$
$97$	4.92979	0.500545	0.250272	+	0.968175i	$0.419480\pi$
$98$	17.5683	1.77467
$99$	0	0
$100$	−12.1442	−1.21442
$101$	−13.7082	−1.36402	−0.682010	−	0.731343i	$-0.738894\pi$
$101$	−13.7082	−1.36402	−0.682010	+	0.731343i	$0.738894\pi$
$102$	0	0
$103$	−4.44209	−0.437692	−0.218846	−	0.975759i	$-0.570229\pi$
$103$	−4.44209	−0.437692	−0.218846	+	0.975759i	$0.570229\pi$
$104$	50.5333	4.95520
$105$	0	0
$106$	8.35286	0.811301
$107$	8.45050	0.816940	0.408470	−	0.912772i	$-0.366062\pi$
$107$	8.45050	0.816940	0.408470	+	0.912772i	$0.366062\pi$
$108$	0	0
$109$	9.51522	0.911393	0.455697	−	0.890135i	$-0.349390\pi$
$109$	9.51522	0.911393	0.455697	+	0.890135i	$0.349390\pi$
$110$	−17.6387	−1.68178
$111$	0	0
$112$	−6.86787	−0.648953
$113$	10.6467	1.00156	0.500781	−	0.865574i	$-0.333046\pi$
$113$	10.6467	1.00156	0.500781	+	0.865574i	$0.333046\pi$
$114$	0	0
$115$	0	0
$116$	−15.5096	−1.44003
$117$	0	0
$118$	−8.65779	−0.797014
$119$	3.50677	0.321465
$120$	0	0
$121$	6.12009	0.556372
$122$	−19.3638	−1.75311
$123$	0	0
$124$	−19.8140	−1.77935
$125$	11.9220	1.06634
$126$	0	0
$127$	−9.02496	−0.800836	−0.400418	−	0.916333i	$-0.631135\pi$
$127$	−9.02496	−0.800836	−0.400418	+	0.916333i	$0.631135\pi$
$128$	−9.45076	−0.835337
$129$	0	0
$130$	−26.8220	−2.35244
$131$	−8.03040	−0.701619	−0.350810	−	0.936447i	$-0.614094\pi$
$131$	−8.03040	−0.701619	−0.350810	+	0.936447i	$0.614094\pi$
$132$	0	0
$133$	−3.13249	−0.271621
$134$	30.5024	2.63501
$135$	0	0
$136$	46.0755	3.95094
$137$	−17.3960	−1.48624	−0.743121	−	0.669158i	$-0.766655\pi$
$137$	−17.3960	−1.48624	−0.743121	+	0.669158i	$0.766655\pi$
$138$	0	0
$139$	−1.63378	−0.138576	−0.0692878	−	0.997597i	$-0.522073\pi$
$139$	−1.63378	−0.138576	−0.0692878	+	0.997597i	$0.522073\pi$
$140$	4.94313	0.417771
$141$	0	0
$142$	35.1575	2.95036
$143$	26.0334	2.17702
$144$	0	0
$145$	4.95850	0.411781
$146$	9.87582	0.817329
$147$	0	0
$148$	−9.17498	−0.754179
$149$	20.1851	1.65362	0.826812	−	0.562478i	$-0.190152\pi$
$149$	20.1851	1.65362	0.826812	+	0.562478i	$0.190152\pi$
$150$	0	0
$151$	−2.42542	−0.197378	−0.0986889	−	0.995118i	$-0.531465\pi$
$151$	−2.42542	−0.197378	−0.0986889	+	0.995118i	$0.531465\pi$
$152$	−41.1578	−3.33834
$153$	0	0
$154$	−6.70573	−0.540363
$155$	6.33464	0.508810
$156$	0	0
$157$	−11.4520	−0.913972	−0.456986	−	0.889474i	$-0.651071\pi$
$157$	−11.4520	−0.913972	−0.456986	+	0.889474i	$0.651071\pi$
$158$	0.963441	0.0766472
$159$	0	0
$160$	22.0682	1.74464
$161$	0	0
$162$	0	0
$163$	−8.24476	−0.645780	−0.322890	−	0.946437i	$-0.604654\pi$
$163$	−8.24476	−0.645780	−0.322890	+	0.946437i	$0.604654\pi$
$164$	−6.94973	−0.542683
$165$	0	0
$166$	−2.95195	−0.229116
$167$	0.244144	0.0188924	0.00944622	−	0.999955i	$-0.496993\pi$
$167$	0.244144	0.0188924	0.00944622	+	0.999955i	$0.496993\pi$
$168$	0	0
$169$	26.5873	2.04518
$170$	−24.4559	−1.87568
$171$	0	0
$172$	−39.7076	−3.02768
$173$	−0.537459	−0.0408622	−0.0204311	−	0.999791i	$-0.506504\pi$
$173$	−0.537459	−0.0408622	−0.0204311	+	0.999791i	$0.506504\pi$
$174$	0	0
$175$	1.47603	0.111578
$176$	−46.4877	−3.50415
$177$	0	0
$178$	4.38159	0.328414
$179$	−0.520660	−0.0389160	−0.0194580	−	0.999811i	$-0.506194\pi$
$179$	−0.520660	−0.0389160	−0.0194580	+	0.999811i	$0.506194\pi$
$180$	0	0
$181$	19.0968	1.41945	0.709726	−	0.704478i	$-0.248819\pi$
$181$	19.0968	1.41945	0.709726	+	0.704478i	$0.248819\pi$
$182$	−10.1970	−0.755849
$183$	0	0
$184$	0	0
$185$	2.93328	0.215659
$186$	0	0
$187$	23.7369	1.73581
$188$	−24.8644	−1.81342
$189$	0	0
$190$	21.8457	1.58485
$191$	−20.7782	−1.50346	−0.751728	−	0.659473i	$-0.770779\pi$
$191$	−20.7782	−1.50346	−0.751728	+	0.659473i	$0.770779\pi$
$192$	0	0
$193$	−11.0869	−0.798050	−0.399025	−	0.916940i	$-0.630651\pi$
$193$	−11.0869	−0.798050	−0.399025	+	0.916940i	$0.630651\pi$
$194$	−13.0703	−0.938391
$195$	0	0
$196$	−33.3259	−2.38042
$197$	14.1574	1.00867	0.504337	−	0.863507i	$-0.331737\pi$
$197$	14.1574	1.00867	0.504337	+	0.863507i	$0.331737\pi$
$198$	0	0
$199$	5.45266	0.386529	0.193264	−	0.981147i	$-0.438093\pi$
$199$	5.45266	0.386529	0.193264	+	0.981147i	$0.438093\pi$
$200$	19.3936	1.37134
$201$	0	0
$202$	36.3444	2.55719
$203$	1.88508	0.132307
$204$	0	0
$205$	2.22186	0.155182
$206$	11.7772	0.820559
$207$	0	0
$208$	−70.6909	−4.90153
$209$	−21.2034	−1.46667
$210$	0	0
$211$	−13.0650	−0.899431	−0.449716	−	0.893172i	$-0.648475\pi$
$211$	−13.0650	−0.899431	−0.449716	+	0.893172i	$0.648475\pi$
$212$	−15.8448	−1.08823
$213$	0	0
$214$	−22.4047	−1.53155
$215$	12.6947	0.865772
$216$	0	0
$217$	2.40825	0.163483
$218$	−25.2276	−1.70863
$219$	0	0
$220$	33.4594	2.25583
$221$	36.0951	2.42802
$222$	0	0
$223$	−19.3440	−1.29537	−0.647683	−	0.761910i	$-0.724262\pi$
$223$	−19.3440	−1.29537	−0.647683	+	0.761910i	$0.724262\pi$
$224$	8.38970	0.560560
$225$	0	0
$226$	−28.2276	−1.87767
$227$	−23.4258	−1.55483	−0.777413	−	0.628990i	$-0.783469\pi$
$227$	−23.4258	−1.55483	−0.777413	+	0.628990i	$0.783469\pi$
$228$	0	0
$229$	−14.4168	−0.952687	−0.476344	−	0.879259i	$-0.658038\pi$
$229$	−14.4168	−0.952687	−0.476344	+	0.879259i	$0.658038\pi$
$230$	0	0
$231$	0	0
$232$	24.7681	1.62611
$233$	25.2797	1.65613	0.828063	−	0.560635i	$-0.189443\pi$
$233$	25.2797	1.65613	0.828063	+	0.560635i	$0.189443\pi$
$234$	0	0
$235$	7.94926	0.518552
$236$	16.4232	1.06906
$237$	0	0
$238$	−9.29744	−0.602663
$239$	22.3911	1.44836	0.724179	−	0.689612i	$-0.242219\pi$
$239$	22.3911	1.44836	0.724179	+	0.689612i	$0.242219\pi$
$240$	0	0
$241$	2.28067	0.146911	0.0734555	−	0.997298i	$-0.476597\pi$
$241$	2.28067	0.146911	0.0734555	+	0.997298i	$0.476597\pi$
$242$	−16.2261	−1.04305
$243$	0	0
$244$	36.7317	2.35151
$245$	10.6544	0.680688
$246$	0	0
$247$	−32.2427	−2.05155
$248$	31.6420	2.00927
$249$	0	0
$250$	−31.6086	−1.99911
$251$	7.06056	0.445659	0.222829	−	0.974857i	$-0.428471\pi$
$251$	7.06056	0.445659	0.222829	+	0.974857i	$0.428471\pi$
$252$	0	0
$253$	0	0
$254$	23.9277	1.50136
$255$	0	0
$256$	−2.77936	−0.173710
$257$	13.2890	0.828946	0.414473	−	0.910062i	$-0.363966\pi$
$257$	13.2890	0.828946	0.414473	+	0.910062i	$0.363966\pi$
$258$	0	0
$259$	1.11515	0.0692921
$260$	50.8795	3.15541
$261$	0	0
$262$	21.2909	1.31535
$263$	14.1343	0.871557	0.435779	−	0.900054i	$-0.356473\pi$
$263$	14.1343	0.871557	0.435779	+	0.900054i	$0.356473\pi$
$264$	0	0
$265$	5.06566	0.311181
$266$	8.30512	0.509220
$267$	0	0
$268$	−57.8610	−3.53442
$269$	13.2283	0.806544	0.403272	−	0.915080i	$-0.367873\pi$
$269$	13.2283	0.806544	0.403272	+	0.915080i	$0.367873\pi$
$270$	0	0
$271$	7.58059	0.460488	0.230244	−	0.973133i	$-0.426048\pi$
$271$	7.58059	0.460488	0.230244	+	0.973133i	$0.426048\pi$
$272$	−64.4549	−3.90815
$273$	0	0
$274$	46.1218	2.78632
$275$	9.99109	0.602485
$276$	0	0
$277$	−12.5567	−0.754461	−0.377230	−	0.926119i	$-0.623123\pi$
$277$	−12.5567	−0.754461	−0.377230	+	0.926119i	$0.623123\pi$
$278$	4.33162	0.259793
$279$	0	0
$280$	−7.89393	−0.471753
$281$	−12.2091	−0.728335	−0.364168	−	0.931333i	$-0.618647\pi$
$281$	−12.2091	−0.728335	−0.364168	+	0.931333i	$0.618647\pi$
$282$	0	0
$283$	5.06964	0.301359	0.150679	−	0.988583i	$-0.451854\pi$
$283$	5.06964	0.301359	0.150679	+	0.988583i	$0.451854\pi$
$284$	−66.6915	−3.95741
$285$	0	0
$286$	−69.0219	−4.08135
$287$	0.844689	0.0498604
$288$	0	0
$289$	15.9110	0.935939
$290$	−13.1464	−0.771983
$291$	0	0
$292$	−18.7338	−1.09631
$293$	−30.5501	−1.78475	−0.892377	−	0.451292i	$-0.850963\pi$
$293$	−30.5501	−1.78475	−0.892377	+	0.451292i	$0.850963\pi$
$294$	0	0
$295$	−5.25059	−0.305701
$296$	14.6520	0.851630
$297$	0	0
$298$	−53.5163	−3.10012
$299$	0	0
$300$	0	0
$301$	4.82617	0.278176
$302$	6.43048	0.370032
$303$	0	0
$304$	57.5756	3.30219
$305$	−11.7433	−0.672420
$306$	0	0
$307$	−24.4388	−1.39480	−0.697398	−	0.716684i	$-0.745659\pi$
$307$	−24.4388	−1.39480	−0.697398	+	0.716684i	$0.745659\pi$
$308$	12.7203	0.724807
$309$	0	0
$310$	−16.7949	−0.953887
$311$	−23.5389	−1.33477	−0.667385	−	0.744713i	$-0.732586\pi$
$311$	−23.5389	−1.33477	−0.667385	+	0.744713i	$0.732586\pi$
$312$	0	0
$313$	−29.0455	−1.64175	−0.820875	−	0.571108i	$-0.806514\pi$
$313$	−29.0455	−1.64175	−0.820875	+	0.571108i	$0.806514\pi$
$314$	30.3626	1.71346
$315$	0	0
$316$	−1.82758	−0.102810
$317$	−7.73221	−0.434284	−0.217142	−	0.976140i	$-0.569674\pi$
$317$	−7.73221	−0.434284	−0.217142	+	0.976140i	$0.569674\pi$
$318$	0	0
$319$	12.7599	0.714416
$320$	−22.3787	−1.25101
$321$	0	0
$322$	0	0
$323$	−29.3984	−1.63577
$324$	0	0
$325$	15.1928	0.842745
$326$	21.8592	1.21067
$327$	0	0
$328$	11.0984	0.612806
$329$	3.02208	0.166613
$330$	0	0
$331$	−10.7783	−0.592427	−0.296214	−	0.955122i	$-0.595724\pi$
$331$	−10.7783	−0.592427	−0.296214	+	0.955122i	$0.595724\pi$
$332$	5.59965	0.307321
$333$	0	0
$334$	−0.647295	−0.0354184
$335$	18.4984	1.01068
$336$	0	0
$337$	16.5590	0.902028	0.451014	−	0.892517i	$-0.351062\pi$
$337$	16.5590	0.902028	0.451014	+	0.892517i	$0.351062\pi$
$338$	−70.4904	−3.83417
$339$	0	0
$340$	46.3912	2.51592
$341$	16.3011	0.882756
$342$	0	0
$343$	8.32944	0.449748
$344$	63.4111	3.41890
$345$	0	0
$346$	1.42496	0.0766061
$347$	−12.8417	−0.689377	−0.344688	−	0.938717i	$-0.612015\pi$
$347$	−12.8417	−0.689377	−0.344688	+	0.938717i	$0.612015\pi$
$348$	0	0
$349$	−15.3266	−0.820412	−0.410206	−	0.911993i	$-0.634543\pi$
$349$	−15.3266	−0.820412	−0.410206	+	0.911993i	$0.634543\pi$
$350$	−3.91339	−0.209179
$351$	0	0
$352$	56.7888	3.02685
$353$	16.8522	0.896952	0.448476	−	0.893795i	$-0.351967\pi$
$353$	16.8522	0.896952	0.448476	+	0.893795i	$0.351967\pi$
$354$	0	0
$355$	21.3216	1.13163
$356$	−8.31157	−0.440512
$357$	0	0
$358$	1.38042	0.0729574
$359$	4.71983	0.249103	0.124552	−	0.992213i	$-0.460251\pi$
$359$	4.71983	0.249103	0.124552	+	0.992213i	$0.460251\pi$
$360$	0	0
$361$	7.26067	0.382141
$362$	−50.6310	−2.66111
$363$	0	0
$364$	19.3430	1.01385
$365$	5.98927	0.313493
$366$	0	0
$367$	26.2863	1.37213	0.686066	−	0.727539i	$-0.259336\pi$
$367$	26.2863	1.37213	0.686066	+	0.727539i	$0.259336\pi$
$368$	0	0
$369$	0	0
$370$	−7.77697	−0.404305
$371$	1.92582	0.0999835
$372$	0	0
$373$	8.66662	0.448740	0.224370	−	0.974504i	$-0.427968\pi$
$373$	8.66662	0.448740	0.224370	+	0.974504i	$0.427968\pi$
$374$	−62.9332	−3.25420
$375$	0	0
$376$	39.7072	2.04774
$377$	19.4031	0.999311
$378$	0	0
$379$	−14.1345	−0.726038	−0.363019	−	0.931782i	$-0.618254\pi$
$379$	−14.1345	−0.726038	−0.363019	+	0.931782i	$0.618254\pi$
$380$	−41.4398	−2.12582
$381$	0	0
$382$	55.0889	2.81859
$383$	−1.01939	−0.0520885	−0.0260443	−	0.999661i	$-0.508291\pi$
$383$	−1.01939	−0.0520885	−0.0260443	+	0.999661i	$0.508291\pi$
$384$	0	0
$385$	−4.06674	−0.207260
$386$	29.3944	1.49614
$387$	0	0
$388$	24.7934	1.25870
$389$	9.25243	0.469117	0.234559	−	0.972102i	$-0.424636\pi$
$389$	9.25243	0.469117	0.234559	+	0.972102i	$0.424636\pi$
$390$	0	0
$391$	0	0
$392$	53.2198	2.68801
$393$	0	0
$394$	−37.5353	−1.89100
$395$	0.584286	0.0293986
$396$	0	0
$397$	14.1244	0.708882	0.354441	−	0.935078i	$-0.384671\pi$
$397$	14.1244	0.708882	0.354441	+	0.935078i	$0.384671\pi$
$398$	−14.4565	−0.724641
$399$	0	0
$400$	−27.1297	−1.35649
$401$	−25.7085	−1.28382	−0.641910	−	0.766780i	$-0.721858\pi$
$401$	−25.7085	−1.28382	−0.641910	+	0.766780i	$0.721858\pi$
$402$	0	0
$403$	24.7881	1.23478
$404$	−68.9429	−3.43004
$405$	0	0
$406$	−4.99789	−0.248041
$407$	7.54832	0.374156
$408$	0	0
$409$	−24.8420	−1.22836	−0.614180	−	0.789166i	$-0.710513\pi$
$409$	−24.8420	−1.22836	−0.614180	+	0.789166i	$0.710513\pi$
$410$	−5.89079	−0.290925
$411$	0	0
$412$	−22.3406	−1.10064
$413$	−1.99612	−0.0982228
$414$	0	0
$415$	−1.79023	−0.0878792
$416$	86.3550	4.23390
$417$	0	0
$418$	56.2163	2.74963
$419$	−0.663091	−0.0323941	−0.0161971	−	0.999869i	$-0.505156\pi$
$419$	−0.663091	−0.0323941	−0.0161971	+	0.999869i	$0.505156\pi$
$420$	0	0
$421$	−16.0064	−0.780103	−0.390051	−	0.920793i	$-0.627543\pi$
$421$	−16.0064	−0.780103	−0.390051	+	0.920793i	$0.627543\pi$
$422$	34.6390	1.68620
$423$	0	0
$424$	25.3034	1.22884
$425$	13.8526	0.671948
$426$	0	0
$427$	−4.46447	−0.216051
$428$	42.5001	2.05432
$429$	0	0
$430$	−33.6573	−1.62310
$431$	1.57522	0.0758759	0.0379379	−	0.999280i	$-0.487921\pi$
$431$	1.57522	0.0758759	0.0379379	+	0.999280i	$0.487921\pi$
$432$	0	0
$433$	34.0227	1.63502	0.817512	−	0.575911i	$-0.195353\pi$
$433$	34.0227	1.63502	0.817512	+	0.575911i	$0.195353\pi$
$434$	−6.38495	−0.306487
$435$	0	0
$436$	47.8550	2.29184
$437$	0	0
$438$	0	0
$439$	17.6400	0.841912	0.420956	−	0.907081i	$-0.361695\pi$
$439$	17.6400	0.841912	0.420956	+	0.907081i	$0.361695\pi$
$440$	−53.4330	−2.54732
$441$	0	0
$442$	−95.6983	−4.55191
$443$	14.3204	0.680382	0.340191	−	0.940356i	$-0.389508\pi$
$443$	14.3204	0.680382	0.340191	+	0.940356i	$0.389508\pi$
$444$	0	0
$445$	2.65725	0.125966
$446$	51.2863	2.42848
$447$	0	0
$448$	−8.50773	−0.401952
$449$	−14.6903	−0.693280	−0.346640	−	0.937998i	$-0.612677\pi$
$449$	−14.6903	−0.693280	−0.346640	+	0.937998i	$0.612677\pi$
$450$	0	0
$451$	5.71759	0.269231
$452$	53.5458	2.51858
$453$	0	0
$454$	62.1085	2.91490
$455$	−6.18403	−0.289912
$456$	0	0
$457$	−7.48051	−0.349923	−0.174962	−	0.984575i	$-0.555980\pi$
$457$	−7.48051	−0.349923	−0.174962	+	0.984575i	$0.555980\pi$
$458$	38.2230	1.78604
$459$	0	0
$460$	0	0
$461$	−21.0428	−0.980061	−0.490031	−	0.871705i	$-0.663014\pi$
$461$	−21.0428	−0.980061	−0.490031	+	0.871705i	$0.663014\pi$
$462$	0	0
$463$	19.0247	0.884151	0.442075	−	0.896978i	$-0.354242\pi$
$463$	19.0247	0.884151	0.442075	+	0.896978i	$0.354242\pi$
$464$	−34.6480	−1.60850
$465$	0	0
$466$	−67.0236	−3.10481
$467$	−12.3086	−0.569574	−0.284787	−	0.958591i	$-0.591923\pi$
$467$	−12.3086	−0.569574	−0.284787	+	0.958591i	$0.591923\pi$
$468$	0	0
$469$	7.03258	0.324734
$470$	−21.0757	−0.972151
$471$	0	0
$472$	−26.2271	−1.20720
$473$	32.6677	1.50206
$474$	0	0
$475$	−12.3741	−0.567762
$476$	17.6366	0.808373
$477$	0	0
$478$	−59.3651	−2.71530
$479$	24.9899	1.14182	0.570908	−	0.821014i	$-0.306591\pi$
$479$	24.9899	1.14182	0.570908	+	0.821014i	$0.306591\pi$
$480$	0	0
$481$	11.4782	0.523362
$482$	−6.04671	−0.275420
$483$	0	0
$484$	30.7798	1.39908
$485$	−7.92658	−0.359927
$486$	0	0
$487$	−17.9553	−0.813632	−0.406816	−	0.913510i	$-0.633361\pi$
$487$	−17.9553	−0.813632	−0.406816	+	0.913510i	$0.633361\pi$
$488$	−58.6588	−2.65536
$489$	0	0
$490$	−28.2480	−1.27611
$491$	25.8920	1.16849	0.584245	−	0.811578i	$-0.301391\pi$
$491$	25.8920	1.16849	0.584245	+	0.811578i	$0.301391\pi$
$492$	0	0
$493$	17.6915	0.796783
$494$	85.4844	3.84613
$495$	0	0
$496$	−44.2639	−1.98751
$497$	8.10586	0.363597
$498$	0	0
$499$	30.3532	1.35880	0.679398	−	0.733770i	$-0.262241\pi$
$499$	30.3532	1.35880	0.679398	+	0.733770i	$0.262241\pi$
$500$	59.9595	2.68147
$501$	0	0
$502$	−18.7195	−0.835494
$503$	28.3608	1.26454	0.632272	−	0.774746i	$-0.282122\pi$
$503$	28.3608	1.26454	0.632272	+	0.774746i	$0.282122\pi$
$504$	0	0
$505$	22.0414	0.980828
$506$	0	0
$507$	0	0
$508$	−45.3893	−2.01382
$509$	−25.0862	−1.11193	−0.555963	−	0.831207i	$-0.687650\pi$
$509$	−25.0862	−1.11193	−0.555963	+	0.831207i	$0.687650\pi$
$510$	0	0
$511$	2.27695	0.100726
$512$	26.2704	1.16100
$513$	0	0
$514$	−35.2330	−1.55406
$515$	7.14240	0.314732
$516$	0	0
$517$	20.4561	0.899658
$518$	−2.95658	−0.129905
$519$	0	0
$520$	−81.2521	−3.56314
$521$	24.6690	1.08077	0.540383	−	0.841419i	$-0.318279\pi$
$521$	24.6690	1.08077	0.540383	+	0.841419i	$0.318279\pi$
$522$	0	0
$523$	17.4775	0.764238	0.382119	−	0.924113i	$-0.375194\pi$
$523$	17.4775	0.764238	0.382119	+	0.924113i	$0.375194\pi$
$524$	−40.3873	−1.76433
$525$	0	0
$526$	−37.4740	−1.63394
$527$	22.6014	0.984531
$528$	0	0
$529$	0	0
$530$	−13.4305	−0.583383
$531$	0	0
$532$	−15.7543	−0.683033
$533$	8.69436	0.376595
$534$	0	0
$535$	−13.5875	−0.587438
$536$	92.4012	3.99112
$537$	0	0
$538$	−35.0720	−1.51206
$539$	27.4174	1.18095
$540$	0	0
$541$	−11.9307	−0.512941	−0.256470	−	0.966552i	$-0.582560\pi$
$541$	−11.9307	−0.512941	−0.256470	+	0.966552i	$0.582560\pi$
$542$	−20.0983	−0.863295
$543$	0	0
$544$	78.7372	3.37583
$545$	−15.2995	−0.655357
$546$	0	0
$547$	22.8621	0.977514	0.488757	−	0.872420i	$-0.337450\pi$
$547$	22.8621	0.977514	0.488757	+	0.872420i	$0.337450\pi$
$548$	−87.4899	−3.73738
$549$	0	0
$550$	−26.4892	−1.12950
$551$	−15.8032	−0.673241
$552$	0	0
$553$	0.222129	0.00944589
$554$	33.2915	1.41442
$555$	0	0
$556$	−8.21680	−0.348470
$557$	22.0421	0.933952	0.466976	−	0.884270i	$-0.345343\pi$
$557$	22.0421	0.933952	0.466976	+	0.884270i	$0.345343\pi$
$558$	0	0
$559$	49.6757	2.10106
$560$	11.0428	0.466643
$561$	0	0
$562$	32.3699	1.36544
$563$	29.5460	1.24521	0.622607	−	0.782535i	$-0.286073\pi$
$563$	29.5460	1.24521	0.622607	+	0.782535i	$0.286073\pi$
$564$	0	0
$565$	−17.1188	−0.720195
$566$	−13.4411	−0.564970
$567$	0	0
$568$	106.503	4.46877
$569$	25.4085	1.06518	0.532589	−	0.846374i	$-0.321219\pi$
$569$	25.4085	1.06518	0.532589	+	0.846374i	$0.321219\pi$
$570$	0	0
$571$	32.5030	1.36021	0.680104	−	0.733116i	$-0.261935\pi$
$571$	32.5030	1.36021	0.680104	+	0.733116i	$0.261935\pi$
$572$	130.930	5.47446
$573$	0	0
$574$	−2.23951	−0.0934753
$575$	0	0
$576$	0	0
$577$	−6.48892	−0.270137	−0.135069	−	0.990836i	$-0.543125\pi$
$577$	−6.48892	−0.270137	−0.135069	+	0.990836i	$0.543125\pi$
$578$	−42.1845	−1.75464
$579$	0	0
$580$	24.9378	1.03549
$581$	−0.680596	−0.0282359
$582$	0	0
$583$	13.0356	0.539880
$584$	29.9169	1.23797
$585$	0	0
$586$	80.9969	3.34595
$587$	−27.8239	−1.14842	−0.574208	−	0.818709i	$-0.694690\pi$
$587$	−27.8239	−1.14842	−0.574208	+	0.818709i	$0.694690\pi$
$588$	0	0
$589$	−20.1891	−0.831879
$590$	13.9208	0.573110
$591$	0	0
$592$	−20.4966	−0.842406
$593$	−41.7172	−1.71312	−0.856559	−	0.516049i	$-0.827402\pi$
$593$	−41.7172	−1.71312	−0.856559	+	0.516049i	$0.827402\pi$
$594$	0	0
$595$	−5.63850	−0.231156
$596$	101.517	4.15829
$597$	0	0
$598$	0	0
$599$	36.3784	1.48638	0.743191	−	0.669079i	$-0.233311\pi$
$599$	36.3784	1.48638	0.743191	+	0.669079i	$0.233311\pi$
$600$	0	0
$601$	−0.695773	−0.0283812	−0.0141906	−	0.999899i	$-0.504517\pi$
$601$	−0.695773	−0.0283812	−0.0141906	+	0.999899i	$0.504517\pi$
$602$	−12.7955	−0.521508
$603$	0	0
$604$	−12.1982	−0.496337
$605$	−9.84045	−0.400071
$606$	0	0
$607$	36.8112	1.49412	0.747061	−	0.664755i	$-0.231464\pi$
$607$	36.8112	1.49412	0.747061	+	0.664755i	$0.231464\pi$
$608$	−70.3336	−2.85240
$609$	0	0
$610$	31.1348	1.26061
$611$	31.1062	1.25842
$612$	0	0
$613$	25.9425	1.04781	0.523903	−	0.851778i	$-0.324475\pi$
$613$	25.9425	1.04781	0.523903	+	0.851778i	$0.324475\pi$
$614$	64.7942	2.61488
$615$	0	0
$616$	−20.3137	−0.818463
$617$	30.4956	1.22771	0.613854	−	0.789419i	$-0.289618\pi$
$617$	30.4956	1.22771	0.613854	+	0.789419i	$0.289618\pi$
$618$	0	0
$619$	−39.8655	−1.60233	−0.801165	−	0.598444i	$-0.795786\pi$
$619$	−39.8655	−1.60233	−0.801165	+	0.598444i	$0.795786\pi$
$620$	31.8588	1.27948
$621$	0	0
$622$	62.4083	2.50235
$623$	1.01021	0.0404732
$624$	0	0
$625$	−7.09590	−0.283836
$626$	77.0079	3.07786
$627$	0	0
$628$	−57.5958	−2.29832
$629$	10.4657	0.417294
$630$	0	0
$631$	−6.16964	−0.245610	−0.122805	−	0.992431i	$-0.539189\pi$
$631$	−6.16964	−0.245610	−0.122805	+	0.992431i	$0.539189\pi$
$632$	2.91856	0.116094
$633$	0	0
$634$	20.5003	0.814171
$635$	14.5112	0.575858
$636$	0	0
$637$	41.6919	1.65189
$638$	−33.8301	−1.33934
$639$	0	0
$640$	15.1958	0.600667
$641$	−36.3673	−1.43642	−0.718211	−	0.695825i	$-0.755039\pi$
$641$	−36.3673	−1.43642	−0.718211	+	0.695825i	$0.755039\pi$
$642$	0	0
$643$	−42.0938	−1.66002	−0.830008	−	0.557751i	$-0.811664\pi$
$643$	−42.0938	−1.66002	−0.830008	+	0.557751i	$0.811664\pi$
$644$	0	0
$645$	0	0
$646$	77.9434	3.06664
$647$	−3.58737	−0.141034	−0.0705170	−	0.997511i	$-0.522465\pi$
$647$	−3.58737	−0.141034	−0.0705170	+	0.997511i	$0.522465\pi$
$648$	0	0
$649$	−13.5115	−0.530373
$650$	−40.2804	−1.57993
$651$	0	0
$652$	−41.4654	−1.62391
$653$	−15.6262	−0.611500	−0.305750	−	0.952112i	$-0.598907\pi$
$653$	−15.6262	−0.611500	−0.305750	+	0.952112i	$0.598907\pi$
$654$	0	0
$655$	12.9120	0.504514
$656$	−15.5255	−0.606169
$657$	0	0
$658$	−8.01240	−0.312356
$659$	37.7808	1.47173	0.735866	−	0.677127i	$-0.236775\pi$
$659$	37.7808	1.47173	0.735866	+	0.677127i	$0.236775\pi$
$660$	0	0
$661$	−17.7475	−0.690299	−0.345150	−	0.938548i	$-0.612172\pi$
$661$	−17.7475	−0.690299	−0.345150	+	0.938548i	$0.612172\pi$
$662$	28.5763	1.11065
$663$	0	0
$664$	−8.94237	−0.347031
$665$	5.03671	0.195315
$666$	0	0
$667$	0	0
$668$	1.22788	0.0475079
$669$	0	0
$670$	−49.0446	−1.89476
$671$	−30.2194	−1.16661
$672$	0	0
$673$	−21.0197	−0.810251	−0.405125	−	0.914261i	$-0.632772\pi$
$673$	−21.0197	−0.810251	−0.405125	+	0.914261i	$0.632772\pi$
$674$	−43.9027	−1.69107
$675$	0	0
$676$	133.716	5.14291
$677$	−15.5670	−0.598290	−0.299145	−	0.954208i	$-0.596701\pi$
$677$	−15.5670	−0.598290	−0.299145	+	0.954208i	$0.596701\pi$
$678$	0	0
$679$	−3.01346	−0.115646
$680$	−74.0844	−2.84101
$681$	0	0
$682$	−43.2189	−1.65494
$683$	−41.6964	−1.59547	−0.797735	−	0.603008i	$-0.793969\pi$
$683$	−41.6964	−1.59547	−0.797735	+	0.603008i	$0.793969\pi$
$684$	0	0
$685$	27.9709	1.06871
$686$	−22.0837	−0.843160
$687$	0	0
$688$	−88.7057	−3.38187
$689$	19.8224	0.755174
$690$	0	0
$691$	9.19006	0.349606	0.174803	−	0.984603i	$-0.444071\pi$
$691$	9.19006	0.349606	0.174803	+	0.984603i	$0.444071\pi$
$692$	−2.70305	−0.102754
$693$	0	0
$694$	34.0469	1.29240
$695$	2.62695	0.0996458
$696$	0	0
$697$	7.92739	0.300271
$698$	40.6351	1.53806
$699$	0	0
$700$	7.42343	0.280579
$701$	−13.7408	−0.518982	−0.259491	−	0.965746i	$-0.583555\pi$
$701$	−13.7408	−0.518982	−0.259491	+	0.965746i	$0.583555\pi$
$702$	0	0
$703$	−9.34867	−0.352592
$704$	−57.5877	−2.17042
$705$	0	0
$706$	−44.6799	−1.68155
$707$	8.37950	0.315144
$708$	0	0
$709$	2.89133	0.108586	0.0542931	−	0.998525i	$-0.482709\pi$
$709$	2.89133	0.108586	0.0542931	+	0.998525i	$0.482709\pi$
$710$	−56.5296	−2.12152
$711$	0	0
$712$	13.2732	0.497433
$713$	0	0
$714$	0	0
$715$	−41.8589	−1.56543
$716$	−2.61856	−0.0978602
$717$	0	0
$718$	−12.5136	−0.467004
$719$	−32.6186	−1.21647	−0.608235	−	0.793757i	$-0.708122\pi$
$719$	−32.6186	−1.21647	−0.608235	+	0.793757i	$0.708122\pi$
$720$	0	0
$721$	2.71534	0.101125
$722$	−19.2501	−0.716415
$723$	0	0
$724$	96.0436	3.56943
$725$	7.44652	0.276557
$726$	0	0
$727$	2.97117	0.110194	0.0550972	−	0.998481i	$-0.482453\pi$
$727$	2.97117	0.110194	0.0550972	+	0.998481i	$0.482453\pi$
$728$	−30.8897	−1.14485
$729$	0	0
$730$	−15.8793	−0.587718
$731$	45.2935	1.67524
$732$	0	0
$733$	−26.4210	−0.975882	−0.487941	−	0.872877i	$-0.662252\pi$
$733$	−26.4210	−0.975882	−0.487941	+	0.872877i	$0.662252\pi$
$734$	−69.6924	−2.57239
$735$	0	0
$736$	0	0
$737$	47.6026	1.75347
$738$	0	0
$739$	6.74530	0.248130	0.124065	−	0.992274i	$-0.460407\pi$
$739$	6.74530	0.248130	0.124065	+	0.992274i	$0.460407\pi$
$740$	14.7524	0.542308
$741$	0	0
$742$	−5.10589	−0.187443
$743$	−3.00267	−0.110157	−0.0550786	−	0.998482i	$-0.517541\pi$
$743$	−3.00267	−0.110157	−0.0550786	+	0.998482i	$0.517541\pi$
$744$	0	0
$745$	−32.4554	−1.18907
$746$	−22.9777	−0.841272
$747$	0	0
$748$	119.380	4.36496
$749$	−5.16558	−0.188746
$750$	0	0
$751$	53.8719	1.96581	0.982907	−	0.184104i	$-0.0589384\pi$
$751$	53.8719	1.96581	0.982907	+	0.184104i	$0.0589384\pi$
$752$	−55.5463	−2.02556
$753$	0	0
$754$	−51.4431	−1.87345
$755$	3.89981	0.141929
$756$	0	0
$757$	2.00018	0.0726976	0.0363488	−	0.999339i	$-0.488427\pi$
$757$	2.00018	0.0726976	0.0363488	+	0.999339i	$0.488427\pi$
$758$	37.4745	1.36113
$759$	0	0
$760$	66.1774	2.40051
$761$	−28.3501	−1.02769	−0.513845	−	0.857883i	$-0.671779\pi$
$761$	−28.3501	−1.02769	−0.513845	+	0.857883i	$0.671779\pi$
$762$	0	0
$763$	−5.81642	−0.210569
$764$	−104.500	−3.78067
$765$	0	0
$766$	2.70270	0.0976524
$767$	−20.5461	−0.741875
$768$	0	0
$769$	1.58169	0.0570370	0.0285185	−	0.999593i	$-0.490921\pi$
$769$	1.58169	0.0570370	0.0285185	+	0.999593i	$0.490921\pi$
$770$	10.7821	0.388560
$771$	0	0
$772$	−55.7593	−2.00682
$773$	13.3204	0.479102	0.239551	−	0.970884i	$-0.423000\pi$
$773$	13.3204	0.479102	0.239551	+	0.970884i	$0.423000\pi$
$774$	0	0
$775$	9.51315	0.341723
$776$	−39.5939	−1.42134
$777$	0	0
$778$	−24.5308	−0.879473
$779$	−7.08130	−0.253714
$780$	0	0
$781$	54.8675	1.96331
$782$	0	0
$783$	0	0
$784$	−74.4491	−2.65890
$785$	18.4136	0.657211
$786$	0	0
$787$	48.5774	1.73160	0.865799	−	0.500393i	$-0.166811\pi$
$787$	48.5774	1.73160	0.865799	+	0.500393i	$0.166811\pi$
$788$	71.2019	2.53646
$789$	0	0
$790$	−1.54911	−0.0551148
$791$	−6.50809	−0.231401
$792$	0	0
$793$	−45.9527	−1.63183
$794$	−37.4477	−1.32897
$795$	0	0
$796$	27.4231	0.971986
$797$	11.0499	0.391408	0.195704	−	0.980663i	$-0.437301\pi$
$797$	11.0499	0.391408	0.195704	+	0.980663i	$0.437301\pi$
$798$	0	0
$799$	28.3622	1.00338
$800$	33.1413	1.17172
$801$	0	0
$802$	68.1605	2.40683
$803$	15.4124	0.543892
$804$	0	0
$805$	0	0
$806$	−65.7202	−2.31489
$807$	0	0
$808$	110.098	3.87325
$809$	48.7671	1.71456	0.857280	−	0.514850i	$-0.172152\pi$
$809$	48.7671	1.71456	0.857280	+	0.514850i	$0.172152\pi$
$810$	0	0
$811$	17.5703	0.616976	0.308488	−	0.951228i	$-0.400177\pi$
$811$	17.5703	0.616976	0.308488	+	0.951228i	$0.400177\pi$
$812$	9.48065	0.332706
$813$	0	0
$814$	−20.0127	−0.701446
$815$	13.2567	0.464362
$816$	0	0
$817$	−40.4593	−1.41549
$818$	65.8633	2.30286
$819$	0	0
$820$	11.1744	0.390228
$821$	11.1999	0.390880	0.195440	−	0.980716i	$-0.437386\pi$
$821$	11.1999	0.390880	0.195440	+	0.980716i	$0.437386\pi$
$822$	0	0
$823$	30.8540	1.07550	0.537752	−	0.843103i	$-0.319274\pi$
$823$	30.8540	1.07550	0.537752	+	0.843103i	$0.319274\pi$
$824$	35.6769	1.24286
$825$	0	0
$826$	5.29229	0.184142
$827$	−6.07065	−0.211097	−0.105549	−	0.994414i	$-0.533660\pi$
$827$	−6.07065	−0.211097	−0.105549	+	0.994414i	$0.533660\pi$
$828$	0	0
$829$	−30.4401	−1.05723	−0.528615	−	0.848862i	$-0.677288\pi$
$829$	−30.4401	−1.05723	−0.528615	+	0.848862i	$0.677288\pi$
$830$	4.74642	0.164751
$831$	0	0
$832$	−87.5699	−3.03594
$833$	38.0141	1.31711
$834$	0	0
$835$	−0.392557	−0.0135850
$836$	−106.639	−3.68817
$837$	0	0
$838$	1.75804	0.0607306
$839$	−15.2573	−0.526741	−0.263371	−	0.964695i	$-0.584834\pi$
$839$	−15.2573	−0.526741	−0.263371	+	0.964695i	$0.584834\pi$
$840$	0	0
$841$	−19.4899	−0.672064
$842$	42.4374	1.46249
$843$	0	0
$844$	−65.7079	−2.26176
$845$	−42.7495	−1.47063
$846$	0	0
$847$	−3.74106	−0.128544
$848$	−35.3968	−1.21553
$849$	0	0
$850$	−36.7271	−1.25973
$851$	0	0
$852$	0	0
$853$	−47.0711	−1.61168	−0.805841	−	0.592131i	$-0.798287\pi$
$853$	−47.0711	−1.61168	−0.805841	+	0.592131i	$0.798287\pi$
$854$	11.8366	0.405040
$855$	0	0
$856$	−67.8706	−2.31977
$857$	30.2445	1.03313	0.516565	−	0.856248i	$-0.327210\pi$
$857$	30.2445	1.03313	0.516565	+	0.856248i	$0.327210\pi$
$858$	0	0
$859$	22.5294	0.768692	0.384346	−	0.923189i	$-0.374427\pi$
$859$	22.5294	0.768692	0.384346	+	0.923189i	$0.374427\pi$
$860$	63.8456	2.17712
$861$	0	0
$862$	−4.17637	−0.142248
$863$	−23.2173	−0.790327	−0.395164	−	0.918611i	$-0.629312\pi$
$863$	−23.2173	−0.790327	−0.395164	+	0.918611i	$0.629312\pi$
$864$	0	0
$865$	0.864176	0.0293829
$866$	−90.2037	−3.06525
$867$	0	0
$868$	12.1118	0.411102
$869$	1.50356	0.0510049
$870$	0	0
$871$	72.3862	2.45271
$872$	−76.4220	−2.58798
$873$	0	0
$874$	0	0
$875$	−7.28763	−0.246367
$876$	0	0
$877$	8.95872	0.302514	0.151257	−	0.988494i	$-0.451668\pi$
$877$	8.95872	0.302514	0.151257	+	0.988494i	$0.451668\pi$
$878$	−46.7687	−1.57837
$879$	0	0
$880$	74.7473	2.51973
$881$	18.8863	0.636295	0.318148	−	0.948041i	$-0.396939\pi$
$881$	18.8863	0.636295	0.318148	+	0.948041i	$0.396939\pi$
$882$	0	0
$883$	−53.0978	−1.78688	−0.893442	−	0.449178i	$-0.851717\pi$
$883$	−53.0978	−1.78688	−0.893442	+	0.449178i	$0.851717\pi$
$884$	181.533	6.10563
$885$	0	0
$886$	−37.9674	−1.27554
$887$	−14.3946	−0.483323	−0.241661	−	0.970361i	$-0.577692\pi$
$887$	−14.3946	−0.483323	−0.241661	+	0.970361i	$0.577692\pi$
$888$	0	0
$889$	5.51673	0.185025
$890$	−7.04512	−0.236153
$891$	0	0
$892$	−97.2867	−3.25740
$893$	−25.3351	−0.847807
$894$	0	0
$895$	0.837166	0.0279834
$896$	5.77702	0.192997
$897$	0	0
$898$	38.9483	1.29972
$899$	12.1495	0.405208
$900$	0	0
$901$	18.0738	0.602125
$902$	−15.1590	−0.504738
$903$	0	0
$904$	−85.5100	−2.84402
$905$	−30.7056	−1.02069
$906$	0	0
$907$	−52.0522	−1.72837	−0.864183	−	0.503178i	$-0.832164\pi$
$907$	−52.0522	−1.72837	−0.864183	+	0.503178i	$0.832164\pi$
$908$	−117.816	−3.90985
$909$	0	0
$910$	16.3956	0.543509
$911$	−11.3022	−0.374459	−0.187229	−	0.982316i	$-0.559951\pi$
$911$	−11.3022	−0.374459	−0.187229	+	0.982316i	$0.559951\pi$
$912$	0	0
$913$	−4.60687	−0.152465
$914$	19.8330	0.656016
$915$	0	0
$916$	−72.5064	−2.39568
$917$	4.90878	0.162102
$918$	0	0
$919$	16.8713	0.556533	0.278266	−	0.960504i	$-0.410240\pi$
$919$	16.8713	0.556533	0.278266	+	0.960504i	$0.410240\pi$
$920$	0	0
$921$	0	0
$922$	55.7904	1.83736
$923$	83.4334	2.74625
$924$	0	0
$925$	4.40511	0.144839
$926$	−50.4398	−1.65755
$927$	0	0
$928$	42.3256	1.38941
$929$	37.1781	1.21977	0.609887	−	0.792488i	$-0.291215\pi$
$929$	37.1781	1.21977	0.609887	+	0.792488i	$0.291215\pi$
$930$	0	0
$931$	−33.9568	−1.11289
$932$	127.139	4.16459
$933$	0	0
$934$	32.6336	1.06780
$935$	−38.1663	−1.24817
$936$	0	0
$937$	−13.4773	−0.440284	−0.220142	−	0.975468i	$-0.570652\pi$
$937$	−13.4773	−0.440284	−0.220142	+	0.975468i	$0.570652\pi$
$938$	−18.6454	−0.608793
$939$	0	0
$940$	39.9793	1.30398
$941$	−19.5740	−0.638094	−0.319047	−	0.947739i	$-0.603363\pi$
$941$	−19.5740	−0.638094	−0.319047	+	0.947739i	$0.603363\pi$
$942$	0	0
$943$	0	0
$944$	36.6890	1.19413
$945$	0	0
$946$	−86.6114	−2.81598
$947$	4.32109	0.140416	0.0702082	−	0.997532i	$-0.477634\pi$
$947$	4.32109	0.140416	0.0702082	+	0.997532i	$0.477634\pi$
$948$	0	0
$949$	23.4366	0.760785
$950$	32.8072	1.06441
$951$	0	0
$952$	−28.1648	−0.912826
$953$	−3.26202	−0.105667	−0.0528336	−	0.998603i	$-0.516825\pi$
$953$	−3.26202	−0.105667	−0.0528336	+	0.998603i	$0.516825\pi$
$954$	0	0
$955$	33.4091	1.08109
$956$	112.612	3.64212
$957$	0	0
$958$	−66.2552	−2.14061
$959$	10.6337	0.343382
$960$	0	0
$961$	−15.4787	−0.499312
$962$	−30.4320	−0.981168
$963$	0	0
$964$	11.4702	0.369430
$965$	17.8265	0.573855
$966$	0	0
$967$	−13.9480	−0.448536	−0.224268	−	0.974527i	$-0.571999\pi$
$967$	−13.9480	−0.448536	−0.224268	+	0.974527i	$0.571999\pi$
$968$	−49.1539	−1.57987
$969$	0	0
$970$	21.0156	0.674770
$971$	−57.6483	−1.85002	−0.925011	−	0.379939i	$-0.875945\pi$
$971$	−57.6483	−1.85002	−0.925011	+	0.379939i	$0.875945\pi$
$972$	0	0
$973$	0.998691	0.0320166
$974$	47.6046	1.52535
$975$	0	0
$976$	82.0576	2.62660
$977$	−13.9380	−0.445916	−0.222958	−	0.974828i	$-0.571571\pi$
$977$	−13.9380	−0.445916	−0.222958	+	0.974828i	$0.571571\pi$
$978$	0	0
$979$	6.83798	0.218543
$980$	53.5845	1.71169
$981$	0	0
$982$	−68.6470	−2.19061
$983$	−29.5018	−0.940963	−0.470481	−	0.882410i	$-0.655920\pi$
$983$	−29.5018	−0.940963	−0.470481	+	0.882410i	$0.655920\pi$
$984$	0	0
$985$	−22.7636	−0.725308
$986$	−46.9051	−1.49376
$987$	0	0
$988$	−162.158	−5.15894
$989$	0	0
$990$	0	0
$991$	13.8941	0.441362	0.220681	−	0.975346i	$-0.429172\pi$
$991$	13.8941	0.441362	0.220681	+	0.975346i	$0.429172\pi$
$992$	54.0722	1.71679
$993$	0	0
$994$	−21.4909	−0.681651
$995$	−8.76729	−0.277942
$996$	0	0
$997$	15.6153	0.494541	0.247271	−	0.968946i	$-0.420466\pi$
$997$	15.6153	0.494541	0.247271	+	0.968946i	$0.420466\pi$
$998$	−80.4750	−2.54739
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.bx.1.1		20
3.2	odd	2	inner	4761.2.a.bx.1.20		20
23.17	odd	22		207.2.i.e.82.4	yes	40
23.19	odd	22		207.2.i.e.154.4	yes	40
23.22	odd	2		4761.2.a.bw.1.1		20
69.17	even	22		207.2.i.e.82.1	✓	40
69.65	even	22		207.2.i.e.154.1	yes	40
69.68	even	2		4761.2.a.bw.1.20		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.2.i.e.82.1	✓	40	69.17	even	22
207.2.i.e.82.4	yes	40	23.17	odd	22
207.2.i.e.154.1	yes	40	69.65	even	22
207.2.i.e.154.4	yes	40	23.19	odd	22
4761.2.a.bw.1.1		20	23.22	odd	2
4761.2.a.bw.1.20		20	69.68	even	2
4761.2.a.bx.1.1		20	1.1	even	1	trivial
4761.2.a.bx.1.20		20	3.2	odd	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:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q-2.65128 q^{2} +5.02931 q^{4} -1.60789 q^{5} -0.611275 q^{7} -8.03155 q^{8} +O(q^{10})\)	\(q-2.65128 q^{2} +5.02931 q^{4} -1.60789 q^{5} -0.611275 q^{7} -8.03155 q^{8} +4.26298 q^{10} -4.13764 q^{11} -6.29184 q^{13} +1.62066 q^{14} +11.2353 q^{16} -5.73681 q^{17} +5.12452 q^{19} -8.08659 q^{20} +10.9701 q^{22} -2.41468 q^{25} +16.6815 q^{26} -3.07429 q^{28} -3.08385 q^{29} -3.93971 q^{31} -13.7249 q^{32} +15.2099 q^{34} +0.982865 q^{35} -1.82430 q^{37} -13.5866 q^{38} +12.9139 q^{40} -1.38185 q^{41} -7.89525 q^{43} -20.8095 q^{44} -4.94390 q^{47} -6.62634 q^{49} +6.40201 q^{50} -31.6436 q^{52} -3.15049 q^{53} +6.65289 q^{55} +4.90949 q^{56} +8.17616 q^{58} +3.26551 q^{59} +7.30354 q^{61} +10.4453 q^{62} +13.9180 q^{64} +10.1166 q^{65} -11.5048 q^{67} -28.8522 q^{68} -2.60585 q^{70} -13.2606 q^{71} -3.72492 q^{73} +4.83675 q^{74} +25.7728 q^{76} +2.52924 q^{77} -0.363386 q^{79} -18.0652 q^{80} +3.66367 q^{82} +1.11340 q^{83} +9.22417 q^{85} +20.9326 q^{86} +33.2317 q^{88} -1.65263 q^{89} +3.84605 q^{91} +13.1077 q^{94} -8.23967 q^{95} +4.92979 q^{97} +17.5683 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{4} + 18 q^{7}+O(q^{10}) \) 20 * q + 20 * q^4 + 18 * q^7	\( 20 q + 20 q^{4} + 18 q^{7} + 22 q^{10} + 16 q^{16} + 40 q^{19} + 14 q^{22} + 20 q^{25} + 32 q^{28} - 22 q^{31} + 60 q^{34} + 18 q^{37} + 74 q^{40} + 32 q^{43} + 2 q^{49} - 52 q^{55} - 24 q^{58} + 70 q^{61} + 36 q^{64} + 64 q^{67} + 48 q^{70} + 40 q^{73} + 82 q^{76} + 106 q^{79} - 36 q^{82} + 2 q^{85} + 58 q^{88} + 88 q^{91} + 40 q^{94} + 24 q^{97}+O(q^{100}) \) 20 * q + 20 * q^4 + 18 * q^7 + 22 * q^10 + 16 * q^16 + 40 * q^19 + 14 * q^22 + 20 * q^25 + 32 * q^28 - 22 * q^31 + 60 * q^34 + 18 * q^37 + 74 * q^40 + 32 * q^43 + 2 * q^49 - 52 * q^55 - 24 * q^58 + 70 * q^61 + 36 * q^64 + 64 * q^67 + 48 * q^70 + 40 * q^73 + 82 * q^76 + 106 * q^79 - 36 * q^82 + 2 * q^85 + 58 * q^88 + 88 * q^91 + 40 * q^94 + 24 * q^97

Introduction

L-functions

Modular forms

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