LMFDB - Embedded newform 9025.2.a.bu.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9025,2,Mod(1,9025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9025.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,2,0,12,0,0,8,0,2,0,0,22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.4227136.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 6x^{4} + 7x^{2} - 1 $ x^6 - 6x^4 + 7x^2 - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.407132$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.45620 q^{2} +1.56104 q^{3} +4.03293 q^{4} +3.83424 q^{6} +4.50527 q^{7} +4.99330 q^{8} -0.563139 q^{9} +2.19869 q^{11} +6.29559 q^{12} -3.75849 q^{13} +11.0659 q^{14} +4.19869 q^{16} +0.665886 q^{17} -1.38318 q^{18} +7.03293 q^{21} +5.40043 q^{22} -0.488026 q^{23} +7.79476 q^{24} -9.23163 q^{26} -5.56222 q^{27} +18.1695 q^{28} +3.59607 q^{29} +6.83424 q^{31} +0.326239 q^{32} +3.43226 q^{33} +1.63555 q^{34} -2.27110 q^{36} -3.01171 q^{37} -5.86718 q^{39} +0.0724126 q^{41} +17.2743 q^{42} +0.420541 q^{43} +8.86718 q^{44} -1.19869 q^{46} +5.02278 q^{47} +6.55434 q^{48} +13.2975 q^{49} +1.03948 q^{51} -15.1578 q^{52} +2.61799 q^{53} -13.6619 q^{54} +22.4962 q^{56} +8.83269 q^{58} +12.5357 q^{59} +7.06587 q^{61} +16.7863 q^{62} -2.53710 q^{63} -7.59607 q^{64} +8.43032 q^{66} +5.72667 q^{67} +2.68548 q^{68} -0.761831 q^{69} +6.97252 q^{71} -2.81192 q^{72} -2.95764 q^{73} -7.39738 q^{74} +9.90571 q^{77} -14.4110 q^{78} +11.3370 q^{79} -6.99346 q^{81} +0.177860 q^{82} -15.6999 q^{83} +28.3634 q^{84} +1.03293 q^{86} +5.61363 q^{87} +10.9787 q^{88} -1.33697 q^{89} -16.9330 q^{91} -1.96818 q^{92} +10.6686 q^{93} +12.3370 q^{94} +0.509273 q^{96} +4.38638 q^{97} +32.6613 q^{98} -1.23817 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{4} + 12 q^{6} + 8 q^{9} + 2 q^{11} + 22 q^{14} + 14 q^{16} + 20 q^{21} + 2 q^{24} - 22 q^{26} - 12 q^{29} + 30 q^{31} + 10 q^{34} - 14 q^{36} - 2 q^{39} + 12 q^{41} + 20 q^{44} + 4 q^{46} + 2 q^{49}+ \cdots - 30 q^{99}+O(q^{100}) $ 6 * q + 2 * q^4 + 12 * q^6 + 8 * q^9 + 2 * q^11 + 22 * q^14 + 14 * q^16 + 20 * q^21 + 2 * q^24 - 22 * q^26 - 12 * q^29 + 30 * q^31 + 10 * q^34 - 14 * q^36 - 2 * q^39 + 12 * q^41 + 20 * q^44 + 4 * q^46 + 2 * q^49 + 40 * q^51 - 4 * q^54 + 46 * q^56 + 20 * q^59 - 2 * q^61 - 12 * q^64 + 6 * q^66 + 18 * q^69 - 2 * q^71 - 22 * q^74 + 24 * q^79 + 14 * q^81 + 48 * q^84 - 16 * q^86 + 36 * q^89 - 24 * q^91 + 30 * q^94 + 26 * q^96 - 30 * q^99

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 $

$2$	2.45620	1.73680	0.868399	−	0.495866i	$-0.165149\pi$
$2$	2.45620	1.73680	0.868399	+	0.495866i	$0.165149\pi$
$3$	1.56104	0.901270	0.450635	−	0.892708i	$-0.351198\pi$
$3$	1.56104	0.901270	0.450635	+	0.892708i	$0.351198\pi$
$4$	4.03293	2.01647
$5$	0	0
$5$	0	0
$6$	3.83424	1.56532
$7$	4.50527	1.70283	0.851417	−	0.524490i	$-0.175744\pi$
$7$	4.50527	1.70283	0.851417	+	0.524490i	$0.175744\pi$
$8$	4.99330	1.76540
$9$	−0.563139	−0.187713
$10$	0	0
$11$	2.19869	0.662930	0.331465	−	0.943467i	$-0.392457\pi$
$11$	2.19869	0.662930	0.331465	+	0.943467i	$0.392457\pi$
$12$	6.29559	1.81738
$13$	−3.75849	−1.04242	−0.521209	−	0.853429i	$-0.674519\pi$
$13$	−3.75849	−1.04242	−0.521209	+	0.853429i	$0.674519\pi$
$14$	11.0659	2.95748
$15$	0	0
$16$	4.19869	1.04967
$17$	0.665886	0.161501	0.0807506	−	0.996734i	$-0.474268\pi$
$17$	0.665886	0.161501	0.0807506	+	0.996734i	$0.474268\pi$
$18$	−1.38318	−0.326020
$19$	0	0
$19$	0	0
$20$	0	0
$21$	7.03293	1.53471
$22$	5.40043	1.15138
$23$	−0.488026	−0.101760	−0.0508802	−	0.998705i	$-0.516203\pi$
$23$	−0.488026	−0.101760	−0.0508802	+	0.998705i	$0.516203\pi$
$24$	7.79476	1.59110
$25$	0	0
$26$	−9.23163	−1.81047
$27$	−5.56222	−1.07045
$28$	18.1695	3.43371
$29$	3.59607	0.667774	0.333887	−	0.942613i	$-0.391640\pi$
$29$	3.59607	0.667774	0.333887	+	0.942613i	$0.391640\pi$
$30$	0	0
$31$	6.83424	1.22747	0.613733	−	0.789514i	$-0.289667\pi$
$31$	6.83424	1.22747	0.613733	+	0.789514i	$0.289667\pi$
$32$	0.326239	0.0576714
$33$	3.43226	0.597479
$34$	1.63555	0.280495
$35$	0	0
$36$	−2.27110	−0.378517
$37$	−3.01171	−0.495123	−0.247561	−	0.968872i	$-0.579629\pi$
$37$	−3.01171	−0.495123	−0.247561	+	0.968872i	$0.579629\pi$
$38$	0	0
$39$	−5.86718	−0.939500
$40$	0	0
$41$	0.0724126	0.0113090	0.00565448	−	0.999984i	$-0.498200\pi$
$41$	0.0724126	0.0113090	0.00565448	+	0.999984i	$0.498200\pi$
$42$	17.2743	2.66548
$43$	0.420541	0.0641319	0.0320660	−	0.999486i	$-0.489791\pi$
$43$	0.420541	0.0641319	0.0320660	+	0.999486i	$0.489791\pi$
$44$	8.86718	1.33678
$45$	0	0
$46$	−1.19869	−0.176737
$47$	5.02278	0.732648	0.366324	−	0.930487i	$-0.380616\pi$
$47$	5.02278	0.732648	0.366324	+	0.930487i	$0.380616\pi$
$48$	6.55434	0.946038
$49$	13.2975	1.89964
$50$	0	0
$51$	1.03948	0.145556
$52$	−15.1578	−2.10200
$53$	2.61799	0.359609	0.179804	−	0.983702i	$-0.442454\pi$
$53$	2.61799	0.359609	0.179804	+	0.983702i	$0.442454\pi$
$54$	−13.6619	−1.85915
$55$	0	0
$56$	22.4962	3.00618
$57$	0	0
$58$	8.83269	1.15979
$59$	12.5357	1.63200	0.816002	−	0.578049i	$-0.196186\pi$
$59$	12.5357	1.63200	0.816002	+	0.578049i	$0.196186\pi$
$60$	0	0
$61$	7.06587	0.904692	0.452346	−	0.891843i	$-0.350587\pi$
$61$	7.06587	0.904692	0.452346	+	0.891843i	$0.350587\pi$
$62$	16.7863	2.13186
$63$	−2.53710	−0.319644
$64$	−7.59607	−0.949509
$65$	0	0
$66$	8.43032	1.03770
$67$	5.72667	0.699624	0.349812	−	0.936820i	$-0.386245\pi$
$67$	5.72667	0.699624	0.349812	+	0.936820i	$0.386245\pi$
$68$	2.68548	0.325662
$69$	−0.761831	−0.0917136
$70$	0	0
$71$	6.97252	0.827486	0.413743	−	0.910394i	$-0.364221\pi$
$71$	6.97252	0.827486	0.413743	+	0.910394i	$0.364221\pi$
$72$	−2.81192	−0.331388
$73$	−2.95764	−0.346165	−0.173083	−	0.984907i	$-0.555373\pi$
$73$	−2.95764	−0.346165	−0.173083	+	0.984907i	$0.555373\pi$
$74$	−7.39738	−0.859928
$75$	0	0
$76$	0	0
$77$	9.90571	1.12886
$78$	−14.4110	−1.63172
$79$	11.3370	1.27551	0.637755	−	0.770240i	$-0.279863\pi$
$79$	11.3370	1.27551	0.637755	+	0.770240i	$0.279863\pi$
$80$	0	0
$81$	−6.99346	−0.777051
$82$	0.177860	0.0196414
$83$	−15.6999	−1.72328	−0.861642	−	0.507517i	$-0.830564\pi$
$83$	−15.6999	−1.72328	−0.861642	+	0.507517i	$0.830564\pi$
$84$	28.3634	3.09470
$85$	0	0
$86$	1.03293	0.111384
$87$	5.61363	0.601845
$88$	10.9787	1.17034
$89$	−1.33697	−0.141719	−0.0708594	−	0.997486i	$-0.522574\pi$
$89$	−1.33697	−0.141719	−0.0708594	+	0.997486i	$0.522574\pi$
$90$	0	0
$91$	−16.9330	−1.77507
$92$	−1.96818	−0.205197
$93$	10.6686	1.10628
$94$	12.3370	1.27246
$95$	0	0
$96$	0.509273	0.0519775
$97$	4.38638	0.445369	0.222685	−	0.974891i	$-0.428518\pi$
$97$	4.38638	0.445369	0.222685	+	0.974891i	$0.428518\pi$
$98$	32.6613	3.29929
$99$	−1.23817	−0.124441
$100$	0	0
$101$	−10.5686	−1.05161	−0.525807	−	0.850604i	$-0.676237\pi$
$101$	−10.5686	−1.05161	−0.525807	+	0.850604i	$0.676237\pi$
$102$	2.55317	0.252801
$103$	5.75615	0.567171	0.283585	−	0.958947i	$-0.408476\pi$
$103$	5.75615	0.567171	0.283585	+	0.958947i	$0.408476\pi$
$104$	−18.7673	−1.84028
$105$	0	0
$106$	6.43032	0.624568
$107$	−1.30229	−0.125897	−0.0629486	−	0.998017i	$-0.520050\pi$
$107$	−1.30229	−0.125897	−0.0629486	+	0.998017i	$0.520050\pi$
$108$	−22.4321	−2.15853
$109$	−12.0329	−1.15255	−0.576273	−	0.817257i	$-0.695494\pi$
$109$	−12.0329	−1.15255	−0.576273	+	0.817257i	$0.695494\pi$
$110$	0	0
$111$	−4.70142	−0.446239
$112$	18.9163	1.78742
$113$	7.74626	0.728707	0.364353	−	0.931261i	$-0.381290\pi$
$113$	7.74626	0.728707	0.364353	+	0.931261i	$0.381290\pi$
$114$	0	0
$115$	0	0
$116$	14.5027	1.34654
$117$	2.11656	0.195676
$118$	30.7901	2.83446
$119$	3.00000	0.275010
$120$	0	0
$121$	−6.16576	−0.560523
$122$	17.3552	1.57127
$123$	0.113039	0.0101924
$124$	27.5621	2.47515
$125$	0	0
$126$	−6.23163	−0.555157
$127$	−3.93635	−0.349295	−0.174647	−	0.984631i	$-0.555879\pi$
$127$	−3.93635	−0.349295	−0.174647	+	0.984631i	$0.555879\pi$
$128$	−19.3100	−1.70678
$129$	0.656483	0.0578001
$130$	0	0
$131$	−16.3250	−1.42632	−0.713160	−	0.701002i	$-0.752737\pi$
$131$	−16.3250	−1.42632	−0.713160	+	0.701002i	$0.752737\pi$
$132$	13.8421	1.20480
$133$	0	0
$134$	14.0659	1.21511
$135$	0	0
$136$	3.32497	0.285114
$137$	−16.4306	−1.40376	−0.701879	−	0.712296i	$-0.747655\pi$
$137$	−16.4306	−1.40376	−0.701879	+	0.712296i	$0.747655\pi$
$138$	−1.87121	−0.159288
$139$	−2.66849	−0.226338	−0.113169	−	0.993576i	$-0.536100\pi$
$139$	−2.66849	−0.226338	−0.113169	+	0.993576i	$0.536100\pi$
$140$	0	0
$141$	7.84079	0.660313
$142$	17.1259	1.43718
$143$	−8.26377	−0.691051
$144$	−2.36445	−0.197037
$145$	0	0
$146$	−7.26456	−0.601219
$147$	20.7580	1.71209
$148$	−12.1460	−0.998399
$149$	−17.9660	−1.47183	−0.735915	−	0.677074i	$-0.763248\pi$
$149$	−17.9660	−1.47183	−0.735915	+	0.677074i	$0.763248\pi$
$150$	0	0
$151$	−12.7344	−1.03631	−0.518154	−	0.855288i	$-0.673380\pi$
$151$	−12.7344	−1.03631	−0.518154	+	0.855288i	$0.673380\pi$
$152$	0	0
$153$	−0.374987	−0.0303159
$154$	24.3304	1.96060
$155$	0	0
$156$	−23.6619	−1.89447
$157$	−20.0541	−1.60049	−0.800245	−	0.599673i	$-0.795298\pi$
$157$	−20.0541	−1.60049	−0.800245	+	0.599673i	$0.795298\pi$
$158$	27.8459	2.21530
$159$	4.08680	0.324104
$160$	0	0
$161$	−2.19869	−0.173281
$162$	−17.1773	−1.34958
$163$	−14.2331	−1.11482	−0.557412	−	0.830236i	$-0.688206\pi$
$163$	−14.2331	−1.11482	−0.557412	+	0.830236i	$0.688206\pi$
$164$	0.292035	0.0228041
$165$	0	0
$166$	−38.5621	−2.99300
$167$	5.61630	0.434602	0.217301	−	0.976105i	$-0.430275\pi$
$167$	5.61630	0.434602	0.217301	+	0.976105i	$0.430275\pi$
$168$	35.1176	2.70938
$169$	1.12628	0.0866368
$170$	0	0
$171$	0	0
$172$	1.69601	0.129320
$173$	−10.6905	−0.812783	−0.406391	−	0.913699i	$-0.633213\pi$
$173$	−10.6905	−0.812783	−0.406391	+	0.913699i	$0.633213\pi$
$174$	13.7882	1.04528
$175$	0	0
$176$	9.23163	0.695860
$177$	19.5687	1.47088
$178$	−3.28388	−0.246137
$179$	7.68942	0.574734	0.287367	−	0.957821i	$-0.407220\pi$
$179$	7.68942	0.574734	0.287367	+	0.957821i	$0.407220\pi$
$180$	0	0
$181$	−6.12628	−0.455363	−0.227681	−	0.973736i	$-0.573114\pi$
$181$	−6.12628	−0.455363	−0.227681	+	0.973736i	$0.573114\pi$
$182$	−41.5910	−3.08293
$183$	11.0301	0.815371
$184$	−2.43686	−0.179648
$185$	0	0
$186$	26.2042	1.92138
$187$	1.46408	0.107064
$188$	20.2565	1.47736
$189$	−25.0593	−1.82280
$190$	0	0
$191$	5.85517	0.423666	0.211833	−	0.977306i	$-0.432057\pi$
$191$	5.85517	0.423666	0.211833	+	0.977306i	$0.432057\pi$
$192$	−11.8578	−0.855764
$193$	2.59117	0.186517	0.0932584	−	0.995642i	$-0.470272\pi$
$193$	2.59117	0.186517	0.0932584	+	0.995642i	$0.470272\pi$
$194$	10.7738	0.773516
$195$	0	0
$196$	53.6279	3.83057
$197$	−19.8628	−1.41517	−0.707584	−	0.706629i	$-0.750215\pi$
$197$	−19.8628	−1.41517	−0.707584	+	0.706629i	$0.750215\pi$
$198$	−3.04120	−0.216128
$199$	−12.7618	−0.904662	−0.452331	−	0.891850i	$-0.649407\pi$
$199$	−12.7618	−0.904662	−0.452331	+	0.891850i	$0.649407\pi$
$200$	0	0
$201$	8.93959	0.630550
$202$	−25.9586	−1.82644
$203$	16.2013	1.13711
$204$	4.19215	0.293509
$205$	0	0
$206$	14.1383	0.985061
$207$	0.274827	0.0191018
$208$	−15.7808	−1.09420
$209$	0	0
$210$	0	0
$211$	13.8552	0.953830	0.476915	−	0.878950i	$-0.341755\pi$
$211$	13.8552	0.953830	0.476915	+	0.878950i	$0.341755\pi$
$212$	10.5582	0.725139
$213$	10.8844	0.745788
$214$	−3.19869	−0.218658
$215$	0	0
$216$	−27.7738	−1.88977
$217$	30.7901	2.09017
$218$	−29.5553	−2.00174
$219$	−4.61701	−0.311988
$220$	0	0
$221$	−2.50273	−0.168352
$222$	−11.5476	−0.775027
$223$	−21.6960	−1.45287	−0.726437	−	0.687233i	$-0.758825\pi$
$223$	−21.6960	−1.45287	−0.726437	+	0.687233i	$0.758825\pi$
$224$	1.46980	0.0982048
$225$	0	0
$226$	19.0264	1.26562
$227$	−8.19628	−0.544006	−0.272003	−	0.962296i	$-0.587686\pi$
$227$	−8.19628	−0.544006	−0.272003	+	0.962296i	$0.587686\pi$
$228$	0	0
$229$	−16.6619	−1.10105	−0.550526	−	0.834818i	$-0.685573\pi$
$229$	−16.6619	−1.10105	−0.550526	+	0.834818i	$0.685573\pi$
$230$	0	0
$231$	15.4633	1.01741
$232$	17.9563	1.17889
$233$	−12.2135	−0.800135	−0.400068	−	0.916486i	$-0.631013\pi$
$233$	−12.2135	−0.800135	−0.400068	+	0.916486i	$0.631013\pi$
$234$	5.19869	0.339849
$235$	0	0
$236$	50.5555	3.29088
$237$	17.6975	1.14958
$238$	7.36861	0.477636
$239$	2.03948	0.131923	0.0659614	−	0.997822i	$-0.478989\pi$
$239$	2.03948	0.131923	0.0659614	+	0.997822i	$0.478989\pi$
$240$	0	0
$241$	17.5237	1.12880	0.564399	−	0.825502i	$-0.309108\pi$
$241$	17.5237	1.12880	0.564399	+	0.825502i	$0.309108\pi$
$242$	−15.1444	−0.973516
$243$	5.76956	0.370118
$244$	28.4962	1.82428
$245$	0	0
$246$	0.277648	0.0177022
$247$	0	0
$248$	34.1254	2.16697
$249$	−24.5082	−1.55314
$250$	0	0
$251$	−3.33806	−0.210696	−0.105348	−	0.994435i	$-0.533596\pi$
$251$	−3.33806	−0.210696	−0.105348	+	0.994435i	$0.533596\pi$
$252$	−10.2319	−0.644552
$253$	−1.07302	−0.0674601
$254$	−9.66849	−0.606655
$255$	0	0
$256$	−32.2371	−2.01482
$257$	27.5845	1.72067	0.860337	−	0.509726i	$-0.170253\pi$
$257$	27.5845	1.72067	0.860337	+	0.509726i	$0.170253\pi$
$258$	1.61246	0.100387
$259$	−13.5686	−0.843112
$260$	0	0
$261$	−2.02509	−0.125350
$262$	−40.0974	−2.47723
$263$	13.6262	0.840227	0.420114	−	0.907471i	$-0.361990\pi$
$263$	13.6262	0.840227	0.420114	+	0.907471i	$0.361990\pi$
$264$	17.1383	1.05479
$265$	0	0
$266$	0	0
$267$	−2.08707	−0.127727
$268$	23.0953	1.41077
$269$	−3.60808	−0.219988	−0.109994	−	0.993932i	$-0.535083\pi$
$269$	−3.60808	−0.219988	−0.109994	+	0.993932i	$0.535083\pi$
$270$	0	0
$271$	10.5631	0.641665	0.320833	−	0.947136i	$-0.396037\pi$
$271$	10.5631	0.641665	0.320833	+	0.947136i	$0.396037\pi$
$272$	2.79585	0.169523
$273$	−26.4332	−1.59981
$274$	−40.3568	−2.43804
$275$	0	0
$276$	−3.07241	−0.184938
$277$	6.73487	0.404659	0.202330	−	0.979317i	$-0.435149\pi$
$277$	6.73487	0.404659	0.202330	+	0.979317i	$0.435149\pi$
$278$	−6.55434	−0.393103
$279$	−3.84863	−0.230412
$280$	0	0
$281$	23.4303	1.39774	0.698868	−	0.715251i	$-0.253688\pi$
$281$	23.4303	1.39774	0.698868	+	0.715251i	$0.253688\pi$
$282$	19.2586	1.14683
$283$	14.6831	0.872822	0.436411	−	0.899747i	$-0.356249\pi$
$283$	14.6831	0.872822	0.436411	+	0.899747i	$0.356249\pi$
$284$	28.1197	1.66860
$285$	0	0
$286$	−20.2975	−1.20022
$287$	0.326239	0.0192573
$288$	−0.183718	−0.0108257
$289$	−16.5566	−0.973917
$290$	0	0
$291$	6.84733	0.401398
$292$	−11.9280	−0.698031
$293$	18.1855	1.06241	0.531206	−	0.847243i	$-0.321739\pi$
$293$	18.1855	1.06241	0.531206	+	0.847243i	$0.321739\pi$
$294$	50.9858	2.97355
$295$	0	0
$296$	−15.0384	−0.874089
$297$	−12.2296	−0.709634
$298$	−44.1281	−2.55627
$299$	1.83424	0.106077
$300$	0	0
$301$	1.89465	0.109206
$302$	−31.2782	−1.79986
$303$	−16.4981	−0.947788
$304$	0	0
$305$	0	0
$306$	−0.921044	−0.0526526
$307$	−29.3801	−1.67681	−0.838406	−	0.545045i	$-0.816512\pi$
$307$	−29.3801	−1.67681	−0.838406	+	0.545045i	$0.816512\pi$
$308$	39.9491	2.27631
$309$	8.98561	0.511174
$310$	0	0
$311$	−0.193232	−0.0109572	−0.00547859	−	0.999985i	$-0.501744\pi$
$311$	−0.193232	−0.0109572	−0.00547859	+	0.999985i	$0.501744\pi$
$312$	−29.2966	−1.65859
$313$	21.2755	1.20256	0.601281	−	0.799038i	$-0.294657\pi$
$313$	21.2755	1.20256	0.601281	+	0.799038i	$0.294657\pi$
$314$	−49.2569	−2.77973
$315$	0	0
$316$	45.7213	2.57202
$317$	−19.1701	−1.07670	−0.538351	−	0.842721i	$-0.680952\pi$
$317$	−19.1701	−1.07670	−0.538351	+	0.842721i	$0.680952\pi$
$318$	10.0380	0.562904
$319$	7.90666	0.442688
$320$	0	0
$321$	−2.03293	−0.113467
$322$	−5.40043	−0.300954
$323$	0	0
$324$	−28.2042	−1.56690
$325$	0	0
$326$	−34.9594	−1.93622
$327$	−18.7839	−1.03875
$328$	0.361578	0.0199648
$329$	22.6290	1.24758
$330$	0	0
$331$	−20.6070	−1.13266	−0.566331	−	0.824178i	$-0.691638\pi$
$331$	−20.6070	−1.13266	−0.566331	+	0.824178i	$0.691638\pi$
$332$	−63.3165	−3.47495
$333$	1.69601	0.0929410
$334$	13.7948	0.754816
$335$	0	0
$336$	29.5291	1.61095
$337$	10.2074	0.556030	0.278015	−	0.960577i	$-0.410324\pi$
$337$	10.2074	0.556030	0.278015	+	0.960577i	$0.410324\pi$
$338$	2.76637	0.150471
$339$	12.0923	0.656761
$340$	0	0
$341$	15.0264	0.813725
$342$	0	0
$343$	28.3719	1.53194
$344$	2.09989	0.113218
$345$	0	0
$346$	−26.2580	−1.41164
$347$	−5.44110	−0.292094	−0.146047	−	0.989278i	$-0.546655\pi$
$347$	−5.44110	−0.292094	−0.146047	+	0.989278i	$0.546655\pi$
$348$	22.6394	1.21360
$349$	−1.55114	−0.0830304	−0.0415152	−	0.999138i	$-0.513219\pi$
$349$	−1.55114	−0.0830304	−0.0415152	+	0.999138i	$0.513219\pi$
$350$	0	0
$351$	20.9056	1.11586
$352$	0.717298	0.0382321
$353$	32.9335	1.75287	0.876437	−	0.481517i	$-0.159914\pi$
$353$	32.9335	1.75287	0.876437	+	0.481517i	$0.159914\pi$
$354$	48.0648	2.55461
$355$	0	0
$356$	−5.39192	−0.285771
$357$	4.68313	0.247858
$358$	18.8868	0.998197
$359$	−3.49727	−0.184579	−0.0922894	−	0.995732i	$-0.529419\pi$
$359$	−3.49727	−0.184579	−0.0922894	+	0.995732i	$0.529419\pi$
$360$	0	0
$361$	0	0
$362$	−15.0474	−0.790873
$363$	−9.62502	−0.505183
$364$	−68.2899	−3.57936
$365$	0	0
$366$	27.0923	1.41614
$367$	−19.5039	−1.01810	−0.509048	−	0.860738i	$-0.670002\pi$
$367$	−19.5039	−1.01810	−0.509048	+	0.860738i	$0.670002\pi$
$368$	−2.04907	−0.106815
$369$	−0.0407784	−0.00212284
$370$	0	0
$371$	11.7948	0.612354
$372$	43.0256	2.23077
$373$	−23.5158	−1.21760	−0.608802	−	0.793322i	$-0.708350\pi$
$373$	−23.5158	−1.21760	−0.608802	+	0.793322i	$0.708350\pi$
$374$	3.59607	0.185949
$375$	0	0
$376$	25.0803	1.29342
$377$	−13.5158	−0.696100
$378$	−61.5508	−3.16583
$379$	−7.05148	−0.362210	−0.181105	−	0.983464i	$-0.557967\pi$
$379$	−7.05148	−0.362210	−0.181105	+	0.983464i	$0.557967\pi$
$380$	0	0
$381$	−6.14483	−0.314809
$382$	14.3815	0.735822
$383$	−3.08409	−0.157589	−0.0787947	−	0.996891i	$-0.525107\pi$
$383$	−3.08409	−0.157589	−0.0787947	+	0.996891i	$0.525107\pi$
$384$	−30.1437	−1.53827
$385$	0	0
$386$	6.36445	0.323942
$387$	−0.236823	−0.0120384
$388$	17.6900	0.898072
$389$	9.39084	0.476134	0.238067	−	0.971249i	$-0.423486\pi$
$389$	9.39084	0.476134	0.238067	+	0.971249i	$0.423486\pi$
$390$	0	0
$391$	−0.324970	−0.0164344
$392$	66.3984	3.35362
$393$	−25.4840	−1.28550
$394$	−48.7871	−2.45786
$395$	0	0
$396$	−4.99346	−0.250931
$397$	26.7595	1.34302	0.671510	−	0.740996i	$-0.265646\pi$
$397$	26.7595	1.34302	0.671510	+	0.740996i	$0.265646\pi$
$398$	−31.3456	−1.57122
$399$	0	0
$400$	0	0
$401$	−25.1701	−1.25694	−0.628468	−	0.777835i	$-0.716318\pi$
$401$	−25.1701	−1.25694	−0.628468	+	0.777835i	$0.716318\pi$
$402$	21.9575	1.09514
$403$	−25.6865	−1.27953
$404$	−42.6225	−2.12055
$405$	0	0
$406$	39.7937	1.97493
$407$	−6.62183	−0.328232
$408$	5.19043	0.256964
$409$	−28.3215	−1.40041	−0.700204	−	0.713943i	$-0.746908\pi$
$409$	−28.3215	−1.40041	−0.700204	+	0.713943i	$0.746908\pi$
$410$	0	0
$411$	−25.6489	−1.26516
$412$	23.2142	1.14368
$413$	56.4766	2.77903
$414$	0.675030	0.0331759
$415$	0	0
$416$	−1.22617	−0.0601178
$417$	−4.16563	−0.203992
$418$	0	0
$419$	13.0449	0.637287	0.318643	−	0.947875i	$-0.396773\pi$
$419$	13.0449	0.637287	0.318643	+	0.947875i	$0.396773\pi$
$420$	0	0
$421$	3.32497	0.162049	0.0810246	−	0.996712i	$-0.474181\pi$
$421$	3.32497	0.162049	0.0810246	+	0.996712i	$0.474181\pi$
$422$	34.0311	1.65661
$423$	−2.82853	−0.137528
$424$	13.0724	0.634852
$425$	0	0
$426$	26.7344	1.29528
$427$	31.8337	1.54054
$428$	−5.25205	−0.253868
$429$	−12.9001	−0.622823
$430$	0	0
$431$	0.0484069	0.00233168	0.00116584	−	0.999999i	$-0.499629\pi$
$431$	0.0484069	0.00233168	0.00116584	+	0.999999i	$0.499629\pi$
$432$	−23.3540	−1.12362
$433$	9.56872	0.459844	0.229922	−	0.973209i	$-0.426153\pi$
$433$	9.56872	0.459844	0.229922	+	0.973209i	$0.426153\pi$
$434$	75.6268	3.63020
$435$	0	0
$436$	−48.5280	−2.32407
$437$	0	0
$438$	−11.3403	−0.541861
$439$	−22.2515	−1.06200	−0.531002	−	0.847370i	$-0.678184\pi$
$439$	−22.2515	−1.06200	−0.531002	+	0.847370i	$0.678184\pi$
$440$	0	0
$441$	−7.48834	−0.356588
$442$	−6.14721	−0.292393
$443$	−20.1157	−0.955727	−0.477863	−	0.878434i	$-0.658589\pi$
$443$	−20.1157	−0.955727	−0.477863	+	0.878434i	$0.658589\pi$
$444$	−18.9605	−0.899827
$445$	0	0
$446$	−53.2899	−2.52335
$447$	−28.0457	−1.32652
$448$	−34.2224	−1.61686
$449$	−12.4973	−0.589783	−0.294891	−	0.955531i	$-0.595283\pi$
$449$	−12.4973	−0.589783	−0.294891	+	0.955531i	$0.595283\pi$
$450$	0	0
$451$	0.159213	0.00749705
$452$	31.2402	1.46941
$453$	−19.8789	−0.933992
$454$	−20.1317	−0.944829
$455$	0	0
$456$	0	0
$457$	28.3179	1.32465	0.662327	−	0.749215i	$-0.269569\pi$
$457$	28.3179	1.32465	0.662327	+	0.749215i	$0.269569\pi$
$458$	−40.9251	−1.91231
$459$	−3.70381	−0.172879
$460$	0	0
$461$	5.31951	0.247754	0.123877	−	0.992298i	$-0.460467\pi$
$461$	5.31951	0.247754	0.123877	+	0.992298i	$0.460467\pi$
$462$	37.9809	1.76703
$463$	17.9327	0.833401	0.416701	−	0.909044i	$-0.363186\pi$
$463$	17.9327	0.833401	0.416701	+	0.909044i	$0.363186\pi$
$464$	15.0988	0.700944
$465$	0	0
$466$	−29.9989	−1.38967
$467$	−28.7791	−1.33174	−0.665868	−	0.746069i	$-0.731939\pi$
$467$	−28.7791	−1.33174	−0.665868	+	0.746069i	$0.731939\pi$
$468$	8.53593	0.394574
$469$	25.8002	1.19134
$470$	0	0
$471$	−31.3053	−1.44247
$472$	62.5943	2.88114
$473$	0.924640	0.0425150
$474$	43.4687	1.99658
$475$	0	0
$476$	12.0988	0.554548
$477$	−1.47429	−0.0675033
$478$	5.00937	0.229123
$479$	−8.05148	−0.367882	−0.183941	−	0.982937i	$-0.558885\pi$
$479$	−8.05148	−0.367882	−0.183941	+	0.982937i	$0.558885\pi$
$480$	0	0
$481$	11.3195	0.516125
$482$	43.0417	1.96049
$483$	−3.43226	−0.156173
$484$	−24.8661	−1.13028
$485$	0	0
$486$	14.1712	0.642819
$487$	−1.09761	−0.0497376	−0.0248688	−	0.999691i	$-0.507917\pi$
$487$	−1.09761	−0.0497376	−0.0248688	+	0.999691i	$0.507917\pi$
$488$	35.2820	1.59714
$489$	−22.2185	−1.00476
$490$	0	0
$491$	13.1053	0.591436	0.295718	−	0.955275i	$-0.404441\pi$
$491$	13.1053	0.591436	0.295718	+	0.955275i	$0.404441\pi$
$492$	0.455880	0.0205527
$493$	2.39458	0.107846
$494$	0	0
$495$	0	0
$496$	28.6949	1.28844
$497$	31.4131	1.40907
$498$	−60.1971	−2.69750
$499$	24.1407	1.08068	0.540342	−	0.841445i	$-0.318295\pi$
$499$	24.1407	1.08068	0.540342	+	0.841445i	$0.318295\pi$
$500$	0	0
$501$	8.76729	0.391694
$502$	−8.19895	−0.365937
$503$	18.9618	0.845465	0.422733	−	0.906254i	$-0.361071\pi$
$503$	18.9618	0.845465	0.422733	+	0.906254i	$0.361071\pi$
$504$	−12.6685	−0.564299
$505$	0	0
$506$	−2.63555	−0.117165
$507$	1.75817	0.0780831
$508$	−15.8751	−0.704342
$509$	21.9605	0.973383	0.486692	−	0.873574i	$-0.338204\pi$
$509$	21.9605	0.973383	0.486692	+	0.873574i	$0.338204\pi$
$510$	0	0
$511$	−13.3250	−0.589462
$512$	−40.5609	−1.79255
$513$	0	0
$514$	67.7531	2.98846
$515$	0	0
$516$	2.64755	0.116552
$517$	11.0435	0.485695
$518$	−33.3272	−1.46431
$519$	−16.6883	−0.732537
$520$	0	0
$521$	6.56968	0.287823	0.143912	−	0.989591i	$-0.454032\pi$
$521$	6.56968	0.287823	0.143912	+	0.989591i	$0.454032\pi$
$522$	−4.97403	−0.217708
$523$	−4.19777	−0.183556	−0.0917779	−	0.995780i	$-0.529255\pi$
$523$	−4.19777	−0.183556	−0.0917779	+	0.995780i	$0.529255\pi$
$524$	−65.8375	−2.87613
$525$	0	0
$526$	33.4687	1.45931
$527$	4.55083	0.198237
$528$	14.4110	0.627157
$529$	−22.7618	−0.989645
$530$	0	0
$531$	−7.05933	−0.306349
$532$	0	0
$533$	−0.272162	−0.0117887
$534$	−5.12628	−0.221836
$535$	0	0
$536$	28.5950	1.23512
$537$	12.0035	0.517990
$538$	−8.86217	−0.382075
$539$	29.2371	1.25933
$540$	0	0
$541$	−4.63009	−0.199063	−0.0995316	−	0.995034i	$-0.531734\pi$
$541$	−4.63009	−0.199063	−0.0995316	+	0.995034i	$0.531734\pi$
$542$	25.9452	1.11444
$543$	−9.56340	−0.410405
$544$	0.217238	0.00931400
$545$	0	0
$546$	−64.9254	−2.77855
$547$	38.0399	1.62647	0.813233	−	0.581938i	$-0.197705\pi$
$547$	38.0399	1.62647	0.813233	+	0.581938i	$0.197705\pi$
$548$	−66.2634	−2.83063
$549$	−3.97907	−0.169823
$550$	0	0
$551$	0	0
$552$	−3.80405	−0.161911
$553$	51.0762	2.17198
$554$	16.5422	0.702811
$555$	0	0
$556$	−10.7618	−0.456403
$557$	−36.4306	−1.54361	−0.771807	−	0.635857i	$-0.780647\pi$
$557$	−36.4306	−1.54361	−0.771807	+	0.635857i	$0.780647\pi$
$558$	−9.45302	−0.400178
$559$	−1.58060	−0.0668523
$560$	0	0
$561$	2.28549	0.0964935
$562$	57.5496	2.42758
$563$	−20.6856	−0.871795	−0.435897	−	0.899996i	$-0.643569\pi$
$563$	−20.6856	−0.871795	−0.435897	+	0.899996i	$0.643569\pi$
$564$	31.6214	1.33150
$565$	0	0
$566$	36.0648	1.51592
$567$	−31.5074	−1.32319
$568$	34.8159	1.46084
$569$	−27.1132	−1.13664	−0.568322	−	0.822806i	$-0.692407\pi$
$569$	−27.1132	−1.13664	−0.568322	+	0.822806i	$0.692407\pi$
$570$	0	0
$571$	46.4687	1.94466	0.972328	−	0.233622i	$-0.0750578\pi$
$571$	46.4687	1.94466	0.972328	+	0.233622i	$0.0750578\pi$
$572$	−33.3272	−1.39348
$573$	9.14019	0.381837
$574$	0.801309	0.0334460
$575$	0	0
$576$	4.27765	0.178235
$577$	−18.0398	−0.751008	−0.375504	−	0.926821i	$-0.622530\pi$
$577$	−18.0398	−0.751008	−0.375504	+	0.926821i	$0.622530\pi$
$578$	−40.6664	−1.69150
$579$	4.04494	0.168102
$580$	0	0
$581$	−70.7322	−2.93447
$582$	16.8184	0.697147
$583$	5.75615	0.238395
$584$	−14.7684	−0.611120
$585$	0	0
$586$	44.6674	1.84519
$587$	−26.0337	−1.07452	−0.537262	−	0.843415i	$-0.680541\pi$
$587$	−26.0337	−1.07452	−0.537262	+	0.843415i	$0.680541\pi$
$588$	83.7156	3.45237
$589$	0	0
$590$	0	0
$591$	−31.0068	−1.27545
$592$	−12.6453	−0.519717
$593$	18.1561	0.745580	0.372790	−	0.927916i	$-0.378401\pi$
$593$	18.1561	0.745580	0.372790	+	0.927916i	$0.378401\pi$
$594$	−30.0384	−1.23249
$595$	0	0
$596$	−72.4556	−2.96790
$597$	−19.9218	−0.815345
$598$	4.50527	0.184234
$599$	40.0713	1.63727	0.818635	−	0.574314i	$-0.194731\pi$
$599$	40.0713	1.63727	0.818635	+	0.574314i	$0.194731\pi$
$600$	0	0
$601$	15.0473	0.613793	0.306897	−	0.951743i	$-0.400709\pi$
$601$	15.0473	0.613793	0.306897	+	0.951743i	$0.400709\pi$
$602$	4.65365	0.189669
$603$	−3.22491	−0.131329
$604$	−51.3568	−2.08968
$605$	0	0
$606$	−40.5226	−1.64612
$607$	29.3860	1.19274	0.596370	−	0.802709i	$-0.296609\pi$
$607$	29.3860	1.19274	0.596370	+	0.802709i	$0.296609\pi$
$608$	0	0
$609$	25.2910	1.02484
$610$	0	0
$611$	−18.8781	−0.763726
$612$	−1.51230	−0.0611310
$613$	34.7078	1.40183	0.700917	−	0.713243i	$-0.252775\pi$
$613$	34.7078	1.40183	0.700917	+	0.713243i	$0.252775\pi$
$614$	−72.1636	−2.91229
$615$	0	0
$616$	49.4622	1.99289
$617$	−10.7275	−0.431874	−0.215937	−	0.976407i	$-0.569281\pi$
$617$	−10.7275	−0.431874	−0.215937	+	0.976407i	$0.569281\pi$
$618$	22.0705	0.887805
$619$	−36.1437	−1.45274	−0.726370	−	0.687304i	$-0.758794\pi$
$619$	−36.1437	−1.45274	−0.726370	+	0.687304i	$0.758794\pi$
$620$	0	0
$621$	2.71451	0.108929
$622$	−0.474617	−0.0190304
$623$	−6.02343	−0.241324
$624$	−24.6345	−0.986168
$625$	0	0
$626$	52.2569	2.08861
$627$	0	0
$628$	−80.8768	−3.22734
$629$	−2.00546	−0.0799629
$630$	0	0
$631$	−31.5764	−1.25704	−0.628519	−	0.777794i	$-0.716339\pi$
$631$	−31.5764	−1.25704	−0.628519	+	0.777794i	$0.716339\pi$
$632$	56.6089	2.25178
$633$	21.6285	0.859658
$634$	−47.0857	−1.87001
$635$	0	0
$636$	16.4818	0.653546
$637$	−49.9786	−1.98022
$638$	19.4204	0.768859
$639$	−3.92650	−0.155330
$640$	0	0
$641$	19.8266	0.783104	0.391552	−	0.920156i	$-0.371938\pi$
$641$	19.8266	0.783104	0.391552	+	0.920156i	$0.371938\pi$
$642$	−4.99330	−0.197070
$643$	−13.4569	−0.530687	−0.265343	−	0.964154i	$-0.585485\pi$
$643$	−13.4569	−0.530687	−0.265343	+	0.964154i	$0.585485\pi$
$644$	−8.86718	−0.349416
$645$	0	0
$646$	0	0
$647$	−4.19511	−0.164927	−0.0824634	−	0.996594i	$-0.526279\pi$
$647$	−4.19511	−0.164927	−0.0824634	+	0.996594i	$0.526279\pi$
$648$	−34.9204	−1.37180
$649$	27.5621	1.08191
$650$	0	0
$651$	48.0648	1.88381
$652$	−57.4012	−2.24801
$653$	12.1680	0.476170	0.238085	−	0.971244i	$-0.423480\pi$
$653$	12.1680	0.476170	0.238085	+	0.971244i	$0.423480\pi$
$654$	−46.1372	−1.80411
$655$	0	0
$656$	0.304038	0.0118707
$657$	1.66556	0.0649798
$658$	55.5814	2.16679
$659$	8.24471	0.321168	0.160584	−	0.987022i	$-0.448662\pi$
$659$	8.24471	0.321168	0.160584	+	0.987022i	$0.448662\pi$
$660$	0	0
$661$	21.1197	0.821462	0.410731	−	0.911756i	$-0.365273\pi$
$661$	21.1197	0.821462	0.410731	+	0.911756i	$0.365273\pi$
$662$	−50.6150	−1.96721
$663$	−3.90687	−0.151730
$664$	−78.3941	−3.04228
$665$	0	0
$666$	4.16576	0.161420
$667$	−1.75498	−0.0679530
$668$	22.6502	0.876361
$669$	−33.8685	−1.30943
$670$	0	0
$671$	15.5357	0.599748
$672$	2.29442	0.0885090
$673$	30.2802	1.16722	0.583608	−	0.812036i	$-0.301641\pi$
$673$	30.2802	1.16722	0.583608	+	0.812036i	$0.301641\pi$
$674$	25.0713	0.965711
$675$	0	0
$676$	4.54221	0.174700
$677$	−49.9003	−1.91783	−0.958913	−	0.283701i	$-0.908438\pi$
$677$	−49.9003	−1.91783	−0.958913	+	0.283701i	$0.908438\pi$
$678$	29.7010	1.14066
$679$	19.7618	0.758389
$680$	0	0
$681$	−12.7948	−0.490296
$682$	36.9079	1.41328
$683$	−3.11357	−0.119137	−0.0595687	−	0.998224i	$-0.518973\pi$
$683$	−3.11357	−0.119137	−0.0595687	+	0.998224i	$0.518973\pi$
$684$	0	0
$685$	0	0
$686$	69.6872	2.66067
$687$	−26.0100	−0.992345
$688$	1.76572	0.0673175
$689$	−9.83970	−0.374863
$690$	0	0
$691$	−30.2831	−1.15202	−0.576012	−	0.817441i	$-0.695392\pi$
$691$	−30.2831	−1.15202	−0.576012	+	0.817441i	$0.695392\pi$
$692$	−43.1140	−1.63895
$693$	−5.57829	−0.211902
$694$	−13.3644	−0.507308
$695$	0	0
$696$	28.0305	1.06250
$697$	0.0482186	0.00182641
$698$	−3.80991	−0.144207
$699$	−19.0659	−0.721137
$700$	0	0
$701$	44.5699	1.68338	0.841691	−	0.539960i	$-0.181561\pi$
$701$	44.5699	1.68338	0.841691	+	0.539960i	$0.181561\pi$
$702$	51.3483	1.93802
$703$	0	0
$704$	−16.7014	−0.629458
$705$	0	0
$706$	80.8914	3.04439
$707$	−47.6144	−1.79073
$708$	78.9194	2.96597
$709$	9.34352	0.350903	0.175452	−	0.984488i	$-0.443861\pi$
$709$	9.34352	0.350903	0.175452	+	0.984488i	$0.443861\pi$
$710$	0	0
$711$	−6.38429	−0.239430
$712$	−6.67591	−0.250190
$713$	−3.33529	−0.124908
$714$	11.5027	0.430479
$715$	0	0
$716$	31.0109	1.15893
$717$	3.18372	0.118898
$718$	−8.59001	−0.320576
$719$	25.3974	0.947163	0.473581	−	0.880750i	$-0.342961\pi$
$719$	25.3974	0.947163	0.473581	+	0.880750i	$0.342961\pi$
$720$	0	0
$721$	25.9330	0.965797
$722$	0	0
$723$	27.3552	1.01735
$724$	−24.7069	−0.918224
$725$	0	0
$726$	−23.6410	−0.877400
$727$	31.4377	1.16596	0.582980	−	0.812486i	$-0.301886\pi$
$727$	31.4377	1.16596	0.582980	+	0.812486i	$0.301886\pi$
$728$	−84.5518	−3.13370
$729$	29.9869	1.11063
$730$	0	0
$731$	0.280033	0.0103574
$732$	44.4838	1.64417
$733$	25.3946	0.937971	0.468985	−	0.883206i	$-0.344620\pi$
$733$	25.3946	0.937971	0.468985	+	0.883206i	$0.344620\pi$
$734$	−47.9056	−1.76823
$735$	0	0
$736$	−0.159213	−0.00586867
$737$	12.5912	0.463802
$738$	−0.100160	−0.00368694
$739$	35.5082	1.30619	0.653095	−	0.757276i	$-0.273470\pi$
$739$	35.5082	1.30619	0.653095	+	0.757276i	$0.273470\pi$
$740$	0	0
$741$	0	0
$742$	28.9703	1.06353
$743$	−17.1099	−0.627700	−0.313850	−	0.949473i	$-0.601619\pi$
$743$	−17.1099	−0.627700	−0.313850	+	0.949473i	$0.601619\pi$
$744$	53.2713	1.95302
$745$	0	0
$746$	−57.7597	−2.11473
$747$	8.84121	0.323483
$748$	5.90453	0.215891
$749$	−5.86718	−0.214382
$750$	0	0
$751$	−7.45125	−0.271900	−0.135950	−	0.990716i	$-0.543409\pi$
$751$	−7.45125	−0.271900	−0.135950	+	0.990716i	$0.543409\pi$
$752$	21.0891	0.769041
$753$	−5.21086	−0.189894
$754$	−33.1976	−1.20899
$755$	0	0
$756$	−101.063	−3.67561
$757$	−53.7675	−1.95421	−0.977107	−	0.212750i	$-0.931758\pi$
$757$	−53.7675	−1.95421	−0.977107	+	0.212750i	$0.931758\pi$
$758$	−17.3199	−0.629086
$759$	−1.67503	−0.0607997
$760$	0	0
$761$	23.3939	0.848029	0.424014	−	0.905655i	$-0.360621\pi$
$761$	23.3939	0.848029	0.424014	+	0.905655i	$0.360621\pi$
$762$	−15.0929	−0.546760
$763$	−54.2117	−1.96259
$764$	23.6135	0.854308
$765$	0	0
$766$	−7.57514	−0.273701
$767$	−47.1152	−1.70123
$768$	−50.3235	−1.81589
$769$	−13.2447	−0.477617	−0.238808	−	0.971067i	$-0.576757\pi$
$769$	−13.2447	−0.477617	−0.238808	+	0.971067i	$0.576757\pi$
$770$	0	0
$771$	43.0606	1.55079
$772$	10.4500	0.376105
$773$	17.8052	0.640410	0.320205	−	0.947348i	$-0.396248\pi$
$773$	17.8052	0.640410	0.320205	+	0.947348i	$0.396248\pi$
$774$	−0.581686	−0.0209083
$775$	0	0
$776$	21.9025	0.786254
$777$	−21.1812	−0.759871
$778$	23.0658	0.826949
$779$	0	0
$780$	0	0
$781$	15.3304	0.548566
$782$	−0.798192	−0.0285433
$783$	−20.0022	−0.714819
$784$	55.8321	1.99400
$785$	0	0
$786$	−62.5939	−2.23265
$787$	43.5779	1.55339	0.776693	−	0.629880i	$-0.216896\pi$
$787$	43.5779	1.55339	0.776693	+	0.629880i	$0.216896\pi$
$788$	−80.1055	−2.85364
$789$	21.2711	0.757271
$790$	0	0
$791$	34.8990	1.24087
$792$	−6.18255	−0.219687
$793$	−26.5570	−0.943068
$794$	65.7267	2.33255
$795$	0	0
$796$	−51.4676	−1.82422
$797$	16.1311	0.571395	0.285697	−	0.958320i	$-0.407775\pi$
$797$	16.1311	0.571395	0.285697	+	0.958320i	$0.407775\pi$
$798$	0	0
$799$	3.34460	0.118323
$800$	0	0
$801$	0.752902	0.0266025
$802$	−61.8230	−2.18304
$803$	−6.50293	−0.229484
$804$	36.0528	1.27148
$805$	0	0
$806$	−63.0912	−2.22229
$807$	−5.63237	−0.198269
$808$	−52.7722	−1.85652
$809$	28.7134	1.00951	0.504755	−	0.863263i	$-0.331583\pi$
$809$	28.7134	1.00951	0.504755	+	0.863263i	$0.331583\pi$
$810$	0	0
$811$	21.5422	0.756449	0.378225	−	0.925714i	$-0.376535\pi$
$811$	21.5422	0.756449	0.378225	+	0.925714i	$0.376535\pi$
$812$	65.3388	2.29294
$813$	16.4895	0.578313
$814$	−16.2646	−0.570073
$815$	0	0
$816$	4.36445	0.152786
$817$	0	0
$818$	−69.5634	−2.43223
$819$	9.53566	0.333203
$820$	0	0
$821$	33.3204	1.16289	0.581445	−	0.813586i	$-0.302487\pi$
$821$	33.3204	1.16289	0.581445	+	0.813586i	$0.302487\pi$
$822$	−62.9988	−2.19734
$823$	−37.2390	−1.29807	−0.649035	−	0.760759i	$-0.724827\pi$
$823$	−37.2390	−1.29807	−0.649035	+	0.760759i	$0.724827\pi$
$824$	28.7422	1.00128
$825$	0	0
$826$	138.718	4.82662
$827$	−29.0512	−1.01021	−0.505105	−	0.863058i	$-0.668546\pi$
$827$	−29.0512	−1.01021	−0.505105	+	0.863058i	$0.668546\pi$
$828$	1.10836	0.0385181
$829$	34.5380	1.19956	0.599778	−	0.800166i	$-0.295256\pi$
$829$	34.5380	1.19956	0.599778	+	0.800166i	$0.295256\pi$
$830$	0	0
$831$	10.5134	0.364707
$832$	28.5498	0.989786
$833$	8.85462	0.306794
$834$	−10.2316	−0.354292
$835$	0	0
$836$	0	0
$837$	−38.0136	−1.31394
$838$	32.0410	1.10684
$839$	−44.7243	−1.54406	−0.772028	−	0.635589i	$-0.780757\pi$
$839$	−44.7243	−1.54406	−0.772028	+	0.635589i	$0.780757\pi$
$840$	0	0
$841$	−16.0683	−0.554078
$842$	8.16680	0.281447
$843$	36.5758	1.25974
$844$	55.8770	1.92337
$845$	0	0
$846$	−6.94743	−0.238858
$847$	−27.7784	−0.954478
$848$	10.9921	0.377471
$849$	22.9210	0.786648
$850$	0	0
$851$	1.46980	0.0503839
$852$	43.8962	1.50386
$853$	−16.2603	−0.556741	−0.278370	−	0.960474i	$-0.589794\pi$
$853$	−16.2603	−0.556741	−0.278370	+	0.960474i	$0.589794\pi$
$854$	78.1900	2.67561
$855$	0	0
$856$	−6.50273	−0.222259
$857$	−16.6379	−0.568340	−0.284170	−	0.958774i	$-0.591718\pi$
$857$	−16.6379	−0.568340	−0.284170	+	0.958774i	$0.591718\pi$
$858$	−31.6853	−1.08172
$859$	40.2724	1.37408	0.687038	−	0.726621i	$-0.258910\pi$
$859$	40.2724	1.37408	0.687038	+	0.726621i	$0.258910\pi$
$860$	0	0
$861$	0.509273	0.0173560
$862$	0.118897	0.00404965
$863$	−32.9634	−1.12209	−0.561044	−	0.827786i	$-0.689600\pi$
$863$	−32.9634	−1.12209	−0.561044	+	0.827786i	$0.689600\pi$
$864$	−1.81461	−0.0617343
$865$	0	0
$866$	23.5027	0.798655
$867$	−25.8456	−0.877762
$868$	124.175	4.21476
$869$	24.9265	0.845574
$870$	0	0
$871$	−21.5237	−0.729301
$872$	−60.0841	−2.03470
$873$	−2.47014	−0.0836016
$874$	0	0
$875$	0	0
$876$	−18.6201	−0.629114
$877$	0.213199	0.00719923	0.00359962	−	0.999994i	$-0.498854\pi$
$877$	0.213199	0.00719923	0.00359962	+	0.999994i	$0.498854\pi$
$878$	−54.6541	−1.84449
$879$	28.3885	0.957519
$880$	0	0
$881$	51.8661	1.74741	0.873707	−	0.486453i	$-0.161710\pi$
$881$	51.8661	1.74741	0.873707	+	0.486453i	$0.161710\pi$
$882$	−18.3929	−0.619321
$883$	41.6451	1.40147	0.700734	−	0.713422i	$-0.252856\pi$
$883$	41.6451	1.40147	0.700734	+	0.713422i	$0.252856\pi$
$884$	−10.0933	−0.339476
$885$	0	0
$886$	−49.4083	−1.65990
$887$	4.09325	0.137438	0.0687190	−	0.997636i	$-0.478109\pi$
$887$	4.09325	0.137438	0.0687190	+	0.997636i	$0.478109\pi$
$888$	−23.4756	−0.787790
$889$	−17.7344	−0.594791
$890$	0	0
$891$	−15.3765	−0.515130
$892$	−87.4987	−2.92967
$893$	0	0
$894$	−68.8859	−2.30389
$895$	0	0
$896$	−86.9967	−2.90636
$897$	2.86334	0.0956040
$898$	−30.6958	−1.02433
$899$	24.5764	0.819670
$900$	0	0
$901$	1.74328	0.0580772
$902$	0.391060	0.0130209
$903$	2.95764	0.0984240
$904$	38.6794	1.28646
$905$	0	0
$906$	−48.8266	−1.62216
$907$	8.40726	0.279158	0.139579	−	0.990211i	$-0.455425\pi$
$907$	8.40726	0.279158	0.139579	+	0.990211i	$0.455425\pi$
$908$	−33.0551	−1.09697
$909$	5.95159	0.197402
$910$	0	0
$911$	8.87918	0.294180	0.147090	−	0.989123i	$-0.453009\pi$
$911$	8.87918	0.294180	0.147090	+	0.989123i	$0.453009\pi$
$912$	0	0
$913$	−34.5192	−1.14242
$914$	69.5544	2.30066
$915$	0	0
$916$	−67.1965	−2.22024
$917$	−73.5485	−2.42878
$918$	−9.09730	−0.300256
$919$	−45.9834	−1.51685	−0.758427	−	0.651758i	$-0.774032\pi$
$919$	−45.9834	−1.51685	−0.758427	+	0.651758i	$0.774032\pi$
$920$	0	0
$921$	−45.8637	−1.51126
$922$	13.0658	0.430299
$923$	−26.2062	−0.862587
$924$	62.3623	2.05157
$925$	0	0
$926$	44.0462	1.44745
$927$	−3.24152	−0.106465
$928$	1.17318	0.0385115
$929$	−22.1778	−0.727629	−0.363814	−	0.931471i	$-0.618526\pi$
$929$	−22.1778	−0.727629	−0.363814	+	0.931471i	$0.618526\pi$
$930$	0	0
$931$	0	0
$932$	−49.2564	−1.61345
$933$	−0.301644	−0.00987538
$934$	−70.6872	−2.31296
$935$	0	0
$936$	10.5686	0.345445
$937$	34.3274	1.12143	0.560714	−	0.828009i	$-0.310527\pi$
$937$	34.3274	1.12143	0.560714	+	0.828009i	$0.310527\pi$
$938$	63.3706	2.06912
$939$	33.2120	1.08383
$940$	0	0
$941$	33.7213	1.09928	0.549641	−	0.835401i	$-0.314765\pi$
$941$	33.7213	1.09928	0.549641	+	0.835401i	$0.314765\pi$
$942$	−76.8923	−2.50529
$943$	−0.0353393	−0.00115080
$944$	52.6334	1.71307
$945$	0	0
$946$	2.27110	0.0738400
$947$	−22.1465	−0.719664	−0.359832	−	0.933017i	$-0.617166\pi$
$947$	−22.1465	−0.719664	−0.359832	+	0.933017i	$0.617166\pi$
$948$	71.3729	2.31809
$949$	11.1163	0.360849
$950$	0	0
$951$	−29.9254	−0.970398
$952$	14.9799	0.485501
$953$	−37.2967	−1.20816	−0.604079	−	0.796924i	$-0.706459\pi$
$953$	−37.2967	−1.20816	−0.604079	+	0.796924i	$0.706459\pi$
$954$	−3.62116	−0.117240
$955$	0	0
$956$	8.22508	0.266018
$957$	12.3426	0.398981
$958$	−19.7761	−0.638936
$959$	−74.0242	−2.39037
$960$	0	0
$961$	15.7069	0.506674
$962$	27.8030	0.896405
$963$	0.733371	0.0236326
$964$	70.6718	2.27618
$965$	0	0
$966$	−8.43032	−0.271241
$967$	−18.6736	−0.600502	−0.300251	−	0.953860i	$-0.597070\pi$
$967$	−18.6736	−0.600502	−0.300251	+	0.953860i	$0.597070\pi$
$968$	−30.7875	−0.989547
$969$	0	0
$970$	0	0
$971$	28.5806	0.917195	0.458598	−	0.888644i	$-0.348352\pi$
$971$	28.5806	0.917195	0.458598	+	0.888644i	$0.348352\pi$
$972$	23.2683	0.746330
$973$	−12.0223	−0.385416
$974$	−2.69596	−0.0863842
$975$	0	0
$976$	29.6674	0.949630
$977$	58.1909	1.86169	0.930846	−	0.365411i	$-0.119072\pi$
$977$	58.1909	1.86169	0.930846	+	0.365411i	$0.119072\pi$
$978$	−54.5732	−1.74506
$979$	−2.93959	−0.0939497
$980$	0	0
$981$	6.77622	0.216348
$982$	32.1894	1.02720
$983$	48.0725	1.53328	0.766638	−	0.642080i	$-0.221928\pi$
$983$	48.0725	1.53328	0.766638	+	0.642080i	$0.221928\pi$
$984$	0.564439	0.0179937
$985$	0	0
$986$	5.88157	0.187307
$987$	35.3249	1.12440
$988$	0	0
$989$	−0.205235	−0.00652609
$990$	0	0
$991$	38.3952	1.21966	0.609832	−	0.792531i	$-0.291237\pi$
$991$	38.3952	1.21966	0.609832	+	0.792531i	$0.291237\pi$
$992$	2.22960	0.0707897
$993$	−32.1684	−1.02083
$994$	77.1570	2.44727
$995$	0	0
$996$	−98.8399	−3.13186
$997$	37.8218	1.19783	0.598914	−	0.800814i	$-0.295599\pi$
$997$	37.8218	1.19783	0.598914	+	0.800814i	$0.295599\pi$
$998$	59.2944	1.87693
$999$	16.7518	0.530004

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bu.1.6		6
5.2	odd	4		1805.2.b.f.1084.6		6
5.3	odd	4		1805.2.b.f.1084.1		6
5.4	even	2	inner	9025.2.a.bu.1.1		6
19.7	even	3		475.2.e.g.201.1		12
19.11	even	3		475.2.e.g.26.1		12
19.18	odd	2		9025.2.a.bt.1.1		6
95.7	odd	12		95.2.i.b.49.1	✓	12
95.18	even	4		1805.2.b.g.1084.6		6
95.37	even	4		1805.2.b.g.1084.1		6
95.49	even	6		475.2.e.g.26.6		12
95.64	even	6		475.2.e.g.201.6		12
95.68	odd	12		95.2.i.b.64.1	yes	12
95.83	odd	12		95.2.i.b.49.6	yes	12
95.87	odd	12		95.2.i.b.64.6	yes	12
95.94	odd	2		9025.2.a.bt.1.6		6
285.68	even	12		855.2.be.d.64.6		12
285.83	even	12		855.2.be.d.334.1		12
285.182	even	12		855.2.be.d.64.1		12
285.197	even	12		855.2.be.d.334.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.i.b.49.1	✓	12	95.7	odd	12
95.2.i.b.49.6	yes	12	95.83	odd	12
95.2.i.b.64.1	yes	12	95.68	odd	12
95.2.i.b.64.6	yes	12	95.87	odd	12
475.2.e.g.26.1		12	19.11	even	3
475.2.e.g.26.6		12	95.49	even	6
475.2.e.g.201.1		12	19.7	even	3
475.2.e.g.201.6		12	95.64	even	6
855.2.be.d.64.1		12	285.182	even	12
855.2.be.d.64.6		12	285.68	even	12
855.2.be.d.334.1		12	285.83	even	12
855.2.be.d.334.6		12	285.197	even	12
1805.2.b.f.1084.1		6	5.3	odd	4
1805.2.b.f.1084.6		6	5.2	odd	4
1805.2.b.g.1084.1		6	95.37	even	4
1805.2.b.g.1084.6		6	95.18	even	4
9025.2.a.bt.1.1		6	19.18	odd	2
9025.2.a.bt.1.6		6	95.94	odd	2
9025.2.a.bu.1.1		6	5.4	even	2	inner
9025.2.a.bu.1.6		6	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.45620 q^{2} +1.56104 q^{3} +4.03293 q^{4} +3.83424 q^{6} +4.50527 q^{7} +4.99330 q^{8} -0.563139 q^{9} +2.19869 q^{11} +6.29559 q^{12} -3.75849 q^{13} +11.0659 q^{14} +4.19869 q^{16} +0.665886 q^{17} -1.38318 q^{18} +7.03293 q^{21} +5.40043 q^{22} -0.488026 q^{23} +7.79476 q^{24} -9.23163 q^{26} -5.56222 q^{27} +18.1695 q^{28} +3.59607 q^{29} +6.83424 q^{31} +0.326239 q^{32} +3.43226 q^{33} +1.63555 q^{34} -2.27110 q^{36} -3.01171 q^{37} -5.86718 q^{39} +0.0724126 q^{41} +17.2743 q^{42} +0.420541 q^{43} +8.86718 q^{44} -1.19869 q^{46} +5.02278 q^{47} +6.55434 q^{48} +13.2975 q^{49} +1.03948 q^{51} -15.1578 q^{52} +2.61799 q^{53} -13.6619 q^{54} +22.4962 q^{56} +8.83269 q^{58} +12.5357 q^{59} +7.06587 q^{61} +16.7863 q^{62} -2.53710 q^{63} -7.59607 q^{64} +8.43032 q^{66} +5.72667 q^{67} +2.68548 q^{68} -0.761831 q^{69} +6.97252 q^{71} -2.81192 q^{72} -2.95764 q^{73} -7.39738 q^{74} +9.90571 q^{77} -14.4110 q^{78} +11.3370 q^{79} -6.99346 q^{81} +0.177860 q^{82} -15.6999 q^{83} +28.3634 q^{84} +1.03293 q^{86} +5.61363 q^{87} +10.9787 q^{88} -1.33697 q^{89} -16.9330 q^{91} -1.96818 q^{92} +10.6686 q^{93} +12.3370 q^{94} +0.509273 q^{96} +4.38638 q^{97} +32.6613 q^{98} -1.23817 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{4} + 12 q^{6} + 8 q^{9} + 2 q^{11} + 22 q^{14} + 14 q^{16} + 20 q^{21} + 2 q^{24} - 22 q^{26} - 12 q^{29} + 30 q^{31} + 10 q^{34} - 14 q^{36} - 2 q^{39} + 12 q^{41} + 20 q^{44} + 4 q^{46} + 2 q^{49}+ \cdots - 30 q^{99}+O(q^{100}) \) 6 * q + 2 * q^4 + 12 * q^6 + 8 * q^9 + 2 * q^11 + 22 * q^14 + 14 * q^16 + 20 * q^21 + 2 * q^24 - 22 * q^26 - 12 * q^29 + 30 * q^31 + 10 * q^34 - 14 * q^36 - 2 * q^39 + 12 * q^41 + 20 * q^44 + 4 * q^46 + 2 * q^49 + 40 * q^51 - 4 * q^54 + 46 * q^56 + 20 * q^59 - 2 * q^61 - 12 * q^64 + 6 * q^66 + 18 * q^69 - 2 * q^71 - 22 * q^74 + 24 * q^79 + 14 * q^81 + 48 * q^84 - 16 * q^86 + 36 * q^89 - 24 * q^91 + 30 * q^94 + 26 * q^96 - 30 * q^99

Introduction

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