LMFDB - Embedded newform 9025.2.a.cv.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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)

$f(q)$	$=$	\(q-0.717155 q^{2} -1.56975 q^{3} -1.48569 q^{4} +1.12576 q^{6} +2.51917 q^{7} +2.49978 q^{8} -0.535883 q^{9} -5.85392 q^{11} +2.33216 q^{12} +0.791698 q^{13} -1.80664 q^{14} +1.17865 q^{16} +0.651447 q^{17} +0.384311 q^{18} -3.95447 q^{21} +4.19817 q^{22} -4.88134 q^{23} -3.92403 q^{24} -0.567770 q^{26} +5.55045 q^{27} -3.74270 q^{28} -4.83251 q^{29} -6.73907 q^{31} -5.84483 q^{32} +9.18920 q^{33} -0.467189 q^{34} +0.796155 q^{36} -0.741957 q^{37} -1.24277 q^{39} +8.04494 q^{41} +2.83597 q^{42} -0.761041 q^{43} +8.69710 q^{44} +3.50068 q^{46} +11.3005 q^{47} -1.85018 q^{48} -0.653781 q^{49} -1.02261 q^{51} -1.17622 q^{52} -12.8983 q^{53} -3.98054 q^{54} +6.29737 q^{56} +3.46566 q^{58} -2.14576 q^{59} -6.75915 q^{61} +4.83296 q^{62} -1.34998 q^{63} +1.83436 q^{64} -6.59008 q^{66} -13.8808 q^{67} -0.967847 q^{68} +7.66249 q^{69} -6.05037 q^{71} -1.33959 q^{72} +11.1309 q^{73} +0.532098 q^{74} -14.7470 q^{77} +0.891258 q^{78} +15.7669 q^{79} -7.10518 q^{81} -5.76948 q^{82} +3.26719 q^{83} +5.87511 q^{84} +0.545785 q^{86} +7.58584 q^{87} -14.6335 q^{88} +1.07484 q^{89} +1.99442 q^{91} +7.25215 q^{92} +10.5787 q^{93} -8.10419 q^{94} +9.17493 q^{96} -10.1175 q^{97} +0.468863 q^{98} +3.13702 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 48 q^{4} + 20 q^{6} + 52 q^{9} + 20 q^{11} + 40 q^{16} + 92 q^{24} + 76 q^{26} + 156 q^{36} + 80 q^{39} + 48 q^{44} + 72 q^{49} + 32 q^{54} + 80 q^{61} + 72 q^{64} + 16 q^{66} + 100 q^{74} + 40 q^{81}+ \cdots + 128 q^{99}+O(q^{100}) $ 40 * q + 48 * q^4 + 20 * q^6 + 52 * q^9 + 20 * q^11 + 40 * q^16 + 92 * q^24 + 76 * q^26 + 156 * q^36 + 80 * q^39 + 48 * q^44 + 72 * q^49 + 32 * q^54 + 80 * q^61 + 72 * q^64 + 16 * q^66 + 100 * q^74 + 40 * q^81 + 380 * q^96 + 128 * 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 $

$2$	−0.717155	−0.507105	−0.253553	−	0.967322i	$-0.581599\pi$
$2$	−0.717155	−0.507105	−0.253553	+	0.967322i	$0.581599\pi$
$3$	−1.56975	−0.906296	−0.453148	−	0.891435i	$-0.649699\pi$
$3$	−1.56975	−0.906296	−0.453148	+	0.891435i	$0.649699\pi$
$4$	−1.48569	−0.742844
$5$	0	0
$5$	0	0
$6$	1.12576	0.459588
$7$	2.51917	0.952157	0.476078	−	0.879403i	$-0.342058\pi$
$7$	2.51917	0.952157	0.476078	+	0.879403i	$0.342058\pi$
$8$	2.49978	0.883806
$9$	−0.535883	−0.178628
$10$	0	0
$11$	−5.85392	−1.76502	−0.882512	−	0.470290i	$-0.844149\pi$
$11$	−5.85392	−1.76502	−0.882512	+	0.470290i	$0.844149\pi$
$12$	2.33216	0.673237
$13$	0.791698	0.219577	0.109789	−	0.993955i	$-0.464983\pi$
$13$	0.791698	0.219577	0.109789	+	0.993955i	$0.464983\pi$
$14$	−1.80664	−0.482844
$15$	0	0
$16$	1.17865	0.294661
$17$	0.651447	0.157999	0.0789996	−	0.996875i	$-0.474827\pi$
$17$	0.651447	0.157999	0.0789996	+	0.996875i	$0.474827\pi$
$18$	0.384311	0.0905831
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−3.95447	−0.862936
$22$	4.19817	0.895053
$23$	−4.88134	−1.01783	−0.508915	−	0.860817i	$-0.669953\pi$
$23$	−4.88134	−1.01783	−0.508915	+	0.860817i	$0.669953\pi$
$24$	−3.92403	−0.800990
$25$	0	0
$26$	−0.567770	−0.111349
$27$	5.55045	1.06819
$28$	−3.74270	−0.707304
$29$	−4.83251	−0.897375	−0.448687	−	0.893689i	$-0.648108\pi$
$29$	−4.83251	−0.897375	−0.448687	+	0.893689i	$0.648108\pi$
$30$	0	0
$31$	−6.73907	−1.21037	−0.605186	−	0.796084i	$-0.706901\pi$
$31$	−6.73907	−1.21037	−0.605186	+	0.796084i	$0.706901\pi$
$32$	−5.84483	−1.03323
$33$	9.18920	1.59963
$34$	−0.467189	−0.0801222
$35$	0	0
$36$	0.796155	0.132693
$37$	−0.741957	−0.121977	−0.0609885	−	0.998138i	$-0.519425\pi$
$37$	−0.741957	−0.121977	−0.0609885	+	0.998138i	$0.519425\pi$
$38$	0	0
$39$	−1.24277	−0.199002
$40$	0	0
$41$	8.04494	1.25641	0.628205	−	0.778048i	$-0.283790\pi$
$41$	8.04494	1.25641	0.628205	+	0.778048i	$0.283790\pi$
$42$	2.83597	0.437600
$43$	−0.761041	−0.116058	−0.0580289	−	0.998315i	$-0.518482\pi$
$43$	−0.761041	−0.116058	−0.0580289	+	0.998315i	$0.518482\pi$
$44$	8.69710	1.31114
$45$	0	0
$46$	3.50068	0.516147
$47$	11.3005	1.64834	0.824171	−	0.566342i	$-0.191642\pi$
$47$	11.3005	1.64834	0.824171	+	0.566342i	$0.191642\pi$
$48$	−1.85018	−0.267050
$49$	−0.653781	−0.0933973
$50$	0	0
$51$	−1.02261	−0.143194
$52$	−1.17622	−0.163112
$53$	−12.8983	−1.77172	−0.885860	−	0.463953i	$-0.846431\pi$
$53$	−12.8983	−1.77172	−0.885860	+	0.463953i	$0.846431\pi$
$54$	−3.98054	−0.541683
$55$	0	0
$56$	6.29737	0.841522
$57$	0	0
$58$	3.46566	0.455064
$59$	−2.14576	−0.279354	−0.139677	−	0.990197i	$-0.544606\pi$
$59$	−2.14576	−0.279354	−0.139677	+	0.990197i	$0.544606\pi$
$60$	0	0
$61$	−6.75915	−0.865421	−0.432710	−	0.901533i	$-0.642443\pi$
$61$	−6.75915	−0.865421	−0.432710	+	0.901533i	$0.642443\pi$
$62$	4.83296	0.613787
$63$	−1.34998	−0.170082
$64$	1.83436	0.229295
$65$	0	0
$66$	−6.59008	−0.811183
$67$	−13.8808	−1.69581	−0.847903	−	0.530152i	$-0.822135\pi$
$67$	−13.8808	−1.69581	−0.847903	+	0.530152i	$0.822135\pi$
$68$	−0.967847	−0.117369
$69$	7.66249	0.922455
$70$	0	0
$71$	−6.05037	−0.718046	−0.359023	−	0.933329i	$-0.616890\pi$
$71$	−6.05037	−0.718046	−0.359023	+	0.933329i	$0.616890\pi$
$72$	−1.33959	−0.157872
$73$	11.1309	1.30277	0.651387	−	0.758746i	$-0.274188\pi$
$73$	11.1309	1.30277	0.651387	+	0.758746i	$0.274188\pi$
$74$	0.532098	0.0618552
$75$	0	0
$76$	0	0
$77$	−14.7470	−1.68058
$78$	0.891258	0.100915
$79$	15.7669	1.77391	0.886955	−	0.461856i	$-0.152816\pi$
$79$	15.7669	1.77391	0.886955	+	0.461856i	$0.152816\pi$
$80$	0	0
$81$	−7.10518	−0.789464
$82$	−5.76948	−0.637132
$83$	3.26719	0.358621	0.179310	−	0.983793i	$-0.442613\pi$
$83$	3.26719	0.358621	0.179310	+	0.983793i	$0.442613\pi$
$84$	5.87511	0.641027
$85$	0	0
$86$	0.545785	0.0588535
$87$	7.58584	0.813287
$88$	−14.6335	−1.55994
$89$	1.07484	0.113932	0.0569662	−	0.998376i	$-0.481857\pi$
$89$	1.07484	0.113932	0.0569662	+	0.998376i	$0.481857\pi$
$90$	0	0
$91$	1.99442	0.209072
$92$	7.25215	0.756089
$93$	10.5787	1.09696
$94$	−8.10419	−0.835883
$95$	0	0
$96$	9.17493	0.936412
$97$	−10.1175	−1.02728	−0.513641	−	0.858005i	$-0.671704\pi$
$97$	−10.1175	−1.02728	−0.513641	+	0.858005i	$0.671704\pi$
$98$	0.468863	0.0473623
$99$	3.13702	0.315282
$100$	0	0
$101$	5.63114	0.560320	0.280160	−	0.959953i	$-0.409613\pi$
$101$	5.63114	0.560320	0.280160	+	0.959953i	$0.409613\pi$
$102$	0.733370	0.0726145
$103$	−0.459634	−0.0452891	−0.0226446	−	0.999744i	$-0.507209\pi$
$103$	−0.459634	−0.0452891	−0.0226446	+	0.999744i	$0.507209\pi$
$104$	1.97907	0.194064
$105$	0	0
$106$	9.25010	0.898449
$107$	5.11103	0.494102	0.247051	−	0.969002i	$-0.420538\pi$
$107$	5.11103	0.494102	0.247051	+	0.969002i	$0.420538\pi$
$108$	−8.24624	−0.793495
$109$	−10.1536	−0.972537	−0.486269	−	0.873809i	$-0.661642\pi$
$109$	−10.1536	−0.972537	−0.486269	+	0.873809i	$0.661642\pi$
$110$	0	0
$111$	1.16469	0.110547
$112$	2.96921	0.280564
$113$	−5.61000	−0.527745	−0.263872	−	0.964558i	$-0.585000\pi$
$113$	−5.61000	−0.527745	−0.263872	+	0.964558i	$0.585000\pi$
$114$	0	0
$115$	0	0
$116$	7.17960	0.666609
$117$	−0.424257	−0.0392226
$118$	1.53884	0.141662
$119$	1.64111	0.150440
$120$	0	0
$121$	23.2684	2.11531
$122$	4.84736	0.438860
$123$	−12.6286	−1.13868
$124$	10.0122	0.899118
$125$	0	0
$126$	0.968146	0.0862493
$127$	−10.8609	−0.963747	−0.481873	−	0.876241i	$-0.660044\pi$
$127$	−10.8609	−0.963747	−0.481873	+	0.876241i	$0.660044\pi$
$128$	10.3741	0.916953
$129$	1.19465	0.105183
$130$	0	0
$131$	16.3484	1.42837	0.714184	−	0.699958i	$-0.246798\pi$
$131$	16.3484	1.42837	0.714184	+	0.699958i	$0.246798\pi$
$132$	−13.6523	−1.18828
$133$	0	0
$134$	9.95467	0.859952
$135$	0	0
$136$	1.62847	0.139641
$137$	18.3333	1.56632	0.783162	−	0.621818i	$-0.213606\pi$
$137$	18.3333	1.56632	0.783162	+	0.621818i	$0.213606\pi$
$138$	−5.49519	−0.467782
$139$	−9.37358	−0.795057	−0.397529	−	0.917590i	$-0.630132\pi$
$139$	−9.37358	−0.795057	−0.397529	+	0.917590i	$0.630132\pi$
$140$	0	0
$141$	−17.7389	−1.49389
$142$	4.33905	0.364125
$143$	−4.63454	−0.387559
$144$	−0.631616	−0.0526347
$145$	0	0
$146$	−7.98258	−0.660643
$147$	1.02627	0.0846456
$148$	1.10232	0.0906099
$149$	−7.69847	−0.630683	−0.315342	−	0.948978i	$-0.602119\pi$
$149$	−7.69847	−0.630683	−0.315342	+	0.948978i	$0.602119\pi$
$150$	0	0
$151$	−0.167275	−0.0136126	−0.00680631	−	0.999977i	$-0.502167\pi$
$151$	−0.167275	−0.0136126	−0.00680631	+	0.999977i	$0.502167\pi$
$152$	0	0
$153$	−0.349100	−0.0282230
$154$	10.5759	0.852231
$155$	0	0
$156$	1.84637	0.147828
$157$	−14.8860	−1.18803	−0.594016	−	0.804453i	$-0.702458\pi$
$157$	−14.8860	−1.18803	−0.594016	+	0.804453i	$0.702458\pi$
$158$	−11.3073	−0.899559
$159$	20.2471	1.60570
$160$	0	0
$161$	−12.2969	−0.969133
$162$	5.09552	0.400342
$163$	−18.7516	−1.46874	−0.734369	−	0.678751i	$-0.762522\pi$
$163$	−18.7516	−1.46874	−0.734369	+	0.678751i	$0.762522\pi$
$164$	−11.9523	−0.933316
$165$	0	0
$166$	−2.34308	−0.181859
$167$	−11.1787	−0.865036	−0.432518	−	0.901625i	$-0.642375\pi$
$167$	−11.1787	−0.865036	−0.432518	+	0.901625i	$0.642375\pi$
$168$	−9.88530	−0.762668
$169$	−12.3732	−0.951786
$170$	0	0
$171$	0	0
$172$	1.13067	0.0862128
$173$	−5.09894	−0.387665	−0.193833	−	0.981035i	$-0.562092\pi$
$173$	−5.09894	−0.387665	−0.193833	+	0.981035i	$0.562092\pi$
$174$	−5.44022	−0.412422
$175$	0	0
$176$	−6.89970	−0.520084
$177$	3.36831	0.253178
$178$	−0.770825	−0.0577757
$179$	0.474937	0.0354984	0.0177492	−	0.999842i	$-0.494350\pi$
$179$	0.474937	0.0354984	0.0177492	+	0.999842i	$0.494350\pi$
$180$	0	0
$181$	−1.41389	−0.105093	−0.0525466	−	0.998618i	$-0.516734\pi$
$181$	−1.41389	−0.105093	−0.0525466	+	0.998618i	$0.516734\pi$
$182$	−1.43031	−0.106022
$183$	10.6102	0.784327
$184$	−12.2023	−0.899564
$185$	0	0
$186$	−7.58654	−0.556272
$187$	−3.81352	−0.278872
$188$	−16.7890	−1.22446
$189$	13.9825	1.01708
$190$	0	0
$191$	12.6109	0.912496	0.456248	−	0.889853i	$-0.349193\pi$
$191$	12.6109	0.912496	0.456248	+	0.889853i	$0.349193\pi$
$192$	−2.87949	−0.207809
$193$	−12.6096	−0.907660	−0.453830	−	0.891088i	$-0.649943\pi$
$193$	−12.6096	−0.907660	−0.453830	+	0.891088i	$0.649943\pi$
$194$	7.25585	0.520940
$195$	0	0
$196$	0.971315	0.0693796
$197$	−1.50078	−0.106926	−0.0534631	−	0.998570i	$-0.517026\pi$
$197$	−1.50078	−0.106926	−0.0534631	+	0.998570i	$0.517026\pi$
$198$	−2.24973	−0.159881
$199$	−15.6650	−1.11046	−0.555231	−	0.831696i	$-0.687370\pi$
$199$	−15.6650	−1.11046	−0.555231	+	0.831696i	$0.687370\pi$
$200$	0	0
$201$	21.7893	1.53690
$202$	−4.03841	−0.284141
$203$	−12.1739	−0.854441
$204$	1.51928	0.106371
$205$	0	0
$206$	0.329629	0.0229664
$207$	2.61583	0.181813
$208$	0.933131	0.0647010
$209$	0	0
$210$	0	0
$211$	17.8162	1.22652	0.613259	−	0.789882i	$-0.289858\pi$
$211$	17.8162	1.22652	0.613259	+	0.789882i	$0.289858\pi$
$212$	19.1629	1.31611
$213$	9.49757	0.650763
$214$	−3.66540	−0.250562
$215$	0	0
$216$	13.8749	0.944068
$217$	−16.9769	−1.15246
$218$	7.28170	0.493179
$219$	−17.4727	−1.18070
$220$	0	0
$221$	0.515749	0.0346930
$222$	−0.835262	−0.0560591
$223$	−18.4288	−1.23408	−0.617041	−	0.786931i	$-0.711669\pi$
$223$	−18.4288	−1.23408	−0.617041	+	0.786931i	$0.711669\pi$
$224$	−14.7241	−0.983797
$225$	0	0
$226$	4.02325	0.267622
$227$	4.13839	0.274674	0.137337	−	0.990524i	$-0.456146\pi$
$227$	4.13839	0.274674	0.137337	+	0.990524i	$0.456146\pi$
$228$	0	0
$229$	−1.37089	−0.0905907	−0.0452953	−	0.998974i	$-0.514423\pi$
$229$	−1.37089	−0.0905907	−0.0452953	+	0.998974i	$0.514423\pi$
$230$	0	0
$231$	23.1492	1.52310
$232$	−12.0802	−0.793105
$233$	1.01441	0.0664564	0.0332282	−	0.999448i	$-0.489421\pi$
$233$	1.01441	0.0664564	0.0332282	+	0.999448i	$0.489421\pi$
$234$	0.304258	0.0198900
$235$	0	0
$236$	3.18793	0.207517
$237$	−24.7500	−1.60769
$238$	−1.17693	−0.0762889
$239$	5.54538	0.358701	0.179351	−	0.983785i	$-0.442600\pi$
$239$	5.54538	0.358701	0.179351	+	0.983785i	$0.442600\pi$
$240$	0	0
$241$	−25.0078	−1.61089	−0.805445	−	0.592670i	$-0.798074\pi$
$241$	−25.0078	−1.61089	−0.805445	+	0.592670i	$0.798074\pi$
$242$	−16.6871	−1.07268
$243$	−5.49800	−0.352697
$244$	10.0420	0.642873
$245$	0	0
$246$	9.05664	0.577430
$247$	0	0
$248$	−16.8462	−1.06973
$249$	−5.12867	−0.325016
$250$	0	0
$251$	10.9588	0.691713	0.345857	−	0.938287i	$-0.387588\pi$
$251$	10.9588	0.691713	0.345857	+	0.938287i	$0.387588\pi$
$252$	2.00565	0.126344
$253$	28.5750	1.79649
$254$	7.78894	0.488721
$255$	0	0
$256$	−11.1086	−0.694287
$257$	−15.0295	−0.937514	−0.468757	−	0.883327i	$-0.655298\pi$
$257$	−15.0295	−0.937514	−0.468757	+	0.883327i	$0.655298\pi$
$258$	−0.856746	−0.0533387
$259$	−1.86912	−0.116141
$260$	0	0
$261$	2.58966	0.160296
$262$	−11.7244	−0.724334
$263$	14.8709	0.916980	0.458490	−	0.888700i	$-0.348390\pi$
$263$	14.8709	0.916980	0.458490	+	0.888700i	$0.348390\pi$
$264$	22.9710	1.41377
$265$	0	0
$266$	0	0
$267$	−1.68722	−0.103256
$268$	20.6225	1.25972
$269$	3.34506	0.203952	0.101976	−	0.994787i	$-0.467484\pi$
$269$	3.34506	0.203952	0.101976	+	0.994787i	$0.467484\pi$
$270$	0	0
$271$	9.34209	0.567492	0.283746	−	0.958900i	$-0.408423\pi$
$271$	9.34209	0.567492	0.283746	+	0.958900i	$0.408423\pi$
$272$	0.767825	0.0465562
$273$	−3.13074	−0.189481
$274$	−13.1479	−0.794291
$275$	0	0
$276$	−11.3841	−0.685240
$277$	−2.79175	−0.167740	−0.0838700	−	0.996477i	$-0.526728\pi$
$277$	−2.79175	−0.167740	−0.0838700	+	0.996477i	$0.526728\pi$
$278$	6.72232	0.403178
$279$	3.61135	0.216206
$280$	0	0
$281$	27.7534	1.65563	0.827816	−	0.561000i	$-0.189583\pi$
$281$	27.7534	1.65563	0.827816	+	0.561000i	$0.189583\pi$
$282$	12.7215	0.757557
$283$	1.13648	0.0675569	0.0337784	−	0.999429i	$-0.489246\pi$
$283$	1.13648	0.0675569	0.0337784	+	0.999429i	$0.489246\pi$
$284$	8.98896	0.533396
$285$	0	0
$286$	3.32368	0.196533
$287$	20.2666	1.19630
$288$	3.13215	0.184563
$289$	−16.5756	−0.975036
$290$	0	0
$291$	15.8820	0.931021
$292$	−16.5370	−0.967757
$293$	−25.0362	−1.46263	−0.731314	−	0.682041i	$-0.761092\pi$
$293$	−25.0362	−1.46263	−0.731314	+	0.682041i	$0.761092\pi$
$294$	−0.735997	−0.0429242
$295$	0	0
$296$	−1.85473	−0.107804
$297$	−32.4919	−1.88537
$298$	5.52100	0.319823
$299$	−3.86454	−0.223492
$300$	0	0
$301$	−1.91719	−0.110505
$302$	0.119962	0.00690304
$303$	−8.83949	−0.507816
$304$	0	0
$305$	0	0
$306$	0.250359	0.0143120
$307$	−18.8926	−1.07826	−0.539130	−	0.842223i	$-0.681247\pi$
$307$	−18.8926	−1.07826	−0.539130	+	0.842223i	$0.681247\pi$
$308$	21.9095	1.24841
$309$	0.721512	0.0410454
$310$	0	0
$311$	13.9570	0.791430	0.395715	−	0.918373i	$-0.370497\pi$
$311$	13.9570	0.791430	0.395715	+	0.918373i	$0.370497\pi$
$312$	−3.10665	−0.175879
$313$	0.212626	0.0120183	0.00600917	−	0.999982i	$-0.498087\pi$
$313$	0.212626	0.0120183	0.00600917	+	0.999982i	$0.498087\pi$
$314$	10.6756	0.602457
$315$	0	0
$316$	−23.4246	−1.31774
$317$	7.50284	0.421402	0.210701	−	0.977551i	$-0.432425\pi$
$317$	7.50284	0.421402	0.210701	+	0.977551i	$0.432425\pi$
$318$	−14.5203	−0.814261
$319$	28.2891	1.58389
$320$	0	0
$321$	−8.02305	−0.447803
$322$	8.81881	0.491453
$323$	0	0
$324$	10.5561	0.586449
$325$	0	0
$326$	13.4478	0.744805
$327$	15.9386	0.881406
$328$	20.1106	1.11042
$329$	28.4678	1.56948
$330$	0	0
$331$	3.22616	0.177326	0.0886629	−	0.996062i	$-0.471741\pi$
$331$	3.22616	0.177326	0.0886629	+	0.996062i	$0.471741\pi$
$332$	−4.85403	−0.266399
$333$	0.397602	0.0217885
$334$	8.01688	0.438664
$335$	0	0
$336$	−4.66092	−0.254274
$337$	34.5994	1.88475	0.942375	−	0.334560i	$-0.108588\pi$
$337$	34.5994	1.88475	0.942375	+	0.334560i	$0.108588\pi$
$338$	8.87352	0.482656
$339$	8.80631	0.478293
$340$	0	0
$341$	39.4500	2.13634
$342$	0	0
$343$	−19.2812	−1.04109
$344$	−1.90244	−0.102572
$345$	0	0
$346$	3.65673	0.196587
$347$	0.0848311	0.00455397	0.00227698	−	0.999997i	$-0.499275\pi$
$347$	0.0848311	0.00455397	0.00227698	+	0.999997i	$0.499275\pi$
$348$	−11.2702	−0.604145
$349$	−17.8090	−0.953295	−0.476648	−	0.879094i	$-0.658148\pi$
$349$	−17.8090	−0.953295	−0.476648	+	0.879094i	$0.658148\pi$
$350$	0	0
$351$	4.39428	0.234549
$352$	34.2152	1.82368
$353$	15.1330	0.805451	0.402725	−	0.915321i	$-0.368063\pi$
$353$	15.1330	0.805451	0.402725	+	0.915321i	$0.368063\pi$
$354$	−2.41560	−0.128388
$355$	0	0
$356$	−1.59687	−0.0846340
$357$	−2.57613	−0.136343
$358$	−0.340603	−0.0180015
$359$	9.00023	0.475014	0.237507	−	0.971386i	$-0.423670\pi$
$359$	9.00023	0.475014	0.237507	+	0.971386i	$0.423670\pi$
$360$	0	0
$361$	0	0
$362$	1.01398	0.0532934
$363$	−36.5256	−1.91710
$364$	−2.96309	−0.155308
$365$	0	0
$366$	−7.60915	−0.397737
$367$	−5.86064	−0.305923	−0.152962	−	0.988232i	$-0.548881\pi$
$367$	−5.86064	−0.305923	−0.152962	+	0.988232i	$0.548881\pi$
$368$	−5.75337	−0.299915
$369$	−4.31115	−0.224429
$370$	0	0
$371$	−32.4931	−1.68696
$372$	−15.7166	−0.814867
$373$	24.6228	1.27492	0.637460	−	0.770484i	$-0.279985\pi$
$373$	24.6228	1.27492	0.637460	+	0.770484i	$0.279985\pi$
$374$	2.73489	0.141418
$375$	0	0
$376$	28.2487	1.45681
$377$	−3.82589	−0.197043
$378$	−10.0277	−0.515767
$379$	2.67753	0.137536	0.0687679	−	0.997633i	$-0.478093\pi$
$379$	2.67753	0.137536	0.0687679	+	0.997633i	$0.478093\pi$
$380$	0	0
$381$	17.0489	0.873440
$382$	−9.04401	−0.462732
$383$	27.8484	1.42299	0.711494	−	0.702692i	$-0.248019\pi$
$383$	27.8484	1.42299	0.711494	+	0.702692i	$0.248019\pi$
$384$	−16.2848	−0.831031
$385$	0	0
$386$	9.04306	0.460280
$387$	0.407829	0.0207311
$388$	15.0315	0.763110
$389$	14.8564	0.753249	0.376624	−	0.926366i	$-0.377085\pi$
$389$	14.8564	0.753249	0.376624	+	0.926366i	$0.377085\pi$
$390$	0	0
$391$	−3.17994	−0.160816
$392$	−1.63431	−0.0825451
$393$	−25.6630	−1.29452
$394$	1.07629	0.0542229
$395$	0	0
$396$	−4.66063	−0.234205
$397$	−11.1189	−0.558043	−0.279022	−	0.960285i	$-0.590010\pi$
$397$	−11.1189	−0.558043	−0.279022	+	0.960285i	$0.590010\pi$
$398$	11.2342	0.563121
$399$	0	0
$400$	0	0
$401$	−28.4510	−1.42078	−0.710388	−	0.703810i	$-0.751481\pi$
$401$	−28.4510	−1.42078	−0.710388	+	0.703810i	$0.751481\pi$
$402$	−15.6263	−0.779371
$403$	−5.33531	−0.265771
$404$	−8.36612	−0.416230
$405$	0	0
$406$	8.73059	0.433292
$407$	4.34336	0.215292
$408$	−2.55630	−0.126556
$409$	−15.8493	−0.783695	−0.391847	−	0.920030i	$-0.628164\pi$
$409$	−15.8493	−0.783695	−0.391847	+	0.920030i	$0.628164\pi$
$410$	0	0
$411$	−28.7788	−1.41955
$412$	0.682873	0.0336428
$413$	−5.40554	−0.265989
$414$	−1.87595	−0.0921981
$415$	0	0
$416$	−4.62734	−0.226874
$417$	14.7142	0.720557
$418$	0	0
$419$	−10.2300	−0.499769	−0.249885	−	0.968276i	$-0.580393\pi$
$419$	−10.2300	−0.499769	−0.249885	+	0.968276i	$0.580393\pi$
$420$	0	0
$421$	31.1606	1.51868	0.759338	−	0.650696i	$-0.225523\pi$
$421$	31.1606	1.51868	0.759338	+	0.650696i	$0.225523\pi$
$422$	−12.7770	−0.621974
$423$	−6.05572	−0.294439
$424$	−32.2430	−1.56586
$425$	0	0
$426$	−6.81123	−0.330005
$427$	−17.0275	−0.824016
$428$	−7.59340	−0.367041
$429$	7.27506	0.351243
$430$	0	0
$431$	−9.60728	−0.462767	−0.231383	−	0.972863i	$-0.574325\pi$
$431$	−9.60728	−0.462767	−0.231383	+	0.972863i	$0.574325\pi$
$432$	6.54202	0.314753
$433$	20.8460	1.00179	0.500897	−	0.865507i	$-0.333004\pi$
$433$	20.8460	1.00179	0.500897	+	0.865507i	$0.333004\pi$
$434$	12.1751	0.584421
$435$	0	0
$436$	15.0851	0.722443
$437$	0	0
$438$	12.5307	0.598738
$439$	1.11124	0.0530364	0.0265182	−	0.999648i	$-0.491558\pi$
$439$	1.11124	0.0530364	0.0265182	+	0.999648i	$0.491558\pi$
$440$	0	0
$441$	0.350350	0.0166833
$442$	−0.369872	−0.0175930
$443$	28.7487	1.36589	0.682947	−	0.730468i	$-0.260698\pi$
$443$	28.7487	1.36589	0.682947	+	0.730468i	$0.260698\pi$
$444$	−1.73036	−0.0821194
$445$	0	0
$446$	13.2163	0.625810
$447$	12.0847	0.571586
$448$	4.62107	0.218325
$449$	−31.9695	−1.50873	−0.754367	−	0.656452i	$-0.772056\pi$
$449$	−31.9695	−1.50873	−0.754367	+	0.656452i	$0.772056\pi$
$450$	0	0
$451$	−47.0945	−2.21759
$452$	8.33472	0.392032
$453$	0.262580	0.0123371
$454$	−2.96787	−0.139289
$455$	0	0
$456$	0	0
$457$	12.0035	0.561498	0.280749	−	0.959781i	$-0.409417\pi$
$457$	12.0035	0.561498	0.280749	+	0.959781i	$0.409417\pi$
$458$	0.983138	0.0459390
$459$	3.61583	0.168772
$460$	0	0
$461$	−33.9752	−1.58238	−0.791191	−	0.611569i	$-0.790539\pi$
$461$	−33.9752	−1.58238	−0.791191	+	0.611569i	$0.790539\pi$
$462$	−16.6015	−0.772374
$463$	24.1005	1.12004	0.560022	−	0.828478i	$-0.310793\pi$
$463$	24.1005	1.12004	0.560022	+	0.828478i	$0.310793\pi$
$464$	−5.69582	−0.264422
$465$	0	0
$466$	−0.727492	−0.0337004
$467$	−0.621098	−0.0287410	−0.0143705	−	0.999897i	$-0.504574\pi$
$467$	−0.621098	−0.0287410	−0.0143705	+	0.999897i	$0.504574\pi$
$468$	0.630314	0.0291363
$469$	−34.9680	−1.61467
$470$	0	0
$471$	23.3673	1.07671
$472$	−5.36393	−0.246895
$473$	4.45508	0.204845
$474$	17.7496	0.815267
$475$	0	0
$476$	−2.43817	−0.111753
$477$	6.91199	0.316478
$478$	−3.97690	−0.181899
$479$	31.0733	1.41978	0.709888	−	0.704314i	$-0.248745\pi$
$479$	31.0733	1.41978	0.709888	+	0.704314i	$0.248745\pi$
$480$	0	0
$481$	−0.587405	−0.0267834
$482$	17.9344	0.816892
$483$	19.3031	0.878322
$484$	−34.5696	−1.57134
$485$	0	0
$486$	3.94292	0.178855
$487$	37.2206	1.68663	0.843313	−	0.537423i	$-0.180602\pi$
$487$	37.2206	1.68663	0.843313	+	0.537423i	$0.180602\pi$
$488$	−16.8964	−0.764864
$489$	29.4353	1.33111
$490$	0	0
$491$	38.5493	1.73971	0.869853	−	0.493310i	$-0.164213\pi$
$491$	38.5493	1.73971	0.869853	+	0.493310i	$0.164213\pi$
$492$	18.7621	0.845861
$493$	−3.14813	−0.141784
$494$	0	0
$495$	0	0
$496$	−7.94298	−0.356650
$497$	−15.2419	−0.683693
$498$	3.67806	0.164818
$499$	−5.38788	−0.241194	−0.120597	−	0.992702i	$-0.538481\pi$
$499$	−5.38788	−0.241194	−0.120597	+	0.992702i	$0.538481\pi$
$500$	0	0
$501$	17.5478	0.783978
$502$	−7.85916	−0.350771
$503$	26.5719	1.18478	0.592392	−	0.805650i	$-0.298184\pi$
$503$	26.5719	1.18478	0.592392	+	0.805650i	$0.298184\pi$
$504$	−3.37465	−0.150319
$505$	0	0
$506$	−20.4927	−0.911012
$507$	19.4229	0.862600
$508$	16.1359	0.715914
$509$	28.1487	1.24767	0.623836	−	0.781556i	$-0.285573\pi$
$509$	28.1487	1.24767	0.623836	+	0.781556i	$0.285573\pi$
$510$	0	0
$511$	28.0406	1.24044
$512$	−12.7817	−0.564876
$513$	0	0
$514$	10.7785	0.475418
$515$	0	0
$516$	−1.77487	−0.0781343
$517$	−66.1520	−2.90936
$518$	1.34045	0.0588958
$519$	8.00406	0.351339
$520$	0	0
$521$	9.50660	0.416492	0.208246	−	0.978077i	$-0.433225\pi$
$521$	9.50660	0.416492	0.208246	+	0.978077i	$0.433225\pi$
$522$	−1.85719	−0.0812869
$523$	−17.3457	−0.758473	−0.379237	−	0.925300i	$-0.623813\pi$
$523$	−17.3457	−0.758473	−0.379237	+	0.925300i	$0.623813\pi$
$524$	−24.2887	−1.06106
$525$	0	0
$526$	−10.6648	−0.465006
$527$	−4.39015	−0.191238
$528$	10.8308	0.471350
$529$	0.827475	0.0359772
$530$	0	0
$531$	1.14988	0.0499004
$532$	0	0
$533$	6.36916	0.275879
$534$	1.21000	0.0523619
$535$	0	0
$536$	−34.6989	−1.49876
$537$	−0.745532	−0.0321721
$538$	−2.39893	−0.103425
$539$	3.82718	0.164848
$540$	0	0
$541$	−18.5619	−0.798039	−0.399020	−	0.916942i	$-0.630649\pi$
$541$	−18.5619	−0.798039	−0.399020	+	0.916942i	$0.630649\pi$
$542$	−6.69973	−0.287778
$543$	2.21945	0.0952456
$544$	−3.80760	−0.163249
$545$	0	0
$546$	2.24523	0.0960870
$547$	5.56689	0.238023	0.119011	−	0.992893i	$-0.462027\pi$
$547$	5.56689	0.238023	0.119011	+	0.992893i	$0.462027\pi$
$548$	−27.2376	−1.16353
$549$	3.62211	0.154588
$550$	0	0
$551$	0	0
$552$	19.1545	0.815271
$553$	39.7194	1.68904
$554$	2.00212	0.0850618
$555$	0	0
$556$	13.9262	0.590603
$557$	25.5819	1.08394	0.541970	−	0.840398i	$-0.317679\pi$
$557$	25.5819	1.08394	0.541970	+	0.840398i	$0.317679\pi$
$558$	−2.58990	−0.109639
$559$	−0.602515	−0.0254837
$560$	0	0
$561$	5.98628	0.252741
$562$	−19.9035	−0.839580
$563$	12.8773	0.542712	0.271356	−	0.962479i	$-0.412528\pi$
$563$	12.8773	0.542712	0.271356	+	0.962479i	$0.412528\pi$
$564$	26.3545	1.10972
$565$	0	0
$566$	−0.815034	−0.0342585
$567$	−17.8992	−0.751694
$568$	−15.1246	−0.634614
$569$	11.4205	0.478773	0.239387	−	0.970924i	$-0.423054\pi$
$569$	11.4205	0.478773	0.239387	+	0.970924i	$0.423054\pi$
$570$	0	0
$571$	11.1881	0.468208	0.234104	−	0.972212i	$-0.424784\pi$
$571$	11.1881	0.468208	0.234104	+	0.972212i	$0.424784\pi$
$572$	6.88547	0.287896
$573$	−19.7960	−0.826991
$574$	−14.5343	−0.606650
$575$	0	0
$576$	−0.983004	−0.0409585
$577$	0.256987	0.0106985	0.00534924	−	0.999986i	$-0.498297\pi$
$577$	0.256987	0.0106985	0.00534924	+	0.999986i	$0.498297\pi$
$578$	11.8873	0.494446
$579$	19.7940	0.822609
$580$	0	0
$581$	8.23061	0.341463
$582$	−11.3899	−0.472126
$583$	75.5057	3.12713
$584$	27.8248	1.15140
$585$	0	0
$586$	17.9548	0.741706
$587$	2.50510	0.103397	0.0516983	−	0.998663i	$-0.483537\pi$
$587$	2.50510	0.103397	0.0516983	+	0.998663i	$0.483537\pi$
$588$	−1.52472	−0.0628785
$589$	0	0
$590$	0	0
$591$	2.35585	0.0969068
$592$	−0.874504	−0.0359419
$593$	−27.9054	−1.14594	−0.572969	−	0.819577i	$-0.694209\pi$
$593$	−27.9054	−1.14594	−0.572969	+	0.819577i	$0.694209\pi$
$594$	23.3018	0.956083
$595$	0	0
$596$	11.4375	0.468499
$597$	24.5901	1.00641
$598$	2.77148	0.113334
$599$	33.3364	1.36209	0.681045	−	0.732242i	$-0.261526\pi$
$599$	33.3364	1.36209	0.681045	+	0.732242i	$0.261526\pi$
$600$	0	0
$601$	33.2223	1.35516	0.677582	−	0.735447i	$-0.263028\pi$
$601$	33.2223	1.35516	0.677582	+	0.735447i	$0.263028\pi$
$602$	1.37493	0.0560378
$603$	7.43847	0.302918
$604$	0.248518	0.0101121
$605$	0	0
$606$	6.33929	0.257516
$607$	20.6372	0.837640	0.418820	−	0.908069i	$-0.362444\pi$
$607$	20.6372	0.837640	0.418820	+	0.908069i	$0.362444\pi$
$608$	0	0
$609$	19.1100	0.774377
$610$	0	0
$611$	8.94655	0.361938
$612$	0.518653	0.0209653
$613$	27.6480	1.11669	0.558346	−	0.829608i	$-0.311436\pi$
$613$	27.6480	1.11669	0.558346	+	0.829608i	$0.311436\pi$
$614$	13.5490	0.546791
$615$	0	0
$616$	−36.8643	−1.48531
$617$	19.4964	0.784896	0.392448	−	0.919774i	$-0.371628\pi$
$617$	19.4964	0.784896	0.392448	+	0.919774i	$0.371628\pi$
$618$	−0.517436	−0.0208143
$619$	26.9685	1.08396	0.541978	−	0.840392i	$-0.317675\pi$
$619$	26.9685	1.08396	0.541978	+	0.840392i	$0.317675\pi$
$620$	0	0
$621$	−27.0937	−1.08723
$622$	−10.0094	−0.401338
$623$	2.70770	0.108482
$624$	−1.46478	−0.0586382
$625$	0	0
$626$	−0.152486	−0.00609456
$627$	0	0
$628$	22.1159	0.882522
$629$	−0.483346	−0.0192723
$630$	0	0
$631$	−0.601618	−0.0239500	−0.0119750	−	0.999928i	$-0.503812\pi$
$631$	−0.601618	−0.0239500	−0.0119750	+	0.999928i	$0.503812\pi$
$632$	39.4137	1.56779
$633$	−27.9670	−1.11159
$634$	−5.38070	−0.213695
$635$	0	0
$636$	−30.0809	−1.19279
$637$	−0.517597	−0.0205079
$638$	−20.2877	−0.803198
$639$	3.24229	0.128263
$640$	0	0
$641$	7.42208	0.293154	0.146577	−	0.989199i	$-0.453174\pi$
$641$	7.42208	0.293154	0.146577	+	0.989199i	$0.453174\pi$
$642$	5.75377	0.227083
$643$	6.20548	0.244720	0.122360	−	0.992486i	$-0.460954\pi$
$643$	6.20548	0.244720	0.122360	+	0.992486i	$0.460954\pi$
$644$	18.2694	0.719915
$645$	0	0
$646$	0	0
$647$	24.3051	0.955530	0.477765	−	0.878488i	$-0.341447\pi$
$647$	24.3051	0.955530	0.477765	+	0.878488i	$0.341447\pi$
$648$	−17.7614	−0.697733
$649$	12.5611	0.493067
$650$	0	0
$651$	26.6494	1.04447
$652$	27.8590	1.09104
$653$	−1.38625	−0.0542483	−0.0271242	−	0.999632i	$-0.508635\pi$
$653$	−1.38625	−0.0542483	−0.0271242	+	0.999632i	$0.508635\pi$
$654$	−11.4305	−0.446966
$655$	0	0
$656$	9.48214	0.370215
$657$	−5.96486	−0.232711
$658$	−20.4158	−0.795892
$659$	−30.4152	−1.18481	−0.592405	−	0.805641i	$-0.701821\pi$
$659$	−30.4152	−1.18481	−0.592405	+	0.805641i	$0.701821\pi$
$660$	0	0
$661$	−18.4422	−0.717317	−0.358659	−	0.933469i	$-0.616766\pi$
$661$	−18.4422	−0.717317	−0.358659	+	0.933469i	$0.616766\pi$
$662$	−2.31366	−0.0899228
$663$	−0.809598	−0.0314422
$664$	8.16726	0.316951
$665$	0	0
$666$	−0.285143	−0.0110490
$667$	23.5891	0.913374
$668$	16.6081	0.642587
$669$	28.9286	1.11844
$670$	0	0
$671$	39.5675	1.52749
$672$	23.1132	0.891611
$673$	−28.6496	−1.10436	−0.552181	−	0.833724i	$-0.686204\pi$
$673$	−28.6496	−1.10436	−0.552181	+	0.833724i	$0.686204\pi$
$674$	−24.8131	−0.955767
$675$	0	0
$676$	18.3827	0.707028
$677$	5.02624	0.193174	0.0965870	−	0.995325i	$-0.469207\pi$
$677$	5.02624	0.193174	0.0965870	+	0.995325i	$0.469207\pi$
$678$	−6.31549	−0.242545
$679$	−25.4878	−0.978133
$680$	0	0
$681$	−6.49624	−0.248936
$682$	−28.2918	−1.08335
$683$	−12.9236	−0.494509	−0.247254	−	0.968951i	$-0.579528\pi$
$683$	−12.9236	−0.494509	−0.247254	+	0.968951i	$0.579528\pi$
$684$	0	0
$685$	0	0
$686$	13.8276	0.527940
$687$	2.15195	0.0821020
$688$	−0.896998	−0.0341977
$689$	−10.2116	−0.389030
$690$	0	0
$691$	24.0901	0.916432	0.458216	−	0.888841i	$-0.348489\pi$
$691$	24.0901	0.916432	0.458216	+	0.888841i	$0.348489\pi$
$692$	7.57544	0.287975
$693$	7.90268	0.300198
$694$	−0.0608371	−0.00230934
$695$	0	0
$696$	18.9629	0.718788
$697$	5.24086	0.198512
$698$	12.7718	0.483421
$699$	−1.59238	−0.0602292
$700$	0	0
$701$	−6.98543	−0.263836	−0.131918	−	0.991261i	$-0.542114\pi$
$701$	−6.98543	−0.263836	−0.131918	+	0.991261i	$0.542114\pi$
$702$	−3.15138	−0.118941
$703$	0	0
$704$	−10.7382	−0.404712
$705$	0	0
$706$	−10.8527	−0.408448
$707$	14.1858	0.533512
$708$	−5.00426	−0.188072
$709$	−0.256274	−0.00962456	−0.00481228	−	0.999988i	$-0.501532\pi$
$709$	−0.256274	−0.00962456	−0.00481228	+	0.999988i	$0.501532\pi$
$710$	0	0
$711$	−8.44919	−0.316869
$712$	2.68685	0.100694
$713$	32.8957	1.23195
$714$	1.84748	0.0691404
$715$	0	0
$716$	−0.705608	−0.0263698
$717$	−8.70487	−0.325089
$718$	−6.45456	−0.240882
$719$	18.5286	0.690999	0.345500	−	0.938419i	$-0.387709\pi$
$719$	18.5286	0.690999	0.345500	+	0.938419i	$0.387709\pi$
$720$	0	0
$721$	−1.15790	−0.0431224
$722$	0	0
$723$	39.2559	1.45994
$724$	2.10059	0.0780679
$725$	0	0
$726$	26.1945	0.972170
$727$	−35.1390	−1.30323	−0.651617	−	0.758548i	$-0.725909\pi$
$727$	−35.1390	−1.30323	−0.651617	+	0.758548i	$0.725909\pi$
$728$	4.98561	0.184779
$729$	29.9460	1.10911
$730$	0	0
$731$	−0.495778	−0.0183370
$732$	−15.7634	−0.582633
$733$	34.9872	1.29228	0.646142	−	0.763217i	$-0.276382\pi$
$733$	34.9872	1.29228	0.646142	+	0.763217i	$0.276382\pi$
$734$	4.20299	0.155135
$735$	0	0
$736$	28.5306	1.05165
$737$	81.2569	2.99314
$738$	3.09176	0.113809
$739$	−4.73129	−0.174043	−0.0870216	−	0.996206i	$-0.527735\pi$
$739$	−4.73129	−0.174043	−0.0870216	+	0.996206i	$0.527735\pi$
$740$	0	0
$741$	0	0
$742$	23.3026	0.855464
$743$	−6.77084	−0.248398	−0.124199	−	0.992257i	$-0.539636\pi$
$743$	−6.77084	−0.248398	−0.124199	+	0.992257i	$0.539636\pi$
$744$	26.4443	0.969496
$745$	0	0
$746$	−17.6584	−0.646519
$747$	−1.75083	−0.0640596
$748$	5.66570	0.207159
$749$	12.8756	0.470463
$750$	0	0
$751$	2.77014	0.101084	0.0505419	−	0.998722i	$-0.483905\pi$
$751$	2.77014	0.101084	0.0505419	+	0.998722i	$0.483905\pi$
$752$	13.3192	0.485702
$753$	−17.2026	−0.626897
$754$	2.74376	0.0999217
$755$	0	0
$756$	−20.7737	−0.755532
$757$	1.03606	0.0376561	0.0188280	−	0.999823i	$-0.494006\pi$
$757$	1.03606	0.0376561	0.0188280	+	0.999823i	$0.494006\pi$
$758$	−1.92021	−0.0697451
$759$	−44.8556	−1.62815
$760$	0	0
$761$	32.4025	1.17459	0.587296	−	0.809373i	$-0.300193\pi$
$761$	32.4025	1.17459	0.587296	+	0.809373i	$0.300193\pi$
$762$	−12.2267	−0.442926
$763$	−25.5786	−0.926008
$764$	−18.7359	−0.677842
$765$	0	0
$766$	−19.9717	−0.721605
$767$	−1.69879	−0.0613399
$768$	17.4377	0.629230
$769$	48.0188	1.73160	0.865801	−	0.500388i	$-0.166809\pi$
$769$	48.0188	1.73160	0.865801	+	0.500388i	$0.166809\pi$
$770$	0	0
$771$	23.5925	0.849665
$772$	18.7340	0.674250
$773$	10.6208	0.382004	0.191002	−	0.981590i	$-0.438826\pi$
$773$	10.6208	0.382004	0.191002	+	0.981590i	$0.438826\pi$
$774$	−0.292477	−0.0105129
$775$	0	0
$776$	−25.2916	−0.907917
$777$	2.93405	0.105258
$778$	−10.6543	−0.381976
$779$	0	0
$780$	0	0
$781$	35.4184	1.26737
$782$	2.28051	0.0815508
$783$	−26.8226	−0.958562
$784$	−0.770576	−0.0275206
$785$	0	0
$786$	18.4043	0.656461
$787$	−26.1251	−0.931259	−0.465629	−	0.884980i	$-0.654172\pi$
$787$	−26.1251	−0.931259	−0.465629	+	0.884980i	$0.654172\pi$
$788$	2.22969	0.0794295
$789$	−23.3436	−0.831055
$790$	0	0
$791$	−14.1326	−0.502496
$792$	7.84185	0.278648
$793$	−5.35120	−0.190027
$794$	7.97400	0.282987
$795$	0	0
$796$	23.2733	0.824899
$797$	−23.4923	−0.832139	−0.416069	−	0.909333i	$-0.636593\pi$
$797$	−23.4923	−0.832139	−0.416069	+	0.909333i	$0.636593\pi$
$798$	0	0
$799$	7.36165	0.260437
$800$	0	0
$801$	−0.575987	−0.0203515
$802$	20.4038	0.720483
$803$	−65.1594	−2.29943
$804$	−32.3722	−1.14168
$805$	0	0
$806$	3.82624	0.134774
$807$	−5.25091	−0.184841
$808$	14.0766	0.495214
$809$	18.6786	0.656706	0.328353	−	0.944555i	$-0.393506\pi$
$809$	18.6786	0.656706	0.328353	+	0.944555i	$0.393506\pi$
$810$	0	0
$811$	16.1813	0.568202	0.284101	−	0.958794i	$-0.408305\pi$
$811$	16.1813	0.568202	0.284101	+	0.958794i	$0.408305\pi$
$812$	18.0866	0.634717
$813$	−14.6648	−0.514315
$814$	−3.11486	−0.109176
$815$	0	0
$816$	−1.20529	−0.0421937
$817$	0	0
$818$	11.3664	0.397416
$819$	−1.06878	−0.0373461
$820$	0	0
$821$	31.3762	1.09504	0.547519	−	0.836793i	$-0.315572\pi$
$821$	31.3762	1.09504	0.547519	+	0.836793i	$0.315572\pi$
$822$	20.6389	0.719863
$823$	−12.2385	−0.426608	−0.213304	−	0.976986i	$-0.568422\pi$
$823$	−12.2385	−0.426608	−0.213304	+	0.976986i	$0.568422\pi$
$824$	−1.14899	−0.0400268
$825$	0	0
$826$	3.87661	0.134885
$827$	−6.21619	−0.216158	−0.108079	−	0.994142i	$-0.534470\pi$
$827$	−6.21619	−0.216158	−0.108079	+	0.994142i	$0.534470\pi$
$828$	−3.88630	−0.135058
$829$	52.2819	1.81583	0.907913	−	0.419158i	$-0.137675\pi$
$829$	52.2819	1.81583	0.907913	+	0.419158i	$0.137675\pi$
$830$	0	0
$831$	4.38235	0.152022
$832$	1.45226	0.0503481
$833$	−0.425904	−0.0147567
$834$	−10.5524	−0.365398
$835$	0	0
$836$	0	0
$837$	−37.4049	−1.29290
$838$	7.33651	0.253436
$839$	−44.5398	−1.53768	−0.768842	−	0.639439i	$-0.779167\pi$
$839$	−44.5398	−1.53768	−0.768842	+	0.639439i	$0.779167\pi$
$840$	0	0
$841$	−5.64685	−0.194719
$842$	−22.3470	−0.770129
$843$	−43.5660	−1.50049
$844$	−26.4693	−0.911112
$845$	0	0
$846$	4.34290	0.149312
$847$	58.6171	2.01411
$848$	−15.2025	−0.522057
$849$	−1.78399	−0.0612265
$850$	0	0
$851$	3.62174	0.124152
$852$	−14.1104	−0.483415
$853$	40.0057	1.36977	0.684885	−	0.728651i	$-0.259852\pi$
$853$	40.0057	1.36977	0.684885	+	0.728651i	$0.259852\pi$
$854$	12.2113	0.417863
$855$	0	0
$856$	12.7765	0.436690
$857$	−27.8347	−0.950817	−0.475408	−	0.879765i	$-0.657700\pi$
$857$	−27.8347	−0.950817	−0.475408	+	0.879765i	$0.657700\pi$
$858$	−5.21735	−0.178117
$859$	−1.48811	−0.0507737	−0.0253869	−	0.999678i	$-0.508082\pi$
$859$	−1.48811	−0.0507737	−0.0253869	+	0.999678i	$0.508082\pi$
$860$	0	0
$861$	−31.8135	−1.08420
$862$	6.88991	0.234671
$863$	13.3429	0.454196	0.227098	−	0.973872i	$-0.427076\pi$
$863$	13.3429	0.454196	0.227098	+	0.973872i	$0.427076\pi$
$864$	−32.4415	−1.10368
$865$	0	0
$866$	−14.9498	−0.508015
$867$	26.0196	0.883671
$868$	25.2223	0.856102
$869$	−92.2979	−3.13099
$870$	0	0
$871$	−10.9894	−0.372361
$872$	−25.3817	−0.859534
$873$	5.42182	0.183501
$874$	0	0
$875$	0	0
$876$	25.9590	0.877074
$877$	−0.167258	−0.00564790	−0.00282395	−	0.999996i	$-0.500899\pi$
$877$	−0.167258	−0.00564790	−0.00282395	+	0.999996i	$0.500899\pi$
$878$	−0.796928	−0.0268950
$879$	39.3005	1.32557
$880$	0	0
$881$	−28.5156	−0.960715	−0.480357	−	0.877073i	$-0.659493\pi$
$881$	−28.5156	−0.960715	−0.480357	+	0.877073i	$0.659493\pi$
$882$	−0.251255	−0.00846021
$883$	−34.8335	−1.17224	−0.586121	−	0.810224i	$-0.699346\pi$
$883$	−34.8335	−1.17224	−0.586121	+	0.810224i	$0.699346\pi$
$884$	−0.766242	−0.0257715
$885$	0	0
$886$	−20.6173	−0.692652
$887$	25.7718	0.865332	0.432666	−	0.901554i	$-0.357573\pi$
$887$	25.7718	0.865332	0.432666	+	0.901554i	$0.357573\pi$
$888$	2.91146	0.0977023
$889$	−27.3604	−0.917638
$890$	0	0
$891$	41.5932	1.39342
$892$	27.3794	0.916731
$893$	0	0
$894$	−8.66659	−0.289854
$895$	0	0
$896$	26.1342	0.873083
$897$	6.06637	0.202550
$898$	22.9271	0.765088
$899$	32.5666	1.08616
$900$	0	0
$901$	−8.40257	−0.279930
$902$	33.7741	1.12455
$903$	3.00951	0.100150
$904$	−14.0238	−0.466424
$905$	0	0
$906$	−0.188310	−0.00625619
$907$	−23.4162	−0.777521	−0.388761	−	0.921339i	$-0.627097\pi$
$907$	−23.4162	−0.777521	−0.388761	+	0.921339i	$0.627097\pi$
$908$	−6.14835	−0.204040
$909$	−3.01763	−0.100089
$910$	0	0
$911$	−7.45626	−0.247037	−0.123518	−	0.992342i	$-0.539418\pi$
$911$	−7.45626	−0.247037	−0.123518	+	0.992342i	$0.539418\pi$
$912$	0	0
$913$	−19.1259	−0.632974
$914$	−8.60834	−0.284739
$915$	0	0
$916$	2.03671	0.0672948
$917$	41.1845	1.36003
$918$	−2.59311	−0.0855854
$919$	40.0009	1.31951	0.659754	−	0.751481i	$-0.270660\pi$
$919$	40.0009	1.31951	0.659754	+	0.751481i	$0.270660\pi$
$920$	0	0
$921$	29.6567	0.977223
$922$	24.3655	0.802434
$923$	−4.79006	−0.157667
$924$	−34.3924	−1.13143
$925$	0	0
$926$	−17.2838	−0.567981
$927$	0.246310	0.00808989
$928$	28.2452	0.927194
$929$	−20.1755	−0.661936	−0.330968	−	0.943642i	$-0.607375\pi$
$929$	−20.1755	−0.661936	−0.330968	+	0.943642i	$0.607375\pi$
$930$	0	0
$931$	0	0
$932$	−1.50710	−0.0493667
$933$	−21.9090	−0.717270
$934$	0.445424	0.0145747
$935$	0	0
$936$	−1.06055	−0.0346652
$937$	34.1398	1.11530	0.557649	−	0.830077i	$-0.311703\pi$
$937$	34.1398	1.11530	0.557649	+	0.830077i	$0.311703\pi$
$938$	25.0775	0.818810
$939$	−0.333770	−0.0108922
$940$	0	0
$941$	26.7549	0.872184	0.436092	−	0.899902i	$-0.356362\pi$
$941$	26.7549	0.872184	0.436092	+	0.899902i	$0.356362\pi$
$942$	−16.7580	−0.546005
$943$	−39.2701	−1.27881
$944$	−2.52909	−0.0823149
$945$	0	0
$946$	−3.19498	−0.103878
$947$	15.6563	0.508763	0.254381	−	0.967104i	$-0.418128\pi$
$947$	15.6563	0.508763	0.254381	+	0.967104i	$0.418128\pi$
$948$	36.7708	1.19426
$949$	8.81230	0.286060
$950$	0	0
$951$	−11.7776	−0.381915
$952$	4.10241	0.132960
$953$	−13.5072	−0.437541	−0.218771	−	0.975776i	$-0.570205\pi$
$953$	−13.5072	−0.437541	−0.218771	+	0.975776i	$0.570205\pi$
$954$	−4.95697	−0.160488
$955$	0	0
$956$	−8.23871	−0.266459
$957$	−44.4069	−1.43547
$958$	−22.2844	−0.719977
$959$	46.1848	1.49139
$960$	0	0
$961$	14.4151	0.465003
$962$	0.421261	0.0135820
$963$	−2.73892	−0.0882603
$964$	37.1537	1.19664
$965$	0	0
$966$	−13.8433	−0.445402
$967$	−21.9060	−0.704450	−0.352225	−	0.935915i	$-0.614575\pi$
$967$	−21.9060	−0.704450	−0.352225	+	0.935915i	$0.614575\pi$
$968$	58.1659	1.86952
$969$	0	0
$970$	0	0
$971$	−30.9663	−0.993756	−0.496878	−	0.867821i	$-0.665520\pi$
$971$	−30.9663	−0.993756	−0.496878	+	0.867821i	$0.665520\pi$
$972$	8.16832	0.261999
$973$	−23.6137	−0.757019
$974$	−26.6929	−0.855297
$975$	0	0
$976$	−7.96664	−0.255006
$977$	9.89768	0.316655	0.158327	−	0.987387i	$-0.449390\pi$
$977$	9.89768	0.316655	0.158327	+	0.987387i	$0.449390\pi$
$978$	−21.1097	−0.675014
$979$	−6.29201	−0.201093
$980$	0	0
$981$	5.44113	0.173722
$982$	−27.6459	−0.882215
$983$	−10.3096	−0.328827	−0.164413	−	0.986392i	$-0.552573\pi$
$983$	−10.3096	−0.328827	−0.164413	+	0.986392i	$0.552573\pi$
$984$	−31.5686	−1.00637
$985$	0	0
$986$	2.25770	0.0718997
$987$	−44.6873	−1.42241
$988$	0	0
$989$	3.71490	0.118127
$990$	0	0
$991$	20.4090	0.648314	0.324157	−	0.946003i	$-0.394919\pi$
$991$	20.4090	0.648314	0.324157	+	0.946003i	$0.394919\pi$
$992$	39.3887	1.25059
$993$	−5.06426	−0.160710
$994$	10.9308	0.346704
$995$	0	0
$996$	7.61961	0.241437
$997$	47.0764	1.49092	0.745462	−	0.666549i	$-0.232229\pi$
$997$	47.0764	1.49092	0.745462	+	0.666549i	$0.232229\pi$
$998$	3.86394	0.122311
$999$	−4.11820	−0.130294

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.cv.1.18		40
5.2	odd	4		1805.2.b.m.1084.18	yes	40
5.3	odd	4		1805.2.b.m.1084.23	yes	40
5.4	even	2	inner	9025.2.a.cv.1.23		40
19.18	odd	2	inner	9025.2.a.cv.1.24		40
95.18	even	4		1805.2.b.m.1084.17	✓	40
95.37	even	4		1805.2.b.m.1084.24	yes	40
95.94	odd	2	inner	9025.2.a.cv.1.17		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.b.m.1084.17	✓	40	95.18	even	4
1805.2.b.m.1084.18	yes	40	5.2	odd	4
1805.2.b.m.1084.23	yes	40	5.3	odd	4
1805.2.b.m.1084.24	yes	40	95.37	even	4
9025.2.a.cv.1.17		40	95.94	odd	2	inner
9025.2.a.cv.1.18		40	1.1	even	1	trivial
9025.2.a.cv.1.23		40	5.4	even	2	inner
9025.2.a.cv.1.24		40	19.18	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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.717155 q^{2} -1.56975 q^{3} -1.48569 q^{4} +1.12576 q^{6} +2.51917 q^{7} +2.49978 q^{8} -0.535883 q^{9} -5.85392 q^{11} +2.33216 q^{12} +0.791698 q^{13} -1.80664 q^{14} +1.17865 q^{16} +0.651447 q^{17} +0.384311 q^{18} -3.95447 q^{21} +4.19817 q^{22} -4.88134 q^{23} -3.92403 q^{24} -0.567770 q^{26} +5.55045 q^{27} -3.74270 q^{28} -4.83251 q^{29} -6.73907 q^{31} -5.84483 q^{32} +9.18920 q^{33} -0.467189 q^{34} +0.796155 q^{36} -0.741957 q^{37} -1.24277 q^{39} +8.04494 q^{41} +2.83597 q^{42} -0.761041 q^{43} +8.69710 q^{44} +3.50068 q^{46} +11.3005 q^{47} -1.85018 q^{48} -0.653781 q^{49} -1.02261 q^{51} -1.17622 q^{52} -12.8983 q^{53} -3.98054 q^{54} +6.29737 q^{56} +3.46566 q^{58} -2.14576 q^{59} -6.75915 q^{61} +4.83296 q^{62} -1.34998 q^{63} +1.83436 q^{64} -6.59008 q^{66} -13.8808 q^{67} -0.967847 q^{68} +7.66249 q^{69} -6.05037 q^{71} -1.33959 q^{72} +11.1309 q^{73} +0.532098 q^{74} -14.7470 q^{77} +0.891258 q^{78} +15.7669 q^{79} -7.10518 q^{81} -5.76948 q^{82} +3.26719 q^{83} +5.87511 q^{84} +0.545785 q^{86} +7.58584 q^{87} -14.6335 q^{88} +1.07484 q^{89} +1.99442 q^{91} +7.25215 q^{92} +10.5787 q^{93} -8.10419 q^{94} +9.17493 q^{96} -10.1175 q^{97} +0.468863 q^{98} +3.13702 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 48 q^{4} + 20 q^{6} + 52 q^{9} + 20 q^{11} + 40 q^{16} + 92 q^{24} + 76 q^{26} + 156 q^{36} + 80 q^{39} + 48 q^{44} + 72 q^{49} + 32 q^{54} + 80 q^{61} + 72 q^{64} + 16 q^{66} + 100 q^{74} + 40 q^{81}+ \cdots + 128 q^{99}+O(q^{100}) \) 40 * q + 48 * q^4 + 20 * q^6 + 52 * q^9 + 20 * q^11 + 40 * q^16 + 92 * q^24 + 76 * q^26 + 156 * q^36 + 80 * q^39 + 48 * q^44 + 72 * q^49 + 32 * q^54 + 80 * q^61 + 72 * q^64 + 16 * q^66 + 100 * q^74 + 40 * q^81 + 380 * q^96 + 128 * q^99

Introduction

L-functions

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