LMFDB - Embedded newform 5625.2.a.bb.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, 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("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.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:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45 $ x^8 - 15x^6 + 70x^4 - 105*x^2 + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.856773$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.856773 q^{2} -1.26594 q^{4} -0.647907 q^{7} -2.79817 q^{8} +O(q^{10})$	$q+0.856773 q^{2} -1.26594 q^{4} -0.647907 q^{7} -2.79817 q^{8} +5.91382 q^{11} -2.88397 q^{13} -0.555109 q^{14} +0.134488 q^{16} +4.20027 q^{17} -1.64791 q^{19} +5.06680 q^{22} +6.42752 q^{23} -2.47091 q^{26} +0.820212 q^{28} -2.25888 q^{29} -5.28440 q^{31} +5.71156 q^{32} +3.59868 q^{34} +1.71560 q^{37} -1.41188 q^{38} +0.176662 q^{41} -7.95077 q^{43} -7.48654 q^{44} +5.50692 q^{46} +1.05903 q^{47} -6.58022 q^{49} +3.65094 q^{52} -4.48612 q^{53} +1.81295 q^{56} -1.93534 q^{58} -9.74542 q^{59} -11.2844 q^{61} -4.52753 q^{62} +4.62453 q^{64} +12.6171 q^{67} -5.31730 q^{68} +6.09048 q^{71} +7.08312 q^{73} +1.46988 q^{74} +2.08615 q^{76} -3.83160 q^{77} +1.58111 q^{79} +0.151359 q^{82} +11.5102 q^{83} -6.81200 q^{86} -16.5479 q^{88} +15.5918 q^{89} +1.86855 q^{91} -8.13685 q^{92} +0.907347 q^{94} +7.55034 q^{97} -5.63775 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{4} + 10 q^{7}+O(q^{10}) $ 8 * q + 14 * q^4 + 10 * q^7	$ 8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} + 22 q^{16} + 2 q^{19} + 10 q^{22} + 70 q^{28} - 6 q^{31} - 50 q^{34} + 50 q^{37} + 14 q^{49} + 80 q^{52} - 30 q^{58} - 54 q^{61} + 36 q^{64} + 10 q^{67} + 30 q^{73} + 56 q^{76} + 28 q^{79} + 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100}) $ 8 * q + 14 * q^4 + 10 * q^7 + 10 * q^13 + 22 * q^16 + 2 * q^19 + 10 * q^22 + 70 * q^28 - 6 * q^31 - 50 * q^34 + 50 * q^37 + 14 * q^49 + 80 * q^52 - 30 * q^58 - 54 * q^61 + 36 * q^64 + 10 * q^67 + 30 * q^73 + 56 * q^76 + 28 * q^79 + 20 * q^88 + 60 * q^91 - 40 * q^94

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.856773	0.605830	0.302915	−	0.953018i	$-0.402040\pi$
$2$	0.856773	0.605830	0.302915	+	0.953018i	$0.402040\pi$
$3$	0	0
$3$	0	0
$4$	−1.26594	−0.632970
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.647907	−0.244886	−0.122443	−	0.992476i	$-0.539073\pi$
$7$	−0.647907	−0.244886	−0.122443	+	0.992476i	$0.539073\pi$
$8$	−2.79817	−0.989302
$9$	0	0
$10$	0	0
$11$	5.91382	1.78308	0.891542	−	0.452939i	$-0.149624\pi$
$11$	5.91382	1.78308	0.891542	+	0.452939i	$0.149624\pi$
$12$	0	0
$13$	−2.88397	−0.799871	−0.399935	−	0.916543i	$-0.630967\pi$
$13$	−2.88397	−0.799871	−0.399935	+	0.916543i	$0.630967\pi$
$14$	−0.555109	−0.148359
$15$	0	0
$16$	0.134488	0.0336219
$17$	4.20027	1.01872	0.509358	−	0.860555i	$-0.329883\pi$
$17$	4.20027	1.01872	0.509358	+	0.860555i	$0.329883\pi$
$18$	0	0
$19$	−1.64791	−0.378056	−0.189028	−	0.981972i	$-0.560534\pi$
$19$	−1.64791	−0.378056	−0.189028	+	0.981972i	$0.560534\pi$
$20$	0	0
$21$	0	0
$22$	5.06680	1.08024
$23$	6.42752	1.34023	0.670115	−	0.742257i	$-0.266245\pi$
$23$	6.42752	1.34023	0.670115	+	0.742257i	$0.266245\pi$
$24$	0	0
$25$	0	0
$26$	−2.47091	−0.484585
$27$	0	0
$28$	0.820212	0.155005
$29$	−2.25888	−0.419463	−0.209732	−	0.977759i	$-0.567259\pi$
$29$	−2.25888	−0.419463	−0.209732	+	0.977759i	$0.567259\pi$
$30$	0	0
$31$	−5.28440	−0.949107	−0.474553	−	0.880227i	$-0.657390\pi$
$31$	−5.28440	−0.949107	−0.474553	+	0.880227i	$0.657390\pi$
$32$	5.71156	1.00967
$33$	0	0
$34$	3.59868	0.617168
$35$	0	0
$36$	0	0
$37$	1.71560	0.282042	0.141021	−	0.990007i	$-0.454961\pi$
$37$	1.71560	0.282042	0.141021	+	0.990007i	$0.454961\pi$
$38$	−1.41188	−0.229037
$39$	0	0
$40$	0	0
$41$	0.176662	0.0275900	0.0137950	−	0.999905i	$-0.495609\pi$
$41$	0.176662	0.0275900	0.0137950	+	0.999905i	$0.495609\pi$
$42$	0	0
$43$	−7.95077	−1.21248	−0.606241	−	0.795281i	$-0.707323\pi$
$43$	−7.95077	−1.21248	−0.606241	+	0.795281i	$0.707323\pi$
$44$	−7.48654	−1.12864
$45$	0	0
$46$	5.50692	0.811951
$47$	1.05903	0.154475	0.0772376	−	0.997013i	$-0.475390\pi$
$47$	1.05903	0.154475	0.0772376	+	0.997013i	$0.475390\pi$
$48$	0	0
$49$	−6.58022	−0.940031
$50$	0	0
$51$	0	0
$52$	3.65094	0.506294
$53$	−4.48612	−0.616216	−0.308108	−	0.951351i	$-0.599696\pi$
$53$	−4.48612	−0.616216	−0.308108	+	0.951351i	$0.599696\pi$
$54$	0	0
$55$	0	0
$56$	1.81295	0.242266
$57$	0	0
$58$	−1.93534	−0.254123
$59$	−9.74542	−1.26875	−0.634373	−	0.773027i	$-0.718742\pi$
$59$	−9.74542	−1.26875	−0.634373	+	0.773027i	$0.718742\pi$
$60$	0	0
$61$	−11.2844	−1.44482	−0.722410	−	0.691465i	$-0.756966\pi$
$61$	−11.2844	−1.44482	−0.722410	+	0.691465i	$0.756966\pi$
$62$	−4.52753	−0.574997
$63$	0	0
$64$	4.62453	0.578067
$65$	0	0
$66$	0	0
$67$	12.6171	1.54143	0.770715	−	0.637181i	$-0.219899\pi$
$67$	12.6171	1.54143	0.770715	+	0.637181i	$0.219899\pi$
$68$	−5.31730	−0.644817
$69$	0	0
$70$	0	0
$71$	6.09048	0.722807	0.361404	−	0.932409i	$-0.382298\pi$
$71$	6.09048	0.722807	0.361404	+	0.932409i	$0.382298\pi$
$72$	0	0
$73$	7.08312	0.829016	0.414508	−	0.910046i	$-0.363954\pi$
$73$	7.08312	0.829016	0.414508	+	0.910046i	$0.363954\pi$
$74$	1.46988	0.170870
$75$	0	0
$76$	2.08615	0.239298
$77$	−3.83160	−0.436652
$78$	0	0
$79$	1.58111	0.177889	0.0889443	−	0.996037i	$-0.471651\pi$
$79$	1.58111	0.177889	0.0889443	+	0.996037i	$0.471651\pi$
$80$	0	0
$81$	0	0
$82$	0.151359	0.0167149
$83$	11.5102	1.26340	0.631702	−	0.775211i	$-0.282357\pi$
$83$	11.5102	1.26340	0.631702	+	0.775211i	$0.282357\pi$
$84$	0	0
$85$	0	0
$86$	−6.81200	−0.734557
$87$	0	0
$88$	−16.5479	−1.76401
$89$	15.5918	1.65272	0.826362	−	0.563140i	$-0.190407\pi$
$89$	15.5918	1.65272	0.826362	+	0.563140i	$0.190407\pi$
$90$	0	0
$91$	1.86855	0.195877
$92$	−8.13685	−0.848326
$93$	0	0
$94$	0.907347	0.0935857
$95$	0	0
$96$	0	0
$97$	7.55034	0.766621	0.383311	−	0.923619i	$-0.374784\pi$
$97$	7.55034	0.766621	0.383311	+	0.923619i	$0.374784\pi$
$98$	−5.63775	−0.569499
$99$	0	0
$100$	0	0
$101$	17.1645	1.70793	0.853965	−	0.520330i	$-0.174191\pi$
$101$	17.1645	1.70793	0.853965	+	0.520330i	$0.174191\pi$
$102$	0	0
$103$	17.3561	1.71015	0.855074	−	0.518506i	$-0.173511\pi$
$103$	17.3561	1.71015	0.855074	+	0.518506i	$0.173511\pi$
$104$	8.06985	0.791314
$105$	0	0
$106$	−3.84358	−0.373322
$107$	−16.2046	−1.56656	−0.783278	−	0.621672i	$-0.786454\pi$
$107$	−16.2046	−1.56656	−0.783278	+	0.621672i	$0.786454\pi$
$108$	0	0
$109$	7.12709	0.682652	0.341326	−	0.939945i	$-0.389124\pi$
$109$	7.12709	0.682652	0.341326	+	0.939945i	$0.389124\pi$
$110$	0	0
$111$	0	0
$112$	−0.0871355	−0.00823353
$113$	14.9372	1.40518	0.702589	−	0.711596i	$-0.252027\pi$
$113$	14.9372	1.40518	0.702589	+	0.711596i	$0.252027\pi$
$114$	0	0
$115$	0	0
$116$	2.85961	0.265508
$117$	0	0
$118$	−8.34961	−0.768644
$119$	−2.72139	−0.249469
$120$	0	0
$121$	23.9733	2.17939
$122$	−9.66817	−0.875315
$123$	0	0
$124$	6.68974	0.600757
$125$	0	0
$126$	0	0
$127$	9.30722	0.825883	0.412941	−	0.910758i	$-0.364501\pi$
$127$	9.30722	0.825883	0.412941	+	0.910758i	$0.364501\pi$
$128$	−7.46095	−0.659461
$129$	0	0
$130$	0	0
$131$	3.47828	0.303899	0.151949	−	0.988388i	$-0.451445\pi$
$131$	3.47828	0.303899	0.151949	+	0.988388i	$0.451445\pi$
$132$	0	0
$133$	1.06769	0.0925805
$134$	10.8100	0.933844
$135$	0	0
$136$	−11.7531	−1.00782
$137$	8.17270	0.698241	0.349120	−	0.937078i	$-0.386480\pi$
$137$	8.17270	0.698241	0.349120	+	0.937078i	$0.386480\pi$
$138$	0	0
$139$	2.93970	0.249342	0.124671	−	0.992198i	$-0.460212\pi$
$139$	2.93970	0.249342	0.124671	+	0.992198i	$0.460212\pi$
$140$	0	0
$141$	0	0
$142$	5.21816	0.437898
$143$	−17.0553	−1.42624
$144$	0	0
$145$	0	0
$146$	6.06862	0.502243
$147$	0	0
$148$	−2.17184	−0.178524
$149$	15.4826	1.26838	0.634191	−	0.773176i	$-0.281333\pi$
$149$	15.4826	1.26838	0.634191	+	0.773176i	$0.281333\pi$
$150$	0	0
$151$	10.8233	0.880791	0.440395	−	0.897804i	$-0.354838\pi$
$151$	10.8233	0.880791	0.440395	+	0.897804i	$0.354838\pi$
$152$	4.61112	0.374011
$153$	0	0
$154$	−3.28281	−0.264537
$155$	0	0
$156$	0	0
$157$	16.9086	1.34945	0.674726	−	0.738068i	$-0.264262\pi$
$157$	16.9086	1.34945	0.674726	+	0.738068i	$0.264262\pi$
$158$	1.35465	0.107770
$159$	0	0
$160$	0	0
$161$	−4.16443	−0.328203
$162$	0	0
$163$	−16.7750	−1.31392	−0.656960	−	0.753926i	$-0.728158\pi$
$163$	−16.7750	−1.31392	−0.656960	+	0.753926i	$0.728158\pi$
$164$	−0.223644	−0.0174637
$165$	0	0
$166$	9.86159	0.765407
$167$	−10.1916	−0.788653	−0.394326	−	0.918970i	$-0.629022\pi$
$167$	−10.1916	−0.788653	−0.394326	+	0.918970i	$0.629022\pi$
$168$	0	0
$169$	−4.68269	−0.360207
$170$	0	0
$171$	0	0
$172$	10.0652	0.767465
$173$	14.3724	1.09271	0.546355	−	0.837554i	$-0.316015\pi$
$173$	14.3724	1.09271	0.546355	+	0.837554i	$0.316015\pi$
$174$	0	0
$175$	0	0
$176$	0.795336	0.0599507
$177$	0	0
$178$	13.3586	1.00127
$179$	−7.59573	−0.567731	−0.283866	−	0.958864i	$-0.591617\pi$
$179$	−7.59573	−0.567731	−0.283866	+	0.958864i	$0.591617\pi$
$180$	0	0
$181$	−21.8203	−1.62189	−0.810945	−	0.585122i	$-0.801047\pi$
$181$	−21.8203	−1.62189	−0.810945	+	0.585122i	$0.801047\pi$
$182$	1.60092	0.118668
$183$	0	0
$184$	−17.9853	−1.32589
$185$	0	0
$186$	0	0
$187$	24.8397	1.81646
$188$	−1.34067	−0.0977783
$189$	0	0
$190$	0	0
$191$	19.0283	1.37684	0.688421	−	0.725311i	$-0.258304\pi$
$191$	19.0283	1.37684	0.688421	+	0.725311i	$0.258304\pi$
$192$	0	0
$193$	−8.57675	−0.617368	−0.308684	−	0.951165i	$-0.599889\pi$
$193$	−8.57675	−0.617368	−0.308684	+	0.951165i	$0.599889\pi$
$194$	6.46893	0.464442
$195$	0	0
$196$	8.33016	0.595012
$197$	18.5726	1.32325	0.661623	−	0.749837i	$-0.269868\pi$
$197$	18.5726	1.32325	0.661623	+	0.749837i	$0.269868\pi$
$198$	0	0
$199$	−7.32927	−0.519558	−0.259779	−	0.965668i	$-0.583650\pi$
$199$	−7.32927	−0.519558	−0.259779	+	0.965668i	$0.583650\pi$
$200$	0	0
$201$	0	0
$202$	14.7061	1.03471
$203$	1.46354	0.102721
$204$	0	0
$205$	0	0
$206$	14.8702	1.03606
$207$	0	0
$208$	−0.387859	−0.0268932
$209$	−9.74542	−0.674105
$210$	0	0
$211$	−14.5553	−1.00203	−0.501013	−	0.865440i	$-0.667039\pi$
$211$	−14.5553	−1.00203	−0.501013	+	0.865440i	$0.667039\pi$
$212$	5.67916	0.390046
$213$	0	0
$214$	−13.8836	−0.949066
$215$	0	0
$216$	0	0
$217$	3.42380	0.232423
$218$	6.10630	0.413571
$219$	0	0
$220$	0	0
$221$	−12.1135	−0.814841
$222$	0	0
$223$	26.8711	1.79942	0.899712	−	0.436485i	$-0.143777\pi$
$223$	26.8711	1.79942	0.899712	+	0.436485i	$0.143777\pi$
$224$	−3.70056	−0.247254
$225$	0	0
$226$	12.7978	0.851298
$227$	−4.20027	−0.278782	−0.139391	−	0.990237i	$-0.544514\pi$
$227$	−4.20027	−0.278782	−0.139391	+	0.990237i	$0.544514\pi$
$228$	0	0
$229$	8.30722	0.548957	0.274478	−	0.961593i	$-0.411495\pi$
$229$	8.30722	0.548957	0.274478	+	0.961593i	$0.411495\pi$
$230$	0	0
$231$	0	0
$232$	6.32072	0.414976
$233$	−22.6158	−1.48161	−0.740805	−	0.671720i	$-0.765556\pi$
$233$	−22.6158	−1.48161	−0.740805	+	0.671720i	$0.765556\pi$
$234$	0	0
$235$	0	0
$236$	12.3371	0.803079
$237$	0	0
$238$	−2.33161	−0.151136
$239$	−1.28688	−0.0832413	−0.0416207	−	0.999133i	$-0.513252\pi$
$239$	−1.28688	−0.0832413	−0.0416207	+	0.999133i	$0.513252\pi$
$240$	0	0
$241$	−4.91599	−0.316667	−0.158333	−	0.987386i	$-0.550612\pi$
$241$	−4.91599	−0.316667	−0.158333	+	0.987386i	$0.550612\pi$
$242$	20.5396	1.32034
$243$	0	0
$244$	14.2854	0.914528
$245$	0	0
$246$	0	0
$247$	4.75252	0.302396
$248$	14.7867	0.938953
$249$	0	0
$250$	0	0
$251$	−18.9609	−1.19680	−0.598399	−	0.801198i	$-0.704196\pi$
$251$	−18.9609	−1.19680	−0.598399	+	0.801198i	$0.704196\pi$
$252$	0	0
$253$	38.0112	2.38974
$254$	7.97417	0.500344
$255$	0	0
$256$	−15.6414	−0.977588
$257$	24.1690	1.50762	0.753810	−	0.657093i	$-0.228214\pi$
$257$	24.1690	1.50762	0.753810	+	0.657093i	$0.228214\pi$
$258$	0	0
$259$	−1.11155	−0.0690682
$260$	0	0
$261$	0	0
$262$	2.98009	0.184111
$263$	−17.6518	−1.08846	−0.544229	−	0.838937i	$-0.683178\pi$
$263$	−17.6518	−1.08846	−0.544229	+	0.838937i	$0.683178\pi$
$264$	0	0
$265$	0	0
$266$	0.914768	0.0560880
$267$	0	0
$268$	−15.9726	−0.975679
$269$	−26.5149	−1.61664	−0.808320	−	0.588743i	$-0.799623\pi$
$269$	−26.5149	−1.61664	−0.808320	+	0.588743i	$0.799623\pi$
$270$	0	0
$271$	20.4180	1.24031	0.620153	−	0.784481i	$-0.287071\pi$
$271$	20.4180	1.24031	0.620153	+	0.784481i	$0.287071\pi$
$272$	0.564885	0.0342512
$273$	0	0
$274$	7.00214	0.423015
$275$	0	0
$276$	0	0
$277$	−14.0643	−0.845043	−0.422521	−	0.906353i	$-0.638855\pi$
$277$	−14.0643	−0.845043	−0.422521	+	0.906353i	$0.638855\pi$
$278$	2.51866	0.151059
$279$	0	0
$280$	0	0
$281$	−6.19966	−0.369841	−0.184920	−	0.982753i	$-0.559203\pi$
$281$	−6.19966	−0.369841	−0.184920	+	0.982753i	$0.559203\pi$
$282$	0	0
$283$	12.9801	0.771586	0.385793	−	0.922585i	$-0.373928\pi$
$283$	12.9801	0.771586	0.385793	+	0.922585i	$0.373928\pi$
$284$	−7.71019	−0.457516
$285$	0	0
$286$	−14.6125	−0.864056
$287$	−0.114461	−0.00675640
$288$	0	0
$289$	0.642299	0.0377823
$290$	0	0
$291$	0	0
$292$	−8.96681	−0.524743
$293$	18.5563	1.08407	0.542037	−	0.840355i	$-0.317653\pi$
$293$	18.5563	1.08407	0.542037	+	0.840355i	$0.317653\pi$
$294$	0	0
$295$	0	0
$296$	−4.80053	−0.279025
$297$	0	0
$298$	13.2650	0.768424
$299$	−18.5368	−1.07201
$300$	0	0
$301$	5.15136	0.296919
$302$	9.27314	0.533609
$303$	0	0
$304$	−0.221623	−0.0127110
$305$	0	0
$306$	0	0
$307$	24.0862	1.37467	0.687337	−	0.726338i	$-0.258779\pi$
$307$	24.0862	1.37467	0.687337	+	0.726338i	$0.258779\pi$
$308$	4.85058	0.276388
$309$	0	0
$310$	0	0
$311$	25.3372	1.43674	0.718370	−	0.695661i	$-0.244889\pi$
$311$	25.3372	1.43674	0.718370	+	0.695661i	$0.244889\pi$
$312$	0	0
$313$	−21.8224	−1.23348	−0.616739	−	0.787168i	$-0.711546\pi$
$313$	−21.8224	−1.23348	−0.616739	+	0.787168i	$0.711546\pi$
$314$	14.4868	0.817539
$315$	0	0
$316$	−2.00159	−0.112598
$317$	1.21940	0.0684884	0.0342442	−	0.999413i	$-0.489098\pi$
$317$	1.21940	0.0684884	0.0342442	+	0.999413i	$0.489098\pi$
$318$	0	0
$319$	−13.3586	−0.747938
$320$	0	0
$321$	0	0
$322$	−3.56797	−0.198835
$323$	−6.92166	−0.385131
$324$	0	0
$325$	0	0
$326$	−14.3724	−0.796011
$327$	0	0
$328$	−0.494331	−0.0272949
$329$	−0.686152	−0.0378288
$330$	0	0
$331$	−5.97470	−0.328399	−0.164200	−	0.986427i	$-0.552504\pi$
$331$	−5.97470	−0.328399	−0.164200	+	0.986427i	$0.552504\pi$
$332$	−14.5712	−0.799697
$333$	0	0
$334$	−8.73192	−0.477789
$335$	0	0
$336$	0	0
$337$	27.1718	1.48014	0.740072	−	0.672527i	$-0.234791\pi$
$337$	27.1718	1.48014	0.740072	+	0.672527i	$0.234791\pi$
$338$	−4.01200	−0.218224
$339$	0	0
$340$	0	0
$341$	−31.2510	−1.69234
$342$	0	0
$343$	8.79871	0.475086
$344$	22.2476	1.19951
$345$	0	0
$346$	12.3138	0.661996
$347$	−12.4242	−0.666964	−0.333482	−	0.942757i	$-0.608224\pi$
$347$	−12.4242	−0.666964	−0.333482	+	0.942757i	$0.608224\pi$
$348$	0	0
$349$	20.2826	1.08570	0.542852	−	0.839828i	$-0.317345\pi$
$349$	20.2826	1.08570	0.542852	+	0.839828i	$0.317345\pi$
$350$	0	0
$351$	0	0
$352$	33.7771	1.80033
$353$	−11.9880	−0.638057	−0.319029	−	0.947745i	$-0.603357\pi$
$353$	−11.9880	−0.638057	−0.319029	+	0.947745i	$0.603357\pi$
$354$	0	0
$355$	0	0
$356$	−19.7382	−1.04613
$357$	0	0
$358$	−6.50781	−0.343949
$359$	3.54576	0.187138	0.0935690	−	0.995613i	$-0.470172\pi$
$359$	3.54576	0.187138	0.0935690	+	0.995613i	$0.470172\pi$
$360$	0	0
$361$	−16.2844	−0.857074
$362$	−18.6950	−0.982589
$363$	0	0
$364$	−2.36547	−0.123984
$365$	0	0
$366$	0	0
$367$	31.3333	1.63558	0.817792	−	0.575514i	$-0.195198\pi$
$367$	31.3333	1.63558	0.817792	+	0.575514i	$0.195198\pi$
$368$	0.864421	0.0450611
$369$	0	0
$370$	0	0
$371$	2.90659	0.150902
$372$	0	0
$373$	−0.133595	−0.00691730	−0.00345865	−	0.999994i	$-0.501101\pi$
$373$	−0.133595	−0.00691730	−0.00345865	+	0.999994i	$0.501101\pi$
$374$	21.2819	1.10046
$375$	0	0
$376$	−2.96334	−0.152823
$377$	6.51455	0.335516
$378$	0	0
$379$	−8.04343	−0.413163	−0.206582	−	0.978429i	$-0.566234\pi$
$379$	−8.04343	−0.413163	−0.206582	+	0.978429i	$0.566234\pi$
$380$	0	0
$381$	0	0
$382$	16.3030	0.834132
$383$	12.2322	0.625034	0.312517	−	0.949912i	$-0.398828\pi$
$383$	12.2322	0.625034	0.312517	+	0.949912i	$0.398828\pi$
$384$	0	0
$385$	0	0
$386$	−7.34832	−0.374020
$387$	0	0
$388$	−9.55829	−0.485249
$389$	−33.0004	−1.67319	−0.836593	−	0.547825i	$-0.815456\pi$
$389$	−33.0004	−1.67319	−0.836593	+	0.547825i	$0.815456\pi$
$390$	0	0
$391$	26.9973	1.36531
$392$	18.4126	0.929974
$393$	0	0
$394$	15.9125	0.801661
$395$	0	0
$396$	0	0
$397$	−13.9249	−0.698872	−0.349436	−	0.936960i	$-0.613627\pi$
$397$	−13.9249	−0.698872	−0.349436	+	0.936960i	$0.613627\pi$
$398$	−6.27952	−0.314764
$399$	0	0
$400$	0	0
$401$	27.9494	1.39573	0.697863	−	0.716231i	$-0.254134\pi$
$401$	27.9494	1.39573	0.697863	+	0.716231i	$0.254134\pi$
$402$	0	0
$403$	15.2401	0.759163
$404$	−21.7292	−1.08107
$405$	0	0
$406$	1.25392	0.0622311
$407$	10.1457	0.502905
$408$	0	0
$409$	25.9289	1.28210	0.641052	−	0.767498i	$-0.278498\pi$
$409$	25.9289	1.28210	0.641052	+	0.767498i	$0.278498\pi$
$410$	0	0
$411$	0	0
$412$	−21.9718	−1.08247
$413$	6.31413	0.310698
$414$	0	0
$415$	0	0
$416$	−16.4720	−0.807606
$417$	0	0
$418$	−8.34961	−0.408393
$419$	1.21614	0.0594123	0.0297061	−	0.999559i	$-0.490543\pi$
$419$	1.21614	0.0594123	0.0297061	+	0.999559i	$0.490543\pi$
$420$	0	0
$421$	20.4626	0.997286	0.498643	−	0.866807i	$-0.333832\pi$
$421$	20.4626	0.997286	0.498643	+	0.866807i	$0.333832\pi$
$422$	−12.4705	−0.607056
$423$	0	0
$424$	12.5529	0.609624
$425$	0	0
$426$	0	0
$427$	7.31124	0.353816
$428$	20.5140	0.991583
$429$	0	0
$430$	0	0
$431$	−21.4381	−1.03264	−0.516319	−	0.856397i	$-0.672698\pi$
$431$	−21.4381	−1.03264	−0.516319	+	0.856397i	$0.672698\pi$
$432$	0	0
$433$	23.2844	1.11898	0.559489	−	0.828838i	$-0.310998\pi$
$433$	23.2844	1.11898	0.559489	+	0.828838i	$0.310998\pi$
$434$	2.93342	0.140809
$435$	0	0
$436$	−9.02248	−0.432098
$437$	−10.5919	−0.506681
$438$	0	0
$439$	−12.5355	−0.598285	−0.299143	−	0.954208i	$-0.596701\pi$
$439$	−12.5355	−0.598285	−0.299143	+	0.954208i	$0.596701\pi$
$440$	0	0
$441$	0	0
$442$	−10.3785	−0.493655
$443$	−13.3228	−0.632986	−0.316493	−	0.948595i	$-0.602505\pi$
$443$	−13.3228	−0.632986	−0.316493	+	0.948595i	$0.602505\pi$
$444$	0	0
$445$	0	0
$446$	23.0224	1.09014
$447$	0	0
$448$	−2.99627	−0.141560
$449$	−30.2500	−1.42758	−0.713792	−	0.700358i	$-0.753024\pi$
$449$	−30.2500	−1.42758	−0.713792	+	0.700358i	$0.753024\pi$
$450$	0	0
$451$	1.04475	0.0491953
$452$	−18.9097	−0.889436
$453$	0	0
$454$	−3.59868	−0.168894
$455$	0	0
$456$	0	0
$457$	−12.8391	−0.600588	−0.300294	−	0.953847i	$-0.597085\pi$
$457$	−12.8391	−0.600588	−0.300294	+	0.953847i	$0.597085\pi$
$458$	7.11740	0.332574
$459$	0	0
$460$	0	0
$461$	−27.8435	−1.29680	−0.648400	−	0.761300i	$-0.724561\pi$
$461$	−27.8435	−1.29680	−0.648400	+	0.761300i	$0.724561\pi$
$462$	0	0
$463$	3.84616	0.178746	0.0893731	−	0.995998i	$-0.471514\pi$
$463$	3.84616	0.178746	0.0893731	+	0.995998i	$0.471514\pi$
$464$	−0.303791	−0.0141031
$465$	0	0
$466$	−19.3766	−0.897603
$467$	−40.6594	−1.88149	−0.940746	−	0.339112i	$-0.889874\pi$
$467$	−40.6594	−1.88149	−0.940746	+	0.339112i	$0.889874\pi$
$468$	0	0
$469$	−8.17473	−0.377474
$470$	0	0
$471$	0	0
$472$	27.2693	1.25517
$473$	−47.0194	−2.16196
$474$	0	0
$475$	0	0
$476$	3.44511	0.157907
$477$	0	0
$478$	−1.10256	−0.0504301
$479$	15.2642	0.697440	0.348720	−	0.937227i	$-0.386616\pi$
$479$	15.2642	0.697440	0.348720	+	0.937227i	$0.386616\pi$
$480$	0	0
$481$	−4.94774	−0.225597
$482$	−4.21189	−0.191846
$483$	0	0
$484$	−30.3487	−1.37949
$485$	0	0
$486$	0	0
$487$	8.06466	0.365444	0.182722	−	0.983165i	$-0.441509\pi$
$487$	8.06466	0.365444	0.182722	+	0.983165i	$0.441509\pi$
$488$	31.5757	1.42936
$489$	0	0
$490$	0	0
$491$	9.50128	0.428787	0.214393	−	0.976747i	$-0.431223\pi$
$491$	9.50128	0.428787	0.214393	+	0.976747i	$0.431223\pi$
$492$	0	0
$493$	−9.48790	−0.427314
$494$	4.07183	0.183200
$495$	0	0
$496$	−0.710687	−0.0319108
$497$	−3.94606	−0.177005
$498$	0	0
$499$	−19.6405	−0.879230	−0.439615	−	0.898186i	$-0.644885\pi$
$499$	−19.6405	−0.879230	−0.439615	+	0.898186i	$0.644885\pi$
$500$	0	0
$501$	0	0
$502$	−16.2451	−0.725056
$503$	20.6428	0.920415	0.460208	−	0.887811i	$-0.347775\pi$
$503$	20.6428	0.920415	0.460208	+	0.887811i	$0.347775\pi$
$504$	0	0
$505$	0	0
$506$	32.5669	1.44778
$507$	0	0
$508$	−11.7824	−0.522759
$509$	31.9372	1.41559	0.707795	−	0.706418i	$-0.249690\pi$
$509$	31.9372	1.41559	0.707795	+	0.706418i	$0.249690\pi$
$510$	0	0
$511$	−4.58920	−0.203014
$512$	1.52077	0.0672093
$513$	0	0
$514$	20.7073	0.913360
$515$	0	0
$516$	0	0
$517$	6.26291	0.275442
$518$	−0.952343	−0.0418435
$519$	0	0
$520$	0	0
$521$	8.88261	0.389154	0.194577	−	0.980887i	$-0.437667\pi$
$521$	8.88261	0.389154	0.194577	+	0.980887i	$0.437667\pi$
$522$	0	0
$523$	28.3512	1.23971	0.619856	−	0.784716i	$-0.287191\pi$
$523$	28.3512	1.23971	0.619856	+	0.784716i	$0.287191\pi$
$524$	−4.40329	−0.192359
$525$	0	0
$526$	−15.1236	−0.659420
$527$	−22.1959	−0.966870
$528$	0	0
$529$	18.3130	0.796215
$530$	0	0
$531$	0	0
$532$	−1.35163	−0.0586007
$533$	−0.509490	−0.0220685
$534$	0	0
$535$	0	0
$536$	−35.3049	−1.52494
$537$	0	0
$538$	−22.7172	−0.979409
$539$	−38.9142	−1.67615
$540$	0	0
$541$	−13.8190	−0.594124	−0.297062	−	0.954858i	$-0.596007\pi$
$541$	−13.8190	−0.594124	−0.297062	+	0.954858i	$0.596007\pi$
$542$	17.4936	0.751414
$543$	0	0
$544$	23.9901	1.02857
$545$	0	0
$546$	0	0
$547$	−12.0831	−0.516637	−0.258318	−	0.966060i	$-0.583168\pi$
$547$	−12.0831	−0.516637	−0.258318	+	0.966060i	$0.583168\pi$
$548$	−10.3461	−0.441966
$549$	0	0
$550$	0	0
$551$	3.72242	0.158580
$552$	0	0
$553$	−1.02441	−0.0435624
$554$	−12.0499	−0.511952
$555$	0	0
$556$	−3.72149	−0.157826
$557$	−15.5918	−0.660644	−0.330322	−	0.943868i	$-0.607157\pi$
$557$	−15.5918	−0.660644	−0.330322	+	0.943868i	$0.607157\pi$
$558$	0	0
$559$	22.9298	0.969828
$560$	0	0
$561$	0	0
$562$	−5.31170	−0.224061
$563$	−19.1854	−0.808570	−0.404285	−	0.914633i	$-0.632480\pi$
$563$	−19.1854	−0.808570	−0.404285	+	0.914633i	$0.632480\pi$
$564$	0	0
$565$	0	0
$566$	11.1210	0.467450
$567$	0	0
$568$	−17.0422	−0.715074
$569$	30.1698	1.26478	0.632392	−	0.774648i	$-0.282073\pi$
$569$	30.1698	1.26478	0.632392	+	0.774648i	$0.282073\pi$
$570$	0	0
$571$	−25.2616	−1.05716	−0.528582	−	0.848882i	$-0.677276\pi$
$571$	−25.2616	−1.05716	−0.528582	+	0.848882i	$0.677276\pi$
$572$	21.5910	0.902765
$573$	0	0
$574$	−0.0980668	−0.00409323
$575$	0	0
$576$	0	0
$577$	−9.00948	−0.375070	−0.187535	−	0.982258i	$-0.560050\pi$
$577$	−9.00948	−0.375070	−0.187535	+	0.982258i	$0.560050\pi$
$578$	0.550304	0.0228896
$579$	0	0
$580$	0	0
$581$	−7.45751	−0.309390
$582$	0	0
$583$	−26.5301	−1.09876
$584$	−19.8198	−0.820147
$585$	0	0
$586$	15.8986	0.656764
$587$	11.5059	0.474901	0.237451	−	0.971400i	$-0.423688\pi$
$587$	11.5059	0.474901	0.237451	+	0.971400i	$0.423688\pi$
$588$	0	0
$589$	8.70820	0.358815
$590$	0	0
$591$	0	0
$592$	0.230727	0.00948280
$593$	23.0392	0.946107	0.473053	−	0.881034i	$-0.343152\pi$
$593$	23.0392	0.946107	0.473053	+	0.881034i	$0.343152\pi$
$594$	0	0
$595$	0	0
$596$	−19.6000	−0.802848
$597$	0	0
$598$	−15.8818	−0.649456
$599$	−2.07694	−0.0848614	−0.0424307	−	0.999099i	$-0.513510\pi$
$599$	−2.07694	−0.0848614	−0.0424307	+	0.999099i	$0.513510\pi$
$600$	0	0
$601$	28.7922	1.17446	0.587230	−	0.809420i	$-0.300218\pi$
$601$	28.7922	1.17446	0.587230	+	0.809420i	$0.300218\pi$
$602$	4.41354	0.179883
$603$	0	0
$604$	−13.7017	−0.557514
$605$	0	0
$606$	0	0
$607$	−6.99573	−0.283948	−0.141974	−	0.989870i	$-0.545345\pi$
$607$	−6.99573	−0.283948	−0.141974	+	0.989870i	$0.545345\pi$
$608$	−9.41212	−0.381712
$609$	0	0
$610$	0	0
$611$	−3.05421	−0.123560
$612$	0	0
$613$	34.6552	1.39971	0.699855	−	0.714285i	$-0.253248\pi$
$613$	34.6552	1.39971	0.699855	+	0.714285i	$0.253248\pi$
$614$	20.6364	0.832819
$615$	0	0
$616$	10.7215	0.431980
$617$	−14.9852	−0.603280	−0.301640	−	0.953422i	$-0.597534\pi$
$617$	−14.9852	−0.603280	−0.301640	+	0.953422i	$0.597534\pi$
$618$	0	0
$619$	−39.8216	−1.60056	−0.800282	−	0.599624i	$-0.795317\pi$
$619$	−39.8216	−1.60056	−0.800282	+	0.599624i	$0.795317\pi$
$620$	0	0
$621$	0	0
$622$	21.7082	0.870420
$623$	−10.1020	−0.404728
$624$	0	0
$625$	0	0
$626$	−18.6969	−0.747277
$627$	0	0
$628$	−21.4053	−0.854164
$629$	7.20598	0.287321
$630$	0	0
$631$	1.66278	0.0661944	0.0330972	−	0.999452i	$-0.489463\pi$
$631$	1.66278	0.0661944	0.0330972	+	0.999452i	$0.489463\pi$
$632$	−4.42421	−0.175986
$633$	0	0
$634$	1.04475	0.0414923
$635$	0	0
$636$	0	0
$637$	18.9772	0.751903
$638$	−11.4453	−0.453123
$639$	0	0
$640$	0	0
$641$	−6.77663	−0.267661	−0.133830	−	0.991004i	$-0.542728\pi$
$641$	−6.77663	−0.267661	−0.133830	+	0.991004i	$0.542728\pi$
$642$	0	0
$643$	20.2108	0.797035	0.398517	−	0.917161i	$-0.369525\pi$
$643$	20.2108	0.797035	0.398517	+	0.917161i	$0.369525\pi$
$644$	5.27192	0.207743
$645$	0	0
$646$	−5.93029	−0.233324
$647$	5.05100	0.198575	0.0992877	−	0.995059i	$-0.468344\pi$
$647$	5.05100	0.198575	0.0992877	+	0.995059i	$0.468344\pi$
$648$	0	0
$649$	−57.6327	−2.26228
$650$	0	0
$651$	0	0
$652$	21.2362	0.831672
$653$	−9.64210	−0.377325	−0.188662	−	0.982042i	$-0.560415\pi$
$653$	−9.64210	−0.377325	−0.188662	+	0.982042i	$0.560415\pi$
$654$	0	0
$655$	0	0
$656$	0.0237589	0.000927629 0
$657$	0	0
$658$	−0.587876	−0.0229178
$659$	26.7332	1.04138	0.520690	−	0.853746i	$-0.325675\pi$
$659$	26.7332	1.04138	0.520690	+	0.853746i	$0.325675\pi$
$660$	0	0
$661$	−2.00159	−0.0778529	−0.0389264	−	0.999242i	$-0.512394\pi$
$661$	−2.00159	−0.0778529	−0.0389264	+	0.999242i	$0.512394\pi$
$662$	−5.11896	−0.198954
$663$	0	0
$664$	−32.2074	−1.24989
$665$	0	0
$666$	0	0
$667$	−14.5190	−0.562177
$668$	12.9020	0.499194
$669$	0	0
$670$	0	0
$671$	−66.7339	−2.57623
$672$	0	0
$673$	−31.2362	−1.20407	−0.602033	−	0.798471i	$-0.705642\pi$
$673$	−31.2362	−1.20407	−0.602033	+	0.798471i	$0.705642\pi$
$674$	23.2801	0.896716
$675$	0	0
$676$	5.92801	0.228000
$677$	42.4874	1.63292	0.816462	−	0.577400i	$-0.195933\pi$
$677$	42.4874	1.63292	0.816462	+	0.577400i	$0.195933\pi$
$678$	0	0
$679$	−4.89192	−0.187735
$680$	0	0
$681$	0	0
$682$	−26.7750	−1.02527
$683$	−5.20811	−0.199283	−0.0996415	−	0.995023i	$-0.531770\pi$
$683$	−5.20811	−0.199283	−0.0996415	+	0.995023i	$0.531770\pi$
$684$	0	0
$685$	0	0
$686$	7.53850	0.287821
$687$	0	0
$688$	−1.06928	−0.0407659
$689$	12.9379	0.492893
$690$	0	0
$691$	8.07419	0.307157	0.153578	−	0.988136i	$-0.450920\pi$
$691$	8.07419	0.307157	0.153578	+	0.988136i	$0.450920\pi$
$692$	−18.1946	−0.691653
$693$	0	0
$694$	−10.6447	−0.404066
$695$	0	0
$696$	0	0
$697$	0.742030	0.0281064
$698$	17.3776	0.657752
$699$	0	0
$700$	0	0
$701$	−31.7552	−1.19938	−0.599689	−	0.800233i	$-0.704709\pi$
$701$	−31.7552	−1.19938	−0.599689	+	0.800233i	$0.704709\pi$
$702$	0	0
$703$	−2.82714	−0.106628
$704$	27.3487	1.03074
$705$	0	0
$706$	−10.2710	−0.386554
$707$	−11.1210	−0.418248
$708$	0	0
$709$	9.71630	0.364903	0.182452	−	0.983215i	$-0.441597\pi$
$709$	9.71630	0.364903	0.182452	+	0.983215i	$0.441597\pi$
$710$	0	0
$711$	0	0
$712$	−43.6284	−1.63504
$713$	−33.9656	−1.27202
$714$	0	0
$715$	0	0
$716$	9.61574	0.359357
$717$	0	0
$718$	3.03791	0.113374
$719$	20.4661	0.763257	0.381628	−	0.924316i	$-0.375363\pi$
$719$	20.4661	0.763257	0.381628	+	0.924316i	$0.375363\pi$
$720$	0	0
$721$	−11.2451	−0.418791
$722$	−13.9520	−0.519241
$723$	0	0
$724$	27.6232	1.02661
$725$	0	0
$726$	0	0
$727$	−50.2337	−1.86306	−0.931532	−	0.363659i	$-0.881527\pi$
$727$	−50.2337	−1.86306	−0.931532	+	0.363659i	$0.881527\pi$
$728$	−5.22851	−0.193781
$729$	0	0
$730$	0	0
$731$	−33.3954	−1.23517
$732$	0	0
$733$	37.8776	1.39904	0.699520	−	0.714613i	$-0.253397\pi$
$733$	37.8776	1.39904	0.699520	+	0.714613i	$0.253397\pi$
$734$	26.8455	0.990886
$735$	0	0
$736$	36.7112	1.35319
$737$	74.6155	2.74850
$738$	0	0
$739$	44.9709	1.65428	0.827141	−	0.561995i	$-0.189966\pi$
$739$	44.9709	1.65428	0.827141	+	0.561995i	$0.189966\pi$
$740$	0	0
$741$	0	0
$742$	2.49028	0.0914212
$743$	29.4546	1.08059	0.540293	−	0.841477i	$-0.318313\pi$
$743$	29.4546	1.08059	0.540293	+	0.841477i	$0.318313\pi$
$744$	0	0
$745$	0	0
$746$	−0.114461	−0.00419071
$747$	0	0
$748$	−31.4455	−1.14976
$749$	10.4991	0.383627
$750$	0	0
$751$	−34.1341	−1.24557	−0.622786	−	0.782392i	$-0.713999\pi$
$751$	−34.1341	−1.24557	−0.622786	+	0.782392i	$0.713999\pi$
$752$	0.142426	0.00519375
$753$	0	0
$754$	5.58148	0.203266
$755$	0	0
$756$	0	0
$757$	−15.2875	−0.555635	−0.277817	−	0.960634i	$-0.589611\pi$
$757$	−15.2875	−0.555635	−0.277817	+	0.960634i	$0.589611\pi$
$758$	−6.89139	−0.250306
$759$	0	0
$760$	0	0
$761$	24.0119	0.870429	0.435215	−	0.900327i	$-0.356672\pi$
$761$	24.0119	0.870429	0.435215	+	0.900327i	$0.356672\pi$
$762$	0	0
$763$	−4.61769	−0.167172
$764$	−24.0887	−0.871500
$765$	0	0
$766$	10.4802	0.378664
$767$	28.1056	1.01483
$768$	0	0
$769$	−0.656954	−0.0236904	−0.0118452	−	0.999930i	$-0.503771\pi$
$769$	−0.656954	−0.0236904	−0.0118452	+	0.999930i	$0.503771\pi$
$770$	0	0
$771$	0	0
$772$	10.8577	0.390776
$773$	28.1609	1.01288	0.506439	−	0.862276i	$-0.330961\pi$
$773$	28.1609	1.01288	0.506439	+	0.862276i	$0.330961\pi$
$774$	0	0
$775$	0	0
$776$	−21.1271	−0.758420
$777$	0	0
$778$	−28.2738	−1.01367
$779$	−0.291123	−0.0104306
$780$	0	0
$781$	36.0180	1.28883
$782$	23.1306	0.827147
$783$	0	0
$784$	−0.884958	−0.0316056
$785$	0	0
$786$	0	0
$787$	7.66560	0.273249	0.136625	−	0.990623i	$-0.456375\pi$
$787$	7.66560	0.273249	0.136625	+	0.990623i	$0.456375\pi$
$788$	−23.5119	−0.837575
$789$	0	0
$790$	0	0
$791$	−9.67794	−0.344108
$792$	0	0
$793$	32.5439	1.15567
$794$	−11.9305	−0.423397
$795$	0	0
$796$	9.27842	0.328865
$797$	34.7609	1.23129	0.615647	−	0.788022i	$-0.288895\pi$
$797$	34.7609	1.23129	0.615647	+	0.788022i	$0.288895\pi$
$798$	0	0
$799$	4.44821	0.157366
$800$	0	0
$801$	0	0
$802$	23.9463	0.845572
$803$	41.8883	1.47821
$804$	0	0
$805$	0	0
$806$	13.0573	0.459923
$807$	0	0
$808$	−48.0291	−1.68966
$809$	−42.5017	−1.49428	−0.747140	−	0.664667i	$-0.768573\pi$
$809$	−42.5017	−1.49428	−0.747140	+	0.664667i	$0.768573\pi$
$810$	0	0
$811$	39.2810	1.37934	0.689672	−	0.724122i	$-0.257755\pi$
$811$	39.2810	1.37934	0.689672	+	0.724122i	$0.257755\pi$
$812$	−1.85276	−0.0650191
$813$	0	0
$814$	8.69258	0.304675
$815$	0	0
$816$	0	0
$817$	13.1021	0.458386
$818$	22.2152	0.776736
$819$	0	0
$820$	0	0
$821$	−0.709911	−0.0247761	−0.0123880	−	0.999923i	$-0.503943\pi$
$821$	−0.709911	−0.0247761	−0.0123880	+	0.999923i	$0.503943\pi$
$822$	0	0
$823$	10.6260	0.370398	0.185199	−	0.982701i	$-0.440707\pi$
$823$	10.6260	0.370398	0.185199	+	0.982701i	$0.440707\pi$
$824$	−48.5653	−1.69185
$825$	0	0
$826$	5.40977	0.188230
$827$	11.0597	0.384585	0.192292	−	0.981338i	$-0.438408\pi$
$827$	11.0597	0.384585	0.192292	+	0.981338i	$0.438408\pi$
$828$	0	0
$829$	−1.64791	−0.0572342	−0.0286171	−	0.999590i	$-0.509110\pi$
$829$	−1.64791	−0.0572342	−0.0286171	+	0.999590i	$0.509110\pi$
$830$	0	0
$831$	0	0
$832$	−13.3370	−0.462379
$833$	−27.6387	−0.957625
$834$	0	0
$835$	0	0
$836$	12.3371	0.426688
$837$	0	0
$838$	1.04195	0.0359937
$839$	23.8789	0.824392	0.412196	−	0.911095i	$-0.364762\pi$
$839$	23.8789	0.824392	0.412196	+	0.911095i	$0.364762\pi$
$840$	0	0
$841$	−23.8975	−0.824051
$842$	17.5318	0.604186
$843$	0	0
$844$	18.4261	0.634252
$845$	0	0
$846$	0	0
$847$	−15.5324	−0.533701
$848$	−0.603328	−0.0207184
$849$	0	0
$850$	0	0
$851$	11.0270	0.378002
$852$	0	0
$853$	−23.3570	−0.799729	−0.399864	−	0.916574i	$-0.630943\pi$
$853$	−23.3570	−0.799729	−0.399864	+	0.916574i	$0.630943\pi$
$854$	6.26407	0.214352
$855$	0	0
$856$	45.3431	1.54980
$857$	−2.70183	−0.0922929	−0.0461464	−	0.998935i	$-0.514694\pi$
$857$	−2.70183	−0.0922929	−0.0461464	+	0.998935i	$0.514694\pi$
$858$	0	0
$859$	23.5781	0.804474	0.402237	−	0.915536i	$-0.368233\pi$
$859$	23.5781	0.804474	0.402237	+	0.915536i	$0.368233\pi$
$860$	0	0
$861$	0	0
$862$	−18.3676	−0.625602
$863$	26.2965	0.895144	0.447572	−	0.894248i	$-0.352289\pi$
$863$	26.2965	0.895144	0.447572	+	0.894248i	$0.352289\pi$
$864$	0	0
$865$	0	0
$866$	19.9494	0.677909
$867$	0	0
$868$	−4.33433	−0.147117
$869$	9.35039	0.317190
$870$	0	0
$871$	−36.3875	−1.23294
$872$	−19.9428	−0.675349
$873$	0	0
$874$	−9.07489	−0.306963
$875$	0	0
$876$	0	0
$877$	−33.4164	−1.12839	−0.564196	−	0.825641i	$-0.690814\pi$
$877$	−33.4164	−1.12839	−0.564196	+	0.825641i	$0.690814\pi$
$878$	−10.7400	−0.362459
$879$	0	0
$880$	0	0
$881$	21.0398	0.708849	0.354425	−	0.935085i	$-0.384677\pi$
$881$	21.0398	0.708849	0.354425	+	0.935085i	$0.384677\pi$
$882$	0	0
$883$	−14.5435	−0.489428	−0.244714	−	0.969595i	$-0.578694\pi$
$883$	−14.5435	−0.489428	−0.244714	+	0.969595i	$0.578694\pi$
$884$	15.3350	0.515770
$885$	0	0
$886$	−11.4146	−0.383482
$887$	−55.3525	−1.85855	−0.929277	−	0.369382i	$-0.879569\pi$
$887$	−55.3525	−1.85855	−0.929277	+	0.369382i	$0.879569\pi$
$888$	0	0
$889$	−6.03021	−0.202247
$890$	0	0
$891$	0	0
$892$	−34.0172	−1.13898
$893$	−1.74518	−0.0584003
$894$	0	0
$895$	0	0
$896$	4.83400	0.161493
$897$	0	0
$898$	−25.9173	−0.864873
$899$	11.9368	0.398115
$900$	0	0
$901$	−18.8429	−0.627749
$902$	0.895112	0.0298040
$903$	0	0
$904$	−41.7969	−1.39015
$905$	0	0
$906$	0	0
$907$	−25.9003	−0.860006	−0.430003	−	0.902828i	$-0.641487\pi$
$907$	−25.9003	−0.860006	−0.430003	+	0.902828i	$0.641487\pi$
$908$	5.31730	0.176461
$909$	0	0
$910$	0	0
$911$	−10.6499	−0.352848	−0.176424	−	0.984314i	$-0.556453\pi$
$911$	−10.6499	−0.352848	−0.176424	+	0.984314i	$0.556453\pi$
$912$	0	0
$913$	68.0690	2.25275
$914$	−11.0002	−0.363854
$915$	0	0
$916$	−10.5165	−0.347473
$917$	−2.25360	−0.0744204
$918$	0	0
$919$	11.1958	0.369314	0.184657	−	0.982803i	$-0.440883\pi$
$919$	11.1958	0.369314	0.184657	+	0.982803i	$0.440883\pi$
$920$	0	0
$921$	0	0
$922$	−23.8555	−0.785640
$923$	−17.5648	−0.578152
$924$	0	0
$925$	0	0
$926$	3.29528	0.108290
$927$	0	0
$928$	−12.9017	−0.423520
$929$	−58.0987	−1.90616	−0.953078	−	0.302723i	$-0.902104\pi$
$929$	−58.0987	−1.90616	−0.953078	+	0.302723i	$0.902104\pi$
$930$	0	0
$931$	10.8436	0.355384
$932$	28.6303	0.937815
$933$	0	0
$934$	−34.8359	−1.13986
$935$	0	0
$936$	0	0
$937$	−40.7577	−1.33150	−0.665749	−	0.746176i	$-0.731888\pi$
$937$	−40.7577	−1.33150	−0.665749	+	0.746176i	$0.731888\pi$
$938$	−7.00389	−0.228685
$939$	0	0
$940$	0	0
$941$	2.47724	0.0807559	0.0403779	−	0.999184i	$-0.487144\pi$
$941$	2.47724	0.0807559	0.0403779	+	0.999184i	$0.487144\pi$
$942$	0	0
$943$	1.13550	0.0369770
$944$	−1.31064	−0.0426577
$945$	0	0
$946$	−40.2850	−1.30978
$947$	18.3448	0.596125	0.298063	−	0.954546i	$-0.403660\pi$
$947$	18.3448	0.596125	0.298063	+	0.954546i	$0.403660\pi$
$948$	0	0
$949$	−20.4275	−0.663106
$950$	0	0
$951$	0	0
$952$	7.61490	0.246800
$953$	13.8466	0.448535	0.224267	−	0.974528i	$-0.428001\pi$
$953$	13.8466	0.448535	0.224267	+	0.974528i	$0.428001\pi$
$954$	0	0
$955$	0	0
$956$	1.62911	0.0526893
$957$	0	0
$958$	13.0780	0.422530
$959$	−5.29515	−0.170989
$960$	0	0
$961$	−3.07508	−0.0991962
$962$	−4.23909	−0.136674
$963$	0	0
$964$	6.22335	0.200441
$965$	0	0
$966$	0	0
$967$	7.61678	0.244939	0.122470	−	0.992472i	$-0.460919\pi$
$967$	7.61678	0.244939	0.122470	+	0.992472i	$0.460919\pi$
$968$	−67.0812	−2.15607
$969$	0	0
$970$	0	0
$971$	40.9964	1.31564	0.657819	−	0.753176i	$-0.271479\pi$
$971$	40.9964	1.31564	0.657819	+	0.753176i	$0.271479\pi$
$972$	0	0
$973$	−1.90465	−0.0610604
$974$	6.90958	0.221397
$975$	0	0
$976$	−1.51761	−0.0485776
$977$	−57.8650	−1.85127	−0.925633	−	0.378423i	$-0.876466\pi$
$977$	−57.8650	−1.85127	−0.925633	+	0.378423i	$0.876466\pi$
$978$	0	0
$979$	92.2069	2.94694
$980$	0	0
$981$	0	0
$982$	8.14044	0.259772
$983$	−8.89795	−0.283801	−0.141900	−	0.989881i	$-0.545321\pi$
$983$	−8.89795	−0.283801	−0.141900	+	0.989881i	$0.545321\pi$
$984$	0	0
$985$	0	0
$986$	−8.12898	−0.258879
$987$	0	0
$988$	−6.01641	−0.191408
$989$	−51.1037	−1.62500
$990$	0	0
$991$	−7.01945	−0.222980	−0.111490	−	0.993766i	$-0.535562\pi$
$991$	−7.01945	−0.222980	−0.111490	+	0.993766i	$0.535562\pi$
$992$	−30.1822	−0.958286
$993$	0	0
$994$	−3.38088	−0.107235
$995$	0	0
$996$	0	0
$997$	−49.6542	−1.57256	−0.786281	−	0.617868i	$-0.787996\pi$
$997$	−49.6542	−1.57256	−0.786281	+	0.617868i	$0.787996\pi$
$998$	−16.8275	−0.532664
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.bb.1.5	yes	8
3.2	odd	2	inner	5625.2.a.bb.1.4	yes	8
5.4	even	2		5625.2.a.z.1.4	✓	8
15.14	odd	2		5625.2.a.z.1.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5625.2.a.z.1.4	✓	8	5.4	even	2
5625.2.a.z.1.5	yes	8	15.14	odd	2
5625.2.a.bb.1.4	yes	8	3.2	odd	2	inner
5625.2.a.bb.1.5	yes	8	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q+0.856773 q^{2} -1.26594 q^{4} -0.647907 q^{7} -2.79817 q^{8} +O(q^{10})\)	\(q+0.856773 q^{2} -1.26594 q^{4} -0.647907 q^{7} -2.79817 q^{8} +5.91382 q^{11} -2.88397 q^{13} -0.555109 q^{14} +0.134488 q^{16} +4.20027 q^{17} -1.64791 q^{19} +5.06680 q^{22} +6.42752 q^{23} -2.47091 q^{26} +0.820212 q^{28} -2.25888 q^{29} -5.28440 q^{31} +5.71156 q^{32} +3.59868 q^{34} +1.71560 q^{37} -1.41188 q^{38} +0.176662 q^{41} -7.95077 q^{43} -7.48654 q^{44} +5.50692 q^{46} +1.05903 q^{47} -6.58022 q^{49} +3.65094 q^{52} -4.48612 q^{53} +1.81295 q^{56} -1.93534 q^{58} -9.74542 q^{59} -11.2844 q^{61} -4.52753 q^{62} +4.62453 q^{64} +12.6171 q^{67} -5.31730 q^{68} +6.09048 q^{71} +7.08312 q^{73} +1.46988 q^{74} +2.08615 q^{76} -3.83160 q^{77} +1.58111 q^{79} +0.151359 q^{82} +11.5102 q^{83} -6.81200 q^{86} -16.5479 q^{88} +15.5918 q^{89} +1.86855 q^{91} -8.13685 q^{92} +0.907347 q^{94} +7.55034 q^{97} -5.63775 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{4} + 10 q^{7}+O(q^{10}) \) 8 * q + 14 * q^4 + 10 * q^7	\( 8 q + 14 q^{4} + 10 q^{7} + 10 q^{13} + 22 q^{16} + 2 q^{19} + 10 q^{22} + 70 q^{28} - 6 q^{31} - 50 q^{34} + 50 q^{37} + 14 q^{49} + 80 q^{52} - 30 q^{58} - 54 q^{61} + 36 q^{64} + 10 q^{67} + 30 q^{73} + 56 q^{76} + 28 q^{79} + 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100}) \) 8 * q + 14 * q^4 + 10 * q^7 + 10 * q^13 + 22 * q^16 + 2 * q^19 + 10 * q^22 + 70 * q^28 - 6 * q^31 - 50 * q^34 + 50 * q^37 + 14 * q^49 + 80 * q^52 - 30 * q^58 - 54 * q^61 + 36 * q^64 + 10 * q^67 + 30 * q^73 + 56 * q^76 + 28 * q^79 + 20 * q^88 + 60 * q^91 - 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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