LMFDB - Embedded newform 7225.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, 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("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17 $ x^12 - 4x^11 - 10x^10 + 52x^9 + 21x^8 - 232x^7 + 44x^6 + 424x^5 - 137x^4 - 316x^3 + 46x^2 + 92*x + 17
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.04505$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.04505 q^{2} -3.19566 q^{3} +2.18224 q^{4} +6.53528 q^{6} +1.17743 q^{7} -0.372688 q^{8} +7.21221 q^{9} +4.92251 q^{11} -6.97368 q^{12} +2.46296 q^{13} -2.40791 q^{14} -3.60231 q^{16} -14.7494 q^{18} +2.04173 q^{19} -3.76267 q^{21} -10.0668 q^{22} -0.119023 q^{23} +1.19098 q^{24} -5.03689 q^{26} -13.4608 q^{27} +2.56944 q^{28} +1.06541 q^{29} +2.79577 q^{31} +8.11229 q^{32} -15.7307 q^{33} +15.7388 q^{36} +2.31477 q^{37} -4.17544 q^{38} -7.87078 q^{39} +0.717574 q^{41} +7.69485 q^{42} +10.0958 q^{43} +10.7421 q^{44} +0.243408 q^{46} +3.39482 q^{47} +11.5117 q^{48} -5.61365 q^{49} +5.37477 q^{52} +13.9241 q^{53} +27.5280 q^{54} -0.438815 q^{56} -6.52466 q^{57} -2.17882 q^{58} +1.51711 q^{59} +8.08120 q^{61} -5.71749 q^{62} +8.49190 q^{63} -9.38544 q^{64} +32.1700 q^{66} +4.92534 q^{67} +0.380355 q^{69} -6.63635 q^{71} -2.68790 q^{72} +3.66716 q^{73} -4.73382 q^{74} +4.45554 q^{76} +5.79593 q^{77} +16.0962 q^{78} +8.18051 q^{79} +21.3794 q^{81} -1.46748 q^{82} +6.08874 q^{83} -8.21104 q^{84} -20.6464 q^{86} -3.40468 q^{87} -1.83456 q^{88} +8.46170 q^{89} +2.89997 q^{91} -0.259736 q^{92} -8.93431 q^{93} -6.94259 q^{94} -25.9241 q^{96} +4.48946 q^{97} +11.4802 q^{98} +35.5022 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{2} - 8 q^{3} + 12 q^{4} + 8 q^{6} - 16 q^{7} + 12 q^{8} + 12 q^{9} + 16 q^{11} - 16 q^{12} + 8 q^{13} - 16 q^{14} + 12 q^{16} - 4 q^{18} + 16 q^{21} - 16 q^{22} - 16 q^{23} + 16 q^{26} - 32 q^{27}+ \cdots + 56 q^{99}+O(q^{100}) $ 12 * q + 4 * q^2 - 8 * q^3 + 12 * q^4 + 8 * q^6 - 16 * q^7 + 12 * q^8 + 12 * q^9 + 16 * q^11 - 16 * q^12 + 8 * q^13 - 16 * q^14 + 12 * q^16 - 4 * q^18 + 16 * q^21 - 16 * q^22 - 16 * q^23 + 16 * q^26 - 32 * q^27 - 40 * q^28 + 16 * q^29 + 24 * q^31 + 28 * q^32 + 12 * q^36 - 24 * q^37 + 24 * q^38 + 8 * q^39 + 8 * q^41 + 16 * q^43 + 8 * q^44 + 40 * q^46 + 32 * q^47 + 24 * q^48 + 20 * q^49 + 24 * q^52 + 8 * q^54 - 24 * q^56 - 32 * q^57 + 16 * q^58 + 8 * q^59 + 24 * q^61 - 8 * q^62 - 48 * q^63 + 36 * q^64 + 40 * q^66 + 8 * q^67 + 48 * q^69 - 16 * q^71 - 12 * q^72 - 16 * q^73 + 16 * q^76 - 24 * q^77 + 24 * q^78 + 40 * q^79 + 36 * q^81 + 16 * q^82 + 40 * q^83 + 32 * q^84 - 8 * q^86 + 32 * q^87 - 48 * q^88 + 8 * q^89 + 72 * q^91 + 8 * q^92 - 24 * q^93 - 16 * q^94 + 8 * q^96 - 32 * q^97 + 60 * q^98 + 56 * 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 $

$2$	−2.04505	−1.44607	−0.723035	−	0.690811i	$-0.757254\pi$
$2$	−2.04505	−1.44607	−0.723035	+	0.690811i	$0.757254\pi$
$3$	−3.19566	−1.84501	−0.922506	−	0.385982i	$-0.873863\pi$
$3$	−3.19566	−1.84501	−0.922506	+	0.385982i	$0.873863\pi$
$4$	2.18224	1.09112
$5$	0	0
$5$	0	0
$6$	6.53528	2.66802
$7$	1.17743	0.445028	0.222514	−	0.974930i	$-0.428574\pi$
$7$	1.17743	0.445028	0.222514	+	0.974930i	$0.428574\pi$
$8$	−0.372688	−0.131765
$9$	7.21221	2.40407
$10$	0	0
$11$	4.92251	1.48419	0.742097	−	0.670293i	$-0.233831\pi$
$11$	4.92251	1.48419	0.742097	+	0.670293i	$0.233831\pi$
$12$	−6.97368	−2.01313
$13$	2.46296	0.683103	0.341551	−	0.939863i	$-0.389048\pi$
$13$	2.46296	0.683103	0.341551	+	0.939863i	$0.389048\pi$
$14$	−2.40791	−0.643541
$15$	0	0
$16$	−3.60231	−0.900578
$17$	0	0
$17$	0	0
$18$	−14.7494	−3.47646
$19$	2.04173	0.468405	0.234202	−	0.972188i	$-0.424752\pi$
$19$	2.04173	0.468405	0.234202	+	0.972188i	$0.424752\pi$
$20$	0	0
$21$	−3.76267	−0.821082
$22$	−10.0668	−2.14625
$23$	−0.119023	−0.0248179	−0.0124090	−	0.999923i	$-0.503950\pi$
$23$	−0.119023	−0.0248179	−0.0124090	+	0.999923i	$0.503950\pi$
$24$	1.19098	0.243108
$25$	0	0
$26$	−5.03689	−0.987815
$27$	−13.4608	−2.59053
$28$	2.56944	0.485578
$29$	1.06541	0.197841	0.0989207	−	0.995095i	$-0.468461\pi$
$29$	1.06541	0.197841	0.0989207	+	0.995095i	$0.468461\pi$
$30$	0	0
$31$	2.79577	0.502135	0.251067	−	0.967970i	$-0.419218\pi$
$31$	2.79577	0.502135	0.251067	+	0.967970i	$0.419218\pi$
$32$	8.11229	1.43406
$33$	−15.7307	−2.73836
$34$	0	0
$35$	0	0
$36$	15.7388	2.62313
$37$	2.31477	0.380545	0.190273	−	0.981731i	$-0.439063\pi$
$37$	2.31477	0.380545	0.190273	+	0.981731i	$0.439063\pi$
$38$	−4.17544	−0.677346
$39$	−7.87078	−1.26033
$40$	0	0
$41$	0.717574	0.112066	0.0560331	−	0.998429i	$-0.482155\pi$
$41$	0.717574	0.112066	0.0560331	+	0.998429i	$0.482155\pi$
$42$	7.69485	1.18734
$43$	10.0958	1.53960	0.769798	−	0.638288i	$-0.220357\pi$
$43$	10.0958	1.53960	0.769798	+	0.638288i	$0.220357\pi$
$44$	10.7421	1.61943
$45$	0	0
$46$	0.243408	0.0358885
$47$	3.39482	0.495186	0.247593	−	0.968864i	$-0.420360\pi$
$47$	3.39482	0.495186	0.247593	+	0.968864i	$0.420360\pi$
$48$	11.5117	1.66158
$49$	−5.61365	−0.801950
$50$	0	0
$51$	0	0
$52$	5.37477	0.745347
$53$	13.9241	1.91262	0.956310	−	0.292355i	$-0.0944390\pi$
$53$	13.9241	1.91262	0.956310	+	0.292355i	$0.0944390\pi$
$54$	27.5280	3.74609
$55$	0	0
$56$	−0.438815	−0.0586391
$57$	−6.52466	−0.864213
$58$	−2.17882	−0.286093
$59$	1.51711	0.197511	0.0987555	−	0.995112i	$-0.468514\pi$
$59$	1.51711	0.197511	0.0987555	+	0.995112i	$0.468514\pi$
$60$	0	0
$61$	8.08120	1.03469	0.517346	−	0.855776i	$-0.326920\pi$
$61$	8.08120	1.03469	0.517346	+	0.855776i	$0.326920\pi$
$62$	−5.71749	−0.726122
$63$	8.49190	1.06988
$64$	−9.38544	−1.17318
$65$	0	0
$66$	32.1700	3.95986
$67$	4.92534	0.601726	0.300863	−	0.953667i	$-0.402725\pi$
$67$	4.92534	0.601726	0.300863	+	0.953667i	$0.402725\pi$
$68$	0	0
$69$	0.380355	0.0457894
$70$	0	0
$71$	−6.63635	−0.787590	−0.393795	−	0.919198i	$-0.628838\pi$
$71$	−6.63635	−0.787590	−0.393795	+	0.919198i	$0.628838\pi$
$72$	−2.68790	−0.316773
$73$	3.66716	0.429209	0.214605	−	0.976701i	$-0.431154\pi$
$73$	3.66716	0.429209	0.214605	+	0.976701i	$0.431154\pi$
$74$	−4.73382	−0.550295
$75$	0	0
$76$	4.45554	0.511086
$77$	5.79593	0.660507
$78$	16.0962	1.82253
$79$	8.18051	0.920380	0.460190	−	0.887821i	$-0.347781\pi$
$79$	8.18051	0.920380	0.460190	+	0.887821i	$0.347781\pi$
$80$	0	0
$81$	21.3794	2.37549
$82$	−1.46748	−0.162056
$83$	6.08874	0.668326	0.334163	−	0.942515i	$-0.391546\pi$
$83$	6.08874	0.668326	0.334163	+	0.942515i	$0.391546\pi$
$84$	−8.21104	−0.895898
$85$	0	0
$86$	−20.6464	−2.22636
$87$	−3.40468	−0.365020
$88$	−1.83456	−0.195565
$89$	8.46170	0.896938	0.448469	−	0.893798i	$-0.351969\pi$
$89$	8.46170	0.896938	0.448469	+	0.893798i	$0.351969\pi$
$90$	0	0
$91$	2.89997	0.304000
$92$	−0.259736	−0.0270793
$93$	−8.93431	−0.926445
$94$	−6.94259	−0.716074
$95$	0	0
$96$	−25.9241	−2.64587
$97$	4.48946	0.455836	0.227918	−	0.973680i	$-0.426808\pi$
$97$	4.48946	0.455836	0.227918	+	0.973680i	$0.426808\pi$
$98$	11.4802	1.15968
$99$	35.5022	3.56811
$100$	0	0
$101$	17.7855	1.76972	0.884861	−	0.465854i	$-0.154253\pi$
$101$	17.7855	1.76972	0.884861	+	0.465854i	$0.154253\pi$
$102$	0	0
$103$	11.3352	1.11689	0.558445	−	0.829542i	$-0.311398\pi$
$103$	11.3352	1.11689	0.558445	+	0.829542i	$0.311398\pi$
$104$	−0.917916	−0.0900091
$105$	0	0
$106$	−28.4755	−2.76578
$107$	−12.8865	−1.24579	−0.622893	−	0.782307i	$-0.714043\pi$
$107$	−12.8865	−1.24579	−0.622893	+	0.782307i	$0.714043\pi$
$108$	−29.3747	−2.82658
$109$	−11.3210	−1.08436	−0.542179	−	0.840263i	$-0.682401\pi$
$109$	−11.3210	−1.08436	−0.542179	+	0.840263i	$0.682401\pi$
$110$	0	0
$111$	−7.39720	−0.702111
$112$	−4.24148	−0.400782
$113$	5.85089	0.550405	0.275203	−	0.961386i	$-0.411255\pi$
$113$	5.85089	0.550405	0.275203	+	0.961386i	$0.411255\pi$
$114$	13.3433	1.24971
$115$	0	0
$116$	2.32498	0.215869
$117$	17.7634	1.64223
$118$	−3.10257	−0.285615
$119$	0	0
$120$	0	0
$121$	13.2311	1.20283
$122$	−16.5265	−1.49624
$123$	−2.29312	−0.206764
$124$	6.10103	0.547889
$125$	0	0
$126$	−17.3664	−1.54712
$127$	6.80398	0.603755	0.301878	−	0.953347i	$-0.402387\pi$
$127$	6.80398	0.603755	0.301878	+	0.953347i	$0.402387\pi$
$128$	2.96912	0.262436
$129$	−32.2627	−2.84057
$130$	0	0
$131$	1.90025	0.166026	0.0830128	−	0.996548i	$-0.473546\pi$
$131$	1.90025	0.166026	0.0830128	+	0.996548i	$0.473546\pi$
$132$	−34.3281	−2.98787
$133$	2.40400	0.208453
$134$	−10.0726	−0.870138
$135$	0	0
$136$	0	0
$137$	−19.3637	−1.65435	−0.827174	−	0.561946i	$-0.810053\pi$
$137$	−19.3637	−1.65435	−0.827174	+	0.561946i	$0.810053\pi$
$138$	−0.777847	−0.0662147
$139$	−8.27692	−0.702039	−0.351020	−	0.936368i	$-0.614165\pi$
$139$	−8.27692	−0.702039	−0.351020	+	0.936368i	$0.614165\pi$
$140$	0	0
$141$	−10.8487	−0.913624
$142$	13.5717	1.13891
$143$	12.1240	1.01386
$144$	−25.9806	−2.16505
$145$	0	0
$146$	−7.49954	−0.620666
$147$	17.9393	1.47961
$148$	5.05137	0.415220
$149$	4.93485	0.404279	0.202139	−	0.979357i	$-0.435211\pi$
$149$	4.93485	0.404279	0.202139	+	0.979357i	$0.435211\pi$
$150$	0	0
$151$	−0.712752	−0.0580029	−0.0290015	−	0.999579i	$-0.509233\pi$
$151$	−0.712752	−0.0580029	−0.0290015	+	0.999579i	$0.509233\pi$
$152$	−0.760928	−0.0617194
$153$	0	0
$154$	−11.8530	−0.955140
$155$	0	0
$156$	−17.1759	−1.37517
$157$	−5.88566	−0.469727	−0.234863	−	0.972028i	$-0.575464\pi$
$157$	−5.88566	−0.469727	−0.234863	+	0.972028i	$0.575464\pi$
$158$	−16.7296	−1.33093
$159$	−44.4966	−3.52881
$160$	0	0
$161$	−0.140141	−0.0110447
$162$	−43.7220	−3.43512
$163$	−5.89619	−0.461826	−0.230913	−	0.972974i	$-0.574171\pi$
$163$	−5.89619	−0.461826	−0.230913	+	0.972974i	$0.574171\pi$
$164$	1.56592	0.122278
$165$	0	0
$166$	−12.4518	−0.966446
$167$	10.7684	0.833284	0.416642	−	0.909071i	$-0.363207\pi$
$167$	10.7684	0.833284	0.416642	+	0.909071i	$0.363207\pi$
$168$	1.40230	0.108190
$169$	−6.93382	−0.533371
$170$	0	0
$171$	14.7254	1.12608
$172$	22.0315	1.67988
$173$	9.69698	0.737248	0.368624	−	0.929579i	$-0.379829\pi$
$173$	9.69698	0.737248	0.368624	+	0.929579i	$0.379829\pi$
$174$	6.96274	0.527844
$175$	0	0
$176$	−17.7324	−1.33663
$177$	−4.84816	−0.364410
$178$	−17.3046	−1.29704
$179$	15.7156	1.17464	0.587319	−	0.809356i	$-0.300183\pi$
$179$	15.7156	1.17464	0.587319	+	0.809356i	$0.300183\pi$
$180$	0	0
$181$	−3.98916	−0.296512	−0.148256	−	0.988949i	$-0.547366\pi$
$181$	−3.98916	−0.296512	−0.148256	+	0.988949i	$0.547366\pi$
$182$	−5.93059	−0.439605
$183$	−25.8247	−1.90902
$184$	0.0443583	0.00327014
$185$	0	0
$186$	18.2711	1.33970
$187$	0	0
$188$	7.40832	0.540307
$189$	−15.8492	−1.15286
$190$	0	0
$191$	19.2007	1.38931	0.694655	−	0.719343i	$-0.255557\pi$
$191$	19.2007	1.38931	0.694655	+	0.719343i	$0.255557\pi$
$192$	29.9926	2.16453
$193$	−11.7595	−0.846466	−0.423233	−	0.906021i	$-0.639105\pi$
$193$	−11.7595	−0.846466	−0.423233	+	0.906021i	$0.639105\pi$
$194$	−9.18118	−0.659170
$195$	0	0
$196$	−12.2503	−0.875024
$197$	−19.1390	−1.36360	−0.681799	−	0.731540i	$-0.738802\pi$
$197$	−19.1390	−1.36360	−0.681799	+	0.731540i	$0.738802\pi$
$198$	−72.6039	−5.15974
$199$	10.4492	0.740723	0.370362	−	0.928888i	$-0.379234\pi$
$199$	10.4492	0.740723	0.370362	+	0.928888i	$0.379234\pi$
$200$	0	0
$201$	−15.7397	−1.11019
$202$	−36.3723	−2.55914
$203$	1.25445	0.0880449
$204$	0	0
$205$	0	0
$206$	−23.1811	−1.61510
$207$	−0.858417	−0.0596641
$208$	−8.87236	−0.615187
$209$	10.0504	0.695204
$210$	0	0
$211$	21.7830	1.49960	0.749801	−	0.661663i	$-0.230149\pi$
$211$	21.7830	1.49960	0.749801	+	0.661663i	$0.230149\pi$
$212$	30.3857	2.08690
$213$	21.2075	1.45311
$214$	26.3536	1.80149
$215$	0	0
$216$	5.01667	0.341341
$217$	3.29183	0.223464
$218$	23.1521	1.56806
$219$	−11.7190	−0.791896
$220$	0	0
$221$	0	0
$222$	15.1277	1.01530
$223$	−20.0185	−1.34054	−0.670270	−	0.742117i	$-0.733822\pi$
$223$	−20.0185	−1.34054	−0.670270	+	0.742117i	$0.733822\pi$
$224$	9.55167	0.638198
$225$	0	0
$226$	−11.9654	−0.795924
$227$	13.7803	0.914630	0.457315	−	0.889305i	$-0.348811\pi$
$227$	13.7803	0.914630	0.457315	+	0.889305i	$0.348811\pi$
$228$	−14.2384	−0.942960
$229$	3.64022	0.240553	0.120276	−	0.992740i	$-0.461622\pi$
$229$	3.64022	0.240553	0.120276	+	0.992740i	$0.461622\pi$
$230$	0	0
$231$	−18.5218	−1.21864
$232$	−0.397065	−0.0260686
$233$	−14.1608	−0.927703	−0.463851	−	0.885913i	$-0.653533\pi$
$233$	−14.1608	−0.927703	−0.463851	+	0.885913i	$0.653533\pi$
$234$	−36.3271	−2.37478
$235$	0	0
$236$	3.31070	0.215508
$237$	−26.1421	−1.69811
$238$	0	0
$239$	−22.6621	−1.46589	−0.732944	−	0.680289i	$-0.761854\pi$
$239$	−22.6621	−1.46589	−0.732944	+	0.680289i	$0.761854\pi$
$240$	0	0
$241$	−6.04498	−0.389391	−0.194696	−	0.980864i	$-0.562372\pi$
$241$	−6.04498	−0.389391	−0.194696	+	0.980864i	$0.562372\pi$
$242$	−27.0584	−1.73938
$243$	−27.9388	−1.79228
$244$	17.6351	1.12897
$245$	0	0
$246$	4.68955	0.298995
$247$	5.02870	0.319969
$248$	−1.04195	−0.0661638
$249$	−19.4575	−1.23307
$250$	0	0
$251$	−0.706952	−0.0446224	−0.0223112	−	0.999751i	$-0.507102\pi$
$251$	−0.706952	−0.0446224	−0.0223112	+	0.999751i	$0.507102\pi$
$252$	18.5313	1.16736
$253$	−0.585891	−0.0368346
$254$	−13.9145	−0.873073
$255$	0	0
$256$	12.6989	0.793679
$257$	15.0724	0.940193	0.470096	−	0.882615i	$-0.344219\pi$
$257$	15.0724	0.940193	0.470096	+	0.882615i	$0.344219\pi$
$258$	65.9789	4.10767
$259$	2.72548	0.169353
$260$	0	0
$261$	7.68395	0.475625
$262$	−3.88611	−0.240085
$263$	14.5292	0.895911	0.447955	−	0.894056i	$-0.352152\pi$
$263$	14.5292	0.895911	0.447955	+	0.894056i	$0.352152\pi$
$264$	5.86262	0.360820
$265$	0	0
$266$	−4.91630	−0.301438
$267$	−27.0407	−1.65486
$268$	10.7483	0.656555
$269$	−28.3603	−1.72916	−0.864579	−	0.502497i	$-0.832415\pi$
$269$	−28.3603	−1.72916	−0.864579	+	0.502497i	$0.832415\pi$
$270$	0	0
$271$	9.29673	0.564736	0.282368	−	0.959306i	$-0.408880\pi$
$271$	9.29673	0.564736	0.282368	+	0.959306i	$0.408880\pi$
$272$	0	0
$273$	−9.26731	−0.560883
$274$	39.5997	2.39230
$275$	0	0
$276$	0.830026	0.0499617
$277$	−22.4063	−1.34627	−0.673133	−	0.739522i	$-0.735052\pi$
$277$	−22.4063	−1.34627	−0.673133	+	0.739522i	$0.735052\pi$
$278$	16.9267	1.01520
$279$	20.1637	1.20717
$280$	0	0
$281$	31.2511	1.86428	0.932142	−	0.362093i	$-0.117938\pi$
$281$	31.2511	1.86428	0.932142	+	0.362093i	$0.117938\pi$
$282$	22.1861	1.32117
$283$	−11.7455	−0.698197	−0.349098	−	0.937086i	$-0.613512\pi$
$283$	−11.7455	−0.698197	−0.349098	+	0.937086i	$0.613512\pi$
$284$	−14.4821	−0.859355
$285$	0	0
$286$	−24.7941	−1.46611
$287$	0.844895	0.0498726
$288$	58.5076	3.44759
$289$	0	0
$290$	0	0
$291$	−14.3468	−0.841023
$292$	8.00263	0.468318
$293$	26.5905	1.55344	0.776718	−	0.629848i	$-0.216883\pi$
$293$	26.5905	1.55344	0.776718	+	0.629848i	$0.216883\pi$
$294$	−36.6868	−2.13962
$295$	0	0
$296$	−0.862685	−0.0501426
$297$	−66.2609	−3.84485
$298$	−10.0920	−0.584616
$299$	−0.293148	−0.0169532
$300$	0	0
$301$	11.8871	0.685163
$302$	1.45761	0.0838763
$303$	−56.8363	−3.26516
$304$	−7.35495	−0.421835
$305$	0	0
$306$	0	0
$307$	−18.6981	−1.06715	−0.533577	−	0.845751i	$-0.679153\pi$
$307$	−18.6981	−1.06715	−0.533577	+	0.845751i	$0.679153\pi$
$308$	12.6481	0.720692
$309$	−36.2234	−2.06068
$310$	0	0
$311$	−7.96633	−0.451729	−0.225865	−	0.974159i	$-0.572521\pi$
$311$	−7.96633	−0.451729	−0.225865	+	0.974159i	$0.572521\pi$
$312$	2.93334	0.166068
$313$	−7.98760	−0.451486	−0.225743	−	0.974187i	$-0.572481\pi$
$313$	−7.98760	−0.451486	−0.225743	+	0.974187i	$0.572481\pi$
$314$	12.0365	0.679258
$315$	0	0
$316$	17.8518	1.00424
$317$	−15.6320	−0.877981	−0.438991	−	0.898492i	$-0.644664\pi$
$317$	−15.6320	−0.877981	−0.438991	+	0.898492i	$0.644664\pi$
$318$	90.9978	5.10290
$319$	5.24449	0.293635
$320$	0	0
$321$	41.1809	2.29849
$322$	0.286596	0.0159714
$323$	0	0
$324$	46.6550	2.59194
$325$	0	0
$326$	12.0580	0.667832
$327$	36.1781	2.00065
$328$	−0.267431	−0.0147664
$329$	3.99718	0.220371
$330$	0	0
$331$	−4.43051	−0.243523	−0.121761	−	0.992559i	$-0.538854\pi$
$331$	−4.43051	−0.243523	−0.121761	+	0.992559i	$0.538854\pi$
$332$	13.2871	0.729223
$333$	16.6946	0.914858
$334$	−22.0219	−1.20499
$335$	0	0
$336$	13.5543	0.739448
$337$	−6.03607	−0.328806	−0.164403	−	0.986393i	$-0.552570\pi$
$337$	−6.03607	−0.328806	−0.164403	+	0.986393i	$0.552570\pi$
$338$	14.1800	0.771291
$339$	−18.6974	−1.01550
$340$	0	0
$341$	13.7622	0.745266
$342$	−30.1142	−1.62839
$343$	−14.8517	−0.801918
$344$	−3.76258	−0.202865
$345$	0	0
$346$	−19.8308	−1.06611
$347$	0.0930963	0.00499767	0.00249883	−	0.999997i	$-0.499205\pi$
$347$	0.0930963	0.00499767	0.00249883	+	0.999997i	$0.499205\pi$
$348$	−7.42982	−0.398280
$349$	9.54221	0.510783	0.255391	−	0.966838i	$-0.417796\pi$
$349$	9.54221	0.510783	0.255391	+	0.966838i	$0.417796\pi$
$350$	0	0
$351$	−33.1534	−1.76960
$352$	39.9329	2.12843
$353$	−17.2138	−0.916197	−0.458099	−	0.888901i	$-0.651469\pi$
$353$	−17.2138	−0.916197	−0.458099	+	0.888901i	$0.651469\pi$
$354$	9.91475	0.526963
$355$	0	0
$356$	18.4655	0.978667
$357$	0	0
$358$	−32.1392	−1.69861
$359$	−17.7449	−0.936538	−0.468269	−	0.883586i	$-0.655122\pi$
$359$	−17.7449	−0.936538	−0.468269	+	0.883586i	$0.655122\pi$
$360$	0	0
$361$	−14.8313	−0.780597
$362$	8.15803	0.428777
$363$	−42.2822	−2.21924
$364$	6.32843	0.331700
$365$	0	0
$366$	52.8129	2.76058
$367$	36.3041	1.89506	0.947529	−	0.319669i	$-0.103572\pi$
$367$	36.3041	1.89506	0.947529	+	0.319669i	$0.103572\pi$
$368$	0.428757	0.0223505
$369$	5.17530	0.269415
$370$	0	0
$371$	16.3947	0.851169
$372$	−19.4968	−1.01086
$373$	−5.89146	−0.305048	−0.152524	−	0.988300i	$-0.548740\pi$
$373$	−5.89146	−0.305048	−0.152524	+	0.988300i	$0.548740\pi$
$374$	0	0
$375$	0	0
$376$	−1.26521	−0.0652482
$377$	2.62406	0.135146
$378$	32.4124	1.66711
$379$	−29.2735	−1.50368	−0.751838	−	0.659347i	$-0.770833\pi$
$379$	−29.2735	−1.50368	−0.751838	+	0.659347i	$0.770833\pi$
$380$	0	0
$381$	−21.7432	−1.11394
$382$	−39.2663	−2.00904
$383$	−12.3062	−0.628816	−0.314408	−	0.949288i	$-0.601806\pi$
$383$	−12.3062	−0.628816	−0.314408	+	0.949288i	$0.601806\pi$
$384$	−9.48830	−0.484198
$385$	0	0
$386$	24.0488	1.22405
$387$	72.8131	3.70130
$388$	9.79708	0.497371
$389$	−2.12579	−0.107782	−0.0538910	−	0.998547i	$-0.517162\pi$
$389$	−2.12579	−0.107782	−0.0538910	+	0.998547i	$0.517162\pi$
$390$	0	0
$391$	0	0
$392$	2.09214	0.105669
$393$	−6.07254	−0.306319
$394$	39.1403	1.97186
$395$	0	0
$396$	77.4743	3.89323
$397$	−25.1983	−1.26467	−0.632334	−	0.774696i	$-0.717903\pi$
$397$	−25.1983	−1.26467	−0.632334	+	0.774696i	$0.717903\pi$
$398$	−21.3691	−1.07114
$399$	−7.68235	−0.384599
$400$	0	0
$401$	32.8681	1.64135	0.820677	−	0.571392i	$-0.193596\pi$
$401$	32.8681	1.64135	0.820677	+	0.571392i	$0.193596\pi$
$402$	32.1885	1.60542
$403$	6.88587	0.343010
$404$	38.8122	1.93098
$405$	0	0
$406$	−2.56541	−0.127319
$407$	11.3945	0.564803
$408$	0	0
$409$	−16.5721	−0.819437	−0.409719	−	0.912212i	$-0.634373\pi$
$409$	−16.5721	−0.819437	−0.409719	+	0.912212i	$0.634373\pi$
$410$	0	0
$411$	61.8796	3.05229
$412$	24.7361	1.21866
$413$	1.78630	0.0878979
$414$	1.75551	0.0862785
$415$	0	0
$416$	19.9803	0.979613
$417$	26.4502	1.29527
$418$	−20.5537	−1.00531
$419$	5.31866	0.259833	0.129917	−	0.991525i	$-0.458529\pi$
$419$	5.31866	0.259833	0.129917	+	0.991525i	$0.458529\pi$
$420$	0	0
$421$	3.72116	0.181358	0.0906791	−	0.995880i	$-0.471096\pi$
$421$	3.72116	0.181358	0.0906791	+	0.995880i	$0.471096\pi$
$422$	−44.5473	−2.16853
$423$	24.4842	1.19046
$424$	−5.18933	−0.252016
$425$	0	0
$426$	−43.3705	−2.10131
$427$	9.51507	0.460466
$428$	−28.1215	−1.35930
$429$	−38.7440	−1.87058
$430$	0	0
$431$	−0.652621	−0.0314357	−0.0157178	−	0.999876i	$-0.505003\pi$
$431$	−0.652621	−0.0314357	−0.0157178	+	0.999876i	$0.505003\pi$
$432$	48.4900	2.33297
$433$	−5.31477	−0.255411	−0.127706	−	0.991812i	$-0.540761\pi$
$433$	−5.31477	−0.255411	−0.127706	+	0.991812i	$0.540761\pi$
$434$	−6.73196	−0.323145
$435$	0	0
$436$	−24.7052	−1.18316
$437$	−0.243012	−0.0116248
$438$	23.9660	1.14514
$439$	−5.53592	−0.264215	−0.132107	−	0.991235i	$-0.542174\pi$
$439$	−5.53592	−0.264215	−0.132107	+	0.991235i	$0.542174\pi$
$440$	0	0
$441$	−40.4869	−1.92795
$442$	0	0
$443$	−14.9894	−0.712169	−0.356084	−	0.934454i	$-0.615888\pi$
$443$	−14.9894	−0.712169	−0.356084	+	0.934454i	$0.615888\pi$
$444$	−16.1425	−0.766087
$445$	0	0
$446$	40.9390	1.93852
$447$	−15.7701	−0.745900
$448$	−11.0507	−0.522097
$449$	20.7816	0.980743	0.490371	−	0.871514i	$-0.336861\pi$
$449$	20.7816	0.980743	0.490371	+	0.871514i	$0.336861\pi$
$450$	0	0
$451$	3.53227	0.166328
$452$	12.7680	0.600558
$453$	2.27771	0.107016
$454$	−28.1814	−1.32262
$455$	0	0
$456$	2.43166	0.113873
$457$	−15.6444	−0.731816	−0.365908	−	0.930651i	$-0.619242\pi$
$457$	−15.6444	−0.731816	−0.365908	+	0.930651i	$0.619242\pi$
$458$	−7.44445	−0.347856
$459$	0	0
$460$	0	0
$461$	−7.39362	−0.344355	−0.172178	−	0.985066i	$-0.555080\pi$
$461$	−7.39362	−0.344355	−0.172178	+	0.985066i	$0.555080\pi$
$462$	37.8780	1.76225
$463$	−17.2150	−0.800049	−0.400025	−	0.916504i	$-0.630998\pi$
$463$	−17.2150	−0.800049	−0.400025	+	0.916504i	$0.630998\pi$
$464$	−3.83793	−0.178172
$465$	0	0
$466$	28.9595	1.34152
$467$	36.3391	1.68157	0.840787	−	0.541367i	$-0.182093\pi$
$467$	36.3391	1.68157	0.840787	+	0.541367i	$0.182093\pi$
$468$	38.7640	1.79187
$469$	5.79925	0.267785
$470$	0	0
$471$	18.8085	0.866651
$472$	−0.565409	−0.0260250
$473$	49.6967	2.28506
$474$	53.4620	2.45559
$475$	0	0
$476$	0	0
$477$	100.423	4.59807
$478$	46.3451	2.11978
$479$	−30.3678	−1.38754	−0.693770	−	0.720196i	$-0.744052\pi$
$479$	−30.3678	−1.38754	−0.693770	+	0.720196i	$0.744052\pi$
$480$	0	0
$481$	5.70118	0.259952
$482$	12.3623	0.563087
$483$	0.447843	0.0203776
$484$	28.8735	1.31243
$485$	0	0
$486$	57.1364	2.59176
$487$	−25.2832	−1.14569	−0.572846	−	0.819663i	$-0.694161\pi$
$487$	−25.2832	−1.14569	−0.572846	+	0.819663i	$0.694161\pi$
$488$	−3.01176	−0.136336
$489$	18.8422	0.852074
$490$	0	0
$491$	−28.7941	−1.29946	−0.649730	−	0.760165i	$-0.725118\pi$
$491$	−28.7941	−1.29946	−0.649730	+	0.760165i	$0.725118\pi$
$492$	−5.00413	−0.225604
$493$	0	0
$494$	−10.2840	−0.462697
$495$	0	0
$496$	−10.0712	−0.452212
$497$	−7.81386	−0.350500
$498$	39.7916	1.78311
$499$	14.9561	0.669527	0.334764	−	0.942302i	$-0.391344\pi$
$499$	14.9561	0.669527	0.334764	+	0.942302i	$0.391344\pi$
$500$	0	0
$501$	−34.4121	−1.53742
$502$	1.44575	0.0645272
$503$	32.5784	1.45260	0.726300	−	0.687378i	$-0.241239\pi$
$503$	32.5784	1.45260	0.726300	+	0.687378i	$0.241239\pi$
$504$	−3.16483	−0.140973
$505$	0	0
$506$	1.19818	0.0532655
$507$	22.1581	0.984075
$508$	14.8479	0.658769
$509$	−33.1868	−1.47098	−0.735490	−	0.677535i	$-0.763048\pi$
$509$	−33.1868	−1.47098	−0.735490	+	0.677535i	$0.763048\pi$
$510$	0	0
$511$	4.31784	0.191010
$512$	−31.9081	−1.41015
$513$	−27.4833	−1.21342
$514$	−30.8239	−1.35958
$515$	0	0
$516$	−70.4049	−3.09940
$517$	16.7111	0.734952
$518$	−5.57375	−0.244897
$519$	−30.9882	−1.36023
$520$	0	0
$521$	28.6670	1.25593	0.627963	−	0.778244i	$-0.283889\pi$
$521$	28.6670	1.25593	0.627963	+	0.778244i	$0.283889\pi$
$522$	−15.7141	−0.687787
$523$	42.6141	1.86338	0.931692	−	0.363248i	$-0.118332\pi$
$523$	42.6141	1.86338	0.931692	+	0.363248i	$0.118332\pi$
$524$	4.14680	0.181154
$525$	0	0
$526$	−29.7130	−1.29555
$527$	0	0
$528$	56.6667	2.46610
$529$	−22.9858	−0.999384
$530$	0	0
$531$	10.9417	0.474831
$532$	5.24610	0.227447
$533$	1.76736	0.0765528
$534$	55.2996	2.39305
$535$	0	0
$536$	−1.83561	−0.0792864
$537$	−50.2216	−2.16722
$538$	57.9983	2.50048
$539$	−27.6333	−1.19025
$540$	0	0
$541$	−22.0963	−0.949993	−0.474997	−	0.879988i	$-0.657551\pi$
$541$	−22.0963	−0.949993	−0.474997	+	0.879988i	$0.657551\pi$
$542$	−19.0123	−0.816648
$543$	12.7480	0.547068
$544$	0	0
$545$	0	0
$546$	18.9521	0.811077
$547$	−38.6850	−1.65405	−0.827026	−	0.562163i	$-0.809969\pi$
$547$	−38.6850	−1.65405	−0.827026	+	0.562163i	$0.809969\pi$
$548$	−42.2561	−1.80509
$549$	58.2834	2.48747
$550$	0	0
$551$	2.17528	0.0926699
$552$	−0.141754	−0.00603344
$553$	9.63200	0.409594
$554$	45.8221	1.94679
$555$	0	0
$556$	−18.0622	−0.766009
$557$	−33.9638	−1.43909	−0.719546	−	0.694445i	$-0.755650\pi$
$557$	−33.9638	−1.43909	−0.719546	+	0.694445i	$0.755650\pi$
$558$	−41.2358	−1.74565
$559$	24.8656	1.05170
$560$	0	0
$561$	0	0
$562$	−63.9101	−2.69589
$563$	2.27374	0.0958267	0.0479133	−	0.998851i	$-0.484743\pi$
$563$	2.27374	0.0958267	0.0479133	+	0.998851i	$0.484743\pi$
$564$	−23.6744	−0.996873
$565$	0	0
$566$	24.0201	1.00964
$567$	25.1728	1.05716
$568$	2.47329	0.103777
$569$	−5.68027	−0.238129	−0.119065	−	0.992887i	$-0.537990\pi$
$569$	−5.68027	−0.238129	−0.119065	+	0.992887i	$0.537990\pi$
$570$	0	0
$571$	12.9627	0.542471	0.271236	−	0.962513i	$-0.412568\pi$
$571$	12.9627	0.542471	0.271236	+	0.962513i	$0.412568\pi$
$572$	26.4574	1.10624
$573$	−61.3587	−2.56330
$574$	−1.72785	−0.0721193
$575$	0	0
$576$	−67.6898	−2.82041
$577$	−27.7816	−1.15656	−0.578281	−	0.815838i	$-0.696276\pi$
$577$	−27.7816	−1.15656	−0.578281	+	0.815838i	$0.696276\pi$
$578$	0	0
$579$	37.5793	1.56174
$580$	0	0
$581$	7.16908	0.297424
$582$	29.3399	1.21618
$583$	68.5415	2.83870
$584$	−1.36671	−0.0565547
$585$	0	0
$586$	−54.3791	−2.24638
$587$	6.89170	0.284451	0.142226	−	0.989834i	$-0.454574\pi$
$587$	6.89170	0.284451	0.142226	+	0.989834i	$0.454574\pi$
$588$	39.1478	1.61443
$589$	5.70820	0.235202
$590$	0	0
$591$	61.1617	2.51585
$592$	−8.33851	−0.342711
$593$	13.9274	0.571930	0.285965	−	0.958240i	$-0.407686\pi$
$593$	13.9274	0.571930	0.285965	+	0.958240i	$0.407686\pi$
$594$	135.507	5.55992
$595$	0	0
$596$	10.7690	0.441117
$597$	−33.3920	−1.36664
$598$	0.599504	0.0245155
$599$	19.3270	0.789681	0.394841	−	0.918750i	$-0.370800\pi$
$599$	19.3270	0.789681	0.394841	+	0.918750i	$0.370800\pi$
$600$	0	0
$601$	42.1759	1.72039	0.860195	−	0.509965i	$-0.170342\pi$
$601$	42.1759	1.72039	0.860195	+	0.509965i	$0.170342\pi$
$602$	−24.3098	−0.990793
$603$	35.5226	1.44659
$604$	−1.55539	−0.0632881
$605$	0	0
$606$	116.233	4.72165
$607$	−18.8481	−0.765021	−0.382511	−	0.923951i	$-0.624940\pi$
$607$	−18.8481	−0.765021	−0.382511	+	0.923951i	$0.624940\pi$
$608$	16.5631	0.671723
$609$	−4.00878	−0.162444
$610$	0	0
$611$	8.36132	0.338263
$612$	0	0
$613$	−16.1284	−0.651420	−0.325710	−	0.945470i	$-0.605603\pi$
$613$	−16.1284	−0.651420	−0.325710	+	0.945470i	$0.605603\pi$
$614$	38.2385	1.54318
$615$	0	0
$616$	−2.16007	−0.0870318
$617$	37.7508	1.51979	0.759894	−	0.650046i	$-0.225251\pi$
$617$	37.7508	1.51979	0.759894	+	0.650046i	$0.225251\pi$
$618$	74.0787	2.97988
$619$	−48.7190	−1.95818	−0.979091	−	0.203425i	$-0.934793\pi$
$619$	−48.7190	−1.95818	−0.979091	+	0.203425i	$0.934793\pi$
$620$	0	0
$621$	1.60214	0.0642916
$622$	16.2916	0.653232
$623$	9.96308	0.399162
$624$	28.3530	1.13503
$625$	0	0
$626$	16.3351	0.652880
$627$	−32.1178	−1.28266
$628$	−12.8439	−0.512528
$629$	0	0
$630$	0	0
$631$	−39.1517	−1.55861	−0.779303	−	0.626647i	$-0.784427\pi$
$631$	−39.1517	−1.55861	−0.779303	+	0.626647i	$0.784427\pi$
$632$	−3.04878	−0.121274
$633$	−69.6109	−2.76678
$634$	31.9683	1.26962
$635$	0	0
$636$	−97.1021	−3.85035
$637$	−13.8262	−0.547815
$638$	−10.7253	−0.424617
$639$	−47.8628	−1.89342
$640$	0	0
$641$	−49.9352	−1.97232	−0.986161	−	0.165793i	$-0.946982\pi$
$641$	−49.9352	−1.97232	−0.986161	+	0.165793i	$0.946982\pi$
$642$	−84.2170	−3.32378
$643$	15.6383	0.616714	0.308357	−	0.951271i	$-0.400221\pi$
$643$	15.6383	0.616714	0.308357	+	0.951271i	$0.400221\pi$
$644$	−0.305821	−0.0120511
$645$	0	0
$646$	0	0
$647$	24.8287	0.976118	0.488059	−	0.872811i	$-0.337705\pi$
$647$	24.8287	0.976118	0.488059	+	0.872811i	$0.337705\pi$
$648$	−7.96784	−0.313006
$649$	7.46800	0.293145
$650$	0	0
$651$	−10.5196	−0.412294
$652$	−12.8669	−0.503907
$653$	16.8005	0.657453	0.328727	−	0.944425i	$-0.393381\pi$
$653$	16.8005	0.657453	0.328727	+	0.944425i	$0.393381\pi$
$654$	−73.9862	−2.89309
$655$	0	0
$656$	−2.58493	−0.100924
$657$	26.4484	1.03185
$658$	−8.17443	−0.318673
$659$	13.1974	0.514097	0.257048	−	0.966399i	$-0.417250\pi$
$659$	13.1974	0.514097	0.257048	+	0.966399i	$0.417250\pi$
$660$	0	0
$661$	−38.1892	−1.48539	−0.742694	−	0.669631i	$-0.766452\pi$
$661$	−38.1892	−1.48539	−0.742694	+	0.669631i	$0.766452\pi$
$662$	9.06062	0.352151
$663$	0	0
$664$	−2.26920	−0.0880620
$665$	0	0
$666$	−34.1413	−1.32295
$667$	−0.126808	−0.00491001
$668$	23.4992	0.909212
$669$	63.9724	2.47331
$670$	0	0
$671$	39.7798	1.53568
$672$	−30.5239	−1.17748
$673$	24.3655	0.939219	0.469610	−	0.882874i	$-0.344395\pi$
$673$	24.3655	0.939219	0.469610	+	0.882874i	$0.344395\pi$
$674$	12.3441	0.475476
$675$	0	0
$676$	−15.1312	−0.581971
$677$	−46.2667	−1.77817	−0.889087	−	0.457737i	$-0.848660\pi$
$677$	−46.2667	−1.77817	−0.889087	+	0.457737i	$0.848660\pi$
$678$	38.2372	1.46849
$679$	5.28604	0.202859
$680$	0	0
$681$	−44.0371	−1.68750
$682$	−28.1444	−1.07771
$683$	16.4922	0.631056	0.315528	−	0.948916i	$-0.397818\pi$
$683$	16.4922	0.631056	0.315528	+	0.948916i	$0.397818\pi$
$684$	32.1343	1.22869
$685$	0	0
$686$	30.3726	1.15963
$687$	−11.6329	−0.443823
$688$	−36.3682	−1.38653
$689$	34.2945	1.30652
$690$	0	0
$691$	−25.8554	−0.983584	−0.491792	−	0.870713i	$-0.663658\pi$
$691$	−25.8554	−0.983584	−0.491792	+	0.870713i	$0.663658\pi$
$692$	21.1611	0.804425
$693$	41.8015	1.58791
$694$	−0.190387	−0.00722698
$695$	0	0
$696$	1.26888	0.0480968
$697$	0	0
$698$	−19.5143	−0.738628
$699$	45.2529	1.71162
$700$	0	0
$701$	8.67606	0.327690	0.163845	−	0.986486i	$-0.447610\pi$
$701$	8.67606	0.327690	0.163845	+	0.986486i	$0.447610\pi$
$702$	67.8005	2.55896
$703$	4.72613	0.178249
$704$	−46.1999	−1.74123
$705$	0	0
$706$	35.2031	1.32489
$707$	20.9412	0.787576
$708$	−10.5799	−0.397615
$709$	35.9232	1.34912	0.674562	−	0.738218i	$-0.264333\pi$
$709$	35.9232	1.34912	0.674562	+	0.738218i	$0.264333\pi$
$710$	0	0
$711$	58.9996	2.21266
$712$	−3.15357	−0.118185
$713$	−0.332760	−0.0124620
$714$	0	0
$715$	0	0
$716$	34.2952	1.28167
$717$	72.4202	2.70458
$718$	36.2892	1.35430
$719$	9.75299	0.363725	0.181863	−	0.983324i	$-0.441787\pi$
$719$	9.75299	0.363725	0.181863	+	0.983324i	$0.441787\pi$
$720$	0	0
$721$	13.3464	0.497047
$722$	30.3309	1.12880
$723$	19.3177	0.718432
$724$	−8.70529	−0.323530
$725$	0	0
$726$	86.4693	3.20918
$727$	11.0694	0.410542	0.205271	−	0.978705i	$-0.434192\pi$
$727$	11.0694	0.410542	0.205271	+	0.978705i	$0.434192\pi$
$728$	−1.08078	−0.0400565
$729$	25.1447	0.931285
$730$	0	0
$731$	0	0
$732$	−56.3557	−2.08297
$733$	−42.1634	−1.55734	−0.778670	−	0.627434i	$-0.784105\pi$
$733$	−42.1634	−1.55734	−0.778670	+	0.627434i	$0.784105\pi$
$734$	−74.2438	−2.74039
$735$	0	0
$736$	−0.965546	−0.0355905
$737$	24.2451	0.893078
$738$	−10.5838	−0.389593
$739$	47.1617	1.73487	0.867435	−	0.497550i	$-0.165767\pi$
$739$	47.1617	1.73487	0.867435	+	0.497550i	$0.165767\pi$
$740$	0	0
$741$	−16.0700	−0.590346
$742$	−33.5279	−1.23085
$743$	−31.7495	−1.16478	−0.582388	−	0.812911i	$-0.697882\pi$
$743$	−31.7495	−1.16478	−0.582388	+	0.812911i	$0.697882\pi$
$744$	3.32971	0.122073
$745$	0	0
$746$	12.0483	0.441121
$747$	43.9133	1.60670
$748$	0	0
$749$	−15.1730	−0.554409
$750$	0	0
$751$	22.1885	0.809668	0.404834	−	0.914390i	$-0.367329\pi$
$751$	22.1885	0.809668	0.404834	+	0.914390i	$0.367329\pi$
$752$	−12.2292	−0.445954
$753$	2.25918	0.0823289
$754$	−5.36634	−0.195431
$755$	0	0
$756$	−34.5867	−1.25791
$757$	13.0640	0.474821	0.237410	−	0.971409i	$-0.423701\pi$
$757$	13.0640	0.474821	0.237410	+	0.971409i	$0.423701\pi$
$758$	59.8658	2.17442
$759$	1.87230	0.0679604
$760$	0	0
$761$	18.9811	0.688065	0.344033	−	0.938958i	$-0.388207\pi$
$761$	18.9811	0.688065	0.344033	+	0.938958i	$0.388207\pi$
$762$	44.4659	1.61083
$763$	−13.3298	−0.482569
$764$	41.9004	1.51590
$765$	0	0
$766$	25.1668	0.909313
$767$	3.73659	0.134920
$768$	−40.5812	−1.46435
$769$	−22.0057	−0.793544	−0.396772	−	0.917917i	$-0.629870\pi$
$769$	−22.0057	−0.793544	−0.396772	+	0.917917i	$0.629870\pi$
$770$	0	0
$771$	−48.1663	−1.73467
$772$	−25.6620	−0.923596
$773$	13.2410	0.476247	0.238123	−	0.971235i	$-0.423468\pi$
$773$	13.2410	0.476247	0.238123	+	0.971235i	$0.423468\pi$
$774$	−148.907	−5.35234
$775$	0	0
$776$	−1.67317	−0.0600632
$777$	−8.70970	−0.312459
$778$	4.34736	0.155860
$779$	1.46509	0.0524924
$780$	0	0
$781$	−32.6676	−1.16894
$782$	0	0
$783$	−14.3412	−0.512514
$784$	20.2221	0.722219
$785$	0	0
$786$	12.4187	0.442959
$787$	−11.9863	−0.427266	−0.213633	−	0.976914i	$-0.568530\pi$
$787$	−11.9863	−0.427266	−0.213633	+	0.976914i	$0.568530\pi$
$788$	−41.7659	−1.48785
$789$	−46.4304	−1.65297
$790$	0	0
$791$	6.88902	0.244945
$792$	−13.2312	−0.470152
$793$	19.9037	0.706801
$794$	51.5319	1.82880
$795$	0	0
$796$	22.8026	0.808218
$797$	23.3187	0.825989	0.412995	−	0.910734i	$-0.364483\pi$
$797$	23.3187	0.825989	0.412995	+	0.910734i	$0.364483\pi$
$798$	15.7108	0.556157
$799$	0	0
$800$	0	0
$801$	61.0276	2.15630
$802$	−67.2170	−2.37351
$803$	18.0517	0.637030
$804$	−34.3478	−1.21135
$805$	0	0
$806$	−14.0820	−0.496016
$807$	90.6298	3.19032
$808$	−6.62844	−0.233188
$809$	−30.8141	−1.08337	−0.541683	−	0.840583i	$-0.682213\pi$
$809$	−30.8141	−1.08337	−0.541683	+	0.840583i	$0.682213\pi$
$810$	0	0
$811$	38.6069	1.35567	0.677836	−	0.735213i	$-0.262918\pi$
$811$	38.6069	1.35567	0.677836	+	0.735213i	$0.262918\pi$
$812$	2.73750	0.0960675
$813$	−29.7091	−1.04194
$814$	−23.3023	−0.816745
$815$	0	0
$816$	0	0
$817$	20.6129	0.721154
$818$	33.8908	1.18496
$819$	20.9152	0.730837
$820$	0	0
$821$	−21.5899	−0.753493	−0.376747	−	0.926316i	$-0.622957\pi$
$821$	−21.5899	−0.753493	−0.376747	+	0.926316i	$0.622957\pi$
$822$	−126.547	−4.41383
$823$	−25.4053	−0.885573	−0.442787	−	0.896627i	$-0.646010\pi$
$823$	−25.4053	−0.885573	−0.442787	+	0.896627i	$0.646010\pi$
$824$	−4.22449	−0.147167
$825$	0	0
$826$	−3.65307	−0.127107
$827$	−14.8822	−0.517506	−0.258753	−	0.965943i	$-0.583312\pi$
$827$	−14.8822	−0.517506	−0.258753	+	0.965943i	$0.583312\pi$
$828$	−1.87327	−0.0651007
$829$	−29.8580	−1.03701	−0.518505	−	0.855075i	$-0.673511\pi$
$829$	−29.8580	−1.03701	−0.518505	+	0.855075i	$0.673511\pi$
$830$	0	0
$831$	71.6029	2.48388
$832$	−23.1160	−0.801402
$833$	0	0
$834$	−54.0920	−1.87305
$835$	0	0
$836$	21.9325	0.758550
$837$	−37.6333	−1.30080
$838$	−10.8769	−0.375737
$839$	32.2189	1.11232	0.556161	−	0.831075i	$-0.312274\pi$
$839$	32.2189	1.11232	0.556161	+	0.831075i	$0.312274\pi$
$840$	0	0
$841$	−27.8649	−0.960859
$842$	−7.60996	−0.262257
$843$	−99.8677	−3.43963
$844$	47.5357	1.63624
$845$	0	0
$846$	−50.0715	−1.72149
$847$	15.5788	0.535293
$848$	−50.1589	−1.72246
$849$	37.5345	1.28818
$850$	0	0
$851$	−0.275510	−0.00944435
$852$	46.2798	1.58552
$853$	−6.64528	−0.227530	−0.113765	−	0.993508i	$-0.536291\pi$
$853$	−6.64528	−0.227530	−0.113765	+	0.993508i	$0.536291\pi$
$854$	−19.4588	−0.665867
$855$	0	0
$856$	4.80265	0.164151
$857$	−34.5296	−1.17951	−0.589754	−	0.807583i	$-0.700775\pi$
$857$	−34.5296	−1.17951	−0.589754	+	0.807583i	$0.700775\pi$
$858$	79.2336	2.70499
$859$	46.7639	1.59556	0.797782	−	0.602946i	$-0.206007\pi$
$859$	46.7639	1.59556	0.797782	+	0.602946i	$0.206007\pi$
$860$	0	0
$861$	−2.69999	−0.0920155
$862$	1.33464	0.0454582
$863$	40.3468	1.37342	0.686711	−	0.726930i	$-0.259054\pi$
$863$	40.3468	1.37342	0.686711	+	0.726930i	$0.259054\pi$
$864$	−109.198	−3.71499
$865$	0	0
$866$	10.8690	0.369343
$867$	0	0
$868$	7.18356	0.243826
$869$	40.2687	1.36602
$870$	0	0
$871$	12.1309	0.411041
$872$	4.21921	0.142880
$873$	32.3790	1.09586
$874$	0.496972	0.0168103
$875$	0	0
$876$	−25.5736	−0.864053
$877$	−15.5336	−0.524531	−0.262266	−	0.964996i	$-0.584470\pi$
$877$	−15.5336	−0.524531	−0.262266	+	0.964996i	$0.584470\pi$
$878$	11.3212	0.382073
$879$	−84.9742	−2.86611
$880$	0	0
$881$	−33.1990	−1.11850	−0.559251	−	0.828998i	$-0.688911\pi$
$881$	−33.1990	−1.11850	−0.559251	+	0.828998i	$0.688911\pi$
$882$	82.7978	2.78795
$883$	−34.1078	−1.14782	−0.573909	−	0.818919i	$-0.694574\pi$
$883$	−34.1078	−1.14782	−0.573909	+	0.818919i	$0.694574\pi$
$884$	0	0
$885$	0	0
$886$	30.6541	1.02985
$887$	32.3508	1.08623	0.543116	−	0.839658i	$-0.317244\pi$
$887$	32.3508	1.08623	0.543116	+	0.839658i	$0.317244\pi$
$888$	2.75685	0.0925137
$889$	8.01122	0.268688
$890$	0	0
$891$	105.240	3.52569
$892$	−43.6852	−1.46269
$893$	6.93131	0.231948
$894$	32.2507	1.07862
$895$	0	0
$896$	3.49594	0.116791
$897$	0.936801	0.0312789
$898$	−42.4994	−1.41822
$899$	2.97864	0.0993431
$900$	0	0
$901$	0	0
$902$	−7.22367	−0.240522
$903$	−37.9872	−1.26413
$904$	−2.18055	−0.0725241
$905$	0	0
$906$	−4.65803	−0.154753
$907$	27.9582	0.928336	0.464168	−	0.885747i	$-0.346353\pi$
$907$	27.9582	0.928336	0.464168	+	0.885747i	$0.346353\pi$
$908$	30.0719	0.997970
$909$	128.273	4.25454
$910$	0	0
$911$	12.3093	0.407826	0.203913	−	0.978989i	$-0.434634\pi$
$911$	12.3093	0.407826	0.203913	+	0.978989i	$0.434634\pi$
$912$	23.5039	0.778291
$913$	29.9719	0.991925
$914$	31.9937	1.05826
$915$	0	0
$916$	7.94384	0.262472
$917$	2.23742	0.0738860
$918$	0	0
$919$	20.0893	0.662683	0.331342	−	0.943511i	$-0.392499\pi$
$919$	20.0893	0.662683	0.331342	+	0.943511i	$0.392499\pi$
$920$	0	0
$921$	59.7526	1.96891
$922$	15.1203	0.497962
$923$	−16.3451	−0.538005
$924$	−40.4190	−1.32969
$925$	0	0
$926$	35.2056	1.15693
$927$	81.7518	2.68508
$928$	8.64290	0.283717
$929$	6.06146	0.198870	0.0994350	−	0.995044i	$-0.468296\pi$
$929$	6.06146	0.198870	0.0994350	+	0.995044i	$0.468296\pi$
$930$	0	0
$931$	−11.4616	−0.375638
$932$	−30.9022	−1.01223
$933$	25.4577	0.833446
$934$	−74.3154	−2.43167
$935$	0	0
$936$	−6.62021	−0.216388
$937$	34.1763	1.11649	0.558246	−	0.829675i	$-0.311474\pi$
$937$	34.1763	1.11649	0.558246	+	0.829675i	$0.311474\pi$
$938$	−11.8598	−0.387236
$939$	25.5256	0.832997
$940$	0	0
$941$	−36.0225	−1.17430	−0.587149	−	0.809479i	$-0.699750\pi$
$941$	−36.0225	−1.17430	−0.587149	+	0.809479i	$0.699750\pi$
$942$	−38.4644	−1.25324
$943$	−0.0854076	−0.00278125
$944$	−5.46511	−0.177874
$945$	0	0
$946$	−101.632	−3.30435
$947$	39.4153	1.28083	0.640413	−	0.768030i	$-0.278763\pi$
$947$	39.4153	1.28083	0.640413	+	0.768030i	$0.278763\pi$
$948$	−57.0483	−1.85284
$949$	9.03209	0.293194
$950$	0	0
$951$	49.9545	1.61989
$952$	0	0
$953$	6.36027	0.206029	0.103015	−	0.994680i	$-0.467151\pi$
$953$	6.36027	0.206029	0.103015	+	0.994680i	$0.467151\pi$
$954$	−205.371	−6.64914
$955$	0	0
$956$	−49.4540	−1.59946
$957$	−16.7596	−0.541760
$958$	62.1038	2.00648
$959$	−22.7994	−0.736231
$960$	0	0
$961$	−23.1837	−0.747861
$962$	−11.6592	−0.375908
$963$	−92.9403	−2.99496
$964$	−13.1916	−0.424872
$965$	0	0
$966$	−0.915862	−0.0294674
$967$	−34.3424	−1.10438	−0.552189	−	0.833719i	$-0.686207\pi$
$967$	−34.3424	−1.10438	−0.552189	+	0.833719i	$0.686207\pi$
$968$	−4.93109	−0.158491
$969$	0	0
$970$	0	0
$971$	−7.98605	−0.256285	−0.128142	−	0.991756i	$-0.540901\pi$
$971$	−7.98605	−0.256285	−0.128142	+	0.991756i	$0.540901\pi$
$972$	−60.9692	−1.95559
$973$	−9.74552	−0.312427
$974$	51.7055	1.65675
$975$	0	0
$976$	−29.1110	−0.931821
$977$	15.0989	0.483056	0.241528	−	0.970394i	$-0.422351\pi$
$977$	15.0989	0.483056	0.241528	+	0.970394i	$0.422351\pi$
$978$	−38.5333	−1.23216
$979$	41.6528	1.33123
$980$	0	0
$981$	−81.6497	−2.60688
$982$	58.8855	1.87911
$983$	−50.2932	−1.60410	−0.802052	−	0.597254i	$-0.796258\pi$
$983$	−50.2932	−1.60410	−0.802052	+	0.597254i	$0.796258\pi$
$984$	0.854618	0.0272442
$985$	0	0
$986$	0	0
$987$	−12.7736	−0.406588
$988$	10.9738	0.349124
$989$	−1.20163	−0.0382096
$990$	0	0
$991$	13.0116	0.413326	0.206663	−	0.978412i	$-0.433740\pi$
$991$	13.0116	0.413326	0.206663	+	0.978412i	$0.433740\pi$
$992$	22.6801	0.720094
$993$	14.1584	0.449302
$994$	15.9798	0.506847
$995$	0	0
$996$	−42.4609	−1.34543
$997$	−9.34885	−0.296081	−0.148040	−	0.988981i	$-0.547297\pi$
$997$	−9.34885	−0.296081	−0.148040	+	0.988981i	$0.547297\pi$
$998$	−30.5860	−0.968183
$999$	−31.1586	−0.985814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bq.1.2		12
5.4	even	2		1445.2.a.q.1.11		12
17.5	odd	16		425.2.m.b.76.6		24
17.7	odd	16		425.2.m.b.151.6		24
17.16	even	2		7225.2.a.bs.1.2		12
85.4	even	4		1445.2.d.j.866.4		24
85.7	even	16		425.2.n.f.49.1		24
85.22	even	16		425.2.n.c.399.6		24
85.24	odd	16		85.2.l.a.66.1	✓	24
85.39	odd	16		85.2.l.a.76.1	yes	24
85.58	even	16		425.2.n.c.49.6		24
85.64	even	4		1445.2.d.j.866.3		24
85.73	even	16		425.2.n.f.399.1		24
85.84	even	2		1445.2.a.p.1.11		12
255.194	even	16		765.2.be.b.406.6		24
255.209	even	16		765.2.be.b.586.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.l.a.66.1	✓	24	85.24	odd	16
85.2.l.a.76.1	yes	24	85.39	odd	16
425.2.m.b.76.6		24	17.5	odd	16
425.2.m.b.151.6		24	17.7	odd	16
425.2.n.c.49.6		24	85.58	even	16
425.2.n.c.399.6		24	85.22	even	16
425.2.n.f.49.1		24	85.7	even	16
425.2.n.f.399.1		24	85.73	even	16
765.2.be.b.406.6		24	255.194	even	16
765.2.be.b.586.6		24	255.209	even	16
1445.2.a.p.1.11		12	85.84	even	2
1445.2.a.q.1.11		12	5.4	even	2
1445.2.d.j.866.3		24	85.64	even	4
1445.2.d.j.866.4		24	85.4	even	4
7225.2.a.bq.1.2		12	1.1	even	1	trivial
7225.2.a.bs.1.2		12	17.16	even	2

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:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q-2.04505 q^{2} -3.19566 q^{3} +2.18224 q^{4} +6.53528 q^{6} +1.17743 q^{7} -0.372688 q^{8} +7.21221 q^{9} +4.92251 q^{11} -6.97368 q^{12} +2.46296 q^{13} -2.40791 q^{14} -3.60231 q^{16} -14.7494 q^{18} +2.04173 q^{19} -3.76267 q^{21} -10.0668 q^{22} -0.119023 q^{23} +1.19098 q^{24} -5.03689 q^{26} -13.4608 q^{27} +2.56944 q^{28} +1.06541 q^{29} +2.79577 q^{31} +8.11229 q^{32} -15.7307 q^{33} +15.7388 q^{36} +2.31477 q^{37} -4.17544 q^{38} -7.87078 q^{39} +0.717574 q^{41} +7.69485 q^{42} +10.0958 q^{43} +10.7421 q^{44} +0.243408 q^{46} +3.39482 q^{47} +11.5117 q^{48} -5.61365 q^{49} +5.37477 q^{52} +13.9241 q^{53} +27.5280 q^{54} -0.438815 q^{56} -6.52466 q^{57} -2.17882 q^{58} +1.51711 q^{59} +8.08120 q^{61} -5.71749 q^{62} +8.49190 q^{63} -9.38544 q^{64} +32.1700 q^{66} +4.92534 q^{67} +0.380355 q^{69} -6.63635 q^{71} -2.68790 q^{72} +3.66716 q^{73} -4.73382 q^{74} +4.45554 q^{76} +5.79593 q^{77} +16.0962 q^{78} +8.18051 q^{79} +21.3794 q^{81} -1.46748 q^{82} +6.08874 q^{83} -8.21104 q^{84} -20.6464 q^{86} -3.40468 q^{87} -1.83456 q^{88} +8.46170 q^{89} +2.89997 q^{91} -0.259736 q^{92} -8.93431 q^{93} -6.94259 q^{94} -25.9241 q^{96} +4.48946 q^{97} +11.4802 q^{98} +35.5022 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{2} - 8 q^{3} + 12 q^{4} + 8 q^{6} - 16 q^{7} + 12 q^{8} + 12 q^{9} + 16 q^{11} - 16 q^{12} + 8 q^{13} - 16 q^{14} + 12 q^{16} - 4 q^{18} + 16 q^{21} - 16 q^{22} - 16 q^{23} + 16 q^{26} - 32 q^{27}+ \cdots + 56 q^{99}+O(q^{100}) \) 12 * q + 4 * q^2 - 8 * q^3 + 12 * q^4 + 8 * q^6 - 16 * q^7 + 12 * q^8 + 12 * q^9 + 16 * q^11 - 16 * q^12 + 8 * q^13 - 16 * q^14 + 12 * q^16 - 4 * q^18 + 16 * q^21 - 16 * q^22 - 16 * q^23 + 16 * q^26 - 32 * q^27 - 40 * q^28 + 16 * q^29 + 24 * q^31 + 28 * q^32 + 12 * q^36 - 24 * q^37 + 24 * q^38 + 8 * q^39 + 8 * q^41 + 16 * q^43 + 8 * q^44 + 40 * q^46 + 32 * q^47 + 24 * q^48 + 20 * q^49 + 24 * q^52 + 8 * q^54 - 24 * q^56 - 32 * q^57 + 16 * q^58 + 8 * q^59 + 24 * q^61 - 8 * q^62 - 48 * q^63 + 36 * q^64 + 40 * q^66 + 8 * q^67 + 48 * q^69 - 16 * q^71 - 12 * q^72 - 16 * q^73 + 16 * q^76 - 24 * q^77 + 24 * q^78 + 40 * q^79 + 36 * q^81 + 16 * q^82 + 40 * q^83 + 32 * q^84 - 8 * q^86 + 32 * q^87 - 48 * q^88 + 8 * q^89 + 72 * q^91 + 8 * q^92 - 24 * q^93 - 16 * q^94 + 8 * q^96 - 32 * q^97 + 60 * q^98 + 56 * q^99

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