LMFDB - Embedded newform 4225.2.a.br.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, 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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.199374400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} + 10x^{2} - 1 $ x^6 - 8x^4 + 10x^2 - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.330837$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.330837 q^{2} -2.69180 q^{3} -1.89055 q^{4} -0.890547 q^{6} +3.35348 q^{7} -1.28714 q^{8} +4.24581 q^{9} +O(q^{10})$	\(q+0.330837 q^{2} -2.69180 q^{3} -1.89055 q^{4} -0.890547 q^{6} +3.35348 q^{7} -1.28714 q^{8} +4.24581 q^{9} -3.24581 q^{11} +5.08898 q^{12} +1.10945 q^{14} +3.35526 q^{16} -1.94881 q^{17} +1.40467 q^{18} -1.24581 q^{19} -9.02690 q^{21} -1.07383 q^{22} +2.69180 q^{23} +3.46472 q^{24} -3.35348 q^{27} -6.33991 q^{28} +3.00000 q^{29} +3.78109 q^{31} +3.68431 q^{32} +8.73709 q^{33} -0.644737 q^{34} -8.02690 q^{36} +1.94881 q^{37} -0.412160 q^{38} +2.78109 q^{41} -2.98643 q^{42} -8.73709 q^{43} +6.13636 q^{44} +0.890547 q^{46} -6.86960 q^{47} -9.03171 q^{48} +4.24581 q^{49} +5.24581 q^{51} -12.8336 q^{53} -1.10945 q^{54} -4.31638 q^{56} +3.35348 q^{57} +0.992510 q^{58} -2.53528 q^{59} -7.49162 q^{61} +1.25092 q^{62} +14.2382 q^{63} -5.49162 q^{64} +2.89055 q^{66} -4.01515 q^{67} +3.68431 q^{68} -7.24581 q^{69} +5.24581 q^{71} -5.46493 q^{72} -5.46493 q^{73} +0.644737 q^{74} +2.35526 q^{76} -10.8848 q^{77} +13.7811 q^{79} -3.71053 q^{81} +0.920088 q^{82} +8.61955 q^{83} +17.0658 q^{84} -2.89055 q^{86} -8.07541 q^{87} +4.17780 q^{88} +10.3164 q^{89} -5.08898 q^{92} -10.1780 q^{93} -2.27271 q^{94} -9.91745 q^{96} -5.26607 q^{97} +1.40467 q^{98} -13.7811 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4} + 10 q^{6} + 6 q^{9}+O(q^{10}) $ 6 * q + 4 * q^4 + 10 * q^6 + 6 * q^9	$ 6 q + 4 q^{4} + 10 q^{6} + 6 q^{9} + 22 q^{14} + 16 q^{16} + 12 q^{19} - 4 q^{21} + 32 q^{24} + 18 q^{29} - 8 q^{31} - 8 q^{34} + 2 q^{36} - 14 q^{41} + 2 q^{44} - 10 q^{46} + 6 q^{49} + 12 q^{51} - 22 q^{54} + 16 q^{56} - 4 q^{59} - 6 q^{61} + 6 q^{64} + 2 q^{66} - 24 q^{69} + 12 q^{71} + 8 q^{74} + 10 q^{76} + 52 q^{79} - 14 q^{81} + 90 q^{84} - 2 q^{86} + 20 q^{89} + 56 q^{94} + 6 q^{96} - 52 q^{99}+O(q^{100}) $ 6 * q + 4 * q^4 + 10 * q^6 + 6 * q^9 + 22 * q^14 + 16 * q^16 + 12 * q^19 - 4 * q^21 + 32 * q^24 + 18 * q^29 - 8 * q^31 - 8 * q^34 + 2 * q^36 - 14 * q^41 + 2 * q^44 - 10 * q^46 + 6 * q^49 + 12 * q^51 - 22 * q^54 + 16 * q^56 - 4 * q^59 - 6 * q^61 + 6 * q^64 + 2 * q^66 - 24 * q^69 + 12 * q^71 + 8 * q^74 + 10 * q^76 + 52 * q^79 - 14 * q^81 + 90 * q^84 - 2 * q^86 + 20 * q^89 + 56 * q^94 + 6 * q^96 - 52 * q^99

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 $

$2$	0.330837	0.233937	0.116968	−	0.993136i	$-0.462682\pi$
$2$	0.330837	0.233937	0.116968	+	0.993136i	$0.462682\pi$
$3$	−2.69180	−1.55411	−0.777057	−	0.629430i	$-0.783288\pi$
$3$	−2.69180	−1.55411	−0.777057	+	0.629430i	$0.783288\pi$
$4$	−1.89055	−0.945274
$5$	0	0
$5$	0	0
$6$	−0.890547	−0.363564
$7$	3.35348	1.26750	0.633748	−	0.773540i	$-0.281516\pi$
$7$	3.35348	1.26750	0.633748	+	0.773540i	$0.281516\pi$
$8$	−1.28714	−0.455071
$9$	4.24581	1.41527
$10$	0	0
$11$	−3.24581	−0.978649	−0.489324	−	0.872102i	$-0.662757\pi$
$11$	−3.24581	−0.978649	−0.489324	+	0.872102i	$0.662757\pi$
$12$	5.08898	1.46906
$13$	0	0
$13$	0	0
$14$	1.10945	0.296514
$15$	0	0
$16$	3.35526	0.838816
$17$	−1.94881	−0.472655	−0.236328	−	0.971673i	$-0.575944\pi$
$17$	−1.94881	−0.472655	−0.236328	+	0.971673i	$0.575944\pi$
$18$	1.40467	0.331084
$19$	−1.24581	−0.285808	−0.142904	−	0.989737i	$-0.545644\pi$
$19$	−1.24581	−0.285808	−0.142904	+	0.989737i	$0.545644\pi$
$20$	0	0
$21$	−9.02690	−1.96983
$22$	−1.07383	−0.228942
$23$	2.69180	0.561280	0.280640	−	0.959813i	$-0.409453\pi$
$23$	2.69180	0.561280	0.280640	+	0.959813i	$0.409453\pi$
$24$	3.46472	0.707232
$25$	0	0
$26$	0	0
$27$	−3.35348	−0.645377
$28$	−6.33991	−1.19813
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	3.78109	0.679105	0.339552	−	0.940587i	$-0.389724\pi$
$31$	3.78109	0.679105	0.339552	+	0.940587i	$0.389724\pi$
$32$	3.68431	0.651301
$33$	8.73709	1.52093
$34$	−0.644737	−0.110571
$35$	0	0
$36$	−8.02690	−1.33782
$37$	1.94881	0.320382	0.160191	−	0.987086i	$-0.448789\pi$
$37$	1.94881	0.320382	0.160191	+	0.987086i	$0.448789\pi$
$38$	−0.412160	−0.0668611
$39$	0	0
$40$	0	0
$41$	2.78109	0.434334	0.217167	−	0.976134i	$-0.430318\pi$
$41$	2.78109	0.434334	0.217167	+	0.976134i	$0.430318\pi$
$42$	−2.98643	−0.460816
$43$	−8.73709	−1.33239	−0.666197	−	0.745776i	$-0.732079\pi$
$43$	−8.73709	−1.33239	−0.666197	+	0.745776i	$0.732079\pi$
$44$	6.13636	0.925091
$45$	0	0
$46$	0.890547	0.131304
$47$	−6.86960	−1.00203	−0.501017	−	0.865437i	$-0.667041\pi$
$47$	−6.86960	−1.00203	−0.501017	+	0.865437i	$0.667041\pi$
$48$	−9.03171	−1.30362
$49$	4.24581	0.606544
$50$	0	0
$51$	5.24581	0.734560
$52$	0	0
$53$	−12.8336	−1.76282	−0.881412	−	0.472347i	$-0.843407\pi$
$53$	−12.8336	−1.76282	−0.881412	+	0.472347i	$0.843407\pi$
$54$	−1.10945	−0.150977
$55$	0	0
$56$	−4.31638	−0.576800
$57$	3.35348	0.444179
$58$	0.992510	0.130323
$59$	−2.53528	−0.330066	−0.165033	−	0.986288i	$-0.552773\pi$
$59$	−2.53528	−0.330066	−0.165033	+	0.986288i	$0.552773\pi$
$60$	0	0
$61$	−7.49162	−0.959204	−0.479602	−	0.877486i	$-0.659219\pi$
$61$	−7.49162	−0.959204	−0.479602	+	0.877486i	$0.659219\pi$
$62$	1.25092	0.158868
$63$	14.2382	1.79385
$64$	−5.49162	−0.686453
$65$	0	0
$66$	2.89055	0.355802
$67$	−4.01515	−0.490529	−0.245264	−	0.969456i	$-0.578875\pi$
$67$	−4.01515	−0.490529	−0.245264	+	0.969456i	$0.578875\pi$
$68$	3.68431	0.446789
$69$	−7.24581	−0.872293
$70$	0	0
$71$	5.24581	0.622563	0.311282	−	0.950318i	$-0.399242\pi$
$71$	5.24581	0.622563	0.311282	+	0.950318i	$0.399242\pi$
$72$	−5.46493	−0.644048
$73$	−5.46493	−0.639622	−0.319811	−	0.947481i	$-0.603619\pi$
$73$	−5.46493	−0.639622	−0.319811	+	0.947481i	$0.603619\pi$
$74$	0.644737	0.0749491
$75$	0	0
$76$	2.35526	0.270167
$77$	−10.8848	−1.24043
$78$	0	0
$79$	13.7811	1.55049	0.775247	−	0.631658i	$-0.217625\pi$
$79$	13.7811	1.55049	0.775247	+	0.631658i	$0.217625\pi$
$80$	0	0
$81$	−3.71053	−0.412281
$82$	0.920088	0.101607
$83$	8.61955	0.946119	0.473059	−	0.881031i	$-0.343150\pi$
$83$	8.61955	0.946119	0.473059	+	0.881031i	$0.343150\pi$
$84$	17.0658	1.86203
$85$	0	0
$86$	−2.89055	−0.311696
$87$	−8.07541	−0.865775
$88$	4.17780	0.445355
$89$	10.3164	1.09353	0.546767	−	0.837285i	$-0.315858\pi$
$89$	10.3164	1.09353	0.546767	+	0.837285i	$0.315858\pi$
$90$	0	0
$91$	0	0
$92$	−5.08898	−0.530563
$93$	−10.1780	−1.05541
$94$	−2.27271	−0.234413
$95$	0	0
$96$	−9.91745	−1.01220
$97$	−5.26607	−0.534689	−0.267344	−	0.963601i	$-0.586146\pi$
$97$	−5.26607	−0.534689	−0.267344	+	0.963601i	$0.586146\pi$
$98$	1.40467	0.141893
$99$	−13.7811	−1.38505
$100$	0	0
$101$	5.71053	0.568219	0.284109	−	0.958792i	$-0.408302\pi$
$101$	5.71053	0.568219	0.284109	+	0.958792i	$0.408302\pi$
$102$	1.73551	0.171841
$103$	−7.36863	−0.726052	−0.363026	−	0.931779i	$-0.618256\pi$
$103$	−7.36863	−0.726052	−0.363026	+	0.931779i	$0.618256\pi$
$104$	0	0
$105$	0	0
$106$	−4.24581	−0.412390
$107$	8.57444	0.828922	0.414461	−	0.910067i	$-0.363970\pi$
$107$	8.57444	0.828922	0.414461	+	0.910067i	$0.363970\pi$
$108$	6.33991	0.610058
$109$	−8.49162	−0.813350	−0.406675	−	0.913573i	$-0.633312\pi$
$109$	−8.49162	−0.813350	−0.406675	+	0.913573i	$0.633312\pi$
$110$	0	0
$111$	−5.24581	−0.497910
$112$	11.2518	1.06320
$113$	−7.33242	−0.689776	−0.344888	−	0.938644i	$-0.612083\pi$
$113$	−7.33242	−0.689776	−0.344888	+	0.938644i	$0.612083\pi$
$114$	1.10945	0.103910
$115$	0	0
$116$	−5.67164	−0.526599
$117$	0	0
$118$	−0.838765	−0.0772145
$119$	−6.53528	−0.599089
$120$	0	0
$121$	−0.464716	−0.0422469
$122$	−2.47850	−0.224393
$123$	−7.48616	−0.675004
$124$	−7.14834	−0.641940
$125$	0	0
$126$	4.71053	0.419647
$127$	−9.16369	−0.813146	−0.406573	−	0.913618i	$-0.633276\pi$
$127$	−9.16369	−0.813146	−0.406573	+	0.913618i	$0.633276\pi$
$128$	−9.18546	−0.811887
$129$	23.5185	2.07069
$130$	0	0
$131$	10.0000	0.873704	0.436852	−	0.899533i	$-0.356093\pi$
$131$	10.0000	0.873704	0.436852	+	0.899533i	$0.356093\pi$
$132$	−16.5179	−1.43770
$133$	−4.17780	−0.362261
$134$	−1.32836	−0.114753
$135$	0	0
$136$	2.50838	0.215092
$137$	16.8487	1.43948	0.719741	−	0.694242i	$-0.244260\pi$
$137$	16.8487	1.43948	0.719741	+	0.694242i	$0.244260\pi$
$138$	−2.39718	−0.204061
$139$	−1.02690	−0.0871009	−0.0435505	−	0.999051i	$-0.513867\pi$
$139$	−1.02690	−0.0871009	−0.0435505	+	0.999051i	$0.513867\pi$
$140$	0	0
$141$	18.4916	1.55728
$142$	1.73551	0.145640
$143$	0	0
$144$	14.2458	1.18715
$145$	0	0
$146$	−1.80800	−0.149631
$147$	−11.4289	−0.942639
$148$	−3.68431	−0.302849
$149$	15.8517	1.29862	0.649309	−	0.760524i	$-0.275058\pi$
$149$	15.8517	1.29862	0.649309	+	0.760524i	$0.275058\pi$
$150$	0	0
$151$	14.5454	1.18369	0.591845	−	0.806052i	$-0.298400\pi$
$151$	14.5454	1.18369	0.591845	+	0.806052i	$0.298400\pi$
$152$	1.60353	0.130063
$153$	−8.27427	−0.668935
$154$	−3.60107	−0.290183
$155$	0	0
$156$	0	0
$157$	10.9210	0.871588	0.435794	−	0.900047i	$-0.356468\pi$
$157$	10.9210	0.871588	0.435794	+	0.900047i	$0.356468\pi$
$158$	4.55929	0.362718
$159$	34.5454	2.73963
$160$	0	0
$161$	9.02690	0.711420
$162$	−1.22758	−0.0964476
$163$	−4.17780	−0.327230	−0.163615	−	0.986524i	$-0.552316\pi$
$163$	−4.17780	−0.327230	−0.163615	+	0.986524i	$0.552316\pi$
$164$	−5.25779	−0.410564
$165$	0	0
$166$	2.85166	0.221332
$167$	−3.35348	−0.259500	−0.129750	−	0.991547i	$-0.541417\pi$
$167$	−3.35348	−0.259500	−0.129750	+	0.991547i	$0.541417\pi$
$168$	11.6188	0.896413
$169$	0	0
$170$	0	0
$171$	−5.28947	−0.404496
$172$	16.5179	1.25948
$173$	8.73709	0.664268	0.332134	−	0.943232i	$-0.392231\pi$
$173$	8.73709	0.664268	0.332134	+	0.943232i	$0.392231\pi$
$174$	−2.67164	−0.202537
$175$	0	0
$176$	−10.8905	−0.820906
$177$	6.82449	0.512960
$178$	3.41303	0.255818
$179$	−18.0101	−1.34614	−0.673071	−	0.739578i	$-0.735025\pi$
$179$	−18.0101	−1.34614	−0.673071	+	0.739578i	$0.735025\pi$
$180$	0	0
$181$	1.04366	0.0775749	0.0387875	−	0.999247i	$-0.487650\pi$
$181$	1.04366	0.0775749	0.0387875	+	0.999247i	$0.487650\pi$
$182$	0	0
$183$	20.1660	1.49071
$184$	−3.46472	−0.255422
$185$	0	0
$186$	−3.36724	−0.246898
$187$	6.32546	0.462564
$188$	12.9873	0.947197
$189$	−11.2458	−0.818012
$190$	0	0
$191$	25.5185	1.84646	0.923228	−	0.384253i	$-0.125541\pi$
$191$	25.5185	1.84646	0.923228	+	0.384253i	$0.125541\pi$
$192$	14.7824	1.06683
$193$	−19.8207	−1.42672	−0.713362	−	0.700795i	$-0.752829\pi$
$193$	−19.8207	−1.42672	−0.713362	+	0.700795i	$0.752829\pi$
$194$	−1.74221	−0.125083
$195$	0	0
$196$	−8.02690	−0.573350
$197$	21.6520	1.54264	0.771319	−	0.636448i	$-0.219597\pi$
$197$	21.6520	1.54264	0.771319	+	0.636448i	$0.219597\pi$
$198$	−4.55929	−0.324015
$199$	18.2291	1.29222	0.646112	−	0.763243i	$-0.276394\pi$
$199$	18.2291	1.29222	0.646112	+	0.763243i	$0.276394\pi$
$200$	0	0
$201$	10.8080	0.762337
$202$	1.88925	0.132927
$203$	10.0604	0.706104
$204$	−9.91745	−0.694361
$205$	0	0
$206$	−2.43781	−0.169850
$207$	11.4289	0.794363
$208$	0	0
$209$	4.04366	0.279706
$210$	0	0
$211$	19.2996	1.32864	0.664320	−	0.747448i	$-0.268721\pi$
$211$	19.2996	1.32864	0.664320	+	0.747448i	$0.268721\pi$
$212$	24.2624	1.66635
$213$	−14.1207	−0.967534
$214$	2.83674	0.193915
$215$	0	0
$216$	4.31638	0.293692
$217$	12.6798	0.860762
$218$	−2.80934	−0.190272
$219$	14.7105	0.994045
$220$	0	0
$221$	0	0
$222$	−1.73551	−0.116480
$223$	12.3707	0.828406	0.414203	−	0.910184i	$-0.364060\pi$
$223$	12.3707	0.828406	0.414203	+	0.910184i	$0.364060\pi$
$224$	12.3553	0.825521
$225$	0	0
$226$	−2.42583	−0.161364
$227$	−6.16282	−0.409040	−0.204520	−	0.978862i	$-0.565563\pi$
$227$	−6.16282	−0.409040	−0.204520	+	0.978862i	$0.565563\pi$
$228$	−6.33991	−0.419871
$229$	−26.9832	−1.78310	−0.891551	−	0.452920i	$-0.850382\pi$
$229$	−26.9832	−1.78310	−0.891551	+	0.452920i	$0.850382\pi$
$230$	0	0
$231$	29.2996	1.92777
$232$	−3.86141	−0.253514
$233$	0.824319	0.0540029	0.0270015	−	0.999635i	$-0.491404\pi$
$233$	0.824319	0.0540029	0.0270015	+	0.999635i	$0.491404\pi$
$234$	0	0
$235$	0	0
$236$	4.79307	0.312003
$237$	−37.0960	−2.40964
$238$	−2.16211	−0.140149
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	22.6938	1.46183	0.730917	−	0.682466i	$-0.239093\pi$
$241$	22.6938	1.46183	0.730917	+	0.682466i	$0.239093\pi$
$242$	−0.153745	−0.00988310
$243$	20.0484	1.28611
$244$	14.1633	0.906710
$245$	0	0
$246$	−2.47670	−0.157908
$247$	0	0
$248$	−4.86678	−0.309041
$249$	−23.2021	−1.47038
$250$	0	0
$251$	19.0269	1.20097	0.600484	−	0.799637i	$-0.294975\pi$
$251$	19.0269	1.20097	0.600484	+	0.799637i	$0.294975\pi$
$252$	−26.9180	−1.69568
$253$	−8.73709	−0.549296
$254$	−3.03168	−0.190225
$255$	0	0
$256$	7.94436	0.496522
$257$	−2.11145	−0.131709	−0.0658544	−	0.997829i	$-0.520977\pi$
$257$	−2.11145	−0.131709	−0.0658544	+	0.997829i	$0.520977\pi$
$258$	7.78079	0.484411
$259$	6.53528	0.406083
$260$	0	0
$261$	12.7374	0.788427
$262$	3.30837	0.204391
$263$	29.9173	1.84478	0.922391	−	0.386257i	$-0.126232\pi$
$263$	29.9173	1.84478	0.922391	+	0.386257i	$0.126232\pi$
$264$	−11.2458	−0.692132
$265$	0	0
$266$	−1.38217	−0.0847461
$267$	−27.7697	−1.69948
$268$	7.59083	0.463684
$269$	18.5891	1.13340	0.566699	−	0.823925i	$-0.308220\pi$
$269$	18.5891	1.13340	0.566699	+	0.823925i	$0.308220\pi$
$270$	0	0
$271$	5.82476	0.353829	0.176914	−	0.984226i	$-0.443388\pi$
$271$	5.82476	0.353829	0.176914	+	0.984226i	$0.443388\pi$
$272$	−6.53876	−0.396471
$273$	0	0
$274$	5.57417	0.336748
$275$	0	0
$276$	13.6985	0.824556
$277$	−13.5403	−0.813560	−0.406780	−	0.913526i	$-0.633349\pi$
$277$	−13.5403	−0.813560	−0.406780	+	0.913526i	$0.633349\pi$
$278$	−0.339738	−0.0203761
$279$	16.0538	0.961116
$280$	0	0
$281$	−0.464716	−0.0277226	−0.0138613	−	0.999904i	$-0.504412\pi$
$281$	−0.464716	−0.0277226	−0.0138613	+	0.999904i	$0.504412\pi$
$282$	6.11770	0.364304
$283$	−10.0604	−0.598031	−0.299015	−	0.954248i	$-0.596658\pi$
$283$	−10.0604	−0.598031	−0.299015	+	0.954248i	$0.596658\pi$
$284$	−9.91745	−0.588492
$285$	0	0
$286$	0	0
$287$	9.32634	0.550516
$288$	15.6429	0.921767
$289$	−13.2021	−0.776597
$290$	0	0
$291$	14.1752	0.830967
$292$	10.3317	0.604618
$293$	−13.4501	−0.785764	−0.392882	−	0.919589i	$-0.628522\pi$
$293$	−13.4501	−0.785764	−0.392882	+	0.919589i	$0.628522\pi$
$294$	−3.78109	−0.220518
$295$	0	0
$296$	−2.50838	−0.145797
$297$	10.8848	0.631597
$298$	5.24431	0.303795
$299$	0	0
$300$	0	0
$301$	−29.2996	−1.68880
$302$	4.81216	0.276909
$303$	−15.3716	−0.883076
$304$	−4.18002	−0.239741
$305$	0	0
$306$	−2.73743	−0.156488
$307$	−24.6077	−1.40444	−0.702219	−	0.711961i	$-0.747807\pi$
$307$	−24.6077	−1.40444	−0.702219	+	0.711961i	$0.747807\pi$
$308$	20.5781	1.17255
$309$	19.8349	1.12837
$310$	0	0
$311$	2.43781	0.138236	0.0691178	−	0.997609i	$-0.477982\pi$
$311$	2.43781	0.138236	0.0691178	+	0.997609i	$0.477982\pi$
$312$	0	0
$313$	19.2965	1.09071	0.545353	−	0.838207i	$-0.316396\pi$
$313$	19.2965	1.09071	0.545353	+	0.838207i	$0.316396\pi$
$314$	3.61305	0.203896
$315$	0	0
$316$	−26.0538	−1.46564
$317$	28.8217	1.61879	0.809395	−	0.587265i	$-0.199795\pi$
$317$	28.8217	1.61879	0.809395	+	0.587265i	$0.199795\pi$
$318$	11.4289	0.640900
$319$	−9.73743	−0.545191
$320$	0	0
$321$	−23.0807	−1.28824
$322$	2.98643	0.166427
$323$	2.42785	0.135089
$324$	7.01492	0.389718
$325$	0	0
$326$	−1.38217	−0.0765512
$327$	22.8578	1.26404
$328$	−3.57964	−0.197653
$329$	−23.0371	−1.27007
$330$	0	0
$331$	−2.97310	−0.163416	−0.0817081	−	0.996656i	$-0.526038\pi$
$331$	−2.97310	−0.163416	−0.0817081	+	0.996656i	$0.526038\pi$
$332$	−16.2957	−0.894341
$333$	8.27427	0.453427
$334$	−1.10945	−0.0607066
$335$	0	0
$336$	−30.2876	−1.65233
$337$	−1.90370	−0.103701	−0.0518505	−	0.998655i	$-0.516512\pi$
$337$	−1.90370	−0.103701	−0.0518505	+	0.998655i	$0.516512\pi$
$338$	0	0
$339$	19.7374	1.07199
$340$	0	0
$341$	−12.2727	−0.664605
$342$	−1.74995	−0.0946265
$343$	−9.23611	−0.498703
$344$	11.2458	0.606333
$345$	0	0
$346$	2.89055	0.155397
$347$	12.6347	0.678266	0.339133	−	0.940738i	$-0.389866\pi$
$347$	12.6347	0.678266	0.339133	+	0.940738i	$0.389866\pi$
$348$	15.2669	0.818394
$349$	8.97310	0.480319	0.240159	−	0.970733i	$-0.422800\pi$
$349$	8.97310	0.480319	0.240159	+	0.970733i	$0.422800\pi$
$350$	0	0
$351$	0	0
$352$	−11.9586	−0.637395
$353$	34.2505	1.82297	0.911484	−	0.411336i	$-0.134938\pi$
$353$	34.2505	1.82297	0.911484	+	0.411336i	$0.134938\pi$
$354$	2.25779	0.120000
$355$	0	0
$356$	−19.5036	−1.03369
$357$	17.5917	0.931052
$358$	−5.95841	−0.314912
$359$	22.4043	1.18245	0.591227	−	0.806505i	$-0.298644\pi$
$359$	22.4043	1.18245	0.591227	+	0.806505i	$0.298644\pi$
$360$	0	0
$361$	−17.4480	−0.918314
$362$	0.345282	0.0181476
$363$	1.25092	0.0656565
$364$	0	0
$365$	0	0
$366$	6.67164	0.348732
$367$	13.1951	0.688777	0.344388	−	0.938827i	$-0.388086\pi$
$367$	13.1951	0.688777	0.344388	+	0.938827i	$0.388086\pi$
$368$	9.03171	0.470811
$369$	11.8080	0.614700
$370$	0	0
$371$	−43.0371	−2.23437
$372$	19.2419	0.997647
$373$	15.2614	0.790206	0.395103	−	0.918637i	$-0.370709\pi$
$373$	15.2614	0.790206	0.395103	+	0.918637i	$0.370709\pi$
$374$	2.09269	0.108211
$375$	0	0
$376$	8.84210	0.455997
$377$	0	0
$378$	−3.72052	−0.191363
$379$	18.2291	0.936363	0.468182	−	0.883632i	$-0.344909\pi$
$379$	18.2291	0.936363	0.468182	+	0.883632i	$0.344909\pi$
$380$	0	0
$381$	24.6669	1.26372
$382$	8.44246	0.431954
$383$	−1.44088	−0.0736255	−0.0368128	−	0.999322i	$-0.511721\pi$
$383$	−1.44088	−0.0736255	−0.0368128	+	0.999322i	$0.511721\pi$
$384$	24.7255	1.26177
$385$	0	0
$386$	−6.55741	−0.333763
$387$	−37.0960	−1.88570
$388$	9.95576	0.505427
$389$	−18.7912	−0.952754	−0.476377	−	0.879241i	$-0.658050\pi$
$389$	−18.7912	−0.952754	−0.476377	+	0.879241i	$0.658050\pi$
$390$	0	0
$391$	−5.24581	−0.265292
$392$	−5.46493	−0.276021
$393$	−26.9180	−1.35784
$394$	7.16326	0.360880
$395$	0	0
$396$	26.0538	1.30925
$397$	17.0927	0.857857	0.428928	−	0.903338i	$-0.358891\pi$
$397$	17.0927	0.857857	0.428928	+	0.903338i	$0.358891\pi$
$398$	6.03084	0.302298
$399$	11.2458	0.562995
$400$	0	0
$401$	22.2021	1.10872	0.554361	−	0.832276i	$-0.312963\pi$
$401$	22.2021	1.10872	0.554361	+	0.832276i	$0.312963\pi$
$402$	3.57568	0.178339
$403$	0	0
$404$	−10.7960	−0.537122
$405$	0	0
$406$	3.32836	0.165184
$407$	−6.32546	−0.313541
$408$	−6.75207	−0.334277
$409$	−9.63276	−0.476309	−0.238155	−	0.971227i	$-0.576542\pi$
$409$	−9.63276	−0.476309	−0.238155	+	0.971227i	$0.576542\pi$
$410$	0	0
$411$	−45.3534	−2.23712
$412$	13.9307	0.686318
$413$	−8.50202	−0.418357
$414$	3.78109	0.185831
$415$	0	0
$416$	0	0
$417$	2.76423	0.135365
$418$	1.33779	0.0654335
$419$	−1.95634	−0.0955733	−0.0477866	−	0.998858i	$-0.515217\pi$
$419$	−1.95634	−0.0955733	−0.0477866	+	0.998858i	$0.515217\pi$
$420$	0	0
$421$	−12.0807	−0.588778	−0.294389	−	0.955686i	$-0.595116\pi$
$421$	−12.0807	−0.588778	−0.294389	+	0.955686i	$0.595116\pi$
$422$	6.38502	0.310818
$423$	−29.1670	−1.41815
$424$	16.5185	0.802210
$425$	0	0
$426$	−4.67164	−0.226342
$427$	−25.1230	−1.21579
$428$	−16.2104	−0.783558
$429$	0	0
$430$	0	0
$431$	24.5891	1.18441	0.592207	−	0.805786i	$-0.298257\pi$
$431$	24.5891	1.18441	0.592207	+	0.805786i	$0.298257\pi$
$432$	−11.2518	−0.541352
$433$	36.0728	1.73355	0.866775	−	0.498700i	$-0.166189\pi$
$433$	36.0728	1.73355	0.866775	+	0.498700i	$0.166189\pi$
$434$	4.19495	0.201364
$435$	0	0
$436$	16.0538	0.768838
$437$	−3.35348	−0.160419
$438$	4.86678	0.232544
$439$	2.53528	0.121003	0.0605013	−	0.998168i	$-0.480730\pi$
$439$	2.53528	0.121003	0.0605013	+	0.998168i	$0.480730\pi$
$440$	0	0
$441$	18.0269	0.858424
$442$	0	0
$443$	19.3579	0.919721	0.459860	−	0.887991i	$-0.347899\pi$
$443$	19.3579	0.919721	0.459860	+	0.887991i	$0.347899\pi$
$444$	9.91745	0.470661
$445$	0	0
$446$	4.09269	0.193795
$447$	−42.6696	−2.01820
$448$	−18.4160	−0.870075
$449$	−24.8080	−1.17076	−0.585381	−	0.810758i	$-0.699055\pi$
$449$	−24.8080	−1.17076	−0.585381	+	0.810758i	$0.699055\pi$
$450$	0	0
$451$	−9.02690	−0.425060
$452$	13.8623	0.652027
$453$	−39.1534	−1.83959
$454$	−2.03888	−0.0956896
$455$	0	0
$456$	−4.31638	−0.202133
$457$	−7.56748	−0.353992	−0.176996	−	0.984212i	$-0.556638\pi$
$457$	−7.56748	−0.353992	−0.176996	+	0.984212i	$0.556638\pi$
$458$	−8.92704	−0.417133
$459$	6.53528	0.305041
$460$	0	0
$461$	−12.3433	−0.574884	−0.287442	−	0.957798i	$-0.592805\pi$
$461$	−12.3433	−0.574884	−0.287442	+	0.957798i	$0.592805\pi$
$462$	9.69338	0.450977
$463$	−22.8578	−1.06229	−0.531146	−	0.847281i	$-0.678238\pi$
$463$	−22.8578	−1.06229	−0.531146	+	0.847281i	$0.678238\pi$
$464$	10.0658	0.467293
$465$	0	0
$466$	0.272715	0.0126333
$467$	15.2976	0.707889	0.353945	−	0.935266i	$-0.384840\pi$
$467$	15.2976	0.707889	0.353945	+	0.935266i	$0.384840\pi$
$468$	0	0
$469$	−13.4647	−0.621743
$470$	0	0
$471$	−29.3971	−1.35455
$472$	3.26325	0.150203
$473$	28.3589	1.30394
$474$	−12.2727	−0.563704
$475$	0	0
$476$	12.3553	0.566303
$477$	−54.4889	−2.49487
$478$	1.32335	0.0605284
$479$	24.2829	1.10951	0.554756	−	0.832013i	$-0.312812\pi$
$479$	24.2829	1.10951	0.554756	+	0.832013i	$0.312812\pi$
$480$	0	0
$481$	0	0
$482$	7.50793	0.341977
$483$	−24.2987	−1.10563
$484$	0.878567	0.0399349
$485$	0	0
$486$	6.63276	0.300868
$487$	−36.8883	−1.67157	−0.835783	−	0.549060i	$-0.814986\pi$
$487$	−36.8883	−1.67157	−0.835783	+	0.549060i	$0.814986\pi$
$488$	9.64273	0.436506
$489$	11.2458	0.508553
$490$	0	0
$491$	−35.3534	−1.59548	−0.797739	−	0.603003i	$-0.793971\pi$
$491$	−35.3534	−1.59548	−0.797739	+	0.603003i	$0.793971\pi$
$492$	14.1529	0.638064
$493$	−5.84642	−0.263310
$494$	0	0
$495$	0	0
$496$	12.6866	0.569644
$497$	17.5917	0.789096
$498$	−7.67612	−0.343975
$499$	16.2189	0.726058	0.363029	−	0.931778i	$-0.381743\pi$
$499$	16.2189	0.726058	0.363029	+	0.931778i	$0.381743\pi$
$500$	0	0
$501$	9.02690	0.403292
$502$	6.29480	0.280950
$503$	−20.2384	−0.902386	−0.451193	−	0.892427i	$-0.649001\pi$
$503$	−20.2384	−0.902386	−0.451193	+	0.892427i	$0.649001\pi$
$504$	−18.3265	−0.816328
$505$	0	0
$506$	−2.89055	−0.128500
$507$	0	0
$508$	17.3244	0.768646
$509$	−20.0371	−0.888127	−0.444063	−	0.895995i	$-0.646463\pi$
$509$	−20.0371	−0.888127	−0.444063	+	0.895995i	$0.646463\pi$
$510$	0	0
$511$	−18.3265	−0.810718
$512$	20.9992	0.928042
$513$	4.17780	0.184454
$514$	−0.698546	−0.0308116
$515$	0	0
$516$	−44.4629	−1.95737
$517$	22.2974	0.980639
$518$	2.16211	0.0949977
$519$	−23.5185	−1.03235
$520$	0	0
$521$	16.0269	0.702151	0.351076	−	0.936347i	$-0.385816\pi$
$521$	16.0269	0.702151	0.351076	+	0.936347i	$0.385816\pi$
$522$	4.21401	0.184442
$523$	11.7380	0.513265	0.256633	−	0.966509i	$-0.417387\pi$
$523$	11.7380	0.513265	0.256633	+	0.966509i	$0.417387\pi$
$524$	−18.9055	−0.825889
$525$	0	0
$526$	9.89775	0.431562
$527$	−7.36863	−0.320982
$528$	29.3152	1.27578
$529$	−15.7542	−0.684965
$530$	0	0
$531$	−10.7643	−0.467132
$532$	7.89832	0.342436
$533$	0	0
$534$	−9.18722	−0.397570
$535$	0	0
$536$	5.16804	0.223225
$537$	48.4798	2.09206
$538$	6.14995	0.265143
$539$	−13.7811	−0.593594
$540$	0	0
$541$	−21.8080	−0.937599	−0.468800	−	0.883305i	$-0.655313\pi$
$541$	−21.8080	−0.937599	−0.468800	+	0.883305i	$0.655313\pi$
$542$	1.92704	0.0827736
$543$	−2.80934	−0.120560
$544$	−7.18002	−0.307841
$545$	0	0
$546$	0	0
$547$	6.30924	0.269764	0.134882	−	0.990862i	$-0.456935\pi$
$547$	6.30924	0.269764	0.134882	+	0.990862i	$0.456935\pi$
$548$	−31.8533	−1.36070
$549$	−31.8080	−1.35753
$550$	0	0
$551$	−3.73743	−0.159220
$552$	9.32634	0.396955
$553$	46.2146	1.96524
$554$	−4.47964	−0.190322
$555$	0	0
$556$	1.94141	0.0823342
$557$	−35.8378	−1.51849	−0.759247	−	0.650802i	$-0.774433\pi$
$557$	−35.8378	−1.51849	−0.759247	+	0.650802i	$0.774433\pi$
$558$	5.31119	0.224840
$559$	0	0
$560$	0	0
$561$	−17.0269	−0.718876
$562$	−0.153745	−0.00648534
$563$	−5.00212	−0.210814	−0.105407	−	0.994429i	$-0.533615\pi$
$563$	−5.00212	−0.210814	−0.105407	+	0.994429i	$0.533615\pi$
$564$	−34.9593	−1.47205
$565$	0	0
$566$	−3.32836	−0.139901
$567$	−12.4432	−0.522564
$568$	−6.75207	−0.283310
$569$	−13.1680	−0.552033	−0.276017	−	0.961153i	$-0.589014\pi$
$569$	−13.1680	−0.552033	−0.276017	+	0.961153i	$0.589014\pi$
$570$	0	0
$571$	19.8349	0.830065	0.415032	−	0.909807i	$-0.363770\pi$
$571$	19.8349	0.830065	0.415032	+	0.909807i	$0.363770\pi$
$572$	0	0
$573$	−68.6909	−2.86960
$574$	3.08549	0.128786
$575$	0	0
$576$	−23.3164	−0.971516
$577$	−10.9210	−0.454646	−0.227323	−	0.973819i	$-0.572997\pi$
$577$	−10.9210	−0.454646	−0.227323	+	0.973819i	$0.572997\pi$
$578$	−4.36775	−0.181675
$579$	53.3534	2.21729
$580$	0	0
$581$	28.9055	1.19920
$582$	4.68969	0.194394
$583$	41.6553	1.72519
$584$	7.03411	0.291073
$585$	0	0
$586$	−4.44979	−0.183819
$587$	−40.4495	−1.66953	−0.834764	−	0.550607i	$-0.814396\pi$
$587$	−40.4495	−1.66953	−0.834764	+	0.550607i	$0.814396\pi$
$588$	21.6069	0.891052
$589$	−4.71053	−0.194094
$590$	0	0
$591$	−58.2829	−2.39744
$592$	6.53876	0.268742
$593$	−1.47709	−0.0606569	−0.0303284	−	0.999540i	$-0.509655\pi$
$593$	−1.47709	−0.0606569	−0.0303284	+	0.999540i	$0.509655\pi$
$594$	3.60107	0.147754
$595$	0	0
$596$	−29.9683	−1.22755
$597$	−49.0690	−2.00826
$598$	0	0
$599$	2.27271	0.0928606	0.0464303	−	0.998922i	$-0.485215\pi$
$599$	2.27271	0.0928606	0.0464303	+	0.998922i	$0.485215\pi$
$600$	0	0
$601$	7.40429	0.302027	0.151014	−	0.988532i	$-0.451746\pi$
$601$	7.40429	0.302027	0.151014	+	0.988532i	$0.451746\pi$
$602$	−9.69338	−0.395073
$603$	−17.0476	−0.694231
$604$	−27.4988	−1.11891
$605$	0	0
$606$	−5.08549	−0.206584
$607$	10.6932	0.434024	0.217012	−	0.976169i	$-0.430369\pi$
$607$	10.6932	0.434024	0.217012	+	0.976169i	$0.430369\pi$
$608$	−4.58996	−0.186147
$609$	−27.0807	−1.09737
$610$	0	0
$611$	0	0
$612$	15.6429	0.632327
$613$	−6.08149	−0.245629	−0.122815	−	0.992430i	$-0.539192\pi$
$613$	−6.08149	−0.245629	−0.122815	+	0.992430i	$0.539192\pi$
$614$	−8.14114	−0.328550
$615$	0	0
$616$	14.0101	0.564485
$617$	−31.8388	−1.28178	−0.640892	−	0.767631i	$-0.721435\pi$
$617$	−31.8388	−1.28178	−0.640892	+	0.767631i	$0.721435\pi$
$618$	6.56211	0.263967
$619$	−26.4043	−1.06128	−0.530639	−	0.847598i	$-0.678048\pi$
$619$	−26.4043	−1.06128	−0.530639	+	0.847598i	$0.678048\pi$
$620$	0	0
$621$	−9.02690	−0.362237
$622$	0.806517	0.0323384
$623$	34.5957	1.38605
$624$	0	0
$625$	0	0
$626$	6.38400	0.255156
$627$	−10.8848	−0.434695
$628$	−20.6466	−0.823889
$629$	−3.79785	−0.151430
$630$	0	0
$631$	35.1680	1.40002	0.700009	−	0.714134i	$-0.253179\pi$
$631$	35.1680	1.40002	0.700009	+	0.714134i	$0.253179\pi$
$632$	−17.7381	−0.705585
$633$	−51.9508	−2.06486
$634$	9.53528	0.378695
$635$	0	0
$636$	−65.3098	−2.58970
$637$	0	0
$638$	−3.22150	−0.127540
$639$	22.2727	0.881095
$640$	0	0
$641$	5.52514	0.218230	0.109115	−	0.994029i	$-0.465198\pi$
$641$	5.52514	0.218230	0.109115	+	0.994029i	$0.465198\pi$
$642$	−7.63594	−0.301367
$643$	32.1552	1.26808	0.634039	−	0.773301i	$-0.281396\pi$
$643$	32.1552	1.26808	0.634039	+	0.773301i	$0.281396\pi$
$644$	−17.0658	−0.672486
$645$	0	0
$646$	0.803220	0.0316023
$647$	13.7843	0.541917	0.270959	−	0.962591i	$-0.412659\pi$
$647$	13.7843	0.541917	0.270959	+	0.962591i	$0.412659\pi$
$648$	4.77595	0.187617
$649$	8.22905	0.323019
$650$	0	0
$651$	−34.1316	−1.33772
$652$	7.89832	0.309322
$653$	−8.50202	−0.332710	−0.166355	−	0.986066i	$-0.553200\pi$
$653$	−8.50202	−0.332710	−0.166355	+	0.986066i	$0.553200\pi$
$654$	7.56219	0.295705
$655$	0	0
$656$	9.33130	0.364326
$657$	−23.2031	−0.905238
$658$	−7.62150	−0.297117
$659$	4.04366	0.157519	0.0787594	−	0.996894i	$-0.474904\pi$
$659$	4.04366	0.157519	0.0787594	+	0.996894i	$0.474904\pi$
$660$	0	0
$661$	31.2727	1.21637	0.608184	−	0.793796i	$-0.291898\pi$
$661$	31.2727	1.21637	0.608184	+	0.793796i	$0.291898\pi$
$662$	−0.983609	−0.0382290
$663$	0	0
$664$	−11.0945	−0.430551
$665$	0	0
$666$	2.73743	0.106073
$667$	8.07541	0.312681
$668$	6.33991	0.245298
$669$	−33.2996	−1.28744
$670$	0	0
$671$	24.3164	0.938723
$672$	−33.2579	−1.28295
$673$	32.0739	1.23636	0.618179	−	0.786037i	$-0.287871\pi$
$673$	32.0739	1.23636	0.618179	+	0.786037i	$0.287871\pi$
$674$	−0.629812	−0.0242595
$675$	0	0
$676$	0	0
$677$	−14.2382	−0.547220	−0.273610	−	0.961841i	$-0.588218\pi$
$677$	−14.2382	−0.547220	−0.273610	+	0.961841i	$0.588218\pi$
$678$	6.52986	0.250778
$679$	−17.6597	−0.677716
$680$	0	0
$681$	16.5891	0.635695
$682$	−4.06026	−0.155475
$683$	25.7847	0.986622	0.493311	−	0.869853i	$-0.335786\pi$
$683$	25.7847	0.986622	0.493311	+	0.869853i	$0.335786\pi$
$684$	10.0000	0.382360
$685$	0	0
$686$	−3.05564	−0.116665
$687$	72.6336	2.77114
$688$	−29.3152	−1.11763
$689$	0	0
$690$	0	0
$691$	−0.0436636	−0.00166104	−0.000830522	−	1.00000i	$-0.500264\pi$
$691$	−0.0436636	−0.00166104	−0.000830522		1.00000i	$0.500264\pi$
$692$	−16.5179	−0.627915
$693$	−46.2146	−1.75555
$694$	4.18002	0.158671
$695$	0	0
$696$	10.3941	0.393989
$697$	−5.41982	−0.205290
$698$	2.96863	0.112364
$699$	−2.21891	−0.0839267
$700$	0	0
$701$	−14.5454	−0.549373	−0.274687	−	0.961534i	$-0.588574\pi$
$701$	−14.5454	−0.549373	−0.274687	+	0.961534i	$0.588574\pi$
$702$	0	0
$703$	−2.42785	−0.0915679
$704$	17.8248	0.671796
$705$	0	0
$706$	11.3313	0.426459
$707$	19.1501	0.720214
$708$	−12.9020	−0.484888
$709$	19.6328	0.737324	0.368662	−	0.929564i	$-0.379816\pi$
$709$	19.6328	0.737324	0.368662	+	0.929564i	$0.379816\pi$
$710$	0	0
$711$	58.5119	2.19437
$712$	−13.2786	−0.497636
$713$	10.1780	0.381168
$714$	5.81998	0.217807
$715$	0	0
$716$	34.0490	1.27247
$717$	−10.7672	−0.402109
$718$	7.41216	0.276619
$719$	47.4312	1.76889	0.884443	−	0.466649i	$-0.154539\pi$
$719$	47.4312	1.76889	0.884443	+	0.466649i	$0.154539\pi$
$720$	0	0
$721$	−24.7105	−0.920268
$722$	−5.77242	−0.214827
$723$	−61.0872	−2.27186
$724$	−1.97310	−0.0733295
$725$	0	0
$726$	0.413851	0.0153595
$727$	−34.0951	−1.26452	−0.632259	−	0.774757i	$-0.717872\pi$
$727$	−34.0951	−1.26452	−0.632259	+	0.774757i	$0.717872\pi$
$728$	0	0
$729$	−42.8349	−1.58648
$730$	0	0
$731$	17.0269	0.629763
$732$	−38.1247	−1.40913
$733$	14.3920	0.531580	0.265790	−	0.964031i	$-0.414367\pi$
$733$	14.3920	0.531580	0.265790	+	0.964031i	$0.414367\pi$
$734$	4.36541	0.161130
$735$	0	0
$736$	9.91745	0.365562
$737$	13.0324	0.480055
$738$	3.90652	0.143801
$739$	−34.4480	−1.26719	−0.633594	−	0.773665i	$-0.718421\pi$
$739$	−34.4480	−1.26719	−0.633594	+	0.773665i	$0.718421\pi$
$740$	0	0
$741$	0	0
$742$	−14.2382	−0.522702
$743$	−40.7134	−1.49363	−0.746816	−	0.665031i	$-0.768418\pi$
$743$	−40.7134	−1.49363	−0.746816	+	0.665031i	$0.768418\pi$
$744$	13.1004	0.480285
$745$	0	0
$746$	5.04903	0.184858
$747$	36.5970	1.33901
$748$	−11.9586	−0.437249
$749$	28.7542	1.05066
$750$	0	0
$751$	32.5018	1.18601	0.593003	−	0.805200i	$-0.297942\pi$
$751$	32.5018	1.18601	0.593003	+	0.805200i	$0.297942\pi$
$752$	−23.0493	−0.840522
$753$	−51.2167	−1.86644
$754$	0	0
$755$	0	0
$756$	21.2607	0.773245
$757$	13.0324	0.473671	0.236836	−	0.971550i	$-0.423890\pi$
$757$	13.0324	0.473671	0.236836	+	0.971550i	$0.423890\pi$
$758$	6.03084	0.219050
$759$	23.5185	0.853668
$760$	0	0
$761$	−3.98985	−0.144632	−0.0723161	−	0.997382i	$-0.523039\pi$
$761$	−3.98985	−0.144632	−0.0723161	+	0.997382i	$0.523039\pi$
$762$	8.16070	0.295631
$763$	−28.4765	−1.03092
$764$	−48.2440	−1.74541
$765$	0	0
$766$	−0.476696	−0.0172237
$767$	0	0
$768$	−21.3847	−0.771652
$769$	6.66686	0.240413	0.120207	−	0.992749i	$-0.461644\pi$
$769$	6.66686	0.240413	0.120207	+	0.992749i	$0.461644\pi$
$770$	0	0
$771$	5.68362	0.204691
$772$	37.4720	1.34865
$773$	−48.3349	−1.73849	−0.869244	−	0.494384i	$-0.835394\pi$
$773$	−48.3349	−1.73849	−0.869244	+	0.494384i	$0.835394\pi$
$774$	−12.2727	−0.441134
$775$	0	0
$776$	6.77815	0.243321
$777$	−17.5917	−0.631099
$778$	−6.21683	−0.222884
$779$	−3.46472	−0.124136
$780$	0	0
$781$	−17.0269	−0.609271
$782$	−1.73551	−0.0620616
$783$	−10.0604	−0.359531
$784$	14.2458	0.508779
$785$	0	0
$786$	−8.90547	−0.317648
$787$	27.9612	0.996709	0.498355	−	0.866973i	$-0.333938\pi$
$787$	27.9612	0.996709	0.498355	+	0.866973i	$0.333938\pi$
$788$	−40.9341	−1.45822
$789$	−80.5316	−2.86700
$790$	0	0
$791$	−24.5891	−0.874288
$792$	17.7381	0.630297
$793$	0	0
$794$	5.65488	0.200684
$795$	0	0
$796$	−34.4629	−1.22150
$797$	−37.2424	−1.31919	−0.659597	−	0.751619i	$-0.729273\pi$
$797$	−37.2424	−1.31919	−0.659597	+	0.751619i	$0.729273\pi$
$798$	3.72052	0.131705
$799$	13.3875	0.473617
$800$	0	0
$801$	43.8014	1.54765
$802$	7.34528	0.259371
$803$	17.7381	0.625965
$804$	−20.4330	−0.720617
$805$	0	0
$806$	0	0
$807$	−50.0382	−1.76143
$808$	−7.35022	−0.258580
$809$	14.5287	0.510801	0.255400	−	0.966835i	$-0.417793\pi$
$809$	14.5287	0.510801	0.255400	+	0.966835i	$0.417793\pi$
$810$	0	0
$811$	44.0538	1.54694	0.773469	−	0.633834i	$-0.218520\pi$
$811$	44.0538	1.54694	0.773469	+	0.633834i	$0.218520\pi$
$812$	−19.0197	−0.667461
$813$	−15.6791	−0.549890
$814$	−2.09269	−0.0733489
$815$	0	0
$816$	17.6011	0.616161
$817$	10.8848	0.380809
$818$	−3.18687	−0.111426
$819$	0	0
$820$	0	0
$821$	27.0269	0.943245	0.471623	−	0.881800i	$-0.343668\pi$
$821$	27.0269	0.943245	0.471623	+	0.881800i	$0.343668\pi$
$822$	−15.0046	−0.523345
$823$	39.5963	1.38024	0.690120	−	0.723695i	$-0.257558\pi$
$823$	39.5963	1.38024	0.690120	+	0.723695i	$0.257558\pi$
$824$	9.48442	0.330405
$825$	0	0
$826$	−2.81278	−0.0978691
$827$	26.5639	0.923716	0.461858	−	0.886954i	$-0.347183\pi$
$827$	26.5639	0.923716	0.461858	+	0.886954i	$0.347183\pi$
$828$	−21.6069	−0.750890
$829$	13.9832	0.485658	0.242829	−	0.970069i	$-0.421925\pi$
$829$	13.9832	0.485658	0.242829	+	0.970069i	$0.421925\pi$
$830$	0	0
$831$	36.4480	1.26437
$832$	0	0
$833$	−8.27427	−0.286686
$834$	0.914507	0.0316668
$835$	0	0
$836$	−7.64474	−0.264399
$837$	−12.6798	−0.438278
$838$	−0.647228	−0.0223581
$839$	14.3941	0.496941	0.248471	−	0.968639i	$-0.420072\pi$
$839$	14.3941	0.496941	0.248471	+	0.968639i	$0.420072\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−3.99674	−0.137737
$843$	1.25092	0.0430841
$844$	−36.4868	−1.25593
$845$	0	0
$846$	−9.64952	−0.331757
$847$	−1.55841	−0.0535477
$848$	−43.0600	−1.47869
$849$	27.0807	0.929408
$850$	0	0
$851$	5.24581	0.179824
$852$	26.6958	0.914584
$853$	27.2633	0.933478	0.466739	−	0.884395i	$-0.345429\pi$
$853$	27.2633	0.933478	0.466739	+	0.884395i	$0.345429\pi$
$854$	−8.31160	−0.284417
$855$	0	0
$856$	−11.0365	−0.377219
$857$	50.6201	1.72915	0.864575	−	0.502503i	$-0.167587\pi$
$857$	50.6201	1.72915	0.864575	+	0.502503i	$0.167587\pi$
$858$	0	0
$859$	1.27992	0.0436702	0.0218351	−	0.999762i	$-0.493049\pi$
$859$	1.27992	0.0436702	0.0218351	+	0.999762i	$0.493049\pi$
$860$	0	0
$861$	−25.1047	−0.855565
$862$	8.13497	0.277078
$863$	−8.38448	−0.285411	−0.142706	−	0.989765i	$-0.545580\pi$
$863$	−8.38448	−0.285411	−0.142706	+	0.989765i	$0.545580\pi$
$864$	−12.3553	−0.420335
$865$	0	0
$866$	11.9342	0.405541
$867$	35.5376	1.20692
$868$	−23.9718	−0.813655
$869$	−44.7308	−1.51739
$870$	0	0
$871$	0	0
$872$	10.9299	0.370132
$873$	−22.3588	−0.756729
$874$	−1.10945	−0.0375278
$875$	0	0
$876$	−27.8109	−0.939645
$877$	55.8862	1.88714	0.943572	−	0.331169i	$-0.107443\pi$
$877$	55.8862	1.88714	0.943572	+	0.331169i	$0.107443\pi$
$878$	0.838765	0.0283069
$879$	36.2051	1.22117
$880$	0	0
$881$	25.1949	0.848839	0.424420	−	0.905466i	$-0.360478\pi$
$881$	25.1949	0.848839	0.424420	+	0.905466i	$0.360478\pi$
$882$	5.96396	0.200817
$883$	30.7868	1.03606	0.518029	−	0.855363i	$-0.326666\pi$
$883$	30.7868	1.03606	0.518029	+	0.855363i	$0.326666\pi$
$884$	0	0
$885$	0	0
$886$	6.40429	0.215156
$887$	12.4721	0.418771	0.209385	−	0.977833i	$-0.432854\pi$
$887$	12.4721	0.418771	0.209385	+	0.977833i	$0.432854\pi$
$888$	6.75207	0.226585
$889$	−30.7302	−1.03066
$890$	0	0
$891$	12.0437	0.403478
$892$	−23.3875	−0.783071
$893$	8.55822	0.286390
$894$	−14.1167	−0.472132
$895$	0	0
$896$	−30.8032	−1.02906
$897$	0	0
$898$	−8.20739	−0.273884
$899$	11.3433	0.378320
$900$	0	0
$901$	25.0101	0.833209
$902$	−2.98643	−0.0994372
$903$	78.8688	2.62459
$904$	9.43781	0.313897
$905$	0	0
$906$	−12.9534	−0.430348
$907$	38.8911	1.29136	0.645678	−	0.763609i	$-0.276575\pi$
$907$	38.8911	1.29136	0.645678	+	0.763609i	$0.276575\pi$
$908$	11.6511	0.386655
$909$	24.2458	0.804183
$910$	0	0
$911$	0.165096	0.00546989	0.00273494	−	0.999996i	$-0.499129\pi$
$911$	0.165096	0.00546989	0.00273494	+	0.999996i	$0.499129\pi$
$912$	11.2518	0.372584
$913$	−27.9774	−0.925918
$914$	−2.50360	−0.0828117
$915$	0	0
$916$	51.0131	1.68552
$917$	33.5348	1.10742
$918$	2.16211	0.0713603
$919$	0.895326	0.0295341	0.0147670	−	0.999891i	$-0.495299\pi$
$919$	0.895326	0.0295341	0.0147670	+	0.999891i	$0.495299\pi$
$920$	0	0
$921$	66.2392	2.18266
$922$	−4.08361	−0.134486
$923$	0	0
$924$	−55.3923	−1.82227
$925$	0	0
$926$	−7.56219	−0.248509
$927$	−31.2858	−1.02756
$928$	11.0529	0.362831
$929$	−12.2895	−0.403205	−0.201602	−	0.979467i	$-0.564615\pi$
$929$	−12.2895	−0.403205	−0.201602	+	0.979467i	$0.564615\pi$
$930$	0	0
$931$	−5.28947	−0.173356
$932$	−1.55841	−0.0510476
$933$	−6.56211	−0.214834
$934$	5.06101	0.165601
$935$	0	0
$936$	0	0
$937$	5.77242	0.188577	0.0942884	−	0.995545i	$-0.469942\pi$
$937$	5.77242	0.188577	0.0942884	+	0.995545i	$0.469942\pi$
$938$	−4.45462	−0.145448
$939$	−51.9425	−1.69508
$940$	0	0
$941$	−55.8887	−1.82192	−0.910960	−	0.412495i	$-0.864658\pi$
$941$	−55.8887	−1.82192	−0.910960	+	0.412495i	$0.864658\pi$
$942$	−9.72563	−0.316878
$943$	7.48616	0.243783
$944$	−8.50655	−0.276864
$945$	0	0
$946$	9.38217	0.305041
$947$	1.89638	0.0616239	0.0308120	−	0.999525i	$-0.490191\pi$
$947$	1.89638	0.0616239	0.0308120	+	0.999525i	$0.490191\pi$
$948$	70.1318	2.27777
$949$	0	0
$950$	0	0
$951$	−77.5825	−2.51578
$952$	8.41179	0.272628
$953$	−33.6523	−1.09011	−0.545053	−	0.838402i	$-0.683490\pi$
$953$	−33.6523	−1.09011	−0.545053	+	0.838402i	$0.683490\pi$
$954$	−18.0269	−0.583643
$955$	0	0
$956$	−7.56219	−0.244579
$957$	26.2113	0.847290
$958$	8.03366	0.259556
$959$	56.5018	1.82454
$960$	0	0
$961$	−16.7033	−0.538817
$962$	0	0
$963$	36.4054	1.17315
$964$	−42.9036	−1.38183
$965$	0	0
$966$	−8.03888	−0.258647
$967$	23.0493	0.741216	0.370608	−	0.928789i	$-0.379149\pi$
$967$	23.0493	0.741216	0.370608	+	0.928789i	$0.379149\pi$
$968$	0.598152	0.0192253
$969$	−6.53528	−0.209944
$970$	0	0
$971$	14.9193	0.478783	0.239391	−	0.970923i	$-0.423052\pi$
$971$	14.9193	0.478783	0.239391	+	0.970923i	$0.423052\pi$
$972$	−37.9025	−1.21572
$973$	−3.44370	−0.110400
$974$	−12.2040	−0.391041
$975$	0	0
$976$	−25.1364	−0.804595
$977$	23.3641	0.747485	0.373742	−	0.927533i	$-0.378074\pi$
$977$	23.3641	0.747485	0.373742	+	0.927533i	$0.378074\pi$
$978$	3.72052	0.118969
$979$	−33.4850	−1.07019
$980$	0	0
$981$	−36.0538	−1.15111
$982$	−11.6962	−0.373241
$983$	−5.31119	−0.169401	−0.0847003	−	0.996406i	$-0.526993\pi$
$983$	−5.31119	−0.169401	−0.0847003	+	0.996406i	$0.526993\pi$
$984$	9.63570	0.307175
$985$	0	0
$986$	−1.93421	−0.0615978
$987$	62.0112	1.97384
$988$	0	0
$989$	−23.5185	−0.747846
$990$	0	0
$991$	−24.0879	−0.765178	−0.382589	−	0.923919i	$-0.624967\pi$
$991$	−24.0879	−0.765178	−0.382589	+	0.923919i	$0.624967\pi$
$992$	13.9307	0.442301
$993$	8.00299	0.253967
$994$	5.81998	0.184599
$995$	0	0
$996$	43.8648	1.38991
$997$	−20.4099	−0.646389	−0.323195	−	0.946332i	$-0.604757\pi$
$997$	−20.4099	−0.646389	−0.323195	+	0.946332i	$0.604757\pi$
$998$	5.36581	0.169852
$999$	−6.53528	−0.206767

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.br.1.4		6
5.2	odd	4		845.2.b.d.339.4		6
5.3	odd	4		845.2.b.d.339.3		6
5.4	even	2	inner	4225.2.a.br.1.3		6
13.3	even	3		325.2.e.e.126.3		12
13.9	even	3		325.2.e.e.276.3		12
13.12	even	2		4225.2.a.bq.1.3		6
65.2	even	12		845.2.l.f.654.5		24
65.3	odd	12		65.2.n.a.9.4	yes	12
65.7	even	12		845.2.l.f.699.8		24
65.8	even	4		845.2.d.d.844.7		12
65.9	even	6		325.2.e.e.276.4		12
65.12	odd	4		845.2.b.e.339.3		6
65.17	odd	12		845.2.n.e.484.3		12
65.18	even	4		845.2.d.d.844.5		12
65.22	odd	12		65.2.n.a.29.4	yes	12
65.23	odd	12		845.2.n.e.529.3		12
65.28	even	12		845.2.l.f.654.8		24
65.29	even	6		325.2.e.e.126.4		12
65.32	even	12		845.2.l.f.699.6		24
65.33	even	12		845.2.l.f.699.5		24
65.37	even	12		845.2.l.f.654.7		24
65.38	odd	4		845.2.b.e.339.4		6
65.42	odd	12		65.2.n.a.9.3	✓	12
65.43	odd	12		845.2.n.e.484.4		12
65.47	even	4		845.2.d.d.844.6		12
65.48	odd	12		65.2.n.a.29.3	yes	12
65.57	even	4		845.2.d.d.844.8		12
65.58	even	12		845.2.l.f.699.7		24
65.62	odd	12		845.2.n.e.529.4		12
65.63	even	12		845.2.l.f.654.6		24
65.64	even	2		4225.2.a.bq.1.4		6
195.68	even	12		585.2.bs.a.334.3		12
195.107	even	12		585.2.bs.a.334.4		12
195.113	even	12		585.2.bs.a.289.4		12
195.152	even	12		585.2.bs.a.289.3		12
260.3	even	12		1040.2.dh.a.529.1		12
260.87	even	12		1040.2.dh.a.289.1		12
260.107	even	12		1040.2.dh.a.529.6		12
260.243	even	12		1040.2.dh.a.289.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.n.a.9.3	✓	12	65.42	odd	12
65.2.n.a.9.4	yes	12	65.3	odd	12
65.2.n.a.29.3	yes	12	65.48	odd	12
65.2.n.a.29.4	yes	12	65.22	odd	12
325.2.e.e.126.3		12	13.3	even	3
325.2.e.e.126.4		12	65.29	even	6
325.2.e.e.276.3		12	13.9	even	3
325.2.e.e.276.4		12	65.9	even	6
585.2.bs.a.289.3		12	195.152	even	12
585.2.bs.a.289.4		12	195.113	even	12
585.2.bs.a.334.3		12	195.68	even	12
585.2.bs.a.334.4		12	195.107	even	12
845.2.b.d.339.3		6	5.3	odd	4
845.2.b.d.339.4		6	5.2	odd	4
845.2.b.e.339.3		6	65.12	odd	4
845.2.b.e.339.4		6	65.38	odd	4
845.2.d.d.844.5		12	65.18	even	4
845.2.d.d.844.6		12	65.47	even	4
845.2.d.d.844.7		12	65.8	even	4
845.2.d.d.844.8		12	65.57	even	4
845.2.l.f.654.5		24	65.2	even	12
845.2.l.f.654.6		24	65.63	even	12
845.2.l.f.654.7		24	65.37	even	12
845.2.l.f.654.8		24	65.28	even	12
845.2.l.f.699.5		24	65.33	even	12
845.2.l.f.699.6		24	65.32	even	12
845.2.l.f.699.7		24	65.58	even	12
845.2.l.f.699.8		24	65.7	even	12
845.2.n.e.484.3		12	65.17	odd	12
845.2.n.e.484.4		12	65.43	odd	12
845.2.n.e.529.3		12	65.23	odd	12
845.2.n.e.529.4		12	65.62	odd	12
1040.2.dh.a.289.1		12	260.87	even	12
1040.2.dh.a.289.6		12	260.243	even	12
1040.2.dh.a.529.1		12	260.3	even	12
1040.2.dh.a.529.6		12	260.107	even	12
4225.2.a.bq.1.3		6	13.12	even	2
4225.2.a.bq.1.4		6	65.64	even	2
4225.2.a.br.1.3		6	5.4	even	2	inner
4225.2.a.br.1.4		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.330837 q^{2} -2.69180 q^{3} -1.89055 q^{4} -0.890547 q^{6} +3.35348 q^{7} -1.28714 q^{8} +4.24581 q^{9} +O(q^{10})\)	\(q+0.330837 q^{2} -2.69180 q^{3} -1.89055 q^{4} -0.890547 q^{6} +3.35348 q^{7} -1.28714 q^{8} +4.24581 q^{9} -3.24581 q^{11} +5.08898 q^{12} +1.10945 q^{14} +3.35526 q^{16} -1.94881 q^{17} +1.40467 q^{18} -1.24581 q^{19} -9.02690 q^{21} -1.07383 q^{22} +2.69180 q^{23} +3.46472 q^{24} -3.35348 q^{27} -6.33991 q^{28} +3.00000 q^{29} +3.78109 q^{31} +3.68431 q^{32} +8.73709 q^{33} -0.644737 q^{34} -8.02690 q^{36} +1.94881 q^{37} -0.412160 q^{38} +2.78109 q^{41} -2.98643 q^{42} -8.73709 q^{43} +6.13636 q^{44} +0.890547 q^{46} -6.86960 q^{47} -9.03171 q^{48} +4.24581 q^{49} +5.24581 q^{51} -12.8336 q^{53} -1.10945 q^{54} -4.31638 q^{56} +3.35348 q^{57} +0.992510 q^{58} -2.53528 q^{59} -7.49162 q^{61} +1.25092 q^{62} +14.2382 q^{63} -5.49162 q^{64} +2.89055 q^{66} -4.01515 q^{67} +3.68431 q^{68} -7.24581 q^{69} +5.24581 q^{71} -5.46493 q^{72} -5.46493 q^{73} +0.644737 q^{74} +2.35526 q^{76} -10.8848 q^{77} +13.7811 q^{79} -3.71053 q^{81} +0.920088 q^{82} +8.61955 q^{83} +17.0658 q^{84} -2.89055 q^{86} -8.07541 q^{87} +4.17780 q^{88} +10.3164 q^{89} -5.08898 q^{92} -10.1780 q^{93} -2.27271 q^{94} -9.91745 q^{96} -5.26607 q^{97} +1.40467 q^{98} -13.7811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} + 10 q^{6} + 6 q^{9}+O(q^{10}) \) 6 * q + 4 * q^4 + 10 * q^6 + 6 * q^9	\( 6 q + 4 q^{4} + 10 q^{6} + 6 q^{9} + 22 q^{14} + 16 q^{16} + 12 q^{19} - 4 q^{21} + 32 q^{24} + 18 q^{29} - 8 q^{31} - 8 q^{34} + 2 q^{36} - 14 q^{41} + 2 q^{44} - 10 q^{46} + 6 q^{49} + 12 q^{51} - 22 q^{54} + 16 q^{56} - 4 q^{59} - 6 q^{61} + 6 q^{64} + 2 q^{66} - 24 q^{69} + 12 q^{71} + 8 q^{74} + 10 q^{76} + 52 q^{79} - 14 q^{81} + 90 q^{84} - 2 q^{86} + 20 q^{89} + 56 q^{94} + 6 q^{96} - 52 q^{99}+O(q^{100}) \) 6 * q + 4 * q^4 + 10 * q^6 + 6 * q^9 + 22 * q^14 + 16 * q^16 + 12 * q^19 - 4 * q^21 + 32 * q^24 + 18 * q^29 - 8 * q^31 - 8 * q^34 + 2 * q^36 - 14 * q^41 + 2 * q^44 - 10 * q^46 + 6 * q^49 + 12 * q^51 - 22 * q^54 + 16 * q^56 - 4 * q^59 - 6 * q^61 + 6 * q^64 + 2 * q^66 - 24 * q^69 + 12 * q^71 + 8 * q^74 + 10 * q^76 + 52 * q^79 - 14 * q^81 + 90 * q^84 - 2 * q^86 + 20 * q^89 + 56 * q^94 + 6 * q^96 - 52 * q^99

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