LMFDB - Embedded newform 5225.2.a.bc.1.29

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [30,0,0,42,0,12,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.29
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.77451 q^{2} -2.74433 q^{3} +5.69792 q^{4} -7.61418 q^{6} -3.20844 q^{7} +10.2599 q^{8} +4.53135 q^{9} +1.00000 q^{11} -15.6370 q^{12} -3.87053 q^{13} -8.90186 q^{14} +17.0705 q^{16} -5.86570 q^{17} +12.5723 q^{18} +1.00000 q^{19} +8.80502 q^{21} +2.77451 q^{22} +6.11891 q^{23} -28.1566 q^{24} -10.7388 q^{26} -4.20253 q^{27} -18.2814 q^{28} +6.40440 q^{29} +1.66215 q^{31} +26.8423 q^{32} -2.74433 q^{33} -16.2745 q^{34} +25.8193 q^{36} -0.251830 q^{37} +2.77451 q^{38} +10.6220 q^{39} +10.7565 q^{41} +24.4296 q^{42} +2.17536 q^{43} +5.69792 q^{44} +16.9770 q^{46} -2.68163 q^{47} -46.8470 q^{48} +3.29408 q^{49} +16.0974 q^{51} -22.0540 q^{52} +8.89909 q^{53} -11.6600 q^{54} -32.9183 q^{56} -2.74433 q^{57} +17.7691 q^{58} -4.09447 q^{59} +4.52319 q^{61} +4.61167 q^{62} -14.5386 q^{63} +40.3335 q^{64} -7.61418 q^{66} -2.61295 q^{67} -33.4223 q^{68} -16.7923 q^{69} -5.51722 q^{71} +46.4913 q^{72} +2.73480 q^{73} -0.698705 q^{74} +5.69792 q^{76} -3.20844 q^{77} +29.4709 q^{78} +8.17104 q^{79} -2.06093 q^{81} +29.8440 q^{82} +12.2337 q^{83} +50.1703 q^{84} +6.03555 q^{86} -17.5758 q^{87} +10.2599 q^{88} -1.84048 q^{89} +12.4184 q^{91} +34.8651 q^{92} -4.56150 q^{93} -7.44022 q^{94} -73.6642 q^{96} +2.65645 q^{97} +9.13947 q^{98} +4.53135 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9} + 30 q^{11} - 4 q^{14} + 66 q^{16} + 30 q^{19} + 14 q^{21} + 22 q^{24} + 30 q^{29} + 26 q^{31} + 12 q^{34} + 78 q^{36} + 64 q^{39} + 22 q^{41} + 42 q^{44} + 28 q^{46}+ \cdots + 40 q^{99}+O(q^{100}) $ 30 * q + 42 * q^4 + 12 * q^6 + 40 * q^9 + 30 * q^11 - 4 * q^14 + 66 * q^16 + 30 * q^19 + 14 * q^21 + 22 * q^24 + 30 * q^29 + 26 * q^31 + 12 * q^34 + 78 * q^36 + 64 * q^39 + 22 * q^41 + 42 * q^44 + 28 * q^46 + 60 * q^49 + 64 * q^51 + 62 * q^54 - 32 * q^56 - 14 * q^59 + 78 * q^61 + 90 * q^64 + 12 * q^66 - 28 * q^69 + 20 * q^71 + 42 * q^74 + 42 * q^76 + 102 * q^79 + 42 * q^81 + 98 * q^84 - 52 * q^86 - 8 * q^89 + 56 * q^91 + 40 * q^94 - 74 * q^96 + 40 * 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 $

$2$	2.77451	1.96188	0.980938	−	0.194319i	$-0.0622498\pi$
$2$	2.77451	1.96188	0.980938	+	0.194319i	$0.0622498\pi$
$3$	−2.74433	−1.58444	−0.792220	−	0.610236i	$-0.791075\pi$
$3$	−2.74433	−1.58444	−0.792220	+	0.610236i	$0.791075\pi$
$4$	5.69792	2.84896
$5$	0	0
$5$	0	0
$6$	−7.61418	−3.10848
$7$	−3.20844	−1.21268	−0.606338	−	0.795207i	$-0.707362\pi$
$7$	−3.20844	−1.21268	−0.606338	+	0.795207i	$0.707362\pi$
$8$	10.2599	3.62743
$9$	4.53135	1.51045
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−15.6370	−4.51401
$13$	−3.87053	−1.07349	−0.536746	−	0.843744i	$-0.680347\pi$
$13$	−3.87053	−1.07349	−0.536746	+	0.843744i	$0.680347\pi$
$14$	−8.90186	−2.37912
$15$	0	0
$16$	17.0705	4.26761
$17$	−5.86570	−1.42264	−0.711321	−	0.702867i	$-0.751903\pi$
$17$	−5.86570	−1.42264	−0.711321	+	0.702867i	$0.751903\pi$
$18$	12.5723	2.96332
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	8.80502	1.92141
$22$	2.77451	0.591528
$23$	6.11891	1.27588	0.637940	−	0.770086i	$-0.279787\pi$
$23$	6.11891	1.27588	0.637940	+	0.770086i	$0.279787\pi$
$24$	−28.1566	−5.74745
$25$	0	0
$26$	−10.7388	−2.10606
$27$	−4.20253	−0.808777
$28$	−18.2814	−3.45487
$29$	6.40440	1.18927	0.594634	−	0.803997i	$-0.297297\pi$
$29$	6.40440	1.18927	0.594634	+	0.803997i	$0.297297\pi$
$30$	0	0
$31$	1.66215	0.298532	0.149266	−	0.988797i	$-0.452309\pi$
$31$	1.66215	0.298532	0.149266	+	0.988797i	$0.452309\pi$
$32$	26.8423	4.74510
$33$	−2.74433	−0.477727
$34$	−16.2745	−2.79105
$35$	0	0
$36$	25.8193	4.30321
$37$	−0.251830	−0.0414006	−0.0207003	−	0.999786i	$-0.506590\pi$
$37$	−0.251830	−0.0414006	−0.0207003	+	0.999786i	$0.506590\pi$
$38$	2.77451	0.450085
$39$	10.6220	1.70088
$40$	0	0
$41$	10.7565	1.67988	0.839941	−	0.542677i	$-0.182589\pi$
$41$	10.7565	1.67988	0.839941	+	0.542677i	$0.182589\pi$
$42$	24.4296	3.76957
$43$	2.17536	0.331739	0.165869	−	0.986148i	$-0.446957\pi$
$43$	2.17536	0.331739	0.165869	+	0.986148i	$0.446957\pi$
$44$	5.69792	0.858994
$45$	0	0
$46$	16.9770	2.50312
$47$	−2.68163	−0.391156	−0.195578	−	0.980688i	$-0.562658\pi$
$47$	−2.68163	−0.391156	−0.195578	+	0.980688i	$0.562658\pi$
$48$	−46.8470	−6.76178
$49$	3.29408	0.470583
$50$	0	0
$51$	16.0974	2.25409
$52$	−22.0540	−3.05834
$53$	8.89909	1.22238	0.611192	−	0.791483i	$-0.290690\pi$
$53$	8.89909	1.22238	0.611192	+	0.791483i	$0.290690\pi$
$54$	−11.6600	−1.58672
$55$	0	0
$56$	−32.9183	−4.39890
$57$	−2.74433	−0.363495
$58$	17.7691	2.33320
$59$	−4.09447	−0.533054	−0.266527	−	0.963827i	$-0.585876\pi$
$59$	−4.09447	−0.533054	−0.266527	+	0.963827i	$0.585876\pi$
$60$	0	0
$61$	4.52319	0.579135	0.289567	−	0.957158i	$-0.406489\pi$
$61$	4.52319	0.579135	0.289567	+	0.957158i	$0.406489\pi$
$62$	4.61167	0.585682
$63$	−14.5386	−1.83169
$64$	40.3335	5.04169
$65$	0	0
$66$	−7.61418	−0.937241
$67$	−2.61295	−0.319222	−0.159611	−	0.987180i	$-0.551024\pi$
$67$	−2.61295	−0.319222	−0.159611	+	0.987180i	$0.551024\pi$
$68$	−33.4223	−4.05305
$69$	−16.7923	−2.02156
$70$	0	0
$71$	−5.51722	−0.654774	−0.327387	−	0.944890i	$-0.606168\pi$
$71$	−5.51722	−0.654774	−0.327387	+	0.944890i	$0.606168\pi$
$72$	46.4913	5.47905
$73$	2.73480	0.320084	0.160042	−	0.987110i	$-0.448837\pi$
$73$	2.73480	0.320084	0.160042	+	0.987110i	$0.448837\pi$
$74$	−0.698705	−0.0812229
$75$	0	0
$76$	5.69792	0.653596
$77$	−3.20844	−0.365636
$78$	29.4709	3.33692
$79$	8.17104	0.919313	0.459657	−	0.888097i	$-0.347972\pi$
$79$	8.17104	0.919313	0.459657	+	0.888097i	$0.347972\pi$
$80$	0	0
$81$	−2.06093	−0.228992
$82$	29.8440	3.29572
$83$	12.2337	1.34283	0.671413	−	0.741083i	$-0.265688\pi$
$83$	12.2337	1.34283	0.671413	+	0.741083i	$0.265688\pi$
$84$	50.1703	5.47403
$85$	0	0
$86$	6.03555	0.650830
$87$	−17.5758	−1.88432
$88$	10.2599	1.09371
$89$	−1.84048	−0.195091	−0.0975454	−	0.995231i	$-0.531099\pi$
$89$	−1.84048	−0.195091	−0.0975454	+	0.995231i	$0.531099\pi$
$90$	0	0
$91$	12.4184	1.30180
$92$	34.8651	3.63493
$93$	−4.56150	−0.473005
$94$	−7.44022	−0.767400
$95$	0	0
$96$	−73.6642	−7.51832
$97$	2.65645	0.269722	0.134861	−	0.990865i	$-0.456941\pi$
$97$	2.65645	0.269722	0.134861	+	0.990865i	$0.456941\pi$
$98$	9.13947	0.923226
$99$	4.53135	0.455418
$100$	0	0
$101$	3.36522	0.334852	0.167426	−	0.985885i	$-0.446454\pi$
$101$	3.36522	0.334852	0.167426	+	0.985885i	$0.446454\pi$
$102$	44.6625	4.42225
$103$	19.1222	1.88416	0.942082	−	0.335381i	$-0.108865\pi$
$103$	19.1222	1.88416	0.942082	+	0.335381i	$0.108865\pi$
$104$	−39.7114	−3.89402
$105$	0	0
$106$	24.6906	2.39817
$107$	19.1096	1.84739	0.923697	−	0.383125i	$-0.125152\pi$
$107$	19.1096	1.84739	0.923697	+	0.383125i	$0.125152\pi$
$108$	−23.9457	−2.30417
$109$	−2.04217	−0.195605	−0.0978023	−	0.995206i	$-0.531181\pi$
$109$	−2.04217	−0.195605	−0.0978023	+	0.995206i	$0.531181\pi$
$110$	0	0
$111$	0.691105	0.0655968
$112$	−54.7695	−5.17523
$113$	−2.01233	−0.189305	−0.0946523	−	0.995510i	$-0.530174\pi$
$113$	−2.01233	−0.189305	−0.0946523	+	0.995510i	$0.530174\pi$
$114$	−7.61418	−0.713133
$115$	0	0
$116$	36.4918	3.38818
$117$	−17.5387	−1.62146
$118$	−11.3601	−1.04579
$119$	18.8197	1.72520
$120$	0	0
$121$	1.00000	0.0909091
$122$	12.5496	1.13619
$123$	−29.5194	−2.66167
$124$	9.47082	0.850505
$125$	0	0
$126$	−40.3374	−3.59354
$127$	−17.6725	−1.56818	−0.784090	−	0.620648i	$-0.786870\pi$
$127$	−17.6725	−1.56818	−0.784090	+	0.620648i	$0.786870\pi$
$128$	58.2211	5.14607
$129$	−5.96990	−0.525620
$130$	0	0
$131$	−4.85389	−0.424086	−0.212043	−	0.977260i	$-0.568012\pi$
$131$	−4.85389	−0.424086	−0.212043	+	0.977260i	$0.568012\pi$
$132$	−15.6370	−1.36102
$133$	−3.20844	−0.278207
$134$	−7.24965	−0.626274
$135$	0	0
$136$	−60.1817	−5.16054
$137$	0.470694	0.0402141	0.0201071	−	0.999798i	$-0.493599\pi$
$137$	0.470694	0.0402141	0.0201071	+	0.999798i	$0.493599\pi$
$138$	−46.5905	−3.96604
$139$	10.2256	0.867325	0.433663	−	0.901075i	$-0.357221\pi$
$139$	10.2256	0.867325	0.433663	+	0.901075i	$0.357221\pi$
$140$	0	0
$141$	7.35929	0.619764
$142$	−15.3076	−1.28459
$143$	−3.87053	−0.323670
$144$	77.3522	6.44601
$145$	0	0
$146$	7.58774	0.627965
$147$	−9.04005	−0.745611
$148$	−1.43491	−0.117949
$149$	−19.2747	−1.57905	−0.789523	−	0.613721i	$-0.789672\pi$
$149$	−19.2747	−1.57905	−0.789523	+	0.613721i	$0.789672\pi$
$150$	0	0
$151$	−9.24566	−0.752401	−0.376200	−	0.926538i	$-0.622770\pi$
$151$	−9.24566	−0.752401	−0.376200	+	0.926538i	$0.622770\pi$
$152$	10.2599	0.832190
$153$	−26.5795	−2.14883
$154$	−8.90186	−0.717332
$155$	0	0
$156$	60.5234	4.84575
$157$	−9.10320	−0.726514	−0.363257	−	0.931689i	$-0.618335\pi$
$157$	−9.10320	−0.726514	−0.363257	+	0.931689i	$0.618335\pi$
$158$	22.6706	1.80358
$159$	−24.4220	−1.93679
$160$	0	0
$161$	−19.6321	−1.54723
$162$	−5.71806	−0.449253
$163$	−13.1874	−1.03292	−0.516458	−	0.856313i	$-0.672750\pi$
$163$	−13.1874	−1.03292	−0.516458	+	0.856313i	$0.672750\pi$
$164$	61.2897	4.78592
$165$	0	0
$166$	33.9426	2.63446
$167$	7.56734	0.585578	0.292789	−	0.956177i	$-0.405417\pi$
$167$	7.56734	0.585578	0.292789	+	0.956177i	$0.405417\pi$
$168$	90.3388	6.96979
$169$	1.98102	0.152386
$170$	0	0
$171$	4.53135	0.346521
$172$	12.3950	0.945110
$173$	12.3483	0.938825	0.469412	−	0.882979i	$-0.344466\pi$
$173$	12.3483	0.938825	0.469412	+	0.882979i	$0.344466\pi$
$174$	−48.7643	−3.69681
$175$	0	0
$176$	17.0705	1.28673
$177$	11.2366	0.844592
$178$	−5.10644	−0.382744
$179$	9.75082	0.728810	0.364405	−	0.931241i	$-0.381272\pi$
$179$	9.75082	0.728810	0.364405	+	0.931241i	$0.381272\pi$
$180$	0	0
$181$	−3.06931	−0.228140	−0.114070	−	0.993473i	$-0.536389\pi$
$181$	−3.06931	−0.228140	−0.114070	+	0.993473i	$0.536389\pi$
$182$	34.4549	2.55397
$183$	−12.4131	−0.917604
$184$	62.7796	4.62817
$185$	0	0
$186$	−12.6559	−0.927978
$187$	−5.86570	−0.428943
$188$	−15.2797	−1.11439
$189$	13.4836	0.980784
$190$	0	0
$191$	11.5267	0.834040	0.417020	−	0.908897i	$-0.363075\pi$
$191$	11.5267	0.834040	0.417020	+	0.908897i	$0.363075\pi$
$192$	−110.688	−7.98825
$193$	−6.25518	−0.450258	−0.225129	−	0.974329i	$-0.572280\pi$
$193$	−6.25518	−0.450258	−0.225129	+	0.974329i	$0.572280\pi$
$194$	7.37036	0.529161
$195$	0	0
$196$	18.7694	1.34067
$197$	6.53160	0.465357	0.232678	−	0.972554i	$-0.425251\pi$
$197$	6.53160	0.465357	0.232678	+	0.972554i	$0.425251\pi$
$198$	12.5723	0.893473
$199$	11.4237	0.809808	0.404904	−	0.914359i	$-0.367305\pi$
$199$	11.4237	0.809808	0.404904	+	0.914359i	$0.367305\pi$
$200$	0	0
$201$	7.17079	0.505788
$202$	9.33686	0.656939
$203$	−20.5481	−1.44220
$204$	91.7218	6.42181
$205$	0	0
$206$	53.0547	3.69650
$207$	27.7269	1.92715
$208$	−66.0717	−4.58125
$209$	1.00000	0.0691714
$210$	0	0
$211$	−0.159122	−0.0109544	−0.00547722	−	0.999985i	$-0.501743\pi$
$211$	−0.159122	−0.0109544	−0.00547722	+	0.999985i	$0.501743\pi$
$212$	50.7063	3.48252
$213$	15.1411	1.03745
$214$	53.0198	3.62436
$215$	0	0
$216$	−43.1176	−2.93378
$217$	−5.33292	−0.362022
$218$	−5.66603	−0.383752
$219$	−7.50519	−0.507154
$220$	0	0
$221$	22.7034	1.52720
$222$	1.91748	0.128693
$223$	0.537456	0.0359907	0.0179954	−	0.999838i	$-0.494272\pi$
$223$	0.537456	0.0359907	0.0179954	+	0.999838i	$0.494272\pi$
$224$	−86.1220	−5.75427
$225$	0	0
$226$	−5.58325	−0.371392
$227$	13.5232	0.897565	0.448783	−	0.893641i	$-0.351858\pi$
$227$	13.5232	0.897565	0.448783	+	0.893641i	$0.351858\pi$
$228$	−15.6370	−1.03558
$229$	−10.6187	−0.701704	−0.350852	−	0.936431i	$-0.614108\pi$
$229$	−10.6187	−0.701704	−0.350852	+	0.936431i	$0.614108\pi$
$230$	0	0
$231$	8.80502	0.579328
$232$	65.7087	4.31399
$233$	7.83409	0.513228	0.256614	−	0.966514i	$-0.417393\pi$
$233$	7.83409	0.513228	0.256614	+	0.966514i	$0.417393\pi$
$234$	−48.6614	−3.18110
$235$	0	0
$236$	−23.3299	−1.51865
$237$	−22.4240	−1.45660
$238$	52.2156	3.38464
$239$	25.3402	1.63912	0.819560	−	0.572994i	$-0.194218\pi$
$239$	25.3402	1.63912	0.819560	+	0.572994i	$0.194218\pi$
$240$	0	0
$241$	−30.0559	−1.93607	−0.968035	−	0.250814i	$-0.919302\pi$
$241$	−30.0559	−1.93607	−0.968035	+	0.250814i	$0.919302\pi$
$242$	2.77451	0.178352
$243$	18.2634	1.17160
$244$	25.7728	1.64993
$245$	0	0
$246$	−81.9019	−5.22188
$247$	−3.87053	−0.246276
$248$	17.0536	1.08290
$249$	−33.5734	−2.12763
$250$	0	0
$251$	24.0428	1.51757	0.758783	−	0.651344i	$-0.225794\pi$
$251$	24.0428	1.51757	0.758783	+	0.651344i	$0.225794\pi$
$252$	−82.8395	−5.21840
$253$	6.11891	0.384693
$254$	−49.0325	−3.07657
$255$	0	0
$256$	80.8682	5.05426
$257$	−6.58227	−0.410591	−0.205295	−	0.978700i	$-0.565815\pi$
$257$	−6.58227	−0.410591	−0.205295	+	0.978700i	$0.565815\pi$
$258$	−16.5635	−1.03120
$259$	0.807981	0.0502055
$260$	0	0
$261$	29.0206	1.79633
$262$	−13.4672	−0.832005
$263$	−16.0460	−0.989442	−0.494721	−	0.869052i	$-0.664730\pi$
$263$	−16.0460	−0.989442	−0.494721	+	0.869052i	$0.664730\pi$
$264$	−28.1566	−1.73292
$265$	0	0
$266$	−8.90186	−0.545808
$267$	5.05089	0.309110
$268$	−14.8884	−0.909451
$269$	−17.0574	−1.04001	−0.520005	−	0.854163i	$-0.674070\pi$
$269$	−17.0574	−1.04001	−0.520005	+	0.854163i	$0.674070\pi$
$270$	0	0
$271$	14.8456	0.901807	0.450904	−	0.892573i	$-0.351102\pi$
$271$	14.8456	0.901807	0.450904	+	0.892573i	$0.351102\pi$
$272$	−100.130	−6.07128
$273$	−34.0801	−2.06262
$274$	1.30595	0.0788951
$275$	0	0
$276$	−95.6812	−5.75933
$277$	−8.59398	−0.516362	−0.258181	−	0.966097i	$-0.583123\pi$
$277$	−8.59398	−0.516362	−0.258181	+	0.966097i	$0.583123\pi$
$278$	28.3711	1.70158
$279$	7.53180	0.450917
$280$	0	0
$281$	−13.4669	−0.803368	−0.401684	−	0.915778i	$-0.631575\pi$
$281$	−13.4669	−0.803368	−0.401684	+	0.915778i	$0.631575\pi$
$282$	20.4184	1.21590
$283$	−14.2887	−0.849374	−0.424687	−	0.905340i	$-0.639616\pi$
$283$	−14.2887	−0.849374	−0.424687	+	0.905340i	$0.639616\pi$
$284$	−31.4367	−1.86542
$285$	0	0
$286$	−10.7388	−0.635001
$287$	−34.5116	−2.03715
$288$	121.632	7.16723
$289$	17.4065	1.02391
$290$	0	0
$291$	−7.29019	−0.427358
$292$	15.5827	0.911907
$293$	29.5456	1.72607	0.863036	−	0.505143i	$-0.168560\pi$
$293$	29.5456	1.72607	0.863036	+	0.505143i	$0.168560\pi$
$294$	−25.0817	−1.46280
$295$	0	0
$296$	−2.58376	−0.150178
$297$	−4.20253	−0.243855
$298$	−53.4779	−3.09789
$299$	−23.6834	−1.36965
$300$	0	0
$301$	−6.97950	−0.402292
$302$	−25.6522	−1.47612
$303$	−9.23529	−0.530553
$304$	17.0705	0.979058
$305$	0	0
$306$	−73.7453	−4.21574
$307$	21.6031	1.23295	0.616477	−	0.787373i	$-0.288559\pi$
$307$	21.6031	1.23295	0.616477	+	0.787373i	$0.288559\pi$
$308$	−18.2814	−1.04168
$309$	−52.4776	−2.98535
$310$	0	0
$311$	21.4045	1.21374	0.606870	−	0.794801i	$-0.292425\pi$
$311$	21.4045	1.21374	0.606870	+	0.794801i	$0.292425\pi$
$312$	108.981	6.16984
$313$	22.7332	1.28496	0.642478	−	0.766304i	$-0.277906\pi$
$313$	22.7332	1.28496	0.642478	+	0.766304i	$0.277906\pi$
$314$	−25.2569	−1.42533
$315$	0	0
$316$	46.5579	2.61909
$317$	1.29414	0.0726859	0.0363430	−	0.999339i	$-0.488429\pi$
$317$	1.29414	0.0726859	0.0363430	+	0.999339i	$0.488429\pi$
$318$	−67.7592	−3.79975
$319$	6.40440	0.358578
$320$	0	0
$321$	−52.4430	−2.92708
$322$	−54.4696	−3.03547
$323$	−5.86570	−0.326376
$324$	−11.7430	−0.652388
$325$	0	0
$326$	−36.5885	−2.02645
$327$	5.60440	0.309924
$328$	110.361	6.09366
$329$	8.60386	0.474346
$330$	0	0
$331$	−22.3804	−1.23014	−0.615068	−	0.788474i	$-0.710872\pi$
$331$	−22.3804	−1.23014	−0.615068	+	0.788474i	$0.710872\pi$
$332$	69.7068	3.82566
$333$	−1.14113	−0.0625335
$334$	20.9957	1.14883
$335$	0	0
$336$	150.306	8.19984
$337$	19.7453	1.07560	0.537798	−	0.843074i	$-0.319256\pi$
$337$	19.7453	1.07560	0.537798	+	0.843074i	$0.319256\pi$
$338$	5.49636	0.298962
$339$	5.52251	0.299942
$340$	0	0
$341$	1.66215	0.0900107
$342$	12.5723	0.679831
$343$	11.8902	0.642011
$344$	22.3190	1.20336
$345$	0	0
$346$	34.2605	1.84186
$347$	−21.6561	−1.16256	−0.581279	−	0.813704i	$-0.697448\pi$
$347$	−21.6561	−1.16256	−0.581279	+	0.813704i	$0.697448\pi$
$348$	−100.145	−5.36836
$349$	4.59915	0.246187	0.123093	−	0.992395i	$-0.460719\pi$
$349$	4.59915	0.246187	0.123093	+	0.992395i	$0.460719\pi$
$350$	0	0
$351$	16.2660	0.868216
$352$	26.8423	1.43070
$353$	−0.690423	−0.0367475	−0.0183737	−	0.999831i	$-0.505849\pi$
$353$	−0.690423	−0.0367475	−0.0183737	+	0.999831i	$0.505849\pi$
$354$	31.1760	1.65699
$355$	0	0
$356$	−10.4869	−0.555806
$357$	−51.6476	−2.73348
$358$	27.0538	1.42984
$359$	0.484355	0.0255633	0.0127816	−	0.999918i	$-0.495931\pi$
$359$	0.484355	0.0255633	0.0127816	+	0.999918i	$0.495931\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−8.51583	−0.447582
$363$	−2.74433	−0.144040
$364$	70.7589	3.70877
$365$	0	0
$366$	−34.4403	−1.80023
$367$	10.4333	0.544614	0.272307	−	0.962210i	$-0.412213\pi$
$367$	10.4333	0.544614	0.272307	+	0.962210i	$0.412213\pi$
$368$	104.453	5.44497
$369$	48.7414	2.53738
$370$	0	0
$371$	−28.5522	−1.48236
$372$	−25.9911	−1.34757
$373$	−12.4034	−0.642226	−0.321113	−	0.947041i	$-0.604057\pi$
$373$	−12.4034	−0.642226	−0.321113	+	0.947041i	$0.604057\pi$
$374$	−16.2745	−0.841533
$375$	0	0
$376$	−27.5134	−1.41889
$377$	−24.7884	−1.27667
$378$	37.4103	1.92418
$379$	−17.1068	−0.878719	−0.439359	−	0.898311i	$-0.644795\pi$
$379$	−17.1068	−0.878719	−0.439359	+	0.898311i	$0.644795\pi$
$380$	0	0
$381$	48.4991	2.48469
$382$	31.9809	1.63628
$383$	−27.5259	−1.40651	−0.703253	−	0.710939i	$-0.748270\pi$
$383$	−27.5259	−1.40651	−0.703253	+	0.710939i	$0.748270\pi$
$384$	−159.778	−8.15363
$385$	0	0
$386$	−17.3551	−0.883350
$387$	9.85730	0.501075
$388$	15.1363	0.768427
$389$	−29.4568	−1.49352	−0.746760	−	0.665094i	$-0.768391\pi$
$389$	−29.4568	−1.49352	−0.746760	+	0.665094i	$0.768391\pi$
$390$	0	0
$391$	−35.8917	−1.81512
$392$	33.7970	1.70701
$393$	13.3207	0.671939
$394$	18.1220	0.912973
$395$	0	0
$396$	25.8193	1.29747
$397$	−0.725541	−0.0364138	−0.0182069	−	0.999834i	$-0.505796\pi$
$397$	−0.725541	−0.0364138	−0.0182069	+	0.999834i	$0.505796\pi$
$398$	31.6953	1.58874
$399$	8.80502	0.440802
$400$	0	0
$401$	25.0438	1.25063	0.625314	−	0.780373i	$-0.284971\pi$
$401$	25.0438	1.25063	0.625314	+	0.780373i	$0.284971\pi$
$402$	19.8954	0.992294
$403$	−6.43342	−0.320471
$404$	19.1748	0.953981
$405$	0	0
$406$	−57.0111	−2.82941
$407$	−0.251830	−0.0124827
$408$	165.158	8.17656
$409$	−38.9407	−1.92550	−0.962748	−	0.270401i	$-0.912844\pi$
$409$	−38.9407	−1.92550	−0.962748	+	0.270401i	$0.912844\pi$
$410$	0	0
$411$	−1.29174	−0.0637168
$412$	108.957	5.36791
$413$	13.1368	0.646422
$414$	76.9287	3.78084
$415$	0	0
$416$	−103.894	−5.09383
$417$	−28.0625	−1.37422
$418$	2.77451	0.135706
$419$	35.0064	1.71018	0.855088	−	0.518482i	$-0.173503\pi$
$419$	35.0064	1.71018	0.855088	+	0.518482i	$0.173503\pi$
$420$	0	0
$421$	−6.05787	−0.295242	−0.147621	−	0.989044i	$-0.547162\pi$
$421$	−6.05787	−0.295242	−0.147621	+	0.989044i	$0.547162\pi$
$422$	−0.441487	−0.0214913
$423$	−12.1514	−0.590822
$424$	91.3040	4.43411
$425$	0	0
$426$	42.0091	2.03535
$427$	−14.5124	−0.702303
$428$	108.885	5.26315
$429$	10.6220	0.512836
$430$	0	0
$431$	−9.48475	−0.456864	−0.228432	−	0.973560i	$-0.573360\pi$
$431$	−9.48475	−0.456864	−0.228432	+	0.973560i	$0.573360\pi$
$432$	−71.7390	−3.45155
$433$	25.8710	1.24328	0.621641	−	0.783302i	$-0.286466\pi$
$433$	25.8710	1.24328	0.621641	+	0.783302i	$0.286466\pi$
$434$	−14.7963	−0.710243
$435$	0	0
$436$	−11.6361	−0.557270
$437$	6.11891	0.292707
$438$	−20.8233	−0.994974
$439$	−1.01956	−0.0486612	−0.0243306	−	0.999704i	$-0.507745\pi$
$439$	−1.01956	−0.0486612	−0.0243306	+	0.999704i	$0.507745\pi$
$440$	0	0
$441$	14.9266	0.710792
$442$	62.9908	2.99617
$443$	17.3007	0.821982	0.410991	−	0.911639i	$-0.365183\pi$
$443$	17.3007	0.821982	0.410991	+	0.911639i	$0.365183\pi$
$444$	3.93786	0.186883
$445$	0	0
$446$	1.49118	0.0706094
$447$	52.8962	2.50190
$448$	−129.408	−6.11393
$449$	−30.9651	−1.46133	−0.730667	−	0.682734i	$-0.760791\pi$
$449$	−30.9651	−1.46133	−0.730667	+	0.682734i	$0.760791\pi$
$450$	0	0
$451$	10.7565	0.506504
$452$	−11.4661	−0.539321
$453$	25.3731	1.19213
$454$	37.5202	1.76091
$455$	0	0
$456$	−28.1566	−1.31855
$457$	−38.9037	−1.81984	−0.909920	−	0.414784i	$-0.863857\pi$
$457$	−38.9037	−1.81984	−0.909920	+	0.414784i	$0.863857\pi$
$458$	−29.4618	−1.37666
$459$	24.6508	1.15060
$460$	0	0
$461$	−24.5284	−1.14240	−0.571200	−	0.820811i	$-0.693522\pi$
$461$	−24.5284	−1.14240	−0.571200	+	0.820811i	$0.693522\pi$
$462$	24.4296	1.13657
$463$	5.44017	0.252826	0.126413	−	0.991978i	$-0.459654\pi$
$463$	5.44017	0.252826	0.126413	+	0.991978i	$0.459654\pi$
$464$	109.326	5.07533
$465$	0	0
$466$	21.7358	1.00689
$467$	−14.4461	−0.668487	−0.334243	−	0.942487i	$-0.608481\pi$
$467$	−14.4461	−0.668487	−0.334243	+	0.942487i	$0.608481\pi$
$468$	−99.9343	−4.61946
$469$	8.38348	0.387113
$470$	0	0
$471$	24.9822	1.15112
$472$	−42.0089	−1.93362
$473$	2.17536	0.100023
$474$	−62.2157	−2.85766
$475$	0	0
$476$	107.233	4.91504
$477$	40.3249	1.84635
$478$	70.3066	3.21575
$479$	25.0982	1.14677	0.573383	−	0.819288i	$-0.305631\pi$
$479$	25.0982	1.14677	0.573383	+	0.819288i	$0.305631\pi$
$480$	0	0
$481$	0.974716	0.0444432
$482$	−83.3905	−3.79833
$483$	53.8771	2.45149
$484$	5.69792	0.258996
$485$	0	0
$486$	50.6721	2.29854
$487$	−27.8647	−1.26267	−0.631336	−	0.775510i	$-0.717493\pi$
$487$	−27.8647	−1.26267	−0.631336	+	0.775510i	$0.717493\pi$
$488$	46.4075	2.10077
$489$	36.1905	1.63659
$490$	0	0
$491$	−5.39887	−0.243648	−0.121824	−	0.992552i	$-0.538874\pi$
$491$	−5.39887	−0.243648	−0.121824	+	0.992552i	$0.538874\pi$
$492$	−168.199	−7.58300
$493$	−37.5663	−1.69190
$494$	−10.7388	−0.483163
$495$	0	0
$496$	28.3737	1.27402
$497$	17.7017	0.794028
$498$	−93.1498	−4.17414
$499$	−40.0954	−1.79492	−0.897458	−	0.441100i	$-0.854589\pi$
$499$	−40.0954	−1.79492	−0.897458	+	0.441100i	$0.854589\pi$
$500$	0	0
$501$	−20.7673	−0.927813
$502$	66.7069	2.97728
$503$	−19.3594	−0.863193	−0.431596	−	0.902067i	$-0.642050\pi$
$503$	−19.3594	−0.863193	−0.431596	+	0.902067i	$0.642050\pi$
$504$	−149.165	−6.64432
$505$	0	0
$506$	16.9770	0.754719
$507$	−5.43657	−0.241446
$508$	−100.696	−4.46768
$509$	−29.1426	−1.29172	−0.645862	−	0.763454i	$-0.723502\pi$
$509$	−29.1426	−1.29172	−0.645862	+	0.763454i	$0.723502\pi$
$510$	0	0
$511$	−8.77444	−0.388158
$512$	107.928	4.76977
$513$	−4.20253	−0.185546
$514$	−18.2626	−0.805528
$515$	0	0
$516$	−34.0160	−1.49747
$517$	−2.68163	−0.117938
$518$	2.24175	0.0984970
$519$	−33.8878	−1.48751
$520$	0	0
$521$	36.9334	1.61808	0.809042	−	0.587751i	$-0.199987\pi$
$521$	36.9334	1.61808	0.809042	+	0.587751i	$0.199987\pi$
$522$	80.5180	3.52418
$523$	−43.8433	−1.91713	−0.958567	−	0.284867i	$-0.908050\pi$
$523$	−43.8433	−1.91713	−0.958567	+	0.284867i	$0.908050\pi$
$524$	−27.6571	−1.20820
$525$	0	0
$526$	−44.5200	−1.94116
$527$	−9.74970	−0.424704
$528$	−46.8470	−2.03875
$529$	14.4410	0.627872
$530$	0	0
$531$	−18.5535	−0.805151
$532$	−18.2814	−0.792601
$533$	−41.6334	−1.80334
$534$	14.0138	0.606435
$535$	0	0
$536$	−26.8086	−1.15796
$537$	−26.7595	−1.15476
$538$	−47.3260	−2.04037
$539$	3.29408	0.141886
$540$	0	0
$541$	26.3462	1.13271	0.566355	−	0.824161i	$-0.308353\pi$
$541$	26.3462	1.13271	0.566355	+	0.824161i	$0.308353\pi$
$542$	41.1894	1.76923
$543$	8.42319	0.361474
$544$	−157.449	−6.75058
$545$	0	0
$546$	−94.5557	−4.04661
$547$	13.6728	0.584609	0.292304	−	0.956325i	$-0.405578\pi$
$547$	13.6728	0.584609	0.292304	+	0.956325i	$0.405578\pi$
$548$	2.68198	0.114568
$549$	20.4961	0.874754
$550$	0	0
$551$	6.40440	0.272837
$552$	−172.288	−7.33306
$553$	−26.2163	−1.11483
$554$	−23.8441	−1.01304
$555$	0	0
$556$	58.2647	2.47097
$557$	−13.0278	−0.552007	−0.276004	−	0.961157i	$-0.589010\pi$
$557$	−13.0278	−0.552007	−0.276004	+	0.961157i	$0.589010\pi$
$558$	20.8971	0.884644
$559$	−8.41978	−0.356119
$560$	0	0
$561$	16.0974	0.679634
$562$	−37.3641	−1.57611
$563$	−26.1689	−1.10289	−0.551445	−	0.834211i	$-0.685923\pi$
$563$	−26.1689	−1.10289	−0.551445	+	0.834211i	$0.685923\pi$
$564$	41.9326	1.76568
$565$	0	0
$566$	−39.6441	−1.66637
$567$	6.61235	0.277693
$568$	−56.6063	−2.37515
$569$	24.2618	1.01711	0.508554	−	0.861030i	$-0.330180\pi$
$569$	24.2618	1.01711	0.508554	+	0.861030i	$0.330180\pi$
$570$	0	0
$571$	6.25197	0.261637	0.130818	−	0.991406i	$-0.458240\pi$
$571$	6.25197	0.261637	0.130818	+	0.991406i	$0.458240\pi$
$572$	−22.0540	−0.922123
$573$	−31.6330	−1.32149
$574$	−95.7528	−3.99664
$575$	0	0
$576$	182.765	7.61521
$577$	5.41416	0.225395	0.112697	−	0.993629i	$-0.464051\pi$
$577$	5.41416	0.225395	0.112697	+	0.993629i	$0.464051\pi$
$578$	48.2944	2.00878
$579$	17.1663	0.713406
$580$	0	0
$581$	−39.2512	−1.62841
$582$	−20.2267	−0.838424
$583$	8.89909	0.368562
$584$	28.0588	1.16108
$585$	0	0
$586$	81.9746	3.38634
$587$	29.2069	1.20550	0.602749	−	0.797931i	$-0.294072\pi$
$587$	29.2069	1.20550	0.602749	+	0.797931i	$0.294072\pi$
$588$	−51.5095	−2.12422
$589$	1.66215	0.0684879
$590$	0	0
$591$	−17.9249	−0.737330
$592$	−4.29885	−0.176682
$593$	−13.3606	−0.548653	−0.274326	−	0.961637i	$-0.588455\pi$
$593$	−13.3606	−0.548653	−0.274326	+	0.961637i	$0.588455\pi$
$594$	−11.6600	−0.478414
$595$	0	0
$596$	−109.826	−4.49864
$597$	−31.3505	−1.28309
$598$	−65.7100	−2.68708
$599$	7.63307	0.311879	0.155939	−	0.987767i	$-0.450160\pi$
$599$	7.63307	0.311879	0.155939	+	0.987767i	$0.450160\pi$
$600$	0	0
$601$	−33.4520	−1.36454	−0.682269	−	0.731102i	$-0.739006\pi$
$601$	−33.4520	−1.36454	−0.682269	+	0.731102i	$0.739006\pi$
$602$	−19.3647	−0.789246
$603$	−11.8402	−0.482169
$604$	−52.6810	−2.14356
$605$	0	0
$606$	−25.6234	−1.04088
$607$	−1.87063	−0.0759265	−0.0379633	−	0.999279i	$-0.512087\pi$
$607$	−1.87063	−0.0759265	−0.0379633	+	0.999279i	$0.512087\pi$
$608$	26.8423	1.08860
$609$	56.3909	2.28507
$610$	0	0
$611$	10.3793	0.419903
$612$	−151.448	−6.12193
$613$	−30.9265	−1.24911	−0.624556	−	0.780980i	$-0.714720\pi$
$613$	−30.9265	−1.24911	−0.624556	+	0.780980i	$0.714720\pi$
$614$	59.9381	2.41890
$615$	0	0
$616$	−32.9183	−1.32632
$617$	1.25799	0.0506447	0.0253224	−	0.999679i	$-0.491939\pi$
$617$	1.25799	0.0506447	0.0253224	+	0.999679i	$0.491939\pi$
$618$	−145.600	−5.85688
$619$	15.5642	0.625578	0.312789	−	0.949823i	$-0.398737\pi$
$619$	15.5642	0.625578	0.312789	+	0.949823i	$0.398737\pi$
$620$	0	0
$621$	−25.7149	−1.03190
$622$	59.3871	2.38121
$623$	5.90508	0.236582
$624$	181.323	7.25872
$625$	0	0
$626$	63.0736	2.52093
$627$	−2.74433	−0.109598
$628$	−51.8693	−2.06981
$629$	1.47716	0.0588982
$630$	0	0
$631$	−33.5188	−1.33436	−0.667182	−	0.744895i	$-0.732500\pi$
$631$	−33.5188	−1.33436	−0.667182	+	0.744895i	$0.732500\pi$
$632$	83.8342	3.33475
$633$	0.436685	0.0173567
$634$	3.59060	0.142601
$635$	0	0
$636$	−139.155	−5.51785
$637$	−12.7499	−0.505168
$638$	17.7691	0.703485
$639$	−25.0005	−0.989003
$640$	0	0
$641$	1.89938	0.0750211	0.0375106	−	0.999296i	$-0.488057\pi$
$641$	1.89938	0.0750211	0.0375106	+	0.999296i	$0.488057\pi$
$642$	−145.504	−5.74258
$643$	−19.4308	−0.766277	−0.383138	−	0.923691i	$-0.625157\pi$
$643$	−19.4308	−0.766277	−0.383138	+	0.923691i	$0.625157\pi$
$644$	−111.862	−4.40800
$645$	0	0
$646$	−16.2745	−0.640310
$647$	−7.80761	−0.306949	−0.153474	−	0.988153i	$-0.549046\pi$
$647$	−7.80761	−0.306949	−0.153474	+	0.988153i	$0.549046\pi$
$648$	−21.1449	−0.830652
$649$	−4.09447	−0.160722
$650$	0	0
$651$	14.6353	0.573602
$652$	−75.1406	−2.94273
$653$	−21.0194	−0.822553	−0.411277	−	0.911511i	$-0.634917\pi$
$653$	−21.0194	−0.822553	−0.411277	+	0.911511i	$0.634917\pi$
$654$	15.5495	0.608032
$655$	0	0
$656$	183.618	7.16909
$657$	12.3923	0.483471
$658$	23.8715	0.930608
$659$	32.1192	1.25119	0.625594	−	0.780149i	$-0.284857\pi$
$659$	32.1192	1.25119	0.625594	+	0.780149i	$0.284857\pi$
$660$	0	0
$661$	−26.8022	−1.04248	−0.521242	−	0.853409i	$-0.674531\pi$
$661$	−26.8022	−1.04248	−0.521242	+	0.853409i	$0.674531\pi$
$662$	−62.0946	−2.41338
$663$	−62.3056	−2.41975
$664$	125.517	4.87101
$665$	0	0
$666$	−3.16608	−0.122683
$667$	39.1880	1.51736
$668$	43.1181	1.66829
$669$	−1.47496	−0.0570252
$670$	0	0
$671$	4.52319	0.174616
$672$	236.347	9.11729
$673$	−13.3416	−0.514281	−0.257140	−	0.966374i	$-0.582780\pi$
$673$	−13.3416	−0.514281	−0.257140	+	0.966374i	$0.582780\pi$
$674$	54.7837	2.11019
$675$	0	0
$676$	11.2877	0.434142
$677$	35.2641	1.35531	0.677654	−	0.735381i	$-0.262997\pi$
$677$	35.2641	1.35531	0.677654	+	0.735381i	$0.262997\pi$
$678$	15.3223	0.588449
$679$	−8.52307	−0.327085
$680$	0	0
$681$	−37.1121	−1.42214
$682$	4.61167	0.176590
$683$	22.6519	0.866753	0.433376	−	0.901213i	$-0.357322\pi$
$683$	22.6519	0.866753	0.433376	+	0.901213i	$0.357322\pi$
$684$	25.8193	0.987224
$685$	0	0
$686$	32.9895	1.25955
$687$	29.1413	1.11181
$688$	37.1343	1.41573
$689$	−34.4442	−1.31222
$690$	0	0
$691$	13.6411	0.518930	0.259465	−	0.965752i	$-0.416454\pi$
$691$	13.6411	0.518930	0.259465	+	0.965752i	$0.416454\pi$
$692$	70.3597	2.67467
$693$	−14.5386	−0.552274
$694$	−60.0850	−2.28080
$695$	0	0
$696$	−180.326	−6.83525
$697$	−63.0944	−2.38987
$698$	12.7604	0.482988
$699$	−21.4993	−0.813179
$700$	0	0
$701$	5.88833	0.222399	0.111200	−	0.993798i	$-0.464531\pi$
$701$	5.88833	0.222399	0.111200	+	0.993798i	$0.464531\pi$
$702$	45.1303	1.70333
$703$	−0.251830	−0.00949795
$704$	40.3335	1.52013
$705$	0	0
$706$	−1.91559	−0.0720940
$707$	−10.7971	−0.406067
$708$	64.0251	2.40621
$709$	−20.0746	−0.753919	−0.376960	−	0.926230i	$-0.623030\pi$
$709$	−20.0746	−0.753919	−0.376960	+	0.926230i	$0.623030\pi$
$710$	0	0
$711$	37.0258	1.38858
$712$	−18.8832	−0.707678
$713$	10.1706	0.380891
$714$	−143.297	−5.36275
$715$	0	0
$716$	55.5594	2.07635
$717$	−69.5418	−2.59709
$718$	1.34385	0.0501520
$719$	−19.9007	−0.742171	−0.371086	−	0.928599i	$-0.621014\pi$
$719$	−19.9007	−0.742171	−0.371086	+	0.928599i	$0.621014\pi$
$720$	0	0
$721$	−61.3524	−2.28488
$722$	2.77451	0.103257
$723$	82.4833	3.06759
$724$	−17.4887	−0.649961
$725$	0	0
$726$	−7.61418	−0.282589
$727$	−2.48983	−0.0923426	−0.0461713	−	0.998934i	$-0.514702\pi$
$727$	−2.48983	−0.0923426	−0.0461713	+	0.998934i	$0.514702\pi$
$728$	127.412	4.72219
$729$	−43.9381	−1.62734
$730$	0	0
$731$	−12.7600	−0.471945
$732$	−70.7289	−2.61422
$733$	1.67302	0.0617945	0.0308973	−	0.999523i	$-0.490164\pi$
$733$	1.67302	0.0617945	0.0308973	+	0.999523i	$0.490164\pi$
$734$	28.9473	1.06847
$735$	0	0
$736$	164.246	6.05418
$737$	−2.61295	−0.0962491
$738$	135.234	4.97802
$739$	12.3807	0.455432	0.227716	−	0.973728i	$-0.426874\pi$
$739$	12.3807	0.455432	0.227716	+	0.973728i	$0.426874\pi$
$740$	0	0
$741$	10.6220	0.390210
$742$	−79.2184	−2.90820
$743$	−2.41474	−0.0885881	−0.0442940	−	0.999019i	$-0.514104\pi$
$743$	−2.41474	−0.0885881	−0.0442940	+	0.999019i	$0.514104\pi$
$744$	−46.8006	−1.71579
$745$	0	0
$746$	−34.4135	−1.25997
$747$	55.4353	2.02827
$748$	−33.4223	−1.22204
$749$	−61.3119	−2.24029
$750$	0	0
$751$	7.67280	0.279984	0.139992	−	0.990153i	$-0.455292\pi$
$751$	7.67280	0.279984	0.139992	+	0.990153i	$0.455292\pi$
$752$	−45.7767	−1.66930
$753$	−65.9813	−2.40449
$754$	−68.7759	−2.50467
$755$	0	0
$756$	76.8282	2.79421
$757$	7.03423	0.255663	0.127832	−	0.991796i	$-0.459198\pi$
$757$	7.03423	0.255663	0.127832	+	0.991796i	$0.459198\pi$
$758$	−47.4631	−1.72394
$759$	−16.7923	−0.609522
$760$	0	0
$761$	8.98118	0.325567	0.162784	−	0.986662i	$-0.447953\pi$
$761$	8.98118	0.325567	0.162784	+	0.986662i	$0.447953\pi$
$762$	134.561	4.87465
$763$	6.55219	0.237205
$764$	65.6780	2.37615
$765$	0	0
$766$	−76.3709	−2.75939
$767$	15.8478	0.572230
$768$	−221.929	−8.00817
$769$	47.2240	1.70294	0.851470	−	0.524404i	$-0.175712\pi$
$769$	47.2240	1.70294	0.851470	+	0.524404i	$0.175712\pi$
$770$	0	0
$771$	18.0639	0.650556
$772$	−35.6415	−1.28277
$773$	−30.3348	−1.09107	−0.545533	−	0.838089i	$-0.683673\pi$
$773$	−30.3348	−1.09107	−0.545533	+	0.838089i	$0.683673\pi$
$774$	27.3492	0.983046
$775$	0	0
$776$	27.2550	0.978398
$777$	−2.21737	−0.0795476
$778$	−81.7283	−2.93010
$779$	10.7565	0.385392
$780$	0	0
$781$	−5.51722	−0.197422
$782$	−99.5820	−3.56104
$783$	−26.9147	−0.961852
$784$	56.2315	2.00827
$785$	0	0
$786$	36.9584	1.31826
$787$	−3.82340	−0.136289	−0.0681447	−	0.997675i	$-0.521708\pi$
$787$	−3.82340	−0.136289	−0.0681447	+	0.997675i	$0.521708\pi$
$788$	37.2165	1.32578
$789$	44.0356	1.56771
$790$	0	0
$791$	6.45645	0.229565
$792$	46.4913	1.65200
$793$	−17.5071	−0.621697
$794$	−2.01302	−0.0714395
$795$	0	0
$796$	65.0916	2.30711
$797$	−32.7856	−1.16133	−0.580663	−	0.814144i	$-0.697207\pi$
$797$	−32.7856	−1.16133	−0.580663	+	0.814144i	$0.697207\pi$
$798$	24.4296	0.864800
$799$	15.7297	0.556475
$800$	0	0
$801$	−8.33987	−0.294675
$802$	69.4844	2.45358
$803$	2.73480	0.0965090
$804$	40.8586	1.44097
$805$	0	0
$806$	−17.8496	−0.628725
$807$	46.8112	1.64783
$808$	34.5270	1.21465
$809$	24.8694	0.874361	0.437181	−	0.899374i	$-0.355977\pi$
$809$	24.8694	0.874361	0.437181	+	0.899374i	$0.355977\pi$
$810$	0	0
$811$	−20.4421	−0.717821	−0.358910	−	0.933372i	$-0.616852\pi$
$811$	−20.4421	−0.717821	−0.358910	+	0.933372i	$0.616852\pi$
$812$	−117.082	−4.10876
$813$	−40.7413	−1.42886
$814$	−0.698705	−0.0244896
$815$	0	0
$816$	274.790	9.61958
$817$	2.17536	0.0761061
$818$	−108.042	−3.77758
$819$	56.2719	1.96630
$820$	0	0
$821$	53.8539	1.87951	0.939757	−	0.341843i	$-0.111051\pi$
$821$	53.8539	1.87951	0.939757	+	0.341843i	$0.111051\pi$
$822$	−3.58395	−0.125005
$823$	40.2952	1.40460	0.702301	−	0.711880i	$-0.252156\pi$
$823$	40.2952	1.40460	0.702301	+	0.711880i	$0.252156\pi$
$824$	196.192	6.83468
$825$	0	0
$826$	36.4484	1.26820
$827$	−33.1923	−1.15421	−0.577105	−	0.816670i	$-0.695818\pi$
$827$	−33.1923	−1.15421	−0.577105	+	0.816670i	$0.695818\pi$
$828$	157.986	5.49038
$829$	28.6178	0.993937	0.496969	−	0.867769i	$-0.334446\pi$
$829$	28.6178	0.993937	0.496969	+	0.867769i	$0.334446\pi$
$830$	0	0
$831$	23.5847	0.818145
$832$	−156.112	−5.41221
$833$	−19.3221	−0.669471
$834$	−77.8596	−2.69606
$835$	0	0
$836$	5.69792	0.197067
$837$	−6.98525	−0.241445
$838$	97.1258	3.35516
$839$	−5.00397	−0.172756	−0.0863781	−	0.996262i	$-0.527529\pi$
$839$	−5.00397	−0.172756	−0.0863781	+	0.996262i	$0.527529\pi$
$840$	0	0
$841$	12.0164	0.414358
$842$	−16.8076	−0.579229
$843$	36.9576	1.27289
$844$	−0.906667	−0.0312088
$845$	0	0
$846$	−33.7142	−1.15912
$847$	−3.20844	−0.110243
$848$	151.911	5.21666
$849$	39.2129	1.34578
$850$	0	0
$851$	−1.54092	−0.0528222
$852$	86.2726	2.95565
$853$	14.8401	0.508117	0.254058	−	0.967189i	$-0.418234\pi$
$853$	14.8401	0.508117	0.254058	+	0.967189i	$0.418234\pi$
$854$	−40.2647	−1.37783
$855$	0	0
$856$	196.063	6.70129
$857$	−9.89677	−0.338067	−0.169034	−	0.985610i	$-0.554065\pi$
$857$	−9.89677	−0.338067	−0.169034	+	0.985610i	$0.554065\pi$
$858$	29.4709	1.00612
$859$	−21.7867	−0.743352	−0.371676	−	0.928363i	$-0.621217\pi$
$859$	−21.7867	−0.743352	−0.371676	+	0.928363i	$0.621217\pi$
$860$	0	0
$861$	94.7112	3.22775
$862$	−26.3155	−0.896311
$863$	−24.4284	−0.831553	−0.415776	−	0.909467i	$-0.636490\pi$
$863$	−24.4284	−0.831553	−0.415776	+	0.909467i	$0.636490\pi$
$864$	−112.806	−3.83772
$865$	0	0
$866$	71.7795	2.43917
$867$	−47.7691	−1.62232
$868$	−30.3865	−1.03139
$869$	8.17104	0.277183
$870$	0	0
$871$	10.1135	0.342682
$872$	−20.9525	−0.709542
$873$	12.0373	0.407401
$874$	16.9770	0.574255
$875$	0	0
$876$	−42.7640	−1.44486
$877$	24.9962	0.844062	0.422031	−	0.906581i	$-0.361317\pi$
$877$	24.9962	0.844062	0.422031	+	0.906581i	$0.361317\pi$
$878$	−2.82879	−0.0954672
$879$	−81.0828	−2.73486
$880$	0	0
$881$	−25.3991	−0.855719	−0.427859	−	0.903845i	$-0.640732\pi$
$881$	−25.3991	−0.855719	−0.427859	+	0.903845i	$0.640732\pi$
$882$	41.4141	1.39449
$883$	48.3559	1.62731	0.813653	−	0.581351i	$-0.197476\pi$
$883$	48.3559	1.62731	0.813653	+	0.581351i	$0.197476\pi$
$884$	129.362	4.35092
$885$	0	0
$886$	48.0010	1.61263
$887$	−22.8073	−0.765793	−0.382897	−	0.923791i	$-0.625073\pi$
$887$	−22.8073	−0.765793	−0.382897	+	0.923791i	$0.625073\pi$
$888$	7.09068	0.237948
$889$	56.7011	1.90169
$890$	0	0
$891$	−2.06093	−0.0690436
$892$	3.06238	0.102536
$893$	−2.68163	−0.0897374
$894$	146.761	4.90843
$895$	0	0
$896$	−186.799	−6.24051
$897$	64.9952	2.17013
$898$	−85.9131	−2.86696
$899$	10.6451	0.355034
$900$	0	0
$901$	−52.1994	−1.73901
$902$	29.8440	0.993698
$903$	19.1540	0.637407
$904$	−20.6464	−0.686689
$905$	0	0
$906$	70.3981	2.33882
$907$	−4.89927	−0.162678	−0.0813388	−	0.996687i	$-0.525920\pi$
$907$	−4.89927	−0.162678	−0.0813388	+	0.996687i	$0.525920\pi$
$908$	77.0540	2.55713
$909$	15.2490	0.505778
$910$	0	0
$911$	14.2078	0.470727	0.235364	−	0.971907i	$-0.424372\pi$
$911$	14.2078	0.470727	0.235364	+	0.971907i	$0.424372\pi$
$912$	−46.8470	−1.55126
$913$	12.2337	0.404877
$914$	−107.939	−3.57030
$915$	0	0
$916$	−60.5046	−1.99913
$917$	15.5734	0.514279
$918$	68.3939	2.25733
$919$	37.3984	1.23366	0.616830	−	0.787096i	$-0.288417\pi$
$919$	37.3984	1.23366	0.616830	+	0.787096i	$0.288417\pi$
$920$	0	0
$921$	−59.2861	−1.95354
$922$	−68.0542	−2.24125
$923$	21.3546	0.702895
$924$	50.1703	1.65048
$925$	0	0
$926$	15.0938	0.496013
$927$	86.6493	2.84594
$928$	171.909	5.64319
$929$	18.4186	0.604294	0.302147	−	0.953261i	$-0.402297\pi$
$929$	18.4186	0.604294	0.302147	+	0.953261i	$0.402297\pi$
$930$	0	0
$931$	3.29408	0.107959
$932$	44.6380	1.46217
$933$	−58.7411	−1.92310
$934$	−40.0810	−1.31149
$935$	0	0
$936$	−179.946	−5.88172
$937$	59.8435	1.95500	0.977501	−	0.210930i	$-0.0676493\pi$
$937$	59.8435	1.95500	0.977501	+	0.210930i	$0.0676493\pi$
$938$	23.2601	0.759468
$939$	−62.3874	−2.03594
$940$	0	0
$941$	14.5570	0.474545	0.237272	−	0.971443i	$-0.423747\pi$
$941$	14.5570	0.474545	0.237272	+	0.971443i	$0.423747\pi$
$942$	69.3134	2.25835
$943$	65.8180	2.14333
$944$	−69.8944	−2.27487
$945$	0	0
$946$	6.03555	0.196233
$947$	15.6623	0.508955	0.254478	−	0.967079i	$-0.418097\pi$
$947$	15.6623	0.508955	0.254478	+	0.967079i	$0.418097\pi$
$948$	−127.770	−4.14979
$949$	−10.5851	−0.343608
$950$	0	0
$951$	−3.55154	−0.115166
$952$	193.089	6.25806
$953$	−27.6656	−0.896176	−0.448088	−	0.893990i	$-0.647895\pi$
$953$	−27.6656	−0.896176	−0.448088	+	0.893990i	$0.647895\pi$
$954$	111.882	3.62231
$955$	0	0
$956$	144.386	4.66979
$957$	−17.5758	−0.568145
$958$	69.6352	2.24981
$959$	−1.51019	−0.0487667
$960$	0	0
$961$	−28.2372	−0.910879
$962$	2.70436	0.0871921
$963$	86.5922	2.79039
$964$	−171.256	−5.51579
$965$	0	0
$966$	149.483	4.80953
$967$	−2.39400	−0.0769857	−0.0384929	−	0.999259i	$-0.512256\pi$
$967$	−2.39400	−0.0769857	−0.0384929	+	0.999259i	$0.512256\pi$
$968$	10.2599	0.329766
$969$	16.0974	0.517124
$970$	0	0
$971$	9.52917	0.305806	0.152903	−	0.988241i	$-0.451138\pi$
$971$	9.52917	0.305806	0.152903	+	0.988241i	$0.451138\pi$
$972$	104.064	3.33784
$973$	−32.8083	−1.05178
$974$	−77.3111	−2.47721
$975$	0	0
$976$	77.2128	2.47152
$977$	17.6663	0.565194	0.282597	−	0.959239i	$-0.408804\pi$
$977$	17.6663	0.565194	0.282597	+	0.959239i	$0.408804\pi$
$978$	100.411	3.21079
$979$	−1.84048	−0.0588221
$980$	0	0
$981$	−9.25380	−0.295451
$982$	−14.9792	−0.478007
$983$	3.24817	0.103601	0.0518003	−	0.998657i	$-0.483504\pi$
$983$	3.24817	0.103601	0.0518003	+	0.998657i	$0.483504\pi$
$984$	−302.867	−9.65504
$985$	0	0
$986$	−104.228	−3.31930
$987$	−23.6118	−0.751573
$988$	−22.0540	−0.701631
$989$	13.3108	0.423259
$990$	0	0
$991$	−15.8274	−0.502774	−0.251387	−	0.967887i	$-0.580887\pi$
$991$	−15.8274	−0.502774	−0.251387	+	0.967887i	$0.580887\pi$
$992$	44.6161	1.41656
$993$	61.4191	1.94908
$994$	49.1135	1.55779
$995$	0	0
$996$	−191.298	−6.06152
$997$	−47.5564	−1.50613	−0.753063	−	0.657948i	$-0.771425\pi$
$997$	−47.5564	−1.50613	−0.753063	+	0.657948i	$0.771425\pi$
$998$	−111.245	−3.52140
$999$	1.05832	0.0334838

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.bc.1.29		30
5.2	odd	4		1045.2.b.e.419.29	yes	30
5.3	odd	4		1045.2.b.e.419.2	✓	30
5.4	even	2	inner	5225.2.a.bc.1.2		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.b.e.419.2	✓	30	5.3	odd	4
1045.2.b.e.419.29	yes	30	5.2	odd	4
5225.2.a.bc.1.2		30	5.4	even	2	inner
5225.2.a.bc.1.29		30	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.77451 q^{2} -2.74433 q^{3} +5.69792 q^{4} -7.61418 q^{6} -3.20844 q^{7} +10.2599 q^{8} +4.53135 q^{9} +1.00000 q^{11} -15.6370 q^{12} -3.87053 q^{13} -8.90186 q^{14} +17.0705 q^{16} -5.86570 q^{17} +12.5723 q^{18} +1.00000 q^{19} +8.80502 q^{21} +2.77451 q^{22} +6.11891 q^{23} -28.1566 q^{24} -10.7388 q^{26} -4.20253 q^{27} -18.2814 q^{28} +6.40440 q^{29} +1.66215 q^{31} +26.8423 q^{32} -2.74433 q^{33} -16.2745 q^{34} +25.8193 q^{36} -0.251830 q^{37} +2.77451 q^{38} +10.6220 q^{39} +10.7565 q^{41} +24.4296 q^{42} +2.17536 q^{43} +5.69792 q^{44} +16.9770 q^{46} -2.68163 q^{47} -46.8470 q^{48} +3.29408 q^{49} +16.0974 q^{51} -22.0540 q^{52} +8.89909 q^{53} -11.6600 q^{54} -32.9183 q^{56} -2.74433 q^{57} +17.7691 q^{58} -4.09447 q^{59} +4.52319 q^{61} +4.61167 q^{62} -14.5386 q^{63} +40.3335 q^{64} -7.61418 q^{66} -2.61295 q^{67} -33.4223 q^{68} -16.7923 q^{69} -5.51722 q^{71} +46.4913 q^{72} +2.73480 q^{73} -0.698705 q^{74} +5.69792 q^{76} -3.20844 q^{77} +29.4709 q^{78} +8.17104 q^{79} -2.06093 q^{81} +29.8440 q^{82} +12.2337 q^{83} +50.1703 q^{84} +6.03555 q^{86} -17.5758 q^{87} +10.2599 q^{88} -1.84048 q^{89} +12.4184 q^{91} +34.8651 q^{92} -4.56150 q^{93} -7.44022 q^{94} -73.6642 q^{96} +2.65645 q^{97} +9.13947 q^{98} +4.53135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9} + 30 q^{11} - 4 q^{14} + 66 q^{16} + 30 q^{19} + 14 q^{21} + 22 q^{24} + 30 q^{29} + 26 q^{31} + 12 q^{34} + 78 q^{36} + 64 q^{39} + 22 q^{41} + 42 q^{44} + 28 q^{46}+ \cdots + 40 q^{99}+O(q^{100}) \) 30 * q + 42 * q^4 + 12 * q^6 + 40 * q^9 + 30 * q^11 - 4 * q^14 + 66 * q^16 + 30 * q^19 + 14 * q^21 + 22 * q^24 + 30 * q^29 + 26 * q^31 + 12 * q^34 + 78 * q^36 + 64 * q^39 + 22 * q^41 + 42 * q^44 + 28 * q^46 + 60 * q^49 + 64 * q^51 + 62 * q^54 - 32 * q^56 - 14 * q^59 + 78 * q^61 + 90 * q^64 + 12 * q^66 - 28 * q^69 + 20 * q^71 + 42 * q^74 + 42 * q^76 + 102 * q^79 + 42 * q^81 + 98 * q^84 - 52 * q^86 - 8 * q^89 + 56 * q^91 + 40 * q^94 - 74 * q^96 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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