LMFDB - Embedded newform 5225.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.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:	$41.7218350561$
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			1.2
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)

$f(q)$	$=$	$q-2.77451 q^{2} +2.74433 q^{3} +5.69792 q^{4} -7.61418 q^{6} +3.20844 q^{7} -10.2599 q^{8} +4.53135 q^{9} +O(q^{10})$	\(q-2.77451 q^{2} +2.74433 q^{3} +5.69792 q^{4} -7.61418 q^{6} +3.20844 q^{7} -10.2599 q^{8} +4.53135 q^{9} +1.00000 q^{11} +15.6370 q^{12} +3.87053 q^{13} -8.90186 q^{14} +17.0705 q^{16} +5.86570 q^{17} -12.5723 q^{18} +1.00000 q^{19} +8.80502 q^{21} -2.77451 q^{22} -6.11891 q^{23} -28.1566 q^{24} -10.7388 q^{26} +4.20253 q^{27} +18.2814 q^{28} +6.40440 q^{29} +1.66215 q^{31} -26.8423 q^{32} +2.74433 q^{33} -16.2745 q^{34} +25.8193 q^{36} +0.251830 q^{37} -2.77451 q^{38} +10.6220 q^{39} +10.7565 q^{41} -24.4296 q^{42} -2.17536 q^{43} +5.69792 q^{44} +16.9770 q^{46} +2.68163 q^{47} +46.8470 q^{48} +3.29408 q^{49} +16.0974 q^{51} +22.0540 q^{52} -8.89909 q^{53} -11.6600 q^{54} -32.9183 q^{56} +2.74433 q^{57} -17.7691 q^{58} -4.09447 q^{59} +4.52319 q^{61} -4.61167 q^{62} +14.5386 q^{63} +40.3335 q^{64} -7.61418 q^{66} +2.61295 q^{67} +33.4223 q^{68} -16.7923 q^{69} -5.51722 q^{71} -46.4913 q^{72} -2.73480 q^{73} -0.698705 q^{74} +5.69792 q^{76} +3.20844 q^{77} -29.4709 q^{78} +8.17104 q^{79} -2.06093 q^{81} -29.8440 q^{82} -12.2337 q^{83} +50.1703 q^{84} +6.03555 q^{86} +17.5758 q^{87} -10.2599 q^{88} -1.84048 q^{89} +12.4184 q^{91} -34.8651 q^{92} +4.56150 q^{93} -7.44022 q^{94} -73.6642 q^{96} -2.65645 q^{97} -9.13947 q^{98} +4.53135 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9}+O(q^{10}) $ 30 * q + 42 * q^4 + 12 * q^6 + 40 * q^9	$ 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}+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

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 $

$2$	−2.77451	−1.96188	−0.980938	−	0.194319i	$-0.937750\pi$
$2$	−2.77451	−1.96188	−0.980938	+	0.194319i	$0.937750\pi$
$3$	2.74433	1.58444	0.792220	−	0.610236i	$-0.208925\pi$
$3$	2.74433	1.58444	0.792220	+	0.610236i	$0.208925\pi$
$4$	5.69792	2.84896
$5$	0	0
$5$	0	0
$6$	−7.61418	−3.10848
$7$	3.20844	1.21268	0.606338	−	0.795207i	$-0.292638\pi$
$7$	3.20844	1.21268	0.606338	+	0.795207i	$0.292638\pi$
$8$	−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	$-0.319653\pi$
$13$	3.87053	1.07349	0.536746	+	0.843744i	$0.319653\pi$
$14$	−8.90186	−2.37912
$15$	0	0
$16$	17.0705	4.26761
$17$	5.86570	1.42264	0.711321	−	0.702867i	$-0.248097\pi$
$17$	5.86570	1.42264	0.711321	+	0.702867i	$0.248097\pi$
$18$	−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	$-0.720213\pi$
$23$	−6.11891	−1.27588	−0.637940	+	0.770086i	$0.720213\pi$
$24$	−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	$-0.493410\pi$
$37$	0.251830	0.0414006	0.0207003	+	0.999786i	$0.493410\pi$
$38$	−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	$-0.553043\pi$
$43$	−2.17536	−0.331739	−0.165869	+	0.986148i	$0.553043\pi$
$44$	5.69792	0.858994
$45$	0	0
$46$	16.9770	2.50312
$47$	2.68163	0.391156	0.195578	−	0.980688i	$-0.437342\pi$
$47$	2.68163	0.391156	0.195578	+	0.980688i	$0.437342\pi$
$48$	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	$-0.709310\pi$
$53$	−8.89909	−1.22238	−0.611192	+	0.791483i	$0.709310\pi$
$54$	−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	$-0.448976\pi$
$67$	2.61295	0.319222	0.159611	+	0.987180i	$0.448976\pi$
$68$	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	$-0.551163\pi$
$73$	−2.73480	−0.320084	−0.160042	+	0.987110i	$0.551163\pi$
$74$	−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	$-0.734312\pi$
$83$	−12.2337	−1.34283	−0.671413	+	0.741083i	$0.734312\pi$
$84$	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	$-0.543059\pi$
$97$	−2.65645	−0.269722	−0.134861	+	0.990865i	$0.543059\pi$
$98$	−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	$-0.891135\pi$
$103$	−19.1222	−1.88416	−0.942082	+	0.335381i	$0.891135\pi$
$104$	−39.7114	−3.89402
$105$	0	0
$106$	24.6906	2.39817
$107$	−19.1096	−1.84739	−0.923697	−	0.383125i	$-0.874848\pi$
$107$	−19.1096	−1.84739	−0.923697	+	0.383125i	$0.874848\pi$
$108$	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	$-0.469826\pi$
$113$	2.01233	0.189305	0.0946523	+	0.995510i	$0.469826\pi$
$114$	−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	$-0.213130\pi$
$127$	17.6725	1.56818	0.784090	+	0.620648i	$0.213130\pi$
$128$	−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	$-0.506401\pi$
$137$	−0.470694	−0.0402141	−0.0201071	+	0.999798i	$0.506401\pi$
$138$	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	$-0.381665\pi$
$157$	9.10320	0.726514	0.363257	+	0.931689i	$0.381665\pi$
$158$	−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	$-0.327250\pi$
$163$	13.1874	1.03292	0.516458	+	0.856313i	$0.327250\pi$
$164$	61.2897	4.78592
$165$	0	0
$166$	33.9426	2.63446
$167$	−7.56734	−0.585578	−0.292789	−	0.956177i	$-0.594583\pi$
$167$	−7.56734	−0.585578	−0.292789	+	0.956177i	$0.594583\pi$
$168$	−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	$-0.655534\pi$
$173$	−12.3483	−0.938825	−0.469412	+	0.882979i	$0.655534\pi$
$174$	−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	$-0.427720\pi$
$193$	6.25518	0.450258	0.225129	+	0.974329i	$0.427720\pi$
$194$	7.37036	0.529161
$195$	0	0
$196$	18.7694	1.34067
$197$	−6.53160	−0.465357	−0.232678	−	0.972554i	$-0.574749\pi$
$197$	−6.53160	−0.465357	−0.232678	+	0.972554i	$0.574749\pi$
$198$	−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	$-0.505728\pi$
$223$	−0.537456	−0.0359907	−0.0179954	+	0.999838i	$0.505728\pi$
$224$	−86.1220	−5.75427
$225$	0	0
$226$	−5.58325	−0.371392
$227$	−13.5232	−0.897565	−0.448783	−	0.893641i	$-0.648142\pi$
$227$	−13.5232	−0.897565	−0.448783	+	0.893641i	$0.648142\pi$
$228$	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	$-0.582607\pi$
$233$	−7.83409	−0.513228	−0.256614	+	0.966514i	$0.582607\pi$
$234$	−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	$-0.434185\pi$
$257$	6.58227	0.410591	0.205295	+	0.978700i	$0.434185\pi$
$258$	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	$-0.335270\pi$
$263$	16.0460	0.989442	0.494721	+	0.869052i	$0.335270\pi$
$264$	−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	$-0.416877\pi$
$277$	8.59398	0.516362	0.258181	+	0.966097i	$0.416877\pi$
$278$	−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	$-0.360384\pi$
$283$	14.2887	0.849374	0.424687	+	0.905340i	$0.360384\pi$
$284$	−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	$-0.831440\pi$
$293$	−29.5456	−1.72607	−0.863036	+	0.505143i	$0.831440\pi$
$294$	−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	$-0.711441\pi$
$307$	−21.6031	−1.23295	−0.616477	+	0.787373i	$0.711441\pi$
$308$	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	$-0.722094\pi$
$313$	−22.7332	−1.28496	−0.642478	+	0.766304i	$0.722094\pi$
$314$	−25.2569	−1.42533
$315$	0	0
$316$	46.5579	2.61909
$317$	−1.29414	−0.0726859	−0.0363430	−	0.999339i	$-0.511571\pi$
$317$	−1.29414	−0.0726859	−0.0363430	+	0.999339i	$0.511571\pi$
$318$	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	$-0.680744\pi$
$337$	−19.7453	−1.07560	−0.537798	+	0.843074i	$0.680744\pi$
$338$	−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	$-0.302552\pi$
$347$	21.6561	1.16256	0.581279	+	0.813704i	$0.302552\pi$
$348$	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	$-0.494151\pi$
$353$	0.690423	0.0367475	0.0183737	+	0.999831i	$0.494151\pi$
$354$	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	$-0.587787\pi$
$367$	−10.4333	−0.544614	−0.272307	+	0.962210i	$0.587787\pi$
$368$	−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	$-0.395943\pi$
$373$	12.4034	0.642226	0.321113	+	0.947041i	$0.395943\pi$
$374$	−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	$-0.251730\pi$
$383$	27.5259	1.40651	0.703253	+	0.710939i	$0.251730\pi$
$384$	−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	$-0.494204\pi$
$397$	0.725541	0.0364138	0.0182069	+	0.999834i	$0.494204\pi$
$398$	−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	$-0.713534\pi$
$433$	−25.8710	−1.24328	−0.621641	+	0.783302i	$0.713534\pi$
$434$	−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	$-0.634817\pi$
$443$	−17.3007	−0.821982	−0.410991	+	0.911639i	$0.634817\pi$
$444$	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	$-0.136143\pi$
$457$	38.9037	1.81984	0.909920	+	0.414784i	$0.136143\pi$
$458$	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	$-0.540346\pi$
$463$	−5.44017	−0.252826	−0.126413	+	0.991978i	$0.540346\pi$
$464$	109.326	5.07533
$465$	0	0
$466$	21.7358	1.00689
$467$	14.4461	0.668487	0.334243	−	0.942487i	$-0.391519\pi$
$467$	14.4461	0.668487	0.334243	+	0.942487i	$0.391519\pi$
$468$	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	$-0.282507\pi$
$487$	27.8647	1.26267	0.631336	+	0.775510i	$0.282507\pi$
$488$	−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	$-0.357950\pi$
$503$	19.3594	0.863193	0.431596	+	0.902067i	$0.357950\pi$
$504$	−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	$-0.0919495\pi$
$523$	43.8433	1.91713	0.958567	+	0.284867i	$0.0919495\pi$
$524$	−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	$-0.594422\pi$
$547$	−13.6728	−0.584609	−0.292304	+	0.956325i	$0.594422\pi$
$548$	−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	$-0.410990\pi$
$557$	13.0278	0.552007	0.276004	+	0.961157i	$0.410990\pi$
$558$	−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	$-0.314077\pi$
$563$	26.1689	1.10289	0.551445	+	0.834211i	$0.314077\pi$
$564$	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	$-0.535949\pi$
$577$	−5.41416	−0.225395	−0.112697	+	0.993629i	$0.535949\pi$
$578$	−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	$-0.705928\pi$
$587$	−29.2069	−1.20550	−0.602749	+	0.797931i	$0.705928\pi$
$588$	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	$-0.411545\pi$
$593$	13.3606	0.548653	0.274326	+	0.961637i	$0.411545\pi$
$594$	−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	$-0.487913\pi$
$607$	1.87063	0.0759265	0.0379633	+	0.999279i	$0.487913\pi$
$608$	−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	$-0.285280\pi$
$613$	30.9265	1.24911	0.624556	+	0.780980i	$0.285280\pi$
$614$	59.9381	2.41890
$615$	0	0
$616$	−32.9183	−1.32632
$617$	−1.25799	−0.0506447	−0.0253224	−	0.999679i	$-0.508061\pi$
$617$	−1.25799	−0.0506447	−0.0253224	+	0.999679i	$0.508061\pi$
$618$	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	$-0.374843\pi$
$643$	19.4308	0.766277	0.383138	+	0.923691i	$0.374843\pi$
$644$	−111.862	−4.40800
$645$	0	0
$646$	−16.2745	−0.640310
$647$	7.80761	0.306949	0.153474	−	0.988153i	$-0.450954\pi$
$647$	7.80761	0.306949	0.153474	+	0.988153i	$0.450954\pi$
$648$	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	$-0.365083\pi$
$653$	21.0194	0.822553	0.411277	+	0.911511i	$0.365083\pi$
$654$	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	$-0.417220\pi$
$673$	13.3416	0.514281	0.257140	+	0.966374i	$0.417220\pi$
$674$	54.7837	2.11019
$675$	0	0
$676$	11.2877	0.434142
$677$	−35.2641	−1.35531	−0.677654	−	0.735381i	$-0.737003\pi$
$677$	−35.2641	−1.35531	−0.677654	+	0.735381i	$0.737003\pi$
$678$	−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	$-0.642678\pi$
$683$	−22.6519	−0.866753	−0.433376	+	0.901213i	$0.642678\pi$
$684$	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	$-0.485298\pi$
$727$	2.48983	0.0923426	0.0461713	+	0.998934i	$0.485298\pi$
$728$	−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	$-0.509836\pi$
$733$	−1.67302	−0.0617945	−0.0308973	+	0.999523i	$0.509836\pi$
$734$	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	$-0.485896\pi$
$743$	2.41474	0.0885881	0.0442940	+	0.999019i	$0.485896\pi$
$744$	−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	$-0.540802\pi$
$757$	−7.03423	−0.255663	−0.127832	+	0.991796i	$0.540802\pi$
$758$	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	$-0.316327\pi$
$773$	30.3348	1.09107	0.545533	+	0.838089i	$0.316327\pi$
$774$	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	$-0.478292\pi$
$787$	3.82340	0.136289	0.0681447	+	0.997675i	$0.478292\pi$
$788$	−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	$-0.302793\pi$
$797$	32.7856	1.16133	0.580663	+	0.814144i	$0.302793\pi$
$798$	−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	$-0.747844\pi$
$823$	−40.2952	−1.40460	−0.702301	+	0.711880i	$0.747844\pi$
$824$	196.192	6.83468
$825$	0	0
$826$	36.4484	1.26820
$827$	33.1923	1.15421	0.577105	−	0.816670i	$-0.304182\pi$
$827$	33.1923	1.15421	0.577105	+	0.816670i	$0.304182\pi$
$828$	−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	$-0.581766\pi$
$853$	−14.8401	−0.508117	−0.254058	+	0.967189i	$0.581766\pi$
$854$	−40.2647	−1.37783
$855$	0	0
$856$	196.063	6.70129
$857$	9.89677	0.338067	0.169034	−	0.985610i	$-0.445935\pi$
$857$	9.89677	0.338067	0.169034	+	0.985610i	$0.445935\pi$
$858$	−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	$-0.363510\pi$
$863$	24.4284	0.831553	0.415776	+	0.909467i	$0.363510\pi$
$864$	−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	$-0.638683\pi$
$877$	−24.9962	−0.844062	−0.422031	+	0.906581i	$0.638683\pi$
$878$	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	$-0.802524\pi$
$883$	−48.3559	−1.62731	−0.813653	+	0.581351i	$0.802524\pi$
$884$	129.362	4.35092
$885$	0	0
$886$	48.0010	1.61263
$887$	22.8073	0.765793	0.382897	−	0.923791i	$-0.374927\pi$
$887$	22.8073	0.765793	0.382897	+	0.923791i	$0.374927\pi$
$888$	−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	$-0.474080\pi$
$907$	4.89927	0.162678	0.0813388	+	0.996687i	$0.474080\pi$
$908$	−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	$-0.932351\pi$
$937$	−59.8435	−1.95500	−0.977501	+	0.210930i	$0.932351\pi$
$938$	−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	$-0.581903\pi$
$947$	−15.6623	−0.508955	−0.254478	+	0.967079i	$0.581903\pi$
$948$	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	$-0.352105\pi$
$953$	27.6656	0.896176	0.448088	+	0.893990i	$0.352105\pi$
$954$	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	$-0.487744\pi$
$967$	2.39400	0.0769857	0.0384929	+	0.999259i	$0.487744\pi$
$968$	−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	$-0.591196\pi$
$977$	−17.6663	−0.565194	−0.282597	+	0.959239i	$0.591196\pi$
$978$	−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	$-0.516496\pi$
$983$	−3.24817	−0.103601	−0.0518003	+	0.998657i	$0.516496\pi$
$984$	−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	$-0.228575\pi$
$997$	47.5564	1.50613	0.753063	+	0.657948i	$0.228575\pi$
$998$	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.1.2		30
5.2	odd	4		1045.2.b.e.419.2	✓	30
5.3	odd	4		1045.2.b.e.419.29	yes	30
5.4	even	2	inner	5225.2.a.bc.1.29		30

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.77451 q^{2} +2.74433 q^{3} +5.69792 q^{4} -7.61418 q^{6} +3.20844 q^{7} -10.2599 q^{8} +4.53135 q^{9} +O(q^{10})\)	\(q-2.77451 q^{2} +2.74433 q^{3} +5.69792 q^{4} -7.61418 q^{6} +3.20844 q^{7} -10.2599 q^{8} +4.53135 q^{9} +1.00000 q^{11} +15.6370 q^{12} +3.87053 q^{13} -8.90186 q^{14} +17.0705 q^{16} +5.86570 q^{17} -12.5723 q^{18} +1.00000 q^{19} +8.80502 q^{21} -2.77451 q^{22} -6.11891 q^{23} -28.1566 q^{24} -10.7388 q^{26} +4.20253 q^{27} +18.2814 q^{28} +6.40440 q^{29} +1.66215 q^{31} -26.8423 q^{32} +2.74433 q^{33} -16.2745 q^{34} +25.8193 q^{36} +0.251830 q^{37} -2.77451 q^{38} +10.6220 q^{39} +10.7565 q^{41} -24.4296 q^{42} -2.17536 q^{43} +5.69792 q^{44} +16.9770 q^{46} +2.68163 q^{47} +46.8470 q^{48} +3.29408 q^{49} +16.0974 q^{51} +22.0540 q^{52} -8.89909 q^{53} -11.6600 q^{54} -32.9183 q^{56} +2.74433 q^{57} -17.7691 q^{58} -4.09447 q^{59} +4.52319 q^{61} -4.61167 q^{62} +14.5386 q^{63} +40.3335 q^{64} -7.61418 q^{66} +2.61295 q^{67} +33.4223 q^{68} -16.7923 q^{69} -5.51722 q^{71} -46.4913 q^{72} -2.73480 q^{73} -0.698705 q^{74} +5.69792 q^{76} +3.20844 q^{77} -29.4709 q^{78} +8.17104 q^{79} -2.06093 q^{81} -29.8440 q^{82} -12.2337 q^{83} +50.1703 q^{84} +6.03555 q^{86} +17.5758 q^{87} -10.2599 q^{88} -1.84048 q^{89} +12.4184 q^{91} -34.8651 q^{92} +4.56150 q^{93} -7.44022 q^{94} -73.6642 q^{96} -2.65645 q^{97} -9.13947 q^{98} +4.53135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9}+O(q^{10}) \) 30 * q + 42 * q^4 + 12 * q^6 + 40 * q^9	\( 30 q + 42 q^{4} + 12 q^{6} + 40 q^{9} + 30 q^{11} - 4 q^{14} + 66 q^{16} + 30 q^{19} + 14 q^{21} + 22 q^{24} + 30 q^{29} + 26 q^{31} + 12 q^{34} + 78 q^{36} + 64 q^{39} + 22 q^{41} + 42 q^{44} + 28 q^{46} + 60 q^{49} + 64 q^{51} + 62 q^{54} - 32 q^{56} - 14 q^{59} + 78 q^{61} + 90 q^{64} + 12 q^{66} - 28 q^{69} + 20 q^{71} + 42 q^{74} + 42 q^{76} + 102 q^{79} + 42 q^{81} + 98 q^{84} - 52 q^{86} - 8 q^{89} + 56 q^{91} + 40 q^{94} - 74 q^{96} + 40 q^{99}+O(q^{100}) \) 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

Representations

Groups

Database

Properties

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