LMFDB - Embedded newform 1881.2.a.p.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1881,2,Mod(1,1881)] 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("1881.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1881, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1881 = 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1881.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,1,0,15,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$15.0198606202$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 $ x^7 - x^6 - 14x^5 + 10x^4 + 59x^3 - 27x^2 - 66*x + 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.55401$ of defining polynomial
Character	$\chi$	$=$	1881.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.55401 q^{2} +4.52299 q^{4} -0.244850 q^{5} +4.42321 q^{7} -6.44376 q^{8} +0.625350 q^{10} +1.00000 q^{11} -5.89690 q^{13} -11.2969 q^{14} +7.41147 q^{16} +2.93922 q^{17} +1.00000 q^{19} -1.10745 q^{20} -2.55401 q^{22} +0.372904 q^{23} -4.94005 q^{25} +15.0608 q^{26} +20.0061 q^{28} +3.47526 q^{29} +6.37391 q^{31} -6.04149 q^{32} -7.50681 q^{34} -1.08302 q^{35} +0.926528 q^{37} -2.55401 q^{38} +1.57775 q^{40} +6.67861 q^{41} +2.12805 q^{43} +4.52299 q^{44} -0.952402 q^{46} +1.72093 q^{47} +12.5648 q^{49} +12.6170 q^{50} -26.6716 q^{52} +1.44022 q^{53} -0.244850 q^{55} -28.5021 q^{56} -8.87587 q^{58} -7.71718 q^{59} +4.16881 q^{61} -16.2791 q^{62} +0.607107 q^{64} +1.44385 q^{65} -11.3778 q^{67} +13.2941 q^{68} +2.76605 q^{70} -2.40794 q^{71} +14.4653 q^{73} -2.36637 q^{74} +4.52299 q^{76} +4.42321 q^{77} +1.67368 q^{79} -1.81470 q^{80} -17.0573 q^{82} +6.47897 q^{83} -0.719667 q^{85} -5.43508 q^{86} -6.44376 q^{88} +4.95706 q^{89} -26.0832 q^{91} +1.68664 q^{92} -4.39529 q^{94} -0.244850 q^{95} +8.24006 q^{97} -32.0906 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 15 q^{4} - 2 q^{5} + 10 q^{7} + 9 q^{8} - 6 q^{10} + 7 q^{11} - 4 q^{13} - 6 q^{14} + 27 q^{16} - 2 q^{17} + 7 q^{19} + 4 q^{20} + q^{22} - 10 q^{23} + 9 q^{25} + 8 q^{26} + 26 q^{28} + 18 q^{29}+ \cdots - 19 q^{98}+O(q^{100}) $ 7 * q + q^2 + 15 * q^4 - 2 * q^5 + 10 * q^7 + 9 * q^8 - 6 * q^10 + 7 * q^11 - 4 * q^13 - 6 * q^14 + 27 * q^16 - 2 * q^17 + 7 * q^19 + 4 * q^20 + q^22 - 10 * q^23 + 9 * q^25 + 8 * q^26 + 26 * q^28 + 18 * q^29 + 24 * q^31 + 49 * q^32 - 6 * q^34 - 8 * q^35 + q^38 - 2 * q^40 + 12 * q^41 + 2 * q^43 + 15 * q^44 - 4 * q^46 - 8 * q^47 + 17 * q^49 + 33 * q^50 - 60 * q^52 - 2 * q^53 - 2 * q^55 - 26 * q^56 - 8 * q^58 + 10 * q^59 + 14 * q^61 - 14 * q^62 + 55 * q^64 + 14 * q^65 + 8 * q^67 + 18 * q^68 - 66 * q^70 - 10 * q^71 - 6 * q^73 - 26 * q^74 + 15 * q^76 + 10 * q^77 + 52 * q^79 + 12 * q^80 + 24 * q^82 + 10 * q^83 - 12 * q^85 - 8 * q^86 + 9 * q^88 + 12 * q^91 + 24 * q^94 - 2 * q^95 - 24 * q^97 - 19 * q^98

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 $

$2$	−2.55401	−1.80596	−0.902981	−	0.429681i	$-0.858626\pi$
$2$	−2.55401	−1.80596	−0.902981	+	0.429681i	$0.858626\pi$
$3$	0	0
$3$	0	0
$4$	4.52299	2.26150
$5$	−0.244850	−0.109500	−0.0547500	−	0.998500i	$-0.517436\pi$
$5$	−0.244850	−0.109500	−0.0547500	+	0.998500i	$0.517436\pi$
$6$	0	0
$7$	4.42321	1.67182	0.835908	−	0.548869i	$-0.184942\pi$
$7$	4.42321	1.67182	0.835908	+	0.548869i	$0.184942\pi$
$8$	−6.44376	−2.27821
$9$	0	0
$10$	0.625350	0.197753
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.89690	−1.63551	−0.817753	−	0.575570i	$-0.804780\pi$
$13$	−5.89690	−1.63551	−0.817753	+	0.575570i	$0.804780\pi$
$14$	−11.2969	−3.01924
$15$	0	0
$16$	7.41147	1.85287
$17$	2.93922	0.712865	0.356433	−	0.934321i	$-0.383993\pi$
$17$	2.93922	0.712865	0.356433	+	0.934321i	$0.383993\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.10745	−0.247634
$21$	0	0
$22$	−2.55401	−0.544518
$23$	0.372904	0.0777558	0.0388779	−	0.999244i	$-0.487622\pi$
$23$	0.372904	0.0777558	0.0388779	+	0.999244i	$0.487622\pi$
$24$	0	0
$25$	−4.94005	−0.988010
$26$	15.0608	2.95366
$27$	0	0
$28$	20.0061	3.78081
$29$	3.47526	0.645340	0.322670	−	0.946512i	$-0.395420\pi$
$29$	3.47526	0.645340	0.322670	+	0.946512i	$0.395420\pi$
$30$	0	0
$31$	6.37391	1.14479	0.572394	−	0.819979i	$-0.306015\pi$
$31$	6.37391	1.14479	0.572394	+	0.819979i	$0.306015\pi$
$32$	−6.04149	−1.06799
$33$	0	0
$34$	−7.50681	−1.28741
$35$	−1.08302	−0.183064
$36$	0	0
$37$	0.926528	0.152320	0.0761601	−	0.997096i	$-0.475734\pi$
$37$	0.926528	0.152320	0.0761601	+	0.997096i	$0.475734\pi$
$38$	−2.55401	−0.414316
$39$	0	0
$40$	1.57775	0.249464
$41$	6.67861	1.04302	0.521512	−	0.853244i	$-0.325368\pi$
$41$	6.67861	1.04302	0.521512	+	0.853244i	$0.325368\pi$
$42$	0	0
$43$	2.12805	0.324525	0.162263	−	0.986748i	$-0.448121\pi$
$43$	2.12805	0.324525	0.162263	+	0.986748i	$0.448121\pi$
$44$	4.52299	0.681867
$45$	0	0
$46$	−0.952402	−0.140424
$47$	1.72093	0.251024	0.125512	−	0.992092i	$-0.459943\pi$
$47$	1.72093	0.251024	0.125512	+	0.992092i	$0.459943\pi$
$48$	0	0
$49$	12.5648	1.79497
$50$	12.6170	1.78431
$51$	0	0
$52$	−26.6716	−3.69869
$53$	1.44022	0.197830	0.0989150	−	0.995096i	$-0.468463\pi$
$53$	1.44022	0.197830	0.0989150	+	0.995096i	$0.468463\pi$
$54$	0	0
$55$	−0.244850	−0.0330155
$56$	−28.5021	−3.80875
$57$	0	0
$58$	−8.87587	−1.16546
$59$	−7.71718	−1.00469	−0.502346	−	0.864667i	$-0.667530\pi$
$59$	−7.71718	−1.00469	−0.502346	+	0.864667i	$0.667530\pi$
$60$	0	0
$61$	4.16881	0.533762	0.266881	−	0.963730i	$-0.414007\pi$
$61$	4.16881	0.533762	0.266881	+	0.963730i	$0.414007\pi$
$62$	−16.2791	−2.06744
$63$	0	0
$64$	0.607107	0.0758884
$65$	1.44385	0.179088
$66$	0	0
$67$	−11.3778	−1.39002	−0.695012	−	0.718998i	$-0.744601\pi$
$67$	−11.3778	−1.39002	−0.695012	+	0.718998i	$0.744601\pi$
$68$	13.2941	1.61214
$69$	0	0
$70$	2.76605	0.330607
$71$	−2.40794	−0.285770	−0.142885	−	0.989739i	$-0.545638\pi$
$71$	−2.40794	−0.285770	−0.142885	+	0.989739i	$0.545638\pi$
$72$	0	0
$73$	14.4653	1.69303	0.846515	−	0.532365i	$-0.178697\pi$
$73$	14.4653	1.69303	0.846515	+	0.532365i	$0.178697\pi$
$74$	−2.36637	−0.275084
$75$	0	0
$76$	4.52299	0.518823
$77$	4.42321	0.504072
$78$	0	0
$79$	1.67368	0.188304	0.0941521	−	0.995558i	$-0.469986\pi$
$79$	1.67368	0.188304	0.0941521	+	0.995558i	$0.469986\pi$
$80$	−1.81470	−0.202889
$81$	0	0
$82$	−17.0573	−1.88366
$83$	6.47897	0.711159	0.355580	−	0.934646i	$-0.384283\pi$
$83$	6.47897	0.711159	0.355580	+	0.934646i	$0.384283\pi$
$84$	0	0
$85$	−0.719667	−0.0780588
$86$	−5.43508	−0.586080
$87$	0	0
$88$	−6.44376	−0.686907
$89$	4.95706	0.525448	0.262724	−	0.964871i	$-0.415379\pi$
$89$	4.95706	0.525448	0.262724	+	0.964871i	$0.415379\pi$
$90$	0	0
$91$	−26.0832	−2.73426
$92$	1.68664	0.175845
$93$	0	0
$94$	−4.39529	−0.453339
$95$	−0.244850	−0.0251210
$96$	0	0
$97$	8.24006	0.836651	0.418325	−	0.908297i	$-0.362617\pi$
$97$	8.24006	0.836651	0.418325	+	0.908297i	$0.362617\pi$
$98$	−32.0906	−3.24164
$99$	0	0
$100$	−22.3438	−2.23438
$101$	−3.51207	−0.349464	−0.174732	−	0.984616i	$-0.555906\pi$
$101$	−3.51207	−0.349464	−0.174732	+	0.984616i	$0.555906\pi$
$102$	0	0
$103$	−2.43194	−0.239626	−0.119813	−	0.992796i	$-0.538230\pi$
$103$	−2.43194	−0.239626	−0.119813	+	0.992796i	$0.538230\pi$
$104$	37.9982	3.72603
$105$	0	0
$106$	−3.67835	−0.357273
$107$	−16.9865	−1.64214	−0.821072	−	0.570825i	$-0.806624\pi$
$107$	−16.9865	−1.64214	−0.821072	+	0.570825i	$0.806624\pi$
$108$	0	0
$109$	3.83586	0.367409	0.183704	−	0.982982i	$-0.441191\pi$
$109$	3.83586	0.367409	0.183704	+	0.982982i	$0.441191\pi$
$110$	0.625350	0.0596247
$111$	0	0
$112$	32.7825	3.09765
$113$	10.7909	1.01512	0.507562	−	0.861615i	$-0.330547\pi$
$113$	10.7909	1.01512	0.507562	+	0.861615i	$0.330547\pi$
$114$	0	0
$115$	−0.0913054	−0.00851427
$116$	15.7186	1.45943
$117$	0	0
$118$	19.7098	1.81443
$119$	13.0008	1.19178
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.6472	−0.963953
$123$	0	0
$124$	28.8291	2.58893
$125$	2.43382	0.217687
$126$	0	0
$127$	−12.0969	−1.07342	−0.536712	−	0.843765i	$-0.680334\pi$
$127$	−12.0969	−1.07342	−0.536712	+	0.843765i	$0.680334\pi$
$128$	10.5324	0.930942
$129$	0	0
$130$	−3.68762	−0.323426
$131$	7.99822	0.698807	0.349404	−	0.936972i	$-0.386384\pi$
$131$	7.99822	0.698807	0.349404	+	0.936972i	$0.386384\pi$
$132$	0	0
$133$	4.42321	0.383541
$134$	29.0592	2.51033
$135$	0	0
$136$	−18.9396	−1.62406
$137$	−15.9191	−1.36006	−0.680030	−	0.733184i	$-0.738033\pi$
$137$	−15.9191	−1.36006	−0.680030	+	0.733184i	$0.738033\pi$
$138$	0	0
$139$	14.7278	1.24920	0.624599	−	0.780946i	$-0.285263\pi$
$139$	14.7278	1.24920	0.624599	+	0.780946i	$0.285263\pi$
$140$	−4.89850	−0.413999
$141$	0	0
$142$	6.14992	0.516090
$143$	−5.89690	−0.493124
$144$	0	0
$145$	−0.850917	−0.0706648
$146$	−36.9445	−3.05755
$147$	0	0
$148$	4.19068	0.344472
$149$	−7.35723	−0.602727	−0.301364	−	0.953509i	$-0.597442\pi$
$149$	−7.35723	−0.602727	−0.301364	+	0.953509i	$0.597442\pi$
$150$	0	0
$151$	10.5197	0.856083	0.428042	−	0.903759i	$-0.359204\pi$
$151$	10.5197	0.856083	0.428042	+	0.903759i	$0.359204\pi$
$152$	−6.44376	−0.522658
$153$	0	0
$154$	−11.2969	−0.910334
$155$	−1.56065	−0.125354
$156$	0	0
$157$	−16.1669	−1.29026	−0.645131	−	0.764072i	$-0.723197\pi$
$157$	−16.1669	−1.29026	−0.645131	+	0.764072i	$0.723197\pi$
$158$	−4.27461	−0.340070
$159$	0	0
$160$	1.47926	0.116945
$161$	1.64943	0.129993
$162$	0	0
$163$	13.6643	1.07027	0.535136	−	0.844766i	$-0.320260\pi$
$163$	13.6643	1.07027	0.535136	+	0.844766i	$0.320260\pi$
$164$	30.2073	2.35879
$165$	0	0
$166$	−16.5474	−1.28433
$167$	−10.3060	−0.797501	−0.398750	−	0.917060i	$-0.630556\pi$
$167$	−10.3060	−0.797501	−0.398750	+	0.917060i	$0.630556\pi$
$168$	0	0
$169$	21.7734	1.67488
$170$	1.83804	0.140971
$171$	0	0
$172$	9.62517	0.733912
$173$	25.3276	1.92562	0.962811	−	0.270174i	$-0.0870813\pi$
$173$	25.3276	1.92562	0.962811	+	0.270174i	$0.0870813\pi$
$174$	0	0
$175$	−21.8509	−1.65177
$176$	7.41147	0.558661
$177$	0	0
$178$	−12.6604	−0.948938
$179$	−22.8968	−1.71139	−0.855695	−	0.517481i	$-0.826870\pi$
$179$	−22.8968	−1.71139	−0.855695	+	0.517481i	$0.826870\pi$
$180$	0	0
$181$	8.05994	0.599091	0.299545	−	0.954082i	$-0.403165\pi$
$181$	8.05994	0.599091	0.299545	+	0.954082i	$0.403165\pi$
$182$	66.6169	4.93798
$183$	0	0
$184$	−2.40290	−0.177144
$185$	−0.226860	−0.0166791
$186$	0	0
$187$	2.93922	0.214937
$188$	7.78376	0.567689
$189$	0	0
$190$	0.625350	0.0453676
$191$	17.2143	1.24559	0.622793	−	0.782387i	$-0.285998\pi$
$191$	17.2143	1.24559	0.622793	+	0.782387i	$0.285998\pi$
$192$	0	0
$193$	−8.31588	−0.598590	−0.299295	−	0.954161i	$-0.596751\pi$
$193$	−8.31588	−0.598590	−0.299295	+	0.954161i	$0.596751\pi$
$194$	−21.0452	−1.51096
$195$	0	0
$196$	56.8304	4.05932
$197$	−4.38913	−0.312713	−0.156356	−	0.987701i	$-0.549975\pi$
$197$	−4.38913	−0.312713	−0.156356	+	0.987701i	$0.549975\pi$
$198$	0	0
$199$	19.8924	1.41014	0.705068	−	0.709140i	$-0.250917\pi$
$199$	19.8924	1.41014	0.705068	+	0.709140i	$0.250917\pi$
$200$	31.8325	2.25090
$201$	0	0
$202$	8.96988	0.631119
$203$	15.3718	1.07889
$204$	0	0
$205$	−1.63526	−0.114211
$206$	6.21121	0.432755
$207$	0	0
$208$	−43.7047	−3.03038
$209$	1.00000	0.0691714
$210$	0	0
$211$	20.8162	1.43304	0.716522	−	0.697564i	$-0.245733\pi$
$211$	20.8162	1.43304	0.716522	+	0.697564i	$0.245733\pi$
$212$	6.51412	0.447391
$213$	0	0
$214$	43.3837	2.96565
$215$	−0.521053	−0.0355355
$216$	0	0
$217$	28.1931	1.91387
$218$	−9.79684	−0.663525
$219$	0	0
$220$	−1.10745	−0.0746645
$221$	−17.3323	−1.16590
$222$	0	0
$223$	11.4967	0.769874	0.384937	−	0.922943i	$-0.374223\pi$
$223$	11.4967	0.769874	0.384937	+	0.922943i	$0.374223\pi$
$224$	−26.7228	−1.78549
$225$	0	0
$226$	−27.5601	−1.83327
$227$	17.5175	1.16268	0.581339	−	0.813662i	$-0.302529\pi$
$227$	17.5175	1.16268	0.581339	+	0.813662i	$0.302529\pi$
$228$	0	0
$229$	−6.64251	−0.438949	−0.219475	−	0.975618i	$-0.570434\pi$
$229$	−6.64251	−0.438949	−0.219475	+	0.975618i	$0.570434\pi$
$230$	0.233195	0.0153764
$231$	0	0
$232$	−22.3937	−1.47022
$233$	12.5323	0.821020	0.410510	−	0.911856i	$-0.365351\pi$
$233$	12.5323	0.821020	0.410510	+	0.911856i	$0.365351\pi$
$234$	0	0
$235$	−0.421370	−0.0274871
$236$	−34.9048	−2.27211
$237$	0	0
$238$	−33.2042	−2.15231
$239$	27.3445	1.76877	0.884386	−	0.466757i	$-0.154578\pi$
$239$	27.3445	1.76877	0.884386	+	0.466757i	$0.154578\pi$
$240$	0	0
$241$	23.5681	1.51816	0.759079	−	0.650999i	$-0.225650\pi$
$241$	23.5681	1.51816	0.759079	+	0.650999i	$0.225650\pi$
$242$	−2.55401	−0.164178
$243$	0	0
$244$	18.8555	1.20710
$245$	−3.07648	−0.196549
$246$	0	0
$247$	−5.89690	−0.375211
$248$	−41.0719	−2.60807
$249$	0	0
$250$	−6.21601	−0.393135
$251$	2.04497	0.129078	0.0645388	−	0.997915i	$-0.479442\pi$
$251$	2.04497	0.129078	0.0645388	+	0.997915i	$0.479442\pi$
$252$	0	0
$253$	0.372904	0.0234443
$254$	30.8956	1.93856
$255$	0	0
$256$	−28.1141	−1.75713
$257$	−21.0705	−1.31434	−0.657171	−	0.753741i	$-0.728247\pi$
$257$	−21.0705	−1.31434	−0.657171	+	0.753741i	$0.728247\pi$
$258$	0	0
$259$	4.09823	0.254651
$260$	6.53054	0.405007
$261$	0	0
$262$	−20.4276	−1.26202
$263$	21.5153	1.32669	0.663345	−	0.748313i	$-0.269136\pi$
$263$	21.5153	1.32669	0.663345	+	0.748313i	$0.269136\pi$
$264$	0	0
$265$	−0.352638	−0.0216624
$266$	−11.2969	−0.692660
$267$	0	0
$268$	−51.4618	−3.14353
$269$	−8.56257	−0.522069	−0.261034	−	0.965329i	$-0.584064\pi$
$269$	−8.56257	−0.522069	−0.261034	+	0.965329i	$0.584064\pi$
$270$	0	0
$271$	−12.5375	−0.761602	−0.380801	−	0.924657i	$-0.624352\pi$
$271$	−12.5375	−0.761602	−0.380801	+	0.924657i	$0.624352\pi$
$272$	21.7839	1.32084
$273$	0	0
$274$	40.6576	2.45622
$275$	−4.94005	−0.297896
$276$	0	0
$277$	11.2455	0.675679	0.337840	−	0.941204i	$-0.390304\pi$
$277$	11.2455	0.675679	0.337840	+	0.941204i	$0.390304\pi$
$278$	−37.6151	−2.25600
$279$	0	0
$280$	6.97873	0.417059
$281$	20.3272	1.21262	0.606310	−	0.795228i	$-0.292649\pi$
$281$	20.3272	1.21262	0.606310	+	0.795228i	$0.292649\pi$
$282$	0	0
$283$	−26.7277	−1.58880	−0.794400	−	0.607395i	$-0.792214\pi$
$283$	−26.7277	−1.58880	−0.794400	+	0.607395i	$0.792214\pi$
$284$	−10.8911	−0.646268
$285$	0	0
$286$	15.0608	0.890562
$287$	29.5409	1.74374
$288$	0	0
$289$	−8.36099	−0.491823
$290$	2.17325	0.127618
$291$	0	0
$292$	65.4262	3.82878
$293$	9.95580	0.581624	0.290812	−	0.956780i	$-0.406075\pi$
$293$	9.95580	0.581624	0.290812	+	0.956780i	$0.406075\pi$
$294$	0	0
$295$	1.88955	0.110014
$296$	−5.97032	−0.347018
$297$	0	0
$298$	18.7905	1.08850
$299$	−2.19898	−0.127170
$300$	0	0
$301$	9.41283	0.542547
$302$	−26.8675	−1.54605
$303$	0	0
$304$	7.41147	0.425077
$305$	−1.02073	−0.0584469
$306$	0	0
$307$	3.58808	0.204783	0.102391	−	0.994744i	$-0.467351\pi$
$307$	3.58808	0.204783	0.102391	+	0.994744i	$0.467351\pi$
$308$	20.0061	1.13996
$309$	0	0
$310$	3.98592	0.226385
$311$	13.8469	0.785187	0.392594	−	0.919712i	$-0.371578\pi$
$311$	13.8469	0.785187	0.392594	+	0.919712i	$0.371578\pi$
$312$	0	0
$313$	−15.3982	−0.870360	−0.435180	−	0.900344i	$-0.643315\pi$
$313$	−15.3982	−0.870360	−0.435180	+	0.900344i	$0.643315\pi$
$314$	41.2906	2.33016
$315$	0	0
$316$	7.57006	0.425849
$317$	5.96359	0.334949	0.167474	−	0.985876i	$-0.446439\pi$
$317$	5.96359	0.334949	0.167474	+	0.985876i	$0.446439\pi$
$318$	0	0
$319$	3.47526	0.194577
$320$	−0.148650	−0.00830979
$321$	0	0
$322$	−4.21267	−0.234763
$323$	2.93922	0.163542
$324$	0	0
$325$	29.1310	1.61590
$326$	−34.8989	−1.93287
$327$	0	0
$328$	−43.0354	−2.37623
$329$	7.61204	0.419665
$330$	0	0
$331$	−5.58507	−0.306983	−0.153492	−	0.988150i	$-0.549052\pi$
$331$	−5.58507	−0.306983	−0.153492	+	0.988150i	$0.549052\pi$
$332$	29.3043	1.60828
$333$	0	0
$334$	26.3216	1.44026
$335$	2.78586	0.152208
$336$	0	0
$337$	−29.0465	−1.58227	−0.791133	−	0.611645i	$-0.790508\pi$
$337$	−29.0465	−1.58227	−0.791133	+	0.611645i	$0.790508\pi$
$338$	−55.6097	−3.02477
$339$	0	0
$340$	−3.25505	−0.176530
$341$	6.37391	0.345166
$342$	0	0
$343$	24.6142	1.32904
$344$	−13.7127	−0.739337
$345$	0	0
$346$	−64.6871	−3.47760
$347$	6.25081	0.335561	0.167781	−	0.985824i	$-0.446340\pi$
$347$	6.25081	0.335561	0.167781	+	0.985824i	$0.446340\pi$
$348$	0	0
$349$	7.81752	0.418462	0.209231	−	0.977866i	$-0.432904\pi$
$349$	7.81752	0.418462	0.209231	+	0.977866i	$0.432904\pi$
$350$	55.8074	2.98303
$351$	0	0
$352$	−6.04149	−0.322012
$353$	−10.0965	−0.537380	−0.268690	−	0.963227i	$-0.586591\pi$
$353$	−10.0965	−0.537380	−0.268690	+	0.963227i	$0.586591\pi$
$354$	0	0
$355$	0.589584	0.0312919
$356$	22.4208	1.18830
$357$	0	0
$358$	58.4788	3.09070
$359$	−29.3266	−1.54780	−0.773899	−	0.633309i	$-0.781696\pi$
$359$	−29.3266	−1.54780	−0.773899	+	0.633309i	$0.781696\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−20.5852	−1.08193
$363$	0	0
$364$	−117.974	−6.18353
$365$	−3.54181	−0.185387
$366$	0	0
$367$	−13.7848	−0.719558	−0.359779	−	0.933037i	$-0.617148\pi$
$367$	−13.7848	−0.719558	−0.359779	+	0.933037i	$0.617148\pi$
$368$	2.76377	0.144071
$369$	0	0
$370$	0.579404	0.0301218
$371$	6.37041	0.330735
$372$	0	0
$373$	16.4169	0.850036	0.425018	−	0.905185i	$-0.360268\pi$
$373$	16.4169	0.850036	0.425018	+	0.905185i	$0.360268\pi$
$374$	−7.50681	−0.388168
$375$	0	0
$376$	−11.0893	−0.571885
$377$	−20.4933	−1.05546
$378$	0	0
$379$	6.93533	0.356244	0.178122	−	0.984008i	$-0.442998\pi$
$379$	6.93533	0.356244	0.178122	+	0.984008i	$0.442998\pi$
$380$	−1.10745	−0.0568111
$381$	0	0
$382$	−43.9657	−2.24948
$383$	−1.47147	−0.0751884	−0.0375942	−	0.999293i	$-0.511969\pi$
$383$	−1.47147	−0.0751884	−0.0375942	+	0.999293i	$0.511969\pi$
$384$	0	0
$385$	−1.08302	−0.0551959
$386$	21.2389	1.08103
$387$	0	0
$388$	37.2697	1.89208
$389$	−6.79609	−0.344575	−0.172288	−	0.985047i	$-0.555116\pi$
$389$	−6.79609	−0.344575	−0.172288	+	0.985047i	$0.555116\pi$
$390$	0	0
$391$	1.09605	0.0554294
$392$	−80.9644	−4.08932
$393$	0	0
$394$	11.2099	0.564747
$395$	−0.409801	−0.0206193
$396$	0	0
$397$	−12.3808	−0.621373	−0.310687	−	0.950512i	$-0.600559\pi$
$397$	−12.3808	−0.621373	−0.310687	+	0.950512i	$0.600559\pi$
$398$	−50.8055	−2.54665
$399$	0	0
$400$	−36.6130	−1.83065
$401$	0.0361352	0.00180451	0.000902254	−	1.00000i	$-0.499713\pi$
$401$	0.0361352	0.00180451	0.000902254		1.00000i	$0.499713\pi$
$402$	0	0
$403$	−37.5863	−1.87231
$404$	−15.8851	−0.790312
$405$	0	0
$406$	−39.2598	−1.94843
$407$	0.926528	0.0459263
$408$	0	0
$409$	−9.94955	−0.491973	−0.245987	−	0.969273i	$-0.579112\pi$
$409$	−9.94955	−0.491973	−0.245987	+	0.969273i	$0.579112\pi$
$410$	4.17647	0.206261
$411$	0	0
$412$	−10.9996	−0.541913
$413$	−34.1347	−1.67966
$414$	0	0
$415$	−1.58637	−0.0778720
$416$	35.6260	1.74671
$417$	0	0
$418$	−2.55401	−0.124921
$419$	−27.2560	−1.33154	−0.665771	−	0.746156i	$-0.731897\pi$
$419$	−27.2560	−1.33154	−0.665771	+	0.746156i	$0.731897\pi$
$420$	0	0
$421$	16.5342	0.805828	0.402914	−	0.915238i	$-0.367997\pi$
$421$	16.5342	0.805828	0.402914	+	0.915238i	$0.367997\pi$
$422$	−53.1648	−2.58802
$423$	0	0
$424$	−9.28045	−0.450699
$425$	−14.5199	−0.704318
$426$	0	0
$427$	18.4395	0.892351
$428$	−76.8296	−3.71370
$429$	0	0
$430$	1.33078	0.0641758
$431$	−33.9550	−1.63556	−0.817778	−	0.575533i	$-0.804794\pi$
$431$	−33.9550	−1.63556	−0.817778	+	0.575533i	$0.804794\pi$
$432$	0	0
$433$	−3.55114	−0.170657	−0.0853285	−	0.996353i	$-0.527194\pi$
$433$	−3.55114	−0.170657	−0.0853285	+	0.996353i	$0.527194\pi$
$434$	−72.0057	−3.45638
$435$	0	0
$436$	17.3496	0.830893
$437$	0.372904	0.0178384
$438$	0	0
$439$	6.28232	0.299839	0.149919	−	0.988698i	$-0.452099\pi$
$439$	6.28232	0.299839	0.149919	+	0.988698i	$0.452099\pi$
$440$	1.57775	0.0752164
$441$	0	0
$442$	44.2669	2.10556
$443$	−26.3969	−1.25415	−0.627077	−	0.778957i	$-0.715749\pi$
$443$	−26.3969	−1.25415	−0.627077	+	0.778957i	$0.715749\pi$
$444$	0	0
$445$	−1.21373	−0.0575366
$446$	−29.3627	−1.39036
$447$	0	0
$448$	2.68536	0.126872
$449$	0.649120	0.0306339	0.0153169	−	0.999883i	$-0.495124\pi$
$449$	0.649120	0.0306339	0.0153169	+	0.999883i	$0.495124\pi$
$450$	0	0
$451$	6.67861	0.314484
$452$	48.8072	2.29570
$453$	0	0
$454$	−44.7400	−2.09975
$455$	6.38647	0.299402
$456$	0	0
$457$	−4.74769	−0.222087	−0.111044	−	0.993816i	$-0.535419\pi$
$457$	−4.74769	−0.222087	−0.111044	+	0.993816i	$0.535419\pi$
$458$	16.9651	0.792725
$459$	0	0
$460$	−0.412974	−0.0192550
$461$	−13.9711	−0.650701	−0.325350	−	0.945594i	$-0.605482\pi$
$461$	−13.9711	−0.650701	−0.325350	+	0.945594i	$0.605482\pi$
$462$	0	0
$463$	−14.7822	−0.686985	−0.343492	−	0.939155i	$-0.611610\pi$
$463$	−14.7822	−0.686985	−0.343492	+	0.939155i	$0.611610\pi$
$464$	25.7568	1.19573
$465$	0	0
$466$	−32.0077	−1.48273
$467$	22.5444	1.04323	0.521615	−	0.853181i	$-0.325330\pi$
$467$	22.5444	1.04323	0.521615	+	0.853181i	$0.325330\pi$
$468$	0	0
$469$	−50.3265	−2.32386
$470$	1.07618	0.0496407
$471$	0	0
$472$	49.7277	2.28890
$473$	2.12805	0.0978480
$474$	0	0
$475$	−4.94005	−0.226665
$476$	58.8024	2.69520
$477$	0	0
$478$	−69.8384	−3.19433
$479$	21.7949	0.995835	0.497918	−	0.867224i	$-0.334098\pi$
$479$	21.7949	0.995835	0.497918	+	0.867224i	$0.334098\pi$
$480$	0	0
$481$	−5.46364	−0.249121
$482$	−60.1934	−2.74173
$483$	0	0
$484$	4.52299	0.205591
$485$	−2.01757	−0.0916133
$486$	0	0
$487$	−5.40569	−0.244955	−0.122478	−	0.992471i	$-0.539084\pi$
$487$	−5.40569	−0.244955	−0.122478	+	0.992471i	$0.539084\pi$
$488$	−26.8628	−1.21602
$489$	0	0
$490$	7.85738	0.354960
$491$	41.4597	1.87105	0.935524	−	0.353263i	$-0.114928\pi$
$491$	41.4597	1.87105	0.935524	+	0.353263i	$0.114928\pi$
$492$	0	0
$493$	10.2146	0.460040
$494$	15.0608	0.677616
$495$	0	0
$496$	47.2400	2.12114
$497$	−10.6508	−0.477755
$498$	0	0
$499$	18.0596	0.808459	0.404229	−	0.914658i	$-0.367540\pi$
$499$	18.0596	0.808459	0.404229	+	0.914658i	$0.367540\pi$
$500$	11.0081	0.492299
$501$	0	0
$502$	−5.22289	−0.233109
$503$	−33.5764	−1.49710	−0.748549	−	0.663080i	$-0.769249\pi$
$503$	−33.5764	−1.49710	−0.748549	+	0.663080i	$0.769249\pi$
$504$	0	0
$505$	0.859929	0.0382664
$506$	−0.952402	−0.0423394
$507$	0	0
$508$	−54.7141	−2.42755
$509$	19.8963	0.881889	0.440944	−	0.897534i	$-0.354644\pi$
$509$	19.8963	0.881889	0.440944	+	0.897534i	$0.354644\pi$
$510$	0	0
$511$	63.9829	2.83044
$512$	50.7391	2.24237
$513$	0	0
$514$	53.8144	2.37365
$515$	0.595459	0.0262391
$516$	0	0
$517$	1.72093	0.0756865
$518$	−10.4669	−0.459891
$519$	0	0
$520$	−9.30384	−0.408001
$521$	22.1379	0.969878	0.484939	−	0.874548i	$-0.338842\pi$
$521$	22.1379	0.969878	0.484939	+	0.874548i	$0.338842\pi$
$522$	0	0
$523$	16.0313	0.701001	0.350500	−	0.936563i	$-0.386012\pi$
$523$	16.0313	0.701001	0.350500	+	0.936563i	$0.386012\pi$
$524$	36.1759	1.58035
$525$	0	0
$526$	−54.9504	−2.39595
$527$	18.7343	0.816079
$528$	0	0
$529$	−22.8609	−0.993954
$530$	0.900643	0.0391214
$531$	0	0
$532$	20.0061	0.867376
$533$	−39.3831	−1.70587
$534$	0	0
$535$	4.15913	0.179815
$536$	73.3160	3.16677
$537$	0	0
$538$	21.8689	0.942836
$539$	12.5648	0.541204
$540$	0	0
$541$	−30.5422	−1.31311	−0.656556	−	0.754278i	$-0.727987\pi$
$541$	−30.5422	−1.31311	−0.656556	+	0.754278i	$0.727987\pi$
$542$	32.0211	1.37542
$543$	0	0
$544$	−17.7572	−0.761336
$545$	−0.939208	−0.0402313
$546$	0	0
$547$	0.0830197	0.00354967	0.00177483	−	0.999998i	$-0.499435\pi$
$547$	0.0830197	0.00354967	0.00177483	+	0.999998i	$0.499435\pi$
$548$	−72.0019	−3.07577
$549$	0	0
$550$	12.6170	0.537989
$551$	3.47526	0.148051
$552$	0	0
$553$	7.40305	0.314810
$554$	−28.7213	−1.22025
$555$	0	0
$556$	66.6138	2.82505
$557$	6.33324	0.268348	0.134174	−	0.990958i	$-0.457162\pi$
$557$	6.33324	0.268348	0.134174	+	0.990958i	$0.457162\pi$
$558$	0	0
$559$	−12.5489	−0.530763
$560$	−8.02678	−0.339193
$561$	0	0
$562$	−51.9160	−2.18995
$563$	26.4539	1.11490	0.557451	−	0.830210i	$-0.311780\pi$
$563$	26.4539	1.11490	0.557451	+	0.830210i	$0.311780\pi$
$564$	0	0
$565$	−2.64215	−0.111156
$566$	68.2630	2.86931
$567$	0	0
$568$	15.5162	0.651045
$569$	23.9553	1.00426	0.502128	−	0.864793i	$-0.332550\pi$
$569$	23.9553	1.00426	0.502128	+	0.864793i	$0.332550\pi$
$570$	0	0
$571$	8.33187	0.348678	0.174339	−	0.984686i	$-0.444221\pi$
$571$	8.33187	0.348678	0.174339	+	0.984686i	$0.444221\pi$
$572$	−26.6716	−1.11520
$573$	0	0
$574$	−75.4479	−3.14913
$575$	−1.84216	−0.0768235
$576$	0	0
$577$	21.7197	0.904203	0.452101	−	0.891967i	$-0.350674\pi$
$577$	21.7197	0.904203	0.452101	+	0.891967i	$0.350674\pi$
$578$	21.3541	0.888214
$579$	0	0
$580$	−3.84869	−0.159808
$581$	28.6578	1.18893
$582$	0	0
$583$	1.44022	0.0596480
$584$	−93.2106	−3.85708
$585$	0	0
$586$	−25.4273	−1.05039
$587$	−38.3274	−1.58194	−0.790971	−	0.611853i	$-0.790424\pi$
$587$	−38.3274	−1.58194	−0.790971	+	0.611853i	$0.790424\pi$
$588$	0	0
$589$	6.37391	0.262632
$590$	−4.82594	−0.198681
$591$	0	0
$592$	6.86693	0.282229
$593$	−25.1482	−1.03271	−0.516357	−	0.856373i	$-0.672712\pi$
$593$	−25.1482	−1.03271	−0.516357	+	0.856373i	$0.672712\pi$
$594$	0	0
$595$	−3.18324	−0.130500
$596$	−33.2767	−1.36307
$597$	0	0
$598$	5.61622	0.229664
$599$	5.97251	0.244030	0.122015	−	0.992528i	$-0.461064\pi$
$599$	5.97251	0.244030	0.122015	+	0.992528i	$0.461064\pi$
$600$	0	0
$601$	−1.61426	−0.0658469	−0.0329234	−	0.999458i	$-0.510482\pi$
$601$	−1.61426	−0.0658469	−0.0329234	+	0.999458i	$0.510482\pi$
$602$	−24.0405	−0.979818
$603$	0	0
$604$	47.5806	1.93603
$605$	−0.244850	−0.00995455
$606$	0	0
$607$	−43.1541	−1.75157	−0.875786	−	0.482700i	$-0.839656\pi$
$607$	−43.1541	−1.75157	−0.875786	+	0.482700i	$0.839656\pi$
$608$	−6.04149	−0.245015
$609$	0	0
$610$	2.60696	0.105553
$611$	−10.1482	−0.410551
$612$	0	0
$613$	−15.2118	−0.614400	−0.307200	−	0.951645i	$-0.599392\pi$
$613$	−15.2118	−0.614400	−0.307200	+	0.951645i	$0.599392\pi$
$614$	−9.16402	−0.369830
$615$	0	0
$616$	−28.5021	−1.14838
$617$	10.1846	0.410016	0.205008	−	0.978760i	$-0.434278\pi$
$617$	10.1846	0.410016	0.205008	+	0.978760i	$0.434278\pi$
$618$	0	0
$619$	23.7636	0.955139	0.477569	−	0.878594i	$-0.341518\pi$
$619$	23.7636	0.955139	0.477569	+	0.878594i	$0.341518\pi$
$620$	−7.05880	−0.283488
$621$	0	0
$622$	−35.3653	−1.41802
$623$	21.9261	0.878452
$624$	0	0
$625$	24.1043	0.964173
$626$	39.3273	1.57184
$627$	0	0
$628$	−73.1229	−2.91792
$629$	2.72327	0.108584
$630$	0	0
$631$	36.1097	1.43751	0.718753	−	0.695266i	$-0.244713\pi$
$631$	36.1097	1.43751	0.718753	+	0.695266i	$0.244713\pi$
$632$	−10.7848	−0.428997
$633$	0	0
$634$	−15.2311	−0.604904
$635$	2.96192	0.117540
$636$	0	0
$637$	−74.0933	−2.93568
$638$	−8.87587	−0.351399
$639$	0	0
$640$	−2.57886	−0.101938
$641$	34.7607	1.37297	0.686483	−	0.727146i	$-0.259154\pi$
$641$	34.7607	1.37297	0.686483	+	0.727146i	$0.259154\pi$
$642$	0	0
$643$	−32.5548	−1.28384	−0.641919	−	0.766773i	$-0.721861\pi$
$643$	−32.5548	−1.28384	−0.641919	+	0.766773i	$0.721861\pi$
$644$	7.46037	0.293980
$645$	0	0
$646$	−7.50681	−0.295351
$647$	0.726588	0.0285651	0.0142826	−	0.999898i	$-0.495454\pi$
$647$	0.726588	0.0285651	0.0142826	+	0.999898i	$0.495454\pi$
$648$	0	0
$649$	−7.71718	−0.302926
$650$	−74.4009	−2.91824
$651$	0	0
$652$	61.8036	2.42041
$653$	2.07531	0.0812132	0.0406066	−	0.999175i	$-0.487071\pi$
$653$	2.07531	0.0812132	0.0406066	+	0.999175i	$0.487071\pi$
$654$	0	0
$655$	−1.95836	−0.0765195
$656$	49.4983	1.93259
$657$	0	0
$658$	−19.4413	−0.757900
$659$	6.83577	0.266284	0.133142	−	0.991097i	$-0.457493\pi$
$659$	6.83577	0.266284	0.133142	+	0.991097i	$0.457493\pi$
$660$	0	0
$661$	26.6223	1.03549	0.517744	−	0.855536i	$-0.326772\pi$
$661$	26.6223	1.03549	0.517744	+	0.855536i	$0.326772\pi$
$662$	14.2644	0.554400
$663$	0	0
$664$	−41.7489	−1.62017
$665$	−1.08302	−0.0419978
$666$	0	0
$667$	1.29594	0.0501790
$668$	−46.6139	−1.80354
$669$	0	0
$670$	−7.11512	−0.274881
$671$	4.16881	0.160935
$672$	0	0
$673$	26.9513	1.03890	0.519449	−	0.854502i	$-0.326137\pi$
$673$	26.9513	1.03890	0.519449	+	0.854502i	$0.326137\pi$
$674$	74.1853	2.85751
$675$	0	0
$676$	98.4810	3.78773
$677$	7.83263	0.301033	0.150516	−	0.988608i	$-0.451906\pi$
$677$	7.83263	0.301033	0.150516	+	0.988608i	$0.451906\pi$
$678$	0	0
$679$	36.4475	1.39873
$680$	4.63736	0.177835
$681$	0	0
$682$	−16.2791	−0.623357
$683$	−3.91204	−0.149690	−0.0748451	−	0.997195i	$-0.523846\pi$
$683$	−3.91204	−0.149690	−0.0748451	+	0.997195i	$0.523846\pi$
$684$	0	0
$685$	3.89779	0.148927
$686$	−62.8651	−2.40020
$687$	0	0
$688$	15.7720	0.601302
$689$	−8.49285	−0.323552
$690$	0	0
$691$	−34.8112	−1.32428	−0.662140	−	0.749380i	$-0.730352\pi$
$691$	−34.8112	−1.32428	−0.662140	+	0.749380i	$0.730352\pi$
$692$	114.557	4.35479
$693$	0	0
$694$	−15.9647	−0.606010
$695$	−3.60610	−0.136787
$696$	0	0
$697$	19.6299	0.743536
$698$	−19.9661	−0.755726
$699$	0	0
$700$	−98.8313	−3.73547
$701$	20.0313	0.756572	0.378286	−	0.925689i	$-0.376514\pi$
$701$	20.0313	0.756572	0.378286	+	0.925689i	$0.376514\pi$
$702$	0	0
$703$	0.926528	0.0349447
$704$	0.607107	0.0228812
$705$	0	0
$706$	25.7865	0.970488
$707$	−15.5346	−0.584240
$708$	0	0
$709$	20.5264	0.770886	0.385443	−	0.922732i	$-0.374049\pi$
$709$	20.5264	0.770886	0.385443	+	0.922732i	$0.374049\pi$
$710$	−1.50581	−0.0565119
$711$	0	0
$712$	−31.9421	−1.19708
$713$	2.37686	0.0890139
$714$	0	0
$715$	1.44385	0.0539971
$716$	−103.562	−3.87030
$717$	0	0
$718$	74.9005	2.79526
$719$	3.14609	0.117329	0.0586647	−	0.998278i	$-0.481316\pi$
$719$	3.14609	0.117329	0.0586647	+	0.998278i	$0.481316\pi$
$720$	0	0
$721$	−10.7570	−0.400611
$722$	−2.55401	−0.0950506
$723$	0	0
$724$	36.4550	1.35484
$725$	−17.1680	−0.637602
$726$	0	0
$727$	36.6121	1.35787	0.678934	−	0.734199i	$-0.262442\pi$
$727$	36.6121	1.35787	0.678934	+	0.734199i	$0.262442\pi$
$728$	168.074	6.22924
$729$	0	0
$730$	9.04584	0.334802
$731$	6.25482	0.231343
$732$	0	0
$733$	−1.44149	−0.0532428	−0.0266214	−	0.999646i	$-0.508475\pi$
$733$	−1.44149	−0.0532428	−0.0266214	+	0.999646i	$0.508475\pi$
$734$	35.2065	1.29949
$735$	0	0
$736$	−2.25289	−0.0830428
$737$	−11.3778	−0.419108
$738$	0	0
$739$	−27.8090	−1.02297	−0.511485	−	0.859292i	$-0.670904\pi$
$739$	−27.8090	−1.02297	−0.511485	+	0.859292i	$0.670904\pi$
$740$	−1.02609	−0.0377197
$741$	0	0
$742$	−16.2701	−0.597295
$743$	−1.15020	−0.0421967	−0.0210983	−	0.999777i	$-0.506716\pi$
$743$	−1.15020	−0.0421967	−0.0210983	+	0.999777i	$0.506716\pi$
$744$	0	0
$745$	1.80141	0.0659987
$746$	−41.9291	−1.53513
$747$	0	0
$748$	13.2941	0.486079
$749$	−75.1347	−2.74536
$750$	0	0
$751$	43.4326	1.58488	0.792439	−	0.609951i	$-0.208811\pi$
$751$	43.4326	1.58488	0.792439	+	0.609951i	$0.208811\pi$
$752$	12.7546	0.465114
$753$	0	0
$754$	52.3401	1.90611
$755$	−2.57575	−0.0937412
$756$	0	0
$757$	−22.5876	−0.820962	−0.410481	−	0.911869i	$-0.634639\pi$
$757$	−22.5876	−0.820962	−0.410481	+	0.911869i	$0.634639\pi$
$758$	−17.7129	−0.643362
$759$	0	0
$760$	1.57775	0.0572311
$761$	−23.0477	−0.835479	−0.417739	−	0.908567i	$-0.637177\pi$
$761$	−23.0477	−0.835479	−0.417739	+	0.908567i	$0.637177\pi$
$762$	0	0
$763$	16.9668	0.614239
$764$	77.8603	2.81689
$765$	0	0
$766$	3.75814	0.135787
$767$	45.5075	1.64318
$768$	0	0
$769$	−26.7350	−0.964088	−0.482044	−	0.876147i	$-0.660105\pi$
$769$	−26.7350	−0.964088	−0.482044	+	0.876147i	$0.660105\pi$
$770$	2.76605	0.0996816
$771$	0	0
$772$	−37.6126	−1.35371
$773$	−28.4074	−1.02174	−0.510872	−	0.859657i	$-0.670677\pi$
$773$	−28.4074	−1.02174	−0.510872	+	0.859657i	$0.670677\pi$
$774$	0	0
$775$	−31.4874	−1.13106
$776$	−53.0969	−1.90607
$777$	0	0
$778$	17.3573	0.622290
$779$	6.67861	0.239286
$780$	0	0
$781$	−2.40794	−0.0861630
$782$	−2.79932	−0.100103
$783$	0	0
$784$	93.1235	3.32584
$785$	3.95847	0.141284
$786$	0	0
$787$	−6.60374	−0.235398	−0.117699	−	0.993049i	$-0.537552\pi$
$787$	−6.60374	−0.235398	−0.117699	+	0.993049i	$0.537552\pi$
$788$	−19.8520	−0.707199
$789$	0	0
$790$	1.04664	0.0372377
$791$	47.7305	1.69710
$792$	0	0
$793$	−24.5831	−0.872970
$794$	31.6207	1.12218
$795$	0	0
$796$	89.9732	3.18901
$797$	−45.9061	−1.62608	−0.813038	−	0.582210i	$-0.802188\pi$
$797$	−45.9061	−1.62608	−0.813038	+	0.582210i	$0.802188\pi$
$798$	0	0
$799$	5.05820	0.178946
$800$	29.8452	1.05519
$801$	0	0
$802$	−0.0922899	−0.00325887
$803$	14.4653	0.510468
$804$	0	0
$805$	−0.403863	−0.0142343
$806$	95.9960	3.38131
$807$	0	0
$808$	22.6309	0.796154
$809$	19.0359	0.669266	0.334633	−	0.942349i	$-0.391388\pi$
$809$	19.0359	0.669266	0.334633	+	0.942349i	$0.391388\pi$
$810$	0	0
$811$	−56.0480	−1.96811	−0.984056	−	0.177860i	$-0.943083\pi$
$811$	−56.0480	−1.96811	−0.984056	+	0.177860i	$0.943083\pi$
$812$	69.5266	2.43990
$813$	0	0
$814$	−2.36637	−0.0829411
$815$	−3.34570	−0.117195
$816$	0	0
$817$	2.12805	0.0744512
$818$	25.4113	0.888485
$819$	0	0
$820$	−7.39625	−0.258288
$821$	−3.28817	−0.114758	−0.0573789	−	0.998352i	$-0.518274\pi$
$821$	−3.28817	−0.114758	−0.0573789	+	0.998352i	$0.518274\pi$
$822$	0	0
$823$	−5.84050	−0.203587	−0.101793	−	0.994806i	$-0.532458\pi$
$823$	−5.84050	−0.203587	−0.101793	+	0.994806i	$0.532458\pi$
$824$	15.6708	0.545919
$825$	0	0
$826$	87.1806	3.03340
$827$	14.4956	0.504061	0.252030	−	0.967719i	$-0.418902\pi$
$827$	14.4956	0.504061	0.252030	+	0.967719i	$0.418902\pi$
$828$	0	0
$829$	−52.7194	−1.83102	−0.915511	−	0.402293i	$-0.868213\pi$
$829$	−52.7194	−1.83102	−0.915511	+	0.402293i	$0.868213\pi$
$830$	4.05162	0.140634
$831$	0	0
$832$	−3.58005	−0.124116
$833$	36.9306	1.27957
$834$	0	0
$835$	2.52342	0.0873264
$836$	4.52299	0.156431
$837$	0	0
$838$	69.6122	2.40471
$839$	28.9506	0.999485	0.499742	−	0.866174i	$-0.333428\pi$
$839$	28.9506	0.999485	0.499742	+	0.866174i	$0.333428\pi$
$840$	0	0
$841$	−16.9226	−0.583536
$842$	−42.2286	−1.45529
$843$	0	0
$844$	94.1514	3.24082
$845$	−5.33122	−0.183399
$846$	0	0
$847$	4.42321	0.151983
$848$	10.6742	0.366553
$849$	0	0
$850$	37.0840	1.27197
$851$	0.345506	0.0118438
$852$	0	0
$853$	−17.2571	−0.590873	−0.295436	−	0.955362i	$-0.595465\pi$
$853$	−17.2571	−0.590873	−0.295436	+	0.955362i	$0.595465\pi$
$854$	−47.0948	−1.61155
$855$	0	0
$856$	109.457	3.74115
$857$	−43.2863	−1.47863	−0.739315	−	0.673359i	$-0.764851\pi$
$857$	−43.2863	−1.47863	−0.739315	+	0.673359i	$0.764851\pi$
$858$	0	0
$859$	−41.5981	−1.41931	−0.709654	−	0.704550i	$-0.751149\pi$
$859$	−41.5981	−1.41931	−0.709654	+	0.704550i	$0.751149\pi$
$860$	−2.35672	−0.0803635
$861$	0	0
$862$	86.7217	2.95375
$863$	31.0676	1.05755	0.528777	−	0.848761i	$-0.322651\pi$
$863$	31.0676	1.05755	0.528777	+	0.848761i	$0.322651\pi$
$864$	0	0
$865$	−6.20146	−0.210856
$866$	9.06967	0.308200
$867$	0	0
$868$	127.517	4.32822
$869$	1.67368	0.0567758
$870$	0	0
$871$	67.0939	2.27339
$872$	−24.7173	−0.837035
$873$	0	0
$874$	−0.952402	−0.0322155
$875$	10.7653	0.363933
$876$	0	0
$877$	0.708001	0.0239075	0.0119537	−	0.999929i	$-0.496195\pi$
$877$	0.708001	0.0239075	0.0119537	+	0.999929i	$0.496195\pi$
$878$	−16.0451	−0.541497
$879$	0	0
$880$	−1.81470	−0.0611734
$881$	35.1566	1.18446	0.592229	−	0.805770i	$-0.298248\pi$
$881$	35.1566	1.18446	0.592229	+	0.805770i	$0.298248\pi$
$882$	0	0
$883$	−1.50768	−0.0507374	−0.0253687	−	0.999678i	$-0.508076\pi$
$883$	−1.50768	−0.0507374	−0.0253687	+	0.999678i	$0.508076\pi$
$884$	−78.3937	−2.63667
$885$	0	0
$886$	67.4180	2.26495
$887$	−46.0348	−1.54570	−0.772849	−	0.634590i	$-0.781169\pi$
$887$	−46.0348	−1.54570	−0.772849	+	0.634590i	$0.781169\pi$
$888$	0	0
$889$	−53.5070	−1.79457
$890$	3.09990	0.103909
$891$	0	0
$892$	51.9993	1.74107
$893$	1.72093	0.0575888
$894$	0	0
$895$	5.60628	0.187397
$896$	46.5871	1.55636
$897$	0	0
$898$	−1.65786	−0.0553236
$899$	22.1510	0.738777
$900$	0	0
$901$	4.23313	0.141026
$902$	−17.0573	−0.567945
$903$	0	0
$904$	−69.5340	−2.31267
$905$	−1.97347	−0.0656005
$906$	0	0
$907$	0.994096	0.0330084	0.0165042	−	0.999864i	$-0.494746\pi$
$907$	0.994096	0.0330084	0.0165042	+	0.999864i	$0.494746\pi$
$908$	79.2315	2.62939
$909$	0	0
$910$	−16.3111	−0.540709
$911$	6.86962	0.227601	0.113800	−	0.993504i	$-0.463698\pi$
$911$	6.86962	0.227601	0.113800	+	0.993504i	$0.463698\pi$
$912$	0	0
$913$	6.47897	0.214423
$914$	12.1257	0.401081
$915$	0	0
$916$	−30.0440	−0.992682
$917$	35.3778	1.16828
$918$	0	0
$919$	12.5820	0.415043	0.207521	−	0.978230i	$-0.433460\pi$
$919$	12.5820	0.415043	0.207521	+	0.978230i	$0.433460\pi$
$920$	0.588350	0.0193973
$921$	0	0
$922$	35.6825	1.17514
$923$	14.1994	0.467379
$924$	0	0
$925$	−4.57709	−0.150494
$926$	37.7538	1.24067
$927$	0	0
$928$	−20.9957	−0.689219
$929$	−43.3197	−1.42127	−0.710636	−	0.703560i	$-0.751593\pi$
$929$	−43.3197	−1.42127	−0.710636	+	0.703560i	$0.751593\pi$
$930$	0	0
$931$	12.5648	0.411794
$932$	56.6836	1.85673
$933$	0	0
$934$	−57.5787	−1.88403
$935$	−0.719667	−0.0235356
$936$	0	0
$937$	−44.3170	−1.44777	−0.723887	−	0.689918i	$-0.757646\pi$
$937$	−44.3170	−1.44777	−0.723887	+	0.689918i	$0.757646\pi$
$938$	128.535	4.19681
$939$	0	0
$940$	−1.90585	−0.0621620
$941$	−58.0333	−1.89183	−0.945916	−	0.324410i	$-0.894834\pi$
$941$	−58.0333	−1.89183	−0.945916	+	0.324410i	$0.894834\pi$
$942$	0	0
$943$	2.49048	0.0811012
$944$	−57.1957	−1.86156
$945$	0	0
$946$	−5.43508	−0.176710
$947$	−27.2005	−0.883897	−0.441949	−	0.897040i	$-0.645713\pi$
$947$	−27.2005	−0.883897	−0.441949	+	0.897040i	$0.645713\pi$
$948$	0	0
$949$	−85.3002	−2.76896
$950$	12.6170	0.409348
$951$	0	0
$952$	−83.7739	−2.71513
$953$	7.27158	0.235550	0.117775	−	0.993040i	$-0.462424\pi$
$953$	7.27158	0.235550	0.117775	+	0.993040i	$0.462424\pi$
$954$	0	0
$955$	−4.21493	−0.136392
$956$	123.679	4.00007
$957$	0	0
$958$	−55.6646	−1.79844
$959$	−70.4135	−2.27377
$960$	0	0
$961$	9.62671	0.310539
$962$	13.9542	0.449902
$963$	0	0
$964$	106.599	3.43331
$965$	2.03614	0.0655457
$966$	0	0
$967$	8.85046	0.284612	0.142306	−	0.989823i	$-0.454548\pi$
$967$	8.85046	0.284612	0.142306	+	0.989823i	$0.454548\pi$
$968$	−6.44376	−0.207110
$969$	0	0
$970$	5.15292	0.165450
$971$	37.1779	1.19310	0.596548	−	0.802577i	$-0.296538\pi$
$971$	37.1779	1.19310	0.596548	+	0.802577i	$0.296538\pi$
$972$	0	0
$973$	65.1442	2.08843
$974$	13.8062	0.442380
$975$	0	0
$976$	30.8970	0.988989
$977$	−46.5181	−1.48825	−0.744123	−	0.668043i	$-0.767132\pi$
$977$	−46.5181	−1.48825	−0.744123	+	0.668043i	$0.767132\pi$
$978$	0	0
$979$	4.95706	0.158428
$980$	−13.9149	−0.444495
$981$	0	0
$982$	−105.889	−3.37904
$983$	−24.6218	−0.785313	−0.392656	−	0.919685i	$-0.628444\pi$
$983$	−24.6218	−0.785313	−0.392656	+	0.919685i	$0.628444\pi$
$984$	0	0
$985$	1.07468	0.0342421
$986$	−26.0881	−0.830815
$987$	0	0
$988$	−26.6716	−0.848538
$989$	0.793560	0.0252337
$990$	0	0
$991$	40.2121	1.27738	0.638690	−	0.769464i	$-0.279477\pi$
$991$	40.2121	1.27738	0.638690	+	0.769464i	$0.279477\pi$
$992$	−38.5079	−1.22263
$993$	0	0
$994$	27.2024	0.862807
$995$	−4.87065	−0.154410
$996$	0	0
$997$	21.2470	0.672900	0.336450	−	0.941701i	$-0.390774\pi$
$997$	21.2470	0.672900	0.336450	+	0.941701i	$0.390774\pi$
$998$	−46.1245	−1.46004
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.p.1.1		7
3.2	odd	2		209.2.a.d.1.7	✓	7
12.11	even	2		3344.2.a.ba.1.7		7
15.14	odd	2		5225.2.a.n.1.1		7
33.32	even	2		2299.2.a.q.1.1		7
57.56	even	2		3971.2.a.i.1.1		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.d.1.7	✓	7	3.2	odd	2
1881.2.a.p.1.1		7	1.1	even	1	trivial
2299.2.a.q.1.1		7	33.32	even	2
3344.2.a.ba.1.7		7	12.11	even	2
3971.2.a.i.1.1		7	57.56	even	2
5225.2.a.n.1.1		7	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1881.a (trivial)

\(f(q)\)	\(=\)	\(q-2.55401 q^{2} +4.52299 q^{4} -0.244850 q^{5} +4.42321 q^{7} -6.44376 q^{8} +0.625350 q^{10} +1.00000 q^{11} -5.89690 q^{13} -11.2969 q^{14} +7.41147 q^{16} +2.93922 q^{17} +1.00000 q^{19} -1.10745 q^{20} -2.55401 q^{22} +0.372904 q^{23} -4.94005 q^{25} +15.0608 q^{26} +20.0061 q^{28} +3.47526 q^{29} +6.37391 q^{31} -6.04149 q^{32} -7.50681 q^{34} -1.08302 q^{35} +0.926528 q^{37} -2.55401 q^{38} +1.57775 q^{40} +6.67861 q^{41} +2.12805 q^{43} +4.52299 q^{44} -0.952402 q^{46} +1.72093 q^{47} +12.5648 q^{49} +12.6170 q^{50} -26.6716 q^{52} +1.44022 q^{53} -0.244850 q^{55} -28.5021 q^{56} -8.87587 q^{58} -7.71718 q^{59} +4.16881 q^{61} -16.2791 q^{62} +0.607107 q^{64} +1.44385 q^{65} -11.3778 q^{67} +13.2941 q^{68} +2.76605 q^{70} -2.40794 q^{71} +14.4653 q^{73} -2.36637 q^{74} +4.52299 q^{76} +4.42321 q^{77} +1.67368 q^{79} -1.81470 q^{80} -17.0573 q^{82} +6.47897 q^{83} -0.719667 q^{85} -5.43508 q^{86} -6.44376 q^{88} +4.95706 q^{89} -26.0832 q^{91} +1.68664 q^{92} -4.39529 q^{94} -0.244850 q^{95} +8.24006 q^{97} -32.0906 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 15 q^{4} - 2 q^{5} + 10 q^{7} + 9 q^{8} - 6 q^{10} + 7 q^{11} - 4 q^{13} - 6 q^{14} + 27 q^{16} - 2 q^{17} + 7 q^{19} + 4 q^{20} + q^{22} - 10 q^{23} + 9 q^{25} + 8 q^{26} + 26 q^{28} + 18 q^{29}+ \cdots - 19 q^{98}+O(q^{100}) \) 7 * q + q^2 + 15 * q^4 - 2 * q^5 + 10 * q^7 + 9 * q^8 - 6 * q^10 + 7 * q^11 - 4 * q^13 - 6 * q^14 + 27 * q^16 - 2 * q^17 + 7 * q^19 + 4 * q^20 + q^22 - 10 * q^23 + 9 * q^25 + 8 * q^26 + 26 * q^28 + 18 * q^29 + 24 * q^31 + 49 * q^32 - 6 * q^34 - 8 * q^35 + q^38 - 2 * q^40 + 12 * q^41 + 2 * q^43 + 15 * q^44 - 4 * q^46 - 8 * q^47 + 17 * q^49 + 33 * q^50 - 60 * q^52 - 2 * q^53 - 2 * q^55 - 26 * q^56 - 8 * q^58 + 10 * q^59 + 14 * q^61 - 14 * q^62 + 55 * q^64 + 14 * q^65 + 8 * q^67 + 18 * q^68 - 66 * q^70 - 10 * q^71 - 6 * q^73 - 26 * q^74 + 15 * q^76 + 10 * q^77 + 52 * q^79 + 12 * q^80 + 24 * q^82 + 10 * q^83 - 12 * q^85 - 8 * q^86 + 9 * q^88 + 12 * q^91 + 24 * q^94 - 2 * q^95 - 24 * q^97 - 19 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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