LMFDB - Embedded newform 1881.2.a.p.1.4

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			1.4
Root			$0.456669$ 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)

$f(q)$	$=$	\(q+0.456669 q^{2} -1.79145 q^{4} -0.221953 q^{5} +4.69915 q^{7} -1.73144 q^{8} -0.101359 q^{10} +1.00000 q^{11} +5.89016 q^{13} +2.14596 q^{14} +2.79221 q^{16} -7.06513 q^{17} +1.00000 q^{19} +0.397618 q^{20} +0.456669 q^{22} -1.06348 q^{23} -4.95074 q^{25} +2.68985 q^{26} -8.41832 q^{28} +7.62662 q^{29} +0.901295 q^{31} +4.73799 q^{32} -3.22642 q^{34} -1.04299 q^{35} -2.71758 q^{37} +0.456669 q^{38} +0.384298 q^{40} -0.788714 q^{41} +0.714571 q^{43} -1.79145 q^{44} -0.485656 q^{46} -3.96368 q^{47} +15.0820 q^{49} -2.26085 q^{50} -10.5519 q^{52} +9.69714 q^{53} -0.221953 q^{55} -8.13629 q^{56} +3.48284 q^{58} +7.33476 q^{59} +8.15179 q^{61} +0.411593 q^{62} -3.42073 q^{64} -1.30734 q^{65} +7.86697 q^{67} +12.6569 q^{68} -0.476301 q^{70} +3.13400 q^{71} -6.49076 q^{73} -1.24103 q^{74} -1.79145 q^{76} +4.69915 q^{77} +12.0148 q^{79} -0.619739 q^{80} -0.360181 q^{82} -16.3902 q^{83} +1.56812 q^{85} +0.326322 q^{86} -1.73144 q^{88} +12.9487 q^{89} +27.6788 q^{91} +1.90517 q^{92} -1.81009 q^{94} -0.221953 q^{95} +8.14414 q^{97} +6.88750 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.456669	0.322914	0.161457	−	0.986880i	$-0.448381\pi$
$2$	0.456669	0.322914	0.161457	+	0.986880i	$0.448381\pi$
$3$	0	0
$3$	0	0
$4$	−1.79145	−0.895727
$5$	−0.221953	−0.0992603	−0.0496301	−	0.998768i	$-0.515804\pi$
$5$	−0.221953	−0.0992603	−0.0496301	+	0.998768i	$0.515804\pi$
$6$	0	0
$7$	4.69915	1.77611	0.888057	−	0.459734i	$-0.152055\pi$
$7$	4.69915	1.77611	0.888057	+	0.459734i	$0.152055\pi$
$8$	−1.73144	−0.612156
$9$	0	0
$10$	−0.101359	−0.0320525
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.89016	1.63364	0.816818	−	0.576895i	$-0.195736\pi$
$13$	5.89016	1.63364	0.816818	+	0.576895i	$0.195736\pi$
$14$	2.14596	0.573531
$15$	0	0
$16$	2.79221	0.698053
$17$	−7.06513	−1.71355	−0.856773	−	0.515694i	$-0.827534\pi$
$17$	−7.06513	−1.71355	−0.856773	+	0.515694i	$0.827534\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0.397618	0.0889101
$21$	0	0
$22$	0.456669	0.0973621
$23$	−1.06348	−0.221750	−0.110875	−	0.993834i	$-0.535365\pi$
$23$	−1.06348	−0.221750	−0.110875	+	0.993834i	$0.535365\pi$
$24$	0	0
$25$	−4.95074	−0.990147
$26$	2.68985	0.527523
$27$	0	0
$28$	−8.41832	−1.59091
$29$	7.62662	1.41623	0.708114	−	0.706098i	$-0.249546\pi$
$29$	7.62662	1.41623	0.708114	+	0.706098i	$0.249546\pi$
$30$	0	0
$31$	0.901295	0.161877	0.0809387	−	0.996719i	$-0.474208\pi$
$31$	0.901295	0.161877	0.0809387	+	0.996719i	$0.474208\pi$
$32$	4.73799	0.837567
$33$	0	0
$34$	−3.22642	−0.553327
$35$	−1.04299	−0.176297
$36$	0	0
$37$	−2.71758	−0.446768	−0.223384	−	0.974731i	$-0.571710\pi$
$37$	−2.71758	−0.446768	−0.223384	+	0.974731i	$0.571710\pi$
$38$	0.456669	0.0740815
$39$	0	0
$40$	0.384298	0.0607628
$41$	−0.788714	−0.123176	−0.0615882	−	0.998102i	$-0.519617\pi$
$41$	−0.788714	−0.123176	−0.0615882	+	0.998102i	$0.519617\pi$
$42$	0	0
$43$	0.714571	0.108971	0.0544855	−	0.998515i	$-0.482648\pi$
$43$	0.714571	0.108971	0.0544855	+	0.998515i	$0.482648\pi$
$44$	−1.79145	−0.270072
$45$	0	0
$46$	−0.485656	−0.0716061
$47$	−3.96368	−0.578163	−0.289081	−	0.957305i	$-0.593350\pi$
$47$	−3.96368	−0.578163	−0.289081	+	0.957305i	$0.593350\pi$
$48$	0	0
$49$	15.0820	2.15458
$50$	−2.26085	−0.319732
$51$	0	0
$52$	−10.5519	−1.46329
$53$	9.69714	1.33200	0.666002	−	0.745950i	$-0.268004\pi$
$53$	9.69714	1.33200	0.666002	+	0.745950i	$0.268004\pi$
$54$	0	0
$55$	−0.221953	−0.0299281
$56$	−8.13629	−1.08726
$57$	0	0
$58$	3.48284	0.457319
$59$	7.33476	0.954904	0.477452	−	0.878658i	$-0.341560\pi$
$59$	7.33476	0.954904	0.477452	+	0.878658i	$0.341560\pi$
$60$	0	0
$61$	8.15179	1.04373	0.521865	−	0.853028i	$-0.325236\pi$
$61$	8.15179	1.04373	0.521865	+	0.853028i	$0.325236\pi$
$62$	0.411593	0.0522724
$63$	0	0
$64$	−3.42073	−0.427592
$65$	−1.30734	−0.162155
$66$	0	0
$67$	7.86697	0.961104	0.480552	−	0.876966i	$-0.340436\pi$
$67$	7.86697	0.961104	0.480552	+	0.876966i	$0.340436\pi$
$68$	12.6569	1.53487
$69$	0	0
$70$	−0.476301	−0.0569289
$71$	3.13400	0.371937	0.185969	−	0.982556i	$-0.440458\pi$
$71$	3.13400	0.371937	0.185969	+	0.982556i	$0.440458\pi$
$72$	0	0
$73$	−6.49076	−0.759687	−0.379843	−	0.925051i	$-0.624022\pi$
$73$	−6.49076	−0.759687	−0.379843	+	0.925051i	$0.624022\pi$
$74$	−1.24103	−0.144267
$75$	0	0
$76$	−1.79145	−0.205494
$77$	4.69915	0.535518
$78$	0	0
$79$	12.0148	1.35177	0.675884	−	0.737008i	$-0.263762\pi$
$79$	12.0148	1.35177	0.675884	+	0.737008i	$0.263762\pi$
$80$	−0.619739	−0.0692890
$81$	0	0
$82$	−0.360181	−0.0397753
$83$	−16.3902	−1.79906	−0.899528	−	0.436863i	$-0.856089\pi$
$83$	−16.3902	−1.79906	−0.899528	+	0.436863i	$0.856089\pi$
$84$	0	0
$85$	1.56812	0.170087
$86$	0.326322	0.0351882
$87$	0	0
$88$	−1.73144	−0.184572
$89$	12.9487	1.37256	0.686281	−	0.727336i	$-0.259242\pi$
$89$	12.9487	1.37256	0.686281	+	0.727336i	$0.259242\pi$
$90$	0	0
$91$	27.6788	2.90152
$92$	1.90517	0.198628
$93$	0	0
$94$	−1.81009	−0.186697
$95$	−0.221953	−0.0227719
$96$	0	0
$97$	8.14414	0.826912	0.413456	−	0.910524i	$-0.364321\pi$
$97$	8.14414	0.826912	0.413456	+	0.910524i	$0.364321\pi$
$98$	6.88750	0.695742
$99$	0	0
$100$	8.86902	0.886902
$101$	15.3713	1.52950	0.764750	−	0.644327i	$-0.222862\pi$
$101$	15.3713	1.52950	0.764750	+	0.644327i	$0.222862\pi$
$102$	0	0
$103$	−8.83682	−0.870718	−0.435359	−	0.900257i	$-0.643378\pi$
$103$	−8.83682	−0.870718	−0.435359	+	0.900257i	$0.643378\pi$
$104$	−10.1984	−1.00004
$105$	0	0
$106$	4.42838	0.430122
$107$	9.04360	0.874277	0.437139	−	0.899394i	$-0.355992\pi$
$107$	9.04360	0.874277	0.437139	+	0.899394i	$0.355992\pi$
$108$	0	0
$109$	−18.5950	−1.78107	−0.890537	−	0.454911i	$-0.849671\pi$
$109$	−18.5950	−1.78107	−0.890537	+	0.454911i	$0.849671\pi$
$110$	−0.101359	−0.00966419
$111$	0	0
$112$	13.1210	1.23982
$113$	4.42423	0.416196	0.208098	−	0.978108i	$-0.433273\pi$
$113$	4.42423	0.416196	0.208098	+	0.978108i	$0.433273\pi$
$114$	0	0
$115$	0.236042	0.0220110
$116$	−13.6627	−1.26855
$117$	0	0
$118$	3.34955	0.308352
$119$	−33.2001	−3.04345
$120$	0	0
$121$	1.00000	0.0909091
$122$	3.72267	0.337035
$123$	0	0
$124$	−1.61463	−0.144998
$125$	2.20859	0.197543
$126$	0	0
$127$	−0.338657	−0.0300510	−0.0150255	−	0.999887i	$-0.504783\pi$
$127$	−0.338657	−0.0300510	−0.0150255	+	0.999887i	$0.504783\pi$
$128$	−11.0381	−0.975642
$129$	0	0
$130$	−0.597020	−0.0523621
$131$	4.55211	0.397720	0.198860	−	0.980028i	$-0.436276\pi$
$131$	4.55211	0.397720	0.198860	+	0.980028i	$0.436276\pi$
$132$	0	0
$133$	4.69915	0.407468
$134$	3.59260	0.310353
$135$	0	0
$136$	12.2328	1.04896
$137$	0.654263	0.0558974	0.0279487	−	0.999609i	$-0.491102\pi$
$137$	0.654263	0.0558974	0.0279487	+	0.999609i	$0.491102\pi$
$138$	0	0
$139$	11.0479	0.937072	0.468536	−	0.883444i	$-0.344782\pi$
$139$	11.0479	0.937072	0.468536	+	0.883444i	$0.344782\pi$
$140$	1.86847	0.157914
$141$	0	0
$142$	1.43120	0.120104
$143$	5.89016	0.492560
$144$	0	0
$145$	−1.69275	−0.140575
$146$	−2.96413	−0.245313
$147$	0	0
$148$	4.86842	0.400182
$149$	7.57743	0.620767	0.310384	−	0.950611i	$-0.399543\pi$
$149$	7.57743	0.620767	0.310384	+	0.950611i	$0.399543\pi$
$150$	0	0
$151$	−6.95296	−0.565824	−0.282912	−	0.959146i	$-0.591300\pi$
$151$	−6.95296	−0.565824	−0.282912	+	0.959146i	$0.591300\pi$
$152$	−1.73144	−0.140438
$153$	0	0
$154$	2.14596	0.172926
$155$	−0.200045	−0.0160680
$156$	0	0
$157$	−14.7427	−1.17659	−0.588297	−	0.808645i	$-0.700201\pi$
$157$	−14.7427	−1.17659	−0.588297	+	0.808645i	$0.700201\pi$
$158$	5.48678	0.436504
$159$	0	0
$160$	−1.05161	−0.0831371
$161$	−4.99744	−0.393853
$162$	0	0
$163$	−4.94015	−0.386942	−0.193471	−	0.981106i	$-0.561975\pi$
$163$	−4.94015	−0.386942	−0.193471	+	0.981106i	$0.561975\pi$
$164$	1.41294	0.110332
$165$	0	0
$166$	−7.48488	−0.580940
$167$	−22.6032	−1.74909	−0.874544	−	0.484946i	$-0.838839\pi$
$167$	−22.6032	−1.74909	−0.874544	+	0.484946i	$0.838839\pi$
$168$	0	0
$169$	21.6940	1.66877
$170$	0.716114	0.0549234
$171$	0	0
$172$	−1.28012	−0.0976083
$173$	8.18620	0.622385	0.311193	−	0.950347i	$-0.399272\pi$
$173$	8.18620	0.622385	0.311193	+	0.950347i	$0.399272\pi$
$174$	0	0
$175$	−23.2643	−1.75861
$176$	2.79221	0.210471
$177$	0	0
$178$	5.91328	0.443219
$179$	14.0830	1.05261	0.526305	−	0.850296i	$-0.323577\pi$
$179$	14.0830	1.05261	0.526305	+	0.850296i	$0.323577\pi$
$180$	0	0
$181$	−14.3260	−1.06484	−0.532422	−	0.846479i	$-0.678718\pi$
$181$	−14.3260	−1.06484	−0.532422	+	0.846479i	$0.678718\pi$
$182$	12.6400	0.936941
$183$	0	0
$184$	1.84134	0.135746
$185$	0.603175	0.0443463
$186$	0	0
$187$	−7.06513	−0.516653
$188$	7.10076	0.517876
$189$	0	0
$190$	−0.101359	−0.00735335
$191$	−0.394967	−0.0285788	−0.0142894	−	0.999898i	$-0.504549\pi$
$191$	−0.394967	−0.0285788	−0.0142894	+	0.999898i	$0.504549\pi$
$192$	0	0
$193$	9.03423	0.650298	0.325149	−	0.945663i	$-0.394586\pi$
$193$	9.03423	0.650298	0.325149	+	0.945663i	$0.394586\pi$
$194$	3.71918	0.267021
$195$	0	0
$196$	−27.0188	−1.92991
$197$	−7.85313	−0.559512	−0.279756	−	0.960071i	$-0.590254\pi$
$197$	−7.85313	−0.559512	−0.279756	+	0.960071i	$0.590254\pi$
$198$	0	0
$199$	7.81540	0.554019	0.277009	−	0.960867i	$-0.410657\pi$
$199$	7.81540	0.554019	0.277009	+	0.960867i	$0.410657\pi$
$200$	8.57190	0.606125
$201$	0	0
$202$	7.01959	0.493897
$203$	35.8386	2.51538
$204$	0	0
$205$	0.175057	0.0122265
$206$	−4.03550	−0.281167
$207$	0	0
$208$	16.4466	1.14037
$209$	1.00000	0.0691714
$210$	0	0
$211$	−21.5955	−1.48670	−0.743348	−	0.668905i	$-0.766763\pi$
$211$	−21.5955	−1.48670	−0.743348	+	0.668905i	$0.766763\pi$
$212$	−17.3720	−1.19311
$213$	0	0
$214$	4.12993	0.282316
$215$	−0.158601	−0.0108165
$216$	0	0
$217$	4.23533	0.287513
$218$	−8.49174	−0.575133
$219$	0	0
$220$	0.397618	0.0268074
$221$	−41.6147	−2.79931
$222$	0	0
$223$	−6.95854	−0.465979	−0.232989	−	0.972479i	$-0.574851\pi$
$223$	−6.95854	−0.465979	−0.232989	+	0.972479i	$0.574851\pi$
$224$	22.2646	1.48761
$225$	0	0
$226$	2.02041	0.134395
$227$	−7.67819	−0.509619	−0.254810	−	0.966991i	$-0.582013\pi$
$227$	−7.67819	−0.509619	−0.254810	+	0.966991i	$0.582013\pi$
$228$	0	0
$229$	5.83925	0.385869	0.192934	−	0.981212i	$-0.438200\pi$
$229$	5.83925	0.385869	0.192934	+	0.981212i	$0.438200\pi$
$230$	0.107793	0.00710765
$231$	0	0
$232$	−13.2050	−0.866952
$233$	21.8431	1.43099	0.715494	−	0.698619i	$-0.246202\pi$
$233$	21.8431	1.43099	0.715494	+	0.698619i	$0.246202\pi$
$234$	0	0
$235$	0.879751	0.0573886
$236$	−13.1399	−0.855333
$237$	0	0
$238$	−15.1615	−0.982772
$239$	−4.21004	−0.272325	−0.136162	−	0.990687i	$-0.543477\pi$
$239$	−4.21004	−0.272325	−0.136162	+	0.990687i	$0.543477\pi$
$240$	0	0
$241$	−17.4276	−1.12261	−0.561304	−	0.827610i	$-0.689700\pi$
$241$	−17.4276	−1.12261	−0.561304	+	0.827610i	$0.689700\pi$
$242$	0.456669	0.0293558
$243$	0	0
$244$	−14.6036	−0.934897
$245$	−3.34750	−0.213864
$246$	0	0
$247$	5.89016	0.374782
$248$	−1.56054	−0.0990942
$249$	0	0
$250$	1.00860	0.0637892
$251$	−2.14594	−0.135451	−0.0677254	−	0.997704i	$-0.521574\pi$
$251$	−2.14594	−0.135451	−0.0677254	+	0.997704i	$0.521574\pi$
$252$	0	0
$253$	−1.06348	−0.0668602
$254$	−0.154654	−0.00970387
$255$	0	0
$256$	1.80070	0.112544
$257$	5.23899	0.326799	0.163399	−	0.986560i	$-0.447754\pi$
$257$	5.23899	0.326799	0.163399	+	0.986560i	$0.447754\pi$
$258$	0	0
$259$	−12.7703	−0.793510
$260$	2.34203	0.145247
$261$	0	0
$262$	2.07881	0.128429
$263$	−12.2597	−0.755967	−0.377983	−	0.925812i	$-0.623382\pi$
$263$	−12.2597	−0.755967	−0.377983	+	0.925812i	$0.623382\pi$
$264$	0	0
$265$	−2.15231	−0.132215
$266$	2.14596	0.131577
$267$	0	0
$268$	−14.0933	−0.860886
$269$	9.37273	0.571465	0.285733	−	0.958309i	$-0.407763\pi$
$269$	9.37273	0.571465	0.285733	+	0.958309i	$0.407763\pi$
$270$	0	0
$271$	−13.3131	−0.808716	−0.404358	−	0.914601i	$-0.632505\pi$
$271$	−13.3131	−0.808716	−0.404358	+	0.914601i	$0.632505\pi$
$272$	−19.7273	−1.19615
$273$	0	0
$274$	0.298781	0.0180500
$275$	−4.95074	−0.298541
$276$	0	0
$277$	−14.5641	−0.875073	−0.437537	−	0.899201i	$-0.644149\pi$
$277$	−14.5641	−0.875073	−0.437537	+	0.899201i	$0.644149\pi$
$278$	5.04524	0.302593
$279$	0	0
$280$	1.80587	0.107922
$281$	−20.2411	−1.20748	−0.603741	−	0.797180i	$-0.706324\pi$
$281$	−20.2411	−1.20748	−0.603741	+	0.797180i	$0.706324\pi$
$282$	0	0
$283$	14.6230	0.869248	0.434624	−	0.900612i	$-0.356881\pi$
$283$	14.6230	0.869248	0.434624	+	0.900612i	$0.356881\pi$
$284$	−5.61442	−0.333154
$285$	0	0
$286$	2.68985	0.159054
$287$	−3.70629	−0.218775
$288$	0	0
$289$	32.9161	1.93624
$290$	−0.773026	−0.0453936
$291$	0	0
$292$	11.6279	0.680472
$293$	−27.6524	−1.61547	−0.807737	−	0.589543i	$-0.799308\pi$
$293$	−27.6524	−1.61547	−0.807737	+	0.589543i	$0.799308\pi$
$294$	0	0
$295$	−1.62797	−0.0947841
$296$	4.70533	0.273491
$297$	0	0
$298$	3.46037	0.200454
$299$	−6.26404	−0.362259
$300$	0	0
$301$	3.35788	0.193545
$302$	−3.17520	−0.182712
$303$	0	0
$304$	2.79221	0.160144
$305$	−1.80931	−0.103601
$306$	0	0
$307$	−0.756065	−0.0431509	−0.0215755	−	0.999767i	$-0.506868\pi$
$307$	−0.756065	−0.0431509	−0.0215755	+	0.999767i	$0.506868\pi$
$308$	−8.41832	−0.479678
$309$	0	0
$310$	−0.0913543	−0.00518858
$311$	−1.13352	−0.0642761	−0.0321381	−	0.999483i	$-0.510232\pi$
$311$	−1.13352	−0.0642761	−0.0321381	+	0.999483i	$0.510232\pi$
$312$	0	0
$313$	−10.9012	−0.616172	−0.308086	−	0.951359i	$-0.599688\pi$
$313$	−10.9012	−0.616172	−0.308086	+	0.951359i	$0.599688\pi$
$314$	−6.73252	−0.379938
$315$	0	0
$316$	−21.5239	−1.21082
$317$	−8.32326	−0.467481	−0.233741	−	0.972299i	$-0.575097\pi$
$317$	−8.32326	−0.467481	−0.233741	+	0.972299i	$0.575097\pi$
$318$	0	0
$319$	7.62662	0.427009
$320$	0.759241	0.0424429
$321$	0	0
$322$	−2.28217	−0.127181
$323$	−7.06513	−0.393114
$324$	0	0
$325$	−29.1606	−1.61754
$326$	−2.25601	−0.124949
$327$	0	0
$328$	1.36561	0.0754031
$329$	−18.6260	−1.02688
$330$	0	0
$331$	−2.47466	−0.136020	−0.0680098	−	0.997685i	$-0.521665\pi$
$331$	−2.47466	−0.136020	−0.0680098	+	0.997685i	$0.521665\pi$
$332$	29.3622	1.61146
$333$	0	0
$334$	−10.3222	−0.564804
$335$	−1.74610	−0.0953994
$336$	0	0
$337$	−30.3242	−1.65187	−0.825933	−	0.563768i	$-0.809351\pi$
$337$	−30.3242	−1.65187	−0.825933	+	0.563768i	$0.809351\pi$
$338$	9.90696	0.538867
$339$	0	0
$340$	−2.80922	−0.152352
$341$	0.901295	0.0488079
$342$	0	0
$343$	37.9788	2.05066
$344$	−1.23724	−0.0667073
$345$	0	0
$346$	3.73838	0.200977
$347$	10.4332	0.560081	0.280041	−	0.959988i	$-0.409652\pi$
$347$	10.4332	0.560081	0.280041	+	0.959988i	$0.409652\pi$
$348$	0	0
$349$	4.89049	0.261782	0.130891	−	0.991397i	$-0.458216\pi$
$349$	4.89049	0.261782	0.130891	+	0.991397i	$0.458216\pi$
$350$	−10.6241	−0.567880
$351$	0	0
$352$	4.73799	0.252536
$353$	−34.1254	−1.81631	−0.908155	−	0.418634i	$-0.862509\pi$
$353$	−34.1254	−1.81631	−0.908155	+	0.418634i	$0.862509\pi$
$354$	0	0
$355$	−0.695600	−0.0369186
$356$	−23.1970	−1.22944
$357$	0	0
$358$	6.43125	0.339902
$359$	−10.5366	−0.556103	−0.278051	−	0.960566i	$-0.589689\pi$
$359$	−10.5366	−0.556103	−0.278051	+	0.960566i	$0.589689\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−6.54224	−0.343853
$363$	0	0
$364$	−49.5852	−2.59897
$365$	1.44064	0.0754067
$366$	0	0
$367$	31.1277	1.62485	0.812427	−	0.583062i	$-0.198146\pi$
$367$	31.1277	1.62485	0.812427	+	0.583062i	$0.198146\pi$
$368$	−2.96945	−0.154793
$369$	0	0
$370$	0.275451	0.0143200
$371$	45.5684	2.36579
$372$	0	0
$373$	28.9881	1.50095	0.750473	−	0.660901i	$-0.229826\pi$
$373$	28.9881	1.50095	0.750473	+	0.660901i	$0.229826\pi$
$374$	−3.22642	−0.166834
$375$	0	0
$376$	6.86288	0.353926
$377$	44.9220	2.31360
$378$	0	0
$379$	−26.4204	−1.35712	−0.678561	−	0.734544i	$-0.737396\pi$
$379$	−26.4204	−1.35712	−0.678561	+	0.734544i	$0.737396\pi$
$380$	0.397618	0.0203974
$381$	0	0
$382$	−0.180369	−0.00922849
$383$	13.6280	0.696360	0.348180	−	0.937428i	$-0.386800\pi$
$383$	13.6280	0.696360	0.348180	+	0.937428i	$0.386800\pi$
$384$	0	0
$385$	−1.04299	−0.0531557
$386$	4.12565	0.209990
$387$	0	0
$388$	−14.5899	−0.740688
$389$	15.7277	0.797428	0.398714	−	0.917075i	$-0.369457\pi$
$389$	15.7277	0.797428	0.398714	+	0.917075i	$0.369457\pi$
$390$	0	0
$391$	7.51360	0.379979
$392$	−26.1136	−1.31894
$393$	0	0
$394$	−3.58628	−0.180674
$395$	−2.66671	−0.134177
$396$	0	0
$397$	16.8523	0.845790	0.422895	−	0.906179i	$-0.361014\pi$
$397$	16.8523	0.845790	0.422895	+	0.906179i	$0.361014\pi$
$398$	3.56905	0.178900
$399$	0	0
$400$	−13.8235	−0.691176
$401$	−30.3744	−1.51682	−0.758412	−	0.651775i	$-0.774025\pi$
$401$	−30.3744	−1.51682	−0.758412	+	0.651775i	$0.774025\pi$
$402$	0	0
$403$	5.30877	0.264449
$404$	−27.5370	−1.37001
$405$	0	0
$406$	16.3664	0.812250
$407$	−2.71758	−0.134706
$408$	0	0
$409$	−1.31844	−0.0651925	−0.0325962	−	0.999469i	$-0.510378\pi$
$409$	−1.31844	−0.0651925	−0.0325962	+	0.999469i	$0.510378\pi$
$410$	0.0799431	0.00394811
$411$	0	0
$412$	15.8307	0.779925
$413$	34.4672	1.69602
$414$	0	0
$415$	3.63785	0.178575
$416$	27.9075	1.36828
$417$	0	0
$418$	0.456669	0.0223364
$419$	30.3257	1.48151	0.740753	−	0.671778i	$-0.234469\pi$
$419$	30.3257	1.48151	0.740753	+	0.671778i	$0.234469\pi$
$420$	0	0
$421$	8.27204	0.403154	0.201577	−	0.979473i	$-0.435393\pi$
$421$	8.27204	0.403154	0.201577	+	0.979473i	$0.435393\pi$
$422$	−9.86200	−0.480075
$423$	0	0
$424$	−16.7900	−0.815395
$425$	34.9776	1.69666
$426$	0	0
$427$	38.3065	1.85378
$428$	−16.2012	−0.783114
$429$	0	0
$430$	−0.0724281	−0.00349279
$431$	−22.9574	−1.10582	−0.552909	−	0.833241i	$-0.686482\pi$
$431$	−22.9574	−1.10582	−0.552909	+	0.833241i	$0.686482\pi$
$432$	0	0
$433$	11.4207	0.548846	0.274423	−	0.961609i	$-0.411513\pi$
$433$	11.4207	0.548846	0.274423	+	0.961609i	$0.411513\pi$
$434$	1.93414	0.0928417
$435$	0	0
$436$	33.3120	1.59536
$437$	−1.06348	−0.0508730
$438$	0	0
$439$	5.43642	0.259466	0.129733	−	0.991549i	$-0.458588\pi$
$439$	5.43642	0.259466	0.129733	+	0.991549i	$0.458588\pi$
$440$	0.384298	0.0183207
$441$	0	0
$442$	−19.0041	−0.903935
$443$	34.7494	1.65100	0.825498	−	0.564405i	$-0.190894\pi$
$443$	34.7494	1.65100	0.825498	+	0.564405i	$0.190894\pi$
$444$	0	0
$445$	−2.87401	−0.136241
$446$	−3.17775	−0.150471
$447$	0	0
$448$	−16.0745	−0.759451
$449$	11.3505	0.535662	0.267831	−	0.963466i	$-0.413693\pi$
$449$	11.3505	0.535662	0.267831	+	0.963466i	$0.413693\pi$
$450$	0	0
$451$	−0.788714	−0.0371391
$452$	−7.92580	−0.372798
$453$	0	0
$454$	−3.50639	−0.164563
$455$	−6.14338	−0.288006
$456$	0	0
$457$	6.89453	0.322513	0.161256	−	0.986913i	$-0.448445\pi$
$457$	6.89453	0.322513	0.161256	+	0.986913i	$0.448445\pi$
$458$	2.66660	0.124602
$459$	0	0
$460$	−0.422857	−0.0197158
$461$	−13.5620	−0.631643	−0.315822	−	0.948819i	$-0.602280\pi$
$461$	−13.5620	−0.631643	−0.315822	+	0.948819i	$0.602280\pi$
$462$	0	0
$463$	−18.6923	−0.868704	−0.434352	−	0.900743i	$-0.643023\pi$
$463$	−18.6923	−0.868704	−0.434352	+	0.900743i	$0.643023\pi$
$464$	21.2951	0.988602
$465$	0	0
$466$	9.97505	0.462085
$467$	14.7156	0.680957	0.340479	−	0.940252i	$-0.389411\pi$
$467$	14.7156	0.680957	0.340479	+	0.940252i	$0.389411\pi$
$468$	0	0
$469$	36.9681	1.70703
$470$	0.401755	0.0185316
$471$	0	0
$472$	−12.6997	−0.584550
$473$	0.714571	0.0328560
$474$	0	0
$475$	−4.95074	−0.227155
$476$	59.4765	2.72610
$477$	0	0
$478$	−1.92259	−0.0879374
$479$	16.3467	0.746901	0.373450	−	0.927650i	$-0.378175\pi$
$479$	16.3467	0.746901	0.373450	+	0.927650i	$0.378175\pi$
$480$	0	0
$481$	−16.0070	−0.729856
$482$	−7.95863	−0.362505
$483$	0	0
$484$	−1.79145	−0.0814297
$485$	−1.80762	−0.0820796
$486$	0	0
$487$	−24.8625	−1.12663	−0.563314	−	0.826243i	$-0.690474\pi$
$487$	−24.8625	−1.12663	−0.563314	+	0.826243i	$0.690474\pi$
$488$	−14.1143	−0.638926
$489$	0	0
$490$	−1.52870	−0.0690596
$491$	−17.5025	−0.789878	−0.394939	−	0.918707i	$-0.629234\pi$
$491$	−17.5025	−0.789878	−0.394939	+	0.918707i	$0.629234\pi$
$492$	0	0
$493$	−53.8830	−2.42677
$494$	2.68985	0.121022
$495$	0	0
$496$	2.51661	0.112999
$497$	14.7271	0.660603
$498$	0	0
$499$	−3.13006	−0.140121	−0.0700603	−	0.997543i	$-0.522319\pi$
$499$	−3.13006	−0.140121	−0.0700603	+	0.997543i	$0.522319\pi$
$500$	−3.95659	−0.176944
$501$	0	0
$502$	−0.979986	−0.0437389
$503$	8.98965	0.400829	0.200414	−	0.979711i	$-0.435771\pi$
$503$	8.98965	0.400829	0.200414	+	0.979711i	$0.435771\pi$
$504$	0	0
$505$	−3.41170	−0.151819
$506$	−0.485656	−0.0215901
$507$	0	0
$508$	0.606689	0.0269175
$509$	−23.6416	−1.04790	−0.523949	−	0.851750i	$-0.675542\pi$
$509$	−23.6416	−1.04790	−0.523949	+	0.851750i	$0.675542\pi$
$510$	0	0
$511$	−30.5011	−1.34929
$512$	22.8986	1.01198
$513$	0	0
$514$	2.39248	0.105528
$515$	1.96136	0.0864277
$516$	0	0
$517$	−3.96368	−0.174323
$518$	−5.83181	−0.256235
$519$	0	0
$520$	2.26357	0.0992643
$521$	−36.0036	−1.57735	−0.788673	−	0.614813i	$-0.789232\pi$
$521$	−36.0036	−1.57735	−0.788673	+	0.614813i	$0.789232\pi$
$522$	0	0
$523$	7.08081	0.309622	0.154811	−	0.987944i	$-0.450523\pi$
$523$	7.08081	0.309622	0.154811	+	0.987944i	$0.450523\pi$
$524$	−8.15489	−0.356248
$525$	0	0
$526$	−5.59863	−0.244112
$527$	−6.36777	−0.277384
$528$	0	0
$529$	−21.8690	−0.950827
$530$	−0.982892	−0.0426941
$531$	0	0
$532$	−8.41832	−0.364980
$533$	−4.64565	−0.201225
$534$	0	0
$535$	−2.00725	−0.0867810
$536$	−13.6212	−0.588345
$537$	0	0
$538$	4.28023	0.184534
$539$	15.0820	0.649630
$540$	0	0
$541$	−4.23609	−0.182124	−0.0910619	−	0.995845i	$-0.529026\pi$
$541$	−4.23609	−0.182124	−0.0910619	+	0.995845i	$0.529026\pi$
$542$	−6.07970	−0.261145
$543$	0	0
$544$	−33.4745	−1.43521
$545$	4.12720	0.176790
$546$	0	0
$547$	9.53317	0.407609	0.203805	−	0.979012i	$-0.434669\pi$
$547$	9.53317	0.407609	0.203805	+	0.979012i	$0.434669\pi$
$548$	−1.17208	−0.0500688
$549$	0	0
$550$	−2.26085	−0.0964028
$551$	7.62662	0.324905
$552$	0	0
$553$	56.4593	2.40089
$554$	−6.65098	−0.282573
$555$	0	0
$556$	−19.7918	−0.839361
$557$	−2.85948	−0.121160	−0.0605800	−	0.998163i	$-0.519295\pi$
$557$	−2.85948	−0.121160	−0.0605800	+	0.998163i	$0.519295\pi$
$558$	0	0
$559$	4.20894	0.178019
$560$	−2.91225	−0.123065
$561$	0	0
$562$	−9.24347	−0.389912
$563$	−12.8091	−0.539841	−0.269920	−	0.962883i	$-0.586997\pi$
$563$	−12.8091	−0.539841	−0.269920	+	0.962883i	$0.586997\pi$
$564$	0	0
$565$	−0.981969	−0.0413118
$566$	6.67787	0.280692
$567$	0	0
$568$	−5.42633	−0.227684
$569$	10.5273	0.441327	0.220664	−	0.975350i	$-0.429178\pi$
$569$	10.5273	0.441327	0.220664	+	0.975350i	$0.429178\pi$
$570$	0	0
$571$	−24.9055	−1.04226	−0.521132	−	0.853476i	$-0.674490\pi$
$571$	−24.9055	−1.04226	−0.521132	+	0.853476i	$0.674490\pi$
$572$	−10.5519	−0.441199
$573$	0	0
$574$	−1.69255	−0.0706455
$575$	5.26499	0.219565
$576$	0	0
$577$	−26.7799	−1.11486	−0.557430	−	0.830224i	$-0.688213\pi$
$577$	−26.7799	−1.11486	−0.557430	+	0.830224i	$0.688213\pi$
$578$	15.0317	0.625238
$579$	0	0
$580$	3.03248	0.125917
$581$	−77.0200	−3.19533
$582$	0	0
$583$	9.69714	0.401615
$584$	11.2384	0.465047
$585$	0	0
$586$	−12.6280	−0.521658
$587$	−18.7991	−0.775920	−0.387960	−	0.921676i	$-0.626820\pi$
$587$	−18.7991	−0.775920	−0.387960	+	0.921676i	$0.626820\pi$
$588$	0	0
$589$	0.901295	0.0371372
$590$	−0.743443	−0.0306071
$591$	0	0
$592$	−7.58807	−0.311868
$593$	3.68216	0.151208	0.0756041	−	0.997138i	$-0.475911\pi$
$593$	3.68216	0.151208	0.0756041	+	0.997138i	$0.475911\pi$
$594$	0	0
$595$	7.36886	0.302094
$596$	−13.5746	−0.556038
$597$	0	0
$598$	−2.86059	−0.116978
$599$	−43.2659	−1.76780	−0.883899	−	0.467677i	$-0.845091\pi$
$599$	−43.2659	−1.76780	−0.883899	+	0.467677i	$0.845091\pi$
$600$	0	0
$601$	−14.4072	−0.587683	−0.293842	−	0.955854i	$-0.594934\pi$
$601$	−14.4072	−0.587683	−0.293842	+	0.955854i	$0.594934\pi$
$602$	1.53344	0.0624983
$603$	0	0
$604$	12.4559	0.506824
$605$	−0.221953	−0.00902366
$606$	0	0
$607$	−6.06496	−0.246169	−0.123084	−	0.992396i	$-0.539279\pi$
$607$	−6.06496	−0.246169	−0.123084	+	0.992396i	$0.539279\pi$
$608$	4.73799	0.192151
$609$	0	0
$610$	−0.826257	−0.0334542
$611$	−23.3467	−0.944508
$612$	0	0
$613$	−3.51855	−0.142113	−0.0710565	−	0.997472i	$-0.522637\pi$
$613$	−3.51855	−0.142113	−0.0710565	+	0.997472i	$0.522637\pi$
$614$	−0.345271	−0.0139340
$615$	0	0
$616$	−8.13629	−0.327821
$617$	−7.17642	−0.288912	−0.144456	−	0.989511i	$-0.546143\pi$
$617$	−7.17642	−0.288912	−0.144456	+	0.989511i	$0.546143\pi$
$618$	0	0
$619$	−8.27421	−0.332568	−0.166284	−	0.986078i	$-0.553177\pi$
$619$	−8.27421	−0.332568	−0.166284	+	0.986078i	$0.553177\pi$
$620$	0.358371	0.0143925
$621$	0	0
$622$	−0.517644	−0.0207556
$623$	60.8480	2.43783
$624$	0	0
$625$	24.2635	0.970539
$626$	−4.97824	−0.198970
$627$	0	0
$628$	26.4108	1.05391
$629$	19.2001	0.765557
$630$	0	0
$631$	−9.09560	−0.362090	−0.181045	−	0.983475i	$-0.557948\pi$
$631$	−9.09560	−0.362090	−0.181045	+	0.983475i	$0.557948\pi$
$632$	−20.8029	−0.827493
$633$	0	0
$634$	−3.80097	−0.150956
$635$	0.0751660	0.00298287
$636$	0	0
$637$	88.8356	3.51980
$638$	3.48284	0.137887
$639$	0	0
$640$	2.44994	0.0968425
$641$	−16.4386	−0.649285	−0.324642	−	0.945837i	$-0.605244\pi$
$641$	−16.4386	−0.649285	−0.324642	+	0.945837i	$0.605244\pi$
$642$	0	0
$643$	7.04594	0.277865	0.138932	−	0.990302i	$-0.455633\pi$
$643$	7.04594	0.277865	0.138932	+	0.990302i	$0.455633\pi$
$644$	8.95268	0.352785
$645$	0	0
$646$	−3.22642	−0.126942
$647$	−18.1024	−0.711679	−0.355840	−	0.934547i	$-0.615805\pi$
$647$	−18.1024	−0.711679	−0.355840	+	0.934547i	$0.615805\pi$
$648$	0	0
$649$	7.33476	0.287914
$650$	−13.3167	−0.522326
$651$	0	0
$652$	8.85005	0.346595
$653$	−37.2985	−1.45960	−0.729802	−	0.683659i	$-0.760388\pi$
$653$	−37.2985	−1.45960	−0.729802	+	0.683659i	$0.760388\pi$
$654$	0	0
$655$	−1.01035	−0.0394778
$656$	−2.20226	−0.0859837
$657$	0	0
$658$	−8.50590	−0.331594
$659$	−13.2223	−0.515068	−0.257534	−	0.966269i	$-0.582910\pi$
$659$	−13.2223	−0.515068	−0.257534	+	0.966269i	$0.582910\pi$
$660$	0	0
$661$	−17.6212	−0.685385	−0.342692	−	0.939448i	$-0.611339\pi$
$661$	−17.6212	−0.685385	−0.342692	+	0.939448i	$0.611339\pi$
$662$	−1.13010	−0.0439226
$663$	0	0
$664$	28.3786	1.10130
$665$	−1.04299	−0.0404454
$666$	0	0
$667$	−8.11073	−0.314049
$668$	40.4926	1.56671
$669$	0	0
$670$	−0.797388	−0.0308058
$671$	8.15179	0.314696
$672$	0	0
$673$	32.2335	1.24251	0.621255	−	0.783609i	$-0.286623\pi$
$673$	32.2335	1.24251	0.621255	+	0.783609i	$0.286623\pi$
$674$	−13.8481	−0.533410
$675$	0	0
$676$	−38.8637	−1.49476
$677$	10.2788	0.395045	0.197523	−	0.980298i	$-0.436710\pi$
$677$	10.2788	0.395045	0.197523	+	0.980298i	$0.436710\pi$
$678$	0	0
$679$	38.2706	1.46869
$680$	−2.71511	−0.104120
$681$	0	0
$682$	0.411593	0.0157607
$683$	−25.4018	−0.971973	−0.485986	−	0.873966i	$-0.661540\pi$
$683$	−25.4018	−0.971973	−0.485986	+	0.873966i	$0.661540\pi$
$684$	0	0
$685$	−0.145215	−0.00554840
$686$	17.3437	0.662186
$687$	0	0
$688$	1.99523	0.0760676
$689$	57.1177	2.17601
$690$	0	0
$691$	−2.58092	−0.0981828	−0.0490914	−	0.998794i	$-0.515633\pi$
$691$	−2.58092	−0.0981828	−0.0490914	+	0.998794i	$0.515633\pi$
$692$	−14.6652	−0.557487
$693$	0	0
$694$	4.76450	0.180858
$695$	−2.45212	−0.0930141
$696$	0	0
$697$	5.57236	0.211068
$698$	2.23334	0.0845330
$699$	0	0
$700$	41.6769	1.57524
$701$	30.0612	1.13540	0.567698	−	0.823237i	$-0.307834\pi$
$701$	30.0612	1.13540	0.567698	+	0.823237i	$0.307834\pi$
$702$	0	0
$703$	−2.71758	−0.102496
$704$	−3.42073	−0.128924
$705$	0	0
$706$	−15.5840	−0.586511
$707$	72.2321	2.71657
$708$	0	0
$709$	−6.90683	−0.259391	−0.129696	−	0.991554i	$-0.541400\pi$
$709$	−6.90683	−0.259391	−0.129696	+	0.991554i	$0.541400\pi$
$710$	−0.317659	−0.0119215
$711$	0	0
$712$	−22.4199	−0.840222
$713$	−0.958506	−0.0358963
$714$	0	0
$715$	−1.30734	−0.0488916
$716$	−25.2290	−0.942851
$717$	0	0
$718$	−4.81175	−0.179573
$719$	−13.1525	−0.490506	−0.245253	−	0.969459i	$-0.578871\pi$
$719$	−13.1525	−0.490506	−0.245253	+	0.969459i	$0.578871\pi$
$720$	0	0
$721$	−41.5256	−1.54649
$722$	0.456669	0.0169955
$723$	0	0
$724$	25.6644	0.953810
$725$	−37.7574	−1.40227
$726$	0	0
$727$	−46.8434	−1.73733	−0.868663	−	0.495404i	$-0.835020\pi$
$727$	−46.8434	−1.73733	−0.868663	+	0.495404i	$0.835020\pi$
$728$	−47.9241	−1.77618
$729$	0	0
$730$	0.657897	0.0243498
$731$	−5.04854	−0.186727
$732$	0	0
$733$	−31.9162	−1.17885	−0.589426	−	0.807822i	$-0.700646\pi$
$733$	−31.9162	−1.17885	−0.589426	+	0.807822i	$0.700646\pi$
$734$	14.2151	0.524688
$735$	0	0
$736$	−5.03874	−0.185731
$737$	7.86697	0.289784
$738$	0	0
$739$	30.0302	1.10468	0.552339	−	0.833620i	$-0.313736\pi$
$739$	30.0302	1.10468	0.552339	+	0.833620i	$0.313736\pi$
$740$	−1.08056	−0.0397222
$741$	0	0
$742$	20.8096	0.763946
$743$	42.6871	1.56604	0.783019	−	0.621998i	$-0.213679\pi$
$743$	42.6871	1.56604	0.783019	+	0.621998i	$0.213679\pi$
$744$	0	0
$745$	−1.68183	−0.0616175
$746$	13.2380	0.484676
$747$	0	0
$748$	12.6569	0.462780
$749$	42.4972	1.55282
$750$	0	0
$751$	30.7211	1.12103	0.560514	−	0.828145i	$-0.310604\pi$
$751$	30.7211	1.12103	0.560514	+	0.828145i	$0.310604\pi$
$752$	−11.0675	−0.403589
$753$	0	0
$754$	20.5145	0.747093
$755$	1.54323	0.0561638
$756$	0	0
$757$	−49.0714	−1.78353	−0.891765	−	0.452498i	$-0.850533\pi$
$757$	−49.0714	−1.78353	−0.891765	+	0.452498i	$0.850533\pi$
$758$	−12.0654	−0.438233
$759$	0	0
$760$	0.384298	0.0139399
$761$	24.8178	0.899645	0.449822	−	0.893118i	$-0.351487\pi$
$761$	24.8178	0.899645	0.449822	+	0.893118i	$0.351487\pi$
$762$	0	0
$763$	−87.3806	−3.16339
$764$	0.707565	0.0255988
$765$	0	0
$766$	6.22350	0.224864
$767$	43.2029	1.55997
$768$	0	0
$769$	−38.9862	−1.40588	−0.702940	−	0.711250i	$-0.748130\pi$
$769$	−38.9862	−1.40588	−0.702940	+	0.711250i	$0.748130\pi$
$770$	−0.476301	−0.0171647
$771$	0	0
$772$	−16.1844	−0.582489
$773$	−0.998099	−0.0358991	−0.0179496	−	0.999839i	$-0.505714\pi$
$773$	−0.998099	−0.0358991	−0.0179496	+	0.999839i	$0.505714\pi$
$774$	0	0
$775$	−4.46208	−0.160283
$776$	−14.1011	−0.506199
$777$	0	0
$778$	7.18237	0.257500
$779$	−0.788714	−0.0282586
$780$	0	0
$781$	3.13400	0.112143
$782$	3.43123	0.122700
$783$	0	0
$784$	42.1123	1.50401
$785$	3.27218	0.116789
$786$	0	0
$787$	18.7420	0.668082	0.334041	−	0.942559i	$-0.391588\pi$
$787$	18.7420	0.668082	0.334041	+	0.942559i	$0.391588\pi$
$788$	14.0685	0.501170
$789$	0	0
$790$	−1.21781	−0.0433276
$791$	20.7901	0.739211
$792$	0	0
$793$	48.0153	1.70507
$794$	7.69590	0.273117
$795$	0	0
$796$	−14.0009	−0.496250
$797$	−5.05128	−0.178925	−0.0894627	−	0.995990i	$-0.528515\pi$
$797$	−5.05128	−0.178925	−0.0894627	+	0.995990i	$0.528515\pi$
$798$	0	0
$799$	28.0039	0.990708
$800$	−23.4566	−0.829315
$801$	0	0
$802$	−13.8710	−0.489803
$803$	−6.49076	−0.229054
$804$	0	0
$805$	1.10920	0.0390940
$806$	2.42435	0.0853941
$807$	0	0
$808$	−26.6144	−0.936293
$809$	4.98214	0.175163	0.0875813	−	0.996157i	$-0.472086\pi$
$809$	4.98214	0.175163	0.0875813	+	0.996157i	$0.472086\pi$
$810$	0	0
$811$	25.3145	0.888913	0.444456	−	0.895800i	$-0.353397\pi$
$811$	25.3145	0.888913	0.444456	+	0.895800i	$0.353397\pi$
$812$	−64.2033	−2.25309
$813$	0	0
$814$	−1.24103	−0.0434982
$815$	1.09648	0.0384080
$816$	0	0
$817$	0.714571	0.0249997
$818$	−0.602089	−0.0210515
$819$	0	0
$820$	−0.313607	−0.0109516
$821$	−5.81543	−0.202960	−0.101480	−	0.994838i	$-0.532358\pi$
$821$	−5.81543	−0.202960	−0.101480	+	0.994838i	$0.532358\pi$
$822$	0	0
$823$	47.7881	1.66579	0.832895	−	0.553431i	$-0.186682\pi$
$823$	47.7881	1.66579	0.832895	+	0.553431i	$0.186682\pi$
$824$	15.3004	0.533015
$825$	0	0
$826$	15.7401	0.547667
$827$	39.0692	1.35857	0.679284	−	0.733875i	$-0.262290\pi$
$827$	39.0692	1.35857	0.679284	+	0.733875i	$0.262290\pi$
$828$	0	0
$829$	−7.57440	−0.263070	−0.131535	−	0.991312i	$-0.541991\pi$
$829$	−7.57440	−0.263070	−0.131535	+	0.991312i	$0.541991\pi$
$830$	1.66129	0.0576642
$831$	0	0
$832$	−20.1487	−0.698529
$833$	−106.557	−3.69197
$834$	0	0
$835$	5.01684	0.173615
$836$	−1.79145	−0.0619587
$837$	0	0
$838$	13.8488	0.478398
$839$	28.0343	0.967850	0.483925	−	0.875110i	$-0.339211\pi$
$839$	28.0343	0.967850	0.483925	+	0.875110i	$0.339211\pi$
$840$	0	0
$841$	29.1653	1.00570
$842$	3.77758	0.130184
$843$	0	0
$844$	38.6874	1.33167
$845$	−4.81504	−0.165642
$846$	0	0
$847$	4.69915	0.161465
$848$	27.0765	0.929811
$849$	0	0
$850$	15.9732	0.547875
$851$	2.89008	0.0990708
$852$	0	0
$853$	−27.9605	−0.957349	−0.478674	−	0.877992i	$-0.658883\pi$
$853$	−27.9605	−0.957349	−0.478674	+	0.877992i	$0.658883\pi$
$854$	17.4934	0.598612
$855$	0	0
$856$	−15.6584	−0.535194
$857$	−17.8805	−0.610787	−0.305393	−	0.952226i	$-0.598788\pi$
$857$	−17.8805	−0.610787	−0.305393	+	0.952226i	$0.598788\pi$
$858$	0	0
$859$	7.76017	0.264774	0.132387	−	0.991198i	$-0.457736\pi$
$859$	7.76017	0.264774	0.132387	+	0.991198i	$0.457736\pi$
$860$	0.284126	0.00968863
$861$	0	0
$862$	−10.4839	−0.357084
$863$	−31.9699	−1.08827	−0.544135	−	0.838998i	$-0.683142\pi$
$863$	−31.9699	−1.08827	−0.544135	+	0.838998i	$0.683142\pi$
$864$	0	0
$865$	−1.81695	−0.0617782
$866$	5.21549	0.177230
$867$	0	0
$868$	−7.58739	−0.257533
$869$	12.0148	0.407574
$870$	0	0
$871$	46.3377	1.57009
$872$	32.1960	1.09029
$873$	0	0
$874$	−0.485656	−0.0164276
$875$	10.3785	0.350858
$876$	0	0
$877$	−2.12224	−0.0716630	−0.0358315	−	0.999358i	$-0.511408\pi$
$877$	−2.12224	−0.0716630	−0.0358315	+	0.999358i	$0.511408\pi$
$878$	2.48264	0.0837852
$879$	0	0
$880$	−0.619739	−0.0208914
$881$	15.1859	0.511627	0.255814	−	0.966726i	$-0.417657\pi$
$881$	15.1859	0.511627	0.255814	+	0.966726i	$0.417657\pi$
$882$	0	0
$883$	29.5964	0.995999	0.498000	−	0.867177i	$-0.334068\pi$
$883$	29.5964	0.995999	0.498000	+	0.867177i	$0.334068\pi$
$884$	74.5509	2.50742
$885$	0	0
$886$	15.8690	0.533129
$887$	−23.2340	−0.780121	−0.390061	−	0.920789i	$-0.627546\pi$
$887$	−23.2340	−0.780121	−0.390061	+	0.920789i	$0.627546\pi$
$888$	0	0
$889$	−1.59140	−0.0533740
$890$	−1.31247	−0.0439940
$891$	0	0
$892$	12.4659	0.417389
$893$	−3.96368	−0.132640
$894$	0	0
$895$	−3.12575	−0.104482
$896$	−51.8699	−1.73285
$897$	0	0
$898$	5.18340	0.172972
$899$	6.87384	0.229255
$900$	0	0
$901$	−68.5116	−2.28245
$902$	−0.360181	−0.0119927
$903$	0	0
$904$	−7.66028	−0.254777
$905$	3.17970	0.105697
$906$	0	0
$907$	−35.4985	−1.17871	−0.589354	−	0.807875i	$-0.700618\pi$
$907$	−35.4985	−1.17871	−0.589354	+	0.807875i	$0.700618\pi$
$908$	13.7551	0.456480
$909$	0	0
$910$	−2.80549	−0.0930010
$911$	38.5075	1.27581	0.637905	−	0.770115i	$-0.279801\pi$
$911$	38.5075	1.27581	0.637905	+	0.770115i	$0.279801\pi$
$912$	0	0
$913$	−16.3902	−0.542436
$914$	3.14852	0.104144
$915$	0	0
$916$	−10.4608	−0.345633
$917$	21.3911	0.706395
$918$	0	0
$919$	20.9347	0.690572	0.345286	−	0.938497i	$-0.387782\pi$
$919$	20.9347	0.690572	0.345286	+	0.938497i	$0.387782\pi$
$920$	−0.408691	−0.0134742
$921$	0	0
$922$	−6.19332	−0.203966
$923$	18.4598	0.607610
$924$	0	0
$925$	13.4540	0.442366
$926$	−8.53618	−0.280516
$927$	0	0
$928$	36.1349	1.18619
$929$	0.536958	0.0176170	0.00880851	−	0.999961i	$-0.497196\pi$
$929$	0.536958	0.0176170	0.00880851	+	0.999961i	$0.497196\pi$
$930$	0	0
$931$	15.0820	0.494294
$932$	−39.1309	−1.28177
$933$	0	0
$934$	6.72016	0.219890
$935$	1.56812	0.0512832
$936$	0	0
$937$	34.4984	1.12701	0.563507	−	0.826112i	$-0.309452\pi$
$937$	34.4984	1.12701	0.563507	+	0.826112i	$0.309452\pi$
$938$	16.8822	0.551223
$939$	0	0
$940$	−1.57603	−0.0514045
$941$	−17.8959	−0.583391	−0.291695	−	0.956511i	$-0.594219\pi$
$941$	−17.8959	−0.583391	−0.291695	+	0.956511i	$0.594219\pi$
$942$	0	0
$943$	0.838778	0.0273144
$944$	20.4802	0.666574
$945$	0	0
$946$	0.326322	0.0106097
$947$	−16.0979	−0.523111	−0.261556	−	0.965188i	$-0.584235\pi$
$947$	−16.0979	−0.523111	−0.261556	+	0.965188i	$0.584235\pi$
$948$	0	0
$949$	−38.2316	−1.24105
$950$	−2.26085	−0.0733516
$951$	0	0
$952$	57.4840	1.86307
$953$	−29.2828	−0.948562	−0.474281	−	0.880374i	$-0.657292\pi$
$953$	−29.2828	−0.948562	−0.474281	+	0.880374i	$0.657292\pi$
$954$	0	0
$955$	0.0876640	0.00283674
$956$	7.54210	0.243929
$957$	0	0
$958$	7.46504	0.241184
$959$	3.07448	0.0992802
$960$	0	0
$961$	−30.1877	−0.973796
$962$	−7.30989	−0.235680
$963$	0	0
$964$	31.2207	1.00555
$965$	−2.00517	−0.0645488
$966$	0	0
$967$	−39.4644	−1.26909	−0.634544	−	0.772887i	$-0.718812\pi$
$967$	−39.4644	−1.26909	−0.634544	+	0.772887i	$0.718812\pi$
$968$	−1.73144	−0.0556505
$969$	0	0
$970$	−0.825481	−0.0265046
$971$	−27.0245	−0.867257	−0.433628	−	0.901092i	$-0.642767\pi$
$971$	−27.0245	−0.867257	−0.433628	+	0.901092i	$0.642767\pi$
$972$	0	0
$973$	51.9159	1.66435
$974$	−11.3539	−0.363804
$975$	0	0
$976$	22.7615	0.728579
$977$	55.2922	1.76895	0.884477	−	0.466584i	$-0.154516\pi$
$977$	55.2922	1.76895	0.884477	+	0.466584i	$0.154516\pi$
$978$	0	0
$979$	12.9487	0.413843
$980$	5.99689	0.191564
$981$	0	0
$982$	−7.99286	−0.255062
$983$	−36.0221	−1.14893	−0.574463	−	0.818531i	$-0.694789\pi$
$983$	−36.0221	−1.14893	−0.574463	+	0.818531i	$0.694789\pi$
$984$	0	0
$985$	1.74302	0.0555373
$986$	−24.6067	−0.783637
$987$	0	0
$988$	−10.5519	−0.335702
$989$	−0.759929	−0.0241643
$990$	0	0
$991$	50.7944	1.61354	0.806768	−	0.590868i	$-0.201215\pi$
$991$	50.7944	1.61354	0.806768	+	0.590868i	$0.201215\pi$
$992$	4.27033	0.135583
$993$	0	0
$994$	6.72543	0.213318
$995$	−1.73465	−0.0549921
$996$	0	0
$997$	−41.8525	−1.32548	−0.662741	−	0.748849i	$-0.730607\pi$
$997$	−41.8525	−1.32548	−0.662741	+	0.748849i	$0.730607\pi$
$998$	−1.42940	−0.0452469
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.p.1.4		7
3.2	odd	2		209.2.a.d.1.4	✓	7
12.11	even	2		3344.2.a.ba.1.5		7
15.14	odd	2		5225.2.a.n.1.4		7
33.32	even	2		2299.2.a.q.1.4		7
57.56	even	2		3971.2.a.i.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.d.1.4	✓	7	3.2	odd	2
1881.2.a.p.1.4		7	1.1	even	1	trivial
2299.2.a.q.1.4		7	33.32	even	2
3344.2.a.ba.1.5		7	12.11	even	2
3971.2.a.i.1.4		7	57.56	even	2
5225.2.a.n.1.4		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)

\(f(q)\)	\(=\)	\(q+0.456669 q^{2} -1.79145 q^{4} -0.221953 q^{5} +4.69915 q^{7} -1.73144 q^{8} -0.101359 q^{10} +1.00000 q^{11} +5.89016 q^{13} +2.14596 q^{14} +2.79221 q^{16} -7.06513 q^{17} +1.00000 q^{19} +0.397618 q^{20} +0.456669 q^{22} -1.06348 q^{23} -4.95074 q^{25} +2.68985 q^{26} -8.41832 q^{28} +7.62662 q^{29} +0.901295 q^{31} +4.73799 q^{32} -3.22642 q^{34} -1.04299 q^{35} -2.71758 q^{37} +0.456669 q^{38} +0.384298 q^{40} -0.788714 q^{41} +0.714571 q^{43} -1.79145 q^{44} -0.485656 q^{46} -3.96368 q^{47} +15.0820 q^{49} -2.26085 q^{50} -10.5519 q^{52} +9.69714 q^{53} -0.221953 q^{55} -8.13629 q^{56} +3.48284 q^{58} +7.33476 q^{59} +8.15179 q^{61} +0.411593 q^{62} -3.42073 q^{64} -1.30734 q^{65} +7.86697 q^{67} +12.6569 q^{68} -0.476301 q^{70} +3.13400 q^{71} -6.49076 q^{73} -1.24103 q^{74} -1.79145 q^{76} +4.69915 q^{77} +12.0148 q^{79} -0.619739 q^{80} -0.360181 q^{82} -16.3902 q^{83} +1.56812 q^{85} +0.326322 q^{86} -1.73144 q^{88} +12.9487 q^{89} +27.6788 q^{91} +1.90517 q^{92} -1.81009 q^{94} -0.221953 q^{95} +8.14414 q^{97} +6.88750 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 15 q^{4} - 2 q^{5} + 10 q^{7} + 9 q^{8} - 6 q^{10} + 7 q^{11} - 4 q^{13} - 6 q^{14} + 27 q^{16} - 2 q^{17} + 7 q^{19} + 4 q^{20} + q^{22} - 10 q^{23} + 9 q^{25} + 8 q^{26} + 26 q^{28} + 18 q^{29}+ \cdots - 19 q^{98}+O(q^{100}) \) 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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Representations

Groups

Database

Properties

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