LMFDB - Embedded newform 1881.2.a.p.1.6

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			1.6
Root			$2.61330$ 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)

$f(q)$	$=$	\(q+2.61330 q^{2} +4.82936 q^{4} -4.07680 q^{5} +3.61829 q^{7} +7.39397 q^{8} -10.6539 q^{10} +1.00000 q^{11} -1.47857 q^{13} +9.45570 q^{14} +9.66398 q^{16} +3.27003 q^{17} +1.00000 q^{19} -19.6883 q^{20} +2.61330 q^{22} +7.45793 q^{23} +11.6203 q^{25} -3.86395 q^{26} +17.4740 q^{28} -1.02535 q^{29} +1.64921 q^{31} +10.4670 q^{32} +8.54558 q^{34} -14.7511 q^{35} -6.71293 q^{37} +2.61330 q^{38} -30.1438 q^{40} +3.92451 q^{41} +5.38113 q^{43} +4.82936 q^{44} +19.4898 q^{46} +3.71597 q^{47} +6.09205 q^{49} +30.3674 q^{50} -7.14054 q^{52} +0.102902 q^{53} -4.07680 q^{55} +26.7536 q^{56} -2.67955 q^{58} -13.2986 q^{59} -6.49664 q^{61} +4.30989 q^{62} +8.02543 q^{64} +6.02783 q^{65} -3.70989 q^{67} +15.7921 q^{68} -38.5490 q^{70} -6.32968 q^{71} -1.37759 q^{73} -17.5429 q^{74} +4.82936 q^{76} +3.61829 q^{77} +13.6725 q^{79} -39.3981 q^{80} +10.2559 q^{82} -5.44061 q^{83} -13.3313 q^{85} +14.0625 q^{86} +7.39397 q^{88} -12.1357 q^{89} -5.34990 q^{91} +36.0170 q^{92} +9.71096 q^{94} -4.07680 q^{95} -13.7910 q^{97} +15.9204 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.61330	1.84789	0.923943	−	0.382531i	$-0.124948\pi$
$2$	2.61330	1.84789	0.923943	+	0.382531i	$0.124948\pi$
$3$	0	0
$3$	0	0
$4$	4.82936	2.41468
$5$	−4.07680	−1.82320	−0.911600	−	0.411078i	$-0.865153\pi$
$5$	−4.07680	−1.82320	−0.911600	+	0.411078i	$0.865153\pi$
$6$	0	0
$7$	3.61829	1.36759	0.683793	−	0.729676i	$-0.260329\pi$
$7$	3.61829	1.36759	0.683793	+	0.729676i	$0.260329\pi$
$8$	7.39397	2.61416
$9$	0	0
$10$	−10.6539	−3.36907
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.47857	−0.410081	−0.205041	−	0.978753i	$-0.565733\pi$
$13$	−1.47857	−0.410081	−0.205041	+	0.978753i	$0.565733\pi$
$14$	9.45570	2.52714
$15$	0	0
$16$	9.66398	2.41600
$17$	3.27003	0.793099	0.396549	−	0.918013i	$-0.370208\pi$
$17$	3.27003	0.793099	0.396549	+	0.918013i	$0.370208\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−19.6883	−4.40244
$21$	0	0
$22$	2.61330	0.557158
$23$	7.45793	1.55509	0.777543	−	0.628830i	$-0.216466\pi$
$23$	7.45793	1.55509	0.777543	+	0.628830i	$0.216466\pi$
$24$	0	0
$25$	11.6203	2.32406
$26$	−3.86395	−0.757783
$27$	0	0
$28$	17.4740	3.30228
$29$	−1.02535	−0.190403	−0.0952013	−	0.995458i	$-0.530349\pi$
$29$	−1.02535	−0.190403	−0.0952013	+	0.995458i	$0.530349\pi$
$30$	0	0
$31$	1.64921	0.296207	0.148104	−	0.988972i	$-0.452683\pi$
$31$	1.64921	0.296207	0.148104	+	0.988972i	$0.452683\pi$
$32$	10.4670	1.85032
$33$	0	0
$34$	8.54558	1.46556
$35$	−14.7511	−2.49338
$36$	0	0
$37$	−6.71293	−1.10360	−0.551799	−	0.833977i	$-0.686059\pi$
$37$	−6.71293	−1.10360	−0.551799	+	0.833977i	$0.686059\pi$
$38$	2.61330	0.423934
$39$	0	0
$40$	−30.1438	−4.76615
$41$	3.92451	0.612905	0.306453	−	0.951886i	$-0.400858\pi$
$41$	3.92451	0.612905	0.306453	+	0.951886i	$0.400858\pi$
$42$	0	0
$43$	5.38113	0.820614	0.410307	−	0.911947i	$-0.365422\pi$
$43$	5.38113	0.820614	0.410307	+	0.911947i	$0.365422\pi$
$44$	4.82936	0.728053
$45$	0	0
$46$	19.4898	2.87362
$47$	3.71597	0.542030	0.271015	−	0.962575i	$-0.412641\pi$
$47$	3.71597	0.542030	0.271015	+	0.962575i	$0.412641\pi$
$48$	0	0
$49$	6.09205	0.870292
$50$	30.3674	4.29460
$51$	0	0
$52$	−7.14054	−0.990215
$53$	0.102902	0.0141347	0.00706733	−	0.999975i	$-0.497750\pi$
$53$	0.102902	0.0141347	0.00706733	+	0.999975i	$0.497750\pi$
$54$	0	0
$55$	−4.07680	−0.549716
$56$	26.7536	3.57509
$57$	0	0
$58$	−2.67955	−0.351842
$59$	−13.2986	−1.73134	−0.865668	−	0.500619i	$-0.833106\pi$
$59$	−13.2986	−1.73134	−0.865668	+	0.500619i	$0.833106\pi$
$60$	0	0
$61$	−6.49664	−0.831809	−0.415905	−	0.909408i	$-0.636535\pi$
$61$	−6.49664	−0.831809	−0.415905	+	0.909408i	$0.636535\pi$
$62$	4.30989	0.547357
$63$	0	0
$64$	8.02543	1.00318
$65$	6.02783	0.747661
$66$	0	0
$67$	−3.70989	−0.453235	−0.226618	−	0.973984i	$-0.572767\pi$
$67$	−3.70989	−0.453235	−0.226618	+	0.973984i	$0.572767\pi$
$68$	15.7921	1.91508
$69$	0	0
$70$	−38.5490	−4.60749
$71$	−6.32968	−0.751194	−0.375597	−	0.926783i	$-0.622562\pi$
$71$	−6.32968	−0.751194	−0.375597	+	0.926783i	$0.622562\pi$
$72$	0	0
$73$	−1.37759	−0.161235	−0.0806173	−	0.996745i	$-0.525689\pi$
$73$	−1.37759	−0.161235	−0.0806173	+	0.996745i	$0.525689\pi$
$74$	−17.5429	−2.03932
$75$	0	0
$76$	4.82936	0.553965
$77$	3.61829	0.412343
$78$	0	0
$79$	13.6725	1.53828	0.769141	−	0.639079i	$-0.220684\pi$
$79$	13.6725	1.53828	0.769141	+	0.639079i	$0.220684\pi$
$80$	−39.3981	−4.40484
$81$	0	0
$82$	10.2559	1.13258
$83$	−5.44061	−0.597184	−0.298592	−	0.954381i	$-0.596517\pi$
$83$	−5.44061	−0.597184	−0.298592	+	0.954381i	$0.596517\pi$
$84$	0	0
$85$	−13.3313	−1.44598
$86$	14.0625	1.51640
$87$	0	0
$88$	7.39397	0.788200
$89$	−12.1357	−1.28638	−0.643191	−	0.765706i	$-0.722390\pi$
$89$	−12.1357	−1.28638	−0.643191	+	0.765706i	$0.722390\pi$
$90$	0	0
$91$	−5.34990	−0.560822
$92$	36.0170	3.75503
$93$	0	0
$94$	9.71096	1.00161
$95$	−4.07680	−0.418271
$96$	0	0
$97$	−13.7910	−1.40026	−0.700131	−	0.714014i	$-0.746875\pi$
$97$	−13.7910	−1.40026	−0.700131	+	0.714014i	$0.746875\pi$
$98$	15.9204	1.60820
$99$	0	0
$100$	56.1186	5.61186
$101$	11.0029	1.09483	0.547413	−	0.836863i	$-0.315613\pi$
$101$	11.0029	1.09483	0.547413	+	0.836863i	$0.315613\pi$
$102$	0	0
$103$	−4.99191	−0.491867	−0.245934	−	0.969287i	$-0.579094\pi$
$103$	−4.99191	−0.491867	−0.245934	+	0.969287i	$0.579094\pi$
$104$	−10.9325	−1.07202
$105$	0	0
$106$	0.268914	0.0261192
$107$	−7.31345	−0.707018	−0.353509	−	0.935431i	$-0.615012\pi$
$107$	−7.31345	−0.707018	−0.353509	+	0.935431i	$0.615012\pi$
$108$	0	0
$109$	−1.44482	−0.138389	−0.0691944	−	0.997603i	$-0.522043\pi$
$109$	−1.44482	−0.138389	−0.0691944	+	0.997603i	$0.522043\pi$
$110$	−10.6539	−1.01581
$111$	0	0
$112$	34.9671	3.30408
$113$	12.0369	1.13234	0.566169	−	0.824289i	$-0.308425\pi$
$113$	12.0369	1.13234	0.566169	+	0.824289i	$0.308425\pi$
$114$	0	0
$115$	−30.4045	−2.83523
$116$	−4.95178	−0.459761
$117$	0	0
$118$	−34.7534	−3.19931
$119$	11.8319	1.08463
$120$	0	0
$121$	1.00000	0.0909091
$122$	−16.9777	−1.53709
$123$	0	0
$124$	7.96463	0.715245
$125$	−26.9897	−2.41403
$126$	0	0
$127$	−4.69692	−0.416784	−0.208392	−	0.978045i	$-0.566823\pi$
$127$	−4.69692	−0.416784	−0.208392	+	0.978045i	$0.566823\pi$
$128$	0.0389415	0.00344197
$129$	0	0
$130$	15.7526	1.38159
$131$	−3.74466	−0.327173	−0.163586	−	0.986529i	$-0.552306\pi$
$131$	−3.74466	−0.327173	−0.163586	+	0.986529i	$0.552306\pi$
$132$	0	0
$133$	3.61829	0.313746
$134$	−9.69507	−0.837526
$135$	0	0
$136$	24.1785	2.07329
$137$	15.7595	1.34643	0.673213	−	0.739449i	$-0.264914\pi$
$137$	15.7595	1.34643	0.673213	+	0.739449i	$0.264914\pi$
$138$	0	0
$139$	−2.52822	−0.214440	−0.107220	−	0.994235i	$-0.534195\pi$
$139$	−2.52822	−0.214440	−0.107220	+	0.994235i	$0.534195\pi$
$140$	−71.2381	−6.02072
$141$	0	0
$142$	−16.5414	−1.38812
$143$	−1.47857	−0.123644
$144$	0	0
$145$	4.18015	0.347142
$146$	−3.60006	−0.297943
$147$	0	0
$148$	−32.4191	−2.66484
$149$	−1.84902	−0.151477	−0.0757387	−	0.997128i	$-0.524131\pi$
$149$	−1.84902	−0.151477	−0.0757387	+	0.997128i	$0.524131\pi$
$150$	0	0
$151$	−15.3184	−1.24659	−0.623296	−	0.781986i	$-0.714207\pi$
$151$	−15.3184	−1.24659	−0.623296	+	0.781986i	$0.714207\pi$
$152$	7.39397	0.599730
$153$	0	0
$154$	9.45570	0.761962
$155$	−6.72351	−0.540045
$156$	0	0
$157$	24.6631	1.96833	0.984165	−	0.177254i	$-0.0567215\pi$
$157$	24.6631	1.96833	0.984165	+	0.177254i	$0.0567215\pi$
$158$	35.7305	2.84257
$159$	0	0
$160$	−42.6718	−3.37350
$161$	26.9850	2.12671
$162$	0	0
$163$	−3.72149	−0.291490	−0.145745	−	0.989322i	$-0.546558\pi$
$163$	−3.72149	−0.291490	−0.145745	+	0.989322i	$0.546558\pi$
$164$	18.9529	1.47997
$165$	0	0
$166$	−14.2180	−1.10353
$167$	−2.64758	−0.204876	−0.102438	−	0.994739i	$-0.532664\pi$
$167$	−2.64758	−0.204876	−0.102438	+	0.994739i	$0.532664\pi$
$168$	0	0
$169$	−10.8138	−0.831833
$170$	−34.8386	−2.67200
$171$	0	0
$172$	25.9874	1.98152
$173$	−7.34552	−0.558469	−0.279235	−	0.960223i	$-0.590081\pi$
$173$	−7.34552	−0.558469	−0.279235	+	0.960223i	$0.590081\pi$
$174$	0	0
$175$	42.0457	3.17835
$176$	9.66398	0.728450
$177$	0	0
$178$	−31.7143	−2.37708
$179$	−9.55394	−0.714095	−0.357047	−	0.934086i	$-0.616217\pi$
$179$	−9.55394	−0.714095	−0.357047	+	0.934086i	$0.616217\pi$
$180$	0	0
$181$	6.02638	0.447937	0.223969	−	0.974596i	$-0.428099\pi$
$181$	6.02638	0.447937	0.223969	+	0.974596i	$0.428099\pi$
$182$	−13.9809	−1.03633
$183$	0	0
$184$	55.1437	4.06525
$185$	27.3673	2.01208
$186$	0	0
$187$	3.27003	0.239128
$188$	17.9458	1.30883
$189$	0	0
$190$	−10.6539	−0.772917
$191$	−17.7069	−1.28123	−0.640613	−	0.767864i	$-0.721320\pi$
$191$	−17.7069	−1.28123	−0.640613	+	0.767864i	$0.721320\pi$
$192$	0	0
$193$	3.69348	0.265863	0.132931	−	0.991125i	$-0.457561\pi$
$193$	3.69348	0.265863	0.132931	+	0.991125i	$0.457561\pi$
$194$	−36.0400	−2.58752
$195$	0	0
$196$	29.4207	2.10148
$197$	25.7789	1.83667	0.918336	−	0.395802i	$-0.129533\pi$
$197$	25.7789	1.83667	0.918336	+	0.395802i	$0.129533\pi$
$198$	0	0
$199$	18.8953	1.33945	0.669726	−	0.742608i	$-0.266411\pi$
$199$	18.8953	1.33945	0.669726	+	0.742608i	$0.266411\pi$
$200$	85.9202	6.07548
$201$	0	0
$202$	28.7538	2.02311
$203$	−3.71002	−0.260392
$204$	0	0
$205$	−15.9994	−1.11745
$206$	−13.0454	−0.908914
$207$	0	0
$208$	−14.2889	−0.990755
$209$	1.00000	0.0691714
$210$	0	0
$211$	−10.1993	−0.702150	−0.351075	−	0.936347i	$-0.614184\pi$
$211$	−10.1993	−0.702150	−0.351075	+	0.936347i	$0.614184\pi$
$212$	0.496950	0.0341306
$213$	0	0
$214$	−19.1123	−1.30649
$215$	−21.9378	−1.49614
$216$	0	0
$217$	5.96733	0.405089
$218$	−3.77576	−0.255727
$219$	0	0
$220$	−19.6883	−1.32739
$221$	−4.83497	−0.325235
$222$	0	0
$223$	−0.262700	−0.0175917	−0.00879584	−	0.999961i	$-0.502800\pi$
$223$	−0.262700	−0.0175917	−0.00879584	+	0.999961i	$0.502800\pi$
$224$	37.8726	2.53047
$225$	0	0
$226$	31.4561	2.09243
$227$	−27.3256	−1.81367	−0.906834	−	0.421489i	$-0.861508\pi$
$227$	−27.3256	−1.81367	−0.906834	+	0.421489i	$0.861508\pi$
$228$	0	0
$229$	−5.53171	−0.365546	−0.182773	−	0.983155i	$-0.558507\pi$
$229$	−5.53171	−0.365546	−0.182773	+	0.983155i	$0.558507\pi$
$230$	−79.4562	−5.23918
$231$	0	0
$232$	−7.58141	−0.497744
$233$	−27.4733	−1.79984	−0.899918	−	0.436059i	$-0.856374\pi$
$233$	−27.4733	−1.79984	−0.899918	+	0.436059i	$0.856374\pi$
$234$	0	0
$235$	−15.1493	−0.988229
$236$	−64.2239	−4.18062
$237$	0	0
$238$	30.9204	2.00427
$239$	−1.40339	−0.0907781	−0.0453890	−	0.998969i	$-0.514453\pi$
$239$	−1.40339	−0.0907781	−0.0453890	+	0.998969i	$0.514453\pi$
$240$	0	0
$241$	−20.7696	−1.33789	−0.668944	−	0.743312i	$-0.733254\pi$
$241$	−20.7696	−1.33789	−0.668944	+	0.743312i	$0.733254\pi$
$242$	2.61330	0.167990
$243$	0	0
$244$	−31.3746	−2.00855
$245$	−24.8361	−1.58672
$246$	0	0
$247$	−1.47857	−0.0940791
$248$	12.1942	0.774334
$249$	0	0
$250$	−70.5322	−4.46085
$251$	1.29936	0.0820148	0.0410074	−	0.999159i	$-0.486943\pi$
$251$	1.29936	0.0820148	0.0410074	+	0.999159i	$0.486943\pi$
$252$	0	0
$253$	7.45793	0.468876
$254$	−12.2745	−0.770169
$255$	0	0
$256$	−15.9491	−0.996819
$257$	−3.41219	−0.212847	−0.106423	−	0.994321i	$-0.533940\pi$
$257$	−3.41219	−0.212847	−0.106423	+	0.994321i	$0.533940\pi$
$258$	0	0
$259$	−24.2893	−1.50927
$260$	29.1106	1.80536
$261$	0	0
$262$	−9.78594	−0.604577
$263$	−14.2418	−0.878189	−0.439094	−	0.898441i	$-0.644701\pi$
$263$	−14.2418	−0.878189	−0.439094	+	0.898441i	$0.644701\pi$
$264$	0	0
$265$	−0.419510	−0.0257703
$266$	9.45570	0.579766
$267$	0	0
$268$	−17.9164	−1.09442
$269$	−5.39477	−0.328925	−0.164462	−	0.986383i	$-0.552589\pi$
$269$	−5.39477	−0.328925	−0.164462	+	0.986383i	$0.552589\pi$
$270$	0	0
$271$	−11.0624	−0.671995	−0.335998	−	0.941863i	$-0.609073\pi$
$271$	−11.0624	−0.671995	−0.335998	+	0.941863i	$0.609073\pi$
$272$	31.6015	1.91612
$273$	0	0
$274$	41.1844	2.48804
$275$	11.6203	0.700731
$276$	0	0
$277$	14.0808	0.846036	0.423018	−	0.906121i	$-0.360971\pi$
$277$	14.0808	0.846036	0.423018	+	0.906121i	$0.360971\pi$
$278$	−6.60700	−0.396261
$279$	0	0
$280$	−109.069	−6.51812
$281$	10.8199	0.645459	0.322729	−	0.946491i	$-0.395400\pi$
$281$	10.8199	0.645459	0.322729	+	0.946491i	$0.395400\pi$
$282$	0	0
$283$	1.90947	0.113506	0.0567532	−	0.998388i	$-0.481925\pi$
$283$	1.90947	0.113506	0.0567532	+	0.998388i	$0.481925\pi$
$284$	−30.5683	−1.81389
$285$	0	0
$286$	−3.86395	−0.228480
$287$	14.2000	0.838201
$288$	0	0
$289$	−6.30690	−0.370994
$290$	10.9240	0.641479
$291$	0	0
$292$	−6.65287	−0.389330
$293$	−3.63550	−0.212388	−0.106194	−	0.994345i	$-0.533867\pi$
$293$	−3.63550	−0.212388	−0.106194	+	0.994345i	$0.533867\pi$
$294$	0	0
$295$	54.2159	3.15657
$296$	−49.6352	−2.88499
$297$	0	0
$298$	−4.83205	−0.279913
$299$	−11.0271	−0.637712
$300$	0	0
$301$	19.4705	1.12226
$302$	−40.0316	−2.30356
$303$	0	0
$304$	9.66398	0.554267
$305$	26.4855	1.51656
$306$	0	0
$307$	−23.5329	−1.34310	−0.671548	−	0.740961i	$-0.734370\pi$
$307$	−23.5329	−1.34310	−0.671548	+	0.740961i	$0.734370\pi$
$308$	17.4740	0.995675
$309$	0	0
$310$	−17.5706	−0.997941
$311$	16.0026	0.907425	0.453713	−	0.891148i	$-0.350099\pi$
$311$	16.0026	0.907425	0.453713	+	0.891148i	$0.350099\pi$
$312$	0	0
$313$	16.0034	0.904566	0.452283	−	0.891875i	$-0.350610\pi$
$313$	16.0034	0.904566	0.452283	+	0.891875i	$0.350610\pi$
$314$	64.4522	3.63725
$315$	0	0
$316$	66.0296	3.71446
$317$	20.8766	1.17255	0.586273	−	0.810113i	$-0.300595\pi$
$317$	20.8766	1.17255	0.586273	+	0.810113i	$0.300595\pi$
$318$	0	0
$319$	−1.02535	−0.0574086
$320$	−32.7181	−1.82900
$321$	0	0
$322$	70.5199	3.92992
$323$	3.27003	0.181949
$324$	0	0
$325$	−17.1814	−0.953054
$326$	−9.72539	−0.538640
$327$	0	0
$328$	29.0177	1.60224
$329$	13.4455	0.741273
$330$	0	0
$331$	15.1136	0.830721	0.415360	−	0.909657i	$-0.363655\pi$
$331$	15.1136	0.830721	0.415360	+	0.909657i	$0.363655\pi$
$332$	−26.2746	−1.44201
$333$	0	0
$334$	−6.91893	−0.378587
$335$	15.1245	0.826339
$336$	0	0
$337$	−12.2766	−0.668751	−0.334376	−	0.942440i	$-0.608525\pi$
$337$	−12.2766	−0.668751	−0.334376	+	0.942440i	$0.608525\pi$
$338$	−28.2598	−1.53713
$339$	0	0
$340$	−64.3814	−3.49157
$341$	1.64921	0.0893098
$342$	0	0
$343$	−3.28525	−0.177387
$344$	39.7879	2.14522
$345$	0	0
$346$	−19.1961	−1.03199
$347$	30.9067	1.65916	0.829580	−	0.558387i	$-0.188580\pi$
$347$	30.9067	1.65916	0.829580	+	0.558387i	$0.188580\pi$
$348$	0	0
$349$	−23.9024	−1.27947	−0.639733	−	0.768597i	$-0.720955\pi$
$349$	−23.9024	−1.27947	−0.639733	+	0.768597i	$0.720955\pi$
$350$	109.878	5.87323
$351$	0	0
$352$	10.4670	0.557892
$353$	−15.0158	−0.799210	−0.399605	−	0.916687i	$-0.630853\pi$
$353$	−15.0158	−0.799210	−0.399605	+	0.916687i	$0.630853\pi$
$354$	0	0
$355$	25.8048	1.36958
$356$	−58.6076	−3.10620
$357$	0	0
$358$	−24.9673	−1.31957
$359$	−14.0826	−0.743251	−0.371626	−	0.928383i	$-0.621200\pi$
$359$	−14.0826	−0.743251	−0.371626	+	0.928383i	$0.621200\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	15.7488	0.827737
$363$	0	0
$364$	−25.8366	−1.35420
$365$	5.61616	0.293963
$366$	0	0
$367$	21.3142	1.11259	0.556296	−	0.830984i	$-0.312222\pi$
$367$	21.3142	1.11259	0.556296	+	0.830984i	$0.312222\pi$
$368$	72.0733	3.75708
$369$	0	0
$370$	71.5190	3.71810
$371$	0.372329	0.0193304
$372$	0	0
$373$	−2.42088	−0.125349	−0.0626743	−	0.998034i	$-0.519963\pi$
$373$	−2.42088	−0.125349	−0.0626743	+	0.998034i	$0.519963\pi$
$374$	8.54558	0.441882
$375$	0	0
$376$	27.4758	1.41696
$377$	1.51605	0.0780806
$378$	0	0
$379$	8.87535	0.455896	0.227948	−	0.973673i	$-0.426798\pi$
$379$	8.87535	0.455896	0.227948	+	0.973673i	$0.426798\pi$
$380$	−19.6883	−1.00999
$381$	0	0
$382$	−46.2735	−2.36756
$383$	3.54065	0.180919	0.0904595	−	0.995900i	$-0.471166\pi$
$383$	3.54065	0.180919	0.0904595	+	0.995900i	$0.471166\pi$
$384$	0	0
$385$	−14.7511	−0.751784
$386$	9.65219	0.491284
$387$	0	0
$388$	−66.6016	−3.38118
$389$	16.7041	0.846933	0.423466	−	0.905912i	$-0.360813\pi$
$389$	16.7041	0.846933	0.423466	+	0.905912i	$0.360813\pi$
$390$	0	0
$391$	24.3877	1.23334
$392$	45.0444	2.27509
$393$	0	0
$394$	67.3681	3.39396
$395$	−55.7403	−2.80460
$396$	0	0
$397$	28.2073	1.41568	0.707841	−	0.706372i	$-0.249669\pi$
$397$	28.2073	1.41568	0.707841	+	0.706372i	$0.249669\pi$
$398$	49.3792	2.47515
$399$	0	0
$400$	112.298	5.61492
$401$	36.2078	1.80813	0.904065	−	0.427395i	$-0.140569\pi$
$401$	36.2078	1.80813	0.904065	+	0.427395i	$0.140569\pi$
$402$	0	0
$403$	−2.43847	−0.121469
$404$	53.1368	2.64365
$405$	0	0
$406$	−9.69540	−0.481175
$407$	−6.71293	−0.332748
$408$	0	0
$409$	36.9236	1.82576	0.912878	−	0.408233i	$-0.133855\pi$
$409$	36.9236	1.82576	0.912878	+	0.408233i	$0.133855\pi$
$410$	−41.8114	−2.06492
$411$	0	0
$412$	−24.1077	−1.18770
$413$	−48.1184	−2.36775
$414$	0	0
$415$	22.1803	1.08879
$416$	−15.4762	−0.758781
$417$	0	0
$418$	2.61330	0.127821
$419$	−18.0690	−0.882726	−0.441363	−	0.897329i	$-0.645505\pi$
$419$	−18.0690	−0.882726	−0.441363	+	0.897329i	$0.645505\pi$
$420$	0	0
$421$	−8.57629	−0.417983	−0.208991	−	0.977918i	$-0.567018\pi$
$421$	−8.57629	−0.417983	−0.208991	+	0.977918i	$0.567018\pi$
$422$	−26.6539	−1.29749
$423$	0	0
$424$	0.760853	0.0369503
$425$	37.9987	1.84321
$426$	0	0
$427$	−23.5067	−1.13757
$428$	−35.3193	−1.70722
$429$	0	0
$430$	−57.3301	−2.76470
$431$	−4.28147	−0.206231	−0.103116	−	0.994669i	$-0.532881\pi$
$431$	−4.28147	−0.206231	−0.103116	+	0.994669i	$0.532881\pi$
$432$	0	0
$433$	18.2035	0.874804	0.437402	−	0.899266i	$-0.355899\pi$
$433$	18.2035	0.874804	0.437402	+	0.899266i	$0.355899\pi$
$434$	15.5945	0.748558
$435$	0	0
$436$	−6.97756	−0.334165
$437$	7.45793	0.356761
$438$	0	0
$439$	29.4442	1.40529	0.702647	−	0.711538i	$-0.252001\pi$
$439$	29.4442	1.40529	0.702647	+	0.711538i	$0.252001\pi$
$440$	−30.1438	−1.43705
$441$	0	0
$442$	−12.6352	−0.600997
$443$	24.3633	1.15753	0.578767	−	0.815493i	$-0.303534\pi$
$443$	24.3633	1.15753	0.578767	+	0.815493i	$0.303534\pi$
$444$	0	0
$445$	49.4748	2.34533
$446$	−0.686514	−0.0325074
$447$	0	0
$448$	29.0384	1.37193
$449$	−38.7776	−1.83003	−0.915015	−	0.403421i	$-0.867821\pi$
$449$	−38.7776	−1.83003	−0.915015	+	0.403421i	$0.867821\pi$
$450$	0	0
$451$	3.92451	0.184798
$452$	58.1306	2.73423
$453$	0	0
$454$	−71.4102	−3.35145
$455$	21.8105	1.02249
$456$	0	0
$457$	−7.47672	−0.349746	−0.174873	−	0.984591i	$-0.555952\pi$
$457$	−7.47672	−0.349746	−0.174873	+	0.984591i	$0.555952\pi$
$458$	−14.4560	−0.675487
$459$	0	0
$460$	−146.834	−6.84618
$461$	−15.9782	−0.744181	−0.372091	−	0.928196i	$-0.621359\pi$
$461$	−15.9782	−0.744181	−0.372091	+	0.928196i	$0.621359\pi$
$462$	0	0
$463$	−6.10221	−0.283594	−0.141797	−	0.989896i	$-0.545288\pi$
$463$	−6.10221	−0.283594	−0.141797	+	0.989896i	$0.545288\pi$
$464$	−9.90896	−0.460012
$465$	0	0
$466$	−71.7961	−3.32589
$467$	26.6373	1.23263	0.616313	−	0.787502i	$-0.288626\pi$
$467$	26.6373	1.23263	0.616313	+	0.787502i	$0.288626\pi$
$468$	0	0
$469$	−13.4235	−0.619838
$470$	−39.5896	−1.82613
$471$	0	0
$472$	−98.3298	−4.52599
$473$	5.38113	0.247424
$474$	0	0
$475$	11.6203	0.533176
$476$	57.1406	2.61904
$477$	0	0
$478$	−3.66750	−0.167747
$479$	16.2094	0.740626	0.370313	−	0.928907i	$-0.379250\pi$
$479$	16.2094	0.740626	0.370313	+	0.928907i	$0.379250\pi$
$480$	0	0
$481$	9.92553	0.452565
$482$	−54.2773	−2.47226
$483$	0	0
$484$	4.82936	0.219516
$485$	56.2231	2.55296
$486$	0	0
$487$	−4.82448	−0.218618	−0.109309	−	0.994008i	$-0.534864\pi$
$487$	−4.82448	−0.218618	−0.109309	+	0.994008i	$0.534864\pi$
$488$	−48.0360	−2.17449
$489$	0	0
$490$	−64.9042	−2.93207
$491$	8.53579	0.385215	0.192607	−	0.981276i	$-0.438306\pi$
$491$	8.53579	0.385215	0.192607	+	0.981276i	$0.438306\pi$
$492$	0	0
$493$	−3.35293	−0.151008
$494$	−3.86395	−0.173847
$495$	0	0
$496$	15.9380	0.715635
$497$	−22.9026	−1.02732
$498$	0	0
$499$	−13.4325	−0.601319	−0.300660	−	0.953732i	$-0.597207\pi$
$499$	−13.4325	−0.601319	−0.300660	+	0.953732i	$0.597207\pi$
$500$	−130.343	−5.82910
$501$	0	0
$502$	3.39562	0.151554
$503$	1.66487	0.0742327	0.0371163	−	0.999311i	$-0.488183\pi$
$503$	1.66487	0.0742327	0.0371163	+	0.999311i	$0.488183\pi$
$504$	0	0
$505$	−44.8565	−1.99609
$506$	19.4898	0.866429
$507$	0	0
$508$	−22.6831	−1.00640
$509$	14.9684	0.663463	0.331731	−	0.943374i	$-0.392367\pi$
$509$	14.9684	0.663463	0.331731	+	0.943374i	$0.392367\pi$
$510$	0	0
$511$	−4.98452	−0.220502
$512$	−41.7577	−1.84545
$513$	0	0
$514$	−8.91709	−0.393316
$515$	20.3510	0.896772
$516$	0	0
$517$	3.71597	0.163428
$518$	−63.4754	−2.78895
$519$	0	0
$520$	44.5696	1.95451
$521$	7.89123	0.345721	0.172861	−	0.984946i	$-0.444699\pi$
$521$	7.89123	0.345721	0.172861	+	0.984946i	$0.444699\pi$
$522$	0	0
$523$	−22.3062	−0.975381	−0.487690	−	0.873017i	$-0.662160\pi$
$523$	−22.3062	−0.975381	−0.487690	+	0.873017i	$0.662160\pi$
$524$	−18.0843	−0.790017
$525$	0	0
$526$	−37.2182	−1.62279
$527$	5.39297	0.234922
$528$	0	0
$529$	32.6207	1.41829
$530$	−1.09631	−0.0476206
$531$	0	0
$532$	17.4740	0.757595
$533$	−5.80266	−0.251341
$534$	0	0
$535$	29.8155	1.28904
$536$	−27.4308	−1.18483
$537$	0	0
$538$	−14.0982	−0.607815
$539$	6.09205	0.262403
$540$	0	0
$541$	38.9694	1.67542	0.837712	−	0.546112i	$-0.183893\pi$
$541$	38.9694	1.67542	0.837712	+	0.546112i	$0.183893\pi$
$542$	−28.9095	−1.24177
$543$	0	0
$544$	34.2273	1.46749
$545$	5.89025	0.252311
$546$	0	0
$547$	37.7503	1.61409	0.807043	−	0.590492i	$-0.201066\pi$
$547$	37.7503	1.61409	0.807043	+	0.590492i	$0.201066\pi$
$548$	76.1083	3.25119
$549$	0	0
$550$	30.3674	1.29487
$551$	−1.02535	−0.0436814
$552$	0	0
$553$	49.4713	2.10373
$554$	36.7975	1.56338
$555$	0	0
$556$	−12.2097	−0.517805
$557$	3.17436	0.134502	0.0672511	−	0.997736i	$-0.478577\pi$
$557$	3.17436	0.134502	0.0672511	+	0.997736i	$0.478577\pi$
$558$	0	0
$559$	−7.95637	−0.336519
$560$	−142.554	−6.02401
$561$	0	0
$562$	28.2756	1.19273
$563$	19.9431	0.840503	0.420252	−	0.907408i	$-0.361942\pi$
$563$	19.9431	0.840503	0.420252	+	0.907408i	$0.361942\pi$
$564$	0	0
$565$	−49.0721	−2.06448
$566$	4.99004	0.209747
$567$	0	0
$568$	−46.8015	−1.96375
$569$	−36.6424	−1.53613	−0.768064	−	0.640374i	$-0.778780\pi$
$569$	−36.6424	−1.53613	−0.768064	+	0.640374i	$0.778780\pi$
$570$	0	0
$571$	11.5300	0.482515	0.241258	−	0.970461i	$-0.422440\pi$
$571$	11.5300	0.482515	0.241258	+	0.970461i	$0.422440\pi$
$572$	−7.14054	−0.298561
$573$	0	0
$574$	37.1090	1.54890
$575$	86.6634	3.61411
$576$	0	0
$577$	28.5590	1.18893	0.594463	−	0.804123i	$-0.297365\pi$
$577$	28.5590	1.18893	0.594463	+	0.804123i	$0.297365\pi$
$578$	−16.4818	−0.685554
$579$	0	0
$580$	20.1874	0.838237
$581$	−19.6857	−0.816701
$582$	0	0
$583$	0.102902	0.00426176
$584$	−10.1859	−0.421494
$585$	0	0
$586$	−9.50067	−0.392469
$587$	−18.1461	−0.748969	−0.374484	−	0.927233i	$-0.622180\pi$
$587$	−18.1461	−0.748969	−0.374484	+	0.927233i	$0.622180\pi$
$588$	0	0
$589$	1.64921	0.0679546
$590$	141.683	5.83298
$591$	0	0
$592$	−64.8736	−2.66629
$593$	−15.3085	−0.628644	−0.314322	−	0.949316i	$-0.601777\pi$
$593$	−15.3085	−0.628644	−0.314322	+	0.949316i	$0.601777\pi$
$594$	0	0
$595$	−48.2364	−1.97750
$596$	−8.92957	−0.365769
$597$	0	0
$598$	−28.8171	−1.17842
$599$	−17.8806	−0.730580	−0.365290	−	0.930894i	$-0.619030\pi$
$599$	−17.8806	−0.730580	−0.365290	+	0.930894i	$0.619030\pi$
$600$	0	0
$601$	−32.5803	−1.32898	−0.664489	−	0.747298i	$-0.731351\pi$
$601$	−32.5803	−1.32898	−0.664489	+	0.747298i	$0.731351\pi$
$602$	50.8823	2.07381
$603$	0	0
$604$	−73.9779	−3.01012
$605$	−4.07680	−0.165746
$606$	0	0
$607$	43.6494	1.77167	0.885837	−	0.463997i	$-0.153585\pi$
$607$	43.6494	1.77167	0.885837	+	0.463997i	$0.153585\pi$
$608$	10.4670	0.424492
$609$	0	0
$610$	69.2147	2.80242
$611$	−5.49432	−0.222276
$612$	0	0
$613$	0.843061	0.0340509	0.0170254	−	0.999855i	$-0.494580\pi$
$613$	0.843061	0.0340509	0.0170254	+	0.999855i	$0.494580\pi$
$614$	−61.4987	−2.48189
$615$	0	0
$616$	26.7536	1.07793
$617$	−4.89882	−0.197219	−0.0986094	−	0.995126i	$-0.531439\pi$
$617$	−4.89882	−0.197219	−0.0986094	+	0.995126i	$0.531439\pi$
$618$	0	0
$619$	−14.8704	−0.597691	−0.298846	−	0.954301i	$-0.596602\pi$
$619$	−14.8704	−0.597691	−0.298846	+	0.954301i	$0.596602\pi$
$620$	−32.4702	−1.30404
$621$	0	0
$622$	41.8197	1.67682
$623$	−43.9105	−1.75924
$624$	0	0
$625$	51.9299	2.07720
$626$	41.8217	1.67153
$627$	0	0
$628$	119.107	4.75289
$629$	−21.9515	−0.875263
$630$	0	0
$631$	2.83922	0.113027	0.0565137	−	0.998402i	$-0.482002\pi$
$631$	2.83922	0.113027	0.0565137	+	0.998402i	$0.482002\pi$
$632$	101.094	4.02132
$633$	0	0
$634$	54.5569	2.16673
$635$	19.1484	0.759881
$636$	0	0
$637$	−9.00751	−0.356891
$638$	−2.67955	−0.106084
$639$	0	0
$640$	−0.158757	−0.00627541
$641$	−20.6746	−0.816599	−0.408299	−	0.912848i	$-0.633878\pi$
$641$	−20.6746	−0.816599	−0.408299	+	0.912848i	$0.633878\pi$
$642$	0	0
$643$	−24.6254	−0.971130	−0.485565	−	0.874201i	$-0.661386\pi$
$643$	−24.6254	−0.971130	−0.485565	+	0.874201i	$0.661386\pi$
$644$	130.320	5.13533
$645$	0	0
$646$	8.54558	0.336222
$647$	−45.3626	−1.78339	−0.891693	−	0.452640i	$-0.850482\pi$
$647$	−45.3626	−1.78339	−0.891693	+	0.452640i	$0.850482\pi$
$648$	0	0
$649$	−13.2986	−0.522017
$650$	−44.9003	−1.76113
$651$	0	0
$652$	−17.9724	−0.703854
$653$	−26.1012	−1.02142	−0.510710	−	0.859753i	$-0.670617\pi$
$653$	−26.1012	−1.02142	−0.510710	+	0.859753i	$0.670617\pi$
$654$	0	0
$655$	15.2662	0.596501
$656$	37.9264	1.48078
$657$	0	0
$658$	35.1371	1.36979
$659$	45.8507	1.78609	0.893045	−	0.449967i	$-0.148564\pi$
$659$	45.8507	1.78609	0.893045	+	0.449967i	$0.148564\pi$
$660$	0	0
$661$	−15.4225	−0.599864	−0.299932	−	0.953961i	$-0.596964\pi$
$661$	−15.4225	−0.599864	−0.299932	+	0.953961i	$0.596964\pi$
$662$	39.4965	1.53508
$663$	0	0
$664$	−40.2277	−1.56114
$665$	−14.7511	−0.572022
$666$	0	0
$667$	−7.64698	−0.296092
$668$	−12.7861	−0.494709
$669$	0	0
$670$	39.5249	1.52698
$671$	−6.49664	−0.250800
$672$	0	0
$673$	−37.8633	−1.45952	−0.729762	−	0.683702i	$-0.760369\pi$
$673$	−37.8633	−1.45952	−0.729762	+	0.683702i	$0.760369\pi$
$674$	−32.0826	−1.23578
$675$	0	0
$676$	−52.2239	−2.00861
$677$	25.3510	0.974320	0.487160	−	0.873313i	$-0.338033\pi$
$677$	25.3510	0.974320	0.487160	+	0.873313i	$0.338033\pi$
$678$	0	0
$679$	−49.8998	−1.91498
$680$	−98.5710	−3.78003
$681$	0	0
$682$	4.30989	0.165034
$683$	21.7513	0.832290	0.416145	−	0.909298i	$-0.363381\pi$
$683$	21.7513	0.832290	0.416145	+	0.909298i	$0.363381\pi$
$684$	0	0
$685$	−64.2484	−2.45480
$686$	−8.58534	−0.327790
$687$	0	0
$688$	52.0031	1.98260
$689$	−0.152147	−0.00579636
$690$	0	0
$691$	17.3051	0.658318	0.329159	−	0.944275i	$-0.393235\pi$
$691$	17.3051	0.658318	0.329159	+	0.944275i	$0.393235\pi$
$692$	−35.4741	−1.34852
$693$	0	0
$694$	80.7687	3.06594
$695$	10.3070	0.390968
$696$	0	0
$697$	12.8333	0.486095
$698$	−62.4643	−2.36431
$699$	0	0
$700$	203.054	7.67470
$701$	29.6923	1.12146	0.560732	−	0.827997i	$-0.310520\pi$
$701$	29.6923	1.12146	0.560732	+	0.827997i	$0.310520\pi$
$702$	0	0
$703$	−6.71293	−0.253183
$704$	8.02543	0.302470
$705$	0	0
$706$	−39.2409	−1.47685
$707$	39.8116	1.49727
$708$	0	0
$709$	−21.4898	−0.807067	−0.403534	−	0.914965i	$-0.632218\pi$
$709$	−21.4898	−0.807067	−0.403534	+	0.914965i	$0.632218\pi$
$710$	67.4359	2.53082
$711$	0	0
$712$	−89.7310	−3.36281
$713$	12.2997	0.460627
$714$	0	0
$715$	6.02783	0.225428
$716$	−46.1394	−1.72431
$717$	0	0
$718$	−36.8021	−1.37344
$719$	−9.61388	−0.358537	−0.179269	−	0.983800i	$-0.557373\pi$
$719$	−9.61388	−0.358537	−0.179269	+	0.983800i	$0.557373\pi$
$720$	0	0
$721$	−18.0622	−0.672671
$722$	2.61330	0.0972571
$723$	0	0
$724$	29.1036	1.08163
$725$	−11.9149	−0.442507
$726$	0	0
$727$	6.84046	0.253699	0.126849	−	0.991922i	$-0.459514\pi$
$727$	6.84046	0.253699	0.126849	+	0.991922i	$0.459514\pi$
$728$	−39.5570	−1.46608
$729$	0	0
$730$	14.6767	0.543210
$731$	17.5965	0.650828
$732$	0	0
$733$	38.2277	1.41197	0.705987	−	0.708225i	$-0.250504\pi$
$733$	38.2277	1.41197	0.705987	+	0.708225i	$0.250504\pi$
$734$	55.7005	2.05594
$735$	0	0
$736$	78.0620	2.87740
$737$	−3.70989	−0.136656
$738$	0	0
$739$	−17.7157	−0.651682	−0.325841	−	0.945425i	$-0.605647\pi$
$739$	−17.7157	−0.651682	−0.325841	+	0.945425i	$0.605647\pi$
$740$	132.166	4.85853
$741$	0	0
$742$	0.973009	0.0357203
$743$	−21.3028	−0.781525	−0.390763	−	0.920491i	$-0.627789\pi$
$743$	−21.3028	−0.781525	−0.390763	+	0.920491i	$0.627789\pi$
$744$	0	0
$745$	7.53808	0.276174
$746$	−6.32650	−0.231630
$747$	0	0
$748$	15.7921	0.577418
$749$	−26.4622	−0.966908
$750$	0	0
$751$	25.1552	0.917926	0.458963	−	0.888455i	$-0.348221\pi$
$751$	25.1552	0.917926	0.458963	+	0.888455i	$0.348221\pi$
$752$	35.9111	1.30954
$753$	0	0
$754$	3.96190	0.144284
$755$	62.4499	2.27279
$756$	0	0
$757$	15.5116	0.563780	0.281890	−	0.959447i	$-0.409039\pi$
$757$	15.5116	0.563780	0.281890	+	0.959447i	$0.409039\pi$
$758$	23.1940	0.842443
$759$	0	0
$760$	−30.1438	−1.09343
$761$	−35.2085	−1.27631	−0.638154	−	0.769908i	$-0.720302\pi$
$761$	−35.2085	−1.27631	−0.638154	+	0.769908i	$0.720302\pi$
$762$	0	0
$763$	−5.22779	−0.189259
$764$	−85.5129	−3.09375
$765$	0	0
$766$	9.25280	0.334317
$767$	19.6630	0.709988
$768$	0	0
$769$	−5.57667	−0.201100	−0.100550	−	0.994932i	$-0.532060\pi$
$769$	−5.57667	−0.201100	−0.100550	+	0.994932i	$0.532060\pi$
$770$	−38.5490	−1.38921
$771$	0	0
$772$	17.8371	0.641973
$773$	−35.9175	−1.29186	−0.645931	−	0.763396i	$-0.723531\pi$
$773$	−35.9175	−1.29186	−0.645931	+	0.763396i	$0.723531\pi$
$774$	0	0
$775$	19.1643	0.688403
$776$	−101.970	−3.66052
$777$	0	0
$778$	43.6530	1.56503
$779$	3.92451	0.140610
$780$	0	0
$781$	−6.32968	−0.226494
$782$	63.7324	2.27906
$783$	0	0
$784$	58.8734	2.10262
$785$	−100.547	−3.58866
$786$	0	0
$787$	7.53242	0.268502	0.134251	−	0.990947i	$-0.457137\pi$
$787$	7.53242	0.268502	0.134251	+	0.990947i	$0.457137\pi$
$788$	124.496	4.43497
$789$	0	0
$790$	−145.666	−5.18257
$791$	43.5531	1.54857
$792$	0	0
$793$	9.60573	0.341110
$794$	73.7141	2.61602
$795$	0	0
$796$	91.2522	3.23435
$797$	49.3837	1.74926	0.874629	−	0.484792i	$-0.161105\pi$
$797$	49.3837	1.74926	0.874629	+	0.484792i	$0.161105\pi$
$798$	0	0
$799$	12.1513	0.429883
$800$	121.629	4.30025
$801$	0	0
$802$	94.6219	3.34122
$803$	−1.37759	−0.0486141
$804$	0	0
$805$	−110.012	−3.87743
$806$	−6.37247	−0.224461
$807$	0	0
$808$	81.3549	2.86205
$809$	−34.4637	−1.21168	−0.605840	−	0.795587i	$-0.707163\pi$
$809$	−34.4637	−1.21168	−0.605840	+	0.795587i	$0.707163\pi$
$810$	0	0
$811$	20.0278	0.703272	0.351636	−	0.936137i	$-0.385625\pi$
$811$	20.0278	0.703272	0.351636	+	0.936137i	$0.385625\pi$
$812$	−17.9170	−0.628763
$813$	0	0
$814$	−17.5429	−0.614879
$815$	15.1718	0.531444
$816$	0	0
$817$	5.38113	0.188262
$818$	96.4926	3.37379
$819$	0	0
$820$	−77.2670	−2.69828
$821$	−6.45245	−0.225192	−0.112596	−	0.993641i	$-0.535917\pi$
$821$	−6.45245	−0.225192	−0.112596	+	0.993641i	$0.535917\pi$
$822$	0	0
$823$	−43.0190	−1.49955	−0.749774	−	0.661694i	$-0.769838\pi$
$823$	−43.0190	−1.49955	−0.749774	+	0.661694i	$0.769838\pi$
$824$	−36.9100	−1.28582
$825$	0	0
$826$	−125.748	−4.37533
$827$	43.1557	1.50067	0.750335	−	0.661058i	$-0.229892\pi$
$827$	43.1557	1.50067	0.750335	+	0.661058i	$0.229892\pi$
$828$	0	0
$829$	13.4937	0.468656	0.234328	−	0.972158i	$-0.424711\pi$
$829$	13.4937	0.468656	0.234328	+	0.972158i	$0.424711\pi$
$830$	57.9638	2.01195
$831$	0	0
$832$	−11.8662	−0.411385
$833$	19.9212	0.690228
$834$	0	0
$835$	10.7937	0.373530
$836$	4.82936	0.167027
$837$	0	0
$838$	−47.2197	−1.63118
$839$	14.6851	0.506988	0.253494	−	0.967337i	$-0.418420\pi$
$839$	14.6851	0.506988	0.253494	+	0.967337i	$0.418420\pi$
$840$	0	0
$841$	−27.9487	−0.963747
$842$	−22.4124	−0.772384
$843$	0	0
$844$	−49.2562	−1.69547
$845$	44.0858	1.51660
$846$	0	0
$847$	3.61829	0.124326
$848$	0.994441	0.0341493
$849$	0	0
$850$	99.3023	3.40604
$851$	−50.0645	−1.71619
$852$	0	0
$853$	−51.5775	−1.76598	−0.882990	−	0.469392i	$-0.844473\pi$
$853$	−51.5775	−1.76598	−0.882990	+	0.469392i	$0.844473\pi$
$854$	−61.4303	−2.10210
$855$	0	0
$856$	−54.0755	−1.84826
$857$	46.6355	1.59304	0.796520	−	0.604612i	$-0.206672\pi$
$857$	46.6355	1.59304	0.796520	+	0.604612i	$0.206672\pi$
$858$	0	0
$859$	−18.6711	−0.637048	−0.318524	−	0.947915i	$-0.603187\pi$
$859$	−18.6711	−0.637048	−0.318524	+	0.947915i	$0.603187\pi$
$860$	−105.945	−3.61271
$861$	0	0
$862$	−11.1888	−0.381091
$863$	−43.0160	−1.46428	−0.732141	−	0.681153i	$-0.761479\pi$
$863$	−43.0160	−1.46428	−0.732141	+	0.681153i	$0.761479\pi$
$864$	0	0
$865$	29.9462	1.01820
$866$	47.5712	1.61654
$867$	0	0
$868$	28.8184	0.978160
$869$	13.6725	0.463809
$870$	0	0
$871$	5.48533	0.185863
$872$	−10.6830	−0.361771
$873$	0	0
$874$	19.4898	0.659253
$875$	−97.6565	−3.30139
$876$	0	0
$877$	56.0429	1.89243	0.946217	−	0.323534i	$-0.104871\pi$
$877$	56.0429	1.89243	0.946217	+	0.323534i	$0.104871\pi$
$878$	76.9466	2.59682
$879$	0	0
$880$	−39.3981	−1.32811
$881$	−17.9947	−0.606257	−0.303128	−	0.952950i	$-0.598031\pi$
$881$	−17.9947	−0.606257	−0.303128	+	0.952950i	$0.598031\pi$
$882$	0	0
$883$	1.99550	0.0671540	0.0335770	−	0.999436i	$-0.489310\pi$
$883$	1.99550	0.0671540	0.0335770	+	0.999436i	$0.489310\pi$
$884$	−23.3498	−0.785339
$885$	0	0
$886$	63.6687	2.13899
$887$	−7.20927	−0.242063	−0.121032	−	0.992649i	$-0.538620\pi$
$887$	−7.20927	−0.242063	−0.121032	+	0.992649i	$0.538620\pi$
$888$	0	0
$889$	−16.9948	−0.569988
$890$	129.293	4.33390
$891$	0	0
$892$	−1.26867	−0.0424782
$893$	3.71597	0.124350
$894$	0	0
$895$	38.9495	1.30194
$896$	0.140902	0.00470720
$897$	0	0
$898$	−101.338	−3.38168
$899$	−1.69102	−0.0563986
$900$	0	0
$901$	0.336492	0.0112102
$902$	10.2559	0.341485
$903$	0	0
$904$	89.0006	2.96012
$905$	−24.5684	−0.816680
$906$	0	0
$907$	−25.7832	−0.856116	−0.428058	−	0.903751i	$-0.640802\pi$
$907$	−25.7832	−0.856116	−0.428058	+	0.903751i	$0.640802\pi$
$908$	−131.965	−4.37942
$909$	0	0
$910$	56.9974	1.88945
$911$	−31.1334	−1.03149	−0.515747	−	0.856741i	$-0.672486\pi$
$911$	−31.1334	−1.03149	−0.515747	+	0.856741i	$0.672486\pi$
$912$	0	0
$913$	−5.44061	−0.180058
$914$	−19.5389	−0.646291
$915$	0	0
$916$	−26.7146	−0.882676
$917$	−13.5493	−0.447437
$918$	0	0
$919$	5.42725	0.179029	0.0895143	−	0.995986i	$-0.471469\pi$
$919$	5.42725	0.179029	0.0895143	+	0.995986i	$0.471469\pi$
$920$	−224.810	−7.41176
$921$	0	0
$922$	−41.7560	−1.37516
$923$	9.35887	0.308051
$924$	0	0
$925$	−78.0063	−2.56483
$926$	−15.9469	−0.524049
$927$	0	0
$928$	−10.7323	−0.352305
$929$	21.6025	0.708756	0.354378	−	0.935102i	$-0.384693\pi$
$929$	21.6025	0.708756	0.354378	+	0.935102i	$0.384693\pi$
$930$	0	0
$931$	6.09205	0.199659
$932$	−132.678	−4.34603
$933$	0	0
$934$	69.6113	2.27775
$935$	−13.3313	−0.435979
$936$	0	0
$937$	31.6840	1.03507	0.517536	−	0.855661i	$-0.326849\pi$
$937$	31.6840	1.03507	0.517536	+	0.855661i	$0.326849\pi$
$938$	−35.0796	−1.14539
$939$	0	0
$940$	−73.1612	−2.38626
$941$	5.63987	0.183854	0.0919272	−	0.995766i	$-0.470697\pi$
$941$	5.63987	0.183854	0.0919272	+	0.995766i	$0.470697\pi$
$942$	0	0
$943$	29.2687	0.953120
$944$	−128.518	−4.18290
$945$	0	0
$946$	14.0625	0.457212
$947$	51.7369	1.68122	0.840611	−	0.541639i	$-0.182196\pi$
$947$	51.7369	1.68122	0.840611	+	0.541639i	$0.182196\pi$
$948$	0	0
$949$	2.03686	0.0661194
$950$	30.3674	0.985248
$951$	0	0
$952$	87.4850	2.83540
$953$	16.2553	0.526561	0.263281	−	0.964719i	$-0.415196\pi$
$953$	16.2553	0.526561	0.263281	+	0.964719i	$0.415196\pi$
$954$	0	0
$955$	72.1874	2.33593
$956$	−6.77750	−0.219200
$957$	0	0
$958$	42.3601	1.36859
$959$	57.0225	1.84135
$960$	0	0
$961$	−28.2801	−0.912261
$962$	25.9384	0.836289
$963$	0	0
$964$	−100.304	−3.23057
$965$	−15.0576	−0.484721
$966$	0	0
$967$	−54.5004	−1.75261	−0.876307	−	0.481754i	$-0.840000\pi$
$967$	−54.5004	−1.75261	−0.876307	+	0.481754i	$0.840000\pi$
$968$	7.39397	0.237651
$969$	0	0
$970$	146.928	4.71758
$971$	34.2679	1.09971	0.549855	−	0.835260i	$-0.314683\pi$
$971$	34.2679	1.09971	0.549855	+	0.835260i	$0.314683\pi$
$972$	0	0
$973$	−9.14783	−0.293266
$974$	−12.6078	−0.403981
$975$	0	0
$976$	−62.7834	−2.00965
$977$	−55.5644	−1.77766	−0.888831	−	0.458234i	$-0.848482\pi$
$977$	−55.5644	−1.77766	−0.888831	+	0.458234i	$0.848482\pi$
$978$	0	0
$979$	−12.1357	−0.387858
$980$	−119.942	−3.83141
$981$	0	0
$982$	22.3066	0.711832
$983$	41.3971	1.32036	0.660181	−	0.751106i	$-0.270479\pi$
$983$	41.3971	1.32036	0.660181	+	0.751106i	$0.270479\pi$
$984$	0	0
$985$	−105.095	−3.34862
$986$	−8.76221	−0.279046
$987$	0	0
$988$	−7.14054	−0.227171
$989$	40.1321	1.27613
$990$	0	0
$991$	60.8219	1.93207	0.966036	−	0.258406i	$-0.0831973\pi$
$991$	60.8219	1.93207	0.966036	+	0.258406i	$0.0831973\pi$
$992$	17.2623	0.548077
$993$	0	0
$994$	−59.8515	−1.89837
$995$	−77.0324	−2.44209
$996$	0	0
$997$	26.0627	0.825414	0.412707	−	0.910864i	$-0.364583\pi$
$997$	26.0627	0.825414	0.412707	+	0.910864i	$0.364583\pi$
$998$	−35.1031	−1.11117
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.p.1.6		7
3.2	odd	2		209.2.a.d.1.2	✓	7
12.11	even	2		3344.2.a.ba.1.4		7
15.14	odd	2		5225.2.a.n.1.6		7
33.32	even	2		2299.2.a.q.1.6		7
57.56	even	2		3971.2.a.i.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.d.1.2	✓	7	3.2	odd	2
1881.2.a.p.1.6		7	1.1	even	1	trivial
2299.2.a.q.1.6		7	33.32	even	2
3344.2.a.ba.1.4		7	12.11	even	2
3971.2.a.i.1.6		7	57.56	even	2
5225.2.a.n.1.6		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)

\(f(q)\)	\(=\)	\(q+2.61330 q^{2} +4.82936 q^{4} -4.07680 q^{5} +3.61829 q^{7} +7.39397 q^{8} -10.6539 q^{10} +1.00000 q^{11} -1.47857 q^{13} +9.45570 q^{14} +9.66398 q^{16} +3.27003 q^{17} +1.00000 q^{19} -19.6883 q^{20} +2.61330 q^{22} +7.45793 q^{23} +11.6203 q^{25} -3.86395 q^{26} +17.4740 q^{28} -1.02535 q^{29} +1.64921 q^{31} +10.4670 q^{32} +8.54558 q^{34} -14.7511 q^{35} -6.71293 q^{37} +2.61330 q^{38} -30.1438 q^{40} +3.92451 q^{41} +5.38113 q^{43} +4.82936 q^{44} +19.4898 q^{46} +3.71597 q^{47} +6.09205 q^{49} +30.3674 q^{50} -7.14054 q^{52} +0.102902 q^{53} -4.07680 q^{55} +26.7536 q^{56} -2.67955 q^{58} -13.2986 q^{59} -6.49664 q^{61} +4.30989 q^{62} +8.02543 q^{64} +6.02783 q^{65} -3.70989 q^{67} +15.7921 q^{68} -38.5490 q^{70} -6.32968 q^{71} -1.37759 q^{73} -17.5429 q^{74} +4.82936 q^{76} +3.61829 q^{77} +13.6725 q^{79} -39.3981 q^{80} +10.2559 q^{82} -5.44061 q^{83} -13.3313 q^{85} +14.0625 q^{86} +7.39397 q^{88} -12.1357 q^{89} -5.34990 q^{91} +36.0170 q^{92} +9.71096 q^{94} -4.07680 q^{95} -13.7910 q^{97} +15.9204 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 15 q^{4} - 2 q^{5} + 10 q^{7} + 9 q^{8} - 6 q^{10} + 7 q^{11} - 4 q^{13} - 6 q^{14} + 27 q^{16} - 2 q^{17} + 7 q^{19} + 4 q^{20} + q^{22} - 10 q^{23} + 9 q^{25} + 8 q^{26} + 26 q^{28} + 18 q^{29}+ \cdots - 19 q^{98}+O(q^{100}) \) 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

Fields

Representations

Groups

Database

Properties

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