LMFDB - Embedded newform 2541.2.a.br.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2541,2,Mod(1,2541)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2541, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2541.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2541 = 3 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2541.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$20.2899871536$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 19x^{8} - x^{7} + 124x^{6} + 6x^{5} - 316x^{4} + 17x^{3} + 253x^{2} - 70x - 11 $ x^10 - 19x^8 - x^7 + 124x^6 + 6x^5 - 316x^4 + 17x^3 + 253x^2 - 70*x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.871604$ of defining polynomial
Character	$\chi$	$=$	2541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.871604 q^{2} +1.00000 q^{3} -1.24031 q^{4} -4.06436 q^{5} -0.871604 q^{6} +1.00000 q^{7} +2.82426 q^{8} +1.00000 q^{9} +3.54252 q^{10} -1.24031 q^{12} +6.09006 q^{13} -0.871604 q^{14} -4.06436 q^{15} +0.0189705 q^{16} -1.70162 q^{17} -0.871604 q^{18} -5.45712 q^{19} +5.04106 q^{20} +1.00000 q^{21} -6.39153 q^{23} +2.82426 q^{24} +11.5191 q^{25} -5.30813 q^{26} +1.00000 q^{27} -1.24031 q^{28} -4.74935 q^{29} +3.54252 q^{30} -4.36309 q^{31} -5.66506 q^{32} +1.48314 q^{34} -4.06436 q^{35} -1.24031 q^{36} +2.43952 q^{37} +4.75645 q^{38} +6.09006 q^{39} -11.4788 q^{40} +3.21037 q^{41} -0.871604 q^{42} -0.127191 q^{43} -4.06436 q^{45} +5.57089 q^{46} +8.46784 q^{47} +0.0189705 q^{48} +1.00000 q^{49} -10.0401 q^{50} -1.70162 q^{51} -7.55354 q^{52} -4.71404 q^{53} -0.871604 q^{54} +2.82426 q^{56} -5.45712 q^{57} +4.13955 q^{58} +4.63368 q^{59} +5.04106 q^{60} -5.31081 q^{61} +3.80289 q^{62} +1.00000 q^{63} +4.89975 q^{64} -24.7522 q^{65} +14.0686 q^{67} +2.11054 q^{68} -6.39153 q^{69} +3.54252 q^{70} +2.21883 q^{71} +2.82426 q^{72} +5.57393 q^{73} -2.12630 q^{74} +11.5191 q^{75} +6.76849 q^{76} -5.30813 q^{78} -6.27134 q^{79} -0.0771030 q^{80} +1.00000 q^{81} -2.79817 q^{82} +0.127722 q^{83} -1.24031 q^{84} +6.91602 q^{85} +0.110860 q^{86} -4.74935 q^{87} -8.12736 q^{89} +3.54252 q^{90} +6.09006 q^{91} +7.92745 q^{92} -4.36309 q^{93} -7.38061 q^{94} +22.1797 q^{95} -5.66506 q^{96} -2.59721 q^{97} -0.871604 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{3} + 18 q^{4} + 5 q^{5} + 10 q^{7} - 3 q^{8} + 10 q^{9} - 6 q^{10} + 18 q^{12} + 6 q^{13} + 5 q^{15} + 38 q^{16} + 8 q^{17} + 7 q^{20} + 10 q^{21} - 3 q^{24} + 31 q^{25} + q^{26} + 10 q^{27}+ \cdots + 24 q^{97}+O(q^{100}) $ 10 * q + 10 * q^3 + 18 * q^4 + 5 * q^5 + 10 * q^7 - 3 * q^8 + 10 * q^9 - 6 * q^10 + 18 * q^12 + 6 * q^13 + 5 * q^15 + 38 * q^16 + 8 * q^17 + 7 * q^20 + 10 * q^21 - 3 * q^24 + 31 * q^25 + q^26 + 10 * q^27 + 18 * q^28 - 14 * q^29 - 6 * q^30 + 26 * q^31 - 41 * q^32 + 21 * q^34 + 5 * q^35 + 18 * q^36 + 24 * q^37 + 8 * q^38 + 6 * q^39 - 5 * q^40 + 19 * q^41 - 6 * q^43 + 5 * q^45 - q^46 + 15 * q^47 + 38 * q^48 + 10 * q^49 - q^50 + 8 * q^51 - 25 * q^52 - q^53 - 3 * q^56 + 11 * q^58 + 23 * q^59 + 7 * q^60 + 11 * q^62 + 10 * q^63 + 53 * q^64 - 29 * q^65 + 38 * q^67 + 87 * q^68 - 6 * q^70 + 26 * q^71 - 3 * q^72 - q^73 - 39 * q^74 + 31 * q^75 - 2 * q^76 + q^78 + 5 * q^79 + 6 * q^80 + 10 * q^81 + 5 * q^82 + 6 * q^83 + 18 * q^84 - q^85 - 41 * q^86 - 14 * q^87 - 9 * q^89 - 6 * q^90 + 6 * q^91 - 48 * q^92 + 26 * q^93 - 42 * q^95 - 41 * q^96 + 24 * q^97

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 $

$2$	−0.871604	−0.616317	−0.308159	−	0.951335i	$-0.599713\pi$
$2$	−0.871604	−0.616317	−0.308159	+	0.951335i	$0.599713\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.24031	−0.620153
$5$	−4.06436	−1.81764	−0.908820	−	0.417189i	$-0.863015\pi$
$5$	−4.06436	−1.81764	−0.908820	+	0.417189i	$0.863015\pi$
$6$	−0.871604	−0.355831
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.82426	0.998528
$9$	1.00000	0.333333
$10$	3.54252	1.12024
$11$	0	0
$11$	0	0
$12$	−1.24031	−0.358045
$13$	6.09006	1.68908	0.844540	−	0.535492i	$-0.179874\pi$
$13$	6.09006	1.68908	0.844540	+	0.535492i	$0.179874\pi$
$14$	−0.871604	−0.232946
$15$	−4.06436	−1.04941
$16$	0.0189705	0.00474262
$17$	−1.70162	−0.412705	−0.206352	−	0.978478i	$-0.566159\pi$
$17$	−1.70162	−0.412705	−0.206352	+	0.978478i	$0.566159\pi$
$18$	−0.871604	−0.205439
$19$	−5.45712	−1.25195	−0.625974	−	0.779844i	$-0.715298\pi$
$19$	−5.45712	−1.25195	−0.625974	+	0.779844i	$0.715298\pi$
$20$	5.04106	1.12721
$21$	1.00000	0.218218
$22$	0	0
$23$	−6.39153	−1.33273	−0.666363	−	0.745627i	$-0.732150\pi$
$23$	−6.39153	−1.33273	−0.666363	+	0.745627i	$0.732150\pi$
$24$	2.82426	0.576501
$25$	11.5191	2.30381
$26$	−5.30813	−1.04101
$27$	1.00000	0.192450
$28$	−1.24031	−0.234396
$29$	−4.74935	−0.881931	−0.440966	−	0.897524i	$-0.645364\pi$
$29$	−4.74935	−0.881931	−0.440966	+	0.897524i	$0.645364\pi$
$30$	3.54252	0.646772
$31$	−4.36309	−0.783634	−0.391817	−	0.920043i	$-0.628153\pi$
$31$	−4.36309	−0.783634	−0.391817	+	0.920043i	$0.628153\pi$
$32$	−5.66506	−1.00145
$33$	0	0
$34$	1.48314	0.254357
$35$	−4.06436	−0.687003
$36$	−1.24031	−0.206718
$37$	2.43952	0.401055	0.200528	−	0.979688i	$-0.435734\pi$
$37$	2.43952	0.401055	0.200528	+	0.979688i	$0.435734\pi$
$38$	4.75645	0.771598
$39$	6.09006	0.975191
$40$	−11.4788	−1.81496
$41$	3.21037	0.501375	0.250688	−	0.968068i	$-0.419343\pi$
$41$	3.21037	0.501375	0.250688	+	0.968068i	$0.419343\pi$
$42$	−0.871604	−0.134491
$43$	−0.127191	−0.0193964	−0.00969821	−	0.999953i	$-0.503087\pi$
$43$	−0.127191	−0.0193964	−0.00969821	+	0.999953i	$0.503087\pi$
$44$	0	0
$45$	−4.06436	−0.605880
$46$	5.57089	0.821382
$47$	8.46784	1.23516	0.617581	−	0.786507i	$-0.288113\pi$
$47$	8.46784	1.23516	0.617581	+	0.786507i	$0.288113\pi$
$48$	0.0189705	0.00273815
$49$	1.00000	0.142857
$50$	−10.0401	−1.41988
$51$	−1.70162	−0.238275
$52$	−7.55354	−1.04749
$53$	−4.71404	−0.647523	−0.323761	−	0.946139i	$-0.604948\pi$
$53$	−4.71404	−0.647523	−0.323761	+	0.946139i	$0.604948\pi$
$54$	−0.871604	−0.118610
$55$	0	0
$56$	2.82426	0.377408
$57$	−5.45712	−0.722813
$58$	4.13955	0.543550
$59$	4.63368	0.603253	0.301627	−	0.953426i	$-0.402470\pi$
$59$	4.63368	0.603253	0.301627	+	0.953426i	$0.402470\pi$
$60$	5.04106	0.650797
$61$	−5.31081	−0.679980	−0.339990	−	0.940429i	$-0.610424\pi$
$61$	−5.31081	−0.679980	−0.339990	+	0.940429i	$0.610424\pi$
$62$	3.80289	0.482967
$63$	1.00000	0.125988
$64$	4.89975	0.612469
$65$	−24.7522	−3.07014
$66$	0	0
$67$	14.0686	1.71876	0.859379	−	0.511340i	$-0.170851\pi$
$67$	14.0686	1.71876	0.859379	+	0.511340i	$0.170851\pi$
$68$	2.11054	0.255940
$69$	−6.39153	−0.769450
$70$	3.54252	0.423412
$71$	2.21883	0.263327	0.131664	−	0.991294i	$-0.457968\pi$
$71$	2.21883	0.263327	0.131664	+	0.991294i	$0.457968\pi$
$72$	2.82426	0.332843
$73$	5.57393	0.652379	0.326189	−	0.945304i	$-0.394235\pi$
$73$	5.57393	0.652379	0.326189	+	0.945304i	$0.394235\pi$
$74$	−2.12630	−0.247177
$75$	11.5191	1.33011
$76$	6.76849	0.776400
$77$	0	0
$78$	−5.30813	−0.601027
$79$	−6.27134	−0.705581	−0.352790	−	0.935702i	$-0.614767\pi$
$79$	−6.27134	−0.705581	−0.352790	+	0.935702i	$0.614767\pi$
$80$	−0.0771030	−0.00862037
$81$	1.00000	0.111111
$82$	−2.79817	−0.309006
$83$	0.127722	0.0140193	0.00700965	−	0.999975i	$-0.497769\pi$
$83$	0.127722	0.0140193	0.00700965	+	0.999975i	$0.497769\pi$
$84$	−1.24031	−0.135328
$85$	6.91602	0.750148
$86$	0.110860	0.0119543
$87$	−4.74935	−0.509183
$88$	0	0
$89$	−8.12736	−0.861498	−0.430749	−	0.902472i	$-0.641751\pi$
$89$	−8.12736	−0.861498	−0.430749	+	0.902472i	$0.641751\pi$
$90$	3.54252	0.373414
$91$	6.09006	0.638412
$92$	7.92745	0.826494
$93$	−4.36309	−0.452431
$94$	−7.38061	−0.761252
$95$	22.1797	2.27559
$96$	−5.66506	−0.578188
$97$	−2.59721	−0.263706	−0.131853	−	0.991269i	$-0.542093\pi$
$97$	−2.59721	−0.263706	−0.131853	+	0.991269i	$0.542093\pi$
$98$	−0.871604	−0.0880453
$99$	0	0
$100$	−14.2872	−1.42872
$101$	6.02573	0.599583	0.299791	−	0.954005i	$-0.403083\pi$
$101$	6.02573	0.599583	0.299791	+	0.954005i	$0.403083\pi$
$102$	1.48314	0.146853
$103$	12.8938	1.27046	0.635232	−	0.772322i	$-0.280905\pi$
$103$	12.8938	1.27046	0.635232	+	0.772322i	$0.280905\pi$
$104$	17.2000	1.68659
$105$	−4.06436	−0.396641
$106$	4.10878	0.399080
$107$	5.34760	0.516972	0.258486	−	0.966015i	$-0.416776\pi$
$107$	5.34760	0.516972	0.258486	+	0.966015i	$0.416776\pi$
$108$	−1.24031	−0.119348
$109$	1.44173	0.138093	0.0690465	−	0.997613i	$-0.478004\pi$
$109$	1.44173	0.138093	0.0690465	+	0.997613i	$0.478004\pi$
$110$	0	0
$111$	2.43952	0.231549
$112$	0.0189705	0.00179254
$113$	19.5174	1.83604	0.918021	−	0.396532i	$-0.129786\pi$
$113$	19.5174	1.83604	0.918021	+	0.396532i	$0.129786\pi$
$114$	4.75645	0.445482
$115$	25.9775	2.42242
$116$	5.89064	0.546932
$117$	6.09006	0.563027
$118$	−4.03873	−0.371795
$119$	−1.70162	−0.155988
$120$	−11.4788	−1.04787
$121$	0	0
$122$	4.62893	0.419083
$123$	3.21037	0.289469
$124$	5.41156	0.485973
$125$	−26.4958	−2.36986
$126$	−0.871604	−0.0776487
$127$	0.641713	0.0569428	0.0284714	−	0.999595i	$-0.490936\pi$
$127$	0.641713	0.0569428	0.0284714	+	0.999595i	$0.490936\pi$
$128$	7.05948	0.623976
$129$	−0.127191	−0.0111985
$130$	21.5742	1.89218
$131$	13.9987	1.22307	0.611537	−	0.791216i	$-0.290552\pi$
$131$	13.9987	1.22307	0.611537	+	0.791216i	$0.290552\pi$
$132$	0	0
$133$	−5.45712	−0.473192
$134$	−12.2623	−1.05930
$135$	−4.06436	−0.349805
$136$	−4.80584	−0.412097
$137$	−2.70017	−0.230691	−0.115346	−	0.993325i	$-0.536798\pi$
$137$	−2.70017	−0.230691	−0.115346	+	0.993325i	$0.536798\pi$
$138$	5.57089	0.474225
$139$	2.06192	0.174890	0.0874448	−	0.996169i	$-0.472130\pi$
$139$	2.06192	0.174890	0.0874448	+	0.996169i	$0.472130\pi$
$140$	5.04106	0.426047
$141$	8.46784	0.713121
$142$	−1.93395	−0.162293
$143$	0	0
$144$	0.0189705	0.00158087
$145$	19.3031	1.60303
$146$	−4.85826	−0.402072
$147$	1.00000	0.0824786
$148$	−3.02576	−0.248716
$149$	−10.0829	−0.826027	−0.413013	−	0.910725i	$-0.635524\pi$
$149$	−10.0829	−0.826027	−0.413013	+	0.910725i	$0.635524\pi$
$150$	−10.0401	−0.819768
$151$	14.9639	1.21775	0.608873	−	0.793268i	$-0.291622\pi$
$151$	14.9639	1.21775	0.608873	+	0.793268i	$0.291622\pi$
$152$	−15.4123	−1.25011
$153$	−1.70162	−0.137568
$154$	0	0
$155$	17.7332	1.42436
$156$	−7.55354	−0.604767
$157$	19.4540	1.55260	0.776300	−	0.630364i	$-0.217094\pi$
$157$	19.4540	1.55260	0.776300	+	0.630364i	$0.217094\pi$
$158$	5.46613	0.434862
$159$	−4.71404	−0.373848
$160$	23.0249	1.82028
$161$	−6.39153	−0.503723
$162$	−0.871604	−0.0684797
$163$	12.1104	0.948562	0.474281	−	0.880374i	$-0.342708\pi$
$163$	12.1104	0.948562	0.474281	+	0.880374i	$0.342708\pi$
$164$	−3.98184	−0.310929
$165$	0	0
$166$	−0.111323	−0.00864034
$167$	13.1680	1.01897	0.509484	−	0.860480i	$-0.329836\pi$
$167$	13.1680	1.01897	0.509484	+	0.860480i	$0.329836\pi$
$168$	2.82426	0.217897
$169$	24.0889	1.85299
$170$	−6.02804	−0.462329
$171$	−5.45712	−0.417316
$172$	0.157755	0.0120287
$173$	14.7179	1.11898	0.559492	−	0.828836i	$-0.310996\pi$
$173$	14.7179	1.11898	0.559492	+	0.828836i	$0.310996\pi$
$174$	4.13955	0.313819
$175$	11.5191	0.870759
$176$	0	0
$177$	4.63368	0.348288
$178$	7.08384	0.530956
$179$	−11.6904	−0.873780	−0.436890	−	0.899515i	$-0.643920\pi$
$179$	−11.6904	−0.873780	−0.436890	+	0.899515i	$0.643920\pi$
$180$	5.04106	0.375738
$181$	19.7002	1.46431	0.732154	−	0.681139i	$-0.238515\pi$
$181$	19.7002	1.46431	0.732154	+	0.681139i	$0.238515\pi$
$182$	−5.30813	−0.393465
$183$	−5.31081	−0.392586
$184$	−18.0514	−1.33077
$185$	−9.91511	−0.728974
$186$	3.80289	0.278841
$187$	0	0
$188$	−10.5027	−0.765989
$189$	1.00000	0.0727393
$190$	−19.3319	−1.40249
$191$	6.96194	0.503748	0.251874	−	0.967760i	$-0.418953\pi$
$191$	6.96194	0.503748	0.251874	+	0.967760i	$0.418953\pi$
$192$	4.89975	0.353609
$193$	−16.8633	−1.21385	−0.606923	−	0.794761i	$-0.707596\pi$
$193$	−16.8633	−1.21385	−0.606923	+	0.794761i	$0.707596\pi$
$194$	2.26374	0.162527
$195$	−24.7522	−1.77254
$196$	−1.24031	−0.0885933
$197$	−6.42145	−0.457510	−0.228755	−	0.973484i	$-0.573465\pi$
$197$	−6.42145	−0.457510	−0.228755	+	0.973484i	$0.573465\pi$
$198$	0	0
$199$	−13.2645	−0.940297	−0.470149	−	0.882587i	$-0.655800\pi$
$199$	−13.2645	−0.940297	−0.470149	+	0.882587i	$0.655800\pi$
$200$	32.5329	2.30042
$201$	14.0686	0.992325
$202$	−5.25205	−0.369533
$203$	−4.74935	−0.333339
$204$	2.11054	0.147767
$205$	−13.0481	−0.911319
$206$	−11.2383	−0.783008
$207$	−6.39153	−0.444242
$208$	0.115531	0.00801067
$209$	0	0
$210$	3.54252	0.244457
$211$	−2.91158	−0.200442	−0.100221	−	0.994965i	$-0.531955\pi$
$211$	−2.91158	−0.200442	−0.100221	+	0.994965i	$0.531955\pi$
$212$	5.84685	0.401563
$213$	2.21883	0.152032
$214$	−4.66099	−0.318619
$215$	0.516950	0.0352557
$216$	2.82426	0.192167
$217$	−4.36309	−0.296186
$218$	−1.25662	−0.0851091
$219$	5.57393	0.376651
$220$	0	0
$221$	−10.3630	−0.697091
$222$	−2.12630	−0.142708
$223$	−3.35570	−0.224714	−0.112357	−	0.993668i	$-0.535840\pi$
$223$	−3.35570	−0.224714	−0.112357	+	0.993668i	$0.535840\pi$
$224$	−5.66506	−0.378513
$225$	11.5191	0.767937
$226$	−17.0114	−1.13158
$227$	−3.33791	−0.221545	−0.110772	−	0.993846i	$-0.535332\pi$
$227$	−3.33791	−0.221545	−0.110772	+	0.993846i	$0.535332\pi$
$228$	6.76849	0.448254
$229$	6.80143	0.449451	0.224726	−	0.974422i	$-0.427851\pi$
$229$	6.80143	0.449451	0.224726	+	0.974422i	$0.427851\pi$
$230$	−22.6421	−1.49298
$231$	0	0
$232$	−13.4134	−0.880634
$233$	−6.18553	−0.405228	−0.202614	−	0.979259i	$-0.564944\pi$
$233$	−6.18553	−0.405228	−0.202614	+	0.979259i	$0.564944\pi$
$234$	−5.30813	−0.347003
$235$	−34.4164	−2.24508
$236$	−5.74718	−0.374109
$237$	−6.27134	−0.407367
$238$	1.48314	0.0961379
$239$	−20.4784	−1.32464	−0.662319	−	0.749222i	$-0.730427\pi$
$239$	−20.4784	−1.32464	−0.662319	+	0.749222i	$0.730427\pi$
$240$	−0.0771030	−0.00497697
$241$	−1.04113	−0.0670653	−0.0335326	−	0.999438i	$-0.510676\pi$
$241$	−1.04113	−0.0670653	−0.0335326	+	0.999438i	$0.510676\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	6.58703	0.421691
$245$	−4.06436	−0.259663
$246$	−2.79817	−0.178405
$247$	−33.2342	−2.11464
$248$	−12.3225	−0.782481
$249$	0.127722	0.00809405
$250$	23.0939	1.46059
$251$	6.63300	0.418672	0.209336	−	0.977844i	$-0.432870\pi$
$251$	6.63300	0.418672	0.209336	+	0.977844i	$0.432870\pi$
$252$	−1.24031	−0.0781319
$253$	0	0
$254$	−0.559320	−0.0350948
$255$	6.91602	0.433098
$256$	−15.9526	−0.997036
$257$	2.88246	0.179803	0.0899015	−	0.995951i	$-0.471345\pi$
$257$	2.88246	0.179803	0.0899015	+	0.995951i	$0.471345\pi$
$258$	0.110860	0.00690184
$259$	2.43952	0.151585
$260$	30.7004	1.90396
$261$	−4.74935	−0.293977
$262$	−12.2013	−0.753801
$263$	21.1018	1.30119	0.650595	−	0.759425i	$-0.274520\pi$
$263$	21.1018	1.30119	0.650595	+	0.759425i	$0.274520\pi$
$264$	0	0
$265$	19.1596	1.17696
$266$	4.75645	0.291636
$267$	−8.12736	−0.497386
$268$	−17.4494	−1.06589
$269$	−17.1315	−1.04453	−0.522263	−	0.852784i	$-0.674912\pi$
$269$	−17.1315	−1.04453	−0.522263	+	0.852784i	$0.674912\pi$
$270$	3.54252	0.215591
$271$	−16.3599	−0.993790	−0.496895	−	0.867811i	$-0.665527\pi$
$271$	−16.3599	−0.993790	−0.496895	+	0.867811i	$0.665527\pi$
$272$	−0.0322806	−0.00195730
$273$	6.09006	0.368587
$274$	2.35348	0.142179
$275$	0	0
$276$	7.92745	0.477177
$277$	2.47852	0.148920	0.0744599	−	0.997224i	$-0.476277\pi$
$277$	2.47852	0.148920	0.0744599	+	0.997224i	$0.476277\pi$
$278$	−1.79718	−0.107787
$279$	−4.36309	−0.261211
$280$	−11.4788	−0.685992
$281$	−24.1948	−1.44334	−0.721671	−	0.692236i	$-0.756626\pi$
$281$	−24.1948	−1.44334	−0.721671	+	0.692236i	$0.756626\pi$
$282$	−7.38061	−0.439509
$283$	29.1859	1.73492	0.867460	−	0.497508i	$-0.165751\pi$
$283$	29.1859	1.73492	0.867460	+	0.497508i	$0.165751\pi$
$284$	−2.75203	−0.163303
$285$	22.1797	1.31381
$286$	0	0
$287$	3.21037	0.189502
$288$	−5.66506	−0.333817
$289$	−14.1045	−0.829675
$290$	−16.8246	−0.987977
$291$	−2.59721	−0.152251
$292$	−6.91338	−0.404575
$293$	27.1918	1.58856	0.794280	−	0.607551i	$-0.207848\pi$
$293$	27.1918	1.58856	0.794280	+	0.607551i	$0.207848\pi$
$294$	−0.871604	−0.0508330
$295$	−18.8330	−1.09650
$296$	6.88986	0.400465
$297$	0	0
$298$	8.78834	0.509095
$299$	−38.9248	−2.25108
$300$	−14.2872	−0.824869
$301$	−0.127191	−0.00733115
$302$	−13.0426	−0.750518
$303$	6.02573	0.346169
$304$	−0.103524	−0.00593752
$305$	21.5851	1.23596
$306$	1.48314	0.0847857
$307$	9.95364	0.568084	0.284042	−	0.958812i	$-0.408324\pi$
$307$	9.95364	0.568084	0.284042	+	0.958812i	$0.408324\pi$
$308$	0	0
$309$	12.8938	0.733502
$310$	−15.4563	−0.877860
$311$	16.2407	0.920923	0.460462	−	0.887680i	$-0.347684\pi$
$311$	16.2407	0.920923	0.460462	+	0.887680i	$0.347684\pi$
$312$	17.2000	0.973756
$313$	16.6148	0.939125	0.469562	−	0.882899i	$-0.344412\pi$
$313$	16.6148	0.939125	0.469562	+	0.882899i	$0.344412\pi$
$314$	−16.9562	−0.956894
$315$	−4.06436	−0.229001
$316$	7.77838	0.437568
$317$	8.10002	0.454942	0.227471	−	0.973785i	$-0.426954\pi$
$317$	8.10002	0.454942	0.227471	+	0.973785i	$0.426954\pi$
$318$	4.10878	0.230409
$319$	0	0
$320$	−19.9144	−1.11325
$321$	5.34760	0.298474
$322$	5.57089	0.310453
$323$	9.28597	0.516685
$324$	−1.24031	−0.0689059
$325$	70.1518	3.89132
$326$	−10.5555	−0.584615
$327$	1.44173	0.0797280
$328$	9.06693	0.500637
$329$	8.46784	0.466847
$330$	0	0
$331$	−6.64605	−0.365300	−0.182650	−	0.983178i	$-0.558468\pi$
$331$	−6.64605	−0.365300	−0.182650	+	0.983178i	$0.558468\pi$
$332$	−0.158414	−0.00869411
$333$	2.43952	0.133685
$334$	−11.4773	−0.628008
$335$	−57.1801	−3.12408
$336$	0.0189705	0.00103492
$337$	20.9521	1.14133	0.570665	−	0.821183i	$-0.306685\pi$
$337$	20.9521	1.14133	0.570665	+	0.821183i	$0.306685\pi$
$338$	−20.9960	−1.14203
$339$	19.5174	1.06004
$340$	−8.57798	−0.465207
$341$	0	0
$342$	4.75645	0.257199
$343$	1.00000	0.0539949
$344$	−0.359220	−0.0193679
$345$	25.9775	1.39858
$346$	−12.8282	−0.689649
$347$	17.7871	0.954862	0.477431	−	0.878669i	$-0.341568\pi$
$347$	17.7871	0.954862	0.477431	+	0.878669i	$0.341568\pi$
$348$	5.89064	0.315772
$349$	−11.7341	−0.628110	−0.314055	−	0.949405i	$-0.601688\pi$
$349$	−11.7341	−0.628110	−0.314055	+	0.949405i	$0.601688\pi$
$350$	−10.0401	−0.536664
$351$	6.09006	0.325064
$352$	0	0
$353$	15.2682	0.812642	0.406321	−	0.913730i	$-0.366811\pi$
$353$	15.2682	0.812642	0.406321	+	0.913730i	$0.366811\pi$
$354$	−4.03873	−0.214656
$355$	−9.01815	−0.478634
$356$	10.0804	0.534261
$357$	−1.70162	−0.0900595
$358$	10.1894	0.538526
$359$	−6.74198	−0.355828	−0.177914	−	0.984046i	$-0.556935\pi$
$359$	−6.74198	−0.355828	−0.177914	+	0.984046i	$0.556935\pi$
$360$	−11.4788	−0.604988
$361$	10.7801	0.567375
$362$	−17.1708	−0.902478
$363$	0	0
$364$	−7.55354	−0.395913
$365$	−22.6545	−1.18579
$366$	4.62893	0.241958
$367$	−20.9743	−1.09485	−0.547424	−	0.836855i	$-0.684392\pi$
$367$	−20.9743	−1.09485	−0.547424	+	0.836855i	$0.684392\pi$
$368$	−0.121250	−0.00632062
$369$	3.21037	0.167125
$370$	8.64205	0.449279
$371$	−4.71404	−0.244741
$372$	5.41156	0.280577
$373$	−29.1253	−1.50805	−0.754027	−	0.656844i	$-0.771891\pi$
$373$	−29.1253	−1.50805	−0.754027	+	0.656844i	$0.771891\pi$
$374$	0	0
$375$	−26.4958	−1.36824
$376$	23.9154	1.23334
$377$	−28.9238	−1.48965
$378$	−0.871604	−0.0448305
$379$	31.9353	1.64041	0.820203	−	0.572072i	$-0.193860\pi$
$379$	31.9353	1.64041	0.820203	+	0.572072i	$0.193860\pi$
$380$	−27.5096	−1.41121
$381$	0.641713	0.0328759
$382$	−6.06806	−0.310469
$383$	3.58968	0.183424	0.0917121	−	0.995786i	$-0.470766\pi$
$383$	3.58968	0.183424	0.0917121	+	0.995786i	$0.470766\pi$
$384$	7.05948	0.360253
$385$	0	0
$386$	14.6981	0.748115
$387$	−0.127191	−0.00646547
$388$	3.22133	0.163538
$389$	10.4710	0.530901	0.265451	−	0.964124i	$-0.414479\pi$
$389$	10.4710	0.530901	0.265451	+	0.964124i	$0.414479\pi$
$390$	21.5742	1.09245
$391$	10.8760	0.550022
$392$	2.82426	0.142647
$393$	13.9987	0.706142
$394$	5.59697	0.281971
$395$	25.4890	1.28249
$396$	0	0
$397$	17.7269	0.889686	0.444843	−	0.895609i	$-0.353259\pi$
$397$	17.7269	0.889686	0.444843	+	0.895609i	$0.353259\pi$
$398$	11.5614	0.579522
$399$	−5.45712	−0.273198
$400$	0.218522	0.0109261
$401$	8.08035	0.403513	0.201757	−	0.979436i	$-0.435335\pi$
$401$	8.08035	0.403513	0.201757	+	0.979436i	$0.435335\pi$
$402$	−12.2623	−0.611587
$403$	−26.5715	−1.32362
$404$	−7.47375	−0.371833
$405$	−4.06436	−0.201960
$406$	4.13955	0.205442
$407$	0	0
$408$	−4.80584	−0.237924
$409$	−39.6889	−1.96249	−0.981245	−	0.192765i	$-0.938254\pi$
$409$	−39.6889	−1.96249	−0.981245	+	0.192765i	$0.938254\pi$
$410$	11.3728	0.561662
$411$	−2.70017	−0.133190
$412$	−15.9922	−0.787881
$413$	4.63368	0.228008
$414$	5.57089	0.273794
$415$	−0.519109	−0.0254820
$416$	−34.5006	−1.69153
$417$	2.06192	0.100973
$418$	0	0
$419$	16.1715	0.790028	0.395014	−	0.918675i	$-0.370740\pi$
$419$	16.1715	0.790028	0.395014	+	0.918675i	$0.370740\pi$
$420$	5.04106	0.245978
$421$	24.8084	1.20909	0.604544	−	0.796572i	$-0.293355\pi$
$421$	24.8084	1.20909	0.604544	+	0.796572i	$0.293355\pi$
$422$	2.53775	0.123536
$423$	8.46784	0.411721
$424$	−13.3137	−0.646570
$425$	−19.6011	−0.950794
$426$	−1.93395	−0.0937000
$427$	−5.31081	−0.257008
$428$	−6.63265	−0.320601
$429$	0	0
$430$	−0.450576	−0.0217287
$431$	−21.8821	−1.05402	−0.527012	−	0.849858i	$-0.676688\pi$
$431$	−21.8821	−1.05402	−0.527012	+	0.849858i	$0.676688\pi$
$432$	0.0189705	0.000912718 0
$433$	39.3439	1.89075	0.945374	−	0.325989i	$-0.105697\pi$
$433$	39.3439	1.89075	0.945374	+	0.325989i	$0.105697\pi$
$434$	3.80289	0.182544
$435$	19.3031	0.925512
$436$	−1.78819	−0.0856388
$437$	34.8793	1.66850
$438$	−4.85826	−0.232137
$439$	−9.59194	−0.457799	−0.228899	−	0.973450i	$-0.573513\pi$
$439$	−9.59194	−0.457799	−0.228899	+	0.973450i	$0.573513\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	9.03244	0.429629
$443$	−0.629136	−0.0298911	−0.0149456	−	0.999888i	$-0.504758\pi$
$443$	−0.629136	−0.0298911	−0.0149456	+	0.999888i	$0.504758\pi$
$444$	−3.02576	−0.143596
$445$	33.0325	1.56589
$446$	2.92484	0.138495
$447$	−10.0829	−0.476907
$448$	4.89975	0.231492
$449$	−17.3364	−0.818154	−0.409077	−	0.912500i	$-0.634149\pi$
$449$	−17.3364	−0.818154	−0.409077	+	0.912500i	$0.634149\pi$
$450$	−10.0401	−0.473293
$451$	0	0
$452$	−24.2075	−1.13863
$453$	14.9639	0.703066
$454$	2.90934	0.136542
$455$	−24.7522	−1.16040
$456$	−15.4123	−0.721749
$457$	29.1154	1.36196	0.680980	−	0.732302i	$-0.261554\pi$
$457$	29.1154	1.36196	0.680980	+	0.732302i	$0.261554\pi$
$458$	−5.92816	−0.277005
$459$	−1.70162	−0.0794250
$460$	−32.2201	−1.50227
$461$	−15.1850	−0.707236	−0.353618	−	0.935390i	$-0.615049\pi$
$461$	−15.1850	−0.707236	−0.353618	+	0.935390i	$0.615049\pi$
$462$	0	0
$463$	17.8886	0.831355	0.415677	−	0.909512i	$-0.363545\pi$
$463$	17.8886	0.831355	0.415677	+	0.909512i	$0.363545\pi$
$464$	−0.0900974	−0.00418267
$465$	17.7332	0.822357
$466$	5.39134	0.249749
$467$	−30.3101	−1.40259	−0.701293	−	0.712874i	$-0.747393\pi$
$467$	−30.3101	−1.40259	−0.701293	+	0.712874i	$0.747393\pi$
$468$	−7.55354	−0.349163
$469$	14.0686	0.649629
$470$	29.9975	1.38368
$471$	19.4540	0.896394
$472$	13.0867	0.602366
$473$	0	0
$474$	5.46613	0.251068
$475$	−62.8609	−2.88425
$476$	2.11054	0.0967362
$477$	−4.71404	−0.215841
$478$	17.8491	0.816397
$479$	7.94750	0.363130	0.181565	−	0.983379i	$-0.441884\pi$
$479$	7.94750	0.363130	0.181565	+	0.983379i	$0.441884\pi$
$480$	23.0249	1.05094
$481$	14.8569	0.677414
$482$	0.907456	0.0413335
$483$	−6.39153	−0.290825
$484$	0	0
$485$	10.5560	0.479323
$486$	−0.871604	−0.0395368
$487$	−10.1711	−0.460896	−0.230448	−	0.973085i	$-0.574019\pi$
$487$	−10.1711	−0.460896	−0.230448	+	0.973085i	$0.574019\pi$
$488$	−14.9991	−0.678979
$489$	12.1104	0.547652
$490$	3.54252	0.160035
$491$	−1.07736	−0.0486206	−0.0243103	−	0.999704i	$-0.507739\pi$
$491$	−1.07736	−0.0486206	−0.0243103	+	0.999704i	$0.507739\pi$
$492$	−3.98184	−0.179515
$493$	8.08161	0.363977
$494$	28.9671	1.30329
$495$	0	0
$496$	−0.0827699	−0.00371648
$497$	2.21883	0.0995283
$498$	−0.111323	−0.00498850
$499$	34.6159	1.54962	0.774810	−	0.632194i	$-0.217845\pi$
$499$	34.6159	1.54962	0.774810	+	0.632194i	$0.217845\pi$
$500$	32.8629	1.46968
$501$	13.1680	0.588302
$502$	−5.78136	−0.258035
$503$	−2.75919	−0.123026	−0.0615130	−	0.998106i	$-0.519593\pi$
$503$	−2.75919	−0.123026	−0.0615130	+	0.998106i	$0.519593\pi$
$504$	2.82426	0.125803
$505$	−24.4908	−1.08982
$506$	0	0
$507$	24.0889	1.06983
$508$	−0.795920	−0.0353132
$509$	14.5914	0.646753	0.323377	−	0.946270i	$-0.395182\pi$
$509$	14.5914	0.646753	0.323377	+	0.946270i	$0.395182\pi$
$510$	−6.02804	−0.266926
$511$	5.57393	0.246576
$512$	−0.214624	−0.00948514
$513$	−5.45712	−0.240938
$514$	−2.51237	−0.110816
$515$	−52.4051	−2.30924
$516$	0.157755	0.00694480
$517$	0	0
$518$	−2.12630	−0.0934242
$519$	14.7179	0.646045
$520$	−69.9069	−3.06562
$521$	−13.7317	−0.601599	−0.300799	−	0.953687i	$-0.597253\pi$
$521$	−13.7317	−0.601599	−0.300799	+	0.953687i	$0.597253\pi$
$522$	4.13955	0.181183
$523$	−7.54609	−0.329967	−0.164984	−	0.986296i	$-0.552757\pi$
$523$	−7.54609	−0.329967	−0.164984	+	0.986296i	$0.552757\pi$
$524$	−17.3627	−0.758493
$525$	11.5191	0.502733
$526$	−18.3924	−0.801946
$527$	7.42434	0.323409
$528$	0	0
$529$	17.8517	0.776160
$530$	−16.6996	−0.725383
$531$	4.63368	0.201084
$532$	6.76849	0.293451
$533$	19.5513	0.846863
$534$	7.08384	0.306548
$535$	−21.7346	−0.939668
$536$	39.7335	1.71623
$537$	−11.6904	−0.504477
$538$	14.9319	0.643760
$539$	0	0
$540$	5.04106	0.216932
$541$	2.01679	0.0867084	0.0433542	−	0.999060i	$-0.486196\pi$
$541$	2.01679	0.0867084	0.0433542	+	0.999060i	$0.486196\pi$
$542$	14.2593	0.612490
$543$	19.7002	0.845418
$544$	9.63981	0.413304
$545$	−5.85973	−0.251003
$546$	−5.30813	−0.227167
$547$	−41.1514	−1.75951	−0.879754	−	0.475429i	$-0.842293\pi$
$547$	−41.1514	−1.75951	−0.879754	+	0.475429i	$0.842293\pi$
$548$	3.34904	0.143064
$549$	−5.31081	−0.226660
$550$	0	0
$551$	25.9177	1.10413
$552$	−18.0514	−0.768318
$553$	−6.27134	−0.266684
$554$	−2.16029	−0.0917818
$555$	−9.91511	−0.420873
$556$	−2.55741	−0.108458
$557$	18.4609	0.782214	0.391107	−	0.920345i	$-0.372092\pi$
$557$	18.4609	0.782214	0.391107	+	0.920345i	$0.372092\pi$
$558$	3.80289	0.160989
$559$	−0.774600	−0.0327621
$560$	−0.0771030	−0.00325819
$561$	0	0
$562$	21.0883	0.889557
$563$	28.5154	1.20178	0.600890	−	0.799332i	$-0.294813\pi$
$563$	28.5154	1.20178	0.600890	+	0.799332i	$0.294813\pi$
$564$	−10.5027	−0.442244
$565$	−79.3258	−3.33726
$566$	−25.4385	−1.06926
$567$	1.00000	0.0419961
$568$	6.26658	0.262940
$569$	−17.7384	−0.743631	−0.371816	−	0.928307i	$-0.621265\pi$
$569$	−17.7384	−0.743631	−0.371816	+	0.928307i	$0.621265\pi$
$570$	−19.3319	−0.809726
$571$	20.8488	0.872496	0.436248	−	0.899827i	$-0.356307\pi$
$571$	20.8488	0.872496	0.436248	+	0.899827i	$0.356307\pi$
$572$	0	0
$573$	6.96194	0.290839
$574$	−2.79817	−0.116793
$575$	−73.6244	−3.07035
$576$	4.89975	0.204156
$577$	21.2988	0.886680	0.443340	−	0.896353i	$-0.353793\pi$
$577$	21.2988	0.886680	0.443340	+	0.896353i	$0.353793\pi$
$578$	12.2935	0.511343
$579$	−16.8633	−0.700814
$580$	−23.9417	−0.994126
$581$	0.127722	0.00529880
$582$	2.26374	0.0938349
$583$	0	0
$584$	15.7422	0.651419
$585$	−24.7522	−1.02338
$586$	−23.7005	−0.979058
$587$	29.1874	1.20469	0.602346	−	0.798235i	$-0.294233\pi$
$587$	29.1874	1.20469	0.602346	+	0.798235i	$0.294233\pi$
$588$	−1.24031	−0.0511494
$589$	23.8099	0.981069
$590$	16.4149	0.675790
$591$	−6.42145	−0.264143
$592$	0.0462789	0.00190205
$593$	37.3094	1.53212	0.766058	−	0.642772i	$-0.222216\pi$
$593$	37.3094	1.53212	0.766058	+	0.642772i	$0.222216\pi$
$594$	0	0
$595$	6.91602	0.283529
$596$	12.5059	0.512263
$597$	−13.2645	−0.542881
$598$	33.9271	1.38738
$599$	−4.95368	−0.202402	−0.101201	−	0.994866i	$-0.532269\pi$
$599$	−4.95368	−0.202402	−0.101201	+	0.994866i	$0.532269\pi$
$600$	32.5329	1.32815
$601$	−29.4572	−1.20158	−0.600792	−	0.799405i	$-0.705148\pi$
$601$	−29.4572	−1.20158	−0.600792	+	0.799405i	$0.705148\pi$
$602$	0.110860	0.00451832
$603$	14.0686	0.572919
$604$	−18.5598	−0.755189
$605$	0	0
$606$	−5.25205	−0.213350
$607$	26.3426	1.06921	0.534606	−	0.845101i	$-0.320460\pi$
$607$	26.3426	1.06921	0.534606	+	0.845101i	$0.320460\pi$
$608$	30.9149	1.25377
$609$	−4.74935	−0.192453
$610$	−18.8136	−0.761742
$611$	51.5697	2.08629
$612$	2.11054	0.0853133
$613$	−48.2089	−1.94714	−0.973569	−	0.228391i	$-0.926653\pi$
$613$	−48.2089	−1.94714	−0.973569	+	0.228391i	$0.926653\pi$
$614$	−8.67563	−0.350120
$615$	−13.0481	−0.526150
$616$	0	0
$617$	−29.2376	−1.17706	−0.588532	−	0.808474i	$-0.700294\pi$
$617$	−29.2376	−1.17706	−0.588532	+	0.808474i	$0.700294\pi$
$618$	−11.2383	−0.452070
$619$	6.14388	0.246943	0.123472	−	0.992348i	$-0.460597\pi$
$619$	6.14388	0.246943	0.123472	+	0.992348i	$0.460597\pi$
$620$	−21.9946	−0.883323
$621$	−6.39153	−0.256483
$622$	−14.1554	−0.567581
$623$	−8.12736	−0.325616
$624$	0.115531	0.00462496
$625$	50.0934	2.00374
$626$	−14.4815	−0.578799
$627$	0	0
$628$	−24.1289	−0.962849
$629$	−4.15115	−0.165517
$630$	3.54252	0.141137
$631$	40.5646	1.61485	0.807425	−	0.589971i	$-0.200861\pi$
$631$	40.5646	1.61485	0.807425	+	0.589971i	$0.200861\pi$
$632$	−17.7119	−0.704542
$633$	−2.91158	−0.115725
$634$	−7.06001	−0.280389
$635$	−2.60815	−0.103501
$636$	5.84685	0.231843
$637$	6.09006	0.241297
$638$	0	0
$639$	2.21883	0.0877757
$640$	−28.6923	−1.13416
$641$	23.5266	0.929244	0.464622	−	0.885509i	$-0.346190\pi$
$641$	23.5266	0.929244	0.464622	+	0.885509i	$0.346190\pi$
$642$	−4.66099	−0.183955
$643$	−21.0143	−0.828724	−0.414362	−	0.910112i	$-0.635995\pi$
$643$	−21.0143	−0.828724	−0.414362	+	0.910112i	$0.635995\pi$
$644$	7.92745	0.312385
$645$	0.516950	0.0203549
$646$	−8.09369	−0.318442
$647$	0.452722	0.0177983	0.00889917	−	0.999960i	$-0.497167\pi$
$647$	0.452722	0.0177983	0.00889917	+	0.999960i	$0.497167\pi$
$648$	2.82426	0.110948
$649$	0	0
$650$	−61.1446	−2.39829
$651$	−4.36309	−0.171003
$652$	−15.0206	−0.588253
$653$	−22.0408	−0.862522	−0.431261	−	0.902227i	$-0.641931\pi$
$653$	−22.0408	−0.862522	−0.431261	+	0.902227i	$0.641931\pi$
$654$	−1.25662	−0.0491378
$655$	−56.8959	−2.22311
$656$	0.0609022	0.00237783
$657$	5.57393	0.217460
$658$	−7.38061	−0.287726
$659$	−18.3859	−0.716215	−0.358107	−	0.933680i	$-0.616578\pi$
$659$	−18.3859	−0.716215	−0.358107	+	0.933680i	$0.616578\pi$
$660$	0	0
$661$	−45.9605	−1.78766	−0.893829	−	0.448408i	$-0.851991\pi$
$661$	−45.9605	−1.78766	−0.893829	+	0.448408i	$0.851991\pi$
$662$	5.79273	0.225141
$663$	−10.3630	−0.402466
$664$	0.360721	0.0139987
$665$	22.1797	0.860092
$666$	−2.12630	−0.0823924
$667$	30.3556	1.17537
$668$	−16.3323	−0.631917
$669$	−3.35570	−0.129739
$670$	49.8384	1.92542
$671$	0	0
$672$	−5.66506	−0.218535
$673$	−29.7402	−1.14640	−0.573200	−	0.819415i	$-0.694298\pi$
$673$	−29.7402	−1.14640	−0.573200	+	0.819415i	$0.694298\pi$
$674$	−18.2619	−0.703422
$675$	11.5191	0.443369
$676$	−29.8776	−1.14914
$677$	−27.7828	−1.06778	−0.533889	−	0.845554i	$-0.679270\pi$
$677$	−27.7828	−1.06778	−0.533889	+	0.845554i	$0.679270\pi$
$678$	−17.0114	−0.653321
$679$	−2.59721	−0.0996716
$680$	19.5327	0.749044
$681$	−3.33791	−0.127909
$682$	0	0
$683$	40.8297	1.56231	0.781153	−	0.624339i	$-0.214632\pi$
$683$	40.8297	1.56231	0.781153	+	0.624339i	$0.214632\pi$
$684$	6.76849	0.258800
$685$	10.9745	0.419313
$686$	−0.871604	−0.0332780
$687$	6.80143	0.259491
$688$	−0.00241287	−9.19898e−5 0
$689$	−28.7088	−1.09372
$690$	−22.6421	−0.861971
$691$	37.8156	1.43857	0.719286	−	0.694714i	$-0.244469\pi$
$691$	37.8156	1.43857	0.719286	+	0.694714i	$0.244469\pi$
$692$	−18.2547	−0.693941
$693$	0	0
$694$	−15.5033	−0.588498
$695$	−8.38039	−0.317886
$696$	−13.4134	−0.508434
$697$	−5.46284	−0.206920
$698$	10.2275	0.387115
$699$	−6.18553	−0.233958
$700$	−14.2872	−0.540004
$701$	−27.4256	−1.03585	−0.517926	−	0.855426i	$-0.673296\pi$
$701$	−27.4256	−1.03585	−0.517926	+	0.855426i	$0.673296\pi$
$702$	−5.30813	−0.200342
$703$	−13.3128	−0.502100
$704$	0	0
$705$	−34.4164	−1.29620
$706$	−13.3078	−0.500845
$707$	6.02573	0.226621
$708$	−5.74718	−0.215992
$709$	−31.8439	−1.19592	−0.597962	−	0.801525i	$-0.704022\pi$
$709$	−31.8439	−1.19592	−0.597962	+	0.801525i	$0.704022\pi$
$710$	7.86026	0.294990
$711$	−6.27134	−0.235194
$712$	−22.9538	−0.860230
$713$	27.8868	1.04437
$714$	1.48314	0.0555053
$715$	0	0
$716$	14.4996	0.541877
$717$	−20.4784	−0.764780
$718$	5.87633	0.219303
$719$	−15.2468	−0.568608	−0.284304	−	0.958734i	$-0.591763\pi$
$719$	−15.2468	−0.568608	−0.284304	+	0.958734i	$0.591763\pi$
$720$	−0.0771030	−0.00287346
$721$	12.8938	0.480190
$722$	−9.39600	−0.349683
$723$	−1.04113	−0.0387201
$724$	−24.4343	−0.908095
$725$	−54.7080	−2.03180
$726$	0	0
$727$	−33.7158	−1.25045	−0.625225	−	0.780445i	$-0.714993\pi$
$727$	−33.7158	−1.25045	−0.625225	+	0.780445i	$0.714993\pi$
$728$	17.2000	0.637473
$729$	1.00000	0.0370370
$730$	19.7457	0.730823
$731$	0.216431	0.00800499
$732$	6.58703	0.243464
$733$	4.51648	0.166820	0.0834101	−	0.996515i	$-0.473419\pi$
$733$	4.51648	0.166820	0.0834101	+	0.996515i	$0.473419\pi$
$734$	18.2813	0.674774
$735$	−4.06436	−0.149916
$736$	36.2084	1.33466
$737$	0	0
$738$	−2.79817	−0.103002
$739$	−10.0400	−0.369327	−0.184664	−	0.982802i	$-0.559120\pi$
$739$	−10.0400	−0.369327	−0.184664	+	0.982802i	$0.559120\pi$
$740$	12.2978	0.452075
$741$	−33.2342	−1.22089
$742$	4.10878	0.150838
$743$	15.7027	0.576076	0.288038	−	0.957619i	$-0.406997\pi$
$743$	15.7027	0.576076	0.288038	+	0.957619i	$0.406997\pi$
$744$	−12.3225	−0.451765
$745$	40.9808	1.50142
$746$	25.3858	0.929439
$747$	0.127722	0.00467310
$748$	0	0
$749$	5.34760	0.195397
$750$	23.0939	0.843269
$751$	19.2780	0.703466	0.351733	−	0.936100i	$-0.385593\pi$
$751$	19.2780	0.703466	0.351733	+	0.936100i	$0.385593\pi$
$752$	0.160639	0.00585790
$753$	6.63300	0.241720
$754$	25.2101	0.918099
$755$	−60.8188	−2.21342
$756$	−1.24031	−0.0451095
$757$	37.5431	1.36453	0.682263	−	0.731106i	$-0.260996\pi$
$757$	37.5431	1.36453	0.682263	+	0.731106i	$0.260996\pi$
$758$	−27.8350	−1.01101
$759$	0	0
$760$	62.6414	2.27224
$761$	38.7469	1.40458	0.702288	−	0.711893i	$-0.252162\pi$
$761$	38.7469	1.40458	0.702288	+	0.711893i	$0.252162\pi$
$762$	−0.559320	−0.0202620
$763$	1.44173	0.0521942
$764$	−8.63494	−0.312401
$765$	6.91602	0.250049
$766$	−3.12878	−0.113048
$767$	28.2194	1.01894
$768$	−15.9526	−0.575639
$769$	24.0014	0.865513	0.432756	−	0.901511i	$-0.357541\pi$
$769$	24.0014	0.865513	0.432756	+	0.901511i	$0.357541\pi$
$770$	0	0
$771$	2.88246	0.103809
$772$	20.9156	0.752770
$773$	−39.2714	−1.41250	−0.706248	−	0.707965i	$-0.749613\pi$
$773$	−39.2714	−1.41250	−0.706248	+	0.707965i	$0.749613\pi$
$774$	0.110860	0.00398478
$775$	−50.2587	−1.80534
$776$	−7.33520	−0.263318
$777$	2.43952	0.0875174
$778$	−9.12658	−0.327204
$779$	−17.5194	−0.627696
$780$	30.7004	1.09925
$781$	0	0
$782$	−9.47956	−0.338988
$783$	−4.74935	−0.169728
$784$	0.0189705	0.000677517 0
$785$	−79.0682	−2.82207
$786$	−12.2013	−0.435207
$787$	−25.4235	−0.906251	−0.453125	−	0.891447i	$-0.649691\pi$
$787$	−25.4235	−0.906251	−0.453125	+	0.891447i	$0.649691\pi$
$788$	7.96457	0.283726
$789$	21.1018	0.751242
$790$	−22.2163	−0.790422
$791$	19.5174	0.693959
$792$	0	0
$793$	−32.3432	−1.14854
$794$	−15.4508	−0.548329
$795$	19.1596	0.679520
$796$	16.4521	0.583128
$797$	−36.7626	−1.30220	−0.651099	−	0.758993i	$-0.725692\pi$
$797$	−36.7626	−1.30220	−0.651099	+	0.758993i	$0.725692\pi$
$798$	4.75645	0.168376
$799$	−14.4091	−0.509757
$800$	−65.2562	−2.30716
$801$	−8.12736	−0.287166
$802$	−7.04287	−0.248692
$803$	0	0
$804$	−17.4494	−0.615393
$805$	25.9775	0.915587
$806$	23.1598	0.815770
$807$	−17.1315	−0.603058
$808$	17.0183	0.598700
$809$	40.4785	1.42315	0.711574	−	0.702612i	$-0.247983\pi$
$809$	40.4785	1.42315	0.711574	+	0.702612i	$0.247983\pi$
$810$	3.54252	0.124471
$811$	9.55021	0.335353	0.167677	−	0.985842i	$-0.446374\pi$
$811$	9.55021	0.335353	0.167677	+	0.985842i	$0.446374\pi$
$812$	5.89064	0.206721
$813$	−16.3599	−0.573765
$814$	0	0
$815$	−49.2212	−1.72414
$816$	−0.0322806	−0.00113005
$817$	0.694095	0.0242833
$818$	34.5930	1.20952
$819$	6.09006	0.212804
$820$	16.1836	0.565157
$821$	36.2589	1.26544	0.632722	−	0.774379i	$-0.281938\pi$
$821$	36.2589	1.26544	0.632722	+	0.774379i	$0.281938\pi$
$822$	2.35348	0.0820871
$823$	−41.7383	−1.45491	−0.727454	−	0.686157i	$-0.759296\pi$
$823$	−41.7383	−1.45491	−0.727454	+	0.686157i	$0.759296\pi$
$824$	36.4155	1.26859
$825$	0	0
$826$	−4.03873	−0.140525
$827$	−28.2659	−0.982901	−0.491450	−	0.870906i	$-0.663533\pi$
$827$	−28.2659	−0.982901	−0.491450	+	0.870906i	$0.663533\pi$
$828$	7.92745	0.275498
$829$	−2.01036	−0.0698226	−0.0349113	−	0.999390i	$-0.511115\pi$
$829$	−2.01036	−0.0698226	−0.0349113	+	0.999390i	$0.511115\pi$
$830$	0.452457	0.0157050
$831$	2.47852	0.0859789
$832$	29.8398	1.03451
$833$	−1.70162	−0.0589578
$834$	−1.79718	−0.0622311
$835$	−53.5195	−1.85212
$836$	0	0
$837$	−4.36309	−0.150810
$838$	−14.0951	−0.486908
$839$	−24.2099	−0.835818	−0.417909	−	0.908489i	$-0.637237\pi$
$839$	−24.2099	−0.835818	−0.417909	+	0.908489i	$0.637237\pi$
$840$	−11.4788	−0.396058
$841$	−6.44371	−0.222197
$842$	−21.6231	−0.745182
$843$	−24.1948	−0.833314
$844$	3.61125	0.124304
$845$	−97.9060	−3.36807
$846$	−7.38061	−0.253751
$847$	0	0
$848$	−0.0894276	−0.00307096
$849$	29.1859	1.00166
$850$	17.0844	0.585991
$851$	−15.5923	−0.534497
$852$	−2.75203	−0.0942831
$853$	−20.1429	−0.689680	−0.344840	−	0.938662i	$-0.612067\pi$
$853$	−20.1429	−0.689680	−0.344840	+	0.938662i	$0.612067\pi$
$854$	4.62893	0.158399
$855$	22.1797	0.758530
$856$	15.1030	0.516211
$857$	26.6092	0.908953	0.454477	−	0.890759i	$-0.349826\pi$
$857$	26.6092	0.908953	0.454477	+	0.890759i	$0.349826\pi$
$858$	0	0
$859$	14.4645	0.493522	0.246761	−	0.969076i	$-0.420634\pi$
$859$	14.4645	0.493522	0.246761	+	0.969076i	$0.420634\pi$
$860$	−0.641176	−0.0218639
$861$	3.21037	0.109409
$862$	19.0725	0.649614
$863$	28.3624	0.965466	0.482733	−	0.875768i	$-0.339644\pi$
$863$	28.3624	0.965466	0.482733	+	0.875768i	$0.339644\pi$
$864$	−5.66506	−0.192729
$865$	−59.8190	−2.03391
$866$	−34.2923	−1.16530
$867$	−14.1045	−0.479013
$868$	5.41156	0.183680
$869$	0	0
$870$	−16.8246	−0.570409
$871$	85.6789	2.90312
$872$	4.07184	0.137890
$873$	−2.59721	−0.0879021
$874$	−30.4010	−1.02833
$875$	−26.4958	−0.895723
$876$	−6.91338	−0.233581
$877$	−10.1417	−0.342461	−0.171231	−	0.985231i	$-0.554774\pi$
$877$	−10.1417	−0.342461	−0.171231	+	0.985231i	$0.554774\pi$
$878$	8.36038	0.282149
$879$	27.1918	0.917156
$880$	0	0
$881$	−3.71883	−0.125291	−0.0626453	−	0.998036i	$-0.519954\pi$
$881$	−3.71883	−0.125291	−0.0626453	+	0.998036i	$0.519954\pi$
$882$	−0.871604	−0.0293484
$883$	17.3305	0.583218	0.291609	−	0.956538i	$-0.405809\pi$
$883$	17.3305	0.583218	0.291609	+	0.956538i	$0.405809\pi$
$884$	12.8533	0.432303
$885$	−18.8330	−0.633063
$886$	0.548358	0.0184224
$887$	5.40696	0.181548	0.0907740	−	0.995872i	$-0.471066\pi$
$887$	5.40696	0.181548	0.0907740	+	0.995872i	$0.471066\pi$
$888$	6.88986	0.231209
$889$	0.641713	0.0215224
$890$	−28.7913	−0.965087
$891$	0	0
$892$	4.16210	0.139357
$893$	−46.2100	−1.54636
$894$	8.78834	0.293926
$895$	47.5139	1.58822
$896$	7.05948	0.235841
$897$	−38.9248	−1.29966
$898$	15.1105	0.504242
$899$	20.7218	0.691111
$900$	−14.2872	−0.476239
$901$	8.02152	0.267236
$902$	0	0
$903$	−0.127191	−0.00423264
$904$	55.1223	1.83334
$905$	−80.0690	−2.66158
$906$	−13.0426	−0.433312
$907$	−10.7020	−0.355353	−0.177677	−	0.984089i	$-0.556858\pi$
$907$	−10.7020	−0.355353	−0.177677	+	0.984089i	$0.556858\pi$
$908$	4.14003	0.137392
$909$	6.02573	0.199861
$910$	21.5742	0.715177
$911$	−27.5708	−0.913463	−0.456732	−	0.889605i	$-0.650980\pi$
$911$	−27.5708	−0.913463	−0.456732	+	0.889605i	$0.650980\pi$
$912$	−0.103524	−0.00342803
$913$	0	0
$914$	−25.3771	−0.839400
$915$	21.5851	0.713580
$916$	−8.43586	−0.278729
$917$	13.9987	0.462278
$918$	1.48314	0.0489510
$919$	−36.4021	−1.20080	−0.600398	−	0.799702i	$-0.704991\pi$
$919$	−36.4021	−1.20080	−0.600398	+	0.799702i	$0.704991\pi$
$920$	73.3674	2.41885
$921$	9.95364	0.327983
$922$	13.2353	0.435882
$923$	13.5128	0.444781
$924$	0	0
$925$	28.1010	0.923956
$926$	−15.5918	−0.512378
$927$	12.8938	0.423488
$928$	26.9054	0.883211
$929$	26.2401	0.860910	0.430455	−	0.902612i	$-0.358353\pi$
$929$	26.2401	0.860910	0.430455	+	0.902612i	$0.358353\pi$
$930$	−15.4563	−0.506833
$931$	−5.45712	−0.178850
$932$	7.67195	0.251303
$933$	16.2407	0.531695
$934$	26.4184	0.864437
$935$	0	0
$936$	17.2000	0.562198
$937$	−23.8799	−0.780121	−0.390061	−	0.920789i	$-0.627546\pi$
$937$	−23.8799	−0.780121	−0.390061	+	0.920789i	$0.627546\pi$
$938$	−12.2623	−0.400378
$939$	16.6148	0.542204
$940$	42.6869	1.39229
$941$	−17.7793	−0.579588	−0.289794	−	0.957089i	$-0.593587\pi$
$941$	−17.7793	−0.579588	−0.289794	+	0.957089i	$0.593587\pi$
$942$	−16.9562	−0.552463
$943$	−20.5192	−0.668196
$944$	0.0879031	0.00286100
$945$	−4.06436	−0.132214
$946$	0	0
$947$	31.7071	1.03034	0.515171	−	0.857088i	$-0.327729\pi$
$947$	31.7071	1.03034	0.515171	+	0.857088i	$0.327729\pi$
$948$	7.77838	0.252630
$949$	33.9456	1.10192
$950$	54.7898	1.77762
$951$	8.10002	0.262661
$952$	−4.80584	−0.155758
$953$	3.28213	0.106319	0.0531593	−	0.998586i	$-0.483071\pi$
$953$	3.28213	0.106319	0.0531593	+	0.998586i	$0.483071\pi$
$954$	4.10878	0.133027
$955$	−28.2959	−0.915633
$956$	25.3995	0.821478
$957$	0	0
$958$	−6.92707	−0.223804
$959$	−2.70017	−0.0871931
$960$	−19.9144	−0.642734
$961$	−11.9635	−0.385918
$962$	−12.9493	−0.417502
$963$	5.34760	0.172324
$964$	1.29132	0.0415907
$965$	68.5386	2.20633
$966$	5.57089	0.179240
$967$	3.83683	0.123384	0.0616921	−	0.998095i	$-0.480350\pi$
$967$	3.83683	0.123384	0.0616921	+	0.998095i	$0.480350\pi$
$968$	0	0
$969$	9.28597	0.298308
$970$	−9.20065	−0.295415
$971$	44.3268	1.42251	0.711257	−	0.702932i	$-0.248126\pi$
$971$	44.3268	1.42251	0.711257	+	0.702932i	$0.248126\pi$
$972$	−1.24031	−0.0397828
$973$	2.06192	0.0661021
$974$	8.86517	0.284058
$975$	70.1518	2.24666
$976$	−0.100749	−0.00322489
$977$	9.69774	0.310258	0.155129	−	0.987894i	$-0.450421\pi$
$977$	9.69774	0.310258	0.155129	+	0.987894i	$0.450421\pi$
$978$	−10.5555	−0.337528
$979$	0	0
$980$	5.04106	0.161031
$981$	1.44173	0.0460310
$982$	0.939032	0.0299657
$983$	28.6752	0.914597	0.457298	−	0.889313i	$-0.348817\pi$
$983$	28.6752	0.914597	0.457298	+	0.889313i	$0.348817\pi$
$984$	9.06693	0.289043
$985$	26.0991	0.831587
$986$	−7.04396	−0.224325
$987$	8.46784	0.269534
$988$	41.2206	1.31140
$989$	0.812944	0.0258501
$990$	0	0
$991$	−14.0455	−0.446170	−0.223085	−	0.974799i	$-0.571613\pi$
$991$	−14.0455	−0.446170	−0.223085	+	0.974799i	$0.571613\pi$
$992$	24.7172	0.784771
$993$	−6.64605	−0.210906
$994$	−1.93395	−0.0613410
$995$	53.9119	1.70912
$996$	−0.158414	−0.00501955
$997$	36.2102	1.14679	0.573394	−	0.819280i	$-0.305626\pi$
$997$	36.2102	1.14679	0.573394	+	0.819280i	$0.305626\pi$
$998$	−30.1714	−0.955058
$999$	2.43952	0.0771831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2541.2.a.br.1.4		10
3.2	odd	2		7623.2.a.cy.1.7		10
11.2	odd	10		231.2.j.g.169.2	✓	20
11.6	odd	10		231.2.j.g.190.2	yes	20
11.10	odd	2		2541.2.a.bq.1.7		10
33.2	even	10		693.2.m.j.631.4		20
33.17	even	10		693.2.m.j.190.4		20
33.32	even	2		7623.2.a.cx.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.g.169.2	✓	20	11.2	odd	10
231.2.j.g.190.2	yes	20	11.6	odd	10
693.2.m.j.190.4		20	33.17	even	10
693.2.m.j.631.4		20	33.2	even	10
2541.2.a.bq.1.7		10	11.10	odd	2
2541.2.a.br.1.4		10	1.1	even	1	trivial
7623.2.a.cx.1.4		10	33.32	even	2
7623.2.a.cy.1.7		10	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2541.a (trivial)

\(f(q)\)	\(=\)	\(q-0.871604 q^{2} +1.00000 q^{3} -1.24031 q^{4} -4.06436 q^{5} -0.871604 q^{6} +1.00000 q^{7} +2.82426 q^{8} +1.00000 q^{9} +3.54252 q^{10} -1.24031 q^{12} +6.09006 q^{13} -0.871604 q^{14} -4.06436 q^{15} +0.0189705 q^{16} -1.70162 q^{17} -0.871604 q^{18} -5.45712 q^{19} +5.04106 q^{20} +1.00000 q^{21} -6.39153 q^{23} +2.82426 q^{24} +11.5191 q^{25} -5.30813 q^{26} +1.00000 q^{27} -1.24031 q^{28} -4.74935 q^{29} +3.54252 q^{30} -4.36309 q^{31} -5.66506 q^{32} +1.48314 q^{34} -4.06436 q^{35} -1.24031 q^{36} +2.43952 q^{37} +4.75645 q^{38} +6.09006 q^{39} -11.4788 q^{40} +3.21037 q^{41} -0.871604 q^{42} -0.127191 q^{43} -4.06436 q^{45} +5.57089 q^{46} +8.46784 q^{47} +0.0189705 q^{48} +1.00000 q^{49} -10.0401 q^{50} -1.70162 q^{51} -7.55354 q^{52} -4.71404 q^{53} -0.871604 q^{54} +2.82426 q^{56} -5.45712 q^{57} +4.13955 q^{58} +4.63368 q^{59} +5.04106 q^{60} -5.31081 q^{61} +3.80289 q^{62} +1.00000 q^{63} +4.89975 q^{64} -24.7522 q^{65} +14.0686 q^{67} +2.11054 q^{68} -6.39153 q^{69} +3.54252 q^{70} +2.21883 q^{71} +2.82426 q^{72} +5.57393 q^{73} -2.12630 q^{74} +11.5191 q^{75} +6.76849 q^{76} -5.30813 q^{78} -6.27134 q^{79} -0.0771030 q^{80} +1.00000 q^{81} -2.79817 q^{82} +0.127722 q^{83} -1.24031 q^{84} +6.91602 q^{85} +0.110860 q^{86} -4.74935 q^{87} -8.12736 q^{89} +3.54252 q^{90} +6.09006 q^{91} +7.92745 q^{92} -4.36309 q^{93} -7.38061 q^{94} +22.1797 q^{95} -5.66506 q^{96} -2.59721 q^{97} -0.871604 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} + 18 q^{4} + 5 q^{5} + 10 q^{7} - 3 q^{8} + 10 q^{9} - 6 q^{10} + 18 q^{12} + 6 q^{13} + 5 q^{15} + 38 q^{16} + 8 q^{17} + 7 q^{20} + 10 q^{21} - 3 q^{24} + 31 q^{25} + q^{26} + 10 q^{27}+ \cdots + 24 q^{97}+O(q^{100}) \) 10 * q + 10 * q^3 + 18 * q^4 + 5 * q^5 + 10 * q^7 - 3 * q^8 + 10 * q^9 - 6 * q^10 + 18 * q^12 + 6 * q^13 + 5 * q^15 + 38 * q^16 + 8 * q^17 + 7 * q^20 + 10 * q^21 - 3 * q^24 + 31 * q^25 + q^26 + 10 * q^27 + 18 * q^28 - 14 * q^29 - 6 * q^30 + 26 * q^31 - 41 * q^32 + 21 * q^34 + 5 * q^35 + 18 * q^36 + 24 * q^37 + 8 * q^38 + 6 * q^39 - 5 * q^40 + 19 * q^41 - 6 * q^43 + 5 * q^45 - q^46 + 15 * q^47 + 38 * q^48 + 10 * q^49 - q^50 + 8 * q^51 - 25 * q^52 - q^53 - 3 * q^56 + 11 * q^58 + 23 * q^59 + 7 * q^60 + 11 * q^62 + 10 * q^63 + 53 * q^64 - 29 * q^65 + 38 * q^67 + 87 * q^68 - 6 * q^70 + 26 * q^71 - 3 * q^72 - q^73 - 39 * q^74 + 31 * q^75 - 2 * q^76 + q^78 + 5 * q^79 + 6 * q^80 + 10 * q^81 + 5 * q^82 + 6 * q^83 + 18 * q^84 - q^85 - 41 * q^86 - 14 * q^87 - 9 * q^89 - 6 * q^90 + 6 * q^91 - 48 * q^92 + 26 * q^93 - 42 * q^95 - 41 * q^96 + 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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