LMFDB - Embedded newform 6069.2.a.be.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6069,2,Mod(1,6069)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6069, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6069.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6069 = 3 \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6069.a (trivial)

Newform invariants

comment:select newform

sage:traces = [10,4,10,12,-6,4,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$48.4612089867$
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} - 4x^{9} - 8x^{8} + 44x^{7} + 5x^{6} - 144x^{5} + 48x^{4} + 160x^{3} - 44x^{2} - 64x - 2 $ x^10 - 4x^9 - 8x^8 + 44x^7 + 5x^6 - 144x^5 + 48x^4 + 160x^3 - 44x^2 - 64*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 357)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.901740$ of defining polynomial
Character	$\chi$	$=$	6069.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.901740 q^{2} +1.00000 q^{3} -1.18686 q^{4} +0.631060 q^{5} -0.901740 q^{6} -1.00000 q^{7} +2.87372 q^{8} +1.00000 q^{9} -0.569052 q^{10} -0.369425 q^{11} -1.18686 q^{12} -4.13176 q^{13} +0.901740 q^{14} +0.631060 q^{15} -0.217625 q^{16} -0.901740 q^{18} -5.75446 q^{19} -0.748982 q^{20} -1.00000 q^{21} +0.333125 q^{22} +6.44827 q^{23} +2.87372 q^{24} -4.60176 q^{25} +3.72577 q^{26} +1.00000 q^{27} +1.18686 q^{28} +0.663950 q^{29} -0.569052 q^{30} -9.71295 q^{31} -5.55121 q^{32} -0.369425 q^{33} -0.631060 q^{35} -1.18686 q^{36} +0.965956 q^{37} +5.18903 q^{38} -4.13176 q^{39} +1.81349 q^{40} +6.52081 q^{41} +0.901740 q^{42} +4.21798 q^{43} +0.438457 q^{44} +0.631060 q^{45} -5.81467 q^{46} +8.03481 q^{47} -0.217625 q^{48} +1.00000 q^{49} +4.14960 q^{50} +4.90383 q^{52} +7.76866 q^{53} -0.901740 q^{54} -0.233129 q^{55} -2.87372 q^{56} -5.75446 q^{57} -0.598711 q^{58} -8.34114 q^{59} -0.748982 q^{60} -12.6862 q^{61} +8.75856 q^{62} -1.00000 q^{63} +5.44100 q^{64} -2.60739 q^{65} +0.333125 q^{66} +3.38378 q^{67} +6.44827 q^{69} +0.569052 q^{70} +9.27556 q^{71} +2.87372 q^{72} +13.4598 q^{73} -0.871041 q^{74} -4.60176 q^{75} +6.82976 q^{76} +0.369425 q^{77} +3.72577 q^{78} +4.51167 q^{79} -0.137335 q^{80} +1.00000 q^{81} -5.88008 q^{82} +0.439943 q^{83} +1.18686 q^{84} -3.80353 q^{86} +0.663950 q^{87} -1.06163 q^{88} -2.28393 q^{89} -0.569052 q^{90} +4.13176 q^{91} -7.65322 q^{92} -9.71295 q^{93} -7.24532 q^{94} -3.63141 q^{95} -5.55121 q^{96} +4.28369 q^{97} -0.901740 q^{98} -0.369425 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{2} + 10 q^{3} + 12 q^{4} - 6 q^{5} + 4 q^{6} - 10 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{11} + 12 q^{12} + 6 q^{13} - 4 q^{14} - 6 q^{15} + 20 q^{16} + 4 q^{18} + 6 q^{19} - 16 q^{20} - 10 q^{21}+ \cdots - 2 q^{99}+O(q^{100}) $ 10 * q + 4 * q^2 + 10 * q^3 + 12 * q^4 - 6 * q^5 + 4 * q^6 - 10 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^11 + 12 * q^12 + 6 * q^13 - 4 * q^14 - 6 * q^15 + 20 * q^16 + 4 * q^18 + 6 * q^19 - 16 * q^20 - 10 * q^21 + 24 * q^22 + 18 * q^23 + 12 * q^24 + 4 * q^25 + 12 * q^26 + 10 * q^27 - 12 * q^28 - 12 * q^29 - 8 * q^31 + 28 * q^32 - 2 * q^33 + 6 * q^35 + 12 * q^36 + 24 * q^37 + 32 * q^38 + 6 * q^39 - 36 * q^40 - 2 * q^41 - 4 * q^42 + 6 * q^43 + 28 * q^44 - 6 * q^45 - 28 * q^46 + 16 * q^47 + 20 * q^48 + 10 * q^49 + 52 * q^50 + 24 * q^52 - 12 * q^53 + 4 * q^54 + 18 * q^55 - 12 * q^56 + 6 * q^57 + 20 * q^58 + 20 * q^59 - 16 * q^60 - 4 * q^61 - 24 * q^62 - 10 * q^63 + 56 * q^64 - 38 * q^65 + 24 * q^66 + 20 * q^67 + 18 * q^69 - 20 * q^71 + 12 * q^72 + 20 * q^73 + 36 * q^74 + 4 * q^75 - 8 * q^76 + 2 * q^77 + 12 * q^78 + 20 * q^79 - 92 * q^80 + 10 * q^81 + 8 * q^82 + 36 * q^83 - 12 * q^84 - 12 * q^87 + 92 * q^88 + 32 * q^89 - 6 * q^91 + 40 * q^92 - 8 * q^93 + 60 * q^94 - 14 * q^95 + 28 * q^96 + 36 * q^97 + 4 * q^98 - 2 * q^99

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 $

$2$	−0.901740	−0.637627	−0.318813	−	0.947818i	$-0.603284\pi$
$2$	−0.901740	−0.637627	−0.318813	+	0.947818i	$0.603284\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.18686	−0.593432
$5$	0.631060	0.282219	0.141109	−	0.989994i	$-0.454933\pi$
$5$	0.631060	0.282219	0.141109	+	0.989994i	$0.454933\pi$
$6$	−0.901740	−0.368134
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.87372	1.01601
$9$	1.00000	0.333333
$10$	−0.569052	−0.179950
$11$	−0.369425	−0.111386	−0.0556929	−	0.998448i	$-0.517737\pi$
$11$	−0.369425	−0.111386	−0.0556929	+	0.998448i	$0.517737\pi$
$12$	−1.18686	−0.342618
$13$	−4.13176	−1.14594	−0.572971	−	0.819575i	$-0.694209\pi$
$13$	−4.13176	−1.14594	−0.572971	+	0.819575i	$0.694209\pi$
$14$	0.901740	0.241000
$15$	0.631060	0.162939
$16$	−0.217625	−0.0544063
$17$	0	0
$17$	0	0
$18$	−0.901740	−0.212542
$19$	−5.75446	−1.32016	−0.660081	−	0.751194i	$-0.729478\pi$
$19$	−5.75446	−1.32016	−0.660081	+	0.751194i	$0.729478\pi$
$20$	−0.748982	−0.167478
$21$	−1.00000	−0.218218
$22$	0.333125	0.0710226
$23$	6.44827	1.34456	0.672279	−	0.740298i	$-0.265316\pi$
$23$	6.44827	1.34456	0.672279	+	0.740298i	$0.265316\pi$
$24$	2.87372	0.586597
$25$	−4.60176	−0.920353
$26$	3.72577	0.730684
$27$	1.00000	0.192450
$28$	1.18686	0.224296
$29$	0.663950	0.123292	0.0616462	−	0.998098i	$-0.480365\pi$
$29$	0.663950	0.123292	0.0616462	+	0.998098i	$0.480365\pi$
$30$	−0.569052	−0.103894
$31$	−9.71295	−1.74450	−0.872249	−	0.489063i	$-0.837339\pi$
$31$	−9.71295	−1.74450	−0.872249	+	0.489063i	$0.837339\pi$
$32$	−5.55121	−0.981324
$33$	−0.369425	−0.0643086
$34$	0	0
$35$	−0.631060	−0.106669
$36$	−1.18686	−0.197811
$37$	0.965956	0.158802	0.0794011	−	0.996843i	$-0.474699\pi$
$37$	0.965956	0.158802	0.0794011	+	0.996843i	$0.474699\pi$
$38$	5.18903	0.841771
$39$	−4.13176	−0.661610
$40$	1.81349	0.286738
$41$	6.52081	1.01838	0.509189	−	0.860654i	$-0.329945\pi$
$41$	6.52081	1.01838	0.509189	+	0.860654i	$0.329945\pi$
$42$	0.901740	0.139142
$43$	4.21798	0.643236	0.321618	−	0.946869i	$-0.395773\pi$
$43$	4.21798	0.643236	0.321618	+	0.946869i	$0.395773\pi$
$44$	0.438457	0.0660999
$45$	0.631060	0.0940729
$46$	−5.81467	−0.857326
$47$	8.03481	1.17200	0.585999	−	0.810312i	$-0.300702\pi$
$47$	8.03481	1.17200	0.585999	+	0.810312i	$0.300702\pi$
$48$	−0.217625	−0.0314115
$49$	1.00000	0.142857
$50$	4.14960	0.586842
$51$	0	0
$52$	4.90383	0.680039
$53$	7.76866	1.06711	0.533553	−	0.845766i	$-0.320856\pi$
$53$	7.76866	1.06711	0.533553	+	0.845766i	$0.320856\pi$
$54$	−0.901740	−0.122711
$55$	−0.233129	−0.0314351
$56$	−2.87372	−0.384018
$57$	−5.75446	−0.762196
$58$	−0.598711	−0.0786145
$59$	−8.34114	−1.08592	−0.542962	−	0.839757i	$-0.682697\pi$
$59$	−8.34114	−1.08592	−0.542962	+	0.839757i	$0.682697\pi$
$60$	−0.748982	−0.0966932
$61$	−12.6862	−1.62430	−0.812152	−	0.583446i	$-0.801704\pi$
$61$	−12.6862	−1.62430	−0.812152	+	0.583446i	$0.801704\pi$
$62$	8.75856	1.11234
$63$	−1.00000	−0.125988
$64$	5.44100	0.680125
$65$	−2.60739	−0.323406
$66$	0.333125	0.0410049
$67$	3.38378	0.413394	0.206697	−	0.978405i	$-0.433728\pi$
$67$	3.38378	0.413394	0.206697	+	0.978405i	$0.433728\pi$
$68$	0	0
$69$	6.44827	0.776281
$70$	0.569052	0.0680148
$71$	9.27556	1.10081	0.550403	−	0.834899i	$-0.314474\pi$
$71$	9.27556	1.10081	0.550403	+	0.834899i	$0.314474\pi$
$72$	2.87372	0.338672
$73$	13.4598	1.57536	0.787678	−	0.616087i	$-0.211283\pi$
$73$	13.4598	1.57536	0.787678	+	0.616087i	$0.211283\pi$
$74$	−0.871041	−0.101257
$75$	−4.60176	−0.531366
$76$	6.82976	0.783427
$77$	0.369425	0.0420999
$78$	3.72577	0.421861
$79$	4.51167	0.507602	0.253801	−	0.967256i	$-0.418319\pi$
$79$	4.51167	0.507602	0.253801	+	0.967256i	$0.418319\pi$
$80$	−0.137335	−0.0153545
$81$	1.00000	0.111111
$82$	−5.88008	−0.649346
$83$	0.439943	0.0482900	0.0241450	−	0.999708i	$-0.492314\pi$
$83$	0.439943	0.0482900	0.0241450	+	0.999708i	$0.492314\pi$
$84$	1.18686	0.129497
$85$	0	0
$86$	−3.80353	−0.410145
$87$	0.663950	0.0711829
$88$	−1.06163	−0.113170
$89$	−2.28393	−0.242096	−0.121048	−	0.992647i	$-0.538625\pi$
$89$	−2.28393	−0.242096	−0.121048	+	0.992647i	$0.538625\pi$
$90$	−0.569052	−0.0599834
$91$	4.13176	0.433126
$92$	−7.65322	−0.797904
$93$	−9.71295	−1.00719
$94$	−7.24532	−0.747297
$95$	−3.63141	−0.372575
$96$	−5.55121	−0.566568
$97$	4.28369	0.434943	0.217471	−	0.976067i	$-0.430219\pi$
$97$	4.28369	0.434943	0.217471	+	0.976067i	$0.430219\pi$
$98$	−0.901740	−0.0910895
$99$	−0.369425	−0.0371286
$100$	5.46167	0.546167
$101$	−1.87512	−0.186581	−0.0932906	−	0.995639i	$-0.529739\pi$
$101$	−1.87512	−0.186581	−0.0932906	+	0.995639i	$0.529739\pi$
$102$	0	0
$103$	−10.5644	−1.04094	−0.520470	−	0.853880i	$-0.674243\pi$
$103$	−10.5644	−1.04094	−0.520470	+	0.853880i	$0.674243\pi$
$104$	−11.8735	−1.16430
$105$	−0.631060	−0.0615851
$106$	−7.00531	−0.680416
$107$	20.2072	1.95351	0.976754	−	0.214363i	$-0.0687676\pi$
$107$	20.2072	1.95351	0.976754	+	0.214363i	$0.0687676\pi$
$108$	−1.18686	−0.114206
$109$	−2.47901	−0.237446	−0.118723	−	0.992927i	$-0.537880\pi$
$109$	−2.47901	−0.237446	−0.118723	+	0.992927i	$0.537880\pi$
$110$	0.210222	0.0200439
$111$	0.965956	0.0916845
$112$	0.217625	0.0205637
$113$	7.51457	0.706911	0.353456	−	0.935451i	$-0.385007\pi$
$113$	7.51457	0.706911	0.353456	+	0.935451i	$0.385007\pi$
$114$	5.18903	0.485997
$115$	4.06925	0.379459
$116$	−0.788018	−0.0731657
$117$	−4.13176	−0.381981
$118$	7.52154	0.692414
$119$	0	0
$120$	1.81349	0.165548
$121$	−10.8635	−0.987593
$122$	11.4397	1.03570
$123$	6.52081	0.587961
$124$	11.5279	1.03524
$125$	−6.05929	−0.541959
$126$	0.901740	0.0803334
$127$	−0.422967	−0.0375322	−0.0187661	−	0.999824i	$-0.505974\pi$
$127$	−0.422967	−0.0375322	−0.0187661	+	0.999824i	$0.505974\pi$
$128$	6.19605	0.547658
$129$	4.21798	0.371373
$130$	2.35119	0.206213
$131$	−16.7422	−1.46278	−0.731388	−	0.681962i	$-0.761127\pi$
$131$	−16.7422	−1.46278	−0.731388	+	0.681962i	$0.761127\pi$
$132$	0.438457	0.0381628
$133$	5.75446	0.498975
$134$	−3.05129	−0.263591
$135$	0.631060	0.0543130
$136$	0	0
$137$	22.4668	1.91947	0.959734	−	0.280911i	$-0.0906366\pi$
$137$	22.4668	1.91947	0.959734	+	0.280911i	$0.0906366\pi$
$138$	−5.81467	−0.494978
$139$	15.4411	1.30969	0.654847	−	0.755761i	$-0.272733\pi$
$139$	15.4411	1.30969	0.654847	+	0.755761i	$0.272733\pi$
$140$	0.748982	0.0633006
$141$	8.03481	0.676653
$142$	−8.36415	−0.701904
$143$	1.52637	0.127642
$144$	−0.217625	−0.0181354
$145$	0.418992	0.0347954
$146$	−12.1373	−1.00449
$147$	1.00000	0.0824786
$148$	−1.14646	−0.0942383
$149$	−3.31998	−0.271983	−0.135992	−	0.990710i	$-0.543422\pi$
$149$	−3.31998	−0.271983	−0.135992	+	0.990710i	$0.543422\pi$
$150$	4.14960	0.338813
$151$	−14.9512	−1.21671	−0.608356	−	0.793664i	$-0.708171\pi$
$151$	−14.9512	−1.21671	−0.608356	+	0.793664i	$0.708171\pi$
$152$	−16.5367	−1.34131
$153$	0	0
$154$	−0.333125	−0.0268440
$155$	−6.12945	−0.492330
$156$	4.90383	0.392621
$157$	19.4634	1.55335	0.776673	−	0.629904i	$-0.216906\pi$
$157$	19.4634	1.55335	0.776673	+	0.629904i	$0.216906\pi$
$158$	−4.06835	−0.323661
$159$	7.76866	0.616094
$160$	−3.50314	−0.276948
$161$	−6.44827	−0.508195
$162$	−0.901740	−0.0708474
$163$	21.1986	1.66041	0.830203	−	0.557462i	$-0.188225\pi$
$163$	21.1986	1.66041	0.830203	+	0.557462i	$0.188225\pi$
$164$	−7.73931	−0.604339
$165$	−0.233129	−0.0181491
$166$	−0.396715	−0.0307910
$167$	−1.58681	−0.122791	−0.0613956	−	0.998114i	$-0.519555\pi$
$167$	−1.58681	−0.122791	−0.0613956	+	0.998114i	$0.519555\pi$
$168$	−2.87372	−0.221713
$169$	4.07141	0.313185
$170$	0	0
$171$	−5.75446	−0.440054
$172$	−5.00617	−0.381717
$173$	−1.09964	−0.0836043	−0.0418021	−	0.999126i	$-0.513310\pi$
$173$	−1.09964	−0.0836043	−0.0418021	+	0.999126i	$0.513310\pi$
$174$	−0.598711	−0.0453881
$175$	4.60176	0.347861
$176$	0.0803962	0.00606009
$177$	−8.34114	−0.626958
$178$	2.05951	0.154367
$179$	6.30828	0.471503	0.235751	−	0.971813i	$-0.424245\pi$
$179$	6.30828	0.471503	0.235751	+	0.971813i	$0.424245\pi$
$180$	−0.748982	−0.0558259
$181$	−6.43953	−0.478646	−0.239323	−	0.970940i	$-0.576926\pi$
$181$	−6.43953	−0.478646	−0.239323	+	0.970940i	$0.576926\pi$
$182$	−3.72577	−0.276173
$183$	−12.6862	−0.937792
$184$	18.5306	1.36609
$185$	0.609576	0.0448169
$186$	8.75856	0.642209
$187$	0	0
$188$	−9.53623	−0.695501
$189$	−1.00000	−0.0727393
$190$	3.27459	0.237563
$191$	20.9490	1.51582	0.757908	−	0.652361i	$-0.226222\pi$
$191$	20.9490	1.51582	0.757908	+	0.652361i	$0.226222\pi$
$192$	5.44100	0.392670
$193$	−7.87040	−0.566523	−0.283262	−	0.959043i	$-0.591417\pi$
$193$	−7.87040	−0.566523	−0.283262	+	0.959043i	$0.591417\pi$
$194$	−3.86277	−0.277331
$195$	−2.60739	−0.186719
$196$	−1.18686	−0.0847760
$197$	−12.1738	−0.867345	−0.433672	−	0.901071i	$-0.642782\pi$
$197$	−12.1738	−0.867345	−0.433672	+	0.901071i	$0.642782\pi$
$198$	0.333125	0.0236742
$199$	−22.1369	−1.56924	−0.784622	−	0.619974i	$-0.787143\pi$
$199$	−22.1369	−1.56924	−0.784622	+	0.619974i	$0.787143\pi$
$200$	−13.2242	−0.935092
$201$	3.38378	0.238673
$202$	1.69087	0.118969
$203$	−0.663950	−0.0466001
$204$	0	0
$205$	4.11502	0.287405
$206$	9.52633	0.663731
$207$	6.44827	0.448186
$208$	0.899175	0.0623466
$209$	2.12584	0.147047
$210$	0.569052	0.0392683
$211$	10.8176	0.744715	0.372357	−	0.928089i	$-0.378550\pi$
$211$	10.8176	0.744715	0.372357	+	0.928089i	$0.378550\pi$
$212$	−9.22034	−0.633255
$213$	9.27556	0.635551
$214$	−18.2217	−1.24561
$215$	2.66180	0.181533
$216$	2.87372	0.195532
$217$	9.71295	0.659358
$218$	2.23543	0.151402
$219$	13.4598	0.909532
$220$	0.276693	0.0186546
$221$	0	0
$222$	−0.871041	−0.0584605
$223$	−24.2559	−1.62430	−0.812148	−	0.583452i	$-0.801702\pi$
$223$	−24.2559	−1.62430	−0.812148	+	0.583452i	$0.801702\pi$
$224$	5.55121	0.370906
$225$	−4.60176	−0.306784
$226$	−6.77619	−0.450746
$227$	−4.57003	−0.303324	−0.151662	−	0.988432i	$-0.548462\pi$
$227$	−4.57003	−0.303324	−0.151662	+	0.988432i	$0.548462\pi$
$228$	6.82976	0.452312
$229$	9.40610	0.621573	0.310786	−	0.950480i	$-0.399408\pi$
$229$	9.40610	0.621573	0.310786	+	0.950480i	$0.399408\pi$
$230$	−3.66941	−0.241953
$231$	0.369425	0.0243064
$232$	1.90801	0.125267
$233$	9.21166	0.603476	0.301738	−	0.953391i	$-0.402433\pi$
$233$	9.21166	0.603476	0.301738	+	0.953391i	$0.402433\pi$
$234$	3.72577	0.243561
$235$	5.07045	0.330760
$236$	9.89980	0.644422
$237$	4.51167	0.293064
$238$	0	0
$239$	28.3078	1.83108	0.915539	−	0.402230i	$-0.131765\pi$
$239$	28.3078	1.83108	0.915539	+	0.402230i	$0.131765\pi$
$240$	−0.137335	−0.00886491
$241$	−18.0223	−1.16092	−0.580459	−	0.814289i	$-0.697127\pi$
$241$	−18.0223	−1.16092	−0.580459	+	0.814289i	$0.697127\pi$
$242$	9.79608	0.629716
$243$	1.00000	0.0641500
$244$	15.0568	0.963914
$245$	0.631060	0.0403169
$246$	−5.88008	−0.374900
$247$	23.7760	1.51283
$248$	−27.9123	−1.77244
$249$	0.439943	0.0278803
$250$	5.46391	0.345568
$251$	11.2948	0.712919	0.356460	−	0.934311i	$-0.383984\pi$
$251$	11.2948	0.712919	0.356460	+	0.934311i	$0.383984\pi$
$252$	1.18686	0.0747654
$253$	−2.38215	−0.149765
$254$	0.381406	0.0239316
$255$	0	0
$256$	−16.4692	−1.02933
$257$	7.97952	0.497749	0.248874	−	0.968536i	$-0.419939\pi$
$257$	7.97952	0.497749	0.248874	+	0.968536i	$0.419939\pi$
$258$	−3.80353	−0.236797
$259$	−0.965956	−0.0600216
$260$	3.09461	0.191920
$261$	0.663950	0.0410975
$262$	15.0971	0.932705
$263$	15.5763	0.960477	0.480238	−	0.877138i	$-0.340550\pi$
$263$	15.5763	0.960477	0.480238	+	0.877138i	$0.340550\pi$
$264$	−1.06163	−0.0653385
$265$	4.90249	0.301157
$266$	−5.18903	−0.318160
$267$	−2.28393	−0.139774
$268$	−4.01609	−0.245322
$269$	−2.41968	−0.147530	−0.0737652	−	0.997276i	$-0.523502\pi$
$269$	−2.41968	−0.147530	−0.0737652	+	0.997276i	$0.523502\pi$
$270$	−0.569052	−0.0346314
$271$	14.7878	0.898295	0.449147	−	0.893458i	$-0.351728\pi$
$271$	14.7878	0.898295	0.449147	+	0.893458i	$0.351728\pi$
$272$	0	0
$273$	4.13176	0.250065
$274$	−20.2592	−1.22390
$275$	1.70001	0.102514
$276$	−7.65322	−0.460670
$277$	29.8845	1.79559	0.897793	−	0.440417i	$-0.145170\pi$
$277$	29.8845	1.79559	0.897793	+	0.440417i	$0.145170\pi$
$278$	−13.9238	−0.835096
$279$	−9.71295	−0.581499
$280$	−1.81349	−0.108377
$281$	−15.9618	−0.952202	−0.476101	−	0.879391i	$-0.657950\pi$
$281$	−15.9618	−0.952202	−0.476101	+	0.879391i	$0.657950\pi$
$282$	−7.24532	−0.431452
$283$	6.74526	0.400964	0.200482	−	0.979697i	$-0.435749\pi$
$283$	6.74526	0.400964	0.200482	+	0.979697i	$0.435749\pi$
$284$	−11.0088	−0.653254
$285$	−3.63141	−0.215106
$286$	−1.37639	−0.0813878
$287$	−6.52081	−0.384911
$288$	−5.55121	−0.327108
$289$	0	0
$290$	−0.377822	−0.0221865
$291$	4.28369	0.251114
$292$	−15.9750	−0.934867
$293$	1.07630	0.0628779	0.0314389	−	0.999506i	$-0.489991\pi$
$293$	1.07630	0.0628779	0.0314389	+	0.999506i	$0.489991\pi$
$294$	−0.901740	−0.0525906
$295$	−5.26376	−0.306468
$296$	2.77589	0.161345
$297$	−0.369425	−0.0214362
$298$	2.99376	0.173424
$299$	−26.6427	−1.54079
$300$	5.46167	0.315330
$301$	−4.21798	−0.243121
$302$	13.4821	0.775808
$303$	−1.87512	−0.107723
$304$	1.25232	0.0718252
$305$	−8.00576	−0.458409
$306$	0	0
$307$	4.09877	0.233929	0.116964	−	0.993136i	$-0.462684\pi$
$307$	4.09877	0.233929	0.116964	+	0.993136i	$0.462684\pi$
$308$	−0.438457	−0.0249834
$309$	−10.5644	−0.600987
$310$	5.52718	0.313922
$311$	−26.2434	−1.48813	−0.744064	−	0.668108i	$-0.767104\pi$
$311$	−26.2434	−1.48813	−0.744064	+	0.668108i	$0.767104\pi$
$312$	−11.8735	−0.672206
$313$	−9.58780	−0.541934	−0.270967	−	0.962589i	$-0.587343\pi$
$313$	−9.58780	−0.541934	−0.270967	+	0.962589i	$0.587343\pi$
$314$	−17.5509	−0.990455
$315$	−0.631060	−0.0355562
$316$	−5.35473	−0.301227
$317$	16.0367	0.900709	0.450355	−	0.892850i	$-0.351298\pi$
$317$	16.0367	0.900709	0.450355	+	0.892850i	$0.351298\pi$
$318$	−7.00531	−0.392838
$319$	−0.245280	−0.0137330
$320$	3.43360	0.191944
$321$	20.2072	1.12786
$322$	5.81467	0.324039
$323$	0	0
$324$	−1.18686	−0.0659369
$325$	19.0134	1.05467
$326$	−19.1157	−1.05872
$327$	−2.47901	−0.137090
$328$	18.7390	1.03469
$329$	−8.03481	−0.442974
$330$	0.210222	0.0115723
$331$	18.2970	1.00570	0.502848	−	0.864375i	$-0.332286\pi$
$331$	18.2970	1.00570	0.502848	+	0.864375i	$0.332286\pi$
$332$	−0.522153	−0.0286569
$333$	0.965956	0.0529341
$334$	1.43089	0.0782950
$335$	2.13537	0.116668
$336$	0.217625	0.0118724
$337$	7.29509	0.397389	0.198694	−	0.980061i	$-0.436330\pi$
$337$	7.29509	0.397389	0.198694	+	0.980061i	$0.436330\pi$
$338$	−3.67135	−0.199695
$339$	7.51457	0.408135
$340$	0	0
$341$	3.58820	0.194312
$342$	5.18903	0.280590
$343$	−1.00000	−0.0539949
$344$	12.1213	0.653538
$345$	4.06925	0.219081
$346$	0.991592	0.0533083
$347$	−22.5231	−1.20910	−0.604551	−	0.796567i	$-0.706647\pi$
$347$	−22.5231	−1.20910	−0.604551	+	0.796567i	$0.706647\pi$
$348$	−0.788018	−0.0422422
$349$	26.9175	1.44086	0.720429	−	0.693528i	$-0.243945\pi$
$349$	26.9175	1.44086	0.720429	+	0.693528i	$0.243945\pi$
$350$	−4.14960	−0.221805
$351$	−4.13176	−0.220537
$352$	2.05075	0.109306
$353$	20.1149	1.07061	0.535304	−	0.844660i	$-0.320197\pi$
$353$	20.1149	1.07061	0.535304	+	0.844660i	$0.320197\pi$
$354$	7.52154	0.399765
$355$	5.85344	0.310668
$356$	2.71071	0.143667
$357$	0	0
$358$	−5.68843	−0.300643
$359$	6.48908	0.342481	0.171240	−	0.985229i	$-0.445223\pi$
$359$	6.48908	0.342481	0.171240	+	0.985229i	$0.445223\pi$
$360$	1.81349	0.0955794
$361$	14.1138	0.742830
$362$	5.80678	0.305198
$363$	−10.8635	−0.570187
$364$	−4.90383	−0.257031
$365$	8.49397	0.444595
$366$	11.4397	0.597961
$367$	6.74119	0.351887	0.175944	−	0.984400i	$-0.443702\pi$
$367$	6.74119	0.351887	0.175944	+	0.984400i	$0.443702\pi$
$368$	−1.40331	−0.0731525
$369$	6.52081	0.339460
$370$	−0.549679	−0.0285765
$371$	−7.76866	−0.403329
$372$	11.5279	0.597696
$373$	13.2838	0.687807	0.343904	−	0.939005i	$-0.388251\pi$
$373$	13.2838	0.687807	0.343904	+	0.939005i	$0.388251\pi$
$374$	0	0
$375$	−6.05929	−0.312900
$376$	23.0898	1.19077
$377$	−2.74328	−0.141286
$378$	0.901740	0.0463805
$379$	13.4136	0.689011	0.344505	−	0.938784i	$-0.388047\pi$
$379$	13.4136	0.689011	0.344505	+	0.938784i	$0.388047\pi$
$380$	4.30999	0.221098
$381$	−0.422967	−0.0216693
$382$	−18.8906	−0.966525
$383$	28.2085	1.44138	0.720692	−	0.693255i	$-0.243824\pi$
$383$	28.2085	1.44138	0.720692	+	0.693255i	$0.243824\pi$
$384$	6.19605	0.316191
$385$	0.233129	0.0118814
$386$	7.09705	0.361231
$387$	4.21798	0.214412
$388$	−5.08415	−0.258109
$389$	−2.24283	−0.113716	−0.0568581	−	0.998382i	$-0.518108\pi$
$389$	−2.24283	−0.113716	−0.0568581	+	0.998382i	$0.518108\pi$
$390$	2.35119	0.119057
$391$	0	0
$392$	2.87372	0.145145
$393$	−16.7422	−0.844534
$394$	10.9776	0.553042
$395$	2.84713	0.143255
$396$	0.438457	0.0220333
$397$	12.0970	0.607129	0.303564	−	0.952811i	$-0.401823\pi$
$397$	12.0970	0.607129	0.303564	+	0.952811i	$0.401823\pi$
$398$	19.9618	1.00059
$399$	5.75446	0.288083
$400$	1.00146	0.0500730
$401$	15.6350	0.780777	0.390388	−	0.920650i	$-0.372341\pi$
$401$	15.6350	0.780777	0.390388	+	0.920650i	$0.372341\pi$
$402$	−3.05129	−0.152185
$403$	40.1315	1.99909
$404$	2.22551	0.110723
$405$	0.631060	0.0313576
$406$	0.598711	0.0297135
$407$	−0.356848	−0.0176883
$408$	0	0
$409$	16.9806	0.839639	0.419819	−	0.907608i	$-0.362093\pi$
$409$	16.9806	0.839639	0.419819	+	0.907608i	$0.362093\pi$
$410$	−3.71068	−0.183257
$411$	22.4668	1.10821
$412$	12.5385	0.617727
$413$	8.34114	0.410440
$414$	−5.81467	−0.285775
$415$	0.277631	0.0136283
$416$	22.9362	1.12454
$417$	15.4411	0.756152
$418$	−1.91696	−0.0937613
$419$	19.8074	0.967653	0.483827	−	0.875164i	$-0.339247\pi$
$419$	19.8074	0.967653	0.483827	+	0.875164i	$0.339247\pi$
$420$	0.748982	0.0365466
$421$	−21.4954	−1.04762	−0.523810	−	0.851835i	$-0.675490\pi$
$421$	−21.4954	−1.04762	−0.523810	+	0.851835i	$0.675490\pi$
$422$	−9.75468	−0.474850
$423$	8.03481	0.390666
$424$	22.3250	1.08420
$425$	0	0
$426$	−8.36415	−0.405244
$427$	12.6862	0.613929
$428$	−23.9833	−1.15927
$429$	1.52637	0.0736940
$430$	−2.40025	−0.115750
$431$	−9.78836	−0.471489	−0.235744	−	0.971815i	$-0.575753\pi$
$431$	−9.78836	−0.471489	−0.235744	+	0.971815i	$0.575753\pi$
$432$	−0.217625	−0.0104705
$433$	−28.7229	−1.38033	−0.690166	−	0.723651i	$-0.742463\pi$
$433$	−28.7229	−1.38033	−0.690166	+	0.723651i	$0.742463\pi$
$434$	−8.75856	−0.420424
$435$	0.418992	0.0200891
$436$	2.94225	0.140908
$437$	−37.1063	−1.77504
$438$	−12.1373	−0.579942
$439$	18.1335	0.865463	0.432732	−	0.901523i	$-0.357550\pi$
$439$	18.1335	0.865463	0.432732	+	0.901523i	$0.357550\pi$
$440$	−0.669949	−0.0319386
$441$	1.00000	0.0476190
$442$	0	0
$443$	30.4981	1.44901	0.724504	−	0.689270i	$-0.242069\pi$
$443$	30.4981	1.44901	0.724504	+	0.689270i	$0.242069\pi$
$444$	−1.14646	−0.0544085
$445$	−1.44129	−0.0683239
$446$	21.8725	1.03569
$447$	−3.31998	−0.157030
$448$	−5.44100	−0.257063
$449$	14.3487	0.677159	0.338579	−	0.940938i	$-0.390054\pi$
$449$	14.3487	0.677159	0.338579	+	0.940938i	$0.390054\pi$
$450$	4.14960	0.195614
$451$	−2.40895	−0.113433
$452$	−8.91877	−0.419504
$453$	−14.9512	−0.702469
$454$	4.12098	0.193407
$455$	2.60739	0.122236
$456$	−16.5367	−0.774403
$457$	−11.9231	−0.557739	−0.278869	−	0.960329i	$-0.589960\pi$
$457$	−11.9231	−0.557739	−0.278869	+	0.960329i	$0.589960\pi$
$458$	−8.48186	−0.396331
$459$	0	0
$460$	−4.82964	−0.225183
$461$	17.3571	0.808399	0.404200	−	0.914671i	$-0.367550\pi$
$461$	17.3571	0.808399	0.404200	+	0.914671i	$0.367550\pi$
$462$	−0.333125	−0.0154984
$463$	14.5685	0.677054	0.338527	−	0.940957i	$-0.390071\pi$
$463$	14.5685	0.677054	0.338527	+	0.940957i	$0.390071\pi$
$464$	−0.144492	−0.00670789
$465$	−6.12945	−0.284247
$466$	−8.30653	−0.384793
$467$	−13.1901	−0.610363	−0.305182	−	0.952294i	$-0.598717\pi$
$467$	−13.1901	−0.610363	−0.305182	+	0.952294i	$0.598717\pi$
$468$	4.90383	0.226680
$469$	−3.38378	−0.156248
$470$	−4.57223	−0.210901
$471$	19.4634	0.896825
$472$	−23.9701	−1.10331
$473$	−1.55823	−0.0716474
$474$	−4.06835	−0.186866
$475$	26.4806	1.21502
$476$	0	0
$477$	7.76866	0.355702
$478$	−25.5263	−1.16754
$479$	17.8906	0.817440	0.408720	−	0.912660i	$-0.365975\pi$
$479$	17.8906	0.817440	0.408720	+	0.912660i	$0.365975\pi$
$480$	−3.50314	−0.159896
$481$	−3.99109	−0.181978
$482$	16.2514	0.740233
$483$	−6.44827	−0.293407
$484$	12.8935	0.586069
$485$	2.70326	0.122749
$486$	−0.901740	−0.0409038
$487$	−9.54854	−0.432686	−0.216343	−	0.976317i	$-0.569413\pi$
$487$	−9.54854	−0.432686	−0.216343	+	0.976317i	$0.569413\pi$
$488$	−36.4567	−1.65032
$489$	21.1986	0.958636
$490$	−0.569052	−0.0257072
$491$	−36.1797	−1.63277	−0.816384	−	0.577510i	$-0.804025\pi$
$491$	−36.1797	−1.63277	−0.816384	+	0.577510i	$0.804025\pi$
$492$	−7.73931	−0.348915
$493$	0	0
$494$	−21.4398	−0.964622
$495$	−0.233129	−0.0104784
$496$	2.11378	0.0949117
$497$	−9.27556	−0.416066
$498$	−0.396715	−0.0177772
$499$	−3.97400	−0.177901	−0.0889504	−	0.996036i	$-0.528351\pi$
$499$	−3.97400	−0.177901	−0.0889504	+	0.996036i	$0.528351\pi$
$500$	7.19155	0.321616
$501$	−1.58681	−0.0708936
$502$	−10.1849	−0.454576
$503$	−18.5350	−0.826434	−0.413217	−	0.910633i	$-0.635595\pi$
$503$	−18.5350	−0.826434	−0.413217	+	0.910633i	$0.635595\pi$
$504$	−2.87372	−0.128006
$505$	−1.18331	−0.0526567
$506$	2.14808	0.0954940
$507$	4.07141	0.180818
$508$	0.502004	0.0222728
$509$	12.0683	0.534916	0.267458	−	0.963569i	$-0.413816\pi$
$509$	12.0683	0.534916	0.267458	+	0.963569i	$0.413816\pi$
$510$	0	0
$511$	−13.4598	−0.595429
$512$	2.45887	0.108668
$513$	−5.75446	−0.254065
$514$	−7.19545	−0.317378
$515$	−6.66676	−0.293773
$516$	−5.00617	−0.220384
$517$	−2.96826	−0.130544
$518$	0.871041	0.0382714
$519$	−1.09964	−0.0482690
$520$	−7.49291	−0.328586
$521$	−14.7173	−0.644777	−0.322388	−	0.946608i	$-0.604486\pi$
$521$	−14.7173	−0.644777	−0.322388	+	0.946608i	$0.604486\pi$
$522$	−0.598711	−0.0262048
$523$	8.29706	0.362805	0.181403	−	0.983409i	$-0.441936\pi$
$523$	8.29706	0.362805	0.181403	+	0.983409i	$0.441936\pi$
$524$	19.8708	0.868058
$525$	4.60176	0.200837
$526$	−14.0458	−0.612426
$527$	0	0
$528$	0.0803962	0.00349880
$529$	18.5802	0.807836
$530$	−4.42077	−0.192026
$531$	−8.34114	−0.361974
$532$	−6.82976	−0.296108
$533$	−26.9424	−1.16700
$534$	2.05951	0.0891237
$535$	12.7520	0.551316
$536$	9.72405	0.420015
$537$	6.30828	0.272222
$538$	2.18192	0.0940693
$539$	−0.369425	−0.0159123
$540$	−0.748982	−0.0322311
$541$	−4.45360	−0.191475	−0.0957375	−	0.995407i	$-0.530521\pi$
$541$	−4.45360	−0.191475	−0.0957375	+	0.995407i	$0.530521\pi$
$542$	−13.3348	−0.572777
$543$	−6.43953	−0.276347
$544$	0	0
$545$	−1.56441	−0.0670118
$546$	−3.72577	−0.159448
$547$	−33.2590	−1.42205	−0.711025	−	0.703167i	$-0.751769\pi$
$547$	−33.2590	−1.42205	−0.711025	+	0.703167i	$0.751769\pi$
$548$	−26.6650	−1.13907
$549$	−12.6862	−0.541434
$550$	−1.53296	−0.0653658
$551$	−3.82067	−0.162766
$552$	18.5306	0.788713
$553$	−4.51167	−0.191856
$554$	−26.9481	−1.14491
$555$	0.609576	0.0258751
$556$	−18.3265	−0.777215
$557$	29.5275	1.25112	0.625561	−	0.780176i	$-0.284870\pi$
$557$	29.5275	1.25112	0.625561	+	0.780176i	$0.284870\pi$
$558$	8.75856	0.370779
$559$	−17.4277	−0.737112
$560$	0.137335	0.00580345
$561$	0	0
$562$	14.3934	0.607149
$563$	−3.07367	−0.129540	−0.0647700	−	0.997900i	$-0.520631\pi$
$563$	−3.07367	−0.129540	−0.0647700	+	0.997900i	$0.520631\pi$
$564$	−9.53623	−0.401548
$565$	4.74214	0.199503
$566$	−6.08247	−0.255665
$567$	−1.00000	−0.0419961
$568$	26.6554	1.11844
$569$	−37.5949	−1.57606	−0.788030	−	0.615637i	$-0.788899\pi$
$569$	−37.5949	−1.57606	−0.788030	+	0.615637i	$0.788899\pi$
$570$	3.27459	0.137157
$571$	−11.4605	−0.479606	−0.239803	−	0.970822i	$-0.577083\pi$
$571$	−11.4605	−0.479606	−0.239803	+	0.970822i	$0.577083\pi$
$572$	−1.81160	−0.0757467
$573$	20.9490	0.875157
$574$	5.88008	0.245430
$575$	−29.6734	−1.23747
$576$	5.44100	0.226708
$577$	23.5806	0.981672	0.490836	−	0.871252i	$-0.336691\pi$
$577$	23.5806	0.981672	0.490836	+	0.871252i	$0.336691\pi$
$578$	0	0
$579$	−7.87040	−0.327082
$580$	−0.497287	−0.0206487
$581$	−0.439943	−0.0182519
$582$	−3.86277	−0.160117
$583$	−2.86993	−0.118861
$584$	38.6799	1.60059
$585$	−2.60739	−0.107802
$586$	−0.970540	−0.0400926
$587$	40.1399	1.65675	0.828376	−	0.560172i	$-0.189265\pi$
$587$	40.1399	1.65675	0.828376	+	0.560172i	$0.189265\pi$
$588$	−1.18686	−0.0489455
$589$	55.8927	2.30302
$590$	4.74654	0.195412
$591$	−12.1738	−0.500762
$592$	−0.210217	−0.00863984
$593$	−26.7909	−1.10017	−0.550085	−	0.835108i	$-0.685405\pi$
$593$	−26.7909	−1.10017	−0.550085	+	0.835108i	$0.685405\pi$
$594$	0.333125	0.0136683
$595$	0	0
$596$	3.94036	0.161404
$597$	−22.1369	−0.906004
$598$	24.0248	0.982447
$599$	−15.6195	−0.638196	−0.319098	−	0.947722i	$-0.603380\pi$
$599$	−15.6195	−0.638196	−0.319098	+	0.947722i	$0.603380\pi$
$600$	−13.2242	−0.539876
$601$	14.9065	0.608048	0.304024	−	0.952664i	$-0.401670\pi$
$601$	14.9065	0.608048	0.304024	+	0.952664i	$0.401670\pi$
$602$	3.80353	0.155020
$603$	3.38378	0.137798
$604$	17.7451	0.722036
$605$	−6.85554	−0.278717
$606$	1.69087	0.0686869
$607$	−17.7082	−0.718754	−0.359377	−	0.933192i	$-0.617011\pi$
$607$	−17.7082	−0.718754	−0.359377	+	0.933192i	$0.617011\pi$
$608$	31.9442	1.29551
$609$	−0.663950	−0.0269046
$610$	7.21912	0.292294
$611$	−33.1979	−1.34304
$612$	0	0
$613$	−6.55292	−0.264670	−0.132335	−	0.991205i	$-0.542247\pi$
$613$	−6.55292	−0.264670	−0.132335	+	0.991205i	$0.542247\pi$
$614$	−3.69602	−0.149159
$615$	4.11502	0.165934
$616$	1.06163	0.0427741
$617$	−26.9798	−1.08617	−0.543084	−	0.839679i	$-0.682743\pi$
$617$	−26.9798	−1.08617	−0.543084	+	0.839679i	$0.682743\pi$
$618$	9.52633	0.383205
$619$	−23.7257	−0.953615	−0.476808	−	0.879008i	$-0.658206\pi$
$619$	−23.7257	−0.953615	−0.476808	+	0.879008i	$0.658206\pi$
$620$	7.27483	0.292164
$621$	6.44827	0.258760
$622$	23.6648	0.948871
$623$	2.28393	0.0915036
$624$	0.899175	0.0359958
$625$	19.1850	0.767402
$626$	8.64570	0.345552
$627$	2.12584	0.0848978
$628$	−23.1004	−0.921805
$629$	0	0
$630$	0.569052	0.0226716
$631$	14.2900	0.568875	0.284437	−	0.958695i	$-0.408193\pi$
$631$	14.2900	0.568875	0.284437	+	0.958695i	$0.408193\pi$
$632$	12.9653	0.515731
$633$	10.8176	0.429961
$634$	−14.4609	−0.574316
$635$	−0.266917	−0.0105923
$636$	−9.22034	−0.365610
$637$	−4.13176	−0.163706
$638$	0.221179	0.00875654
$639$	9.27556	0.366936
$640$	3.91008	0.154559
$641$	−13.6147	−0.537749	−0.268874	−	0.963175i	$-0.586652\pi$
$641$	−13.6147	−0.537749	−0.268874	+	0.963175i	$0.586652\pi$
$642$	−18.2217	−0.719153
$643$	11.8861	0.468744	0.234372	−	0.972147i	$-0.424697\pi$
$643$	11.8861	0.468744	0.234372	+	0.972147i	$0.424697\pi$
$644$	7.65322	0.301579
$645$	2.66180	0.104808
$646$	0	0
$647$	−4.26169	−0.167544	−0.0837722	−	0.996485i	$-0.526697\pi$
$647$	−4.26169	−0.167544	−0.0837722	+	0.996485i	$0.526697\pi$
$648$	2.87372	0.112891
$649$	3.08142	0.120956
$650$	−17.1451	−0.672487
$651$	9.71295	0.380680
$652$	−25.1599	−0.985338
$653$	−8.89111	−0.347936	−0.173968	−	0.984751i	$-0.555659\pi$
$653$	−8.89111	−0.347936	−0.173968	+	0.984751i	$0.555659\pi$
$654$	2.23543	0.0874121
$655$	−10.5654	−0.412822
$656$	−1.41909	−0.0554063
$657$	13.4598	0.525119
$658$	7.24532	0.282452
$659$	30.2214	1.17726	0.588629	−	0.808403i	$-0.299668\pi$
$659$	30.2214	1.17726	0.588629	+	0.808403i	$0.299668\pi$
$660$	0.276693	0.0107702
$661$	46.7834	1.81966	0.909832	−	0.414976i	$-0.136210\pi$
$661$	46.7834	1.81966	0.909832	+	0.414976i	$0.136210\pi$
$662$	−16.4992	−0.641259
$663$	0	0
$664$	1.26428	0.0490634
$665$	3.63141	0.140820
$666$	−0.871041	−0.0337522
$667$	4.28133	0.165774
$668$	1.88333	0.0728683
$669$	−24.2559	−0.937788
$670$	−1.92555	−0.0743904
$671$	4.68660	0.180924
$672$	5.55121	0.214142
$673$	41.7293	1.60854	0.804272	−	0.594261i	$-0.202555\pi$
$673$	41.7293	1.60854	0.804272	+	0.594261i	$0.202555\pi$
$674$	−6.57828	−0.253386
$675$	−4.60176	−0.177122
$676$	−4.83221	−0.185854
$677$	29.1207	1.11920	0.559601	−	0.828762i	$-0.310955\pi$
$677$	29.1207	1.11920	0.559601	+	0.828762i	$0.310955\pi$
$678$	−6.77619	−0.260238
$679$	−4.28369	−0.164393
$680$	0	0
$681$	−4.57003	−0.175124
$682$	−3.23563	−0.123899
$683$	−41.1086	−1.57298	−0.786489	−	0.617605i	$-0.788103\pi$
$683$	−41.1086	−1.57298	−0.786489	+	0.617605i	$0.788103\pi$
$684$	6.82976	0.261142
$685$	14.1779	0.541709
$686$	0.901740	0.0344286
$687$	9.40610	0.358865
$688$	−0.917940	−0.0349961
$689$	−32.0982	−1.22284
$690$	−3.66941	−0.139692
$691$	−4.06354	−0.154584	−0.0772922	−	0.997008i	$-0.524627\pi$
$691$	−4.06354	−0.154584	−0.0772922	+	0.997008i	$0.524627\pi$
$692$	1.30513	0.0496135
$693$	0.369425	0.0140333
$694$	20.3100	0.770955
$695$	9.74424	0.369620
$696$	1.90801	0.0723229
$697$	0	0
$698$	−24.2726	−0.918730
$699$	9.21166	0.348417
$700$	−5.46167	−0.206432
$701$	25.4533	0.961359	0.480680	−	0.876896i	$-0.340390\pi$
$701$	25.4533	0.961359	0.480680	+	0.876896i	$0.340390\pi$
$702$	3.72577	0.140620
$703$	−5.55855	−0.209645
$704$	−2.01004	−0.0757562
$705$	5.07045	0.190964
$706$	−18.1384	−0.682648
$707$	1.87512	0.0705211
$708$	9.89980	0.372057
$709$	4.59158	0.172440	0.0862202	−	0.996276i	$-0.472521\pi$
$709$	4.59158	0.172440	0.0862202	+	0.996276i	$0.472521\pi$
$710$	−5.27828	−0.198090
$711$	4.51167	0.169201
$712$	−6.56337	−0.245973
$713$	−62.6318	−2.34558
$714$	0	0
$715$	0.963233	0.0360229
$716$	−7.48707	−0.279805
$717$	28.3078	1.05717
$718$	−5.85147	−0.218375
$719$	16.3658	0.610340	0.305170	−	0.952298i	$-0.401287\pi$
$719$	16.3658	0.610340	0.305170	+	0.952298i	$0.401287\pi$
$720$	−0.137335	−0.00511816
$721$	10.5644	0.393438
$722$	−12.7270	−0.473648
$723$	−18.0223	−0.670257
$724$	7.64284	0.284044
$725$	−3.05534	−0.113472
$726$	9.79608	0.363567
$727$	4.97474	0.184503	0.0922515	−	0.995736i	$-0.470594\pi$
$727$	4.97474	0.184503	0.0922515	+	0.995736i	$0.470594\pi$
$728$	11.8735	0.440062
$729$	1.00000	0.0370370
$730$	−7.65936	−0.283486
$731$	0	0
$732$	15.0568	0.556516
$733$	24.9209	0.920476	0.460238	−	0.887796i	$-0.347764\pi$
$733$	24.9209	0.920476	0.460238	+	0.887796i	$0.347764\pi$
$734$	−6.07880	−0.224373
$735$	0.631060	0.0232770
$736$	−35.7957	−1.31945
$737$	−1.25005	−0.0460463
$738$	−5.88008	−0.216449
$739$	23.4573	0.862890	0.431445	−	0.902139i	$-0.358004\pi$
$739$	23.4573	0.862890	0.431445	+	0.902139i	$0.358004\pi$
$740$	−0.723484	−0.0265958
$741$	23.7760	0.873434
$742$	7.00531	0.257173
$743$	5.87063	0.215373	0.107686	−	0.994185i	$-0.465656\pi$
$743$	5.87063	0.215373	0.107686	+	0.994185i	$0.465656\pi$
$744$	−27.9123	−1.02332
$745$	−2.09511	−0.0767588
$746$	−11.9785	−0.438564
$747$	0.439943	0.0160967
$748$	0	0
$749$	−20.2072	−0.738357
$750$	5.46391	0.199514
$751$	−11.6601	−0.425481	−0.212741	−	0.977109i	$-0.568239\pi$
$751$	−11.6601	−0.425481	−0.212741	+	0.977109i	$0.568239\pi$
$752$	−1.74858	−0.0637641
$753$	11.2948	0.411604
$754$	2.47373	0.0900878
$755$	−9.43511	−0.343379
$756$	1.18686	0.0431658
$757$	−38.8880	−1.41341	−0.706704	−	0.707510i	$-0.749819\pi$
$757$	−38.8880	−1.41341	−0.706704	+	0.707510i	$0.749819\pi$
$758$	−12.0956	−0.439332
$759$	−2.38215	−0.0864667
$760$	−10.4357	−0.378541
$761$	−17.4871	−0.633908	−0.316954	−	0.948441i	$-0.602660\pi$
$761$	−17.4871	−0.633908	−0.316954	+	0.948441i	$0.602660\pi$
$762$	0.381406	0.0138169
$763$	2.47901	0.0897463
$764$	−24.8636	−0.899534
$765$	0	0
$766$	−25.4367	−0.919066
$767$	34.4635	1.24441
$768$	−16.4692	−0.594282
$769$	−14.3626	−0.517928	−0.258964	−	0.965887i	$-0.583381\pi$
$769$	−14.3626	−0.517928	−0.258964	+	0.965887i	$0.583381\pi$
$770$	−0.210222	−0.00757588
$771$	7.97952	0.287375
$772$	9.34109	0.336193
$773$	17.7643	0.638937	0.319469	−	0.947597i	$-0.396496\pi$
$773$	17.7643	0.638937	0.319469	+	0.947597i	$0.396496\pi$
$774$	−3.80353	−0.136715
$775$	44.6967	1.60555
$776$	12.3101	0.441908
$777$	−0.965956	−0.0346535
$778$	2.02245	0.0725085
$779$	−37.5237	−1.34443
$780$	3.09461	0.110805
$781$	−3.42662	−0.122614
$782$	0	0
$783$	0.663950	0.0237276
$784$	−0.217625	−0.00777233
$785$	12.2826	0.438383
$786$	15.0971	0.538497
$787$	38.9112	1.38703	0.693517	−	0.720440i	$-0.256060\pi$
$787$	38.9112	1.38703	0.693517	+	0.720440i	$0.256060\pi$
$788$	14.4486	0.514710
$789$	15.5763	0.554532
$790$	−2.56737	−0.0913431
$791$	−7.51457	−0.267187
$792$	−1.06163	−0.0377232
$793$	52.4163	1.86136
$794$	−10.9083	−0.387122
$795$	4.90249	0.173873
$796$	26.2735	0.931240
$797$	32.8439	1.16339	0.581696	−	0.813406i	$-0.302389\pi$
$797$	32.8439	1.16339	0.581696	+	0.813406i	$0.302389\pi$
$798$	−5.18903	−0.183690
$799$	0	0
$800$	25.5453	0.903164
$801$	−2.28393	−0.0806986
$802$	−14.0987	−0.497844
$803$	−4.97240	−0.175472
$804$	−4.01609	−0.141636
$805$	−4.06925	−0.143422
$806$	−36.1882	−1.27468
$807$	−2.41968	−0.0851767
$808$	−5.38857	−0.189569
$809$	−2.13795	−0.0751665	−0.0375832	−	0.999294i	$-0.511966\pi$
$809$	−2.13795	−0.0751665	−0.0375832	+	0.999294i	$0.511966\pi$
$810$	−0.569052	−0.0199945
$811$	44.7014	1.56968	0.784839	−	0.619700i	$-0.212746\pi$
$811$	44.7014	1.56968	0.784839	+	0.619700i	$0.212746\pi$
$812$	0.788018	0.0276540
$813$	14.7878	0.518631
$814$	0.321784	0.0112785
$815$	13.3776	0.468597
$816$	0	0
$817$	−24.2722	−0.849177
$818$	−15.3121	−0.535376
$819$	4.13176	0.144375
$820$	−4.88397	−0.170556
$821$	−46.4962	−1.62273	−0.811364	−	0.584542i	$-0.801274\pi$
$821$	−46.4962	−1.62273	−0.811364	+	0.584542i	$0.801274\pi$
$822$	−20.2592	−0.706621
$823$	−12.4481	−0.433913	−0.216957	−	0.976181i	$-0.569613\pi$
$823$	−12.4481	−0.433913	−0.216957	+	0.976181i	$0.569613\pi$
$824$	−30.3591	−1.05761
$825$	1.70001	0.0591866
$826$	−7.52154	−0.261708
$827$	1.54463	0.0537121	0.0268560	−	0.999639i	$-0.491450\pi$
$827$	1.54463	0.0537121	0.0268560	+	0.999639i	$0.491450\pi$
$828$	−7.65322	−0.265968
$829$	−30.5764	−1.06196	−0.530981	−	0.847384i	$-0.678177\pi$
$829$	−30.5764	−1.06196	−0.530981	+	0.847384i	$0.678177\pi$
$830$	−0.250351	−0.00868980
$831$	29.8845	1.03668
$832$	−22.4809	−0.779384
$833$	0	0
$834$	−13.9238	−0.482143
$835$	−1.00137	−0.0346540
$836$	−2.52308	−0.0872626
$837$	−9.71295	−0.335729
$838$	−17.8611	−0.617001
$839$	21.9508	0.757826	0.378913	−	0.925432i	$-0.376298\pi$
$839$	21.9508	0.757826	0.378913	+	0.925432i	$0.376298\pi$
$840$	−1.81349	−0.0625714
$841$	−28.5592	−0.984799
$842$	19.3833	0.667991
$843$	−15.9618	−0.549754
$844$	−12.8390	−0.441938
$845$	2.56930	0.0883867
$846$	−7.24532	−0.249099
$847$	10.8635	0.373275
$848$	−1.69066	−0.0580574
$849$	6.74526	0.231497
$850$	0	0
$851$	6.22875	0.213519
$852$	−11.0088	−0.377156
$853$	26.9316	0.922121	0.461061	−	0.887369i	$-0.347469\pi$
$853$	26.9316	0.922121	0.461061	+	0.887369i	$0.347469\pi$
$854$	−11.4397	−0.391458
$855$	−3.63141	−0.124192
$856$	58.0701	1.98479
$857$	−2.61387	−0.0892881	−0.0446441	−	0.999003i	$-0.514215\pi$
$857$	−2.61387	−0.0892881	−0.0446441	+	0.999003i	$0.514215\pi$
$858$	−1.37639	−0.0469893
$859$	6.38797	0.217955	0.108977	−	0.994044i	$-0.465242\pi$
$859$	6.38797	0.217955	0.108977	+	0.994044i	$0.465242\pi$
$860$	−3.15920	−0.107728
$861$	−6.52081	−0.222228
$862$	8.82656	0.300634
$863$	47.3237	1.61092	0.805459	−	0.592652i	$-0.201919\pi$
$863$	47.3237	1.61092	0.805459	+	0.592652i	$0.201919\pi$
$864$	−5.55121	−0.188856
$865$	−0.693940	−0.0235947
$866$	25.9006	0.880137
$867$	0	0
$868$	−11.5279	−0.391284
$869$	−1.66672	−0.0565396
$870$	−0.377822	−0.0128094
$871$	−13.9810	−0.473727
$872$	−7.12400	−0.241249
$873$	4.28369	0.144981
$874$	33.4603	1.13181
$875$	6.05929	0.204841
$876$	−15.9750	−0.539746
$877$	7.62042	0.257323	0.128662	−	0.991689i	$-0.458932\pi$
$877$	7.62042	0.257323	0.128662	+	0.991689i	$0.458932\pi$
$878$	−16.3517	−0.551843
$879$	1.07630	0.0363026
$880$	0.0507348	0.00171027
$881$	−21.2295	−0.715241	−0.357621	−	0.933867i	$-0.616412\pi$
$881$	−21.2295	−0.715241	−0.357621	+	0.933867i	$0.616412\pi$
$882$	−0.901740	−0.0303632
$883$	43.4020	1.46059	0.730297	−	0.683130i	$-0.239382\pi$
$883$	43.4020	1.46059	0.730297	+	0.683130i	$0.239382\pi$
$884$	0	0
$885$	−5.26376	−0.176939
$886$	−27.5014	−0.923927
$887$	2.77055	0.0930261	0.0465130	−	0.998918i	$-0.485189\pi$
$887$	2.77055	0.0930261	0.0465130	+	0.998918i	$0.485189\pi$
$888$	2.77589	0.0931528
$889$	0.422967	0.0141859
$890$	1.29967	0.0435652
$891$	−0.369425	−0.0123762
$892$	28.7885	0.963909
$893$	−46.2360	−1.54723
$894$	2.99376	0.100126
$895$	3.98090	0.133067
$896$	−6.19605	−0.206995
$897$	−26.6427	−0.889574
$898$	−12.9388	−0.431775
$899$	−6.44891	−0.215083
$900$	5.46167	0.182056
$901$	0	0
$902$	2.17225	0.0723279
$903$	−4.21798	−0.140366
$904$	21.5948	0.718232
$905$	−4.06373	−0.135083
$906$	13.4821	0.447913
$907$	−49.9548	−1.65872	−0.829361	−	0.558714i	$-0.811295\pi$
$907$	−49.9548	−1.65872	−0.829361	+	0.558714i	$0.811295\pi$
$908$	5.42401	0.180002
$909$	−1.87512	−0.0621937
$910$	−2.35119	−0.0779410
$911$	53.0869	1.75885	0.879423	−	0.476041i	$-0.157929\pi$
$911$	53.0869	1.75885	0.879423	+	0.476041i	$0.157929\pi$
$912$	1.25232	0.0414683
$913$	−0.162526	−0.00537882
$914$	10.7515	0.355629
$915$	−8.00576	−0.264662
$916$	−11.1638	−0.368861
$917$	16.7422	0.552877
$918$	0	0
$919$	−2.55712	−0.0843518	−0.0421759	−	0.999110i	$-0.513429\pi$
$919$	−2.55712	−0.0843518	−0.0421759	+	0.999110i	$0.513429\pi$
$920$	11.6939	0.385536
$921$	4.09877	0.135059
$922$	−15.6516	−0.515457
$923$	−38.3244	−1.26146
$924$	−0.438457	−0.0144242
$925$	−4.44510	−0.146154
$926$	−13.1370	−0.431708
$927$	−10.5644	−0.346980
$928$	−3.68572	−0.120990
$929$	−10.3456	−0.339428	−0.169714	−	0.985493i	$-0.554284\pi$
$929$	−10.3456	−0.339428	−0.169714	+	0.985493i	$0.554284\pi$
$930$	5.52718	0.181243
$931$	−5.75446	−0.188595
$932$	−10.9330	−0.358122
$933$	−26.2434	−0.859172
$934$	11.8940	0.389184
$935$	0	0
$936$	−11.8735	−0.388098
$937$	−14.3793	−0.469752	−0.234876	−	0.972025i	$-0.575468\pi$
$937$	−14.3793	−0.469752	−0.234876	+	0.972025i	$0.575468\pi$
$938$	3.05129	0.0996282
$939$	−9.58780	−0.312886
$940$	−6.01793	−0.196283
$941$	−41.0815	−1.33922	−0.669609	−	0.742713i	$-0.733539\pi$
$941$	−41.0815	−1.33922	−0.669609	+	0.742713i	$0.733539\pi$
$942$	−17.5509	−0.571839
$943$	42.0479	1.36927
$944$	1.81524	0.0590811
$945$	−0.631060	−0.0205284
$946$	1.40512	0.0456843
$947$	7.20008	0.233971	0.116986	−	0.993134i	$-0.462677\pi$
$947$	7.20008	0.233971	0.116986	+	0.993134i	$0.462677\pi$
$948$	−5.35473	−0.173914
$949$	−55.6128	−1.80527
$950$	−23.8787	−0.774726
$951$	16.0367	0.520025
$952$	0	0
$953$	54.6671	1.77084	0.885421	−	0.464790i	$-0.153870\pi$
$953$	54.6671	1.77084	0.885421	+	0.464790i	$0.153870\pi$
$954$	−7.00531	−0.226805
$955$	13.2201	0.427792
$956$	−33.5975	−1.08662
$957$	−0.245280	−0.00792876
$958$	−16.1326	−0.521222
$959$	−22.4668	−0.725491
$960$	3.43360	0.110819
$961$	63.3414	2.04327
$962$	3.59893	0.116034
$963$	20.2072	0.651169
$964$	21.3900	0.688927
$965$	−4.96669	−0.159883
$966$	5.81467	0.187084
$967$	−8.09878	−0.260439	−0.130220	−	0.991485i	$-0.541568\pi$
$967$	−8.09878	−0.260439	−0.130220	+	0.991485i	$0.541568\pi$
$968$	−31.2188	−1.00341
$969$	0	0
$970$	−2.43764	−0.0782680
$971$	54.9130	1.76224	0.881121	−	0.472891i	$-0.156790\pi$
$971$	54.9130	1.76224	0.881121	+	0.472891i	$0.156790\pi$
$972$	−1.18686	−0.0380687
$973$	−15.4411	−0.495018
$974$	8.61031	0.275892
$975$	19.0134	0.608915
$976$	2.76084	0.0883724
$977$	−0.689908	−0.0220721	−0.0110361	−	0.999939i	$-0.503513\pi$
$977$	−0.689908	−0.0220721	−0.0110361	+	0.999939i	$0.503513\pi$
$978$	−19.1157	−0.611252
$979$	0.843739	0.0269660
$980$	−0.748982	−0.0239254
$981$	−2.47901	−0.0791488
$982$	32.6247	1.04110
$983$	−42.3641	−1.35120	−0.675602	−	0.737266i	$-0.736116\pi$
$983$	−42.3641	−1.35120	−0.675602	+	0.737266i	$0.736116\pi$
$984$	18.7390	0.597378
$985$	−7.68238	−0.244781
$986$	0	0
$987$	−8.03481	−0.255751
$988$	−28.2189	−0.897763
$989$	27.1987	0.864869
$990$	0.210222	0.00668130
$991$	10.9091	0.346540	0.173270	−	0.984874i	$-0.444567\pi$
$991$	10.9091	0.346540	0.173270	+	0.984874i	$0.444567\pi$
$992$	53.9186	1.71192
$993$	18.2970	0.580639
$994$	8.36415	0.265295
$995$	−13.9697	−0.442870
$996$	−0.522153	−0.0165450
$997$	−48.6167	−1.53971	−0.769853	−	0.638222i	$-0.779670\pi$
$997$	−48.6167	−1.53971	−0.769853	+	0.638222i	$0.779670\pi$
$998$	3.58352	0.113434
$999$	0.965956	0.0305615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6069.2.a.be.1.3		10
17.2	even	8		357.2.k.b.106.3	yes	20
17.9	even	8		357.2.k.b.64.8	✓	20
17.16	even	2		6069.2.a.bd.1.3		10
51.2	odd	8		1071.2.n.b.820.8		20
51.26	odd	8		1071.2.n.b.64.3		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.k.b.64.8	✓	20	17.9	even	8
357.2.k.b.106.3	yes	20	17.2	even	8
1071.2.n.b.64.3		20	51.26	odd	8
1071.2.n.b.820.8		20	51.2	odd	8
6069.2.a.bd.1.3		10	17.16	even	2
6069.2.a.be.1.3		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6069 = 3 \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6069.a (trivial)

\(f(q)\)	\(=\)	\(q-0.901740 q^{2} +1.00000 q^{3} -1.18686 q^{4} +0.631060 q^{5} -0.901740 q^{6} -1.00000 q^{7} +2.87372 q^{8} +1.00000 q^{9} -0.569052 q^{10} -0.369425 q^{11} -1.18686 q^{12} -4.13176 q^{13} +0.901740 q^{14} +0.631060 q^{15} -0.217625 q^{16} -0.901740 q^{18} -5.75446 q^{19} -0.748982 q^{20} -1.00000 q^{21} +0.333125 q^{22} +6.44827 q^{23} +2.87372 q^{24} -4.60176 q^{25} +3.72577 q^{26} +1.00000 q^{27} +1.18686 q^{28} +0.663950 q^{29} -0.569052 q^{30} -9.71295 q^{31} -5.55121 q^{32} -0.369425 q^{33} -0.631060 q^{35} -1.18686 q^{36} +0.965956 q^{37} +5.18903 q^{38} -4.13176 q^{39} +1.81349 q^{40} +6.52081 q^{41} +0.901740 q^{42} +4.21798 q^{43} +0.438457 q^{44} +0.631060 q^{45} -5.81467 q^{46} +8.03481 q^{47} -0.217625 q^{48} +1.00000 q^{49} +4.14960 q^{50} +4.90383 q^{52} +7.76866 q^{53} -0.901740 q^{54} -0.233129 q^{55} -2.87372 q^{56} -5.75446 q^{57} -0.598711 q^{58} -8.34114 q^{59} -0.748982 q^{60} -12.6862 q^{61} +8.75856 q^{62} -1.00000 q^{63} +5.44100 q^{64} -2.60739 q^{65} +0.333125 q^{66} +3.38378 q^{67} +6.44827 q^{69} +0.569052 q^{70} +9.27556 q^{71} +2.87372 q^{72} +13.4598 q^{73} -0.871041 q^{74} -4.60176 q^{75} +6.82976 q^{76} +0.369425 q^{77} +3.72577 q^{78} +4.51167 q^{79} -0.137335 q^{80} +1.00000 q^{81} -5.88008 q^{82} +0.439943 q^{83} +1.18686 q^{84} -3.80353 q^{86} +0.663950 q^{87} -1.06163 q^{88} -2.28393 q^{89} -0.569052 q^{90} +4.13176 q^{91} -7.65322 q^{92} -9.71295 q^{93} -7.24532 q^{94} -3.63141 q^{95} -5.55121 q^{96} +4.28369 q^{97} -0.901740 q^{98} -0.369425 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 10 q^{3} + 12 q^{4} - 6 q^{5} + 4 q^{6} - 10 q^{7} + 12 q^{8} + 10 q^{9} - 2 q^{11} + 12 q^{12} + 6 q^{13} - 4 q^{14} - 6 q^{15} + 20 q^{16} + 4 q^{18} + 6 q^{19} - 16 q^{20} - 10 q^{21}+ \cdots - 2 q^{99}+O(q^{100}) \) 10 * q + 4 * q^2 + 10 * q^3 + 12 * q^4 - 6 * q^5 + 4 * q^6 - 10 * q^7 + 12 * q^8 + 10 * q^9 - 2 * q^11 + 12 * q^12 + 6 * q^13 - 4 * q^14 - 6 * q^15 + 20 * q^16 + 4 * q^18 + 6 * q^19 - 16 * q^20 - 10 * q^21 + 24 * q^22 + 18 * q^23 + 12 * q^24 + 4 * q^25 + 12 * q^26 + 10 * q^27 - 12 * q^28 - 12 * q^29 - 8 * q^31 + 28 * q^32 - 2 * q^33 + 6 * q^35 + 12 * q^36 + 24 * q^37 + 32 * q^38 + 6 * q^39 - 36 * q^40 - 2 * q^41 - 4 * q^42 + 6 * q^43 + 28 * q^44 - 6 * q^45 - 28 * q^46 + 16 * q^47 + 20 * q^48 + 10 * q^49 + 52 * q^50 + 24 * q^52 - 12 * q^53 + 4 * q^54 + 18 * q^55 - 12 * q^56 + 6 * q^57 + 20 * q^58 + 20 * q^59 - 16 * q^60 - 4 * q^61 - 24 * q^62 - 10 * q^63 + 56 * q^64 - 38 * q^65 + 24 * q^66 + 20 * q^67 + 18 * q^69 - 20 * q^71 + 12 * q^72 + 20 * q^73 + 36 * q^74 + 4 * q^75 - 8 * q^76 + 2 * q^77 + 12 * q^78 + 20 * q^79 - 92 * q^80 + 10 * q^81 + 8 * q^82 + 36 * q^83 - 12 * q^84 - 12 * q^87 + 92 * q^88 + 32 * q^89 - 6 * q^91 + 40 * q^92 - 8 * q^93 + 60 * q^94 - 14 * q^95 + 28 * q^96 + 36 * q^97 + 4 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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