LMFDB - Embedded newform 667.2.a.c.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [667,2,Mod(1,667)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(667, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("667.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 667 = 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	667.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:	$5.32602181482$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 58 x^{10} + 24 x^{9} - 298 x^{8} + 97 x^{7} + 641 x^{6} - 402 x^{5} + \cdots - 40 $ x^13 - 4x^12 - 11x^11 + 58x^10 + 24x^9 - 298x^8 + 97x^7 + 641x^6 - 402x^5 - 547x^4 + 352x^3 + 219x^2 - 88x - 40
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.30511$ of defining polynomial
Character	$\chi$	$=$	667.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.30511 q^{2} +3.02097 q^{3} -0.296685 q^{4} +0.572849 q^{5} +3.94270 q^{6} +1.21746 q^{7} -2.99743 q^{8} +6.12628 q^{9} +O(q^{10})$	\(q+1.30511 q^{2} +3.02097 q^{3} -0.296685 q^{4} +0.572849 q^{5} +3.94270 q^{6} +1.21746 q^{7} -2.99743 q^{8} +6.12628 q^{9} +0.747631 q^{10} -3.82178 q^{11} -0.896278 q^{12} +1.55748 q^{13} +1.58892 q^{14} +1.73056 q^{15} -3.31861 q^{16} +3.24470 q^{17} +7.99547 q^{18} +7.65905 q^{19} -0.169956 q^{20} +3.67790 q^{21} -4.98785 q^{22} -1.00000 q^{23} -9.05515 q^{24} -4.67184 q^{25} +2.03268 q^{26} +9.44440 q^{27} -0.361201 q^{28} +1.00000 q^{29} +2.25857 q^{30} -9.54445 q^{31} +1.66371 q^{32} -11.5455 q^{33} +4.23469 q^{34} +0.697418 q^{35} -1.81758 q^{36} -5.96920 q^{37} +9.99591 q^{38} +4.70510 q^{39} -1.71707 q^{40} -2.82463 q^{41} +4.80007 q^{42} -11.8023 q^{43} +1.13387 q^{44} +3.50943 q^{45} -1.30511 q^{46} +0.392305 q^{47} -10.0254 q^{48} -5.51780 q^{49} -6.09728 q^{50} +9.80214 q^{51} -0.462081 q^{52} +11.2317 q^{53} +12.3260 q^{54} -2.18930 q^{55} -3.64924 q^{56} +23.1378 q^{57} +1.30511 q^{58} -13.0256 q^{59} -0.513432 q^{60} +9.25323 q^{61} -12.4566 q^{62} +7.45847 q^{63} +8.80854 q^{64} +0.892200 q^{65} -15.0682 q^{66} -10.1462 q^{67} -0.962653 q^{68} -3.02097 q^{69} +0.910208 q^{70} +10.3791 q^{71} -18.3631 q^{72} -2.55044 q^{73} -7.79047 q^{74} -14.1135 q^{75} -2.27233 q^{76} -4.65285 q^{77} +6.14068 q^{78} +0.852395 q^{79} -1.90106 q^{80} +10.1524 q^{81} -3.68645 q^{82} -0.606412 q^{83} -1.09118 q^{84} +1.85872 q^{85} -15.4033 q^{86} +3.02097 q^{87} +11.4555 q^{88} +3.82257 q^{89} +4.58019 q^{90} +1.89616 q^{91} +0.296685 q^{92} -28.8335 q^{93} +0.512002 q^{94} +4.38748 q^{95} +5.02602 q^{96} +5.13466 q^{97} -7.20134 q^{98} -23.4133 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 4 q^{2} + 3 q^{3} + 12 q^{4} + 16 q^{5} + q^{7} + 6 q^{8} + 14 q^{9}+O(q^{10}) $ 13 * q + 4 * q^2 + 3 * q^3 + 12 * q^4 + 16 * q^5 + q^7 + 6 * q^8 + 14 * q^9	\( 13 q + 4 q^{2} + 3 q^{3} + 12 q^{4} + 16 q^{5} + q^{7} + 6 q^{8} + 14 q^{9} + 10 q^{10} + 10 q^{11} + 3 q^{12} + 7 q^{13} - 12 q^{14} + 8 q^{15} + 2 q^{16} + 26 q^{17} + 7 q^{18} + 25 q^{20} + 11 q^{21} - 15 q^{22} - 13 q^{23} + 10 q^{24} + 19 q^{25} - 15 q^{26} + 12 q^{27} + 5 q^{28} + 13 q^{29} + 7 q^{30} - 6 q^{31} + 16 q^{32} + 3 q^{33} + 11 q^{34} + q^{35} - 22 q^{36} + 15 q^{37} + 8 q^{38} - 4 q^{39} + 14 q^{40} + 9 q^{41} - 34 q^{42} + q^{43} + 29 q^{44} + 16 q^{45} - 4 q^{46} + 15 q^{47} - 15 q^{48} + 4 q^{49} + 31 q^{50} - 14 q^{51} - 8 q^{52} + 43 q^{53} - 35 q^{54} - 3 q^{55} - 5 q^{56} + 6 q^{57} + 4 q^{58} - 9 q^{59} + 3 q^{60} + 20 q^{61} + 11 q^{62} + 21 q^{63} - 16 q^{64} - 25 q^{65} - q^{66} + q^{67} + 21 q^{68} - 3 q^{69} - 2 q^{70} + 17 q^{71} - 59 q^{72} + 26 q^{73} + 11 q^{74} - 43 q^{75} + 8 q^{76} + 17 q^{77} - 10 q^{78} + 5 q^{79} + 10 q^{80} - 3 q^{81} - 25 q^{82} + 4 q^{83} - 78 q^{84} + 20 q^{85} - 13 q^{86} + 3 q^{87} - 32 q^{88} + 48 q^{89} + 17 q^{90} - 9 q^{91} - 12 q^{92} - 8 q^{93} - 65 q^{94} + 8 q^{95} + 8 q^{96} + 30 q^{97} + 2 q^{98} + 4 q^{99}+O(q^{100}) \) 13 * q + 4 * q^2 + 3 * q^3 + 12 * q^4 + 16 * q^5 + q^7 + 6 * q^8 + 14 * q^9 + 10 * q^10 + 10 * q^11 + 3 * q^12 + 7 * q^13 - 12 * q^14 + 8 * q^15 + 2 * q^16 + 26 * q^17 + 7 * q^18 + 25 * q^20 + 11 * q^21 - 15 * q^22 - 13 * q^23 + 10 * q^24 + 19 * q^25 - 15 * q^26 + 12 * q^27 + 5 * q^28 + 13 * q^29 + 7 * q^30 - 6 * q^31 + 16 * q^32 + 3 * q^33 + 11 * q^34 + q^35 - 22 * q^36 + 15 * q^37 + 8 * q^38 - 4 * q^39 + 14 * q^40 + 9 * q^41 - 34 * q^42 + q^43 + 29 * q^44 + 16 * q^45 - 4 * q^46 + 15 * q^47 - 15 * q^48 + 4 * q^49 + 31 * q^50 - 14 * q^51 - 8 * q^52 + 43 * q^53 - 35 * q^54 - 3 * q^55 - 5 * q^56 + 6 * q^57 + 4 * q^58 - 9 * q^59 + 3 * q^60 + 20 * q^61 + 11 * q^62 + 21 * q^63 - 16 * q^64 - 25 * q^65 - q^66 + q^67 + 21 * q^68 - 3 * q^69 - 2 * q^70 + 17 * q^71 - 59 * q^72 + 26 * q^73 + 11 * q^74 - 43 * q^75 + 8 * q^76 + 17 * q^77 - 10 * q^78 + 5 * q^79 + 10 * q^80 - 3 * q^81 - 25 * q^82 + 4 * q^83 - 78 * q^84 + 20 * q^85 - 13 * q^86 + 3 * q^87 - 32 * q^88 + 48 * q^89 + 17 * q^90 - 9 * q^91 - 12 * q^92 - 8 * q^93 - 65 * q^94 + 8 * q^95 + 8 * q^96 + 30 * q^97 + 2 * q^98 + 4 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	1.30511	0.922853	0.461426	−	0.887178i	$-0.347338\pi$
$2$	1.30511	0.922853	0.461426	+	0.887178i	$0.347338\pi$
$3$	3.02097	1.74416	0.872080	−	0.489364i	$-0.162771\pi$
$3$	3.02097	1.74416	0.872080	+	0.489364i	$0.162771\pi$
$4$	−0.296685	−0.148343
$5$	0.572849	0.256186	0.128093	−	0.991762i	$-0.459114\pi$
$5$	0.572849	0.256186	0.128093	+	0.991762i	$0.459114\pi$
$6$	3.94270	1.60960
$7$	1.21746	0.460155	0.230078	−	0.973172i	$-0.426102\pi$
$7$	1.21746	0.460155	0.230078	+	0.973172i	$0.426102\pi$
$8$	−2.99743	−1.05975
$9$	6.12628	2.04209
$10$	0.747631	0.236422
$11$	−3.82178	−1.15231	−0.576155	−	0.817340i	$-0.695448\pi$
$11$	−3.82178	−1.15231	−0.576155	+	0.817340i	$0.695448\pi$
$12$	−0.896278	−0.258733
$13$	1.55748	0.431967	0.215984	−	0.976397i	$-0.430704\pi$
$13$	1.55748	0.431967	0.215984	+	0.976397i	$0.430704\pi$
$14$	1.58892	0.424656
$15$	1.73056	0.446829
$16$	−3.31861	−0.829652
$17$	3.24470	0.786954	0.393477	−	0.919334i	$-0.371272\pi$
$17$	3.24470	0.786954	0.393477	+	0.919334i	$0.371272\pi$
$18$	7.99547	1.88455
$19$	7.65905	1.75711	0.878553	−	0.477644i	$-0.158509\pi$
$19$	7.65905	1.75711	0.878553	+	0.477644i	$0.158509\pi$
$20$	−0.169956	−0.0380033
$21$	3.67790	0.802584
$22$	−4.98785	−1.06341
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−9.05515	−1.84838
$25$	−4.67184	−0.934369
$26$	2.03268	0.398642
$27$	9.44440	1.81758
$28$	−0.361201	−0.0682606
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	2.25857	0.412357
$31$	−9.54445	−1.71423	−0.857117	−	0.515123i	$-0.827746\pi$
$31$	−9.54445	−1.71423	−0.857117	+	0.515123i	$0.827746\pi$
$32$	1.66371	0.294105
$33$	−11.5455	−2.00981
$34$	4.23469	0.726243
$35$	0.697418	0.117885
$36$	−1.81758	−0.302929
$37$	−5.96920	−0.981330	−0.490665	−	0.871348i	$-0.663246\pi$
$37$	−5.96920	−0.981330	−0.490665	+	0.871348i	$0.663246\pi$
$38$	9.99591	1.62155
$39$	4.70510	0.753420
$40$	−1.71707	−0.271493
$41$	−2.82463	−0.441133	−0.220566	−	0.975372i	$-0.570791\pi$
$41$	−2.82463	−0.441133	−0.220566	+	0.975372i	$0.570791\pi$
$42$	4.80007	0.740667
$43$	−11.8023	−1.79983	−0.899914	−	0.436067i	$-0.856371\pi$
$43$	−11.8023	−1.79983	−0.899914	+	0.436067i	$0.856371\pi$
$44$	1.13387	0.170937
$45$	3.50943	0.523155
$46$	−1.30511	−0.192428
$47$	0.392305	0.0572236	0.0286118	−	0.999591i	$-0.490891\pi$
$47$	0.392305	0.0572236	0.0286118	+	0.999591i	$0.490891\pi$
$48$	−10.0254	−1.44705
$49$	−5.51780	−0.788257
$50$	−6.09728	−0.862285
$51$	9.80214	1.37257
$52$	−0.462081	−0.0640792
$53$	11.2317	1.54279	0.771394	−	0.636358i	$-0.219560\pi$
$53$	11.2317	1.54279	0.771394	+	0.636358i	$0.219560\pi$
$54$	12.3260	1.67735
$55$	−2.18930	−0.295205
$56$	−3.64924	−0.487650
$57$	23.1378	3.06467
$58$	1.30511	0.171369
$59$	−13.0256	−1.69579	−0.847896	−	0.530162i	$-0.822131\pi$
$59$	−13.0256	−1.69579	−0.847896	+	0.530162i	$0.822131\pi$
$60$	−0.513432	−0.0662837
$61$	9.25323	1.18475	0.592377	−	0.805661i	$-0.298190\pi$
$61$	9.25323	1.18475	0.592377	+	0.805661i	$0.298190\pi$
$62$	−12.4566	−1.58198
$63$	7.45847	0.939679
$64$	8.80854	1.10107
$65$	0.892200	0.110664
$66$	−15.0682	−1.85476
$67$	−10.1462	−1.23955	−0.619776	−	0.784779i	$-0.712777\pi$
$67$	−10.1462	−1.23955	−0.619776	+	0.784779i	$0.712777\pi$
$68$	−0.962653	−0.116739
$69$	−3.02097	−0.363682
$70$	0.910208	0.108791
$71$	10.3791	1.23178	0.615889	−	0.787833i	$-0.288797\pi$
$71$	10.3791	1.23178	0.615889	+	0.787833i	$0.288797\pi$
$72$	−18.3631	−2.16411
$73$	−2.55044	−0.298506	−0.149253	−	0.988799i	$-0.547687\pi$
$73$	−2.55044	−0.298506	−0.149253	+	0.988799i	$0.547687\pi$
$74$	−7.79047	−0.905624
$75$	−14.1135	−1.62969
$76$	−2.27233	−0.260654
$77$	−4.65285	−0.530241
$78$	6.14068	0.695296
$79$	0.852395	0.0959019	0.0479510	−	0.998850i	$-0.484731\pi$
$79$	0.852395	0.0959019	0.0479510	+	0.998850i	$0.484731\pi$
$80$	−1.90106	−0.212545
$81$	10.1524	1.12805
$82$	−3.68645	−0.407101
$83$	−0.606412	−0.0665624	−0.0332812	−	0.999446i	$-0.510596\pi$
$83$	−0.606412	−0.0665624	−0.0332812	+	0.999446i	$0.510596\pi$
$84$	−1.09118	−0.119057
$85$	1.85872	0.201606
$86$	−15.4033	−1.66098
$87$	3.02097	0.323882
$88$	11.4555	1.22116
$89$	3.82257	0.405192	0.202596	−	0.979262i	$-0.435062\pi$
$89$	3.82257	0.405192	0.202596	+	0.979262i	$0.435062\pi$
$90$	4.58019	0.482795
$91$	1.89616	0.198772
$92$	0.296685	0.0309316
$93$	−28.8335	−2.98990
$94$	0.512002	0.0528090
$95$	4.38748	0.450146
$96$	5.02602	0.512966
$97$	5.13466	0.521346	0.260673	−	0.965427i	$-0.416056\pi$
$97$	5.13466	0.521346	0.260673	+	0.965427i	$0.416056\pi$
$98$	−7.20134	−0.727445
$99$	−23.4133	−2.35312
$100$	1.38607	0.138607
$101$	−11.9061	−1.18470	−0.592349	−	0.805682i	$-0.701799\pi$
$101$	−11.9061	−1.18470	−0.592349	+	0.805682i	$0.701799\pi$
$102$	12.7929	1.26668
$103$	3.70761	0.365322	0.182661	−	0.983176i	$-0.441529\pi$
$103$	3.70761	0.365322	0.182661	+	0.983176i	$0.441529\pi$
$104$	−4.66844	−0.457778
$105$	2.10688	0.205611
$106$	14.6586	1.42377
$107$	9.65894	0.933765	0.466882	−	0.884319i	$-0.345377\pi$
$107$	9.65894	0.933765	0.466882	+	0.884319i	$0.345377\pi$
$108$	−2.80201	−0.269624
$109$	10.5404	1.00959	0.504794	−	0.863240i	$-0.331568\pi$
$109$	10.5404	1.00959	0.504794	+	0.863240i	$0.331568\pi$
$110$	−2.85728	−0.272431
$111$	−18.0328	−1.71160
$112$	−4.04026	−0.381769
$113$	17.8383	1.67809	0.839045	−	0.544062i	$-0.183114\pi$
$113$	17.8383	1.67809	0.839045	+	0.544062i	$0.183114\pi$
$114$	30.1974	2.82824
$115$	−0.572849	−0.0534184
$116$	−0.296685	−0.0275465
$117$	9.54155	0.882117
$118$	−16.9999	−1.56497
$119$	3.95028	0.362121
$120$	−5.18723	−0.473527
$121$	3.60600	0.327818
$122$	12.0765	1.09335
$123$	−8.53312	−0.769406
$124$	2.83170	0.254294
$125$	−5.54050	−0.495558
$126$	9.73414	0.867186
$127$	8.21224	0.728718	0.364359	−	0.931258i	$-0.381288\pi$
$127$	8.21224	0.728718	0.364359	+	0.931258i	$0.381288\pi$
$128$	8.16870	0.722018
$129$	−35.6543	−3.13919
$130$	1.16442	0.102126
$131$	20.3674	1.77951	0.889754	−	0.456441i	$-0.150876\pi$
$131$	20.3674	1.77951	0.889754	+	0.456441i	$0.150876\pi$
$132$	3.42538	0.298141
$133$	9.32456	0.808542
$134$	−13.2419	−1.14392
$135$	5.41021	0.465637
$136$	−9.72575	−0.833976
$137$	3.66808	0.313385	0.156693	−	0.987647i	$-0.449917\pi$
$137$	3.66808	0.313385	0.156693	+	0.987647i	$0.449917\pi$
$138$	−3.94270	−0.335625
$139$	−9.77997	−0.829526	−0.414763	−	0.909929i	$-0.636136\pi$
$139$	−9.77997	−0.829526	−0.414763	+	0.909929i	$0.636136\pi$
$140$	−0.206914	−0.0174874
$141$	1.18514	0.0998071
$142$	13.5459	1.13675
$143$	−5.95235	−0.497760
$144$	−20.3307	−1.69423
$145$	0.572849	0.0475725
$146$	−3.32860	−0.275477
$147$	−16.6691	−1.37485
$148$	1.77097	0.145573
$149$	−14.7501	−1.20837	−0.604187	−	0.796843i	$-0.706502\pi$
$149$	−14.7501	−1.20837	−0.604187	+	0.796843i	$0.706502\pi$
$150$	−18.4197	−1.50396
$151$	2.30674	0.187720	0.0938601	−	0.995585i	$-0.470079\pi$
$151$	2.30674	0.187720	0.0938601	+	0.995585i	$0.470079\pi$
$152$	−22.9575	−1.86210
$153$	19.8779	1.60703
$154$	−6.07249	−0.489335
$155$	−5.46752	−0.439162
$156$	−1.39594	−0.111764
$157$	−8.57476	−0.684341	−0.342170	−	0.939638i	$-0.611162\pi$
$157$	−8.57476	−0.684341	−0.342170	+	0.939638i	$0.611162\pi$
$158$	1.11247	0.0885034
$159$	33.9305	2.69087
$160$	0.953053	0.0753454
$161$	−1.21746	−0.0959490
$162$	13.2501	1.04102
$163$	8.09710	0.634214	0.317107	−	0.948390i	$-0.397289\pi$
$163$	8.09710	0.634214	0.317107	+	0.948390i	$0.397289\pi$
$164$	0.838025	0.0654388
$165$	−6.61382	−0.514885
$166$	−0.791435	−0.0614273
$167$	16.3123	1.26228	0.631141	−	0.775668i	$-0.282587\pi$
$167$	16.3123	1.26228	0.631141	+	0.775668i	$0.282587\pi$
$168$	−11.0243	−0.850540
$169$	−10.5743	−0.813404
$170$	2.42584	0.186053
$171$	46.9215	3.58817
$172$	3.50156	0.266991
$173$	8.46918	0.643900	0.321950	−	0.946757i	$-0.395662\pi$
$173$	8.46918	0.643900	0.321950	+	0.946757i	$0.395662\pi$
$174$	3.94270	0.298896
$175$	−5.68777	−0.429955
$176$	12.6830	0.956016
$177$	−39.3501	−2.95773
$178$	4.98888	0.373932
$179$	11.0527	0.826120	0.413060	−	0.910704i	$-0.364460\pi$
$179$	11.0527	0.826120	0.413060	+	0.910704i	$0.364460\pi$
$180$	−1.04120	−0.0776062
$181$	22.3254	1.65943	0.829716	−	0.558185i	$-0.188502\pi$
$181$	22.3254	1.65943	0.829716	+	0.558185i	$0.188502\pi$
$182$	2.47470	0.183437
$183$	27.9538	2.06640
$184$	2.99743	0.220973
$185$	−3.41945	−0.251403
$186$	−37.6309	−2.75923
$187$	−12.4005	−0.906815
$188$	−0.116391	−0.00848870
$189$	11.4981	0.836367
$190$	5.72614	0.415418
$191$	−0.112731	−0.00815694	−0.00407847	−	0.999992i	$-0.501298\pi$
$191$	−0.112731	−0.00815694	−0.00407847	+	0.999992i	$0.501298\pi$
$192$	26.6104	1.92044
$193$	−14.9894	−1.07896	−0.539480	−	0.841998i	$-0.681379\pi$
$193$	−14.9894	−1.07896	−0.539480	+	0.841998i	$0.681379\pi$
$194$	6.70131	0.481126
$195$	2.69531	0.193015
$196$	1.63705	0.116932
$197$	−14.4378	−1.02865	−0.514326	−	0.857595i	$-0.671958\pi$
$197$	−14.4378	−1.02865	−0.514326	+	0.857595i	$0.671958\pi$
$198$	−30.5569	−2.17159
$199$	−18.3497	−1.30078	−0.650389	−	0.759601i	$-0.725394\pi$
$199$	−18.3497	−1.30078	−0.650389	+	0.759601i	$0.725394\pi$
$200$	14.0035	0.990199
$201$	−30.6513	−2.16198
$202$	−15.5387	−1.09330
$203$	1.21746	0.0854487
$204$	−2.90815	−0.203611
$205$	−1.61808	−0.113012
$206$	4.83884	0.337138
$207$	−6.12628	−0.425806
$208$	−5.16866	−0.358382
$209$	−29.2712	−2.02473
$210$	2.74971	0.189748
$211$	8.48555	0.584170	0.292085	−	0.956392i	$-0.405651\pi$
$211$	8.48555	0.584170	0.292085	+	0.956392i	$0.405651\pi$
$212$	−3.33227	−0.228861
$213$	31.3551	2.14842
$214$	12.6060	0.861727
$215$	−6.76091	−0.461090
$216$	−28.3089	−1.92618
$217$	−11.6199	−0.788813
$218$	13.7564	0.931701
$219$	−7.70480	−0.520642
$220$	0.649533	0.0437915
$221$	5.05355	0.339938
$222$	−23.5348	−1.57955
$223$	7.93065	0.531076	0.265538	−	0.964100i	$-0.414450\pi$
$223$	7.93065	0.531076	0.265538	+	0.964100i	$0.414450\pi$
$224$	2.02549	0.135334
$225$	−28.6210	−1.90807
$226$	23.2810	1.54863
$227$	−26.6251	−1.76717	−0.883584	−	0.468273i	$-0.844877\pi$
$227$	−26.6251	−1.76717	−0.883584	+	0.468273i	$0.844877\pi$
$228$	−6.86464	−0.454622
$229$	22.1498	1.46370	0.731850	−	0.681465i	$-0.238657\pi$
$229$	22.1498	1.46370	0.731850	+	0.681465i	$0.238657\pi$
$230$	−0.747631	−0.0492973
$231$	−14.0561	−0.924826
$232$	−2.99743	−0.196791
$233$	4.06717	0.266449	0.133225	−	0.991086i	$-0.457467\pi$
$233$	4.06717	0.266449	0.133225	+	0.991086i	$0.457467\pi$
$234$	12.4528	0.814064
$235$	0.224732	0.0146599
$236$	3.86451	0.251558
$237$	2.57506	0.167268
$238$	5.15555	0.334185
$239$	7.55588	0.488749	0.244375	−	0.969681i	$-0.421417\pi$
$239$	7.55588	0.488749	0.244375	+	0.969681i	$0.421417\pi$
$240$	−5.74305	−0.370712
$241$	6.65618	0.428762	0.214381	−	0.976750i	$-0.431227\pi$
$241$	6.65618	0.428762	0.214381	+	0.976750i	$0.431227\pi$
$242$	4.70623	0.302528
$243$	2.33705	0.149922
$244$	−2.74530	−0.175750
$245$	−3.16086	−0.201940
$246$	−11.1367	−0.710048
$247$	11.9288	0.759012
$248$	28.6088	1.81666
$249$	−1.83195	−0.116095
$250$	−7.23097	−0.457327
$251$	−3.90001	−0.246166	−0.123083	−	0.992396i	$-0.539278\pi$
$251$	−3.90001	−0.246166	−0.123083	+	0.992396i	$0.539278\pi$
$252$	−2.21282	−0.139395
$253$	3.82178	0.240273
$254$	10.7179	0.672500
$255$	5.61514	0.351634
$256$	−6.95601	−0.434751
$257$	−9.77946	−0.610026	−0.305013	−	0.952348i	$-0.598661\pi$
$257$	−9.77946	−0.610026	−0.305013	+	0.952348i	$0.598661\pi$
$258$	−46.5328	−2.89701
$259$	−7.26724	−0.451564
$260$	−0.264703	−0.0164162
$261$	6.12628	0.379207
$262$	26.5817	1.64222
$263$	14.2353	0.877784	0.438892	−	0.898540i	$-0.355371\pi$
$263$	14.2353	0.877784	0.438892	+	0.898540i	$0.355371\pi$
$264$	34.6068	2.12990
$265$	6.43404	0.395240
$266$	12.1696	0.746165
$267$	11.5479	0.706719
$268$	3.01022	0.183878
$269$	−19.0080	−1.15894	−0.579470	−	0.814993i	$-0.696741\pi$
$269$	−19.0080	−1.15894	−0.579470	+	0.814993i	$0.696741\pi$
$270$	7.06093	0.429714
$271$	−26.6085	−1.61635	−0.808176	−	0.588941i	$-0.799545\pi$
$271$	−26.6085	−1.61635	−0.808176	+	0.588941i	$0.799545\pi$
$272$	−10.7679	−0.652898
$273$	5.72826	0.346690
$274$	4.78725	0.289208
$275$	17.8548	1.07668
$276$	0.896278	0.0539496
$277$	−10.9717	−0.659224	−0.329612	−	0.944116i	$-0.606918\pi$
$277$	−10.9717	−0.659224	−0.329612	+	0.944116i	$0.606918\pi$
$278$	−12.7639	−0.765530
$279$	−58.4719	−3.50062
$280$	−2.09046	−0.124929
$281$	28.0886	1.67563	0.837813	−	0.545957i	$-0.183834\pi$
$281$	28.0886	1.67563	0.837813	+	0.545957i	$0.183834\pi$
$282$	1.54674	0.0921073
$283$	−8.51705	−0.506286	−0.253143	−	0.967429i	$-0.581464\pi$
$283$	−8.51705	−0.506286	−0.253143	+	0.967429i	$0.581464\pi$
$284$	−3.07934	−0.182725
$285$	13.2544	0.785126
$286$	−7.76847	−0.459359
$287$	−3.43886	−0.202989
$288$	10.1923	0.600589
$289$	−6.47195	−0.380703
$290$	0.747631	0.0439024
$291$	15.5117	0.909311
$292$	0.756677	0.0442812
$293$	25.2762	1.47665	0.738326	−	0.674444i	$-0.235617\pi$
$293$	25.2762	1.47665	0.738326	+	0.674444i	$0.235617\pi$
$294$	−21.7551	−1.26878
$295$	−7.46171	−0.434438
$296$	17.8923	1.03997
$297$	−36.0944	−2.09441
$298$	−19.2505	−1.11515
$299$	−1.55748	−0.0900714
$300$	4.18727	0.241752
$301$	−14.3687	−0.828200
$302$	3.01056	0.173238
$303$	−35.9679	−2.06630
$304$	−25.4174	−1.45779
$305$	5.30070	0.303517
$306$	25.9429	1.48306
$307$	−10.8941	−0.621759	−0.310880	−	0.950449i	$-0.600624\pi$
$307$	−10.8941	−0.621759	−0.310880	+	0.950449i	$0.600624\pi$
$308$	1.38043	0.0786574
$309$	11.2006	0.637179
$310$	−7.13572	−0.405282
$311$	15.0844	0.855358	0.427679	−	0.903931i	$-0.359331\pi$
$311$	15.0844	0.855358	0.427679	+	0.903931i	$0.359331\pi$
$312$	−14.1032	−0.798438
$313$	3.08345	0.174287	0.0871434	−	0.996196i	$-0.472226\pi$
$313$	3.08345	0.174287	0.0871434	+	0.996196i	$0.472226\pi$
$314$	−11.1910	−0.631546
$315$	4.27258	0.240732
$316$	−0.252893	−0.0142263
$317$	20.7123	1.16332	0.581659	−	0.813433i	$-0.302404\pi$
$317$	20.7123	1.16332	0.581659	+	0.813433i	$0.302404\pi$
$318$	44.2831	2.48327
$319$	−3.82178	−0.213979
$320$	5.04596	0.282078
$321$	29.1794	1.62863
$322$	−1.58892	−0.0885468
$323$	24.8513	1.38276
$324$	−3.01208	−0.167338
$325$	−7.27630	−0.403617
$326$	10.5676	0.585286
$327$	31.8423	1.76088
$328$	8.46662	0.467491
$329$	0.477615	0.0263317
$330$	−8.63177	−0.475163
$331$	31.9172	1.75433	0.877165	−	0.480189i	$-0.159432\pi$
$331$	31.9172	1.75433	0.877165	+	0.480189i	$0.159432\pi$
$332$	0.179913	0.00987404
$333$	−36.5690	−2.00397
$334$	21.2893	1.16490
$335$	−5.81222	−0.317556
$336$	−12.2055	−0.665865
$337$	7.63778	0.416056	0.208028	−	0.978123i	$-0.433295\pi$
$337$	7.63778	0.416056	0.208028	+	0.978123i	$0.433295\pi$
$338$	−13.8006	−0.750652
$339$	53.8891	2.92686
$340$	−0.551455	−0.0299068
$341$	36.4768	1.97533
$342$	61.2377	3.31136
$343$	−15.2399	−0.822876
$344$	35.3764	1.90737
$345$	−1.73056	−0.0931702
$346$	11.0532	0.594225
$347$	−6.79169	−0.364597	−0.182298	−	0.983243i	$-0.558354\pi$
$347$	−6.79169	−0.364597	−0.182298	+	0.983243i	$0.558354\pi$
$348$	−0.896278	−0.0480456
$349$	13.0456	0.698316	0.349158	−	0.937064i	$-0.386468\pi$
$349$	13.0456	0.698316	0.349158	+	0.937064i	$0.386468\pi$
$350$	−7.42317	−0.396785
$351$	14.7095	0.785133
$352$	−6.35832	−0.338900
$353$	−17.8334	−0.949176	−0.474588	−	0.880208i	$-0.657403\pi$
$353$	−17.8334	−0.949176	−0.474588	+	0.880208i	$0.657403\pi$
$354$	−51.3562	−2.72955
$355$	5.94568	0.315564
$356$	−1.13410	−0.0601072
$357$	11.9337	0.631597
$358$	14.4250	0.762387
$359$	−0.664618	−0.0350772	−0.0175386	−	0.999846i	$-0.505583\pi$
$359$	−0.664618	−0.0350772	−0.0175386	+	0.999846i	$0.505583\pi$
$360$	−10.5193	−0.554414
$361$	39.6610	2.08742
$362$	29.1371	1.53141
$363$	10.8936	0.571767
$364$	−0.562564	−0.0294864
$365$	−1.46101	−0.0764730
$366$	36.4827	1.90698
$367$	−7.77513	−0.405859	−0.202929	−	0.979193i	$-0.565046\pi$
$367$	−7.77513	−0.405859	−0.202929	+	0.979193i	$0.565046\pi$
$368$	3.31861	0.172994
$369$	−17.3044	−0.900834
$370$	−4.46276	−0.232008
$371$	13.6741	0.709922
$372$	8.55448	0.443529
$373$	−6.25735	−0.323993	−0.161997	−	0.986791i	$-0.551793\pi$
$373$	−6.25735	−0.323993	−0.161997	+	0.986791i	$0.551793\pi$
$374$	−16.1840	−0.836857
$375$	−16.7377	−0.864332
$376$	−1.17591	−0.0606428
$377$	1.55748	0.0802143
$378$	15.0064	0.771843
$379$	−16.9416	−0.870231	−0.435116	−	0.900375i	$-0.643293\pi$
$379$	−16.9416	−0.870231	−0.435116	+	0.900375i	$0.643293\pi$
$380$	−1.30170	−0.0667758
$381$	24.8090	1.27100
$382$	−0.147127	−0.00752765
$383$	−8.01241	−0.409415	−0.204707	−	0.978823i	$-0.565624\pi$
$383$	−8.01241	−0.409415	−0.204707	+	0.978823i	$0.565624\pi$
$384$	24.6774	1.25932
$385$	−2.66538	−0.135840
$386$	−19.5628	−0.995722
$387$	−72.3039	−3.67542
$388$	−1.52338	−0.0773379
$389$	11.0961	0.562595	0.281297	−	0.959621i	$-0.409235\pi$
$389$	11.0961	0.562595	0.281297	+	0.959621i	$0.409235\pi$
$390$	3.51768	0.178125
$391$	−3.24470	−0.164091
$392$	16.5392	0.835357
$393$	61.5294	3.10375
$394$	−18.8429	−0.949294
$395$	0.488293	0.0245687
$396$	6.94638	0.349069
$397$	18.7848	0.942780	0.471390	−	0.881925i	$-0.343752\pi$
$397$	18.7848	0.942780	0.471390	+	0.881925i	$0.343752\pi$
$398$	−23.9484	−1.20043
$399$	28.1692	1.41023
$400$	15.5040	0.775201
$401$	−8.32239	−0.415600	−0.207800	−	0.978171i	$-0.566630\pi$
$401$	−8.32239	−0.415600	−0.207800	+	0.978171i	$0.566630\pi$
$402$	−40.0034	−1.99519
$403$	−14.8653	−0.740493
$404$	3.53235	0.175741
$405$	5.81581	0.288990
$406$	1.58892	0.0788566
$407$	22.8130	1.13080
$408$	−29.3812	−1.45459
$409$	−21.5434	−1.06525	−0.532627	−	0.846350i	$-0.678795\pi$
$409$	−21.5434	−1.06525	−0.532627	+	0.846350i	$0.678795\pi$
$410$	−2.11178	−0.104293
$411$	11.0812	0.546594
$412$	−1.09999	−0.0541928
$413$	−15.8581	−0.780328
$414$	−7.99547	−0.392956
$415$	−0.347382	−0.0170523
$416$	2.59119	0.127044
$417$	−29.5450	−1.44683
$418$	−38.2022	−1.86853
$419$	−12.6654	−0.618746	−0.309373	−	0.950941i	$-0.600119\pi$
$419$	−12.6654	−0.618746	−0.309373	+	0.950941i	$0.600119\pi$
$420$	−0.625081	−0.0305008
$421$	0.796006	0.0387950	0.0193975	−	0.999812i	$-0.493825\pi$
$421$	0.796006	0.0387950	0.0193975	+	0.999812i	$0.493825\pi$
$422$	11.0746	0.539103
$423$	2.40337	0.116856
$424$	−33.6661	−1.63497
$425$	−15.1587	−0.735306
$426$	40.9219	1.98267
$427$	11.2654	0.545171
$428$	−2.86566	−0.138517
$429$	−17.9819	−0.868173
$430$	−8.82374	−0.425518
$431$	32.6597	1.57316	0.786580	−	0.617488i	$-0.211849\pi$
$431$	32.6597	1.57316	0.786580	+	0.617488i	$0.211849\pi$
$432$	−31.3423	−1.50795
$433$	10.5971	0.509266	0.254633	−	0.967038i	$-0.418045\pi$
$433$	10.5971	0.509266	0.254633	+	0.967038i	$0.418045\pi$
$434$	−15.1653	−0.727959
$435$	1.73056	0.0829740
$436$	−3.12718	−0.149765
$437$	−7.65905	−0.366382
$438$	−10.0556	−0.480476
$439$	29.6100	1.41321	0.706604	−	0.707609i	$-0.250226\pi$
$439$	29.6100	1.41321	0.706604	+	0.707609i	$0.250226\pi$
$440$	6.56228	0.312844
$441$	−33.8036	−1.60969
$442$	6.59544	0.313713
$443$	−34.0175	−1.61622	−0.808111	−	0.589031i	$-0.799510\pi$
$443$	−34.0175	−1.61622	−0.808111	+	0.589031i	$0.799510\pi$
$444$	5.35006	0.253903
$445$	2.18975	0.103804
$446$	10.3504	0.490105
$447$	−44.5596	−2.10760
$448$	10.7240	0.506662
$449$	38.7800	1.83014	0.915071	−	0.403292i	$-0.132134\pi$
$449$	38.7800	1.83014	0.915071	+	0.403292i	$0.132134\pi$
$450$	−37.3536	−1.76087
$451$	10.7951	0.508322
$452$	−5.29237	−0.248932
$453$	6.96861	0.327414
$454$	−34.7487	−1.63084
$455$	1.08621	0.0509225
$456$	−69.3539	−3.24779
$457$	−20.0321	−0.937064	−0.468532	−	0.883447i	$-0.655217\pi$
$457$	−20.0321	−0.937064	−0.468532	+	0.883447i	$0.655217\pi$
$458$	28.9080	1.35078
$459$	30.6442	1.43035
$460$	0.169956	0.00792423
$461$	−12.2868	−0.572252	−0.286126	−	0.958192i	$-0.592368\pi$
$461$	−12.2868	−0.572252	−0.286126	+	0.958192i	$0.592368\pi$
$462$	−18.3448	−0.853478
$463$	−37.8812	−1.76049	−0.880245	−	0.474520i	$-0.842622\pi$
$463$	−37.8812	−1.76049	−0.880245	+	0.474520i	$0.842622\pi$
$464$	−3.31861	−0.154062
$465$	−16.5172	−0.765969
$466$	5.30811	0.245893
$467$	0.782271	0.0361992	0.0180996	−	0.999836i	$-0.494238\pi$
$467$	0.782271	0.0361992	0.0180996	+	0.999836i	$0.494238\pi$
$468$	−2.83084	−0.130856
$469$	−12.3525	−0.570386
$470$	0.293300	0.0135289
$471$	−25.9041	−1.19360
$472$	39.0434	1.79712
$473$	45.1057	2.07396
$474$	3.36074	0.154364
$475$	−35.7819	−1.64179
$476$	−1.17199	−0.0537180
$477$	68.8083	3.15051
$478$	9.86126	0.451044
$479$	7.13285	0.325908	0.162954	−	0.986634i	$-0.447898\pi$
$479$	7.13285	0.325908	0.162954	+	0.986634i	$0.447898\pi$
$480$	2.87915	0.131414
$481$	−9.29691	−0.423903
$482$	8.68705	0.395684
$483$	−3.67790	−0.167350
$484$	−1.06985	−0.0486294
$485$	2.94139	0.133561
$486$	3.05012	0.138356
$487$	−18.0659	−0.818644	−0.409322	−	0.912390i	$-0.634235\pi$
$487$	−18.0659	−0.818644	−0.409322	+	0.912390i	$0.634235\pi$
$488$	−27.7359	−1.25555
$489$	24.4611	1.10617
$490$	−4.12528	−0.186361
$491$	−1.19496	−0.0539278	−0.0269639	−	0.999636i	$-0.508584\pi$
$491$	−1.19496	−0.0539278	−0.0269639	+	0.999636i	$0.508584\pi$
$492$	2.53165	0.114136
$493$	3.24470	0.146134
$494$	15.5684	0.700457
$495$	−13.4123	−0.602837
$496$	31.6743	1.42222
$497$	12.6362	0.566809
$498$	−2.39090	−0.107139
$499$	22.2895	0.997815	0.498907	−	0.866655i	$-0.333735\pi$
$499$	22.2895	0.997815	0.498907	+	0.866655i	$0.333735\pi$
$500$	1.64379	0.0735123
$501$	49.2790	2.20162
$502$	−5.08994	−0.227175
$503$	−13.3652	−0.595924	−0.297962	−	0.954578i	$-0.596307\pi$
$503$	−13.3652	−0.595924	−0.297962	+	0.954578i	$0.596307\pi$
$504$	−22.3562	−0.995827
$505$	−6.82037	−0.303503
$506$	4.98785	0.221737
$507$	−31.9445	−1.41871
$508$	−2.43645	−0.108100
$509$	1.42116	0.0629917	0.0314958	−	0.999504i	$-0.489973\pi$
$509$	1.42116	0.0629917	0.0314958	+	0.999504i	$0.489973\pi$
$510$	7.32838	0.324506
$511$	−3.10505	−0.137359
$512$	−25.4158	−1.12323
$513$	72.3351	3.19367
$514$	−12.7633	−0.562964
$515$	2.12390	0.0935901
$516$	10.5781	0.465675
$517$	−1.49930	−0.0659393
$518$	−9.48456	−0.416727
$519$	25.5852	1.12306
$520$	−2.67431	−0.117276
$521$	−0.459592	−0.0201351	−0.0100675	−	0.999949i	$-0.503205\pi$
$521$	−0.459592	−0.0201351	−0.0100675	+	0.999949i	$0.503205\pi$
$522$	7.99547	0.349952
$523$	−28.7937	−1.25906	−0.629530	−	0.776976i	$-0.716753\pi$
$523$	−28.7937	−1.25906	−0.629530	+	0.776976i	$0.716753\pi$
$524$	−6.04271	−0.263977
$525$	−17.1826	−0.749910
$526$	18.5786	0.810066
$527$	−30.9688	−1.34902
$528$	38.3150	1.66744
$529$	1.00000	0.0434783
$530$	8.39714	0.364748
$531$	−79.7986	−3.46297
$532$	−2.76646	−0.119941
$533$	−4.39930	−0.190555
$534$	15.0713	0.652198
$535$	5.53311	0.239217
$536$	30.4124	1.31362
$537$	33.3900	1.44089
$538$	−24.8076	−1.06953
$539$	21.0878	0.908317
$540$	−1.60513	−0.0690738
$541$	−10.0629	−0.432639	−0.216320	−	0.976323i	$-0.569405\pi$
$541$	−10.0629	−0.432639	−0.216320	+	0.976323i	$0.569405\pi$
$542$	−34.7271	−1.49165
$543$	67.4444	2.89432
$544$	5.39822	0.231447
$545$	6.03806	0.258642
$546$	7.47601	0.319944
$547$	37.1617	1.58892	0.794460	−	0.607317i	$-0.207754\pi$
$547$	37.1617	1.58892	0.794460	+	0.607317i	$0.207754\pi$
$548$	−1.08827	−0.0464884
$549$	56.6878	2.41938
$550$	23.3024	0.993620
$551$	7.65905	0.326287
$552$	9.05515	0.385413
$553$	1.03775	0.0441298
$554$	−14.3193	−0.608367
$555$	−10.3301	−0.438487
$556$	2.90157	0.123054
$557$	23.4235	0.992486	0.496243	−	0.868184i	$-0.334713\pi$
$557$	23.4235	0.992486	0.496243	+	0.868184i	$0.334713\pi$
$558$	−76.3123	−3.23056
$559$	−18.3818	−0.777467
$560$	−2.31446	−0.0978037
$561$	−37.4616	−1.58163
$562$	36.6588	1.54636
$563$	21.1477	0.891267	0.445634	−	0.895215i	$-0.352978\pi$
$563$	21.1477	0.891267	0.445634	+	0.895215i	$0.352978\pi$
$564$	−0.351615	−0.0148056
$565$	10.2187	0.429903
$566$	−11.1157	−0.467228
$567$	12.3602	0.519078
$568$	−31.1107	−1.30538
$569$	−7.90059	−0.331210	−0.165605	−	0.986192i	$-0.552958\pi$
$569$	−7.90059	−0.331210	−0.165605	+	0.986192i	$0.552958\pi$
$570$	17.2985	0.724555
$571$	20.5706	0.860851	0.430425	−	0.902626i	$-0.358364\pi$
$571$	20.5706	0.860851	0.430425	+	0.902626i	$0.358364\pi$
$572$	1.76597	0.0738391
$573$	−0.340558	−0.0142270
$574$	−4.48809	−0.187329
$575$	4.67184	0.194829
$576$	53.9635	2.24848
$577$	34.1557	1.42192	0.710959	−	0.703233i	$-0.248261\pi$
$577$	34.1557	1.42192	0.710959	+	0.703233i	$0.248261\pi$
$578$	−8.44661	−0.351333
$579$	−45.2826	−1.88188
$580$	−0.169956	−0.00705703
$581$	−0.738280	−0.0306290
$582$	20.2445	0.839160
$583$	−42.9249	−1.77777
$584$	7.64476	0.316342
$585$	5.46587	0.225986
$586$	32.9883	1.36273
$587$	−26.8588	−1.10858	−0.554290	−	0.832323i	$-0.687010\pi$
$587$	−26.8588	−1.10858	−0.554290	+	0.832323i	$0.687010\pi$
$588$	4.94548	0.203948
$589$	−73.1014	−3.01209
$590$	−9.73837	−0.400922
$591$	−43.6162	−1.79413
$592$	19.8094	0.814163
$593$	−31.3103	−1.28576	−0.642881	−	0.765966i	$-0.722261\pi$
$593$	−31.3103	−1.28576	−0.642881	+	0.765966i	$0.722261\pi$
$594$	−47.1072	−1.93283
$595$	2.26291	0.0927703
$596$	4.37613	0.179253
$597$	−55.4341	−2.26877
$598$	−2.03268	−0.0831226
$599$	36.5278	1.49248	0.746242	−	0.665675i	$-0.231856\pi$
$599$	36.5278	1.49248	0.746242	+	0.665675i	$0.231856\pi$
$600$	42.3043	1.72706
$601$	19.7339	0.804964	0.402482	−	0.915428i	$-0.368148\pi$
$601$	19.7339	0.804964	0.402482	+	0.915428i	$0.368148\pi$
$602$	−18.7528	−0.764307
$603$	−62.1583	−2.53128
$604$	−0.684377	−0.0278469
$605$	2.06569	0.0839823
$606$	−46.9421	−1.90689
$607$	−22.6610	−0.919782	−0.459891	−	0.887975i	$-0.652112\pi$
$607$	−22.6610	−0.919782	−0.459891	+	0.887975i	$0.652112\pi$
$608$	12.7424	0.516773
$609$	3.67790	0.149036
$610$	6.91800	0.280102
$611$	0.611008	0.0247187
$612$	−5.89748	−0.238392
$613$	−10.0972	−0.407821	−0.203911	−	0.978989i	$-0.565365\pi$
$613$	−10.0972	−0.407821	−0.203911	+	0.978989i	$0.565365\pi$
$614$	−14.2180	−0.573792
$615$	−4.88819	−0.197111
$616$	13.9466	0.561924
$617$	−10.9420	−0.440509	−0.220254	−	0.975442i	$-0.570689\pi$
$617$	−10.9420	−0.440509	−0.220254	+	0.975442i	$0.570689\pi$
$618$	14.6180	0.588022
$619$	−36.8923	−1.48283	−0.741413	−	0.671049i	$-0.765844\pi$
$619$	−36.8923	−1.48283	−0.741413	+	0.671049i	$0.765844\pi$
$620$	1.62213	0.0651464
$621$	−9.44440	−0.378991
$622$	19.6868	0.789369
$623$	4.65381	0.186451
$624$	−15.6144	−0.625076
$625$	20.1854	0.807414
$626$	4.02424	0.160841
$627$	−88.4275	−3.53145
$628$	2.54401	0.101517
$629$	−19.3682	−0.772262
$630$	5.57619	0.222161
$631$	−6.52074	−0.259587	−0.129793	−	0.991541i	$-0.541431\pi$
$631$	−6.52074	−0.259587	−0.129793	+	0.991541i	$0.541431\pi$
$632$	−2.55499	−0.101632
$633$	25.6346	1.01889
$634$	27.0318	1.07357
$635$	4.70437	0.186687
$636$	−10.0667	−0.399170
$637$	−8.59386	−0.340501
$638$	−4.98785	−0.197471
$639$	63.5855	2.51540
$640$	4.67943	0.184971
$641$	−27.5826	−1.08945	−0.544723	−	0.838616i	$-0.683365\pi$
$641$	−27.5826	−1.08945	−0.544723	+	0.838616i	$0.683365\pi$
$642$	38.0823	1.50299
$643$	13.6254	0.537333	0.268666	−	0.963233i	$-0.413417\pi$
$643$	13.6254	0.537333	0.268666	+	0.963233i	$0.413417\pi$
$644$	0.361201	0.0142333
$645$	−20.4245	−0.804215
$646$	32.4337	1.27609
$647$	−21.2032	−0.833583	−0.416792	−	0.909002i	$-0.636846\pi$
$647$	−21.2032	−0.833583	−0.416792	+	0.909002i	$0.636846\pi$
$648$	−30.4312	−1.19545
$649$	49.7811	1.95408
$650$	−9.49638	−0.372479
$651$	−35.1035	−1.37582
$652$	−2.40229	−0.0940810
$653$	17.3572	0.679239	0.339620	−	0.940563i	$-0.389702\pi$
$653$	17.3572	0.679239	0.339620	+	0.940563i	$0.389702\pi$
$654$	41.5577	1.62504
$655$	11.6674	0.455884
$656$	9.37383	0.365987
$657$	−15.6247	−0.609577
$658$	0.623340	0.0243003
$659$	−19.9119	−0.775658	−0.387829	−	0.921731i	$-0.626775\pi$
$659$	−19.9119	−0.775658	−0.387829	+	0.921731i	$0.626775\pi$
$660$	1.96222	0.0763794
$661$	6.21683	0.241807	0.120903	−	0.992664i	$-0.461421\pi$
$661$	6.21683	0.241807	0.120903	+	0.992664i	$0.461421\pi$
$662$	41.6555	1.61899
$663$	15.2666	0.592907
$664$	1.81768	0.0705395
$665$	5.34156	0.207137
$666$	−47.7266	−1.84937
$667$	−1.00000	−0.0387202
$668$	−4.83962	−0.187250
$669$	23.9583	0.926281
$670$	−7.58559	−0.293057
$671$	−35.3638	−1.36520
$672$	6.11895	0.236044
$673$	−30.8251	−1.18822	−0.594109	−	0.804384i	$-0.702495\pi$
$673$	−30.8251	−1.18822	−0.594109	+	0.804384i	$0.702495\pi$
$674$	9.96815	0.383959
$675$	−44.1228	−1.69829
$676$	3.13723	0.120663
$677$	2.78347	0.106977	0.0534887	−	0.998568i	$-0.482966\pi$
$677$	2.78347	0.106977	0.0534887	+	0.998568i	$0.482966\pi$
$678$	70.3313	2.70106
$679$	6.25123	0.239900
$680$	−5.57138	−0.213653
$681$	−80.4336	−3.08222
$682$	47.6062	1.82294
$683$	−32.3027	−1.23603	−0.618013	−	0.786168i	$-0.712062\pi$
$683$	−32.3027	−1.23603	−0.618013	+	0.786168i	$0.712062\pi$
$684$	−13.9209	−0.532279
$685$	2.10125	0.0802848
$686$	−19.8897	−0.759393
$687$	66.9140	2.55293
$688$	39.1671	1.49323
$689$	17.4931	0.666434
$690$	−2.25857	−0.0859824
$691$	−28.1794	−1.07200	−0.535998	−	0.844219i	$-0.680065\pi$
$691$	−28.1794	−1.07200	−0.535998	+	0.844219i	$0.680065\pi$
$692$	−2.51268	−0.0955179
$693$	−28.5046	−1.08280
$694$	−8.86391	−0.336469
$695$	−5.60244	−0.212513
$696$	−9.05515	−0.343235
$697$	−9.16506	−0.347151
$698$	17.0260	0.644443
$699$	12.2868	0.464730
$700$	1.68748	0.0637806
$701$	18.3177	0.691850	0.345925	−	0.938262i	$-0.387565\pi$
$701$	18.3177	0.691850	0.345925	+	0.938262i	$0.387565\pi$
$702$	19.1975	0.724562
$703$	−45.7184	−1.72430
$704$	−33.6643	−1.26877
$705$	0.678908	0.0255692
$706$	−23.2746	−0.875950
$707$	−14.4951	−0.545145
$708$	11.6746	0.438758
$709$	−27.7519	−1.04224	−0.521122	−	0.853482i	$-0.674486\pi$
$709$	−27.7519	−1.04224	−0.521122	+	0.853482i	$0.674486\pi$
$710$	7.75977	0.291219
$711$	5.22201	0.195841
$712$	−11.4579	−0.429402
$713$	9.54445	0.357442
$714$	15.5748	0.582871
$715$	−3.40979	−0.127519
$716$	−3.27918	−0.122549
$717$	22.8261	0.852457
$718$	−0.867400	−0.0323711
$719$	35.7583	1.33356	0.666780	−	0.745255i	$-0.267672\pi$
$719$	35.7583	1.33356	0.666780	+	0.745255i	$0.267672\pi$
$720$	−11.6464	−0.434036
$721$	4.51385	0.168105
$722$	51.7621	1.92638
$723$	20.1081	0.747829
$724$	−6.62361	−0.246165
$725$	−4.67184	−0.173508
$726$	14.2174	0.527657
$727$	−17.5411	−0.650564	−0.325282	−	0.945617i	$-0.605459\pi$
$727$	−17.5411	−0.650564	−0.325282	+	0.945617i	$0.605459\pi$
$728$	−5.68362	−0.210649
$729$	−23.3971	−0.866561
$730$	−1.90679	−0.0705733
$731$	−38.2948	−1.41638
$732$	−8.29347	−0.306535
$733$	−2.71604	−0.100319	−0.0501595	−	0.998741i	$-0.515973\pi$
$733$	−2.71604	−0.100319	−0.0501595	+	0.998741i	$0.515973\pi$
$734$	−10.1474	−0.374548
$735$	−9.54888	−0.352216
$736$	−1.66371	−0.0613251
$737$	38.7764	1.42835
$738$	−22.5842	−0.831337
$739$	−50.4189	−1.85469	−0.927344	−	0.374210i	$-0.877914\pi$
$739$	−50.4189	−1.85469	−0.927344	+	0.374210i	$0.877914\pi$
$740$	1.01450	0.0372938
$741$	36.0366	1.32384
$742$	17.8462	0.655153
$743$	−22.6523	−0.831033	−0.415517	−	0.909586i	$-0.636399\pi$
$743$	−22.6523	−0.831033	−0.415517	+	0.909586i	$0.636399\pi$
$744$	86.4264	3.16855
$745$	−8.44956	−0.309568
$746$	−8.16654	−0.298998
$747$	−3.71505	−0.135926
$748$	3.67905	0.134519
$749$	11.7593	0.429677
$750$	−21.8446	−0.797651
$751$	−12.3305	−0.449945	−0.224972	−	0.974365i	$-0.572229\pi$
$751$	−12.3305	−0.449945	−0.224972	+	0.974365i	$0.572229\pi$
$752$	−1.30191	−0.0474757
$753$	−11.7818	−0.429353
$754$	2.03268	0.0740260
$755$	1.32141	0.0480912
$756$	−3.41133	−0.124069
$757$	−5.85052	−0.212641	−0.106320	−	0.994332i	$-0.533907\pi$
$757$	−5.85052	−0.212641	−0.106320	+	0.994332i	$0.533907\pi$
$758$	−22.1107	−0.803095
$759$	11.5455	0.419075
$760$	−13.1511	−0.477042
$761$	−15.9306	−0.577484	−0.288742	−	0.957407i	$-0.593237\pi$
$761$	−15.9306	−0.577484	−0.288742	+	0.957407i	$0.593237\pi$
$762$	32.3784	1.17295
$763$	12.8325	0.464567
$764$	0.0334457	0.00121002
$765$	11.3870	0.411699
$766$	−10.4571	−0.377830
$767$	−20.2872	−0.732527
$768$	−21.0139	−0.758274
$769$	−13.5190	−0.487506	−0.243753	−	0.969837i	$-0.578379\pi$
$769$	−13.5190	−0.487506	−0.243753	+	0.969837i	$0.578379\pi$
$770$	−3.47861	−0.125361
$771$	−29.5435	−1.06398
$772$	4.44714	0.160056
$773$	48.7672	1.75403	0.877017	−	0.480460i	$-0.159530\pi$
$773$	48.7672	1.75403	0.877017	+	0.480460i	$0.159530\pi$
$774$	−94.3647	−3.39187
$775$	44.5902	1.60173
$776$	−15.3908	−0.552497
$777$	−21.9541	−0.787600
$778$	14.4816	0.519192
$779$	−21.6340	−0.775117
$780$	−0.799660	−0.0286324
$781$	−39.6668	−1.41939
$782$	−4.23469	−0.151432
$783$	9.44440	0.337515
$784$	18.3114	0.653979
$785$	−4.91204	−0.175318
$786$	80.3026	2.86430
$787$	6.25934	0.223122	0.111561	−	0.993758i	$-0.464415\pi$
$787$	6.25934	0.223122	0.111561	+	0.993758i	$0.464415\pi$
$788$	4.28349	0.152593
$789$	43.0044	1.53100
$790$	0.637277	0.0226733
$791$	21.7174	0.772182
$792$	70.1797	2.49373
$793$	14.4117	0.511775
$794$	24.5162	0.870047
$795$	19.4371	0.689362
$796$	5.44410	0.192961
$797$	27.7945	0.984530	0.492265	−	0.870445i	$-0.336169\pi$
$797$	27.7945	0.984530	0.492265	+	0.870445i	$0.336169\pi$
$798$	36.7640	1.30143
$799$	1.27291	0.0450324
$800$	−7.77258	−0.274802
$801$	23.4181	0.827439
$802$	−10.8616	−0.383538
$803$	9.74721	0.343972
$804$	9.09379	0.320713
$805$	−0.697418	−0.0245808
$806$	−19.4008	−0.683366
$807$	−57.4228	−2.02138
$808$	35.6876	1.25549
$809$	7.21091	0.253522	0.126761	−	0.991933i	$-0.459542\pi$
$809$	7.21091	0.253522	0.126761	+	0.991933i	$0.459542\pi$
$810$	7.59028	0.266695
$811$	5.68402	0.199593	0.0997964	−	0.995008i	$-0.468181\pi$
$811$	5.68402	0.199593	0.0997964	+	0.995008i	$0.468181\pi$
$812$	−0.361201	−0.0126757
$813$	−80.3836	−2.81918
$814$	29.7735	1.04356
$815$	4.63841	0.162477
$816$	−32.5294	−1.13876
$817$	−90.3941	−3.16249
$818$	−28.1166	−0.983073
$819$	11.6164	0.405911
$820$	0.480062	0.0167645
$821$	17.6765	0.616915	0.308458	−	0.951238i	$-0.400187\pi$
$821$	17.6765	0.616915	0.308458	+	0.951238i	$0.400187\pi$
$822$	14.4622	0.504426
$823$	−48.5960	−1.69395	−0.846975	−	0.531633i	$-0.821578\pi$
$823$	−48.5960	−1.69395	−0.846975	+	0.531633i	$0.821578\pi$
$824$	−11.1133	−0.387150
$825$	53.9387	1.87791
$826$	−20.6966	−0.720128
$827$	−48.9103	−1.70078	−0.850389	−	0.526154i	$-0.823633\pi$
$827$	−48.9103	−1.70078	−0.850389	+	0.526154i	$0.823633\pi$
$828$	1.81758	0.0631651
$829$	38.9522	1.35287	0.676434	−	0.736504i	$-0.263525\pi$
$829$	38.9522	1.35287	0.676434	+	0.736504i	$0.263525\pi$
$830$	−0.453372	−0.0157368
$831$	−33.1451	−1.14979
$832$	13.7191	0.475625
$833$	−17.9036	−0.620322
$834$	−38.5595	−1.33521
$835$	9.34447	0.323379
$836$	8.68433	0.300354
$837$	−90.1416	−3.11575
$838$	−16.5298	−0.571012
$839$	27.9458	0.964797	0.482399	−	0.875952i	$-0.339766\pi$
$839$	27.9458	0.964797	0.482399	+	0.875952i	$0.339766\pi$
$840$	−6.31523	−0.217896
$841$	1.00000	0.0344828
$842$	1.03888	0.0358020
$843$	84.8550	2.92256
$844$	−2.51754	−0.0866573
$845$	−6.05745	−0.208383
$846$	3.13667	0.107841
$847$	4.39015	0.150847
$848$	−37.2735	−1.27998
$849$	−25.7298	−0.883044
$850$	−19.7838	−0.678579
$851$	5.96920	0.204622
$852$	−9.30260	−0.318702
$853$	19.3309	0.661878	0.330939	−	0.943652i	$-0.392635\pi$
$853$	19.3309	0.661878	0.330939	+	0.943652i	$0.392635\pi$
$854$	14.7026	0.503113
$855$	26.8789	0.919239
$856$	−28.9520	−0.989558
$857$	−42.6767	−1.45781	−0.728904	−	0.684616i	$-0.759970\pi$
$857$	−42.6767	−1.45781	−0.728904	+	0.684616i	$0.759970\pi$
$858$	−23.4683	−0.801196
$859$	−10.8383	−0.369800	−0.184900	−	0.982757i	$-0.559196\pi$
$859$	−10.8383	−0.369800	−0.184900	+	0.982757i	$0.559196\pi$
$860$	2.00586	0.0683993
$861$	−10.3887	−0.354046
$862$	42.6245	1.45180
$863$	35.3269	1.20254	0.601271	−	0.799045i	$-0.294661\pi$
$863$	35.3269	1.20254	0.601271	+	0.799045i	$0.294661\pi$
$864$	15.7127	0.534558
$865$	4.85156	0.164958
$866$	13.8304	0.469977
$867$	−19.5516	−0.664007
$868$	3.44747	0.117015
$869$	−3.25767	−0.110509
$870$	2.25857	0.0765728
$871$	−15.8025	−0.535446
$872$	−31.5941	−1.06991
$873$	31.4564	1.06464
$874$	−9.99591	−0.338117
$875$	−6.74532	−0.228033
$876$	2.28590	0.0772335
$877$	16.6037	0.560667	0.280333	−	0.959903i	$-0.409555\pi$
$877$	16.6037	0.560667	0.280333	+	0.959903i	$0.409555\pi$
$878$	38.6443	1.30418
$879$	76.3588	2.57552
$880$	7.26543	0.244918
$881$	23.2009	0.781659	0.390830	−	0.920463i	$-0.372188\pi$
$881$	23.2009	0.781659	0.390830	+	0.920463i	$0.372188\pi$
$882$	−44.1174	−1.48551
$883$	25.1560	0.846568	0.423284	−	0.905997i	$-0.360877\pi$
$883$	25.1560	0.846568	0.423284	+	0.905997i	$0.360877\pi$
$884$	−1.49931	−0.0504274
$885$	−22.5416	−0.757729
$886$	−44.3966	−1.49153
$887$	−17.2591	−0.579502	−0.289751	−	0.957102i	$-0.593573\pi$
$887$	−17.2591	−0.579502	−0.289751	+	0.957102i	$0.593573\pi$
$888$	54.0520	1.81387
$889$	9.99804	0.335324
$890$	2.85787	0.0957961
$891$	−38.8004	−1.29986
$892$	−2.35291	−0.0787812
$893$	3.00469	0.100548
$894$	−58.1552	−1.94500
$895$	6.33154	0.211640
$896$	9.94504	0.332240
$897$	−4.70510	−0.157099
$898$	50.6122	1.68895
$899$	−9.54445	−0.318325
$900$	8.49143	0.283048
$901$	36.4433	1.21410
$902$	14.0888	0.469106
$903$	−43.4076	−1.44451
$904$	−53.4692	−1.77836
$905$	12.7891	0.425123
$906$	9.09481	0.302155
$907$	−28.5679	−0.948581	−0.474290	−	0.880368i	$-0.657295\pi$
$907$	−28.5679	−0.948581	−0.474290	+	0.880368i	$0.657295\pi$
$908$	7.89927	0.262146
$909$	−72.9399	−2.41926
$910$	1.41763	0.0469940
$911$	13.1206	0.434705	0.217353	−	0.976093i	$-0.430258\pi$
$911$	13.1206	0.434705	0.217353	+	0.976093i	$0.430258\pi$
$912$	−76.7852	−2.54261
$913$	2.31757	0.0767005
$914$	−26.1442	−0.864772
$915$	16.0133	0.529382
$916$	−6.57152	−0.217129
$917$	24.7964	0.818850
$918$	39.9941	1.32000
$919$	−2.49407	−0.0822717	−0.0411358	−	0.999154i	$-0.513098\pi$
$919$	−2.49407	−0.0822717	−0.0411358	+	0.999154i	$0.513098\pi$
$920$	1.71707	0.0566102
$921$	−32.9108	−1.08445
$922$	−16.0356	−0.528104
$923$	16.1653	0.532088
$924$	4.17025	0.137191
$925$	27.8872	0.916925
$926$	−49.4392	−1.62467
$927$	22.7138	0.746020
$928$	1.66371	0.0546139
$929$	−14.5108	−0.476083	−0.238042	−	0.971255i	$-0.576505\pi$
$929$	−14.5108	−0.476083	−0.238042	+	0.971255i	$0.576505\pi$
$930$	−21.5568	−0.706876
$931$	−42.2611	−1.38505
$932$	−1.20667	−0.0395258
$933$	45.5696	1.49188
$934$	1.02095	0.0334065
$935$	−7.10362	−0.232313
$936$	−28.6001	−0.934825
$937$	−36.4431	−1.19054	−0.595271	−	0.803525i	$-0.702955\pi$
$937$	−36.4431	−1.19054	−0.595271	+	0.803525i	$0.702955\pi$
$938$	−16.1214	−0.526383
$939$	9.31501	0.303984
$940$	−0.0666745	−0.00217468
$941$	−6.09402	−0.198659	−0.0993296	−	0.995055i	$-0.531670\pi$
$941$	−6.09402	−0.198659	−0.0993296	+	0.995055i	$0.531670\pi$
$942$	−33.8078	−1.10152
$943$	2.82463	0.0919825
$944$	43.2270	1.40692
$945$	6.58669	0.214265
$946$	58.8679	1.91396
$947$	−11.8081	−0.383711	−0.191855	−	0.981423i	$-0.561450\pi$
$947$	−11.8081	−0.383711	−0.191855	+	0.981423i	$0.561450\pi$
$948$	−0.763983	−0.0248130
$949$	−3.97226	−0.128945
$950$	−46.6993	−1.51513
$951$	62.5712	2.02901
$952$	−11.8407	−0.383758
$953$	28.4697	0.922225	0.461112	−	0.887342i	$-0.347451\pi$
$953$	28.4697	0.922225	0.461112	+	0.887342i	$0.347451\pi$
$954$	89.8024	2.90746
$955$	−0.0645779	−0.00208969
$956$	−2.24172	−0.0725023
$957$	−11.5455	−0.373213
$958$	9.30916	0.300765
$959$	4.46573	0.144206
$960$	15.2437	0.491988
$961$	60.0964	1.93860
$962$	−12.1335	−0.391200
$963$	59.1733	1.90683
$964$	−1.97479	−0.0636037
$965$	−8.58666	−0.276414
$966$	−4.80007	−0.154440
$967$	13.8752	0.446196	0.223098	−	0.974796i	$-0.428383\pi$
$967$	13.8752	0.446196	0.223098	+	0.974796i	$0.428383\pi$
$968$	−10.8087	−0.347406
$969$	75.0751	2.41176
$970$	3.83883	0.123258
$971$	−13.7112	−0.440013	−0.220007	−	0.975498i	$-0.570608\pi$
$971$	−13.7112	−0.440013	−0.220007	+	0.975498i	$0.570608\pi$
$972$	−0.693370	−0.0222398
$973$	−11.9067	−0.381711
$974$	−23.5780	−0.755488
$975$	−21.9815	−0.703972
$976$	−30.7078	−0.982934
$977$	35.8383	1.14657	0.573284	−	0.819356i	$-0.305669\pi$
$977$	35.8383	1.14657	0.573284	+	0.819356i	$0.305669\pi$
$978$	31.9245	1.02083
$979$	−14.6090	−0.466906
$980$	0.937782	0.0299563
$981$	64.5735	2.06167
$982$	−1.55956	−0.0497675
$983$	9.92432	0.316537	0.158268	−	0.987396i	$-0.449409\pi$
$983$	9.92432	0.316537	0.158268	+	0.987396i	$0.449409\pi$
$984$	25.5774	0.815379
$985$	−8.27068	−0.263526
$986$	4.23469	0.134860
$987$	1.44286	0.0459268
$988$	−3.53910	−0.112594
$989$	11.8023	0.375290
$990$	−17.5045	−0.556329
$991$	52.4826	1.66716	0.833582	−	0.552396i	$-0.186286\pi$
$991$	52.4826	1.66716	0.833582	+	0.552396i	$0.186286\pi$
$992$	−15.8792	−0.504164
$993$	96.4211	3.05983
$994$	16.4916	0.523081
$995$	−10.5116	−0.333241
$996$	0.543514	0.0172219
$997$	3.72613	0.118008	0.0590038	−	0.998258i	$-0.481208\pi$
$997$	3.72613	0.118008	0.0590038	+	0.998258i	$0.481208\pi$
$998$	29.0903	0.920836
$999$	−56.3755	−1.78364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	667.2.a.c.1.9	✓	13
3.2	odd	2		6003.2.a.o.1.5		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
667.2.a.c.1.9	✓	13	1.1	even	1	trivial
6003.2.a.o.1.5		13	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:	\( N \)	\(=\)	\( 667 = 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	667.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30511 q^{2} +3.02097 q^{3} -0.296685 q^{4} +0.572849 q^{5} +3.94270 q^{6} +1.21746 q^{7} -2.99743 q^{8} +6.12628 q^{9} +O(q^{10})\)	\(q+1.30511 q^{2} +3.02097 q^{3} -0.296685 q^{4} +0.572849 q^{5} +3.94270 q^{6} +1.21746 q^{7} -2.99743 q^{8} +6.12628 q^{9} +0.747631 q^{10} -3.82178 q^{11} -0.896278 q^{12} +1.55748 q^{13} +1.58892 q^{14} +1.73056 q^{15} -3.31861 q^{16} +3.24470 q^{17} +7.99547 q^{18} +7.65905 q^{19} -0.169956 q^{20} +3.67790 q^{21} -4.98785 q^{22} -1.00000 q^{23} -9.05515 q^{24} -4.67184 q^{25} +2.03268 q^{26} +9.44440 q^{27} -0.361201 q^{28} +1.00000 q^{29} +2.25857 q^{30} -9.54445 q^{31} +1.66371 q^{32} -11.5455 q^{33} +4.23469 q^{34} +0.697418 q^{35} -1.81758 q^{36} -5.96920 q^{37} +9.99591 q^{38} +4.70510 q^{39} -1.71707 q^{40} -2.82463 q^{41} +4.80007 q^{42} -11.8023 q^{43} +1.13387 q^{44} +3.50943 q^{45} -1.30511 q^{46} +0.392305 q^{47} -10.0254 q^{48} -5.51780 q^{49} -6.09728 q^{50} +9.80214 q^{51} -0.462081 q^{52} +11.2317 q^{53} +12.3260 q^{54} -2.18930 q^{55} -3.64924 q^{56} +23.1378 q^{57} +1.30511 q^{58} -13.0256 q^{59} -0.513432 q^{60} +9.25323 q^{61} -12.4566 q^{62} +7.45847 q^{63} +8.80854 q^{64} +0.892200 q^{65} -15.0682 q^{66} -10.1462 q^{67} -0.962653 q^{68} -3.02097 q^{69} +0.910208 q^{70} +10.3791 q^{71} -18.3631 q^{72} -2.55044 q^{73} -7.79047 q^{74} -14.1135 q^{75} -2.27233 q^{76} -4.65285 q^{77} +6.14068 q^{78} +0.852395 q^{79} -1.90106 q^{80} +10.1524 q^{81} -3.68645 q^{82} -0.606412 q^{83} -1.09118 q^{84} +1.85872 q^{85} -15.4033 q^{86} +3.02097 q^{87} +11.4555 q^{88} +3.82257 q^{89} +4.58019 q^{90} +1.89616 q^{91} +0.296685 q^{92} -28.8335 q^{93} +0.512002 q^{94} +4.38748 q^{95} +5.02602 q^{96} +5.13466 q^{97} -7.20134 q^{98} -23.4133 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{2} + 3 q^{3} + 12 q^{4} + 16 q^{5} + q^{7} + 6 q^{8} + 14 q^{9}+O(q^{10}) \) 13 * q + 4 * q^2 + 3 * q^3 + 12 * q^4 + 16 * q^5 + q^7 + 6 * q^8 + 14 * q^9	\( 13 q + 4 q^{2} + 3 q^{3} + 12 q^{4} + 16 q^{5} + q^{7} + 6 q^{8} + 14 q^{9} + 10 q^{10} + 10 q^{11} + 3 q^{12} + 7 q^{13} - 12 q^{14} + 8 q^{15} + 2 q^{16} + 26 q^{17} + 7 q^{18} + 25 q^{20} + 11 q^{21} - 15 q^{22} - 13 q^{23} + 10 q^{24} + 19 q^{25} - 15 q^{26} + 12 q^{27} + 5 q^{28} + 13 q^{29} + 7 q^{30} - 6 q^{31} + 16 q^{32} + 3 q^{33} + 11 q^{34} + q^{35} - 22 q^{36} + 15 q^{37} + 8 q^{38} - 4 q^{39} + 14 q^{40} + 9 q^{41} - 34 q^{42} + q^{43} + 29 q^{44} + 16 q^{45} - 4 q^{46} + 15 q^{47} - 15 q^{48} + 4 q^{49} + 31 q^{50} - 14 q^{51} - 8 q^{52} + 43 q^{53} - 35 q^{54} - 3 q^{55} - 5 q^{56} + 6 q^{57} + 4 q^{58} - 9 q^{59} + 3 q^{60} + 20 q^{61} + 11 q^{62} + 21 q^{63} - 16 q^{64} - 25 q^{65} - q^{66} + q^{67} + 21 q^{68} - 3 q^{69} - 2 q^{70} + 17 q^{71} - 59 q^{72} + 26 q^{73} + 11 q^{74} - 43 q^{75} + 8 q^{76} + 17 q^{77} - 10 q^{78} + 5 q^{79} + 10 q^{80} - 3 q^{81} - 25 q^{82} + 4 q^{83} - 78 q^{84} + 20 q^{85} - 13 q^{86} + 3 q^{87} - 32 q^{88} + 48 q^{89} + 17 q^{90} - 9 q^{91} - 12 q^{92} - 8 q^{93} - 65 q^{94} + 8 q^{95} + 8 q^{96} + 30 q^{97} + 2 q^{98} + 4 q^{99}+O(q^{100}) \) 13 * q + 4 * q^2 + 3 * q^3 + 12 * q^4 + 16 * q^5 + q^7 + 6 * q^8 + 14 * q^9 + 10 * q^10 + 10 * q^11 + 3 * q^12 + 7 * q^13 - 12 * q^14 + 8 * q^15 + 2 * q^16 + 26 * q^17 + 7 * q^18 + 25 * q^20 + 11 * q^21 - 15 * q^22 - 13 * q^23 + 10 * q^24 + 19 * q^25 - 15 * q^26 + 12 * q^27 + 5 * q^28 + 13 * q^29 + 7 * q^30 - 6 * q^31 + 16 * q^32 + 3 * q^33 + 11 * q^34 + q^35 - 22 * q^36 + 15 * q^37 + 8 * q^38 - 4 * q^39 + 14 * q^40 + 9 * q^41 - 34 * q^42 + q^43 + 29 * q^44 + 16 * q^45 - 4 * q^46 + 15 * q^47 - 15 * q^48 + 4 * q^49 + 31 * q^50 - 14 * q^51 - 8 * q^52 + 43 * q^53 - 35 * q^54 - 3 * q^55 - 5 * q^56 + 6 * q^57 + 4 * q^58 - 9 * q^59 + 3 * q^60 + 20 * q^61 + 11 * q^62 + 21 * q^63 - 16 * q^64 - 25 * q^65 - q^66 + q^67 + 21 * q^68 - 3 * q^69 - 2 * q^70 + 17 * q^71 - 59 * q^72 + 26 * q^73 + 11 * q^74 - 43 * q^75 + 8 * q^76 + 17 * q^77 - 10 * q^78 + 5 * q^79 + 10 * q^80 - 3 * q^81 - 25 * q^82 + 4 * q^83 - 78 * q^84 + 20 * q^85 - 13 * q^86 + 3 * q^87 - 32 * q^88 + 48 * q^89 + 17 * q^90 - 9 * q^91 - 12 * q^92 - 8 * q^93 - 65 * q^94 + 8 * q^95 + 8 * q^96 + 30 * q^97 + 2 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more