LMFDB - Embedded newform 8031.2.a.c.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8031,2,Mod(1,8031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8031, 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("8031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8031 = 3 \cdot 2677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8031.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:	$64.1278578633$
Analytic rank:	$0$
Dimension:	$121$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36975 q^{2} -1.00000 q^{3} +3.61573 q^{4} -1.85498 q^{5} +2.36975 q^{6} +4.52062 q^{7} -3.82888 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.36975 q^{2} -1.00000 q^{3} +3.61573 q^{4} -1.85498 q^{5} +2.36975 q^{6} +4.52062 q^{7} -3.82888 q^{8} +1.00000 q^{9} +4.39585 q^{10} +6.02531 q^{11} -3.61573 q^{12} +0.342912 q^{13} -10.7128 q^{14} +1.85498 q^{15} +1.84204 q^{16} +3.63212 q^{17} -2.36975 q^{18} -5.51306 q^{19} -6.70711 q^{20} -4.52062 q^{21} -14.2785 q^{22} +8.93696 q^{23} +3.82888 q^{24} -1.55904 q^{25} -0.812617 q^{26} -1.00000 q^{27} +16.3453 q^{28} -3.02109 q^{29} -4.39585 q^{30} -10.3201 q^{31} +3.29258 q^{32} -6.02531 q^{33} -8.60724 q^{34} -8.38567 q^{35} +3.61573 q^{36} +2.59244 q^{37} +13.0646 q^{38} -0.342912 q^{39} +7.10250 q^{40} +1.23624 q^{41} +10.7128 q^{42} +1.73880 q^{43} +21.7859 q^{44} -1.85498 q^{45} -21.1784 q^{46} +2.29467 q^{47} -1.84204 q^{48} +13.4360 q^{49} +3.69454 q^{50} -3.63212 q^{51} +1.23988 q^{52} -3.34590 q^{53} +2.36975 q^{54} -11.1768 q^{55} -17.3089 q^{56} +5.51306 q^{57} +7.15924 q^{58} +4.54531 q^{59} +6.70711 q^{60} -3.91440 q^{61} +24.4560 q^{62} +4.52062 q^{63} -11.4867 q^{64} -0.636096 q^{65} +14.2785 q^{66} -10.0401 q^{67} +13.1328 q^{68} -8.93696 q^{69} +19.8720 q^{70} -9.22735 q^{71} -3.82888 q^{72} +15.7162 q^{73} -6.14345 q^{74} +1.55904 q^{75} -19.9337 q^{76} +27.2381 q^{77} +0.812617 q^{78} +8.09234 q^{79} -3.41695 q^{80} +1.00000 q^{81} -2.92957 q^{82} +8.15333 q^{83} -16.3453 q^{84} -6.73753 q^{85} -4.12052 q^{86} +3.02109 q^{87} -23.0702 q^{88} +13.3609 q^{89} +4.39585 q^{90} +1.55018 q^{91} +32.3136 q^{92} +10.3201 q^{93} -5.43779 q^{94} +10.2266 q^{95} -3.29258 q^{96} +14.3947 q^{97} -31.8400 q^{98} +6.02531 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 121 q + 7 q^{2} - 121 q^{3} + 123 q^{4} + 24 q^{5} - 7 q^{6} - 14 q^{7} + 18 q^{8} + 121 q^{9}+O(q^{10}) $ 121 * q + 7 * q^2 - 121 * q^3 + 123 * q^4 + 24 * q^5 - 7 * q^6 - 14 * q^7 + 18 * q^8 + 121 * q^9	\( 121 q + 7 q^{2} - 121 q^{3} + 123 q^{4} + 24 q^{5} - 7 q^{6} - 14 q^{7} + 18 q^{8} + 121 q^{9} + 18 q^{10} + 32 q^{11} - 123 q^{12} + 2 q^{13} + 37 q^{14} - 24 q^{15} + 131 q^{16} + 87 q^{17} + 7 q^{18} - 10 q^{19} + 60 q^{20} + 14 q^{21} - 22 q^{22} + 31 q^{23} - 18 q^{24} + 147 q^{25} + 37 q^{26} - 121 q^{27} - 29 q^{28} + 68 q^{29} - 18 q^{30} + 25 q^{31} + 43 q^{32} - 32 q^{33} + 27 q^{34} + 51 q^{35} + 123 q^{36} - 4 q^{37} + 36 q^{38} - 2 q^{39} + 61 q^{40} + 132 q^{41} - 37 q^{42} - 91 q^{43} + 94 q^{44} + 24 q^{45} + 39 q^{47} - 131 q^{48} + 217 q^{49} + 54 q^{50} - 87 q^{51} - 12 q^{52} + 55 q^{53} - 7 q^{54} + 7 q^{55} + 104 q^{56} + 10 q^{57} - 3 q^{58} + 58 q^{59} - 60 q^{60} + 126 q^{61} + 74 q^{62} - 14 q^{63} + 122 q^{64} + 128 q^{65} + 22 q^{66} - 139 q^{67} + 190 q^{68} - 31 q^{69} - 18 q^{70} + 37 q^{71} + 18 q^{72} + 84 q^{73} + 79 q^{74} - 147 q^{75} + 23 q^{76} + 95 q^{77} - 37 q^{78} - 14 q^{79} + 145 q^{80} + 121 q^{81} + 9 q^{82} + 58 q^{83} + 29 q^{84} + 32 q^{85} + 28 q^{86} - 68 q^{87} - 84 q^{88} + 198 q^{89} + 18 q^{90} + 5 q^{91} + 98 q^{92} - 25 q^{93} + 9 q^{94} + 42 q^{95} - 43 q^{96} + 73 q^{97} + 69 q^{98} + 32 q^{99}+O(q^{100}) \) 121 * q + 7 * q^2 - 121 * q^3 + 123 * q^4 + 24 * q^5 - 7 * q^6 - 14 * q^7 + 18 * q^8 + 121 * q^9 + 18 * q^10 + 32 * q^11 - 123 * q^12 + 2 * q^13 + 37 * q^14 - 24 * q^15 + 131 * q^16 + 87 * q^17 + 7 * q^18 - 10 * q^19 + 60 * q^20 + 14 * q^21 - 22 * q^22 + 31 * q^23 - 18 * q^24 + 147 * q^25 + 37 * q^26 - 121 * q^27 - 29 * q^28 + 68 * q^29 - 18 * q^30 + 25 * q^31 + 43 * q^32 - 32 * q^33 + 27 * q^34 + 51 * q^35 + 123 * q^36 - 4 * q^37 + 36 * q^38 - 2 * q^39 + 61 * q^40 + 132 * q^41 - 37 * q^42 - 91 * q^43 + 94 * q^44 + 24 * q^45 + 39 * q^47 - 131 * q^48 + 217 * q^49 + 54 * q^50 - 87 * q^51 - 12 * q^52 + 55 * q^53 - 7 * q^54 + 7 * q^55 + 104 * q^56 + 10 * q^57 - 3 * q^58 + 58 * q^59 - 60 * q^60 + 126 * q^61 + 74 * q^62 - 14 * q^63 + 122 * q^64 + 128 * q^65 + 22 * q^66 - 139 * q^67 + 190 * q^68 - 31 * q^69 - 18 * q^70 + 37 * q^71 + 18 * q^72 + 84 * q^73 + 79 * q^74 - 147 * q^75 + 23 * q^76 + 95 * q^77 - 37 * q^78 - 14 * q^79 + 145 * q^80 + 121 * q^81 + 9 * q^82 + 58 * q^83 + 29 * q^84 + 32 * q^85 + 28 * q^86 - 68 * q^87 - 84 * q^88 + 198 * q^89 + 18 * q^90 + 5 * q^91 + 98 * q^92 - 25 * q^93 + 9 * q^94 + 42 * q^95 - 43 * q^96 + 73 * q^97 + 69 * q^98 + 32 * 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 $

$2$	−2.36975	−1.67567	−0.837834	−	0.545925i	$-0.816178\pi$
$2$	−2.36975	−1.67567	−0.837834	+	0.545925i	$0.816178\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	3.61573	1.80786
$5$	−1.85498	−0.829573	−0.414787	−	0.909919i	$-0.636144\pi$
$5$	−1.85498	−0.829573	−0.414787	+	0.909919i	$0.636144\pi$
$6$	2.36975	0.967448
$7$	4.52062	1.70863	0.854317	−	0.519752i	$-0.173976\pi$
$7$	4.52062	1.70863	0.854317	+	0.519752i	$0.173976\pi$
$8$	−3.82888	−1.35371
$9$	1.00000	0.333333
$10$	4.39585	1.39009
$11$	6.02531	1.81670	0.908350	−	0.418212i	$-0.137343\pi$
$11$	6.02531	1.81670	0.908350	+	0.418212i	$0.137343\pi$
$12$	−3.61573	−1.04377
$13$	0.342912	0.0951067	0.0475533	−	0.998869i	$-0.484858\pi$
$13$	0.342912	0.0951067	0.0475533	+	0.998869i	$0.484858\pi$
$14$	−10.7128	−2.86310
$15$	1.85498	0.478954
$16$	1.84204	0.460510
$17$	3.63212	0.880920	0.440460	−	0.897772i	$-0.354815\pi$
$17$	3.63212	0.880920	0.440460	+	0.897772i	$0.354815\pi$
$18$	−2.36975	−0.558556
$19$	−5.51306	−1.26478	−0.632392	−	0.774649i	$-0.717927\pi$
$19$	−5.51306	−1.26478	−0.632392	+	0.774649i	$0.717927\pi$
$20$	−6.70711	−1.49976
$21$	−4.52062	−0.986480
$22$	−14.2785	−3.04419
$23$	8.93696	1.86348	0.931742	−	0.363120i	$-0.118288\pi$
$23$	8.93696	1.86348	0.931742	+	0.363120i	$0.118288\pi$
$24$	3.82888	0.781567
$25$	−1.55904	−0.311808
$26$	−0.812617	−0.159367
$27$	−1.00000	−0.192450
$28$	16.3453	3.08898
$29$	−3.02109	−0.561003	−0.280501	−	0.959854i	$-0.590501\pi$
$29$	−3.02109	−0.561003	−0.280501	+	0.959854i	$0.590501\pi$
$30$	−4.39585	−0.802569
$31$	−10.3201	−1.85354	−0.926770	−	0.375629i	$-0.877427\pi$
$31$	−10.3201	−1.85354	−0.926770	+	0.375629i	$0.877427\pi$
$32$	3.29258	0.582051
$33$	−6.02531	−1.04887
$34$	−8.60724	−1.47613
$35$	−8.38567	−1.41744
$36$	3.61573	0.602622
$37$	2.59244	0.426195	0.213097	−	0.977031i	$-0.431645\pi$
$37$	2.59244	0.426195	0.213097	+	0.977031i	$0.431645\pi$
$38$	13.0646	2.11936
$39$	−0.342912	−0.0549099
$40$	7.10250	1.12300
$41$	1.23624	0.193068	0.0965338	−	0.995330i	$-0.469224\pi$
$41$	1.23624	0.193068	0.0965338	+	0.995330i	$0.469224\pi$
$42$	10.7128	1.65301
$43$	1.73880	0.265164	0.132582	−	0.991172i	$-0.457673\pi$
$43$	1.73880	0.265164	0.132582	+	0.991172i	$0.457673\pi$
$44$	21.7859	3.28435
$45$	−1.85498	−0.276524
$46$	−21.1784	−3.12258
$47$	2.29467	0.334712	0.167356	−	0.985897i	$-0.446477\pi$
$47$	2.29467	0.334712	0.167356	+	0.985897i	$0.446477\pi$
$48$	−1.84204	−0.265875
$49$	13.4360	1.91943
$50$	3.69454	0.522487
$51$	−3.63212	−0.508599
$52$	1.23988	0.171940
$53$	−3.34590	−0.459595	−0.229797	−	0.973238i	$-0.573806\pi$
$53$	−3.34590	−0.459595	−0.229797	+	0.973238i	$0.573806\pi$
$54$	2.36975	0.322483
$55$	−11.1768	−1.50709
$56$	−17.3089	−2.31300
$57$	5.51306	0.730223
$58$	7.15924	0.940055
$59$	4.54531	0.591750	0.295875	−	0.955227i	$-0.404389\pi$
$59$	4.54531	0.591750	0.295875	+	0.955227i	$0.404389\pi$
$60$	6.70711	0.865885
$61$	−3.91440	−0.501188	−0.250594	−	0.968092i	$-0.580626\pi$
$61$	−3.91440	−0.501188	−0.250594	+	0.968092i	$0.580626\pi$
$62$	24.4560	3.10592
$63$	4.52062	0.569545
$64$	−11.4867	−1.43584
$65$	−0.636096	−0.0788980
$66$	14.2785	1.75756
$67$	−10.0401	−1.22659	−0.613297	−	0.789853i	$-0.710157\pi$
$67$	−10.0401	−1.22659	−0.613297	+	0.789853i	$0.710157\pi$
$68$	13.1328	1.59258
$69$	−8.93696	−1.07588
$70$	19.8720	2.37516
$71$	−9.22735	−1.09508	−0.547542	−	0.836778i	$-0.684436\pi$
$71$	−9.22735	−1.09508	−0.547542	+	0.836778i	$0.684436\pi$
$72$	−3.82888	−0.451238
$73$	15.7162	1.83945	0.919724	−	0.392567i	$-0.128413\pi$
$73$	15.7162	1.83945	0.919724	+	0.392567i	$0.128413\pi$
$74$	−6.14345	−0.714161
$75$	1.55904	0.180023
$76$	−19.9337	−2.28656
$77$	27.2381	3.10407
$78$	0.812617	0.0920107
$79$	8.09234	0.910459	0.455229	−	0.890374i	$-0.349557\pi$
$79$	8.09234	0.910459	0.455229	+	0.890374i	$0.349557\pi$
$80$	−3.41695	−0.382027
$81$	1.00000	0.111111
$82$	−2.92957	−0.323517
$83$	8.15333	0.894945	0.447472	−	0.894298i	$-0.352324\pi$
$83$	8.15333	0.894945	0.447472	+	0.894298i	$0.352324\pi$
$84$	−16.3453	−1.78342
$85$	−6.73753	−0.730787
$86$	−4.12052	−0.444327
$87$	3.02109	0.323895
$88$	−23.0702	−2.45929
$89$	13.3609	1.41625	0.708124	−	0.706088i	$-0.249542\pi$
$89$	13.3609	1.41625	0.708124	+	0.706088i	$0.249542\pi$
$90$	4.39585	0.463363
$91$	1.55018	0.162503
$92$	32.3136	3.36893
$93$	10.3201	1.07014
$94$	−5.43779	−0.560866
$95$	10.2266	1.04923
$96$	−3.29258	−0.336048
$97$	14.3947	1.46157	0.730783	−	0.682610i	$-0.239155\pi$
$97$	14.3947	1.46157	0.730783	+	0.682610i	$0.239155\pi$
$98$	−31.8400	−3.21633
$99$	6.02531	0.605566
$100$	−5.63707	−0.563707
$101$	−5.42912	−0.540218	−0.270109	−	0.962830i	$-0.587060\pi$
$101$	−5.42912	−0.540218	−0.270109	+	0.962830i	$0.587060\pi$
$102$	8.60724	0.852244
$103$	9.05652	0.892365	0.446183	−	0.894942i	$-0.352783\pi$
$103$	9.05652	0.892365	0.446183	+	0.894942i	$0.352783\pi$
$104$	−1.31297	−0.128747
$105$	8.38567	0.818358
$106$	7.92896	0.770129
$107$	−4.14785	−0.400988	−0.200494	−	0.979695i	$-0.564255\pi$
$107$	−4.14785	−0.400988	−0.200494	+	0.979695i	$0.564255\pi$
$108$	−3.61573	−0.347924
$109$	−12.7225	−1.21860	−0.609299	−	0.792940i	$-0.708549\pi$
$109$	−12.7225	−1.21860	−0.609299	+	0.792940i	$0.708549\pi$
$110$	26.4864	2.52538
$111$	−2.59244	−0.246064
$112$	8.32716	0.786843
$113$	−1.84518	−0.173580	−0.0867900	−	0.996227i	$-0.527661\pi$
$113$	−1.84518	−0.173580	−0.0867900	+	0.996227i	$0.527661\pi$
$114$	−13.0646	−1.22361
$115$	−16.5779	−1.54590
$116$	−10.9235	−1.01422
$117$	0.342912	0.0317022
$118$	−10.7713	−0.991576
$119$	16.4195	1.50517
$120$	−7.10250	−0.648367
$121$	25.3044	2.30040
$122$	9.27616	0.839824
$123$	−1.23624	−0.111468
$124$	−37.3146	−3.35095
$125$	12.1669	1.08824
$126$	−10.7128	−0.954368
$127$	9.33323	0.828190	0.414095	−	0.910234i	$-0.364098\pi$
$127$	9.33323	0.828190	0.414095	+	0.910234i	$0.364098\pi$
$128$	20.6354	1.82393
$129$	−1.73880	−0.153093
$130$	1.50739	0.132207
$131$	4.90669	0.428700	0.214350	−	0.976757i	$-0.431237\pi$
$131$	4.90669	0.428700	0.214350	+	0.976757i	$0.431237\pi$
$132$	−21.7859	−1.89622
$133$	−24.9225	−2.16105
$134$	23.7926	2.05536
$135$	1.85498	0.159651
$136$	−13.9070	−1.19251
$137$	−4.46839	−0.381760	−0.190880	−	0.981613i	$-0.561134\pi$
$137$	−4.46839	−0.381760	−0.190880	+	0.981613i	$0.561134\pi$
$138$	21.1784	1.80282
$139$	13.6422	1.15712	0.578560	−	0.815640i	$-0.303615\pi$
$139$	13.6422	1.15712	0.578560	+	0.815640i	$0.303615\pi$
$140$	−30.3203	−2.56253
$141$	−2.29467	−0.193246
$142$	21.8665	1.83500
$143$	2.06615	0.172780
$144$	1.84204	0.153503
$145$	5.60407	0.465393
$146$	−37.2436	−3.08230
$147$	−13.4360	−1.10818
$148$	9.37357	0.770503
$149$	6.45712	0.528988	0.264494	−	0.964387i	$-0.414795\pi$
$149$	6.45712	0.528988	0.264494	+	0.964387i	$0.414795\pi$
$150$	−3.69454	−0.301658
$151$	−10.6388	−0.865771	−0.432885	−	0.901449i	$-0.642505\pi$
$151$	−10.6388	−0.865771	−0.432885	+	0.901449i	$0.642505\pi$
$152$	21.1088	1.71215
$153$	3.63212	0.293640
$154$	−64.5477	−5.20140
$155$	19.1436	1.53765
$156$	−1.23988	−0.0992696
$157$	−13.6004	−1.08543	−0.542715	−	0.839917i	$-0.682604\pi$
$157$	−13.6004	−1.08543	−0.542715	+	0.839917i	$0.682604\pi$
$158$	−19.1768	−1.52563
$159$	3.34590	0.265347
$160$	−6.10768	−0.482854
$161$	40.4006	3.18401
$162$	−2.36975	−0.186185
$163$	−12.3680	−0.968737	−0.484369	−	0.874864i	$-0.660951\pi$
$163$	−12.3680	−0.968737	−0.484369	+	0.874864i	$0.660951\pi$
$164$	4.46989	0.349040
$165$	11.1768	0.870116
$166$	−19.3214	−1.49963
$167$	−6.79775	−0.526025	−0.263013	−	0.964792i	$-0.584716\pi$
$167$	−6.79775	−0.526025	−0.263013	+	0.964792i	$0.584716\pi$
$168$	17.3089	1.33541
$169$	−12.8824	−0.990955
$170$	15.9663	1.22456
$171$	−5.51306	−0.421594
$172$	6.28702	0.479381
$173$	−7.54100	−0.573331	−0.286666	−	0.958031i	$-0.592547\pi$
$173$	−7.54100	−0.573331	−0.286666	+	0.958031i	$0.592547\pi$
$174$	−7.15924	−0.542741
$175$	−7.04783	−0.532766
$176$	11.0989	0.836608
$177$	−4.54531	−0.341647
$178$	−31.6619	−2.37316
$179$	24.3578	1.82059	0.910295	−	0.413959i	$-0.135854\pi$
$179$	24.3578	1.82059	0.910295	+	0.413959i	$0.135854\pi$
$180$	−6.70711	−0.499919
$181$	7.58998	0.564159	0.282079	−	0.959391i	$-0.408976\pi$
$181$	7.58998	0.564159	0.282079	+	0.959391i	$0.408976\pi$
$182$	−3.67353	−0.272300
$183$	3.91440	0.289361
$184$	−34.2185	−2.52262
$185$	−4.80894	−0.353560
$186$	−24.4560	−1.79320
$187$	21.8847	1.60037
$188$	8.29689	0.605113
$189$	−4.52062	−0.328827
$190$	−24.2346	−1.75816
$191$	−1.52583	−0.110405	−0.0552025	−	0.998475i	$-0.517580\pi$
$191$	−1.52583	−0.110405	−0.0552025	+	0.998475i	$0.517580\pi$
$192$	11.4867	0.828980
$193$	−20.2154	−1.45513	−0.727567	−	0.686037i	$-0.759349\pi$
$193$	−20.2154	−1.45513	−0.727567	+	0.686037i	$0.759349\pi$
$194$	−34.1120	−2.44910
$195$	0.636096	0.0455518
$196$	48.5810	3.47007
$197$	11.5556	0.823303	0.411651	−	0.911341i	$-0.364952\pi$
$197$	11.5556	0.823303	0.411651	+	0.911341i	$0.364952\pi$
$198$	−14.2785	−1.01473
$199$	3.60189	0.255332	0.127666	−	0.991817i	$-0.459252\pi$
$199$	3.60189	0.255332	0.127666	+	0.991817i	$0.459252\pi$
$200$	5.96938	0.422099
$201$	10.0401	0.708174
$202$	12.8657	0.905226
$203$	−13.6572	−0.958549
$204$	−13.1328	−0.919478
$205$	−2.29319	−0.160164
$206$	−21.4617	−1.49531
$207$	8.93696	0.621162
$208$	0.631657	0.0437976
$209$	−33.2179	−2.29773
$210$	−19.8720	−1.37130
$211$	19.0974	1.31472	0.657361	−	0.753576i	$-0.271673\pi$
$211$	19.0974	1.31472	0.657361	+	0.753576i	$0.271673\pi$
$212$	−12.0979	−0.830885
$213$	9.22735	0.632247
$214$	9.82938	0.671922
$215$	−3.22544	−0.219973
$216$	3.82888	0.260522
$217$	−46.6531	−3.16702
$218$	30.1493	2.04197
$219$	−15.7162	−1.06201
$220$	−40.4124	−2.72461
$221$	1.24550	0.0837813
$222$	6.14345	0.412321
$223$	2.92292	0.195733	0.0978667	−	0.995200i	$-0.468798\pi$
$223$	2.92292	0.195733	0.0978667	+	0.995200i	$0.468798\pi$
$224$	14.8845	0.994513
$225$	−1.55904	−0.103936
$226$	4.37262	0.290862
$227$	10.2014	0.677088	0.338544	−	0.940951i	$-0.390066\pi$
$227$	10.2014	0.677088	0.338544	+	0.940951i	$0.390066\pi$
$228$	19.9337	1.32014
$229$	3.62179	0.239335	0.119667	−	0.992814i	$-0.461817\pi$
$229$	3.62179	0.239335	0.119667	+	0.992814i	$0.461817\pi$
$230$	39.2855	2.59041
$231$	−27.2381	−1.79214
$232$	11.5674	0.759437
$233$	−25.5545	−1.67413	−0.837065	−	0.547104i	$-0.815730\pi$
$233$	−25.5545	−1.67413	−0.837065	+	0.547104i	$0.815730\pi$
$234$	−0.812617	−0.0531224
$235$	−4.25657	−0.277668
$236$	16.4346	1.06980
$237$	−8.09234	−0.525654
$238$	−38.9101	−2.52216
$239$	9.41158	0.608785	0.304392	−	0.952547i	$-0.401547\pi$
$239$	9.41158	0.608785	0.304392	+	0.952547i	$0.401547\pi$
$240$	3.41695	0.220563
$241$	26.1600	1.68511	0.842556	−	0.538609i	$-0.181050\pi$
$241$	26.1600	1.68511	0.842556	+	0.538609i	$0.181050\pi$
$242$	−59.9651	−3.85470
$243$	−1.00000	−0.0641500
$244$	−14.1534	−0.906079
$245$	−24.9236	−1.59231
$246$	2.92957	0.186783
$247$	−1.89050	−0.120289
$248$	39.5143	2.50916
$249$	−8.15333	−0.516697
$250$	−28.8326	−1.82353
$251$	19.3396	1.22070	0.610351	−	0.792131i	$-0.291028\pi$
$251$	19.3396	1.22070	0.610351	+	0.792131i	$0.291028\pi$
$252$	16.3453	1.02966
$253$	53.8480	3.38539
$254$	−22.1174	−1.38777
$255$	6.73753	0.421920
$256$	−25.9275	−1.62047
$257$	4.33810	0.270604	0.135302	−	0.990804i	$-0.456800\pi$
$257$	4.33810	0.270604	0.135302	+	0.990804i	$0.456800\pi$
$258$	4.12052	0.256532
$259$	11.7195	0.728211
$260$	−2.29995	−0.142637
$261$	−3.02109	−0.187001
$262$	−11.6276	−0.718359
$263$	−7.88972	−0.486501	−0.243250	−	0.969964i	$-0.578214\pi$
$263$	−7.88972	−0.486501	−0.243250	+	0.969964i	$0.578214\pi$
$264$	23.0702	1.41987
$265$	6.20659	0.381268
$266$	59.0601	3.62121
$267$	−13.3609	−0.817671
$268$	−36.3023	−2.21751
$269$	17.1402	1.04506	0.522528	−	0.852622i	$-0.324989\pi$
$269$	17.1402	1.04506	0.522528	+	0.852622i	$0.324989\pi$
$270$	−4.39585	−0.267523
$271$	−2.42594	−0.147366	−0.0736828	−	0.997282i	$-0.523475\pi$
$271$	−2.42594	−0.147366	−0.0736828	+	0.997282i	$0.523475\pi$
$272$	6.69052	0.405672
$273$	−1.55018	−0.0938209
$274$	10.5890	0.639703
$275$	−9.39371	−0.566462
$276$	−32.3136	−1.94505
$277$	−18.2386	−1.09585	−0.547924	−	0.836528i	$-0.684582\pi$
$277$	−18.2386	−1.09585	−0.547924	+	0.836528i	$0.684582\pi$
$278$	−32.3288	−1.93895
$279$	−10.3201	−0.617847
$280$	32.1077	1.91880
$281$	3.31214	0.197586	0.0987930	−	0.995108i	$-0.468502\pi$
$281$	3.31214	0.197586	0.0987930	+	0.995108i	$0.468502\pi$
$282$	5.43779	0.323816
$283$	−7.61963	−0.452940	−0.226470	−	0.974018i	$-0.572719\pi$
$283$	−7.61963	−0.452940	−0.226470	+	0.974018i	$0.572719\pi$
$284$	−33.3636	−1.97976
$285$	−10.2266	−0.605773
$286$	−4.89627	−0.289522
$287$	5.58855	0.329882
$288$	3.29258	0.194017
$289$	−3.80767	−0.223981
$290$	−13.2803	−0.779844
$291$	−14.3947	−0.843835
$292$	56.8257	3.32547
$293$	−25.0747	−1.46488	−0.732441	−	0.680830i	$-0.761619\pi$
$293$	−25.0747	−1.46488	−0.732441	+	0.680830i	$0.761619\pi$
$294$	31.8400	1.85695
$295$	−8.43148	−0.490900
$296$	−9.92615	−0.576946
$297$	−6.02531	−0.349624
$298$	−15.3018	−0.886409
$299$	3.06459	0.177230
$300$	5.63707	0.325456
$301$	7.86045	0.453069
$302$	25.2113	1.45074
$303$	5.42912	0.311895
$304$	−10.1553	−0.582445
$305$	7.26114	0.415772
$306$	−8.60724	−0.492043
$307$	−10.0776	−0.575161	−0.287580	−	0.957756i	$-0.592851\pi$
$307$	−10.0776	−0.575161	−0.287580	+	0.957756i	$0.592851\pi$
$308$	98.4858	5.61175
$309$	−9.05652	−0.515207
$310$	−45.3655	−2.57659
$311$	9.21570	0.522574	0.261287	−	0.965261i	$-0.415853\pi$
$311$	9.21570	0.522574	0.261287	+	0.965261i	$0.415853\pi$
$312$	1.31297	0.0743322
$313$	−4.63443	−0.261953	−0.130977	−	0.991385i	$-0.541811\pi$
$313$	−4.63443	−0.261953	−0.130977	+	0.991385i	$0.541811\pi$
$314$	32.2296	1.81882
$315$	−8.38567	−0.472479
$316$	29.2597	1.64599
$317$	−23.2863	−1.30789	−0.653943	−	0.756544i	$-0.726887\pi$
$317$	−23.2863	−1.30789	−0.653943	+	0.756544i	$0.726887\pi$
$318$	−7.92896	−0.444634
$319$	−18.2030	−1.01917
$320$	21.3076	1.19113
$321$	4.14785	0.231510
$322$	−95.7395	−5.33535
$323$	−20.0241	−1.11417
$324$	3.61573	0.200874
$325$	−0.534614	−0.0296550
$326$	29.3091	1.62328
$327$	12.7225	0.703558
$328$	−4.73340	−0.261358
$329$	10.3733	0.571900
$330$	−26.4864	−1.45803
$331$	18.4728	1.01536	0.507678	−	0.861547i	$-0.330504\pi$
$331$	18.4728	1.01536	0.507678	+	0.861547i	$0.330504\pi$
$332$	29.4802	1.61794
$333$	2.59244	0.142065
$334$	16.1090	0.881444
$335$	18.6242	1.01755
$336$	−8.32716	−0.454284
$337$	4.89669	0.266739	0.133370	−	0.991066i	$-0.457420\pi$
$337$	4.89669	0.266739	0.133370	+	0.991066i	$0.457420\pi$
$338$	30.5281	1.66051
$339$	1.84518	0.100216
$340$	−24.3611	−1.32116
$341$	−62.1816	−3.36732
$342$	13.0646	0.706452
$343$	29.0948	1.57097
$344$	−6.65765	−0.358956
$345$	16.5779	0.892524
$346$	17.8703	0.960713
$347$	25.1618	1.35075	0.675377	−	0.737472i	$-0.263981\pi$
$347$	25.1618	1.35075	0.675377	+	0.737472i	$0.263981\pi$
$348$	10.9235	0.585559
$349$	−12.0316	−0.644037	−0.322018	−	0.946733i	$-0.604361\pi$
$349$	−12.0316	−0.644037	−0.322018	+	0.946733i	$0.604361\pi$
$350$	16.7016	0.892739
$351$	−0.342912	−0.0183033
$352$	19.8388	1.05741
$353$	10.2157	0.543726	0.271863	−	0.962336i	$-0.412360\pi$
$353$	10.2157	0.543726	0.271863	+	0.962336i	$0.412360\pi$
$354$	10.7713	0.572487
$355$	17.1166	0.908453
$356$	48.3093	2.56039
$357$	−16.4195	−0.869010
$358$	−57.7221	−3.05071
$359$	−21.1664	−1.11712	−0.558559	−	0.829465i	$-0.688646\pi$
$359$	−21.1664	−1.11712	−0.558559	+	0.829465i	$0.688646\pi$
$360$	7.10250	0.374335
$361$	11.3938	0.599676
$362$	−17.9864	−0.945343
$363$	−25.3044	−1.32813
$364$	5.60502	0.293783
$365$	−29.1534	−1.52596
$366$	−9.27616	−0.484873
$367$	7.07985	0.369565	0.184783	−	0.982779i	$-0.440842\pi$
$367$	7.07985	0.369565	0.184783	+	0.982779i	$0.440842\pi$
$368$	16.4622	0.858153
$369$	1.23624	0.0643558
$370$	11.3960	0.592449
$371$	−15.1256	−0.785280
$372$	37.3146	1.93467
$373$	25.0635	1.29774	0.648870	−	0.760899i	$-0.275242\pi$
$373$	25.0635	1.29774	0.648870	+	0.760899i	$0.275242\pi$
$374$	−51.8613	−2.68168
$375$	−12.1669	−0.628296
$376$	−8.78600	−0.453104
$377$	−1.03597	−0.0533551
$378$	10.7128	0.551005
$379$	−4.13491	−0.212396	−0.106198	−	0.994345i	$-0.533868\pi$
$379$	−4.13491	−0.212396	−0.106198	+	0.994345i	$0.533868\pi$
$380$	36.9767	1.89687
$381$	−9.33323	−0.478156
$382$	3.61584	0.185002
$383$	3.54609	0.181197	0.0905984	−	0.995888i	$-0.471122\pi$
$383$	3.54609	0.181197	0.0905984	+	0.995888i	$0.471122\pi$
$384$	−20.6354	−1.05305
$385$	−50.5263	−2.57506
$386$	47.9054	2.43832
$387$	1.73880	0.0883881
$388$	52.0475	2.64231
$389$	−36.7037	−1.86095	−0.930475	−	0.366354i	$-0.880606\pi$
$389$	−36.7037	−1.86095	−0.930475	+	0.366354i	$0.880606\pi$
$390$	−1.50739	−0.0763296
$391$	32.4601	1.64158
$392$	−51.4449	−2.59836
$393$	−4.90669	−0.247510
$394$	−27.3839	−1.37958
$395$	−15.0111	−0.755292
$396$	21.7859	1.09478
$397$	22.8813	1.14838	0.574189	−	0.818723i	$-0.305318\pi$
$397$	22.8813	1.14838	0.574189	+	0.818723i	$0.305318\pi$
$398$	−8.53560	−0.427851
$399$	24.9225	1.24768
$400$	−2.87181	−0.143591
$401$	−32.1612	−1.60605	−0.803026	−	0.595944i	$-0.796778\pi$
$401$	−32.1612	−1.60605	−0.803026	+	0.595944i	$0.796778\pi$
$402$	−23.7926	−1.18666
$403$	−3.53888	−0.176284
$404$	−19.6302	−0.976640
$405$	−1.85498	−0.0921748
$406$	32.3642	1.60621
$407$	15.6203	0.774268
$408$	13.9070	0.688497
$409$	32.2052	1.59245	0.796224	−	0.605003i	$-0.206828\pi$
$409$	32.2052	1.59245	0.796224	+	0.605003i	$0.206828\pi$
$410$	5.43431	0.268381
$411$	4.46839	0.220409
$412$	32.7459	1.61328
$413$	20.5476	1.01108
$414$	−21.1784	−1.04086
$415$	−15.1243	−0.742422
$416$	1.12907	0.0553570
$417$	−13.6422	−0.668064
$418$	78.7182	3.85023
$419$	−3.21440	−0.157034	−0.0785168	−	0.996913i	$-0.525018\pi$
$419$	−3.21440	−0.157034	−0.0785168	+	0.996913i	$0.525018\pi$
$420$	30.3203	1.47948
$421$	−39.0301	−1.90221	−0.951106	−	0.308865i	$-0.900051\pi$
$421$	−39.0301	−1.90221	−0.951106	+	0.308865i	$0.900051\pi$
$422$	−45.2562	−2.20304
$423$	2.29467	0.111571
$424$	12.8111	0.622160
$425$	−5.66263	−0.274678
$426$	−21.8665	−1.05944
$427$	−17.6955	−0.856346
$428$	−14.9975	−0.724931
$429$	−2.06615	−0.0997547
$430$	7.64349	0.368602
$431$	29.6407	1.42774	0.713872	−	0.700276i	$-0.246940\pi$
$431$	29.6407	1.42774	0.713872	+	0.700276i	$0.246940\pi$
$432$	−1.84204	−0.0886252
$433$	−1.20181	−0.0577555	−0.0288778	−	0.999583i	$-0.509193\pi$
$433$	−1.20181	−0.0577555	−0.0288778	+	0.999583i	$0.509193\pi$
$434$	110.556	5.30688
$435$	−5.60407	−0.268695
$436$	−46.0013	−2.20306
$437$	−49.2700	−2.35690
$438$	37.2436	1.77957
$439$	−33.7640	−1.61147	−0.805733	−	0.592279i	$-0.798228\pi$
$439$	−33.7640	−1.61147	−0.805733	+	0.592279i	$0.798228\pi$
$440$	42.7948	2.04016
$441$	13.4360	0.639810
$442$	−2.95153	−0.140390
$443$	21.6130	1.02686	0.513431	−	0.858131i	$-0.328374\pi$
$443$	21.6130	1.02686	0.513431	+	0.858131i	$0.328374\pi$
$444$	−9.37357	−0.444850
$445$	−24.7842	−1.17488
$446$	−6.92661	−0.327984
$447$	−6.45712	−0.305411
$448$	−51.9269	−2.45332
$449$	38.6688	1.82489	0.912447	−	0.409194i	$-0.134190\pi$
$449$	38.6688	1.82489	0.912447	+	0.409194i	$0.134190\pi$
$450$	3.69454	0.174162
$451$	7.44870	0.350746
$452$	−6.67167	−0.313809
$453$	10.6388	0.499853
$454$	−24.1747	−1.13457
$455$	−2.87555	−0.134808
$456$	−21.1088	−0.988512
$457$	17.8222	0.833687	0.416843	−	0.908978i	$-0.363136\pi$
$457$	17.8222	0.833687	0.416843	+	0.908978i	$0.363136\pi$
$458$	−8.58275	−0.401045
$459$	−3.63212	−0.169533
$460$	−59.9412	−2.79477
$461$	−10.9963	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$461$	−10.9963	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$462$	64.5477	3.00303
$463$	9.77947	0.454490	0.227245	−	0.973838i	$-0.427028\pi$
$463$	9.77947	0.454490	0.227245	+	0.973838i	$0.427028\pi$
$464$	−5.56497	−0.258347
$465$	−19.1436	−0.887761
$466$	60.5578	2.80529
$467$	−21.3322	−0.987136	−0.493568	−	0.869707i	$-0.664308\pi$
$467$	−21.3322	−0.987136	−0.493568	+	0.869707i	$0.664308\pi$
$468$	1.23988	0.0573133
$469$	−45.3875	−2.09580
$470$	10.0870	0.465279
$471$	13.6004	0.626674
$472$	−17.4035	−0.801059
$473$	10.4768	0.481724
$474$	19.1768	0.880821
$475$	8.59509	0.394370
$476$	59.3683	2.72114
$477$	−3.34590	−0.153198
$478$	−22.3031	−1.02012
$479$	42.6959	1.95082	0.975412	−	0.220387i	$-0.0707321\pi$
$479$	42.6959	1.95082	0.975412	+	0.220387i	$0.0707321\pi$
$480$	6.10768	0.278776
$481$	0.888980	0.0405340
$482$	−61.9926	−2.82369
$483$	−40.4006	−1.83829
$484$	91.4937	4.15881
$485$	−26.7020	−1.21248
$486$	2.36975	0.107494
$487$	2.79653	0.126723	0.0633615	−	0.997991i	$-0.479818\pi$
$487$	2.79653	0.126723	0.0633615	+	0.997991i	$0.479818\pi$
$488$	14.9878	0.678464
$489$	12.3680	0.559301
$490$	59.0627	2.66818
$491$	42.0588	1.89809	0.949044	−	0.315143i	$-0.102053\pi$
$491$	42.0588	1.89809	0.949044	+	0.315143i	$0.102053\pi$
$492$	−4.46989	−0.201518
$493$	−10.9730	−0.494198
$494$	4.48001	0.201565
$495$	−11.1768	−0.502362
$496$	−19.0100	−0.853573
$497$	−41.7133	−1.87110
$498$	19.3214	0.865812
$499$	7.23418	0.323846	0.161923	−	0.986803i	$-0.448230\pi$
$499$	7.23418	0.323846	0.161923	+	0.986803i	$0.448230\pi$
$500$	43.9922	1.96739
$501$	6.79775	0.303701
$502$	−45.8300	−2.04549
$503$	6.91375	0.308269	0.154134	−	0.988050i	$-0.450741\pi$
$503$	6.91375	0.308269	0.154134	+	0.988050i	$0.450741\pi$
$504$	−17.3089	−0.771000
$505$	10.0709	0.448150
$506$	−127.606	−5.67279
$507$	12.8824	0.572128
$508$	33.7464	1.49726
$509$	−12.9405	−0.573579	−0.286790	−	0.957994i	$-0.592588\pi$
$509$	−12.9405	−0.573579	−0.286790	+	0.957994i	$0.592588\pi$
$510$	−15.9663	−0.706998
$511$	71.0472	3.14294
$512$	20.1710	0.891439
$513$	5.51306	0.243408
$514$	−10.2802	−0.453442
$515$	−16.7997	−0.740282
$516$	−6.28702	−0.276771
$517$	13.8261	0.608070
$518$	−27.7722	−1.22024
$519$	7.54100	0.331013
$520$	2.43553	0.106805
$521$	19.8682	0.870442	0.435221	−	0.900324i	$-0.356670\pi$
$521$	19.8682	0.870442	0.435221	+	0.900324i	$0.356670\pi$
$522$	7.15924	0.313352
$523$	−30.3338	−1.32641	−0.663203	−	0.748440i	$-0.730803\pi$
$523$	−30.3338	−1.32641	−0.663203	+	0.748440i	$0.730803\pi$
$524$	17.7413	0.775031
$525$	7.04783	0.307593
$526$	18.6967	0.815214
$527$	−37.4838	−1.63282
$528$	−11.0989	−0.483016
$529$	56.8693	2.47258
$530$	−14.7081	−0.638878
$531$	4.54531	0.197250
$532$	−90.1129	−3.90689
$533$	0.423920	0.0183620
$534$	31.6619	1.37015
$535$	7.69419	0.332649
$536$	38.4423	1.66046
$537$	−24.3578	−1.05112
$538$	−40.6180	−1.75117
$539$	80.9562	3.48703
$540$	6.70711	0.288628
$541$	−27.2465	−1.17142	−0.585709	−	0.810522i	$-0.699184\pi$
$541$	−27.2465	−1.17142	−0.585709	+	0.810522i	$0.699184\pi$
$542$	5.74888	0.246936
$543$	−7.58998	−0.325717
$544$	11.9591	0.512741
$545$	23.6001	1.01092
$546$	3.67353	0.157213
$547$	−4.73892	−0.202622	−0.101311	−	0.994855i	$-0.532304\pi$
$547$	−4.73892	−0.202622	−0.101311	+	0.994855i	$0.532304\pi$
$548$	−16.1565	−0.690171
$549$	−3.91440	−0.167063
$550$	22.2608	0.949202
$551$	16.6555	0.709547
$552$	34.2185	1.45644
$553$	36.5824	1.55564
$554$	43.2209	1.83628
$555$	4.80894	0.204128
$556$	49.3267	2.09192
$557$	−17.6027	−0.745850	−0.372925	−	0.927862i	$-0.621645\pi$
$557$	−17.6027	−0.745850	−0.372925	+	0.927862i	$0.621645\pi$
$558$	24.4560	1.03531
$559$	0.596255	0.0252189
$560$	−15.4467	−0.652744
$561$	−21.8847	−0.923972
$562$	−7.84896	−0.331089
$563$	21.3586	0.900156	0.450078	−	0.892989i	$-0.351396\pi$
$563$	21.3586	0.900156	0.450078	+	0.892989i	$0.351396\pi$
$564$	−8.29689	−0.349362
$565$	3.42278	0.143997
$566$	18.0566	0.758977
$567$	4.52062	0.189848
$568$	35.3304	1.48243
$569$	−26.8414	−1.12525	−0.562625	−	0.826713i	$-0.690208\pi$
$569$	−26.8414	−1.12525	−0.562625	+	0.826713i	$0.690208\pi$
$570$	24.2346	1.01508
$571$	14.9810	0.626937	0.313468	−	0.949599i	$-0.398509\pi$
$571$	14.9810	0.626937	0.313468	+	0.949599i	$0.398509\pi$
$572$	7.47064	0.312363
$573$	1.52583	0.0637424
$574$	−13.2435	−0.552773
$575$	−13.9331	−0.581050
$576$	−11.4867	−0.478612
$577$	−28.8786	−1.20223	−0.601116	−	0.799161i	$-0.705277\pi$
$577$	−28.8786	−1.20223	−0.601116	+	0.799161i	$0.705277\pi$
$578$	9.02324	0.375317
$579$	20.2154	0.840122
$580$	20.2628	0.841367
$581$	36.8581	1.52913
$582$	34.1120	1.41399
$583$	−20.1601	−0.834946
$584$	−60.1756	−2.49008
$585$	−0.636096	−0.0262993
$586$	59.4210	2.45466
$587$	−39.9095	−1.64724	−0.823621	−	0.567141i	$-0.808050\pi$
$587$	−39.9095	−1.64724	−0.823621	+	0.567141i	$0.808050\pi$
$588$	−48.5810	−2.00345
$589$	56.8952	2.34433
$590$	19.9805	0.822585
$591$	−11.5556	−0.475334
$592$	4.77538	0.196267
$593$	13.3346	0.547588	0.273794	−	0.961788i	$-0.411721\pi$
$593$	13.3346	0.547588	0.273794	+	0.961788i	$0.411721\pi$
$594$	14.2785	0.585854
$595$	−30.4578	−1.24865
$596$	23.3472	0.956339
$597$	−3.60189	−0.147416
$598$	−7.26232	−0.296979
$599$	−13.8971	−0.567821	−0.283911	−	0.958851i	$-0.591632\pi$
$599$	−13.8971	−0.567821	−0.283911	+	0.958851i	$0.591632\pi$
$600$	−5.96938	−0.243699
$601$	17.6534	0.720098	0.360049	−	0.932933i	$-0.382760\pi$
$601$	17.6534	0.720098	0.360049	+	0.932933i	$0.382760\pi$
$602$	−18.6273	−0.759193
$603$	−10.0401	−0.408864
$604$	−38.4669	−1.56520
$605$	−46.9391	−1.90835
$606$	−12.8657	−0.522632
$607$	12.8426	0.521266	0.260633	−	0.965438i	$-0.416069\pi$
$607$	12.8426	0.521266	0.260633	+	0.965438i	$0.416069\pi$
$608$	−18.1522	−0.736169
$609$	13.6572	0.553418
$610$	−17.2071	−0.696696
$611$	0.786869	0.0318333
$612$	13.1328	0.530861
$613$	36.7108	1.48273	0.741367	−	0.671099i	$-0.234178\pi$
$613$	36.7108	1.48273	0.741367	+	0.671099i	$0.234178\pi$
$614$	23.8815	0.963779
$615$	2.29319	0.0924705
$616$	−104.292	−4.20203
$617$	23.9574	0.964490	0.482245	−	0.876036i	$-0.339821\pi$
$617$	23.9574	0.964490	0.482245	+	0.876036i	$0.339821\pi$
$618$	21.4617	0.863317
$619$	−21.5631	−0.866694	−0.433347	−	0.901227i	$-0.642668\pi$
$619$	−21.5631	−0.866694	−0.433347	+	0.901227i	$0.642668\pi$
$620$	69.2179	2.77986
$621$	−8.93696	−0.358628
$622$	−21.8389	−0.875661
$623$	60.3994	2.41985
$624$	−0.631657	−0.0252865
$625$	−14.7742	−0.590967
$626$	10.9824	0.438947
$627$	33.2179	1.32660
$628$	−49.1754	−1.96231
$629$	9.41608	0.375443
$630$	19.8720	0.791718
$631$	33.7872	1.34505	0.672524	−	0.740076i	$-0.265210\pi$
$631$	33.7872	1.34505	0.672524	+	0.740076i	$0.265210\pi$
$632$	−30.9846	−1.23250
$633$	−19.0974	−0.759055
$634$	55.1827	2.19158
$635$	−17.3130	−0.687044
$636$	12.0979	0.479712
$637$	4.60737	0.182551
$638$	43.1367	1.70780
$639$	−9.22735	−0.365028
$640$	−38.2784	−1.51309
$641$	4.40611	0.174031	0.0870155	−	0.996207i	$-0.472267\pi$
$641$	4.40611	0.174031	0.0870155	+	0.996207i	$0.472267\pi$
$642$	−9.82938	−0.387935
$643$	1.25672	0.0495602	0.0247801	−	0.999693i	$-0.492111\pi$
$643$	1.25672	0.0495602	0.0247801	+	0.999693i	$0.492111\pi$
$644$	146.078	5.75627
$645$	3.22544	0.127002
$646$	47.4522	1.86698
$647$	−7.53778	−0.296341	−0.148170	−	0.988962i	$-0.547338\pi$
$647$	−7.53778	−0.296341	−0.148170	+	0.988962i	$0.547338\pi$
$648$	−3.82888	−0.150413
$649$	27.3869	1.07503
$650$	1.26690	0.0496920
$651$	46.6531	1.82848
$652$	−44.7194	−1.75135
$653$	23.4990	0.919585	0.459793	−	0.888026i	$-0.347924\pi$
$653$	23.4990	0.919585	0.459793	+	0.888026i	$0.347924\pi$
$654$	−30.1493	−1.17893
$655$	−9.10183	−0.355638
$656$	2.27719	0.0889095
$657$	15.7162	0.613149
$658$	−24.5822	−0.958314
$659$	−44.6342	−1.73870	−0.869350	−	0.494196i	$-0.835462\pi$
$659$	−44.6342	−1.73870	−0.869350	+	0.494196i	$0.835462\pi$
$660$	40.4124	1.57305
$661$	1.76954	0.0688271	0.0344135	−	0.999408i	$-0.489044\pi$
$661$	1.76954	0.0688271	0.0344135	+	0.999408i	$0.489044\pi$
$662$	−43.7759	−1.70140
$663$	−1.24550	−0.0483712
$664$	−31.2181	−1.21150
$665$	46.2307	1.79275
$666$	−6.14345	−0.238054
$667$	−26.9994	−1.04542
$668$	−24.5788	−0.950983
$669$	−2.92292	−0.113007
$670$	−44.1348	−1.70507
$671$	−23.5855	−0.910507
$672$	−14.8845	−0.574182
$673$	1.32364	0.0510225	0.0255113	−	0.999675i	$-0.491879\pi$
$673$	1.32364	0.0510225	0.0255113	+	0.999675i	$0.491879\pi$
$674$	−11.6039	−0.446967
$675$	1.55904	0.0600075
$676$	−46.5793	−1.79151
$677$	−26.6555	−1.02446	−0.512228	−	0.858850i	$-0.671180\pi$
$677$	−26.6555	−1.02446	−0.512228	+	0.858850i	$0.671180\pi$
$678$	−4.37262	−0.167929
$679$	65.0732	2.49728
$680$	25.7972	0.989276
$681$	−10.2014	−0.390917
$682$	147.355	5.64252
$683$	20.1718	0.771854	0.385927	−	0.922529i	$-0.373882\pi$
$683$	20.1718	0.771854	0.385927	+	0.922529i	$0.373882\pi$
$684$	−19.9337	−0.762186
$685$	8.28878	0.316698
$686$	−68.9475	−2.63243
$687$	−3.62179	−0.138180
$688$	3.20293	0.122111
$689$	−1.14735	−0.0437106
$690$	−39.2855	−1.49557
$691$	−50.3443	−1.91519	−0.957594	−	0.288122i	$-0.906969\pi$
$691$	−50.3443	−1.91519	−0.957594	+	0.288122i	$0.906969\pi$
$692$	−27.2662	−1.03651
$693$	27.2381	1.03469
$694$	−59.6272	−2.26342
$695$	−25.3061	−0.959916
$696$	−11.5674	−0.438461
$697$	4.49016	0.170077
$698$	28.5119	1.07919
$699$	25.5545	0.966559
$700$	−25.4831	−0.963169
$701$	−22.0420	−0.832513	−0.416257	−	0.909247i	$-0.636658\pi$
$701$	−22.0420	−0.832513	−0.416257	+	0.909247i	$0.636658\pi$
$702$	0.812617	0.0306702
$703$	−14.2923	−0.539044
$704$	−69.2108	−2.60848
$705$	4.25657	0.160312
$706$	−24.2086	−0.911104
$707$	−24.5430	−0.923034
$708$	−16.4346	−0.617651
$709$	−12.1771	−0.457322	−0.228661	−	0.973506i	$-0.573435\pi$
$709$	−12.1771	−0.457322	−0.228661	+	0.973506i	$0.573435\pi$
$710$	−40.5620	−1.52227
$711$	8.09234	0.303486
$712$	−51.1571	−1.91719
$713$	−92.2301	−3.45404
$714$	38.9101	1.45617
$715$	−3.83267	−0.143334
$716$	88.0714	3.29138
$717$	−9.41158	−0.351482
$718$	50.1590	1.87192
$719$	4.72589	0.176246	0.0881229	−	0.996110i	$-0.471913\pi$
$719$	4.72589	0.176246	0.0881229	+	0.996110i	$0.471913\pi$
$720$	−3.41695	−0.127342
$721$	40.9411	1.52473
$722$	−27.0006	−1.00486
$723$	−26.1600	−0.972899
$724$	27.4433	1.01992
$725$	4.71001	0.174925
$726$	59.9651	2.22551
$727$	−1.39753	−0.0518315	−0.0259157	−	0.999664i	$-0.508250\pi$
$727$	−1.39753	−0.0518315	−0.0259157	+	0.999664i	$0.508250\pi$
$728$	−5.93543	−0.219982
$729$	1.00000	0.0370370
$730$	69.0862	2.55700
$731$	6.31553	0.233588
$732$	14.1534	0.523125
$733$	5.67222	0.209508	0.104754	−	0.994498i	$-0.466594\pi$
$733$	5.67222	0.209508	0.104754	+	0.994498i	$0.466594\pi$
$734$	−16.7775	−0.619269
$735$	24.9236	0.919320
$736$	29.4257	1.08464
$737$	−60.4947	−2.22835
$738$	−2.92957	−0.107839
$739$	−33.8376	−1.24474	−0.622369	−	0.782724i	$-0.713829\pi$
$739$	−33.8376	−1.24474	−0.622369	+	0.782724i	$0.713829\pi$
$740$	−17.3878	−0.639189
$741$	1.89050	0.0694491
$742$	35.8438	1.31587
$743$	−32.7649	−1.20203	−0.601014	−	0.799238i	$-0.705236\pi$
$743$	−32.7649	−1.20203	−0.601014	+	0.799238i	$0.705236\pi$
$744$	−39.5143	−1.44867
$745$	−11.9778	−0.438834
$746$	−59.3944	−2.17458
$747$	8.15333	0.298315
$748$	79.1291	2.89325
$749$	−18.7509	−0.685141
$750$	28.8326	1.05282
$751$	9.86334	0.359919	0.179959	−	0.983674i	$-0.442403\pi$
$751$	9.86334	0.359919	0.179959	+	0.983674i	$0.442403\pi$
$752$	4.22687	0.154138
$753$	−19.3396	−0.704773
$754$	2.45499	0.0894055
$755$	19.7347	0.718220
$756$	−16.3453	−0.594474
$757$	−12.1818	−0.442756	−0.221378	−	0.975188i	$-0.571055\pi$
$757$	−12.1818	−0.442756	−0.221378	+	0.975188i	$0.571055\pi$
$758$	9.79873	0.355906
$759$	−53.8480	−1.95456
$760$	−39.1565	−1.42036
$761$	4.36379	0.158187	0.0790936	−	0.996867i	$-0.474797\pi$
$761$	4.36379	0.158187	0.0790936	+	0.996867i	$0.474797\pi$
$762$	22.1174	0.801231
$763$	−57.5138	−2.08214
$764$	−5.51698	−0.199597
$765$	−6.73753	−0.243596
$766$	−8.40336	−0.303626
$767$	1.55864	0.0562793
$768$	25.9275	0.935579
$769$	13.6366	0.491749	0.245874	−	0.969302i	$-0.420925\pi$
$769$	13.6366	0.491749	0.245874	+	0.969302i	$0.420925\pi$
$770$	119.735	4.31494
$771$	−4.33810	−0.156233
$772$	−73.0933	−2.63069
$773$	24.7297	0.889464	0.444732	−	0.895664i	$-0.353299\pi$
$773$	24.7297	0.889464	0.444732	+	0.895664i	$0.353299\pi$
$774$	−4.12052	−0.148109
$775$	16.0894	0.577949
$776$	−55.1158	−1.97854
$777$	−11.7195	−0.420433
$778$	86.9787	3.11834
$779$	−6.81544	−0.244189
$780$	2.29995	0.0823514
$781$	−55.5976	−1.98944
$782$	−76.9225	−2.75074
$783$	3.02109	0.107965
$784$	24.7497	0.883917
$785$	25.2285	0.900444
$786$	11.6276	0.414744
$787$	−44.7389	−1.59477	−0.797384	−	0.603472i	$-0.793783\pi$
$787$	−44.7389	−1.59477	−0.797384	+	0.603472i	$0.793783\pi$
$788$	41.7819	1.48842
$789$	7.88972	0.280881
$790$	35.5727	1.26562
$791$	−8.34136	−0.296585
$792$	−23.0702	−0.819763
$793$	−1.34229	−0.0476663
$794$	−54.2229	−1.92430
$795$	−6.20659	−0.220125
$796$	13.0235	0.461605
$797$	−46.0791	−1.63220	−0.816102	−	0.577908i	$-0.803869\pi$
$797$	−46.0791	−1.63220	−0.816102	+	0.577908i	$0.803869\pi$
$798$	−59.0601	−2.09070
$799$	8.33452	0.294854
$800$	−5.13327	−0.181488
$801$	13.3609	0.472083
$802$	76.2140	2.69121
$803$	94.6952	3.34172
$804$	36.3023	1.28028
$805$	−74.9424	−2.64137
$806$	8.38627	0.295394
$807$	−17.1402	−0.603363
$808$	20.7874	0.731300
$809$	1.37361	0.0482937	0.0241468	−	0.999708i	$-0.492313\pi$
$809$	1.37361	0.0482937	0.0241468	+	0.999708i	$0.492313\pi$
$810$	4.39585	0.154454
$811$	−7.28086	−0.255666	−0.127833	−	0.991796i	$-0.540802\pi$
$811$	−7.28086	−0.255666	−0.127833	+	0.991796i	$0.540802\pi$
$812$	−49.3808	−1.73293
$813$	2.42594	0.0850815
$814$	−37.0162	−1.29742
$815$	22.9424	0.803639
$816$	−6.69052	−0.234215
$817$	−9.58610	−0.335375
$818$	−76.3185	−2.66841
$819$	1.55018	0.0541675
$820$	−8.29157	−0.289554
$821$	11.3553	0.396301	0.198151	−	0.980172i	$-0.436506\pi$
$821$	11.3553	0.396301	0.198151	+	0.980172i	$0.436506\pi$
$822$	−10.5890	−0.369333
$823$	18.3850	0.640860	0.320430	−	0.947272i	$-0.396173\pi$
$823$	18.3850	0.640860	0.320430	+	0.947272i	$0.396173\pi$
$824$	−34.6763	−1.20801
$825$	9.39371	0.327047
$826$	−48.6928	−1.69424
$827$	39.9348	1.38867	0.694334	−	0.719652i	$-0.255699\pi$
$827$	39.9348	1.38867	0.694334	+	0.719652i	$0.255699\pi$
$828$	32.3136	1.12298
$829$	25.1979	0.875160	0.437580	−	0.899179i	$-0.355836\pi$
$829$	25.1979	0.875160	0.437580	+	0.899179i	$0.355836\pi$
$830$	35.8408	1.24405
$831$	18.2386	0.632688
$832$	−3.93892	−0.136558
$833$	48.8013	1.69086
$834$	32.3288	1.11945
$835$	12.6097	0.436377
$836$	−120.107	−4.15399
$837$	10.3201	0.356714
$838$	7.61733	0.263136
$839$	17.2282	0.594785	0.297392	−	0.954755i	$-0.403883\pi$
$839$	17.2282	0.594785	0.297392	+	0.954755i	$0.403883\pi$
$840$	−32.1077	−1.10782
$841$	−19.8730	−0.685276
$842$	92.4917	3.18748
$843$	−3.31214	−0.114076
$844$	69.0512	2.37684
$845$	23.8966	0.822070
$846$	−5.43779	−0.186955
$847$	114.391	3.93054
$848$	−6.16328	−0.211648
$849$	7.61963	0.261505
$850$	13.4190	0.460269
$851$	23.1686	0.794208
$852$	33.3636	1.14302
$853$	48.4066	1.65741	0.828705	−	0.559685i	$-0.189078\pi$
$853$	48.4066	1.65741	0.828705	+	0.559685i	$0.189078\pi$
$854$	41.9340	1.43495
$855$	10.2266	0.349743
$856$	15.8816	0.542822
$857$	22.3685	0.764094	0.382047	−	0.924143i	$-0.375219\pi$
$857$	22.3685	0.764094	0.382047	+	0.924143i	$0.375219\pi$
$858$	4.89627	0.167156
$859$	22.8945	0.781149	0.390575	−	0.920571i	$-0.372276\pi$
$859$	22.8945	0.781149	0.390575	+	0.920571i	$0.372276\pi$
$860$	−11.6623	−0.397682
$861$	−5.58855	−0.190457
$862$	−70.2412	−2.39243
$863$	36.0271	1.22638	0.613188	−	0.789937i	$-0.289887\pi$
$863$	36.0271	1.22638	0.613188	+	0.789937i	$0.289887\pi$
$864$	−3.29258	−0.112016
$865$	13.9884	0.475620
$866$	2.84800	0.0967791
$867$	3.80767	0.129315
$868$	−168.685	−5.72555
$869$	48.7588	1.65403
$870$	13.2803	0.450243
$871$	−3.44287	−0.116657
$872$	48.7131	1.64963
$873$	14.3947	0.487188
$874$	116.758	3.94939
$875$	55.0020	1.85941
$876$	−56.8257	−1.91996
$877$	5.75335	0.194277	0.0971384	−	0.995271i	$-0.469031\pi$
$877$	5.75335	0.194277	0.0971384	+	0.995271i	$0.469031\pi$
$878$	80.0122	2.70028
$879$	25.0747	0.845750
$880$	−20.5882	−0.694028
$881$	−22.1816	−0.747316	−0.373658	−	0.927566i	$-0.621897\pi$
$881$	−22.1816	−0.747316	−0.373658	+	0.927566i	$0.621897\pi$
$882$	−31.8400	−1.07211
$883$	41.7781	1.40595	0.702973	−	0.711216i	$-0.251855\pi$
$883$	41.7781	1.40595	0.702973	+	0.711216i	$0.251855\pi$
$884$	4.50339	0.151465
$885$	8.43148	0.283421
$886$	−51.2174	−1.72068
$887$	−20.2605	−0.680281	−0.340140	−	0.940375i	$-0.610475\pi$
$887$	−20.2605	−0.680281	−0.340140	+	0.940375i	$0.610475\pi$
$888$	9.92615	0.333100
$889$	42.1920	1.41507
$890$	58.7323	1.96871
$891$	6.02531	0.201855
$892$	10.5685	0.353859
$893$	−12.6506	−0.423338
$894$	15.3018	0.511768
$895$	−45.1834	−1.51031
$896$	93.2850	3.11643
$897$	−3.06459	−0.102324
$898$	−91.6356	−3.05792
$899$	31.1779	1.03984
$900$	−5.63707	−0.187902
$901$	−12.1527	−0.404866
$902$	−17.6516	−0.587733
$903$	−7.86045	−0.261579
$904$	7.06497	0.234977
$905$	−14.0793	−0.468011
$906$	−25.2113	−0.837588
$907$	37.5160	1.24570	0.622849	−	0.782342i	$-0.285975\pi$
$907$	37.5160	1.24570	0.622849	+	0.782342i	$0.285975\pi$
$908$	36.8853	1.22408
$909$	−5.42912	−0.180073
$910$	6.81434	0.225893
$911$	9.77989	0.324022	0.162011	−	0.986789i	$-0.448202\pi$
$911$	9.77989	0.324022	0.162011	+	0.986789i	$0.448202\pi$
$912$	10.1553	0.336275
$913$	49.1264	1.62585
$914$	−42.2342	−1.39698
$915$	−7.26114	−0.240046
$916$	13.0954	0.432684
$917$	22.1813	0.732491
$918$	8.60724	0.284081
$919$	31.6565	1.04425	0.522125	−	0.852869i	$-0.325139\pi$
$919$	31.6565	1.04425	0.522125	+	0.852869i	$0.325139\pi$
$920$	63.4748	2.09270
$921$	10.0776	0.332069
$922$	26.0585	0.858190
$923$	−3.16417	−0.104150
$924$	−98.4858	−3.23994
$925$	−4.04172	−0.132891
$926$	−23.1749	−0.761575
$927$	9.05652	0.297455
$928$	−9.94719	−0.326533
$929$	−31.1754	−1.02283	−0.511416	−	0.859333i	$-0.670879\pi$
$929$	−31.1754	−1.02283	−0.511416	+	0.859333i	$0.670879\pi$
$930$	45.3655	1.48759
$931$	−74.0736	−2.42766
$932$	−92.3981	−3.02660
$933$	−9.21570	−0.301708
$934$	50.5520	1.65411
$935$	−40.5957	−1.32762
$936$	−1.31297	−0.0429157
$937$	24.0352	0.785197	0.392599	−	0.919710i	$-0.371576\pi$
$937$	24.0352	0.785197	0.392599	+	0.919710i	$0.371576\pi$
$938$	107.557	3.51186
$939$	4.63443	0.151239
$940$	−15.3906	−0.501986
$941$	53.4068	1.74101	0.870506	−	0.492158i	$-0.163792\pi$
$941$	53.4068	1.74101	0.870506	+	0.492158i	$0.163792\pi$
$942$	−32.2296	−1.05010
$943$	11.0482	0.359778
$944$	8.37265	0.272506
$945$	8.38567	0.272786
$946$	−24.8274	−0.807209
$947$	6.46099	0.209954	0.104977	−	0.994475i	$-0.466523\pi$
$947$	6.46099	0.209954	0.104977	+	0.994475i	$0.466523\pi$
$948$	−29.2597	−0.950311
$949$	5.38929	0.174944
$950$	−20.3682	−0.660833
$951$	23.2863	0.755109
$952$	−62.8681	−2.03757
$953$	36.9703	1.19758	0.598792	−	0.800905i	$-0.295648\pi$
$953$	36.9703	1.19758	0.598792	+	0.800905i	$0.295648\pi$
$954$	7.92896	0.256710
$955$	2.83038	0.0915891
$956$	34.0297	1.10060
$957$	18.2030	0.588420
$958$	−101.179	−3.26894
$959$	−20.1999	−0.652288
$960$	−21.3076	−0.687699
$961$	75.5039	2.43561
$962$	−2.10666	−0.0679215
$963$	−4.14785	−0.133663
$964$	94.5873	3.04645
$965$	37.4991	1.20714
$966$	95.7395	3.08037
$967$	−17.5110	−0.563115	−0.281557	−	0.959544i	$-0.590851\pi$
$967$	−17.5110	−0.563115	−0.281557	+	0.959544i	$0.590851\pi$
$968$	−96.8873	−3.11408
$969$	20.0241	0.643268
$970$	63.2771	2.03171
$971$	17.7790	0.570555	0.285278	−	0.958445i	$-0.407914\pi$
$971$	17.7790	0.570555	0.285278	+	0.958445i	$0.407914\pi$
$972$	−3.61573	−0.115975
$973$	61.6714	1.97710
$974$	−6.62710	−0.212346
$975$	0.534614	0.0171213
$976$	−7.21048	−0.230802
$977$	33.4786	1.07107	0.535537	−	0.844512i	$-0.320109\pi$
$977$	33.4786	1.07107	0.535537	+	0.844512i	$0.320109\pi$
$978$	−29.3091	−0.937203
$979$	80.5033	2.57290
$980$	−90.1169	−2.87868
$981$	−12.7225	−0.406200
$982$	−99.6690	−3.18057
$983$	17.0746	0.544595	0.272297	−	0.962213i	$-0.412217\pi$
$983$	17.0746	0.544595	0.272297	+	0.962213i	$0.412217\pi$
$984$	4.73340	0.150895
$985$	−21.4354	−0.682990
$986$	26.0033	0.828113
$987$	−10.3733	−0.330186
$988$	−6.83552	−0.217467
$989$	15.5396	0.494129
$990$	26.4864	0.841792
$991$	−17.6932	−0.562041	−0.281021	−	0.959702i	$-0.590673\pi$
$991$	−17.6932	−0.562041	−0.281021	+	0.959702i	$0.590673\pi$
$992$	−33.9797	−1.07886
$993$	−18.4728	−0.586216
$994$	98.8503	3.13534
$995$	−6.68145	−0.211816
$996$	−29.4802	−0.934117
$997$	−9.11560	−0.288694	−0.144347	−	0.989527i	$-0.546108\pi$
$997$	−9.11560	−0.288694	−0.144347	+	0.989527i	$0.546108\pi$
$998$	−17.1432	−0.542659
$999$	−2.59244	−0.0820213

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8031.2.a.c.1.12	✓	121

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8031.2.a.c.1.12	✓	121	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8031 = 3 \cdot 2677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.36975 q^{2} -1.00000 q^{3} +3.61573 q^{4} -1.85498 q^{5} +2.36975 q^{6} +4.52062 q^{7} -3.82888 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.36975 q^{2} -1.00000 q^{3} +3.61573 q^{4} -1.85498 q^{5} +2.36975 q^{6} +4.52062 q^{7} -3.82888 q^{8} +1.00000 q^{9} +4.39585 q^{10} +6.02531 q^{11} -3.61573 q^{12} +0.342912 q^{13} -10.7128 q^{14} +1.85498 q^{15} +1.84204 q^{16} +3.63212 q^{17} -2.36975 q^{18} -5.51306 q^{19} -6.70711 q^{20} -4.52062 q^{21} -14.2785 q^{22} +8.93696 q^{23} +3.82888 q^{24} -1.55904 q^{25} -0.812617 q^{26} -1.00000 q^{27} +16.3453 q^{28} -3.02109 q^{29} -4.39585 q^{30} -10.3201 q^{31} +3.29258 q^{32} -6.02531 q^{33} -8.60724 q^{34} -8.38567 q^{35} +3.61573 q^{36} +2.59244 q^{37} +13.0646 q^{38} -0.342912 q^{39} +7.10250 q^{40} +1.23624 q^{41} +10.7128 q^{42} +1.73880 q^{43} +21.7859 q^{44} -1.85498 q^{45} -21.1784 q^{46} +2.29467 q^{47} -1.84204 q^{48} +13.4360 q^{49} +3.69454 q^{50} -3.63212 q^{51} +1.23988 q^{52} -3.34590 q^{53} +2.36975 q^{54} -11.1768 q^{55} -17.3089 q^{56} +5.51306 q^{57} +7.15924 q^{58} +4.54531 q^{59} +6.70711 q^{60} -3.91440 q^{61} +24.4560 q^{62} +4.52062 q^{63} -11.4867 q^{64} -0.636096 q^{65} +14.2785 q^{66} -10.0401 q^{67} +13.1328 q^{68} -8.93696 q^{69} +19.8720 q^{70} -9.22735 q^{71} -3.82888 q^{72} +15.7162 q^{73} -6.14345 q^{74} +1.55904 q^{75} -19.9337 q^{76} +27.2381 q^{77} +0.812617 q^{78} +8.09234 q^{79} -3.41695 q^{80} +1.00000 q^{81} -2.92957 q^{82} +8.15333 q^{83} -16.3453 q^{84} -6.73753 q^{85} -4.12052 q^{86} +3.02109 q^{87} -23.0702 q^{88} +13.3609 q^{89} +4.39585 q^{90} +1.55018 q^{91} +32.3136 q^{92} +10.3201 q^{93} -5.43779 q^{94} +10.2266 q^{95} -3.29258 q^{96} +14.3947 q^{97} -31.8400 q^{98} +6.02531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 121 q + 7 q^{2} - 121 q^{3} + 123 q^{4} + 24 q^{5} - 7 q^{6} - 14 q^{7} + 18 q^{8} + 121 q^{9}+O(q^{10}) \) 121 * q + 7 * q^2 - 121 * q^3 + 123 * q^4 + 24 * q^5 - 7 * q^6 - 14 * q^7 + 18 * q^8 + 121 * q^9	\( 121 q + 7 q^{2} - 121 q^{3} + 123 q^{4} + 24 q^{5} - 7 q^{6} - 14 q^{7} + 18 q^{8} + 121 q^{9} + 18 q^{10} + 32 q^{11} - 123 q^{12} + 2 q^{13} + 37 q^{14} - 24 q^{15} + 131 q^{16} + 87 q^{17} + 7 q^{18} - 10 q^{19} + 60 q^{20} + 14 q^{21} - 22 q^{22} + 31 q^{23} - 18 q^{24} + 147 q^{25} + 37 q^{26} - 121 q^{27} - 29 q^{28} + 68 q^{29} - 18 q^{30} + 25 q^{31} + 43 q^{32} - 32 q^{33} + 27 q^{34} + 51 q^{35} + 123 q^{36} - 4 q^{37} + 36 q^{38} - 2 q^{39} + 61 q^{40} + 132 q^{41} - 37 q^{42} - 91 q^{43} + 94 q^{44} + 24 q^{45} + 39 q^{47} - 131 q^{48} + 217 q^{49} + 54 q^{50} - 87 q^{51} - 12 q^{52} + 55 q^{53} - 7 q^{54} + 7 q^{55} + 104 q^{56} + 10 q^{57} - 3 q^{58} + 58 q^{59} - 60 q^{60} + 126 q^{61} + 74 q^{62} - 14 q^{63} + 122 q^{64} + 128 q^{65} + 22 q^{66} - 139 q^{67} + 190 q^{68} - 31 q^{69} - 18 q^{70} + 37 q^{71} + 18 q^{72} + 84 q^{73} + 79 q^{74} - 147 q^{75} + 23 q^{76} + 95 q^{77} - 37 q^{78} - 14 q^{79} + 145 q^{80} + 121 q^{81} + 9 q^{82} + 58 q^{83} + 29 q^{84} + 32 q^{85} + 28 q^{86} - 68 q^{87} - 84 q^{88} + 198 q^{89} + 18 q^{90} + 5 q^{91} + 98 q^{92} - 25 q^{93} + 9 q^{94} + 42 q^{95} - 43 q^{96} + 73 q^{97} + 69 q^{98} + 32 q^{99}+O(q^{100}) \) 121 * q + 7 * q^2 - 121 * q^3 + 123 * q^4 + 24 * q^5 - 7 * q^6 - 14 * q^7 + 18 * q^8 + 121 * q^9 + 18 * q^10 + 32 * q^11 - 123 * q^12 + 2 * q^13 + 37 * q^14 - 24 * q^15 + 131 * q^16 + 87 * q^17 + 7 * q^18 - 10 * q^19 + 60 * q^20 + 14 * q^21 - 22 * q^22 + 31 * q^23 - 18 * q^24 + 147 * q^25 + 37 * q^26 - 121 * q^27 - 29 * q^28 + 68 * q^29 - 18 * q^30 + 25 * q^31 + 43 * q^32 - 32 * q^33 + 27 * q^34 + 51 * q^35 + 123 * q^36 - 4 * q^37 + 36 * q^38 - 2 * q^39 + 61 * q^40 + 132 * q^41 - 37 * q^42 - 91 * q^43 + 94 * q^44 + 24 * q^45 + 39 * q^47 - 131 * q^48 + 217 * q^49 + 54 * q^50 - 87 * q^51 - 12 * q^52 + 55 * q^53 - 7 * q^54 + 7 * q^55 + 104 * q^56 + 10 * q^57 - 3 * q^58 + 58 * q^59 - 60 * q^60 + 126 * q^61 + 74 * q^62 - 14 * q^63 + 122 * q^64 + 128 * q^65 + 22 * q^66 - 139 * q^67 + 190 * q^68 - 31 * q^69 - 18 * q^70 + 37 * q^71 + 18 * q^72 + 84 * q^73 + 79 * q^74 - 147 * q^75 + 23 * q^76 + 95 * q^77 - 37 * q^78 - 14 * q^79 + 145 * q^80 + 121 * q^81 + 9 * q^82 + 58 * q^83 + 29 * q^84 + 32 * q^85 + 28 * q^86 - 68 * q^87 - 84 * q^88 + 198 * q^89 + 18 * q^90 + 5 * q^91 + 98 * q^92 - 25 * q^93 + 9 * q^94 + 42 * q^95 - 43 * q^96 + 73 * q^97 + 69 * q^98 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more