LMFDB - Embedded newform 6223.2.a.j.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6223,2,Mod(1,6223)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6223, 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("6223.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6223 = 7^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6223.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:	$49.6909051778$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 22 x^{13} + 186 x^{11} - 763 x^{9} - 7 x^{8} + 1588 x^{7} + 64 x^{6} - 1625 x^{5} - 185 x^{4} + \cdots - 13 $ x^15 - 22x^13 + 186x^11 - 763x^9 - 7x^8 + 1588x^7 + 64x^6 - 1625x^5 - 185x^4 + 726x^3 + 145x^2 - 83*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.601235$ of defining polynomial
Character	$\chi$	$=$	6223.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.601235 q^{2} -0.551784 q^{3} -1.63852 q^{4} -4.40463 q^{5} -0.331752 q^{6} -2.18760 q^{8} -2.69553 q^{9} +O(q^{10})$	\(q+0.601235 q^{2} -0.551784 q^{3} -1.63852 q^{4} -4.40463 q^{5} -0.331752 q^{6} -2.18760 q^{8} -2.69553 q^{9} -2.64822 q^{10} +4.30434 q^{11} +0.904107 q^{12} +1.91150 q^{13} +2.43041 q^{15} +1.96177 q^{16} -7.97603 q^{17} -1.62065 q^{18} -2.00095 q^{19} +7.21707 q^{20} +2.58792 q^{22} +2.11447 q^{23} +1.20708 q^{24} +14.4008 q^{25} +1.14926 q^{26} +3.14270 q^{27} -4.21397 q^{29} +1.46125 q^{30} +2.87794 q^{31} +5.55469 q^{32} -2.37507 q^{33} -4.79547 q^{34} +4.41668 q^{36} -2.54141 q^{37} -1.20304 q^{38} -1.05473 q^{39} +9.63559 q^{40} +6.34847 q^{41} +10.2731 q^{43} -7.05274 q^{44} +11.8728 q^{45} +1.27129 q^{46} -10.7858 q^{47} -1.08247 q^{48} +8.65827 q^{50} +4.40104 q^{51} -3.13202 q^{52} -4.18365 q^{53} +1.88950 q^{54} -18.9591 q^{55} +1.10409 q^{57} -2.53358 q^{58} +14.3788 q^{59} -3.98226 q^{60} +8.44301 q^{61} +1.73032 q^{62} -0.583865 q^{64} -8.41946 q^{65} -1.42797 q^{66} -7.52093 q^{67} +13.0689 q^{68} -1.16673 q^{69} +10.3166 q^{71} +5.89676 q^{72} +3.45838 q^{73} -1.52799 q^{74} -7.94613 q^{75} +3.27860 q^{76} -0.634143 q^{78} +14.3417 q^{79} -8.64088 q^{80} +6.35251 q^{81} +3.81692 q^{82} +5.58186 q^{83} +35.1315 q^{85} +6.17654 q^{86} +2.32520 q^{87} -9.41619 q^{88} +7.39046 q^{89} +7.13837 q^{90} -3.46459 q^{92} -1.58800 q^{93} -6.48482 q^{94} +8.81347 q^{95} -3.06499 q^{96} -13.7121 q^{97} -11.6025 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 4 q^{3} + 14 q^{4} - 7 q^{5} - 8 q^{6} + 9 q^{9}+O(q^{10}) $ 15 * q - 4 * q^3 + 14 * q^4 - 7 * q^5 - 8 * q^6 + 9 * q^9	$ 15 q - 4 q^{3} + 14 q^{4} - 7 q^{5} - 8 q^{6} + 9 q^{9} - 10 q^{10} + 14 q^{11} - 10 q^{12} - 6 q^{13} + 6 q^{15} + 20 q^{16} - 10 q^{17} + q^{18} - 13 q^{19} - 8 q^{20} - 11 q^{22} + 15 q^{23} - 34 q^{24} - 22 q^{26} - 22 q^{27} + 16 q^{29} + 7 q^{30} - 22 q^{31} - 14 q^{33} + 15 q^{34} + 20 q^{36} - 14 q^{37} + 6 q^{38} + 29 q^{39} - 22 q^{40} - 19 q^{41} - q^{43} + 25 q^{44} + 8 q^{45} - 28 q^{46} - 49 q^{47} + 14 q^{48} + 24 q^{50} - 8 q^{51} + 17 q^{52} - 28 q^{53} - 13 q^{54} - 39 q^{55} - 12 q^{57} - 10 q^{58} - 43 q^{59} - 60 q^{60} - 27 q^{61} - 14 q^{62} + 18 q^{64} - 8 q^{65} + 36 q^{66} + 3 q^{67} - 13 q^{68} + 17 q^{69} + 55 q^{71} - 21 q^{72} + 3 q^{73} - 12 q^{74} - 8 q^{75} + 20 q^{76} - 6 q^{78} + 18 q^{79} - 29 q^{80} - 17 q^{81} - 14 q^{82} - 17 q^{83} + 7 q^{85} + 4 q^{86} - 35 q^{87} - 114 q^{88} - 36 q^{89} + 39 q^{90} + 45 q^{92} + 15 q^{93} + 15 q^{94} + 59 q^{95} - 85 q^{96} + 2 q^{97} + 35 q^{99}+O(q^{100}) $ 15 * q - 4 * q^3 + 14 * q^4 - 7 * q^5 - 8 * q^6 + 9 * q^9 - 10 * q^10 + 14 * q^11 - 10 * q^12 - 6 * q^13 + 6 * q^15 + 20 * q^16 - 10 * q^17 + q^18 - 13 * q^19 - 8 * q^20 - 11 * q^22 + 15 * q^23 - 34 * q^24 - 22 * q^26 - 22 * q^27 + 16 * q^29 + 7 * q^30 - 22 * q^31 - 14 * q^33 + 15 * q^34 + 20 * q^36 - 14 * q^37 + 6 * q^38 + 29 * q^39 - 22 * q^40 - 19 * q^41 - q^43 + 25 * q^44 + 8 * q^45 - 28 * q^46 - 49 * q^47 + 14 * q^48 + 24 * q^50 - 8 * q^51 + 17 * q^52 - 28 * q^53 - 13 * q^54 - 39 * q^55 - 12 * q^57 - 10 * q^58 - 43 * q^59 - 60 * q^60 - 27 * q^61 - 14 * q^62 + 18 * q^64 - 8 * q^65 + 36 * q^66 + 3 * q^67 - 13 * q^68 + 17 * q^69 + 55 * q^71 - 21 * q^72 + 3 * q^73 - 12 * q^74 - 8 * q^75 + 20 * q^76 - 6 * q^78 + 18 * q^79 - 29 * q^80 - 17 * q^81 - 14 * q^82 - 17 * q^83 + 7 * q^85 + 4 * q^86 - 35 * q^87 - 114 * q^88 - 36 * q^89 + 39 * q^90 + 45 * q^92 + 15 * q^93 + 15 * q^94 + 59 * q^95 - 85 * q^96 + 2 * q^97 + 35 * 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 $

$2$	0.601235	0.425137	0.212569	−	0.977146i	$-0.431817\pi$
$2$	0.601235	0.425137	0.212569	+	0.977146i	$0.431817\pi$
$3$	−0.551784	−0.318573	−0.159286	−	0.987232i	$-0.550919\pi$
$3$	−0.551784	−0.318573	−0.159286	+	0.987232i	$0.550919\pi$
$4$	−1.63852	−0.819258
$5$	−4.40463	−1.96981	−0.984906	−	0.173088i	$-0.944625\pi$
$5$	−4.40463	−1.96981	−0.984906	+	0.173088i	$0.944625\pi$
$6$	−0.331752	−0.135437
$7$	0	0
$7$	0	0
$8$	−2.18760	−0.773435
$9$	−2.69553	−0.898512
$10$	−2.64822	−0.837441
$11$	4.30434	1.29781	0.648904	−	0.760870i	$-0.275228\pi$
$11$	4.30434	1.29781	0.648904	+	0.760870i	$0.275228\pi$
$12$	0.904107	0.260993
$13$	1.91150	0.530154	0.265077	−	0.964227i	$-0.414603\pi$
$13$	1.91150	0.530154	0.265077	+	0.964227i	$0.414603\pi$
$14$	0	0
$15$	2.43041	0.627528
$16$	1.96177	0.490442
$17$	−7.97603	−1.93447	−0.967235	−	0.253881i	$-0.918293\pi$
$17$	−7.97603	−1.93447	−0.967235	+	0.253881i	$0.918293\pi$
$18$	−1.62065	−0.381991
$19$	−2.00095	−0.459050	−0.229525	−	0.973303i	$-0.573717\pi$
$19$	−2.00095	−0.459050	−0.229525	+	0.973303i	$0.573717\pi$
$20$	7.21707	1.61379
$21$	0	0
$22$	2.58792	0.551747
$23$	2.11447	0.440897	0.220448	−	0.975399i	$-0.429248\pi$
$23$	2.11447	0.440897	0.220448	+	0.975399i	$0.429248\pi$
$24$	1.20708	0.246395
$25$	14.4008	2.88016
$26$	1.14926	0.225388
$27$	3.14270	0.604814
$28$	0	0
$29$	−4.21397	−0.782514	−0.391257	−	0.920281i	$-0.627960\pi$
$29$	−4.21397	−0.782514	−0.391257	+	0.920281i	$0.627960\pi$
$30$	1.46125	0.266786
$31$	2.87794	0.516894	0.258447	−	0.966025i	$-0.416789\pi$
$31$	2.87794	0.516894	0.258447	+	0.966025i	$0.416789\pi$
$32$	5.55469	0.981940
$33$	−2.37507	−0.413446
$34$	−4.79547	−0.822416
$35$	0	0
$36$	4.41668	0.736113
$37$	−2.54141	−0.417805	−0.208903	−	0.977936i	$-0.566989\pi$
$37$	−2.54141	−0.417805	−0.208903	+	0.977936i	$0.566989\pi$
$38$	−1.20304	−0.195159
$39$	−1.05473	−0.168893
$40$	9.63559	1.52352
$41$	6.34847	0.991464	0.495732	−	0.868475i	$-0.334900\pi$
$41$	6.34847	0.991464	0.495732	+	0.868475i	$0.334900\pi$
$42$	0	0
$43$	10.2731	1.56663	0.783316	−	0.621624i	$-0.213527\pi$
$43$	10.2731	1.56663	0.783316	+	0.621624i	$0.213527\pi$
$44$	−7.05274	−1.06324
$45$	11.8728	1.76990
$46$	1.27129	0.187442
$47$	−10.7858	−1.57328	−0.786638	−	0.617415i	$-0.788180\pi$
$47$	−10.7858	−1.57328	−0.786638	+	0.617415i	$0.788180\pi$
$48$	−1.08247	−0.156241
$49$	0	0
$50$	8.65827	1.22446
$51$	4.40104	0.616269
$52$	−3.13202	−0.434333
$53$	−4.18365	−0.574668	−0.287334	−	0.957830i	$-0.592769\pi$
$53$	−4.18365	−0.574668	−0.287334	+	0.957830i	$0.592769\pi$
$54$	1.88950	0.257129
$55$	−18.9591	−2.55644
$56$	0	0
$57$	1.10409	0.146241
$58$	−2.53358	−0.332676
$59$	14.3788	1.87196	0.935982	−	0.352049i	$-0.114515\pi$
$59$	14.3788	1.87196	0.935982	+	0.352049i	$0.114515\pi$
$60$	−3.98226	−0.514108
$61$	8.44301	1.08102	0.540509	−	0.841338i	$-0.318232\pi$
$61$	8.44301	1.08102	0.540509	+	0.841338i	$0.318232\pi$
$62$	1.73032	0.219751
$63$	0	0
$64$	−0.583865	−0.0729831
$65$	−8.41946	−1.04430
$66$	−1.42797	−0.175771
$67$	−7.52093	−0.918828	−0.459414	−	0.888222i	$-0.651941\pi$
$67$	−7.52093	−0.918828	−0.459414	+	0.888222i	$0.651941\pi$
$68$	13.0689	1.58483
$69$	−1.16673	−0.140458
$70$	0	0
$71$	10.3166	1.22436	0.612179	−	0.790720i	$-0.290293\pi$
$71$	10.3166	1.22436	0.612179	+	0.790720i	$0.290293\pi$
$72$	5.89676	0.694940
$73$	3.45838	0.404773	0.202387	−	0.979306i	$-0.435130\pi$
$73$	3.45838	0.404773	0.202387	+	0.979306i	$0.435130\pi$
$74$	−1.52799	−0.177625
$75$	−7.94613	−0.917540
$76$	3.27860	0.376081
$77$	0	0
$78$	−0.634143	−0.0718026
$79$	14.3417	1.61356	0.806782	−	0.590850i	$-0.201207\pi$
$79$	14.3417	1.61356	0.806782	+	0.590850i	$0.201207\pi$
$80$	−8.64088	−0.966080
$81$	6.35251	0.705835
$82$	3.81692	0.421508
$83$	5.58186	0.612688	0.306344	−	0.951921i	$-0.400894\pi$
$83$	5.58186	0.612688	0.306344	+	0.951921i	$0.400894\pi$
$84$	0	0
$85$	35.1315	3.81054
$86$	6.17654	0.666034
$87$	2.32520	0.249288
$88$	−9.41619	−1.00377
$89$	7.39046	0.783387	0.391693	−	0.920096i	$-0.371889\pi$
$89$	7.39046	0.783387	0.391693	+	0.920096i	$0.371889\pi$
$90$	7.13837	0.752450
$91$	0	0
$92$	−3.46459	−0.361208
$93$	−1.58800	−0.164668
$94$	−6.48482	−0.668858
$95$	8.81347	0.904243
$96$	−3.06499	−0.312819
$97$	−13.7121	−1.39226	−0.696128	−	0.717917i	$-0.745096\pi$
$97$	−13.7121	−1.39226	−0.696128	+	0.717917i	$0.745096\pi$
$98$	0	0
$99$	−11.6025	−1.16610
$100$	−23.5960	−2.35960
$101$	−1.24756	−0.124137	−0.0620683	−	0.998072i	$-0.519770\pi$
$101$	−1.24756	−0.124137	−0.0620683	+	0.998072i	$0.519770\pi$
$102$	2.64606	0.261999
$103$	−5.12235	−0.504720	−0.252360	−	0.967633i	$-0.581207\pi$
$103$	−5.12235	−0.504720	−0.252360	+	0.967633i	$0.581207\pi$
$104$	−4.18160	−0.410040
$105$	0	0
$106$	−2.51536	−0.244313
$107$	−16.0637	−1.55293	−0.776467	−	0.630157i	$-0.782990\pi$
$107$	−16.0637	−1.55293	−0.776467	+	0.630157i	$0.782990\pi$
$108$	−5.14937	−0.495499
$109$	−8.30523	−0.795496	−0.397748	−	0.917495i	$-0.630208\pi$
$109$	−8.30523	−0.795496	−0.397748	+	0.917495i	$0.630208\pi$
$110$	−11.3988	−1.08684
$111$	1.40231	0.133101
$112$	0	0
$113$	−17.1150	−1.61004	−0.805020	−	0.593247i	$-0.797846\pi$
$113$	−17.1150	−1.61004	−0.805020	+	0.593247i	$0.797846\pi$
$114$	0.663820	0.0621724
$115$	−9.31345	−0.868484
$116$	6.90466	0.641081
$117$	−5.15251	−0.476350
$118$	8.64505	0.795841
$119$	0	0
$120$	−5.31676	−0.485352
$121$	7.52736	0.684306
$122$	5.07624	0.459581
$123$	−3.50298	−0.315853
$124$	−4.71556	−0.423470
$125$	−41.4071	−3.70357
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−11.4604	−1.01297
$129$	−5.66853	−0.499086
$130$	−5.06207	−0.443973
$131$	−4.01168	−0.350502	−0.175251	−	0.984524i	$-0.556074\pi$
$131$	−4.01168	−0.350502	−0.175251	+	0.984524i	$0.556074\pi$
$132$	3.89159	0.338719
$133$	0	0
$134$	−4.52185	−0.390628
$135$	−13.8425	−1.19137
$136$	17.4484	1.49619
$137$	−6.88474	−0.588203	−0.294102	−	0.955774i	$-0.595020\pi$
$137$	−6.88474	−0.588203	−0.294102	+	0.955774i	$0.595020\pi$
$138$	−0.701478	−0.0597137
$139$	6.93504	0.588222	0.294111	−	0.955771i	$-0.404976\pi$
$139$	6.93504	0.588222	0.294111	+	0.955771i	$0.404976\pi$
$140$	0	0
$141$	5.95145	0.501203
$142$	6.20271	0.520520
$143$	8.22775	0.688039
$144$	−5.28802	−0.440668
$145$	18.5610	1.54141
$146$	2.07930	0.172084
$147$	0	0
$148$	4.16414	0.342291
$149$	−14.3035	−1.17179	−0.585896	−	0.810386i	$-0.699257\pi$
$149$	−14.3035	−1.17179	−0.585896	+	0.810386i	$0.699257\pi$
$150$	−4.77749	−0.390081
$151$	3.87716	0.315519	0.157760	−	0.987478i	$-0.449573\pi$
$151$	3.87716	0.315519	0.157760	+	0.987478i	$0.449573\pi$
$152$	4.37729	0.355045
$153$	21.4997	1.73814
$154$	0	0
$155$	−12.6763	−1.01818
$156$	1.72820	0.138367
$157$	20.8139	1.66113	0.830565	−	0.556921i	$-0.188017\pi$
$157$	20.8139	1.66113	0.830565	+	0.556921i	$0.188017\pi$
$158$	8.62271	0.685986
$159$	2.30847	0.183074
$160$	−24.4664	−1.93424
$161$	0	0
$162$	3.81935	0.300077
$163$	−24.2929	−1.90277	−0.951383	−	0.308011i	$-0.900337\pi$
$163$	−24.2929	−1.90277	−0.951383	+	0.308011i	$0.900337\pi$
$164$	−10.4021	−0.812265
$165$	10.4613	0.814411
$166$	3.35601	0.260477
$167$	−4.02158	−0.311200	−0.155600	−	0.987820i	$-0.549731\pi$
$167$	−4.02158	−0.311200	−0.155600	+	0.987820i	$0.549731\pi$
$168$	0	0
$169$	−9.34617	−0.718936
$170$	21.1223	1.62000
$171$	5.39364	0.412462
$172$	−16.8326	−1.28348
$173$	5.91524	0.449727	0.224864	−	0.974390i	$-0.427806\pi$
$173$	5.91524	0.449727	0.224864	+	0.974390i	$0.427806\pi$
$174$	1.39799	0.105981
$175$	0	0
$176$	8.44413	0.636500
$177$	−7.93400	−0.596356
$178$	4.44340	0.333047
$179$	−1.75316	−0.131038	−0.0655188	−	0.997851i	$-0.520870\pi$
$179$	−1.75316	−0.131038	−0.0655188	+	0.997851i	$0.520870\pi$
$180$	−19.4539	−1.45000
$181$	−1.44921	−0.107719	−0.0538593	−	0.998549i	$-0.517152\pi$
$181$	−1.44921	−0.107719	−0.0538593	+	0.998549i	$0.517152\pi$
$182$	0	0
$183$	−4.65872	−0.344382
$184$	−4.62561	−0.341005
$185$	11.1940	0.822998
$186$	−0.954763	−0.0700066
$187$	−34.3316	−2.51057
$188$	17.6728	1.28892
$189$	0	0
$190$	5.29897	0.384428
$191$	2.95252	0.213637	0.106818	−	0.994279i	$-0.465934\pi$
$191$	2.95252	0.213637	0.106818	+	0.994279i	$0.465934\pi$
$192$	0.322167	0.0232504
$193$	19.5764	1.40914	0.704570	−	0.709634i	$-0.251140\pi$
$193$	19.5764	1.40914	0.704570	+	0.709634i	$0.251140\pi$
$194$	−8.24422	−0.591900
$195$	4.64572	0.332687
$196$	0	0
$197$	9.65828	0.688124	0.344062	−	0.938947i	$-0.388197\pi$
$197$	9.65828	0.688124	0.344062	+	0.938947i	$0.388197\pi$
$198$	−6.97583	−0.495751
$199$	17.5931	1.24714	0.623570	−	0.781768i	$-0.285682\pi$
$199$	17.5931	1.24714	0.623570	+	0.781768i	$0.285682\pi$
$200$	−31.5033	−2.22762
$201$	4.14993	0.292713
$202$	−0.750075	−0.0527751
$203$	0	0
$204$	−7.21118	−0.504884
$205$	−27.9627	−1.95300
$206$	−3.07973	−0.214575
$207$	−5.69962	−0.396151
$208$	3.74992	0.260010
$209$	−8.61279	−0.595759
$210$	0	0
$211$	−3.30191	−0.227313	−0.113656	−	0.993520i	$-0.536256\pi$
$211$	−3.30191	−0.227313	−0.113656	+	0.993520i	$0.536256\pi$
$212$	6.85498	0.470802
$213$	−5.69254	−0.390047
$214$	−9.65805	−0.660210
$215$	−45.2492	−3.08597
$216$	−6.87499	−0.467784
$217$	0	0
$218$	−4.99339	−0.338195
$219$	−1.90828	−0.128950
$220$	31.0647	2.09438
$221$	−15.2462	−1.02557
$222$	0.843117	0.0565863
$223$	−16.9302	−1.13373	−0.566866	−	0.823810i	$-0.691844\pi$
$223$	−16.9302	−1.13373	−0.566866	+	0.823810i	$0.691844\pi$
$224$	0	0
$225$	−38.8179	−2.58786
$226$	−10.2901	−0.684488
$227$	−6.21508	−0.412509	−0.206255	−	0.978498i	$-0.566128\pi$
$227$	−6.21508	−0.412509	−0.206255	+	0.978498i	$0.566128\pi$
$228$	−1.80908	−0.119809
$229$	1.17160	0.0774215	0.0387108	−	0.999250i	$-0.487675\pi$
$229$	1.17160	0.0774215	0.0387108	+	0.999250i	$0.487675\pi$
$230$	−5.59957	−0.369225
$231$	0	0
$232$	9.21849	0.605224
$233$	−23.8516	−1.56257	−0.781286	−	0.624173i	$-0.785436\pi$
$233$	−23.8516	−1.56257	−0.781286	+	0.624173i	$0.785436\pi$
$234$	−3.09787	−0.202514
$235$	47.5077	3.09906
$236$	−23.5599	−1.53362
$237$	−7.91350	−0.514037
$238$	0	0
$239$	14.9800	0.968977	0.484488	−	0.874798i	$-0.339006\pi$
$239$	14.9800	0.968977	0.484488	+	0.874798i	$0.339006\pi$
$240$	4.76790	0.307766
$241$	10.1108	0.651291	0.325646	−	0.945492i	$-0.394418\pi$
$241$	10.1108	0.651291	0.325646	+	0.945492i	$0.394418\pi$
$242$	4.52571	0.290924
$243$	−12.9333	−0.829673
$244$	−13.8340	−0.885632
$245$	0	0
$246$	−2.10612	−0.134281
$247$	−3.82482	−0.243368
$248$	−6.29580	−0.399784
$249$	−3.07998	−0.195186
$250$	−24.8954	−1.57452
$251$	−2.23745	−0.141227	−0.0706135	−	0.997504i	$-0.522496\pi$
$251$	−2.23745	−0.141227	−0.0706135	+	0.997504i	$0.522496\pi$
$252$	0	0
$253$	9.10138	0.572199
$254$	0.601235	0.0377248
$255$	−19.3850	−1.21393
$256$	−5.72268	−0.357667
$257$	−9.85947	−0.615017	−0.307508	−	0.951545i	$-0.599495\pi$
$257$	−9.85947	−0.615017	−0.307508	+	0.951545i	$0.599495\pi$
$258$	−3.40812	−0.212180
$259$	0	0
$260$	13.7954	0.855555
$261$	11.3589	0.703098
$262$	−2.41196	−0.149012
$263$	−8.92520	−0.550352	−0.275176	−	0.961394i	$-0.588736\pi$
$263$	−8.92520	−0.550352	−0.275176	+	0.961394i	$0.588736\pi$
$264$	5.19570	0.319773
$265$	18.4274	1.13199
$266$	0	0
$267$	−4.07793	−0.249566
$268$	12.3232	0.752758
$269$	−0.300991	−0.0183518	−0.00917588	−	0.999958i	$-0.502921\pi$
$269$	−0.300991	−0.0183518	−0.00917588	+	0.999958i	$0.502921\pi$
$270$	−8.32257	−0.506496
$271$	4.19940	0.255095	0.127548	−	0.991832i	$-0.459289\pi$
$271$	4.19940	0.255095	0.127548	+	0.991832i	$0.459289\pi$
$272$	−15.6471	−0.948746
$273$	0	0
$274$	−4.13935	−0.250067
$275$	61.9860	3.73790
$276$	1.91170	0.115071
$277$	15.0056	0.901600	0.450800	−	0.892625i	$-0.351139\pi$
$277$	15.0056	0.901600	0.450800	+	0.892625i	$0.351139\pi$
$278$	4.16959	0.250075
$279$	−7.75760	−0.464435
$280$	0	0
$281$	−5.79057	−0.345437	−0.172718	−	0.984971i	$-0.555255\pi$
$281$	−5.79057	−0.345437	−0.172718	+	0.984971i	$0.555255\pi$
$282$	3.57822	0.213080
$283$	−23.2225	−1.38043	−0.690217	−	0.723603i	$-0.742485\pi$
$283$	−23.2225	−1.38043	−0.690217	+	0.723603i	$0.742485\pi$
$284$	−16.9039	−1.00306
$285$	−4.86313	−0.288067
$286$	4.94681	0.292511
$287$	0	0
$288$	−14.9729	−0.882284
$289$	46.6170	2.74218
$290$	11.1595	0.655309
$291$	7.56614	0.443535
$292$	−5.66662	−0.331614
$293$	20.2897	1.18534	0.592668	−	0.805447i	$-0.298075\pi$
$293$	20.2897	1.18534	0.592668	+	0.805447i	$0.298075\pi$
$294$	0	0
$295$	−63.3335	−3.68742
$296$	5.55960	0.323145
$297$	13.5273	0.784932
$298$	−8.59979	−0.498173
$299$	4.04180	0.233743
$300$	13.0199	0.751703
$301$	0	0
$302$	2.33109	0.134139
$303$	0.688382	0.0395465
$304$	−3.92541	−0.225138
$305$	−37.1884	−2.12940
$306$	12.9263	0.738950
$307$	−16.9061	−0.964883	−0.482442	−	0.875928i	$-0.660250\pi$
$307$	−16.9061	−0.964883	−0.482442	+	0.875928i	$0.660250\pi$
$308$	0	0
$309$	2.82643	0.160790
$310$	−7.62143	−0.432868
$311$	−10.4239	−0.591085	−0.295543	−	0.955330i	$-0.595500\pi$
$311$	−10.4239	−0.591085	−0.295543	+	0.955330i	$0.595500\pi$
$312$	2.30734	0.130627
$313$	21.5137	1.21603	0.608013	−	0.793927i	$-0.291967\pi$
$313$	21.5137	1.21603	0.608013	+	0.793927i	$0.291967\pi$
$314$	12.5140	0.706209
$315$	0	0
$316$	−23.4991	−1.32193
$317$	−4.79749	−0.269454	−0.134727	−	0.990883i	$-0.543016\pi$
$317$	−4.79749	−0.269454	−0.134727	+	0.990883i	$0.543016\pi$
$318$	1.38793	0.0778314
$319$	−18.1384	−1.01555
$320$	2.57171	0.143763
$321$	8.86368	0.494722
$322$	0	0
$323$	15.9597	0.888020
$324$	−10.4087	−0.578261
$325$	27.5271	1.52693
$326$	−14.6057	−0.808937
$327$	4.58269	0.253423
$328$	−13.8879	−0.766833
$329$	0	0
$330$	6.28970	0.346237
$331$	−20.8566	−1.14638	−0.573190	−	0.819422i	$-0.694294\pi$
$331$	−20.8566	−1.14638	−0.573190	+	0.819422i	$0.694294\pi$
$332$	−9.14596	−0.501950
$333$	6.85046	0.375403
$334$	−2.41792	−0.132303
$335$	33.1270	1.80992
$336$	0	0
$337$	0.154395	0.00841045	0.00420522	−	0.999991i	$-0.498661\pi$
$337$	0.154395	0.00841045	0.00420522	+	0.999991i	$0.498661\pi$
$338$	−5.61924	−0.305647
$339$	9.44376	0.512915
$340$	−57.5635	−3.12182
$341$	12.3877	0.670829
$342$	3.24285	0.175353
$343$	0	0
$344$	−22.4735	−1.21169
$345$	5.13901	0.276675
$346$	3.55645	0.191196
$347$	−17.7217	−0.951350	−0.475675	−	0.879621i	$-0.657796\pi$
$347$	−17.7217	−0.951350	−0.475675	+	0.879621i	$0.657796\pi$
$348$	−3.80988	−0.204231
$349$	−12.2962	−0.658202	−0.329101	−	0.944295i	$-0.606746\pi$
$349$	−12.2962	−0.658202	−0.329101	+	0.944295i	$0.606746\pi$
$350$	0	0
$351$	6.00728	0.320645
$352$	23.9093	1.27437
$353$	−16.1440	−0.859258	−0.429629	−	0.903005i	$-0.641356\pi$
$353$	−16.1440	−0.859258	−0.429629	+	0.903005i	$0.641356\pi$
$354$	−4.77020	−0.253533
$355$	−45.4409	−2.41175
$356$	−12.1094	−0.641796
$357$	0	0
$358$	−1.05406	−0.0557089
$359$	10.9875	0.579900	0.289950	−	0.957042i	$-0.406361\pi$
$359$	10.9875	0.579900	0.289950	+	0.957042i	$0.406361\pi$
$360$	−25.9731	−1.36890
$361$	−14.9962	−0.789273
$362$	−0.871314	−0.0457952
$363$	−4.15348	−0.218001
$364$	0	0
$365$	−15.2329	−0.797327
$366$	−2.80098	−0.146410
$367$	−15.7238	−0.820778	−0.410389	−	0.911911i	$-0.634607\pi$
$367$	−15.7238	−0.820778	−0.410389	+	0.911911i	$0.634607\pi$
$368$	4.14809	0.216234
$369$	−17.1125	−0.890842
$370$	6.73022	0.349887
$371$	0	0
$372$	2.60197	0.134906
$373$	3.75838	0.194602	0.0973008	−	0.995255i	$-0.468979\pi$
$373$	3.75838	0.194602	0.0973008	+	0.995255i	$0.468979\pi$
$374$	−20.6413	−1.06734
$375$	22.8478	1.17985
$376$	23.5951	1.21683
$377$	−8.05500	−0.414853
$378$	0	0
$379$	9.46136	0.485997	0.242999	−	0.970027i	$-0.421869\pi$
$379$	9.46136	0.485997	0.242999	+	0.970027i	$0.421869\pi$
$380$	−14.4410	−0.740809
$381$	−0.551784	−0.0282687
$382$	1.77516	0.0908249
$383$	10.3557	0.529150	0.264575	−	0.964365i	$-0.414768\pi$
$383$	10.3557	0.529150	0.264575	+	0.964365i	$0.414768\pi$
$384$	6.32368	0.322704
$385$	0	0
$386$	11.7700	0.599078
$387$	−27.6915	−1.40764
$388$	22.4676	1.14062
$389$	17.0981	0.866907	0.433453	−	0.901176i	$-0.357295\pi$
$389$	17.0981	0.866907	0.433453	+	0.901176i	$0.357295\pi$
$390$	2.79317	0.141438
$391$	−16.8650	−0.852902
$392$	0	0
$393$	2.21358	0.111660
$394$	5.80689	0.292547
$395$	−63.1698	−3.17842
$396$	19.0109	0.955333
$397$	22.1102	1.10968	0.554840	−	0.831957i	$-0.312780\pi$
$397$	22.1102	1.10968	0.554840	+	0.831957i	$0.312780\pi$
$398$	10.5776	0.530205
$399$	0	0
$400$	28.2511	1.41255
$401$	38.1546	1.90535	0.952675	−	0.303990i	$-0.0983191\pi$
$401$	38.1546	1.90535	0.952675	+	0.303990i	$0.0983191\pi$
$402$	2.49508	0.124443
$403$	5.50119	0.274034
$404$	2.04414	0.101700
$405$	−27.9805	−1.39036
$406$	0	0
$407$	−10.9391	−0.542231
$408$	−9.62774	−0.476644
$409$	−9.50413	−0.469949	−0.234974	−	0.972002i	$-0.575501\pi$
$409$	−9.50413	−0.469949	−0.234974	+	0.972002i	$0.575501\pi$
$410$	−16.8121	−0.830293
$411$	3.79889	0.187385
$412$	8.39305	0.413496
$413$	0	0
$414$	−3.42681	−0.168418
$415$	−24.5860	−1.20688
$416$	10.6178	0.520580
$417$	−3.82664	−0.187391
$418$	−5.17831	−0.253279
$419$	−5.66006	−0.276512	−0.138256	−	0.990397i	$-0.544150\pi$
$419$	−5.66006	−0.276512	−0.138256	+	0.990397i	$0.544150\pi$
$420$	0	0
$421$	−10.0019	−0.487462	−0.243731	−	0.969843i	$-0.578371\pi$
$421$	−10.0019	−0.487462	−0.243731	+	0.969843i	$0.578371\pi$
$422$	−1.98522	−0.0966391
$423$	29.0736	1.41361
$424$	9.15216	0.444468
$425$	−114.861	−5.57159
$426$	−3.42255	−0.165823
$427$	0	0
$428$	26.3206	1.27225
$429$	−4.53994	−0.219190
$430$	−27.2054	−1.31196
$431$	2.95645	0.142407	0.0712036	−	0.997462i	$-0.477316\pi$
$431$	2.95645	0.142407	0.0712036	+	0.997462i	$0.477316\pi$
$432$	6.16526	0.296626
$433$	38.2222	1.83684	0.918420	−	0.395607i	$-0.129466\pi$
$433$	38.2222	1.83684	0.918420	+	0.395607i	$0.129466\pi$
$434$	0	0
$435$	−10.2417	−0.491050
$436$	13.6083	0.651717
$437$	−4.23095	−0.202394
$438$	−1.14732	−0.0548213
$439$	−3.57924	−0.170828	−0.0854140	−	0.996346i	$-0.527221\pi$
$439$	−3.57924	−0.170828	−0.0854140	+	0.996346i	$0.527221\pi$
$440$	41.4749	1.97724
$441$	0	0
$442$	−9.16653	−0.436007
$443$	−1.21879	−0.0579064	−0.0289532	−	0.999581i	$-0.509217\pi$
$443$	−1.21879	−0.0579064	−0.0289532	+	0.999581i	$0.509217\pi$
$444$	−2.29771	−0.109044
$445$	−32.5523	−1.54313
$446$	−10.1790	−0.481992
$447$	7.89247	0.373301
$448$	0	0
$449$	36.5812	1.72637	0.863186	−	0.504885i	$-0.168465\pi$
$449$	36.5812	1.72637	0.863186	+	0.504885i	$0.168465\pi$
$450$	−23.3387	−1.10020
$451$	27.3260	1.28673
$452$	28.0432	1.31904
$453$	−2.13936	−0.100516
$454$	−3.73672	−0.175373
$455$	0	0
$456$	−2.41532	−0.113108
$457$	−0.708080	−0.0331226	−0.0165613	−	0.999863i	$-0.505272\pi$
$457$	−0.708080	−0.0331226	−0.0165613	+	0.999863i	$0.505272\pi$
$458$	0.704407	0.0329148
$459$	−25.0663	−1.16999
$460$	15.2602	0.711512
$461$	7.44049	0.346538	0.173269	−	0.984875i	$-0.444567\pi$
$461$	7.44049	0.346538	0.173269	+	0.984875i	$0.444567\pi$
$462$	0	0
$463$	22.9967	1.06875	0.534374	−	0.845248i	$-0.320548\pi$
$463$	22.9967	1.06875	0.534374	+	0.845248i	$0.320548\pi$
$464$	−8.26683	−0.383778
$465$	6.99457	0.324366
$466$	−14.3404	−0.664308
$467$	23.3688	1.08138	0.540691	−	0.841222i	$-0.318163\pi$
$467$	23.3688	1.08138	0.540691	+	0.841222i	$0.318163\pi$
$468$	8.44248	0.390254
$469$	0	0
$470$	28.5633	1.31753
$471$	−11.4848	−0.529191
$472$	−31.4552	−1.44784
$473$	44.2189	2.03319
$474$	−4.75787	−0.218536
$475$	−28.8154	−1.32214
$476$	0	0
$477$	11.2772	0.516346
$478$	9.00651	0.411948
$479$	−10.9284	−0.499333	−0.249667	−	0.968332i	$-0.580321\pi$
$479$	−10.9284	−0.499333	−0.249667	+	0.968332i	$0.580321\pi$
$480$	13.5002	0.616195
$481$	−4.85790	−0.221501
$482$	6.07894	0.276888
$483$	0	0
$484$	−12.3337	−0.560623
$485$	60.3970	2.74248
$486$	−7.77597	−0.352725
$487$	−0.259244	−0.0117475	−0.00587374	−	0.999983i	$-0.501870\pi$
$487$	−0.259244	−0.0117475	−0.00587374	+	0.999983i	$0.501870\pi$
$488$	−18.4700	−0.836096
$489$	13.4044	0.606169
$490$	0	0
$491$	13.2585	0.598346	0.299173	−	0.954199i	$-0.403289\pi$
$491$	13.2585	0.598346	0.299173	+	0.954199i	$0.403289\pi$
$492$	5.73970	0.258765
$493$	33.6107	1.51375
$494$	−2.29962	−0.103465
$495$	51.1048	2.29699
$496$	5.64586	0.253507
$497$	0	0
$498$	−1.85179	−0.0829807
$499$	−32.5419	−1.45677	−0.728387	−	0.685166i	$-0.759730\pi$
$499$	−32.5419	−1.45677	−0.728387	+	0.685166i	$0.759730\pi$
$500$	67.8463	3.03418
$501$	2.21904	0.0991396
$502$	−1.34524	−0.0600408
$503$	−39.1034	−1.74354	−0.871768	−	0.489920i	$-0.837026\pi$
$503$	−39.1034	−1.74354	−0.871768	+	0.489920i	$0.837026\pi$
$504$	0	0
$505$	5.49503	0.244526
$506$	5.47207	0.243263
$507$	5.15707	0.229033
$508$	−1.63852	−0.0726974
$509$	−16.7722	−0.743414	−0.371707	−	0.928350i	$-0.621227\pi$
$509$	−16.7722	−0.743414	−0.371707	+	0.928350i	$0.621227\pi$
$510$	−11.6549	−0.516089
$511$	0	0
$512$	19.4802	0.860910
$513$	−6.28841	−0.277640
$514$	−5.92786	−0.261467
$515$	22.5621	0.994203
$516$	9.28798	0.408880
$517$	−46.4259	−2.04181
$518$	0	0
$519$	−3.26393	−0.143271
$520$	18.4184	0.807701
$521$	7.87075	0.344824	0.172412	−	0.985025i	$-0.444844\pi$
$521$	7.87075	0.344824	0.172412	+	0.985025i	$0.444844\pi$
$522$	6.82937	0.298913
$523$	−15.7546	−0.688899	−0.344450	−	0.938805i	$-0.611934\pi$
$523$	−15.7546	−0.688899	−0.344450	+	0.938805i	$0.611934\pi$
$524$	6.57321	0.287152
$525$	0	0
$526$	−5.36614	−0.233975
$527$	−22.9546	−0.999916
$528$	−4.65933	−0.202771
$529$	−18.5290	−0.805610
$530$	11.0792	0.481251
$531$	−38.7586	−1.68198
$532$	0	0
$533$	12.1351	0.525629
$534$	−2.45180	−0.106100
$535$	70.7547	3.05899
$536$	16.4528	0.710653
$537$	0.967367	0.0417450
$538$	−0.180966	−0.00780202
$539$	0	0
$540$	22.6811	0.976039
$541$	24.5723	1.05645	0.528223	−	0.849106i	$-0.322858\pi$
$541$	24.5723	1.05645	0.528223	+	0.849106i	$0.322858\pi$
$542$	2.52482	0.108451
$543$	0.799649	0.0343162
$544$	−44.3044	−1.89953
$545$	36.5815	1.56698
$546$	0	0
$547$	18.1043	0.774083	0.387041	−	0.922062i	$-0.373497\pi$
$547$	18.1043	0.774083	0.387041	+	0.922062i	$0.373497\pi$
$548$	11.2808	0.481890
$549$	−22.7584	−0.971306
$550$	37.2682	1.58912
$551$	8.43196	0.359213
$552$	2.55234	0.108635
$553$	0	0
$554$	9.02190	0.383304
$555$	−6.17666	−0.262185
$556$	−11.3632	−0.481906
$557$	0.484330	0.0205217	0.0102609	−	0.999947i	$-0.496734\pi$
$557$	0.484330	0.0205217	0.0102609	+	0.999947i	$0.496734\pi$
$558$	−4.66414	−0.197449
$559$	19.6370	0.830557
$560$	0	0
$561$	18.9436	0.799799
$562$	−3.48149	−0.146858
$563$	−25.0910	−1.05746	−0.528731	−	0.848790i	$-0.677332\pi$
$563$	−25.0910	−1.05746	−0.528731	+	0.848790i	$0.677332\pi$
$564$	−9.75155	−0.410614
$565$	75.3852	3.17148
$566$	−13.9622	−0.586874
$567$	0	0
$568$	−22.5687	−0.946960
$569$	−31.4579	−1.31878	−0.659392	−	0.751799i	$-0.729186\pi$
$569$	−31.4579	−1.31878	−0.659392	+	0.751799i	$0.729186\pi$
$570$	−2.92388	−0.122468
$571$	20.3592	0.852007	0.426004	−	0.904721i	$-0.359921\pi$
$571$	20.3592	0.852007	0.426004	+	0.904721i	$0.359921\pi$
$572$	−13.4813	−0.563681
$573$	−1.62915	−0.0680587
$574$	0	0
$575$	30.4500	1.26985
$576$	1.57383	0.0655762
$577$	1.95608	0.0814326	0.0407163	−	0.999171i	$-0.487036\pi$
$577$	1.95608	0.0814326	0.0407163	+	0.999171i	$0.487036\pi$
$578$	28.0278	1.16580
$579$	−10.8019	−0.448913
$580$	−30.4125	−1.26281
$581$	0	0
$582$	4.54903	0.188563
$583$	−18.0079	−0.745809
$584$	−7.56557	−0.313065
$585$	22.6949	0.938320
$586$	12.1989	0.503930
$587$	17.7291	0.731760	0.365880	−	0.930662i	$-0.380768\pi$
$587$	17.7291	0.731760	0.365880	+	0.930662i	$0.380768\pi$
$588$	0	0
$589$	−5.75863	−0.237280
$590$	−38.0783	−1.56766
$591$	−5.32928	−0.219217
$592$	−4.98566	−0.204909
$593$	42.1260	1.72991	0.864954	−	0.501852i	$-0.167348\pi$
$593$	42.1260	1.72991	0.864954	+	0.501852i	$0.167348\pi$
$594$	8.13307	0.333704
$595$	0	0
$596$	23.4366	0.960000
$597$	−9.70757	−0.397304
$598$	2.43007	0.0993730
$599$	−8.55477	−0.349538	−0.174769	−	0.984609i	$-0.555918\pi$
$599$	−8.55477	−0.349538	−0.174769	+	0.984609i	$0.555918\pi$
$600$	17.3830	0.709657
$601$	−5.49228	−0.224035	−0.112017	−	0.993706i	$-0.535731\pi$
$601$	−5.49228	−0.224035	−0.112017	+	0.993706i	$0.535731\pi$
$602$	0	0
$603$	20.2729	0.825578
$604$	−6.35280	−0.258492
$605$	−33.1553	−1.34795
$606$	0.413879	0.0168127
$607$	13.0024	0.527749	0.263875	−	0.964557i	$-0.414999\pi$
$607$	13.0024	0.527749	0.263875	+	0.964557i	$0.414999\pi$
$608$	−11.1147	−0.450760
$609$	0	0
$610$	−22.3590	−0.905288
$611$	−20.6171	−0.834079
$612$	−35.2275	−1.42399
$613$	−9.24230	−0.373293	−0.186646	−	0.982427i	$-0.559762\pi$
$613$	−9.24230	−0.373293	−0.186646	+	0.982427i	$0.559762\pi$
$614$	−10.1645	−0.410208
$615$	15.4294	0.622172
$616$	0	0
$617$	33.9563	1.36703	0.683515	−	0.729936i	$-0.260450\pi$
$617$	33.9563	1.36703	0.683515	+	0.729936i	$0.260450\pi$
$618$	1.69935	0.0683578
$619$	22.8026	0.916515	0.458258	−	0.888819i	$-0.348474\pi$
$619$	22.8026	0.916515	0.458258	+	0.888819i	$0.348474\pi$
$620$	20.7703	0.834156
$621$	6.64514	0.266660
$622$	−6.26721	−0.251292
$623$	0	0
$624$	−2.06915	−0.0828321
$625$	110.379	4.41517
$626$	12.9348	0.516978
$627$	4.75240	0.189793
$628$	−34.1039	−1.36090
$629$	20.2704	0.808232
$630$	0	0
$631$	2.92274	0.116352	0.0581762	−	0.998306i	$-0.481471\pi$
$631$	2.92274	0.116352	0.0581762	+	0.998306i	$0.481471\pi$
$632$	−31.3739	−1.24799
$633$	1.82194	0.0724156
$634$	−2.88442	−0.114555
$635$	−4.40463	−0.174793
$636$	−3.78247	−0.149985
$637$	0	0
$638$	−10.9054	−0.431750
$639$	−27.8088	−1.10010
$640$	50.4790	1.99536
$641$	−47.5449	−1.87791	−0.938955	−	0.344040i	$-0.888205\pi$
$641$	−47.5449	−1.87791	−0.938955	+	0.344040i	$0.888205\pi$
$642$	5.32915	0.210325
$643$	4.78341	0.188639	0.0943197	−	0.995542i	$-0.469932\pi$
$643$	4.78341	0.188639	0.0943197	+	0.995542i	$0.469932\pi$
$644$	0	0
$645$	24.9678	0.983106
$646$	9.59551	0.377530
$647$	−27.4077	−1.07751	−0.538753	−	0.842464i	$-0.681104\pi$
$647$	−27.4077	−1.07751	−0.538753	+	0.842464i	$0.681104\pi$
$648$	−13.8968	−0.545917
$649$	61.8914	2.42945
$650$	16.5503	0.649155
$651$	0	0
$652$	39.8043	1.55886
$653$	−1.65336	−0.0647009	−0.0323504	−	0.999477i	$-0.510299\pi$
$653$	−1.65336	−0.0647009	−0.0323504	+	0.999477i	$0.510299\pi$
$654$	2.75527	0.107740
$655$	17.6700	0.690424
$656$	12.4542	0.486256
$657$	−9.32219	−0.363693
$658$	0	0
$659$	21.2478	0.827697	0.413848	−	0.910346i	$-0.364184\pi$
$659$	21.2478	0.827697	0.413848	+	0.910346i	$0.364184\pi$
$660$	−17.1410	−0.667213
$661$	−33.1581	−1.28970	−0.644850	−	0.764309i	$-0.723080\pi$
$661$	−33.1581	−1.28970	−0.644850	+	0.764309i	$0.723080\pi$
$662$	−12.5397	−0.487369
$663$	8.41259	0.326718
$664$	−12.2109	−0.473874
$665$	0	0
$666$	4.11874	0.159598
$667$	−8.91029	−0.345008
$668$	6.58943	0.254953
$669$	9.34183	0.361176
$670$	19.9171	0.769464
$671$	36.3416	1.40295
$672$	0	0
$673$	11.3989	0.439394	0.219697	−	0.975568i	$-0.429493\pi$
$673$	11.3989	0.439394	0.219697	+	0.975568i	$0.429493\pi$
$674$	0.0928278	0.00357559
$675$	45.2575	1.74196
$676$	15.3139	0.588994
$677$	18.8081	0.722852	0.361426	−	0.932401i	$-0.382290\pi$
$677$	18.8081	0.722852	0.361426	+	0.932401i	$0.382290\pi$
$678$	5.67792	0.218059
$679$	0	0
$680$	−76.8538	−2.94721
$681$	3.42938	0.131414
$682$	7.44789	0.285195
$683$	−22.0996	−0.845619	−0.422810	−	0.906219i	$-0.638956\pi$
$683$	−22.0996	−0.845619	−0.422810	+	0.906219i	$0.638956\pi$
$684$	−8.83757	−0.337913
$685$	30.3248	1.15865
$686$	0	0
$687$	−0.646470	−0.0246644
$688$	20.1534	0.768343
$689$	−7.99704	−0.304663
$690$	3.08975	0.117625
$691$	10.0595	0.382682	0.191341	−	0.981524i	$-0.438716\pi$
$691$	10.0595	0.382682	0.191341	+	0.981524i	$0.438716\pi$
$692$	−9.69222	−0.368443
$693$	0	0
$694$	−10.6549	−0.404454
$695$	−30.5463	−1.15869
$696$	−5.08661	−0.192808
$697$	−50.6356	−1.91796
$698$	−7.39292	−0.279826
$699$	13.1609	0.497793
$700$	0	0
$701$	4.48043	0.169224	0.0846118	−	0.996414i	$-0.473035\pi$
$701$	4.48043	0.169224	0.0846118	+	0.996414i	$0.473035\pi$
$702$	3.61178	0.136318
$703$	5.08525	0.191794
$704$	−2.51316	−0.0947181
$705$	−26.2140	−0.987275
$706$	−9.70633	−0.365303
$707$	0	0
$708$	13.0000	0.488570
$709$	16.8448	0.632618	0.316309	−	0.948656i	$-0.397556\pi$
$709$	16.8448	0.632618	0.316309	+	0.948656i	$0.397556\pi$
$710$	−27.3207	−1.02533
$711$	−38.6585	−1.44981
$712$	−16.1674	−0.605898
$713$	6.08531	0.227897
$714$	0	0
$715$	−36.2402	−1.35531
$716$	2.87259	0.107354
$717$	−8.26573	−0.308689
$718$	6.60609	0.246537
$719$	−12.2620	−0.457295	−0.228647	−	0.973509i	$-0.573430\pi$
$719$	−12.2620	−0.457295	−0.228647	+	0.973509i	$0.573430\pi$
$720$	23.2918	0.868034
$721$	0	0
$722$	−9.01623	−0.335549
$723$	−5.57895	−0.207484
$724$	2.37455	0.0882494
$725$	−60.6845	−2.25377
$726$	−2.49722	−0.0926803
$727$	−50.2692	−1.86438	−0.932190	−	0.361969i	$-0.882105\pi$
$727$	−50.2692	−1.86438	−0.932190	+	0.361969i	$0.882105\pi$
$728$	0	0
$729$	−11.9211	−0.441523
$730$	−9.15856	−0.338973
$731$	−81.9385	−3.03060
$732$	7.63339	0.282138
$733$	−22.1666	−0.818742	−0.409371	−	0.912368i	$-0.634252\pi$
$733$	−22.1666	−0.818742	−0.409371	+	0.912368i	$0.634252\pi$
$734$	−9.45372	−0.348943
$735$	0	0
$736$	11.7452	0.432934
$737$	−32.3727	−1.19246
$738$	−10.2886	−0.378730
$739$	−13.9133	−0.511808	−0.255904	−	0.966702i	$-0.582373\pi$
$739$	−13.9133	−0.511808	−0.255904	+	0.966702i	$0.582373\pi$
$740$	−18.3415	−0.674248
$741$	2.11047	0.0775302
$742$	0	0
$743$	−42.1511	−1.54637	−0.773187	−	0.634178i	$-0.781339\pi$
$743$	−42.1511	−1.54637	−0.773187	+	0.634178i	$0.781339\pi$
$744$	3.47392	0.127360
$745$	63.0019	2.30821
$746$	2.25967	0.0827324
$747$	−15.0461	−0.550507
$748$	56.2528	2.05681
$749$	0	0
$750$	13.7369	0.501600
$751$	−7.52694	−0.274662	−0.137331	−	0.990525i	$-0.543852\pi$
$751$	−7.52694	−0.274662	−0.137331	+	0.990525i	$0.543852\pi$
$752$	−21.1593	−0.771601
$753$	1.23459	0.0449910
$754$	−4.84294	−0.176370
$755$	−17.0775	−0.621513
$756$	0	0
$757$	−25.7235	−0.934936	−0.467468	−	0.884010i	$-0.654834\pi$
$757$	−25.7235	−0.934936	−0.467468	+	0.884010i	$0.654834\pi$
$758$	5.68850	0.206616
$759$	−5.02200	−0.182287
$760$	−19.2804	−0.699373
$761$	10.1165	0.366723	0.183361	−	0.983046i	$-0.441302\pi$
$761$	10.1165	0.366723	0.183361	+	0.983046i	$0.441302\pi$
$762$	−0.331752	−0.0120181
$763$	0	0
$764$	−4.83775	−0.175023
$765$	−94.6981	−3.42382
$766$	6.22619	0.224961
$767$	27.4851	0.992430
$768$	3.15768	0.113943
$769$	6.87069	0.247763	0.123882	−	0.992297i	$-0.460466\pi$
$769$	6.87069	0.247763	0.123882	+	0.992297i	$0.460466\pi$
$770$	0	0
$771$	5.44030	0.195928
$772$	−32.0763	−1.15445
$773$	−17.8361	−0.641520	−0.320760	−	0.947160i	$-0.603938\pi$
$773$	−17.8361	−0.641520	−0.320760	+	0.947160i	$0.603938\pi$
$774$	−16.6491	−0.598439
$775$	41.4447	1.48874
$776$	29.9967	1.07682
$777$	0	0
$778$	10.2800	0.368554
$779$	−12.7030	−0.455132
$780$	−7.61209	−0.272556
$781$	44.4062	1.58898
$782$	−10.1398	−0.362600
$783$	−13.2433	−0.473275
$784$	0	0
$785$	−91.6777	−3.27212
$786$	1.33088	0.0474710
$787$	15.3891	0.548562	0.274281	−	0.961650i	$-0.411560\pi$
$787$	15.3891	0.548562	0.274281	+	0.961650i	$0.411560\pi$
$788$	−15.8252	−0.563751
$789$	4.92478	0.175327
$790$	−37.9799	−1.35126
$791$	0	0
$792$	25.3817	0.901899
$793$	16.1388	0.573106
$794$	13.2934	0.471766
$795$	−10.1680	−0.360621
$796$	−28.8265	−1.02173
$797$	39.1163	1.38557	0.692786	−	0.721143i	$-0.256383\pi$
$797$	39.1163	1.38557	0.692786	+	0.721143i	$0.256383\pi$
$798$	0	0
$799$	86.0281	3.04346
$800$	79.9920	2.82815
$801$	−19.9212	−0.703882
$802$	22.9399	0.810035
$803$	14.8861	0.525318
$804$	−6.79973	−0.239808
$805$	0	0
$806$	3.30751	0.116502
$807$	0.166082	0.00584637
$808$	2.72916	0.0960115
$809$	−6.83182	−0.240194	−0.120097	−	0.992762i	$-0.538321\pi$
$809$	−6.83182	−0.240194	−0.120097	+	0.992762i	$0.538321\pi$
$810$	−16.8228	−0.591095
$811$	−36.2145	−1.27166	−0.635832	−	0.771828i	$-0.719343\pi$
$811$	−36.2145	−1.27166	−0.635832	+	0.771828i	$0.719343\pi$
$812$	0	0
$813$	−2.31716	−0.0812663
$814$	−6.57697	−0.230523
$815$	107.001	3.74809
$816$	8.63383	0.302245
$817$	−20.5560	−0.719163
$818$	−5.71421	−0.199793
$819$	0	0
$820$	45.8173	1.60001
$821$	14.8622	0.518693	0.259346	−	0.965784i	$-0.416493\pi$
$821$	14.8622	0.518693	0.259346	+	0.965784i	$0.416493\pi$
$822$	2.28403	0.0796645
$823$	15.4230	0.537611	0.268805	−	0.963195i	$-0.413371\pi$
$823$	15.4230	0.537611	0.268805	+	0.963195i	$0.413371\pi$
$824$	11.2057	0.390368
$825$	−34.2029	−1.19079
$826$	0	0
$827$	−35.9214	−1.24911	−0.624554	−	0.780981i	$-0.714719\pi$
$827$	−35.9214	−1.24911	−0.624554	+	0.780981i	$0.714719\pi$
$828$	9.33892	0.324550
$829$	−31.5985	−1.09746	−0.548730	−	0.836000i	$-0.684889\pi$
$829$	−31.5985	−1.09746	−0.548730	+	0.836000i	$0.684889\pi$
$830$	−14.7820	−0.513090
$831$	−8.27986	−0.287225
$832$	−1.11606	−0.0386923
$833$	0	0
$834$	−2.30071	−0.0796671
$835$	17.7136	0.613005
$836$	14.1122	0.488081
$837$	9.04453	0.312625
$838$	−3.40302	−0.117556
$839$	39.5472	1.36532	0.682660	−	0.730736i	$-0.260823\pi$
$839$	39.5472	1.36532	0.682660	+	0.730736i	$0.260823\pi$
$840$	0	0
$841$	−11.2425	−0.387671
$842$	−6.01349	−0.207238
$843$	3.19514	0.110047
$844$	5.41023	0.186228
$845$	41.1665	1.41617
$846$	17.4801	0.600977
$847$	0	0
$848$	−8.20735	−0.281842
$849$	12.8138	0.439768
$850$	−69.0586	−2.36869
$851$	−5.37373	−0.184209
$852$	9.32732	0.319549
$853$	5.00383	0.171328	0.0856639	−	0.996324i	$-0.472699\pi$
$853$	5.00383	0.171328	0.0856639	+	0.996324i	$0.472699\pi$
$854$	0	0
$855$	−23.7570	−0.812473
$856$	35.1410	1.20109
$857$	−41.4504	−1.41592	−0.707959	−	0.706254i	$-0.750384\pi$
$857$	−41.4504	−1.41592	−0.707959	+	0.706254i	$0.750384\pi$
$858$	−2.72957	−0.0931859
$859$	−27.5845	−0.941171	−0.470585	−	0.882354i	$-0.655957\pi$
$859$	−27.5845	−0.941171	−0.470585	+	0.882354i	$0.655957\pi$
$860$	74.1416	2.52821
$861$	0	0
$862$	1.77752	0.0605426
$863$	39.6613	1.35009	0.675043	−	0.737778i	$-0.264125\pi$
$863$	39.6613	1.35009	0.675043	+	0.737778i	$0.264125\pi$
$864$	17.4567	0.593891
$865$	−26.0545	−0.885879
$866$	22.9805	0.780909
$867$	−25.7225	−0.873582
$868$	0	0
$869$	61.7314	2.09410
$870$	−6.15764	−0.208764
$871$	−14.3763	−0.487121
$872$	18.1685	0.615264
$873$	36.9615	1.25096
$874$	−2.54379	−0.0860451
$875$	0	0
$876$	3.12675	0.105643
$877$	−4.96526	−0.167665	−0.0838324	−	0.996480i	$-0.526716\pi$
$877$	−4.96526	−0.167665	−0.0838324	+	0.996480i	$0.526716\pi$
$878$	−2.15197	−0.0726253
$879$	−11.1955	−0.377615
$880$	−37.1933	−1.25379
$881$	−32.2888	−1.08784	−0.543919	−	0.839138i	$-0.683060\pi$
$881$	−32.2888	−1.08784	−0.543919	+	0.839138i	$0.683060\pi$
$882$	0	0
$883$	−19.3735	−0.651969	−0.325984	−	0.945375i	$-0.605696\pi$
$883$	−19.3735	−0.651969	−0.325984	+	0.945375i	$0.605696\pi$
$884$	24.9811	0.840205
$885$	34.9464	1.17471
$886$	−0.732779	−0.0246182
$887$	2.87426	0.0965082	0.0482541	−	0.998835i	$-0.484634\pi$
$887$	2.87426	0.0965082	0.0482541	+	0.998835i	$0.484634\pi$
$888$	−3.06770	−0.102945
$889$	0	0
$890$	−19.5716	−0.656040
$891$	27.3434	0.916038
$892$	27.7405	0.928819
$893$	21.5820	0.722213
$894$	4.74523	0.158704
$895$	7.72204	0.258119
$896$	0	0
$897$	−2.23020	−0.0744642
$898$	21.9939	0.733946
$899$	−12.1276	−0.404477
$900$	63.6037	2.12012
$901$	33.3689	1.11168
$902$	16.4293	0.547037
$903$	0	0
$904$	37.4408	1.24526
$905$	6.38323	0.212186
$906$	−1.28626	−0.0427330
$907$	−31.5100	−1.04627	−0.523137	−	0.852249i	$-0.675238\pi$
$907$	−31.5100	−1.04627	−0.523137	+	0.852249i	$0.675238\pi$
$908$	10.1835	0.337952
$909$	3.36283	0.111538
$910$	0	0
$911$	8.79160	0.291279	0.145639	−	0.989338i	$-0.453476\pi$
$911$	8.79160	0.291279	0.145639	+	0.989338i	$0.453476\pi$
$912$	2.16598	0.0717227
$913$	24.0262	0.795152
$914$	−0.425722	−0.0140816
$915$	20.5200	0.678369
$916$	−1.91969	−0.0634282
$917$	0	0
$918$	−15.0707	−0.497408
$919$	11.9465	0.394079	0.197040	−	0.980396i	$-0.436867\pi$
$919$	11.9465	0.394079	0.197040	+	0.980396i	$0.436867\pi$
$920$	20.3741	0.671715
$921$	9.32852	0.307385
$922$	4.47348	0.147326
$923$	19.7202	0.649098
$924$	0	0
$925$	−36.5984	−1.20335
$926$	13.8264	0.454364
$927$	13.8075	0.453497
$928$	−23.4073	−0.768382
$929$	−14.7987	−0.485530	−0.242765	−	0.970085i	$-0.578054\pi$
$929$	−14.7987	−0.485530	−0.242765	+	0.970085i	$0.578054\pi$
$930$	4.20538	0.137900
$931$	0	0
$932$	39.0813	1.28015
$933$	5.75174	0.188304
$934$	14.0502	0.459736
$935$	151.218	4.94536
$936$	11.2717	0.368425
$937$	−30.4352	−0.994274	−0.497137	−	0.867672i	$-0.665615\pi$
$937$	−30.4352	−0.994274	−0.497137	+	0.867672i	$0.665615\pi$
$938$	0	0
$939$	−11.8709	−0.387393
$940$	−77.8421	−2.53893
$941$	−57.3813	−1.87058	−0.935288	−	0.353887i	$-0.884860\pi$
$941$	−57.3813	−1.87058	−0.935288	+	0.353887i	$0.884860\pi$
$942$	−6.90505	−0.224979
$943$	13.4236	0.437133
$944$	28.2079	0.918090
$945$	0	0
$946$	26.5860	0.864384
$947$	−28.7119	−0.933012	−0.466506	−	0.884518i	$-0.654487\pi$
$947$	−28.7119	−0.933012	−0.466506	+	0.884518i	$0.654487\pi$
$948$	12.9664	0.421129
$949$	6.61069	0.214592
$950$	−17.3248	−0.562091
$951$	2.64718	0.0858406
$952$	0	0
$953$	−33.7610	−1.09363	−0.546814	−	0.837254i	$-0.684159\pi$
$953$	−33.7610	−1.09363	−0.546814	+	0.837254i	$0.684159\pi$
$954$	6.78023	0.219518
$955$	−13.0048	−0.420824
$956$	−24.5450	−0.793842
$957$	10.0085	0.323527
$958$	−6.57056	−0.212285
$959$	0	0
$960$	−1.41903	−0.0457990
$961$	−22.7174	−0.732821
$962$	−2.92074	−0.0941685
$963$	43.3002	1.39533
$964$	−16.5666	−0.533576
$965$	−86.2269	−2.77574
$966$	0	0
$967$	28.9226	0.930089	0.465045	−	0.885287i	$-0.346038\pi$
$967$	28.9226	0.930089	0.465045	+	0.885287i	$0.346038\pi$
$968$	−16.4669	−0.529266
$969$	−8.80629	−0.282899
$970$	36.3128	1.16593
$971$	−0.773763	−0.0248312	−0.0124156	−	0.999923i	$-0.503952\pi$
$971$	−0.773763	−0.0248312	−0.0124156	+	0.999923i	$0.503952\pi$
$972$	21.1915	0.679717
$973$	0	0
$974$	−0.155867	−0.00499430
$975$	−15.1890	−0.486438
$976$	16.5633	0.530177
$977$	−18.8567	−0.603278	−0.301639	−	0.953422i	$-0.597534\pi$
$977$	−18.8567	−0.603278	−0.301639	+	0.953422i	$0.597534\pi$
$978$	8.05921	0.257705
$979$	31.8111	1.01669
$980$	0	0
$981$	22.3870	0.714763
$982$	7.97145	0.254379
$983$	−4.28493	−0.136668	−0.0683340	−	0.997663i	$-0.521768\pi$
$983$	−4.28493	−0.136668	−0.0683340	+	0.997663i	$0.521768\pi$
$984$	7.66314	0.244292
$985$	−42.5412	−1.35547
$986$	20.2079	0.643552
$987$	0	0
$988$	6.26703	0.199381
$989$	21.7221	0.690723
$990$	30.7260	0.976536
$991$	1.52397	0.0484105	0.0242052	−	0.999707i	$-0.492294\pi$
$991$	1.52397	0.0484105	0.0242052	+	0.999707i	$0.492294\pi$
$992$	15.9861	0.507559
$993$	11.5083	0.365205
$994$	0	0
$995$	−77.4910	−2.45663
$996$	5.04659	0.159907
$997$	35.2873	1.11756	0.558780	−	0.829316i	$-0.311270\pi$
$997$	35.2873	1.11756	0.558780	+	0.829316i	$0.311270\pi$
$998$	−19.5653	−0.619329
$999$	−7.98690	−0.252694

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6223.2.a.j.1.9		15
7.6	odd	2		889.2.a.b.1.9	✓	15
21.20	even	2		8001.2.a.q.1.7		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.b.1.9	✓	15	7.6	odd	2
6223.2.a.j.1.9		15	1.1	even	1	trivial
8001.2.a.q.1.7		15	21.20	even	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:	\( N \)	\(=\)	\( 6223 = 7^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6223.a (trivial)

\(f(q)\)	\(=\)	\(q+0.601235 q^{2} -0.551784 q^{3} -1.63852 q^{4} -4.40463 q^{5} -0.331752 q^{6} -2.18760 q^{8} -2.69553 q^{9} +O(q^{10})\)	\(q+0.601235 q^{2} -0.551784 q^{3} -1.63852 q^{4} -4.40463 q^{5} -0.331752 q^{6} -2.18760 q^{8} -2.69553 q^{9} -2.64822 q^{10} +4.30434 q^{11} +0.904107 q^{12} +1.91150 q^{13} +2.43041 q^{15} +1.96177 q^{16} -7.97603 q^{17} -1.62065 q^{18} -2.00095 q^{19} +7.21707 q^{20} +2.58792 q^{22} +2.11447 q^{23} +1.20708 q^{24} +14.4008 q^{25} +1.14926 q^{26} +3.14270 q^{27} -4.21397 q^{29} +1.46125 q^{30} +2.87794 q^{31} +5.55469 q^{32} -2.37507 q^{33} -4.79547 q^{34} +4.41668 q^{36} -2.54141 q^{37} -1.20304 q^{38} -1.05473 q^{39} +9.63559 q^{40} +6.34847 q^{41} +10.2731 q^{43} -7.05274 q^{44} +11.8728 q^{45} +1.27129 q^{46} -10.7858 q^{47} -1.08247 q^{48} +8.65827 q^{50} +4.40104 q^{51} -3.13202 q^{52} -4.18365 q^{53} +1.88950 q^{54} -18.9591 q^{55} +1.10409 q^{57} -2.53358 q^{58} +14.3788 q^{59} -3.98226 q^{60} +8.44301 q^{61} +1.73032 q^{62} -0.583865 q^{64} -8.41946 q^{65} -1.42797 q^{66} -7.52093 q^{67} +13.0689 q^{68} -1.16673 q^{69} +10.3166 q^{71} +5.89676 q^{72} +3.45838 q^{73} -1.52799 q^{74} -7.94613 q^{75} +3.27860 q^{76} -0.634143 q^{78} +14.3417 q^{79} -8.64088 q^{80} +6.35251 q^{81} +3.81692 q^{82} +5.58186 q^{83} +35.1315 q^{85} +6.17654 q^{86} +2.32520 q^{87} -9.41619 q^{88} +7.39046 q^{89} +7.13837 q^{90} -3.46459 q^{92} -1.58800 q^{93} -6.48482 q^{94} +8.81347 q^{95} -3.06499 q^{96} -13.7121 q^{97} -11.6025 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 4 q^{3} + 14 q^{4} - 7 q^{5} - 8 q^{6} + 9 q^{9}+O(q^{10}) \) 15 * q - 4 * q^3 + 14 * q^4 - 7 * q^5 - 8 * q^6 + 9 * q^9	\( 15 q - 4 q^{3} + 14 q^{4} - 7 q^{5} - 8 q^{6} + 9 q^{9} - 10 q^{10} + 14 q^{11} - 10 q^{12} - 6 q^{13} + 6 q^{15} + 20 q^{16} - 10 q^{17} + q^{18} - 13 q^{19} - 8 q^{20} - 11 q^{22} + 15 q^{23} - 34 q^{24} - 22 q^{26} - 22 q^{27} + 16 q^{29} + 7 q^{30} - 22 q^{31} - 14 q^{33} + 15 q^{34} + 20 q^{36} - 14 q^{37} + 6 q^{38} + 29 q^{39} - 22 q^{40} - 19 q^{41} - q^{43} + 25 q^{44} + 8 q^{45} - 28 q^{46} - 49 q^{47} + 14 q^{48} + 24 q^{50} - 8 q^{51} + 17 q^{52} - 28 q^{53} - 13 q^{54} - 39 q^{55} - 12 q^{57} - 10 q^{58} - 43 q^{59} - 60 q^{60} - 27 q^{61} - 14 q^{62} + 18 q^{64} - 8 q^{65} + 36 q^{66} + 3 q^{67} - 13 q^{68} + 17 q^{69} + 55 q^{71} - 21 q^{72} + 3 q^{73} - 12 q^{74} - 8 q^{75} + 20 q^{76} - 6 q^{78} + 18 q^{79} - 29 q^{80} - 17 q^{81} - 14 q^{82} - 17 q^{83} + 7 q^{85} + 4 q^{86} - 35 q^{87} - 114 q^{88} - 36 q^{89} + 39 q^{90} + 45 q^{92} + 15 q^{93} + 15 q^{94} + 59 q^{95} - 85 q^{96} + 2 q^{97} + 35 q^{99}+O(q^{100}) \) 15 * q - 4 * q^3 + 14 * q^4 - 7 * q^5 - 8 * q^6 + 9 * q^9 - 10 * q^10 + 14 * q^11 - 10 * q^12 - 6 * q^13 + 6 * q^15 + 20 * q^16 - 10 * q^17 + q^18 - 13 * q^19 - 8 * q^20 - 11 * q^22 + 15 * q^23 - 34 * q^24 - 22 * q^26 - 22 * q^27 + 16 * q^29 + 7 * q^30 - 22 * q^31 - 14 * q^33 + 15 * q^34 + 20 * q^36 - 14 * q^37 + 6 * q^38 + 29 * q^39 - 22 * q^40 - 19 * q^41 - q^43 + 25 * q^44 + 8 * q^45 - 28 * q^46 - 49 * q^47 + 14 * q^48 + 24 * q^50 - 8 * q^51 + 17 * q^52 - 28 * q^53 - 13 * q^54 - 39 * q^55 - 12 * q^57 - 10 * q^58 - 43 * q^59 - 60 * q^60 - 27 * q^61 - 14 * q^62 + 18 * q^64 - 8 * q^65 + 36 * q^66 + 3 * q^67 - 13 * q^68 + 17 * q^69 + 55 * q^71 - 21 * q^72 + 3 * q^73 - 12 * q^74 - 8 * q^75 + 20 * q^76 - 6 * q^78 + 18 * q^79 - 29 * q^80 - 17 * q^81 - 14 * q^82 - 17 * q^83 + 7 * q^85 + 4 * q^86 - 35 * q^87 - 114 * q^88 - 36 * q^89 + 39 * q^90 + 45 * q^92 + 15 * q^93 + 15 * q^94 + 59 * q^95 - 85 * q^96 + 2 * q^97 + 35 * q^99

Introduction

L-functions

Modular forms

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Database

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