LMFDB - Embedded newform 543.2.a.d.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [543,2,Mod(1,543)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("543.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(543, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 543 = 3 \cdot 181 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	543.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,3,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.33587682976$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 6x^{6} + 21x^{5} + 5x^{4} - 35x^{3} + 10x^{2} + 4x - 1 $ x^8 - 3x^7 - 6x^6 + 21x^5 + 5x^4 - 35x^3 + 10x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.48495$ of defining polynomial
Character	$\chi$	$=$	543.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.48495 q^{2} -1.00000 q^{3} +4.17495 q^{4} +0.370811 q^{5} -2.48495 q^{6} -1.96328 q^{7} +5.40464 q^{8} +1.00000 q^{9} +0.921444 q^{10} +4.04845 q^{11} -4.17495 q^{12} +4.06041 q^{13} -4.87863 q^{14} -0.370811 q^{15} +5.08033 q^{16} +3.62919 q^{17} +2.48495 q^{18} +5.64676 q^{19} +1.54812 q^{20} +1.96328 q^{21} +10.0602 q^{22} -8.92443 q^{23} -5.40464 q^{24} -4.86250 q^{25} +10.0899 q^{26} -1.00000 q^{27} -8.19659 q^{28} -6.74722 q^{29} -0.921444 q^{30} -8.80518 q^{31} +1.81507 q^{32} -4.04845 q^{33} +9.01834 q^{34} -0.728004 q^{35} +4.17495 q^{36} -0.921480 q^{37} +14.0319 q^{38} -4.06041 q^{39} +2.00410 q^{40} -0.316916 q^{41} +4.87863 q^{42} -2.22243 q^{43} +16.9021 q^{44} +0.370811 q^{45} -22.1767 q^{46} -4.93727 q^{47} -5.08033 q^{48} -3.14555 q^{49} -12.0830 q^{50} -3.62919 q^{51} +16.9520 q^{52} +14.1879 q^{53} -2.48495 q^{54} +1.50121 q^{55} -10.6108 q^{56} -5.64676 q^{57} -16.7665 q^{58} -0.537828 q^{59} -1.54812 q^{60} -2.35027 q^{61} -21.8804 q^{62} -1.96328 q^{63} -5.65033 q^{64} +1.50564 q^{65} -10.0602 q^{66} +10.6251 q^{67} +15.1517 q^{68} +8.92443 q^{69} -1.80905 q^{70} +1.33418 q^{71} +5.40464 q^{72} +8.35531 q^{73} -2.28983 q^{74} +4.86250 q^{75} +23.5749 q^{76} -7.94822 q^{77} -10.0899 q^{78} -15.5212 q^{79} +1.88384 q^{80} +1.00000 q^{81} -0.787518 q^{82} +9.55110 q^{83} +8.19659 q^{84} +1.34574 q^{85} -5.52261 q^{86} +6.74722 q^{87} +21.8804 q^{88} +3.36562 q^{89} +0.921444 q^{90} -7.97171 q^{91} -37.2591 q^{92} +8.80518 q^{93} -12.2688 q^{94} +2.09388 q^{95} -1.81507 q^{96} +12.3388 q^{97} -7.81652 q^{98} +4.04845 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{2} - 8 q^{3} + 5 q^{4} + 5 q^{5} - 3 q^{6} - 3 q^{7} + 6 q^{8} + 8 q^{9} + 2 q^{10} + 4 q^{11} - 5 q^{12} + 13 q^{13} + 5 q^{14} - 5 q^{15} + 3 q^{16} + 27 q^{17} + 3 q^{18} - 10 q^{19} + 21 q^{20}+ \cdots + 4 q^{99}+O(q^{100}) $ 8 * q + 3 * q^2 - 8 * q^3 + 5 * q^4 + 5 * q^5 - 3 * q^6 - 3 * q^7 + 6 * q^8 + 8 * q^9 + 2 * q^10 + 4 * q^11 - 5 * q^12 + 13 * q^13 + 5 * q^14 - 5 * q^15 + 3 * q^16 + 27 * q^17 + 3 * q^18 - 10 * q^19 + 21 * q^20 + 3 * q^21 + 11 * q^22 + 3 * q^23 - 6 * q^24 + 13 * q^25 + 11 * q^26 - 8 * q^27 - 3 * q^28 + 16 * q^29 - 2 * q^30 - q^31 + 10 * q^32 - 4 * q^33 + 10 * q^34 + 14 * q^35 + 5 * q^36 + 3 * q^37 + 14 * q^38 - 13 * q^39 + 11 * q^40 + 20 * q^41 - 5 * q^42 - 4 * q^43 + 9 * q^44 + 5 * q^45 - 17 * q^46 + 12 * q^47 - 3 * q^48 + 5 * q^49 + 2 * q^50 - 27 * q^51 + 21 * q^52 + 23 * q^53 - 3 * q^54 - 13 * q^55 - 8 * q^56 + 10 * q^57 - 21 * q^58 + 7 * q^59 - 21 * q^60 + q^61 + q^62 - 3 * q^63 - 32 * q^64 + 10 * q^65 - 11 * q^66 - 25 * q^67 - q^68 - 3 * q^69 - 10 * q^70 + 11 * q^71 + 6 * q^72 + 23 * q^73 + 2 * q^74 - 13 * q^75 - 19 * q^76 + 20 * q^77 - 11 * q^78 - 26 * q^79 - 15 * q^80 + 8 * q^81 - 22 * q^82 + 16 * q^83 + 3 * q^84 - 33 * q^85 - 21 * q^86 - 16 * q^87 - q^88 + 14 * q^89 + 2 * q^90 - 35 * q^91 - 18 * q^92 + q^93 - 52 * q^94 + 10 * q^95 - 10 * q^96 + 28 * q^97 - 51 * q^98 + 4 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.48495	1.75712	0.878561	−	0.477630i	$-0.158504\pi$
$2$	2.48495	1.75712	0.878561	+	0.477630i	$0.158504\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.17495	2.08748
$5$	0.370811	0.165832	0.0829158	−	0.996557i	$-0.473577\pi$
$5$	0.370811	0.165832	0.0829158	+	0.996557i	$0.473577\pi$
$6$	−2.48495	−1.01447
$7$	−1.96328	−0.742048	−0.371024	−	0.928623i	$-0.620993\pi$
$7$	−1.96328	−0.742048	−0.371024	+	0.928623i	$0.620993\pi$
$8$	5.40464	1.91083
$9$	1.00000	0.333333
$10$	0.921444	0.291386
$11$	4.04845	1.22065	0.610326	−	0.792150i	$-0.291038\pi$
$11$	4.04845	1.22065	0.610326	+	0.792150i	$0.291038\pi$
$12$	−4.17495	−1.20521
$13$	4.06041	1.12616	0.563078	−	0.826404i	$-0.309617\pi$
$13$	4.06041	1.12616	0.563078	+	0.826404i	$0.309617\pi$
$14$	−4.87863	−1.30387
$15$	−0.370811	−0.0957429
$16$	5.08033	1.27008
$17$	3.62919	0.880208	0.440104	−	0.897947i	$-0.354942\pi$
$17$	3.62919	0.880208	0.440104	+	0.897947i	$0.354942\pi$
$18$	2.48495	0.585707
$19$	5.64676	1.29545	0.647727	−	0.761872i	$-0.275720\pi$
$19$	5.64676	1.29545	0.647727	+	0.761872i	$0.275720\pi$
$20$	1.54812	0.346170
$21$	1.96328	0.428422
$22$	10.0602	2.14484
$23$	−8.92443	−1.86087	−0.930436	−	0.366454i	$-0.880572\pi$
$23$	−8.92443	−1.86087	−0.930436	+	0.366454i	$0.880572\pi$
$24$	−5.40464	−1.10322
$25$	−4.86250	−0.972500
$26$	10.0899	1.97879
$27$	−1.00000	−0.192450
$28$	−8.19659	−1.54901
$29$	−6.74722	−1.25293	−0.626464	−	0.779451i	$-0.715498\pi$
$29$	−6.74722	−1.25293	−0.626464	+	0.779451i	$0.715498\pi$
$30$	−0.921444	−0.168232
$31$	−8.80518	−1.58146	−0.790729	−	0.612166i	$-0.790298\pi$
$31$	−8.80518	−1.58146	−0.790729	+	0.612166i	$0.790298\pi$
$32$	1.81507	0.320861
$33$	−4.04845	−0.704744
$34$	9.01834	1.54663
$35$	−0.728004	−0.123055
$36$	4.17495	0.695826
$37$	−0.921480	−0.151490	−0.0757452	−	0.997127i	$-0.524134\pi$
$37$	−0.921480	−0.151490	−0.0757452	+	0.997127i	$0.524134\pi$
$38$	14.0319	2.27627
$39$	−4.06041	−0.650186
$40$	2.00410	0.316876
$41$	−0.316916	−0.0494939	−0.0247470	−	0.999694i	$-0.507878\pi$
$41$	−0.316916	−0.0494939	−0.0247470	+	0.999694i	$0.507878\pi$
$42$	4.87863	0.752789
$43$	−2.22243	−0.338917	−0.169458	−	0.985537i	$-0.554202\pi$
$43$	−2.22243	−0.338917	−0.169458	+	0.985537i	$0.554202\pi$
$44$	16.9021	2.54808
$45$	0.370811	0.0552772
$46$	−22.1767	−3.26978
$47$	−4.93727	−0.720174	−0.360087	−	0.932919i	$-0.617253\pi$
$47$	−4.93727	−0.720174	−0.360087	+	0.932919i	$0.617253\pi$
$48$	−5.08033	−0.733283
$49$	−3.14555	−0.449364
$50$	−12.0830	−1.70880
$51$	−3.62919	−0.508188
$52$	16.9520	2.35082
$53$	14.1879	1.94885	0.974426	−	0.224711i	$-0.0721437\pi$
$53$	14.1879	1.94885	0.974426	+	0.224711i	$0.0721437\pi$
$54$	−2.48495	−0.338158
$55$	1.50121	0.202423
$56$	−10.6108	−1.41793
$57$	−5.64676	−0.747931
$58$	−16.7665	−2.20155
$59$	−0.537828	−0.0700192	−0.0350096	−	0.999387i	$-0.511146\pi$
$59$	−0.537828	−0.0700192	−0.0350096	+	0.999387i	$0.511146\pi$
$60$	−1.54812	−0.199861
$61$	−2.35027	−0.300921	−0.150461	−	0.988616i	$-0.548076\pi$
$61$	−2.35027	−0.300921	−0.150461	+	0.988616i	$0.548076\pi$
$62$	−21.8804	−2.77881
$63$	−1.96328	−0.247349
$64$	−5.65033	−0.706291
$65$	1.50564	0.186752
$66$	−10.0602	−1.23832
$67$	10.6251	1.29806	0.649032	−	0.760761i	$-0.275174\pi$
$67$	10.6251	1.29806	0.649032	+	0.760761i	$0.275174\pi$
$68$	15.1517	1.83741
$69$	8.92443	1.07438
$70$	−1.80905	−0.216223
$71$	1.33418	0.158338	0.0791691	−	0.996861i	$-0.474773\pi$
$71$	1.33418	0.158338	0.0791691	+	0.996861i	$0.474773\pi$
$72$	5.40464	0.636943
$73$	8.35531	0.977915	0.488958	−	0.872308i	$-0.337377\pi$
$73$	8.35531	0.977915	0.488958	+	0.872308i	$0.337377\pi$
$74$	−2.28983	−0.266187
$75$	4.86250	0.561473
$76$	23.5749	2.70423
$77$	−7.94822	−0.905783
$78$	−10.0899	−1.14246
$79$	−15.5212	−1.74627	−0.873137	−	0.487475i	$-0.837918\pi$
$79$	−15.5212	−1.74627	−0.873137	+	0.487475i	$0.837918\pi$
$80$	1.88384	0.210620
$81$	1.00000	0.111111
$82$	−0.787518	−0.0869669
$83$	9.55110	1.04837	0.524185	−	0.851604i	$-0.324370\pi$
$83$	9.55110	1.04837	0.524185	+	0.851604i	$0.324370\pi$
$84$	8.19659	0.894321
$85$	1.34574	0.145966
$86$	−5.52261	−0.595518
$87$	6.74722	0.723378
$88$	21.8804	2.33246
$89$	3.36562	0.356755	0.178378	−	0.983962i	$-0.442915\pi$
$89$	3.36562	0.356755	0.178378	+	0.983962i	$0.442915\pi$
$90$	0.921444	0.0971288
$91$	−7.97171	−0.835662
$92$	−37.2591	−3.88453
$93$	8.80518	0.913055
$94$	−12.2688	−1.26543
$95$	2.09388	0.214827
$96$	−1.81507	−0.185249
$97$	12.3388	1.25281	0.626407	−	0.779496i	$-0.284525\pi$
$97$	12.3388	1.25281	0.626407	+	0.779496i	$0.284525\pi$
$98$	−7.81652	−0.789587
$99$	4.04845	0.406884
$100$	−20.3007	−2.03007
$101$	1.21813	0.121208	0.0606042	−	0.998162i	$-0.480697\pi$
$101$	1.21813	0.121208	0.0606042	+	0.998162i	$0.480697\pi$
$102$	−9.01834	−0.892948
$103$	−1.05825	−0.104272	−0.0521361	−	0.998640i	$-0.516603\pi$
$103$	−1.05825	−0.104272	−0.0521361	+	0.998640i	$0.516603\pi$
$104$	21.9451	2.15189
$105$	0.728004	0.0710459
$106$	35.2560	3.42437
$107$	−3.95524	−0.382368	−0.191184	−	0.981554i	$-0.561233\pi$
$107$	−3.95524	−0.382368	−0.191184	+	0.981554i	$0.561233\pi$
$108$	−4.17495	−0.401735
$109$	−1.34588	−0.128912	−0.0644562	−	0.997921i	$-0.520531\pi$
$109$	−1.34588	−0.128912	−0.0644562	+	0.997921i	$0.520531\pi$
$110$	3.73042	0.355681
$111$	0.921480	0.0874630
$112$	−9.97409	−0.942463
$113$	−2.24544	−0.211233	−0.105617	−	0.994407i	$-0.533682\pi$
$113$	−2.24544	−0.211233	−0.105617	+	0.994407i	$0.533682\pi$
$114$	−14.0319	−1.31421
$115$	−3.30927	−0.308591
$116$	−28.1693	−2.61546
$117$	4.06041	0.375385
$118$	−1.33647	−0.123032
$119$	−7.12510	−0.653157
$120$	−2.00410	−0.182948
$121$	5.38992	0.489993
$122$	−5.84030	−0.528756
$123$	0.316916	0.0285753
$124$	−36.7612	−3.30126
$125$	−3.65712	−0.327103
$126$	−4.87863	−0.434623
$127$	−17.6937	−1.57006	−0.785032	−	0.619455i	$-0.787354\pi$
$127$	−17.6937	−1.57006	−0.785032	+	0.619455i	$0.787354\pi$
$128$	−17.6709	−1.56190
$129$	2.22243	0.195674
$130$	3.74144	0.328146
$131$	13.4044	1.17115	0.585575	−	0.810618i	$-0.300869\pi$
$131$	13.4044	1.17115	0.585575	+	0.810618i	$0.300869\pi$
$132$	−16.9021	−1.47114
$133$	−11.0861	−0.961290
$134$	26.4028	2.28086
$135$	−0.370811	−0.0319143
$136$	19.6145	1.68193
$137$	5.68019	0.485292	0.242646	−	0.970115i	$-0.421985\pi$
$137$	5.68019	0.485292	0.242646	+	0.970115i	$0.421985\pi$
$138$	22.1767	1.88781
$139$	−17.0855	−1.44917	−0.724585	−	0.689185i	$-0.757969\pi$
$139$	−17.0855	−1.44917	−0.724585	+	0.689185i	$0.757969\pi$
$140$	−3.03938	−0.256875
$141$	4.93727	0.415793
$142$	3.31537	0.278220
$143$	16.4384	1.37465
$144$	5.08033	0.423361
$145$	−2.50194	−0.207775
$146$	20.7625	1.71832
$147$	3.14555	0.259440
$148$	−3.84714	−0.316233
$149$	−12.9363	−1.05978	−0.529890	−	0.848067i	$-0.677767\pi$
$149$	−12.9363	−1.05978	−0.529890	+	0.848067i	$0.677767\pi$
$150$	12.0830	0.986577
$151$	−10.7174	−0.872173	−0.436087	−	0.899905i	$-0.643636\pi$
$151$	−10.7174	−0.872173	−0.436087	+	0.899905i	$0.643636\pi$
$152$	30.5187	2.47539
$153$	3.62919	0.293403
$154$	−19.7509	−1.59157
$155$	−3.26506	−0.262256
$156$	−16.9520	−1.35725
$157$	−3.06175	−0.244354	−0.122177	−	0.992508i	$-0.538988\pi$
$157$	−3.06175	−0.244354	−0.122177	+	0.992508i	$0.538988\pi$
$158$	−38.5694	−3.06842
$159$	−14.1879	−1.12517
$160$	0.673046	0.0532089
$161$	17.5211	1.38086
$162$	2.48495	0.195236
$163$	5.50865	0.431471	0.215735	−	0.976452i	$-0.430785\pi$
$163$	5.50865	0.431471	0.215735	+	0.976452i	$0.430785\pi$
$164$	−1.32311	−0.103317
$165$	−1.50121	−0.116869
$166$	23.7340	1.84211
$167$	20.2718	1.56868	0.784341	−	0.620330i	$-0.213001\pi$
$167$	20.2718	1.56868	0.784341	+	0.620330i	$0.213001\pi$
$168$	10.6108	0.818641
$169$	3.48696	0.268227
$170$	3.34410	0.256480
$171$	5.64676	0.431818
$172$	−9.27853	−0.707481
$173$	19.9597	1.51751	0.758753	−	0.651378i	$-0.225809\pi$
$173$	19.9597	1.51751	0.758753	+	0.651378i	$0.225809\pi$
$174$	16.7665	1.27106
$175$	9.54643	0.721642
$176$	20.5675	1.55033
$177$	0.537828	0.0404256
$178$	8.36339	0.626862
$179$	−2.45329	−0.183367	−0.0916837	−	0.995788i	$-0.529225\pi$
$179$	−2.45329	−0.183367	−0.0916837	+	0.995788i	$0.529225\pi$
$180$	1.54812	0.115390
$181$	1.00000	0.0743294
$181$	1.00000	0.0743294
$182$	−19.8093	−1.46836
$183$	2.35027	0.173737
$184$	−48.2334	−3.55581
$185$	−0.341695	−0.0251219
$186$	21.8804	1.60435
$187$	14.6926	1.07443
$188$	−20.6129	−1.50335
$189$	1.96328	0.142807
$190$	5.20317	0.377478
$191$	26.6172	1.92595	0.962975	−	0.269590i	$-0.0868880\pi$
$191$	26.6172	1.92595	0.962975	+	0.269590i	$0.0868880\pi$
$192$	5.65033	0.407777
$193$	9.38507	0.675552	0.337776	−	0.941226i	$-0.390325\pi$
$193$	9.38507	0.675552	0.337776	+	0.941226i	$0.390325\pi$
$194$	30.6612	2.20135
$195$	−1.50564	−0.107821
$196$	−13.1325	−0.938037
$197$	−2.17420	−0.154905	−0.0774527	−	0.996996i	$-0.524679\pi$
$197$	−2.17420	−0.154905	−0.0774527	+	0.996996i	$0.524679\pi$
$198$	10.0602	0.714945
$199$	−19.5191	−1.38367	−0.691835	−	0.722055i	$-0.743198\pi$
$199$	−19.5191	−1.38367	−0.691835	+	0.722055i	$0.743198\pi$
$200$	−26.2801	−1.85828
$201$	−10.6251	−0.749438
$202$	3.02699	0.212978
$203$	13.2467	0.929733
$204$	−15.1517	−1.06083
$205$	−0.117516	−0.00820766
$206$	−2.62969	−0.183219
$207$	−8.92443	−0.620291
$208$	20.6282	1.43031
$209$	22.8606	1.58130
$210$	1.80905	0.124836
$211$	−8.94618	−0.615881	−0.307940	−	0.951406i	$-0.599640\pi$
$211$	−8.94618	−0.615881	−0.307940	+	0.951406i	$0.599640\pi$
$212$	59.2336	4.06818
$213$	−1.33418	−0.0914166
$214$	−9.82857	−0.671867
$215$	−0.824100	−0.0562031
$216$	−5.40464	−0.367739
$217$	17.2870	1.17352
$218$	−3.34445	−0.226515
$219$	−8.35531	−0.564600
$220$	6.26747	0.422553
$221$	14.7360	0.991251
$222$	2.28983	0.153683
$223$	−17.4305	−1.16723	−0.583615	−	0.812030i	$-0.698362\pi$
$223$	−17.4305	−1.16723	−0.583615	+	0.812030i	$0.698362\pi$
$224$	−3.56347	−0.238095
$225$	−4.86250	−0.324167
$226$	−5.57980	−0.371163
$227$	5.26176	0.349235	0.174618	−	0.984636i	$-0.444131\pi$
$227$	5.26176	0.349235	0.174618	+	0.984636i	$0.444131\pi$
$228$	−23.5749	−1.56129
$229$	24.2475	1.60232	0.801159	−	0.598451i	$-0.204217\pi$
$229$	24.2475	1.60232	0.801159	+	0.598451i	$0.204217\pi$
$230$	−8.22337	−0.542233
$231$	7.94822	0.522954
$232$	−36.4663	−2.39413
$233$	12.1328	0.794845	0.397422	−	0.917636i	$-0.369905\pi$
$233$	12.1328	0.794845	0.397422	+	0.917636i	$0.369905\pi$
$234$	10.0899	0.659598
$235$	−1.83079	−0.119428
$236$	−2.24541	−0.146164
$237$	15.5212	1.00821
$238$	−17.7055	−1.14768
$239$	15.1685	0.981172	0.490586	−	0.871393i	$-0.336783\pi$
$239$	15.1685	0.981172	0.490586	+	0.871393i	$0.336783\pi$
$240$	−1.88384	−0.121601
$241$	−18.2872	−1.17798	−0.588990	−	0.808141i	$-0.700474\pi$
$241$	−18.2872	−1.17798	−0.588990	+	0.808141i	$0.700474\pi$
$242$	13.3937	0.860977
$243$	−1.00000	−0.0641500
$244$	−9.81228	−0.628167
$245$	−1.16640	−0.0745188
$246$	0.787518	0.0502103
$247$	22.9282	1.45888
$248$	−47.5889	−3.02190
$249$	−9.55110	−0.605277
$250$	−9.08774	−0.574759
$251$	−1.81307	−0.114440	−0.0572199	−	0.998362i	$-0.518224\pi$
$251$	−1.81307	−0.114440	−0.0572199	+	0.998362i	$0.518224\pi$
$252$	−8.19659	−0.516336
$253$	−36.1301	−2.27148
$254$	−43.9679	−2.75879
$255$	−1.34574	−0.0842736
$256$	−32.6105	−2.03816
$257$	−6.67435	−0.416335	−0.208167	−	0.978093i	$-0.566750\pi$
$257$	−6.67435	−0.416335	−0.208167	+	0.978093i	$0.566750\pi$
$258$	5.52261	0.343823
$259$	1.80912	0.112413
$260$	6.28600	0.389841
$261$	−6.74722	−0.417642
$262$	33.3093	2.05785
$263$	−21.0546	−1.29828	−0.649140	−	0.760669i	$-0.724871\pi$
$263$	−21.0546	−1.29828	−0.649140	+	0.760669i	$0.724871\pi$
$264$	−21.8804	−1.34665
$265$	5.26101	0.323181
$266$	−27.5484	−1.68910
$267$	−3.36562	−0.205973
$268$	44.3594	2.70968
$269$	5.45589	0.332652	0.166326	−	0.986071i	$-0.446810\pi$
$269$	5.45589	0.332652	0.166326	+	0.986071i	$0.446810\pi$
$270$	−0.921444	−0.0560773
$271$	−4.00206	−0.243108	−0.121554	−	0.992585i	$-0.538788\pi$
$271$	−4.00206	−0.243108	−0.121554	+	0.992585i	$0.538788\pi$
$272$	18.4375	1.11794
$273$	7.97171	0.482470
$274$	14.1150	0.852716
$275$	−19.6856	−1.18708
$276$	37.2591	2.24273
$277$	22.5668	1.35591	0.677954	−	0.735105i	$-0.262867\pi$
$277$	22.5668	1.35591	0.677954	+	0.735105i	$0.262867\pi$
$278$	−42.4565	−2.54637
$279$	−8.80518	−0.527153
$280$	−3.93460	−0.235137
$281$	−9.87322	−0.588987	−0.294494	−	0.955653i	$-0.595151\pi$
$281$	−9.87322	−0.588987	−0.294494	+	0.955653i	$0.595151\pi$
$282$	12.2688	0.730599
$283$	−29.5858	−1.75869	−0.879347	−	0.476181i	$-0.842021\pi$
$283$	−29.5858	−1.75869	−0.879347	+	0.476181i	$0.842021\pi$
$284$	5.57015	0.330527
$285$	−2.09388	−0.124031
$286$	40.8484	2.41542
$287$	0.622193	0.0367269
$288$	1.81507	0.106954
$289$	−3.82899	−0.225234
$290$	−6.21719	−0.365086
$291$	−12.3388	−0.723312
$292$	34.8830	2.04138
$293$	−16.2215	−0.947668	−0.473834	−	0.880614i	$-0.657130\pi$
$293$	−16.2215	−0.947668	−0.473834	+	0.880614i	$0.657130\pi$
$294$	7.81652	0.455868
$295$	−0.199432	−0.0116114
$296$	−4.98027	−0.289472
$297$	−4.04845	−0.234915
$298$	−32.1459	−1.86216
$299$	−36.2369	−2.09563
$300$	20.3007	1.17206
$301$	4.36324	0.251493
$302$	−26.6323	−1.53251
$303$	−1.21813	−0.0699798
$304$	28.6874	1.64534
$305$	−0.871506	−0.0499023
$306$	9.01834	0.515544
$307$	13.1246	0.749063	0.374531	−	0.927214i	$-0.377804\pi$
$307$	13.1246	0.749063	0.374531	+	0.927214i	$0.377804\pi$
$308$	−33.1834	−1.89080
$309$	1.05825	0.0602016
$310$	−8.11349	−0.460815
$311$	18.3176	1.03869	0.519347	−	0.854563i	$-0.326175\pi$
$311$	18.3176	1.03869	0.519347	+	0.854563i	$0.326175\pi$
$312$	−21.9451	−1.24240
$313$	25.8600	1.46170	0.730848	−	0.682540i	$-0.239125\pi$
$313$	25.8600	1.46170	0.730848	+	0.682540i	$0.239125\pi$
$314$	−7.60829	−0.429360
$315$	−0.728004	−0.0410184
$316$	−64.8004	−3.64531
$317$	18.3141	1.02862	0.514312	−	0.857603i	$-0.328047\pi$
$317$	18.3141	1.02862	0.514312	+	0.857603i	$0.328047\pi$
$318$	−35.2560	−1.97706
$319$	−27.3158	−1.52939
$320$	−2.09520	−0.117125
$321$	3.95524	0.220760
$322$	43.5390	2.42634
$323$	20.4931	1.14027
$324$	4.17495	0.231942
$325$	−19.7438	−1.09519
$326$	13.6887	0.758147
$327$	1.34588	0.0744276
$328$	−1.71282	−0.0945745
$329$	9.69321	0.534404
$330$	−3.73042	−0.205353
$331$	14.6270	0.803971	0.401985	−	0.915646i	$-0.368320\pi$
$331$	14.6270	0.803971	0.401985	+	0.915646i	$0.368320\pi$
$332$	39.8754	2.18845
$333$	−0.921480	−0.0504968
$334$	50.3744	2.75637
$335$	3.93991	0.215260
$336$	9.97409	0.544131
$337$	−5.19389	−0.282929	−0.141465	−	0.989943i	$-0.545181\pi$
$337$	−5.19389	−0.282929	−0.141465	+	0.989943i	$0.545181\pi$
$338$	8.66489	0.471308
$339$	2.24544	0.121956
$340$	5.61841	0.304701
$341$	−35.6473	−1.93041
$342$	14.0319	0.758757
$343$	19.9185	1.07550
$344$	−12.0114	−0.647612
$345$	3.30927	0.178165
$346$	49.5987	2.66644
$347$	18.0979	0.971545	0.485773	−	0.874085i	$-0.338538\pi$
$347$	18.0979	0.971545	0.485773	+	0.874085i	$0.338538\pi$
$348$	28.1693	1.51003
$349$	21.5687	1.15455	0.577274	−	0.816551i	$-0.304117\pi$
$349$	21.5687	1.15455	0.577274	+	0.816551i	$0.304117\pi$
$350$	23.7224	1.26801
$351$	−4.06041	−0.216729
$352$	7.34820	0.391660
$353$	−12.6910	−0.675472	−0.337736	−	0.941241i	$-0.609661\pi$
$353$	−12.6910	−0.675472	−0.337736	+	0.941241i	$0.609661\pi$
$354$	1.33647	0.0710327
$355$	0.494729	0.0262575
$356$	14.0513	0.744718
$357$	7.12510	0.377100
$358$	−6.09629	−0.322199
$359$	−2.09266	−0.110446	−0.0552231	−	0.998474i	$-0.517587\pi$
$359$	−2.09266	−0.110446	−0.0552231	+	0.998474i	$0.517587\pi$
$360$	2.00410	0.105625
$361$	12.8858	0.678203
$362$	2.48495	0.130606
$363$	−5.38992	−0.282897
$364$	−33.2815	−1.74443
$365$	3.09824	0.162169
$366$	5.84030	0.305277
$367$	27.7795	1.45008	0.725039	−	0.688708i	$-0.241822\pi$
$367$	27.7795	1.45008	0.725039	+	0.688708i	$0.241822\pi$
$368$	−45.3391	−2.36346
$369$	−0.316916	−0.0164980
$370$	−0.849093	−0.0441422
$371$	−27.8547	−1.44614
$372$	36.7612	1.90598
$373$	−19.1864	−0.993433	−0.496716	−	0.867913i	$-0.665461\pi$
$373$	−19.1864	−0.993433	−0.496716	+	0.867913i	$0.665461\pi$
$374$	36.5103	1.88790
$375$	3.65712	0.188853
$376$	−26.6842	−1.37613
$377$	−27.3965	−1.41099
$378$	4.87863	0.250930
$379$	−1.30800	−0.0671872	−0.0335936	−	0.999436i	$-0.510695\pi$
$379$	−1.30800	−0.0671872	−0.0335936	+	0.999436i	$0.510695\pi$
$380$	8.74184	0.448447
$381$	17.6937	0.906477
$382$	66.1422	3.38413
$383$	32.9356	1.68293	0.841466	−	0.540310i	$-0.181693\pi$
$383$	32.9356	1.68293	0.841466	+	0.540310i	$0.181693\pi$
$384$	17.6709	0.901764
$385$	−2.94728	−0.150208
$386$	23.3214	1.18703
$387$	−2.22243	−0.112972
$388$	51.5138	2.61522
$389$	−14.6600	−0.743291	−0.371645	−	0.928375i	$-0.621206\pi$
$389$	−14.6600	−0.743291	−0.371645	+	0.928375i	$0.621206\pi$
$390$	−3.74144	−0.189455
$391$	−32.3884	−1.63795
$392$	−17.0006	−0.858658
$393$	−13.4044	−0.676164
$394$	−5.40277	−0.272188
$395$	−5.75544	−0.289587
$396$	16.9021	0.849361
$397$	7.01622	0.352134	0.176067	−	0.984378i	$-0.443662\pi$
$397$	7.01622	0.352134	0.176067	+	0.984378i	$0.443662\pi$
$398$	−48.5038	−2.43128
$399$	11.0861	0.555001
$400$	−24.7031	−1.23516
$401$	21.5827	1.07779	0.538895	−	0.842373i	$-0.318842\pi$
$401$	21.5827	1.07779	0.538895	+	0.842373i	$0.318842\pi$
$402$	−26.4028	−1.31685
$403$	−35.7527	−1.78097
$404$	5.08564	0.253020
$405$	0.370811	0.0184257
$406$	32.9172	1.63365
$407$	−3.73056	−0.184917
$408$	−19.6145	−0.971061
$409$	−5.30477	−0.262304	−0.131152	−	0.991362i	$-0.541868\pi$
$409$	−5.30477	−0.262304	−0.131152	+	0.991362i	$0.541868\pi$
$410$	−0.292020	−0.0144219
$411$	−5.68019	−0.280183
$412$	−4.41814	−0.217666
$413$	1.05590	0.0519577
$414$	−22.1767	−1.08993
$415$	3.54165	0.173853
$416$	7.36992	0.361340
$417$	17.0855	0.836679
$418$	56.8073	2.77854
$419$	−30.8119	−1.50526	−0.752630	−	0.658443i	$-0.771215\pi$
$419$	−30.8119	−1.50526	−0.752630	+	0.658443i	$0.771215\pi$
$420$	3.03938	0.148307
$421$	22.1309	1.07860	0.539298	−	0.842115i	$-0.318690\pi$
$421$	22.1309	1.07860	0.539298	+	0.842115i	$0.318690\pi$
$422$	−22.2308	−1.08218
$423$	−4.93727	−0.240058
$424$	76.6803	3.72392
$425$	−17.6469	−0.856002
$426$	−3.31537	−0.160630
$427$	4.61423	0.223298
$428$	−16.5130	−0.798184
$429$	−16.4384	−0.793652
$430$	−2.04784	−0.0987558
$431$	21.0033	1.01169	0.505846	−	0.862624i	$-0.331180\pi$
$431$	21.0033	1.01169	0.505846	+	0.862624i	$0.331180\pi$
$432$	−5.08033	−0.244428
$433$	−8.63171	−0.414814	−0.207407	−	0.978255i	$-0.566502\pi$
$433$	−8.63171	−0.414814	−0.207407	+	0.978255i	$0.566502\pi$
$434$	42.9573	2.06201
$435$	2.50194	0.119959
$436$	−5.61900	−0.269102
$437$	−50.3941	−2.41068
$438$	−20.7625	−0.992070
$439$	−36.9023	−1.76125	−0.880624	−	0.473815i	$-0.842877\pi$
$439$	−36.9023	−1.76125	−0.880624	+	0.473815i	$0.842877\pi$
$440$	8.11349	0.386795
$441$	−3.14555	−0.149788
$442$	36.6182	1.74175
$443$	−38.0863	−1.80953	−0.904767	−	0.425906i	$-0.859955\pi$
$443$	−38.0863	−1.80953	−0.904767	+	0.425906i	$0.859955\pi$
$444$	3.84714	0.182577
$445$	1.24801	0.0591613
$446$	−43.3138	−2.05097
$447$	12.9363	0.611864
$448$	11.0931	0.524102
$449$	10.5002	0.495534	0.247767	−	0.968820i	$-0.420303\pi$
$449$	10.5002	0.495534	0.247767	+	0.968820i	$0.420303\pi$
$450$	−12.0830	−0.569600
$451$	−1.28302	−0.0604149
$452$	−9.37461	−0.440945
$453$	10.7174	0.503549
$454$	13.0752	0.613649
$455$	−2.95600	−0.138579
$456$	−30.5187	−1.42917
$457$	−14.4330	−0.675147	−0.337574	−	0.941299i	$-0.609606\pi$
$457$	−14.4330	−0.675147	−0.337574	+	0.941299i	$0.609606\pi$
$458$	60.2537	2.81547
$459$	−3.62919	−0.169396
$460$	−13.8161	−0.644178
$461$	24.9671	1.16284	0.581418	−	0.813605i	$-0.302498\pi$
$461$	24.9671	1.16284	0.581418	+	0.813605i	$0.302498\pi$
$462$	19.7509	0.918894
$463$	32.3333	1.50266	0.751329	−	0.659928i	$-0.229413\pi$
$463$	32.3333	1.50266	0.751329	+	0.659928i	$0.229413\pi$
$464$	−34.2781	−1.59132
$465$	3.26506	0.151413
$466$	30.1493	1.39664
$467$	32.8660	1.52086	0.760428	−	0.649422i	$-0.224989\pi$
$467$	32.8660	1.52086	0.760428	+	0.649422i	$0.224989\pi$
$468$	16.9520	0.783608
$469$	−20.8600	−0.963227
$470$	−4.54942	−0.209849
$471$	3.06175	0.141078
$472$	−2.90677	−0.133795
$473$	−8.99738	−0.413700
$474$	38.5694	1.77155
$475$	−27.4573	−1.25983
$476$	−29.7470	−1.36345
$477$	14.1879	0.649617
$478$	37.6930	1.72404
$479$	10.2308	0.467455	0.233728	−	0.972302i	$-0.424908\pi$
$479$	10.2308	0.467455	0.233728	+	0.972302i	$0.424908\pi$
$480$	−0.673046	−0.0307202
$481$	−3.74159	−0.170602
$482$	−45.4426	−2.06985
$483$	−17.5211	−0.797239
$484$	22.5027	1.02285
$485$	4.57535	0.207756
$486$	−2.48495	−0.112719
$487$	7.93179	0.359424	0.179712	−	0.983719i	$-0.442483\pi$
$487$	7.93179	0.359424	0.179712	+	0.983719i	$0.442483\pi$
$488$	−12.7024	−0.575010
$489$	−5.50865	−0.249110
$490$	−2.89845	−0.130939
$491$	39.3721	1.77684	0.888420	−	0.459032i	$-0.151804\pi$
$491$	39.3721	1.77684	0.888420	+	0.459032i	$0.151804\pi$
$492$	1.32311	0.0596504
$493$	−24.4869	−1.10284
$494$	56.9752	2.56344
$495$	1.50121	0.0674743
$496$	−44.7333	−2.00858
$497$	−2.61937	−0.117495
$498$	−23.7340	−1.06354
$499$	1.78195	0.0797712	0.0398856	−	0.999204i	$-0.487301\pi$
$499$	1.78195	0.0797712	0.0398856	+	0.999204i	$0.487301\pi$
$500$	−15.2683	−0.682820
$501$	−20.2718	−0.905679
$502$	−4.50537	−0.201085
$503$	−37.1023	−1.65431	−0.827156	−	0.561973i	$-0.810042\pi$
$503$	−37.1023	−1.65431	−0.827156	+	0.561973i	$0.810042\pi$
$504$	−10.6108	−0.472643
$505$	0.451696	0.0201002
$506$	−89.7813	−3.99126
$507$	−3.48696	−0.154861
$508$	−73.8705	−3.27747
$509$	−18.0275	−0.799054	−0.399527	−	0.916722i	$-0.630826\pi$
$509$	−18.0275	−0.799054	−0.399527	+	0.916722i	$0.630826\pi$
$510$	−3.34410	−0.148079
$511$	−16.4038	−0.725660
$512$	−45.6936	−2.01939
$513$	−5.64676	−0.249310
$514$	−16.5854	−0.731551
$515$	−0.392410	−0.0172916
$516$	9.27853	0.408465
$517$	−19.9883	−0.879083
$518$	4.49556	0.197524
$519$	−19.9597	−0.876133
$520$	8.13747	0.356852
$521$	20.9237	0.916683	0.458342	−	0.888776i	$-0.348444\pi$
$521$	20.9237	0.916683	0.458342	+	0.888776i	$0.348444\pi$
$522$	−16.7665	−0.733849
$523$	43.3377	1.89502	0.947512	−	0.319720i	$-0.103589\pi$
$523$	43.3377	1.89502	0.947512	+	0.319720i	$0.103589\pi$
$524$	55.9629	2.44475
$525$	−9.54643	−0.416640
$526$	−52.3195	−2.28124
$527$	−31.9557	−1.39201
$528$	−20.5675	−0.895084
$529$	56.6455	2.46285
$530$	13.0733	0.567869
$531$	−0.537828	−0.0233397
$532$	−46.2841	−2.00667
$533$	−1.28681	−0.0557379
$534$	−8.36339	−0.361919
$535$	−1.46665	−0.0634087
$536$	57.4250	2.48038
$537$	2.45329	0.105867
$538$	13.5576	0.584509
$539$	−12.7346	−0.548517
$540$	−1.54812	−0.0666204
$541$	29.2960	1.25953	0.629767	−	0.776784i	$-0.283150\pi$
$541$	29.2960	1.25953	0.629767	+	0.776784i	$0.283150\pi$
$542$	−9.94490	−0.427170
$543$	−1.00000	−0.0429141
$544$	6.58722	0.282425
$545$	−0.499068	−0.0213777
$546$	19.8093	0.847758
$547$	−38.8865	−1.66267	−0.831334	−	0.555773i	$-0.812422\pi$
$547$	−38.8865	−1.66267	−0.831334	+	0.555773i	$0.812422\pi$
$548$	23.7145	1.01304
$549$	−2.35027	−0.100307
$550$	−48.9176	−2.08585
$551$	−38.0999	−1.62311
$552$	48.2334	2.05295
$553$	30.4724	1.29582
$554$	56.0773	2.38249
$555$	0.341695	0.0145041
$556$	−71.3310	−3.02511
$557$	41.8529	1.77336	0.886682	−	0.462379i	$-0.153004\pi$
$557$	41.8529	1.77336	0.886682	+	0.462379i	$0.153004\pi$
$558$	−21.8804	−0.926271
$559$	−9.02397	−0.381673
$560$	−3.69850	−0.156290
$561$	−14.6926	−0.620321
$562$	−24.5344	−1.03492
$563$	−2.68310	−0.113079	−0.0565396	−	0.998400i	$-0.518007\pi$
$563$	−2.68310	−0.113079	−0.0565396	+	0.998400i	$0.518007\pi$
$564$	20.6129	0.867958
$565$	−0.832633	−0.0350291
$566$	−73.5191	−3.09024
$567$	−1.96328	−0.0824498
$568$	7.21078	0.302557
$569$	−9.17154	−0.384491	−0.192246	−	0.981347i	$-0.561577\pi$
$569$	−9.17154	−0.384491	−0.192246	+	0.981347i	$0.561577\pi$
$570$	−5.20317	−0.217937
$571$	−26.6381	−1.11477	−0.557385	−	0.830254i	$-0.688195\pi$
$571$	−26.6381	−1.11477	−0.557385	+	0.830254i	$0.688195\pi$
$572$	68.6294	2.86954
$573$	−26.6172	−1.11195
$574$	1.54612	0.0645336
$575$	43.3950	1.80970
$576$	−5.65033	−0.235430
$577$	−10.2555	−0.426941	−0.213470	−	0.976950i	$-0.568477\pi$
$577$	−10.2555	−0.426941	−0.213470	+	0.976950i	$0.568477\pi$
$578$	−9.51482	−0.395764
$579$	−9.38507	−0.390030
$580$	−10.4455	−0.433725
$581$	−18.7515	−0.777941
$582$	−30.6612	−1.27095
$583$	57.4388	2.37887
$584$	45.1575	1.86863
$585$	1.50564	0.0622507
$586$	−40.3095	−1.66517
$587$	−12.5974	−0.519949	−0.259974	−	0.965616i	$-0.583714\pi$
$587$	−12.5974	−0.519949	−0.259974	+	0.965616i	$0.583714\pi$
$588$	13.1325	0.541576
$589$	−49.7207	−2.04871
$590$	−0.495578	−0.0204026
$591$	2.17420	0.0894347
$592$	−4.68143	−0.192405
$593$	19.2129	0.788979	0.394490	−	0.918900i	$-0.370921\pi$
$593$	19.2129	0.788979	0.394490	+	0.918900i	$0.370921\pi$
$594$	−10.0602	−0.412774
$595$	−2.64206	−0.108314
$596$	−54.0083	−2.21226
$597$	19.5191	0.798863
$598$	−90.0467	−3.68228
$599$	−20.6159	−0.842345	−0.421172	−	0.906981i	$-0.638381\pi$
$599$	−20.6159	−0.842345	−0.421172	+	0.906981i	$0.638381\pi$
$600$	26.2801	1.07288
$601$	14.6688	0.598354	0.299177	−	0.954198i	$-0.403288\pi$
$601$	14.6688	0.598354	0.299177	+	0.954198i	$0.403288\pi$
$602$	10.8424	0.441903
$603$	10.6251	0.432688
$604$	−44.7448	−1.82064
$605$	1.99864	0.0812563
$606$	−3.02699	−0.122963
$607$	18.7059	0.759250	0.379625	−	0.925140i	$-0.376053\pi$
$607$	18.7059	0.759250	0.379625	+	0.925140i	$0.376053\pi$
$608$	10.2492	0.415661
$609$	−13.2467	−0.536781
$610$	−2.16564	−0.0876844
$611$	−20.0473	−0.811029
$612$	15.1517	0.612471
$613$	−2.57586	−0.104038	−0.0520191	−	0.998646i	$-0.516566\pi$
$613$	−2.57586	−0.104038	−0.0520191	+	0.998646i	$0.516566\pi$
$614$	32.6140	1.31619
$615$	0.117516	0.00473869
$616$	−42.9573	−1.73080
$617$	−19.1221	−0.769828	−0.384914	−	0.922952i	$-0.625769\pi$
$617$	−19.1221	−0.769828	−0.384914	+	0.922952i	$0.625769\pi$
$618$	2.62969	0.105782
$619$	−11.0897	−0.445733	−0.222867	−	0.974849i	$-0.571541\pi$
$619$	−11.0897	−0.445733	−0.222867	+	0.974849i	$0.571541\pi$
$620$	−13.6315	−0.547453
$621$	8.92443	0.358125
$622$	45.5182	1.82511
$623$	−6.60765	−0.264730
$624$	−20.6282	−0.825791
$625$	22.9564	0.918256
$626$	64.2608	2.56838
$627$	−22.8606	−0.912964
$628$	−12.7827	−0.510084
$629$	−3.34423	−0.133343
$630$	−1.80905	−0.0720743
$631$	28.4699	1.13337	0.566685	−	0.823935i	$-0.308226\pi$
$631$	28.4699	1.13337	0.566685	+	0.823935i	$0.308226\pi$
$632$	−83.8866	−3.33683
$633$	8.94618	0.355579
$634$	45.5096	1.80742
$635$	−6.56102	−0.260366
$636$	−59.2336	−2.34877
$637$	−12.7722	−0.506054
$638$	−67.8782	−2.68732
$639$	1.33418	0.0527794
$640$	−6.55255	−0.259012
$641$	−15.8650	−0.626632	−0.313316	−	0.949649i	$-0.601440\pi$
$641$	−15.8650	−0.626632	−0.313316	+	0.949649i	$0.601440\pi$
$642$	9.82857	0.387903
$643$	5.72748	0.225870	0.112935	−	0.993602i	$-0.463975\pi$
$643$	5.72748	0.225870	0.112935	+	0.993602i	$0.463975\pi$
$644$	73.1499	2.88251
$645$	0.824100	0.0324489
$646$	50.9243	2.00359
$647$	5.88463	0.231349	0.115674	−	0.993287i	$-0.463097\pi$
$647$	5.88463	0.231349	0.115674	+	0.993287i	$0.463097\pi$
$648$	5.40464	0.212314
$649$	−2.17737	−0.0854691
$650$	−49.0622	−1.92438
$651$	−17.2870	−0.677531
$652$	22.9984	0.900686
$653$	−41.1149	−1.60895	−0.804476	−	0.593985i	$-0.797554\pi$
$653$	−41.1149	−1.60895	−0.804476	+	0.593985i	$0.797554\pi$
$654$	3.34445	0.130778
$655$	4.97050	0.194214
$656$	−1.61004	−0.0628614
$657$	8.35531	0.325972
$658$	24.0871	0.939013
$659$	−12.1465	−0.473162	−0.236581	−	0.971612i	$-0.576027\pi$
$659$	−12.1465	−0.473162	−0.236581	+	0.971612i	$0.576027\pi$
$660$	−6.26747	−0.243961
$661$	−15.9501	−0.620389	−0.310194	−	0.950673i	$-0.600394\pi$
$661$	−15.9501	−0.620389	−0.310194	+	0.950673i	$0.600394\pi$
$662$	36.3472	1.41267
$663$	−14.7360	−0.572299
$664$	51.6203	2.00326
$665$	−4.11086	−0.159412
$666$	−2.28983	−0.0887290
$667$	60.2151	2.33154
$668$	84.6340	3.27459
$669$	17.4305	0.673901
$670$	9.79045	0.378238
$671$	−9.51495	−0.367321
$672$	3.56347	0.137464
$673$	−28.3034	−1.09102	−0.545508	−	0.838106i	$-0.683663\pi$
$673$	−28.3034	−1.09102	−0.545508	+	0.838106i	$0.683663\pi$
$674$	−12.9065	−0.497141
$675$	4.86250	0.187158
$676$	14.5579	0.559918
$677$	5.99704	0.230485	0.115242	−	0.993337i	$-0.463236\pi$
$677$	5.99704	0.230485	0.115242	+	0.993337i	$0.463236\pi$
$678$	5.57980	0.214291
$679$	−24.2244	−0.929648
$680$	7.27326	0.278917
$681$	−5.26176	−0.201631
$682$	−88.5816	−3.39197
$683$	33.7218	1.29033	0.645165	−	0.764043i	$-0.276789\pi$
$683$	33.7218	1.29033	0.645165	+	0.764043i	$0.276789\pi$
$684$	23.5749	0.901411
$685$	2.10628	0.0804767
$686$	49.4964	1.88978
$687$	−24.2475	−0.925099
$688$	−11.2907	−0.430453
$689$	57.6085	2.19471
$690$	8.22337	0.313058
$691$	8.64738	0.328962	0.164481	−	0.986380i	$-0.447405\pi$
$691$	8.64738	0.328962	0.164481	+	0.986380i	$0.447405\pi$
$692$	83.3308	3.16776
$693$	−7.94822	−0.301928
$694$	44.9722	1.70712
$695$	−6.33548	−0.240318
$696$	36.4663	1.38225
$697$	−1.15015	−0.0435649
$698$	53.5971	2.02868
$699$	−12.1328	−0.458904
$700$	39.8559	1.50641
$701$	12.2904	0.464202	0.232101	−	0.972692i	$-0.425440\pi$
$701$	12.2904	0.464202	0.232101	+	0.972692i	$0.425440\pi$
$702$	−10.0899	−0.380819
$703$	−5.20337	−0.196249
$704$	−22.8750	−0.862136
$705$	1.83079	0.0689516
$706$	−31.5364	−1.18689
$707$	−2.39153	−0.0899426
$708$	2.24541	0.0843875
$709$	2.80892	0.105491	0.0527457	−	0.998608i	$-0.483203\pi$
$709$	2.80892	0.105491	0.0527457	+	0.998608i	$0.483203\pi$
$710$	1.22937	0.0461376
$711$	−15.5212	−0.582091
$712$	18.1900	0.681698
$713$	78.5813	2.94289
$714$	17.7055	0.662611
$715$	6.09552	0.227960
$716$	−10.2424	−0.382775
$717$	−15.1685	−0.566480
$718$	−5.20013	−0.194067
$719$	−17.4798	−0.651888	−0.325944	−	0.945389i	$-0.605682\pi$
$719$	−17.4798	−0.651888	−0.325944	+	0.945389i	$0.605682\pi$
$720$	1.88384	0.0702066
$721$	2.07763	0.0773751
$722$	32.0206	1.19168
$723$	18.2872	0.680106
$724$	4.17495	0.155161
$725$	32.8084	1.21847
$726$	−13.3937	−0.497085
$727$	−26.2700	−0.974299	−0.487149	−	0.873319i	$-0.661963\pi$
$727$	−26.2700	−0.974299	−0.487149	+	0.873319i	$0.661963\pi$
$728$	−43.0842	−1.59681
$729$	1.00000	0.0370370
$730$	7.69895	0.284951
$731$	−8.06561	−0.298317
$732$	9.81228	0.362672
$733$	−29.3942	−1.08570	−0.542849	−	0.839830i	$-0.682655\pi$
$733$	−29.3942	−1.08570	−0.542849	+	0.839830i	$0.682655\pi$
$734$	69.0305	2.54796
$735$	1.16640	0.0430234
$736$	−16.1984	−0.597082
$737$	43.0152	1.58449
$738$	−0.787518	−0.0289890
$739$	−2.53688	−0.0933208	−0.0466604	−	0.998911i	$-0.514858\pi$
$739$	−2.53688	−0.0933208	−0.0466604	+	0.998911i	$0.514858\pi$
$740$	−1.42656	−0.0524414
$741$	−22.9282	−0.842287
$742$	−69.2173	−2.54105
$743$	−37.6157	−1.37998	−0.689992	−	0.723817i	$-0.742386\pi$
$743$	−37.6157	−1.37998	−0.689992	+	0.723817i	$0.742386\pi$
$744$	47.5889	1.74469
$745$	−4.79690	−0.175745
$746$	−47.6771	−1.74558
$747$	9.55110	0.349457
$748$	61.3408	2.24284
$749$	7.76524	0.283736
$750$	9.08774	0.331838
$751$	19.3798	0.707180	0.353590	−	0.935401i	$-0.384961\pi$
$751$	19.3798	0.707180	0.353590	+	0.935401i	$0.384961\pi$
$752$	−25.0830	−0.914681
$753$	1.81307	0.0660718
$754$	−68.0788	−2.47928
$755$	−3.97414	−0.144634
$756$	8.19659	0.298107
$757$	−0.893853	−0.0324877	−0.0162438	−	0.999868i	$-0.505171\pi$
$757$	−0.893853	−0.0324877	−0.0162438	+	0.999868i	$0.505171\pi$
$758$	−3.25030	−0.118056
$759$	36.1301	1.31144
$760$	11.3167	0.410498
$761$	−20.9488	−0.759392	−0.379696	−	0.925111i	$-0.623971\pi$
$761$	−20.9488	−0.759392	−0.379696	+	0.925111i	$0.623971\pi$
$762$	43.9679	1.59279
$763$	2.64234	0.0956592
$764$	111.125	4.02038
$765$	1.34574	0.0486554
$766$	81.8432	2.95712
$767$	−2.18380	−0.0788526
$768$	32.6105	1.17673
$769$	−52.9737	−1.91028	−0.955141	−	0.296153i	$-0.904296\pi$
$769$	−52.9737	−1.91028	−0.955141	+	0.296153i	$0.904296\pi$
$770$	−7.32384	−0.263933
$771$	6.67435	0.240371
$772$	39.1823	1.41020
$773$	−4.57701	−0.164624	−0.0823118	−	0.996607i	$-0.526230\pi$
$773$	−4.57701	−0.164624	−0.0823118	+	0.996607i	$0.526230\pi$
$774$	−5.52261	−0.198506
$775$	42.8152	1.53797
$776$	66.6867	2.39391
$777$	−1.80912	−0.0649018
$778$	−36.4293	−1.30605
$779$	−1.78955	−0.0641171
$780$	−6.28600	−0.225075
$781$	5.40136	0.193276
$782$	−80.4835	−2.87809
$783$	6.74722	0.241126
$784$	−15.9804	−0.570730
$785$	−1.13533	−0.0405217
$786$	−33.3093	−1.18810
$787$	10.6325	0.379008	0.189504	−	0.981880i	$-0.439312\pi$
$787$	10.6325	0.379008	0.189504	+	0.981880i	$0.439312\pi$
$788$	−9.07719	−0.323361
$789$	21.0546	0.749563
$790$	−14.3019	−0.508840
$791$	4.40842	0.156745
$792$	21.8804	0.777486
$793$	−9.54307	−0.338885
$794$	17.4349	0.618743
$795$	−5.26101	−0.186589
$796$	−81.4912	−2.88838
$797$	−56.1843	−1.99015	−0.995076	−	0.0991189i	$-0.968398\pi$
$797$	−56.1843	−1.99015	−0.995076	+	0.0991189i	$0.968398\pi$
$798$	27.5484	0.975205
$799$	−17.9183	−0.633903
$800$	−8.82575	−0.312038
$801$	3.36562	0.118918
$802$	53.6319	1.89381
$803$	33.8260	1.19369
$804$	−44.3594	−1.56443
$805$	6.49702	0.228990
$806$	−88.8435	−3.12938
$807$	−5.45589	−0.192057
$808$	6.58356	0.231609
$809$	11.6538	0.409726	0.204863	−	0.978791i	$-0.434325\pi$
$809$	11.6538	0.409726	0.204863	+	0.978791i	$0.434325\pi$
$810$	0.921444	0.0323763
$811$	−16.4662	−0.578206	−0.289103	−	0.957298i	$-0.593357\pi$
$811$	−16.4662	−0.578206	−0.289103	+	0.957298i	$0.593357\pi$
$812$	55.3042	1.94080
$813$	4.00206	0.140358
$814$	−9.27025	−0.324922
$815$	2.04267	0.0715515
$816$	−18.4375	−0.645441
$817$	−12.5495	−0.439051
$818$	−13.1821	−0.460900
$819$	−7.97171	−0.278554
$820$	−0.490623	−0.0171333
$821$	43.9803	1.53492	0.767462	−	0.641094i	$-0.221519\pi$
$821$	43.9803	1.53492	0.767462	+	0.641094i	$0.221519\pi$
$822$	−14.1150	−0.492316
$823$	15.3338	0.534504	0.267252	−	0.963627i	$-0.413884\pi$
$823$	15.3338	0.534504	0.267252	+	0.963627i	$0.413884\pi$
$824$	−5.71945	−0.199246
$825$	19.6856	0.685364
$826$	2.62386	0.0912959
$827$	−16.9750	−0.590280	−0.295140	−	0.955454i	$-0.595366\pi$
$827$	−16.9750	−0.590280	−0.295140	+	0.955454i	$0.595366\pi$
$828$	−37.2591	−1.29484
$829$	27.5204	0.955823	0.477911	−	0.878408i	$-0.341394\pi$
$829$	27.5204	0.955823	0.477911	+	0.878408i	$0.341394\pi$
$830$	8.80081	0.305481
$831$	−22.5668	−0.782833
$832$	−22.9427	−0.795394
$833$	−11.4158	−0.395534
$834$	42.4565	1.47015
$835$	7.51702	0.260137
$836$	95.4419	3.30093
$837$	8.80518	0.304352
$838$	−76.5659	−2.64493
$839$	15.2470	0.526384	0.263192	−	0.964744i	$-0.415225\pi$
$839$	15.2470	0.526384	0.263192	+	0.964744i	$0.415225\pi$
$840$	3.93460	0.135757
$841$	16.5250	0.569827
$842$	54.9942	1.89522
$843$	9.87322	0.340052
$844$	−37.3499	−1.28564
$845$	1.29300	0.0444806
$846$	−12.2688	−0.421811
$847$	−10.5819	−0.363598
$848$	72.0790	2.47520
$849$	29.5858	1.01538
$850$	−43.8517	−1.50410
$851$	8.22369	0.281904
$852$	−5.57015	−0.190830
$853$	−23.7441	−0.812982	−0.406491	−	0.913655i	$-0.633248\pi$
$853$	−23.7441	−0.812982	−0.406491	+	0.913655i	$0.633248\pi$
$854$	11.4661	0.392362
$855$	2.09388	0.0716091
$856$	−21.3767	−0.730640
$857$	4.28871	0.146500	0.0732498	−	0.997314i	$-0.476663\pi$
$857$	4.28871	0.146500	0.0732498	+	0.997314i	$0.476663\pi$
$858$	−40.8484	−1.39454
$859$	1.15611	0.0394458	0.0197229	−	0.999805i	$-0.493722\pi$
$859$	1.15611	0.0394458	0.0197229	+	0.999805i	$0.493722\pi$
$860$	−3.44058	−0.117323
$861$	−0.622193	−0.0212043
$862$	52.1920	1.77767
$863$	−49.4010	−1.68163	−0.840815	−	0.541323i	$-0.817924\pi$
$863$	−49.4010	−1.68163	−0.840815	+	0.541323i	$0.817924\pi$
$864$	−1.81507	−0.0617498
$865$	7.40126	0.251651
$866$	−21.4493	−0.728878
$867$	3.82899	0.130039
$868$	72.1725	2.44969
$869$	−62.8368	−2.13159
$870$	6.21719	0.210782
$871$	43.1424	1.46182
$872$	−7.27402	−0.246329
$873$	12.3388	0.417604
$874$	−125.227	−4.23585
$875$	7.17994	0.242726
$876$	−34.8830	−1.17859
$877$	43.4495	1.46719	0.733593	−	0.679589i	$-0.237842\pi$
$877$	43.4495	1.46719	0.733593	+	0.679589i	$0.237842\pi$
$878$	−91.7001	−3.09473
$879$	16.2215	0.547137
$880$	7.62663	0.257094
$881$	−51.9644	−1.75073	−0.875363	−	0.483466i	$-0.839378\pi$
$881$	−51.9644	−1.75073	−0.875363	+	0.483466i	$0.839378\pi$
$882$	−7.81652	−0.263196
$883$	−35.2566	−1.18648	−0.593239	−	0.805026i	$-0.702151\pi$
$883$	−35.2566	−1.18648	−0.593239	+	0.805026i	$0.702151\pi$
$884$	61.5222	2.06921
$885$	0.199432	0.00670384
$886$	−94.6424	−3.17957
$887$	−24.7631	−0.831464	−0.415732	−	0.909487i	$-0.636475\pi$
$887$	−24.7631	−0.831464	−0.415732	+	0.909487i	$0.636475\pi$
$888$	4.98027	0.167127
$889$	34.7377	1.16506
$890$	3.10123	0.103954
$891$	4.04845	0.135628
$892$	−72.7714	−2.43657
$893$	−27.8795	−0.932953
$894$	32.1459	1.07512
$895$	−0.909706	−0.0304081
$896$	34.6928	1.15901
$897$	36.2369	1.20991
$898$	26.0924	0.870714
$899$	59.4105	1.98145
$900$	−20.3007	−0.676690
$901$	51.4904	1.71539
$902$	−3.18823	−0.106156
$903$	−4.36324	−0.145199
$904$	−12.1358	−0.403631
$905$	0.370811	0.0123262
$906$	26.6323	0.884798
$907$	−55.6804	−1.84884	−0.924419	−	0.381377i	$-0.875450\pi$
$907$	−55.6804	−1.84884	−0.924419	+	0.381377i	$0.875450\pi$
$908$	21.9676	0.729021
$909$	1.21813	0.0404028
$910$	−7.34549	−0.243501
$911$	39.9314	1.32299	0.661493	−	0.749951i	$-0.269923\pi$
$911$	39.9314	1.32299	0.661493	+	0.749951i	$0.269923\pi$
$912$	−28.6874	−0.949935
$913$	38.6671	1.27970
$914$	−35.8652	−1.18632
$915$	0.871506	0.0288111
$916$	101.232	3.34480
$917$	−26.3166	−0.869050
$918$	−9.01834	−0.297649
$919$	27.4981	0.907080	0.453540	−	0.891236i	$-0.350161\pi$
$919$	27.4981	0.907080	0.453540	+	0.891236i	$0.350161\pi$
$920$	−17.8854	−0.589666
$921$	−13.1246	−0.432472
$922$	62.0420	2.04324
$923$	5.41733	0.178314
$924$	33.1834	1.09166
$925$	4.48070	0.147324
$926$	80.3466	2.64035
$927$	−1.05825	−0.0347574
$928$	−12.2466	−0.402016
$929$	−45.6591	−1.49803	−0.749013	−	0.662556i	$-0.769472\pi$
$929$	−45.6591	−1.49803	−0.749013	+	0.662556i	$0.769472\pi$
$930$	8.11349	0.266052
$931$	−17.7621	−0.582131
$932$	50.6538	1.65922
$933$	−18.3176	−0.599690
$934$	81.6702	2.67233
$935$	5.44817	0.178174
$936$	21.9451	0.717297
$937$	−10.2634	−0.335291	−0.167646	−	0.985847i	$-0.553616\pi$
$937$	−10.2634	−0.335291	−0.167646	+	0.985847i	$0.553616\pi$
$938$	−51.8361	−1.69251
$939$	−25.8600	−0.843910
$940$	−7.64347	−0.249302
$941$	29.9236	0.975482	0.487741	−	0.872988i	$-0.337821\pi$
$941$	29.9236	0.975482	0.487741	+	0.872988i	$0.337821\pi$
$942$	7.60829	0.247891
$943$	2.82829	0.0921019
$944$	−2.73234	−0.0889302
$945$	0.728004	0.0236820
$946$	−22.3580	−0.726921
$947$	16.8152	0.546421	0.273211	−	0.961954i	$-0.411914\pi$
$947$	16.8152	0.546421	0.273211	+	0.961954i	$0.411914\pi$
$948$	64.8004	2.10462
$949$	33.9260	1.10129
$950$	−68.2300	−2.21367
$951$	−18.3141	−0.593876
$952$	−38.5086	−1.24807
$953$	−12.5451	−0.406374	−0.203187	−	0.979140i	$-0.565130\pi$
$953$	−12.5451	−0.406374	−0.203187	+	0.979140i	$0.565130\pi$
$954$	35.2560	1.14146
$955$	9.86993	0.319383
$956$	63.3280	2.04817
$957$	27.3158	0.882993
$958$	25.4229	0.821376
$959$	−11.1518	−0.360110
$960$	2.09520	0.0676223
$961$	46.5313	1.50101
$962$	−9.29765	−0.299768
$963$	−3.95524	−0.127456
$964$	−76.3480	−2.45900
$965$	3.48009	0.112028
$966$	−43.5390	−1.40085
$967$	−8.59093	−0.276266	−0.138133	−	0.990414i	$-0.544110\pi$
$967$	−8.59093	−0.276266	−0.138133	+	0.990414i	$0.544110\pi$
$968$	29.1306	0.936292
$969$	−20.4931	−0.658335
$970$	11.3695	0.365053
$971$	23.9170	0.767535	0.383767	−	0.923430i	$-0.374626\pi$
$971$	23.9170	0.767535	0.383767	+	0.923430i	$0.374626\pi$
$972$	−4.17495	−0.133912
$973$	33.5435	1.07535
$974$	19.7101	0.631551
$975$	19.7438	0.632306
$976$	−11.9402	−0.382195
$977$	−21.5036	−0.687963	−0.343981	−	0.938977i	$-0.611776\pi$
$977$	−21.5036	−0.687963	−0.343981	+	0.938977i	$0.611776\pi$
$978$	−13.6887	−0.437716
$979$	13.6255	0.435474
$980$	−4.86968	−0.155556
$981$	−1.34588	−0.0429708
$982$	97.8376	3.12212
$983$	23.9678	0.764456	0.382228	−	0.924068i	$-0.375157\pi$
$983$	23.9678	0.764456	0.382228	+	0.924068i	$0.375157\pi$
$984$	1.71282	0.0546026
$985$	−0.806217	−0.0256882
$986$	−60.8487	−1.93782
$987$	−9.69321	−0.308538
$988$	95.7240	3.04539
$989$	19.8339	0.630681
$990$	3.73042	0.118560
$991$	9.12524	0.289873	0.144936	−	0.989441i	$-0.453702\pi$
$991$	9.12524	0.289873	0.144936	+	0.989441i	$0.453702\pi$
$992$	−15.9820	−0.507429
$993$	−14.6270	−0.464173
$994$	−6.50898	−0.206452
$995$	−7.23788	−0.229456
$996$	−39.8754	−1.26350
$997$	−23.5653	−0.746321	−0.373161	−	0.927767i	$-0.621726\pi$
$997$	−23.5653	−0.746321	−0.373161	+	0.927767i	$0.621726\pi$
$998$	4.42806	0.140168
$999$	0.921480	0.0291543

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	543.2.a.d.1.8	✓	8
3.2	odd	2		1629.2.a.e.1.1		8
4.3	odd	2		8688.2.a.bf.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
543.2.a.d.1.8	✓	8	1.1	even	1	trivial
1629.2.a.e.1.1		8	3.2	odd	2
8688.2.a.bf.1.4		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 543 = 3 \cdot 181 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	543.a (trivial)

\(f(q)\)	\(=\)	\(q+2.48495 q^{2} -1.00000 q^{3} +4.17495 q^{4} +0.370811 q^{5} -2.48495 q^{6} -1.96328 q^{7} +5.40464 q^{8} +1.00000 q^{9} +0.921444 q^{10} +4.04845 q^{11} -4.17495 q^{12} +4.06041 q^{13} -4.87863 q^{14} -0.370811 q^{15} +5.08033 q^{16} +3.62919 q^{17} +2.48495 q^{18} +5.64676 q^{19} +1.54812 q^{20} +1.96328 q^{21} +10.0602 q^{22} -8.92443 q^{23} -5.40464 q^{24} -4.86250 q^{25} +10.0899 q^{26} -1.00000 q^{27} -8.19659 q^{28} -6.74722 q^{29} -0.921444 q^{30} -8.80518 q^{31} +1.81507 q^{32} -4.04845 q^{33} +9.01834 q^{34} -0.728004 q^{35} +4.17495 q^{36} -0.921480 q^{37} +14.0319 q^{38} -4.06041 q^{39} +2.00410 q^{40} -0.316916 q^{41} +4.87863 q^{42} -2.22243 q^{43} +16.9021 q^{44} +0.370811 q^{45} -22.1767 q^{46} -4.93727 q^{47} -5.08033 q^{48} -3.14555 q^{49} -12.0830 q^{50} -3.62919 q^{51} +16.9520 q^{52} +14.1879 q^{53} -2.48495 q^{54} +1.50121 q^{55} -10.6108 q^{56} -5.64676 q^{57} -16.7665 q^{58} -0.537828 q^{59} -1.54812 q^{60} -2.35027 q^{61} -21.8804 q^{62} -1.96328 q^{63} -5.65033 q^{64} +1.50564 q^{65} -10.0602 q^{66} +10.6251 q^{67} +15.1517 q^{68} +8.92443 q^{69} -1.80905 q^{70} +1.33418 q^{71} +5.40464 q^{72} +8.35531 q^{73} -2.28983 q^{74} +4.86250 q^{75} +23.5749 q^{76} -7.94822 q^{77} -10.0899 q^{78} -15.5212 q^{79} +1.88384 q^{80} +1.00000 q^{81} -0.787518 q^{82} +9.55110 q^{83} +8.19659 q^{84} +1.34574 q^{85} -5.52261 q^{86} +6.74722 q^{87} +21.8804 q^{88} +3.36562 q^{89} +0.921444 q^{90} -7.97171 q^{91} -37.2591 q^{92} +8.80518 q^{93} -12.2688 q^{94} +2.09388 q^{95} -1.81507 q^{96} +12.3388 q^{97} -7.81652 q^{98} +4.04845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{2} - 8 q^{3} + 5 q^{4} + 5 q^{5} - 3 q^{6} - 3 q^{7} + 6 q^{8} + 8 q^{9} + 2 q^{10} + 4 q^{11} - 5 q^{12} + 13 q^{13} + 5 q^{14} - 5 q^{15} + 3 q^{16} + 27 q^{17} + 3 q^{18} - 10 q^{19} + 21 q^{20}+ \cdots + 4 q^{99}+O(q^{100}) \) 8 * q + 3 * q^2 - 8 * q^3 + 5 * q^4 + 5 * q^5 - 3 * q^6 - 3 * q^7 + 6 * q^8 + 8 * q^9 + 2 * q^10 + 4 * q^11 - 5 * q^12 + 13 * q^13 + 5 * q^14 - 5 * q^15 + 3 * q^16 + 27 * q^17 + 3 * q^18 - 10 * q^19 + 21 * q^20 + 3 * q^21 + 11 * q^22 + 3 * q^23 - 6 * q^24 + 13 * q^25 + 11 * q^26 - 8 * q^27 - 3 * q^28 + 16 * q^29 - 2 * q^30 - q^31 + 10 * q^32 - 4 * q^33 + 10 * q^34 + 14 * q^35 + 5 * q^36 + 3 * q^37 + 14 * q^38 - 13 * q^39 + 11 * q^40 + 20 * q^41 - 5 * q^42 - 4 * q^43 + 9 * q^44 + 5 * q^45 - 17 * q^46 + 12 * q^47 - 3 * q^48 + 5 * q^49 + 2 * q^50 - 27 * q^51 + 21 * q^52 + 23 * q^53 - 3 * q^54 - 13 * q^55 - 8 * q^56 + 10 * q^57 - 21 * q^58 + 7 * q^59 - 21 * q^60 + q^61 + q^62 - 3 * q^63 - 32 * q^64 + 10 * q^65 - 11 * q^66 - 25 * q^67 - q^68 - 3 * q^69 - 10 * q^70 + 11 * q^71 + 6 * q^72 + 23 * q^73 + 2 * q^74 - 13 * q^75 - 19 * q^76 + 20 * q^77 - 11 * q^78 - 26 * q^79 - 15 * q^80 + 8 * q^81 - 22 * q^82 + 16 * q^83 + 3 * q^84 - 33 * q^85 - 21 * q^86 - 16 * q^87 - q^88 + 14 * q^89 + 2 * q^90 - 35 * q^91 - 18 * q^92 + q^93 - 52 * q^94 + 10 * q^95 - 10 * q^96 + 28 * q^97 - 51 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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