LMFDB - Embedded newform 161.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [161,2,Mod(1,161)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(161, 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("161.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 161 = 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	161.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:	$1.28559147254$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - 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			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -3.23607 q^{5} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} -2.00000 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -3.23607 q^{5} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} -2.00000 q^{9} -2.00000 q^{10} +4.47214 q^{11} +1.61803 q^{12} +0.236068 q^{13} -0.618034 q^{14} +3.23607 q^{15} +1.85410 q^{16} -1.23607 q^{18} -7.23607 q^{19} +5.23607 q^{20} +1.00000 q^{21} +2.76393 q^{22} -1.00000 q^{23} +2.23607 q^{24} +5.47214 q^{25} +0.145898 q^{26} +5.00000 q^{27} +1.61803 q^{28} -1.47214 q^{29} +2.00000 q^{30} -9.00000 q^{31} +5.61803 q^{32} -4.47214 q^{33} +3.23607 q^{35} +3.23607 q^{36} -5.70820 q^{37} -4.47214 q^{38} -0.236068 q^{39} +7.23607 q^{40} -2.23607 q^{41} +0.618034 q^{42} +2.47214 q^{43} -7.23607 q^{44} +6.47214 q^{45} -0.618034 q^{46} -3.47214 q^{47} -1.85410 q^{48} +1.00000 q^{49} +3.38197 q^{50} -0.381966 q^{52} +11.2361 q^{53} +3.09017 q^{54} -14.4721 q^{55} +2.23607 q^{56} +7.23607 q^{57} -0.909830 q^{58} -1.52786 q^{59} -5.23607 q^{60} +13.4164 q^{61} -5.56231 q^{62} +2.00000 q^{63} -0.236068 q^{64} -0.763932 q^{65} -2.76393 q^{66} -12.1803 q^{67} +1.00000 q^{69} +2.00000 q^{70} -10.2361 q^{71} +4.47214 q^{72} -6.70820 q^{73} -3.52786 q^{74} -5.47214 q^{75} +11.7082 q^{76} -4.47214 q^{77} -0.145898 q^{78} -7.23607 q^{79} -6.00000 q^{80} +1.00000 q^{81} -1.38197 q^{82} +6.47214 q^{83} -1.61803 q^{84} +1.52786 q^{86} +1.47214 q^{87} -10.0000 q^{88} +8.94427 q^{89} +4.00000 q^{90} -0.236068 q^{91} +1.61803 q^{92} +9.00000 q^{93} -2.14590 q^{94} +23.4164 q^{95} -5.61803 q^{96} +3.70820 q^{97} +0.618034 q^{98} -8.94427 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{5} + q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q - q^2 - 2 * q^3 - q^4 - 2 * q^5 + q^6 - 2 * q^7 - 4 * q^9	$ 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{5} + q^{6} - 2 q^{7} - 4 q^{9} - 4 q^{10} + q^{12} - 4 q^{13} + q^{14} + 2 q^{15} - 3 q^{16} + 2 q^{18} - 10 q^{19} + 6 q^{20} + 2 q^{21} + 10 q^{22} - 2 q^{23} + 2 q^{25} + 7 q^{26} + 10 q^{27} + q^{28} + 6 q^{29} + 4 q^{30} - 18 q^{31} + 9 q^{32} + 2 q^{35} + 2 q^{36} + 2 q^{37} + 4 q^{39} + 10 q^{40} - q^{42} - 4 q^{43} - 10 q^{44} + 4 q^{45} + q^{46} + 2 q^{47} + 3 q^{48} + 2 q^{49} + 9 q^{50} - 3 q^{52} + 18 q^{53} - 5 q^{54} - 20 q^{55} + 10 q^{57} - 13 q^{58} - 12 q^{59} - 6 q^{60} + 9 q^{62} + 4 q^{63} + 4 q^{64} - 6 q^{65} - 10 q^{66} - 2 q^{67} + 2 q^{69} + 4 q^{70} - 16 q^{71} - 16 q^{74} - 2 q^{75} + 10 q^{76} - 7 q^{78} - 10 q^{79} - 12 q^{80} + 2 q^{81} - 5 q^{82} + 4 q^{83} - q^{84} + 12 q^{86} - 6 q^{87} - 20 q^{88} + 8 q^{90} + 4 q^{91} + q^{92} + 18 q^{93} - 11 q^{94} + 20 q^{95} - 9 q^{96} - 6 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 - 2 * q^3 - q^4 - 2 * q^5 + q^6 - 2 * q^7 - 4 * q^9 - 4 * q^10 + q^12 - 4 * q^13 + q^14 + 2 * q^15 - 3 * q^16 + 2 * q^18 - 10 * q^19 + 6 * q^20 + 2 * q^21 + 10 * q^22 - 2 * q^23 + 2 * q^25 + 7 * q^26 + 10 * q^27 + q^28 + 6 * q^29 + 4 * q^30 - 18 * q^31 + 9 * q^32 + 2 * q^35 + 2 * q^36 + 2 * q^37 + 4 * q^39 + 10 * q^40 - q^42 - 4 * q^43 - 10 * q^44 + 4 * q^45 + q^46 + 2 * q^47 + 3 * q^48 + 2 * q^49 + 9 * q^50 - 3 * q^52 + 18 * q^53 - 5 * q^54 - 20 * q^55 + 10 * q^57 - 13 * q^58 - 12 * q^59 - 6 * q^60 + 9 * q^62 + 4 * q^63 + 4 * q^64 - 6 * q^65 - 10 * q^66 - 2 * q^67 + 2 * q^69 + 4 * q^70 - 16 * q^71 - 16 * q^74 - 2 * q^75 + 10 * q^76 - 7 * q^78 - 10 * q^79 - 12 * q^80 + 2 * q^81 - 5 * q^82 + 4 * q^83 - q^84 + 12 * q^86 - 6 * q^87 - 20 * q^88 + 8 * q^90 + 4 * q^91 + q^92 + 18 * q^93 - 11 * q^94 + 20 * q^95 - 9 * q^96 - 6 * q^97 - q^98

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 $

$2$	0.618034	0.437016	0.218508	−	0.975835i	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	−1.61803	−0.809017
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	−0.618034	−0.252311
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.23607	−0.790569
$9$	−2.00000	−0.666667
$10$	−2.00000	−0.632456
$11$	4.47214	1.34840	0.674200	−	0.738549i	$-0.264489\pi$
$11$	4.47214	1.34840	0.674200	+	0.738549i	$0.264489\pi$
$12$	1.61803	0.467086
$13$	0.236068	0.0654735	0.0327367	−	0.999464i	$-0.489578\pi$
$13$	0.236068	0.0654735	0.0327367	+	0.999464i	$0.489578\pi$
$14$	−0.618034	−0.165177
$15$	3.23607	0.835549
$16$	1.85410	0.463525
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−1.23607	−0.291344
$19$	−7.23607	−1.66007	−0.830034	−	0.557713i	$-0.811679\pi$
$19$	−7.23607	−1.66007	−0.830034	+	0.557713i	$0.811679\pi$
$20$	5.23607	1.17082
$21$	1.00000	0.218218
$22$	2.76393	0.589272
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	2.23607	0.456435
$25$	5.47214	1.09443
$26$	0.145898	0.0286130
$27$	5.00000	0.962250
$28$	1.61803	0.305780
$29$	−1.47214	−0.273369	−0.136684	−	0.990615i	$-0.543645\pi$
$29$	−1.47214	−0.273369	−0.136684	+	0.990615i	$0.543645\pi$
$30$	2.00000	0.365148
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	5.61803	0.993137
$33$	−4.47214	−0.778499
$34$	0	0
$35$	3.23607	0.546995
$36$	3.23607	0.539345
$37$	−5.70820	−0.938423	−0.469211	−	0.883086i	$-0.655462\pi$
$37$	−5.70820	−0.938423	−0.469211	+	0.883086i	$0.655462\pi$
$38$	−4.47214	−0.725476
$39$	−0.236068	−0.0378011
$40$	7.23607	1.14412
$41$	−2.23607	−0.349215	−0.174608	−	0.984638i	$-0.555866\pi$
$41$	−2.23607	−0.349215	−0.174608	+	0.984638i	$0.555866\pi$
$42$	0.618034	0.0953647
$43$	2.47214	0.376997	0.188499	−	0.982073i	$-0.439638\pi$
$43$	2.47214	0.376997	0.188499	+	0.982073i	$0.439638\pi$
$44$	−7.23607	−1.09088
$45$	6.47214	0.964809
$46$	−0.618034	−0.0911241
$47$	−3.47214	−0.506463	−0.253232	−	0.967406i	$-0.581493\pi$
$47$	−3.47214	−0.506463	−0.253232	+	0.967406i	$0.581493\pi$
$48$	−1.85410	−0.267617
$49$	1.00000	0.142857
$50$	3.38197	0.478282
$51$	0	0
$52$	−0.381966	−0.0529692
$53$	11.2361	1.54339	0.771696	−	0.635991i	$-0.219409\pi$
$53$	11.2361	1.54339	0.771696	+	0.635991i	$0.219409\pi$
$54$	3.09017	0.420519
$55$	−14.4721	−1.95142
$56$	2.23607	0.298807
$57$	7.23607	0.958441
$58$	−0.909830	−0.119467
$59$	−1.52786	−0.198911	−0.0994555	−	0.995042i	$-0.531710\pi$
$59$	−1.52786	−0.198911	−0.0994555	+	0.995042i	$0.531710\pi$
$60$	−5.23607	−0.675973
$61$	13.4164	1.71780	0.858898	−	0.512148i	$-0.171150\pi$
$61$	13.4164	1.71780	0.858898	+	0.512148i	$0.171150\pi$
$62$	−5.56231	−0.706414
$63$	2.00000	0.251976
$64$	−0.236068	−0.0295085
$65$	−0.763932	−0.0947541
$66$	−2.76393	−0.340217
$67$	−12.1803	−1.48807	−0.744033	−	0.668143i	$-0.767089\pi$
$67$	−12.1803	−1.48807	−0.744033	+	0.668143i	$0.767089\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	2.00000	0.239046
$71$	−10.2361	−1.21480	−0.607399	−	0.794397i	$-0.707787\pi$
$71$	−10.2361	−1.21480	−0.607399	+	0.794397i	$0.707787\pi$
$72$	4.47214	0.527046
$73$	−6.70820	−0.785136	−0.392568	−	0.919723i	$-0.628413\pi$
$73$	−6.70820	−0.785136	−0.392568	+	0.919723i	$0.628413\pi$
$74$	−3.52786	−0.410106
$75$	−5.47214	−0.631868
$76$	11.7082	1.34302
$77$	−4.47214	−0.509647
$78$	−0.145898	−0.0165197
$79$	−7.23607	−0.814121	−0.407061	−	0.913401i	$-0.633446\pi$
$79$	−7.23607	−0.814121	−0.407061	+	0.913401i	$0.633446\pi$
$80$	−6.00000	−0.670820
$81$	1.00000	0.111111
$82$	−1.38197	−0.152613
$83$	6.47214	0.710409	0.355205	−	0.934789i	$-0.384411\pi$
$83$	6.47214	0.710409	0.355205	+	0.934789i	$0.384411\pi$
$84$	−1.61803	−0.176542
$85$	0	0
$86$	1.52786	0.164754
$87$	1.47214	0.157830
$88$	−10.0000	−1.06600
$89$	8.94427	0.948091	0.474045	−	0.880500i	$-0.342793\pi$
$89$	8.94427	0.948091	0.474045	+	0.880500i	$0.342793\pi$
$90$	4.00000	0.421637
$91$	−0.236068	−0.0247466
$92$	1.61803	0.168692
$93$	9.00000	0.933257
$94$	−2.14590	−0.221332
$95$	23.4164	2.40247
$96$	−5.61803	−0.573388
$97$	3.70820	0.376511	0.188256	−	0.982120i	$-0.439717\pi$
$97$	3.70820	0.376511	0.188256	+	0.982120i	$0.439717\pi$
$98$	0.618034	0.0624309
$99$	−8.94427	−0.898933
$100$	−8.85410	−0.885410
$101$	−13.4164	−1.33498	−0.667491	−	0.744618i	$-0.732632\pi$
$101$	−13.4164	−1.33498	−0.667491	+	0.744618i	$0.732632\pi$
$102$	0	0
$103$	−15.7082	−1.54778	−0.773888	−	0.633323i	$-0.781691\pi$
$103$	−15.7082	−1.54778	−0.773888	+	0.633323i	$0.781691\pi$
$104$	−0.527864	−0.0517613
$105$	−3.23607	−0.315808
$106$	6.94427	0.674487
$107$	11.2361	1.08623	0.543116	−	0.839658i	$-0.317244\pi$
$107$	11.2361	1.08623	0.543116	+	0.839658i	$0.317244\pi$
$108$	−8.09017	−0.778477
$109$	−15.4164	−1.47662	−0.738312	−	0.674459i	$-0.764377\pi$
$109$	−15.4164	−1.47662	−0.738312	+	0.674459i	$0.764377\pi$
$110$	−8.94427	−0.852803
$111$	5.70820	0.541799
$112$	−1.85410	−0.175196
$113$	2.47214	0.232559	0.116279	−	0.993217i	$-0.462903\pi$
$113$	2.47214	0.232559	0.116279	+	0.993217i	$0.462903\pi$
$114$	4.47214	0.418854
$115$	3.23607	0.301765
$116$	2.38197	0.221160
$117$	−0.472136	−0.0436490
$118$	−0.944272	−0.0869273
$119$	0	0
$120$	−7.23607	−0.660560
$121$	9.00000	0.818182
$122$	8.29180	0.750704
$123$	2.23607	0.201619
$124$	14.5623	1.30773
$125$	−1.52786	−0.136656
$126$	1.23607	0.110118
$127$	−2.70820	−0.240314	−0.120157	−	0.992755i	$-0.538340\pi$
$127$	−2.70820	−0.240314	−0.120157	+	0.992755i	$0.538340\pi$
$128$	−11.3820	−1.00603
$129$	−2.47214	−0.217659
$130$	−0.472136	−0.0414091
$131$	3.94427	0.344613	0.172306	−	0.985043i	$-0.444878\pi$
$131$	3.94427	0.344613	0.172306	+	0.985043i	$0.444878\pi$
$132$	7.23607	0.629819
$133$	7.23607	0.627447
$134$	−7.52786	−0.650308
$135$	−16.1803	−1.39258
$136$	0	0
$137$	15.7082	1.34204	0.671021	−	0.741438i	$-0.265856\pi$
$137$	15.7082	1.34204	0.671021	+	0.741438i	$0.265856\pi$
$138$	0.618034	0.0526105
$139$	2.52786	0.214411	0.107205	−	0.994237i	$-0.465810\pi$
$139$	2.52786	0.214411	0.107205	+	0.994237i	$0.465810\pi$
$140$	−5.23607	−0.442529
$141$	3.47214	0.292407
$142$	−6.32624	−0.530886
$143$	1.05573	0.0882844
$144$	−3.70820	−0.309017
$145$	4.76393	0.395623
$146$	−4.14590	−0.343117
$147$	−1.00000	−0.0824786
$148$	9.23607	0.759200
$149$	15.2361	1.24819	0.624094	−	0.781350i	$-0.285468\pi$
$149$	15.2361	1.24819	0.624094	+	0.781350i	$0.285468\pi$
$150$	−3.38197	−0.276136
$151$	−15.1803	−1.23536	−0.617679	−	0.786430i	$-0.711927\pi$
$151$	−15.1803	−1.23536	−0.617679	+	0.786430i	$0.711927\pi$
$152$	16.1803	1.31240
$153$	0	0
$154$	−2.76393	−0.222724
$155$	29.1246	2.33935
$156$	0.381966	0.0305818
$157$	15.4164	1.23036	0.615182	−	0.788385i	$-0.289083\pi$
$157$	15.4164	1.23036	0.615182	+	0.788385i	$0.289083\pi$
$158$	−4.47214	−0.355784
$159$	−11.2361	−0.891078
$160$	−18.1803	−1.43728
$161$	1.00000	0.0788110
$162$	0.618034	0.0485573
$163$	19.1803	1.50232	0.751160	−	0.660120i	$-0.229495\pi$
$163$	19.1803	1.50232	0.751160	+	0.660120i	$0.229495\pi$
$164$	3.61803	0.282521
$165$	14.4721	1.12665
$166$	4.00000	0.310460
$167$	21.8885	1.69379	0.846893	−	0.531763i	$-0.178470\pi$
$167$	21.8885	1.69379	0.846893	+	0.531763i	$0.178470\pi$
$168$	−2.23607	−0.172516
$169$	−12.9443	−0.995713
$170$	0	0
$171$	14.4721	1.10671
$172$	−4.00000	−0.304997
$173$	−3.52786	−0.268219	−0.134109	−	0.990967i	$-0.542817\pi$
$173$	−3.52786	−0.268219	−0.134109	+	0.990967i	$0.542817\pi$
$174$	0.909830	0.0689740
$175$	−5.47214	−0.413655
$176$	8.29180	0.625018
$177$	1.52786	0.114841
$178$	5.52786	0.414331
$179$	−18.7082	−1.39832	−0.699158	−	0.714967i	$-0.746442\pi$
$179$	−18.7082	−1.39832	−0.699158	+	0.714967i	$0.746442\pi$
$180$	−10.4721	−0.780547
$181$	−5.05573	−0.375789	−0.187895	−	0.982189i	$-0.560166\pi$
$181$	−5.05573	−0.375789	−0.187895	+	0.982189i	$0.560166\pi$
$182$	−0.145898	−0.0108147
$183$	−13.4164	−0.991769
$184$	2.23607	0.164845
$185$	18.4721	1.35810
$186$	5.56231	0.407848
$187$	0	0
$188$	5.61803	0.409737
$189$	−5.00000	−0.363696
$190$	14.4721	1.04992
$191$	−24.1803	−1.74963	−0.874814	−	0.484459i	$-0.839016\pi$
$191$	−24.1803	−1.74963	−0.874814	+	0.484459i	$0.839016\pi$
$192$	0.236068	0.0170367
$193$	4.41641	0.317900	0.158950	−	0.987287i	$-0.449189\pi$
$193$	4.41641	0.317900	0.158950	+	0.987287i	$0.449189\pi$
$194$	2.29180	0.164541
$195$	0.763932	0.0547063
$196$	−1.61803	−0.115574
$197$	−21.4721	−1.52983	−0.764913	−	0.644133i	$-0.777218\pi$
$197$	−21.4721	−1.52983	−0.764913	+	0.644133i	$0.777218\pi$
$198$	−5.52786	−0.392848
$199$	15.8885	1.12631	0.563155	−	0.826352i	$-0.309588\pi$
$199$	15.8885	1.12631	0.563155	+	0.826352i	$0.309588\pi$
$200$	−12.2361	−0.865221
$201$	12.1803	0.859135
$202$	−8.29180	−0.583409
$203$	1.47214	0.103324
$204$	0	0
$205$	7.23607	0.505389
$206$	−9.70820	−0.676403
$207$	2.00000	0.139010
$208$	0.437694	0.0303486
$209$	−32.3607	−2.23844
$210$	−2.00000	−0.138013
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−18.1803	−1.24863
$213$	10.2361	0.701364
$214$	6.94427	0.474701
$215$	−8.00000	−0.545595
$216$	−11.1803	−0.760726
$217$	9.00000	0.610960
$218$	−9.52786	−0.645308
$219$	6.70820	0.453298
$220$	23.4164	1.57873
$221$	0	0
$222$	3.52786	0.236775
$223$	3.41641	0.228780	0.114390	−	0.993436i	$-0.463509\pi$
$223$	3.41641	0.228780	0.114390	+	0.993436i	$0.463509\pi$
$224$	−5.61803	−0.375371
$225$	−10.9443	−0.729618
$226$	1.52786	0.101632
$227$	−5.88854	−0.390836	−0.195418	−	0.980720i	$-0.562606\pi$
$227$	−5.88854	−0.390836	−0.195418	+	0.980720i	$0.562606\pi$
$228$	−11.7082	−0.775395
$229$	14.7639	0.975628	0.487814	−	0.872948i	$-0.337794\pi$
$229$	14.7639	0.975628	0.487814	+	0.872948i	$0.337794\pi$
$230$	2.00000	0.131876
$231$	4.47214	0.294245
$232$	3.29180	0.216117
$233$	−11.4721	−0.751565	−0.375782	−	0.926708i	$-0.622626\pi$
$233$	−11.4721	−0.751565	−0.375782	+	0.926708i	$0.622626\pi$
$234$	−0.291796	−0.0190753
$235$	11.2361	0.732960
$236$	2.47214	0.160922
$237$	7.23607	0.470033
$238$	0	0
$239$	−15.7639	−1.01968	−0.509842	−	0.860268i	$-0.670296\pi$
$239$	−15.7639	−1.01968	−0.509842	+	0.860268i	$0.670296\pi$
$240$	6.00000	0.387298
$241$	−21.4164	−1.37955	−0.689776	−	0.724023i	$-0.742291\pi$
$241$	−21.4164	−1.37955	−0.689776	+	0.724023i	$0.742291\pi$
$242$	5.56231	0.357559
$243$	−16.0000	−1.02640
$244$	−21.7082	−1.38973
$245$	−3.23607	−0.206745
$246$	1.38197	0.0881109
$247$	−1.70820	−0.108690
$248$	20.1246	1.27791
$249$	−6.47214	−0.410155
$250$	−0.944272	−0.0597210
$251$	8.29180	0.523374	0.261687	−	0.965153i	$-0.415721\pi$
$251$	8.29180	0.523374	0.261687	+	0.965153i	$0.415721\pi$
$252$	−3.23607	−0.203853
$253$	−4.47214	−0.281161
$254$	−1.67376	−0.105021
$255$	0	0
$256$	−6.56231	−0.410144
$257$	4.23607	0.264239	0.132119	−	0.991234i	$-0.457822\pi$
$257$	4.23607	0.264239	0.132119	+	0.991234i	$0.457822\pi$
$258$	−1.52786	−0.0951207
$259$	5.70820	0.354691
$260$	1.23607	0.0766577
$261$	2.94427	0.182246
$262$	2.43769	0.150601
$263$	−26.9443	−1.66145	−0.830727	−	0.556679i	$-0.812075\pi$
$263$	−26.9443	−1.66145	−0.830727	+	0.556679i	$0.812075\pi$
$264$	10.0000	0.615457
$265$	−36.3607	−2.23362
$266$	4.47214	0.274204
$267$	−8.94427	−0.547381
$268$	19.7082	1.20387
$269$	−9.18034	−0.559735	−0.279868	−	0.960039i	$-0.590291\pi$
$269$	−9.18034	−0.559735	−0.279868	+	0.960039i	$0.590291\pi$
$270$	−10.0000	−0.608581
$271$	−16.9443	−1.02929	−0.514646	−	0.857403i	$-0.672076\pi$
$271$	−16.9443	−1.02929	−0.514646	+	0.857403i	$0.672076\pi$
$272$	0	0
$273$	0.236068	0.0142875
$274$	9.70820	0.586494
$275$	24.4721	1.47573
$276$	−1.61803	−0.0973942
$277$	20.4164	1.22670	0.613352	−	0.789810i	$-0.289821\pi$
$277$	20.4164	1.22670	0.613352	+	0.789810i	$0.289821\pi$
$278$	1.56231	0.0937009
$279$	18.0000	1.07763
$280$	−7.23607	−0.432438
$281$	3.70820	0.221213	0.110606	−	0.993864i	$-0.464721\pi$
$281$	3.70820	0.221213	0.110606	+	0.993864i	$0.464721\pi$
$282$	2.14590	0.127786
$283$	18.9443	1.12612	0.563060	−	0.826416i	$-0.309624\pi$
$283$	18.9443	1.12612	0.563060	+	0.826416i	$0.309624\pi$
$284$	16.5623	0.982792
$285$	−23.4164	−1.38707
$286$	0.652476	0.0385817
$287$	2.23607	0.131991
$288$	−11.2361	−0.662092
$289$	−17.0000	−1.00000
$290$	2.94427	0.172894
$291$	−3.70820	−0.217379
$292$	10.8541	0.635188
$293$	−15.7082	−0.917683	−0.458842	−	0.888518i	$-0.651735\pi$
$293$	−15.7082	−0.917683	−0.458842	+	0.888518i	$0.651735\pi$
$294$	−0.618034	−0.0360445
$295$	4.94427	0.287867
$296$	12.7639	0.741888
$297$	22.3607	1.29750
$298$	9.41641	0.545478
$299$	−0.236068	−0.0136522
$300$	8.85410	0.511192
$301$	−2.47214	−0.142492
$302$	−9.38197	−0.539871
$303$	13.4164	0.770752
$304$	−13.4164	−0.769484
$305$	−43.4164	−2.48602
$306$	0	0
$307$	−11.4164	−0.651569	−0.325784	−	0.945444i	$-0.605628\pi$
$307$	−11.4164	−0.651569	−0.325784	+	0.945444i	$0.605628\pi$
$308$	7.23607	0.412313
$309$	15.7082	0.893609
$310$	18.0000	1.02233
$311$	−2.88854	−0.163794	−0.0818971	−	0.996641i	$-0.526098\pi$
$311$	−2.88854	−0.163794	−0.0818971	+	0.996641i	$0.526098\pi$
$312$	0.527864	0.0298844
$313$	2.76393	0.156227	0.0781133	−	0.996944i	$-0.475110\pi$
$313$	2.76393	0.156227	0.0781133	+	0.996944i	$0.475110\pi$
$314$	9.52786	0.537688
$315$	−6.47214	−0.364664
$316$	11.7082	0.658638
$317$	31.3050	1.75826	0.879131	−	0.476581i	$-0.158124\pi$
$317$	31.3050	1.75826	0.879131	+	0.476581i	$0.158124\pi$
$318$	−6.94427	−0.389415
$319$	−6.58359	−0.368610
$320$	0.763932	0.0427051
$321$	−11.2361	−0.627136
$322$	0.618034	0.0344417
$323$	0	0
$324$	−1.61803	−0.0898908
$325$	1.29180	0.0716560
$326$	11.8541	0.656538
$327$	15.4164	0.852529
$328$	5.00000	0.276079
$329$	3.47214	0.191425
$330$	8.94427	0.492366
$331$	0.708204	0.0389264	0.0194632	−	0.999811i	$-0.493804\pi$
$331$	0.708204	0.0389264	0.0194632	+	0.999811i	$0.493804\pi$
$332$	−10.4721	−0.574733
$333$	11.4164	0.625615
$334$	13.5279	0.740212
$335$	39.4164	2.15355
$336$	1.85410	0.101150
$337$	−17.5967	−0.958556	−0.479278	−	0.877663i	$-0.659101\pi$
$337$	−17.5967	−0.958556	−0.479278	+	0.877663i	$0.659101\pi$
$338$	−8.00000	−0.435143
$339$	−2.47214	−0.134268
$340$	0	0
$341$	−40.2492	−2.17962
$342$	8.94427	0.483651
$343$	−1.00000	−0.0539949
$344$	−5.52786	−0.298042
$345$	−3.23607	−0.174224
$346$	−2.18034	−0.117216
$347$	−5.88854	−0.316114	−0.158057	−	0.987430i	$-0.550523\pi$
$347$	−5.88854	−0.316114	−0.158057	+	0.987430i	$0.550523\pi$
$348$	−2.38197	−0.127687
$349$	1.29180	0.0691483	0.0345741	−	0.999402i	$-0.488993\pi$
$349$	1.29180	0.0691483	0.0345741	+	0.999402i	$0.488993\pi$
$350$	−3.38197	−0.180774
$351$	1.18034	0.0630019
$352$	25.1246	1.33915
$353$	−1.76393	−0.0938846	−0.0469423	−	0.998898i	$-0.514948\pi$
$353$	−1.76393	−0.0938846	−0.0469423	+	0.998898i	$0.514948\pi$
$354$	0.944272	0.0501875
$355$	33.1246	1.75807
$356$	−14.4721	−0.767022
$357$	0	0
$358$	−11.5623	−0.611087
$359$	4.76393	0.251431	0.125715	−	0.992066i	$-0.459877\pi$
$359$	4.76393	0.251431	0.125715	+	0.992066i	$0.459877\pi$
$360$	−14.4721	−0.762749
$361$	33.3607	1.75583
$362$	−3.12461	−0.164226
$363$	−9.00000	−0.472377
$364$	0.381966	0.0200205
$365$	21.7082	1.13626
$366$	−8.29180	−0.433419
$367$	−4.58359	−0.239262	−0.119631	−	0.992818i	$-0.538171\pi$
$367$	−4.58359	−0.239262	−0.119631	+	0.992818i	$0.538171\pi$
$368$	−1.85410	−0.0966517
$369$	4.47214	0.232810
$370$	11.4164	0.593511
$371$	−11.2361	−0.583348
$372$	−14.5623	−0.755020
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	1.52786	0.0788986
$376$	7.76393	0.400394
$377$	−0.347524	−0.0178984
$378$	−3.09017	−0.158941
$379$	8.29180	0.425921	0.212960	−	0.977061i	$-0.431689\pi$
$379$	8.29180	0.425921	0.212960	+	0.977061i	$0.431689\pi$
$380$	−37.8885	−1.94364
$381$	2.70820	0.138745
$382$	−14.9443	−0.764615
$383$	−15.7082	−0.802652	−0.401326	−	0.915935i	$-0.631451\pi$
$383$	−15.7082	−0.802652	−0.401326	+	0.915935i	$0.631451\pi$
$384$	11.3820	0.580834
$385$	14.4721	0.737568
$386$	2.72949	0.138927
$387$	−4.94427	−0.251331
$388$	−6.00000	−0.304604
$389$	−7.41641	−0.376027	−0.188013	−	0.982166i	$-0.560205\pi$
$389$	−7.41641	−0.376027	−0.188013	+	0.982166i	$0.560205\pi$
$390$	0.472136	0.0239075
$391$	0	0
$392$	−2.23607	−0.112938
$393$	−3.94427	−0.198962
$394$	−13.2705	−0.668559
$395$	23.4164	1.17821
$396$	14.4721	0.727252
$397$	−25.6525	−1.28746	−0.643730	−	0.765252i	$-0.722614\pi$
$397$	−25.6525	−1.28746	−0.643730	+	0.765252i	$0.722614\pi$
$398$	9.81966	0.492215
$399$	−7.23607	−0.362257
$400$	10.1459	0.507295
$401$	−19.8885	−0.993186	−0.496593	−	0.867983i	$-0.665416\pi$
$401$	−19.8885	−0.993186	−0.496593	+	0.867983i	$0.665416\pi$
$402$	7.52786	0.375456
$403$	−2.12461	−0.105834
$404$	21.7082	1.08002
$405$	−3.23607	−0.160802
$406$	0.909830	0.0451541
$407$	−25.5279	−1.26537
$408$	0	0
$409$	4.12461	0.203949	0.101974	−	0.994787i	$-0.467484\pi$
$409$	4.12461	0.203949	0.101974	+	0.994787i	$0.467484\pi$
$410$	4.47214	0.220863
$411$	−15.7082	−0.774829
$412$	25.4164	1.25218
$413$	1.52786	0.0751813
$414$	1.23607	0.0607494
$415$	−20.9443	−1.02811
$416$	1.32624	0.0650242
$417$	−2.52786	−0.123790
$418$	−20.0000	−0.978232
$419$	26.4721	1.29325	0.646624	−	0.762809i	$-0.276180\pi$
$419$	26.4721	1.29325	0.646624	+	0.762809i	$0.276180\pi$
$420$	5.23607	0.255494
$421$	−8.58359	−0.418339	−0.209169	−	0.977879i	$-0.567076\pi$
$421$	−8.58359	−0.418339	−0.209169	+	0.977879i	$0.567076\pi$
$422$	−7.41641	−0.361025
$423$	6.94427	0.337642
$424$	−25.1246	−1.22016
$425$	0	0
$426$	6.32624	0.306507
$427$	−13.4164	−0.649265
$428$	−18.1803	−0.878780
$429$	−1.05573	−0.0509710
$430$	−4.94427	−0.238434
$431$	18.1803	0.875716	0.437858	−	0.899044i	$-0.355737\pi$
$431$	18.1803	0.875716	0.437858	+	0.899044i	$0.355737\pi$
$432$	9.27051	0.446028
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	5.56231	0.266999
$435$	−4.76393	−0.228413
$436$	24.9443	1.19461
$437$	7.23607	0.346148
$438$	4.14590	0.198099
$439$	30.3050	1.44638	0.723188	−	0.690651i	$-0.242676\pi$
$439$	30.3050	1.44638	0.723188	+	0.690651i	$0.242676\pi$
$440$	32.3607	1.54273
$441$	−2.00000	−0.0952381
$442$	0	0
$443$	−7.18034	−0.341148	−0.170574	−	0.985345i	$-0.554562\pi$
$443$	−7.18034	−0.341148	−0.170574	+	0.985345i	$0.554562\pi$
$444$	−9.23607	−0.438324
$445$	−28.9443	−1.37209
$446$	2.11146	0.0999803
$447$	−15.2361	−0.720641
$448$	0.236068	0.0111532
$449$	32.8328	1.54948	0.774738	−	0.632282i	$-0.217882\pi$
$449$	32.8328	1.54948	0.774738	+	0.632282i	$0.217882\pi$
$450$	−6.76393	−0.318855
$451$	−10.0000	−0.470882
$452$	−4.00000	−0.188144
$453$	15.1803	0.713235
$454$	−3.63932	−0.170802
$455$	0.763932	0.0358137
$456$	−16.1803	−0.757714
$457$	14.4721	0.676978	0.338489	−	0.940970i	$-0.390084\pi$
$457$	14.4721	0.676978	0.338489	+	0.940970i	$0.390084\pi$
$458$	9.12461	0.426365
$459$	0	0
$460$	−5.23607	−0.244133
$461$	−25.7639	−1.19995	−0.599973	−	0.800020i	$-0.704822\pi$
$461$	−25.7639	−1.19995	−0.599973	+	0.800020i	$0.704822\pi$
$462$	2.76393	0.128590
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	−2.72949	−0.126713
$465$	−29.1246	−1.35062
$466$	−7.09017	−0.328446
$467$	−11.5279	−0.533446	−0.266723	−	0.963773i	$-0.585941\pi$
$467$	−11.5279	−0.533446	−0.266723	+	0.963773i	$0.585941\pi$
$468$	0.763932	0.0353128
$469$	12.1803	0.562436
$470$	6.94427	0.320315
$471$	−15.4164	−0.710351
$472$	3.41641	0.157253
$473$	11.0557	0.508343
$474$	4.47214	0.205412
$475$	−39.5967	−1.81682
$476$	0	0
$477$	−22.4721	−1.02893
$478$	−9.74265	−0.445618
$479$	−23.1246	−1.05659	−0.528295	−	0.849061i	$-0.677169\pi$
$479$	−23.1246	−1.05659	−0.528295	+	0.849061i	$0.677169\pi$
$480$	18.1803	0.829815
$481$	−1.34752	−0.0614418
$482$	−13.2361	−0.602886
$483$	−1.00000	−0.0455016
$484$	−14.5623	−0.661923
$485$	−12.0000	−0.544892
$486$	−9.88854	−0.448553
$487$	20.1246	0.911933	0.455967	−	0.889997i	$-0.349294\pi$
$487$	20.1246	0.911933	0.455967	+	0.889997i	$0.349294\pi$
$488$	−30.0000	−1.35804
$489$	−19.1803	−0.867365
$490$	−2.00000	−0.0903508
$491$	11.7639	0.530899	0.265449	−	0.964125i	$-0.414480\pi$
$491$	11.7639	0.530899	0.265449	+	0.964125i	$0.414480\pi$
$492$	−3.61803	−0.163114
$493$	0	0
$494$	−1.05573	−0.0474995
$495$	28.9443	1.30095
$496$	−16.6869	−0.749265
$497$	10.2361	0.459150
$498$	−4.00000	−0.179244
$499$	−36.7082	−1.64328	−0.821642	−	0.570003i	$-0.806942\pi$
$499$	−36.7082	−1.64328	−0.821642	+	0.570003i	$0.806942\pi$
$500$	2.47214	0.110557
$501$	−21.8885	−0.977908
$502$	5.12461	0.228723
$503$	−15.5967	−0.695425	−0.347712	−	0.937601i	$-0.613041\pi$
$503$	−15.5967	−0.695425	−0.347712	+	0.937601i	$0.613041\pi$
$504$	−4.47214	−0.199205
$505$	43.4164	1.93200
$506$	−2.76393	−0.122872
$507$	12.9443	0.574875
$508$	4.38197	0.194418
$509$	−12.8197	−0.568221	−0.284111	−	0.958791i	$-0.591698\pi$
$509$	−12.8197	−0.568221	−0.284111	+	0.958791i	$0.591698\pi$
$510$	0	0
$511$	6.70820	0.296753
$512$	18.7082	0.826794
$513$	−36.1803	−1.59740
$514$	2.61803	0.115477
$515$	50.8328	2.23996
$516$	4.00000	0.176090
$517$	−15.5279	−0.682915
$518$	3.52786	0.155005
$519$	3.52786	0.154856
$520$	1.70820	0.0749097
$521$	21.3050	0.933387	0.466693	−	0.884419i	$-0.345445\pi$
$521$	21.3050	0.933387	0.466693	+	0.884419i	$0.345445\pi$
$522$	1.81966	0.0796444
$523$	−7.70820	−0.337056	−0.168528	−	0.985697i	$-0.553901\pi$
$523$	−7.70820	−0.337056	−0.168528	+	0.985697i	$0.553901\pi$
$524$	−6.38197	−0.278797
$525$	5.47214	0.238824
$526$	−16.6525	−0.726082
$527$	0	0
$528$	−8.29180	−0.360854
$529$	1.00000	0.0434783
$530$	−22.4721	−0.976127
$531$	3.05573	0.132607
$532$	−11.7082	−0.507615
$533$	−0.527864	−0.0228643
$534$	−5.52786	−0.239214
$535$	−36.3607	−1.57201
$536$	27.2361	1.17642
$537$	18.7082	0.807319
$538$	−5.67376	−0.244613
$539$	4.47214	0.192629
$540$	26.1803	1.12662
$541$	21.0000	0.902861	0.451430	−	0.892306i	$-0.350914\pi$
$541$	21.0000	0.902861	0.451430	+	0.892306i	$0.350914\pi$
$542$	−10.4721	−0.449817
$543$	5.05573	0.216962
$544$	0	0
$545$	49.8885	2.13699
$546$	0.145898	0.00624386
$547$	43.1803	1.84626	0.923129	−	0.384490i	$-0.125623\pi$
$547$	43.1803	1.84626	0.923129	+	0.384490i	$0.125623\pi$
$548$	−25.4164	−1.08574
$549$	−26.8328	−1.14520
$550$	15.1246	0.644916
$551$	10.6525	0.453811
$552$	−2.23607	−0.0951734
$553$	7.23607	0.307709
$554$	12.6180	0.536089
$555$	−18.4721	−0.784099
$556$	−4.09017	−0.173462
$557$	−26.7639	−1.13402	−0.567012	−	0.823709i	$-0.691901\pi$
$557$	−26.7639	−1.13402	−0.567012	+	0.823709i	$0.691901\pi$
$558$	11.1246	0.470942
$559$	0.583592	0.0246833
$560$	6.00000	0.253546
$561$	0	0
$562$	2.29180	0.0966736
$563$	−9.59675	−0.404455	−0.202227	−	0.979339i	$-0.564818\pi$
$563$	−9.59675	−0.404455	−0.202227	+	0.979339i	$0.564818\pi$
$564$	−5.61803	−0.236562
$565$	−8.00000	−0.336563
$566$	11.7082	0.492133
$567$	−1.00000	−0.0419961
$568$	22.8885	0.960382
$569$	−8.18034	−0.342938	−0.171469	−	0.985190i	$-0.554851\pi$
$569$	−8.18034	−0.342938	−0.171469	+	0.985190i	$0.554851\pi$
$570$	−14.4721	−0.606171
$571$	16.2918	0.681790	0.340895	−	0.940101i	$-0.389270\pi$
$571$	16.2918	0.681790	0.340895	+	0.940101i	$0.389270\pi$
$572$	−1.70820	−0.0714236
$573$	24.1803	1.01015
$574$	1.38197	0.0576821
$575$	−5.47214	−0.228204
$576$	0.472136	0.0196723
$577$	−15.2918	−0.636606	−0.318303	−	0.947989i	$-0.603113\pi$
$577$	−15.2918	−0.636606	−0.318303	+	0.947989i	$0.603113\pi$
$578$	−10.5066	−0.437016
$579$	−4.41641	−0.183540
$580$	−7.70820	−0.320066
$581$	−6.47214	−0.268509
$582$	−2.29180	−0.0949980
$583$	50.2492	2.08111
$584$	15.0000	0.620704
$585$	1.52786	0.0631694
$586$	−9.70820	−0.401042
$587$	20.8885	0.862162	0.431081	−	0.902313i	$-0.358132\pi$
$587$	20.8885	0.862162	0.431081	+	0.902313i	$0.358132\pi$
$588$	1.61803	0.0667266
$589$	65.1246	2.68341
$590$	3.05573	0.125802
$591$	21.4721	0.883246
$592$	−10.5836	−0.434983
$593$	34.3607	1.41102	0.705512	−	0.708698i	$-0.250717\pi$
$593$	34.3607	1.41102	0.705512	+	0.708698i	$0.250717\pi$
$594$	13.8197	0.567028
$595$	0	0
$596$	−24.6525	−1.00980
$597$	−15.8885	−0.650275
$598$	−0.145898	−0.00596621
$599$	−2.47214	−0.101009	−0.0505044	−	0.998724i	$-0.516083\pi$
$599$	−2.47214	−0.101009	−0.0505044	+	0.998724i	$0.516083\pi$
$600$	12.2361	0.499535
$601$	22.2361	0.907028	0.453514	−	0.891249i	$-0.350170\pi$
$601$	22.2361	0.907028	0.453514	+	0.891249i	$0.350170\pi$
$602$	−1.52786	−0.0622711
$603$	24.3607	0.992044
$604$	24.5623	0.999426
$605$	−29.1246	−1.18408
$606$	8.29180	0.336831
$607$	−33.3050	−1.35181	−0.675903	−	0.736990i	$-0.736246\pi$
$607$	−33.3050	−1.35181	−0.675903	+	0.736990i	$0.736246\pi$
$608$	−40.6525	−1.64868
$609$	−1.47214	−0.0596540
$610$	−26.8328	−1.08643
$611$	−0.819660	−0.0331599
$612$	0	0
$613$	−22.1803	−0.895855	−0.447928	−	0.894070i	$-0.647838\pi$
$613$	−22.1803	−0.895855	−0.447928	+	0.894070i	$0.647838\pi$
$614$	−7.05573	−0.284746
$615$	−7.23607	−0.291786
$616$	10.0000	0.402911
$617$	35.2361	1.41855	0.709275	−	0.704932i	$-0.249022\pi$
$617$	35.2361	1.41855	0.709275	+	0.704932i	$0.249022\pi$
$618$	9.70820	0.390521
$619$	−47.4853	−1.90860	−0.954298	−	0.298858i	$-0.903394\pi$
$619$	−47.4853	−1.90860	−0.954298	+	0.298858i	$0.903394\pi$
$620$	−47.1246	−1.89257
$621$	−5.00000	−0.200643
$622$	−1.78522	−0.0715807
$623$	−8.94427	−0.358345
$624$	−0.437694	−0.0175218
$625$	−22.4164	−0.896656
$626$	1.70820	0.0682736
$627$	32.3607	1.29236
$628$	−24.9443	−0.995385
$629$	0	0
$630$	−4.00000	−0.159364
$631$	3.41641	0.136005	0.0680025	−	0.997685i	$-0.478337\pi$
$631$	3.41641	0.136005	0.0680025	+	0.997685i	$0.478337\pi$
$632$	16.1803	0.643619
$633$	12.0000	0.476957
$634$	19.3475	0.768388
$635$	8.76393	0.347786
$636$	18.1803	0.720897
$637$	0.236068	0.00935335
$638$	−4.06888	−0.161089
$639$	20.4721	0.809865
$640$	36.8328	1.45594
$641$	7.05573	0.278685	0.139342	−	0.990244i	$-0.455501\pi$
$641$	7.05573	0.278685	0.139342	+	0.990244i	$0.455501\pi$
$642$	−6.94427	−0.274069
$643$	−17.7082	−0.698343	−0.349172	−	0.937059i	$-0.613537\pi$
$643$	−17.7082	−0.698343	−0.349172	+	0.937059i	$0.613537\pi$
$644$	−1.61803	−0.0637595
$645$	8.00000	0.315000
$646$	0	0
$647$	−18.5279	−0.728405	−0.364203	−	0.931320i	$-0.618658\pi$
$647$	−18.5279	−0.728405	−0.364203	+	0.931320i	$0.618658\pi$
$648$	−2.23607	−0.0878410
$649$	−6.83282	−0.268211
$650$	0.798374	0.0313148
$651$	−9.00000	−0.352738
$652$	−31.0344	−1.21540
$653$	24.5279	0.959849	0.479925	−	0.877310i	$-0.340664\pi$
$653$	24.5279	0.959849	0.479925	+	0.877310i	$0.340664\pi$
$654$	9.52786	0.372569
$655$	−12.7639	−0.498728
$656$	−4.14590	−0.161870
$657$	13.4164	0.523424
$658$	2.14590	0.0836558
$659$	6.76393	0.263485	0.131743	−	0.991284i	$-0.457943\pi$
$659$	6.76393	0.263485	0.131743	+	0.991284i	$0.457943\pi$
$660$	−23.4164	−0.911482
$661$	−41.7082	−1.62226	−0.811131	−	0.584865i	$-0.801147\pi$
$661$	−41.7082	−1.62226	−0.811131	+	0.584865i	$0.801147\pi$
$662$	0.437694	0.0170115
$663$	0	0
$664$	−14.4721	−0.561628
$665$	−23.4164	−0.908049
$666$	7.05573	0.273404
$667$	1.47214	0.0570013
$668$	−35.4164	−1.37030
$669$	−3.41641	−0.132086
$670$	24.3607	0.941135
$671$	60.0000	2.31627
$672$	5.61803	0.216720
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	−10.8754	−0.418904
$675$	27.3607	1.05311
$676$	20.9443	0.805549
$677$	23.2361	0.893035	0.446517	−	0.894775i	$-0.352664\pi$
$677$	23.2361	0.893035	0.446517	+	0.894775i	$0.352664\pi$
$678$	−1.52786	−0.0586773
$679$	−3.70820	−0.142308
$680$	0	0
$681$	5.88854	0.225649
$682$	−24.8754	−0.952528
$683$	−5.18034	−0.198220	−0.0991101	−	0.995076i	$-0.531600\pi$
$683$	−5.18034	−0.198220	−0.0991101	+	0.995076i	$0.531600\pi$
$684$	−23.4164	−0.895349
$685$	−50.8328	−1.94222
$686$	−0.618034	−0.0235966
$687$	−14.7639	−0.563279
$688$	4.58359	0.174748
$689$	2.65248	0.101051
$690$	−2.00000	−0.0761387
$691$	42.8328	1.62944	0.814719	−	0.579857i	$-0.196891\pi$
$691$	42.8328	1.62944	0.814719	+	0.579857i	$0.196891\pi$
$692$	5.70820	0.216993
$693$	8.94427	0.339765
$694$	−3.63932	−0.138147
$695$	−8.18034	−0.310298
$696$	−3.29180	−0.124775
$697$	0	0
$698$	0.798374	0.0302189
$699$	11.4721	0.433916
$700$	8.85410	0.334654
$701$	−22.7639	−0.859782	−0.429891	−	0.902881i	$-0.641448\pi$
$701$	−22.7639	−0.859782	−0.429891	+	0.902881i	$0.641448\pi$
$702$	0.729490	0.0275328
$703$	41.3050	1.55785
$704$	−1.05573	−0.0397892
$705$	−11.2361	−0.423175
$706$	−1.09017	−0.0410291
$707$	13.4164	0.504576
$708$	−2.47214	−0.0929086
$709$	34.2492	1.28626	0.643128	−	0.765758i	$-0.277636\pi$
$709$	34.2492	1.28626	0.643128	+	0.765758i	$0.277636\pi$
$710$	20.4721	0.768306
$711$	14.4721	0.542748
$712$	−20.0000	−0.749532
$713$	9.00000	0.337053
$714$	0	0
$715$	−3.41641	−0.127766
$716$	30.2705	1.13126
$717$	15.7639	0.588715
$718$	2.94427	0.109879
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	12.0000	0.447214
$721$	15.7082	0.585004
$722$	20.6180	0.767324
$723$	21.4164	0.796485
$724$	8.18034	0.304020
$725$	−8.05573	−0.299182
$726$	−5.56231	−0.206437
$727$	3.23607	0.120019	0.0600096	−	0.998198i	$-0.480887\pi$
$727$	3.23607	0.120019	0.0600096	+	0.998198i	$0.480887\pi$
$728$	0.527864	0.0195639
$729$	13.0000	0.481481
$730$	13.4164	0.496564
$731$	0	0
$732$	21.7082	0.802358
$733$	12.9443	0.478108	0.239054	−	0.971006i	$-0.423163\pi$
$733$	12.9443	0.478108	0.239054	+	0.971006i	$0.423163\pi$
$734$	−2.83282	−0.104561
$735$	3.23607	0.119364
$736$	−5.61803	−0.207083
$737$	−54.4721	−2.00651
$738$	2.76393	0.101742
$739$	8.70820	0.320336	0.160168	−	0.987090i	$-0.448796\pi$
$739$	8.70820	0.320336	0.160168	+	0.987090i	$0.448796\pi$
$740$	−29.8885	−1.09872
$741$	1.70820	0.0627524
$742$	−6.94427	−0.254932
$743$	−25.5279	−0.936527	−0.468263	−	0.883589i	$-0.655120\pi$
$743$	−25.5279	−0.936527	−0.468263	+	0.883589i	$0.655120\pi$
$744$	−20.1246	−0.737804
$745$	−49.3050	−1.80639
$746$	16.0689	0.588324
$747$	−12.9443	−0.473606
$748$	0	0
$749$	−11.2361	−0.410557
$750$	0.944272	0.0344799
$751$	−8.58359	−0.313220	−0.156610	−	0.987661i	$-0.550057\pi$
$751$	−8.58359	−0.313220	−0.156610	+	0.987661i	$0.550057\pi$
$752$	−6.43769	−0.234759
$753$	−8.29180	−0.302170
$754$	−0.214782	−0.00782189
$755$	49.1246	1.78783
$756$	8.09017	0.294237
$757$	−28.8328	−1.04795	−0.523973	−	0.851735i	$-0.675551\pi$
$757$	−28.8328	−1.04795	−0.523973	+	0.851735i	$0.675551\pi$
$758$	5.12461	0.186134
$759$	4.47214	0.162328
$760$	−52.3607	−1.89932
$761$	−31.6525	−1.14740	−0.573701	−	0.819065i	$-0.694493\pi$
$761$	−31.6525	−1.14740	−0.573701	+	0.819065i	$0.694493\pi$
$762$	1.67376	0.0606340
$763$	15.4164	0.558111
$764$	39.1246	1.41548
$765$	0	0
$766$	−9.70820	−0.350772
$767$	−0.360680	−0.0130234
$768$	6.56231	0.236797
$769$	−43.2361	−1.55913	−0.779566	−	0.626320i	$-0.784560\pi$
$769$	−43.2361	−1.55913	−0.779566	+	0.626320i	$0.784560\pi$
$770$	8.94427	0.322329
$771$	−4.23607	−0.152558
$772$	−7.14590	−0.257186
$773$	40.6525	1.46217	0.731084	−	0.682288i	$-0.239015\pi$
$773$	40.6525	1.46217	0.731084	+	0.682288i	$0.239015\pi$
$774$	−3.05573	−0.109836
$775$	−49.2492	−1.76908
$776$	−8.29180	−0.297658
$777$	−5.70820	−0.204781
$778$	−4.58359	−0.164330
$779$	16.1803	0.579721
$780$	−1.23607	−0.0442583
$781$	−45.7771	−1.63803
$782$	0	0
$783$	−7.36068	−0.263049
$784$	1.85410	0.0662179
$785$	−49.8885	−1.78060
$786$	−2.43769	−0.0869497
$787$	−0.360680	−0.0128568	−0.00642842	−	0.999979i	$-0.502046\pi$
$787$	−0.360680	−0.0128568	−0.00642842	+	0.999979i	$0.502046\pi$
$788$	34.7426	1.23766
$789$	26.9443	0.959241
$790$	14.4721	0.514895
$791$	−2.47214	−0.0878990
$792$	20.0000	0.710669
$793$	3.16718	0.112470
$794$	−15.8541	−0.562641
$795$	36.3607	1.28958
$796$	−25.7082	−0.911203
$797$	−36.7639	−1.30225	−0.651123	−	0.758973i	$-0.725702\pi$
$797$	−36.7639	−1.30225	−0.651123	+	0.758973i	$0.725702\pi$
$798$	−4.47214	−0.158312
$799$	0	0
$800$	30.7426	1.08692
$801$	−17.8885	−0.632061
$802$	−12.2918	−0.434038
$803$	−30.0000	−1.05868
$804$	−19.7082	−0.695055
$805$	−3.23607	−0.114056
$806$	−1.31308	−0.0462514
$807$	9.18034	0.323163
$808$	30.0000	1.05540
$809$	−32.8328	−1.15434	−0.577170	−	0.816624i	$-0.695843\pi$
$809$	−32.8328	−1.15434	−0.577170	+	0.816624i	$0.695843\pi$
$810$	−2.00000	−0.0702728
$811$	−45.3607	−1.59283	−0.796414	−	0.604751i	$-0.793273\pi$
$811$	−45.3607	−1.59283	−0.796414	+	0.604751i	$0.793273\pi$
$812$	−2.38197	−0.0835906
$813$	16.9443	0.594262
$814$	−15.7771	−0.552987
$815$	−62.0689	−2.17418
$816$	0	0
$817$	−17.8885	−0.625841
$818$	2.54915	0.0891289
$819$	0.472136	0.0164978
$820$	−11.7082	−0.408868
$821$	22.5836	0.788173	0.394086	−	0.919073i	$-0.371061\pi$
$821$	22.5836	0.788173	0.394086	+	0.919073i	$0.371061\pi$
$822$	−9.70820	−0.338612
$823$	25.5410	0.890304	0.445152	−	0.895455i	$-0.353150\pi$
$823$	25.5410	0.890304	0.445152	+	0.895455i	$0.353150\pi$
$824$	35.1246	1.22362
$825$	−24.4721	−0.852010
$826$	0.944272	0.0328554
$827$	−17.3050	−0.601752	−0.300876	−	0.953663i	$-0.597279\pi$
$827$	−17.3050	−0.601752	−0.300876	+	0.953663i	$0.597279\pi$
$828$	−3.23607	−0.112461
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	−12.9443	−0.449302
$831$	−20.4164	−0.708237
$832$	−0.0557281	−0.00193202
$833$	0	0
$834$	−1.56231	−0.0540982
$835$	−70.8328	−2.45127
$836$	52.3607	1.81093
$837$	−45.0000	−1.55543
$838$	16.3607	0.565170
$839$	20.0689	0.692855	0.346427	−	0.938077i	$-0.387395\pi$
$839$	20.0689	0.692855	0.346427	+	0.938077i	$0.387395\pi$
$840$	7.23607	0.249668
$841$	−26.8328	−0.925270
$842$	−5.30495	−0.182821
$843$	−3.70820	−0.127717
$844$	19.4164	0.668340
$845$	41.8885	1.44101
$846$	4.29180	0.147555
$847$	−9.00000	−0.309244
$848$	20.8328	0.715402
$849$	−18.9443	−0.650166
$850$	0	0
$851$	5.70820	0.195675
$852$	−16.5623	−0.567415
$853$	44.8328	1.53505	0.767523	−	0.641021i	$-0.221489\pi$
$853$	44.8328	1.53505	0.767523	+	0.641021i	$0.221489\pi$
$854$	−8.29180	−0.283739
$855$	−46.8328	−1.60165
$856$	−25.1246	−0.858742
$857$	9.54102	0.325915	0.162958	−	0.986633i	$-0.447897\pi$
$857$	9.54102	0.325915	0.162958	+	0.986633i	$0.447897\pi$
$858$	−0.652476	−0.0222752
$859$	11.5836	0.395227	0.197614	−	0.980280i	$-0.436681\pi$
$859$	11.5836	0.395227	0.197614	+	0.980280i	$0.436681\pi$
$860$	12.9443	0.441396
$861$	−2.23607	−0.0762050
$862$	11.2361	0.382702
$863$	2.23607	0.0761166	0.0380583	−	0.999276i	$-0.487883\pi$
$863$	2.23607	0.0761166	0.0380583	+	0.999276i	$0.487883\pi$
$864$	28.0902	0.955647
$865$	11.4164	0.388170
$866$	8.65248	0.294023
$867$	17.0000	0.577350
$868$	−14.5623	−0.494277
$869$	−32.3607	−1.09776
$870$	−2.94427	−0.0998202
$871$	−2.87539	−0.0974288
$872$	34.4721	1.16737
$873$	−7.41641	−0.251007
$874$	4.47214	0.151272
$875$	1.52786	0.0516512
$876$	−10.8541	−0.366726
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	18.7295	0.632090
$879$	15.7082	0.529825
$880$	−26.8328	−0.904534
$881$	25.5967	0.862376	0.431188	−	0.902262i	$-0.358095\pi$
$881$	25.5967	0.862376	0.431188	+	0.902262i	$0.358095\pi$
$882$	−1.23607	−0.0416206
$883$	9.88854	0.332776	0.166388	−	0.986060i	$-0.446790\pi$
$883$	9.88854	0.332776	0.166388	+	0.986060i	$0.446790\pi$
$884$	0	0
$885$	−4.94427	−0.166200
$886$	−4.43769	−0.149087
$887$	16.8885	0.567062	0.283531	−	0.958963i	$-0.408494\pi$
$887$	16.8885	0.567062	0.283531	+	0.958963i	$0.408494\pi$
$888$	−12.7639	−0.428330
$889$	2.70820	0.0908302
$890$	−17.8885	−0.599625
$891$	4.47214	0.149822
$892$	−5.52786	−0.185087
$893$	25.1246	0.840763
$894$	−9.41641	−0.314932
$895$	60.5410	2.02366
$896$	11.3820	0.380245
$897$	0.236068	0.00788208
$898$	20.2918	0.677146
$899$	13.2492	0.441886
$900$	17.7082	0.590273
$901$	0	0
$902$	−6.18034	−0.205783
$903$	2.47214	0.0822675
$904$	−5.52786	−0.183854
$905$	16.3607	0.543847
$906$	9.38197	0.311695
$907$	12.5410	0.416418	0.208209	−	0.978084i	$-0.433237\pi$
$907$	12.5410	0.416418	0.208209	+	0.978084i	$0.433237\pi$
$908$	9.52786	0.316193
$909$	26.8328	0.889988
$910$	0.472136	0.0156512
$911$	−21.7771	−0.721507	−0.360754	−	0.932661i	$-0.617480\pi$
$911$	−21.7771	−0.721507	−0.360754	+	0.932661i	$0.617480\pi$
$912$	13.4164	0.444262
$913$	28.9443	0.957916
$914$	8.94427	0.295850
$915$	43.4164	1.43530
$916$	−23.8885	−0.789300
$917$	−3.94427	−0.130251
$918$	0	0
$919$	3.70820	0.122322	0.0611612	−	0.998128i	$-0.480520\pi$
$919$	3.70820	0.122322	0.0611612	+	0.998128i	$0.480520\pi$
$920$	−7.23607	−0.238566
$921$	11.4164	0.376183
$922$	−15.9230	−0.524396
$923$	−2.41641	−0.0795370
$924$	−7.23607	−0.238049
$925$	−31.2361	−1.02704
$926$	0	0
$927$	31.4164	1.03185
$928$	−8.27051	−0.271493
$929$	51.6525	1.69466	0.847331	−	0.531065i	$-0.178208\pi$
$929$	51.6525	1.69466	0.847331	+	0.531065i	$0.178208\pi$
$930$	−18.0000	−0.590243
$931$	−7.23607	−0.237153
$932$	18.5623	0.608029
$933$	2.88854	0.0945667
$934$	−7.12461	−0.233124
$935$	0	0
$936$	1.05573	0.0345076
$937$	−57.1246	−1.86618	−0.933090	−	0.359643i	$-0.882898\pi$
$937$	−57.1246	−1.86618	−0.933090	+	0.359643i	$0.882898\pi$
$938$	7.52786	0.245793
$939$	−2.76393	−0.0901975
$940$	−18.1803	−0.592977
$941$	53.0132	1.72818	0.864090	−	0.503338i	$-0.167895\pi$
$941$	53.0132	1.72818	0.864090	+	0.503338i	$0.167895\pi$
$942$	−9.52786	−0.310435
$943$	2.23607	0.0728164
$944$	−2.83282	−0.0922003
$945$	16.1803	0.526346
$946$	6.83282	0.222154
$947$	−22.5967	−0.734296	−0.367148	−	0.930163i	$-0.619666\pi$
$947$	−22.5967	−0.734296	−0.367148	+	0.930163i	$0.619666\pi$
$948$	−11.7082	−0.380265
$949$	−1.58359	−0.0514056
$950$	−24.4721	−0.793981
$951$	−31.3050	−1.01513
$952$	0	0
$953$	12.1115	0.392329	0.196164	−	0.980571i	$-0.437151\pi$
$953$	12.1115	0.392329	0.196164	+	0.980571i	$0.437151\pi$
$954$	−13.8885	−0.449658
$955$	78.2492	2.53209
$956$	25.5066	0.824942
$957$	6.58359	0.212817
$958$	−14.2918	−0.461747
$959$	−15.7082	−0.507244
$960$	−0.763932	−0.0246558
$961$	50.0000	1.61290
$962$	−0.832816	−0.0268511
$963$	−22.4721	−0.724154
$964$	34.6525	1.11608
$965$	−14.2918	−0.460069
$966$	−0.618034	−0.0198849
$967$	−31.0689	−0.999108	−0.499554	−	0.866283i	$-0.666503\pi$
$967$	−31.0689	−0.999108	−0.499554	+	0.866283i	$0.666503\pi$
$968$	−20.1246	−0.646830
$969$	0	0
$970$	−7.41641	−0.238127
$971$	33.7082	1.08175	0.540874	−	0.841104i	$-0.318094\pi$
$971$	33.7082	1.08175	0.540874	+	0.841104i	$0.318094\pi$
$972$	25.8885	0.830375
$973$	−2.52786	−0.0810396
$974$	12.4377	0.398529
$975$	−1.29180	−0.0413706
$976$	24.8754	0.796242
$977$	−18.6525	−0.596746	−0.298373	−	0.954449i	$-0.596444\pi$
$977$	−18.6525	−0.596746	−0.298373	+	0.954449i	$0.596444\pi$
$978$	−11.8541	−0.379052
$979$	40.0000	1.27841
$980$	5.23607	0.167260
$981$	30.8328	0.984416
$982$	7.27051	0.232011
$983$	−4.18034	−0.133332	−0.0666661	−	0.997775i	$-0.521236\pi$
$983$	−4.18034	−0.133332	−0.0666661	+	0.997775i	$0.521236\pi$
$984$	−5.00000	−0.159394
$985$	69.4853	2.21399
$986$	0	0
$987$	−3.47214	−0.110519
$988$	2.76393	0.0879324
$989$	−2.47214	−0.0786094
$990$	17.8885	0.568535
$991$	−48.9443	−1.55477	−0.777383	−	0.629028i	$-0.783453\pi$
$991$	−48.9443	−1.55477	−0.777383	+	0.629028i	$0.783453\pi$
$992$	−50.5623	−1.60535
$993$	−0.708204	−0.0224742
$994$	6.32624	0.200656
$995$	−51.4164	−1.63001
$996$	10.4721	0.331822
$997$	−27.3050	−0.864756	−0.432378	−	0.901692i	$-0.642325\pi$
$997$	−27.3050	−0.864756	−0.432378	+	0.901692i	$0.642325\pi$
$998$	−22.6869	−0.718142
$999$	−28.5410	−0.902998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	161.2.a.b.1.2	✓	2
3.2	odd	2		1449.2.a.i.1.1		2
4.3	odd	2		2576.2.a.s.1.1		2
5.4	even	2		4025.2.a.i.1.1		2
7.6	odd	2		1127.2.a.d.1.2		2
23.22	odd	2		3703.2.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.b.1.2	✓	2	1.1	even	1	trivial
1127.2.a.d.1.2		2	7.6	odd	2
1449.2.a.i.1.1		2	3.2	odd	2
2576.2.a.s.1.1		2	4.3	odd	2
3703.2.a.b.1.2		2	23.22	odd	2
4025.2.a.i.1.1		2	5.4	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:	\( N \)	\(=\)	\( 161 = 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	161.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -3.23607 q^{5} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} -1.00000 q^{3} -1.61803 q^{4} -3.23607 q^{5} -0.618034 q^{6} -1.00000 q^{7} -2.23607 q^{8} -2.00000 q^{9} -2.00000 q^{10} +4.47214 q^{11} +1.61803 q^{12} +0.236068 q^{13} -0.618034 q^{14} +3.23607 q^{15} +1.85410 q^{16} -1.23607 q^{18} -7.23607 q^{19} +5.23607 q^{20} +1.00000 q^{21} +2.76393 q^{22} -1.00000 q^{23} +2.23607 q^{24} +5.47214 q^{25} +0.145898 q^{26} +5.00000 q^{27} +1.61803 q^{28} -1.47214 q^{29} +2.00000 q^{30} -9.00000 q^{31} +5.61803 q^{32} -4.47214 q^{33} +3.23607 q^{35} +3.23607 q^{36} -5.70820 q^{37} -4.47214 q^{38} -0.236068 q^{39} +7.23607 q^{40} -2.23607 q^{41} +0.618034 q^{42} +2.47214 q^{43} -7.23607 q^{44} +6.47214 q^{45} -0.618034 q^{46} -3.47214 q^{47} -1.85410 q^{48} +1.00000 q^{49} +3.38197 q^{50} -0.381966 q^{52} +11.2361 q^{53} +3.09017 q^{54} -14.4721 q^{55} +2.23607 q^{56} +7.23607 q^{57} -0.909830 q^{58} -1.52786 q^{59} -5.23607 q^{60} +13.4164 q^{61} -5.56231 q^{62} +2.00000 q^{63} -0.236068 q^{64} -0.763932 q^{65} -2.76393 q^{66} -12.1803 q^{67} +1.00000 q^{69} +2.00000 q^{70} -10.2361 q^{71} +4.47214 q^{72} -6.70820 q^{73} -3.52786 q^{74} -5.47214 q^{75} +11.7082 q^{76} -4.47214 q^{77} -0.145898 q^{78} -7.23607 q^{79} -6.00000 q^{80} +1.00000 q^{81} -1.38197 q^{82} +6.47214 q^{83} -1.61803 q^{84} +1.52786 q^{86} +1.47214 q^{87} -10.0000 q^{88} +8.94427 q^{89} +4.00000 q^{90} -0.236068 q^{91} +1.61803 q^{92} +9.00000 q^{93} -2.14590 q^{94} +23.4164 q^{95} -5.61803 q^{96} +3.70820 q^{97} +0.618034 q^{98} -8.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{5} + q^{6} - 2 q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q - q^2 - 2 * q^3 - q^4 - 2 * q^5 + q^6 - 2 * q^7 - 4 * q^9	\( 2 q - q^{2} - 2 q^{3} - q^{4} - 2 q^{5} + q^{6} - 2 q^{7} - 4 q^{9} - 4 q^{10} + q^{12} - 4 q^{13} + q^{14} + 2 q^{15} - 3 q^{16} + 2 q^{18} - 10 q^{19} + 6 q^{20} + 2 q^{21} + 10 q^{22} - 2 q^{23} + 2 q^{25} + 7 q^{26} + 10 q^{27} + q^{28} + 6 q^{29} + 4 q^{30} - 18 q^{31} + 9 q^{32} + 2 q^{35} + 2 q^{36} + 2 q^{37} + 4 q^{39} + 10 q^{40} - q^{42} - 4 q^{43} - 10 q^{44} + 4 q^{45} + q^{46} + 2 q^{47} + 3 q^{48} + 2 q^{49} + 9 q^{50} - 3 q^{52} + 18 q^{53} - 5 q^{54} - 20 q^{55} + 10 q^{57} - 13 q^{58} - 12 q^{59} - 6 q^{60} + 9 q^{62} + 4 q^{63} + 4 q^{64} - 6 q^{65} - 10 q^{66} - 2 q^{67} + 2 q^{69} + 4 q^{70} - 16 q^{71} - 16 q^{74} - 2 q^{75} + 10 q^{76} - 7 q^{78} - 10 q^{79} - 12 q^{80} + 2 q^{81} - 5 q^{82} + 4 q^{83} - q^{84} + 12 q^{86} - 6 q^{87} - 20 q^{88} + 8 q^{90} + 4 q^{91} + q^{92} + 18 q^{93} - 11 q^{94} + 20 q^{95} - 9 q^{96} - 6 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 - q^4 - 2 * q^5 + q^6 - 2 * q^7 - 4 * q^9 - 4 * q^10 + q^12 - 4 * q^13 + q^14 + 2 * q^15 - 3 * q^16 + 2 * q^18 - 10 * q^19 + 6 * q^20 + 2 * q^21 + 10 * q^22 - 2 * q^23 + 2 * q^25 + 7 * q^26 + 10 * q^27 + q^28 + 6 * q^29 + 4 * q^30 - 18 * q^31 + 9 * q^32 + 2 * q^35 + 2 * q^36 + 2 * q^37 + 4 * q^39 + 10 * q^40 - q^42 - 4 * q^43 - 10 * q^44 + 4 * q^45 + q^46 + 2 * q^47 + 3 * q^48 + 2 * q^49 + 9 * q^50 - 3 * q^52 + 18 * q^53 - 5 * q^54 - 20 * q^55 + 10 * q^57 - 13 * q^58 - 12 * q^59 - 6 * q^60 + 9 * q^62 + 4 * q^63 + 4 * q^64 - 6 * q^65 - 10 * q^66 - 2 * q^67 + 2 * q^69 + 4 * q^70 - 16 * q^71 - 16 * q^74 - 2 * q^75 + 10 * q^76 - 7 * q^78 - 10 * q^79 - 12 * q^80 + 2 * q^81 - 5 * q^82 + 4 * q^83 - q^84 + 12 * q^86 - 6 * q^87 - 20 * q^88 + 8 * q^90 + 4 * q^91 + q^92 + 18 * q^93 - 11 * q^94 + 20 * q^95 - 9 * q^96 - 6 * q^97 - q^98

Introduction

L-functions

Modular forms

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Groups

Database

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