LMFDB - Embedded newform 1155.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1155,2,Mod(1,1155)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1155, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1155.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1155 = 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1155.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:	$9.22272143346$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
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			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1155.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{2} -1.00000 q^{3} -1.46410 q^{4} -1.00000 q^{5} +0.732051 q^{6} +1.00000 q^{7} +2.53590 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.732051 q^{2} -1.00000 q^{3} -1.46410 q^{4} -1.00000 q^{5} +0.732051 q^{6} +1.00000 q^{7} +2.53590 q^{8} +1.00000 q^{9} +0.732051 q^{10} -1.00000 q^{11} +1.46410 q^{12} -4.19615 q^{13} -0.732051 q^{14} +1.00000 q^{15} +1.07180 q^{16} -6.46410 q^{17} -0.732051 q^{18} -1.53590 q^{19} +1.46410 q^{20} -1.00000 q^{21} +0.732051 q^{22} +8.46410 q^{23} -2.53590 q^{24} +1.00000 q^{25} +3.07180 q^{26} -1.00000 q^{27} -1.46410 q^{28} +5.73205 q^{29} -0.732051 q^{30} -5.26795 q^{31} -5.85641 q^{32} +1.00000 q^{33} +4.73205 q^{34} -1.00000 q^{35} -1.46410 q^{36} -4.19615 q^{37} +1.12436 q^{38} +4.19615 q^{39} -2.53590 q^{40} -7.66025 q^{41} +0.732051 q^{42} +1.73205 q^{43} +1.46410 q^{44} -1.00000 q^{45} -6.19615 q^{46} +1.26795 q^{47} -1.07180 q^{48} +1.00000 q^{49} -0.732051 q^{50} +6.46410 q^{51} +6.14359 q^{52} +4.46410 q^{53} +0.732051 q^{54} +1.00000 q^{55} +2.53590 q^{56} +1.53590 q^{57} -4.19615 q^{58} +13.7321 q^{59} -1.46410 q^{60} +11.3923 q^{61} +3.85641 q^{62} +1.00000 q^{63} +2.14359 q^{64} +4.19615 q^{65} -0.732051 q^{66} +4.92820 q^{67} +9.46410 q^{68} -8.46410 q^{69} +0.732051 q^{70} +7.66025 q^{71} +2.53590 q^{72} +8.39230 q^{73} +3.07180 q^{74} -1.00000 q^{75} +2.24871 q^{76} -1.00000 q^{77} -3.07180 q^{78} +3.66025 q^{79} -1.07180 q^{80} +1.00000 q^{81} +5.60770 q^{82} +2.46410 q^{83} +1.46410 q^{84} +6.46410 q^{85} -1.26795 q^{86} -5.73205 q^{87} -2.53590 q^{88} +14.6603 q^{89} +0.732051 q^{90} -4.19615 q^{91} -12.3923 q^{92} +5.26795 q^{93} -0.928203 q^{94} +1.53590 q^{95} +5.85641 q^{96} -2.26795 q^{97} -0.732051 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 12 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 12 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{15} + 16 q^{16} - 6 q^{17} + 2 q^{18} - 10 q^{19} - 4 q^{20} - 2 q^{21} - 2 q^{22} + 10 q^{23} - 12 q^{24} + 2 q^{25} + 20 q^{26} - 2 q^{27} + 4 q^{28} + 8 q^{29} + 2 q^{30} - 14 q^{31} + 16 q^{32} + 2 q^{33} + 6 q^{34} - 2 q^{35} + 4 q^{36} + 2 q^{37} - 22 q^{38} - 2 q^{39} - 12 q^{40} + 2 q^{41} - 2 q^{42} - 4 q^{44} - 2 q^{45} - 2 q^{46} + 6 q^{47} - 16 q^{48} + 2 q^{49} + 2 q^{50} + 6 q^{51} + 40 q^{52} + 2 q^{53} - 2 q^{54} + 2 q^{55} + 12 q^{56} + 10 q^{57} + 2 q^{58} + 24 q^{59} + 4 q^{60} + 2 q^{61} - 20 q^{62} + 2 q^{63} + 32 q^{64} - 2 q^{65} + 2 q^{66} - 4 q^{67} + 12 q^{68} - 10 q^{69} - 2 q^{70} - 2 q^{71} + 12 q^{72} - 4 q^{73} + 20 q^{74} - 2 q^{75} - 44 q^{76} - 2 q^{77} - 20 q^{78} - 10 q^{79} - 16 q^{80} + 2 q^{81} + 32 q^{82} - 2 q^{83} - 4 q^{84} + 6 q^{85} - 6 q^{86} - 8 q^{87} - 12 q^{88} + 12 q^{89} - 2 q^{90} + 2 q^{91} - 4 q^{92} + 14 q^{93} + 12 q^{94} + 10 q^{95} - 16 q^{96} - 8 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 12 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 4 * q^12 + 2 * q^13 + 2 * q^14 + 2 * q^15 + 16 * q^16 - 6 * q^17 + 2 * q^18 - 10 * q^19 - 4 * q^20 - 2 * q^21 - 2 * q^22 + 10 * q^23 - 12 * q^24 + 2 * q^25 + 20 * q^26 - 2 * q^27 + 4 * q^28 + 8 * q^29 + 2 * q^30 - 14 * q^31 + 16 * q^32 + 2 * q^33 + 6 * q^34 - 2 * q^35 + 4 * q^36 + 2 * q^37 - 22 * q^38 - 2 * q^39 - 12 * q^40 + 2 * q^41 - 2 * q^42 - 4 * q^44 - 2 * q^45 - 2 * q^46 + 6 * q^47 - 16 * q^48 + 2 * q^49 + 2 * q^50 + 6 * q^51 + 40 * q^52 + 2 * q^53 - 2 * q^54 + 2 * q^55 + 12 * q^56 + 10 * q^57 + 2 * q^58 + 24 * q^59 + 4 * q^60 + 2 * q^61 - 20 * q^62 + 2 * q^63 + 32 * q^64 - 2 * q^65 + 2 * q^66 - 4 * q^67 + 12 * q^68 - 10 * q^69 - 2 * q^70 - 2 * q^71 + 12 * q^72 - 4 * q^73 + 20 * q^74 - 2 * q^75 - 44 * q^76 - 2 * q^77 - 20 * q^78 - 10 * q^79 - 16 * q^80 + 2 * q^81 + 32 * q^82 - 2 * q^83 - 4 * q^84 + 6 * q^85 - 6 * q^86 - 8 * q^87 - 12 * q^88 + 12 * q^89 - 2 * q^90 + 2 * q^91 - 4 * q^92 + 14 * q^93 + 12 * q^94 + 10 * q^95 - 16 * q^96 - 8 * q^97 + 2 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.732051	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$2$	−0.732051	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.46410	−0.732051
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0.732051	0.298858
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.53590	0.896575
$9$	1.00000	0.333333
$10$	0.732051	0.231495
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.46410	0.422650
$13$	−4.19615	−1.16380	−0.581902	−	0.813259i	$-0.697691\pi$
$13$	−4.19615	−1.16380	−0.581902	+	0.813259i	$0.697691\pi$
$14$	−0.732051	−0.195649
$15$	1.00000	0.258199
$16$	1.07180	0.267949
$17$	−6.46410	−1.56777	−0.783887	−	0.620903i	$-0.786766\pi$
$17$	−6.46410	−1.56777	−0.783887	+	0.620903i	$0.786766\pi$
$18$	−0.732051	−0.172546
$19$	−1.53590	−0.352359	−0.176180	−	0.984358i	$-0.556374\pi$
$19$	−1.53590	−0.352359	−0.176180	+	0.984358i	$0.556374\pi$
$20$	1.46410	0.327383
$21$	−1.00000	−0.218218
$22$	0.732051	0.156074
$23$	8.46410	1.76489	0.882444	−	0.470418i	$-0.155897\pi$
$23$	8.46410	1.76489	0.882444	+	0.470418i	$0.155897\pi$
$24$	−2.53590	−0.517638
$25$	1.00000	0.200000
$26$	3.07180	0.602429
$27$	−1.00000	−0.192450
$28$	−1.46410	−0.276689
$29$	5.73205	1.06442	0.532208	−	0.846614i	$-0.321363\pi$
$29$	5.73205	1.06442	0.532208	+	0.846614i	$0.321363\pi$
$30$	−0.732051	−0.133654
$31$	−5.26795	−0.946152	−0.473076	−	0.881022i	$-0.656856\pi$
$31$	−5.26795	−0.946152	−0.473076	+	0.881022i	$0.656856\pi$
$32$	−5.85641	−1.03528
$33$	1.00000	0.174078
$34$	4.73205	0.811540
$35$	−1.00000	−0.169031
$36$	−1.46410	−0.244017
$37$	−4.19615	−0.689843	−0.344922	−	0.938631i	$-0.612095\pi$
$37$	−4.19615	−0.689843	−0.344922	+	0.938631i	$0.612095\pi$
$38$	1.12436	0.182395
$39$	4.19615	0.671922
$40$	−2.53590	−0.400961
$41$	−7.66025	−1.19633	−0.598165	−	0.801373i	$-0.704103\pi$
$41$	−7.66025	−1.19633	−0.598165	+	0.801373i	$0.704103\pi$
$42$	0.732051	0.112958
$43$	1.73205	0.264135	0.132068	−	0.991241i	$-0.457838\pi$
$43$	1.73205	0.264135	0.132068	+	0.991241i	$0.457838\pi$
$44$	1.46410	0.220722
$45$	−1.00000	−0.149071
$46$	−6.19615	−0.913573
$47$	1.26795	0.184949	0.0924747	−	0.995715i	$-0.470522\pi$
$47$	1.26795	0.184949	0.0924747	+	0.995715i	$0.470522\pi$
$48$	−1.07180	−0.154701
$49$	1.00000	0.142857
$50$	−0.732051	−0.103528
$51$	6.46410	0.905155
$52$	6.14359	0.851963
$53$	4.46410	0.613192	0.306596	−	0.951840i	$-0.400810\pi$
$53$	4.46410	0.613192	0.306596	+	0.951840i	$0.400810\pi$
$54$	0.732051	0.0996195
$55$	1.00000	0.134840
$56$	2.53590	0.338874
$57$	1.53590	0.203435
$58$	−4.19615	−0.550982
$59$	13.7321	1.78776	0.893880	−	0.448306i	$-0.147972\pi$
$59$	13.7321	1.78776	0.893880	+	0.448306i	$0.147972\pi$
$60$	−1.46410	−0.189015
$61$	11.3923	1.45864	0.729318	−	0.684175i	$-0.239838\pi$
$61$	11.3923	1.45864	0.729318	+	0.684175i	$0.239838\pi$
$62$	3.85641	0.489764
$63$	1.00000	0.125988
$64$	2.14359	0.267949
$65$	4.19615	0.520469
$66$	−0.732051	−0.0901092
$67$	4.92820	0.602076	0.301038	−	0.953612i	$-0.402667\pi$
$67$	4.92820	0.602076	0.301038	+	0.953612i	$0.402667\pi$
$68$	9.46410	1.14769
$69$	−8.46410	−1.01896
$70$	0.732051	0.0874968
$71$	7.66025	0.909105	0.454552	−	0.890720i	$-0.349799\pi$
$71$	7.66025	0.909105	0.454552	+	0.890720i	$0.349799\pi$
$72$	2.53590	0.298858
$73$	8.39230	0.982245	0.491122	−	0.871091i	$-0.336587\pi$
$73$	8.39230	0.982245	0.491122	+	0.871091i	$0.336587\pi$
$74$	3.07180	0.357089
$75$	−1.00000	−0.115470
$76$	2.24871	0.257945
$77$	−1.00000	−0.113961
$78$	−3.07180	−0.347812
$79$	3.66025	0.411811	0.205905	−	0.978572i	$-0.433986\pi$
$79$	3.66025	0.411811	0.205905	+	0.978572i	$0.433986\pi$
$80$	−1.07180	−0.119831
$81$	1.00000	0.111111
$82$	5.60770	0.619266
$83$	2.46410	0.270470	0.135235	−	0.990814i	$-0.456821\pi$
$83$	2.46410	0.270470	0.135235	+	0.990814i	$0.456821\pi$
$84$	1.46410	0.159747
$85$	6.46410	0.701130
$86$	−1.26795	−0.136726
$87$	−5.73205	−0.614540
$88$	−2.53590	−0.270328
$89$	14.6603	1.55398	0.776992	−	0.629511i	$-0.216745\pi$
$89$	14.6603	1.55398	0.776992	+	0.629511i	$0.216745\pi$
$90$	0.732051	0.0771649
$91$	−4.19615	−0.439876
$92$	−12.3923	−1.29199
$93$	5.26795	0.546261
$94$	−0.928203	−0.0957369
$95$	1.53590	0.157580
$96$	5.85641	0.597717
$97$	−2.26795	−0.230275	−0.115138	−	0.993350i	$-0.536731\pi$
$97$	−2.26795	−0.230275	−0.115138	+	0.993350i	$0.536731\pi$
$98$	−0.732051	−0.0739483
$99$	−1.00000	−0.100504
$100$	−1.46410	−0.146410
$101$	16.9282	1.68442	0.842210	−	0.539150i	$-0.181255\pi$
$101$	16.9282	1.68442	0.842210	+	0.539150i	$0.181255\pi$
$102$	−4.73205	−0.468543
$103$	−14.2679	−1.40586	−0.702931	−	0.711258i	$-0.748126\pi$
$103$	−14.2679	−1.40586	−0.702931	+	0.711258i	$0.748126\pi$
$104$	−10.6410	−1.04344
$105$	1.00000	0.0975900
$106$	−3.26795	−0.317411
$107$	5.26795	0.509272	0.254636	−	0.967037i	$-0.418044\pi$
$107$	5.26795	0.509272	0.254636	+	0.967037i	$0.418044\pi$
$108$	1.46410	0.140883
$109$	−5.26795	−0.504578	−0.252289	−	0.967652i	$-0.581183\pi$
$109$	−5.26795	−0.504578	−0.252289	+	0.967652i	$0.581183\pi$
$110$	−0.732051	−0.0697983
$111$	4.19615	0.398281
$112$	1.07180	0.101275
$113$	10.8564	1.02128	0.510642	−	0.859793i	$-0.329408\pi$
$113$	10.8564	1.02128	0.510642	+	0.859793i	$0.329408\pi$
$114$	−1.12436	−0.105306
$115$	−8.46410	−0.789282
$116$	−8.39230	−0.779206
$117$	−4.19615	−0.387934
$118$	−10.0526	−0.925413
$119$	−6.46410	−0.592563
$120$	2.53590	0.231495
$121$	1.00000	0.0909091
$122$	−8.33975	−0.755045
$123$	7.66025	0.690702
$124$	7.71281	0.692631
$125$	−1.00000	−0.0894427
$126$	−0.732051	−0.0652163
$127$	−22.1244	−1.96322	−0.981610	−	0.190900i	$-0.938859\pi$
$127$	−22.1244	−1.96322	−0.981610	+	0.190900i	$0.938859\pi$
$128$	10.1436	0.896575
$129$	−1.73205	−0.152499
$130$	−3.07180	−0.269414
$131$	16.1962	1.41506	0.707532	−	0.706681i	$-0.249808\pi$
$131$	16.1962	1.41506	0.707532	+	0.706681i	$0.249808\pi$
$132$	−1.46410	−0.127434
$133$	−1.53590	−0.133179
$134$	−3.60770	−0.311657
$135$	1.00000	0.0860663
$136$	−16.3923	−1.40563
$137$	−6.53590	−0.558399	−0.279200	−	0.960233i	$-0.590069\pi$
$137$	−6.53590	−0.558399	−0.279200	+	0.960233i	$0.590069\pi$
$138$	6.19615	0.527452
$139$	11.4641	0.972372	0.486186	−	0.873855i	$-0.338388\pi$
$139$	11.4641	0.972372	0.486186	+	0.873855i	$0.338388\pi$
$140$	1.46410	0.123739
$141$	−1.26795	−0.106781
$142$	−5.60770	−0.470587
$143$	4.19615	0.350900
$144$	1.07180	0.0893164
$145$	−5.73205	−0.476021
$146$	−6.14359	−0.508447
$147$	−1.00000	−0.0824786
$148$	6.14359	0.505000
$149$	15.4641	1.26687	0.633434	−	0.773796i	$-0.281645\pi$
$149$	15.4641	1.26687	0.633434	+	0.773796i	$0.281645\pi$
$150$	0.732051	0.0597717
$151$	−11.8564	−0.964861	−0.482430	−	0.875934i	$-0.660246\pi$
$151$	−11.8564	−0.964861	−0.482430	+	0.875934i	$0.660246\pi$
$152$	−3.89488	−0.315917
$153$	−6.46410	−0.522592
$154$	0.732051	0.0589903
$155$	5.26795	0.423132
$156$	−6.14359	−0.491881
$157$	−14.1244	−1.12725	−0.563623	−	0.826032i	$-0.690593\pi$
$157$	−14.1244	−1.12725	−0.563623	+	0.826032i	$0.690593\pi$
$158$	−2.67949	−0.213169
$159$	−4.46410	−0.354026
$160$	5.85641	0.462990
$161$	8.46410	0.667065
$162$	−0.732051	−0.0575153
$163$	19.1244	1.49794	0.748968	−	0.662607i	$-0.230550\pi$
$163$	19.1244	1.49794	0.748968	+	0.662607i	$0.230550\pi$
$164$	11.2154	0.875775
$165$	−1.00000	−0.0778499
$166$	−1.80385	−0.140006
$167$	−10.3923	−0.804181	−0.402090	−	0.915600i	$-0.631716\pi$
$167$	−10.3923	−0.804181	−0.402090	+	0.915600i	$0.631716\pi$
$168$	−2.53590	−0.195649
$169$	4.60770	0.354438
$170$	−4.73205	−0.362932
$171$	−1.53590	−0.117453
$172$	−2.53590	−0.193360
$173$	−25.4641	−1.93600	−0.968000	−	0.250951i	$-0.919257\pi$
$173$	−25.4641	−1.93600	−0.968000	+	0.250951i	$0.919257\pi$
$174$	4.19615	0.318109
$175$	1.00000	0.0755929
$176$	−1.07180	−0.0807897
$177$	−13.7321	−1.03216
$178$	−10.7321	−0.804401
$179$	−20.0526	−1.49880	−0.749399	−	0.662118i	$-0.769658\pi$
$179$	−20.0526	−1.49880	−0.749399	+	0.662118i	$0.769658\pi$
$180$	1.46410	0.109128
$181$	7.07180	0.525643	0.262821	−	0.964845i	$-0.415347\pi$
$181$	7.07180	0.525643	0.262821	+	0.964845i	$0.415347\pi$
$182$	3.07180	0.227697
$183$	−11.3923	−0.842143
$184$	21.4641	1.58235
$185$	4.19615	0.308507
$186$	−3.85641	−0.282765
$187$	6.46410	0.472702
$188$	−1.85641	−0.135392
$189$	−1.00000	−0.0727393
$190$	−1.12436	−0.0815693
$191$	−1.07180	−0.0775525	−0.0387762	−	0.999248i	$-0.512346\pi$
$191$	−1.07180	−0.0775525	−0.0387762	+	0.999248i	$0.512346\pi$
$192$	−2.14359	−0.154701
$193$	−7.46410	−0.537278	−0.268639	−	0.963241i	$-0.586574\pi$
$193$	−7.46410	−0.537278	−0.268639	+	0.963241i	$0.586574\pi$
$194$	1.66025	0.119199
$195$	−4.19615	−0.300493
$196$	−1.46410	−0.104579
$197$	5.46410	0.389301	0.194651	−	0.980873i	$-0.437643\pi$
$197$	5.46410	0.389301	0.194651	+	0.980873i	$0.437643\pi$
$198$	0.732051	0.0520246
$199$	−14.9282	−1.05823	−0.529116	−	0.848549i	$-0.677476\pi$
$199$	−14.9282	−1.05823	−0.529116	+	0.848549i	$0.677476\pi$
$200$	2.53590	0.179315
$201$	−4.92820	−0.347609
$202$	−12.3923	−0.871920
$203$	5.73205	0.402311
$204$	−9.46410	−0.662620
$205$	7.66025	0.535015
$206$	10.4449	0.727728
$207$	8.46410	0.588296
$208$	−4.49742	−0.311840
$209$	1.53590	0.106240
$210$	−0.732051	−0.0505163
$211$	4.53590	0.312264	0.156132	−	0.987736i	$-0.450097\pi$
$211$	4.53590	0.312264	0.156132	+	0.987736i	$0.450097\pi$
$212$	−6.53590	−0.448887
$213$	−7.66025	−0.524872
$214$	−3.85641	−0.263619
$215$	−1.73205	−0.118125
$216$	−2.53590	−0.172546
$217$	−5.26795	−0.357612
$218$	3.85641	0.261189
$219$	−8.39230	−0.567099
$220$	−1.46410	−0.0987097
$221$	27.1244	1.82458
$222$	−3.07180	−0.206166
$223$	0.803848	0.0538296	0.0269148	−	0.999638i	$-0.491432\pi$
$223$	0.803848	0.0538296	0.0269148	+	0.999638i	$0.491432\pi$
$224$	−5.85641	−0.391298
$225$	1.00000	0.0666667
$226$	−7.94744	−0.528656
$227$	−5.39230	−0.357900	−0.178950	−	0.983858i	$-0.557270\pi$
$227$	−5.39230	−0.357900	−0.178950	+	0.983858i	$0.557270\pi$
$228$	−2.24871	−0.148925
$229$	28.1962	1.86325	0.931627	−	0.363416i	$-0.118390\pi$
$229$	28.1962	1.86325	0.931627	+	0.363416i	$0.118390\pi$
$230$	6.19615	0.408562
$231$	1.00000	0.0657952
$232$	14.5359	0.954328
$233$	−10.1962	−0.667972	−0.333986	−	0.942578i	$-0.608394\pi$
$233$	−10.1962	−0.667972	−0.333986	+	0.942578i	$0.608394\pi$
$234$	3.07180	0.200810
$235$	−1.26795	−0.0827119
$236$	−20.1051	−1.30873
$237$	−3.66025	−0.237759
$238$	4.73205	0.306733
$239$	18.1244	1.17237	0.586184	−	0.810178i	$-0.300630\pi$
$239$	18.1244	1.17237	0.586184	+	0.810178i	$0.300630\pi$
$240$	1.07180	0.0691842
$241$	26.9282	1.73460	0.867299	−	0.497788i	$-0.165854\pi$
$241$	26.9282	1.73460	0.867299	+	0.497788i	$0.165854\pi$
$242$	−0.732051	−0.0470580
$243$	−1.00000	−0.0641500
$244$	−16.6795	−1.06780
$245$	−1.00000	−0.0638877
$246$	−5.60770	−0.357534
$247$	6.44486	0.410077
$248$	−13.3590	−0.848296
$249$	−2.46410	−0.156156
$250$	0.732051	0.0462990
$251$	−6.92820	−0.437304	−0.218652	−	0.975803i	$-0.570166\pi$
$251$	−6.92820	−0.437304	−0.218652	+	0.975803i	$0.570166\pi$
$252$	−1.46410	−0.0922297
$253$	−8.46410	−0.532134
$254$	16.1962	1.01624
$255$	−6.46410	−0.404798
$256$	−11.7128	−0.732051
$257$	13.8038	0.861060	0.430530	−	0.902576i	$-0.358327\pi$
$257$	13.8038	0.861060	0.430530	+	0.902576i	$0.358327\pi$
$258$	1.26795	0.0789391
$259$	−4.19615	−0.260736
$260$	−6.14359	−0.381009
$261$	5.73205	0.354805
$262$	−11.8564	−0.732491
$263$	11.3205	0.698052	0.349026	−	0.937113i	$-0.386512\pi$
$263$	11.3205	0.698052	0.349026	+	0.937113i	$0.386512\pi$
$264$	2.53590	0.156074
$265$	−4.46410	−0.274228
$266$	1.12436	0.0689387
$267$	−14.6603	−0.897193
$268$	−7.21539	−0.440750
$269$	7.58846	0.462676	0.231338	−	0.972873i	$-0.425690\pi$
$269$	7.58846	0.462676	0.231338	+	0.972873i	$0.425690\pi$
$270$	−0.732051	−0.0445512
$271$	−17.9282	−1.08906	−0.544530	−	0.838741i	$-0.683292\pi$
$271$	−17.9282	−1.08906	−0.544530	+	0.838741i	$0.683292\pi$
$272$	−6.92820	−0.420084
$273$	4.19615	0.253963
$274$	4.78461	0.289049
$275$	−1.00000	−0.0603023
$276$	12.3923	0.745929
$277$	−9.07180	−0.545071	−0.272536	−	0.962146i	$-0.587862\pi$
$277$	−9.07180	−0.545071	−0.272536	+	0.962146i	$0.587862\pi$
$278$	−8.39230	−0.503337
$279$	−5.26795	−0.315384
$280$	−2.53590	−0.151549
$281$	17.0718	1.01842	0.509209	−	0.860643i	$-0.329938\pi$
$281$	17.0718	1.01842	0.509209	+	0.860643i	$0.329938\pi$
$282$	0.928203	0.0552737
$283$	−0.196152	−0.0116601	−0.00583003	−	0.999983i	$-0.501856\pi$
$283$	−0.196152	−0.0116601	−0.00583003	+	0.999983i	$0.501856\pi$
$284$	−11.2154	−0.665511
$285$	−1.53590	−0.0909788
$286$	−3.07180	−0.181639
$287$	−7.66025	−0.452170
$288$	−5.85641	−0.345092
$289$	24.7846	1.45792
$290$	4.19615	0.246407
$291$	2.26795	0.132950
$292$	−12.2872	−0.719053
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0.732051	0.0426941
$295$	−13.7321	−0.799511
$296$	−10.6410	−0.618497
$297$	1.00000	0.0580259
$298$	−11.3205	−0.655779
$299$	−35.5167	−2.05398
$300$	1.46410	0.0845299
$301$	1.73205	0.0998337
$302$	8.67949	0.499449
$303$	−16.9282	−0.972500
$304$	−1.64617	−0.0944144
$305$	−11.3923	−0.652321
$306$	4.73205	0.270513
$307$	−10.3923	−0.593120	−0.296560	−	0.955014i	$-0.595840\pi$
$307$	−10.3923	−0.593120	−0.296560	+	0.955014i	$0.595840\pi$
$308$	1.46410	0.0834249
$309$	14.2679	0.811675
$310$	−3.85641	−0.219029
$311$	7.32051	0.415108	0.207554	−	0.978224i	$-0.433450\pi$
$311$	7.32051	0.415108	0.207554	+	0.978224i	$0.433450\pi$
$312$	10.6410	0.602429
$313$	1.73205	0.0979013	0.0489506	−	0.998801i	$-0.484412\pi$
$313$	1.73205	0.0979013	0.0489506	+	0.998801i	$0.484412\pi$
$314$	10.3397	0.583506
$315$	−1.00000	−0.0563436
$316$	−5.35898	−0.301466
$317$	18.9282	1.06311	0.531557	−	0.847023i	$-0.321607\pi$
$317$	18.9282	1.06311	0.531557	+	0.847023i	$0.321607\pi$
$318$	3.26795	0.183257
$319$	−5.73205	−0.320933
$320$	−2.14359	−0.119831
$321$	−5.26795	−0.294028
$322$	−6.19615	−0.345298
$323$	9.92820	0.552420
$324$	−1.46410	−0.0813390
$325$	−4.19615	−0.232761
$326$	−14.0000	−0.775388
$327$	5.26795	0.291318
$328$	−19.4256	−1.07260
$329$	1.26795	0.0699043
$330$	0.732051	0.0402981
$331$	8.32051	0.457336	0.228668	−	0.973504i	$-0.426563\pi$
$331$	8.32051	0.457336	0.228668	+	0.973504i	$0.426563\pi$
$332$	−3.60770	−0.197998
$333$	−4.19615	−0.229948
$334$	7.60770	0.416275
$335$	−4.92820	−0.269257
$336$	−1.07180	−0.0584713
$337$	−10.6603	−0.580701	−0.290351	−	0.956920i	$-0.593772\pi$
$337$	−10.6603	−0.580701	−0.290351	+	0.956920i	$0.593772\pi$
$338$	−3.37307	−0.183471
$339$	−10.8564	−0.589639
$340$	−9.46410	−0.513263
$341$	5.26795	0.285275
$342$	1.12436	0.0607982
$343$	1.00000	0.0539949
$344$	4.39230	0.236817
$345$	8.46410	0.455692
$346$	18.6410	1.00215
$347$	−1.46410	−0.0785971	−0.0392985	−	0.999228i	$-0.512512\pi$
$347$	−1.46410	−0.0785971	−0.0392985	+	0.999228i	$0.512512\pi$
$348$	8.39230	0.449875
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	−0.732051	−0.0391298
$351$	4.19615	0.223974
$352$	5.85641	0.312148
$353$	15.4641	0.823071	0.411536	−	0.911394i	$-0.364993\pi$
$353$	15.4641	0.823071	0.411536	+	0.911394i	$0.364993\pi$
$354$	10.0526	0.534287
$355$	−7.66025	−0.406564
$356$	−21.4641	−1.13760
$357$	6.46410	0.342117
$358$	14.6795	0.775835
$359$	−34.5167	−1.82172	−0.910860	−	0.412716i	$-0.864580\pi$
$359$	−34.5167	−1.82172	−0.910860	+	0.412716i	$0.864580\pi$
$360$	−2.53590	−0.133654
$361$	−16.6410	−0.875843
$362$	−5.17691	−0.272093
$363$	−1.00000	−0.0524864
$364$	6.14359	0.322012
$365$	−8.39230	−0.439273
$366$	8.33975	0.435926
$367$	16.8038	0.877154	0.438577	−	0.898694i	$-0.355483\pi$
$367$	16.8038	0.877154	0.438577	+	0.898694i	$0.355483\pi$
$368$	9.07180	0.472900
$369$	−7.66025	−0.398777
$370$	−3.07180	−0.159695
$371$	4.46410	0.231765
$372$	−7.71281	−0.399891
$373$	29.1962	1.51172	0.755860	−	0.654734i	$-0.227219\pi$
$373$	29.1962	1.51172	0.755860	+	0.654734i	$0.227219\pi$
$374$	−4.73205	−0.244689
$375$	1.00000	0.0516398
$376$	3.21539	0.165821
$377$	−24.0526	−1.23877
$378$	0.732051	0.0376526
$379$	20.8564	1.07132	0.535661	−	0.844433i	$-0.320063\pi$
$379$	20.8564	1.07132	0.535661	+	0.844433i	$0.320063\pi$
$380$	−2.24871	−0.115356
$381$	22.1244	1.13347
$382$	0.784610	0.0401441
$383$	32.7321	1.67253	0.836265	−	0.548326i	$-0.184735\pi$
$383$	32.7321	1.67253	0.836265	+	0.548326i	$0.184735\pi$
$384$	−10.1436	−0.517638
$385$	1.00000	0.0509647
$386$	5.46410	0.278115
$387$	1.73205	0.0880451
$388$	3.32051	0.168573
$389$	−15.5167	−0.786726	−0.393363	−	0.919383i	$-0.628688\pi$
$389$	−15.5167	−0.786726	−0.393363	+	0.919383i	$0.628688\pi$
$390$	3.07180	0.155546
$391$	−54.7128	−2.76695
$392$	2.53590	0.128082
$393$	−16.1962	−0.816988
$394$	−4.00000	−0.201517
$395$	−3.66025	−0.184167
$396$	1.46410	0.0735739
$397$	37.8564	1.89996	0.949979	−	0.312313i	$-0.101104\pi$
$397$	37.8564	1.89996	0.949979	+	0.312313i	$0.101104\pi$
$398$	10.9282	0.547781
$399$	1.53590	0.0768911
$400$	1.07180	0.0535898
$401$	−33.1244	−1.65415	−0.827076	−	0.562091i	$-0.809997\pi$
$401$	−33.1244	−1.65415	−0.827076	+	0.562091i	$0.809997\pi$
$402$	3.60770	0.179935
$403$	22.1051	1.10113
$404$	−24.7846	−1.23308
$405$	−1.00000	−0.0496904
$406$	−4.19615	−0.208252
$407$	4.19615	0.207996
$408$	16.3923	0.811540
$409$	−9.07180	−0.448571	−0.224286	−	0.974523i	$-0.572005\pi$
$409$	−9.07180	−0.448571	−0.224286	+	0.974523i	$0.572005\pi$
$410$	−5.60770	−0.276944
$411$	6.53590	0.322392
$412$	20.8897	1.02916
$413$	13.7321	0.675710
$414$	−6.19615	−0.304524
$415$	−2.46410	−0.120958
$416$	24.5744	1.20486
$417$	−11.4641	−0.561399
$418$	−1.12436	−0.0549940
$419$	29.7321	1.45251	0.726253	−	0.687428i	$-0.241260\pi$
$419$	29.7321	1.45251	0.726253	+	0.687428i	$0.241260\pi$
$420$	−1.46410	−0.0714408
$421$	24.7128	1.20443	0.602214	−	0.798334i	$-0.294285\pi$
$421$	24.7128	1.20443	0.602214	+	0.798334i	$0.294285\pi$
$422$	−3.32051	−0.161640
$423$	1.26795	0.0616498
$424$	11.3205	0.549772
$425$	−6.46410	−0.313555
$426$	5.60770	0.271694
$427$	11.3923	0.551312
$428$	−7.71281	−0.372813
$429$	−4.19615	−0.202592
$430$	1.26795	0.0611459
$431$	−24.3923	−1.17494	−0.587468	−	0.809247i	$-0.699875\pi$
$431$	−24.3923	−1.17494	−0.587468	+	0.809247i	$0.699875\pi$
$432$	−1.07180	−0.0515668
$433$	9.32051	0.447915	0.223958	−	0.974599i	$-0.428102\pi$
$433$	9.32051	0.447915	0.223958	+	0.974599i	$0.428102\pi$
$434$	3.85641	0.185113
$435$	5.73205	0.274831
$436$	7.71281	0.369377
$437$	−13.0000	−0.621874
$438$	6.14359	0.293552
$439$	−26.7128	−1.27493	−0.637467	−	0.770478i	$-0.720018\pi$
$439$	−26.7128	−1.27493	−0.637467	+	0.770478i	$0.720018\pi$
$440$	2.53590	0.120894
$441$	1.00000	0.0476190
$442$	−19.8564	−0.944473
$443$	−23.4641	−1.11481	−0.557407	−	0.830240i	$-0.688204\pi$
$443$	−23.4641	−1.11481	−0.557407	+	0.830240i	$0.688204\pi$
$444$	−6.14359	−0.291562
$445$	−14.6603	−0.694963
$446$	−0.588457	−0.0278643
$447$	−15.4641	−0.731427
$448$	2.14359	0.101275
$449$	16.0526	0.757567	0.378784	−	0.925485i	$-0.376342\pi$
$449$	16.0526	0.757567	0.378784	+	0.925485i	$0.376342\pi$
$450$	−0.732051	−0.0345092
$451$	7.66025	0.360707
$452$	−15.8949	−0.747632
$453$	11.8564	0.557063
$454$	3.94744	0.185263
$455$	4.19615	0.196719
$456$	3.89488	0.182395
$457$	−2.66025	−0.124441	−0.0622207	−	0.998062i	$-0.519818\pi$
$457$	−2.66025	−0.124441	−0.0622207	+	0.998062i	$0.519818\pi$
$458$	−20.6410	−0.964491
$459$	6.46410	0.301718
$460$	12.3923	0.577794
$461$	29.3205	1.36559	0.682796	−	0.730609i	$-0.260764\pi$
$461$	29.3205	1.36559	0.682796	+	0.730609i	$0.260764\pi$
$462$	−0.732051	−0.0340581
$463$	12.5359	0.582593	0.291296	−	0.956633i	$-0.405913\pi$
$463$	12.5359	0.582593	0.291296	+	0.956633i	$0.405913\pi$
$464$	6.14359	0.285209
$465$	−5.26795	−0.244295
$466$	7.46410	0.345768
$467$	−27.3205	−1.26424	−0.632121	−	0.774870i	$-0.717816\pi$
$467$	−27.3205	−1.26424	−0.632121	+	0.774870i	$0.717816\pi$
$468$	6.14359	0.283988
$469$	4.92820	0.227563
$470$	0.928203	0.0428148
$471$	14.1244	0.650816
$472$	34.8231	1.60286
$473$	−1.73205	−0.0796398
$474$	2.67949	0.123073
$475$	−1.53590	−0.0704719
$476$	9.46410	0.433786
$477$	4.46410	0.204397
$478$	−13.2679	−0.606862
$479$	−6.73205	−0.307595	−0.153798	−	0.988102i	$-0.549150\pi$
$479$	−6.73205	−0.307595	−0.153798	+	0.988102i	$0.549150\pi$
$480$	−5.85641	−0.267307
$481$	17.6077	0.802842
$482$	−19.7128	−0.897894
$483$	−8.46410	−0.385130
$484$	−1.46410	−0.0665501
$485$	2.26795	0.102982
$486$	0.732051	0.0332065
$487$	0.535898	0.0242839	0.0121419	−	0.999926i	$-0.496135\pi$
$487$	0.535898	0.0242839	0.0121419	+	0.999926i	$0.496135\pi$
$488$	28.8897	1.30778
$489$	−19.1244	−0.864833
$490$	0.732051	0.0330707
$491$	3.33975	0.150721	0.0753603	−	0.997156i	$-0.475989\pi$
$491$	3.33975	0.150721	0.0753603	+	0.997156i	$0.475989\pi$
$492$	−11.2154	−0.505629
$493$	−37.0526	−1.66876
$494$	−4.71797	−0.212271
$495$	1.00000	0.0449467
$496$	−5.64617	−0.253521
$497$	7.66025	0.343609
$498$	1.80385	0.0808323
$499$	29.3923	1.31578	0.657890	−	0.753114i	$-0.271449\pi$
$499$	29.3923	1.31578	0.657890	+	0.753114i	$0.271449\pi$
$500$	1.46410	0.0654766
$501$	10.3923	0.464294
$502$	5.07180	0.226365
$503$	−23.0000	−1.02552	−0.512760	−	0.858532i	$-0.671377\pi$
$503$	−23.0000	−1.02552	−0.512760	+	0.858532i	$0.671377\pi$
$504$	2.53590	0.112958
$505$	−16.9282	−0.753295
$506$	6.19615	0.275453
$507$	−4.60770	−0.204635
$508$	32.3923	1.43718
$509$	−17.1962	−0.762206	−0.381103	−	0.924533i	$-0.624456\pi$
$509$	−17.1962	−0.762206	−0.381103	+	0.924533i	$0.624456\pi$
$510$	4.73205	0.209539
$511$	8.39230	0.371254
$512$	−11.7128	−0.517638
$513$	1.53590	0.0678116
$514$	−10.1051	−0.445718
$515$	14.2679	0.628721
$516$	2.53590	0.111637
$517$	−1.26795	−0.0557643
$518$	3.07180	0.134967
$519$	25.4641	1.11775
$520$	10.6410	0.466639
$521$	−9.05256	−0.396600	−0.198300	−	0.980141i	$-0.563542\pi$
$521$	−9.05256	−0.396600	−0.198300	+	0.980141i	$0.563542\pi$
$522$	−4.19615	−0.183661
$523$	−27.3205	−1.19464	−0.597321	−	0.802002i	$-0.703768\pi$
$523$	−27.3205	−1.19464	−0.597321	+	0.802002i	$0.703768\pi$
$524$	−23.7128	−1.03590
$525$	−1.00000	−0.0436436
$526$	−8.28719	−0.361339
$527$	34.0526	1.48335
$528$	1.07180	0.0466440
$529$	48.6410	2.11483
$530$	3.26795	0.141951
$531$	13.7321	0.595920
$532$	2.24871	0.0974940
$533$	32.1436	1.39229
$534$	10.7321	0.464421
$535$	−5.26795	−0.227753
$536$	12.4974	0.539806
$537$	20.0526	0.865332
$538$	−5.55514	−0.239499
$539$	−1.00000	−0.0430730
$540$	−1.46410	−0.0630049
$541$	−34.0526	−1.46403	−0.732017	−	0.681286i	$-0.761421\pi$
$541$	−34.0526	−1.46403	−0.732017	+	0.681286i	$0.761421\pi$
$542$	13.1244	0.563739
$543$	−7.07180	−0.303480
$544$	37.8564	1.62308
$545$	5.26795	0.225654
$546$	−3.07180	−0.131461
$547$	23.5885	1.00857	0.504285	−	0.863537i	$-0.331756\pi$
$547$	23.5885	1.00857	0.504285	+	0.863537i	$0.331756\pi$
$548$	9.56922	0.408777
$549$	11.3923	0.486212
$550$	0.732051	0.0312148
$551$	−8.80385	−0.375057
$552$	−21.4641	−0.913573
$553$	3.66025	0.155650
$554$	6.64102	0.282150
$555$	−4.19615	−0.178117
$556$	−16.7846	−0.711826
$557$	8.53590	0.361678	0.180839	−	0.983513i	$-0.442119\pi$
$557$	8.53590	0.361678	0.180839	+	0.983513i	$0.442119\pi$
$558$	3.85641	0.163255
$559$	−7.26795	−0.307401
$560$	−1.07180	−0.0452917
$561$	−6.46410	−0.272915
$562$	−12.4974	−0.527172
$563$	−24.9282	−1.05060	−0.525299	−	0.850918i	$-0.676047\pi$
$563$	−24.9282	−1.05060	−0.525299	+	0.850918i	$0.676047\pi$
$564$	1.85641	0.0781688
$565$	−10.8564	−0.456732
$566$	0.143594	0.00603569
$567$	1.00000	0.0419961
$568$	19.4256	0.815081
$569$	−4.66025	−0.195368	−0.0976840	−	0.995217i	$-0.531143\pi$
$569$	−4.66025	−0.195368	−0.0976840	+	0.995217i	$0.531143\pi$
$570$	1.12436	0.0470941
$571$	17.1244	0.716632	0.358316	−	0.933600i	$-0.383351\pi$
$571$	17.1244	0.716632	0.358316	+	0.933600i	$0.383351\pi$
$572$	−6.14359	−0.256877
$573$	1.07180	0.0447750
$574$	5.60770	0.234061
$575$	8.46410	0.352977
$576$	2.14359	0.0893164
$577$	−7.07180	−0.294403	−0.147201	−	0.989107i	$-0.547027\pi$
$577$	−7.07180	−0.294403	−0.147201	+	0.989107i	$0.547027\pi$
$578$	−18.1436	−0.754674
$579$	7.46410	0.310197
$580$	8.39230	0.348471
$581$	2.46410	0.102228
$582$	−1.66025	−0.0688197
$583$	−4.46410	−0.184884
$584$	21.2820	0.880657
$585$	4.19615	0.173490
$586$	−6.58846	−0.272167
$587$	−18.5885	−0.767228	−0.383614	−	0.923494i	$-0.625321\pi$
$587$	−18.5885	−0.767228	−0.383614	+	0.923494i	$0.625321\pi$
$588$	1.46410	0.0603785
$589$	8.09103	0.333385
$590$	10.0526	0.413857
$591$	−5.46410	−0.224763
$592$	−4.49742	−0.184843
$593$	16.0000	0.657041	0.328521	−	0.944497i	$-0.393450\pi$
$593$	16.0000	0.657041	0.328521	+	0.944497i	$0.393450\pi$
$594$	−0.732051	−0.0300364
$595$	6.46410	0.265002
$596$	−22.6410	−0.927412
$597$	14.9282	0.610971
$598$	26.0000	1.06322
$599$	28.2487	1.15421	0.577106	−	0.816670i	$-0.304182\pi$
$599$	28.2487	1.15421	0.577106	+	0.816670i	$0.304182\pi$
$600$	−2.53590	−0.103528
$601$	17.7846	0.725449	0.362725	−	0.931896i	$-0.381847\pi$
$601$	17.7846	0.725449	0.362725	+	0.931896i	$0.381847\pi$
$602$	−1.26795	−0.0516778
$603$	4.92820	0.200692
$604$	17.3590	0.706327
$605$	−1.00000	−0.0406558
$606$	12.3923	0.503403
$607$	23.1769	0.940722	0.470361	−	0.882474i	$-0.344124\pi$
$607$	23.1769	0.940722	0.470361	+	0.882474i	$0.344124\pi$
$608$	8.99485	0.364789
$609$	−5.73205	−0.232274
$610$	8.33975	0.337666
$611$	−5.32051	−0.215245
$612$	9.46410	0.382564
$613$	34.7846	1.40494	0.702469	−	0.711715i	$-0.252081\pi$
$613$	34.7846	1.40494	0.702469	+	0.711715i	$0.252081\pi$
$614$	7.60770	0.307022
$615$	−7.66025	−0.308891
$616$	−2.53590	−0.102174
$617$	−46.6410	−1.87770	−0.938848	−	0.344331i	$-0.888106\pi$
$617$	−46.6410	−1.87770	−0.938848	+	0.344331i	$0.888106\pi$
$618$	−10.4449	−0.420154
$619$	−24.9282	−1.00195	−0.500975	−	0.865462i	$-0.667025\pi$
$619$	−24.9282	−1.00195	−0.500975	+	0.865462i	$0.667025\pi$
$620$	−7.71281	−0.309754
$621$	−8.46410	−0.339653
$622$	−5.35898	−0.214876
$623$	14.6603	0.587351
$624$	4.49742	0.180041
$625$	1.00000	0.0400000
$626$	−1.26795	−0.0506774
$627$	−1.53590	−0.0613379
$628$	20.6795	0.825202
$629$	27.1244	1.08152
$630$	0.732051	0.0291656
$631$	−16.7128	−0.665327	−0.332663	−	0.943046i	$-0.607947\pi$
$631$	−16.7128	−0.665327	−0.332663	+	0.943046i	$0.607947\pi$
$632$	9.28203	0.369219
$633$	−4.53590	−0.180286
$634$	−13.8564	−0.550308
$635$	22.1244	0.877978
$636$	6.53590	0.259165
$637$	−4.19615	−0.166258
$638$	4.19615	0.166127
$639$	7.66025	0.303035
$640$	−10.1436	−0.400961
$641$	−28.6410	−1.13125	−0.565626	−	0.824662i	$-0.691365\pi$
$641$	−28.6410	−1.13125	−0.565626	+	0.824662i	$0.691365\pi$
$642$	3.85641	0.152200
$643$	−6.80385	−0.268318	−0.134159	−	0.990960i	$-0.542833\pi$
$643$	−6.80385	−0.268318	−0.134159	+	0.990960i	$0.542833\pi$
$644$	−12.3923	−0.488325
$645$	1.73205	0.0681994
$646$	−7.26795	−0.285954
$647$	44.1051	1.73395	0.866976	−	0.498351i	$-0.166061\pi$
$647$	44.1051	1.73395	0.866976	+	0.498351i	$0.166061\pi$
$648$	2.53590	0.0996195
$649$	−13.7321	−0.539030
$650$	3.07180	0.120486
$651$	5.26795	0.206467
$652$	−28.0000	−1.09656
$653$	−29.5359	−1.15583	−0.577915	−	0.816097i	$-0.696133\pi$
$653$	−29.5359	−1.15583	−0.577915	+	0.816097i	$0.696133\pi$
$654$	−3.85641	−0.150797
$655$	−16.1962	−0.632836
$656$	−8.21024	−0.320556
$657$	8.39230	0.327415
$658$	−0.928203	−0.0361851
$659$	−30.1244	−1.17348	−0.586739	−	0.809776i	$-0.699589\pi$
$659$	−30.1244	−1.17348	−0.586739	+	0.809776i	$0.699589\pi$
$660$	1.46410	0.0569901
$661$	−2.73205	−0.106264	−0.0531322	−	0.998587i	$-0.516920\pi$
$661$	−2.73205	−0.106264	−0.0531322	+	0.998587i	$0.516920\pi$
$662$	−6.09103	−0.236735
$663$	−27.1244	−1.05342
$664$	6.24871	0.242497
$665$	1.53590	0.0595596
$666$	3.07180	0.119030
$667$	48.5167	1.87857
$668$	15.2154	0.588701
$669$	−0.803848	−0.0310785
$670$	3.60770	0.139377
$671$	−11.3923	−0.439795
$672$	5.85641	0.225916
$673$	−17.5885	−0.677985	−0.338993	−	0.940789i	$-0.610086\pi$
$673$	−17.5885	−0.677985	−0.338993	+	0.940789i	$0.610086\pi$
$674$	7.80385	0.300593
$675$	−1.00000	−0.0384900
$676$	−6.74613	−0.259467
$677$	−39.2487	−1.50845	−0.754225	−	0.656616i	$-0.771987\pi$
$677$	−39.2487	−1.50845	−0.754225	+	0.656616i	$0.771987\pi$
$678$	7.94744	0.305220
$679$	−2.26795	−0.0870359
$680$	16.3923	0.628616
$681$	5.39230	0.206634
$682$	−3.85641	−0.147669
$683$	−46.9282	−1.79566	−0.897829	−	0.440344i	$-0.854856\pi$
$683$	−46.9282	−1.79566	−0.897829	+	0.440344i	$0.854856\pi$
$684$	2.24871	0.0859816
$685$	6.53590	0.249724
$686$	−0.732051	−0.0279498
$687$	−28.1962	−1.07575
$688$	1.85641	0.0707748
$689$	−18.7321	−0.713634
$690$	−6.19615	−0.235883
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	37.2820	1.41725
$693$	−1.00000	−0.0379869
$694$	1.07180	0.0406848
$695$	−11.4641	−0.434858
$696$	−14.5359	−0.550982
$697$	49.5167	1.87558
$698$	10.9808	0.415628
$699$	10.1962	0.385654
$700$	−1.46410	−0.0553378
$701$	33.1962	1.25380	0.626901	−	0.779099i	$-0.284323\pi$
$701$	33.1962	1.25380	0.626901	+	0.779099i	$0.284323\pi$
$702$	−3.07180	−0.115937
$703$	6.44486	0.243073
$704$	−2.14359	−0.0807897
$705$	1.26795	0.0477537
$706$	−11.3205	−0.426053
$707$	16.9282	0.636651
$708$	20.1051	0.755597
$709$	36.4641	1.36944	0.684719	−	0.728807i	$-0.259925\pi$
$709$	36.4641	1.36944	0.684719	+	0.728807i	$0.259925\pi$
$710$	5.60770	0.210453
$711$	3.66025	0.137270
$712$	37.1769	1.39326
$713$	−44.5885	−1.66985
$714$	−4.73205	−0.177093
$715$	−4.19615	−0.156927
$716$	29.3590	1.09720
$717$	−18.1244	−0.676866
$718$	25.2679	0.942991
$719$	32.6603	1.21802	0.609011	−	0.793162i	$-0.291567\pi$
$719$	32.6603	1.21802	0.609011	+	0.793162i	$0.291567\pi$
$720$	−1.07180	−0.0399435
$721$	−14.2679	−0.531366
$722$	12.1821	0.453370
$723$	−26.9282	−1.00147
$724$	−10.3538	−0.384797
$725$	5.73205	0.212883
$726$	0.732051	0.0271690
$727$	42.9090	1.59141	0.795703	−	0.605687i	$-0.207102\pi$
$727$	42.9090	1.59141	0.795703	+	0.605687i	$0.207102\pi$
$728$	−10.6410	−0.394382
$729$	1.00000	0.0370370
$730$	6.14359	0.227385
$731$	−11.1962	−0.414105
$732$	16.6795	0.616492
$733$	51.1769	1.89026	0.945131	−	0.326691i	$-0.105934\pi$
$733$	51.1769	1.89026	0.945131	+	0.326691i	$0.105934\pi$
$734$	−12.3013	−0.454048
$735$	1.00000	0.0368856
$736$	−49.5692	−1.82715
$737$	−4.92820	−0.181533
$738$	5.60770	0.206422
$739$	8.53590	0.313998	0.156999	−	0.987599i	$-0.449818\pi$
$739$	8.53590	0.313998	0.156999	+	0.987599i	$0.449818\pi$
$740$	−6.14359	−0.225843
$741$	−6.44486	−0.236758
$742$	−3.26795	−0.119970
$743$	19.1769	0.703533	0.351766	−	0.936088i	$-0.385581\pi$
$743$	19.1769	0.703533	0.351766	+	0.936088i	$0.385581\pi$
$744$	13.3590	0.489764
$745$	−15.4641	−0.566561
$746$	−21.3731	−0.782524
$747$	2.46410	0.0901568
$748$	−9.46410	−0.346042
$749$	5.26795	0.192487
$750$	−0.732051	−0.0267307
$751$	16.3205	0.595544	0.297772	−	0.954637i	$-0.403757\pi$
$751$	16.3205	0.595544	0.297772	+	0.954637i	$0.403757\pi$
$752$	1.35898	0.0495570
$753$	6.92820	0.252478
$754$	17.6077	0.641234
$755$	11.8564	0.431499
$756$	1.46410	0.0532489
$757$	8.53590	0.310243	0.155121	−	0.987895i	$-0.450423\pi$
$757$	8.53590	0.310243	0.155121	+	0.987895i	$0.450423\pi$
$758$	−15.2679	−0.554557
$759$	8.46410	0.307227
$760$	3.89488	0.141282
$761$	−11.0718	−0.401352	−0.200676	−	0.979658i	$-0.564314\pi$
$761$	−11.0718	−0.401352	−0.200676	+	0.979658i	$0.564314\pi$
$762$	−16.1962	−0.586725
$763$	−5.26795	−0.190713
$764$	1.56922	0.0567724
$765$	6.46410	0.233710
$766$	−23.9615	−0.865765
$767$	−57.6218	−2.08060
$768$	11.7128	0.422650
$769$	−23.7846	−0.857695	−0.428847	−	0.903377i	$-0.641080\pi$
$769$	−23.7846	−0.857695	−0.428847	+	0.903377i	$0.641080\pi$
$770$	−0.732051	−0.0263813
$771$	−13.8038	−0.497133
$772$	10.9282	0.393315
$773$	36.2487	1.30378	0.651888	−	0.758315i	$-0.273977\pi$
$773$	36.2487	1.30378	0.651888	+	0.758315i	$0.273977\pi$
$774$	−1.26795	−0.0455755
$775$	−5.26795	−0.189230
$776$	−5.75129	−0.206459
$777$	4.19615	0.150536
$778$	11.3590	0.407239
$779$	11.7654	0.421538
$780$	6.14359	0.219976
$781$	−7.66025	−0.274105
$782$	40.0526	1.43228
$783$	−5.73205	−0.204847
$784$	1.07180	0.0382785
$785$	14.1244	0.504120
$786$	11.8564	0.422904
$787$	−42.6410	−1.51999	−0.759994	−	0.649930i	$-0.774798\pi$
$787$	−42.6410	−1.51999	−0.759994	+	0.649930i	$0.774798\pi$
$788$	−8.00000	−0.284988
$789$	−11.3205	−0.403021
$790$	2.67949	0.0953320
$791$	10.8564	0.386009
$792$	−2.53590	−0.0901092
$793$	−47.8038	−1.69756
$794$	−27.7128	−0.983491
$795$	4.46410	0.158325
$796$	21.8564	0.774680
$797$	−11.1769	−0.395907	−0.197953	−	0.980211i	$-0.563429\pi$
$797$	−11.1769	−0.395907	−0.197953	+	0.980211i	$0.563429\pi$
$798$	−1.12436	−0.0398018
$799$	−8.19615	−0.289959
$800$	−5.85641	−0.207055
$801$	14.6603	0.517995
$802$	24.2487	0.856252
$803$	−8.39230	−0.296158
$804$	7.21539	0.254467
$805$	−8.46410	−0.298320
$806$	−16.1821	−0.569989
$807$	−7.58846	−0.267126
$808$	42.9282	1.51021
$809$	8.53590	0.300106	0.150053	−	0.988678i	$-0.452056\pi$
$809$	8.53590	0.300106	0.150053	+	0.988678i	$0.452056\pi$
$810$	0.732051	0.0257216
$811$	4.78461	0.168010	0.0840052	−	0.996465i	$-0.473229\pi$
$811$	4.78461	0.168010	0.0840052	+	0.996465i	$0.473229\pi$
$812$	−8.39230	−0.294512
$813$	17.9282	0.628770
$814$	−3.07180	−0.107666
$815$	−19.1244	−0.669897
$816$	6.92820	0.242536
$817$	−2.66025	−0.0930705
$818$	6.64102	0.232198
$819$	−4.19615	−0.146625
$820$	−11.2154	−0.391658
$821$	−9.98076	−0.348331	−0.174165	−	0.984716i	$-0.555723\pi$
$821$	−9.98076	−0.348331	−0.174165	+	0.984716i	$0.555723\pi$
$822$	−4.78461	−0.166882
$823$	18.4449	0.642948	0.321474	−	0.946918i	$-0.395822\pi$
$823$	18.4449	0.642948	0.321474	+	0.946918i	$0.395822\pi$
$824$	−36.1821	−1.26046
$825$	1.00000	0.0348155
$826$	−10.0526	−0.349773
$827$	13.4641	0.468193	0.234096	−	0.972213i	$-0.424787\pi$
$827$	13.4641	0.468193	0.234096	+	0.972213i	$0.424787\pi$
$828$	−12.3923	−0.430662
$829$	−35.1769	−1.22174	−0.610872	−	0.791729i	$-0.709181\pi$
$829$	−35.1769	−1.22174	−0.610872	+	0.791729i	$0.709181\pi$
$830$	1.80385	0.0626125
$831$	9.07180	0.314697
$832$	−8.99485	−0.311840
$833$	−6.46410	−0.223968
$834$	8.39230	0.290602
$835$	10.3923	0.359641
$836$	−2.24871	−0.0777733
$837$	5.26795	0.182087
$838$	−21.7654	−0.751872
$839$	15.3397	0.529587	0.264793	−	0.964305i	$-0.414696\pi$
$839$	15.3397	0.529587	0.264793	+	0.964305i	$0.414696\pi$
$840$	2.53590	0.0874968
$841$	3.85641	0.132980
$842$	−18.0910	−0.623458
$843$	−17.0718	−0.587984
$844$	−6.64102	−0.228593
$845$	−4.60770	−0.158510
$846$	−0.928203	−0.0319123
$847$	1.00000	0.0343604
$848$	4.78461	0.164304
$849$	0.196152	0.00673193
$850$	4.73205	0.162308
$851$	−35.5167	−1.21750
$852$	11.2154	0.384233
$853$	10.8756	0.372375	0.186187	−	0.982514i	$-0.440387\pi$
$853$	10.8756	0.372375	0.186187	+	0.982514i	$0.440387\pi$
$854$	−8.33975	−0.285380
$855$	1.53590	0.0525266
$856$	13.3590	0.456601
$857$	49.7128	1.69816	0.849079	−	0.528266i	$-0.177158\pi$
$857$	49.7128	1.69816	0.849079	+	0.528266i	$0.177158\pi$
$858$	3.07180	0.104869
$859$	−48.5885	−1.65782	−0.828908	−	0.559384i	$-0.811038\pi$
$859$	−48.5885	−1.65782	−0.828908	+	0.559384i	$0.811038\pi$
$860$	2.53590	0.0864734
$861$	7.66025	0.261061
$862$	17.8564	0.608192
$863$	−25.5359	−0.869252	−0.434626	−	0.900611i	$-0.643119\pi$
$863$	−25.5359	−0.869252	−0.434626	+	0.900611i	$0.643119\pi$
$864$	5.85641	0.199239
$865$	25.4641	0.865805
$866$	−6.82309	−0.231858
$867$	−24.7846	−0.841729
$868$	7.71281	0.261790
$869$	−3.66025	−0.124166
$870$	−4.19615	−0.142263
$871$	−20.6795	−0.700698
$872$	−13.3590	−0.452392
$873$	−2.26795	−0.0767585
$874$	9.51666	0.321906
$875$	−1.00000	−0.0338062
$876$	12.2872	0.415146
$877$	18.3731	0.620414	0.310207	−	0.950669i	$-0.399602\pi$
$877$	18.3731	0.620414	0.310207	+	0.950669i	$0.399602\pi$
$878$	19.5551	0.659954
$879$	−9.00000	−0.303562
$880$	1.07180	0.0361303
$881$	16.9090	0.569678	0.284839	−	0.958575i	$-0.408060\pi$
$881$	16.9090	0.569678	0.284839	+	0.958575i	$0.408060\pi$
$882$	−0.732051	−0.0246494
$883$	−37.1769	−1.25110	−0.625551	−	0.780183i	$-0.715126\pi$
$883$	−37.1769	−1.25110	−0.625551	+	0.780183i	$0.715126\pi$
$884$	−39.7128	−1.33569
$885$	13.7321	0.461598
$886$	17.1769	0.577070
$887$	−21.3923	−0.718283	−0.359142	−	0.933283i	$-0.616931\pi$
$887$	−21.3923	−0.718283	−0.359142	+	0.933283i	$0.616931\pi$
$888$	10.6410	0.357089
$889$	−22.1244	−0.742027
$890$	10.7321	0.359739
$891$	−1.00000	−0.0335013
$892$	−1.17691	−0.0394060
$893$	−1.94744	−0.0651686
$894$	11.3205	0.378614
$895$	20.0526	0.670283
$896$	10.1436	0.338874
$897$	35.5167	1.18587
$898$	−11.7513	−0.392146
$899$	−30.1962	−1.00710
$900$	−1.46410	−0.0488034
$901$	−28.8564	−0.961346
$902$	−5.60770	−0.186716
$903$	−1.73205	−0.0576390
$904$	27.5307	0.915659
$905$	−7.07180	−0.235074
$906$	−8.67949	−0.288357
$907$	18.3397	0.608961	0.304481	−	0.952519i	$-0.401517\pi$
$907$	18.3397	0.608961	0.304481	+	0.952519i	$0.401517\pi$
$908$	7.89488	0.262001
$909$	16.9282	0.561473
$910$	−3.07180	−0.101829
$911$	31.6077	1.04721	0.523605	−	0.851961i	$-0.324587\pi$
$911$	31.6077	1.04721	0.523605	+	0.851961i	$0.324587\pi$
$912$	1.64617	0.0545102
$913$	−2.46410	−0.0815499
$914$	1.94744	0.0644156
$915$	11.3923	0.376618
$916$	−41.2820	−1.36400
$917$	16.1962	0.534844
$918$	−4.73205	−0.156181
$919$	−46.4974	−1.53381	−0.766904	−	0.641762i	$-0.778204\pi$
$919$	−46.4974	−1.53381	−0.766904	+	0.641762i	$0.778204\pi$
$920$	−21.4641	−0.707650
$921$	10.3923	0.342438
$922$	−21.4641	−0.706883
$923$	−32.1436	−1.05802
$924$	−1.46410	−0.0481654
$925$	−4.19615	−0.137969
$926$	−9.17691	−0.301572
$927$	−14.2679	−0.468621
$928$	−33.5692	−1.10196
$929$	−15.7128	−0.515521	−0.257760	−	0.966209i	$-0.582984\pi$
$929$	−15.7128	−0.515521	−0.257760	+	0.966209i	$0.582984\pi$
$930$	3.85641	0.126457
$931$	−1.53590	−0.0503370
$932$	14.9282	0.488990
$933$	−7.32051	−0.239663
$934$	20.0000	0.654420
$935$	−6.46410	−0.211399
$936$	−10.6410	−0.347812
$937$	26.9808	0.881423	0.440712	−	0.897649i	$-0.354726\pi$
$937$	26.9808	0.881423	0.440712	+	0.897649i	$0.354726\pi$
$938$	−3.60770	−0.117795
$939$	−1.73205	−0.0565233
$940$	1.85641	0.0605493
$941$	26.8372	0.874867	0.437433	−	0.899251i	$-0.355888\pi$
$941$	26.8372	0.874867	0.437433	+	0.899251i	$0.355888\pi$
$942$	−10.3397	−0.336887
$943$	−64.8372	−2.11139
$944$	14.7180	0.479029
$945$	1.00000	0.0325300
$946$	1.26795	0.0412246
$947$	−12.7128	−0.413111	−0.206555	−	0.978435i	$-0.566225\pi$
$947$	−12.7128	−0.413111	−0.206555	+	0.978435i	$0.566225\pi$
$948$	5.35898	0.174052
$949$	−35.2154	−1.14314
$950$	1.12436	0.0364789
$951$	−18.9282	−0.613789
$952$	−16.3923	−0.531278
$953$	8.92820	0.289213	0.144606	−	0.989489i	$-0.453808\pi$
$953$	8.92820	0.289213	0.144606	+	0.989489i	$0.453808\pi$
$954$	−3.26795	−0.105804
$955$	1.07180	0.0346825
$956$	−26.5359	−0.858232
$957$	5.73205	0.185291
$958$	4.92820	0.159223
$959$	−6.53590	−0.211055
$960$	2.14359	0.0691842
$961$	−3.24871	−0.104797
$962$	−12.8897	−0.415581
$963$	5.26795	0.169757
$964$	−39.4256	−1.26981
$965$	7.46410	0.240278
$966$	6.19615	0.199358
$967$	30.5167	0.981350	0.490675	−	0.871343i	$-0.336750\pi$
$967$	30.5167	0.981350	0.490675	+	0.871343i	$0.336750\pi$
$968$	2.53590	0.0815069
$969$	−9.92820	−0.318940
$970$	−1.66025	−0.0533075
$971$	42.6603	1.36903	0.684516	−	0.728998i	$-0.260013\pi$
$971$	42.6603	1.36903	0.684516	+	0.728998i	$0.260013\pi$
$972$	1.46410	0.0469611
$973$	11.4641	0.367522
$974$	−0.392305	−0.0125703
$975$	4.19615	0.134384
$976$	12.2102	0.390840
$977$	11.3923	0.364472	0.182236	−	0.983255i	$-0.441666\pi$
$977$	11.3923	0.364472	0.182236	+	0.983255i	$0.441666\pi$
$978$	14.0000	0.447671
$979$	−14.6603	−0.468544
$980$	1.46410	0.0467690
$981$	−5.26795	−0.168193
$982$	−2.44486	−0.0780187
$983$	10.6795	0.340623	0.170311	−	0.985390i	$-0.445523\pi$
$983$	10.6795	0.340623	0.170311	+	0.985390i	$0.445523\pi$
$984$	19.4256	0.619266
$985$	−5.46410	−0.174101
$986$	27.1244	0.863815
$987$	−1.26795	−0.0403593
$988$	−9.43594	−0.300197
$989$	14.6603	0.466169
$990$	−0.732051	−0.0232661
$991$	−30.5692	−0.971063	−0.485532	−	0.874219i	$-0.661374\pi$
$991$	−30.5692	−0.971063	−0.485532	+	0.874219i	$0.661374\pi$
$992$	30.8513	0.979528
$993$	−8.32051	−0.264043
$994$	−5.60770	−0.177865
$995$	14.9282	0.473256
$996$	3.60770	0.114314
$997$	−29.7128	−0.941014	−0.470507	−	0.882396i	$-0.655929\pi$
$997$	−29.7128	−0.941014	−0.470507	+	0.882396i	$0.655929\pi$
$998$	−21.5167	−0.681098
$999$	4.19615	0.132760

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1155.2.a.r.1.1	✓	2
3.2	odd	2		3465.2.a.v.1.2		2
5.4	even	2		5775.2.a.bc.1.2		2
7.6	odd	2		8085.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.r.1.1	✓	2	1.1	even	1	trivial
3465.2.a.v.1.2		2	3.2	odd	2
5775.2.a.bc.1.2		2	5.4	even	2
8085.2.a.bh.1.1		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1155.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{2} -1.00000 q^{3} -1.46410 q^{4} -1.00000 q^{5} +0.732051 q^{6} +1.00000 q^{7} +2.53590 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.732051 q^{2} -1.00000 q^{3} -1.46410 q^{4} -1.00000 q^{5} +0.732051 q^{6} +1.00000 q^{7} +2.53590 q^{8} +1.00000 q^{9} +0.732051 q^{10} -1.00000 q^{11} +1.46410 q^{12} -4.19615 q^{13} -0.732051 q^{14} +1.00000 q^{15} +1.07180 q^{16} -6.46410 q^{17} -0.732051 q^{18} -1.53590 q^{19} +1.46410 q^{20} -1.00000 q^{21} +0.732051 q^{22} +8.46410 q^{23} -2.53590 q^{24} +1.00000 q^{25} +3.07180 q^{26} -1.00000 q^{27} -1.46410 q^{28} +5.73205 q^{29} -0.732051 q^{30} -5.26795 q^{31} -5.85641 q^{32} +1.00000 q^{33} +4.73205 q^{34} -1.00000 q^{35} -1.46410 q^{36} -4.19615 q^{37} +1.12436 q^{38} +4.19615 q^{39} -2.53590 q^{40} -7.66025 q^{41} +0.732051 q^{42} +1.73205 q^{43} +1.46410 q^{44} -1.00000 q^{45} -6.19615 q^{46} +1.26795 q^{47} -1.07180 q^{48} +1.00000 q^{49} -0.732051 q^{50} +6.46410 q^{51} +6.14359 q^{52} +4.46410 q^{53} +0.732051 q^{54} +1.00000 q^{55} +2.53590 q^{56} +1.53590 q^{57} -4.19615 q^{58} +13.7321 q^{59} -1.46410 q^{60} +11.3923 q^{61} +3.85641 q^{62} +1.00000 q^{63} +2.14359 q^{64} +4.19615 q^{65} -0.732051 q^{66} +4.92820 q^{67} +9.46410 q^{68} -8.46410 q^{69} +0.732051 q^{70} +7.66025 q^{71} +2.53590 q^{72} +8.39230 q^{73} +3.07180 q^{74} -1.00000 q^{75} +2.24871 q^{76} -1.00000 q^{77} -3.07180 q^{78} +3.66025 q^{79} -1.07180 q^{80} +1.00000 q^{81} +5.60770 q^{82} +2.46410 q^{83} +1.46410 q^{84} +6.46410 q^{85} -1.26795 q^{86} -5.73205 q^{87} -2.53590 q^{88} +14.6603 q^{89} +0.732051 q^{90} -4.19615 q^{91} -12.3923 q^{92} +5.26795 q^{93} -0.928203 q^{94} +1.53590 q^{95} +5.85641 q^{96} -2.26795 q^{97} -0.732051 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 12 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 12 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 4 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{15} + 16 q^{16} - 6 q^{17} + 2 q^{18} - 10 q^{19} - 4 q^{20} - 2 q^{21} - 2 q^{22} + 10 q^{23} - 12 q^{24} + 2 q^{25} + 20 q^{26} - 2 q^{27} + 4 q^{28} + 8 q^{29} + 2 q^{30} - 14 q^{31} + 16 q^{32} + 2 q^{33} + 6 q^{34} - 2 q^{35} + 4 q^{36} + 2 q^{37} - 22 q^{38} - 2 q^{39} - 12 q^{40} + 2 q^{41} - 2 q^{42} - 4 q^{44} - 2 q^{45} - 2 q^{46} + 6 q^{47} - 16 q^{48} + 2 q^{49} + 2 q^{50} + 6 q^{51} + 40 q^{52} + 2 q^{53} - 2 q^{54} + 2 q^{55} + 12 q^{56} + 10 q^{57} + 2 q^{58} + 24 q^{59} + 4 q^{60} + 2 q^{61} - 20 q^{62} + 2 q^{63} + 32 q^{64} - 2 q^{65} + 2 q^{66} - 4 q^{67} + 12 q^{68} - 10 q^{69} - 2 q^{70} - 2 q^{71} + 12 q^{72} - 4 q^{73} + 20 q^{74} - 2 q^{75} - 44 q^{76} - 2 q^{77} - 20 q^{78} - 10 q^{79} - 16 q^{80} + 2 q^{81} + 32 q^{82} - 2 q^{83} - 4 q^{84} + 6 q^{85} - 6 q^{86} - 8 q^{87} - 12 q^{88} + 12 q^{89} - 2 q^{90} + 2 q^{91} - 4 q^{92} + 14 q^{93} + 12 q^{94} + 10 q^{95} - 16 q^{96} - 8 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^7 + 12 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^11 - 4 * q^12 + 2 * q^13 + 2 * q^14 + 2 * q^15 + 16 * q^16 - 6 * q^17 + 2 * q^18 - 10 * q^19 - 4 * q^20 - 2 * q^21 - 2 * q^22 + 10 * q^23 - 12 * q^24 + 2 * q^25 + 20 * q^26 - 2 * q^27 + 4 * q^28 + 8 * q^29 + 2 * q^30 - 14 * q^31 + 16 * q^32 + 2 * q^33 + 6 * q^34 - 2 * q^35 + 4 * q^36 + 2 * q^37 - 22 * q^38 - 2 * q^39 - 12 * q^40 + 2 * q^41 - 2 * q^42 - 4 * q^44 - 2 * q^45 - 2 * q^46 + 6 * q^47 - 16 * q^48 + 2 * q^49 + 2 * q^50 + 6 * q^51 + 40 * q^52 + 2 * q^53 - 2 * q^54 + 2 * q^55 + 12 * q^56 + 10 * q^57 + 2 * q^58 + 24 * q^59 + 4 * q^60 + 2 * q^61 - 20 * q^62 + 2 * q^63 + 32 * q^64 - 2 * q^65 + 2 * q^66 - 4 * q^67 + 12 * q^68 - 10 * q^69 - 2 * q^70 - 2 * q^71 + 12 * q^72 - 4 * q^73 + 20 * q^74 - 2 * q^75 - 44 * q^76 - 2 * q^77 - 20 * q^78 - 10 * q^79 - 16 * q^80 + 2 * q^81 + 32 * q^82 - 2 * q^83 - 4 * q^84 + 6 * q^85 - 6 * q^86 - 8 * q^87 - 12 * q^88 + 12 * q^89 - 2 * q^90 + 2 * q^91 - 4 * q^92 + 14 * q^93 + 12 * q^94 + 10 * q^95 - 16 * q^96 - 8 * q^97 + 2 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more