LMFDB - Embedded newform 4830.2.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4830,2,Mod(1,4830)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4830, 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("4830.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4830.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:	$38.5677441763$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	4830.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.12311 q^{11} +1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -5.12311 q^{17} -1.00000 q^{18} -7.12311 q^{19} +1.00000 q^{20} -1.00000 q^{21} -5.12311 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} +8.24621 q^{31} -1.00000 q^{32} +5.12311 q^{33} +5.12311 q^{34} -1.00000 q^{35} +1.00000 q^{36} -3.12311 q^{37} +7.12311 q^{38} -4.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +1.00000 q^{42} -6.24621 q^{43} +5.12311 q^{44} +1.00000 q^{45} +1.00000 q^{46} -11.1231 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -5.12311 q^{51} -4.00000 q^{52} +3.12311 q^{53} -1.00000 q^{54} +5.12311 q^{55} +1.00000 q^{56} -7.12311 q^{57} -2.00000 q^{58} -1.12311 q^{59} +1.00000 q^{60} -6.00000 q^{61} -8.24621 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -5.12311 q^{66} -4.00000 q^{67} -5.12311 q^{68} -1.00000 q^{69} +1.00000 q^{70} -4.00000 q^{71} -1.00000 q^{72} -4.87689 q^{73} +3.12311 q^{74} +1.00000 q^{75} -7.12311 q^{76} -5.12311 q^{77} +4.00000 q^{78} -7.36932 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -0.876894 q^{83} -1.00000 q^{84} -5.12311 q^{85} +6.24621 q^{86} +2.00000 q^{87} -5.12311 q^{88} +1.12311 q^{89} -1.00000 q^{90} +4.00000 q^{91} -1.00000 q^{92} +8.24621 q^{93} +11.1231 q^{94} -7.12311 q^{95} -1.00000 q^{96} -6.87689 q^{97} -1.00000 q^{98} +5.12311 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} - 8 q^{13} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 6 q^{19} + 2 q^{20} - 2 q^{21} - 2 q^{22} - 2 q^{23} - 2 q^{24} + 2 q^{25} + 8 q^{26} + 2 q^{27} - 2 q^{28} + 4 q^{29} - 2 q^{30} - 2 q^{32} + 2 q^{33} + 2 q^{34} - 2 q^{35} + 2 q^{36} + 2 q^{37} + 6 q^{38} - 8 q^{39} - 2 q^{40} - 4 q^{41} + 2 q^{42} + 4 q^{43} + 2 q^{44} + 2 q^{45} + 2 q^{46} - 14 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} - 2 q^{51} - 8 q^{52} - 2 q^{53} - 2 q^{54} + 2 q^{55} + 2 q^{56} - 6 q^{57} - 4 q^{58} + 6 q^{59} + 2 q^{60} - 12 q^{61} - 2 q^{63} + 2 q^{64} - 8 q^{65} - 2 q^{66} - 8 q^{67} - 2 q^{68} - 2 q^{69} + 2 q^{70} - 8 q^{71} - 2 q^{72} - 18 q^{73} - 2 q^{74} + 2 q^{75} - 6 q^{76} - 2 q^{77} + 8 q^{78} + 10 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} - 10 q^{83} - 2 q^{84} - 2 q^{85} - 4 q^{86} + 4 q^{87} - 2 q^{88} - 6 q^{89} - 2 q^{90} + 8 q^{91} - 2 q^{92} + 14 q^{94} - 6 q^{95} - 2 q^{96} - 22 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^11 + 2 * q^12 - 8 * q^13 + 2 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 6 * q^19 + 2 * q^20 - 2 * q^21 - 2 * q^22 - 2 * q^23 - 2 * q^24 + 2 * q^25 + 8 * q^26 + 2 * q^27 - 2 * q^28 + 4 * q^29 - 2 * q^30 - 2 * q^32 + 2 * q^33 + 2 * q^34 - 2 * q^35 + 2 * q^36 + 2 * q^37 + 6 * q^38 - 8 * q^39 - 2 * q^40 - 4 * q^41 + 2 * q^42 + 4 * q^43 + 2 * q^44 + 2 * q^45 + 2 * q^46 - 14 * q^47 + 2 * q^48 + 2 * q^49 - 2 * q^50 - 2 * q^51 - 8 * q^52 - 2 * q^53 - 2 * q^54 + 2 * q^55 + 2 * q^56 - 6 * q^57 - 4 * q^58 + 6 * q^59 + 2 * q^60 - 12 * q^61 - 2 * q^63 + 2 * q^64 - 8 * q^65 - 2 * q^66 - 8 * q^67 - 2 * q^68 - 2 * q^69 + 2 * q^70 - 8 * q^71 - 2 * q^72 - 18 * q^73 - 2 * q^74 + 2 * q^75 - 6 * q^76 - 2 * q^77 + 8 * q^78 + 10 * q^79 + 2 * q^80 + 2 * q^81 + 4 * q^82 - 10 * q^83 - 2 * q^84 - 2 * q^85 - 4 * q^86 + 4 * q^87 - 2 * q^88 - 6 * q^89 - 2 * q^90 + 8 * q^91 - 2 * q^92 + 14 * q^94 - 6 * q^95 - 2 * q^96 - 22 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	5.12311	1.54467	0.772337	−	0.635213i	$-0.219088\pi$
$11$	5.12311	1.54467	0.772337	+	0.635213i	$0.219088\pi$
$12$	1.00000	0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	1.00000	0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−5.12311	−1.24254	−0.621268	−	0.783598i	$-0.713382\pi$
$17$	−5.12311	−1.24254	−0.621268	+	0.783598i	$0.713382\pi$
$18$	−1.00000	−0.235702
$19$	−7.12311	−1.63415	−0.817076	−	0.576530i	$-0.804407\pi$
$19$	−7.12311	−1.63415	−0.817076	+	0.576530i	$0.804407\pi$
$20$	1.00000	0.223607
$21$	−1.00000	−0.218218
$22$	−5.12311	−1.09225
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	4.00000	0.784465
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	−1.00000	−0.182574
$31$	8.24621	1.48106	0.740532	−	0.672022i	$-0.234574\pi$
$31$	8.24621	1.48106	0.740532	+	0.672022i	$0.234574\pi$
$32$	−1.00000	−0.176777
$33$	5.12311	0.891818
$34$	5.12311	0.878605
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	−3.12311	−0.513435	−0.256718	−	0.966486i	$-0.582641\pi$
$37$	−3.12311	−0.513435	−0.256718	+	0.966486i	$0.582641\pi$
$38$	7.12311	1.15552
$39$	−4.00000	−0.640513
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	1.00000	0.154303
$43$	−6.24621	−0.952538	−0.476269	−	0.879300i	$-0.658011\pi$
$43$	−6.24621	−0.952538	−0.476269	+	0.879300i	$0.658011\pi$
$44$	5.12311	0.772337
$45$	1.00000	0.149071
$46$	1.00000	0.147442
$47$	−11.1231	−1.62247	−0.811236	−	0.584719i	$-0.801205\pi$
$47$	−11.1231	−1.62247	−0.811236	+	0.584719i	$0.801205\pi$
$48$	1.00000	0.144338
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−5.12311	−0.717378
$52$	−4.00000	−0.554700
$53$	3.12311	0.428992	0.214496	−	0.976725i	$-0.431189\pi$
$53$	3.12311	0.428992	0.214496	+	0.976725i	$0.431189\pi$
$54$	−1.00000	−0.136083
$55$	5.12311	0.690799
$56$	1.00000	0.133631
$57$	−7.12311	−0.943478
$58$	−2.00000	−0.262613
$59$	−1.12311	−0.146216	−0.0731079	−	0.997324i	$-0.523292\pi$
$59$	−1.12311	−0.146216	−0.0731079	+	0.997324i	$0.523292\pi$
$60$	1.00000	0.129099
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	−8.24621	−1.04727
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	−5.12311	−0.630611
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−5.12311	−0.621268
$69$	−1.00000	−0.120386
$70$	1.00000	0.119523
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	−1.00000	−0.117851
$73$	−4.87689	−0.570797	−0.285399	−	0.958409i	$-0.592126\pi$
$73$	−4.87689	−0.570797	−0.285399	+	0.958409i	$0.592126\pi$
$74$	3.12311	0.363054
$75$	1.00000	0.115470
$76$	−7.12311	−0.817076
$77$	−5.12311	−0.583832
$78$	4.00000	0.452911
$79$	−7.36932	−0.829113	−0.414556	−	0.910024i	$-0.636063\pi$
$79$	−7.36932	−0.829113	−0.414556	+	0.910024i	$0.636063\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	2.00000	0.220863
$83$	−0.876894	−0.0962517	−0.0481258	−	0.998841i	$-0.515325\pi$
$83$	−0.876894	−0.0962517	−0.0481258	+	0.998841i	$0.515325\pi$
$84$	−1.00000	−0.109109
$85$	−5.12311	−0.555679
$86$	6.24621	0.673546
$87$	2.00000	0.214423
$88$	−5.12311	−0.546125
$89$	1.12311	0.119049	0.0595245	−	0.998227i	$-0.481042\pi$
$89$	1.12311	0.119049	0.0595245	+	0.998227i	$0.481042\pi$
$90$	−1.00000	−0.105409
$91$	4.00000	0.419314
$92$	−1.00000	−0.104257
$93$	8.24621	0.855092
$94$	11.1231	1.14726
$95$	−7.12311	−0.730815
$96$	−1.00000	−0.102062
$97$	−6.87689	−0.698243	−0.349121	−	0.937077i	$-0.613520\pi$
$97$	−6.87689	−0.698243	−0.349121	+	0.937077i	$0.613520\pi$
$98$	−1.00000	−0.101015
$99$	5.12311	0.514891
$100$	1.00000	0.100000
$101$	5.12311	0.509768	0.254884	−	0.966972i	$-0.417963\pi$
$101$	5.12311	0.509768	0.254884	+	0.966972i	$0.417963\pi$
$102$	5.12311	0.507263
$103$	−14.2462	−1.40372	−0.701860	−	0.712314i	$-0.747647\pi$
$103$	−14.2462	−1.40372	−0.701860	+	0.712314i	$0.747647\pi$
$104$	4.00000	0.392232
$105$	−1.00000	−0.0975900
$106$	−3.12311	−0.303343
$107$	2.24621	0.217149	0.108575	−	0.994088i	$-0.465371\pi$
$107$	2.24621	0.217149	0.108575	+	0.994088i	$0.465371\pi$
$108$	1.00000	0.0962250
$109$	0.246211	0.0235828	0.0117914	−	0.999930i	$-0.496247\pi$
$109$	0.246211	0.0235828	0.0117914	+	0.999930i	$0.496247\pi$
$110$	−5.12311	−0.488469
$111$	−3.12311	−0.296432
$112$	−1.00000	−0.0944911
$113$	9.36932	0.881391	0.440696	−	0.897657i	$-0.354732\pi$
$113$	9.36932	0.881391	0.440696	+	0.897657i	$0.354732\pi$
$114$	7.12311	0.667140
$115$	−1.00000	−0.0932505
$116$	2.00000	0.185695
$117$	−4.00000	−0.369800
$118$	1.12311	0.103390
$119$	5.12311	0.469634
$120$	−1.00000	−0.0912871
$121$	15.2462	1.38602
$122$	6.00000	0.543214
$123$	−2.00000	−0.180334
$124$	8.24621	0.740532
$125$	1.00000	0.0894427
$126$	1.00000	0.0890871
$127$	22.2462	1.97403	0.987016	−	0.160622i	$-0.0513500\pi$
$127$	22.2462	1.97403	0.987016	+	0.160622i	$0.0513500\pi$
$128$	−1.00000	−0.0883883
$129$	−6.24621	−0.549948
$130$	4.00000	0.350823
$131$	−17.1231	−1.49605	−0.748026	−	0.663669i	$-0.768998\pi$
$131$	−17.1231	−1.49605	−0.748026	+	0.663669i	$0.768998\pi$
$132$	5.12311	0.445909
$133$	7.12311	0.617652
$134$	4.00000	0.345547
$135$	1.00000	0.0860663
$136$	5.12311	0.439303
$137$	−15.1231	−1.29205	−0.646027	−	0.763315i	$-0.723571\pi$
$137$	−15.1231	−1.29205	−0.646027	+	0.763315i	$0.723571\pi$
$138$	1.00000	0.0851257
$139$	5.12311	0.434536	0.217268	−	0.976112i	$-0.430285\pi$
$139$	5.12311	0.434536	0.217268	+	0.976112i	$0.430285\pi$
$140$	−1.00000	−0.0845154
$141$	−11.1231	−0.936734
$142$	4.00000	0.335673
$143$	−20.4924	−1.71366
$144$	1.00000	0.0833333
$145$	2.00000	0.166091
$146$	4.87689	0.403615
$147$	1.00000	0.0824786
$148$	−3.12311	−0.256718
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	−1.00000	−0.0816497
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	7.12311	0.577760
$153$	−5.12311	−0.414179
$154$	5.12311	0.412832
$155$	8.24621	0.662352
$156$	−4.00000	−0.320256
$157$	6.00000	0.478852	0.239426	−	0.970915i	$-0.423041\pi$
$157$	6.00000	0.478852	0.239426	+	0.970915i	$0.423041\pi$
$158$	7.36932	0.586271
$159$	3.12311	0.247678
$160$	−1.00000	−0.0790569
$161$	1.00000	0.0788110
$162$	−1.00000	−0.0785674
$163$	8.49242	0.665178	0.332589	−	0.943072i	$-0.392078\pi$
$163$	8.49242	0.665178	0.332589	+	0.943072i	$0.392078\pi$
$164$	−2.00000	−0.156174
$165$	5.12311	0.398833
$166$	0.876894	0.0680602
$167$	−12.8769	−0.996444	−0.498222	−	0.867050i	$-0.666014\pi$
$167$	−12.8769	−0.996444	−0.498222	+	0.867050i	$0.666014\pi$
$168$	1.00000	0.0771517
$169$	3.00000	0.230769
$170$	5.12311	0.392924
$171$	−7.12311	−0.544718
$172$	−6.24621	−0.476269
$173$	−8.00000	−0.608229	−0.304114	−	0.952636i	$-0.598361\pi$
$173$	−8.00000	−0.608229	−0.304114	+	0.952636i	$0.598361\pi$
$174$	−2.00000	−0.151620
$175$	−1.00000	−0.0755929
$176$	5.12311	0.386169
$177$	−1.12311	−0.0844178
$178$	−1.12311	−0.0841803
$179$	−14.2462	−1.06481	−0.532406	−	0.846489i	$-0.678712\pi$
$179$	−14.2462	−1.06481	−0.532406	+	0.846489i	$0.678712\pi$
$180$	1.00000	0.0745356
$181$	20.2462	1.50489	0.752445	−	0.658656i	$-0.228875\pi$
$181$	20.2462	1.50489	0.752445	+	0.658656i	$0.228875\pi$
$182$	−4.00000	−0.296500
$183$	−6.00000	−0.443533
$184$	1.00000	0.0737210
$185$	−3.12311	−0.229615
$186$	−8.24621	−0.604642
$187$	−26.2462	−1.91931
$188$	−11.1231	−0.811236
$189$	−1.00000	−0.0727393
$190$	7.12311	0.516764
$191$	−7.36932	−0.533225	−0.266613	−	0.963804i	$-0.585904\pi$
$191$	−7.36932	−0.533225	−0.266613	+	0.963804i	$0.585904\pi$
$192$	1.00000	0.0721688
$193$	26.4924	1.90697	0.953483	−	0.301446i	$-0.0974694\pi$
$193$	26.4924	1.90697	0.953483	+	0.301446i	$0.0974694\pi$
$194$	6.87689	0.493732
$195$	−4.00000	−0.286446
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	−5.12311	−0.364083
$199$	−6.24621	−0.442782	−0.221391	−	0.975185i	$-0.571060\pi$
$199$	−6.24621	−0.442782	−0.221391	+	0.975185i	$0.571060\pi$
$200$	−1.00000	−0.0707107
$201$	−4.00000	−0.282138
$202$	−5.12311	−0.360460
$203$	−2.00000	−0.140372
$204$	−5.12311	−0.358689
$205$	−2.00000	−0.139686
$206$	14.2462	0.992581
$207$	−1.00000	−0.0695048
$208$	−4.00000	−0.277350
$209$	−36.4924	−2.52423
$210$	1.00000	0.0690066
$211$	−6.24621	−0.430007	−0.215003	−	0.976613i	$-0.568976\pi$
$211$	−6.24621	−0.430007	−0.215003	+	0.976613i	$0.568976\pi$
$212$	3.12311	0.214496
$213$	−4.00000	−0.274075
$214$	−2.24621	−0.153548
$215$	−6.24621	−0.425988
$216$	−1.00000	−0.0680414
$217$	−8.24621	−0.559789
$218$	−0.246211	−0.0166755
$219$	−4.87689	−0.329550
$220$	5.12311	0.345400
$221$	20.4924	1.37847
$222$	3.12311	0.209609
$223$	−15.6155	−1.04569	−0.522847	−	0.852427i	$-0.675130\pi$
$223$	−15.6155	−1.04569	−0.522847	+	0.852427i	$0.675130\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	−9.36932	−0.623238
$227$	5.36932	0.356374	0.178187	−	0.983997i	$-0.442977\pi$
$227$	5.36932	0.356374	0.178187	+	0.983997i	$0.442977\pi$
$228$	−7.12311	−0.471739
$229$	−18.4924	−1.22201	−0.611007	−	0.791625i	$-0.709235\pi$
$229$	−18.4924	−1.22201	−0.611007	+	0.791625i	$0.709235\pi$
$230$	1.00000	0.0659380
$231$	−5.12311	−0.337076
$232$	−2.00000	−0.131306
$233$	−4.24621	−0.278179	−0.139089	−	0.990280i	$-0.544417\pi$
$233$	−4.24621	−0.278179	−0.139089	+	0.990280i	$0.544417\pi$
$234$	4.00000	0.261488
$235$	−11.1231	−0.725591
$236$	−1.12311	−0.0731079
$237$	−7.36932	−0.478689
$238$	−5.12311	−0.332082
$239$	16.4924	1.06681	0.533403	−	0.845861i	$-0.320913\pi$
$239$	16.4924	1.06681	0.533403	+	0.845861i	$0.320913\pi$
$240$	1.00000	0.0645497
$241$	29.1231	1.87598	0.937992	−	0.346657i	$-0.112683\pi$
$241$	29.1231	1.87598	0.937992	+	0.346657i	$0.112683\pi$
$242$	−15.2462	−0.980064
$243$	1.00000	0.0641500
$244$	−6.00000	−0.384111
$245$	1.00000	0.0638877
$246$	2.00000	0.127515
$247$	28.4924	1.81293
$248$	−8.24621	−0.523635
$249$	−0.876894	−0.0555709
$250$	−1.00000	−0.0632456
$251$	−3.12311	−0.197129	−0.0985643	−	0.995131i	$-0.531425\pi$
$251$	−3.12311	−0.197129	−0.0985643	+	0.995131i	$0.531425\pi$
$252$	−1.00000	−0.0629941
$253$	−5.12311	−0.322087
$254$	−22.2462	−1.39585
$255$	−5.12311	−0.320821
$256$	1.00000	0.0625000
$257$	−8.87689	−0.553725	−0.276863	−	0.960909i	$-0.589295\pi$
$257$	−8.87689	−0.553725	−0.276863	+	0.960909i	$0.589295\pi$
$258$	6.24621	0.388872
$259$	3.12311	0.194060
$260$	−4.00000	−0.248069
$261$	2.00000	0.123797
$262$	17.1231	1.05787
$263$	−10.2462	−0.631808	−0.315904	−	0.948791i	$-0.602308\pi$
$263$	−10.2462	−0.631808	−0.315904	+	0.948791i	$0.602308\pi$
$264$	−5.12311	−0.315305
$265$	3.12311	0.191851
$266$	−7.12311	−0.436746
$267$	1.12311	0.0687329
$268$	−4.00000	−0.244339
$269$	−18.8769	−1.15094	−0.575472	−	0.817821i	$-0.695182\pi$
$269$	−18.8769	−1.15094	−0.575472	+	0.817821i	$0.695182\pi$
$270$	−1.00000	−0.0608581
$271$	−20.2462	−1.22987	−0.614935	−	0.788578i	$-0.710818\pi$
$271$	−20.2462	−1.22987	−0.614935	+	0.788578i	$0.710818\pi$
$272$	−5.12311	−0.310634
$273$	4.00000	0.242091
$274$	15.1231	0.913620
$275$	5.12311	0.308935
$276$	−1.00000	−0.0601929
$277$	3.75379	0.225543	0.112772	−	0.993621i	$-0.464027\pi$
$277$	3.75379	0.225543	0.112772	+	0.993621i	$0.464027\pi$
$278$	−5.12311	−0.307263
$279$	8.24621	0.493688
$280$	1.00000	0.0597614
$281$	−30.4924	−1.81903	−0.909513	−	0.415676i	$-0.863545\pi$
$281$	−30.4924	−1.81903	−0.909513	+	0.415676i	$0.863545\pi$
$282$	11.1231	0.662371
$283$	8.87689	0.527677	0.263838	−	0.964567i	$-0.415011\pi$
$283$	8.87689	0.527677	0.263838	+	0.964567i	$0.415011\pi$
$284$	−4.00000	−0.237356
$285$	−7.12311	−0.421936
$286$	20.4924	1.21174
$287$	2.00000	0.118056
$288$	−1.00000	−0.0589256
$289$	9.24621	0.543895
$290$	−2.00000	−0.117444
$291$	−6.87689	−0.403131
$292$	−4.87689	−0.285399
$293$	−14.4924	−0.846656	−0.423328	−	0.905976i	$-0.639138\pi$
$293$	−14.4924	−0.846656	−0.423328	+	0.905976i	$0.639138\pi$
$294$	−1.00000	−0.0583212
$295$	−1.12311	−0.0653897
$296$	3.12311	0.181527
$297$	5.12311	0.297273
$298$	14.0000	0.810998
$299$	4.00000	0.231326
$300$	1.00000	0.0577350
$301$	6.24621	0.360026
$302$	−12.0000	−0.690522
$303$	5.12311	0.294315
$304$	−7.12311	−0.408538
$305$	−6.00000	−0.343559
$306$	5.12311	0.292868
$307$	2.24621	0.128198	0.0640990	−	0.997944i	$-0.479583\pi$
$307$	2.24621	0.128198	0.0640990	+	0.997944i	$0.479583\pi$
$308$	−5.12311	−0.291916
$309$	−14.2462	−0.810439
$310$	−8.24621	−0.468353
$311$	26.0000	1.47432	0.737162	−	0.675716i	$-0.236165\pi$
$311$	26.0000	1.47432	0.737162	+	0.675716i	$0.236165\pi$
$312$	4.00000	0.226455
$313$	5.12311	0.289575	0.144788	−	0.989463i	$-0.453750\pi$
$313$	5.12311	0.289575	0.144788	+	0.989463i	$0.453750\pi$
$314$	−6.00000	−0.338600
$315$	−1.00000	−0.0563436
$316$	−7.36932	−0.414556
$317$	18.4924	1.03864	0.519319	−	0.854580i	$-0.326186\pi$
$317$	18.4924	1.03864	0.519319	+	0.854580i	$0.326186\pi$
$318$	−3.12311	−0.175135
$319$	10.2462	0.573678
$320$	1.00000	0.0559017
$321$	2.24621	0.125371
$322$	−1.00000	−0.0557278
$323$	36.4924	2.03049
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	−8.49242	−0.470352
$327$	0.246211	0.0136155
$328$	2.00000	0.110432
$329$	11.1231	0.613237
$330$	−5.12311	−0.282018
$331$	32.4924	1.78595	0.892973	−	0.450111i	$-0.148616\pi$
$331$	32.4924	1.78595	0.892973	+	0.450111i	$0.148616\pi$
$332$	−0.876894	−0.0481258
$333$	−3.12311	−0.171145
$334$	12.8769	0.704592
$335$	−4.00000	−0.218543
$336$	−1.00000	−0.0545545
$337$	−2.63068	−0.143302	−0.0716512	−	0.997430i	$-0.522827\pi$
$337$	−2.63068	−0.143302	−0.0716512	+	0.997430i	$0.522827\pi$
$338$	−3.00000	−0.163178
$339$	9.36932	0.508871
$340$	−5.12311	−0.277839
$341$	42.2462	2.28776
$342$	7.12311	0.385173
$343$	−1.00000	−0.0539949
$344$	6.24621	0.336773
$345$	−1.00000	−0.0538382
$346$	8.00000	0.430083
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	2.00000	0.107211
$349$	−8.63068	−0.461990	−0.230995	−	0.972955i	$-0.574198\pi$
$349$	−8.63068	−0.461990	−0.230995	+	0.972955i	$0.574198\pi$
$350$	1.00000	0.0534522
$351$	−4.00000	−0.213504
$352$	−5.12311	−0.273062
$353$	3.12311	0.166226	0.0831131	−	0.996540i	$-0.473514\pi$
$353$	3.12311	0.166226	0.0831131	+	0.996540i	$0.473514\pi$
$354$	1.12311	0.0596924
$355$	−4.00000	−0.212298
$356$	1.12311	0.0595245
$357$	5.12311	0.271144
$358$	14.2462	0.752936
$359$	−17.6155	−0.929712	−0.464856	−	0.885386i	$-0.653894\pi$
$359$	−17.6155	−0.929712	−0.464856	+	0.885386i	$0.653894\pi$
$360$	−1.00000	−0.0527046
$361$	31.7386	1.67045
$362$	−20.2462	−1.06412
$363$	15.2462	0.800219
$364$	4.00000	0.209657
$365$	−4.87689	−0.255268
$366$	6.00000	0.313625
$367$	−14.2462	−0.743646	−0.371823	−	0.928304i	$-0.621267\pi$
$367$	−14.2462	−0.743646	−0.371823	+	0.928304i	$0.621267\pi$
$368$	−1.00000	−0.0521286
$369$	−2.00000	−0.104116
$370$	3.12311	0.162363
$371$	−3.12311	−0.162144
$372$	8.24621	0.427546
$373$	25.3693	1.31357	0.656787	−	0.754076i	$-0.271915\pi$
$373$	25.3693	1.31357	0.656787	+	0.754076i	$0.271915\pi$
$374$	26.2462	1.35716
$375$	1.00000	0.0516398
$376$	11.1231	0.573630
$377$	−8.00000	−0.412021
$378$	1.00000	0.0514344
$379$	25.6155	1.31578	0.657891	−	0.753113i	$-0.271449\pi$
$379$	25.6155	1.31578	0.657891	+	0.753113i	$0.271449\pi$
$380$	−7.12311	−0.365408
$381$	22.2462	1.13971
$382$	7.36932	0.377047
$383$	20.4924	1.04711	0.523557	−	0.851991i	$-0.324605\pi$
$383$	20.4924	1.04711	0.523557	+	0.851991i	$0.324605\pi$
$384$	−1.00000	−0.0510310
$385$	−5.12311	−0.261098
$386$	−26.4924	−1.34843
$387$	−6.24621	−0.317513
$388$	−6.87689	−0.349121
$389$	−12.2462	−0.620908	−0.310454	−	0.950588i	$-0.600481\pi$
$389$	−12.2462	−0.620908	−0.310454	+	0.950588i	$0.600481\pi$
$390$	4.00000	0.202548
$391$	5.12311	0.259087
$392$	−1.00000	−0.0505076
$393$	−17.1231	−0.863746
$394$	−6.00000	−0.302276
$395$	−7.36932	−0.370791
$396$	5.12311	0.257446
$397$	17.7538	0.891037	0.445519	−	0.895273i	$-0.353019\pi$
$397$	17.7538	0.891037	0.445519	+	0.895273i	$0.353019\pi$
$398$	6.24621	0.313094
$399$	7.12311	0.356601
$400$	1.00000	0.0500000
$401$	−32.7386	−1.63489	−0.817445	−	0.576007i	$-0.804610\pi$
$401$	−32.7386	−1.63489	−0.817445	+	0.576007i	$0.804610\pi$
$402$	4.00000	0.199502
$403$	−32.9848	−1.64309
$404$	5.12311	0.254884
$405$	1.00000	0.0496904
$406$	2.00000	0.0992583
$407$	−16.0000	−0.793091
$408$	5.12311	0.253632
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	2.00000	0.0987730
$411$	−15.1231	−0.745968
$412$	−14.2462	−0.701860
$413$	1.12311	0.0552644
$414$	1.00000	0.0491473
$415$	−0.876894	−0.0430451
$416$	4.00000	0.196116
$417$	5.12311	0.250880
$418$	36.4924	1.78490
$419$	−13.3693	−0.653134	−0.326567	−	0.945174i	$-0.605892\pi$
$419$	−13.3693	−0.653134	−0.326567	+	0.945174i	$0.605892\pi$
$420$	−1.00000	−0.0487950
$421$	−14.4924	−0.706317	−0.353159	−	0.935563i	$-0.614892\pi$
$421$	−14.4924	−0.706317	−0.353159	+	0.935563i	$0.614892\pi$
$422$	6.24621	0.304061
$423$	−11.1231	−0.540824
$424$	−3.12311	−0.151671
$425$	−5.12311	−0.248507
$426$	4.00000	0.193801
$427$	6.00000	0.290360
$428$	2.24621	0.108575
$429$	−20.4924	−0.989383
$430$	6.24621	0.301219
$431$	−0.630683	−0.0303789	−0.0151895	−	0.999885i	$-0.504835\pi$
$431$	−0.630683	−0.0303789	−0.0151895	+	0.999885i	$0.504835\pi$
$432$	1.00000	0.0481125
$433$	−3.36932	−0.161919	−0.0809595	−	0.996717i	$-0.525798\pi$
$433$	−3.36932	−0.161919	−0.0809595	+	0.996717i	$0.525798\pi$
$434$	8.24621	0.395831
$435$	2.00000	0.0958927
$436$	0.246211	0.0117914
$437$	7.12311	0.340744
$438$	4.87689	0.233027
$439$	−28.7386	−1.37162	−0.685810	−	0.727781i	$-0.740552\pi$
$439$	−28.7386	−1.37162	−0.685810	+	0.727781i	$0.740552\pi$
$440$	−5.12311	−0.244234
$441$	1.00000	0.0476190
$442$	−20.4924	−0.974725
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	−3.12311	−0.148216
$445$	1.12311	0.0532403
$446$	15.6155	0.739417
$447$	−14.0000	−0.662177
$448$	−1.00000	−0.0472456
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	−1.00000	−0.0471405
$451$	−10.2462	−0.482475
$452$	9.36932	0.440696
$453$	12.0000	0.563809
$454$	−5.36932	−0.251995
$455$	4.00000	0.187523
$456$	7.12311	0.333570
$457$	−31.6155	−1.47891	−0.739456	−	0.673205i	$-0.764917\pi$
$457$	−31.6155	−1.47891	−0.739456	+	0.673205i	$0.764917\pi$
$458$	18.4924	0.864094
$459$	−5.12311	−0.239126
$460$	−1.00000	−0.0466252
$461$	−31.3693	−1.46101	−0.730507	−	0.682905i	$-0.760716\pi$
$461$	−31.3693	−1.46101	−0.730507	+	0.682905i	$0.760716\pi$
$462$	5.12311	0.238348
$463$	3.50758	0.163011	0.0815055	−	0.996673i	$-0.474027\pi$
$463$	3.50758	0.163011	0.0815055	+	0.996673i	$0.474027\pi$
$464$	2.00000	0.0928477
$465$	8.24621	0.382409
$466$	4.24621	0.196702
$467$	33.8617	1.56693	0.783467	−	0.621433i	$-0.213449\pi$
$467$	33.8617	1.56693	0.783467	+	0.621433i	$0.213449\pi$
$468$	−4.00000	−0.184900
$469$	4.00000	0.184703
$470$	11.1231	0.513071
$471$	6.00000	0.276465
$472$	1.12311	0.0516951
$473$	−32.0000	−1.47136
$474$	7.36932	0.338484
$475$	−7.12311	−0.326831
$476$	5.12311	0.234817
$477$	3.12311	0.142997
$478$	−16.4924	−0.754346
$479$	40.4924	1.85015	0.925073	−	0.379789i	$-0.124004\pi$
$479$	40.4924	1.85015	0.925073	+	0.379789i	$0.124004\pi$
$480$	−1.00000	−0.0456435
$481$	12.4924	0.569605
$482$	−29.1231	−1.32652
$483$	1.00000	0.0455016
$484$	15.2462	0.693010
$485$	−6.87689	−0.312264
$486$	−1.00000	−0.0453609
$487$	3.50758	0.158944	0.0794718	−	0.996837i	$-0.474677\pi$
$487$	3.50758	0.158944	0.0794718	+	0.996837i	$0.474677\pi$
$488$	6.00000	0.271607
$489$	8.49242	0.384041
$490$	−1.00000	−0.0451754
$491$	1.75379	0.0791474	0.0395737	−	0.999217i	$-0.487400\pi$
$491$	1.75379	0.0791474	0.0395737	+	0.999217i	$0.487400\pi$
$492$	−2.00000	−0.0901670
$493$	−10.2462	−0.461466
$494$	−28.4924	−1.28193
$495$	5.12311	0.230266
$496$	8.24621	0.370266
$497$	4.00000	0.179425
$498$	0.876894	0.0392946
$499$	−10.7386	−0.480727	−0.240364	−	0.970683i	$-0.577267\pi$
$499$	−10.7386	−0.480727	−0.240364	+	0.970683i	$0.577267\pi$
$500$	1.00000	0.0447214
$501$	−12.8769	−0.575297
$502$	3.12311	0.139391
$503$	−3.50758	−0.156395	−0.0781976	−	0.996938i	$-0.524916\pi$
$503$	−3.50758	−0.156395	−0.0781976	+	0.996938i	$0.524916\pi$
$504$	1.00000	0.0445435
$505$	5.12311	0.227975
$506$	5.12311	0.227750
$507$	3.00000	0.133235
$508$	22.2462	0.987016
$509$	10.8769	0.482110	0.241055	−	0.970511i	$-0.422507\pi$
$509$	10.8769	0.482110	0.241055	+	0.970511i	$0.422507\pi$
$510$	5.12311	0.226855
$511$	4.87689	0.215741
$512$	−1.00000	−0.0441942
$513$	−7.12311	−0.314493
$514$	8.87689	0.391543
$515$	−14.2462	−0.627763
$516$	−6.24621	−0.274974
$517$	−56.9848	−2.50619
$518$	−3.12311	−0.137221
$519$	−8.00000	−0.351161
$520$	4.00000	0.175412
$521$	−15.8617	−0.694915	−0.347458	−	0.937696i	$-0.612955\pi$
$521$	−15.8617	−0.694915	−0.347458	+	0.937696i	$0.612955\pi$
$522$	−2.00000	−0.0875376
$523$	−10.6307	−0.464847	−0.232424	−	0.972615i	$-0.574666\pi$
$523$	−10.6307	−0.464847	−0.232424	+	0.972615i	$0.574666\pi$
$524$	−17.1231	−0.748026
$525$	−1.00000	−0.0436436
$526$	10.2462	0.446756
$527$	−42.2462	−1.84027
$528$	5.12311	0.222955
$529$	1.00000	0.0434783
$530$	−3.12311	−0.135659
$531$	−1.12311	−0.0487386
$532$	7.12311	0.308826
$533$	8.00000	0.346518
$534$	−1.12311	−0.0486015
$535$	2.24621	0.0971122
$536$	4.00000	0.172774
$537$	−14.2462	−0.614769
$538$	18.8769	0.813841
$539$	5.12311	0.220668
$540$	1.00000	0.0430331
$541$	−45.2311	−1.94463	−0.972317	−	0.233664i	$-0.924929\pi$
$541$	−45.2311	−1.94463	−0.972317	+	0.233664i	$0.924929\pi$
$542$	20.2462	0.869649
$543$	20.2462	0.868848
$544$	5.12311	0.219651
$545$	0.246211	0.0105465
$546$	−4.00000	−0.171184
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−15.1231	−0.646027
$549$	−6.00000	−0.256074
$550$	−5.12311	−0.218450
$551$	−14.2462	−0.606909
$552$	1.00000	0.0425628
$553$	7.36932	0.313375
$554$	−3.75379	−0.159483
$555$	−3.12311	−0.132568
$556$	5.12311	0.217268
$557$	−43.1231	−1.82718	−0.913592	−	0.406631i	$-0.866703\pi$
$557$	−43.1231	−1.82718	−0.913592	+	0.406631i	$0.866703\pi$
$558$	−8.24621	−0.349090
$559$	24.9848	1.05675
$560$	−1.00000	−0.0422577
$561$	−26.2462	−1.10812
$562$	30.4924	1.28625
$563$	−8.87689	−0.374116	−0.187058	−	0.982349i	$-0.559895\pi$
$563$	−8.87689	−0.374116	−0.187058	+	0.982349i	$0.559895\pi$
$564$	−11.1231	−0.468367
$565$	9.36932	0.394170
$566$	−8.87689	−0.373124
$567$	−1.00000	−0.0419961
$568$	4.00000	0.167836
$569$	−40.7386	−1.70785	−0.853926	−	0.520394i	$-0.825785\pi$
$569$	−40.7386	−1.70785	−0.853926	+	0.520394i	$0.825785\pi$
$570$	7.12311	0.298354
$571$	−9.12311	−0.381790	−0.190895	−	0.981610i	$-0.561139\pi$
$571$	−9.12311	−0.381790	−0.190895	+	0.981610i	$0.561139\pi$
$572$	−20.4924	−0.856831
$573$	−7.36932	−0.307858
$574$	−2.00000	−0.0834784
$575$	−1.00000	−0.0417029
$576$	1.00000	0.0416667
$577$	−12.8769	−0.536072	−0.268036	−	0.963409i	$-0.586375\pi$
$577$	−12.8769	−0.536072	−0.268036	+	0.963409i	$0.586375\pi$
$578$	−9.24621	−0.384592
$579$	26.4924	1.10099
$580$	2.00000	0.0830455
$581$	0.876894	0.0363797
$582$	6.87689	0.285056
$583$	16.0000	0.662652
$584$	4.87689	0.201807
$585$	−4.00000	−0.165380
$586$	14.4924	0.598676
$587$	22.7386	0.938524	0.469262	−	0.883059i	$-0.344520\pi$
$587$	22.7386	0.938524	0.469262	+	0.883059i	$0.344520\pi$
$588$	1.00000	0.0412393
$589$	−58.7386	−2.42028
$590$	1.12311	0.0462375
$591$	6.00000	0.246807
$592$	−3.12311	−0.128359
$593$	21.8617	0.897754	0.448877	−	0.893594i	$-0.351824\pi$
$593$	21.8617	0.897754	0.448877	+	0.893594i	$0.351824\pi$
$594$	−5.12311	−0.210204
$595$	5.12311	0.210027
$596$	−14.0000	−0.573462
$597$	−6.24621	−0.255640
$598$	−4.00000	−0.163572
$599$	−8.00000	−0.326871	−0.163436	−	0.986554i	$-0.552258\pi$
$599$	−8.00000	−0.326871	−0.163436	+	0.986554i	$0.552258\pi$
$600$	−1.00000	−0.0408248
$601$	26.4924	1.08065	0.540324	−	0.841457i	$-0.318302\pi$
$601$	26.4924	1.08065	0.540324	+	0.841457i	$0.318302\pi$
$602$	−6.24621	−0.254577
$603$	−4.00000	−0.162893
$604$	12.0000	0.488273
$605$	15.2462	0.619847
$606$	−5.12311	−0.208112
$607$	21.3693	0.867354	0.433677	−	0.901068i	$-0.357216\pi$
$607$	21.3693	0.867354	0.433677	+	0.901068i	$0.357216\pi$
$608$	7.12311	0.288880
$609$	−2.00000	−0.0810441
$610$	6.00000	0.242933
$611$	44.4924	1.79997
$612$	−5.12311	−0.207089
$613$	9.86174	0.398312	0.199156	−	0.979968i	$-0.436180\pi$
$613$	9.86174	0.398312	0.199156	+	0.979968i	$0.436180\pi$
$614$	−2.24621	−0.0906497
$615$	−2.00000	−0.0806478
$616$	5.12311	0.206416
$617$	19.1231	0.769867	0.384934	−	0.922944i	$-0.374224\pi$
$617$	19.1231	0.769867	0.384934	+	0.922944i	$0.374224\pi$
$618$	14.2462	0.573067
$619$	−17.8617	−0.717924	−0.358962	−	0.933352i	$-0.616869\pi$
$619$	−17.8617	−0.717924	−0.358962	+	0.933352i	$0.616869\pi$
$620$	8.24621	0.331176
$621$	−1.00000	−0.0401286
$622$	−26.0000	−1.04251
$623$	−1.12311	−0.0449963
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−5.12311	−0.204760
$627$	−36.4924	−1.45737
$628$	6.00000	0.239426
$629$	16.0000	0.637962
$630$	1.00000	0.0398410
$631$	17.6155	0.701263	0.350632	−	0.936513i	$-0.385967\pi$
$631$	17.6155	0.701263	0.350632	+	0.936513i	$0.385967\pi$
$632$	7.36932	0.293136
$633$	−6.24621	−0.248265
$634$	−18.4924	−0.734428
$635$	22.2462	0.882814
$636$	3.12311	0.123839
$637$	−4.00000	−0.158486
$638$	−10.2462	−0.405651
$639$	−4.00000	−0.158238
$640$	−1.00000	−0.0395285
$641$	6.00000	0.236986	0.118493	−	0.992955i	$-0.462194\pi$
$641$	6.00000	0.236986	0.118493	+	0.992955i	$0.462194\pi$
$642$	−2.24621	−0.0886509
$643$	−13.3693	−0.527234	−0.263617	−	0.964627i	$-0.584916\pi$
$643$	−13.3693	−0.527234	−0.263617	+	0.964627i	$0.584916\pi$
$644$	1.00000	0.0394055
$645$	−6.24621	−0.245944
$646$	−36.4924	−1.43578
$647$	41.8617	1.64575	0.822877	−	0.568219i	$-0.192367\pi$
$647$	41.8617	1.64575	0.822877	+	0.568219i	$0.192367\pi$
$648$	−1.00000	−0.0392837
$649$	−5.75379	−0.225856
$650$	4.00000	0.156893
$651$	−8.24621	−0.323195
$652$	8.49242	0.332589
$653$	−12.7386	−0.498501	−0.249251	−	0.968439i	$-0.580184\pi$
$653$	−12.7386	−0.498501	−0.249251	+	0.968439i	$0.580184\pi$
$654$	−0.246211	−0.00962762
$655$	−17.1231	−0.669055
$656$	−2.00000	−0.0780869
$657$	−4.87689	−0.190266
$658$	−11.1231	−0.433624
$659$	1.12311	0.0437500	0.0218750	−	0.999761i	$-0.493036\pi$
$659$	1.12311	0.0437500	0.0218750	+	0.999761i	$0.493036\pi$
$660$	5.12311	0.199417
$661$	44.2462	1.72098	0.860489	−	0.509469i	$-0.170158\pi$
$661$	44.2462	1.72098	0.860489	+	0.509469i	$0.170158\pi$
$662$	−32.4924	−1.26285
$663$	20.4924	0.795860
$664$	0.876894	0.0340301
$665$	7.12311	0.276222
$666$	3.12311	0.121018
$667$	−2.00000	−0.0774403
$668$	−12.8769	−0.498222
$669$	−15.6155	−0.603731
$670$	4.00000	0.154533
$671$	−30.7386	−1.18665
$672$	1.00000	0.0385758
$673$	−38.4924	−1.48377	−0.741887	−	0.670525i	$-0.766069\pi$
$673$	−38.4924	−1.48377	−0.741887	+	0.670525i	$0.766069\pi$
$674$	2.63068	0.101330
$675$	1.00000	0.0384900
$676$	3.00000	0.115385
$677$	−36.2462	−1.39306	−0.696528	−	0.717530i	$-0.745273\pi$
$677$	−36.2462	−1.39306	−0.696528	+	0.717530i	$0.745273\pi$
$678$	−9.36932	−0.359826
$679$	6.87689	0.263911
$680$	5.12311	0.196462
$681$	5.36932	0.205753
$682$	−42.2462	−1.61769
$683$	−32.4924	−1.24329	−0.621644	−	0.783300i	$-0.713535\pi$
$683$	−32.4924	−1.24329	−0.621644	+	0.783300i	$0.713535\pi$
$684$	−7.12311	−0.272359
$685$	−15.1231	−0.577824
$686$	1.00000	0.0381802
$687$	−18.4924	−0.705530
$688$	−6.24621	−0.238135
$689$	−12.4924	−0.475923
$690$	1.00000	0.0380693
$691$	8.63068	0.328327	0.164163	−	0.986433i	$-0.447508\pi$
$691$	8.63068	0.328327	0.164163	+	0.986433i	$0.447508\pi$
$692$	−8.00000	−0.304114
$693$	−5.12311	−0.194611
$694$	12.0000	0.455514
$695$	5.12311	0.194330
$696$	−2.00000	−0.0758098
$697$	10.2462	0.388103
$698$	8.63068	0.326676
$699$	−4.24621	−0.160606
$700$	−1.00000	−0.0377964
$701$	37.2311	1.40620	0.703099	−	0.711092i	$-0.251799\pi$
$701$	37.2311	1.40620	0.703099	+	0.711092i	$0.251799\pi$
$702$	4.00000	0.150970
$703$	22.2462	0.839032
$704$	5.12311	0.193084
$705$	−11.1231	−0.418920
$706$	−3.12311	−0.117540
$707$	−5.12311	−0.192674
$708$	−1.12311	−0.0422089
$709$	45.2311	1.69869	0.849344	−	0.527840i	$-0.176998\pi$
$709$	45.2311	1.69869	0.849344	+	0.527840i	$0.176998\pi$
$710$	4.00000	0.150117
$711$	−7.36932	−0.276371
$712$	−1.12311	−0.0420902
$713$	−8.24621	−0.308823
$714$	−5.12311	−0.191727
$715$	−20.4924	−0.766373
$716$	−14.2462	−0.532406
$717$	16.4924	0.615921
$718$	17.6155	0.657406
$719$	34.4924	1.28635	0.643175	−	0.765719i	$-0.277617\pi$
$719$	34.4924	1.28635	0.643175	+	0.765719i	$0.277617\pi$
$720$	1.00000	0.0372678
$721$	14.2462	0.530557
$722$	−31.7386	−1.18119
$723$	29.1231	1.08310
$724$	20.2462	0.752445
$725$	2.00000	0.0742781
$726$	−15.2462	−0.565840
$727$	−38.2462	−1.41847	−0.709237	−	0.704970i	$-0.750960\pi$
$727$	−38.2462	−1.41847	−0.709237	+	0.704970i	$0.750960\pi$
$728$	−4.00000	−0.148250
$729$	1.00000	0.0370370
$730$	4.87689	0.180502
$731$	32.0000	1.18356
$732$	−6.00000	−0.221766
$733$	40.2462	1.48653	0.743264	−	0.668998i	$-0.233277\pi$
$733$	40.2462	1.48653	0.743264	+	0.668998i	$0.233277\pi$
$734$	14.2462	0.525837
$735$	1.00000	0.0368856
$736$	1.00000	0.0368605
$737$	−20.4924	−0.754848
$738$	2.00000	0.0736210
$739$	−50.7386	−1.86645	−0.933225	−	0.359291i	$-0.883018\pi$
$739$	−50.7386	−1.86645	−0.933225	+	0.359291i	$0.883018\pi$
$740$	−3.12311	−0.114808
$741$	28.4924	1.04670
$742$	3.12311	0.114653
$743$	44.4924	1.63227	0.816134	−	0.577862i	$-0.196113\pi$
$743$	44.4924	1.63227	0.816134	+	0.577862i	$0.196113\pi$
$744$	−8.24621	−0.302321
$745$	−14.0000	−0.512920
$746$	−25.3693	−0.928837
$747$	−0.876894	−0.0320839
$748$	−26.2462	−0.959657
$749$	−2.24621	−0.0820748
$750$	−1.00000	−0.0365148
$751$	41.6155	1.51857	0.759286	−	0.650757i	$-0.225548\pi$
$751$	41.6155	1.51857	0.759286	+	0.650757i	$0.225548\pi$
$752$	−11.1231	−0.405618
$753$	−3.12311	−0.113812
$754$	8.00000	0.291343
$755$	12.0000	0.436725
$756$	−1.00000	−0.0363696
$757$	25.8617	0.939961	0.469980	−	0.882677i	$-0.344261\pi$
$757$	25.8617	0.939961	0.469980	+	0.882677i	$0.344261\pi$
$758$	−25.6155	−0.930398
$759$	−5.12311	−0.185957
$760$	7.12311	0.258382
$761$	−34.4924	−1.25035	−0.625175	−	0.780485i	$-0.714972\pi$
$761$	−34.4924	−1.25035	−0.625175	+	0.780485i	$0.714972\pi$
$762$	−22.2462	−0.805895
$763$	−0.246211	−0.00891345
$764$	−7.36932	−0.266613
$765$	−5.12311	−0.185226
$766$	−20.4924	−0.740421
$767$	4.49242	0.162212
$768$	1.00000	0.0360844
$769$	25.6155	0.923720	0.461860	−	0.886953i	$-0.347182\pi$
$769$	25.6155	0.923720	0.461860	+	0.886953i	$0.347182\pi$
$770$	5.12311	0.184624
$771$	−8.87689	−0.319694
$772$	26.4924	0.953483
$773$	22.4924	0.808996	0.404498	−	0.914539i	$-0.367446\pi$
$773$	22.4924	0.808996	0.404498	+	0.914539i	$0.367446\pi$
$774$	6.24621	0.224515
$775$	8.24621	0.296213
$776$	6.87689	0.246866
$777$	3.12311	0.112041
$778$	12.2462	0.439048
$779$	14.2462	0.510423
$780$	−4.00000	−0.143223
$781$	−20.4924	−0.733277
$782$	−5.12311	−0.183202
$783$	2.00000	0.0714742
$784$	1.00000	0.0357143
$785$	6.00000	0.214149
$786$	17.1231	0.610761
$787$	−13.3693	−0.476565	−0.238282	−	0.971196i	$-0.576584\pi$
$787$	−13.3693	−0.476565	−0.238282	+	0.971196i	$0.576584\pi$
$788$	6.00000	0.213741
$789$	−10.2462	−0.364775
$790$	7.36932	0.262189
$791$	−9.36932	−0.333135
$792$	−5.12311	−0.182042
$793$	24.0000	0.852265
$794$	−17.7538	−0.630058
$795$	3.12311	0.110765
$796$	−6.24621	−0.221391
$797$	−32.2462	−1.14222	−0.571110	−	0.820874i	$-0.693487\pi$
$797$	−32.2462	−1.14222	−0.571110	+	0.820874i	$0.693487\pi$
$798$	−7.12311	−0.252155
$799$	56.9848	2.01598
$800$	−1.00000	−0.0353553
$801$	1.12311	0.0396830
$802$	32.7386	1.15604
$803$	−24.9848	−0.881696
$804$	−4.00000	−0.141069
$805$	1.00000	0.0352454
$806$	32.9848	1.16184
$807$	−18.8769	−0.664498
$808$	−5.12311	−0.180230
$809$	28.7386	1.01040	0.505198	−	0.863003i	$-0.331419\pi$
$809$	28.7386	1.01040	0.505198	+	0.863003i	$0.331419\pi$
$810$	−1.00000	−0.0351364
$811$	−25.1231	−0.882192	−0.441096	−	0.897460i	$-0.645410\pi$
$811$	−25.1231	−0.882192	−0.441096	+	0.897460i	$0.645410\pi$
$812$	−2.00000	−0.0701862
$813$	−20.2462	−0.710066
$814$	16.0000	0.560800
$815$	8.49242	0.297477
$816$	−5.12311	−0.179345
$817$	44.4924	1.55659
$818$	22.0000	0.769212
$819$	4.00000	0.139771
$820$	−2.00000	−0.0698430
$821$	50.4924	1.76220	0.881099	−	0.472932i	$-0.156804\pi$
$821$	50.4924	1.76220	0.881099	+	0.472932i	$0.156804\pi$
$822$	15.1231	0.527479
$823$	26.7386	0.932050	0.466025	−	0.884772i	$-0.345686\pi$
$823$	26.7386	0.932050	0.466025	+	0.884772i	$0.345686\pi$
$824$	14.2462	0.496290
$825$	5.12311	0.178364
$826$	−1.12311	−0.0390778
$827$	−46.7386	−1.62526	−0.812631	−	0.582779i	$-0.801965\pi$
$827$	−46.7386	−1.62526	−0.812631	+	0.582779i	$0.801965\pi$
$828$	−1.00000	−0.0347524
$829$	6.87689	0.238844	0.119422	−	0.992844i	$-0.461896\pi$
$829$	6.87689	0.238844	0.119422	+	0.992844i	$0.461896\pi$
$830$	0.876894	0.0304374
$831$	3.75379	0.130217
$832$	−4.00000	−0.138675
$833$	−5.12311	−0.177505
$834$	−5.12311	−0.177399
$835$	−12.8769	−0.445623
$836$	−36.4924	−1.26212
$837$	8.24621	0.285031
$838$	13.3693	0.461835
$839$	35.2311	1.21631	0.608156	−	0.793818i	$-0.291910\pi$
$839$	35.2311	1.21631	0.608156	+	0.793818i	$0.291910\pi$
$840$	1.00000	0.0345033
$841$	−25.0000	−0.862069
$842$	14.4924	0.499442
$843$	−30.4924	−1.05021
$844$	−6.24621	−0.215003
$845$	3.00000	0.103203
$846$	11.1231	0.382420
$847$	−15.2462	−0.523866
$848$	3.12311	0.107248
$849$	8.87689	0.304654
$850$	5.12311	0.175721
$851$	3.12311	0.107059
$852$	−4.00000	−0.137038
$853$	−14.2462	−0.487781	−0.243890	−	0.969803i	$-0.578424\pi$
$853$	−14.2462	−0.487781	−0.243890	+	0.969803i	$0.578424\pi$
$854$	−6.00000	−0.205316
$855$	−7.12311	−0.243605
$856$	−2.24621	−0.0767739
$857$	−21.8617	−0.746783	−0.373391	−	0.927674i	$-0.621805\pi$
$857$	−21.8617	−0.746783	−0.373391	+	0.927674i	$0.621805\pi$
$858$	20.4924	0.699600
$859$	8.63068	0.294475	0.147238	−	0.989101i	$-0.452962\pi$
$859$	8.63068	0.294475	0.147238	+	0.989101i	$0.452962\pi$
$860$	−6.24621	−0.212994
$861$	2.00000	0.0681598
$862$	0.630683	0.0214812
$863$	−9.75379	−0.332023	−0.166011	−	0.986124i	$-0.553089\pi$
$863$	−9.75379	−0.332023	−0.166011	+	0.986124i	$0.553089\pi$
$864$	−1.00000	−0.0340207
$865$	−8.00000	−0.272008
$866$	3.36932	0.114494
$867$	9.24621	0.314018
$868$	−8.24621	−0.279895
$869$	−37.7538	−1.28071
$870$	−2.00000	−0.0678064
$871$	16.0000	0.542139
$872$	−0.246211	−0.00833777
$873$	−6.87689	−0.232748
$874$	−7.12311	−0.240943
$875$	−1.00000	−0.0338062
$876$	−4.87689	−0.164775
$877$	44.2462	1.49409	0.747044	−	0.664774i	$-0.231472\pi$
$877$	44.2462	1.49409	0.747044	+	0.664774i	$0.231472\pi$
$878$	28.7386	0.969882
$879$	−14.4924	−0.488817
$880$	5.12311	0.172700
$881$	−34.8769	−1.17503	−0.587516	−	0.809212i	$-0.699894\pi$
$881$	−34.8769	−1.17503	−0.587516	+	0.809212i	$0.699894\pi$
$882$	−1.00000	−0.0336718
$883$	−28.0000	−0.942275	−0.471138	−	0.882060i	$-0.656156\pi$
$883$	−28.0000	−0.942275	−0.471138	+	0.882060i	$0.656156\pi$
$884$	20.4924	0.689235
$885$	−1.12311	−0.0377528
$886$	−4.00000	−0.134383
$887$	−15.1231	−0.507784	−0.253892	−	0.967233i	$-0.581711\pi$
$887$	−15.1231	−0.507784	−0.253892	+	0.967233i	$0.581711\pi$
$888$	3.12311	0.104805
$889$	−22.2462	−0.746114
$890$	−1.12311	−0.0376466
$891$	5.12311	0.171630
$892$	−15.6155	−0.522847
$893$	79.2311	2.65137
$894$	14.0000	0.468230
$895$	−14.2462	−0.476198
$896$	1.00000	0.0334077
$897$	4.00000	0.133556
$898$	18.0000	0.600668
$899$	16.4924	0.550053
$900$	1.00000	0.0333333
$901$	−16.0000	−0.533037
$902$	10.2462	0.341162
$903$	6.24621	0.207861
$904$	−9.36932	−0.311619
$905$	20.2462	0.673007
$906$	−12.0000	−0.398673
$907$	19.5076	0.647738	0.323869	−	0.946102i	$-0.395016\pi$
$907$	19.5076	0.647738	0.323869	+	0.946102i	$0.395016\pi$
$908$	5.36932	0.178187
$909$	5.12311	0.169923
$910$	−4.00000	−0.132599
$911$	51.8617	1.71826	0.859128	−	0.511761i	$-0.171007\pi$
$911$	51.8617	1.71826	0.859128	+	0.511761i	$0.171007\pi$
$912$	−7.12311	−0.235870
$913$	−4.49242	−0.148677
$914$	31.6155	1.04575
$915$	−6.00000	−0.198354
$916$	−18.4924	−0.611007
$917$	17.1231	0.565455
$918$	5.12311	0.169088
$919$	19.8617	0.655178	0.327589	−	0.944820i	$-0.393764\pi$
$919$	19.8617	0.655178	0.327589	+	0.944820i	$0.393764\pi$
$920$	1.00000	0.0329690
$921$	2.24621	0.0740152
$922$	31.3693	1.03309
$923$	16.0000	0.526646
$924$	−5.12311	−0.168538
$925$	−3.12311	−0.102687
$926$	−3.50758	−0.115266
$927$	−14.2462	−0.467907
$928$	−2.00000	−0.0656532
$929$	35.7538	1.17304	0.586522	−	0.809933i	$-0.300497\pi$
$929$	35.7538	1.17304	0.586522	+	0.809933i	$0.300497\pi$
$930$	−8.24621	−0.270404
$931$	−7.12311	−0.233450
$932$	−4.24621	−0.139089
$933$	26.0000	0.851202
$934$	−33.8617	−1.10799
$935$	−26.2462	−0.858343
$936$	4.00000	0.130744
$937$	−53.1231	−1.73546	−0.867728	−	0.497039i	$-0.834421\pi$
$937$	−53.1231	−1.73546	−0.867728	+	0.497039i	$0.834421\pi$
$938$	−4.00000	−0.130605
$939$	5.12311	0.167186
$940$	−11.1231	−0.362796
$941$	−32.7386	−1.06725	−0.533624	−	0.845722i	$-0.679170\pi$
$941$	−32.7386	−1.06725	−0.533624	+	0.845722i	$0.679170\pi$
$942$	−6.00000	−0.195491
$943$	2.00000	0.0651290
$944$	−1.12311	−0.0365540
$945$	−1.00000	−0.0325300
$946$	32.0000	1.04041
$947$	28.9848	0.941881	0.470940	−	0.882165i	$-0.343915\pi$
$947$	28.9848	0.941881	0.470940	+	0.882165i	$0.343915\pi$
$948$	−7.36932	−0.239344
$949$	19.5076	0.633243
$950$	7.12311	0.231104
$951$	18.4924	0.599658
$952$	−5.12311	−0.166041
$953$	−5.36932	−0.173929	−0.0869646	−	0.996211i	$-0.527717\pi$
$953$	−5.36932	−0.173929	−0.0869646	+	0.996211i	$0.527717\pi$
$954$	−3.12311	−0.101114
$955$	−7.36932	−0.238465
$956$	16.4924	0.533403
$957$	10.2462	0.331213
$958$	−40.4924	−1.30825
$959$	15.1231	0.488351
$960$	1.00000	0.0322749
$961$	37.0000	1.19355
$962$	−12.4924	−0.402772
$963$	2.24621	0.0723831
$964$	29.1231	0.937992
$965$	26.4924	0.852821
$966$	−1.00000	−0.0321745
$967$	19.5076	0.627321	0.313661	−	0.949535i	$-0.398445\pi$
$967$	19.5076	0.627321	0.313661	+	0.949535i	$0.398445\pi$
$968$	−15.2462	−0.490032
$969$	36.4924	1.17231
$970$	6.87689	0.220804
$971$	36.8769	1.18344	0.591718	−	0.806145i	$-0.298450\pi$
$971$	36.8769	1.18344	0.591718	+	0.806145i	$0.298450\pi$
$972$	1.00000	0.0320750
$973$	−5.12311	−0.164239
$974$	−3.50758	−0.112390
$975$	−4.00000	−0.128103
$976$	−6.00000	−0.192055
$977$	−29.8617	−0.955362	−0.477681	−	0.878533i	$-0.658522\pi$
$977$	−29.8617	−0.955362	−0.477681	+	0.878533i	$0.658522\pi$
$978$	−8.49242	−0.271558
$979$	5.75379	0.183892
$980$	1.00000	0.0319438
$981$	0.246211	0.00786092
$982$	−1.75379	−0.0559656
$983$	−4.49242	−0.143286	−0.0716430	−	0.997430i	$-0.522824\pi$
$983$	−4.49242	−0.143286	−0.0716430	+	0.997430i	$0.522824\pi$
$984$	2.00000	0.0637577
$985$	6.00000	0.191176
$986$	10.2462	0.326306
$987$	11.1231	0.354052
$988$	28.4924	0.906465
$989$	6.24621	0.198618
$990$	−5.12311	−0.162823
$991$	−32.4924	−1.03216	−0.516078	−	0.856542i	$-0.672609\pi$
$991$	−32.4924	−1.03216	−0.516078	+	0.856542i	$0.672609\pi$
$992$	−8.24621	−0.261817
$993$	32.4924	1.03112
$994$	−4.00000	−0.126872
$995$	−6.24621	−0.198018
$996$	−0.876894	−0.0277855
$997$	−16.4924	−0.522320	−0.261160	−	0.965295i	$-0.584105\pi$
$997$	−16.4924	−0.522320	−0.261160	+	0.965295i	$0.584105\pi$
$998$	10.7386	0.339926
$999$	−3.12311	−0.0988107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4830.2.a.br.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4830.2.a.br.1.2	✓	2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4830 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4830.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +5.12311 q^{11} +1.00000 q^{12} -4.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -5.12311 q^{17} -1.00000 q^{18} -7.12311 q^{19} +1.00000 q^{20} -1.00000 q^{21} -5.12311 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +4.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} +2.00000 q^{29} -1.00000 q^{30} +8.24621 q^{31} -1.00000 q^{32} +5.12311 q^{33} +5.12311 q^{34} -1.00000 q^{35} +1.00000 q^{36} -3.12311 q^{37} +7.12311 q^{38} -4.00000 q^{39} -1.00000 q^{40} -2.00000 q^{41} +1.00000 q^{42} -6.24621 q^{43} +5.12311 q^{44} +1.00000 q^{45} +1.00000 q^{46} -11.1231 q^{47} +1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -5.12311 q^{51} -4.00000 q^{52} +3.12311 q^{53} -1.00000 q^{54} +5.12311 q^{55} +1.00000 q^{56} -7.12311 q^{57} -2.00000 q^{58} -1.12311 q^{59} +1.00000 q^{60} -6.00000 q^{61} -8.24621 q^{62} -1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -5.12311 q^{66} -4.00000 q^{67} -5.12311 q^{68} -1.00000 q^{69} +1.00000 q^{70} -4.00000 q^{71} -1.00000 q^{72} -4.87689 q^{73} +3.12311 q^{74} +1.00000 q^{75} -7.12311 q^{76} -5.12311 q^{77} +4.00000 q^{78} -7.36932 q^{79} +1.00000 q^{80} +1.00000 q^{81} +2.00000 q^{82} -0.876894 q^{83} -1.00000 q^{84} -5.12311 q^{85} +6.24621 q^{86} +2.00000 q^{87} -5.12311 q^{88} +1.12311 q^{89} -1.00000 q^{90} +4.00000 q^{91} -1.00000 q^{92} +8.24621 q^{93} +11.1231 q^{94} -7.12311 q^{95} -1.00000 q^{96} -6.87689 q^{97} -1.00000 q^{98} +5.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} - 2 q^{10} + 2 q^{11} + 2 q^{12} - 8 q^{13} + 2 q^{14} + 2 q^{15} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 6 q^{19} + 2 q^{20} - 2 q^{21} - 2 q^{22} - 2 q^{23} - 2 q^{24} + 2 q^{25} + 8 q^{26} + 2 q^{27} - 2 q^{28} + 4 q^{29} - 2 q^{30} - 2 q^{32} + 2 q^{33} + 2 q^{34} - 2 q^{35} + 2 q^{36} + 2 q^{37} + 6 q^{38} - 8 q^{39} - 2 q^{40} - 4 q^{41} + 2 q^{42} + 4 q^{43} + 2 q^{44} + 2 q^{45} + 2 q^{46} - 14 q^{47} + 2 q^{48} + 2 q^{49} - 2 q^{50} - 2 q^{51} - 8 q^{52} - 2 q^{53} - 2 q^{54} + 2 q^{55} + 2 q^{56} - 6 q^{57} - 4 q^{58} + 6 q^{59} + 2 q^{60} - 12 q^{61} - 2 q^{63} + 2 q^{64} - 8 q^{65} - 2 q^{66} - 8 q^{67} - 2 q^{68} - 2 q^{69} + 2 q^{70} - 8 q^{71} - 2 q^{72} - 18 q^{73} - 2 q^{74} + 2 q^{75} - 6 q^{76} - 2 q^{77} + 8 q^{78} + 10 q^{79} + 2 q^{80} + 2 q^{81} + 4 q^{82} - 10 q^{83} - 2 q^{84} - 2 q^{85} - 4 q^{86} + 4 q^{87} - 2 q^{88} - 6 q^{89} - 2 q^{90} + 8 q^{91} - 2 q^{92} + 14 q^{94} - 6 q^{95} - 2 q^{96} - 22 q^{97} - 2 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 - 2 * q^6 - 2 * q^7 - 2 * q^8 + 2 * q^9 - 2 * q^10 + 2 * q^11 + 2 * q^12 - 8 * q^13 + 2 * q^14 + 2 * q^15 + 2 * q^16 - 2 * q^17 - 2 * q^18 - 6 * q^19 + 2 * q^20 - 2 * q^21 - 2 * q^22 - 2 * q^23 - 2 * q^24 + 2 * q^25 + 8 * q^26 + 2 * q^27 - 2 * q^28 + 4 * q^29 - 2 * q^30 - 2 * q^32 + 2 * q^33 + 2 * q^34 - 2 * q^35 + 2 * q^36 + 2 * q^37 + 6 * q^38 - 8 * q^39 - 2 * q^40 - 4 * q^41 + 2 * q^42 + 4 * q^43 + 2 * q^44 + 2 * q^45 + 2 * q^46 - 14 * q^47 + 2 * q^48 + 2 * q^49 - 2 * q^50 - 2 * q^51 - 8 * q^52 - 2 * q^53 - 2 * q^54 + 2 * q^55 + 2 * q^56 - 6 * q^57 - 4 * q^58 + 6 * q^59 + 2 * q^60 - 12 * q^61 - 2 * q^63 + 2 * q^64 - 8 * q^65 - 2 * q^66 - 8 * q^67 - 2 * q^68 - 2 * q^69 + 2 * q^70 - 8 * q^71 - 2 * q^72 - 18 * q^73 - 2 * q^74 + 2 * q^75 - 6 * q^76 - 2 * q^77 + 8 * q^78 + 10 * q^79 + 2 * q^80 + 2 * q^81 + 4 * q^82 - 10 * q^83 - 2 * q^84 - 2 * q^85 - 4 * q^86 + 4 * q^87 - 2 * q^88 - 6 * q^89 - 2 * q^90 + 8 * q^91 - 2 * q^92 + 14 * q^94 - 6 * q^95 - 2 * q^96 - 22 * q^97 - 2 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

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