LMFDB - Embedded newform 4002.2.a.bj.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 26x^{6} + 4x^{5} + 209x^{4} + 113x^{3} - 436x^{2} - 360x - 16 $ x^8 - x^7 - 26x^6 + 4x^5 + 209x^4 + 113x^3 - 436x^2 - 360x - 16
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			1.5
Root			$-0.0471692$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.0471692 q^{5} -1.00000 q^{6} -3.98374 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.0471692 q^{5} -1.00000 q^{6} -3.98374 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.0471692 q^{10} -5.70443 q^{11} -1.00000 q^{12} -2.39975 q^{13} -3.98374 q^{14} +0.0471692 q^{15} +1.00000 q^{16} +1.20250 q^{17} +1.00000 q^{18} -3.98374 q^{19} -0.0471692 q^{20} +3.98374 q^{21} -5.70443 q^{22} +1.00000 q^{23} -1.00000 q^{24} -4.99778 q^{25} -2.39975 q^{26} -1.00000 q^{27} -3.98374 q^{28} +1.00000 q^{29} +0.0471692 q^{30} +7.22454 q^{31} +1.00000 q^{32} +5.70443 q^{33} +1.20250 q^{34} +0.187910 q^{35} +1.00000 q^{36} +6.50564 q^{37} -3.98374 q^{38} +2.39975 q^{39} -0.0471692 q^{40} +7.53340 q^{41} +3.98374 q^{42} +9.23926 q^{43} -5.70443 q^{44} -0.0471692 q^{45} +1.00000 q^{46} +5.96195 q^{47} -1.00000 q^{48} +8.87020 q^{49} -4.99778 q^{50} -1.20250 q^{51} -2.39975 q^{52} +5.81391 q^{53} -1.00000 q^{54} +0.269073 q^{55} -3.98374 q^{56} +3.98374 q^{57} +1.00000 q^{58} +4.24309 q^{59} +0.0471692 q^{60} +0.600036 q^{61} +7.22454 q^{62} -3.98374 q^{63} +1.00000 q^{64} +0.113194 q^{65} +5.70443 q^{66} -8.33828 q^{67} +1.20250 q^{68} -1.00000 q^{69} +0.187910 q^{70} +1.23529 q^{71} +1.00000 q^{72} +5.54398 q^{73} +6.50564 q^{74} +4.99778 q^{75} -3.98374 q^{76} +22.7250 q^{77} +2.39975 q^{78} -11.1562 q^{79} -0.0471692 q^{80} +1.00000 q^{81} +7.53340 q^{82} +3.18624 q^{83} +3.98374 q^{84} -0.0567210 q^{85} +9.23926 q^{86} -1.00000 q^{87} -5.70443 q^{88} -11.3402 q^{89} -0.0471692 q^{90} +9.55998 q^{91} +1.00000 q^{92} -7.22454 q^{93} +5.96195 q^{94} +0.187910 q^{95} -1.00000 q^{96} -8.49728 q^{97} +8.87020 q^{98} -5.70443 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} + q^{10} + 3 q^{11} - 8 q^{12} + 9 q^{13} - q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + q^{20} + 3 q^{22} + 8 q^{23} - 8 q^{24} + 13 q^{25} + 9 q^{26} - 8 q^{27} + 8 q^{29} - q^{30} - 9 q^{31} + 8 q^{32} - 3 q^{33} + 8 q^{34} - 2 q^{35} + 8 q^{36} + 7 q^{37} - 9 q^{39} + q^{40} + 3 q^{41} + 16 q^{43} + 3 q^{44} + q^{45} + 8 q^{46} + 24 q^{47} - 8 q^{48} + 6 q^{49} + 13 q^{50} - 8 q^{51} + 9 q^{52} + 8 q^{53} - 8 q^{54} + 13 q^{55} + 8 q^{58} - 3 q^{59} - q^{60} + 31 q^{61} - 9 q^{62} + 8 q^{64} + 13 q^{65} - 3 q^{66} - 11 q^{67} + 8 q^{68} - 8 q^{69} - 2 q^{70} + 7 q^{71} + 8 q^{72} + 14 q^{73} + 7 q^{74} - 13 q^{75} + 10 q^{77} - 9 q^{78} + 12 q^{79} + q^{80} + 8 q^{81} + 3 q^{82} - 8 q^{83} + 22 q^{85} + 16 q^{86} - 8 q^{87} + 3 q^{88} - 12 q^{89} + q^{90} + 28 q^{91} + 8 q^{92} + 9 q^{93} + 24 q^{94} - 2 q^{95} - 8 q^{96} + 16 q^{97} + 6 q^{98} + 3 q^{99}+O(q^{100}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9 + q^10 + 3 * q^11 - 8 * q^12 + 9 * q^13 - q^15 + 8 * q^16 + 8 * q^17 + 8 * q^18 + q^20 + 3 * q^22 + 8 * q^23 - 8 * q^24 + 13 * q^25 + 9 * q^26 - 8 * q^27 + 8 * q^29 - q^30 - 9 * q^31 + 8 * q^32 - 3 * q^33 + 8 * q^34 - 2 * q^35 + 8 * q^36 + 7 * q^37 - 9 * q^39 + q^40 + 3 * q^41 + 16 * q^43 + 3 * q^44 + q^45 + 8 * q^46 + 24 * q^47 - 8 * q^48 + 6 * q^49 + 13 * q^50 - 8 * q^51 + 9 * q^52 + 8 * q^53 - 8 * q^54 + 13 * q^55 + 8 * q^58 - 3 * q^59 - q^60 + 31 * q^61 - 9 * q^62 + 8 * q^64 + 13 * q^65 - 3 * q^66 - 11 * q^67 + 8 * q^68 - 8 * q^69 - 2 * q^70 + 7 * q^71 + 8 * q^72 + 14 * q^73 + 7 * q^74 - 13 * q^75 + 10 * q^77 - 9 * q^78 + 12 * q^79 + q^80 + 8 * q^81 + 3 * q^82 - 8 * q^83 + 22 * q^85 + 16 * q^86 - 8 * q^87 + 3 * q^88 - 12 * q^89 + q^90 + 28 * q^91 + 8 * q^92 + 9 * q^93 + 24 * q^94 - 2 * q^95 - 8 * q^96 + 16 * q^97 + 6 * q^98 + 3 * 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
$5$	−0.0471692	−0.0210947	−0.0105473	−	0.999944i	$-0.503357\pi$
$5$	−0.0471692	−0.0210947	−0.0105473	+	0.999944i	$0.503357\pi$
$6$	−1.00000	−0.408248
$7$	−3.98374	−1.50571	−0.752857	−	0.658185i	$-0.771325\pi$
$7$	−3.98374	−1.50571	−0.752857	+	0.658185i	$0.771325\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−0.0471692	−0.0149162
$11$	−5.70443	−1.71995	−0.859975	−	0.510336i	$-0.829521\pi$
$11$	−5.70443	−1.71995	−0.859975	+	0.510336i	$0.829521\pi$
$12$	−1.00000	−0.288675
$13$	−2.39975	−0.665570	−0.332785	−	0.943003i	$-0.607988\pi$
$13$	−2.39975	−0.665570	−0.332785	+	0.943003i	$0.607988\pi$
$14$	−3.98374	−1.06470
$15$	0.0471692	0.0121790
$16$	1.00000	0.250000
$17$	1.20250	0.291650	0.145825	−	0.989310i	$-0.453416\pi$
$17$	1.20250	0.291650	0.145825	+	0.989310i	$0.453416\pi$
$18$	1.00000	0.235702
$19$	−3.98374	−0.913933	−0.456967	−	0.889484i	$-0.651064\pi$
$19$	−3.98374	−0.913933	−0.456967	+	0.889484i	$0.651064\pi$
$20$	−0.0471692	−0.0105473
$21$	3.98374	0.869324
$22$	−5.70443	−1.21619
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	−4.99778	−0.999555
$26$	−2.39975	−0.470629
$27$	−1.00000	−0.192450
$28$	−3.98374	−0.752857
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0.0471692	0.00861187
$31$	7.22454	1.29757	0.648783	−	0.760973i	$-0.275278\pi$
$31$	7.22454	1.29757	0.648783	+	0.760973i	$0.275278\pi$
$32$	1.00000	0.176777
$33$	5.70443	0.993014
$34$	1.20250	0.206227
$35$	0.187910	0.0317625
$36$	1.00000	0.166667
$37$	6.50564	1.06952	0.534760	−	0.845004i	$-0.320402\pi$
$37$	6.50564	1.06952	0.534760	+	0.845004i	$0.320402\pi$
$38$	−3.98374	−0.646248
$39$	2.39975	0.384267
$40$	−0.0471692	−0.00745810
$41$	7.53340	1.17652	0.588260	−	0.808672i	$-0.299813\pi$
$41$	7.53340	1.17652	0.588260	+	0.808672i	$0.299813\pi$
$42$	3.98374	0.614705
$43$	9.23926	1.40897	0.704487	−	0.709717i	$-0.251177\pi$
$43$	9.23926	1.40897	0.704487	+	0.709717i	$0.251177\pi$
$44$	−5.70443	−0.859975
$45$	−0.0471692	−0.00703156
$46$	1.00000	0.147442
$47$	5.96195	0.869640	0.434820	−	0.900517i	$-0.356812\pi$
$47$	5.96195	0.869640	0.434820	+	0.900517i	$0.356812\pi$
$48$	−1.00000	−0.144338
$49$	8.87020	1.26717
$50$	−4.99778	−0.706792
$51$	−1.20250	−0.168384
$52$	−2.39975	−0.332785
$53$	5.81391	0.798602	0.399301	−	0.916820i	$-0.369253\pi$
$53$	5.81391	0.798602	0.399301	+	0.916820i	$0.369253\pi$
$54$	−1.00000	−0.136083
$55$	0.269073	0.0362818
$56$	−3.98374	−0.532350
$57$	3.98374	0.527660
$58$	1.00000	0.131306
$59$	4.24309	0.552404	0.276202	−	0.961100i	$-0.410924\pi$
$59$	4.24309	0.552404	0.276202	+	0.961100i	$0.410924\pi$
$60$	0.0471692	0.00608951
$61$	0.600036	0.0768268	0.0384134	−	0.999262i	$-0.487770\pi$
$61$	0.600036	0.0768268	0.0384134	+	0.999262i	$0.487770\pi$
$62$	7.22454	0.917518
$63$	−3.98374	−0.501904
$64$	1.00000	0.125000
$65$	0.113194	0.0140400
$66$	5.70443	0.702167
$67$	−8.33828	−1.01868	−0.509342	−	0.860564i	$-0.670111\pi$
$67$	−8.33828	−1.01868	−0.509342	+	0.860564i	$0.670111\pi$
$68$	1.20250	0.145825
$69$	−1.00000	−0.120386
$70$	0.187910	0.0224595
$71$	1.23529	0.146602	0.0733012	−	0.997310i	$-0.476647\pi$
$71$	1.23529	0.146602	0.0733012	+	0.997310i	$0.476647\pi$
$72$	1.00000	0.117851
$73$	5.54398	0.648874	0.324437	−	0.945907i	$-0.394825\pi$
$73$	5.54398	0.648874	0.324437	+	0.945907i	$0.394825\pi$
$74$	6.50564	0.756265
$75$	4.99778	0.577093
$76$	−3.98374	−0.456967
$77$	22.7250	2.58975
$78$	2.39975	0.271718
$79$	−11.1562	−1.25517	−0.627586	−	0.778548i	$-0.715957\pi$
$79$	−11.1562	−1.25517	−0.627586	+	0.778548i	$0.715957\pi$
$80$	−0.0471692	−0.00527367
$81$	1.00000	0.111111
$82$	7.53340	0.831925
$83$	3.18624	0.349736	0.174868	−	0.984592i	$-0.444050\pi$
$83$	3.18624	0.349736	0.174868	+	0.984592i	$0.444050\pi$
$84$	3.98374	0.434662
$85$	−0.0567210	−0.00615226
$86$	9.23926	0.996295
$87$	−1.00000	−0.107211
$88$	−5.70443	−0.608094
$89$	−11.3402	−1.20206	−0.601031	−	0.799226i	$-0.705243\pi$
$89$	−11.3402	−1.20206	−0.601031	+	0.799226i	$0.705243\pi$
$90$	−0.0471692	−0.00497207
$91$	9.55998	1.00216
$92$	1.00000	0.104257
$93$	−7.22454	−0.749150
$94$	5.96195	0.614928
$95$	0.187910	0.0192791
$96$	−1.00000	−0.102062
$97$	−8.49728	−0.862768	−0.431384	−	0.902168i	$-0.641975\pi$
$97$	−8.49728	−0.862768	−0.431384	+	0.902168i	$0.641975\pi$
$98$	8.87020	0.896026
$99$	−5.70443	−0.573317
$100$	−4.99778	−0.499778
$101$	−9.59106	−0.954346	−0.477173	−	0.878809i	$-0.658339\pi$
$101$	−9.59106	−0.954346	−0.477173	+	0.878809i	$0.658339\pi$
$102$	−1.20250	−0.119065
$103$	2.06525	0.203495	0.101748	−	0.994810i	$-0.467557\pi$
$103$	2.06525	0.203495	0.101748	+	0.994810i	$0.467557\pi$
$104$	−2.39975	−0.235315
$105$	−0.187910	−0.0183381
$106$	5.81391	0.564697
$107$	−5.58471	−0.539894	−0.269947	−	0.962875i	$-0.587006\pi$
$107$	−5.58471	−0.539894	−0.269947	+	0.962875i	$0.587006\pi$
$108$	−1.00000	−0.0962250
$109$	15.2049	1.45637	0.728183	−	0.685382i	$-0.240365\pi$
$109$	15.2049	1.45637	0.728183	+	0.685382i	$0.240365\pi$
$110$	0.269073	0.0256551
$111$	−6.50564	−0.617488
$112$	−3.98374	−0.376428
$113$	3.46632	0.326084	0.163042	−	0.986619i	$-0.447869\pi$
$113$	3.46632	0.326084	0.163042	+	0.986619i	$0.447869\pi$
$114$	3.98374	0.373112
$115$	−0.0471692	−0.00439855
$116$	1.00000	0.0928477
$117$	−2.39975	−0.221857
$118$	4.24309	0.390608
$119$	−4.79046	−0.439141
$120$	0.0471692	0.00430593
$121$	21.5405	1.95823
$122$	0.600036	0.0543248
$123$	−7.53340	−0.679264
$124$	7.22454	0.648783
$125$	0.471587	0.0421800
$126$	−3.98374	−0.354900
$127$	−15.1327	−1.34281	−0.671403	−	0.741092i	$-0.734308\pi$
$127$	−15.1327	−1.34281	−0.671403	+	0.741092i	$0.734308\pi$
$128$	1.00000	0.0883883
$129$	−9.23926	−0.813471
$130$	0.113194	0.00992778
$131$	0.279421	0.0244131	0.0122066	−	0.999925i	$-0.496114\pi$
$131$	0.279421	0.0244131	0.0122066	+	0.999925i	$0.496114\pi$
$132$	5.70443	0.496507
$133$	15.8702	1.37612
$134$	−8.33828	−0.720318
$135$	0.0471692	0.00405967
$136$	1.20250	0.103114
$137$	8.70145	0.743415	0.371707	−	0.928350i	$-0.378772\pi$
$137$	8.70145	0.743415	0.371707	+	0.928350i	$0.378772\pi$
$138$	−1.00000	−0.0851257
$139$	19.8165	1.68081	0.840405	−	0.541959i	$-0.182317\pi$
$139$	19.8165	1.68081	0.840405	+	0.541959i	$0.182317\pi$
$140$	0.187910	0.0158813
$141$	−5.96195	−0.502087
$142$	1.23529	0.103664
$143$	13.6892	1.14475
$144$	1.00000	0.0833333
$145$	−0.0471692	−0.00391718
$146$	5.54398	0.458823
$147$	−8.87020	−0.731602
$148$	6.50564	0.534760
$149$	18.7745	1.53807	0.769033	−	0.639209i	$-0.220738\pi$
$149$	18.7745	1.53807	0.769033	+	0.639209i	$0.220738\pi$
$150$	4.99778	0.408067
$151$	18.2077	1.48172	0.740862	−	0.671657i	$-0.234417\pi$
$151$	18.2077	1.48172	0.740862	+	0.671657i	$0.234417\pi$
$152$	−3.98374	−0.323124
$153$	1.20250	0.0972165
$154$	22.7250	1.83123
$155$	−0.340776	−0.0273718
$156$	2.39975	0.192134
$157$	1.10961	0.0885565	0.0442782	−	0.999019i	$-0.485901\pi$
$157$	1.10961	0.0885565	0.0442782	+	0.999019i	$0.485901\pi$
$158$	−11.1562	−0.887540
$159$	−5.81391	−0.461073
$160$	−0.0471692	−0.00372905
$161$	−3.98374	−0.313963
$162$	1.00000	0.0785674
$163$	−14.7307	−1.15380	−0.576898	−	0.816816i	$-0.695737\pi$
$163$	−14.7307	−1.15380	−0.576898	+	0.816816i	$0.695737\pi$
$164$	7.53340	0.588260
$165$	−0.269073	−0.0209473
$166$	3.18624	0.247301
$167$	−20.5488	−1.59011	−0.795056	−	0.606536i	$-0.792559\pi$
$167$	−20.5488	−1.59011	−0.795056	+	0.606536i	$0.792559\pi$
$168$	3.98374	0.307352
$169$	−7.24121	−0.557016
$170$	−0.0567210	−0.00435030
$171$	−3.98374	−0.304644
$172$	9.23926	0.704487
$173$	5.83964	0.443979	0.221990	−	0.975049i	$-0.428745\pi$
$173$	5.83964	0.443979	0.221990	+	0.975049i	$0.428745\pi$
$174$	−1.00000	−0.0758098
$175$	19.9098	1.50504
$176$	−5.70443	−0.429988
$177$	−4.24309	−0.318931
$178$	−11.3402	−0.849986
$179$	6.33267	0.473326	0.236663	−	0.971592i	$-0.423946\pi$
$179$	6.33267	0.473326	0.236663	+	0.971592i	$0.423946\pi$
$180$	−0.0471692	−0.00351578
$181$	10.8151	0.803878	0.401939	−	0.915666i	$-0.368336\pi$
$181$	10.8151	0.803878	0.401939	+	0.915666i	$0.368336\pi$
$182$	9.55998	0.708633
$183$	−0.600036	−0.0443560
$184$	1.00000	0.0737210
$185$	−0.306865	−0.0225612
$186$	−7.22454	−0.529729
$187$	−6.85959	−0.501623
$188$	5.96195	0.434820
$189$	3.98374	0.289775
$190$	0.187910	0.0136324
$191$	−10.0860	−0.729796	−0.364898	−	0.931047i	$-0.618896\pi$
$191$	−10.0860	−0.729796	−0.364898	+	0.931047i	$0.618896\pi$
$192$	−1.00000	−0.0721688
$193$	−6.94152	−0.499661	−0.249831	−	0.968290i	$-0.580375\pi$
$193$	−6.94152	−0.499661	−0.249831	+	0.968290i	$0.580375\pi$
$194$	−8.49728	−0.610069
$195$	−0.113194	−0.00810600
$196$	8.87020	0.633586
$197$	4.61572	0.328857	0.164428	−	0.986389i	$-0.447422\pi$
$197$	4.61572	0.328857	0.164428	+	0.986389i	$0.447422\pi$
$198$	−5.70443	−0.405396
$199$	−21.2502	−1.50639	−0.753193	−	0.657799i	$-0.771488\pi$
$199$	−21.2502	−1.50639	−0.753193	+	0.657799i	$0.771488\pi$
$200$	−4.99778	−0.353396
$201$	8.33828	0.588137
$202$	−9.59106	−0.674825
$203$	−3.98374	−0.279604
$204$	−1.20250	−0.0841920
$205$	−0.355344	−0.0248183
$206$	2.06525	0.143893
$207$	1.00000	0.0695048
$208$	−2.39975	−0.166393
$209$	22.7250	1.57192
$210$	−0.187910	−0.0129670
$211$	8.96262	0.617013	0.308506	−	0.951222i	$-0.400171\pi$
$211$	8.96262	0.617013	0.308506	+	0.951222i	$0.400171\pi$
$212$	5.81391	0.399301
$213$	−1.23529	−0.0846410
$214$	−5.58471	−0.381763
$215$	−0.435808	−0.0297219
$216$	−1.00000	−0.0680414
$217$	−28.7807	−1.95376
$218$	15.2049	1.02981
$219$	−5.54398	−0.374628
$220$	0.269073	0.0181409
$221$	−2.88570	−0.194113
$222$	−6.50564	−0.436630
$223$	6.76946	0.453317	0.226658	−	0.973974i	$-0.427220\pi$
$223$	6.76946	0.453317	0.226658	+	0.973974i	$0.427220\pi$
$224$	−3.98374	−0.266175
$225$	−4.99778	−0.333185
$226$	3.46632	0.230576
$227$	−8.37927	−0.556152	−0.278076	−	0.960559i	$-0.589697\pi$
$227$	−8.37927	−0.556152	−0.278076	+	0.960559i	$0.589697\pi$
$228$	3.98374	0.263830
$229$	−6.01277	−0.397335	−0.198667	−	0.980067i	$-0.563661\pi$
$229$	−6.01277	−0.397335	−0.198667	+	0.980067i	$0.563661\pi$
$230$	−0.0471692	−0.00311024
$231$	−22.7250	−1.49519
$232$	1.00000	0.0656532
$233$	0.614151	0.0402343	0.0201172	−	0.999798i	$-0.493596\pi$
$233$	0.614151	0.0402343	0.0201172	+	0.999798i	$0.493596\pi$
$234$	−2.39975	−0.156876
$235$	−0.281220	−0.0183448
$236$	4.24309	0.276202
$237$	11.1562	0.724673
$238$	−4.79046	−0.310519
$239$	−13.0144	−0.841830	−0.420915	−	0.907100i	$-0.638291\pi$
$239$	−13.0144	−0.841830	−0.420915	+	0.907100i	$0.638291\pi$
$240$	0.0471692	0.00304476
$241$	6.36598	0.410069	0.205034	−	0.978755i	$-0.434269\pi$
$241$	6.36598	0.410069	0.205034	+	0.978755i	$0.434269\pi$
$242$	21.5405	1.38468
$243$	−1.00000	−0.0641500
$244$	0.600036	0.0384134
$245$	−0.418400	−0.0267306
$246$	−7.53340	−0.480312
$247$	9.55998	0.608287
$248$	7.22454	0.458759
$249$	−3.18624	−0.201920
$250$	0.471587	0.0298258
$251$	4.22847	0.266899	0.133449	−	0.991056i	$-0.457395\pi$
$251$	4.22847	0.266899	0.133449	+	0.991056i	$0.457395\pi$
$252$	−3.98374	−0.250952
$253$	−5.70443	−0.358634
$254$	−15.1327	−0.949507
$255$	0.0567210	0.00355201
$256$	1.00000	0.0625000
$257$	28.1058	1.75319	0.876595	−	0.481229i	$-0.159809\pi$
$257$	28.1058	1.75319	0.876595	+	0.481229i	$0.159809\pi$
$258$	−9.23926	−0.575211
$259$	−25.9168	−1.61039
$260$	0.113194	0.00702000
$261$	1.00000	0.0618984
$262$	0.279421	0.0172627
$263$	−18.6270	−1.14859	−0.574293	−	0.818650i	$-0.694723\pi$
$263$	−18.6270	−1.14859	−0.574293	+	0.818650i	$0.694723\pi$
$264$	5.70443	0.351083
$265$	−0.274237	−0.0168463
$266$	15.8702	0.973065
$267$	11.3402	0.694011
$268$	−8.33828	−0.509342
$269$	12.5030	0.762319	0.381159	−	0.924509i	$-0.375525\pi$
$269$	12.5030	0.762319	0.381159	+	0.924509i	$0.375525\pi$
$270$	0.0471692	0.00287062
$271$	6.98179	0.424114	0.212057	−	0.977257i	$-0.431984\pi$
$271$	6.98179	0.424114	0.212057	+	0.977257i	$0.431984\pi$
$272$	1.20250	0.0729124
$273$	−9.55998	−0.578596
$274$	8.70145	0.525674
$275$	28.5095	1.71918
$276$	−1.00000	−0.0601929
$277$	27.6482	1.66122	0.830611	−	0.556854i	$-0.187992\pi$
$277$	27.6482	1.66122	0.830611	+	0.556854i	$0.187992\pi$
$278$	19.8165	1.18851
$279$	7.22454	0.432522
$280$	0.187910	0.0112298
$281$	−13.4925	−0.804896	−0.402448	−	0.915443i	$-0.631841\pi$
$281$	−13.4925	−0.804896	−0.402448	+	0.915443i	$0.631841\pi$
$282$	−5.96195	−0.355029
$283$	−25.5438	−1.51842	−0.759212	−	0.650844i	$-0.774415\pi$
$283$	−25.5438	−1.51842	−0.759212	+	0.650844i	$0.774415\pi$
$284$	1.23529	0.0733012
$285$	−0.187910	−0.0111308
$286$	13.6892	0.809459
$287$	−30.0111	−1.77150
$288$	1.00000	0.0589256
$289$	−15.5540	−0.914941
$290$	−0.0471692	−0.00276987
$291$	8.49728	0.498119
$292$	5.54398	0.324437
$293$	0.841428	0.0491568	0.0245784	−	0.999698i	$-0.492176\pi$
$293$	0.841428	0.0491568	0.0245784	+	0.999698i	$0.492176\pi$
$294$	−8.87020	−0.517321
$295$	−0.200143	−0.0116528
$296$	6.50564	0.378132
$297$	5.70443	0.331005
$298$	18.7745	1.08758
$299$	−2.39975	−0.138781
$300$	4.99778	0.288547
$301$	−36.8068	−2.12151
$302$	18.2077	1.04774
$303$	9.59106	0.550992
$304$	−3.98374	−0.228483
$305$	−0.0283032	−0.00162064
$306$	1.20250	0.0687425
$307$	6.38392	0.364349	0.182175	−	0.983266i	$-0.441686\pi$
$307$	6.38392	0.364349	0.182175	+	0.983266i	$0.441686\pi$
$308$	22.7250	1.29488
$309$	−2.06525	−0.117488
$310$	−0.340776	−0.0193548
$311$	−34.1081	−1.93409	−0.967047	−	0.254596i	$-0.918057\pi$
$311$	−34.1081	−1.93409	−0.967047	+	0.254596i	$0.918057\pi$
$312$	2.39975	0.135859
$313$	22.1066	1.24954	0.624769	−	0.780810i	$-0.285193\pi$
$313$	22.1066	1.24954	0.624769	+	0.780810i	$0.285193\pi$
$314$	1.10961	0.0626189
$315$	0.187910	0.0105875
$316$	−11.1562	−0.627586
$317$	21.0975	1.18495	0.592476	−	0.805588i	$-0.298151\pi$
$317$	21.0975	1.18495	0.592476	+	0.805588i	$0.298151\pi$
$318$	−5.81391	−0.326028
$319$	−5.70443	−0.319387
$320$	−0.0471692	−0.00263684
$321$	5.58471	0.311708
$322$	−3.98374	−0.222005
$323$	−4.79046	−0.266548
$324$	1.00000	0.0555556
$325$	11.9934	0.665274
$326$	−14.7307	−0.815857
$327$	−15.2049	−0.840834
$328$	7.53340	0.415962
$329$	−23.7509	−1.30943
$330$	−0.269073	−0.0148120
$331$	31.9935	1.75852	0.879262	−	0.476339i	$-0.158037\pi$
$331$	31.9935	1.75852	0.879262	+	0.476339i	$0.158037\pi$
$332$	3.18624	0.174868
$333$	6.50564	0.356507
$334$	−20.5488	−1.12438
$335$	0.393310	0.0214888
$336$	3.98374	0.217331
$337$	30.9989	1.68862	0.844309	−	0.535856i	$-0.180011\pi$
$337$	30.9989	1.68862	0.844309	+	0.535856i	$0.180011\pi$
$338$	−7.24121	−0.393870
$339$	−3.46632	−0.188265
$340$	−0.0567210	−0.00307613
$341$	−41.2119	−2.23175
$342$	−3.98374	−0.215416
$343$	−7.45041	−0.402284
$344$	9.23926	0.498147
$345$	0.0471692	0.00253950
$346$	5.83964	0.313941
$347$	16.6987	0.896432	0.448216	−	0.893925i	$-0.352060\pi$
$347$	16.6987	0.896432	0.448216	+	0.893925i	$0.352060\pi$
$348$	−1.00000	−0.0536056
$349$	14.1244	0.756063	0.378032	−	0.925793i	$-0.376601\pi$
$349$	14.1244	0.756063	0.378032	+	0.925793i	$0.376601\pi$
$350$	19.9098	1.06423
$351$	2.39975	0.128089
$352$	−5.70443	−0.304047
$353$	−14.6511	−0.779798	−0.389899	−	0.920858i	$-0.627490\pi$
$353$	−14.6511	−0.779798	−0.389899	+	0.920858i	$0.627490\pi$
$354$	−4.24309	−0.225518
$355$	−0.0582678	−0.00309253
$356$	−11.3402	−0.601031
$357$	4.79046	0.253538
$358$	6.33267	0.334692
$359$	29.1141	1.53658	0.768291	−	0.640101i	$-0.221107\pi$
$359$	29.1141	1.53658	0.768291	+	0.640101i	$0.221107\pi$
$360$	−0.0471692	−0.00248603
$361$	−3.12980	−0.164726
$362$	10.8151	0.568428
$363$	−21.5405	−1.13058
$364$	9.55998	0.501079
$365$	−0.261505	−0.0136878
$366$	−0.600036	−0.0313644
$367$	37.6825	1.96701	0.983504	−	0.180884i	$-0.0578958\pi$
$367$	37.6825	1.96701	0.983504	+	0.180884i	$0.0578958\pi$
$368$	1.00000	0.0521286
$369$	7.53340	0.392173
$370$	−0.306865	−0.0159532
$371$	−23.1611	−1.20247
$372$	−7.22454	−0.374575
$373$	33.3640	1.72752	0.863760	−	0.503903i	$-0.168103\pi$
$373$	33.3640	1.72752	0.863760	+	0.503903i	$0.168103\pi$
$374$	−6.85959	−0.354701
$375$	−0.471587	−0.0243526
$376$	5.96195	0.307464
$377$	−2.39975	−0.123593
$378$	3.98374	0.204902
$379$	−15.2846	−0.785118	−0.392559	−	0.919727i	$-0.628410\pi$
$379$	−15.2846	−0.785118	−0.392559	+	0.919727i	$0.628410\pi$
$380$	0.187910	0.00963957
$381$	15.1327	0.775269
$382$	−10.0860	−0.516044
$383$	14.6477	0.748462	0.374231	−	0.927336i	$-0.377907\pi$
$383$	14.6477	0.748462	0.374231	+	0.927336i	$0.377907\pi$
$384$	−1.00000	−0.0510310
$385$	−1.07192	−0.0546300
$386$	−6.94152	−0.353314
$387$	9.23926	0.469658
$388$	−8.49728	−0.431384
$389$	7.55026	0.382813	0.191407	−	0.981511i	$-0.438695\pi$
$389$	7.55026	0.382813	0.191407	+	0.981511i	$0.438695\pi$
$390$	−0.113194	−0.00573181
$391$	1.20250	0.0608131
$392$	8.87020	0.448013
$393$	−0.279421	−0.0140949
$394$	4.61572	0.232537
$395$	0.526229	0.0264774
$396$	−5.70443	−0.286658
$397$	31.4583	1.57884	0.789422	−	0.613850i	$-0.210380\pi$
$397$	31.4583	1.57884	0.789422	+	0.613850i	$0.210380\pi$
$398$	−21.2502	−1.06518
$399$	−15.8702	−0.794504
$400$	−4.99778	−0.249889
$401$	−6.03789	−0.301518	−0.150759	−	0.988571i	$-0.548172\pi$
$401$	−6.03789	−0.301518	−0.150759	+	0.988571i	$0.548172\pi$
$402$	8.33828	0.415876
$403$	−17.3371	−0.863622
$404$	−9.59106	−0.477173
$405$	−0.0471692	−0.00234385
$406$	−3.98374	−0.197710
$407$	−37.1109	−1.83952
$408$	−1.20250	−0.0595327
$409$	12.9171	0.638708	0.319354	−	0.947635i	$-0.396534\pi$
$409$	12.9171	0.638708	0.319354	+	0.947635i	$0.396534\pi$
$410$	−0.355344	−0.0175492
$411$	−8.70145	−0.429211
$412$	2.06525	0.101748
$413$	−16.9034	−0.831762
$414$	1.00000	0.0491473
$415$	−0.150292	−0.00737757
$416$	−2.39975	−0.117657
$417$	−19.8165	−0.970416
$418$	22.7250	1.11151
$419$	3.02514	0.147788	0.0738939	−	0.997266i	$-0.476457\pi$
$419$	3.02514	0.147788	0.0738939	+	0.997266i	$0.476457\pi$
$420$	−0.187910	−0.00916906
$421$	−7.15551	−0.348738	−0.174369	−	0.984680i	$-0.555789\pi$
$421$	−7.15551	−0.348738	−0.174369	+	0.984680i	$0.555789\pi$
$422$	8.96262	0.436294
$423$	5.96195	0.289880
$424$	5.81391	0.282349
$425$	−6.00983	−0.291520
$426$	−1.23529	−0.0598502
$427$	−2.39039	−0.115679
$428$	−5.58471	−0.269947
$429$	−13.6892	−0.660920
$430$	−0.435808	−0.0210165
$431$	21.4910	1.03518	0.517591	−	0.855628i	$-0.326829\pi$
$431$	21.4910	1.03518	0.517591	+	0.855628i	$0.326829\pi$
$432$	−1.00000	−0.0481125
$433$	−24.1259	−1.15942	−0.579708	−	0.814824i	$-0.696833\pi$
$433$	−24.1259	−1.15942	−0.579708	+	0.814824i	$0.696833\pi$
$434$	−28.7807	−1.38152
$435$	0.0471692	0.00226159
$436$	15.2049	0.728183
$437$	−3.98374	−0.190568
$438$	−5.54398	−0.264902
$439$	−1.46054	−0.0697077	−0.0348538	−	0.999392i	$-0.511097\pi$
$439$	−1.46054	−0.0697077	−0.0348538	+	0.999392i	$0.511097\pi$
$440$	0.269073	0.0128276
$441$	8.87020	0.422391
$442$	−2.88570	−0.137259
$443$	−12.9692	−0.616184	−0.308092	−	0.951357i	$-0.599691\pi$
$443$	−12.9692	−0.616184	−0.308092	+	0.951357i	$0.599691\pi$
$444$	−6.50564	−0.308744
$445$	0.534909	0.0253571
$446$	6.76946	0.320543
$447$	−18.7745	−0.888003
$448$	−3.98374	−0.188214
$449$	−9.26165	−0.437084	−0.218542	−	0.975828i	$-0.570130\pi$
$449$	−9.26165	−0.437084	−0.218542	+	0.975828i	$0.570130\pi$
$450$	−4.99778	−0.235597
$451$	−42.9737	−2.02355
$452$	3.46632	0.163042
$453$	−18.2077	−0.855474
$454$	−8.37927	−0.393259
$455$	−0.450936	−0.0211402
$456$	3.98374	0.186556
$457$	−40.6437	−1.90123	−0.950615	−	0.310372i	$-0.899546\pi$
$457$	−40.6437	−1.90123	−0.950615	+	0.310372i	$0.899546\pi$
$458$	−6.01277	−0.280958
$459$	−1.20250	−0.0561280
$460$	−0.0471692	−0.00219927
$461$	26.0054	1.21119	0.605595	−	0.795773i	$-0.292935\pi$
$461$	26.0054	1.21119	0.605595	+	0.795773i	$0.292935\pi$
$462$	−22.7250	−1.05726
$463$	6.26747	0.291274	0.145637	−	0.989338i	$-0.453477\pi$
$463$	6.26747	0.291274	0.145637	+	0.989338i	$0.453477\pi$
$464$	1.00000	0.0464238
$465$	0.340776	0.0158031
$466$	0.614151	0.0284500
$467$	−11.3865	−0.526906	−0.263453	−	0.964672i	$-0.584861\pi$
$467$	−11.3865	−0.526906	−0.263453	+	0.964672i	$0.584861\pi$
$468$	−2.39975	−0.110928
$469$	33.2176	1.53385
$470$	−0.281220	−0.0129717
$471$	−1.10961	−0.0511281
$472$	4.24309	0.195304
$473$	−52.7047	−2.42336
$474$	11.1562	0.512422
$475$	19.9098	0.913527
$476$	−4.79046	−0.219570
$477$	5.81391	0.266201
$478$	−13.0144	−0.595264
$479$	−23.0530	−1.05332	−0.526660	−	0.850076i	$-0.676556\pi$
$479$	−23.0530	−1.05332	−0.526660	+	0.850076i	$0.676556\pi$
$480$	0.0471692	0.00215297
$481$	−15.6119	−0.711841
$482$	6.36598	0.289962
$483$	3.98374	0.181267
$484$	21.5405	0.979114
$485$	0.400810	0.0181998
$486$	−1.00000	−0.0453609
$487$	43.3328	1.96360	0.981799	−	0.189922i	$-0.0608234\pi$
$487$	43.3328	1.96360	0.981799	+	0.189922i	$0.0608234\pi$
$488$	0.600036	0.0271624
$489$	14.7307	0.666145
$490$	−0.418400	−0.0189014
$491$	26.9428	1.21591	0.607957	−	0.793970i	$-0.291989\pi$
$491$	26.9428	1.21591	0.607957	+	0.793970i	$0.291989\pi$
$492$	−7.53340	−0.339632
$493$	1.20250	0.0541580
$494$	9.55998	0.430124
$495$	0.269073	0.0120939
$496$	7.22454	0.324392
$497$	−4.92109	−0.220741
$498$	−3.18624	−0.142779
$499$	25.0716	1.12236	0.561179	−	0.827695i	$-0.310348\pi$
$499$	25.0716	1.12236	0.561179	+	0.827695i	$0.310348\pi$
$500$	0.471587	0.0210900
$501$	20.5488	0.918052
$502$	4.22847	0.188726
$503$	12.7211	0.567208	0.283604	−	0.958942i	$-0.408470\pi$
$503$	12.7211	0.567208	0.283604	+	0.958942i	$0.408470\pi$
$504$	−3.98374	−0.177450
$505$	0.452402	0.0201316
$506$	−5.70443	−0.253593
$507$	7.24121	0.321593
$508$	−15.1327	−0.671403
$509$	−37.0035	−1.64015	−0.820076	−	0.572255i	$-0.806069\pi$
$509$	−37.0035	−1.64015	−0.820076	+	0.572255i	$0.806069\pi$
$510$	0.0567210	0.00251165
$511$	−22.0858	−0.977018
$512$	1.00000	0.0441942
$513$	3.98374	0.175887
$514$	28.1058	1.23969
$515$	−0.0974161	−0.00429267
$516$	−9.23926	−0.406736
$517$	−34.0095	−1.49574
$518$	−25.9168	−1.13872
$519$	−5.83964	−0.256332
$520$	0.113194	0.00496389
$521$	−25.4787	−1.11624	−0.558121	−	0.829759i	$-0.688478\pi$
$521$	−25.4787	−1.11624	−0.558121	+	0.829759i	$0.688478\pi$
$522$	1.00000	0.0437688
$523$	8.58965	0.375599	0.187800	−	0.982207i	$-0.439864\pi$
$523$	8.58965	0.375599	0.187800	+	0.982207i	$0.439864\pi$
$524$	0.279421	0.0122066
$525$	−19.9098	−0.868937
$526$	−18.6270	−0.812174
$527$	8.68753	0.378435
$528$	5.70443	0.248253
$529$	1.00000	0.0434783
$530$	−0.274237	−0.0119121
$531$	4.24309	0.184135
$532$	15.8702	0.688061
$533$	−18.0783	−0.783056
$534$	11.3402	0.490740
$535$	0.263426	0.0113889
$536$	−8.33828	−0.360159
$537$	−6.33267	−0.273275
$538$	12.5030	0.539041
$539$	−50.5994	−2.17947
$540$	0.0471692	0.00202984
$541$	17.6881	0.760470	0.380235	−	0.924890i	$-0.375843\pi$
$541$	17.6881	0.760470	0.380235	+	0.924890i	$0.375843\pi$
$542$	6.98179	0.299894
$543$	−10.8151	−0.464119
$544$	1.20250	0.0515568
$545$	−0.717203	−0.0307216
$546$	−9.55998	−0.409129
$547$	−31.1030	−1.32987	−0.664934	−	0.746902i	$-0.731541\pi$
$547$	−31.1030	−1.32987	−0.664934	+	0.746902i	$0.731541\pi$
$548$	8.70145	0.371707
$549$	0.600036	0.0256089
$550$	28.5095	1.21565
$551$	−3.98374	−0.169713
$552$	−1.00000	−0.0425628
$553$	44.4435	1.88993
$554$	27.6482	1.17466
$555$	0.306865	0.0130257
$556$	19.8165	0.840405
$557$	−32.6077	−1.38163	−0.690817	−	0.723030i	$-0.742749\pi$
$557$	−32.6077	−1.38163	−0.690817	+	0.723030i	$0.742749\pi$
$558$	7.22454	0.305839
$559$	−22.1719	−0.937771
$560$	0.187910	0.00794064
$561$	6.85959	0.289612
$562$	−13.4925	−0.569147
$563$	25.2416	1.06381	0.531904	−	0.846805i	$-0.321477\pi$
$563$	25.2416	1.06381	0.531904	+	0.846805i	$0.321477\pi$
$564$	−5.96195	−0.251043
$565$	−0.163503	−0.00687864
$566$	−25.5438	−1.07369
$567$	−3.98374	−0.167301
$568$	1.23529	0.0518318
$569$	19.5499	0.819573	0.409787	−	0.912181i	$-0.365603\pi$
$569$	19.5499	0.819573	0.409787	+	0.912181i	$0.365603\pi$
$570$	−0.187910	−0.00787067
$571$	8.80481	0.368470	0.184235	−	0.982882i	$-0.441019\pi$
$571$	8.80481	0.368470	0.184235	+	0.982882i	$0.441019\pi$
$572$	13.6892	0.572374
$573$	10.0860	0.421348
$574$	−30.0111	−1.25264
$575$	−4.99778	−0.208422
$576$	1.00000	0.0416667
$577$	−17.8464	−0.742955	−0.371478	−	0.928442i	$-0.621149\pi$
$577$	−17.8464	−0.742955	−0.371478	+	0.928442i	$0.621149\pi$
$578$	−15.5540	−0.646961
$579$	6.94152	0.288480
$580$	−0.0471692	−0.00195859
$581$	−12.6932	−0.526602
$582$	8.49728	0.352224
$583$	−33.1651	−1.37356
$584$	5.54398	0.229412
$585$	0.113194	0.00468000
$586$	0.841428	0.0347591
$587$	34.4737	1.42288	0.711441	−	0.702746i	$-0.248043\pi$
$587$	34.4737	1.42288	0.711441	+	0.702746i	$0.248043\pi$
$588$	−8.87020	−0.365801
$589$	−28.7807	−1.18589
$590$	−0.200143	−0.00823976
$591$	−4.61572	−0.189865
$592$	6.50564	0.267380
$593$	−20.5710	−0.844749	−0.422374	−	0.906421i	$-0.638803\pi$
$593$	−20.5710	−0.844749	−0.422374	+	0.906421i	$0.638803\pi$
$594$	5.70443	0.234056
$595$	0.225962	0.00926353
$596$	18.7745	0.769033
$597$	21.2502	0.869713
$598$	−2.39975	−0.0981330
$599$	−10.2437	−0.418545	−0.209273	−	0.977857i	$-0.567110\pi$
$599$	−10.2437	−0.418545	−0.209273	+	0.977857i	$0.567110\pi$
$600$	4.99778	0.204033
$601$	22.5214	0.918666	0.459333	−	0.888264i	$-0.348088\pi$
$601$	22.5214	0.918666	0.459333	+	0.888264i	$0.348088\pi$
$602$	−36.8068	−1.50013
$603$	−8.33828	−0.339561
$604$	18.2077	0.740862
$605$	−1.01605	−0.0413082
$606$	9.59106	0.389610
$607$	22.9823	0.932821	0.466411	−	0.884568i	$-0.345547\pi$
$607$	22.9823	0.932821	0.466411	+	0.884568i	$0.345547\pi$
$608$	−3.98374	−0.161562
$609$	3.98374	0.161429
$610$	−0.0283032	−0.00114596
$611$	−14.3072	−0.578807
$612$	1.20250	0.0486083
$613$	30.7928	1.24371	0.621855	−	0.783132i	$-0.286379\pi$
$613$	30.7928	1.24371	0.621855	+	0.783132i	$0.286379\pi$
$614$	6.38392	0.257634
$615$	0.355344	0.0143289
$616$	22.7250	0.915615
$617$	44.1552	1.77762	0.888810	−	0.458276i	$-0.151533\pi$
$617$	44.1552	1.77762	0.888810	+	0.458276i	$0.151533\pi$
$618$	−2.06525	−0.0830765
$619$	21.9641	0.882810	0.441405	−	0.897308i	$-0.354480\pi$
$619$	21.9641	0.882810	0.441405	+	0.897308i	$0.354480\pi$
$620$	−0.340776	−0.0136859
$621$	−1.00000	−0.0401286
$622$	−34.1081	−1.36761
$623$	45.1766	1.80996
$624$	2.39975	0.0960668
$625$	24.9666	0.998665
$626$	22.1066	0.883556
$627$	−22.7250	−0.907548
$628$	1.10961	0.0442782
$629$	7.82304	0.311925
$630$	0.187910	0.00748650
$631$	−19.4098	−0.772692	−0.386346	−	0.922354i	$-0.626263\pi$
$631$	−19.4098	−0.772692	−0.386346	+	0.922354i	$0.626263\pi$
$632$	−11.1562	−0.443770
$633$	−8.96262	−0.356232
$634$	21.0975	0.837887
$635$	0.713795	0.0283261
$636$	−5.81391	−0.230537
$637$	−21.2863	−0.843392
$638$	−5.70443	−0.225841
$639$	1.23529	0.0488675
$640$	−0.0471692	−0.00186452
$641$	−25.1325	−0.992674	−0.496337	−	0.868130i	$-0.665322\pi$
$641$	−25.1325	−0.992674	−0.496337	+	0.868130i	$0.665322\pi$
$642$	5.58471	0.220411
$643$	42.1776	1.66332	0.831661	−	0.555283i	$-0.187390\pi$
$643$	42.1776	1.66332	0.831661	+	0.555283i	$0.187390\pi$
$644$	−3.98374	−0.156981
$645$	0.435808	0.0171599
$646$	−4.79046	−0.188478
$647$	−19.4752	−0.765650	−0.382825	−	0.923821i	$-0.625049\pi$
$647$	−19.4752	−0.765650	−0.382825	+	0.923821i	$0.625049\pi$
$648$	1.00000	0.0392837
$649$	−24.2044	−0.950107
$650$	11.9934	0.470420
$651$	28.7807	1.12801
$652$	−14.7307	−0.576898
$653$	47.1808	1.84633	0.923164	−	0.384406i	$-0.125594\pi$
$653$	47.1808	1.84633	0.923164	+	0.384406i	$0.125594\pi$
$654$	−15.2049	−0.594559
$655$	−0.0131801	−0.000514987 0
$656$	7.53340	0.294130
$657$	5.54398	0.216291
$658$	−23.7509	−0.925906
$659$	−42.2343	−1.64521	−0.822607	−	0.568611i	$-0.807481\pi$
$659$	−42.2343	−1.64521	−0.822607	+	0.568611i	$0.807481\pi$
$660$	−0.269073	−0.0104737
$661$	−35.5894	−1.38427	−0.692134	−	0.721769i	$-0.743329\pi$
$661$	−35.5894	−1.38427	−0.692134	+	0.721769i	$0.743329\pi$
$662$	31.9935	1.24346
$663$	2.88570	0.112071
$664$	3.18624	0.123650
$665$	−0.748584	−0.0290288
$666$	6.50564	0.252088
$667$	1.00000	0.0387202
$668$	−20.5488	−0.795056
$669$	−6.76946	−0.261722
$670$	0.393310	0.0151949
$671$	−3.42287	−0.132138
$672$	3.98374	0.153676
$673$	22.1444	0.853603	0.426801	−	0.904345i	$-0.359640\pi$
$673$	22.1444	0.853603	0.426801	+	0.904345i	$0.359640\pi$
$674$	30.9989	1.19403
$675$	4.99778	0.192364
$676$	−7.24121	−0.278508
$677$	−31.8409	−1.22375	−0.611873	−	0.790956i	$-0.709584\pi$
$677$	−31.8409	−1.22375	−0.611873	+	0.790956i	$0.709584\pi$
$678$	−3.46632	−0.133123
$679$	33.8510	1.29908
$680$	−0.0567210	−0.00217515
$681$	8.37927	0.321094
$682$	−41.2119	−1.57809
$683$	−17.4312	−0.666986	−0.333493	−	0.942753i	$-0.608227\pi$
$683$	−17.4312	−0.666986	−0.333493	+	0.942753i	$0.608227\pi$
$684$	−3.98374	−0.152322
$685$	−0.410440	−0.0156821
$686$	−7.45041	−0.284458
$687$	6.01277	0.229401
$688$	9.23926	0.352243
$689$	−13.9519	−0.531526
$690$	0.0471692	0.00179570
$691$	14.4512	0.549751	0.274875	−	0.961480i	$-0.411363\pi$
$691$	14.4512	0.549751	0.274875	+	0.961480i	$0.411363\pi$
$692$	5.83964	0.221990
$693$	22.7250	0.863250
$694$	16.6987	0.633873
$695$	−0.934725	−0.0354561
$696$	−1.00000	−0.0379049
$697$	9.05893	0.343131
$698$	14.1244	0.534617
$699$	−0.614151	−0.0232293
$700$	19.9098	0.752522
$701$	−40.9802	−1.54780	−0.773900	−	0.633308i	$-0.781697\pi$
$701$	−40.9802	−1.54780	−0.773900	+	0.633308i	$0.781697\pi$
$702$	2.39975	0.0905727
$703$	−25.9168	−0.977470
$704$	−5.70443	−0.214994
$705$	0.281220	0.0105914
$706$	−14.6511	−0.551401
$707$	38.2083	1.43697
$708$	−4.24309	−0.159465
$709$	−22.0370	−0.827617	−0.413809	−	0.910364i	$-0.635802\pi$
$709$	−22.0370	−0.827617	−0.413809	+	0.910364i	$0.635802\pi$
$710$	−0.0582678	−0.00218675
$711$	−11.1562	−0.418390
$712$	−11.3402	−0.424993
$713$	7.22454	0.270561
$714$	4.79046	0.179278
$715$	−0.645708	−0.0241481
$716$	6.33267	0.236663
$717$	13.0144	0.486031
$718$	29.1141	1.08653
$719$	−41.3642	−1.54262	−0.771312	−	0.636457i	$-0.780399\pi$
$719$	−41.3642	−1.54262	−0.771312	+	0.636457i	$0.780399\pi$
$720$	−0.0471692	−0.00175789
$721$	−8.22742	−0.306405
$722$	−3.12980	−0.116479
$723$	−6.36598	−0.236753
$724$	10.8151	0.401939
$725$	−4.99778	−0.185613
$726$	−21.5405	−0.799443
$727$	−28.0007	−1.03849	−0.519244	−	0.854626i	$-0.673787\pi$
$727$	−28.0007	−1.03849	−0.519244	+	0.854626i	$0.673787\pi$
$728$	9.55998	0.354316
$729$	1.00000	0.0370370
$730$	−0.261505	−0.00967873
$731$	11.1102	0.410926
$732$	−0.600036	−0.0221780
$733$	−26.8052	−0.990074	−0.495037	−	0.868872i	$-0.664845\pi$
$733$	−26.8052	−0.990074	−0.495037	+	0.868872i	$0.664845\pi$
$734$	37.6825	1.39089
$735$	0.418400	0.0154329
$736$	1.00000	0.0368605
$737$	47.5652	1.75208
$738$	7.53340	0.277308
$739$	−6.34850	−0.233533	−0.116767	−	0.993159i	$-0.537253\pi$
$739$	−6.34850	−0.233533	−0.116767	+	0.993159i	$0.537253\pi$
$740$	−0.306865	−0.0112806
$741$	−9.55998	−0.351195
$742$	−23.1611	−0.850272
$743$	18.2606	0.669915	0.334958	−	0.942233i	$-0.391278\pi$
$743$	18.2606	0.669915	0.334958	+	0.942233i	$0.391278\pi$
$744$	−7.22454	−0.264865
$745$	−0.885576	−0.0324450
$746$	33.3640	1.22154
$747$	3.18624	0.116579
$748$	−6.85959	−0.250811
$749$	22.2481	0.812926
$750$	−0.471587	−0.0172199
$751$	−8.82044	−0.321863	−0.160931	−	0.986966i	$-0.551450\pi$
$751$	−8.82044	−0.321863	−0.160931	+	0.986966i	$0.551450\pi$
$752$	5.96195	0.217410
$753$	−4.22847	−0.154094
$754$	−2.39975	−0.0873937
$755$	−0.858843	−0.0312565
$756$	3.98374	0.144887
$757$	49.8837	1.81305	0.906527	−	0.422149i	$-0.138724\pi$
$757$	49.8837	1.81305	0.906527	+	0.422149i	$0.138724\pi$
$758$	−15.2846	−0.555162
$759$	5.70443	0.207058
$760$	0.187910	0.00681620
$761$	6.19382	0.224526	0.112263	−	0.993679i	$-0.464190\pi$
$761$	6.19382	0.224526	0.112263	+	0.993679i	$0.464190\pi$
$762$	15.1327	0.548198
$763$	−60.5725	−2.19287
$764$	−10.0860	−0.364898
$765$	−0.0567210	−0.00205075
$766$	14.6477	0.529242
$767$	−10.1824	−0.367664
$768$	−1.00000	−0.0360844
$769$	36.1325	1.30297	0.651485	−	0.758661i	$-0.274146\pi$
$769$	36.1325	1.30297	0.651485	+	0.758661i	$0.274146\pi$
$770$	−1.07192	−0.0386292
$771$	−28.1058	−1.01220
$772$	−6.94152	−0.249831
$773$	−45.8623	−1.64955	−0.824776	−	0.565460i	$-0.808699\pi$
$773$	−45.8623	−1.64955	−0.824776	+	0.565460i	$0.808699\pi$
$774$	9.23926	0.332098
$775$	−36.1066	−1.29699
$776$	−8.49728	−0.305035
$777$	25.9168	0.929759
$778$	7.55026	0.270690
$779$	−30.0111	−1.07526
$780$	−0.113194	−0.00405300
$781$	−7.04665	−0.252149
$782$	1.20250	0.0430014
$783$	−1.00000	−0.0357371
$784$	8.87020	0.316793
$785$	−0.0523393	−0.00186807
$786$	−0.279421	−0.00996662
$787$	−36.5541	−1.30301	−0.651507	−	0.758643i	$-0.725863\pi$
$787$	−36.5541	−1.30301	−0.651507	+	0.758643i	$0.725863\pi$
$788$	4.61572	0.164428
$789$	18.6270	0.663137
$790$	0.526229	0.0187224
$791$	−13.8089	−0.490989
$792$	−5.70443	−0.202698
$793$	−1.43994	−0.0511336
$794$	31.4583	1.11641
$795$	0.274237	0.00972620
$796$	−21.2502	−0.753193
$797$	−50.2146	−1.77869	−0.889347	−	0.457233i	$-0.848841\pi$
$797$	−50.2146	−1.77869	−0.889347	+	0.457233i	$0.848841\pi$
$798$	−15.8702	−0.561799
$799$	7.16926	0.253630
$800$	−4.99778	−0.176698
$801$	−11.3402	−0.400687
$802$	−6.03789	−0.213205
$803$	−31.6253	−1.11603
$804$	8.33828	0.294069
$805$	0.187910	0.00662295
$806$	−17.3371	−0.610673
$807$	−12.5030	−0.440125
$808$	−9.59106	−0.337412
$809$	−31.4683	−1.10637	−0.553184	−	0.833059i	$-0.686587\pi$
$809$	−31.4683	−1.10637	−0.553184	+	0.833059i	$0.686587\pi$
$810$	−0.0471692	−0.00165736
$811$	14.9322	0.524342	0.262171	−	0.965021i	$-0.415562\pi$
$811$	14.9322	0.524342	0.262171	+	0.965021i	$0.415562\pi$
$812$	−3.98374	−0.139802
$813$	−6.98179	−0.244862
$814$	−37.1109	−1.30074
$815$	0.694834	0.0243390
$816$	−1.20250	−0.0420960
$817$	−36.8068	−1.28771
$818$	12.9171	0.451635
$819$	9.55998	0.334053
$820$	−0.355344	−0.0124092
$821$	−2.07085	−0.0722733	−0.0361366	−	0.999347i	$-0.511505\pi$
$821$	−2.07085	−0.0722733	−0.0361366	+	0.999347i	$0.511505\pi$
$822$	−8.70145	−0.303498
$823$	11.9294	0.415833	0.207916	−	0.978147i	$-0.433332\pi$
$823$	11.9294	0.415833	0.207916	+	0.978147i	$0.433332\pi$
$824$	2.06525	0.0719464
$825$	−28.5095	−0.992572
$826$	−16.9034	−0.588144
$827$	20.4446	0.710928	0.355464	−	0.934690i	$-0.384323\pi$
$827$	20.4446	0.710928	0.355464	+	0.934690i	$0.384323\pi$
$828$	1.00000	0.0347524
$829$	−46.3047	−1.60823	−0.804114	−	0.594475i	$-0.797360\pi$
$829$	−46.3047	−1.60823	−0.804114	+	0.594475i	$0.797360\pi$
$830$	−0.150292	−0.00521673
$831$	−27.6482	−0.959107
$832$	−2.39975	−0.0831963
$833$	10.6664	0.369570
$834$	−19.8165	−0.686188
$835$	0.969269	0.0335429
$836$	22.7250	0.785960
$837$	−7.22454	−0.249717
$838$	3.02514	0.104502
$839$	25.2076	0.870263	0.435131	−	0.900367i	$-0.356702\pi$
$839$	25.2076	0.870263	0.435131	+	0.900367i	$0.356702\pi$
$840$	−0.187910	−0.00648350
$841$	1.00000	0.0344828
$842$	−7.15551	−0.246595
$843$	13.4925	0.464707
$844$	8.96262	0.308506
$845$	0.341562	0.0117501
$846$	5.96195	0.204976
$847$	−85.8118	−2.94853
$848$	5.81391	0.199651
$849$	25.5438	0.876662
$850$	−6.00983	−0.206136
$851$	6.50564	0.223010
$852$	−1.23529	−0.0423205
$853$	20.3941	0.698280	0.349140	−	0.937071i	$-0.386474\pi$
$853$	20.3941	0.698280	0.349140	+	0.937071i	$0.386474\pi$
$854$	−2.39039	−0.0817975
$855$	0.187910	0.00642638
$856$	−5.58471	−0.190882
$857$	−26.3314	−0.899462	−0.449731	−	0.893164i	$-0.648480\pi$
$857$	−26.3314	−0.899462	−0.449731	+	0.893164i	$0.648480\pi$
$858$	−13.6892	−0.467341
$859$	−4.09853	−0.139840	−0.0699200	−	0.997553i	$-0.522274\pi$
$859$	−4.09853	−0.139840	−0.0699200	+	0.997553i	$0.522274\pi$
$860$	−0.435808	−0.0148609
$861$	30.0111	1.02278
$862$	21.4910	0.731985
$863$	8.85252	0.301343	0.150672	−	0.988584i	$-0.451856\pi$
$863$	8.85252	0.301343	0.150672	+	0.988584i	$0.451856\pi$
$864$	−1.00000	−0.0340207
$865$	−0.275451	−0.00936561
$866$	−24.1259	−0.819831
$867$	15.5540	0.528241
$868$	−28.7807	−0.976881
$869$	63.6398	2.15883
$870$	0.0471692	0.00159918
$871$	20.0098	0.678006
$872$	15.2049	0.514903
$873$	−8.49728	−0.287589
$874$	−3.98374	−0.134752
$875$	−1.87868	−0.0635110
$876$	−5.54398	−0.187314
$877$	9.98305	0.337104	0.168552	−	0.985693i	$-0.446091\pi$
$877$	9.98305	0.337104	0.168552	+	0.985693i	$0.446091\pi$
$878$	−1.46054	−0.0492908
$879$	−0.841428	−0.0283807
$880$	0.269073	0.00907045
$881$	1.41954	0.0478257	0.0239128	−	0.999714i	$-0.492388\pi$
$881$	1.41954	0.0478257	0.0239128	+	0.999714i	$0.492388\pi$
$882$	8.87020	0.298675
$883$	−32.8590	−1.10579	−0.552897	−	0.833249i	$-0.686478\pi$
$883$	−32.8590	−1.10579	−0.552897	+	0.833249i	$0.686478\pi$
$884$	−2.88570	−0.0970567
$885$	0.200143	0.00672774
$886$	−12.9692	−0.435708
$887$	42.1790	1.41623	0.708116	−	0.706096i	$-0.249545\pi$
$887$	42.1790	1.41623	0.708116	+	0.706096i	$0.249545\pi$
$888$	−6.50564	−0.218315
$889$	60.2846	2.02188
$890$	0.534909	0.0179302
$891$	−5.70443	−0.191106
$892$	6.76946	0.226658
$893$	−23.7509	−0.794793
$894$	−18.7745	−0.627913
$895$	−0.298707	−0.00998466
$896$	−3.98374	−0.133087
$897$	2.39975	0.0801253
$898$	−9.26165	−0.309065
$899$	7.22454	0.240952
$900$	−4.99778	−0.166593
$901$	6.99124	0.232912
$902$	−42.9737	−1.43087
$903$	36.8068	1.22485
$904$	3.46632	0.115288
$905$	−0.510138	−0.0169576
$906$	−18.2077	−0.604911
$907$	−1.11575	−0.0370478	−0.0185239	−	0.999828i	$-0.505897\pi$
$907$	−1.11575	−0.0370478	−0.0185239	+	0.999828i	$0.505897\pi$
$908$	−8.37927	−0.278076
$909$	−9.59106	−0.318115
$910$	−0.450936	−0.0149484
$911$	−11.8425	−0.392359	−0.196179	−	0.980568i	$-0.562853\pi$
$911$	−11.8425	−0.392359	−0.196179	+	0.980568i	$0.562853\pi$
$912$	3.98374	0.131915
$913$	−18.1757	−0.601528
$914$	−40.6437	−1.34437
$915$	0.0283032	0.000935675 0
$916$	−6.01277	−0.198667
$917$	−1.11314	−0.0367592
$918$	−1.20250	−0.0396885
$919$	46.0889	1.52033	0.760167	−	0.649728i	$-0.225117\pi$
$919$	46.0889	1.52033	0.760167	+	0.649728i	$0.225117\pi$
$920$	−0.0471692	−0.00155512
$921$	−6.38392	−0.210357
$922$	26.0054	0.856441
$923$	−2.96440	−0.0975743
$924$	−22.7250	−0.747597
$925$	−32.5137	−1.06904
$926$	6.26747	0.205962
$927$	2.06525	0.0678317
$928$	1.00000	0.0328266
$929$	−40.4772	−1.32801	−0.664007	−	0.747727i	$-0.731145\pi$
$929$	−40.4772	−1.32801	−0.664007	+	0.747727i	$0.731145\pi$
$930$	0.340776	0.0111745
$931$	−35.3366	−1.15811
$932$	0.614151	0.0201172
$933$	34.1081	1.11665
$934$	−11.3865	−0.372579
$935$	0.323561	0.0105816
$936$	−2.39975	−0.0784382
$937$	19.1838	0.626707	0.313354	−	0.949637i	$-0.398547\pi$
$937$	19.1838	0.626707	0.313354	+	0.949637i	$0.398547\pi$
$938$	33.2176	1.08459
$939$	−22.1066	−0.721421
$940$	−0.281220	−0.00917239
$941$	−36.6604	−1.19509	−0.597547	−	0.801834i	$-0.703858\pi$
$941$	−36.6604	−1.19509	−0.597547	+	0.801834i	$0.703858\pi$
$942$	−1.10961	−0.0361530
$943$	7.53340	0.245321
$944$	4.24309	0.138101
$945$	−0.187910	−0.00611270
$946$	−52.7047	−1.71358
$947$	39.1801	1.27318	0.636590	−	0.771202i	$-0.280344\pi$
$947$	39.1801	1.27318	0.636590	+	0.771202i	$0.280344\pi$
$948$	11.1562	0.362337
$949$	−13.3042	−0.431871
$950$	19.9098	0.645961
$951$	−21.0975	−0.684132
$952$	−4.79046	−0.155260
$953$	15.2499	0.493993	0.246996	−	0.969016i	$-0.420556\pi$
$953$	15.2499	0.493993	0.246996	+	0.969016i	$0.420556\pi$
$954$	5.81391	0.188232
$955$	0.475747	0.0153948
$956$	−13.0144	−0.420915
$957$	5.70443	0.184398
$958$	−23.0530	−0.744810
$959$	−34.6643	−1.11937
$960$	0.0471692	0.00152238
$961$	21.1940	0.683679
$962$	−15.6119	−0.503348
$963$	−5.58471	−0.179965
$964$	6.36598	0.205034
$965$	0.327425	0.0105402
$966$	3.98374	0.128175
$967$	11.0047	0.353887	0.176944	−	0.984221i	$-0.443379\pi$
$967$	11.0047	0.353887	0.176944	+	0.984221i	$0.443379\pi$
$968$	21.5405	0.692338
$969$	4.79046	0.153892
$970$	0.400810	0.0128692
$971$	0.117521	0.00377142	0.00188571	−	0.999998i	$-0.499400\pi$
$971$	0.117521	0.00377142	0.00188571	+	0.999998i	$0.499400\pi$
$972$	−1.00000	−0.0320750
$973$	−78.9436	−2.53082
$974$	43.3328	1.38847
$975$	−11.9934	−0.384096
$976$	0.600036	0.0192067
$977$	2.53688	0.0811619	0.0405810	−	0.999176i	$-0.487079\pi$
$977$	2.53688	0.0811619	0.0405810	+	0.999176i	$0.487079\pi$
$978$	14.7307	0.471035
$979$	64.6896	2.06749
$980$	−0.418400	−0.0133653
$981$	15.2049	0.485456
$982$	26.9428	0.859780
$983$	−17.7674	−0.566692	−0.283346	−	0.959018i	$-0.591445\pi$
$983$	−17.7674	−0.566692	−0.283346	+	0.959018i	$0.591445\pi$
$984$	−7.53340	−0.240156
$985$	−0.217720	−0.00693713
$986$	1.20250	0.0382955
$987$	23.7509	0.755999
$988$	9.55998	0.304143
$989$	9.23926	0.293791
$990$	0.269073	0.00855170
$991$	22.4508	0.713174	0.356587	−	0.934262i	$-0.383940\pi$
$991$	22.4508	0.713174	0.356587	+	0.934262i	$0.383940\pi$
$992$	7.22454	0.229380
$993$	−31.9935	−1.01528
$994$	−4.92109	−0.156088
$995$	1.00235	0.0317768
$996$	−3.18624	−0.100960
$997$	12.9797	0.411072	0.205536	−	0.978650i	$-0.434106\pi$
$997$	12.9797	0.411072	0.205536	+	0.978650i	$0.434106\pi$
$998$	25.0716	0.793626
$999$	−6.50564	−0.205829

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bj.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bj.1.5	✓	8	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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.0471692 q^{5} -1.00000 q^{6} -3.98374 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.0471692 q^{5} -1.00000 q^{6} -3.98374 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.0471692 q^{10} -5.70443 q^{11} -1.00000 q^{12} -2.39975 q^{13} -3.98374 q^{14} +0.0471692 q^{15} +1.00000 q^{16} +1.20250 q^{17} +1.00000 q^{18} -3.98374 q^{19} -0.0471692 q^{20} +3.98374 q^{21} -5.70443 q^{22} +1.00000 q^{23} -1.00000 q^{24} -4.99778 q^{25} -2.39975 q^{26} -1.00000 q^{27} -3.98374 q^{28} +1.00000 q^{29} +0.0471692 q^{30} +7.22454 q^{31} +1.00000 q^{32} +5.70443 q^{33} +1.20250 q^{34} +0.187910 q^{35} +1.00000 q^{36} +6.50564 q^{37} -3.98374 q^{38} +2.39975 q^{39} -0.0471692 q^{40} +7.53340 q^{41} +3.98374 q^{42} +9.23926 q^{43} -5.70443 q^{44} -0.0471692 q^{45} +1.00000 q^{46} +5.96195 q^{47} -1.00000 q^{48} +8.87020 q^{49} -4.99778 q^{50} -1.20250 q^{51} -2.39975 q^{52} +5.81391 q^{53} -1.00000 q^{54} +0.269073 q^{55} -3.98374 q^{56} +3.98374 q^{57} +1.00000 q^{58} +4.24309 q^{59} +0.0471692 q^{60} +0.600036 q^{61} +7.22454 q^{62} -3.98374 q^{63} +1.00000 q^{64} +0.113194 q^{65} +5.70443 q^{66} -8.33828 q^{67} +1.20250 q^{68} -1.00000 q^{69} +0.187910 q^{70} +1.23529 q^{71} +1.00000 q^{72} +5.54398 q^{73} +6.50564 q^{74} +4.99778 q^{75} -3.98374 q^{76} +22.7250 q^{77} +2.39975 q^{78} -11.1562 q^{79} -0.0471692 q^{80} +1.00000 q^{81} +7.53340 q^{82} +3.18624 q^{83} +3.98374 q^{84} -0.0567210 q^{85} +9.23926 q^{86} -1.00000 q^{87} -5.70443 q^{88} -11.3402 q^{89} -0.0471692 q^{90} +9.55998 q^{91} +1.00000 q^{92} -7.22454 q^{93} +5.96195 q^{94} +0.187910 q^{95} -1.00000 q^{96} -8.49728 q^{97} +8.87020 q^{98} -5.70443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + q^{5} - 8 q^{6} + 8 q^{8} + 8 q^{9} + q^{10} + 3 q^{11} - 8 q^{12} + 9 q^{13} - q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + q^{20} + 3 q^{22} + 8 q^{23} - 8 q^{24} + 13 q^{25} + 9 q^{26} - 8 q^{27} + 8 q^{29} - q^{30} - 9 q^{31} + 8 q^{32} - 3 q^{33} + 8 q^{34} - 2 q^{35} + 8 q^{36} + 7 q^{37} - 9 q^{39} + q^{40} + 3 q^{41} + 16 q^{43} + 3 q^{44} + q^{45} + 8 q^{46} + 24 q^{47} - 8 q^{48} + 6 q^{49} + 13 q^{50} - 8 q^{51} + 9 q^{52} + 8 q^{53} - 8 q^{54} + 13 q^{55} + 8 q^{58} - 3 q^{59} - q^{60} + 31 q^{61} - 9 q^{62} + 8 q^{64} + 13 q^{65} - 3 q^{66} - 11 q^{67} + 8 q^{68} - 8 q^{69} - 2 q^{70} + 7 q^{71} + 8 q^{72} + 14 q^{73} + 7 q^{74} - 13 q^{75} + 10 q^{77} - 9 q^{78} + 12 q^{79} + q^{80} + 8 q^{81} + 3 q^{82} - 8 q^{83} + 22 q^{85} + 16 q^{86} - 8 q^{87} + 3 q^{88} - 12 q^{89} + q^{90} + 28 q^{91} + 8 q^{92} + 9 q^{93} + 24 q^{94} - 2 q^{95} - 8 q^{96} + 16 q^{97} + 6 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + q^5 - 8 * q^6 + 8 * q^8 + 8 * q^9 + q^10 + 3 * q^11 - 8 * q^12 + 9 * q^13 - q^15 + 8 * q^16 + 8 * q^17 + 8 * q^18 + q^20 + 3 * q^22 + 8 * q^23 - 8 * q^24 + 13 * q^25 + 9 * q^26 - 8 * q^27 + 8 * q^29 - q^30 - 9 * q^31 + 8 * q^32 - 3 * q^33 + 8 * q^34 - 2 * q^35 + 8 * q^36 + 7 * q^37 - 9 * q^39 + q^40 + 3 * q^41 + 16 * q^43 + 3 * q^44 + q^45 + 8 * q^46 + 24 * q^47 - 8 * q^48 + 6 * q^49 + 13 * q^50 - 8 * q^51 + 9 * q^52 + 8 * q^53 - 8 * q^54 + 13 * q^55 + 8 * q^58 - 3 * q^59 - q^60 + 31 * q^61 - 9 * q^62 + 8 * q^64 + 13 * q^65 - 3 * q^66 - 11 * q^67 + 8 * q^68 - 8 * q^69 - 2 * q^70 + 7 * q^71 + 8 * q^72 + 14 * q^73 + 7 * q^74 - 13 * q^75 + 10 * q^77 - 9 * q^78 + 12 * q^79 + q^80 + 8 * q^81 + 3 * q^82 - 8 * q^83 + 22 * q^85 + 16 * q^86 - 8 * q^87 + 3 * q^88 - 12 * q^89 + q^90 + 28 * q^91 + 8 * q^92 + 9 * q^93 + 24 * q^94 - 2 * q^95 - 8 * q^96 + 16 * q^97 + 6 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more