LMFDB - Embedded newform 4002.2.a.s.1.2

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:	$2$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} +1.00000 q^{6} -0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} +1.00000 q^{6} -0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.561553 q^{10} -2.00000 q^{11} -1.00000 q^{12} +3.12311 q^{13} +0.561553 q^{14} -0.561553 q^{15} +1.00000 q^{16} -6.56155 q^{17} -1.00000 q^{18} +2.56155 q^{19} +0.561553 q^{20} +0.561553 q^{21} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -4.68466 q^{25} -3.12311 q^{26} -1.00000 q^{27} -0.561553 q^{28} +1.00000 q^{29} +0.561553 q^{30} -1.00000 q^{32} +2.00000 q^{33} +6.56155 q^{34} -0.315342 q^{35} +1.00000 q^{36} +2.56155 q^{37} -2.56155 q^{38} -3.12311 q^{39} -0.561553 q^{40} +0.561553 q^{41} -0.561553 q^{42} +11.6847 q^{43} -2.00000 q^{44} +0.561553 q^{45} +1.00000 q^{46} +6.56155 q^{47} -1.00000 q^{48} -6.68466 q^{49} +4.68466 q^{50} +6.56155 q^{51} +3.12311 q^{52} -2.00000 q^{53} +1.00000 q^{54} -1.12311 q^{55} +0.561553 q^{56} -2.56155 q^{57} -1.00000 q^{58} +2.56155 q^{59} -0.561553 q^{60} +5.12311 q^{61} -0.561553 q^{63} +1.00000 q^{64} +1.75379 q^{65} -2.00000 q^{66} -6.56155 q^{68} +1.00000 q^{69} +0.315342 q^{70} -8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} -2.56155 q^{74} +4.68466 q^{75} +2.56155 q^{76} +1.12311 q^{77} +3.12311 q^{78} -16.2462 q^{79} +0.561553 q^{80} +1.00000 q^{81} -0.561553 q^{82} -11.1231 q^{83} +0.561553 q^{84} -3.68466 q^{85} -11.6847 q^{86} -1.00000 q^{87} +2.00000 q^{88} +5.12311 q^{89} -0.561553 q^{90} -1.75379 q^{91} -1.00000 q^{92} -6.56155 q^{94} +1.43845 q^{95} +1.00000 q^{96} +4.87689 q^{97} +6.68466 q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{10} - 4 q^{11} - 2 q^{12} - 2 q^{13} - 3 q^{14} + 3 q^{15} + 2 q^{16} - 9 q^{17} - 2 q^{18} + q^{19} - 3 q^{20} - 3 q^{21} + 4 q^{22} - 2 q^{23} + 2 q^{24} + 3 q^{25} + 2 q^{26} - 2 q^{27} + 3 q^{28} + 2 q^{29} - 3 q^{30} - 2 q^{32} + 4 q^{33} + 9 q^{34} - 13 q^{35} + 2 q^{36} + q^{37} - q^{38} + 2 q^{39} + 3 q^{40} - 3 q^{41} + 3 q^{42} + 11 q^{43} - 4 q^{44} - 3 q^{45} + 2 q^{46} + 9 q^{47} - 2 q^{48} - q^{49} - 3 q^{50} + 9 q^{51} - 2 q^{52} - 4 q^{53} + 2 q^{54} + 6 q^{55} - 3 q^{56} - q^{57} - 2 q^{58} + q^{59} + 3 q^{60} + 2 q^{61} + 3 q^{63} + 2 q^{64} + 20 q^{65} - 4 q^{66} - 9 q^{68} + 2 q^{69} + 13 q^{70} - 16 q^{71} - 2 q^{72} + 12 q^{73} - q^{74} - 3 q^{75} + q^{76} - 6 q^{77} - 2 q^{78} - 16 q^{79} - 3 q^{80} + 2 q^{81} + 3 q^{82} - 14 q^{83} - 3 q^{84} + 5 q^{85} - 11 q^{86} - 2 q^{87} + 4 q^{88} + 2 q^{89} + 3 q^{90} - 20 q^{91} - 2 q^{92} - 9 q^{94} + 7 q^{95} + 2 q^{96} + 18 q^{97} + q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 + 3 * q^10 - 4 * q^11 - 2 * q^12 - 2 * q^13 - 3 * q^14 + 3 * q^15 + 2 * q^16 - 9 * q^17 - 2 * q^18 + q^19 - 3 * q^20 - 3 * q^21 + 4 * q^22 - 2 * q^23 + 2 * q^24 + 3 * q^25 + 2 * q^26 - 2 * q^27 + 3 * q^28 + 2 * q^29 - 3 * q^30 - 2 * q^32 + 4 * q^33 + 9 * q^34 - 13 * q^35 + 2 * q^36 + q^37 - q^38 + 2 * q^39 + 3 * q^40 - 3 * q^41 + 3 * q^42 + 11 * q^43 - 4 * q^44 - 3 * q^45 + 2 * q^46 + 9 * q^47 - 2 * q^48 - q^49 - 3 * q^50 + 9 * q^51 - 2 * q^52 - 4 * q^53 + 2 * q^54 + 6 * q^55 - 3 * q^56 - q^57 - 2 * q^58 + q^59 + 3 * q^60 + 2 * q^61 + 3 * q^63 + 2 * q^64 + 20 * q^65 - 4 * q^66 - 9 * q^68 + 2 * q^69 + 13 * q^70 - 16 * q^71 - 2 * q^72 + 12 * q^73 - q^74 - 3 * q^75 + q^76 - 6 * q^77 - 2 * q^78 - 16 * q^79 - 3 * q^80 + 2 * q^81 + 3 * q^82 - 14 * q^83 - 3 * q^84 + 5 * q^85 - 11 * q^86 - 2 * q^87 + 4 * q^88 + 2 * q^89 + 3 * q^90 - 20 * q^91 - 2 * q^92 - 9 * q^94 + 7 * q^95 + 2 * q^96 + 18 * q^97 + q^98 - 4 * 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
$5$	0.561553	0.251134	0.125567	−	0.992085i	$-0.459925\pi$
$5$	0.561553	0.251134	0.125567	+	0.992085i	$0.459925\pi$
$6$	1.00000	0.408248
$7$	−0.561553	−0.212247	−0.106124	−	0.994353i	$-0.533844\pi$
$7$	−0.561553	−0.212247	−0.106124	+	0.994353i	$0.533844\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−0.561553	−0.177579
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.00000	−0.288675
$13$	3.12311	0.866194	0.433097	−	0.901347i	$-0.357421\pi$
$13$	3.12311	0.866194	0.433097	+	0.901347i	$0.357421\pi$
$14$	0.561553	0.150081
$15$	−0.561553	−0.144992
$16$	1.00000	0.250000
$17$	−6.56155	−1.59141	−0.795705	−	0.605684i	$-0.792900\pi$
$17$	−6.56155	−1.59141	−0.795705	+	0.605684i	$0.792900\pi$
$18$	−1.00000	−0.235702
$19$	2.56155	0.587661	0.293830	−	0.955858i	$-0.405070\pi$
$19$	2.56155	0.587661	0.293830	+	0.955858i	$0.405070\pi$
$20$	0.561553	0.125567
$21$	0.561553	0.122541
$22$	2.00000	0.426401
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	−4.68466	−0.936932
$26$	−3.12311	−0.612491
$27$	−1.00000	−0.192450
$28$	−0.561553	−0.106124
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0.561553	0.102525
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	6.56155	1.12530
$35$	−0.315342	−0.0533025
$36$	1.00000	0.166667
$37$	2.56155	0.421117	0.210558	−	0.977581i	$-0.432472\pi$
$37$	2.56155	0.421117	0.210558	+	0.977581i	$0.432472\pi$
$38$	−2.56155	−0.415539
$39$	−3.12311	−0.500097
$40$	−0.561553	−0.0887893
$41$	0.561553	0.0876998	0.0438499	−	0.999038i	$-0.486038\pi$
$41$	0.561553	0.0876998	0.0438499	+	0.999038i	$0.486038\pi$
$42$	−0.561553	−0.0866495
$43$	11.6847	1.78189	0.890947	−	0.454108i	$-0.150042\pi$
$43$	11.6847	1.78189	0.890947	+	0.454108i	$0.150042\pi$
$44$	−2.00000	−0.301511
$45$	0.561553	0.0837114
$46$	1.00000	0.147442
$47$	6.56155	0.957101	0.478550	−	0.878060i	$-0.341162\pi$
$47$	6.56155	0.957101	0.478550	+	0.878060i	$0.341162\pi$
$48$	−1.00000	−0.144338
$49$	−6.68466	−0.954951
$50$	4.68466	0.662511
$51$	6.56155	0.918801
$52$	3.12311	0.433097
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	−1.12311	−0.151440
$56$	0.561553	0.0750407
$57$	−2.56155	−0.339286
$58$	−1.00000	−0.131306
$59$	2.56155	0.333486	0.166743	−	0.986000i	$-0.446675\pi$
$59$	2.56155	0.333486	0.166743	+	0.986000i	$0.446675\pi$
$60$	−0.561553	−0.0724962
$61$	5.12311	0.655946	0.327973	−	0.944687i	$-0.393634\pi$
$61$	5.12311	0.655946	0.327973	+	0.944687i	$0.393634\pi$
$62$	0	0
$63$	−0.561553	−0.0707490
$64$	1.00000	0.125000
$65$	1.75379	0.217531
$66$	−2.00000	−0.246183
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−6.56155	−0.795705
$69$	1.00000	0.120386
$70$	0.315342	0.0376905
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−1.00000	−0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−2.56155	−0.297774
$75$	4.68466	0.540938
$76$	2.56155	0.293830
$77$	1.12311	0.127990
$78$	3.12311	0.353622
$79$	−16.2462	−1.82784	−0.913921	−	0.405893i	$-0.866961\pi$
$79$	−16.2462	−1.82784	−0.913921	+	0.405893i	$0.866961\pi$
$80$	0.561553	0.0627835
$81$	1.00000	0.111111
$82$	−0.561553	−0.0620131
$83$	−11.1231	−1.22092	−0.610460	−	0.792047i	$-0.709015\pi$
$83$	−11.1231	−1.22092	−0.610460	+	0.792047i	$0.709015\pi$
$84$	0.561553	0.0612704
$85$	−3.68466	−0.399657
$86$	−11.6847	−1.25999
$87$	−1.00000	−0.107211
$88$	2.00000	0.213201
$89$	5.12311	0.543048	0.271524	−	0.962432i	$-0.412472\pi$
$89$	5.12311	0.543048	0.271524	+	0.962432i	$0.412472\pi$
$90$	−0.561553	−0.0591929
$91$	−1.75379	−0.183847
$92$	−1.00000	−0.104257
$93$	0	0
$94$	−6.56155	−0.676772
$95$	1.43845	0.147582
$96$	1.00000	0.102062
$97$	4.87689	0.495174	0.247587	−	0.968866i	$-0.420362\pi$
$97$	4.87689	0.495174	0.247587	+	0.968866i	$0.420362\pi$
$98$	6.68466	0.675252
$99$	−2.00000	−0.201008
$100$	−4.68466	−0.468466
$101$	12.2462	1.21854	0.609272	−	0.792961i	$-0.291462\pi$
$101$	12.2462	1.21854	0.609272	+	0.792961i	$0.291462\pi$
$102$	−6.56155	−0.649691
$103$	14.8078	1.45905	0.729526	−	0.683953i	$-0.239741\pi$
$103$	14.8078	1.45905	0.729526	+	0.683953i	$0.239741\pi$
$104$	−3.12311	−0.306246
$105$	0.315342	0.0307742
$106$	2.00000	0.194257
$107$	15.9309	1.54010	0.770048	−	0.637986i	$-0.220232\pi$
$107$	15.9309	1.54010	0.770048	+	0.637986i	$0.220232\pi$
$108$	−1.00000	−0.0962250
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	1.12311	0.107084
$111$	−2.56155	−0.243132
$112$	−0.561553	−0.0530618
$113$	−13.9309	−1.31051	−0.655253	−	0.755410i	$-0.727438\pi$
$113$	−13.9309	−1.31051	−0.655253	+	0.755410i	$0.727438\pi$
$114$	2.56155	0.239911
$115$	−0.561553	−0.0523651
$116$	1.00000	0.0928477
$117$	3.12311	0.288731
$118$	−2.56155	−0.235810
$119$	3.68466	0.337772
$120$	0.561553	0.0512625
$121$	−7.00000	−0.636364
$122$	−5.12311	−0.463824
$123$	−0.561553	−0.0506335
$124$	0	0
$125$	−5.43845	−0.486430
$126$	0.561553	0.0500271
$127$	17.1231	1.51943	0.759715	−	0.650256i	$-0.225338\pi$
$127$	17.1231	1.51943	0.759715	+	0.650256i	$0.225338\pi$
$128$	−1.00000	−0.0883883
$129$	−11.6847	−1.02878
$130$	−1.75379	−0.153817
$131$	10.2462	0.895216	0.447608	−	0.894230i	$-0.352276\pi$
$131$	10.2462	0.895216	0.447608	+	0.894230i	$0.352276\pi$
$132$	2.00000	0.174078
$133$	−1.43845	−0.124729
$134$	0	0
$135$	−0.561553	−0.0483308
$136$	6.56155	0.562649
$137$	11.3693	0.971346	0.485673	−	0.874140i	$-0.338575\pi$
$137$	11.3693	0.971346	0.485673	+	0.874140i	$0.338575\pi$
$138$	−1.00000	−0.0851257
$139$	−6.87689	−0.583291	−0.291645	−	0.956527i	$-0.594203\pi$
$139$	−6.87689	−0.583291	−0.291645	+	0.956527i	$0.594203\pi$
$140$	−0.315342	−0.0266512
$141$	−6.56155	−0.552582
$142$	8.00000	0.671345
$143$	−6.24621	−0.522334
$144$	1.00000	0.0833333
$145$	0.561553	0.0466344
$146$	−6.00000	−0.496564
$147$	6.68466	0.551341
$148$	2.56155	0.210558
$149$	−2.31534	−0.189680	−0.0948401	−	0.995493i	$-0.530234\pi$
$149$	−2.31534	−0.189680	−0.0948401	+	0.995493i	$0.530234\pi$
$150$	−4.68466	−0.382501
$151$	−15.6847	−1.27640	−0.638200	−	0.769871i	$-0.720321\pi$
$151$	−15.6847	−1.27640	−0.638200	+	0.769871i	$0.720321\pi$
$152$	−2.56155	−0.207769
$153$	−6.56155	−0.530470
$154$	−1.12311	−0.0905024
$155$	0	0
$156$	−3.12311	−0.250049
$157$	−7.68466	−0.613303	−0.306651	−	0.951822i	$-0.599209\pi$
$157$	−7.68466	−0.613303	−0.306651	+	0.951822i	$0.599209\pi$
$158$	16.2462	1.29248
$159$	2.00000	0.158610
$160$	−0.561553	−0.0443946
$161$	0.561553	0.0442566
$162$	−1.00000	−0.0785674
$163$	15.6847	1.22852	0.614259	−	0.789105i	$-0.289455\pi$
$163$	15.6847	1.22852	0.614259	+	0.789105i	$0.289455\pi$
$164$	0.561553	0.0438499
$165$	1.12311	0.0874337
$166$	11.1231	0.863320
$167$	−2.87689	−0.222621	−0.111310	−	0.993786i	$-0.535505\pi$
$167$	−2.87689	−0.222621	−0.111310	+	0.993786i	$0.535505\pi$
$168$	−0.561553	−0.0433247
$169$	−3.24621	−0.249709
$170$	3.68466	0.282600
$171$	2.56155	0.195887
$172$	11.6847	0.890947
$173$	−21.0540	−1.60070	−0.800352	−	0.599530i	$-0.795354\pi$
$173$	−21.0540	−1.60070	−0.800352	+	0.599530i	$0.795354\pi$
$174$	1.00000	0.0758098
$175$	2.63068	0.198861
$176$	−2.00000	−0.150756
$177$	−2.56155	−0.192538
$178$	−5.12311	−0.383993
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0.561553	0.0418557
$181$	−6.24621	−0.464277	−0.232139	−	0.972683i	$-0.574572\pi$
$181$	−6.24621	−0.464277	−0.232139	+	0.972683i	$0.574572\pi$
$182$	1.75379	0.129999
$183$	−5.12311	−0.378711
$184$	1.00000	0.0737210
$185$	1.43845	0.105757
$186$	0	0
$187$	13.1231	0.959657
$188$	6.56155	0.478550
$189$	0.561553	0.0408470
$190$	−1.43845	−0.104356
$191$	17.4384	1.26180	0.630901	−	0.775863i	$-0.282685\pi$
$191$	17.4384	1.26180	0.630901	+	0.775863i	$0.282685\pi$
$192$	−1.00000	−0.0721688
$193$	0.876894	0.0631202	0.0315601	−	0.999502i	$-0.489952\pi$
$193$	0.876894	0.0631202	0.0315601	+	0.999502i	$0.489952\pi$
$194$	−4.87689	−0.350141
$195$	−1.75379	−0.125591
$196$	−6.68466	−0.477476
$197$	−9.68466	−0.690003	−0.345002	−	0.938602i	$-0.612122\pi$
$197$	−9.68466	−0.690003	−0.345002	+	0.938602i	$0.612122\pi$
$198$	2.00000	0.142134
$199$	17.3693	1.23128	0.615639	−	0.788028i	$-0.288898\pi$
$199$	17.3693	1.23128	0.615639	+	0.788028i	$0.288898\pi$
$200$	4.68466	0.331255
$201$	0	0
$202$	−12.2462	−0.861640
$203$	−0.561553	−0.0394133
$204$	6.56155	0.459401
$205$	0.315342	0.0220244
$206$	−14.8078	−1.03171
$207$	−1.00000	−0.0695048
$208$	3.12311	0.216548
$209$	−5.12311	−0.354373
$210$	−0.315342	−0.0217606
$211$	18.5616	1.27783	0.638915	−	0.769277i	$-0.279384\pi$
$211$	18.5616	1.27783	0.638915	+	0.769277i	$0.279384\pi$
$212$	−2.00000	−0.137361
$213$	8.00000	0.548151
$214$	−15.9309	−1.08901
$215$	6.56155	0.447494
$216$	1.00000	0.0680414
$217$	0	0
$218$	−12.0000	−0.812743
$219$	−6.00000	−0.405442
$220$	−1.12311	−0.0757198
$221$	−20.4924	−1.37847
$222$	2.56155	0.171920
$223$	1.75379	0.117442	0.0587212	−	0.998274i	$-0.481298\pi$
$223$	1.75379	0.117442	0.0587212	+	0.998274i	$0.481298\pi$
$224$	0.561553	0.0375203
$225$	−4.68466	−0.312311
$226$	13.9309	0.926668
$227$	21.6847	1.43926	0.719631	−	0.694357i	$-0.244311\pi$
$227$	21.6847	1.43926	0.719631	+	0.694357i	$0.244311\pi$
$228$	−2.56155	−0.169643
$229$	5.43845	0.359383	0.179691	−	0.983723i	$-0.442490\pi$
$229$	5.43845	0.359383	0.179691	+	0.983723i	$0.442490\pi$
$230$	0.561553	0.0370277
$231$	−1.12311	−0.0738949
$232$	−1.00000	−0.0656532
$233$	−3.12311	−0.204601	−0.102301	−	0.994754i	$-0.532620\pi$
$233$	−3.12311	−0.204601	−0.102301	+	0.994754i	$0.532620\pi$
$234$	−3.12311	−0.204164
$235$	3.68466	0.240361
$236$	2.56155	0.166743
$237$	16.2462	1.05530
$238$	−3.68466	−0.238841
$239$	−7.36932	−0.476681	−0.238341	−	0.971182i	$-0.576604\pi$
$239$	−7.36932	−0.476681	−0.238341	+	0.971182i	$0.576604\pi$
$240$	−0.561553	−0.0362481
$241$	−25.0540	−1.61387	−0.806934	−	0.590641i	$-0.798875\pi$
$241$	−25.0540	−1.61387	−0.806934	+	0.590641i	$0.798875\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	5.12311	0.327973
$245$	−3.75379	−0.239821
$246$	0.561553	0.0358033
$247$	8.00000	0.509028
$248$	0	0
$249$	11.1231	0.704898
$250$	5.43845	0.343958
$251$	−7.12311	−0.449606	−0.224803	−	0.974404i	$-0.572174\pi$
$251$	−7.12311	−0.449606	−0.224803	+	0.974404i	$0.572174\pi$
$252$	−0.561553	−0.0353745
$253$	2.00000	0.125739
$254$	−17.1231	−1.07440
$255$	3.68466	0.230742
$256$	1.00000	0.0625000
$257$	15.1231	0.943353	0.471677	−	0.881772i	$-0.343649\pi$
$257$	15.1231	0.943353	0.471677	+	0.881772i	$0.343649\pi$
$258$	11.6847	0.727455
$259$	−1.43845	−0.0893808
$260$	1.75379	0.108765
$261$	1.00000	0.0618984
$262$	−10.2462	−0.633013
$263$	−20.1771	−1.24417	−0.622086	−	0.782949i	$-0.713715\pi$
$263$	−20.1771	−1.24417	−0.622086	+	0.782949i	$0.713715\pi$
$264$	−2.00000	−0.123091
$265$	−1.12311	−0.0689918
$266$	1.43845	0.0881969
$267$	−5.12311	−0.313529
$268$	0	0
$269$	12.2462	0.746665	0.373332	−	0.927698i	$-0.378215\pi$
$269$	12.2462	0.746665	0.373332	+	0.927698i	$0.378215\pi$
$270$	0.561553	0.0341750
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	−6.56155	−0.397853
$273$	1.75379	0.106144
$274$	−11.3693	−0.686846
$275$	9.36932	0.564991
$276$	1.00000	0.0601929
$277$	22.4924	1.35144	0.675719	−	0.737159i	$-0.263833\pi$
$277$	22.4924	1.35144	0.675719	+	0.737159i	$0.263833\pi$
$278$	6.87689	0.412449
$279$	0	0
$280$	0.315342	0.0188453
$281$	−27.8617	−1.66209	−0.831046	−	0.556204i	$-0.812257\pi$
$281$	−27.8617	−1.66209	−0.831046	+	0.556204i	$0.812257\pi$
$282$	6.56155	0.390735
$283$	19.3693	1.15139	0.575693	−	0.817666i	$-0.304732\pi$
$283$	19.3693	1.15139	0.575693	+	0.817666i	$0.304732\pi$
$284$	−8.00000	−0.474713
$285$	−1.43845	−0.0852063
$286$	6.24621	0.369346
$287$	−0.315342	−0.0186140
$288$	−1.00000	−0.0589256
$289$	26.0540	1.53259
$290$	−0.561553	−0.0329755
$291$	−4.87689	−0.285889
$292$	6.00000	0.351123
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	−6.68466	−0.389857
$295$	1.43845	0.0837496
$296$	−2.56155	−0.148887
$297$	2.00000	0.116052
$298$	2.31534	0.134124
$299$	−3.12311	−0.180614
$300$	4.68466	0.270469
$301$	−6.56155	−0.378202
$302$	15.6847	0.902551
$303$	−12.2462	−0.703526
$304$	2.56155	0.146915
$305$	2.87689	0.164730
$306$	6.56155	0.375099
$307$	24.4924	1.39786	0.698928	−	0.715192i	$-0.253661\pi$
$307$	24.4924	1.39786	0.698928	+	0.715192i	$0.253661\pi$
$308$	1.12311	0.0639949
$309$	−14.8078	−0.842384
$310$	0	0
$311$	27.0540	1.53409	0.767045	−	0.641593i	$-0.221726\pi$
$311$	27.0540	1.53409	0.767045	+	0.641593i	$0.221726\pi$
$312$	3.12311	0.176811
$313$	30.8078	1.74136	0.870679	−	0.491852i	$-0.163680\pi$
$313$	30.8078	1.74136	0.870679	+	0.491852i	$0.163680\pi$
$314$	7.68466	0.433670
$315$	−0.315342	−0.0177675
$316$	−16.2462	−0.913921
$317$	16.8769	0.947901	0.473950	−	0.880552i	$-0.342828\pi$
$317$	16.8769	0.947901	0.473950	+	0.880552i	$0.342828\pi$
$318$	−2.00000	−0.112154
$319$	−2.00000	−0.111979
$320$	0.561553	0.0313918
$321$	−15.9309	−0.889174
$322$	−0.561553	−0.0312941
$323$	−16.8078	−0.935209
$324$	1.00000	0.0555556
$325$	−14.6307	−0.811564
$326$	−15.6847	−0.868693
$327$	−12.0000	−0.663602
$328$	−0.561553	−0.0310066
$329$	−3.68466	−0.203142
$330$	−1.12311	−0.0618249
$331$	21.4384	1.17836	0.589182	−	0.808000i	$-0.299450\pi$
$331$	21.4384	1.17836	0.589182	+	0.808000i	$0.299450\pi$
$332$	−11.1231	−0.610460
$333$	2.56155	0.140372
$334$	2.87689	0.157417
$335$	0	0
$336$	0.561553	0.0306352
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	3.24621	0.176571
$339$	13.9309	0.756621
$340$	−3.68466	−0.199829
$341$	0	0
$342$	−2.56155	−0.138513
$343$	7.68466	0.414933
$344$	−11.6847	−0.629995
$345$	0.561553	0.0302330
$346$	21.0540	1.13187
$347$	−8.80776	−0.472826	−0.236413	−	0.971653i	$-0.575972\pi$
$347$	−8.80776	−0.472826	−0.236413	+	0.971653i	$0.575972\pi$
$348$	−1.00000	−0.0536056
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	−2.63068	−0.140616
$351$	−3.12311	−0.166699
$352$	2.00000	0.106600
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	2.56155	0.136145
$355$	−4.49242	−0.238433
$356$	5.12311	0.271524
$357$	−3.68466	−0.195013
$358$	−16.0000	−0.845626
$359$	16.8078	0.887080	0.443540	−	0.896255i	$-0.353722\pi$
$359$	16.8078	0.887080	0.443540	+	0.896255i	$0.353722\pi$
$360$	−0.561553	−0.0295964
$361$	−12.4384	−0.654655
$362$	6.24621	0.328294
$363$	7.00000	0.367405
$364$	−1.75379	−0.0919235
$365$	3.36932	0.176358
$366$	5.12311	0.267789
$367$	16.7386	0.873750	0.436875	−	0.899522i	$-0.356085\pi$
$367$	16.7386	0.873750	0.436875	+	0.899522i	$0.356085\pi$
$368$	−1.00000	−0.0521286
$369$	0.561553	0.0292333
$370$	−1.43845	−0.0747813
$371$	1.12311	0.0583087
$372$	0	0
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	−13.1231	−0.678580
$375$	5.43845	0.280840
$376$	−6.56155	−0.338386
$377$	3.12311	0.160848
$378$	−0.561553	−0.0288832
$379$	32.4924	1.66902	0.834512	−	0.550990i	$-0.185750\pi$
$379$	32.4924	1.66902	0.834512	+	0.550990i	$0.185750\pi$
$380$	1.43845	0.0737908
$381$	−17.1231	−0.877243
$382$	−17.4384	−0.892229
$383$	11.3693	0.580945	0.290472	−	0.956883i	$-0.406188\pi$
$383$	11.3693	0.580945	0.290472	+	0.956883i	$0.406188\pi$
$384$	1.00000	0.0510310
$385$	0.630683	0.0321426
$386$	−0.876894	−0.0446327
$387$	11.6847	0.593965
$388$	4.87689	0.247587
$389$	−7.61553	−0.386123	−0.193061	−	0.981187i	$-0.561842\pi$
$389$	−7.61553	−0.386123	−0.193061	+	0.981187i	$0.561842\pi$
$390$	1.75379	0.0888065
$391$	6.56155	0.331832
$392$	6.68466	0.337626
$393$	−10.2462	−0.516853
$394$	9.68466	0.487906
$395$	−9.12311	−0.459033
$396$	−2.00000	−0.100504
$397$	26.4924	1.32962	0.664808	−	0.747014i	$-0.268513\pi$
$397$	26.4924	1.32962	0.664808	+	0.747014i	$0.268513\pi$
$398$	−17.3693	−0.870645
$399$	1.43845	0.0720124
$400$	−4.68466	−0.234233
$401$	15.3693	0.767507	0.383754	−	0.923436i	$-0.374631\pi$
$401$	15.3693	0.767507	0.383754	+	0.923436i	$0.374631\pi$
$402$	0	0
$403$	0	0
$404$	12.2462	0.609272
$405$	0.561553	0.0279038
$406$	0.561553	0.0278694
$407$	−5.12311	−0.253943
$408$	−6.56155	−0.324845
$409$	19.1231	0.945577	0.472788	−	0.881176i	$-0.343248\pi$
$409$	19.1231	0.945577	0.472788	+	0.881176i	$0.343248\pi$
$410$	−0.315342	−0.0155736
$411$	−11.3693	−0.560807
$412$	14.8078	0.729526
$413$	−1.43845	−0.0707814
$414$	1.00000	0.0491473
$415$	−6.24621	−0.306614
$416$	−3.12311	−0.153123
$417$	6.87689	0.336763
$418$	5.12311	0.250579
$419$	−3.93087	−0.192036	−0.0960178	−	0.995380i	$-0.530611\pi$
$419$	−3.93087	−0.192036	−0.0960178	+	0.995380i	$0.530611\pi$
$420$	0.315342	0.0153871
$421$	17.6155	0.858528	0.429264	−	0.903179i	$-0.358773\pi$
$421$	17.6155	0.858528	0.429264	+	0.903179i	$0.358773\pi$
$422$	−18.5616	−0.903562
$423$	6.56155	0.319034
$424$	2.00000	0.0971286
$425$	30.7386	1.49104
$426$	−8.00000	−0.387601
$427$	−2.87689	−0.139223
$428$	15.9309	0.770048
$429$	6.24621	0.301570
$430$	−6.56155	−0.316426
$431$	−4.00000	−0.192673	−0.0963366	−	0.995349i	$-0.530713\pi$
$431$	−4.00000	−0.192673	−0.0963366	+	0.995349i	$0.530713\pi$
$432$	−1.00000	−0.0481125
$433$	10.4924	0.504234	0.252117	−	0.967697i	$-0.418873\pi$
$433$	10.4924	0.504234	0.252117	+	0.967697i	$0.418873\pi$
$434$	0	0
$435$	−0.561553	−0.0269244
$436$	12.0000	0.574696
$437$	−2.56155	−0.122536
$438$	6.00000	0.286691
$439$	−6.56155	−0.313166	−0.156583	−	0.987665i	$-0.550048\pi$
$439$	−6.56155	−0.313166	−0.156583	+	0.987665i	$0.550048\pi$
$440$	1.12311	0.0535420
$441$	−6.68466	−0.318317
$442$	20.4924	0.974725
$443$	−22.2462	−1.05695	−0.528475	−	0.848949i	$-0.677236\pi$
$443$	−22.2462	−1.05695	−0.528475	+	0.848949i	$0.677236\pi$
$444$	−2.56155	−0.121566
$445$	2.87689	0.136378
$446$	−1.75379	−0.0830443
$447$	2.31534	0.109512
$448$	−0.561553	−0.0265309
$449$	−7.43845	−0.351042	−0.175521	−	0.984476i	$-0.556161\pi$
$449$	−7.43845	−0.351042	−0.175521	+	0.984476i	$0.556161\pi$
$450$	4.68466	0.220837
$451$	−1.12311	−0.0528850
$452$	−13.9309	−0.655253
$453$	15.6847	0.736930
$454$	−21.6847	−1.01771
$455$	−0.984845	−0.0461702
$456$	2.56155	0.119956
$457$	24.5616	1.14894	0.574470	−	0.818525i	$-0.305208\pi$
$457$	24.5616	1.14894	0.574470	+	0.818525i	$0.305208\pi$
$458$	−5.43845	−0.254122
$459$	6.56155	0.306267
$460$	−0.561553	−0.0261825
$461$	−36.1080	−1.68171	−0.840857	−	0.541257i	$-0.817949\pi$
$461$	−36.1080	−1.68171	−0.840857	+	0.541257i	$0.817949\pi$
$462$	1.12311	0.0522516
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	3.12311	0.144675
$467$	−25.3693	−1.17395	−0.586976	−	0.809604i	$-0.699682\pi$
$467$	−25.3693	−1.17395	−0.586976	+	0.809604i	$0.699682\pi$
$468$	3.12311	0.144366
$469$	0	0
$470$	−3.68466	−0.169961
$471$	7.68466	0.354090
$472$	−2.56155	−0.117905
$473$	−23.3693	−1.07452
$474$	−16.2462	−0.746213
$475$	−12.0000	−0.550598
$476$	3.68466	0.168886
$477$	−2.00000	−0.0915737
$478$	7.36932	0.337065
$479$	3.50758	0.160265	0.0801327	−	0.996784i	$-0.474466\pi$
$479$	3.50758	0.160265	0.0801327	+	0.996784i	$0.474466\pi$
$480$	0.561553	0.0256313
$481$	8.00000	0.364769
$482$	25.0540	1.14118
$483$	−0.561553	−0.0255515
$484$	−7.00000	−0.318182
$485$	2.73863	0.124355
$486$	1.00000	0.0453609
$487$	1.43845	0.0651823	0.0325911	−	0.999469i	$-0.489624\pi$
$487$	1.43845	0.0651823	0.0325911	+	0.999469i	$0.489624\pi$
$488$	−5.12311	−0.231912
$489$	−15.6847	−0.709285
$490$	3.75379	0.169579
$491$	0.630683	0.0284623	0.0142312	−	0.999899i	$-0.495470\pi$
$491$	0.630683	0.0284623	0.0142312	+	0.999899i	$0.495470\pi$
$492$	−0.561553	−0.0253168
$493$	−6.56155	−0.295517
$494$	−8.00000	−0.359937
$495$	−1.12311	−0.0504798
$496$	0	0
$497$	4.49242	0.201513
$498$	−11.1231	−0.498438
$499$	37.6155	1.68390	0.841951	−	0.539554i	$-0.181407\pi$
$499$	37.6155	1.68390	0.841951	+	0.539554i	$0.181407\pi$
$500$	−5.43845	−0.243215
$501$	2.87689	0.128530
$502$	7.12311	0.317920
$503$	−15.0540	−0.671224	−0.335612	−	0.942000i	$-0.608943\pi$
$503$	−15.0540	−0.671224	−0.335612	+	0.942000i	$0.608943\pi$
$504$	0.561553	0.0250136
$505$	6.87689	0.306018
$506$	−2.00000	−0.0889108
$507$	3.24621	0.144169
$508$	17.1231	0.759715
$509$	−45.0540	−1.99698	−0.998491	−	0.0549125i	$-0.982512\pi$
$509$	−45.0540	−1.99698	−0.998491	+	0.0549125i	$0.982512\pi$
$510$	−3.68466	−0.163159
$511$	−3.36932	−0.149050
$512$	−1.00000	−0.0441942
$513$	−2.56155	−0.113095
$514$	−15.1231	−0.667052
$515$	8.31534	0.366418
$516$	−11.6847	−0.514388
$517$	−13.1231	−0.577154
$518$	1.43845	0.0632017
$519$	21.0540	0.924167
$520$	−1.75379	−0.0769087
$521$	23.8617	1.04540	0.522701	−	0.852516i	$-0.324924\pi$
$521$	23.8617	1.04540	0.522701	+	0.852516i	$0.324924\pi$
$522$	−1.00000	−0.0437688
$523$	21.1231	0.923649	0.461824	−	0.886971i	$-0.347195\pi$
$523$	21.1231	0.923649	0.461824	+	0.886971i	$0.347195\pi$
$524$	10.2462	0.447608
$525$	−2.63068	−0.114812
$526$	20.1771	0.879763
$527$	0	0
$528$	2.00000	0.0870388
$529$	1.00000	0.0434783
$530$	1.12311	0.0487846
$531$	2.56155	0.111162
$532$	−1.43845	−0.0623646
$533$	1.75379	0.0759650
$534$	5.12311	0.221698
$535$	8.94602	0.386770
$536$	0	0
$537$	−16.0000	−0.690451
$538$	−12.2462	−0.527972
$539$	13.3693	0.575857
$540$	−0.561553	−0.0241654
$541$	−26.3153	−1.13138	−0.565692	−	0.824616i	$-0.691391\pi$
$541$	−26.3153	−1.13138	−0.565692	+	0.824616i	$0.691391\pi$
$542$	−4.00000	−0.171815
$543$	6.24621	0.268051
$544$	6.56155	0.281324
$545$	6.73863	0.288651
$546$	−1.75379	−0.0750552
$547$	11.3693	0.486117	0.243058	−	0.970012i	$-0.421849\pi$
$547$	11.3693	0.486117	0.243058	+	0.970012i	$0.421849\pi$
$548$	11.3693	0.485673
$549$	5.12311	0.218649
$550$	−9.36932	−0.399509
$551$	2.56155	0.109126
$552$	−1.00000	−0.0425628
$553$	9.12311	0.387954
$554$	−22.4924	−0.955611
$555$	−1.43845	−0.0610587
$556$	−6.87689	−0.291645
$557$	32.5616	1.37968	0.689839	−	0.723963i	$-0.257681\pi$
$557$	32.5616	1.37968	0.689839	+	0.723963i	$0.257681\pi$
$558$	0	0
$559$	36.4924	1.54347
$560$	−0.315342	−0.0133256
$561$	−13.1231	−0.554058
$562$	27.8617	1.17528
$563$	42.9848	1.81160	0.905798	−	0.423711i	$-0.139273\pi$
$563$	42.9848	1.81160	0.905798	+	0.423711i	$0.139273\pi$
$564$	−6.56155	−0.276291
$565$	−7.82292	−0.329113
$566$	−19.3693	−0.814153
$567$	−0.561553	−0.0235830
$568$	8.00000	0.335673
$569$	−35.6847	−1.49598	−0.747989	−	0.663711i	$-0.768981\pi$
$569$	−35.6847	−1.49598	−0.747989	+	0.663711i	$0.768981\pi$
$570$	1.43845	0.0602499
$571$	34.8769	1.45955	0.729776	−	0.683686i	$-0.239624\pi$
$571$	34.8769	1.45955	0.729776	+	0.683686i	$0.239624\pi$
$572$	−6.24621	−0.261167
$573$	−17.4384	−0.728502
$574$	0.315342	0.0131621
$575$	4.68466	0.195364
$576$	1.00000	0.0416667
$577$	−32.8769	−1.36868	−0.684342	−	0.729162i	$-0.739910\pi$
$577$	−32.8769	−1.36868	−0.684342	+	0.729162i	$0.739910\pi$
$578$	−26.0540	−1.08370
$579$	−0.876894	−0.0364425
$580$	0.561553	0.0233172
$581$	6.24621	0.259137
$582$	4.87689	0.202154
$583$	4.00000	0.165663
$584$	−6.00000	−0.248282
$585$	1.75379	0.0725102
$586$	−22.0000	−0.908812
$587$	−32.8078	−1.35412	−0.677061	−	0.735927i	$-0.736747\pi$
$587$	−32.8078	−1.35412	−0.677061	+	0.735927i	$0.736747\pi$
$588$	6.68466	0.275671
$589$	0	0
$590$	−1.43845	−0.0592199
$591$	9.68466	0.398374
$592$	2.56155	0.105279
$593$	10.6307	0.436550	0.218275	−	0.975887i	$-0.429957\pi$
$593$	10.6307	0.436550	0.218275	+	0.975887i	$0.429957\pi$
$594$	−2.00000	−0.0820610
$595$	2.06913	0.0848261
$596$	−2.31534	−0.0948401
$597$	−17.3693	−0.710879
$598$	3.12311	0.127713
$599$	26.2462	1.07239	0.536196	−	0.844094i	$-0.319861\pi$
$599$	26.2462	1.07239	0.536196	+	0.844094i	$0.319861\pi$
$600$	−4.68466	−0.191250
$601$	7.12311	0.290558	0.145279	−	0.989391i	$-0.453592\pi$
$601$	7.12311	0.290558	0.145279	+	0.989391i	$0.453592\pi$
$602$	6.56155	0.267429
$603$	0	0
$604$	−15.6847	−0.638200
$605$	−3.93087	−0.159813
$606$	12.2462	0.497468
$607$	−26.7386	−1.08529	−0.542644	−	0.839963i	$-0.682577\pi$
$607$	−26.7386	−1.08529	−0.542644	+	0.839963i	$0.682577\pi$
$608$	−2.56155	−0.103885
$609$	0.561553	0.0227553
$610$	−2.87689	−0.116482
$611$	20.4924	0.829035
$612$	−6.56155	−0.265235
$613$	−31.3693	−1.26699	−0.633497	−	0.773745i	$-0.718381\pi$
$613$	−31.3693	−1.26699	−0.633497	+	0.773745i	$0.718381\pi$
$614$	−24.4924	−0.988434
$615$	−0.315342	−0.0127158
$616$	−1.12311	−0.0452512
$617$	24.8078	0.998723	0.499361	−	0.866394i	$-0.333568\pi$
$617$	24.8078	0.998723	0.499361	+	0.866394i	$0.333568\pi$
$618$	14.8078	0.595656
$619$	−2.56155	−0.102958	−0.0514788	−	0.998674i	$-0.516393\pi$
$619$	−2.56155	−0.102958	−0.0514788	+	0.998674i	$0.516393\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	−27.0540	−1.08477
$623$	−2.87689	−0.115260
$624$	−3.12311	−0.125024
$625$	20.3693	0.814773
$626$	−30.8078	−1.23133
$627$	5.12311	0.204597
$628$	−7.68466	−0.306651
$629$	−16.8078	−0.670169
$630$	0.315342	0.0125635
$631$	−26.1771	−1.04209	−0.521047	−	0.853528i	$-0.674458\pi$
$631$	−26.1771	−1.04209	−0.521047	+	0.853528i	$0.674458\pi$
$632$	16.2462	0.646240
$633$	−18.5616	−0.737755
$634$	−16.8769	−0.670267
$635$	9.61553	0.381581
$636$	2.00000	0.0793052
$637$	−20.8769	−0.827173
$638$	2.00000	0.0791808
$639$	−8.00000	−0.316475
$640$	−0.561553	−0.0221973
$641$	22.5616	0.891128	0.445564	−	0.895250i	$-0.353003\pi$
$641$	22.5616	0.891128	0.445564	+	0.895250i	$0.353003\pi$
$642$	15.9309	0.628741
$643$	−36.4924	−1.43912	−0.719560	−	0.694430i	$-0.755657\pi$
$643$	−36.4924	−1.43912	−0.719560	+	0.694430i	$0.755657\pi$
$644$	0.561553	0.0221283
$645$	−6.56155	−0.258361
$646$	16.8078	0.661293
$647$	5.12311	0.201410	0.100705	−	0.994916i	$-0.467890\pi$
$647$	5.12311	0.201410	0.100705	+	0.994916i	$0.467890\pi$
$648$	−1.00000	−0.0392837
$649$	−5.12311	−0.201099
$650$	14.6307	0.573863
$651$	0	0
$652$	15.6847	0.614259
$653$	13.5076	0.528592	0.264296	−	0.964442i	$-0.414860\pi$
$653$	13.5076	0.528592	0.264296	+	0.964442i	$0.414860\pi$
$654$	12.0000	0.469237
$655$	5.75379	0.224819
$656$	0.561553	0.0219250
$657$	6.00000	0.234082
$658$	3.68466	0.143643
$659$	−2.63068	−0.102477	−0.0512384	−	0.998686i	$-0.516317\pi$
$659$	−2.63068	−0.102477	−0.0512384	+	0.998686i	$0.516317\pi$
$660$	1.12311	0.0437168
$661$	−34.2462	−1.33202	−0.666012	−	0.745941i	$-0.732000\pi$
$661$	−34.2462	−1.33202	−0.666012	+	0.745941i	$0.732000\pi$
$662$	−21.4384	−0.833229
$663$	20.4924	0.795860
$664$	11.1231	0.431660
$665$	−0.807764	−0.0313237
$666$	−2.56155	−0.0992582
$667$	−1.00000	−0.0387202
$668$	−2.87689	−0.111310
$669$	−1.75379	−0.0678054
$670$	0	0
$671$	−10.2462	−0.395551
$672$	−0.561553	−0.0216624
$673$	−43.9309	−1.69341	−0.846705	−	0.532062i	$-0.821417\pi$
$673$	−43.9309	−1.69341	−0.846705	+	0.532062i	$0.821417\pi$
$674$	−6.00000	−0.231111
$675$	4.68466	0.180313
$676$	−3.24621	−0.124854
$677$	16.7386	0.643318	0.321659	−	0.946856i	$-0.395760\pi$
$677$	16.7386	0.643318	0.321659	+	0.946856i	$0.395760\pi$
$678$	−13.9309	−0.535012
$679$	−2.73863	−0.105099
$680$	3.68466	0.141300
$681$	−21.6847	−0.830958
$682$	0	0
$683$	−10.4233	−0.398836	−0.199418	−	0.979914i	$-0.563905\pi$
$683$	−10.4233	−0.398836	−0.199418	+	0.979914i	$0.563905\pi$
$684$	2.56155	0.0979434
$685$	6.38447	0.243938
$686$	−7.68466	−0.293402
$687$	−5.43845	−0.207490
$688$	11.6847	0.445473
$689$	−6.24621	−0.237962
$690$	−0.561553	−0.0213780
$691$	24.4924	0.931736	0.465868	−	0.884854i	$-0.345742\pi$
$691$	24.4924	0.931736	0.465868	+	0.884854i	$0.345742\pi$
$692$	−21.0540	−0.800352
$693$	1.12311	0.0426633
$694$	8.80776	0.334338
$695$	−3.86174	−0.146484
$696$	1.00000	0.0379049
$697$	−3.68466	−0.139566
$698$	22.0000	0.832712
$699$	3.12311	0.118127
$700$	2.63068	0.0994305
$701$	11.3002	0.426802	0.213401	−	0.976965i	$-0.431546\pi$
$701$	11.3002	0.426802	0.213401	+	0.976965i	$0.431546\pi$
$702$	3.12311	0.117874
$703$	6.56155	0.247474
$704$	−2.00000	−0.0753778
$705$	−3.68466	−0.138772
$706$	−14.0000	−0.526897
$707$	−6.87689	−0.258632
$708$	−2.56155	−0.0962690
$709$	−41.6155	−1.56290	−0.781452	−	0.623965i	$-0.785521\pi$
$709$	−41.6155	−1.56290	−0.781452	+	0.623965i	$0.785521\pi$
$710$	4.49242	0.168598
$711$	−16.2462	−0.609281
$712$	−5.12311	−0.191997
$713$	0	0
$714$	3.68466	0.137895
$715$	−3.50758	−0.131176
$716$	16.0000	0.597948
$717$	7.36932	0.275212
$718$	−16.8078	−0.627260
$719$	−26.2462	−0.978819	−0.489409	−	0.872054i	$-0.662788\pi$
$719$	−26.2462	−0.978819	−0.489409	+	0.872054i	$0.662788\pi$
$720$	0.561553	0.0209278
$721$	−8.31534	−0.309680
$722$	12.4384	0.462911
$723$	25.0540	0.931767
$724$	−6.24621	−0.232139
$725$	−4.68466	−0.173984
$726$	−7.00000	−0.259794
$727$	−53.2311	−1.97423	−0.987115	−	0.160011i	$-0.948847\pi$
$727$	−53.2311	−1.97423	−0.987115	+	0.160011i	$0.948847\pi$
$728$	1.75379	0.0649997
$729$	1.00000	0.0370370
$730$	−3.36932	−0.124704
$731$	−76.6695	−2.83572
$732$	−5.12311	−0.189355
$733$	−17.6155	−0.650644	−0.325322	−	0.945603i	$-0.605473\pi$
$733$	−17.6155	−0.650644	−0.325322	+	0.945603i	$0.605473\pi$
$734$	−16.7386	−0.617834
$735$	3.75379	0.138461
$736$	1.00000	0.0368605
$737$	0	0
$738$	−0.561553	−0.0206710
$739$	−50.7386	−1.86645	−0.933225	−	0.359291i	$-0.883018\pi$
$739$	−50.7386	−1.86645	−0.933225	+	0.359291i	$0.883018\pi$
$740$	1.43845	0.0528784
$741$	−8.00000	−0.293887
$742$	−1.12311	−0.0412305
$743$	44.1771	1.62070	0.810350	−	0.585946i	$-0.199277\pi$
$743$	44.1771	1.62070	0.810350	+	0.585946i	$0.199277\pi$
$744$	0	0
$745$	−1.30019	−0.0476352
$746$	−20.0000	−0.732252
$747$	−11.1231	−0.406973
$748$	13.1231	0.479828
$749$	−8.94602	−0.326881
$750$	−5.43845	−0.198584
$751$	−30.0000	−1.09472	−0.547358	−	0.836899i	$-0.684366\pi$
$751$	−30.0000	−1.09472	−0.547358	+	0.836899i	$0.684366\pi$
$752$	6.56155	0.239275
$753$	7.12311	0.259580
$754$	−3.12311	−0.113737
$755$	−8.80776	−0.320547
$756$	0.561553	0.0204235
$757$	13.4384	0.488429	0.244214	−	0.969721i	$-0.421470\pi$
$757$	13.4384	0.488429	0.244214	+	0.969721i	$0.421470\pi$
$758$	−32.4924	−1.18018
$759$	−2.00000	−0.0725954
$760$	−1.43845	−0.0521780
$761$	−10.6307	−0.385362	−0.192681	−	0.981261i	$-0.561718\pi$
$761$	−10.6307	−0.385362	−0.192681	+	0.981261i	$0.561718\pi$
$762$	17.1231	0.620305
$763$	−6.73863	−0.243955
$764$	17.4384	0.630901
$765$	−3.68466	−0.133219
$766$	−11.3693	−0.410790
$767$	8.00000	0.288863
$768$	−1.00000	−0.0360844
$769$	−6.49242	−0.234123	−0.117061	−	0.993125i	$-0.537347\pi$
$769$	−6.49242	−0.234123	−0.117061	+	0.993125i	$0.537347\pi$
$770$	−0.630683	−0.0227282
$771$	−15.1231	−0.544645
$772$	0.876894	0.0315601
$773$	−36.2462	−1.30369	−0.651843	−	0.758354i	$-0.726004\pi$
$773$	−36.2462	−1.30369	−0.651843	+	0.758354i	$0.726004\pi$
$774$	−11.6847	−0.419996
$775$	0	0
$776$	−4.87689	−0.175070
$777$	1.43845	0.0516040
$778$	7.61553	0.273030
$779$	1.43845	0.0515377
$780$	−1.75379	−0.0627957
$781$	16.0000	0.572525
$782$	−6.56155	−0.234641
$783$	−1.00000	−0.0357371
$784$	−6.68466	−0.238738
$785$	−4.31534	−0.154021
$786$	10.2462	0.365470
$787$	34.2462	1.22075	0.610373	−	0.792114i	$-0.291020\pi$
$787$	34.2462	1.22075	0.610373	+	0.792114i	$0.291020\pi$
$788$	−9.68466	−0.345002
$789$	20.1771	0.718323
$790$	9.12311	0.324586
$791$	7.82292	0.278151
$792$	2.00000	0.0710669
$793$	16.0000	0.568177
$794$	−26.4924	−0.940181
$795$	1.12311	0.0398325
$796$	17.3693	0.615639
$797$	−16.8769	−0.597810	−0.298905	−	0.954283i	$-0.596621\pi$
$797$	−16.8769	−0.597810	−0.298905	+	0.954283i	$0.596621\pi$
$798$	−1.43845	−0.0509205
$799$	−43.0540	−1.52314
$800$	4.68466	0.165628
$801$	5.12311	0.181016
$802$	−15.3693	−0.542709
$803$	−12.0000	−0.423471
$804$	0	0
$805$	0.315342	0.0111143
$806$	0	0
$807$	−12.2462	−0.431087
$808$	−12.2462	−0.430820
$809$	−30.4924	−1.07206	−0.536028	−	0.844200i	$-0.680076\pi$
$809$	−30.4924	−1.07206	−0.536028	+	0.844200i	$0.680076\pi$
$810$	−0.561553	−0.0197310
$811$	18.7386	0.658002	0.329001	−	0.944329i	$-0.393288\pi$
$811$	18.7386	0.658002	0.329001	+	0.944329i	$0.393288\pi$
$812$	−0.561553	−0.0197066
$813$	−4.00000	−0.140286
$814$	5.12311	0.179565
$815$	8.80776	0.308523
$816$	6.56155	0.229700
$817$	29.9309	1.04715
$818$	−19.1231	−0.668624
$819$	−1.75379	−0.0612823
$820$	0.315342	0.0110122
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	11.3693	0.396550
$823$	−22.1080	−0.770635	−0.385317	−	0.922784i	$-0.625908\pi$
$823$	−22.1080	−0.770635	−0.385317	+	0.922784i	$0.625908\pi$
$824$	−14.8078	−0.515853
$825$	−9.36932	−0.326198
$826$	1.43845	0.0500500
$827$	−9.36932	−0.325803	−0.162902	−	0.986642i	$-0.552085\pi$
$827$	−9.36932	−0.325803	−0.162902	+	0.986642i	$0.552085\pi$
$828$	−1.00000	−0.0347524
$829$	−14.8078	−0.514295	−0.257147	−	0.966372i	$-0.582783\pi$
$829$	−14.8078	−0.514295	−0.257147	+	0.966372i	$0.582783\pi$
$830$	6.24621	0.216809
$831$	−22.4924	−0.780253
$832$	3.12311	0.108274
$833$	43.8617	1.51972
$834$	−6.87689	−0.238127
$835$	−1.61553	−0.0559077
$836$	−5.12311	−0.177186
$837$	0	0
$838$	3.93087	0.135790
$839$	9.30019	0.321078	0.160539	−	0.987029i	$-0.448677\pi$
$839$	9.30019	0.321078	0.160539	+	0.987029i	$0.448677\pi$
$840$	−0.315342	−0.0108803
$841$	1.00000	0.0344828
$842$	−17.6155	−0.607071
$843$	27.8617	0.959609
$844$	18.5616	0.638915
$845$	−1.82292	−0.0627103
$846$	−6.56155	−0.225591
$847$	3.93087	0.135066
$848$	−2.00000	−0.0686803
$849$	−19.3693	−0.664753
$850$	−30.7386	−1.05433
$851$	−2.56155	−0.0878089
$852$	8.00000	0.274075
$853$	9.05398	0.310002	0.155001	−	0.987914i	$-0.450462\pi$
$853$	9.05398	0.310002	0.155001	+	0.987914i	$0.450462\pi$
$854$	2.87689	0.0984453
$855$	1.43845	0.0491939
$856$	−15.9309	−0.544506
$857$	38.4924	1.31488	0.657438	−	0.753509i	$-0.271640\pi$
$857$	38.4924	1.31488	0.657438	+	0.753509i	$0.271640\pi$
$858$	−6.24621	−0.213242
$859$	−20.8078	−0.709952	−0.354976	−	0.934875i	$-0.615511\pi$
$859$	−20.8078	−0.709952	−0.354976	+	0.934875i	$0.615511\pi$
$860$	6.56155	0.223747
$861$	0.315342	0.0107468
$862$	4.00000	0.136241
$863$	11.5076	0.391722	0.195861	−	0.980632i	$-0.437250\pi$
$863$	11.5076	0.391722	0.195861	+	0.980632i	$0.437250\pi$
$864$	1.00000	0.0340207
$865$	−11.8229	−0.401991
$866$	−10.4924	−0.356547
$867$	−26.0540	−0.884839
$868$	0	0
$869$	32.4924	1.10223
$870$	0.561553	0.0190384
$871$	0	0
$872$	−12.0000	−0.406371
$873$	4.87689	0.165058
$874$	2.56155	0.0866458
$875$	3.05398	0.103243
$876$	−6.00000	−0.202721
$877$	13.8617	0.468078	0.234039	−	0.972227i	$-0.424806\pi$
$877$	13.8617	0.468078	0.234039	+	0.972227i	$0.424806\pi$
$878$	6.56155	0.221442
$879$	−22.0000	−0.742042
$880$	−1.12311	−0.0378599
$881$	35.8617	1.20821	0.604106	−	0.796904i	$-0.293530\pi$
$881$	35.8617	1.20821	0.604106	+	0.796904i	$0.293530\pi$
$882$	6.68466	0.225084
$883$	−40.4924	−1.36268	−0.681339	−	0.731968i	$-0.738602\pi$
$883$	−40.4924	−1.36268	−0.681339	+	0.731968i	$0.738602\pi$
$884$	−20.4924	−0.689235
$885$	−1.43845	−0.0483529
$886$	22.2462	0.747376
$887$	−48.9848	−1.64475	−0.822375	−	0.568946i	$-0.807351\pi$
$887$	−48.9848	−1.64475	−0.822375	+	0.568946i	$0.807351\pi$
$888$	2.56155	0.0859601
$889$	−9.61553	−0.322494
$890$	−2.87689	−0.0964337
$891$	−2.00000	−0.0670025
$892$	1.75379	0.0587212
$893$	16.8078	0.562450
$894$	−2.31534	−0.0774366
$895$	8.98485	0.300330
$896$	0.561553	0.0187602
$897$	3.12311	0.104277
$898$	7.43845	0.248224
$899$	0	0
$900$	−4.68466	−0.156155
$901$	13.1231	0.437194
$902$	1.12311	0.0373953
$903$	6.56155	0.218355
$904$	13.9309	0.463334
$905$	−3.50758	−0.116596
$906$	−15.6847	−0.521088
$907$	4.00000	0.132818	0.0664089	−	0.997792i	$-0.478846\pi$
$907$	4.00000	0.132818	0.0664089	+	0.997792i	$0.478846\pi$
$908$	21.6847	0.719631
$909$	12.2462	0.406181
$910$	0.984845	0.0326473
$911$	39.2311	1.29978	0.649891	−	0.760027i	$-0.274814\pi$
$911$	39.2311	1.29978	0.649891	+	0.760027i	$0.274814\pi$
$912$	−2.56155	−0.0848215
$913$	22.2462	0.736242
$914$	−24.5616	−0.812424
$915$	−2.87689	−0.0951072
$916$	5.43845	0.179691
$917$	−5.75379	−0.190007
$918$	−6.56155	−0.216564
$919$	21.6847	0.715311	0.357655	−	0.933854i	$-0.383576\pi$
$919$	21.6847	0.715311	0.357655	+	0.933854i	$0.383576\pi$
$920$	0.561553	0.0185138
$921$	−24.4924	−0.807053
$922$	36.1080	1.18915
$923$	−24.9848	−0.822386
$924$	−1.12311	−0.0369475
$925$	−12.0000	−0.394558
$926$	8.00000	0.262896
$927$	14.8078	0.486351
$928$	−1.00000	−0.0328266
$929$	6.49242	0.213009	0.106505	−	0.994312i	$-0.466034\pi$
$929$	6.49242	0.213009	0.106505	+	0.994312i	$0.466034\pi$
$930$	0	0
$931$	−17.1231	−0.561187
$932$	−3.12311	−0.102301
$933$	−27.0540	−0.885707
$934$	25.3693	0.830109
$935$	7.36932	0.241002
$936$	−3.12311	−0.102082
$937$	23.3002	0.761184	0.380592	−	0.924743i	$-0.375720\pi$
$937$	23.3002	0.761184	0.380592	+	0.924743i	$0.375720\pi$
$938$	0	0
$939$	−30.8078	−1.00537
$940$	3.68466	0.120180
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	−7.68466	−0.250380
$943$	−0.561553	−0.0182867
$944$	2.56155	0.0833714
$945$	0.315342	0.0102581
$946$	23.3693	0.759802
$947$	−27.5076	−0.893876	−0.446938	−	0.894565i	$-0.647486\pi$
$947$	−27.5076	−0.893876	−0.446938	+	0.894565i	$0.647486\pi$
$948$	16.2462	0.527652
$949$	18.7386	0.608282
$950$	12.0000	0.389331
$951$	−16.8769	−0.547271
$952$	−3.68466	−0.119420
$953$	35.2311	1.14125	0.570623	−	0.821212i	$-0.306702\pi$
$953$	35.2311	1.14125	0.570623	+	0.821212i	$0.306702\pi$
$954$	2.00000	0.0647524
$955$	9.79261	0.316881
$956$	−7.36932	−0.238341
$957$	2.00000	0.0646508
$958$	−3.50758	−0.113325
$959$	−6.38447	−0.206165
$960$	−0.561553	−0.0181240
$961$	−31.0000	−1.00000
$962$	−8.00000	−0.257930
$963$	15.9309	0.513365
$964$	−25.0540	−0.806934
$965$	0.492423	0.0158516
$966$	0.561553	0.0180677
$967$	21.7538	0.699555	0.349777	−	0.936833i	$-0.386257\pi$
$967$	21.7538	0.699555	0.349777	+	0.936833i	$0.386257\pi$
$968$	7.00000	0.224989
$969$	16.8078	0.539943
$970$	−2.73863	−0.0879322
$971$	20.2462	0.649732	0.324866	−	0.945760i	$-0.394681\pi$
$971$	20.2462	0.649732	0.324866	+	0.945760i	$0.394681\pi$
$972$	−1.00000	−0.0320750
$973$	3.86174	0.123802
$974$	−1.43845	−0.0460908
$975$	14.6307	0.468557
$976$	5.12311	0.163987
$977$	−36.9848	−1.18325	−0.591625	−	0.806213i	$-0.701513\pi$
$977$	−36.9848	−1.18325	−0.591625	+	0.806213i	$0.701513\pi$
$978$	15.6847	0.501540
$979$	−10.2462	−0.327470
$980$	−3.75379	−0.119910
$981$	12.0000	0.383131
$982$	−0.630683	−0.0201259
$983$	10.7386	0.342509	0.171255	−	0.985227i	$-0.445218\pi$
$983$	10.7386	0.342509	0.171255	+	0.985227i	$0.445218\pi$
$984$	0.561553	0.0179016
$985$	−5.43845	−0.173283
$986$	6.56155	0.208962
$987$	3.68466	0.117284
$988$	8.00000	0.254514
$989$	−11.6847	−0.371551
$990$	1.12311	0.0356946
$991$	50.4233	1.60175	0.800874	−	0.598832i	$-0.204368\pi$
$991$	50.4233	1.60175	0.800874	+	0.598832i	$0.204368\pi$
$992$	0	0
$993$	−21.4384	−0.680329
$994$	−4.49242	−0.142491
$995$	9.75379	0.309216
$996$	11.1231	0.352449
$997$	−42.6695	−1.35136	−0.675678	−	0.737197i	$-0.736149\pi$
$997$	−42.6695	−1.35136	−0.675678	+	0.737197i	$0.736149\pi$
$998$	−37.6155	−1.19070
$999$	−2.56155	−0.0810439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.s.1.2	✓	2

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} +1.00000 q^{6} -0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +0.561553 q^{5} +1.00000 q^{6} -0.561553 q^{7} -1.00000 q^{8} +1.00000 q^{9} -0.561553 q^{10} -2.00000 q^{11} -1.00000 q^{12} +3.12311 q^{13} +0.561553 q^{14} -0.561553 q^{15} +1.00000 q^{16} -6.56155 q^{17} -1.00000 q^{18} +2.56155 q^{19} +0.561553 q^{20} +0.561553 q^{21} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -4.68466 q^{25} -3.12311 q^{26} -1.00000 q^{27} -0.561553 q^{28} +1.00000 q^{29} +0.561553 q^{30} -1.00000 q^{32} +2.00000 q^{33} +6.56155 q^{34} -0.315342 q^{35} +1.00000 q^{36} +2.56155 q^{37} -2.56155 q^{38} -3.12311 q^{39} -0.561553 q^{40} +0.561553 q^{41} -0.561553 q^{42} +11.6847 q^{43} -2.00000 q^{44} +0.561553 q^{45} +1.00000 q^{46} +6.56155 q^{47} -1.00000 q^{48} -6.68466 q^{49} +4.68466 q^{50} +6.56155 q^{51} +3.12311 q^{52} -2.00000 q^{53} +1.00000 q^{54} -1.12311 q^{55} +0.561553 q^{56} -2.56155 q^{57} -1.00000 q^{58} +2.56155 q^{59} -0.561553 q^{60} +5.12311 q^{61} -0.561553 q^{63} +1.00000 q^{64} +1.75379 q^{65} -2.00000 q^{66} -6.56155 q^{68} +1.00000 q^{69} +0.315342 q^{70} -8.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} -2.56155 q^{74} +4.68466 q^{75} +2.56155 q^{76} +1.12311 q^{77} +3.12311 q^{78} -16.2462 q^{79} +0.561553 q^{80} +1.00000 q^{81} -0.561553 q^{82} -11.1231 q^{83} +0.561553 q^{84} -3.68466 q^{85} -11.6847 q^{86} -1.00000 q^{87} +2.00000 q^{88} +5.12311 q^{89} -0.561553 q^{90} -1.75379 q^{91} -1.00000 q^{92} -6.56155 q^{94} +1.43845 q^{95} +1.00000 q^{96} +4.87689 q^{97} +6.68466 q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 3 q^{5} + 2 q^{6} + 3 q^{7} - 2 q^{8} + 2 q^{9} + 3 q^{10} - 4 q^{11} - 2 q^{12} - 2 q^{13} - 3 q^{14} + 3 q^{15} + 2 q^{16} - 9 q^{17} - 2 q^{18} + q^{19} - 3 q^{20} - 3 q^{21} + 4 q^{22} - 2 q^{23} + 2 q^{24} + 3 q^{25} + 2 q^{26} - 2 q^{27} + 3 q^{28} + 2 q^{29} - 3 q^{30} - 2 q^{32} + 4 q^{33} + 9 q^{34} - 13 q^{35} + 2 q^{36} + q^{37} - q^{38} + 2 q^{39} + 3 q^{40} - 3 q^{41} + 3 q^{42} + 11 q^{43} - 4 q^{44} - 3 q^{45} + 2 q^{46} + 9 q^{47} - 2 q^{48} - q^{49} - 3 q^{50} + 9 q^{51} - 2 q^{52} - 4 q^{53} + 2 q^{54} + 6 q^{55} - 3 q^{56} - q^{57} - 2 q^{58} + q^{59} + 3 q^{60} + 2 q^{61} + 3 q^{63} + 2 q^{64} + 20 q^{65} - 4 q^{66} - 9 q^{68} + 2 q^{69} + 13 q^{70} - 16 q^{71} - 2 q^{72} + 12 q^{73} - q^{74} - 3 q^{75} + q^{76} - 6 q^{77} - 2 q^{78} - 16 q^{79} - 3 q^{80} + 2 q^{81} + 3 q^{82} - 14 q^{83} - 3 q^{84} + 5 q^{85} - 11 q^{86} - 2 q^{87} + 4 q^{88} + 2 q^{89} + 3 q^{90} - 20 q^{91} - 2 q^{92} - 9 q^{94} + 7 q^{95} + 2 q^{96} + 18 q^{97} + q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 3 * q^5 + 2 * q^6 + 3 * q^7 - 2 * q^8 + 2 * q^9 + 3 * q^10 - 4 * q^11 - 2 * q^12 - 2 * q^13 - 3 * q^14 + 3 * q^15 + 2 * q^16 - 9 * q^17 - 2 * q^18 + q^19 - 3 * q^20 - 3 * q^21 + 4 * q^22 - 2 * q^23 + 2 * q^24 + 3 * q^25 + 2 * q^26 - 2 * q^27 + 3 * q^28 + 2 * q^29 - 3 * q^30 - 2 * q^32 + 4 * q^33 + 9 * q^34 - 13 * q^35 + 2 * q^36 + q^37 - q^38 + 2 * q^39 + 3 * q^40 - 3 * q^41 + 3 * q^42 + 11 * q^43 - 4 * q^44 - 3 * q^45 + 2 * q^46 + 9 * q^47 - 2 * q^48 - q^49 - 3 * q^50 + 9 * q^51 - 2 * q^52 - 4 * q^53 + 2 * q^54 + 6 * q^55 - 3 * q^56 - q^57 - 2 * q^58 + q^59 + 3 * q^60 + 2 * q^61 + 3 * q^63 + 2 * q^64 + 20 * q^65 - 4 * q^66 - 9 * q^68 + 2 * q^69 + 13 * q^70 - 16 * q^71 - 2 * q^72 + 12 * q^73 - q^74 - 3 * q^75 + q^76 - 6 * q^77 - 2 * q^78 - 16 * q^79 - 3 * q^80 + 2 * q^81 + 3 * q^82 - 14 * q^83 - 3 * q^84 + 5 * q^85 - 11 * q^86 - 2 * q^87 + 4 * q^88 + 2 * q^89 + 3 * q^90 - 20 * q^91 - 2 * q^92 - 9 * q^94 + 7 * q^95 + 2 * q^96 + 18 * q^97 + q^98 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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