LMFDB - Embedded newform 4002.2.a.w.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:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.73205$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +0.732051 q^{11} +1.00000 q^{12} +4.73205 q^{13} +3.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} +0.464102 q^{17} +1.00000 q^{18} -4.46410 q^{19} +3.00000 q^{20} +3.00000 q^{21} +0.732051 q^{22} +1.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +4.73205 q^{26} +1.00000 q^{27} +3.00000 q^{28} -1.00000 q^{29} +3.00000 q^{30} -2.73205 q^{31} +1.00000 q^{32} +0.732051 q^{33} +0.464102 q^{34} +9.00000 q^{35} +1.00000 q^{36} +3.19615 q^{37} -4.46410 q^{38} +4.73205 q^{39} +3.00000 q^{40} -12.1244 q^{41} +3.00000 q^{42} -11.9282 q^{43} +0.732051 q^{44} +3.00000 q^{45} +1.00000 q^{46} -9.39230 q^{47} +1.00000 q^{48} +2.00000 q^{49} +4.00000 q^{50} +0.464102 q^{51} +4.73205 q^{52} +2.53590 q^{53} +1.00000 q^{54} +2.19615 q^{55} +3.00000 q^{56} -4.46410 q^{57} -1.00000 q^{58} +10.4641 q^{59} +3.00000 q^{60} -14.3923 q^{61} -2.73205 q^{62} +3.00000 q^{63} +1.00000 q^{64} +14.1962 q^{65} +0.732051 q^{66} -2.53590 q^{67} +0.464102 q^{68} +1.00000 q^{69} +9.00000 q^{70} -11.6603 q^{71} +1.00000 q^{72} +0.928203 q^{73} +3.19615 q^{74} +4.00000 q^{75} -4.46410 q^{76} +2.19615 q^{77} +4.73205 q^{78} +7.12436 q^{79} +3.00000 q^{80} +1.00000 q^{81} -12.1244 q^{82} -11.4641 q^{83} +3.00000 q^{84} +1.39230 q^{85} -11.9282 q^{86} -1.00000 q^{87} +0.732051 q^{88} +1.46410 q^{89} +3.00000 q^{90} +14.1962 q^{91} +1.00000 q^{92} -2.73205 q^{93} -9.39230 q^{94} -13.3923 q^{95} +1.00000 q^{96} +4.00000 q^{97} +2.00000 q^{98} +0.732051 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 6 q^{5} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 6 * q^5 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 6 q^{5} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} + 6 q^{10} - 2 q^{11} + 2 q^{12} + 6 q^{13} + 6 q^{14} + 6 q^{15} + 2 q^{16} - 6 q^{17} + 2 q^{18} - 2 q^{19} + 6 q^{20} + 6 q^{21} - 2 q^{22} + 2 q^{23} + 2 q^{24} + 8 q^{25} + 6 q^{26} + 2 q^{27} + 6 q^{28} - 2 q^{29} + 6 q^{30} - 2 q^{31} + 2 q^{32} - 2 q^{33} - 6 q^{34} + 18 q^{35} + 2 q^{36} - 4 q^{37} - 2 q^{38} + 6 q^{39} + 6 q^{40} + 6 q^{42} - 10 q^{43} - 2 q^{44} + 6 q^{45} + 2 q^{46} + 2 q^{47} + 2 q^{48} + 4 q^{49} + 8 q^{50} - 6 q^{51} + 6 q^{52} + 12 q^{53} + 2 q^{54} - 6 q^{55} + 6 q^{56} - 2 q^{57} - 2 q^{58} + 14 q^{59} + 6 q^{60} - 8 q^{61} - 2 q^{62} + 6 q^{63} + 2 q^{64} + 18 q^{65} - 2 q^{66} - 12 q^{67} - 6 q^{68} + 2 q^{69} + 18 q^{70} - 6 q^{71} + 2 q^{72} - 12 q^{73} - 4 q^{74} + 8 q^{75} - 2 q^{76} - 6 q^{77} + 6 q^{78} - 10 q^{79} + 6 q^{80} + 2 q^{81} - 16 q^{83} + 6 q^{84} - 18 q^{85} - 10 q^{86} - 2 q^{87} - 2 q^{88} - 4 q^{89} + 6 q^{90} + 18 q^{91} + 2 q^{92} - 2 q^{93} + 2 q^{94} - 6 q^{95} + 2 q^{96} + 8 q^{97} + 4 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 6 * q^5 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9 + 6 * q^10 - 2 * q^11 + 2 * q^12 + 6 * q^13 + 6 * q^14 + 6 * q^15 + 2 * q^16 - 6 * q^17 + 2 * q^18 - 2 * q^19 + 6 * q^20 + 6 * q^21 - 2 * q^22 + 2 * q^23 + 2 * q^24 + 8 * q^25 + 6 * q^26 + 2 * q^27 + 6 * q^28 - 2 * q^29 + 6 * q^30 - 2 * q^31 + 2 * q^32 - 2 * q^33 - 6 * q^34 + 18 * q^35 + 2 * q^36 - 4 * q^37 - 2 * q^38 + 6 * q^39 + 6 * q^40 + 6 * q^42 - 10 * q^43 - 2 * q^44 + 6 * q^45 + 2 * q^46 + 2 * q^47 + 2 * q^48 + 4 * q^49 + 8 * q^50 - 6 * q^51 + 6 * q^52 + 12 * q^53 + 2 * q^54 - 6 * q^55 + 6 * q^56 - 2 * q^57 - 2 * q^58 + 14 * q^59 + 6 * q^60 - 8 * q^61 - 2 * q^62 + 6 * q^63 + 2 * q^64 + 18 * q^65 - 2 * q^66 - 12 * q^67 - 6 * q^68 + 2 * q^69 + 18 * q^70 - 6 * q^71 + 2 * q^72 - 12 * q^73 - 4 * q^74 + 8 * q^75 - 2 * q^76 - 6 * q^77 + 6 * q^78 - 10 * q^79 + 6 * q^80 + 2 * q^81 - 16 * q^83 + 6 * q^84 - 18 * q^85 - 10 * q^86 - 2 * q^87 - 2 * q^88 - 4 * q^89 + 6 * q^90 + 18 * q^91 + 2 * q^92 - 2 * q^93 + 2 * q^94 - 6 * q^95 + 2 * q^96 + 8 * q^97 + 4 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	1.00000	0.408248
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.00000	0.948683
$11$	0.732051	0.220722	0.110361	−	0.993892i	$-0.464799\pi$
$11$	0.732051	0.220722	0.110361	+	0.993892i	$0.464799\pi$
$12$	1.00000	0.288675
$13$	4.73205	1.31243	0.656217	−	0.754572i	$-0.272155\pi$
$13$	4.73205	1.31243	0.656217	+	0.754572i	$0.272155\pi$
$14$	3.00000	0.801784
$15$	3.00000	0.774597
$16$	1.00000	0.250000
$17$	0.464102	0.112561	0.0562806	−	0.998415i	$-0.482076\pi$
$17$	0.464102	0.112561	0.0562806	+	0.998415i	$0.482076\pi$
$18$	1.00000	0.235702
$19$	−4.46410	−1.02414	−0.512068	−	0.858945i	$-0.671120\pi$
$19$	−4.46410	−1.02414	−0.512068	+	0.858945i	$0.671120\pi$
$20$	3.00000	0.670820
$21$	3.00000	0.654654
$22$	0.732051	0.156074
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	4.00000	0.800000
$26$	4.73205	0.928032
$27$	1.00000	0.192450
$28$	3.00000	0.566947
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	3.00000	0.547723
$31$	−2.73205	−0.490691	−0.245345	−	0.969436i	$-0.578901\pi$
$31$	−2.73205	−0.490691	−0.245345	+	0.969436i	$0.578901\pi$
$32$	1.00000	0.176777
$33$	0.732051	0.127434
$34$	0.464102	0.0795928
$35$	9.00000	1.52128
$36$	1.00000	0.166667
$37$	3.19615	0.525444	0.262722	−	0.964872i	$-0.415380\pi$
$37$	3.19615	0.525444	0.262722	+	0.964872i	$0.415380\pi$
$38$	−4.46410	−0.724173
$39$	4.73205	0.757735
$40$	3.00000	0.474342
$41$	−12.1244	−1.89351	−0.946753	−	0.321960i	$-0.895658\pi$
$41$	−12.1244	−1.89351	−0.946753	+	0.321960i	$0.895658\pi$
$42$	3.00000	0.462910
$43$	−11.9282	−1.81903	−0.909517	−	0.415667i	$-0.863548\pi$
$43$	−11.9282	−1.81903	−0.909517	+	0.415667i	$0.863548\pi$
$44$	0.732051	0.110361
$45$	3.00000	0.447214
$46$	1.00000	0.147442
$47$	−9.39230	−1.37001	−0.685004	−	0.728539i	$-0.740200\pi$
$47$	−9.39230	−1.37001	−0.685004	+	0.728539i	$0.740200\pi$
$48$	1.00000	0.144338
$49$	2.00000	0.285714
$50$	4.00000	0.565685
$51$	0.464102	0.0649872
$52$	4.73205	0.656217
$53$	2.53590	0.348332	0.174166	−	0.984716i	$-0.444277\pi$
$53$	2.53590	0.348332	0.174166	+	0.984716i	$0.444277\pi$
$54$	1.00000	0.136083
$55$	2.19615	0.296129
$56$	3.00000	0.400892
$57$	−4.46410	−0.591285
$58$	−1.00000	−0.131306
$59$	10.4641	1.36231	0.681155	−	0.732139i	$-0.261478\pi$
$59$	10.4641	1.36231	0.681155	+	0.732139i	$0.261478\pi$
$60$	3.00000	0.387298
$61$	−14.3923	−1.84275	−0.921373	−	0.388680i	$-0.872931\pi$
$61$	−14.3923	−1.84275	−0.921373	+	0.388680i	$0.872931\pi$
$62$	−2.73205	−0.346971
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	14.1962	1.76082
$66$	0.732051	0.0901092
$67$	−2.53590	−0.309809	−0.154905	−	0.987929i	$-0.549507\pi$
$67$	−2.53590	−0.309809	−0.154905	+	0.987929i	$0.549507\pi$
$68$	0.464102	0.0562806
$69$	1.00000	0.120386
$70$	9.00000	1.07571
$71$	−11.6603	−1.38382	−0.691909	−	0.721985i	$-0.743230\pi$
$71$	−11.6603	−1.38382	−0.691909	+	0.721985i	$0.743230\pi$
$72$	1.00000	0.117851
$73$	0.928203	0.108638	0.0543190	−	0.998524i	$-0.482701\pi$
$73$	0.928203	0.108638	0.0543190	+	0.998524i	$0.482701\pi$
$74$	3.19615	0.371545
$75$	4.00000	0.461880
$76$	−4.46410	−0.512068
$77$	2.19615	0.250275
$78$	4.73205	0.535799
$79$	7.12436	0.801553	0.400776	−	0.916176i	$-0.368740\pi$
$79$	7.12436	0.801553	0.400776	+	0.916176i	$0.368740\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	−12.1244	−1.33891
$83$	−11.4641	−1.25835	−0.629174	−	0.777264i	$-0.716607\pi$
$83$	−11.4641	−1.25835	−0.629174	+	0.777264i	$0.716607\pi$
$84$	3.00000	0.327327
$85$	1.39230	0.151017
$86$	−11.9282	−1.28625
$87$	−1.00000	−0.107211
$88$	0.732051	0.0780369
$89$	1.46410	0.155194	0.0775972	−	0.996985i	$-0.475275\pi$
$89$	1.46410	0.155194	0.0775972	+	0.996985i	$0.475275\pi$
$90$	3.00000	0.316228
$91$	14.1962	1.48816
$92$	1.00000	0.104257
$93$	−2.73205	−0.283300
$94$	−9.39230	−0.968742
$95$	−13.3923	−1.37402
$96$	1.00000	0.102062
$97$	4.00000	0.406138	0.203069	−	0.979164i	$-0.434908\pi$
$97$	4.00000	0.406138	0.203069	+	0.979164i	$0.434908\pi$
$98$	2.00000	0.202031
$99$	0.732051	0.0735739
$100$	4.00000	0.400000
$101$	16.9282	1.68442	0.842210	−	0.539150i	$-0.181255\pi$
$101$	16.9282	1.68442	0.842210	+	0.539150i	$0.181255\pi$
$102$	0.464102	0.0459529
$103$	−12.8564	−1.26678	−0.633390	−	0.773833i	$-0.718337\pi$
$103$	−12.8564	−1.26678	−0.633390	+	0.773833i	$0.718337\pi$
$104$	4.73205	0.464016
$105$	9.00000	0.878310
$106$	2.53590	0.246308
$107$	9.19615	0.889026	0.444513	−	0.895772i	$-0.353377\pi$
$107$	9.19615	0.889026	0.444513	+	0.895772i	$0.353377\pi$
$108$	1.00000	0.0962250
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	2.19615	0.209395
$111$	3.19615	0.303365
$112$	3.00000	0.283473
$113$	11.3923	1.07170	0.535849	−	0.844314i	$-0.319992\pi$
$113$	11.3923	1.07170	0.535849	+	0.844314i	$0.319992\pi$
$114$	−4.46410	−0.418101
$115$	3.00000	0.279751
$116$	−1.00000	−0.0928477
$117$	4.73205	0.437478
$118$	10.4641	0.963299
$119$	1.39230	0.127632
$120$	3.00000	0.273861
$121$	−10.4641	−0.951282
$122$	−14.3923	−1.30302
$123$	−12.1244	−1.09322
$124$	−2.73205	−0.245345
$125$	−3.00000	−0.268328
$126$	3.00000	0.267261
$127$	11.4641	1.01727	0.508637	−	0.860981i	$-0.330149\pi$
$127$	11.4641	1.01727	0.508637	+	0.860981i	$0.330149\pi$
$128$	1.00000	0.0883883
$129$	−11.9282	−1.05022
$130$	14.1962	1.24508
$131$	−18.1962	−1.58981	−0.794903	−	0.606737i	$-0.792478\pi$
$131$	−18.1962	−1.58981	−0.794903	+	0.606737i	$0.792478\pi$
$132$	0.732051	0.0637168
$133$	−13.3923	−1.16126
$134$	−2.53590	−0.219068
$135$	3.00000	0.258199
$136$	0.464102	0.0397964
$137$	−12.9282	−1.10453	−0.552265	−	0.833668i	$-0.686237\pi$
$137$	−12.9282	−1.10453	−0.552265	+	0.833668i	$0.686237\pi$
$138$	1.00000	0.0851257
$139$	−19.8564	−1.68420	−0.842099	−	0.539323i	$-0.818680\pi$
$139$	−19.8564	−1.68420	−0.842099	+	0.539323i	$0.818680\pi$
$140$	9.00000	0.760639
$141$	−9.39230	−0.790975
$142$	−11.6603	−0.978507
$143$	3.46410	0.289683
$144$	1.00000	0.0833333
$145$	−3.00000	−0.249136
$146$	0.928203	0.0768186
$147$	2.00000	0.164957
$148$	3.19615	0.262722
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	4.00000	0.326599
$151$	14.1244	1.14942	0.574712	−	0.818356i	$-0.305114\pi$
$151$	14.1244	1.14942	0.574712	+	0.818356i	$0.305114\pi$
$152$	−4.46410	−0.362086
$153$	0.464102	0.0375204
$154$	2.19615	0.176971
$155$	−8.19615	−0.658331
$156$	4.73205	0.378867
$157$	−0.267949	−0.0213847	−0.0106923	−	0.999943i	$-0.503404\pi$
$157$	−0.267949	−0.0213847	−0.0106923	+	0.999943i	$0.503404\pi$
$158$	7.12436	0.566783
$159$	2.53590	0.201110
$160$	3.00000	0.237171
$161$	3.00000	0.236433
$162$	1.00000	0.0785674
$163$	−20.2679	−1.58751	−0.793754	−	0.608239i	$-0.791876\pi$
$163$	−20.2679	−1.58751	−0.793754	+	0.608239i	$0.791876\pi$
$164$	−12.1244	−0.946753
$165$	2.19615	0.170970
$166$	−11.4641	−0.889787
$167$	19.8564	1.53653	0.768267	−	0.640129i	$-0.221119\pi$
$167$	19.8564	1.53653	0.768267	+	0.640129i	$0.221119\pi$
$168$	3.00000	0.231455
$169$	9.39230	0.722485
$170$	1.39230	0.106785
$171$	−4.46410	−0.341378
$172$	−11.9282	−0.909517
$173$	8.26795	0.628601	0.314300	−	0.949324i	$-0.398230\pi$
$173$	8.26795	0.628601	0.314300	+	0.949324i	$0.398230\pi$
$174$	−1.00000	−0.0758098
$175$	12.0000	0.907115
$176$	0.732051	0.0551804
$177$	10.4641	0.786530
$178$	1.46410	0.109739
$179$	10.3923	0.776757	0.388379	−	0.921500i	$-0.373035\pi$
$179$	10.3923	0.776757	0.388379	+	0.921500i	$0.373035\pi$
$180$	3.00000	0.223607
$181$	19.1244	1.42150	0.710751	−	0.703444i	$-0.248355\pi$
$181$	19.1244	1.42150	0.710751	+	0.703444i	$0.248355\pi$
$182$	14.1962	1.05229
$183$	−14.3923	−1.06391
$184$	1.00000	0.0737210
$185$	9.58846	0.704957
$186$	−2.73205	−0.200324
$187$	0.339746	0.0248447
$188$	−9.39230	−0.685004
$189$	3.00000	0.218218
$190$	−13.3923	−0.971580
$191$	−1.87564	−0.135717	−0.0678584	−	0.997695i	$-0.521617\pi$
$191$	−1.87564	−0.135717	−0.0678584	+	0.997695i	$0.521617\pi$
$192$	1.00000	0.0721688
$193$	9.46410	0.681241	0.340620	−	0.940201i	$-0.389363\pi$
$193$	9.46410	0.681241	0.340620	+	0.940201i	$0.389363\pi$
$194$	4.00000	0.287183
$195$	14.1962	1.01661
$196$	2.00000	0.142857
$197$	17.7321	1.26336	0.631678	−	0.775231i	$-0.282366\pi$
$197$	17.7321	1.26336	0.631678	+	0.775231i	$0.282366\pi$
$198$	0.732051	0.0520246
$199$	17.3205	1.22782	0.613909	−	0.789377i	$-0.289596\pi$
$199$	17.3205	1.22782	0.613909	+	0.789377i	$0.289596\pi$
$200$	4.00000	0.282843
$201$	−2.53590	−0.178868
$202$	16.9282	1.19106
$203$	−3.00000	−0.210559
$204$	0.464102	0.0324936
$205$	−36.3731	−2.54041
$206$	−12.8564	−0.895748
$207$	1.00000	0.0695048
$208$	4.73205	0.328109
$209$	−3.26795	−0.226049
$210$	9.00000	0.621059
$211$	11.1962	0.770775	0.385387	−	0.922755i	$-0.374068\pi$
$211$	11.1962	0.770775	0.385387	+	0.922755i	$0.374068\pi$
$212$	2.53590	0.174166
$213$	−11.6603	−0.798947
$214$	9.19615	0.628636
$215$	−35.7846	−2.44049
$216$	1.00000	0.0680414
$217$	−8.19615	−0.556391
$218$	−8.00000	−0.541828
$219$	0.928203	0.0627222
$220$	2.19615	0.148065
$221$	2.19615	0.147729
$222$	3.19615	0.214512
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	3.00000	0.200446
$225$	4.00000	0.266667
$226$	11.3923	0.757805
$227$	−25.5885	−1.69837	−0.849183	−	0.528099i	$-0.822905\pi$
$227$	−25.5885	−1.69837	−0.849183	+	0.528099i	$0.822905\pi$
$228$	−4.46410	−0.295642
$229$	13.1962	0.872026	0.436013	−	0.899940i	$-0.356390\pi$
$229$	13.1962	0.872026	0.436013	+	0.899940i	$0.356390\pi$
$230$	3.00000	0.197814
$231$	2.19615	0.144496
$232$	−1.00000	−0.0656532
$233$	23.3205	1.52778	0.763889	−	0.645348i	$-0.223288\pi$
$233$	23.3205	1.52778	0.763889	+	0.645348i	$0.223288\pi$
$234$	4.73205	0.309344
$235$	−28.1769	−1.83806
$236$	10.4641	0.681155
$237$	7.12436	0.462777
$238$	1.39230	0.0902497
$239$	−7.85641	−0.508189	−0.254094	−	0.967179i	$-0.581777\pi$
$239$	−7.85641	−0.508189	−0.254094	+	0.967179i	$0.581777\pi$
$240$	3.00000	0.193649
$241$	13.7321	0.884559	0.442280	−	0.896877i	$-0.354170\pi$
$241$	13.7321	0.884559	0.442280	+	0.896877i	$0.354170\pi$
$242$	−10.4641	−0.672658
$243$	1.00000	0.0641500
$244$	−14.3923	−0.921373
$245$	6.00000	0.383326
$246$	−12.1244	−0.773021
$247$	−21.1244	−1.34411
$248$	−2.73205	−0.173485
$249$	−11.4641	−0.726508
$250$	−3.00000	−0.189737
$251$	1.46410	0.0924133	0.0462066	−	0.998932i	$-0.485287\pi$
$251$	1.46410	0.0924133	0.0462066	+	0.998932i	$0.485287\pi$
$252$	3.00000	0.188982
$253$	0.732051	0.0460236
$254$	11.4641	0.719322
$255$	1.39230	0.0871895
$256$	1.00000	0.0625000
$257$	−22.7321	−1.41799	−0.708993	−	0.705215i	$-0.750850\pi$
$257$	−22.7321	−1.41799	−0.708993	+	0.705215i	$0.750850\pi$
$258$	−11.9282	−0.742617
$259$	9.58846	0.595798
$260$	14.1962	0.880408
$261$	−1.00000	−0.0618984
$262$	−18.1962	−1.12416
$263$	29.0526	1.79146	0.895729	−	0.444601i	$-0.146654\pi$
$263$	29.0526	1.79146	0.895729	+	0.444601i	$0.146654\pi$
$264$	0.732051	0.0450546
$265$	7.60770	0.467337
$266$	−13.3923	−0.821135
$267$	1.46410	0.0896016
$268$	−2.53590	−0.154905
$269$	−11.6603	−0.710938	−0.355469	−	0.934688i	$-0.615679\pi$
$269$	−11.6603	−0.710938	−0.355469	+	0.934688i	$0.615679\pi$
$270$	3.00000	0.182574
$271$	24.7846	1.50556	0.752779	−	0.658273i	$-0.228713\pi$
$271$	24.7846	1.50556	0.752779	+	0.658273i	$0.228713\pi$
$272$	0.464102	0.0281403
$273$	14.1962	0.859190
$274$	−12.9282	−0.781021
$275$	2.92820	0.176577
$276$	1.00000	0.0601929
$277$	19.1244	1.14907	0.574536	−	0.818480i	$-0.305183\pi$
$277$	19.1244	1.14907	0.574536	+	0.818480i	$0.305183\pi$
$278$	−19.8564	−1.19091
$279$	−2.73205	−0.163564
$280$	9.00000	0.537853
$281$	27.4641	1.63837	0.819185	−	0.573529i	$-0.194426\pi$
$281$	27.4641	1.63837	0.819185	+	0.573529i	$0.194426\pi$
$282$	−9.39230	−0.559304
$283$	8.58846	0.510531	0.255265	−	0.966871i	$-0.417837\pi$
$283$	8.58846	0.510531	0.255265	+	0.966871i	$0.417837\pi$
$284$	−11.6603	−0.691909
$285$	−13.3923	−0.793292
$286$	3.46410	0.204837
$287$	−36.3731	−2.14703
$288$	1.00000	0.0589256
$289$	−16.7846	−0.987330
$290$	−3.00000	−0.176166
$291$	4.00000	0.234484
$292$	0.928203	0.0543190
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	2.00000	0.116642
$295$	31.3923	1.82773
$296$	3.19615	0.185773
$297$	0.732051	0.0424779
$298$	5.00000	0.289642
$299$	4.73205	0.273662
$300$	4.00000	0.230940
$301$	−35.7846	−2.06259
$302$	14.1244	0.812765
$303$	16.9282	0.972500
$304$	−4.46410	−0.256034
$305$	−43.1769	−2.47230
$306$	0.464102	0.0265309
$307$	−9.07180	−0.517755	−0.258877	−	0.965910i	$-0.583353\pi$
$307$	−9.07180	−0.517755	−0.258877	+	0.965910i	$0.583353\pi$
$308$	2.19615	0.125137
$309$	−12.8564	−0.731375
$310$	−8.19615	−0.465510
$311$	−6.32051	−0.358403	−0.179202	−	0.983812i	$-0.557351\pi$
$311$	−6.32051	−0.358403	−0.179202	+	0.983812i	$0.557351\pi$
$312$	4.73205	0.267900
$313$	−20.1244	−1.13750	−0.568748	−	0.822512i	$-0.692572\pi$
$313$	−20.1244	−1.13750	−0.568748	+	0.822512i	$0.692572\pi$
$314$	−0.267949	−0.0151212
$315$	9.00000	0.507093
$316$	7.12436	0.400776
$317$	10.3397	0.580738	0.290369	−	0.956915i	$-0.406222\pi$
$317$	10.3397	0.580738	0.290369	+	0.956915i	$0.406222\pi$
$318$	2.53590	0.142206
$319$	−0.732051	−0.0409870
$320$	3.00000	0.167705
$321$	9.19615	0.513279
$322$	3.00000	0.167183
$323$	−2.07180	−0.115278
$324$	1.00000	0.0555556
$325$	18.9282	1.04995
$326$	−20.2679	−1.12254
$327$	−8.00000	−0.442401
$328$	−12.1244	−0.669456
$329$	−28.1769	−1.55344
$330$	2.19615	0.120894
$331$	29.4449	1.61844	0.809218	−	0.587508i	$-0.199891\pi$
$331$	29.4449	1.61844	0.809218	+	0.587508i	$0.199891\pi$
$332$	−11.4641	−0.629174
$333$	3.19615	0.175148
$334$	19.8564	1.08649
$335$	−7.60770	−0.415653
$336$	3.00000	0.163663
$337$	−20.9282	−1.14003	−0.570016	−	0.821634i	$-0.693063\pi$
$337$	−20.9282	−1.14003	−0.570016	+	0.821634i	$0.693063\pi$
$338$	9.39230	0.510874
$339$	11.3923	0.618745
$340$	1.39230	0.0755083
$341$	−2.00000	−0.108306
$342$	−4.46410	−0.241391
$343$	−15.0000	−0.809924
$344$	−11.9282	−0.643126
$345$	3.00000	0.161515
$346$	8.26795	0.444488
$347$	18.8564	1.01226	0.506132	−	0.862456i	$-0.331075\pi$
$347$	18.8564	1.01226	0.506132	+	0.862456i	$0.331075\pi$
$348$	−1.00000	−0.0536056
$349$	−6.39230	−0.342172	−0.171086	−	0.985256i	$-0.554728\pi$
$349$	−6.39230	−0.342172	−0.171086	+	0.985256i	$0.554728\pi$
$350$	12.0000	0.641427
$351$	4.73205	0.252578
$352$	0.732051	0.0390184
$353$	−11.4641	−0.610173	−0.305086	−	0.952325i	$-0.598685\pi$
$353$	−11.4641	−0.610173	−0.305086	+	0.952325i	$0.598685\pi$
$354$	10.4641	0.556161
$355$	−34.9808	−1.85659
$356$	1.46410	0.0775972
$357$	1.39230	0.0736886
$358$	10.3923	0.549250
$359$	−4.26795	−0.225254	−0.112627	−	0.993637i	$-0.535926\pi$
$359$	−4.26795	−0.225254	−0.112627	+	0.993637i	$0.535926\pi$
$360$	3.00000	0.158114
$361$	0.928203	0.0488528
$362$	19.1244	1.00515
$363$	−10.4641	−0.549223
$364$	14.1962	0.744081
$365$	2.78461	0.145753
$366$	−14.3923	−0.752298
$367$	14.3397	0.748529	0.374264	−	0.927322i	$-0.377895\pi$
$367$	14.3397	0.748529	0.374264	+	0.927322i	$0.377895\pi$
$368$	1.00000	0.0521286
$369$	−12.1244	−0.631169
$370$	9.58846	0.498480
$371$	7.60770	0.394972
$372$	−2.73205	−0.141650
$373$	30.9282	1.60140	0.800701	−	0.599064i	$-0.204461\pi$
$373$	30.9282	1.60140	0.800701	+	0.599064i	$0.204461\pi$
$374$	0.339746	0.0175678
$375$	−3.00000	−0.154919
$376$	−9.39230	−0.484371
$377$	−4.73205	−0.243713
$378$	3.00000	0.154303
$379$	3.85641	0.198090	0.0990451	−	0.995083i	$-0.468421\pi$
$379$	3.85641	0.198090	0.0990451	+	0.995083i	$0.468421\pi$
$380$	−13.3923	−0.687011
$381$	11.4641	0.587324
$382$	−1.87564	−0.0959663
$383$	9.80385	0.500953	0.250477	−	0.968123i	$-0.419413\pi$
$383$	9.80385	0.500953	0.250477	+	0.968123i	$0.419413\pi$
$384$	1.00000	0.0510310
$385$	6.58846	0.335779
$386$	9.46410	0.481710
$387$	−11.9282	−0.606345
$388$	4.00000	0.203069
$389$	13.8564	0.702548	0.351274	−	0.936273i	$-0.385749\pi$
$389$	13.8564	0.702548	0.351274	+	0.936273i	$0.385749\pi$
$390$	14.1962	0.718850
$391$	0.464102	0.0234706
$392$	2.00000	0.101015
$393$	−18.1962	−0.917874
$394$	17.7321	0.893328
$395$	21.3731	1.07540
$396$	0.732051	0.0367869
$397$	17.6603	0.886343	0.443171	−	0.896437i	$-0.353853\pi$
$397$	17.6603	0.886343	0.443171	+	0.896437i	$0.353853\pi$
$398$	17.3205	0.868199
$399$	−13.3923	−0.670454
$400$	4.00000	0.200000
$401$	29.7128	1.48379	0.741894	−	0.670518i	$-0.233928\pi$
$401$	29.7128	1.48379	0.741894	+	0.670518i	$0.233928\pi$
$402$	−2.53590	−0.126479
$403$	−12.9282	−0.644000
$404$	16.9282	0.842210
$405$	3.00000	0.149071
$406$	−3.00000	−0.148888
$407$	2.33975	0.115977
$408$	0.464102	0.0229765
$409$	−5.12436	−0.253383	−0.126692	−	0.991942i	$-0.540436\pi$
$409$	−5.12436	−0.253383	−0.126692	+	0.991942i	$0.540436\pi$
$410$	−36.3731	−1.79634
$411$	−12.9282	−0.637701
$412$	−12.8564	−0.633390
$413$	31.3923	1.54471
$414$	1.00000	0.0491473
$415$	−34.3923	−1.68825
$416$	4.73205	0.232008
$417$	−19.8564	−0.972372
$418$	−3.26795	−0.159841
$419$	21.5885	1.05467	0.527333	−	0.849659i	$-0.323192\pi$
$419$	21.5885	1.05467	0.527333	+	0.849659i	$0.323192\pi$
$420$	9.00000	0.439155
$421$	9.60770	0.468250	0.234125	−	0.972206i	$-0.424777\pi$
$421$	9.60770	0.468250	0.234125	+	0.972206i	$0.424777\pi$
$422$	11.1962	0.545020
$423$	−9.39230	−0.456669
$424$	2.53590	0.123154
$425$	1.85641	0.0900489
$426$	−11.6603	−0.564941
$427$	−43.1769	−2.08948
$428$	9.19615	0.444513
$429$	3.46410	0.167248
$430$	−35.7846	−1.72569
$431$	19.8038	0.953918	0.476959	−	0.878926i	$-0.341739\pi$
$431$	19.8038	0.953918	0.476959	+	0.878926i	$0.341739\pi$
$432$	1.00000	0.0481125
$433$	−33.5167	−1.61071	−0.805354	−	0.592794i	$-0.798025\pi$
$433$	−33.5167	−1.61071	−0.805354	+	0.592794i	$0.798025\pi$
$434$	−8.19615	−0.393428
$435$	−3.00000	−0.143839
$436$	−8.00000	−0.383131
$437$	−4.46410	−0.213547
$438$	0.928203	0.0443513
$439$	10.8038	0.515640	0.257820	−	0.966193i	$-0.416996\pi$
$439$	10.8038	0.515640	0.257820	+	0.966193i	$0.416996\pi$
$440$	2.19615	0.104697
$441$	2.00000	0.0952381
$442$	2.19615	0.104460
$443$	−37.1769	−1.76633	−0.883164	−	0.469064i	$-0.844591\pi$
$443$	−37.1769	−1.76633	−0.883164	+	0.469064i	$0.844591\pi$
$444$	3.19615	0.151683
$445$	4.39230	0.208215
$446$	8.00000	0.378811
$447$	5.00000	0.236492
$448$	3.00000	0.141737
$449$	−8.66025	−0.408703	−0.204351	−	0.978898i	$-0.565508\pi$
$449$	−8.66025	−0.408703	−0.204351	+	0.978898i	$0.565508\pi$
$450$	4.00000	0.188562
$451$	−8.87564	−0.417938
$452$	11.3923	0.535849
$453$	14.1244	0.663620
$454$	−25.5885	−1.20093
$455$	42.5885	1.99658
$456$	−4.46410	−0.209051
$457$	−34.9090	−1.63297	−0.816486	−	0.577365i	$-0.804081\pi$
$457$	−34.9090	−1.63297	−0.816486	+	0.577365i	$0.804081\pi$
$458$	13.1962	0.616616
$459$	0.464102	0.0216624
$460$	3.00000	0.139876
$461$	30.0526	1.39969	0.699844	−	0.714296i	$-0.253253\pi$
$461$	30.0526	1.39969	0.699844	+	0.714296i	$0.253253\pi$
$462$	2.19615	0.102174
$463$	31.1769	1.44891	0.724457	−	0.689320i	$-0.242091\pi$
$463$	31.1769	1.44891	0.724457	+	0.689320i	$0.242091\pi$
$464$	−1.00000	−0.0464238
$465$	−8.19615	−0.380087
$466$	23.3205	1.08030
$467$	11.8038	0.546217	0.273108	−	0.961983i	$-0.411948\pi$
$467$	11.8038	0.546217	0.273108	+	0.961983i	$0.411948\pi$
$468$	4.73205	0.218739
$469$	−7.60770	−0.351291
$470$	−28.1769	−1.29970
$471$	−0.267949	−0.0123464
$472$	10.4641	0.481649
$473$	−8.73205	−0.401500
$474$	7.12436	0.327232
$475$	−17.8564	−0.819308
$476$	1.39230	0.0638162
$477$	2.53590	0.116111
$478$	−7.85641	−0.359344
$479$	25.8564	1.18141	0.590705	−	0.806888i	$-0.298850\pi$
$479$	25.8564	1.18141	0.590705	+	0.806888i	$0.298850\pi$
$480$	3.00000	0.136931
$481$	15.1244	0.689611
$482$	13.7321	0.625478
$483$	3.00000	0.136505
$484$	−10.4641	−0.475641
$485$	12.0000	0.544892
$486$	1.00000	0.0453609
$487$	−32.5167	−1.47347	−0.736735	−	0.676181i	$-0.763634\pi$
$487$	−32.5167	−1.47347	−0.736735	+	0.676181i	$0.763634\pi$
$488$	−14.3923	−0.651509
$489$	−20.2679	−0.916548
$490$	6.00000	0.271052
$491$	−6.33975	−0.286109	−0.143054	−	0.989715i	$-0.545692\pi$
$491$	−6.33975	−0.286109	−0.143054	+	0.989715i	$0.545692\pi$
$492$	−12.1244	−0.546608
$493$	−0.464102	−0.0209021
$494$	−21.1244	−0.950430
$495$	2.19615	0.0987097
$496$	−2.73205	−0.122673
$497$	−34.9808	−1.56910
$498$	−11.4641	−0.513719
$499$	20.0526	0.897676	0.448838	−	0.893613i	$-0.351838\pi$
$499$	20.0526	0.897676	0.448838	+	0.893613i	$0.351838\pi$
$500$	−3.00000	−0.134164
$501$	19.8564	0.887119
$502$	1.46410	0.0653461
$503$	8.66025	0.386142	0.193071	−	0.981185i	$-0.438155\pi$
$503$	8.66025	0.386142	0.193071	+	0.981185i	$0.438155\pi$
$504$	3.00000	0.133631
$505$	50.7846	2.25989
$506$	0.732051	0.0325436
$507$	9.39230	0.417127
$508$	11.4641	0.508637
$509$	−10.1244	−0.448754	−0.224377	−	0.974502i	$-0.572035\pi$
$509$	−10.1244	−0.448754	−0.224377	+	0.974502i	$0.572035\pi$
$510$	1.39230	0.0616523
$511$	2.78461	0.123184
$512$	1.00000	0.0441942
$513$	−4.46410	−0.197095
$514$	−22.7321	−1.00267
$515$	−38.5692	−1.69956
$516$	−11.9282	−0.525110
$517$	−6.87564	−0.302390
$518$	9.58846	0.421293
$519$	8.26795	0.362923
$520$	14.1962	0.622542
$521$	39.5167	1.73126	0.865628	−	0.500687i	$-0.166919\pi$
$521$	39.5167	1.73126	0.865628	+	0.500687i	$0.166919\pi$
$522$	−1.00000	−0.0437688
$523$	−44.2487	−1.93486	−0.967431	−	0.253137i	$-0.918538\pi$
$523$	−44.2487	−1.93486	−0.967431	+	0.253137i	$0.918538\pi$
$524$	−18.1962	−0.794903
$525$	12.0000	0.523723
$526$	29.0526	1.26675
$527$	−1.26795	−0.0552327
$528$	0.732051	0.0318584
$529$	1.00000	0.0434783
$530$	7.60770	0.330457
$531$	10.4641	0.454103
$532$	−13.3923	−0.580630
$533$	−57.3731	−2.48510
$534$	1.46410	0.0633579
$535$	27.5885	1.19275
$536$	−2.53590	−0.109534
$537$	10.3923	0.448461
$538$	−11.6603	−0.502709
$539$	1.46410	0.0630633
$540$	3.00000	0.129099
$541$	−6.32051	−0.271740	−0.135870	−	0.990727i	$-0.543383\pi$
$541$	−6.32051	−0.271740	−0.135870	+	0.990727i	$0.543383\pi$
$542$	24.7846	1.06459
$543$	19.1244	0.820705
$544$	0.464102	0.0198982
$545$	−24.0000	−1.02805
$546$	14.1962	0.607539
$547$	−33.3205	−1.42468	−0.712341	−	0.701834i	$-0.752365\pi$
$547$	−33.3205	−1.42468	−0.712341	+	0.701834i	$0.752365\pi$
$548$	−12.9282	−0.552265
$549$	−14.3923	−0.614249
$550$	2.92820	0.124859
$551$	4.46410	0.190177
$552$	1.00000	0.0425628
$553$	21.3731	0.908875
$554$	19.1244	0.812516
$555$	9.58846	0.407007
$556$	−19.8564	−0.842099
$557$	4.07180	0.172528	0.0862638	−	0.996272i	$-0.472507\pi$
$557$	4.07180	0.172528	0.0862638	+	0.996272i	$0.472507\pi$
$558$	−2.73205	−0.115657
$559$	−56.4449	−2.38736
$560$	9.00000	0.380319
$561$	0.339746	0.0143441
$562$	27.4641	1.15850
$563$	37.3205	1.57287	0.786436	−	0.617672i	$-0.211924\pi$
$563$	37.3205	1.57287	0.786436	+	0.617672i	$0.211924\pi$
$564$	−9.39230	−0.395487
$565$	34.1769	1.43783
$566$	8.58846	0.361000
$567$	3.00000	0.125988
$568$	−11.6603	−0.489253
$569$	−10.6077	−0.444698	−0.222349	−	0.974967i	$-0.571372\pi$
$569$	−10.6077	−0.444698	−0.222349	+	0.974967i	$0.571372\pi$
$570$	−13.3923	−0.560942
$571$	−23.9090	−1.00056	−0.500280	−	0.865864i	$-0.666769\pi$
$571$	−23.9090	−1.00056	−0.500280	+	0.865864i	$0.666769\pi$
$572$	3.46410	0.144841
$573$	−1.87564	−0.0783562
$574$	−36.3731	−1.51818
$575$	4.00000	0.166812
$576$	1.00000	0.0416667
$577$	24.4449	1.01765	0.508826	−	0.860869i	$-0.330079\pi$
$577$	24.4449	1.01765	0.508826	+	0.860869i	$0.330079\pi$
$578$	−16.7846	−0.698148
$579$	9.46410	0.393315
$580$	−3.00000	−0.124568
$581$	−34.3923	−1.42683
$582$	4.00000	0.165805
$583$	1.85641	0.0768845
$584$	0.928203	0.0384093
$585$	14.1962	0.586939
$586$	6.00000	0.247858
$587$	28.1769	1.16299	0.581493	−	0.813552i	$-0.302469\pi$
$587$	28.1769	1.16299	0.581493	+	0.813552i	$0.302469\pi$
$588$	2.00000	0.0824786
$589$	12.1962	0.502534
$590$	31.3923	1.29240
$591$	17.7321	0.729399
$592$	3.19615	0.131361
$593$	33.4641	1.37421	0.687103	−	0.726560i	$-0.258882\pi$
$593$	33.4641	1.37421	0.687103	+	0.726560i	$0.258882\pi$
$594$	0.732051	0.0300364
$595$	4.17691	0.171237
$596$	5.00000	0.204808
$597$	17.3205	0.708881
$598$	4.73205	0.193508
$599$	−42.3923	−1.73210	−0.866051	−	0.499955i	$-0.833350\pi$
$599$	−42.3923	−1.73210	−0.866051	+	0.499955i	$0.833350\pi$
$600$	4.00000	0.163299
$601$	20.7846	0.847822	0.423911	−	0.905704i	$-0.360657\pi$
$601$	20.7846	0.847822	0.423911	+	0.905704i	$0.360657\pi$
$602$	−35.7846	−1.45847
$603$	−2.53590	−0.103270
$604$	14.1244	0.574712
$605$	−31.3923	−1.27628
$606$	16.9282	0.687661
$607$	7.12436	0.289169	0.144584	−	0.989492i	$-0.453816\pi$
$607$	7.12436	0.289169	0.144584	+	0.989492i	$0.453816\pi$
$608$	−4.46410	−0.181043
$609$	−3.00000	−0.121566
$610$	−43.1769	−1.74818
$611$	−44.4449	−1.79805
$612$	0.464102	0.0187602
$613$	−12.1436	−0.490475	−0.245238	−	0.969463i	$-0.578866\pi$
$613$	−12.1436	−0.490475	−0.245238	+	0.969463i	$0.578866\pi$
$614$	−9.07180	−0.366108
$615$	−36.3731	−1.46670
$616$	2.19615	0.0884855
$617$	−21.1436	−0.851209	−0.425605	−	0.904909i	$-0.639939\pi$
$617$	−21.1436	−0.851209	−0.425605	+	0.904909i	$0.639939\pi$
$618$	−12.8564	−0.517161
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	−8.19615	−0.329165
$621$	1.00000	0.0401286
$622$	−6.32051	−0.253429
$623$	4.39230	0.175974
$624$	4.73205	0.189434
$625$	−29.0000	−1.16000
$626$	−20.1244	−0.804331
$627$	−3.26795	−0.130509
$628$	−0.267949	−0.0106923
$629$	1.48334	0.0591446
$630$	9.00000	0.358569
$631$	−17.0000	−0.676759	−0.338380	−	0.941010i	$-0.609879\pi$
$631$	−17.0000	−0.676759	−0.338380	+	0.941010i	$0.609879\pi$
$632$	7.12436	0.283392
$633$	11.1962	0.445007
$634$	10.3397	0.410644
$635$	34.3923	1.36482
$636$	2.53590	0.100555
$637$	9.46410	0.374981
$638$	−0.732051	−0.0289822
$639$	−11.6603	−0.461273
$640$	3.00000	0.118585
$641$	8.46410	0.334312	0.167156	−	0.985930i	$-0.446542\pi$
$641$	8.46410	0.334312	0.167156	+	0.985930i	$0.446542\pi$
$642$	9.19615	0.362943
$643$	−29.9090	−1.17949	−0.589747	−	0.807588i	$-0.700773\pi$
$643$	−29.9090	−1.17949	−0.589747	+	0.807588i	$0.700773\pi$
$644$	3.00000	0.118217
$645$	−35.7846	−1.40902
$646$	−2.07180	−0.0815138
$647$	−16.5885	−0.652160	−0.326080	−	0.945342i	$-0.605728\pi$
$647$	−16.5885	−0.652160	−0.326080	+	0.945342i	$0.605728\pi$
$648$	1.00000	0.0392837
$649$	7.66025	0.300691
$650$	18.9282	0.742425
$651$	−8.19615	−0.321233
$652$	−20.2679	−0.793754
$653$	17.6077	0.689042	0.344521	−	0.938779i	$-0.388041\pi$
$653$	17.6077	0.689042	0.344521	+	0.938779i	$0.388041\pi$
$654$	−8.00000	−0.312825
$655$	−54.5885	−2.13295
$656$	−12.1244	−0.473377
$657$	0.928203	0.0362127
$658$	−28.1769	−1.09845
$659$	32.0000	1.24654	0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000	1.24654	0.623272	+	0.782006i	$0.285803\pi$
$660$	2.19615	0.0854851
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	29.4449	1.14441
$663$	2.19615	0.0852915
$664$	−11.4641	−0.444893
$665$	−40.1769	−1.55799
$666$	3.19615	0.123848
$667$	−1.00000	−0.0387202
$668$	19.8564	0.768267
$669$	8.00000	0.309298
$670$	−7.60770	−0.293911
$671$	−10.5359	−0.406734
$672$	3.00000	0.115728
$673$	19.0000	0.732396	0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000	0.732396	0.366198	+	0.930537i	$0.380659\pi$
$674$	−20.9282	−0.806124
$675$	4.00000	0.153960
$676$	9.39230	0.361242
$677$	6.78461	0.260754	0.130377	−	0.991465i	$-0.458381\pi$
$677$	6.78461	0.260754	0.130377	+	0.991465i	$0.458381\pi$
$678$	11.3923	0.437519
$679$	12.0000	0.460518
$680$	1.39230	0.0533925
$681$	−25.5885	−0.980552
$682$	−2.00000	−0.0765840
$683$	−21.5359	−0.824048	−0.412024	−	0.911173i	$-0.635178\pi$
$683$	−21.5359	−0.824048	−0.412024	+	0.911173i	$0.635178\pi$
$684$	−4.46410	−0.170689
$685$	−38.7846	−1.48188
$686$	−15.0000	−0.572703
$687$	13.1962	0.503465
$688$	−11.9282	−0.454758
$689$	12.0000	0.457164
$690$	3.00000	0.114208
$691$	−39.5167	−1.50328	−0.751642	−	0.659571i	$-0.770738\pi$
$691$	−39.5167	−1.50328	−0.751642	+	0.659571i	$0.770738\pi$
$692$	8.26795	0.314300
$693$	2.19615	0.0834249
$694$	18.8564	0.715779
$695$	−59.5692	−2.25959
$696$	−1.00000	−0.0379049
$697$	−5.62693	−0.213135
$698$	−6.39230	−0.241952
$699$	23.3205	0.882063
$700$	12.0000	0.453557
$701$	−23.7846	−0.898332	−0.449166	−	0.893448i	$-0.648279\pi$
$701$	−23.7846	−0.898332	−0.449166	+	0.893448i	$0.648279\pi$
$702$	4.73205	0.178600
$703$	−14.2679	−0.538126
$704$	0.732051	0.0275902
$705$	−28.1769	−1.06120
$706$	−11.4641	−0.431457
$707$	50.7846	1.90995
$708$	10.4641	0.393265
$709$	4.33975	0.162983	0.0814913	−	0.996674i	$-0.474032\pi$
$709$	4.33975	0.162983	0.0814913	+	0.996674i	$0.474032\pi$
$710$	−34.9808	−1.31280
$711$	7.12436	0.267184
$712$	1.46410	0.0548695
$713$	−2.73205	−0.102316
$714$	1.39230	0.0521057
$715$	10.3923	0.388650
$716$	10.3923	0.388379
$717$	−7.85641	−0.293403
$718$	−4.26795	−0.159278
$719$	38.1962	1.42448	0.712238	−	0.701938i	$-0.247682\pi$
$719$	38.1962	1.42448	0.712238	+	0.701938i	$0.247682\pi$
$720$	3.00000	0.111803
$721$	−38.5692	−1.43639
$722$	0.928203	0.0345441
$723$	13.7321	0.510700
$724$	19.1244	0.710751
$725$	−4.00000	−0.148556
$726$	−10.4641	−0.388359
$727$	−20.2487	−0.750983	−0.375492	−	0.926826i	$-0.622526\pi$
$727$	−20.2487	−0.750983	−0.375492	+	0.926826i	$0.622526\pi$
$728$	14.1962	0.526144
$729$	1.00000	0.0370370
$730$	2.78461	0.103063
$731$	−5.53590	−0.204753
$732$	−14.3923	−0.531955
$733$	−2.92820	−0.108156	−0.0540778	−	0.998537i	$-0.517222\pi$
$733$	−2.92820	−0.108156	−0.0540778	+	0.998537i	$0.517222\pi$
$734$	14.3397	0.529290
$735$	6.00000	0.221313
$736$	1.00000	0.0368605
$737$	−1.85641	−0.0683816
$738$	−12.1244	−0.446304
$739$	−15.6077	−0.574138	−0.287069	−	0.957910i	$-0.592681\pi$
$739$	−15.6077	−0.574138	−0.287069	+	0.957910i	$0.592681\pi$
$740$	9.58846	0.352479
$741$	−21.1244	−0.776023
$742$	7.60770	0.279287
$743$	−47.4449	−1.74058	−0.870292	−	0.492537i	$-0.836070\pi$
$743$	−47.4449	−1.74058	−0.870292	+	0.492537i	$0.836070\pi$
$744$	−2.73205	−0.100162
$745$	15.0000	0.549557
$746$	30.9282	1.13236
$747$	−11.4641	−0.419450
$748$	0.339746	0.0124223
$749$	27.5885	1.00806
$750$	−3.00000	−0.109545
$751$	−29.3205	−1.06992	−0.534960	−	0.844877i	$-0.679673\pi$
$751$	−29.3205	−1.06992	−0.534960	+	0.844877i	$0.679673\pi$
$752$	−9.39230	−0.342502
$753$	1.46410	0.0533548
$754$	−4.73205	−0.172331
$755$	42.3731	1.54211
$756$	3.00000	0.109109
$757$	−1.19615	−0.0434749	−0.0217374	−	0.999764i	$-0.506920\pi$
$757$	−1.19615	−0.0434749	−0.0217374	+	0.999764i	$0.506920\pi$
$758$	3.85641	0.140071
$759$	0.732051	0.0265718
$760$	−13.3923	−0.485790
$761$	11.8038	0.427889	0.213945	−	0.976846i	$-0.431369\pi$
$761$	11.8038	0.427889	0.213945	+	0.976846i	$0.431369\pi$
$762$	11.4641	0.415301
$763$	−24.0000	−0.868858
$764$	−1.87564	−0.0678584
$765$	1.39230	0.0503389
$766$	9.80385	0.354227
$767$	49.5167	1.78794
$768$	1.00000	0.0360844
$769$	7.94744	0.286592	0.143296	−	0.989680i	$-0.454230\pi$
$769$	7.94744	0.286592	0.143296	+	0.989680i	$0.454230\pi$
$770$	6.58846	0.237432
$771$	−22.7321	−0.818675
$772$	9.46410	0.340620
$773$	24.9808	0.898496	0.449248	−	0.893407i	$-0.351692\pi$
$773$	24.9808	0.898496	0.449248	+	0.893407i	$0.351692\pi$
$774$	−11.9282	−0.428750
$775$	−10.9282	−0.392553
$776$	4.00000	0.143592
$777$	9.58846	0.343984
$778$	13.8564	0.496776
$779$	54.1244	1.93921
$780$	14.1962	0.508304
$781$	−8.53590	−0.305438
$782$	0.464102	0.0165962
$783$	−1.00000	−0.0357371
$784$	2.00000	0.0714286
$785$	−0.803848	−0.0286906
$786$	−18.1962	−0.649035
$787$	−5.80385	−0.206885	−0.103442	−	0.994635i	$-0.532986\pi$
$787$	−5.80385	−0.206885	−0.103442	+	0.994635i	$0.532986\pi$
$788$	17.7321	0.631678
$789$	29.0526	1.03430
$790$	21.3731	0.760420
$791$	34.1769	1.21519
$792$	0.732051	0.0260123
$793$	−68.1051	−2.41848
$794$	17.6603	0.626739
$795$	7.60770	0.269817
$796$	17.3205	0.613909
$797$	−53.2295	−1.88548	−0.942742	−	0.333522i	$-0.891763\pi$
$797$	−53.2295	−1.88548	−0.942742	+	0.333522i	$0.891763\pi$
$798$	−13.3923	−0.474082
$799$	−4.35898	−0.154210
$800$	4.00000	0.141421
$801$	1.46410	0.0517315
$802$	29.7128	1.04920
$803$	0.679492	0.0239787
$804$	−2.53590	−0.0894342
$805$	9.00000	0.317208
$806$	−12.9282	−0.455377
$807$	−11.6603	−0.410460
$808$	16.9282	0.595532
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	3.00000	0.105409
$811$	−53.5692	−1.88107	−0.940535	−	0.339697i	$-0.889675\pi$
$811$	−53.5692	−1.88107	−0.940535	+	0.339697i	$0.889675\pi$
$812$	−3.00000	−0.105279
$813$	24.7846	0.869234
$814$	2.33975	0.0820081
$815$	−60.8038	−2.12987
$816$	0.464102	0.0162468
$817$	53.2487	1.86294
$818$	−5.12436	−0.179169
$819$	14.1962	0.496054
$820$	−36.3731	−1.27020
$821$	7.46410	0.260499	0.130249	−	0.991481i	$-0.458422\pi$
$821$	7.46410	0.260499	0.130249	+	0.991481i	$0.458422\pi$
$822$	−12.9282	−0.450923
$823$	46.3013	1.61396	0.806980	−	0.590578i	$-0.201100\pi$
$823$	46.3013	1.61396	0.806980	+	0.590578i	$0.201100\pi$
$824$	−12.8564	−0.447874
$825$	2.92820	0.101947
$826$	31.3923	1.09228
$827$	−21.0718	−0.732738	−0.366369	−	0.930470i	$-0.619399\pi$
$827$	−21.0718	−0.732738	−0.366369	+	0.930470i	$0.619399\pi$
$828$	1.00000	0.0347524
$829$	−14.6077	−0.507346	−0.253673	−	0.967290i	$-0.581639\pi$
$829$	−14.6077	−0.507346	−0.253673	+	0.967290i	$0.581639\pi$
$830$	−34.3923	−1.19377
$831$	19.1244	0.663417
$832$	4.73205	0.164054
$833$	0.928203	0.0321603
$834$	−19.8564	−0.687571
$835$	59.5692	2.06148
$836$	−3.26795	−0.113024
$837$	−2.73205	−0.0944335
$838$	21.5885	0.745761
$839$	20.2679	0.699727	0.349864	−	0.936801i	$-0.386228\pi$
$839$	20.2679	0.699727	0.349864	+	0.936801i	$0.386228\pi$
$840$	9.00000	0.310530
$841$	1.00000	0.0344828
$842$	9.60770	0.331103
$843$	27.4641	0.945914
$844$	11.1962	0.385387
$845$	28.1769	0.969315
$846$	−9.39230	−0.322914
$847$	−31.3923	−1.07865
$848$	2.53590	0.0870831
$849$	8.58846	0.294755
$850$	1.85641	0.0636742
$851$	3.19615	0.109563
$852$	−11.6603	−0.399474
$853$	2.21539	0.0758535	0.0379268	−	0.999281i	$-0.487925\pi$
$853$	2.21539	0.0758535	0.0379268	+	0.999281i	$0.487925\pi$
$854$	−43.1769	−1.47748
$855$	−13.3923	−0.458007
$856$	9.19615	0.314318
$857$	32.4449	1.10830	0.554148	−	0.832418i	$-0.313044\pi$
$857$	32.4449	1.10830	0.554148	+	0.832418i	$0.313044\pi$
$858$	3.46410	0.118262
$859$	5.58846	0.190676	0.0953379	−	0.995445i	$-0.469607\pi$
$859$	5.58846	0.190676	0.0953379	+	0.995445i	$0.469607\pi$
$860$	−35.7846	−1.22025
$861$	−36.3731	−1.23959
$862$	19.8038	0.674522
$863$	44.0526	1.49957	0.749783	−	0.661683i	$-0.230158\pi$
$863$	44.0526	1.49957	0.749783	+	0.661683i	$0.230158\pi$
$864$	1.00000	0.0340207
$865$	24.8038	0.843356
$866$	−33.5167	−1.13894
$867$	−16.7846	−0.570035
$868$	−8.19615	−0.278196
$869$	5.21539	0.176920
$870$	−3.00000	−0.101710
$871$	−12.0000	−0.406604
$872$	−8.00000	−0.270914
$873$	4.00000	0.135379
$874$	−4.46410	−0.151000
$875$	−9.00000	−0.304256
$876$	0.928203	0.0313611
$877$	−13.1244	−0.443178	−0.221589	−	0.975140i	$-0.571124\pi$
$877$	−13.1244	−0.443178	−0.221589	+	0.975140i	$0.571124\pi$
$878$	10.8038	0.364612
$879$	6.00000	0.202375
$880$	2.19615	0.0740323
$881$	14.9282	0.502944	0.251472	−	0.967865i	$-0.419085\pi$
$881$	14.9282	0.502944	0.251472	+	0.967865i	$0.419085\pi$
$882$	2.00000	0.0673435
$883$	−35.5167	−1.19523	−0.597615	−	0.801783i	$-0.703885\pi$
$883$	−35.5167	−1.19523	−0.597615	+	0.801783i	$0.703885\pi$
$884$	2.19615	0.0738646
$885$	31.3923	1.05524
$886$	−37.1769	−1.24898
$887$	−8.53590	−0.286607	−0.143304	−	0.989679i	$-0.545773\pi$
$887$	−8.53590	−0.286607	−0.143304	+	0.989679i	$0.545773\pi$
$888$	3.19615	0.107256
$889$	34.3923	1.15348
$890$	4.39230	0.147230
$891$	0.732051	0.0245246
$892$	8.00000	0.267860
$893$	41.9282	1.40307
$894$	5.00000	0.167225
$895$	31.1769	1.04213
$896$	3.00000	0.100223
$897$	4.73205	0.157999
$898$	−8.66025	−0.288996
$899$	2.73205	0.0911190
$900$	4.00000	0.133333
$901$	1.17691	0.0392087
$902$	−8.87564	−0.295527
$903$	−35.7846	−1.19084
$904$	11.3923	0.378902
$905$	57.3731	1.90715
$906$	14.1244	0.469250
$907$	−24.1436	−0.801675	−0.400837	−	0.916149i	$-0.631281\pi$
$907$	−24.1436	−0.801675	−0.400837	+	0.916149i	$0.631281\pi$
$908$	−25.5885	−0.849183
$909$	16.9282	0.561473
$910$	42.5885	1.41179
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	−4.46410	−0.147821
$913$	−8.39230	−0.277745
$914$	−34.9090	−1.15469
$915$	−43.1769	−1.42738
$916$	13.1962	0.436013
$917$	−54.5885	−1.80267
$918$	0.464102	0.0153176
$919$	32.6077	1.07563	0.537814	−	0.843063i	$-0.319250\pi$
$919$	32.6077	1.07563	0.537814	+	0.843063i	$0.319250\pi$
$920$	3.00000	0.0989071
$921$	−9.07180	−0.298926
$922$	30.0526	0.989728
$923$	−55.1769	−1.81617
$924$	2.19615	0.0722481
$925$	12.7846	0.420355
$926$	31.1769	1.02454
$927$	−12.8564	−0.422260
$928$	−1.00000	−0.0328266
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	−8.19615	−0.268762
$931$	−8.92820	−0.292610
$932$	23.3205	0.763889
$933$	−6.32051	−0.206924
$934$	11.8038	0.386234
$935$	1.01924	0.0333326
$936$	4.73205	0.154672
$937$	−55.5885	−1.81600	−0.907998	−	0.418975i	$-0.862390\pi$
$937$	−55.5885	−1.81600	−0.907998	+	0.418975i	$0.862390\pi$
$938$	−7.60770	−0.248400
$939$	−20.1244	−0.656734
$940$	−28.1769	−0.919030
$941$	−37.1769	−1.21193	−0.605966	−	0.795490i	$-0.707213\pi$
$941$	−37.1769	−1.21193	−0.605966	+	0.795490i	$0.707213\pi$
$942$	−0.267949	−0.00873026
$943$	−12.1244	−0.394823
$944$	10.4641	0.340577
$945$	9.00000	0.292770
$946$	−8.73205	−0.283903
$947$	−28.0526	−0.911586	−0.455793	−	0.890086i	$-0.650644\pi$
$947$	−28.0526	−0.911586	−0.455793	+	0.890086i	$0.650644\pi$
$948$	7.12436	0.231388
$949$	4.39230	0.142580
$950$	−17.8564	−0.579338
$951$	10.3397	0.335289
$952$	1.39230	0.0451249
$953$	−38.5359	−1.24830	−0.624150	−	0.781304i	$-0.714555\pi$
$953$	−38.5359	−1.24830	−0.624150	+	0.781304i	$0.714555\pi$
$954$	2.53590	0.0821027
$955$	−5.62693	−0.182083
$956$	−7.85641	−0.254094
$957$	−0.732051	−0.0236638
$958$	25.8564	0.835383
$959$	−38.7846	−1.25242
$960$	3.00000	0.0968246
$961$	−23.5359	−0.759223
$962$	15.1244	0.487629
$963$	9.19615	0.296342
$964$	13.7321	0.442280
$965$	28.3923	0.913981
$966$	3.00000	0.0965234
$967$	−39.1244	−1.25815	−0.629077	−	0.777343i	$-0.716567\pi$
$967$	−39.1244	−1.25815	−0.629077	+	0.777343i	$0.716567\pi$
$968$	−10.4641	−0.336329
$969$	−2.07180	−0.0665557
$970$	12.0000	0.385297
$971$	−9.41154	−0.302031	−0.151015	−	0.988531i	$-0.548254\pi$
$971$	−9.41154	−0.302031	−0.151015	+	0.988531i	$0.548254\pi$
$972$	1.00000	0.0320750
$973$	−59.5692	−1.90970
$974$	−32.5167	−1.04190
$975$	18.9282	0.606188
$976$	−14.3923	−0.460686
$977$	18.9808	0.607248	0.303624	−	0.952792i	$-0.401803\pi$
$977$	18.9808	0.607248	0.303624	+	0.952792i	$0.401803\pi$
$978$	−20.2679	−0.648098
$979$	1.07180	0.0342548
$980$	6.00000	0.191663
$981$	−8.00000	−0.255420
$982$	−6.33975	−0.202309
$983$	−26.1051	−0.832624	−0.416312	−	0.909222i	$-0.636678\pi$
$983$	−26.1051	−0.832624	−0.416312	+	0.909222i	$0.636678\pi$
$984$	−12.1244	−0.386510
$985$	53.1962	1.69497
$986$	−0.464102	−0.0147800
$987$	−28.1769	−0.896881
$988$	−21.1244	−0.672055
$989$	−11.9282	−0.379295
$990$	2.19615	0.0697983
$991$	−13.4449	−0.427090	−0.213545	−	0.976933i	$-0.568501\pi$
$991$	−13.4449	−0.427090	−0.213545	+	0.976933i	$0.568501\pi$
$992$	−2.73205	−0.0867427
$993$	29.4449	0.934405
$994$	−34.9808	−1.10952
$995$	51.9615	1.64729
$996$	−11.4641	−0.363254
$997$	−53.4974	−1.69428	−0.847140	−	0.531369i	$-0.821678\pi$
$997$	−53.4974	−1.69428	−0.847140	+	0.531369i	$0.821678\pi$
$998$	20.0526	0.634753
$999$	3.19615	0.101122

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.w.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.00000 q^{5} +1.00000 q^{6} +3.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.00000 q^{10} +0.732051 q^{11} +1.00000 q^{12} +4.73205 q^{13} +3.00000 q^{14} +3.00000 q^{15} +1.00000 q^{16} +0.464102 q^{17} +1.00000 q^{18} -4.46410 q^{19} +3.00000 q^{20} +3.00000 q^{21} +0.732051 q^{22} +1.00000 q^{23} +1.00000 q^{24} +4.00000 q^{25} +4.73205 q^{26} +1.00000 q^{27} +3.00000 q^{28} -1.00000 q^{29} +3.00000 q^{30} -2.73205 q^{31} +1.00000 q^{32} +0.732051 q^{33} +0.464102 q^{34} +9.00000 q^{35} +1.00000 q^{36} +3.19615 q^{37} -4.46410 q^{38} +4.73205 q^{39} +3.00000 q^{40} -12.1244 q^{41} +3.00000 q^{42} -11.9282 q^{43} +0.732051 q^{44} +3.00000 q^{45} +1.00000 q^{46} -9.39230 q^{47} +1.00000 q^{48} +2.00000 q^{49} +4.00000 q^{50} +0.464102 q^{51} +4.73205 q^{52} +2.53590 q^{53} +1.00000 q^{54} +2.19615 q^{55} +3.00000 q^{56} -4.46410 q^{57} -1.00000 q^{58} +10.4641 q^{59} +3.00000 q^{60} -14.3923 q^{61} -2.73205 q^{62} +3.00000 q^{63} +1.00000 q^{64} +14.1962 q^{65} +0.732051 q^{66} -2.53590 q^{67} +0.464102 q^{68} +1.00000 q^{69} +9.00000 q^{70} -11.6603 q^{71} +1.00000 q^{72} +0.928203 q^{73} +3.19615 q^{74} +4.00000 q^{75} -4.46410 q^{76} +2.19615 q^{77} +4.73205 q^{78} +7.12436 q^{79} +3.00000 q^{80} +1.00000 q^{81} -12.1244 q^{82} -11.4641 q^{83} +3.00000 q^{84} +1.39230 q^{85} -11.9282 q^{86} -1.00000 q^{87} +0.732051 q^{88} +1.46410 q^{89} +3.00000 q^{90} +14.1962 q^{91} +1.00000 q^{92} -2.73205 q^{93} -9.39230 q^{94} -13.3923 q^{95} +1.00000 q^{96} +4.00000 q^{97} +2.00000 q^{98} +0.732051 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 6 q^{5} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 6 * q^5 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 6 q^{5} + 2 q^{6} + 6 q^{7} + 2 q^{8} + 2 q^{9} + 6 q^{10} - 2 q^{11} + 2 q^{12} + 6 q^{13} + 6 q^{14} + 6 q^{15} + 2 q^{16} - 6 q^{17} + 2 q^{18} - 2 q^{19} + 6 q^{20} + 6 q^{21} - 2 q^{22} + 2 q^{23} + 2 q^{24} + 8 q^{25} + 6 q^{26} + 2 q^{27} + 6 q^{28} - 2 q^{29} + 6 q^{30} - 2 q^{31} + 2 q^{32} - 2 q^{33} - 6 q^{34} + 18 q^{35} + 2 q^{36} - 4 q^{37} - 2 q^{38} + 6 q^{39} + 6 q^{40} + 6 q^{42} - 10 q^{43} - 2 q^{44} + 6 q^{45} + 2 q^{46} + 2 q^{47} + 2 q^{48} + 4 q^{49} + 8 q^{50} - 6 q^{51} + 6 q^{52} + 12 q^{53} + 2 q^{54} - 6 q^{55} + 6 q^{56} - 2 q^{57} - 2 q^{58} + 14 q^{59} + 6 q^{60} - 8 q^{61} - 2 q^{62} + 6 q^{63} + 2 q^{64} + 18 q^{65} - 2 q^{66} - 12 q^{67} - 6 q^{68} + 2 q^{69} + 18 q^{70} - 6 q^{71} + 2 q^{72} - 12 q^{73} - 4 q^{74} + 8 q^{75} - 2 q^{76} - 6 q^{77} + 6 q^{78} - 10 q^{79} + 6 q^{80} + 2 q^{81} - 16 q^{83} + 6 q^{84} - 18 q^{85} - 10 q^{86} - 2 q^{87} - 2 q^{88} - 4 q^{89} + 6 q^{90} + 18 q^{91} + 2 q^{92} - 2 q^{93} + 2 q^{94} - 6 q^{95} + 2 q^{96} + 8 q^{97} + 4 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 6 * q^5 + 2 * q^6 + 6 * q^7 + 2 * q^8 + 2 * q^9 + 6 * q^10 - 2 * q^11 + 2 * q^12 + 6 * q^13 + 6 * q^14 + 6 * q^15 + 2 * q^16 - 6 * q^17 + 2 * q^18 - 2 * q^19 + 6 * q^20 + 6 * q^21 - 2 * q^22 + 2 * q^23 + 2 * q^24 + 8 * q^25 + 6 * q^26 + 2 * q^27 + 6 * q^28 - 2 * q^29 + 6 * q^30 - 2 * q^31 + 2 * q^32 - 2 * q^33 - 6 * q^34 + 18 * q^35 + 2 * q^36 - 4 * q^37 - 2 * q^38 + 6 * q^39 + 6 * q^40 + 6 * q^42 - 10 * q^43 - 2 * q^44 + 6 * q^45 + 2 * q^46 + 2 * q^47 + 2 * q^48 + 4 * q^49 + 8 * q^50 - 6 * q^51 + 6 * q^52 + 12 * q^53 + 2 * q^54 - 6 * q^55 + 6 * q^56 - 2 * q^57 - 2 * q^58 + 14 * q^59 + 6 * q^60 - 8 * q^61 - 2 * q^62 + 6 * q^63 + 2 * q^64 + 18 * q^65 - 2 * q^66 - 12 * q^67 - 6 * q^68 + 2 * q^69 + 18 * q^70 - 6 * q^71 + 2 * q^72 - 12 * q^73 - 4 * q^74 + 8 * q^75 - 2 * q^76 - 6 * q^77 + 6 * q^78 - 10 * q^79 + 6 * q^80 + 2 * q^81 - 16 * q^83 + 6 * q^84 - 18 * q^85 - 10 * q^86 - 2 * q^87 - 2 * q^88 - 4 * q^89 + 6 * q^90 + 18 * q^91 + 2 * q^92 - 2 * q^93 + 2 * q^94 - 6 * q^95 + 2 * q^96 + 8 * q^97 + 4 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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