LMFDB - Embedded newform 8034.2.a.bc.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 48 x^{13} + 44 x^{12} + 872 x^{11} - 707 x^{10} - 7580 x^{9} + 5112 x^{8} + \cdots + 2048 $ x^15 - x^14 - 48x^13 + 44x^12 + 872x^11 - 707x^10 - 7580x^9 + 5112x^8 + 33191x^7 - 16428x^6 - 71361x^5 + 21747x^4 + 65434x^3 - 11840x^2 - 17600*x + 2048
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.676512$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.676512 q^{5} -1.00000 q^{6} -2.72085 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.676512 q^{5} -1.00000 q^{6} -2.72085 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.676512 q^{10} -5.27216 q^{11} -1.00000 q^{12} -1.00000 q^{13} -2.72085 q^{14} +0.676512 q^{15} +1.00000 q^{16} +1.01679 q^{17} +1.00000 q^{18} +4.84704 q^{19} -0.676512 q^{20} +2.72085 q^{21} -5.27216 q^{22} -5.02811 q^{23} -1.00000 q^{24} -4.54233 q^{25} -1.00000 q^{26} -1.00000 q^{27} -2.72085 q^{28} +6.30498 q^{29} +0.676512 q^{30} -3.04264 q^{31} +1.00000 q^{32} +5.27216 q^{33} +1.01679 q^{34} +1.84069 q^{35} +1.00000 q^{36} -1.00182 q^{37} +4.84704 q^{38} +1.00000 q^{39} -0.676512 q^{40} +9.67843 q^{41} +2.72085 q^{42} -8.83597 q^{43} -5.27216 q^{44} -0.676512 q^{45} -5.02811 q^{46} -8.64110 q^{47} -1.00000 q^{48} +0.402999 q^{49} -4.54233 q^{50} -1.01679 q^{51} -1.00000 q^{52} -2.36350 q^{53} -1.00000 q^{54} +3.56668 q^{55} -2.72085 q^{56} -4.84704 q^{57} +6.30498 q^{58} -13.7006 q^{59} +0.676512 q^{60} +11.9378 q^{61} -3.04264 q^{62} -2.72085 q^{63} +1.00000 q^{64} +0.676512 q^{65} +5.27216 q^{66} +2.97048 q^{67} +1.01679 q^{68} +5.02811 q^{69} +1.84069 q^{70} +15.6217 q^{71} +1.00000 q^{72} +1.34277 q^{73} -1.00182 q^{74} +4.54233 q^{75} +4.84704 q^{76} +14.3447 q^{77} +1.00000 q^{78} -5.10293 q^{79} -0.676512 q^{80} +1.00000 q^{81} +9.67843 q^{82} -0.744233 q^{83} +2.72085 q^{84} -0.687874 q^{85} -8.83597 q^{86} -6.30498 q^{87} -5.27216 q^{88} +13.8862 q^{89} -0.676512 q^{90} +2.72085 q^{91} -5.02811 q^{92} +3.04264 q^{93} -8.64110 q^{94} -3.27908 q^{95} -1.00000 q^{96} -2.57742 q^{97} +0.402999 q^{98} -5.27216 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{2} - 15 q^{3} + 15 q^{4} - q^{5} - 15 q^{6} + 5 q^{7} + 15 q^{8} + 15 q^{9}+O(q^{10}) $ 15 * q + 15 * q^2 - 15 * q^3 + 15 * q^4 - q^5 - 15 * q^6 + 5 * q^7 + 15 * q^8 + 15 * q^9	\( 15 q + 15 q^{2} - 15 q^{3} + 15 q^{4} - q^{5} - 15 q^{6} + 5 q^{7} + 15 q^{8} + 15 q^{9} - q^{10} + 3 q^{11} - 15 q^{12} - 15 q^{13} + 5 q^{14} + q^{15} + 15 q^{16} - 2 q^{17} + 15 q^{18} + 8 q^{19} - q^{20} - 5 q^{21} + 3 q^{22} + 3 q^{23} - 15 q^{24} + 22 q^{25} - 15 q^{26} - 15 q^{27} + 5 q^{28} + 26 q^{29} + q^{30} + 15 q^{32} - 3 q^{33} - 2 q^{34} - 8 q^{35} + 15 q^{36} + 25 q^{37} + 8 q^{38} + 15 q^{39} - q^{40} - q^{41} - 5 q^{42} + 10 q^{43} + 3 q^{44} - q^{45} + 3 q^{46} - 3 q^{47} - 15 q^{48} + 32 q^{49} + 22 q^{50} + 2 q^{51} - 15 q^{52} + 13 q^{53} - 15 q^{54} - 2 q^{55} + 5 q^{56} - 8 q^{57} + 26 q^{58} + 28 q^{59} + q^{60} + 22 q^{61} + 5 q^{63} + 15 q^{64} + q^{65} - 3 q^{66} + 29 q^{67} - 2 q^{68} - 3 q^{69} - 8 q^{70} + 18 q^{71} + 15 q^{72} + 23 q^{73} + 25 q^{74} - 22 q^{75} + 8 q^{76} + 17 q^{77} + 15 q^{78} + 27 q^{79} - q^{80} + 15 q^{81} - q^{82} + 7 q^{83} - 5 q^{84} + 43 q^{85} + 10 q^{86} - 26 q^{87} + 3 q^{88} + 35 q^{89} - q^{90} - 5 q^{91} + 3 q^{92} - 3 q^{94} + 6 q^{95} - 15 q^{96} + 19 q^{97} + 32 q^{98} + 3 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 - 15 * q^3 + 15 * q^4 - q^5 - 15 * q^6 + 5 * q^7 + 15 * q^8 + 15 * q^9 - q^10 + 3 * q^11 - 15 * q^12 - 15 * q^13 + 5 * q^14 + q^15 + 15 * q^16 - 2 * q^17 + 15 * q^18 + 8 * q^19 - q^20 - 5 * q^21 + 3 * q^22 + 3 * q^23 - 15 * q^24 + 22 * q^25 - 15 * q^26 - 15 * q^27 + 5 * q^28 + 26 * q^29 + q^30 + 15 * q^32 - 3 * q^33 - 2 * q^34 - 8 * q^35 + 15 * q^36 + 25 * q^37 + 8 * q^38 + 15 * q^39 - q^40 - q^41 - 5 * q^42 + 10 * q^43 + 3 * q^44 - q^45 + 3 * q^46 - 3 * q^47 - 15 * q^48 + 32 * q^49 + 22 * q^50 + 2 * q^51 - 15 * q^52 + 13 * q^53 - 15 * q^54 - 2 * q^55 + 5 * q^56 - 8 * q^57 + 26 * q^58 + 28 * q^59 + q^60 + 22 * q^61 + 5 * q^63 + 15 * q^64 + q^65 - 3 * q^66 + 29 * q^67 - 2 * q^68 - 3 * q^69 - 8 * q^70 + 18 * q^71 + 15 * q^72 + 23 * q^73 + 25 * q^74 - 22 * q^75 + 8 * q^76 + 17 * q^77 + 15 * q^78 + 27 * q^79 - q^80 + 15 * q^81 - q^82 + 7 * q^83 - 5 * q^84 + 43 * q^85 + 10 * q^86 - 26 * q^87 + 3 * q^88 + 35 * q^89 - q^90 - 5 * q^91 + 3 * q^92 - 3 * q^94 + 6 * q^95 - 15 * q^96 + 19 * q^97 + 32 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−0.676512	−0.302546	−0.151273	−	0.988492i	$-0.548337\pi$
$5$	−0.676512	−0.302546	−0.151273	+	0.988492i	$0.548337\pi$
$6$	−1.00000	−0.408248
$7$	−2.72085	−1.02838	−0.514191	−	0.857675i	$-0.671908\pi$
$7$	−2.72085	−1.02838	−0.514191	+	0.857675i	$0.671908\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−0.676512	−0.213932
$11$	−5.27216	−1.58962	−0.794809	−	0.606860i	$-0.792429\pi$
$11$	−5.27216	−1.58962	−0.794809	+	0.606860i	$0.792429\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.72085	−0.727176
$15$	0.676512	0.174675
$16$	1.00000	0.250000
$17$	1.01679	0.246609	0.123304	−	0.992369i	$-0.460651\pi$
$17$	1.01679	0.246609	0.123304	+	0.992369i	$0.460651\pi$
$18$	1.00000	0.235702
$19$	4.84704	1.11199	0.555994	−	0.831186i	$-0.312338\pi$
$19$	4.84704	1.11199	0.555994	+	0.831186i	$0.312338\pi$
$20$	−0.676512	−0.151273
$21$	2.72085	0.593737
$22$	−5.27216	−1.12403
$23$	−5.02811	−1.04843	−0.524217	−	0.851585i	$-0.675642\pi$
$23$	−5.02811	−1.04843	−0.524217	+	0.851585i	$0.675642\pi$
$24$	−1.00000	−0.204124
$25$	−4.54233	−0.908466
$26$	−1.00000	−0.196116
$27$	−1.00000	−0.192450
$28$	−2.72085	−0.514191
$29$	6.30498	1.17081	0.585403	−	0.810743i	$-0.300936\pi$
$29$	6.30498	1.17081	0.585403	+	0.810743i	$0.300936\pi$
$30$	0.676512	0.123514
$31$	−3.04264	−0.546474	−0.273237	−	0.961947i	$-0.588094\pi$
$31$	−3.04264	−0.546474	−0.273237	+	0.961947i	$0.588094\pi$
$32$	1.00000	0.176777
$33$	5.27216	0.917766
$34$	1.01679	0.174379
$35$	1.84069	0.311133
$36$	1.00000	0.166667
$37$	−1.00182	−0.164698	−0.0823489	−	0.996604i	$-0.526242\pi$
$37$	−1.00182	−0.164698	−0.0823489	+	0.996604i	$0.526242\pi$
$38$	4.84704	0.786294
$39$	1.00000	0.160128
$40$	−0.676512	−0.106966
$41$	9.67843	1.51152	0.755759	−	0.654850i	$-0.227268\pi$
$41$	9.67843	1.51152	0.755759	+	0.654850i	$0.227268\pi$
$42$	2.72085	0.419836
$43$	−8.83597	−1.34747	−0.673737	−	0.738972i	$-0.735312\pi$
$43$	−8.83597	−1.34747	−0.673737	+	0.738972i	$0.735312\pi$
$44$	−5.27216	−0.794809
$45$	−0.676512	−0.100849
$46$	−5.02811	−0.741355
$47$	−8.64110	−1.26043	−0.630217	−	0.776419i	$-0.717034\pi$
$47$	−8.64110	−1.26043	−0.630217	+	0.776419i	$0.717034\pi$
$48$	−1.00000	−0.144338
$49$	0.402999	0.0575712
$50$	−4.54233	−0.642383
$51$	−1.01679	−0.142380
$52$	−1.00000	−0.138675
$53$	−2.36350	−0.324652	−0.162326	−	0.986737i	$-0.551900\pi$
$53$	−2.36350	−0.324652	−0.162326	+	0.986737i	$0.551900\pi$
$54$	−1.00000	−0.136083
$55$	3.56668	0.480932
$56$	−2.72085	−0.363588
$57$	−4.84704	−0.642006
$58$	6.30498	0.827884
$59$	−13.7006	−1.78367	−0.891833	−	0.452365i	$-0.850580\pi$
$59$	−13.7006	−1.78367	−0.891833	+	0.452365i	$0.850580\pi$
$60$	0.676512	0.0873374
$61$	11.9378	1.52848	0.764242	−	0.644930i	$-0.223113\pi$
$61$	11.9378	1.52848	0.764242	+	0.644930i	$0.223113\pi$
$62$	−3.04264	−0.386416
$63$	−2.72085	−0.342794
$64$	1.00000	0.125000
$65$	0.676512	0.0839110
$66$	5.27216	0.648959
$67$	2.97048	0.362902	0.181451	−	0.983400i	$-0.441921\pi$
$67$	2.97048	0.362902	0.181451	+	0.983400i	$0.441921\pi$
$68$	1.01679	0.123304
$69$	5.02811	0.605314
$70$	1.84069	0.220004
$71$	15.6217	1.85395	0.926977	−	0.375119i	$-0.122398\pi$
$71$	15.6217	1.85395	0.926977	+	0.375119i	$0.122398\pi$
$72$	1.00000	0.117851
$73$	1.34277	0.157159	0.0785796	−	0.996908i	$-0.474962\pi$
$73$	1.34277	0.157159	0.0785796	+	0.996908i	$0.474962\pi$
$74$	−1.00182	−0.116459
$75$	4.54233	0.524503
$76$	4.84704	0.555994
$77$	14.3447	1.63474
$78$	1.00000	0.113228
$79$	−5.10293	−0.574125	−0.287062	−	0.957912i	$-0.592679\pi$
$79$	−5.10293	−0.574125	−0.287062	+	0.957912i	$0.592679\pi$
$80$	−0.676512	−0.0756364
$81$	1.00000	0.111111
$82$	9.67843	1.06880
$83$	−0.744233	−0.0816902	−0.0408451	−	0.999165i	$-0.513005\pi$
$83$	−0.744233	−0.0816902	−0.0408451	+	0.999165i	$0.513005\pi$
$84$	2.72085	0.296869
$85$	−0.687874	−0.0746104
$86$	−8.83597	−0.952807
$87$	−6.30498	−0.675965
$88$	−5.27216	−0.562015
$89$	13.8862	1.47194	0.735969	−	0.677016i	$-0.236727\pi$
$89$	13.8862	1.47194	0.735969	+	0.677016i	$0.236727\pi$
$90$	−0.676512	−0.0713107
$91$	2.72085	0.285222
$92$	−5.02811	−0.524217
$93$	3.04264	0.315507
$94$	−8.64110	−0.891262
$95$	−3.27908	−0.336427
$96$	−1.00000	−0.102062
$97$	−2.57742	−0.261697	−0.130849	−	0.991402i	$-0.541770\pi$
$97$	−2.57742	−0.261697	−0.130849	+	0.991402i	$0.541770\pi$
$98$	0.402999	0.0407090
$99$	−5.27216	−0.529873
$100$	−4.54233	−0.454233
$101$	8.50857	0.846634	0.423317	−	0.905982i	$-0.360866\pi$
$101$	8.50857	0.846634	0.423317	+	0.905982i	$0.360866\pi$
$102$	−1.01679	−0.100678
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−1.00000	−0.0980581
$105$	−1.84069	−0.179633
$106$	−2.36350	−0.229564
$107$	−17.8523	−1.72585	−0.862925	−	0.505332i	$-0.831370\pi$
$107$	−17.8523	−1.72585	−0.862925	+	0.505332i	$0.831370\pi$
$108$	−1.00000	−0.0962250
$109$	−0.493205	−0.0472405	−0.0236202	−	0.999721i	$-0.507519\pi$
$109$	−0.493205	−0.0472405	−0.0236202	+	0.999721i	$0.507519\pi$
$110$	3.56668	0.340070
$111$	1.00182	0.0950883
$112$	−2.72085	−0.257096
$113$	−10.3192	−0.970746	−0.485373	−	0.874307i	$-0.661316\pi$
$113$	−10.3192	−0.970746	−0.485373	+	0.874307i	$0.661316\pi$
$114$	−4.84704	−0.453967
$115$	3.40158	0.317199
$116$	6.30498	0.585403
$117$	−1.00000	−0.0924500
$118$	−13.7006	−1.26124
$119$	−2.76654	−0.253608
$120$	0.676512	0.0617568
$121$	16.7957	1.52688
$122$	11.9378	1.08080
$123$	−9.67843	−0.872675
$124$	−3.04264	−0.273237
$125$	6.45551	0.577398
$126$	−2.72085	−0.242392
$127$	14.1251	1.25340	0.626698	−	0.779262i	$-0.284406\pi$
$127$	14.1251	1.25340	0.626698	+	0.779262i	$0.284406\pi$
$128$	1.00000	0.0883883
$129$	8.83597	0.777964
$130$	0.676512	0.0593341
$131$	−6.29354	−0.549870	−0.274935	−	0.961463i	$-0.588656\pi$
$131$	−6.29354	−0.549870	−0.274935	+	0.961463i	$0.588656\pi$
$132$	5.27216	0.458883
$133$	−13.1881	−1.14355
$134$	2.97048	0.256611
$135$	0.676512	0.0582249
$136$	1.01679	0.0871893
$137$	−6.42562	−0.548978	−0.274489	−	0.961590i	$-0.588509\pi$
$137$	−6.42562	−0.548978	−0.274489	+	0.961590i	$0.588509\pi$
$138$	5.02811	0.428021
$139$	18.3900	1.55982	0.779909	−	0.625893i	$-0.215265\pi$
$139$	18.3900	1.55982	0.779909	+	0.625893i	$0.215265\pi$
$140$	1.84069	0.155566
$141$	8.64110	0.727712
$142$	15.6217	1.31094
$143$	5.27216	0.440881
$144$	1.00000	0.0833333
$145$	−4.26540	−0.354222
$146$	1.34277	0.111128
$147$	−0.402999	−0.0332388
$148$	−1.00182	−0.0823489
$149$	14.0749	1.15306	0.576531	−	0.817075i	$-0.304406\pi$
$149$	14.0749	1.15306	0.576531	+	0.817075i	$0.304406\pi$
$150$	4.54233	0.370880
$151$	23.4183	1.90575	0.952877	−	0.303358i	$-0.0981077\pi$
$151$	23.4183	1.90575	0.952877	+	0.303358i	$0.0981077\pi$
$152$	4.84704	0.393147
$153$	1.01679	0.0822029
$154$	14.3447	1.15593
$155$	2.05838	0.165333
$156$	1.00000	0.0800641
$157$	4.71578	0.376360	0.188180	−	0.982135i	$-0.439741\pi$
$157$	4.71578	0.376360	0.188180	+	0.982135i	$0.439741\pi$
$158$	−5.10293	−0.405967
$159$	2.36350	0.187438
$160$	−0.676512	−0.0534830
$161$	13.6807	1.07819
$162$	1.00000	0.0785674
$163$	14.3157	1.12129	0.560647	−	0.828055i	$-0.310552\pi$
$163$	14.3157	1.12129	0.560647	+	0.828055i	$0.310552\pi$
$164$	9.67843	0.755759
$165$	−3.56668	−0.277666
$166$	−0.744233	−0.0577637
$167$	5.09109	0.393960	0.196980	−	0.980407i	$-0.436887\pi$
$167$	5.09109	0.393960	0.196980	+	0.980407i	$0.436887\pi$
$168$	2.72085	0.209918
$169$	1.00000	0.0769231
$170$	−0.687874	−0.0527575
$171$	4.84704	0.370663
$172$	−8.83597	−0.673737
$173$	8.23408	0.626025	0.313013	−	0.949749i	$-0.398662\pi$
$173$	8.23408	0.626025	0.313013	+	0.949749i	$0.398662\pi$
$174$	−6.30498	−0.477979
$175$	12.3590	0.934251
$176$	−5.27216	−0.397404
$177$	13.7006	1.02980
$178$	13.8862	1.04082
$179$	21.3025	1.59223	0.796113	−	0.605148i	$-0.206886\pi$
$179$	21.3025	1.59223	0.796113	+	0.605148i	$0.206886\pi$
$180$	−0.676512	−0.0504243
$181$	8.88993	0.660783	0.330392	−	0.943844i	$-0.392819\pi$
$181$	8.88993	0.660783	0.330392	+	0.943844i	$0.392819\pi$
$182$	2.72085	0.201682
$183$	−11.9378	−0.882471
$184$	−5.02811	−0.370677
$185$	0.677742	0.0498286
$186$	3.04264	0.223097
$187$	−5.36070	−0.392014
$188$	−8.64110	−0.630217
$189$	2.72085	0.197912
$190$	−3.27908	−0.237890
$191$	−11.8651	−0.858532	−0.429266	−	0.903178i	$-0.641228\pi$
$191$	−11.8651	−0.858532	−0.429266	+	0.903178i	$0.641228\pi$
$192$	−1.00000	−0.0721688
$193$	10.6974	0.770019	0.385009	−	0.922913i	$-0.374198\pi$
$193$	10.6974	0.770019	0.385009	+	0.922913i	$0.374198\pi$
$194$	−2.57742	−0.185048
$195$	−0.676512	−0.0484461
$196$	0.402999	0.0287856
$197$	−27.1432	−1.93387	−0.966936	−	0.255021i	$-0.917918\pi$
$197$	−27.1432	−1.93387	−0.966936	+	0.255021i	$0.917918\pi$
$198$	−5.27216	−0.374676
$199$	16.0741	1.13946	0.569731	−	0.821831i	$-0.307047\pi$
$199$	16.0741	1.13946	0.569731	+	0.821831i	$0.307047\pi$
$200$	−4.54233	−0.321191
$201$	−2.97048	−0.209522
$202$	8.50857	0.598661
$203$	−17.1549	−1.20404
$204$	−1.01679	−0.0711898
$205$	−6.54758	−0.457303
$206$	−1.00000	−0.0696733
$207$	−5.02811	−0.349478
$208$	−1.00000	−0.0693375
$209$	−25.5544	−1.76764
$210$	−1.84069	−0.127019
$211$	13.8759	0.955258	0.477629	−	0.878562i	$-0.341496\pi$
$211$	13.8759	0.955258	0.477629	+	0.878562i	$0.341496\pi$
$212$	−2.36350	−0.162326
$213$	−15.6217	−1.07038
$214$	−17.8523	−1.22036
$215$	5.97765	0.407672
$216$	−1.00000	−0.0680414
$217$	8.27855	0.561985
$218$	−0.493205	−0.0334041
$219$	−1.34277	−0.0907359
$220$	3.56668	0.240466
$221$	−1.01679	−0.0683969
$222$	1.00182	0.0672376
$223$	3.62805	0.242952	0.121476	−	0.992594i	$-0.461237\pi$
$223$	3.62805	0.242952	0.121476	+	0.992594i	$0.461237\pi$
$224$	−2.72085	−0.181794
$225$	−4.54233	−0.302822
$226$	−10.3192	−0.686421
$227$	12.8428	0.852406	0.426203	−	0.904627i	$-0.359851\pi$
$227$	12.8428	0.852406	0.426203	+	0.904627i	$0.359851\pi$
$228$	−4.84704	−0.321003
$229$	16.1697	1.06852	0.534261	−	0.845320i	$-0.320590\pi$
$229$	16.1697	1.06852	0.534261	+	0.845320i	$0.320590\pi$
$230$	3.40158	0.224294
$231$	−14.3447	−0.943815
$232$	6.30498	0.413942
$233$	−3.08086	−0.201834	−0.100917	−	0.994895i	$-0.532178\pi$
$233$	−3.08086	−0.201834	−0.100917	+	0.994895i	$0.532178\pi$
$234$	−1.00000	−0.0653720
$235$	5.84581	0.381339
$236$	−13.7006	−0.891833
$237$	5.10293	0.331471
$238$	−2.76654	−0.179328
$239$	16.9163	1.09422	0.547111	−	0.837060i	$-0.315728\pi$
$239$	16.9163	1.09422	0.547111	+	0.837060i	$0.315728\pi$
$240$	0.676512	0.0436687
$241$	−29.6909	−1.91256	−0.956281	−	0.292450i	$-0.905530\pi$
$241$	−29.6909	−1.91256	−0.956281	+	0.292450i	$0.905530\pi$
$242$	16.7957	1.07967
$243$	−1.00000	−0.0641500
$244$	11.9378	0.764242
$245$	−0.272634	−0.0174179
$246$	−9.67843	−0.617074
$247$	−4.84704	−0.308410
$248$	−3.04264	−0.193208
$249$	0.744233	0.0471638
$250$	6.45551	0.408282
$251$	−3.96047	−0.249983	−0.124991	−	0.992158i	$-0.539890\pi$
$251$	−3.96047	−0.249983	−0.124991	+	0.992158i	$0.539890\pi$
$252$	−2.72085	−0.171397
$253$	26.5090	1.66661
$254$	14.1251	0.886285
$255$	0.687874	0.0430763
$256$	1.00000	0.0625000
$257$	28.3377	1.76766	0.883828	−	0.467812i	$-0.154957\pi$
$257$	28.3377	1.76766	0.883828	+	0.467812i	$0.154957\pi$
$258$	8.83597	0.550104
$259$	2.72579	0.169372
$260$	0.676512	0.0419555
$261$	6.30498	0.390268
$262$	−6.29354	−0.388817
$263$	−5.13158	−0.316427	−0.158213	−	0.987405i	$-0.550573\pi$
$263$	−5.13158	−0.316427	−0.158213	+	0.987405i	$0.550573\pi$
$264$	5.27216	0.324479
$265$	1.59894	0.0982221
$266$	−13.1881	−0.808611
$267$	−13.8862	−0.849823
$268$	2.97048	0.181451
$269$	4.33267	0.264168	0.132084	−	0.991239i	$-0.457833\pi$
$269$	4.33267	0.264168	0.132084	+	0.991239i	$0.457833\pi$
$270$	0.676512	0.0411712
$271$	−15.4148	−0.936385	−0.468193	−	0.883626i	$-0.655095\pi$
$271$	−15.4148	−0.936385	−0.468193	+	0.883626i	$0.655095\pi$
$272$	1.01679	0.0616522
$273$	−2.72085	−0.164673
$274$	−6.42562	−0.388186
$275$	23.9479	1.44411
$276$	5.02811	0.302657
$277$	−26.4488	−1.58915	−0.794576	−	0.607164i	$-0.792307\pi$
$277$	−26.4488	−1.58915	−0.794576	+	0.607164i	$0.792307\pi$
$278$	18.3900	1.10296
$279$	−3.04264	−0.182158
$280$	1.84069	0.110002
$281$	−1.70942	−0.101976	−0.0509879	−	0.998699i	$-0.516237\pi$
$281$	−1.70942	−0.101976	−0.0509879	+	0.998699i	$0.516237\pi$
$282$	8.64110	0.514570
$283$	−4.39481	−0.261244	−0.130622	−	0.991432i	$-0.541697\pi$
$283$	−4.39481	−0.261244	−0.130622	+	0.991432i	$0.541697\pi$
$284$	15.6217	0.926977
$285$	3.27908	0.194236
$286$	5.27216	0.311750
$287$	−26.3335	−1.55442
$288$	1.00000	0.0589256
$289$	−15.9661	−0.939184
$290$	−4.26540	−0.250473
$291$	2.57742	0.151091
$292$	1.34277	0.0785796
$293$	−2.45624	−0.143495	−0.0717475	−	0.997423i	$-0.522858\pi$
$293$	−2.45624	−0.143495	−0.0717475	+	0.997423i	$0.522858\pi$
$294$	−0.402999	−0.0235034
$295$	9.26862	0.539640
$296$	−1.00182	−0.0582295
$297$	5.27216	0.305922
$298$	14.0749	0.815338
$299$	5.02811	0.290783
$300$	4.54233	0.262252
$301$	24.0413	1.38572
$302$	23.4183	1.34757
$303$	−8.50857	−0.488804
$304$	4.84704	0.277997
$305$	−8.07610	−0.462436
$306$	1.01679	0.0581262
$307$	−11.7261	−0.669242	−0.334621	−	0.942353i	$-0.608608\pi$
$307$	−11.7261	−0.669242	−0.334621	+	0.942353i	$0.608608\pi$
$308$	14.3447	0.817368
$309$	1.00000	0.0568880
$310$	2.05838	0.116908
$311$	−1.75340	−0.0994264	−0.0497132	−	0.998764i	$-0.515831\pi$
$311$	−1.75340	−0.0994264	−0.0497132	+	0.998764i	$0.515831\pi$
$312$	1.00000	0.0566139
$313$	21.0420	1.18936	0.594681	−	0.803962i	$-0.297278\pi$
$313$	21.0420	1.18936	0.594681	+	0.803962i	$0.297278\pi$
$314$	4.71578	0.266127
$315$	1.84069	0.103711
$316$	−5.10293	−0.287062
$317$	25.9164	1.45561	0.727804	−	0.685785i	$-0.240541\pi$
$317$	25.9164	1.45561	0.727804	+	0.685785i	$0.240541\pi$
$318$	2.36350	0.132539
$319$	−33.2409	−1.86113
$320$	−0.676512	−0.0378182
$321$	17.8523	0.996420
$322$	13.6807	0.762397
$323$	4.92844	0.274226
$324$	1.00000	0.0555556
$325$	4.54233	0.251963
$326$	14.3157	0.792875
$327$	0.493205	0.0272743
$328$	9.67843	0.534402
$329$	23.5111	1.29621
$330$	−3.56668	−0.196340
$331$	−18.0656	−0.992974	−0.496487	−	0.868044i	$-0.665377\pi$
$331$	−18.0656	−0.992974	−0.496487	+	0.868044i	$0.665377\pi$
$332$	−0.744233	−0.0408451
$333$	−1.00182	−0.0548993
$334$	5.09109	0.278572
$335$	−2.00957	−0.109794
$336$	2.72085	0.148434
$337$	1.94031	0.105696	0.0528478	−	0.998603i	$-0.483170\pi$
$337$	1.94031	0.105696	0.0528478	+	0.998603i	$0.483170\pi$
$338$	1.00000	0.0543928
$339$	10.3192	0.560460
$340$	−0.687874	−0.0373052
$341$	16.0413	0.868685
$342$	4.84704	0.262098
$343$	17.9494	0.969178
$344$	−8.83597	−0.476404
$345$	−3.40158	−0.183135
$346$	8.23408	0.442667
$347$	0.229716	0.0123318	0.00616590	−	0.999981i	$-0.498037\pi$
$347$	0.229716	0.0123318	0.00616590	+	0.999981i	$0.498037\pi$
$348$	−6.30498	−0.337982
$349$	19.8807	1.06419	0.532095	−	0.846684i	$-0.321405\pi$
$349$	19.8807	1.06419	0.532095	+	0.846684i	$0.321405\pi$
$350$	12.3590	0.660615
$351$	1.00000	0.0533761
$352$	−5.27216	−0.281007
$353$	−10.2245	−0.544195	−0.272098	−	0.962270i	$-0.587717\pi$
$353$	−10.2245	−0.544195	−0.272098	+	0.962270i	$0.587717\pi$
$354$	13.7006	0.728178
$355$	−10.5683	−0.560905
$356$	13.8862	0.735969
$357$	2.76654	0.146421
$358$	21.3025	1.12587
$359$	−4.10661	−0.216739	−0.108369	−	0.994111i	$-0.534563\pi$
$359$	−4.10661	−0.216739	−0.108369	+	0.994111i	$0.534563\pi$
$360$	−0.676512	−0.0356553
$361$	4.49382	0.236517
$362$	8.88993	0.467244
$363$	−16.7957	−0.881547
$364$	2.72085	0.142611
$365$	−0.908400	−0.0475478
$366$	−11.9378	−0.624001
$367$	24.2806	1.26744	0.633718	−	0.773564i	$-0.281528\pi$
$367$	24.2806	1.26744	0.633718	+	0.773564i	$0.281528\pi$
$368$	−5.02811	−0.262109
$369$	9.67843	0.503839
$370$	0.677742	0.0352341
$371$	6.43073	0.333867
$372$	3.04264	0.157754
$373$	−2.09879	−0.108671	−0.0543355	−	0.998523i	$-0.517304\pi$
$373$	−2.09879	−0.108671	−0.0543355	+	0.998523i	$0.517304\pi$
$374$	−5.36070	−0.277195
$375$	−6.45551	−0.333361
$376$	−8.64110	−0.445631
$377$	−6.30498	−0.324723
$378$	2.72085	0.139945
$379$	−2.52123	−0.129507	−0.0647534	−	0.997901i	$-0.520626\pi$
$379$	−2.52123	−0.129507	−0.0647534	+	0.997901i	$0.520626\pi$
$380$	−3.27908	−0.168213
$381$	−14.1251	−0.723649
$382$	−11.8651	−0.607074
$383$	2.79825	0.142984	0.0714918	−	0.997441i	$-0.477224\pi$
$383$	2.79825	0.142984	0.0714918	+	0.997441i	$0.477224\pi$
$384$	−1.00000	−0.0510310
$385$	−9.70440	−0.494582
$386$	10.6974	0.544485
$387$	−8.83597	−0.449158
$388$	−2.57742	−0.130849
$389$	−21.9755	−1.11420	−0.557102	−	0.830444i	$-0.688087\pi$
$389$	−21.9755	−1.11420	−0.557102	+	0.830444i	$0.688087\pi$
$390$	−0.676512	−0.0342565
$391$	−5.11255	−0.258553
$392$	0.402999	0.0203545
$393$	6.29354	0.317467
$394$	−27.1432	−1.36745
$395$	3.45220	0.173699
$396$	−5.27216	−0.264936
$397$	−18.4038	−0.923658	−0.461829	−	0.886969i	$-0.652807\pi$
$397$	−18.4038	−0.923658	−0.461829	+	0.886969i	$0.652807\pi$
$398$	16.0741	0.805721
$399$	13.1881	0.660228
$400$	−4.54233	−0.227117
$401$	9.25701	0.462273	0.231136	−	0.972921i	$-0.425756\pi$
$401$	9.25701	0.462273	0.231136	+	0.972921i	$0.425756\pi$
$402$	−2.97048	−0.148154
$403$	3.04264	0.151565
$404$	8.50857	0.423317
$405$	−0.676512	−0.0336162
$406$	−17.1549	−0.851382
$407$	5.28175	0.261807
$408$	−1.01679	−0.0503388
$409$	−6.19590	−0.306367	−0.153184	−	0.988198i	$-0.548953\pi$
$409$	−6.19590	−0.306367	−0.153184	+	0.988198i	$0.548953\pi$
$410$	−6.54758	−0.323362
$411$	6.42562	0.316952
$412$	−1.00000	−0.0492665
$413$	37.2772	1.83429
$414$	−5.02811	−0.247118
$415$	0.503483	0.0247150
$416$	−1.00000	−0.0490290
$417$	−18.3900	−0.900562
$418$	−25.5544	−1.24991
$419$	15.4153	0.753087	0.376543	−	0.926399i	$-0.377113\pi$
$419$	15.4153	0.753087	0.376543	+	0.926399i	$0.377113\pi$
$420$	−1.84069	−0.0898163
$421$	−25.8241	−1.25859	−0.629294	−	0.777167i	$-0.716656\pi$
$421$	−25.8241	−1.25859	−0.629294	+	0.777167i	$0.716656\pi$
$422$	13.8759	0.675469
$423$	−8.64110	−0.420145
$424$	−2.36350	−0.114782
$425$	−4.61861	−0.224036
$426$	−15.6217	−0.756873
$427$	−32.4810	−1.57187
$428$	−17.8523	−0.862925
$429$	−5.27216	−0.254543
$430$	5.97765	0.288268
$431$	−7.57974	−0.365103	−0.182552	−	0.983196i	$-0.558436\pi$
$431$	−7.57974	−0.365103	−0.182552	+	0.983196i	$0.558436\pi$
$432$	−1.00000	−0.0481125
$433$	−16.0633	−0.771954	−0.385977	−	0.922508i	$-0.626136\pi$
$433$	−16.0633	−0.771954	−0.385977	+	0.922508i	$0.626136\pi$
$434$	8.27855	0.397383
$435$	4.26540	0.204510
$436$	−0.493205	−0.0236202
$437$	−24.3715	−1.16585
$438$	−1.34277	−0.0641600
$439$	34.5792	1.65037	0.825187	−	0.564859i	$-0.191069\pi$
$439$	34.5792	1.65037	0.825187	+	0.564859i	$0.191069\pi$
$440$	3.56668	0.170035
$441$	0.402999	0.0191904
$442$	−1.01679	−0.0483639
$443$	21.2435	1.00931	0.504654	−	0.863322i	$-0.331620\pi$
$443$	21.2435	1.00931	0.504654	+	0.863322i	$0.331620\pi$
$444$	1.00182	0.0475442
$445$	−9.39420	−0.445328
$446$	3.62805	0.171793
$447$	−14.0749	−0.665720
$448$	−2.72085	−0.128548
$449$	35.5472	1.67757	0.838787	−	0.544460i	$-0.183265\pi$
$449$	35.5472	1.67757	0.838787	+	0.544460i	$0.183265\pi$
$450$	−4.54233	−0.214128
$451$	−51.0263	−2.40273
$452$	−10.3192	−0.485373
$453$	−23.4183	−1.10029
$454$	12.8428	0.602742
$455$	−1.84069	−0.0862927
$456$	−4.84704	−0.226984
$457$	24.4777	1.14502	0.572509	−	0.819898i	$-0.305970\pi$
$457$	24.4777	1.14502	0.572509	+	0.819898i	$0.305970\pi$
$458$	16.1697	0.755559
$459$	−1.01679	−0.0474599
$460$	3.40158	0.158600
$461$	−26.6751	−1.24238	−0.621191	−	0.783659i	$-0.713351\pi$
$461$	−26.6751	−1.24238	−0.621191	+	0.783659i	$0.713351\pi$
$462$	−14.3447	−0.667378
$463$	7.77420	0.361298	0.180649	−	0.983548i	$-0.442180\pi$
$463$	7.77420	0.361298	0.180649	+	0.983548i	$0.442180\pi$
$464$	6.30498	0.292701
$465$	−2.05838	−0.0954553
$466$	−3.08086	−0.142718
$467$	22.4004	1.03657	0.518283	−	0.855209i	$-0.326571\pi$
$467$	22.4004	1.03657	0.518283	+	0.855209i	$0.326571\pi$
$468$	−1.00000	−0.0462250
$469$	−8.08222	−0.373202
$470$	5.84581	0.269647
$471$	−4.71578	−0.217292
$472$	−13.7006	−0.630621
$473$	46.5847	2.14197
$474$	5.10293	0.234385
$475$	−22.0169	−1.01020
$476$	−2.76654	−0.126804
$477$	−2.36350	−0.108217
$478$	16.9163	0.773731
$479$	−38.0641	−1.73919	−0.869597	−	0.493762i	$-0.835621\pi$
$479$	−38.0641	−1.73919	−0.869597	+	0.493762i	$0.835621\pi$
$480$	0.676512	0.0308784
$481$	1.00182	0.0456790
$482$	−29.6909	−1.35239
$483$	−13.6807	−0.622494
$484$	16.7957	0.763442
$485$	1.74366	0.0791754
$486$	−1.00000	−0.0453609
$487$	17.7607	0.804815	0.402408	−	0.915461i	$-0.368173\pi$
$487$	17.7607	0.804815	0.402408	+	0.915461i	$0.368173\pi$
$488$	11.9378	0.540401
$489$	−14.3157	−0.647380
$490$	−0.272634	−0.0123163
$491$	−31.1190	−1.40438	−0.702191	−	0.711989i	$-0.747795\pi$
$491$	−31.1190	−1.40438	−0.702191	+	0.711989i	$0.747795\pi$
$492$	−9.67843	−0.436337
$493$	6.41086	0.288731
$494$	−4.84704	−0.218079
$495$	3.56668	0.160311
$496$	−3.04264	−0.136619
$497$	−42.5042	−1.90657
$498$	0.744233	0.0333499
$499$	15.2463	0.682517	0.341258	−	0.939970i	$-0.389147\pi$
$499$	15.2463	0.682517	0.341258	+	0.939970i	$0.389147\pi$
$500$	6.45551	0.288699
$501$	−5.09109	−0.227453
$502$	−3.96047	−0.176764
$503$	−42.7565	−1.90642	−0.953210	−	0.302310i	$-0.902242\pi$
$503$	−42.7565	−1.90642	−0.953210	+	0.302310i	$0.902242\pi$
$504$	−2.72085	−0.121196
$505$	−5.75615	−0.256145
$506$	26.5090	1.17847
$507$	−1.00000	−0.0444116
$508$	14.1251	0.626698
$509$	−11.1467	−0.494069	−0.247035	−	0.969007i	$-0.579456\pi$
$509$	−11.1467	−0.494069	−0.247035	+	0.969007i	$0.579456\pi$
$510$	0.687874	0.0304596
$511$	−3.65347	−0.161620
$512$	1.00000	0.0441942
$513$	−4.84704	−0.214002
$514$	28.3377	1.24992
$515$	0.676512	0.0298107
$516$	8.83597	0.388982
$517$	45.5573	2.00361
$518$	2.72579	0.119764
$519$	−8.23408	−0.361436
$520$	0.676512	0.0296670
$521$	31.2470	1.36896	0.684479	−	0.729033i	$-0.260030\pi$
$521$	31.2470	1.36896	0.684479	+	0.729033i	$0.260030\pi$
$522$	6.30498	0.275961
$523$	8.46338	0.370078	0.185039	−	0.982731i	$-0.440759\pi$
$523$	8.46338	0.370078	0.185039	+	0.982731i	$0.440759\pi$
$524$	−6.29354	−0.274935
$525$	−12.3590	−0.539390
$526$	−5.13158	−0.223748
$527$	−3.09374	−0.134765
$528$	5.27216	0.229442
$529$	2.28192	0.0992141
$530$	1.59894	0.0694535
$531$	−13.7006	−0.594555
$532$	−13.1881	−0.571775
$533$	−9.67843	−0.419219
$534$	−13.8862	−0.600916
$535$	12.0773	0.522148
$536$	2.97048	0.128305
$537$	−21.3025	−0.919272
$538$	4.33267	0.186795
$539$	−2.12468	−0.0915162
$540$	0.676512	0.0291125
$541$	−12.8504	−0.552480	−0.276240	−	0.961089i	$-0.589088\pi$
$541$	−12.8504	−0.552480	−0.276240	+	0.961089i	$0.589088\pi$
$542$	−15.4148	−0.662124
$543$	−8.88993	−0.381503
$544$	1.01679	0.0435947
$545$	0.333659	0.0142924
$546$	−2.72085	−0.116441
$547$	−11.5043	−0.491888	−0.245944	−	0.969284i	$-0.579098\pi$
$547$	−11.5043	−0.491888	−0.245944	+	0.969284i	$0.579098\pi$
$548$	−6.42562	−0.274489
$549$	11.9378	0.509495
$550$	23.9479	1.02114
$551$	30.5605	1.30192
$552$	5.02811	0.214011
$553$	13.8843	0.590420
$554$	−26.4488	−1.12370
$555$	−0.677742	−0.0287686
$556$	18.3900	0.779909
$557$	−37.2435	−1.57806	−0.789028	−	0.614357i	$-0.789416\pi$
$557$	−37.2435	−1.57806	−0.789028	+	0.614357i	$0.789416\pi$
$558$	−3.04264	−0.128805
$559$	8.83597	0.373722
$560$	1.84069	0.0777832
$561$	5.36070	0.226329
$562$	−1.70942	−0.0721077
$563$	−13.2837	−0.559841	−0.279920	−	0.960023i	$-0.590308\pi$
$563$	−13.2837	−0.559841	−0.279920	+	0.960023i	$0.590308\pi$
$564$	8.64110	0.363856
$565$	6.98104	0.293695
$566$	−4.39481	−0.184728
$567$	−2.72085	−0.114265
$568$	15.6217	0.655472
$569$	−4.29722	−0.180149	−0.0900745	−	0.995935i	$-0.528711\pi$
$569$	−4.29722	−0.180149	−0.0900745	+	0.995935i	$0.528711\pi$
$570$	3.27908	0.137346
$571$	43.0999	1.80368	0.901838	−	0.432074i	$-0.142218\pi$
$571$	43.0999	1.80368	0.901838	+	0.432074i	$0.142218\pi$
$572$	5.27216	0.220440
$573$	11.8651	0.495674
$574$	−26.3335	−1.09914
$575$	22.8394	0.952467
$576$	1.00000	0.0416667
$577$	17.0048	0.707918	0.353959	−	0.935261i	$-0.384835\pi$
$577$	17.0048	0.707918	0.353959	+	0.935261i	$0.384835\pi$
$578$	−15.9661	−0.664103
$579$	−10.6974	−0.444570
$580$	−4.26540	−0.177111
$581$	2.02494	0.0840088
$582$	2.57742	0.106838
$583$	12.4608	0.516073
$584$	1.34277	0.0555642
$585$	0.676512	0.0279703
$586$	−2.45624	−0.101466
$587$	−0.310300	−0.0128075	−0.00640373	−	0.999979i	$-0.502038\pi$
$587$	−0.310300	−0.0128075	−0.00640373	+	0.999979i	$0.502038\pi$
$588$	−0.402999	−0.0166194
$589$	−14.7478	−0.607673
$590$	9.26862	0.381583
$591$	27.1432	1.11652
$592$	−1.00182	−0.0411745
$593$	34.7530	1.42714	0.713568	−	0.700586i	$-0.247078\pi$
$593$	34.7530	1.42714	0.713568	+	0.700586i	$0.247078\pi$
$594$	5.27216	0.216320
$595$	1.87160	0.0767280
$596$	14.0749	0.576531
$597$	−16.0741	−0.657868
$598$	5.02811	0.205615
$599$	−19.2572	−0.786827	−0.393414	−	0.919362i	$-0.628706\pi$
$599$	−19.2572	−0.786827	−0.393414	+	0.919362i	$0.628706\pi$
$600$	4.54233	0.185440
$601$	−16.5213	−0.673917	−0.336959	−	0.941519i	$-0.609398\pi$
$601$	−16.5213	−0.673917	−0.336959	+	0.941519i	$0.609398\pi$
$602$	24.0413	0.979851
$603$	2.97048	0.120967
$604$	23.4183	0.952877
$605$	−11.3625	−0.461952
$606$	−8.50857	−0.345637
$607$	−1.62313	−0.0658809	−0.0329404	−	0.999457i	$-0.510487\pi$
$607$	−1.62313	−0.0658809	−0.0329404	+	0.999457i	$0.510487\pi$
$608$	4.84704	0.196574
$609$	17.1549	0.695151
$610$	−8.07610	−0.326992
$611$	8.64110	0.349582
$612$	1.01679	0.0411015
$613$	29.4693	1.19025	0.595127	−	0.803632i	$-0.297102\pi$
$613$	29.4693	1.19025	0.595127	+	0.803632i	$0.297102\pi$
$614$	−11.7261	−0.473225
$615$	6.54758	0.264024
$616$	14.3447	0.577966
$617$	−9.07021	−0.365153	−0.182576	−	0.983192i	$-0.558444\pi$
$617$	−9.07021	−0.365153	−0.182576	+	0.983192i	$0.558444\pi$
$618$	1.00000	0.0402259
$619$	−4.18897	−0.168369	−0.0841845	−	0.996450i	$-0.526829\pi$
$619$	−4.18897	−0.168369	−0.0841845	+	0.996450i	$0.526829\pi$
$620$	2.05838	0.0826667
$621$	5.02811	0.201771
$622$	−1.75340	−0.0703051
$623$	−37.7823	−1.51371
$624$	1.00000	0.0400320
$625$	18.3444	0.733777
$626$	21.0420	0.841006
$627$	25.5544	1.02054
$628$	4.71578	0.188180
$629$	−1.01864	−0.0406159
$630$	1.84069	0.0733347
$631$	−15.6492	−0.622986	−0.311493	−	0.950248i	$-0.600829\pi$
$631$	−15.6492	−0.622986	−0.311493	+	0.950248i	$0.600829\pi$
$632$	−5.10293	−0.202984
$633$	−13.8759	−0.551519
$634$	25.9164	1.02927
$635$	−9.55578	−0.379210
$636$	2.36350	0.0937191
$637$	−0.402999	−0.0159674
$638$	−33.2409	−1.31602
$639$	15.6217	0.617985
$640$	−0.676512	−0.0267415
$641$	−6.08581	−0.240375	−0.120187	−	0.992751i	$-0.538350\pi$
$641$	−6.08581	−0.240375	−0.120187	+	0.992751i	$0.538350\pi$
$642$	17.8523	0.704575
$643$	20.2998	0.800545	0.400273	−	0.916396i	$-0.368915\pi$
$643$	20.2998	0.800545	0.400273	+	0.916396i	$0.368915\pi$
$644$	13.6807	0.539096
$645$	−5.97765	−0.235370
$646$	4.92844	0.193907
$647$	−40.1434	−1.57820	−0.789100	−	0.614264i	$-0.789453\pi$
$647$	−40.1434	−1.57820	−0.789100	+	0.614264i	$0.789453\pi$
$648$	1.00000	0.0392837
$649$	72.2318	2.83535
$650$	4.54233	0.178165
$651$	−8.27855	−0.324462
$652$	14.3157	0.560647
$653$	−9.92143	−0.388256	−0.194128	−	0.980976i	$-0.562188\pi$
$653$	−9.92143	−0.388256	−0.194128	+	0.980976i	$0.562188\pi$
$654$	0.493205	0.0192858
$655$	4.25766	0.166361
$656$	9.67843	0.377879
$657$	1.34277	0.0523864
$658$	23.5111	0.916558
$659$	−3.80652	−0.148281	−0.0741405	−	0.997248i	$-0.523621\pi$
$659$	−3.80652	−0.148281	−0.0741405	+	0.997248i	$0.523621\pi$
$660$	−3.56668	−0.138833
$661$	−0.560567	−0.0218035	−0.0109018	−	0.999941i	$-0.503470\pi$
$661$	−0.560567	−0.0218035	−0.0109018	+	0.999941i	$0.503470\pi$
$662$	−18.0656	−0.702138
$663$	1.01679	0.0394890
$664$	−0.744233	−0.0288818
$665$	8.92188	0.345976
$666$	−1.00182	−0.0388197
$667$	−31.7022	−1.22751
$668$	5.09109	0.196980
$669$	−3.62805	−0.140269
$670$	−2.00957	−0.0776364
$671$	−62.9383	−2.42970
$672$	2.72085	0.104959
$673$	13.2680	0.511444	0.255722	−	0.966750i	$-0.417687\pi$
$673$	13.2680	0.511444	0.255722	+	0.966750i	$0.417687\pi$
$674$	1.94031	0.0747381
$675$	4.54233	0.174834
$676$	1.00000	0.0384615
$677$	28.8453	1.10861	0.554307	−	0.832312i	$-0.312983\pi$
$677$	28.8453	1.10861	0.554307	+	0.832312i	$0.312983\pi$
$678$	10.3192	0.396305
$679$	7.01276	0.269125
$680$	−0.687874	−0.0263787
$681$	−12.8428	−0.492137
$682$	16.0413	0.614253
$683$	6.72659	0.257386	0.128693	−	0.991684i	$-0.458922\pi$
$683$	6.72659	0.257386	0.128693	+	0.991684i	$0.458922\pi$
$684$	4.84704	0.185331
$685$	4.34701	0.166091
$686$	17.9494	0.685312
$687$	−16.1697	−0.616911
$688$	−8.83597	−0.336868
$689$	2.36350	0.0900424
$690$	−3.40158	−0.129496
$691$	11.4509	0.435612	0.217806	−	0.975992i	$-0.430110\pi$
$691$	11.4509	0.435612	0.217806	+	0.975992i	$0.430110\pi$
$692$	8.23408	0.313013
$693$	14.3447	0.544912
$694$	0.229716	0.00871990
$695$	−12.4411	−0.471916
$696$	−6.30498	−0.238990
$697$	9.84097	0.372753
$698$	19.8807	0.752497
$699$	3.08086	0.116529
$700$	12.3590	0.467126
$701$	41.0431	1.55018	0.775088	−	0.631853i	$-0.217705\pi$
$701$	41.0431	1.55018	0.775088	+	0.631853i	$0.217705\pi$
$702$	1.00000	0.0377426
$703$	−4.85585	−0.183142
$704$	−5.27216	−0.198702
$705$	−5.84581	−0.220166
$706$	−10.2245	−0.384804
$707$	−23.1505	−0.870664
$708$	13.7006	0.514900
$709$	25.2166	0.947030	0.473515	−	0.880786i	$-0.342985\pi$
$709$	25.2166	0.947030	0.473515	+	0.880786i	$0.342985\pi$
$710$	−10.5683	−0.396620
$711$	−5.10293	−0.191375
$712$	13.8862	0.520408
$713$	15.2987	0.572942
$714$	2.76654	0.103535
$715$	−3.56668	−0.133386
$716$	21.3025	0.796113
$717$	−16.9163	−0.631749
$718$	−4.10661	−0.153257
$719$	−22.2351	−0.829231	−0.414616	−	0.909997i	$-0.636084\pi$
$719$	−22.2351	−0.829231	−0.414616	+	0.909997i	$0.636084\pi$
$720$	−0.676512	−0.0252121
$721$	2.72085	0.101330
$722$	4.49382	0.167243
$723$	29.6909	1.10422
$724$	8.88993	0.330392
$725$	−28.6393	−1.06364
$726$	−16.7957	−0.623348
$727$	−30.4872	−1.13071	−0.565354	−	0.824848i	$-0.691260\pi$
$727$	−30.4872	−1.13071	−0.565354	+	0.824848i	$0.691260\pi$
$728$	2.72085	0.100841
$729$	1.00000	0.0370370
$730$	−0.908400	−0.0336214
$731$	−8.98436	−0.332299
$732$	−11.9378	−0.441235
$733$	−41.4654	−1.53156	−0.765780	−	0.643103i	$-0.777647\pi$
$733$	−41.4654	−1.53156	−0.765780	+	0.643103i	$0.777647\pi$
$734$	24.2806	0.896212
$735$	0.272634	0.0100562
$736$	−5.02811	−0.185339
$737$	−15.6609	−0.576876
$738$	9.67843	0.356268
$739$	50.7684	1.86755	0.933773	−	0.357866i	$-0.116495\pi$
$739$	50.7684	1.86755	0.933773	+	0.357866i	$0.116495\pi$
$740$	0.677742	0.0249143
$741$	4.84704	0.178061
$742$	6.43073	0.236080
$743$	15.2433	0.559223	0.279611	−	0.960113i	$-0.409794\pi$
$743$	15.2433	0.559223	0.279611	+	0.960113i	$0.409794\pi$
$744$	3.04264	0.111549
$745$	−9.52185	−0.348854
$746$	−2.09879	−0.0768421
$747$	−0.744233	−0.0272301
$748$	−5.36070	−0.196007
$749$	48.5734	1.77483
$750$	−6.45551	−0.235722
$751$	12.7402	0.464898	0.232449	−	0.972609i	$-0.425326\pi$
$751$	12.7402	0.464898	0.232449	+	0.972609i	$0.425326\pi$
$752$	−8.64110	−0.315109
$753$	3.96047	0.144328
$754$	−6.30498	−0.229614
$755$	−15.8428	−0.576577
$756$	2.72085	0.0989562
$757$	15.1143	0.549337	0.274668	−	0.961539i	$-0.411432\pi$
$757$	15.1143	0.549337	0.274668	+	0.961539i	$0.411432\pi$
$758$	−2.52123	−0.0915752
$759$	−26.5090	−0.962217
$760$	−3.27908	−0.118945
$761$	7.05638	0.255794	0.127897	−	0.991787i	$-0.459177\pi$
$761$	7.05638	0.255794	0.127897	+	0.991787i	$0.459177\pi$
$762$	−14.1251	−0.511697
$763$	1.34193	0.0485813
$764$	−11.8651	−0.429266
$765$	−0.687874	−0.0248701
$766$	2.79825	0.101105
$767$	13.7006	0.494700
$768$	−1.00000	−0.0360844
$769$	15.4429	0.556887	0.278443	−	0.960453i	$-0.410182\pi$
$769$	15.4429	0.556887	0.278443	+	0.960453i	$0.410182\pi$
$770$	−9.70440	−0.349722
$771$	−28.3377	−1.02056
$772$	10.6974	0.385009
$773$	−8.93806	−0.321480	−0.160740	−	0.986997i	$-0.551388\pi$
$773$	−8.93806	−0.321480	−0.160740	+	0.986997i	$0.551388\pi$
$774$	−8.83597	−0.317602
$775$	13.8207	0.496453
$776$	−2.57742	−0.0925240
$777$	−2.72579	−0.0977872
$778$	−21.9755	−0.787861
$779$	46.9118	1.68079
$780$	−0.676512	−0.0242230
$781$	−82.3601	−2.94708
$782$	−5.11255	−0.182825
$783$	−6.30498	−0.225322
$784$	0.402999	0.0143928
$785$	−3.19028	−0.113866
$786$	6.29354	0.224483
$787$	54.3035	1.93571	0.967856	−	0.251507i	$-0.0809261\pi$
$787$	54.3035	1.93571	0.967856	+	0.251507i	$0.0809261\pi$
$788$	−27.1432	−0.966936
$789$	5.13158	0.182689
$790$	3.45220	0.122824
$791$	28.0769	0.998298
$792$	−5.27216	−0.187338
$793$	−11.9378	−0.423925
$794$	−18.4038	−0.653125
$795$	−1.59894	−0.0567086
$796$	16.0741	0.569731
$797$	38.1146	1.35009	0.675044	−	0.737778i	$-0.264125\pi$
$797$	38.1146	1.35009	0.675044	+	0.737778i	$0.264125\pi$
$798$	13.1881	0.466852
$799$	−8.78622	−0.310834
$800$	−4.54233	−0.160596
$801$	13.8862	0.490646
$802$	9.25701	0.326876
$803$	−7.07930	−0.249823
$804$	−2.97048	−0.104761
$805$	−9.25518	−0.326202
$806$	3.04264	0.107172
$807$	−4.33267	−0.152517
$808$	8.50857	0.299330
$809$	−39.7580	−1.39782	−0.698909	−	0.715211i	$-0.746331\pi$
$809$	−39.7580	−1.39782	−0.698909	+	0.715211i	$0.746331\pi$
$810$	−0.676512	−0.0237702
$811$	37.2157	1.30682	0.653410	−	0.757004i	$-0.273338\pi$
$811$	37.2157	1.30682	0.653410	+	0.757004i	$0.273338\pi$
$812$	−17.1549	−0.602018
$813$	15.4148	0.540622
$814$	5.28175	0.185125
$815$	−9.68477	−0.339243
$816$	−1.01679	−0.0355949
$817$	−42.8283	−1.49837
$818$	−6.19590	−0.216635
$819$	2.72085	0.0950740
$820$	−6.54758	−0.228651
$821$	21.1159	0.736952	0.368476	−	0.929637i	$-0.379880\pi$
$821$	21.1159	0.736952	0.368476	+	0.929637i	$0.379880\pi$
$822$	6.42562	0.224119
$823$	−33.4694	−1.16667	−0.583335	−	0.812232i	$-0.698252\pi$
$823$	−33.4694	−1.16667	−0.583335	+	0.812232i	$0.698252\pi$
$824$	−1.00000	−0.0348367
$825$	−23.9479	−0.833759
$826$	37.2772	1.29704
$827$	−25.7208	−0.894401	−0.447201	−	0.894434i	$-0.647579\pi$
$827$	−25.7208	−0.894401	−0.447201	+	0.894434i	$0.647579\pi$
$828$	−5.02811	−0.174739
$829$	−22.7542	−0.790286	−0.395143	−	0.918620i	$-0.629305\pi$
$829$	−22.7542	−0.790286	−0.395143	+	0.918620i	$0.629305\pi$
$830$	0.503483	0.0174761
$831$	26.4488	0.917498
$832$	−1.00000	−0.0346688
$833$	0.409766	0.0141976
$834$	−18.3900	−0.636793
$835$	−3.44419	−0.119191
$836$	−25.5544	−0.883818
$837$	3.04264	0.105169
$838$	15.4153	0.532513
$839$	19.5982	0.676605	0.338303	−	0.941037i	$-0.390147\pi$
$839$	19.5982	0.676605	0.338303	+	0.941037i	$0.390147\pi$
$840$	−1.84069	−0.0635097
$841$	10.7528	0.370785
$842$	−25.8241	−0.889957
$843$	1.70942	0.0588757
$844$	13.8759	0.477629
$845$	−0.676512	−0.0232727
$846$	−8.64110	−0.297087
$847$	−45.6986	−1.57022
$848$	−2.36350	−0.0811631
$849$	4.39481	0.150829
$850$	−4.61861	−0.158417
$851$	5.03725	0.172675
$852$	−15.6217	−0.535190
$853$	39.9423	1.36760	0.683799	−	0.729670i	$-0.260327\pi$
$853$	39.9423	1.36760	0.683799	+	0.729670i	$0.260327\pi$
$854$	−32.4810	−1.11148
$855$	−3.27908	−0.112142
$856$	−17.8523	−0.610180
$857$	−54.7153	−1.86904	−0.934519	−	0.355912i	$-0.884170\pi$
$857$	−54.7153	−1.86904	−0.934519	+	0.355912i	$0.884170\pi$
$858$	−5.27216	−0.179989
$859$	−30.0969	−1.02689	−0.513446	−	0.858122i	$-0.671632\pi$
$859$	−30.0969	−1.02689	−0.513446	+	0.858122i	$0.671632\pi$
$860$	5.97765	0.203836
$861$	26.3335	0.897444
$862$	−7.57974	−0.258167
$863$	30.9324	1.05295	0.526475	−	0.850191i	$-0.323513\pi$
$863$	30.9324	1.05295	0.526475	+	0.850191i	$0.323513\pi$
$864$	−1.00000	−0.0340207
$865$	−5.57046	−0.189401
$866$	−16.0633	−0.545854
$867$	15.9661	0.542238
$868$	8.27855	0.280992
$869$	26.9035	0.912639
$870$	4.26540	0.144610
$871$	−2.97048	−0.100651
$872$	−0.493205	−0.0167020
$873$	−2.57742	−0.0872325
$874$	−24.3715	−0.824378
$875$	−17.5644	−0.593786
$876$	−1.34277	−0.0453680
$877$	32.5460	1.09900	0.549500	−	0.835494i	$-0.314818\pi$
$877$	32.5460	1.09900	0.549500	+	0.835494i	$0.314818\pi$
$878$	34.5792	1.16699
$879$	2.45624	0.0828469
$880$	3.56668	0.120233
$881$	7.14872	0.240847	0.120423	−	0.992723i	$-0.461575\pi$
$881$	7.14872	0.240847	0.120423	+	0.992723i	$0.461575\pi$
$882$	0.402999	0.0135697
$883$	−7.51123	−0.252773	−0.126387	−	0.991981i	$-0.540338\pi$
$883$	−7.51123	−0.252773	−0.126387	+	0.991981i	$0.540338\pi$
$884$	−1.01679	−0.0341985
$885$	−9.26862	−0.311561
$886$	21.2435	0.713688
$887$	7.79271	0.261654	0.130827	−	0.991405i	$-0.458237\pi$
$887$	7.79271	0.261654	0.130827	+	0.991405i	$0.458237\pi$
$888$	1.00182	0.0336188
$889$	−38.4321	−1.28897
$890$	−9.39420	−0.314894
$891$	−5.27216	−0.176624
$892$	3.62805	0.121476
$893$	−41.8838	−1.40159
$894$	−14.0749	−0.470735
$895$	−14.4114	−0.481721
$896$	−2.72085	−0.0908971
$897$	−5.02811	−0.167884
$898$	35.5472	1.18622
$899$	−19.1838	−0.639815
$900$	−4.54233	−0.151411
$901$	−2.40320	−0.0800621
$902$	−51.0263	−1.69899
$903$	−24.0413	−0.800045
$904$	−10.3192	−0.343210
$905$	−6.01415	−0.199917
$906$	−23.4183	−0.778021
$907$	−2.50049	−0.0830273	−0.0415136	−	0.999138i	$-0.513218\pi$
$907$	−2.50049	−0.0830273	−0.0415136	+	0.999138i	$0.513218\pi$
$908$	12.8428	0.426203
$909$	8.50857	0.282211
$910$	−1.84069	−0.0610181
$911$	−23.9600	−0.793831	−0.396915	−	0.917855i	$-0.629919\pi$
$911$	−23.9600	−0.793831	−0.396915	+	0.917855i	$0.629919\pi$
$912$	−4.84704	−0.160502
$913$	3.92372	0.129856
$914$	24.4777	0.809650
$915$	8.07610	0.266988
$916$	16.1697	0.534261
$917$	17.1238	0.565476
$918$	−1.01679	−0.0335592
$919$	−52.1127	−1.71904	−0.859519	−	0.511104i	$-0.829237\pi$
$919$	−52.1127	−1.71904	−0.859519	+	0.511104i	$0.829237\pi$
$920$	3.40158	0.112147
$921$	11.7261	0.386387
$922$	−26.6751	−0.878496
$923$	−15.6217	−0.514194
$924$	−14.3447	−0.471907
$925$	4.55059	0.149622
$926$	7.77420	0.255476
$927$	−1.00000	−0.0328443
$928$	6.30498	0.206971
$929$	16.1845	0.530995	0.265498	−	0.964112i	$-0.414464\pi$
$929$	16.1845	0.530995	0.265498	+	0.964112i	$0.414464\pi$
$930$	−2.05838	−0.0674971
$931$	1.95335	0.0640185
$932$	−3.08086	−0.100917
$933$	1.75340	0.0574039
$934$	22.4004	0.732963
$935$	3.62658	0.118602
$936$	−1.00000	−0.0326860
$937$	12.5147	0.408837	0.204419	−	0.978884i	$-0.434470\pi$
$937$	12.5147	0.408837	0.204419	+	0.978884i	$0.434470\pi$
$938$	−8.08222	−0.263894
$939$	−21.0420	−0.686679
$940$	5.84581	0.190669
$941$	−51.5353	−1.68000	−0.840002	−	0.542583i	$-0.817446\pi$
$941$	−51.5353	−1.68000	−0.840002	+	0.542583i	$0.817446\pi$
$942$	−4.71578	−0.153648
$943$	−48.6643	−1.58473
$944$	−13.7006	−0.445916
$945$	−1.84069	−0.0598775
$946$	46.5847	1.51460
$947$	−42.8955	−1.39392	−0.696959	−	0.717111i	$-0.745464\pi$
$947$	−42.8955	−1.39392	−0.696959	+	0.717111i	$0.745464\pi$
$948$	5.10293	0.165736
$949$	−1.34277	−0.0435881
$950$	−22.0169	−0.714322
$951$	−25.9164	−0.840396
$952$	−2.76654	−0.0896640
$953$	13.4076	0.434314	0.217157	−	0.976137i	$-0.430322\pi$
$953$	13.4076	0.434314	0.217157	+	0.976137i	$0.430322\pi$
$954$	−2.36350	−0.0765213
$955$	8.02692	0.259745
$956$	16.9163	0.547111
$957$	33.2409	1.07453
$958$	−38.0641	−1.22980
$959$	17.4831	0.564559
$960$	0.676512	0.0218343
$961$	−21.7423	−0.701366
$962$	1.00182	0.0322999
$963$	−17.8523	−0.575283
$964$	−29.6909	−0.956281
$965$	−7.23695	−0.232966
$966$	−13.6807	−0.440170
$967$	44.3002	1.42460	0.712299	−	0.701876i	$-0.247654\pi$
$967$	44.3002	1.42460	0.712299	+	0.701876i	$0.247654\pi$
$968$	16.7957	0.539835
$969$	−4.92844	−0.158324
$970$	1.74366	0.0559855
$971$	4.94382	0.158655	0.0793274	−	0.996849i	$-0.474723\pi$
$971$	4.94382	0.158655	0.0793274	+	0.996849i	$0.474723\pi$
$972$	−1.00000	−0.0320750
$973$	−50.0363	−1.60409
$974$	17.7607	0.569090
$975$	−4.54233	−0.145471
$976$	11.9378	0.382121
$977$	44.7955	1.43313	0.716567	−	0.697518i	$-0.245712\pi$
$977$	44.7955	1.43313	0.716567	+	0.697518i	$0.245712\pi$
$978$	−14.3157	−0.457767
$979$	−73.2105	−2.33982
$980$	−0.272634	−0.00870896
$981$	−0.493205	−0.0157468
$982$	−31.1190	−0.993048
$983$	8.01746	0.255717	0.127859	−	0.991792i	$-0.459190\pi$
$983$	8.01746	0.255717	0.127859	+	0.991792i	$0.459190\pi$
$984$	−9.67843	−0.308537
$985$	18.3627	0.585084
$986$	6.41086	0.204163
$987$	−23.5111	−0.748367
$988$	−4.84704	−0.154205
$989$	44.4283	1.41274
$990$	3.56668	0.113357
$991$	−46.1093	−1.46471	−0.732356	−	0.680922i	$-0.761579\pi$
$991$	−46.1093	−1.46471	−0.732356	+	0.680922i	$0.761579\pi$
$992$	−3.04264	−0.0966039
$993$	18.0656	0.573294
$994$	−42.5042	−1.34815
$995$	−10.8743	−0.344739
$996$	0.744233	0.0235819
$997$	6.11084	0.193532	0.0967662	−	0.995307i	$-0.469150\pi$
$997$	6.11084	0.193532	0.0967662	+	0.995307i	$0.469150\pi$
$998$	15.2463	0.482612
$999$	1.00182	0.0316961

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.bc.1.7	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.bc.1.7	✓	15	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:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.676512 q^{5} -1.00000 q^{6} -2.72085 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.676512 q^{5} -1.00000 q^{6} -2.72085 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.676512 q^{10} -5.27216 q^{11} -1.00000 q^{12} -1.00000 q^{13} -2.72085 q^{14} +0.676512 q^{15} +1.00000 q^{16} +1.01679 q^{17} +1.00000 q^{18} +4.84704 q^{19} -0.676512 q^{20} +2.72085 q^{21} -5.27216 q^{22} -5.02811 q^{23} -1.00000 q^{24} -4.54233 q^{25} -1.00000 q^{26} -1.00000 q^{27} -2.72085 q^{28} +6.30498 q^{29} +0.676512 q^{30} -3.04264 q^{31} +1.00000 q^{32} +5.27216 q^{33} +1.01679 q^{34} +1.84069 q^{35} +1.00000 q^{36} -1.00182 q^{37} +4.84704 q^{38} +1.00000 q^{39} -0.676512 q^{40} +9.67843 q^{41} +2.72085 q^{42} -8.83597 q^{43} -5.27216 q^{44} -0.676512 q^{45} -5.02811 q^{46} -8.64110 q^{47} -1.00000 q^{48} +0.402999 q^{49} -4.54233 q^{50} -1.01679 q^{51} -1.00000 q^{52} -2.36350 q^{53} -1.00000 q^{54} +3.56668 q^{55} -2.72085 q^{56} -4.84704 q^{57} +6.30498 q^{58} -13.7006 q^{59} +0.676512 q^{60} +11.9378 q^{61} -3.04264 q^{62} -2.72085 q^{63} +1.00000 q^{64} +0.676512 q^{65} +5.27216 q^{66} +2.97048 q^{67} +1.01679 q^{68} +5.02811 q^{69} +1.84069 q^{70} +15.6217 q^{71} +1.00000 q^{72} +1.34277 q^{73} -1.00182 q^{74} +4.54233 q^{75} +4.84704 q^{76} +14.3447 q^{77} +1.00000 q^{78} -5.10293 q^{79} -0.676512 q^{80} +1.00000 q^{81} +9.67843 q^{82} -0.744233 q^{83} +2.72085 q^{84} -0.687874 q^{85} -8.83597 q^{86} -6.30498 q^{87} -5.27216 q^{88} +13.8862 q^{89} -0.676512 q^{90} +2.72085 q^{91} -5.02811 q^{92} +3.04264 q^{93} -8.64110 q^{94} -3.27908 q^{95} -1.00000 q^{96} -2.57742 q^{97} +0.402999 q^{98} -5.27216 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{2} - 15 q^{3} + 15 q^{4} - q^{5} - 15 q^{6} + 5 q^{7} + 15 q^{8} + 15 q^{9}+O(q^{10}) \) 15 * q + 15 * q^2 - 15 * q^3 + 15 * q^4 - q^5 - 15 * q^6 + 5 * q^7 + 15 * q^8 + 15 * q^9	\( 15 q + 15 q^{2} - 15 q^{3} + 15 q^{4} - q^{5} - 15 q^{6} + 5 q^{7} + 15 q^{8} + 15 q^{9} - q^{10} + 3 q^{11} - 15 q^{12} - 15 q^{13} + 5 q^{14} + q^{15} + 15 q^{16} - 2 q^{17} + 15 q^{18} + 8 q^{19} - q^{20} - 5 q^{21} + 3 q^{22} + 3 q^{23} - 15 q^{24} + 22 q^{25} - 15 q^{26} - 15 q^{27} + 5 q^{28} + 26 q^{29} + q^{30} + 15 q^{32} - 3 q^{33} - 2 q^{34} - 8 q^{35} + 15 q^{36} + 25 q^{37} + 8 q^{38} + 15 q^{39} - q^{40} - q^{41} - 5 q^{42} + 10 q^{43} + 3 q^{44} - q^{45} + 3 q^{46} - 3 q^{47} - 15 q^{48} + 32 q^{49} + 22 q^{50} + 2 q^{51} - 15 q^{52} + 13 q^{53} - 15 q^{54} - 2 q^{55} + 5 q^{56} - 8 q^{57} + 26 q^{58} + 28 q^{59} + q^{60} + 22 q^{61} + 5 q^{63} + 15 q^{64} + q^{65} - 3 q^{66} + 29 q^{67} - 2 q^{68} - 3 q^{69} - 8 q^{70} + 18 q^{71} + 15 q^{72} + 23 q^{73} + 25 q^{74} - 22 q^{75} + 8 q^{76} + 17 q^{77} + 15 q^{78} + 27 q^{79} - q^{80} + 15 q^{81} - q^{82} + 7 q^{83} - 5 q^{84} + 43 q^{85} + 10 q^{86} - 26 q^{87} + 3 q^{88} + 35 q^{89} - q^{90} - 5 q^{91} + 3 q^{92} - 3 q^{94} + 6 q^{95} - 15 q^{96} + 19 q^{97} + 32 q^{98} + 3 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 - 15 * q^3 + 15 * q^4 - q^5 - 15 * q^6 + 5 * q^7 + 15 * q^8 + 15 * q^9 - q^10 + 3 * q^11 - 15 * q^12 - 15 * q^13 + 5 * q^14 + q^15 + 15 * q^16 - 2 * q^17 + 15 * q^18 + 8 * q^19 - q^20 - 5 * q^21 + 3 * q^22 + 3 * q^23 - 15 * q^24 + 22 * q^25 - 15 * q^26 - 15 * q^27 + 5 * q^28 + 26 * q^29 + q^30 + 15 * q^32 - 3 * q^33 - 2 * q^34 - 8 * q^35 + 15 * q^36 + 25 * q^37 + 8 * q^38 + 15 * q^39 - q^40 - q^41 - 5 * q^42 + 10 * q^43 + 3 * q^44 - q^45 + 3 * q^46 - 3 * q^47 - 15 * q^48 + 32 * q^49 + 22 * q^50 + 2 * q^51 - 15 * q^52 + 13 * q^53 - 15 * q^54 - 2 * q^55 + 5 * q^56 - 8 * q^57 + 26 * q^58 + 28 * q^59 + q^60 + 22 * q^61 + 5 * q^63 + 15 * q^64 + q^65 - 3 * q^66 + 29 * q^67 - 2 * q^68 - 3 * q^69 - 8 * q^70 + 18 * q^71 + 15 * q^72 + 23 * q^73 + 25 * q^74 - 22 * q^75 + 8 * q^76 + 17 * q^77 + 15 * q^78 + 27 * q^79 - q^80 + 15 * q^81 - q^82 + 7 * q^83 - 5 * q^84 + 43 * q^85 + 10 * q^86 - 26 * q^87 + 3 * q^88 + 35 * q^89 - q^90 - 5 * q^91 + 3 * q^92 - 3 * q^94 + 6 * q^95 - 15 * q^96 + 19 * q^97 + 32 * q^98 + 3 * q^99

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