LMFDB - Embedded newform 8034.2.a.q.1.4

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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 11x^{6} + 21x^{5} + 23x^{4} - 29x^{3} - 27x^{2} + x + 3 $ x^8 - 2x^7 - 11x^6 + 21x^5 + 23x^4 - 29x^3 - 27x^2 + x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.501271$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.50127 q^{5} +1.00000 q^{6} +1.44399 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.50127 q^{5} +1.00000 q^{6} +1.44399 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.50127 q^{10} -2.94272 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.44399 q^{14} -1.50127 q^{15} +1.00000 q^{16} +4.34586 q^{17} +1.00000 q^{18} -4.58431 q^{19} -1.50127 q^{20} +1.44399 q^{21} -2.94272 q^{22} -8.17758 q^{23} +1.00000 q^{24} -2.74619 q^{25} -1.00000 q^{26} +1.00000 q^{27} +1.44399 q^{28} -6.77674 q^{29} -1.50127 q^{30} +7.21512 q^{31} +1.00000 q^{32} -2.94272 q^{33} +4.34586 q^{34} -2.16782 q^{35} +1.00000 q^{36} +2.49161 q^{37} -4.58431 q^{38} -1.00000 q^{39} -1.50127 q^{40} +2.44182 q^{41} +1.44399 q^{42} -1.24658 q^{43} -2.94272 q^{44} -1.50127 q^{45} -8.17758 q^{46} -6.82549 q^{47} +1.00000 q^{48} -4.91489 q^{49} -2.74619 q^{50} +4.34586 q^{51} -1.00000 q^{52} +6.28858 q^{53} +1.00000 q^{54} +4.41782 q^{55} +1.44399 q^{56} -4.58431 q^{57} -6.77674 q^{58} +2.04358 q^{59} -1.50127 q^{60} +0.559740 q^{61} +7.21512 q^{62} +1.44399 q^{63} +1.00000 q^{64} +1.50127 q^{65} -2.94272 q^{66} +3.43251 q^{67} +4.34586 q^{68} -8.17758 q^{69} -2.16782 q^{70} -11.2772 q^{71} +1.00000 q^{72} -4.58425 q^{73} +2.49161 q^{74} -2.74619 q^{75} -4.58431 q^{76} -4.24926 q^{77} -1.00000 q^{78} -0.911183 q^{79} -1.50127 q^{80} +1.00000 q^{81} +2.44182 q^{82} -7.71954 q^{83} +1.44399 q^{84} -6.52431 q^{85} -1.24658 q^{86} -6.77674 q^{87} -2.94272 q^{88} -12.0347 q^{89} -1.50127 q^{90} -1.44399 q^{91} -8.17758 q^{92} +7.21512 q^{93} -6.82549 q^{94} +6.88230 q^{95} +1.00000 q^{96} -7.00623 q^{97} -4.91489 q^{98} -2.94272 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 6 q^{5} + 8 q^{6} - 3 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 6 * q^5 + 8 * q^6 - 3 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 6 q^{5} + 8 q^{6} - 3 q^{7} + 8 q^{8} + 8 q^{9} - 6 q^{10} - 15 q^{11} + 8 q^{12} - 8 q^{13} - 3 q^{14} - 6 q^{15} + 8 q^{16} - 11 q^{17} + 8 q^{18} - 15 q^{19} - 6 q^{20} - 3 q^{21} - 15 q^{22} + q^{23} + 8 q^{24} - 10 q^{25} - 8 q^{26} + 8 q^{27} - 3 q^{28} - 10 q^{29} - 6 q^{30} - 3 q^{31} + 8 q^{32} - 15 q^{33} - 11 q^{34} - 12 q^{35} + 8 q^{36} - 26 q^{37} - 15 q^{38} - 8 q^{39} - 6 q^{40} - 12 q^{41} - 3 q^{42} - 4 q^{43} - 15 q^{44} - 6 q^{45} + q^{46} - 6 q^{47} + 8 q^{48} - 5 q^{49} - 10 q^{50} - 11 q^{51} - 8 q^{52} - 4 q^{53} + 8 q^{54} - 3 q^{56} - 15 q^{57} - 10 q^{58} - 19 q^{59} - 6 q^{60} - 14 q^{61} - 3 q^{62} - 3 q^{63} + 8 q^{64} + 6 q^{65} - 15 q^{66} - 13 q^{67} - 11 q^{68} + q^{69} - 12 q^{70} - 31 q^{71} + 8 q^{72} - 27 q^{73} - 26 q^{74} - 10 q^{75} - 15 q^{76} - 30 q^{77} - 8 q^{78} - 13 q^{79} - 6 q^{80} + 8 q^{81} - 12 q^{82} - 28 q^{83} - 3 q^{84} + 15 q^{85} - 4 q^{86} - 10 q^{87} - 15 q^{88} - 2 q^{89} - 6 q^{90} + 3 q^{91} + q^{92} - 3 q^{93} - 6 q^{94} - 18 q^{95} + 8 q^{96} - 30 q^{97} - 5 q^{98} - 15 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 6 * q^5 + 8 * q^6 - 3 * q^7 + 8 * q^8 + 8 * q^9 - 6 * q^10 - 15 * q^11 + 8 * q^12 - 8 * q^13 - 3 * q^14 - 6 * q^15 + 8 * q^16 - 11 * q^17 + 8 * q^18 - 15 * q^19 - 6 * q^20 - 3 * q^21 - 15 * q^22 + q^23 + 8 * q^24 - 10 * q^25 - 8 * q^26 + 8 * q^27 - 3 * q^28 - 10 * q^29 - 6 * q^30 - 3 * q^31 + 8 * q^32 - 15 * q^33 - 11 * q^34 - 12 * q^35 + 8 * q^36 - 26 * q^37 - 15 * q^38 - 8 * q^39 - 6 * q^40 - 12 * q^41 - 3 * q^42 - 4 * q^43 - 15 * q^44 - 6 * q^45 + q^46 - 6 * q^47 + 8 * q^48 - 5 * q^49 - 10 * q^50 - 11 * q^51 - 8 * q^52 - 4 * q^53 + 8 * q^54 - 3 * q^56 - 15 * q^57 - 10 * q^58 - 19 * q^59 - 6 * q^60 - 14 * q^61 - 3 * q^62 - 3 * q^63 + 8 * q^64 + 6 * q^65 - 15 * q^66 - 13 * q^67 - 11 * q^68 + q^69 - 12 * q^70 - 31 * q^71 + 8 * q^72 - 27 * q^73 - 26 * q^74 - 10 * q^75 - 15 * q^76 - 30 * q^77 - 8 * q^78 - 13 * q^79 - 6 * q^80 + 8 * q^81 - 12 * q^82 - 28 * q^83 - 3 * q^84 + 15 * q^85 - 4 * q^86 - 10 * q^87 - 15 * q^88 - 2 * q^89 - 6 * q^90 + 3 * q^91 + q^92 - 3 * q^93 - 6 * q^94 - 18 * q^95 + 8 * q^96 - 30 * q^97 - 5 * q^98 - 15 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.50127	−0.671389	−0.335694	−	0.941971i	$-0.608971\pi$
$5$	−1.50127	−0.671389	−0.335694	+	0.941971i	$0.608971\pi$
$6$	1.00000	0.408248
$7$	1.44399	0.545777	0.272889	−	0.962046i	$-0.412021\pi$
$7$	1.44399	0.545777	0.272889	+	0.962046i	$0.412021\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.50127	−0.474744
$11$	−2.94272	−0.887263	−0.443632	−	0.896209i	$-0.646310\pi$
$11$	−2.94272	−0.887263	−0.443632	+	0.896209i	$0.646310\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.44399	0.385923
$15$	−1.50127	−0.387627
$16$	1.00000	0.250000
$17$	4.34586	1.05402	0.527012	−	0.849858i	$-0.323312\pi$
$17$	4.34586	1.05402	0.527012	+	0.849858i	$0.323312\pi$
$18$	1.00000	0.235702
$19$	−4.58431	−1.05171	−0.525857	−	0.850573i	$-0.676255\pi$
$19$	−4.58431	−1.05171	−0.525857	+	0.850573i	$0.676255\pi$
$20$	−1.50127	−0.335694
$21$	1.44399	0.315105
$22$	−2.94272	−0.627390
$23$	−8.17758	−1.70514	−0.852572	−	0.522610i	$-0.824958\pi$
$23$	−8.17758	−1.70514	−0.852572	+	0.522610i	$0.824958\pi$
$24$	1.00000	0.204124
$25$	−2.74619	−0.549237
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	1.44399	0.272889
$29$	−6.77674	−1.25841	−0.629204	−	0.777240i	$-0.716619\pi$
$29$	−6.77674	−1.25841	−0.629204	+	0.777240i	$0.716619\pi$
$30$	−1.50127	−0.274093
$31$	7.21512	1.29587	0.647937	−	0.761694i	$-0.275632\pi$
$31$	7.21512	1.29587	0.647937	+	0.761694i	$0.275632\pi$
$32$	1.00000	0.176777
$33$	−2.94272	−0.512262
$34$	4.34586	0.745308
$35$	−2.16782	−0.366429
$36$	1.00000	0.166667
$37$	2.49161	0.409619	0.204809	−	0.978802i	$-0.434343\pi$
$37$	2.49161	0.409619	0.204809	+	0.978802i	$0.434343\pi$
$38$	−4.58431	−0.743674
$39$	−1.00000	−0.160128
$40$	−1.50127	−0.237372
$41$	2.44182	0.381348	0.190674	−	0.981653i	$-0.438933\pi$
$41$	2.44182	0.381348	0.190674	+	0.981653i	$0.438933\pi$
$42$	1.44399	0.222813
$43$	−1.24658	−0.190102	−0.0950510	−	0.995472i	$-0.530301\pi$
$43$	−1.24658	−0.190102	−0.0950510	+	0.995472i	$0.530301\pi$
$44$	−2.94272	−0.443632
$45$	−1.50127	−0.223796
$46$	−8.17758	−1.20572
$47$	−6.82549	−0.995600	−0.497800	−	0.867292i	$-0.665859\pi$
$47$	−6.82549	−0.995600	−0.497800	+	0.867292i	$0.665859\pi$
$48$	1.00000	0.144338
$49$	−4.91489	−0.702127
$50$	−2.74619	−0.388369
$51$	4.34586	0.608542
$52$	−1.00000	−0.138675
$53$	6.28858	0.863802	0.431901	−	0.901921i	$-0.357843\pi$
$53$	6.28858	0.863802	0.431901	+	0.901921i	$0.357843\pi$
$54$	1.00000	0.136083
$55$	4.41782	0.595699
$56$	1.44399	0.192961
$57$	−4.58431	−0.607207
$58$	−6.77674	−0.889829
$59$	2.04358	0.266052	0.133026	−	0.991113i	$-0.457531\pi$
$59$	2.04358	0.266052	0.133026	+	0.991113i	$0.457531\pi$
$60$	−1.50127	−0.193813
$61$	0.559740	0.0716674	0.0358337	−	0.999358i	$-0.488591\pi$
$61$	0.559740	0.0716674	0.0358337	+	0.999358i	$0.488591\pi$
$62$	7.21512	0.916321
$63$	1.44399	0.181926
$64$	1.00000	0.125000
$65$	1.50127	0.186210
$66$	−2.94272	−0.362224
$67$	3.43251	0.419348	0.209674	−	0.977771i	$-0.432760\pi$
$67$	3.43251	0.419348	0.209674	+	0.977771i	$0.432760\pi$
$68$	4.34586	0.527012
$69$	−8.17758	−0.984465
$70$	−2.16782	−0.259104
$71$	−11.2772	−1.33835	−0.669177	−	0.743103i	$-0.733353\pi$
$71$	−11.2772	−1.33835	−0.669177	+	0.743103i	$0.733353\pi$
$72$	1.00000	0.117851
$73$	−4.58425	−0.536546	−0.268273	−	0.963343i	$-0.586453\pi$
$73$	−4.58425	−0.536546	−0.268273	+	0.963343i	$0.586453\pi$
$74$	2.49161	0.289644
$75$	−2.74619	−0.317102
$76$	−4.58431	−0.525857
$77$	−4.24926	−0.484248
$78$	−1.00000	−0.113228
$79$	−0.911183	−0.102516	−0.0512581	−	0.998685i	$-0.516323\pi$
$79$	−0.911183	−0.102516	−0.0512581	+	0.998685i	$0.516323\pi$
$80$	−1.50127	−0.167847
$81$	1.00000	0.111111
$82$	2.44182	0.269654
$83$	−7.71954	−0.847329	−0.423665	−	0.905819i	$-0.639256\pi$
$83$	−7.71954	−0.847329	−0.423665	+	0.905819i	$0.639256\pi$
$84$	1.44399	0.157552
$85$	−6.52431	−0.707660
$86$	−1.24658	−0.134422
$87$	−6.77674	−0.726543
$88$	−2.94272	−0.313695
$89$	−12.0347	−1.27567	−0.637836	−	0.770173i	$-0.720170\pi$
$89$	−12.0347	−1.27567	−0.637836	+	0.770173i	$0.720170\pi$
$90$	−1.50127	−0.158248
$91$	−1.44399	−0.151371
$92$	−8.17758	−0.852572
$93$	7.21512	0.748173
$94$	−6.82549	−0.703996
$95$	6.88230	0.706109
$96$	1.00000	0.102062
$97$	−7.00623	−0.711375	−0.355687	−	0.934605i	$-0.615753\pi$
$97$	−7.00623	−0.711375	−0.355687	+	0.934605i	$0.615753\pi$
$98$	−4.91489	−0.496479
$99$	−2.94272	−0.295754
$100$	−2.74619	−0.274619
$101$	−18.2763	−1.81856	−0.909280	−	0.416185i	$-0.863367\pi$
$101$	−18.2763	−1.81856	−0.909280	+	0.416185i	$0.863367\pi$
$102$	4.34586	0.430304
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−1.00000	−0.0980581
$105$	−2.16782	−0.211558
$106$	6.28858	0.610800
$107$	−15.2371	−1.47302	−0.736511	−	0.676426i	$-0.763528\pi$
$107$	−15.2371	−1.47302	−0.736511	+	0.676426i	$0.763528\pi$
$108$	1.00000	0.0962250
$109$	−10.2209	−0.978985	−0.489492	−	0.872008i	$-0.662818\pi$
$109$	−10.2209	−0.978985	−0.489492	+	0.872008i	$0.662818\pi$
$110$	4.41782	0.421223
$111$	2.49161	0.236494
$112$	1.44399	0.136444
$113$	4.57132	0.430033	0.215017	−	0.976610i	$-0.431019\pi$
$113$	4.57132	0.430033	0.215017	+	0.976610i	$0.431019\pi$
$114$	−4.58431	−0.429360
$115$	12.2768	1.14481
$116$	−6.77674	−0.629204
$117$	−1.00000	−0.0924500
$118$	2.04358	0.188127
$119$	6.27537	0.575263
$120$	−1.50127	−0.137047
$121$	−2.34040	−0.212764
$122$	0.559740	0.0506765
$123$	2.44182	0.220171
$124$	7.21512	0.647937
$125$	11.6291	1.04014
$126$	1.44399	0.128641
$127$	19.0793	1.69301	0.846506	−	0.532378i	$-0.178702\pi$
$127$	19.0793	1.69301	0.846506	+	0.532378i	$0.178702\pi$
$128$	1.00000	0.0883883
$129$	−1.24658	−0.109755
$130$	1.50127	0.131670
$131$	−6.26756	−0.547599	−0.273800	−	0.961787i	$-0.588281\pi$
$131$	−6.26756	−0.547599	−0.273800	+	0.961787i	$0.588281\pi$
$132$	−2.94272	−0.256131
$133$	−6.61970	−0.574001
$134$	3.43251	0.296524
$135$	−1.50127	−0.129209
$136$	4.34586	0.372654
$137$	17.8235	1.52277	0.761384	−	0.648301i	$-0.224520\pi$
$137$	17.8235	1.52277	0.761384	+	0.648301i	$0.224520\pi$
$138$	−8.17758	−0.696122
$139$	18.3218	1.55404	0.777018	−	0.629478i	$-0.216731\pi$
$139$	18.3218	1.55404	0.777018	+	0.629478i	$0.216731\pi$
$140$	−2.16782	−0.183214
$141$	−6.82549	−0.574810
$142$	−11.2772	−0.946360
$143$	2.94272	0.246083
$144$	1.00000	0.0833333
$145$	10.1737	0.844882
$146$	−4.58425	−0.379395
$147$	−4.91489	−0.405373
$148$	2.49161	0.204809
$149$	15.2040	1.24556	0.622782	−	0.782395i	$-0.286002\pi$
$149$	15.2040	1.24556	0.622782	+	0.782395i	$0.286002\pi$
$150$	−2.74619	−0.224225
$151$	−11.1476	−0.907180	−0.453590	−	0.891210i	$-0.649857\pi$
$151$	−11.1476	−0.907180	−0.453590	+	0.891210i	$0.649857\pi$
$152$	−4.58431	−0.371837
$153$	4.34586	0.351342
$154$	−4.24926	−0.342415
$155$	−10.8318	−0.870035
$156$	−1.00000	−0.0800641
$157$	−16.2929	−1.30032	−0.650159	−	0.759798i	$-0.725298\pi$
$157$	−16.2929	−1.30032	−0.650159	+	0.759798i	$0.725298\pi$
$158$	−0.911183	−0.0724898
$159$	6.28858	0.498716
$160$	−1.50127	−0.118686
$161$	−11.8084	−0.930628
$162$	1.00000	0.0785674
$163$	1.63262	0.127876	0.0639382	−	0.997954i	$-0.479634\pi$
$163$	1.63262	0.127876	0.0639382	+	0.997954i	$0.479634\pi$
$164$	2.44182	0.190674
$165$	4.41782	0.343927
$166$	−7.71954	−0.599152
$167$	−17.8060	−1.37787	−0.688936	−	0.724822i	$-0.741922\pi$
$167$	−17.8060	−1.37787	−0.688936	+	0.724822i	$0.741922\pi$
$168$	1.44399	0.111406
$169$	1.00000	0.0769231
$170$	−6.52431	−0.500392
$171$	−4.58431	−0.350571
$172$	−1.24658	−0.0950510
$173$	6.55111	0.498072	0.249036	−	0.968494i	$-0.419886\pi$
$173$	6.55111	0.498072	0.249036	+	0.968494i	$0.419886\pi$
$174$	−6.77674	−0.513743
$175$	−3.96547	−0.299761
$176$	−2.94272	−0.221816
$177$	2.04358	0.153605
$178$	−12.0347	−0.902036
$179$	24.1051	1.80170	0.900849	−	0.434133i	$-0.142945\pi$
$179$	24.1051	1.80170	0.900849	+	0.434133i	$0.142945\pi$
$180$	−1.50127	−0.111898
$181$	−2.51598	−0.187012	−0.0935058	−	0.995619i	$-0.529807\pi$
$181$	−2.51598	−0.187012	−0.0935058	+	0.995619i	$0.529807\pi$
$182$	−1.44399	−0.107036
$183$	0.559740	0.0413772
$184$	−8.17758	−0.602859
$185$	−3.74059	−0.275014
$186$	7.21512	0.529038
$187$	−12.7886	−0.935198
$188$	−6.82549	−0.497800
$189$	1.44399	0.105035
$190$	6.88230	0.499294
$191$	21.3970	1.54823	0.774115	−	0.633045i	$-0.218195\pi$
$191$	21.3970	1.54823	0.774115	+	0.633045i	$0.218195\pi$
$192$	1.00000	0.0721688
$193$	−18.8435	−1.35638	−0.678192	−	0.734885i	$-0.737236\pi$
$193$	−18.8435	−1.35638	−0.678192	+	0.734885i	$0.737236\pi$
$194$	−7.00623	−0.503018
$195$	1.50127	0.107508
$196$	−4.91489	−0.351064
$197$	−2.50485	−0.178463	−0.0892316	−	0.996011i	$-0.528441\pi$
$197$	−2.50485	−0.178463	−0.0892316	+	0.996011i	$0.528441\pi$
$198$	−2.94272	−0.209130
$199$	−4.13150	−0.292874	−0.146437	−	0.989220i	$-0.546781\pi$
$199$	−4.13150	−0.292874	−0.146437	+	0.989220i	$0.546781\pi$
$200$	−2.74619	−0.194185
$201$	3.43251	0.242111
$202$	−18.2763	−1.28592
$203$	−9.78555	−0.686811
$204$	4.34586	0.304271
$205$	−3.66583	−0.256033
$206$	−1.00000	−0.0696733
$207$	−8.17758	−0.568381
$208$	−1.00000	−0.0693375
$209$	13.4903	0.933147
$210$	−2.16782	−0.149594
$211$	7.96741	0.548499	0.274250	−	0.961659i	$-0.411571\pi$
$211$	7.96741	0.548499	0.274250	+	0.961659i	$0.411571\pi$
$212$	6.28858	0.431901
$213$	−11.2772	−0.772699
$214$	−15.2371	−1.04158
$215$	1.87146	0.127632
$216$	1.00000	0.0680414
$217$	10.4186	0.707258
$218$	−10.2209	−0.692247
$219$	−4.58425	−0.309775
$220$	4.41782	0.297849
$221$	−4.34586	−0.292334
$222$	2.49161	0.167226
$223$	26.0255	1.74280	0.871398	−	0.490576i	$-0.163214\pi$
$223$	26.0255	1.74280	0.871398	+	0.490576i	$0.163214\pi$
$224$	1.44399	0.0964807
$225$	−2.74619	−0.183079
$226$	4.57132	0.304080
$227$	−0.492444	−0.0326847	−0.0163423	−	0.999866i	$-0.505202\pi$
$227$	−0.492444	−0.0326847	−0.0163423	+	0.999866i	$0.505202\pi$
$228$	−4.58431	−0.303604
$229$	2.92563	0.193331	0.0966656	−	0.995317i	$-0.469182\pi$
$229$	2.92563	0.193331	0.0966656	+	0.995317i	$0.469182\pi$
$230$	12.2768	0.809506
$231$	−4.24926	−0.279581
$232$	−6.77674	−0.444915
$233$	−24.5166	−1.60614	−0.803068	−	0.595888i	$-0.796800\pi$
$233$	−24.5166	−1.60614	−0.803068	+	0.595888i	$0.796800\pi$
$234$	−1.00000	−0.0653720
$235$	10.2469	0.668435
$236$	2.04358	0.133026
$237$	−0.911183	−0.0591877
$238$	6.27537	0.406772
$239$	16.8929	1.09271	0.546356	−	0.837553i	$-0.316015\pi$
$239$	16.8929	1.09271	0.546356	+	0.837553i	$0.316015\pi$
$240$	−1.50127	−0.0969066
$241$	−16.5141	−1.06376	−0.531882	−	0.846818i	$-0.678515\pi$
$241$	−16.5141	−1.06376	−0.531882	+	0.846818i	$0.678515\pi$
$242$	−2.34040	−0.150447
$243$	1.00000	0.0641500
$244$	0.559740	0.0358337
$245$	7.37858	0.471400
$246$	2.44182	0.155685
$247$	4.58431	0.291693
$248$	7.21512	0.458160
$249$	−7.71954	−0.489206
$250$	11.6291	0.735490
$251$	−9.34241	−0.589688	−0.294844	−	0.955545i	$-0.595268\pi$
$251$	−9.34241	−0.589688	−0.294844	+	0.955545i	$0.595268\pi$
$252$	1.44399	0.0909629
$253$	24.0643	1.51291
$254$	19.0793	1.19714
$255$	−6.52431	−0.408568
$256$	1.00000	0.0625000
$257$	1.37996	0.0860793	0.0430397	−	0.999073i	$-0.486296\pi$
$257$	1.37996	0.0860793	0.0430397	+	0.999073i	$0.486296\pi$
$258$	−1.24658	−0.0776088
$259$	3.59787	0.223561
$260$	1.50127	0.0931049
$261$	−6.77674	−0.419470
$262$	−6.26756	−0.387211
$263$	−11.9740	−0.738350	−0.369175	−	0.929360i	$-0.620360\pi$
$263$	−11.9740	−0.738350	−0.369175	+	0.929360i	$0.620360\pi$
$264$	−2.94272	−0.181112
$265$	−9.44086	−0.579947
$266$	−6.61970	−0.405880
$267$	−12.0347	−0.736509
$268$	3.43251	0.209674
$269$	−7.97608	−0.486310	−0.243155	−	0.969987i	$-0.578182\pi$
$269$	−7.97608	−0.486310	−0.243155	+	0.969987i	$0.578182\pi$
$270$	−1.50127	−0.0913644
$271$	30.2559	1.83792	0.918958	−	0.394356i	$-0.129032\pi$
$271$	30.2559	1.83792	0.918958	+	0.394356i	$0.129032\pi$
$272$	4.34586	0.263506
$273$	−1.44399	−0.0873943
$274$	17.8235	1.07676
$275$	8.08125	0.487318
$276$	−8.17758	−0.492233
$277$	7.30468	0.438896	0.219448	−	0.975624i	$-0.429574\pi$
$277$	7.30468	0.438896	0.219448	+	0.975624i	$0.429574\pi$
$278$	18.3218	1.09887
$279$	7.21512	0.431958
$280$	−2.16782	−0.129552
$281$	−24.6754	−1.47201	−0.736005	−	0.676976i	$-0.763290\pi$
$281$	−24.6754	−1.47201	−0.736005	+	0.676976i	$0.763290\pi$
$282$	−6.82549	−0.406452
$283$	−26.0399	−1.54791	−0.773954	−	0.633242i	$-0.781724\pi$
$283$	−26.0399	−1.54791	−0.773954	+	0.633242i	$0.781724\pi$
$284$	−11.2772	−0.669177
$285$	6.88230	0.407672
$286$	2.94272	0.174007
$287$	3.52596	0.208131
$288$	1.00000	0.0589256
$289$	1.88646	0.110968
$290$	10.1737	0.597422
$291$	−7.00623	−0.410712
$292$	−4.58425	−0.268273
$293$	−12.0095	−0.701602	−0.350801	−	0.936450i	$-0.614091\pi$
$293$	−12.0095	−0.701602	−0.350801	+	0.936450i	$0.614091\pi$
$294$	−4.91489	−0.286642
$295$	−3.06797	−0.178624
$296$	2.49161	0.144822
$297$	−2.94272	−0.170754
$298$	15.2040	0.880747
$299$	8.17758	0.472922
$300$	−2.74619	−0.158551
$301$	−1.80005	−0.103753
$302$	−11.1476	−0.641473
$303$	−18.2763	−1.04995
$304$	−4.58431	−0.262928
$305$	−0.840322	−0.0481167
$306$	4.34586	0.248436
$307$	−1.78627	−0.101948	−0.0509740	−	0.998700i	$-0.516233\pi$
$307$	−1.78627	−0.101948	−0.0509740	+	0.998700i	$0.516233\pi$
$308$	−4.24926	−0.242124
$309$	−1.00000	−0.0568880
$310$	−10.8318	−0.615207
$311$	−1.24835	−0.0707874	−0.0353937	−	0.999373i	$-0.511269\pi$
$311$	−1.24835	−0.0707874	−0.0353937	+	0.999373i	$0.511269\pi$
$312$	−1.00000	−0.0566139
$313$	32.4911	1.83651	0.918254	−	0.395993i	$-0.129599\pi$
$313$	32.4911	1.83651	0.918254	+	0.395993i	$0.129599\pi$
$314$	−16.2929	−0.919463
$315$	−2.16782	−0.122143
$316$	−0.911183	−0.0512581
$317$	−9.71122	−0.545437	−0.272718	−	0.962094i	$-0.587923\pi$
$317$	−9.71122	−0.545437	−0.272718	+	0.962094i	$0.587923\pi$
$318$	6.28858	0.352646
$319$	19.9420	1.11654
$320$	−1.50127	−0.0839236
$321$	−15.2371	−0.850449
$322$	−11.8084	−0.658054
$323$	−19.9228	−1.10853
$324$	1.00000	0.0555556
$325$	2.74619	0.152331
$326$	1.63262	0.0904222
$327$	−10.2209	−0.565217
$328$	2.44182	0.134827
$329$	−9.85594	−0.543376
$330$	4.41782	0.243193
$331$	−7.69458	−0.422932	−0.211466	−	0.977385i	$-0.567824\pi$
$331$	−7.69458	−0.422932	−0.211466	+	0.977385i	$0.567824\pi$
$332$	−7.71954	−0.423665
$333$	2.49161	0.136540
$334$	−17.8060	−0.974303
$335$	−5.15313	−0.281546
$336$	1.44399	0.0787761
$337$	17.5988	0.958668	0.479334	−	0.877633i	$-0.340878\pi$
$337$	17.5988	0.958668	0.479334	+	0.877633i	$0.340878\pi$
$338$	1.00000	0.0543928
$339$	4.57132	0.248280
$340$	−6.52431	−0.353830
$341$	−21.2321	−1.14978
$342$	−4.58431	−0.247891
$343$	−17.2050	−0.928982
$344$	−1.24658	−0.0672112
$345$	12.2768	0.660959
$346$	6.55111	0.352190
$347$	−17.5637	−0.942868	−0.471434	−	0.881901i	$-0.656264\pi$
$347$	−17.5637	−0.942868	−0.471434	+	0.881901i	$0.656264\pi$
$348$	−6.77674	−0.363271
$349$	−7.84509	−0.419938	−0.209969	−	0.977708i	$-0.567336\pi$
$349$	−7.84509	−0.419938	−0.209969	+	0.977708i	$0.567336\pi$
$350$	−3.96547	−0.211963
$351$	−1.00000	−0.0533761
$352$	−2.94272	−0.156847
$353$	−12.1804	−0.648298	−0.324149	−	0.946006i	$-0.605078\pi$
$353$	−12.1804	−0.648298	−0.324149	+	0.946006i	$0.605078\pi$
$354$	2.04358	0.108615
$355$	16.9301	0.898556
$356$	−12.0347	−0.637836
$357$	6.27537	0.332128
$358$	24.1051	1.27399
$359$	0.0253071	0.00133566	0.000667829	−	1.00000i	$-0.499787\pi$
$359$	0.0253071	0.00133566	0.000667829		1.00000i	$0.499787\pi$
$360$	−1.50127	−0.0791239
$361$	2.01593	0.106101
$362$	−2.51598	−0.132237
$363$	−2.34040	−0.122839
$364$	−1.44399	−0.0756857
$365$	6.88220	0.360231
$366$	0.559740	0.0292581
$367$	−25.2702	−1.31910	−0.659548	−	0.751662i	$-0.729252\pi$
$367$	−25.2702	−1.31910	−0.659548	+	0.751662i	$0.729252\pi$
$368$	−8.17758	−0.426286
$369$	2.44182	0.127116
$370$	−3.74059	−0.194464
$371$	9.08064	0.471443
$372$	7.21512	0.374086
$373$	14.3435	0.742676	0.371338	−	0.928498i	$-0.378899\pi$
$373$	14.3435	0.742676	0.371338	+	0.928498i	$0.378899\pi$
$374$	−12.7886	−0.661285
$375$	11.6291	0.600525
$376$	−6.82549	−0.351998
$377$	6.77674	0.349020
$378$	1.44399	0.0742709
$379$	26.5230	1.36239	0.681197	−	0.732100i	$-0.261460\pi$
$379$	26.5230	1.36239	0.681197	+	0.732100i	$0.261460\pi$
$380$	6.88230	0.353054
$381$	19.0793	0.977461
$382$	21.3970	1.09476
$383$	7.51601	0.384050	0.192025	−	0.981390i	$-0.438495\pi$
$383$	7.51601	0.384050	0.192025	+	0.981390i	$0.438495\pi$
$384$	1.00000	0.0510310
$385$	6.37929	0.325119
$386$	−18.8435	−0.959108
$387$	−1.24658	−0.0633673
$388$	−7.00623	−0.355687
$389$	2.17413	0.110233	0.0551164	−	0.998480i	$-0.482447\pi$
$389$	2.17413	0.110233	0.0551164	+	0.998480i	$0.482447\pi$
$390$	1.50127	0.0760198
$391$	−35.5386	−1.79726
$392$	−4.91489	−0.248239
$393$	−6.26756	−0.316157
$394$	−2.50485	−0.126193
$395$	1.36793	0.0688282
$396$	−2.94272	−0.147877
$397$	10.0997	0.506890	0.253445	−	0.967350i	$-0.418436\pi$
$397$	10.0997	0.506890	0.253445	+	0.967350i	$0.418436\pi$
$398$	−4.13150	−0.207094
$399$	−6.61970	−0.331400
$400$	−2.74619	−0.137309
$401$	−16.9294	−0.845412	−0.422706	−	0.906267i	$-0.638920\pi$
$401$	−16.9294	−0.845412	−0.422706	+	0.906267i	$0.638920\pi$
$402$	3.43251	0.171198
$403$	−7.21512	−0.359411
$404$	−18.2763	−0.909280
$405$	−1.50127	−0.0745988
$406$	−9.78555	−0.485649
$407$	−7.33212	−0.363440
$408$	4.34586	0.215152
$409$	−13.8422	−0.684454	−0.342227	−	0.939617i	$-0.611181\pi$
$409$	−13.8422	−0.684454	−0.342227	+	0.939617i	$0.611181\pi$
$410$	−3.66583	−0.181042
$411$	17.8235	0.879170
$412$	−1.00000	−0.0492665
$413$	2.95092	0.145205
$414$	−8.17758	−0.401906
$415$	11.5891	0.568887
$416$	−1.00000	−0.0490290
$417$	18.3218	0.897223
$418$	13.4903	0.659834
$419$	−6.80728	−0.332557	−0.166279	−	0.986079i	$-0.553175\pi$
$419$	−6.80728	−0.332557	−0.166279	+	0.986079i	$0.553175\pi$
$420$	−2.16782	−0.105779
$421$	−9.38076	−0.457190	−0.228595	−	0.973522i	$-0.573413\pi$
$421$	−9.38076	−0.457190	−0.228595	+	0.973522i	$0.573413\pi$
$422$	7.96741	0.387848
$423$	−6.82549	−0.331867
$424$	6.28858	0.305400
$425$	−11.9345	−0.578909
$426$	−11.2772	−0.546381
$427$	0.808260	0.0391144
$428$	−15.2371	−0.736511
$429$	2.94272	0.142076
$430$	1.87146	0.0902497
$431$	−13.9944	−0.674085	−0.337042	−	0.941489i	$-0.609427\pi$
$431$	−13.9944	−0.674085	−0.337042	+	0.941489i	$0.609427\pi$
$432$	1.00000	0.0481125
$433$	23.2108	1.11544	0.557720	−	0.830029i	$-0.311676\pi$
$433$	23.2108	1.11544	0.557720	+	0.830029i	$0.311676\pi$
$434$	10.4186	0.500107
$435$	10.1737	0.487793
$436$	−10.2209	−0.489492
$437$	37.4886	1.79332
$438$	−4.58425	−0.219044
$439$	0.818220	0.0390515	0.0195258	−	0.999809i	$-0.493784\pi$
$439$	0.818220	0.0390515	0.0195258	+	0.999809i	$0.493784\pi$
$440$	4.41782	0.210611
$441$	−4.91489	−0.234042
$442$	−4.34586	−0.206711
$443$	−1.54162	−0.0732448	−0.0366224	−	0.999329i	$-0.511660\pi$
$443$	−1.54162	−0.0732448	−0.0366224	+	0.999329i	$0.511660\pi$
$444$	2.49161	0.118247
$445$	18.0673	0.856471
$446$	26.0255	1.23234
$447$	15.2040	0.719127
$448$	1.44399	0.0682221
$449$	16.7284	0.789462	0.394731	−	0.918797i	$-0.370838\pi$
$449$	16.7284	0.789462	0.394731	+	0.918797i	$0.370838\pi$
$450$	−2.74619	−0.129456
$451$	−7.18558	−0.338356
$452$	4.57132	0.215017
$453$	−11.1476	−0.523761
$454$	−0.492444	−0.0231116
$455$	2.16782	0.101629
$456$	−4.58431	−0.214680
$457$	−14.5479	−0.680524	−0.340262	−	0.940331i	$-0.610516\pi$
$457$	−14.5479	−0.680524	−0.340262	+	0.940331i	$0.610516\pi$
$458$	2.92563	0.136706
$459$	4.34586	0.202847
$460$	12.2768	0.572407
$461$	−17.1561	−0.799039	−0.399520	−	0.916725i	$-0.630823\pi$
$461$	−17.1561	−0.799039	−0.399520	+	0.916725i	$0.630823\pi$
$462$	−4.24926	−0.197693
$463$	22.9630	1.06718	0.533590	−	0.845744i	$-0.320843\pi$
$463$	22.9630	1.06718	0.533590	+	0.845744i	$0.320843\pi$
$464$	−6.77674	−0.314602
$465$	−10.8318	−0.502315
$466$	−24.5166	−1.13571
$467$	−20.5164	−0.949387	−0.474693	−	0.880151i	$-0.657441\pi$
$467$	−20.5164	−0.949387	−0.474693	+	0.880151i	$0.657441\pi$
$468$	−1.00000	−0.0462250
$469$	4.95652	0.228871
$470$	10.2469	0.472655
$471$	−16.2929	−0.750739
$472$	2.04358	0.0940636
$473$	3.66834	0.168671
$474$	−0.911183	−0.0418520
$475$	12.5894	0.577640
$476$	6.27537	0.287631
$477$	6.28858	0.287934
$478$	16.8929	0.772664
$479$	−39.2449	−1.79314	−0.896572	−	0.442898i	$-0.853950\pi$
$479$	−39.2449	−1.79314	−0.896572	+	0.442898i	$0.853950\pi$
$480$	−1.50127	−0.0685233
$481$	−2.49161	−0.113608
$482$	−16.5141	−0.752195
$483$	−11.8084	−0.537299
$484$	−2.34040	−0.106382
$485$	10.5183	0.477609
$486$	1.00000	0.0453609
$487$	4.15066	0.188084	0.0940422	−	0.995568i	$-0.470021\pi$
$487$	4.15066	0.188084	0.0940422	+	0.995568i	$0.470021\pi$
$488$	0.559740	0.0253383
$489$	1.63262	0.0738294
$490$	7.37858	0.333330
$491$	−2.73037	−0.123220	−0.0616099	−	0.998100i	$-0.519623\pi$
$491$	−2.73037	−0.123220	−0.0616099	+	0.998100i	$0.519623\pi$
$492$	2.44182	0.110086
$493$	−29.4507	−1.32639
$494$	4.58431	0.206258
$495$	4.41782	0.198566
$496$	7.21512	0.323968
$497$	−16.2841	−0.730443
$498$	−7.71954	−0.345921
$499$	33.3507	1.49298	0.746492	−	0.665394i	$-0.231737\pi$
$499$	33.3507	1.49298	0.746492	+	0.665394i	$0.231737\pi$
$500$	11.6291	0.520070
$501$	−17.8060	−0.795515
$502$	−9.34241	−0.416972
$503$	4.40985	0.196625	0.0983127	−	0.995156i	$-0.468655\pi$
$503$	4.40985	0.196625	0.0983127	+	0.995156i	$0.468655\pi$
$504$	1.44399	0.0643205
$505$	27.4377	1.22096
$506$	24.0643	1.06979
$507$	1.00000	0.0444116
$508$	19.0793	0.846506
$509$	4.59545	0.203690	0.101845	−	0.994800i	$-0.467525\pi$
$509$	4.59545	0.203690	0.101845	+	0.994800i	$0.467525\pi$
$510$	−6.52431	−0.288901
$511$	−6.61961	−0.292834
$512$	1.00000	0.0441942
$513$	−4.58431	−0.202402
$514$	1.37996	0.0608673
$515$	1.50127	0.0661539
$516$	−1.24658	−0.0548777
$517$	20.0855	0.883359
$518$	3.59787	0.158081
$519$	6.55111	0.287562
$520$	1.50127	0.0658351
$521$	−5.62050	−0.246239	−0.123119	−	0.992392i	$-0.539290\pi$
$521$	−5.62050	−0.246239	−0.123119	+	0.992392i	$0.539290\pi$
$522$	−6.77674	−0.296610
$523$	32.4326	1.41818	0.709089	−	0.705119i	$-0.249107\pi$
$523$	32.4326	1.41818	0.709089	+	0.705119i	$0.249107\pi$
$524$	−6.26756	−0.273800
$525$	−3.96547	−0.173067
$526$	−11.9740	−0.522092
$527$	31.3559	1.36588
$528$	−2.94272	−0.128065
$529$	43.8729	1.90752
$530$	−9.44086	−0.410085
$531$	2.04358	0.0886840
$532$	−6.61970	−0.287001
$533$	−2.44182	−0.105767
$534$	−12.0347	−0.520791
$535$	22.8749	0.988970
$536$	3.43251	0.148262
$537$	24.1051	1.04021
$538$	−7.97608	−0.343873
$539$	14.4631	0.622972
$540$	−1.50127	−0.0646044
$541$	−15.8097	−0.679712	−0.339856	−	0.940477i	$-0.610378\pi$
$541$	−15.8097	−0.679712	−0.339856	+	0.940477i	$0.610378\pi$
$542$	30.2559	1.29960
$543$	−2.51598	−0.107971
$544$	4.34586	0.186327
$545$	15.3443	0.657279
$546$	−1.44399	−0.0617971
$547$	−9.72607	−0.415857	−0.207928	−	0.978144i	$-0.566672\pi$
$547$	−9.72607	−0.415857	−0.207928	+	0.978144i	$0.566672\pi$
$548$	17.8235	0.761384
$549$	0.559740	0.0238891
$550$	8.08125	0.344586
$551$	31.0667	1.32349
$552$	−8.17758	−0.348061
$553$	−1.31574	−0.0559510
$554$	7.30468	0.310346
$555$	−3.74059	−0.158779
$556$	18.3218	0.777018
$557$	−15.2775	−0.647327	−0.323664	−	0.946172i	$-0.604915\pi$
$557$	−15.2775	−0.647327	−0.323664	+	0.946172i	$0.604915\pi$
$558$	7.21512	0.305440
$559$	1.24658	0.0527248
$560$	−2.16782	−0.0916072
$561$	−12.7886	−0.539937
$562$	−24.6754	−1.04087
$563$	3.59576	0.151543	0.0757716	−	0.997125i	$-0.475858\pi$
$563$	3.59576	0.151543	0.0757716	+	0.997125i	$0.475858\pi$
$564$	−6.82549	−0.287405
$565$	−6.86279	−0.288720
$566$	−26.0399	−1.09454
$567$	1.44399	0.0606419
$568$	−11.2772	−0.473180
$569$	34.0468	1.42732	0.713659	−	0.700494i	$-0.247037\pi$
$569$	34.0468	1.42732	0.713659	+	0.700494i	$0.247037\pi$
$570$	6.88230	0.288268
$571$	8.31462	0.347956	0.173978	−	0.984750i	$-0.444338\pi$
$571$	8.31462	0.347956	0.173978	+	0.984750i	$0.444338\pi$
$572$	2.94272	0.123041
$573$	21.3970	0.893871
$574$	3.52596	0.147171
$575$	22.4572	0.936528
$576$	1.00000	0.0416667
$577$	−15.1127	−0.629151	−0.314576	−	0.949232i	$-0.601862\pi$
$577$	−15.1127	−0.629151	−0.314576	+	0.949232i	$0.601862\pi$
$578$	1.88646	0.0784664
$579$	−18.8435	−0.783108
$580$	10.1737	0.422441
$581$	−11.1469	−0.462453
$582$	−7.00623	−0.290418
$583$	−18.5055	−0.766420
$584$	−4.58425	−0.189698
$585$	1.50127	0.0620699
$586$	−12.0095	−0.496107
$587$	−15.1131	−0.623783	−0.311892	−	0.950118i	$-0.600963\pi$
$587$	−15.1131	−0.623783	−0.311892	+	0.950118i	$0.600963\pi$
$588$	−4.91489	−0.202687
$589$	−33.0763	−1.36289
$590$	−3.06797	−0.126306
$591$	−2.50485	−0.103036
$592$	2.49161	0.102405
$593$	30.8559	1.26710	0.633550	−	0.773702i	$-0.281597\pi$
$593$	30.8559	1.26710	0.633550	+	0.773702i	$0.281597\pi$
$594$	−2.94272	−0.120741
$595$	−9.42104	−0.386225
$596$	15.2040	0.622782
$597$	−4.13150	−0.169091
$598$	8.17758	0.334406
$599$	−21.0251	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$599$	−21.0251	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$600$	−2.74619	−0.112113
$601$	27.5460	1.12363	0.561813	−	0.827264i	$-0.310104\pi$
$601$	27.5460	1.12363	0.561813	+	0.827264i	$0.310104\pi$
$602$	−1.80005	−0.0733647
$603$	3.43251	0.139783
$604$	−11.1476	−0.453590
$605$	3.51358	0.142847
$606$	−18.2763	−0.742424
$607$	32.9239	1.33634	0.668169	−	0.744009i	$-0.267078\pi$
$607$	32.9239	1.33634	0.668169	+	0.744009i	$0.267078\pi$
$608$	−4.58431	−0.185918
$609$	−9.78555	−0.396530
$610$	−0.840322	−0.0340236
$611$	6.82549	0.276130
$612$	4.34586	0.175671
$613$	−28.2443	−1.14077	−0.570387	−	0.821376i	$-0.693207\pi$
$613$	−28.2443	−1.14077	−0.570387	+	0.821376i	$0.693207\pi$
$614$	−1.78627	−0.0720881
$615$	−3.66583	−0.147821
$616$	−4.24926	−0.171208
$617$	−1.17239	−0.0471985	−0.0235992	−	0.999721i	$-0.507513\pi$
$617$	−1.17239	−0.0471985	−0.0235992	+	0.999721i	$0.507513\pi$
$618$	−1.00000	−0.0402259
$619$	−39.8918	−1.60339	−0.801693	−	0.597736i	$-0.796067\pi$
$619$	−39.8918	−1.60339	−0.801693	+	0.597736i	$0.796067\pi$
$620$	−10.8318	−0.435017
$621$	−8.17758	−0.328155
$622$	−1.24835	−0.0500543
$623$	−17.3779	−0.696232
$624$	−1.00000	−0.0400320
$625$	−3.72754	−0.149102
$626$	32.4911	1.29861
$627$	13.4903	0.538753
$628$	−16.2929	−0.650159
$629$	10.8282	0.431748
$630$	−2.16782	−0.0863681
$631$	21.8510	0.869875	0.434937	−	0.900461i	$-0.356771\pi$
$631$	21.8510	0.869875	0.434937	+	0.900461i	$0.356771\pi$
$632$	−0.911183	−0.0362449
$633$	7.96741	0.316676
$634$	−9.71122	−0.385682
$635$	−28.6432	−1.13667
$636$	6.28858	0.249358
$637$	4.91489	0.194735
$638$	19.9420	0.789513
$639$	−11.2772	−0.446118
$640$	−1.50127	−0.0593429
$641$	11.9995	0.473951	0.236976	−	0.971516i	$-0.423844\pi$
$641$	11.9995	0.473951	0.236976	+	0.971516i	$0.423844\pi$
$642$	−15.2371	−0.601359
$643$	−44.7984	−1.76668	−0.883339	−	0.468735i	$-0.844710\pi$
$643$	−44.7984	−1.76668	−0.883339	+	0.468735i	$0.844710\pi$
$644$	−11.8084	−0.465314
$645$	1.87146	0.0736886
$646$	−19.9228	−0.783851
$647$	4.02388	0.158195	0.0790975	−	0.996867i	$-0.474796\pi$
$647$	4.02388	0.158195	0.0790975	+	0.996867i	$0.474796\pi$
$648$	1.00000	0.0392837
$649$	−6.01370	−0.236058
$650$	2.74619	0.107714
$651$	10.4186	0.408336
$652$	1.63262	0.0639382
$653$	3.93783	0.154099	0.0770496	−	0.997027i	$-0.475450\pi$
$653$	3.93783	0.154099	0.0770496	+	0.997027i	$0.475450\pi$
$654$	−10.2209	−0.399669
$655$	9.40931	0.367652
$656$	2.44182	0.0953369
$657$	−4.58425	−0.178849
$658$	−9.85594	−0.384225
$659$	23.2337	0.905055	0.452528	−	0.891750i	$-0.350522\pi$
$659$	23.2337	0.905055	0.452528	+	0.891750i	$0.350522\pi$
$660$	4.41782	0.171963
$661$	−15.7186	−0.611382	−0.305691	−	0.952131i	$-0.598888\pi$
$661$	−15.7186	−0.611382	−0.305691	+	0.952131i	$0.598888\pi$
$662$	−7.69458	−0.299058
$663$	−4.34586	−0.168779
$664$	−7.71954	−0.299576
$665$	9.93797	0.385378
$666$	2.49161	0.0965481
$667$	55.4173	2.14577
$668$	−17.8060	−0.688936
$669$	26.0255	1.00620
$670$	−5.15313	−0.199083
$671$	−1.64716	−0.0635879
$672$	1.44399	0.0557031
$673$	−8.70623	−0.335600	−0.167800	−	0.985821i	$-0.553666\pi$
$673$	−8.70623	−0.335600	−0.167800	+	0.985821i	$0.553666\pi$
$674$	17.5988	0.677881
$675$	−2.74619	−0.105701
$676$	1.00000	0.0384615
$677$	22.7977	0.876188	0.438094	−	0.898929i	$-0.355654\pi$
$677$	22.7977	0.876188	0.438094	+	0.898929i	$0.355654\pi$
$678$	4.57132	0.175560
$679$	−10.1169	−0.388252
$680$	−6.52431	−0.250196
$681$	−0.492444	−0.0188705
$682$	−21.2321	−0.813018
$683$	−34.2713	−1.31135	−0.655677	−	0.755042i	$-0.727617\pi$
$683$	−34.2713	−1.31135	−0.655677	+	0.755042i	$0.727617\pi$
$684$	−4.58431	−0.175286
$685$	−26.7580	−1.02237
$686$	−17.2050	−0.656890
$687$	2.92563	0.111620
$688$	−1.24658	−0.0475255
$689$	−6.28858	−0.239576
$690$	12.2768	0.467369
$691$	−11.5634	−0.439891	−0.219946	−	0.975512i	$-0.570588\pi$
$691$	−11.5634	−0.439891	−0.219946	+	0.975512i	$0.570588\pi$
$692$	6.55111	0.249036
$693$	−4.24926	−0.161416
$694$	−17.5637	−0.666709
$695$	−27.5060	−1.04336
$696$	−6.77674	−0.256872
$697$	10.6118	0.401950
$698$	−7.84509	−0.296941
$699$	−24.5166	−0.927303
$700$	−3.96547	−0.149881
$701$	−3.79813	−0.143453	−0.0717267	−	0.997424i	$-0.522851\pi$
$701$	−3.79813	−0.143453	−0.0717267	+	0.997424i	$0.522851\pi$
$702$	−1.00000	−0.0377426
$703$	−11.4223	−0.430802
$704$	−2.94272	−0.110908
$705$	10.2469	0.385921
$706$	−12.1804	−0.458416
$707$	−26.3908	−0.992529
$708$	2.04358	0.0768026
$709$	−6.94212	−0.260717	−0.130358	−	0.991467i	$-0.541613\pi$
$709$	−6.94212	−0.260717	−0.130358	+	0.991467i	$0.541613\pi$
$710$	16.9301	0.635375
$711$	−0.911183	−0.0341720
$712$	−12.0347	−0.451018
$713$	−59.0022	−2.20965
$714$	6.27537	0.234850
$715$	−4.41782	−0.165217
$716$	24.1051	0.900849
$717$	16.8929	0.630878
$718$	0.0253071	0.000944452 0
$719$	26.0867	0.972870	0.486435	−	0.873717i	$-0.338297\pi$
$719$	26.0867	0.972870	0.486435	+	0.873717i	$0.338297\pi$
$720$	−1.50127	−0.0559491
$721$	−1.44399	−0.0537770
$722$	2.01593	0.0750250
$723$	−16.5141	−0.614165
$724$	−2.51598	−0.0935058
$725$	18.6102	0.691165
$726$	−2.34040	−0.0868605
$727$	12.7471	0.472765	0.236383	−	0.971660i	$-0.424038\pi$
$727$	12.7471	0.472765	0.236383	+	0.971660i	$0.424038\pi$
$728$	−1.44399	−0.0535179
$729$	1.00000	0.0370370
$730$	6.88220	0.254722
$731$	−5.41747	−0.200372
$732$	0.559740	0.0206886
$733$	−4.40804	−0.162815	−0.0814074	−	0.996681i	$-0.525941\pi$
$733$	−4.40804	−0.162815	−0.0814074	+	0.996681i	$0.525941\pi$
$734$	−25.2702	−0.932742
$735$	7.37858	0.272163
$736$	−8.17758	−0.301430
$737$	−10.1009	−0.372072
$738$	2.44182	0.0898845
$739$	−32.3066	−1.18842	−0.594208	−	0.804311i	$-0.702534\pi$
$739$	−32.3066	−1.18842	−0.594208	+	0.804311i	$0.702534\pi$
$740$	−3.74059	−0.137507
$741$	4.58431	0.168409
$742$	9.08064	0.333361
$743$	42.2576	1.55028	0.775141	−	0.631789i	$-0.217679\pi$
$743$	42.2576	1.55028	0.775141	+	0.631789i	$0.217679\pi$
$744$	7.21512	0.264519
$745$	−22.8254	−0.836258
$746$	14.3435	0.525152
$747$	−7.71954	−0.282443
$748$	−12.7886	−0.467599
$749$	−22.0022	−0.803941
$750$	11.6291	0.424636
$751$	−2.62428	−0.0957613	−0.0478806	−	0.998853i	$-0.515247\pi$
$751$	−2.62428	−0.0957613	−0.0478806	+	0.998853i	$0.515247\pi$
$752$	−6.82549	−0.248900
$753$	−9.34241	−0.340456
$754$	6.77674	0.246794
$755$	16.7356	0.609071
$756$	1.44399	0.0525174
$757$	20.2740	0.736872	0.368436	−	0.929653i	$-0.379893\pi$
$757$	20.2740	0.736872	0.368436	+	0.929653i	$0.379893\pi$
$758$	26.5230	0.963358
$759$	24.0643	0.873480
$760$	6.88230	0.249647
$761$	−8.49352	−0.307890	−0.153945	−	0.988079i	$-0.549198\pi$
$761$	−8.49352	−0.307890	−0.153945	+	0.988079i	$0.549198\pi$
$762$	19.0793	0.691170
$763$	−14.7589	−0.534307
$764$	21.3970	0.774115
$765$	−6.52431	−0.235887
$766$	7.51601	0.271564
$767$	−2.04358	−0.0737896
$768$	1.00000	0.0360844
$769$	50.2444	1.81186	0.905929	−	0.423429i	$-0.139174\pi$
$769$	50.2444	1.81186	0.905929	+	0.423429i	$0.139174\pi$
$770$	6.37929	0.229894
$771$	1.37996	0.0496979
$772$	−18.8435	−0.678192
$773$	−4.35820	−0.156753	−0.0783767	−	0.996924i	$-0.524974\pi$
$773$	−4.35820	−0.156753	−0.0783767	+	0.996924i	$0.524974\pi$
$774$	−1.24658	−0.0448075
$775$	−19.8140	−0.711741
$776$	−7.00623	−0.251509
$777$	3.59787	0.129073
$778$	2.17413	0.0779464
$779$	−11.1941	−0.401069
$780$	1.50127	0.0537541
$781$	33.1856	1.18747
$782$	−35.5386	−1.27086
$783$	−6.77674	−0.242181
$784$	−4.91489	−0.175532
$785$	24.4601	0.873019
$786$	−6.26756	−0.223556
$787$	13.1624	0.469188	0.234594	−	0.972093i	$-0.424624\pi$
$787$	13.1624	0.469188	0.234594	+	0.972093i	$0.424624\pi$
$788$	−2.50485	−0.0892316
$789$	−11.9740	−0.426286
$790$	1.36793	0.0486689
$791$	6.60094	0.234702
$792$	−2.94272	−0.104565
$793$	−0.559740	−0.0198770
$794$	10.0997	0.358425
$795$	−9.44086	−0.334833
$796$	−4.13150	−0.146437
$797$	29.6540	1.05040	0.525200	−	0.850979i	$-0.323990\pi$
$797$	29.6540	1.05040	0.525200	+	0.850979i	$0.323990\pi$
$798$	−6.61970	−0.234335
$799$	−29.6626	−1.04939
$800$	−2.74619	−0.0970923
$801$	−12.0347	−0.425224
$802$	−16.9294	−0.597796
$803$	13.4902	0.476057
$804$	3.43251	0.121055
$805$	17.7275	0.624814
$806$	−7.21512	−0.254142
$807$	−7.97608	−0.280771
$808$	−18.2763	−0.642958
$809$	22.8389	0.802974	0.401487	−	0.915865i	$-0.368493\pi$
$809$	22.8389	0.802974	0.401487	+	0.915865i	$0.368493\pi$
$810$	−1.50127	−0.0527493
$811$	−20.7348	−0.728096	−0.364048	−	0.931380i	$-0.618606\pi$
$811$	−20.7348	−0.728096	−0.364048	+	0.931380i	$0.618606\pi$
$812$	−9.78555	−0.343405
$813$	30.2559	1.06112
$814$	−7.33212	−0.256991
$815$	−2.45100	−0.0858547
$816$	4.34586	0.152135
$817$	5.71472	0.199933
$818$	−13.8422	−0.483982
$819$	−1.44399	−0.0504571
$820$	−3.66583	−0.128016
$821$	−23.6464	−0.825264	−0.412632	−	0.910898i	$-0.635390\pi$
$821$	−23.6464	−0.825264	−0.412632	+	0.910898i	$0.635390\pi$
$822$	17.8235	0.621667
$823$	0.359761	0.0125405	0.00627025	−	0.999980i	$-0.498004\pi$
$823$	0.359761	0.0125405	0.00627025	+	0.999980i	$0.498004\pi$
$824$	−1.00000	−0.0348367
$825$	8.08125	0.281353
$826$	2.95092	0.102676
$827$	−42.1917	−1.46715	−0.733574	−	0.679609i	$-0.762149\pi$
$827$	−42.1917	−1.46715	−0.733574	+	0.679609i	$0.762149\pi$
$828$	−8.17758	−0.284191
$829$	33.4538	1.16190	0.580949	−	0.813940i	$-0.302682\pi$
$829$	33.4538	1.16190	0.580949	+	0.813940i	$0.302682\pi$
$830$	11.5891	0.402264
$831$	7.30468	0.253397
$832$	−1.00000	−0.0346688
$833$	−21.3594	−0.740060
$834$	18.3218	0.634433
$835$	26.7317	0.925088
$836$	13.4903	0.466573
$837$	7.21512	0.249391
$838$	−6.80728	−0.235154
$839$	34.9326	1.20601	0.603004	−	0.797738i	$-0.293970\pi$
$839$	34.9326	1.20601	0.603004	+	0.797738i	$0.293970\pi$
$840$	−2.16782	−0.0747969
$841$	16.9242	0.583593
$842$	−9.38076	−0.323282
$843$	−24.6754	−0.849865
$844$	7.96741	0.274250
$845$	−1.50127	−0.0516453
$846$	−6.82549	−0.234665
$847$	−3.37952	−0.116122
$848$	6.28858	0.215951
$849$	−26.0399	−0.893685
$850$	−11.9345	−0.409351
$851$	−20.3754	−0.698459
$852$	−11.2772	−0.386350
$853$	−23.1655	−0.793173	−0.396586	−	0.917997i	$-0.629805\pi$
$853$	−23.1655	−0.793173	−0.396586	+	0.917997i	$0.629805\pi$
$854$	0.808260	0.0276581
$855$	6.88230	0.235370
$856$	−15.2371	−0.520792
$857$	49.3743	1.68660	0.843298	−	0.537447i	$-0.180611\pi$
$857$	49.3743	1.68660	0.843298	+	0.537447i	$0.180611\pi$
$858$	2.94272	0.100463
$859$	23.4263	0.799294	0.399647	−	0.916669i	$-0.369133\pi$
$859$	23.4263	0.799294	0.399647	+	0.916669i	$0.369133\pi$
$860$	1.87146	0.0638162
$861$	3.52596	0.120164
$862$	−13.9944	−0.476650
$863$	18.0490	0.614396	0.307198	−	0.951646i	$-0.400609\pi$
$863$	18.0490	0.614396	0.307198	+	0.951646i	$0.400609\pi$
$864$	1.00000	0.0340207
$865$	−9.83500	−0.334400
$866$	23.2108	0.788735
$867$	1.88646	0.0640676
$868$	10.4186	0.353629
$869$	2.68136	0.0909588
$870$	10.1737	0.344921
$871$	−3.43251	−0.116306
$872$	−10.2209	−0.346123
$873$	−7.00623	−0.237125
$874$	37.4886	1.26807
$875$	16.7923	0.567685
$876$	−4.58425	−0.154887
$877$	14.7812	0.499126	0.249563	−	0.968359i	$-0.419713\pi$
$877$	14.7812	0.499126	0.249563	+	0.968359i	$0.419713\pi$
$878$	0.818220	0.0276136
$879$	−12.0095	−0.405070
$880$	4.41782	0.148925
$881$	32.6195	1.09898	0.549490	−	0.835500i	$-0.314822\pi$
$881$	32.6195	1.09898	0.549490	+	0.835500i	$0.314822\pi$
$882$	−4.91489	−0.165493
$883$	53.7619	1.80923	0.904615	−	0.426229i	$-0.140158\pi$
$883$	53.7619	1.80923	0.904615	+	0.426229i	$0.140158\pi$
$884$	−4.34586	−0.146167
$885$	−3.06797	−0.103129
$886$	−1.54162	−0.0517919
$887$	24.1792	0.811857	0.405929	−	0.913905i	$-0.366948\pi$
$887$	24.1792	0.811857	0.405929	+	0.913905i	$0.366948\pi$
$888$	2.49161	0.0836131
$889$	27.5503	0.924008
$890$	18.0673	0.605617
$891$	−2.94272	−0.0985848
$892$	26.0255	0.871398
$893$	31.2902	1.04709
$894$	15.2040	0.508499
$895$	−36.1882	−1.20964
$896$	1.44399	0.0482403
$897$	8.17758	0.273042
$898$	16.7284	0.558234
$899$	−48.8950	−1.63074
$900$	−2.74619	−0.0915395
$901$	27.3292	0.910469
$902$	−7.18558	−0.239254
$903$	−1.80005	−0.0599020
$904$	4.57132	0.152040
$905$	3.77718	0.125558
$906$	−11.1476	−0.370355
$907$	−18.1330	−0.602097	−0.301048	−	0.953609i	$-0.597337\pi$
$907$	−18.1330	−0.602097	−0.301048	+	0.953609i	$0.597337\pi$
$908$	−0.492444	−0.0163423
$909$	−18.2763	−0.606187
$910$	2.16782	0.0718626
$911$	19.4308	0.643773	0.321886	−	0.946778i	$-0.395683\pi$
$911$	19.4308	0.643773	0.321886	+	0.946778i	$0.395683\pi$
$912$	−4.58431	−0.151802
$913$	22.7164	0.751804
$914$	−14.5479	−0.481203
$915$	−0.840322	−0.0277802
$916$	2.92563	0.0966656
$917$	−9.05030	−0.298867
$918$	4.34586	0.143435
$919$	−10.1426	−0.334573	−0.167286	−	0.985908i	$-0.553500\pi$
$919$	−10.1426	−0.334573	−0.167286	+	0.985908i	$0.553500\pi$
$920$	12.2768	0.404753
$921$	−1.78627	−0.0588597
$922$	−17.1561	−0.565006
$923$	11.2772	0.371193
$924$	−4.24926	−0.139790
$925$	−6.84243	−0.224978
$926$	22.9630	0.754610
$927$	−1.00000	−0.0328443
$928$	−6.77674	−0.222457
$929$	48.5139	1.59169	0.795845	−	0.605501i	$-0.207027\pi$
$929$	48.5139	1.59169	0.795845	+	0.605501i	$0.207027\pi$
$930$	−10.8318	−0.355190
$931$	22.5314	0.738437
$932$	−24.5166	−0.803068
$933$	−1.24835	−0.0408691
$934$	−20.5164	−0.671318
$935$	19.1992	0.627881
$936$	−1.00000	−0.0326860
$937$	−4.89644	−0.159960	−0.0799799	−	0.996796i	$-0.525486\pi$
$937$	−4.89644	−0.159960	−0.0799799	+	0.996796i	$0.525486\pi$
$938$	4.95652	0.161836
$939$	32.4911	1.06031
$940$	10.2469	0.334217
$941$	60.7948	1.98186	0.990928	−	0.134396i	$-0.0429096\pi$
$941$	60.7948	1.98186	0.990928	+	0.134396i	$0.0429096\pi$
$942$	−16.2929	−0.530852
$943$	−19.9682	−0.650253
$944$	2.04358	0.0665130
$945$	−2.16782	−0.0705192
$946$	3.66834	0.119268
$947$	−24.3547	−0.791420	−0.395710	−	0.918376i	$-0.629501\pi$
$947$	−24.3547	−0.791420	−0.395710	+	0.918376i	$0.629501\pi$
$948$	−0.911183	−0.0295939
$949$	4.58425	0.148811
$950$	12.5894	0.408453
$951$	−9.71122	−0.314908
$952$	6.27537	0.203386
$953$	−56.4551	−1.82876	−0.914380	−	0.404858i	$-0.867321\pi$
$953$	−56.4551	−1.82876	−0.914380	+	0.404858i	$0.867321\pi$
$954$	6.28858	0.203600
$955$	−32.1226	−1.03946
$956$	16.8929	0.546356
$957$	19.9420	0.644635
$958$	−39.2449	−1.26794
$959$	25.7370	0.831092
$960$	−1.50127	−0.0484533
$961$	21.0579	0.679287
$962$	−2.49161	−0.0803329
$963$	−15.2371	−0.491007
$964$	−16.5141	−0.531882
$965$	28.2892	0.910661
$966$	−11.8084	−0.379927
$967$	55.8495	1.79600	0.897999	−	0.439997i	$-0.145020\pi$
$967$	55.8495	1.79600	0.897999	+	0.439997i	$0.145020\pi$
$968$	−2.34040	−0.0752234
$969$	−19.9228	−0.640011
$970$	10.5183	0.337721
$971$	−24.1103	−0.773736	−0.386868	−	0.922135i	$-0.626443\pi$
$971$	−24.1103	−0.773736	−0.386868	+	0.922135i	$0.626443\pi$
$972$	1.00000	0.0320750
$973$	26.4565	0.848158
$974$	4.15066	0.132996
$975$	2.74619	0.0879483
$976$	0.559740	0.0179169
$977$	−38.8640	−1.24337	−0.621684	−	0.783268i	$-0.713551\pi$
$977$	−38.8640	−1.24337	−0.621684	+	0.783268i	$0.713551\pi$
$978$	1.63262	0.0522053
$979$	35.4146	1.13186
$980$	7.37858	0.235700
$981$	−10.2209	−0.326328
$982$	−2.73037	−0.0871296
$983$	−35.8844	−1.14454	−0.572268	−	0.820067i	$-0.693936\pi$
$983$	−35.8844	−1.14454	−0.572268	+	0.820067i	$0.693936\pi$
$984$	2.44182	0.0778423
$985$	3.76046	0.119818
$986$	−29.4507	−0.937902
$987$	−9.85594	−0.313718
$988$	4.58431	0.145846
$989$	10.1940	0.324151
$990$	4.41782	0.140408
$991$	13.2335	0.420377	0.210188	−	0.977661i	$-0.432592\pi$
$991$	13.2335	0.420377	0.210188	+	0.977661i	$0.432592\pi$
$992$	7.21512	0.229080
$993$	−7.69458	−0.244180
$994$	−16.2841	−0.516501
$995$	6.20251	0.196633
$996$	−7.71954	−0.244603
$997$	17.1205	0.542213	0.271106	−	0.962549i	$-0.412611\pi$
$997$	17.1205	0.542213	0.271106	+	0.962549i	$0.412611\pi$
$998$	33.3507	1.05570
$999$	2.49161	0.0788312

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.q.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.q.1.4	✓	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.50127 q^{5} +1.00000 q^{6} +1.44399 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.50127 q^{5} +1.00000 q^{6} +1.44399 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.50127 q^{10} -2.94272 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.44399 q^{14} -1.50127 q^{15} +1.00000 q^{16} +4.34586 q^{17} +1.00000 q^{18} -4.58431 q^{19} -1.50127 q^{20} +1.44399 q^{21} -2.94272 q^{22} -8.17758 q^{23} +1.00000 q^{24} -2.74619 q^{25} -1.00000 q^{26} +1.00000 q^{27} +1.44399 q^{28} -6.77674 q^{29} -1.50127 q^{30} +7.21512 q^{31} +1.00000 q^{32} -2.94272 q^{33} +4.34586 q^{34} -2.16782 q^{35} +1.00000 q^{36} +2.49161 q^{37} -4.58431 q^{38} -1.00000 q^{39} -1.50127 q^{40} +2.44182 q^{41} +1.44399 q^{42} -1.24658 q^{43} -2.94272 q^{44} -1.50127 q^{45} -8.17758 q^{46} -6.82549 q^{47} +1.00000 q^{48} -4.91489 q^{49} -2.74619 q^{50} +4.34586 q^{51} -1.00000 q^{52} +6.28858 q^{53} +1.00000 q^{54} +4.41782 q^{55} +1.44399 q^{56} -4.58431 q^{57} -6.77674 q^{58} +2.04358 q^{59} -1.50127 q^{60} +0.559740 q^{61} +7.21512 q^{62} +1.44399 q^{63} +1.00000 q^{64} +1.50127 q^{65} -2.94272 q^{66} +3.43251 q^{67} +4.34586 q^{68} -8.17758 q^{69} -2.16782 q^{70} -11.2772 q^{71} +1.00000 q^{72} -4.58425 q^{73} +2.49161 q^{74} -2.74619 q^{75} -4.58431 q^{76} -4.24926 q^{77} -1.00000 q^{78} -0.911183 q^{79} -1.50127 q^{80} +1.00000 q^{81} +2.44182 q^{82} -7.71954 q^{83} +1.44399 q^{84} -6.52431 q^{85} -1.24658 q^{86} -6.77674 q^{87} -2.94272 q^{88} -12.0347 q^{89} -1.50127 q^{90} -1.44399 q^{91} -8.17758 q^{92} +7.21512 q^{93} -6.82549 q^{94} +6.88230 q^{95} +1.00000 q^{96} -7.00623 q^{97} -4.91489 q^{98} -2.94272 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 6 q^{5} + 8 q^{6} - 3 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 6 * q^5 + 8 * q^6 - 3 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} - 6 q^{5} + 8 q^{6} - 3 q^{7} + 8 q^{8} + 8 q^{9} - 6 q^{10} - 15 q^{11} + 8 q^{12} - 8 q^{13} - 3 q^{14} - 6 q^{15} + 8 q^{16} - 11 q^{17} + 8 q^{18} - 15 q^{19} - 6 q^{20} - 3 q^{21} - 15 q^{22} + q^{23} + 8 q^{24} - 10 q^{25} - 8 q^{26} + 8 q^{27} - 3 q^{28} - 10 q^{29} - 6 q^{30} - 3 q^{31} + 8 q^{32} - 15 q^{33} - 11 q^{34} - 12 q^{35} + 8 q^{36} - 26 q^{37} - 15 q^{38} - 8 q^{39} - 6 q^{40} - 12 q^{41} - 3 q^{42} - 4 q^{43} - 15 q^{44} - 6 q^{45} + q^{46} - 6 q^{47} + 8 q^{48} - 5 q^{49} - 10 q^{50} - 11 q^{51} - 8 q^{52} - 4 q^{53} + 8 q^{54} - 3 q^{56} - 15 q^{57} - 10 q^{58} - 19 q^{59} - 6 q^{60} - 14 q^{61} - 3 q^{62} - 3 q^{63} + 8 q^{64} + 6 q^{65} - 15 q^{66} - 13 q^{67} - 11 q^{68} + q^{69} - 12 q^{70} - 31 q^{71} + 8 q^{72} - 27 q^{73} - 26 q^{74} - 10 q^{75} - 15 q^{76} - 30 q^{77} - 8 q^{78} - 13 q^{79} - 6 q^{80} + 8 q^{81} - 12 q^{82} - 28 q^{83} - 3 q^{84} + 15 q^{85} - 4 q^{86} - 10 q^{87} - 15 q^{88} - 2 q^{89} - 6 q^{90} + 3 q^{91} + q^{92} - 3 q^{93} - 6 q^{94} - 18 q^{95} + 8 q^{96} - 30 q^{97} - 5 q^{98} - 15 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 - 6 * q^5 + 8 * q^6 - 3 * q^7 + 8 * q^8 + 8 * q^9 - 6 * q^10 - 15 * q^11 + 8 * q^12 - 8 * q^13 - 3 * q^14 - 6 * q^15 + 8 * q^16 - 11 * q^17 + 8 * q^18 - 15 * q^19 - 6 * q^20 - 3 * q^21 - 15 * q^22 + q^23 + 8 * q^24 - 10 * q^25 - 8 * q^26 + 8 * q^27 - 3 * q^28 - 10 * q^29 - 6 * q^30 - 3 * q^31 + 8 * q^32 - 15 * q^33 - 11 * q^34 - 12 * q^35 + 8 * q^36 - 26 * q^37 - 15 * q^38 - 8 * q^39 - 6 * q^40 - 12 * q^41 - 3 * q^42 - 4 * q^43 - 15 * q^44 - 6 * q^45 + q^46 - 6 * q^47 + 8 * q^48 - 5 * q^49 - 10 * q^50 - 11 * q^51 - 8 * q^52 - 4 * q^53 + 8 * q^54 - 3 * q^56 - 15 * q^57 - 10 * q^58 - 19 * q^59 - 6 * q^60 - 14 * q^61 - 3 * q^62 - 3 * q^63 + 8 * q^64 + 6 * q^65 - 15 * q^66 - 13 * q^67 - 11 * q^68 + q^69 - 12 * q^70 - 31 * q^71 + 8 * q^72 - 27 * q^73 - 26 * q^74 - 10 * q^75 - 15 * q^76 - 30 * q^77 - 8 * q^78 - 13 * q^79 - 6 * q^80 + 8 * q^81 - 12 * q^82 - 28 * q^83 - 3 * q^84 + 15 * q^85 - 4 * q^86 - 10 * q^87 - 15 * q^88 - 2 * q^89 - 6 * q^90 + 3 * q^91 + q^92 - 3 * q^93 - 6 * q^94 - 18 * q^95 + 8 * q^96 - 30 * q^97 - 5 * q^98 - 15 * q^99

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