LMFDB - Embedded newform 4134.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4134,2,Mod(1,4134)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4134, 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("4134.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4134 = 2 \cdot 3 \cdot 13 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4134.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:	$33.0101561956$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 21x^{6} + 43x^{5} + 96x^{4} - 235x^{3} + 136x^{2} - 8x - 8 $ x^8 - 2x^7 - 21x^6 + 43x^5 + 96x^4 - 235x^3 + 136x^2 - 8*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.37280$ of defining polynomial
Character	$\chi$	$=$	4134.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.37280 q^{5} +1.00000 q^{6} -2.39442 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.37280 q^{5} +1.00000 q^{6} -2.39442 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.37280 q^{10} -1.38948 q^{11} +1.00000 q^{12} -1.00000 q^{13} -2.39442 q^{14} -3.37280 q^{15} +1.00000 q^{16} -7.03761 q^{17} +1.00000 q^{18} +6.87878 q^{19} -3.37280 q^{20} -2.39442 q^{21} -1.38948 q^{22} -0.863362 q^{23} +1.00000 q^{24} +6.37577 q^{25} -1.00000 q^{26} +1.00000 q^{27} -2.39442 q^{28} +6.51660 q^{29} -3.37280 q^{30} +4.73018 q^{31} +1.00000 q^{32} -1.38948 q^{33} -7.03761 q^{34} +8.07591 q^{35} +1.00000 q^{36} +5.79249 q^{37} +6.87878 q^{38} -1.00000 q^{39} -3.37280 q^{40} +2.85180 q^{41} -2.39442 q^{42} +5.25630 q^{43} -1.38948 q^{44} -3.37280 q^{45} -0.863362 q^{46} -5.32245 q^{47} +1.00000 q^{48} -1.26674 q^{49} +6.37577 q^{50} -7.03761 q^{51} -1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} +4.68643 q^{55} -2.39442 q^{56} +6.87878 q^{57} +6.51660 q^{58} +10.1783 q^{59} -3.37280 q^{60} +10.2797 q^{61} +4.73018 q^{62} -2.39442 q^{63} +1.00000 q^{64} +3.37280 q^{65} -1.38948 q^{66} +3.49739 q^{67} -7.03761 q^{68} -0.863362 q^{69} +8.07591 q^{70} -13.9245 q^{71} +1.00000 q^{72} +5.42613 q^{73} +5.79249 q^{74} +6.37577 q^{75} +6.87878 q^{76} +3.32700 q^{77} -1.00000 q^{78} +9.25522 q^{79} -3.37280 q^{80} +1.00000 q^{81} +2.85180 q^{82} +13.1597 q^{83} -2.39442 q^{84} +23.7364 q^{85} +5.25630 q^{86} +6.51660 q^{87} -1.38948 q^{88} +9.86398 q^{89} -3.37280 q^{90} +2.39442 q^{91} -0.863362 q^{92} +4.73018 q^{93} -5.32245 q^{94} -23.2007 q^{95} +1.00000 q^{96} +4.70083 q^{97} -1.26674 q^{98} -1.38948 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 2 q^{5} + 8 q^{6} + 7 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 2 * q^5 + 8 * q^6 + 7 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 2 q^{5} + 8 q^{6} + 7 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 7 q^{11} + 8 q^{12} - 8 q^{13} + 7 q^{14} + 2 q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + 11 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 5 q^{23} + 8 q^{24} + 6 q^{25} - 8 q^{26} + 8 q^{27} + 7 q^{28} + 13 q^{29} + 2 q^{30} + 12 q^{31} + 8 q^{32} + 7 q^{33} + 8 q^{34} + 14 q^{35} + 8 q^{36} + 10 q^{37} + 11 q^{38} - 8 q^{39} + 2 q^{40} + 19 q^{41} + 7 q^{42} + 10 q^{43} + 7 q^{44} + 2 q^{45} + 5 q^{46} + 6 q^{47} + 8 q^{48} + 15 q^{49} + 6 q^{50} + 8 q^{51} - 8 q^{52} + 8 q^{53} + 8 q^{54} + 5 q^{55} + 7 q^{56} + 11 q^{57} + 13 q^{58} + 11 q^{59} + 2 q^{60} + 6 q^{61} + 12 q^{62} + 7 q^{63} + 8 q^{64} - 2 q^{65} + 7 q^{66} + 5 q^{67} + 8 q^{68} + 5 q^{69} + 14 q^{70} - 6 q^{71} + 8 q^{72} + 18 q^{73} + 10 q^{74} + 6 q^{75} + 11 q^{76} - 8 q^{77} - 8 q^{78} + 17 q^{79} + 2 q^{80} + 8 q^{81} + 19 q^{82} + 16 q^{83} + 7 q^{84} + 25 q^{85} + 10 q^{86} + 13 q^{87} + 7 q^{88} - 8 q^{89} + 2 q^{90} - 7 q^{91} + 5 q^{92} + 12 q^{93} + 6 q^{94} - 30 q^{95} + 8 q^{96} + 11 q^{97} + 15 q^{98} + 7 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 2 * q^5 + 8 * q^6 + 7 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 7 * q^11 + 8 * q^12 - 8 * q^13 + 7 * q^14 + 2 * q^15 + 8 * q^16 + 8 * q^17 + 8 * q^18 + 11 * q^19 + 2 * q^20 + 7 * q^21 + 7 * q^22 + 5 * q^23 + 8 * q^24 + 6 * q^25 - 8 * q^26 + 8 * q^27 + 7 * q^28 + 13 * q^29 + 2 * q^30 + 12 * q^31 + 8 * q^32 + 7 * q^33 + 8 * q^34 + 14 * q^35 + 8 * q^36 + 10 * q^37 + 11 * q^38 - 8 * q^39 + 2 * q^40 + 19 * q^41 + 7 * q^42 + 10 * q^43 + 7 * q^44 + 2 * q^45 + 5 * q^46 + 6 * q^47 + 8 * q^48 + 15 * q^49 + 6 * q^50 + 8 * q^51 - 8 * q^52 + 8 * q^53 + 8 * q^54 + 5 * q^55 + 7 * q^56 + 11 * q^57 + 13 * q^58 + 11 * q^59 + 2 * q^60 + 6 * q^61 + 12 * q^62 + 7 * q^63 + 8 * q^64 - 2 * q^65 + 7 * q^66 + 5 * q^67 + 8 * q^68 + 5 * q^69 + 14 * q^70 - 6 * q^71 + 8 * q^72 + 18 * q^73 + 10 * q^74 + 6 * q^75 + 11 * q^76 - 8 * q^77 - 8 * q^78 + 17 * q^79 + 2 * q^80 + 8 * q^81 + 19 * q^82 + 16 * q^83 + 7 * q^84 + 25 * q^85 + 10 * q^86 + 13 * q^87 + 7 * q^88 - 8 * q^89 + 2 * q^90 - 7 * q^91 + 5 * q^92 + 12 * q^93 + 6 * q^94 - 30 * q^95 + 8 * q^96 + 11 * q^97 + 15 * q^98 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−3.37280	−1.50836	−0.754181	−	0.656667i	$-0.771966\pi$
$5$	−3.37280	−1.50836	−0.754181	+	0.656667i	$0.771966\pi$
$6$	1.00000	0.408248
$7$	−2.39442	−0.905007	−0.452503	−	0.891763i	$-0.649469\pi$
$7$	−2.39442	−0.905007	−0.452503	+	0.891763i	$0.649469\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−3.37280	−1.06657
$11$	−1.38948	−0.418943	−0.209472	−	0.977815i	$-0.567174\pi$
$11$	−1.38948	−0.418943	−0.209472	+	0.977815i	$0.567174\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.39442	−0.639937
$15$	−3.37280	−0.870853
$16$	1.00000	0.250000
$17$	−7.03761	−1.70687	−0.853435	−	0.521199i	$-0.825485\pi$
$17$	−7.03761	−1.70687	−0.853435	+	0.521199i	$0.825485\pi$
$18$	1.00000	0.235702
$19$	6.87878	1.57810	0.789050	−	0.614329i	$-0.210573\pi$
$19$	6.87878	1.57810	0.789050	+	0.614329i	$0.210573\pi$
$20$	−3.37280	−0.754181
$21$	−2.39442	−0.522506
$22$	−1.38948	−0.296238
$23$	−0.863362	−0.180023	−0.0900117	−	0.995941i	$-0.528690\pi$
$23$	−0.863362	−0.180023	−0.0900117	+	0.995941i	$0.528690\pi$
$24$	1.00000	0.204124
$25$	6.37577	1.27515
$26$	−1.00000	−0.196116
$27$	1.00000	0.192450
$28$	−2.39442	−0.452503
$29$	6.51660	1.21010	0.605051	−	0.796186i	$-0.293153\pi$
$29$	6.51660	1.21010	0.605051	+	0.796186i	$0.293153\pi$
$30$	−3.37280	−0.615786
$31$	4.73018	0.849566	0.424783	−	0.905295i	$-0.360350\pi$
$31$	4.73018	0.849566	0.424783	+	0.905295i	$0.360350\pi$
$32$	1.00000	0.176777
$33$	−1.38948	−0.241877
$34$	−7.03761	−1.20694
$35$	8.07591	1.36508
$36$	1.00000	0.166667
$37$	5.79249	0.952280	0.476140	−	0.879369i	$-0.342036\pi$
$37$	5.79249	0.952280	0.476140	+	0.879369i	$0.342036\pi$
$38$	6.87878	1.11589
$39$	−1.00000	−0.160128
$40$	−3.37280	−0.533286
$41$	2.85180	0.445376	0.222688	−	0.974890i	$-0.428517\pi$
$41$	2.85180	0.445376	0.222688	+	0.974890i	$0.428517\pi$
$42$	−2.39442	−0.369468
$43$	5.25630	0.801578	0.400789	−	0.916170i	$-0.368736\pi$
$43$	5.25630	0.801578	0.400789	+	0.916170i	$0.368736\pi$
$44$	−1.38948	−0.209472
$45$	−3.37280	−0.502787
$46$	−0.863362	−0.127296
$47$	−5.32245	−0.776359	−0.388179	−	0.921584i	$-0.626896\pi$
$47$	−5.32245	−0.776359	−0.388179	+	0.921584i	$0.626896\pi$
$48$	1.00000	0.144338
$49$	−1.26674	−0.180962
$50$	6.37577	0.901671
$51$	−7.03761	−0.985462
$52$	−1.00000	−0.138675
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	1.00000	0.136083
$55$	4.68643	0.631918
$56$	−2.39442	−0.319968
$57$	6.87878	0.911116
$58$	6.51660	0.855672
$59$	10.1783	1.32511	0.662553	−	0.749015i	$-0.269473\pi$
$59$	10.1783	1.32511	0.662553	+	0.749015i	$0.269473\pi$
$60$	−3.37280	−0.435426
$61$	10.2797	1.31619	0.658093	−	0.752937i	$-0.271363\pi$
$61$	10.2797	1.31619	0.658093	+	0.752937i	$0.271363\pi$
$62$	4.73018	0.600734
$63$	−2.39442	−0.301669
$64$	1.00000	0.125000
$65$	3.37280	0.418344
$66$	−1.38948	−0.171033
$67$	3.49739	0.427274	0.213637	−	0.976913i	$-0.431469\pi$
$67$	3.49739	0.427274	0.213637	+	0.976913i	$0.431469\pi$
$68$	−7.03761	−0.853435
$69$	−0.863362	−0.103937
$70$	8.07591	0.965256
$71$	−13.9245	−1.65253	−0.826265	−	0.563281i	$-0.809539\pi$
$71$	−13.9245	−1.65253	−0.826265	+	0.563281i	$0.809539\pi$
$72$	1.00000	0.117851
$73$	5.42613	0.635080	0.317540	−	0.948245i	$-0.397143\pi$
$73$	5.42613	0.635080	0.317540	+	0.948245i	$0.397143\pi$
$74$	5.79249	0.673364
$75$	6.37577	0.736211
$76$	6.87878	0.789050
$77$	3.32700	0.379147
$78$	−1.00000	−0.113228
$79$	9.25522	1.04129	0.520647	−	0.853772i	$-0.325691\pi$
$79$	9.25522	1.04129	0.520647	+	0.853772i	$0.325691\pi$
$80$	−3.37280	−0.377090
$81$	1.00000	0.111111
$82$	2.85180	0.314928
$83$	13.1597	1.44446	0.722231	−	0.691652i	$-0.243117\pi$
$83$	13.1597	1.44446	0.722231	+	0.691652i	$0.243117\pi$
$84$	−2.39442	−0.261253
$85$	23.7364	2.57458
$86$	5.25630	0.566801
$87$	6.51660	0.698653
$88$	−1.38948	−0.148119
$89$	9.86398	1.04558	0.522790	−	0.852462i	$-0.324891\pi$
$89$	9.86398	1.04558	0.522790	+	0.852462i	$0.324891\pi$
$90$	−3.37280	−0.355524
$91$	2.39442	0.251004
$92$	−0.863362	−0.0900117
$93$	4.73018	0.490497
$94$	−5.32245	−0.548968
$95$	−23.2007	−2.38035
$96$	1.00000	0.102062
$97$	4.70083	0.477297	0.238648	−	0.971106i	$-0.423296\pi$
$97$	4.70083	0.477297	0.238648	+	0.971106i	$0.423296\pi$
$98$	−1.26674	−0.127960
$99$	−1.38948	−0.139648
$100$	6.37577	0.637577
$101$	0.838031	0.0833872	0.0416936	−	0.999130i	$-0.486725\pi$
$101$	0.838031	0.0833872	0.0416936	+	0.999130i	$0.486725\pi$
$102$	−7.03761	−0.696827
$103$	−14.2040	−1.39956	−0.699779	−	0.714359i	$-0.746718\pi$
$103$	−14.2040	−1.39956	−0.699779	+	0.714359i	$0.746718\pi$
$104$	−1.00000	−0.0980581
$105$	8.07591	0.788128
$106$	1.00000	0.0971286
$107$	−12.3498	−1.19390	−0.596949	−	0.802279i	$-0.703621\pi$
$107$	−12.3498	−1.19390	−0.596949	+	0.802279i	$0.703621\pi$
$108$	1.00000	0.0962250
$109$	−8.29246	−0.794274	−0.397137	−	0.917759i	$-0.629996\pi$
$109$	−8.29246	−0.794274	−0.397137	+	0.917759i	$0.629996\pi$
$110$	4.68643	0.446834
$111$	5.79249	0.549799
$112$	−2.39442	−0.226252
$113$	−4.07034	−0.382905	−0.191453	−	0.981502i	$-0.561320\pi$
$113$	−4.07034	−0.382905	−0.191453	+	0.981502i	$0.561320\pi$
$114$	6.87878	0.644257
$115$	2.91195	0.271541
$116$	6.51660	0.605051
$117$	−1.00000	−0.0924500
$118$	10.1783	0.936991
$119$	16.8510	1.54473
$120$	−3.37280	−0.307893
$121$	−9.06935	−0.824486
$122$	10.2797	0.930684
$123$	2.85180	0.257138
$124$	4.73018	0.424783
$125$	−4.64021	−0.415033
$126$	−2.39442	−0.213312
$127$	−13.7606	−1.22106	−0.610530	−	0.791993i	$-0.709043\pi$
$127$	−13.7606	−1.22106	−0.610530	+	0.791993i	$0.709043\pi$
$128$	1.00000	0.0883883
$129$	5.25630	0.462791
$130$	3.37280	0.295814
$131$	4.81852	0.420996	0.210498	−	0.977594i	$-0.432491\pi$
$131$	4.81852	0.420996	0.210498	+	0.977594i	$0.432491\pi$
$132$	−1.38948	−0.120939
$133$	−16.4707	−1.42819
$134$	3.49739	0.302128
$135$	−3.37280	−0.290284
$136$	−7.03761	−0.603470
$137$	12.5550	1.07264	0.536321	−	0.844014i	$-0.319814\pi$
$137$	12.5550	1.07264	0.536321	+	0.844014i	$0.319814\pi$
$138$	−0.863362	−0.0734943
$139$	−0.373348	−0.0316669	−0.0158335	−	0.999875i	$-0.505040\pi$
$139$	−0.373348	−0.0316669	−0.0158335	+	0.999875i	$0.505040\pi$
$140$	8.07591	0.682539
$141$	−5.32245	−0.448231
$142$	−13.9245	−1.16852
$143$	1.38948	0.116194
$144$	1.00000	0.0833333
$145$	−21.9792	−1.82527
$146$	5.42613	0.449069
$147$	−1.26674	−0.104479
$148$	5.79249	0.476140
$149$	3.81906	0.312870	0.156435	−	0.987688i	$-0.450000\pi$
$149$	3.81906	0.312870	0.156435	+	0.987688i	$0.450000\pi$
$150$	6.37577	0.520580
$151$	−1.28574	−0.104632	−0.0523159	−	0.998631i	$-0.516660\pi$
$151$	−1.28574	−0.104632	−0.0523159	+	0.998631i	$0.516660\pi$
$152$	6.87878	0.557943
$153$	−7.03761	−0.568957
$154$	3.32700	0.268097
$155$	−15.9540	−1.28145
$156$	−1.00000	−0.0800641
$157$	3.78692	0.302229	0.151115	−	0.988516i	$-0.451714\pi$
$157$	3.78692	0.302229	0.151115	+	0.988516i	$0.451714\pi$
$158$	9.25522	0.736306
$159$	1.00000	0.0793052
$160$	−3.37280	−0.266643
$161$	2.06725	0.162923
$162$	1.00000	0.0785674
$163$	2.88241	0.225767	0.112884	−	0.993608i	$-0.463991\pi$
$163$	2.88241	0.225767	0.112884	+	0.993608i	$0.463991\pi$
$164$	2.85180	0.222688
$165$	4.68643	0.364838
$166$	13.1597	1.02139
$167$	5.21788	0.403772	0.201886	−	0.979409i	$-0.435293\pi$
$167$	5.21788	0.403772	0.201886	+	0.979409i	$0.435293\pi$
$168$	−2.39442	−0.184734
$169$	1.00000	0.0769231
$170$	23.7364	1.82050
$171$	6.87878	0.526033
$172$	5.25630	0.400789
$173$	−2.29305	−0.174338	−0.0871688	−	0.996194i	$-0.527782\pi$
$173$	−2.29305	−0.174338	−0.0871688	+	0.996194i	$0.527782\pi$
$174$	6.51660	0.494022
$175$	−15.2663	−1.15402
$176$	−1.38948	−0.104736
$177$	10.1783	0.765050
$178$	9.86398	0.739336
$179$	9.33950	0.698067	0.349034	−	0.937110i	$-0.386510\pi$
$179$	9.33950	0.698067	0.349034	+	0.937110i	$0.386510\pi$
$180$	−3.37280	−0.251394
$181$	1.19361	0.0887206	0.0443603	−	0.999016i	$-0.485875\pi$
$181$	1.19361	0.0887206	0.0443603	+	0.999016i	$0.485875\pi$
$182$	2.39442	0.177486
$183$	10.2797	0.759900
$184$	−0.863362	−0.0636479
$185$	−19.5369	−1.43638
$186$	4.73018	0.346834
$187$	9.77860	0.715082
$188$	−5.32245	−0.388179
$189$	−2.39442	−0.174169
$190$	−23.2007	−1.68316
$191$	−22.8247	−1.65154	−0.825769	−	0.564009i	$-0.809258\pi$
$191$	−22.8247	−1.65154	−0.825769	+	0.564009i	$0.809258\pi$
$192$	1.00000	0.0721688
$193$	23.9543	1.72427	0.862133	−	0.506681i	$-0.169128\pi$
$193$	23.9543	1.72427	0.862133	+	0.506681i	$0.169128\pi$
$194$	4.70083	0.337500
$195$	3.37280	0.241531
$196$	−1.26674	−0.0904812
$197$	10.5679	0.752934	0.376467	−	0.926430i	$-0.377139\pi$
$197$	10.5679	0.752934	0.376467	+	0.926430i	$0.377139\pi$
$198$	−1.38948	−0.0987459
$199$	18.2958	1.29695	0.648476	−	0.761235i	$-0.275407\pi$
$199$	18.2958	1.29695	0.648476	+	0.761235i	$0.275407\pi$
$200$	6.37577	0.450835
$201$	3.49739	0.246687
$202$	0.838031	0.0589636
$203$	−15.6035	−1.09515
$204$	−7.03761	−0.492731
$205$	−9.61854	−0.671788
$206$	−14.2040	−0.989638
$207$	−0.863362	−0.0600078
$208$	−1.00000	−0.0693375
$209$	−9.55791	−0.661135
$210$	8.07591	0.557291
$211$	17.7591	1.22259	0.611294	−	0.791404i	$-0.290649\pi$
$211$	17.7591	1.22259	0.611294	+	0.791404i	$0.290649\pi$
$212$	1.00000	0.0686803
$213$	−13.9245	−0.954089
$214$	−12.3498	−0.844214
$215$	−17.7284	−1.20907
$216$	1.00000	0.0680414
$217$	−11.3261	−0.768863
$218$	−8.29246	−0.561636
$219$	5.42613	0.366664
$220$	4.68643	0.315959
$221$	7.03761	0.473401
$222$	5.79249	0.388767
$223$	0.460517	0.0308385	0.0154192	−	0.999881i	$-0.495092\pi$
$223$	0.460517	0.0308385	0.0154192	+	0.999881i	$0.495092\pi$
$224$	−2.39442	−0.159984
$225$	6.37577	0.425052
$226$	−4.07034	−0.270755
$227$	17.1274	1.13679	0.568394	−	0.822756i	$-0.307565\pi$
$227$	17.1274	1.13679	0.568394	+	0.822756i	$0.307565\pi$
$228$	6.87878	0.455558
$229$	−7.33367	−0.484623	−0.242311	−	0.970199i	$-0.577906\pi$
$229$	−7.33367	−0.484623	−0.242311	+	0.970199i	$0.577906\pi$
$230$	2.91195	0.192008
$231$	3.32700	0.218900
$232$	6.51660	0.427836
$233$	−3.14374	−0.205953	−0.102977	−	0.994684i	$-0.532837\pi$
$233$	−3.14374	−0.205953	−0.102977	+	0.994684i	$0.532837\pi$
$234$	−1.00000	−0.0653720
$235$	17.9515	1.17103
$236$	10.1783	0.662553
$237$	9.25522	0.601191
$238$	16.8510	1.09229
$239$	10.2068	0.660225	0.330113	−	0.943942i	$-0.392913\pi$
$239$	10.2068	0.660225	0.330113	+	0.943942i	$0.392913\pi$
$240$	−3.37280	−0.217713
$241$	−7.05656	−0.454553	−0.227276	−	0.973830i	$-0.572982\pi$
$241$	−7.05656	−0.454553	−0.227276	+	0.973830i	$0.572982\pi$
$242$	−9.06935	−0.583000
$243$	1.00000	0.0641500
$244$	10.2797	0.658093
$245$	4.27245	0.272957
$246$	2.85180	0.181824
$247$	−6.87878	−0.437686
$248$	4.73018	0.300367
$249$	13.1597	0.833961
$250$	−4.64021	−0.293472
$251$	−0.0699600	−0.00441584	−0.00220792	−	0.999998i	$-0.500703\pi$
$251$	−0.0699600	−0.00441584	−0.00220792	+	0.999998i	$0.500703\pi$
$252$	−2.39442	−0.150834
$253$	1.19962	0.0754197
$254$	−13.7606	−0.863419
$255$	23.7364	1.48643
$256$	1.00000	0.0625000
$257$	−5.08811	−0.317388	−0.158694	−	0.987328i	$-0.550728\pi$
$257$	−5.08811	−0.317388	−0.158694	+	0.987328i	$0.550728\pi$
$258$	5.25630	0.327243
$259$	−13.8697	−0.861820
$260$	3.37280	0.209172
$261$	6.51660	0.403368
$262$	4.81852	0.297689
$263$	−16.8283	−1.03768	−0.518839	−	0.854872i	$-0.673636\pi$
$263$	−16.8283	−1.03768	−0.518839	+	0.854872i	$0.673636\pi$
$264$	−1.38948	−0.0855165
$265$	−3.37280	−0.207189
$266$	−16.4707	−1.00988
$267$	9.86398	0.603666
$268$	3.49739	0.213637
$269$	10.5684	0.644367	0.322184	−	0.946677i	$-0.395583\pi$
$269$	10.5684	0.644367	0.322184	+	0.946677i	$0.395583\pi$
$270$	−3.37280	−0.205262
$271$	6.55650	0.398279	0.199139	−	0.979971i	$-0.436185\pi$
$271$	6.55650	0.398279	0.199139	+	0.979971i	$0.436185\pi$
$272$	−7.03761	−0.426718
$273$	2.39442	0.144917
$274$	12.5550	0.758473
$275$	−8.85900	−0.534218
$276$	−0.863362	−0.0519683
$277$	−9.46563	−0.568734	−0.284367	−	0.958715i	$-0.591783\pi$
$277$	−9.46563	−0.568734	−0.284367	+	0.958715i	$0.591783\pi$
$278$	−0.373348	−0.0223919
$279$	4.73018	0.283189
$280$	8.07591	0.482628
$281$	−23.7050	−1.41412	−0.707062	−	0.707151i	$-0.749980\pi$
$281$	−23.7050	−1.41412	−0.707062	+	0.707151i	$0.749980\pi$
$282$	−5.32245	−0.316947
$283$	−13.8980	−0.826150	−0.413075	−	0.910697i	$-0.635545\pi$
$283$	−13.8980	−0.826150	−0.413075	+	0.910697i	$0.635545\pi$
$284$	−13.9245	−0.826265
$285$	−23.2007	−1.37429
$286$	1.38948	0.0821616
$287$	−6.82841	−0.403068
$288$	1.00000	0.0589256
$289$	32.5279	1.91341
$290$	−21.9792	−1.29066
$291$	4.70083	0.275568
$292$	5.42613	0.317540
$293$	−4.66646	−0.272618	−0.136309	−	0.990666i	$-0.543524\pi$
$293$	−4.66646	−0.272618	−0.136309	+	0.990666i	$0.543524\pi$
$294$	−1.26674	−0.0738776
$295$	−34.3294	−1.99874
$296$	5.79249	0.336682
$297$	−1.38948	−0.0806257
$298$	3.81906	0.221232
$299$	0.863362	0.0499295
$300$	6.37577	0.368105
$301$	−12.5858	−0.725434
$302$	−1.28574	−0.0739859
$303$	0.838031	0.0481436
$304$	6.87878	0.394525
$305$	−34.6715	−1.98528
$306$	−7.03761	−0.402313
$307$	−4.51687	−0.257792	−0.128896	−	0.991658i	$-0.541143\pi$
$307$	−4.51687	−0.257792	−0.128896	+	0.991658i	$0.541143\pi$
$308$	3.32700	0.189573
$309$	−14.2040	−0.808036
$310$	−15.9540	−0.906124
$311$	33.4085	1.89442	0.947210	−	0.320613i	$-0.103889\pi$
$311$	33.4085	1.89442	0.947210	+	0.320613i	$0.103889\pi$
$312$	−1.00000	−0.0566139
$313$	18.1041	1.02331	0.511653	−	0.859192i	$-0.329033\pi$
$313$	18.1041	1.02331	0.511653	+	0.859192i	$0.329033\pi$
$314$	3.78692	0.213708
$315$	8.07591	0.455026
$316$	9.25522	0.520647
$317$	4.00480	0.224932	0.112466	−	0.993656i	$-0.464125\pi$
$317$	4.00480	0.224932	0.112466	+	0.993656i	$0.464125\pi$
$318$	1.00000	0.0560772
$319$	−9.05468	−0.506965
$320$	−3.37280	−0.188545
$321$	−12.3498	−0.689298
$322$	2.06725	0.115204
$323$	−48.4101	−2.69361
$324$	1.00000	0.0555556
$325$	−6.37577	−0.353664
$326$	2.88241	0.159642
$327$	−8.29246	−0.458574
$328$	2.85180	0.157464
$329$	12.7442	0.702610
$330$	4.68643	0.257980
$331$	19.9515	1.09664	0.548318	−	0.836270i	$-0.315268\pi$
$331$	19.9515	1.09664	0.548318	+	0.836270i	$0.315268\pi$
$332$	13.1597	0.722231
$333$	5.79249	0.317427
$334$	5.21788	0.285510
$335$	−11.7960	−0.644484
$336$	−2.39442	−0.130627
$337$	−11.4068	−0.621371	−0.310685	−	0.950513i	$-0.600559\pi$
$337$	−11.4068	−0.621371	−0.310685	+	0.950513i	$0.600559\pi$
$338$	1.00000	0.0543928
$339$	−4.07034	−0.221070
$340$	23.7364	1.28729
$341$	−6.57248	−0.355920
$342$	6.87878	0.371962
$343$	19.7941	1.06878
$344$	5.25630	0.283401
$345$	2.91195	0.156774
$346$	−2.29305	−0.123275
$347$	−23.4359	−1.25810	−0.629052	−	0.777363i	$-0.716557\pi$
$347$	−23.4359	−1.25810	−0.629052	+	0.777363i	$0.716557\pi$
$348$	6.51660	0.349327
$349$	15.3427	0.821275	0.410637	−	0.911799i	$-0.365306\pi$
$349$	15.3427	0.821275	0.410637	+	0.911799i	$0.365306\pi$
$350$	−15.2663	−0.816018
$351$	−1.00000	−0.0533761
$352$	−1.38948	−0.0740594
$353$	3.68436	0.196099	0.0980493	−	0.995182i	$-0.468740\pi$
$353$	3.68436	0.196099	0.0980493	+	0.995182i	$0.468740\pi$
$354$	10.1783	0.540972
$355$	46.9644	2.49261
$356$	9.86398	0.522790
$357$	16.8510	0.891850
$358$	9.33950	0.493608
$359$	−29.6867	−1.56681	−0.783403	−	0.621514i	$-0.786518\pi$
$359$	−29.6867	−1.56681	−0.783403	+	0.621514i	$0.786518\pi$
$360$	−3.37280	−0.177762
$361$	28.3176	1.49040
$362$	1.19361	0.0627349
$363$	−9.06935	−0.476017
$364$	2.39442	0.125502
$365$	−18.3012	−0.957930
$366$	10.2797	0.537331
$367$	−4.90051	−0.255805	−0.127902	−	0.991787i	$-0.540824\pi$
$367$	−4.90051	−0.255805	−0.127902	+	0.991787i	$0.540824\pi$
$368$	−0.863362	−0.0450059
$369$	2.85180	0.148459
$370$	−19.5369	−1.01568
$371$	−2.39442	−0.124312
$372$	4.73018	0.245249
$373$	−11.3211	−0.586185	−0.293092	−	0.956084i	$-0.594684\pi$
$373$	−11.3211	−0.586185	−0.293092	+	0.956084i	$0.594684\pi$
$374$	9.77860	0.505639
$375$	−4.64021	−0.239619
$376$	−5.32245	−0.274484
$377$	−6.51660	−0.335622
$378$	−2.39442	−0.123156
$379$	−8.38734	−0.430829	−0.215414	−	0.976523i	$-0.569110\pi$
$379$	−8.38734	−0.430829	−0.215414	+	0.976523i	$0.569110\pi$
$380$	−23.2007	−1.19017
$381$	−13.7606	−0.704979
$382$	−22.8247	−1.16781
$383$	19.2692	0.984608	0.492304	−	0.870423i	$-0.336155\pi$
$383$	19.2692	0.984608	0.492304	+	0.870423i	$0.336155\pi$
$384$	1.00000	0.0510310
$385$	−11.2213	−0.571890
$386$	23.9543	1.21924
$387$	5.25630	0.267193
$388$	4.70083	0.238648
$389$	14.0545	0.712594	0.356297	−	0.934373i	$-0.384039\pi$
$389$	14.0545	0.712594	0.356297	+	0.934373i	$0.384039\pi$
$390$	3.37280	0.170788
$391$	6.07600	0.307277
$392$	−1.26674	−0.0639799
$393$	4.81852	0.243062
$394$	10.5679	0.532405
$395$	−31.2160	−1.57065
$396$	−1.38948	−0.0698239
$397$	5.84217	0.293210	0.146605	−	0.989195i	$-0.453165\pi$
$397$	5.84217	0.293210	0.146605	+	0.989195i	$0.453165\pi$
$398$	18.2958	0.917084
$399$	−16.4707	−0.824567
$400$	6.37577	0.318789
$401$	−6.79705	−0.339428	−0.169714	−	0.985493i	$-0.554284\pi$
$401$	−6.79705	−0.339428	−0.169714	+	0.985493i	$0.554284\pi$
$402$	3.49739	0.174434
$403$	−4.73018	−0.235627
$404$	0.838031	0.0416936
$405$	−3.37280	−0.167596
$406$	−15.6035	−0.774389
$407$	−8.04854	−0.398951
$408$	−7.03761	−0.348413
$409$	14.1699	0.700659	0.350329	−	0.936627i	$-0.386070\pi$
$409$	14.1699	0.700659	0.350329	+	0.936627i	$0.386070\pi$
$410$	−9.61854	−0.475026
$411$	12.5550	0.619291
$412$	−14.2040	−0.699779
$413$	−24.3712	−1.19923
$414$	−0.863362	−0.0424319
$415$	−44.3849	−2.17877
$416$	−1.00000	−0.0490290
$417$	−0.373348	−0.0182829
$418$	−9.55791	−0.467493
$419$	28.5589	1.39519	0.697597	−	0.716490i	$-0.254253\pi$
$419$	28.5589	1.39519	0.697597	+	0.716490i	$0.254253\pi$
$420$	8.07591	0.394064
$421$	−17.4979	−0.852798	−0.426399	−	0.904535i	$-0.640218\pi$
$421$	−17.4979	−0.852798	−0.426399	+	0.904535i	$0.640218\pi$
$422$	17.7591	0.864500
$423$	−5.32245	−0.258786
$424$	1.00000	0.0485643
$425$	−44.8702	−2.17652
$426$	−13.9245	−0.674643
$427$	−24.6141	−1.19116
$428$	−12.3498	−0.596949
$429$	1.38948	0.0670846
$430$	−17.7284	−0.854941
$431$	−31.1734	−1.50157	−0.750784	−	0.660548i	$-0.770324\pi$
$431$	−31.1734	−1.50157	−0.750784	+	0.660548i	$0.770324\pi$
$432$	1.00000	0.0481125
$433$	−21.0657	−1.01235	−0.506176	−	0.862430i	$-0.668942\pi$
$433$	−21.0657	−1.01235	−0.506176	+	0.862430i	$0.668942\pi$
$434$	−11.3261	−0.543668
$435$	−21.9792	−1.05382
$436$	−8.29246	−0.397137
$437$	−5.93888	−0.284095
$438$	5.42613	0.259270
$439$	19.1928	0.916022	0.458011	−	0.888947i	$-0.348562\pi$
$439$	19.1928	0.916022	0.458011	+	0.888947i	$0.348562\pi$
$440$	4.68643	0.223417
$441$	−1.26674	−0.0603208
$442$	7.03761	0.334745
$443$	−24.6912	−1.17311	−0.586557	−	0.809908i	$-0.699517\pi$
$443$	−24.6912	−1.17311	−0.586557	+	0.809908i	$0.699517\pi$
$444$	5.79249	0.274900
$445$	−33.2692	−1.57711
$446$	0.460517	0.0218061
$447$	3.81906	0.180635
$448$	−2.39442	−0.113126
$449$	29.0104	1.36909	0.684543	−	0.728972i	$-0.260002\pi$
$449$	29.0104	1.36909	0.684543	+	0.728972i	$0.260002\pi$
$450$	6.37577	0.300557
$451$	−3.96251	−0.186587
$452$	−4.07034	−0.191453
$453$	−1.28574	−0.0604093
$454$	17.1274	0.803831
$455$	−8.07591	−0.378604
$456$	6.87878	0.322128
$457$	−6.82621	−0.319317	−0.159658	−	0.987172i	$-0.551039\pi$
$457$	−6.82621	−0.319317	−0.159658	+	0.987172i	$0.551039\pi$
$458$	−7.33367	−0.342680
$459$	−7.03761	−0.328487
$460$	2.91195	0.135770
$461$	18.2443	0.849722	0.424861	−	0.905259i	$-0.360323\pi$
$461$	18.2443	0.849722	0.424861	+	0.905259i	$0.360323\pi$
$462$	3.32700	0.154786
$463$	6.89121	0.320262	0.160131	−	0.987096i	$-0.448808\pi$
$463$	6.89121	0.320262	0.160131	+	0.987096i	$0.448808\pi$
$464$	6.51660	0.302526
$465$	−15.9540	−0.739847
$466$	−3.14374	−0.145631
$467$	−5.30516	−0.245494	−0.122747	−	0.992438i	$-0.539170\pi$
$467$	−5.30516	−0.245494	−0.122747	+	0.992438i	$0.539170\pi$
$468$	−1.00000	−0.0462250
$469$	−8.37423	−0.386686
$470$	17.9515	0.828043
$471$	3.78692	0.174492
$472$	10.1783	0.468495
$473$	−7.30351	−0.335816
$474$	9.25522	0.425107
$475$	43.8575	2.01232
$476$	16.8510	0.772365
$477$	1.00000	0.0457869
$478$	10.2068	0.466850
$479$	−26.0306	−1.18937	−0.594685	−	0.803959i	$-0.702723\pi$
$479$	−26.0306	−1.18937	−0.594685	+	0.803959i	$0.702723\pi$
$480$	−3.37280	−0.153947
$481$	−5.79249	−0.264115
$482$	−7.05656	−0.321417
$483$	2.06725	0.0940634
$484$	−9.06935	−0.412243
$485$	−15.8550	−0.719936
$486$	1.00000	0.0453609
$487$	12.0053	0.544013	0.272006	−	0.962295i	$-0.412313\pi$
$487$	12.0053	0.544013	0.272006	+	0.962295i	$0.412313\pi$
$488$	10.2797	0.465342
$489$	2.88241	0.130347
$490$	4.27245	0.193010
$491$	−11.6935	−0.527719	−0.263860	−	0.964561i	$-0.584996\pi$
$491$	−11.6935	−0.527719	−0.263860	+	0.964561i	$0.584996\pi$
$492$	2.85180	0.128569
$493$	−45.8613	−2.06549
$494$	−6.87878	−0.309491
$495$	4.68643	0.210639
$496$	4.73018	0.212391
$497$	33.3411	1.49555
$498$	13.1597	0.589699
$499$	10.9629	0.490765	0.245382	−	0.969426i	$-0.421087\pi$
$499$	10.9629	0.490765	0.245382	+	0.969426i	$0.421087\pi$
$500$	−4.64021	−0.207516
$501$	5.21788	0.233118
$502$	−0.0699600	−0.00312247
$503$	−3.58735	−0.159952	−0.0799759	−	0.996797i	$-0.525484\pi$
$503$	−3.58735	−0.159952	−0.0799759	+	0.996797i	$0.525484\pi$
$504$	−2.39442	−0.106656
$505$	−2.82651	−0.125778
$506$	1.19962	0.0533298
$507$	1.00000	0.0444116
$508$	−13.7606	−0.610530
$509$	25.9612	1.15071	0.575356	−	0.817903i	$-0.304864\pi$
$509$	25.9612	1.15071	0.575356	+	0.817903i	$0.304864\pi$
$510$	23.7364	1.05107
$511$	−12.9924	−0.574752
$512$	1.00000	0.0441942
$513$	6.87878	0.303705
$514$	−5.08811	−0.224427
$515$	47.9071	2.11104
$516$	5.25630	0.231396
$517$	7.39542	0.325250
$518$	−13.8697	−0.609399
$519$	−2.29305	−0.100654
$520$	3.37280	0.147907
$521$	16.5316	0.724261	0.362130	−	0.932127i	$-0.382050\pi$
$521$	16.5316	0.724261	0.362130	+	0.932127i	$0.382050\pi$
$522$	6.51660	0.285224
$523$	11.1090	0.485762	0.242881	−	0.970056i	$-0.421907\pi$
$523$	11.1090	0.485762	0.242881	+	0.970056i	$0.421907\pi$
$524$	4.81852	0.210498
$525$	−15.2663	−0.666276
$526$	−16.8283	−0.733749
$527$	−33.2892	−1.45010
$528$	−1.38948	−0.0604693
$529$	−22.2546	−0.967592
$530$	−3.37280	−0.146505
$531$	10.1783	0.441702
$532$	−16.4707	−0.714096
$533$	−2.85180	−0.123525
$534$	9.86398	0.426856
$535$	41.6533	1.80083
$536$	3.49739	0.151064
$537$	9.33950	0.403029
$538$	10.5684	0.455637
$539$	1.76010	0.0758130
$540$	−3.37280	−0.145142
$541$	7.38394	0.317460	0.158730	−	0.987322i	$-0.449260\pi$
$541$	7.38394	0.317460	0.158730	+	0.987322i	$0.449260\pi$
$542$	6.55650	0.281626
$543$	1.19361	0.0512229
$544$	−7.03761	−0.301735
$545$	27.9688	1.19805
$546$	2.39442	0.102472
$547$	−1.90777	−0.0815704	−0.0407852	−	0.999168i	$-0.512986\pi$
$547$	−1.90777	−0.0815704	−0.0407852	+	0.999168i	$0.512986\pi$
$548$	12.5550	0.536321
$549$	10.2797	0.438729
$550$	−8.85900	−0.377749
$551$	44.8263	1.90966
$552$	−0.863362	−0.0367471
$553$	−22.1609	−0.942378
$554$	−9.46563	−0.402156
$555$	−19.5369	−0.829296
$556$	−0.373348	−0.0158335
$557$	−30.3719	−1.28690	−0.643450	−	0.765488i	$-0.722497\pi$
$557$	−30.3719	−1.28690	−0.643450	+	0.765488i	$0.722497\pi$
$558$	4.73018	0.200245
$559$	−5.25630	−0.222318
$560$	8.07591	0.341269
$561$	9.77860	0.412853
$562$	−23.7050	−0.999937
$563$	30.5608	1.28798	0.643992	−	0.765032i	$-0.277277\pi$
$563$	30.5608	1.28798	0.643992	+	0.765032i	$0.277277\pi$
$564$	−5.32245	−0.224115
$565$	13.7284	0.577560
$566$	−13.8980	−0.584176
$567$	−2.39442	−0.100556
$568$	−13.9245	−0.584258
$569$	30.4098	1.27485	0.637423	−	0.770514i	$-0.280000\pi$
$569$	30.4098	1.27485	0.637423	+	0.770514i	$0.280000\pi$
$570$	−23.2007	−0.971772
$571$	−23.0558	−0.964855	−0.482428	−	0.875936i	$-0.660245\pi$
$571$	−23.0558	−0.964855	−0.482428	+	0.875936i	$0.660245\pi$
$572$	1.38948	0.0580970
$573$	−22.8247	−0.953516
$574$	−6.82841	−0.285012
$575$	−5.50460	−0.229558
$576$	1.00000	0.0416667
$577$	−4.81242	−0.200344	−0.100172	−	0.994970i	$-0.531939\pi$
$577$	−4.81242	−0.200344	−0.100172	+	0.994970i	$0.531939\pi$
$578$	32.5279	1.35298
$579$	23.9543	0.995506
$580$	−21.9792	−0.912636
$581$	−31.5098	−1.30725
$582$	4.70083	0.194856
$583$	−1.38948	−0.0575463
$584$	5.42613	0.224535
$585$	3.37280	0.139448
$586$	−4.66646	−0.192770
$587$	−38.2565	−1.57901	−0.789507	−	0.613741i	$-0.789664\pi$
$587$	−38.2565	−1.57901	−0.789507	+	0.613741i	$0.789664\pi$
$588$	−1.26674	−0.0522393
$589$	32.5379	1.34070
$590$	−34.3294	−1.41332
$591$	10.5679	0.434707
$592$	5.79249	0.238070
$593$	−31.5579	−1.29593	−0.647963	−	0.761672i	$-0.724379\pi$
$593$	−31.5579	−1.29593	−0.647963	+	0.761672i	$0.724379\pi$
$594$	−1.38948	−0.0570110
$595$	−56.8351	−2.33001
$596$	3.81906	0.156435
$597$	18.2958	0.748796
$598$	0.863362	0.0353055
$599$	45.6725	1.86613	0.933065	−	0.359709i	$-0.117124\pi$
$599$	45.6725	1.86613	0.933065	+	0.359709i	$0.117124\pi$
$600$	6.37577	0.260290
$601$	9.18154	0.374523	0.187261	−	0.982310i	$-0.440039\pi$
$601$	9.18154	0.374523	0.187261	+	0.982310i	$0.440039\pi$
$602$	−12.5858	−0.512959
$603$	3.49739	0.142425
$604$	−1.28574	−0.0523159
$605$	30.5891	1.24362
$606$	0.838031	0.0340427
$607$	41.7501	1.69458	0.847292	−	0.531128i	$-0.178232\pi$
$607$	41.7501	1.69458	0.847292	+	0.531128i	$0.178232\pi$
$608$	6.87878	0.278971
$609$	−15.6035	−0.632286
$610$	−34.6715	−1.40381
$611$	5.32245	0.215323
$612$	−7.03761	−0.284478
$613$	−23.2139	−0.937602	−0.468801	−	0.883304i	$-0.655314\pi$
$613$	−23.2139	−0.937602	−0.468801	+	0.883304i	$0.655314\pi$
$614$	−4.51687	−0.182286
$615$	−9.61854	−0.387857
$616$	3.32700	0.134049
$617$	13.4037	0.539615	0.269807	−	0.962914i	$-0.413040\pi$
$617$	13.4037	0.539615	0.269807	+	0.962914i	$0.413040\pi$
$618$	−14.2040	−0.571368
$619$	−39.4819	−1.58691	−0.793456	−	0.608627i	$-0.791720\pi$
$619$	−39.4819	−1.58691	−0.793456	+	0.608627i	$0.791720\pi$
$620$	−15.9540	−0.640726
$621$	−0.863362	−0.0346455
$622$	33.4085	1.33956
$623$	−23.6185	−0.946257
$624$	−1.00000	−0.0400320
$625$	−16.2284	−0.649135
$626$	18.1041	0.723586
$627$	−9.55791	−0.381706
$628$	3.78692	0.151115
$629$	−40.7653	−1.62542
$630$	8.07591	0.321752
$631$	45.8505	1.82528	0.912640	−	0.408764i	$-0.134040\pi$
$631$	45.8505	1.82528	0.912640	+	0.408764i	$0.134040\pi$
$632$	9.25522	0.368153
$633$	17.7591	0.705861
$634$	4.00480	0.159051
$635$	46.4119	1.84180
$636$	1.00000	0.0396526
$637$	1.26674	0.0501899
$638$	−9.05468	−0.358478
$639$	−13.9245	−0.550844
$640$	−3.37280	−0.133322
$641$	−12.3841	−0.489143	−0.244572	−	0.969631i	$-0.578647\pi$
$641$	−12.3841	−0.489143	−0.244572	+	0.969631i	$0.578647\pi$
$642$	−12.3498	−0.487407
$643$	29.9669	1.18178	0.590890	−	0.806752i	$-0.298777\pi$
$643$	29.9669	1.18178	0.590890	+	0.806752i	$0.298777\pi$
$644$	2.06725	0.0814613
$645$	−17.7284	−0.698056
$646$	−48.4101	−1.90467
$647$	−35.5131	−1.39616	−0.698082	−	0.716018i	$-0.745963\pi$
$647$	−35.5131	−1.39616	−0.698082	+	0.716018i	$0.745963\pi$
$648$	1.00000	0.0392837
$649$	−14.1426	−0.555144
$650$	−6.37577	−0.250078
$651$	−11.3261	−0.443903
$652$	2.88241	0.112884
$653$	−18.8967	−0.739487	−0.369743	−	0.929134i	$-0.620554\pi$
$653$	−18.8967	−0.739487	−0.369743	+	0.929134i	$0.620554\pi$
$654$	−8.29246	−0.324261
$655$	−16.2519	−0.635014
$656$	2.85180	0.111344
$657$	5.42613	0.211693
$658$	12.7442	0.496820
$659$	−2.26269	−0.0881419	−0.0440709	−	0.999028i	$-0.514033\pi$
$659$	−2.26269	−0.0881419	−0.0440709	+	0.999028i	$0.514033\pi$
$660$	4.68643	0.182419
$661$	−12.1525	−0.472678	−0.236339	−	0.971671i	$-0.575948\pi$
$661$	−12.1525	−0.472678	−0.236339	+	0.971671i	$0.575948\pi$
$662$	19.9515	0.775439
$663$	7.03761	0.273318
$664$	13.1597	0.510694
$665$	55.5524	2.15423
$666$	5.79249	0.224455
$667$	−5.62619	−0.217847
$668$	5.21788	0.201886
$669$	0.460517	0.0178046
$670$	−11.7960	−0.455719
$671$	−14.2835	−0.551408
$672$	−2.39442	−0.0923669
$673$	37.6399	1.45091	0.725455	−	0.688270i	$-0.241629\pi$
$673$	37.6399	1.45091	0.725455	+	0.688270i	$0.241629\pi$
$674$	−11.4068	−0.439375
$675$	6.37577	0.245404
$676$	1.00000	0.0384615
$677$	30.1375	1.15828	0.579139	−	0.815229i	$-0.303389\pi$
$677$	30.1375	1.15828	0.579139	+	0.815229i	$0.303389\pi$
$678$	−4.07034	−0.156320
$679$	−11.2558	−0.431957
$680$	23.7364	0.910251
$681$	17.1274	0.656325
$682$	−6.57248	−0.251673
$683$	44.8563	1.71638	0.858189	−	0.513335i	$-0.171590\pi$
$683$	44.8563	1.71638	0.858189	+	0.513335i	$0.171590\pi$
$684$	6.87878	0.263017
$685$	−42.3454	−1.61793
$686$	19.7941	0.755741
$687$	−7.33367	−0.279797
$688$	5.25630	0.200394
$689$	−1.00000	−0.0380970
$690$	2.91195	0.110856
$691$	−6.62598	−0.252064	−0.126032	−	0.992026i	$-0.540224\pi$
$691$	−6.62598	−0.252064	−0.126032	+	0.992026i	$0.540224\pi$
$692$	−2.29305	−0.0871688
$693$	3.32700	0.126382
$694$	−23.4359	−0.889614
$695$	1.25923	0.0477652
$696$	6.51660	0.247011
$697$	−20.0698	−0.760199
$698$	15.3427	0.580729
$699$	−3.14374	−0.118907
$700$	−15.2663	−0.577012
$701$	−34.6084	−1.30714	−0.653571	−	0.756865i	$-0.726730\pi$
$701$	−34.6084	−1.30714	−0.653571	+	0.756865i	$0.726730\pi$
$702$	−1.00000	−0.0377426
$703$	39.8453	1.50279
$704$	−1.38948	−0.0523679
$705$	17.9515	0.676094
$706$	3.68436	0.138663
$707$	−2.00660	−0.0754660
$708$	10.1783	0.382525
$709$	36.8229	1.38291	0.691456	−	0.722418i	$-0.256969\pi$
$709$	36.8229	1.38291	0.691456	+	0.722418i	$0.256969\pi$
$710$	46.9644	1.76254
$711$	9.25522	0.347098
$712$	9.86398	0.369668
$713$	−4.08386	−0.152942
$714$	16.8510	0.630633
$715$	−4.68643	−0.175263
$716$	9.33950	0.349034
$717$	10.2068	0.381181
$718$	−29.6867	−1.10790
$719$	−30.7016	−1.14498	−0.572489	−	0.819912i	$-0.694022\pi$
$719$	−30.7016	−1.14498	−0.572489	+	0.819912i	$0.694022\pi$
$720$	−3.37280	−0.125697
$721$	34.0103	1.26661
$722$	28.3176	1.05387
$723$	−7.05656	−0.262436
$724$	1.19361	0.0443603
$725$	41.5484	1.54307
$726$	−9.06935	−0.336595
$727$	−42.4152	−1.57309	−0.786545	−	0.617532i	$-0.788132\pi$
$727$	−42.4152	−1.57309	−0.786545	+	0.617532i	$0.788132\pi$
$728$	2.39442	0.0887432
$729$	1.00000	0.0370370
$730$	−18.3012	−0.677359
$731$	−36.9918	−1.36819
$732$	10.2797	0.379950
$733$	−44.1669	−1.63134	−0.815671	−	0.578516i	$-0.803632\pi$
$733$	−44.1669	−1.63134	−0.815671	+	0.578516i	$0.803632\pi$
$734$	−4.90051	−0.180881
$735$	4.27245	0.157592
$736$	−0.863362	−0.0318240
$737$	−4.85954	−0.179004
$738$	2.85180	0.104976
$739$	−17.8651	−0.657178	−0.328589	−	0.944473i	$-0.606573\pi$
$739$	−17.8651	−0.657178	−0.328589	+	0.944473i	$0.606573\pi$
$740$	−19.5369	−0.718191
$741$	−6.87878	−0.252698
$742$	−2.39442	−0.0879020
$743$	−37.9102	−1.39079	−0.695396	−	0.718627i	$-0.744771\pi$
$743$	−37.9102	−1.39079	−0.695396	+	0.718627i	$0.744771\pi$
$744$	4.73018	0.173417
$745$	−12.8809	−0.471921
$746$	−11.3211	−0.414495
$747$	13.1597	0.481487
$748$	9.77860	0.357541
$749$	29.5706	1.08049
$750$	−4.64021	−0.169436
$751$	−42.2900	−1.54318	−0.771592	−	0.636118i	$-0.780539\pi$
$751$	−42.2900	−1.54318	−0.771592	+	0.636118i	$0.780539\pi$
$752$	−5.32245	−0.194090
$753$	−0.0699600	−0.00254948
$754$	−6.51660	−0.237321
$755$	4.33654	0.157823
$756$	−2.39442	−0.0870843
$757$	49.3628	1.79412	0.897061	−	0.441906i	$-0.145698\pi$
$757$	49.3628	1.79412	0.897061	+	0.441906i	$0.145698\pi$
$758$	−8.38734	−0.304642
$759$	1.19962	0.0435436
$760$	−23.2007	−0.841579
$761$	31.1689	1.12987	0.564936	−	0.825135i	$-0.308901\pi$
$761$	31.1689	1.12987	0.564936	+	0.825135i	$0.308901\pi$
$762$	−13.7606	−0.498495
$763$	19.8557	0.718823
$764$	−22.8247	−0.825769
$765$	23.7364	0.858192
$766$	19.2692	0.696223
$767$	−10.1783	−0.367518
$768$	1.00000	0.0360844
$769$	18.2743	0.658987	0.329494	−	0.944158i	$-0.393122\pi$
$769$	18.2743	0.658987	0.329494	+	0.944158i	$0.393122\pi$
$770$	−11.2213	−0.404388
$771$	−5.08811	−0.183244
$772$	23.9543	0.862133
$773$	−3.48116	−0.125209	−0.0626043	−	0.998038i	$-0.519941\pi$
$773$	−3.48116	−0.125209	−0.0626043	+	0.998038i	$0.519941\pi$
$774$	5.25630	0.188934
$775$	30.1586	1.08333
$776$	4.70083	0.168750
$777$	−13.8697	−0.497572
$778$	14.0545	0.503880
$779$	19.6169	0.702847
$780$	3.37280	0.120766
$781$	19.3477	0.692317
$782$	6.07600	0.217277
$783$	6.51660	0.232884
$784$	−1.26674	−0.0452406
$785$	−12.7725	−0.455871
$786$	4.81852	0.171871
$787$	23.1237	0.824272	0.412136	−	0.911122i	$-0.364783\pi$
$787$	23.1237	0.824272	0.412136	+	0.911122i	$0.364783\pi$
$788$	10.5679	0.376467
$789$	−16.8283	−0.599104
$790$	−31.2160	−1.11062
$791$	9.74612	0.346532
$792$	−1.38948	−0.0493730
$793$	−10.2797	−0.365044
$794$	5.84217	0.207331
$795$	−3.37280	−0.119621
$796$	18.2958	0.648476
$797$	3.74260	0.132570	0.0662848	−	0.997801i	$-0.478885\pi$
$797$	3.74260	0.132570	0.0662848	+	0.997801i	$0.478885\pi$
$798$	−16.4707	−0.583057
$799$	37.4573	1.32514
$800$	6.37577	0.225418
$801$	9.86398	0.348527
$802$	−6.79705	−0.240012
$803$	−7.53948	−0.266063
$804$	3.49739	0.123343
$805$	−6.97244	−0.245746
$806$	−4.73018	−0.166614
$807$	10.5684	0.372026
$808$	0.838031	0.0294818
$809$	−21.3377	−0.750195	−0.375097	−	0.926985i	$-0.622391\pi$
$809$	−21.3377	−0.750195	−0.375097	+	0.926985i	$0.622391\pi$
$810$	−3.37280	−0.118508
$811$	41.0551	1.44164	0.720819	−	0.693123i	$-0.243766\pi$
$811$	41.0551	1.44164	0.720819	+	0.693123i	$0.243766\pi$
$812$	−15.6035	−0.547576
$813$	6.55650	0.229946
$814$	−8.04854	−0.282101
$815$	−9.72177	−0.340539
$816$	−7.03761	−0.246365
$817$	36.1569	1.26497
$818$	14.1699	0.495440
$819$	2.39442	0.0836679
$820$	−9.61854	−0.335894
$821$	11.8691	0.414235	0.207118	−	0.978316i	$-0.433592\pi$
$821$	11.8691	0.414235	0.207118	+	0.978316i	$0.433592\pi$
$822$	12.5550	0.437905
$823$	45.6718	1.59202	0.796010	−	0.605283i	$-0.206940\pi$
$823$	45.6718	1.59202	0.796010	+	0.605283i	$0.206940\pi$
$824$	−14.2040	−0.494819
$825$	−8.85900	−0.308431
$826$	−24.3712	−0.847983
$827$	3.18117	0.110620	0.0553101	−	0.998469i	$-0.482385\pi$
$827$	3.18117	0.110620	0.0553101	+	0.998469i	$0.482385\pi$
$828$	−0.863362	−0.0300039
$829$	−40.8193	−1.41771	−0.708856	−	0.705353i	$-0.750788\pi$
$829$	−40.8193	−1.41771	−0.708856	+	0.705353i	$0.750788\pi$
$830$	−44.3849	−1.54062
$831$	−9.46563	−0.328359
$832$	−1.00000	−0.0346688
$833$	8.91479	0.308879
$834$	−0.373348	−0.0129280
$835$	−17.5989	−0.609034
$836$	−9.55791	−0.330567
$837$	4.73018	0.163499
$838$	28.5589	0.986551
$839$	−25.5872	−0.883368	−0.441684	−	0.897171i	$-0.645619\pi$
$839$	−25.5872	−0.883368	−0.441684	+	0.897171i	$0.645619\pi$
$840$	8.07591	0.278645
$841$	13.4661	0.464349
$842$	−17.4979	−0.603019
$843$	−23.7050	−0.816445
$844$	17.7591	0.611294
$845$	−3.37280	−0.116028
$846$	−5.32245	−0.182989
$847$	21.7159	0.746166
$848$	1.00000	0.0343401
$849$	−13.8980	−0.476978
$850$	−44.8702	−1.53903
$851$	−5.00102	−0.171433
$852$	−13.9245	−0.477045
$853$	9.97783	0.341634	0.170817	−	0.985303i	$-0.445359\pi$
$853$	9.97783	0.341634	0.170817	+	0.985303i	$0.445359\pi$
$854$	−24.6141	−0.842276
$855$	−23.2007	−0.793448
$856$	−12.3498	−0.422107
$857$	43.9442	1.50110	0.750552	−	0.660811i	$-0.229787\pi$
$857$	43.9442	1.50110	0.750552	+	0.660811i	$0.229787\pi$
$858$	1.38948	0.0474360
$859$	7.46783	0.254799	0.127400	−	0.991851i	$-0.459337\pi$
$859$	7.46783	0.254799	0.127400	+	0.991851i	$0.459337\pi$
$860$	−17.7284	−0.604535
$861$	−6.82841	−0.232712
$862$	−31.1734	−1.06177
$863$	8.62812	0.293705	0.146852	−	0.989158i	$-0.453086\pi$
$863$	8.62812	0.293705	0.146852	+	0.989158i	$0.453086\pi$
$864$	1.00000	0.0340207
$865$	7.73400	0.262964
$866$	−21.0657	−0.715841
$867$	32.5279	1.10471
$868$	−11.3261	−0.384431
$869$	−12.8599	−0.436243
$870$	−21.9792	−0.745164
$871$	−3.49739	−0.118504
$872$	−8.29246	−0.280818
$873$	4.70083	0.159099
$874$	−5.93888	−0.200886
$875$	11.1106	0.375608
$876$	5.42613	0.183332
$877$	17.6088	0.594608	0.297304	−	0.954783i	$-0.403913\pi$
$877$	17.6088	0.594608	0.297304	+	0.954783i	$0.403913\pi$
$878$	19.1928	0.647725
$879$	−4.66646	−0.157396
$880$	4.68643	0.157980
$881$	33.3557	1.12378	0.561890	−	0.827212i	$-0.310074\pi$
$881$	33.3557	1.12378	0.561890	+	0.827212i	$0.310074\pi$
$882$	−1.26674	−0.0426532
$883$	31.1918	1.04969	0.524844	−	0.851198i	$-0.324124\pi$
$883$	31.1918	1.04969	0.524844	+	0.851198i	$0.324124\pi$
$884$	7.03761	0.236700
$885$	−34.3294	−1.15397
$886$	−24.6912	−0.829516
$887$	−34.6407	−1.16312	−0.581561	−	0.813503i	$-0.697558\pi$
$887$	−34.6407	−1.16312	−0.581561	+	0.813503i	$0.697558\pi$
$888$	5.79249	0.194383
$889$	32.9488	1.10507
$890$	−33.2692	−1.11519
$891$	−1.38948	−0.0465493
$892$	0.460517	0.0154192
$893$	−36.6119	−1.22517
$894$	3.81906	0.127729
$895$	−31.5003	−1.05294
$896$	−2.39442	−0.0799921
$897$	0.863362	0.0288268
$898$	29.0104	0.968091
$899$	30.8247	1.02806
$900$	6.37577	0.212526
$901$	−7.03761	−0.234457
$902$	−3.96251	−0.131937
$903$	−12.5858	−0.418829
$904$	−4.07034	−0.135377
$905$	−4.02582	−0.133823
$906$	−1.28574	−0.0427158
$907$	14.8419	0.492817	0.246409	−	0.969166i	$-0.420749\pi$
$907$	14.8419	0.492817	0.246409	+	0.969166i	$0.420749\pi$
$908$	17.1274	0.568394
$909$	0.838031	0.0277957
$910$	−8.07591	−0.267714
$911$	−14.9787	−0.496267	−0.248134	−	0.968726i	$-0.579817\pi$
$911$	−14.9787	−0.496267	−0.248134	+	0.968726i	$0.579817\pi$
$912$	6.87878	0.227779
$913$	−18.2851	−0.605148
$914$	−6.82621	−0.225791
$915$	−34.6715	−1.14620
$916$	−7.33367	−0.242311
$917$	−11.5376	−0.381004
$918$	−7.03761	−0.232276
$919$	12.0470	0.397393	0.198696	−	0.980061i	$-0.436329\pi$
$919$	12.0470	0.397393	0.198696	+	0.980061i	$0.436329\pi$
$920$	2.91195	0.0960041
$921$	−4.51687	−0.148836
$922$	18.2443	0.600844
$923$	13.9245	0.458330
$924$	3.32700	0.109450
$925$	36.9316	1.21430
$926$	6.89121	0.226459
$927$	−14.2040	−0.466520
$928$	6.51660	0.213918
$929$	15.4465	0.506784	0.253392	−	0.967364i	$-0.418454\pi$
$929$	15.4465	0.506784	0.253392	+	0.967364i	$0.418454\pi$
$930$	−15.9540	−0.523151
$931$	−8.71360	−0.285577
$932$	−3.14374	−0.102977
$933$	33.4085	1.09374
$934$	−5.30516	−0.173590
$935$	−32.9813	−1.07860
$936$	−1.00000	−0.0326860
$937$	−33.2408	−1.08593	−0.542964	−	0.839756i	$-0.682698\pi$
$937$	−33.2408	−1.08593	−0.542964	+	0.839756i	$0.682698\pi$
$938$	−8.37423	−0.273428
$939$	18.1041	0.590806
$940$	17.9515	0.585515
$941$	37.3954	1.21905	0.609527	−	0.792765i	$-0.291359\pi$
$941$	37.3954	1.21905	0.609527	+	0.792765i	$0.291359\pi$
$942$	3.78692	0.123385
$943$	−2.46213	−0.0801781
$944$	10.1783	0.331276
$945$	8.07591	0.262709
$946$	−7.30351	−0.237458
$947$	58.6679	1.90645	0.953225	−	0.302260i	$-0.0977412\pi$
$947$	58.6679	1.90645	0.953225	+	0.302260i	$0.0977412\pi$
$948$	9.25522	0.300596
$949$	−5.42613	−0.176139
$950$	43.8575	1.42293
$951$	4.00480	0.129864
$952$	16.8510	0.546144
$953$	−57.9841	−1.87829	−0.939145	−	0.343521i	$-0.888380\pi$
$953$	−57.9841	−1.87829	−0.939145	+	0.343521i	$0.888380\pi$
$954$	1.00000	0.0323762
$955$	76.9831	2.49112
$956$	10.2068	0.330113
$957$	−9.05468	−0.292696
$958$	−26.0306	−0.841011
$959$	−30.0619	−0.970749
$960$	−3.37280	−0.108857
$961$	−8.62538	−0.278238
$962$	−5.79249	−0.186757
$963$	−12.3498	−0.397966
$964$	−7.05656	−0.227276
$965$	−80.7930	−2.60082
$966$	2.06725	0.0665128
$967$	36.0612	1.15965	0.579825	−	0.814741i	$-0.303121\pi$
$967$	36.0612	1.15965	0.579825	+	0.814741i	$0.303121\pi$
$968$	−9.06935	−0.291500
$969$	−48.4101	−1.55516
$970$	−15.8550	−0.509072
$971$	4.64329	0.149010	0.0745052	−	0.997221i	$-0.476262\pi$
$971$	4.64329	0.149010	0.0745052	+	0.997221i	$0.476262\pi$
$972$	1.00000	0.0320750
$973$	0.893952	0.0286588
$974$	12.0053	0.384675
$975$	−6.37577	−0.204188
$976$	10.2797	0.329047
$977$	35.9592	1.15044	0.575219	−	0.818000i	$-0.304917\pi$
$977$	35.9592	1.15044	0.575219	+	0.818000i	$0.304917\pi$
$978$	2.88241	0.0921692
$979$	−13.7058	−0.438039
$980$	4.27245	0.136478
$981$	−8.29246	−0.264758
$982$	−11.6935	−0.373154
$983$	5.89004	0.187863	0.0939316	−	0.995579i	$-0.470057\pi$
$983$	5.89004	0.187863	0.0939316	+	0.995579i	$0.470057\pi$
$984$	2.85180	0.0909119
$985$	−35.6435	−1.13570
$986$	−45.8613	−1.46052
$987$	12.7442	0.405652
$988$	−6.87878	−0.218843
$989$	−4.53809	−0.144303
$990$	4.68643	0.148945
$991$	9.31381	0.295863	0.147931	−	0.988998i	$-0.452738\pi$
$991$	9.31381	0.295863	0.147931	+	0.988998i	$0.452738\pi$
$992$	4.73018	0.150183
$993$	19.9515	0.633143
$994$	33.3411	1.05751
$995$	−61.7079	−1.95627
$996$	13.1597	0.416980
$997$	7.12751	0.225731	0.112865	−	0.993610i	$-0.463997\pi$
$997$	7.12751	0.225731	0.112865	+	0.993610i	$0.463997\pi$
$998$	10.9629	0.347023
$999$	5.79249	0.183266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4134.2.a.x.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4134.2.a.x.1.1	✓	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:	\( N \)	\(=\)	\( 4134 = 2 \cdot 3 \cdot 13 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4134.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.37280 q^{5} +1.00000 q^{6} -2.39442 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.37280 q^{5} +1.00000 q^{6} -2.39442 q^{7} +1.00000 q^{8} +1.00000 q^{9} -3.37280 q^{10} -1.38948 q^{11} +1.00000 q^{12} -1.00000 q^{13} -2.39442 q^{14} -3.37280 q^{15} +1.00000 q^{16} -7.03761 q^{17} +1.00000 q^{18} +6.87878 q^{19} -3.37280 q^{20} -2.39442 q^{21} -1.38948 q^{22} -0.863362 q^{23} +1.00000 q^{24} +6.37577 q^{25} -1.00000 q^{26} +1.00000 q^{27} -2.39442 q^{28} +6.51660 q^{29} -3.37280 q^{30} +4.73018 q^{31} +1.00000 q^{32} -1.38948 q^{33} -7.03761 q^{34} +8.07591 q^{35} +1.00000 q^{36} +5.79249 q^{37} +6.87878 q^{38} -1.00000 q^{39} -3.37280 q^{40} +2.85180 q^{41} -2.39442 q^{42} +5.25630 q^{43} -1.38948 q^{44} -3.37280 q^{45} -0.863362 q^{46} -5.32245 q^{47} +1.00000 q^{48} -1.26674 q^{49} +6.37577 q^{50} -7.03761 q^{51} -1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} +4.68643 q^{55} -2.39442 q^{56} +6.87878 q^{57} +6.51660 q^{58} +10.1783 q^{59} -3.37280 q^{60} +10.2797 q^{61} +4.73018 q^{62} -2.39442 q^{63} +1.00000 q^{64} +3.37280 q^{65} -1.38948 q^{66} +3.49739 q^{67} -7.03761 q^{68} -0.863362 q^{69} +8.07591 q^{70} -13.9245 q^{71} +1.00000 q^{72} +5.42613 q^{73} +5.79249 q^{74} +6.37577 q^{75} +6.87878 q^{76} +3.32700 q^{77} -1.00000 q^{78} +9.25522 q^{79} -3.37280 q^{80} +1.00000 q^{81} +2.85180 q^{82} +13.1597 q^{83} -2.39442 q^{84} +23.7364 q^{85} +5.25630 q^{86} +6.51660 q^{87} -1.38948 q^{88} +9.86398 q^{89} -3.37280 q^{90} +2.39442 q^{91} -0.863362 q^{92} +4.73018 q^{93} -5.32245 q^{94} -23.2007 q^{95} +1.00000 q^{96} +4.70083 q^{97} -1.26674 q^{98} -1.38948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 2 q^{5} + 8 q^{6} + 7 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 2 * q^5 + 8 * q^6 + 7 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} + 8 q^{3} + 8 q^{4} + 2 q^{5} + 8 q^{6} + 7 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 7 q^{11} + 8 q^{12} - 8 q^{13} + 7 q^{14} + 2 q^{15} + 8 q^{16} + 8 q^{17} + 8 q^{18} + 11 q^{19} + 2 q^{20} + 7 q^{21} + 7 q^{22} + 5 q^{23} + 8 q^{24} + 6 q^{25} - 8 q^{26} + 8 q^{27} + 7 q^{28} + 13 q^{29} + 2 q^{30} + 12 q^{31} + 8 q^{32} + 7 q^{33} + 8 q^{34} + 14 q^{35} + 8 q^{36} + 10 q^{37} + 11 q^{38} - 8 q^{39} + 2 q^{40} + 19 q^{41} + 7 q^{42} + 10 q^{43} + 7 q^{44} + 2 q^{45} + 5 q^{46} + 6 q^{47} + 8 q^{48} + 15 q^{49} + 6 q^{50} + 8 q^{51} - 8 q^{52} + 8 q^{53} + 8 q^{54} + 5 q^{55} + 7 q^{56} + 11 q^{57} + 13 q^{58} + 11 q^{59} + 2 q^{60} + 6 q^{61} + 12 q^{62} + 7 q^{63} + 8 q^{64} - 2 q^{65} + 7 q^{66} + 5 q^{67} + 8 q^{68} + 5 q^{69} + 14 q^{70} - 6 q^{71} + 8 q^{72} + 18 q^{73} + 10 q^{74} + 6 q^{75} + 11 q^{76} - 8 q^{77} - 8 q^{78} + 17 q^{79} + 2 q^{80} + 8 q^{81} + 19 q^{82} + 16 q^{83} + 7 q^{84} + 25 q^{85} + 10 q^{86} + 13 q^{87} + 7 q^{88} - 8 q^{89} + 2 q^{90} - 7 q^{91} + 5 q^{92} + 12 q^{93} + 6 q^{94} - 30 q^{95} + 8 q^{96} + 11 q^{97} + 15 q^{98} + 7 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^3 + 8 * q^4 + 2 * q^5 + 8 * q^6 + 7 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 7 * q^11 + 8 * q^12 - 8 * q^13 + 7 * q^14 + 2 * q^15 + 8 * q^16 + 8 * q^17 + 8 * q^18 + 11 * q^19 + 2 * q^20 + 7 * q^21 + 7 * q^22 + 5 * q^23 + 8 * q^24 + 6 * q^25 - 8 * q^26 + 8 * q^27 + 7 * q^28 + 13 * q^29 + 2 * q^30 + 12 * q^31 + 8 * q^32 + 7 * q^33 + 8 * q^34 + 14 * q^35 + 8 * q^36 + 10 * q^37 + 11 * q^38 - 8 * q^39 + 2 * q^40 + 19 * q^41 + 7 * q^42 + 10 * q^43 + 7 * q^44 + 2 * q^45 + 5 * q^46 + 6 * q^47 + 8 * q^48 + 15 * q^49 + 6 * q^50 + 8 * q^51 - 8 * q^52 + 8 * q^53 + 8 * q^54 + 5 * q^55 + 7 * q^56 + 11 * q^57 + 13 * q^58 + 11 * q^59 + 2 * q^60 + 6 * q^61 + 12 * q^62 + 7 * q^63 + 8 * q^64 - 2 * q^65 + 7 * q^66 + 5 * q^67 + 8 * q^68 + 5 * q^69 + 14 * q^70 - 6 * q^71 + 8 * q^72 + 18 * q^73 + 10 * q^74 + 6 * q^75 + 11 * q^76 - 8 * q^77 - 8 * q^78 + 17 * q^79 + 2 * q^80 + 8 * q^81 + 19 * q^82 + 16 * q^83 + 7 * q^84 + 25 * q^85 + 10 * q^86 + 13 * q^87 + 7 * q^88 - 8 * q^89 + 2 * q^90 - 7 * q^91 + 5 * q^92 + 12 * q^93 + 6 * q^94 - 30 * q^95 + 8 * q^96 + 11 * q^97 + 15 * q^98 + 7 * q^99

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