LMFDB - Embedded newform 8038.2.a.b.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8038,2,Mod(1,8038)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8038, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8038.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8038 = 2 \cdot 4019 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8038.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:	$64.1837531447$
Analytic rank:	$0$
Dimension:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	8038.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.12762 q^{3} +1.00000 q^{4} -1.48235 q^{5} +2.12762 q^{6} +0.377312 q^{7} -1.00000 q^{8} +1.52675 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.12762 q^{3} +1.00000 q^{4} -1.48235 q^{5} +2.12762 q^{6} +0.377312 q^{7} -1.00000 q^{8} +1.52675 q^{9} +1.48235 q^{10} +4.19165 q^{11} -2.12762 q^{12} -1.24880 q^{13} -0.377312 q^{14} +3.15387 q^{15} +1.00000 q^{16} -6.36865 q^{17} -1.52675 q^{18} -5.31515 q^{19} -1.48235 q^{20} -0.802775 q^{21} -4.19165 q^{22} +5.73594 q^{23} +2.12762 q^{24} -2.80264 q^{25} +1.24880 q^{26} +3.13451 q^{27} +0.377312 q^{28} -3.43651 q^{29} -3.15387 q^{30} +6.62925 q^{31} -1.00000 q^{32} -8.91822 q^{33} +6.36865 q^{34} -0.559308 q^{35} +1.52675 q^{36} +0.794189 q^{37} +5.31515 q^{38} +2.65697 q^{39} +1.48235 q^{40} -5.19867 q^{41} +0.802775 q^{42} +0.575801 q^{43} +4.19165 q^{44} -2.26318 q^{45} -5.73594 q^{46} +5.01935 q^{47} -2.12762 q^{48} -6.85764 q^{49} +2.80264 q^{50} +13.5500 q^{51} -1.24880 q^{52} -14.3821 q^{53} -3.13451 q^{54} -6.21349 q^{55} -0.377312 q^{56} +11.3086 q^{57} +3.43651 q^{58} +8.60376 q^{59} +3.15387 q^{60} +4.91679 q^{61} -6.62925 q^{62} +0.576061 q^{63} +1.00000 q^{64} +1.85116 q^{65} +8.91822 q^{66} +0.405979 q^{67} -6.36865 q^{68} -12.2039 q^{69} +0.559308 q^{70} -5.78987 q^{71} -1.52675 q^{72} -8.43147 q^{73} -0.794189 q^{74} +5.96294 q^{75} -5.31515 q^{76} +1.58156 q^{77} -2.65697 q^{78} -5.69568 q^{79} -1.48235 q^{80} -11.2493 q^{81} +5.19867 q^{82} +8.24824 q^{83} -0.802775 q^{84} +9.44056 q^{85} -0.575801 q^{86} +7.31158 q^{87} -4.19165 q^{88} +8.97961 q^{89} +2.26318 q^{90} -0.471188 q^{91} +5.73594 q^{92} -14.1045 q^{93} -5.01935 q^{94} +7.87891 q^{95} +2.12762 q^{96} +15.9882 q^{97} +6.85764 q^{98} +6.39960 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) $ 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * 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
$3$	−2.12762	−1.22838	−0.614190	−	0.789158i	$-0.710517\pi$
$3$	−2.12762	−1.22838	−0.614190	+	0.789158i	$0.710517\pi$
$4$	1.00000	0.500000
$5$	−1.48235	−0.662927	−0.331463	−	0.943468i	$-0.607542\pi$
$5$	−1.48235	−0.662927	−0.331463	+	0.943468i	$0.607542\pi$
$6$	2.12762	0.868596
$7$	0.377312	0.142611	0.0713053	−	0.997455i	$-0.477284\pi$
$7$	0.377312	0.142611	0.0713053	+	0.997455i	$0.477284\pi$
$8$	−1.00000	−0.353553
$9$	1.52675	0.508917
$10$	1.48235	0.468760
$11$	4.19165	1.26383	0.631915	−	0.775038i	$-0.282269\pi$
$11$	4.19165	1.26383	0.631915	+	0.775038i	$0.282269\pi$
$12$	−2.12762	−0.614190
$13$	−1.24880	−0.346355	−0.173178	−	0.984891i	$-0.555403\pi$
$13$	−1.24880	−0.346355	−0.173178	+	0.984891i	$0.555403\pi$
$14$	−0.377312	−0.100841
$15$	3.15387	0.814326
$16$	1.00000	0.250000
$17$	−6.36865	−1.54462	−0.772312	−	0.635243i	$-0.780900\pi$
$17$	−6.36865	−1.54462	−0.772312	+	0.635243i	$0.780900\pi$
$18$	−1.52675	−0.359858
$19$	−5.31515	−1.21938	−0.609689	−	0.792640i	$-0.708706\pi$
$19$	−5.31515	−1.21938	−0.609689	+	0.792640i	$0.708706\pi$
$20$	−1.48235	−0.331463
$21$	−0.802775	−0.175180
$22$	−4.19165	−0.893663
$23$	5.73594	1.19603	0.598013	−	0.801486i	$-0.295957\pi$
$23$	5.73594	1.19603	0.598013	+	0.801486i	$0.295957\pi$
$24$	2.12762	0.434298
$25$	−2.80264	−0.560528
$26$	1.24880	0.244910
$27$	3.13451	0.603237
$28$	0.377312	0.0713053
$29$	−3.43651	−0.638144	−0.319072	−	0.947730i	$-0.603371\pi$
$29$	−3.43651	−0.638144	−0.319072	+	0.947730i	$0.603371\pi$
$30$	−3.15387	−0.575815
$31$	6.62925	1.19065	0.595325	−	0.803485i	$-0.297024\pi$
$31$	6.62925	1.19065	0.595325	+	0.803485i	$0.297024\pi$
$32$	−1.00000	−0.176777
$33$	−8.91822	−1.55246
$34$	6.36865	1.09221
$35$	−0.559308	−0.0945404
$36$	1.52675	0.254458
$37$	0.794189	0.130564	0.0652820	−	0.997867i	$-0.479205\pi$
$37$	0.794189	0.130564	0.0652820	+	0.997867i	$0.479205\pi$
$38$	5.31515	0.862231
$39$	2.65697	0.425455
$40$	1.48235	0.234380
$41$	−5.19867	−0.811896	−0.405948	−	0.913896i	$-0.633059\pi$
$41$	−5.19867	−0.811896	−0.405948	+	0.913896i	$0.633059\pi$
$42$	0.802775	0.123871
$43$	0.575801	0.0878089	0.0439045	−	0.999036i	$-0.486020\pi$
$43$	0.575801	0.0878089	0.0439045	+	0.999036i	$0.486020\pi$
$44$	4.19165	0.631915
$45$	−2.26318	−0.337375
$46$	−5.73594	−0.845718
$47$	5.01935	0.732147	0.366073	−	0.930586i	$-0.380702\pi$
$47$	5.01935	0.732147	0.366073	+	0.930586i	$0.380702\pi$
$48$	−2.12762	−0.307095
$49$	−6.85764	−0.979662
$50$	2.80264	0.396353
$51$	13.5500	1.89738
$52$	−1.24880	−0.173178
$53$	−14.3821	−1.97553	−0.987767	−	0.155938i	$-0.950160\pi$
$53$	−14.3821	−1.97553	−0.987767	+	0.155938i	$0.950160\pi$
$54$	−3.13451	−0.426553
$55$	−6.21349	−0.837827
$56$	−0.377312	−0.0504204
$57$	11.3086	1.49786
$58$	3.43651	0.451236
$59$	8.60376	1.12011	0.560057	−	0.828454i	$-0.310779\pi$
$59$	8.60376	1.12011	0.560057	+	0.828454i	$0.310779\pi$
$60$	3.15387	0.407163
$61$	4.91679	0.629530	0.314765	−	0.949170i	$-0.398074\pi$
$61$	4.91679	0.629530	0.314765	+	0.949170i	$0.398074\pi$
$62$	−6.62925	−0.841916
$63$	0.576061	0.0725769
$64$	1.00000	0.125000
$65$	1.85116	0.229608
$66$	8.91822	1.09776
$67$	0.405979	0.0495982	0.0247991	−	0.999692i	$-0.492105\pi$
$67$	0.405979	0.0495982	0.0247991	+	0.999692i	$0.492105\pi$
$68$	−6.36865	−0.772312
$69$	−12.2039	−1.46917
$70$	0.559308	0.0668501
$71$	−5.78987	−0.687131	−0.343565	−	0.939129i	$-0.611635\pi$
$71$	−5.78987	−0.687131	−0.343565	+	0.939129i	$0.611635\pi$
$72$	−1.52675	−0.179929
$73$	−8.43147	−0.986829	−0.493415	−	0.869794i	$-0.664251\pi$
$73$	−8.43147	−0.986829	−0.493415	+	0.869794i	$0.664251\pi$
$74$	−0.794189	−0.0923227
$75$	5.96294	0.688541
$76$	−5.31515	−0.609689
$77$	1.58156	0.180235
$78$	−2.65697	−0.300842
$79$	−5.69568	−0.640815	−0.320407	−	0.947280i	$-0.603820\pi$
$79$	−5.69568	−0.640815	−0.320407	+	0.947280i	$0.603820\pi$
$80$	−1.48235	−0.165732
$81$	−11.2493	−1.24992
$82$	5.19867	0.574097
$83$	8.24824	0.905362	0.452681	−	0.891673i	$-0.350468\pi$
$83$	8.24824	0.905362	0.452681	+	0.891673i	$0.350468\pi$
$84$	−0.802775	−0.0875899
$85$	9.44056	1.02397
$86$	−0.575801	−0.0620903
$87$	7.31158	0.783884
$88$	−4.19165	−0.446831
$89$	8.97961	0.951837	0.475919	−	0.879489i	$-0.342116\pi$
$89$	8.97961	0.951837	0.475919	+	0.879489i	$0.342116\pi$
$90$	2.26318	0.238560
$91$	−0.471188	−0.0493939
$92$	5.73594	0.598013
$93$	−14.1045	−1.46257
$94$	−5.01935	−0.517706
$95$	7.87891	0.808359
$96$	2.12762	0.217149
$97$	15.9882	1.62336	0.811679	−	0.584103i	$-0.198554\pi$
$97$	15.9882	1.62336	0.811679	+	0.584103i	$0.198554\pi$
$98$	6.85764	0.692726
$99$	6.39960	0.643184
$100$	−2.80264	−0.280264
$101$	−13.2435	−1.31778	−0.658891	−	0.752239i	$-0.728974\pi$
$101$	−13.2435	−1.31778	−0.658891	+	0.752239i	$0.728974\pi$
$102$	−13.5500	−1.34165
$103$	16.9939	1.67446	0.837230	−	0.546851i	$-0.184174\pi$
$103$	16.9939	1.67446	0.837230	+	0.546851i	$0.184174\pi$
$104$	1.24880	0.122455
$105$	1.18999	0.116131
$106$	14.3821	1.39691
$107$	14.4306	1.39506	0.697530	−	0.716555i	$-0.254282\pi$
$107$	14.4306	1.39506	0.697530	+	0.716555i	$0.254282\pi$
$108$	3.13451	0.301618
$109$	−10.4878	−1.00455	−0.502277	−	0.864707i	$-0.667504\pi$
$109$	−10.4878	−1.00455	−0.502277	+	0.864707i	$0.667504\pi$
$110$	6.21349	0.592433
$111$	−1.68973	−0.160382
$112$	0.377312	0.0356526
$113$	0.772892	0.0727075	0.0363538	−	0.999339i	$-0.488426\pi$
$113$	0.772892	0.0727075	0.0363538	+	0.999339i	$0.488426\pi$
$114$	−11.3086	−1.05915
$115$	−8.50267	−0.792878
$116$	−3.43651	−0.319072
$117$	−1.90661	−0.176266
$118$	−8.60376	−0.792040
$119$	−2.40297	−0.220280
$120$	−3.15387	−0.287908
$121$	6.56993	0.597266
$122$	−4.91679	−0.445145
$123$	11.0608	0.997317
$124$	6.62925	0.595325
$125$	11.5662	1.03452
$126$	−0.576061	−0.0513196
$127$	−7.84609	−0.696228	−0.348114	−	0.937452i	$-0.613178\pi$
$127$	−7.84609	−0.696228	−0.348114	+	0.937452i	$0.613178\pi$
$128$	−1.00000	−0.0883883
$129$	−1.22508	−0.107863
$130$	−1.85116	−0.162357
$131$	−21.9608	−1.91872	−0.959360	−	0.282185i	$-0.908941\pi$
$131$	−21.9608	−1.91872	−0.959360	+	0.282185i	$0.908941\pi$
$132$	−8.91822	−0.776232
$133$	−2.00547	−0.173896
$134$	−0.405979	−0.0350712
$135$	−4.64644	−0.399902
$136$	6.36865	0.546107
$137$	−13.5494	−1.15760	−0.578802	−	0.815468i	$-0.696479\pi$
$137$	−13.5494	−1.15760	−0.578802	+	0.815468i	$0.696479\pi$
$138$	12.2039	1.03886
$139$	−5.39244	−0.457380	−0.228690	−	0.973499i	$-0.573444\pi$
$139$	−5.39244	−0.457380	−0.228690	+	0.973499i	$0.573444\pi$
$140$	−0.559308	−0.0472702
$141$	−10.6792	−0.899354
$142$	5.78987	0.485875
$143$	−5.23454	−0.437734
$144$	1.52675	0.127229
$145$	5.09411	0.423043
$146$	8.43147	0.697794
$147$	14.5904	1.20340
$148$	0.794189	0.0652820
$149$	3.48390	0.285412	0.142706	−	0.989765i	$-0.454420\pi$
$149$	3.48390	0.285412	0.142706	+	0.989765i	$0.454420\pi$
$150$	−5.96294	−0.486872
$151$	−23.5301	−1.91486	−0.957428	−	0.288672i	$-0.906786\pi$
$151$	−23.5301	−1.91486	−0.957428	+	0.288672i	$0.906786\pi$
$152$	5.31515	0.431116
$153$	−9.72333	−0.786085
$154$	−1.58156	−0.127446
$155$	−9.82687	−0.789313
$156$	2.65697	0.212728
$157$	23.2253	1.85358	0.926790	−	0.375580i	$-0.122557\pi$
$157$	23.2253	1.85358	0.926790	+	0.375580i	$0.122557\pi$
$158$	5.69568	0.453124
$159$	30.5996	2.42671
$160$	1.48235	0.117190
$161$	2.16424	0.170566
$162$	11.2493	0.883827
$163$	−8.82778	−0.691445	−0.345723	−	0.938337i	$-0.612366\pi$
$163$	−8.82778	−0.691445	−0.345723	+	0.938337i	$0.612366\pi$
$164$	−5.19867	−0.405948
$165$	13.2199	1.02917
$166$	−8.24824	−0.640187
$167$	8.23111	0.636942	0.318471	−	0.947933i	$-0.396831\pi$
$167$	8.23111	0.636942	0.318471	+	0.947933i	$0.396831\pi$
$168$	0.802775	0.0619354
$169$	−11.4405	−0.880038
$170$	−9.44056	−0.724058
$171$	−8.11490	−0.620562
$172$	0.575801	0.0439045
$173$	−9.29803	−0.706916	−0.353458	−	0.935450i	$-0.614994\pi$
$173$	−9.29803	−0.706916	−0.353458	+	0.935450i	$0.614994\pi$
$174$	−7.31158	−0.554289
$175$	−1.05747	−0.0799372
$176$	4.19165	0.315958
$177$	−18.3055	−1.37593
$178$	−8.97961	−0.673051
$179$	11.4719	0.857453	0.428726	−	0.903434i	$-0.358962\pi$
$179$	11.4719	0.857453	0.428726	+	0.903434i	$0.358962\pi$
$180$	−2.26318	−0.168687
$181$	−20.9512	−1.55729	−0.778644	−	0.627465i	$-0.784092\pi$
$181$	−20.9512	−1.55729	−0.778644	+	0.627465i	$0.784092\pi$
$182$	0.471188	0.0349267
$183$	−10.4610	−0.773302
$184$	−5.73594	−0.422859
$185$	−1.17727	−0.0865544
$186$	14.1045	1.03419
$187$	−26.6951	−1.95214
$188$	5.01935	0.366073
$189$	1.18269	0.0860279
$190$	−7.87891	−0.571596
$191$	−1.83849	−0.133028	−0.0665141	−	0.997785i	$-0.521188\pi$
$191$	−1.83849	−0.133028	−0.0665141	+	0.997785i	$0.521188\pi$
$192$	−2.12762	−0.153547
$193$	−3.98432	−0.286797	−0.143399	−	0.989665i	$-0.545803\pi$
$193$	−3.98432	−0.286797	−0.143399	+	0.989665i	$0.545803\pi$
$194$	−15.9882	−1.14789
$195$	−3.93856	−0.282046
$196$	−6.85764	−0.489831
$197$	5.54617	0.395148	0.197574	−	0.980288i	$-0.436694\pi$
$197$	5.54617	0.395148	0.197574	+	0.980288i	$0.436694\pi$
$198$	−6.39960	−0.454800
$199$	−10.7507	−0.762095	−0.381048	−	0.924555i	$-0.624437\pi$
$199$	−10.7507	−0.762095	−0.381048	+	0.924555i	$0.624437\pi$
$200$	2.80264	0.198177
$201$	−0.863767	−0.0609254
$202$	13.2435	0.931812
$203$	−1.29664	−0.0910061
$204$	13.5500	0.948692
$205$	7.70625	0.538228
$206$	−16.9939	−1.18402
$207$	8.75734	0.608677
$208$	−1.24880	−0.0865888
$209$	−22.2792	−1.54109
$210$	−1.18999	−0.0821173
$211$	−26.8916	−1.85129	−0.925646	−	0.378391i	$-0.876477\pi$
$211$	−26.8916	−1.85129	−0.925646	+	0.378391i	$0.876477\pi$
$212$	−14.3821	−0.987767
$213$	12.3186	0.844057
$214$	−14.4306	−0.986457
$215$	−0.853539	−0.0582109
$216$	−3.13451	−0.213276
$217$	2.50130	0.169799
$218$	10.4878	0.710327
$219$	17.9389	1.21220
$220$	−6.21349	−0.418913
$221$	7.95317	0.534988
$222$	1.68973	0.113407
$223$	0.580779	0.0388918	0.0194459	−	0.999811i	$-0.493810\pi$
$223$	0.580779	0.0388918	0.0194459	+	0.999811i	$0.493810\pi$
$224$	−0.377312	−0.0252102
$225$	−4.27893	−0.285262
$226$	−0.772892	−0.0514120
$227$	20.8825	1.38602	0.693011	−	0.720927i	$-0.256284\pi$
$227$	20.8825	1.38602	0.693011	+	0.720927i	$0.256284\pi$
$228$	11.3086	0.748930
$229$	−0.325195	−0.0214895	−0.0107448	−	0.999942i	$-0.503420\pi$
$229$	−0.325195	−0.0214895	−0.0107448	+	0.999942i	$0.503420\pi$
$230$	8.50267	0.560649
$231$	−3.36495	−0.221398
$232$	3.43651	0.225618
$233$	9.04156	0.592332	0.296166	−	0.955136i	$-0.404292\pi$
$233$	9.04156	0.592332	0.296166	+	0.955136i	$0.404292\pi$
$234$	1.90661	0.124639
$235$	−7.44042	−0.485360
$236$	8.60376	0.560057
$237$	12.1182	0.787164
$238$	2.40297	0.155761
$239$	7.79457	0.504189	0.252094	−	0.967703i	$-0.418881\pi$
$239$	7.79457	0.504189	0.252094	+	0.967703i	$0.418881\pi$
$240$	3.15387	0.203581
$241$	2.68169	0.172743	0.0863714	−	0.996263i	$-0.472473\pi$
$241$	2.68169	0.172743	0.0863714	+	0.996263i	$0.472473\pi$
$242$	−6.56993	−0.422331
$243$	14.5306	0.932140
$244$	4.91679	0.314765
$245$	10.1654	0.649444
$246$	−11.0608	−0.705210
$247$	6.63756	0.422338
$248$	−6.62925	−0.420958
$249$	−17.5491	−1.11213
$250$	−11.5662	−0.731513
$251$	−23.3736	−1.47533	−0.737664	−	0.675168i	$-0.764071\pi$
$251$	−23.3736	−1.47533	−0.737664	+	0.675168i	$0.764071\pi$
$252$	0.576061	0.0362884
$253$	24.0430	1.51157
$254$	7.84609	0.492308
$255$	−20.0859	−1.25783
$256$	1.00000	0.0625000
$257$	−19.4505	−1.21329	−0.606644	−	0.794974i	$-0.707484\pi$
$257$	−19.4505	−1.21329	−0.606644	+	0.794974i	$0.707484\pi$
$258$	1.22508	0.0762704
$259$	0.299657	0.0186198
$260$	1.85116	0.114804
$261$	−5.24670	−0.324762
$262$	21.9608	1.35674
$263$	−24.2670	−1.49636	−0.748182	−	0.663493i	$-0.769073\pi$
$263$	−24.2670	−1.49636	−0.748182	+	0.663493i	$0.769073\pi$
$264$	8.91822	0.548879
$265$	21.3193	1.30963
$266$	2.00547	0.122963
$267$	−19.1052	−1.16922
$268$	0.405979	0.0247991
$269$	17.7533	1.08244	0.541220	−	0.840881i	$-0.317963\pi$
$269$	17.7533	1.08244	0.541220	+	0.840881i	$0.317963\pi$
$270$	4.64644	0.282773
$271$	13.0681	0.793830	0.396915	−	0.917855i	$-0.370081\pi$
$271$	13.0681	0.793830	0.396915	+	0.917855i	$0.370081\pi$
$272$	−6.36865	−0.386156
$273$	1.00251	0.0606744
$274$	13.5494	0.818549
$275$	−11.7477	−0.708412
$276$	−12.2039	−0.734587
$277$	24.4828	1.47103	0.735515	−	0.677509i	$-0.236940\pi$
$277$	24.4828	1.47103	0.735515	+	0.677509i	$0.236940\pi$
$278$	5.39244	0.323417
$279$	10.1212	0.605941
$280$	0.559308	0.0334251
$281$	−3.28618	−0.196037	−0.0980187	−	0.995185i	$-0.531250\pi$
$281$	−3.28618	−0.196037	−0.0980187	+	0.995185i	$0.531250\pi$
$282$	10.6792	0.635939
$283$	25.7893	1.53301	0.766507	−	0.642236i	$-0.221993\pi$
$283$	25.7893	1.53301	0.766507	+	0.642236i	$0.221993\pi$
$284$	−5.78987	−0.343565
$285$	−16.7633	−0.992972
$286$	5.23454	0.309525
$287$	−1.96152	−0.115785
$288$	−1.52675	−0.0899646
$289$	23.5597	1.38586
$290$	−5.09411	−0.299137
$291$	−34.0168	−1.99410
$292$	−8.43147	−0.493415
$293$	−15.5073	−0.905943	−0.452972	−	0.891525i	$-0.649636\pi$
$293$	−15.5073	−0.905943	−0.452972	+	0.891525i	$0.649636\pi$
$294$	−14.5904	−0.850930
$295$	−12.7538	−0.742554
$296$	−0.794189	−0.0461613
$297$	13.1388	0.762389
$298$	−3.48390	−0.201817
$299$	−7.16304	−0.414250
$300$	5.96294	0.344271
$301$	0.217257	0.0125225
$302$	23.5301	1.35401
$303$	28.1772	1.61874
$304$	−5.31515	−0.304845
$305$	−7.28840	−0.417333
$306$	9.72333	0.555846
$307$	2.80217	0.159928	0.0799640	−	0.996798i	$-0.474519\pi$
$307$	2.80217	0.159928	0.0799640	+	0.996798i	$0.474519\pi$
$308$	1.58156	0.0901177
$309$	−36.1565	−2.05687
$310$	9.82687	0.558129
$311$	−3.69355	−0.209442	−0.104721	−	0.994502i	$-0.533395\pi$
$311$	−3.69355	−0.209442	−0.104721	+	0.994502i	$0.533395\pi$
$312$	−2.65697	−0.150421
$313$	−22.7685	−1.28695	−0.643477	−	0.765466i	$-0.722509\pi$
$313$	−22.7685	−1.28695	−0.643477	+	0.765466i	$0.722509\pi$
$314$	−23.2253	−1.31068
$315$	−0.853924	−0.0481132
$316$	−5.69568	−0.320407
$317$	−15.7652	−0.885464	−0.442732	−	0.896654i	$-0.645991\pi$
$317$	−15.7652	−0.885464	−0.442732	+	0.896654i	$0.645991\pi$
$318$	−30.5996	−1.71594
$319$	−14.4047	−0.806506
$320$	−1.48235	−0.0828659
$321$	−30.7028	−1.71366
$322$	−2.16424	−0.120608
$323$	33.8503	1.88348
$324$	−11.2493	−0.624960
$325$	3.49994	0.194142
$326$	8.82778	0.488926
$327$	22.3141	1.23397
$328$	5.19867	0.287049
$329$	1.89386	0.104412
$330$	−13.2199	−0.727733
$331$	−14.4117	−0.792138	−0.396069	−	0.918221i	$-0.629626\pi$
$331$	−14.4117	−0.792138	−0.396069	+	0.918221i	$0.629626\pi$
$332$	8.24824	0.452681
$333$	1.21253	0.0664462
$334$	−8.23111	−0.450386
$335$	−0.601802	−0.0328800
$336$	−0.802775	−0.0437950
$337$	10.5687	0.575711	0.287856	−	0.957674i	$-0.407058\pi$
$337$	10.5687	0.575711	0.287856	+	0.957674i	$0.407058\pi$
$338$	11.4405	0.622281
$339$	−1.64442	−0.0893125
$340$	9.44056	0.511986
$341$	27.7875	1.50478
$342$	8.11490	0.438804
$343$	−5.22865	−0.282321
$344$	−0.575801	−0.0310451
$345$	18.0904	0.973955
$346$	9.29803	0.499865
$347$	29.0338	1.55861	0.779307	−	0.626642i	$-0.215571\pi$
$347$	29.0338	1.55861	0.779307	+	0.626642i	$0.215571\pi$
$348$	7.31158	0.391942
$349$	−6.42440	−0.343890	−0.171945	−	0.985107i	$-0.555005\pi$
$349$	−6.42440	−0.343890	−0.171945	+	0.985107i	$0.555005\pi$
$350$	1.05747	0.0565241
$351$	−3.91438	−0.208934
$352$	−4.19165	−0.223416
$353$	6.68886	0.356012	0.178006	−	0.984029i	$-0.443035\pi$
$353$	6.68886	0.356012	0.178006	+	0.984029i	$0.443035\pi$
$354$	18.3055	0.972926
$355$	8.58261	0.455517
$356$	8.97961	0.475919
$357$	5.11259	0.270587
$358$	−11.4719	−0.606310
$359$	30.3425	1.60142	0.800709	−	0.599054i	$-0.204456\pi$
$359$	30.3425	1.60142	0.800709	+	0.599054i	$0.204456\pi$
$360$	2.26318	0.119280
$361$	9.25081	0.486885
$362$	20.9512	1.10117
$363$	−13.9783	−0.733670
$364$	−0.471188	−0.0246969
$365$	12.4984	0.654196
$366$	10.4610	0.546807
$367$	−16.4358	−0.857939	−0.428970	−	0.903319i	$-0.641123\pi$
$367$	−16.4358	−0.857939	−0.428970	+	0.903319i	$0.641123\pi$
$368$	5.73594	0.299006
$369$	−7.93707	−0.413188
$370$	1.17727	0.0612032
$371$	−5.42654	−0.281732
$372$	−14.1045	−0.731285
$373$	−7.68877	−0.398109	−0.199055	−	0.979988i	$-0.563787\pi$
$373$	−7.68877	−0.398109	−0.199055	+	0.979988i	$0.563787\pi$
$374$	26.6951	1.38037
$375$	−24.6085	−1.27078
$376$	−5.01935	−0.258853
$377$	4.29152	0.221025
$378$	−1.18269	−0.0608309
$379$	3.88286	0.199449	0.0997245	−	0.995015i	$-0.468204\pi$
$379$	3.88286	0.199449	0.0997245	+	0.995015i	$0.468204\pi$
$380$	7.87891	0.404179
$381$	16.6935	0.855232
$382$	1.83849	0.0940651
$383$	11.7798	0.601921	0.300961	−	0.953637i	$-0.402693\pi$
$383$	11.7798	0.601921	0.300961	+	0.953637i	$0.402693\pi$
$384$	2.12762	0.108574
$385$	−2.34442	−0.119483
$386$	3.98432	0.202796
$387$	0.879105	0.0446874
$388$	15.9882	0.811679
$389$	27.5550	1.39710	0.698548	−	0.715563i	$-0.253830\pi$
$389$	27.5550	1.39710	0.698548	+	0.715563i	$0.253830\pi$
$390$	3.93856	0.199437
$391$	−36.5302	−1.84741
$392$	6.85764	0.346363
$393$	46.7240	2.35692
$394$	−5.54617	−0.279412
$395$	8.44300	0.424813
$396$	6.39960	0.321592
$397$	−9.74282	−0.488978	−0.244489	−	0.969652i	$-0.578620\pi$
$397$	−9.74282	−0.488978	−0.244489	+	0.969652i	$0.578620\pi$
$398$	10.7507	0.538883
$399$	4.26687	0.213611
$400$	−2.80264	−0.140132
$401$	34.9247	1.74406	0.872029	−	0.489454i	$-0.162804\pi$
$401$	34.9247	1.74406	0.872029	+	0.489454i	$0.162804\pi$
$402$	0.863767	0.0430808
$403$	−8.27862	−0.412387
$404$	−13.2435	−0.658891
$405$	16.6754	0.828606
$406$	1.29664	0.0643510
$407$	3.32896	0.165011
$408$	−13.5500	−0.670827
$409$	26.0387	1.28753	0.643766	−	0.765222i	$-0.277371\pi$
$409$	26.0387	1.28753	0.643766	+	0.765222i	$0.277371\pi$
$410$	−7.70625	−0.380585
$411$	28.8279	1.42198
$412$	16.9939	0.837230
$413$	3.24630	0.159740
$414$	−8.75734	−0.430400
$415$	−12.2268	−0.600189
$416$	1.24880	0.0612275
$417$	11.4730	0.561837
$418$	22.2792	1.08971
$419$	38.0615	1.85943	0.929714	−	0.368282i	$-0.120054\pi$
$419$	38.0615	1.85943	0.929714	+	0.368282i	$0.120054\pi$
$420$	1.18999	0.0580657
$421$	9.63932	0.469792	0.234896	−	0.972021i	$-0.424525\pi$
$421$	9.63932	0.469792	0.234896	+	0.972021i	$0.424525\pi$
$422$	26.8916	1.30906
$423$	7.66329	0.372602
$424$	14.3821	0.698457
$425$	17.8490	0.865805
$426$	−12.3186	−0.596839
$427$	1.85516	0.0897777
$428$	14.4306	0.697530
$429$	11.1371	0.537703
$430$	0.853539	0.0411613
$431$	4.28060	0.206189	0.103095	−	0.994672i	$-0.467126\pi$
$431$	4.28060	0.206189	0.103095	+	0.994672i	$0.467126\pi$
$432$	3.13451	0.150809
$433$	−28.4294	−1.36623	−0.683114	−	0.730312i	$-0.739375\pi$
$433$	−28.4294	−1.36623	−0.683114	+	0.730312i	$0.739375\pi$
$434$	−2.50130	−0.120066
$435$	−10.8383	−0.519658
$436$	−10.4878	−0.502277
$437$	−30.4874	−1.45841
$438$	−17.9389	−0.857156
$439$	−0.0248163	−0.00118442	−0.000592208	−	1.00000i	$-0.500189\pi$
$439$	−0.0248163	−0.00118442	−0.000592208		1.00000i	$0.500189\pi$
$440$	6.21349	0.296217
$441$	−10.4699	−0.498566
$442$	−7.95317	−0.378294
$443$	8.12479	0.386020	0.193010	−	0.981197i	$-0.438175\pi$
$443$	8.12479	0.386020	0.193010	+	0.981197i	$0.438175\pi$
$444$	−1.68973	−0.0801911
$445$	−13.3109	−0.630998
$446$	−0.580779	−0.0275007
$447$	−7.41240	−0.350594
$448$	0.377312	0.0178263
$449$	3.14956	0.148637	0.0743183	−	0.997235i	$-0.476322\pi$
$449$	3.14956	0.148637	0.0743183	+	0.997235i	$0.476322\pi$
$450$	4.27893	0.201711
$451$	−21.7910	−1.02610
$452$	0.772892	0.0363538
$453$	50.0631	2.35217
$454$	−20.8825	−0.980066
$455$	0.698465	0.0327445
$456$	−11.3086	−0.529574
$457$	8.58695	0.401681	0.200840	−	0.979624i	$-0.435633\pi$
$457$	8.58695	0.401681	0.200840	+	0.979624i	$0.435633\pi$
$458$	0.325195	0.0151954
$459$	−19.9626	−0.931774
$460$	−8.50267	−0.396439
$461$	0.233620	0.0108808	0.00544039	−	0.999985i	$-0.498268\pi$
$461$	0.233620	0.0108808	0.00544039	+	0.999985i	$0.498268\pi$
$462$	3.36495	0.156552
$463$	37.1694	1.72741	0.863704	−	0.503999i	$-0.168138\pi$
$463$	37.1694	1.72741	0.863704	+	0.503999i	$0.168138\pi$
$464$	−3.43651	−0.159536
$465$	20.9078	0.969576
$466$	−9.04156	−0.418842
$467$	−2.32017	−0.107365	−0.0536824	−	0.998558i	$-0.517096\pi$
$467$	−2.32017	−0.107365	−0.0536824	+	0.998558i	$0.517096\pi$
$468$	−1.90661	−0.0881329
$469$	0.153181	0.00707323
$470$	7.44042	0.343201
$471$	−49.4145	−2.27690
$472$	−8.60376	−0.396020
$473$	2.41356	0.110976
$474$	−12.1182	−0.556609
$475$	14.8964	0.683496
$476$	−2.40297	−0.110140
$477$	−21.9579	−1.00538
$478$	−7.79457	−0.356515
$479$	10.6150	0.485011	0.242505	−	0.970150i	$-0.422031\pi$
$479$	10.6150	0.485011	0.242505	+	0.970150i	$0.422031\pi$
$480$	−3.15387	−0.143954
$481$	−0.991784	−0.0452215
$482$	−2.68169	−0.122148
$483$	−4.60467	−0.209520
$484$	6.56993	0.298633
$485$	−23.7001	−1.07617
$486$	−14.5306	−0.659123
$487$	4.90525	0.222278	0.111139	−	0.993805i	$-0.464550\pi$
$487$	4.90525	0.222278	0.111139	+	0.993805i	$0.464550\pi$
$488$	−4.91679	−0.222573
$489$	18.7821	0.849357
$490$	−10.1654	−0.459227
$491$	25.1741	1.13609	0.568046	−	0.822997i	$-0.307700\pi$
$491$	25.1741	1.13609	0.568046	+	0.822997i	$0.307700\pi$
$492$	11.0608	0.498658
$493$	21.8859	0.985693
$494$	−6.63756	−0.298638
$495$	−9.48645	−0.426384
$496$	6.62925	0.297662
$497$	−2.18459	−0.0979921
$498$	17.5491	0.786393
$499$	36.8975	1.65176	0.825879	−	0.563847i	$-0.190679\pi$
$499$	36.8975	1.65176	0.825879	+	0.563847i	$0.190679\pi$
$500$	11.5662	0.517258
$501$	−17.5126	−0.782407
$502$	23.3736	1.04321
$503$	7.27689	0.324461	0.162230	−	0.986753i	$-0.448131\pi$
$503$	7.27689	0.324461	0.162230	+	0.986753i	$0.448131\pi$
$504$	−0.576061	−0.0256598
$505$	19.6315	0.873592
$506$	−24.0430	−1.06884
$507$	24.3410	1.08102
$508$	−7.84609	−0.348114
$509$	18.3365	0.812749	0.406375	−	0.913707i	$-0.366793\pi$
$509$	18.3365	0.812749	0.406375	+	0.913707i	$0.366793\pi$
$510$	20.0859	0.889418
$511$	−3.18130	−0.140732
$512$	−1.00000	−0.0441942
$513$	−16.6604	−0.735574
$514$	19.4505	0.857924
$515$	−25.1909	−1.11004
$516$	−1.22508	−0.0539313
$517$	21.0393	0.925309
$518$	−0.299657	−0.0131662
$519$	19.7826	0.868362
$520$	−1.85116	−0.0811787
$521$	37.6448	1.64925	0.824625	−	0.565680i	$-0.191386\pi$
$521$	37.6448	1.64925	0.824625	+	0.565680i	$0.191386\pi$
$522$	5.24670	0.229642
$523$	29.6623	1.29704	0.648520	−	0.761197i	$-0.275388\pi$
$523$	29.6623	1.29704	0.648520	+	0.761197i	$0.275388\pi$
$524$	−21.9608	−0.959360
$525$	2.24989	0.0981932
$526$	24.2670	1.05809
$527$	−42.2194	−1.83911
$528$	−8.91822	−0.388116
$529$	9.90099	0.430478
$530$	−21.3193	−0.926051
$531$	13.1358	0.570045
$532$	−2.00547	−0.0869481
$533$	6.49211	0.281204
$534$	19.1052	0.826762
$535$	−21.3912	−0.924823
$536$	−0.405979	−0.0175356
$537$	−24.4079	−1.05328
$538$	−17.7533	−0.765400
$539$	−28.7448	−1.23813
$540$	−4.64644	−0.199951
$541$	−27.2031	−1.16955	−0.584776	−	0.811195i	$-0.698818\pi$
$541$	−27.2031	−1.16955	−0.584776	+	0.811195i	$0.698818\pi$
$542$	−13.0681	−0.561322
$543$	44.5761	1.91294
$544$	6.36865	0.273054
$545$	15.5467	0.665946
$546$	−1.00251	−0.0429033
$547$	1.09970	0.0470196	0.0235098	−	0.999724i	$-0.492516\pi$
$547$	1.09970	0.0470196	0.0235098	+	0.999724i	$0.492516\pi$
$548$	−13.5494	−0.578802
$549$	7.50671	0.320379
$550$	11.7477	0.500923
$551$	18.2656	0.778140
$552$	12.2039	0.519431
$553$	−2.14905	−0.0913869
$554$	−24.4828	−1.04017
$555$	2.50477	0.106322
$556$	−5.39244	−0.228690
$557$	−13.6180	−0.577011	−0.288506	−	0.957478i	$-0.593158\pi$
$557$	−13.6180	−0.577011	−0.288506	+	0.957478i	$0.593158\pi$
$558$	−10.1212	−0.428465
$559$	−0.719061	−0.0304131
$560$	−0.559308	−0.0236351
$561$	56.7970	2.39797
$562$	3.28618	0.138619
$563$	−4.94946	−0.208595	−0.104297	−	0.994546i	$-0.533259\pi$
$563$	−4.94946	−0.208595	−0.104297	+	0.994546i	$0.533259\pi$
$564$	−10.6792	−0.449677
$565$	−1.14570	−0.0481998
$566$	−25.7893	−1.08400
$567$	−4.24449	−0.178252
$568$	5.78987	0.242937
$569$	−34.3153	−1.43857	−0.719286	−	0.694714i	$-0.755531\pi$
$569$	−34.3153	−1.43857	−0.719286	+	0.694714i	$0.755531\pi$
$570$	16.7633	0.702137
$571$	−5.45432	−0.228256	−0.114128	−	0.993466i	$-0.536407\pi$
$571$	−5.45432	−0.228256	−0.114128	+	0.993466i	$0.536407\pi$
$572$	−5.23454	−0.218867
$573$	3.91159	0.163409
$574$	1.96152	0.0818723
$575$	−16.0758	−0.670406
$576$	1.52675	0.0636146
$577$	39.7668	1.65551	0.827757	−	0.561087i	$-0.189617\pi$
$577$	39.7668	1.65551	0.827757	+	0.561087i	$0.189617\pi$
$578$	−23.5597	−0.979953
$579$	8.47709	0.352296
$580$	5.09411	0.211522
$581$	3.11216	0.129114
$582$	34.0168	1.41004
$583$	−60.2847	−2.49674
$584$	8.43147	0.348897
$585$	2.82626	0.116851
$586$	15.5073	0.640599
$587$	−26.0103	−1.07356	−0.536779	−	0.843723i	$-0.680359\pi$
$587$	−26.0103	−1.07356	−0.536779	+	0.843723i	$0.680359\pi$
$588$	14.5904	0.601699
$589$	−35.2355	−1.45185
$590$	12.7538	0.525065
$591$	−11.8001	−0.485392
$592$	0.794189	0.0326410
$593$	23.1609	0.951105	0.475553	−	0.879687i	$-0.342248\pi$
$593$	23.1609	0.951105	0.475553	+	0.879687i	$0.342248\pi$
$594$	−13.1388	−0.539090
$595$	3.56204	0.146029
$596$	3.48390	0.142706
$597$	22.8733	0.936142
$598$	7.16304	0.292919
$599$	−1.52975	−0.0625038	−0.0312519	−	0.999512i	$-0.509949\pi$
$599$	−1.52975	−0.0625038	−0.0312519	+	0.999512i	$0.509949\pi$
$600$	−5.96294	−0.243436
$601$	−37.7790	−1.54104	−0.770518	−	0.637418i	$-0.780002\pi$
$601$	−37.7790	−1.54104	−0.770518	+	0.637418i	$0.780002\pi$
$602$	−0.217257	−0.00885473
$603$	0.619828	0.0252414
$604$	−23.5301	−0.957428
$605$	−9.73893	−0.395944
$606$	−28.1772	−1.14462
$607$	−17.2024	−0.698225	−0.349113	−	0.937081i	$-0.613517\pi$
$607$	−17.2024	−0.698225	−0.349113	+	0.937081i	$0.613517\pi$
$608$	5.31515	0.215558
$609$	2.75875	0.111790
$610$	7.28840	0.295099
$611$	−6.26816	−0.253583
$612$	−9.72333	−0.393042
$613$	−2.73774	−0.110576	−0.0552881	−	0.998470i	$-0.517608\pi$
$613$	−2.73774	−0.110576	−0.0552881	+	0.998470i	$0.517608\pi$
$614$	−2.80217	−0.113086
$615$	−16.3959	−0.661148
$616$	−1.58156	−0.0637229
$617$	7.99119	0.321713	0.160857	−	0.986978i	$-0.448574\pi$
$617$	7.99119	0.321713	0.160857	+	0.986978i	$0.448574\pi$
$618$	36.1565	1.45443
$619$	0.999024	0.0401542	0.0200771	−	0.999798i	$-0.493609\pi$
$619$	0.999024	0.0401542	0.0200771	+	0.999798i	$0.493609\pi$
$620$	−9.82687	−0.394657
$621$	17.9794	0.721487
$622$	3.69355	0.148098
$623$	3.38812	0.135742
$624$	2.65697	0.106364
$625$	−3.13201	−0.125280
$626$	22.7685	0.910014
$627$	47.4017	1.89304
$628$	23.2253	0.926790
$629$	−5.05791	−0.201672
$630$	0.853924	0.0340211
$631$	−11.9612	−0.476166	−0.238083	−	0.971245i	$-0.576519\pi$
$631$	−11.9612	−0.476166	−0.238083	+	0.971245i	$0.576519\pi$
$632$	5.69568	0.226562
$633$	57.2149	2.27409
$634$	15.7652	0.626118
$635$	11.6306	0.461548
$636$	30.5996	1.21335
$637$	8.56382	0.339311
$638$	14.4047	0.570286
$639$	−8.83968	−0.349692
$640$	1.48235	0.0585950
$641$	6.52790	0.257836	0.128918	−	0.991655i	$-0.458850\pi$
$641$	6.52790	0.257836	0.128918	+	0.991655i	$0.458850\pi$
$642$	30.7028	1.21174
$643$	48.4989	1.91261	0.956305	−	0.292370i	$-0.0944440\pi$
$643$	48.4989	1.91261	0.956305	+	0.292370i	$0.0944440\pi$
$644$	2.16424	0.0852829
$645$	1.81600	0.0715051
$646$	−33.8503	−1.33182
$647$	11.2012	0.440364	0.220182	−	0.975459i	$-0.429335\pi$
$647$	11.2012	0.440364	0.220182	+	0.975459i	$0.429335\pi$
$648$	11.2493	0.441914
$649$	36.0639	1.41563
$650$	−3.49994	−0.137279
$651$	−5.32180	−0.208578
$652$	−8.82778	−0.345723
$653$	−8.62009	−0.337330	−0.168665	−	0.985673i	$-0.553946\pi$
$653$	−8.62009	−0.337330	−0.168665	+	0.985673i	$0.553946\pi$
$654$	−22.3141	−0.872551
$655$	32.5535	1.27197
$656$	−5.19867	−0.202974
$657$	−12.8728	−0.502214
$658$	−1.89386	−0.0738303
$659$	8.28941	0.322910	0.161455	−	0.986880i	$-0.448381\pi$
$659$	8.28941	0.322910	0.161455	+	0.986880i	$0.448381\pi$
$660$	13.2199	0.514585
$661$	−40.6113	−1.57960	−0.789798	−	0.613367i	$-0.789815\pi$
$661$	−40.6113	−1.57960	−0.789798	+	0.613367i	$0.789815\pi$
$662$	14.4117	0.560126
$663$	−16.9213	−0.657169
$664$	−8.24824	−0.320094
$665$	2.97281	0.115281
$666$	−1.21253	−0.0469845
$667$	−19.7116	−0.763237
$668$	8.23111	0.318471
$669$	−1.23567	−0.0477740
$670$	0.601802	0.0232497
$671$	20.6095	0.795619
$672$	0.802775	0.0309677
$673$	−43.3084	−1.66942	−0.834708	−	0.550693i	$-0.814363\pi$
$673$	−43.3084	−1.66942	−0.834708	+	0.550693i	$0.814363\pi$
$674$	−10.5687	−0.407089
$675$	−8.78490	−0.338131
$676$	−11.4405	−0.440019
$677$	15.2181	0.584880	0.292440	−	0.956284i	$-0.405533\pi$
$677$	15.2181	0.584880	0.292440	+	0.956284i	$0.405533\pi$
$678$	1.64442	0.0631535
$679$	6.03255	0.231508
$680$	−9.44056	−0.362029
$681$	−44.4300	−1.70256
$682$	−27.7875	−1.06404
$683$	33.9100	1.29753	0.648766	−	0.760988i	$-0.275286\pi$
$683$	33.9100	1.29753	0.648766	+	0.760988i	$0.275286\pi$
$684$	−8.11490	−0.310281
$685$	20.0849	0.767406
$686$	5.22865	0.199631
$687$	0.691891	0.0263973
$688$	0.575801	0.0219522
$689$	17.9604	0.684236
$690$	−18.0904	−0.688690
$691$	−11.0194	−0.419198	−0.209599	−	0.977787i	$-0.567216\pi$
$691$	−11.0194	−0.419198	−0.209599	+	0.977787i	$0.567216\pi$
$692$	−9.29803	−0.353458
$693$	2.41465	0.0917248
$694$	−29.0338	−1.10211
$695$	7.99348	0.303210
$696$	−7.31158	−0.277145
$697$	33.1085	1.25407
$698$	6.42440	0.243167
$699$	−19.2370	−0.727609
$700$	−1.05747	−0.0399686
$701$	10.2386	0.386708	0.193354	−	0.981129i	$-0.438063\pi$
$701$	10.2386	0.386708	0.193354	+	0.981129i	$0.438063\pi$
$702$	3.91438	0.147739
$703$	−4.22124	−0.159207
$704$	4.19165	0.157979
$705$	15.8304	0.596206
$706$	−6.68886	−0.251739
$707$	−4.99695	−0.187929
$708$	−18.3055	−0.687963
$709$	−2.43391	−0.0914074	−0.0457037	−	0.998955i	$-0.514553\pi$
$709$	−2.43391	−0.0914074	−0.0457037	+	0.998955i	$0.514553\pi$
$710$	−8.58261	−0.322099
$711$	−8.69589	−0.326121
$712$	−8.97961	−0.336525
$713$	38.0250	1.42405
$714$	−5.11259	−0.191334
$715$	7.75941	0.290186
$716$	11.4719	0.428726
$717$	−16.5838	−0.619335
$718$	−30.3425	−1.13237
$719$	−21.4743	−0.800854	−0.400427	−	0.916329i	$-0.631138\pi$
$719$	−21.4743	−0.800854	−0.400427	+	0.916329i	$0.631138\pi$
$720$	−2.26318	−0.0843436
$721$	6.41201	0.238796
$722$	−9.25081	−0.344279
$723$	−5.70560	−0.212194
$724$	−20.9512	−0.778644
$725$	9.63131	0.357698
$726$	13.9783	0.518783
$727$	12.7312	0.472175	0.236088	−	0.971732i	$-0.424135\pi$
$727$	12.7312	0.472175	0.236088	+	0.971732i	$0.424135\pi$
$728$	0.471188	0.0174634
$729$	2.83226	0.104898
$730$	−12.4984	−0.462586
$731$	−3.66708	−0.135632
$732$	−10.4610	−0.386651
$733$	36.5391	1.34960	0.674801	−	0.738000i	$-0.264229\pi$
$733$	36.5391	1.34960	0.674801	+	0.738000i	$0.264229\pi$
$734$	16.4358	0.606655
$735$	−21.6281	−0.797764
$736$	−5.73594	−0.211429
$737$	1.70172	0.0626837
$738$	7.93707	0.292168
$739$	13.3959	0.492777	0.246389	−	0.969171i	$-0.420756\pi$
$739$	13.3959	0.492777	0.246389	+	0.969171i	$0.420756\pi$
$740$	−1.17727	−0.0432772
$741$	−14.1222	−0.518791
$742$	5.42654	0.199215
$743$	50.6761	1.85913	0.929563	−	0.368664i	$-0.120185\pi$
$743$	50.6761	1.85913	0.929563	+	0.368664i	$0.120185\pi$
$744$	14.1045	0.517096
$745$	−5.16436	−0.189207
$746$	7.68877	0.281506
$747$	12.5930	0.460754
$748$	−26.6951	−0.976071
$749$	5.44484	0.198950
$750$	24.6085	0.898576
$751$	−31.4043	−1.14596	−0.572979	−	0.819570i	$-0.694212\pi$
$751$	−31.4043	−1.14596	−0.572979	+	0.819570i	$0.694212\pi$
$752$	5.01935	0.183037
$753$	49.7300	1.81226
$754$	−4.29152	−0.156288
$755$	34.8799	1.26941
$756$	1.18269	0.0430140
$757$	−2.17272	−0.0789690	−0.0394845	−	0.999220i	$-0.512572\pi$
$757$	−2.17272	−0.0789690	−0.0394845	+	0.999220i	$0.512572\pi$
$758$	−3.88286	−0.141032
$759$	−51.1544	−1.85679
$760$	−7.87891	−0.285798
$761$	25.4110	0.921148	0.460574	−	0.887621i	$-0.347644\pi$
$761$	25.4110	0.921148	0.460574	+	0.887621i	$0.347644\pi$
$762$	−16.6935	−0.604741
$763$	−3.95719	−0.143260
$764$	−1.83849	−0.0665141
$765$	14.4134	0.521117
$766$	−11.7798	−0.425623
$767$	−10.7444	−0.387957
$768$	−2.12762	−0.0767737
$769$	36.5345	1.31747	0.658734	−	0.752376i	$-0.271092\pi$
$769$	36.5345	1.31747	0.658734	+	0.752376i	$0.271092\pi$
$770$	2.34442	0.0844872
$771$	41.3831	1.49038
$772$	−3.98432	−0.143399
$773$	23.6573	0.850895	0.425447	−	0.904983i	$-0.360117\pi$
$773$	23.6573	0.850895	0.425447	+	0.904983i	$0.360117\pi$
$774$	−0.879105	−0.0315988
$775$	−18.5794	−0.667392
$776$	−15.9882	−0.573944
$777$	−0.637556	−0.0228722
$778$	−27.5550	−0.987896
$779$	27.6317	0.990009
$780$	−3.93856	−0.141023
$781$	−24.2691	−0.868416
$782$	36.5302	1.30632
$783$	−10.7718	−0.384952
$784$	−6.85764	−0.244916
$785$	−34.4280	−1.22879
$786$	−46.7240	−1.66659
$787$	−11.1154	−0.396222	−0.198111	−	0.980180i	$-0.563481\pi$
$787$	−11.1154	−0.396222	−0.198111	+	0.980180i	$0.563481\pi$
$788$	5.54617	0.197574
$789$	51.6308	1.83810
$790$	−8.44300	−0.300388
$791$	0.291621	0.0103689
$792$	−6.39960	−0.227400
$793$	−6.14009	−0.218041
$794$	9.74282	0.345760
$795$	−45.3593	−1.60873
$796$	−10.7507	−0.381048
$797$	−1.63168	−0.0577972	−0.0288986	−	0.999582i	$-0.509200\pi$
$797$	−1.63168	−0.0577972	−0.0288986	+	0.999582i	$0.509200\pi$
$798$	−4.26687	−0.151046
$799$	−31.9664	−1.13089
$800$	2.80264	0.0990883
$801$	13.7096	0.484406
$802$	−34.9247	−1.23324
$803$	−35.3418	−1.24718
$804$	−0.863767	−0.0304627
$805$	−3.20816	−0.113073
$806$	8.27862	0.291602
$807$	−37.7723	−1.32965
$808$	13.2435	0.465906
$809$	40.2780	1.41610	0.708050	−	0.706162i	$-0.249575\pi$
$809$	40.2780	1.41610	0.708050	+	0.706162i	$0.249575\pi$
$810$	−16.6754	−0.585913
$811$	48.4734	1.70213	0.851066	−	0.525058i	$-0.175957\pi$
$811$	48.4734	1.70213	0.851066	+	0.525058i	$0.175957\pi$
$812$	−1.29664	−0.0455031
$813$	−27.8039	−0.975124
$814$	−3.32896	−0.116680
$815$	13.0859	0.458378
$816$	13.5500	0.474346
$817$	−3.06047	−0.107072
$818$	−26.0387	−0.910423
$819$	−0.719386	−0.0251374
$820$	7.70625	0.269114
$821$	−19.3667	−0.675903	−0.337951	−	0.941164i	$-0.609734\pi$
$821$	−19.3667	−0.675903	−0.337951	+	0.941164i	$0.609734\pi$
$822$	−28.8279	−1.00549
$823$	−19.9110	−0.694054	−0.347027	−	0.937855i	$-0.612809\pi$
$823$	−19.9110	−0.694054	−0.347027	+	0.937855i	$0.612809\pi$
$824$	−16.9939	−0.592011
$825$	24.9946	0.870199
$826$	−3.24630	−0.112953
$827$	−10.4292	−0.362659	−0.181329	−	0.983422i	$-0.558040\pi$
$827$	−10.4292	−0.362659	−0.181329	+	0.983422i	$0.558040\pi$
$828$	8.75734	0.304339
$829$	35.0861	1.21859	0.609295	−	0.792944i	$-0.291453\pi$
$829$	35.0861	1.21859	0.609295	+	0.792944i	$0.291453\pi$
$830$	12.2268	0.424397
$831$	−52.0900	−1.80698
$832$	−1.24880	−0.0432944
$833$	43.6739	1.51321
$834$	−11.4730	−0.397279
$835$	−12.2014	−0.422246
$836$	−22.2792	−0.770544
$837$	20.7795	0.718243
$838$	−38.0615	−1.31481
$839$	45.3661	1.56621	0.783105	−	0.621889i	$-0.213635\pi$
$839$	45.3661	1.56621	0.783105	+	0.621889i	$0.213635\pi$
$840$	−1.18999	−0.0410587
$841$	−17.1904	−0.592772
$842$	−9.63932	−0.332193
$843$	6.99174	0.240808
$844$	−26.8916	−0.925646
$845$	16.9588	0.583401
$846$	−7.66329	−0.263469
$847$	2.47891	0.0851765
$848$	−14.3821	−0.493883
$849$	−54.8697	−1.88312
$850$	−17.8490	−0.612217
$851$	4.55542	0.156158
$852$	12.3186	0.422029
$853$	0.0708612	0.00242624	0.00121312	−	0.999999i	$-0.499614\pi$
$853$	0.0708612	0.00242624	0.00121312	+	0.999999i	$0.499614\pi$
$854$	−1.85516	−0.0634824
$855$	12.0291	0.411387
$856$	−14.4306	−0.493228
$857$	11.7709	0.402088	0.201044	−	0.979582i	$-0.435567\pi$
$857$	11.7709	0.402088	0.201044	+	0.979582i	$0.435567\pi$
$858$	−11.1371	−0.380214
$859$	35.4981	1.21118	0.605590	−	0.795777i	$-0.292937\pi$
$859$	35.4981	1.21118	0.605590	+	0.795777i	$0.292937\pi$
$860$	−0.853539	−0.0291054
$861$	4.17337	0.142228
$862$	−4.28060	−0.145798
$863$	48.2069	1.64098	0.820491	−	0.571659i	$-0.193700\pi$
$863$	48.2069	1.64098	0.820491	+	0.571659i	$0.193700\pi$
$864$	−3.13451	−0.106638
$865$	13.7829	0.468634
$866$	28.4294	0.966069
$867$	−50.1259	−1.70237
$868$	2.50130	0.0848996
$869$	−23.8743	−0.809881
$870$	10.8383	0.367453
$871$	−0.506987	−0.0171786
$872$	10.4878	0.355163
$873$	24.4100	0.826154
$874$	30.4874	1.03125
$875$	4.36408	0.147533
$876$	17.9389	0.606101
$877$	−35.7204	−1.20619	−0.603096	−	0.797669i	$-0.706066\pi$
$877$	−35.7204	−1.20619	−0.603096	+	0.797669i	$0.706066\pi$
$878$	0.0248163	0.000837508 0
$879$	32.9935	1.11284
$880$	−6.21349	−0.209457
$881$	−47.9444	−1.61529	−0.807643	−	0.589672i	$-0.799257\pi$
$881$	−47.9444	−1.61529	−0.807643	+	0.589672i	$0.799257\pi$
$882$	10.4699	0.352540
$883$	12.1439	0.408674	0.204337	−	0.978901i	$-0.434496\pi$
$883$	12.1439	0.408674	0.204337	+	0.978901i	$0.434496\pi$
$884$	7.95317	0.267494
$885$	27.1351	0.912138
$886$	−8.12479	−0.272958
$887$	43.5804	1.46329	0.731643	−	0.681688i	$-0.238754\pi$
$887$	43.5804	1.46329	0.731643	+	0.681688i	$0.238754\pi$
$888$	1.68973	0.0567036
$889$	−2.96042	−0.0992894
$890$	13.3109	0.446183
$891$	−47.1531	−1.57969
$892$	0.580779	0.0194459
$893$	−26.6786	−0.892764
$894$	7.41240	0.247908
$895$	−17.0054	−0.568428
$896$	−0.377312	−0.0126051
$897$	15.2402	0.508856
$898$	−3.14956	−0.105102
$899$	−22.7815	−0.759806
$900$	−4.27893	−0.142631
$901$	91.5945	3.05146
$902$	21.7910	0.725562
$903$	−0.462239	−0.0153824
$904$	−0.772892	−0.0257060
$905$	31.0570	1.03237
$906$	−50.0631	−1.66324
$907$	33.5839	1.11514	0.557568	−	0.830132i	$-0.311735\pi$
$907$	33.5839	1.11514	0.557568	+	0.830132i	$0.311735\pi$
$908$	20.8825	0.693011
$909$	−20.2196	−0.670641
$910$	−0.698465	−0.0231539
$911$	−4.64446	−0.153878	−0.0769388	−	0.997036i	$-0.524515\pi$
$911$	−4.64446	−0.153878	−0.0769388	+	0.997036i	$0.524515\pi$
$912$	11.3086	0.374465
$913$	34.5737	1.14422
$914$	−8.58695	−0.284031
$915$	15.5069	0.512643
$916$	−0.325195	−0.0107448
$917$	−8.28606	−0.273630
$918$	19.9626	0.658864
$919$	−8.48691	−0.279957	−0.139979	−	0.990155i	$-0.544703\pi$
$919$	−8.48691	−0.279957	−0.139979	+	0.990155i	$0.544703\pi$
$920$	8.50267	0.280325
$921$	−5.96193	−0.196452
$922$	−0.233620	−0.00769388
$923$	7.23039	0.237991
$924$	−3.36495	−0.110699
$925$	−2.22583	−0.0731847
$926$	−37.1694	−1.22146
$927$	25.9455	0.852160
$928$	3.43651	0.112809
$929$	−47.6416	−1.56307	−0.781536	−	0.623861i	$-0.785563\pi$
$929$	−47.6416	−1.56307	−0.781536	+	0.623861i	$0.785563\pi$
$930$	−20.9078	−0.685594
$931$	36.4494	1.19458
$932$	9.04156	0.296166
$933$	7.85845	0.257274
$934$	2.32017	0.0759183
$935$	39.5715	1.29413
$936$	1.90661	0.0623194
$937$	8.93774	0.291983	0.145992	−	0.989286i	$-0.453363\pi$
$937$	8.93774	0.291983	0.145992	+	0.989286i	$0.453363\pi$
$938$	−0.153181	−0.00500153
$939$	48.4427	1.58087
$940$	−7.44042	−0.242680
$941$	42.9900	1.40143	0.700716	−	0.713440i	$-0.252864\pi$
$941$	42.9900	1.40143	0.700716	+	0.713440i	$0.252864\pi$
$942$	49.4145	1.61001
$943$	−29.8193	−0.971049
$944$	8.60376	0.280029
$945$	−1.75316	−0.0570302
$946$	−2.41356	−0.0784715
$947$	33.3873	1.08494	0.542470	−	0.840075i	$-0.317489\pi$
$947$	33.3873	1.08494	0.542470	+	0.840075i	$0.317489\pi$
$948$	12.1182	0.393582
$949$	10.5292	0.341793
$950$	−14.8964	−0.483305
$951$	33.5424	1.08769
$952$	2.40297	0.0778806
$953$	−27.1924	−0.880849	−0.440424	−	0.897790i	$-0.645172\pi$
$953$	−27.1924	−0.880849	−0.440424	+	0.897790i	$0.645172\pi$
$954$	21.9579	0.710912
$955$	2.72528	0.0881879
$956$	7.79457	0.252094
$957$	30.6476	0.990696
$958$	−10.6150	−0.342954
$959$	−5.11235	−0.165086
$960$	3.15387	0.101791
$961$	12.9470	0.417645
$962$	0.991784	0.0319764
$963$	22.0319	0.709969
$964$	2.68169	0.0863714
$965$	5.90615	0.190126
$966$	4.60467	0.148153
$967$	−36.0500	−1.15929	−0.579645	−	0.814869i	$-0.696809\pi$
$967$	−36.0500	−1.15929	−0.579645	+	0.814869i	$0.696809\pi$
$968$	−6.56993	−0.211166
$969$	−72.0205	−2.31363
$970$	23.7001	0.760966
$971$	40.7098	1.30644	0.653220	−	0.757168i	$-0.273418\pi$
$971$	40.7098	1.30644	0.653220	+	0.757168i	$0.273418\pi$
$972$	14.5306	0.466070
$973$	−2.03463	−0.0652273
$974$	−4.90525	−0.157174
$975$	−7.44653	−0.238480
$976$	4.91679	0.157383
$977$	−12.9290	−0.413635	−0.206817	−	0.978380i	$-0.566311\pi$
$977$	−12.9290	−0.413635	−0.206817	+	0.978380i	$0.566311\pi$
$978$	−18.7821	−0.600586
$979$	37.6394	1.20296
$980$	10.1654	0.324722
$981$	−16.0123	−0.511234
$982$	−25.1741	−0.803338
$983$	32.8765	1.04860	0.524299	−	0.851534i	$-0.324327\pi$
$983$	32.8765	1.04860	0.524299	+	0.851534i	$0.324327\pi$
$984$	−11.0608	−0.352605
$985$	−8.22137	−0.261955
$986$	−21.8859	−0.696990
$987$	−4.02941	−0.128257
$988$	6.63756	0.211169
$989$	3.30276	0.105022
$990$	9.48645	0.301499
$991$	−4.26750	−0.135562	−0.0677808	−	0.997700i	$-0.521592\pi$
$991$	−4.26750	−0.135562	−0.0677808	+	0.997700i	$0.521592\pi$
$992$	−6.62925	−0.210479
$993$	30.6625	0.973046
$994$	2.18459	0.0692909
$995$	15.9363	0.505213
$996$	−17.5491	−0.556064
$997$	22.3174	0.706800	0.353400	−	0.935472i	$-0.385025\pi$
$997$	22.3174	0.706800	0.353400	+	0.935472i	$0.385025\pi$
$998$	−36.8975	−1.16797
$999$	2.48940	0.0787610

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8038.2.a.b.1.15	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8038.2.a.b.1.15	✓	83	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:	\( N \)	\(=\)	\( 8038 = 2 \cdot 4019 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8038.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.12762 q^{3} +1.00000 q^{4} -1.48235 q^{5} +2.12762 q^{6} +0.377312 q^{7} -1.00000 q^{8} +1.52675 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.12762 q^{3} +1.00000 q^{4} -1.48235 q^{5} +2.12762 q^{6} +0.377312 q^{7} -1.00000 q^{8} +1.52675 q^{9} +1.48235 q^{10} +4.19165 q^{11} -2.12762 q^{12} -1.24880 q^{13} -0.377312 q^{14} +3.15387 q^{15} +1.00000 q^{16} -6.36865 q^{17} -1.52675 q^{18} -5.31515 q^{19} -1.48235 q^{20} -0.802775 q^{21} -4.19165 q^{22} +5.73594 q^{23} +2.12762 q^{24} -2.80264 q^{25} +1.24880 q^{26} +3.13451 q^{27} +0.377312 q^{28} -3.43651 q^{29} -3.15387 q^{30} +6.62925 q^{31} -1.00000 q^{32} -8.91822 q^{33} +6.36865 q^{34} -0.559308 q^{35} +1.52675 q^{36} +0.794189 q^{37} +5.31515 q^{38} +2.65697 q^{39} +1.48235 q^{40} -5.19867 q^{41} +0.802775 q^{42} +0.575801 q^{43} +4.19165 q^{44} -2.26318 q^{45} -5.73594 q^{46} +5.01935 q^{47} -2.12762 q^{48} -6.85764 q^{49} +2.80264 q^{50} +13.5500 q^{51} -1.24880 q^{52} -14.3821 q^{53} -3.13451 q^{54} -6.21349 q^{55} -0.377312 q^{56} +11.3086 q^{57} +3.43651 q^{58} +8.60376 q^{59} +3.15387 q^{60} +4.91679 q^{61} -6.62925 q^{62} +0.576061 q^{63} +1.00000 q^{64} +1.85116 q^{65} +8.91822 q^{66} +0.405979 q^{67} -6.36865 q^{68} -12.2039 q^{69} +0.559308 q^{70} -5.78987 q^{71} -1.52675 q^{72} -8.43147 q^{73} -0.794189 q^{74} +5.96294 q^{75} -5.31515 q^{76} +1.58156 q^{77} -2.65697 q^{78} -5.69568 q^{79} -1.48235 q^{80} -11.2493 q^{81} +5.19867 q^{82} +8.24824 q^{83} -0.802775 q^{84} +9.44056 q^{85} -0.575801 q^{86} +7.31158 q^{87} -4.19165 q^{88} +8.97961 q^{89} +2.26318 q^{90} -0.471188 q^{91} +5.73594 q^{92} -14.1045 q^{93} -5.01935 q^{94} +7.87891 q^{95} +2.12762 q^{96} +15.9882 q^{97} +6.85764 q^{98} +6.39960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9}+O(q^{10}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9	\( 83 q - 83 q^{2} + 20 q^{3} + 83 q^{4} + 31 q^{5} - 20 q^{6} - 3 q^{7} - 83 q^{8} + 91 q^{9} - 31 q^{10} + 3 q^{11} + 20 q^{12} + 28 q^{13} + 3 q^{14} + 12 q^{15} + 83 q^{16} + 36 q^{17} - 91 q^{18} - 38 q^{19} + 31 q^{20} + 21 q^{21} - 3 q^{22} + 50 q^{23} - 20 q^{24} + 94 q^{25} - 28 q^{26} + 74 q^{27} - 3 q^{28} + 48 q^{29} - 12 q^{30} - 41 q^{31} - 83 q^{32} + 40 q^{33} - 36 q^{34} + 40 q^{35} + 91 q^{36} + 37 q^{37} + 38 q^{38} + q^{39} - 31 q^{40} + 44 q^{41} - 21 q^{42} - 21 q^{43} + 3 q^{44} + 98 q^{45} - 50 q^{46} + 62 q^{47} + 20 q^{48} + 74 q^{49} - 94 q^{50} + 11 q^{51} + 28 q^{52} + 99 q^{53} - 74 q^{54} - 20 q^{55} + 3 q^{56} + 24 q^{57} - 48 q^{58} + 33 q^{59} + 12 q^{60} + 38 q^{61} + 41 q^{62} + 43 q^{63} + 83 q^{64} + 85 q^{65} - 40 q^{66} + q^{67} + 36 q^{68} + 73 q^{69} - 40 q^{70} + 46 q^{71} - 91 q^{72} - 4 q^{73} - 37 q^{74} + 89 q^{75} - 38 q^{76} + 118 q^{77} - q^{78} - 29 q^{79} + 31 q^{80} + 115 q^{81} - 44 q^{82} + 69 q^{83} + 21 q^{84} + 20 q^{85} + 21 q^{86} + 57 q^{87} - 3 q^{88} + 78 q^{89} - 98 q^{90} - 37 q^{91} + 50 q^{92} + 61 q^{93} - 62 q^{94} + 49 q^{95} - 20 q^{96} + 21 q^{97} - 74 q^{98} - 20 q^{99}+O(q^{100}) \) 83 * q - 83 * q^2 + 20 * q^3 + 83 * q^4 + 31 * q^5 - 20 * q^6 - 3 * q^7 - 83 * q^8 + 91 * q^9 - 31 * q^10 + 3 * q^11 + 20 * q^12 + 28 * q^13 + 3 * q^14 + 12 * q^15 + 83 * q^16 + 36 * q^17 - 91 * q^18 - 38 * q^19 + 31 * q^20 + 21 * q^21 - 3 * q^22 + 50 * q^23 - 20 * q^24 + 94 * q^25 - 28 * q^26 + 74 * q^27 - 3 * q^28 + 48 * q^29 - 12 * q^30 - 41 * q^31 - 83 * q^32 + 40 * q^33 - 36 * q^34 + 40 * q^35 + 91 * q^36 + 37 * q^37 + 38 * q^38 + q^39 - 31 * q^40 + 44 * q^41 - 21 * q^42 - 21 * q^43 + 3 * q^44 + 98 * q^45 - 50 * q^46 + 62 * q^47 + 20 * q^48 + 74 * q^49 - 94 * q^50 + 11 * q^51 + 28 * q^52 + 99 * q^53 - 74 * q^54 - 20 * q^55 + 3 * q^56 + 24 * q^57 - 48 * q^58 + 33 * q^59 + 12 * q^60 + 38 * q^61 + 41 * q^62 + 43 * q^63 + 83 * q^64 + 85 * q^65 - 40 * q^66 + q^67 + 36 * q^68 + 73 * q^69 - 40 * q^70 + 46 * q^71 - 91 * q^72 - 4 * q^73 - 37 * q^74 + 89 * q^75 - 38 * q^76 + 118 * q^77 - q^78 - 29 * q^79 + 31 * q^80 + 115 * q^81 - 44 * q^82 + 69 * q^83 + 21 * q^84 + 20 * q^85 + 21 * q^86 + 57 * q^87 - 3 * q^88 + 78 * q^89 - 98 * q^90 - 37 * q^91 + 50 * q^92 + 61 * q^93 - 62 * q^94 + 49 * q^95 - 20 * q^96 + 21 * q^97 - 74 * q^98 - 20 * q^99

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