LMFDB - Embedded newform 8042.2.a.c.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$0$
Dimension:	$86$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -2.59394 q^{3} +1.00000 q^{4} -0.859547 q^{5} +2.59394 q^{6} -3.29568 q^{7} -1.00000 q^{8} +3.72855 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -2.59394 q^{3} +1.00000 q^{4} -0.859547 q^{5} +2.59394 q^{6} -3.29568 q^{7} -1.00000 q^{8} +3.72855 q^{9} +0.859547 q^{10} -0.214484 q^{11} -2.59394 q^{12} +0.577971 q^{13} +3.29568 q^{14} +2.22962 q^{15} +1.00000 q^{16} +6.01944 q^{17} -3.72855 q^{18} +0.272395 q^{19} -0.859547 q^{20} +8.54880 q^{21} +0.214484 q^{22} +2.84049 q^{23} +2.59394 q^{24} -4.26118 q^{25} -0.577971 q^{26} -1.88982 q^{27} -3.29568 q^{28} +3.81514 q^{29} -2.22962 q^{30} +8.62269 q^{31} -1.00000 q^{32} +0.556360 q^{33} -6.01944 q^{34} +2.83279 q^{35} +3.72855 q^{36} +1.26944 q^{37} -0.272395 q^{38} -1.49922 q^{39} +0.859547 q^{40} -8.40030 q^{41} -8.54880 q^{42} -0.438888 q^{43} -0.214484 q^{44} -3.20486 q^{45} -2.84049 q^{46} +7.89612 q^{47} -2.59394 q^{48} +3.86149 q^{49} +4.26118 q^{50} -15.6141 q^{51} +0.577971 q^{52} +12.2211 q^{53} +1.88982 q^{54} +0.184359 q^{55} +3.29568 q^{56} -0.706577 q^{57} -3.81514 q^{58} -3.37745 q^{59} +2.22962 q^{60} -9.85667 q^{61} -8.62269 q^{62} -12.2881 q^{63} +1.00000 q^{64} -0.496793 q^{65} -0.556360 q^{66} +6.77627 q^{67} +6.01944 q^{68} -7.36806 q^{69} -2.83279 q^{70} +14.8404 q^{71} -3.72855 q^{72} -1.64480 q^{73} -1.26944 q^{74} +11.0533 q^{75} +0.272395 q^{76} +0.706870 q^{77} +1.49922 q^{78} +2.37919 q^{79} -0.859547 q^{80} -6.28357 q^{81} +8.40030 q^{82} -0.683145 q^{83} +8.54880 q^{84} -5.17399 q^{85} +0.438888 q^{86} -9.89626 q^{87} +0.214484 q^{88} -16.0029 q^{89} +3.20486 q^{90} -1.90480 q^{91} +2.84049 q^{92} -22.3668 q^{93} -7.89612 q^{94} -0.234136 q^{95} +2.59394 q^{96} +3.79190 q^{97} -3.86149 q^{98} -0.799715 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) $ 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * 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
$3$	−2.59394	−1.49761	−0.748807	−	0.662788i	$-0.769373\pi$
$3$	−2.59394	−1.49761	−0.748807	+	0.662788i	$0.769373\pi$
$4$	1.00000	0.500000
$5$	−0.859547	−0.384401	−0.192201	−	0.981356i	$-0.561562\pi$
$5$	−0.859547	−0.384401	−0.192201	+	0.981356i	$0.561562\pi$
$6$	2.59394	1.05897
$7$	−3.29568	−1.24565	−0.622824	−	0.782362i	$-0.714015\pi$
$7$	−3.29568	−1.24565	−0.622824	+	0.782362i	$0.714015\pi$
$8$	−1.00000	−0.353553
$9$	3.72855	1.24285
$10$	0.859547	0.271813
$11$	−0.214484	−0.0646694	−0.0323347	−	0.999477i	$-0.510294\pi$
$11$	−0.214484	−0.0646694	−0.0323347	+	0.999477i	$0.510294\pi$
$12$	−2.59394	−0.748807
$13$	0.577971	0.160300	0.0801501	−	0.996783i	$-0.474460\pi$
$13$	0.577971	0.160300	0.0801501	+	0.996783i	$0.474460\pi$
$14$	3.29568	0.880807
$15$	2.22962	0.575685
$16$	1.00000	0.250000
$17$	6.01944	1.45993	0.729964	−	0.683485i	$-0.239537\pi$
$17$	6.01944	1.45993	0.729964	+	0.683485i	$0.239537\pi$
$18$	−3.72855	−0.878828
$19$	0.272395	0.0624917	0.0312458	−	0.999512i	$-0.490053\pi$
$19$	0.272395	0.0624917	0.0312458	+	0.999512i	$0.490053\pi$
$20$	−0.859547	−0.192201
$21$	8.54880	1.86550
$22$	0.214484	0.0457282
$23$	2.84049	0.592282	0.296141	−	0.955144i	$-0.404300\pi$
$23$	2.84049	0.592282	0.296141	+	0.955144i	$0.404300\pi$
$24$	2.59394	0.529487
$25$	−4.26118	−0.852236
$26$	−0.577971	−0.113349
$27$	−1.88982	−0.363696
$28$	−3.29568	−0.622824
$29$	3.81514	0.708453	0.354227	−	0.935160i	$-0.384744\pi$
$29$	3.81514	0.708453	0.354227	+	0.935160i	$0.384744\pi$
$30$	−2.22962	−0.407071
$31$	8.62269	1.54868	0.774341	−	0.632769i	$-0.218082\pi$
$31$	8.62269	1.54868	0.774341	+	0.632769i	$0.218082\pi$
$32$	−1.00000	−0.176777
$33$	0.556360	0.0968498
$34$	−6.01944	−1.03233
$35$	2.83279	0.478829
$36$	3.72855	0.621425
$37$	1.26944	0.208695	0.104347	−	0.994541i	$-0.466725\pi$
$37$	1.26944	0.208695	0.104347	+	0.994541i	$0.466725\pi$
$38$	−0.272395	−0.0441883
$39$	−1.49922	−0.240068
$40$	0.859547	0.135906
$41$	−8.40030	−1.31191	−0.655953	−	0.754801i	$-0.727733\pi$
$41$	−8.40030	−1.31191	−0.655953	+	0.754801i	$0.727733\pi$
$42$	−8.54880	−1.31911
$43$	−0.438888	−0.0669299	−0.0334649	−	0.999440i	$-0.510654\pi$
$43$	−0.438888	−0.0669299	−0.0334649	+	0.999440i	$0.510654\pi$
$44$	−0.214484	−0.0323347
$45$	−3.20486	−0.477753
$46$	−2.84049	−0.418807
$47$	7.89612	1.15177	0.575884	−	0.817531i	$-0.304658\pi$
$47$	7.89612	1.15177	0.575884	+	0.817531i	$0.304658\pi$
$48$	−2.59394	−0.374404
$49$	3.86149	0.551641
$50$	4.26118	0.602622
$51$	−15.6141	−2.18641
$52$	0.577971	0.0801501
$53$	12.2211	1.67869	0.839347	−	0.543596i	$-0.182938\pi$
$53$	12.2211	1.67869	0.839347	+	0.543596i	$0.182938\pi$
$54$	1.88982	0.257172
$55$	0.184359	0.0248590
$56$	3.29568	0.440403
$57$	−0.706577	−0.0935885
$58$	−3.81514	−0.500952
$59$	−3.37745	−0.439706	−0.219853	−	0.975533i	$-0.570558\pi$
$59$	−3.37745	−0.439706	−0.219853	+	0.975533i	$0.570558\pi$
$60$	2.22962	0.287842
$61$	−9.85667	−1.26202	−0.631008	−	0.775776i	$-0.717359\pi$
$61$	−9.85667	−1.26202	−0.631008	+	0.775776i	$0.717359\pi$
$62$	−8.62269	−1.09508
$63$	−12.2881	−1.54815
$64$	1.00000	0.125000
$65$	−0.496793	−0.0616196
$66$	−0.556360	−0.0684832
$67$	6.77627	0.827854	0.413927	−	0.910310i	$-0.364157\pi$
$67$	6.77627	0.827854	0.413927	+	0.910310i	$0.364157\pi$
$68$	6.01944	0.729964
$69$	−7.36806	−0.887011
$70$	−2.83279	−0.338583
$71$	14.8404	1.76123	0.880617	−	0.473829i	$-0.157128\pi$
$71$	14.8404	1.76123	0.880617	+	0.473829i	$0.157128\pi$
$72$	−3.72855	−0.439414
$73$	−1.64480	−0.192509	−0.0962546	−	0.995357i	$-0.530686\pi$
$73$	−1.64480	−0.192509	−0.0962546	+	0.995357i	$0.530686\pi$
$74$	−1.26944	−0.147569
$75$	11.0533	1.27632
$76$	0.272395	0.0312458
$77$	0.706870	0.0805553
$78$	1.49922	0.169754
$79$	2.37919	0.267679	0.133840	−	0.991003i	$-0.457269\pi$
$79$	2.37919	0.267679	0.133840	+	0.991003i	$0.457269\pi$
$80$	−0.859547	−0.0961003
$81$	−6.28357	−0.698174
$82$	8.40030	0.927658
$83$	−0.683145	−0.0749849	−0.0374925	−	0.999297i	$-0.511937\pi$
$83$	−0.683145	−0.0749849	−0.0374925	+	0.999297i	$0.511937\pi$
$84$	8.54880	0.932751
$85$	−5.17399	−0.561198
$86$	0.438888	0.0473266
$87$	−9.89626	−1.06099
$88$	0.214484	0.0228641
$89$	−16.0029	−1.69630	−0.848151	−	0.529755i	$-0.822284\pi$
$89$	−16.0029	−1.69630	−0.848151	+	0.529755i	$0.822284\pi$
$90$	3.20486	0.337822
$91$	−1.90480	−0.199678
$92$	2.84049	0.296141
$93$	−22.3668	−2.31933
$94$	−7.89612	−0.814423
$95$	−0.234136	−0.0240219
$96$	2.59394	0.264743
$97$	3.79190	0.385009	0.192505	−	0.981296i	$-0.438339\pi$
$97$	3.79190	0.385009	0.192505	+	0.981296i	$0.438339\pi$
$98$	−3.86149	−0.390069
$99$	−0.799715	−0.0803743
$100$	−4.26118	−0.426118
$101$	−7.52717	−0.748981	−0.374491	−	0.927231i	$-0.622182\pi$
$101$	−7.52717	−0.748981	−0.374491	+	0.927231i	$0.622182\pi$
$102$	15.6141	1.54603
$103$	−0.884886	−0.0871904	−0.0435952	−	0.999049i	$-0.513881\pi$
$103$	−0.884886	−0.0871904	−0.0435952	+	0.999049i	$0.513881\pi$
$104$	−0.577971	−0.0566747
$105$	−7.34810	−0.717101
$106$	−12.2211	−1.18702
$107$	−6.22409	−0.601705	−0.300853	−	0.953671i	$-0.597271\pi$
$107$	−6.22409	−0.601705	−0.300853	+	0.953671i	$0.597271\pi$
$108$	−1.88982	−0.181848
$109$	−16.8282	−1.61185	−0.805924	−	0.592020i	$-0.798331\pi$
$109$	−16.8282	−1.61185	−0.805924	+	0.592020i	$0.798331\pi$
$110$	−0.184359	−0.0175780
$111$	−3.29286	−0.312544
$112$	−3.29568	−0.311412
$113$	−10.5751	−0.994823	−0.497412	−	0.867515i	$-0.665716\pi$
$113$	−10.5751	−0.994823	−0.497412	+	0.867515i	$0.665716\pi$
$114$	0.706577	0.0661770
$115$	−2.44153	−0.227674
$116$	3.81514	0.354227
$117$	2.15499	0.199229
$118$	3.37745	0.310919
$119$	−19.8381	−1.81856
$120$	−2.22962	−0.203535
$121$	−10.9540	−0.995818
$122$	9.85667	0.892381
$123$	21.7899	1.96473
$124$	8.62269	0.774341
$125$	7.96042	0.712001
$126$	12.2881	1.09471
$127$	0.441878	0.0392103	0.0196052	−	0.999808i	$-0.493759\pi$
$127$	0.441878	0.0392103	0.0196052	+	0.999808i	$0.493759\pi$
$128$	−1.00000	−0.0883883
$129$	1.13845	0.100235
$130$	0.496793	0.0435716
$131$	11.5681	1.01071	0.505353	−	0.862913i	$-0.331362\pi$
$131$	11.5681	1.01071	0.505353	+	0.862913i	$0.331362\pi$
$132$	0.556360	0.0484249
$133$	−0.897726	−0.0778427
$134$	−6.77627	−0.585381
$135$	1.62439	0.139805
$136$	−6.01944	−0.516163
$137$	−0.498277	−0.0425707	−0.0212853	−	0.999773i	$-0.506776\pi$
$137$	−0.498277	−0.0425707	−0.0212853	+	0.999773i	$0.506776\pi$
$138$	7.36806	0.627211
$139$	16.9212	1.43524	0.717619	−	0.696436i	$-0.245232\pi$
$139$	16.9212	1.43524	0.717619	+	0.696436i	$0.245232\pi$
$140$	2.83279	0.239414
$141$	−20.4821	−1.72490
$142$	−14.8404	−1.24538
$143$	−0.123965	−0.0103665
$144$	3.72855	0.310712
$145$	−3.27929	−0.272330
$146$	1.64480	0.136125
$147$	−10.0165	−0.826145
$148$	1.26944	0.104347
$149$	16.0277	1.31304	0.656520	−	0.754309i	$-0.272028\pi$
$149$	16.0277	1.31304	0.656520	+	0.754309i	$0.272028\pi$
$150$	−11.0533	−0.902495
$151$	7.91714	0.644288	0.322144	−	0.946691i	$-0.395597\pi$
$151$	7.91714	0.644288	0.322144	+	0.946691i	$0.395597\pi$
$152$	−0.272395	−0.0220941
$153$	22.4438	1.81447
$154$	−0.706870	−0.0569612
$155$	−7.41161	−0.595315
$156$	−1.49922	−0.120034
$157$	−5.37384	−0.428879	−0.214440	−	0.976737i	$-0.568793\pi$
$157$	−5.37384	−0.428879	−0.214440	+	0.976737i	$0.568793\pi$
$158$	−2.37919	−0.189278
$159$	−31.7008	−2.51404
$160$	0.859547	0.0679531
$161$	−9.36132	−0.737776
$162$	6.28357	0.493684
$163$	12.4404	0.974410	0.487205	−	0.873288i	$-0.338016\pi$
$163$	12.4404	0.974410	0.487205	+	0.873288i	$0.338016\pi$
$164$	−8.40030	−0.655953
$165$	−0.478217	−0.0372292
$166$	0.683145	0.0530223
$167$	−20.0060	−1.54811	−0.774057	−	0.633116i	$-0.781776\pi$
$167$	−20.0060	−1.54811	−0.774057	+	0.633116i	$0.781776\pi$
$168$	−8.54880	−0.659555
$169$	−12.6660	−0.974304
$170$	5.17399	0.396827
$171$	1.01564	0.0776678
$172$	−0.438888	−0.0334649
$173$	18.5547	1.41068	0.705342	−	0.708867i	$-0.250793\pi$
$173$	18.5547	1.41068	0.705342	+	0.708867i	$0.250793\pi$
$174$	9.89626	0.750233
$175$	14.0435	1.06159
$176$	−0.214484	−0.0161673
$177$	8.76091	0.658510
$178$	16.0029	1.19947
$179$	13.2578	0.990933	0.495466	−	0.868627i	$-0.334997\pi$
$179$	13.2578	0.990933	0.495466	+	0.868627i	$0.334997\pi$
$180$	−3.20486	−0.238876
$181$	−3.73076	−0.277305	−0.138653	−	0.990341i	$-0.544277\pi$
$181$	−3.73076	−0.277305	−0.138653	+	0.990341i	$0.544277\pi$
$182$	1.90480	0.141193
$183$	25.5676	1.89001
$184$	−2.84049	−0.209403
$185$	−1.09114	−0.0802224
$186$	22.3668	1.64001
$187$	−1.29107	−0.0944127
$188$	7.89612	0.575884
$189$	6.22823	0.453037
$190$	0.234136	0.0169860
$191$	16.8251	1.21742	0.608709	−	0.793393i	$-0.291688\pi$
$191$	16.8251	1.21742	0.608709	+	0.793393i	$0.291688\pi$
$192$	−2.59394	−0.187202
$193$	−2.81052	−0.202306	−0.101153	−	0.994871i	$-0.532253\pi$
$193$	−2.81052	−0.202306	−0.101153	+	0.994871i	$0.532253\pi$
$194$	−3.79190	−0.272242
$195$	1.28865	0.0922824
$196$	3.86149	0.275820
$197$	−16.4126	−1.16935	−0.584676	−	0.811267i	$-0.698778\pi$
$197$	−16.4126	−1.16935	−0.584676	+	0.811267i	$0.698778\pi$
$198$	0.799715	0.0568332
$199$	14.8153	1.05023	0.525116	−	0.851031i	$-0.324022\pi$
$199$	14.8153	1.05023	0.525116	+	0.851031i	$0.324022\pi$
$200$	4.26118	0.301311
$201$	−17.5773	−1.23981
$202$	7.52717	0.529610
$203$	−12.5735	−0.882484
$204$	−15.6141	−1.09320
$205$	7.22045	0.504298
$206$	0.884886	0.0616530
$207$	10.5909	0.736118
$208$	0.577971	0.0400750
$209$	−0.0584244	−0.00404130
$210$	7.34810	0.507067
$211$	−15.7474	−1.08410	−0.542049	−	0.840347i	$-0.682351\pi$
$211$	−15.7474	−1.08410	−0.542049	+	0.840347i	$0.682351\pi$
$212$	12.2211	0.839347
$213$	−38.4952	−2.63765
$214$	6.22409	0.425470
$215$	0.377245	0.0257279
$216$	1.88982	0.128586
$217$	−28.4176	−1.92911
$218$	16.8282	1.13975
$219$	4.26652	0.288305
$220$	0.184359	0.0124295
$221$	3.47906	0.234027
$222$	3.29286	0.221002
$223$	7.69950	0.515597	0.257798	−	0.966199i	$-0.417003\pi$
$223$	7.69950	0.515597	0.257798	+	0.966199i	$0.417003\pi$
$224$	3.29568	0.220202
$225$	−15.8880	−1.05920
$226$	10.5751	0.703446
$227$	12.9742	0.861129	0.430565	−	0.902560i	$-0.358314\pi$
$227$	12.9742	0.861129	0.430565	+	0.902560i	$0.358314\pi$
$228$	−0.706577	−0.0467942
$229$	−4.83978	−0.319822	−0.159911	−	0.987131i	$-0.551121\pi$
$229$	−4.83978	−0.319822	−0.159911	+	0.987131i	$0.551121\pi$
$230$	2.44153	0.160990
$231$	−1.83358	−0.120641
$232$	−3.81514	−0.250476
$233$	12.1250	0.794333	0.397166	−	0.917747i	$-0.369994\pi$
$233$	12.1250	0.794333	0.397166	+	0.917747i	$0.369994\pi$
$234$	−2.15499	−0.140876
$235$	−6.78709	−0.442741
$236$	−3.37745	−0.219853
$237$	−6.17148	−0.400881
$238$	19.8381	1.28591
$239$	−28.8855	−1.86845	−0.934225	−	0.356684i	$-0.883907\pi$
$239$	−28.8855	−1.86845	−0.934225	+	0.356684i	$0.883907\pi$
$240$	2.22962	0.143921
$241$	14.6062	0.940866	0.470433	−	0.882436i	$-0.344098\pi$
$241$	14.6062	0.940866	0.470433	+	0.882436i	$0.344098\pi$
$242$	10.9540	0.704150
$243$	21.9687	1.40929
$244$	−9.85667	−0.631008
$245$	−3.31913	−0.212051
$246$	−21.7899	−1.38927
$247$	0.157436	0.0100174
$248$	−8.62269	−0.547541
$249$	1.77204	0.112298
$250$	−7.96042	−0.503461
$251$	−29.9159	−1.88827	−0.944137	−	0.329553i	$-0.893102\pi$
$251$	−29.9159	−1.88827	−0.944137	+	0.329553i	$0.893102\pi$
$252$	−12.2881	−0.774077
$253$	−0.609239	−0.0383025
$254$	−0.441878	−0.0277259
$255$	13.4210	0.840458
$256$	1.00000	0.0625000
$257$	−18.0859	−1.12817	−0.564084	−	0.825718i	$-0.690770\pi$
$257$	−18.0859	−1.12817	−0.564084	+	0.825718i	$0.690770\pi$
$258$	−1.13845	−0.0708769
$259$	−4.18366	−0.259960
$260$	−0.496793	−0.0308098
$261$	14.2249	0.880501
$262$	−11.5681	−0.714677
$263$	−15.2492	−0.940304	−0.470152	−	0.882585i	$-0.655801\pi$
$263$	−15.2492	−0.940304	−0.470152	+	0.882585i	$0.655801\pi$
$264$	−0.556360	−0.0342416
$265$	−10.5046	−0.645292
$266$	0.897726	0.0550431
$267$	41.5106	2.54041
$268$	6.77627	0.413927
$269$	−15.9496	−0.972465	−0.486233	−	0.873829i	$-0.661629\pi$
$269$	−15.9496	−0.972465	−0.486233	+	0.873829i	$0.661629\pi$
$270$	−1.62439	−0.0988571
$271$	24.3920	1.48171	0.740855	−	0.671665i	$-0.234420\pi$
$271$	24.3920	1.48171	0.740855	+	0.671665i	$0.234420\pi$
$272$	6.01944	0.364982
$273$	4.94096	0.299040
$274$	0.498277	0.0301020
$275$	0.913955	0.0551136
$276$	−7.36806	−0.443505
$277$	−6.85269	−0.411738	−0.205869	−	0.978580i	$-0.566002\pi$
$277$	−6.85269	−0.411738	−0.205869	+	0.978580i	$0.566002\pi$
$278$	−16.9212	−1.01487
$279$	32.1501	1.92478
$280$	−2.83279	−0.169291
$281$	2.93915	0.175335	0.0876676	−	0.996150i	$-0.472059\pi$
$281$	2.93915	0.175335	0.0876676	+	0.996150i	$0.472059\pi$
$282$	20.4821	1.21969
$283$	−22.1499	−1.31667	−0.658336	−	0.752724i	$-0.728740\pi$
$283$	−22.1499	−1.31667	−0.658336	+	0.752724i	$0.728740\pi$
$284$	14.8404	0.880617
$285$	0.607336	0.0359755
$286$	0.123965	0.00733023
$287$	27.6847	1.63418
$288$	−3.72855	−0.219707
$289$	19.2336	1.13139
$290$	3.27929	0.192567
$291$	−9.83598	−0.576595
$292$	−1.64480	−0.0962546
$293$	−2.73354	−0.159695	−0.0798476	−	0.996807i	$-0.525443\pi$
$293$	−2.73354	−0.159695	−0.0798476	+	0.996807i	$0.525443\pi$
$294$	10.0165	0.584173
$295$	2.90307	0.169023
$296$	−1.26944	−0.0737847
$297$	0.405336	0.0235200
$298$	−16.0277	−0.928459
$299$	1.64172	0.0949430
$300$	11.0533	0.638160
$301$	1.44643	0.0833711
$302$	−7.91714	−0.455580
$303$	19.5251	1.12169
$304$	0.272395	0.0156229
$305$	8.47227	0.485121
$306$	−22.4438	−1.28303
$307$	9.36898	0.534716	0.267358	−	0.963597i	$-0.413849\pi$
$307$	9.36898	0.534716	0.267358	+	0.963597i	$0.413849\pi$
$308$	0.706870	0.0402777
$309$	2.29535	0.130578
$310$	7.41161	0.420951
$311$	−25.2611	−1.43242	−0.716211	−	0.697883i	$-0.754125\pi$
$311$	−25.2611	−1.43242	−0.716211	+	0.697883i	$0.754125\pi$
$312$	1.49922	0.0848768
$313$	−4.99421	−0.282289	−0.141145	−	0.989989i	$-0.545078\pi$
$313$	−4.99421	−0.282289	−0.141145	+	0.989989i	$0.545078\pi$
$314$	5.37384	0.303263
$315$	10.5622	0.595112
$316$	2.37919	0.133840
$317$	13.3623	0.750503	0.375251	−	0.926923i	$-0.377556\pi$
$317$	13.3623	0.750503	0.375251	+	0.926923i	$0.377556\pi$
$318$	31.7008	1.77769
$319$	−0.818286	−0.0458152
$320$	−0.859547	−0.0480501
$321$	16.1449	0.901123
$322$	9.36132	0.521686
$323$	1.63966	0.0912334
$324$	−6.28357	−0.349087
$325$	−2.46284	−0.136614
$326$	−12.4404	−0.689012
$327$	43.6514	2.41393
$328$	8.40030	0.463829
$329$	−26.0231	−1.43470
$330$	0.478217	0.0263250
$331$	−16.1125	−0.885624	−0.442812	−	0.896614i	$-0.646019\pi$
$331$	−16.1125	−0.885624	−0.442812	+	0.896614i	$0.646019\pi$
$332$	−0.683145	−0.0374925
$333$	4.73317	0.259376
$334$	20.0060	1.09468
$335$	−5.82452	−0.318228
$336$	8.54880	0.466375
$337$	−0.830310	−0.0452299	−0.0226149	−	0.999744i	$-0.507199\pi$
$337$	−0.830310	−0.0452299	−0.0226149	+	0.999744i	$0.507199\pi$
$338$	12.6660	0.688937
$339$	27.4313	1.48986
$340$	−5.17399	−0.280599
$341$	−1.84943	−0.100152
$342$	−1.01564	−0.0549194
$343$	10.3435	0.558498
$344$	0.438888	0.0236633
$345$	6.33320	0.340968
$346$	−18.5547	−0.997505
$347$	−5.79706	−0.311202	−0.155601	−	0.987820i	$-0.549731\pi$
$347$	−5.79706	−0.311202	−0.155601	+	0.987820i	$0.549731\pi$
$348$	−9.89626	−0.530495
$349$	13.0665	0.699432	0.349716	−	0.936856i	$-0.386278\pi$
$349$	13.0665	0.699432	0.349716	+	0.936856i	$0.386278\pi$
$350$	−14.0435	−0.750655
$351$	−1.09226	−0.0583005
$352$	0.214484	0.0114320
$353$	28.8089	1.53334	0.766672	−	0.642038i	$-0.221911\pi$
$353$	28.8089	1.53334	0.766672	+	0.642038i	$0.221911\pi$
$354$	−8.76091	−0.465637
$355$	−12.7560	−0.677020
$356$	−16.0029	−0.848151
$357$	51.4590	2.72350
$358$	−13.2578	−0.700695
$359$	17.9892	0.949432	0.474716	−	0.880139i	$-0.342551\pi$
$359$	17.9892	0.949432	0.474716	+	0.880139i	$0.342551\pi$
$360$	3.20486	0.168911
$361$	−18.9258	−0.996095
$362$	3.73076	0.196084
$363$	28.4141	1.49135
$364$	−1.90480	−0.0998389
$365$	1.41378	0.0740008
$366$	−25.5676	−1.33644
$367$	−12.6204	−0.658778	−0.329389	−	0.944194i	$-0.606843\pi$
$367$	−12.6204	−0.658778	−0.329389	+	0.944194i	$0.606843\pi$
$368$	2.84049	0.148071
$369$	−31.3209	−1.63050
$370$	1.09114	0.0567258
$371$	−40.2767	−2.09106
$372$	−22.3668	−1.15966
$373$	31.7521	1.64406	0.822031	−	0.569443i	$-0.192841\pi$
$373$	31.7521	1.64406	0.822031	+	0.569443i	$0.192841\pi$
$374$	1.29107	0.0667598
$375$	−20.6489	−1.06630
$376$	−7.89612	−0.407212
$377$	2.20504	0.113565
$378$	−6.22823	−0.320346
$379$	−36.0652	−1.85254	−0.926272	−	0.376855i	$-0.877006\pi$
$379$	−36.0652	−1.85254	−0.926272	+	0.376855i	$0.877006\pi$
$380$	−0.234136	−0.0120109
$381$	−1.14621	−0.0587220
$382$	−16.8251	−0.860845
$383$	−30.2081	−1.54356	−0.771781	−	0.635889i	$-0.780634\pi$
$383$	−30.2081	−1.54356	−0.771781	+	0.635889i	$0.780634\pi$
$384$	2.59394	0.132372
$385$	−0.607588	−0.0309656
$386$	2.81052	0.143052
$387$	−1.63642	−0.0831838
$388$	3.79190	0.192505
$389$	20.5431	1.04158	0.520789	−	0.853686i	$-0.325638\pi$
$389$	20.5431	1.04158	0.520789	+	0.853686i	$0.325638\pi$
$390$	−1.28865	−0.0652535
$391$	17.0981	0.864690
$392$	−3.86149	−0.195034
$393$	−30.0069	−1.51365
$394$	16.4126	0.826856
$395$	−2.04502	−0.102896
$396$	−0.799715	−0.0401872
$397$	21.4893	1.07852	0.539258	−	0.842141i	$-0.318705\pi$
$397$	21.4893	1.07852	0.539258	+	0.842141i	$0.318705\pi$
$398$	−14.8153	−0.742626
$399$	2.32865	0.116578
$400$	−4.26118	−0.213059
$401$	−37.0212	−1.84875	−0.924376	−	0.381483i	$-0.875413\pi$
$401$	−37.0212	−1.84875	−0.924376	+	0.381483i	$0.875413\pi$
$402$	17.5773	0.876675
$403$	4.98366	0.248254
$404$	−7.52717	−0.374491
$405$	5.40102	0.268379
$406$	12.5735	0.624010
$407$	−0.272275	−0.0134962
$408$	15.6141	0.773013
$409$	−20.3727	−1.00736	−0.503682	−	0.863889i	$-0.668022\pi$
$409$	−20.3727	−1.00736	−0.503682	+	0.863889i	$0.668022\pi$
$410$	−7.22045	−0.356593
$411$	1.29250	0.0637545
$412$	−0.884886	−0.0435952
$413$	11.1310	0.547719
$414$	−10.5909	−0.520514
$415$	0.587195	0.0288243
$416$	−0.577971	−0.0283373
$417$	−43.8927	−2.14943
$418$	0.0584244	0.00285763
$419$	28.1112	1.37332	0.686661	−	0.726978i	$-0.259076\pi$
$419$	28.1112	1.37332	0.686661	+	0.726978i	$0.259076\pi$
$420$	−7.34810	−0.358550
$421$	−0.202907	−0.00988910	−0.00494455	−	0.999988i	$-0.501574\pi$
$421$	−0.202907	−0.00988910	−0.00494455	+	0.999988i	$0.501574\pi$
$422$	15.7474	0.766573
$423$	29.4411	1.43147
$424$	−12.2211	−0.593508
$425$	−25.6499	−1.24420
$426$	38.4952	1.86510
$427$	32.4844	1.57203
$428$	−6.22409	−0.300853
$429$	0.321560	0.0155250
$430$	−0.377245	−0.0181924
$431$	12.4226	0.598376	0.299188	−	0.954194i	$-0.403284\pi$
$431$	12.4226	0.598376	0.299188	+	0.954194i	$0.403284\pi$
$432$	−1.88982	−0.0909239
$433$	31.2682	1.50266	0.751328	−	0.659929i	$-0.229414\pi$
$433$	31.2682	1.50266	0.751328	+	0.659929i	$0.229414\pi$
$434$	28.4176	1.36409
$435$	8.50630	0.407846
$436$	−16.8282	−0.805924
$437$	0.773734	0.0370127
$438$	−4.26652	−0.203862
$439$	20.9378	0.999304	0.499652	−	0.866226i	$-0.333461\pi$
$439$	20.9378	0.999304	0.499652	+	0.866226i	$0.333461\pi$
$440$	−0.184359	−0.00878898
$441$	14.3977	0.685607
$442$	−3.47906	−0.165482
$443$	12.4567	0.591836	0.295918	−	0.955213i	$-0.404375\pi$
$443$	12.4567	0.591836	0.295918	+	0.955213i	$0.404375\pi$
$444$	−3.29286	−0.156272
$445$	13.7552	0.652060
$446$	−7.69950	−0.364582
$447$	−41.5749	−1.96643
$448$	−3.29568	−0.155706
$449$	5.45129	0.257262	0.128631	−	0.991693i	$-0.458942\pi$
$449$	5.45129	0.257262	0.128631	+	0.991693i	$0.458942\pi$
$450$	15.8880	0.748968
$451$	1.80173	0.0848402
$452$	−10.5751	−0.497412
$453$	−20.5366	−0.964895
$454$	−12.9742	−0.608910
$455$	1.63727	0.0767563
$456$	0.706577	0.0330885
$457$	36.5002	1.70741	0.853703	−	0.520760i	$-0.174351\pi$
$457$	36.5002	1.70741	0.853703	+	0.520760i	$0.174351\pi$
$458$	4.83978	0.226148
$459$	−11.3756	−0.530970
$460$	−2.44153	−0.113837
$461$	16.5147	0.769167	0.384583	−	0.923090i	$-0.374345\pi$
$461$	16.5147	0.769167	0.384583	+	0.923090i	$0.374345\pi$
$462$	1.83358	0.0853060
$463$	12.7582	0.592923	0.296461	−	0.955045i	$-0.404193\pi$
$463$	12.7582	0.592923	0.296461	+	0.955045i	$0.404193\pi$
$464$	3.81514	0.177113
$465$	19.2253	0.891552
$466$	−12.1250	−0.561678
$467$	5.22843	0.241943	0.120971	−	0.992656i	$-0.461399\pi$
$467$	5.22843	0.241943	0.120971	+	0.992656i	$0.461399\pi$
$468$	2.15499	0.0996145
$469$	−22.3324	−1.03121
$470$	6.78709	0.313065
$471$	13.9394	0.642296
$472$	3.37745	0.155460
$473$	0.0941346	0.00432831
$474$	6.17148	0.283465
$475$	−1.16072	−0.0532577
$476$	−19.8381	−0.909279
$477$	45.5669	2.08636
$478$	28.8855	1.32119
$479$	−0.185788	−0.00848889	−0.00424445	−	0.999991i	$-0.501351\pi$
$479$	−0.185788	−0.00848889	−0.00424445	+	0.999991i	$0.501351\pi$
$480$	−2.22962	−0.101768
$481$	0.733699	0.0334538
$482$	−14.6062	−0.665293
$483$	24.2828	1.10490
$484$	−10.9540	−0.497909
$485$	−3.25932	−0.147998
$486$	−21.9687	−0.996519
$487$	−5.11940	−0.231982	−0.115991	−	0.993250i	$-0.537004\pi$
$487$	−5.11940	−0.231982	−0.115991	+	0.993250i	$0.537004\pi$
$488$	9.85667	0.446190
$489$	−32.2698	−1.45929
$490$	3.31913	0.149943
$491$	18.1467	0.818951	0.409476	−	0.912321i	$-0.365712\pi$
$491$	18.1467	0.818951	0.409476	+	0.912321i	$0.365712\pi$
$492$	21.7899	0.982365
$493$	22.9650	1.03429
$494$	−0.157436	−0.00708339
$495$	0.687392	0.0308960
$496$	8.62269	0.387170
$497$	−48.9092	−2.19388
$498$	−1.77204	−0.0794070
$499$	−20.2236	−0.905335	−0.452667	−	0.891679i	$-0.649528\pi$
$499$	−20.2236	−0.905335	−0.452667	+	0.891679i	$0.649528\pi$
$500$	7.96042	0.356001
$501$	51.8946	2.31848
$502$	29.9159	1.33521
$503$	11.2702	0.502513	0.251257	−	0.967920i	$-0.419156\pi$
$503$	11.2702	0.502513	0.251257	+	0.967920i	$0.419156\pi$
$504$	12.2881	0.547355
$505$	6.46995	0.287909
$506$	0.609239	0.0270840
$507$	32.8548	1.45913
$508$	0.441878	0.0196052
$509$	−11.5245	−0.510814	−0.255407	−	0.966834i	$-0.582209\pi$
$509$	−11.5245	−0.510814	−0.255407	+	0.966834i	$0.582209\pi$
$510$	−13.4210	−0.594294
$511$	5.42073	0.239799
$512$	−1.00000	−0.0441942
$513$	−0.514777	−0.0227280
$514$	18.0859	0.797735
$515$	0.760601	0.0335161
$516$	1.13845	0.0501176
$517$	−1.69359	−0.0744841
$518$	4.18366	0.183820
$519$	−48.1298	−2.11266
$520$	0.496793	0.0217858
$521$	31.5858	1.38380	0.691900	−	0.721993i	$-0.256774\pi$
$521$	31.5858	1.38380	0.691900	+	0.721993i	$0.256774\pi$
$522$	−14.2249	−0.622608
$523$	−4.78787	−0.209359	−0.104679	−	0.994506i	$-0.533382\pi$
$523$	−4.78787	−0.209359	−0.104679	+	0.994506i	$0.533382\pi$
$524$	11.5681	0.505353
$525$	−36.4280	−1.58985
$526$	15.2492	0.664896
$527$	51.9038	2.26096
$528$	0.556360	0.0242125
$529$	−14.9316	−0.649202
$530$	10.5046	0.456290
$531$	−12.5930	−0.546489
$532$	−0.897726	−0.0389213
$533$	−4.85513	−0.210299
$534$	−41.5106	−1.79634
$535$	5.34990	0.231296
$536$	−6.77627	−0.292690
$537$	−34.3899	−1.48404
$538$	15.9496	0.687637
$539$	−0.828227	−0.0356743
$540$	1.62439	0.0699025
$541$	6.62763	0.284944	0.142472	−	0.989799i	$-0.454495\pi$
$541$	6.62763	0.284944	0.142472	+	0.989799i	$0.454495\pi$
$542$	−24.3920	−1.04773
$543$	9.67738	0.415296
$544$	−6.01944	−0.258081
$545$	14.4646	0.619596
$546$	−4.94096	−0.211453
$547$	32.1201	1.37335	0.686677	−	0.726962i	$-0.259068\pi$
$547$	32.1201	1.37335	0.686677	+	0.726962i	$0.259068\pi$
$548$	−0.498277	−0.0212853
$549$	−36.7511	−1.56850
$550$	−0.913955	−0.0389712
$551$	1.03922	0.0442724
$552$	7.36806	0.313606
$553$	−7.84103	−0.333435
$554$	6.85269	0.291143
$555$	2.83037	0.120142
$556$	16.9212	0.717619
$557$	33.9454	1.43831	0.719157	−	0.694848i	$-0.244528\pi$
$557$	33.9454	1.43831	0.719157	+	0.694848i	$0.244528\pi$
$558$	−32.1501	−1.36102
$559$	−0.253665	−0.0107289
$560$	2.83279	0.119707
$561$	3.34897	0.141394
$562$	−2.93915	−0.123981
$563$	−16.5135	−0.695961	−0.347981	−	0.937502i	$-0.613132\pi$
$563$	−16.5135	−0.695961	−0.347981	+	0.937502i	$0.613132\pi$
$564$	−20.4821	−0.862452
$565$	9.08981	0.382411
$566$	22.1499	0.931028
$567$	20.7086	0.869680
$568$	−14.8404	−0.622690
$569$	−31.1502	−1.30588	−0.652942	−	0.757408i	$-0.726465\pi$
$569$	−31.1502	−1.30588	−0.652942	+	0.757408i	$0.726465\pi$
$570$	−0.607336	−0.0254385
$571$	−15.4768	−0.647684	−0.323842	−	0.946111i	$-0.604975\pi$
$571$	−15.4768	−0.647684	−0.323842	+	0.946111i	$0.604975\pi$
$572$	−0.123965	−0.00518326
$573$	−43.6433	−1.82322
$574$	−27.6847	−1.15554
$575$	−12.1038	−0.504764
$576$	3.72855	0.155356
$577$	24.4444	1.01763	0.508817	−	0.860875i	$-0.330083\pi$
$577$	24.4444	1.01763	0.508817	+	0.860875i	$0.330083\pi$
$578$	−19.2336	−0.800014
$579$	7.29033	0.302976
$580$	−3.27929	−0.136165
$581$	2.25142	0.0934048
$582$	9.83598	0.407714
$583$	−2.62123	−0.108560
$584$	1.64480	0.0680623
$585$	−1.85232	−0.0765839
$586$	2.73354	0.112922
$587$	32.8941	1.35769	0.678843	−	0.734283i	$-0.262482\pi$
$587$	32.8941	1.35769	0.678843	+	0.734283i	$0.262482\pi$
$588$	−10.0165	−0.413073
$589$	2.34878	0.0967797
$590$	−2.90307	−0.119518
$591$	42.5735	1.75124
$592$	1.26944	0.0521737
$593$	−5.82300	−0.239122	−0.119561	−	0.992827i	$-0.538149\pi$
$593$	−5.82300	−0.239122	−0.119561	+	0.992827i	$0.538149\pi$
$594$	−0.405336	−0.0166311
$595$	17.0518	0.699055
$596$	16.0277	0.656520
$597$	−38.4302	−1.57284
$598$	−1.64172	−0.0671348
$599$	−27.9900	−1.14364	−0.571820	−	0.820379i	$-0.693762\pi$
$599$	−27.9900	−1.14364	−0.571820	+	0.820379i	$0.693762\pi$
$600$	−11.0533	−0.451248
$601$	−10.4177	−0.424947	−0.212474	−	0.977167i	$-0.568152\pi$
$601$	−10.4177	−0.424947	−0.212474	+	0.977167i	$0.568152\pi$
$602$	−1.44643	−0.0589523
$603$	25.2657	1.02890
$604$	7.91714	0.322144
$605$	9.41547	0.382793
$606$	−19.5251	−0.793151
$607$	34.1311	1.38534	0.692670	−	0.721255i	$-0.256434\pi$
$607$	34.1311	1.38534	0.692670	+	0.721255i	$0.256434\pi$
$608$	−0.272395	−0.0110471
$609$	32.6149	1.32162
$610$	−8.47227	−0.343032
$611$	4.56373	0.184629
$612$	22.4438	0.907236
$613$	47.1809	1.90562	0.952810	−	0.303567i	$-0.0981776\pi$
$613$	47.1809	1.90562	0.952810	+	0.303567i	$0.0981776\pi$
$614$	−9.36898	−0.378101
$615$	−18.7295	−0.755245
$616$	−0.706870	−0.0284806
$617$	0.729747	0.0293785	0.0146893	−	0.999892i	$-0.495324\pi$
$617$	0.729747	0.0293785	0.0146893	+	0.999892i	$0.495324\pi$
$618$	−2.29535	−0.0923324
$619$	3.58031	0.143905	0.0719524	−	0.997408i	$-0.477077\pi$
$619$	3.58031	0.143905	0.0719524	+	0.997408i	$0.477077\pi$
$620$	−7.41161	−0.297657
$621$	−5.36800	−0.215410
$622$	25.2611	1.01288
$623$	52.7403	2.11300
$624$	−1.49922	−0.0600170
$625$	14.4635	0.578542
$626$	4.99421	0.199609
$627$	0.151550	0.00605231
$628$	−5.37384	−0.214440
$629$	7.64132	0.304679
$630$	−10.5622	−0.420808
$631$	−3.61541	−0.143927	−0.0719636	−	0.997407i	$-0.522927\pi$
$631$	−3.61541	−0.143927	−0.0719636	+	0.997407i	$0.522927\pi$
$632$	−2.37919	−0.0946390
$633$	40.8480	1.62356
$634$	−13.3623	−0.530686
$635$	−0.379815	−0.0150725
$636$	−31.7008	−1.25702
$637$	2.23182	0.0884281
$638$	0.818286	0.0323963
$639$	55.3332	2.18895
$640$	0.859547	0.0339766
$641$	−25.7428	−1.01678	−0.508389	−	0.861128i	$-0.669759\pi$
$641$	−25.7428	−1.01678	−0.508389	+	0.861128i	$0.669759\pi$
$642$	−16.1449	−0.637190
$643$	36.3175	1.43222	0.716110	−	0.697987i	$-0.245921\pi$
$643$	36.3175	1.43222	0.716110	+	0.697987i	$0.245921\pi$
$644$	−9.36132	−0.368888
$645$	−0.978553	−0.0385305
$646$	−1.63966	−0.0645117
$647$	0.676228	0.0265853	0.0132926	−	0.999912i	$-0.495769\pi$
$647$	0.676228	0.0265853	0.0132926	+	0.999912i	$0.495769\pi$
$648$	6.28357	0.246842
$649$	0.724408	0.0284355
$650$	2.46284	0.0966004
$651$	73.7137	2.88907
$652$	12.4404	0.487205
$653$	−35.6870	−1.39654	−0.698269	−	0.715835i	$-0.746046\pi$
$653$	−35.6870	−1.39654	−0.698269	+	0.715835i	$0.746046\pi$
$654$	−43.6514	−1.70690
$655$	−9.94328	−0.388516
$656$	−8.40030	−0.327977
$657$	−6.13272	−0.239260
$658$	26.0231	1.01449
$659$	36.3846	1.41734	0.708672	−	0.705538i	$-0.249295\pi$
$659$	36.3846	1.41734	0.708672	+	0.705538i	$0.249295\pi$
$660$	−0.478217	−0.0186146
$661$	−4.21235	−0.163841	−0.0819207	−	0.996639i	$-0.526105\pi$
$661$	−4.21235	−0.163841	−0.0819207	+	0.996639i	$0.526105\pi$
$662$	16.1125	0.626231
$663$	−9.02448	−0.350482
$664$	0.683145	0.0265112
$665$	0.771637	0.0299228
$666$	−4.73317	−0.183407
$667$	10.8368	0.419604
$668$	−20.0060	−0.774057
$669$	−19.9721	−0.772165
$670$	5.82452	0.225021
$671$	2.11410	0.0816138
$672$	−8.54880	−0.329777
$673$	40.4786	1.56033	0.780167	−	0.625571i	$-0.215134\pi$
$673$	40.4786	1.56033	0.780167	+	0.625571i	$0.215134\pi$
$674$	0.830310	0.0319824
$675$	8.05285	0.309954
$676$	−12.6660	−0.487152
$677$	−7.86358	−0.302222	−0.151111	−	0.988517i	$-0.548285\pi$
$677$	−7.86358	−0.302222	−0.151111	+	0.988517i	$0.548285\pi$
$678$	−27.4313	−1.05349
$679$	−12.4969	−0.479586
$680$	5.17399	0.198413
$681$	−33.6544	−1.28964
$682$	1.84943	0.0708183
$683$	15.3894	0.588858	0.294429	−	0.955673i	$-0.404871\pi$
$683$	15.3894	0.588858	0.294429	+	0.955673i	$0.404871\pi$
$684$	1.01564	0.0388339
$685$	0.428292	0.0163642
$686$	−10.3435	−0.394918
$687$	12.5541	0.478969
$688$	−0.438888	−0.0167325
$689$	7.06342	0.269095
$690$	−6.33320	−0.241101
$691$	−15.2466	−0.580007	−0.290004	−	0.957026i	$-0.593657\pi$
$691$	−15.2466	−0.580007	−0.290004	+	0.957026i	$0.593657\pi$
$692$	18.5547	0.705342
$693$	2.63560	0.100118
$694$	5.79706	0.220053
$695$	−14.5446	−0.551707
$696$	9.89626	0.375117
$697$	−50.5651	−1.91529
$698$	−13.0665	−0.494573
$699$	−31.4515	−1.18960
$700$	14.0435	0.530793
$701$	40.4465	1.52764	0.763822	−	0.645427i	$-0.223321\pi$
$701$	40.4465	1.52764	0.763822	+	0.645427i	$0.223321\pi$
$702$	1.09226	0.0412247
$703$	0.345789	0.0130417
$704$	−0.214484	−0.00808367
$705$	17.6053	0.663055
$706$	−28.8089	−1.08424
$707$	24.8071	0.932967
$708$	8.76091	0.329255
$709$	−27.7344	−1.04159	−0.520794	−	0.853683i	$-0.674364\pi$
$709$	−27.7344	−1.04159	−0.520794	+	0.853683i	$0.674364\pi$
$710$	12.7560	0.478726
$711$	8.87092	0.332685
$712$	16.0029	0.599733
$713$	24.4926	0.917256
$714$	−51.4590	−1.92580
$715$	0.106554	0.00398490
$716$	13.2578	0.495466
$717$	74.9275	2.79822
$718$	−17.9892	−0.671350
$719$	−42.3104	−1.57791	−0.788956	−	0.614450i	$-0.789378\pi$
$719$	−42.3104	−1.57791	−0.788956	+	0.614450i	$0.789378\pi$
$720$	−3.20486	−0.119438
$721$	2.91630	0.108609
$722$	18.9258	0.704345
$723$	−37.8876	−1.40906
$724$	−3.73076	−0.138653
$725$	−16.2570	−0.603769
$726$	−28.4141	−1.05454
$727$	−44.2633	−1.64164	−0.820818	−	0.571190i	$-0.806482\pi$
$727$	−44.2633	−1.64164	−0.820818	+	0.571190i	$0.806482\pi$
$728$	1.90480	0.0705967
$729$	−38.1348	−1.41240
$730$	−1.41378	−0.0523264
$731$	−2.64186	−0.0977128
$732$	25.5676	0.945007
$733$	14.1990	0.524451	0.262226	−	0.965007i	$-0.415544\pi$
$733$	14.1990	0.524451	0.262226	+	0.965007i	$0.415544\pi$
$734$	12.6204	0.465826
$735$	8.60963	0.317571
$736$	−2.84049	−0.104702
$737$	−1.45340	−0.0535368
$738$	31.3209	1.15294
$739$	41.2777	1.51842	0.759212	−	0.650843i	$-0.225584\pi$
$739$	41.2777	1.51842	0.759212	+	0.650843i	$0.225584\pi$
$740$	−1.09114	−0.0401112
$741$	−0.408381	−0.0150022
$742$	40.2767	1.47860
$743$	−19.6006	−0.719075	−0.359537	−	0.933131i	$-0.617066\pi$
$743$	−19.6006	−0.719075	−0.359537	+	0.933131i	$0.617066\pi$
$744$	22.3668	0.820006
$745$	−13.7766	−0.504734
$746$	−31.7521	−1.16253
$747$	−2.54714	−0.0931950
$748$	−1.29107	−0.0472063
$749$	20.5126	0.749514
$750$	20.6489	0.753991
$751$	39.8357	1.45363	0.726813	−	0.686835i	$-0.241001\pi$
$751$	39.8357	1.45363	0.726813	+	0.686835i	$0.241001\pi$
$752$	7.89612	0.287942
$753$	77.6002	2.82791
$754$	−2.20504	−0.0803027
$755$	−6.80515	−0.247665
$756$	6.22823	0.226519
$757$	14.5589	0.529151	0.264575	−	0.964365i	$-0.414768\pi$
$757$	14.5589	0.529151	0.264575	+	0.964365i	$0.414768\pi$
$758$	36.0652	1.30995
$759$	1.58033	0.0573624
$760$	0.234136	0.00849301
$761$	−49.5646	−1.79671	−0.898357	−	0.439266i	$-0.855239\pi$
$761$	−49.5646	−1.79671	−0.898357	+	0.439266i	$0.855239\pi$
$762$	1.14621	0.0415227
$763$	55.4602	2.00780
$764$	16.8251	0.608709
$765$	−19.2915	−0.697485
$766$	30.2081	1.09146
$767$	−1.95206	−0.0704850
$768$	−2.59394	−0.0936009
$769$	50.9074	1.83577	0.917885	−	0.396847i	$-0.129896\pi$
$769$	50.9074	1.83577	0.917885	+	0.396847i	$0.129896\pi$
$770$	0.607588	0.0218960
$771$	46.9138	1.68956
$772$	−2.81052	−0.101153
$773$	−42.4254	−1.52594	−0.762968	−	0.646436i	$-0.776259\pi$
$773$	−42.4254	−1.52594	−0.762968	+	0.646436i	$0.776259\pi$
$774$	1.63642	0.0588198
$775$	−36.7428	−1.31984
$776$	−3.79190	−0.136121
$777$	10.8522	0.389320
$778$	−20.5431	−0.736507
$779$	−2.28820	−0.0819833
$780$	1.28865	0.0461412
$781$	−3.18303	−0.113898
$782$	−17.0981	−0.611428
$783$	−7.20992	−0.257661
$784$	3.86149	0.137910
$785$	4.61907	0.164862
$786$	30.0069	1.07031
$787$	8.75741	0.312168	0.156084	−	0.987744i	$-0.450113\pi$
$787$	8.75741	0.312168	0.156084	+	0.987744i	$0.450113\pi$
$788$	−16.4126	−0.584676
$789$	39.5555	1.40821
$790$	2.04502	0.0727586
$791$	34.8522	1.23920
$792$	0.799715	0.0284166
$793$	−5.69686	−0.202302
$794$	−21.4893	−0.762626
$795$	27.2483	0.966398
$796$	14.8153	0.525116
$797$	2.31530	0.0820123	0.0410061	−	0.999159i	$-0.486944\pi$
$797$	2.31530	0.0820123	0.0410061	+	0.999159i	$0.486944\pi$
$798$	−2.32865	−0.0824333
$799$	47.5302	1.68150
$800$	4.26118	0.150655
$801$	−59.6675	−2.10825
$802$	37.0212	1.30726
$803$	0.352783	0.0124495
$804$	−17.5773	−0.619903
$805$	8.04650	0.283602
$806$	−4.98366	−0.175542
$807$	41.3724	1.45638
$808$	7.52717	0.264805
$809$	52.1642	1.83399	0.916997	−	0.398893i	$-0.130606\pi$
$809$	52.1642	1.83399	0.916997	+	0.398893i	$0.130606\pi$
$810$	−5.40102	−0.189772
$811$	−44.2360	−1.55334	−0.776668	−	0.629910i	$-0.783092\pi$
$811$	−44.2360	−1.55334	−0.776668	+	0.629910i	$0.783092\pi$
$812$	−12.5735	−0.441242
$813$	−63.2716	−2.21903
$814$	0.272275	0.00954322
$815$	−10.6931	−0.374564
$816$	−15.6141	−0.546602
$817$	−0.119551	−0.00418256
$818$	20.3727	0.712314
$819$	−7.10216	−0.248169
$820$	7.22045	0.252149
$821$	28.3011	0.987714	0.493857	−	0.869543i	$-0.335587\pi$
$821$	28.3011	0.987714	0.493857	+	0.869543i	$0.335587\pi$
$822$	−1.29250	−0.0450812
$823$	16.7863	0.585134	0.292567	−	0.956245i	$-0.405490\pi$
$823$	16.7863	0.585134	0.292567	+	0.956245i	$0.405490\pi$
$824$	0.884886	0.0308265
$825$	−2.37075	−0.0825389
$826$	−11.1310	−0.387296
$827$	34.8798	1.21289	0.606445	−	0.795126i	$-0.292595\pi$
$827$	34.8798	1.21289	0.606445	+	0.795126i	$0.292595\pi$
$828$	10.5909	0.368059
$829$	28.2720	0.981927	0.490964	−	0.871180i	$-0.336645\pi$
$829$	28.2720	0.981927	0.490964	+	0.871180i	$0.336645\pi$
$830$	−0.587195	−0.0203818
$831$	17.7755	0.616625
$832$	0.577971	0.0200375
$833$	23.2440	0.805356
$834$	43.8927	1.51988
$835$	17.1961	0.595097
$836$	−0.0584244	−0.00202065
$837$	−16.2953	−0.563249
$838$	−28.1112	−0.971085
$839$	−33.8765	−1.16955	−0.584773	−	0.811197i	$-0.698816\pi$
$839$	−33.8765	−1.16955	−0.584773	+	0.811197i	$0.698816\pi$
$840$	7.34810	0.253533
$841$	−14.4447	−0.498094
$842$	0.202907	0.00699265
$843$	−7.62400	−0.262584
$844$	−15.7474	−0.542049
$845$	10.8870	0.374523
$846$	−29.4411	−1.01221
$847$	36.1008	1.24044
$848$	12.2211	0.419673
$849$	57.4555	1.97187
$850$	25.6499	0.879784
$851$	3.60583	0.123606
$852$	−38.4952	−1.31882
$853$	16.7073	0.572049	0.286024	−	0.958222i	$-0.407666\pi$
$853$	16.7073	0.572049	0.286024	+	0.958222i	$0.407666\pi$
$854$	−32.4844	−1.11159
$855$	−0.872989	−0.0298556
$856$	6.22409	0.212735
$857$	−35.9797	−1.22904	−0.614522	−	0.788900i	$-0.710651\pi$
$857$	−35.9797	−1.22904	−0.614522	+	0.788900i	$0.710651\pi$
$858$	−0.321560	−0.0109779
$859$	−26.9581	−0.919800	−0.459900	−	0.887971i	$-0.652115\pi$
$859$	−26.9581	−0.919800	−0.459900	+	0.887971i	$0.652115\pi$
$860$	0.377245	0.0128640
$861$	−71.8125	−2.44736
$862$	−12.4226	−0.423116
$863$	−13.0122	−0.442940	−0.221470	−	0.975167i	$-0.571085\pi$
$863$	−13.0122	−0.442940	−0.221470	+	0.975167i	$0.571085\pi$
$864$	1.88982	0.0642929
$865$	−15.9486	−0.542269
$866$	−31.2682	−1.06254
$867$	−49.8910	−1.69439
$868$	−28.4176	−0.964556
$869$	−0.510298	−0.0173107
$870$	−8.50630	−0.288391
$871$	3.91649	0.132705
$872$	16.8282	0.569874
$873$	14.1383	0.478508
$874$	−0.773734	−0.0261719
$875$	−26.2350	−0.886904
$876$	4.26652	0.144152
$877$	43.3050	1.46231	0.731153	−	0.682213i	$-0.238983\pi$
$877$	43.3050	1.46231	0.731153	+	0.682213i	$0.238983\pi$
$878$	−20.9378	−0.706615
$879$	7.09065	0.239162
$880$	0.184359	0.00621474
$881$	−9.17571	−0.309137	−0.154569	−	0.987982i	$-0.549399\pi$
$881$	−9.17571	−0.309137	−0.154569	+	0.987982i	$0.549399\pi$
$882$	−14.3977	−0.484797
$883$	−50.5421	−1.70088	−0.850438	−	0.526076i	$-0.823663\pi$
$883$	−50.5421	−1.70088	−0.850438	+	0.526076i	$0.823663\pi$
$884$	3.47906	0.117013
$885$	−7.53041	−0.253132
$886$	−12.4567	−0.418491
$887$	−9.05723	−0.304112	−0.152056	−	0.988372i	$-0.548589\pi$
$887$	−9.05723	−0.304112	−0.152056	+	0.988372i	$0.548589\pi$
$888$	3.29286	0.110501
$889$	−1.45629	−0.0488423
$890$	−13.7552	−0.461076
$891$	1.34772	0.0451505
$892$	7.69950	0.257798
$893$	2.15086	0.0719759
$894$	41.5749	1.39047
$895$	−11.3957	−0.380916
$896$	3.29568	0.110101
$897$	−4.25852	−0.142188
$898$	−5.45129	−0.181912
$899$	32.8968	1.09717
$900$	−15.8880	−0.529601
$901$	73.5640	2.45077
$902$	−1.80173	−0.0599911
$903$	−3.75197	−0.124858
$904$	10.5751	0.351723
$905$	3.20676	0.106596
$906$	20.5366	0.682284
$907$	−35.2174	−1.16937	−0.584687	−	0.811259i	$-0.698783\pi$
$907$	−35.2174	−1.16937	−0.584687	+	0.811259i	$0.698783\pi$
$908$	12.9742	0.430565
$909$	−28.0654	−0.930871
$910$	−1.63727	−0.0542749
$911$	48.0050	1.59048	0.795238	−	0.606297i	$-0.207346\pi$
$911$	48.0050	1.59048	0.795238	+	0.606297i	$0.207346\pi$
$912$	−0.706577	−0.0233971
$913$	0.146524	0.00484923
$914$	−36.5002	−1.20732
$915$	−21.9766	−0.726524
$916$	−4.83978	−0.159911
$917$	−38.1246	−1.25898
$918$	11.3756	0.375452
$919$	−40.4524	−1.33440	−0.667200	−	0.744878i	$-0.732507\pi$
$919$	−40.4524	−1.33440	−0.667200	+	0.744878i	$0.732507\pi$
$920$	2.44153	0.0804949
$921$	−24.3026	−0.800799
$922$	−16.5147	−0.543883
$923$	8.57733	0.282326
$924$	−1.83358	−0.0603204
$925$	−5.40931	−0.177857
$926$	−12.7582	−0.419260
$927$	−3.29934	−0.108365
$928$	−3.81514	−0.125238
$929$	−4.72986	−0.155182	−0.0775909	−	0.996985i	$-0.524723\pi$
$929$	−4.72986	−0.155182	−0.0775909	+	0.996985i	$0.524723\pi$
$930$	−19.2253	−0.630422
$931$	1.05185	0.0344730
$932$	12.1250	0.397166
$933$	65.5258	2.14522
$934$	−5.22843	−0.171079
$935$	1.10974	0.0362923
$936$	−2.15499	−0.0704381
$937$	40.5497	1.32470	0.662350	−	0.749195i	$-0.269559\pi$
$937$	40.5497	1.32470	0.662350	+	0.749195i	$0.269559\pi$
$938$	22.3324	0.729179
$939$	12.9547	0.422761
$940$	−6.78709	−0.221370
$941$	−19.6747	−0.641378	−0.320689	−	0.947184i	$-0.603914\pi$
$941$	−19.6747	−0.641378	−0.320689	+	0.947184i	$0.603914\pi$
$942$	−13.9394	−0.454172
$943$	−23.8609	−0.777019
$944$	−3.37745	−0.109926
$945$	−5.35345	−0.174148
$946$	−0.0941346	−0.00306058
$947$	−22.7788	−0.740212	−0.370106	−	0.928989i	$-0.620679\pi$
$947$	−22.7788	−0.740212	−0.370106	+	0.928989i	$0.620679\pi$
$948$	−6.17148	−0.200440
$949$	−0.950646	−0.0308593
$950$	1.16072	0.0376588
$951$	−34.6611	−1.12396
$952$	19.8381	0.642957
$953$	2.67313	0.0865913	0.0432957	−	0.999062i	$-0.486214\pi$
$953$	2.67313	0.0865913	0.0432957	+	0.999062i	$0.486214\pi$
$954$	−45.5669	−1.47528
$955$	−14.4619	−0.467977
$956$	−28.8855	−0.934225
$957$	2.12259	0.0686136
$958$	0.185788	0.00600255
$959$	1.64216	0.0530281
$960$	2.22962	0.0719606
$961$	43.3508	1.39841
$962$	−0.733699	−0.0236554
$963$	−23.2068	−0.747830
$964$	14.6062	0.470433
$965$	2.41577	0.0777665
$966$	−24.2828	−0.781285
$967$	59.0292	1.89825	0.949126	−	0.314896i	$-0.101970\pi$
$967$	59.0292	1.89825	0.949126	+	0.314896i	$0.101970\pi$
$968$	10.9540	0.352075
$969$	−4.25320	−0.136632
$970$	3.25932	0.104650
$971$	−38.8931	−1.24814	−0.624069	−	0.781369i	$-0.714522\pi$
$971$	−38.8931	−1.24814	−0.624069	+	0.781369i	$0.714522\pi$
$972$	21.9687	0.704646
$973$	−55.7668	−1.78780
$974$	5.11940	0.164036
$975$	6.38846	0.204594
$976$	−9.85667	−0.315504
$977$	−32.4213	−1.03725	−0.518625	−	0.855002i	$-0.673556\pi$
$977$	−32.4213	−1.03725	−0.518625	+	0.855002i	$0.673556\pi$
$978$	32.2698	1.03187
$979$	3.43236	0.109699
$980$	−3.31913	−0.106026
$981$	−62.7447	−2.00328
$982$	−18.1467	−0.579086
$983$	47.4266	1.51267	0.756337	−	0.654182i	$-0.226987\pi$
$983$	47.4266	1.51267	0.756337	+	0.654182i	$0.226987\pi$
$984$	−21.7899	−0.694637
$985$	14.1074	0.449500
$986$	−22.9650	−0.731354
$987$	67.5024	2.14863
$988$	0.157436	0.00500871
$989$	−1.24666	−0.0396414
$990$	−0.687392	−0.0218468
$991$	19.9541	0.633863	0.316931	−	0.948448i	$-0.397348\pi$
$991$	19.9541	0.633863	0.316931	+	0.948448i	$0.397348\pi$
$992$	−8.62269	−0.273771
$993$	41.7950	1.32632
$994$	48.9092	1.55131
$995$	−12.7345	−0.403710
$996$	1.77204	0.0561492
$997$	−36.4276	−1.15367	−0.576836	−	0.816860i	$-0.695713\pi$
$997$	−36.4276	−1.15367	−0.576836	+	0.816860i	$0.695713\pi$
$998$	20.2236	0.640168
$999$	−2.39901	−0.0759013

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.c.1.7	✓	86

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.c.1.7	✓	86	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -2.59394 q^{3} +1.00000 q^{4} -0.859547 q^{5} +2.59394 q^{6} -3.29568 q^{7} -1.00000 q^{8} +3.72855 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -2.59394 q^{3} +1.00000 q^{4} -0.859547 q^{5} +2.59394 q^{6} -3.29568 q^{7} -1.00000 q^{8} +3.72855 q^{9} +0.859547 q^{10} -0.214484 q^{11} -2.59394 q^{12} +0.577971 q^{13} +3.29568 q^{14} +2.22962 q^{15} +1.00000 q^{16} +6.01944 q^{17} -3.72855 q^{18} +0.272395 q^{19} -0.859547 q^{20} +8.54880 q^{21} +0.214484 q^{22} +2.84049 q^{23} +2.59394 q^{24} -4.26118 q^{25} -0.577971 q^{26} -1.88982 q^{27} -3.29568 q^{28} +3.81514 q^{29} -2.22962 q^{30} +8.62269 q^{31} -1.00000 q^{32} +0.556360 q^{33} -6.01944 q^{34} +2.83279 q^{35} +3.72855 q^{36} +1.26944 q^{37} -0.272395 q^{38} -1.49922 q^{39} +0.859547 q^{40} -8.40030 q^{41} -8.54880 q^{42} -0.438888 q^{43} -0.214484 q^{44} -3.20486 q^{45} -2.84049 q^{46} +7.89612 q^{47} -2.59394 q^{48} +3.86149 q^{49} +4.26118 q^{50} -15.6141 q^{51} +0.577971 q^{52} +12.2211 q^{53} +1.88982 q^{54} +0.184359 q^{55} +3.29568 q^{56} -0.706577 q^{57} -3.81514 q^{58} -3.37745 q^{59} +2.22962 q^{60} -9.85667 q^{61} -8.62269 q^{62} -12.2881 q^{63} +1.00000 q^{64} -0.496793 q^{65} -0.556360 q^{66} +6.77627 q^{67} +6.01944 q^{68} -7.36806 q^{69} -2.83279 q^{70} +14.8404 q^{71} -3.72855 q^{72} -1.64480 q^{73} -1.26944 q^{74} +11.0533 q^{75} +0.272395 q^{76} +0.706870 q^{77} +1.49922 q^{78} +2.37919 q^{79} -0.859547 q^{80} -6.28357 q^{81} +8.40030 q^{82} -0.683145 q^{83} +8.54880 q^{84} -5.17399 q^{85} +0.438888 q^{86} -9.89626 q^{87} +0.214484 q^{88} -16.0029 q^{89} +3.20486 q^{90} -1.90480 q^{91} +2.84049 q^{92} -22.3668 q^{93} -7.89612 q^{94} -0.234136 q^{95} +2.59394 q^{96} +3.79190 q^{97} -3.86149 q^{98} -0.799715 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9	\( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 86 * q - 86 * q^2 + 12 * q^3 + 86 * q^4 - 4 * q^5 - 12 * q^6 + 35 * q^7 - 86 * q^8 + 72 * q^9 + 4 * q^10 + 13 * q^11 + 12 * q^12 + 45 * q^13 - 35 * q^14 + 17 * q^15 + 86 * q^16 + 5 * q^17 - 72 * q^18 + 47 * q^19 - 4 * q^20 + 15 * q^21 - 13 * q^22 + 6 * q^23 - 12 * q^24 + 112 * q^25 - 45 * q^26 + 51 * q^27 + 35 * q^28 - 14 * q^29 - 17 * q^30 + 24 * q^31 - 86 * q^32 + 43 * q^33 - 5 * q^34 + 42 * q^35 + 72 * q^36 + 61 * q^37 - 47 * q^38 + 20 * q^39 + 4 * q^40 - 16 * q^41 - 15 * q^42 + 72 * q^43 + 13 * q^44 + 6 * q^45 - 6 * q^46 + 11 * q^47 + 12 * q^48 + 89 * q^49 - 112 * q^50 + 56 * q^51 + 45 * q^52 - 7 * q^53 - 51 * q^54 + 48 * q^55 - 35 * q^56 + 65 * q^57 + 14 * q^58 + 24 * q^59 + 17 * q^60 + 31 * q^61 - 24 * q^62 + 98 * q^63 + 86 * q^64 - 9 * q^65 - 43 * q^66 + 157 * q^67 + 5 * q^68 + q^69 - 42 * q^70 - 11 * q^71 - 72 * q^72 + 74 * q^73 - 61 * q^74 + 76 * q^75 + 47 * q^76 - 13 * q^77 - 20 * q^78 + 57 * q^79 - 4 * q^80 + 34 * q^81 + 16 * q^82 + 65 * q^83 + 15 * q^84 + 102 * q^85 - 72 * q^86 + 49 * q^87 - 13 * q^88 - 34 * q^89 - 6 * q^90 + 91 * q^91 + 6 * q^92 + 57 * q^93 - 11 * q^94 - 13 * q^95 - 12 * q^96 + 64 * q^97 - 89 * q^98 + 56 * q^99

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