LMFDB - Embedded newform 4030.2.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4030,2,Mod(1,4030)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4030, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4030.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4030 = 2 \cdot 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4030.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:	$32.1797120146$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 16x^{6} + 18x^{5} + 64x^{4} - 84x^{3} - 19x^{2} + 22x + 4 $ x^8 - x^7 - 16x^6 + 18x^5 + 64x^4 - 84x^3 - 19x^2 + 22x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.635022$ of defining polynomial
Character	$\chi$	$=$	4030.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -0.635022 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.635022 q^{6} +1.23180 q^{7} +1.00000 q^{8} -2.59675 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -0.635022 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.635022 q^{6} +1.23180 q^{7} +1.00000 q^{8} -2.59675 q^{9} -1.00000 q^{10} +0.272596 q^{11} -0.635022 q^{12} -1.00000 q^{13} +1.23180 q^{14} +0.635022 q^{15} +1.00000 q^{16} +7.11122 q^{17} -2.59675 q^{18} -1.95394 q^{19} -1.00000 q^{20} -0.782218 q^{21} +0.272596 q^{22} -1.59230 q^{23} -0.635022 q^{24} +1.00000 q^{25} -1.00000 q^{26} +3.55406 q^{27} +1.23180 q^{28} +0.671264 q^{29} +0.635022 q^{30} -1.00000 q^{31} +1.00000 q^{32} -0.173104 q^{33} +7.11122 q^{34} -1.23180 q^{35} -2.59675 q^{36} +6.46612 q^{37} -1.95394 q^{38} +0.635022 q^{39} -1.00000 q^{40} -11.0790 q^{41} -0.782218 q^{42} +6.68387 q^{43} +0.272596 q^{44} +2.59675 q^{45} -1.59230 q^{46} +2.10016 q^{47} -0.635022 q^{48} -5.48268 q^{49} +1.00000 q^{50} -4.51578 q^{51} -1.00000 q^{52} +7.92467 q^{53} +3.55406 q^{54} -0.272596 q^{55} +1.23180 q^{56} +1.24080 q^{57} +0.671264 q^{58} +8.12564 q^{59} +0.635022 q^{60} -8.58732 q^{61} -1.00000 q^{62} -3.19866 q^{63} +1.00000 q^{64} +1.00000 q^{65} -0.173104 q^{66} +4.35909 q^{67} +7.11122 q^{68} +1.01115 q^{69} -1.23180 q^{70} +5.14505 q^{71} -2.59675 q^{72} +10.1523 q^{73} +6.46612 q^{74} -0.635022 q^{75} -1.95394 q^{76} +0.335782 q^{77} +0.635022 q^{78} +13.5679 q^{79} -1.00000 q^{80} +5.53334 q^{81} -11.0790 q^{82} -9.00122 q^{83} -0.782218 q^{84} -7.11122 q^{85} +6.68387 q^{86} -0.426267 q^{87} +0.272596 q^{88} +8.93093 q^{89} +2.59675 q^{90} -1.23180 q^{91} -1.59230 q^{92} +0.635022 q^{93} +2.10016 q^{94} +1.95394 q^{95} -0.635022 q^{96} +15.7222 q^{97} -5.48268 q^{98} -0.707862 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} - q^{6} + q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 - q^6 + q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} - q^{6} + q^{7} + 8 q^{8} + 9 q^{9} - 8 q^{10} + 4 q^{11} - q^{12} - 8 q^{13} + q^{14} + q^{15} + 8 q^{16} - 5 q^{17} + 9 q^{18} + 2 q^{19} - 8 q^{20} + 17 q^{21} + 4 q^{22} + 4 q^{23} - q^{24} + 8 q^{25} - 8 q^{26} + 11 q^{27} + q^{28} + 11 q^{29} + q^{30} - 8 q^{31} + 8 q^{32} + 10 q^{33} - 5 q^{34} - q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} + q^{39} - 8 q^{40} + 10 q^{41} + 17 q^{42} + 19 q^{43} + 4 q^{44} - 9 q^{45} + 4 q^{46} + 11 q^{47} - q^{48} + 11 q^{49} + 8 q^{50} + 7 q^{51} - 8 q^{52} + 8 q^{53} + 11 q^{54} - 4 q^{55} + q^{56} - 11 q^{57} + 11 q^{58} + 28 q^{59} + q^{60} - 12 q^{61} - 8 q^{62} + 20 q^{63} + 8 q^{64} + 8 q^{65} + 10 q^{66} + 24 q^{67} - 5 q^{68} + 30 q^{69} - q^{70} + 18 q^{71} + 9 q^{72} - 3 q^{73} + 19 q^{74} - q^{75} + 2 q^{76} - 7 q^{77} + q^{78} + 22 q^{79} - 8 q^{80} + 24 q^{81} + 10 q^{82} + 17 q^{83} + 17 q^{84} + 5 q^{85} + 19 q^{86} + 11 q^{87} + 4 q^{88} + 17 q^{89} - 9 q^{90} - q^{91} + 4 q^{92} + q^{93} + 11 q^{94} - 2 q^{95} - q^{96} - 24 q^{97} + 11 q^{98} + 23 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 - q^6 + q^7 + 8 * q^8 + 9 * q^9 - 8 * q^10 + 4 * q^11 - q^12 - 8 * q^13 + q^14 + q^15 + 8 * q^16 - 5 * q^17 + 9 * q^18 + 2 * q^19 - 8 * q^20 + 17 * q^21 + 4 * q^22 + 4 * q^23 - q^24 + 8 * q^25 - 8 * q^26 + 11 * q^27 + q^28 + 11 * q^29 + q^30 - 8 * q^31 + 8 * q^32 + 10 * q^33 - 5 * q^34 - q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 + q^39 - 8 * q^40 + 10 * q^41 + 17 * q^42 + 19 * q^43 + 4 * q^44 - 9 * q^45 + 4 * q^46 + 11 * q^47 - q^48 + 11 * q^49 + 8 * q^50 + 7 * q^51 - 8 * q^52 + 8 * q^53 + 11 * q^54 - 4 * q^55 + q^56 - 11 * q^57 + 11 * q^58 + 28 * q^59 + q^60 - 12 * q^61 - 8 * q^62 + 20 * q^63 + 8 * q^64 + 8 * q^65 + 10 * q^66 + 24 * q^67 - 5 * q^68 + 30 * q^69 - q^70 + 18 * q^71 + 9 * q^72 - 3 * q^73 + 19 * q^74 - q^75 + 2 * q^76 - 7 * q^77 + q^78 + 22 * q^79 - 8 * q^80 + 24 * q^81 + 10 * q^82 + 17 * q^83 + 17 * q^84 + 5 * q^85 + 19 * q^86 + 11 * q^87 + 4 * q^88 + 17 * q^89 - 9 * q^90 - q^91 + 4 * q^92 + q^93 + 11 * q^94 - 2 * q^95 - q^96 - 24 * q^97 + 11 * q^98 + 23 * 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
$3$	−0.635022	−0.366630	−0.183315	−	0.983054i	$-0.558683\pi$
$3$	−0.635022	−0.366630	−0.183315	+	0.983054i	$0.558683\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.635022	−0.259247
$7$	1.23180	0.465575	0.232788	−	0.972528i	$-0.425215\pi$
$7$	1.23180	0.465575	0.232788	+	0.972528i	$0.425215\pi$
$8$	1.00000	0.353553
$9$	−2.59675	−0.865582
$10$	−1.00000	−0.316228
$11$	0.272596	0.0821907	0.0410953	−	0.999155i	$-0.486915\pi$
$11$	0.272596	0.0821907	0.0410953	+	0.999155i	$0.486915\pi$
$12$	−0.635022	−0.183315
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.23180	0.329211
$15$	0.635022	0.163962
$16$	1.00000	0.250000
$17$	7.11122	1.72473	0.862363	−	0.506291i	$-0.168984\pi$
$17$	7.11122	1.72473	0.862363	+	0.506291i	$0.168984\pi$
$18$	−2.59675	−0.612059
$19$	−1.95394	−0.448265	−0.224133	−	0.974559i	$-0.571955\pi$
$19$	−1.95394	−0.448265	−0.224133	+	0.974559i	$0.571955\pi$
$20$	−1.00000	−0.223607
$21$	−0.782218	−0.170694
$22$	0.272596	0.0581176
$23$	−1.59230	−0.332018	−0.166009	−	0.986124i	$-0.553088\pi$
$23$	−1.59230	−0.332018	−0.166009	+	0.986124i	$0.553088\pi$
$24$	−0.635022	−0.129623
$25$	1.00000	0.200000
$26$	−1.00000	−0.196116
$27$	3.55406	0.683979
$28$	1.23180	0.232788
$29$	0.671264	0.124651	0.0623253	−	0.998056i	$-0.480148\pi$
$29$	0.671264	0.124651	0.0623253	+	0.998056i	$0.480148\pi$
$30$	0.635022	0.115939
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	−0.173104	−0.0301336
$34$	7.11122	1.21956
$35$	−1.23180	−0.208212
$36$	−2.59675	−0.432791
$37$	6.46612	1.06302	0.531512	−	0.847051i	$-0.321624\pi$
$37$	6.46612	1.06302	0.531512	+	0.847051i	$0.321624\pi$
$38$	−1.95394	−0.316972
$39$	0.635022	0.101685
$40$	−1.00000	−0.158114
$41$	−11.0790	−1.73025	−0.865126	−	0.501554i	$-0.832762\pi$
$41$	−11.0790	−1.73025	−0.865126	+	0.501554i	$0.832762\pi$
$42$	−0.782218	−0.120699
$43$	6.68387	1.01928	0.509640	−	0.860387i	$-0.329778\pi$
$43$	6.68387	1.01928	0.509640	+	0.860387i	$0.329778\pi$
$44$	0.272596	0.0410953
$45$	2.59675	0.387100
$46$	−1.59230	−0.234772
$47$	2.10016	0.306340	0.153170	−	0.988200i	$-0.451052\pi$
$47$	2.10016	0.306340	0.153170	+	0.988200i	$0.451052\pi$
$48$	−0.635022	−0.0916575
$49$	−5.48268	−0.783240
$50$	1.00000	0.141421
$51$	−4.51578	−0.632336
$52$	−1.00000	−0.138675
$53$	7.92467	1.08854	0.544269	−	0.838911i	$-0.316807\pi$
$53$	7.92467	1.08854	0.544269	+	0.838911i	$0.316807\pi$
$54$	3.55406	0.483646
$55$	−0.272596	−0.0367568
$56$	1.23180	0.164606
$57$	1.24080	0.164348
$58$	0.671264	0.0881413
$59$	8.12564	1.05787	0.528934	−	0.848663i	$-0.322592\pi$
$59$	8.12564	1.05787	0.528934	+	0.848663i	$0.322592\pi$
$60$	0.635022	0.0819810
$61$	−8.58732	−1.09949	−0.549747	−	0.835331i	$-0.685276\pi$
$61$	−8.58732	−1.09949	−0.549747	+	0.835331i	$0.685276\pi$
$62$	−1.00000	−0.127000
$63$	−3.19866	−0.402994
$64$	1.00000	0.125000
$65$	1.00000	0.124035
$66$	−0.173104	−0.0213077
$67$	4.35909	0.532547	0.266274	−	0.963897i	$-0.414208\pi$
$67$	4.35909	0.532547	0.266274	+	0.963897i	$0.414208\pi$
$68$	7.11122	0.862363
$69$	1.01115	0.121728
$70$	−1.23180	−0.147228
$71$	5.14505	0.610606	0.305303	−	0.952255i	$-0.401242\pi$
$71$	5.14505	0.610606	0.305303	+	0.952255i	$0.401242\pi$
$72$	−2.59675	−0.306030
$73$	10.1523	1.18824	0.594118	−	0.804378i	$-0.297501\pi$
$73$	10.1523	1.18824	0.594118	+	0.804378i	$0.297501\pi$
$74$	6.46612	0.751671
$75$	−0.635022	−0.0733260
$76$	−1.95394	−0.224133
$77$	0.335782	0.0382660
$78$	0.635022	0.0719021
$79$	13.5679	1.52650	0.763252	−	0.646101i	$-0.223602\pi$
$79$	13.5679	1.52650	0.763252	+	0.646101i	$0.223602\pi$
$80$	−1.00000	−0.111803
$81$	5.53334	0.614815
$82$	−11.0790	−1.22347
$83$	−9.00122	−0.988012	−0.494006	−	0.869459i	$-0.664468\pi$
$83$	−9.00122	−0.988012	−0.494006	+	0.869459i	$0.664468\pi$
$84$	−0.782218	−0.0853469
$85$	−7.11122	−0.771320
$86$	6.68387	0.720741
$87$	−0.426267	−0.0457007
$88$	0.272596	0.0290588
$89$	8.93093	0.946677	0.473338	−	0.880881i	$-0.343049\pi$
$89$	8.93093	0.946677	0.473338	+	0.880881i	$0.343049\pi$
$90$	2.59675	0.273721
$91$	−1.23180	−0.129127
$92$	−1.59230	−0.166009
$93$	0.635022	0.0658487
$94$	2.10016	0.216615
$95$	1.95394	0.200470
$96$	−0.635022	−0.0648116
$97$	15.7222	1.59634	0.798172	−	0.602430i	$-0.205801\pi$
$97$	15.7222	1.59634	0.798172	+	0.602430i	$0.205801\pi$
$98$	−5.48268	−0.553834
$99$	−0.707862	−0.0711428
$100$	1.00000	0.100000
$101$	7.64601	0.760806	0.380403	−	0.924821i	$-0.375785\pi$
$101$	7.64601	0.760806	0.380403	+	0.924821i	$0.375785\pi$
$102$	−4.51578	−0.447129
$103$	−5.75324	−0.566884	−0.283442	−	0.958989i	$-0.591476\pi$
$103$	−5.75324	−0.566884	−0.283442	+	0.958989i	$0.591476\pi$
$104$	−1.00000	−0.0980581
$105$	0.782218	0.0763366
$106$	7.92467	0.769712
$107$	−4.45612	−0.430789	−0.215394	−	0.976527i	$-0.569104\pi$
$107$	−4.45612	−0.430789	−0.215394	+	0.976527i	$0.569104\pi$
$108$	3.55406	0.341989
$109$	8.27284	0.792394	0.396197	−	0.918165i	$-0.370330\pi$
$109$	8.27284	0.792394	0.396197	+	0.918165i	$0.370330\pi$
$110$	−0.272596	−0.0259910
$111$	−4.10613	−0.389736
$112$	1.23180	0.116394
$113$	9.85052	0.926659	0.463330	−	0.886186i	$-0.346655\pi$
$113$	9.85052	0.926659	0.463330	+	0.886186i	$0.346655\pi$
$114$	1.24080	0.116211
$115$	1.59230	0.148483
$116$	0.671264	0.0623253
$117$	2.59675	0.240069
$118$	8.12564	0.748026
$119$	8.75958	0.802989
$120$	0.635022	0.0579693
$121$	−10.9257	−0.993245
$122$	−8.58732	−0.777459
$123$	7.03542	0.634363
$124$	−1.00000	−0.0898027
$125$	−1.00000	−0.0894427
$126$	−3.19866	−0.284960
$127$	−17.2692	−1.53239	−0.766195	−	0.642608i	$-0.777852\pi$
$127$	−17.2692	−1.53239	−0.766195	+	0.642608i	$0.777852\pi$
$128$	1.00000	0.0883883
$129$	−4.24441	−0.373699
$130$	1.00000	0.0877058
$131$	6.28322	0.548968	0.274484	−	0.961592i	$-0.411493\pi$
$131$	6.28322	0.548968	0.274484	+	0.961592i	$0.411493\pi$
$132$	−0.173104	−0.0150668
$133$	−2.40686	−0.208701
$134$	4.35909	0.376568
$135$	−3.55406	−0.305885
$136$	7.11122	0.609782
$137$	−5.33968	−0.456200	−0.228100	−	0.973638i	$-0.573251\pi$
$137$	−5.33968	−0.456200	−0.228100	+	0.973638i	$0.573251\pi$
$138$	1.01115	0.0860746
$139$	12.2006	1.03484	0.517422	−	0.855731i	$-0.326892\pi$
$139$	12.2006	1.03484	0.517422	+	0.855731i	$0.326892\pi$
$140$	−1.23180	−0.104106
$141$	−1.33365	−0.112314
$142$	5.14505	0.431763
$143$	−0.272596	−0.0227956
$144$	−2.59675	−0.216396
$145$	−0.671264	−0.0557454
$146$	10.1523	0.840210
$147$	3.48162	0.287159
$148$	6.46612	0.531512
$149$	−7.79203	−0.638348	−0.319174	−	0.947696i	$-0.603405\pi$
$149$	−7.79203	−0.638348	−0.319174	+	0.947696i	$0.603405\pi$
$150$	−0.635022	−0.0518493
$151$	18.9468	1.54187	0.770935	−	0.636914i	$-0.219789\pi$
$151$	18.9468	1.54187	0.770935	+	0.636914i	$0.219789\pi$
$152$	−1.95394	−0.158486
$153$	−18.4660	−1.49289
$154$	0.335782	0.0270581
$155$	1.00000	0.0803219
$156$	0.635022	0.0508424
$157$	7.62496	0.608538	0.304269	−	0.952586i	$-0.401588\pi$
$157$	7.62496	0.608538	0.304269	+	0.952586i	$0.401588\pi$
$158$	13.5679	1.07940
$159$	−5.03234	−0.399090
$160$	−1.00000	−0.0790569
$161$	−1.96139	−0.154580
$162$	5.53334	0.434740
$163$	−8.43039	−0.660319	−0.330159	−	0.943925i	$-0.607103\pi$
$163$	−8.43039	−0.660319	−0.330159	+	0.943925i	$0.607103\pi$
$164$	−11.0790	−0.865126
$165$	0.173104	0.0134761
$166$	−9.00122	−0.698630
$167$	24.1193	1.86641	0.933203	−	0.359351i	$-0.117002\pi$
$167$	24.1193	1.86641	0.933203	+	0.359351i	$0.117002\pi$
$168$	−0.782218	−0.0603494
$169$	1.00000	0.0769231
$170$	−7.11122	−0.545406
$171$	5.07390	0.388011
$172$	6.68387	0.509640
$173$	1.59651	0.121380	0.0606900	−	0.998157i	$-0.480670\pi$
$173$	1.59651	0.121380	0.0606900	+	0.998157i	$0.480670\pi$
$174$	−0.426267	−0.0323152
$175$	1.23180	0.0931151
$176$	0.272596	0.0205477
$177$	−5.15996	−0.387846
$178$	8.93093	0.669401
$179$	10.8435	0.810483	0.405242	−	0.914210i	$-0.367187\pi$
$179$	10.8435	0.810483	0.405242	+	0.914210i	$0.367187\pi$
$180$	2.59675	0.193550
$181$	−0.0526882	−0.00391628	−0.00195814	−	0.999998i	$-0.500623\pi$
$181$	−0.0526882	−0.00391628	−0.00195814	+	0.999998i	$0.500623\pi$
$182$	−1.23180	−0.0913068
$183$	5.45313	0.403107
$184$	−1.59230	−0.117386
$185$	−6.46612	−0.475398
$186$	0.635022	0.0465621
$187$	1.93849	0.141756
$188$	2.10016	0.153170
$189$	4.37787	0.318444
$190$	1.95394	0.141754
$191$	−1.31127	−0.0948801	−0.0474401	−	0.998874i	$-0.515106\pi$
$191$	−1.31127	−0.0948801	−0.0474401	+	0.998874i	$0.515106\pi$
$192$	−0.635022	−0.0458288
$193$	5.15787	0.371272	0.185636	−	0.982619i	$-0.440566\pi$
$193$	5.15787	0.371272	0.185636	+	0.982619i	$0.440566\pi$
$194$	15.7222	1.12879
$195$	−0.635022	−0.0454749
$196$	−5.48268	−0.391620
$197$	−0.571208	−0.0406969	−0.0203485	−	0.999793i	$-0.506478\pi$
$197$	−0.571208	−0.0406969	−0.0203485	+	0.999793i	$0.506478\pi$
$198$	−0.707862	−0.0503056
$199$	0.766897	0.0543639	0.0271820	−	0.999631i	$-0.491347\pi$
$199$	0.766897	0.0543639	0.0271820	+	0.999631i	$0.491347\pi$
$200$	1.00000	0.0707107
$201$	−2.76812	−0.195248
$202$	7.64601	0.537971
$203$	0.826861	0.0580342
$204$	−4.51578	−0.316168
$205$	11.0790	0.773792
$206$	−5.75324	−0.400847
$207$	4.13481	0.287389
$208$	−1.00000	−0.0693375
$209$	−0.532637	−0.0368432
$210$	0.782218	0.0539781
$211$	−9.86552	−0.679170	−0.339585	−	0.940575i	$-0.610287\pi$
$211$	−9.86552	−0.679170	−0.339585	+	0.940575i	$0.610287\pi$
$212$	7.92467	0.544269
$213$	−3.26722	−0.223866
$214$	−4.45612	−0.304614
$215$	−6.68387	−0.455836
$216$	3.55406	0.241823
$217$	−1.23180	−0.0836198
$218$	8.27284	0.560307
$219$	−6.44693	−0.435643
$220$	−0.272596	−0.0183784
$221$	−7.11122	−0.478353
$222$	−4.10613	−0.275585
$223$	−2.04843	−0.137173	−0.0685865	−	0.997645i	$-0.521849\pi$
$223$	−2.04843	−0.137173	−0.0685865	+	0.997645i	$0.521849\pi$
$224$	1.23180	0.0823029
$225$	−2.59675	−0.173116
$226$	9.85052	0.655247
$227$	−10.9084	−0.724019	−0.362010	−	0.932174i	$-0.617909\pi$
$227$	−10.9084	−0.724019	−0.362010	+	0.932174i	$0.617909\pi$
$228$	1.24080	0.0821738
$229$	−3.97344	−0.262572	−0.131286	−	0.991345i	$-0.541911\pi$
$229$	−3.97344	−0.262572	−0.131286	+	0.991345i	$0.541911\pi$
$230$	1.59230	0.104993
$231$	−0.213229	−0.0140294
$232$	0.671264	0.0440706
$233$	−0.177298	−0.0116152	−0.00580759	−	0.999983i	$-0.501849\pi$
$233$	−0.177298	−0.0116152	−0.00580759	+	0.999983i	$0.501849\pi$
$234$	2.59675	0.169755
$235$	−2.10016	−0.137000
$236$	8.12564	0.528934
$237$	−8.61589	−0.559662
$238$	8.75958	0.567799
$239$	−18.7030	−1.20980	−0.604898	−	0.796303i	$-0.706786\pi$
$239$	−18.7030	−1.20980	−0.604898	+	0.796303i	$0.706786\pi$
$240$	0.635022	0.0409905
$241$	14.8399	0.955920	0.477960	−	0.878382i	$-0.341376\pi$
$241$	14.8399	0.955920	0.477960	+	0.878382i	$0.341376\pi$
$242$	−10.9257	−0.702330
$243$	−14.1760	−0.909388
$244$	−8.58732	−0.549747
$245$	5.48268	0.350275
$246$	7.03542	0.448562
$247$	1.95394	0.124326
$248$	−1.00000	−0.0635001
$249$	5.71597	0.362235
$250$	−1.00000	−0.0632456
$251$	6.18551	0.390426	0.195213	−	0.980761i	$-0.437460\pi$
$251$	6.18551	0.390426	0.195213	+	0.980761i	$0.437460\pi$
$252$	−3.19866	−0.201497
$253$	−0.434055	−0.0272888
$254$	−17.2692	−1.08356
$255$	4.51578	0.282789
$256$	1.00000	0.0625000
$257$	−5.66129	−0.353141	−0.176571	−	0.984288i	$-0.556500\pi$
$257$	−5.66129	−0.353141	−0.176571	+	0.984288i	$0.556500\pi$
$258$	−4.24441	−0.264245
$259$	7.96494	0.494917
$260$	1.00000	0.0620174
$261$	−1.74310	−0.107895
$262$	6.28322	0.388179
$263$	9.59564	0.591692	0.295846	−	0.955236i	$-0.404398\pi$
$263$	9.59564	0.591692	0.295846	+	0.955236i	$0.404398\pi$
$264$	−0.173104	−0.0106538
$265$	−7.92467	−0.486809
$266$	−2.40686	−0.147574
$267$	−5.67134	−0.347080
$268$	4.35909	0.266274
$269$	−20.4741	−1.24833	−0.624165	−	0.781292i	$-0.714561\pi$
$269$	−20.4741	−1.24833	−0.624165	+	0.781292i	$0.714561\pi$
$270$	−3.55406	−0.216293
$271$	−5.75652	−0.349683	−0.174842	−	0.984597i	$-0.555941\pi$
$271$	−5.75652	−0.349683	−0.174842	+	0.984597i	$0.555941\pi$
$272$	7.11122	0.431181
$273$	0.782218	0.0473420
$274$	−5.33968	−0.322582
$275$	0.272596	0.0164381
$276$	1.01115	0.0608639
$277$	10.7117	0.643603	0.321802	−	0.946807i	$-0.395712\pi$
$277$	10.7117	0.643603	0.321802	+	0.946807i	$0.395712\pi$
$278$	12.2006	0.731745
$279$	2.59675	0.155463
$280$	−1.23180	−0.0736139
$281$	5.43868	0.324444	0.162222	−	0.986754i	$-0.448134\pi$
$281$	5.43868	0.324444	0.162222	+	0.986754i	$0.448134\pi$
$282$	−1.33365	−0.0794177
$283$	−17.9430	−1.06660	−0.533301	−	0.845926i	$-0.679049\pi$
$283$	−17.9430	−1.06660	−0.533301	+	0.845926i	$0.679049\pi$
$284$	5.14505	0.305303
$285$	−1.24080	−0.0734985
$286$	−0.272596	−0.0161189
$287$	−13.6471	−0.805563
$288$	−2.59675	−0.153015
$289$	33.5695	1.97468
$290$	−0.671264	−0.0394180
$291$	−9.98392	−0.585268
$292$	10.1523	0.594118
$293$	6.11034	0.356970	0.178485	−	0.983943i	$-0.442880\pi$
$293$	6.11034	0.356970	0.178485	+	0.983943i	$0.442880\pi$
$294$	3.48162	0.203052
$295$	−8.12564	−0.473093
$296$	6.46612	0.375835
$297$	0.968820	0.0562167
$298$	−7.79203	−0.451380
$299$	1.59230	0.0920853
$300$	−0.635022	−0.0366630
$301$	8.23317	0.474552
$302$	18.9468	1.09027
$303$	−4.85538	−0.278934
$304$	−1.95394	−0.112066
$305$	8.58732	0.491708
$306$	−18.4660	−1.05563
$307$	9.38551	0.535659	0.267830	−	0.963466i	$-0.413694\pi$
$307$	9.38551	0.535659	0.267830	+	0.963466i	$0.413694\pi$
$308$	0.335782	0.0191330
$309$	3.65343	0.207837
$310$	1.00000	0.0567962
$311$	−33.2713	−1.88664	−0.943322	−	0.331880i	$-0.892317\pi$
$311$	−33.2713	−1.88664	−0.943322	+	0.331880i	$0.892317\pi$
$312$	0.635022	0.0359510
$313$	−15.2607	−0.862585	−0.431293	−	0.902212i	$-0.641942\pi$
$313$	−15.2607	−0.862585	−0.431293	+	0.902212i	$0.641942\pi$
$314$	7.62496	0.430301
$315$	3.19866	0.180224
$316$	13.5679	0.763252
$317$	19.3544	1.08705	0.543525	−	0.839393i	$-0.317089\pi$
$317$	19.3544	1.08705	0.543525	+	0.839393i	$0.317089\pi$
$318$	−5.03234	−0.282200
$319$	0.182984	0.0102451
$320$	−1.00000	−0.0559017
$321$	2.82973	0.157940
$322$	−1.96139	−0.109304
$323$	−13.8949	−0.773135
$324$	5.53334	0.307408
$325$	−1.00000	−0.0554700
$326$	−8.43039	−0.466916
$327$	−5.25343	−0.290515
$328$	−11.0790	−0.611737
$329$	2.58698	0.142625
$330$	0.173104	0.00952907
$331$	10.1890	0.560038	0.280019	−	0.959994i	$-0.409659\pi$
$331$	10.1890	0.560038	0.280019	+	0.959994i	$0.409659\pi$
$332$	−9.00122	−0.494006
$333$	−16.7909	−0.920134
$334$	24.1193	1.31975
$335$	−4.35909	−0.238162
$336$	−0.782218	−0.0426735
$337$	−13.0282	−0.709690	−0.354845	−	0.934925i	$-0.615466\pi$
$337$	−13.0282	−0.709690	−0.354845	+	0.934925i	$0.615466\pi$
$338$	1.00000	0.0543928
$339$	−6.25530	−0.339741
$340$	−7.11122	−0.385660
$341$	−0.272596	−0.0147619
$342$	5.07390	0.274365
$343$	−15.3761	−0.830232
$344$	6.68387	0.360370
$345$	−1.01115	−0.0544384
$346$	1.59651	0.0858286
$347$	16.6741	0.895110	0.447555	−	0.894256i	$-0.352295\pi$
$347$	16.6741	0.895110	0.447555	+	0.894256i	$0.352295\pi$
$348$	−0.426267	−0.0228503
$349$	7.19094	0.384922	0.192461	−	0.981305i	$-0.438353\pi$
$349$	7.19094	0.384922	0.192461	+	0.981305i	$0.438353\pi$
$350$	1.23180	0.0658423
$351$	−3.55406	−0.189702
$352$	0.272596	0.0145294
$353$	3.90993	0.208105	0.104052	−	0.994572i	$-0.466819\pi$
$353$	3.90993	0.208105	0.104052	+	0.994572i	$0.466819\pi$
$354$	−5.15996	−0.274249
$355$	−5.14505	−0.273071
$356$	8.93093	0.473338
$357$	−5.56252	−0.294400
$358$	10.8435	0.573098
$359$	11.1264	0.587231	0.293616	−	0.955924i	$-0.405141\pi$
$359$	11.1264	0.587231	0.293616	+	0.955924i	$0.405141\pi$
$360$	2.59675	0.136861
$361$	−15.1821	−0.799058
$362$	−0.0526882	−0.00276923
$363$	6.93805	0.364153
$364$	−1.23180	−0.0645637
$365$	−10.1523	−0.531395
$366$	5.45313	0.285040
$367$	6.59930	0.344481	0.172240	−	0.985055i	$-0.444899\pi$
$367$	6.59930	0.344481	0.172240	+	0.985055i	$0.444899\pi$
$368$	−1.59230	−0.0830046
$369$	28.7694	1.49768
$370$	−6.46612	−0.336157
$371$	9.76158	0.506796
$372$	0.635022	0.0329243
$373$	24.2258	1.25436	0.627181	−	0.778874i	$-0.284209\pi$
$373$	24.2258	1.25436	0.627181	+	0.778874i	$0.284209\pi$
$374$	1.93849	0.100237
$375$	0.635022	0.0327924
$376$	2.10016	0.108308
$377$	−0.671264	−0.0345719
$378$	4.37787	0.225174
$379$	0.870831	0.0447316	0.0223658	−	0.999750i	$-0.492880\pi$
$379$	0.870831	0.0447316	0.0223658	+	0.999750i	$0.492880\pi$
$380$	1.95394	0.100235
$381$	10.9663	0.561820
$382$	−1.31127	−0.0670904
$383$	18.4271	0.941581	0.470791	−	0.882245i	$-0.343969\pi$
$383$	18.4271	0.941581	0.470791	+	0.882245i	$0.343969\pi$
$384$	−0.635022	−0.0324058
$385$	−0.335782	−0.0171131
$386$	5.15787	0.262529
$387$	−17.3563	−0.882272
$388$	15.7222	0.798172
$389$	21.8910	1.10992	0.554959	−	0.831878i	$-0.312734\pi$
$389$	21.8910	1.10992	0.554959	+	0.831878i	$0.312734\pi$
$390$	−0.635022	−0.0321556
$391$	−11.3232	−0.572640
$392$	−5.48268	−0.276917
$393$	−3.98998	−0.201268
$394$	−0.571208	−0.0287771
$395$	−13.5679	−0.682673
$396$	−0.707862	−0.0355714
$397$	−20.1091	−1.00925	−0.504624	−	0.863339i	$-0.668369\pi$
$397$	−20.1091	−1.00925	−0.504624	+	0.863339i	$0.668369\pi$
$398$	0.766897	0.0384411
$399$	1.52841	0.0765162
$400$	1.00000	0.0500000
$401$	1.96722	0.0982385	0.0491193	−	0.998793i	$-0.484359\pi$
$401$	1.96722	0.0982385	0.0491193	+	0.998793i	$0.484359\pi$
$402$	−2.76812	−0.138061
$403$	1.00000	0.0498135
$404$	7.64601	0.380403
$405$	−5.53334	−0.274954
$406$	0.826861	0.0410364
$407$	1.76264	0.0873706
$408$	−4.51578	−0.223565
$409$	8.94583	0.442343	0.221171	−	0.975235i	$-0.429012\pi$
$409$	8.94583	0.442343	0.221171	+	0.975235i	$0.429012\pi$
$410$	11.0790	0.547154
$411$	3.39081	0.167257
$412$	−5.75324	−0.283442
$413$	10.0091	0.492517
$414$	4.13481	0.203215
$415$	9.00122	0.441852
$416$	−1.00000	−0.0490290
$417$	−7.74766	−0.379405
$418$	−0.532637	−0.0260521
$419$	26.9093	1.31461	0.657303	−	0.753626i	$-0.271697\pi$
$419$	26.9093	1.31461	0.657303	+	0.753626i	$0.271697\pi$
$420$	0.782218	0.0381683
$421$	−7.55904	−0.368405	−0.184203	−	0.982888i	$-0.558970\pi$
$421$	−7.55904	−0.368405	−0.184203	+	0.982888i	$0.558970\pi$
$422$	−9.86552	−0.480246
$423$	−5.45360	−0.265163
$424$	7.92467	0.384856
$425$	7.11122	0.344945
$426$	−3.26722	−0.158297
$427$	−10.5778	−0.511897
$428$	−4.45612	−0.215394
$429$	0.173104	0.00835755
$430$	−6.68387	−0.322325
$431$	6.12573	0.295066	0.147533	−	0.989057i	$-0.452867\pi$
$431$	6.12573	0.295066	0.147533	+	0.989057i	$0.452867\pi$
$432$	3.55406	0.170995
$433$	−32.8758	−1.57991	−0.789955	−	0.613165i	$-0.789896\pi$
$433$	−32.8758	−1.57991	−0.789955	+	0.613165i	$0.789896\pi$
$434$	−1.23180	−0.0591281
$435$	0.426267	0.0204380
$436$	8.27284	0.396197
$437$	3.11127	0.148832
$438$	−6.44693	−0.308046
$439$	−0.617160	−0.0294554	−0.0147277	−	0.999892i	$-0.504688\pi$
$439$	−0.617160	−0.0294554	−0.0147277	+	0.999892i	$0.504688\pi$
$440$	−0.272596	−0.0129955
$441$	14.2371	0.677958
$442$	−7.11122	−0.338246
$443$	25.0624	1.19075	0.595375	−	0.803448i	$-0.297003\pi$
$443$	25.0624	1.19075	0.595375	+	0.803448i	$0.297003\pi$
$444$	−4.10613	−0.194868
$445$	−8.93093	−0.423367
$446$	−2.04843	−0.0969960
$447$	4.94811	0.234037
$448$	1.23180	0.0581969
$449$	−27.1106	−1.27943	−0.639715	−	0.768612i	$-0.720948\pi$
$449$	−27.1106	−1.27943	−0.639715	+	0.768612i	$0.720948\pi$
$450$	−2.59675	−0.122412
$451$	−3.02009	−0.142211
$452$	9.85052	0.463330
$453$	−12.0316	−0.565296
$454$	−10.9084	−0.511959
$455$	1.23180	0.0577475
$456$	1.24080	0.0581056
$457$	−31.6368	−1.47991	−0.739954	−	0.672657i	$-0.765153\pi$
$457$	−31.6368	−1.47991	−0.739954	+	0.672657i	$0.765153\pi$
$458$	−3.97344	−0.185667
$459$	25.2737	1.17967
$460$	1.59230	0.0742416
$461$	−24.8568	−1.15770	−0.578848	−	0.815436i	$-0.696497\pi$
$461$	−24.8568	−1.15770	−0.578848	+	0.815436i	$0.696497\pi$
$462$	−0.213229	−0.00992032
$463$	−12.7600	−0.593007	−0.296503	−	0.955032i	$-0.595821\pi$
$463$	−12.7600	−0.593007	−0.296503	+	0.955032i	$0.595821\pi$
$464$	0.671264	0.0311626
$465$	−0.635022	−0.0294484
$466$	−0.177298	−0.00821318
$467$	−0.528860	−0.0244727	−0.0122364	−	0.999925i	$-0.503895\pi$
$467$	−0.528860	−0.0244727	−0.0122364	+	0.999925i	$0.503895\pi$
$468$	2.59675	0.120035
$469$	5.36951	0.247941
$470$	−2.10016	−0.0968734
$471$	−4.84201	−0.223108
$472$	8.12564	0.374013
$473$	1.82199	0.0837754
$474$	−8.61589	−0.395741
$475$	−1.95394	−0.0896531
$476$	8.75958	0.401495
$477$	−20.5784	−0.942219
$478$	−18.7030	−0.855455
$479$	3.07938	0.140701	0.0703503	−	0.997522i	$-0.477588\pi$
$479$	3.07938	0.140701	0.0703503	+	0.997522i	$0.477588\pi$
$480$	0.635022	0.0289846
$481$	−6.46612	−0.294830
$482$	14.8399	0.675938
$483$	1.24553	0.0566735
$484$	−10.9257	−0.496622
$485$	−15.7222	−0.713907
$486$	−14.1760	−0.643035
$487$	−8.04516	−0.364561	−0.182280	−	0.983247i	$-0.558348\pi$
$487$	−8.04516	−0.364561	−0.182280	+	0.983247i	$0.558348\pi$
$488$	−8.58732	−0.388730
$489$	5.35348	0.242093
$490$	5.48268	0.247682
$491$	18.0102	0.812789	0.406395	−	0.913698i	$-0.366786\pi$
$491$	18.0102	0.812789	0.406395	+	0.913698i	$0.366786\pi$
$492$	7.03542	0.317181
$493$	4.77351	0.214988
$494$	1.95394	0.0879121
$495$	0.707862	0.0318160
$496$	−1.00000	−0.0449013
$497$	6.33766	0.284283
$498$	5.71597	0.256139
$499$	−27.2005	−1.21766	−0.608830	−	0.793300i	$-0.708361\pi$
$499$	−27.2005	−1.21766	−0.608830	+	0.793300i	$0.708361\pi$
$500$	−1.00000	−0.0447214
$501$	−15.3163	−0.684280
$502$	6.18551	0.276073
$503$	16.0727	0.716646	0.358323	−	0.933598i	$-0.383349\pi$
$503$	16.0727	0.716646	0.358323	+	0.933598i	$0.383349\pi$
$504$	−3.19866	−0.142480
$505$	−7.64601	−0.340243
$506$	−0.434055	−0.0192961
$507$	−0.635022	−0.0282023
$508$	−17.2692	−0.766195
$509$	−7.86530	−0.348623	−0.174312	−	0.984691i	$-0.555770\pi$
$509$	−7.86530	−0.348623	−0.174312	+	0.984691i	$0.555770\pi$
$510$	4.51578	0.199962
$511$	12.5056	0.553213
$512$	1.00000	0.0441942
$513$	−6.94443	−0.306604
$514$	−5.66129	−0.249709
$515$	5.75324	0.253518
$516$	−4.24441	−0.186850
$517$	0.572496	0.0251783
$518$	7.96494	0.349959
$519$	−1.01382	−0.0445016
$520$	1.00000	0.0438529
$521$	−4.09836	−0.179553	−0.0897763	−	0.995962i	$-0.528615\pi$
$521$	−4.09836	−0.179553	−0.0897763	+	0.995962i	$0.528615\pi$
$522$	−1.74310	−0.0762935
$523$	−42.2018	−1.84536	−0.922678	−	0.385573i	$-0.874004\pi$
$523$	−42.2018	−1.84536	−0.922678	+	0.385573i	$0.874004\pi$
$524$	6.28322	0.274484
$525$	−0.782218	−0.0341388
$526$	9.59564	0.418390
$527$	−7.11122	−0.309770
$528$	−0.173104	−0.00753339
$529$	−20.4646	−0.889764
$530$	−7.92467	−0.344226
$531$	−21.1002	−0.915672
$532$	−2.40686	−0.104351
$533$	11.0790	0.479886
$534$	−5.67134	−0.245423
$535$	4.45612	0.192655
$536$	4.35909	0.188284
$537$	−6.88588	−0.297147
$538$	−20.4741	−0.882703
$539$	−1.49455	−0.0643750
$540$	−3.55406	−0.152942
$541$	−33.2862	−1.43108	−0.715542	−	0.698570i	$-0.753820\pi$
$541$	−33.2862	−1.43108	−0.715542	+	0.698570i	$0.753820\pi$
$542$	−5.75652	−0.247264
$543$	0.0334582	0.00143583
$544$	7.11122	0.304891
$545$	−8.27284	−0.354369
$546$	0.782218	0.0334758
$547$	1.75889	0.0752045	0.0376023	−	0.999293i	$-0.488028\pi$
$547$	1.75889	0.0752045	0.0376023	+	0.999293i	$0.488028\pi$
$548$	−5.33968	−0.228100
$549$	22.2991	0.951702
$550$	0.272596	0.0116235
$551$	−1.31161	−0.0558765
$552$	1.01115	0.0430373
$553$	16.7128	0.710702
$554$	10.7117	0.455096
$555$	4.10613	0.174295
$556$	12.2006	0.517422
$557$	4.03845	0.171115	0.0855573	−	0.996333i	$-0.472733\pi$
$557$	4.03845	0.171115	0.0855573	+	0.996333i	$0.472733\pi$
$558$	2.59675	0.109929
$559$	−6.68387	−0.282698
$560$	−1.23180	−0.0520529
$561$	−1.23098	−0.0519721
$562$	5.43868	0.229417
$563$	22.4793	0.947388	0.473694	−	0.880690i	$-0.342920\pi$
$563$	22.4793	0.947388	0.473694	+	0.880690i	$0.342920\pi$
$564$	−1.33365	−0.0561568
$565$	−9.85052	−0.414415
$566$	−17.9430	−0.754201
$567$	6.81595	0.286243
$568$	5.14505	0.215882
$569$	−20.2194	−0.847640	−0.423820	−	0.905746i	$-0.639311\pi$
$569$	−20.2194	−0.847640	−0.423820	+	0.905746i	$0.639311\pi$
$570$	−1.24080	−0.0519713
$571$	−23.2457	−0.972803	−0.486401	−	0.873736i	$-0.661691\pi$
$571$	−23.2457	−0.972803	−0.486401	+	0.873736i	$0.661691\pi$
$572$	−0.272596	−0.0113978
$573$	0.832685	0.0347859
$574$	−13.6471	−0.569619
$575$	−1.59230	−0.0664037
$576$	−2.59675	−0.108198
$577$	−38.4366	−1.60014	−0.800069	−	0.599909i	$-0.795204\pi$
$577$	−38.4366	−1.60014	−0.800069	+	0.599909i	$0.795204\pi$
$578$	33.5695	1.39631
$579$	−3.27536	−0.136119
$580$	−0.671264	−0.0278727
$581$	−11.0877	−0.459994
$582$	−9.98392	−0.413847
$583$	2.16023	0.0894676
$584$	10.1523	0.420105
$585$	−2.59675	−0.107362
$586$	6.11034	0.252416
$587$	−12.2666	−0.506296	−0.253148	−	0.967428i	$-0.581466\pi$
$587$	−12.2666	−0.506296	−0.253148	+	0.967428i	$0.581466\pi$
$588$	3.48162	0.143580
$589$	1.95394	0.0805108
$590$	−8.12564	−0.334527
$591$	0.362730	0.0149207
$592$	6.46612	0.265756
$593$	−10.8058	−0.443741	−0.221870	−	0.975076i	$-0.571216\pi$
$593$	−10.8058	−0.443741	−0.221870	+	0.975076i	$0.571216\pi$
$594$	0.968820	0.0397512
$595$	−8.75958	−0.359108
$596$	−7.79203	−0.319174
$597$	−0.486997	−0.0199314
$598$	1.59230	0.0651141
$599$	21.8357	0.892184	0.446092	−	0.894987i	$-0.352815\pi$
$599$	21.8357	0.892184	0.446092	+	0.894987i	$0.352815\pi$
$600$	−0.635022	−0.0259247
$601$	−32.6514	−1.33188	−0.665938	−	0.746007i	$-0.731969\pi$
$601$	−32.6514	−1.33188	−0.665938	+	0.746007i	$0.731969\pi$
$602$	8.23317	0.335559
$603$	−11.3194	−0.460964
$604$	18.9468	0.770935
$605$	10.9257	0.444193
$606$	−4.85538	−0.197236
$607$	18.8305	0.764307	0.382154	−	0.924099i	$-0.375183\pi$
$607$	18.8305	0.764307	0.382154	+	0.924099i	$0.375183\pi$
$608$	−1.95394	−0.0792429
$609$	−0.525075	−0.0212771
$610$	8.58732	0.347690
$611$	−2.10016	−0.0849636
$612$	−18.4660	−0.746446
$613$	12.1312	0.489975	0.244987	−	0.969526i	$-0.421216\pi$
$613$	12.1312	0.489975	0.244987	+	0.969526i	$0.421216\pi$
$614$	9.38551	0.378768
$615$	−7.03542	−0.283696
$616$	0.335782	0.0135291
$617$	−22.3515	−0.899838	−0.449919	−	0.893069i	$-0.648547\pi$
$617$	−22.3515	−0.899838	−0.449919	+	0.893069i	$0.648547\pi$
$618$	3.65343	0.146963
$619$	30.6469	1.23180	0.615901	−	0.787823i	$-0.288792\pi$
$619$	30.6469	1.23180	0.615901	+	0.787823i	$0.288792\pi$
$620$	1.00000	0.0401610
$621$	−5.65914	−0.227093
$622$	−33.2713	−1.33406
$623$	11.0011	0.440749
$624$	0.635022	0.0254212
$625$	1.00000	0.0400000
$626$	−15.2607	−0.609940
$627$	0.338236	0.0135078
$628$	7.62496	0.304269
$629$	45.9820	1.83342
$630$	3.19866	0.127438
$631$	−29.7798	−1.18552	−0.592758	−	0.805381i	$-0.701961\pi$
$631$	−29.7798	−1.18552	−0.592758	+	0.805381i	$0.701961\pi$
$632$	13.5679	0.539700
$633$	6.26482	0.249004
$634$	19.3544	0.768661
$635$	17.2692	0.685306
$636$	−5.03234	−0.199545
$637$	5.48268	0.217232
$638$	0.182984	0.00724439
$639$	−13.3604	−0.528530
$640$	−1.00000	−0.0395285
$641$	19.5760	0.773206	0.386603	−	0.922246i	$-0.373648\pi$
$641$	19.5760	0.773206	0.386603	+	0.922246i	$0.373648\pi$
$642$	2.82973	0.111681
$643$	−33.8021	−1.33302	−0.666512	−	0.745495i	$-0.732213\pi$
$643$	−33.8021	−1.33302	−0.666512	+	0.745495i	$0.732213\pi$
$644$	−1.96139	−0.0772898
$645$	4.24441	0.167123
$646$	−13.8949	−0.546689
$647$	−31.5700	−1.24114	−0.620572	−	0.784149i	$-0.713100\pi$
$647$	−31.5700	−1.24114	−0.620572	+	0.784149i	$0.713100\pi$
$648$	5.53334	0.217370
$649$	2.21501	0.0869469
$650$	−1.00000	−0.0392232
$651$	0.782218	0.0306575
$652$	−8.43039	−0.330159
$653$	−35.7069	−1.39732	−0.698659	−	0.715455i	$-0.746220\pi$
$653$	−35.7069	−1.39732	−0.698659	+	0.715455i	$0.746220\pi$
$654$	−5.25343	−0.205425
$655$	−6.28322	−0.245506
$656$	−11.0790	−0.432563
$657$	−26.3629	−1.02852
$658$	2.58698	0.100851
$659$	24.3706	0.949343	0.474671	−	0.880163i	$-0.342567\pi$
$659$	24.3706	0.949343	0.474671	+	0.880163i	$0.342567\pi$
$660$	0.173104	0.00673807
$661$	8.64646	0.336308	0.168154	−	0.985761i	$-0.446219\pi$
$661$	8.64646	0.336308	0.168154	+	0.985761i	$0.446219\pi$
$662$	10.1890	0.396006
$663$	4.51578	0.175378
$664$	−9.00122	−0.349315
$665$	2.40686	0.0933341
$666$	−16.7909	−0.650633
$667$	−1.06886	−0.0413863
$668$	24.1193	0.933203
$669$	1.30080	0.0502918
$670$	−4.35909	−0.168406
$671$	−2.34087	−0.0903681
$672$	−0.782218	−0.0301747
$673$	−39.9771	−1.54101	−0.770503	−	0.637437i	$-0.779995\pi$
$673$	−39.9771	−1.54101	−0.770503	+	0.637437i	$0.779995\pi$
$674$	−13.0282	−0.501826
$675$	3.55406	0.136796
$676$	1.00000	0.0384615
$677$	27.0593	1.03997	0.519987	−	0.854174i	$-0.325937\pi$
$677$	27.0593	1.03997	0.519987	+	0.854174i	$0.325937\pi$
$678$	−6.25530	−0.240233
$679$	19.3665	0.743218
$680$	−7.11122	−0.272703
$681$	6.92710	0.265447
$682$	−0.272596	−0.0104382
$683$	5.93046	0.226923	0.113461	−	0.993542i	$-0.463806\pi$
$683$	5.93046	0.226923	0.113461	+	0.993542i	$0.463806\pi$
$684$	5.07390	0.194005
$685$	5.33968	0.204019
$686$	−15.3761	−0.587063
$687$	2.52322	0.0962669
$688$	6.68387	0.254820
$689$	−7.92467	−0.301906
$690$	−1.01115	−0.0384937
$691$	5.13122	0.195201	0.0976005	−	0.995226i	$-0.468883\pi$
$691$	5.13122	0.195201	0.0976005	+	0.995226i	$0.468883\pi$
$692$	1.59651	0.0606900
$693$	−0.871942	−0.0331223
$694$	16.6741	0.632938
$695$	−12.2006	−0.462796
$696$	−0.426267	−0.0161576
$697$	−78.7854	−2.98421
$698$	7.19094	0.272181
$699$	0.112588	0.00425848
$700$	1.23180	0.0465575
$701$	0.838202	0.0316584	0.0158292	−	0.999875i	$-0.494961\pi$
$701$	0.838202	0.0316584	0.0158292	+	0.999875i	$0.494961\pi$
$702$	−3.55406	−0.134139
$703$	−12.6344	−0.476517
$704$	0.272596	0.0102738
$705$	1.33365	0.0502282
$706$	3.90993	0.147152
$707$	9.41833	0.354213
$708$	−5.15996	−0.193923
$709$	19.1224	0.718158	0.359079	−	0.933307i	$-0.383091\pi$
$709$	19.1224	0.718158	0.359079	+	0.933307i	$0.383091\pi$
$710$	−5.14505	−0.193090
$711$	−35.2323	−1.32131
$712$	8.93093	0.334701
$713$	1.59230	0.0596323
$714$	−5.56252	−0.208172
$715$	0.272596	0.0101945
$716$	10.8435	0.405242
$717$	11.8768	0.443548
$718$	11.1264	0.415235
$719$	−35.2362	−1.31409	−0.657044	−	0.753853i	$-0.728193\pi$
$719$	−35.2362	−1.31409	−0.657044	+	0.753853i	$0.728193\pi$
$720$	2.59675	0.0967751
$721$	−7.08682	−0.263927
$722$	−15.1821	−0.565019
$723$	−9.42364	−0.350469
$724$	−0.0526882	−0.00195814
$725$	0.671264	0.0249301
$726$	6.93805	0.257495
$727$	12.4556	0.461952	0.230976	−	0.972959i	$-0.425808\pi$
$727$	12.4556	0.461952	0.230976	+	0.972959i	$0.425808\pi$
$728$	−1.23180	−0.0456534
$729$	−7.59797	−0.281406
$730$	−10.1523	−0.375753
$731$	47.5305	1.75798
$732$	5.45313	0.201554
$733$	−12.7415	−0.470620	−0.235310	−	0.971920i	$-0.575610\pi$
$733$	−12.7415	−0.470620	−0.235310	+	0.971920i	$0.575610\pi$
$734$	6.59930	0.243585
$735$	−3.48162	−0.128421
$736$	−1.59230	−0.0586931
$737$	1.18827	0.0437704
$738$	28.7694	1.05902
$739$	32.9455	1.21192	0.605960	−	0.795495i	$-0.292789\pi$
$739$	32.9455	1.21192	0.605960	+	0.795495i	$0.292789\pi$
$740$	−6.46612	−0.237699
$741$	−1.24080	−0.0455818
$742$	9.76158	0.358359
$743$	−6.67402	−0.244846	−0.122423	−	0.992478i	$-0.539066\pi$
$743$	−6.67402	−0.244846	−0.122423	+	0.992478i	$0.539066\pi$
$744$	0.635022	0.0232810
$745$	7.79203	0.285478
$746$	24.2258	0.886968
$747$	23.3739	0.855206
$748$	1.93849	0.0708782
$749$	−5.48903	−0.200565
$750$	0.635022	0.0231877
$751$	20.6764	0.754492	0.377246	−	0.926113i	$-0.376871\pi$
$751$	20.6764	0.754492	0.377246	+	0.926113i	$0.376871\pi$
$752$	2.10016	0.0765851
$753$	−3.92793	−0.143142
$754$	−0.671264	−0.0244460
$755$	−18.9468	−0.689545
$756$	4.37787	0.159222
$757$	−11.1005	−0.403456	−0.201728	−	0.979442i	$-0.564656\pi$
$757$	−11.1005	−0.403456	−0.201728	+	0.979442i	$0.564656\pi$
$758$	0.870831	0.0316300
$759$	0.275634	0.0100049
$760$	1.95394	0.0708770
$761$	19.0219	0.689542	0.344771	−	0.938687i	$-0.387957\pi$
$761$	19.0219	0.689542	0.344771	+	0.938687i	$0.387957\pi$
$762$	10.9663	0.397267
$763$	10.1905	0.368919
$764$	−1.31127	−0.0474401
$765$	18.4660	0.667641
$766$	18.4271	0.665798
$767$	−8.12564	−0.293400
$768$	−0.635022	−0.0229144
$769$	−14.3658	−0.518042	−0.259021	−	0.965872i	$-0.583400\pi$
$769$	−14.3658	−0.518042	−0.259021	+	0.965872i	$0.583400\pi$
$770$	−0.335782	−0.0121008
$771$	3.59504	0.129472
$772$	5.15787	0.185636
$773$	−2.02694	−0.0729038	−0.0364519	−	0.999335i	$-0.511606\pi$
$773$	−2.02694	−0.0729038	−0.0364519	+	0.999335i	$0.511606\pi$
$774$	−17.3563	−0.623860
$775$	−1.00000	−0.0359211
$776$	15.7222	0.564393
$777$	−5.05791	−0.181452
$778$	21.8910	0.784830
$779$	21.6478	0.775612
$780$	−0.635022	−0.0227374
$781$	1.40252	0.0501861
$782$	−11.3232	−0.404918
$783$	2.38571	0.0852583
$784$	−5.48268	−0.195810
$785$	−7.62496	−0.272146
$786$	−3.98998	−0.142318
$787$	−1.92174	−0.0685027	−0.0342514	−	0.999413i	$-0.510905\pi$
$787$	−1.92174	−0.0685027	−0.0342514	+	0.999413i	$0.510905\pi$
$788$	−0.571208	−0.0203485
$789$	−6.09344	−0.216932
$790$	−13.5679	−0.482723
$791$	12.1338	0.431430
$792$	−0.707862	−0.0251528
$793$	8.58732	0.304945
$794$	−20.1091	−0.713646
$795$	5.03234	0.178479
$796$	0.766897	0.0271820
$797$	16.2020	0.573904	0.286952	−	0.957945i	$-0.407358\pi$
$797$	16.2020	0.573904	0.286952	+	0.957945i	$0.407358\pi$
$798$	1.52841	0.0541051
$799$	14.9347	0.528353
$800$	1.00000	0.0353553
$801$	−23.1914	−0.819427
$802$	1.96722	0.0694651
$803$	2.76747	0.0976619
$804$	−2.76812	−0.0976239
$805$	1.96139	0.0691301
$806$	1.00000	0.0352235
$807$	13.0015	0.457675
$808$	7.64601	0.268986
$809$	−21.9073	−0.770221	−0.385111	−	0.922870i	$-0.625837\pi$
$809$	−21.9073	−0.770221	−0.385111	+	0.922870i	$0.625837\pi$
$810$	−5.53334	−0.194422
$811$	52.5171	1.84412	0.922062	−	0.387043i	$-0.126503\pi$
$811$	52.5171	1.84412	0.922062	+	0.387043i	$0.126503\pi$
$812$	0.826861	0.0290171
$813$	3.65551	0.128204
$814$	1.76264	0.0617803
$815$	8.43039	0.295304
$816$	−4.51578	−0.158084
$817$	−13.0599	−0.456908
$818$	8.94583	0.312784
$819$	3.19866	0.111770
$820$	11.0790	0.386896
$821$	13.9807	0.487930	0.243965	−	0.969784i	$-0.421552\pi$
$821$	13.9807	0.487930	0.243965	+	0.969784i	$0.421552\pi$
$822$	3.39081	0.118268
$823$	−6.90815	−0.240803	−0.120402	−	0.992725i	$-0.538418\pi$
$823$	−6.90815	−0.240803	−0.120402	+	0.992725i	$0.538418\pi$
$824$	−5.75324	−0.200424
$825$	−0.173104	−0.00602671
$826$	10.0091	0.348262
$827$	−13.4592	−0.468022	−0.234011	−	0.972234i	$-0.575185\pi$
$827$	−13.4592	−0.468022	−0.234011	+	0.972234i	$0.575185\pi$
$828$	4.13481	0.143695
$829$	−27.2579	−0.946707	−0.473353	−	0.880873i	$-0.656957\pi$
$829$	−27.2579	−0.946707	−0.473353	+	0.880873i	$0.656957\pi$
$830$	9.00122	0.312437
$831$	−6.80216	−0.235964
$832$	−1.00000	−0.0346688
$833$	−38.9885	−1.35087
$834$	−7.74766	−0.268280
$835$	−24.1193	−0.834682
$836$	−0.532637	−0.0184216
$837$	−3.55406	−0.122846
$838$	26.9093	0.929567
$839$	50.0451	1.72775	0.863874	−	0.503708i	$-0.168031\pi$
$839$	50.0451	1.72775	0.863874	+	0.503708i	$0.168031\pi$
$840$	0.782218	0.0269891
$841$	−28.5494	−0.984462
$842$	−7.55904	−0.260502
$843$	−3.45368	−0.118951
$844$	−9.86552	−0.339585
$845$	−1.00000	−0.0344010
$846$	−5.45360	−0.187498
$847$	−13.4582	−0.462430
$848$	7.92467	0.272134
$849$	11.3942	0.391048
$850$	7.11122	0.243913
$851$	−10.2960	−0.352943
$852$	−3.26722	−0.111933
$853$	−11.2662	−0.385747	−0.192873	−	0.981224i	$-0.561781\pi$
$853$	−11.2662	−0.385747	−0.192873	+	0.981224i	$0.561781\pi$
$854$	−10.5778	−0.361966
$855$	−5.07390	−0.173524
$856$	−4.45612	−0.152307
$857$	−53.1205	−1.81456	−0.907280	−	0.420526i	$-0.861845\pi$
$857$	−53.1205	−1.81456	−0.907280	+	0.420526i	$0.861845\pi$
$858$	0.173104	0.00590968
$859$	24.0616	0.820970	0.410485	−	0.911867i	$-0.365359\pi$
$859$	24.0616	0.820970	0.410485	+	0.911867i	$0.365359\pi$
$860$	−6.68387	−0.227918
$861$	8.66621	0.295344
$862$	6.12573	0.208643
$863$	−8.86015	−0.301603	−0.150801	−	0.988564i	$-0.548185\pi$
$863$	−8.86015	−0.301603	−0.150801	+	0.988564i	$0.548185\pi$
$864$	3.55406	0.120911
$865$	−1.59651	−0.0542828
$866$	−32.8758	−1.11717
$867$	−21.3174	−0.723976
$868$	−1.23180	−0.0418099
$869$	3.69854	0.125464
$870$	0.426267	0.0144518
$871$	−4.35909	−0.147702
$872$	8.27284	0.280154
$873$	−40.8265	−1.38177
$874$	3.11127	0.105240
$875$	−1.23180	−0.0416423
$876$	−6.44693	−0.217821
$877$	−28.1897	−0.951900	−0.475950	−	0.879472i	$-0.657896\pi$
$877$	−28.1897	−0.951900	−0.475950	+	0.879472i	$0.657896\pi$
$878$	−0.617160	−0.0208281
$879$	−3.88020	−0.130876
$880$	−0.272596	−0.00918920
$881$	−31.8962	−1.07461	−0.537305	−	0.843388i	$-0.680558\pi$
$881$	−31.8962	−1.07461	−0.537305	+	0.843388i	$0.680558\pi$
$882$	14.2371	0.479389
$883$	39.0760	1.31501	0.657506	−	0.753449i	$-0.271611\pi$
$883$	39.0760	1.31501	0.657506	+	0.753449i	$0.271611\pi$
$884$	−7.11122	−0.239176
$885$	5.15996	0.173450
$886$	25.0624	0.841988
$887$	2.94756	0.0989693	0.0494846	−	0.998775i	$-0.484242\pi$
$887$	2.94756	0.0989693	0.0494846	+	0.998775i	$0.484242\pi$
$888$	−4.10613	−0.137793
$889$	−21.2721	−0.713443
$890$	−8.93093	−0.299365
$891$	1.50836	0.0505321
$892$	−2.04843	−0.0685865
$893$	−4.10360	−0.137322
$894$	4.94811	0.165489
$895$	−10.8435	−0.362459
$896$	1.23180	0.0411514
$897$	−1.01115	−0.0337612
$898$	−27.1106	−0.904694
$899$	−0.671264	−0.0223879
$900$	−2.59675	−0.0865582
$901$	56.3541	1.87743
$902$	−3.02009	−0.100558
$903$	−5.22824	−0.173985
$904$	9.85052	0.327623
$905$	0.0526882	0.00175142
$906$	−12.0316	−0.399725
$907$	40.3728	1.34056	0.670278	−	0.742110i	$-0.266175\pi$
$907$	40.3728	1.34056	0.670278	+	0.742110i	$0.266175\pi$
$908$	−10.9084	−0.362010
$909$	−19.8548	−0.658541
$910$	1.23180	0.0408337
$911$	−14.5546	−0.482214	−0.241107	−	0.970499i	$-0.577511\pi$
$911$	−14.5546	−0.482214	−0.241107	+	0.970499i	$0.577511\pi$
$912$	1.24080	0.0410869
$913$	−2.45369	−0.0812054
$914$	−31.6368	−1.04645
$915$	−5.45313	−0.180275
$916$	−3.97344	−0.131286
$917$	7.73965	0.255586
$918$	25.2737	0.834156
$919$	46.0433	1.51883	0.759414	−	0.650607i	$-0.225486\pi$
$919$	46.0433	1.51883	0.759414	+	0.650607i	$0.225486\pi$
$920$	1.59230	0.0524967
$921$	−5.96000	−0.196389
$922$	−24.8568	−0.818614
$923$	−5.14505	−0.169352
$924$	−0.213229	−0.00701472
$925$	6.46612	0.212605
$926$	−12.7600	−0.419319
$927$	14.9397	0.490685
$928$	0.671264	0.0220353
$929$	−12.2841	−0.403029	−0.201514	−	0.979486i	$-0.564586\pi$
$929$	−12.2841	−0.403029	−0.201514	+	0.979486i	$0.564586\pi$
$930$	−0.635022	−0.0208232
$931$	10.7128	0.351099
$932$	−0.177298	−0.00580759
$933$	21.1280	0.691700
$934$	−0.528860	−0.0173048
$935$	−1.93849	−0.0633954
$936$	2.59675	0.0848773
$937$	−13.8708	−0.453139	−0.226570	−	0.973995i	$-0.572751\pi$
$937$	−13.8708	−0.453139	−0.226570	+	0.973995i	$0.572751\pi$
$938$	5.36951	0.175321
$939$	9.69087	0.316250
$940$	−2.10016	−0.0684998
$941$	−0.552739	−0.0180188	−0.00900939	−	0.999959i	$-0.502868\pi$
$941$	−0.552739	−0.0180188	−0.00900939	+	0.999959i	$0.502868\pi$
$942$	−4.84201	−0.157761
$943$	17.6412	0.574476
$944$	8.12564	0.264467
$945$	−4.37787	−0.142412
$946$	1.82199	0.0592382
$947$	1.18091	0.0383743	0.0191872	−	0.999816i	$-0.493892\pi$
$947$	1.18091	0.0383743	0.0191872	+	0.999816i	$0.493892\pi$
$948$	−8.61589	−0.279831
$949$	−10.1523	−0.329557
$950$	−1.95394	−0.0633943
$951$	−12.2905	−0.398546
$952$	8.75958	0.283900
$953$	60.8222	1.97023	0.985113	−	0.171908i	$-0.0549933\pi$
$953$	60.8222	1.97023	0.985113	+	0.171908i	$0.0549933\pi$
$954$	−20.5784	−0.666249
$955$	1.31127	0.0424317
$956$	−18.7030	−0.604898
$957$	−0.116199	−0.00375617
$958$	3.07938	0.0994904
$959$	−6.57740	−0.212395
$960$	0.635022	0.0204952
$961$	1.00000	0.0322581
$962$	−6.46612	−0.208476
$963$	11.5714	0.372883
$964$	14.8399	0.477960
$965$	−5.15787	−0.166038
$966$	1.24553	0.0400742
$967$	−29.5228	−0.949391	−0.474695	−	0.880150i	$-0.657442\pi$
$967$	−29.5228	−0.949391	−0.474695	+	0.880150i	$0.657442\pi$
$968$	−10.9257	−0.351165
$969$	8.82358	0.283454
$970$	−15.7222	−0.504808
$971$	−19.2480	−0.617699	−0.308849	−	0.951111i	$-0.599944\pi$
$971$	−19.2480	−0.617699	−0.308849	+	0.951111i	$0.599944\pi$
$972$	−14.1760	−0.454694
$973$	15.0287	0.481798
$974$	−8.04516	−0.257783
$975$	0.635022	0.0203370
$976$	−8.58732	−0.274873
$977$	39.7794	1.27265	0.636327	−	0.771419i	$-0.280453\pi$
$977$	39.7794	1.27265	0.636327	+	0.771419i	$0.280453\pi$
$978$	5.35348	0.171185
$979$	2.43453	0.0778080
$980$	5.48268	0.175138
$981$	−21.4825	−0.685882
$982$	18.0102	0.574729
$983$	−1.07234	−0.0342024	−0.0171012	−	0.999854i	$-0.505444\pi$
$983$	−1.07234	−0.0342024	−0.0171012	+	0.999854i	$0.505444\pi$
$984$	7.03542	0.224281
$985$	0.571208	0.0182002
$986$	4.77351	0.152019
$987$	−1.64279	−0.0522904
$988$	1.95394	0.0621632
$989$	−10.6428	−0.338420
$990$	0.707862	0.0224973
$991$	−19.4967	−0.619334	−0.309667	−	0.950845i	$-0.600218\pi$
$991$	−19.4967	−0.619334	−0.309667	+	0.950845i	$0.600218\pi$
$992$	−1.00000	−0.0317500
$993$	−6.47023	−0.205327
$994$	6.33766	0.201018
$995$	−0.766897	−0.0243123
$996$	5.71597	0.181117
$997$	−5.97916	−0.189362	−0.0946810	−	0.995508i	$-0.530183\pi$
$997$	−5.97916	−0.189362	−0.0946810	+	0.995508i	$0.530183\pi$
$998$	−27.2005	−0.861016
$999$	22.9809	0.727085

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4030.2.a.n.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4030.2.a.n.1.4	✓	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4030 = 2 \cdot 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4030.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -0.635022 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.635022 q^{6} +1.23180 q^{7} +1.00000 q^{8} -2.59675 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -0.635022 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.635022 q^{6} +1.23180 q^{7} +1.00000 q^{8} -2.59675 q^{9} -1.00000 q^{10} +0.272596 q^{11} -0.635022 q^{12} -1.00000 q^{13} +1.23180 q^{14} +0.635022 q^{15} +1.00000 q^{16} +7.11122 q^{17} -2.59675 q^{18} -1.95394 q^{19} -1.00000 q^{20} -0.782218 q^{21} +0.272596 q^{22} -1.59230 q^{23} -0.635022 q^{24} +1.00000 q^{25} -1.00000 q^{26} +3.55406 q^{27} +1.23180 q^{28} +0.671264 q^{29} +0.635022 q^{30} -1.00000 q^{31} +1.00000 q^{32} -0.173104 q^{33} +7.11122 q^{34} -1.23180 q^{35} -2.59675 q^{36} +6.46612 q^{37} -1.95394 q^{38} +0.635022 q^{39} -1.00000 q^{40} -11.0790 q^{41} -0.782218 q^{42} +6.68387 q^{43} +0.272596 q^{44} +2.59675 q^{45} -1.59230 q^{46} +2.10016 q^{47} -0.635022 q^{48} -5.48268 q^{49} +1.00000 q^{50} -4.51578 q^{51} -1.00000 q^{52} +7.92467 q^{53} +3.55406 q^{54} -0.272596 q^{55} +1.23180 q^{56} +1.24080 q^{57} +0.671264 q^{58} +8.12564 q^{59} +0.635022 q^{60} -8.58732 q^{61} -1.00000 q^{62} -3.19866 q^{63} +1.00000 q^{64} +1.00000 q^{65} -0.173104 q^{66} +4.35909 q^{67} +7.11122 q^{68} +1.01115 q^{69} -1.23180 q^{70} +5.14505 q^{71} -2.59675 q^{72} +10.1523 q^{73} +6.46612 q^{74} -0.635022 q^{75} -1.95394 q^{76} +0.335782 q^{77} +0.635022 q^{78} +13.5679 q^{79} -1.00000 q^{80} +5.53334 q^{81} -11.0790 q^{82} -9.00122 q^{83} -0.782218 q^{84} -7.11122 q^{85} +6.68387 q^{86} -0.426267 q^{87} +0.272596 q^{88} +8.93093 q^{89} +2.59675 q^{90} -1.23180 q^{91} -1.59230 q^{92} +0.635022 q^{93} +2.10016 q^{94} +1.95394 q^{95} -0.635022 q^{96} +15.7222 q^{97} -5.48268 q^{98} -0.707862 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} - q^{6} + q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 - q^6 + q^7 + 8 * q^8 + 9 * q^9	\( 8 q + 8 q^{2} - q^{3} + 8 q^{4} - 8 q^{5} - q^{6} + q^{7} + 8 q^{8} + 9 q^{9} - 8 q^{10} + 4 q^{11} - q^{12} - 8 q^{13} + q^{14} + q^{15} + 8 q^{16} - 5 q^{17} + 9 q^{18} + 2 q^{19} - 8 q^{20} + 17 q^{21} + 4 q^{22} + 4 q^{23} - q^{24} + 8 q^{25} - 8 q^{26} + 11 q^{27} + q^{28} + 11 q^{29} + q^{30} - 8 q^{31} + 8 q^{32} + 10 q^{33} - 5 q^{34} - q^{35} + 9 q^{36} + 19 q^{37} + 2 q^{38} + q^{39} - 8 q^{40} + 10 q^{41} + 17 q^{42} + 19 q^{43} + 4 q^{44} - 9 q^{45} + 4 q^{46} + 11 q^{47} - q^{48} + 11 q^{49} + 8 q^{50} + 7 q^{51} - 8 q^{52} + 8 q^{53} + 11 q^{54} - 4 q^{55} + q^{56} - 11 q^{57} + 11 q^{58} + 28 q^{59} + q^{60} - 12 q^{61} - 8 q^{62} + 20 q^{63} + 8 q^{64} + 8 q^{65} + 10 q^{66} + 24 q^{67} - 5 q^{68} + 30 q^{69} - q^{70} + 18 q^{71} + 9 q^{72} - 3 q^{73} + 19 q^{74} - q^{75} + 2 q^{76} - 7 q^{77} + q^{78} + 22 q^{79} - 8 q^{80} + 24 q^{81} + 10 q^{82} + 17 q^{83} + 17 q^{84} + 5 q^{85} + 19 q^{86} + 11 q^{87} + 4 q^{88} + 17 q^{89} - 9 q^{90} - q^{91} + 4 q^{92} + q^{93} + 11 q^{94} - 2 q^{95} - q^{96} - 24 q^{97} + 11 q^{98} + 23 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - q^3 + 8 * q^4 - 8 * q^5 - q^6 + q^7 + 8 * q^8 + 9 * q^9 - 8 * q^10 + 4 * q^11 - q^12 - 8 * q^13 + q^14 + q^15 + 8 * q^16 - 5 * q^17 + 9 * q^18 + 2 * q^19 - 8 * q^20 + 17 * q^21 + 4 * q^22 + 4 * q^23 - q^24 + 8 * q^25 - 8 * q^26 + 11 * q^27 + q^28 + 11 * q^29 + q^30 - 8 * q^31 + 8 * q^32 + 10 * q^33 - 5 * q^34 - q^35 + 9 * q^36 + 19 * q^37 + 2 * q^38 + q^39 - 8 * q^40 + 10 * q^41 + 17 * q^42 + 19 * q^43 + 4 * q^44 - 9 * q^45 + 4 * q^46 + 11 * q^47 - q^48 + 11 * q^49 + 8 * q^50 + 7 * q^51 - 8 * q^52 + 8 * q^53 + 11 * q^54 - 4 * q^55 + q^56 - 11 * q^57 + 11 * q^58 + 28 * q^59 + q^60 - 12 * q^61 - 8 * q^62 + 20 * q^63 + 8 * q^64 + 8 * q^65 + 10 * q^66 + 24 * q^67 - 5 * q^68 + 30 * q^69 - q^70 + 18 * q^71 + 9 * q^72 - 3 * q^73 + 19 * q^74 - q^75 + 2 * q^76 - 7 * q^77 + q^78 + 22 * q^79 - 8 * q^80 + 24 * q^81 + 10 * q^82 + 17 * q^83 + 17 * q^84 + 5 * q^85 + 19 * q^86 + 11 * q^87 + 4 * q^88 + 17 * q^89 - 9 * q^90 - q^91 + 4 * q^92 + q^93 + 11 * q^94 - 2 * q^95 - q^96 - 24 * q^97 + 11 * q^98 + 23 * q^99

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