LMFDB - Embedded newform 4002.2.a.bi.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4002,2,Mod(1,4002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4002, 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("4002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4002 = 2 \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4002.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:	$31.9561308889$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 24x^{6} - 3x^{5} + 194x^{4} + 39x^{3} - 607x^{2} - 104x + 600 $ x^8 - 24x^6 - 3x^5 + 194x^4 + 39x^3 - 607x^2 - 104x + 600
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.85533$ of defining polynomial
Character	$\chi$	$=$	4002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.27098 q^{5} -1.00000 q^{6} +3.55617 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.27098 q^{5} -1.00000 q^{6} +3.55617 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.27098 q^{10} -5.15608 q^{11} +1.00000 q^{12} -7.00436 q^{13} -3.55617 q^{14} +1.27098 q^{15} +1.00000 q^{16} +1.35112 q^{17} -1.00000 q^{18} -2.38748 q^{19} +1.27098 q^{20} +3.55617 q^{21} +5.15608 q^{22} -1.00000 q^{23} -1.00000 q^{24} -3.38461 q^{25} +7.00436 q^{26} +1.00000 q^{27} +3.55617 q^{28} +1.00000 q^{29} -1.27098 q^{30} +2.05949 q^{31} -1.00000 q^{32} -5.15608 q^{33} -1.35112 q^{34} +4.51982 q^{35} +1.00000 q^{36} -1.49026 q^{37} +2.38748 q^{38} -7.00436 q^{39} -1.27098 q^{40} -8.70584 q^{41} -3.55617 q^{42} +2.66386 q^{43} -5.15608 q^{44} +1.27098 q^{45} +1.00000 q^{46} -11.8619 q^{47} +1.00000 q^{48} +5.64635 q^{49} +3.38461 q^{50} +1.35112 q^{51} -7.00436 q^{52} +8.79312 q^{53} -1.00000 q^{54} -6.55328 q^{55} -3.55617 q^{56} -2.38748 q^{57} -1.00000 q^{58} -5.12084 q^{59} +1.27098 q^{60} +4.67059 q^{61} -2.05949 q^{62} +3.55617 q^{63} +1.00000 q^{64} -8.90239 q^{65} +5.15608 q^{66} +5.68433 q^{67} +1.35112 q^{68} -1.00000 q^{69} -4.51982 q^{70} -8.29814 q^{71} -1.00000 q^{72} -14.0697 q^{73} +1.49026 q^{74} -3.38461 q^{75} -2.38748 q^{76} -18.3359 q^{77} +7.00436 q^{78} -3.38346 q^{79} +1.27098 q^{80} +1.00000 q^{81} +8.70584 q^{82} -13.3000 q^{83} +3.55617 q^{84} +1.71725 q^{85} -2.66386 q^{86} +1.00000 q^{87} +5.15608 q^{88} -17.9618 q^{89} -1.27098 q^{90} -24.9087 q^{91} -1.00000 q^{92} +2.05949 q^{93} +11.8619 q^{94} -3.03443 q^{95} -1.00000 q^{96} +19.1826 q^{97} -5.64635 q^{98} -5.15608 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9} + 3 q^{10} - 7 q^{11} + 8 q^{12} - 3 q^{13} + 6 q^{14} - 3 q^{15} + 8 q^{16} - 12 q^{17} - 8 q^{18} - 4 q^{19} - 3 q^{20} - 6 q^{21} + 7 q^{22} - 8 q^{23} - 8 q^{24} + 13 q^{25} + 3 q^{26} + 8 q^{27} - 6 q^{28} + 8 q^{29} + 3 q^{30} - q^{31} - 8 q^{32} - 7 q^{33} + 12 q^{34} - 6 q^{35} + 8 q^{36} - 11 q^{37} + 4 q^{38} - 3 q^{39} + 3 q^{40} - 17 q^{41} + 6 q^{42} - 10 q^{43} - 7 q^{44} - 3 q^{45} + 8 q^{46} - 26 q^{47} + 8 q^{48} + 10 q^{49} - 13 q^{50} - 12 q^{51} - 3 q^{52} - 2 q^{53} - 8 q^{54} + q^{55} + 6 q^{56} - 4 q^{57} - 8 q^{58} - 25 q^{59} - 3 q^{60} + 3 q^{61} + q^{62} - 6 q^{63} + 8 q^{64} - 25 q^{65} + 7 q^{66} + q^{67} - 12 q^{68} - 8 q^{69} + 6 q^{70} - 27 q^{71} - 8 q^{72} + 16 q^{73} + 11 q^{74} + 13 q^{75} - 4 q^{76} - 16 q^{77} + 3 q^{78} - 6 q^{79} - 3 q^{80} + 8 q^{81} + 17 q^{82} - 44 q^{83} - 6 q^{84} - 20 q^{85} + 10 q^{86} + 8 q^{87} + 7 q^{88} - 52 q^{89} + 3 q^{90} - 18 q^{91} - 8 q^{92} - q^{93} + 26 q^{94} - 56 q^{95} - 8 q^{96} - 4 q^{97} - 10 q^{98} - 7 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9 + 3 * q^10 - 7 * q^11 + 8 * q^12 - 3 * q^13 + 6 * q^14 - 3 * q^15 + 8 * q^16 - 12 * q^17 - 8 * q^18 - 4 * q^19 - 3 * q^20 - 6 * q^21 + 7 * q^22 - 8 * q^23 - 8 * q^24 + 13 * q^25 + 3 * q^26 + 8 * q^27 - 6 * q^28 + 8 * q^29 + 3 * q^30 - q^31 - 8 * q^32 - 7 * q^33 + 12 * q^34 - 6 * q^35 + 8 * q^36 - 11 * q^37 + 4 * q^38 - 3 * q^39 + 3 * q^40 - 17 * q^41 + 6 * q^42 - 10 * q^43 - 7 * q^44 - 3 * q^45 + 8 * q^46 - 26 * q^47 + 8 * q^48 + 10 * q^49 - 13 * q^50 - 12 * q^51 - 3 * q^52 - 2 * q^53 - 8 * q^54 + q^55 + 6 * q^56 - 4 * q^57 - 8 * q^58 - 25 * q^59 - 3 * q^60 + 3 * q^61 + q^62 - 6 * q^63 + 8 * q^64 - 25 * q^65 + 7 * q^66 + q^67 - 12 * q^68 - 8 * q^69 + 6 * q^70 - 27 * q^71 - 8 * q^72 + 16 * q^73 + 11 * q^74 + 13 * q^75 - 4 * q^76 - 16 * q^77 + 3 * q^78 - 6 * q^79 - 3 * q^80 + 8 * q^81 + 17 * q^82 - 44 * q^83 - 6 * q^84 - 20 * q^85 + 10 * q^86 + 8 * q^87 + 7 * q^88 - 52 * q^89 + 3 * q^90 - 18 * q^91 - 8 * q^92 - q^93 + 26 * q^94 - 56 * q^95 - 8 * q^96 - 4 * q^97 - 10 * q^98 - 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	1.27098	0.568399	0.284200	−	0.958765i	$-0.408272\pi$
$5$	1.27098	0.568399	0.284200	+	0.958765i	$0.408272\pi$
$6$	−1.00000	−0.408248
$7$	3.55617	1.34411	0.672053	−	0.740503i	$-0.265413\pi$
$7$	3.55617	1.34411	0.672053	+	0.740503i	$0.265413\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.27098	−0.401919
$11$	−5.15608	−1.55462	−0.777309	−	0.629119i	$-0.783416\pi$
$11$	−5.15608	−1.55462	−0.777309	+	0.629119i	$0.783416\pi$
$12$	1.00000	0.288675
$13$	−7.00436	−1.94266	−0.971330	−	0.237736i	$-0.923595\pi$
$13$	−7.00436	−1.94266	−0.971330	+	0.237736i	$0.923595\pi$
$14$	−3.55617	−0.950427
$15$	1.27098	0.328165
$16$	1.00000	0.250000
$17$	1.35112	0.327695	0.163848	−	0.986486i	$-0.447609\pi$
$17$	1.35112	0.327695	0.163848	+	0.986486i	$0.447609\pi$
$18$	−1.00000	−0.235702
$19$	−2.38748	−0.547724	−0.273862	−	0.961769i	$-0.588301\pi$
$19$	−2.38748	−0.547724	−0.273862	+	0.961769i	$0.588301\pi$
$20$	1.27098	0.284200
$21$	3.55617	0.776020
$22$	5.15608	1.09928
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	−3.38461	−0.676922
$26$	7.00436	1.37367
$27$	1.00000	0.192450
$28$	3.55617	0.672053
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−1.27098	−0.232048
$31$	2.05949	0.369895	0.184948	−	0.982748i	$-0.440788\pi$
$31$	2.05949	0.369895	0.184948	+	0.982748i	$0.440788\pi$
$32$	−1.00000	−0.176777
$33$	−5.15608	−0.897559
$34$	−1.35112	−0.231716
$35$	4.51982	0.763989
$36$	1.00000	0.166667
$37$	−1.49026	−0.244998	−0.122499	−	0.992469i	$-0.539091\pi$
$37$	−1.49026	−0.244998	−0.122499	+	0.992469i	$0.539091\pi$
$38$	2.38748	0.387300
$39$	−7.00436	−1.12160
$40$	−1.27098	−0.200959
$41$	−8.70584	−1.35962	−0.679812	−	0.733387i	$-0.737939\pi$
$41$	−8.70584	−1.35962	−0.679812	+	0.733387i	$0.737939\pi$
$42$	−3.55617	−0.548729
$43$	2.66386	0.406234	0.203117	−	0.979154i	$-0.434893\pi$
$43$	2.66386	0.406234	0.203117	+	0.979154i	$0.434893\pi$
$44$	−5.15608	−0.777309
$45$	1.27098	0.189466
$46$	1.00000	0.147442
$47$	−11.8619	−1.73024	−0.865121	−	0.501563i	$-0.832759\pi$
$47$	−11.8619	−1.73024	−0.865121	+	0.501563i	$0.832759\pi$
$48$	1.00000	0.144338
$49$	5.64635	0.806621
$50$	3.38461	0.478656
$51$	1.35112	0.189195
$52$	−7.00436	−0.971330
$53$	8.79312	1.20783	0.603914	−	0.797050i	$-0.293607\pi$
$53$	8.79312	1.20783	0.603914	+	0.797050i	$0.293607\pi$
$54$	−1.00000	−0.136083
$55$	−6.55328	−0.883643
$56$	−3.55617	−0.475213
$57$	−2.38748	−0.316229
$58$	−1.00000	−0.131306
$59$	−5.12084	−0.666677	−0.333338	−	0.942807i	$-0.608175\pi$
$59$	−5.12084	−0.666677	−0.333338	+	0.942807i	$0.608175\pi$
$60$	1.27098	0.164083
$61$	4.67059	0.598008	0.299004	−	0.954252i	$-0.403346\pi$
$61$	4.67059	0.598008	0.299004	+	0.954252i	$0.403346\pi$
$62$	−2.05949	−0.261555
$63$	3.55617	0.448035
$64$	1.00000	0.125000
$65$	−8.90239	−1.10421
$66$	5.15608	0.634670
$67$	5.68433	0.694452	0.347226	−	0.937782i	$-0.387124\pi$
$67$	5.68433	0.694452	0.347226	+	0.937782i	$0.387124\pi$
$68$	1.35112	0.163848
$69$	−1.00000	−0.120386
$70$	−4.51982	−0.540222
$71$	−8.29814	−0.984807	−0.492404	−	0.870367i	$-0.663882\pi$
$71$	−8.29814	−0.984807	−0.492404	+	0.870367i	$0.663882\pi$
$72$	−1.00000	−0.117851
$73$	−14.0697	−1.64674	−0.823368	−	0.567507i	$-0.807908\pi$
$73$	−14.0697	−1.64674	−0.823368	+	0.567507i	$0.807908\pi$
$74$	1.49026	0.173240
$75$	−3.38461	−0.390821
$76$	−2.38748	−0.273862
$77$	−18.3359	−2.08957
$78$	7.00436	0.793088
$79$	−3.38346	−0.380669	−0.190334	−	0.981719i	$-0.560957\pi$
$79$	−3.38346	−0.380669	−0.190334	+	0.981719i	$0.560957\pi$
$80$	1.27098	0.142100
$81$	1.00000	0.111111
$82$	8.70584	0.961399
$83$	−13.3000	−1.45987	−0.729934	−	0.683518i	$-0.760449\pi$
$83$	−13.3000	−1.45987	−0.729934	+	0.683518i	$0.760449\pi$
$84$	3.55617	0.388010
$85$	1.71725	0.186262
$86$	−2.66386	−0.287251
$87$	1.00000	0.107211
$88$	5.15608	0.549640
$89$	−17.9618	−1.90395	−0.951974	−	0.306178i	$-0.900950\pi$
$89$	−17.9618	−1.90395	−0.951974	+	0.306178i	$0.900950\pi$
$90$	−1.27098	−0.133973
$91$	−24.9087	−2.61114
$92$	−1.00000	−0.104257
$93$	2.05949	0.213559
$94$	11.8619	1.22347
$95$	−3.03443	−0.311326
$96$	−1.00000	−0.102062
$97$	19.1826	1.94770	0.973849	−	0.227198i	$-0.0729563\pi$
$97$	19.1826	1.94770	0.973849	+	0.227198i	$0.0729563\pi$
$98$	−5.64635	−0.570367
$99$	−5.15608	−0.518206
$100$	−3.38461	−0.338461
$101$	−5.88296	−0.585377	−0.292688	−	0.956208i	$-0.594550\pi$
$101$	−5.88296	−0.585377	−0.292688	+	0.956208i	$0.594550\pi$
$102$	−1.35112	−0.133781
$103$	1.87749	0.184994	0.0924972	−	0.995713i	$-0.470515\pi$
$103$	1.87749	0.184994	0.0924972	+	0.995713i	$0.470515\pi$
$104$	7.00436	0.686834
$105$	4.51982	0.441089
$106$	−8.79312	−0.854063
$107$	17.9346	1.73380	0.866902	−	0.498479i	$-0.166108\pi$
$107$	17.9346	1.73380	0.866902	+	0.498479i	$0.166108\pi$
$108$	1.00000	0.0962250
$109$	−7.36656	−0.705588	−0.352794	−	0.935701i	$-0.614768\pi$
$109$	−7.36656	−0.705588	−0.352794	+	0.935701i	$0.614768\pi$
$110$	6.55328	0.624830
$111$	−1.49026	−0.141449
$112$	3.55617	0.336027
$113$	−14.4153	−1.35608	−0.678038	−	0.735027i	$-0.737170\pi$
$113$	−14.4153	−1.35608	−0.678038	+	0.735027i	$0.737170\pi$
$114$	2.38748	0.223608
$115$	−1.27098	−0.118519
$116$	1.00000	0.0928477
$117$	−7.00436	−0.647553
$118$	5.12084	0.471412
$119$	4.80482	0.440457
$120$	−1.27098	−0.116024
$121$	15.5852	1.41684
$122$	−4.67059	−0.422856
$123$	−8.70584	−0.784979
$124$	2.05949	0.184948
$125$	−10.6567	−0.953161
$126$	−3.55617	−0.316809
$127$	9.72588	0.863032	0.431516	−	0.902105i	$-0.357979\pi$
$127$	9.72588	0.863032	0.431516	+	0.902105i	$0.357979\pi$
$128$	−1.00000	−0.0883883
$129$	2.66386	0.234540
$130$	8.90239	0.780792
$131$	5.33949	0.466513	0.233257	−	0.972415i	$-0.425062\pi$
$131$	5.33949	0.466513	0.233257	+	0.972415i	$0.425062\pi$
$132$	−5.15608	−0.448780
$133$	−8.49027	−0.736200
$134$	−5.68433	−0.491052
$135$	1.27098	0.109388
$136$	−1.35112	−0.115858
$137$	−5.09789	−0.435542	−0.217771	−	0.976000i	$-0.569879\pi$
$137$	−5.09789	−0.435542	−0.217771	+	0.976000i	$0.569879\pi$
$138$	1.00000	0.0851257
$139$	21.8698	1.85497	0.927486	−	0.373857i	$-0.121965\pi$
$139$	21.8698	1.85497	0.927486	+	0.373857i	$0.121965\pi$
$140$	4.51982	0.381994
$141$	−11.8619	−0.998956
$142$	8.29814	0.696364
$143$	36.1151	3.02009
$144$	1.00000	0.0833333
$145$	1.27098	0.105549
$146$	14.0697	1.16442
$147$	5.64635	0.465703
$148$	−1.49026	−0.122499
$149$	17.5495	1.43771	0.718855	−	0.695160i	$-0.244667\pi$
$149$	17.5495	1.43771	0.718855	+	0.695160i	$0.244667\pi$
$150$	3.38461	0.276352
$151$	−2.81294	−0.228914	−0.114457	−	0.993428i	$-0.536513\pi$
$151$	−2.81294	−0.228914	−0.114457	+	0.993428i	$0.536513\pi$
$152$	2.38748	0.193650
$153$	1.35112	0.109232
$154$	18.3359	1.47755
$155$	2.61757	0.210248
$156$	−7.00436	−0.560798
$157$	4.97882	0.397353	0.198677	−	0.980065i	$-0.436336\pi$
$157$	4.97882	0.397353	0.198677	+	0.980065i	$0.436336\pi$
$158$	3.38346	0.269173
$159$	8.79312	0.697340
$160$	−1.27098	−0.100480
$161$	−3.55617	−0.280265
$162$	−1.00000	−0.0785674
$163$	−16.8603	−1.32060	−0.660302	−	0.751000i	$-0.729572\pi$
$163$	−16.8603	−1.32060	−0.660302	+	0.751000i	$0.729572\pi$
$164$	−8.70584	−0.679812
$165$	−6.55328	−0.510172
$166$	13.3000	1.03228
$167$	7.64373	0.591489	0.295745	−	0.955267i	$-0.404432\pi$
$167$	7.64373	0.591489	0.295745	+	0.955267i	$0.404432\pi$
$168$	−3.55617	−0.274364
$169$	36.0611	2.77393
$170$	−1.71725	−0.131707
$171$	−2.38748	−0.182575
$172$	2.66386	0.203117
$173$	−1.24075	−0.0943324	−0.0471662	−	0.998887i	$-0.515019\pi$
$173$	−1.24075	−0.0943324	−0.0471662	+	0.998887i	$0.515019\pi$
$174$	−1.00000	−0.0758098
$175$	−12.0363	−0.909856
$176$	−5.15608	−0.388655
$177$	−5.12084	−0.384906
$178$	17.9618	1.34629
$179$	−24.8582	−1.85799	−0.928995	−	0.370093i	$-0.879326\pi$
$179$	−24.8582	−1.85799	−0.928995	+	0.370093i	$0.879326\pi$
$180$	1.27098	0.0947332
$181$	4.12360	0.306505	0.153252	−	0.988187i	$-0.451025\pi$
$181$	4.12360	0.306505	0.153252	+	0.988187i	$0.451025\pi$
$182$	24.9087	1.84636
$183$	4.67059	0.345260
$184$	1.00000	0.0737210
$185$	−1.89409	−0.139256
$186$	−2.05949	−0.151009
$187$	−6.96650	−0.509441
$188$	−11.8619	−0.865121
$189$	3.55617	0.258673
$190$	3.03443	0.220141
$191$	0.377313	0.0273014	0.0136507	−	0.999907i	$-0.495655\pi$
$191$	0.377313	0.0273014	0.0136507	+	0.999907i	$0.495655\pi$
$192$	1.00000	0.0721688
$193$	−10.6030	−0.763218	−0.381609	−	0.924324i	$-0.624630\pi$
$193$	−10.6030	−0.763218	−0.381609	+	0.924324i	$0.624630\pi$
$194$	−19.1826	−1.37723
$195$	−8.90239	−0.637514
$196$	5.64635	0.403311
$197$	−5.72888	−0.408166	−0.204083	−	0.978954i	$-0.565421\pi$
$197$	−5.72888	−0.408166	−0.204083	+	0.978954i	$0.565421\pi$
$198$	5.15608	0.366427
$199$	15.4695	1.09660	0.548301	−	0.836281i	$-0.315275\pi$
$199$	15.4695	1.09660	0.548301	+	0.836281i	$0.315275\pi$
$200$	3.38461	0.239328
$201$	5.68433	0.400942
$202$	5.88296	0.413924
$203$	3.55617	0.249594
$204$	1.35112	0.0945975
$205$	−11.0649	−0.772809
$206$	−1.87749	−0.130811
$207$	−1.00000	−0.0695048
$208$	−7.00436	−0.485665
$209$	12.3100	0.851502
$210$	−4.51982	−0.311897
$211$	−17.6724	−1.21662	−0.608309	−	0.793700i	$-0.708152\pi$
$211$	−17.6724	−1.21662	−0.608309	+	0.793700i	$0.708152\pi$
$212$	8.79312	0.603914
$213$	−8.29814	−0.568579
$214$	−17.9346	−1.22598
$215$	3.38571	0.230903
$216$	−1.00000	−0.0680414
$217$	7.32390	0.497179
$218$	7.36656	0.498926
$219$	−14.0697	−0.950744
$220$	−6.55328	−0.441822
$221$	−9.46375	−0.636601
$222$	1.49026	0.100020
$223$	−15.8328	−1.06024	−0.530120	−	0.847923i	$-0.677853\pi$
$223$	−15.8328	−1.06024	−0.530120	+	0.847923i	$0.677853\pi$
$224$	−3.55617	−0.237607
$225$	−3.38461	−0.225641
$226$	14.4153	0.958891
$227$	−5.22577	−0.346846	−0.173423	−	0.984847i	$-0.555483\pi$
$227$	−5.22577	−0.346846	−0.173423	+	0.984847i	$0.555483\pi$
$228$	−2.38748	−0.158114
$229$	17.9272	1.18467	0.592333	−	0.805693i	$-0.298207\pi$
$229$	17.9272	1.18467	0.592333	+	0.805693i	$0.298207\pi$
$230$	1.27098	0.0838059
$231$	−18.3359	−1.20641
$232$	−1.00000	−0.0656532
$233$	−3.81246	−0.249763	−0.124881	−	0.992172i	$-0.539855\pi$
$233$	−3.81246	−0.249763	−0.124881	+	0.992172i	$0.539855\pi$
$234$	7.00436	0.457889
$235$	−15.0763	−0.983468
$236$	−5.12084	−0.333338
$237$	−3.38346	−0.219779
$238$	−4.80482	−0.311450
$239$	−8.22242	−0.531864	−0.265932	−	0.963992i	$-0.585680\pi$
$239$	−8.22242	−0.531864	−0.265932	+	0.963992i	$0.585680\pi$
$240$	1.27098	0.0820413
$241$	25.1659	1.62108	0.810539	−	0.585684i	$-0.199174\pi$
$241$	25.1659	1.62108	0.810539	+	0.585684i	$0.199174\pi$
$242$	−15.5852	−1.00186
$243$	1.00000	0.0641500
$244$	4.67059	0.299004
$245$	7.17639	0.458483
$246$	8.70584	0.555064
$247$	16.7227	1.06404
$248$	−2.05949	−0.130778
$249$	−13.3000	−0.842855
$250$	10.6567	0.673987
$251$	22.1256	1.39656	0.698278	−	0.715827i	$-0.253950\pi$
$251$	22.1256	1.39656	0.698278	+	0.715827i	$0.253950\pi$
$252$	3.55617	0.224018
$253$	5.15608	0.324160
$254$	−9.72588	−0.610256
$255$	1.71725	0.107538
$256$	1.00000	0.0625000
$257$	6.83211	0.426175	0.213088	−	0.977033i	$-0.431648\pi$
$257$	6.83211	0.426175	0.213088	+	0.977033i	$0.431648\pi$
$258$	−2.66386	−0.165845
$259$	−5.29963	−0.329303
$260$	−8.90239	−0.552103
$261$	1.00000	0.0618984
$262$	−5.33949	−0.329875
$263$	2.54269	0.156789	0.0783944	−	0.996922i	$-0.475021\pi$
$263$	2.54269	0.156789	0.0783944	+	0.996922i	$0.475021\pi$
$264$	5.15608	0.317335
$265$	11.1759	0.686528
$266$	8.49027	0.520572
$267$	−17.9618	−1.09925
$268$	5.68433	0.347226
$269$	15.2555	0.930146	0.465073	−	0.885272i	$-0.346028\pi$
$269$	15.2555	0.930146	0.465073	+	0.885272i	$0.346028\pi$
$270$	−1.27098	−0.0773493
$271$	−32.3195	−1.96327	−0.981635	−	0.190771i	$-0.938901\pi$
$271$	−32.3195	−1.96327	−0.981635	+	0.190771i	$0.938901\pi$
$272$	1.35112	0.0819239
$273$	−24.9087	−1.50754
$274$	5.09789	0.307975
$275$	17.4513	1.05236
$276$	−1.00000	−0.0601929
$277$	−20.7744	−1.24821	−0.624107	−	0.781339i	$-0.714537\pi$
$277$	−20.7744	−1.24821	−0.624107	+	0.781339i	$0.714537\pi$
$278$	−21.8698	−1.31166
$279$	2.05949	0.123298
$280$	−4.51982	−0.270111
$281$	−19.8392	−1.18351	−0.591754	−	0.806119i	$-0.701564\pi$
$281$	−19.8392	−1.18351	−0.591754	+	0.806119i	$0.701564\pi$
$282$	11.8619	0.706369
$283$	−3.30625	−0.196536	−0.0982680	−	0.995160i	$-0.531330\pi$
$283$	−3.30625	−0.196536	−0.0982680	+	0.995160i	$0.531330\pi$
$284$	−8.29814	−0.492404
$285$	−3.03443	−0.179744
$286$	−36.1151	−2.13553
$287$	−30.9594	−1.82748
$288$	−1.00000	−0.0589256
$289$	−15.1745	−0.892616
$290$	−1.27098	−0.0746345
$291$	19.1826	1.12450
$292$	−14.0697	−0.823368
$293$	−3.83056	−0.223784	−0.111892	−	0.993720i	$-0.535691\pi$
$293$	−3.83056	−0.223784	−0.111892	+	0.993720i	$0.535691\pi$
$294$	−5.64635	−0.329302
$295$	−6.50848	−0.378938
$296$	1.49026	0.0866198
$297$	−5.15608	−0.299186
$298$	−17.5495	−1.01661
$299$	7.00436	0.405073
$300$	−3.38461	−0.195411
$301$	9.47313	0.546022
$302$	2.81294	0.161866
$303$	−5.88296	−0.337967
$304$	−2.38748	−0.136931
$305$	5.93623	0.339907
$306$	−1.35112	−0.0772386
$307$	31.6974	1.80907	0.904533	−	0.426403i	$-0.140219\pi$
$307$	31.6974	1.80907	0.904533	+	0.426403i	$0.140219\pi$
$308$	−18.3359	−1.04479
$309$	1.87749	0.106807
$310$	−2.61757	−0.148668
$311$	29.0819	1.64908	0.824541	−	0.565802i	$-0.191433\pi$
$311$	29.0819	1.64908	0.824541	+	0.565802i	$0.191433\pi$
$312$	7.00436	0.396544
$313$	13.7348	0.776337	0.388169	−	0.921588i	$-0.373108\pi$
$313$	13.7348	0.776337	0.388169	+	0.921588i	$0.373108\pi$
$314$	−4.97882	−0.280971
$315$	4.51982	0.254663
$316$	−3.38346	−0.190334
$317$	31.0575	1.74436	0.872180	−	0.489185i	$-0.162706\pi$
$317$	31.0575	1.74436	0.872180	+	0.489185i	$0.162706\pi$
$318$	−8.79312	−0.493094
$319$	−5.15608	−0.288685
$320$	1.27098	0.0710499
$321$	17.9346	1.00101
$322$	3.55617	0.198178
$323$	−3.22577	−0.179487
$324$	1.00000	0.0555556
$325$	23.7070	1.31503
$326$	16.8603	0.933808
$327$	−7.36656	−0.407372
$328$	8.70584	0.480699
$329$	−42.1831	−2.32563
$330$	6.55328	0.360746
$331$	−5.74186	−0.315601	−0.157800	−	0.987471i	$-0.550440\pi$
$331$	−5.74186	−0.315601	−0.157800	+	0.987471i	$0.550440\pi$
$332$	−13.3000	−0.729934
$333$	−1.49026	−0.0816659
$334$	−7.64373	−0.418246
$335$	7.22467	0.394726
$336$	3.55617	0.194005
$337$	−14.4609	−0.787735	−0.393867	−	0.919167i	$-0.628863\pi$
$337$	−14.4609	−0.787735	−0.393867	+	0.919167i	$0.628863\pi$
$338$	−36.0611	−1.96146
$339$	−14.4153	−0.782931
$340$	1.71725	0.0931309
$341$	−10.6189	−0.575046
$342$	2.38748	0.129100
$343$	−4.81382	−0.259922
$344$	−2.66386	−0.143626
$345$	−1.27098	−0.0684272
$346$	1.24075	0.0667031
$347$	21.1657	1.13623	0.568116	−	0.822949i	$-0.307673\pi$
$347$	21.1657	1.13623	0.568116	+	0.822949i	$0.307673\pi$
$348$	1.00000	0.0536056
$349$	−28.2520	−1.51230	−0.756148	−	0.654401i	$-0.772921\pi$
$349$	−28.2520	−1.51230	−0.756148	+	0.654401i	$0.772921\pi$
$350$	12.0363	0.643365
$351$	−7.00436	−0.373865
$352$	5.15608	0.274820
$353$	−21.8756	−1.16432	−0.582160	−	0.813074i	$-0.697792\pi$
$353$	−21.8756	−1.16432	−0.582160	+	0.813074i	$0.697792\pi$
$354$	5.12084	0.272170
$355$	−10.5468	−0.559764
$356$	−17.9618	−0.951974
$357$	4.80482	0.254298
$358$	24.8582	1.31380
$359$	−34.4973	−1.82070	−0.910348	−	0.413844i	$-0.864186\pi$
$359$	−34.4973	−1.82070	−0.910348	+	0.413844i	$0.864186\pi$
$360$	−1.27098	−0.0669865
$361$	−13.3000	−0.699998
$362$	−4.12360	−0.216731
$363$	15.5852	0.818011
$364$	−24.9087	−1.30557
$365$	−17.8823	−0.936004
$366$	−4.67059	−0.244136
$367$	12.0238	0.627635	0.313817	−	0.949483i	$-0.398392\pi$
$367$	12.0238	0.627635	0.313817	+	0.949483i	$0.398392\pi$
$368$	−1.00000	−0.0521286
$369$	−8.70584	−0.453208
$370$	1.89409	0.0984692
$371$	31.2698	1.62345
$372$	2.05949	0.106780
$373$	34.5695	1.78994	0.894971	−	0.446125i	$-0.147196\pi$
$373$	34.5695	1.78994	0.894971	+	0.446125i	$0.147196\pi$
$374$	6.96650	0.360229
$375$	−10.6567	−0.550308
$376$	11.8619	0.611733
$377$	−7.00436	−0.360743
$378$	−3.55617	−0.182910
$379$	−5.39787	−0.277270	−0.138635	−	0.990344i	$-0.544272\pi$
$379$	−5.39787	−0.277270	−0.138635	+	0.990344i	$0.544272\pi$
$380$	−3.03443	−0.155663
$381$	9.72588	0.498272
$382$	−0.377313	−0.0193050
$383$	−4.79803	−0.245168	−0.122584	−	0.992458i	$-0.539118\pi$
$383$	−4.79803	−0.245168	−0.122584	+	0.992458i	$0.539118\pi$
$384$	−1.00000	−0.0510310
$385$	−23.3046	−1.18771
$386$	10.6030	0.539677
$387$	2.66386	0.135411
$388$	19.1826	0.973849
$389$	18.8338	0.954910	0.477455	−	0.878656i	$-0.341559\pi$
$389$	18.8338	0.954910	0.477455	+	0.878656i	$0.341559\pi$
$390$	8.90239	0.450790
$391$	−1.35112	−0.0683292
$392$	−5.64635	−0.285184
$393$	5.33949	0.269342
$394$	5.72888	0.288617
$395$	−4.30030	−0.216372
$396$	−5.15608	−0.259103
$397$	1.30415	0.0654534	0.0327267	−	0.999464i	$-0.489581\pi$
$397$	1.30415	0.0654534	0.0327267	+	0.999464i	$0.489581\pi$
$398$	−15.4695	−0.775414
$399$	−8.49027	−0.425045
$400$	−3.38461	−0.169231
$401$	−14.8877	−0.743456	−0.371728	−	0.928342i	$-0.621235\pi$
$401$	−14.8877	−0.743456	−0.371728	+	0.928342i	$0.621235\pi$
$402$	−5.68433	−0.283509
$403$	−14.4254	−0.718581
$404$	−5.88296	−0.292688
$405$	1.27098	0.0631555
$406$	−3.55617	−0.176490
$407$	7.68392	0.380878
$408$	−1.35112	−0.0668906
$409$	−0.863003	−0.0426728	−0.0213364	−	0.999772i	$-0.506792\pi$
$409$	−0.863003	−0.0426728	−0.0213364	+	0.999772i	$0.506792\pi$
$410$	11.0649	0.546458
$411$	−5.09789	−0.251460
$412$	1.87749	0.0924972
$413$	−18.2106	−0.896084
$414$	1.00000	0.0491473
$415$	−16.9041	−0.829788
$416$	7.00436	0.343417
$417$	21.8698	1.07097
$418$	−12.3100	−0.602103
$419$	−23.0733	−1.12720	−0.563602	−	0.826047i	$-0.690585\pi$
$419$	−23.0733	−1.12720	−0.563602	+	0.826047i	$0.690585\pi$
$420$	4.51982	0.220545
$421$	24.0550	1.17237	0.586184	−	0.810178i	$-0.300630\pi$
$421$	24.0550	1.17237	0.586184	+	0.810178i	$0.300630\pi$
$422$	17.6724	0.860279
$423$	−11.8619	−0.576748
$424$	−8.79312	−0.427032
$425$	−4.57303	−0.221824
$426$	8.29814	0.402046
$427$	16.6094	0.803786
$428$	17.9346	0.866902
$429$	36.1151	1.74365
$430$	−3.38571	−0.163273
$431$	−13.6485	−0.657423	−0.328711	−	0.944430i	$-0.606614\pi$
$431$	−13.6485	−0.657423	−0.328711	+	0.944430i	$0.606614\pi$
$432$	1.00000	0.0481125
$433$	−40.1283	−1.92845	−0.964223	−	0.265094i	$-0.914597\pi$
$433$	−40.1283	−1.92845	−0.964223	+	0.265094i	$0.914597\pi$
$434$	−7.32390	−0.351558
$435$	1.27098	0.0609388
$436$	−7.36656	−0.352794
$437$	2.38748	0.114208
$438$	14.0697	0.672278
$439$	3.93933	0.188014	0.0940070	−	0.995572i	$-0.470032\pi$
$439$	3.93933	0.188014	0.0940070	+	0.995572i	$0.470032\pi$
$440$	6.55328	0.312415
$441$	5.64635	0.268874
$442$	9.46375	0.450145
$443$	−6.90018	−0.327837	−0.163919	−	0.986474i	$-0.552413\pi$
$443$	−6.90018	−0.327837	−0.163919	+	0.986474i	$0.552413\pi$
$444$	−1.49026	−0.0707247
$445$	−22.8291	−1.08220
$446$	15.8328	0.749703
$447$	17.5495	0.830062
$448$	3.55617	0.168013
$449$	27.3893	1.29258	0.646291	−	0.763091i	$-0.276319\pi$
$449$	27.3893	1.29258	0.646291	+	0.763091i	$0.276319\pi$
$450$	3.38461	0.159552
$451$	44.8880	2.11370
$452$	−14.4153	−0.678038
$453$	−2.81294	−0.132163
$454$	5.22577	0.245258
$455$	−31.6584	−1.48417
$456$	2.38748	0.111804
$457$	−5.21128	−0.243773	−0.121887	−	0.992544i	$-0.538894\pi$
$457$	−5.21128	−0.243773	−0.121887	+	0.992544i	$0.538894\pi$
$458$	−17.9272	−0.837685
$459$	1.35112	0.0630650
$460$	−1.27098	−0.0592597
$461$	−13.9878	−0.651475	−0.325738	−	0.945460i	$-0.605613\pi$
$461$	−13.9878	−0.651475	−0.325738	+	0.945460i	$0.605613\pi$
$462$	18.3359	0.853064
$463$	13.6029	0.632179	0.316090	−	0.948729i	$-0.397630\pi$
$463$	13.6029	0.632179	0.316090	+	0.948729i	$0.397630\pi$
$464$	1.00000	0.0464238
$465$	2.61757	0.121387
$466$	3.81246	0.176609
$467$	7.86813	0.364094	0.182047	−	0.983290i	$-0.441728\pi$
$467$	7.86813	0.364094	0.182047	+	0.983290i	$0.441728\pi$
$468$	−7.00436	−0.323777
$469$	20.2145	0.933417
$470$	15.0763	0.695417
$471$	4.97882	0.229412
$472$	5.12084	0.235706
$473$	−13.7351	−0.631539
$474$	3.38346	0.155407
$475$	8.08068	0.370767
$476$	4.80482	0.220229
$477$	8.79312	0.402609
$478$	8.22242	0.376085
$479$	−5.01456	−0.229121	−0.114561	−	0.993416i	$-0.536546\pi$
$479$	−5.01456	−0.229121	−0.114561	+	0.993416i	$0.536546\pi$
$480$	−1.27098	−0.0580120
$481$	10.4383	0.475947
$482$	−25.1659	−1.14628
$483$	−3.55617	−0.161811
$484$	15.5852	0.708419
$485$	24.3807	1.10707
$486$	−1.00000	−0.0453609
$487$	22.4652	1.01799	0.508997	−	0.860768i	$-0.330016\pi$
$487$	22.4652	1.01799	0.508997	+	0.860768i	$0.330016\pi$
$488$	−4.67059	−0.211428
$489$	−16.8603	−0.762451
$490$	−7.17639	−0.324196
$491$	−13.0664	−0.589679	−0.294840	−	0.955547i	$-0.595266\pi$
$491$	−13.0664	−0.589679	−0.294840	+	0.955547i	$0.595266\pi$
$492$	−8.70584	−0.392489
$493$	1.35112	0.0608515
$494$	−16.7227	−0.752392
$495$	−6.55328	−0.294548
$496$	2.05949	0.0924738
$497$	−29.5096	−1.32369
$498$	13.3000	0.595989
$499$	−16.9156	−0.757247	−0.378623	−	0.925551i	$-0.623602\pi$
$499$	−16.9156	−0.757247	−0.378623	+	0.925551i	$0.623602\pi$
$500$	−10.6567	−0.476581
$501$	7.64373	0.341496
$502$	−22.1256	−0.987514
$503$	−7.27602	−0.324422	−0.162211	−	0.986756i	$-0.551862\pi$
$503$	−7.27602	−0.324422	−0.162211	+	0.986756i	$0.551862\pi$
$504$	−3.55617	−0.158404
$505$	−7.47712	−0.332728
$506$	−5.15608	−0.229216
$507$	36.0611	1.60153
$508$	9.72588	0.431516
$509$	−18.7492	−0.831045	−0.415522	−	0.909583i	$-0.636401\pi$
$509$	−18.7492	−0.831045	−0.415522	+	0.909583i	$0.636401\pi$
$510$	−1.71725	−0.0760411
$511$	−50.0343	−2.21339
$512$	−1.00000	−0.0441942
$513$	−2.38748	−0.105410
$514$	−6.83211	−0.301351
$515$	2.38625	0.105151
$516$	2.66386	0.117270
$517$	61.1612	2.68987
$518$	5.29963	0.232852
$519$	−1.24075	−0.0544629
$520$	8.90239	0.390396
$521$	43.6957	1.91435	0.957173	−	0.289518i	$-0.0934950\pi$
$521$	43.6957	1.91435	0.957173	+	0.289518i	$0.0934950\pi$
$522$	−1.00000	−0.0437688
$523$	18.5994	0.813294	0.406647	−	0.913585i	$-0.366698\pi$
$523$	18.5994	0.813294	0.406647	+	0.913585i	$0.366698\pi$
$524$	5.33949	0.233257
$525$	−12.0363	−0.525305
$526$	−2.54269	−0.110866
$527$	2.78262	0.121213
$528$	−5.15608	−0.224390
$529$	1.00000	0.0434783
$530$	−11.1759	−0.485449
$531$	−5.12084	−0.222226
$532$	−8.49027	−0.368100
$533$	60.9788	2.64129
$534$	17.9618	0.777284
$535$	22.7945	0.985492
$536$	−5.68433	−0.245526
$537$	−24.8582	−1.07271
$538$	−15.2555	−0.657712
$539$	−29.1130	−1.25399
$540$	1.27098	0.0546942
$541$	10.9982	0.472847	0.236424	−	0.971650i	$-0.424025\pi$
$541$	10.9982	0.472847	0.236424	+	0.971650i	$0.424025\pi$
$542$	32.3195	1.38824
$543$	4.12360	0.176961
$544$	−1.35112	−0.0579289
$545$	−9.36274	−0.401056
$546$	24.9087	1.06599
$547$	1.37055	0.0586007	0.0293003	−	0.999571i	$-0.490672\pi$
$547$	1.37055	0.0586007	0.0293003	+	0.999571i	$0.490672\pi$
$548$	−5.09789	−0.217771
$549$	4.67059	0.199336
$550$	−17.4513	−0.744128
$551$	−2.38748	−0.101710
$552$	1.00000	0.0425628
$553$	−12.0321	−0.511659
$554$	20.7744	0.882621
$555$	−1.89409	−0.0803998
$556$	21.8698	0.927486
$557$	−6.17793	−0.261767	−0.130884	−	0.991398i	$-0.541781\pi$
$557$	−6.17793	−0.261767	−0.130884	+	0.991398i	$0.541781\pi$
$558$	−2.05949	−0.0871852
$559$	−18.6586	−0.789175
$560$	4.51982	0.190997
$561$	−6.96650	−0.294126
$562$	19.8392	0.836866
$563$	32.2283	1.35826	0.679131	−	0.734017i	$-0.262357\pi$
$563$	32.2283	1.35826	0.679131	+	0.734017i	$0.262357\pi$
$564$	−11.8619	−0.499478
$565$	−18.3215	−0.770793
$566$	3.30625	0.138972
$567$	3.55617	0.149345
$568$	8.29814	0.348182
$569$	−42.8657	−1.79703	−0.898513	−	0.438947i	$-0.855351\pi$
$569$	−42.8657	−1.79703	−0.898513	+	0.438947i	$0.855351\pi$
$570$	3.03443	0.127098
$571$	−15.2339	−0.637518	−0.318759	−	0.947836i	$-0.603266\pi$
$571$	−15.2339	−0.637518	−0.318759	+	0.947836i	$0.603266\pi$
$572$	36.1151	1.51005
$573$	0.377313	0.0157625
$574$	30.9594	1.29222
$575$	3.38461	0.141148
$576$	1.00000	0.0416667
$577$	13.2858	0.553097	0.276548	−	0.961000i	$-0.410809\pi$
$577$	13.2858	0.553097	0.276548	+	0.961000i	$0.410809\pi$
$578$	15.1745	0.631175
$579$	−10.6030	−0.440644
$580$	1.27098	0.0527745
$581$	−47.2972	−1.96222
$582$	−19.1826	−0.795144
$583$	−45.3381	−1.87771
$584$	14.0697	0.582209
$585$	−8.90239	−0.368069
$586$	3.83056	0.158239
$587$	−5.87126	−0.242333	−0.121166	−	0.992632i	$-0.538663\pi$
$587$	−5.87126	−0.242333	−0.121166	+	0.992632i	$0.538663\pi$
$588$	5.64635	0.232851
$589$	−4.91698	−0.202601
$590$	6.50848	0.267950
$591$	−5.72888	−0.235655
$592$	−1.49026	−0.0612494
$593$	−13.1941	−0.541816	−0.270908	−	0.962605i	$-0.587324\pi$
$593$	−13.1941	−0.541816	−0.270908	+	0.962605i	$0.587324\pi$
$594$	5.15608	0.211557
$595$	6.10683	0.250356
$596$	17.5495	0.718855
$597$	15.4695	0.633123
$598$	−7.00436	−0.286430
$599$	2.41958	0.0988612	0.0494306	−	0.998778i	$-0.484259\pi$
$599$	2.41958	0.0988612	0.0494306	+	0.998778i	$0.484259\pi$
$600$	3.38461	0.138176
$601$	−12.1466	−0.495471	−0.247735	−	0.968828i	$-0.579686\pi$
$601$	−12.1466	−0.495471	−0.247735	+	0.968828i	$0.579686\pi$
$602$	−9.47313	−0.386096
$603$	5.68433	0.231484
$604$	−2.81294	−0.114457
$605$	19.8085	0.805329
$606$	5.88296	0.238979
$607$	−25.5865	−1.03852	−0.519262	−	0.854615i	$-0.673793\pi$
$607$	−25.5865	−1.03852	−0.519262	+	0.854615i	$0.673793\pi$
$608$	2.38748	0.0968249
$609$	3.55617	0.144103
$610$	−5.93623	−0.240351
$611$	83.0853	3.36127
$612$	1.35112	0.0546159
$613$	3.46769	0.140059	0.0700293	−	0.997545i	$-0.477691\pi$
$613$	3.46769	0.140059	0.0700293	+	0.997545i	$0.477691\pi$
$614$	−31.6974	−1.27920
$615$	−11.0649	−0.446181
$616$	18.3359	0.738775
$617$	37.6501	1.51574	0.757868	−	0.652408i	$-0.226241\pi$
$617$	37.6501	1.51574	0.757868	+	0.652408i	$0.226241\pi$
$618$	−1.87749	−0.0755237
$619$	−29.1216	−1.17050	−0.585248	−	0.810855i	$-0.699003\pi$
$619$	−29.1216	−1.17050	−0.585248	+	0.810855i	$0.699003\pi$
$620$	2.61757	0.105124
$621$	−1.00000	−0.0401286
$622$	−29.0819	−1.16608
$623$	−63.8753	−2.55911
$624$	−7.00436	−0.280399
$625$	3.37866	0.135147
$626$	−13.7348	−0.548953
$627$	12.3100	0.491615
$628$	4.97882	0.198677
$629$	−2.01353	−0.0802846
$630$	−4.51982	−0.180074
$631$	7.86312	0.313026	0.156513	−	0.987676i	$-0.449975\pi$
$631$	7.86312	0.313026	0.156513	+	0.987676i	$0.449975\pi$
$632$	3.38346	0.134587
$633$	−17.6724	−0.702415
$634$	−31.0575	−1.23345
$635$	12.3614	0.490547
$636$	8.79312	0.348670
$637$	−39.5490	−1.56699
$638$	5.15608	0.204131
$639$	−8.29814	−0.328269
$640$	−1.27098	−0.0502399
$641$	20.0526	0.792030	0.396015	−	0.918244i	$-0.370393\pi$
$641$	20.0526	0.792030	0.396015	+	0.918244i	$0.370393\pi$
$642$	−17.9346	−0.707822
$643$	9.25033	0.364797	0.182399	−	0.983225i	$-0.441614\pi$
$643$	9.25033	0.364797	0.182399	+	0.983225i	$0.441614\pi$
$644$	−3.55617	−0.140133
$645$	3.38571	0.133312
$646$	3.22577	0.126916
$647$	12.2463	0.481450	0.240725	−	0.970593i	$-0.422615\pi$
$647$	12.2463	0.481450	0.240725	+	0.970593i	$0.422615\pi$
$648$	−1.00000	−0.0392837
$649$	26.4035	1.03643
$650$	−23.7070	−0.929867
$651$	7.32390	0.287046
$652$	−16.8603	−0.660302
$653$	−48.2700	−1.88895	−0.944476	−	0.328579i	$-0.893430\pi$
$653$	−48.2700	−1.88895	−0.944476	+	0.328579i	$0.893430\pi$
$654$	7.36656	0.288055
$655$	6.78638	0.265166
$656$	−8.70584	−0.339906
$657$	−14.0697	−0.548912
$658$	42.1831	1.64447
$659$	−30.9027	−1.20380	−0.601900	−	0.798572i	$-0.705589\pi$
$659$	−30.9027	−1.20380	−0.601900	+	0.798572i	$0.705589\pi$
$660$	−6.55328	−0.255086
$661$	32.8731	1.27862	0.639308	−	0.768951i	$-0.279221\pi$
$661$	32.8731	1.27862	0.639308	+	0.768951i	$0.279221\pi$
$662$	5.74186	0.223164
$663$	−9.46375	−0.367542
$664$	13.3000	0.516141
$665$	−10.7910	−0.418455
$666$	1.49026	0.0577465
$667$	−1.00000	−0.0387202
$668$	7.64373	0.295745
$669$	−15.8328	−0.612130
$670$	−7.22467	−0.279113
$671$	−24.0820	−0.929674
$672$	−3.55617	−0.137182
$673$	39.9735	1.54086	0.770432	−	0.637522i	$-0.220041\pi$
$673$	39.9735	1.54086	0.770432	+	0.637522i	$0.220041\pi$
$674$	14.4609	0.557013
$675$	−3.38461	−0.130274
$676$	36.0611	1.38696
$677$	−2.00192	−0.0769400	−0.0384700	−	0.999260i	$-0.512248\pi$
$677$	−2.00192	−0.0769400	−0.0384700	+	0.999260i	$0.512248\pi$
$678$	14.4153	0.553616
$679$	68.2166	2.61791
$680$	−1.71725	−0.0658535
$681$	−5.22577	−0.200252
$682$	10.6189	0.406619
$683$	−11.2142	−0.429098	−0.214549	−	0.976713i	$-0.568828\pi$
$683$	−11.2142	−0.429098	−0.214549	+	0.976713i	$0.568828\pi$
$684$	−2.38748	−0.0912874
$685$	−6.47931	−0.247562
$686$	4.81382	0.183792
$687$	17.9272	0.683967
$688$	2.66386	0.101559
$689$	−61.5902	−2.34640
$690$	1.27098	0.0483853
$691$	−23.3403	−0.887908	−0.443954	−	0.896050i	$-0.646425\pi$
$691$	−23.3403	−0.887908	−0.443954	+	0.896050i	$0.646425\pi$
$692$	−1.24075	−0.0471662
$693$	−18.3359	−0.696524
$694$	−21.1657	−0.803437
$695$	27.7961	1.05436
$696$	−1.00000	−0.0379049
$697$	−11.7627	−0.445542
$698$	28.2520	1.06935
$699$	−3.81246	−0.144201
$700$	−12.0363	−0.454928
$701$	−26.4159	−0.997715	−0.498858	−	0.866684i	$-0.666247\pi$
$701$	−26.4159	−0.997715	−0.498858	+	0.866684i	$0.666247\pi$
$702$	7.00436	0.264363
$703$	3.55797	0.134191
$704$	−5.15608	−0.194327
$705$	−15.0763	−0.567806
$706$	21.8756	0.823299
$707$	−20.9208	−0.786808
$708$	−5.12084	−0.192453
$709$	10.3261	0.387806	0.193903	−	0.981021i	$-0.437885\pi$
$709$	10.3261	0.387806	0.193903	+	0.981021i	$0.437885\pi$
$710$	10.5468	0.395813
$711$	−3.38346	−0.126890
$712$	17.9618	0.673147
$713$	−2.05949	−0.0771285
$714$	−4.80482	−0.179816
$715$	45.9015	1.71662
$716$	−24.8582	−0.928995
$717$	−8.22242	−0.307072
$718$	34.4973	1.28743
$719$	−6.99482	−0.260863	−0.130431	−	0.991457i	$-0.541636\pi$
$719$	−6.99482	−0.260863	−0.130431	+	0.991457i	$0.541636\pi$
$720$	1.27098	0.0473666
$721$	6.67667	0.248652
$722$	13.3000	0.494973
$723$	25.1659	0.935930
$724$	4.12360	0.153252
$725$	−3.38461	−0.125701
$726$	−15.5852	−0.578421
$727$	−16.9111	−0.627197	−0.313599	−	0.949556i	$-0.601535\pi$
$727$	−16.9111	−0.627197	−0.313599	+	0.949556i	$0.601535\pi$
$728$	24.9087	0.923178
$729$	1.00000	0.0370370
$730$	17.8823	0.661855
$731$	3.59920	0.133121
$732$	4.67059	0.172630
$733$	−19.5462	−0.721956	−0.360978	−	0.932574i	$-0.617557\pi$
$733$	−19.5462	−0.721956	−0.360978	+	0.932574i	$0.617557\pi$
$734$	−12.0238	−0.443805
$735$	7.17639	0.264705
$736$	1.00000	0.0368605
$737$	−29.3089	−1.07961
$738$	8.70584	0.320466
$739$	11.6721	0.429363	0.214682	−	0.976684i	$-0.431129\pi$
$739$	11.6721	0.429363	0.214682	+	0.976684i	$0.431129\pi$
$740$	−1.89409	−0.0696282
$741$	16.7227	0.614325
$742$	−31.2698	−1.14795
$743$	−24.2208	−0.888574	−0.444287	−	0.895885i	$-0.646543\pi$
$743$	−24.2208	−0.888574	−0.444287	+	0.895885i	$0.646543\pi$
$744$	−2.05949	−0.0755046
$745$	22.3050	0.817193
$746$	−34.5695	−1.26568
$747$	−13.3000	−0.486623
$748$	−6.96650	−0.254721
$749$	63.7785	2.33042
$750$	10.6567	0.389126
$751$	−26.9549	−0.983599	−0.491799	−	0.870709i	$-0.663661\pi$
$751$	−26.9549	−0.983599	−0.491799	+	0.870709i	$0.663661\pi$
$752$	−11.8619	−0.432561
$753$	22.1256	0.806301
$754$	7.00436	0.255084
$755$	−3.57518	−0.130114
$756$	3.55617	0.129337
$757$	−4.97252	−0.180729	−0.0903646	−	0.995909i	$-0.528803\pi$
$757$	−4.97252	−0.180729	−0.0903646	+	0.995909i	$0.528803\pi$
$758$	5.39787	0.196060
$759$	5.15608	0.187154
$760$	3.03443	0.110070
$761$	−27.1879	−0.985562	−0.492781	−	0.870154i	$-0.664020\pi$
$761$	−27.1879	−0.985562	−0.492781	+	0.870154i	$0.664020\pi$
$762$	−9.72588	−0.352331
$763$	−26.1967	−0.948386
$764$	0.377313	0.0136507
$765$	1.71725	0.0620873
$766$	4.79803	0.173360
$767$	35.8682	1.29513
$768$	1.00000	0.0360844
$769$	15.4403	0.556792	0.278396	−	0.960466i	$-0.410197\pi$
$769$	15.4403	0.556792	0.278396	+	0.960466i	$0.410197\pi$
$770$	23.3046	0.839838
$771$	6.83211	0.246052
$772$	−10.6030	−0.381609
$773$	24.9178	0.896231	0.448116	−	0.893976i	$-0.352095\pi$
$773$	24.9178	0.896231	0.448116	+	0.893976i	$0.352095\pi$
$774$	−2.66386	−0.0957504
$775$	−6.97058	−0.250390
$776$	−19.1826	−0.688615
$777$	−5.29963	−0.190123
$778$	−18.8338	−0.675223
$779$	20.7850	0.744699
$780$	−8.90239	−0.318757
$781$	42.7859	1.53100
$782$	1.35112	0.0483161
$783$	1.00000	0.0357371
$784$	5.64635	0.201655
$785$	6.32798	0.225855
$786$	−5.33949	−0.190453
$787$	−13.4854	−0.480701	−0.240351	−	0.970686i	$-0.577262\pi$
$787$	−13.4854	−0.480701	−0.240351	+	0.970686i	$0.577262\pi$
$788$	−5.72888	−0.204083
$789$	2.54269	0.0905220
$790$	4.30030	0.152998
$791$	−51.2632	−1.82271
$792$	5.15608	0.183213
$793$	−32.7145	−1.16173
$794$	−1.30415	−0.0462825
$795$	11.1759	0.396367
$796$	15.4695	0.548301
$797$	40.8543	1.44713	0.723566	−	0.690255i	$-0.242502\pi$
$797$	40.8543	1.44713	0.723566	+	0.690255i	$0.242502\pi$
$798$	8.49027	0.300552
$799$	−16.0269	−0.566993
$800$	3.38461	0.119664
$801$	−17.9618	−0.634650
$802$	14.8877	0.525703
$803$	72.5447	2.56005
$804$	5.68433	0.200471
$805$	−4.51982	−0.159303
$806$	14.4254	0.508113
$807$	15.2555	0.537020
$808$	5.88296	0.206962
$809$	−25.7028	−0.903661	−0.451831	−	0.892104i	$-0.649229\pi$
$809$	−25.7028	−0.903661	−0.451831	+	0.892104i	$0.649229\pi$
$810$	−1.27098	−0.0446576
$811$	2.99380	0.105126	0.0525632	−	0.998618i	$-0.483261\pi$
$811$	2.99380	0.105126	0.0525632	+	0.998618i	$0.483261\pi$
$812$	3.55617	0.124797
$813$	−32.3195	−1.13349
$814$	−7.68392	−0.269321
$815$	−21.4291	−0.750630
$816$	1.35112	0.0472988
$817$	−6.35989	−0.222505
$818$	0.863003	0.0301742
$819$	−24.9087	−0.870380
$820$	−11.0649	−0.386404
$821$	32.7287	1.14224	0.571119	−	0.820867i	$-0.306509\pi$
$821$	32.7287	1.14224	0.571119	+	0.820867i	$0.306509\pi$
$822$	5.09789	0.177809
$823$	22.1924	0.773577	0.386788	−	0.922169i	$-0.373584\pi$
$823$	22.1924	0.773577	0.386788	+	0.922169i	$0.373584\pi$
$824$	−1.87749	−0.0654054
$825$	17.4513	0.607578
$826$	18.2106	0.633627
$827$	−47.5339	−1.65291	−0.826457	−	0.563000i	$-0.809647\pi$
$827$	−47.5339	−1.65291	−0.826457	+	0.563000i	$0.809647\pi$
$828$	−1.00000	−0.0347524
$829$	50.0323	1.73769	0.868847	−	0.495081i	$-0.164862\pi$
$829$	50.0323	1.73769	0.868847	+	0.495081i	$0.164862\pi$
$830$	16.9041	0.586748
$831$	−20.7744	−0.720657
$832$	−7.00436	−0.242832
$833$	7.62891	0.264326
$834$	−21.8698	−0.757289
$835$	9.71501	0.336202
$836$	12.3100	0.425751
$837$	2.05949	0.0711864
$838$	23.0733	0.797054
$839$	−6.22684	−0.214974	−0.107487	−	0.994206i	$-0.534280\pi$
$839$	−6.22684	−0.214974	−0.107487	+	0.994206i	$0.534280\pi$
$840$	−4.51982	−0.155949
$841$	1.00000	0.0344828
$842$	−24.0550	−0.828989
$843$	−19.8392	−0.683298
$844$	−17.6724	−0.608309
$845$	45.8328	1.57670
$846$	11.8619	0.407822
$847$	55.4237	1.90438
$848$	8.79312	0.301957
$849$	−3.30625	−0.113470
$850$	4.57303	0.156854
$851$	1.49026	0.0510856
$852$	−8.29814	−0.284289
$853$	−26.5408	−0.908739	−0.454370	−	0.890813i	$-0.650135\pi$
$853$	−26.5408	−0.908739	−0.454370	+	0.890813i	$0.650135\pi$
$854$	−16.6094	−0.568363
$855$	−3.03443	−0.103775
$856$	−17.9346	−0.612992
$857$	−21.7781	−0.743927	−0.371964	−	0.928247i	$-0.621315\pi$
$857$	−21.7781	−0.743927	−0.371964	+	0.928247i	$0.621315\pi$
$858$	−36.1151	−1.23295
$859$	17.6311	0.601567	0.300783	−	0.953692i	$-0.402752\pi$
$859$	17.6311	0.601567	0.300783	+	0.953692i	$0.402752\pi$
$860$	3.38571	0.115452
$861$	−30.9594	−1.05509
$862$	13.6485	0.464868
$863$	47.8029	1.62723	0.813615	−	0.581404i	$-0.197496\pi$
$863$	47.8029	1.62723	0.813615	+	0.581404i	$0.197496\pi$
$864$	−1.00000	−0.0340207
$865$	−1.57697	−0.0536185
$866$	40.1283	1.36362
$867$	−15.1745	−0.515352
$868$	7.32390	0.248589
$869$	17.4454	0.591794
$870$	−1.27098	−0.0430902
$871$	−39.8151	−1.34908
$872$	7.36656	0.249463
$873$	19.1826	0.649232
$874$	−2.38748	−0.0807576
$875$	−37.8969	−1.28115
$876$	−14.0697	−0.475372
$877$	27.4242	0.926049	0.463024	−	0.886346i	$-0.346764\pi$
$877$	27.4242	0.926049	0.463024	+	0.886346i	$0.346764\pi$
$878$	−3.93933	−0.132946
$879$	−3.83056	−0.129202
$880$	−6.55328	−0.220911
$881$	−38.1880	−1.28658	−0.643292	−	0.765621i	$-0.722432\pi$
$881$	−38.1880	−1.28658	−0.643292	+	0.765621i	$0.722432\pi$
$882$	−5.64635	−0.190122
$883$	−43.2440	−1.45528	−0.727638	−	0.685961i	$-0.759382\pi$
$883$	−43.2440	−1.45528	−0.727638	+	0.685961i	$0.759382\pi$
$884$	−9.46375	−0.318300
$885$	−6.50848	−0.218780
$886$	6.90018	0.231816
$887$	−33.9200	−1.13892	−0.569461	−	0.822018i	$-0.692848\pi$
$887$	−33.9200	−1.13892	−0.569461	+	0.822018i	$0.692848\pi$
$888$	1.49026	0.0500099
$889$	34.5869	1.16001
$890$	22.8291	0.765233
$891$	−5.15608	−0.172735
$892$	−15.8328	−0.530120
$893$	28.3201	0.947696
$894$	−17.5495	−0.586943
$895$	−31.5942	−1.05608
$896$	−3.55617	−0.118803
$897$	7.00436	0.233869
$898$	−27.3893	−0.913994
$899$	2.05949	0.0686878
$900$	−3.38461	−0.112820
$901$	11.8806	0.395800
$902$	−44.8880	−1.49461
$903$	9.47313	0.315246
$904$	14.4153	0.479445
$905$	5.24101	0.174217
$906$	2.81294	0.0934536
$907$	−15.8784	−0.527234	−0.263617	−	0.964627i	$-0.584916\pi$
$907$	−15.8784	−0.527234	−0.263617	+	0.964627i	$0.584916\pi$
$908$	−5.22577	−0.173423
$909$	−5.88296	−0.195126
$910$	31.6584	1.04947
$911$	−15.7669	−0.522379	−0.261190	−	0.965287i	$-0.584115\pi$
$911$	−15.7669	−0.522379	−0.261190	+	0.965287i	$0.584115\pi$
$912$	−2.38748	−0.0790572
$913$	68.5761	2.26954
$914$	5.21128	0.172374
$915$	5.93623	0.196246
$916$	17.9272	0.592333
$917$	18.9881	0.627043
$918$	−1.35112	−0.0445937
$919$	19.4736	0.642376	0.321188	−	0.947015i	$-0.395918\pi$
$919$	19.4736	0.642376	0.321188	+	0.947015i	$0.395918\pi$
$920$	1.27098	0.0419029
$921$	31.6974	1.04447
$922$	13.9878	0.460663
$923$	58.1231	1.91315
$924$	−18.3359	−0.603207
$925$	5.04396	0.165844
$926$	−13.6029	−0.447018
$927$	1.87749	0.0616648
$928$	−1.00000	−0.0328266
$929$	26.7217	0.876710	0.438355	−	0.898802i	$-0.355561\pi$
$929$	26.7217	0.876710	0.438355	+	0.898802i	$0.355561\pi$
$930$	−2.61757	−0.0858335
$931$	−13.4805	−0.441806
$932$	−3.81246	−0.124881
$933$	29.0819	0.952098
$934$	−7.86813	−0.257453
$935$	−8.85428	−0.289566
$936$	7.00436	0.228945
$937$	52.7876	1.72450	0.862249	−	0.506485i	$-0.169056\pi$
$937$	52.7876	1.72450	0.862249	+	0.506485i	$0.169056\pi$
$938$	−20.2145	−0.660025
$939$	13.7348	0.448218
$940$	−15.0763	−0.491734
$941$	54.6553	1.78171	0.890855	−	0.454287i	$-0.150106\pi$
$941$	54.6553	1.78171	0.890855	+	0.454287i	$0.150106\pi$
$942$	−4.97882	−0.162219
$943$	8.70584	0.283501
$944$	−5.12084	−0.166669
$945$	4.51982	0.147030
$946$	13.7351	0.446566
$947$	25.7795	0.837721	0.418860	−	0.908051i	$-0.362430\pi$
$947$	25.7795	0.837721	0.418860	+	0.908051i	$0.362430\pi$
$948$	−3.38346	−0.109890
$949$	98.5494	3.19905
$950$	−8.08068	−0.262172
$951$	31.0575	1.00711
$952$	−4.80482	−0.155725
$953$	40.9400	1.32618	0.663089	−	0.748540i	$-0.269245\pi$
$953$	40.9400	1.32618	0.663089	+	0.748540i	$0.269245\pi$
$954$	−8.79312	−0.284688
$955$	0.479556	0.0155181
$956$	−8.22242	−0.265932
$957$	−5.15608	−0.166673
$958$	5.01456	0.162013
$959$	−18.1290	−0.585415
$960$	1.27098	0.0410207
$961$	−26.7585	−0.863177
$962$	−10.4383	−0.336545
$963$	17.9346	0.577935
$964$	25.1659	0.810539
$965$	−13.4761	−0.433812
$966$	3.55617	0.114418
$967$	−44.5876	−1.43384	−0.716920	−	0.697156i	$-0.754449\pi$
$967$	−44.5876	−1.43384	−0.716920	+	0.697156i	$0.754449\pi$
$968$	−15.5852	−0.500928
$969$	−3.22577	−0.103627
$970$	−24.3807	−0.782816
$971$	−7.38344	−0.236946	−0.118473	−	0.992957i	$-0.537800\pi$
$971$	−7.38344	−0.236946	−0.118473	+	0.992957i	$0.537800\pi$
$972$	1.00000	0.0320750
$973$	77.7727	2.49328
$974$	−22.4652	−0.719830
$975$	23.7070	0.759233
$976$	4.67059	0.149502
$977$	4.51062	0.144308	0.0721538	−	0.997394i	$-0.477013\pi$
$977$	4.51062	0.144308	0.0721538	+	0.997394i	$0.477013\pi$
$978$	16.8603	0.539134
$979$	92.6126	2.95991
$980$	7.17639	0.229241
$981$	−7.36656	−0.235196
$982$	13.0664	0.416966
$983$	18.7591	0.598322	0.299161	−	0.954203i	$-0.403293\pi$
$983$	18.7591	0.598322	0.299161	+	0.954203i	$0.403293\pi$
$984$	8.70584	0.277532
$985$	−7.28129	−0.232001
$986$	−1.35112	−0.0430285
$987$	−42.1831	−1.34270
$988$	16.7227	0.532021
$989$	−2.66386	−0.0847057
$990$	6.55328	0.208277
$991$	3.14486	0.0998998	0.0499499	−	0.998752i	$-0.484094\pi$
$991$	3.14486	0.0998998	0.0499499	+	0.998752i	$0.484094\pi$
$992$	−2.05949	−0.0653889
$993$	−5.74186	−0.182212
$994$	29.5096	0.935987
$995$	19.6614	0.623307
$996$	−13.3000	−0.421428
$997$	−40.2660	−1.27524	−0.637619	−	0.770352i	$-0.720081\pi$
$997$	−40.2660	−1.27524	−0.637619	+	0.770352i	$0.720081\pi$
$998$	16.9156	0.535454
$999$	−1.49026	−0.0471498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bi.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bi.1.6	✓	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:	\( N \)	\(=\)	\( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.27098 q^{5} -1.00000 q^{6} +3.55617 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.27098 q^{5} -1.00000 q^{6} +3.55617 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.27098 q^{10} -5.15608 q^{11} +1.00000 q^{12} -7.00436 q^{13} -3.55617 q^{14} +1.27098 q^{15} +1.00000 q^{16} +1.35112 q^{17} -1.00000 q^{18} -2.38748 q^{19} +1.27098 q^{20} +3.55617 q^{21} +5.15608 q^{22} -1.00000 q^{23} -1.00000 q^{24} -3.38461 q^{25} +7.00436 q^{26} +1.00000 q^{27} +3.55617 q^{28} +1.00000 q^{29} -1.27098 q^{30} +2.05949 q^{31} -1.00000 q^{32} -5.15608 q^{33} -1.35112 q^{34} +4.51982 q^{35} +1.00000 q^{36} -1.49026 q^{37} +2.38748 q^{38} -7.00436 q^{39} -1.27098 q^{40} -8.70584 q^{41} -3.55617 q^{42} +2.66386 q^{43} -5.15608 q^{44} +1.27098 q^{45} +1.00000 q^{46} -11.8619 q^{47} +1.00000 q^{48} +5.64635 q^{49} +3.38461 q^{50} +1.35112 q^{51} -7.00436 q^{52} +8.79312 q^{53} -1.00000 q^{54} -6.55328 q^{55} -3.55617 q^{56} -2.38748 q^{57} -1.00000 q^{58} -5.12084 q^{59} +1.27098 q^{60} +4.67059 q^{61} -2.05949 q^{62} +3.55617 q^{63} +1.00000 q^{64} -8.90239 q^{65} +5.15608 q^{66} +5.68433 q^{67} +1.35112 q^{68} -1.00000 q^{69} -4.51982 q^{70} -8.29814 q^{71} -1.00000 q^{72} -14.0697 q^{73} +1.49026 q^{74} -3.38461 q^{75} -2.38748 q^{76} -18.3359 q^{77} +7.00436 q^{78} -3.38346 q^{79} +1.27098 q^{80} +1.00000 q^{81} +8.70584 q^{82} -13.3000 q^{83} +3.55617 q^{84} +1.71725 q^{85} -2.66386 q^{86} +1.00000 q^{87} +5.15608 q^{88} -17.9618 q^{89} -1.27098 q^{90} -24.9087 q^{91} -1.00000 q^{92} +2.05949 q^{93} +11.8619 q^{94} -3.03443 q^{95} -1.00000 q^{96} +19.1826 q^{97} -5.64635 q^{98} -5.15608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 3 q^{5} - 8 q^{6} - 6 q^{7} - 8 q^{8} + 8 q^{9} + 3 q^{10} - 7 q^{11} + 8 q^{12} - 3 q^{13} + 6 q^{14} - 3 q^{15} + 8 q^{16} - 12 q^{17} - 8 q^{18} - 4 q^{19} - 3 q^{20} - 6 q^{21} + 7 q^{22} - 8 q^{23} - 8 q^{24} + 13 q^{25} + 3 q^{26} + 8 q^{27} - 6 q^{28} + 8 q^{29} + 3 q^{30} - q^{31} - 8 q^{32} - 7 q^{33} + 12 q^{34} - 6 q^{35} + 8 q^{36} - 11 q^{37} + 4 q^{38} - 3 q^{39} + 3 q^{40} - 17 q^{41} + 6 q^{42} - 10 q^{43} - 7 q^{44} - 3 q^{45} + 8 q^{46} - 26 q^{47} + 8 q^{48} + 10 q^{49} - 13 q^{50} - 12 q^{51} - 3 q^{52} - 2 q^{53} - 8 q^{54} + q^{55} + 6 q^{56} - 4 q^{57} - 8 q^{58} - 25 q^{59} - 3 q^{60} + 3 q^{61} + q^{62} - 6 q^{63} + 8 q^{64} - 25 q^{65} + 7 q^{66} + q^{67} - 12 q^{68} - 8 q^{69} + 6 q^{70} - 27 q^{71} - 8 q^{72} + 16 q^{73} + 11 q^{74} + 13 q^{75} - 4 q^{76} - 16 q^{77} + 3 q^{78} - 6 q^{79} - 3 q^{80} + 8 q^{81} + 17 q^{82} - 44 q^{83} - 6 q^{84} - 20 q^{85} + 10 q^{86} + 8 q^{87} + 7 q^{88} - 52 q^{89} + 3 q^{90} - 18 q^{91} - 8 q^{92} - q^{93} + 26 q^{94} - 56 q^{95} - 8 q^{96} - 4 q^{97} - 10 q^{98} - 7 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 3 * q^5 - 8 * q^6 - 6 * q^7 - 8 * q^8 + 8 * q^9 + 3 * q^10 - 7 * q^11 + 8 * q^12 - 3 * q^13 + 6 * q^14 - 3 * q^15 + 8 * q^16 - 12 * q^17 - 8 * q^18 - 4 * q^19 - 3 * q^20 - 6 * q^21 + 7 * q^22 - 8 * q^23 - 8 * q^24 + 13 * q^25 + 3 * q^26 + 8 * q^27 - 6 * q^28 + 8 * q^29 + 3 * q^30 - q^31 - 8 * q^32 - 7 * q^33 + 12 * q^34 - 6 * q^35 + 8 * q^36 - 11 * q^37 + 4 * q^38 - 3 * q^39 + 3 * q^40 - 17 * q^41 + 6 * q^42 - 10 * q^43 - 7 * q^44 - 3 * q^45 + 8 * q^46 - 26 * q^47 + 8 * q^48 + 10 * q^49 - 13 * q^50 - 12 * q^51 - 3 * q^52 - 2 * q^53 - 8 * q^54 + q^55 + 6 * q^56 - 4 * q^57 - 8 * q^58 - 25 * q^59 - 3 * q^60 + 3 * q^61 + q^62 - 6 * q^63 + 8 * q^64 - 25 * q^65 + 7 * q^66 + q^67 - 12 * q^68 - 8 * q^69 + 6 * q^70 - 27 * q^71 - 8 * q^72 + 16 * q^73 + 11 * q^74 + 13 * q^75 - 4 * q^76 - 16 * q^77 + 3 * q^78 - 6 * q^79 - 3 * q^80 + 8 * q^81 + 17 * q^82 - 44 * q^83 - 6 * q^84 - 20 * q^85 + 10 * q^86 + 8 * q^87 + 7 * q^88 - 52 * q^89 + 3 * q^90 - 18 * q^91 - 8 * q^92 - q^93 + 26 * q^94 - 56 * q^95 - 8 * q^96 - 4 * q^97 - 10 * q^98 - 7 * q^99

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