LMFDB - Embedded newform 4002.2.a.bk.1.7

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 30x^{6} + 52x^{5} + 267x^{4} - 352x^{3} - 632x^{2} + 240x + 288 $ x^8 - 2x^7 - 30x^6 + 52x^5 + 267x^4 - 352x^3 - 632x^2 + 240*x + 288
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$3.65713$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.65713 q^{5} -1.00000 q^{6} -4.08293 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.65713 q^{5} -1.00000 q^{6} -4.08293 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.65713 q^{10} -2.11001 q^{11} -1.00000 q^{12} +2.59337 q^{13} -4.08293 q^{14} -3.65713 q^{15} +1.00000 q^{16} +1.54712 q^{17} +1.00000 q^{18} +4.60452 q^{19} +3.65713 q^{20} +4.08293 q^{21} -2.11001 q^{22} -1.00000 q^{23} -1.00000 q^{24} +8.37463 q^{25} +2.59337 q^{26} -1.00000 q^{27} -4.08293 q^{28} -1.00000 q^{29} -3.65713 q^{30} +0.201477 q^{31} +1.00000 q^{32} +2.11001 q^{33} +1.54712 q^{34} -14.9318 q^{35} +1.00000 q^{36} +4.67124 q^{37} +4.60452 q^{38} -2.59337 q^{39} +3.65713 q^{40} +1.23729 q^{41} +4.08293 q^{42} -3.50772 q^{43} -2.11001 q^{44} +3.65713 q^{45} -1.00000 q^{46} -2.84477 q^{47} -1.00000 q^{48} +9.67035 q^{49} +8.37463 q^{50} -1.54712 q^{51} +2.59337 q^{52} +7.19876 q^{53} -1.00000 q^{54} -7.71660 q^{55} -4.08293 q^{56} -4.60452 q^{57} -1.00000 q^{58} +9.32047 q^{59} -3.65713 q^{60} +4.11001 q^{61} +0.201477 q^{62} -4.08293 q^{63} +1.00000 q^{64} +9.48430 q^{65} +2.11001 q^{66} +1.70283 q^{67} +1.54712 q^{68} +1.00000 q^{69} -14.9318 q^{70} +4.12267 q^{71} +1.00000 q^{72} +4.28354 q^{73} +4.67124 q^{74} -8.37463 q^{75} +4.60452 q^{76} +8.61504 q^{77} -2.59337 q^{78} -5.51512 q^{79} +3.65713 q^{80} +1.00000 q^{81} +1.23729 q^{82} -0.151639 q^{83} +4.08293 q^{84} +5.65803 q^{85} -3.50772 q^{86} +1.00000 q^{87} -2.11001 q^{88} +8.36672 q^{89} +3.65713 q^{90} -10.5886 q^{91} -1.00000 q^{92} -0.201477 q^{93} -2.84477 q^{94} +16.8393 q^{95} -1.00000 q^{96} -1.09022 q^{97} +9.67035 q^{98} -2.11001 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} - 8 q^{12} + 13 q^{13} + 5 q^{14} - 2 q^{15} + 8 q^{16} + 5 q^{17} + 8 q^{18} + 7 q^{19} + 2 q^{20} - 5 q^{21} + 3 q^{22} - 8 q^{23} - 8 q^{24} + 24 q^{25} + 13 q^{26} - 8 q^{27} + 5 q^{28} - 8 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 3 q^{33} + 5 q^{34} + 9 q^{35} + 8 q^{36} + 22 q^{37} + 7 q^{38} - 13 q^{39} + 2 q^{40} - 8 q^{41} - 5 q^{42} - q^{43} + 3 q^{44} + 2 q^{45} - 8 q^{46} - 9 q^{47} - 8 q^{48} + 33 q^{49} + 24 q^{50} - 5 q^{51} + 13 q^{52} + 14 q^{53} - 8 q^{54} - 17 q^{55} + 5 q^{56} - 7 q^{57} - 8 q^{58} - 4 q^{59} - 2 q^{60} + 13 q^{61} + 15 q^{62} + 5 q^{63} + 8 q^{64} + 21 q^{65} - 3 q^{66} - 3 q^{67} + 5 q^{68} + 8 q^{69} + 9 q^{70} + 7 q^{71} + 8 q^{72} + 16 q^{73} + 22 q^{74} - 24 q^{75} + 7 q^{76} - 13 q^{78} + 14 q^{79} + 2 q^{80} + 8 q^{81} - 8 q^{82} + 36 q^{83} - 5 q^{84} + 47 q^{85} - q^{86} + 8 q^{87} + 3 q^{88} - 12 q^{89} + 2 q^{90} + 20 q^{91} - 8 q^{92} - 15 q^{93} - 9 q^{94} + 7 q^{95} - 8 q^{96} + 10 q^{97} + 33 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 - 8 * q^12 + 13 * q^13 + 5 * q^14 - 2 * q^15 + 8 * q^16 + 5 * q^17 + 8 * q^18 + 7 * q^19 + 2 * q^20 - 5 * q^21 + 3 * q^22 - 8 * q^23 - 8 * q^24 + 24 * q^25 + 13 * q^26 - 8 * q^27 + 5 * q^28 - 8 * q^29 - 2 * q^30 + 15 * q^31 + 8 * q^32 - 3 * q^33 + 5 * q^34 + 9 * q^35 + 8 * q^36 + 22 * q^37 + 7 * q^38 - 13 * q^39 + 2 * q^40 - 8 * q^41 - 5 * q^42 - q^43 + 3 * q^44 + 2 * q^45 - 8 * q^46 - 9 * q^47 - 8 * q^48 + 33 * q^49 + 24 * q^50 - 5 * q^51 + 13 * q^52 + 14 * q^53 - 8 * q^54 - 17 * q^55 + 5 * q^56 - 7 * q^57 - 8 * q^58 - 4 * q^59 - 2 * q^60 + 13 * q^61 + 15 * q^62 + 5 * q^63 + 8 * q^64 + 21 * q^65 - 3 * q^66 - 3 * q^67 + 5 * q^68 + 8 * q^69 + 9 * q^70 + 7 * q^71 + 8 * q^72 + 16 * q^73 + 22 * q^74 - 24 * q^75 + 7 * q^76 - 13 * q^78 + 14 * q^79 + 2 * q^80 + 8 * q^81 - 8 * q^82 + 36 * q^83 - 5 * q^84 + 47 * q^85 - q^86 + 8 * q^87 + 3 * q^88 - 12 * q^89 + 2 * q^90 + 20 * q^91 - 8 * q^92 - 15 * q^93 - 9 * q^94 + 7 * q^95 - 8 * q^96 + 10 * q^97 + 33 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	3.65713	1.63552	0.817760	−	0.575559i	$-0.195216\pi$
$5$	3.65713	1.63552	0.817760	+	0.575559i	$0.195216\pi$
$6$	−1.00000	−0.408248
$7$	−4.08293	−1.54320	−0.771602	−	0.636106i	$-0.780544\pi$
$7$	−4.08293	−1.54320	−0.771602	+	0.636106i	$0.780544\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.65713	1.15649
$11$	−2.11001	−0.636193	−0.318096	−	0.948058i	$-0.603044\pi$
$11$	−2.11001	−0.636193	−0.318096	+	0.948058i	$0.603044\pi$
$12$	−1.00000	−0.288675
$13$	2.59337	0.719272	0.359636	−	0.933093i	$-0.382901\pi$
$13$	2.59337	0.719272	0.359636	+	0.933093i	$0.382901\pi$
$14$	−4.08293	−1.09121
$15$	−3.65713	−0.944268
$16$	1.00000	0.250000
$17$	1.54712	0.375232	0.187616	−	0.982242i	$-0.439924\pi$
$17$	1.54712	0.375232	0.187616	+	0.982242i	$0.439924\pi$
$18$	1.00000	0.235702
$19$	4.60452	1.05635	0.528174	−	0.849136i	$-0.322877\pi$
$19$	4.60452	1.05635	0.528174	+	0.849136i	$0.322877\pi$
$20$	3.65713	0.817760
$21$	4.08293	0.890969
$22$	−2.11001	−0.449856
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	8.37463	1.67493
$26$	2.59337	0.508602
$27$	−1.00000	−0.192450
$28$	−4.08293	−0.771602
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−3.65713	−0.667698
$31$	0.201477	0.0361863	0.0180932	−	0.999836i	$-0.494240\pi$
$31$	0.201477	0.0361863	0.0180932	+	0.999836i	$0.494240\pi$
$32$	1.00000	0.176777
$33$	2.11001	0.367306
$34$	1.54712	0.265329
$35$	−14.9318	−2.52394
$36$	1.00000	0.166667
$37$	4.67124	0.767948	0.383974	−	0.923344i	$-0.374555\pi$
$37$	4.67124	0.767948	0.383974	+	0.923344i	$0.374555\pi$
$38$	4.60452	0.746951
$39$	−2.59337	−0.415272
$40$	3.65713	0.578244
$41$	1.23729	0.193232	0.0966162	−	0.995322i	$-0.469198\pi$
$41$	1.23729	0.193232	0.0966162	+	0.995322i	$0.469198\pi$
$42$	4.08293	0.630010
$43$	−3.50772	−0.534922	−0.267461	−	0.963569i	$-0.586185\pi$
$43$	−3.50772	−0.534922	−0.267461	+	0.963569i	$0.586185\pi$
$44$	−2.11001	−0.318096
$45$	3.65713	0.545173
$46$	−1.00000	−0.147442
$47$	−2.84477	−0.414953	−0.207476	−	0.978240i	$-0.566525\pi$
$47$	−2.84477	−0.414953	−0.207476	+	0.978240i	$0.566525\pi$
$48$	−1.00000	−0.144338
$49$	9.67035	1.38148
$50$	8.37463	1.18435
$51$	−1.54712	−0.216640
$52$	2.59337	0.359636
$53$	7.19876	0.988826	0.494413	−	0.869227i	$-0.335383\pi$
$53$	7.19876	0.988826	0.494413	+	0.869227i	$0.335383\pi$
$54$	−1.00000	−0.136083
$55$	−7.71660	−1.04051
$56$	−4.08293	−0.545605
$57$	−4.60452	−0.609883
$58$	−1.00000	−0.131306
$59$	9.32047	1.21342	0.606711	−	0.794923i	$-0.292489\pi$
$59$	9.32047	1.21342	0.606711	+	0.794923i	$0.292489\pi$
$60$	−3.65713	−0.472134
$61$	4.11001	0.526233	0.263117	−	0.964764i	$-0.415250\pi$
$61$	4.11001	0.526233	0.263117	+	0.964764i	$0.415250\pi$
$62$	0.201477	0.0255876
$63$	−4.08293	−0.514401
$64$	1.00000	0.125000
$65$	9.48430	1.17638
$66$	2.11001	0.259725
$67$	1.70283	0.208034	0.104017	−	0.994576i	$-0.466830\pi$
$67$	1.70283	0.208034	0.104017	+	0.994576i	$0.466830\pi$
$68$	1.54712	0.187616
$69$	1.00000	0.120386
$70$	−14.9318	−1.78470
$71$	4.12267	0.489271	0.244636	−	0.969615i	$-0.421332\pi$
$71$	4.12267	0.489271	0.244636	+	0.969615i	$0.421332\pi$
$72$	1.00000	0.117851
$73$	4.28354	0.501350	0.250675	−	0.968071i	$-0.419347\pi$
$73$	4.28354	0.501350	0.250675	+	0.968071i	$0.419347\pi$
$74$	4.67124	0.543021
$75$	−8.37463	−0.967019
$76$	4.60452	0.528174
$77$	8.61504	0.981775
$78$	−2.59337	−0.293641
$79$	−5.51512	−0.620499	−0.310250	−	0.950655i	$-0.600413\pi$
$79$	−5.51512	−0.620499	−0.310250	+	0.950655i	$0.600413\pi$
$80$	3.65713	0.408880
$81$	1.00000	0.111111
$82$	1.23729	0.136636
$83$	−0.151639	−0.0166446	−0.00832228	−	0.999965i	$-0.502649\pi$
$83$	−0.151639	−0.0166446	−0.00832228	+	0.999965i	$0.502649\pi$
$84$	4.08293	0.445485
$85$	5.65803	0.613700
$86$	−3.50772	−0.378247
$87$	1.00000	0.107211
$88$	−2.11001	−0.224928
$89$	8.36672	0.886870	0.443435	−	0.896306i	$-0.353760\pi$
$89$	8.36672	0.886870	0.443435	+	0.896306i	$0.353760\pi$
$90$	3.65713	0.385496
$91$	−10.5886	−1.10998
$92$	−1.00000	−0.104257
$93$	−0.201477	−0.0208922
$94$	−2.84477	−0.293416
$95$	16.8393	1.72768
$96$	−1.00000	−0.102062
$97$	−1.09022	−0.110695	−0.0553474	−	0.998467i	$-0.517627\pi$
$97$	−1.09022	−0.110695	−0.0553474	+	0.998467i	$0.517627\pi$
$98$	9.67035	0.976853
$99$	−2.11001	−0.212064
$100$	8.37463	0.837463
$101$	−10.4104	−1.03588	−0.517939	−	0.855418i	$-0.673301\pi$
$101$	−10.4104	−1.03588	−0.517939	+	0.855418i	$0.673301\pi$
$102$	−1.54712	−0.153188
$103$	−0.509795	−0.0502316	−0.0251158	−	0.999685i	$-0.507995\pi$
$103$	−0.509795	−0.0502316	−0.0251158	+	0.999685i	$0.507995\pi$
$104$	2.59337	0.254301
$105$	14.9318	1.45720
$106$	7.19876	0.699205
$107$	18.1733	1.75688	0.878438	−	0.477856i	$-0.158586\pi$
$107$	18.1733	1.75688	0.878438	+	0.477856i	$0.158586\pi$
$108$	−1.00000	−0.0962250
$109$	1.26675	0.121333	0.0606665	−	0.998158i	$-0.480677\pi$
$109$	1.26675	0.121333	0.0606665	+	0.998158i	$0.480677\pi$
$110$	−7.71660	−0.735749
$111$	−4.67124	−0.443375
$112$	−4.08293	−0.385801
$113$	10.9139	1.02669	0.513347	−	0.858181i	$-0.328405\pi$
$113$	10.9139	1.02669	0.513347	+	0.858181i	$0.328405\pi$
$114$	−4.60452	−0.431253
$115$	−3.65713	−0.341029
$116$	−1.00000	−0.0928477
$117$	2.59337	0.239757
$118$	9.32047	0.858019
$119$	−6.31680	−0.579060
$120$	−3.65713	−0.333849
$121$	−6.54785	−0.595259
$122$	4.11001	0.372103
$123$	−1.23729	−0.111563
$124$	0.201477	0.0180932
$125$	12.3415	1.10385
$126$	−4.08293	−0.363737
$127$	18.9866	1.68479	0.842395	−	0.538861i	$-0.181145\pi$
$127$	18.9866	1.68479	0.842395	+	0.538861i	$0.181145\pi$
$128$	1.00000	0.0883883
$129$	3.50772	0.308837
$130$	9.48430	0.831828
$131$	9.40989	0.822146	0.411073	−	0.911603i	$-0.365154\pi$
$131$	9.40989	0.822146	0.411073	+	0.911603i	$0.365154\pi$
$132$	2.11001	0.183653
$133$	−18.7999	−1.63016
$134$	1.70283	0.147102
$135$	−3.65713	−0.314756
$136$	1.54712	0.132665
$137$	−3.51512	−0.300317	−0.150158	−	0.988662i	$-0.547978\pi$
$137$	−3.51512	−0.300317	−0.150158	+	0.988662i	$0.547978\pi$
$138$	1.00000	0.0851257
$139$	9.92542	0.841864	0.420932	−	0.907092i	$-0.361703\pi$
$139$	9.92542	0.841864	0.420932	+	0.907092i	$0.361703\pi$
$140$	−14.9318	−1.26197
$141$	2.84477	0.239573
$142$	4.12267	0.345967
$143$	−5.47204	−0.457595
$144$	1.00000	0.0833333
$145$	−3.65713	−0.303708
$146$	4.28354	0.354508
$147$	−9.67035	−0.797597
$148$	4.67124	0.383974
$149$	−19.3702	−1.58687	−0.793435	−	0.608655i	$-0.791710\pi$
$149$	−19.3702	−1.58687	−0.793435	+	0.608655i	$0.791710\pi$
$150$	−8.37463	−0.683785
$151$	−2.04680	−0.166566	−0.0832832	−	0.996526i	$-0.526541\pi$
$151$	−2.04680	−0.166566	−0.0832832	+	0.996526i	$0.526541\pi$
$152$	4.60452	0.373476
$153$	1.54712	0.125077
$154$	8.61504	0.694220
$155$	0.736828	0.0591835
$156$	−2.59337	−0.207636
$157$	1.26651	0.101078	0.0505391	−	0.998722i	$-0.483906\pi$
$157$	1.26651	0.101078	0.0505391	+	0.998722i	$0.483906\pi$
$158$	−5.51512	−0.438759
$159$	−7.19876	−0.570899
$160$	3.65713	0.289122
$161$	4.08293	0.321780
$162$	1.00000	0.0785674
$163$	−6.36364	−0.498439	−0.249219	−	0.968447i	$-0.580174\pi$
$163$	−6.36364	−0.498439	−0.249219	+	0.968447i	$0.580174\pi$
$164$	1.23729	0.0966162
$165$	7.71660	0.600736
$166$	−0.151639	−0.0117695
$167$	10.7745	0.833758	0.416879	−	0.908962i	$-0.363124\pi$
$167$	10.7745	0.833758	0.416879	+	0.908962i	$0.363124\pi$
$168$	4.08293	0.315005
$169$	−6.27443	−0.482648
$170$	5.65803	0.433951
$171$	4.60452	0.352116
$172$	−3.50772	−0.267461
$173$	9.08031	0.690363	0.345181	−	0.938536i	$-0.387817\pi$
$173$	9.08031	0.690363	0.345181	+	0.938536i	$0.387817\pi$
$174$	1.00000	0.0758098
$175$	−34.1930	−2.58475
$176$	−2.11001	−0.159048
$177$	−9.32047	−0.700569
$178$	8.36672	0.627112
$179$	−18.2934	−1.36731	−0.683656	−	0.729804i	$-0.739611\pi$
$179$	−18.2934	−1.36731	−0.683656	+	0.729804i	$0.739611\pi$
$180$	3.65713	0.272587
$181$	2.06839	0.153742	0.0768709	−	0.997041i	$-0.475507\pi$
$181$	2.06839	0.153742	0.0768709	+	0.997041i	$0.475507\pi$
$182$	−10.5886	−0.784876
$183$	−4.11001	−0.303821
$184$	−1.00000	−0.0737210
$185$	17.0834	1.25599
$186$	−0.201477	−0.0147730
$187$	−3.26445	−0.238720
$188$	−2.84477	−0.207476
$189$	4.08293	0.296990
$190$	16.8393	1.22165
$191$	−4.48470	−0.324501	−0.162251	−	0.986750i	$-0.551875\pi$
$191$	−4.48470	−0.324501	−0.162251	+	0.986750i	$0.551875\pi$
$192$	−1.00000	−0.0721688
$193$	−20.9608	−1.50879	−0.754396	−	0.656420i	$-0.772070\pi$
$193$	−20.9608	−1.50879	−0.754396	+	0.656420i	$0.772070\pi$
$194$	−1.09022	−0.0782730
$195$	−9.48430	−0.679185
$196$	9.67035	0.690739
$197$	3.63286	0.258830	0.129415	−	0.991590i	$-0.458690\pi$
$197$	3.63286	0.258830	0.129415	+	0.991590i	$0.458690\pi$
$198$	−2.11001	−0.149952
$199$	−11.5063	−0.815657	−0.407828	−	0.913059i	$-0.633714\pi$
$199$	−11.5063	−0.815657	−0.407828	+	0.913059i	$0.633714\pi$
$200$	8.37463	0.592176
$201$	−1.70283	−0.120108
$202$	−10.4104	−0.732476
$203$	4.08293	0.286566
$204$	−1.54712	−0.108320
$205$	4.52494	0.316035
$206$	−0.509795	−0.0355191
$207$	−1.00000	−0.0695048
$208$	2.59337	0.179818
$209$	−9.71559	−0.672041
$210$	14.9318	1.03039
$211$	−9.82327	−0.676262	−0.338131	−	0.941099i	$-0.609795\pi$
$211$	−9.82327	−0.676262	−0.338131	+	0.941099i	$0.609795\pi$
$212$	7.19876	0.494413
$213$	−4.12267	−0.282481
$214$	18.1733	1.24230
$215$	−12.8282	−0.874876
$216$	−1.00000	−0.0680414
$217$	−0.822617	−0.0558429
$218$	1.26675	0.0857954
$219$	−4.28354	−0.289455
$220$	−7.71660	−0.520253
$221$	4.01226	0.269894
$222$	−4.67124	−0.313513
$223$	22.4585	1.50393	0.751967	−	0.659200i	$-0.229105\pi$
$223$	22.4585	1.50393	0.751967	+	0.659200i	$0.229105\pi$
$224$	−4.08293	−0.272802
$225$	8.37463	0.558308
$226$	10.9139	0.725983
$227$	25.5228	1.69401	0.847003	−	0.531589i	$-0.178405\pi$
$227$	25.5228	1.69401	0.847003	+	0.531589i	$0.178405\pi$
$228$	−4.60452	−0.304942
$229$	−19.7258	−1.30352	−0.651758	−	0.758427i	$-0.725968\pi$
$229$	−19.7258	−1.30352	−0.651758	+	0.758427i	$0.725968\pi$
$230$	−3.65713	−0.241144
$231$	−8.61504	−0.566828
$232$	−1.00000	−0.0656532
$233$	−8.67045	−0.568020	−0.284010	−	0.958821i	$-0.591665\pi$
$233$	−8.67045	−0.568020	−0.284010	+	0.958821i	$0.591665\pi$
$234$	2.59337	0.169534
$235$	−10.4037	−0.678663
$236$	9.32047	0.606711
$237$	5.51512	0.358246
$238$	−6.31680	−0.409457
$239$	−15.4492	−0.999326	−0.499663	−	0.866220i	$-0.666543\pi$
$239$	−15.4492	−0.999326	−0.499663	+	0.866220i	$0.666543\pi$
$240$	−3.65713	−0.236067
$241$	20.8524	1.34322	0.671609	−	0.740906i	$-0.265603\pi$
$241$	20.8524	1.34322	0.671609	+	0.740906i	$0.265603\pi$
$242$	−6.54785	−0.420912
$243$	−1.00000	−0.0641500
$244$	4.11001	0.263117
$245$	35.3658	2.25944
$246$	−1.23729	−0.0788868
$247$	11.9412	0.759802
$248$	0.201477	0.0127938
$249$	0.151639	0.00960974
$250$	12.3415	0.780543
$251$	−3.90973	−0.246780	−0.123390	−	0.992358i	$-0.539377\pi$
$251$	−3.90973	−0.246780	−0.123390	+	0.992358i	$0.539377\pi$
$252$	−4.08293	−0.257201
$253$	2.11001	0.132655
$254$	18.9866	1.19133
$255$	−5.65803	−0.354320
$256$	1.00000	0.0625000
$257$	−1.43065	−0.0892416	−0.0446208	−	0.999004i	$-0.514208\pi$
$257$	−1.43065	−0.0892416	−0.0446208	+	0.999004i	$0.514208\pi$
$258$	3.50772	0.218381
$259$	−19.0724	−1.18510
$260$	9.48430	0.588192
$261$	−1.00000	−0.0618984
$262$	9.40989	0.581345
$263$	1.48582	0.0916194	0.0458097	−	0.998950i	$-0.485413\pi$
$263$	1.48582	0.0916194	0.0458097	+	0.998950i	$0.485413\pi$
$264$	2.11001	0.129862
$265$	26.3268	1.61724
$266$	−18.7999	−1.15270
$267$	−8.36672	−0.512035
$268$	1.70283	0.104017
$269$	−2.44927	−0.149335	−0.0746673	−	0.997208i	$-0.523789\pi$
$269$	−2.44927	−0.149335	−0.0746673	+	0.997208i	$0.523789\pi$
$270$	−3.65713	−0.222566
$271$	26.2881	1.59689	0.798444	−	0.602069i	$-0.205657\pi$
$271$	26.2881	1.59689	0.798444	+	0.602069i	$0.205657\pi$
$272$	1.54712	0.0938080
$273$	10.5886	0.640849
$274$	−3.51512	−0.212356
$275$	−17.6706	−1.06558
$276$	1.00000	0.0601929
$277$	7.90653	0.475057	0.237529	−	0.971381i	$-0.423663\pi$
$277$	7.90653	0.475057	0.237529	+	0.971381i	$0.423663\pi$
$278$	9.92542	0.595287
$279$	0.201477	0.0120621
$280$	−14.9318	−0.892348
$281$	23.8268	1.42139	0.710693	−	0.703503i	$-0.248382\pi$
$281$	23.8268	1.42139	0.710693	+	0.703503i	$0.248382\pi$
$282$	2.84477	0.169404
$283$	−3.90756	−0.232281	−0.116140	−	0.993233i	$-0.537052\pi$
$283$	−3.90756	−0.232281	−0.116140	+	0.993233i	$0.537052\pi$
$284$	4.12267	0.244636
$285$	−16.8393	−0.997476
$286$	−5.47204	−0.323569
$287$	−5.05178	−0.298197
$288$	1.00000	0.0589256
$289$	−14.6064	−0.859201
$290$	−3.65713	−0.214754
$291$	1.09022	0.0639096
$292$	4.28354	0.250675
$293$	24.0151	1.40298	0.701488	−	0.712682i	$-0.252520\pi$
$293$	24.0151	1.40298	0.701488	+	0.712682i	$0.252520\pi$
$294$	−9.67035	−0.563986
$295$	34.0862	1.98458
$296$	4.67124	0.271511
$297$	2.11001	0.122435
$298$	−19.3702	−1.12209
$299$	−2.59337	−0.149978
$300$	−8.37463	−0.483509
$301$	14.3218	0.825494
$302$	−2.04680	−0.117780
$303$	10.4104	0.598064
$304$	4.60452	0.264087
$305$	15.0309	0.860665
$306$	1.54712	0.0884431
$307$	−11.8161	−0.674380	−0.337190	−	0.941437i	$-0.609477\pi$
$307$	−11.8161	−0.674380	−0.337190	+	0.941437i	$0.609477\pi$
$308$	8.61504	0.490887
$309$	0.509795	0.0290012
$310$	0.736828	0.0418490
$311$	−1.39883	−0.0793204	−0.0396602	−	0.999213i	$-0.512628\pi$
$311$	−1.39883	−0.0793204	−0.0396602	+	0.999213i	$0.512628\pi$
$312$	−2.59337	−0.146821
$313$	0.343763	0.0194306	0.00971532	−	0.999953i	$-0.496907\pi$
$313$	0.343763	0.0194306	0.00971532	+	0.999953i	$0.496907\pi$
$314$	1.26651	0.0714731
$315$	−14.9318	−0.841314
$316$	−5.51512	−0.310250
$317$	−23.7395	−1.33334	−0.666671	−	0.745352i	$-0.732281\pi$
$317$	−23.7395	−1.33334	−0.666671	+	0.745352i	$0.732281\pi$
$318$	−7.19876	−0.403686
$319$	2.11001	0.118138
$320$	3.65713	0.204440
$321$	−18.1733	−1.01433
$322$	4.08293	0.227533
$323$	7.12375	0.396376
$324$	1.00000	0.0555556
$325$	21.7185	1.20473
$326$	−6.36364	−0.352449
$327$	−1.26675	−0.0700516
$328$	1.23729	0.0683179
$329$	11.6150	0.640357
$330$	7.71660	0.424785
$331$	−17.2835	−0.949988	−0.474994	−	0.879989i	$-0.657550\pi$
$331$	−17.2835	−0.949988	−0.474994	+	0.879989i	$0.657550\pi$
$332$	−0.151639	−0.00832228
$333$	4.67124	0.255983
$334$	10.7745	0.589556
$335$	6.22747	0.340243
$336$	4.08293	0.222742
$337$	12.8932	0.702339	0.351169	−	0.936312i	$-0.385784\pi$
$337$	12.8932	0.702339	0.351169	+	0.936312i	$0.385784\pi$
$338$	−6.27443	−0.341284
$339$	−10.9139	−0.592762
$340$	5.65803	0.306850
$341$	−0.425119	−0.0230215
$342$	4.60452	0.248984
$343$	−10.9029	−0.588699
$344$	−3.50772	−0.189124
$345$	3.65713	0.196893
$346$	9.08031	0.488160
$347$	11.4217	0.613149	0.306575	−	0.951847i	$-0.400817\pi$
$347$	11.4217	0.613149	0.306575	+	0.951847i	$0.400817\pi$
$348$	1.00000	0.0536056
$349$	−18.0483	−0.966105	−0.483053	−	0.875591i	$-0.660472\pi$
$349$	−18.0483	−0.966105	−0.483053	+	0.875591i	$0.660472\pi$
$350$	−34.1930	−1.82770
$351$	−2.59337	−0.138424
$352$	−2.11001	−0.112464
$353$	−22.8166	−1.21440	−0.607202	−	0.794548i	$-0.707708\pi$
$353$	−22.8166	−1.21440	−0.607202	+	0.794548i	$0.707708\pi$
$354$	−9.32047	−0.495377
$355$	15.0772	0.800213
$356$	8.36672	0.443435
$357$	6.31680	0.334320
$358$	−18.2934	−0.966836
$359$	−13.9448	−0.735981	−0.367990	−	0.929830i	$-0.619954\pi$
$359$	−13.9448	−0.735981	−0.367990	+	0.929830i	$0.619954\pi$
$360$	3.65713	0.192748
$361$	2.20158	0.115873
$362$	2.06839	0.108712
$363$	6.54785	0.343673
$364$	−10.5886	−0.554991
$365$	15.6655	0.819969
$366$	−4.11001	−0.214834
$367$	−2.25192	−0.117549	−0.0587747	−	0.998271i	$-0.518719\pi$
$367$	−2.25192	−0.117549	−0.0587747	+	0.998271i	$0.518719\pi$
$368$	−1.00000	−0.0521286
$369$	1.23729	0.0644108
$370$	17.0834	0.888122
$371$	−29.3921	−1.52596
$372$	−0.201477	−0.0104461
$373$	−10.9144	−0.565126	−0.282563	−	0.959249i	$-0.591185\pi$
$373$	−10.9144	−0.565126	−0.282563	+	0.959249i	$0.591185\pi$
$374$	−3.26445	−0.168800
$375$	−12.3415	−0.637310
$376$	−2.84477	−0.146708
$377$	−2.59337	−0.133565
$378$	4.08293	0.210003
$379$	16.5862	0.851976	0.425988	−	0.904729i	$-0.359927\pi$
$379$	16.5862	0.851976	0.425988	+	0.904729i	$0.359927\pi$
$380$	16.8393	0.863840
$381$	−18.9866	−0.972714
$382$	−4.48470	−0.229457
$383$	−9.82012	−0.501785	−0.250892	−	0.968015i	$-0.580724\pi$
$383$	−9.82012	−0.501785	−0.250892	+	0.968015i	$0.580724\pi$
$384$	−1.00000	−0.0510310
$385$	31.5064	1.60571
$386$	−20.9608	−1.06688
$387$	−3.50772	−0.178307
$388$	−1.09022	−0.0553474
$389$	−28.3887	−1.43936	−0.719682	−	0.694303i	$-0.755713\pi$
$389$	−28.3887	−1.43936	−0.719682	+	0.694303i	$0.755713\pi$
$390$	−9.48430	−0.480256
$391$	−1.54712	−0.0782413
$392$	9.67035	0.488426
$393$	−9.40989	−0.474666
$394$	3.63286	0.183021
$395$	−20.1695	−1.01484
$396$	−2.11001	−0.106032
$397$	0.579579	0.0290882	0.0145441	−	0.999894i	$-0.495370\pi$
$397$	0.579579	0.0290882	0.0145441	+	0.999894i	$0.495370\pi$
$398$	−11.5063	−0.576756
$399$	18.7999	0.941174
$400$	8.37463	0.418731
$401$	8.60296	0.429611	0.214806	−	0.976657i	$-0.431088\pi$
$401$	8.60296	0.429611	0.214806	+	0.976657i	$0.431088\pi$
$402$	−1.70283	−0.0849294
$403$	0.522505	0.0260278
$404$	−10.4104	−0.517939
$405$	3.65713	0.181724
$406$	4.08293	0.202633
$407$	−9.85638	−0.488563
$408$	−1.54712	−0.0765939
$409$	24.7968	1.22612	0.613062	−	0.790035i	$-0.289938\pi$
$409$	24.7968	1.22612	0.613062	+	0.790035i	$0.289938\pi$
$410$	4.52494	0.223471
$411$	3.51512	0.173388
$412$	−0.509795	−0.0251158
$413$	−38.0549	−1.87256
$414$	−1.00000	−0.0491473
$415$	−0.554565	−0.0272225
$416$	2.59337	0.127150
$417$	−9.92542	−0.486050
$418$	−9.71559	−0.475205
$419$	33.2159	1.62271	0.811353	−	0.584557i	$-0.198732\pi$
$419$	33.2159	1.62271	0.811353	+	0.584557i	$0.198732\pi$
$420$	14.9318	0.728599
$421$	6.47029	0.315343	0.157671	−	0.987492i	$-0.449601\pi$
$421$	6.47029	0.315343	0.157671	+	0.987492i	$0.449601\pi$
$422$	−9.82327	−0.478189
$423$	−2.84477	−0.138318
$424$	7.19876	0.349603
$425$	12.9566	0.628486
$426$	−4.12267	−0.199744
$427$	−16.7809	−0.812085
$428$	18.1733	0.878438
$429$	5.47204	0.264193
$430$	−12.8282	−0.618631
$431$	−27.2115	−1.31073	−0.655367	−	0.755311i	$-0.727486\pi$
$431$	−27.2115	−1.31073	−0.655367	+	0.755311i	$0.727486\pi$
$432$	−1.00000	−0.0481125
$433$	35.7153	1.71637	0.858185	−	0.513341i	$-0.171592\pi$
$433$	35.7153	1.71637	0.858185	+	0.513341i	$0.171592\pi$
$434$	−0.822617	−0.0394869
$435$	3.65713	0.175346
$436$	1.26675	0.0606665
$437$	−4.60452	−0.220264
$438$	−4.28354	−0.204675
$439$	6.81905	0.325456	0.162728	−	0.986671i	$-0.447971\pi$
$439$	6.81905	0.325456	0.162728	+	0.986671i	$0.447971\pi$
$440$	−7.71660	−0.367874
$441$	9.67035	0.460493
$442$	4.01226	0.190844
$443$	−10.7646	−0.511442	−0.255721	−	0.966751i	$-0.582313\pi$
$443$	−10.7646	−0.511442	−0.255721	+	0.966751i	$0.582313\pi$
$444$	−4.67124	−0.221687
$445$	30.5982	1.45049
$446$	22.4585	1.06344
$447$	19.3702	0.916180
$448$	−4.08293	−0.192900
$449$	−8.17517	−0.385810	−0.192905	−	0.981217i	$-0.561791\pi$
$449$	−8.17517	−0.385810	−0.192905	+	0.981217i	$0.561791\pi$
$450$	8.37463	0.394784
$451$	−2.61070	−0.122933
$452$	10.9139	0.513347
$453$	2.04680	0.0961672
$454$	25.5228	1.19784
$455$	−38.7238	−1.81540
$456$	−4.60452	−0.215626
$457$	−6.98937	−0.326949	−0.163474	−	0.986548i	$-0.552270\pi$
$457$	−6.98937	−0.326949	−0.163474	+	0.986548i	$0.552270\pi$
$458$	−19.7258	−0.921725
$459$	−1.54712	−0.0722135
$460$	−3.65713	−0.170515
$461$	−34.7564	−1.61876	−0.809382	−	0.587282i	$-0.800198\pi$
$461$	−34.7564	−1.61876	−0.809382	+	0.587282i	$0.800198\pi$
$462$	−8.61504	−0.400808
$463$	5.94660	0.276362	0.138181	−	0.990407i	$-0.455874\pi$
$463$	5.94660	0.276362	0.138181	+	0.990407i	$0.455874\pi$
$464$	−1.00000	−0.0464238
$465$	−0.736828	−0.0341696
$466$	−8.67045	−0.401651
$467$	−32.0474	−1.48298	−0.741488	−	0.670966i	$-0.765880\pi$
$467$	−32.0474	−1.48298	−0.741488	+	0.670966i	$0.765880\pi$
$468$	2.59337	0.119879
$469$	−6.95254	−0.321038
$470$	−10.4037	−0.479887
$471$	−1.26651	−0.0583576
$472$	9.32047	0.429009
$473$	7.40133	0.340314
$474$	5.51512	0.253318
$475$	38.5611	1.76931
$476$	−6.31680	−0.289530
$477$	7.19876	0.329609
$478$	−15.4492	−0.706630
$479$	−27.5533	−1.25894	−0.629471	−	0.777024i	$-0.716728\pi$
$479$	−27.5533	−1.25894	−0.629471	+	0.777024i	$0.716728\pi$
$480$	−3.65713	−0.166925
$481$	12.1143	0.552363
$482$	20.8524	0.949799
$483$	−4.08293	−0.185780
$484$	−6.54785	−0.297629
$485$	−3.98707	−0.181043
$486$	−1.00000	−0.0453609
$487$	−4.70625	−0.213261	−0.106630	−	0.994299i	$-0.534006\pi$
$487$	−4.70625	−0.213261	−0.106630	+	0.994299i	$0.534006\pi$
$488$	4.11001	0.186052
$489$	6.36364	0.287774
$490$	35.3658	1.59766
$491$	3.36669	0.151936	0.0759682	−	0.997110i	$-0.475795\pi$
$491$	3.36669	0.151936	0.0759682	+	0.997110i	$0.475795\pi$
$492$	−1.23729	−0.0557814
$493$	−1.54712	−0.0696789
$494$	11.9412	0.537261
$495$	−7.71660	−0.346835
$496$	0.201477	0.00904658
$497$	−16.8326	−0.755045
$498$	0.151639	0.00679512
$499$	12.4793	0.558650	0.279325	−	0.960197i	$-0.409889\pi$
$499$	12.4793	0.558650	0.279325	+	0.960197i	$0.409889\pi$
$500$	12.3415	0.551927
$501$	−10.7745	−0.481370
$502$	−3.90973	−0.174500
$503$	12.3568	0.550963	0.275481	−	0.961306i	$-0.411163\pi$
$503$	12.3568	0.550963	0.275481	+	0.961306i	$0.411163\pi$
$504$	−4.08293	−0.181868
$505$	−38.0724	−1.69420
$506$	2.11001	0.0938015
$507$	6.27443	0.278657
$508$	18.9866	0.842395
$509$	−30.5996	−1.35630	−0.678152	−	0.734922i	$-0.737219\pi$
$509$	−30.5996	−1.35630	−0.678152	+	0.734922i	$0.737219\pi$
$510$	−5.65803	−0.250542
$511$	−17.4894	−0.773686
$512$	1.00000	0.0441942
$513$	−4.60452	−0.203294
$514$	−1.43065	−0.0631034
$515$	−1.86439	−0.0821548
$516$	3.50772	0.154419
$517$	6.00250	0.263990
$518$	−19.0724	−0.837992
$519$	−9.08031	−0.398581
$520$	9.48430	0.415914
$521$	28.7389	1.25907	0.629536	−	0.776971i	$-0.283245\pi$
$521$	28.7389	1.25907	0.629536	+	0.776971i	$0.283245\pi$
$522$	−1.00000	−0.0437688
$523$	−23.2317	−1.01585	−0.507925	−	0.861401i	$-0.669587\pi$
$523$	−23.2317	−1.01585	−0.507925	+	0.861401i	$0.669587\pi$
$524$	9.40989	0.411073
$525$	34.1930	1.49231
$526$	1.48582	0.0647847
$527$	0.311709	0.0135783
$528$	2.11001	0.0918265
$529$	1.00000	0.0434783
$530$	26.3268	1.14356
$531$	9.32047	0.404474
$532$	−18.7999	−0.815081
$533$	3.20875	0.138987
$534$	−8.36672	−0.362063
$535$	66.4621	2.87341
$536$	1.70283	0.0735510
$537$	18.2934	0.789418
$538$	−2.44927	−0.105596
$539$	−20.4046	−0.878886
$540$	−3.65713	−0.157378
$541$	20.4606	0.879670	0.439835	−	0.898079i	$-0.355037\pi$
$541$	20.4606	0.879670	0.439835	+	0.898079i	$0.355037\pi$
$542$	26.2881	1.12917
$543$	−2.06839	−0.0887629
$544$	1.54712	0.0663323
$545$	4.63269	0.198442
$546$	10.5886	0.453149
$547$	−12.2750	−0.524840	−0.262420	−	0.964954i	$-0.584521\pi$
$547$	−12.2750	−0.524840	−0.262420	+	0.964954i	$0.584521\pi$
$548$	−3.51512	−0.150158
$549$	4.11001	0.175411
$550$	−17.6706	−0.753475
$551$	−4.60452	−0.196159
$552$	1.00000	0.0425628
$553$	22.5179	0.957557
$554$	7.90653	0.335916
$555$	−17.0834	−0.725149
$556$	9.92542	0.420932
$557$	−6.59263	−0.279339	−0.139669	−	0.990198i	$-0.544604\pi$
$557$	−6.59263	−0.279339	−0.139669	+	0.990198i	$0.544604\pi$
$558$	0.201477	0.00852920
$559$	−9.09682	−0.384754
$560$	−14.9318	−0.630985
$561$	3.26445	0.137825
$562$	23.8268	1.00507
$563$	−5.14382	−0.216786	−0.108393	−	0.994108i	$-0.534571\pi$
$563$	−5.14382	−0.216786	−0.108393	+	0.994108i	$0.534571\pi$
$564$	2.84477	0.119787
$565$	39.9136	1.67918
$566$	−3.90756	−0.164247
$567$	−4.08293	−0.171467
$568$	4.12267	0.172983
$569$	−30.0362	−1.25918	−0.629591	−	0.776927i	$-0.716778\pi$
$569$	−30.0362	−1.25918	−0.629591	+	0.776927i	$0.716778\pi$
$570$	−16.8393	−0.705322
$571$	7.05751	0.295348	0.147674	−	0.989036i	$-0.452821\pi$
$571$	7.05751	0.295348	0.147674	+	0.989036i	$0.452821\pi$
$572$	−5.47204	−0.228798
$573$	4.48470	0.187351
$574$	−5.05178	−0.210857
$575$	−8.37463	−0.349246
$576$	1.00000	0.0416667
$577$	12.4598	0.518708	0.259354	−	0.965782i	$-0.416490\pi$
$577$	12.4598	0.518708	0.259354	+	0.965782i	$0.416490\pi$
$578$	−14.6064	−0.607547
$579$	20.9608	0.871101
$580$	−3.65713	−0.151854
$581$	0.619133	0.0256860
$582$	1.09022	0.0451909
$583$	−15.1895	−0.629084
$584$	4.28354	0.177254
$585$	9.48430	0.392128
$586$	24.0151	0.992053
$587$	−36.7752	−1.51787	−0.758937	−	0.651164i	$-0.774281\pi$
$587$	−36.7752	−1.51787	−0.758937	+	0.651164i	$0.774281\pi$
$588$	−9.67035	−0.398798
$589$	0.927704	0.0382254
$590$	34.0862	1.40331
$591$	−3.63286	−0.149436
$592$	4.67124	0.191987
$593$	38.3033	1.57293	0.786464	−	0.617636i	$-0.211910\pi$
$593$	38.3033	1.57293	0.786464	+	0.617636i	$0.211910\pi$
$594$	2.11001	0.0865748
$595$	−23.1014	−0.947064
$596$	−19.3702	−0.793435
$597$	11.5063	0.470920
$598$	−2.59337	−0.106051
$599$	−9.46664	−0.386796	−0.193398	−	0.981120i	$-0.561951\pi$
$599$	−9.46664	−0.386796	−0.193398	+	0.981120i	$0.561951\pi$
$600$	−8.37463	−0.341893
$601$	−16.9441	−0.691164	−0.345582	−	0.938388i	$-0.612319\pi$
$601$	−16.9441	−0.691164	−0.345582	+	0.938388i	$0.612319\pi$
$602$	14.3218	0.583712
$603$	1.70283	0.0693446
$604$	−2.04680	−0.0832832
$605$	−23.9464	−0.973558
$606$	10.4104	0.422895
$607$	−11.4721	−0.465639	−0.232819	−	0.972520i	$-0.574795\pi$
$607$	−11.4721	−0.465639	−0.232819	+	0.972520i	$0.574795\pi$
$608$	4.60452	0.186738
$609$	−4.08293	−0.165449
$610$	15.0309	0.608582
$611$	−7.37755	−0.298464
$612$	1.54712	0.0625387
$613$	−26.5120	−1.07081	−0.535405	−	0.844595i	$-0.679841\pi$
$613$	−26.5120	−1.07081	−0.535405	+	0.844595i	$0.679841\pi$
$614$	−11.8161	−0.476859
$615$	−4.52494	−0.182463
$616$	8.61504	0.347110
$617$	2.27771	0.0916970	0.0458485	−	0.998948i	$-0.485401\pi$
$617$	2.27771	0.0916970	0.0458485	+	0.998948i	$0.485401\pi$
$618$	0.509795	0.0205070
$619$	14.2915	0.574425	0.287212	−	0.957867i	$-0.407271\pi$
$619$	14.2915	0.574425	0.287212	+	0.957867i	$0.407271\pi$
$620$	0.736828	0.0295917
$621$	1.00000	0.0401286
$622$	−1.39883	−0.0560880
$623$	−34.1608	−1.36862
$624$	−2.59337	−0.103818
$625$	3.26125	0.130450
$626$	0.343763	0.0137395
$627$	9.71559	0.388003
$628$	1.26651	0.0505391
$629$	7.22698	0.288159
$630$	−14.9318	−0.594899
$631$	32.0834	1.27722	0.638609	−	0.769531i	$-0.279510\pi$
$631$	32.0834	1.27722	0.638609	+	0.769531i	$0.279510\pi$
$632$	−5.51512	−0.219380
$633$	9.82327	0.390440
$634$	−23.7395	−0.942815
$635$	69.4366	2.75551
$636$	−7.19876	−0.285449
$637$	25.0788	0.993658
$638$	2.11001	0.0835362
$639$	4.12267	0.163090
$640$	3.65713	0.144561
$641$	−45.7450	−1.80682	−0.903410	−	0.428778i	$-0.858944\pi$
$641$	−45.7450	−1.80682	−0.903410	+	0.428778i	$0.858944\pi$
$642$	−18.1733	−0.717242
$643$	−45.9987	−1.81401	−0.907006	−	0.421117i	$-0.861638\pi$
$643$	−45.9987	−1.81401	−0.907006	+	0.421117i	$0.861638\pi$
$644$	4.08293	0.160890
$645$	12.8282	0.505110
$646$	7.12375	0.280280
$647$	−8.35987	−0.328661	−0.164330	−	0.986405i	$-0.552546\pi$
$647$	−8.35987	−0.328661	−0.164330	+	0.986405i	$0.552546\pi$
$648$	1.00000	0.0392837
$649$	−19.6663	−0.771970
$650$	21.7185	0.851870
$651$	0.822617	0.0322409
$652$	−6.36364	−0.249219
$653$	−14.1611	−0.554166	−0.277083	−	0.960846i	$-0.589368\pi$
$653$	−14.1611	−0.554166	−0.277083	+	0.960846i	$0.589368\pi$
$654$	−1.26675	−0.0495340
$655$	34.4132	1.34464
$656$	1.23729	0.0483081
$657$	4.28354	0.167117
$658$	11.6150	0.452800
$659$	35.6093	1.38714	0.693570	−	0.720389i	$-0.256037\pi$
$659$	35.6093	1.38714	0.693570	+	0.720389i	$0.256037\pi$
$660$	7.71660	0.300368
$661$	−20.6433	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$661$	−20.6433	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$662$	−17.2835	−0.671743
$663$	−4.01226	−0.155823
$664$	−0.151639	−0.00588474
$665$	−68.7539	−2.66616
$666$	4.67124	0.181007
$667$	1.00000	0.0387202
$668$	10.7745	0.416879
$669$	−22.4585	−0.868297
$670$	6.22747	0.240588
$671$	−8.67218	−0.334786
$672$	4.08293	0.157503
$673$	−4.51595	−0.174077	−0.0870385	−	0.996205i	$-0.527740\pi$
$673$	−4.51595	−0.174077	−0.0870385	+	0.996205i	$0.527740\pi$
$674$	12.8932	0.496629
$675$	−8.37463	−0.322340
$676$	−6.27443	−0.241324
$677$	9.35757	0.359641	0.179820	−	0.983699i	$-0.442448\pi$
$677$	9.35757	0.359641	0.179820	+	0.983699i	$0.442448\pi$
$678$	−10.9139	−0.419146
$679$	4.45128	0.170825
$680$	5.65803	0.216976
$681$	−25.5228	−0.978034
$682$	−0.425119	−0.0162786
$683$	−12.2723	−0.469586	−0.234793	−	0.972045i	$-0.575441\pi$
$683$	−12.2723	−0.469586	−0.234793	+	0.972045i	$0.575441\pi$
$684$	4.60452	0.176058
$685$	−12.8553	−0.491174
$686$	−10.9029	−0.416273
$687$	19.7258	0.752586
$688$	−3.50772	−0.133731
$689$	18.6691	0.711234
$690$	3.65713	0.139225
$691$	−6.71037	−0.255274	−0.127637	−	0.991821i	$-0.540739\pi$
$691$	−6.71037	−0.255274	−0.127637	+	0.991821i	$0.540739\pi$
$692$	9.08031	0.345181
$693$	8.61504	0.327258
$694$	11.4217	0.433562
$695$	36.2986	1.37688
$696$	1.00000	0.0379049
$697$	1.91424	0.0725070
$698$	−18.0483	−0.683139
$699$	8.67045	0.327947
$700$	−34.1930	−1.29238
$701$	−24.4041	−0.921729	−0.460865	−	0.887470i	$-0.652461\pi$
$701$	−24.4041	−0.921729	−0.460865	+	0.887470i	$0.652461\pi$
$702$	−2.59337	−0.0978805
$703$	21.5088	0.811221
$704$	−2.11001	−0.0795241
$705$	10.4037	0.391826
$706$	−22.8166	−0.858713
$707$	42.5051	1.59857
$708$	−9.32047	−0.350285
$709$	38.9675	1.46346	0.731728	−	0.681596i	$-0.238714\pi$
$709$	38.9675	1.46346	0.731728	+	0.681596i	$0.238714\pi$
$710$	15.0772	0.565836
$711$	−5.51512	−0.206833
$712$	8.36672	0.313556
$713$	−0.201477	−0.00754537
$714$	6.31680	0.236400
$715$	−20.0120	−0.748406
$716$	−18.2934	−0.683656
$717$	15.4492	0.576961
$718$	−13.9448	−0.520417
$719$	−12.8527	−0.479325	−0.239662	−	0.970856i	$-0.577037\pi$
$719$	−12.8527	−0.479325	−0.239662	+	0.970856i	$0.577037\pi$
$720$	3.65713	0.136293
$721$	2.08146	0.0775176
$722$	2.20158	0.0819344
$723$	−20.8524	−0.775508
$724$	2.06839	0.0768709
$725$	−8.37463	−0.311026
$726$	6.54785	0.243013
$727$	−40.9706	−1.51951	−0.759757	−	0.650207i	$-0.774682\pi$
$727$	−40.9706	−1.51951	−0.759757	+	0.650207i	$0.774682\pi$
$728$	−10.5886	−0.392438
$729$	1.00000	0.0370370
$730$	15.6655	0.579805
$731$	−5.42687	−0.200720
$732$	−4.11001	−0.151910
$733$	2.62636	0.0970069	0.0485034	−	0.998823i	$-0.484555\pi$
$733$	2.62636	0.0970069	0.0485034	+	0.998823i	$0.484555\pi$
$734$	−2.25192	−0.0831200
$735$	−35.3658	−1.30449
$736$	−1.00000	−0.0368605
$737$	−3.59299	−0.132349
$738$	1.23729	0.0455453
$739$	−31.8920	−1.17317	−0.586583	−	0.809889i	$-0.699527\pi$
$739$	−31.8920	−1.17317	−0.586583	+	0.809889i	$0.699527\pi$
$740$	17.0834	0.627997
$741$	−11.9412	−0.438672
$742$	−29.3921	−1.07902
$743$	13.8674	0.508744	0.254372	−	0.967106i	$-0.418131\pi$
$743$	13.8674	0.508744	0.254372	+	0.967106i	$0.418131\pi$
$744$	−0.201477	−0.00738651
$745$	−70.8395	−2.59536
$746$	−10.9144	−0.399604
$747$	−0.151639	−0.00554819
$748$	−3.26445	−0.119360
$749$	−74.2002	−2.71122
$750$	−12.3415	−0.450647
$751$	−13.4503	−0.490808	−0.245404	−	0.969421i	$-0.578921\pi$
$751$	−13.4503	−0.490808	−0.245404	+	0.969421i	$0.578921\pi$
$752$	−2.84477	−0.103738
$753$	3.90973	0.142478
$754$	−2.59337	−0.0944450
$755$	−7.48543	−0.272423
$756$	4.08293	0.148495
$757$	−12.8891	−0.468464	−0.234232	−	0.972181i	$-0.575258\pi$
$757$	−12.8891	−0.468464	−0.234232	+	0.972181i	$0.575258\pi$
$758$	16.5862	0.602438
$759$	−2.11001	−0.0765886
$760$	16.8393	0.610827
$761$	−50.9658	−1.84751	−0.923755	−	0.382984i	$-0.874896\pi$
$761$	−50.9658	−1.84751	−0.923755	+	0.382984i	$0.874896\pi$
$762$	−18.9866	−0.687812
$763$	−5.17207	−0.187241
$764$	−4.48470	−0.162251
$765$	5.65803	0.204567
$766$	−9.82012	−0.354815
$767$	24.1714	0.872780
$768$	−1.00000	−0.0360844
$769$	−12.7622	−0.460216	−0.230108	−	0.973165i	$-0.573908\pi$
$769$	−12.7622	−0.460216	−0.230108	+	0.973165i	$0.573908\pi$
$770$	31.5064	1.13541
$771$	1.43065	0.0515237
$772$	−20.9608	−0.754396
$773$	22.8684	0.822520	0.411260	−	0.911518i	$-0.365089\pi$
$773$	22.8684	0.822520	0.411260	+	0.911518i	$0.365089\pi$
$774$	−3.50772	−0.126082
$775$	1.68729	0.0606094
$776$	−1.09022	−0.0391365
$777$	19.0724	0.684218
$778$	−28.3887	−1.01778
$779$	5.69713	0.204121
$780$	−9.48430	−0.339593
$781$	−8.69889	−0.311271
$782$	−1.54712	−0.0553250
$783$	1.00000	0.0357371
$784$	9.67035	0.345370
$785$	4.63179	0.165316
$786$	−9.40989	−0.335640
$787$	−8.46308	−0.301676	−0.150838	−	0.988558i	$-0.548197\pi$
$787$	−8.46308	−0.301676	−0.150838	+	0.988558i	$0.548197\pi$
$788$	3.63286	0.129415
$789$	−1.48582	−0.0528965
$790$	−20.1695	−0.717600
$791$	−44.5608	−1.58440
$792$	−2.11001	−0.0749760
$793$	10.6588	0.378505
$794$	0.579579	0.0205685
$795$	−26.3268	−0.933716
$796$	−11.5063	−0.407828
$797$	−15.3222	−0.542742	−0.271371	−	0.962475i	$-0.587477\pi$
$797$	−15.3222	−0.542742	−0.271371	+	0.962475i	$0.587477\pi$
$798$	18.7999	0.665511
$799$	−4.40121	−0.155704
$800$	8.37463	0.296088
$801$	8.36672	0.295623
$802$	8.60296	0.303781
$803$	−9.03832	−0.318955
$804$	−1.70283	−0.0600542
$805$	14.9318	0.526278
$806$	0.522505	0.0184044
$807$	2.44927	0.0862184
$808$	−10.4104	−0.366238
$809$	−3.02326	−0.106292	−0.0531461	−	0.998587i	$-0.516925\pi$
$809$	−3.02326	−0.106292	−0.0531461	+	0.998587i	$0.516925\pi$
$810$	3.65713	0.128499
$811$	39.4480	1.38521	0.692604	−	0.721318i	$-0.256463\pi$
$811$	39.4480	1.38521	0.692604	+	0.721318i	$0.256463\pi$
$812$	4.08293	0.143283
$813$	−26.2881	−0.921964
$814$	−9.85638	−0.345466
$815$	−23.2727	−0.815206
$816$	−1.54712	−0.0541601
$817$	−16.1514	−0.565064
$818$	24.7968	0.867000
$819$	−10.5886	−0.369994
$820$	4.52494	0.158018
$821$	−8.48901	−0.296269	−0.148134	−	0.988967i	$-0.547327\pi$
$821$	−8.48901	−0.296269	−0.148134	+	0.988967i	$0.547327\pi$
$822$	3.51512	0.122604
$823$	12.5946	0.439021	0.219511	−	0.975610i	$-0.429554\pi$
$823$	12.5946	0.439021	0.219511	+	0.975610i	$0.429554\pi$
$824$	−0.509795	−0.0177595
$825$	17.6706	0.615210
$826$	−38.0549	−1.32410
$827$	−41.3954	−1.43946	−0.719730	−	0.694254i	$-0.755734\pi$
$827$	−41.3954	−1.43946	−0.719730	+	0.694254i	$0.755734\pi$
$828$	−1.00000	−0.0347524
$829$	−30.4236	−1.05666	−0.528328	−	0.849041i	$-0.677181\pi$
$829$	−30.4236	−1.05666	−0.528328	+	0.849041i	$0.677181\pi$
$830$	−0.554565	−0.0192492
$831$	−7.90653	−0.274275
$832$	2.59337	0.0899089
$833$	14.9612	0.518375
$834$	−9.92542	−0.343689
$835$	39.4039	1.36363
$836$	−9.71559	−0.336021
$837$	−0.201477	−0.00696406
$838$	33.2159	1.14743
$839$	6.69594	0.231169	0.115585	−	0.993298i	$-0.463126\pi$
$839$	6.69594	0.231169	0.115585	+	0.993298i	$0.463126\pi$
$840$	14.9318	0.515197
$841$	1.00000	0.0344828
$842$	6.47029	0.222981
$843$	−23.8268	−0.820637
$844$	−9.82327	−0.338131
$845$	−22.9464	−0.789381
$846$	−2.84477	−0.0978053
$847$	26.7344	0.918606
$848$	7.19876	0.247206
$849$	3.90756	0.134107
$850$	12.9566	0.444407
$851$	−4.67124	−0.160128
$852$	−4.12267	−0.141240
$853$	−39.8004	−1.36274	−0.681369	−	0.731940i	$-0.738615\pi$
$853$	−39.8004	−1.36274	−0.681369	+	0.731940i	$0.738615\pi$
$854$	−16.7809	−0.574231
$855$	16.8393	0.575893
$856$	18.1733	0.621150
$857$	−23.2068	−0.792728	−0.396364	−	0.918093i	$-0.629728\pi$
$857$	−23.2068	−0.792728	−0.396364	+	0.918093i	$0.629728\pi$
$858$	5.47204	0.186812
$859$	−7.55081	−0.257630	−0.128815	−	0.991669i	$-0.541117\pi$
$859$	−7.55081	−0.257630	−0.128815	+	0.991669i	$0.541117\pi$
$860$	−12.8282	−0.437438
$861$	5.05178	0.172164
$862$	−27.2115	−0.926829
$863$	−9.53526	−0.324584	−0.162292	−	0.986743i	$-0.551889\pi$
$863$	−9.53526	−0.324584	−0.162292	+	0.986743i	$0.551889\pi$
$864$	−1.00000	−0.0340207
$865$	33.2079	1.12910
$866$	35.7153	1.21366
$867$	14.6064	0.496060
$868$	−0.822617	−0.0279214
$869$	11.6370	0.394757
$870$	3.65713	0.123988
$871$	4.41607	0.149633
$872$	1.26675	0.0428977
$873$	−1.09022	−0.0368982
$874$	−4.60452	−0.155750
$875$	−50.3894	−1.70347
$876$	−4.28354	−0.144727
$877$	1.13663	0.0383811	0.0191906	−	0.999816i	$-0.493891\pi$
$877$	1.13663	0.0383811	0.0191906	+	0.999816i	$0.493891\pi$
$878$	6.81905	0.230132
$879$	−24.0151	−0.810008
$880$	−7.71660	−0.260126
$881$	−50.6912	−1.70783	−0.853915	−	0.520412i	$-0.825778\pi$
$881$	−50.6912	−1.70783	−0.853915	+	0.520412i	$0.825778\pi$
$882$	9.67035	0.325618
$883$	−50.3572	−1.69466	−0.847328	−	0.531071i	$-0.821790\pi$
$883$	−50.3572	−1.69466	−0.847328	+	0.531071i	$0.821790\pi$
$884$	4.01226	0.134947
$885$	−34.0862	−1.14580
$886$	−10.7646	−0.361644
$887$	56.0607	1.88233	0.941166	−	0.337944i	$-0.109731\pi$
$887$	56.0607	1.88233	0.941166	+	0.337944i	$0.109731\pi$
$888$	−4.67124	−0.156757
$889$	−77.5211	−2.59997
$890$	30.5982	1.02565
$891$	−2.11001	−0.0706881
$892$	22.4585	0.751967
$893$	−13.0988	−0.438335
$894$	19.3702	0.647837
$895$	−66.9014	−2.23627
$896$	−4.08293	−0.136401
$897$	2.59337	0.0865901
$898$	−8.17517	−0.272809
$899$	−0.201477	−0.00671963
$900$	8.37463	0.279154
$901$	11.1374	0.371039
$902$	−2.61070	−0.0869267
$903$	−14.3218	−0.476599
$904$	10.9139	0.362991
$905$	7.56436	0.251448
$906$	2.04680	0.0680005
$907$	−12.3708	−0.410766	−0.205383	−	0.978682i	$-0.565844\pi$
$907$	−12.3708	−0.410766	−0.205383	+	0.978682i	$0.565844\pi$
$908$	25.5228	0.847003
$909$	−10.4104	−0.345293
$910$	−38.7238	−1.28368
$911$	16.1750	0.535902	0.267951	−	0.963433i	$-0.413654\pi$
$911$	16.1750	0.535902	0.267951	+	0.963433i	$0.413654\pi$
$912$	−4.60452	−0.152471
$913$	0.319961	0.0105891
$914$	−6.98937	−0.231188
$915$	−15.0309	−0.496905
$916$	−19.7258	−0.651758
$917$	−38.4199	−1.26874
$918$	−1.54712	−0.0510626
$919$	−25.5016	−0.841219	−0.420610	−	0.907242i	$-0.638184\pi$
$919$	−25.5016	−0.841219	−0.420610	+	0.907242i	$0.638184\pi$
$920$	−3.65713	−0.120572
$921$	11.8161	0.389354
$922$	−34.7564	−1.14464
$923$	10.6916	0.351919
$924$	−8.61504	−0.283414
$925$	39.1199	1.28626
$926$	5.94660	0.195418
$927$	−0.509795	−0.0167439
$928$	−1.00000	−0.0328266
$929$	46.7882	1.53507	0.767535	−	0.641007i	$-0.221483\pi$
$929$	46.7882	1.53507	0.767535	+	0.641007i	$0.221483\pi$
$930$	−0.736828	−0.0241616
$931$	44.5273	1.45932
$932$	−8.67045	−0.284010
$933$	1.39883	0.0457956
$934$	−32.0474	−1.04862
$935$	−11.9385	−0.390431
$936$	2.59337	0.0847670
$937$	−11.5413	−0.377038	−0.188519	−	0.982070i	$-0.560369\pi$
$937$	−11.5413	−0.377038	−0.188519	+	0.982070i	$0.560369\pi$
$938$	−6.95254	−0.227008
$939$	−0.343763	−0.0112183
$940$	−10.4037	−0.339332
$941$	−27.8522	−0.907955	−0.453978	−	0.891013i	$-0.649995\pi$
$941$	−27.8522	−0.907955	−0.453978	+	0.891013i	$0.649995\pi$
$942$	−1.26651	−0.0412650
$943$	−1.23729	−0.0402917
$944$	9.32047	0.303355
$945$	14.9318	0.485733
$946$	7.40133	0.240638
$947$	−40.4832	−1.31553	−0.657764	−	0.753224i	$-0.728498\pi$
$947$	−40.4832	−1.31553	−0.657764	+	0.753224i	$0.728498\pi$
$948$	5.51512	0.179123
$949$	11.1088	0.360607
$950$	38.5611	1.25109
$951$	23.7395	0.769805
$952$	−6.31680	−0.204729
$953$	42.9473	1.39120	0.695600	−	0.718429i	$-0.255138\pi$
$953$	42.9473	1.39120	0.695600	+	0.718429i	$0.255138\pi$
$954$	7.19876	0.233068
$955$	−16.4011	−0.530728
$956$	−15.4492	−0.499663
$957$	−2.11001	−0.0682070
$958$	−27.5533	−0.890207
$959$	14.3520	0.463450
$960$	−3.65713	−0.118033
$961$	−30.9594	−0.998691
$962$	12.1143	0.390580
$963$	18.1733	0.585625
$964$	20.8524	0.671609
$965$	−76.6564	−2.46766
$966$	−4.08293	−0.131366
$967$	2.75370	0.0885532	0.0442766	−	0.999019i	$-0.485902\pi$
$967$	2.75370	0.0885532	0.0442766	+	0.999019i	$0.485902\pi$
$968$	−6.54785	−0.210456
$969$	−7.12375	−0.228848
$970$	−3.98707	−0.128017
$971$	−47.3131	−1.51835	−0.759175	−	0.650886i	$-0.774397\pi$
$971$	−47.3131	−1.51835	−0.759175	+	0.650886i	$0.774397\pi$
$972$	−1.00000	−0.0320750
$973$	−40.5248	−1.29917
$974$	−4.70625	−0.150798
$975$	−21.7185	−0.695549
$976$	4.11001	0.131558
$977$	39.4398	1.26179	0.630896	−	0.775867i	$-0.282687\pi$
$977$	39.4398	1.26179	0.630896	+	0.775867i	$0.282687\pi$
$978$	6.36364	0.203487
$979$	−17.6539	−0.564220
$980$	35.3658	1.12972
$981$	1.26675	0.0404443
$982$	3.36669	0.107435
$983$	−62.2788	−1.98639	−0.993193	−	0.116483i	$-0.962838\pi$
$983$	−62.2788	−1.98639	−0.993193	+	0.116483i	$0.962838\pi$
$984$	−1.23729	−0.0394434
$985$	13.2858	0.423322
$986$	−1.54712	−0.0492704
$987$	−11.6150	−0.369710
$988$	11.9412	0.379901
$989$	3.50772	0.111539
$990$	−7.71660	−0.245250
$991$	18.1289	0.575883	0.287942	−	0.957648i	$-0.407029\pi$
$991$	18.1289	0.575883	0.287942	+	0.957648i	$0.407029\pi$
$992$	0.201477	0.00639690
$993$	17.2835	0.548476
$994$	−16.8326	−0.533898
$995$	−42.0799	−1.33402
$996$	0.151639	0.00480487
$997$	29.4684	0.933274	0.466637	−	0.884449i	$-0.345465\pi$
$997$	29.4684	0.933274	0.466637	+	0.884449i	$0.345465\pi$
$998$	12.4793	0.395025
$999$	−4.67124	−0.147792

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.bk.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.bk.1.7	✓	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.65713 q^{5} -1.00000 q^{6} -4.08293 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.65713 q^{5} -1.00000 q^{6} -4.08293 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.65713 q^{10} -2.11001 q^{11} -1.00000 q^{12} +2.59337 q^{13} -4.08293 q^{14} -3.65713 q^{15} +1.00000 q^{16} +1.54712 q^{17} +1.00000 q^{18} +4.60452 q^{19} +3.65713 q^{20} +4.08293 q^{21} -2.11001 q^{22} -1.00000 q^{23} -1.00000 q^{24} +8.37463 q^{25} +2.59337 q^{26} -1.00000 q^{27} -4.08293 q^{28} -1.00000 q^{29} -3.65713 q^{30} +0.201477 q^{31} +1.00000 q^{32} +2.11001 q^{33} +1.54712 q^{34} -14.9318 q^{35} +1.00000 q^{36} +4.67124 q^{37} +4.60452 q^{38} -2.59337 q^{39} +3.65713 q^{40} +1.23729 q^{41} +4.08293 q^{42} -3.50772 q^{43} -2.11001 q^{44} +3.65713 q^{45} -1.00000 q^{46} -2.84477 q^{47} -1.00000 q^{48} +9.67035 q^{49} +8.37463 q^{50} -1.54712 q^{51} +2.59337 q^{52} +7.19876 q^{53} -1.00000 q^{54} -7.71660 q^{55} -4.08293 q^{56} -4.60452 q^{57} -1.00000 q^{58} +9.32047 q^{59} -3.65713 q^{60} +4.11001 q^{61} +0.201477 q^{62} -4.08293 q^{63} +1.00000 q^{64} +9.48430 q^{65} +2.11001 q^{66} +1.70283 q^{67} +1.54712 q^{68} +1.00000 q^{69} -14.9318 q^{70} +4.12267 q^{71} +1.00000 q^{72} +4.28354 q^{73} +4.67124 q^{74} -8.37463 q^{75} +4.60452 q^{76} +8.61504 q^{77} -2.59337 q^{78} -5.51512 q^{79} +3.65713 q^{80} +1.00000 q^{81} +1.23729 q^{82} -0.151639 q^{83} +4.08293 q^{84} +5.65803 q^{85} -3.50772 q^{86} +1.00000 q^{87} -2.11001 q^{88} +8.36672 q^{89} +3.65713 q^{90} -10.5886 q^{91} -1.00000 q^{92} -0.201477 q^{93} -2.84477 q^{94} +16.8393 q^{95} -1.00000 q^{96} -1.09022 q^{97} +9.67035 q^{98} -2.11001 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9	\( 8 q + 8 q^{2} - 8 q^{3} + 8 q^{4} + 2 q^{5} - 8 q^{6} + 5 q^{7} + 8 q^{8} + 8 q^{9} + 2 q^{10} + 3 q^{11} - 8 q^{12} + 13 q^{13} + 5 q^{14} - 2 q^{15} + 8 q^{16} + 5 q^{17} + 8 q^{18} + 7 q^{19} + 2 q^{20} - 5 q^{21} + 3 q^{22} - 8 q^{23} - 8 q^{24} + 24 q^{25} + 13 q^{26} - 8 q^{27} + 5 q^{28} - 8 q^{29} - 2 q^{30} + 15 q^{31} + 8 q^{32} - 3 q^{33} + 5 q^{34} + 9 q^{35} + 8 q^{36} + 22 q^{37} + 7 q^{38} - 13 q^{39} + 2 q^{40} - 8 q^{41} - 5 q^{42} - q^{43} + 3 q^{44} + 2 q^{45} - 8 q^{46} - 9 q^{47} - 8 q^{48} + 33 q^{49} + 24 q^{50} - 5 q^{51} + 13 q^{52} + 14 q^{53} - 8 q^{54} - 17 q^{55} + 5 q^{56} - 7 q^{57} - 8 q^{58} - 4 q^{59} - 2 q^{60} + 13 q^{61} + 15 q^{62} + 5 q^{63} + 8 q^{64} + 21 q^{65} - 3 q^{66} - 3 q^{67} + 5 q^{68} + 8 q^{69} + 9 q^{70} + 7 q^{71} + 8 q^{72} + 16 q^{73} + 22 q^{74} - 24 q^{75} + 7 q^{76} - 13 q^{78} + 14 q^{79} + 2 q^{80} + 8 q^{81} - 8 q^{82} + 36 q^{83} - 5 q^{84} + 47 q^{85} - q^{86} + 8 q^{87} + 3 q^{88} - 12 q^{89} + 2 q^{90} + 20 q^{91} - 8 q^{92} - 15 q^{93} - 9 q^{94} + 7 q^{95} - 8 q^{96} + 10 q^{97} + 33 q^{98} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^2 - 8 * q^3 + 8 * q^4 + 2 * q^5 - 8 * q^6 + 5 * q^7 + 8 * q^8 + 8 * q^9 + 2 * q^10 + 3 * q^11 - 8 * q^12 + 13 * q^13 + 5 * q^14 - 2 * q^15 + 8 * q^16 + 5 * q^17 + 8 * q^18 + 7 * q^19 + 2 * q^20 - 5 * q^21 + 3 * q^22 - 8 * q^23 - 8 * q^24 + 24 * q^25 + 13 * q^26 - 8 * q^27 + 5 * q^28 - 8 * q^29 - 2 * q^30 + 15 * q^31 + 8 * q^32 - 3 * q^33 + 5 * q^34 + 9 * q^35 + 8 * q^36 + 22 * q^37 + 7 * q^38 - 13 * q^39 + 2 * q^40 - 8 * q^41 - 5 * q^42 - q^43 + 3 * q^44 + 2 * q^45 - 8 * q^46 - 9 * q^47 - 8 * q^48 + 33 * q^49 + 24 * q^50 - 5 * q^51 + 13 * q^52 + 14 * q^53 - 8 * q^54 - 17 * q^55 + 5 * q^56 - 7 * q^57 - 8 * q^58 - 4 * q^59 - 2 * q^60 + 13 * q^61 + 15 * q^62 + 5 * q^63 + 8 * q^64 + 21 * q^65 - 3 * q^66 - 3 * q^67 + 5 * q^68 + 8 * q^69 + 9 * q^70 + 7 * q^71 + 8 * q^72 + 16 * q^73 + 22 * q^74 - 24 * q^75 + 7 * q^76 - 13 * q^78 + 14 * q^79 + 2 * q^80 + 8 * q^81 - 8 * q^82 + 36 * q^83 - 5 * q^84 + 47 * q^85 - q^86 + 8 * q^87 + 3 * q^88 - 12 * q^89 + 2 * q^90 + 20 * q^91 - 8 * q^92 - 15 * q^93 - 9 * q^94 + 7 * q^95 - 8 * q^96 + 10 * q^97 + 33 * q^98 + 3 * q^99

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