LMFDB - Embedded newform 4002.2.a.u.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.56155 q^{5} -1.00000 q^{6} +2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.56155 q^{5} -1.00000 q^{6} +2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.56155 q^{10} +5.12311 q^{11} +1.00000 q^{12} +3.12311 q^{13} -2.56155 q^{14} +2.56155 q^{15} +1.00000 q^{16} -2.56155 q^{17} -1.00000 q^{18} -6.56155 q^{19} +2.56155 q^{20} +2.56155 q^{21} -5.12311 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.56155 q^{25} -3.12311 q^{26} +1.00000 q^{27} +2.56155 q^{28} -1.00000 q^{29} -2.56155 q^{30} -1.00000 q^{32} +5.12311 q^{33} +2.56155 q^{34} +6.56155 q^{35} +1.00000 q^{36} +8.56155 q^{37} +6.56155 q^{38} +3.12311 q^{39} -2.56155 q^{40} +8.56155 q^{41} -2.56155 q^{42} +6.56155 q^{43} +5.12311 q^{44} +2.56155 q^{45} +1.00000 q^{46} -3.68466 q^{47} +1.00000 q^{48} -0.438447 q^{49} -1.56155 q^{50} -2.56155 q^{51} +3.12311 q^{52} -1.12311 q^{53} -1.00000 q^{54} +13.1231 q^{55} -2.56155 q^{56} -6.56155 q^{57} +1.00000 q^{58} +3.68466 q^{59} +2.56155 q^{60} -2.00000 q^{61} +2.56155 q^{63} +1.00000 q^{64} +8.00000 q^{65} -5.12311 q^{66} +7.12311 q^{67} -2.56155 q^{68} -1.00000 q^{69} -6.56155 q^{70} +8.00000 q^{71} -1.00000 q^{72} -12.2462 q^{73} -8.56155 q^{74} +1.56155 q^{75} -6.56155 q^{76} +13.1231 q^{77} -3.12311 q^{78} -4.24621 q^{79} +2.56155 q^{80} +1.00000 q^{81} -8.56155 q^{82} -3.12311 q^{83} +2.56155 q^{84} -6.56155 q^{85} -6.56155 q^{86} -1.00000 q^{87} -5.12311 q^{88} -7.36932 q^{89} -2.56155 q^{90} +8.00000 q^{91} -1.00000 q^{92} +3.68466 q^{94} -16.8078 q^{95} -1.00000 q^{96} -8.00000 q^{97} +0.438447 q^{98} +5.12311 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9} - q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{13} - q^{14} + q^{15} + 2 q^{16} - q^{17} - 2 q^{18} - 9 q^{19} + q^{20} + q^{21} - 2 q^{22} - 2 q^{23} - 2 q^{24} - q^{25} + 2 q^{26} + 2 q^{27} + q^{28} - 2 q^{29} - q^{30} - 2 q^{32} + 2 q^{33} + q^{34} + 9 q^{35} + 2 q^{36} + 13 q^{37} + 9 q^{38} - 2 q^{39} - q^{40} + 13 q^{41} - q^{42} + 9 q^{43} + 2 q^{44} + q^{45} + 2 q^{46} + 5 q^{47} + 2 q^{48} - 5 q^{49} + q^{50} - q^{51} - 2 q^{52} + 6 q^{53} - 2 q^{54} + 18 q^{55} - q^{56} - 9 q^{57} + 2 q^{58} - 5 q^{59} + q^{60} - 4 q^{61} + q^{63} + 2 q^{64} + 16 q^{65} - 2 q^{66} + 6 q^{67} - q^{68} - 2 q^{69} - 9 q^{70} + 16 q^{71} - 2 q^{72} - 8 q^{73} - 13 q^{74} - q^{75} - 9 q^{76} + 18 q^{77} + 2 q^{78} + 8 q^{79} + q^{80} + 2 q^{81} - 13 q^{82} + 2 q^{83} + q^{84} - 9 q^{85} - 9 q^{86} - 2 q^{87} - 2 q^{88} + 10 q^{89} - q^{90} + 16 q^{91} - 2 q^{92} - 5 q^{94} - 13 q^{95} - 2 q^{96} - 16 q^{97} + 5 q^{98} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9 - q^10 + 2 * q^11 + 2 * q^12 - 2 * q^13 - q^14 + q^15 + 2 * q^16 - q^17 - 2 * q^18 - 9 * q^19 + q^20 + q^21 - 2 * q^22 - 2 * q^23 - 2 * q^24 - q^25 + 2 * q^26 + 2 * q^27 + q^28 - 2 * q^29 - q^30 - 2 * q^32 + 2 * q^33 + q^34 + 9 * q^35 + 2 * q^36 + 13 * q^37 + 9 * q^38 - 2 * q^39 - q^40 + 13 * q^41 - q^42 + 9 * q^43 + 2 * q^44 + q^45 + 2 * q^46 + 5 * q^47 + 2 * q^48 - 5 * q^49 + q^50 - q^51 - 2 * q^52 + 6 * q^53 - 2 * q^54 + 18 * q^55 - q^56 - 9 * q^57 + 2 * q^58 - 5 * q^59 + q^60 - 4 * q^61 + q^63 + 2 * q^64 + 16 * q^65 - 2 * q^66 + 6 * q^67 - q^68 - 2 * q^69 - 9 * q^70 + 16 * q^71 - 2 * q^72 - 8 * q^73 - 13 * q^74 - q^75 - 9 * q^76 + 18 * q^77 + 2 * q^78 + 8 * q^79 + q^80 + 2 * q^81 - 13 * q^82 + 2 * q^83 + q^84 - 9 * q^85 - 9 * q^86 - 2 * q^87 - 2 * q^88 + 10 * q^89 - q^90 + 16 * q^91 - 2 * q^92 - 5 * q^94 - 13 * q^95 - 2 * q^96 - 16 * q^97 + 5 * q^98 + 2 * 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
$5$	2.56155	1.14556	0.572781	−	0.819709i	$-0.305865\pi$
$5$	2.56155	1.14556	0.572781	+	0.819709i	$0.305865\pi$
$6$	−1.00000	−0.408248
$7$	2.56155	0.968176	0.484088	−	0.875019i	$-0.339151\pi$
$7$	2.56155	0.968176	0.484088	+	0.875019i	$0.339151\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.56155	−0.810034
$11$	5.12311	1.54467	0.772337	−	0.635213i	$-0.219088\pi$
$11$	5.12311	1.54467	0.772337	+	0.635213i	$0.219088\pi$
$12$	1.00000	0.288675
$13$	3.12311	0.866194	0.433097	−	0.901347i	$-0.357421\pi$
$13$	3.12311	0.866194	0.433097	+	0.901347i	$0.357421\pi$
$14$	−2.56155	−0.684604
$15$	2.56155	0.661390
$16$	1.00000	0.250000
$17$	−2.56155	−0.621268	−0.310634	−	0.950530i	$-0.600541\pi$
$17$	−2.56155	−0.621268	−0.310634	+	0.950530i	$0.600541\pi$
$18$	−1.00000	−0.235702
$19$	−6.56155	−1.50532	−0.752662	−	0.658407i	$-0.771230\pi$
$19$	−6.56155	−1.50532	−0.752662	+	0.658407i	$0.771230\pi$
$20$	2.56155	0.572781
$21$	2.56155	0.558977
$22$	−5.12311	−1.09225
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	1.56155	0.312311
$26$	−3.12311	−0.612491
$27$	1.00000	0.192450
$28$	2.56155	0.484088
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−2.56155	−0.467673
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	5.12311	0.891818
$34$	2.56155	0.439303
$35$	6.56155	1.10910
$36$	1.00000	0.166667
$37$	8.56155	1.40751	0.703755	−	0.710442i	$-0.251505\pi$
$37$	8.56155	1.40751	0.703755	+	0.710442i	$0.251505\pi$
$38$	6.56155	1.06442
$39$	3.12311	0.500097
$40$	−2.56155	−0.405017
$41$	8.56155	1.33709	0.668545	−	0.743672i	$-0.266917\pi$
$41$	8.56155	1.33709	0.668545	+	0.743672i	$0.266917\pi$
$42$	−2.56155	−0.395256
$43$	6.56155	1.00063	0.500314	−	0.865844i	$-0.333218\pi$
$43$	6.56155	1.00063	0.500314	+	0.865844i	$0.333218\pi$
$44$	5.12311	0.772337
$45$	2.56155	0.381854
$46$	1.00000	0.147442
$47$	−3.68466	−0.537463	−0.268731	−	0.963215i	$-0.586604\pi$
$47$	−3.68466	−0.537463	−0.268731	+	0.963215i	$0.586604\pi$
$48$	1.00000	0.144338
$49$	−0.438447	−0.0626353
$50$	−1.56155	−0.220837
$51$	−2.56155	−0.358689
$52$	3.12311	0.433097
$53$	−1.12311	−0.154270	−0.0771352	−	0.997021i	$-0.524577\pi$
$53$	−1.12311	−0.154270	−0.0771352	+	0.997021i	$0.524577\pi$
$54$	−1.00000	−0.136083
$55$	13.1231	1.76952
$56$	−2.56155	−0.342302
$57$	−6.56155	−0.869099
$58$	1.00000	0.131306
$59$	3.68466	0.479702	0.239851	−	0.970810i	$-0.422901\pi$
$59$	3.68466	0.479702	0.239851	+	0.970810i	$0.422901\pi$
$60$	2.56155	0.330695
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	2.56155	0.322725
$64$	1.00000	0.125000
$65$	8.00000	0.992278
$66$	−5.12311	−0.630611
$67$	7.12311	0.870226	0.435113	−	0.900376i	$-0.356708\pi$
$67$	7.12311	0.870226	0.435113	+	0.900376i	$0.356708\pi$
$68$	−2.56155	−0.310634
$69$	−1.00000	−0.120386
$70$	−6.56155	−0.784256
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	−1.00000	−0.117851
$73$	−12.2462	−1.43331	−0.716655	−	0.697428i	$-0.754328\pi$
$73$	−12.2462	−1.43331	−0.716655	+	0.697428i	$0.754328\pi$
$74$	−8.56155	−0.995260
$75$	1.56155	0.180313
$76$	−6.56155	−0.752662
$77$	13.1231	1.49552
$78$	−3.12311	−0.353622
$79$	−4.24621	−0.477736	−0.238868	−	0.971052i	$-0.576776\pi$
$79$	−4.24621	−0.477736	−0.238868	+	0.971052i	$0.576776\pi$
$80$	2.56155	0.286390
$81$	1.00000	0.111111
$82$	−8.56155	−0.945465
$83$	−3.12311	−0.342805	−0.171403	−	0.985201i	$-0.554830\pi$
$83$	−3.12311	−0.342805	−0.171403	+	0.985201i	$0.554830\pi$
$84$	2.56155	0.279488
$85$	−6.56155	−0.711700
$86$	−6.56155	−0.707550
$87$	−1.00000	−0.107211
$88$	−5.12311	−0.546125
$89$	−7.36932	−0.781146	−0.390573	−	0.920572i	$-0.627723\pi$
$89$	−7.36932	−0.781146	−0.390573	+	0.920572i	$0.627723\pi$
$90$	−2.56155	−0.270011
$91$	8.00000	0.838628
$92$	−1.00000	−0.104257
$93$	0	0
$94$	3.68466	0.380043
$95$	−16.8078	−1.72444
$96$	−1.00000	−0.102062
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0.438447	0.0442899
$99$	5.12311	0.514891
$100$	1.56155	0.156155
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	2.56155	0.253632
$103$	−10.5616	−1.04066	−0.520330	−	0.853965i	$-0.674191\pi$
$103$	−10.5616	−1.04066	−0.520330	+	0.853965i	$0.674191\pi$
$104$	−3.12311	−0.306246
$105$	6.56155	0.640342
$106$	1.12311	0.109086
$107$	14.8078	1.43152	0.715760	−	0.698346i	$-0.246080\pi$
$107$	14.8078	1.43152	0.715760	+	0.698346i	$0.246080\pi$
$108$	1.00000	0.0962250
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	−13.1231	−1.25124
$111$	8.56155	0.812627
$112$	2.56155	0.242044
$113$	−5.43845	−0.511606	−0.255803	−	0.966729i	$-0.582340\pi$
$113$	−5.43845	−0.511606	−0.255803	+	0.966729i	$0.582340\pi$
$114$	6.56155	0.614546
$115$	−2.56155	−0.238866
$116$	−1.00000	−0.0928477
$117$	3.12311	0.288731
$118$	−3.68466	−0.339200
$119$	−6.56155	−0.601497
$120$	−2.56155	−0.233837
$121$	15.2462	1.38602
$122$	2.00000	0.181071
$123$	8.56155	0.771969
$124$	0	0
$125$	−8.80776	−0.787790
$126$	−2.56155	−0.228201
$127$	−9.12311	−0.809545	−0.404772	−	0.914417i	$-0.632649\pi$
$127$	−9.12311	−0.809545	−0.404772	+	0.914417i	$0.632649\pi$
$128$	−1.00000	−0.0883883
$129$	6.56155	0.577713
$130$	−8.00000	−0.701646
$131$	1.75379	0.153229	0.0766146	−	0.997061i	$-0.475589\pi$
$131$	1.75379	0.153229	0.0766146	+	0.997061i	$0.475589\pi$
$132$	5.12311	0.445909
$133$	−16.8078	−1.45742
$134$	−7.12311	−0.615343
$135$	2.56155	0.220463
$136$	2.56155	0.219651
$137$	−1.12311	−0.0959534	−0.0479767	−	0.998848i	$-0.515277\pi$
$137$	−1.12311	−0.0959534	−0.0479767	+	0.998848i	$0.515277\pi$
$138$	1.00000	0.0851257
$139$	−5.12311	−0.434536	−0.217268	−	0.976112i	$-0.569715\pi$
$139$	−5.12311	−0.434536	−0.217268	+	0.976112i	$0.569715\pi$
$140$	6.56155	0.554552
$141$	−3.68466	−0.310304
$142$	−8.00000	−0.671345
$143$	16.0000	1.33799
$144$	1.00000	0.0833333
$145$	−2.56155	−0.212725
$146$	12.2462	1.01350
$147$	−0.438447	−0.0361625
$148$	8.56155	0.703755
$149$	19.6847	1.61263	0.806315	−	0.591486i	$-0.201459\pi$
$149$	19.6847	1.61263	0.806315	+	0.591486i	$0.201459\pi$
$150$	−1.56155	−0.127500
$151$	−3.68466	−0.299853	−0.149927	−	0.988697i	$-0.547904\pi$
$151$	−3.68466	−0.299853	−0.149927	+	0.988697i	$0.547904\pi$
$152$	6.56155	0.532212
$153$	−2.56155	−0.207089
$154$	−13.1231	−1.05749
$155$	0	0
$156$	3.12311	0.250049
$157$	10.8078	0.862553	0.431277	−	0.902220i	$-0.358063\pi$
$157$	10.8078	0.862553	0.431277	+	0.902220i	$0.358063\pi$
$158$	4.24621	0.337810
$159$	−1.12311	−0.0890681
$160$	−2.56155	−0.202509
$161$	−2.56155	−0.201879
$162$	−1.00000	−0.0785674
$163$	−7.68466	−0.601909	−0.300954	−	0.953639i	$-0.597305\pi$
$163$	−7.68466	−0.601909	−0.300954	+	0.953639i	$0.597305\pi$
$164$	8.56155	0.668545
$165$	13.1231	1.02163
$166$	3.12311	0.242400
$167$	15.3693	1.18931	0.594657	−	0.803980i	$-0.297288\pi$
$167$	15.3693	1.18931	0.594657	+	0.803980i	$0.297288\pi$
$168$	−2.56155	−0.197628
$169$	−3.24621	−0.249709
$170$	6.56155	0.503248
$171$	−6.56155	−0.501774
$172$	6.56155	0.500314
$173$	−1.68466	−0.128082	−0.0640411	−	0.997947i	$-0.520399\pi$
$173$	−1.68466	−0.128082	−0.0640411	+	0.997947i	$0.520399\pi$
$174$	1.00000	0.0758098
$175$	4.00000	0.302372
$176$	5.12311	0.386169
$177$	3.68466	0.276956
$178$	7.36932	0.552354
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	2.56155	0.190927
$181$	−12.0000	−0.891953	−0.445976	−	0.895045i	$-0.647144\pi$
$181$	−12.0000	−0.891953	−0.445976	+	0.895045i	$0.647144\pi$
$182$	−8.00000	−0.592999
$183$	−2.00000	−0.147844
$184$	1.00000	0.0737210
$185$	21.9309	1.61239
$186$	0	0
$187$	−13.1231	−0.959657
$188$	−3.68466	−0.268731
$189$	2.56155	0.186326
$190$	16.8078	1.21936
$191$	6.31534	0.456962	0.228481	−	0.973548i	$-0.426624\pi$
$191$	6.31534	0.456962	0.228481	+	0.973548i	$0.426624\pi$
$192$	1.00000	0.0721688
$193$	8.87689	0.638973	0.319486	−	0.947591i	$-0.396490\pi$
$193$	8.87689	0.638973	0.319486	+	0.947591i	$0.396490\pi$
$194$	8.00000	0.574367
$195$	8.00000	0.572892
$196$	−0.438447	−0.0313177
$197$	−10.3153	−0.734938	−0.367469	−	0.930036i	$-0.619776\pi$
$197$	−10.3153	−0.734938	−0.367469	+	0.930036i	$0.619776\pi$
$198$	−5.12311	−0.364083
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	−1.56155	−0.110418
$201$	7.12311	0.502425
$202$	−2.00000	−0.140720
$203$	−2.56155	−0.179786
$204$	−2.56155	−0.179345
$205$	21.9309	1.53172
$206$	10.5616	0.735858
$207$	−1.00000	−0.0695048
$208$	3.12311	0.216548
$209$	−33.6155	−2.32523
$210$	−6.56155	−0.452790
$211$	9.93087	0.683669	0.341835	−	0.939760i	$-0.388952\pi$
$211$	9.93087	0.683669	0.341835	+	0.939760i	$0.388952\pi$
$212$	−1.12311	−0.0771352
$213$	8.00000	0.548151
$214$	−14.8078	−1.01224
$215$	16.8078	1.14628
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	0	0
$219$	−12.2462	−0.827522
$220$	13.1231	0.884760
$221$	−8.00000	−0.538138
$222$	−8.56155	−0.574614
$223$	−28.4924	−1.90799	−0.953997	−	0.299817i	$-0.903075\pi$
$223$	−28.4924	−1.90799	−0.953997	+	0.299817i	$0.903075\pi$
$224$	−2.56155	−0.171151
$225$	1.56155	0.104104
$226$	5.43845	0.361760
$227$	−7.93087	−0.526390	−0.263195	−	0.964743i	$-0.584776\pi$
$227$	−7.93087	−0.526390	−0.263195	+	0.964743i	$0.584776\pi$
$228$	−6.56155	−0.434549
$229$	−14.8078	−0.978525	−0.489262	−	0.872137i	$-0.662734\pi$
$229$	−14.8078	−0.978525	−0.489262	+	0.872137i	$0.662734\pi$
$230$	2.56155	0.168904
$231$	13.1231	0.863437
$232$	1.00000	0.0656532
$233$	−25.3693	−1.66200	−0.831000	−	0.556273i	$-0.812231\pi$
$233$	−25.3693	−1.66200	−0.831000	+	0.556273i	$0.812231\pi$
$234$	−3.12311	−0.204164
$235$	−9.43845	−0.615696
$236$	3.68466	0.239851
$237$	−4.24621	−0.275821
$238$	6.56155	0.425322
$239$	2.87689	0.186091	0.0930454	−	0.995662i	$-0.470340\pi$
$239$	2.87689	0.186091	0.0930454	+	0.995662i	$0.470340\pi$
$240$	2.56155	0.165348
$241$	13.6847	0.881506	0.440753	−	0.897628i	$-0.354711\pi$
$241$	13.6847	0.881506	0.440753	+	0.897628i	$0.354711\pi$
$242$	−15.2462	−0.980064
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	−1.12311	−0.0717526
$246$	−8.56155	−0.545865
$247$	−20.4924	−1.30390
$248$	0	0
$249$	−3.12311	−0.197919
$250$	8.80776	0.557052
$251$	−14.2462	−0.899213	−0.449606	−	0.893227i	$-0.648436\pi$
$251$	−14.2462	−0.899213	−0.449606	+	0.893227i	$0.648436\pi$
$252$	2.56155	0.161363
$253$	−5.12311	−0.322087
$254$	9.12311	0.572435
$255$	−6.56155	−0.410900
$256$	1.00000	0.0625000
$257$	21.3693	1.33298	0.666491	−	0.745513i	$-0.267796\pi$
$257$	21.3693	1.33298	0.666491	+	0.745513i	$0.267796\pi$
$258$	−6.56155	−0.408504
$259$	21.9309	1.36272
$260$	8.00000	0.496139
$261$	−1.00000	−0.0618984
$262$	−1.75379	−0.108349
$263$	−18.8078	−1.15974	−0.579868	−	0.814710i	$-0.696896\pi$
$263$	−18.8078	−1.15974	−0.579868	+	0.814710i	$0.696896\pi$
$264$	−5.12311	−0.315305
$265$	−2.87689	−0.176726
$266$	16.8078	1.03055
$267$	−7.36932	−0.450995
$268$	7.12311	0.435113
$269$	−7.75379	−0.472757	−0.236378	−	0.971661i	$-0.575960\pi$
$269$	−7.75379	−0.472757	−0.236378	+	0.971661i	$0.575960\pi$
$270$	−2.56155	−0.155891
$271$	−16.4924	−1.00184	−0.500922	−	0.865493i	$-0.667006\pi$
$271$	−16.4924	−1.00184	−0.500922	+	0.865493i	$0.667006\pi$
$272$	−2.56155	−0.155317
$273$	8.00000	0.484182
$274$	1.12311	0.0678493
$275$	8.00000	0.482418
$276$	−1.00000	−0.0601929
$277$	26.0000	1.56219	0.781094	−	0.624413i	$-0.214662\pi$
$277$	26.0000	1.56219	0.781094	+	0.624413i	$0.214662\pi$
$278$	5.12311	0.307263
$279$	0	0
$280$	−6.56155	−0.392128
$281$	30.4924	1.81903	0.909513	−	0.415676i	$-0.136455\pi$
$281$	30.4924	1.81903	0.909513	+	0.415676i	$0.136455\pi$
$282$	3.68466	0.219418
$283$	14.4924	0.861485	0.430743	−	0.902475i	$-0.358252\pi$
$283$	14.4924	0.861485	0.430743	+	0.902475i	$0.358252\pi$
$284$	8.00000	0.474713
$285$	−16.8078	−0.995606
$286$	−16.0000	−0.946100
$287$	21.9309	1.29454
$288$	−1.00000	−0.0589256
$289$	−10.4384	−0.614026
$290$	2.56155	0.150420
$291$	−8.00000	−0.468968
$292$	−12.2462	−0.716655
$293$	−2.49242	−0.145609	−0.0728044	−	0.997346i	$-0.523195\pi$
$293$	−2.49242	−0.145609	−0.0728044	+	0.997346i	$0.523195\pi$
$294$	0.438447	0.0255708
$295$	9.43845	0.549528
$296$	−8.56155	−0.497630
$297$	5.12311	0.297273
$298$	−19.6847	−1.14030
$299$	−3.12311	−0.180614
$300$	1.56155	0.0901563
$301$	16.8078	0.968783
$302$	3.68466	0.212028
$303$	2.00000	0.114897
$304$	−6.56155	−0.376331
$305$	−5.12311	−0.293348
$306$	2.56155	0.146434
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	13.1231	0.747758
$309$	−10.5616	−0.600826
$310$	0	0
$311$	15.0540	0.853633	0.426816	−	0.904338i	$-0.359635\pi$
$311$	15.0540	0.853633	0.426816	+	0.904338i	$0.359635\pi$
$312$	−3.12311	−0.176811
$313$	6.31534	0.356964	0.178482	−	0.983943i	$-0.442881\pi$
$313$	6.31534	0.356964	0.178482	+	0.983943i	$0.442881\pi$
$314$	−10.8078	−0.609917
$315$	6.56155	0.369702
$316$	−4.24621	−0.238868
$317$	8.87689	0.498576	0.249288	−	0.968429i	$-0.419803\pi$
$317$	8.87689	0.498576	0.249288	+	0.968429i	$0.419803\pi$
$318$	1.12311	0.0629806
$319$	−5.12311	−0.286839
$320$	2.56155	0.143195
$321$	14.8078	0.826489
$322$	2.56155	0.142750
$323$	16.8078	0.935209
$324$	1.00000	0.0555556
$325$	4.87689	0.270521
$326$	7.68466	0.425614
$327$	0	0
$328$	−8.56155	−0.472733
$329$	−9.43845	−0.520358
$330$	−13.1231	−0.722403
$331$	−21.4384	−1.17836	−0.589182	−	0.808000i	$-0.700550\pi$
$331$	−21.4384	−1.17836	−0.589182	+	0.808000i	$0.700550\pi$
$332$	−3.12311	−0.171403
$333$	8.56155	0.469170
$334$	−15.3693	−0.840972
$335$	18.2462	0.996897
$336$	2.56155	0.139744
$337$	−6.87689	−0.374608	−0.187304	−	0.982302i	$-0.559975\pi$
$337$	−6.87689	−0.374608	−0.187304	+	0.982302i	$0.559975\pi$
$338$	3.24621	0.176571
$339$	−5.43845	−0.295376
$340$	−6.56155	−0.355850
$341$	0	0
$342$	6.56155	0.354808
$343$	−19.0540	−1.02882
$344$	−6.56155	−0.353775
$345$	−2.56155	−0.137909
$346$	1.68466	0.0905678
$347$	34.5616	1.85536	0.927681	−	0.373375i	$-0.121799\pi$
$347$	34.5616	1.85536	0.927681	+	0.373375i	$0.121799\pi$
$348$	−1.00000	−0.0536056
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	−4.00000	−0.213809
$351$	3.12311	0.166699
$352$	−5.12311	−0.273062
$353$	28.7386	1.52960	0.764802	−	0.644266i	$-0.222837\pi$
$353$	28.7386	1.52960	0.764802	+	0.644266i	$0.222837\pi$
$354$	−3.68466	−0.195837
$355$	20.4924	1.08762
$356$	−7.36932	−0.390573
$357$	−6.56155	−0.347274
$358$	0	0
$359$	35.9309	1.89636	0.948179	−	0.317736i	$-0.102922\pi$
$359$	35.9309	1.89636	0.948179	+	0.317736i	$0.102922\pi$
$360$	−2.56155	−0.135006
$361$	24.0540	1.26600
$362$	12.0000	0.630706
$363$	15.2462	0.800219
$364$	8.00000	0.419314
$365$	−31.3693	−1.64195
$366$	2.00000	0.104542
$367$	−14.4924	−0.756498	−0.378249	−	0.925704i	$-0.623474\pi$
$367$	−14.4924	−0.756498	−0.378249	+	0.925704i	$0.623474\pi$
$368$	−1.00000	−0.0521286
$369$	8.56155	0.445697
$370$	−21.9309	−1.14013
$371$	−2.87689	−0.149361
$372$	0	0
$373$	18.2462	0.944753	0.472377	−	0.881397i	$-0.343396\pi$
$373$	18.2462	0.944753	0.472377	+	0.881397i	$0.343396\pi$
$374$	13.1231	0.678580
$375$	−8.80776	−0.454831
$376$	3.68466	0.190022
$377$	−3.12311	−0.160848
$378$	−2.56155	−0.131752
$379$	18.2462	0.937245	0.468622	−	0.883399i	$-0.344750\pi$
$379$	18.2462	0.937245	0.468622	+	0.883399i	$0.344750\pi$
$380$	−16.8078	−0.862220
$381$	−9.12311	−0.467391
$382$	−6.31534	−0.323121
$383$	−35.3693	−1.80729	−0.903644	−	0.428285i	$-0.859118\pi$
$383$	−35.3693	−1.80729	−0.903644	+	0.428285i	$0.859118\pi$
$384$	−1.00000	−0.0510310
$385$	33.6155	1.71321
$386$	−8.87689	−0.451822
$387$	6.56155	0.333542
$388$	−8.00000	−0.406138
$389$	5.36932	0.272235	0.136118	−	0.990693i	$-0.456538\pi$
$389$	5.36932	0.272235	0.136118	+	0.990693i	$0.456538\pi$
$390$	−8.00000	−0.405096
$391$	2.56155	0.129543
$392$	0.438447	0.0221449
$393$	1.75379	0.0884669
$394$	10.3153	0.519679
$395$	−10.8769	−0.547276
$396$	5.12311	0.257446
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	16.0000	0.802008
$399$	−16.8078	−0.841441
$400$	1.56155	0.0780776
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	−7.12311	−0.355268
$403$	0	0
$404$	2.00000	0.0995037
$405$	2.56155	0.127285
$406$	2.56155	0.127128
$407$	43.8617	2.17415
$408$	2.56155	0.126816
$409$	−7.12311	−0.352215	−0.176107	−	0.984371i	$-0.556351\pi$
$409$	−7.12311	−0.352215	−0.176107	+	0.984371i	$0.556351\pi$
$410$	−21.9309	−1.08309
$411$	−1.12311	−0.0553987
$412$	−10.5616	−0.520330
$413$	9.43845	0.464436
$414$	1.00000	0.0491473
$415$	−8.00000	−0.392705
$416$	−3.12311	−0.153123
$417$	−5.12311	−0.250880
$418$	33.6155	1.64419
$419$	15.9309	0.778274	0.389137	−	0.921180i	$-0.372773\pi$
$419$	15.9309	0.778274	0.389137	+	0.921180i	$0.372773\pi$
$420$	6.56155	0.320171
$421$	−28.7386	−1.40064	−0.700318	−	0.713831i	$-0.746958\pi$
$421$	−28.7386	−1.40064	−0.700318	+	0.713831i	$0.746958\pi$
$422$	−9.93087	−0.483427
$423$	−3.68466	−0.179154
$424$	1.12311	0.0545428
$425$	−4.00000	−0.194029
$426$	−8.00000	−0.387601
$427$	−5.12311	−0.247924
$428$	14.8078	0.715760
$429$	16.0000	0.772487
$430$	−16.8078	−0.810542
$431$	1.75379	0.0844770	0.0422385	−	0.999108i	$-0.486551\pi$
$431$	1.75379	0.0844770	0.0422385	+	0.999108i	$0.486551\pi$
$432$	1.00000	0.0481125
$433$	−3.36932	−0.161919	−0.0809595	−	0.996717i	$-0.525798\pi$
$433$	−3.36932	−0.161919	−0.0809595	+	0.996717i	$0.525798\pi$
$434$	0	0
$435$	−2.56155	−0.122817
$436$	0	0
$437$	6.56155	0.313882
$438$	12.2462	0.585147
$439$	3.68466	0.175859	0.0879296	−	0.996127i	$-0.471975\pi$
$439$	3.68466	0.175859	0.0879296	+	0.996127i	$0.471975\pi$
$440$	−13.1231	−0.625620
$441$	−0.438447	−0.0208784
$442$	8.00000	0.380521
$443$	0.492423	0.0233957	0.0116978	−	0.999932i	$-0.496276\pi$
$443$	0.492423	0.0233957	0.0116978	+	0.999932i	$0.496276\pi$
$444$	8.56155	0.406313
$445$	−18.8769	−0.894851
$446$	28.4924	1.34916
$447$	19.6847	0.931052
$448$	2.56155	0.121022
$449$	−32.4233	−1.53015	−0.765075	−	0.643941i	$-0.777298\pi$
$449$	−32.4233	−1.53015	−0.765075	+	0.643941i	$0.777298\pi$
$450$	−1.56155	−0.0736123
$451$	43.8617	2.06537
$452$	−5.43845	−0.255803
$453$	−3.68466	−0.173120
$454$	7.93087	0.372214
$455$	20.4924	0.960700
$456$	6.56155	0.307273
$457$	−28.4233	−1.32959	−0.664793	−	0.747028i	$-0.731480\pi$
$457$	−28.4233	−1.32959	−0.664793	+	0.747028i	$0.731480\pi$
$458$	14.8078	0.691921
$459$	−2.56155	−0.119563
$460$	−2.56155	−0.119433
$461$	−9.36932	−0.436373	−0.218186	−	0.975907i	$-0.570014\pi$
$461$	−9.36932	−0.436373	−0.218186	+	0.975907i	$0.570014\pi$
$462$	−13.1231	−0.610542
$463$	−4.49242	−0.208781	−0.104390	−	0.994536i	$-0.533289\pi$
$463$	−4.49242	−0.208781	−0.104390	+	0.994536i	$0.533289\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	25.3693	1.17521
$467$	−36.4924	−1.68867	−0.844334	−	0.535817i	$-0.820004\pi$
$467$	−36.4924	−1.68867	−0.844334	+	0.535817i	$0.820004\pi$
$468$	3.12311	0.144366
$469$	18.2462	0.842532
$470$	9.43845	0.435363
$471$	10.8078	0.497995
$472$	−3.68466	−0.169600
$473$	33.6155	1.54564
$474$	4.24621	0.195035
$475$	−10.2462	−0.470128
$476$	−6.56155	−0.300748
$477$	−1.12311	−0.0514235
$478$	−2.87689	−0.131586
$479$	−15.6155	−0.713492	−0.356746	−	0.934201i	$-0.616114\pi$
$479$	−15.6155	−0.713492	−0.356746	+	0.934201i	$0.616114\pi$
$480$	−2.56155	−0.116918
$481$	26.7386	1.21918
$482$	−13.6847	−0.623319
$483$	−2.56155	−0.116555
$484$	15.2462	0.693010
$485$	−20.4924	−0.930513
$486$	−1.00000	−0.0453609
$487$	35.6847	1.61703	0.808513	−	0.588478i	$-0.200273\pi$
$487$	35.6847	1.61703	0.808513	+	0.588478i	$0.200273\pi$
$488$	2.00000	0.0905357
$489$	−7.68466	−0.347512
$490$	1.12311	0.0507367
$491$	29.6155	1.33653	0.668265	−	0.743923i	$-0.267037\pi$
$491$	29.6155	1.33653	0.668265	+	0.743923i	$0.267037\pi$
$492$	8.56155	0.385985
$493$	2.56155	0.115367
$494$	20.4924	0.921998
$495$	13.1231	0.589840
$496$	0	0
$497$	20.4924	0.919211
$498$	3.12311	0.139950
$499$	13.6155	0.609515	0.304757	−	0.952430i	$-0.401425\pi$
$499$	13.6155	0.609515	0.304757	+	0.952430i	$0.401425\pi$
$500$	−8.80776	−0.393895
$501$	15.3693	0.686650
$502$	14.2462	0.635840
$503$	40.5616	1.80855	0.904275	−	0.426950i	$-0.140412\pi$
$503$	40.5616	1.80855	0.904275	+	0.426950i	$0.140412\pi$
$504$	−2.56155	−0.114101
$505$	5.12311	0.227975
$506$	5.12311	0.227750
$507$	−3.24621	−0.144169
$508$	−9.12311	−0.404772
$509$	34.8078	1.54283	0.771414	−	0.636334i	$-0.219550\pi$
$509$	34.8078	1.54283	0.771414	+	0.636334i	$0.219550\pi$
$510$	6.56155	0.290550
$511$	−31.3693	−1.38770
$512$	−1.00000	−0.0441942
$513$	−6.56155	−0.289700
$514$	−21.3693	−0.942560
$515$	−27.0540	−1.19214
$516$	6.56155	0.288856
$517$	−18.8769	−0.830205
$518$	−21.9309	−0.963587
$519$	−1.68466	−0.0739483
$520$	−8.00000	−0.350823
$521$	−20.2462	−0.887003	−0.443501	−	0.896274i	$-0.646264\pi$
$521$	−20.2462	−0.887003	−0.443501	+	0.896274i	$0.646264\pi$
$522$	1.00000	0.0437688
$523$	−1.50758	−0.0659218	−0.0329609	−	0.999457i	$-0.510494\pi$
$523$	−1.50758	−0.0659218	−0.0329609	+	0.999457i	$0.510494\pi$
$524$	1.75379	0.0766146
$525$	4.00000	0.174574
$526$	18.8078	0.820057
$527$	0	0
$528$	5.12311	0.222955
$529$	1.00000	0.0434783
$530$	2.87689	0.124964
$531$	3.68466	0.159901
$532$	−16.8078	−0.728709
$533$	26.7386	1.15818
$534$	7.36932	0.318902
$535$	37.9309	1.63989
$536$	−7.12311	−0.307671
$537$	0	0
$538$	7.75379	0.334290
$539$	−2.24621	−0.0967512
$540$	2.56155	0.110232
$541$	−31.9309	−1.37282	−0.686408	−	0.727217i	$-0.740813\pi$
$541$	−31.9309	−1.37282	−0.686408	+	0.727217i	$0.740813\pi$
$542$	16.4924	0.708410
$543$	−12.0000	−0.514969
$544$	2.56155	0.109826
$545$	0	0
$546$	−8.00000	−0.342368
$547$	14.8769	0.636090	0.318045	−	0.948076i	$-0.396974\pi$
$547$	14.8769	0.636090	0.318045	+	0.948076i	$0.396974\pi$
$548$	−1.12311	−0.0479767
$549$	−2.00000	−0.0853579
$550$	−8.00000	−0.341121
$551$	6.56155	0.279532
$552$	1.00000	0.0425628
$553$	−10.8769	−0.462533
$554$	−26.0000	−1.10463
$555$	21.9309	0.930914
$556$	−5.12311	−0.217268
$557$	12.3153	0.521818	0.260909	−	0.965363i	$-0.415978\pi$
$557$	12.3153	0.521818	0.260909	+	0.965363i	$0.415978\pi$
$558$	0	0
$559$	20.4924	0.866737
$560$	6.56155	0.277276
$561$	−13.1231	−0.554058
$562$	−30.4924	−1.28625
$563$	35.3693	1.49064	0.745319	−	0.666707i	$-0.232297\pi$
$563$	35.3693	1.49064	0.745319	+	0.666707i	$0.232297\pi$
$564$	−3.68466	−0.155152
$565$	−13.9309	−0.586076
$566$	−14.4924	−0.609162
$567$	2.56155	0.107575
$568$	−8.00000	−0.335673
$569$	−37.9309	−1.59014	−0.795072	−	0.606515i	$-0.792567\pi$
$569$	−37.9309	−1.59014	−0.795072	+	0.606515i	$0.792567\pi$
$570$	16.8078	0.704000
$571$	9.50758	0.397880	0.198940	−	0.980012i	$-0.436250\pi$
$571$	9.50758	0.397880	0.198940	+	0.980012i	$0.436250\pi$
$572$	16.0000	0.668994
$573$	6.31534	0.263827
$574$	−21.9309	−0.915377
$575$	−1.56155	−0.0651213
$576$	1.00000	0.0416667
$577$	−35.6155	−1.48269	−0.741347	−	0.671122i	$-0.765813\pi$
$577$	−35.6155	−1.48269	−0.741347	+	0.671122i	$0.765813\pi$
$578$	10.4384	0.434182
$579$	8.87689	0.368911
$580$	−2.56155	−0.106363
$581$	−8.00000	−0.331896
$582$	8.00000	0.331611
$583$	−5.75379	−0.238298
$584$	12.2462	0.506752
$585$	8.00000	0.330759
$586$	2.49242	0.102961
$587$	10.5616	0.435922	0.217961	−	0.975957i	$-0.430059\pi$
$587$	10.5616	0.435922	0.217961	+	0.975957i	$0.430059\pi$
$588$	−0.438447	−0.0180813
$589$	0	0
$590$	−9.43845	−0.388575
$591$	−10.3153	−0.424316
$592$	8.56155	0.351878
$593$	−11.1231	−0.456771	−0.228386	−	0.973571i	$-0.573345\pi$
$593$	−11.1231	−0.456771	−0.228386	+	0.973571i	$0.573345\pi$
$594$	−5.12311	−0.210204
$595$	−16.8078	−0.689051
$596$	19.6847	0.806315
$597$	−16.0000	−0.654836
$598$	3.12311	0.127713
$599$	−28.9848	−1.18429	−0.592144	−	0.805832i	$-0.701718\pi$
$599$	−28.9848	−1.18429	−0.592144	+	0.805832i	$0.701718\pi$
$600$	−1.56155	−0.0637501
$601$	−17.3693	−0.708509	−0.354255	−	0.935149i	$-0.615265\pi$
$601$	−17.3693	−0.708509	−0.354255	+	0.935149i	$0.615265\pi$
$602$	−16.8078	−0.685033
$603$	7.12311	0.290075
$604$	−3.68466	−0.149927
$605$	39.0540	1.58777
$606$	−2.00000	−0.0812444
$607$	−12.0000	−0.487065	−0.243532	−	0.969893i	$-0.578306\pi$
$607$	−12.0000	−0.487065	−0.243532	+	0.969893i	$0.578306\pi$
$608$	6.56155	0.266106
$609$	−2.56155	−0.103799
$610$	5.12311	0.207428
$611$	−11.5076	−0.465547
$612$	−2.56155	−0.103545
$613$	−1.61553	−0.0652506	−0.0326253	−	0.999468i	$-0.510387\pi$
$613$	−1.61553	−0.0652506	−0.0326253	+	0.999468i	$0.510387\pi$
$614$	−4.00000	−0.161427
$615$	21.9309	0.884338
$616$	−13.1231	−0.528745
$617$	33.3002	1.34062	0.670308	−	0.742083i	$-0.266162\pi$
$617$	33.3002	1.34062	0.670308	+	0.742083i	$0.266162\pi$
$618$	10.5616	0.424848
$619$	0.315342	0.0126746	0.00633732	−	0.999980i	$-0.497983\pi$
$619$	0.315342	0.0126746	0.00633732	+	0.999980i	$0.497983\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	−15.0540	−0.603609
$623$	−18.8769	−0.756287
$624$	3.12311	0.125024
$625$	−30.3693	−1.21477
$626$	−6.31534	−0.252412
$627$	−33.6155	−1.34247
$628$	10.8078	0.431277
$629$	−21.9309	−0.874441
$630$	−6.56155	−0.261419
$631$	−2.56155	−0.101974	−0.0509869	−	0.998699i	$-0.516237\pi$
$631$	−2.56155	−0.101974	−0.0509869	+	0.998699i	$0.516237\pi$
$632$	4.24621	0.168905
$633$	9.93087	0.394717
$634$	−8.87689	−0.352547
$635$	−23.3693	−0.927383
$636$	−1.12311	−0.0445340
$637$	−1.36932	−0.0542543
$638$	5.12311	0.202826
$639$	8.00000	0.316475
$640$	−2.56155	−0.101254
$641$	−44.6695	−1.76434	−0.882170	−	0.470932i	$-0.843918\pi$
$641$	−44.6695	−1.76434	−0.882170	+	0.470932i	$0.843918\pi$
$642$	−14.8078	−0.584416
$643$	16.8769	0.665560	0.332780	−	0.943005i	$-0.392013\pi$
$643$	16.8769	0.665560	0.332780	+	0.943005i	$0.392013\pi$
$644$	−2.56155	−0.100939
$645$	16.8078	0.661805
$646$	−16.8078	−0.661293
$647$	23.3693	0.918743	0.459371	−	0.888244i	$-0.348075\pi$
$647$	23.3693	0.918743	0.459371	+	0.888244i	$0.348075\pi$
$648$	−1.00000	−0.0392837
$649$	18.8769	0.740983
$650$	−4.87689	−0.191288
$651$	0	0
$652$	−7.68466	−0.300954
$653$	40.2462	1.57496	0.787478	−	0.616343i	$-0.211386\pi$
$653$	40.2462	1.57496	0.787478	+	0.616343i	$0.211386\pi$
$654$	0	0
$655$	4.49242	0.175533
$656$	8.56155	0.334272
$657$	−12.2462	−0.477770
$658$	9.43845	0.367949
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	13.1231	0.510816
$661$	−20.4924	−0.797063	−0.398531	−	0.917155i	$-0.630480\pi$
$661$	−20.4924	−0.797063	−0.398531	+	0.917155i	$0.630480\pi$
$662$	21.4384	0.833229
$663$	−8.00000	−0.310694
$664$	3.12311	0.121200
$665$	−43.0540	−1.66956
$666$	−8.56155	−0.331753
$667$	1.00000	0.0387202
$668$	15.3693	0.594657
$669$	−28.4924	−1.10158
$670$	−18.2462	−0.704913
$671$	−10.2462	−0.395551
$672$	−2.56155	−0.0988140
$673$	−9.19224	−0.354335	−0.177167	−	0.984181i	$-0.556693\pi$
$673$	−9.19224	−0.354335	−0.177167	+	0.984181i	$0.556693\pi$
$674$	6.87689	0.264888
$675$	1.56155	0.0601042
$676$	−3.24621	−0.124854
$677$	−16.2462	−0.624393	−0.312196	−	0.950018i	$-0.601065\pi$
$677$	−16.2462	−0.624393	−0.312196	+	0.950018i	$0.601065\pi$
$678$	5.43845	0.208862
$679$	−20.4924	−0.786427
$680$	6.56155	0.251624
$681$	−7.93087	−0.303912
$682$	0	0
$683$	36.1771	1.38428	0.692139	−	0.721764i	$-0.256669\pi$
$683$	36.1771	1.38428	0.692139	+	0.721764i	$0.256669\pi$
$684$	−6.56155	−0.250887
$685$	−2.87689	−0.109920
$686$	19.0540	0.727484
$687$	−14.8078	−0.564951
$688$	6.56155	0.250157
$689$	−3.50758	−0.133628
$690$	2.56155	0.0975166
$691$	28.4924	1.08390	0.541951	−	0.840410i	$-0.317686\pi$
$691$	28.4924	1.08390	0.541951	+	0.840410i	$0.317686\pi$
$692$	−1.68466	−0.0640411
$693$	13.1231	0.498506
$694$	−34.5616	−1.31194
$695$	−13.1231	−0.497788
$696$	1.00000	0.0379049
$697$	−21.9309	−0.830691
$698$	14.0000	0.529908
$699$	−25.3693	−0.959556
$700$	4.00000	0.151186
$701$	35.5464	1.34257	0.671284	−	0.741200i	$-0.265743\pi$
$701$	35.5464	1.34257	0.671284	+	0.741200i	$0.265743\pi$
$702$	−3.12311	−0.117874
$703$	−56.1771	−2.11876
$704$	5.12311	0.193084
$705$	−9.43845	−0.355472
$706$	−28.7386	−1.08159
$707$	5.12311	0.192674
$708$	3.68466	0.138478
$709$	1.61553	0.0606724	0.0303362	−	0.999540i	$-0.490342\pi$
$709$	1.61553	0.0606724	0.0303362	+	0.999540i	$0.490342\pi$
$710$	−20.4924	−0.769067
$711$	−4.24621	−0.159245
$712$	7.36932	0.276177
$713$	0	0
$714$	6.56155	0.245560
$715$	40.9848	1.53275
$716$	0	0
$717$	2.87689	0.107440
$718$	−35.9309	−1.34093
$719$	1.26137	0.0470410	0.0235205	−	0.999723i	$-0.492512\pi$
$719$	1.26137	0.0470410	0.0235205	+	0.999723i	$0.492512\pi$
$720$	2.56155	0.0954634
$721$	−27.0540	−1.00754
$722$	−24.0540	−0.895196
$723$	13.6847	0.508938
$724$	−12.0000	−0.445976
$725$	−1.56155	−0.0579946
$726$	−15.2462	−0.565840
$727$	−20.7386	−0.769153	−0.384577	−	0.923093i	$-0.625653\pi$
$727$	−20.7386	−0.769153	−0.384577	+	0.923093i	$0.625653\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	31.3693	1.16103
$731$	−16.8078	−0.621658
$732$	−2.00000	−0.0739221
$733$	−36.2462	−1.33878	−0.669392	−	0.742909i	$-0.733445\pi$
$733$	−36.2462	−1.33878	−0.669392	+	0.742909i	$0.733445\pi$
$734$	14.4924	0.534925
$735$	−1.12311	−0.0414264
$736$	1.00000	0.0368605
$737$	36.4924	1.34422
$738$	−8.56155	−0.315155
$739$	−9.75379	−0.358799	−0.179399	−	0.983776i	$-0.557415\pi$
$739$	−9.75379	−0.358799	−0.179399	+	0.983776i	$0.557415\pi$
$740$	21.9309	0.806195
$741$	−20.4924	−0.752808
$742$	2.87689	0.105614
$743$	−15.9309	−0.584447	−0.292223	−	0.956350i	$-0.594395\pi$
$743$	−15.9309	−0.584447	−0.292223	+	0.956350i	$0.594395\pi$
$744$	0	0
$745$	50.4233	1.84737
$746$	−18.2462	−0.668041
$747$	−3.12311	−0.114268
$748$	−13.1231	−0.479828
$749$	37.9309	1.38596
$750$	8.80776	0.321614
$751$	36.2462	1.32264	0.661322	−	0.750103i	$-0.269996\pi$
$751$	36.2462	1.32264	0.661322	+	0.750103i	$0.269996\pi$
$752$	−3.68466	−0.134366
$753$	−14.2462	−0.519161
$754$	3.12311	0.113737
$755$	−9.43845	−0.343500
$756$	2.56155	0.0931628
$757$	5.68466	0.206612	0.103306	−	0.994650i	$-0.467058\pi$
$757$	5.68466	0.206612	0.103306	+	0.994650i	$0.467058\pi$
$758$	−18.2462	−0.662732
$759$	−5.12311	−0.185957
$760$	16.8078	0.609682
$761$	−49.3693	−1.78964	−0.894818	−	0.446431i	$-0.852695\pi$
$761$	−49.3693	−1.78964	−0.894818	+	0.446431i	$0.852695\pi$
$762$	9.12311	0.330495
$763$	0	0
$764$	6.31534	0.228481
$765$	−6.56155	−0.237233
$766$	35.3693	1.27795
$767$	11.5076	0.415515
$768$	1.00000	0.0360844
$769$	45.6155	1.64494	0.822469	−	0.568810i	$-0.192596\pi$
$769$	45.6155	1.64494	0.822469	+	0.568810i	$0.192596\pi$
$770$	−33.6155	−1.21142
$771$	21.3693	0.769597
$772$	8.87689	0.319486
$773$	−32.2462	−1.15982	−0.579908	−	0.814682i	$-0.696911\pi$
$773$	−32.2462	−1.15982	−0.579908	+	0.814682i	$0.696911\pi$
$774$	−6.56155	−0.235850
$775$	0	0
$776$	8.00000	0.287183
$777$	21.9309	0.786766
$778$	−5.36932	−0.192499
$779$	−56.1771	−2.01275
$780$	8.00000	0.286446
$781$	40.9848	1.46655
$782$	−2.56155	−0.0916009
$783$	−1.00000	−0.0357371
$784$	−0.438447	−0.0156588
$785$	27.6847	0.988108
$786$	−1.75379	−0.0625556
$787$	−23.1231	−0.824250	−0.412125	−	0.911127i	$-0.635213\pi$
$787$	−23.1231	−0.824250	−0.412125	+	0.911127i	$0.635213\pi$
$788$	−10.3153	−0.367469
$789$	−18.8078	−0.669574
$790$	10.8769	0.386983
$791$	−13.9309	−0.495325
$792$	−5.12311	−0.182042
$793$	−6.24621	−0.221809
$794$	2.00000	0.0709773
$795$	−2.87689	−0.102033
$796$	−16.0000	−0.567105
$797$	−44.1080	−1.56238	−0.781192	−	0.624291i	$-0.785388\pi$
$797$	−44.1080	−1.56238	−0.781192	+	0.624291i	$0.785388\pi$
$798$	16.8078	0.594988
$799$	9.43845	0.333908
$800$	−1.56155	−0.0552092
$801$	−7.36932	−0.260382
$802$	10.0000	0.353112
$803$	−62.7386	−2.21400
$804$	7.12311	0.251213
$805$	−6.56155	−0.231264
$806$	0	0
$807$	−7.75379	−0.272946
$808$	−2.00000	−0.0703598
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	−2.56155	−0.0900038
$811$	−36.0000	−1.26413	−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000	−1.26413	−0.632065	+	0.774915i	$0.717793\pi$
$812$	−2.56155	−0.0898929
$813$	−16.4924	−0.578415
$814$	−43.8617	−1.53735
$815$	−19.6847	−0.689524
$816$	−2.56155	−0.0896723
$817$	−43.0540	−1.50627
$818$	7.12311	0.249054
$819$	8.00000	0.279543
$820$	21.9309	0.765859
$821$	−5.50758	−0.192216	−0.0961079	−	0.995371i	$-0.530639\pi$
$821$	−5.50758	−0.192216	−0.0961079	+	0.995371i	$0.530639\pi$
$822$	1.12311	0.0391728
$823$	−22.8769	−0.797438	−0.398719	−	0.917073i	$-0.630545\pi$
$823$	−22.8769	−0.797438	−0.398719	+	0.917073i	$0.630545\pi$
$824$	10.5616	0.367929
$825$	8.00000	0.278524
$826$	−9.43845	−0.328406
$827$	26.2462	0.912670	0.456335	−	0.889808i	$-0.349162\pi$
$827$	26.2462	0.912670	0.456335	+	0.889808i	$0.349162\pi$
$828$	−1.00000	−0.0347524
$829$	2.80776	0.0975177	0.0487588	−	0.998811i	$-0.484473\pi$
$829$	2.80776	0.0975177	0.0487588	+	0.998811i	$0.484473\pi$
$830$	8.00000	0.277684
$831$	26.0000	0.901930
$832$	3.12311	0.108274
$833$	1.12311	0.0389133
$834$	5.12311	0.177399
$835$	39.3693	1.36243
$836$	−33.6155	−1.16262
$837$	0	0
$838$	−15.9309	−0.550323
$839$	35.4384	1.22347	0.611736	−	0.791062i	$-0.290472\pi$
$839$	35.4384	1.22347	0.611736	+	0.791062i	$0.290472\pi$
$840$	−6.56155	−0.226395
$841$	1.00000	0.0344828
$842$	28.7386	0.990399
$843$	30.4924	1.05021
$844$	9.93087	0.341835
$845$	−8.31534	−0.286056
$846$	3.68466	0.126681
$847$	39.0540	1.34191
$848$	−1.12311	−0.0385676
$849$	14.4924	0.497379
$850$	4.00000	0.137199
$851$	−8.56155	−0.293486
$852$	8.00000	0.274075
$853$	−12.0691	−0.413239	−0.206620	−	0.978421i	$-0.566246\pi$
$853$	−12.0691	−0.413239	−0.206620	+	0.978421i	$0.566246\pi$
$854$	5.12311	0.175309
$855$	−16.8078	−0.574813
$856$	−14.8078	−0.506119
$857$	−0.738634	−0.0252312	−0.0126156	−	0.999920i	$-0.504016\pi$
$857$	−0.738634	−0.0252312	−0.0126156	+	0.999920i	$0.504016\pi$
$858$	−16.0000	−0.546231
$859$	45.7926	1.56242	0.781212	−	0.624266i	$-0.214602\pi$
$859$	45.7926	1.56242	0.781212	+	0.624266i	$0.214602\pi$
$860$	16.8078	0.573140
$861$	21.9309	0.747402
$862$	−1.75379	−0.0597343
$863$	−48.0000	−1.63394	−0.816970	−	0.576681i	$-0.804348\pi$
$863$	−48.0000	−1.63394	−0.816970	+	0.576681i	$0.804348\pi$
$864$	−1.00000	−0.0340207
$865$	−4.31534	−0.146726
$866$	3.36932	0.114494
$867$	−10.4384	−0.354508
$868$	0	0
$869$	−21.7538	−0.737947
$870$	2.56155	0.0868448
$871$	22.2462	0.753784
$872$	0	0
$873$	−8.00000	−0.270759
$874$	−6.56155	−0.221948
$875$	−22.5616	−0.762720
$876$	−12.2462	−0.413761
$877$	−21.3693	−0.721591	−0.360795	−	0.932645i	$-0.617495\pi$
$877$	−21.3693	−0.721591	−0.360795	+	0.932645i	$0.617495\pi$
$878$	−3.68466	−0.124351
$879$	−2.49242	−0.0840673
$880$	13.1231	0.442380
$881$	−29.1231	−0.981182	−0.490591	−	0.871390i	$-0.663219\pi$
$881$	−29.1231	−0.981182	−0.490591	+	0.871390i	$0.663219\pi$
$882$	0.438447	0.0147633
$883$	38.7386	1.30366	0.651829	−	0.758366i	$-0.274002\pi$
$883$	38.7386	1.30366	0.651829	+	0.758366i	$0.274002\pi$
$884$	−8.00000	−0.269069
$885$	9.43845	0.317270
$886$	−0.492423	−0.0165433
$887$	2.73863	0.0919543	0.0459772	−	0.998942i	$-0.485360\pi$
$887$	2.73863	0.0919543	0.0459772	+	0.998942i	$0.485360\pi$
$888$	−8.56155	−0.287307
$889$	−23.3693	−0.783782
$890$	18.8769	0.632755
$891$	5.12311	0.171630
$892$	−28.4924	−0.953997
$893$	24.1771	0.809055
$894$	−19.6847	−0.658353
$895$	0	0
$896$	−2.56155	−0.0855755
$897$	−3.12311	−0.104277
$898$	32.4233	1.08198
$899$	0	0
$900$	1.56155	0.0520518
$901$	2.87689	0.0958432
$902$	−43.8617	−1.46044
$903$	16.8078	0.559327
$904$	5.43845	0.180880
$905$	−30.7386	−1.02179
$906$	3.68466	0.122415
$907$	2.24621	0.0745842	0.0372921	−	0.999304i	$-0.488127\pi$
$907$	2.24621	0.0745842	0.0372921	+	0.999304i	$0.488127\pi$
$908$	−7.93087	−0.263195
$909$	2.00000	0.0663358
$910$	−20.4924	−0.679317
$911$	−35.1231	−1.16368	−0.581840	−	0.813303i	$-0.697667\pi$
$911$	−35.1231	−1.16368	−0.581840	+	0.813303i	$0.697667\pi$
$912$	−6.56155	−0.217275
$913$	−16.0000	−0.529523
$914$	28.4233	0.940159
$915$	−5.12311	−0.169365
$916$	−14.8078	−0.489262
$917$	4.49242	0.148353
$918$	2.56155	0.0845438
$919$	35.5464	1.17257	0.586284	−	0.810106i	$-0.300590\pi$
$919$	35.5464	1.17257	0.586284	+	0.810106i	$0.300590\pi$
$920$	2.56155	0.0844519
$921$	4.00000	0.131804
$922$	9.36932	0.308562
$923$	24.9848	0.822386
$924$	13.1231	0.431718
$925$	13.3693	0.439580
$926$	4.49242	0.147630
$927$	−10.5616	−0.346887
$928$	1.00000	0.0328266
$929$	6.49242	0.213009	0.106505	−	0.994312i	$-0.466034\pi$
$929$	6.49242	0.213009	0.106505	+	0.994312i	$0.466034\pi$
$930$	0	0
$931$	2.87689	0.0942864
$932$	−25.3693	−0.831000
$933$	15.0540	0.492845
$934$	36.4924	1.19407
$935$	−33.6155	−1.09935
$936$	−3.12311	−0.102082
$937$	−3.93087	−0.128416	−0.0642080	−	0.997937i	$-0.520452\pi$
$937$	−3.93087	−0.128416	−0.0642080	+	0.997937i	$0.520452\pi$
$938$	−18.2462	−0.595760
$939$	6.31534	0.206093
$940$	−9.43845	−0.307848
$941$	−43.3693	−1.41380	−0.706900	−	0.707314i	$-0.749907\pi$
$941$	−43.3693	−1.41380	−0.706900	+	0.707314i	$0.749907\pi$
$942$	−10.8078	−0.352136
$943$	−8.56155	−0.278803
$944$	3.68466	0.119925
$945$	6.56155	0.213447
$946$	−33.6155	−1.09294
$947$	55.2311	1.79477	0.897384	−	0.441250i	$-0.145465\pi$
$947$	55.2311	1.79477	0.897384	+	0.441250i	$0.145465\pi$
$948$	−4.24621	−0.137911
$949$	−38.2462	−1.24152
$950$	10.2462	0.332431
$951$	8.87689	0.287853
$952$	6.56155	0.212661
$953$	8.87689	0.287551	0.143775	−	0.989610i	$-0.454076\pi$
$953$	8.87689	0.287551	0.143775	+	0.989610i	$0.454076\pi$
$954$	1.12311	0.0363619
$955$	16.1771	0.523478
$956$	2.87689	0.0930454
$957$	−5.12311	−0.165606
$958$	15.6155	0.504515
$959$	−2.87689	−0.0928998
$960$	2.56155	0.0826738
$961$	−31.0000	−1.00000
$962$	−26.7386	−0.862088
$963$	14.8078	0.477174
$964$	13.6847	0.440753
$965$	22.7386	0.731983
$966$	2.56155	0.0824166
$967$	1.75379	0.0563980	0.0281990	−	0.999602i	$-0.491023\pi$
$967$	1.75379	0.0563980	0.0281990	+	0.999602i	$0.491023\pi$
$968$	−15.2462	−0.490032
$969$	16.8078	0.539943
$970$	20.4924	0.657972
$971$	9.12311	0.292774	0.146387	−	0.989227i	$-0.453235\pi$
$971$	9.12311	0.292774	0.146387	+	0.989227i	$0.453235\pi$
$972$	1.00000	0.0320750
$973$	−13.1231	−0.420707
$974$	−35.6847	−1.14341
$975$	4.87689	0.156186
$976$	−2.00000	−0.0640184
$977$	1.36932	0.0438083	0.0219042	−	0.999760i	$-0.493027\pi$
$977$	1.36932	0.0438083	0.0219042	+	0.999760i	$0.493027\pi$
$978$	7.68466	0.245728
$979$	−37.7538	−1.20662
$980$	−1.12311	−0.0358763
$981$	0	0
$982$	−29.6155	−0.945069
$983$	−4.87689	−0.155549	−0.0777744	−	0.996971i	$-0.524781\pi$
$983$	−4.87689	−0.155549	−0.0777744	+	0.996971i	$0.524781\pi$
$984$	−8.56155	−0.272932
$985$	−26.4233	−0.841916
$986$	−2.56155	−0.0815765
$987$	−9.43845	−0.300429
$988$	−20.4924	−0.651951
$989$	−6.56155	−0.208645
$990$	−13.1231	−0.417080
$991$	−16.8078	−0.533916	−0.266958	−	0.963708i	$-0.586019\pi$
$991$	−16.8078	−0.533916	−0.266958	+	0.963708i	$0.586019\pi$
$992$	0	0
$993$	−21.4384	−0.680329
$994$	−20.4924	−0.649980
$995$	−40.9848	−1.29931
$996$	−3.12311	−0.0989594
$997$	54.6695	1.73140	0.865700	−	0.500563i	$-0.166874\pi$
$997$	54.6695	1.73140	0.865700	+	0.500563i	$0.166874\pi$
$998$	−13.6155	−0.430992
$999$	8.56155	0.270876

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4002.2.a.u.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4002.2.a.u.1.2	✓	2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.56155 q^{5} -1.00000 q^{6} +2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.56155 q^{5} -1.00000 q^{6} +2.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.56155 q^{10} +5.12311 q^{11} +1.00000 q^{12} +3.12311 q^{13} -2.56155 q^{14} +2.56155 q^{15} +1.00000 q^{16} -2.56155 q^{17} -1.00000 q^{18} -6.56155 q^{19} +2.56155 q^{20} +2.56155 q^{21} -5.12311 q^{22} -1.00000 q^{23} -1.00000 q^{24} +1.56155 q^{25} -3.12311 q^{26} +1.00000 q^{27} +2.56155 q^{28} -1.00000 q^{29} -2.56155 q^{30} -1.00000 q^{32} +5.12311 q^{33} +2.56155 q^{34} +6.56155 q^{35} +1.00000 q^{36} +8.56155 q^{37} +6.56155 q^{38} +3.12311 q^{39} -2.56155 q^{40} +8.56155 q^{41} -2.56155 q^{42} +6.56155 q^{43} +5.12311 q^{44} +2.56155 q^{45} +1.00000 q^{46} -3.68466 q^{47} +1.00000 q^{48} -0.438447 q^{49} -1.56155 q^{50} -2.56155 q^{51} +3.12311 q^{52} -1.12311 q^{53} -1.00000 q^{54} +13.1231 q^{55} -2.56155 q^{56} -6.56155 q^{57} +1.00000 q^{58} +3.68466 q^{59} +2.56155 q^{60} -2.00000 q^{61} +2.56155 q^{63} +1.00000 q^{64} +8.00000 q^{65} -5.12311 q^{66} +7.12311 q^{67} -2.56155 q^{68} -1.00000 q^{69} -6.56155 q^{70} +8.00000 q^{71} -1.00000 q^{72} -12.2462 q^{73} -8.56155 q^{74} +1.56155 q^{75} -6.56155 q^{76} +13.1231 q^{77} -3.12311 q^{78} -4.24621 q^{79} +2.56155 q^{80} +1.00000 q^{81} -8.56155 q^{82} -3.12311 q^{83} +2.56155 q^{84} -6.56155 q^{85} -6.56155 q^{86} -1.00000 q^{87} -5.12311 q^{88} -7.36932 q^{89} -2.56155 q^{90} +8.00000 q^{91} -1.00000 q^{92} +3.68466 q^{94} -16.8078 q^{95} -1.00000 q^{96} -8.00000 q^{97} +0.438447 q^{98} +5.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + q^{5} - 2 q^{6} + q^{7} - 2 q^{8} + 2 q^{9} - q^{10} + 2 q^{11} + 2 q^{12} - 2 q^{13} - q^{14} + q^{15} + 2 q^{16} - q^{17} - 2 q^{18} - 9 q^{19} + q^{20} + q^{21} - 2 q^{22} - 2 q^{23} - 2 q^{24} - q^{25} + 2 q^{26} + 2 q^{27} + q^{28} - 2 q^{29} - q^{30} - 2 q^{32} + 2 q^{33} + q^{34} + 9 q^{35} + 2 q^{36} + 13 q^{37} + 9 q^{38} - 2 q^{39} - q^{40} + 13 q^{41} - q^{42} + 9 q^{43} + 2 q^{44} + q^{45} + 2 q^{46} + 5 q^{47} + 2 q^{48} - 5 q^{49} + q^{50} - q^{51} - 2 q^{52} + 6 q^{53} - 2 q^{54} + 18 q^{55} - q^{56} - 9 q^{57} + 2 q^{58} - 5 q^{59} + q^{60} - 4 q^{61} + q^{63} + 2 q^{64} + 16 q^{65} - 2 q^{66} + 6 q^{67} - q^{68} - 2 q^{69} - 9 q^{70} + 16 q^{71} - 2 q^{72} - 8 q^{73} - 13 q^{74} - q^{75} - 9 q^{76} + 18 q^{77} + 2 q^{78} + 8 q^{79} + q^{80} + 2 q^{81} - 13 q^{82} + 2 q^{83} + q^{84} - 9 q^{85} - 9 q^{86} - 2 q^{87} - 2 q^{88} + 10 q^{89} - q^{90} + 16 q^{91} - 2 q^{92} - 5 q^{94} - 13 q^{95} - 2 q^{96} - 16 q^{97} + 5 q^{98} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + q^5 - 2 * q^6 + q^7 - 2 * q^8 + 2 * q^9 - q^10 + 2 * q^11 + 2 * q^12 - 2 * q^13 - q^14 + q^15 + 2 * q^16 - q^17 - 2 * q^18 - 9 * q^19 + q^20 + q^21 - 2 * q^22 - 2 * q^23 - 2 * q^24 - q^25 + 2 * q^26 + 2 * q^27 + q^28 - 2 * q^29 - q^30 - 2 * q^32 + 2 * q^33 + q^34 + 9 * q^35 + 2 * q^36 + 13 * q^37 + 9 * q^38 - 2 * q^39 - q^40 + 13 * q^41 - q^42 + 9 * q^43 + 2 * q^44 + q^45 + 2 * q^46 + 5 * q^47 + 2 * q^48 - 5 * q^49 + q^50 - q^51 - 2 * q^52 + 6 * q^53 - 2 * q^54 + 18 * q^55 - q^56 - 9 * q^57 + 2 * q^58 - 5 * q^59 + q^60 - 4 * q^61 + q^63 + 2 * q^64 + 16 * q^65 - 2 * q^66 + 6 * q^67 - q^68 - 2 * q^69 - 9 * q^70 + 16 * q^71 - 2 * q^72 - 8 * q^73 - 13 * q^74 - q^75 - 9 * q^76 + 18 * q^77 + 2 * q^78 + 8 * q^79 + q^80 + 2 * q^81 - 13 * q^82 + 2 * q^83 + q^84 - 9 * q^85 - 9 * q^86 - 2 * q^87 - 2 * q^88 + 10 * q^89 - q^90 + 16 * q^91 - 2 * q^92 - 5 * q^94 - 13 * q^95 - 2 * q^96 - 16 * q^97 + 5 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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