LMFDB - Embedded newform 8619.2.a.bw.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8619,2,Mod(1,8619)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8619, 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("8619.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8619 = 3 \cdot 13^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8619.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:	$68.8230615021$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8619.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.324966 q^{2} -1.00000 q^{3} -1.89440 q^{4} +0.225941 q^{5} -0.324966 q^{6} +3.77265 q^{7} -1.26555 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.324966 q^{2} -1.00000 q^{3} -1.89440 q^{4} +0.225941 q^{5} -0.324966 q^{6} +3.77265 q^{7} -1.26555 q^{8} +1.00000 q^{9} +0.0734231 q^{10} +1.08979 q^{11} +1.89440 q^{12} +1.22598 q^{14} -0.225941 q^{15} +3.37753 q^{16} -1.00000 q^{17} +0.324966 q^{18} +0.928865 q^{19} -0.428022 q^{20} -3.77265 q^{21} +0.354144 q^{22} -5.82354 q^{23} +1.26555 q^{24} -4.94895 q^{25} -1.00000 q^{27} -7.14690 q^{28} +4.43398 q^{29} -0.0734231 q^{30} +8.67716 q^{31} +3.62868 q^{32} -1.08979 q^{33} -0.324966 q^{34} +0.852396 q^{35} -1.89440 q^{36} -6.85589 q^{37} +0.301849 q^{38} -0.285939 q^{40} -6.09033 q^{41} -1.22598 q^{42} -10.3317 q^{43} -2.06449 q^{44} +0.225941 q^{45} -1.89245 q^{46} +4.77366 q^{47} -3.37753 q^{48} +7.23290 q^{49} -1.60824 q^{50} +1.00000 q^{51} -10.0842 q^{53} -0.324966 q^{54} +0.246228 q^{55} -4.77446 q^{56} -0.928865 q^{57} +1.44089 q^{58} -6.75390 q^{59} +0.428022 q^{60} -3.28356 q^{61} +2.81978 q^{62} +3.77265 q^{63} -5.57587 q^{64} -0.354144 q^{66} -1.30265 q^{67} +1.89440 q^{68} +5.82354 q^{69} +0.277000 q^{70} +10.9601 q^{71} -1.26555 q^{72} -5.12081 q^{73} -2.22793 q^{74} +4.94895 q^{75} -1.75964 q^{76} +4.11139 q^{77} -9.16176 q^{79} +0.763123 q^{80} +1.00000 q^{81} -1.97915 q^{82} -1.65443 q^{83} +7.14690 q^{84} -0.225941 q^{85} -3.35744 q^{86} -4.43398 q^{87} -1.37918 q^{88} +14.3740 q^{89} +0.0734231 q^{90} +11.0321 q^{92} -8.67716 q^{93} +1.55128 q^{94} +0.209869 q^{95} -3.62868 q^{96} +18.6019 q^{97} +2.35045 q^{98} +1.08979 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 7 q^{2} - 24 q^{3} + 17 q^{4} - 13 q^{5} - 7 q^{6} - 12 q^{7} + 21 q^{8} + 24 q^{9}+O(q^{10}) $ 24 * q + 7 * q^2 - 24 * q^3 + 17 * q^4 - 13 * q^5 - 7 * q^6 - 12 * q^7 + 21 * q^8 + 24 * q^9	\( 24 q + 7 q^{2} - 24 q^{3} + 17 q^{4} - 13 q^{5} - 7 q^{6} - 12 q^{7} + 21 q^{8} + 24 q^{9} - 4 q^{10} - q^{11} - 17 q^{12} - 18 q^{14} + 13 q^{15} + 35 q^{16} - 24 q^{17} + 7 q^{18} - 16 q^{19} - 18 q^{20} + 12 q^{21} + 8 q^{22} + 3 q^{23} - 21 q^{24} + 23 q^{25} - 24 q^{27} - 30 q^{28} + 5 q^{29} + 4 q^{30} - 46 q^{31} + 44 q^{32} + q^{33} - 7 q^{34} + 11 q^{35} + 17 q^{36} - 38 q^{37} - q^{38} - 59 q^{40} - 31 q^{41} + 18 q^{42} + 3 q^{43} + 3 q^{44} - 13 q^{45} - 36 q^{46} + 49 q^{47} - 35 q^{48} + 30 q^{49} + 15 q^{50} + 24 q^{51} - 11 q^{53} - 7 q^{54} - 10 q^{55} - 38 q^{56} + 16 q^{57} - 53 q^{58} - 23 q^{59} + 18 q^{60} - 2 q^{61} - 26 q^{62} - 12 q^{63} + 57 q^{64} - 8 q^{66} - 17 q^{68} - 3 q^{69} + 10 q^{70} + 29 q^{71} + 21 q^{72} - 48 q^{73} - 66 q^{74} - 23 q^{75} - 38 q^{76} + 51 q^{77} - 26 q^{79} - 52 q^{80} + 24 q^{81} - 45 q^{82} + 8 q^{83} + 30 q^{84} + 13 q^{85} - 38 q^{86} - 5 q^{87} - 34 q^{88} - 42 q^{89} - 4 q^{90} - 10 q^{92} + 46 q^{93} + 107 q^{94} - 44 q^{96} - 60 q^{97} + 19 q^{98} - q^{99}+O(q^{100}) \) 24 * q + 7 * q^2 - 24 * q^3 + 17 * q^4 - 13 * q^5 - 7 * q^6 - 12 * q^7 + 21 * q^8 + 24 * q^9 - 4 * q^10 - q^11 - 17 * q^12 - 18 * q^14 + 13 * q^15 + 35 * q^16 - 24 * q^17 + 7 * q^18 - 16 * q^19 - 18 * q^20 + 12 * q^21 + 8 * q^22 + 3 * q^23 - 21 * q^24 + 23 * q^25 - 24 * q^27 - 30 * q^28 + 5 * q^29 + 4 * q^30 - 46 * q^31 + 44 * q^32 + q^33 - 7 * q^34 + 11 * q^35 + 17 * q^36 - 38 * q^37 - q^38 - 59 * q^40 - 31 * q^41 + 18 * q^42 + 3 * q^43 + 3 * q^44 - 13 * q^45 - 36 * q^46 + 49 * q^47 - 35 * q^48 + 30 * q^49 + 15 * q^50 + 24 * q^51 - 11 * q^53 - 7 * q^54 - 10 * q^55 - 38 * q^56 + 16 * q^57 - 53 * q^58 - 23 * q^59 + 18 * q^60 - 2 * q^61 - 26 * q^62 - 12 * q^63 + 57 * q^64 - 8 * q^66 - 17 * q^68 - 3 * q^69 + 10 * q^70 + 29 * q^71 + 21 * q^72 - 48 * q^73 - 66 * q^74 - 23 * q^75 - 38 * q^76 + 51 * q^77 - 26 * q^79 - 52 * q^80 + 24 * q^81 - 45 * q^82 + 8 * q^83 + 30 * q^84 + 13 * q^85 - 38 * q^86 - 5 * q^87 - 34 * q^88 - 42 * q^89 - 4 * q^90 - 10 * q^92 + 46 * q^93 + 107 * q^94 - 44 * q^96 - 60 * q^97 + 19 * q^98 - 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 $

$2$	0.324966	0.229786	0.114893	−	0.993378i	$-0.463348\pi$
$2$	0.324966	0.229786	0.114893	+	0.993378i	$0.463348\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.89440	−0.947199
$5$	0.225941	0.101044	0.0505219	−	0.998723i	$-0.483912\pi$
$5$	0.225941	0.101044	0.0505219	+	0.998723i	$0.483912\pi$
$6$	−0.324966	−0.132667
$7$	3.77265	1.42593	0.712964	−	0.701201i	$-0.247352\pi$
$7$	3.77265	1.42593	0.712964	+	0.701201i	$0.247352\pi$
$8$	−1.26555	−0.447438
$9$	1.00000	0.333333
$10$	0.0734231	0.0232184
$11$	1.08979	0.328583	0.164292	−	0.986412i	$-0.447466\pi$
$11$	1.08979	0.328583	0.164292	+	0.986412i	$0.447466\pi$
$12$	1.89440	0.546865
$13$	0	0
$13$	0	0
$14$	1.22598	0.327658
$15$	−0.225941	−0.0583377
$16$	3.37753	0.844384
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.324966	0.0765952
$19$	0.928865	0.213096	0.106548	−	0.994308i	$-0.466020\pi$
$19$	0.928865	0.213096	0.106548	+	0.994308i	$0.466020\pi$
$20$	−0.428022	−0.0957086
$21$	−3.77265	−0.823260
$22$	0.354144	0.0755037
$23$	−5.82354	−1.21429	−0.607146	−	0.794590i	$-0.707686\pi$
$23$	−5.82354	−1.21429	−0.607146	+	0.794590i	$0.707686\pi$
$24$	1.26555	0.258329
$25$	−4.94895	−0.989790
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−7.14690	−1.35064
$29$	4.43398	0.823369	0.411685	−	0.911326i	$-0.364941\pi$
$29$	4.43398	0.823369	0.411685	+	0.911326i	$0.364941\pi$
$30$	−0.0734231	−0.0134052
$31$	8.67716	1.55846	0.779232	−	0.626735i	$-0.215609\pi$
$31$	8.67716	1.55846	0.779232	+	0.626735i	$0.215609\pi$
$32$	3.62868	0.641465
$33$	−1.08979	−0.189708
$34$	−0.324966	−0.0557312
$35$	0.852396	0.144081
$36$	−1.89440	−0.315733
$37$	−6.85589	−1.12710	−0.563551	−	0.826081i	$-0.690565\pi$
$37$	−6.85589	−1.12710	−0.563551	+	0.826081i	$0.690565\pi$
$38$	0.301849	0.0489664
$39$	0	0
$40$	−0.285939	−0.0452109
$41$	−6.09033	−0.951150	−0.475575	−	0.879675i	$-0.657760\pi$
$41$	−6.09033	−0.951150	−0.475575	+	0.879675i	$0.657760\pi$
$42$	−1.22598	−0.189173
$43$	−10.3317	−1.57557	−0.787783	−	0.615952i	$-0.788771\pi$
$43$	−10.3317	−1.57557	−0.787783	+	0.615952i	$0.788771\pi$
$44$	−2.06449	−0.311233
$45$	0.225941	0.0336813
$46$	−1.89245	−0.279027
$47$	4.77366	0.696310	0.348155	−	0.937437i	$-0.386808\pi$
$47$	4.77366	0.696310	0.348155	+	0.937437i	$0.386808\pi$
$48$	−3.37753	−0.487505
$49$	7.23290	1.03327
$50$	−1.60824	−0.227440
$51$	1.00000	0.140028
$52$	0	0
$53$	−10.0842	−1.38516	−0.692582	−	0.721339i	$-0.743527\pi$
$53$	−10.0842	−1.38516	−0.692582	+	0.721339i	$0.743527\pi$
$54$	−0.324966	−0.0442223
$55$	0.246228	0.0332013
$56$	−4.77446	−0.638015
$57$	−0.928865	−0.123031
$58$	1.44089	0.189198
$59$	−6.75390	−0.879283	−0.439641	−	0.898173i	$-0.644894\pi$
$59$	−6.75390	−0.879283	−0.439641	+	0.898173i	$0.644894\pi$
$60$	0.428022	0.0552574
$61$	−3.28356	−0.420416	−0.210208	−	0.977657i	$-0.567414\pi$
$61$	−3.28356	−0.420416	−0.210208	+	0.977657i	$0.567414\pi$
$62$	2.81978	0.358113
$63$	3.77265	0.475309
$64$	−5.57587	−0.696984
$65$	0	0
$66$	−0.354144	−0.0435921
$67$	−1.30265	−0.159144	−0.0795720	−	0.996829i	$-0.525355\pi$
$67$	−1.30265	−0.159144	−0.0795720	+	0.996829i	$0.525355\pi$
$68$	1.89440	0.229729
$69$	5.82354	0.701072
$70$	0.277000	0.0331078
$71$	10.9601	1.30072	0.650362	−	0.759624i	$-0.274617\pi$
$71$	10.9601	1.30072	0.650362	+	0.759624i	$0.274617\pi$
$72$	−1.26555	−0.149146
$73$	−5.12081	−0.599345	−0.299673	−	0.954042i	$-0.596877\pi$
$73$	−5.12081	−0.599345	−0.299673	+	0.954042i	$0.596877\pi$
$74$	−2.22793	−0.258992
$75$	4.94895	0.571456
$76$	−1.75964	−0.201844
$77$	4.11139	0.468536
$78$	0	0
$79$	−9.16176	−1.03078	−0.515389	−	0.856956i	$-0.672353\pi$
$79$	−9.16176	−1.03078	−0.515389	+	0.856956i	$0.672353\pi$
$80$	0.763123	0.0853198
$81$	1.00000	0.111111
$82$	−1.97915	−0.218561
$83$	−1.65443	−0.181598	−0.0907988	−	0.995869i	$-0.528942\pi$
$83$	−1.65443	−0.181598	−0.0907988	+	0.995869i	$0.528942\pi$
$84$	7.14690	0.779791
$85$	−0.225941	−0.0245067
$86$	−3.35744	−0.362042
$87$	−4.43398	−0.475372
$88$	−1.37918	−0.147021
$89$	14.3740	1.52364	0.761821	−	0.647788i	$-0.224306\pi$
$89$	14.3740	1.52364	0.761821	+	0.647788i	$0.224306\pi$
$90$	0.0734231	0.00773947
$91$	0	0
$92$	11.0321	1.15018
$93$	−8.67716	−0.899780
$94$	1.55128	0.160002
$95$	0.209869	0.0215321
$96$	−3.62868	−0.370350
$97$	18.6019	1.88874	0.944369	−	0.328888i	$-0.106674\pi$
$97$	18.6019	1.88874	0.944369	+	0.328888i	$0.106674\pi$
$98$	2.35045	0.237431
$99$	1.08979	0.109528
$100$	9.37528	0.937528
$101$	4.22522	0.420425	0.210212	−	0.977656i	$-0.432584\pi$
$101$	4.22522	0.420425	0.210212	+	0.977656i	$0.432584\pi$
$102$	0.324966	0.0321764
$103$	−5.20531	−0.512895	−0.256447	−	0.966558i	$-0.582552\pi$
$103$	−5.20531	−0.512895	−0.256447	+	0.966558i	$0.582552\pi$
$104$	0	0
$105$	−0.852396	−0.0831854
$106$	−3.27701	−0.318291
$107$	−14.7966	−1.43044	−0.715219	−	0.698901i	$-0.753673\pi$
$107$	−14.7966	−1.43044	−0.715219	+	0.698901i	$0.753673\pi$
$108$	1.89440	0.182288
$109$	−2.43374	−0.233110	−0.116555	−	0.993184i	$-0.537185\pi$
$109$	−2.43374	−0.233110	−0.116555	+	0.993184i	$0.537185\pi$
$110$	0.0800155	0.00762918
$111$	6.85589	0.650733
$112$	12.7423	1.20403
$113$	19.8945	1.87152	0.935759	−	0.352640i	$-0.114716\pi$
$113$	19.8945	1.87152	0.935759	+	0.352640i	$0.114716\pi$
$114$	−0.301849	−0.0282708
$115$	−1.31578	−0.122697
$116$	−8.39972	−0.779894
$117$	0	0
$118$	−2.19479	−0.202047
$119$	−3.77265	−0.345838
$120$	0.285939	0.0261025
$121$	−9.81236	−0.892033
$122$	−1.06704	−0.0966056
$123$	6.09033	0.549147
$124$	−16.4380	−1.47618
$125$	−2.24788	−0.201056
$126$	1.22598	0.109219
$127$	−17.0526	−1.51318	−0.756588	−	0.653892i	$-0.773135\pi$
$127$	−17.0526	−1.51318	−0.756588	+	0.653892i	$0.773135\pi$
$128$	−9.06932	−0.801622
$129$	10.3317	0.909654
$130$	0	0
$131$	−4.60541	−0.402376	−0.201188	−	0.979553i	$-0.564480\pi$
$131$	−4.60541	−0.402376	−0.201188	+	0.979553i	$0.564480\pi$
$132$	2.06449	0.179691
$133$	3.50428	0.303860
$134$	−0.423317	−0.0365690
$135$	−0.225941	−0.0194459
$136$	1.26555	0.108520
$137$	15.6978	1.34115	0.670576	−	0.741841i	$-0.266047\pi$
$137$	15.6978	1.34115	0.670576	+	0.741841i	$0.266047\pi$
$138$	1.89245	0.161096
$139$	5.81375	0.493115	0.246558	−	0.969128i	$-0.420701\pi$
$139$	5.81375	0.493115	0.246558	+	0.969128i	$0.420701\pi$
$140$	−1.61478	−0.136474
$141$	−4.77366	−0.402015
$142$	3.56166	0.298888
$143$	0	0
$144$	3.37753	0.281461
$145$	1.00182	0.0831964
$146$	−1.66409	−0.137721
$147$	−7.23290	−0.596560
$148$	12.9878	1.06759
$149$	1.38306	0.113305	0.0566524	−	0.998394i	$-0.481957\pi$
$149$	1.38306	0.113305	0.0566524	+	0.998394i	$0.481957\pi$
$150$	1.60824	0.131312
$151$	−7.04377	−0.573214	−0.286607	−	0.958048i	$-0.592527\pi$
$151$	−7.04377	−0.573214	−0.286607	+	0.958048i	$0.592527\pi$
$152$	−1.17552	−0.0953474
$153$	−1.00000	−0.0808452
$154$	1.33606	0.107663
$155$	1.96053	0.157473
$156$	0	0
$157$	−10.6442	−0.849496	−0.424748	−	0.905312i	$-0.639637\pi$
$157$	−10.6442	−0.849496	−0.424748	+	0.905312i	$0.639637\pi$
$158$	−2.97726	−0.236858
$159$	10.0842	0.799725
$160$	0.819867	0.0648161
$161$	−21.9702	−1.73149
$162$	0.324966	0.0255317
$163$	15.8777	1.24364	0.621820	−	0.783160i	$-0.286393\pi$
$163$	15.8777	1.24364	0.621820	+	0.783160i	$0.286393\pi$
$164$	11.5375	0.900928
$165$	−0.246228	−0.0191688
$166$	−0.537634	−0.0417285
$167$	0.245307	0.0189825	0.00949123	−	0.999955i	$-0.496979\pi$
$167$	0.245307	0.0189825	0.00949123	+	0.999955i	$0.496979\pi$
$168$	4.77446	0.368358
$169$	0	0
$170$	−0.0734231	−0.00563129
$171$	0.928865	0.0710321
$172$	19.5723	1.49237
$173$	−2.38562	−0.181375	−0.0906876	−	0.995879i	$-0.528906\pi$
$173$	−2.38562	−0.181375	−0.0906876	+	0.995879i	$0.528906\pi$
$174$	−1.44089	−0.109234
$175$	−18.6707	−1.41137
$176$	3.68079	0.277450
$177$	6.75390	0.507654
$178$	4.67106	0.350111
$179$	−7.64876	−0.571695	−0.285848	−	0.958275i	$-0.592275\pi$
$179$	−7.64876	−0.571695	−0.285848	+	0.958275i	$0.592275\pi$
$180$	−0.428022	−0.0319029
$181$	−19.2826	−1.43326	−0.716631	−	0.697452i	$-0.754317\pi$
$181$	−19.2826	−1.43326	−0.716631	+	0.697452i	$0.754317\pi$
$182$	0	0
$183$	3.28356	0.242727
$184$	7.36996	0.543321
$185$	−1.54903	−0.113887
$186$	−2.81978	−0.206756
$187$	−1.08979	−0.0796931
$188$	−9.04321	−0.659544
$189$	−3.77265	−0.274420
$190$	0.0682001	0.00494776
$191$	1.22992	0.0889941	0.0444971	−	0.999010i	$-0.485831\pi$
$191$	1.22992	0.0889941	0.0444971	+	0.999010i	$0.485831\pi$
$192$	5.57587	0.402404
$193$	−5.93703	−0.427357	−0.213678	−	0.976904i	$-0.568544\pi$
$193$	−5.93703	−0.427357	−0.213678	+	0.976904i	$0.568544\pi$
$194$	6.04499	0.434005
$195$	0	0
$196$	−13.7020	−0.978713
$197$	−9.13348	−0.650733	−0.325367	−	0.945588i	$-0.605488\pi$
$197$	−9.13348	−0.650733	−0.325367	+	0.945588i	$0.605488\pi$
$198$	0.354144	0.0251679
$199$	2.45748	0.174206	0.0871030	−	0.996199i	$-0.472239\pi$
$199$	2.45748	0.174206	0.0871030	+	0.996199i	$0.472239\pi$
$200$	6.26313	0.442870
$201$	1.30265	0.0918819
$202$	1.37305	0.0966075
$203$	16.7279	1.17407
$204$	−1.89440	−0.132634
$205$	−1.37606	−0.0961079
$206$	−1.69155	−0.117856
$207$	−5.82354	−0.404764
$208$	0	0
$209$	1.01226	0.0700198
$210$	−0.277000	−0.0191148
$211$	−13.1205	−0.903253	−0.451627	−	0.892207i	$-0.649156\pi$
$211$	−13.1205	−0.903253	−0.451627	+	0.892207i	$0.649156\pi$
$212$	19.1034	1.31203
$213$	−10.9601	−0.750973
$214$	−4.80838	−0.328694
$215$	−2.33435	−0.159201
$216$	1.26555	0.0861095
$217$	32.7359	2.22226
$218$	−0.790884	−0.0535654
$219$	5.12081	0.346032
$220$	−0.466453	−0.0314482
$221$	0	0
$222$	2.22793	0.149529
$223$	9.87727	0.661431	0.330716	−	0.943730i	$-0.392710\pi$
$223$	9.87727	0.661431	0.330716	+	0.943730i	$0.392710\pi$
$224$	13.6897	0.914684
$225$	−4.94895	−0.329930
$226$	6.46504	0.430048
$227$	23.3178	1.54766	0.773828	−	0.633396i	$-0.218340\pi$
$227$	23.3178	1.54766	0.773828	+	0.633396i	$0.218340\pi$
$228$	1.75964	0.116535
$229$	9.54104	0.630490	0.315245	−	0.949010i	$-0.397913\pi$
$229$	9.54104	0.630490	0.315245	+	0.949010i	$0.397913\pi$
$230$	−0.427582	−0.0281939
$231$	−4.11139	−0.270509
$232$	−5.61141	−0.368407
$233$	−2.43483	−0.159511	−0.0797555	−	0.996814i	$-0.525414\pi$
$233$	−2.43483	−0.159511	−0.0797555	+	0.996814i	$0.525414\pi$
$234$	0	0
$235$	1.07857	0.0703579
$236$	12.7946	0.832855
$237$	9.16176	0.595120
$238$	−1.22598	−0.0794687
$239$	−28.5855	−1.84904	−0.924522	−	0.381128i	$-0.875536\pi$
$239$	−28.5855	−1.84904	−0.924522	+	0.381128i	$0.875536\pi$
$240$	−0.763123	−0.0492594
$241$	−13.9314	−0.897403	−0.448702	−	0.893682i	$-0.648113\pi$
$241$	−13.9314	−0.897403	−0.448702	+	0.893682i	$0.648113\pi$
$242$	−3.18868	−0.204976
$243$	−1.00000	−0.0641500
$244$	6.22036	0.398218
$245$	1.63421	0.104406
$246$	1.97915	0.126186
$247$	0	0
$248$	−10.9814	−0.697317
$249$	1.65443	0.104845
$250$	−0.730483	−0.0461998
$251$	−14.1826	−0.895197	−0.447598	−	0.894235i	$-0.647721\pi$
$251$	−14.1826	−0.895197	−0.447598	+	0.894235i	$0.647721\pi$
$252$	−7.14690	−0.450212
$253$	−6.34642	−0.398996
$254$	−5.54152	−0.347706
$255$	0.225941	0.0141490
$256$	8.20453	0.512783
$257$	31.1275	1.94168	0.970840	−	0.239729i	$-0.0770585\pi$
$257$	31.1275	1.94168	0.970840	+	0.239729i	$0.0770585\pi$
$258$	3.35744	0.209025
$259$	−25.8649	−1.60717
$260$	0	0
$261$	4.43398	0.274456
$262$	−1.49660	−0.0924603
$263$	23.2206	1.43184	0.715921	−	0.698181i	$-0.246007\pi$
$263$	23.2206	1.43184	0.715921	+	0.698181i	$0.246007\pi$
$264$	1.37918	0.0848824
$265$	−2.27842	−0.139962
$266$	1.13877	0.0698226
$267$	−14.3740	−0.879675
$268$	2.46774	0.150741
$269$	−7.24967	−0.442020	−0.221010	−	0.975272i	$-0.570935\pi$
$269$	−7.24967	−0.442020	−0.221010	+	0.975272i	$0.570935\pi$
$270$	−0.0734231	−0.00446839
$271$	−26.7622	−1.62569	−0.812844	−	0.582482i	$-0.802082\pi$
$271$	−26.7622	−1.62569	−0.812844	+	0.582482i	$0.802082\pi$
$272$	−3.37753	−0.204793
$273$	0	0
$274$	5.10124	0.308177
$275$	−5.39330	−0.325228
$276$	−11.0321	−0.664054
$277$	−1.44045	−0.0865484	−0.0432742	−	0.999063i	$-0.513779\pi$
$277$	−1.44045	−0.0865484	−0.0432742	+	0.999063i	$0.513779\pi$
$278$	1.88927	0.113311
$279$	8.67716	0.519488
$280$	−1.07875	−0.0644675
$281$	−17.2383	−1.02835	−0.514174	−	0.857686i	$-0.671902\pi$
$281$	−17.2383	−1.02835	−0.514174	+	0.857686i	$0.671902\pi$
$282$	−1.55128	−0.0923772
$283$	−7.01641	−0.417083	−0.208541	−	0.978014i	$-0.566872\pi$
$283$	−7.01641	−0.417083	−0.208541	+	0.978014i	$0.566872\pi$
$284$	−20.7628	−1.23204
$285$	−0.209869	−0.0124315
$286$	0	0
$287$	−22.9767	−1.35627
$288$	3.62868	0.213822
$289$	1.00000	0.0588235
$290$	0.325556	0.0191173
$291$	−18.6019	−1.09046
$292$	9.70084	0.567699
$293$	28.6900	1.67609	0.838043	−	0.545605i	$-0.183700\pi$
$293$	28.6900	1.67609	0.838043	+	0.545605i	$0.183700\pi$
$294$	−2.35045	−0.137081
$295$	−1.52598	−0.0888461
$296$	8.67645	0.504308
$297$	−1.08979	−0.0632359
$298$	0.449448	0.0260358
$299$	0	0
$300$	−9.37528	−0.541282
$301$	−38.9778	−2.24664
$302$	−2.28898	−0.131716
$303$	−4.22522	−0.242732
$304$	3.13727	0.179935
$305$	−0.741890	−0.0424805
$306$	−0.324966	−0.0185771
$307$	15.3365	0.875303	0.437651	−	0.899145i	$-0.355810\pi$
$307$	15.3365	0.875303	0.437651	+	0.899145i	$0.355810\pi$
$308$	−7.78860	−0.443797
$309$	5.20531	0.296120
$310$	0.637104	0.0361851
$311$	3.44806	0.195521	0.0977607	−	0.995210i	$-0.468832\pi$
$311$	3.44806	0.195521	0.0977607	+	0.995210i	$0.468832\pi$
$312$	0	0
$313$	17.0650	0.964569	0.482284	−	0.876015i	$-0.339807\pi$
$313$	17.0650	0.964569	0.482284	+	0.876015i	$0.339807\pi$
$314$	−3.45899	−0.195202
$315$	0.852396	0.0480271
$316$	17.3560	0.976352
$317$	−30.8646	−1.73353	−0.866765	−	0.498718i	$-0.833804\pi$
$317$	−30.8646	−1.73353	−0.866765	+	0.498718i	$0.833804\pi$
$318$	3.27701	0.183765
$319$	4.83209	0.270545
$320$	−1.25982	−0.0704260
$321$	14.7966	0.825864
$322$	−7.13956	−0.397872
$323$	−0.928865	−0.0516834
$324$	−1.89440	−0.105244
$325$	0	0
$326$	5.15972	0.285771
$327$	2.43374	0.134586
$328$	7.70760	0.425581
$329$	18.0094	0.992888
$330$	−0.0800155	−0.00440471
$331$	−33.0693	−1.81765	−0.908827	−	0.417174i	$-0.863021\pi$
$331$	−33.0693	−1.81765	−0.908827	+	0.417174i	$0.863021\pi$
$332$	3.13415	0.172009
$333$	−6.85589	−0.375701
$334$	0.0797166	0.00436190
$335$	−0.294322	−0.0160805
$336$	−12.7423	−0.695147
$337$	−5.37261	−0.292665	−0.146332	−	0.989235i	$-0.546747\pi$
$337$	−5.37261	−0.292665	−0.146332	+	0.989235i	$0.546747\pi$
$338$	0	0
$339$	−19.8945	−1.08052
$340$	0.428022	0.0232127
$341$	9.45626	0.512085
$342$	0.301849	0.0163221
$343$	0.878649	0.0474426
$344$	13.0752	0.704969
$345$	1.31578	0.0708390
$346$	−0.775244	−0.0416774
$347$	18.4408	0.989955	0.494978	−	0.868906i	$-0.335176\pi$
$347$	18.4408	0.989955	0.494978	+	0.868906i	$0.335176\pi$
$348$	8.39972	0.450272
$349$	9.23379	0.494274	0.247137	−	0.968981i	$-0.420510\pi$
$349$	9.23379	0.494274	0.247137	+	0.968981i	$0.420510\pi$
$350$	−6.06733	−0.324312
$351$	0	0
$352$	3.95448	0.210775
$353$	−4.57436	−0.243468	−0.121734	−	0.992563i	$-0.538846\pi$
$353$	−4.57436	−0.243468	−0.121734	+	0.992563i	$0.538846\pi$
$354$	2.19479	0.116652
$355$	2.47633	0.131430
$356$	−27.2301	−1.44319
$357$	3.77265	0.199670
$358$	−2.48559	−0.131367
$359$	−13.2078	−0.697080	−0.348540	−	0.937294i	$-0.613322\pi$
$359$	−13.2078	−0.697080	−0.348540	+	0.937294i	$0.613322\pi$
$360$	−0.285939	−0.0150703
$361$	−18.1372	−0.954590
$362$	−6.26618	−0.329343
$363$	9.81236	0.515016
$364$	0	0
$365$	−1.15700	−0.0605601
$366$	1.06704	0.0557753
$367$	−5.17091	−0.269919	−0.134960	−	0.990851i	$-0.543090\pi$
$367$	−5.17091	−0.269919	−0.134960	+	0.990851i	$0.543090\pi$
$368$	−19.6692	−1.02533
$369$	−6.09033	−0.317050
$370$	−0.503381	−0.0261695
$371$	−38.0440	−1.97515
$372$	16.4380	0.852270
$373$	−0.202173	−0.0104681	−0.00523406	−	0.999986i	$-0.501666\pi$
$373$	−0.202173	−0.0104681	−0.00523406	+	0.999986i	$0.501666\pi$
$374$	−0.354144	−0.0183123
$375$	2.24788	0.116080
$376$	−6.04129	−0.311556
$377$	0	0
$378$	−1.22598	−0.0630578
$379$	−8.73235	−0.448551	−0.224275	−	0.974526i	$-0.572001\pi$
$379$	−8.73235	−0.448551	−0.224275	+	0.974526i	$0.572001\pi$
$380$	−0.397574	−0.0203951
$381$	17.0526	0.873632
$382$	0.399683	0.0204496
$383$	−27.7849	−1.41974	−0.709871	−	0.704332i	$-0.751247\pi$
$383$	−27.7849	−1.41974	−0.709871	+	0.704332i	$0.751247\pi$
$384$	9.06932	0.462817
$385$	0.928931	0.0473427
$386$	−1.92933	−0.0982005
$387$	−10.3317	−0.525189
$388$	−35.2394	−1.78901
$389$	4.33195	0.219639	0.109819	−	0.993952i	$-0.464973\pi$
$389$	4.33195	0.219639	0.109819	+	0.993952i	$0.464973\pi$
$390$	0	0
$391$	5.82354	0.294509
$392$	−9.15357	−0.462325
$393$	4.60541	0.232312
$394$	−2.96807	−0.149529
$395$	−2.07002	−0.104154
$396$	−2.06449	−0.103744
$397$	−8.96618	−0.450000	−0.225000	−	0.974359i	$-0.572238\pi$
$397$	−8.96618	−0.450000	−0.225000	+	0.974359i	$0.572238\pi$
$398$	0.798596	0.0400300
$399$	−3.50428	−0.175434
$400$	−16.7153	−0.835763
$401$	−22.0858	−1.10291	−0.551456	−	0.834204i	$-0.685927\pi$
$401$	−22.0858	−1.10291	−0.551456	+	0.834204i	$0.685927\pi$
$402$	0.423317	0.0211131
$403$	0	0
$404$	−8.00424	−0.398226
$405$	0.225941	0.0112271
$406$	5.43598	0.269783
$407$	−7.47146	−0.370347
$408$	−1.26555	−0.0626539
$409$	−1.77540	−0.0877877	−0.0438938	−	0.999036i	$-0.513976\pi$
$409$	−1.77540	−0.0877877	−0.0438938	+	0.999036i	$0.513976\pi$
$410$	−0.447171	−0.0220842
$411$	−15.6978	−0.774314
$412$	9.86093	0.485813
$413$	−25.4801	−1.25379
$414$	−1.89245	−0.0930089
$415$	−0.373804	−0.0183493
$416$	0	0
$417$	−5.81375	−0.284700
$418$	0.328951	0.0160895
$419$	−27.9814	−1.36698	−0.683492	−	0.729958i	$-0.739539\pi$
$419$	−27.9814	−1.36698	−0.683492	+	0.729958i	$0.739539\pi$
$420$	1.61478	0.0787931
$421$	−3.74416	−0.182479	−0.0912395	−	0.995829i	$-0.529083\pi$
$421$	−3.74416	−0.182479	−0.0912395	+	0.995829i	$0.529083\pi$
$422$	−4.26372	−0.207555
$423$	4.77366	0.232103
$424$	12.7620	0.619776
$425$	4.94895	0.240059
$426$	−3.56166	−0.172563
$427$	−12.3877	−0.599483
$428$	28.0306	1.35491
$429$	0	0
$430$	−0.758584	−0.0365822
$431$	15.7223	0.757318	0.378659	−	0.925536i	$-0.376385\pi$
$431$	15.7223	0.757318	0.378659	+	0.925536i	$0.376385\pi$
$432$	−3.37753	−0.162502
$433$	−28.6198	−1.37538	−0.687689	−	0.726005i	$-0.741375\pi$
$433$	−28.6198	−1.37538	−0.687689	+	0.726005i	$0.741375\pi$
$434$	10.6381	0.510643
$435$	−1.00182	−0.0480335
$436$	4.61048	0.220802
$437$	−5.40928	−0.258761
$438$	1.66409	0.0795132
$439$	36.6019	1.74691	0.873456	−	0.486902i	$-0.161873\pi$
$439$	36.6019	1.74691	0.873456	+	0.486902i	$0.161873\pi$
$440$	−0.311612	−0.0148555
$441$	7.23290	0.344424
$442$	0	0
$443$	−35.7424	−1.69817	−0.849087	−	0.528253i	$-0.822847\pi$
$443$	−35.7424	−1.69817	−0.849087	+	0.528253i	$0.822847\pi$
$444$	−12.9878	−0.616373
$445$	3.24768	0.153955
$446$	3.20978	0.151987
$447$	−1.38306	−0.0654165
$448$	−21.0358	−0.993850
$449$	−4.54953	−0.214705	−0.107353	−	0.994221i	$-0.534237\pi$
$449$	−4.54953	−0.214705	−0.107353	+	0.994221i	$0.534237\pi$
$450$	−1.60824	−0.0758132
$451$	−6.63717	−0.312532
$452$	−37.6881	−1.77270
$453$	7.04377	0.330945
$454$	7.57748	0.355629
$455$	0	0
$456$	1.17552	0.0550488
$457$	−21.0829	−0.986218	−0.493109	−	0.869967i	$-0.664140\pi$
$457$	−21.0829	−0.986218	−0.493109	+	0.869967i	$0.664140\pi$
$458$	3.10051	0.144877
$459$	1.00000	0.0466760
$460$	2.49260	0.116218
$461$	−7.61251	−0.354550	−0.177275	−	0.984161i	$-0.556728\pi$
$461$	−7.61251	−0.354550	−0.177275	+	0.984161i	$0.556728\pi$
$462$	−1.33606	−0.0621592
$463$	−11.7147	−0.544430	−0.272215	−	0.962236i	$-0.587756\pi$
$463$	−11.7147	−0.544430	−0.272215	+	0.962236i	$0.587756\pi$
$464$	14.9759	0.695240
$465$	−1.96053	−0.0909172
$466$	−0.791237	−0.0366533
$467$	−11.9720	−0.554000	−0.277000	−	0.960870i	$-0.589340\pi$
$467$	−11.9720	−0.554000	−0.277000	+	0.960870i	$0.589340\pi$
$468$	0	0
$469$	−4.91445	−0.226928
$470$	0.350497	0.0161672
$471$	10.6442	0.490457
$472$	8.54737	0.393425
$473$	−11.2593	−0.517705
$474$	2.97726	0.136750
$475$	−4.59691	−0.210920
$476$	7.14690	0.327578
$477$	−10.0842	−0.461722
$478$	−9.28933	−0.424884
$479$	−8.85206	−0.404461	−0.202231	−	0.979338i	$-0.564819\pi$
$479$	−8.85206	−0.404461	−0.202231	+	0.979338i	$0.564819\pi$
$480$	−0.819867	−0.0374216
$481$	0	0
$482$	−4.52725	−0.206210
$483$	21.9702	0.999678
$484$	18.5885	0.844933
$485$	4.20293	0.190845
$486$	−0.324966	−0.0147408
$487$	0.0340495	0.00154293	0.000771466	−	1.00000i	$-0.499754\pi$
$487$	0.0340495	0.00154293	0.000771466		1.00000i	$0.499754\pi$
$488$	4.15549	0.188110
$489$	−15.8777	−0.718016
$490$	0.531062	0.0239909
$491$	6.57205	0.296592	0.148296	−	0.988943i	$-0.452621\pi$
$491$	6.57205	0.296592	0.148296	+	0.988943i	$0.452621\pi$
$492$	−11.5375	−0.520151
$493$	−4.43398	−0.199696
$494$	0	0
$495$	0.246228	0.0110671
$496$	29.3074	1.31594
$497$	41.3486	1.85474
$498$	0.537634	0.0240920
$499$	5.67761	0.254165	0.127082	−	0.991892i	$-0.459439\pi$
$499$	5.67761	0.254165	0.127082	+	0.991892i	$0.459439\pi$
$500$	4.25837	0.190440
$501$	−0.245307	−0.0109595
$502$	−4.60886	−0.205703
$503$	−4.02904	−0.179646	−0.0898230	−	0.995958i	$-0.528630\pi$
$503$	−4.02904	−0.179646	−0.0898230	+	0.995958i	$0.528630\pi$
$504$	−4.77446	−0.212672
$505$	0.954649	0.0424813
$506$	−2.06237	−0.0916835
$507$	0	0
$508$	32.3044	1.43328
$509$	−8.54616	−0.378802	−0.189401	−	0.981900i	$-0.560655\pi$
$509$	−8.54616	−0.378802	−0.189401	+	0.981900i	$0.560655\pi$
$510$	0.0734231	0.00325123
$511$	−19.3190	−0.854623
$512$	20.8048	0.919452
$513$	−0.928865	−0.0410104
$514$	10.1154	0.446170
$515$	−1.17609	−0.0518249
$516$	−19.5723	−0.861623
$517$	5.20227	0.228796
$518$	−8.40521	−0.369304
$519$	2.38562	0.104717
$520$	0	0
$521$	−21.4632	−0.940318	−0.470159	−	0.882582i	$-0.655803\pi$
$521$	−21.4632	−0.940318	−0.470159	+	0.882582i	$0.655803\pi$
$522$	1.44089	0.0630661
$523$	−21.4966	−0.939980	−0.469990	−	0.882672i	$-0.655742\pi$
$523$	−21.4966	−0.939980	−0.469990	+	0.882672i	$0.655742\pi$
$524$	8.72447	0.381130
$525$	18.6707	0.814855
$526$	7.54590	0.329017
$527$	−8.67716	−0.377983
$528$	−3.68079	−0.160186
$529$	10.9136	0.474505
$530$	−0.740410	−0.0321613
$531$	−6.75390	−0.293094
$532$	−6.63850	−0.287816
$533$	0	0
$534$	−4.67106	−0.202137
$535$	−3.34315	−0.144537
$536$	1.64856	0.0712071
$537$	7.64876	0.330068
$538$	−2.35590	−0.101570
$539$	7.88232	0.339516
$540$	0.428022	0.0184191
$541$	20.1843	0.867793	0.433896	−	0.900963i	$-0.357138\pi$
$541$	20.1843	0.867793	0.433896	+	0.900963i	$0.357138\pi$
$542$	−8.69680	−0.373560
$543$	19.2826	0.827494
$544$	−3.62868	−0.155578
$545$	−0.549882	−0.0235544
$546$	0	0
$547$	30.6953	1.31244	0.656219	−	0.754571i	$-0.272155\pi$
$547$	30.6953	1.31244	0.656219	+	0.754571i	$0.272155\pi$
$548$	−29.7378	−1.27034
$549$	−3.28356	−0.140139
$550$	−1.75264	−0.0747328
$551$	4.11857	0.175457
$552$	−7.36996	−0.313686
$553$	−34.5641	−1.46982
$554$	−0.468098	−0.0198876
$555$	1.54903	0.0657525
$556$	−11.0135	−0.467078
$557$	18.9079	0.801153	0.400577	−	0.916263i	$-0.368810\pi$
$557$	18.9079	0.801153	0.400577	+	0.916263i	$0.368810\pi$
$558$	2.81978	0.119371
$559$	0	0
$560$	2.87900	0.121660
$561$	1.08979	0.0460108
$562$	−5.60185	−0.236300
$563$	−19.3339	−0.814828	−0.407414	−	0.913244i	$-0.633569\pi$
$563$	−19.3339	−0.814828	−0.407414	+	0.913244i	$0.633569\pi$
$564$	9.04321	0.380788
$565$	4.49498	0.189105
$566$	−2.28010	−0.0958396
$567$	3.77265	0.158436
$568$	−13.8705	−0.581994
$569$	−34.6422	−1.45228	−0.726139	−	0.687548i	$-0.758687\pi$
$569$	−34.6422	−1.45228	−0.726139	+	0.687548i	$0.758687\pi$
$570$	−0.0682001	−0.00285659
$571$	34.6805	1.45133	0.725667	−	0.688046i	$-0.241531\pi$
$571$	34.6805	1.45133	0.725667	+	0.688046i	$0.241531\pi$
$572$	0	0
$573$	−1.22992	−0.0513808
$574$	−7.46664	−0.311652
$575$	28.8204	1.20189
$576$	−5.57587	−0.232328
$577$	5.03045	0.209420	0.104710	−	0.994503i	$-0.466609\pi$
$577$	5.03045	0.209420	0.104710	+	0.994503i	$0.466609\pi$
$578$	0.324966	0.0135168
$579$	5.93703	0.246735
$580$	−1.89784	−0.0788035
$581$	−6.24160	−0.258945
$582$	−6.04499	−0.250573
$583$	−10.9896	−0.455142
$584$	6.48062	0.268170
$585$	0	0
$586$	9.32326	0.385140
$587$	15.9851	0.659776	0.329888	−	0.944020i	$-0.392989\pi$
$587$	15.9851	0.659776	0.329888	+	0.944020i	$0.392989\pi$
$588$	13.7020	0.565060
$589$	8.05991	0.332103
$590$	−0.495892	−0.0204156
$591$	9.13348	0.375701
$592$	−23.1560	−0.951707
$593$	12.0510	0.494877	0.247438	−	0.968904i	$-0.420411\pi$
$593$	12.0510	0.494877	0.247438	+	0.968904i	$0.420411\pi$
$594$	−0.354144	−0.0145307
$595$	−0.852396	−0.0349448
$596$	−2.62007	−0.107322
$597$	−2.45748	−0.100578
$598$	0	0
$599$	−35.0622	−1.43260	−0.716302	−	0.697791i	$-0.754167\pi$
$599$	−35.0622	−1.43260	−0.716302	+	0.697791i	$0.754167\pi$
$600$	−6.26313	−0.255691
$601$	−9.60699	−0.391877	−0.195939	−	0.980616i	$-0.562775\pi$
$601$	−9.60699	−0.391877	−0.195939	+	0.980616i	$0.562775\pi$
$602$	−12.6665	−0.516247
$603$	−1.30265	−0.0530480
$604$	13.3437	0.542947
$605$	−2.21701	−0.0901345
$606$	−1.37305	−0.0557764
$607$	32.5335	1.32049	0.660247	−	0.751049i	$-0.270452\pi$
$607$	32.5335	1.32049	0.660247	+	0.751049i	$0.270452\pi$
$608$	3.37055	0.136694
$609$	−16.7279	−0.677847
$610$	−0.241089	−0.00976140
$611$	0	0
$612$	1.89440	0.0765765
$613$	−33.4339	−1.35038	−0.675191	−	0.737643i	$-0.735939\pi$
$613$	−33.4339	−1.35038	−0.675191	+	0.737643i	$0.735939\pi$
$614$	4.98385	0.201132
$615$	1.37606	0.0554879
$616$	−5.20315	−0.209641
$617$	3.50985	0.141301	0.0706506	−	0.997501i	$-0.477492\pi$
$617$	3.50985	0.141301	0.0706506	+	0.997501i	$0.477492\pi$
$618$	1.69155	0.0680441
$619$	−11.5658	−0.464871	−0.232435	−	0.972612i	$-0.574669\pi$
$619$	−11.5658	−0.464871	−0.232435	+	0.972612i	$0.574669\pi$
$620$	−3.71402	−0.149158
$621$	5.82354	0.233691
$622$	1.12050	0.0449280
$623$	54.2281	2.17260
$624$	0	0
$625$	24.2369	0.969475
$626$	5.54553	0.221644
$627$	−1.01226	−0.0404260
$628$	20.1643	0.804642
$629$	6.85589	0.273362
$630$	0.277000	0.0110359
$631$	−29.5050	−1.17457	−0.587287	−	0.809379i	$-0.699804\pi$
$631$	−29.5050	−1.17457	−0.587287	+	0.809379i	$0.699804\pi$
$632$	11.5946	0.461210
$633$	13.1205	0.521494
$634$	−10.0299	−0.398340
$635$	−3.85289	−0.152897
$636$	−19.1034	−0.757499
$637$	0	0
$638$	1.57027	0.0621674
$639$	10.9601	0.433575
$640$	−2.04913	−0.0809990
$641$	−45.8996	−1.81293	−0.906463	−	0.422286i	$-0.861228\pi$
$641$	−45.8996	−1.81293	−0.906463	+	0.422286i	$0.861228\pi$
$642$	4.80838	0.189772
$643$	34.9962	1.38011	0.690057	−	0.723755i	$-0.257585\pi$
$643$	34.9962	1.38011	0.690057	+	0.723755i	$0.257585\pi$
$644$	41.6203	1.64007
$645$	2.33435	0.0919149
$646$	−0.301849	−0.0118761
$647$	−0.414198	−0.0162838	−0.00814191	−	0.999967i	$-0.502592\pi$
$647$	−0.414198	−0.0162838	−0.00814191	+	0.999967i	$0.502592\pi$
$648$	−1.26555	−0.0497154
$649$	−7.36031	−0.288917
$650$	0	0
$651$	−32.7359	−1.28302
$652$	−30.0787	−1.17797
$653$	16.7598	0.655862	0.327931	−	0.944702i	$-0.393649\pi$
$653$	16.7598	0.655862	0.327931	+	0.944702i	$0.393649\pi$
$654$	0.790884	0.0309260
$655$	−1.04055	−0.0406576
$656$	−20.5703	−0.803136
$657$	−5.12081	−0.199782
$658$	5.85243	0.228151
$659$	−37.4711	−1.45967	−0.729834	−	0.683625i	$-0.760403\pi$
$659$	−37.4711	−1.45967	−0.729834	+	0.683625i	$0.760403\pi$
$660$	0.466453	0.0181566
$661$	−12.8558	−0.500034	−0.250017	−	0.968241i	$-0.580436\pi$
$661$	−12.8558	−0.500034	−0.250017	+	0.968241i	$0.580436\pi$
$662$	−10.7464	−0.417671
$663$	0	0
$664$	2.09376	0.0812537
$665$	0.791761	0.0307032
$666$	−2.22793	−0.0863306
$667$	−25.8215	−0.999811
$668$	−0.464710	−0.0179802
$669$	−9.87727	−0.381877
$670$	−0.0956446	−0.00369507
$671$	−3.57838	−0.138142
$672$	−13.6897	−0.528093
$673$	−37.5416	−1.44712	−0.723562	−	0.690260i	$-0.757496\pi$
$673$	−37.5416	−1.44712	−0.723562	+	0.690260i	$0.757496\pi$
$674$	−1.74591	−0.0672501
$675$	4.94895	0.190485
$676$	0	0
$677$	41.9818	1.61349	0.806746	−	0.590899i	$-0.201227\pi$
$677$	41.9818	1.61349	0.806746	+	0.590899i	$0.201227\pi$
$678$	−6.46504	−0.248288
$679$	70.1785	2.69320
$680$	0.285939	0.0109652
$681$	−23.3178	−0.893539
$682$	3.07296	0.117670
$683$	43.6924	1.67184	0.835922	−	0.548848i	$-0.184933\pi$
$683$	43.6924	1.67184	0.835922	+	0.548848i	$0.184933\pi$
$684$	−1.75964	−0.0672815
$685$	3.54677	0.135515
$686$	0.285531	0.0109016
$687$	−9.54104	−0.364013
$688$	−34.8956	−1.33038
$689$	0	0
$690$	0.427582	0.0162778
$691$	−37.9319	−1.44300	−0.721498	−	0.692416i	$-0.756546\pi$
$691$	−37.9319	−1.44300	−0.721498	+	0.692416i	$0.756546\pi$
$692$	4.51931	0.171798
$693$	4.11139	0.156179
$694$	5.99264	0.227477
$695$	1.31356	0.0498263
$696$	5.61141	0.212700
$697$	6.09033	0.230688
$698$	3.00067	0.113577
$699$	2.43483	0.0920937
$700$	35.3697	1.33685
$701$	8.27276	0.312458	0.156229	−	0.987721i	$-0.450066\pi$
$701$	8.27276	0.312458	0.156229	+	0.987721i	$0.450066\pi$
$702$	0	0
$703$	−6.36820	−0.240181
$704$	−6.07652	−0.229017
$705$	−1.07857	−0.0406211
$706$	−1.48651	−0.0559455
$707$	15.9403	0.599495
$708$	−12.7946	−0.480849
$709$	−31.2101	−1.17212	−0.586060	−	0.810268i	$-0.699322\pi$
$709$	−31.2101	−1.17212	−0.586060	+	0.810268i	$0.699322\pi$
$710$	0.804724	0.0302008
$711$	−9.16176	−0.343593
$712$	−18.1910	−0.681736
$713$	−50.5318	−1.89243
$714$	1.22598	0.0458813
$715$	0	0
$716$	14.4898	0.541509
$717$	28.5855	1.06755
$718$	−4.29208	−0.160179
$719$	44.3453	1.65380	0.826900	−	0.562350i	$-0.190103\pi$
$719$	44.3453	1.65380	0.826900	+	0.562350i	$0.190103\pi$
$720$	0.763123	0.0284399
$721$	−19.6378	−0.731351
$722$	−5.89397	−0.219351
$723$	13.9314	0.518116
$724$	36.5288	1.35758
$725$	−21.9435	−0.814963
$726$	3.18868	0.118343
$727$	0.172388	0.00639351	0.00319675	−	0.999995i	$-0.498982\pi$
$727$	0.172388	0.00639351	0.00319675	+	0.999995i	$0.498982\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−0.375986	−0.0139158
$731$	10.3317	0.382131
$732$	−6.22036	−0.229911
$733$	−51.0118	−1.88416	−0.942081	−	0.335385i	$-0.891133\pi$
$733$	−51.0118	−1.88416	−0.942081	+	0.335385i	$0.891133\pi$
$734$	−1.68037	−0.0620235
$735$	−1.63421	−0.0602787
$736$	−21.1317	−0.778926
$737$	−1.41961	−0.0522921
$738$	−1.97915	−0.0728535
$739$	−1.13546	−0.0417686	−0.0208843	−	0.999782i	$-0.506648\pi$
$739$	−1.13546	−0.0417686	−0.0208843	+	0.999782i	$0.506648\pi$
$740$	2.93447	0.107873
$741$	0	0
$742$	−12.3630	−0.453860
$743$	37.2089	1.36506	0.682531	−	0.730856i	$-0.260879\pi$
$743$	37.2089	1.36506	0.682531	+	0.730856i	$0.260879\pi$
$744$	10.9814	0.402596
$745$	0.312490	0.0114488
$746$	−0.0656994	−0.00240542
$747$	−1.65443	−0.0605325
$748$	2.06449	0.0754852
$749$	−55.8223	−2.03970
$750$	0.730483	0.0266735
$751$	36.6011	1.33559	0.667797	−	0.744344i	$-0.267238\pi$
$751$	36.6011	1.33559	0.667797	+	0.744344i	$0.267238\pi$
$752$	16.1232	0.587953
$753$	14.1826	0.516842
$754$	0	0
$755$	−1.59148	−0.0579197
$756$	7.14690	0.259930
$757$	−45.2828	−1.64583	−0.822915	−	0.568164i	$-0.807654\pi$
$757$	−45.2828	−1.64583	−0.822915	+	0.568164i	$0.807654\pi$
$758$	−2.83771	−0.103070
$759$	6.34642	0.230360
$760$	−0.265598	−0.00963427
$761$	23.7674	0.861567	0.430784	−	0.902455i	$-0.358237\pi$
$761$	23.7674	0.861567	0.430784	+	0.902455i	$0.358237\pi$
$762$	5.54152	0.200748
$763$	−9.18167	−0.332399
$764$	−2.32996	−0.0842951
$765$	−0.225941	−0.00816891
$766$	−9.02915	−0.326236
$767$	0	0
$768$	−8.20453	−0.296055
$769$	−24.2817	−0.875621	−0.437811	−	0.899067i	$-0.644246\pi$
$769$	−24.2817	−0.875621	−0.437811	+	0.899067i	$0.644246\pi$
$770$	0.301871	0.0108787
$771$	−31.1275	−1.12103
$772$	11.2471	0.404792
$773$	43.5436	1.56615	0.783077	−	0.621925i	$-0.213649\pi$
$773$	43.5436	1.56615	0.783077	+	0.621925i	$0.213649\pi$
$774$	−3.35744	−0.120681
$775$	−42.9429	−1.54255
$776$	−23.5416	−0.845093
$777$	25.8649	0.927898
$778$	1.40774	0.0504698
$779$	−5.65709	−0.202686
$780$	0	0
$781$	11.9442	0.427396
$782$	1.89245	0.0676739
$783$	−4.43398	−0.158457
$784$	24.4294	0.872478
$785$	−2.40495	−0.0858364
$786$	1.49660	0.0533820
$787$	11.4514	0.408200	0.204100	−	0.978950i	$-0.434573\pi$
$787$	11.4514	0.408200	0.204100	+	0.978950i	$0.434573\pi$
$788$	17.3024	0.616374
$789$	−23.2206	−0.826675
$790$	−0.672685	−0.0239331
$791$	75.0550	2.66865
$792$	−1.37918	−0.0490069
$793$	0	0
$794$	−2.91370	−0.103403
$795$	2.27842	0.0808073
$796$	−4.65544	−0.165008
$797$	−2.39097	−0.0846924	−0.0423462	−	0.999103i	$-0.513483\pi$
$797$	−2.39097	−0.0846924	−0.0423462	+	0.999103i	$0.513483\pi$
$798$	−1.13877	−0.0403121
$799$	−4.77366	−0.168880
$800$	−17.9581	−0.634916
$801$	14.3740	0.507881
$802$	−7.17712	−0.253433
$803$	−5.58059	−0.196935
$804$	−2.46774	−0.0870304
$805$	−4.96396	−0.174957
$806$	0	0
$807$	7.24967	0.255200
$808$	−5.34721	−0.188114
$809$	−25.4260	−0.893929	−0.446965	−	0.894552i	$-0.647495\pi$
$809$	−25.4260	−0.893929	−0.446965	+	0.894552i	$0.647495\pi$
$810$	0.0734231	0.00257982
$811$	−30.8987	−1.08500	−0.542500	−	0.840055i	$-0.682522\pi$
$811$	−30.8987	−1.08500	−0.542500	+	0.840055i	$0.682522\pi$
$812$	−31.6892	−1.11207
$813$	26.7622	0.938591
$814$	−2.42797	−0.0851003
$815$	3.58743	0.125662
$816$	3.37753	0.118237
$817$	−9.59673	−0.335747
$818$	−0.576943	−0.0201723
$819$	0	0
$820$	2.60680	0.0910333
$821$	4.13601	0.144348	0.0721738	−	0.997392i	$-0.477006\pi$
$821$	4.13601	0.144348	0.0721738	+	0.997392i	$0.477006\pi$
$822$	−5.10124	−0.177926
$823$	−1.77016	−0.0617038	−0.0308519	−	0.999524i	$-0.509822\pi$
$823$	−1.77016	−0.0617038	−0.0308519	+	0.999524i	$0.509822\pi$
$824$	6.58756	0.229489
$825$	5.39330	0.187771
$826$	−8.28017	−0.288104
$827$	−37.4781	−1.30324	−0.651620	−	0.758545i	$-0.725910\pi$
$827$	−37.4781	−1.30324	−0.651620	+	0.758545i	$0.725910\pi$
$828$	11.0321	0.383392
$829$	−34.3278	−1.19225	−0.596126	−	0.802891i	$-0.703294\pi$
$829$	−34.3278	−1.19225	−0.596126	+	0.802891i	$0.703294\pi$
$830$	−0.121474	−0.00421641
$831$	1.44045	0.0499688
$832$	0	0
$833$	−7.23290	−0.250605
$834$	−1.88927	−0.0654200
$835$	0.0554250	0.00191806
$836$	−1.91763	−0.0663227
$837$	−8.67716	−0.299927
$838$	−9.09302	−0.314113
$839$	−35.6778	−1.23173	−0.615867	−	0.787850i	$-0.711194\pi$
$839$	−35.6778	−1.23173	−0.615867	+	0.787850i	$0.711194\pi$
$840$	1.07875	0.0372203
$841$	−9.33983	−0.322063
$842$	−1.21672	−0.0419310
$843$	17.2383	0.593717
$844$	24.8555	0.855560
$845$	0	0
$846$	1.55128	0.0533340
$847$	−37.0186	−1.27198
$848$	−34.0596	−1.16961
$849$	7.01641	0.240803
$850$	1.60824	0.0551622
$851$	39.9256	1.36863
$852$	20.7628	0.711321
$853$	2.27992	0.0780630	0.0390315	−	0.999238i	$-0.487573\pi$
$853$	2.27992	0.0780630	0.0390315	+	0.999238i	$0.487573\pi$
$854$	−4.02558	−0.137753
$855$	0.209869	0.00717735
$856$	18.7257	0.640032
$857$	25.6518	0.876250	0.438125	−	0.898914i	$-0.355643\pi$
$857$	25.6518	0.876250	0.438125	+	0.898914i	$0.355643\pi$
$858$	0	0
$859$	6.86970	0.234391	0.117196	−	0.993109i	$-0.462610\pi$
$859$	6.86970	0.234391	0.117196	+	0.993109i	$0.462610\pi$
$860$	4.42219	0.150795
$861$	22.9767	0.783044
$862$	5.10922	0.174021
$863$	4.85992	0.165434	0.0827168	−	0.996573i	$-0.473640\pi$
$863$	4.85992	0.165434	0.0827168	+	0.996573i	$0.473640\pi$
$864$	−3.62868	−0.123450
$865$	−0.539009	−0.0183268
$866$	−9.30045	−0.316042
$867$	−1.00000	−0.0339618
$868$	−62.0148	−2.10492
$869$	−9.98437	−0.338697
$870$	−0.325556	−0.0110374
$871$	0	0
$872$	3.08002	0.104302
$873$	18.6019	0.629579
$874$	−1.75783	−0.0594595
$875$	−8.48045	−0.286692
$876$	−9.70084	−0.327761
$877$	−33.7109	−1.13834	−0.569168	−	0.822221i	$-0.692735\pi$
$877$	−33.7109	−1.13834	−0.569168	+	0.822221i	$0.692735\pi$
$878$	11.8944	0.401415
$879$	−28.6900	−0.967688
$880$	0.831642	0.0280346
$881$	16.3210	0.549870	0.274935	−	0.961463i	$-0.411344\pi$
$881$	16.3210	0.549870	0.274935	+	0.961463i	$0.411344\pi$
$882$	2.35045	0.0791436
$883$	−18.8787	−0.635320	−0.317660	−	0.948205i	$-0.602897\pi$
$883$	−18.8787	−0.635320	−0.317660	+	0.948205i	$0.602897\pi$
$884$	0	0
$885$	1.52598	0.0512953
$886$	−11.6151	−0.390216
$887$	−10.7542	−0.361090	−0.180545	−	0.983567i	$-0.557786\pi$
$887$	−10.7542	−0.361090	−0.180545	+	0.983567i	$0.557786\pi$
$888$	−8.67645	−0.291163
$889$	−64.3336	−2.15768
$890$	1.05538	0.0353766
$891$	1.08979	0.0365092
$892$	−18.7115	−0.626507
$893$	4.43409	0.148381
$894$	−0.449448	−0.0150318
$895$	−1.72817	−0.0577663
$896$	−34.2154	−1.14306
$897$	0	0
$898$	−1.47844	−0.0493362
$899$	38.4744	1.28319
$900$	9.37528	0.312509
$901$	10.0842	0.335952
$902$	−2.15685	−0.0718153
$903$	38.9778	1.29710
$904$	−25.1774	−0.837389
$905$	−4.35672	−0.144822
$906$	2.28898	0.0760464
$907$	40.4092	1.34177	0.670883	−	0.741563i	$-0.265915\pi$
$907$	40.4092	1.34177	0.670883	+	0.741563i	$0.265915\pi$
$908$	−44.1731	−1.46594
$909$	4.22522	0.140142
$910$	0	0
$911$	−29.5118	−0.977769	−0.488885	−	0.872348i	$-0.662596\pi$
$911$	−29.5118	−0.977769	−0.488885	+	0.872348i	$0.662596\pi$
$912$	−3.13727	−0.103885
$913$	−1.80298	−0.0596699
$914$	−6.85124	−0.226619
$915$	0.741890	0.0245261
$916$	−18.0745	−0.597199
$917$	−17.3746	−0.573760
$918$	0.324966	0.0107255
$919$	23.6108	0.778850	0.389425	−	0.921058i	$-0.372674\pi$
$919$	23.6108	0.778850	0.389425	+	0.921058i	$0.372674\pi$
$920$	1.66518	0.0548992
$921$	−15.3365	−0.505356
$922$	−2.47381	−0.0814704
$923$	0	0
$924$	7.78860	0.256226
$925$	33.9295	1.11559
$926$	−3.80689	−0.125102
$927$	−5.20531	−0.170965
$928$	16.0895	0.528163
$929$	13.7415	0.450845	0.225423	−	0.974261i	$-0.427624\pi$
$929$	13.7415	0.450845	0.225423	+	0.974261i	$0.427624\pi$
$930$	−0.637104	−0.0208915
$931$	6.71839	0.220186
$932$	4.61254	0.151089
$933$	−3.44806	−0.112884
$934$	−3.89050	−0.127301
$935$	−0.246228	−0.00805250
$936$	0	0
$937$	47.8323	1.56261	0.781307	−	0.624147i	$-0.214553\pi$
$937$	47.8323	1.56261	0.781307	+	0.624147i	$0.214553\pi$
$938$	−1.59703	−0.0521448
$939$	−17.0650	−0.556894
$940$	−2.04323	−0.0666429
$941$	−0.910077	−0.0296677	−0.0148338	−	0.999890i	$-0.504722\pi$
$941$	−0.910077	−0.0296677	−0.0148338	+	0.999890i	$0.504722\pi$
$942$	3.45899	0.112700
$943$	35.4673	1.15497
$944$	−22.8115	−0.742452
$945$	−0.852396	−0.0277285
$946$	−3.65890	−0.118961
$947$	−47.2080	−1.53406	−0.767028	−	0.641614i	$-0.778265\pi$
$947$	−47.2080	−1.53406	−0.767028	+	0.641614i	$0.778265\pi$
$948$	−17.3560	−0.563697
$949$	0	0
$950$	−1.49384	−0.0484665
$951$	30.8646	1.00085
$952$	4.77446	0.154741
$953$	−60.0273	−1.94448	−0.972238	−	0.233994i	$-0.924820\pi$
$953$	−60.0273	−1.94448	−0.972238	+	0.233994i	$0.924820\pi$
$954$	−3.27701	−0.106097
$955$	0.277890	0.00899231
$956$	54.1524	1.75141
$957$	−4.83209	−0.156199
$958$	−2.87662	−0.0929393
$959$	59.2222	1.91239
$960$	1.25982	0.0406605
$961$	44.2932	1.42881
$962$	0	0
$963$	−14.7966	−0.476813
$964$	26.3917	0.850019
$965$	−1.34142	−0.0431818
$966$	7.13956	0.229712
$967$	11.6650	0.375121	0.187560	−	0.982253i	$-0.439942\pi$
$967$	11.6650	0.375121	0.187560	+	0.982253i	$0.439942\pi$
$968$	12.4180	0.399130
$969$	0.928865	0.0298394
$970$	1.36581	0.0438535
$971$	5.10891	0.163953	0.0819764	−	0.996634i	$-0.473877\pi$
$971$	5.10891	0.163953	0.0819764	+	0.996634i	$0.473877\pi$
$972$	1.89440	0.0607628
$973$	21.9332	0.703147
$974$	0.0110649	0.000354543 0
$975$	0	0
$976$	−11.0903	−0.354993
$977$	18.0013	0.575912	0.287956	−	0.957644i	$-0.407024\pi$
$977$	18.0013	0.575912	0.287956	+	0.957644i	$0.407024\pi$
$978$	−5.15972	−0.164990
$979$	15.6646	0.500643
$980$	−3.09584	−0.0988930
$981$	−2.43374	−0.0777035
$982$	2.13569	0.0681526
$983$	12.0800	0.385292	0.192646	−	0.981268i	$-0.438293\pi$
$983$	12.0800	0.385292	0.192646	+	0.981268i	$0.438293\pi$
$984$	−7.70760	−0.245709
$985$	−2.06363	−0.0657526
$986$	−1.44089	−0.0458873
$987$	−18.0094	−0.573244
$988$	0	0
$989$	60.1669	1.91320
$990$	0.0800155	0.00254306
$991$	15.2798	0.485380	0.242690	−	0.970104i	$-0.421970\pi$
$991$	15.2798	0.485380	0.242690	+	0.970104i	$0.421970\pi$
$992$	31.4866	0.999701
$993$	33.0693	1.04942
$994$	13.4369	0.426192
$995$	0.555245	0.0176024
$996$	−3.13415	−0.0993095
$997$	59.2039	1.87501	0.937503	−	0.347978i	$-0.113132\pi$
$997$	59.2039	1.87501	0.937503	+	0.347978i	$0.113132\pi$
$998$	1.84503	0.0584034
$999$	6.85589	0.216911

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8619.2.a.bw.1.12	yes	24
13.12	even	2		8619.2.a.bt.1.13	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8619.2.a.bt.1.13	✓	24	13.12	even	2
8619.2.a.bw.1.12	yes	24	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:	\( N \)	\(=\)	\( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8619.a (trivial)

\(f(q)\)	\(=\)	\(q+0.324966 q^{2} -1.00000 q^{3} -1.89440 q^{4} +0.225941 q^{5} -0.324966 q^{6} +3.77265 q^{7} -1.26555 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.324966 q^{2} -1.00000 q^{3} -1.89440 q^{4} +0.225941 q^{5} -0.324966 q^{6} +3.77265 q^{7} -1.26555 q^{8} +1.00000 q^{9} +0.0734231 q^{10} +1.08979 q^{11} +1.89440 q^{12} +1.22598 q^{14} -0.225941 q^{15} +3.37753 q^{16} -1.00000 q^{17} +0.324966 q^{18} +0.928865 q^{19} -0.428022 q^{20} -3.77265 q^{21} +0.354144 q^{22} -5.82354 q^{23} +1.26555 q^{24} -4.94895 q^{25} -1.00000 q^{27} -7.14690 q^{28} +4.43398 q^{29} -0.0734231 q^{30} +8.67716 q^{31} +3.62868 q^{32} -1.08979 q^{33} -0.324966 q^{34} +0.852396 q^{35} -1.89440 q^{36} -6.85589 q^{37} +0.301849 q^{38} -0.285939 q^{40} -6.09033 q^{41} -1.22598 q^{42} -10.3317 q^{43} -2.06449 q^{44} +0.225941 q^{45} -1.89245 q^{46} +4.77366 q^{47} -3.37753 q^{48} +7.23290 q^{49} -1.60824 q^{50} +1.00000 q^{51} -10.0842 q^{53} -0.324966 q^{54} +0.246228 q^{55} -4.77446 q^{56} -0.928865 q^{57} +1.44089 q^{58} -6.75390 q^{59} +0.428022 q^{60} -3.28356 q^{61} +2.81978 q^{62} +3.77265 q^{63} -5.57587 q^{64} -0.354144 q^{66} -1.30265 q^{67} +1.89440 q^{68} +5.82354 q^{69} +0.277000 q^{70} +10.9601 q^{71} -1.26555 q^{72} -5.12081 q^{73} -2.22793 q^{74} +4.94895 q^{75} -1.75964 q^{76} +4.11139 q^{77} -9.16176 q^{79} +0.763123 q^{80} +1.00000 q^{81} -1.97915 q^{82} -1.65443 q^{83} +7.14690 q^{84} -0.225941 q^{85} -3.35744 q^{86} -4.43398 q^{87} -1.37918 q^{88} +14.3740 q^{89} +0.0734231 q^{90} +11.0321 q^{92} -8.67716 q^{93} +1.55128 q^{94} +0.209869 q^{95} -3.62868 q^{96} +18.6019 q^{97} +2.35045 q^{98} +1.08979 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 7 q^{2} - 24 q^{3} + 17 q^{4} - 13 q^{5} - 7 q^{6} - 12 q^{7} + 21 q^{8} + 24 q^{9}+O(q^{10}) \) 24 * q + 7 * q^2 - 24 * q^3 + 17 * q^4 - 13 * q^5 - 7 * q^6 - 12 * q^7 + 21 * q^8 + 24 * q^9	\( 24 q + 7 q^{2} - 24 q^{3} + 17 q^{4} - 13 q^{5} - 7 q^{6} - 12 q^{7} + 21 q^{8} + 24 q^{9} - 4 q^{10} - q^{11} - 17 q^{12} - 18 q^{14} + 13 q^{15} + 35 q^{16} - 24 q^{17} + 7 q^{18} - 16 q^{19} - 18 q^{20} + 12 q^{21} + 8 q^{22} + 3 q^{23} - 21 q^{24} + 23 q^{25} - 24 q^{27} - 30 q^{28} + 5 q^{29} + 4 q^{30} - 46 q^{31} + 44 q^{32} + q^{33} - 7 q^{34} + 11 q^{35} + 17 q^{36} - 38 q^{37} - q^{38} - 59 q^{40} - 31 q^{41} + 18 q^{42} + 3 q^{43} + 3 q^{44} - 13 q^{45} - 36 q^{46} + 49 q^{47} - 35 q^{48} + 30 q^{49} + 15 q^{50} + 24 q^{51} - 11 q^{53} - 7 q^{54} - 10 q^{55} - 38 q^{56} + 16 q^{57} - 53 q^{58} - 23 q^{59} + 18 q^{60} - 2 q^{61} - 26 q^{62} - 12 q^{63} + 57 q^{64} - 8 q^{66} - 17 q^{68} - 3 q^{69} + 10 q^{70} + 29 q^{71} + 21 q^{72} - 48 q^{73} - 66 q^{74} - 23 q^{75} - 38 q^{76} + 51 q^{77} - 26 q^{79} - 52 q^{80} + 24 q^{81} - 45 q^{82} + 8 q^{83} + 30 q^{84} + 13 q^{85} - 38 q^{86} - 5 q^{87} - 34 q^{88} - 42 q^{89} - 4 q^{90} - 10 q^{92} + 46 q^{93} + 107 q^{94} - 44 q^{96} - 60 q^{97} + 19 q^{98} - q^{99}+O(q^{100}) \) 24 * q + 7 * q^2 - 24 * q^3 + 17 * q^4 - 13 * q^5 - 7 * q^6 - 12 * q^7 + 21 * q^8 + 24 * q^9 - 4 * q^10 - q^11 - 17 * q^12 - 18 * q^14 + 13 * q^15 + 35 * q^16 - 24 * q^17 + 7 * q^18 - 16 * q^19 - 18 * q^20 + 12 * q^21 + 8 * q^22 + 3 * q^23 - 21 * q^24 + 23 * q^25 - 24 * q^27 - 30 * q^28 + 5 * q^29 + 4 * q^30 - 46 * q^31 + 44 * q^32 + q^33 - 7 * q^34 + 11 * q^35 + 17 * q^36 - 38 * q^37 - q^38 - 59 * q^40 - 31 * q^41 + 18 * q^42 + 3 * q^43 + 3 * q^44 - 13 * q^45 - 36 * q^46 + 49 * q^47 - 35 * q^48 + 30 * q^49 + 15 * q^50 + 24 * q^51 - 11 * q^53 - 7 * q^54 - 10 * q^55 - 38 * q^56 + 16 * q^57 - 53 * q^58 - 23 * q^59 + 18 * q^60 - 2 * q^61 - 26 * q^62 - 12 * q^63 + 57 * q^64 - 8 * q^66 - 17 * q^68 - 3 * q^69 + 10 * q^70 + 29 * q^71 + 21 * q^72 - 48 * q^73 - 66 * q^74 - 23 * q^75 - 38 * q^76 + 51 * q^77 - 26 * q^79 - 52 * q^80 + 24 * q^81 - 45 * q^82 + 8 * q^83 + 30 * q^84 + 13 * q^85 - 38 * q^86 - 5 * q^87 - 34 * q^88 - 42 * q^89 - 4 * q^90 - 10 * q^92 + 46 * q^93 + 107 * q^94 - 44 * q^96 - 60 * q^97 + 19 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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