LMFDB - Embedded newform 6015.2.a.e.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6015,2,Mod(1,6015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6015, 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("6015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6015 = 3 \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6015.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:	$48.0300168158$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	6015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.917882 q^{2} -1.00000 q^{3} -1.15749 q^{4} -1.00000 q^{5} -0.917882 q^{6} -0.377330 q^{7} -2.89821 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.917882 q^{2} -1.00000 q^{3} -1.15749 q^{4} -1.00000 q^{5} -0.917882 q^{6} -0.377330 q^{7} -2.89821 q^{8} +1.00000 q^{9} -0.917882 q^{10} +4.59630 q^{11} +1.15749 q^{12} -0.575903 q^{13} -0.346345 q^{14} +1.00000 q^{15} -0.345224 q^{16} +6.01746 q^{17} +0.917882 q^{18} +0.252196 q^{19} +1.15749 q^{20} +0.377330 q^{21} +4.21886 q^{22} -6.19186 q^{23} +2.89821 q^{24} +1.00000 q^{25} -0.528611 q^{26} -1.00000 q^{27} +0.436757 q^{28} -7.90905 q^{29} +0.917882 q^{30} -5.59230 q^{31} +5.47954 q^{32} -4.59630 q^{33} +5.52332 q^{34} +0.377330 q^{35} -1.15749 q^{36} +1.02420 q^{37} +0.231486 q^{38} +0.575903 q^{39} +2.89821 q^{40} +11.6785 q^{41} +0.346345 q^{42} +2.64149 q^{43} -5.32018 q^{44} -1.00000 q^{45} -5.68339 q^{46} -5.59022 q^{47} +0.345224 q^{48} -6.85762 q^{49} +0.917882 q^{50} -6.01746 q^{51} +0.666604 q^{52} -0.00630436 q^{53} -0.917882 q^{54} -4.59630 q^{55} +1.09358 q^{56} -0.252196 q^{57} -7.25957 q^{58} -8.04531 q^{59} -1.15749 q^{60} +4.43583 q^{61} -5.13307 q^{62} -0.377330 q^{63} +5.72001 q^{64} +0.575903 q^{65} -4.21886 q^{66} -1.74890 q^{67} -6.96517 q^{68} +6.19186 q^{69} +0.346345 q^{70} -3.12057 q^{71} -2.89821 q^{72} +9.35994 q^{73} +0.940091 q^{74} -1.00000 q^{75} -0.291915 q^{76} -1.73432 q^{77} +0.528611 q^{78} +3.50541 q^{79} +0.345224 q^{80} +1.00000 q^{81} +10.7195 q^{82} -14.2352 q^{83} -0.436757 q^{84} -6.01746 q^{85} +2.42457 q^{86} +7.90905 q^{87} -13.3210 q^{88} +4.63984 q^{89} -0.917882 q^{90} +0.217306 q^{91} +7.16703 q^{92} +5.59230 q^{93} -5.13116 q^{94} -0.252196 q^{95} -5.47954 q^{96} +11.5924 q^{97} -6.29449 q^{98} +4.59630 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9}+O(q^{10}) $ 31 * q + 6 * q^2 - 31 * q^3 + 24 * q^4 - 31 * q^5 - 6 * q^6 - 4 * q^7 + 15 * q^8 + 31 * q^9	\( 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9} - 6 q^{10} - q^{11} - 24 q^{12} + 8 q^{13} + 8 q^{14} + 31 q^{15} + 10 q^{16} + 42 q^{17} + 6 q^{18} - 19 q^{19} - 24 q^{20} + 4 q^{21} + 17 q^{22} + 26 q^{23} - 15 q^{24} + 31 q^{25} + 4 q^{26} - 31 q^{27} - 16 q^{28} - 11 q^{29} + 6 q^{30} - 3 q^{31} + 41 q^{32} + q^{33} + 6 q^{34} + 4 q^{35} + 24 q^{36} + 10 q^{37} + 12 q^{38} - 8 q^{39} - 15 q^{40} + 27 q^{41} - 8 q^{42} - 37 q^{43} - 9 q^{44} - 31 q^{45} - 23 q^{46} + 48 q^{47} - 10 q^{48} + 31 q^{49} + 6 q^{50} - 42 q^{51} + 18 q^{52} + 35 q^{53} - 6 q^{54} + q^{55} + 7 q^{56} + 19 q^{57} - 26 q^{58} - 3 q^{59} + 24 q^{60} + 11 q^{61} + 49 q^{62} - 4 q^{63} - 27 q^{64} - 8 q^{65} - 17 q^{66} - 44 q^{67} + 72 q^{68} - 26 q^{69} - 8 q^{70} + 6 q^{71} + 15 q^{72} + 49 q^{73} - 6 q^{74} - 31 q^{75} - 36 q^{76} + 66 q^{77} - 4 q^{78} - 41 q^{79} - 10 q^{80} + 31 q^{81} + 9 q^{82} + 53 q^{83} + 16 q^{84} - 42 q^{85} + 3 q^{86} + 11 q^{87} + 21 q^{88} + 31 q^{89} - 6 q^{90} - 45 q^{91} + 45 q^{92} + 3 q^{93} - 4 q^{94} + 19 q^{95} - 41 q^{96} + 37 q^{97} + 41 q^{98} - q^{99}+O(q^{100}) \) 31 * q + 6 * q^2 - 31 * q^3 + 24 * q^4 - 31 * q^5 - 6 * q^6 - 4 * q^7 + 15 * q^8 + 31 * q^9 - 6 * q^10 - q^11 - 24 * q^12 + 8 * q^13 + 8 * q^14 + 31 * q^15 + 10 * q^16 + 42 * q^17 + 6 * q^18 - 19 * q^19 - 24 * q^20 + 4 * q^21 + 17 * q^22 + 26 * q^23 - 15 * q^24 + 31 * q^25 + 4 * q^26 - 31 * q^27 - 16 * q^28 - 11 * q^29 + 6 * q^30 - 3 * q^31 + 41 * q^32 + q^33 + 6 * q^34 + 4 * q^35 + 24 * q^36 + 10 * q^37 + 12 * q^38 - 8 * q^39 - 15 * q^40 + 27 * q^41 - 8 * q^42 - 37 * q^43 - 9 * q^44 - 31 * q^45 - 23 * q^46 + 48 * q^47 - 10 * q^48 + 31 * q^49 + 6 * q^50 - 42 * q^51 + 18 * q^52 + 35 * q^53 - 6 * q^54 + q^55 + 7 * q^56 + 19 * q^57 - 26 * q^58 - 3 * q^59 + 24 * q^60 + 11 * q^61 + 49 * q^62 - 4 * q^63 - 27 * q^64 - 8 * q^65 - 17 * q^66 - 44 * q^67 + 72 * q^68 - 26 * q^69 - 8 * q^70 + 6 * q^71 + 15 * q^72 + 49 * q^73 - 6 * q^74 - 31 * q^75 - 36 * q^76 + 66 * q^77 - 4 * q^78 - 41 * q^79 - 10 * q^80 + 31 * q^81 + 9 * q^82 + 53 * q^83 + 16 * q^84 - 42 * q^85 + 3 * q^86 + 11 * q^87 + 21 * q^88 + 31 * q^89 - 6 * q^90 - 45 * q^91 + 45 * q^92 + 3 * q^93 - 4 * q^94 + 19 * q^95 - 41 * q^96 + 37 * q^97 + 41 * 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 $

$2$	0.917882	0.649040	0.324520	−	0.945879i	$-0.394797\pi$
$2$	0.917882	0.649040	0.324520	+	0.945879i	$0.394797\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.15749	−0.578747
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.917882	−0.374724
$7$	−0.377330	−0.142617	−0.0713087	−	0.997454i	$-0.522718\pi$
$7$	−0.377330	−0.142617	−0.0713087	+	0.997454i	$0.522718\pi$
$8$	−2.89821	−1.02467
$9$	1.00000	0.333333
$10$	−0.917882	−0.290260
$11$	4.59630	1.38584	0.692918	−	0.721016i	$-0.256325\pi$
$11$	4.59630	1.38584	0.692918	+	0.721016i	$0.256325\pi$
$12$	1.15749	0.334139
$13$	−0.575903	−0.159727	−0.0798634	−	0.996806i	$-0.525448\pi$
$13$	−0.575903	−0.159727	−0.0798634	+	0.996806i	$0.525448\pi$
$14$	−0.346345	−0.0925645
$15$	1.00000	0.258199
$16$	−0.345224	−0.0863059
$17$	6.01746	1.45945	0.729724	−	0.683742i	$-0.239648\pi$
$17$	6.01746	1.45945	0.729724	+	0.683742i	$0.239648\pi$
$18$	0.917882	0.216347
$19$	0.252196	0.0578577	0.0289288	−	0.999581i	$-0.490790\pi$
$19$	0.252196	0.0578577	0.0289288	+	0.999581i	$0.490790\pi$
$20$	1.15749	0.258823
$21$	0.377330	0.0823402
$22$	4.21886	0.899464
$23$	−6.19186	−1.29109	−0.645546	−	0.763722i	$-0.723370\pi$
$23$	−6.19186	−1.29109	−0.645546	+	0.763722i	$0.723370\pi$
$24$	2.89821	0.591594
$25$	1.00000	0.200000
$26$	−0.528611	−0.103669
$27$	−1.00000	−0.192450
$28$	0.436757	0.0825394
$29$	−7.90905	−1.46867	−0.734336	−	0.678786i	$-0.762507\pi$
$29$	−7.90905	−1.46867	−0.734336	+	0.678786i	$0.762507\pi$
$30$	0.917882	0.167582
$31$	−5.59230	−1.00441	−0.502204	−	0.864749i	$-0.667477\pi$
$31$	−5.59230	−1.00441	−0.502204	+	0.864749i	$0.667477\pi$
$32$	5.47954	0.968654
$33$	−4.59630	−0.800113
$34$	5.52332	0.947241
$35$	0.377330	0.0637805
$36$	−1.15749	−0.192916
$37$	1.02420	0.168377	0.0841884	−	0.996450i	$-0.473170\pi$
$37$	1.02420	0.168377	0.0841884	+	0.996450i	$0.473170\pi$
$38$	0.231486	0.0375520
$39$	0.575903	0.0922183
$40$	2.89821	0.458246
$41$	11.6785	1.82388	0.911940	−	0.410324i	$-0.134584\pi$
$41$	11.6785	1.82388	0.911940	+	0.410324i	$0.134584\pi$
$42$	0.346345	0.0534421
$43$	2.64149	0.402823	0.201411	−	0.979507i	$-0.435447\pi$
$43$	2.64149	0.402823	0.201411	+	0.979507i	$0.435447\pi$
$44$	−5.32018	−0.802048
$45$	−1.00000	−0.149071
$46$	−5.68339	−0.837971
$47$	−5.59022	−0.815417	−0.407708	−	0.913112i	$-0.633672\pi$
$47$	−5.59022	−0.815417	−0.407708	+	0.913112i	$0.633672\pi$
$48$	0.345224	0.0498287
$49$	−6.85762	−0.979660
$50$	0.917882	0.129808
$51$	−6.01746	−0.842613
$52$	0.666604	0.0924413
$53$	−0.00630436	−0.000865970 0	−0.000432985	−	1.00000i	$-0.500138\pi$
$53$	−0.00630436	−0.000865970 0	−0.000432985		1.00000i	$0.500138\pi$
$54$	−0.917882	−0.124908
$55$	−4.59630	−0.619765
$56$	1.09358	0.146136
$57$	−0.252196	−0.0334041
$58$	−7.25957	−0.953228
$59$	−8.04531	−1.04741	−0.523705	−	0.851899i	$-0.675451\pi$
$59$	−8.04531	−1.04741	−0.523705	+	0.851899i	$0.675451\pi$
$60$	−1.15749	−0.149432
$61$	4.43583	0.567949	0.283975	−	0.958832i	$-0.408347\pi$
$61$	4.43583	0.567949	0.283975	+	0.958832i	$0.408347\pi$
$62$	−5.13307	−0.651901
$63$	−0.377330	−0.0475391
$64$	5.72001	0.715002
$65$	0.575903	0.0714320
$66$	−4.21886	−0.519306
$67$	−1.74890	−0.213662	−0.106831	−	0.994277i	$-0.534070\pi$
$67$	−1.74890	−0.213662	−0.106831	+	0.994277i	$0.534070\pi$
$68$	−6.96517	−0.844651
$69$	6.19186	0.745412
$70$	0.346345	0.0413961
$71$	−3.12057	−0.370343	−0.185171	−	0.982706i	$-0.559284\pi$
$71$	−3.12057	−0.370343	−0.185171	+	0.982706i	$0.559284\pi$
$72$	−2.89821	−0.341557
$73$	9.35994	1.09550	0.547749	−	0.836643i	$-0.315485\pi$
$73$	9.35994	1.09550	0.547749	+	0.836643i	$0.315485\pi$
$74$	0.940091	0.109283
$75$	−1.00000	−0.115470
$76$	−0.291915	−0.0334849
$77$	−1.73432	−0.197644
$78$	0.528611	0.0598534
$79$	3.50541	0.394390	0.197195	−	0.980364i	$-0.436817\pi$
$79$	3.50541	0.394390	0.197195	+	0.980364i	$0.436817\pi$
$80$	0.345224	0.0385972
$81$	1.00000	0.111111
$82$	10.7195	1.18377
$83$	−14.2352	−1.56252	−0.781258	−	0.624208i	$-0.785422\pi$
$83$	−14.2352	−1.56252	−0.781258	+	0.624208i	$0.785422\pi$
$84$	−0.436757	−0.0476541
$85$	−6.01746	−0.652685
$86$	2.42457	0.261448
$87$	7.90905	0.847939
$88$	−13.3210	−1.42003
$89$	4.63984	0.491822	0.245911	−	0.969292i	$-0.420913\pi$
$89$	4.63984	0.491822	0.245911	+	0.969292i	$0.420913\pi$
$90$	−0.917882	−0.0967532
$91$	0.217306	0.0227798
$92$	7.16703	0.747215
$93$	5.59230	0.579895
$94$	−5.13116	−0.529239
$95$	−0.252196	−0.0258747
$96$	−5.47954	−0.559253
$97$	11.5924	1.17703	0.588513	−	0.808488i	$-0.299714\pi$
$97$	11.5924	1.17703	0.588513	+	0.808488i	$0.299714\pi$
$98$	−6.29449	−0.635839
$99$	4.59630	0.461945
$100$	−1.15749	−0.115749
$101$	3.09006	0.307473	0.153736	−	0.988112i	$-0.450869\pi$
$101$	3.09006	0.307473	0.153736	+	0.988112i	$0.450869\pi$
$102$	−5.52332	−0.546890
$103$	6.55983	0.646359	0.323180	−	0.946338i	$-0.395248\pi$
$103$	6.55983	0.646359	0.323180	+	0.946338i	$0.395248\pi$
$104$	1.66909	0.163667
$105$	−0.377330	−0.0368237
$106$	−0.00578665	−0.000562049 0
$107$	14.3365	1.38596	0.692980	−	0.720957i	$-0.256297\pi$
$107$	14.3365	1.38596	0.692980	+	0.720957i	$0.256297\pi$
$108$	1.15749	0.111380
$109$	4.61958	0.442476	0.221238	−	0.975220i	$-0.428990\pi$
$109$	4.61958	0.442476	0.221238	+	0.975220i	$0.428990\pi$
$110$	−4.21886	−0.402252
$111$	−1.02420	−0.0972124
$112$	0.130263	0.0123087
$113$	4.18273	0.393478	0.196739	−	0.980456i	$-0.436965\pi$
$113$	4.18273	0.393478	0.196739	+	0.980456i	$0.436965\pi$
$114$	−0.231486	−0.0216806
$115$	6.19186	0.577394
$116$	9.15467	0.849989
$117$	−0.575903	−0.0532423
$118$	−7.38465	−0.679812
$119$	−2.27057	−0.208143
$120$	−2.89821	−0.264569
$121$	10.1260	0.920542
$122$	4.07156	0.368622
$123$	−11.6785	−1.05302
$124$	6.47305	0.581297
$125$	−1.00000	−0.0894427
$126$	−0.346345	−0.0308548
$127$	−18.8606	−1.67361	−0.836803	−	0.547503i	$-0.815578\pi$
$127$	−18.8606	−1.67361	−0.836803	+	0.547503i	$0.815578\pi$
$128$	−5.70878	−0.504589
$129$	−2.64149	−0.232570
$130$	0.528611	0.0463622
$131$	−8.53257	−0.745494	−0.372747	−	0.927933i	$-0.621584\pi$
$131$	−8.53257	−0.745494	−0.372747	+	0.927933i	$0.621584\pi$
$132$	5.32018	0.463063
$133$	−0.0951611	−0.00825151
$134$	−1.60528	−0.138675
$135$	1.00000	0.0860663
$136$	−17.4398	−1.49545
$137$	21.6107	1.84633	0.923163	−	0.384408i	$-0.125595\pi$
$137$	21.6107	1.84633	0.923163	+	0.384408i	$0.125595\pi$
$138$	5.68339	0.483803
$139$	−13.5766	−1.15155	−0.575777	−	0.817606i	$-0.695301\pi$
$139$	−13.5766	−1.15155	−0.575777	+	0.817606i	$0.695301\pi$
$140$	−0.436757	−0.0369127
$141$	5.59022	0.470781
$142$	−2.86431	−0.240368
$143$	−2.64702	−0.221355
$144$	−0.345224	−0.0287686
$145$	7.90905	0.656810
$146$	8.59132	0.711022
$147$	6.85762	0.565607
$148$	−1.18550	−0.0974475
$149$	−1.64055	−0.134399	−0.0671993	−	0.997740i	$-0.521406\pi$
$149$	−1.64055	−0.134399	−0.0671993	+	0.997740i	$0.521406\pi$
$150$	−0.917882	−0.0749447
$151$	21.3364	1.73633	0.868167	−	0.496272i	$-0.165298\pi$
$151$	21.3364	1.73633	0.868167	+	0.496272i	$0.165298\pi$
$152$	−0.730915	−0.0592850
$153$	6.01746	0.486483
$154$	−1.59190	−0.128279
$155$	5.59230	0.449185
$156$	−0.666604	−0.0533710
$157$	11.3880	0.908858	0.454429	−	0.890783i	$-0.349843\pi$
$157$	11.3880	0.908858	0.454429	+	0.890783i	$0.349843\pi$
$158$	3.21755	0.255975
$159$	0.00630436	0.000499968 0
$160$	−5.47954	−0.433195
$161$	2.33638	0.184132
$162$	0.917882	0.0721156
$163$	18.6089	1.45756	0.728780	−	0.684748i	$-0.240088\pi$
$163$	18.6089	1.45756	0.728780	+	0.684748i	$0.240088\pi$
$164$	−13.5178	−1.05556
$165$	4.59630	0.357821
$166$	−13.0662	−1.01414
$167$	19.5665	1.51410	0.757050	−	0.653357i	$-0.226640\pi$
$167$	19.5665	1.51410	0.757050	+	0.653357i	$0.226640\pi$
$168$	−1.09358	−0.0843716
$169$	−12.6683	−0.974487
$170$	−5.52332	−0.423619
$171$	0.252196	0.0192859
$172$	−3.05750	−0.233132
$173$	−13.7466	−1.04513	−0.522567	−	0.852598i	$-0.675025\pi$
$173$	−13.7466	−1.04513	−0.522567	+	0.852598i	$0.675025\pi$
$174$	7.25957	0.550346
$175$	−0.377330	−0.0285235
$176$	−1.58675	−0.119606
$177$	8.04531	0.604723
$178$	4.25882	0.319212
$179$	5.17041	0.386455	0.193227	−	0.981154i	$-0.438104\pi$
$179$	5.17041	0.386455	0.193227	+	0.981154i	$0.438104\pi$
$180$	1.15749	0.0862744
$181$	−17.0199	−1.26508	−0.632540	−	0.774528i	$-0.717988\pi$
$181$	−17.0199	−1.26508	−0.632540	+	0.774528i	$0.717988\pi$
$182$	0.199461	0.0147850
$183$	−4.43583	−0.327906
$184$	17.9453	1.32294
$185$	−1.02420	−0.0753004
$186$	5.13307	0.376375
$187$	27.6580	2.02256
$188$	6.47064	0.471920
$189$	0.377330	0.0274467
$190$	−0.231486	−0.0167938
$191$	8.18497	0.592243	0.296122	−	0.955150i	$-0.404307\pi$
$191$	8.18497	0.592243	0.296122	+	0.955150i	$0.404307\pi$
$192$	−5.72001	−0.412806
$193$	23.8965	1.72011	0.860054	−	0.510203i	$-0.170430\pi$
$193$	23.8965	1.72011	0.860054	+	0.510203i	$0.170430\pi$
$194$	10.6404	0.763937
$195$	−0.575903	−0.0412413
$196$	7.93765	0.566975
$197$	−5.33086	−0.379808	−0.189904	−	0.981803i	$-0.560818\pi$
$197$	−5.33086	−0.379808	−0.189904	+	0.981803i	$0.560818\pi$
$198$	4.21886	0.299821
$199$	0.906593	0.0642667	0.0321334	−	0.999484i	$-0.489770\pi$
$199$	0.906593	0.0642667	0.0321334	+	0.999484i	$0.489770\pi$
$200$	−2.89821	−0.204934
$201$	1.74890	0.123358
$202$	2.83631	0.199562
$203$	2.98432	0.209458
$204$	6.96517	0.487659
$205$	−11.6785	−0.815664
$206$	6.02115	0.419513
$207$	−6.19186	−0.430364
$208$	0.198815	0.0137854
$209$	1.15917	0.0801813
$210$	−0.346345	−0.0239000
$211$	16.3179	1.12337	0.561684	−	0.827352i	$-0.310154\pi$
$211$	16.3179	1.12337	0.561684	+	0.827352i	$0.310154\pi$
$212$	0.00729725	0.000501177 0
$213$	3.12057	0.213818
$214$	13.1592	0.899544
$215$	−2.64149	−0.180148
$216$	2.89821	0.197198
$217$	2.11015	0.143246
$218$	4.24023	0.287185
$219$	−9.35994	−0.632486
$220$	5.32018	0.358687
$221$	−3.46547	−0.233113
$222$	−0.940091	−0.0630947
$223$	22.2335	1.48887	0.744433	−	0.667697i	$-0.232720\pi$
$223$	22.2335	1.48887	0.744433	+	0.667697i	$0.232720\pi$
$224$	−2.06760	−0.138147
$225$	1.00000	0.0666667
$226$	3.83925	0.255383
$227$	21.5145	1.42797	0.713984	−	0.700162i	$-0.246889\pi$
$227$	21.5145	1.42797	0.713984	+	0.700162i	$0.246889\pi$
$228$	0.291915	0.0193325
$229$	20.7286	1.36978	0.684892	−	0.728644i	$-0.259849\pi$
$229$	20.7286	1.36978	0.684892	+	0.728644i	$0.259849\pi$
$230$	5.68339	0.374752
$231$	1.73432	0.114110
$232$	22.9220	1.50491
$233$	1.97207	0.129194	0.0645972	−	0.997911i	$-0.479424\pi$
$233$	1.97207	0.129194	0.0645972	+	0.997911i	$0.479424\pi$
$234$	−0.528611	−0.0345564
$235$	5.59022	0.364666
$236$	9.31240	0.606185
$237$	−3.50541	−0.227701
$238$	−2.08411	−0.135093
$239$	−5.08512	−0.328929	−0.164465	−	0.986383i	$-0.552590\pi$
$239$	−5.08512	−0.328929	−0.164465	+	0.986383i	$0.552590\pi$
$240$	−0.345224	−0.0222841
$241$	−27.2885	−1.75781	−0.878905	−	0.476998i	$-0.841725\pi$
$241$	−27.2885	−1.75781	−0.878905	+	0.476998i	$0.841725\pi$
$242$	9.29444	0.597469
$243$	−1.00000	−0.0641500
$244$	−5.13444	−0.328699
$245$	6.85762	0.438117
$246$	−10.7195	−0.683451
$247$	−0.145240	−0.00924142
$248$	16.2076	1.02919
$249$	14.2352	0.902119
$250$	−0.917882	−0.0580519
$251$	23.4946	1.48296	0.741482	−	0.670973i	$-0.234123\pi$
$251$	23.4946	1.48296	0.741482	+	0.670973i	$0.234123\pi$
$252$	0.436757	0.0275131
$253$	−28.4596	−1.78924
$254$	−17.3118	−1.08624
$255$	6.01746	0.376828
$256$	−16.6800	−1.04250
$257$	−20.1479	−1.25679	−0.628397	−	0.777893i	$-0.716288\pi$
$257$	−20.1479	−1.25679	−0.628397	+	0.777893i	$0.716288\pi$
$258$	−2.42457	−0.150947
$259$	−0.386460	−0.0240135
$260$	−0.666604	−0.0413410
$261$	−7.90905	−0.489558
$262$	−7.83189	−0.483856
$263$	24.8714	1.53364	0.766819	−	0.641864i	$-0.221839\pi$
$263$	24.8714	1.53364	0.766819	+	0.641864i	$0.221839\pi$
$264$	13.3210	0.819852
$265$	0.00630436	0.000387273 0
$266$	−0.0873466	−0.00535557
$267$	−4.63984	−0.283954
$268$	2.02434	0.123656
$269$	12.2100	0.744456	0.372228	−	0.928141i	$-0.378594\pi$
$269$	12.2100	0.744456	0.372228	+	0.928141i	$0.378594\pi$
$270$	0.917882	0.0558605
$271$	17.5757	1.06765	0.533825	−	0.845595i	$-0.320754\pi$
$271$	17.5757	1.06765	0.533825	+	0.845595i	$0.320754\pi$
$272$	−2.07737	−0.125959
$273$	−0.217306	−0.0131519
$274$	19.8361	1.19834
$275$	4.59630	0.277167
$276$	−7.16703	−0.431405
$277$	16.8419	1.01193	0.505966	−	0.862554i	$-0.331136\pi$
$277$	16.8419	1.01193	0.505966	+	0.862554i	$0.331136\pi$
$278$	−12.4617	−0.747406
$279$	−5.59230	−0.334802
$280$	−1.09358	−0.0653539
$281$	18.5799	1.10838	0.554191	−	0.832390i	$-0.313028\pi$
$281$	18.5799	1.10838	0.554191	+	0.832390i	$0.313028\pi$
$282$	5.13116	0.305556
$283$	18.3165	1.08881	0.544403	−	0.838824i	$-0.316756\pi$
$283$	18.3165	1.08881	0.544403	+	0.838824i	$0.316756\pi$
$284$	3.61203	0.214335
$285$	0.252196	0.0149388
$286$	−2.42965	−0.143668
$287$	−4.40666	−0.260117
$288$	5.47954	0.322885
$289$	19.2098	1.12999
$290$	7.25957	0.426297
$291$	−11.5924	−0.679556
$292$	−10.8341	−0.634016
$293$	24.4334	1.42741	0.713706	−	0.700445i	$-0.247015\pi$
$293$	24.4334	1.42741	0.713706	+	0.700445i	$0.247015\pi$
$294$	6.29449	0.367102
$295$	8.04531	0.468416
$296$	−2.96833	−0.172531
$297$	−4.59630	−0.266704
$298$	−1.50583	−0.0872302
$299$	3.56591	0.206222
$300$	1.15749	0.0668279
$301$	−0.996713	−0.0574496
$302$	19.5843	1.12695
$303$	−3.09006	−0.177520
$304$	−0.0870639	−0.00499346
$305$	−4.43583	−0.253995
$306$	5.52332	0.315747
$307$	−4.39675	−0.250936	−0.125468	−	0.992098i	$-0.540043\pi$
$307$	−4.39675	−0.250936	−0.125468	+	0.992098i	$0.540043\pi$
$308$	2.00747	0.114386
$309$	−6.55983	−0.373176
$310$	5.13307	0.291539
$311$	−18.3458	−1.04030	−0.520148	−	0.854076i	$-0.674123\pi$
$311$	−18.3458	−1.04030	−0.520148	+	0.854076i	$0.674123\pi$
$312$	−1.66909	−0.0944933
$313$	−31.9081	−1.80355	−0.901776	−	0.432203i	$-0.857736\pi$
$313$	−31.9081	−1.80355	−0.901776	+	0.432203i	$0.857736\pi$
$314$	10.4528	0.589886
$315$	0.377330	0.0212602
$316$	−4.05749	−0.228252
$317$	−9.26440	−0.520340	−0.260170	−	0.965563i	$-0.583779\pi$
$317$	−9.26440	−0.520340	−0.260170	+	0.965563i	$0.583779\pi$
$318$	0.00578665	0.000324499 0
$319$	−36.3523	−2.03534
$320$	−5.72001	−0.319758
$321$	−14.3365	−0.800185
$322$	2.14452	0.119509
$323$	1.51758	0.0844403
$324$	−1.15749	−0.0643052
$325$	−0.575903	−0.0319454
$326$	17.0807	0.946015
$327$	−4.61958	−0.255464
$328$	−33.8468	−1.86888
$329$	2.10936	0.116293
$330$	4.21886	0.232241
$331$	−26.3568	−1.44870	−0.724350	−	0.689433i	$-0.757860\pi$
$331$	−26.3568	−1.44870	−0.724350	+	0.689433i	$0.757860\pi$
$332$	16.4771	0.904301
$333$	1.02420	0.0561256
$334$	17.9597	0.982712
$335$	1.74890	0.0955526
$336$	−0.130263	−0.00710645
$337$	−1.08577	−0.0591456	−0.0295728	−	0.999563i	$-0.509415\pi$
$337$	−1.08577	−0.0591456	−0.0295728	+	0.999563i	$0.509415\pi$
$338$	−11.6280	−0.632482
$339$	−4.18273	−0.227175
$340$	6.96517	0.377739
$341$	−25.7039	−1.39194
$342$	0.231486	0.0125173
$343$	5.22890	0.282334
$344$	−7.65557	−0.412761
$345$	−6.19186	−0.333358
$346$	−12.6177	−0.678334
$347$	−25.8739	−1.38898	−0.694492	−	0.719501i	$-0.744371\pi$
$347$	−25.8739	−1.38898	−0.694492	+	0.719501i	$0.744371\pi$
$348$	−9.15467	−0.490742
$349$	−13.0215	−0.697023	−0.348512	−	0.937304i	$-0.613313\pi$
$349$	−13.0215	−0.697023	−0.348512	+	0.937304i	$0.613313\pi$
$350$	−0.346345	−0.0185129
$351$	0.575903	0.0307394
$352$	25.1856	1.34240
$353$	24.8121	1.32061	0.660306	−	0.750997i	$-0.270427\pi$
$353$	24.8121	1.32061	0.660306	+	0.750997i	$0.270427\pi$
$354$	7.38465	0.392490
$355$	3.12057	0.165622
$356$	−5.37058	−0.284640
$357$	2.27057	0.120171
$358$	4.74583	0.250825
$359$	11.8478	0.625301	0.312650	−	0.949868i	$-0.398783\pi$
$359$	11.8478	0.625301	0.312650	+	0.949868i	$0.398783\pi$
$360$	2.89821	0.152749
$361$	−18.9364	−0.996652
$362$	−15.6223	−0.821088
$363$	−10.1260	−0.531475
$364$	−0.251530	−0.0131837
$365$	−9.35994	−0.489922
$366$	−4.07156	−0.212824
$367$	−23.9267	−1.24896	−0.624481	−	0.781040i	$-0.714689\pi$
$367$	−23.9267	−1.24896	−0.624481	+	0.781040i	$0.714689\pi$
$368$	2.13758	0.111429
$369$	11.6785	0.607960
$370$	−0.940091	−0.0488730
$371$	0.00237882	0.000123502 0
$372$	−6.47305	−0.335612
$373$	−2.13758	−0.110680	−0.0553399	−	0.998468i	$-0.517624\pi$
$373$	−2.13758	−0.110680	−0.0553399	+	0.998468i	$0.517624\pi$
$374$	25.3868	1.31272
$375$	1.00000	0.0516398
$376$	16.2016	0.835534
$377$	4.55484	0.234586
$378$	0.346345	0.0178140
$379$	6.86471	0.352616	0.176308	−	0.984335i	$-0.443584\pi$
$379$	6.86471	0.352616	0.176308	+	0.984335i	$0.443584\pi$
$380$	0.291915	0.0149749
$381$	18.8606	0.966257
$382$	7.51283	0.384390
$383$	18.9613	0.968875	0.484437	−	0.874826i	$-0.339024\pi$
$383$	18.9613	0.968875	0.484437	+	0.874826i	$0.339024\pi$
$384$	5.70878	0.291325
$385$	1.73432	0.0883893
$386$	21.9342	1.11642
$387$	2.64149	0.134274
$388$	−13.4181	−0.681200
$389$	−16.7844	−0.851004	−0.425502	−	0.904957i	$-0.639903\pi$
$389$	−16.7844	−0.851004	−0.425502	+	0.904957i	$0.639903\pi$
$390$	−0.528611	−0.0267673
$391$	−37.2593	−1.88428
$392$	19.8748	1.00383
$393$	8.53257	0.430411
$394$	−4.89310	−0.246511
$395$	−3.50541	−0.176376
$396$	−5.32018	−0.267349
$397$	2.87522	0.144303	0.0721516	−	0.997394i	$-0.477013\pi$
$397$	2.87522	0.144303	0.0721516	+	0.997394i	$0.477013\pi$
$398$	0.832146	0.0417117
$399$	0.0951611	0.00476401
$400$	−0.345224	−0.0172612
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	1.60528	0.0800642
$403$	3.22062	0.160431
$404$	−3.57673	−0.177949
$405$	−1.00000	−0.0496904
$406$	2.73926	0.135947
$407$	4.70751	0.233343
$408$	17.4398	0.863401
$409$	35.5587	1.75826	0.879132	−	0.476578i	$-0.158123\pi$
$409$	35.5587	1.75826	0.879132	+	0.476578i	$0.158123\pi$
$410$	−10.7195	−0.529399
$411$	−21.6107	−1.06598
$412$	−7.59296	−0.374078
$413$	3.03574	0.149379
$414$	−5.68339	−0.279324
$415$	14.2352	0.698778
$416$	−3.15568	−0.154720
$417$	13.5766	0.664850
$418$	1.06398	0.0520409
$419$	−17.2695	−0.843668	−0.421834	−	0.906673i	$-0.638614\pi$
$419$	−17.2695	−0.843668	−0.421834	+	0.906673i	$0.638614\pi$
$420$	0.436757	0.0213116
$421$	27.2333	1.32727	0.663636	−	0.748056i	$-0.269013\pi$
$421$	27.2333	1.32727	0.663636	+	0.748056i	$0.269013\pi$
$422$	14.9779	0.729111
$423$	−5.59022	−0.271806
$424$	0.0182713	0.000887334 0
$425$	6.01746	0.291890
$426$	2.86431	0.138776
$427$	−1.67377	−0.0809995
$428$	−16.5944	−0.802120
$429$	2.64702	0.127799
$430$	−2.42457	−0.116923
$431$	32.8617	1.58289	0.791446	−	0.611239i	$-0.209328\pi$
$431$	32.8617	1.58289	0.791446	+	0.611239i	$0.209328\pi$
$432$	0.345224	0.0166096
$433$	−0.526453	−0.0252997	−0.0126498	−	0.999920i	$-0.504027\pi$
$433$	−0.526453	−0.0252997	−0.0126498	+	0.999920i	$0.504027\pi$
$434$	1.93686	0.0929725
$435$	−7.90905	−0.379210
$436$	−5.34714	−0.256081
$437$	−1.56156	−0.0746996
$438$	−8.59132	−0.410509
$439$	−28.4478	−1.35774	−0.678869	−	0.734259i	$-0.737530\pi$
$439$	−28.4478	−1.35774	−0.678869	+	0.734259i	$0.737530\pi$
$440$	13.3210	0.635055
$441$	−6.85762	−0.326553
$442$	−3.18090	−0.151300
$443$	41.4568	1.96967	0.984837	−	0.173483i	$-0.0555022\pi$
$443$	41.4568	1.96967	0.984837	+	0.173483i	$0.0555022\pi$
$444$	1.18550	0.0562613
$445$	−4.63984	−0.219949
$446$	20.4077	0.966334
$447$	1.64055	0.0775951
$448$	−2.15833	−0.101972
$449$	−2.06723	−0.0975588	−0.0487794	−	0.998810i	$-0.515533\pi$
$449$	−2.06723	−0.0975588	−0.0487794	+	0.998810i	$0.515533\pi$
$450$	0.917882	0.0432694
$451$	53.6780	2.52760
$452$	−4.84148	−0.227724
$453$	−21.3364	−1.00247
$454$	19.7478	0.926808
$455$	−0.217306	−0.0101874
$456$	0.730915	0.0342282
$457$	9.03281	0.422537	0.211268	−	0.977428i	$-0.432241\pi$
$457$	9.03281	0.422537	0.211268	+	0.977428i	$0.432241\pi$
$458$	19.0264	0.889046
$459$	−6.01746	−0.280871
$460$	−7.16703	−0.334165
$461$	19.5191	0.909094	0.454547	−	0.890723i	$-0.349801\pi$
$461$	19.5191	0.909094	0.454547	+	0.890723i	$0.349801\pi$
$462$	1.59190	0.0740620
$463$	−42.1671	−1.95967	−0.979836	−	0.199805i	$-0.935969\pi$
$463$	−42.1671	−1.95967	−0.979836	+	0.199805i	$0.935969\pi$
$464$	2.73039	0.126755
$465$	−5.59230	−0.259337
$466$	1.81012	0.0838524
$467$	−3.56548	−0.164991	−0.0824954	−	0.996591i	$-0.526289\pi$
$467$	−3.56548	−0.164991	−0.0824954	+	0.996591i	$0.526289\pi$
$468$	0.666604	0.0308138
$469$	0.659913	0.0304719
$470$	5.13116	0.236683
$471$	−11.3880	−0.524730
$472$	23.3170	1.07325
$473$	12.1411	0.558247
$474$	−3.21755	−0.147787
$475$	0.252196	0.0115715
$476$	2.62817	0.120462
$477$	−0.00630436	−0.000288657 0
$478$	−4.66754	−0.213488
$479$	12.1787	0.556458	0.278229	−	0.960515i	$-0.410253\pi$
$479$	12.1787	0.556458	0.278229	+	0.960515i	$0.410253\pi$
$480$	5.47954	0.250105
$481$	−0.589837	−0.0268943
$482$	−25.0477	−1.14089
$483$	−2.33638	−0.106309
$484$	−11.7207	−0.532761
$485$	−11.5924	−0.526382
$486$	−0.917882	−0.0416360
$487$	35.9254	1.62794	0.813969	−	0.580909i	$-0.197303\pi$
$487$	35.9254	1.62794	0.813969	+	0.580909i	$0.197303\pi$
$488$	−12.8559	−0.581961
$489$	−18.6089	−0.841522
$490$	6.29449	0.284356
$491$	−26.9205	−1.21491	−0.607453	−	0.794356i	$-0.707809\pi$
$491$	−26.9205	−1.21491	−0.607453	+	0.794356i	$0.707809\pi$
$492$	13.5178	0.609430
$493$	−47.5924	−2.14345
$494$	−0.133313	−0.00599806
$495$	−4.59630	−0.206588
$496$	1.93060	0.0866863
$497$	1.17748	0.0528174
$498$	13.0662	0.585512
$499$	−24.8159	−1.11091	−0.555456	−	0.831546i	$-0.687456\pi$
$499$	−24.8159	−1.11091	−0.555456	+	0.831546i	$0.687456\pi$
$500$	1.15749	0.0517647
$501$	−19.5665	−0.874166
$502$	21.5652	0.962503
$503$	−24.3805	−1.08707	−0.543537	−	0.839386i	$-0.682915\pi$
$503$	−24.3805	−1.08707	−0.543537	+	0.839386i	$0.682915\pi$
$504$	1.09358	0.0487120
$505$	−3.09006	−0.137506
$506$	−26.1226	−1.16129
$507$	12.6683	0.562621
$508$	21.8310	0.968594
$509$	21.4726	0.951758	0.475879	−	0.879511i	$-0.342130\pi$
$509$	21.4726	0.951758	0.475879	+	0.879511i	$0.342130\pi$
$510$	5.52332	0.244577
$511$	−3.53179	−0.156237
$512$	−3.89272	−0.172036
$513$	−0.252196	−0.0111347
$514$	−18.4934	−0.815710
$515$	−6.55983	−0.289061
$516$	3.05750	0.134599
$517$	−25.6943	−1.13003
$518$	−0.354725	−0.0155857
$519$	13.7466	0.603408
$520$	−1.66909	−0.0731942
$521$	2.43852	0.106834	0.0534168	−	0.998572i	$-0.482989\pi$
$521$	2.43852	0.106834	0.0534168	+	0.998572i	$0.482989\pi$
$522$	−7.25957	−0.317743
$523$	−5.38621	−0.235522	−0.117761	−	0.993042i	$-0.537572\pi$
$523$	−5.38621	−0.235522	−0.117761	+	0.993042i	$0.537572\pi$
$524$	9.87640	0.431452
$525$	0.377330	0.0164680
$526$	22.8290	0.995393
$527$	−33.6515	−1.46588
$528$	1.58675	0.0690545
$529$	15.3391	0.666917
$530$	0.00578665	0.000251356 0
$531$	−8.04531	−0.349137
$532$	0.110148	0.00477554
$533$	−6.72570	−0.291322
$534$	−4.25882	−0.184297
$535$	−14.3365	−0.619820
$536$	5.06867	0.218933
$537$	−5.17041	−0.223120
$538$	11.2073	0.483182
$539$	−31.5197	−1.35765
$540$	−1.15749	−0.0498106
$541$	−5.60927	−0.241161	−0.120581	−	0.992704i	$-0.538476\pi$
$541$	−5.60927	−0.241161	−0.120581	+	0.992704i	$0.538476\pi$
$542$	16.1325	0.692948
$543$	17.0199	0.730394
$544$	32.9729	1.41370
$545$	−4.61958	−0.197881
$546$	−0.199461	−0.00853614
$547$	−10.8734	−0.464914	−0.232457	−	0.972607i	$-0.574677\pi$
$547$	−10.8734	−0.464914	−0.232457	+	0.972607i	$0.574677\pi$
$548$	−25.0142	−1.06855
$549$	4.43583	0.189316
$550$	4.21886	0.179893
$551$	−1.99463	−0.0849740
$552$	−17.9453	−0.763802
$553$	−1.32270	−0.0562469
$554$	15.4589	0.656784
$555$	1.02420	0.0434747
$556$	15.7149	0.666458
$557$	−1.59485	−0.0675761	−0.0337880	−	0.999429i	$-0.510757\pi$
$557$	−1.59485	−0.0675761	−0.0337880	+	0.999429i	$0.510757\pi$
$558$	−5.13307	−0.217300
$559$	−1.52124	−0.0643416
$560$	−0.130263	−0.00550463
$561$	−27.6580	−1.16772
$562$	17.0541	0.719384
$563$	−30.0306	−1.26564	−0.632819	−	0.774299i	$-0.718102\pi$
$563$	−30.0306	−1.26564	−0.632819	+	0.774299i	$0.718102\pi$
$564$	−6.47064	−0.272463
$565$	−4.18273	−0.175969
$566$	16.8124	0.706679
$567$	−0.377330	−0.0158464
$568$	9.04404	0.379479
$569$	2.06323	0.0864950	0.0432475	−	0.999064i	$-0.486230\pi$
$569$	2.06323	0.0864950	0.0432475	+	0.999064i	$0.486230\pi$
$570$	0.231486	0.00969588
$571$	41.2778	1.72742	0.863711	−	0.503987i	$-0.168134\pi$
$571$	41.2778	1.72742	0.863711	+	0.503987i	$0.168134\pi$
$572$	3.06391	0.128109
$573$	−8.18497	−0.341932
$574$	−4.04480	−0.168826
$575$	−6.19186	−0.258218
$576$	5.72001	0.238334
$577$	36.9720	1.53917	0.769583	−	0.638547i	$-0.220464\pi$
$577$	36.9720	1.53917	0.769583	+	0.638547i	$0.220464\pi$
$578$	17.6324	0.733409
$579$	−23.8965	−0.993105
$580$	−9.15467	−0.380127
$581$	5.37137	0.222842
$582$	−10.6404	−0.441059
$583$	−0.0289767	−0.00120009
$584$	−27.1270	−1.12252
$585$	0.575903	0.0238107
$586$	22.4269	0.926448
$587$	33.5853	1.38621	0.693107	−	0.720835i	$-0.256241\pi$
$587$	33.5853	1.38621	0.693107	+	0.720835i	$0.256241\pi$
$588$	−7.93765	−0.327343
$589$	−1.41036	−0.0581127
$590$	7.38465	0.304021
$591$	5.33086	0.219282
$592$	−0.353577	−0.0145319
$593$	−23.1689	−0.951432	−0.475716	−	0.879599i	$-0.657811\pi$
$593$	−23.1689	−0.951432	−0.475716	+	0.879599i	$0.657811\pi$
$594$	−4.21886	−0.173102
$595$	2.27057	0.0930843
$596$	1.89892	0.0777828
$597$	−0.906593	−0.0371044
$598$	3.27308	0.133846
$599$	1.49388	0.0610383	0.0305192	−	0.999534i	$-0.490284\pi$
$599$	1.49388	0.0610383	0.0305192	+	0.999534i	$0.490284\pi$
$600$	2.89821	0.118319
$601$	−6.12833	−0.249980	−0.124990	−	0.992158i	$-0.539890\pi$
$601$	−6.12833	−0.249980	−0.124990	+	0.992158i	$0.539890\pi$
$602$	−0.914865	−0.0372871
$603$	−1.74890	−0.0712207
$604$	−24.6968	−1.00490
$605$	−10.1260	−0.411679
$606$	−2.83631	−0.115217
$607$	−24.0660	−0.976810	−0.488405	−	0.872617i	$-0.662421\pi$
$607$	−24.0660	−0.976810	−0.488405	+	0.872617i	$0.662421\pi$
$608$	1.38192	0.0560441
$609$	−2.98432	−0.120931
$610$	−4.07156	−0.164853
$611$	3.21942	0.130244
$612$	−6.96517	−0.281550
$613$	2.10084	0.0848520	0.0424260	−	0.999100i	$-0.486491\pi$
$613$	2.10084	0.0848520	0.0424260	+	0.999100i	$0.486491\pi$
$614$	−4.03569	−0.162867
$615$	11.6785	0.470924
$616$	5.02642	0.202520
$617$	−9.24863	−0.372336	−0.186168	−	0.982518i	$-0.559607\pi$
$617$	−9.24863	−0.372336	−0.186168	+	0.982518i	$0.559607\pi$
$618$	−6.02115	−0.242206
$619$	5.15149	0.207056	0.103528	−	0.994627i	$-0.466987\pi$
$619$	5.15149	0.207056	0.103528	+	0.994627i	$0.466987\pi$
$620$	−6.47305	−0.259964
$621$	6.19186	0.248471
$622$	−16.8393	−0.675194
$623$	−1.75075	−0.0701424
$624$	−0.198815	−0.00795898
$625$	1.00000	0.0400000
$626$	−29.2879	−1.17058
$627$	−1.15917	−0.0462927
$628$	−13.1815	−0.525999
$629$	6.16306	0.245737
$630$	0.346345	0.0137987
$631$	−36.2077	−1.44141	−0.720704	−	0.693243i	$-0.756181\pi$
$631$	−36.2077	−1.44141	−0.720704	+	0.693243i	$0.756181\pi$
$632$	−10.1594	−0.404119
$633$	−16.3179	−0.648576
$634$	−8.50362	−0.337722
$635$	18.8606	0.748460
$636$	−0.00729725	−0.000289355 0
$637$	3.94933	0.156478
$638$	−33.3671	−1.32102
$639$	−3.12057	−0.123448
$640$	5.70878	0.225659
$641$	−41.5433	−1.64086	−0.820430	−	0.571746i	$-0.806266\pi$
$641$	−41.5433	−1.64086	−0.820430	+	0.571746i	$0.806266\pi$
$642$	−13.1592	−0.519352
$643$	−11.7488	−0.463329	−0.231664	−	0.972796i	$-0.574417\pi$
$643$	−11.7488	−0.463329	−0.231664	+	0.972796i	$0.574417\pi$
$644$	−2.70434	−0.106566
$645$	2.64149	0.104008
$646$	1.39296	0.0548052
$647$	−37.4338	−1.47167	−0.735837	−	0.677159i	$-0.763211\pi$
$647$	−37.4338	−1.47167	−0.735837	+	0.677159i	$0.763211\pi$
$648$	−2.89821	−0.113852
$649$	−36.9787	−1.45154
$650$	−0.528611	−0.0207338
$651$	−2.11015	−0.0827031
$652$	−21.5396	−0.843557
$653$	4.16326	0.162921	0.0814605	−	0.996677i	$-0.474042\pi$
$653$	4.16326	0.162921	0.0814605	+	0.996677i	$0.474042\pi$
$654$	−4.24023	−0.165806
$655$	8.53257	0.333395
$656$	−4.03170	−0.157412
$657$	9.35994	0.365166
$658$	1.93614	0.0754787
$659$	15.3104	0.596410	0.298205	−	0.954502i	$-0.403612\pi$
$659$	15.3104	0.596410	0.298205	+	0.954502i	$0.403612\pi$
$660$	−5.32018	−0.207088
$661$	−18.2638	−0.710379	−0.355189	−	0.934794i	$-0.615584\pi$
$661$	−18.2638	−0.710379	−0.355189	+	0.934794i	$0.615584\pi$
$662$	−24.1924	−0.940264
$663$	3.46547	0.134588
$664$	41.2565	1.60106
$665$	0.0951611	0.00369019
$666$	0.940091	0.0364278
$667$	48.9717	1.89619
$668$	−22.6481	−0.876280
$669$	−22.2335	−0.859597
$670$	1.60528	0.0620175
$671$	20.3884	0.787085
$672$	2.06760	0.0797592
$673$	10.1856	0.392626	0.196313	−	0.980541i	$-0.437103\pi$
$673$	10.1856	0.392626	0.196313	+	0.980541i	$0.437103\pi$
$674$	−0.996608	−0.0383879
$675$	−1.00000	−0.0384900
$676$	14.6635	0.563981
$677$	33.6696	1.29403	0.647014	−	0.762478i	$-0.276018\pi$
$677$	33.6696	1.29403	0.647014	+	0.762478i	$0.276018\pi$
$678$	−3.83925	−0.147446
$679$	−4.37415	−0.167864
$680$	17.4398	0.668787
$681$	−21.5145	−0.824437
$682$	−23.5931	−0.903428
$683$	12.4002	0.474480	0.237240	−	0.971451i	$-0.423757\pi$
$683$	12.4002	0.474480	0.237240	+	0.971451i	$0.423757\pi$
$684$	−0.291915	−0.0111616
$685$	−21.6107	−0.825702
$686$	4.79951	0.183246
$687$	−20.7286	−0.790846
$688$	−0.911903	−0.0347660
$689$	0.00363070	0.000138319 0
$690$	−5.68339	−0.216363
$691$	−0.842856	−0.0320638	−0.0160319	−	0.999871i	$-0.505103\pi$
$691$	−0.842856	−0.0320638	−0.0160319	+	0.999871i	$0.505103\pi$
$692$	15.9116	0.604867
$693$	−1.73432	−0.0658815
$694$	−23.7492	−0.901506
$695$	13.5766	0.514991
$696$	−22.9220	−0.868858
$697$	70.2751	2.66186
$698$	−11.9522	−0.452396
$699$	−1.97207	−0.0745904
$700$	0.436757	0.0165079
$701$	22.0672	0.833467	0.416734	−	0.909029i	$-0.363175\pi$
$701$	22.0672	0.833467	0.416734	+	0.909029i	$0.363175\pi$
$702$	0.528611	0.0199511
$703$	0.258298	0.00974189
$704$	26.2909	0.990875
$705$	−5.59022	−0.210540
$706$	22.7745	0.857131
$707$	−1.16597	−0.0438510
$708$	−9.31240	−0.349981
$709$	15.6077	0.586158	0.293079	−	0.956088i	$-0.405320\pi$
$709$	15.6077	0.586158	0.293079	+	0.956088i	$0.405320\pi$
$710$	2.86431	0.107496
$711$	3.50541	0.131463
$712$	−13.4472	−0.503955
$713$	34.6267	1.29678
$714$	2.08411	0.0779960
$715$	2.64702	0.0989930
$716$	−5.98472	−0.223659
$717$	5.08512	0.189907
$718$	10.8748	0.405845
$719$	21.7722	0.811965	0.405983	−	0.913881i	$-0.366929\pi$
$719$	21.7722	0.811965	0.405983	+	0.913881i	$0.366929\pi$
$720$	0.345224	0.0128657
$721$	−2.47522	−0.0921821
$722$	−17.3814	−0.646868
$723$	27.2885	1.01487
$724$	19.7004	0.732161
$725$	−7.90905	−0.293735
$726$	−9.29444	−0.344949
$727$	49.1607	1.82327	0.911635	−	0.411001i	$-0.134821\pi$
$727$	49.1607	1.82327	0.911635	+	0.411001i	$0.134821\pi$
$728$	−0.629796	−0.0233418
$729$	1.00000	0.0370370
$730$	−8.59132	−0.317979
$731$	15.8950	0.587899
$732$	5.13444	0.189774
$733$	2.75506	0.101760	0.0508802	−	0.998705i	$-0.483797\pi$
$733$	2.75506	0.101760	0.0508802	+	0.998705i	$0.483797\pi$
$734$	−21.9618	−0.810627
$735$	−6.85762	−0.252947
$736$	−33.9285	−1.25062
$737$	−8.03846	−0.296101
$738$	10.7195	0.394591
$739$	−0.293751	−0.0108058	−0.00540290	−	0.999985i	$-0.501720\pi$
$739$	−0.293751	−0.0108058	−0.00540290	+	0.999985i	$0.501720\pi$
$740$	1.18550	0.0435798
$741$	0.145240	0.00533554
$742$	0.00218348	8.01581e−5 0
$743$	−3.76972	−0.138298	−0.0691488	−	0.997606i	$-0.522028\pi$
$743$	−3.76972	−0.138298	−0.0691488	+	0.997606i	$0.522028\pi$
$744$	−16.2076	−0.594201
$745$	1.64055	0.0601049
$746$	−1.96205	−0.0718357
$747$	−14.2352	−0.520839
$748$	−32.0140	−1.17055
$749$	−5.40959	−0.197662
$750$	0.917882	0.0335163
$751$	−28.6336	−1.04486	−0.522428	−	0.852684i	$-0.674974\pi$
$751$	−28.6336	−1.04486	−0.522428	+	0.852684i	$0.674974\pi$
$752$	1.92987	0.0703753
$753$	−23.4946	−0.856189
$754$	4.18081	0.152256
$755$	−21.3364	−0.776512
$756$	−0.436757	−0.0158847
$757$	41.4731	1.50737	0.753684	−	0.657237i	$-0.228275\pi$
$757$	41.4731	1.50737	0.753684	+	0.657237i	$0.228275\pi$
$758$	6.30099	0.228862
$759$	28.4596	1.03302
$760$	0.730915	0.0265131
$761$	−1.59582	−0.0578485	−0.0289242	−	0.999582i	$-0.509208\pi$
$761$	−1.59582	−0.0578485	−0.0289242	+	0.999582i	$0.509208\pi$
$762$	17.3118	0.627140
$763$	−1.74311	−0.0631048
$764$	−9.47404	−0.342759
$765$	−6.01746	−0.217562
$766$	17.4042	0.628839
$767$	4.63332	0.167300
$768$	16.6800	0.601888
$769$	25.0321	0.902681	0.451340	−	0.892352i	$-0.350946\pi$
$769$	25.0321	0.902681	0.451340	+	0.892352i	$0.350946\pi$
$770$	1.59190	0.0573682
$771$	20.1479	0.725610
$772$	−27.6600	−0.995507
$773$	28.3560	1.01990	0.509948	−	0.860205i	$-0.329665\pi$
$773$	28.3560	1.01990	0.509948	+	0.860205i	$0.329665\pi$
$774$	2.42457	0.0871494
$775$	−5.59230	−0.200881
$776$	−33.5970	−1.20606
$777$	0.386460	0.0138642
$778$	−15.4061	−0.552336
$779$	2.94528	0.105525
$780$	0.666604	0.0238682
$781$	−14.3431	−0.513235
$782$	−34.1996	−1.22297
$783$	7.90905	0.282646
$784$	2.36741	0.0845505
$785$	−11.3880	−0.406454
$786$	7.83189	0.279354
$787$	−4.30244	−0.153366	−0.0766828	−	0.997056i	$-0.524433\pi$
$787$	−4.30244	−0.153366	−0.0766828	+	0.997056i	$0.524433\pi$
$788$	6.17044	0.219813
$789$	−24.8714	−0.885446
$790$	−3.21755	−0.114475
$791$	−1.57827	−0.0561169
$792$	−13.3210	−0.473342
$793$	−2.55461	−0.0907167
$794$	2.63911	0.0936586
$795$	−0.00630436	−0.000223592 0
$796$	−1.04938	−0.0371941
$797$	−15.5352	−0.550285	−0.275143	−	0.961403i	$-0.588725\pi$
$797$	−15.5352	−0.550285	−0.275143	+	0.961403i	$0.588725\pi$
$798$	0.0873466	0.00309204
$799$	−33.6389	−1.19006
$800$	5.47954	0.193731
$801$	4.63984	0.163941
$802$	0.917882	0.0324115
$803$	43.0211	1.51818
$804$	−2.02434	−0.0713929
$805$	−2.33638	−0.0823464
$806$	2.95615	0.104126
$807$	−12.2100	−0.429812
$808$	−8.95564	−0.315058
$809$	42.2094	1.48400	0.742002	−	0.670398i	$-0.233876\pi$
$809$	42.2094	1.48400	0.742002	+	0.670398i	$0.233876\pi$
$810$	−0.917882	−0.0322511
$811$	−3.91997	−0.137649	−0.0688245	−	0.997629i	$-0.521925\pi$
$811$	−3.91997	−0.137649	−0.0688245	+	0.997629i	$0.521925\pi$
$812$	−3.45433	−0.121223
$813$	−17.5757	−0.616408
$814$	4.32094	0.151449
$815$	−18.6089	−0.651840
$816$	2.07737	0.0727225
$817$	0.666172	0.0233064
$818$	32.6387	1.14118
$819$	0.217306	0.00759327
$820$	13.5178	0.472063
$821$	24.9474	0.870670	0.435335	−	0.900268i	$-0.356630\pi$
$821$	24.9474	0.870670	0.435335	+	0.900268i	$0.356630\pi$
$822$	−19.8361	−0.691862
$823$	23.9008	0.833128	0.416564	−	0.909106i	$-0.363234\pi$
$823$	23.9008	0.833128	0.416564	+	0.909106i	$0.363234\pi$
$824$	−19.0117	−0.662305
$825$	−4.59630	−0.160023
$826$	2.78645	0.0969530
$827$	−9.37916	−0.326145	−0.163073	−	0.986614i	$-0.552140\pi$
$827$	−9.37916	−0.326145	−0.163073	+	0.986614i	$0.552140\pi$
$828$	7.16703	0.249072
$829$	−10.9650	−0.380829	−0.190415	−	0.981704i	$-0.560983\pi$
$829$	−10.9650	−0.380829	−0.190415	+	0.981704i	$0.560983\pi$
$830$	13.0662	0.453535
$831$	−16.8419	−0.584239
$832$	−3.29417	−0.114205
$833$	−41.2655	−1.42976
$834$	12.4617	0.431515
$835$	−19.5665	−0.677126
$836$	−1.34173	−0.0464046
$837$	5.59230	0.193298
$838$	−15.8513	−0.547575
$839$	23.4841	0.810762	0.405381	−	0.914148i	$-0.367139\pi$
$839$	23.4841	0.810762	0.405381	+	0.914148i	$0.367139\pi$
$840$	1.09358	0.0377321
$841$	33.5530	1.15700
$842$	24.9970	0.861453
$843$	−18.5799	−0.639924
$844$	−18.8878	−0.650145
$845$	12.6683	0.435804
$846$	−5.13116	−0.176413
$847$	−3.82083	−0.131285
$848$	0.00217641	7.47383e−5 0
$849$	−18.3165	−0.628622
$850$	5.52332	0.189448
$851$	−6.34167	−0.217390
$852$	−3.61203	−0.123746
$853$	35.7244	1.22318	0.611590	−	0.791175i	$-0.290530\pi$
$853$	35.7244	1.22318	0.611590	+	0.791175i	$0.290530\pi$
$854$	−1.53632	−0.0525719
$855$	−0.252196	−0.00862491
$856$	−41.5501	−1.42015
$857$	48.8055	1.66716	0.833582	−	0.552395i	$-0.186286\pi$
$857$	48.8055	1.66716	0.833582	+	0.552395i	$0.186286\pi$
$858$	2.42965	0.0829470
$859$	38.2408	1.30476	0.652380	−	0.757892i	$-0.273771\pi$
$859$	38.2408	1.30476	0.652380	+	0.757892i	$0.273771\pi$
$860$	3.05750	0.104260
$861$	4.40666	0.150179
$862$	30.1632	1.02736
$863$	−9.91767	−0.337601	−0.168801	−	0.985650i	$-0.553989\pi$
$863$	−9.91767	−0.337601	−0.168801	+	0.985650i	$0.553989\pi$
$864$	−5.47954	−0.186418
$865$	13.7466	0.467398
$866$	−0.483221	−0.0164205
$867$	−19.2098	−0.652400
$868$	−2.44248	−0.0829031
$869$	16.1119	0.546560
$870$	−7.25957	−0.246122
$871$	1.00720	0.0341276
$872$	−13.3885	−0.453392
$873$	11.5924	0.392342
$874$	−1.43333	−0.0484830
$875$	0.377330	0.0127561
$876$	10.8341	0.366049
$877$	−43.2811	−1.46150	−0.730750	−	0.682646i	$-0.760829\pi$
$877$	−43.2811	−1.46150	−0.730750	+	0.682646i	$0.760829\pi$
$878$	−26.1117	−0.881227
$879$	−24.4334	−0.824117
$880$	1.58675	0.0534894
$881$	46.8971	1.58000	0.790001	−	0.613105i	$-0.210080\pi$
$881$	46.8971	1.58000	0.790001	+	0.613105i	$0.210080\pi$
$882$	−6.29449	−0.211946
$883$	−32.3438	−1.08846	−0.544228	−	0.838937i	$-0.683177\pi$
$883$	−32.3438	−1.08846	−0.544228	+	0.838937i	$0.683177\pi$
$884$	4.01126	0.134913
$885$	−8.04531	−0.270440
$886$	38.0525	1.27840
$887$	−12.5677	−0.421981	−0.210991	−	0.977488i	$-0.567669\pi$
$887$	−12.5677	−0.421981	−0.210991	+	0.977488i	$0.567669\pi$
$888$	2.96833	0.0996106
$889$	7.11667	0.238686
$890$	−4.25882	−0.142756
$891$	4.59630	0.153982
$892$	−25.7351	−0.861676
$893$	−1.40983	−0.0471781
$894$	1.50583	0.0503624
$895$	−5.17041	−0.172828
$896$	2.15409	0.0719632
$897$	−3.56591	−0.119062
$898$	−1.89748	−0.0633196
$899$	44.2298	1.47515
$900$	−1.15749	−0.0385831
$901$	−0.0379362	−0.00126384
$902$	49.2701	1.64051
$903$	0.996713	0.0331685
$904$	−12.1224	−0.403185
$905$	17.0199	0.565761
$906$	−19.5843	−0.650646
$907$	46.8928	1.55705	0.778525	−	0.627614i	$-0.215968\pi$
$907$	46.8928	1.55705	0.778525	+	0.627614i	$0.215968\pi$
$908$	−24.9029	−0.826431
$909$	3.09006	0.102491
$910$	−0.199461	−0.00661206
$911$	−35.0077	−1.15986	−0.579929	−	0.814667i	$-0.696920\pi$
$911$	−35.0077	−1.15986	−0.579929	+	0.814667i	$0.696920\pi$
$912$	0.0870639	0.00288297
$913$	−65.4292	−2.16539
$914$	8.29105	0.274243
$915$	4.43583	0.146644
$916$	−23.9932	−0.792758
$917$	3.21960	0.106321
$918$	−5.52332	−0.182297
$919$	−23.6063	−0.778700	−0.389350	−	0.921090i	$-0.627300\pi$
$919$	−23.6063	−0.778700	−0.389350	+	0.921090i	$0.627300\pi$
$920$	−17.9453	−0.591638
$921$	4.39675	0.144878
$922$	17.9162	0.590039
$923$	1.79714	0.0591537
$924$	−2.00747	−0.0660408
$925$	1.02420	0.0336753
$926$	−38.7044	−1.27191
$927$	6.55983	0.215453
$928$	−43.3379	−1.42264
$929$	29.8586	0.979627	0.489814	−	0.871827i	$-0.337065\pi$
$929$	29.8586	0.979627	0.489814	+	0.871827i	$0.337065\pi$
$930$	−5.13307	−0.168320
$931$	−1.72946	−0.0566809
$932$	−2.28265	−0.0747708
$933$	18.3458	0.600615
$934$	−3.27269	−0.107086
$935$	−27.6580	−0.904515
$936$	1.66909	0.0545558
$937$	−20.7681	−0.678465	−0.339232	−	0.940703i	$-0.610167\pi$
$937$	−20.7681	−0.678465	−0.339232	+	0.940703i	$0.610167\pi$
$938$	0.605722	0.0197775
$939$	31.9081	1.04128
$940$	−6.47064	−0.211049
$941$	−41.1803	−1.34244	−0.671220	−	0.741258i	$-0.734230\pi$
$941$	−41.1803	−1.34244	−0.671220	+	0.741258i	$0.734230\pi$
$942$	−10.4528	−0.340571
$943$	−72.3118	−2.35480
$944$	2.77743	0.0903977
$945$	−0.377330	−0.0122746
$946$	11.1441	0.362325
$947$	35.6799	1.15944	0.579721	−	0.814815i	$-0.303161\pi$
$947$	35.6799	1.15944	0.579721	+	0.814815i	$0.303161\pi$
$948$	4.05749	0.131781
$949$	−5.39042	−0.174980
$950$	0.231486	0.00751039
$951$	9.26440	0.300419
$952$	6.58058	0.213278
$953$	−39.8644	−1.29134	−0.645668	−	0.763619i	$-0.723421\pi$
$953$	−39.8644	−1.29134	−0.645668	+	0.763619i	$0.723421\pi$
$954$	−0.00578665	−0.000187350 0
$955$	−8.18497	−0.264859
$956$	5.88600	0.190367
$957$	36.3523	1.17510
$958$	11.1786	0.361164
$959$	−8.15437	−0.263318
$960$	5.72001	0.184613
$961$	0.273862	0.00883426
$962$	−0.541401	−0.0174555
$963$	14.3365	0.461987
$964$	31.5863	1.01733
$965$	−23.8965	−0.769256
$966$	−2.14452	−0.0689987
$967$	−30.2337	−0.972249	−0.486125	−	0.873889i	$-0.661590\pi$
$967$	−30.2337	−0.972249	−0.486125	+	0.873889i	$0.661590\pi$
$968$	−29.3471	−0.943252
$969$	−1.51758	−0.0487516
$970$	−10.6404	−0.341643
$971$	−18.3822	−0.589912	−0.294956	−	0.955511i	$-0.595305\pi$
$971$	−18.3822	−0.589912	−0.294956	+	0.955511i	$0.595305\pi$
$972$	1.15749	0.0371266
$973$	5.12287	0.164232
$974$	32.9753	1.05660
$975$	0.575903	0.0184437
$976$	−1.53135	−0.0490174
$977$	−0.806450	−0.0258006	−0.0129003	−	0.999917i	$-0.504106\pi$
$977$	−0.806450	−0.0258006	−0.0129003	+	0.999917i	$0.504106\pi$
$978$	−17.0807	−0.546182
$979$	21.3261	0.681585
$980$	−7.93765	−0.253559
$981$	4.61958	0.147492
$982$	−24.7098	−0.788523
$983$	26.9537	0.859691	0.429845	−	0.902903i	$-0.358568\pi$
$983$	26.9537	0.859691	0.429845	+	0.902903i	$0.358568\pi$
$984$	33.8468	1.07900
$985$	5.33086	0.169855
$986$	−43.6842	−1.39119
$987$	−2.10936	−0.0671416
$988$	0.168115	0.00534844
$989$	−16.3557	−0.520081
$990$	−4.21886	−0.134084
$991$	−9.77103	−0.310387	−0.155193	−	0.987884i	$-0.549600\pi$
$991$	−9.77103	−0.310387	−0.155193	+	0.987884i	$0.549600\pi$
$992$	−30.6432	−0.972924
$993$	26.3568	0.836407
$994$	1.08079	0.0342806
$995$	−0.906593	−0.0287409
$996$	−16.4771	−0.522098
$997$	−0.104346	−0.00330468	−0.00165234	−	0.999999i	$-0.500526\pi$
$997$	−0.104346	−0.00330468	−0.00165234	+	0.999999i	$0.500526\pi$
$998$	−22.7781	−0.721027
$999$	−1.02420	−0.0324041

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6015.2.a.e.1.19	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6015.2.a.e.1.19	✓	31	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:	\( N \)	\(=\)	\( 6015 = 3 \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6015.a (trivial)

\(f(q)\)	\(=\)	\(q+0.917882 q^{2} -1.00000 q^{3} -1.15749 q^{4} -1.00000 q^{5} -0.917882 q^{6} -0.377330 q^{7} -2.89821 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.917882 q^{2} -1.00000 q^{3} -1.15749 q^{4} -1.00000 q^{5} -0.917882 q^{6} -0.377330 q^{7} -2.89821 q^{8} +1.00000 q^{9} -0.917882 q^{10} +4.59630 q^{11} +1.15749 q^{12} -0.575903 q^{13} -0.346345 q^{14} +1.00000 q^{15} -0.345224 q^{16} +6.01746 q^{17} +0.917882 q^{18} +0.252196 q^{19} +1.15749 q^{20} +0.377330 q^{21} +4.21886 q^{22} -6.19186 q^{23} +2.89821 q^{24} +1.00000 q^{25} -0.528611 q^{26} -1.00000 q^{27} +0.436757 q^{28} -7.90905 q^{29} +0.917882 q^{30} -5.59230 q^{31} +5.47954 q^{32} -4.59630 q^{33} +5.52332 q^{34} +0.377330 q^{35} -1.15749 q^{36} +1.02420 q^{37} +0.231486 q^{38} +0.575903 q^{39} +2.89821 q^{40} +11.6785 q^{41} +0.346345 q^{42} +2.64149 q^{43} -5.32018 q^{44} -1.00000 q^{45} -5.68339 q^{46} -5.59022 q^{47} +0.345224 q^{48} -6.85762 q^{49} +0.917882 q^{50} -6.01746 q^{51} +0.666604 q^{52} -0.00630436 q^{53} -0.917882 q^{54} -4.59630 q^{55} +1.09358 q^{56} -0.252196 q^{57} -7.25957 q^{58} -8.04531 q^{59} -1.15749 q^{60} +4.43583 q^{61} -5.13307 q^{62} -0.377330 q^{63} +5.72001 q^{64} +0.575903 q^{65} -4.21886 q^{66} -1.74890 q^{67} -6.96517 q^{68} +6.19186 q^{69} +0.346345 q^{70} -3.12057 q^{71} -2.89821 q^{72} +9.35994 q^{73} +0.940091 q^{74} -1.00000 q^{75} -0.291915 q^{76} -1.73432 q^{77} +0.528611 q^{78} +3.50541 q^{79} +0.345224 q^{80} +1.00000 q^{81} +10.7195 q^{82} -14.2352 q^{83} -0.436757 q^{84} -6.01746 q^{85} +2.42457 q^{86} +7.90905 q^{87} -13.3210 q^{88} +4.63984 q^{89} -0.917882 q^{90} +0.217306 q^{91} +7.16703 q^{92} +5.59230 q^{93} -5.13116 q^{94} -0.252196 q^{95} -5.47954 q^{96} +11.5924 q^{97} -6.29449 q^{98} +4.59630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9}+O(q^{10}) \) 31 * q + 6 * q^2 - 31 * q^3 + 24 * q^4 - 31 * q^5 - 6 * q^6 - 4 * q^7 + 15 * q^8 + 31 * q^9	\( 31 q + 6 q^{2} - 31 q^{3} + 24 q^{4} - 31 q^{5} - 6 q^{6} - 4 q^{7} + 15 q^{8} + 31 q^{9} - 6 q^{10} - q^{11} - 24 q^{12} + 8 q^{13} + 8 q^{14} + 31 q^{15} + 10 q^{16} + 42 q^{17} + 6 q^{18} - 19 q^{19} - 24 q^{20} + 4 q^{21} + 17 q^{22} + 26 q^{23} - 15 q^{24} + 31 q^{25} + 4 q^{26} - 31 q^{27} - 16 q^{28} - 11 q^{29} + 6 q^{30} - 3 q^{31} + 41 q^{32} + q^{33} + 6 q^{34} + 4 q^{35} + 24 q^{36} + 10 q^{37} + 12 q^{38} - 8 q^{39} - 15 q^{40} + 27 q^{41} - 8 q^{42} - 37 q^{43} - 9 q^{44} - 31 q^{45} - 23 q^{46} + 48 q^{47} - 10 q^{48} + 31 q^{49} + 6 q^{50} - 42 q^{51} + 18 q^{52} + 35 q^{53} - 6 q^{54} + q^{55} + 7 q^{56} + 19 q^{57} - 26 q^{58} - 3 q^{59} + 24 q^{60} + 11 q^{61} + 49 q^{62} - 4 q^{63} - 27 q^{64} - 8 q^{65} - 17 q^{66} - 44 q^{67} + 72 q^{68} - 26 q^{69} - 8 q^{70} + 6 q^{71} + 15 q^{72} + 49 q^{73} - 6 q^{74} - 31 q^{75} - 36 q^{76} + 66 q^{77} - 4 q^{78} - 41 q^{79} - 10 q^{80} + 31 q^{81} + 9 q^{82} + 53 q^{83} + 16 q^{84} - 42 q^{85} + 3 q^{86} + 11 q^{87} + 21 q^{88} + 31 q^{89} - 6 q^{90} - 45 q^{91} + 45 q^{92} + 3 q^{93} - 4 q^{94} + 19 q^{95} - 41 q^{96} + 37 q^{97} + 41 q^{98} - q^{99}+O(q^{100}) \) 31 * q + 6 * q^2 - 31 * q^3 + 24 * q^4 - 31 * q^5 - 6 * q^6 - 4 * q^7 + 15 * q^8 + 31 * q^9 - 6 * q^10 - q^11 - 24 * q^12 + 8 * q^13 + 8 * q^14 + 31 * q^15 + 10 * q^16 + 42 * q^17 + 6 * q^18 - 19 * q^19 - 24 * q^20 + 4 * q^21 + 17 * q^22 + 26 * q^23 - 15 * q^24 + 31 * q^25 + 4 * q^26 - 31 * q^27 - 16 * q^28 - 11 * q^29 + 6 * q^30 - 3 * q^31 + 41 * q^32 + q^33 + 6 * q^34 + 4 * q^35 + 24 * q^36 + 10 * q^37 + 12 * q^38 - 8 * q^39 - 15 * q^40 + 27 * q^41 - 8 * q^42 - 37 * q^43 - 9 * q^44 - 31 * q^45 - 23 * q^46 + 48 * q^47 - 10 * q^48 + 31 * q^49 + 6 * q^50 - 42 * q^51 + 18 * q^52 + 35 * q^53 - 6 * q^54 + q^55 + 7 * q^56 + 19 * q^57 - 26 * q^58 - 3 * q^59 + 24 * q^60 + 11 * q^61 + 49 * q^62 - 4 * q^63 - 27 * q^64 - 8 * q^65 - 17 * q^66 - 44 * q^67 + 72 * q^68 - 26 * q^69 - 8 * q^70 + 6 * q^71 + 15 * q^72 + 49 * q^73 - 6 * q^74 - 31 * q^75 - 36 * q^76 + 66 * q^77 - 4 * q^78 - 41 * q^79 - 10 * q^80 + 31 * q^81 + 9 * q^82 + 53 * q^83 + 16 * q^84 - 42 * q^85 + 3 * q^86 + 11 * q^87 + 21 * q^88 + 31 * q^89 - 6 * q^90 - 45 * q^91 + 45 * q^92 + 3 * q^93 - 4 * q^94 + 19 * q^95 - 41 * q^96 + 37 * q^97 + 41 * q^98 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more