LMFDB - Embedded newform 5239.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5239,2,Mod(1,5239)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5239, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5239.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5239 = 13^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5239.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:	$41.8336256189$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 7x^{6} + 19x^{5} + 21x^{4} - 31x^{3} - 29x^{2} - x + 1 $ x^8 - 3x^7 - 7x^6 + 19x^5 + 21x^4 - 31x^3 - 29x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 403)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.75938$ of defining polynomial
Character	$\chi$	$=$	5239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.75938 q^{2} -1.67414 q^{3} +1.09540 q^{4} +3.67909 q^{5} +2.94544 q^{6} -4.14977 q^{7} +1.59152 q^{8} -0.197253 q^{9} +O(q^{10})$	\(q-1.75938 q^{2} -1.67414 q^{3} +1.09540 q^{4} +3.67909 q^{5} +2.94544 q^{6} -4.14977 q^{7} +1.59152 q^{8} -0.197253 q^{9} -6.47291 q^{10} +5.07739 q^{11} -1.83386 q^{12} +7.30102 q^{14} -6.15932 q^{15} -4.99090 q^{16} -0.643026 q^{17} +0.347042 q^{18} +7.24868 q^{19} +4.03010 q^{20} +6.94731 q^{21} -8.93304 q^{22} +0.685475 q^{23} -2.66443 q^{24} +8.53574 q^{25} +5.35265 q^{27} -4.54568 q^{28} +5.82822 q^{29} +10.8366 q^{30} +1.00000 q^{31} +5.59782 q^{32} -8.50026 q^{33} +1.13132 q^{34} -15.2674 q^{35} -0.216072 q^{36} -8.72827 q^{37} -12.7532 q^{38} +5.85537 q^{40} +11.1619 q^{41} -12.2229 q^{42} -1.81828 q^{43} +5.56180 q^{44} -0.725712 q^{45} -1.20601 q^{46} -6.05989 q^{47} +8.35547 q^{48} +10.2206 q^{49} -15.0176 q^{50} +1.07652 q^{51} -1.56129 q^{53} -9.41733 q^{54} +18.6802 q^{55} -6.60446 q^{56} -12.1353 q^{57} -10.2540 q^{58} +7.14043 q^{59} -6.74695 q^{60} +1.54027 q^{61} -1.75938 q^{62} +0.818555 q^{63} +0.133123 q^{64} +14.9552 q^{66} -2.10412 q^{67} -0.704374 q^{68} -1.14758 q^{69} +26.8611 q^{70} -0.546141 q^{71} -0.313933 q^{72} +9.58155 q^{73} +15.3563 q^{74} -14.2900 q^{75} +7.94024 q^{76} -21.0700 q^{77} -4.15021 q^{79} -18.3620 q^{80} -8.36933 q^{81} -19.6379 q^{82} -6.77183 q^{83} +7.61011 q^{84} -2.36575 q^{85} +3.19904 q^{86} -9.75726 q^{87} +8.08078 q^{88} +14.7212 q^{89} +1.27680 q^{90} +0.750873 q^{92} -1.67414 q^{93} +10.6616 q^{94} +26.6686 q^{95} -9.37154 q^{96} +12.2489 q^{97} -17.9819 q^{98} -1.00153 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{2} - 3 q^{3} + 9 q^{4} + 15 q^{5} + 4 q^{7} + 15 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q + 5 * q^2 - 3 * q^3 + 9 * q^4 + 15 * q^5 + 4 * q^7 + 15 * q^8 + 9 * q^9	\( 8 q + 5 q^{2} - 3 q^{3} + 9 q^{4} + 15 q^{5} + 4 q^{7} + 15 q^{8} + 9 q^{9} + 9 q^{10} + 5 q^{11} - 9 q^{12} + 2 q^{14} - 2 q^{15} + 3 q^{16} - 11 q^{17} + 30 q^{18} + 9 q^{19} + 31 q^{20} + 16 q^{21} - 2 q^{22} - 13 q^{24} + 19 q^{25} - 9 q^{27} - 16 q^{28} - 12 q^{29} - 7 q^{30} + 8 q^{31} + 25 q^{32} + 14 q^{33} - 22 q^{34} + 7 q^{35} + 37 q^{36} + 9 q^{37} - 9 q^{38} + 55 q^{40} + 25 q^{41} - 3 q^{42} + 7 q^{43} + 26 q^{44} + 45 q^{45} - 5 q^{46} + 17 q^{47} - 9 q^{48} + 11 q^{50} - 10 q^{51} - 15 q^{53} - 54 q^{54} + 7 q^{55} - 14 q^{56} + 7 q^{57} + 5 q^{58} + 15 q^{59} - 61 q^{60} + 11 q^{61} + 5 q^{62} + 21 q^{63} + 47 q^{64} + 83 q^{66} - 18 q^{67} - 16 q^{68} - 15 q^{69} + 24 q^{70} + 7 q^{71} + 21 q^{72} - 24 q^{73} + 48 q^{74} - 17 q^{75} + 3 q^{76} - 49 q^{77} + 33 q^{79} + 16 q^{80} + 20 q^{81} - q^{82} + 13 q^{83} + 6 q^{84} - q^{85} - 19 q^{86} + 18 q^{87} + 37 q^{88} + 23 q^{89} + 117 q^{90} + 22 q^{92} - 3 q^{93} + 10 q^{94} + 43 q^{95} - 46 q^{96} + 17 q^{97} - 52 q^{98} - 51 q^{99}+O(q^{100}) \) 8 * q + 5 * q^2 - 3 * q^3 + 9 * q^4 + 15 * q^5 + 4 * q^7 + 15 * q^8 + 9 * q^9 + 9 * q^10 + 5 * q^11 - 9 * q^12 + 2 * q^14 - 2 * q^15 + 3 * q^16 - 11 * q^17 + 30 * q^18 + 9 * q^19 + 31 * q^20 + 16 * q^21 - 2 * q^22 - 13 * q^24 + 19 * q^25 - 9 * q^27 - 16 * q^28 - 12 * q^29 - 7 * q^30 + 8 * q^31 + 25 * q^32 + 14 * q^33 - 22 * q^34 + 7 * q^35 + 37 * q^36 + 9 * q^37 - 9 * q^38 + 55 * q^40 + 25 * q^41 - 3 * q^42 + 7 * q^43 + 26 * q^44 + 45 * q^45 - 5 * q^46 + 17 * q^47 - 9 * q^48 + 11 * q^50 - 10 * q^51 - 15 * q^53 - 54 * q^54 + 7 * q^55 - 14 * q^56 + 7 * q^57 + 5 * q^58 + 15 * q^59 - 61 * q^60 + 11 * q^61 + 5 * q^62 + 21 * q^63 + 47 * q^64 + 83 * q^66 - 18 * q^67 - 16 * q^68 - 15 * q^69 + 24 * q^70 + 7 * q^71 + 21 * q^72 - 24 * q^73 + 48 * q^74 - 17 * q^75 + 3 * q^76 - 49 * q^77 + 33 * q^79 + 16 * q^80 + 20 * q^81 - q^82 + 13 * q^83 + 6 * q^84 - q^85 - 19 * q^86 + 18 * q^87 + 37 * q^88 + 23 * q^89 + 117 * q^90 + 22 * q^92 - 3 * q^93 + 10 * q^94 + 43 * q^95 - 46 * q^96 + 17 * q^97 - 52 * q^98 - 51 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.75938	−1.24407	−0.622033	−	0.782991i	$-0.713693\pi$
$2$	−1.75938	−1.24407	−0.622033	+	0.782991i	$0.713693\pi$
$3$	−1.67414	−0.966566	−0.483283	−	0.875464i	$-0.660556\pi$
$3$	−1.67414	−0.966566	−0.483283	+	0.875464i	$0.660556\pi$
$4$	1.09540	0.547702
$5$	3.67909	1.64534	0.822671	−	0.568518i	$-0.192483\pi$
$5$	3.67909	1.64534	0.822671	+	0.568518i	$0.192483\pi$
$6$	2.94544	1.20247
$7$	−4.14977	−1.56847	−0.784234	−	0.620466i	$-0.786944\pi$
$7$	−4.14977	−1.56847	−0.784234	+	0.620466i	$0.786944\pi$
$8$	1.59152	0.562688
$9$	−0.197253	−0.0657510
$10$	−6.47291	−2.04691
$11$	5.07739	1.53089	0.765445	−	0.643501i	$-0.222519\pi$
$11$	5.07739	1.53089	0.765445	+	0.643501i	$0.222519\pi$
$12$	−1.83386	−0.529390
$13$	0	0
$13$	0	0
$14$	7.30102	1.95128
$15$	−6.15932	−1.59033
$16$	−4.99090	−1.24772
$17$	−0.643026	−0.155957	−0.0779784	−	0.996955i	$-0.524847\pi$
$17$	−0.643026	−0.155957	−0.0779784	+	0.996955i	$0.524847\pi$
$18$	0.347042	0.0817986
$19$	7.24868	1.66296	0.831481	−	0.555554i	$-0.187494\pi$
$19$	7.24868	1.66296	0.831481	+	0.555554i	$0.187494\pi$
$20$	4.03010	0.901157
$21$	6.94731	1.51603
$22$	−8.93304	−1.90453
$23$	0.685475	0.142931	0.0714657	−	0.997443i	$-0.477232\pi$
$23$	0.685475	0.142931	0.0714657	+	0.997443i	$0.477232\pi$
$24$	−2.66443	−0.543875
$25$	8.53574	1.70715
$26$	0	0
$27$	5.35265	1.03012
$28$	−4.54568	−0.859053
$29$	5.82822	1.08227	0.541137	−	0.840935i	$-0.317994\pi$
$29$	5.82822	1.08227	0.541137	+	0.840935i	$0.317994\pi$
$30$	10.8366	1.97848
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	5.59782	0.989564
$33$	−8.50026	−1.47971
$34$	1.13132	0.194021
$35$	−15.2674	−2.58066
$36$	−0.216072	−0.0360120
$37$	−8.72827	−1.43492	−0.717460	−	0.696600i	$-0.754695\pi$
$37$	−8.72827	−1.43492	−0.717460	+	0.696600i	$0.754695\pi$
$38$	−12.7532	−2.06884
$39$	0	0
$40$	5.85537	0.925815
$41$	11.1619	1.74319	0.871596	−	0.490225i	$-0.163086\pi$
$41$	11.1619	1.74319	0.871596	+	0.490225i	$0.163086\pi$
$42$	−12.2229	−1.88604
$43$	−1.81828	−0.277285	−0.138642	−	0.990343i	$-0.544274\pi$
$43$	−1.81828	−0.277285	−0.138642	+	0.990343i	$0.544274\pi$
$44$	5.56180	0.838472
$45$	−0.725712	−0.108183
$46$	−1.20601	−0.177816
$47$	−6.05989	−0.883926	−0.441963	−	0.897033i	$-0.645718\pi$
$47$	−6.05989	−0.883926	−0.441963	+	0.897033i	$0.645718\pi$
$48$	8.35547	1.20601
$49$	10.2206	1.46009
$50$	−15.0176	−2.12381
$51$	1.07652	0.150742
$52$	0	0
$53$	−1.56129	−0.214460	−0.107230	−	0.994234i	$-0.534198\pi$
$53$	−1.56129	−0.214460	−0.107230	+	0.994234i	$0.534198\pi$
$54$	−9.41733	−1.28154
$55$	18.6802	2.51884
$56$	−6.60446	−0.882559
$57$	−12.1353	−1.60736
$58$	−10.2540	−1.34642
$59$	7.14043	0.929604	0.464802	−	0.885415i	$-0.346125\pi$
$59$	7.14043	0.929604	0.464802	+	0.885415i	$0.346125\pi$
$60$	−6.74695	−0.871028
$61$	1.54027	0.197211	0.0986054	−	0.995127i	$-0.468562\pi$
$61$	1.54027	0.197211	0.0986054	+	0.995127i	$0.468562\pi$
$62$	−1.75938	−0.223441
$63$	0.818555	0.103128
$64$	0.133123	0.0166404
$65$	0	0
$66$	14.9552	1.84085
$67$	−2.10412	−0.257059	−0.128530	−	0.991706i	$-0.541026\pi$
$67$	−2.10412	−0.257059	−0.128530	+	0.991706i	$0.541026\pi$
$68$	−0.704374	−0.0854179
$69$	−1.14758	−0.138153
$70$	26.8611	3.21052
$71$	−0.546141	−0.0648150	−0.0324075	−	0.999475i	$-0.510317\pi$
$71$	−0.546141	−0.0648150	−0.0324075	+	0.999475i	$0.510317\pi$
$72$	−0.313933	−0.0369973
$73$	9.58155	1.12144	0.560718	−	0.828007i	$-0.310525\pi$
$73$	9.58155	1.12144	0.560718	+	0.828007i	$0.310525\pi$
$74$	15.3563	1.78514
$75$	−14.2900	−1.65007
$76$	7.94024	0.910808
$77$	−21.0700	−2.40115
$78$	0	0
$79$	−4.15021	−0.466935	−0.233468	−	0.972365i	$-0.575007\pi$
$79$	−4.15021	−0.466935	−0.233468	+	0.972365i	$0.575007\pi$
$80$	−18.3620	−2.05293
$81$	−8.36933	−0.929926
$82$	−19.6379	−2.16865
$83$	−6.77183	−0.743305	−0.371652	−	0.928372i	$-0.621209\pi$
$83$	−6.77183	−0.743305	−0.371652	+	0.928372i	$0.621209\pi$
$84$	7.61011	0.830331
$85$	−2.36575	−0.256602
$86$	3.19904	0.344961
$87$	−9.75726	−1.04609
$88$	8.08078	0.861414
$89$	14.7212	1.56045	0.780224	−	0.625500i	$-0.215105\pi$
$89$	14.7212	1.56045	0.780224	+	0.625500i	$0.215105\pi$
$90$	1.27680	0.134587
$91$	0	0
$92$	0.750873	0.0782839
$93$	−1.67414	−0.173600
$94$	10.6616	1.09966
$95$	26.6686	2.73614
$96$	−9.37154	−0.956479
$97$	12.2489	1.24369	0.621843	−	0.783142i	$-0.286384\pi$
$97$	12.2489	1.24369	0.621843	+	0.783142i	$0.286384\pi$
$98$	−17.9819	−1.81645
$99$	−1.00153	−0.100658
$100$	9.35009	0.935009
$101$	−18.9673	−1.88732	−0.943661	−	0.330914i	$-0.892643\pi$
$101$	−18.9673	−1.88732	−0.943661	+	0.330914i	$0.892643\pi$
$102$	−1.89400	−0.187534
$103$	11.2464	1.10814	0.554068	−	0.832471i	$-0.313075\pi$
$103$	11.2464	1.10814	0.554068	+	0.832471i	$0.313075\pi$
$104$	0	0
$105$	25.5598	2.49438
$106$	2.74690	0.266802
$107$	−13.0590	−1.26246	−0.631232	−	0.775594i	$-0.717450\pi$
$107$	−13.0590	−1.26246	−0.631232	+	0.775594i	$0.717450\pi$
$108$	5.86332	0.564198
$109$	13.8788	1.32935	0.664673	−	0.747134i	$-0.268571\pi$
$109$	13.8788	1.32935	0.664673	+	0.747134i	$0.268571\pi$
$110$	−32.8655	−3.13360
$111$	14.6124	1.38694
$112$	20.7111	1.95702
$113$	0.433714	0.0408004	0.0204002	−	0.999792i	$-0.493506\pi$
$113$	0.433714	0.0408004	0.0204002	+	0.999792i	$0.493506\pi$
$114$	21.3506	1.99966
$115$	2.52193	0.235171
$116$	6.38426	0.592764
$117$	0	0
$118$	−12.5627	−1.15649
$119$	2.66841	0.244613
$120$	−9.80271	−0.894861
$121$	14.7799	1.34363
$122$	−2.70991	−0.245343
$123$	−18.6865	−1.68491
$124$	1.09540	0.0983703
$125$	13.0083	1.16350
$126$	−1.44015	−0.128298
$127$	3.09560	0.274690	0.137345	−	0.990523i	$-0.456143\pi$
$127$	3.09560	0.274690	0.137345	+	0.990523i	$0.456143\pi$
$128$	−11.4299	−1.01027
$129$	3.04405	0.268014
$130$	0	0
$131$	−0.655715	−0.0572901	−0.0286450	−	0.999590i	$-0.509119\pi$
$131$	−0.655715	−0.0572901	−0.0286450	+	0.999590i	$0.509119\pi$
$132$	−9.31123	−0.810439
$133$	−30.0804	−2.60830
$134$	3.70194	0.319799
$135$	19.6929	1.69490
$136$	−1.02339	−0.0877551
$137$	0.444583	0.0379833	0.0189917	−	0.999820i	$-0.493954\pi$
$137$	0.444583	0.0379833	0.0189917	+	0.999820i	$0.493954\pi$
$138$	2.01903	0.171871
$139$	1.98710	0.168544	0.0842718	−	0.996443i	$-0.473144\pi$
$139$	1.98710	0.168544	0.0842718	+	0.996443i	$0.473144\pi$
$140$	−16.7240	−1.41344
$141$	10.1451	0.854372
$142$	0.960867	0.0806342
$143$	0	0
$144$	0.984469	0.0820391
$145$	21.4426	1.78071
$146$	−16.8576	−1.39514
$147$	−17.1108	−1.41127
$148$	−9.56099	−0.785909
$149$	−15.8870	−1.30152	−0.650759	−	0.759285i	$-0.725549\pi$
$149$	−15.8870	−1.30152	−0.650759	+	0.759285i	$0.725549\pi$
$150$	25.1415	2.05280
$151$	−20.0014	−1.62769	−0.813847	−	0.581079i	$-0.802631\pi$
$151$	−20.0014	−1.62769	−0.813847	+	0.581079i	$0.802631\pi$
$152$	11.5364	0.935729
$153$	0.126839	0.0102543
$154$	37.0701	2.98719
$155$	3.67909	0.295512
$156$	0	0
$157$	0.109578	0.00874525	0.00437263	−	0.999990i	$-0.498608\pi$
$157$	0.109578	0.00874525	0.00437263	+	0.999990i	$0.498608\pi$
$158$	7.30178	0.580899
$159$	2.61382	0.207290
$160$	20.5949	1.62817
$161$	−2.84457	−0.224183
$162$	14.7248	1.15689
$163$	5.63385	0.441277	0.220639	−	0.975356i	$-0.429186\pi$
$163$	5.63385	0.441277	0.220639	+	0.975356i	$0.429186\pi$
$164$	12.2268	0.954750
$165$	−31.2733	−2.43462
$166$	11.9142	0.924721
$167$	−9.04293	−0.699763	−0.349881	−	0.936794i	$-0.613778\pi$
$167$	−9.04293	−0.699763	−0.349881	+	0.936794i	$0.613778\pi$
$168$	11.0568	0.853051
$169$	0	0
$170$	4.16225	0.319230
$171$	−1.42982	−0.109341
$172$	−1.99175	−0.151870
$173$	9.53376	0.724838	0.362419	−	0.932015i	$-0.381951\pi$
$173$	9.53376	0.724838	0.362419	+	0.932015i	$0.381951\pi$
$174$	17.1667	1.30140
$175$	−35.4214	−2.67761
$176$	−25.3407	−1.91013
$177$	−11.9541	−0.898523
$178$	−25.9002	−1.94130
$179$	−10.3075	−0.770419	−0.385209	−	0.922829i	$-0.625871\pi$
$179$	−10.3075	−0.770419	−0.385209	+	0.922829i	$0.625871\pi$
$180$	−0.794948	−0.0592520
$181$	3.40674	0.253221	0.126611	−	0.991953i	$-0.459590\pi$
$181$	3.40674	0.253221	0.126611	+	0.991953i	$0.459590\pi$
$182$	0	0
$183$	−2.57862	−0.190617
$184$	1.09095	0.0804259
$185$	−32.1121	−2.36093
$186$	2.94544	0.215970
$187$	−3.26489	−0.238753
$188$	−6.63803	−0.484128
$189$	−22.2123	−1.61571
$190$	−46.9201	−3.40394
$191$	−17.0195	−1.23149	−0.615745	−	0.787946i	$-0.711145\pi$
$191$	−17.0195	−1.23149	−0.615745	+	0.787946i	$0.711145\pi$
$192$	−0.222867	−0.0160841
$193$	0.103108	0.00742191	0.00371095	−	0.999993i	$-0.498819\pi$
$193$	0.103108	0.00742191	0.00371095	+	0.999993i	$0.498819\pi$
$194$	−21.5504	−1.54723
$195$	0	0
$196$	11.1957	0.799695
$197$	2.51791	0.179393	0.0896967	−	0.995969i	$-0.471410\pi$
$197$	2.51791	0.179393	0.0896967	+	0.995969i	$0.471410\pi$
$198$	1.76207	0.125225
$199$	−0.00157334	−0.000111531 0	−5.57657e−5	−	1.00000i	$-0.500018\pi$
$199$	−0.00157334	−0.000111531 0	−5.57657e−5		1.00000i	$0.500018\pi$
$200$	13.5848	0.960592
$201$	3.52259	0.248465
$202$	33.3707	2.34795
$203$	−24.1858	−1.69751
$204$	1.17922	0.0825620
$205$	41.0656	2.86814
$206$	−19.7866	−1.37860
$207$	−0.135212	−0.00939788
$208$	0	0
$209$	36.8044	2.54581
$210$	−44.9693	−3.10318
$211$	−16.9852	−1.16931	−0.584653	−	0.811283i	$-0.698769\pi$
$211$	−16.9852	−1.16931	−0.584653	+	0.811283i	$0.698769\pi$
$212$	−1.71025	−0.117460
$213$	0.914316	0.0626479
$214$	22.9757	1.57059
$215$	−6.68962	−0.456228
$216$	8.51887	0.579636
$217$	−4.14977	−0.281705
$218$	−24.4180	−1.65380
$219$	−16.0409	−1.08394
$220$	20.4624	1.37957
$221$	0	0
$222$	−25.7086	−1.72545
$223$	−10.0015	−0.669748	−0.334874	−	0.942263i	$-0.608694\pi$
$223$	−10.0015	−0.669748	−0.334874	+	0.942263i	$0.608694\pi$
$224$	−23.2297	−1.55210
$225$	−1.68370	−0.112247
$226$	−0.763066	−0.0507584
$227$	15.2887	1.01475	0.507375	−	0.861726i	$-0.330616\pi$
$227$	15.2887	1.01475	0.507375	+	0.861726i	$0.330616\pi$
$228$	−13.2931	−0.880356
$229$	16.6053	1.09731	0.548654	−	0.836049i	$-0.315140\pi$
$229$	16.6053	1.09731	0.548654	+	0.836049i	$0.315140\pi$
$230$	−4.43702	−0.292568
$231$	35.2742	2.32087
$232$	9.27575	0.608983
$233$	−10.3751	−0.679696	−0.339848	−	0.940480i	$-0.610376\pi$
$233$	−10.3751	−0.679696	−0.339848	+	0.940480i	$0.610376\pi$
$234$	0	0
$235$	−22.2949	−1.45436
$236$	7.82166	0.509146
$237$	6.94804	0.451324
$238$	−4.69474	−0.304315
$239$	−26.7111	−1.72780	−0.863900	−	0.503664i	$-0.831985\pi$
$239$	−26.7111	−1.72780	−0.863900	+	0.503664i	$0.831985\pi$
$240$	30.7406	1.98429
$241$	8.83202	0.568920	0.284460	−	0.958688i	$-0.408186\pi$
$241$	8.83202	0.568920	0.284460	+	0.958688i	$0.408186\pi$
$242$	−26.0034	−1.67156
$243$	−2.04651	−0.131284
$244$	1.68721	0.108013
$245$	37.6027	2.40235
$246$	32.8767	2.09614
$247$	0	0
$248$	1.59152	0.101062
$249$	11.3370	0.718453
$250$	−22.8865	−1.44747
$251$	28.0408	1.76992	0.884959	−	0.465668i	$-0.154186\pi$
$251$	28.0408	1.76992	0.884959	+	0.465668i	$0.154186\pi$
$252$	0.896649	0.0564836
$253$	3.48042	0.218812
$254$	−5.44633	−0.341733
$255$	3.96061	0.248023
$256$	19.8432	1.24020
$257$	19.0954	1.19114	0.595568	−	0.803305i	$-0.296927\pi$
$257$	19.0954	1.19114	0.595568	+	0.803305i	$0.296927\pi$
$258$	−5.35564	−0.333427
$259$	36.2204	2.25062
$260$	0	0
$261$	−1.14963	−0.0711605
$262$	1.15365	0.0712727
$263$	−2.07863	−0.128174	−0.0640870	−	0.997944i	$-0.520414\pi$
$263$	−2.07863	−0.128174	−0.0640870	+	0.997944i	$0.520414\pi$
$264$	−13.5284	−0.832614
$265$	−5.74414	−0.352860
$266$	52.9227	3.24490
$267$	−24.6454	−1.50828
$268$	−2.30486	−0.140792
$269$	−19.6631	−1.19888	−0.599440	−	0.800420i	$-0.704610\pi$
$269$	−19.6631	−1.19888	−0.599440	+	0.800420i	$0.704610\pi$
$270$	−34.6472	−2.10856
$271$	7.58175	0.460559	0.230279	−	0.973125i	$-0.426036\pi$
$271$	7.58175	0.460559	0.230279	+	0.973125i	$0.426036\pi$
$272$	3.20928	0.194591
$273$	0	0
$274$	−0.782189	−0.0472538
$275$	43.3393	2.61346
$276$	−1.25707	−0.0756665
$277$	8.83217	0.530674	0.265337	−	0.964156i	$-0.414517\pi$
$277$	8.83217	0.530674	0.265337	+	0.964156i	$0.414517\pi$
$278$	−3.49606	−0.209680
$279$	−0.197253	−0.0118092
$280$	−24.2984	−1.45211
$281$	−3.73446	−0.222779	−0.111390	−	0.993777i	$-0.535530\pi$
$281$	−3.73446	−0.222779	−0.111390	+	0.993777i	$0.535530\pi$
$282$	−17.8491	−1.06290
$283$	−3.46167	−0.205775	−0.102887	−	0.994693i	$-0.532808\pi$
$283$	−3.46167	−0.205775	−0.102887	+	0.994693i	$0.532808\pi$
$284$	−0.598245	−0.0354993
$285$	−44.6470	−2.64466
$286$	0	0
$287$	−46.3192	−2.73414
$288$	−1.10419	−0.0650648
$289$	−16.5865	−0.975677
$290$	−37.7256	−2.21532
$291$	−20.5064	−1.20210
$292$	10.4957	0.614213
$293$	22.9317	1.33969	0.669843	−	0.742502i	$-0.266361\pi$
$293$	22.9317	1.33969	0.669843	+	0.742502i	$0.266361\pi$
$294$	30.1043	1.75572
$295$	26.2703	1.52952
$296$	−13.8913	−0.807413
$297$	27.1775	1.57700
$298$	27.9513	1.61917
$299$	0	0
$300$	−15.6534	−0.903747
$301$	7.54545	0.434912
$302$	35.1901	2.02496
$303$	31.7540	1.82422
$304$	−36.1774	−2.07492
$305$	5.66678	0.324479
$306$	−0.223157	−0.0127570
$307$	5.99871	0.342364	0.171182	−	0.985239i	$-0.445241\pi$
$307$	5.99871	0.342364	0.171182	+	0.985239i	$0.445241\pi$
$308$	−23.0802	−1.31512
$309$	−18.8280	−1.07109
$310$	−6.47291	−0.367637
$311$	−0.121230	−0.00687433	−0.00343716	−	0.999994i	$-0.501094\pi$
$311$	−0.121230	−0.00687433	−0.00343716	+	0.999994i	$0.501094\pi$
$312$	0	0
$313$	−29.3281	−1.65772	−0.828862	−	0.559454i	$-0.811011\pi$
$313$	−29.3281	−1.65772	−0.828862	+	0.559454i	$0.811011\pi$
$314$	−0.192788	−0.0108797
$315$	3.01154	0.169681
$316$	−4.54616	−0.255742
$317$	14.6861	0.824856	0.412428	−	0.910990i	$-0.364681\pi$
$317$	14.6861	0.824856	0.412428	+	0.910990i	$0.364681\pi$
$318$	−4.59870	−0.257882
$319$	29.5922	1.65684
$320$	0.489773	0.0273792
$321$	21.8627	1.22025
$322$	5.00466	0.278899
$323$	−4.66109	−0.259350
$324$	−9.16781	−0.509323
$325$	0	0
$326$	−9.91206	−0.548978
$327$	−23.2350	−1.28490
$328$	17.7644	0.980874
$329$	25.1472	1.38641
$330$	55.0215	3.02883
$331$	−20.0348	−1.10121	−0.550605	−	0.834766i	$-0.685603\pi$
$331$	−20.0348	−1.10121	−0.550605	+	0.834766i	$0.685603\pi$
$332$	−7.41789	−0.407110
$333$	1.72168	0.0943473
$334$	15.9099	0.870552
$335$	−7.74126	−0.422950
$336$	−34.6733	−1.89158
$337$	1.25458	0.0683411	0.0341706	−	0.999416i	$-0.489121\pi$
$337$	1.25458	0.0683411	0.0341706	+	0.999416i	$0.489121\pi$
$338$	0	0
$339$	−0.726098	−0.0394362
$340$	−2.59146	−0.140542
$341$	5.07739	0.274956
$342$	2.51560	0.136028
$343$	−13.3649	−0.721636
$344$	−2.89383	−0.156025
$345$	−4.22206	−0.227308
$346$	−16.7735	−0.901747
$347$	33.4450	1.79542	0.897711	−	0.440586i	$-0.145229\pi$
$347$	33.4450	1.79542	0.897711	+	0.440586i	$0.145229\pi$
$348$	−10.6882	−0.572945
$349$	−16.0739	−0.860417	−0.430208	−	0.902730i	$-0.641560\pi$
$349$	−16.0739	−0.860417	−0.430208	+	0.902730i	$0.641560\pi$
$350$	62.3196	3.33112
$351$	0	0
$352$	28.4223	1.51491
$353$	33.7999	1.79899	0.899493	−	0.436935i	$-0.143936\pi$
$353$	33.7999	1.79899	0.899493	+	0.436935i	$0.143936\pi$
$354$	21.0317	1.11782
$355$	−2.00930	−0.106643
$356$	16.1257	0.854662
$357$	−4.46730	−0.236435
$358$	18.1348	0.958452
$359$	−8.58273	−0.452979	−0.226490	−	0.974014i	$-0.572725\pi$
$359$	−8.58273	−0.452979	−0.226490	+	0.974014i	$0.572725\pi$
$360$	−1.15499	−0.0608732
$361$	33.5434	1.76544
$362$	−5.99374	−0.315024
$363$	−24.7436	−1.29870
$364$	0	0
$365$	35.2514	1.84514
$366$	4.53677	0.237141
$367$	−22.3679	−1.16759	−0.583796	−	0.811900i	$-0.698433\pi$
$367$	−22.3679	−1.16759	−0.583796	+	0.811900i	$0.698433\pi$
$368$	−3.42114	−0.178339
$369$	−2.20171	−0.114616
$370$	56.4973	2.93716
$371$	6.47901	0.336373
$372$	−1.83386	−0.0950813
$373$	14.2950	0.740166	0.370083	−	0.928999i	$-0.379329\pi$
$373$	14.2950	0.740166	0.370083	+	0.928999i	$0.379329\pi$
$374$	5.74418	0.297024
$375$	−21.7778	−1.12460
$376$	−9.64445	−0.497375
$377$	0	0
$378$	39.0798	2.01005
$379$	−3.86964	−0.198770	−0.0993852	−	0.995049i	$-0.531688\pi$
$379$	−3.86964	−0.198770	−0.0993852	+	0.995049i	$0.531688\pi$
$380$	29.2129	1.49859
$381$	−5.18247	−0.265506
$382$	29.9437	1.53206
$383$	−17.3267	−0.885352	−0.442676	−	0.896682i	$-0.645971\pi$
$383$	−17.3267	−0.885352	−0.442676	+	0.896682i	$0.645971\pi$
$384$	19.1352	0.976488
$385$	−77.5186	−3.95071
$386$	−0.181407	−0.00923335
$387$	0.358661	0.0182317
$388$	13.4175	0.681170
$389$	31.8175	1.61321	0.806605	−	0.591091i	$-0.201303\pi$
$389$	31.8175	1.61321	0.806605	+	0.591091i	$0.201303\pi$
$390$	0	0
$391$	−0.440778	−0.0222911
$392$	16.2664	0.821576
$393$	1.09776	0.0553746
$394$	−4.42995	−0.223177
$395$	−15.2690	−0.768268
$396$	−1.09708	−0.0551304
$397$	−9.19501	−0.461484	−0.230742	−	0.973015i	$-0.574115\pi$
$397$	−9.19501	−0.461484	−0.230742	+	0.973015i	$0.574115\pi$
$398$	0.00276811	0.000138753 0
$399$	50.3588	2.52109
$400$	−42.6010	−2.13005
$401$	8.67433	0.433175	0.216588	−	0.976263i	$-0.430507\pi$
$401$	8.67433	0.433175	0.216588	+	0.976263i	$0.430507\pi$
$402$	−6.19757	−0.309107
$403$	0	0
$404$	−20.7769	−1.03369
$405$	−30.7916	−1.53005
$406$	42.5519	2.11182
$407$	−44.3168	−2.19670
$408$	1.71330	0.0848210
$409$	11.7332	0.580171	0.290085	−	0.957001i	$-0.406316\pi$
$409$	11.7332	0.580171	0.290085	+	0.957001i	$0.406316\pi$
$410$	−72.2498	−3.56816
$411$	−0.744295	−0.0367134
$412$	12.3193	0.606929
$413$	−29.6312	−1.45805
$414$	0.237889	0.0116916
$415$	−24.9142	−1.22299
$416$	0	0
$417$	−3.32668	−0.162908
$418$	−64.7527	−3.16716
$419$	20.6599	1.00930	0.504651	−	0.863324i	$-0.331621\pi$
$419$	20.6599	1.00930	0.504651	+	0.863324i	$0.331621\pi$
$420$	27.9983	1.36618
$421$	−4.02970	−0.196396	−0.0981978	−	0.995167i	$-0.531308\pi$
$421$	−4.02970	−0.196396	−0.0981978	+	0.995167i	$0.531308\pi$
$422$	29.8833	1.45469
$423$	1.19533	0.0581190
$424$	−2.48483	−0.120674
$425$	−5.48870	−0.266241
$426$	−1.60863	−0.0779382
$427$	−6.39176	−0.309319
$428$	−14.3049	−0.691455
$429$	0	0
$430$	11.7696	0.567578
$431$	−6.35712	−0.306212	−0.153106	−	0.988210i	$-0.548928\pi$
$431$	−6.35712	−0.306212	−0.153106	+	0.988210i	$0.548928\pi$
$432$	−26.7145	−1.28530
$433$	23.5980	1.13405	0.567024	−	0.823701i	$-0.308095\pi$
$433$	23.5980	1.13405	0.567024	+	0.823701i	$0.308095\pi$
$434$	7.30102	0.350460
$435$	−35.8979	−1.72117
$436$	15.2029	0.728086
$437$	4.96879	0.237689
$438$	28.2219	1.34850
$439$	0.942932	0.0450037	0.0225018	−	0.999747i	$-0.492837\pi$
$439$	0.942932	0.0450037	0.0225018	+	0.999747i	$0.492837\pi$
$440$	29.7300	1.41732
$441$	−2.01605	−0.0960023
$442$	0	0
$443$	−21.8875	−1.03991	−0.519954	−	0.854194i	$-0.674051\pi$
$443$	−21.8875	−1.03991	−0.519954	+	0.854194i	$0.674051\pi$
$444$	16.0064	0.759632
$445$	54.1609	2.56747
$446$	17.5964	0.833212
$447$	26.5972	1.25800
$448$	−0.552432	−0.0260999
$449$	−13.8867	−0.655352	−0.327676	−	0.944790i	$-0.606265\pi$
$449$	−13.8867	−0.655352	−0.327676	+	0.944790i	$0.606265\pi$
$450$	2.96226	0.139642
$451$	56.6732	2.66864
$452$	0.475092	0.0223465
$453$	33.4852	1.57327
$454$	−26.8987	−1.26242
$455$	0	0
$456$	−19.3136	−0.904444
$457$	22.4197	1.04875	0.524374	−	0.851488i	$-0.324299\pi$
$457$	22.4197	1.04875	0.524374	+	0.851488i	$0.324299\pi$
$458$	−29.2150	−1.36513
$459$	−3.44189	−0.160654
$460$	2.76253	0.128804
$461$	−32.1892	−1.49920	−0.749599	−	0.661892i	$-0.769754\pi$
$461$	−32.1892	−1.49920	−0.749599	+	0.661892i	$0.769754\pi$
$462$	−62.0606	−2.88732
$463$	20.1648	0.937138	0.468569	−	0.883427i	$-0.344770\pi$
$463$	20.1648	0.937138	0.468569	+	0.883427i	$0.344770\pi$
$464$	−29.0881	−1.35038
$465$	−6.15932	−0.285632
$466$	18.2537	0.845587
$467$	2.15941	0.0999256	0.0499628	−	0.998751i	$-0.484090\pi$
$467$	2.15941	0.0999256	0.0499628	+	0.998751i	$0.484090\pi$
$468$	0	0
$469$	8.73163	0.403189
$470$	39.2251	1.80932
$471$	−0.183448	−0.00845286
$472$	11.3642	0.523078
$473$	−9.23211	−0.424493
$474$	−12.2242	−0.561477
$475$	61.8728	2.83892
$476$	2.92299	0.133975
$477$	0.307969	0.0141009
$478$	46.9949	2.14950
$479$	43.6098	1.99258	0.996291	−	0.0860496i	$-0.0274244\pi$
$479$	43.6098	1.99258	0.996291	+	0.0860496i	$0.0274244\pi$
$480$	−34.4788	−1.57373
$481$	0	0
$482$	−15.5388	−0.707775
$483$	4.76221	0.216688
$484$	16.1900	0.735907
$485$	45.0648	2.04629
$486$	3.60059	0.163326
$487$	−1.18940	−0.0538970	−0.0269485	−	0.999637i	$-0.508579\pi$
$487$	−1.18940	−0.0538970	−0.0269485	+	0.999637i	$0.508579\pi$
$488$	2.45137	0.110968
$489$	−9.43186	−0.426523
$490$	−66.1572	−2.98868
$491$	−0.630068	−0.0284346	−0.0142173	−	0.999899i	$-0.504526\pi$
$491$	−0.630068	−0.0284346	−0.0142173	+	0.999899i	$0.504526\pi$
$492$	−20.4693	−0.922829
$493$	−3.74770	−0.168788
$494$	0	0
$495$	−3.68472	−0.165616
$496$	−4.99090	−0.224098
$497$	2.26636	0.101660
$498$	−19.9460	−0.893803
$499$	11.4173	0.511106	0.255553	−	0.966795i	$-0.417742\pi$
$499$	11.4173	0.511106	0.255553	+	0.966795i	$0.417742\pi$
$500$	14.2494	0.637251
$501$	15.1391	0.676367
$502$	−49.3343	−2.20190
$503$	25.6104	1.14191	0.570955	−	0.820982i	$-0.306573\pi$
$503$	25.6104	1.14191	0.570955	+	0.820982i	$0.306573\pi$
$504$	1.30275	0.0580291
$505$	−69.7827	−3.10529
$506$	−6.12338	−0.272217
$507$	0	0
$508$	3.39094	0.150448
$509$	37.4325	1.65917	0.829583	−	0.558384i	$-0.188578\pi$
$509$	37.4325	1.65917	0.829583	+	0.558384i	$0.188578\pi$
$510$	−6.96819	−0.308557
$511$	−39.7613	−1.75894
$512$	−12.0519	−0.532623
$513$	38.7997	1.71305
$514$	−33.5959	−1.48185
$515$	41.3764	1.82326
$516$	3.33447	0.146792
$517$	−30.7684	−1.35319
$518$	−63.7253	−2.79993
$519$	−15.9609	−0.700604
$520$	0	0
$521$	37.2529	1.63208	0.816040	−	0.577996i	$-0.196165\pi$
$521$	37.2529	1.63208	0.816040	+	0.577996i	$0.196165\pi$
$522$	2.02264	0.0885285
$523$	14.8086	0.647537	0.323768	−	0.946136i	$-0.395050\pi$
$523$	14.8086	0.647537	0.323768	+	0.946136i	$0.395050\pi$
$524$	−0.718274	−0.0313779
$525$	59.3004	2.58808
$526$	3.65710	0.159457
$527$	−0.643026	−0.0280107
$528$	42.4240	1.84627
$529$	−22.5301	−0.979571
$530$	10.1061	0.438981
$531$	−1.40847	−0.0611224
$532$	−32.9502	−1.42857
$533$	0	0
$534$	43.3606	1.87640
$535$	−48.0454	−2.07718
$536$	−3.34876	−0.144644
$537$	17.2562	0.744660
$538$	34.5948	1.49149
$539$	51.8941	2.23524
$540$	21.5717	0.928299
$541$	27.9305	1.20083	0.600413	−	0.799690i	$-0.295003\pi$
$541$	27.9305	1.20083	0.600413	+	0.799690i	$0.295003\pi$
$542$	−13.3392	−0.572966
$543$	−5.70336	−0.244755
$544$	−3.59954	−0.154329
$545$	51.0614	2.18723
$546$	0	0
$547$	34.4275	1.47202	0.736008	−	0.676973i	$-0.236709\pi$
$547$	34.4275	1.47202	0.736008	+	0.676973i	$0.236709\pi$
$548$	0.486999	0.0208036
$549$	−0.303822	−0.0129668
$550$	−76.2501	−3.25131
$551$	42.2469	1.79978
$552$	−1.82640	−0.0777369
$553$	17.2224	0.732373
$554$	−15.5391	−0.660193
$555$	53.7603	2.28200
$556$	2.17668	0.0923117
$557$	−4.37431	−0.185346	−0.0926728	−	0.995697i	$-0.529541\pi$
$557$	−4.37431	−0.185346	−0.0926728	+	0.995697i	$0.529541\pi$
$558$	0.347042	0.0146915
$559$	0	0
$560$	76.1981	3.21996
$561$	5.46589	0.230770
$562$	6.57033	0.277152
$563$	43.6108	1.83798	0.918988	−	0.394285i	$-0.129008\pi$
$563$	43.6108	1.83798	0.918988	+	0.394285i	$0.129008\pi$
$564$	11.1130	0.467942
$565$	1.59567	0.0671305
$566$	6.09038	0.255998
$567$	34.7308	1.45856
$568$	−0.869196	−0.0364706
$569$	9.54889	0.400310	0.200155	−	0.979764i	$-0.435855\pi$
$569$	9.54889	0.400310	0.200155	+	0.979764i	$0.435855\pi$
$570$	78.5508	3.29013
$571$	−15.7147	−0.657639	−0.328819	−	0.944393i	$-0.606651\pi$
$571$	−15.7147	−0.657639	−0.328819	+	0.944393i	$0.606651\pi$
$572$	0	0
$573$	28.4931	1.19032
$574$	81.4930	3.40145
$575$	5.85104	0.244005
$576$	−0.0262590	−0.00109412
$577$	1.09693	0.0456660	0.0228330	−	0.999739i	$-0.492731\pi$
$577$	1.09693	0.0456660	0.0228330	+	0.999739i	$0.492731\pi$
$578$	29.1819	1.21381
$579$	−0.172618	−0.00717376
$580$	23.4883	0.975299
$581$	28.1016	1.16585
$582$	36.0784	1.49550
$583$	−7.92729	−0.328315
$584$	15.2493	0.631019
$585$	0	0
$586$	−40.3456	−1.66666
$587$	2.59363	0.107051	0.0535253	−	0.998566i	$-0.482954\pi$
$587$	2.59363	0.107051	0.0535253	+	0.998566i	$0.482954\pi$
$588$	−18.7432	−0.772957
$589$	7.24868	0.298677
$590$	−46.2193	−1.90282
$591$	−4.21533	−0.173396
$592$	43.5619	1.79038
$593$	−13.7627	−0.565165	−0.282582	−	0.959243i	$-0.591191\pi$
$593$	−13.7627	−0.565165	−0.282582	+	0.959243i	$0.591191\pi$
$594$	−47.8154	−1.96189
$595$	9.81735	0.402472
$596$	−17.4027	−0.712844
$597$	0.00263400	0.000107802 0
$598$	0	0
$599$	10.5758	0.432114	0.216057	−	0.976381i	$-0.430680\pi$
$599$	10.5758	0.432114	0.216057	+	0.976381i	$0.430680\pi$
$600$	−22.7429	−0.928476
$601$	34.3691	1.40194	0.700972	−	0.713189i	$-0.252750\pi$
$601$	34.3691	1.40194	0.700972	+	0.713189i	$0.252750\pi$
$602$	−13.2753	−0.541060
$603$	0.415044	0.0169019
$604$	−21.9097	−0.891492
$605$	54.3766	2.21072
$606$	−55.8673	−2.26945
$607$	14.5224	0.589444	0.294722	−	0.955583i	$-0.404773\pi$
$607$	14.5224	0.589444	0.294722	+	0.955583i	$0.404773\pi$
$608$	40.5768	1.64561
$609$	40.4904	1.64076
$610$	−9.97000	−0.403674
$611$	0	0
$612$	0.138940	0.00561631
$613$	36.8553	1.48857	0.744287	−	0.667860i	$-0.232790\pi$
$613$	36.8553	1.48857	0.744287	+	0.667860i	$0.232790\pi$
$614$	−10.5540	−0.425924
$615$	−68.7496	−2.77225
$616$	−33.5334	−1.35110
$617$	27.1376	1.09252	0.546259	−	0.837616i	$-0.316051\pi$
$617$	27.1376	1.09252	0.546259	+	0.837616i	$0.316051\pi$
$618$	33.1255	1.33250
$619$	−21.8315	−0.877481	−0.438740	−	0.898614i	$-0.644575\pi$
$619$	−21.8315	−0.877481	−0.438740	+	0.898614i	$0.644575\pi$
$620$	4.03010	0.161853
$621$	3.66911	0.147236
$622$	0.213289	0.00855212
$623$	−61.0899	−2.44751
$624$	0	0
$625$	5.18015	0.207206
$626$	51.5992	2.06232
$627$	−61.6157	−2.46069
$628$	0.120032	0.00478980
$629$	5.61251	0.223785
$630$	−5.29843	−0.211095
$631$	32.3195	1.28662	0.643309	−	0.765607i	$-0.277561\pi$
$631$	32.3195	1.28662	0.643309	+	0.765607i	$0.277561\pi$
$632$	−6.60516	−0.262739
$633$	28.4355	1.13021
$634$	−25.8384	−1.02618
$635$	11.3890	0.451959
$636$	2.86319	0.113533
$637$	0	0
$638$	−52.0637	−2.06122
$639$	0.107728	0.00426165
$640$	−42.0515	−1.66223
$641$	−15.3890	−0.607831	−0.303915	−	0.952699i	$-0.598294\pi$
$641$	−15.3890	−0.607831	−0.303915	+	0.952699i	$0.598294\pi$
$642$	−38.4646	−1.51808
$643$	−10.6498	−0.419986	−0.209993	−	0.977703i	$-0.567344\pi$
$643$	−10.6498	−0.419986	−0.209993	+	0.977703i	$0.567344\pi$
$644$	−3.11595	−0.122786
$645$	11.1994	0.440974
$646$	8.20061	0.322649
$647$	38.8098	1.52577	0.762886	−	0.646533i	$-0.223782\pi$
$647$	38.8098	1.52577	0.762886	+	0.646533i	$0.223782\pi$
$648$	−13.3200	−0.523259
$649$	36.2547	1.42312
$650$	0	0
$651$	6.94731	0.272286
$652$	6.17135	0.241689
$653$	−18.6676	−0.730520	−0.365260	−	0.930906i	$-0.619020\pi$
$653$	−18.6676	−0.730520	−0.365260	+	0.930906i	$0.619020\pi$
$654$	40.8792	1.59850
$655$	−2.41244	−0.0942618
$656$	−55.7078	−2.17502
$657$	−1.88999	−0.0737355
$658$	−44.2433	−1.72478
$659$	−38.2464	−1.48987	−0.744934	−	0.667138i	$-0.767519\pi$
$659$	−38.2464	−1.48987	−0.744934	+	0.667138i	$0.767519\pi$
$660$	−34.2569	−1.33345
$661$	−13.4345	−0.522541	−0.261270	−	0.965266i	$-0.584141\pi$
$661$	−13.4345	−0.522541	−0.261270	+	0.965266i	$0.584141\pi$
$662$	35.2487	1.36998
$663$	0	0
$664$	−10.7775	−0.418249
$665$	−110.669	−4.29154
$666$	−3.02908	−0.117374
$667$	3.99510	0.154691
$668$	−9.90567	−0.383262
$669$	16.7439	0.647356
$670$	13.6198	0.526178
$671$	7.82053	0.301908
$672$	38.8898	1.50021
$673$	−16.8262	−0.648603	−0.324301	−	0.945954i	$-0.605129\pi$
$673$	−16.8262	−0.648603	−0.324301	+	0.945954i	$0.605129\pi$
$674$	−2.20727	−0.0850209
$675$	45.6888	1.75856
$676$	0	0
$677$	−27.1489	−1.04342	−0.521708	−	0.853124i	$-0.674705\pi$
$677$	−27.1489	−1.04342	−0.521708	+	0.853124i	$0.674705\pi$
$678$	1.27748	0.0490613
$679$	−50.8301	−1.95068
$680$	−3.76515	−0.144387
$681$	−25.5955	−0.980822
$682$	−8.93304	−0.342064
$683$	−14.3975	−0.550905	−0.275452	−	0.961315i	$-0.588828\pi$
$683$	−14.3975	−0.550905	−0.275452	+	0.961315i	$0.588828\pi$
$684$	−1.56624	−0.0598865
$685$	1.63566	0.0624955
$686$	23.5139	0.897764
$687$	−27.7996	−1.06062
$688$	9.07484	0.345975
$689$	0	0
$690$	7.42820	0.282787
$691$	30.0423	1.14286	0.571431	−	0.820650i	$-0.306389\pi$
$691$	30.0423	1.14286	0.571431	+	0.820650i	$0.306389\pi$
$692$	10.4433	0.396996
$693$	4.15612	0.157878
$694$	−58.8423	−2.23362
$695$	7.31073	0.277312
$696$	−15.5289	−0.588622
$697$	−7.17737	−0.271862
$698$	28.2801	1.07042
$699$	17.3694	0.656971
$700$	−38.8008	−1.46653
$701$	1.37156	0.0518029	0.0259015	−	0.999665i	$-0.491754\pi$
$701$	1.37156	0.0518029	0.0259015	+	0.999665i	$0.491754\pi$
$702$	0	0
$703$	−63.2685	−2.38622
$704$	0.675919	0.0254747
$705$	37.3248	1.40573
$706$	−59.4667	−2.23806
$707$	78.7102	2.96020
$708$	−13.0946	−0.492123
$709$	−2.70050	−0.101420	−0.0507098	−	0.998713i	$-0.516148\pi$
$709$	−2.70050	−0.101420	−0.0507098	+	0.998713i	$0.516148\pi$
$710$	3.53512	0.132671
$711$	0.818641	0.0307014
$712$	23.4292	0.878047
$713$	0.685475	0.0256712
$714$	7.85966	0.294140
$715$	0	0
$716$	−11.2909	−0.421960
$717$	44.7182	1.67003
$718$	15.1003	0.563537
$719$	11.6253	0.433552	0.216776	−	0.976221i	$-0.430446\pi$
$719$	11.6253	0.433552	0.216776	+	0.976221i	$0.430446\pi$
$720$	3.62195	0.134982
$721$	−46.6698	−1.73808
$722$	−59.0154	−2.19633
$723$	−14.7860	−0.549899
$724$	3.73176	0.138690
$725$	49.7482	1.84760
$726$	43.5333	1.61567
$727$	−3.07478	−0.114037	−0.0570186	−	0.998373i	$-0.518159\pi$
$727$	−3.07478	−0.114037	−0.0570186	+	0.998373i	$0.518159\pi$
$728$	0	0
$729$	28.5341	1.05682
$730$	−62.0205	−2.29548
$731$	1.16920	0.0432444
$732$	−2.82463	−0.104401
$733$	−28.7230	−1.06091	−0.530455	−	0.847713i	$-0.677979\pi$
$733$	−28.7230	−1.06091	−0.530455	+	0.847713i	$0.677979\pi$
$734$	39.3535	1.45256
$735$	−62.9522	−2.32203
$736$	3.83717	0.141440
$737$	−10.6834	−0.393530
$738$	3.87364	0.142591
$739$	41.9629	1.54363	0.771816	−	0.635846i	$-0.219349\pi$
$739$	41.9629	1.54363	0.771816	+	0.635846i	$0.219349\pi$
$740$	−35.1758	−1.29309
$741$	0	0
$742$	−11.3990	−0.418471
$743$	−27.0052	−0.990725	−0.495363	−	0.868686i	$-0.664965\pi$
$743$	−27.0052	−0.990725	−0.495363	+	0.868686i	$0.664965\pi$
$744$	−2.66443	−0.0976829
$745$	−58.4500	−2.14144
$746$	−25.1502	−0.920815
$747$	1.33576	0.0488730
$748$	−3.57638	−0.130765
$749$	54.1920	1.98013
$750$	38.3153	1.39908
$751$	−20.4276	−0.745414	−0.372707	−	0.927949i	$-0.621570\pi$
$751$	−20.4276	−0.745414	−0.372707	+	0.927949i	$0.621570\pi$
$752$	30.2443	1.10290
$753$	−46.9442	−1.71074
$754$	0	0
$755$	−73.5872	−2.67811
$756$	−24.3315	−0.884926
$757$	12.6110	0.458355	0.229177	−	0.973385i	$-0.426396\pi$
$757$	12.6110	0.458355	0.229177	+	0.973385i	$0.426396\pi$
$758$	6.80816	0.247284
$759$	−5.82672	−0.211497
$760$	42.4437	1.53959
$761$	12.6269	0.457726	0.228863	−	0.973459i	$-0.426499\pi$
$761$	12.6269	0.457726	0.228863	+	0.973459i	$0.426499\pi$
$762$	9.11792	0.330307
$763$	−57.5938	−2.08504
$764$	−18.6433	−0.674490
$765$	0.466652	0.0168718
$766$	30.4841	1.10144
$767$	0	0
$768$	−33.2203	−1.19873
$769$	51.9432	1.87312	0.936560	−	0.350507i	$-0.113991\pi$
$769$	51.9432	1.87312	0.936560	+	0.350507i	$0.113991\pi$
$770$	136.384	4.91495
$771$	−31.9683	−1.15131
$772$	0.112945	0.00406500
$773$	19.4706	0.700310	0.350155	−	0.936692i	$-0.386129\pi$
$773$	19.4706	0.700310	0.350155	+	0.936692i	$0.386129\pi$
$774$	−0.631019	−0.0226815
$775$	8.53574	0.306613
$776$	19.4944	0.699808
$777$	−60.6380	−2.17538
$778$	−55.9789	−2.00694
$779$	80.9088	2.89886
$780$	0	0
$781$	−2.77297	−0.0992246
$782$	0.775495	0.0277316
$783$	31.1964	1.11487
$784$	−51.0101	−1.82179
$785$	0.403147	0.0143889
$786$	−1.93137	−0.0688898
$787$	36.0333	1.28445	0.642225	−	0.766516i	$-0.278012\pi$
$787$	36.0333	1.28445	0.642225	+	0.766516i	$0.278012\pi$
$788$	2.75813	0.0982542
$789$	3.47993	0.123889
$790$	26.8640	0.955777
$791$	−1.79981	−0.0639940
$792$	−1.59396	−0.0566388
$793$	0	0
$794$	16.1775	0.574117
$795$	9.61650	0.341062
$796$	−0.00172345	−6.10860e−5 0
$797$	−17.0989	−0.605673	−0.302837	−	0.953042i	$-0.597934\pi$
$797$	−17.0989	−0.605673	−0.302837	+	0.953042i	$0.597934\pi$
$798$	−88.6001	−3.13641
$799$	3.89667	0.137854
$800$	47.7815	1.68933
$801$	−2.90381	−0.102601
$802$	−15.2614	−0.538899
$803$	48.6493	1.71680
$804$	3.85867	0.136085
$805$	−10.4654	−0.368858
$806$	0	0
$807$	32.9188	1.15880
$808$	−30.1870	−1.06197
$809$	−33.4560	−1.17625	−0.588125	−	0.808770i	$-0.700134\pi$
$809$	−33.4560	−1.17625	−0.588125	+	0.808770i	$0.700134\pi$
$810$	54.1740	1.90348
$811$	−6.67383	−0.234350	−0.117175	−	0.993111i	$-0.537384\pi$
$811$	−6.67383	−0.234350	−0.117175	+	0.993111i	$0.537384\pi$
$812$	−26.4933	−0.929731
$813$	−12.6929	−0.445160
$814$	77.9700	2.73285
$815$	20.7275	0.726052
$816$	−5.37278	−0.188085
$817$	−13.1801	−0.461114
$818$	−20.6432	−0.721772
$819$	0	0
$820$	44.9834	1.57089
$821$	17.0481	0.594983	0.297491	−	0.954725i	$-0.403850\pi$
$821$	17.0481	0.594983	0.297491	+	0.954725i	$0.403850\pi$
$822$	1.30950	0.0456739
$823$	−36.2818	−1.26470	−0.632352	−	0.774682i	$-0.717910\pi$
$823$	−36.2818	−1.26470	−0.632352	+	0.774682i	$0.717910\pi$
$824$	17.8988	0.623535
$825$	−72.5560	−2.52608
$826$	52.1324	1.81392
$827$	−14.2203	−0.494488	−0.247244	−	0.968953i	$-0.579525\pi$
$827$	−14.2203	−0.494488	−0.247244	+	0.968953i	$0.579525\pi$
$828$	−0.148112	−0.00514724
$829$	23.0164	0.799393	0.399696	−	0.916648i	$-0.369116\pi$
$829$	23.0164	0.799393	0.399696	+	0.916648i	$0.369116\pi$
$830$	43.8335	1.52148
$831$	−14.7863	−0.512931
$832$	0	0
$833$	−6.57213	−0.227711
$834$	5.85289	0.202669
$835$	−33.2698	−1.15135
$836$	40.3157	1.39435
$837$	5.35265	0.185015
$838$	−36.3485	−1.25564
$839$	−40.5007	−1.39824	−0.699120	−	0.715004i	$-0.746425\pi$
$839$	−40.5007	−1.39824	−0.699120	+	0.715004i	$0.746425\pi$
$840$	40.6790	1.40356
$841$	4.96817	0.171316
$842$	7.08976	0.244329
$843$	6.25202	0.215331
$844$	−18.6056	−0.640432
$845$	0	0
$846$	−2.10304	−0.0723039
$847$	−61.3332	−2.10743
$848$	7.79225	0.267587
$849$	5.79532	0.198895
$850$	9.65669	0.331222
$851$	−5.98301	−0.205095
$852$	1.00155	0.0343124
$853$	−49.0041	−1.67787	−0.838934	−	0.544234i	$-0.816820\pi$
$853$	−49.0041	−1.67787	−0.838934	+	0.544234i	$0.816820\pi$
$854$	11.2455	0.384813
$855$	−5.26045	−0.179904
$856$	−20.7838	−0.710374
$857$	−21.6945	−0.741069	−0.370534	−	0.928819i	$-0.620825\pi$
$857$	−21.6945	−0.741069	−0.370534	+	0.928819i	$0.620825\pi$
$858$	0	0
$859$	36.1931	1.23489	0.617446	−	0.786613i	$-0.288167\pi$
$859$	36.1931	1.23489	0.617446	+	0.786613i	$0.288167\pi$
$860$	−7.32784	−0.249877
$861$	77.5449	2.64272
$862$	11.1846	0.380948
$863$	6.19578	0.210907	0.105453	−	0.994424i	$-0.466371\pi$
$863$	6.19578	0.210907	0.105453	+	0.994424i	$0.466371\pi$
$864$	29.9632	1.01937
$865$	35.0756	1.19261
$866$	−41.5178	−1.41083
$867$	27.7682	0.943056
$868$	−4.54568	−0.154291
$869$	−21.0722	−0.714827
$870$	63.1579	2.14125
$871$	0	0
$872$	22.0884	0.748008
$873$	−2.41613	−0.0817736
$874$	−8.74197	−0.295702
$875$	−53.9816	−1.82491
$876$	−17.5712	−0.593677
$877$	15.6975	0.530067	0.265033	−	0.964239i	$-0.414617\pi$
$877$	15.6975	0.530067	0.265033	+	0.964239i	$0.414617\pi$
$878$	−1.65897	−0.0559876
$879$	−38.3910	−1.29490
$880$	−93.2310	−3.14282
$881$	−10.5437	−0.355225	−0.177613	−	0.984100i	$-0.556837\pi$
$881$	−10.5437	−0.355225	−0.177613	+	0.984100i	$0.556837\pi$
$882$	3.54699	0.119433
$883$	11.6997	0.393727	0.196864	−	0.980431i	$-0.436924\pi$
$883$	11.6997	0.393727	0.196864	+	0.980431i	$0.436924\pi$
$884$	0	0
$885$	−43.9802	−1.47838
$886$	38.5084	1.29372
$887$	23.8193	0.799774	0.399887	−	0.916564i	$-0.369049\pi$
$887$	23.8193	0.799774	0.399887	+	0.916564i	$0.369049\pi$
$888$	23.2559	0.780417
$889$	−12.8460	−0.430843
$890$	−95.2893	−3.19411
$891$	−42.4944	−1.42361
$892$	−10.9557	−0.366823
$893$	−43.9262	−1.46993
$894$	−46.7944	−1.56504
$895$	−37.9223	−1.26760
$896$	47.4313	1.58457
$897$	0	0
$898$	24.4318	0.815301
$899$	5.82822	0.194382
$900$	−1.84433	−0.0614777
$901$	1.00395	0.0334465
$902$	−99.7094	−3.31996
$903$	−12.6321	−0.420371
$904$	0.690266	0.0229579
$905$	12.5337	0.416635
$906$	−58.9131	−1.95726
$907$	−2.24010	−0.0743814	−0.0371907	−	0.999308i	$-0.511841\pi$
$907$	−2.24010	−0.0743814	−0.0371907	+	0.999308i	$0.511841\pi$
$908$	16.7474	0.555781
$909$	3.74136	0.124093
$910$	0	0
$911$	12.9228	0.428152	0.214076	−	0.976817i	$-0.431326\pi$
$911$	12.9228	0.428152	0.214076	+	0.976817i	$0.431326\pi$
$912$	60.5661	2.00554
$913$	−34.3832	−1.13792
$914$	−39.4447	−1.30471
$915$	−9.48699	−0.313630
$916$	18.1895	0.600999
$917$	2.72107	0.0898577
$918$	6.05559	0.199864
$919$	−21.0966	−0.695913	−0.347956	−	0.937511i	$-0.613124\pi$
$919$	−21.0966	−0.695913	−0.347956	+	0.937511i	$0.613124\pi$
$920$	4.01371	0.132328
$921$	−10.0427	−0.330917
$922$	56.6328	1.86510
$923$	0	0
$924$	38.6395	1.27115
$925$	−74.5023	−2.44962
$926$	−35.4775	−1.16586
$927$	−2.21838	−0.0728610
$928$	32.6253	1.07098
$929$	−19.7434	−0.647760	−0.323880	−	0.946098i	$-0.604987\pi$
$929$	−19.7434	−0.647760	−0.323880	+	0.946098i	$0.604987\pi$
$930$	10.8366	0.355345
$931$	74.0861	2.42807
$932$	−11.3649	−0.372271
$933$	0.202956	0.00664449
$934$	−3.79921	−0.124314
$935$	−12.0119	−0.392830
$936$	0	0
$937$	−6.87850	−0.224711	−0.112355	−	0.993668i	$-0.535839\pi$
$937$	−6.87850	−0.224711	−0.112355	+	0.993668i	$0.535839\pi$
$938$	−15.3622	−0.501594
$939$	49.0994	1.60230
$940$	−24.4219	−0.796556
$941$	−16.7174	−0.544973	−0.272486	−	0.962160i	$-0.587846\pi$
$941$	−16.7174	−0.544973	−0.272486	+	0.962160i	$0.587846\pi$
$942$	0.322755	0.0105159
$943$	7.65118	0.249157
$944$	−35.6371	−1.15989
$945$	−81.7211	−2.65839
$946$	16.2428	0.528097
$947$	−42.1651	−1.37018	−0.685091	−	0.728457i	$-0.740238\pi$
$947$	−42.1651	−1.37018	−0.685091	+	0.728457i	$0.740238\pi$
$948$	7.61091	0.247191
$949$	0	0
$950$	−108.858	−3.53181
$951$	−24.5867	−0.797277
$952$	4.24684	0.137641
$953$	−47.4062	−1.53564	−0.767819	−	0.640667i	$-0.778658\pi$
$953$	−47.4062	−1.53564	−0.767819	+	0.640667i	$0.778658\pi$
$954$	−0.541834	−0.0175425
$955$	−62.6164	−2.02622
$956$	−29.2595	−0.946320
$957$	−49.5414	−1.60145
$958$	−76.7260	−2.47890
$959$	−1.84492	−0.0595756
$960$	−0.819949	−0.0264638
$961$	1.00000	0.0322581
$962$	0	0
$963$	2.57593	0.0830082
$964$	9.67464	0.311599
$965$	0.379346	0.0122116
$966$	−8.37851	−0.269574
$967$	−22.3941	−0.720147	−0.360074	−	0.932924i	$-0.617248\pi$
$967$	−22.3941	−0.720147	−0.360074	+	0.932924i	$0.617248\pi$
$968$	23.5225	0.756043
$969$	7.80332	0.250679
$970$	−79.2860	−2.54572
$971$	5.11176	0.164044	0.0820221	−	0.996631i	$-0.473862\pi$
$971$	5.11176	0.164044	0.0820221	+	0.996631i	$0.473862\pi$
$972$	−2.24176	−0.0719045
$973$	−8.24602	−0.264355
$974$	2.09261	0.0670515
$975$	0	0
$976$	−7.68731	−0.246065
$977$	19.3950	0.620500	0.310250	−	0.950655i	$-0.399587\pi$
$977$	19.3950	0.620500	0.310250	+	0.950655i	$0.399587\pi$
$978$	16.5942	0.530624
$979$	74.7455	2.38888
$980$	41.1901	1.31577
$981$	−2.73763	−0.0874058
$982$	1.10853	0.0353745
$983$	40.4106	1.28890	0.644449	−	0.764647i	$-0.277087\pi$
$983$	40.4106	1.28890	0.644449	+	0.764647i	$0.277087\pi$
$984$	−29.7401	−0.948079
$985$	9.26362	0.295163
$986$	6.59361	0.209983
$987$	−42.0999	−1.34005
$988$	0	0
$989$	−1.24638	−0.0396327
$990$	6.48281	0.206037
$991$	−56.3062	−1.78863	−0.894313	−	0.447443i	$-0.852335\pi$
$991$	−56.3062	−1.78863	−0.894313	+	0.447443i	$0.852335\pi$
$992$	5.59782	0.177731
$993$	33.5410	1.06439
$994$	−3.98738	−0.126472
$995$	−0.00578849	−0.000183507 0
$996$	12.4186	0.393498
$997$	27.2985	0.864552	0.432276	−	0.901741i	$-0.357711\pi$
$997$	27.2985	0.864552	0.432276	+	0.901741i	$0.357711\pi$
$998$	−20.0872	−0.635851
$999$	−46.7194	−1.47814

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5239.2.a.j.1.1		8
13.12	even	2		403.2.a.d.1.8	✓	8
39.38	odd	2		3627.2.a.q.1.1		8
52.51	odd	2		6448.2.a.bf.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.a.d.1.8	✓	8	13.12	even	2
3627.2.a.q.1.1		8	39.38	odd	2
5239.2.a.j.1.1		8	1.1	even	1	trivial
6448.2.a.bf.1.6		8	52.51	odd	2

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:	\( N \)	\(=\)	\( 5239 = 13^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5239.a (trivial)

\(f(q)\)	\(=\)	\(q-1.75938 q^{2} -1.67414 q^{3} +1.09540 q^{4} +3.67909 q^{5} +2.94544 q^{6} -4.14977 q^{7} +1.59152 q^{8} -0.197253 q^{9} +O(q^{10})\)	\(q-1.75938 q^{2} -1.67414 q^{3} +1.09540 q^{4} +3.67909 q^{5} +2.94544 q^{6} -4.14977 q^{7} +1.59152 q^{8} -0.197253 q^{9} -6.47291 q^{10} +5.07739 q^{11} -1.83386 q^{12} +7.30102 q^{14} -6.15932 q^{15} -4.99090 q^{16} -0.643026 q^{17} +0.347042 q^{18} +7.24868 q^{19} +4.03010 q^{20} +6.94731 q^{21} -8.93304 q^{22} +0.685475 q^{23} -2.66443 q^{24} +8.53574 q^{25} +5.35265 q^{27} -4.54568 q^{28} +5.82822 q^{29} +10.8366 q^{30} +1.00000 q^{31} +5.59782 q^{32} -8.50026 q^{33} +1.13132 q^{34} -15.2674 q^{35} -0.216072 q^{36} -8.72827 q^{37} -12.7532 q^{38} +5.85537 q^{40} +11.1619 q^{41} -12.2229 q^{42} -1.81828 q^{43} +5.56180 q^{44} -0.725712 q^{45} -1.20601 q^{46} -6.05989 q^{47} +8.35547 q^{48} +10.2206 q^{49} -15.0176 q^{50} +1.07652 q^{51} -1.56129 q^{53} -9.41733 q^{54} +18.6802 q^{55} -6.60446 q^{56} -12.1353 q^{57} -10.2540 q^{58} +7.14043 q^{59} -6.74695 q^{60} +1.54027 q^{61} -1.75938 q^{62} +0.818555 q^{63} +0.133123 q^{64} +14.9552 q^{66} -2.10412 q^{67} -0.704374 q^{68} -1.14758 q^{69} +26.8611 q^{70} -0.546141 q^{71} -0.313933 q^{72} +9.58155 q^{73} +15.3563 q^{74} -14.2900 q^{75} +7.94024 q^{76} -21.0700 q^{77} -4.15021 q^{79} -18.3620 q^{80} -8.36933 q^{81} -19.6379 q^{82} -6.77183 q^{83} +7.61011 q^{84} -2.36575 q^{85} +3.19904 q^{86} -9.75726 q^{87} +8.08078 q^{88} +14.7212 q^{89} +1.27680 q^{90} +0.750873 q^{92} -1.67414 q^{93} +10.6616 q^{94} +26.6686 q^{95} -9.37154 q^{96} +12.2489 q^{97} -17.9819 q^{98} -1.00153 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} - 3 q^{3} + 9 q^{4} + 15 q^{5} + 4 q^{7} + 15 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q + 5 * q^2 - 3 * q^3 + 9 * q^4 + 15 * q^5 + 4 * q^7 + 15 * q^8 + 9 * q^9	\( 8 q + 5 q^{2} - 3 q^{3} + 9 q^{4} + 15 q^{5} + 4 q^{7} + 15 q^{8} + 9 q^{9} + 9 q^{10} + 5 q^{11} - 9 q^{12} + 2 q^{14} - 2 q^{15} + 3 q^{16} - 11 q^{17} + 30 q^{18} + 9 q^{19} + 31 q^{20} + 16 q^{21} - 2 q^{22} - 13 q^{24} + 19 q^{25} - 9 q^{27} - 16 q^{28} - 12 q^{29} - 7 q^{30} + 8 q^{31} + 25 q^{32} + 14 q^{33} - 22 q^{34} + 7 q^{35} + 37 q^{36} + 9 q^{37} - 9 q^{38} + 55 q^{40} + 25 q^{41} - 3 q^{42} + 7 q^{43} + 26 q^{44} + 45 q^{45} - 5 q^{46} + 17 q^{47} - 9 q^{48} + 11 q^{50} - 10 q^{51} - 15 q^{53} - 54 q^{54} + 7 q^{55} - 14 q^{56} + 7 q^{57} + 5 q^{58} + 15 q^{59} - 61 q^{60} + 11 q^{61} + 5 q^{62} + 21 q^{63} + 47 q^{64} + 83 q^{66} - 18 q^{67} - 16 q^{68} - 15 q^{69} + 24 q^{70} + 7 q^{71} + 21 q^{72} - 24 q^{73} + 48 q^{74} - 17 q^{75} + 3 q^{76} - 49 q^{77} + 33 q^{79} + 16 q^{80} + 20 q^{81} - q^{82} + 13 q^{83} + 6 q^{84} - q^{85} - 19 q^{86} + 18 q^{87} + 37 q^{88} + 23 q^{89} + 117 q^{90} + 22 q^{92} - 3 q^{93} + 10 q^{94} + 43 q^{95} - 46 q^{96} + 17 q^{97} - 52 q^{98} - 51 q^{99}+O(q^{100}) \) 8 * q + 5 * q^2 - 3 * q^3 + 9 * q^4 + 15 * q^5 + 4 * q^7 + 15 * q^8 + 9 * q^9 + 9 * q^10 + 5 * q^11 - 9 * q^12 + 2 * q^14 - 2 * q^15 + 3 * q^16 - 11 * q^17 + 30 * q^18 + 9 * q^19 + 31 * q^20 + 16 * q^21 - 2 * q^22 - 13 * q^24 + 19 * q^25 - 9 * q^27 - 16 * q^28 - 12 * q^29 - 7 * q^30 + 8 * q^31 + 25 * q^32 + 14 * q^33 - 22 * q^34 + 7 * q^35 + 37 * q^36 + 9 * q^37 - 9 * q^38 + 55 * q^40 + 25 * q^41 - 3 * q^42 + 7 * q^43 + 26 * q^44 + 45 * q^45 - 5 * q^46 + 17 * q^47 - 9 * q^48 + 11 * q^50 - 10 * q^51 - 15 * q^53 - 54 * q^54 + 7 * q^55 - 14 * q^56 + 7 * q^57 + 5 * q^58 + 15 * q^59 - 61 * q^60 + 11 * q^61 + 5 * q^62 + 21 * q^63 + 47 * q^64 + 83 * q^66 - 18 * q^67 - 16 * q^68 - 15 * q^69 + 24 * q^70 + 7 * q^71 + 21 * q^72 - 24 * q^73 + 48 * q^74 - 17 * q^75 + 3 * q^76 - 49 * q^77 + 33 * q^79 + 16 * q^80 + 20 * q^81 - q^82 + 13 * q^83 + 6 * q^84 - q^85 - 19 * q^86 + 18 * q^87 + 37 * q^88 + 23 * q^89 + 117 * q^90 + 22 * q^92 - 3 * q^93 + 10 * q^94 + 43 * q^95 - 46 * q^96 + 17 * q^97 - 52 * q^98 - 51 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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