LMFDB - Embedded newform 8015.2.a.n.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8015,2,Mod(1,8015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8015, 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("8015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8015 = 5 \cdot 7 \cdot 229 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8015.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:	$64.0000972201$
Analytic rank:	$0$
Dimension:	$68$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.33888 q^{2} -1.00822 q^{3} +3.47038 q^{4} -1.00000 q^{5} +2.35811 q^{6} +1.00000 q^{7} -3.43905 q^{8} -1.98350 q^{9} +O(q^{10})$	\(q-2.33888 q^{2} -1.00822 q^{3} +3.47038 q^{4} -1.00000 q^{5} +2.35811 q^{6} +1.00000 q^{7} -3.43905 q^{8} -1.98350 q^{9} +2.33888 q^{10} +4.79074 q^{11} -3.49890 q^{12} +1.66103 q^{13} -2.33888 q^{14} +1.00822 q^{15} +1.10278 q^{16} +4.37326 q^{17} +4.63917 q^{18} +0.809488 q^{19} -3.47038 q^{20} -1.00822 q^{21} -11.2050 q^{22} +1.74201 q^{23} +3.46732 q^{24} +1.00000 q^{25} -3.88496 q^{26} +5.02445 q^{27} +3.47038 q^{28} +10.2995 q^{29} -2.35811 q^{30} +4.74222 q^{31} +4.29882 q^{32} -4.83012 q^{33} -10.2286 q^{34} -1.00000 q^{35} -6.88348 q^{36} +3.07107 q^{37} -1.89330 q^{38} -1.67468 q^{39} +3.43905 q^{40} -4.37003 q^{41} +2.35811 q^{42} +10.0143 q^{43} +16.6257 q^{44} +1.98350 q^{45} -4.07436 q^{46} -5.66026 q^{47} -1.11185 q^{48} +1.00000 q^{49} -2.33888 q^{50} -4.40921 q^{51} +5.76441 q^{52} +11.1626 q^{53} -11.7516 q^{54} -4.79074 q^{55} -3.43905 q^{56} -0.816140 q^{57} -24.0892 q^{58} +3.35973 q^{59} +3.49890 q^{60} +9.01471 q^{61} -11.0915 q^{62} -1.98350 q^{63} -12.2600 q^{64} -1.66103 q^{65} +11.2971 q^{66} +0.555272 q^{67} +15.1769 q^{68} -1.75633 q^{69} +2.33888 q^{70} -0.423631 q^{71} +6.82134 q^{72} +10.1012 q^{73} -7.18288 q^{74} -1.00822 q^{75} +2.80923 q^{76} +4.79074 q^{77} +3.91689 q^{78} -5.24527 q^{79} -1.10278 q^{80} +0.884738 q^{81} +10.2210 q^{82} +8.74282 q^{83} -3.49890 q^{84} -4.37326 q^{85} -23.4222 q^{86} -10.3841 q^{87} -16.4756 q^{88} -3.18961 q^{89} -4.63917 q^{90} +1.66103 q^{91} +6.04544 q^{92} -4.78119 q^{93} +13.2387 q^{94} -0.809488 q^{95} -4.33415 q^{96} +17.6989 q^{97} -2.33888 q^{98} -9.50242 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9}+O(q^{10}) $ 68 * q + 9 * q^2 + 83 * q^4 - 68 * q^5 + 5 * q^6 + 68 * q^7 + 30 * q^8 + 86 * q^9	\( 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9} - 9 q^{10} + 5 q^{11} + 9 q^{12} + 15 q^{13} + 9 q^{14} + 109 q^{16} + 7 q^{17} + 39 q^{18} + 20 q^{19} - 83 q^{20} + 56 q^{22} + 36 q^{23} + q^{24} + 68 q^{25} + q^{26} + 12 q^{27} + 83 q^{28} - 16 q^{29} - 5 q^{30} + 31 q^{31} + 79 q^{32} + 45 q^{33} + 31 q^{34} - 68 q^{35} + 114 q^{36} + 72 q^{37} + 8 q^{38} + 47 q^{39} - 30 q^{40} + 6 q^{41} + 5 q^{42} + 75 q^{43} + 15 q^{44} - 86 q^{45} + 29 q^{46} - 10 q^{47} + 44 q^{48} + 68 q^{49} + 9 q^{50} + 23 q^{51} + 37 q^{52} + 41 q^{53} + 4 q^{54} - 5 q^{55} + 30 q^{56} + 55 q^{57} + 66 q^{58} - 5 q^{59} - 9 q^{60} - 2 q^{61} + 3 q^{62} + 86 q^{63} + 162 q^{64} - 15 q^{65} - 23 q^{66} + 92 q^{67} + 35 q^{68} - 25 q^{69} - 9 q^{70} - 2 q^{71} + 128 q^{72} + 80 q^{73} + 18 q^{74} + 71 q^{76} + 5 q^{77} + 20 q^{78} + 100 q^{79} - 109 q^{80} + 140 q^{81} + 36 q^{82} - 60 q^{83} + 9 q^{84} - 7 q^{85} - 27 q^{86} + 24 q^{87} + 175 q^{88} + 19 q^{89} - 39 q^{90} + 15 q^{91} + 75 q^{92} + 37 q^{93} + 11 q^{94} - 20 q^{95} + 15 q^{96} + 96 q^{97} + 9 q^{98} + 3 q^{99}+O(q^{100}) \) 68 * q + 9 * q^2 + 83 * q^4 - 68 * q^5 + 5 * q^6 + 68 * q^7 + 30 * q^8 + 86 * q^9 - 9 * q^10 + 5 * q^11 + 9 * q^12 + 15 * q^13 + 9 * q^14 + 109 * q^16 + 7 * q^17 + 39 * q^18 + 20 * q^19 - 83 * q^20 + 56 * q^22 + 36 * q^23 + q^24 + 68 * q^25 + q^26 + 12 * q^27 + 83 * q^28 - 16 * q^29 - 5 * q^30 + 31 * q^31 + 79 * q^32 + 45 * q^33 + 31 * q^34 - 68 * q^35 + 114 * q^36 + 72 * q^37 + 8 * q^38 + 47 * q^39 - 30 * q^40 + 6 * q^41 + 5 * q^42 + 75 * q^43 + 15 * q^44 - 86 * q^45 + 29 * q^46 - 10 * q^47 + 44 * q^48 + 68 * q^49 + 9 * q^50 + 23 * q^51 + 37 * q^52 + 41 * q^53 + 4 * q^54 - 5 * q^55 + 30 * q^56 + 55 * q^57 + 66 * q^58 - 5 * q^59 - 9 * q^60 - 2 * q^61 + 3 * q^62 + 86 * q^63 + 162 * q^64 - 15 * q^65 - 23 * q^66 + 92 * q^67 + 35 * q^68 - 25 * q^69 - 9 * q^70 - 2 * q^71 + 128 * q^72 + 80 * q^73 + 18 * q^74 + 71 * q^76 + 5 * q^77 + 20 * q^78 + 100 * q^79 - 109 * q^80 + 140 * q^81 + 36 * q^82 - 60 * q^83 + 9 * q^84 - 7 * q^85 - 27 * q^86 + 24 * q^87 + 175 * q^88 + 19 * q^89 - 39 * q^90 + 15 * q^91 + 75 * q^92 + 37 * q^93 + 11 * q^94 - 20 * q^95 + 15 * q^96 + 96 * q^97 + 9 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.33888	−1.65384	−0.826921	−	0.562319i	$-0.809909\pi$
$2$	−2.33888	−1.65384	−0.826921	+	0.562319i	$0.809909\pi$
$3$	−1.00822	−0.582095	−0.291048	−	0.956709i	$-0.594004\pi$
$3$	−1.00822	−0.582095	−0.291048	+	0.956709i	$0.594004\pi$
$4$	3.47038	1.73519
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.35811	0.962693
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−3.43905	−1.21589
$9$	−1.98350	−0.661165
$10$	2.33888	0.739620
$11$	4.79074	1.44446	0.722232	−	0.691651i	$-0.243116\pi$
$11$	4.79074	1.44446	0.722232	+	0.691651i	$0.243116\pi$
$12$	−3.49890	−1.01005
$13$	1.66103	0.460687	0.230344	−	0.973109i	$-0.426015\pi$
$13$	1.66103	0.460687	0.230344	+	0.973109i	$0.426015\pi$
$14$	−2.33888	−0.625093
$15$	1.00822	0.260321
$16$	1.10278	0.275696
$17$	4.37326	1.06067	0.530336	−	0.847787i	$-0.322066\pi$
$17$	4.37326	1.06067	0.530336	+	0.847787i	$0.322066\pi$
$18$	4.63917	1.09346
$19$	0.809488	0.185709	0.0928546	−	0.995680i	$-0.470401\pi$
$19$	0.809488	0.185709	0.0928546	+	0.995680i	$0.470401\pi$
$20$	−3.47038	−0.776001
$21$	−1.00822	−0.220011
$22$	−11.2050	−2.38891
$23$	1.74201	0.363234	0.181617	−	0.983369i	$-0.441867\pi$
$23$	1.74201	0.363234	0.181617	+	0.983369i	$0.441867\pi$
$24$	3.46732	0.707763
$25$	1.00000	0.200000
$26$	−3.88496	−0.761904
$27$	5.02445	0.966956
$28$	3.47038	0.655840
$29$	10.2995	1.91256	0.956281	−	0.292450i	$-0.0944704\pi$
$29$	10.2995	1.91256	0.956281	+	0.292450i	$0.0944704\pi$
$30$	−2.35811	−0.430530
$31$	4.74222	0.851728	0.425864	−	0.904787i	$-0.359970\pi$
$31$	4.74222	0.851728	0.425864	+	0.904787i	$0.359970\pi$
$32$	4.29882	0.759931
$33$	−4.83012	−0.840815
$34$	−10.2286	−1.75418
$35$	−1.00000	−0.169031
$36$	−6.88348	−1.14725
$37$	3.07107	0.504881	0.252441	−	0.967612i	$-0.418767\pi$
$37$	3.07107	0.504881	0.252441	+	0.967612i	$0.418767\pi$
$38$	−1.89330	−0.307133
$39$	−1.67468	−0.268164
$40$	3.43905	0.543762
$41$	−4.37003	−0.682484	−0.341242	−	0.939975i	$-0.610848\pi$
$41$	−4.37003	−0.682484	−0.341242	+	0.939975i	$0.610848\pi$
$42$	2.35811	0.363864
$43$	10.0143	1.52716	0.763581	−	0.645713i	$-0.223440\pi$
$43$	10.0143	1.52716	0.763581	+	0.645713i	$0.223440\pi$
$44$	16.6257	2.50642
$45$	1.98350	0.295682
$46$	−4.07436	−0.600732
$47$	−5.66026	−0.825633	−0.412817	−	0.910814i	$-0.635455\pi$
$47$	−5.66026	−0.825633	−0.412817	+	0.910814i	$0.635455\pi$
$48$	−1.11185	−0.160481
$49$	1.00000	0.142857
$50$	−2.33888	−0.330768
$51$	−4.40921	−0.617412
$52$	5.76441	0.799380
$53$	11.1626	1.53331	0.766653	−	0.642061i	$-0.221921\pi$
$53$	11.1626	1.53331	0.766653	+	0.642061i	$0.221921\pi$
$54$	−11.7516	−1.59919
$55$	−4.79074	−0.645984
$56$	−3.43905	−0.459563
$57$	−0.816140	−0.108100
$58$	−24.0892	−3.16307
$59$	3.35973	0.437400	0.218700	−	0.975792i	$-0.429818\pi$
$59$	3.35973	0.437400	0.218700	+	0.975792i	$0.429818\pi$
$60$	3.49890	0.451706
$61$	9.01471	1.15421	0.577107	−	0.816668i	$-0.304181\pi$
$61$	9.01471	1.15421	0.577107	+	0.816668i	$0.304181\pi$
$62$	−11.0915	−1.40862
$63$	−1.98350	−0.249897
$64$	−12.2600	−1.53250
$65$	−1.66103	−0.206026
$66$	11.2971	1.39058
$67$	0.555272	0.0678373	0.0339186	−	0.999425i	$-0.489201\pi$
$67$	0.555272	0.0678373	0.0339186	+	0.999425i	$0.489201\pi$
$68$	15.1769	1.84047
$69$	−1.75633	−0.211437
$70$	2.33888	0.279550
$71$	−0.423631	−0.0502758	−0.0251379	−	0.999684i	$-0.508002\pi$
$71$	−0.423631	−0.0502758	−0.0251379	+	0.999684i	$0.508002\pi$
$72$	6.82134	0.803903
$73$	10.1012	1.18225	0.591127	−	0.806578i	$-0.298683\pi$
$73$	10.1012	1.18225	0.591127	+	0.806578i	$0.298683\pi$
$74$	−7.18288	−0.834993
$75$	−1.00822	−0.116419
$76$	2.80923	0.322241
$77$	4.79074	0.545956
$78$	3.91689	0.443501
$79$	−5.24527	−0.590139	−0.295070	−	0.955476i	$-0.595343\pi$
$79$	−5.24527	−0.590139	−0.295070	+	0.955476i	$0.595343\pi$
$80$	−1.10278	−0.123295
$81$	0.884738	0.0983042
$82$	10.2210	1.12872
$83$	8.74282	0.959649	0.479825	−	0.877364i	$-0.340700\pi$
$83$	8.74282	0.959649	0.479825	+	0.877364i	$0.340700\pi$
$84$	−3.49890	−0.381762
$85$	−4.37326	−0.474347
$86$	−23.4222	−2.52568
$87$	−10.3841	−1.11329
$88$	−16.4756	−1.75631
$89$	−3.18961	−0.338098	−0.169049	−	0.985608i	$-0.554070\pi$
$89$	−3.18961	−0.338098	−0.169049	+	0.985608i	$0.554070\pi$
$90$	−4.63917	−0.489011
$91$	1.66103	0.174123
$92$	6.04544	0.630281
$93$	−4.78119	−0.495787
$94$	13.2387	1.36547
$95$	−0.809488	−0.0830517
$96$	−4.33415	−0.442353
$97$	17.6989	1.79705	0.898527	−	0.438918i	$-0.144638\pi$
$97$	17.6989	1.79705	0.898527	+	0.438918i	$0.144638\pi$
$98$	−2.33888	−0.236263
$99$	−9.50242	−0.955029
$100$	3.47038	0.347038
$101$	−10.3937	−1.03422	−0.517108	−	0.855920i	$-0.672991\pi$
$101$	−10.3937	−1.03422	−0.517108	+	0.855920i	$0.672991\pi$
$102$	10.3126	1.02110
$103$	−1.86508	−0.183771	−0.0918857	−	0.995770i	$-0.529289\pi$
$103$	−1.86508	−0.183771	−0.0918857	+	0.995770i	$0.529289\pi$
$104$	−5.71237	−0.560144
$105$	1.00822	0.0983921
$106$	−26.1081	−2.53585
$107$	5.94626	0.574847	0.287423	−	0.957804i	$-0.407201\pi$
$107$	5.94626	0.574847	0.287423	+	0.957804i	$0.407201\pi$
$108$	17.4368	1.67785
$109$	3.62142	0.346869	0.173434	−	0.984845i	$-0.444514\pi$
$109$	3.62142	0.346869	0.173434	+	0.984845i	$0.444514\pi$
$110$	11.2050	1.06835
$111$	−3.09631	−0.293889
$112$	1.10278	0.104203
$113$	1.59846	0.150370	0.0751851	−	0.997170i	$-0.476045\pi$
$113$	1.59846	0.150370	0.0751851	+	0.997170i	$0.476045\pi$
$114$	1.90886	0.178781
$115$	−1.74201	−0.162443
$116$	35.7431	3.31866
$117$	−3.29465	−0.304590
$118$	−7.85803	−0.723390
$119$	4.37326	0.400896
$120$	−3.46732	−0.316521
$121$	11.9512	1.08647
$122$	−21.0844	−1.90889
$123$	4.40595	0.397271
$124$	16.4573	1.47791
$125$	−1.00000	−0.0894427
$126$	4.63917	0.413290
$127$	−6.93469	−0.615354	−0.307677	−	0.951491i	$-0.599552\pi$
$127$	−6.93469	−0.615354	−0.307677	+	0.951491i	$0.599552\pi$
$128$	20.0771	1.77458
$129$	−10.0966	−0.888953
$130$	3.88496	0.340734
$131$	−18.6484	−1.62932	−0.814659	−	0.579940i	$-0.803076\pi$
$131$	−18.6484	−1.62932	−0.814659	+	0.579940i	$0.803076\pi$
$132$	−16.7623	−1.45898
$133$	0.809488	0.0701915
$134$	−1.29872	−0.112192
$135$	−5.02445	−0.432436
$136$	−15.0399	−1.28966
$137$	5.22595	0.446483	0.223241	−	0.974763i	$-0.428336\pi$
$137$	5.22595	0.446483	0.223241	+	0.974763i	$0.428336\pi$
$138$	4.10785	0.349683
$139$	23.0962	1.95899	0.979496	−	0.201461i	$-0.0645690\pi$
$139$	23.0962	1.95899	0.979496	+	0.201461i	$0.0645690\pi$
$140$	−3.47038	−0.293301
$141$	5.70678	0.480597
$142$	0.990825	0.0831482
$143$	7.95758	0.665446
$144$	−2.18736	−0.182280
$145$	−10.2995	−0.855324
$146$	−23.6255	−1.95526
$147$	−1.00822	−0.0831565
$148$	10.6578	0.876065
$149$	1.51236	0.123898	0.0619488	−	0.998079i	$-0.480268\pi$
$149$	1.51236	0.123898	0.0619488	+	0.998079i	$0.480268\pi$
$150$	2.35811	0.192539
$151$	−7.90899	−0.643625	−0.321812	−	0.946803i	$-0.604292\pi$
$151$	−7.90899	−0.643625	−0.321812	+	0.946803i	$0.604292\pi$
$152$	−2.78387	−0.225802
$153$	−8.67435	−0.701279
$154$	−11.2050	−0.902924
$155$	−4.74222	−0.380904
$156$	−5.81179	−0.465316
$157$	2.53564	0.202366	0.101183	−	0.994868i	$-0.467737\pi$
$157$	2.53564	0.202366	0.101183	+	0.994868i	$0.467737\pi$
$158$	12.2681	0.975997
$159$	−11.2544	−0.892531
$160$	−4.29882	−0.339852
$161$	1.74201	0.137290
$162$	−2.06930	−0.162580
$163$	14.8399	1.16235	0.581175	−	0.813779i	$-0.302593\pi$
$163$	14.8399	1.16235	0.581175	+	0.813779i	$0.302593\pi$
$164$	−15.1657	−1.18424
$165$	4.83012	0.376024
$166$	−20.4485	−1.58711
$167$	−14.8292	−1.14752	−0.573759	−	0.819024i	$-0.694515\pi$
$167$	−14.8292	−1.14752	−0.573759	+	0.819024i	$0.694515\pi$
$168$	3.46732	0.267509
$169$	−10.2410	−0.787767
$170$	10.2286	0.784495
$171$	−1.60561	−0.122784
$172$	34.7533	2.64992
$173$	−22.6578	−1.72264	−0.861321	−	0.508061i	$-0.830363\pi$
$173$	−22.6578	−1.72264	−0.861321	+	0.508061i	$0.830363\pi$
$174$	24.2872	1.84121
$175$	1.00000	0.0755929
$176$	5.28315	0.398232
$177$	−3.38735	−0.254609
$178$	7.46012	0.559160
$179$	15.8614	1.18554	0.592768	−	0.805373i	$-0.298035\pi$
$179$	15.8614	1.18554	0.592768	+	0.805373i	$0.298035\pi$
$180$	6.88348	0.513065
$181$	−6.44724	−0.479220	−0.239610	−	0.970869i	$-0.577020\pi$
$181$	−6.44724	−0.479220	−0.239610	+	0.970869i	$0.577020\pi$
$182$	−3.88496	−0.287973
$183$	−9.08879	−0.671863
$184$	−5.99086	−0.441652
$185$	−3.07107	−0.225790
$186$	11.1827	0.819953
$187$	20.9512	1.53210
$188$	−19.6432	−1.43263
$189$	5.02445	0.365475
$190$	1.89330	0.137354
$191$	−7.11017	−0.514474	−0.257237	−	0.966348i	$-0.582812\pi$
$191$	−7.11017	−0.514474	−0.257237	+	0.966348i	$0.582812\pi$
$192$	12.3608	0.892062
$193$	14.0790	1.01343	0.506715	−	0.862114i	$-0.330860\pi$
$193$	14.0790	1.01343	0.506715	+	0.862114i	$0.330860\pi$
$194$	−41.3958	−2.97204
$195$	1.67468	0.119927
$196$	3.47038	0.247884
$197$	−19.2036	−1.36820	−0.684099	−	0.729389i	$-0.739805\pi$
$197$	−19.2036	−1.36820	−0.684099	+	0.729389i	$0.739805\pi$
$198$	22.2251	1.57947
$199$	−6.05816	−0.429452	−0.214726	−	0.976674i	$-0.568886\pi$
$199$	−6.05816	−0.429452	−0.214726	+	0.976674i	$0.568886\pi$
$200$	−3.43905	−0.243178
$201$	−0.559836	−0.0394878
$202$	24.3097	1.71043
$203$	10.2995	0.722880
$204$	−15.3016	−1.07133
$205$	4.37003	0.305216
$206$	4.36220	0.303929
$207$	−3.45527	−0.240158
$208$	1.83176	0.127010
$209$	3.87805	0.268250
$210$	−2.35811	−0.162725
$211$	11.3469	0.781150	0.390575	−	0.920571i	$-0.372276\pi$
$211$	11.3469	0.781150	0.390575	+	0.920571i	$0.372276\pi$
$212$	38.7386	2.66058
$213$	0.427113	0.0292653
$214$	−13.9076	−0.950705
$215$	−10.0143	−0.682967
$216$	−17.2794	−1.17571
$217$	4.74222	0.321923
$218$	−8.47007	−0.573666
$219$	−10.1842	−0.688185
$220$	−16.6257	−1.12090
$221$	7.26413	0.488638
$222$	7.24192	0.486046
$223$	−10.5779	−0.708347	−0.354173	−	0.935180i	$-0.615238\pi$
$223$	−10.5779	−0.708347	−0.354173	+	0.935180i	$0.615238\pi$
$224$	4.29882	0.287227
$225$	−1.98350	−0.132233
$226$	−3.73861	−0.248688
$227$	8.20307	0.544457	0.272229	−	0.962233i	$-0.412239\pi$
$227$	8.20307	0.544457	0.272229	+	0.962233i	$0.412239\pi$
$228$	−2.83232	−0.187575
$229$	−1.00000	−0.0660819
$229$	−1.00000	−0.0660819
$230$	4.07436	0.268655
$231$	−4.83012	−0.317798
$232$	−35.4204	−2.32546
$233$	13.3046	0.871615	0.435807	−	0.900040i	$-0.356463\pi$
$233$	13.3046	0.871615	0.435807	+	0.900040i	$0.356463\pi$
$234$	7.70580	0.503744
$235$	5.66026	0.369234
$236$	11.6596	0.758973
$237$	5.28838	0.343517
$238$	−10.2286	−0.663019
$239$	−1.74080	−0.112603	−0.0563015	−	0.998414i	$-0.517931\pi$
$239$	−1.74080	−0.112603	−0.0563015	+	0.998414i	$0.517931\pi$
$240$	1.11185	0.0717694
$241$	−22.7496	−1.46543	−0.732714	−	0.680536i	$-0.761747\pi$
$241$	−22.7496	−1.46543	−0.732714	+	0.680536i	$0.761747\pi$
$242$	−27.9525	−1.79686
$243$	−15.9654	−1.02418
$244$	31.2845	2.00278
$245$	−1.00000	−0.0638877
$246$	−10.3050	−0.657023
$247$	1.34458	0.0855539
$248$	−16.3087	−1.03561
$249$	−8.81468	−0.558607
$250$	2.33888	0.147924
$251$	16.6242	1.04931	0.524657	−	0.851314i	$-0.324194\pi$
$251$	16.6242	1.04931	0.524657	+	0.851314i	$0.324194\pi$
$252$	−6.88348	−0.433619
$253$	8.34552	0.524679
$254$	16.2194	1.01770
$255$	4.40921	0.276115
$256$	−22.4380	−1.40238
$257$	−10.9533	−0.683247	−0.341624	−	0.939837i	$-0.610977\pi$
$257$	−10.9533	−0.683247	−0.341624	+	0.939837i	$0.610977\pi$
$258$	23.6147	1.47019
$259$	3.07107	0.190827
$260$	−5.76441	−0.357494
$261$	−20.4289	−1.26452
$262$	43.6165	2.69463
$263$	1.43339	0.0883867	0.0441933	−	0.999023i	$-0.485928\pi$
$263$	1.43339	0.0883867	0.0441933	+	0.999023i	$0.485928\pi$
$264$	16.6110	1.02234
$265$	−11.1626	−0.685716
$266$	−1.89330	−0.116086
$267$	3.21582	0.196805
$268$	1.92701	0.117711
$269$	−4.21299	−0.256870	−0.128435	−	0.991718i	$-0.540995\pi$
$269$	−4.21299	−0.256870	−0.128435	+	0.991718i	$0.540995\pi$
$270$	11.7516	0.715181
$271$	0.532503	0.0323472	0.0161736	−	0.999869i	$-0.494852\pi$
$271$	0.532503	0.0323472	0.0161736	+	0.999869i	$0.494852\pi$
$272$	4.82276	0.292423
$273$	−1.67468	−0.101356
$274$	−12.2229	−0.738412
$275$	4.79074	0.288893
$276$	−6.09512	−0.366883
$277$	−7.82002	−0.469860	−0.234930	−	0.972012i	$-0.575486\pi$
$277$	−7.82002	−0.469860	−0.234930	+	0.972012i	$0.575486\pi$
$278$	−54.0193	−3.23986
$279$	−9.40617	−0.563133
$280$	3.43905	0.205523
$281$	−24.4447	−1.45825	−0.729125	−	0.684381i	$-0.760073\pi$
$281$	−24.4447	−1.45825	−0.729125	+	0.684381i	$0.760073\pi$
$282$	−13.3475	−0.794831
$283$	16.9084	1.00510	0.502549	−	0.864549i	$-0.332396\pi$
$283$	16.9084	1.00510	0.502549	+	0.864549i	$0.332396\pi$
$284$	−1.47016	−0.0872381
$285$	0.816140	0.0483440
$286$	−18.6119	−1.10054
$287$	−4.37003	−0.257955
$288$	−8.52669	−0.502440
$289$	2.12544	0.125026
$290$	24.0892	1.41457
$291$	−17.8444	−1.04606
$292$	35.0550	2.05144
$293$	−8.47147	−0.494909	−0.247454	−	0.968900i	$-0.579594\pi$
$293$	−8.47147	−0.494909	−0.247454	+	0.968900i	$0.579594\pi$
$294$	2.35811	0.137528
$295$	−3.35973	−0.195611
$296$	−10.5616	−0.613879
$297$	24.0709	1.39673
$298$	−3.53724	−0.204907
$299$	2.89353	0.167337
$300$	−3.49890	−0.202009
$301$	10.0143	0.577213
$302$	18.4982	1.06445
$303$	10.4792	0.602012
$304$	0.892689	0.0511992
$305$	−9.01471	−0.516181
$306$	20.2883	1.15980
$307$	28.2778	1.61390	0.806949	−	0.590620i	$-0.201117\pi$
$307$	28.2778	1.61390	0.806949	+	0.590620i	$0.201117\pi$
$308$	16.6257	0.947338
$309$	1.88040	0.106972
$310$	11.0915	0.629955
$311$	−30.3583	−1.72146	−0.860732	−	0.509059i	$-0.829993\pi$
$311$	−30.3583	−1.72146	−0.860732	+	0.509059i	$0.829993\pi$
$312$	5.75932	0.326057
$313$	−16.9421	−0.957623	−0.478812	−	0.877918i	$-0.658932\pi$
$313$	−16.9421	−0.957623	−0.478812	+	0.877918i	$0.658932\pi$
$314$	−5.93058	−0.334682
$315$	1.98350	0.111757
$316$	−18.2031	−1.02400
$317$	−16.6828	−0.937000	−0.468500	−	0.883464i	$-0.655205\pi$
$317$	−16.6828	−0.937000	−0.468500	+	0.883464i	$0.655205\pi$
$318$	26.3227	1.47610
$319$	49.3421	2.76263
$320$	12.2600	0.685356
$321$	−5.99513	−0.334616
$322$	−4.07436	−0.227055
$323$	3.54010	0.196977
$324$	3.07038	0.170576
$325$	1.66103	0.0921375
$326$	−34.7088	−1.92234
$327$	−3.65118	−0.201911
$328$	15.0288	0.829825
$329$	−5.66026	−0.312060
$330$	−11.2971	−0.621884
$331$	4.97735	0.273580	0.136790	−	0.990600i	$-0.456321\pi$
$331$	4.97735	0.273580	0.136790	+	0.990600i	$0.456321\pi$
$332$	30.3409	1.66517
$333$	−6.09145	−0.333810
$334$	34.6838	1.89781
$335$	−0.555272	−0.0303378
$336$	−1.11185	−0.0606562
$337$	15.1624	0.825947	0.412973	−	0.910743i	$-0.364490\pi$
$337$	15.1624	0.825947	0.412973	+	0.910743i	$0.364490\pi$
$338$	23.9525	1.30284
$339$	−1.61159	−0.0875298
$340$	−15.1769	−0.823083
$341$	22.7188	1.23029
$342$	3.75535	0.203066
$343$	1.00000	0.0539949
$344$	−34.4396	−1.85686
$345$	1.75633	0.0945575
$346$	52.9940	2.84898
$347$	−1.12419	−0.0603499	−0.0301749	−	0.999545i	$-0.509606\pi$
$347$	−1.12419	−0.0603499	−0.0301749	+	0.999545i	$0.509606\pi$
$348$	−36.0368	−1.93178
$349$	36.1257	1.93377	0.966883	−	0.255220i	$-0.0821480\pi$
$349$	36.1257	1.93377	0.966883	+	0.255220i	$0.0821480\pi$
$350$	−2.33888	−0.125019
$351$	8.34578	0.445465
$352$	20.5946	1.09769
$353$	−14.2963	−0.760916	−0.380458	−	0.924798i	$-0.624234\pi$
$353$	−14.2963	−0.760916	−0.380458	+	0.924798i	$0.624234\pi$
$354$	7.92261	0.421082
$355$	0.423631	0.0224840
$356$	−11.0692	−0.586664
$357$	−4.40921	−0.233360
$358$	−37.0980	−1.96069
$359$	4.22079	0.222765	0.111382	−	0.993778i	$-0.464472\pi$
$359$	4.22079	0.222765	0.111382	+	0.993778i	$0.464472\pi$
$360$	−6.82134	−0.359516
$361$	−18.3447	−0.965512
$362$	15.0794	0.792553
$363$	−12.0494	−0.632432
$364$	5.76441	0.302137
$365$	−10.1012	−0.528720
$366$	21.2576	1.11115
$367$	−29.1847	−1.52343	−0.761715	−	0.647913i	$-0.775642\pi$
$367$	−29.1847	−1.52343	−0.761715	+	0.647913i	$0.775642\pi$
$368$	1.92106	0.100142
$369$	8.66793	0.451235
$370$	7.18288	0.373420
$371$	11.1626	0.579535
$372$	−16.5926	−0.860285
$373$	−15.5678	−0.806073	−0.403036	−	0.915184i	$-0.632045\pi$
$373$	−15.5678	−0.806073	−0.403036	+	0.915184i	$0.632045\pi$
$374$	−49.0024	−2.53385
$375$	1.00822	0.0520642
$376$	19.4659	1.00388
$377$	17.1077	0.881093
$378$	−11.7516	−0.604438
$379$	−35.4648	−1.82170	−0.910851	−	0.412735i	$-0.864573\pi$
$379$	−35.4648	−1.82170	−0.910851	+	0.412735i	$0.864573\pi$
$380$	−2.80923	−0.144110
$381$	6.99168	0.358195
$382$	16.6299	0.850858
$383$	30.2919	1.54785	0.773923	−	0.633280i	$-0.218292\pi$
$383$	30.2919	1.54785	0.773923	+	0.633280i	$0.218292\pi$
$384$	−20.2421	−1.03298
$385$	−4.79074	−0.244159
$386$	−32.9292	−1.67605
$387$	−19.8632	−1.00971
$388$	61.4220	3.11823
$389$	−23.5448	−1.19377	−0.596885	−	0.802327i	$-0.703595\pi$
$389$	−23.5448	−1.19377	−0.596885	+	0.802327i	$0.703595\pi$
$390$	−3.91689	−0.198339
$391$	7.61827	0.385272
$392$	−3.43905	−0.173698
$393$	18.8017	0.948419
$394$	44.9150	2.26278
$395$	5.24527	0.263918
$396$	−32.9770	−1.65716
$397$	−2.86530	−0.143805	−0.0719025	−	0.997412i	$-0.522907\pi$
$397$	−2.86530	−0.143805	−0.0719025	+	0.997412i	$0.522907\pi$
$398$	14.1693	0.710245
$399$	−0.816140	−0.0408581
$400$	1.10278	0.0551391
$401$	−28.2077	−1.40863	−0.704314	−	0.709889i	$-0.748745\pi$
$401$	−28.2077	−1.40863	−0.704314	+	0.709889i	$0.748745\pi$
$402$	1.30939	0.0653065
$403$	7.87698	0.392380
$404$	−36.0702	−1.79456
$405$	−0.884738	−0.0439630
$406$	−24.0892	−1.19553
$407$	14.7127	0.729282
$408$	15.1635	0.750705
$409$	−13.5951	−0.672233	−0.336116	−	0.941820i	$-0.609114\pi$
$409$	−13.5951	−0.672233	−0.336116	+	0.941820i	$0.609114\pi$
$410$	−10.2210	−0.504779
$411$	−5.26890	−0.259896
$412$	−6.47253	−0.318878
$413$	3.35973	0.165322
$414$	8.08147	0.397183
$415$	−8.74282	−0.429168
$416$	7.14048	0.350091
$417$	−23.2860	−1.14032
$418$	−9.07030	−0.443643
$419$	−40.2148	−1.96462	−0.982311	−	0.187255i	$-0.940041\pi$
$419$	−40.2148	−1.96462	−0.982311	+	0.187255i	$0.940041\pi$
$420$	3.49890	0.170729
$421$	−11.2421	−0.547906	−0.273953	−	0.961743i	$-0.588331\pi$
$421$	−11.2421	−0.547906	−0.273953	+	0.961743i	$0.588331\pi$
$422$	−26.5390	−1.29190
$423$	11.2271	0.545880
$424$	−38.3889	−1.86433
$425$	4.37326	0.212134
$426$	−0.998968	−0.0484002
$427$	9.01471	0.436252
$428$	20.6358	0.997469
$429$	−8.02298	−0.387353
$430$	23.4222	1.12952
$431$	−2.19235	−0.105602	−0.0528008	−	0.998605i	$-0.516815\pi$
$431$	−2.19235	−0.105602	−0.0528008	+	0.998605i	$0.516815\pi$
$432$	5.54088	0.266586
$433$	−18.4249	−0.885445	−0.442723	−	0.896659i	$-0.645987\pi$
$433$	−18.4249	−0.885445	−0.442723	+	0.896659i	$0.645987\pi$
$434$	−11.0915	−0.532409
$435$	10.3841	0.497880
$436$	12.5677	0.601883
$437$	1.41014	0.0674559
$438$	23.8197	1.13815
$439$	−1.96271	−0.0936751	−0.0468376	−	0.998903i	$-0.514914\pi$
$439$	−1.96271	−0.0936751	−0.0468376	+	0.998903i	$0.514914\pi$
$440$	16.4756	0.785444
$441$	−1.98350	−0.0944521
$442$	−16.9900	−0.808130
$443$	39.5696	1.88001	0.940005	−	0.341161i	$-0.110820\pi$
$443$	39.5696	1.88001	0.940005	+	0.341161i	$0.110820\pi$
$444$	−10.7454	−0.509953
$445$	3.18961	0.151202
$446$	24.7404	1.17149
$447$	−1.52479	−0.0721202
$448$	−12.2600	−0.579231
$449$	−29.7326	−1.40317	−0.701584	−	0.712586i	$-0.747524\pi$
$449$	−29.7326	−1.40317	−0.701584	+	0.712586i	$0.747524\pi$
$450$	4.63917	0.218692
$451$	−20.9357	−0.985823
$452$	5.54726	0.260921
$453$	7.97400	0.374651
$454$	−19.1860	−0.900446
$455$	−1.66103	−0.0778704
$456$	2.80675	0.131438
$457$	14.1508	0.661947	0.330973	−	0.943640i	$-0.392623\pi$
$457$	14.1508	0.661947	0.330973	+	0.943640i	$0.392623\pi$
$458$	2.33888	0.109289
$459$	21.9733	1.02562
$460$	−6.04544	−0.281870
$461$	0.251974	0.0117356	0.00586779	−	0.999983i	$-0.498132\pi$
$461$	0.251974	0.0117356	0.00586779	+	0.999983i	$0.498132\pi$
$462$	11.2971	0.525588
$463$	−7.48263	−0.347747	−0.173874	−	0.984768i	$-0.555628\pi$
$463$	−7.48263	−0.347747	−0.173874	+	0.984768i	$0.555628\pi$
$464$	11.3581	0.527285
$465$	4.78119	0.221723
$466$	−31.1180	−1.44151
$467$	−39.0034	−1.80486	−0.902431	−	0.430834i	$-0.858219\pi$
$467$	−39.0034	−1.80486	−0.902431	+	0.430834i	$0.858219\pi$
$468$	−11.4337	−0.528522
$469$	0.555272	0.0256401
$470$	−13.2387	−0.610655
$471$	−2.55648	−0.117797
$472$	−11.5543	−0.531830
$473$	47.9758	2.20593
$474$	−12.3689	−0.568123
$475$	0.809488	0.0371418
$476$	15.1769	0.695632
$477$	−22.1410	−1.01377
$478$	4.07153	0.186227
$479$	0.166757	0.00761934	0.00380967	−	0.999993i	$-0.498787\pi$
$479$	0.166757	0.00761934	0.00380967	+	0.999993i	$0.498787\pi$
$480$	4.33415	0.197826
$481$	5.10115	0.232592
$482$	53.2086	2.42359
$483$	−1.75633	−0.0799157
$484$	41.4753	1.88524
$485$	−17.6989	−0.803667
$486$	37.3412	1.69383
$487$	−9.94582	−0.450688	−0.225344	−	0.974279i	$-0.572351\pi$
$487$	−9.94582	−0.450688	−0.225344	+	0.974279i	$0.572351\pi$
$488$	−31.0020	−1.40340
$489$	−14.9619	−0.676598
$490$	2.33888	0.105660
$491$	−27.9983	−1.26354	−0.631772	−	0.775154i	$-0.717672\pi$
$491$	−27.9983	−1.26354	−0.631772	+	0.775154i	$0.717672\pi$
$492$	15.2903	0.689341
$493$	45.0423	2.02860
$494$	−3.14483	−0.141493
$495$	9.50242	0.427102
$496$	5.22964	0.234818
$497$	−0.423631	−0.0190025
$498$	20.6165	0.923848
$499$	2.79197	0.124986	0.0624928	−	0.998045i	$-0.480095\pi$
$499$	2.79197	0.124986	0.0624928	+	0.998045i	$0.480095\pi$
$500$	−3.47038	−0.155200
$501$	14.9511	0.667964
$502$	−38.8822	−1.73540
$503$	−2.66313	−0.118743	−0.0593715	−	0.998236i	$-0.518910\pi$
$503$	−2.66313	−0.118743	−0.0593715	+	0.998236i	$0.518910\pi$
$504$	6.82134	0.303847
$505$	10.3937	0.462515
$506$	−19.5192	−0.867735
$507$	10.3251	0.458556
$508$	−24.0660	−1.06776
$509$	13.8119	0.612203	0.306102	−	0.951999i	$-0.400975\pi$
$509$	13.8119	0.612203	0.306102	+	0.951999i	$0.400975\pi$
$510$	−10.3126	−0.456651
$511$	10.1012	0.446850
$512$	12.3257	0.544725
$513$	4.06723	0.179573
$514$	25.6185	1.12998
$515$	1.86508	0.0821851
$516$	−35.0389	−1.54250
$517$	−27.1168	−1.19260
$518$	−7.18288	−0.315598
$519$	22.8440	1.00274
$520$	5.71237	0.250504
$521$	4.19608	0.183833	0.0919167	−	0.995767i	$-0.470701\pi$
$521$	4.19608	0.183833	0.0919167	+	0.995767i	$0.470701\pi$
$522$	47.7809	2.09131
$523$	31.4085	1.37340	0.686699	−	0.726942i	$-0.259059\pi$
$523$	31.4085	1.37340	0.686699	+	0.726942i	$0.259059\pi$
$524$	−64.7171	−2.82718
$525$	−1.00822	−0.0440023
$526$	−3.35254	−0.146178
$527$	20.7390	0.903404
$528$	−5.32657	−0.231809
$529$	−19.9654	−0.868061
$530$	26.1081	1.13406
$531$	−6.66402	−0.289194
$532$	2.80923	0.121796
$533$	−7.25876	−0.314412
$534$	−7.52143	−0.325484
$535$	−5.94626	−0.257079
$536$	−1.90961	−0.0824826
$537$	−15.9918	−0.690095
$538$	9.85369	0.424823
$539$	4.79074	0.206352
$540$	−17.4368	−0.750359
$541$	44.1207	1.89690	0.948448	−	0.316932i	$-0.102653\pi$
$541$	44.1207	1.89690	0.948448	+	0.316932i	$0.102653\pi$
$542$	−1.24546	−0.0534972
$543$	6.50023	0.278952
$544$	18.7999	0.806038
$545$	−3.62142	−0.155124
$546$	3.91689	0.167627
$547$	38.8863	1.66266	0.831329	−	0.555780i	$-0.187580\pi$
$547$	38.8863	1.66266	0.831329	+	0.555780i	$0.187580\pi$
$548$	18.1360	0.774733
$549$	−17.8806	−0.763126
$550$	−11.2050	−0.477783
$551$	8.33729	0.355180
$552$	6.04010	0.257084
$553$	−5.24527	−0.223052
$554$	18.2901	0.777073
$555$	3.09631	0.131431
$556$	80.1526	3.39923
$557$	11.1059	0.470571	0.235286	−	0.971926i	$-0.424397\pi$
$557$	11.1059	0.470571	0.235286	+	0.971926i	$0.424397\pi$
$558$	21.9999	0.931332
$559$	16.6340	0.703544
$560$	−1.10278	−0.0466011
$561$	−21.1234	−0.891830
$562$	57.1734	2.41171
$563$	−25.0221	−1.05456	−0.527279	−	0.849692i	$-0.676788\pi$
$563$	−25.0221	−1.05456	−0.527279	+	0.849692i	$0.676788\pi$
$564$	19.8047	0.833928
$565$	−1.59846	−0.0672476
$566$	−39.5467	−1.66227
$567$	0.884738	0.0371555
$568$	1.45689	0.0611297
$569$	−34.4193	−1.44293	−0.721466	−	0.692450i	$-0.756531\pi$
$569$	−34.4193	−1.44293	−0.721466	+	0.692450i	$0.756531\pi$
$570$	−1.90886	−0.0799533
$571$	−9.05737	−0.379039	−0.189520	−	0.981877i	$-0.560693\pi$
$571$	−9.05737	−0.379039	−0.189520	+	0.981877i	$0.560693\pi$
$572$	27.6158	1.15468
$573$	7.16860	0.299473
$574$	10.2210	0.426616
$575$	1.74201	0.0726468
$576$	24.3177	1.01324
$577$	18.0633	0.751984	0.375992	−	0.926623i	$-0.377302\pi$
$577$	18.0633	0.751984	0.375992	+	0.926623i	$0.377302\pi$
$578$	−4.97115	−0.206773
$579$	−14.1947	−0.589913
$580$	−35.7431	−1.48415
$581$	8.74282	0.362713
$582$	41.7360	1.73001
$583$	53.4773	2.21481
$584$	−34.7385	−1.43749
$585$	3.29465	0.136217
$586$	19.8138	0.818500
$587$	−29.3287	−1.21053	−0.605263	−	0.796025i	$-0.706932\pi$
$587$	−29.3287	−1.21053	−0.605263	+	0.796025i	$0.706932\pi$
$588$	−3.49890	−0.144292
$589$	3.83877	0.158174
$590$	7.85803	0.323510
$591$	19.3614	0.796422
$592$	3.38672	0.139194
$593$	0.576871	0.0236892	0.0118446	−	0.999930i	$-0.496230\pi$
$593$	0.576871	0.0236892	0.0118446	+	0.999930i	$0.496230\pi$
$594$	−56.2990	−2.30997
$595$	−4.37326	−0.179286
$596$	5.24847	0.214986
$597$	6.10795	0.249982
$598$	−6.76764	−0.276749
$599$	8.92273	0.364573	0.182286	−	0.983245i	$-0.441650\pi$
$599$	8.92273	0.364573	0.182286	+	0.983245i	$0.441650\pi$
$600$	3.46732	0.141553
$601$	17.2344	0.703008	0.351504	−	0.936186i	$-0.385670\pi$
$601$	17.2344	0.703008	0.351504	+	0.936186i	$0.385670\pi$
$602$	−23.4222	−0.954618
$603$	−1.10138	−0.0448516
$604$	−27.4472	−1.11681
$605$	−11.9512	−0.485886
$606$	−24.5095	−0.995632
$607$	−42.9149	−1.74186	−0.870930	−	0.491407i	$-0.836483\pi$
$607$	−42.9149	−1.74186	−0.870930	+	0.491407i	$0.836483\pi$
$608$	3.47984	0.141126
$609$	−10.3841	−0.420785
$610$	21.0844	0.853681
$611$	−9.40187	−0.380359
$612$	−30.1033	−1.21685
$613$	5.82100	0.235108	0.117554	−	0.993066i	$-0.462495\pi$
$613$	5.82100	0.235108	0.117554	+	0.993066i	$0.462495\pi$
$614$	−66.1385	−2.66913
$615$	−4.40595	−0.177665
$616$	−16.4756	−0.663821
$617$	−27.3325	−1.10037	−0.550183	−	0.835044i	$-0.685442\pi$
$617$	−27.3325	−1.10037	−0.550183	+	0.835044i	$0.685442\pi$
$618$	−4.39805	−0.176916
$619$	−27.0791	−1.08840	−0.544199	−	0.838956i	$-0.683167\pi$
$619$	−27.0791	−1.08840	−0.544199	+	0.838956i	$0.683167\pi$
$620$	−16.4573	−0.660942
$621$	8.75265	0.351232
$622$	71.0046	2.84703
$623$	−3.18961	−0.127789
$624$	−1.84681	−0.0739316
$625$	1.00000	0.0400000
$626$	39.6256	1.58376
$627$	−3.90992	−0.156147
$628$	8.79965	0.351144
$629$	13.4306	0.535513
$630$	−4.63917	−0.184829
$631$	−1.85037	−0.0736619	−0.0368310	−	0.999322i	$-0.511726\pi$
$631$	−1.85037	−0.0736619	−0.0368310	+	0.999322i	$0.511726\pi$
$632$	18.0388	0.717544
$633$	−11.4401	−0.454704
$634$	39.0192	1.54965
$635$	6.93469	0.275195
$636$	−39.0570	−1.54871
$637$	1.66103	0.0658125
$638$	−115.405	−4.56894
$639$	0.840271	0.0332406
$640$	−20.0771	−0.793618
$641$	35.1793	1.38950	0.694750	−	0.719251i	$-0.255515\pi$
$641$	35.1793	1.38950	0.694750	+	0.719251i	$0.255515\pi$
$642$	14.0219	0.553401
$643$	15.6740	0.618121	0.309060	−	0.951042i	$-0.399985\pi$
$643$	15.6740	0.618121	0.309060	+	0.951042i	$0.399985\pi$
$644$	6.04544	0.238224
$645$	10.0966	0.397552
$646$	−8.27989	−0.325768
$647$	−8.43447	−0.331593	−0.165797	−	0.986160i	$-0.553020\pi$
$647$	−8.43447	−0.331593	−0.165797	+	0.986160i	$0.553020\pi$
$648$	−3.04266	−0.119527
$649$	16.0956	0.631809
$650$	−3.88496	−0.152381
$651$	−4.78119	−0.187390
$652$	51.5001	2.01690
$653$	26.2713	1.02808	0.514038	−	0.857767i	$-0.328149\pi$
$653$	26.2713	1.02808	0.514038	+	0.857767i	$0.328149\pi$
$654$	8.53969	0.333928
$655$	18.6484	0.728653
$656$	−4.81919	−0.188158
$657$	−20.0357	−0.781666
$658$	13.2387	0.516098
$659$	26.3640	1.02700	0.513499	−	0.858090i	$-0.328349\pi$
$659$	26.3640	1.02700	0.513499	+	0.858090i	$0.328349\pi$
$660$	16.7623	0.652473
$661$	2.37198	0.0922593	0.0461296	−	0.998935i	$-0.485311\pi$
$661$	2.37198	0.0922593	0.0461296	+	0.998935i	$0.485311\pi$
$662$	−11.6414	−0.452458
$663$	−7.32383	−0.284434
$664$	−30.0670	−1.16683
$665$	−0.809488	−0.0313906
$666$	14.2472	0.552068
$667$	17.9418	0.694708
$668$	−51.4629	−1.99116
$669$	10.6648	0.412325
$670$	1.29872	0.0501738
$671$	43.1871	1.66722
$672$	−4.33415	−0.167194
$673$	32.7666	1.26306	0.631530	−	0.775351i	$-0.282427\pi$
$673$	32.7666	1.26306	0.631530	+	0.775351i	$0.282427\pi$
$674$	−35.4630	−1.36598
$675$	5.02445	0.193391
$676$	−35.5401	−1.36693
$677$	5.38386	0.206919	0.103459	−	0.994634i	$-0.467009\pi$
$677$	5.38386	0.206919	0.103459	+	0.994634i	$0.467009\pi$
$678$	3.76933	0.144760
$679$	17.6989	0.679223
$680$	15.0399	0.576753
$681$	−8.27049	−0.316926
$682$	−53.1366	−2.03470
$683$	−22.8096	−0.872784	−0.436392	−	0.899757i	$-0.643744\pi$
$683$	−22.8096	−0.872784	−0.436392	+	0.899757i	$0.643744\pi$
$684$	−5.57209	−0.213054
$685$	−5.22595	−0.199673
$686$	−2.33888	−0.0892990
$687$	1.00822	0.0384659
$688$	11.0436	0.421032
$689$	18.5415	0.706375
$690$	−4.10785	−0.156383
$691$	1.51713	0.0577144	0.0288572	−	0.999584i	$-0.490813\pi$
$691$	1.51713	0.0577144	0.0288572	+	0.999584i	$0.490813\pi$
$692$	−78.6313	−2.98911
$693$	−9.50242	−0.360967
$694$	2.62936	0.0998091
$695$	−23.0962	−0.876088
$696$	35.7115	1.35364
$697$	−19.1113	−0.723892
$698$	−84.4939	−3.19814
$699$	−13.4140	−0.507363
$700$	3.47038	0.131168
$701$	9.49821	0.358742	0.179371	−	0.983781i	$-0.442594\pi$
$701$	9.49821	0.358742	0.179371	+	0.983781i	$0.442594\pi$
$702$	−19.5198	−0.736728
$703$	2.48599	0.0937610
$704$	−58.7346	−2.21364
$705$	−5.70678	−0.214930
$706$	33.4374	1.25843
$707$	−10.3937	−0.390897
$708$	−11.7554	−0.441794
$709$	23.0766	0.866659	0.433330	−	0.901235i	$-0.357339\pi$
$709$	23.0766	0.866659	0.433330	+	0.901235i	$0.357339\pi$
$710$	−0.990825	−0.0371850
$711$	10.4040	0.390180
$712$	10.9692	0.411089
$713$	8.26100	0.309377
$714$	10.3126	0.385940
$715$	−7.95758	−0.297597
$716$	55.0451	2.05713
$717$	1.75511	0.0655456
$718$	−9.87193	−0.368417
$719$	20.3968	0.760671	0.380335	−	0.924849i	$-0.375809\pi$
$719$	20.3968	0.760671	0.380335	+	0.924849i	$0.375809\pi$
$720$	2.18736	0.0815183
$721$	−1.86508	−0.0694591
$722$	42.9062	1.59680
$723$	22.9365	0.853019
$724$	−22.3744	−0.831537
$725$	10.2995	0.382512
$726$	28.1823	1.04594
$727$	23.6164	0.875885	0.437942	−	0.899003i	$-0.355707\pi$
$727$	23.6164	0.875885	0.437942	+	0.899003i	$0.355707\pi$
$728$	−5.71237	−0.211715
$729$	13.4424	0.497866
$730$	23.6255	0.874420
$731$	43.7950	1.61982
$732$	−31.5416	−1.16581
$733$	38.1886	1.41053	0.705264	−	0.708945i	$-0.250829\pi$
$733$	38.1886	1.41053	0.705264	+	0.708945i	$0.250829\pi$
$734$	68.2597	2.51951
$735$	1.00822	0.0371887
$736$	7.48859	0.276033
$737$	2.66017	0.0979885
$738$	−20.2733	−0.746270
$739$	−0.144713	−0.00532334	−0.00266167	−	0.999996i	$-0.500847\pi$
$739$	−0.144713	−0.00532334	−0.00266167	+	0.999996i	$0.500847\pi$
$740$	−10.6578	−0.391788
$741$	−1.35564	−0.0498005
$742$	−26.1081	−0.958459
$743$	3.35216	0.122979	0.0614893	−	0.998108i	$-0.480415\pi$
$743$	3.35216	0.122979	0.0614893	+	0.998108i	$0.480415\pi$
$744$	16.4428	0.602821
$745$	−1.51236	−0.0554087
$746$	36.4114	1.33312
$747$	−17.3413	−0.634487
$748$	72.7086	2.65849
$749$	5.94626	0.217272
$750$	−2.35811	−0.0861059
$751$	−40.0383	−1.46102	−0.730510	−	0.682902i	$-0.760717\pi$
$751$	−40.0383	−1.46102	−0.730510	+	0.682902i	$0.760717\pi$
$752$	−6.24203	−0.227624
$753$	−16.7609	−0.610800
$754$	−40.0130	−1.45719
$755$	7.90899	0.287838
$756$	17.4368	0.634169
$757$	−46.4851	−1.68953	−0.844764	−	0.535139i	$-0.820259\pi$
$757$	−46.4851	−1.68953	−0.844764	+	0.535139i	$0.820259\pi$
$758$	82.9480	3.01281
$759$	−8.41411	−0.305413
$760$	2.78387	0.100982
$761$	−23.0277	−0.834753	−0.417376	−	0.908734i	$-0.637050\pi$
$761$	−23.0277	−0.834753	−0.417376	+	0.908734i	$0.637050\pi$
$762$	−16.3527	−0.592397
$763$	3.62142	0.131104
$764$	−24.6750	−0.892710
$765$	8.67435	0.313622
$766$	−70.8494	−2.55989
$767$	5.58063	0.201505
$768$	22.6224	0.816317
$769$	19.3973	0.699484	0.349742	−	0.936846i	$-0.386269\pi$
$769$	19.3973	0.699484	0.349742	+	0.936846i	$0.386269\pi$
$770$	11.2050	0.403800
$771$	11.0433	0.397715
$772$	48.8595	1.75849
$773$	11.7863	0.423925	0.211963	−	0.977278i	$-0.432015\pi$
$773$	11.7863	0.423925	0.211963	+	0.977278i	$0.432015\pi$
$774$	46.4578	1.66989
$775$	4.74222	0.170346
$776$	−60.8675	−2.18502
$777$	−3.09631	−0.111080
$778$	55.0686	1.97431
$779$	−3.53749	−0.126744
$780$	5.81179	0.208095
$781$	−2.02951	−0.0726215
$782$	−17.8183	−0.637179
$783$	51.7492	1.84936
$784$	1.10278	0.0393851
$785$	−2.53564	−0.0905010
$786$	−43.9749	−1.56853
$787$	3.14318	0.112042	0.0560212	−	0.998430i	$-0.482159\pi$
$787$	3.14318	0.112042	0.0560212	+	0.998430i	$0.482159\pi$
$788$	−66.6438	−2.37409
$789$	−1.44517	−0.0514495
$790$	−12.2681	−0.436479
$791$	1.59846	0.0568346
$792$	32.6793	1.16121
$793$	14.9737	0.531732
$794$	6.70159	0.237831
$795$	11.2544	0.399152
$796$	−21.0241	−0.745181
$797$	32.8151	1.16237	0.581186	−	0.813771i	$-0.302589\pi$
$797$	32.8151	1.16237	0.581186	+	0.813771i	$0.302589\pi$
$798$	1.90886	0.0675729
$799$	−24.7538	−0.875726
$800$	4.29882	0.151986
$801$	6.32657	0.223538
$802$	65.9747	2.32965
$803$	48.3922	1.70772
$804$	−1.94284	−0.0685188
$805$	−1.74201	−0.0613978
$806$	−18.4233	−0.648935
$807$	4.24761	0.149523
$808$	35.7446	1.25749
$809$	48.3288	1.69915	0.849575	−	0.527467i	$-0.176858\pi$
$809$	48.3288	1.69915	0.849575	+	0.527467i	$0.176858\pi$
$810$	2.06930	0.0727078
$811$	18.7240	0.657489	0.328744	−	0.944419i	$-0.393375\pi$
$811$	18.7240	0.657489	0.328744	+	0.944419i	$0.393375\pi$
$812$	35.7431	1.25434
$813$	−0.536879	−0.0188292
$814$	−34.4113	−1.20612
$815$	−14.8399	−0.519819
$816$	−4.86240	−0.170218
$817$	8.10642	0.283608
$818$	31.7973	1.11177
$819$	−3.29465	−0.115124
$820$	15.1657	0.529608
$821$	48.3164	1.68625	0.843126	−	0.537716i	$-0.180713\pi$
$821$	48.3164	1.68625	0.843126	+	0.537716i	$0.180713\pi$
$822$	12.3233	0.429826
$823$	22.7638	0.793496	0.396748	−	0.917928i	$-0.370139\pi$
$823$	22.7638	0.793496	0.396748	+	0.917928i	$0.370139\pi$
$824$	6.41409	0.223446
$825$	−4.83012	−0.168163
$826$	−7.85803	−0.273416
$827$	9.55711	0.332333	0.166167	−	0.986098i	$-0.446861\pi$
$827$	9.55711	0.332333	0.166167	+	0.986098i	$0.446861\pi$
$828$	−11.9911	−0.416719
$829$	−11.7218	−0.407115	−0.203557	−	0.979063i	$-0.565250\pi$
$829$	−11.7218	−0.407115	−0.203557	+	0.979063i	$0.565250\pi$
$830$	20.4485	0.709776
$831$	7.88429	0.273503
$832$	−20.3643	−0.706004
$833$	4.37326	0.151525
$834$	54.4633	1.88591
$835$	14.8292	0.513185
$836$	13.4583	0.465465
$837$	23.8271	0.823584
$838$	94.0578	3.24917
$839$	8.63128	0.297985	0.148992	−	0.988838i	$-0.452397\pi$
$839$	8.63128	0.297985	0.148992	+	0.988838i	$0.452397\pi$
$840$	−3.46732	−0.119634
$841$	77.0789	2.65789
$842$	26.2939	0.906149
$843$	24.6456	0.848840
$844$	39.3779	1.35544
$845$	10.2410	0.352300
$846$	−26.2589	−0.902798
$847$	11.9512	0.410649
$848$	12.3100	0.422726
$849$	−17.0473	−0.585063
$850$	−10.2286	−0.350837
$851$	5.34984	0.183390
$852$	1.48224	0.0507809
$853$	22.9635	0.786256	0.393128	−	0.919484i	$-0.371393\pi$
$853$	22.9635	0.786256	0.393128	+	0.919484i	$0.371393\pi$
$854$	−21.0844	−0.721492
$855$	1.60561	0.0549109
$856$	−20.4495	−0.698949
$857$	21.1259	0.721648	0.360824	−	0.932634i	$-0.382495\pi$
$857$	21.1259	0.721648	0.360824	+	0.932634i	$0.382495\pi$
$858$	18.7648	0.640620
$859$	1.22656	0.0418498	0.0209249	−	0.999781i	$-0.493339\pi$
$859$	1.22656	0.0418498	0.0209249	+	0.999781i	$0.493339\pi$
$860$	−34.7533	−1.18508
$861$	4.40595	0.150154
$862$	5.12765	0.174648
$863$	30.0848	1.02410	0.512050	−	0.858956i	$-0.328886\pi$
$863$	30.0848	1.02410	0.512050	+	0.858956i	$0.328886\pi$
$864$	21.5992	0.734821
$865$	22.6578	0.770389
$866$	43.0938	1.46439
$867$	−2.14290	−0.0727768
$868$	16.4573	0.558598
$869$	−25.1288	−0.852435
$870$	−24.2872	−0.823414
$871$	0.922325	0.0312518
$872$	−12.4542	−0.421754
$873$	−35.1057	−1.18815
$874$	−3.29814	−0.111561
$875$	−1.00000	−0.0338062
$876$	−35.3431	−1.19413
$877$	25.7024	0.867907	0.433954	−	0.900935i	$-0.357118\pi$
$877$	25.7024	0.867907	0.433954	+	0.900935i	$0.357118\pi$
$878$	4.59056	0.154924
$879$	8.54109	0.288084
$880$	−5.28315	−0.178095
$881$	−19.1552	−0.645356	−0.322678	−	0.946509i	$-0.604583\pi$
$881$	−19.1552	−0.645356	−0.322678	+	0.946509i	$0.604583\pi$
$882$	4.63917	0.156209
$883$	5.99415	0.201719	0.100860	−	0.994901i	$-0.467841\pi$
$883$	5.99415	0.201719	0.100860	+	0.994901i	$0.467841\pi$
$884$	25.2093	0.847881
$885$	3.38735	0.113864
$886$	−92.5488	−3.10924
$887$	−10.7208	−0.359968	−0.179984	−	0.983670i	$-0.557605\pi$
$887$	−10.7208	−0.359968	−0.179984	+	0.983670i	$0.557605\pi$
$888$	10.6484	0.357336
$889$	−6.93469	−0.232582
$890$	−7.46012	−0.250064
$891$	4.23855	0.141997
$892$	−36.7093	−1.22912
$893$	−4.58191	−0.153328
$894$	3.56631	0.119275
$895$	−15.8614	−0.530188
$896$	20.0771	0.670729
$897$	−2.91732	−0.0974063
$898$	69.5412	2.32062
$899$	48.8423	1.62898
$900$	−6.88348	−0.229449
$901$	48.8172	1.62634
$902$	48.9662	1.63040
$903$	−10.0966	−0.335993
$904$	−5.49718	−0.182833
$905$	6.44724	0.214314
$906$	−18.6503	−0.619613
$907$	−40.0909	−1.33120	−0.665598	−	0.746310i	$-0.731824\pi$
$907$	−40.0909	−1.33120	−0.665598	+	0.746310i	$0.731824\pi$
$908$	28.4678	0.944737
$909$	20.6159	0.683787
$910$	3.88496	0.128785
$911$	−37.9462	−1.25721	−0.628607	−	0.777723i	$-0.716374\pi$
$911$	−37.9462	−1.25721	−0.628607	+	0.777723i	$0.716374\pi$
$912$	−0.900026	−0.0298028
$913$	41.8846	1.38618
$914$	−33.0971	−1.09475
$915$	9.08879	0.300466
$916$	−3.47038	−0.114665
$917$	−18.6484	−0.615824
$918$	−51.3929	−1.69622
$919$	14.3238	0.472498	0.236249	−	0.971693i	$-0.424082\pi$
$919$	14.3238	0.472498	0.236249	+	0.971693i	$0.424082\pi$
$920$	5.99086	0.197513
$921$	−28.5102	−0.939443
$922$	−0.589337	−0.0194088
$923$	−0.703665	−0.0231614
$924$	−16.7623	−0.551441
$925$	3.07107	0.100976
$926$	17.5010	0.575119
$927$	3.69937	0.121503
$928$	44.2755	1.45342
$929$	−4.01167	−0.131619	−0.0658094	−	0.997832i	$-0.520963\pi$
$929$	−4.01167	−0.131619	−0.0658094	+	0.997832i	$0.520963\pi$
$930$	−11.1827	−0.366694
$931$	0.809488	0.0265299
$932$	46.1721	1.51242
$933$	30.6078	1.00206
$934$	91.2245	2.98496
$935$	−20.9512	−0.685177
$936$	11.3305	0.370348
$937$	−28.2780	−0.923801	−0.461900	−	0.886932i	$-0.652832\pi$
$937$	−28.2780	−0.923801	−0.461900	+	0.886932i	$0.652832\pi$
$938$	−1.29872	−0.0424046
$939$	17.0813	0.557428
$940$	19.6432	0.640692
$941$	10.8621	0.354095	0.177047	−	0.984202i	$-0.443345\pi$
$941$	10.8621	0.354095	0.177047	+	0.984202i	$0.443345\pi$
$942$	5.97932	0.194817
$943$	−7.61264	−0.247902
$944$	3.70506	0.120589
$945$	−5.02445	−0.163445
$946$	−112.210	−3.64826
$947$	−36.1725	−1.17545	−0.587724	−	0.809062i	$-0.699976\pi$
$947$	−36.1725	−1.17545	−0.587724	+	0.809062i	$0.699976\pi$
$948$	18.3527	0.596068
$949$	16.7784	0.544650
$950$	−1.89330	−0.0614267
$951$	16.8199	0.545423
$952$	−15.0399	−0.487445
$953$	48.6727	1.57666	0.788332	−	0.615250i	$-0.210945\pi$
$953$	48.6727	1.57666	0.788332	+	0.615250i	$0.210945\pi$
$954$	51.7853	1.67661
$955$	7.11017	0.230080
$956$	−6.04124	−0.195388
$957$	−49.7476	−1.60811
$958$	−0.390026	−0.0126012
$959$	5.22595	0.168755
$960$	−12.3608	−0.398942
$961$	−8.51135	−0.274560
$962$	−11.9310	−0.384671
$963$	−11.7944	−0.380069
$964$	−78.9497	−2.54280
$965$	−14.0790	−0.453219
$966$	4.10785	0.132168
$967$	21.7156	0.698326	0.349163	−	0.937062i	$-0.386466\pi$
$967$	21.7156	0.698326	0.349163	+	0.937062i	$0.386466\pi$
$968$	−41.1009	−1.32103
$969$	−3.56920	−0.114659
$970$	41.3958	1.32914
$971$	−4.46802	−0.143385	−0.0716927	−	0.997427i	$-0.522840\pi$
$971$	−4.46802	−0.143385	−0.0716927	+	0.997427i	$0.522840\pi$
$972$	−55.4059	−1.77715
$973$	23.0962	0.740430
$974$	23.2621	0.745367
$975$	−1.67468	−0.0536328
$976$	9.94126	0.318212
$977$	−9.14779	−0.292664	−0.146332	−	0.989236i	$-0.546747\pi$
$977$	−9.14779	−0.292664	−0.146332	+	0.989236i	$0.546747\pi$
$978$	34.9940	1.11899
$979$	−15.2806	−0.488370
$980$	−3.47038	−0.110857
$981$	−7.18306	−0.229337
$982$	65.4847	2.08970
$983$	−61.0742	−1.94796	−0.973982	−	0.226627i	$-0.927230\pi$
$983$	−61.0742	−1.94796	−0.973982	+	0.226627i	$0.927230\pi$
$984$	−15.1523	−0.483037
$985$	19.2036	0.611877
$986$	−105.349	−3.35498
$987$	5.70678	0.181649
$988$	4.66622	0.148452
$989$	17.4450	0.554717
$990$	−22.2251	−0.706359
$991$	0.514960	0.0163583	0.00817913	−	0.999967i	$-0.497396\pi$
$991$	0.514960	0.0163583	0.00817913	+	0.999967i	$0.497396\pi$
$992$	20.3860	0.647255
$993$	−5.01826	−0.159250
$994$	0.990825	0.0314271
$995$	6.05816	0.192057
$996$	−30.5903	−0.969290
$997$	−26.1850	−0.829286	−0.414643	−	0.909984i	$-0.636094\pi$
$997$	−26.1850	−0.829286	−0.414643	+	0.909984i	$0.636094\pi$
$998$	−6.53009	−0.206706
$999$	15.4305	0.488198

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8015.2.a.n.1.7	✓	68

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8015.2.a.n.1.7	✓	68	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:	\( N \)	\(=\)	\( 8015 = 5 \cdot 7 \cdot 229 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8015.a (trivial)

\(f(q)\)	\(=\)	\(q-2.33888 q^{2} -1.00822 q^{3} +3.47038 q^{4} -1.00000 q^{5} +2.35811 q^{6} +1.00000 q^{7} -3.43905 q^{8} -1.98350 q^{9} +O(q^{10})\)	\(q-2.33888 q^{2} -1.00822 q^{3} +3.47038 q^{4} -1.00000 q^{5} +2.35811 q^{6} +1.00000 q^{7} -3.43905 q^{8} -1.98350 q^{9} +2.33888 q^{10} +4.79074 q^{11} -3.49890 q^{12} +1.66103 q^{13} -2.33888 q^{14} +1.00822 q^{15} +1.10278 q^{16} +4.37326 q^{17} +4.63917 q^{18} +0.809488 q^{19} -3.47038 q^{20} -1.00822 q^{21} -11.2050 q^{22} +1.74201 q^{23} +3.46732 q^{24} +1.00000 q^{25} -3.88496 q^{26} +5.02445 q^{27} +3.47038 q^{28} +10.2995 q^{29} -2.35811 q^{30} +4.74222 q^{31} +4.29882 q^{32} -4.83012 q^{33} -10.2286 q^{34} -1.00000 q^{35} -6.88348 q^{36} +3.07107 q^{37} -1.89330 q^{38} -1.67468 q^{39} +3.43905 q^{40} -4.37003 q^{41} +2.35811 q^{42} +10.0143 q^{43} +16.6257 q^{44} +1.98350 q^{45} -4.07436 q^{46} -5.66026 q^{47} -1.11185 q^{48} +1.00000 q^{49} -2.33888 q^{50} -4.40921 q^{51} +5.76441 q^{52} +11.1626 q^{53} -11.7516 q^{54} -4.79074 q^{55} -3.43905 q^{56} -0.816140 q^{57} -24.0892 q^{58} +3.35973 q^{59} +3.49890 q^{60} +9.01471 q^{61} -11.0915 q^{62} -1.98350 q^{63} -12.2600 q^{64} -1.66103 q^{65} +11.2971 q^{66} +0.555272 q^{67} +15.1769 q^{68} -1.75633 q^{69} +2.33888 q^{70} -0.423631 q^{71} +6.82134 q^{72} +10.1012 q^{73} -7.18288 q^{74} -1.00822 q^{75} +2.80923 q^{76} +4.79074 q^{77} +3.91689 q^{78} -5.24527 q^{79} -1.10278 q^{80} +0.884738 q^{81} +10.2210 q^{82} +8.74282 q^{83} -3.49890 q^{84} -4.37326 q^{85} -23.4222 q^{86} -10.3841 q^{87} -16.4756 q^{88} -3.18961 q^{89} -4.63917 q^{90} +1.66103 q^{91} +6.04544 q^{92} -4.78119 q^{93} +13.2387 q^{94} -0.809488 q^{95} -4.33415 q^{96} +17.6989 q^{97} -2.33888 q^{98} -9.50242 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9}+O(q^{10}) \) 68 * q + 9 * q^2 + 83 * q^4 - 68 * q^5 + 5 * q^6 + 68 * q^7 + 30 * q^8 + 86 * q^9	\( 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9} - 9 q^{10} + 5 q^{11} + 9 q^{12} + 15 q^{13} + 9 q^{14} + 109 q^{16} + 7 q^{17} + 39 q^{18} + 20 q^{19} - 83 q^{20} + 56 q^{22} + 36 q^{23} + q^{24} + 68 q^{25} + q^{26} + 12 q^{27} + 83 q^{28} - 16 q^{29} - 5 q^{30} + 31 q^{31} + 79 q^{32} + 45 q^{33} + 31 q^{34} - 68 q^{35} + 114 q^{36} + 72 q^{37} + 8 q^{38} + 47 q^{39} - 30 q^{40} + 6 q^{41} + 5 q^{42} + 75 q^{43} + 15 q^{44} - 86 q^{45} + 29 q^{46} - 10 q^{47} + 44 q^{48} + 68 q^{49} + 9 q^{50} + 23 q^{51} + 37 q^{52} + 41 q^{53} + 4 q^{54} - 5 q^{55} + 30 q^{56} + 55 q^{57} + 66 q^{58} - 5 q^{59} - 9 q^{60} - 2 q^{61} + 3 q^{62} + 86 q^{63} + 162 q^{64} - 15 q^{65} - 23 q^{66} + 92 q^{67} + 35 q^{68} - 25 q^{69} - 9 q^{70} - 2 q^{71} + 128 q^{72} + 80 q^{73} + 18 q^{74} + 71 q^{76} + 5 q^{77} + 20 q^{78} + 100 q^{79} - 109 q^{80} + 140 q^{81} + 36 q^{82} - 60 q^{83} + 9 q^{84} - 7 q^{85} - 27 q^{86} + 24 q^{87} + 175 q^{88} + 19 q^{89} - 39 q^{90} + 15 q^{91} + 75 q^{92} + 37 q^{93} + 11 q^{94} - 20 q^{95} + 15 q^{96} + 96 q^{97} + 9 q^{98} + 3 q^{99}+O(q^{100}) \) 68 * q + 9 * q^2 + 83 * q^4 - 68 * q^5 + 5 * q^6 + 68 * q^7 + 30 * q^8 + 86 * q^9 - 9 * q^10 + 5 * q^11 + 9 * q^12 + 15 * q^13 + 9 * q^14 + 109 * q^16 + 7 * q^17 + 39 * q^18 + 20 * q^19 - 83 * q^20 + 56 * q^22 + 36 * q^23 + q^24 + 68 * q^25 + q^26 + 12 * q^27 + 83 * q^28 - 16 * q^29 - 5 * q^30 + 31 * q^31 + 79 * q^32 + 45 * q^33 + 31 * q^34 - 68 * q^35 + 114 * q^36 + 72 * q^37 + 8 * q^38 + 47 * q^39 - 30 * q^40 + 6 * q^41 + 5 * q^42 + 75 * q^43 + 15 * q^44 - 86 * q^45 + 29 * q^46 - 10 * q^47 + 44 * q^48 + 68 * q^49 + 9 * q^50 + 23 * q^51 + 37 * q^52 + 41 * q^53 + 4 * q^54 - 5 * q^55 + 30 * q^56 + 55 * q^57 + 66 * q^58 - 5 * q^59 - 9 * q^60 - 2 * q^61 + 3 * q^62 + 86 * q^63 + 162 * q^64 - 15 * q^65 - 23 * q^66 + 92 * q^67 + 35 * q^68 - 25 * q^69 - 9 * q^70 - 2 * q^71 + 128 * q^72 + 80 * q^73 + 18 * q^74 + 71 * q^76 + 5 * q^77 + 20 * q^78 + 100 * q^79 - 109 * q^80 + 140 * q^81 + 36 * q^82 - 60 * q^83 + 9 * q^84 - 7 * q^85 - 27 * q^86 + 24 * q^87 + 175 * q^88 + 19 * q^89 - 39 * q^90 + 15 * q^91 + 75 * q^92 + 37 * q^93 + 11 * q^94 - 20 * q^95 + 15 * q^96 + 96 * q^97 + 9 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more