LMFDB - Embedded newform 8005.2.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8005,2,Mod(1,8005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8005, 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("8005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8005 = 5 \cdot 1601 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8005.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:	$63.9202468180$
Analytic rank:	$1$
Dimension:	$127$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69002 q^{2} +2.05113 q^{3} +5.23619 q^{4} -1.00000 q^{5} -5.51758 q^{6} -1.38538 q^{7} -8.70541 q^{8} +1.20715 q^{9} +O(q^{10})$	\(q-2.69002 q^{2} +2.05113 q^{3} +5.23619 q^{4} -1.00000 q^{5} -5.51758 q^{6} -1.38538 q^{7} -8.70541 q^{8} +1.20715 q^{9} +2.69002 q^{10} +1.93988 q^{11} +10.7401 q^{12} +4.32088 q^{13} +3.72670 q^{14} -2.05113 q^{15} +12.9453 q^{16} -4.04578 q^{17} -3.24724 q^{18} -4.73138 q^{19} -5.23619 q^{20} -2.84160 q^{21} -5.21831 q^{22} -0.808388 q^{23} -17.8560 q^{24} +1.00000 q^{25} -11.6232 q^{26} -3.67738 q^{27} -7.25413 q^{28} -2.23470 q^{29} +5.51758 q^{30} +0.820011 q^{31} -17.4123 q^{32} +3.97895 q^{33} +10.8832 q^{34} +1.38538 q^{35} +6.32084 q^{36} +3.61583 q^{37} +12.7275 q^{38} +8.86270 q^{39} +8.70541 q^{40} +5.58513 q^{41} +7.64396 q^{42} +12.1612 q^{43} +10.1576 q^{44} -1.20715 q^{45} +2.17458 q^{46} +6.91380 q^{47} +26.5526 q^{48} -5.08072 q^{49} -2.69002 q^{50} -8.29843 q^{51} +22.6250 q^{52} -12.2952 q^{53} +9.89222 q^{54} -1.93988 q^{55} +12.0603 q^{56} -9.70468 q^{57} +6.01139 q^{58} -7.26412 q^{59} -10.7401 q^{60} -1.38258 q^{61} -2.20584 q^{62} -1.67236 q^{63} +20.9488 q^{64} -4.32088 q^{65} -10.7035 q^{66} +14.3222 q^{67} -21.1845 q^{68} -1.65811 q^{69} -3.72670 q^{70} -2.52449 q^{71} -10.5087 q^{72} +4.31419 q^{73} -9.72666 q^{74} +2.05113 q^{75} -24.7744 q^{76} -2.68748 q^{77} -23.8408 q^{78} -9.69374 q^{79} -12.9453 q^{80} -11.1642 q^{81} -15.0241 q^{82} -7.24012 q^{83} -14.8792 q^{84} +4.04578 q^{85} -32.7138 q^{86} -4.58367 q^{87} -16.8875 q^{88} -6.26494 q^{89} +3.24724 q^{90} -5.98607 q^{91} -4.23287 q^{92} +1.68195 q^{93} -18.5982 q^{94} +4.73138 q^{95} -35.7150 q^{96} -11.5587 q^{97} +13.6672 q^{98} +2.34172 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 127 q - 6 q^{2} - 18 q^{3} + 114 q^{4} - 127 q^{5} - 20 q^{6} + 28 q^{7} - 18 q^{8} + 101 q^{9}+O(q^{10}) $ 127 * q - 6 * q^2 - 18 * q^3 + 114 * q^4 - 127 * q^5 - 20 * q^6 + 28 * q^7 - 18 * q^8 + 101 * q^9	\( 127 q - 6 q^{2} - 18 q^{3} + 114 q^{4} - 127 q^{5} - 20 q^{6} + 28 q^{7} - 18 q^{8} + 101 q^{9} + 6 q^{10} - 45 q^{11} - 30 q^{12} - 53 q^{14} + 18 q^{15} + 84 q^{16} - 36 q^{17} - 10 q^{18} - 49 q^{19} - 114 q^{20} - 48 q^{21} + 13 q^{22} - 29 q^{23} - 63 q^{24} + 127 q^{25} - 55 q^{26} - 75 q^{27} + 44 q^{28} - 45 q^{29} + 20 q^{30} - 49 q^{31} - 32 q^{32} - 8 q^{33} - 52 q^{34} - 28 q^{35} + 44 q^{36} + 36 q^{37} - 65 q^{38} - 52 q^{39} + 18 q^{40} - 66 q^{41} - 18 q^{42} - 5 q^{43} - 93 q^{44} - 101 q^{45} - 25 q^{46} - 32 q^{47} - 54 q^{48} + 77 q^{49} - 6 q^{50} - 102 q^{51} - 13 q^{52} - 67 q^{53} - 53 q^{54} + 45 q^{55} - 158 q^{56} + 16 q^{57} + 35 q^{58} - 213 q^{59} + 30 q^{60} - 62 q^{61} - 33 q^{62} + 59 q^{63} + 34 q^{64} - 60 q^{66} + 10 q^{67} - 94 q^{68} - 93 q^{69} + 53 q^{70} - 118 q^{71} - 24 q^{72} + 35 q^{73} - 107 q^{74} - 18 q^{75} - 98 q^{76} - 93 q^{77} + 21 q^{78} - 64 q^{79} - 84 q^{80} + 15 q^{81} + 15 q^{82} - 187 q^{83} - 118 q^{84} + 36 q^{85} - 126 q^{86} - 53 q^{87} + 15 q^{88} - 138 q^{89} + 10 q^{90} - 138 q^{91} - 86 q^{92} + 23 q^{93} - 60 q^{94} + 49 q^{95} - 92 q^{96} + 9 q^{97} - 67 q^{98} - 147 q^{99}+O(q^{100}) \) 127 * q - 6 * q^2 - 18 * q^3 + 114 * q^4 - 127 * q^5 - 20 * q^6 + 28 * q^7 - 18 * q^8 + 101 * q^9 + 6 * q^10 - 45 * q^11 - 30 * q^12 - 53 * q^14 + 18 * q^15 + 84 * q^16 - 36 * q^17 - 10 * q^18 - 49 * q^19 - 114 * q^20 - 48 * q^21 + 13 * q^22 - 29 * q^23 - 63 * q^24 + 127 * q^25 - 55 * q^26 - 75 * q^27 + 44 * q^28 - 45 * q^29 + 20 * q^30 - 49 * q^31 - 32 * q^32 - 8 * q^33 - 52 * q^34 - 28 * q^35 + 44 * q^36 + 36 * q^37 - 65 * q^38 - 52 * q^39 + 18 * q^40 - 66 * q^41 - 18 * q^42 - 5 * q^43 - 93 * q^44 - 101 * q^45 - 25 * q^46 - 32 * q^47 - 54 * q^48 + 77 * q^49 - 6 * q^50 - 102 * q^51 - 13 * q^52 - 67 * q^53 - 53 * q^54 + 45 * q^55 - 158 * q^56 + 16 * q^57 + 35 * q^58 - 213 * q^59 + 30 * q^60 - 62 * q^61 - 33 * q^62 + 59 * q^63 + 34 * q^64 - 60 * q^66 + 10 * q^67 - 94 * q^68 - 93 * q^69 + 53 * q^70 - 118 * q^71 - 24 * q^72 + 35 * q^73 - 107 * q^74 - 18 * q^75 - 98 * q^76 - 93 * q^77 + 21 * q^78 - 64 * q^79 - 84 * q^80 + 15 * q^81 + 15 * q^82 - 187 * q^83 - 118 * q^84 + 36 * q^85 - 126 * q^86 - 53 * q^87 + 15 * q^88 - 138 * q^89 + 10 * q^90 - 138 * q^91 - 86 * q^92 + 23 * q^93 - 60 * q^94 + 49 * q^95 - 92 * q^96 + 9 * q^97 - 67 * q^98 - 147 * 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 $

$2$	−2.69002	−1.90213	−0.951065	−	0.308992i	$-0.900008\pi$
$2$	−2.69002	−1.90213	−0.951065	+	0.308992i	$0.900008\pi$
$3$	2.05113	1.18422	0.592111	−	0.805856i	$-0.298295\pi$
$3$	2.05113	1.18422	0.592111	+	0.805856i	$0.298295\pi$
$4$	5.23619	2.61810
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−5.51758	−2.25254
$7$	−1.38538	−0.523625	−0.261813	−	0.965119i	$-0.584320\pi$
$7$	−1.38538	−0.523625	−0.261813	+	0.965119i	$0.584320\pi$
$8$	−8.70541	−3.07783
$9$	1.20715	0.402382
$10$	2.69002	0.850658
$11$	1.93988	0.584896	0.292448	−	0.956281i	$-0.405530\pi$
$11$	1.93988	0.584896	0.292448	+	0.956281i	$0.405530\pi$
$12$	10.7401	3.10041
$13$	4.32088	1.19840	0.599198	−	0.800601i	$-0.295486\pi$
$13$	4.32088	1.19840	0.599198	+	0.800601i	$0.295486\pi$
$14$	3.72670	0.996003
$15$	−2.05113	−0.529600
$16$	12.9453	3.23633
$17$	−4.04578	−0.981245	−0.490623	−	0.871372i	$-0.663231\pi$
$17$	−4.04578	−0.981245	−0.490623	+	0.871372i	$0.663231\pi$
$18$	−3.24724	−0.765382
$19$	−4.73138	−1.08545	−0.542726	−	0.839910i	$-0.682608\pi$
$19$	−4.73138	−1.08545	−0.542726	+	0.839910i	$0.682608\pi$
$20$	−5.23619	−1.17085
$21$	−2.84160	−0.620089
$22$	−5.21831	−1.11255
$23$	−0.808388	−0.168561	−0.0842803	−	0.996442i	$-0.526859\pi$
$23$	−0.808388	−0.168561	−0.0842803	+	0.996442i	$0.526859\pi$
$24$	−17.8560	−3.64483
$25$	1.00000	0.200000
$26$	−11.6232	−2.27951
$27$	−3.67738	−0.707713
$28$	−7.25413	−1.37090
$29$	−2.23470	−0.414974	−0.207487	−	0.978238i	$-0.566528\pi$
$29$	−2.23470	−0.414974	−0.207487	+	0.978238i	$0.566528\pi$
$30$	5.51758	1.00737
$31$	0.820011	0.147278	0.0736392	−	0.997285i	$-0.476539\pi$
$31$	0.820011	0.147278	0.0736392	+	0.997285i	$0.476539\pi$
$32$	−17.4123	−3.07809
$33$	3.97895	0.692647
$34$	10.8832	1.86646
$35$	1.38538	0.234172
$36$	6.32084	1.05347
$37$	3.61583	0.594439	0.297220	−	0.954809i	$-0.403941\pi$
$37$	3.61583	0.594439	0.297220	+	0.954809i	$0.403941\pi$
$38$	12.7275	2.06467
$39$	8.86270	1.41917
$40$	8.70541	1.37645
$41$	5.58513	0.872250	0.436125	−	0.899886i	$-0.356351\pi$
$41$	5.58513	0.872250	0.436125	+	0.899886i	$0.356351\pi$
$42$	7.64396	1.17949
$43$	12.1612	1.85457	0.927283	−	0.374361i	$-0.122138\pi$
$43$	12.1612	1.85457	0.927283	+	0.374361i	$0.122138\pi$
$44$	10.1576	1.53131
$45$	−1.20715	−0.179951
$46$	2.17458	0.320624
$47$	6.91380	1.00848	0.504241	−	0.863563i	$-0.331772\pi$
$47$	6.91380	1.00848	0.504241	+	0.863563i	$0.331772\pi$
$48$	26.5526	3.83254
$49$	−5.08072	−0.725816
$50$	−2.69002	−0.380426
$51$	−8.29843	−1.16201
$52$	22.6250	3.13752
$53$	−12.2952	−1.68888	−0.844439	−	0.535652i	$-0.820066\pi$
$53$	−12.2952	−1.68888	−0.844439	+	0.535652i	$0.820066\pi$
$54$	9.89222	1.34616
$55$	−1.93988	−0.261573
$56$	12.0603	1.61163
$57$	−9.70468	−1.28542
$58$	6.01139	0.789334
$59$	−7.26412	−0.945709	−0.472854	−	0.881141i	$-0.656776\pi$
$59$	−7.26412	−0.945709	−0.472854	+	0.881141i	$0.656776\pi$
$60$	−10.7401	−1.38654
$61$	−1.38258	−0.177021	−0.0885103	−	0.996075i	$-0.528211\pi$
$61$	−1.38258	−0.177021	−0.0885103	+	0.996075i	$0.528211\pi$
$62$	−2.20584	−0.280142
$63$	−1.67236	−0.210697
$64$	20.9488	2.61860
$65$	−4.32088	−0.535939
$66$	−10.7035	−1.31750
$67$	14.3222	1.74974	0.874869	−	0.484359i	$-0.160947\pi$
$67$	14.3222	1.74974	0.874869	+	0.484359i	$0.160947\pi$
$68$	−21.1845	−2.56899
$69$	−1.65811	−0.199613
$70$	−3.72670	−0.445426
$71$	−2.52449	−0.299602	−0.149801	−	0.988716i	$-0.547863\pi$
$71$	−2.52449	−0.299602	−0.149801	+	0.988716i	$0.547863\pi$
$72$	−10.5087	−1.23846
$73$	4.31419	0.504938	0.252469	−	0.967605i	$-0.418757\pi$
$73$	4.31419	0.504938	0.252469	+	0.967605i	$0.418757\pi$
$74$	−9.72666	−1.13070
$75$	2.05113	0.236844
$76$	−24.7744	−2.84182
$77$	−2.68748	−0.306266
$78$	−23.8408	−2.69944
$79$	−9.69374	−1.09063	−0.545315	−	0.838231i	$-0.683590\pi$
$79$	−9.69374	−1.09063	−0.545315	+	0.838231i	$0.683590\pi$
$80$	−12.9453	−1.44733
$81$	−11.1642	−1.24047
$82$	−15.0241	−1.65913
$83$	−7.24012	−0.794706	−0.397353	−	0.917666i	$-0.630071\pi$
$83$	−7.24012	−0.794706	−0.397353	+	0.917666i	$0.630071\pi$
$84$	−14.8792	−1.62345
$85$	4.04578	0.438826
$86$	−32.7138	−3.52762
$87$	−4.58367	−0.491421
$88$	−16.8875	−1.80021
$89$	−6.26494	−0.664083	−0.332041	−	0.943265i	$-0.607737\pi$
$89$	−6.26494	−0.664083	−0.332041	+	0.943265i	$0.607737\pi$
$90$	3.24724	0.342289
$91$	−5.98607	−0.627511
$92$	−4.23287	−0.441308
$93$	1.68195	0.174410
$94$	−18.5982	−1.91826
$95$	4.73138	0.485429
$96$	−35.7150	−3.64515
$97$	−11.5587	−1.17361	−0.586803	−	0.809730i	$-0.699614\pi$
$97$	−11.5587	−1.17361	−0.586803	+	0.809730i	$0.699614\pi$
$98$	13.6672	1.38060
$99$	2.34172	0.235351
$100$	5.23619	0.523619
$101$	9.86474	0.981578	0.490789	−	0.871278i	$-0.336709\pi$
$101$	9.86474	0.981578	0.490789	+	0.871278i	$0.336709\pi$
$102$	22.3229	2.21030
$103$	16.2715	1.60328	0.801639	−	0.597808i	$-0.203962\pi$
$103$	16.2715	1.60328	0.801639	+	0.597808i	$0.203962\pi$
$104$	−37.6151	−3.68846
$105$	2.84160	0.277312
$106$	33.0744	3.21247
$107$	19.6177	1.89652	0.948260	−	0.317496i	$-0.102842\pi$
$107$	19.6177	1.89652	0.948260	+	0.317496i	$0.102842\pi$
$108$	−19.2555	−1.85286
$109$	−6.65940	−0.637855	−0.318928	−	0.947779i	$-0.603323\pi$
$109$	−6.65940	−0.637855	−0.318928	+	0.947779i	$0.603323\pi$
$110$	5.21831	0.497547
$111$	7.41655	0.703948
$112$	−17.9342	−1.69463
$113$	−3.63673	−0.342115	−0.171057	−	0.985261i	$-0.554718\pi$
$113$	−3.63673	−0.342115	−0.171057	+	0.985261i	$0.554718\pi$
$114$	26.1058	2.44503
$115$	0.808388	0.0753826
$116$	−11.7013	−1.08644
$117$	5.21593	0.482213
$118$	19.5406	1.79886
$119$	5.60495	0.513805
$120$	17.8560	1.63002
$121$	−7.23686	−0.657897
$122$	3.71915	0.336716
$123$	11.4558	1.03294
$124$	4.29374	0.385589
$125$	−1.00000	−0.0894427
$126$	4.49867	0.400773
$127$	−15.1846	−1.34741	−0.673707	−	0.738999i	$-0.735299\pi$
$127$	−15.1846	−1.34741	−0.673707	+	0.738999i	$0.735299\pi$
$128$	−21.5280	−1.90282
$129$	24.9442	2.19622
$130$	11.6232	1.01943
$131$	4.53262	0.396017	0.198008	−	0.980200i	$-0.436553\pi$
$131$	4.53262	0.396017	0.198008	+	0.980200i	$0.436553\pi$
$132$	20.8346	1.81342
$133$	6.55477	0.568370
$134$	−38.5270	−3.32823
$135$	3.67738	0.316499
$136$	35.2202	3.02010
$137$	−7.45963	−0.637319	−0.318659	−	0.947869i	$-0.603233\pi$
$137$	−7.45963	−0.637319	−0.318659	+	0.947869i	$0.603233\pi$
$138$	4.46035	0.379690
$139$	10.7698	0.913483	0.456742	−	0.889599i	$-0.349016\pi$
$139$	10.7698	0.913483	0.456742	+	0.889599i	$0.349016\pi$
$140$	7.25413	0.613086
$141$	14.1811	1.19427
$142$	6.79092	0.569881
$143$	8.38199	0.700938
$144$	15.6269	1.30224
$145$	2.23470	0.185582
$146$	−11.6052	−0.960457
$147$	−10.4212	−0.859528
$148$	18.9332	1.55630
$149$	−11.6884	−0.957553	−0.478776	−	0.877937i	$-0.658920\pi$
$149$	−11.6884	−0.957553	−0.478776	+	0.877937i	$0.658920\pi$
$150$	−5.51758	−0.450509
$151$	7.71664	0.627971	0.313986	−	0.949428i	$-0.398336\pi$
$151$	7.71664	0.627971	0.313986	+	0.949428i	$0.398336\pi$
$152$	41.1886	3.34084
$153$	−4.88384	−0.394835
$154$	7.22936	0.582558
$155$	−0.820011	−0.0658649
$156$	46.4068	3.71552
$157$	−9.51429	−0.759323	−0.379662	−	0.925125i	$-0.623960\pi$
$157$	−9.51429	−0.759323	−0.379662	+	0.925125i	$0.623960\pi$
$158$	26.0763	2.07452
$159$	−25.2191	−2.00001
$160$	17.4123	1.37657
$161$	1.11993	0.0882626
$162$	30.0320	2.35954
$163$	−12.5666	−0.984291	−0.492146	−	0.870513i	$-0.663787\pi$
$163$	−12.5666	−0.984291	−0.492146	+	0.870513i	$0.663787\pi$
$164$	29.2448	2.28364
$165$	−3.97895	−0.309761
$166$	19.4760	1.51163
$167$	−0.0227980	−0.00176416	−0.000882080	−	1.00000i	$-0.500281\pi$
$167$	−0.0227980	−0.00176416	−0.000882080		1.00000i	$0.500281\pi$
$168$	24.7373	1.90853
$169$	5.67001	0.436155
$170$	−10.8832	−0.834704
$171$	−5.71146	−0.436766
$172$	63.6784	4.85543
$173$	−16.3209	−1.24086	−0.620428	−	0.784263i	$-0.713041\pi$
$173$	−16.3209	−1.24086	−0.620428	+	0.784263i	$0.713041\pi$
$174$	12.3302	0.934747
$175$	−1.38538	−0.104725
$176$	25.1124	1.89292
$177$	−14.8997	−1.11993
$178$	16.8528	1.26317
$179$	−10.1311	−0.757238	−0.378619	−	0.925553i	$-0.623601\pi$
$179$	−10.1311	−0.757238	−0.378619	+	0.925553i	$0.623601\pi$
$180$	−6.32084	−0.471128
$181$	−9.58351	−0.712337	−0.356168	−	0.934422i	$-0.615917\pi$
$181$	−9.58351	−0.712337	−0.356168	+	0.934422i	$0.615917\pi$
$182$	16.1026	1.19361
$183$	−2.83585	−0.209632
$184$	7.03735	0.518800
$185$	−3.61583	−0.265841
$186$	−4.52448	−0.331751
$187$	−7.84833	−0.573926
$188$	36.2020	2.64030
$189$	5.09458	0.370576
$190$	−12.7275	−0.923349
$191$	−19.6684	−1.42316	−0.711578	−	0.702607i	$-0.752019\pi$
$191$	−19.6684	−1.42316	−0.711578	+	0.702607i	$0.752019\pi$
$192$	42.9688	3.10100
$193$	−6.06207	−0.436357	−0.218179	−	0.975909i	$-0.570012\pi$
$193$	−6.06207	−0.436357	−0.218179	+	0.975909i	$0.570012\pi$
$194$	31.0930	2.23235
$195$	−8.86270	−0.634671
$196$	−26.6036	−1.90026
$197$	−3.67433	−0.261785	−0.130893	−	0.991397i	$-0.541784\pi$
$197$	−3.67433	−0.261785	−0.130893	+	0.991397i	$0.541784\pi$
$198$	−6.29926	−0.447669
$199$	−8.24821	−0.584700	−0.292350	−	0.956311i	$-0.594437\pi$
$199$	−8.24821	−0.584700	−0.292350	+	0.956311i	$0.594437\pi$
$200$	−8.70541	−0.615566
$201$	29.3768	2.07208
$202$	−26.5363	−1.86709
$203$	3.09592	0.217291
$204$	−43.4522	−3.04226
$205$	−5.58513	−0.390082
$206$	−43.7706	−3.04964
$207$	−0.975842	−0.0678257
$208$	55.9352	3.87841
$209$	−9.17831	−0.634877
$210$	−7.64396	−0.527483
$211$	−11.2412	−0.773876	−0.386938	−	0.922106i	$-0.626467\pi$
$211$	−11.2412	−0.773876	−0.386938	+	0.922106i	$0.626467\pi$
$212$	−64.3801	−4.42165
$213$	−5.17806	−0.354795
$214$	−52.7721	−3.60742
$215$	−12.1612	−0.829387
$216$	32.0131	2.17822
$217$	−1.13603	−0.0771187
$218$	17.9139	1.21328
$219$	8.84898	0.597959
$220$	−10.1576	−0.684825
$221$	−17.4813	−1.17592
$222$	−19.9507	−1.33900
$223$	20.7832	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$223$	20.7832	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$224$	24.1227	1.61177
$225$	1.20715	0.0804763
$226$	9.78287	0.650747
$227$	−8.26868	−0.548812	−0.274406	−	0.961614i	$-0.588481\pi$
$227$	−8.26868	−0.548812	−0.274406	+	0.961614i	$0.588481\pi$
$228$	−50.8156	−3.36534
$229$	8.19504	0.541544	0.270772	−	0.962644i	$-0.412721\pi$
$229$	8.19504	0.541544	0.270772	+	0.962644i	$0.412721\pi$
$230$	−2.17458	−0.143387
$231$	−5.51237	−0.362687
$232$	19.4540	1.27722
$233$	21.8603	1.43212	0.716058	−	0.698041i	$-0.245945\pi$
$233$	21.8603	1.43212	0.716058	+	0.698041i	$0.245945\pi$
$234$	−14.0309	−0.917231
$235$	−6.91380	−0.451006
$236$	−38.0364	−2.47596
$237$	−19.8831	−1.29155
$238$	−15.0774	−0.977323
$239$	−20.2126	−1.30744	−0.653721	−	0.756736i	$-0.726793\pi$
$239$	−20.2126	−1.30744	−0.653721	+	0.756736i	$0.726793\pi$
$240$	−26.5526	−1.71396
$241$	−9.00590	−0.580121	−0.290060	−	0.957008i	$-0.593675\pi$
$241$	−9.00590	−0.580121	−0.290060	+	0.957008i	$0.593675\pi$
$242$	19.4673	1.25140
$243$	−11.8672	−0.761280
$244$	−7.23943	−0.463457
$245$	5.08072	0.324595
$246$	−30.8164	−1.96478
$247$	−20.4437	−1.30080
$248$	−7.13854	−0.453297
$249$	−14.8504	−0.941109
$250$	2.69002	0.170132
$251$	6.34335	0.400389	0.200194	−	0.979756i	$-0.435843\pi$
$251$	6.34335	0.400389	0.200194	+	0.979756i	$0.435843\pi$
$252$	−8.75679	−0.551626
$253$	−1.56818	−0.0985904
$254$	40.8468	2.56296
$255$	8.29843	0.519668
$256$	16.0130	1.00082
$257$	−18.9673	−1.18315	−0.591573	−	0.806252i	$-0.701493\pi$
$257$	−18.9673	−1.18315	−0.591573	+	0.806252i	$0.701493\pi$
$258$	−67.1004	−4.17749
$259$	−5.00931	−0.311264
$260$	−22.6250	−1.40314
$261$	−2.69761	−0.166978
$262$	−12.1928	−0.753275
$263$	14.8471	0.915513	0.457756	−	0.889078i	$-0.348653\pi$
$263$	14.8471	0.915513	0.457756	+	0.889078i	$0.348653\pi$
$264$	−34.6384	−2.13185
$265$	12.2952	0.755289
$266$	−17.6324	−1.08111
$267$	−12.8502	−0.786421
$268$	74.9939	4.58098
$269$	5.04501	0.307599	0.153800	−	0.988102i	$-0.450849\pi$
$269$	5.04501	0.307599	0.153800	+	0.988102i	$0.450849\pi$
$270$	−9.89222	−0.602022
$271$	15.0955	0.916986	0.458493	−	0.888698i	$-0.348389\pi$
$271$	15.0955	0.916986	0.458493	+	0.888698i	$0.348389\pi$
$272$	−52.3739	−3.17563
$273$	−12.2782	−0.743112
$274$	20.0665	1.21226
$275$	1.93988	0.116979
$276$	−8.68219	−0.522606
$277$	3.27710	0.196902	0.0984509	−	0.995142i	$-0.468611\pi$
$277$	3.27710	0.196902	0.0984509	+	0.995142i	$0.468611\pi$
$278$	−28.9710	−1.73756
$279$	0.989872	0.0592621
$280$	−12.0603	−0.720742
$281$	20.9460	1.24954	0.624768	−	0.780811i	$-0.285194\pi$
$281$	20.9460	1.24954	0.624768	+	0.780811i	$0.285194\pi$
$282$	−38.1474	−2.27165
$283$	−7.20354	−0.428206	−0.214103	−	0.976811i	$-0.568683\pi$
$283$	−7.20354	−0.428206	−0.214103	+	0.976811i	$0.568683\pi$
$284$	−13.2187	−0.784386
$285$	9.70468	0.574856
$286$	−22.5477	−1.33327
$287$	−7.73754	−0.456732
$288$	−21.0192	−1.23857
$289$	−0.631684	−0.0371579
$290$	−6.01139	−0.353001
$291$	−23.7084	−1.38981
$292$	22.5899	1.32198
$293$	27.7493	1.62113	0.810564	−	0.585650i	$-0.199161\pi$
$293$	27.7493	1.62113	0.810564	+	0.585650i	$0.199161\pi$
$294$	28.0333	1.63493
$295$	7.26412	0.422934
$296$	−31.4773	−1.82958
$297$	−7.13369	−0.413938
$298$	31.4421	1.82139
$299$	−3.49295	−0.202002
$300$	10.7401	0.620081
$301$	−16.8479	−0.971098
$302$	−20.7579	−1.19448
$303$	20.2339	1.16241
$304$	−61.2492	−3.51288
$305$	1.38258	0.0791661
$306$	13.1376	0.751027
$307$	10.3046	0.588113	0.294057	−	0.955788i	$-0.404995\pi$
$307$	10.3046	0.588113	0.294057	+	0.955788i	$0.404995\pi$
$308$	−14.0721	−0.801835
$309$	33.3750	1.89864
$310$	2.20584	0.125284
$311$	−15.3433	−0.870038	−0.435019	−	0.900421i	$-0.643258\pi$
$311$	−15.3433	−0.870038	−0.435019	+	0.900421i	$0.643258\pi$
$312$	−77.1535	−4.36795
$313$	10.1903	0.575988	0.287994	−	0.957632i	$-0.407012\pi$
$313$	10.1903	0.575988	0.287994	+	0.957632i	$0.407012\pi$
$314$	25.5936	1.44433
$315$	1.67236	0.0942267
$316$	−50.7583	−2.85538
$317$	−13.4241	−0.753970	−0.376985	−	0.926219i	$-0.623039\pi$
$317$	−13.4241	−0.753970	−0.376985	+	0.926219i	$0.623039\pi$
$318$	67.8399	3.80427
$319$	−4.33506	−0.242717
$320$	−20.9488	−1.17107
$321$	40.2386	2.24590
$322$	−3.01262	−0.167887
$323$	19.1421	1.06509
$324$	−58.4581	−3.24767
$325$	4.32088	0.239679
$326$	33.8043	1.87225
$327$	−13.6593	−0.755362
$328$	−48.6208	−2.68464
$329$	−9.57826	−0.528066
$330$	10.7035	0.589206
$331$	−13.8946	−0.763717	−0.381859	−	0.924221i	$-0.624716\pi$
$331$	−13.8946	−0.763717	−0.381859	+	0.924221i	$0.624716\pi$
$332$	−37.9107	−2.08062
$333$	4.36484	0.239192
$334$	0.0613269	0.00335566
$335$	−14.3222	−0.782507
$336$	−36.7855	−2.00681
$337$	−28.0910	−1.53022	−0.765108	−	0.643902i	$-0.777314\pi$
$337$	−28.0910	−1.53022	−0.765108	+	0.643902i	$0.777314\pi$
$338$	−15.2524	−0.829623
$339$	−7.45941	−0.405140
$340$	21.1845	1.14889
$341$	1.59072	0.0861425
$342$	15.3639	0.830786
$343$	16.7364	0.903681
$344$	−105.868	−5.70804
$345$	1.65811	0.0892697
$346$	43.9035	2.36027
$347$	−15.3942	−0.826404	−0.413202	−	0.910639i	$-0.635590\pi$
$347$	−15.3942	−0.826404	−0.413202	+	0.910639i	$0.635590\pi$
$348$	−24.0010	−1.28659
$349$	30.3553	1.62488	0.812440	−	0.583045i	$-0.198139\pi$
$349$	30.3553	1.62488	0.812440	+	0.583045i	$0.198139\pi$
$350$	3.72670	0.199201
$351$	−15.8895	−0.848121
$352$	−33.7778	−1.80036
$353$	−24.0981	−1.28261	−0.641306	−	0.767285i	$-0.721607\pi$
$353$	−24.0981	−1.28261	−0.641306	+	0.767285i	$0.721607\pi$
$354$	40.0804	2.13025
$355$	2.52449	0.133986
$356$	−32.8044	−1.73863
$357$	11.4965	0.608459
$358$	27.2530	1.44036
$359$	−36.3442	−1.91817	−0.959086	−	0.283113i	$-0.908633\pi$
$359$	−36.3442	−1.91817	−0.959086	+	0.283113i	$0.908633\pi$
$360$	10.5087	0.553857
$361$	3.38592	0.178206
$362$	25.7798	1.35496
$363$	−14.8438	−0.779096
$364$	−31.3442	−1.64288
$365$	−4.31419	−0.225815
$366$	7.62847	0.398747
$367$	−18.3927	−0.960090	−0.480045	−	0.877244i	$-0.659380\pi$
$367$	−18.3927	−0.960090	−0.480045	+	0.877244i	$0.659380\pi$
$368$	−10.4648	−0.545518
$369$	6.74206	0.350978
$370$	9.72666	0.505665
$371$	17.0336	0.884340
$372$	8.80702	0.456623
$373$	−14.1950	−0.734987	−0.367493	−	0.930026i	$-0.619784\pi$
$373$	−14.1950	−0.734987	−0.367493	+	0.930026i	$0.619784\pi$
$374$	21.1121	1.09168
$375$	−2.05113	−0.105920
$376$	−60.1875	−3.10393
$377$	−9.65589	−0.497303
$378$	−13.7045	−0.704884
$379$	16.5377	0.849482	0.424741	−	0.905315i	$-0.360365\pi$
$379$	16.5377	0.849482	0.424741	+	0.905315i	$0.360365\pi$
$380$	24.7744	1.27090
$381$	−31.1456	−1.59564
$382$	52.9083	2.70703
$383$	−23.1431	−1.18256	−0.591279	−	0.806467i	$-0.701377\pi$
$383$	−23.1431	−1.18256	−0.591279	+	0.806467i	$0.701377\pi$
$384$	−44.1567	−2.25336
$385$	2.68748	0.136967
$386$	16.3071	0.830008
$387$	14.6803	0.746243
$388$	−60.5235	−3.07261
$389$	−14.7277	−0.746722	−0.373361	−	0.927686i	$-0.621795\pi$
$389$	−14.7277	−0.746722	−0.373361	+	0.927686i	$0.621795\pi$
$390$	23.8408	1.20723
$391$	3.27056	0.165399
$392$	44.2297	2.23394
$393$	9.29700	0.468972
$394$	9.88402	0.497950
$395$	9.69374	0.487745
$396$	12.2617	0.616173
$397$	−12.3576	−0.620209	−0.310104	−	0.950703i	$-0.600364\pi$
$397$	−12.3576	−0.620209	−0.310104	+	0.950703i	$0.600364\pi$
$398$	22.1878	1.11217
$399$	13.4447	0.673077
$400$	12.9453	0.647266
$401$	−17.7148	−0.884635	−0.442317	−	0.896859i	$-0.645843\pi$
$401$	−17.7148	−0.884635	−0.442317	+	0.896859i	$0.645843\pi$
$402$	−79.0241	−3.94136
$403$	3.54317	0.176498
$404$	51.6537	2.56987
$405$	11.1642	0.554755
$406$	−8.32807	−0.413315
$407$	7.01429	0.347685
$408$	72.2412	3.57647
$409$	−28.8978	−1.42890	−0.714451	−	0.699685i	$-0.753324\pi$
$409$	−28.8978	−1.42890	−0.714451	+	0.699685i	$0.753324\pi$
$410$	15.0241	0.741987
$411$	−15.3007	−0.754727
$412$	85.2007	4.19754
$413$	10.0636	0.495197
$414$	2.62503	0.129013
$415$	7.24012	0.355403
$416$	−75.2366	−3.68878
$417$	22.0903	1.08177
$418$	24.6898	1.20762
$419$	−25.9768	−1.26905	−0.634524	−	0.772903i	$-0.718804\pi$
$419$	−25.9768	−1.26905	−0.634524	+	0.772903i	$0.718804\pi$
$420$	14.8792	0.726030
$421$	−1.28000	−0.0623836	−0.0311918	−	0.999513i	$-0.509930\pi$
$421$	−1.28000	−0.0623836	−0.0311918	+	0.999513i	$0.509930\pi$
$422$	30.2390	1.47201
$423$	8.34596	0.405794
$424$	107.035	5.19808
$425$	−4.04578	−0.196249
$426$	13.9291	0.674866
$427$	1.91540	0.0926925
$428$	102.722	4.96527
$429$	17.1926	0.830066
$430$	32.7138	1.57760
$431$	−13.7996	−0.664705	−0.332352	−	0.943155i	$-0.607842\pi$
$431$	−13.7996	−0.664705	−0.332352	+	0.943155i	$0.607842\pi$
$432$	−47.6049	−2.29039
$433$	25.2891	1.21532	0.607658	−	0.794199i	$-0.292109\pi$
$433$	25.2891	1.21532	0.607658	+	0.794199i	$0.292109\pi$
$434$	3.05594	0.146690
$435$	4.58367	0.219770
$436$	−34.8699	−1.66997
$437$	3.82479	0.182964
$438$	−23.8039	−1.13739
$439$	32.8565	1.56815	0.784077	−	0.620663i	$-0.213137\pi$
$439$	32.8565	1.56815	0.784077	+	0.620663i	$0.213137\pi$
$440$	16.8875	0.805078
$441$	−6.13316	−0.292055
$442$	47.0251	2.23675
$443$	−1.52897	−0.0726437	−0.0363219	−	0.999340i	$-0.511564\pi$
$443$	−1.52897	−0.0726437	−0.0363219	+	0.999340i	$0.511564\pi$
$444$	38.8345	1.84300
$445$	6.26494	0.296987
$446$	−55.9072	−2.64728
$447$	−23.9745	−1.13396
$448$	−29.0221	−1.37117
$449$	10.2547	0.483950	0.241975	−	0.970282i	$-0.422205\pi$
$449$	10.2547	0.483950	0.241975	+	0.970282i	$0.422205\pi$
$450$	−3.24724	−0.153076
$451$	10.8345	0.510176
$452$	−19.0426	−0.895689
$453$	15.8279	0.743658
$454$	22.2429	1.04391
$455$	5.98607	0.280631
$456$	84.4833	3.95629
$457$	−13.4329	−0.628363	−0.314182	−	0.949363i	$-0.601730\pi$
$457$	−13.4329	−0.628363	−0.314182	+	0.949363i	$0.601730\pi$
$458$	−22.0448	−1.03009
$459$	14.8779	0.694440
$460$	4.23287	0.197359
$461$	7.91296	0.368543	0.184272	−	0.982875i	$-0.441007\pi$
$461$	7.91296	0.368543	0.184272	+	0.982875i	$0.441007\pi$
$462$	14.8284	0.689878
$463$	38.0903	1.77021	0.885103	−	0.465395i	$-0.154088\pi$
$463$	38.0903	1.77021	0.885103	+	0.465395i	$0.154088\pi$
$464$	−28.9290	−1.34299
$465$	−1.68195	−0.0779986
$466$	−58.8046	−2.72407
$467$	2.19222	0.101444	0.0507220	−	0.998713i	$-0.483848\pi$
$467$	2.19222	0.101444	0.0507220	+	0.998713i	$0.483848\pi$
$468$	27.3116	1.26248
$469$	−19.8418	−0.916208
$470$	18.5982	0.857873
$471$	−19.5151	−0.899207
$472$	63.2372	2.91073
$473$	23.5913	1.08473
$474$	53.4860	2.45669
$475$	−4.73138	−0.217090
$476$	29.3486	1.34519
$477$	−14.8421	−0.679574
$478$	54.3722	2.48692
$479$	1.42175	0.0649615	0.0324807	−	0.999472i	$-0.489659\pi$
$479$	1.42175	0.0649615	0.0324807	+	0.999472i	$0.489659\pi$
$480$	35.7150	1.63016
$481$	15.6236	0.712374
$482$	24.2260	1.10347
$483$	2.29712	0.104522
$484$	−37.8936	−1.72244
$485$	11.5587	0.524853
$486$	31.9229	1.44805
$487$	−13.7825	−0.624546	−0.312273	−	0.949992i	$-0.601090\pi$
$487$	−13.7825	−0.624546	−0.312273	+	0.949992i	$0.601090\pi$
$488$	12.0359	0.544839
$489$	−25.7757	−1.16562
$490$	−13.6672	−0.617422
$491$	−24.7229	−1.11573	−0.557866	−	0.829931i	$-0.688380\pi$
$491$	−24.7229	−1.11573	−0.557866	+	0.829931i	$0.688380\pi$
$492$	59.9850	2.70433
$493$	9.04111	0.407191
$494$	54.9939	2.47429
$495$	−2.34172	−0.105252
$496$	10.6153	0.476642
$497$	3.49738	0.156879
$498$	39.9479	1.79011
$499$	30.2450	1.35395	0.676977	−	0.736005i	$-0.263290\pi$
$499$	30.2450	1.35395	0.676977	+	0.736005i	$0.263290\pi$
$500$	−5.23619	−0.234170
$501$	−0.0467616	−0.00208916
$502$	−17.0637	−0.761592
$503$	14.4914	0.646138	0.323069	−	0.946375i	$-0.395285\pi$
$503$	14.4914	0.646138	0.323069	+	0.946375i	$0.395285\pi$
$504$	14.5586	0.648490
$505$	−9.86474	−0.438975
$506$	4.21842	0.187532
$507$	11.6299	0.516504
$508$	−79.5094	−3.52766
$509$	−14.3250	−0.634944	−0.317472	−	0.948268i	$-0.602834\pi$
$509$	−14.3250	−0.634944	−0.317472	+	0.948268i	$0.602834\pi$
$510$	−22.3229	−0.988475
$511$	−5.97681	−0.264398
$512$	−0.0193966	−0.000857216 0
$513$	17.3991	0.768188
$514$	51.0223	2.25050
$515$	−16.2715	−0.717008
$516$	130.613	5.74991
$517$	13.4119	0.589857
$518$	13.4751	0.592064
$519$	−33.4764	−1.46945
$520$	37.6151	1.64953
$521$	−28.4226	−1.24522	−0.622609	−	0.782533i	$-0.713927\pi$
$521$	−28.4226	−1.24522	−0.622609	+	0.782533i	$0.713927\pi$
$522$	7.25662	0.317614
$523$	−6.98975	−0.305640	−0.152820	−	0.988254i	$-0.548836\pi$
$523$	−6.98975	−0.305640	−0.152820	+	0.988254i	$0.548836\pi$
$524$	23.7337	1.03681
$525$	−2.84160	−0.124018
$526$	−39.9390	−1.74142
$527$	−3.31758	−0.144516
$528$	51.5088	2.24163
$529$	−22.3465	−0.971587
$530$	−33.0744	−1.43666
$531$	−8.76885	−0.380536
$532$	34.3220	1.48805
$533$	24.1327	1.04530
$534$	34.5673	1.49587
$535$	−19.6177	−0.848149
$536$	−124.681	−5.38539
$537$	−20.7803	−0.896737
$538$	−13.5712	−0.585094
$539$	−9.85598	−0.424527
$540$	19.2555	0.828624
$541$	−10.6041	−0.455905	−0.227952	−	0.973672i	$-0.573203\pi$
$541$	−10.6041	−0.455905	−0.227952	+	0.973672i	$0.573203\pi$
$542$	−40.6071	−1.74423
$543$	−19.6571	−0.843565
$544$	70.4464	3.02036
$545$	6.65940	0.285257
$546$	33.0286	1.41350
$547$	−12.9955	−0.555646	−0.277823	−	0.960632i	$-0.589613\pi$
$547$	−12.9955	−0.555646	−0.277823	+	0.960632i	$0.589613\pi$
$548$	−39.0600	−1.66856
$549$	−1.66897	−0.0712299
$550$	−5.21831	−0.222510
$551$	10.5732	0.450434
$552$	14.4345	0.614375
$553$	13.4295	0.571082
$554$	−8.81545	−0.374533
$555$	−7.41655	−0.314815
$556$	56.3928	2.39159
$557$	38.5264	1.63242	0.816208	−	0.577758i	$-0.196072\pi$
$557$	38.5264	1.63242	0.816208	+	0.577758i	$0.196072\pi$
$558$	−2.66277	−0.112724
$559$	52.5471	2.22251
$560$	17.9342	0.757860
$561$	−16.0980	−0.679656
$562$	−56.3452	−2.37678
$563$	−33.0366	−1.39233	−0.696164	−	0.717883i	$-0.745111\pi$
$563$	−33.0366	−1.39233	−0.696164	+	0.717883i	$0.745111\pi$
$564$	74.2551	3.12670
$565$	3.63673	0.152998
$566$	19.3776	0.814503
$567$	15.4667	0.649542
$568$	21.9767	0.922123
$569$	19.0304	0.797796	0.398898	−	0.916995i	$-0.369393\pi$
$569$	19.0304	0.797796	0.398898	+	0.916995i	$0.369393\pi$
$570$	−26.1058	−1.09345
$571$	−10.8712	−0.454945	−0.227472	−	0.973785i	$-0.573046\pi$
$571$	−10.8712	−0.454945	−0.227472	+	0.973785i	$0.573046\pi$
$572$	43.8897	1.83512
$573$	−40.3425	−1.68533
$574$	20.8141	0.868764
$575$	−0.808388	−0.0337121
$576$	25.2882	1.05368
$577$	17.3881	0.723877	0.361938	−	0.932202i	$-0.382115\pi$
$577$	17.3881	0.723877	0.361938	+	0.932202i	$0.382115\pi$
$578$	1.69924	0.0706791
$579$	−12.4341	−0.516744
$580$	11.7013	0.485872
$581$	10.0303	0.416128
$582$	63.7760	2.64360
$583$	−23.8513	−0.987818
$584$	−37.5568	−1.55411
$585$	−5.21593	−0.215652
$586$	−74.6460	−3.08360
$587$	21.0663	0.869501	0.434750	−	0.900551i	$-0.356837\pi$
$587$	21.0663	0.869501	0.434750	+	0.900551i	$0.356837\pi$
$588$	−54.5675	−2.25033
$589$	−3.87978	−0.159864
$590$	−19.5406	−0.804475
$591$	−7.53654	−0.310012
$592$	46.8082	1.92380
$593$	35.0428	1.43904	0.719518	−	0.694473i	$-0.244363\pi$
$593$	35.0428	1.43904	0.719518	+	0.694473i	$0.244363\pi$
$594$	19.1897	0.787364
$595$	−5.60495	−0.229781
$596$	−61.2028	−2.50697
$597$	−16.9182	−0.692414
$598$	9.39609	0.384235
$599$	−35.8561	−1.46504	−0.732520	−	0.680746i	$-0.761656\pi$
$599$	−35.8561	−1.46504	−0.732520	+	0.680746i	$0.761656\pi$
$600$	−17.8560	−0.728966
$601$	−7.41619	−0.302513	−0.151256	−	0.988495i	$-0.548332\pi$
$601$	−7.41619	−0.302513	−0.151256	+	0.988495i	$0.548332\pi$
$602$	45.3212	1.84715
$603$	17.2890	0.704063
$604$	40.4058	1.64409
$605$	7.23686	0.294220
$606$	−54.4295	−2.21105
$607$	−3.82881	−0.155407	−0.0777034	−	0.996977i	$-0.524759\pi$
$607$	−3.82881	−0.155407	−0.0777034	+	0.996977i	$0.524759\pi$
$608$	82.3843	3.34112
$609$	6.35014	0.257321
$610$	−3.71915	−0.150584
$611$	29.8737	1.20856
$612$	−25.5727	−1.03372
$613$	−40.0343	−1.61697	−0.808485	−	0.588517i	$-0.799712\pi$
$613$	−40.0343	−1.61697	−0.808485	+	0.588517i	$0.799712\pi$
$614$	−27.7195	−1.11867
$615$	−11.4558	−0.461944
$616$	23.3956	0.942636
$617$	−13.6021	−0.547599	−0.273799	−	0.961787i	$-0.588280\pi$
$617$	−13.6021	−0.547599	−0.273799	+	0.961787i	$0.588280\pi$
$618$	−89.7793	−3.61145
$619$	17.5173	0.704082	0.352041	−	0.935985i	$-0.385488\pi$
$619$	17.5173	0.704082	0.352041	+	0.935985i	$0.385488\pi$
$620$	−4.29374	−0.172441
$621$	2.97275	0.119292
$622$	41.2737	1.65492
$623$	8.67934	0.347730
$624$	114.731	4.59290
$625$	1.00000	0.0400000
$626$	−27.4120	−1.09560
$627$	−18.8259	−0.751835
$628$	−49.8187	−1.98798
$629$	−14.6289	−0.583291
$630$	−4.49867	−0.179231
$631$	10.2056	0.406278	0.203139	−	0.979150i	$-0.434886\pi$
$631$	10.2056	0.406278	0.203139	+	0.979150i	$0.434886\pi$
$632$	84.3880	3.35677
$633$	−23.0572	−0.916441
$634$	36.1109	1.43415
$635$	15.1846	0.602582
$636$	−132.052	−5.23621
$637$	−21.9532	−0.869816
$638$	11.6614	0.461678
$639$	−3.04743	−0.120554
$640$	21.5280	0.850968
$641$	33.8750	1.33798	0.668990	−	0.743271i	$-0.266727\pi$
$641$	33.8750	1.33798	0.668990	+	0.743271i	$0.266727\pi$
$642$	−108.243	−4.27199
$643$	−40.9036	−1.61308	−0.806539	−	0.591180i	$-0.798662\pi$
$643$	−40.9036	−1.61308	−0.806539	+	0.591180i	$0.798662\pi$
$644$	5.86415	0.231080
$645$	−24.9442	−0.982178
$646$	−51.4926	−2.02595
$647$	7.99359	0.314261	0.157130	−	0.987578i	$-0.449776\pi$
$647$	7.99359	0.314261	0.157130	+	0.987578i	$0.449776\pi$
$648$	97.1893	3.81796
$649$	−14.0915	−0.553141
$650$	−11.6232	−0.455901
$651$	−2.33015	−0.0913256
$652$	−65.8011	−2.57697
$653$	−18.6089	−0.728224	−0.364112	−	0.931355i	$-0.618628\pi$
$653$	−18.6089	−0.728224	−0.364112	+	0.931355i	$0.618628\pi$
$654$	36.7438	1.43680
$655$	−4.53262	−0.177104
$656$	72.3013	2.82289
$657$	5.20785	0.203178
$658$	25.7657	1.00445
$659$	−18.3532	−0.714941	−0.357470	−	0.933924i	$-0.616361\pi$
$659$	−18.3532	−0.714941	−0.357470	+	0.933924i	$0.616361\pi$
$660$	−20.8346	−0.810984
$661$	8.22147	0.319778	0.159889	−	0.987135i	$-0.448886\pi$
$661$	8.22147	0.319778	0.159889	+	0.987135i	$0.448886\pi$
$662$	37.3768	1.45269
$663$	−35.8565	−1.39255
$664$	63.0282	2.44597
$665$	−6.55477	−0.254183
$666$	−11.7415	−0.454973
$667$	1.80651	0.0699482
$668$	−0.119375	−0.00461874
$669$	42.6292	1.64814
$670$	38.5270	1.48843
$671$	−2.68203	−0.103539
$672$	49.4789	1.90869
$673$	45.9897	1.77277	0.886387	−	0.462945i	$-0.153207\pi$
$673$	45.9897	1.77277	0.886387	+	0.462945i	$0.153207\pi$
$674$	75.5653	2.91067
$675$	−3.67738	−0.141543
$676$	29.6893	1.14190
$677$	−7.63041	−0.293261	−0.146630	−	0.989191i	$-0.546843\pi$
$677$	−7.63041	−0.293261	−0.146630	+	0.989191i	$0.546843\pi$
$678$	20.0660	0.770628
$679$	16.0132	0.614530
$680$	−35.2202	−1.35063
$681$	−16.9602	−0.649915
$682$	−4.27907	−0.163854
$683$	−13.7534	−0.526260	−0.263130	−	0.964760i	$-0.584755\pi$
$683$	−13.7534	−0.526260	−0.263130	+	0.964760i	$0.584755\pi$
$684$	−29.9063	−1.14350
$685$	7.45963	0.285018
$686$	−45.0212	−1.71892
$687$	16.8091	0.641308
$688$	157.431	6.00199
$689$	−53.1262	−2.02395
$690$	−4.46035	−0.169803
$691$	49.5079	1.88337	0.941685	−	0.336497i	$-0.109242\pi$
$691$	49.5079	1.88337	0.941685	+	0.336497i	$0.109242\pi$
$692$	−85.4594	−3.24868
$693$	−3.24418	−0.123236
$694$	41.4107	1.57193
$695$	−10.7698	−0.408522
$696$	39.9028	1.51251
$697$	−22.5962	−0.855891
$698$	−81.6562	−3.09073
$699$	44.8384	1.69594
$700$	−7.25413	−0.274180
$701$	−42.3425	−1.59925	−0.799627	−	0.600497i	$-0.794970\pi$
$701$	−42.3425	−1.59925	−0.799627	+	0.600497i	$0.794970\pi$
$702$	42.7431	1.61324
$703$	−17.1079	−0.645236
$704$	40.6382	1.53161
$705$	−14.1811	−0.534092
$706$	64.8243	2.43969
$707$	−13.6664	−0.513979
$708$	−78.0176	−2.93208
$709$	−0.135807	−0.00510033	−0.00255017	−	0.999997i	$-0.500812\pi$
$709$	−0.135807	−0.00510033	−0.00255017	+	0.999997i	$0.500812\pi$
$710$	−6.79092	−0.254859
$711$	−11.7017	−0.438850
$712$	54.5389	2.04393
$713$	−0.662887	−0.0248253
$714$	−30.9258	−1.15737
$715$	−8.38199	−0.313469
$716$	−53.0486	−1.98252
$717$	−41.4587	−1.54830
$718$	97.7665	3.64861
$719$	−32.2138	−1.20137	−0.600686	−	0.799485i	$-0.705106\pi$
$719$	−32.2138	−1.20137	−0.600686	+	0.799485i	$0.705106\pi$
$720$	−15.6269	−0.582380
$721$	−22.5423	−0.839517
$722$	−9.10819	−0.338972
$723$	−18.4723	−0.686992
$724$	−50.1811	−1.86497
$725$	−2.23470	−0.0829948
$726$	39.9300	1.48194
$727$	−14.0276	−0.520253	−0.260127	−	0.965575i	$-0.583764\pi$
$727$	−14.0276	−0.520253	−0.260127	+	0.965575i	$0.583764\pi$
$728$	52.1112	1.93137
$729$	9.15155	0.338946
$730$	11.6052	0.429529
$731$	−49.2015	−1.81978
$732$	−14.8490	−0.548836
$733$	26.8081	0.990179	0.495089	−	0.868842i	$-0.335135\pi$
$733$	26.8081	0.990179	0.495089	+	0.868842i	$0.335135\pi$
$734$	49.4766	1.82622
$735$	10.4212	0.384393
$736$	14.0759	0.518845
$737$	27.7834	1.02342
$738$	−18.1363	−0.667605
$739$	13.2063	0.485801	0.242901	−	0.970051i	$-0.421901\pi$
$739$	13.2063	0.485801	0.242901	+	0.970051i	$0.421901\pi$
$740$	−18.9332	−0.695998
$741$	−41.9328	−1.54044
$742$	−45.8206	−1.68213
$743$	17.8277	0.654034	0.327017	−	0.945019i	$-0.393957\pi$
$743$	17.8277	0.654034	0.327017	+	0.945019i	$0.393957\pi$
$744$	−14.6421	−0.536805
$745$	11.6884	0.428231
$746$	38.1847	1.39804
$747$	−8.73987	−0.319775
$748$	−41.0953	−1.50259
$749$	−27.1781	−0.993066
$750$	5.51758	0.201474
$751$	−47.5220	−1.73410	−0.867051	−	0.498220i	$-0.833987\pi$
$751$	−47.5220	−1.73410	−0.867051	+	0.498220i	$0.833987\pi$
$752$	89.5014	3.26378
$753$	13.0111	0.474149
$754$	25.9745	0.945935
$755$	−7.71664	−0.280837
$756$	26.6762	0.970205
$757$	51.7824	1.88206	0.941032	−	0.338317i	$-0.109858\pi$
$757$	51.7824	1.88206	0.941032	+	0.338317i	$0.109858\pi$
$758$	−44.4866	−1.61583
$759$	−3.21654	−0.116753
$760$	−41.1886	−1.49407
$761$	−38.8146	−1.40703	−0.703514	−	0.710681i	$-0.748387\pi$
$761$	−38.8146	−1.40703	−0.703514	+	0.710681i	$0.748387\pi$
$762$	83.7822	3.03511
$763$	9.22582	0.333997
$764$	−102.988	−3.72596
$765$	4.88384	0.176576
$766$	62.2554	2.24938
$767$	−31.3874	−1.13333
$768$	32.8449	1.18519
$769$	−8.51187	−0.306946	−0.153473	−	0.988153i	$-0.549046\pi$
$769$	−8.51187	−0.306946	−0.153473	+	0.988153i	$0.549046\pi$
$770$	−7.22936	−0.260528
$771$	−38.9044	−1.40111
$772$	−31.7422	−1.14243
$773$	−41.2723	−1.48446	−0.742230	−	0.670145i	$-0.766232\pi$
$773$	−41.2723	−1.48446	−0.742230	+	0.670145i	$0.766232\pi$
$774$	−39.4904	−1.41945
$775$	0.820011	0.0294557
$776$	100.623	3.61216
$777$	−10.2748	−0.368605
$778$	39.6177	1.42036
$779$	−26.4253	−0.946786
$780$	−46.4068	−1.66163
$781$	−4.89721	−0.175236
$782$	−8.79786	−0.314611
$783$	8.21786	0.293682
$784$	−65.7715	−2.34898
$785$	9.51429	0.339580
$786$	−25.0091	−0.892045
$787$	−22.3318	−0.796043	−0.398021	−	0.917376i	$-0.630303\pi$
$787$	−22.3318	−0.796043	−0.398021	+	0.917376i	$0.630303\pi$
$788$	−19.2395	−0.685379
$789$	30.4534	1.08417
$790$	−26.0763	−0.927754
$791$	5.03826	0.179140
$792$	−20.3856	−0.724372
$793$	−5.97394	−0.212141
$794$	33.2421	1.17972
$795$	25.2191	0.894430
$796$	−43.1892	−1.53080
$797$	43.9194	1.55571	0.777853	−	0.628446i	$-0.216309\pi$
$797$	43.9194	1.55571	0.777853	+	0.628446i	$0.216309\pi$
$798$	−36.1665	−1.28028
$799$	−27.9717	−0.989567
$800$	−17.4123	−0.615619
$801$	−7.56269	−0.267215
$802$	47.6531	1.68269
$803$	8.36902	0.295336
$804$	153.823	5.42490
$805$	−1.11993	−0.0394722
$806$	−9.53119	−0.335722
$807$	10.3480	0.364266
$808$	−85.8766	−3.02113
$809$	−23.2165	−0.816250	−0.408125	−	0.912926i	$-0.633817\pi$
$809$	−23.2165	−0.816250	−0.408125	+	0.912926i	$0.633817\pi$
$810$	−30.0320	−1.05522
$811$	35.9361	1.26189	0.630944	−	0.775829i	$-0.282668\pi$
$811$	35.9361	1.26189	0.630944	+	0.775829i	$0.282668\pi$
$812$	16.2108	0.568888
$813$	30.9629	1.08592
$814$	−18.8686	−0.661342
$815$	12.5666	0.440188
$816$	−107.426	−3.76066
$817$	−57.5392	−2.01304
$818$	77.7355	2.71796
$819$	−7.22606	−0.252499
$820$	−29.2448	−1.02127
$821$	29.5182	1.03019	0.515096	−	0.857133i	$-0.327756\pi$
$821$	29.5182	1.03019	0.515096	+	0.857133i	$0.327756\pi$
$822$	41.1591	1.43559
$823$	13.5509	0.472354	0.236177	−	0.971710i	$-0.424106\pi$
$823$	13.5509	0.472354	0.236177	+	0.971710i	$0.424106\pi$
$824$	−141.650	−4.93462
$825$	3.97895	0.138529
$826$	−27.0712	−0.941929
$827$	−11.5228	−0.400687	−0.200343	−	0.979726i	$-0.564206\pi$
$827$	−11.5228	−0.400687	−0.200343	+	0.979726i	$0.564206\pi$
$828$	−5.10969	−0.177574
$829$	−17.3718	−0.603346	−0.301673	−	0.953411i	$-0.597545\pi$
$829$	−17.3718	−0.603346	−0.301673	+	0.953411i	$0.597545\pi$
$830$	−19.4760	−0.676023
$831$	6.72177	0.233175
$832$	90.5173	3.13812
$833$	20.5554	0.712204
$834$	−59.4233	−2.05766
$835$	0.0227980	0.000788956 0
$836$	−48.0594	−1.66217
$837$	−3.01549	−0.104231
$838$	69.8780	2.41390
$839$	−38.7242	−1.33691	−0.668454	−	0.743754i	$-0.733043\pi$
$839$	−38.7242	−1.33691	−0.668454	+	0.743754i	$0.733043\pi$
$840$	−24.7373	−0.853519
$841$	−24.0061	−0.827797
$842$	3.44324	0.118662
$843$	42.9631	1.47973
$844$	−58.8611	−2.02608
$845$	−5.67001	−0.195054
$846$	−22.4508	−0.771873
$847$	10.0258	0.344491
$848$	−159.166	−5.46577
$849$	−14.7754	−0.507091
$850$	10.8832	0.373291
$851$	−2.92300	−0.100199
$852$	−27.1133	−0.928888
$853$	37.1221	1.27104	0.635518	−	0.772086i	$-0.280786\pi$
$853$	37.1221	1.27104	0.635518	+	0.772086i	$0.280786\pi$
$854$	−5.15245	−0.176313
$855$	5.71146	0.195328
$856$	−170.781	−5.83716
$857$	19.9468	0.681369	0.340684	−	0.940178i	$-0.389341\pi$
$857$	19.9468	0.681369	0.340684	+	0.940178i	$0.389341\pi$
$858$	−46.2483	−1.57889
$859$	16.5871	0.565945	0.282973	−	0.959128i	$-0.408679\pi$
$859$	16.5871	0.565945	0.282973	+	0.959128i	$0.408679\pi$
$860$	−63.6784	−2.17142
$861$	−15.8707	−0.540873
$862$	37.1213	1.26435
$863$	39.3992	1.34117	0.670583	−	0.741835i	$-0.266044\pi$
$863$	39.3992	1.34117	0.670583	+	0.741835i	$0.266044\pi$
$864$	64.0318	2.17841
$865$	16.3209	0.554928
$866$	−68.0280	−2.31169
$867$	−1.29567	−0.0440032
$868$	−5.94847	−0.201904
$869$	−18.8047	−0.637906
$870$	−12.3302	−0.418032
$871$	61.8846	2.09688
$872$	57.9728	1.96321
$873$	−13.9530	−0.472238
$874$	−10.2887	−0.348022
$875$	1.38538	0.0468345
$876$	46.3350	1.56551
$877$	−51.7055	−1.74597	−0.872985	−	0.487747i	$-0.837819\pi$
$877$	−51.7055	−1.74597	−0.872985	+	0.487747i	$0.837819\pi$
$878$	−88.3845	−2.98283
$879$	56.9174	1.91978
$880$	−25.1124	−0.846539
$881$	52.2590	1.76065	0.880325	−	0.474371i	$-0.157324\pi$
$881$	52.2590	1.76065	0.880325	+	0.474371i	$0.157324\pi$
$882$	16.4983	0.555527
$883$	−47.0199	−1.58235	−0.791173	−	0.611593i	$-0.790529\pi$
$883$	−47.0199	−1.58235	−0.791173	+	0.611593i	$0.790529\pi$
$884$	−91.5356	−3.07867
$885$	14.8997	0.500847
$886$	4.11296	0.138178
$887$	−6.17592	−0.207367	−0.103684	−	0.994610i	$-0.533063\pi$
$887$	−6.17592	−0.207367	−0.103684	+	0.994610i	$0.533063\pi$
$888$	−64.5642	−2.16663
$889$	21.0365	0.705540
$890$	−16.8528	−0.564907
$891$	−21.6573	−0.725546
$892$	108.825	3.64373
$893$	−32.7118	−1.09466
$894$	64.4918	2.15693
$895$	10.1311	0.338647
$896$	29.8245	0.996366
$897$	−7.16450	−0.239216
$898$	−27.5854	−0.920536
$899$	−1.83248	−0.0611167
$900$	6.32084	0.210695
$901$	49.7437	1.65720
$902$	−29.1449	−0.970420
$903$	−34.5573	−1.15000
$904$	31.6592	1.05297
$905$	9.58351	0.318567
$906$	−42.5772	−1.41453
$907$	−20.4371	−0.678601	−0.339301	−	0.940678i	$-0.610190\pi$
$907$	−20.4371	−0.678601	−0.339301	+	0.940678i	$0.610190\pi$
$908$	−43.2964	−1.43684
$909$	11.9082	0.394969
$910$	−16.1026	−0.533797
$911$	−25.5361	−0.846048	−0.423024	−	0.906119i	$-0.639031\pi$
$911$	−25.5361	−0.846048	−0.423024	+	0.906119i	$0.639031\pi$
$912$	−125.630	−4.16003
$913$	−14.0450	−0.464821
$914$	36.1347	1.19523
$915$	2.83585	0.0937502
$916$	42.9108	1.41781
$917$	−6.27941	−0.207364
$918$	−40.0217	−1.32091
$919$	6.96133	0.229633	0.114817	−	0.993387i	$-0.463372\pi$
$919$	6.96133	0.229633	0.114817	+	0.993387i	$0.463372\pi$
$920$	−7.03735	−0.232015
$921$	21.1361	0.696457
$922$	−21.2860	−0.701017
$923$	−10.9080	−0.359042
$924$	−28.8638	−0.949551
$925$	3.61583	0.118888
$926$	−102.464	−3.36716
$927$	19.6421	0.645130
$928$	38.9114	1.27733
$929$	20.0553	0.657994	0.328997	−	0.944331i	$-0.393289\pi$
$929$	20.0553	0.657994	0.328997	+	0.944331i	$0.393289\pi$
$930$	4.52448	0.148363
$931$	24.0388	0.787839
$932$	114.465	3.74942
$933$	−31.4711	−1.03032
$934$	−5.89712	−0.192960
$935$	7.84833	0.256668
$936$	−45.4068	−1.48417
$937$	8.45554	0.276230	0.138115	−	0.990416i	$-0.455896\pi$
$937$	8.45554	0.276230	0.138115	+	0.990416i	$0.455896\pi$
$938$	53.3747	1.74275
$939$	20.9016	0.682097
$940$	−36.2020	−1.18078
$941$	31.6188	1.03074	0.515372	−	0.856966i	$-0.327654\pi$
$941$	31.6188	1.03074	0.515372	+	0.856966i	$0.327654\pi$
$942$	52.4959	1.71041
$943$	−4.51495	−0.147027
$944$	−94.0365	−3.06063
$945$	−5.09458	−0.165727
$946$	−63.4610	−2.06329
$947$	37.7937	1.22813	0.614065	−	0.789256i	$-0.289533\pi$
$947$	37.7937	1.22813	0.614065	+	0.789256i	$0.289533\pi$
$948$	−104.112	−3.38140
$949$	18.6411	0.605116
$950$	12.7275	0.412934
$951$	−27.5345	−0.892868
$952$	−48.7934	−1.58140
$953$	21.5020	0.696518	0.348259	−	0.937398i	$-0.386773\pi$
$953$	21.5020	0.696518	0.348259	+	0.937398i	$0.386773\pi$
$954$	39.9255	1.29264
$955$	19.6684	0.636455
$956$	−105.837	−3.42301
$957$	−8.89178	−0.287430
$958$	−3.82453	−0.123565
$959$	10.3344	0.333716
$960$	−42.9688	−1.38681
$961$	−30.3276	−0.978309
$962$	−42.0277	−1.35503
$963$	23.6815	0.763125
$964$	−47.1566	−1.51881
$965$	6.06207	0.195145
$966$	−6.17929	−0.198815
$967$	9.19873	0.295811	0.147906	−	0.989001i	$-0.452747\pi$
$967$	9.19873	0.295811	0.147906	+	0.989001i	$0.452747\pi$
$968$	62.9999	2.02489
$969$	39.2630	1.26131
$970$	−31.0930	−0.998338
$971$	45.4270	1.45782	0.728912	−	0.684608i	$-0.240027\pi$
$971$	45.4270	1.45782	0.728912	+	0.684608i	$0.240027\pi$
$972$	−62.1388	−1.99310
$973$	−14.9203	−0.478323
$974$	37.0753	1.18797
$975$	8.86270	0.283834
$976$	−17.8979	−0.572898
$977$	−12.5457	−0.401374	−0.200687	−	0.979655i	$-0.564317\pi$
$977$	−12.5457	−0.401374	−0.200687	+	0.979655i	$0.564317\pi$
$978$	69.3372	2.21716
$979$	−12.1532	−0.388419
$980$	26.6036	0.849821
$981$	−8.03886	−0.256661
$982$	66.5052	2.12226
$983$	36.1055	1.15159	0.575794	−	0.817595i	$-0.304693\pi$
$983$	36.1055	1.15159	0.575794	+	0.817595i	$0.304693\pi$
$984$	−99.7278	−3.17921
$985$	3.67433	0.117074
$986$	−24.3207	−0.774530
$987$	−19.6463	−0.625348
$988$	−107.047	−3.40563
$989$	−9.83097	−0.312607
$990$	6.29926	0.200204
$991$	−61.7828	−1.96259	−0.981297	−	0.192499i	$-0.938341\pi$
$991$	−61.7828	−1.96259	−0.981297	+	0.192499i	$0.938341\pi$
$992$	−14.2783	−0.453336
$993$	−28.4997	−0.904411
$994$	−9.40802	−0.298404
$995$	8.24821	0.261486
$996$	−77.7598	−2.46391
$997$	−25.8134	−0.817519	−0.408759	−	0.912642i	$-0.634038\pi$
$997$	−25.8134	−0.817519	−0.408759	+	0.912642i	$0.634038\pi$
$998$	−81.3596	−2.57539
$999$	−13.2968	−0.420692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8005.2.a.f.1.4	✓	127

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8005.2.a.f.1.4	✓	127	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:	\( N \)	\(=\)	\( 8005 = 5 \cdot 1601 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8005.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69002 q^{2} +2.05113 q^{3} +5.23619 q^{4} -1.00000 q^{5} -5.51758 q^{6} -1.38538 q^{7} -8.70541 q^{8} +1.20715 q^{9} +O(q^{10})\)	\(q-2.69002 q^{2} +2.05113 q^{3} +5.23619 q^{4} -1.00000 q^{5} -5.51758 q^{6} -1.38538 q^{7} -8.70541 q^{8} +1.20715 q^{9} +2.69002 q^{10} +1.93988 q^{11} +10.7401 q^{12} +4.32088 q^{13} +3.72670 q^{14} -2.05113 q^{15} +12.9453 q^{16} -4.04578 q^{17} -3.24724 q^{18} -4.73138 q^{19} -5.23619 q^{20} -2.84160 q^{21} -5.21831 q^{22} -0.808388 q^{23} -17.8560 q^{24} +1.00000 q^{25} -11.6232 q^{26} -3.67738 q^{27} -7.25413 q^{28} -2.23470 q^{29} +5.51758 q^{30} +0.820011 q^{31} -17.4123 q^{32} +3.97895 q^{33} +10.8832 q^{34} +1.38538 q^{35} +6.32084 q^{36} +3.61583 q^{37} +12.7275 q^{38} +8.86270 q^{39} +8.70541 q^{40} +5.58513 q^{41} +7.64396 q^{42} +12.1612 q^{43} +10.1576 q^{44} -1.20715 q^{45} +2.17458 q^{46} +6.91380 q^{47} +26.5526 q^{48} -5.08072 q^{49} -2.69002 q^{50} -8.29843 q^{51} +22.6250 q^{52} -12.2952 q^{53} +9.89222 q^{54} -1.93988 q^{55} +12.0603 q^{56} -9.70468 q^{57} +6.01139 q^{58} -7.26412 q^{59} -10.7401 q^{60} -1.38258 q^{61} -2.20584 q^{62} -1.67236 q^{63} +20.9488 q^{64} -4.32088 q^{65} -10.7035 q^{66} +14.3222 q^{67} -21.1845 q^{68} -1.65811 q^{69} -3.72670 q^{70} -2.52449 q^{71} -10.5087 q^{72} +4.31419 q^{73} -9.72666 q^{74} +2.05113 q^{75} -24.7744 q^{76} -2.68748 q^{77} -23.8408 q^{78} -9.69374 q^{79} -12.9453 q^{80} -11.1642 q^{81} -15.0241 q^{82} -7.24012 q^{83} -14.8792 q^{84} +4.04578 q^{85} -32.7138 q^{86} -4.58367 q^{87} -16.8875 q^{88} -6.26494 q^{89} +3.24724 q^{90} -5.98607 q^{91} -4.23287 q^{92} +1.68195 q^{93} -18.5982 q^{94} +4.73138 q^{95} -35.7150 q^{96} -11.5587 q^{97} +13.6672 q^{98} +2.34172 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 127 q - 6 q^{2} - 18 q^{3} + 114 q^{4} - 127 q^{5} - 20 q^{6} + 28 q^{7} - 18 q^{8} + 101 q^{9}+O(q^{10}) \) 127 * q - 6 * q^2 - 18 * q^3 + 114 * q^4 - 127 * q^5 - 20 * q^6 + 28 * q^7 - 18 * q^8 + 101 * q^9	\( 127 q - 6 q^{2} - 18 q^{3} + 114 q^{4} - 127 q^{5} - 20 q^{6} + 28 q^{7} - 18 q^{8} + 101 q^{9} + 6 q^{10} - 45 q^{11} - 30 q^{12} - 53 q^{14} + 18 q^{15} + 84 q^{16} - 36 q^{17} - 10 q^{18} - 49 q^{19} - 114 q^{20} - 48 q^{21} + 13 q^{22} - 29 q^{23} - 63 q^{24} + 127 q^{25} - 55 q^{26} - 75 q^{27} + 44 q^{28} - 45 q^{29} + 20 q^{30} - 49 q^{31} - 32 q^{32} - 8 q^{33} - 52 q^{34} - 28 q^{35} + 44 q^{36} + 36 q^{37} - 65 q^{38} - 52 q^{39} + 18 q^{40} - 66 q^{41} - 18 q^{42} - 5 q^{43} - 93 q^{44} - 101 q^{45} - 25 q^{46} - 32 q^{47} - 54 q^{48} + 77 q^{49} - 6 q^{50} - 102 q^{51} - 13 q^{52} - 67 q^{53} - 53 q^{54} + 45 q^{55} - 158 q^{56} + 16 q^{57} + 35 q^{58} - 213 q^{59} + 30 q^{60} - 62 q^{61} - 33 q^{62} + 59 q^{63} + 34 q^{64} - 60 q^{66} + 10 q^{67} - 94 q^{68} - 93 q^{69} + 53 q^{70} - 118 q^{71} - 24 q^{72} + 35 q^{73} - 107 q^{74} - 18 q^{75} - 98 q^{76} - 93 q^{77} + 21 q^{78} - 64 q^{79} - 84 q^{80} + 15 q^{81} + 15 q^{82} - 187 q^{83} - 118 q^{84} + 36 q^{85} - 126 q^{86} - 53 q^{87} + 15 q^{88} - 138 q^{89} + 10 q^{90} - 138 q^{91} - 86 q^{92} + 23 q^{93} - 60 q^{94} + 49 q^{95} - 92 q^{96} + 9 q^{97} - 67 q^{98} - 147 q^{99}+O(q^{100}) \) 127 * q - 6 * q^2 - 18 * q^3 + 114 * q^4 - 127 * q^5 - 20 * q^6 + 28 * q^7 - 18 * q^8 + 101 * q^9 + 6 * q^10 - 45 * q^11 - 30 * q^12 - 53 * q^14 + 18 * q^15 + 84 * q^16 - 36 * q^17 - 10 * q^18 - 49 * q^19 - 114 * q^20 - 48 * q^21 + 13 * q^22 - 29 * q^23 - 63 * q^24 + 127 * q^25 - 55 * q^26 - 75 * q^27 + 44 * q^28 - 45 * q^29 + 20 * q^30 - 49 * q^31 - 32 * q^32 - 8 * q^33 - 52 * q^34 - 28 * q^35 + 44 * q^36 + 36 * q^37 - 65 * q^38 - 52 * q^39 + 18 * q^40 - 66 * q^41 - 18 * q^42 - 5 * q^43 - 93 * q^44 - 101 * q^45 - 25 * q^46 - 32 * q^47 - 54 * q^48 + 77 * q^49 - 6 * q^50 - 102 * q^51 - 13 * q^52 - 67 * q^53 - 53 * q^54 + 45 * q^55 - 158 * q^56 + 16 * q^57 + 35 * q^58 - 213 * q^59 + 30 * q^60 - 62 * q^61 - 33 * q^62 + 59 * q^63 + 34 * q^64 - 60 * q^66 + 10 * q^67 - 94 * q^68 - 93 * q^69 + 53 * q^70 - 118 * q^71 - 24 * q^72 + 35 * q^73 - 107 * q^74 - 18 * q^75 - 98 * q^76 - 93 * q^77 + 21 * q^78 - 64 * q^79 - 84 * q^80 + 15 * q^81 + 15 * q^82 - 187 * q^83 - 118 * q^84 + 36 * q^85 - 126 * q^86 - 53 * q^87 + 15 * q^88 - 138 * q^89 + 10 * q^90 - 138 * q^91 - 86 * q^92 + 23 * q^93 - 60 * q^94 + 49 * q^95 - 92 * q^96 + 9 * q^97 - 67 * q^98 - 147 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more