LMFDB - Embedded newform 8031.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8031,2,Mod(1,8031)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8031, 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("8031.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8031 = 3 \cdot 2677 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8031.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:	$64.1278578633$
Analytic rank:	$1$
Dimension:	$102$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8031.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.71515 q^{2} -1.00000 q^{3} +5.37202 q^{4} +0.694821 q^{5} +2.71515 q^{6} +4.70425 q^{7} -9.15552 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.71515 q^{2} -1.00000 q^{3} +5.37202 q^{4} +0.694821 q^{5} +2.71515 q^{6} +4.70425 q^{7} -9.15552 q^{8} +1.00000 q^{9} -1.88654 q^{10} -2.64356 q^{11} -5.37202 q^{12} +3.34489 q^{13} -12.7727 q^{14} -0.694821 q^{15} +14.1145 q^{16} +2.32220 q^{17} -2.71515 q^{18} -1.75052 q^{19} +3.73259 q^{20} -4.70425 q^{21} +7.17765 q^{22} -0.201020 q^{23} +9.15552 q^{24} -4.51722 q^{25} -9.08187 q^{26} -1.00000 q^{27} +25.2713 q^{28} +5.37029 q^{29} +1.88654 q^{30} -5.14363 q^{31} -20.0120 q^{32} +2.64356 q^{33} -6.30511 q^{34} +3.26861 q^{35} +5.37202 q^{36} -0.104230 q^{37} +4.75293 q^{38} -3.34489 q^{39} -6.36145 q^{40} -4.61915 q^{41} +12.7727 q^{42} -8.09619 q^{43} -14.2012 q^{44} +0.694821 q^{45} +0.545798 q^{46} -9.79368 q^{47} -14.1145 q^{48} +15.1300 q^{49} +12.2649 q^{50} -2.32220 q^{51} +17.9688 q^{52} -7.08677 q^{53} +2.71515 q^{54} -1.83680 q^{55} -43.0699 q^{56} +1.75052 q^{57} -14.5811 q^{58} -1.46332 q^{59} -3.73259 q^{60} +11.9421 q^{61} +13.9657 q^{62} +4.70425 q^{63} +26.1064 q^{64} +2.32410 q^{65} -7.17765 q^{66} +5.44194 q^{67} +12.4749 q^{68} +0.201020 q^{69} -8.87477 q^{70} +6.97388 q^{71} -9.15552 q^{72} -6.65620 q^{73} +0.283000 q^{74} +4.51722 q^{75} -9.40385 q^{76} -12.4360 q^{77} +9.08187 q^{78} +6.48188 q^{79} +9.80708 q^{80} +1.00000 q^{81} +12.5417 q^{82} -0.672215 q^{83} -25.2713 q^{84} +1.61351 q^{85} +21.9823 q^{86} -5.37029 q^{87} +24.2032 q^{88} -8.94870 q^{89} -1.88654 q^{90} +15.7352 q^{91} -1.07988 q^{92} +5.14363 q^{93} +26.5913 q^{94} -1.21630 q^{95} +20.0120 q^{96} -17.9487 q^{97} -41.0801 q^{98} -2.64356 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9}+O(q^{10}) $ 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9} - 16 q^{10} - 28 q^{11} - 96 q^{12} - 2 q^{13} - 41 q^{14} + 20 q^{15} + 88 q^{16} - 77 q^{17} - 6 q^{18} + 10 q^{19} - 50 q^{20} - 12 q^{21} + 24 q^{22} - 29 q^{23} + 21 q^{24} + 74 q^{25} - 45 q^{26} - 102 q^{27} + 19 q^{28} - 68 q^{29} + 16 q^{30} - 29 q^{31} - 48 q^{32} + 28 q^{33} - 19 q^{34} - 49 q^{35} + 96 q^{36} + 4 q^{37} - 44 q^{38} + 2 q^{39} - 41 q^{40} - 122 q^{41} + 41 q^{42} + 85 q^{43} - 86 q^{44} - 20 q^{45} - 28 q^{46} - 39 q^{47} - 88 q^{48} + 24 q^{49} - 37 q^{50} + 77 q^{51} + 8 q^{52} - 37 q^{53} + 6 q^{54} - 13 q^{55} - 130 q^{56} - 10 q^{57} + 17 q^{58} - 58 q^{59} + 50 q^{60} - 114 q^{61} - 64 q^{62} + 12 q^{63} + 47 q^{64} - 92 q^{65} - 24 q^{66} + 121 q^{67} - 138 q^{68} + 29 q^{69} - 2 q^{70} - 67 q^{71} - 21 q^{72} - 72 q^{73} - 111 q^{74} - 74 q^{75} - 17 q^{76} - 57 q^{77} + 45 q^{78} - 24 q^{79} - 97 q^{80} + 102 q^{81} - q^{82} - 78 q^{83} - 19 q^{84} - 24 q^{85} - 80 q^{86} + 68 q^{87} + 54 q^{88} - 176 q^{89} - 16 q^{90} - 3 q^{91} - 82 q^{92} + 29 q^{93} - 41 q^{94} - 90 q^{95} + 48 q^{96} - 77 q^{97} - 48 q^{98} - 28 q^{99}+O(q^{100}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9 - 16 * q^10 - 28 * q^11 - 96 * q^12 - 2 * q^13 - 41 * q^14 + 20 * q^15 + 88 * q^16 - 77 * q^17 - 6 * q^18 + 10 * q^19 - 50 * q^20 - 12 * q^21 + 24 * q^22 - 29 * q^23 + 21 * q^24 + 74 * q^25 - 45 * q^26 - 102 * q^27 + 19 * q^28 - 68 * q^29 + 16 * q^30 - 29 * q^31 - 48 * q^32 + 28 * q^33 - 19 * q^34 - 49 * q^35 + 96 * q^36 + 4 * q^37 - 44 * q^38 + 2 * q^39 - 41 * q^40 - 122 * q^41 + 41 * q^42 + 85 * q^43 - 86 * q^44 - 20 * q^45 - 28 * q^46 - 39 * q^47 - 88 * q^48 + 24 * q^49 - 37 * q^50 + 77 * q^51 + 8 * q^52 - 37 * q^53 + 6 * q^54 - 13 * q^55 - 130 * q^56 - 10 * q^57 + 17 * q^58 - 58 * q^59 + 50 * q^60 - 114 * q^61 - 64 * q^62 + 12 * q^63 + 47 * q^64 - 92 * q^65 - 24 * q^66 + 121 * q^67 - 138 * q^68 + 29 * q^69 - 2 * q^70 - 67 * q^71 - 21 * q^72 - 72 * q^73 - 111 * q^74 - 74 * q^75 - 17 * q^76 - 57 * q^77 + 45 * q^78 - 24 * q^79 - 97 * q^80 + 102 * q^81 - q^82 - 78 * q^83 - 19 * q^84 - 24 * q^85 - 80 * q^86 + 68 * q^87 + 54 * q^88 - 176 * q^89 - 16 * q^90 - 3 * q^91 - 82 * q^92 + 29 * q^93 - 41 * q^94 - 90 * q^95 + 48 * q^96 - 77 * q^97 - 48 * q^98 - 28 * 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 $

$2$	−2.71515	−1.91990	−0.959949	−	0.280175i	$-0.909608\pi$
$2$	−2.71515	−1.91990	−0.959949	+	0.280175i	$0.909608\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.37202	2.68601
$5$	0.694821	0.310734	0.155367	−	0.987857i	$-0.450344\pi$
$5$	0.694821	0.310734	0.155367	+	0.987857i	$0.450344\pi$
$6$	2.71515	1.10845
$7$	4.70425	1.77804	0.889020	−	0.457868i	$-0.151387\pi$
$7$	4.70425	1.77804	0.889020	+	0.457868i	$0.151387\pi$
$8$	−9.15552	−3.23697
$9$	1.00000	0.333333
$10$	−1.88654	−0.596577
$11$	−2.64356	−0.797063	−0.398532	−	0.917155i	$-0.630480\pi$
$11$	−2.64356	−0.797063	−0.398532	+	0.917155i	$0.630480\pi$
$12$	−5.37202	−1.55077
$13$	3.34489	0.927706	0.463853	−	0.885912i	$-0.346467\pi$
$13$	3.34489	0.927706	0.463853	+	0.885912i	$0.346467\pi$
$14$	−12.7727	−3.41366
$15$	−0.694821	−0.179402
$16$	14.1145	3.52863
$17$	2.32220	0.563216	0.281608	−	0.959530i	$-0.409132\pi$
$17$	2.32220	0.563216	0.281608	+	0.959530i	$0.409132\pi$
$18$	−2.71515	−0.639966
$19$	−1.75052	−0.401598	−0.200799	−	0.979632i	$-0.564354\pi$
$19$	−1.75052	−0.401598	−0.200799	+	0.979632i	$0.564354\pi$
$20$	3.73259	0.834633
$21$	−4.70425	−1.02655
$22$	7.17765	1.53028
$23$	−0.201020	−0.0419155	−0.0209578	−	0.999780i	$-0.506672\pi$
$23$	−0.201020	−0.0419155	−0.0209578	+	0.999780i	$0.506672\pi$
$24$	9.15552	1.86886
$25$	−4.51722	−0.903445
$26$	−9.08187	−1.78110
$27$	−1.00000	−0.192450
$28$	25.2713	4.77583
$29$	5.37029	0.997237	0.498619	−	0.866821i	$-0.333841\pi$
$29$	5.37029	0.997237	0.498619	+	0.866821i	$0.333841\pi$
$30$	1.88654	0.344434
$31$	−5.14363	−0.923824	−0.461912	−	0.886926i	$-0.652836\pi$
$31$	−5.14363	−0.923824	−0.461912	+	0.886926i	$0.652836\pi$
$32$	−20.0120	−3.53765
$33$	2.64356	0.460185
$34$	−6.30511	−1.08132
$35$	3.26861	0.552497
$36$	5.37202	0.895336
$37$	−0.104230	−0.0171353	−0.00856765	−	0.999963i	$-0.502727\pi$
$37$	−0.104230	−0.0171353	−0.00856765	+	0.999963i	$0.502727\pi$
$38$	4.75293	0.771027
$39$	−3.34489	−0.535612
$40$	−6.36145	−1.00583
$41$	−4.61915	−0.721389	−0.360695	−	0.932684i	$-0.617460\pi$
$41$	−4.61915	−0.721389	−0.360695	+	0.932684i	$0.617460\pi$
$42$	12.7727	1.97088
$43$	−8.09619	−1.23466	−0.617329	−	0.786705i	$-0.711785\pi$
$43$	−8.09619	−1.23466	−0.617329	+	0.786705i	$0.711785\pi$
$44$	−14.2012	−2.14092
$45$	0.694821	0.103578
$46$	0.545798	0.0804735
$47$	−9.79368	−1.42856	−0.714278	−	0.699862i	$-0.753245\pi$
$47$	−9.79368	−1.42856	−0.714278	+	0.699862i	$0.753245\pi$
$48$	−14.1145	−2.03726
$49$	15.1300	2.16143
$50$	12.2649	1.73452
$51$	−2.32220	−0.325173
$52$	17.9688	2.49183
$53$	−7.08677	−0.973443	−0.486722	−	0.873557i	$-0.661807\pi$
$53$	−7.08677	−0.973443	−0.486722	+	0.873557i	$0.661807\pi$
$54$	2.71515	0.369485
$55$	−1.83680	−0.247674
$56$	−43.0699	−5.75545
$57$	1.75052	0.231863
$58$	−14.5811	−1.91459
$59$	−1.46332	−0.190509	−0.0952543	−	0.995453i	$-0.530366\pi$
$59$	−1.46332	−0.190509	−0.0952543	+	0.995453i	$0.530366\pi$
$60$	−3.73259	−0.481876
$61$	11.9421	1.52903	0.764514	−	0.644607i	$-0.222979\pi$
$61$	11.9421	1.52903	0.764514	+	0.644607i	$0.222979\pi$
$62$	13.9657	1.77365
$63$	4.70425	0.592680
$64$	26.1064	3.26330
$65$	2.32410	0.288270
$66$	−7.17765	−0.883508
$67$	5.44194	0.664839	0.332420	−	0.943132i	$-0.392135\pi$
$67$	5.44194	0.664839	0.332420	+	0.943132i	$0.392135\pi$
$68$	12.4749	1.51280
$69$	0.201020	0.0241999
$70$	−8.87477	−1.06074
$71$	6.97388	0.827647	0.413824	−	0.910357i	$-0.364193\pi$
$71$	6.97388	0.827647	0.413824	+	0.910357i	$0.364193\pi$
$72$	−9.15552	−1.07899
$73$	−6.65620	−0.779049	−0.389524	−	0.921016i	$-0.627361\pi$
$73$	−6.65620	−0.779049	−0.389524	+	0.921016i	$0.627361\pi$
$74$	0.283000	0.0328980
$75$	4.51722	0.521604
$76$	−9.40385	−1.07870
$77$	−12.4360	−1.41721
$78$	9.08187	1.02832
$79$	6.48188	0.729269	0.364634	−	0.931151i	$-0.381194\pi$
$79$	6.48188	0.729269	0.364634	+	0.931151i	$0.381194\pi$
$80$	9.80708	1.09647
$81$	1.00000	0.111111
$82$	12.5417	1.38499
$83$	−0.672215	−0.0737851	−0.0368926	−	0.999319i	$-0.511746\pi$
$83$	−0.672215	−0.0737851	−0.0368926	+	0.999319i	$0.511746\pi$
$84$	−25.2713	−2.75733
$85$	1.61351	0.175010
$86$	21.9823	2.37042
$87$	−5.37029	−0.575755
$88$	24.2032	2.58007
$89$	−8.94870	−0.948560	−0.474280	−	0.880374i	$-0.657292\pi$
$89$	−8.94870	−0.948560	−0.474280	+	0.880374i	$0.657292\pi$
$90$	−1.88654	−0.198859
$91$	15.7352	1.64950
$92$	−1.07988	−0.112585
$93$	5.14363	0.533370
$94$	26.5913	2.74268
$95$	−1.21630	−0.124790
$96$	20.0120	2.04247
$97$	−17.9487	−1.82242	−0.911209	−	0.411944i	$-0.864850\pi$
$97$	−17.9487	−1.82242	−0.911209	+	0.411944i	$0.864850\pi$
$98$	−41.0801	−4.14972
$99$	−2.64356	−0.265688
$100$	−24.2666	−2.42666
$101$	−16.3936	−1.63122	−0.815611	−	0.578601i	$-0.803599\pi$
$101$	−16.3936	−1.63122	−0.815611	+	0.578601i	$0.803599\pi$
$102$	6.30511	0.624299
$103$	6.23122	0.613980	0.306990	−	0.951713i	$-0.400678\pi$
$103$	6.23122	0.613980	0.306990	+	0.951713i	$0.400678\pi$
$104$	−30.6242	−3.00295
$105$	−3.26861	−0.318984
$106$	19.2416	1.86891
$107$	−0.208802	−0.0201857	−0.0100928	−	0.999949i	$-0.503213\pi$
$107$	−0.208802	−0.0201857	−0.0100928	+	0.999949i	$0.503213\pi$
$108$	−5.37202	−0.516923
$109$	−14.3945	−1.37874	−0.689372	−	0.724407i	$-0.742114\pi$
$109$	−14.3945	−1.37874	−0.689372	+	0.724407i	$0.742114\pi$
$110$	4.98718	0.475509
$111$	0.104230	0.00989307
$112$	66.3983	6.27405
$113$	−7.27734	−0.684595	−0.342298	−	0.939592i	$-0.611205\pi$
$113$	−7.27734	−0.684595	−0.342298	+	0.939592i	$0.611205\pi$
$114$	−4.75293	−0.445153
$115$	−0.139673	−0.0130246
$116$	28.8493	2.67859
$117$	3.34489	0.309235
$118$	3.97314	0.365757
$119$	10.9242	1.00142
$120$	6.36145	0.580718
$121$	−4.01159	−0.364690
$122$	−32.4245	−2.93558
$123$	4.61915	0.416494
$124$	−27.6317	−2.48140
$125$	−6.61277	−0.591464
$126$	−12.7727	−1.13789
$127$	−0.992677	−0.0880858	−0.0440429	−	0.999030i	$-0.514024\pi$
$127$	−0.992677	−0.0880858	−0.0440429	+	0.999030i	$0.514024\pi$
$128$	−30.8587	−2.72755
$129$	8.09619	0.712830
$130$	−6.31028	−0.553448
$131$	−10.6911	−0.934087	−0.467044	−	0.884234i	$-0.654681\pi$
$131$	−10.6911	−0.934087	−0.467044	+	0.884234i	$0.654681\pi$
$132$	14.2012	1.23606
$133$	−8.23491	−0.714057
$134$	−14.7757	−1.27642
$135$	−0.694821	−0.0598007
$136$	−21.2609	−1.82311
$137$	−9.20017	−0.786024	−0.393012	−	0.919533i	$-0.628567\pi$
$137$	−9.20017	−0.786024	−0.393012	+	0.919533i	$0.628567\pi$
$138$	−0.545798	−0.0464614
$139$	−5.90304	−0.500690	−0.250345	−	0.968157i	$-0.580544\pi$
$139$	−5.90304	−0.500690	−0.250345	+	0.968157i	$0.580544\pi$
$140$	17.5591	1.48401
$141$	9.79368	0.824777
$142$	−18.9351	−1.58900
$143$	−8.84242	−0.739441
$144$	14.1145	1.17621
$145$	3.73139	0.309875
$146$	18.0725	1.49569
$147$	−15.1300	−1.24790
$148$	−0.559925	−0.0460256
$149$	−17.8884	−1.46547	−0.732737	−	0.680512i	$-0.761757\pi$
$149$	−17.8884	−1.46547	−0.732737	+	0.680512i	$0.761757\pi$
$150$	−12.2649	−1.00143
$151$	23.7802	1.93521	0.967603	−	0.252476i	$-0.0812448\pi$
$151$	23.7802	1.93521	0.967603	+	0.252476i	$0.0812448\pi$
$152$	16.0270	1.29996
$153$	2.32220	0.187739
$154$	33.7655	2.72090
$155$	−3.57391	−0.287063
$156$	−17.9688	−1.43866
$157$	−14.1681	−1.13074	−0.565369	−	0.824838i	$-0.691266\pi$
$157$	−14.1681	−1.13074	−0.565369	+	0.824838i	$0.691266\pi$
$158$	−17.5993	−1.40012
$159$	7.08677	0.562018
$160$	−13.9048	−1.09927
$161$	−0.945648	−0.0745275
$162$	−2.71515	−0.213322
$163$	−4.28033	−0.335262	−0.167631	−	0.985850i	$-0.553612\pi$
$163$	−4.28033	−0.335262	−0.167631	+	0.985850i	$0.553612\pi$
$164$	−24.8141	−1.93766
$165$	1.83680	0.142995
$166$	1.82516	0.141660
$167$	1.26633	0.0979912	0.0489956	−	0.998799i	$-0.484398\pi$
$167$	1.26633	0.0979912	0.0489956	+	0.998799i	$0.484398\pi$
$168$	43.0699	3.32291
$169$	−1.81169	−0.139361
$170$	−4.38093	−0.336002
$171$	−1.75052	−0.133866
$172$	−43.4929	−3.31630
$173$	4.54297	0.345396	0.172698	−	0.984975i	$-0.444752\pi$
$173$	4.54297	0.345396	0.172698	+	0.984975i	$0.444752\pi$
$174$	14.5811	1.10539
$175$	−21.2502	−1.60636
$176$	−37.3126	−2.81254
$177$	1.46332	0.109990
$178$	24.2970	1.82114
$179$	−7.62596	−0.569991	−0.284995	−	0.958529i	$-0.591992\pi$
$179$	−7.62596	−0.569991	−0.284995	+	0.958529i	$0.591992\pi$
$180$	3.73259	0.278211
$181$	−11.8129	−0.878043	−0.439021	−	0.898477i	$-0.644675\pi$
$181$	−11.8129	−0.878043	−0.439021	+	0.898477i	$0.644675\pi$
$182$	−42.7234	−3.16687
$183$	−11.9421	−0.882785
$184$	1.84044	0.135679
$185$	−0.0724212	−0.00532451
$186$	−13.9657	−1.02402
$187$	−6.13887	−0.448919
$188$	−52.6118	−3.83711
$189$	−4.70425	−0.342184
$190$	3.30244	0.239584
$191$	−13.7707	−0.996411	−0.498205	−	0.867059i	$-0.666007\pi$
$191$	−13.7707	−0.996411	−0.498205	+	0.867059i	$0.666007\pi$
$192$	−26.1064	−1.88407
$193$	2.01737	0.145213	0.0726067	−	0.997361i	$-0.476868\pi$
$193$	2.01737	0.145213	0.0726067	+	0.997361i	$0.476868\pi$
$194$	48.7334	3.49886
$195$	−2.32410	−0.166432
$196$	81.2785	5.80561
$197$	20.7633	1.47933	0.739663	−	0.672977i	$-0.234985\pi$
$197$	20.7633	1.47933	0.739663	+	0.672977i	$0.234985\pi$
$198$	7.17765	0.510093
$199$	−20.6345	−1.46274	−0.731370	−	0.681980i	$-0.761119\pi$
$199$	−20.6345	−1.46274	−0.731370	+	0.681980i	$0.761119\pi$
$200$	41.3575	2.92442
$201$	−5.44194	−0.383845
$202$	44.5110	3.13178
$203$	25.2632	1.77313
$204$	−12.4749	−0.873417
$205$	−3.20948	−0.224160
$206$	−16.9187	−1.17878
$207$	−0.201020	−0.0139718
$208$	47.2116	3.27354
$209$	4.62761	0.320099
$210$	8.87477	0.612417
$211$	−23.6283	−1.62664	−0.813319	−	0.581818i	$-0.802342\pi$
$211$	−23.6283	−1.62664	−0.813319	+	0.581818i	$0.802342\pi$
$212$	−38.0703	−2.61468
$213$	−6.97388	−0.477842
$214$	0.566928	0.0387544
$215$	−5.62541	−0.383650
$216$	9.15552	0.622954
$217$	−24.1969	−1.64260
$218$	39.0832	2.64705
$219$	6.65620	0.449784
$220$	−9.86733	−0.665255
$221$	7.76751	0.522499
$222$	−0.283000	−0.0189937
$223$	6.82897	0.457302	0.228651	−	0.973508i	$-0.426569\pi$
$223$	6.82897	0.457302	0.228651	+	0.973508i	$0.426569\pi$
$224$	−94.1415	−6.29009
$225$	−4.51722	−0.301148
$226$	19.7591	1.31435
$227$	28.9229	1.91968	0.959840	−	0.280549i	$-0.0905164\pi$
$227$	28.9229	1.91968	0.959840	+	0.280549i	$0.0905164\pi$
$228$	9.40385	0.622785
$229$	−2.66664	−0.176216	−0.0881082	−	0.996111i	$-0.528082\pi$
$229$	−2.66664	−0.176216	−0.0881082	+	0.996111i	$0.528082\pi$
$230$	0.379232	0.0250058
$231$	12.4360	0.818227
$232$	−49.1678	−3.22802
$233$	−4.95531	−0.324633	−0.162317	−	0.986739i	$-0.551897\pi$
$233$	−4.95531	−0.324633	−0.162317	+	0.986739i	$0.551897\pi$
$234$	−9.08187	−0.593701
$235$	−6.80486	−0.443900
$236$	−7.86100	−0.511708
$237$	−6.48188	−0.421043
$238$	−29.6608	−1.92263
$239$	−3.42150	−0.221318	−0.110659	−	0.993858i	$-0.535296\pi$
$239$	−3.42150	−0.221318	−0.110659	+	0.993858i	$0.535296\pi$
$240$	−9.80708	−0.633045
$241$	−18.5558	−1.19528	−0.597640	−	0.801764i	$-0.703895\pi$
$241$	−18.5558	−1.19528	−0.597640	+	0.801764i	$0.703895\pi$
$242$	10.8921	0.700169
$243$	−1.00000	−0.0641500
$244$	64.1531	4.10698
$245$	10.5126	0.671628
$246$	−12.5417	−0.799627
$247$	−5.85532	−0.372565
$248$	47.0926	2.99039
$249$	0.672215	0.0425999
$250$	17.9546	1.13555
$251$	11.4910	0.725308	0.362654	−	0.931924i	$-0.381871\pi$
$251$	11.4910	0.725308	0.362654	+	0.931924i	$0.381871\pi$
$252$	25.2713	1.59194
$253$	0.531408	0.0334093
$254$	2.69526	0.169116
$255$	−1.61351	−0.101042
$256$	31.5731	1.97332
$257$	−9.69760	−0.604919	−0.302460	−	0.953162i	$-0.597808\pi$
$257$	−9.69760	−0.604919	−0.302460	+	0.953162i	$0.597808\pi$
$258$	−21.9823	−1.36856
$259$	−0.490324	−0.0304673
$260$	12.4851	0.774295
$261$	5.37029	0.332412
$262$	29.0279	1.79335
$263$	3.40660	0.210060	0.105030	−	0.994469i	$-0.466506\pi$
$263$	3.40660	0.210060	0.105030	+	0.994469i	$0.466506\pi$
$264$	−24.2032	−1.48960
$265$	−4.92404	−0.302481
$266$	22.3590	1.37092
$267$	8.94870	0.547651
$268$	29.2342	1.78576
$269$	16.5341	1.00810	0.504052	−	0.863673i	$-0.331842\pi$
$269$	16.5341	1.00810	0.504052	+	0.863673i	$0.331842\pi$
$270$	1.88654	0.114811
$271$	11.8992	0.722824	0.361412	−	0.932406i	$-0.382295\pi$
$271$	11.8992	0.722824	0.361412	+	0.932406i	$0.382295\pi$
$272$	32.7768	1.98738
$273$	−15.7352	−0.952339
$274$	24.9798	1.50909
$275$	11.9415	0.720102
$276$	1.07988	0.0650013
$277$	5.82588	0.350043	0.175022	−	0.984565i	$-0.444000\pi$
$277$	5.82588	0.350043	0.175022	+	0.984565i	$0.444000\pi$
$278$	16.0276	0.961273
$279$	−5.14363	−0.307941
$280$	−29.9259	−1.78841
$281$	−0.240212	−0.0143299	−0.00716493	−	0.999974i	$-0.502281\pi$
$281$	−0.240212	−0.0143299	−0.00716493	+	0.999974i	$0.502281\pi$
$282$	−26.5913	−1.58349
$283$	19.2408	1.14374	0.571872	−	0.820343i	$-0.306217\pi$
$283$	19.2408	1.14374	0.571872	+	0.820343i	$0.306217\pi$
$284$	37.4638	2.22307
$285$	1.21630	0.0720475
$286$	24.0085	1.41965
$287$	−21.7296	−1.28266
$288$	−20.0120	−1.17922
$289$	−11.6074	−0.682788
$290$	−10.1313	−0.594929
$291$	17.9487	1.05217
$292$	−35.7572	−2.09253
$293$	3.98884	0.233031	0.116515	−	0.993189i	$-0.462828\pi$
$293$	3.98884	0.233031	0.116515	+	0.993189i	$0.462828\pi$
$294$	41.0801	2.39584
$295$	−1.01675	−0.0591974
$296$	0.954280	0.0554664
$297$	2.64356	0.153395
$298$	48.5696	2.81356
$299$	−0.672390	−0.0388853
$300$	24.2666	1.40103
$301$	−38.0865	−2.19527
$302$	−64.5668	−3.71540
$303$	16.3936	0.941786
$304$	−24.7078	−1.41709
$305$	8.29762	0.475120
$306$	−6.30511	−0.360439
$307$	11.3947	0.650331	0.325165	−	0.945657i	$-0.394580\pi$
$307$	11.3947	0.650331	0.325165	+	0.945657i	$0.394580\pi$
$308$	−66.8062	−3.80664
$309$	−6.23122	−0.354482
$310$	9.70368	0.551132
$311$	−17.2381	−0.977481	−0.488741	−	0.872429i	$-0.662544\pi$
$311$	−17.2381	−0.977481	−0.488741	+	0.872429i	$0.662544\pi$
$312$	30.6242	1.73376
$313$	−15.0796	−0.852351	−0.426176	−	0.904640i	$-0.640139\pi$
$313$	−15.0796	−0.852351	−0.426176	+	0.904640i	$0.640139\pi$
$314$	38.4685	2.17090
$315$	3.26861	0.184166
$316$	34.8208	1.95882
$317$	2.06313	0.115877	0.0579384	−	0.998320i	$-0.481547\pi$
$317$	2.06313	0.115877	0.0579384	+	0.998320i	$0.481547\pi$
$318$	−19.2416	−1.07902
$319$	−14.1967	−0.794861
$320$	18.1393	1.01402
$321$	0.208802	0.0116542
$322$	2.56757	0.143085
$323$	−4.06507	−0.226186
$324$	5.37202	0.298445
$325$	−15.1096	−0.838131
$326$	11.6217	0.643668
$327$	14.3945	0.796019
$328$	42.2907	2.33511
$329$	−46.0719	−2.54003
$330$	−4.98718	−0.274535
$331$	13.0867	0.719311	0.359655	−	0.933085i	$-0.382894\pi$
$331$	13.0867	0.719311	0.359655	+	0.933085i	$0.382894\pi$
$332$	−3.61115	−0.198188
$333$	−0.104230	−0.00571177
$334$	−3.43826	−0.188133
$335$	3.78118	0.206588
$336$	−66.3983	−3.62233
$337$	31.7844	1.73141	0.865703	−	0.500558i	$-0.166872\pi$
$337$	31.7844	1.73141	0.865703	+	0.500558i	$0.166872\pi$
$338$	4.91900	0.267559
$339$	7.27734	0.395251
$340$	8.66783	0.470079
$341$	13.5975	0.736346
$342$	4.75293	0.257009
$343$	38.2455	2.06506
$344$	74.1248	3.99654
$345$	0.139673	0.00751973
$346$	−12.3348	−0.663124
$347$	−9.62339	−0.516611	−0.258305	−	0.966063i	$-0.583164\pi$
$347$	−9.62339	−0.516611	−0.258305	+	0.966063i	$0.583164\pi$
$348$	−28.8493	−1.54648
$349$	−8.91464	−0.477190	−0.238595	−	0.971119i	$-0.576687\pi$
$349$	−8.91464	−0.477190	−0.238595	+	0.971119i	$0.576687\pi$
$350$	57.6973	3.08405
$351$	−3.34489	−0.178537
$352$	52.9029	2.81973
$353$	−3.61671	−0.192498	−0.0962491	−	0.995357i	$-0.530685\pi$
$353$	−3.61671	−0.192498	−0.0962491	+	0.995357i	$0.530685\pi$
$354$	−3.97314	−0.211170
$355$	4.84560	0.257178
$356$	−48.0726	−2.54784
$357$	−10.9242	−0.578171
$358$	20.7056	1.09432
$359$	35.8872	1.89406	0.947028	−	0.321151i	$-0.104070\pi$
$359$	35.8872	1.89406	0.947028	+	0.321151i	$0.104070\pi$
$360$	−6.36145	−0.335278
$361$	−15.9357	−0.838719
$362$	32.0736	1.68575
$363$	4.01159	0.210554
$364$	84.5299	4.43057
$365$	−4.62487	−0.242077
$366$	32.4245	1.69486
$367$	10.3566	0.540611	0.270305	−	0.962775i	$-0.412875\pi$
$367$	10.3566	0.540611	0.270305	+	0.962775i	$0.412875\pi$
$368$	−2.83730	−0.147905
$369$	−4.61915	−0.240463
$370$	0.196634	0.0102225
$371$	−33.3380	−1.73082
$372$	27.6317	1.43264
$373$	−5.81641	−0.301162	−0.150581	−	0.988598i	$-0.548114\pi$
$373$	−5.81641	−0.301162	−0.150581	+	0.988598i	$0.548114\pi$
$374$	16.6679	0.861878
$375$	6.61277	0.341482
$376$	89.6663	4.62418
$377$	17.9630	0.925143
$378$	12.7727	0.656958
$379$	−8.52681	−0.437993	−0.218996	−	0.975726i	$-0.570278\pi$
$379$	−8.52681	−0.437993	−0.218996	+	0.975726i	$0.570278\pi$
$380$	−6.53399	−0.335187
$381$	0.992677	0.0508564
$382$	37.3894	1.91301
$383$	17.0306	0.870224	0.435112	−	0.900376i	$-0.356709\pi$
$383$	17.0306	0.870224	0.435112	+	0.900376i	$0.356709\pi$
$384$	30.8587	1.57475
$385$	−8.64078	−0.440375
$386$	−5.47745	−0.278795
$387$	−8.09619	−0.411552
$388$	−96.4209	−4.89503
$389$	−29.8811	−1.51503	−0.757516	−	0.652817i	$-0.773587\pi$
$389$	−29.8811	−1.51503	−0.757516	+	0.652817i	$0.773587\pi$
$390$	6.31028	0.319533
$391$	−0.466808	−0.0236075
$392$	−138.523	−6.99646
$393$	10.6911	0.539295
$394$	−56.3755	−2.84016
$395$	4.50375	0.226608
$396$	−14.2012	−0.713640
$397$	−16.5730	−0.831774	−0.415887	−	0.909416i	$-0.636529\pi$
$397$	−16.5730	−0.831774	−0.415887	+	0.909416i	$0.636529\pi$
$398$	56.0257	2.80831
$399$	8.23491	0.412261
$400$	−63.7585	−3.18793
$401$	33.7378	1.68479	0.842393	−	0.538863i	$-0.181146\pi$
$401$	33.7378	1.68479	0.842393	+	0.538863i	$0.181146\pi$
$402$	14.7757	0.736943
$403$	−17.2049	−0.857037
$404$	−88.0666	−4.38148
$405$	0.694821	0.0345260
$406$	−68.5932	−3.40422
$407$	0.275538	0.0136579
$408$	21.2609	1.05257
$409$	−12.5278	−0.619461	−0.309731	−	0.950824i	$-0.600239\pi$
$409$	−12.5278	−0.619461	−0.309731	+	0.950824i	$0.600239\pi$
$410$	8.71421	0.430364
$411$	9.20017	0.453811
$412$	33.4742	1.64916
$413$	−6.88384	−0.338732
$414$	0.545798	0.0268245
$415$	−0.467069	−0.0229275
$416$	−66.9380	−3.28190
$417$	5.90304	0.289073
$418$	−12.5646	−0.614557
$419$	20.9173	1.02188	0.510938	−	0.859618i	$-0.329298\pi$
$419$	20.9173	1.02188	0.510938	+	0.859618i	$0.329298\pi$
$420$	−17.5591	−0.856794
$421$	32.0033	1.55974	0.779872	−	0.625939i	$-0.215284\pi$
$421$	32.0033	1.55974	0.779872	+	0.625939i	$0.215284\pi$
$422$	64.1542	3.12298
$423$	−9.79368	−0.476185
$424$	64.8831	3.15100
$425$	−10.4899	−0.508835
$426$	18.9351	0.917409
$427$	56.1786	2.71867
$428$	−1.12169	−0.0542188
$429$	8.84242	0.426916
$430$	15.2738	0.736568
$431$	34.0717	1.64118	0.820588	−	0.571520i	$-0.193646\pi$
$431$	34.0717	1.64118	0.820588	+	0.571520i	$0.193646\pi$
$432$	−14.1145	−0.679086
$433$	2.23556	0.107434	0.0537171	−	0.998556i	$-0.482893\pi$
$433$	2.23556	0.107434	0.0537171	+	0.998556i	$0.482893\pi$
$434$	65.6982	3.15362
$435$	−3.73139	−0.178906
$436$	−77.3276	−3.70332
$437$	0.351890	0.0168332
$438$	−18.0725	−0.863539
$439$	−30.9655	−1.47790	−0.738951	−	0.673759i	$-0.764679\pi$
$439$	−30.9655	−1.47790	−0.738951	+	0.673759i	$0.764679\pi$
$440$	16.8169	0.801713
$441$	15.1300	0.720475
$442$	−21.0899	−1.00315
$443$	−16.1963	−0.769509	−0.384755	−	0.923019i	$-0.625714\pi$
$443$	−16.1963	−0.769509	−0.384755	+	0.923019i	$0.625714\pi$
$444$	0.559925	0.0265729
$445$	−6.21775	−0.294749
$446$	−18.5417	−0.877973
$447$	17.8884	0.846092
$448$	122.811	5.80228
$449$	37.8328	1.78544	0.892721	−	0.450611i	$-0.148794\pi$
$449$	37.8328	1.78544	0.892721	+	0.450611i	$0.148794\pi$
$450$	12.2649	0.578174
$451$	12.2110	0.574993
$452$	−39.0940	−1.83883
$453$	−23.7802	−1.11729
$454$	−78.5299	−3.68559
$455$	10.9332	0.512555
$456$	−16.0270	−0.750531
$457$	29.5581	1.38267	0.691334	−	0.722536i	$-0.257023\pi$
$457$	29.5581	1.38267	0.691334	+	0.722536i	$0.257023\pi$
$458$	7.24031	0.338318
$459$	−2.32220	−0.108391
$460$	−0.750325	−0.0349841
$461$	−5.28732	−0.246255	−0.123128	−	0.992391i	$-0.539292\pi$
$461$	−5.28732	−0.246255	−0.123128	+	0.992391i	$0.539292\pi$
$462$	−33.7655	−1.57091
$463$	−37.5153	−1.74348	−0.871742	−	0.489966i	$-0.837009\pi$
$463$	−37.5153	−1.74348	−0.871742	+	0.489966i	$0.837009\pi$
$464$	75.7991	3.51889
$465$	3.57391	0.165736
$466$	13.4544	0.623263
$467$	24.9227	1.15329	0.576643	−	0.816996i	$-0.304362\pi$
$467$	24.9227	1.15329	0.576643	+	0.816996i	$0.304362\pi$
$468$	17.9688	0.830609
$469$	25.6003	1.18211
$470$	18.4762	0.852243
$471$	14.1681	0.652832
$472$	13.3975	0.616670
$473$	21.4028	0.984100
$474$	17.5993	0.808361
$475$	7.90751	0.362821
$476$	58.6851	2.68983
$477$	−7.08677	−0.324481
$478$	9.28987	0.424909
$479$	−15.4505	−0.705952	−0.352976	−	0.935632i	$-0.614830\pi$
$479$	−15.4505	−0.705952	−0.352976	+	0.935632i	$0.614830\pi$
$480$	13.9048	0.634663
$481$	−0.348638	−0.0158965
$482$	50.3816	2.29482
$483$	0.945648	0.0430285
$484$	−21.5504	−0.979562
$485$	−12.4712	−0.566287
$486$	2.71515	0.123162
$487$	1.04389	0.0473031	0.0236515	−	0.999720i	$-0.492471\pi$
$487$	1.04389	0.0473031	0.0236515	+	0.999720i	$0.492471\pi$
$488$	−109.336	−4.94941
$489$	4.28033	0.193563
$490$	−28.5433	−1.28946
$491$	−7.80621	−0.352289	−0.176145	−	0.984364i	$-0.556363\pi$
$491$	−7.80621	−0.352289	−0.176145	+	0.984364i	$0.556363\pi$
$492$	24.8141	1.11871
$493$	12.4709	0.561660
$494$	15.8980	0.715287
$495$	−1.83680	−0.0825581
$496$	−72.6000	−3.25984
$497$	32.8069	1.47159
$498$	−1.82516	−0.0817874
$499$	23.9499	1.07214	0.536072	−	0.844172i	$-0.319908\pi$
$499$	23.9499	1.07214	0.536072	+	0.844172i	$0.319908\pi$
$500$	−35.5239	−1.58868
$501$	−1.26633	−0.0565753
$502$	−31.1998	−1.39252
$503$	19.0365	0.848795	0.424397	−	0.905476i	$-0.360486\pi$
$503$	19.0365	0.848795	0.424397	+	0.905476i	$0.360486\pi$
$504$	−43.0699	−1.91848
$505$	−11.3906	−0.506875
$506$	−1.44285	−0.0641425
$507$	1.81169	0.0804600
$508$	−5.33268	−0.236599
$509$	9.11020	0.403802	0.201901	−	0.979406i	$-0.435288\pi$
$509$	9.11020	0.403802	0.201901	+	0.979406i	$0.435288\pi$
$510$	4.38093	0.193991
$511$	−31.3124	−1.38518
$512$	−24.0082	−1.06102
$513$	1.75052	0.0772875
$514$	26.3304	1.16138
$515$	4.32958	0.190784
$516$	43.4929	1.91467
$517$	25.8902	1.13865
$518$	1.33130	0.0584940
$519$	−4.54297	−0.199414
$520$	−21.2784	−0.933118
$521$	8.88422	0.389225	0.194612	−	0.980880i	$-0.437655\pi$
$521$	8.88422	0.389225	0.194612	+	0.980880i	$0.437655\pi$
$522$	−14.5811	−0.638198
$523$	−38.1070	−1.66630	−0.833151	−	0.553046i	$-0.813466\pi$
$523$	−38.1070	−1.66630	−0.833151	+	0.553046i	$0.813466\pi$
$524$	−57.4329	−2.50897
$525$	21.2502	0.927433
$526$	−9.24942	−0.403294
$527$	−11.9445	−0.520312
$528$	37.3126	1.62382
$529$	−22.9596	−0.998243
$530$	13.3695	0.580734
$531$	−1.46332	−0.0635028
$532$	−44.2381	−1.91796
$533$	−15.4506	−0.669238
$534$	−24.2970	−1.05143
$535$	−0.145080	−0.00627236
$536$	−49.8238	−2.15206
$537$	7.62596	0.329084
$538$	−44.8926	−1.93546
$539$	−39.9970	−1.72279
$540$	−3.73259	−0.160625
$541$	8.60339	0.369889	0.184944	−	0.982749i	$-0.440790\pi$
$541$	8.60339	0.369889	0.184944	+	0.982749i	$0.440790\pi$
$542$	−32.3080	−1.38775
$543$	11.8129	0.506938
$544$	−46.4718	−1.99246
$545$	−10.0016	−0.428422
$546$	42.7234	1.82839
$547$	4.56804	0.195315	0.0976576	−	0.995220i	$-0.468865\pi$
$547$	4.56804	0.195315	0.0976576	+	0.995220i	$0.468865\pi$
$548$	−49.4235	−2.11127
$549$	11.9421	0.509676
$550$	−32.4230	−1.38252
$551$	−9.40082	−0.400488
$552$	−1.84044	−0.0783344
$553$	30.4924	1.29667
$554$	−15.8181	−0.672048
$555$	0.0724212	0.00307411
$556$	−31.7113	−1.34486
$557$	28.0010	1.18644	0.593220	−	0.805040i	$-0.297856\pi$
$557$	28.0010	1.18644	0.593220	+	0.805040i	$0.297856\pi$
$558$	13.9657	0.591216
$559$	−27.0809	−1.14540
$560$	46.1350	1.94956
$561$	6.13887	0.259183
$562$	0.652211	0.0275119
$563$	−9.39898	−0.396120	−0.198060	−	0.980190i	$-0.563464\pi$
$563$	−9.39898	−0.396120	−0.198060	+	0.980190i	$0.563464\pi$
$564$	52.6118	2.21536
$565$	−5.05646	−0.212727
$566$	−52.2414	−2.19587
$567$	4.70425	0.197560
$568$	−63.8495	−2.67907
$569$	−23.0858	−0.967806	−0.483903	−	0.875122i	$-0.660781\pi$
$569$	−23.0858	−0.967806	−0.483903	+	0.875122i	$0.660781\pi$
$570$	−3.30244	−0.138324
$571$	4.79264	0.200566	0.100283	−	0.994959i	$-0.468025\pi$
$571$	4.79264	0.200566	0.100283	+	0.994959i	$0.468025\pi$
$572$	−47.5017	−1.98614
$573$	13.7707	0.575278
$574$	58.9991	2.46258
$575$	0.908051	0.0378684
$576$	26.1064	1.08777
$577$	−15.4567	−0.643471	−0.321736	−	0.946830i	$-0.604266\pi$
$577$	−15.4567	−0.643471	−0.321736	+	0.946830i	$0.604266\pi$
$578$	31.5158	1.31088
$579$	−2.01737	−0.0838390
$580$	20.0451	0.832327
$581$	−3.16227	−0.131193
$582$	−48.7334	−2.02007
$583$	18.7343	0.775895
$584$	60.9409	2.52175
$585$	2.32410	0.0960898
$586$	−10.8303	−0.447395
$587$	24.1678	0.997513	0.498757	−	0.866742i	$-0.333790\pi$
$587$	24.1678	0.997513	0.498757	+	0.866742i	$0.333790\pi$
$588$	−81.2785	−3.35187
$589$	9.00405	0.371006
$590$	2.76062	0.113653
$591$	−20.7633	−0.854089
$592$	−1.47116	−0.0604642
$593$	8.53973	0.350685	0.175342	−	0.984508i	$-0.443897\pi$
$593$	8.53973	0.350685	0.175342	+	0.984508i	$0.443897\pi$
$594$	−7.17765	−0.294503
$595$	7.59038	0.311175
$596$	−96.0968	−3.93628
$597$	20.6345	0.844514
$598$	1.82564	0.0746558
$599$	−34.0494	−1.39122	−0.695610	−	0.718419i	$-0.744866\pi$
$599$	−34.0494	−1.39122	−0.695610	+	0.718419i	$0.744866\pi$
$600$	−41.3575	−1.68841
$601$	−36.6314	−1.49423	−0.747114	−	0.664696i	$-0.768561\pi$
$601$	−36.6314	−1.49423	−0.747114	+	0.664696i	$0.768561\pi$
$602$	103.410	4.21470
$603$	5.44194	0.221613
$604$	127.748	5.19798
$605$	−2.78734	−0.113322
$606$	−44.5110	−1.80813
$607$	−16.0934	−0.653212	−0.326606	−	0.945160i	$-0.605905\pi$
$607$	−16.0934	−0.653212	−0.326606	+	0.945160i	$0.605905\pi$
$608$	35.0315	1.42071
$609$	−25.2632	−1.02372
$610$	−22.5293	−0.912183
$611$	−32.7588	−1.32528
$612$	12.4749	0.504268
$613$	30.3571	1.22611	0.613055	−	0.790040i	$-0.289940\pi$
$613$	30.3571	1.22611	0.613055	+	0.790040i	$0.289940\pi$
$614$	−30.9383	−1.24857
$615$	3.20948	0.129419
$616$	113.858	4.58746
$617$	−22.0089	−0.886044	−0.443022	−	0.896511i	$-0.646094\pi$
$617$	−22.0089	−0.886044	−0.443022	+	0.896511i	$0.646094\pi$
$618$	16.9187	0.680569
$619$	−24.4297	−0.981913	−0.490957	−	0.871184i	$-0.663353\pi$
$619$	−24.4297	−0.981913	−0.490957	+	0.871184i	$0.663353\pi$
$620$	−19.1991	−0.771054
$621$	0.201020	0.00806665
$622$	46.8039	1.87666
$623$	−42.0969	−1.68658
$624$	−47.2116	−1.88998
$625$	17.9914	0.719657
$626$	40.9434	1.63643
$627$	−4.62761	−0.184809
$628$	−76.1113	−3.03717
$629$	−0.242043	−0.00965088
$630$	−8.87477	−0.353579
$631$	26.9703	1.07367	0.536835	−	0.843687i	$-0.319620\pi$
$631$	26.9703	1.07367	0.536835	+	0.843687i	$0.319620\pi$
$632$	−59.3450	−2.36062
$633$	23.6283	0.939140
$634$	−5.60170	−0.222472
$635$	−0.689733	−0.0273712
$636$	38.0703	1.50958
$637$	50.6082	2.00517
$638$	38.5460	1.52605
$639$	6.97388	0.275882
$640$	−21.4413	−0.847542
$641$	−4.23018	−0.167082	−0.0835411	−	0.996504i	$-0.526623\pi$
$641$	−4.23018	−0.167082	−0.0835411	+	0.996504i	$0.526623\pi$
$642$	−0.566928	−0.0223749
$643$	5.56928	0.219631	0.109815	−	0.993952i	$-0.464974\pi$
$643$	5.56928	0.219631	0.109815	+	0.993952i	$0.464974\pi$
$644$	−5.08004	−0.200181
$645$	5.62541	0.221500
$646$	11.0372	0.434255
$647$	20.4314	0.803242	0.401621	−	0.915806i	$-0.368447\pi$
$647$	20.4314	0.803242	0.401621	+	0.915806i	$0.368447\pi$
$648$	−9.15552	−0.359663
$649$	3.86838	0.151847
$650$	41.0248	1.60913
$651$	24.1969	0.948353
$652$	−22.9940	−0.900516
$653$	9.19180	0.359703	0.179852	−	0.983694i	$-0.442438\pi$
$653$	9.19180	0.359703	0.179852	+	0.983694i	$0.442438\pi$
$654$	−39.0832	−1.52827
$655$	−7.42842	−0.290252
$656$	−65.1971	−2.54552
$657$	−6.65620	−0.259683
$658$	125.092	4.87660
$659$	34.0811	1.32761	0.663805	−	0.747905i	$-0.268940\pi$
$659$	34.0811	1.32761	0.663805	+	0.747905i	$0.268940\pi$
$660$	9.86733	0.384085
$661$	21.2239	0.825513	0.412757	−	0.910841i	$-0.364566\pi$
$661$	21.2239	0.825513	0.412757	+	0.910841i	$0.364566\pi$
$662$	−35.5323	−1.38100
$663$	−7.76751	−0.301665
$664$	6.15447	0.238840
$665$	−5.72179	−0.221881
$666$	0.283000	0.0109660
$667$	−1.07953	−0.0417997
$668$	6.80272	0.263205
$669$	−6.82897	−0.264023
$670$	−10.2664	−0.396628
$671$	−31.5696	−1.21873
$672$	94.1415	3.63159
$673$	−9.35291	−0.360528	−0.180264	−	0.983618i	$-0.557695\pi$
$673$	−9.35291	−0.360528	−0.180264	+	0.983618i	$0.557695\pi$
$674$	−86.2992	−3.32412
$675$	4.51722	0.173868
$676$	−9.73243	−0.374324
$677$	22.7074	0.872717	0.436358	−	0.899773i	$-0.356268\pi$
$677$	22.7074	0.872717	0.436358	+	0.899773i	$0.356268\pi$
$678$	−19.7591	−0.758842
$679$	−84.4354	−3.24033
$680$	−14.7726	−0.566502
$681$	−28.9229	−1.10833
$682$	−36.9192	−1.41371
$683$	−34.9077	−1.33570	−0.667852	−	0.744294i	$-0.732786\pi$
$683$	−34.9077	−1.33570	−0.667852	+	0.744294i	$0.732786\pi$
$684$	−9.40385	−0.359565
$685$	−6.39248	−0.244244
$686$	−103.842	−3.96471
$687$	2.66664	0.101739
$688$	−114.274	−4.35666
$689$	−23.7045	−0.903069
$690$	−0.379232	−0.0144371
$691$	−32.0515	−1.21930	−0.609648	−	0.792673i	$-0.708689\pi$
$691$	−32.0515	−1.21930	−0.609648	+	0.792673i	$0.708689\pi$
$692$	24.4049	0.927735
$693$	−12.4360	−0.472403
$694$	26.1289	0.991840
$695$	−4.10156	−0.155581
$696$	49.1678	1.86370
$697$	−10.7266	−0.406298
$698$	24.2045	0.916155
$699$	4.95531	0.187427
$700$	−114.156	−4.31470
$701$	−10.5591	−0.398810	−0.199405	−	0.979917i	$-0.563901\pi$
$701$	−10.5591	−0.398810	−0.199405	+	0.979917i	$0.563901\pi$
$702$	9.08187	0.342773
$703$	0.182457	0.00688150
$704$	−69.0138	−2.60106
$705$	6.80486	0.256286
$706$	9.81990	0.369577
$707$	−77.1195	−2.90038
$708$	7.86100	0.295435
$709$	11.5575	0.434049	0.217025	−	0.976166i	$-0.430365\pi$
$709$	11.5575	0.434049	0.217025	+	0.976166i	$0.430365\pi$
$710$	−13.1565	−0.493755
$711$	6.48188	0.243090
$712$	81.9300	3.07046
$713$	1.03397	0.0387226
$714$	29.6608	1.11003
$715$	−6.14390	−0.229769
$716$	−40.9668	−1.53100
$717$	3.42150	0.127778
$718$	−97.4391	−3.63639
$719$	6.12977	0.228602	0.114301	−	0.993446i	$-0.463537\pi$
$719$	6.12977	0.228602	0.114301	+	0.993446i	$0.463537\pi$
$720$	9.80708	0.365488
$721$	29.3132	1.09168
$722$	43.2677	1.61026
$723$	18.5558	0.690096
$724$	−63.4589	−2.35843
$725$	−24.2588	−0.900949
$726$	−10.8921	−0.404242
$727$	29.1765	1.08210	0.541049	−	0.840991i	$-0.318027\pi$
$727$	29.1765	1.08210	0.541049	+	0.840991i	$0.318027\pi$
$728$	−144.064	−5.33937
$729$	1.00000	0.0370370
$730$	12.5572	0.464762
$731$	−18.8010	−0.695379
$732$	−64.1531	−2.37117
$733$	13.5952	0.502148	0.251074	−	0.967968i	$-0.419216\pi$
$733$	13.5952	0.502148	0.251074	+	0.967968i	$0.419216\pi$
$734$	−28.1197	−1.03792
$735$	−10.5126	−0.387764
$736$	4.02281	0.148283
$737$	−14.3861	−0.529919
$738$	12.5417	0.461665
$739$	−9.75815	−0.358959	−0.179480	−	0.983762i	$-0.557441\pi$
$739$	−9.75815	−0.358959	−0.179480	+	0.983762i	$0.557441\pi$
$740$	−0.389048	−0.0143017
$741$	5.85532	0.215100
$742$	90.5174	3.32300
$743$	0.318128	0.0116710	0.00583548	−	0.999983i	$-0.498142\pi$
$743$	0.318128	0.0116710	0.00583548	+	0.999983i	$0.498142\pi$
$744$	−47.0926	−1.72650
$745$	−12.4292	−0.455372
$746$	15.7924	0.578201
$747$	−0.672215	−0.0245950
$748$	−32.9781	−1.20580
$749$	−0.982257	−0.0358909
$750$	−17.9546	−0.655611
$751$	20.4435	0.745992	0.372996	−	0.927833i	$-0.378330\pi$
$751$	20.4435	0.745992	0.372996	+	0.927833i	$0.378330\pi$
$752$	−138.233	−5.04085
$753$	−11.4910	−0.418756
$754$	−48.7723	−1.77618
$755$	16.5230	0.601334
$756$	−25.2713	−0.919109
$757$	28.9140	1.05090	0.525449	−	0.850825i	$-0.323897\pi$
$757$	28.9140	1.05090	0.525449	+	0.850825i	$0.323897\pi$
$758$	23.1515	0.840902
$759$	−0.531408	−0.0192889
$760$	11.1359	0.403941
$761$	27.2253	0.986917	0.493459	−	0.869769i	$-0.335732\pi$
$761$	27.2253	0.986917	0.493459	+	0.869769i	$0.335732\pi$
$762$	−2.69526	−0.0976390
$763$	−67.7154	−2.45146
$764$	−73.9763	−2.67637
$765$	1.61351	0.0583367
$766$	−46.2406	−1.67074
$767$	−4.89466	−0.176736
$768$	−31.5731	−1.13930
$769$	−16.6087	−0.598925	−0.299462	−	0.954108i	$-0.596807\pi$
$769$	−16.6087	−0.598925	−0.299462	+	0.954108i	$0.596807\pi$
$770$	23.4610	0.845475
$771$	9.69760	0.349250
$772$	10.8373	0.390045
$773$	18.0431	0.648964	0.324482	−	0.945892i	$-0.394810\pi$
$773$	18.0431	0.648964	0.324482	+	0.945892i	$0.394810\pi$
$774$	21.9823	0.790139
$775$	23.2349	0.834624
$776$	164.330	5.89910
$777$	0.490324	0.0175903
$778$	81.1316	2.90871
$779$	8.08593	0.289708
$780$	−12.4851	−0.447039
$781$	−18.4359	−0.659687
$782$	1.26745	0.0453240
$783$	−5.37029	−0.191918
$784$	213.553	7.62688
$785$	−9.84430	−0.351358
$786$	−29.0279	−1.03539
$787$	−26.8028	−0.955417	−0.477708	−	0.878519i	$-0.658532\pi$
$787$	−26.8028	−0.955417	−0.477708	+	0.878519i	$0.658532\pi$
$788$	111.541	3.97348
$789$	−3.40660	−0.121278
$790$	−12.2283	−0.435065
$791$	−34.2345	−1.21724
$792$	24.2032	0.860022
$793$	39.9450	1.41849
$794$	44.9981	1.59692
$795$	4.92404	0.174638
$796$	−110.849	−3.92893
$797$	−48.1881	−1.70691	−0.853455	−	0.521166i	$-0.825497\pi$
$797$	−48.1881	−1.70691	−0.853455	+	0.521166i	$0.825497\pi$
$798$	−22.3590	−0.791499
$799$	−22.7429	−0.804585
$800$	90.3986	3.19607
$801$	−8.94870	−0.316187
$802$	−91.6031	−3.23462
$803$	17.5960	0.620951
$804$	−29.2342	−1.03101
$805$	−0.657056	−0.0231582
$806$	46.7138	1.64542
$807$	−16.5341	−0.582029
$808$	150.092	5.28021
$809$	−25.0667	−0.881299	−0.440650	−	0.897679i	$-0.645252\pi$
$809$	−25.0667	−0.881299	−0.440650	+	0.897679i	$0.645252\pi$
$810$	−1.88654	−0.0662863
$811$	−4.29712	−0.150892	−0.0754461	−	0.997150i	$-0.524038\pi$
$811$	−4.29712	−0.150892	−0.0754461	+	0.997150i	$0.524038\pi$
$812$	135.714	4.76264
$813$	−11.8992	−0.417323
$814$	−0.748126	−0.0262218
$815$	−2.97407	−0.104177
$816$	−32.7768	−1.14742
$817$	14.1726	0.495836
$818$	34.0149	1.18930
$819$	15.7352	0.549833
$820$	−17.2414	−0.602096
$821$	−1.14479	−0.0399534	−0.0199767	−	0.999800i	$-0.506359\pi$
$821$	−1.14479	−0.0399534	−0.0199767	+	0.999800i	$0.506359\pi$
$822$	−24.9798	−0.871271
$823$	32.2248	1.12329	0.561643	−	0.827380i	$-0.310170\pi$
$823$	32.2248	1.12329	0.561643	+	0.827380i	$0.310170\pi$
$824$	−57.0501	−1.98743
$825$	−11.9415	−0.415751
$826$	18.6906	0.650331
$827$	37.5829	1.30689	0.653443	−	0.756976i	$-0.273324\pi$
$827$	37.5829	1.30689	0.653443	+	0.756976i	$0.273324\pi$
$828$	−1.07988	−0.0375285
$829$	−44.2771	−1.53781	−0.768904	−	0.639365i	$-0.779197\pi$
$829$	−44.2771	−1.53781	−0.768904	+	0.639365i	$0.779197\pi$
$830$	1.26816	0.0440185
$831$	−5.82588	−0.202098
$832$	87.3231	3.02739
$833$	35.1348	1.21735
$834$	−16.0276	−0.554991
$835$	0.879870	0.0304492
$836$	24.8596	0.859788
$837$	5.14363	0.177790
$838$	−56.7934	−1.96190
$839$	−23.5186	−0.811953	−0.405976	−	0.913884i	$-0.633068\pi$
$839$	−23.5186	−0.811953	−0.405976	+	0.913884i	$0.633068\pi$
$840$	29.9259	1.03254
$841$	−0.160023	−0.00551803
$842$	−86.8936	−2.99455
$843$	0.240212	0.00827335
$844$	−126.932	−4.36916
$845$	−1.25880	−0.0433041
$846$	26.5913	0.914227
$847$	−18.8716	−0.648434
$848$	−100.027	−3.43492
$849$	−19.2408	−0.660341
$850$	28.4816	0.976911
$851$	0.0209523	0.000718235 0
$852$	−37.4638	−1.28349
$853$	11.6289	0.398165	0.199082	−	0.979983i	$-0.436204\pi$
$853$	11.6289	0.398165	0.199082	+	0.979983i	$0.436204\pi$
$854$	−152.533	−5.21958
$855$	−1.21630	−0.0415966
$856$	1.91169	0.0653403
$857$	−35.4647	−1.21145	−0.605725	−	0.795674i	$-0.707117\pi$
$857$	−35.4647	−1.21145	−0.605725	+	0.795674i	$0.707117\pi$
$858$	−24.0085	−0.819636
$859$	10.0422	0.342634	0.171317	−	0.985216i	$-0.445198\pi$
$859$	10.0422	0.342634	0.171317	+	0.985216i	$0.445198\pi$
$860$	−30.2198	−1.03049
$861$	21.7296	0.740544
$862$	−92.5096	−3.15089
$863$	−42.7025	−1.45361	−0.726805	−	0.686844i	$-0.758996\pi$
$863$	−42.7025	−1.45361	−0.726805	+	0.686844i	$0.758996\pi$
$864$	20.0120	0.680822
$865$	3.15655	0.107326
$866$	−6.06987	−0.206263
$867$	11.6074	0.394208
$868$	−129.986	−4.41203
$869$	−17.1352	−0.581273
$870$	10.1313	0.343482
$871$	18.2027	0.616775
$872$	131.789	4.46295
$873$	−17.9487	−0.607473
$874$	−0.955433	−0.0323180
$875$	−31.1081	−1.05165
$876$	35.7572	1.20812
$877$	41.3765	1.39719	0.698593	−	0.715519i	$-0.253810\pi$
$877$	41.3765	1.39719	0.698593	+	0.715519i	$0.253810\pi$
$878$	84.0758	2.83742
$879$	−3.98884	−0.134540
$880$	−25.9256	−0.873952
$881$	46.3699	1.56224	0.781120	−	0.624381i	$-0.214649\pi$
$881$	46.3699	1.56224	0.781120	+	0.624381i	$0.214649\pi$
$882$	−41.0801	−1.38324
$883$	−37.9554	−1.27730	−0.638650	−	0.769497i	$-0.720507\pi$
$883$	−37.9554	−1.27730	−0.638650	+	0.769497i	$0.720507\pi$
$884$	41.7272	1.40344
$885$	1.01675	0.0341776
$886$	43.9753	1.47738
$887$	9.83731	0.330305	0.165152	−	0.986268i	$-0.447188\pi$
$887$	9.83731	0.330305	0.165152	+	0.986268i	$0.447188\pi$
$888$	−0.954280	−0.0320235
$889$	−4.66980	−0.156620
$890$	16.8821	0.565889
$891$	−2.64356	−0.0885626
$892$	36.6854	1.22832
$893$	17.1441	0.573705
$894$	−48.5696	−1.62441
$895$	−5.29868	−0.177115
$896$	−145.167	−4.84969
$897$	0.672390	0.0224504
$898$	−102.722	−3.42787
$899$	−27.6228	−0.921271
$900$	−24.2666	−0.808887
$901$	−16.4569	−0.548259
$902$	−33.1546	−1.10393
$903$	38.0865	1.26744
$904$	66.6279	2.21601
$905$	−8.20783	−0.272837
$906$	64.5668	2.14509
$907$	58.3583	1.93775	0.968877	−	0.247542i	$-0.0796229\pi$
$907$	58.3583	1.93775	0.968877	+	0.247542i	$0.0796229\pi$
$908$	155.374	5.15628
$909$	−16.3936	−0.543741
$910$	−29.6851	−0.984053
$911$	2.57894	0.0854442	0.0427221	−	0.999087i	$-0.486397\pi$
$911$	2.57894	0.0854442	0.0427221	+	0.999087i	$0.486397\pi$
$912$	24.7078	0.818158
$913$	1.77704	0.0588114
$914$	−80.2544	−2.65458
$915$	−8.29762	−0.274311
$916$	−14.3252	−0.473319
$917$	−50.2937	−1.66084
$918$	6.30511	0.208100
$919$	16.3444	0.539152	0.269576	−	0.962979i	$-0.413117\pi$
$919$	16.3444	0.539152	0.269576	+	0.962979i	$0.413117\pi$
$920$	1.27878	0.0421601
$921$	−11.3947	−0.375469
$922$	14.3559	0.472785
$923$	23.3269	0.767814
$924$	66.8062	2.19776
$925$	0.470830	0.0154808
$926$	101.859	3.34731
$927$	6.23122	0.204660
$928$	−107.470	−3.52788
$929$	−5.01148	−0.164421	−0.0822106	−	0.996615i	$-0.526198\pi$
$929$	−5.01148	−0.164421	−0.0822106	+	0.996615i	$0.526198\pi$
$930$	−9.70368	−0.318196
$931$	−26.4854	−0.868024
$932$	−26.6200	−0.871968
$933$	17.2381	0.564349
$934$	−67.6688	−2.21419
$935$	−4.26542	−0.139494
$936$	−30.6242	−1.00098
$937$	−23.4916	−0.767437	−0.383718	−	0.923450i	$-0.625357\pi$
$937$	−23.4916	−0.767437	−0.383718	+	0.923450i	$0.625357\pi$
$938$	−69.5085	−2.26953
$939$	15.0796	0.492105
$940$	−36.5558	−1.19232
$941$	−40.3552	−1.31554	−0.657770	−	0.753218i	$-0.728500\pi$
$941$	−40.3552	−1.31554	−0.657770	+	0.753218i	$0.728500\pi$
$942$	−38.4685	−1.25337
$943$	0.928540	0.0302374
$944$	−20.6541	−0.672235
$945$	−3.26861	−0.106328
$946$	−58.1116	−1.88937
$947$	−14.6798	−0.477028	−0.238514	−	0.971139i	$-0.576660\pi$
$947$	−14.6798	−0.477028	−0.238514	+	0.971139i	$0.576660\pi$
$948$	−34.8208	−1.13093
$949$	−22.2643	−0.722729
$950$	−21.4700	−0.696580
$951$	−2.06313	−0.0669015
$952$	−100.017	−3.24156
$953$	12.0270	0.389593	0.194796	−	0.980844i	$-0.437595\pi$
$953$	12.0270	0.389593	0.194796	+	0.980844i	$0.437595\pi$
$954$	19.2416	0.622970
$955$	−9.56816	−0.309618
$956$	−18.3804	−0.594463
$957$	14.1967	0.458913
$958$	41.9504	1.35536
$959$	−43.2799	−1.39758
$960$	−18.1393	−0.585443
$961$	−4.54304	−0.146550
$962$	0.946603	0.0305197
$963$	−0.208802	−0.00672855
$964$	−99.6818	−3.21054
$965$	1.40171	0.0451227
$966$	−2.56757	−0.0826103
$967$	−50.4236	−1.62151	−0.810756	−	0.585384i	$-0.800944\pi$
$967$	−50.4236	−1.62151	−0.810756	+	0.585384i	$0.800944\pi$
$968$	36.7282	1.18049
$969$	4.06507	0.130589
$970$	33.8610	1.08721
$971$	9.19189	0.294982	0.147491	−	0.989063i	$-0.452880\pi$
$971$	9.19189	0.294982	0.147491	+	0.989063i	$0.452880\pi$
$972$	−5.37202	−0.172308
$973$	−27.7694	−0.890246
$974$	−2.83431	−0.0908171
$975$	15.1096	0.483895
$976$	168.557	5.39538
$977$	−6.24136	−0.199679	−0.0998394	−	0.995004i	$-0.531833\pi$
$977$	−6.24136	−0.199679	−0.0998394	+	0.995004i	$0.531833\pi$
$978$	−11.6217	−0.371622
$979$	23.6564	0.756062
$980$	56.4741	1.80400
$981$	−14.3945	−0.459582
$982$	21.1950	0.676359
$983$	23.2549	0.741717	0.370859	−	0.928689i	$-0.379063\pi$
$983$	23.2549	0.741717	0.370859	+	0.928689i	$0.379063\pi$
$984$	−42.2907	−1.34818
$985$	14.4268	0.459676
$986$	−33.8603	−1.07833
$987$	46.0719	1.46649
$988$	−31.4549	−1.00071
$989$	1.62749	0.0517513
$990$	4.98718	0.158503
$991$	24.0712	0.764645	0.382323	−	0.924029i	$-0.375124\pi$
$991$	24.0712	0.764645	0.382323	+	0.924029i	$0.375124\pi$
$992$	102.934	3.26817
$993$	−13.0867	−0.415294
$994$	−89.0755	−2.82530
$995$	−14.3373	−0.454523
$996$	3.61115	0.114424
$997$	33.9393	1.07487	0.537435	−	0.843305i	$-0.319393\pi$
$997$	33.9393	1.07487	0.537435	+	0.843305i	$0.319393\pi$
$998$	−65.0274	−2.05841
$999$	0.104230	0.00329769

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8031.2.a.b.1.2	✓	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8031.2.a.b.1.2	✓	102	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:	\( N \)	\(=\)	\( 8031 = 3 \cdot 2677 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8031.a (trivial)

\(f(q)\)	\(=\)	\(q-2.71515 q^{2} -1.00000 q^{3} +5.37202 q^{4} +0.694821 q^{5} +2.71515 q^{6} +4.70425 q^{7} -9.15552 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.71515 q^{2} -1.00000 q^{3} +5.37202 q^{4} +0.694821 q^{5} +2.71515 q^{6} +4.70425 q^{7} -9.15552 q^{8} +1.00000 q^{9} -1.88654 q^{10} -2.64356 q^{11} -5.37202 q^{12} +3.34489 q^{13} -12.7727 q^{14} -0.694821 q^{15} +14.1145 q^{16} +2.32220 q^{17} -2.71515 q^{18} -1.75052 q^{19} +3.73259 q^{20} -4.70425 q^{21} +7.17765 q^{22} -0.201020 q^{23} +9.15552 q^{24} -4.51722 q^{25} -9.08187 q^{26} -1.00000 q^{27} +25.2713 q^{28} +5.37029 q^{29} +1.88654 q^{30} -5.14363 q^{31} -20.0120 q^{32} +2.64356 q^{33} -6.30511 q^{34} +3.26861 q^{35} +5.37202 q^{36} -0.104230 q^{37} +4.75293 q^{38} -3.34489 q^{39} -6.36145 q^{40} -4.61915 q^{41} +12.7727 q^{42} -8.09619 q^{43} -14.2012 q^{44} +0.694821 q^{45} +0.545798 q^{46} -9.79368 q^{47} -14.1145 q^{48} +15.1300 q^{49} +12.2649 q^{50} -2.32220 q^{51} +17.9688 q^{52} -7.08677 q^{53} +2.71515 q^{54} -1.83680 q^{55} -43.0699 q^{56} +1.75052 q^{57} -14.5811 q^{58} -1.46332 q^{59} -3.73259 q^{60} +11.9421 q^{61} +13.9657 q^{62} +4.70425 q^{63} +26.1064 q^{64} +2.32410 q^{65} -7.17765 q^{66} +5.44194 q^{67} +12.4749 q^{68} +0.201020 q^{69} -8.87477 q^{70} +6.97388 q^{71} -9.15552 q^{72} -6.65620 q^{73} +0.283000 q^{74} +4.51722 q^{75} -9.40385 q^{76} -12.4360 q^{77} +9.08187 q^{78} +6.48188 q^{79} +9.80708 q^{80} +1.00000 q^{81} +12.5417 q^{82} -0.672215 q^{83} -25.2713 q^{84} +1.61351 q^{85} +21.9823 q^{86} -5.37029 q^{87} +24.2032 q^{88} -8.94870 q^{89} -1.88654 q^{90} +15.7352 q^{91} -1.07988 q^{92} +5.14363 q^{93} +26.5913 q^{94} -1.21630 q^{95} +20.0120 q^{96} -17.9487 q^{97} -41.0801 q^{98} -2.64356 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9}+O(q^{10}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9	\( 102 q - 6 q^{2} - 102 q^{3} + 96 q^{4} - 20 q^{5} + 6 q^{6} + 12 q^{7} - 21 q^{8} + 102 q^{9} - 16 q^{10} - 28 q^{11} - 96 q^{12} - 2 q^{13} - 41 q^{14} + 20 q^{15} + 88 q^{16} - 77 q^{17} - 6 q^{18} + 10 q^{19} - 50 q^{20} - 12 q^{21} + 24 q^{22} - 29 q^{23} + 21 q^{24} + 74 q^{25} - 45 q^{26} - 102 q^{27} + 19 q^{28} - 68 q^{29} + 16 q^{30} - 29 q^{31} - 48 q^{32} + 28 q^{33} - 19 q^{34} - 49 q^{35} + 96 q^{36} + 4 q^{37} - 44 q^{38} + 2 q^{39} - 41 q^{40} - 122 q^{41} + 41 q^{42} + 85 q^{43} - 86 q^{44} - 20 q^{45} - 28 q^{46} - 39 q^{47} - 88 q^{48} + 24 q^{49} - 37 q^{50} + 77 q^{51} + 8 q^{52} - 37 q^{53} + 6 q^{54} - 13 q^{55} - 130 q^{56} - 10 q^{57} + 17 q^{58} - 58 q^{59} + 50 q^{60} - 114 q^{61} - 64 q^{62} + 12 q^{63} + 47 q^{64} - 92 q^{65} - 24 q^{66} + 121 q^{67} - 138 q^{68} + 29 q^{69} - 2 q^{70} - 67 q^{71} - 21 q^{72} - 72 q^{73} - 111 q^{74} - 74 q^{75} - 17 q^{76} - 57 q^{77} + 45 q^{78} - 24 q^{79} - 97 q^{80} + 102 q^{81} - q^{82} - 78 q^{83} - 19 q^{84} - 24 q^{85} - 80 q^{86} + 68 q^{87} + 54 q^{88} - 176 q^{89} - 16 q^{90} - 3 q^{91} - 82 q^{92} + 29 q^{93} - 41 q^{94} - 90 q^{95} + 48 q^{96} - 77 q^{97} - 48 q^{98} - 28 q^{99}+O(q^{100}) \) 102 * q - 6 * q^2 - 102 * q^3 + 96 * q^4 - 20 * q^5 + 6 * q^6 + 12 * q^7 - 21 * q^8 + 102 * q^9 - 16 * q^10 - 28 * q^11 - 96 * q^12 - 2 * q^13 - 41 * q^14 + 20 * q^15 + 88 * q^16 - 77 * q^17 - 6 * q^18 + 10 * q^19 - 50 * q^20 - 12 * q^21 + 24 * q^22 - 29 * q^23 + 21 * q^24 + 74 * q^25 - 45 * q^26 - 102 * q^27 + 19 * q^28 - 68 * q^29 + 16 * q^30 - 29 * q^31 - 48 * q^32 + 28 * q^33 - 19 * q^34 - 49 * q^35 + 96 * q^36 + 4 * q^37 - 44 * q^38 + 2 * q^39 - 41 * q^40 - 122 * q^41 + 41 * q^42 + 85 * q^43 - 86 * q^44 - 20 * q^45 - 28 * q^46 - 39 * q^47 - 88 * q^48 + 24 * q^49 - 37 * q^50 + 77 * q^51 + 8 * q^52 - 37 * q^53 + 6 * q^54 - 13 * q^55 - 130 * q^56 - 10 * q^57 + 17 * q^58 - 58 * q^59 + 50 * q^60 - 114 * q^61 - 64 * q^62 + 12 * q^63 + 47 * q^64 - 92 * q^65 - 24 * q^66 + 121 * q^67 - 138 * q^68 + 29 * q^69 - 2 * q^70 - 67 * q^71 - 21 * q^72 - 72 * q^73 - 111 * q^74 - 74 * q^75 - 17 * q^76 - 57 * q^77 + 45 * q^78 - 24 * q^79 - 97 * q^80 + 102 * q^81 - q^82 - 78 * q^83 - 19 * q^84 - 24 * q^85 - 80 * q^86 + 68 * q^87 + 54 * q^88 - 176 * q^89 - 16 * q^90 - 3 * q^91 - 82 * q^92 + 29 * q^93 - 41 * q^94 - 90 * q^95 + 48 * q^96 - 77 * q^97 - 48 * q^98 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more