LMFDB - Embedded newform 2541.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2541,2,Mod(1,2541)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2541, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2541.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2541 = 3 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2541.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:	$20.2899871536$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	2541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{2} -1.00000 q^{3} -1.46410 q^{4} -0.267949 q^{5} -0.732051 q^{6} +1.00000 q^{7} -2.53590 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.732051 q^{2} -1.00000 q^{3} -1.46410 q^{4} -0.267949 q^{5} -0.732051 q^{6} +1.00000 q^{7} -2.53590 q^{8} +1.00000 q^{9} -0.196152 q^{10} +1.46410 q^{12} -2.73205 q^{13} +0.732051 q^{14} +0.267949 q^{15} +1.07180 q^{16} +3.73205 q^{17} +0.732051 q^{18} +2.19615 q^{19} +0.392305 q^{20} -1.00000 q^{21} +3.26795 q^{23} +2.53590 q^{24} -4.92820 q^{25} -2.00000 q^{26} -1.00000 q^{27} -1.46410 q^{28} -1.26795 q^{29} +0.196152 q^{30} +7.46410 q^{31} +5.85641 q^{32} +2.73205 q^{34} -0.267949 q^{35} -1.46410 q^{36} -9.46410 q^{37} +1.60770 q^{38} +2.73205 q^{39} +0.679492 q^{40} -4.00000 q^{41} -0.732051 q^{42} -0.464102 q^{43} -0.267949 q^{45} +2.39230 q^{46} -9.19615 q^{47} -1.07180 q^{48} +1.00000 q^{49} -3.60770 q^{50} -3.73205 q^{51} +4.00000 q^{52} +2.53590 q^{53} -0.732051 q^{54} -2.53590 q^{56} -2.19615 q^{57} -0.928203 q^{58} -10.1244 q^{59} -0.392305 q^{60} -8.19615 q^{61} +5.46410 q^{62} +1.00000 q^{63} +2.14359 q^{64} +0.732051 q^{65} -7.00000 q^{67} -5.46410 q^{68} -3.26795 q^{69} -0.196152 q^{70} +9.12436 q^{71} -2.53590 q^{72} +6.73205 q^{73} -6.92820 q^{74} +4.92820 q^{75} -3.21539 q^{76} +2.00000 q^{78} +0.535898 q^{79} -0.287187 q^{80} +1.00000 q^{81} -2.92820 q^{82} -4.26795 q^{83} +1.46410 q^{84} -1.00000 q^{85} -0.339746 q^{86} +1.26795 q^{87} -14.6603 q^{89} -0.196152 q^{90} -2.73205 q^{91} -4.78461 q^{92} -7.46410 q^{93} -6.73205 q^{94} -0.588457 q^{95} -5.85641 q^{96} +3.26795 q^{97} +0.732051 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 + 2 * q^7 - 12 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9} + 10 q^{10} - 4 q^{12} - 2 q^{13} - 2 q^{14} + 4 q^{15} + 16 q^{16} + 4 q^{17} - 2 q^{18} - 6 q^{19} - 20 q^{20} - 2 q^{21} + 10 q^{23} + 12 q^{24} + 4 q^{25} - 4 q^{26} - 2 q^{27} + 4 q^{28} - 6 q^{29} - 10 q^{30} + 8 q^{31} - 16 q^{32} + 2 q^{34} - 4 q^{35} + 4 q^{36} - 12 q^{37} + 24 q^{38} + 2 q^{39} + 36 q^{40} - 8 q^{41} + 2 q^{42} + 6 q^{43} - 4 q^{45} - 16 q^{46} - 8 q^{47} - 16 q^{48} + 2 q^{49} - 28 q^{50} - 4 q^{51} + 8 q^{52} + 12 q^{53} + 2 q^{54} - 12 q^{56} + 6 q^{57} + 12 q^{58} + 4 q^{59} + 20 q^{60} - 6 q^{61} + 4 q^{62} + 2 q^{63} + 32 q^{64} - 2 q^{65} - 14 q^{67} - 4 q^{68} - 10 q^{69} + 10 q^{70} - 6 q^{71} - 12 q^{72} + 10 q^{73} - 4 q^{75} - 48 q^{76} + 4 q^{78} + 8 q^{79} - 56 q^{80} + 2 q^{81} + 8 q^{82} - 12 q^{83} - 4 q^{84} - 2 q^{85} - 18 q^{86} + 6 q^{87} - 12 q^{89} + 10 q^{90} - 2 q^{91} + 32 q^{92} - 8 q^{93} - 10 q^{94} + 30 q^{95} + 16 q^{96} + 10 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 + 2 * q^7 - 12 * q^8 + 2 * q^9 + 10 * q^10 - 4 * q^12 - 2 * q^13 - 2 * q^14 + 4 * q^15 + 16 * q^16 + 4 * q^17 - 2 * q^18 - 6 * q^19 - 20 * q^20 - 2 * q^21 + 10 * q^23 + 12 * q^24 + 4 * q^25 - 4 * q^26 - 2 * q^27 + 4 * q^28 - 6 * q^29 - 10 * q^30 + 8 * q^31 - 16 * q^32 + 2 * q^34 - 4 * q^35 + 4 * q^36 - 12 * q^37 + 24 * q^38 + 2 * q^39 + 36 * q^40 - 8 * q^41 + 2 * q^42 + 6 * q^43 - 4 * q^45 - 16 * q^46 - 8 * q^47 - 16 * q^48 + 2 * q^49 - 28 * q^50 - 4 * q^51 + 8 * q^52 + 12 * q^53 + 2 * q^54 - 12 * q^56 + 6 * q^57 + 12 * q^58 + 4 * q^59 + 20 * q^60 - 6 * q^61 + 4 * q^62 + 2 * q^63 + 32 * q^64 - 2 * q^65 - 14 * q^67 - 4 * q^68 - 10 * q^69 + 10 * q^70 - 6 * q^71 - 12 * q^72 + 10 * q^73 - 4 * q^75 - 48 * q^76 + 4 * q^78 + 8 * q^79 - 56 * q^80 + 2 * q^81 + 8 * q^82 - 12 * q^83 - 4 * q^84 - 2 * q^85 - 18 * q^86 + 6 * q^87 - 12 * q^89 + 10 * q^90 - 2 * q^91 + 32 * q^92 - 8 * q^93 - 10 * q^94 + 30 * q^95 + 16 * q^96 + 10 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.732051	0.517638	0.258819	−	0.965926i	$-0.416667\pi$
$2$	0.732051	0.517638	0.258819	+	0.965926i	$0.416667\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.46410	−0.732051
$5$	−0.267949	−0.119831	−0.0599153	−	0.998203i	$-0.519083\pi$
$5$	−0.267949	−0.119831	−0.0599153	+	0.998203i	$0.519083\pi$
$6$	−0.732051	−0.298858
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.53590	−0.896575
$9$	1.00000	0.333333
$10$	−0.196152	−0.0620288
$11$	0	0
$11$	0	0
$12$	1.46410	0.422650
$13$	−2.73205	−0.757735	−0.378867	−	0.925451i	$-0.623686\pi$
$13$	−2.73205	−0.757735	−0.378867	+	0.925451i	$0.623686\pi$
$14$	0.732051	0.195649
$15$	0.267949	0.0691842
$16$	1.07180	0.267949
$17$	3.73205	0.905155	0.452578	−	0.891725i	$-0.350505\pi$
$17$	3.73205	0.905155	0.452578	+	0.891725i	$0.350505\pi$
$18$	0.732051	0.172546
$19$	2.19615	0.503832	0.251916	−	0.967749i	$-0.418939\pi$
$19$	2.19615	0.503832	0.251916	+	0.967749i	$0.418939\pi$
$20$	0.392305	0.0877220
$21$	−1.00000	−0.218218
$22$	0	0
$23$	3.26795	0.681415	0.340707	−	0.940169i	$-0.389334\pi$
$23$	3.26795	0.681415	0.340707	+	0.940169i	$0.389334\pi$
$24$	2.53590	0.517638
$25$	−4.92820	−0.985641
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	−1.46410	−0.276689
$29$	−1.26795	−0.235452	−0.117726	−	0.993046i	$-0.537560\pi$
$29$	−1.26795	−0.235452	−0.117726	+	0.993046i	$0.537560\pi$
$30$	0.196152	0.0358124
$31$	7.46410	1.34059	0.670296	−	0.742094i	$-0.266167\pi$
$31$	7.46410	1.34059	0.670296	+	0.742094i	$0.266167\pi$
$32$	5.85641	1.03528
$33$	0	0
$34$	2.73205	0.468543
$35$	−0.267949	−0.0452917
$36$	−1.46410	−0.244017
$37$	−9.46410	−1.55589	−0.777944	−	0.628333i	$-0.783737\pi$
$37$	−9.46410	−1.55589	−0.777944	+	0.628333i	$0.783737\pi$
$38$	1.60770	0.260803
$39$	2.73205	0.437478
$40$	0.679492	0.107437
$41$	−4.00000	−0.624695	−0.312348	−	0.949968i	$-0.601115\pi$
$41$	−4.00000	−0.624695	−0.312348	+	0.949968i	$0.601115\pi$
$42$	−0.732051	−0.112958
$43$	−0.464102	−0.0707748	−0.0353874	−	0.999374i	$-0.511267\pi$
$43$	−0.464102	−0.0707748	−0.0353874	+	0.999374i	$0.511267\pi$
$44$	0	0
$45$	−0.267949	−0.0399435
$46$	2.39230	0.352726
$47$	−9.19615	−1.34140	−0.670698	−	0.741730i	$-0.734005\pi$
$47$	−9.19615	−1.34140	−0.670698	+	0.741730i	$0.734005\pi$
$48$	−1.07180	−0.154701
$49$	1.00000	0.142857
$50$	−3.60770	−0.510205
$51$	−3.73205	−0.522592
$52$	4.00000	0.554700
$53$	2.53590	0.348332	0.174166	−	0.984716i	$-0.444277\pi$
$53$	2.53590	0.348332	0.174166	+	0.984716i	$0.444277\pi$
$54$	−0.732051	−0.0996195
$55$	0	0
$56$	−2.53590	−0.338874
$57$	−2.19615	−0.290887
$58$	−0.928203	−0.121879
$59$	−10.1244	−1.31808	−0.659039	−	0.752108i	$-0.729037\pi$
$59$	−10.1244	−1.31808	−0.659039	+	0.752108i	$0.729037\pi$
$60$	−0.392305	−0.0506463
$61$	−8.19615	−1.04941	−0.524705	−	0.851284i	$-0.675824\pi$
$61$	−8.19615	−1.04941	−0.524705	+	0.851284i	$0.675824\pi$
$62$	5.46410	0.693942
$63$	1.00000	0.125988
$64$	2.14359	0.267949
$65$	0.732051	0.0907997
$66$	0	0
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−5.46410	−0.662620
$69$	−3.26795	−0.393415
$70$	−0.196152	−0.0234447
$71$	9.12436	1.08286	0.541431	−	0.840745i	$-0.317883\pi$
$71$	9.12436	1.08286	0.541431	+	0.840745i	$0.317883\pi$
$72$	−2.53590	−0.298858
$73$	6.73205	0.787927	0.393963	−	0.919126i	$-0.371104\pi$
$73$	6.73205	0.787927	0.393963	+	0.919126i	$0.371104\pi$
$74$	−6.92820	−0.805387
$75$	4.92820	0.569060
$76$	−3.21539	−0.368831
$77$	0	0
$78$	2.00000	0.226455
$79$	0.535898	0.0602933	0.0301466	−	0.999545i	$-0.490403\pi$
$79$	0.535898	0.0602933	0.0301466	+	0.999545i	$0.490403\pi$
$80$	−0.287187	−0.0321085
$81$	1.00000	0.111111
$82$	−2.92820	−0.323366
$83$	−4.26795	−0.468468	−0.234234	−	0.972180i	$-0.575258\pi$
$83$	−4.26795	−0.468468	−0.234234	+	0.972180i	$0.575258\pi$
$84$	1.46410	0.159747
$85$	−1.00000	−0.108465
$86$	−0.339746	−0.0366357
$87$	1.26795	0.135938
$88$	0	0
$89$	−14.6603	−1.55398	−0.776992	−	0.629511i	$-0.783255\pi$
$89$	−14.6603	−1.55398	−0.776992	+	0.629511i	$0.783255\pi$
$90$	−0.196152	−0.0206763
$91$	−2.73205	−0.286397
$92$	−4.78461	−0.498830
$93$	−7.46410	−0.773991
$94$	−6.73205	−0.694358
$95$	−0.588457	−0.0603744
$96$	−5.85641	−0.597717
$97$	3.26795	0.331810	0.165905	−	0.986142i	$-0.446945\pi$
$97$	3.26795	0.331810	0.165905	+	0.986142i	$0.446945\pi$
$98$	0.732051	0.0739483
$99$	0	0
$100$	7.21539	0.721539
$101$	−9.73205	−0.968375	−0.484188	−	0.874964i	$-0.660885\pi$
$101$	−9.73205	−0.968375	−0.484188	+	0.874964i	$0.660885\pi$
$102$	−2.73205	−0.270513
$103$	4.19615	0.413459	0.206730	−	0.978398i	$-0.433718\pi$
$103$	4.19615	0.413459	0.206730	+	0.978398i	$0.433718\pi$
$104$	6.92820	0.679366
$105$	0.267949	0.0261492
$106$	1.85641	0.180310
$107$	−18.5885	−1.79701	−0.898507	−	0.438959i	$-0.855347\pi$
$107$	−18.5885	−1.79701	−0.898507	+	0.438959i	$0.855347\pi$
$108$	1.46410	0.140883
$109$	−17.3923	−1.66588	−0.832940	−	0.553363i	$-0.813344\pi$
$109$	−17.3923	−1.66588	−0.832940	+	0.553363i	$0.813344\pi$
$110$	0	0
$111$	9.46410	0.898293
$112$	1.07180	0.101275
$113$	8.53590	0.802990	0.401495	−	0.915861i	$-0.368491\pi$
$113$	8.53590	0.802990	0.401495	+	0.915861i	$0.368491\pi$
$114$	−1.60770	−0.150574
$115$	−0.875644	−0.0816543
$116$	1.85641	0.172363
$117$	−2.73205	−0.252578
$118$	−7.41154	−0.682288
$119$	3.73205	0.342117
$120$	−0.679492	−0.0620288
$121$	0	0
$122$	−6.00000	−0.543214
$123$	4.00000	0.360668
$124$	−10.9282	−0.981382
$125$	2.66025	0.237940
$126$	0.732051	0.0652163
$127$	3.53590	0.313760	0.156880	−	0.987618i	$-0.449856\pi$
$127$	3.53590	0.313760	0.156880	+	0.987618i	$0.449856\pi$
$128$	−10.1436	−0.896575
$129$	0.464102	0.0408619
$130$	0.535898	0.0470014
$131$	−5.19615	−0.453990	−0.226995	−	0.973896i	$-0.572890\pi$
$131$	−5.19615	−0.453990	−0.226995	+	0.973896i	$0.572890\pi$
$132$	0	0
$133$	2.19615	0.190431
$134$	−5.12436	−0.442677
$135$	0.267949	0.0230614
$136$	−9.46410	−0.811540
$137$	13.6603	1.16707	0.583537	−	0.812086i	$-0.301668\pi$
$137$	13.6603	1.16707	0.583537	+	0.812086i	$0.301668\pi$
$138$	−2.39230	−0.203647
$139$	−7.80385	−0.661914	−0.330957	−	0.943646i	$-0.607371\pi$
$139$	−7.80385	−0.661914	−0.330957	+	0.943646i	$0.607371\pi$
$140$	0.392305	0.0331558
$141$	9.19615	0.774456
$142$	6.67949	0.560531
$143$	0	0
$144$	1.07180	0.0893164
$145$	0.339746	0.0282144
$146$	4.92820	0.407861
$147$	−1.00000	−0.0824786
$148$	13.8564	1.13899
$149$	2.73205	0.223818	0.111909	−	0.993718i	$-0.464303\pi$
$149$	2.73205	0.223818	0.111909	+	0.993718i	$0.464303\pi$
$150$	3.60770	0.294567
$151$	−9.39230	−0.764335	−0.382167	−	0.924093i	$-0.624822\pi$
$151$	−9.39230	−0.764335	−0.382167	+	0.924093i	$0.624822\pi$
$152$	−5.56922	−0.451723
$153$	3.73205	0.301718
$154$	0	0
$155$	−2.00000	−0.160644
$156$	−4.00000	−0.320256
$157$	−6.53590	−0.521621	−0.260811	−	0.965390i	$-0.583990\pi$
$157$	−6.53590	−0.521621	−0.260811	+	0.965390i	$0.583990\pi$
$158$	0.392305	0.0312101
$159$	−2.53590	−0.201110
$160$	−1.56922	−0.124058
$161$	3.26795	0.257550
$162$	0.732051	0.0575153
$163$	7.46410	0.584634	0.292317	−	0.956322i	$-0.405574\pi$
$163$	7.46410	0.584634	0.292317	+	0.956322i	$0.405574\pi$
$164$	5.85641	0.457309
$165$	0	0
$166$	−3.12436	−0.242497
$167$	−25.0526	−1.93863	−0.969313	−	0.245831i	$-0.920939\pi$
$167$	−25.0526	−1.93863	−0.969313	+	0.245831i	$0.920939\pi$
$168$	2.53590	0.195649
$169$	−5.53590	−0.425838
$170$	−0.732051	−0.0561457
$171$	2.19615	0.167944
$172$	0.679492	0.0518108
$173$	−5.19615	−0.395056	−0.197528	−	0.980297i	$-0.563291\pi$
$173$	−5.19615	−0.395056	−0.197528	+	0.980297i	$0.563291\pi$
$174$	0.928203	0.0703669
$175$	−4.92820	−0.372537
$176$	0	0
$177$	10.1244	0.760993
$178$	−10.7321	−0.804401
$179$	−4.39230	−0.328296	−0.164148	−	0.986436i	$-0.552488\pi$
$179$	−4.39230	−0.328296	−0.164148	+	0.986436i	$0.552488\pi$
$180$	0.392305	0.0292407
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	−2.00000	−0.148250
$183$	8.19615	0.605877
$184$	−8.28719	−0.610940
$185$	2.53590	0.186443
$186$	−5.46410	−0.400647
$187$	0	0
$188$	13.4641	0.981971
$189$	−1.00000	−0.0727393
$190$	−0.430781	−0.0312521
$191$	16.1962	1.17191	0.585956	−	0.810343i	$-0.300719\pi$
$191$	16.1962	1.17191	0.585956	+	0.810343i	$0.300719\pi$
$192$	−2.14359	−0.154701
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	2.39230	0.171757
$195$	−0.732051	−0.0524232
$196$	−1.46410	−0.104579
$197$	−12.5359	−0.893146	−0.446573	−	0.894747i	$-0.647356\pi$
$197$	−12.5359	−0.893146	−0.446573	+	0.894747i	$0.647356\pi$
$198$	0	0
$199$	−20.1962	−1.43167	−0.715834	−	0.698271i	$-0.753953\pi$
$199$	−20.1962	−1.43167	−0.715834	+	0.698271i	$0.753953\pi$
$200$	12.4974	0.883701
$201$	7.00000	0.493742
$202$	−7.12436	−0.501268
$203$	−1.26795	−0.0889926
$204$	5.46410	0.382564
$205$	1.07180	0.0748575
$206$	3.07180	0.214022
$207$	3.26795	0.227138
$208$	−2.92820	−0.203034
$209$	0	0
$210$	0.196152	0.0135358
$211$	17.7846	1.22434	0.612172	−	0.790725i	$-0.290296\pi$
$211$	17.7846	1.22434	0.612172	+	0.790725i	$0.290296\pi$
$212$	−3.71281	−0.254997
$213$	−9.12436	−0.625191
$214$	−13.6077	−0.930203
$215$	0.124356	0.00848099
$216$	2.53590	0.172546
$217$	7.46410	0.506696
$218$	−12.7321	−0.862323
$219$	−6.73205	−0.454910
$220$	0	0
$221$	−10.1962	−0.685867
$222$	6.92820	0.464991
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	5.85641	0.391298
$225$	−4.92820	−0.328547
$226$	6.24871	0.415658
$227$	−14.6603	−0.973035	−0.486518	−	0.873671i	$-0.661733\pi$
$227$	−14.6603	−0.973035	−0.486518	+	0.873671i	$0.661733\pi$
$228$	3.21539	0.212944
$229$	6.92820	0.457829	0.228914	−	0.973447i	$-0.426482\pi$
$229$	6.92820	0.457829	0.228914	+	0.973447i	$0.426482\pi$
$230$	−0.641016	−0.0422674
$231$	0	0
$232$	3.21539	0.211101
$233$	−20.1962	−1.32309	−0.661547	−	0.749904i	$-0.730100\pi$
$233$	−20.1962	−1.32309	−0.661547	+	0.749904i	$0.730100\pi$
$234$	−2.00000	−0.130744
$235$	2.46410	0.160740
$236$	14.8231	0.964901
$237$	−0.535898	−0.0348103
$238$	2.73205	0.177093
$239$	−24.9282	−1.61247	−0.806236	−	0.591594i	$-0.798499\pi$
$239$	−24.9282	−1.61247	−0.806236	+	0.591594i	$0.798499\pi$
$240$	0.287187	0.0185378
$241$	−15.6603	−1.00877	−0.504383	−	0.863480i	$-0.668280\pi$
$241$	−15.6603	−1.00877	−0.504383	+	0.863480i	$0.668280\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	12.0000	0.768221
$245$	−0.267949	−0.0171186
$246$	2.92820	0.186695
$247$	−6.00000	−0.381771
$248$	−18.9282	−1.20194
$249$	4.26795	0.270470
$250$	1.94744	0.123167
$251$	−1.07180	−0.0676512	−0.0338256	−	0.999428i	$-0.510769\pi$
$251$	−1.07180	−0.0676512	−0.0338256	+	0.999428i	$0.510769\pi$
$252$	−1.46410	−0.0922297
$253$	0	0
$254$	2.58846	0.162414
$255$	1.00000	0.0626224
$256$	−11.7128	−0.732051
$257$	4.12436	0.257270	0.128635	−	0.991692i	$-0.458940\pi$
$257$	4.12436	0.257270	0.128635	+	0.991692i	$0.458940\pi$
$258$	0.339746	0.0211517
$259$	−9.46410	−0.588071
$260$	−1.07180	−0.0664700
$261$	−1.26795	−0.0784841
$262$	−3.80385	−0.235002
$263$	−5.80385	−0.357881	−0.178940	−	0.983860i	$-0.557267\pi$
$263$	−5.80385	−0.357881	−0.178940	+	0.983860i	$0.557267\pi$
$264$	0	0
$265$	−0.679492	−0.0417409
$266$	1.60770	0.0985741
$267$	14.6603	0.897193
$268$	10.2487	0.626040
$269$	23.8564	1.45455	0.727275	−	0.686346i	$-0.240786\pi$
$269$	23.8564	1.45455	0.727275	+	0.686346i	$0.240786\pi$
$270$	0.196152	0.0119375
$271$	−3.60770	−0.219152	−0.109576	−	0.993978i	$-0.534949\pi$
$271$	−3.60770	−0.219152	−0.109576	+	0.993978i	$0.534949\pi$
$272$	4.00000	0.242536
$273$	2.73205	0.165351
$274$	10.0000	0.604122
$275$	0	0
$276$	4.78461	0.288000
$277$	25.7846	1.54925	0.774624	−	0.632423i	$-0.217939\pi$
$277$	25.7846	1.54925	0.774624	+	0.632423i	$0.217939\pi$
$278$	−5.71281	−0.342632
$279$	7.46410	0.446864
$280$	0.679492	0.0406074
$281$	27.7128	1.65321	0.826604	−	0.562784i	$-0.190270\pi$
$281$	27.7128	1.65321	0.826604	+	0.562784i	$0.190270\pi$
$282$	6.73205	0.400888
$283$	26.2487	1.56032	0.780162	−	0.625578i	$-0.215137\pi$
$283$	26.2487	1.56032	0.780162	+	0.625578i	$0.215137\pi$
$284$	−13.3590	−0.792710
$285$	0.588457	0.0348572
$286$	0	0
$287$	−4.00000	−0.236113
$288$	5.85641	0.345092
$289$	−3.07180	−0.180694
$290$	0.248711	0.0146048
$291$	−3.26795	−0.191571
$292$	−9.85641	−0.576803
$293$	16.2679	0.950384	0.475192	−	0.879882i	$-0.342379\pi$
$293$	16.2679	0.950384	0.475192	+	0.879882i	$0.342379\pi$
$294$	−0.732051	−0.0426941
$295$	2.71281	0.157946
$296$	24.0000	1.39497
$297$	0	0
$298$	2.00000	0.115857
$299$	−8.92820	−0.516331
$300$	−7.21539	−0.416581
$301$	−0.464102	−0.0267504
$302$	−6.87564	−0.395649
$303$	9.73205	0.559092
$304$	2.35383	0.135001
$305$	2.19615	0.125751
$306$	2.73205	0.156181
$307$	−26.5885	−1.51748	−0.758742	−	0.651392i	$-0.774186\pi$
$307$	−26.5885	−1.51748	−0.758742	+	0.651392i	$0.774186\pi$
$308$	0	0
$309$	−4.19615	−0.238711
$310$	−1.46410	−0.0831554
$311$	19.0526	1.08037	0.540186	−	0.841546i	$-0.318354\pi$
$311$	19.0526	1.08037	0.540186	+	0.841546i	$0.318354\pi$
$312$	−6.92820	−0.392232
$313$	0.392305	0.0221744	0.0110872	−	0.999939i	$-0.496471\pi$
$313$	0.392305	0.0221744	0.0110872	+	0.999939i	$0.496471\pi$
$314$	−4.78461	−0.270011
$315$	−0.267949	−0.0150972
$316$	−0.784610	−0.0441377
$317$	29.5167	1.65782	0.828910	−	0.559381i	$-0.188961\pi$
$317$	29.5167	1.65782	0.828910	+	0.559381i	$0.188961\pi$
$318$	−1.85641	−0.104102
$319$	0	0
$320$	−0.574374	−0.0321085
$321$	18.5885	1.03751
$322$	2.39230	0.133318
$323$	8.19615	0.456046
$324$	−1.46410	−0.0813390
$325$	13.4641	0.746854
$326$	5.46410	0.302629
$327$	17.3923	0.961797
$328$	10.1436	0.560086
$329$	−9.19615	−0.507000
$330$	0	0
$331$	26.1769	1.43881	0.719407	−	0.694589i	$-0.244414\pi$
$331$	26.1769	1.43881	0.719407	+	0.694589i	$0.244414\pi$
$332$	6.24871	0.342943
$333$	−9.46410	−0.518630
$334$	−18.3397	−1.00351
$335$	1.87564	0.102477
$336$	−1.07180	−0.0584713
$337$	28.0000	1.52526	0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000	1.52526	0.762629	+	0.646837i	$0.223908\pi$
$338$	−4.05256	−0.220430
$339$	−8.53590	−0.463606
$340$	1.46410	0.0794021
$341$	0	0
$342$	1.60770	0.0869342
$343$	1.00000	0.0539949
$344$	1.17691	0.0634550
$345$	0.875644	0.0471431
$346$	−3.80385	−0.204496
$347$	−26.0526	−1.39857	−0.699287	−	0.714841i	$-0.746499\pi$
$347$	−26.0526	−1.39857	−0.699287	+	0.714841i	$0.746499\pi$
$348$	−1.85641	−0.0995138
$349$	−1.60770	−0.0860579	−0.0430290	−	0.999074i	$-0.513701\pi$
$349$	−1.60770	−0.0860579	−0.0430290	+	0.999074i	$0.513701\pi$
$350$	−3.60770	−0.192839
$351$	2.73205	0.145826
$352$	0	0
$353$	−11.0718	−0.589292	−0.294646	−	0.955606i	$-0.595202\pi$
$353$	−11.0718	−0.589292	−0.294646	+	0.955606i	$0.595202\pi$
$354$	7.41154	0.393919
$355$	−2.44486	−0.129760
$356$	21.4641	1.13760
$357$	−3.73205	−0.197521
$358$	−3.21539	−0.169939
$359$	−25.1244	−1.32601	−0.663006	−	0.748614i	$-0.730720\pi$
$359$	−25.1244	−1.32601	−0.663006	+	0.748614i	$0.730720\pi$
$360$	0.679492	0.0358124
$361$	−14.1769	−0.746153
$362$	−5.85641	−0.307806
$363$	0	0
$364$	4.00000	0.209657
$365$	−1.80385	−0.0944177
$366$	6.00000	0.313625
$367$	−12.7321	−0.664608	−0.332304	−	0.943172i	$-0.607826\pi$
$367$	−12.7321	−0.664608	−0.332304	+	0.943172i	$0.607826\pi$
$368$	3.50258	0.182584
$369$	−4.00000	−0.208232
$370$	1.85641	0.0965100
$371$	2.53590	0.131657
$372$	10.9282	0.566601
$373$	23.9282	1.23896	0.619478	−	0.785014i	$-0.287344\pi$
$373$	23.9282	1.23896	0.619478	+	0.785014i	$0.287344\pi$
$374$	0	0
$375$	−2.66025	−0.137375
$376$	23.3205	1.20266
$377$	3.46410	0.178410
$378$	−0.732051	−0.0376526
$379$	3.53590	0.181627	0.0908135	−	0.995868i	$-0.471053\pi$
$379$	3.53590	0.181627	0.0908135	+	0.995868i	$0.471053\pi$
$380$	0.861561	0.0441972
$381$	−3.53590	−0.181150
$382$	11.8564	0.606627
$383$	−34.1244	−1.74367	−0.871837	−	0.489797i	$-0.837071\pi$
$383$	−34.1244	−1.74367	−0.871837	+	0.489797i	$0.837071\pi$
$384$	10.1436	0.517638
$385$	0	0
$386$	3.66025	0.186302
$387$	−0.464102	−0.0235916
$388$	−4.78461	−0.242902
$389$	16.3923	0.831123	0.415561	−	0.909565i	$-0.363585\pi$
$389$	16.3923	0.831123	0.415561	+	0.909565i	$0.363585\pi$
$390$	−0.535898	−0.0271363
$391$	12.1962	0.616786
$392$	−2.53590	−0.128082
$393$	5.19615	0.262111
$394$	−9.17691	−0.462326
$395$	−0.143594	−0.00722498
$396$	0	0
$397$	−13.3205	−0.668537	−0.334269	−	0.942478i	$-0.608489\pi$
$397$	−13.3205	−0.668537	−0.334269	+	0.942478i	$0.608489\pi$
$398$	−14.7846	−0.741086
$399$	−2.19615	−0.109945
$400$	−5.28203	−0.264102
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	5.12436	0.255580
$403$	−20.3923	−1.01581
$404$	14.2487	0.708900
$405$	−0.267949	−0.0133145
$406$	−0.928203	−0.0460660
$407$	0	0
$408$	9.46410	0.468543
$409$	23.6603	1.16992	0.584962	−	0.811061i	$-0.301109\pi$
$409$	23.6603	1.16992	0.584962	+	0.811061i	$0.301109\pi$
$410$	0.784610	0.0387491
$411$	−13.6603	−0.673811
$412$	−6.14359	−0.302673
$413$	−10.1244	−0.498187
$414$	2.39230	0.117575
$415$	1.14359	0.0561368
$416$	−16.0000	−0.784465
$417$	7.80385	0.382156
$418$	0	0
$419$	9.73205	0.475442	0.237721	−	0.971334i	$-0.423600\pi$
$419$	9.73205	0.475442	0.237721	+	0.971334i	$0.423600\pi$
$420$	−0.392305	−0.0191425
$421$	38.8564	1.89375	0.946873	−	0.321609i	$-0.104224\pi$
$421$	38.8564	1.89375	0.946873	+	0.321609i	$0.104224\pi$
$422$	13.0192	0.633767
$423$	−9.19615	−0.447132
$424$	−6.43078	−0.312306
$425$	−18.3923	−0.892158
$426$	−6.67949	−0.323622
$427$	−8.19615	−0.396640
$428$	27.2154	1.31551
$429$	0	0
$430$	0.0910347	0.00439008
$431$	−23.3205	−1.12331	−0.561655	−	0.827372i	$-0.689835\pi$
$431$	−23.3205	−1.12331	−0.561655	+	0.827372i	$0.689835\pi$
$432$	−1.07180	−0.0515668
$433$	33.9090	1.62956	0.814780	−	0.579770i	$-0.196857\pi$
$433$	33.9090	1.62956	0.814780	+	0.579770i	$0.196857\pi$
$434$	5.46410	0.262285
$435$	−0.339746	−0.0162896
$436$	25.4641	1.21951
$437$	7.17691	0.343318
$438$	−4.92820	−0.235479
$439$	0.143594	0.00685335	0.00342667	−	0.999994i	$-0.498909\pi$
$439$	0.143594	0.00685335	0.00342667	+	0.999994i	$0.498909\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−7.46410	−0.355031
$443$	−5.80385	−0.275749	−0.137875	−	0.990450i	$-0.544027\pi$
$443$	−5.80385	−0.275749	−0.137875	+	0.990450i	$0.544027\pi$
$444$	−13.8564	−0.657596
$445$	3.92820	0.186215
$446$	11.7128	0.554618
$447$	−2.73205	−0.129222
$448$	2.14359	0.101275
$449$	−16.2487	−0.766824	−0.383412	−	0.923577i	$-0.625251\pi$
$449$	−16.2487	−0.766824	−0.383412	+	0.923577i	$0.625251\pi$
$450$	−3.60770	−0.170068
$451$	0	0
$452$	−12.4974	−0.587829
$453$	9.39230	0.441289
$454$	−10.7321	−0.503680
$455$	0.732051	0.0343191
$456$	5.56922	0.260803
$457$	33.9282	1.58709	0.793547	−	0.608509i	$-0.208232\pi$
$457$	33.9282	1.58709	0.793547	+	0.608509i	$0.208232\pi$
$458$	5.07180	0.236989
$459$	−3.73205	−0.174197
$460$	1.28203	0.0597751
$461$	27.5885	1.28492	0.642461	−	0.766318i	$-0.277913\pi$
$461$	27.5885	1.28492	0.642461	+	0.766318i	$0.277913\pi$
$462$	0	0
$463$	31.4641	1.46226	0.731130	−	0.682238i	$-0.238993\pi$
$463$	31.4641	1.46226	0.731130	+	0.682238i	$0.238993\pi$
$464$	−1.35898	−0.0630892
$465$	2.00000	0.0927478
$466$	−14.7846	−0.684884
$467$	−34.3923	−1.59149	−0.795743	−	0.605634i	$-0.792919\pi$
$467$	−34.3923	−1.59149	−0.795743	+	0.605634i	$0.792919\pi$
$468$	4.00000	0.184900
$469$	−7.00000	−0.323230
$470$	1.80385	0.0832053
$471$	6.53590	0.301158
$472$	25.6743	1.18176
$473$	0	0
$474$	−0.392305	−0.0180192
$475$	−10.8231	−0.496597
$476$	−5.46410	−0.250447
$477$	2.53590	0.116111
$478$	−18.2487	−0.834677
$479$	−32.3731	−1.47916	−0.739582	−	0.673067i	$-0.764977\pi$
$479$	−32.3731	−1.47916	−0.739582	+	0.673067i	$0.764977\pi$
$480$	1.56922	0.0716247
$481$	25.8564	1.17895
$482$	−11.4641	−0.522176
$483$	−3.26795	−0.148697
$484$	0	0
$485$	−0.875644	−0.0397610
$486$	−0.732051	−0.0332065
$487$	42.1769	1.91122	0.955609	−	0.294637i	$-0.0951988\pi$
$487$	42.1769	1.91122	0.955609	+	0.294637i	$0.0951988\pi$
$488$	20.7846	0.940875
$489$	−7.46410	−0.337538
$490$	−0.196152	−0.00886126
$491$	0.0525589	0.00237195	0.00118597	−	0.999999i	$-0.499622\pi$
$491$	0.0525589	0.00237195	0.00118597	+	0.999999i	$0.499622\pi$
$492$	−5.85641	−0.264027
$493$	−4.73205	−0.213121
$494$	−4.39230	−0.197619
$495$	0	0
$496$	8.00000	0.359211
$497$	9.12436	0.409283
$498$	3.12436	0.140006
$499$	40.3205	1.80499	0.902497	−	0.430696i	$-0.141732\pi$
$499$	40.3205	1.80499	0.902497	+	0.430696i	$0.141732\pi$
$500$	−3.89488	−0.174184
$501$	25.0526	1.11927
$502$	−0.784610	−0.0350188
$503$	18.5167	0.825617	0.412809	−	0.910818i	$-0.364548\pi$
$503$	18.5167	0.825617	0.412809	+	0.910818i	$0.364548\pi$
$504$	−2.53590	−0.112958
$505$	2.60770	0.116041
$506$	0	0
$507$	5.53590	0.245858
$508$	−5.17691	−0.229688
$509$	−17.1962	−0.762206	−0.381103	−	0.924533i	$-0.624456\pi$
$509$	−17.1962	−0.762206	−0.381103	+	0.924533i	$0.624456\pi$
$510$	0.732051	0.0324158
$511$	6.73205	0.297808
$512$	11.7128	0.517638
$513$	−2.19615	−0.0969625
$514$	3.01924	0.133173
$515$	−1.12436	−0.0495450
$516$	−0.679492	−0.0299130
$517$	0	0
$518$	−6.92820	−0.304408
$519$	5.19615	0.228086
$520$	−1.85641	−0.0814088
$521$	−19.9808	−0.875373	−0.437687	−	0.899128i	$-0.644202\pi$
$521$	−19.9808	−0.875373	−0.437687	+	0.899128i	$0.644202\pi$
$522$	−0.928203	−0.0406264
$523$	−26.5885	−1.16263	−0.581316	−	0.813678i	$-0.697462\pi$
$523$	−26.5885	−1.16263	−0.581316	+	0.813678i	$0.697462\pi$
$524$	7.60770	0.332344
$525$	4.92820	0.215084
$526$	−4.24871	−0.185253
$527$	27.8564	1.21344
$528$	0	0
$529$	−12.3205	−0.535674
$530$	−0.497423	−0.0216067
$531$	−10.1244	−0.439360
$532$	−3.21539	−0.139405
$533$	10.9282	0.473353
$534$	10.7321	0.464421
$535$	4.98076	0.215337
$536$	17.7513	0.766739
$537$	4.39230	0.189542
$538$	17.4641	0.752931
$539$	0	0
$540$	−0.392305	−0.0168821
$541$	−41.3923	−1.77959	−0.889797	−	0.456356i	$-0.849154\pi$
$541$	−41.3923	−1.77959	−0.889797	+	0.456356i	$0.849154\pi$
$542$	−2.64102	−0.113441
$543$	8.00000	0.343313
$544$	21.8564	0.937086
$545$	4.66025	0.199623
$546$	2.00000	0.0855921
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	−20.0000	−0.854358
$549$	−8.19615	−0.349803
$550$	0	0
$551$	−2.78461	−0.118628
$552$	8.28719	0.352726
$553$	0.535898	0.0227887
$554$	18.8756	0.801949
$555$	−2.53590	−0.107643
$556$	11.4256	0.484554
$557$	−16.9282	−0.717271	−0.358635	−	0.933478i	$-0.616758\pi$
$557$	−16.9282	−0.717271	−0.358635	+	0.933478i	$0.616758\pi$
$558$	5.46410	0.231314
$559$	1.26795	0.0536285
$560$	−0.287187	−0.0121359
$561$	0	0
$562$	20.2872	0.855763
$563$	9.73205	0.410157	0.205079	−	0.978746i	$-0.434255\pi$
$563$	9.73205	0.410157	0.205079	+	0.978746i	$0.434255\pi$
$564$	−13.4641	−0.566941
$565$	−2.28719	−0.0962227
$566$	19.2154	0.807683
$567$	1.00000	0.0419961
$568$	−23.1384	−0.970867
$569$	−25.2679	−1.05929	−0.529644	−	0.848220i	$-0.677674\pi$
$569$	−25.2679	−1.05929	−0.529644	+	0.848220i	$0.677674\pi$
$570$	0.430781	0.0180434
$571$	−1.07180	−0.0448533	−0.0224266	−	0.999748i	$-0.507139\pi$
$571$	−1.07180	−0.0448533	−0.0224266	+	0.999748i	$0.507139\pi$
$572$	0	0
$573$	−16.1962	−0.676604
$574$	−2.92820	−0.122221
$575$	−16.1051	−0.671630
$576$	2.14359	0.0893164
$577$	−14.4449	−0.601348	−0.300674	−	0.953727i	$-0.597212\pi$
$577$	−14.4449	−0.601348	−0.300674	+	0.953727i	$0.597212\pi$
$578$	−2.24871	−0.0935341
$579$	−5.00000	−0.207793
$580$	−0.497423	−0.0206543
$581$	−4.26795	−0.177064
$582$	−2.39230	−0.0991642
$583$	0	0
$584$	−17.0718	−0.706436
$585$	0.732051	0.0302666
$586$	11.9090	0.491955
$587$	−8.12436	−0.335328	−0.167664	−	0.985844i	$-0.553622\pi$
$587$	−8.12436	−0.335328	−0.167664	+	0.985844i	$0.553622\pi$
$588$	1.46410	0.0603785
$589$	16.3923	0.675433
$590$	1.98592	0.0817589
$591$	12.5359	0.515658
$592$	−10.1436	−0.416899
$593$	−35.5885	−1.46144	−0.730721	−	0.682676i	$-0.760816\pi$
$593$	−35.5885	−1.46144	−0.730721	+	0.682676i	$0.760816\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	−4.00000	−0.163846
$597$	20.1962	0.826573
$598$	−6.53590	−0.267273
$599$	23.1769	0.946983	0.473492	−	0.880798i	$-0.342993\pi$
$599$	23.1769	0.946983	0.473492	+	0.880798i	$0.342993\pi$
$600$	−12.4974	−0.510205
$601$	−31.9090	−1.30159	−0.650797	−	0.759252i	$-0.725565\pi$
$601$	−31.9090	−1.30159	−0.650797	+	0.759252i	$0.725565\pi$
$602$	−0.339746	−0.0138470
$603$	−7.00000	−0.285062
$604$	13.7513	0.559532
$605$	0	0
$606$	7.12436	0.289407
$607$	4.58846	0.186240	0.0931199	−	0.995655i	$-0.470316\pi$
$607$	4.58846	0.186240	0.0931199	+	0.995655i	$0.470316\pi$
$608$	12.8616	0.521605
$609$	1.26795	0.0513799
$610$	1.60770	0.0650937
$611$	25.1244	1.01642
$612$	−5.46410	−0.220873
$613$	14.6077	0.589999	0.295000	−	0.955497i	$-0.404680\pi$
$613$	14.6077	0.589999	0.295000	+	0.955497i	$0.404680\pi$
$614$	−19.4641	−0.785507
$615$	−1.07180	−0.0432190
$616$	0	0
$617$	−39.4641	−1.58876	−0.794382	−	0.607418i	$-0.792205\pi$
$617$	−39.4641	−1.58876	−0.794382	+	0.607418i	$0.792205\pi$
$618$	−3.07180	−0.123566
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	2.92820	0.117599
$621$	−3.26795	−0.131138
$622$	13.9474	0.559241
$623$	−14.6603	−0.587351
$624$	2.92820	0.117222
$625$	23.9282	0.957128
$626$	0.287187	0.0114783
$627$	0	0
$628$	9.56922	0.381853
$629$	−35.3205	−1.40832
$630$	−0.196152	−0.00781490
$631$	−0.856406	−0.0340930	−0.0170465	−	0.999855i	$-0.505426\pi$
$631$	−0.856406	−0.0340930	−0.0170465	+	0.999855i	$0.505426\pi$
$632$	−1.35898	−0.0540575
$633$	−17.7846	−0.706875
$634$	21.6077	0.858151
$635$	−0.947441	−0.0375981
$636$	3.71281	0.147223
$637$	−2.73205	−0.108248
$638$	0	0
$639$	9.12436	0.360954
$640$	2.71797	0.107437
$641$	35.9090	1.41832	0.709159	−	0.705048i	$-0.249075\pi$
$641$	35.9090	1.41832	0.709159	+	0.705048i	$0.249075\pi$
$642$	13.6077	0.537053
$643$	−41.8564	−1.65066	−0.825328	−	0.564654i	$-0.809010\pi$
$643$	−41.8564	−1.65066	−0.825328	+	0.564654i	$0.809010\pi$
$644$	−4.78461	−0.188540
$645$	−0.124356	−0.00489650
$646$	6.00000	0.236067
$647$	−27.1962	−1.06919	−0.534596	−	0.845108i	$-0.679536\pi$
$647$	−27.1962	−1.06919	−0.534596	+	0.845108i	$0.679536\pi$
$648$	−2.53590	−0.0996195
$649$	0	0
$650$	9.85641	0.386600
$651$	−7.46410	−0.292541
$652$	−10.9282	−0.427981
$653$	20.9808	0.821041	0.410520	−	0.911851i	$-0.365347\pi$
$653$	20.9808	0.821041	0.410520	+	0.911851i	$0.365347\pi$
$654$	12.7321	0.497863
$655$	1.39230	0.0544019
$656$	−4.28719	−0.167387
$657$	6.73205	0.262642
$658$	−6.73205	−0.262443
$659$	3.51666	0.136990	0.0684948	−	0.997651i	$-0.478180\pi$
$659$	3.51666	0.136990	0.0684948	+	0.997651i	$0.478180\pi$
$660$	0	0
$661$	−45.8564	−1.78361	−0.891804	−	0.452422i	$-0.850560\pi$
$661$	−45.8564	−1.78361	−0.891804	+	0.452422i	$0.850560\pi$
$662$	19.1628	0.744785
$663$	10.1962	0.395986
$664$	10.8231	0.420017
$665$	−0.588457	−0.0228194
$666$	−6.92820	−0.268462
$667$	−4.14359	−0.160441
$668$	36.6795	1.41917
$669$	−16.0000	−0.618596
$670$	1.37307	0.0530462
$671$	0	0
$672$	−5.85641	−0.225916
$673$	8.60770	0.331802	0.165901	−	0.986142i	$-0.446947\pi$
$673$	8.60770	0.331802	0.165901	+	0.986142i	$0.446947\pi$
$674$	20.4974	0.789531
$675$	4.92820	0.189687
$676$	8.10512	0.311735
$677$	−27.5885	−1.06031	−0.530155	−	0.847901i	$-0.677866\pi$
$677$	−27.5885	−1.06031	−0.530155	+	0.847901i	$0.677866\pi$
$678$	−6.24871	−0.239980
$679$	3.26795	0.125412
$680$	2.53590	0.0972473
$681$	14.6603	0.561782
$682$	0	0
$683$	−28.2487	−1.08091	−0.540453	−	0.841374i	$-0.681747\pi$
$683$	−28.2487	−1.08091	−0.540453	+	0.841374i	$0.681747\pi$
$684$	−3.21539	−0.122944
$685$	−3.66025	−0.139851
$686$	0.732051	0.0279498
$687$	−6.92820	−0.264327
$688$	−0.497423	−0.0189641
$689$	−6.92820	−0.263944
$690$	0.641016	0.0244031
$691$	−8.39230	−0.319258	−0.159629	−	0.987177i	$-0.551030\pi$
$691$	−8.39230	−0.319258	−0.159629	+	0.987177i	$0.551030\pi$
$692$	7.60770	0.289201
$693$	0	0
$694$	−19.0718	−0.723956
$695$	2.09103	0.0793175
$696$	−3.21539	−0.121879
$697$	−14.9282	−0.565446
$698$	−1.17691	−0.0445469
$699$	20.1962	0.763889
$700$	7.21539	0.272716
$701$	−13.6603	−0.515941	−0.257970	−	0.966153i	$-0.583054\pi$
$701$	−13.6603	−0.515941	−0.257970	+	0.966153i	$0.583054\pi$
$702$	2.00000	0.0754851
$703$	−20.7846	−0.783906
$704$	0	0
$705$	−2.46410	−0.0928034
$706$	−8.10512	−0.305040
$707$	−9.73205	−0.366011
$708$	−14.8231	−0.557086
$709$	−8.60770	−0.323269	−0.161634	−	0.986851i	$-0.551677\pi$
$709$	−8.60770	−0.323269	−0.161634	+	0.986851i	$0.551677\pi$
$710$	−1.78976	−0.0671687
$711$	0.535898	0.0200978
$712$	37.1769	1.39326
$713$	24.3923	0.913499
$714$	−2.73205	−0.102244
$715$	0	0
$716$	6.43078	0.240330
$717$	24.9282	0.930961
$718$	−18.3923	−0.686395
$719$	−44.1051	−1.64484	−0.822422	−	0.568878i	$-0.807378\pi$
$719$	−44.1051	−1.64484	−0.822422	+	0.568878i	$0.807378\pi$
$720$	−0.287187	−0.0107028
$721$	4.19615	0.156273
$722$	−10.3782	−0.386237
$723$	15.6603	0.582411
$724$	11.7128	0.435303
$725$	6.24871	0.232071
$726$	0	0
$727$	37.5167	1.39142	0.695708	−	0.718325i	$-0.255091\pi$
$727$	37.5167	1.39142	0.695708	+	0.718325i	$0.255091\pi$
$728$	6.92820	0.256776
$729$	1.00000	0.0370370
$730$	−1.32051	−0.0488742
$731$	−1.73205	−0.0640622
$732$	−12.0000	−0.443533
$733$	14.4449	0.533533	0.266767	−	0.963761i	$-0.414045\pi$
$733$	14.4449	0.533533	0.266767	+	0.963761i	$0.414045\pi$
$734$	−9.32051	−0.344026
$735$	0.267949	0.00988345
$736$	19.1384	0.705452
$737$	0	0
$738$	−2.92820	−0.107789
$739$	−1.32051	−0.0485757	−0.0242878	−	0.999705i	$-0.507732\pi$
$739$	−1.32051	−0.0485757	−0.0242878	+	0.999705i	$0.507732\pi$
$740$	−3.71281	−0.136486
$741$	6.00000	0.220416
$742$	1.85641	0.0681508
$743$	17.1244	0.628232	0.314116	−	0.949385i	$-0.398292\pi$
$743$	17.1244	0.628232	0.314116	+	0.949385i	$0.398292\pi$
$744$	18.9282	0.693942
$745$	−0.732051	−0.0268203
$746$	17.5167	0.641331
$747$	−4.26795	−0.156156
$748$	0	0
$749$	−18.5885	−0.679207
$750$	−1.94744	−0.0711105
$751$	11.3923	0.415711	0.207856	−	0.978160i	$-0.433352\pi$
$751$	11.3923	0.415711	0.207856	+	0.978160i	$0.433352\pi$
$752$	−9.85641	−0.359426
$753$	1.07180	0.0390584
$754$	2.53590	0.0923520
$755$	2.51666	0.0915907
$756$	1.46410	0.0532489
$757$	−7.92820	−0.288155	−0.144078	−	0.989566i	$-0.546022\pi$
$757$	−7.92820	−0.288155	−0.144078	+	0.989566i	$0.546022\pi$
$758$	2.58846	0.0940170
$759$	0	0
$760$	1.49227	0.0541302
$761$	−20.9090	−0.757949	−0.378975	−	0.925407i	$-0.623723\pi$
$761$	−20.9090	−0.757949	−0.378975	+	0.925407i	$0.623723\pi$
$762$	−2.58846	−0.0937699
$763$	−17.3923	−0.629644
$764$	−23.7128	−0.857899
$765$	−1.00000	−0.0361551
$766$	−24.9808	−0.902592
$767$	27.6603	0.998754
$768$	11.7128	0.422650
$769$	38.6410	1.39343	0.696715	−	0.717348i	$-0.254644\pi$
$769$	38.6410	1.39343	0.696715	+	0.717348i	$0.254644\pi$
$770$	0	0
$771$	−4.12436	−0.148535
$772$	−7.32051	−0.263471
$773$	−1.73205	−0.0622975	−0.0311488	−	0.999515i	$-0.509917\pi$
$773$	−1.73205	−0.0622975	−0.0311488	+	0.999515i	$0.509917\pi$
$774$	−0.339746	−0.0122119
$775$	−36.7846	−1.32134
$776$	−8.28719	−0.297493
$777$	9.46410	0.339523
$778$	12.0000	0.430221
$779$	−8.78461	−0.314741
$780$	1.07180	0.0383765
$781$	0	0
$782$	8.92820	0.319272
$783$	1.26795	0.0453128
$784$	1.07180	0.0382785
$785$	1.75129	0.0625062
$786$	3.80385	0.135679
$787$	15.1769	0.540999	0.270499	−	0.962720i	$-0.412811\pi$
$787$	15.1769	0.540999	0.270499	+	0.962720i	$0.412811\pi$
$788$	18.3538	0.653828
$789$	5.80385	0.206622
$790$	−0.105118	−0.00373992
$791$	8.53590	0.303502
$792$	0	0
$793$	22.3923	0.795174
$794$	−9.75129	−0.346060
$795$	0.679492	0.0240991
$796$	29.5692	1.04805
$797$	−18.9090	−0.669790	−0.334895	−	0.942255i	$-0.608701\pi$
$797$	−18.9090	−0.669790	−0.334895	+	0.942255i	$0.608701\pi$
$798$	−1.60770	−0.0569118
$799$	−34.3205	−1.21417
$800$	−28.8616	−1.02041
$801$	−14.6603	−0.517995
$802$	−11.7128	−0.413594
$803$	0	0
$804$	−10.2487	−0.361444
$805$	−0.875644	−0.0308624
$806$	−14.9282	−0.525824
$807$	−23.8564	−0.839785
$808$	24.6795	0.868221
$809$	−20.4449	−0.718803	−0.359402	−	0.933183i	$-0.617019\pi$
$809$	−20.4449	−0.718803	−0.359402	+	0.933183i	$0.617019\pi$
$810$	−0.196152	−0.00689209
$811$	−45.9090	−1.61208	−0.806041	−	0.591860i	$-0.798394\pi$
$811$	−45.9090	−1.61208	−0.806041	+	0.591860i	$0.798394\pi$
$812$	1.85641	0.0651471
$813$	3.60770	0.126527
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	−4.00000	−0.140028
$817$	−1.01924	−0.0356586
$818$	17.3205	0.605597
$819$	−2.73205	−0.0954656
$820$	−1.56922	−0.0547995
$821$	−23.8564	−0.832594	−0.416297	−	0.909229i	$-0.636672\pi$
$821$	−23.8564	−0.832594	−0.416297	+	0.909229i	$0.636672\pi$
$822$	−10.0000	−0.348790
$823$	−29.3205	−1.02205	−0.511024	−	0.859566i	$-0.670734\pi$
$823$	−29.3205	−1.02205	−0.511024	+	0.859566i	$0.670734\pi$
$824$	−10.6410	−0.370697
$825$	0	0
$826$	−7.41154	−0.257881
$827$	29.5167	1.02639	0.513197	−	0.858271i	$-0.328461\pi$
$827$	29.5167	1.02639	0.513197	+	0.858271i	$0.328461\pi$
$828$	−4.78461	−0.166277
$829$	31.8038	1.10459	0.552297	−	0.833648i	$-0.313752\pi$
$829$	31.8038	1.10459	0.552297	+	0.833648i	$0.313752\pi$
$830$	0.837169	0.0290585
$831$	−25.7846	−0.894458
$832$	−5.85641	−0.203034
$833$	3.73205	0.129308
$834$	5.71281	0.197819
$835$	6.71281	0.232306
$836$	0	0
$837$	−7.46410	−0.257997
$838$	7.12436	0.246107
$839$	48.2295	1.66507	0.832533	−	0.553975i	$-0.186890\pi$
$839$	48.2295	1.66507	0.832533	+	0.553975i	$0.186890\pi$
$840$	−0.679492	−0.0234447
$841$	−27.3923	−0.944562
$842$	28.4449	0.980275
$843$	−27.7128	−0.954480
$844$	−26.0385	−0.896281
$845$	1.48334	0.0510284
$846$	−6.73205	−0.231453
$847$	0	0
$848$	2.71797	0.0933354
$849$	−26.2487	−0.900853
$850$	−13.4641	−0.461815
$851$	−30.9282	−1.06021
$852$	13.3590	0.457671
$853$	4.00000	0.136957	0.0684787	−	0.997653i	$-0.478185\pi$
$853$	4.00000	0.136957	0.0684787	+	0.997653i	$0.478185\pi$
$854$	−6.00000	−0.205316
$855$	−0.588457	−0.0201248
$856$	47.1384	1.61116
$857$	−26.1244	−0.892391	−0.446195	−	0.894936i	$-0.647221\pi$
$857$	−26.1244	−0.892391	−0.446195	+	0.894936i	$0.647221\pi$
$858$	0	0
$859$	−30.0000	−1.02359	−0.511793	−	0.859109i	$-0.671019\pi$
$859$	−30.0000	−1.02359	−0.511793	+	0.859109i	$0.671019\pi$
$860$	−0.182069	−0.00620851
$861$	4.00000	0.136320
$862$	−17.0718	−0.581468
$863$	−0.928203	−0.0315964	−0.0157982	−	0.999875i	$-0.505029\pi$
$863$	−0.928203	−0.0315964	−0.0157982	+	0.999875i	$0.505029\pi$
$864$	−5.85641	−0.199239
$865$	1.39230	0.0473398
$866$	24.8231	0.843523
$867$	3.07180	0.104324
$868$	−10.9282	−0.370927
$869$	0	0
$870$	−0.248711	−0.00843210
$871$	19.1244	0.648004
$872$	44.1051	1.49359
$873$	3.26795	0.110603
$874$	5.25387	0.177715
$875$	2.66025	0.0899330
$876$	9.85641	0.333017
$877$	−25.1769	−0.850164	−0.425082	−	0.905155i	$-0.639755\pi$
$877$	−25.1769	−0.850164	−0.425082	+	0.905155i	$0.639755\pi$
$878$	0.105118	0.00354755
$879$	−16.2679	−0.548704
$880$	0	0
$881$	39.8564	1.34280	0.671398	−	0.741097i	$-0.265694\pi$
$881$	39.8564	1.34280	0.671398	+	0.741097i	$0.265694\pi$
$882$	0.732051	0.0246494
$883$	27.7846	0.935027	0.467513	−	0.883986i	$-0.345150\pi$
$883$	27.7846	0.935027	0.467513	+	0.883986i	$0.345150\pi$
$884$	14.9282	0.502090
$885$	−2.71281	−0.0911902
$886$	−4.24871	−0.142738
$887$	10.1244	0.339943	0.169971	−	0.985449i	$-0.445632\pi$
$887$	10.1244	0.339943	0.169971	+	0.985449i	$0.445632\pi$
$888$	−24.0000	−0.805387
$889$	3.53590	0.118590
$890$	2.87564	0.0963918
$891$	0	0
$892$	−23.4256	−0.784348
$893$	−20.1962	−0.675838
$894$	−2.00000	−0.0668900
$895$	1.17691	0.0393399
$896$	−10.1436	−0.338874
$897$	8.92820	0.298104
$898$	−11.8949	−0.396937
$899$	−9.46410	−0.315645
$900$	7.21539	0.240513
$901$	9.46410	0.315295
$902$	0	0
$903$	0.464102	0.0154443
$904$	−21.6462	−0.719941
$905$	2.14359	0.0712555
$906$	6.87564	0.228428
$907$	2.78461	0.0924614	0.0462307	−	0.998931i	$-0.485279\pi$
$907$	2.78461	0.0924614	0.0462307	+	0.998931i	$0.485279\pi$
$908$	21.4641	0.712311
$909$	−9.73205	−0.322792
$910$	0.535898	0.0177649
$911$	26.7846	0.887414	0.443707	−	0.896172i	$-0.353663\pi$
$911$	26.7846	0.887414	0.443707	+	0.896172i	$0.353663\pi$
$912$	−2.35383	−0.0779431
$913$	0	0
$914$	24.8372	0.821541
$915$	−2.19615	−0.0726026
$916$	−10.1436	−0.335154
$917$	−5.19615	−0.171592
$918$	−2.73205	−0.0901711
$919$	6.39230	0.210863	0.105431	−	0.994427i	$-0.466378\pi$
$919$	6.39230	0.210863	0.105431	+	0.994427i	$0.466378\pi$
$920$	2.22055	0.0732092
$921$	26.5885	0.876119
$922$	20.1962	0.665125
$923$	−24.9282	−0.820522
$924$	0	0
$925$	46.6410	1.53355
$926$	23.0333	0.756922
$927$	4.19615	0.137820
$928$	−7.42563	−0.243758
$929$	14.6603	0.480987	0.240494	−	0.970651i	$-0.422691\pi$
$929$	14.6603	0.480987	0.240494	+	0.970651i	$0.422691\pi$
$930$	1.46410	0.0480098
$931$	2.19615	0.0719760
$932$	29.5692	0.968572
$933$	−19.0526	−0.623753
$934$	−25.1769	−0.823814
$935$	0	0
$936$	6.92820	0.226455
$937$	48.4449	1.58262	0.791312	−	0.611412i	$-0.209398\pi$
$937$	48.4449	1.58262	0.791312	+	0.611412i	$0.209398\pi$
$938$	−5.12436	−0.167316
$939$	−0.392305	−0.0128024
$940$	−3.60770	−0.117670
$941$	13.6077	0.443598	0.221799	−	0.975092i	$-0.428807\pi$
$941$	13.6077	0.443598	0.221799	+	0.975092i	$0.428807\pi$
$942$	4.78461	0.155891
$943$	−13.0718	−0.425676
$944$	−10.8513	−0.353178
$945$	0.267949	0.00871639
$946$	0	0
$947$	48.9282	1.58995	0.794976	−	0.606640i	$-0.207483\pi$
$947$	48.9282	1.58995	0.794976	+	0.606640i	$0.207483\pi$
$948$	0.784610	0.0254829
$949$	−18.3923	−0.597039
$950$	−7.92305	−0.257058
$951$	−29.5167	−0.957143
$952$	−9.46410	−0.306733
$953$	49.3205	1.59765	0.798824	−	0.601565i	$-0.205456\pi$
$953$	49.3205	1.59765	0.798824	+	0.601565i	$0.205456\pi$
$954$	1.85641	0.0601034
$955$	−4.33975	−0.140431
$956$	36.4974	1.18041
$957$	0	0
$958$	−23.6987	−0.765671
$959$	13.6603	0.441113
$960$	0.574374	0.0185378
$961$	24.7128	0.797188
$962$	18.9282	0.610270
$963$	−18.5885	−0.599005
$964$	22.9282	0.738468
$965$	−1.33975	−0.0431279
$966$	−2.39230	−0.0769711
$967$	−29.3923	−0.945193	−0.472596	−	0.881279i	$-0.656683\pi$
$967$	−29.3923	−0.945193	−0.472596	+	0.881279i	$0.656683\pi$
$968$	0	0
$969$	−8.19615	−0.263298
$970$	−0.641016	−0.0205818
$971$	−21.8756	−0.702023	−0.351011	−	0.936371i	$-0.614162\pi$
$971$	−21.8756	−0.702023	−0.351011	+	0.936371i	$0.614162\pi$
$972$	1.46410	0.0469611
$973$	−7.80385	−0.250180
$974$	30.8756	0.989319
$975$	−13.4641	−0.431196
$976$	−8.78461	−0.281189
$977$	50.1962	1.60592	0.802959	−	0.596035i	$-0.203258\pi$
$977$	50.1962	1.60592	0.802959	+	0.596035i	$0.203258\pi$
$978$	−5.46410	−0.174723
$979$	0	0
$980$	0.392305	0.0125317
$981$	−17.3923	−0.555294
$982$	0.0384758	0.00122781
$983$	8.78461	0.280186	0.140093	−	0.990138i	$-0.455260\pi$
$983$	8.78461	0.280186	0.140093	+	0.990138i	$0.455260\pi$
$984$	−10.1436	−0.323366
$985$	3.35898	0.107026
$986$	−3.46410	−0.110319
$987$	9.19615	0.292717
$988$	8.78461	0.279476
$989$	−1.51666	−0.0482270
$990$	0	0
$991$	17.6077	0.559327	0.279663	−	0.960098i	$-0.409777\pi$
$991$	17.6077	0.559327	0.279663	+	0.960098i	$0.409777\pi$
$992$	43.7128	1.38788
$993$	−26.1769	−0.830699
$994$	6.67949	0.211861
$995$	5.41154	0.171557
$996$	−6.24871	−0.197998
$997$	61.0333	1.93294	0.966472	−	0.256771i	$-0.0826585\pi$
$997$	61.0333	1.93294	0.966472	+	0.256771i	$0.0826585\pi$
$998$	29.5167	0.934334
$999$	9.46410	0.299431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2541.2.a.o.1.2	✓	2
3.2	odd	2		7623.2.a.bv.1.1		2
11.10	odd	2		2541.2.a.be.1.1	yes	2
33.32	even	2		7623.2.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.o.1.2	✓	2	1.1	even	1	trivial
2541.2.a.be.1.1	yes	2	11.10	odd	2
7623.2.a.x.1.2		2	33.32	even	2
7623.2.a.bv.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2541 = 3 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2541.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{2} -1.00000 q^{3} -1.46410 q^{4} -0.267949 q^{5} -0.732051 q^{6} +1.00000 q^{7} -2.53590 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.732051 q^{2} -1.00000 q^{3} -1.46410 q^{4} -0.267949 q^{5} -0.732051 q^{6} +1.00000 q^{7} -2.53590 q^{8} +1.00000 q^{9} -0.196152 q^{10} +1.46410 q^{12} -2.73205 q^{13} +0.732051 q^{14} +0.267949 q^{15} +1.07180 q^{16} +3.73205 q^{17} +0.732051 q^{18} +2.19615 q^{19} +0.392305 q^{20} -1.00000 q^{21} +3.26795 q^{23} +2.53590 q^{24} -4.92820 q^{25} -2.00000 q^{26} -1.00000 q^{27} -1.46410 q^{28} -1.26795 q^{29} +0.196152 q^{30} +7.46410 q^{31} +5.85641 q^{32} +2.73205 q^{34} -0.267949 q^{35} -1.46410 q^{36} -9.46410 q^{37} +1.60770 q^{38} +2.73205 q^{39} +0.679492 q^{40} -4.00000 q^{41} -0.732051 q^{42} -0.464102 q^{43} -0.267949 q^{45} +2.39230 q^{46} -9.19615 q^{47} -1.07180 q^{48} +1.00000 q^{49} -3.60770 q^{50} -3.73205 q^{51} +4.00000 q^{52} +2.53590 q^{53} -0.732051 q^{54} -2.53590 q^{56} -2.19615 q^{57} -0.928203 q^{58} -10.1244 q^{59} -0.392305 q^{60} -8.19615 q^{61} +5.46410 q^{62} +1.00000 q^{63} +2.14359 q^{64} +0.732051 q^{65} -7.00000 q^{67} -5.46410 q^{68} -3.26795 q^{69} -0.196152 q^{70} +9.12436 q^{71} -2.53590 q^{72} +6.73205 q^{73} -6.92820 q^{74} +4.92820 q^{75} -3.21539 q^{76} +2.00000 q^{78} +0.535898 q^{79} -0.287187 q^{80} +1.00000 q^{81} -2.92820 q^{82} -4.26795 q^{83} +1.46410 q^{84} -1.00000 q^{85} -0.339746 q^{86} +1.26795 q^{87} -14.6603 q^{89} -0.196152 q^{90} -2.73205 q^{91} -4.78461 q^{92} -7.46410 q^{93} -6.73205 q^{94} -0.588457 q^{95} -5.85641 q^{96} +3.26795 q^{97} +0.732051 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 + 2 * q^7 - 12 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{7} - 12 q^{8} + 2 q^{9} + 10 q^{10} - 4 q^{12} - 2 q^{13} - 2 q^{14} + 4 q^{15} + 16 q^{16} + 4 q^{17} - 2 q^{18} - 6 q^{19} - 20 q^{20} - 2 q^{21} + 10 q^{23} + 12 q^{24} + 4 q^{25} - 4 q^{26} - 2 q^{27} + 4 q^{28} - 6 q^{29} - 10 q^{30} + 8 q^{31} - 16 q^{32} + 2 q^{34} - 4 q^{35} + 4 q^{36} - 12 q^{37} + 24 q^{38} + 2 q^{39} + 36 q^{40} - 8 q^{41} + 2 q^{42} + 6 q^{43} - 4 q^{45} - 16 q^{46} - 8 q^{47} - 16 q^{48} + 2 q^{49} - 28 q^{50} - 4 q^{51} + 8 q^{52} + 12 q^{53} + 2 q^{54} - 12 q^{56} + 6 q^{57} + 12 q^{58} + 4 q^{59} + 20 q^{60} - 6 q^{61} + 4 q^{62} + 2 q^{63} + 32 q^{64} - 2 q^{65} - 14 q^{67} - 4 q^{68} - 10 q^{69} + 10 q^{70} - 6 q^{71} - 12 q^{72} + 10 q^{73} - 4 q^{75} - 48 q^{76} + 4 q^{78} + 8 q^{79} - 56 q^{80} + 2 q^{81} + 8 q^{82} - 12 q^{83} - 4 q^{84} - 2 q^{85} - 18 q^{86} + 6 q^{87} - 12 q^{89} + 10 q^{90} - 2 q^{91} + 32 q^{92} - 8 q^{93} - 10 q^{94} + 30 q^{95} + 16 q^{96} + 10 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 + 2 * q^7 - 12 * q^8 + 2 * q^9 + 10 * q^10 - 4 * q^12 - 2 * q^13 - 2 * q^14 + 4 * q^15 + 16 * q^16 + 4 * q^17 - 2 * q^18 - 6 * q^19 - 20 * q^20 - 2 * q^21 + 10 * q^23 + 12 * q^24 + 4 * q^25 - 4 * q^26 - 2 * q^27 + 4 * q^28 - 6 * q^29 - 10 * q^30 + 8 * q^31 - 16 * q^32 + 2 * q^34 - 4 * q^35 + 4 * q^36 - 12 * q^37 + 24 * q^38 + 2 * q^39 + 36 * q^40 - 8 * q^41 + 2 * q^42 + 6 * q^43 - 4 * q^45 - 16 * q^46 - 8 * q^47 - 16 * q^48 + 2 * q^49 - 28 * q^50 - 4 * q^51 + 8 * q^52 + 12 * q^53 + 2 * q^54 - 12 * q^56 + 6 * q^57 + 12 * q^58 + 4 * q^59 + 20 * q^60 - 6 * q^61 + 4 * q^62 + 2 * q^63 + 32 * q^64 - 2 * q^65 - 14 * q^67 - 4 * q^68 - 10 * q^69 + 10 * q^70 - 6 * q^71 - 12 * q^72 + 10 * q^73 - 4 * q^75 - 48 * q^76 + 4 * q^78 + 8 * q^79 - 56 * q^80 + 2 * q^81 + 8 * q^82 - 12 * q^83 - 4 * q^84 - 2 * q^85 - 18 * q^86 + 6 * q^87 - 12 * q^89 + 10 * q^90 - 2 * q^91 + 32 * q^92 - 8 * q^93 - 10 * q^94 + 30 * q^95 + 16 * q^96 + 10 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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