LMFDB - Embedded newform 2541.2.a.be.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:	$0$
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)

$f(q)$	$=$	$q+2.73205 q^{2} -1.00000 q^{3} +5.46410 q^{4} -3.73205 q^{5} -2.73205 q^{6} -1.00000 q^{7} +9.46410 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.73205 q^{2} -1.00000 q^{3} +5.46410 q^{4} -3.73205 q^{5} -2.73205 q^{6} -1.00000 q^{7} +9.46410 q^{8} +1.00000 q^{9} -10.1962 q^{10} -5.46410 q^{12} -0.732051 q^{13} -2.73205 q^{14} +3.73205 q^{15} +14.9282 q^{16} -0.267949 q^{17} +2.73205 q^{18} +8.19615 q^{19} -20.3923 q^{20} +1.00000 q^{21} +6.73205 q^{23} -9.46410 q^{24} +8.92820 q^{25} -2.00000 q^{26} -1.00000 q^{27} -5.46410 q^{28} +4.73205 q^{29} +10.1962 q^{30} +0.535898 q^{31} +21.8564 q^{32} -0.732051 q^{34} +3.73205 q^{35} +5.46410 q^{36} -2.53590 q^{37} +22.3923 q^{38} +0.732051 q^{39} -35.3205 q^{40} +4.00000 q^{41} +2.73205 q^{42} -6.46410 q^{43} -3.73205 q^{45} +18.3923 q^{46} +1.19615 q^{47} -14.9282 q^{48} +1.00000 q^{49} +24.3923 q^{50} +0.267949 q^{51} -4.00000 q^{52} +9.46410 q^{53} -2.73205 q^{54} -9.46410 q^{56} -8.19615 q^{57} +12.9282 q^{58} +14.1244 q^{59} +20.3923 q^{60} -2.19615 q^{61} +1.46410 q^{62} -1.00000 q^{63} +29.8564 q^{64} +2.73205 q^{65} -7.00000 q^{67} -1.46410 q^{68} -6.73205 q^{69} +10.1962 q^{70} -15.1244 q^{71} +9.46410 q^{72} -3.26795 q^{73} -6.92820 q^{74} -8.92820 q^{75} +44.7846 q^{76} +2.00000 q^{78} -7.46410 q^{79} -55.7128 q^{80} +1.00000 q^{81} +10.9282 q^{82} +7.73205 q^{83} +5.46410 q^{84} +1.00000 q^{85} -17.6603 q^{86} -4.73205 q^{87} +2.66025 q^{89} -10.1962 q^{90} +0.732051 q^{91} +36.7846 q^{92} -0.535898 q^{93} +3.26795 q^{94} -30.5885 q^{95} -21.8564 q^{96} +6.73205 q^{97} +2.73205 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.73205	1.93185	0.965926	−	0.258819i	$-0.0833333\pi$
$2$	2.73205	1.93185	0.965926	+	0.258819i	$0.0833333\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.46410	2.73205
$5$	−3.73205	−1.66902	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−3.73205	−1.66902	−0.834512	+	0.550990i	$0.814250\pi$
$6$	−2.73205	−1.11536
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	9.46410	3.34607
$9$	1.00000	0.333333
$10$	−10.1962	−3.22431
$11$	0	0
$11$	0	0
$12$	−5.46410	−1.57735
$13$	−0.732051	−0.203034	−0.101517	−	0.994834i	$-0.532370\pi$
$13$	−0.732051	−0.203034	−0.101517	+	0.994834i	$0.532370\pi$
$14$	−2.73205	−0.730171
$15$	3.73205	0.963611
$16$	14.9282	3.73205
$17$	−0.267949	−0.0649872	−0.0324936	−	0.999472i	$-0.510345\pi$
$17$	−0.267949	−0.0649872	−0.0324936	+	0.999472i	$0.510345\pi$
$18$	2.73205	0.643951
$19$	8.19615	1.88033	0.940163	−	0.340725i	$-0.110672\pi$
$19$	8.19615	1.88033	0.940163	+	0.340725i	$0.110672\pi$
$20$	−20.3923	−4.55986
$21$	1.00000	0.218218
$22$	0	0
$23$	6.73205	1.40373	0.701865	−	0.712310i	$-0.252351\pi$
$23$	6.73205	1.40373	0.701865	+	0.712310i	$0.252351\pi$
$24$	−9.46410	−1.93185
$25$	8.92820	1.78564
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	−5.46410	−1.03262
$29$	4.73205	0.878720	0.439360	−	0.898311i	$-0.355205\pi$
$29$	4.73205	0.878720	0.439360	+	0.898311i	$0.355205\pi$
$30$	10.1962	1.86155
$31$	0.535898	0.0962502	0.0481251	−	0.998841i	$-0.484675\pi$
$31$	0.535898	0.0962502	0.0481251	+	0.998841i	$0.484675\pi$
$32$	21.8564	3.86370
$33$	0	0
$34$	−0.732051	−0.125546
$35$	3.73205	0.630832
$36$	5.46410	0.910684
$37$	−2.53590	−0.416899	−0.208450	−	0.978033i	$-0.566842\pi$
$37$	−2.53590	−0.416899	−0.208450	+	0.978033i	$0.566842\pi$
$38$	22.3923	3.63251
$39$	0.732051	0.117222
$40$	−35.3205	−5.58466
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	2.73205	0.421565
$43$	−6.46410	−0.985766	−0.492883	−	0.870096i	$-0.664057\pi$
$43$	−6.46410	−0.985766	−0.492883	+	0.870096i	$0.664057\pi$
$44$	0	0
$45$	−3.73205	−0.556341
$46$	18.3923	2.71180
$47$	1.19615	0.174477	0.0872384	−	0.996187i	$-0.472196\pi$
$47$	1.19615	0.174477	0.0872384	+	0.996187i	$0.472196\pi$
$48$	−14.9282	−2.15470
$49$	1.00000	0.142857
$50$	24.3923	3.44959
$51$	0.267949	0.0375204
$52$	−4.00000	−0.554700
$53$	9.46410	1.29999	0.649997	−	0.759937i	$-0.274770\pi$
$53$	9.46410	1.29999	0.649997	+	0.759937i	$0.274770\pi$
$54$	−2.73205	−0.371785
$55$	0	0
$56$	−9.46410	−1.26469
$57$	−8.19615	−1.08561
$58$	12.9282	1.69756
$59$	14.1244	1.83883	0.919417	−	0.393284i	$-0.128661\pi$
$59$	14.1244	1.83883	0.919417	+	0.393284i	$0.128661\pi$
$60$	20.3923	2.63264
$61$	−2.19615	−0.281189	−0.140594	−	0.990067i	$-0.544901\pi$
$61$	−2.19615	−0.281189	−0.140594	+	0.990067i	$0.544901\pi$
$62$	1.46410	0.185941
$63$	−1.00000	−0.125988
$64$	29.8564	3.73205
$65$	2.73205	0.338869
$66$	0	0
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	−1.46410	−0.177548
$69$	−6.73205	−0.810444
$70$	10.1962	1.21867
$71$	−15.1244	−1.79493	−0.897465	−	0.441085i	$-0.854594\pi$
$71$	−15.1244	−1.79493	−0.897465	+	0.441085i	$0.854594\pi$
$72$	9.46410	1.11536
$73$	−3.26795	−0.382485	−0.191242	−	0.981543i	$-0.561252\pi$
$73$	−3.26795	−0.382485	−0.191242	+	0.981543i	$0.561252\pi$
$74$	−6.92820	−0.805387
$75$	−8.92820	−1.03094
$76$	44.7846	5.13715
$77$	0	0
$78$	2.00000	0.226455
$79$	−7.46410	−0.839777	−0.419889	−	0.907576i	$-0.637931\pi$
$79$	−7.46410	−0.839777	−0.419889	+	0.907576i	$0.637931\pi$
$80$	−55.7128	−6.22888
$81$	1.00000	0.111111
$82$	10.9282	1.20682
$83$	7.73205	0.848703	0.424351	−	0.905498i	$-0.360502\pi$
$83$	7.73205	0.848703	0.424351	+	0.905498i	$0.360502\pi$
$84$	5.46410	0.596182
$85$	1.00000	0.108465
$86$	−17.6603	−1.90435
$87$	−4.73205	−0.507329
$88$	0	0
$89$	2.66025	0.281986	0.140993	−	0.990011i	$-0.454970\pi$
$89$	2.66025	0.281986	0.140993	+	0.990011i	$0.454970\pi$
$90$	−10.1962	−1.07477
$91$	0.732051	0.0767398
$92$	36.7846	3.83506
$93$	−0.535898	−0.0555701
$94$	3.26795	0.337063
$95$	−30.5885	−3.13831
$96$	−21.8564	−2.23071
$97$	6.73205	0.683536	0.341768	−	0.939784i	$-0.388974\pi$
$97$	6.73205	0.683536	0.341768	+	0.939784i	$0.388974\pi$
$98$	2.73205	0.275979
$99$	0	0
$100$	48.7846	4.87846
$101$	6.26795	0.623684	0.311842	−	0.950134i	$-0.399054\pi$
$101$	6.26795	0.623684	0.311842	+	0.950134i	$0.399054\pi$
$102$	0.732051	0.0724838
$103$	−6.19615	−0.610525	−0.305263	−	0.952268i	$-0.598744\pi$
$103$	−6.19615	−0.610525	−0.305263	+	0.952268i	$0.598744\pi$
$104$	−6.92820	−0.679366
$105$	−3.73205	−0.364211
$106$	25.8564	2.51140
$107$	−12.5885	−1.21697	−0.608486	−	0.793565i	$-0.708223\pi$
$107$	−12.5885	−1.21697	−0.608486	+	0.793565i	$0.708223\pi$
$108$	−5.46410	−0.525783
$109$	−3.39230	−0.324924	−0.162462	−	0.986715i	$-0.551943\pi$
$109$	−3.39230	−0.324924	−0.162462	+	0.986715i	$0.551943\pi$
$110$	0	0
$111$	2.53590	0.240697
$112$	−14.9282	−1.41058
$113$	15.4641	1.45474	0.727370	−	0.686245i	$-0.240742\pi$
$113$	15.4641	1.45474	0.727370	+	0.686245i	$0.240742\pi$
$114$	−22.3923	−2.09723
$115$	−25.1244	−2.34286
$116$	25.8564	2.40071
$117$	−0.732051	−0.0676781
$118$	38.5885	3.55236
$119$	0.267949	0.0245629
$120$	35.3205	3.22431
$121$	0	0
$122$	−6.00000	−0.543214
$123$	−4.00000	−0.360668
$124$	2.92820	0.262960
$125$	−14.6603	−1.31125
$126$	−2.73205	−0.243390
$127$	−10.4641	−0.928539	−0.464269	−	0.885694i	$-0.653683\pi$
$127$	−10.4641	−0.928539	−0.464269	+	0.885694i	$0.653683\pi$
$128$	37.8564	3.34607
$129$	6.46410	0.569132
$130$	7.46410	0.654645
$131$	−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
$133$	−8.19615	−0.710697
$134$	−19.1244	−1.65209
$135$	3.73205	0.321204
$136$	−2.53590	−0.217451
$137$	−3.66025	−0.312717	−0.156358	−	0.987700i	$-0.549975\pi$
$137$	−3.66025	−0.312717	−0.156358	+	0.987700i	$0.549975\pi$
$138$	−18.3923	−1.56566
$139$	18.1962	1.54338	0.771689	−	0.636000i	$-0.219412\pi$
$139$	18.1962	1.54338	0.771689	+	0.636000i	$0.219412\pi$
$140$	20.3923	1.72346
$141$	−1.19615	−0.100734
$142$	−41.3205	−3.46754
$143$	0	0
$144$	14.9282	1.24402
$145$	−17.6603	−1.46660
$146$	−8.92820	−0.738903
$147$	−1.00000	−0.0824786
$148$	−13.8564	−1.13899
$149$	0.732051	0.0599719	0.0299860	−	0.999550i	$-0.490454\pi$
$149$	0.732051	0.0599719	0.0299860	+	0.999550i	$0.490454\pi$
$150$	−24.3923	−1.99162
$151$	−11.3923	−0.927093	−0.463546	−	0.886073i	$-0.653423\pi$
$151$	−11.3923	−0.927093	−0.463546	+	0.886073i	$0.653423\pi$
$152$	77.5692	6.29169
$153$	−0.267949	−0.0216624
$154$	0	0
$155$	−2.00000	−0.160644
$156$	4.00000	0.320256
$157$	−13.4641	−1.07455	−0.537276	−	0.843406i	$-0.680547\pi$
$157$	−13.4641	−1.07455	−0.537276	+	0.843406i	$0.680547\pi$
$158$	−20.3923	−1.62232
$159$	−9.46410	−0.750552
$160$	−81.5692	−6.44861
$161$	−6.73205	−0.530560
$162$	2.73205	0.214650
$163$	0.535898	0.0419748	0.0209874	−	0.999780i	$-0.493319\pi$
$163$	0.535898	0.0419748	0.0209874	+	0.999780i	$0.493319\pi$
$164$	21.8564	1.70670
$165$	0	0
$166$	21.1244	1.63957
$167$	−13.0526	−1.01004	−0.505019	−	0.863108i	$-0.668514\pi$
$167$	−13.0526	−1.01004	−0.505019	+	0.863108i	$0.668514\pi$
$168$	9.46410	0.730171
$169$	−12.4641	−0.958777
$170$	2.73205	0.209539
$171$	8.19615	0.626775
$172$	−35.3205	−2.69316
$173$	−5.19615	−0.395056	−0.197528	−	0.980297i	$-0.563291\pi$
$173$	−5.19615	−0.395056	−0.197528	+	0.980297i	$0.563291\pi$
$174$	−12.9282	−0.980085
$175$	−8.92820	−0.674909
$176$	0	0
$177$	−14.1244	−1.06165
$178$	7.26795	0.544756
$179$	16.3923	1.22522	0.612609	−	0.790386i	$-0.290120\pi$
$179$	16.3923	1.22522	0.612609	+	0.790386i	$0.290120\pi$
$180$	−20.3923	−1.51995
$181$	−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
$183$	2.19615	0.162344
$184$	63.7128	4.69697
$185$	9.46410	0.695815
$186$	−1.46410	−0.107353
$187$	0	0
$188$	6.53590	0.476679
$189$	1.00000	0.0727393
$190$	−83.5692	−6.06275
$191$	5.80385	0.419952	0.209976	−	0.977707i	$-0.432661\pi$
$191$	5.80385	0.419952	0.209976	+	0.977707i	$0.432661\pi$
$192$	−29.8564	−2.15470
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	18.3923	1.32049
$195$	−2.73205	−0.195646
$196$	5.46410	0.390293
$197$	19.4641	1.38676	0.693380	−	0.720572i	$-0.256121\pi$
$197$	19.4641	1.38676	0.693380	+	0.720572i	$0.256121\pi$
$198$	0	0
$199$	−9.80385	−0.694976	−0.347488	−	0.937684i	$-0.612965\pi$
$199$	−9.80385	−0.694976	−0.347488	+	0.937684i	$0.612965\pi$
$200$	84.4974	5.97487
$201$	7.00000	0.493742
$202$	17.1244	1.20487
$203$	−4.73205	−0.332125
$204$	1.46410	0.102508
$205$	−14.9282	−1.04263
$206$	−16.9282	−1.17944
$207$	6.73205	0.467910
$208$	−10.9282	−0.757735
$209$	0	0
$210$	−10.1962	−0.703601
$211$	23.7846	1.63740	0.818700	−	0.574221i	$-0.194695\pi$
$211$	23.7846	1.63740	0.818700	+	0.574221i	$0.194695\pi$
$212$	51.7128	3.55165
$213$	15.1244	1.03630
$214$	−34.3923	−2.35101
$215$	24.1244	1.64527
$216$	−9.46410	−0.643951
$217$	−0.535898	−0.0363792
$218$	−9.26795	−0.627705
$219$	3.26795	0.220828
$220$	0	0
$221$	0.196152	0.0131946
$222$	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	$0.320040\pi$
$224$	−21.8564	−1.46034
$225$	8.92820	0.595214
$226$	42.2487	2.81034
$227$	−2.66025	−0.176567	−0.0882836	−	0.996095i	$-0.528138\pi$
$227$	−2.66025	−0.176567	−0.0882836	+	0.996095i	$0.528138\pi$
$228$	−44.7846	−2.96593
$229$	−6.92820	−0.457829	−0.228914	−	0.973447i	$-0.573518\pi$
$229$	−6.92820	−0.457829	−0.228914	+	0.973447i	$0.573518\pi$
$230$	−68.6410	−4.52605
$231$	0	0
$232$	44.7846	2.94025
$233$	9.80385	0.642271	0.321136	−	0.947033i	$-0.395935\pi$
$233$	9.80385	0.642271	0.321136	+	0.947033i	$0.395935\pi$
$234$	−2.00000	−0.130744
$235$	−4.46410	−0.291206
$236$	77.1769	5.02379
$237$	7.46410	0.484846
$238$	0.732051	0.0474518
$239$	11.0718	0.716175	0.358087	−	0.933688i	$-0.383429\pi$
$239$	11.0718	0.716175	0.358087	+	0.933688i	$0.383429\pi$
$240$	55.7128	3.59625
$241$	−1.66025	−0.106946	−0.0534732	−	0.998569i	$-0.517029\pi$
$241$	−1.66025	−0.106946	−0.0534732	+	0.998569i	$0.517029\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−12.0000	−0.768221
$245$	−3.73205	−0.238432
$246$	−10.9282	−0.696757
$247$	−6.00000	−0.381771
$248$	5.07180	0.322059
$249$	−7.73205	−0.489999
$250$	−40.0526	−2.53315
$251$	−14.9282	−0.942260	−0.471130	−	0.882064i	$-0.656154\pi$
$251$	−14.9282	−0.942260	−0.471130	+	0.882064i	$0.656154\pi$
$252$	−5.46410	−0.344206
$253$	0	0
$254$	−28.5885	−1.79380
$255$	−1.00000	−0.0626224
$256$	43.7128	2.73205
$257$	−20.1244	−1.25532	−0.627661	−	0.778486i	$-0.715988\pi$
$257$	−20.1244	−1.25532	−0.627661	+	0.778486i	$0.715988\pi$
$258$	17.6603	1.09948
$259$	2.53590	0.157573
$260$	14.9282	0.925808
$261$	4.73205	0.292907
$262$	−14.1962	−0.877041
$263$	16.1962	0.998698	0.499349	−	0.866401i	$-0.333573\pi$
$263$	16.1962	0.998698	0.499349	+	0.866401i	$0.333573\pi$
$264$	0	0
$265$	−35.3205	−2.16972
$266$	−22.3923	−1.37296
$267$	−2.66025	−0.162805
$268$	−38.2487	−2.33641
$269$	−3.85641	−0.235129	−0.117565	−	0.993065i	$-0.537509\pi$
$269$	−3.85641	−0.235129	−0.117565	+	0.993065i	$0.537509\pi$
$270$	10.1962	0.620518
$271$	24.3923	1.48173	0.740863	−	0.671656i	$-0.234417\pi$
$271$	24.3923	1.48173	0.740863	+	0.671656i	$0.234417\pi$
$272$	−4.00000	−0.242536
$273$	−0.732051	−0.0443057
$274$	−10.0000	−0.604122
$275$	0	0
$276$	−36.7846	−2.21417
$277$	15.7846	0.948405	0.474203	−	0.880416i	$-0.342736\pi$
$277$	15.7846	0.948405	0.474203	+	0.880416i	$0.342736\pi$
$278$	49.7128	2.98158
$279$	0.535898	0.0320834
$280$	35.3205	2.11080
$281$	27.7128	1.65321	0.826604	−	0.562784i	$-0.190270\pi$
$281$	27.7128	1.65321	0.826604	+	0.562784i	$0.190270\pi$
$282$	−3.26795	−0.194604
$283$	22.2487	1.32255	0.661274	−	0.750144i	$-0.270016\pi$
$283$	22.2487	1.32255	0.661274	+	0.750144i	$0.270016\pi$
$284$	−82.6410	−4.90384
$285$	30.5885	1.81190
$286$	0	0
$287$	−4.00000	−0.236113
$288$	21.8564	1.28790
$289$	−16.9282	−0.995777
$290$	−48.2487	−2.83326
$291$	−6.73205	−0.394640
$292$	−17.8564	−1.04497
$293$	−19.7321	−1.15276	−0.576379	−	0.817182i	$-0.695535\pi$
$293$	−19.7321	−1.15276	−0.576379	+	0.817182i	$0.695535\pi$
$294$	−2.73205	−0.159336
$295$	−52.7128	−3.06906
$296$	−24.0000	−1.39497
$297$	0	0
$298$	2.00000	0.115857
$299$	−4.92820	−0.285005
$300$	−48.7846	−2.81658
$301$	6.46410	0.372585
$302$	−31.1244	−1.79101
$303$	−6.26795	−0.360084
$304$	122.354	7.01747
$305$	8.19615	0.469310
$306$	−0.732051	−0.0418486
$307$	−4.58846	−0.261877	−0.130939	−	0.991390i	$-0.541799\pi$
$307$	−4.58846	−0.261877	−0.130939	+	0.991390i	$0.541799\pi$
$308$	0	0
$309$	6.19615	0.352487
$310$	−5.46410	−0.310340
$311$	−19.0526	−1.08037	−0.540186	−	0.841546i	$-0.681646\pi$
$311$	−19.0526	−1.08037	−0.540186	+	0.841546i	$0.681646\pi$
$312$	6.92820	0.392232
$313$	−20.3923	−1.15264	−0.576321	−	0.817224i	$-0.695512\pi$
$313$	−20.3923	−1.15264	−0.576321	+	0.817224i	$0.695512\pi$
$314$	−36.7846	−2.07588
$315$	3.73205	0.210277
$316$	−40.7846	−2.29431
$317$	−15.5167	−0.871502	−0.435751	−	0.900067i	$-0.643517\pi$
$317$	−15.5167	−0.871502	−0.435751	+	0.900067i	$0.643517\pi$
$318$	−25.8564	−1.44996
$319$	0	0
$320$	−111.426	−6.22888
$321$	12.5885	0.702619
$322$	−18.3923	−1.02496
$323$	−2.19615	−0.122197
$324$	5.46410	0.303561
$325$	−6.53590	−0.362546
$326$	1.46410	0.0810891
$327$	3.39230	0.187595
$328$	37.8564	2.09027
$329$	−1.19615	−0.0659460
$330$	0	0
$331$	−36.1769	−1.98846	−0.994232	−	0.107255i	$-0.965794\pi$
$331$	−36.1769	−1.98846	−0.994232	+	0.107255i	$0.965794\pi$
$332$	42.2487	2.31870
$333$	−2.53590	−0.138966
$334$	−35.6603	−1.95124
$335$	26.1244	1.42733
$336$	14.9282	0.814400
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	−34.0526	−1.85222
$339$	−15.4641	−0.839895
$340$	5.46410	0.296333
$341$	0	0
$342$	22.3923	1.21084
$343$	−1.00000	−0.0539949
$344$	−61.1769	−3.29844
$345$	25.1244	1.35265
$346$	−14.1962	−0.763190
$347$	−12.0526	−0.647015	−0.323508	−	0.946226i	$-0.604862\pi$
$347$	−12.0526	−0.647015	−0.323508	+	0.946226i	$0.604862\pi$
$348$	−25.8564	−1.38605
$349$	22.3923	1.19863	0.599316	−	0.800512i	$-0.295439\pi$
$349$	22.3923	1.19863	0.599316	+	0.800512i	$0.295439\pi$
$350$	−24.3923	−1.30382
$351$	0.732051	0.0390740
$352$	0	0
$353$	−24.9282	−1.32679	−0.663397	−	0.748267i	$-0.730886\pi$
$353$	−24.9282	−1.32679	−0.663397	+	0.748267i	$0.730886\pi$
$354$	−38.5885	−2.05095
$355$	56.4449	2.99578
$356$	14.5359	0.770401
$357$	−0.267949	−0.0141814
$358$	44.7846	2.36694
$359$	0.875644	0.0462147	0.0231074	−	0.999733i	$-0.492644\pi$
$359$	0.875644	0.0462147	0.0231074	+	0.999733i	$0.492644\pi$
$360$	−35.3205	−1.86155
$361$	48.1769	2.53563
$362$	−21.8564	−1.14875
$363$	0	0
$364$	4.00000	0.209657
$365$	12.1962	0.638376
$366$	6.00000	0.313625
$367$	−9.26795	−0.483783	−0.241892	−	0.970303i	$-0.577768\pi$
$367$	−9.26795	−0.483783	−0.241892	+	0.970303i	$0.577768\pi$
$368$	100.497	5.23879
$369$	4.00000	0.208232
$370$	25.8564	1.34421
$371$	−9.46410	−0.491352
$372$	−2.92820	−0.151820
$373$	−10.0718	−0.521498	−0.260749	−	0.965407i	$-0.583969\pi$
$373$	−10.0718	−0.521498	−0.260749	+	0.965407i	$0.583969\pi$
$374$	0	0
$375$	14.6603	0.757052
$376$	11.3205	0.583811
$377$	−3.46410	−0.178410
$378$	2.73205	0.140522
$379$	10.4641	0.537505	0.268752	−	0.963209i	$-0.413389\pi$
$379$	10.4641	0.537505	0.268752	+	0.963209i	$0.413389\pi$
$380$	−167.138	−8.57402
$381$	10.4641	0.536092
$382$	15.8564	0.811284
$383$	−9.87564	−0.504622	−0.252311	−	0.967646i	$-0.581191\pi$
$383$	−9.87564	−0.504622	−0.252311	+	0.967646i	$0.581191\pi$
$384$	−37.8564	−1.93185
$385$	0	0
$386$	−13.6603	−0.695289
$387$	−6.46410	−0.328589
$388$	36.7846	1.86746
$389$	−4.39230	−0.222699	−0.111349	−	0.993781i	$-0.535517\pi$
$389$	−4.39230	−0.222699	−0.111349	+	0.993781i	$0.535517\pi$
$390$	−7.46410	−0.377959
$391$	−1.80385	−0.0912245
$392$	9.46410	0.478009
$393$	5.19615	0.262111
$394$	53.1769	2.67901
$395$	27.8564	1.40161
$396$	0	0
$397$	21.3205	1.07005	0.535023	−	0.844838i	$-0.320303\pi$
$397$	21.3205	1.07005	0.535023	+	0.844838i	$0.320303\pi$
$398$	−26.7846	−1.34259
$399$	8.19615	0.410321
$400$	133.282	6.66410
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	19.1244	0.953836
$403$	−0.392305	−0.0195421
$404$	34.2487	1.70394
$405$	−3.73205	−0.185447
$406$	−12.9282	−0.641616
$407$	0	0
$408$	2.53590	0.125546
$409$	−6.33975	−0.313480	−0.156740	−	0.987640i	$-0.550099\pi$
$409$	−6.33975	−0.313480	−0.156740	+	0.987640i	$0.550099\pi$
$410$	−40.7846	−2.01421
$411$	3.66025	0.180547
$412$	−33.8564	−1.66799
$413$	−14.1244	−0.695014
$414$	18.3923	0.903932
$415$	−28.8564	−1.41651
$416$	−16.0000	−0.784465
$417$	−18.1962	−0.891069
$418$	0	0
$419$	6.26795	0.306209	0.153105	−	0.988210i	$-0.451073\pi$
$419$	6.26795	0.306209	0.153105	+	0.988210i	$0.451073\pi$
$420$	−20.3923	−0.995043
$421$	11.1436	0.543106	0.271553	−	0.962424i	$-0.412463\pi$
$421$	11.1436	0.543106	0.271553	+	0.962424i	$0.412463\pi$
$422$	64.9808	3.16321
$423$	1.19615	0.0581589
$424$	89.5692	4.34987
$425$	−2.39230	−0.116044
$426$	41.3205	2.00199
$427$	2.19615	0.106279
$428$	−68.7846	−3.32483
$429$	0	0
$430$	65.9090	3.17841
$431$	−11.3205	−0.545290	−0.272645	−	0.962115i	$-0.587898\pi$
$431$	−11.3205	−0.545290	−0.272645	+	0.962115i	$0.587898\pi$
$432$	−14.9282	−0.718234
$433$	−31.9090	−1.53345	−0.766724	−	0.641977i	$-0.778114\pi$
$433$	−31.9090	−1.53345	−0.766724	+	0.641977i	$0.778114\pi$
$434$	−1.46410	−0.0702791
$435$	17.6603	0.846744
$436$	−18.5359	−0.887709
$437$	55.1769	2.63947
$438$	8.92820	0.426606
$439$	−27.8564	−1.32951	−0.664757	−	0.747060i	$-0.731465\pi$
$439$	−27.8564	−1.32951	−0.664757	+	0.747060i	$0.731465\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0.535898	0.0254901
$443$	−16.1962	−0.769502	−0.384751	−	0.923020i	$-0.625713\pi$
$443$	−16.1962	−0.769502	−0.384751	+	0.923020i	$0.625713\pi$
$444$	13.8564	0.657596
$445$	−9.92820	−0.470642
$446$	43.7128	2.06986
$447$	−0.732051	−0.0346248
$448$	−29.8564	−1.41058
$449$	32.2487	1.52191	0.760955	−	0.648804i	$-0.224731\pi$
$449$	32.2487	1.52191	0.760955	+	0.648804i	$0.224731\pi$
$450$	24.3923	1.14986
$451$	0	0
$452$	84.4974	3.97442
$453$	11.3923	0.535257
$454$	−7.26795	−0.341102
$455$	−2.73205	−0.128081
$456$	−77.5692	−3.63251
$457$	−20.0718	−0.938919	−0.469460	−	0.882954i	$-0.655551\pi$
$457$	−20.0718	−0.938919	−0.469460	+	0.882954i	$0.655551\pi$
$458$	−18.9282	−0.884457
$459$	0.267949	0.0125068
$460$	−137.282	−6.40081
$461$	3.58846	0.167131	0.0835656	−	0.996502i	$-0.473369\pi$
$461$	3.58846	0.167131	0.0835656	+	0.996502i	$0.473369\pi$
$462$	0	0
$463$	24.5359	1.14028	0.570140	−	0.821548i	$-0.306889\pi$
$463$	24.5359	1.14028	0.570140	+	0.821548i	$0.306889\pi$
$464$	70.6410	3.27943
$465$	2.00000	0.0927478
$466$	26.7846	1.24077
$467$	−13.6077	−0.629689	−0.314845	−	0.949143i	$-0.601952\pi$
$467$	−13.6077	−0.629689	−0.314845	+	0.949143i	$0.601952\pi$
$468$	−4.00000	−0.184900
$469$	7.00000	0.323230
$470$	−12.1962	−0.562567
$471$	13.4641	0.620393
$472$	133.674	6.15286
$473$	0	0
$474$	20.3923	0.936650
$475$	73.1769	3.35759
$476$	1.46410	0.0671070
$477$	9.46410	0.433331
$478$	30.2487	1.38354
$479$	−40.3731	−1.84469	−0.922346	−	0.386364i	$-0.873731\pi$
$479$	−40.3731	−1.84469	−0.922346	+	0.386364i	$0.873731\pi$
$480$	81.5692	3.72311
$481$	1.85641	0.0846448
$482$	−4.53590	−0.206605
$483$	6.73205	0.306319
$484$	0	0
$485$	−25.1244	−1.14084
$486$	−2.73205	−0.123928
$487$	−20.1769	−0.914303	−0.457152	−	0.889389i	$-0.651130\pi$
$487$	−20.1769	−0.914303	−0.457152	+	0.889389i	$0.651130\pi$
$488$	−20.7846	−0.940875
$489$	−0.535898	−0.0242342
$490$	−10.1962	−0.460615
$491$	38.0526	1.71729	0.858644	−	0.512572i	$-0.171307\pi$
$491$	38.0526	1.71729	0.858644	+	0.512572i	$0.171307\pi$
$492$	−21.8564	−0.985363
$493$	−1.26795	−0.0571056
$494$	−16.3923	−0.737525
$495$	0	0
$496$	8.00000	0.359211
$497$	15.1244	0.678420
$498$	−21.1244	−0.946605
$499$	5.67949	0.254249	0.127124	−	0.991887i	$-0.459425\pi$
$499$	5.67949	0.254249	0.127124	+	0.991887i	$0.459425\pi$
$500$	−80.1051	−3.58241
$501$	13.0526	0.583145
$502$	−40.7846	−1.82031
$503$	26.5167	1.18232	0.591160	−	0.806555i	$-0.298670\pi$
$503$	26.5167	1.18232	0.591160	+	0.806555i	$0.298670\pi$
$504$	−9.46410	−0.421565
$505$	−23.3923	−1.04094
$506$	0	0
$507$	12.4641	0.553550
$508$	−57.1769	−2.53682
$509$	−6.80385	−0.301575	−0.150788	−	0.988566i	$-0.548181\pi$
$509$	−6.80385	−0.301575	−0.150788	+	0.988566i	$0.548181\pi$
$510$	−2.73205	−0.120977
$511$	3.26795	0.144566
$512$	43.7128	1.93185
$513$	−8.19615	−0.361869
$514$	−54.9808	−2.42510
$515$	23.1244	1.01898
$516$	35.3205	1.55490
$517$	0	0
$518$	6.92820	0.304408
$519$	5.19615	0.228086
$520$	25.8564	1.13388
$521$	31.9808	1.40110	0.700551	−	0.713602i	$-0.252937\pi$
$521$	31.9808	1.40110	0.700551	+	0.713602i	$0.252937\pi$
$522$	12.9282	0.565852
$523$	−4.58846	−0.200639	−0.100320	−	0.994955i	$-0.531987\pi$
$523$	−4.58846	−0.200639	−0.100320	+	0.994955i	$0.531987\pi$
$524$	−28.3923	−1.24032
$525$	8.92820	0.389659
$526$	44.2487	1.92934
$527$	−0.143594	−0.00625503
$528$	0	0
$529$	22.3205	0.970457
$530$	−96.4974	−4.19158
$531$	14.1244	0.612945
$532$	−44.7846	−1.94166
$533$	−2.92820	−0.126835
$534$	−7.26795	−0.314515
$535$	46.9808	2.03116
$536$	−66.2487	−2.86151
$537$	−16.3923	−0.707380
$538$	−10.5359	−0.454235
$539$	0	0
$540$	20.3923	0.877545
$541$	20.6077	0.885994	0.442997	−	0.896523i	$-0.353915\pi$
$541$	20.6077	0.885994	0.442997	+	0.896523i	$0.353915\pi$
$542$	66.6410	2.86248
$543$	8.00000	0.343313
$544$	−5.85641	−0.251091
$545$	12.6603	0.542306
$546$	−2.00000	−0.0855921
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	−20.0000	−0.854358
$549$	−2.19615	−0.0937295
$550$	0	0
$551$	38.7846	1.65228
$552$	−63.7128	−2.71180
$553$	7.46410	0.317406
$554$	43.1244	1.83218
$555$	−9.46410	−0.401729
$556$	99.4256	4.21659
$557$	3.07180	0.130156	0.0650781	−	0.997880i	$-0.479270\pi$
$557$	3.07180	0.130156	0.0650781	+	0.997880i	$0.479270\pi$
$558$	1.46410	0.0619804
$559$	4.73205	0.200144
$560$	55.7128	2.35430
$561$	0	0
$562$	75.7128	3.19375
$563$	−6.26795	−0.264163	−0.132081	−	0.991239i	$-0.542166\pi$
$563$	−6.26795	−0.264163	−0.132081	+	0.991239i	$0.542166\pi$
$564$	−6.53590	−0.275211
$565$	−57.7128	−2.42800
$566$	60.7846	2.55497
$567$	−1.00000	−0.0419961
$568$	−143.138	−6.00596
$569$	28.7321	1.20451	0.602255	−	0.798304i	$-0.294269\pi$
$569$	28.7321	1.20451	0.602255	+	0.798304i	$0.294269\pi$
$570$	83.5692	3.50033
$571$	14.9282	0.624726	0.312363	−	0.949963i	$-0.398880\pi$
$571$	14.9282	0.624726	0.312363	+	0.949963i	$0.398880\pi$
$572$	0	0
$573$	−5.80385	−0.242459
$574$	−10.9282	−0.456134
$575$	60.1051	2.50656
$576$	29.8564	1.24402
$577$	44.4449	1.85026	0.925132	−	0.379646i	$-0.123954\pi$
$577$	44.4449	1.85026	0.925132	+	0.379646i	$0.123954\pi$
$578$	−46.2487	−1.92369
$579$	5.00000	0.207793
$580$	−96.4974	−4.00684
$581$	−7.73205	−0.320780
$582$	−18.3923	−0.762386
$583$	0	0
$584$	−30.9282	−1.27982
$585$	2.73205	0.112956
$586$	−53.9090	−2.22696
$587$	16.1244	0.665523	0.332762	−	0.943011i	$-0.392020\pi$
$587$	16.1244	0.665523	0.332762	+	0.943011i	$0.392020\pi$
$588$	−5.46410	−0.225336
$589$	4.39230	0.180982
$590$	−144.014	−5.92897
$591$	−19.4641	−0.800646
$592$	−37.8564	−1.55589
$593$	4.41154	0.181160	0.0905802	−	0.995889i	$-0.471128\pi$
$593$	4.41154	0.181160	0.0905802	+	0.995889i	$0.471128\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	4.00000	0.163846
$597$	9.80385	0.401245
$598$	−13.4641	−0.550588
$599$	−39.1769	−1.60073	−0.800363	−	0.599516i	$-0.795360\pi$
$599$	−39.1769	−1.60073	−0.800363	+	0.599516i	$0.795360\pi$
$600$	−84.4974	−3.44959
$601$	−33.9090	−1.38318	−0.691588	−	0.722292i	$-0.743088\pi$
$601$	−33.9090	−1.38318	−0.691588	+	0.722292i	$0.743088\pi$
$602$	17.6603	0.719778
$603$	−7.00000	−0.285062
$604$	−62.2487	−2.53286
$605$	0	0
$606$	−17.1244	−0.695629
$607$	26.5885	1.07919	0.539596	−	0.841924i	$-0.318577\pi$
$607$	26.5885	1.07919	0.539596	+	0.841924i	$0.318577\pi$
$608$	179.138	7.26502
$609$	4.73205	0.191752
$610$	22.3923	0.906638
$611$	−0.875644	−0.0354248
$612$	−1.46410	−0.0591828
$613$	−35.3923	−1.42948	−0.714741	−	0.699389i	$-0.753455\pi$
$613$	−35.3923	−1.42948	−0.714741	+	0.699389i	$0.753455\pi$
$614$	−12.5359	−0.505908
$615$	14.9282	0.601963
$616$	0	0
$617$	−32.5359	−1.30985	−0.654923	−	0.755696i	$-0.727299\pi$
$617$	−32.5359	−1.30985	−0.654923	+	0.755696i	$0.727299\pi$
$618$	16.9282	0.680952
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	−10.9282	−0.438887
$621$	−6.73205	−0.270148
$622$	−52.0526	−2.08712
$623$	−2.66025	−0.106581
$624$	10.9282	0.437478
$625$	10.0718	0.402872
$626$	−55.7128	−2.22673
$627$	0	0
$628$	−73.5692	−2.93573
$629$	0.679492	0.0270931
$630$	10.1962	0.406224
$631$	26.8564	1.06914	0.534568	−	0.845125i	$-0.320474\pi$
$631$	26.8564	1.06914	0.534568	+	0.845125i	$0.320474\pi$
$632$	−70.6410	−2.80995
$633$	−23.7846	−0.945353
$634$	−42.3923	−1.68361
$635$	39.0526	1.54975
$636$	−51.7128	−2.05055
$637$	−0.732051	−0.0290049
$638$	0	0
$639$	−15.1244	−0.598310
$640$	−141.282	−5.58466
$641$	−29.9090	−1.18133	−0.590667	−	0.806916i	$-0.701135\pi$
$641$	−29.9090	−1.18133	−0.590667	+	0.806916i	$0.701135\pi$
$642$	34.3923	1.35736
$643$	−14.1436	−0.557769	−0.278884	−	0.960325i	$-0.589965\pi$
$643$	−14.1436	−0.557769	−0.278884	+	0.960325i	$0.589965\pi$
$644$	−36.7846	−1.44952
$645$	−24.1244	−0.949896
$646$	−6.00000	−0.236067
$647$	−16.8038	−0.660627	−0.330314	−	0.943871i	$-0.607155\pi$
$647$	−16.8038	−0.660627	−0.330314	+	0.943871i	$0.607155\pi$
$648$	9.46410	0.371785
$649$	0	0
$650$	−17.8564	−0.700386
$651$	0.535898	0.0210035
$652$	2.92820	0.114677
$653$	−30.9808	−1.21237	−0.606185	−	0.795323i	$-0.707301\pi$
$653$	−30.9808	−1.21237	−0.606185	+	0.795323i	$0.707301\pi$
$654$	9.26795	0.362405
$655$	19.3923	0.757720
$656$	59.7128	2.33139
$657$	−3.26795	−0.127495
$658$	−3.26795	−0.127398
$659$	41.5167	1.61726	0.808630	−	0.588318i	$-0.200210\pi$
$659$	41.5167	1.61726	0.808630	+	0.588318i	$0.200210\pi$
$660$	0	0
$661$	−18.1436	−0.705704	−0.352852	−	0.935679i	$-0.614788\pi$
$661$	−18.1436	−0.705704	−0.352852	+	0.935679i	$0.614788\pi$
$662$	−98.8372	−3.84142
$663$	−0.196152	−0.00761793
$664$	73.1769	2.83982
$665$	30.5885	1.18617
$666$	−6.92820	−0.268462
$667$	31.8564	1.23348
$668$	−71.3205	−2.75947
$669$	−16.0000	−0.618596
$670$	71.3731	2.75738
$671$	0	0
$672$	21.8564	0.843129
$673$	−29.3923	−1.13299	−0.566495	−	0.824065i	$-0.691701\pi$
$673$	−29.3923	−1.13299	−0.566495	+	0.824065i	$0.691701\pi$
$674$	−76.4974	−2.94657
$675$	−8.92820	−0.343647
$676$	−68.1051	−2.61943
$677$	−3.58846	−0.137916	−0.0689578	−	0.997620i	$-0.521967\pi$
$677$	−3.58846	−0.137916	−0.0689578	+	0.997620i	$0.521967\pi$
$678$	−42.2487	−1.62255
$679$	−6.73205	−0.258352
$680$	9.46410	0.362932
$681$	2.66025	0.101941
$682$	0	0
$683$	20.2487	0.774795	0.387398	−	0.921913i	$-0.373374\pi$
$683$	20.2487	0.774795	0.387398	+	0.921913i	$0.373374\pi$
$684$	44.7846	1.71238
$685$	13.6603	0.521931
$686$	−2.73205	−0.104310
$687$	6.92820	0.264327
$688$	−96.4974	−3.67893
$689$	−6.92820	−0.263944
$690$	68.6410	2.61312
$691$	12.3923	0.471425	0.235713	−	0.971823i	$-0.424258\pi$
$691$	12.3923	0.471425	0.235713	+	0.971823i	$0.424258\pi$
$692$	−28.3923	−1.07931
$693$	0	0
$694$	−32.9282	−1.24994
$695$	−67.9090	−2.57593
$696$	−44.7846	−1.69756
$697$	−1.07180	−0.0405972
$698$	61.1769	2.31558
$699$	−9.80385	−0.370816
$700$	−48.7846	−1.84388
$701$	−3.66025	−0.138246	−0.0691229	−	0.997608i	$-0.522020\pi$
$701$	−3.66025	−0.138246	−0.0691229	+	0.997608i	$0.522020\pi$
$702$	2.00000	0.0754851
$703$	−20.7846	−0.783906
$704$	0	0
$705$	4.46410	0.168128
$706$	−68.1051	−2.56317
$707$	−6.26795	−0.235730
$708$	−77.1769	−2.90049
$709$	−29.3923	−1.10385	−0.551926	−	0.833893i	$-0.686107\pi$
$709$	−29.3923	−1.10385	−0.551926	+	0.833893i	$0.686107\pi$
$710$	154.210	5.78741
$711$	−7.46410	−0.279926
$712$	25.1769	0.943545
$713$	3.60770	0.135109
$714$	−0.732051	−0.0273963
$715$	0	0
$716$	89.5692	3.34736
$717$	−11.0718	−0.413484
$718$	2.39230	0.0892800
$719$	32.1051	1.19732	0.598659	−	0.801004i	$-0.295700\pi$
$719$	32.1051	1.19732	0.598659	+	0.801004i	$0.295700\pi$
$720$	−55.7128	−2.07629
$721$	6.19615	0.230757
$722$	131.622	4.89846
$723$	1.66025	0.0617455
$724$	−43.7128	−1.62457
$725$	42.2487	1.56908
$726$	0	0
$727$	−7.51666	−0.278778	−0.139389	−	0.990238i	$-0.544514\pi$
$727$	−7.51666	−0.278778	−0.139389	+	0.990238i	$0.544514\pi$
$728$	6.92820	0.256776
$729$	1.00000	0.0370370
$730$	33.3205	1.23325
$731$	1.73205	0.0640622
$732$	12.0000	0.443533
$733$	44.4449	1.64161	0.820804	−	0.571210i	$-0.193526\pi$
$733$	44.4449	1.64161	0.820804	+	0.571210i	$0.193526\pi$
$734$	−25.3205	−0.934597
$735$	3.73205	0.137659
$736$	147.138	5.42359
$737$	0	0
$738$	10.9282	0.402273
$739$	−33.3205	−1.22571	−0.612857	−	0.790194i	$-0.709980\pi$
$739$	−33.3205	−1.22571	−0.612857	+	0.790194i	$0.709980\pi$
$740$	51.7128	1.90100
$741$	6.00000	0.220416
$742$	−25.8564	−0.949219
$743$	7.12436	0.261367	0.130684	−	0.991424i	$-0.458283\pi$
$743$	7.12436	0.261367	0.130684	+	0.991424i	$0.458283\pi$
$744$	−5.07180	−0.185941
$745$	−2.73205	−0.100095
$746$	−27.5167	−1.00746
$747$	7.73205	0.282901
$748$	0	0
$749$	12.5885	0.459972
$750$	40.0526	1.46251
$751$	−9.39230	−0.342730	−0.171365	−	0.985208i	$-0.554818\pi$
$751$	−9.39230	−0.342730	−0.171365	+	0.985208i	$0.554818\pi$
$752$	17.8564	0.651156
$753$	14.9282	0.544014
$754$	−9.46410	−0.344662
$755$	42.5167	1.54734
$756$	5.46410	0.198727
$757$	5.92820	0.215464	0.107732	−	0.994180i	$-0.465641\pi$
$757$	5.92820	0.215464	0.107732	+	0.994180i	$0.465641\pi$
$758$	28.5885	1.03838
$759$	0	0
$760$	−289.492	−10.5010
$761$	−44.9090	−1.62795	−0.813974	−	0.580901i	$-0.802700\pi$
$761$	−44.9090	−1.62795	−0.813974	+	0.580901i	$0.802700\pi$
$762$	28.5885	1.03565
$763$	3.39230	0.122810
$764$	31.7128	1.14733
$765$	1.00000	0.0361551
$766$	−26.9808	−0.974855
$767$	−10.3397	−0.373347
$768$	−43.7128	−1.57735
$769$	30.6410	1.10494	0.552472	−	0.833532i	$-0.313685\pi$
$769$	30.6410	1.10494	0.552472	+	0.833532i	$0.313685\pi$
$770$	0	0
$771$	20.1244	0.724761
$772$	−27.3205	−0.983287
$773$	1.73205	0.0622975	0.0311488	−	0.999515i	$-0.490083\pi$
$773$	1.73205	0.0622975	0.0311488	+	0.999515i	$0.490083\pi$
$774$	−17.6603	−0.634785
$775$	4.78461	0.171868
$776$	63.7128	2.28716
$777$	−2.53590	−0.0909748
$778$	−12.0000	−0.430221
$779$	32.7846	1.17463
$780$	−14.9282	−0.534515
$781$	0	0
$782$	−4.92820	−0.176232
$783$	−4.73205	−0.169110
$784$	14.9282	0.533150
$785$	50.2487	1.79345
$786$	14.1962	0.506360
$787$	47.1769	1.68168	0.840838	−	0.541287i	$-0.182063\pi$
$787$	47.1769	1.68168	0.840838	+	0.541287i	$0.182063\pi$
$788$	106.354	3.78870
$789$	−16.1962	−0.576598
$790$	76.1051	2.70770
$791$	−15.4641	−0.549840
$792$	0	0
$793$	1.60770	0.0570909
$794$	58.2487	2.06717
$795$	35.3205	1.25269
$796$	−53.5692	−1.89871
$797$	46.9090	1.66160	0.830800	−	0.556570i	$-0.187883\pi$
$797$	46.9090	1.66160	0.830800	+	0.556570i	$0.187883\pi$
$798$	22.3923	0.792679
$799$	−0.320508	−0.0113388
$800$	195.138	6.89919
$801$	2.66025	0.0939955
$802$	−43.7128	−1.54355
$803$	0	0
$804$	38.2487	1.34893
$805$	25.1244	0.885517
$806$	−1.07180	−0.0377524
$807$	3.85641	0.135752
$808$	59.3205	2.08689
$809$	−38.4449	−1.35165	−0.675825	−	0.737062i	$-0.736212\pi$
$809$	−38.4449	−1.35165	−0.675825	+	0.737062i	$0.736212\pi$
$810$	−10.1962	−0.358256
$811$	−19.9090	−0.699098	−0.349549	−	0.936918i	$-0.613665\pi$
$811$	−19.9090	−0.699098	−0.349549	+	0.936918i	$0.613665\pi$
$812$	−25.8564	−0.907382
$813$	−24.3923	−0.855475
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	4.00000	0.140028
$817$	−52.9808	−1.85356
$818$	−17.3205	−0.605597
$819$	0.732051	0.0255799
$820$	−81.5692	−2.84852
$821$	−3.85641	−0.134590	−0.0672948	−	0.997733i	$-0.521437\pi$
$821$	−3.85641	−0.134590	−0.0672948	+	0.997733i	$0.521437\pi$
$822$	10.0000	0.348790
$823$	5.32051	0.185461	0.0927306	−	0.995691i	$-0.470440\pi$
$823$	5.32051	0.185461	0.0927306	+	0.995691i	$0.470440\pi$
$824$	−58.6410	−2.04286
$825$	0	0
$826$	−38.5885	−1.34266
$827$	15.5167	0.539567	0.269784	−	0.962921i	$-0.413048\pi$
$827$	15.5167	0.539567	0.269784	+	0.962921i	$0.413048\pi$
$828$	36.7846	1.27835
$829$	42.1962	1.46553	0.732766	−	0.680480i	$-0.238229\pi$
$829$	42.1962	1.46553	0.732766	+	0.680480i	$0.238229\pi$
$830$	−78.8372	−2.73648
$831$	−15.7846	−0.547562
$832$	−21.8564	−0.757735
$833$	−0.267949	−0.00928389
$834$	−49.7128	−1.72141
$835$	48.7128	1.68578
$836$	0	0
$837$	−0.535898	−0.0185234
$838$	17.1244	0.591551
$839$	−52.2295	−1.80316	−0.901581	−	0.432611i	$-0.857592\pi$
$839$	−52.2295	−1.80316	−0.901581	+	0.432611i	$0.857592\pi$
$840$	−35.3205	−1.21867
$841$	−6.60770	−0.227852
$842$	30.4449	1.04920
$843$	−27.7128	−0.954480
$844$	129.962	4.47346
$845$	46.5167	1.60022
$846$	3.26795	0.112354
$847$	0	0
$848$	141.282	4.85164
$849$	−22.2487	−0.763574
$850$	−6.53590	−0.224179
$851$	−17.0718	−0.585214
$852$	82.6410	2.83123
$853$	−4.00000	−0.136957	−0.0684787	−	0.997653i	$-0.521815\pi$
$853$	−4.00000	−0.136957	−0.0684787	+	0.997653i	$0.521815\pi$
$854$	6.00000	0.205316
$855$	−30.5885	−1.04610
$856$	−119.138	−4.07207
$857$	1.87564	0.0640708	0.0320354	−	0.999487i	$-0.489801\pi$
$857$	1.87564	0.0640708	0.0320354	+	0.999487i	$0.489801\pi$
$858$	0	0
$859$	−30.0000	−1.02359	−0.511793	−	0.859109i	$-0.671019\pi$
$859$	−30.0000	−1.02359	−0.511793	+	0.859109i	$0.671019\pi$
$860$	131.818	4.49495
$861$	4.00000	0.136320
$862$	−30.9282	−1.05342
$863$	12.9282	0.440081	0.220041	−	0.975491i	$-0.429381\pi$
$863$	12.9282	0.440081	0.220041	+	0.975491i	$0.429381\pi$
$864$	−21.8564	−0.743570
$865$	19.3923	0.659358
$866$	−87.1769	−2.96239
$867$	16.9282	0.574912
$868$	−2.92820	−0.0993897
$869$	0	0
$870$	48.2487	1.63578
$871$	5.12436	0.173632
$872$	−32.1051	−1.08722
$873$	6.73205	0.227845
$874$	150.746	5.09906
$875$	14.6603	0.495607
$876$	17.8564	0.603312
$877$	−37.1769	−1.25538	−0.627688	−	0.778465i	$-0.715998\pi$
$877$	−37.1769	−1.25538	−0.627688	+	0.778465i	$0.715998\pi$
$878$	−76.1051	−2.56842
$879$	19.7321	0.665546
$880$	0	0
$881$	12.1436	0.409128	0.204564	−	0.978853i	$-0.434422\pi$
$881$	12.1436	0.409128	0.204564	+	0.978853i	$0.434422\pi$
$882$	2.73205	0.0919929
$883$	−13.7846	−0.463889	−0.231945	−	0.972729i	$-0.574509\pi$
$883$	−13.7846	−0.463889	−0.231945	+	0.972729i	$0.574509\pi$
$884$	1.07180	0.0360484
$885$	52.7128	1.77192
$886$	−44.2487	−1.48656
$887$	14.1244	0.474249	0.237125	−	0.971479i	$-0.423795\pi$
$887$	14.1244	0.474249	0.237125	+	0.971479i	$0.423795\pi$
$888$	24.0000	0.805387
$889$	10.4641	0.350955
$890$	−27.1244	−0.909210
$891$	0	0
$892$	87.4256	2.92723
$893$	9.80385	0.328073
$894$	−2.00000	−0.0668900
$895$	−61.1769	−2.04492
$896$	−37.8564	−1.26469
$897$	4.92820	0.164548
$898$	88.1051	2.94011
$899$	2.53590	0.0845769
$900$	48.7846	1.62615
$901$	−2.53590	−0.0844830
$902$	0	0
$903$	−6.46410	−0.215112
$904$	146.354	4.86766
$905$	29.8564	0.992461
$906$	31.1244	1.03404
$907$	−38.7846	−1.28782	−0.643911	−	0.765100i	$-0.722689\pi$
$907$	−38.7846	−1.28782	−0.643911	+	0.765100i	$0.722689\pi$
$908$	−14.5359	−0.482391
$909$	6.26795	0.207895
$910$	−7.46410	−0.247433
$911$	−14.7846	−0.489836	−0.244918	−	0.969544i	$-0.578761\pi$
$911$	−14.7846	−0.489836	−0.244918	+	0.969544i	$0.578761\pi$
$912$	−122.354	−4.05154
$913$	0	0
$914$	−54.8372	−1.81385
$915$	−8.19615	−0.270956
$916$	−37.8564	−1.25081
$917$	5.19615	0.171592
$918$	0.732051	0.0241613
$919$	14.3923	0.474758	0.237379	−	0.971417i	$-0.423712\pi$
$919$	14.3923	0.474758	0.237379	+	0.971417i	$0.423712\pi$
$920$	−237.779	−7.83936
$921$	4.58846	0.151195
$922$	9.80385	0.322873
$923$	11.0718	0.364433
$924$	0	0
$925$	−22.6410	−0.744432
$926$	67.0333	2.20285
$927$	−6.19615	−0.203508
$928$	103.426	3.39511
$929$	−2.66025	−0.0872801	−0.0436401	−	0.999047i	$-0.513895\pi$
$929$	−2.66025	−0.0872801	−0.0436401	+	0.999047i	$0.513895\pi$
$930$	5.46410	0.179175
$931$	8.19615	0.268618
$932$	53.5692	1.75472
$933$	19.0526	0.623753
$934$	−37.1769	−1.21647
$935$	0	0
$936$	−6.92820	−0.226455
$937$	10.4449	0.341219	0.170609	−	0.985339i	$-0.445426\pi$
$937$	10.4449	0.341219	0.170609	+	0.985339i	$0.445426\pi$
$938$	19.1244	0.624432
$939$	20.3923	0.665478
$940$	−24.3923	−0.795589
$941$	−34.3923	−1.12116	−0.560579	−	0.828101i	$-0.689421\pi$
$941$	−34.3923	−1.12116	−0.560579	+	0.828101i	$0.689421\pi$
$942$	36.7846	1.19851
$943$	26.9282	0.876903
$944$	210.851	6.86262
$945$	−3.73205	−0.121404
$946$	0	0
$947$	35.0718	1.13968	0.569840	−	0.821756i	$-0.307005\pi$
$947$	35.0718	1.13968	0.569840	+	0.821756i	$0.307005\pi$
$948$	40.7846	1.32462
$949$	2.39230	0.0776575
$950$	199.923	6.48636
$951$	15.5167	0.503162
$952$	2.53590	0.0821889
$953$	−14.6795	−0.475515	−0.237758	−	0.971324i	$-0.576412\pi$
$953$	−14.6795	−0.475515	−0.237758	+	0.971324i	$0.576412\pi$
$954$	25.8564	0.837132
$955$	−21.6603	−0.700909
$956$	60.4974	1.95663
$957$	0	0
$958$	−110.301	−3.56367
$959$	3.66025	0.118196
$960$	111.426	3.59625
$961$	−30.7128	−0.990736
$962$	5.07180	0.163521
$963$	−12.5885	−0.405657
$964$	−9.07180	−0.292183
$965$	18.6603	0.600695
$966$	18.3923	0.591763
$967$	8.60770	0.276805	0.138402	−	0.990376i	$-0.455803\pi$
$967$	8.60770	0.276805	0.138402	+	0.990376i	$0.455803\pi$
$968$	0	0
$969$	2.19615	0.0705506
$970$	−68.6410	−2.20393
$971$	−46.1244	−1.48020	−0.740101	−	0.672496i	$-0.765222\pi$
$971$	−46.1244	−1.48020	−0.740101	+	0.672496i	$0.765222\pi$
$972$	−5.46410	−0.175261
$973$	−18.1962	−0.583342
$974$	−55.1244	−1.76630
$975$	6.53590	0.209316
$976$	−32.7846	−1.04941
$977$	39.8038	1.27344	0.636719	−	0.771096i	$-0.280291\pi$
$977$	39.8038	1.27344	0.636719	+	0.771096i	$0.280291\pi$
$978$	−1.46410	−0.0468168
$979$	0	0
$980$	−20.3923	−0.651408
$981$	−3.39230	−0.108308
$982$	103.962	3.31755
$983$	−32.7846	−1.04567	−0.522833	−	0.852435i	$-0.675125\pi$
$983$	−32.7846	−1.04567	−0.522833	+	0.852435i	$0.675125\pi$
$984$	−37.8564	−1.20682
$985$	−72.6410	−2.31454
$986$	−3.46410	−0.110319
$987$	1.19615	0.0380740
$988$	−32.7846	−1.04302
$989$	−43.5167	−1.38375
$990$	0	0
$991$	38.3923	1.21957	0.609786	−	0.792566i	$-0.291255\pi$
$991$	38.3923	1.21957	0.609786	+	0.792566i	$0.291255\pi$
$992$	11.7128	0.371882
$993$	36.1769	1.14804
$994$	41.3205	1.31061
$995$	36.5885	1.15993
$996$	−42.2487	−1.33870
$997$	29.0333	0.919495	0.459747	−	0.888050i	$-0.347940\pi$
$997$	29.0333	0.919495	0.459747	+	0.888050i	$0.347940\pi$
$998$	15.5167	0.491171
$999$	2.53590	0.0802323

Twists

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

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

\(f(q)\)	\(=\)	\(q+2.73205 q^{2} -1.00000 q^{3} +5.46410 q^{4} -3.73205 q^{5} -2.73205 q^{6} -1.00000 q^{7} +9.46410 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.73205 q^{2} -1.00000 q^{3} +5.46410 q^{4} -3.73205 q^{5} -2.73205 q^{6} -1.00000 q^{7} +9.46410 q^{8} +1.00000 q^{9} -10.1962 q^{10} -5.46410 q^{12} -0.732051 q^{13} -2.73205 q^{14} +3.73205 q^{15} +14.9282 q^{16} -0.267949 q^{17} +2.73205 q^{18} +8.19615 q^{19} -20.3923 q^{20} +1.00000 q^{21} +6.73205 q^{23} -9.46410 q^{24} +8.92820 q^{25} -2.00000 q^{26} -1.00000 q^{27} -5.46410 q^{28} +4.73205 q^{29} +10.1962 q^{30} +0.535898 q^{31} +21.8564 q^{32} -0.732051 q^{34} +3.73205 q^{35} +5.46410 q^{36} -2.53590 q^{37} +22.3923 q^{38} +0.732051 q^{39} -35.3205 q^{40} +4.00000 q^{41} +2.73205 q^{42} -6.46410 q^{43} -3.73205 q^{45} +18.3923 q^{46} +1.19615 q^{47} -14.9282 q^{48} +1.00000 q^{49} +24.3923 q^{50} +0.267949 q^{51} -4.00000 q^{52} +9.46410 q^{53} -2.73205 q^{54} -9.46410 q^{56} -8.19615 q^{57} +12.9282 q^{58} +14.1244 q^{59} +20.3923 q^{60} -2.19615 q^{61} +1.46410 q^{62} -1.00000 q^{63} +29.8564 q^{64} +2.73205 q^{65} -7.00000 q^{67} -1.46410 q^{68} -6.73205 q^{69} +10.1962 q^{70} -15.1244 q^{71} +9.46410 q^{72} -3.26795 q^{73} -6.92820 q^{74} -8.92820 q^{75} +44.7846 q^{76} +2.00000 q^{78} -7.46410 q^{79} -55.7128 q^{80} +1.00000 q^{81} +10.9282 q^{82} +7.73205 q^{83} +5.46410 q^{84} +1.00000 q^{85} -17.6603 q^{86} -4.73205 q^{87} +2.66025 q^{89} -10.1962 q^{90} +0.732051 q^{91} +36.7846 q^{92} -0.535898 q^{93} +3.26795 q^{94} -30.5885 q^{95} -21.8564 q^{96} +6.73205 q^{97} +2.73205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9}+O(q^{10}) \) 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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Representations

Groups

Database

Properties

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