LMFDB - Embedded newform 2541.2.a.bn.1.3

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:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.737640$ 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)

$f(q)$	$=$	$q+1.45589 q^{2} -1.00000 q^{3} +0.119606 q^{4} +2.38705 q^{5} -1.45589 q^{6} -1.00000 q^{7} -2.73764 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.45589 q^{2} -1.00000 q^{3} +0.119606 q^{4} +2.38705 q^{5} -1.45589 q^{6} -1.00000 q^{7} -2.73764 q^{8} +1.00000 q^{9} +3.47528 q^{10} -0.119606 q^{12} -0.911774 q^{13} -1.45589 q^{14} -2.38705 q^{15} -4.22491 q^{16} -2.18663 q^{17} +1.45589 q^{18} +0.711349 q^{19} +0.285507 q^{20} +1.00000 q^{21} -7.80466 q^{23} +2.73764 q^{24} +0.698028 q^{25} -1.32744 q^{26} -1.00000 q^{27} -0.119606 q^{28} +8.86742 q^{29} -3.47528 q^{30} -6.62627 q^{31} -0.675706 q^{32} -3.18348 q^{34} -2.38705 q^{35} +0.119606 q^{36} -4.20551 q^{37} +1.03564 q^{38} +0.911774 q^{39} -6.53490 q^{40} -11.9576 q^{41} +1.45589 q^{42} +7.51601 q^{43} +2.38705 q^{45} -11.3627 q^{46} +6.27171 q^{47} +4.22491 q^{48} +1.00000 q^{49} +1.01625 q^{50} +2.18663 q^{51} -0.109054 q^{52} -6.56545 q^{53} -1.45589 q^{54} +2.73764 q^{56} -0.711349 q^{57} +12.9100 q^{58} -9.62312 q^{59} -0.285507 q^{60} -0.0627598 q^{61} -9.64709 q^{62} -1.00000 q^{63} +7.46606 q^{64} -2.17645 q^{65} -11.3294 q^{67} -0.261535 q^{68} +7.80466 q^{69} -3.47528 q^{70} -5.96945 q^{71} -2.73764 q^{72} -4.38705 q^{73} -6.12275 q^{74} -0.698028 q^{75} +0.0850818 q^{76} +1.32744 q^{78} -4.85725 q^{79} -10.0851 q^{80} +1.00000 q^{81} -17.4089 q^{82} +10.0490 q^{83} +0.119606 q^{84} -5.21960 q^{85} +10.9425 q^{86} -8.86742 q^{87} -0.261535 q^{89} +3.47528 q^{90} +0.911774 q^{91} -0.933487 q^{92} +6.62627 q^{93} +9.13090 q^{94} +1.69803 q^{95} +0.675706 q^{96} +9.88466 q^{97} +1.45589 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{5} - q^{6} - 4 q^{7} - 9 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q + q^2 - 4 * q^3 + 3 * q^4 - 4 * q^5 - q^6 - 4 * q^7 - 9 * q^8 + 4 * q^9	$ 4 q + q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{5} - q^{6} - 4 q^{7} - 9 q^{8} + 4 q^{9} + 10 q^{10} - 3 q^{12} + 6 q^{13} - q^{14} + 4 q^{15} - 3 q^{16} + 8 q^{17} + q^{18} - 10 q^{19} + 4 q^{21} - 10 q^{23} + 9 q^{24} + 12 q^{25} - 20 q^{26} - 4 q^{27} - 3 q^{28} - 10 q^{30} - 18 q^{31} - 2 q^{32} + 18 q^{34} + 4 q^{35} + 3 q^{36} - 2 q^{37} - 8 q^{38} - 6 q^{39} + 6 q^{40} + 10 q^{41} + q^{42} - 4 q^{43} - 4 q^{45} + 11 q^{46} + 4 q^{47} + 3 q^{48} + 4 q^{49} - 9 q^{50} - 8 q^{51} + 20 q^{52} - q^{54} + 9 q^{56} + 10 q^{57} + 14 q^{58} - 16 q^{59} + 14 q^{61} - 4 q^{63} - 11 q^{64} - 28 q^{65} - 28 q^{67} - 16 q^{68} + 10 q^{69} - 10 q^{70} - 18 q^{71} - 9 q^{72} - 4 q^{73} - 41 q^{74} - 12 q^{75} - 4 q^{76} + 20 q^{78} - 20 q^{79} - 36 q^{80} + 4 q^{81} - 24 q^{82} + 6 q^{83} + 3 q^{84} - 20 q^{85} - 20 q^{86} - 16 q^{89} + 10 q^{90} - 6 q^{91} - 22 q^{92} + 18 q^{93} - 16 q^{94} + 16 q^{95} + 2 q^{96} + 32 q^{97} + q^{98}+O(q^{100}) $ 4 * q + q^2 - 4 * q^3 + 3 * q^4 - 4 * q^5 - q^6 - 4 * q^7 - 9 * q^8 + 4 * q^9 + 10 * q^10 - 3 * q^12 + 6 * q^13 - q^14 + 4 * q^15 - 3 * q^16 + 8 * q^17 + q^18 - 10 * q^19 + 4 * q^21 - 10 * q^23 + 9 * q^24 + 12 * q^25 - 20 * q^26 - 4 * q^27 - 3 * q^28 - 10 * q^30 - 18 * q^31 - 2 * q^32 + 18 * q^34 + 4 * q^35 + 3 * q^36 - 2 * q^37 - 8 * q^38 - 6 * q^39 + 6 * q^40 + 10 * q^41 + q^42 - 4 * q^43 - 4 * q^45 + 11 * q^46 + 4 * q^47 + 3 * q^48 + 4 * q^49 - 9 * q^50 - 8 * q^51 + 20 * q^52 - q^54 + 9 * q^56 + 10 * q^57 + 14 * q^58 - 16 * q^59 + 14 * q^61 - 4 * q^63 - 11 * q^64 - 28 * q^65 - 28 * q^67 - 16 * q^68 + 10 * q^69 - 10 * q^70 - 18 * q^71 - 9 * q^72 - 4 * q^73 - 41 * q^74 - 12 * q^75 - 4 * q^76 + 20 * q^78 - 20 * q^79 - 36 * q^80 + 4 * q^81 - 24 * q^82 + 6 * q^83 + 3 * q^84 - 20 * q^85 - 20 * q^86 - 16 * q^89 + 10 * q^90 - 6 * q^91 - 22 * q^92 + 18 * q^93 - 16 * q^94 + 16 * q^95 + 2 * q^96 + 32 * q^97 + 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 $

$2$	1.45589	1.02947	0.514734	−	0.857350i	$-0.327891\pi$
$2$	1.45589	1.02947	0.514734	+	0.857350i	$0.327891\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.119606	0.0598032
$5$	2.38705	1.06752	0.533762	−	0.845635i	$-0.320778\pi$
$5$	2.38705	1.06752	0.533762	+	0.845635i	$0.320778\pi$
$6$	−1.45589	−0.594363
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.73764	−0.967902
$9$	1.00000	0.333333
$10$	3.47528	1.09898
$11$	0	0
$11$	0	0
$12$	−0.119606	−0.0345274
$13$	−0.911774	−0.252880	−0.126440	−	0.991974i	$-0.540355\pi$
$13$	−0.911774	−0.252880	−0.126440	+	0.991974i	$0.540355\pi$
$14$	−1.45589	−0.389102
$15$	−2.38705	−0.616335
$16$	−4.22491	−1.05623
$17$	−2.18663	−0.530335	−0.265168	−	0.964202i	$-0.585427\pi$
$17$	−2.18663	−0.530335	−0.265168	+	0.964202i	$0.585427\pi$
$18$	1.45589	0.343156
$19$	0.711349	0.163195	0.0815973	−	0.996665i	$-0.473998\pi$
$19$	0.711349	0.163195	0.0815973	+	0.996665i	$0.473998\pi$
$20$	0.285507	0.0638413
$21$	1.00000	0.218218
$22$	0	0
$23$	−7.80466	−1.62738	−0.813692	−	0.581296i	$-0.802546\pi$
$23$	−7.80466	−1.62738	−0.813692	+	0.581296i	$0.802546\pi$
$24$	2.73764	0.558818
$25$	0.698028	0.139606
$26$	−1.32744	−0.260332
$27$	−1.00000	−0.192450
$28$	−0.119606	−0.0226035
$29$	8.86742	1.64664	0.823320	−	0.567578i	$-0.192120\pi$
$29$	8.86742	1.64664	0.823320	+	0.567578i	$0.192120\pi$
$30$	−3.47528	−0.634497
$31$	−6.62627	−1.19011	−0.595056	−	0.803684i	$-0.702870\pi$
$31$	−6.62627	−1.19011	−0.595056	+	0.803684i	$0.702870\pi$
$32$	−0.675706	−0.119449
$33$	0	0
$34$	−3.18348	−0.545963
$35$	−2.38705	−0.403486
$36$	0.119606	0.0199344
$37$	−4.20551	−0.691382	−0.345691	−	0.938348i	$-0.612355\pi$
$37$	−4.20551	−0.691382	−0.345691	+	0.938348i	$0.612355\pi$
$38$	1.03564	0.168003
$39$	0.911774	0.146001
$40$	−6.53490	−1.03326
$41$	−11.9576	−1.86746	−0.933731	−	0.357975i	$-0.883467\pi$
$41$	−11.9576	−1.86746	−0.933731	+	0.357975i	$0.883467\pi$
$42$	1.45589	0.224648
$43$	7.51601	1.14618	0.573091	−	0.819492i	$-0.305744\pi$
$43$	7.51601	1.14618	0.573091	+	0.819492i	$0.305744\pi$
$44$	0	0
$45$	2.38705	0.355841
$46$	−11.3627	−1.67534
$47$	6.27171	0.914823	0.457412	−	0.889255i	$-0.348777\pi$
$47$	6.27171	0.914823	0.457412	+	0.889255i	$0.348777\pi$
$48$	4.22491	0.609813
$49$	1.00000	0.142857
$50$	1.01625	0.143719
$51$	2.18663	0.306189
$52$	−0.109054	−0.0151231
$53$	−6.56545	−0.901834	−0.450917	−	0.892566i	$-0.648903\pi$
$53$	−6.56545	−0.901834	−0.450917	+	0.892566i	$0.648903\pi$
$54$	−1.45589	−0.198121
$55$	0	0
$56$	2.73764	0.365833
$57$	−0.711349	−0.0942204
$58$	12.9100	1.69516
$59$	−9.62312	−1.25282	−0.626412	−	0.779492i	$-0.715477\pi$
$59$	−9.62312	−1.25282	−0.626412	+	0.779492i	$0.715477\pi$
$60$	−0.285507	−0.0368588
$61$	−0.0627598	−0.00803556	−0.00401778	−	0.999992i	$-0.501279\pi$
$61$	−0.0627598	−0.00803556	−0.00401778	+	0.999992i	$0.501279\pi$
$62$	−9.64709	−1.22518
$63$	−1.00000	−0.125988
$64$	7.46606	0.933258
$65$	−2.17645	−0.269956
$66$	0	0
$67$	−11.3294	−1.38410	−0.692052	−	0.721847i	$-0.743293\pi$
$67$	−11.3294	−1.38410	−0.692052	+	0.721847i	$0.743293\pi$
$68$	−0.261535	−0.0317157
$69$	7.80466	0.939571
$70$	−3.47528	−0.415375
$71$	−5.96945	−0.708443	−0.354221	−	0.935162i	$-0.615254\pi$
$71$	−5.96945	−0.708443	−0.354221	+	0.935162i	$0.615254\pi$
$72$	−2.73764	−0.322634
$73$	−4.38705	−0.513466	−0.256733	−	0.966482i	$-0.582646\pi$
$73$	−4.38705	−0.513466	−0.256733	+	0.966482i	$0.582646\pi$
$74$	−6.12275	−0.711755
$75$	−0.698028	−0.0806013
$76$	0.0850818	0.00975955
$77$	0	0
$78$	1.32744	0.150303
$79$	−4.85725	−0.546483	−0.273241	−	0.961945i	$-0.588096\pi$
$79$	−4.85725	−0.546483	−0.273241	+	0.961945i	$0.588096\pi$
$80$	−10.0851	−1.12755
$81$	1.00000	0.111111
$82$	−17.4089	−1.92249
$83$	10.0490	1.10302	0.551509	−	0.834169i	$-0.314052\pi$
$83$	10.0490	1.10302	0.551509	+	0.834169i	$0.314052\pi$
$84$	0.119606	0.0130501
$85$	−5.21960	−0.566145
$86$	10.9425	1.17996
$87$	−8.86742	−0.950688
$88$	0	0
$89$	−0.261535	−0.0277226	−0.0138613	−	0.999904i	$-0.504412\pi$
$89$	−0.261535	−0.0277226	−0.0138613	+	0.999904i	$0.504412\pi$
$90$	3.47528	0.366327
$91$	0.911774	0.0955798
$92$	−0.933487	−0.0973227
$93$	6.62627	0.687112
$94$	9.13090	0.941781
$95$	1.69803	0.174214
$96$	0.675706	0.0689639
$97$	9.88466	1.00363	0.501817	−	0.864974i	$-0.332665\pi$
$97$	9.88466	1.00363	0.501817	+	0.864974i	$0.332665\pi$
$98$	1.45589	0.147067
$99$	0	0
$100$	0.0834885	0.00834885
$101$	9.94742	0.989805	0.494902	−	0.868949i	$-0.335204\pi$
$101$	9.94742	0.989805	0.494902	+	0.868949i	$0.335204\pi$
$102$	3.18348	0.315212
$103$	−0.658765	−0.0649101	−0.0324550	−	0.999473i	$-0.510333\pi$
$103$	−0.658765	−0.0649101	−0.0324550	+	0.999473i	$0.510333\pi$
$104$	2.49611	0.244764
$105$	2.38705	0.232953
$106$	−9.55855	−0.928409
$107$	0.283088	0.0273672	0.0136836	−	0.999906i	$-0.495644\pi$
$107$	0.283088	0.0273672	0.0136836	+	0.999906i	$0.495644\pi$
$108$	−0.119606	−0.0115091
$109$	−8.97962	−0.860092	−0.430046	−	0.902807i	$-0.641503\pi$
$109$	−8.97962	−0.860092	−0.430046	+	0.902807i	$0.641503\pi$
$110$	0	0
$111$	4.20551	0.399170
$112$	4.22491	0.399216
$113$	1.32938	0.125058	0.0625289	−	0.998043i	$-0.480083\pi$
$113$	1.32938	0.125058	0.0625289	+	0.998043i	$0.480083\pi$
$114$	−1.03564	−0.0969969
$115$	−18.6302	−1.73727
$116$	1.06060	0.0984742
$117$	−0.911774	−0.0842935
$118$	−14.0102	−1.28974
$119$	2.18663	0.200448
$120$	6.53490	0.596552
$121$	0	0
$122$	−0.0913711	−0.00827235
$123$	11.9576	1.07818
$124$	−0.792543	−0.0711725
$125$	−10.2690	−0.918491
$126$	−1.45589	−0.129701
$127$	−21.8372	−1.93773	−0.968867	−	0.247580i	$-0.920365\pi$
$127$	−21.8372	−1.93773	−0.968867	+	0.247580i	$0.920365\pi$
$128$	12.2212	1.08021
$129$	−7.51601	−0.661748
$130$	−3.16867	−0.277911
$131$	−9.22227	−0.805754	−0.402877	−	0.915254i	$-0.631990\pi$
$131$	−9.22227	−0.805754	−0.402877	+	0.915254i	$0.631990\pi$
$132$	0	0
$133$	−0.711349	−0.0616817
$134$	−16.4943	−1.42489
$135$	−2.38705	−0.205445
$136$	5.98620	0.513313
$137$	−2.04750	−0.174929	−0.0874647	−	0.996168i	$-0.527876\pi$
$137$	−2.04750	−0.174929	−0.0874647	+	0.996168i	$0.527876\pi$
$138$	11.3627	0.967258
$139$	−12.8454	−1.08953	−0.544766	−	0.838588i	$-0.683382\pi$
$139$	−12.8454	−1.08953	−0.544766	+	0.838588i	$0.683382\pi$
$140$	−0.285507	−0.0241297
$141$	−6.27171	−0.528173
$142$	−8.69084	−0.729319
$143$	0	0
$144$	−4.22491	−0.352076
$145$	21.1670	1.75783
$146$	−6.38705	−0.528596
$147$	−1.00000	−0.0824786
$148$	−0.503006	−0.0413468
$149$	15.3451	1.25712	0.628561	−	0.777761i	$-0.283644\pi$
$149$	15.3451	1.25712	0.628561	+	0.777761i	$0.283644\pi$
$150$	−1.01625	−0.0829764
$151$	−6.90936	−0.562275	−0.281138	−	0.959667i	$-0.590712\pi$
$151$	−6.90936	−0.562275	−0.281138	+	0.959667i	$0.590712\pi$
$152$	−1.94742	−0.157956
$153$	−2.18663	−0.176778
$154$	0	0
$155$	−15.8173	−1.27047
$156$	0.109054	0.00873130
$157$	−9.33762	−0.745223	−0.372611	−	0.927987i	$-0.621538\pi$
$157$	−9.33762	−0.745223	−0.372611	+	0.927987i	$0.621538\pi$
$158$	−7.07160	−0.562586
$159$	6.56545	0.520674
$160$	−1.61295	−0.127515
$161$	7.80466	0.615094
$162$	1.45589	0.114385
$163$	10.4309	0.817013	0.408507	−	0.912755i	$-0.366050\pi$
$163$	10.4309	0.817013	0.408507	+	0.912755i	$0.366050\pi$
$164$	−1.43020	−0.111680
$165$	0	0
$166$	14.6302	1.13552
$167$	1.35141	0.104575	0.0522877	−	0.998632i	$-0.483349\pi$
$167$	1.35141	0.104575	0.0522877	+	0.998632i	$0.483349\pi$
$168$	−2.73764	−0.211214
$169$	−12.1687	−0.936051
$170$	−7.59915	−0.582828
$171$	0.711349	0.0543982
$172$	0.898962	0.0685452
$173$	19.4855	1.48145	0.740726	−	0.671807i	$-0.234482\pi$
$173$	19.4855	1.48145	0.740726	+	0.671807i	$0.234482\pi$
$174$	−12.9100	−0.978702
$175$	−0.698028	−0.0527659
$176$	0	0
$177$	9.62312	0.723318
$178$	−0.380765	−0.0285395
$179$	−12.8541	−0.960761	−0.480380	−	0.877060i	$-0.659501\pi$
$179$	−12.8541	−0.960761	−0.480380	+	0.877060i	$0.659501\pi$
$180$	0.285507	0.0212804
$181$	14.3749	1.06848	0.534239	−	0.845333i	$-0.320598\pi$
$181$	14.3749	1.06848	0.534239	+	0.845333i	$0.320598\pi$
$182$	1.32744	0.0983963
$183$	0.0627598	0.00463933
$184$	21.3664	1.57515
$185$	−10.0388	−0.738066
$186$	9.64709	0.707359
$187$	0	0
$188$	0.750136	0.0547093
$189$	1.00000	0.0727393
$190$	2.47214	0.179348
$191$	−23.9520	−1.73311	−0.866554	−	0.499083i	$-0.833670\pi$
$191$	−23.9520	−1.73311	−0.866554	+	0.499083i	$0.833670\pi$
$192$	−7.46606	−0.538817
$193$	0.139609	0.0100493	0.00502463	−	0.999987i	$-0.498401\pi$
$193$	0.139609	0.0100493	0.00502463	+	0.999987i	$0.498401\pi$
$194$	14.3909	1.03321
$195$	2.17645	0.155859
$196$	0.119606	0.00854331
$197$	14.6113	1.04101	0.520505	−	0.853859i	$-0.325744\pi$
$197$	14.6113	1.04101	0.520505	+	0.853859i	$0.325744\pi$
$198$	0	0
$199$	−26.4702	−1.87642	−0.938210	−	0.346066i	$-0.887517\pi$
$199$	−26.4702	−1.87642	−0.938210	+	0.346066i	$0.887517\pi$
$200$	−1.91095	−0.135124
$201$	11.3294	0.799113
$202$	14.4823	1.01897
$203$	−8.86742	−0.622371
$204$	0.261535	0.0183111
$205$	−28.5434	−1.99356
$206$	−0.959087	−0.0668228
$207$	−7.80466	−0.542462
$208$	3.85216	0.267099
$209$	0	0
$210$	3.47528	0.239817
$211$	17.5618	1.20901	0.604503	−	0.796603i	$-0.293372\pi$
$211$	17.5618	1.20901	0.604503	+	0.796603i	$0.293372\pi$
$212$	−0.785269	−0.0539325
$213$	5.96945	0.409020
$214$	0.412145	0.0281736
$215$	17.9411	1.22357
$216$	2.73764	0.186273
$217$	6.62627	0.449820
$218$	−13.0733	−0.885436
$219$	4.38705	0.296450
$220$	0	0
$221$	1.99371	0.134111
$222$	6.12275	0.410932
$223$	22.5839	1.51233	0.756164	−	0.654383i	$-0.227071\pi$
$223$	22.5839	1.51233	0.756164	+	0.654383i	$0.227071\pi$
$224$	0.675706	0.0451475
$225$	0.698028	0.0465352
$226$	1.93543	0.128743
$227$	21.9714	1.45829	0.729146	−	0.684358i	$-0.239917\pi$
$227$	21.9714	1.45829	0.729146	+	0.684358i	$0.239917\pi$
$228$	−0.0850818	−0.00563468
$229$	23.3803	1.54501	0.772507	−	0.635007i	$-0.219003\pi$
$229$	23.3803	1.54501	0.772507	+	0.635007i	$0.219003\pi$
$230$	−27.1234	−1.78846
$231$	0	0
$232$	−24.2758	−1.59379
$233$	−1.04315	−0.0683390	−0.0341695	−	0.999416i	$-0.510879\pi$
$233$	−1.04315	−0.0683390	−0.0341695	+	0.999416i	$0.510879\pi$
$234$	−1.32744	−0.0867774
$235$	14.9709	0.976595
$236$	−1.15099	−0.0749228
$237$	4.85725	0.315512
$238$	3.18348	0.206355
$239$	−12.4140	−0.802994	−0.401497	−	0.915860i	$-0.631510\pi$
$239$	−12.4140	−0.802994	−0.401497	+	0.915860i	$0.631510\pi$
$240$	10.0851	0.650989
$241$	9.42435	0.607076	0.303538	−	0.952819i	$-0.401832\pi$
$241$	9.42435	0.607076	0.303538	+	0.952819i	$0.401832\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	−0.00750646	−0.000480552 0
$245$	2.38705	0.152503
$246$	17.4089	1.10995
$247$	−0.648589	−0.0412687
$248$	18.1403	1.15191
$249$	−10.0490	−0.636827
$250$	−14.9506	−0.945557
$251$	−3.07219	−0.193915	−0.0969576	−	0.995289i	$-0.530911\pi$
$251$	−3.07219	−0.193915	−0.0969576	+	0.995289i	$0.530911\pi$
$252$	−0.119606	−0.00753449
$253$	0	0
$254$	−31.7924	−1.99483
$255$	5.21960	0.326864
$256$	2.86049	0.178781
$257$	0.626267	0.0390654	0.0195327	−	0.999809i	$-0.493782\pi$
$257$	0.626267	0.0390654	0.0195327	+	0.999809i	$0.493782\pi$
$258$	−10.9425	−0.681248
$259$	4.20551	0.261318
$260$	−0.260318	−0.0161442
$261$	8.86742	0.548880
$262$	−13.4266	−0.829497
$263$	1.02517	0.0632149	0.0316074	−	0.999500i	$-0.489937\pi$
$263$	1.02517	0.0632149	0.0316074	+	0.999500i	$0.489937\pi$
$264$	0	0
$265$	−15.6721	−0.962729
$266$	−1.03564	−0.0634993
$267$	0.261535	0.0160057
$268$	−1.35507	−0.0827738
$269$	32.0066	1.95147	0.975737	−	0.218945i	$-0.0702617\pi$
$269$	32.0066	1.95147	0.975737	+	0.218945i	$0.0702617\pi$
$270$	−3.47528	−0.211499
$271$	5.95536	0.361762	0.180881	−	0.983505i	$-0.442105\pi$
$271$	5.95536	0.361762	0.180881	+	0.983505i	$0.442105\pi$
$272$	9.23830	0.560155
$273$	−0.911774	−0.0551830
$274$	−2.98092	−0.180084
$275$	0	0
$276$	0.933487	0.0561893
$277$	−1.27771	−0.0767700	−0.0383850	−	0.999263i	$-0.512221\pi$
$277$	−1.27771	−0.0767700	−0.0383850	+	0.999263i	$0.512221\pi$
$278$	−18.7014	−1.12164
$279$	−6.62627	−0.396704
$280$	6.53490	0.390535
$281$	−4.45011	−0.265471	−0.132736	−	0.991151i	$-0.542376\pi$
$281$	−4.45011	−0.265471	−0.132736	+	0.991151i	$0.542376\pi$
$282$	−9.13090	−0.543737
$283$	22.7964	1.35511	0.677554	−	0.735473i	$-0.263040\pi$
$283$	22.7964	1.35511	0.677554	+	0.735473i	$0.263040\pi$
$284$	−0.713983	−0.0423671
$285$	−1.69803	−0.100582
$286$	0	0
$287$	11.9576	0.705834
$288$	−0.675706	−0.0398163
$289$	−12.2187	−0.718744
$290$	30.8168	1.80962
$291$	−9.88466	−0.579449
$292$	−0.524719	−0.0307069
$293$	−8.39334	−0.490344	−0.245172	−	0.969480i	$-0.578844\pi$
$293$	−8.39334	−0.490344	−0.245172	+	0.969480i	$0.578844\pi$
$294$	−1.45589	−0.0849090
$295$	−22.9709	−1.33742
$296$	11.5132	0.669190
$297$	0	0
$298$	22.3408	1.29417
$299$	7.11609	0.411534
$300$	−0.0834885	−0.00482021
$301$	−7.51601	−0.433216
$302$	−10.0592	−0.578844
$303$	−9.94742	−0.571464
$304$	−3.00538	−0.172370
$305$	−0.149811	−0.00857815
$306$	−3.18348	−0.181988
$307$	11.0560	0.630999	0.315500	−	0.948926i	$-0.397828\pi$
$307$	11.0560	0.630999	0.315500	+	0.948926i	$0.397828\pi$
$308$	0	0
$309$	0.658765	0.0374758
$310$	−23.0281	−1.30791
$311$	−22.5914	−1.28104	−0.640519	−	0.767942i	$-0.721281\pi$
$311$	−22.5914	−1.28104	−0.640519	+	0.767942i	$0.721281\pi$
$312$	−2.49611	−0.141314
$313$	2.54869	0.144061	0.0720303	−	0.997402i	$-0.477052\pi$
$313$	2.54869	0.144061	0.0720303	+	0.997402i	$0.477052\pi$
$314$	−13.5945	−0.767183
$315$	−2.38705	−0.134495
$316$	−0.580957	−0.0326814
$317$	31.3434	1.76042	0.880212	−	0.474581i	$-0.157401\pi$
$317$	31.3434	1.76042	0.880212	+	0.474581i	$0.157401\pi$
$318$	9.55855	0.536017
$319$	0	0
$320$	17.8219	0.996274
$321$	−0.283088	−0.0158005
$322$	11.3627	0.633219
$323$	−1.55546	−0.0865479
$324$	0.119606	0.00664480
$325$	−0.636443	−0.0353035
$326$	15.1863	0.841089
$327$	8.97962	0.496574
$328$	32.7356	1.80752
$329$	−6.27171	−0.345771
$330$	0	0
$331$	−4.82789	−0.265365	−0.132682	−	0.991159i	$-0.542359\pi$
$331$	−4.82789	−0.265365	−0.132682	+	0.991159i	$0.542359\pi$
$332$	1.20192	0.0659639
$333$	−4.20551	−0.230461
$334$	1.96750	0.107657
$335$	−27.0438	−1.47756
$336$	−4.22491	−0.230488
$337$	21.1153	1.15022	0.575112	−	0.818074i	$-0.304958\pi$
$337$	21.1153	1.15022	0.575112	+	0.818074i	$0.304958\pi$
$338$	−17.7162	−0.963634
$339$	−1.32938	−0.0722022
$340$	−0.624297	−0.0338573
$341$	0	0
$342$	1.03564	0.0560012
$343$	−1.00000	−0.0539949
$344$	−20.5761	−1.10939
$345$	18.6302	1.00301
$346$	28.3686	1.52511
$347$	−12.4251	−0.667015	−0.333508	−	0.942747i	$-0.608232\pi$
$347$	−12.4251	−0.667015	−0.333508	+	0.942747i	$0.608232\pi$
$348$	−1.06060	−0.0568541
$349$	16.9682	0.908289	0.454145	−	0.890928i	$-0.349945\pi$
$349$	16.9682	0.908289	0.454145	+	0.890928i	$0.349945\pi$
$350$	−1.01625	−0.0543208
$351$	0.911774	0.0486669
$352$	0	0
$353$	−4.57101	−0.243291	−0.121645	−	0.992574i	$-0.538817\pi$
$353$	−4.57101	−0.243291	−0.121645	+	0.992574i	$0.538817\pi$
$354$	14.0102	0.744632
$355$	−14.2494	−0.756279
$356$	−0.0312812	−0.00165790
$357$	−2.18663	−0.115729
$358$	−18.7141	−0.989072
$359$	−30.9733	−1.63471	−0.817355	−	0.576134i	$-0.804560\pi$
$359$	−30.9733	−1.63471	−0.817355	+	0.576134i	$0.804560\pi$
$360$	−6.53490	−0.344419
$361$	−18.4940	−0.973368
$362$	20.9282	1.09996
$363$	0	0
$364$	0.109054	0.00571598
$365$	−10.4721	−0.548137
$366$	0.0913711	0.00477604
$367$	14.7422	0.769536	0.384768	−	0.923013i	$-0.374281\pi$
$367$	14.7422	0.769536	0.384768	+	0.923013i	$0.374281\pi$
$368$	32.9740	1.71889
$369$	−11.9576	−0.622487
$370$	−14.6153	−0.759815
$371$	6.56545	0.340861
$372$	0.792543	0.0410915
$373$	−12.5701	−0.650853	−0.325427	−	0.945567i	$-0.605508\pi$
$373$	−12.5701	−0.650853	−0.325427	+	0.945567i	$0.605508\pi$
$374$	0	0
$375$	10.2690	0.530291
$376$	−17.1697	−0.885459
$377$	−8.08508	−0.416403
$378$	1.45589	0.0748827
$379$	16.0340	0.823610	0.411805	−	0.911272i	$-0.364899\pi$
$379$	16.0340	0.823610	0.411805	+	0.911272i	$0.364899\pi$
$380$	0.203095	0.0104185
$381$	21.8372	1.11875
$382$	−34.8714	−1.78418
$383$	−2.81023	−0.143596	−0.0717979	−	0.997419i	$-0.522874\pi$
$383$	−2.81023	−0.143596	−0.0717979	+	0.997419i	$0.522874\pi$
$384$	−12.2212	−0.623658
$385$	0	0
$386$	0.203254	0.0103454
$387$	7.51601	0.382060
$388$	1.18227	0.0600205
$389$	−20.6050	−1.04471	−0.522357	−	0.852727i	$-0.674947\pi$
$389$	−20.6050	−1.04471	−0.522357	+	0.852727i	$0.674947\pi$
$390$	3.16867	0.160452
$391$	17.0659	0.863060
$392$	−2.73764	−0.138272
$393$	9.22227	0.465202
$394$	21.2724	1.07169
$395$	−11.5945	−0.583383
$396$	0	0
$397$	28.6588	1.43834	0.719171	−	0.694833i	$-0.244522\pi$
$397$	28.6588	1.43834	0.719171	+	0.694833i	$0.244522\pi$
$398$	−38.5376	−1.93171
$399$	0.711349	0.0356120
$400$	−2.94910	−0.147455
$401$	−22.3156	−1.11439	−0.557194	−	0.830383i	$-0.688122\pi$
$401$	−22.3156	−1.11439	−0.557194	+	0.830383i	$0.688122\pi$
$402$	16.4943	0.822661
$403$	6.04165	0.300956
$404$	1.18977	0.0591935
$405$	2.38705	0.118614
$406$	−12.9100	−0.640711
$407$	0	0
$408$	−5.98620	−0.296361
$409$	16.4360	0.812709	0.406354	−	0.913716i	$-0.366800\pi$
$409$	16.4360	0.812709	0.406354	+	0.913716i	$0.366800\pi$
$410$	−41.5560	−2.05230
$411$	2.04750	0.100995
$412$	−0.0787925	−0.00388183
$413$	9.62312	0.473523
$414$	−11.3627	−0.558447
$415$	23.9874	1.17750
$416$	0.616090	0.0302063
$417$	12.8454	0.629042
$418$	0	0
$419$	−12.7725	−0.623975	−0.311988	−	0.950086i	$-0.600995\pi$
$419$	−12.7725	−0.623975	−0.311988	+	0.950086i	$0.600995\pi$
$420$	0.285507	0.0139313
$421$	10.8032	0.526514	0.263257	−	0.964726i	$-0.415203\pi$
$421$	10.8032	0.526514	0.263257	+	0.964726i	$0.415203\pi$
$422$	25.5680	1.24463
$423$	6.27171	0.304941
$424$	17.9738	0.872887
$425$	−1.52633	−0.0740378
$426$	8.69084	0.421072
$427$	0.0627598	0.00303716
$428$	0.0338592	0.00163664
$429$	0	0
$430$	26.1202	1.25963
$431$	27.5395	1.32653	0.663266	−	0.748384i	$-0.269170\pi$
$431$	27.5395	1.32653	0.663266	+	0.748384i	$0.269170\pi$
$432$	4.22491	0.203271
$433$	15.2600	0.733351	0.366675	−	0.930349i	$-0.380496\pi$
$433$	15.2600	0.733351	0.366675	+	0.930349i	$0.380496\pi$
$434$	9.64709	0.463075
$435$	−21.1670	−1.01488
$436$	−1.07402	−0.0514362
$437$	−5.55184	−0.265580
$438$	6.38705	0.305185
$439$	−28.5500	−1.36262	−0.681308	−	0.731997i	$-0.738589\pi$
$439$	−28.5500	−1.36262	−0.681308	+	0.731997i	$0.738589\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	2.90262	0.138063
$443$	11.3310	0.538354	0.269177	−	0.963091i	$-0.413248\pi$
$443$	11.3310	0.538354	0.269177	+	0.963091i	$0.413248\pi$
$444$	0.503006	0.0238716
$445$	−0.624297	−0.0295945
$446$	32.8795	1.55689
$447$	−15.3451	−0.725799
$448$	−7.46606	−0.352738
$449$	−26.8968	−1.26934	−0.634669	−	0.772784i	$-0.718863\pi$
$449$	−26.8968	−1.26934	−0.634669	+	0.772784i	$0.718863\pi$
$450$	1.01625	0.0479064
$451$	0	0
$452$	0.159003	0.00747885
$453$	6.90936	0.324630
$454$	31.9879	1.50126
$455$	2.17645	0.102034
$456$	1.94742	0.0911961
$457$	−28.9694	−1.35513	−0.677567	−	0.735461i	$-0.736966\pi$
$457$	−28.9694	−1.35513	−0.677567	+	0.735461i	$0.736966\pi$
$458$	34.0391	1.59054
$459$	2.18663	0.102063
$460$	−2.22828	−0.103894
$461$	−41.9375	−1.95322	−0.976612	−	0.215008i	$-0.931022\pi$
$461$	−41.9375	−1.95322	−0.976612	+	0.215008i	$0.931022\pi$
$462$	0	0
$463$	−1.14904	−0.0534005	−0.0267003	−	0.999643i	$-0.508500\pi$
$463$	−1.14904	−0.0534005	−0.0267003	+	0.999643i	$0.508500\pi$
$464$	−37.4640	−1.73922
$465$	15.8173	0.733508
$466$	−1.51871	−0.0703528
$467$	29.5887	1.36920	0.684600	−	0.728919i	$-0.259977\pi$
$467$	29.5887	1.36920	0.684600	+	0.728919i	$0.259977\pi$
$468$	−0.109054	−0.00504102
$469$	11.3294	0.523142
$470$	21.7960	1.00537
$471$	9.33762	0.430255
$472$	26.3446	1.21261
$473$	0	0
$474$	7.07160	0.324809
$475$	0.496541	0.0227829
$476$	0.261535	0.0119874
$477$	−6.56545	−0.300611
$478$	−18.0734	−0.826656
$479$	−10.1728	−0.464809	−0.232404	−	0.972619i	$-0.574659\pi$
$479$	−10.1728	−0.464809	−0.232404	+	0.972619i	$0.574659\pi$
$480$	1.61295	0.0736206
$481$	3.83448	0.174837
$482$	13.7208	0.624965
$483$	−7.80466	−0.355124
$484$	0	0
$485$	23.5952	1.07140
$486$	−1.45589	−0.0660404
$487$	43.8320	1.98622	0.993110	−	0.117187i	$-0.0373878\pi$
$487$	43.8320	1.98622	0.993110	+	0.117187i	$0.0373878\pi$
$488$	0.171814	0.00777764
$489$	−10.4309	−0.471703
$490$	3.47528	0.156997
$491$	19.1207	0.862906	0.431453	−	0.902135i	$-0.358001\pi$
$491$	19.1207	0.862906	0.431453	+	0.902135i	$0.358001\pi$
$492$	1.43020	0.0644786
$493$	−19.3898	−0.873271
$494$	−0.944272	−0.0424848
$495$	0	0
$496$	27.9954	1.25703
$497$	5.96945	0.267766
$498$	−14.6302	−0.655593
$499$	20.4971	0.917576	0.458788	−	0.888546i	$-0.348284\pi$
$499$	20.4971	0.917576	0.458788	+	0.888546i	$0.348284\pi$
$500$	−1.22824	−0.0549287
$501$	−1.35141	−0.0603766
$502$	−4.47277	−0.199629
$503$	18.9794	0.846251	0.423126	−	0.906071i	$-0.360933\pi$
$503$	18.9794	0.846251	0.423126	+	0.906071i	$0.360933\pi$
$504$	2.73764	0.121944
$505$	23.7450	1.05664
$506$	0	0
$507$	12.1687	0.540430
$508$	−2.61186	−0.115883
$509$	23.5548	1.04405	0.522023	−	0.852931i	$-0.325177\pi$
$509$	23.5548	1.04405	0.522023	+	0.852931i	$0.325177\pi$
$510$	7.59915	0.336496
$511$	4.38705	0.194072
$512$	−20.2778	−0.896159
$513$	−0.711349	−0.0314068
$514$	0.911774	0.0402166
$515$	−1.57251	−0.0692930
$516$	−0.898962	−0.0395746
$517$	0	0
$518$	6.12275	0.269018
$519$	−19.4855	−0.855317
$520$	5.95835	0.261291
$521$	−5.09046	−0.223017	−0.111509	−	0.993763i	$-0.535568\pi$
$521$	−5.09046	−0.223017	−0.111509	+	0.993763i	$0.535568\pi$
$522$	12.9100	0.565054
$523$	22.7768	0.995959	0.497979	−	0.867189i	$-0.334075\pi$
$523$	22.7768	0.995959	0.497979	+	0.867189i	$0.334075\pi$
$524$	−1.10304	−0.0481866
$525$	0.698028	0.0304644
$526$	1.49254	0.0650777
$527$	14.4892	0.631159
$528$	0	0
$529$	37.9128	1.64838
$530$	−22.8168	−0.991098
$531$	−9.62312	−0.417608
$532$	−0.0850818	−0.00368876
$533$	10.9026	0.472245
$534$	0.380765	0.0164773
$535$	0.675747	0.0292151
$536$	31.0158	1.33968
$537$	12.8541	0.554695
$538$	46.5979	2.00898
$539$	0	0
$540$	−0.285507	−0.0122863
$541$	−11.4416	−0.491912	−0.245956	−	0.969281i	$-0.579102\pi$
$541$	−11.4416	−0.491912	−0.245956	+	0.969281i	$0.579102\pi$
$542$	8.67032	0.372422
$543$	−14.3749	−0.616886
$544$	1.47752	0.0633480
$545$	−21.4348	−0.918168
$546$	−1.32744	−0.0568091
$547$	3.79691	0.162344	0.0811720	−	0.996700i	$-0.474134\pi$
$547$	3.79691	0.162344	0.0811720	+	0.996700i	$0.474134\pi$
$548$	−0.244893	−0.0104613
$549$	−0.0627598	−0.00267852
$550$	0	0
$551$	6.30783	0.268723
$552$	−21.3664	−0.909413
$553$	4.85725	0.206551
$554$	−1.86020	−0.0790322
$555$	10.0388	0.426123
$556$	−1.53639	−0.0651575
$557$	−3.48937	−0.147849	−0.0739247	−	0.997264i	$-0.523552\pi$
$557$	−3.48937	−0.147849	−0.0739247	+	0.997264i	$0.523552\pi$
$558$	−9.64709	−0.408394
$559$	−6.85290	−0.289847
$560$	10.0851	0.426172
$561$	0	0
$562$	−6.47885	−0.273294
$563$	−28.1619	−1.18688	−0.593441	−	0.804877i	$-0.702231\pi$
$563$	−28.1619	−1.18688	−0.593441	+	0.804877i	$0.702231\pi$
$564$	−0.750136	−0.0315864
$565$	3.17331	0.133502
$566$	33.1890	1.39504
$567$	−1.00000	−0.0419961
$568$	16.3422	0.685703
$569$	21.5992	0.905483	0.452742	−	0.891642i	$-0.350446\pi$
$569$	21.5992	0.905483	0.452742	+	0.891642i	$0.350446\pi$
$570$	−2.47214	−0.103546
$571$	14.4959	0.606636	0.303318	−	0.952889i	$-0.401906\pi$
$571$	14.4959	0.606636	0.303318	+	0.952889i	$0.401906\pi$
$572$	0	0
$573$	23.9520	1.00061
$574$	17.4089	0.726634
$575$	−5.44787	−0.227192
$576$	7.46606	0.311086
$577$	5.07081	0.211101	0.105550	−	0.994414i	$-0.466340\pi$
$577$	5.07081	0.211101	0.105550	+	0.994414i	$0.466340\pi$
$578$	−17.7890	−0.739924
$579$	−0.139609	−0.00580194
$580$	2.53171	0.105124
$581$	−10.0490	−0.416901
$582$	−14.3909	−0.596524
$583$	0	0
$584$	12.0102	0.496985
$585$	−2.17645	−0.0899853
$586$	−12.2198	−0.504794
$587$	−43.3905	−1.79092	−0.895458	−	0.445146i	$-0.853152\pi$
$587$	−43.3905	−1.79092	−0.895458	+	0.445146i	$0.853152\pi$
$588$	−0.119606	−0.00493248
$589$	−4.71359	−0.194220
$590$	−33.4430	−1.37683
$591$	−14.6113	−0.601027
$592$	17.7679	0.730256
$593$	−31.4532	−1.29163	−0.645815	−	0.763494i	$-0.723482\pi$
$593$	−31.4532	−1.29163	−0.645815	+	0.763494i	$0.723482\pi$
$594$	0	0
$595$	5.21960	0.213983
$596$	1.83537	0.0751798
$597$	26.4702	1.08335
$598$	10.3602	0.423661
$599$	−22.1842	−0.906422	−0.453211	−	0.891403i	$-0.649722\pi$
$599$	−22.1842	−0.906422	−0.453211	+	0.891403i	$0.649722\pi$
$600$	1.91095	0.0780141
$601$	32.2182	1.31421	0.657103	−	0.753800i	$-0.271781\pi$
$601$	32.2182	1.31421	0.657103	+	0.753800i	$0.271781\pi$
$602$	−10.9425	−0.445981
$603$	−11.3294	−0.461368
$604$	−0.826403	−0.0336258
$605$	0	0
$606$	−14.4823	−0.588304
$607$	30.2533	1.22794	0.613971	−	0.789328i	$-0.289571\pi$
$607$	30.2533	1.22794	0.613971	+	0.789328i	$0.289571\pi$
$608$	−0.480662	−0.0194934
$609$	8.86742	0.359326
$610$	−0.218108	−0.00883092
$611$	−5.71838	−0.231341
$612$	−0.261535	−0.0105719
$613$	−4.18959	−0.169216	−0.0846080	−	0.996414i	$-0.526964\pi$
$613$	−4.18959	−0.169216	−0.0846080	+	0.996414i	$0.526964\pi$
$614$	16.0963	0.649593
$615$	28.5434	1.15098
$616$	0	0
$617$	−31.1457	−1.25388	−0.626939	−	0.779068i	$-0.715693\pi$
$617$	−31.1457	−1.25388	−0.626939	+	0.779068i	$0.715693\pi$
$618$	0.959087	0.0385802
$619$	−47.8439	−1.92301	−0.961504	−	0.274790i	$-0.911392\pi$
$619$	−47.8439	−1.92301	−0.961504	+	0.274790i	$0.911392\pi$
$620$	−1.89184	−0.0759783
$621$	7.80466	0.313190
$622$	−32.8905	−1.31879
$623$	0.261535	0.0104782
$624$	−3.85216	−0.154210
$625$	−28.0029	−1.12012
$626$	3.71061	0.148306
$627$	0	0
$628$	−1.11684	−0.0445667
$629$	9.19590	0.366664
$630$	−3.47528	−0.138458
$631$	4.01691	0.159911	0.0799554	−	0.996798i	$-0.474522\pi$
$631$	4.01691	0.159911	0.0799554	+	0.996798i	$0.474522\pi$
$632$	13.2974	0.528942
$633$	−17.5618	−0.698020
$634$	45.6325	1.81230
$635$	−52.1265	−2.06858
$636$	0.785269	0.0311380
$637$	−0.911774	−0.0361258
$638$	0	0
$639$	−5.96945	−0.236148
$640$	29.1726	1.15315
$641$	27.6852	1.09350	0.546750	−	0.837296i	$-0.315865\pi$
$641$	27.6852	1.09350	0.546750	+	0.837296i	$0.315865\pi$
$642$	−0.412145	−0.0162661
$643$	−4.25792	−0.167916	−0.0839579	−	0.996469i	$-0.526756\pi$
$643$	−4.25792	−0.167916	−0.0839579	+	0.996469i	$0.526756\pi$
$644$	0.933487	0.0367845
$645$	−17.9411	−0.706431
$646$	−2.26457	−0.0890982
$647$	5.24503	0.206203	0.103102	−	0.994671i	$-0.467123\pi$
$647$	5.24503	0.206203	0.103102	+	0.994671i	$0.467123\pi$
$648$	−2.73764	−0.107545
$649$	0	0
$650$	−0.926589	−0.0363438
$651$	−6.62627	−0.259704
$652$	1.24761	0.0488600
$653$	−9.83716	−0.384958	−0.192479	−	0.981301i	$-0.561653\pi$
$653$	−9.83716	−0.384958	−0.192479	+	0.981301i	$0.561653\pi$
$654$	13.0733	0.511207
$655$	−22.0141	−0.860161
$656$	50.5197	1.97246
$657$	−4.38705	−0.171155
$658$	−9.13090	−0.355960
$659$	50.5575	1.96944	0.984720	−	0.174145i	$-0.0557162\pi$
$659$	50.5575	1.96944	0.984720	+	0.174145i	$0.0557162\pi$
$660$	0	0
$661$	22.9751	0.893629	0.446814	−	0.894627i	$-0.352558\pi$
$661$	22.9751	0.893629	0.446814	+	0.894627i	$0.352558\pi$
$662$	−7.02887	−0.273185
$663$	−1.99371	−0.0774293
$664$	−27.5104	−1.06761
$665$	−1.69803	−0.0658467
$666$	−6.12275	−0.237252
$667$	−69.2072	−2.67972
$668$	0.161637	0.00625394
$669$	−22.5839	−0.873142
$670$	−39.3728	−1.52110
$671$	0	0
$672$	−0.675706	−0.0260659
$673$	−21.8706	−0.843048	−0.421524	−	0.906817i	$-0.638505\pi$
$673$	−21.8706	−0.843048	−0.421524	+	0.906817i	$0.638505\pi$
$674$	30.7415	1.18412
$675$	−0.698028	−0.0268671
$676$	−1.45545	−0.0559788
$677$	−14.6716	−0.563876	−0.281938	−	0.959433i	$-0.590977\pi$
$677$	−14.6716	−0.563876	−0.281938	+	0.959433i	$0.590977\pi$
$678$	−1.93543	−0.0743298
$679$	−9.88466	−0.379338
$680$	14.2894	0.547973
$681$	−21.9714	−0.841945
$682$	0	0
$683$	−9.70579	−0.371382	−0.185691	−	0.982608i	$-0.559452\pi$
$683$	−9.70579	−0.371382	−0.185691	+	0.982608i	$0.559452\pi$
$684$	0.0850818	0.00325318
$685$	−4.88748	−0.186741
$686$	−1.45589	−0.0555860
$687$	−23.3803	−0.892014
$688$	−31.7545	−1.21063
$689$	5.98620	0.228056
$690$	27.1234	1.03257
$691$	−43.6828	−1.66177	−0.830887	−	0.556441i	$-0.812166\pi$
$691$	−43.6828	−1.66177	−0.830887	+	0.556441i	$0.812166\pi$
$692$	2.33058	0.0885955
$693$	0	0
$694$	−18.0896	−0.686670
$695$	−30.6627	−1.16310
$696$	24.2758	0.920172
$697$	26.1468	0.990381
$698$	24.7038	0.935054
$699$	1.04315	0.0394555
$700$	−0.0834885	−0.00315557
$701$	−20.7227	−0.782687	−0.391343	−	0.920245i	$-0.627989\pi$
$701$	−20.7227	−0.782687	−0.391343	+	0.920245i	$0.627989\pi$
$702$	1.32744	0.0501010
$703$	−2.99159	−0.112830
$704$	0	0
$705$	−14.9709	−0.563837
$706$	−6.65488	−0.250460
$707$	−9.94742	−0.374111
$708$	1.15099	0.0432567
$709$	−19.2327	−0.722301	−0.361150	−	0.932508i	$-0.617616\pi$
$709$	−19.2327	−0.722301	−0.361150	+	0.932508i	$0.617616\pi$
$710$	−20.7455	−0.778565
$711$	−4.85725	−0.182161
$712$	0.715988	0.0268328
$713$	51.7158	1.93677
$714$	−3.18348	−0.119139
$715$	0	0
$716$	−1.53743	−0.0574565
$717$	12.4140	0.463609
$718$	−45.0937	−1.68288
$719$	−5.36985	−0.200261	−0.100131	−	0.994974i	$-0.531926\pi$
$719$	−5.36985	−0.200261	−0.100131	+	0.994974i	$0.531926\pi$
$720$	−10.0851	−0.375849
$721$	0.658765	0.0245337
$722$	−26.9251	−1.00205
$723$	−9.42435	−0.350495
$724$	1.71933	0.0638984
$725$	6.18971	0.229880
$726$	0	0
$727$	8.19052	0.303769	0.151885	−	0.988398i	$-0.451466\pi$
$727$	8.19052	0.303769	0.151885	+	0.988398i	$0.451466\pi$
$728$	−2.49611	−0.0925119
$729$	1.00000	0.0370370
$730$	−15.2462	−0.564289
$731$	−16.4347	−0.607860
$732$	0.00750646	0.000277447 0
$733$	24.3265	0.898521	0.449260	−	0.893401i	$-0.351688\pi$
$733$	24.3265	0.898521	0.449260	+	0.893401i	$0.351688\pi$
$734$	21.4630	0.792213
$735$	−2.38705	−0.0880478
$736$	5.27365	0.194389
$737$	0	0
$738$	−17.4089	−0.640831
$739$	−18.9692	−0.697795	−0.348897	−	0.937161i	$-0.613444\pi$
$739$	−18.9692	−0.697795	−0.348897	+	0.937161i	$0.613444\pi$
$740$	−1.20070	−0.0441387
$741$	0.648589	0.0238265
$742$	9.55855	0.350906
$743$	−16.8997	−0.619988	−0.309994	−	0.950738i	$-0.600327\pi$
$743$	−16.8997	−0.619988	−0.309994	+	0.950738i	$0.600327\pi$
$744$	−18.1403	−0.665057
$745$	36.6296	1.34201
$746$	−18.3006	−0.670032
$747$	10.0490	0.367672
$748$	0	0
$749$	−0.283088	−0.0103438
$750$	14.9506	0.545917
$751$	−22.5942	−0.824475	−0.412237	−	0.911076i	$-0.635253\pi$
$751$	−22.5942	−0.824475	−0.412237	+	0.911076i	$0.635253\pi$
$752$	−26.4974	−0.966261
$753$	3.07219	0.111957
$754$	−11.7710	−0.428673
$755$	−16.4930	−0.600242
$756$	0.119606	0.00435004
$757$	3.87744	0.140928	0.0704640	−	0.997514i	$-0.477552\pi$
$757$	3.87744	0.140928	0.0704640	+	0.997514i	$0.477552\pi$
$758$	23.3436	0.847879
$759$	0	0
$760$	−4.64859	−0.168622
$761$	37.4591	1.35789	0.678945	−	0.734189i	$-0.262437\pi$
$761$	37.4591	1.35789	0.678945	+	0.734189i	$0.262437\pi$
$762$	31.7924	1.15172
$763$	8.97962	0.325084
$764$	−2.86481	−0.103645
$765$	−5.21960	−0.188715
$766$	−4.09137	−0.147827
$767$	8.77411	0.316815
$768$	−2.86049	−0.103219
$769$	−20.6137	−0.743349	−0.371674	−	0.928363i	$-0.621216\pi$
$769$	−20.6137	−0.743349	−0.371674	+	0.928363i	$0.621216\pi$
$770$	0	0
$771$	−0.626267	−0.0225544
$772$	0.0166981	0.000600977 0
$773$	9.03686	0.325033	0.162517	−	0.986706i	$-0.448039\pi$
$773$	9.03686	0.325033	0.162517	+	0.986706i	$0.448039\pi$
$774$	10.9425	0.393319
$775$	−4.62532	−0.166146
$776$	−27.0606	−0.971420
$777$	−4.20551	−0.150872
$778$	−29.9985	−1.07550
$779$	−8.50602	−0.304760
$780$	0.260318	0.00932086
$781$	0	0
$782$	24.8460	0.888492
$783$	−8.86742	−0.316896
$784$	−4.22491	−0.150890
$785$	−22.2894	−0.795543
$786$	13.4266	0.478910
$787$	−33.4585	−1.19267	−0.596333	−	0.802737i	$-0.703376\pi$
$787$	−33.4585	−1.19267	−0.596333	+	0.802737i	$0.703376\pi$
$788$	1.74760	0.0622557
$789$	−1.02517	−0.0364971
$790$	−16.8803	−0.600574
$791$	−1.32938	−0.0472674
$792$	0	0
$793$	0.0572227	0.00203204
$794$	41.7239	1.48073
$795$	15.6721	0.555832
$796$	−3.16600	−0.112216
$797$	0.766169	0.0271391	0.0135695	−	0.999908i	$-0.495681\pi$
$797$	0.766169	0.0271391	0.0135695	+	0.999908i	$0.495681\pi$
$798$	1.03564	0.0366614
$799$	−13.7139	−0.485163
$800$	−0.471661	−0.0166757
$801$	−0.261535	−0.00924087
$802$	−32.4890	−1.14723
$803$	0	0
$804$	1.35507	0.0477895
$805$	18.6302	0.656627
$806$	8.79597	0.309825
$807$	−32.0066	−1.12668
$808$	−27.2324	−0.958034
$809$	−10.2311	−0.359708	−0.179854	−	0.983693i	$-0.557562\pi$
$809$	−10.2311	−0.359708	−0.179854	+	0.983693i	$0.557562\pi$
$810$	3.47528	0.122109
$811$	10.4196	0.365880	0.182940	−	0.983124i	$-0.441439\pi$
$811$	10.4196	0.365880	0.182940	+	0.983124i	$0.441439\pi$
$812$	−1.06060	−0.0372198
$813$	−5.95536	−0.208863
$814$	0	0
$815$	24.8992	0.872181
$816$	−9.23830	−0.323405
$817$	5.34650	0.187051
$818$	23.9290	0.836657
$819$	0.911774	0.0318599
$820$	−3.41397	−0.119221
$821$	27.7513	0.968527	0.484264	−	0.874922i	$-0.339088\pi$
$821$	27.7513	0.968527	0.484264	+	0.874922i	$0.339088\pi$
$822$	2.98092	0.103972
$823$	−18.7419	−0.653302	−0.326651	−	0.945145i	$-0.605920\pi$
$823$	−18.7419	−0.653302	−0.326651	+	0.945145i	$0.605920\pi$
$824$	1.80346	0.0628266
$825$	0	0
$826$	14.0102	0.487476
$827$	−55.6432	−1.93490	−0.967451	−	0.253058i	$-0.918564\pi$
$827$	−55.6432	−1.93490	−0.967451	+	0.253058i	$0.918564\pi$
$828$	−0.933487	−0.0324409
$829$	−37.5600	−1.30451	−0.652257	−	0.757998i	$-0.726178\pi$
$829$	−37.5600	−1.30451	−0.652257	+	0.757998i	$0.726178\pi$
$830$	34.9230	1.21219
$831$	1.27771	0.0443232
$832$	−6.80736	−0.236003
$833$	−2.18663	−0.0757622
$834$	18.7014	0.647578
$835$	3.22589	0.111637
$836$	0	0
$837$	6.62627	0.229037
$838$	−18.5953	−0.642362
$839$	−44.6850	−1.54270	−0.771348	−	0.636413i	$-0.780417\pi$
$839$	−44.6850	−1.54270	−0.771348	+	0.636413i	$0.780417\pi$
$840$	−6.53490	−0.225475
$841$	49.6312	1.71142
$842$	15.7282	0.542029
$843$	4.45011	0.153270
$844$	2.10051	0.0723024
$845$	−29.0473	−0.999257
$846$	9.13090	0.313927
$847$	0	0
$848$	27.7384	0.952541
$849$	−22.7964	−0.782371
$850$	−2.22216	−0.0762195
$851$	32.8226	1.12514
$852$	0.713983	0.0244607
$853$	55.6861	1.90666	0.953329	−	0.301934i	$-0.0976323\pi$
$853$	55.6861	1.90666	0.953329	+	0.301934i	$0.0976323\pi$
$854$	0.0913711	0.00312665
$855$	1.69803	0.0580713
$856$	−0.774994	−0.0264888
$857$	−37.5988	−1.28435	−0.642176	−	0.766557i	$-0.721968\pi$
$857$	−37.5988	−1.28435	−0.642176	+	0.766557i	$0.721968\pi$
$858$	0	0
$859$	44.1084	1.50496	0.752479	−	0.658616i	$-0.228858\pi$
$859$	44.1084	1.50496	0.752479	+	0.658616i	$0.228858\pi$
$860$	2.14587	0.0731736
$861$	−11.9576	−0.407514
$862$	40.0944	1.36562
$863$	−4.51049	−0.153539	−0.0767694	−	0.997049i	$-0.524461\pi$
$863$	−4.51049	−0.153539	−0.0767694	+	0.997049i	$0.524461\pi$
$864$	0.675706	0.0229880
$865$	46.5128	1.58148
$866$	22.2169	0.754961
$867$	12.2187	0.414967
$868$	0.792543	0.0269007
$869$	0	0
$870$	−30.8168	−1.04479
$871$	10.3298	0.350013
$872$	24.5830	0.832485
$873$	9.88466	0.334545
$874$	−8.08284	−0.273406
$875$	10.2690	0.347157
$876$	0.524719	0.0177286
$877$	−36.0877	−1.21860	−0.609298	−	0.792941i	$-0.708549\pi$
$877$	−36.0877	−1.21860	−0.609298	+	0.792941i	$0.708549\pi$
$878$	−41.5655	−1.40277
$879$	8.39334	0.283100
$880$	0	0
$881$	−12.8240	−0.432052	−0.216026	−	0.976388i	$-0.569310\pi$
$881$	−12.8240	−0.432052	−0.216026	+	0.976388i	$0.569310\pi$
$882$	1.45589	0.0490223
$883$	−9.36217	−0.315062	−0.157531	−	0.987514i	$-0.550353\pi$
$883$	−9.36217	−0.315062	−0.157531	+	0.987514i	$0.550353\pi$
$884$	0.238460	0.00802029
$885$	22.9709	0.772159
$886$	16.4967	0.554217
$887$	−9.91449	−0.332896	−0.166448	−	0.986050i	$-0.553230\pi$
$887$	−9.91449	−0.332896	−0.166448	+	0.986050i	$0.553230\pi$
$888$	−11.5132	−0.386357
$889$	21.8372	0.732395
$890$	−0.908906	−0.0304666
$891$	0	0
$892$	2.70117	0.0904419
$893$	4.46137	0.149294
$894$	−22.3408	−0.747187
$895$	−30.6834	−1.02563
$896$	−12.2212	−0.408280
$897$	−7.11609	−0.237599
$898$	−39.1587	−1.30674
$899$	−58.7579	−1.95969
$900$	0.0834885	0.00278295
$901$	14.3562	0.478275
$902$	0	0
$903$	7.51601	0.250117
$904$	−3.63937	−0.121044
$905$	34.3137	1.14063
$906$	10.0592	0.334196
$907$	−13.9881	−0.464469	−0.232234	−	0.972660i	$-0.574604\pi$
$907$	−13.9881	−0.464469	−0.232234	+	0.972660i	$0.574604\pi$
$908$	2.62792	0.0872105
$909$	9.94742	0.329935
$910$	3.16867	0.105040
$911$	−27.1168	−0.898419	−0.449210	−	0.893426i	$-0.648294\pi$
$911$	−27.1168	−0.898419	−0.449210	+	0.893426i	$0.648294\pi$
$912$	3.00538	0.0995181
$913$	0	0
$914$	−42.1762	−1.39507
$915$	0.149811	0.00495260
$916$	2.79643	0.0923967
$917$	9.22227	0.304546
$918$	3.18348	0.105071
$919$	4.42881	0.146093	0.0730464	−	0.997329i	$-0.476728\pi$
$919$	4.42881	0.146093	0.0730464	+	0.997329i	$0.476728\pi$
$920$	51.0027	1.68151
$921$	−11.0560	−0.364307
$922$	−61.0563	−2.01078
$923$	5.44278	0.179151
$924$	0	0
$925$	−2.93556	−0.0965208
$926$	−1.67288	−0.0549741
$927$	−0.658765	−0.0216367
$928$	−5.99177	−0.196689
$929$	−56.7744	−1.86271	−0.931353	−	0.364116i	$-0.881371\pi$
$929$	−56.7744	−1.86271	−0.931353	+	0.364116i	$0.881371\pi$
$930$	23.0281	0.755122
$931$	0.711349	0.0233135
$932$	−0.124767	−0.00408689
$933$	22.5914	0.739608
$934$	43.0777	1.40955
$935$	0	0
$936$	2.49611	0.0815878
$937$	−2.85797	−0.0933659	−0.0466830	−	0.998910i	$-0.514865\pi$
$937$	−2.85797	−0.0933659	−0.0466830	+	0.998910i	$0.514865\pi$
$938$	16.4943	0.538558
$939$	−2.54869	−0.0831734
$940$	1.79062	0.0584034
$941$	24.2516	0.790580	0.395290	−	0.918556i	$-0.370644\pi$
$941$	24.2516	0.790580	0.395290	+	0.918556i	$0.370644\pi$
$942$	13.5945	0.442933
$943$	93.3250	3.03908
$944$	40.6568	1.32327
$945$	2.38705	0.0776509
$946$	0	0
$947$	−45.0901	−1.46523	−0.732616	−	0.680642i	$-0.761701\pi$
$947$	−45.0901	−1.46523	−0.732616	+	0.680642i	$0.761701\pi$
$948$	0.580957	0.0188686
$949$	4.00000	0.129845
$950$	0.722907	0.0234542
$951$	−31.3434	−1.01638
$952$	−5.98620	−0.194014
$953$	−37.7101	−1.22155	−0.610775	−	0.791804i	$-0.709142\pi$
$953$	−37.7101	−1.22155	−0.610775	+	0.791804i	$0.709142\pi$
$954$	−9.55855	−0.309470
$955$	−57.1748	−1.85013
$956$	−1.48479	−0.0480216
$957$	0	0
$958$	−14.8105	−0.478505
$959$	2.04750	0.0661171
$960$	−17.8219	−0.575199
$961$	12.9074	0.416368
$962$	5.58256	0.179989
$963$	0.283088	0.00912240
$964$	1.12721	0.0363050
$965$	0.333254	0.0107278
$966$	−11.3627	−0.365589
$967$	−60.7131	−1.95240	−0.976201	−	0.216870i	$-0.930415\pi$
$967$	−60.7131	−1.95240	−0.976201	+	0.216870i	$0.930415\pi$
$968$	0	0
$969$	1.55546	0.0499684
$970$	34.3520	1.10297
$971$	−29.4859	−0.946247	−0.473124	−	0.880996i	$-0.656874\pi$
$971$	−29.4859	−0.946247	−0.473124	+	0.880996i	$0.656874\pi$
$972$	−0.119606	−0.00383637
$973$	12.8454	0.411804
$974$	63.8145	2.04475
$975$	0.636443	0.0203825
$976$	0.265154	0.00848738
$977$	4.55016	0.145573	0.0727863	−	0.997348i	$-0.476811\pi$
$977$	4.55016	0.145573	0.0727863	+	0.997348i	$0.476811\pi$
$978$	−15.1863	−0.485603
$979$	0	0
$980$	0.285507	0.00912018
$981$	−8.97962	−0.286697
$982$	27.8376	0.888334
$983$	−36.9736	−1.17927	−0.589637	−	0.807668i	$-0.700729\pi$
$983$	−36.9736	−1.17927	−0.589637	+	0.807668i	$0.700729\pi$
$984$	−32.7356	−1.04357
$985$	34.8779	1.11130
$986$	−28.2293	−0.899004
$987$	6.27171	0.199631
$988$	−0.0775753	−0.00246800
$989$	−58.6599	−1.86528
$990$	0	0
$991$	30.5701	0.971090	0.485545	−	0.874212i	$-0.338621\pi$
$991$	30.5701	0.971090	0.485545	+	0.874212i	$0.338621\pi$
$992$	4.47741	0.142158
$993$	4.82789	0.153209
$994$	8.69084	0.275657
$995$	−63.1857	−2.00312
$996$	−1.20192	−0.0380843
$997$	−2.65445	−0.0840671	−0.0420336	−	0.999116i	$-0.513384\pi$
$997$	−2.65445	−0.0840671	−0.0420336	+	0.999116i	$0.513384\pi$
$998$	29.8415	0.944615
$999$	4.20551	0.133057

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2541.2.a.bn.1.3		4
3.2	odd	2		7623.2.a.ci.1.2		4
11.3	even	5		231.2.j.f.64.2	✓	8
11.4	even	5		231.2.j.f.148.2	yes	8
11.10	odd	2		2541.2.a.bm.1.2		4
33.14	odd	10		693.2.m.f.64.1		8
33.26	odd	10		693.2.m.f.379.1		8
33.32	even	2		7623.2.a.cl.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.j.f.64.2	✓	8	11.3	even	5
231.2.j.f.148.2	yes	8	11.4	even	5
693.2.m.f.64.1		8	33.14	odd	10
693.2.m.f.379.1		8	33.26	odd	10
2541.2.a.bm.1.2		4	11.10	odd	2
2541.2.a.bn.1.3		4	1.1	even	1	trivial
7623.2.a.ci.1.2		4	3.2	odd	2
7623.2.a.cl.1.3		4	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)

\(f(q)\)	\(=\)	\(q+1.45589 q^{2} -1.00000 q^{3} +0.119606 q^{4} +2.38705 q^{5} -1.45589 q^{6} -1.00000 q^{7} -2.73764 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.45589 q^{2} -1.00000 q^{3} +0.119606 q^{4} +2.38705 q^{5} -1.45589 q^{6} -1.00000 q^{7} -2.73764 q^{8} +1.00000 q^{9} +3.47528 q^{10} -0.119606 q^{12} -0.911774 q^{13} -1.45589 q^{14} -2.38705 q^{15} -4.22491 q^{16} -2.18663 q^{17} +1.45589 q^{18} +0.711349 q^{19} +0.285507 q^{20} +1.00000 q^{21} -7.80466 q^{23} +2.73764 q^{24} +0.698028 q^{25} -1.32744 q^{26} -1.00000 q^{27} -0.119606 q^{28} +8.86742 q^{29} -3.47528 q^{30} -6.62627 q^{31} -0.675706 q^{32} -3.18348 q^{34} -2.38705 q^{35} +0.119606 q^{36} -4.20551 q^{37} +1.03564 q^{38} +0.911774 q^{39} -6.53490 q^{40} -11.9576 q^{41} +1.45589 q^{42} +7.51601 q^{43} +2.38705 q^{45} -11.3627 q^{46} +6.27171 q^{47} +4.22491 q^{48} +1.00000 q^{49} +1.01625 q^{50} +2.18663 q^{51} -0.109054 q^{52} -6.56545 q^{53} -1.45589 q^{54} +2.73764 q^{56} -0.711349 q^{57} +12.9100 q^{58} -9.62312 q^{59} -0.285507 q^{60} -0.0627598 q^{61} -9.64709 q^{62} -1.00000 q^{63} +7.46606 q^{64} -2.17645 q^{65} -11.3294 q^{67} -0.261535 q^{68} +7.80466 q^{69} -3.47528 q^{70} -5.96945 q^{71} -2.73764 q^{72} -4.38705 q^{73} -6.12275 q^{74} -0.698028 q^{75} +0.0850818 q^{76} +1.32744 q^{78} -4.85725 q^{79} -10.0851 q^{80} +1.00000 q^{81} -17.4089 q^{82} +10.0490 q^{83} +0.119606 q^{84} -5.21960 q^{85} +10.9425 q^{86} -8.86742 q^{87} -0.261535 q^{89} +3.47528 q^{90} +0.911774 q^{91} -0.933487 q^{92} +6.62627 q^{93} +9.13090 q^{94} +1.69803 q^{95} +0.675706 q^{96} +9.88466 q^{97} +1.45589 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{5} - q^{6} - 4 q^{7} - 9 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q + q^2 - 4 * q^3 + 3 * q^4 - 4 * q^5 - q^6 - 4 * q^7 - 9 * q^8 + 4 * q^9	\( 4 q + q^{2} - 4 q^{3} + 3 q^{4} - 4 q^{5} - q^{6} - 4 q^{7} - 9 q^{8} + 4 q^{9} + 10 q^{10} - 3 q^{12} + 6 q^{13} - q^{14} + 4 q^{15} - 3 q^{16} + 8 q^{17} + q^{18} - 10 q^{19} + 4 q^{21} - 10 q^{23} + 9 q^{24} + 12 q^{25} - 20 q^{26} - 4 q^{27} - 3 q^{28} - 10 q^{30} - 18 q^{31} - 2 q^{32} + 18 q^{34} + 4 q^{35} + 3 q^{36} - 2 q^{37} - 8 q^{38} - 6 q^{39} + 6 q^{40} + 10 q^{41} + q^{42} - 4 q^{43} - 4 q^{45} + 11 q^{46} + 4 q^{47} + 3 q^{48} + 4 q^{49} - 9 q^{50} - 8 q^{51} + 20 q^{52} - q^{54} + 9 q^{56} + 10 q^{57} + 14 q^{58} - 16 q^{59} + 14 q^{61} - 4 q^{63} - 11 q^{64} - 28 q^{65} - 28 q^{67} - 16 q^{68} + 10 q^{69} - 10 q^{70} - 18 q^{71} - 9 q^{72} - 4 q^{73} - 41 q^{74} - 12 q^{75} - 4 q^{76} + 20 q^{78} - 20 q^{79} - 36 q^{80} + 4 q^{81} - 24 q^{82} + 6 q^{83} + 3 q^{84} - 20 q^{85} - 20 q^{86} - 16 q^{89} + 10 q^{90} - 6 q^{91} - 22 q^{92} + 18 q^{93} - 16 q^{94} + 16 q^{95} + 2 q^{96} + 32 q^{97} + q^{98}+O(q^{100}) \) 4 * q + q^2 - 4 * q^3 + 3 * q^4 - 4 * q^5 - q^6 - 4 * q^7 - 9 * q^8 + 4 * q^9 + 10 * q^10 - 3 * q^12 + 6 * q^13 - q^14 + 4 * q^15 - 3 * q^16 + 8 * q^17 + q^18 - 10 * q^19 + 4 * q^21 - 10 * q^23 + 9 * q^24 + 12 * q^25 - 20 * q^26 - 4 * q^27 - 3 * q^28 - 10 * q^30 - 18 * q^31 - 2 * q^32 + 18 * q^34 + 4 * q^35 + 3 * q^36 - 2 * q^37 - 8 * q^38 - 6 * q^39 + 6 * q^40 + 10 * q^41 + q^42 - 4 * q^43 - 4 * q^45 + 11 * q^46 + 4 * q^47 + 3 * q^48 + 4 * q^49 - 9 * q^50 - 8 * q^51 + 20 * q^52 - q^54 + 9 * q^56 + 10 * q^57 + 14 * q^58 - 16 * q^59 + 14 * q^61 - 4 * q^63 - 11 * q^64 - 28 * q^65 - 28 * q^67 - 16 * q^68 + 10 * q^69 - 10 * q^70 - 18 * q^71 - 9 * q^72 - 4 * q^73 - 41 * q^74 - 12 * q^75 - 4 * q^76 + 20 * q^78 - 20 * q^79 - 36 * q^80 + 4 * q^81 - 24 * q^82 + 6 * q^83 + 3 * q^84 - 20 * q^85 - 20 * q^86 - 16 * q^89 + 10 * q^90 - 6 * q^91 - 22 * q^92 + 18 * q^93 - 16 * q^94 + 16 * q^95 + 2 * q^96 + 32 * q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more