LMFDB - Embedded newform 8007.2.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, 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("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$46$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58291 q^{2} +1.00000 q^{3} +4.67145 q^{4} +2.37262 q^{5} -2.58291 q^{6} -1.51271 q^{7} -6.90012 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.58291 q^{2} +1.00000 q^{3} +4.67145 q^{4} +2.37262 q^{5} -2.58291 q^{6} -1.51271 q^{7} -6.90012 q^{8} +1.00000 q^{9} -6.12828 q^{10} +0.0286762 q^{11} +4.67145 q^{12} -2.28200 q^{13} +3.90720 q^{14} +2.37262 q^{15} +8.47952 q^{16} -1.00000 q^{17} -2.58291 q^{18} +3.73826 q^{19} +11.0836 q^{20} -1.51271 q^{21} -0.0740682 q^{22} +0.0338407 q^{23} -6.90012 q^{24} +0.629343 q^{25} +5.89421 q^{26} +1.00000 q^{27} -7.06654 q^{28} -5.28971 q^{29} -6.12828 q^{30} +5.40462 q^{31} -8.10165 q^{32} +0.0286762 q^{33} +2.58291 q^{34} -3.58909 q^{35} +4.67145 q^{36} -4.32358 q^{37} -9.65561 q^{38} -2.28200 q^{39} -16.3714 q^{40} +9.63055 q^{41} +3.90720 q^{42} -6.66478 q^{43} +0.133959 q^{44} +2.37262 q^{45} -0.0874076 q^{46} -7.80572 q^{47} +8.47952 q^{48} -4.71171 q^{49} -1.62554 q^{50} -1.00000 q^{51} -10.6602 q^{52} -4.66126 q^{53} -2.58291 q^{54} +0.0680378 q^{55} +10.4379 q^{56} +3.73826 q^{57} +13.6629 q^{58} -6.95607 q^{59} +11.0836 q^{60} +12.0288 q^{61} -13.9597 q^{62} -1.51271 q^{63} +3.96681 q^{64} -5.41433 q^{65} -0.0740682 q^{66} -10.8834 q^{67} -4.67145 q^{68} +0.0338407 q^{69} +9.27031 q^{70} -14.3426 q^{71} -6.90012 q^{72} +6.50034 q^{73} +11.1674 q^{74} +0.629343 q^{75} +17.4631 q^{76} -0.0433788 q^{77} +5.89421 q^{78} +7.88568 q^{79} +20.1187 q^{80} +1.00000 q^{81} -24.8749 q^{82} -6.05338 q^{83} -7.06654 q^{84} -2.37262 q^{85} +17.2145 q^{86} -5.28971 q^{87} -0.197869 q^{88} +14.4772 q^{89} -6.12828 q^{90} +3.45200 q^{91} +0.158085 q^{92} +5.40462 q^{93} +20.1615 q^{94} +8.86949 q^{95} -8.10165 q^{96} -17.9042 q^{97} +12.1699 q^{98} +0.0286762 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9}+O(q^{10}) $ 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9} - 10 q^{10} - 25 q^{11} + 43 q^{12} - 8 q^{13} - 28 q^{14} - 19 q^{15} + 33 q^{16} - 46 q^{17} - 5 q^{18} - 2 q^{19} - 56 q^{20} + q^{21} - 19 q^{22} - 64 q^{23} - 18 q^{24} + 11 q^{25} - 13 q^{26} + 46 q^{27} - 38 q^{28} - 51 q^{29} - 10 q^{30} - 19 q^{31} - 61 q^{32} - 25 q^{33} + 5 q^{34} - 39 q^{35} + 43 q^{36} - 46 q^{37} - 48 q^{38} - 8 q^{39} - 10 q^{40} - 53 q^{41} - 28 q^{42} - 33 q^{43} - 62 q^{44} - 19 q^{45} + 2 q^{46} - 45 q^{47} + 33 q^{48} + 21 q^{49} - 60 q^{50} - 46 q^{51} - 63 q^{52} - 47 q^{53} - 5 q^{54} + 5 q^{55} - 82 q^{56} - 2 q^{57} - 21 q^{58} - 65 q^{59} - 56 q^{60} - 37 q^{61} - 46 q^{62} + q^{63} + 74 q^{64} - 85 q^{65} - 19 q^{66} - 52 q^{67} - 43 q^{68} - 64 q^{69} - 20 q^{70} - 48 q^{71} - 18 q^{72} - 39 q^{73} - 16 q^{74} + 11 q^{75} + 42 q^{76} - 78 q^{77} - 13 q^{78} - 26 q^{79} - 78 q^{80} + 46 q^{81} + 3 q^{82} - 47 q^{83} - 38 q^{84} + 19 q^{85} - 6 q^{86} - 51 q^{87} - 58 q^{88} - 58 q^{89} - 10 q^{90} - 43 q^{91} - 68 q^{92} - 19 q^{93} - 78 q^{95} - 61 q^{96} - 44 q^{97} - 4 q^{98} - 25 q^{99}+O(q^{100}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9 - 10 * q^10 - 25 * q^11 + 43 * q^12 - 8 * q^13 - 28 * q^14 - 19 * q^15 + 33 * q^16 - 46 * q^17 - 5 * q^18 - 2 * q^19 - 56 * q^20 + q^21 - 19 * q^22 - 64 * q^23 - 18 * q^24 + 11 * q^25 - 13 * q^26 + 46 * q^27 - 38 * q^28 - 51 * q^29 - 10 * q^30 - 19 * q^31 - 61 * q^32 - 25 * q^33 + 5 * q^34 - 39 * q^35 + 43 * q^36 - 46 * q^37 - 48 * q^38 - 8 * q^39 - 10 * q^40 - 53 * q^41 - 28 * q^42 - 33 * q^43 - 62 * q^44 - 19 * q^45 + 2 * q^46 - 45 * q^47 + 33 * q^48 + 21 * q^49 - 60 * q^50 - 46 * q^51 - 63 * q^52 - 47 * q^53 - 5 * q^54 + 5 * q^55 - 82 * q^56 - 2 * q^57 - 21 * q^58 - 65 * q^59 - 56 * q^60 - 37 * q^61 - 46 * q^62 + q^63 + 74 * q^64 - 85 * q^65 - 19 * q^66 - 52 * q^67 - 43 * q^68 - 64 * q^69 - 20 * q^70 - 48 * q^71 - 18 * q^72 - 39 * q^73 - 16 * q^74 + 11 * q^75 + 42 * q^76 - 78 * q^77 - 13 * q^78 - 26 * q^79 - 78 * q^80 + 46 * q^81 + 3 * q^82 - 47 * q^83 - 38 * q^84 + 19 * q^85 - 6 * q^86 - 51 * q^87 - 58 * q^88 - 58 * q^89 - 10 * q^90 - 43 * q^91 - 68 * q^92 - 19 * q^93 - 78 * q^95 - 61 * q^96 - 44 * q^97 - 4 * q^98 - 25 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.58291	−1.82640	−0.913198	−	0.407516i	$-0.866395\pi$
$2$	−2.58291	−1.82640	−0.913198	+	0.407516i	$0.866395\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.67145	2.33572
$5$	2.37262	1.06107	0.530535	−	0.847663i	$-0.321991\pi$
$5$	2.37262	1.06107	0.530535	+	0.847663i	$0.321991\pi$
$6$	−2.58291	−1.05447
$7$	−1.51271	−0.571750	−0.285875	−	0.958267i	$-0.592284\pi$
$7$	−1.51271	−0.571750	−0.285875	+	0.958267i	$0.592284\pi$
$8$	−6.90012	−2.43956
$9$	1.00000	0.333333
$10$	−6.12828	−1.93793
$11$	0.0286762	0.00864620	0.00432310	−	0.999991i	$-0.498624\pi$
$11$	0.0286762	0.00864620	0.00432310	+	0.999991i	$0.498624\pi$
$12$	4.67145	1.34853
$13$	−2.28200	−0.632913	−0.316457	−	0.948607i	$-0.602493\pi$
$13$	−2.28200	−0.632913	−0.316457	+	0.948607i	$0.602493\pi$
$14$	3.90720	1.04424
$15$	2.37262	0.612609
$16$	8.47952	2.11988
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−2.58291	−0.608799
$19$	3.73826	0.857616	0.428808	−	0.903396i	$-0.358934\pi$
$19$	3.73826	0.857616	0.428808	+	0.903396i	$0.358934\pi$
$20$	11.0836	2.47837
$21$	−1.51271	−0.330100
$22$	−0.0740682	−0.0157914
$23$	0.0338407	0.00705627	0.00352814	−	0.999994i	$-0.498877\pi$
$23$	0.0338407	0.00705627	0.00352814	+	0.999994i	$0.498877\pi$
$24$	−6.90012	−1.40848
$25$	0.629343	0.125869
$26$	5.89421	1.15595
$27$	1.00000	0.192450
$28$	−7.06654	−1.33545
$29$	−5.28971	−0.982275	−0.491138	−	0.871082i	$-0.663419\pi$
$29$	−5.28971	−0.982275	−0.491138	+	0.871082i	$0.663419\pi$
$30$	−6.12828	−1.11887
$31$	5.40462	0.970699	0.485350	−	0.874320i	$-0.338692\pi$
$31$	5.40462	0.970699	0.485350	+	0.874320i	$0.338692\pi$
$32$	−8.10165	−1.43218
$33$	0.0286762	0.00499189
$34$	2.58291	0.442966
$35$	−3.58909	−0.606667
$36$	4.67145	0.778575
$37$	−4.32358	−0.710791	−0.355396	−	0.934716i	$-0.615654\pi$
$37$	−4.32358	−0.710791	−0.355396	+	0.934716i	$0.615654\pi$
$38$	−9.65561	−1.56635
$39$	−2.28200	−0.365413
$40$	−16.3714	−2.58854
$41$	9.63055	1.50404	0.752020	−	0.659140i	$-0.229080\pi$
$41$	9.63055	1.50404	0.752020	+	0.659140i	$0.229080\pi$
$42$	3.90720	0.602894
$43$	−6.66478	−1.01637	−0.508184	−	0.861248i	$-0.669683\pi$
$43$	−6.66478	−1.01637	−0.508184	+	0.861248i	$0.669683\pi$
$44$	0.133959	0.0201951
$45$	2.37262	0.353690
$46$	−0.0874076	−0.0128876
$47$	−7.80572	−1.13858	−0.569291	−	0.822136i	$-0.692782\pi$
$47$	−7.80572	−1.13858	−0.569291	+	0.822136i	$0.692782\pi$
$48$	8.47952	1.22391
$49$	−4.71171	−0.673102
$50$	−1.62554	−0.229886
$51$	−1.00000	−0.140028
$52$	−10.6602	−1.47831
$53$	−4.66126	−0.640273	−0.320137	−	0.947371i	$-0.603729\pi$
$53$	−4.66126	−0.640273	−0.320137	+	0.947371i	$0.603729\pi$
$54$	−2.58291	−0.351490
$55$	0.0680378	0.00917422
$56$	10.4379	1.39482
$57$	3.73826	0.495145
$58$	13.6629	1.79402
$59$	−6.95607	−0.905603	−0.452802	−	0.891611i	$-0.649575\pi$
$59$	−6.95607	−0.905603	−0.452802	+	0.891611i	$0.649575\pi$
$60$	11.0836	1.43088
$61$	12.0288	1.54013	0.770066	−	0.637964i	$-0.220223\pi$
$61$	12.0288	1.54013	0.770066	+	0.637964i	$0.220223\pi$
$62$	−13.9597	−1.77288
$63$	−1.51271	−0.190583
$64$	3.96681	0.495851
$65$	−5.41433	−0.671565
$66$	−0.0740682	−0.00911716
$67$	−10.8834	−1.32962	−0.664811	−	0.747012i	$-0.731488\pi$
$67$	−10.8834	−1.32962	−0.664811	+	0.747012i	$0.731488\pi$
$68$	−4.67145	−0.566496
$69$	0.0338407	0.00407394
$70$	9.27031	1.10801
$71$	−14.3426	−1.70215	−0.851076	−	0.525043i	$-0.824049\pi$
$71$	−14.3426	−1.70215	−0.851076	+	0.525043i	$0.824049\pi$
$72$	−6.90012	−0.813187
$73$	6.50034	0.760807	0.380404	−	0.924821i	$-0.375785\pi$
$73$	6.50034	0.760807	0.380404	+	0.924821i	$0.375785\pi$
$74$	11.1674	1.29819
$75$	0.629343	0.0726702
$76$	17.4631	2.00315
$77$	−0.0433788	−0.00494347
$78$	5.89421	0.667388
$79$	7.88568	0.887209	0.443604	−	0.896223i	$-0.353700\pi$
$79$	7.88568	0.887209	0.443604	+	0.896223i	$0.353700\pi$
$80$	20.1187	2.24934
$81$	1.00000	0.111111
$82$	−24.8749	−2.74697
$83$	−6.05338	−0.664445	−0.332222	−	0.943201i	$-0.607798\pi$
$83$	−6.05338	−0.664445	−0.332222	+	0.943201i	$0.607798\pi$
$84$	−7.06654	−0.771023
$85$	−2.37262	−0.257347
$86$	17.2145	1.85629
$87$	−5.28971	−0.567117
$88$	−0.197869	−0.0210929
$89$	14.4772	1.53458	0.767291	−	0.641299i	$-0.221604\pi$
$89$	14.4772	1.53458	0.767291	+	0.641299i	$0.221604\pi$
$90$	−6.12828	−0.645978
$91$	3.45200	0.361868
$92$	0.158085	0.0164815
$93$	5.40462	0.560433
$94$	20.1615	2.07950
$95$	8.86949	0.909990
$96$	−8.10165	−0.826871
$97$	−17.9042	−1.81790	−0.908951	−	0.416904i	$-0.863115\pi$
$97$	−17.9042	−1.81790	−0.908951	+	0.416904i	$0.863115\pi$
$98$	12.1699	1.22935
$99$	0.0286762	0.00288207
$100$	2.93994	0.293994
$101$	3.95845	0.393881	0.196940	−	0.980415i	$-0.436899\pi$
$101$	3.95845	0.393881	0.196940	+	0.980415i	$0.436899\pi$
$102$	2.58291	0.255747
$103$	7.76336	0.764947	0.382473	−	0.923967i	$-0.375072\pi$
$103$	7.76336	0.764947	0.382473	+	0.923967i	$0.375072\pi$
$104$	15.7461	1.54403
$105$	−3.58909	−0.350259
$106$	12.0396	1.16939
$107$	1.85870	0.179687	0.0898436	−	0.995956i	$-0.471363\pi$
$107$	1.85870	0.179687	0.0898436	+	0.995956i	$0.471363\pi$
$108$	4.67145	0.449510
$109$	9.08366	0.870057	0.435028	−	0.900417i	$-0.356738\pi$
$109$	9.08366	0.870057	0.435028	+	0.900417i	$0.356738\pi$
$110$	−0.175736	−0.0167558
$111$	−4.32358	−0.410376
$112$	−12.8271	−1.21204
$113$	−14.5043	−1.36445	−0.682224	−	0.731143i	$-0.738987\pi$
$113$	−14.5043	−1.36445	−0.682224	+	0.731143i	$0.738987\pi$
$114$	−9.65561	−0.904331
$115$	0.0802912	0.00748720
$116$	−24.7106	−2.29432
$117$	−2.28200	−0.210971
$118$	17.9669	1.65399
$119$	1.51271	0.138670
$120$	−16.3714	−1.49450
$121$	−10.9992	−0.999925
$122$	−31.0694	−2.81289
$123$	9.63055	0.868358
$124$	25.2474	2.26728
$125$	−10.3699	−0.927514
$126$	3.90720	0.348081
$127$	8.52240	0.756241	0.378120	−	0.925756i	$-0.376571\pi$
$127$	8.52240	0.756241	0.378120	+	0.925756i	$0.376571\pi$
$128$	5.95736	0.526561
$129$	−6.66478	−0.586801
$130$	13.9848	1.22654
$131$	3.65276	0.319143	0.159572	−	0.987186i	$-0.448989\pi$
$131$	3.65276	0.319143	0.159572	+	0.987186i	$0.448989\pi$
$132$	0.133959	0.0116597
$133$	−5.65490	−0.490342
$134$	28.1110	2.42842
$135$	2.37262	0.204203
$136$	6.90012	0.591680
$137$	4.43424	0.378842	0.189421	−	0.981896i	$-0.439339\pi$
$137$	4.43424	0.378842	0.189421	+	0.981896i	$0.439339\pi$
$138$	−0.0874076	−0.00744063
$139$	15.4863	1.31353	0.656766	−	0.754094i	$-0.271924\pi$
$139$	15.4863	1.31353	0.656766	+	0.754094i	$0.271924\pi$
$140$	−16.7662	−1.41701
$141$	−7.80572	−0.657360
$142$	37.0457	3.10880
$143$	−0.0654391	−0.00547230
$144$	8.47952	0.706627
$145$	−12.5505	−1.04226
$146$	−16.7898	−1.38954
$147$	−4.71171	−0.388615
$148$	−20.1974	−1.66021
$149$	−24.1740	−1.98041	−0.990207	−	0.139605i	$-0.955417\pi$
$149$	−24.1740	−1.98041	−0.990207	+	0.139605i	$0.955417\pi$
$150$	−1.62554	−0.132725
$151$	−8.74504	−0.711662	−0.355831	−	0.934550i	$-0.615802\pi$
$151$	−8.74504	−0.711662	−0.355831	+	0.934550i	$0.615802\pi$
$152$	−25.7944	−2.09221
$153$	−1.00000	−0.0808452
$154$	0.112044	0.00902873
$155$	12.8231	1.02998
$156$	−10.6602	−0.853503
$157$	1.00000	0.0798087
$157$	1.00000	0.0798087
$158$	−20.3680	−1.62039
$159$	−4.66126	−0.369662
$160$	−19.2222	−1.51964
$161$	−0.0511911	−0.00403443
$162$	−2.58291	−0.202933
$163$	8.83357	0.691898	0.345949	−	0.938253i	$-0.387557\pi$
$163$	8.83357	0.691898	0.345949	+	0.938253i	$0.387557\pi$
$164$	44.9886	3.51302
$165$	0.0680378	0.00529674
$166$	15.6354	1.21354
$167$	0.442060	0.0342077	0.0171038	−	0.999854i	$-0.494555\pi$
$167$	0.442060	0.0342077	0.0171038	+	0.999854i	$0.494555\pi$
$168$	10.4379	0.805299
$169$	−7.79247	−0.599421
$170$	6.12828	0.470018
$171$	3.73826	0.285872
$172$	−31.1341	−2.37396
$173$	−20.4052	−1.55138	−0.775688	−	0.631117i	$-0.782597\pi$
$173$	−20.4052	−1.55138	−0.775688	+	0.631117i	$0.782597\pi$
$174$	13.6629	1.03578
$175$	−0.952012	−0.0719654
$176$	0.243161	0.0183289
$177$	−6.95607	−0.522850
$178$	−37.3934	−2.80275
$179$	−7.46180	−0.557721	−0.278860	−	0.960332i	$-0.589957\pi$
$179$	−7.46180	−0.557721	−0.278860	+	0.960332i	$0.589957\pi$
$180$	11.0836	0.826122
$181$	12.9590	0.963237	0.481619	−	0.876381i	$-0.340049\pi$
$181$	12.9590	0.963237	0.481619	+	0.876381i	$0.340049\pi$
$182$	−8.91623	−0.660915
$183$	12.0288	0.889196
$184$	−0.233505	−0.0172142
$185$	−10.2582	−0.754199
$186$	−13.9597	−1.02357
$187$	−0.0286762	−0.00209701
$188$	−36.4640	−2.65941
$189$	−1.51271	−0.110033
$190$	−22.9091	−1.66200
$191$	11.3965	0.824624	0.412312	−	0.911043i	$-0.364721\pi$
$191$	11.3965	0.824624	0.412312	+	0.911043i	$0.364721\pi$
$192$	3.96681	0.286280
$193$	20.0172	1.44087	0.720435	−	0.693523i	$-0.243942\pi$
$193$	20.0172	1.44087	0.720435	+	0.693523i	$0.243942\pi$
$194$	46.2451	3.32021
$195$	−5.41433	−0.387728
$196$	−22.0105	−1.57218
$197$	−22.2550	−1.58560	−0.792801	−	0.609480i	$-0.791378\pi$
$197$	−22.2550	−1.58560	−0.792801	+	0.609480i	$0.791378\pi$
$198$	−0.0740682	−0.00526380
$199$	0.846276	0.0599909	0.0299955	−	0.999550i	$-0.490451\pi$
$199$	0.846276	0.0599909	0.0299955	+	0.999550i	$0.490451\pi$
$200$	−4.34254	−0.307064
$201$	−10.8834	−0.767658
$202$	−10.2243	−0.719382
$203$	8.00180	0.561616
$204$	−4.67145	−0.327067
$205$	22.8497	1.59589
$206$	−20.0521	−1.39710
$207$	0.0338407	0.00235209
$208$	−19.3503	−1.34170
$209$	0.107199	0.00741512
$210$	9.27031	0.639712
$211$	−25.3900	−1.74792	−0.873961	−	0.485996i	$-0.838457\pi$
$211$	−25.3900	−1.74792	−0.873961	+	0.485996i	$0.838457\pi$
$212$	−21.7748	−1.49550
$213$	−14.3426	−0.982738
$214$	−4.80086	−0.328180
$215$	−15.8130	−1.07844
$216$	−6.90012	−0.469494
$217$	−8.17562	−0.554998
$218$	−23.4623	−1.58907
$219$	6.50034	0.439252
$220$	0.317835	0.0214284
$221$	2.28200	0.153504
$222$	11.1674	0.749508
$223$	2.62729	0.175936	0.0879682	−	0.996123i	$-0.471963\pi$
$223$	2.62729	0.175936	0.0879682	+	0.996123i	$0.471963\pi$
$224$	12.2554	0.818850
$225$	0.629343	0.0419562
$226$	37.4633	2.49202
$227$	14.8856	0.987990	0.493995	−	0.869465i	$-0.335536\pi$
$227$	14.8856	0.987990	0.493995	+	0.869465i	$0.335536\pi$
$228$	17.4631	1.15652
$229$	−13.0204	−0.860413	−0.430206	−	0.902731i	$-0.641559\pi$
$229$	−13.0204	−0.860413	−0.430206	+	0.902731i	$0.641559\pi$
$230$	−0.207385	−0.0136746
$231$	−0.0433788	−0.00285411
$232$	36.4997	2.39632
$233$	−21.9985	−1.44117	−0.720583	−	0.693368i	$-0.756126\pi$
$233$	−21.9985	−1.44117	−0.720583	+	0.693368i	$0.756126\pi$
$234$	5.89421	0.385317
$235$	−18.5200	−1.20811
$236$	−32.4949	−2.11524
$237$	7.88568	0.512230
$238$	−3.90720	−0.253266
$239$	−11.3861	−0.736505	−0.368253	−	0.929726i	$-0.620044\pi$
$239$	−11.3861	−0.736505	−0.368253	+	0.929726i	$0.620044\pi$
$240$	20.1187	1.29866
$241$	4.02368	0.259188	0.129594	−	0.991567i	$-0.458633\pi$
$241$	4.02368	0.259188	0.129594	+	0.991567i	$0.458633\pi$
$242$	28.4099	1.82626
$243$	1.00000	0.0641500
$244$	56.1920	3.59732
$245$	−11.1791	−0.714208
$246$	−24.8749	−1.58597
$247$	−8.53072	−0.542797
$248$	−37.2926	−2.36808
$249$	−6.05338	−0.383617
$250$	26.7846	1.69401
$251$	−19.8345	−1.25194	−0.625971	−	0.779847i	$-0.715297\pi$
$251$	−19.8345	−1.25194	−0.625971	+	0.779847i	$0.715297\pi$
$252$	−7.06654	−0.445150
$253$	0.000970423 0	6.10100e−5 0
$254$	−22.0126	−1.38120
$255$	−2.37262	−0.148579
$256$	−23.3210	−1.45756
$257$	−22.4275	−1.39899	−0.699495	−	0.714637i	$-0.746592\pi$
$257$	−22.4275	−1.39899	−0.699495	+	0.714637i	$0.746592\pi$
$258$	17.2145	1.07173
$259$	6.54031	0.406395
$260$	−25.2928	−1.56859
$261$	−5.28971	−0.327425
$262$	−9.43478	−0.582882
$263$	−16.1298	−0.994607	−0.497304	−	0.867577i	$-0.665677\pi$
$263$	−16.1298	−0.994607	−0.497304	+	0.867577i	$0.665677\pi$
$264$	−0.197869	−0.0121780
$265$	−11.0594	−0.679374
$266$	14.6061	0.895559
$267$	14.4772	0.885991
$268$	−50.8413	−3.10563
$269$	−16.2546	−0.991062	−0.495531	−	0.868590i	$-0.665026\pi$
$269$	−16.2546	−0.991062	−0.495531	+	0.868590i	$0.665026\pi$
$270$	−6.12828	−0.372955
$271$	−21.3796	−1.29872	−0.649360	−	0.760481i	$-0.724963\pi$
$271$	−21.3796	−1.29872	−0.649360	+	0.760481i	$0.724963\pi$
$272$	−8.47952	−0.514147
$273$	3.45200	0.208925
$274$	−11.4533	−0.691917
$275$	0.0180472	0.00108828
$276$	0.158085	0.00951560
$277$	21.3761	1.28436	0.642182	−	0.766552i	$-0.278029\pi$
$277$	21.3761	1.28436	0.642182	+	0.766552i	$0.278029\pi$
$278$	−39.9999	−2.39903
$279$	5.40462	0.323566
$280$	24.7651	1.48000
$281$	−11.0237	−0.657620	−0.328810	−	0.944396i	$-0.606648\pi$
$281$	−11.0237	−0.657620	−0.328810	+	0.944396i	$0.606648\pi$
$282$	20.1615	1.20060
$283$	11.1859	0.664931	0.332466	−	0.943115i	$-0.392119\pi$
$283$	11.1859	0.664931	0.332466	+	0.943115i	$0.392119\pi$
$284$	−67.0006	−3.97576
$285$	8.86949	0.525383
$286$	0.169024	0.00999458
$287$	−14.5682	−0.859935
$288$	−8.10165	−0.477394
$289$	1.00000	0.0588235
$290$	32.4169	1.90358
$291$	−17.9042	−1.04957
$292$	30.3660	1.77704
$293$	−14.6330	−0.854870	−0.427435	−	0.904046i	$-0.640583\pi$
$293$	−14.6330	−0.854870	−0.427435	+	0.904046i	$0.640583\pi$
$294$	12.1699	0.709766
$295$	−16.5041	−0.960908
$296$	29.8332	1.73402
$297$	0.0286762	0.00166396
$298$	62.4395	3.61702
$299$	−0.0772245	−0.00446601
$300$	2.93994	0.169738
$301$	10.0819	0.581109
$302$	22.5877	1.29978
$303$	3.95845	0.227407
$304$	31.6987	1.81804
$305$	28.5399	1.63419
$306$	2.58291	0.147655
$307$	−17.6730	−1.00865	−0.504327	−	0.863513i	$-0.668259\pi$
$307$	−17.6730	−1.00865	−0.504327	+	0.863513i	$0.668259\pi$
$308$	−0.202642	−0.0115466
$309$	7.76336	0.441642
$310$	−33.1211	−1.88115
$311$	11.9994	0.680421	0.340210	−	0.940349i	$-0.389502\pi$
$311$	11.9994	0.680421	0.340210	+	0.940349i	$0.389502\pi$
$312$	15.7461	0.891446
$313$	23.4022	1.32277	0.661384	−	0.750047i	$-0.269969\pi$
$313$	23.4022	1.32277	0.661384	+	0.750047i	$0.269969\pi$
$314$	−2.58291	−0.145762
$315$	−3.58909	−0.202222
$316$	36.8376	2.07227
$317$	30.6745	1.72285	0.861427	−	0.507882i	$-0.169571\pi$
$317$	30.6745	1.72285	0.861427	+	0.507882i	$0.169571\pi$
$318$	12.0396	0.675149
$319$	−0.151689	−0.00849295
$320$	9.41175	0.526133
$321$	1.85870	0.103742
$322$	0.132222	0.00736846
$323$	−3.73826	−0.208002
$324$	4.67145	0.259525
$325$	−1.43616	−0.0796639
$326$	−22.8163	−1.26368
$327$	9.08366	0.502328
$328$	−66.4520	−3.66920
$329$	11.8078	0.650984
$330$	−0.175736	−0.00967394
$331$	−5.21080	−0.286412	−0.143206	−	0.989693i	$-0.545741\pi$
$331$	−5.21080	−0.286412	−0.143206	+	0.989693i	$0.545741\pi$
$332$	−28.2780	−1.55196
$333$	−4.32358	−0.236930
$334$	−1.14180	−0.0624768
$335$	−25.8223	−1.41082
$336$	−12.8271	−0.699773
$337$	30.7138	1.67309	0.836544	−	0.547900i	$-0.184573\pi$
$337$	30.7138	1.67309	0.836544	+	0.547900i	$0.184573\pi$
$338$	20.1273	1.09478
$339$	−14.5043	−0.787765
$340$	−11.0836	−0.601092
$341$	0.154984	0.00839286
$342$	−9.65561	−0.522115
$343$	17.7164	0.956596
$344$	45.9877	2.47949
$345$	0.0802912	0.00432274
$346$	52.7048	2.83343
$347$	22.9647	1.23281	0.616406	−	0.787429i	$-0.288588\pi$
$347$	22.9647	1.23281	0.616406	+	0.787429i	$0.288588\pi$
$348$	−24.7106	−1.32463
$349$	−17.6342	−0.943935	−0.471968	−	0.881616i	$-0.656456\pi$
$349$	−17.6342	−0.943935	−0.471968	+	0.881616i	$0.656456\pi$
$350$	2.45897	0.131437
$351$	−2.28200	−0.121804
$352$	−0.232324	−0.0123829
$353$	−18.5631	−0.988012	−0.494006	−	0.869459i	$-0.664468\pi$
$353$	−18.5631	−0.988012	−0.494006	+	0.869459i	$0.664468\pi$
$354$	17.9669	0.954932
$355$	−34.0296	−1.80610
$356$	67.6295	3.58436
$357$	1.51271	0.0800610
$358$	19.2732	1.01862
$359$	−22.1744	−1.17032	−0.585160	−	0.810918i	$-0.698968\pi$
$359$	−22.1744	−1.17032	−0.585160	+	0.810918i	$0.698968\pi$
$360$	−16.3714	−0.862848
$361$	−5.02541	−0.264495
$362$	−33.4721	−1.75925
$363$	−10.9992	−0.577307
$364$	16.1259	0.845224
$365$	15.4229	0.807270
$366$	−31.0694	−1.62402
$367$	36.6906	1.91524	0.957618	−	0.288043i	$-0.0930045\pi$
$367$	36.6906	1.91524	0.957618	+	0.288043i	$0.0930045\pi$
$368$	0.286953	0.0149585
$369$	9.63055	0.501347
$370$	26.4961	1.37747
$371$	7.05113	0.366076
$372$	25.2474	1.30902
$373$	−18.7946	−0.973148	−0.486574	−	0.873639i	$-0.661754\pi$
$373$	−18.7946	−0.973148	−0.486574	+	0.873639i	$0.661754\pi$
$374$	0.0740682	0.00382997
$375$	−10.3699	−0.535501
$376$	53.8604	2.77764
$377$	12.0711	0.621695
$378$	3.90720	0.200965
$379$	−19.8808	−1.02121	−0.510605	−	0.859816i	$-0.670578\pi$
$379$	−19.8808	−1.02121	−0.510605	+	0.859816i	$0.670578\pi$
$380$	41.4333	2.12549
$381$	8.52240	0.436616
$382$	−29.4363	−1.50609
$383$	20.6523	1.05528	0.527642	−	0.849467i	$-0.323076\pi$
$383$	20.6523	1.05528	0.527642	+	0.849467i	$0.323076\pi$
$384$	5.95736	0.304010
$385$	−0.102921	−0.00524536
$386$	−51.7027	−2.63160
$387$	−6.66478	−0.338790
$388$	−83.6388	−4.24611
$389$	18.9814	0.962397	0.481199	−	0.876612i	$-0.340202\pi$
$389$	18.9814	0.962397	0.481199	+	0.876612i	$0.340202\pi$
$390$	13.9848	0.708145
$391$	−0.0338407	−0.00171140
$392$	32.5114	1.64207
$393$	3.65276	0.184258
$394$	57.4828	2.89594
$395$	18.7098	0.941390
$396$	0.133959	0.00673171
$397$	−9.79340	−0.491517	−0.245758	−	0.969331i	$-0.579037\pi$
$397$	−9.79340	−0.491517	−0.245758	+	0.969331i	$0.579037\pi$
$398$	−2.18586	−0.109567
$399$	−5.65490	−0.283099
$400$	5.33653	0.266826
$401$	−0.655997	−0.0327589	−0.0163795	−	0.999866i	$-0.505214\pi$
$401$	−0.655997	−0.0327589	−0.0163795	+	0.999866i	$0.505214\pi$
$402$	28.1110	1.40205
$403$	−12.3334	−0.614368
$404$	18.4917	0.919997
$405$	2.37262	0.117897
$406$	−20.6680	−1.02573
$407$	−0.123984	−0.00614565
$408$	6.90012	0.341607
$409$	35.9707	1.77864	0.889319	−	0.457287i	$-0.151179\pi$
$409$	35.9707	1.77864	0.889319	+	0.457287i	$0.151179\pi$
$410$	−59.0188	−2.91473
$411$	4.43424	0.218725
$412$	36.2661	1.78670
$413$	10.5225	0.517779
$414$	−0.0874076	−0.00429585
$415$	−14.3624	−0.705022
$416$	18.4880	0.906447
$417$	15.4863	0.758369
$418$	−0.276886	−0.0135429
$419$	6.54048	0.319523	0.159762	−	0.987156i	$-0.448927\pi$
$419$	6.54048	0.319523	0.159762	+	0.987156i	$0.448927\pi$
$420$	−16.7662	−0.818109
$421$	1.25924	0.0613718	0.0306859	−	0.999529i	$-0.490231\pi$
$421$	1.25924	0.0613718	0.0306859	+	0.999529i	$0.490231\pi$
$422$	65.5803	3.19240
$423$	−7.80572	−0.379527
$424$	32.1632	1.56199
$425$	−0.629343	−0.0305276
$426$	37.0457	1.79487
$427$	−18.1961	−0.880571
$428$	8.68281	0.419700
$429$	−0.0654391	−0.00315943
$430$	40.8436	1.96965
$431$	35.8060	1.72471	0.862357	−	0.506301i	$-0.168988\pi$
$431$	35.8060	1.72471	0.862357	+	0.506301i	$0.168988\pi$
$432$	8.47952	0.407971
$433$	−20.8729	−1.00309	−0.501545	−	0.865132i	$-0.667235\pi$
$433$	−20.8729	−1.00309	−0.501545	+	0.865132i	$0.667235\pi$
$434$	21.1169	1.01365
$435$	−12.5505	−0.601750
$436$	42.4338	2.03221
$437$	0.126505	0.00605157
$438$	−16.7898	−0.802249
$439$	−36.6084	−1.74722	−0.873612	−	0.486622i	$-0.838229\pi$
$439$	−36.6084	−1.74722	−0.873612	+	0.486622i	$0.838229\pi$
$440$	−0.469469	−0.0223811
$441$	−4.71171	−0.224367
$442$	−5.89421	−0.280359
$443$	12.2761	0.583257	0.291629	−	0.956532i	$-0.405803\pi$
$443$	12.2761	0.583257	0.291629	+	0.956532i	$0.405803\pi$
$444$	−20.1974	−0.958524
$445$	34.3490	1.62830
$446$	−6.78607	−0.321330
$447$	−24.1740	−1.14339
$448$	−6.00063	−0.283503
$449$	−41.6897	−1.96746	−0.983730	−	0.179651i	$-0.942503\pi$
$449$	−41.6897	−1.96746	−0.983730	+	0.179651i	$0.942503\pi$
$450$	−1.62554	−0.0766286
$451$	0.276168	0.0130042
$452$	−67.7560	−3.18698
$453$	−8.74504	−0.410878
$454$	−38.4482	−1.80446
$455$	8.19031	0.383967
$456$	−25.7944	−1.20794
$457$	25.6731	1.20094	0.600468	−	0.799649i	$-0.294981\pi$
$457$	25.6731	1.20094	0.600468	+	0.799649i	$0.294981\pi$
$458$	33.6306	1.57145
$459$	−1.00000	−0.0466760
$460$	0.375076	0.0174880
$461$	27.7166	1.29089	0.645446	−	0.763806i	$-0.276671\pi$
$461$	27.7166	1.29089	0.645446	+	0.763806i	$0.276671\pi$
$462$	0.112044	0.00521274
$463$	−0.997341	−0.0463504	−0.0231752	−	0.999731i	$-0.507378\pi$
$463$	−0.997341	−0.0463504	−0.0231752	+	0.999731i	$0.507378\pi$
$464$	−44.8543	−2.08231
$465$	12.8231	0.594659
$466$	56.8201	2.63214
$467$	−41.6728	−1.92839	−0.964194	−	0.265199i	$-0.914562\pi$
$467$	−41.6728	−1.92839	−0.964194	+	0.265199i	$0.914562\pi$
$468$	−10.6602	−0.492770
$469$	16.4635	0.760212
$470$	47.8357	2.20649
$471$	1.00000	0.0460776
$472$	47.9977	2.20927
$473$	−0.191120	−0.00878773
$474$	−20.3680	−0.935535
$475$	2.35265	0.107947
$476$	7.06654	0.323894
$477$	−4.66126	−0.213424
$478$	29.4093	1.34515
$479$	4.86314	0.222202	0.111101	−	0.993809i	$-0.464562\pi$
$479$	4.86314	0.222202	0.111101	+	0.993809i	$0.464562\pi$
$480$	−19.2222	−0.877367
$481$	9.86641	0.449869
$482$	−10.3928	−0.473380
$483$	−0.0511911	−0.00232928
$484$	−51.3821	−2.33555
$485$	−42.4800	−1.92892
$486$	−2.58291	−0.117163
$487$	−20.0544	−0.908753	−0.454376	−	0.890810i	$-0.650138\pi$
$487$	−20.0544	−0.908753	−0.454376	+	0.890810i	$0.650138\pi$
$488$	−83.0003	−3.75725
$489$	8.83357	0.399468
$490$	28.8747	1.30443
$491$	−7.57996	−0.342079	−0.171039	−	0.985264i	$-0.554713\pi$
$491$	−7.57996	−0.342079	−0.171039	+	0.985264i	$0.554713\pi$
$492$	44.9886	2.02824
$493$	5.28971	0.238237
$494$	22.0341	0.991362
$495$	0.0680378	0.00305807
$496$	45.8286	2.05777
$497$	21.6962	0.973206
$498$	15.6354	0.700637
$499$	−21.0943	−0.944311	−0.472155	−	0.881515i	$-0.656524\pi$
$499$	−21.0943	−0.944311	−0.472155	+	0.881515i	$0.656524\pi$
$500$	−48.4426	−2.16642
$501$	0.442060	0.0197498
$502$	51.2308	2.28654
$503$	−20.9533	−0.934260	−0.467130	−	0.884189i	$-0.654712\pi$
$503$	−20.9533	−0.934260	−0.467130	+	0.884189i	$0.654712\pi$
$504$	10.4379	0.464940
$505$	9.39192	0.417935
$506$	−0.00250652	−0.000111428 0
$507$	−7.79247	−0.346076
$508$	39.8119	1.76637
$509$	28.6678	1.27068	0.635339	−	0.772233i	$-0.280860\pi$
$509$	28.6678	1.27068	0.635339	+	0.772233i	$0.280860\pi$
$510$	6.12828	0.271365
$511$	−9.83312	−0.434992
$512$	48.3214	2.13552
$513$	3.73826	0.165048
$514$	57.9284	2.55511
$515$	18.4195	0.811662
$516$	−31.1341	−1.37060
$517$	−0.223838	−0.00984440
$518$	−16.8931	−0.742239
$519$	−20.4052	−0.895687
$520$	37.3595	1.63832
$521$	−27.3387	−1.19773	−0.598864	−	0.800851i	$-0.704381\pi$
$521$	−27.3387	−1.19773	−0.598864	+	0.800851i	$0.704381\pi$
$522$	13.6629	0.598008
$523$	41.8747	1.83105	0.915526	−	0.402260i	$-0.131775\pi$
$523$	41.8747	1.83105	0.915526	+	0.402260i	$0.131775\pi$
$524$	17.0637	0.745431
$525$	−0.952012	−0.0415492
$526$	41.6619	1.81655
$527$	−5.40462	−0.235429
$528$	0.243161	0.0105822
$529$	−22.9989	−0.999950
$530$	28.5655	1.24081
$531$	−6.95607	−0.301868
$532$	−26.4166	−1.14530
$533$	−21.9769	−0.951927
$534$	−37.3934	−1.61817
$535$	4.40999	0.190661
$536$	75.0969	3.24369
$537$	−7.46180	−0.322000
$538$	41.9843	1.81007
$539$	−0.135114	−0.00581977
$540$	11.0836	0.476962
$541$	24.8749	1.06945	0.534727	−	0.845025i	$-0.320414\pi$
$541$	24.8749	1.06945	0.534727	+	0.845025i	$0.320414\pi$
$542$	55.2218	2.37198
$543$	12.9590	0.556125
$544$	8.10165	0.347355
$545$	21.5521	0.923191
$546$	−8.91623	−0.381579
$547$	−3.06881	−0.131213	−0.0656063	−	0.997846i	$-0.520898\pi$
$547$	−3.06881	−0.131213	−0.0656063	+	0.997846i	$0.520898\pi$
$548$	20.7143	0.884871
$549$	12.0288	0.513377
$550$	−0.0466143	−0.00198764
$551$	−19.7743	−0.842415
$552$	−0.233505	−0.00993863
$553$	−11.9287	−0.507262
$554$	−55.2126	−2.34576
$555$	−10.2582	−0.435437
$556$	72.3435	3.06805
$557$	−4.80336	−0.203525	−0.101762	−	0.994809i	$-0.532448\pi$
$557$	−4.80336	−0.203525	−0.101762	+	0.994809i	$0.532448\pi$
$558$	−13.9597	−0.590960
$559$	15.2090	0.643273
$560$	−30.4338	−1.28606
$561$	−0.0286762	−0.00121071
$562$	28.4733	1.20107
$563$	−16.8888	−0.711776	−0.355888	−	0.934529i	$-0.615822\pi$
$563$	−16.8888	−0.711776	−0.355888	+	0.934529i	$0.615822\pi$
$564$	−36.4640	−1.53541
$565$	−34.4132	−1.44777
$566$	−28.8922	−1.21443
$567$	−1.51271	−0.0635278
$568$	98.9656	4.15250
$569$	19.9842	0.837780	0.418890	−	0.908037i	$-0.362419\pi$
$569$	19.9842	0.837780	0.418890	+	0.908037i	$0.362419\pi$
$570$	−22.9091	−0.959558
$571$	8.52893	0.356925	0.178462	−	0.983947i	$-0.442888\pi$
$571$	8.52893	0.356925	0.178462	+	0.983947i	$0.442888\pi$
$572$	−0.305695	−0.0127818
$573$	11.3965	0.476097
$574$	37.6285	1.57058
$575$	0.0212974	0.000888163 0
$576$	3.96681	0.165284
$577$	−18.7931	−0.782366	−0.391183	−	0.920313i	$-0.627934\pi$
$577$	−18.7931	−0.782366	−0.391183	+	0.920313i	$0.627934\pi$
$578$	−2.58291	−0.107435
$579$	20.0172	0.831886
$580$	−58.6290	−2.43444
$581$	9.15700	0.379896
$582$	46.2451	1.91692
$583$	−0.133667	−0.00553593
$584$	−44.8531	−1.85604
$585$	−5.41433	−0.223855
$586$	37.7958	1.56133
$587$	5.70930	0.235648	0.117824	−	0.993034i	$-0.462408\pi$
$587$	5.70930	0.235648	0.117824	+	0.993034i	$0.462408\pi$
$588$	−22.0105	−0.907698
$589$	20.2039	0.832487
$590$	42.6288	1.75500
$591$	−22.2550	−0.915448
$592$	−36.6619	−1.50679
$593$	−31.6183	−1.29841	−0.649204	−	0.760614i	$-0.724898\pi$
$593$	−31.6183	−1.29841	−0.649204	+	0.760614i	$0.724898\pi$
$594$	−0.0740682	−0.00303905
$595$	3.58909	0.147138
$596$	−112.928	−4.62570
$597$	0.846276	0.0346358
$598$	0.199464	0.00815670
$599$	−27.0048	−1.10339	−0.551693	−	0.834048i	$-0.686018\pi$
$599$	−27.0048	−1.10339	−0.551693	+	0.834048i	$0.686018\pi$
$600$	−4.34254	−0.177283
$601$	−4.78688	−0.195261	−0.0976304	−	0.995223i	$-0.531126\pi$
$601$	−4.78688	−0.195261	−0.0976304	+	0.995223i	$0.531126\pi$
$602$	−26.0406	−1.06134
$603$	−10.8834	−0.443207
$604$	−40.8520	−1.66224
$605$	−26.0969	−1.06099
$606$	−10.2243	−0.415336
$607$	−28.2056	−1.14483	−0.572415	−	0.819964i	$-0.693993\pi$
$607$	−28.2056	−1.14483	−0.572415	+	0.819964i	$0.693993\pi$
$608$	−30.2861	−1.22826
$609$	8.00180	0.324249
$610$	−73.7160	−2.98467
$611$	17.8127	0.720623
$612$	−4.67145	−0.188832
$613$	4.49092	0.181387	0.0906933	−	0.995879i	$-0.471092\pi$
$613$	4.49092	0.181387	0.0906933	+	0.995879i	$0.471092\pi$
$614$	45.6479	1.84220
$615$	22.8497	0.921388
$616$	0.299319	0.0120599
$617$	−15.9359	−0.641557	−0.320778	−	0.947154i	$-0.603945\pi$
$617$	−15.9359	−0.641557	−0.320778	+	0.947154i	$0.603945\pi$
$618$	−20.0521	−0.806614
$619$	47.2496	1.89912	0.949560	−	0.313585i	$-0.101530\pi$
$619$	47.2496	1.89912	0.949560	+	0.313585i	$0.101530\pi$
$620$	59.9026	2.40575
$621$	0.0338407	0.00135798
$622$	−30.9933	−1.24272
$623$	−21.8998	−0.877397
$624$	−19.3503	−0.774631
$625$	−27.7506	−1.11003
$626$	−60.4458	−2.41590
$627$	0.107199	0.00428112
$628$	4.67145	0.186411
$629$	4.32358	0.172392
$630$	9.27031	0.369338
$631$	43.4005	1.72775	0.863874	−	0.503709i	$-0.168032\pi$
$631$	43.4005	1.72775	0.863874	+	0.503709i	$0.168032\pi$
$632$	−54.4122	−2.16440
$633$	−25.3900	−1.00916
$634$	−79.2297	−3.14661
$635$	20.2205	0.802424
$636$	−21.7748	−0.863428
$637$	10.7521	0.426015
$638$	0.391799	0.0155115
$639$	−14.3426	−0.567384
$640$	14.1346	0.558718
$641$	43.2254	1.70730	0.853650	−	0.520847i	$-0.174384\pi$
$641$	43.2254	1.70730	0.853650	+	0.520847i	$0.174384\pi$
$642$	−4.80086	−0.189475
$643$	−31.2128	−1.23091	−0.615457	−	0.788170i	$-0.711029\pi$
$643$	−31.2128	−1.23091	−0.615457	+	0.788170i	$0.711029\pi$
$644$	−0.239137	−0.00942330
$645$	−15.8130	−0.622636
$646$	9.65561	0.379895
$647$	−29.6126	−1.16419	−0.582096	−	0.813120i	$-0.697767\pi$
$647$	−29.6126	−1.16419	−0.582096	+	0.813120i	$0.697767\pi$
$648$	−6.90012	−0.271062
$649$	−0.199474	−0.00783003
$650$	3.70948	0.145498
$651$	−8.17562	−0.320428
$652$	41.2655	1.61608
$653$	19.9254	0.779743	0.389871	−	0.920869i	$-0.372519\pi$
$653$	19.9254	0.779743	0.389871	+	0.920869i	$0.372519\pi$
$654$	−23.4623	−0.917449
$655$	8.66663	0.338633
$656$	81.6625	3.18839
$657$	6.50034	0.253602
$658$	−30.4985	−1.18895
$659$	22.6520	0.882395	0.441197	−	0.897410i	$-0.354554\pi$
$659$	22.6520	0.882395	0.441197	+	0.897410i	$0.354554\pi$
$660$	0.317835	0.0123717
$661$	9.50529	0.369713	0.184857	−	0.982766i	$-0.440818\pi$
$661$	9.50529	0.369713	0.184857	+	0.982766i	$0.440818\pi$
$662$	13.4591	0.523101
$663$	2.28200	0.0886256
$664$	41.7690	1.62095
$665$	−13.4170	−0.520287
$666$	11.1674	0.432729
$667$	−0.179008	−0.00693120
$668$	2.06506	0.0798997
$669$	2.62729	0.101577
$670$	66.6967	2.57672
$671$	0.344941	0.0133163
$672$	12.2554	0.472764
$673$	−39.6317	−1.52769	−0.763846	−	0.645399i	$-0.776691\pi$
$673$	−39.6317	−1.52769	−0.763846	+	0.645399i	$0.776691\pi$
$674$	−79.3311	−3.05572
$675$	0.629343	0.0242234
$676$	−36.4021	−1.40008
$677$	14.6513	0.563096	0.281548	−	0.959547i	$-0.409152\pi$
$677$	14.6513	0.563096	0.281548	+	0.959547i	$0.409152\pi$
$678$	37.4633	1.43877
$679$	27.0839	1.03939
$680$	16.3714	0.627814
$681$	14.8856	0.570417
$682$	−0.400311	−0.0153287
$683$	23.4988	0.899157	0.449579	−	0.893241i	$-0.351574\pi$
$683$	23.4988	0.899157	0.449579	+	0.893241i	$0.351574\pi$
$684$	17.4631	0.667718
$685$	10.5208	0.401978
$686$	−45.7600	−1.74712
$687$	−13.0204	−0.496760
$688$	−56.5141	−2.15458
$689$	10.6370	0.405237
$690$	−0.207385	−0.00789503
$691$	15.5103	0.590041	0.295020	−	0.955491i	$-0.404674\pi$
$691$	15.5103	0.590041	0.295020	+	0.955491i	$0.404674\pi$
$692$	−95.3216	−3.62358
$693$	−0.0433788	−0.00164782
$694$	−59.3159	−2.25160
$695$	36.7432	1.39375
$696$	36.4997	1.38352
$697$	−9.63055	−0.364783
$698$	45.5475	1.72400
$699$	−21.9985	−0.832058
$700$	−4.44728	−0.168091
$701$	0.516175	0.0194957	0.00974783	−	0.999952i	$-0.496897\pi$
$701$	0.516175	0.0194957	0.00974783	+	0.999952i	$0.496897\pi$
$702$	5.89421	0.222463
$703$	−16.1627	−0.609586
$704$	0.113753	0.00428723
$705$	−18.5200	−0.697505
$706$	47.9468	1.80450
$707$	−5.98799	−0.225201
$708$	−32.4949	−1.22123
$709$	40.0680	1.50478	0.752392	−	0.658715i	$-0.228900\pi$
$709$	40.0680	1.50478	0.752392	+	0.658715i	$0.228900\pi$
$710$	87.8954	3.29866
$711$	7.88568	0.295736
$712$	−99.8945	−3.74370
$713$	0.182896	0.00684952
$714$	−3.90720	−0.146223
$715$	−0.155262	−0.00580649
$716$	−34.8574	−1.30268
$717$	−11.3861	−0.425222
$718$	57.2745	2.13747
$719$	−36.8765	−1.37526	−0.687631	−	0.726060i	$-0.741349\pi$
$719$	−36.8765	−1.37526	−0.687631	+	0.726060i	$0.741349\pi$
$720$	20.1187	0.749780
$721$	−11.7437	−0.437358
$722$	12.9802	0.483073
$723$	4.02368	0.149642
$724$	60.5374	2.24986
$725$	−3.32904	−0.123638
$726$	28.4099	1.05439
$727$	−30.7527	−1.14056	−0.570278	−	0.821452i	$-0.693164\pi$
$727$	−30.7527	−1.14056	−0.570278	+	0.821452i	$0.693164\pi$
$728$	−23.8192	−0.882800
$729$	1.00000	0.0370370
$730$	−39.8359	−1.47439
$731$	6.66478	0.246506
$732$	56.1920	2.07692
$733$	−24.1240	−0.891042	−0.445521	−	0.895271i	$-0.646982\pi$
$733$	−24.1240	−0.891042	−0.445521	+	0.895271i	$0.646982\pi$
$734$	−94.7688	−3.49798
$735$	−11.1791	−0.412348
$736$	−0.274165	−0.0101059
$737$	−0.312095	−0.0114962
$738$	−24.8749	−0.915658
$739$	−30.5703	−1.12455	−0.562273	−	0.826951i	$-0.690073\pi$
$739$	−30.5703	−1.12455	−0.562273	+	0.826951i	$0.690073\pi$
$740$	−47.9207	−1.76160
$741$	−8.53072	−0.313384
$742$	−18.2125	−0.668601
$743$	21.5567	0.790840	0.395420	−	0.918500i	$-0.370599\pi$
$743$	21.5567	0.790840	0.395420	+	0.918500i	$0.370599\pi$
$744$	−37.2926	−1.36721
$745$	−57.3559	−2.10136
$746$	48.5449	1.77735
$747$	−6.05338	−0.221482
$748$	−0.133959	−0.00489804
$749$	−2.81167	−0.102736
$750$	26.7846	0.978036
$751$	34.4229	1.25611	0.628054	−	0.778170i	$-0.283852\pi$
$751$	34.4229	1.25611	0.628054	+	0.778170i	$0.283852\pi$
$752$	−66.1888	−2.41366
$753$	−19.8345	−0.722809
$754$	−31.1787	−1.13546
$755$	−20.7487	−0.755122
$756$	−7.06654	−0.257008
$757$	−33.0431	−1.20097	−0.600486	−	0.799635i	$-0.705026\pi$
$757$	−33.0431	−1.20097	−0.600486	+	0.799635i	$0.705026\pi$
$758$	51.3505	1.86513
$759$	0.000970423 0	3.52241e−5 0
$760$	−61.2005	−2.21998
$761$	−13.5002	−0.489382	−0.244691	−	0.969601i	$-0.578687\pi$
$761$	−13.5002	−0.489382	−0.244691	+	0.969601i	$0.578687\pi$
$762$	−22.0126	−0.797434
$763$	−13.7409	−0.497455
$764$	53.2383	1.92609
$765$	−2.37262	−0.0857824
$766$	−53.3432	−1.92737
$767$	15.8738	0.573168
$768$	−23.3210	−0.841523
$769$	38.0526	1.37221	0.686106	−	0.727502i	$-0.259319\pi$
$769$	38.0526	1.37221	0.686106	+	0.727502i	$0.259319\pi$
$770$	0.265837	0.00958011
$771$	−22.4275	−0.807708
$772$	93.5093	3.36547
$773$	−5.98815	−0.215379	−0.107689	−	0.994185i	$-0.534345\pi$
$773$	−5.98815	−0.215379	−0.107689	+	0.994185i	$0.534345\pi$
$774$	17.2145	0.618764
$775$	3.40136	0.122180
$776$	123.541	4.43488
$777$	6.54031	0.234632
$778$	−49.0274	−1.75772
$779$	36.0015	1.28989
$780$	−25.2928	−0.905626
$781$	−0.411291	−0.0147171
$782$	0.0874076	0.00312569
$783$	−5.28971	−0.189039
$784$	−39.9531	−1.42690
$785$	2.37262	0.0846826
$786$	−9.43478	−0.336527
$787$	32.4237	1.15578	0.577890	−	0.816115i	$-0.303876\pi$
$787$	32.4237	1.15578	0.577890	+	0.816115i	$0.303876\pi$
$788$	−103.963	−3.70353
$789$	−16.1298	−0.574237
$790$	−48.3257	−1.71935
$791$	21.9408	0.780124
$792$	−0.197869	−0.00703098
$793$	−27.4498	−0.974770
$794$	25.2955	0.897705
$795$	−11.0594	−0.392237
$796$	3.95333	0.140122
$797$	9.29495	0.329244	0.164622	−	0.986357i	$-0.447360\pi$
$797$	9.29495	0.329244	0.164622	+	0.986357i	$0.447360\pi$
$798$	14.6061	0.517051
$799$	7.80572	0.276146
$800$	−5.09871	−0.180267
$801$	14.4772	0.511527
$802$	1.69438	0.0598308
$803$	0.186405	0.00657809
$804$	−50.8413	−1.79304
$805$	−0.121457	−0.00428081
$806$	31.8560	1.12208
$807$	−16.2546	−0.572190
$808$	−27.3138	−0.960896
$809$	−47.0654	−1.65473	−0.827366	−	0.561663i	$-0.810162\pi$
$809$	−47.0654	−1.65473	−0.827366	+	0.561663i	$0.810162\pi$
$810$	−6.12828	−0.215326
$811$	−49.1822	−1.72702	−0.863510	−	0.504331i	$-0.831739\pi$
$811$	−49.1822	−1.72702	−0.863510	+	0.504331i	$0.831739\pi$
$812$	37.3800	1.31178
$813$	−21.3796	−0.749817
$814$	0.320239	0.0112244
$815$	20.9587	0.734152
$816$	−8.47952	−0.296843
$817$	−24.9147	−0.871654
$818$	−92.9093	−3.24850
$819$	3.45200	0.120623
$820$	106.741	3.72756
$821$	8.57676	0.299331	0.149666	−	0.988737i	$-0.452180\pi$
$821$	8.57676	0.299331	0.149666	+	0.988737i	$0.452180\pi$
$822$	−11.4533	−0.399478
$823$	1.25206	0.0436442	0.0218221	−	0.999762i	$-0.493053\pi$
$823$	1.25206	0.0436442	0.0218221	+	0.999762i	$0.493053\pi$
$824$	−53.5681	−1.86613
$825$	0.0180472	0.000628321 0
$826$	−27.1787	−0.945669
$827$	38.9584	1.35472	0.677358	−	0.735654i	$-0.263125\pi$
$827$	38.9584	1.35472	0.677358	+	0.735654i	$0.263125\pi$
$828$	0.158085	0.00549383
$829$	10.3655	0.360008	0.180004	−	0.983666i	$-0.442389\pi$
$829$	10.3655	0.360008	0.180004	+	0.983666i	$0.442389\pi$
$830$	37.0968	1.28765
$831$	21.3761	0.741528
$832$	−9.05226	−0.313831
$833$	4.71171	0.163251
$834$	−39.9999	−1.38508
$835$	1.04884	0.0362967
$836$	0.500775	0.0173197
$837$	5.40462	0.186811
$838$	−16.8935	−0.583576
$839$	−16.4567	−0.568147	−0.284073	−	0.958803i	$-0.591686\pi$
$839$	−16.4567	−0.568147	−0.284073	+	0.958803i	$0.591686\pi$
$840$	24.7651	0.854479
$841$	−1.01893	−0.0351355
$842$	−3.25252	−0.112089
$843$	−11.0237	−0.379677
$844$	−118.608	−4.08266
$845$	−18.4886	−0.636027
$846$	20.1615	0.693167
$847$	16.6386	0.571708
$848$	−39.5253	−1.35730
$849$	11.1859	0.383898
$850$	1.62554	0.0557555
$851$	−0.146313	−0.00501554
$852$	−67.0006	−2.29540
$853$	−14.1523	−0.484566	−0.242283	−	0.970206i	$-0.577896\pi$
$853$	−14.1523	−0.484566	−0.242283	+	0.970206i	$0.577896\pi$
$854$	46.9990	1.60827
$855$	8.86949	0.303330
$856$	−12.8252	−0.438358
$857$	−35.3956	−1.20909	−0.604545	−	0.796571i	$-0.706645\pi$
$857$	−35.3956	−1.20909	−0.604545	+	0.796571i	$0.706645\pi$
$858$	0.169024	0.00577037
$859$	−7.80266	−0.266223	−0.133112	−	0.991101i	$-0.542497\pi$
$859$	−7.80266	−0.266223	−0.133112	+	0.991101i	$0.542497\pi$
$860$	−73.8696	−2.51893
$861$	−14.5682	−0.496484
$862$	−92.4838	−3.15001
$863$	−43.0230	−1.46452	−0.732260	−	0.681026i	$-0.761534\pi$
$863$	−43.0230	−1.46452	−0.732260	+	0.681026i	$0.761534\pi$
$864$	−8.10165	−0.275624
$865$	−48.4138	−1.64612
$866$	53.9130	1.83204
$867$	1.00000	0.0339618
$868$	−38.1920	−1.29632
$869$	0.226131	0.00767098
$870$	32.4169	1.09903
$871$	24.8360	0.841536
$872$	−62.6783	−2.12256
$873$	−17.9042	−0.605967
$874$	−0.326753	−0.0110526
$875$	15.6867	0.530307
$876$	30.3660	1.02597
$877$	−11.9359	−0.403048	−0.201524	−	0.979484i	$-0.564589\pi$
$877$	−11.9359	−0.403048	−0.201524	+	0.979484i	$0.564589\pi$
$878$	94.5564	3.19113
$879$	−14.6330	−0.493559
$880$	0.576928	0.0194483
$881$	−57.4822	−1.93662	−0.968312	−	0.249742i	$-0.919654\pi$
$881$	−57.4822	−1.93662	−0.968312	+	0.249742i	$0.919654\pi$
$882$	12.1699	0.409783
$883$	3.38181	0.113807	0.0569035	−	0.998380i	$-0.481877\pi$
$883$	3.38181	0.113807	0.0569035	+	0.998380i	$0.481877\pi$
$884$	10.6602	0.358543
$885$	−16.5041	−0.554780
$886$	−31.7082	−1.06526
$887$	−21.2891	−0.714818	−0.357409	−	0.933948i	$-0.616340\pi$
$887$	−21.2891	−0.714818	−0.357409	+	0.933948i	$0.616340\pi$
$888$	29.8332	1.00114
$889$	−12.8919	−0.432381
$890$	−88.7205	−2.97392
$891$	0.0286762	0.000960689 0
$892$	12.2733	0.410939
$893$	−29.1798	−0.976465
$894$	62.4395	2.08829
$895$	−17.7040	−0.591781
$896$	−9.01175	−0.301062
$897$	−0.0772245	−0.00257845
$898$	107.681	3.59336
$899$	−28.5889	−0.953494
$900$	2.93994	0.0979980
$901$	4.66126	0.155289
$902$	−0.713318	−0.0237509
$903$	10.0819	0.335503
$904$	100.081	3.32866
$905$	30.7469	1.02206
$906$	22.5877	0.750426
$907$	−5.16508	−0.171504	−0.0857518	−	0.996317i	$-0.527329\pi$
$907$	−5.16508	−0.171504	−0.0857518	+	0.996317i	$0.527329\pi$
$908$	69.5372	2.30767
$909$	3.95845	0.131294
$910$	−21.1549	−0.701277
$911$	26.6273	0.882203	0.441101	−	0.897457i	$-0.354588\pi$
$911$	26.6273	0.882203	0.441101	+	0.897457i	$0.354588\pi$
$912$	31.6987	1.04965
$913$	−0.173588	−0.00574492
$914$	−66.3113	−2.19338
$915$	28.5399	0.943499
$916$	−60.8241	−2.00969
$917$	−5.52557	−0.182470
$918$	2.58291	0.0852489
$919$	16.5329	0.545372	0.272686	−	0.962103i	$-0.412088\pi$
$919$	16.5329	0.545372	0.272686	+	0.962103i	$0.412088\pi$
$920$	−0.554019	−0.0182655
$921$	−17.6730	−0.582346
$922$	−71.5897	−2.35768
$923$	32.7298	1.07731
$924$	−0.202642	−0.00666642
$925$	−2.72101	−0.0894663
$926$	2.57605	0.0846542
$927$	7.76336	0.254982
$928$	42.8554	1.40680
$929$	−27.5828	−0.904963	−0.452481	−	0.891774i	$-0.649461\pi$
$929$	−27.5828	−0.904963	−0.452481	+	0.891774i	$0.649461\pi$
$930$	−33.1211	−1.08608
$931$	−17.6136	−0.577263
$932$	−102.765	−3.36617
$933$	11.9994	0.392841
$934$	107.637	3.52200
$935$	−0.0680378	−0.00222508
$936$	15.7461	0.514677
$937$	16.6308	0.543304	0.271652	−	0.962396i	$-0.412430\pi$
$937$	16.6308	0.543304	0.271652	+	0.962396i	$0.412430\pi$
$938$	−42.5237	−1.38845
$939$	23.4022	0.763701
$940$	−86.5153	−2.82182
$941$	−36.7704	−1.19868	−0.599341	−	0.800494i	$-0.704571\pi$
$941$	−36.7704	−1.19868	−0.599341	+	0.800494i	$0.704571\pi$
$942$	−2.58291	−0.0841559
$943$	0.325905	0.0106129
$944$	−58.9842	−1.91977
$945$	−3.58909	−0.116753
$946$	0.493648	0.0160499
$947$	48.1935	1.56608	0.783039	−	0.621972i	$-0.213668\pi$
$947$	48.1935	1.56608	0.783039	+	0.621972i	$0.213668\pi$
$948$	36.8376	1.19643
$949$	−14.8338	−0.481525
$950$	−6.07669	−0.197154
$951$	30.6745	0.994690
$952$	−10.4379	−0.338293
$953$	21.7547	0.704704	0.352352	−	0.935868i	$-0.385382\pi$
$953$	21.7547	0.704704	0.352352	+	0.935868i	$0.385382\pi$
$954$	12.0396	0.389798
$955$	27.0397	0.874983
$956$	−53.1896	−1.72027
$957$	−0.151689	−0.00490341
$958$	−12.5611	−0.405830
$959$	−6.70771	−0.216603
$960$	9.41175	0.303763
$961$	−1.79004	−0.0577431
$962$	−25.4841	−0.821640
$963$	1.85870	0.0598957
$964$	18.7964	0.605392
$965$	47.4933	1.52886
$966$	0.132222	0.00425418
$967$	−17.3147	−0.556805	−0.278402	−	0.960465i	$-0.589805\pi$
$967$	−17.3147	−0.556805	−0.278402	+	0.960465i	$0.589805\pi$
$968$	75.8956	2.43938
$969$	−3.73826	−0.120090
$970$	109.722	3.52297
$971$	9.92492	0.318506	0.159253	−	0.987238i	$-0.449091\pi$
$971$	9.92492	0.318506	0.159253	+	0.987238i	$0.449091\pi$
$972$	4.67145	0.149837
$973$	−23.4263	−0.751013
$974$	51.7989	1.65974
$975$	−1.43616	−0.0459940
$976$	101.999	3.26490
$977$	13.8017	0.441556	0.220778	−	0.975324i	$-0.429140\pi$
$977$	13.8017	0.441556	0.220778	+	0.975324i	$0.429140\pi$
$978$	−22.8163	−0.729586
$979$	0.415152	0.0132683
$980$	−52.2227	−1.66819
$981$	9.08366	0.290019
$982$	19.5784	0.624772
$983$	28.3122	0.903018	0.451509	−	0.892267i	$-0.350886\pi$
$983$	28.3122	0.903018	0.451509	+	0.892267i	$0.350886\pi$
$984$	−66.4520	−2.11841
$985$	−52.8027	−1.68243
$986$	−13.6629	−0.435115
$987$	11.8078	0.375846
$988$	−39.8508	−1.26782
$989$	−0.225541	−0.00717178
$990$	−0.175736	−0.00558525
$991$	20.1767	0.640933	0.320467	−	0.947260i	$-0.396160\pi$
$991$	20.1767	0.640933	0.320467	+	0.947260i	$0.396160\pi$
$992$	−43.7863	−1.39022
$993$	−5.21080	−0.165360
$994$	−56.0393	−1.77746
$995$	2.00789	0.0636545
$996$	−28.2780	−0.896024
$997$	−33.4910	−1.06067	−0.530335	−	0.847788i	$-0.677934\pi$
$997$	−33.4910	−1.06067	−0.530335	+	0.847788i	$0.677934\pi$
$998$	54.4848	1.72469
$999$	−4.32358	−0.136792

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.e.1.4	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.e.1.4	✓	46	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58291 q^{2} +1.00000 q^{3} +4.67145 q^{4} +2.37262 q^{5} -2.58291 q^{6} -1.51271 q^{7} -6.90012 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.58291 q^{2} +1.00000 q^{3} +4.67145 q^{4} +2.37262 q^{5} -2.58291 q^{6} -1.51271 q^{7} -6.90012 q^{8} +1.00000 q^{9} -6.12828 q^{10} +0.0286762 q^{11} +4.67145 q^{12} -2.28200 q^{13} +3.90720 q^{14} +2.37262 q^{15} +8.47952 q^{16} -1.00000 q^{17} -2.58291 q^{18} +3.73826 q^{19} +11.0836 q^{20} -1.51271 q^{21} -0.0740682 q^{22} +0.0338407 q^{23} -6.90012 q^{24} +0.629343 q^{25} +5.89421 q^{26} +1.00000 q^{27} -7.06654 q^{28} -5.28971 q^{29} -6.12828 q^{30} +5.40462 q^{31} -8.10165 q^{32} +0.0286762 q^{33} +2.58291 q^{34} -3.58909 q^{35} +4.67145 q^{36} -4.32358 q^{37} -9.65561 q^{38} -2.28200 q^{39} -16.3714 q^{40} +9.63055 q^{41} +3.90720 q^{42} -6.66478 q^{43} +0.133959 q^{44} +2.37262 q^{45} -0.0874076 q^{46} -7.80572 q^{47} +8.47952 q^{48} -4.71171 q^{49} -1.62554 q^{50} -1.00000 q^{51} -10.6602 q^{52} -4.66126 q^{53} -2.58291 q^{54} +0.0680378 q^{55} +10.4379 q^{56} +3.73826 q^{57} +13.6629 q^{58} -6.95607 q^{59} +11.0836 q^{60} +12.0288 q^{61} -13.9597 q^{62} -1.51271 q^{63} +3.96681 q^{64} -5.41433 q^{65} -0.0740682 q^{66} -10.8834 q^{67} -4.67145 q^{68} +0.0338407 q^{69} +9.27031 q^{70} -14.3426 q^{71} -6.90012 q^{72} +6.50034 q^{73} +11.1674 q^{74} +0.629343 q^{75} +17.4631 q^{76} -0.0433788 q^{77} +5.89421 q^{78} +7.88568 q^{79} +20.1187 q^{80} +1.00000 q^{81} -24.8749 q^{82} -6.05338 q^{83} -7.06654 q^{84} -2.37262 q^{85} +17.2145 q^{86} -5.28971 q^{87} -0.197869 q^{88} +14.4772 q^{89} -6.12828 q^{90} +3.45200 q^{91} +0.158085 q^{92} +5.40462 q^{93} +20.1615 q^{94} +8.86949 q^{95} -8.10165 q^{96} -17.9042 q^{97} +12.1699 q^{98} +0.0286762 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9}+O(q^{10}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9	\( 46 q - 5 q^{2} + 46 q^{3} + 43 q^{4} - 19 q^{5} - 5 q^{6} + q^{7} - 18 q^{8} + 46 q^{9} - 10 q^{10} - 25 q^{11} + 43 q^{12} - 8 q^{13} - 28 q^{14} - 19 q^{15} + 33 q^{16} - 46 q^{17} - 5 q^{18} - 2 q^{19} - 56 q^{20} + q^{21} - 19 q^{22} - 64 q^{23} - 18 q^{24} + 11 q^{25} - 13 q^{26} + 46 q^{27} - 38 q^{28} - 51 q^{29} - 10 q^{30} - 19 q^{31} - 61 q^{32} - 25 q^{33} + 5 q^{34} - 39 q^{35} + 43 q^{36} - 46 q^{37} - 48 q^{38} - 8 q^{39} - 10 q^{40} - 53 q^{41} - 28 q^{42} - 33 q^{43} - 62 q^{44} - 19 q^{45} + 2 q^{46} - 45 q^{47} + 33 q^{48} + 21 q^{49} - 60 q^{50} - 46 q^{51} - 63 q^{52} - 47 q^{53} - 5 q^{54} + 5 q^{55} - 82 q^{56} - 2 q^{57} - 21 q^{58} - 65 q^{59} - 56 q^{60} - 37 q^{61} - 46 q^{62} + q^{63} + 74 q^{64} - 85 q^{65} - 19 q^{66} - 52 q^{67} - 43 q^{68} - 64 q^{69} - 20 q^{70} - 48 q^{71} - 18 q^{72} - 39 q^{73} - 16 q^{74} + 11 q^{75} + 42 q^{76} - 78 q^{77} - 13 q^{78} - 26 q^{79} - 78 q^{80} + 46 q^{81} + 3 q^{82} - 47 q^{83} - 38 q^{84} + 19 q^{85} - 6 q^{86} - 51 q^{87} - 58 q^{88} - 58 q^{89} - 10 q^{90} - 43 q^{91} - 68 q^{92} - 19 q^{93} - 78 q^{95} - 61 q^{96} - 44 q^{97} - 4 q^{98} - 25 q^{99}+O(q^{100}) \) 46 * q - 5 * q^2 + 46 * q^3 + 43 * q^4 - 19 * q^5 - 5 * q^6 + q^7 - 18 * q^8 + 46 * q^9 - 10 * q^10 - 25 * q^11 + 43 * q^12 - 8 * q^13 - 28 * q^14 - 19 * q^15 + 33 * q^16 - 46 * q^17 - 5 * q^18 - 2 * q^19 - 56 * q^20 + q^21 - 19 * q^22 - 64 * q^23 - 18 * q^24 + 11 * q^25 - 13 * q^26 + 46 * q^27 - 38 * q^28 - 51 * q^29 - 10 * q^30 - 19 * q^31 - 61 * q^32 - 25 * q^33 + 5 * q^34 - 39 * q^35 + 43 * q^36 - 46 * q^37 - 48 * q^38 - 8 * q^39 - 10 * q^40 - 53 * q^41 - 28 * q^42 - 33 * q^43 - 62 * q^44 - 19 * q^45 + 2 * q^46 - 45 * q^47 + 33 * q^48 + 21 * q^49 - 60 * q^50 - 46 * q^51 - 63 * q^52 - 47 * q^53 - 5 * q^54 + 5 * q^55 - 82 * q^56 - 2 * q^57 - 21 * q^58 - 65 * q^59 - 56 * q^60 - 37 * q^61 - 46 * q^62 + q^63 + 74 * q^64 - 85 * q^65 - 19 * q^66 - 52 * q^67 - 43 * q^68 - 64 * q^69 - 20 * q^70 - 48 * q^71 - 18 * q^72 - 39 * q^73 - 16 * q^74 + 11 * q^75 + 42 * q^76 - 78 * q^77 - 13 * q^78 - 26 * q^79 - 78 * q^80 + 46 * q^81 + 3 * q^82 - 47 * q^83 - 38 * q^84 + 19 * q^85 - 6 * q^86 - 51 * q^87 - 58 * q^88 - 58 * q^89 - 10 * q^90 - 43 * q^91 - 68 * q^92 - 19 * q^93 - 78 * q^95 - 61 * q^96 - 44 * q^97 - 4 * q^98 - 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more