LMFDB - Embedded newform 8041.2.a.j.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8041,2,Mod(1,8041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, 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("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8041 = 11 \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8041.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$64.2077082653$
Analytic rank:	$0$
Dimension:	$82$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.34655 q^{2} +3.28622 q^{3} +3.50630 q^{4} +3.73933 q^{5} -7.71129 q^{6} +3.13477 q^{7} -3.53461 q^{8} +7.79927 q^{9} +O(q^{10})$	\(q-2.34655 q^{2} +3.28622 q^{3} +3.50630 q^{4} +3.73933 q^{5} -7.71129 q^{6} +3.13477 q^{7} -3.53461 q^{8} +7.79927 q^{9} -8.77452 q^{10} +1.00000 q^{11} +11.5225 q^{12} +0.157450 q^{13} -7.35589 q^{14} +12.2883 q^{15} +1.28155 q^{16} +1.00000 q^{17} -18.3014 q^{18} -5.83875 q^{19} +13.1112 q^{20} +10.3015 q^{21} -2.34655 q^{22} +0.802293 q^{23} -11.6155 q^{24} +8.98258 q^{25} -0.369464 q^{26} +15.7715 q^{27} +10.9914 q^{28} +1.68678 q^{29} -28.8351 q^{30} +0.923264 q^{31} +4.06201 q^{32} +3.28622 q^{33} -2.34655 q^{34} +11.7219 q^{35} +27.3466 q^{36} +7.42808 q^{37} +13.7009 q^{38} +0.517416 q^{39} -13.2171 q^{40} -5.49809 q^{41} -24.1731 q^{42} +1.00000 q^{43} +3.50630 q^{44} +29.1640 q^{45} -1.88262 q^{46} -6.58511 q^{47} +4.21145 q^{48} +2.82675 q^{49} -21.0781 q^{50} +3.28622 q^{51} +0.552067 q^{52} -4.16982 q^{53} -37.0086 q^{54} +3.73933 q^{55} -11.0802 q^{56} -19.1874 q^{57} -3.95812 q^{58} -0.00333073 q^{59} +43.0864 q^{60} +7.96347 q^{61} -2.16649 q^{62} +24.4489 q^{63} -12.0948 q^{64} +0.588757 q^{65} -7.71129 q^{66} -2.69073 q^{67} +3.50630 q^{68} +2.63651 q^{69} -27.5061 q^{70} -0.778687 q^{71} -27.5674 q^{72} -13.6854 q^{73} -17.4304 q^{74} +29.5188 q^{75} -20.4724 q^{76} +3.13477 q^{77} -1.21414 q^{78} -7.37260 q^{79} +4.79212 q^{80} +28.4308 q^{81} +12.9016 q^{82} +6.85470 q^{83} +36.1203 q^{84} +3.73933 q^{85} -2.34655 q^{86} +5.54314 q^{87} -3.53461 q^{88} -9.62399 q^{89} -68.4349 q^{90} +0.493568 q^{91} +2.81308 q^{92} +3.03405 q^{93} +15.4523 q^{94} -21.8330 q^{95} +13.3487 q^{96} +5.96564 q^{97} -6.63312 q^{98} +7.79927 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9}+O(q^{10}) $ 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9	\( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9} + q^{10} + 82 q^{11} + 3 q^{12} + 26 q^{13} + 17 q^{14} + 66 q^{15} + 122 q^{16} + 82 q^{17} + 18 q^{18} + 12 q^{19} + 9 q^{20} + 22 q^{21} + 8 q^{22} + 50 q^{23} + 15 q^{24} + 117 q^{25} + 36 q^{26} + 30 q^{27} + 11 q^{28} + 33 q^{29} - 26 q^{30} + 40 q^{31} + 58 q^{32} + 6 q^{33} + 8 q^{34} + 16 q^{35} + 160 q^{36} + 31 q^{37} + 18 q^{38} + 41 q^{39} - 29 q^{40} + 42 q^{41} - 51 q^{42} + 82 q^{43} + 98 q^{44} - 2 q^{45} - 19 q^{46} + 84 q^{47} - 46 q^{48} + 136 q^{49} + 59 q^{50} + 6 q^{51} + 45 q^{52} + 83 q^{53} + 24 q^{54} + 11 q^{55} + 21 q^{56} + 23 q^{57} + 14 q^{58} + 96 q^{59} + 184 q^{60} - 6 q^{61} - 23 q^{62} + 8 q^{63} + 148 q^{64} + 5 q^{65} + 10 q^{66} + 78 q^{67} + 98 q^{68} + 61 q^{69} - 3 q^{70} + 155 q^{71} + 50 q^{72} - 23 q^{73} + 10 q^{74} - 19 q^{75} + 44 q^{76} + 8 q^{77} - 27 q^{78} + 31 q^{79} + 19 q^{80} + 150 q^{81} - 12 q^{82} + 54 q^{83} + 8 q^{84} + 11 q^{85} + 8 q^{86} + 20 q^{87} + 30 q^{88} + 25 q^{89} - 81 q^{90} - 14 q^{91} + 60 q^{92} + 36 q^{93} + 19 q^{94} + 111 q^{95} - 6 q^{96} + 2 q^{97} - 5 q^{98} + 108 q^{99}+O(q^{100}) \) 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9 + q^10 + 82 * q^11 + 3 * q^12 + 26 * q^13 + 17 * q^14 + 66 * q^15 + 122 * q^16 + 82 * q^17 + 18 * q^18 + 12 * q^19 + 9 * q^20 + 22 * q^21 + 8 * q^22 + 50 * q^23 + 15 * q^24 + 117 * q^25 + 36 * q^26 + 30 * q^27 + 11 * q^28 + 33 * q^29 - 26 * q^30 + 40 * q^31 + 58 * q^32 + 6 * q^33 + 8 * q^34 + 16 * q^35 + 160 * q^36 + 31 * q^37 + 18 * q^38 + 41 * q^39 - 29 * q^40 + 42 * q^41 - 51 * q^42 + 82 * q^43 + 98 * q^44 - 2 * q^45 - 19 * q^46 + 84 * q^47 - 46 * q^48 + 136 * q^49 + 59 * q^50 + 6 * q^51 + 45 * q^52 + 83 * q^53 + 24 * q^54 + 11 * q^55 + 21 * q^56 + 23 * q^57 + 14 * q^58 + 96 * q^59 + 184 * q^60 - 6 * q^61 - 23 * q^62 + 8 * q^63 + 148 * q^64 + 5 * q^65 + 10 * q^66 + 78 * q^67 + 98 * q^68 + 61 * q^69 - 3 * q^70 + 155 * q^71 + 50 * q^72 - 23 * q^73 + 10 * q^74 - 19 * q^75 + 44 * q^76 + 8 * q^77 - 27 * q^78 + 31 * q^79 + 19 * q^80 + 150 * q^81 - 12 * q^82 + 54 * q^83 + 8 * q^84 + 11 * q^85 + 8 * q^86 + 20 * q^87 + 30 * q^88 + 25 * q^89 - 81 * q^90 - 14 * q^91 + 60 * q^92 + 36 * q^93 + 19 * q^94 + 111 * q^95 - 6 * q^96 + 2 * q^97 - 5 * q^98 + 108 * 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 $

$2$	−2.34655	−1.65926	−0.829631	−	0.558312i	$-0.811449\pi$
$2$	−2.34655	−1.65926	−0.829631	+	0.558312i	$0.811449\pi$
$3$	3.28622	1.89730	0.948651	−	0.316324i	$-0.102449\pi$
$3$	3.28622	1.89730	0.948651	+	0.316324i	$0.102449\pi$
$4$	3.50630	1.75315
$5$	3.73933	1.67228	0.836139	−	0.548517i	$-0.184807\pi$
$5$	3.73933	1.67228	0.836139	+	0.548517i	$0.184807\pi$
$6$	−7.71129	−3.14812
$7$	3.13477	1.18483	0.592415	−	0.805633i	$-0.298175\pi$
$7$	3.13477	1.18483	0.592415	+	0.805633i	$0.298175\pi$
$8$	−3.53461	−1.24967
$9$	7.79927	2.59976
$10$	−8.77452	−2.77475
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	11.5225	3.32626
$13$	0.157450	0.0436687	0.0218344	−	0.999762i	$-0.493049\pi$
$13$	0.157450	0.0436687	0.0218344	+	0.999762i	$0.493049\pi$
$14$	−7.35589	−1.96594
$15$	12.2883	3.17282
$16$	1.28155	0.320386
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−18.3014	−4.31368
$19$	−5.83875	−1.33950	−0.669750	−	0.742587i	$-0.733599\pi$
$19$	−5.83875	−1.33950	−0.669750	+	0.742587i	$0.733599\pi$
$20$	13.1112	2.93176
$21$	10.3015	2.24798
$22$	−2.34655	−0.500286
$23$	0.802293	0.167290	0.0836448	−	0.996496i	$-0.473344\pi$
$23$	0.802293	0.167290	0.0836448	+	0.996496i	$0.473344\pi$
$24$	−11.6155	−2.37101
$25$	8.98258	1.79652
$26$	−0.369464	−0.0724579
$27$	15.7715	3.03522
$28$	10.9914	2.07719
$29$	1.68678	0.313227	0.156614	−	0.987660i	$-0.449942\pi$
$29$	1.68678	0.313227	0.156614	+	0.987660i	$0.449942\pi$
$30$	−28.8351	−5.26454
$31$	0.923264	0.165823	0.0829116	−	0.996557i	$-0.473578\pi$
$31$	0.923264	0.165823	0.0829116	+	0.996557i	$0.473578\pi$
$32$	4.06201	0.718069
$33$	3.28622	0.572058
$34$	−2.34655	−0.402430
$35$	11.7219	1.98137
$36$	27.3466	4.55776
$37$	7.42808	1.22117	0.610585	−	0.791951i	$-0.290935\pi$
$37$	7.42808	1.22117	0.610585	+	0.791951i	$0.290935\pi$
$38$	13.7009	2.22258
$39$	0.517416	0.0828528
$40$	−13.2171	−2.08980
$41$	−5.49809	−0.858658	−0.429329	−	0.903148i	$-0.641250\pi$
$41$	−5.49809	−0.858658	−0.429329	+	0.903148i	$0.641250\pi$
$42$	−24.1731	−3.72999
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	3.50630	0.528595
$45$	29.1640	4.34752
$46$	−1.88262	−0.277577
$47$	−6.58511	−0.960537	−0.480269	−	0.877121i	$-0.659461\pi$
$47$	−6.58511	−0.960537	−0.480269	+	0.877121i	$0.659461\pi$
$48$	4.21145	0.607870
$49$	2.82675	0.403822
$50$	−21.0781	−2.98089
$51$	3.28622	0.460163
$52$	0.552067	0.0765579
$53$	−4.16982	−0.572768	−0.286384	−	0.958115i	$-0.592453\pi$
$53$	−4.16982	−0.572768	−0.286384	+	0.958115i	$0.592453\pi$
$54$	−37.0086	−5.03623
$55$	3.73933	0.504211
$56$	−11.0802	−1.48065
$57$	−19.1874	−2.54144
$58$	−3.95812	−0.519726
$59$	−0.00333073	−0.000433624 0	−0.000216812	−	1.00000i	$-0.500069\pi$
$59$	−0.00333073	−0.000433624 0	−0.000216812		1.00000i	$0.500069\pi$
$60$	43.0864	5.56243
$61$	7.96347	1.01962	0.509809	−	0.860288i	$-0.329716\pi$
$61$	7.96347	1.01962	0.509809	+	0.860288i	$0.329716\pi$
$62$	−2.16649	−0.275144
$63$	24.4489	3.08027
$64$	−12.0948	−1.51185
$65$	0.588757	0.0730263
$66$	−7.71129	−0.949194
$67$	−2.69073	−0.328724	−0.164362	−	0.986400i	$-0.552557\pi$
$67$	−2.69073	−0.328724	−0.164362	+	0.986400i	$0.552557\pi$
$68$	3.50630	0.425201
$69$	2.63651	0.317399
$70$	−27.5061	−3.28760
$71$	−0.778687	−0.0924131	−0.0462066	−	0.998932i	$-0.514713\pi$
$71$	−0.778687	−0.0924131	−0.0462066	+	0.998932i	$0.514713\pi$
$72$	−27.5674	−3.24885
$73$	−13.6854	−1.60175	−0.800875	−	0.598831i	$-0.795632\pi$
$73$	−13.6854	−1.60175	−0.800875	+	0.598831i	$0.795632\pi$
$74$	−17.4304	−2.02624
$75$	29.5188	3.40853
$76$	−20.4724	−2.34835
$77$	3.13477	0.357240
$78$	−1.21414	−0.137475
$79$	−7.37260	−0.829482	−0.414741	−	0.909940i	$-0.636128\pi$
$79$	−7.37260	−0.829482	−0.414741	+	0.909940i	$0.636128\pi$
$80$	4.79212	0.535775
$81$	28.4308	3.15897
$82$	12.9016	1.42474
$83$	6.85470	0.752401	0.376200	−	0.926538i	$-0.377230\pi$
$83$	6.85470	0.752401	0.376200	+	0.926538i	$0.377230\pi$
$84$	36.1203	3.94105
$85$	3.73933	0.405587
$86$	−2.34655	−0.253035
$87$	5.54314	0.594287
$88$	−3.53461	−0.376791
$89$	−9.62399	−1.02014	−0.510070	−	0.860133i	$-0.670381\pi$
$89$	−9.62399	−1.02014	−0.510070	+	0.860133i	$0.670381\pi$
$90$	−68.4349	−7.21367
$91$	0.493568	0.0517400
$92$	2.81308	0.293284
$93$	3.03405	0.314617
$94$	15.4523	1.59378
$95$	−21.8330	−2.24002
$96$	13.3487	1.36239
$97$	5.96564	0.605719	0.302860	−	0.953035i	$-0.402059\pi$
$97$	5.96564	0.605719	0.302860	+	0.953035i	$0.402059\pi$
$98$	−6.63312	−0.670046
$99$	7.79927	0.783856
$100$	31.4956	3.14956
$101$	−17.7077	−1.76198	−0.880989	−	0.473136i	$-0.843122\pi$
$101$	−17.7077	−1.76198	−0.880989	+	0.473136i	$0.843122\pi$
$102$	−7.71129	−0.763532
$103$	2.17627	0.214435	0.107217	−	0.994236i	$-0.465806\pi$
$103$	2.17627	0.214435	0.107217	+	0.994236i	$0.465806\pi$
$104$	−0.556524	−0.0545717
$105$	38.5208	3.75925
$106$	9.78469	0.950373
$107$	−13.6880	−1.32327	−0.661634	−	0.749827i	$-0.730137\pi$
$107$	−13.6880	−1.32327	−0.661634	+	0.749827i	$0.730137\pi$
$108$	55.2995	5.32120
$109$	−5.90381	−0.565482	−0.282741	−	0.959196i	$-0.591244\pi$
$109$	−5.90381	−0.565482	−0.282741	+	0.959196i	$0.591244\pi$
$110$	−8.77452	−0.836618
$111$	24.4103	2.31693
$112$	4.01735	0.379603
$113$	11.0631	1.04073	0.520365	−	0.853944i	$-0.325796\pi$
$113$	11.0631	1.04073	0.520365	+	0.853944i	$0.325796\pi$
$114$	45.0243	4.21691
$115$	3.00004	0.279755
$116$	5.91436	0.549135
$117$	1.22799	0.113528
$118$	0.00781573	0.000719496 0
$119$	3.13477	0.287363
$120$	−43.4343	−3.96499
$121$	1.00000	0.0909091
$122$	−18.6867	−1.69181
$123$	−18.0680	−1.62913
$124$	3.23724	0.290713
$125$	14.8922	1.33200
$126$	−57.3705	−5.11097
$127$	−4.94357	−0.438671	−0.219336	−	0.975649i	$-0.570389\pi$
$127$	−4.94357	−0.438671	−0.219336	+	0.975649i	$0.570389\pi$
$128$	20.2571	1.79049
$129$	3.28622	0.289336
$130$	−1.38155	−0.121170
$131$	8.20749	0.717092	0.358546	−	0.933512i	$-0.383273\pi$
$131$	8.20749	0.717092	0.358546	+	0.933512i	$0.383273\pi$
$132$	11.5225	1.00290
$133$	−18.3031	−1.58708
$134$	6.31392	0.545440
$135$	58.9747	5.07573
$136$	−3.53461	−0.303091
$137$	13.7948	1.17857	0.589286	−	0.807925i	$-0.299409\pi$
$137$	13.7948	1.17857	0.589286	+	0.807925i	$0.299409\pi$
$138$	−6.18671	−0.526648
$139$	−18.4278	−1.56303	−0.781515	−	0.623887i	$-0.785553\pi$
$139$	−18.4278	−1.56303	−0.781515	+	0.623887i	$0.785553\pi$
$140$	41.1006	3.47363
$141$	−21.6402	−1.82243
$142$	1.82723	0.153338
$143$	0.157450	0.0131666
$144$	9.99512	0.832927
$145$	6.30743	0.523803
$146$	32.1134	2.65772
$147$	9.28935	0.766172
$148$	26.0451	2.14089
$149$	−4.72476	−0.387067	−0.193534	−	0.981094i	$-0.561995\pi$
$149$	−4.72476	−0.387067	−0.193534	+	0.981094i	$0.561995\pi$
$150$	−69.2673	−5.65565
$151$	13.3970	1.09024	0.545118	−	0.838359i	$-0.316485\pi$
$151$	13.3970	1.09024	0.545118	+	0.838359i	$0.316485\pi$
$152$	20.6377	1.67394
$153$	7.79927	0.630533
$154$	−7.35589	−0.592754
$155$	3.45239	0.277302
$156$	1.81422	0.145253
$157$	−7.86577	−0.627757	−0.313879	−	0.949463i	$-0.601628\pi$
$157$	−7.86577	−0.627757	−0.313879	+	0.949463i	$0.601628\pi$
$158$	17.3002	1.37633
$159$	−13.7029	−1.08671
$160$	15.1892	1.20081
$161$	2.51500	0.198210
$162$	−66.7143	−5.24157
$163$	−10.0524	−0.787365	−0.393683	−	0.919246i	$-0.628799\pi$
$163$	−10.0524	−0.787365	−0.393683	+	0.919246i	$0.628799\pi$
$164$	−19.2780	−1.50536
$165$	12.2883	0.956641
$166$	−16.0849	−1.24843
$167$	−4.54613	−0.351790	−0.175895	−	0.984409i	$-0.556282\pi$
$167$	−4.54613	−0.351790	−0.175895	+	0.984409i	$0.556282\pi$
$168$	−36.4120	−2.80924
$169$	−12.9752	−0.998093
$170$	−8.77452	−0.672975
$171$	−45.5379	−3.48237
$172$	3.50630	0.267353
$173$	−22.8399	−1.73649	−0.868244	−	0.496138i	$-0.834751\pi$
$173$	−22.8399	−1.73649	−0.868244	+	0.496138i	$0.834751\pi$
$174$	−13.0073	−0.986078
$175$	28.1583	2.12857
$176$	1.28155	0.0966002
$177$	−0.0109455	−0.000822716 0
$178$	22.5832	1.69268
$179$	−17.7864	−1.32942	−0.664709	−	0.747102i	$-0.731445\pi$
$179$	−17.7864	−1.32942	−0.664709	+	0.747102i	$0.731445\pi$
$180$	102.258	7.62185
$181$	−6.89009	−0.512137	−0.256068	−	0.966659i	$-0.582427\pi$
$181$	−6.89009	−0.512137	−0.256068	+	0.966659i	$0.582427\pi$
$182$	−1.15818	−0.0858503
$183$	26.1698	1.93452
$184$	−2.83579	−0.209057
$185$	27.7760	2.04213
$186$	−7.11956	−0.522031
$187$	1.00000	0.0731272
$188$	−23.0894	−1.68397
$189$	49.4399	3.59622
$190$	51.2322	3.71678
$191$	−5.22557	−0.378109	−0.189055	−	0.981967i	$-0.560542\pi$
$191$	−5.22557	−0.378109	−0.189055	+	0.981967i	$0.560542\pi$
$192$	−39.7463	−2.86844
$193$	12.8224	0.922978	0.461489	−	0.887146i	$-0.347315\pi$
$193$	12.8224	0.922978	0.461489	+	0.887146i	$0.347315\pi$
$194$	−13.9987	−1.00505
$195$	1.93479	0.138553
$196$	9.91145	0.707961
$197$	6.03779	0.430175	0.215087	−	0.976595i	$-0.430996\pi$
$197$	6.03779	0.430175	0.215087	+	0.976595i	$0.430996\pi$
$198$	−18.3014	−1.30062
$199$	9.39589	0.666057	0.333028	−	0.942917i	$-0.391930\pi$
$199$	9.39589	0.666057	0.333028	+	0.942917i	$0.391930\pi$
$200$	−31.7499	−2.24506
$201$	−8.84233	−0.623690
$202$	41.5519	2.92358
$203$	5.28766	0.371121
$204$	11.5225	0.806736
$205$	−20.5592	−1.43591
$206$	−5.10674	−0.355803
$207$	6.25729	0.434912
$208$	0.201779	0.0139909
$209$	−5.83875	−0.403874
$210$	−90.3911	−6.23758
$211$	27.6908	1.90631	0.953157	−	0.302477i	$-0.0978136\pi$
$211$	27.6908	1.90631	0.953157	+	0.302477i	$0.0978136\pi$
$212$	−14.6206	−1.00415
$213$	−2.55894	−0.175336
$214$	32.1196	2.19565
$215$	3.73933	0.255020
$216$	−55.7460	−3.79304
$217$	2.89422	0.196472
$218$	13.8536	0.938284
$219$	−44.9732	−3.03900
$220$	13.1112	0.883958
$221$	0.157450	0.0105912
$222$	−57.2801	−3.84439
$223$	1.64128	0.109908	0.0549541	−	0.998489i	$-0.482499\pi$
$223$	1.64128	0.109908	0.0549541	+	0.998489i	$0.482499\pi$
$224$	12.7335	0.850790
$225$	70.0575	4.67050
$226$	−25.9602	−1.72684
$227$	12.2734	0.814612	0.407306	−	0.913292i	$-0.366468\pi$
$227$	12.2734	0.814612	0.407306	+	0.913292i	$0.366468\pi$
$228$	−67.2769	−4.45552
$229$	−14.6293	−0.966730	−0.483365	−	0.875419i	$-0.660586\pi$
$229$	−14.6293	−0.966730	−0.483365	+	0.875419i	$0.660586\pi$
$230$	−7.03974	−0.464186
$231$	10.3015	0.677792
$232$	−5.96212	−0.391432
$233$	17.3190	1.13460	0.567301	−	0.823510i	$-0.307988\pi$
$233$	17.3190	1.13460	0.567301	+	0.823510i	$0.307988\pi$
$234$	−2.88155	−0.188373
$235$	−24.6239	−1.60629
$236$	−0.0116785	−0.000760209 0
$237$	−24.2280	−1.57378
$238$	−7.35589	−0.476811
$239$	24.2707	1.56994	0.784969	−	0.619535i	$-0.212679\pi$
$239$	24.2707	1.56994	0.784969	+	0.619535i	$0.212679\pi$
$240$	15.7480	1.01653
$241$	2.60739	0.167957	0.0839783	−	0.996468i	$-0.473237\pi$
$241$	2.60739	0.167957	0.0839783	+	0.996468i	$0.473237\pi$
$242$	−2.34655	−0.150842
$243$	46.1155	2.95831
$244$	27.9223	1.78754
$245$	10.5702	0.675303
$246$	42.3974	2.70316
$247$	−0.919310	−0.0584943
$248$	−3.26338	−0.207225
$249$	22.5261	1.42753
$250$	−34.9452	−2.21013
$251$	−17.4595	−1.10203	−0.551016	−	0.834495i	$-0.685760\pi$
$251$	−17.4595	−1.10203	−0.551016	+	0.834495i	$0.685760\pi$
$252$	85.7251	5.40017
$253$	0.802293	0.0504397
$254$	11.6003	0.727870
$255$	12.2883	0.769521
$256$	−23.3446	−1.45904
$257$	25.9458	1.61845	0.809227	−	0.587496i	$-0.199886\pi$
$257$	25.9458	1.61845	0.809227	+	0.587496i	$0.199886\pi$
$258$	−7.71129	−0.480084
$259$	23.2853	1.44688
$260$	2.06436	0.128026
$261$	13.1557	0.814314
$262$	−19.2593	−1.18984
$263$	17.4395	1.07536	0.537682	−	0.843148i	$-0.319300\pi$
$263$	17.4395	1.07536	0.537682	+	0.843148i	$0.319300\pi$
$264$	−11.6155	−0.714886
$265$	−15.5923	−0.957828
$266$	42.9491	2.63338
$267$	−31.6266	−1.93552
$268$	−9.43449	−0.576303
$269$	1.26835	0.0773325	0.0386662	−	0.999252i	$-0.487689\pi$
$269$	1.26835	0.0773325	0.0386662	+	0.999252i	$0.487689\pi$
$270$	−138.387	−8.42197
$271$	−9.05410	−0.549997	−0.274999	−	0.961445i	$-0.588677\pi$
$271$	−9.05410	−0.549997	−0.274999	+	0.961445i	$0.588677\pi$
$272$	1.28155	0.0777051
$273$	1.62198	0.0981665
$274$	−32.3703	−1.95556
$275$	8.98258	0.541670
$276$	9.24441	0.556448
$277$	−25.0642	−1.50596	−0.752982	−	0.658042i	$-0.771385\pi$
$277$	−25.0642	−1.50596	−0.752982	+	0.658042i	$0.771385\pi$
$278$	43.2419	2.59348
$279$	7.20078	0.431100
$280$	−41.4324	−2.47606
$281$	−26.0912	−1.55647	−0.778235	−	0.627973i	$-0.783885\pi$
$281$	−26.0912	−1.55647	−0.778235	+	0.627973i	$0.783885\pi$
$282$	50.7797	3.02389
$283$	13.6420	0.810932	0.405466	−	0.914110i	$-0.367109\pi$
$283$	13.6420	0.810932	0.405466	+	0.914110i	$0.367109\pi$
$284$	−2.73031	−0.162014
$285$	−71.7481	−4.24999
$286$	−0.369464	−0.0218469
$287$	−17.2352	−1.01736
$288$	31.6807	1.86680
$289$	1.00000	0.0588235
$290$	−14.8007	−0.869127
$291$	19.6044	1.14923
$292$	−47.9850	−2.80811
$293$	22.7130	1.32691	0.663453	−	0.748218i	$-0.269090\pi$
$293$	22.7130	1.32691	0.663453	+	0.748218i	$0.269090\pi$
$294$	−21.7979	−1.27128
$295$	−0.0124547	−0.000725141 0
$296$	−26.2554	−1.52606
$297$	15.7715	0.915153
$298$	11.0869	0.642246
$299$	0.126321	0.00730533
$300$	103.502	5.97567
$301$	3.13477	0.180685
$302$	−31.4368	−1.80899
$303$	−58.1913	−3.34301
$304$	−7.48262	−0.429158
$305$	29.7780	1.70509
$306$	−18.3014	−1.04622
$307$	3.04953	0.174046	0.0870231	−	0.996206i	$-0.472265\pi$
$307$	3.04953	0.174046	0.0870231	+	0.996206i	$0.472265\pi$
$308$	10.9914	0.626295
$309$	7.15172	0.406847
$310$	−8.10120	−0.460117
$311$	9.36660	0.531131	0.265565	−	0.964093i	$-0.414441\pi$
$311$	9.36660	0.531131	0.265565	+	0.964093i	$0.414441\pi$
$312$	−1.82886	−0.103539
$313$	−25.8421	−1.46068	−0.730342	−	0.683082i	$-0.760639\pi$
$313$	−25.8421	−1.46068	−0.730342	+	0.683082i	$0.760639\pi$
$314$	18.4574	1.04161
$315$	91.4224	5.15107
$316$	−25.8505	−1.45421
$317$	21.8487	1.22714	0.613572	−	0.789639i	$-0.289732\pi$
$317$	21.8487	1.22714	0.613572	+	0.789639i	$0.289732\pi$
$318$	32.1547	1.80314
$319$	1.68678	0.0944416
$320$	−45.2265	−2.52824
$321$	−44.9818	−2.51064
$322$	−5.90157	−0.328882
$323$	−5.83875	−0.324876
$324$	99.6869	5.53816
$325$	1.41431	0.0784516
$326$	23.5885	1.30645
$327$	−19.4012	−1.07289
$328$	19.4336	1.07304
$329$	−20.6428	−1.13807
$330$	−28.8351	−1.58732
$331$	−21.5290	−1.18334	−0.591670	−	0.806180i	$-0.701531\pi$
$331$	−21.5290	−1.18334	−0.591670	+	0.806180i	$0.701531\pi$
$332$	24.0346	1.31907
$333$	57.9336	3.17474
$334$	10.6677	0.583711
$335$	−10.0615	−0.549719
$336$	13.2019	0.720223
$337$	−18.2913	−0.996391	−0.498195	−	0.867065i	$-0.666004\pi$
$337$	−18.2913	−0.996391	−0.498195	+	0.867065i	$0.666004\pi$
$338$	30.4470	1.65610
$339$	36.3559	1.97458
$340$	13.1112	0.711055
$341$	0.923264	0.0499975
$342$	106.857	5.77817
$343$	−13.0821	−0.706370
$344$	−3.53461	−0.190574
$345$	9.85879	0.530779
$346$	53.5951	2.88129
$347$	19.3085	1.03653	0.518266	−	0.855219i	$-0.326578\pi$
$347$	19.3085	1.03653	0.518266	+	0.855219i	$0.326578\pi$
$348$	19.4359	1.04187
$349$	−3.10537	−0.166226	−0.0831132	−	0.996540i	$-0.526486\pi$
$349$	−3.10537	−0.166226	−0.0831132	+	0.996540i	$0.526486\pi$
$350$	−66.0748	−3.53185
$351$	2.48322	0.132544
$352$	4.06201	0.216506
$353$	8.28086	0.440746	0.220373	−	0.975416i	$-0.429273\pi$
$353$	8.28086	0.440746	0.220373	+	0.975416i	$0.429273\pi$
$354$	0.0256842	0.00136510
$355$	−2.91177	−0.154540
$356$	−33.7446	−1.78846
$357$	10.3015	0.545215
$358$	41.7367	2.20585
$359$	18.7141	0.987693	0.493846	−	0.869549i	$-0.335590\pi$
$359$	18.7141	0.987693	0.493846	+	0.869549i	$0.335590\pi$
$360$	−103.084	−5.43298
$361$	15.0909	0.794260
$362$	16.1680	0.849769
$363$	3.28622	0.172482
$364$	1.73060	0.0907081
$365$	−51.1741	−2.67857
$366$	−61.4087	−3.20988
$367$	2.88918	0.150814	0.0754069	−	0.997153i	$-0.475974\pi$
$367$	2.88918	0.150814	0.0754069	+	0.997153i	$0.475974\pi$
$368$	1.02817	0.0535973
$369$	−42.8811	−2.23230
$370$	−65.1779	−3.38844
$371$	−13.0714	−0.678633
$372$	10.6383	0.551570
$373$	0.725098	0.0375442	0.0187721	−	0.999824i	$-0.494024\pi$
$373$	0.725098	0.0375442	0.0187721	+	0.999824i	$0.494024\pi$
$374$	−2.34655	−0.121337
$375$	48.9390	2.52720
$376$	23.2758	1.20036
$377$	0.265583	0.0136782
$378$	−116.013	−5.96707
$379$	18.9756	0.974711	0.487355	−	0.873204i	$-0.337962\pi$
$379$	18.9756	0.974711	0.487355	+	0.873204i	$0.337962\pi$
$380$	−76.5530	−3.92709
$381$	−16.2457	−0.832292
$382$	12.2621	0.627382
$383$	19.1017	0.976049	0.488025	−	0.872830i	$-0.337718\pi$
$383$	19.1017	0.976049	0.488025	+	0.872830i	$0.337718\pi$
$384$	66.5692	3.39710
$385$	11.7219	0.597404
$386$	−30.0885	−1.53146
$387$	7.79927	0.396459
$388$	20.9173	1.06192
$389$	−36.8471	−1.86822	−0.934111	−	0.356984i	$-0.883805\pi$
$389$	−36.8471	−1.86822	−0.934111	+	0.356984i	$0.883805\pi$
$390$	−4.54008	−0.229896
$391$	0.802293	0.0405737
$392$	−9.99148	−0.504646
$393$	26.9717	1.36054
$394$	−14.1680	−0.713772
$395$	−27.5686	−1.38712
$396$	27.3466	1.37422
$397$	30.6800	1.53978	0.769892	−	0.638174i	$-0.220310\pi$
$397$	30.6800	1.53978	0.769892	+	0.638174i	$0.220310\pi$
$398$	−22.0479	−1.10516
$399$	−60.1481	−3.01117
$400$	11.5116	0.575579
$401$	−25.1078	−1.25382	−0.626911	−	0.779091i	$-0.715681\pi$
$401$	−25.1078	−1.25382	−0.626911	+	0.779091i	$0.715681\pi$
$402$	20.7490	1.03486
$403$	0.145368	0.00724129
$404$	−62.0884	−3.08901
$405$	106.312	5.28269
$406$	−12.4078	−0.615787
$407$	7.42808	0.368196
$408$	−11.6155	−0.575054
$409$	30.7734	1.52165	0.760823	−	0.648960i	$-0.224796\pi$
$409$	30.7734	1.52165	0.760823	+	0.648960i	$0.224796\pi$
$410$	48.2431	2.38256
$411$	45.3329	2.23611
$412$	7.63067	0.375936
$413$	−0.0104411	−0.000513771 0
$414$	−14.6831	−0.721633
$415$	25.6320	1.25822
$416$	0.639563	0.0313572
$417$	−60.5580	−2.96554
$418$	13.7009	0.670134
$419$	35.2436	1.72176	0.860882	−	0.508805i	$-0.169913\pi$
$419$	35.2436	1.72176	0.860882	+	0.508805i	$0.169913\pi$
$420$	135.066	6.59053
$421$	−40.2018	−1.95931	−0.979657	−	0.200680i	$-0.935685\pi$
$421$	−40.2018	−1.95931	−0.979657	+	0.200680i	$0.935685\pi$
$422$	−64.9779	−3.16307
$423$	−51.3591	−2.49716
$424$	14.7387	0.715774
$425$	8.98258	0.435719
$426$	6.00468	0.290928
$427$	24.9636	1.20807
$428$	−47.9942	−2.31989
$429$	0.517416	0.0249811
$430$	−8.77452	−0.423145
$431$	24.9486	1.20173	0.600867	−	0.799349i	$-0.294822\pi$
$431$	24.9486	1.20173	0.600867	+	0.799349i	$0.294822\pi$
$432$	20.2119	0.972444
$433$	−14.7226	−0.707525	−0.353763	−	0.935335i	$-0.615098\pi$
$433$	−14.7226	−0.707525	−0.353763	+	0.935335i	$0.615098\pi$
$434$	−6.79143	−0.325999
$435$	20.7276	0.993813
$436$	−20.7005	−0.991376
$437$	−4.68438	−0.224084
$438$	105.532	5.04250
$439$	−40.9925	−1.95646	−0.978232	−	0.207514i	$-0.933463\pi$
$439$	−40.9925	−1.95646	−0.978232	+	0.207514i	$0.933463\pi$
$440$	−13.2171	−0.630099
$441$	22.0466	1.04984
$442$	−0.369464	−0.0175736
$443$	−22.5415	−1.07098	−0.535490	−	0.844541i	$-0.679873\pi$
$443$	−22.5415	−1.07098	−0.535490	+	0.844541i	$0.679873\pi$
$444$	85.5900	4.06192
$445$	−35.9873	−1.70596
$446$	−3.85134	−0.182366
$447$	−15.5266	−0.734383
$448$	−37.9144	−1.79129
$449$	16.0252	0.756275	0.378137	−	0.925749i	$-0.376565\pi$
$449$	16.0252	0.756275	0.378137	+	0.925749i	$0.376565\pi$
$450$	−164.394	−7.74959
$451$	−5.49809	−0.258895
$452$	38.7906	1.82456
$453$	44.0257	2.06851
$454$	−28.8001	−1.35165
$455$	1.84561	0.0865238
$456$	67.8201	3.17597
$457$	−7.64623	−0.357675	−0.178838	−	0.983879i	$-0.557234\pi$
$457$	−7.64623	−0.357675	−0.178838	+	0.983879i	$0.557234\pi$
$458$	34.3283	1.60406
$459$	15.7715	0.736149
$460$	10.5190	0.490452
$461$	−25.2790	−1.17736	−0.588680	−	0.808366i	$-0.700352\pi$
$461$	−25.2790	−1.17736	−0.588680	+	0.808366i	$0.700352\pi$
$462$	−24.1731	−1.12463
$463$	1.44153	0.0669935	0.0334968	−	0.999439i	$-0.489336\pi$
$463$	1.44153	0.0669935	0.0334968	+	0.999439i	$0.489336\pi$
$464$	2.16169	0.100354
$465$	11.3453	0.526127
$466$	−40.6398	−1.88260
$467$	41.1549	1.90442	0.952210	−	0.305445i	$-0.0988052\pi$
$467$	41.1549	1.90442	0.952210	+	0.305445i	$0.0988052\pi$
$468$	4.30572	0.199032
$469$	−8.43479	−0.389483
$470$	57.7812	2.66525
$471$	−25.8487	−1.19104
$472$	0.0117728	0.000541889 0
$473$	1.00000	0.0459800
$474$	56.8522	2.61131
$475$	−52.4470	−2.40643
$476$	10.9914	0.503791
$477$	−32.5215	−1.48906
$478$	−56.9523	−2.60494
$479$	−32.5566	−1.48755	−0.743774	−	0.668432i	$-0.766966\pi$
$479$	−32.5566	−1.48755	−0.743774	+	0.668432i	$0.766966\pi$
$480$	49.9151	2.27830
$481$	1.16955	0.0533269
$482$	−6.11837	−0.278684
$483$	8.26485	0.376064
$484$	3.50630	0.159377
$485$	22.3075	1.01293
$486$	−108.212	−4.90861
$487$	−26.9786	−1.22251	−0.611257	−	0.791432i	$-0.709336\pi$
$487$	−26.9786	−1.22251	−0.611257	+	0.791432i	$0.709336\pi$
$488$	−28.1478	−1.27419
$489$	−33.0345	−1.49387
$490$	−24.8034	−1.12050
$491$	8.77555	0.396035	0.198018	−	0.980198i	$-0.436550\pi$
$491$	8.77555	0.396035	0.198018	+	0.980198i	$0.436550\pi$
$492$	−63.3517	−2.85612
$493$	1.68678	0.0759688
$494$	2.15721	0.0970573
$495$	29.1640	1.31083
$496$	1.18321	0.0531275
$497$	−2.44100	−0.109494
$498$	−52.8586	−2.36865
$499$	1.75861	0.0787261	0.0393630	−	0.999225i	$-0.487467\pi$
$499$	1.75861	0.0787261	0.0393630	+	0.999225i	$0.487467\pi$
$500$	52.2164	2.33519
$501$	−14.9396	−0.667452
$502$	40.9695	1.82856
$503$	−42.9646	−1.91570	−0.957849	−	0.287274i	$-0.907251\pi$
$503$	−42.9646	−1.91570	−0.957849	+	0.287274i	$0.907251\pi$
$504$	−86.4173	−3.84933
$505$	−66.2148	−2.94652
$506$	−1.88262	−0.0836927
$507$	−42.6394	−1.89368
$508$	−17.3337	−0.769057
$509$	28.3483	1.25652	0.628258	−	0.778005i	$-0.283768\pi$
$509$	28.3483	1.25652	0.628258	+	0.778005i	$0.283768\pi$
$510$	−28.8351	−1.27684
$511$	−42.9004	−1.89780
$512$	14.2652	0.630438
$513$	−92.0856	−4.06568
$514$	−60.8831	−2.68544
$515$	8.13780	0.358594
$516$	11.5225	0.507249
$517$	−6.58511	−0.289613
$518$	−54.6401	−2.40075
$519$	−75.0571	−3.29464
$520$	−2.08103	−0.0912591
$521$	2.17196	0.0951553	0.0475777	−	0.998868i	$-0.484850\pi$
$521$	2.17196	0.0951553	0.0475777	+	0.998868i	$0.484850\pi$
$522$	−30.8704	−1.35116
$523$	20.5959	0.900595	0.450298	−	0.892878i	$-0.351318\pi$
$523$	20.5959	0.900595	0.450298	+	0.892878i	$0.351318\pi$
$524$	28.7779	1.25717
$525$	92.5344	4.03853
$526$	−40.9226	−1.78431
$527$	0.923264	0.0402180
$528$	4.21145	0.183280
$529$	−22.3563	−0.972014
$530$	36.5882	1.58929
$531$	−0.0259773	−0.00112732
$532$	−64.1762	−2.78239
$533$	−0.865674	−0.0374965
$534$	74.2134	3.21153
$535$	−51.1839	−2.21287
$536$	9.51067	0.410798
$537$	−58.4501	−2.52231
$538$	−2.97624	−0.128315
$539$	2.82675	0.121757
$540$	206.783	8.89853
$541$	16.0554	0.690277	0.345139	−	0.938552i	$-0.387832\pi$
$541$	16.0554	0.690277	0.345139	+	0.938552i	$0.387832\pi$
$542$	21.2459	0.912590
$543$	−22.6424	−0.971678
$544$	4.06201	0.174157
$545$	−22.0763	−0.945644
$546$	−3.80605	−0.162884
$547$	8.15397	0.348639	0.174319	−	0.984689i	$-0.444227\pi$
$547$	8.15397	0.348639	0.174319	+	0.984689i	$0.444227\pi$
$548$	48.3688	2.06621
$549$	62.1093	2.65076
$550$	−21.0781	−0.898772
$551$	−9.84868	−0.419568
$552$	−9.31905	−0.396645
$553$	−23.1114	−0.982795
$554$	58.8145	2.49879
$555$	91.2783	3.87455
$556$	−64.6136	−2.74023
$557$	−17.6992	−0.749940	−0.374970	−	0.927037i	$-0.622347\pi$
$557$	−17.6992	−0.749940	−0.374970	+	0.927037i	$0.622347\pi$
$558$	−16.8970	−0.715307
$559$	0.157450	0.00665942
$560$	15.0222	0.634803
$561$	3.28622	0.138744
$562$	61.2243	2.58259
$563$	5.34456	0.225246	0.112623	−	0.993638i	$-0.464075\pi$
$563$	5.34456	0.225246	0.112623	+	0.993638i	$0.464075\pi$
$564$	−75.8769	−3.19499
$565$	41.3686	1.74039
$566$	−32.0116	−1.34555
$567$	89.1238	3.74285
$568$	2.75236	0.115486
$569$	23.1252	0.969457	0.484729	−	0.874665i	$-0.338918\pi$
$569$	23.1252	0.969457	0.484729	+	0.874665i	$0.338918\pi$
$570$	168.361	7.05185
$571$	−34.6881	−1.45165	−0.725825	−	0.687879i	$-0.758542\pi$
$571$	−34.6881	−1.45165	−0.725825	+	0.687879i	$0.758542\pi$
$572$	0.552067	0.0230831
$573$	−17.1724	−0.717388
$574$	40.4433	1.68807
$575$	7.20666	0.300538
$576$	−94.3307	−3.93044
$577$	4.35115	0.181141	0.0905705	−	0.995890i	$-0.471131\pi$
$577$	4.35115	0.181141	0.0905705	+	0.995890i	$0.471131\pi$
$578$	−2.34655	−0.0976037
$579$	42.1374	1.75117
$580$	22.1157	0.918306
$581$	21.4879	0.891467
$582$	−46.0028	−1.90688
$583$	−4.16982	−0.172696
$584$	48.3724	2.00167
$585$	4.59187	0.189851
$586$	−53.2971	−2.20168
$587$	13.1832	0.544130	0.272065	−	0.962279i	$-0.412293\pi$
$587$	13.1832	0.544130	0.272065	+	0.962279i	$0.412293\pi$
$588$	32.5712	1.34322
$589$	−5.39070	−0.222120
$590$	0.0292256	0.00120320
$591$	19.8415	0.816171
$592$	9.51943	0.391246
$593$	−12.5264	−0.514398	−0.257199	−	0.966358i	$-0.582800\pi$
$593$	−12.5264	−0.514398	−0.257199	+	0.966358i	$0.582800\pi$
$594$	−37.0086	−1.51848
$595$	11.7219	0.480552
$596$	−16.5664	−0.678587
$597$	30.8770	1.26371
$598$	−0.296418	−0.0121215
$599$	44.7035	1.82654	0.913268	−	0.407358i	$-0.133550\pi$
$599$	44.7035	1.82654	0.913268	+	0.407358i	$0.133550\pi$
$600$	−104.337	−4.25956
$601$	3.71184	0.151409	0.0757045	−	0.997130i	$-0.475879\pi$
$601$	3.71184	0.151409	0.0757045	+	0.997130i	$0.475879\pi$
$602$	−7.35589	−0.299804
$603$	−20.9857	−0.854603
$604$	46.9741	1.91135
$605$	3.73933	0.152025
$606$	136.549	5.54692
$607$	48.1253	1.95334	0.976672	−	0.214735i	$-0.0688887\pi$
$607$	48.1253	1.95334	0.976672	+	0.214735i	$0.0688887\pi$
$608$	−23.7171	−0.961854
$609$	17.3764	0.704129
$610$	−69.8757	−2.82918
$611$	−1.03683	−0.0419455
$612$	27.3466	1.10542
$613$	−21.7624	−0.878976	−0.439488	−	0.898248i	$-0.644840\pi$
$613$	−21.7624	−0.878976	−0.439488	+	0.898248i	$0.644840\pi$
$614$	−7.15589	−0.288788
$615$	−67.5620	−2.72436
$616$	−11.0802	−0.446433
$617$	18.1265	0.729745	0.364872	−	0.931058i	$-0.381113\pi$
$617$	18.1265	0.729745	0.364872	+	0.931058i	$0.381113\pi$
$618$	−16.7819	−0.675066
$619$	−40.9507	−1.64595	−0.822973	−	0.568080i	$-0.807686\pi$
$619$	−40.9507	−1.64595	−0.822973	+	0.568080i	$0.807686\pi$
$620$	12.1051	0.486153
$621$	12.6533	0.507761
$622$	−21.9792	−0.881285
$623$	−30.1689	−1.20869
$624$	0.663092	0.0265449
$625$	10.7738	0.430953
$626$	60.6399	2.42366
$627$	−19.1874	−0.766272
$628$	−27.5798	−1.10055
$629$	7.42808	0.296177
$630$	−214.527	−8.54697
$631$	−12.3015	−0.489714	−0.244857	−	0.969559i	$-0.578741\pi$
$631$	−12.3015	−0.489714	−0.244857	+	0.969559i	$0.578741\pi$
$632$	26.0593	1.03658
$633$	90.9982	3.61685
$634$	−51.2690	−2.03615
$635$	−18.4856	−0.733580
$636$	−48.0467	−1.90517
$637$	0.445072	0.0176344
$638$	−3.95812	−0.156703
$639$	−6.07319	−0.240252
$640$	75.7478	2.99420
$641$	34.9917	1.38209	0.691044	−	0.722813i	$-0.257151\pi$
$641$	34.9917	1.38209	0.691044	+	0.722813i	$0.257151\pi$
$642$	105.552	4.16581
$643$	−17.7621	−0.700468	−0.350234	−	0.936662i	$-0.613898\pi$
$643$	−17.7621	−0.700468	−0.350234	+	0.936662i	$0.613898\pi$
$644$	8.81834	0.347491
$645$	12.2883	0.483850
$646$	13.7009	0.539055
$647$	−8.70847	−0.342365	−0.171183	−	0.985239i	$-0.554759\pi$
$647$	−8.70847	−0.342365	−0.171183	+	0.985239i	$0.554759\pi$
$648$	−100.492	−3.94769
$649$	−0.00333073	−0.000130743 0
$650$	−3.31874	−0.130172
$651$	9.51104	0.372767
$652$	−35.2468	−1.38037
$653$	−17.5676	−0.687472	−0.343736	−	0.939066i	$-0.611693\pi$
$653$	−17.5676	−0.687472	−0.343736	+	0.939066i	$0.611693\pi$
$654$	45.5260	1.78021
$655$	30.6905	1.19918
$656$	−7.04606	−0.275102
$657$	−106.736	−4.16416
$658$	48.4393	1.88836
$659$	0.809846	0.0315471	0.0157736	−	0.999876i	$-0.494979\pi$
$659$	0.809846	0.0315471	0.0157736	+	0.999876i	$0.494979\pi$
$660$	43.0864	1.67714
$661$	−27.3586	−1.06412	−0.532062	−	0.846705i	$-0.678583\pi$
$661$	−27.3586	−1.06412	−0.532062	+	0.846705i	$0.678583\pi$
$662$	50.5188	1.96347
$663$	0.517416	0.0200948
$664$	−24.2287	−0.940256
$665$	−68.4413	−2.65404
$666$	−135.944	−5.26773
$667$	1.35329	0.0523997
$668$	−15.9401	−0.616740
$669$	5.39361	0.208529
$670$	23.6098	0.912128
$671$	7.96347	0.307426
$672$	41.8450	1.61421
$673$	0.243284	0.00937789	0.00468895	−	0.999989i	$-0.498507\pi$
$673$	0.243284	0.00937789	0.00468895	+	0.999989i	$0.498507\pi$
$674$	42.9215	1.65327
$675$	141.668	5.45282
$676$	−45.4950	−1.74981
$677$	−48.9083	−1.87970	−0.939849	−	0.341591i	$-0.889034\pi$
$677$	−48.9083	−1.87970	−0.939849	+	0.341591i	$0.889034\pi$
$678$	−85.3109	−3.27635
$679$	18.7009	0.717674
$680$	−13.2171	−0.506852
$681$	40.3330	1.54556
$682$	−2.16649	−0.0829590
$683$	0.445464	0.0170452	0.00852260	−	0.999964i	$-0.497287\pi$
$683$	0.445464	0.0170452	0.00852260	+	0.999964i	$0.497287\pi$
$684$	−159.670	−6.10512
$685$	51.5834	1.97090
$686$	30.6979	1.17205
$687$	−48.0751	−1.83418
$688$	1.28155	0.0488585
$689$	−0.656537	−0.0250121
$690$	−23.1342	−0.880702
$691$	−29.9516	−1.13941	−0.569706	−	0.821849i	$-0.692943\pi$
$691$	−29.9516	−1.13941	−0.569706	+	0.821849i	$0.692943\pi$
$692$	−80.0837	−3.04432
$693$	24.4489	0.928736
$694$	−45.3083	−1.71988
$695$	−68.9078	−2.61382
$696$	−19.5928	−0.742665
$697$	−5.49809	−0.208255
$698$	7.28690	0.275813
$699$	56.9140	2.15268
$700$	98.7314	3.73170
$701$	16.9462	0.640048	0.320024	−	0.947409i	$-0.396309\pi$
$701$	16.9462	0.640048	0.320024	+	0.947409i	$0.396309\pi$
$702$	−5.82699	−0.219926
$703$	−43.3707	−1.63576
$704$	−12.0948	−0.455840
$705$	−80.9196	−3.04761
$706$	−19.4315	−0.731312
$707$	−55.5094	−2.08764
$708$	−0.0383783	−0.00144235
$709$	1.38862	0.0521508	0.0260754	−	0.999660i	$-0.491699\pi$
$709$	1.38862	0.0521508	0.0260754	+	0.999660i	$0.491699\pi$
$710$	6.83261	0.256423
$711$	−57.5009	−2.15645
$712$	34.0171	1.27484
$713$	0.740728	0.0277405
$714$	−24.1731	−0.904655
$715$	0.588757	0.0220183
$716$	−62.3645	−2.33067
$717$	79.7588	2.97865
$718$	−43.9136	−1.63884
$719$	10.8162	0.403377	0.201688	−	0.979450i	$-0.435357\pi$
$719$	10.8162	0.403377	0.201688	+	0.979450i	$0.435357\pi$
$720$	37.3750	1.39289
$721$	6.82211	0.254069
$722$	−35.4117	−1.31789
$723$	8.56846	0.318664
$724$	−24.1587	−0.897853
$725$	15.1516	0.562718
$726$	−7.71129	−0.286193
$727$	45.9183	1.70301	0.851507	−	0.524343i	$-0.175689\pi$
$727$	45.9183	1.70301	0.851507	+	0.524343i	$0.175689\pi$
$728$	−1.74457	−0.0646582
$729$	66.2535	2.45383
$730$	120.083	4.44445
$731$	1.00000	0.0369863
$732$	91.7590	3.39151
$733$	39.0237	1.44137	0.720687	−	0.693260i	$-0.243826\pi$
$733$	39.0237	1.44137	0.720687	+	0.693260i	$0.243826\pi$
$734$	−6.77960	−0.250240
$735$	34.7359	1.28125
$736$	3.25892	0.120125
$737$	−2.69073	−0.0991141
$738$	100.623	3.70397
$739$	30.7395	1.13077	0.565385	−	0.824827i	$-0.308728\pi$
$739$	30.7395	1.13077	0.565385	+	0.824827i	$0.308728\pi$
$740$	97.3912	3.58017
$741$	−3.02106	−0.110981
$742$	30.6727	1.12603
$743$	11.9117	0.436999	0.218499	−	0.975837i	$-0.429884\pi$
$743$	11.9117	0.436999	0.218499	+	0.975837i	$0.429884\pi$
$744$	−10.7242	−0.393168
$745$	−17.6674	−0.647284
$746$	−1.70148	−0.0622956
$747$	53.4616	1.95606
$748$	3.50630	0.128203
$749$	−42.9087	−1.56785
$750$	−114.838	−4.19329
$751$	−19.7493	−0.720660	−0.360330	−	0.932825i	$-0.617336\pi$
$751$	−19.7493	−0.720660	−0.360330	+	0.932825i	$0.617336\pi$
$752$	−8.43912	−0.307743
$753$	−57.3757	−2.09089
$754$	−0.623205	−0.0226958
$755$	50.0960	1.82318
$756$	173.351	6.30472
$757$	0.0478783	0.00174017	0.000870084	−	1.00000i	$-0.499723\pi$
$757$	0.0478783	0.00174017	0.000870084		1.00000i	$0.499723\pi$
$758$	−44.5272	−1.61730
$759$	2.63651	0.0956994
$760$	77.1711	2.79929
$761$	45.9562	1.66591	0.832955	−	0.553340i	$-0.186647\pi$
$761$	45.9562	1.66591	0.832955	+	0.553340i	$0.186647\pi$
$762$	38.1213	1.38099
$763$	−18.5071	−0.670001
$764$	−18.3224	−0.662882
$765$	29.1640	1.05443
$766$	−44.8230	−1.61952
$767$	−0.000524423 0	−1.89358e−5 0
$768$	−76.7156	−2.76824
$769$	−27.1374	−0.978600	−0.489300	−	0.872116i	$-0.662748\pi$
$769$	−27.1374	−0.978600	−0.489300	+	0.872116i	$0.662748\pi$
$770$	−27.5061	−0.991250
$771$	85.2637	3.07070
$772$	44.9593	1.61812
$773$	−41.1983	−1.48180	−0.740899	−	0.671616i	$-0.765600\pi$
$773$	−41.1983	−1.48180	−0.740899	+	0.671616i	$0.765600\pi$
$774$	−18.3014	−0.657829
$775$	8.29329	0.297904
$776$	−21.0862	−0.756951
$777$	76.5207	2.74516
$778$	86.4636	3.09987
$779$	32.1020	1.15017
$780$	6.78395	0.242904
$781$	−0.778687	−0.0278636
$782$	−1.88262	−0.0673224
$783$	26.6030	0.950714
$784$	3.62261	0.129379
$785$	−29.4127	−1.04978
$786$	−63.2904	−2.25749
$787$	12.6871	0.452247	0.226124	−	0.974099i	$-0.427395\pi$
$787$	12.6871	0.452247	0.226124	+	0.974099i	$0.427395\pi$
$788$	21.1703	0.754161
$789$	57.3100	2.04029
$790$	64.6910	2.30160
$791$	34.6803	1.23309
$792$	−27.5674	−0.979565
$793$	1.25385	0.0445255
$794$	−71.9922	−2.55491
$795$	−51.2398	−1.81729
$796$	32.9448	1.16770
$797$	8.09034	0.286575	0.143287	−	0.989681i	$-0.454233\pi$
$797$	8.09034	0.286575	0.143287	+	0.989681i	$0.454233\pi$
$798$	141.141	4.99632
$799$	−6.58511	−0.232965
$800$	36.4873	1.29002
$801$	−75.0601	−2.65212
$802$	58.9167	2.08042
$803$	−13.6854	−0.482946
$804$	−31.0039	−1.09342
$805$	9.40441	0.331462
$806$	−0.341113	−0.0120152
$807$	4.16807	0.146723
$808$	62.5897	2.20190
$809$	−8.82110	−0.310134	−0.155067	−	0.987904i	$-0.549559\pi$
$809$	−8.82110	−0.310134	−0.155067	+	0.987904i	$0.549559\pi$
$810$	−249.467	−8.76536
$811$	2.71442	0.0953163	0.0476581	−	0.998864i	$-0.484824\pi$
$811$	2.71442	0.0953163	0.0476581	+	0.998864i	$0.484824\pi$
$812$	18.5401	0.650631
$813$	−29.7538	−1.04351
$814$	−17.4304	−0.610934
$815$	−37.5893	−1.31669
$816$	4.21145	0.147430
$817$	−5.83875	−0.204272
$818$	−72.2113	−2.52481
$819$	3.84947	0.134511
$820$	−72.0866	−2.51737
$821$	−25.3175	−0.883588	−0.441794	−	0.897116i	$-0.645658\pi$
$821$	−25.3175	−0.883588	−0.441794	+	0.897116i	$0.645658\pi$
$822$	−106.376	−3.71029
$823$	45.2475	1.57723	0.788614	−	0.614889i	$-0.210799\pi$
$823$	45.2475	1.57723	0.788614	+	0.614889i	$0.210799\pi$
$824$	−7.69228	−0.267973
$825$	29.5188	1.02771
$826$	0.0245005	0.000852481 0
$827$	27.4728	0.955322	0.477661	−	0.878544i	$-0.341485\pi$
$827$	27.4728	0.955322	0.477661	+	0.878544i	$0.341485\pi$
$828$	21.9400	0.762466
$829$	10.2452	0.355832	0.177916	−	0.984046i	$-0.443065\pi$
$829$	10.2452	0.355832	0.177916	+	0.984046i	$0.443065\pi$
$830$	−60.1467	−2.08772
$831$	−82.3666	−2.85727
$832$	−1.90433	−0.0660207
$833$	2.82675	0.0979412
$834$	142.102	4.92061
$835$	−16.9995	−0.588291
$836$	−20.4724	−0.708053
$837$	14.5612	0.503310
$838$	−82.7009	−2.85686
$839$	9.46425	0.326742	0.163371	−	0.986565i	$-0.447763\pi$
$839$	9.46425	0.326742	0.163371	+	0.986565i	$0.447763\pi$
$840$	−136.156	−4.69784
$841$	−26.1548	−0.901889
$842$	94.3355	3.25101
$843$	−85.7415	−2.95309
$844$	97.0923	3.34205
$845$	−48.5186	−1.66909
$846$	120.517	4.14345
$847$	3.13477	0.107712
$848$	−5.34381	−0.183507
$849$	44.8306	1.53858
$850$	−21.0781	−0.722972
$851$	5.95949	0.204289
$852$	−8.97241	−0.307390
$853$	18.6804	0.639605	0.319802	−	0.947484i	$-0.396383\pi$
$853$	18.6804	0.639605	0.319802	+	0.947484i	$0.396383\pi$
$854$	−58.5784	−2.00451
$855$	−170.281	−5.82350
$856$	48.3818	1.65365
$857$	−20.8450	−0.712050	−0.356025	−	0.934476i	$-0.615868\pi$
$857$	−20.8450	−0.712050	−0.356025	+	0.934476i	$0.615868\pi$
$858$	−1.21414	−0.0414501
$859$	−25.1809	−0.859161	−0.429581	−	0.903028i	$-0.641339\pi$
$859$	−25.1809	−0.859161	−0.429581	+	0.903028i	$0.641339\pi$
$860$	13.1112	0.447089
$861$	−56.6388	−1.93025
$862$	−58.5432	−1.99399
$863$	−40.0323	−1.36272	−0.681358	−	0.731950i	$-0.738610\pi$
$863$	−40.0323	−1.36272	−0.681358	+	0.731950i	$0.738610\pi$
$864$	64.0639	2.17950
$865$	−85.4060	−2.90389
$866$	34.5474	1.17397
$867$	3.28622	0.111606
$868$	10.1480	0.344445
$869$	−7.37260	−0.250098
$870$	−48.6384	−1.64900
$871$	−0.423654	−0.0143550
$872$	20.8677	0.706669
$873$	46.5276	1.57472
$874$	10.9921	0.371815
$875$	46.6834	1.57819
$876$	−157.689	−5.32783
$877$	1.55572	0.0525328	0.0262664	−	0.999655i	$-0.491638\pi$
$877$	1.55572	0.0525328	0.0262664	+	0.999655i	$0.491638\pi$
$878$	96.1909	3.24629
$879$	74.6399	2.51754
$880$	4.79212	0.161542
$881$	17.1664	0.578351	0.289175	−	0.957276i	$-0.406619\pi$
$881$	17.1664	0.578351	0.289175	+	0.957276i	$0.406619\pi$
$882$	−51.7335	−1.74196
$883$	−55.0298	−1.85190	−0.925949	−	0.377647i	$-0.876733\pi$
$883$	−55.0298	−1.85190	−0.925949	+	0.377647i	$0.876733\pi$
$884$	0.552067	0.0185680
$885$	−0.0409289	−0.00137581
$886$	52.8948	1.77704
$887$	−1.81992	−0.0611070	−0.0305535	−	0.999533i	$-0.509727\pi$
$887$	−1.81992	−0.0611070	−0.0305535	+	0.999533i	$0.509727\pi$
$888$	−86.2811	−2.89540
$889$	−15.4969	−0.519751
$890$	84.4459	2.83063
$891$	28.4308	0.952467
$892$	5.75482	0.192686
$893$	38.4488	1.28664
$894$	36.4340	1.21853
$895$	−66.5092	−2.22316
$896$	63.5011	2.12142
$897$	0.415119	0.0138604
$898$	−37.6039	−1.25486
$899$	1.55734	0.0519403
$900$	245.643	8.18809
$901$	−4.16982	−0.138917
$902$	12.9016	0.429575
$903$	10.3015	0.342814
$904$	−39.1038	−1.30057
$905$	−25.7643	−0.856435
$906$	−103.309	−3.43220
$907$	40.4476	1.34304	0.671520	−	0.740986i	$-0.265642\pi$
$907$	40.4476	1.34304	0.671520	+	0.740986i	$0.265642\pi$
$908$	43.0341	1.42814
$909$	−138.107	−4.58071
$910$	−4.33083	−0.143566
$911$	18.5804	0.615597	0.307798	−	0.951452i	$-0.400408\pi$
$911$	18.5804	0.615597	0.307798	+	0.951452i	$0.400408\pi$
$912$	−24.5896	−0.814242
$913$	6.85470	0.226857
$914$	17.9423	0.593477
$915$	97.8573	3.23506
$916$	−51.2946	−1.69482
$917$	25.7286	0.849632
$918$	−37.0086	−1.22146
$919$	43.0843	1.42122	0.710610	−	0.703586i	$-0.248419\pi$
$919$	43.0843	1.42122	0.710610	+	0.703586i	$0.248419\pi$
$920$	−10.6040	−0.349602
$921$	10.0215	0.330218
$922$	59.3184	1.95355
$923$	−0.122604	−0.00403557
$924$	36.1203	1.18827
$925$	66.7233	2.19385
$926$	−3.38262	−0.111160
$927$	16.9733	0.557478
$928$	6.85172	0.224919
$929$	−18.1144	−0.594314	−0.297157	−	0.954829i	$-0.596039\pi$
$929$	−18.1144	−0.594314	−0.297157	+	0.954829i	$0.596039\pi$
$930$	−26.6224	−0.872982
$931$	−16.5047	−0.540920
$932$	60.7255	1.98913
$933$	30.7807	1.00772
$934$	−96.5720	−3.15993
$935$	3.73933	0.122289
$936$	−4.34048	−0.141873
$937$	57.6138	1.88216	0.941081	−	0.338182i	$-0.109812\pi$
$937$	57.6138	1.88216	0.941081	+	0.338182i	$0.109812\pi$
$938$	19.7927	0.646254
$939$	−84.9230	−2.77136
$940$	−86.3388	−2.81606
$941$	28.0969	0.915932	0.457966	−	0.888970i	$-0.348578\pi$
$941$	28.0969	0.915932	0.457966	+	0.888970i	$0.348578\pi$
$942$	60.6553	1.97626
$943$	−4.41108	−0.143644
$944$	−0.00426849	−0.000138927 0
$945$	184.872	6.01388
$946$	−2.34655	−0.0762930
$947$	−36.2630	−1.17839	−0.589195	−	0.807991i	$-0.700555\pi$
$947$	−36.2630	−1.17839	−0.589195	+	0.807991i	$0.700555\pi$
$948$	−84.9507	−2.75907
$949$	−2.15476	−0.0699464
$950$	123.070	3.99290
$951$	71.7996	2.32826
$952$	−11.0802	−0.359111
$953$	−3.68408	−0.119339	−0.0596695	−	0.998218i	$-0.519005\pi$
$953$	−3.68408	−0.119339	−0.0596695	+	0.998218i	$0.519005\pi$
$954$	76.3134	2.47074
$955$	−19.5401	−0.632304
$956$	85.1003	2.75234
$957$	5.54314	0.179184
$958$	76.3956	2.46823
$959$	43.2435	1.39641
$960$	−148.624	−4.79683
$961$	−30.1476	−0.972503
$962$	−2.74441	−0.0884833
$963$	−106.756	−3.44018
$964$	9.14229	0.294453
$965$	47.9473	1.54348
$966$	−19.3939	−0.623988
$967$	56.4893	1.81657	0.908286	−	0.418350i	$-0.137391\pi$
$967$	56.4893	1.81657	0.908286	+	0.418350i	$0.137391\pi$
$968$	−3.53461	−0.113607
$969$	−19.1874	−0.616389
$970$	−52.3457	−1.68072
$971$	55.3772	1.77714	0.888570	−	0.458741i	$-0.151699\pi$
$971$	55.3772	1.77714	0.888570	+	0.458741i	$0.151699\pi$
$972$	161.695	5.18636
$973$	−57.7670	−1.85192
$974$	63.3066	2.02847
$975$	4.64773	0.148846
$976$	10.2056	0.326672
$977$	25.9519	0.830276	0.415138	−	0.909759i	$-0.363733\pi$
$977$	25.9519	0.830276	0.415138	+	0.909759i	$0.363733\pi$
$978$	77.5170	2.47872
$979$	−9.62399	−0.307584
$980$	37.0622	1.18391
$981$	−46.0454	−1.47012
$982$	−20.5923	−0.657126
$983$	47.2667	1.50757	0.753787	−	0.657118i	$-0.228225\pi$
$983$	47.2667	1.50757	0.753787	+	0.657118i	$0.228225\pi$
$984$	63.8632	2.03589
$985$	22.5773	0.719372
$986$	−3.95812	−0.126052
$987$	−67.8368	−2.15927
$988$	−3.22338	−0.102549
$989$	0.802293	0.0255114
$990$	−68.4349	−2.17500
$991$	4.73933	0.150550	0.0752748	−	0.997163i	$-0.476017\pi$
$991$	4.73933	0.150550	0.0752748	+	0.997163i	$0.476017\pi$
$992$	3.75031	0.119072
$993$	−70.7490	−2.24515
$994$	5.72793	0.181679
$995$	35.1343	1.11383
$996$	78.9832	2.50268
$997$	42.3144	1.34011	0.670055	−	0.742312i	$-0.266271\pi$
$997$	42.3144	1.34011	0.670055	+	0.742312i	$0.266271\pi$
$998$	−4.12666	−0.130627
$999$	117.152	3.70652

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.j.1.8	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.j.1.8	✓	82	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:	\( N \)	\(=\)	\( 8041 = 11 \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8041.a (trivial)

\(f(q)\)	\(=\)	\(q-2.34655 q^{2} +3.28622 q^{3} +3.50630 q^{4} +3.73933 q^{5} -7.71129 q^{6} +3.13477 q^{7} -3.53461 q^{8} +7.79927 q^{9} +O(q^{10})\)	\(q-2.34655 q^{2} +3.28622 q^{3} +3.50630 q^{4} +3.73933 q^{5} -7.71129 q^{6} +3.13477 q^{7} -3.53461 q^{8} +7.79927 q^{9} -8.77452 q^{10} +1.00000 q^{11} +11.5225 q^{12} +0.157450 q^{13} -7.35589 q^{14} +12.2883 q^{15} +1.28155 q^{16} +1.00000 q^{17} -18.3014 q^{18} -5.83875 q^{19} +13.1112 q^{20} +10.3015 q^{21} -2.34655 q^{22} +0.802293 q^{23} -11.6155 q^{24} +8.98258 q^{25} -0.369464 q^{26} +15.7715 q^{27} +10.9914 q^{28} +1.68678 q^{29} -28.8351 q^{30} +0.923264 q^{31} +4.06201 q^{32} +3.28622 q^{33} -2.34655 q^{34} +11.7219 q^{35} +27.3466 q^{36} +7.42808 q^{37} +13.7009 q^{38} +0.517416 q^{39} -13.2171 q^{40} -5.49809 q^{41} -24.1731 q^{42} +1.00000 q^{43} +3.50630 q^{44} +29.1640 q^{45} -1.88262 q^{46} -6.58511 q^{47} +4.21145 q^{48} +2.82675 q^{49} -21.0781 q^{50} +3.28622 q^{51} +0.552067 q^{52} -4.16982 q^{53} -37.0086 q^{54} +3.73933 q^{55} -11.0802 q^{56} -19.1874 q^{57} -3.95812 q^{58} -0.00333073 q^{59} +43.0864 q^{60} +7.96347 q^{61} -2.16649 q^{62} +24.4489 q^{63} -12.0948 q^{64} +0.588757 q^{65} -7.71129 q^{66} -2.69073 q^{67} +3.50630 q^{68} +2.63651 q^{69} -27.5061 q^{70} -0.778687 q^{71} -27.5674 q^{72} -13.6854 q^{73} -17.4304 q^{74} +29.5188 q^{75} -20.4724 q^{76} +3.13477 q^{77} -1.21414 q^{78} -7.37260 q^{79} +4.79212 q^{80} +28.4308 q^{81} +12.9016 q^{82} +6.85470 q^{83} +36.1203 q^{84} +3.73933 q^{85} -2.34655 q^{86} +5.54314 q^{87} -3.53461 q^{88} -9.62399 q^{89} -68.4349 q^{90} +0.493568 q^{91} +2.81308 q^{92} +3.03405 q^{93} +15.4523 q^{94} -21.8330 q^{95} +13.3487 q^{96} +5.96564 q^{97} -6.63312 q^{98} +7.79927 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9}+O(q^{10}) \) 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9	\( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9} + q^{10} + 82 q^{11} + 3 q^{12} + 26 q^{13} + 17 q^{14} + 66 q^{15} + 122 q^{16} + 82 q^{17} + 18 q^{18} + 12 q^{19} + 9 q^{20} + 22 q^{21} + 8 q^{22} + 50 q^{23} + 15 q^{24} + 117 q^{25} + 36 q^{26} + 30 q^{27} + 11 q^{28} + 33 q^{29} - 26 q^{30} + 40 q^{31} + 58 q^{32} + 6 q^{33} + 8 q^{34} + 16 q^{35} + 160 q^{36} + 31 q^{37} + 18 q^{38} + 41 q^{39} - 29 q^{40} + 42 q^{41} - 51 q^{42} + 82 q^{43} + 98 q^{44} - 2 q^{45} - 19 q^{46} + 84 q^{47} - 46 q^{48} + 136 q^{49} + 59 q^{50} + 6 q^{51} + 45 q^{52} + 83 q^{53} + 24 q^{54} + 11 q^{55} + 21 q^{56} + 23 q^{57} + 14 q^{58} + 96 q^{59} + 184 q^{60} - 6 q^{61} - 23 q^{62} + 8 q^{63} + 148 q^{64} + 5 q^{65} + 10 q^{66} + 78 q^{67} + 98 q^{68} + 61 q^{69} - 3 q^{70} + 155 q^{71} + 50 q^{72} - 23 q^{73} + 10 q^{74} - 19 q^{75} + 44 q^{76} + 8 q^{77} - 27 q^{78} + 31 q^{79} + 19 q^{80} + 150 q^{81} - 12 q^{82} + 54 q^{83} + 8 q^{84} + 11 q^{85} + 8 q^{86} + 20 q^{87} + 30 q^{88} + 25 q^{89} - 81 q^{90} - 14 q^{91} + 60 q^{92} + 36 q^{93} + 19 q^{94} + 111 q^{95} - 6 q^{96} + 2 q^{97} - 5 q^{98} + 108 q^{99}+O(q^{100}) \) 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9 + q^10 + 82 * q^11 + 3 * q^12 + 26 * q^13 + 17 * q^14 + 66 * q^15 + 122 * q^16 + 82 * q^17 + 18 * q^18 + 12 * q^19 + 9 * q^20 + 22 * q^21 + 8 * q^22 + 50 * q^23 + 15 * q^24 + 117 * q^25 + 36 * q^26 + 30 * q^27 + 11 * q^28 + 33 * q^29 - 26 * q^30 + 40 * q^31 + 58 * q^32 + 6 * q^33 + 8 * q^34 + 16 * q^35 + 160 * q^36 + 31 * q^37 + 18 * q^38 + 41 * q^39 - 29 * q^40 + 42 * q^41 - 51 * q^42 + 82 * q^43 + 98 * q^44 - 2 * q^45 - 19 * q^46 + 84 * q^47 - 46 * q^48 + 136 * q^49 + 59 * q^50 + 6 * q^51 + 45 * q^52 + 83 * q^53 + 24 * q^54 + 11 * q^55 + 21 * q^56 + 23 * q^57 + 14 * q^58 + 96 * q^59 + 184 * q^60 - 6 * q^61 - 23 * q^62 + 8 * q^63 + 148 * q^64 + 5 * q^65 + 10 * q^66 + 78 * q^67 + 98 * q^68 + 61 * q^69 - 3 * q^70 + 155 * q^71 + 50 * q^72 - 23 * q^73 + 10 * q^74 - 19 * q^75 + 44 * q^76 + 8 * q^77 - 27 * q^78 + 31 * q^79 + 19 * q^80 + 150 * q^81 - 12 * q^82 + 54 * q^83 + 8 * q^84 + 11 * q^85 + 8 * q^86 + 20 * q^87 + 30 * q^88 + 25 * q^89 - 81 * q^90 - 14 * q^91 + 60 * q^92 + 36 * q^93 + 19 * q^94 + 111 * q^95 - 6 * q^96 + 2 * q^97 - 5 * q^98 + 108 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more