LMFDB - Embedded newform 8041.2.a.c.1.18

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:	$1$
Dimension:	$60$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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)

$f(q)$	$=$	$q-1.33974 q^{2} +3.18054 q^{3} -0.205105 q^{4} +0.454728 q^{5} -4.26109 q^{6} -3.38234 q^{7} +2.95426 q^{8} +7.11585 q^{9} +O(q^{10})$	\(q-1.33974 q^{2} +3.18054 q^{3} -0.205105 q^{4} +0.454728 q^{5} -4.26109 q^{6} -3.38234 q^{7} +2.95426 q^{8} +7.11585 q^{9} -0.609216 q^{10} +1.00000 q^{11} -0.652344 q^{12} -0.363807 q^{13} +4.53144 q^{14} +1.44628 q^{15} -3.54772 q^{16} +1.00000 q^{17} -9.53336 q^{18} +3.53233 q^{19} -0.0932669 q^{20} -10.7577 q^{21} -1.33974 q^{22} -7.09441 q^{23} +9.39615 q^{24} -4.79322 q^{25} +0.487405 q^{26} +13.0906 q^{27} +0.693733 q^{28} -9.91792 q^{29} -1.93764 q^{30} -1.62727 q^{31} -1.15550 q^{32} +3.18054 q^{33} -1.33974 q^{34} -1.53804 q^{35} -1.45949 q^{36} -3.43458 q^{37} -4.73239 q^{38} -1.15710 q^{39} +1.34339 q^{40} +10.6751 q^{41} +14.4124 q^{42} -1.00000 q^{43} -0.205105 q^{44} +3.23578 q^{45} +9.50464 q^{46} +5.57798 q^{47} -11.2837 q^{48} +4.44020 q^{49} +6.42166 q^{50} +3.18054 q^{51} +0.0746184 q^{52} -10.9510 q^{53} -17.5380 q^{54} +0.454728 q^{55} -9.99230 q^{56} +11.2347 q^{57} +13.2874 q^{58} -4.77559 q^{59} -0.296639 q^{60} +8.94141 q^{61} +2.18012 q^{62} -24.0682 q^{63} +8.64352 q^{64} -0.165433 q^{65} -4.26109 q^{66} -9.73412 q^{67} -0.205105 q^{68} -22.5641 q^{69} +2.06058 q^{70} -9.56012 q^{71} +21.0221 q^{72} +2.62661 q^{73} +4.60143 q^{74} -15.2450 q^{75} -0.724496 q^{76} -3.38234 q^{77} +1.55021 q^{78} -1.94778 q^{79} -1.61325 q^{80} +20.2877 q^{81} -14.3018 q^{82} +7.37085 q^{83} +2.20645 q^{84} +0.454728 q^{85} +1.33974 q^{86} -31.5444 q^{87} +2.95426 q^{88} -1.08449 q^{89} -4.33509 q^{90} +1.23052 q^{91} +1.45510 q^{92} -5.17560 q^{93} -7.47302 q^{94} +1.60625 q^{95} -3.67513 q^{96} +4.37233 q^{97} -5.94870 q^{98} +7.11585 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 60 q - 9 q^{2} - 6 q^{3} + 43 q^{4} - 15 q^{5} - 4 q^{6} - 17 q^{7} - 21 q^{8} + 40 q^{9}+O(q^{10}) $ 60 * q - 9 * q^2 - 6 * q^3 + 43 * q^4 - 15 * q^5 - 4 * q^6 - 17 * q^7 - 21 * q^8 + 40 * q^9	\( 60 q - 9 q^{2} - 6 q^{3} + 43 q^{4} - 15 q^{5} - 4 q^{6} - 17 q^{7} - 21 q^{8} + 40 q^{9} + q^{10} + 60 q^{11} - 13 q^{12} - 2 q^{13} - 15 q^{14} - 32 q^{15} + 21 q^{16} + 60 q^{17} - 41 q^{18} - 24 q^{19} - 33 q^{20} - 12 q^{21} - 9 q^{22} - 20 q^{23} + 9 q^{24} + 25 q^{25} - 40 q^{26} - 18 q^{27} - 3 q^{28} - 54 q^{29} - 18 q^{30} - 50 q^{31} - 51 q^{32} - 6 q^{33} - 9 q^{34} - 30 q^{35} + 27 q^{36} - 63 q^{37} - 38 q^{38} - 55 q^{39} - 5 q^{40} - 53 q^{41} - 47 q^{42} - 60 q^{43} + 43 q^{44} - 36 q^{45} - 27 q^{46} - 47 q^{47} - 38 q^{48} + 5 q^{49} - 4 q^{50} - 6 q^{51} - 13 q^{52} - 24 q^{53} - 58 q^{54} - 15 q^{55} - 45 q^{56} + 23 q^{57} + 16 q^{58} - 57 q^{59} - 4 q^{60} - 8 q^{61} - 3 q^{62} - 57 q^{63} - 5 q^{64} - 27 q^{65} - 4 q^{66} - 49 q^{67} + 43 q^{68} - 57 q^{69} - 7 q^{70} - 151 q^{71} - 29 q^{72} - 12 q^{73} + 18 q^{74} - 23 q^{75} - 38 q^{76} - 17 q^{77} + 37 q^{78} - 11 q^{79} - 21 q^{80} + 28 q^{81} + 44 q^{82} - 42 q^{83} + 16 q^{84} - 15 q^{85} + 9 q^{86} - 56 q^{87} - 21 q^{88} - 88 q^{89} + 47 q^{90} - 60 q^{91} + 26 q^{92} - 16 q^{93} + 37 q^{94} - 57 q^{95} + 108 q^{96} - 22 q^{97} + 8 q^{98} + 40 q^{99}+O(q^{100}) \) 60 * q - 9 * q^2 - 6 * q^3 + 43 * q^4 - 15 * q^5 - 4 * q^6 - 17 * q^7 - 21 * q^8 + 40 * q^9 + q^10 + 60 * q^11 - 13 * q^12 - 2 * q^13 - 15 * q^14 - 32 * q^15 + 21 * q^16 + 60 * q^17 - 41 * q^18 - 24 * q^19 - 33 * q^20 - 12 * q^21 - 9 * q^22 - 20 * q^23 + 9 * q^24 + 25 * q^25 - 40 * q^26 - 18 * q^27 - 3 * q^28 - 54 * q^29 - 18 * q^30 - 50 * q^31 - 51 * q^32 - 6 * q^33 - 9 * q^34 - 30 * q^35 + 27 * q^36 - 63 * q^37 - 38 * q^38 - 55 * q^39 - 5 * q^40 - 53 * q^41 - 47 * q^42 - 60 * q^43 + 43 * q^44 - 36 * q^45 - 27 * q^46 - 47 * q^47 - 38 * q^48 + 5 * q^49 - 4 * q^50 - 6 * q^51 - 13 * q^52 - 24 * q^53 - 58 * q^54 - 15 * q^55 - 45 * q^56 + 23 * q^57 + 16 * q^58 - 57 * q^59 - 4 * q^60 - 8 * q^61 - 3 * q^62 - 57 * q^63 - 5 * q^64 - 27 * q^65 - 4 * q^66 - 49 * q^67 + 43 * q^68 - 57 * q^69 - 7 * q^70 - 151 * q^71 - 29 * q^72 - 12 * q^73 + 18 * q^74 - 23 * q^75 - 38 * q^76 - 17 * q^77 + 37 * q^78 - 11 * q^79 - 21 * q^80 + 28 * q^81 + 44 * q^82 - 42 * q^83 + 16 * q^84 - 15 * q^85 + 9 * q^86 - 56 * q^87 - 21 * q^88 - 88 * q^89 + 47 * q^90 - 60 * q^91 + 26 * q^92 - 16 * q^93 + 37 * q^94 - 57 * q^95 + 108 * q^96 - 22 * q^97 + 8 * q^98 + 40 * 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 $

$2$	−1.33974	−0.947337	−0.473669	−	0.880703i	$-0.657071\pi$
$2$	−1.33974	−0.947337	−0.473669	+	0.880703i	$0.657071\pi$
$3$	3.18054	1.83629	0.918143	−	0.396248i	$-0.129688\pi$
$3$	3.18054	1.83629	0.918143	+	0.396248i	$0.129688\pi$
$4$	−0.205105	−0.102552
$5$	0.454728	0.203361	0.101680	−	0.994817i	$-0.467578\pi$
$5$	0.454728	0.203361	0.101680	+	0.994817i	$0.467578\pi$
$6$	−4.26109	−1.73958
$7$	−3.38234	−1.27840	−0.639202	−	0.769039i	$-0.720735\pi$
$7$	−3.38234	−1.27840	−0.639202	+	0.769039i	$0.720735\pi$
$8$	2.95426	1.04449
$9$	7.11585	2.37195
$10$	−0.609216	−0.192651
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.652344	−0.188315
$13$	−0.363807	−0.100902	−0.0504509	−	0.998727i	$-0.516066\pi$
$13$	−0.363807	−0.100902	−0.0504509	+	0.998727i	$0.516066\pi$
$14$	4.53144	1.21108
$15$	1.44628	0.373429
$16$	−3.54772	−0.886931
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−9.53336	−2.24704
$19$	3.53233	0.810371	0.405186	−	0.914234i	$-0.367207\pi$
$19$	3.53233	0.810371	0.405186	+	0.914234i	$0.367207\pi$
$20$	−0.0932669	−0.0208551
$21$	−10.7577	−2.34751
$22$	−1.33974	−0.285633
$23$	−7.09441	−1.47929	−0.739643	−	0.672999i	$-0.765006\pi$
$23$	−7.09441	−1.47929	−0.739643	+	0.672999i	$0.765006\pi$
$24$	9.39615	1.91798
$25$	−4.79322	−0.958644
$26$	0.487405	0.0955880
$27$	13.0906	2.51929
$28$	0.693733	0.131103
$29$	−9.91792	−1.84171	−0.920856	−	0.389903i	$-0.872509\pi$
$29$	−9.91792	−1.84171	−0.920856	+	0.389903i	$0.872509\pi$
$30$	−1.93764	−0.353763
$31$	−1.62727	−0.292266	−0.146133	−	0.989265i	$-0.546683\pi$
$31$	−1.62727	−0.292266	−0.146133	+	0.989265i	$0.546683\pi$
$32$	−1.15550	−0.204266
$33$	3.18054	0.553661
$34$	−1.33974	−0.229763
$35$	−1.53804	−0.259977
$36$	−1.45949	−0.243249
$37$	−3.43458	−0.564642	−0.282321	−	0.959320i	$-0.591104\pi$
$37$	−3.43458	−0.564642	−0.282321	+	0.959320i	$0.591104\pi$
$38$	−4.73239	−0.767695
$39$	−1.15710	−0.185285
$40$	1.34339	0.212408
$41$	10.6751	1.66716	0.833582	−	0.552396i	$-0.186286\pi$
$41$	10.6751	1.66716	0.833582	+	0.552396i	$0.186286\pi$
$42$	14.4124	2.22389
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	−0.205105	−0.0309207
$45$	3.23578	0.482361
$46$	9.50464	1.40138
$47$	5.57798	0.813632	0.406816	−	0.913510i	$-0.366639\pi$
$47$	5.57798	0.813632	0.406816	+	0.913510i	$0.366639\pi$
$48$	−11.2837	−1.62866
$49$	4.44020	0.634315
$50$	6.42166	0.908159
$51$	3.18054	0.445365
$52$	0.0746184	0.0103477
$53$	−10.9510	−1.50424	−0.752118	−	0.659029i	$-0.770967\pi$
$53$	−10.9510	−1.50424	−0.752118	+	0.659029i	$0.770967\pi$
$54$	−17.5380	−2.38662
$55$	0.454728	0.0613156
$56$	−9.99230	−1.33528
$57$	11.2347	1.48807
$58$	13.2874	1.74472
$59$	−4.77559	−0.621729	−0.310865	−	0.950454i	$-0.600619\pi$
$59$	−4.77559	−0.621729	−0.310865	+	0.950454i	$0.600619\pi$
$60$	−0.296639	−0.0382960
$61$	8.94141	1.14483	0.572415	−	0.819964i	$-0.306007\pi$
$61$	8.94141	1.14483	0.572415	+	0.819964i	$0.306007\pi$
$62$	2.18012	0.276875
$63$	−24.0682	−3.03231
$64$	8.64352	1.08044
$65$	−0.165433	−0.0205195
$66$	−4.26109	−0.524504
$67$	−9.73412	−1.18921	−0.594606	−	0.804017i	$-0.702692\pi$
$67$	−9.73412	−1.18921	−0.594606	+	0.804017i	$0.702692\pi$
$68$	−0.205105	−0.0248726
$69$	−22.5641	−2.71639
$70$	2.06058	0.246286
$71$	−9.56012	−1.13458	−0.567289	−	0.823519i	$-0.692008\pi$
$71$	−9.56012	−1.13458	−0.567289	+	0.823519i	$0.692008\pi$
$72$	21.0221	2.47747
$73$	2.62661	0.307422	0.153711	−	0.988116i	$-0.450878\pi$
$73$	2.62661	0.307422	0.153711	+	0.988116i	$0.450878\pi$
$74$	4.60143	0.534906
$75$	−15.2450	−1.76035
$76$	−0.724496	−0.0831054
$77$	−3.38234	−0.385453
$78$	1.55021	0.175527
$79$	−1.94778	−0.219143	−0.109571	−	0.993979i	$-0.534948\pi$
$79$	−1.94778	−0.219143	−0.109571	+	0.993979i	$0.534948\pi$
$80$	−1.61325	−0.180367
$81$	20.2877	2.25419
$82$	−14.3018	−1.57937
$83$	7.37085	0.809056	0.404528	−	0.914526i	$-0.367436\pi$
$83$	7.37085	0.809056	0.404528	+	0.914526i	$0.367436\pi$
$84$	2.20645	0.240743
$85$	0.454728	0.0493222
$86$	1.33974	0.144468
$87$	−31.5444	−3.38191
$88$	2.95426	0.314925
$89$	−1.08449	−0.114956	−0.0574779	−	0.998347i	$-0.518306\pi$
$89$	−1.08449	−0.114956	−0.0574779	+	0.998347i	$0.518306\pi$
$90$	−4.33509	−0.456959
$91$	1.23052	0.128993
$92$	1.45510	0.151704
$93$	−5.17560	−0.536685
$94$	−7.47302	−0.770783
$95$	1.60625	0.164798
$96$	−3.67513	−0.375092
$97$	4.37233	0.443943	0.221972	−	0.975053i	$-0.428751\pi$
$97$	4.37233	0.443943	0.221972	+	0.975053i	$0.428751\pi$
$98$	−5.94870	−0.600910
$99$	7.11585	0.715169
$100$	0.983112	0.0983112
$101$	−10.1659	−1.01155	−0.505774	−	0.862666i	$-0.668793\pi$
$101$	−10.1659	−1.01155	−0.505774	+	0.862666i	$0.668793\pi$
$102$	−4.26109	−0.421911
$103$	−17.3571	−1.71024	−0.855121	−	0.518429i	$-0.826517\pi$
$103$	−17.3571	−1.71024	−0.855121	+	0.518429i	$0.826517\pi$
$104$	−1.07478	−0.105391
$105$	−4.89181	−0.477392
$106$	14.6715	1.42502
$107$	−9.75071	−0.942636	−0.471318	−	0.881963i	$-0.656222\pi$
$107$	−9.75071	−0.942636	−0.471318	+	0.881963i	$0.656222\pi$
$108$	−2.68495	−0.258359
$109$	−15.8175	−1.51505	−0.757523	−	0.652809i	$-0.773591\pi$
$109$	−15.8175	−1.51505	−0.757523	+	0.652809i	$0.773591\pi$
$110$	−0.609216	−0.0580865
$111$	−10.9238	−1.03684
$112$	11.9996	1.13386
$113$	15.6875	1.47575	0.737876	−	0.674936i	$-0.235829\pi$
$113$	15.6875	1.47575	0.737876	+	0.674936i	$0.235829\pi$
$114$	−15.0516	−1.40971
$115$	−3.22603	−0.300829
$116$	2.03421	0.188872
$117$	−2.58879	−0.239334
$118$	6.39804	0.588987
$119$	−3.38234	−0.310058
$120$	4.27269	0.390042
$121$	1.00000	0.0909091
$122$	−11.9791	−1.08454
$123$	33.9525	3.06139
$124$	0.333761	0.0299726
$125$	−4.45326	−0.398311
$126$	32.2450	2.87262
$127$	−1.75198	−0.155463	−0.0777316	−	0.996974i	$-0.524768\pi$
$127$	−1.75198	−0.155463	−0.0777316	+	0.996974i	$0.524768\pi$
$128$	−9.26903	−0.819274
$129$	−3.18054	−0.280031
$130$	0.221637	0.0194388
$131$	1.44102	0.125903	0.0629514	−	0.998017i	$-0.479949\pi$
$131$	1.44102	0.125903	0.0629514	+	0.998017i	$0.479949\pi$
$132$	−0.652344	−0.0567792
$133$	−11.9475	−1.03598
$134$	13.0412	1.12658
$135$	5.95268	0.512325
$136$	2.95426	0.253326
$137$	−5.73142	−0.489668	−0.244834	−	0.969565i	$-0.578733\pi$
$137$	−5.73142	−0.489668	−0.244834	+	0.969565i	$0.578733\pi$
$138$	30.2299	2.57334
$139$	−18.6648	−1.58312	−0.791562	−	0.611089i	$-0.790732\pi$
$139$	−18.6648	−1.58312	−0.791562	+	0.611089i	$0.790732\pi$
$140$	0.315460	0.0266612
$141$	17.7410	1.49406
$142$	12.8080	1.07483
$143$	−0.363807	−0.0304230
$144$	−25.2451	−2.10375
$145$	−4.50996	−0.374532
$146$	−3.51897	−0.291232
$147$	14.1223	1.16478
$148$	0.704448	0.0579053
$149$	18.7357	1.53488	0.767442	−	0.641118i	$-0.221529\pi$
$149$	18.7357	1.53488	0.767442	+	0.641118i	$0.221529\pi$
$150$	20.4244	1.66764
$151$	−2.35525	−0.191668	−0.0958340	−	0.995397i	$-0.530552\pi$
$151$	−2.35525	−0.191668	−0.0958340	+	0.995397i	$0.530552\pi$
$152$	10.4354	0.846423
$153$	7.11585	0.575282
$154$	4.53144	0.365154
$155$	−0.739966	−0.0594355
$156$	0.237327	0.0190014
$157$	−11.8953	−0.949348	−0.474674	−	0.880162i	$-0.657434\pi$
$157$	−11.8953	−0.949348	−0.474674	+	0.880162i	$0.657434\pi$
$158$	2.60951	0.207602
$159$	−34.8301	−2.76221
$160$	−0.525441	−0.0415397
$161$	23.9957	1.89112
$162$	−27.1802	−2.13548
$163$	−15.1839	−1.18929	−0.594647	−	0.803987i	$-0.702708\pi$
$163$	−15.1839	−1.18929	−0.594647	+	0.803987i	$0.702708\pi$
$164$	−2.18950	−0.170971
$165$	1.44628	0.112593
$166$	−9.87500	−0.766449
$167$	7.10092	0.549486	0.274743	−	0.961518i	$-0.411407\pi$
$167$	7.10092	0.549486	0.274743	+	0.961518i	$0.411407\pi$
$168$	−31.7809	−2.45195
$169$	−12.8676	−0.989819
$170$	−0.609216	−0.0467248
$171$	25.1355	1.92216
$172$	0.205105	0.0156391
$173$	16.1314	1.22645	0.613224	−	0.789909i	$-0.289872\pi$
$173$	16.1314	1.22645	0.613224	+	0.789909i	$0.289872\pi$
$174$	42.2612	3.20381
$175$	16.2123	1.22553
$176$	−3.54772	−0.267420
$177$	−15.1890	−1.14167
$178$	1.45293	0.108902
$179$	−23.1515	−1.73042	−0.865211	−	0.501409i	$-0.832815\pi$
$179$	−23.1515	−1.73042	−0.865211	+	0.501409i	$0.832815\pi$
$180$	−0.663673	−0.0494672
$181$	6.54202	0.486265	0.243132	−	0.969993i	$-0.421825\pi$
$181$	6.54202	0.486265	0.243132	+	0.969993i	$0.421825\pi$
$182$	−1.64857	−0.122200
$183$	28.4385	2.10224
$184$	−20.9587	−1.54510
$185$	−1.56180	−0.114826
$186$	6.93395	0.508422
$187$	1.00000	0.0731272
$188$	−1.14407	−0.0834398
$189$	−44.2769	−3.22067
$190$	−2.15195	−0.156119
$191$	6.50927	0.470994	0.235497	−	0.971875i	$-0.424328\pi$
$191$	6.50927	0.470994	0.235497	+	0.971875i	$0.424328\pi$
$192$	27.4911	1.98400
$193$	14.7553	1.06211	0.531057	−	0.847336i	$-0.321795\pi$
$193$	14.7553	1.06211	0.531057	+	0.847336i	$0.321795\pi$
$194$	−5.85778	−0.420564
$195$	−0.526167	−0.0376796
$196$	−0.910706	−0.0650504
$197$	16.1112	1.14788	0.573939	−	0.818898i	$-0.305415\pi$
$197$	16.1112	1.14788	0.573939	+	0.818898i	$0.305415\pi$
$198$	−9.53336	−0.677507
$199$	18.4615	1.30870	0.654350	−	0.756192i	$-0.272942\pi$
$199$	18.4615	1.30870	0.654350	+	0.756192i	$0.272942\pi$
$200$	−14.1604	−1.00129
$201$	−30.9598	−2.18373
$202$	13.6197	0.958278
$203$	33.5458	2.35445
$204$	−0.652344	−0.0456732
$205$	4.85425	0.339036
$206$	23.2539	1.62017
$207$	−50.4827	−3.50879
$208$	1.29069	0.0894929
$209$	3.53233	0.244336
$210$	6.55375	0.452251
$211$	−9.02130	−0.621052	−0.310526	−	0.950565i	$-0.600505\pi$
$211$	−9.02130	−0.621052	−0.310526	+	0.950565i	$0.600505\pi$
$212$	2.24610	0.154263
$213$	−30.4064	−2.08341
$214$	13.0634	0.892994
$215$	−0.454728	−0.0310122
$216$	38.6731	2.63137
$217$	5.50398	0.373634
$218$	21.1914	1.43526
$219$	8.35405	0.564514
$220$	−0.0932669	−0.00628805
$221$	−0.363807	−0.0244723
$222$	14.6351	0.982241
$223$	2.87288	0.192382	0.0961911	−	0.995363i	$-0.469334\pi$
$223$	2.87288	0.192382	0.0961911	+	0.995363i	$0.469334\pi$
$224$	3.90831	0.261135
$225$	−34.1078	−2.27386
$226$	−21.0171	−1.39803
$227$	10.9656	0.727812	0.363906	−	0.931436i	$-0.381443\pi$
$227$	10.9656	0.727812	0.363906	+	0.931436i	$0.381443\pi$
$228$	−2.30429	−0.152605
$229$	12.2995	0.812773	0.406387	−	0.913701i	$-0.366789\pi$
$229$	12.2995	0.812773	0.406387	+	0.913701i	$0.366789\pi$
$230$	4.32203	0.284986
$231$	−10.7577	−0.707802
$232$	−29.3001	−1.92365
$233$	−4.27920	−0.280340	−0.140170	−	0.990127i	$-0.544765\pi$
$233$	−4.27920	−0.280340	−0.140170	+	0.990127i	$0.544765\pi$
$234$	3.46830	0.226730
$235$	2.53646	0.165461
$236$	0.979496	0.0637597
$237$	−6.19500	−0.402409
$238$	4.53144	0.293730
$239$	12.7502	0.824742	0.412371	−	0.911016i	$-0.364701\pi$
$239$	12.7502	0.824742	0.412371	+	0.911016i	$0.364701\pi$
$240$	−5.13101	−0.331205
$241$	−6.56600	−0.422953	−0.211477	−	0.977383i	$-0.567827\pi$
$241$	−6.56600	−0.422953	−0.211477	+	0.977383i	$0.567827\pi$
$242$	−1.33974	−0.0861216
$243$	25.2541	1.62005
$244$	−1.83393	−0.117405
$245$	2.01909	0.128995
$246$	−45.4874	−2.90017
$247$	−1.28508	−0.0817679
$248$	−4.80738	−0.305269
$249$	23.4433	1.48566
$250$	5.96619	0.377335
$251$	−1.09967	−0.0694105	−0.0347052	−	0.999398i	$-0.511049\pi$
$251$	−1.09967	−0.0694105	−0.0347052	+	0.999398i	$0.511049\pi$
$252$	4.93650	0.310970
$253$	−7.09441	−0.446022
$254$	2.34719	0.147276
$255$	1.44628	0.0905697
$256$	−4.86897	−0.304311
$257$	20.9046	1.30399	0.651995	−	0.758223i	$-0.273932\pi$
$257$	20.9046	1.30399	0.651995	+	0.758223i	$0.273932\pi$
$258$	4.26109	0.265284
$259$	11.6169	0.721840
$260$	0.0339311	0.00210432
$261$	−70.5744	−4.36845
$262$	−1.93059	−0.119272
$263$	2.35788	0.145393	0.0726965	−	0.997354i	$-0.476840\pi$
$263$	2.35788	0.145393	0.0726965	+	0.997354i	$0.476840\pi$
$264$	9.39615	0.578293
$265$	−4.97973	−0.305902
$266$	16.0065	0.981423
$267$	−3.44927	−0.211092
$268$	1.99651	0.121956
$269$	−1.13140	−0.0689825	−0.0344913	−	0.999405i	$-0.510981\pi$
$269$	−1.13140	−0.0689825	−0.0344913	+	0.999405i	$0.510981\pi$
$270$	−7.97502	−0.485344
$271$	−12.6849	−0.770550	−0.385275	−	0.922802i	$-0.625893\pi$
$271$	−12.6849	−0.770550	−0.385275	+	0.922802i	$0.625893\pi$
$272$	−3.54772	−0.215112
$273$	3.91371	0.236868
$274$	7.67859	0.463881
$275$	−4.79322	−0.289042
$276$	4.62799	0.278572
$277$	−19.3801	−1.16444	−0.582220	−	0.813031i	$-0.697816\pi$
$277$	−19.3801	−1.16444	−0.582220	+	0.813031i	$0.697816\pi$
$278$	25.0059	1.49975
$279$	−11.5794	−0.693241
$280$	−4.54378	−0.271543
$281$	−17.0650	−1.01801	−0.509007	−	0.860762i	$-0.669987\pi$
$281$	−17.0650	−1.01801	−0.509007	+	0.860762i	$0.669987\pi$
$282$	−23.7683	−1.41538
$283$	15.4406	0.917846	0.458923	−	0.888476i	$-0.348235\pi$
$283$	15.4406	0.917846	0.458923	+	0.888476i	$0.348235\pi$
$284$	1.96082	0.116354
$285$	5.10874	0.302616
$286$	0.487405	0.0288209
$287$	−36.1066	−2.13131
$288$	−8.22240	−0.484509
$289$	1.00000	0.0588235
$290$	6.04216	0.354808
$291$	13.9064	0.815207
$292$	−0.538730	−0.0315268
$293$	−9.35143	−0.546316	−0.273158	−	0.961969i	$-0.588068\pi$
$293$	−9.35143	−0.546316	−0.273158	+	0.961969i	$0.588068\pi$
$294$	−18.9201	−1.10344
$295$	−2.17160	−0.126435
$296$	−10.1466	−0.589762
$297$	13.0906	0.759595
$298$	−25.1009	−1.45405
$299$	2.58099	0.149263
$300$	3.12683	0.180528
$301$	3.38234	0.194955
$302$	3.15542	0.181574
$303$	−32.3332	−1.85749
$304$	−12.5317	−0.718743
$305$	4.06591	0.232814
$306$	−9.53336	−0.544986
$307$	0.775325	0.0442502	0.0221251	−	0.999755i	$-0.492957\pi$
$307$	0.775325	0.0442502	0.0221251	+	0.999755i	$0.492957\pi$
$308$	0.693733	0.0395291
$309$	−55.2048	−3.14049
$310$	0.991360	0.0563055
$311$	13.3235	0.755505	0.377753	−	0.925907i	$-0.376697\pi$
$311$	13.3235	0.755505	0.377753	+	0.925907i	$0.376697\pi$
$312$	−3.41838	−0.193528
$313$	−29.1692	−1.64874	−0.824370	−	0.566052i	$-0.808470\pi$
$313$	−29.1692	−1.64874	−0.824370	+	0.566052i	$0.808470\pi$
$314$	15.9366	0.899353
$315$	−10.9445	−0.616652
$316$	0.399499	0.0224736
$317$	−15.2934	−0.858965	−0.429483	−	0.903075i	$-0.641304\pi$
$317$	−15.2934	−0.858965	−0.429483	+	0.903075i	$0.641304\pi$
$318$	46.6632	2.61674
$319$	−9.91792	−0.555297
$320$	3.93045	0.219719
$321$	−31.0125	−1.73095
$322$	−32.1479	−1.79153
$323$	3.53233	0.196544
$324$	−4.16111	−0.231173
$325$	1.74381	0.0967290
$326$	20.3424	1.12666
$327$	−50.3084	−2.78206
$328$	31.5369	1.74133
$329$	−18.8666	−1.04015
$330$	−1.93764	−0.106663
$331$	27.1009	1.48960	0.744800	−	0.667288i	$-0.232545\pi$
$331$	27.1009	1.48960	0.744800	+	0.667288i	$0.232545\pi$
$332$	−1.51179	−0.0829705
$333$	−24.4399	−1.33930
$334$	−9.51337	−0.520548
$335$	−4.42638	−0.241839
$336$	38.1652	2.08208
$337$	−7.32796	−0.399179	−0.199590	−	0.979880i	$-0.563961\pi$
$337$	−7.32796	−0.399179	−0.199590	+	0.979880i	$0.563961\pi$
$338$	17.2393	0.937692
$339$	49.8946	2.70990
$340$	−0.0932669	−0.00505811
$341$	−1.62727	−0.0881217
$342$	−33.6749	−1.82093
$343$	8.65810	0.467493
$344$	−2.95426	−0.159283
$345$	−10.2605	−0.552408
$346$	−21.6118	−1.16186
$347$	5.98643	0.321368	0.160684	−	0.987006i	$-0.448630\pi$
$347$	5.98643	0.321368	0.160684	+	0.987006i	$0.448630\pi$
$348$	6.46990	0.346823
$349$	32.9434	1.76342	0.881710	−	0.471791i	$-0.156392\pi$
$349$	32.9434	1.76342	0.881710	+	0.471791i	$0.156392\pi$
$350$	−21.7202	−1.16099
$351$	−4.76246	−0.254201
$352$	−1.15550	−0.0615886
$353$	−1.41781	−0.0754626	−0.0377313	−	0.999288i	$-0.512013\pi$
$353$	−1.41781	−0.0754626	−0.0377313	+	0.999288i	$0.512013\pi$
$354$	20.3492	1.08155
$355$	−4.34726	−0.230728
$356$	0.222434	0.0117890
$357$	−10.7577	−0.569356
$358$	31.0169	1.63929
$359$	−28.3613	−1.49685	−0.748426	−	0.663218i	$-0.769190\pi$
$359$	−28.3613	−1.49685	−0.748426	+	0.663218i	$0.769190\pi$
$360$	9.55933	0.503821
$361$	−6.52268	−0.343299
$362$	−8.76459	−0.460657
$363$	3.18054	0.166935
$364$	−0.252385	−0.0132285
$365$	1.19439	0.0625175
$366$	−38.1002	−1.99153
$367$	−26.1698	−1.36605	−0.683026	−	0.730394i	$-0.739336\pi$
$367$	−26.1698	−1.36605	−0.683026	+	0.730394i	$0.739336\pi$
$368$	25.1690	1.31202
$369$	75.9621	3.95443
$370$	2.09240	0.108779
$371$	37.0400	1.92302
$372$	1.06154	0.0550383
$373$	−13.7307	−0.710951	−0.355475	−	0.934686i	$-0.615681\pi$
$373$	−13.7307	−0.710951	−0.355475	+	0.934686i	$0.615681\pi$
$374$	−1.33974	−0.0692762
$375$	−14.1638	−0.731414
$376$	16.4788	0.849829
$377$	3.60821	0.185832
$378$	59.3194	3.05106
$379$	6.70266	0.344293	0.172146	−	0.985071i	$-0.444930\pi$
$379$	6.70266	0.344293	0.172146	+	0.985071i	$0.444930\pi$
$380$	−0.329449	−0.0169004
$381$	−5.57225	−0.285475
$382$	−8.72071	−0.446190
$383$	34.6423	1.77014	0.885069	−	0.465460i	$-0.154111\pi$
$383$	34.6423	1.77014	0.885069	+	0.465460i	$0.154111\pi$
$384$	−29.4805	−1.50442
$385$	−1.53804	−0.0783860
$386$	−19.7683	−1.00618
$387$	−7.11585	−0.361719
$388$	−0.896786	−0.0455274
$389$	−11.0721	−0.561377	−0.280689	−	0.959799i	$-0.590563\pi$
$389$	−11.0721	−0.561377	−0.280689	+	0.959799i	$0.590563\pi$
$390$	0.704926	0.0356953
$391$	−7.09441	−0.358780
$392$	13.1175	0.662535
$393$	4.58323	0.231194
$394$	−21.5848	−1.08743
$395$	−0.885711	−0.0445650
$396$	−1.45949	−0.0733423
$397$	−5.14570	−0.258255	−0.129128	−	0.991628i	$-0.541218\pi$
$397$	−5.14570	−0.258255	−0.129128	+	0.991628i	$0.541218\pi$
$398$	−24.7335	−1.23978
$399$	−37.9996	−1.90236
$400$	17.0050	0.850251
$401$	−32.1141	−1.60370	−0.801851	−	0.597524i	$-0.796151\pi$
$401$	−32.1141	−1.60370	−0.801851	+	0.597524i	$0.796151\pi$
$402$	41.4779	2.06873
$403$	0.592012	0.0294902
$404$	2.08508	0.103737
$405$	9.22541	0.458414
$406$	−44.9425	−2.23046
$407$	−3.43458	−0.170246
$408$	9.39615	0.465179
$409$	−10.3666	−0.512594	−0.256297	−	0.966598i	$-0.582502\pi$
$409$	−10.3666	−0.512594	−0.256297	+	0.966598i	$0.582502\pi$
$410$	−6.50342	−0.321181
$411$	−18.2290	−0.899171
$412$	3.56001	0.175389
$413$	16.1527	0.794820
$414$	67.6336	3.32401
$415$	3.35173	0.164530
$416$	0.420380	0.0206108
$417$	−59.3641	−2.90707
$418$	−4.73239	−0.231469
$419$	−37.7103	−1.84227	−0.921135	−	0.389243i	$-0.872737\pi$
$419$	−37.7103	−1.84227	−0.921135	+	0.389243i	$0.872737\pi$
$420$	1.00333	0.0489577
$421$	7.34649	0.358046	0.179023	−	0.983845i	$-0.442706\pi$
$421$	7.34649	0.358046	0.179023	+	0.983845i	$0.442706\pi$
$422$	12.0862	0.588346
$423$	39.6920	1.92989
$424$	−32.3521	−1.57116
$425$	−4.79322	−0.232505
$426$	40.7365	1.97369
$427$	−30.2429	−1.46356
$428$	1.99991	0.0966695
$429$	−1.15710	−0.0558654
$430$	0.609216	0.0293790
$431$	−34.8022	−1.67636	−0.838181	−	0.545391i	$-0.816381\pi$
$431$	−34.8022	−1.67636	−0.838181	+	0.545391i	$0.816381\pi$
$432$	−46.4419	−2.23444
$433$	6.12722	0.294456	0.147228	−	0.989103i	$-0.452965\pi$
$433$	6.12722	0.294456	0.147228	+	0.989103i	$0.452965\pi$
$434$	−7.37388	−0.353958
$435$	−14.3441	−0.687748
$436$	3.24425	0.155371
$437$	−25.0598	−1.19877
$438$	−11.1922	−0.534785
$439$	4.40017	0.210008	0.105004	−	0.994472i	$-0.466514\pi$
$439$	4.40017	0.210008	0.105004	+	0.994472i	$0.466514\pi$
$440$	1.34339	0.0640434
$441$	31.5958	1.50456
$442$	0.487405	0.0231835
$443$	28.6867	1.36295	0.681473	−	0.731843i	$-0.261340\pi$
$443$	28.6867	1.36295	0.681473	+	0.731843i	$0.261340\pi$
$444$	2.24053	0.106331
$445$	−0.493149	−0.0233775
$446$	−3.84890	−0.182251
$447$	59.5895	2.81849
$448$	−29.2353	−1.38124
$449$	−14.8943	−0.702907	−0.351453	−	0.936205i	$-0.614312\pi$
$449$	−14.8943	−0.702907	−0.351453	+	0.936205i	$0.614312\pi$
$450$	45.6955	2.15411
$451$	10.6751	0.502669
$452$	−3.21757	−0.151342
$453$	−7.49099	−0.351957
$454$	−14.6910	−0.689483
$455$	0.559551	0.0262321
$456$	33.1903	1.55428
$457$	13.2441	0.619531	0.309765	−	0.950813i	$-0.399749\pi$
$457$	13.2441	0.619531	0.309765	+	0.950813i	$0.399749\pi$
$458$	−16.4781	−0.769970
$459$	13.0906	0.611018
$460$	0.661673	0.0308507
$461$	−31.4082	−1.46282	−0.731412	−	0.681935i	$-0.761138\pi$
$461$	−31.4082	−1.46282	−0.731412	+	0.681935i	$0.761138\pi$
$462$	14.4124	0.670527
$463$	−6.43607	−0.299110	−0.149555	−	0.988753i	$-0.547784\pi$
$463$	−6.43607	−0.299110	−0.149555	+	0.988753i	$0.547784\pi$
$464$	35.1860	1.63347
$465$	−2.35349	−0.109141
$466$	5.73301	0.265576
$467$	−28.6463	−1.32559	−0.662796	−	0.748800i	$-0.730630\pi$
$467$	−28.6463	−1.32559	−0.662796	+	0.748800i	$0.730630\pi$
$468$	0.530973	0.0245442
$469$	32.9241	1.52029
$470$	−3.39820	−0.156747
$471$	−37.8335	−1.74328
$472$	−14.1083	−0.649389
$473$	−1.00000	−0.0459800
$474$	8.29967	0.381217
$475$	−16.9312	−0.776858
$476$	0.693733	0.0317972
$477$	−77.9256	−3.56797
$478$	−17.0819	−0.781309
$479$	−30.7527	−1.40513	−0.702564	−	0.711620i	$-0.747962\pi$
$479$	−30.7527	−1.40513	−0.702564	+	0.711620i	$0.747962\pi$
$480$	−1.67119	−0.0762789
$481$	1.24952	0.0569734
$482$	8.79671	0.400679
$483$	76.3193	3.47265
$484$	−0.205105	−0.00932294
$485$	1.98822	0.0902806
$486$	−33.8339	−1.53474
$487$	−30.3257	−1.37419	−0.687094	−	0.726568i	$-0.741114\pi$
$487$	−30.3257	−1.37419	−0.687094	+	0.726568i	$0.741114\pi$
$488$	26.4153	1.19576
$489$	−48.2930	−2.18389
$490$	−2.70504	−0.122201
$491$	21.5481	0.972454	0.486227	−	0.873833i	$-0.338373\pi$
$491$	21.5481	0.972454	0.486227	+	0.873833i	$0.338373\pi$
$492$	−6.96381	−0.313953
$493$	−9.91792	−0.446681
$494$	1.72167	0.0774618
$495$	3.23578	0.145437
$496$	5.77311	0.259220
$497$	32.3355	1.45045
$498$	−31.4078	−1.40742
$499$	−17.9627	−0.804122	−0.402061	−	0.915613i	$-0.631706\pi$
$499$	−17.9627	−0.804122	−0.402061	+	0.915613i	$0.631706\pi$
$500$	0.913383	0.0408477
$501$	22.5848	1.00901
$502$	1.47327	0.0657551
$503$	−22.6201	−1.00858	−0.504290	−	0.863534i	$-0.668246\pi$
$503$	−22.6201	−1.00858	−0.504290	+	0.863534i	$0.668246\pi$
$504$	−71.1037	−3.16721
$505$	−4.62274	−0.205709
$506$	9.50464	0.422533
$507$	−40.9261	−1.81759
$508$	0.359339	0.0159431
$509$	−21.6174	−0.958173	−0.479086	−	0.877768i	$-0.659032\pi$
$509$	−21.6174	−0.958173	−0.479086	+	0.877768i	$0.659032\pi$
$510$	−1.93764	−0.0858001
$511$	−8.88409	−0.393009
$512$	25.0612	1.10756
$513$	46.2403	2.04156
$514$	−28.0066	−1.23532
$515$	−7.89274	−0.347796
$516$	0.652344	0.0287178
$517$	5.57798	0.245319
$518$	−15.5636	−0.683825
$519$	51.3066	2.25211
$520$	−0.488733	−0.0214323
$521$	25.0232	1.09628	0.548142	−	0.836385i	$-0.315335\pi$
$521$	25.0232	1.09628	0.548142	+	0.836385i	$0.315335\pi$
$522$	94.5512	4.13839
$523$	43.9456	1.92161	0.960803	−	0.277233i	$-0.0894175\pi$
$523$	43.9456	1.92161	0.960803	+	0.277233i	$0.0894175\pi$
$524$	−0.295560	−0.0129116
$525$	51.5639	2.25043
$526$	−3.15894	−0.137736
$527$	−1.62727	−0.0708850
$528$	−11.2837	−0.491059
$529$	27.3306	1.18829
$530$	6.67153	0.289793
$531$	−33.9824	−1.47471
$532$	2.45049	0.106242
$533$	−3.88366	−0.168220
$534$	4.62111	0.199975
$535$	−4.43392	−0.191695
$536$	−28.7571	−1.24212
$537$	−73.6342	−3.17755
$538$	1.51578	0.0653497
$539$	4.44020	0.191253
$540$	−1.22092	−0.0525401
$541$	−29.6146	−1.27323	−0.636615	−	0.771181i	$-0.719666\pi$
$541$	−29.6146	−1.27323	−0.636615	+	0.771181i	$0.719666\pi$
$542$	16.9944	0.729970
$543$	20.8072	0.892922
$544$	−1.15550	−0.0495419
$545$	−7.19269	−0.308101
$546$	−5.24334	−0.224394
$547$	28.0154	1.19785	0.598926	−	0.800804i	$-0.295594\pi$
$547$	28.0154	1.19785	0.598926	+	0.800804i	$0.295594\pi$
$548$	1.17554	0.0502166
$549$	63.6257	2.71548
$550$	6.42166	0.273820
$551$	−35.0333	−1.49247
$552$	−66.6601	−2.83724
$553$	6.58805	0.280153
$554$	25.9643	1.10312
$555$	−4.96737	−0.210853
$556$	3.82823	0.162353
$557$	−14.2431	−0.603502	−0.301751	−	0.953387i	$-0.597571\pi$
$557$	−14.2431	−0.603502	−0.301751	+	0.953387i	$0.597571\pi$
$558$	15.5134	0.656733
$559$	0.363807	0.0153874
$560$	5.45656	0.230582
$561$	3.18054	0.134283
$562$	22.8626	0.964402
$563$	14.7689	0.622435	0.311217	−	0.950339i	$-0.399263\pi$
$563$	14.7689	0.622435	0.311217	+	0.950339i	$0.399263\pi$
$564$	−3.63876	−0.153219
$565$	7.13353	0.300110
$566$	−20.6863	−0.869510
$567$	−68.6199	−2.88177
$568$	−28.2431	−1.18505
$569$	−0.628253	−0.0263377	−0.0131689	−	0.999913i	$-0.504192\pi$
$569$	−0.628253	−0.0263377	−0.0131689	+	0.999913i	$0.504192\pi$
$570$	−6.84437	−0.286679
$571$	3.87314	0.162086	0.0810429	−	0.996711i	$-0.474175\pi$
$571$	3.87314	0.162086	0.0810429	+	0.996711i	$0.474175\pi$
$572$	0.0746184	0.00311995
$573$	20.7030	0.864880
$574$	48.3734	2.01907
$575$	34.0051	1.41811
$576$	61.5060	2.56275
$577$	7.71170	0.321042	0.160521	−	0.987032i	$-0.448683\pi$
$577$	7.71170	0.321042	0.160521	+	0.987032i	$0.448683\pi$
$578$	−1.33974	−0.0557257
$579$	46.9300	1.95034
$580$	0.925014	0.0384091
$581$	−24.9307	−1.03430
$582$	−18.6309	−0.772276
$583$	−10.9510	−0.453544
$584$	7.75970	0.321099
$585$	−1.17720	−0.0486711
$586$	12.5285	0.517546
$587$	−28.9472	−1.19478	−0.597390	−	0.801951i	$-0.703796\pi$
$587$	−28.9472	−1.19478	−0.597390	+	0.801951i	$0.703796\pi$
$588$	−2.89654	−0.119451
$589$	−5.74805	−0.236844
$590$	2.90937	0.119777
$591$	51.2425	2.10783
$592$	12.1849	0.500798
$593$	−16.8254	−0.690937	−0.345469	−	0.938430i	$-0.612280\pi$
$593$	−16.8254	−0.690937	−0.345469	+	0.938430i	$0.612280\pi$
$594$	−17.5380	−0.719593
$595$	−1.53804	−0.0630537
$596$	−3.84277	−0.157406
$597$	58.7175	2.40315
$598$	−3.45785	−0.141402
$599$	−6.48600	−0.265011	−0.132505	−	0.991182i	$-0.542302\pi$
$599$	−6.48600	−0.265011	−0.132505	+	0.991182i	$0.542302\pi$
$600$	−45.0378	−1.83866
$601$	−27.1885	−1.10904	−0.554520	−	0.832170i	$-0.687098\pi$
$601$	−27.1885	−1.10904	−0.554520	+	0.832170i	$0.687098\pi$
$602$	−4.53144	−0.184688
$603$	−69.2665	−2.82075
$604$	0.483074	0.0196560
$605$	0.454728	0.0184873
$606$	43.3180	1.75967
$607$	−25.0333	−1.01607	−0.508035	−	0.861336i	$-0.669628\pi$
$607$	−25.0333	−1.01607	−0.508035	+	0.861336i	$0.669628\pi$
$608$	−4.08162	−0.165532
$609$	106.694	4.32345
$610$	−5.44726	−0.220553
$611$	−2.02931	−0.0820969
$612$	−1.45949	−0.0589965
$613$	16.2501	0.656334	0.328167	−	0.944620i	$-0.393569\pi$
$613$	16.2501	0.656334	0.328167	+	0.944620i	$0.393569\pi$
$614$	−1.03873	−0.0419198
$615$	15.4391	0.622566
$616$	−9.99230	−0.402601
$617$	28.2145	1.13587	0.567937	−	0.823072i	$-0.307742\pi$
$617$	28.2145	1.13587	0.567937	+	0.823072i	$0.307742\pi$
$618$	73.9600	2.97511
$619$	10.9539	0.440276	0.220138	−	0.975469i	$-0.429349\pi$
$619$	10.9539	0.440276	0.220138	+	0.975469i	$0.429349\pi$
$620$	0.151770	0.00609525
$621$	−92.8702	−3.72675
$622$	−17.8500	−0.715718
$623$	3.66811	0.146960
$624$	4.10508	0.164335
$625$	21.9411	0.877644
$626$	39.0790	1.56191
$627$	11.2347	0.448671
$628$	2.43978	0.0973578
$629$	−3.43458	−0.136946
$630$	14.6627	0.584177
$631$	18.1931	0.724256	0.362128	−	0.932128i	$-0.382050\pi$
$631$	18.1931	0.724256	0.362128	+	0.932128i	$0.382050\pi$
$632$	−5.75425	−0.228892
$633$	−28.6926	−1.14043
$634$	20.4892	0.813730
$635$	−0.796676	−0.0316151
$636$	7.14382	0.283271
$637$	−1.61538	−0.0640035
$638$	13.2874	0.526054
$639$	−68.0284	−2.69116
$640$	−4.21489	−0.166608
$641$	−15.9555	−0.630206	−0.315103	−	0.949057i	$-0.602039\pi$
$641$	−15.9555	−0.630206	−0.315103	+	0.949057i	$0.602039\pi$
$642$	41.5486	1.63979
$643$	11.3028	0.445737	0.222869	−	0.974848i	$-0.428458\pi$
$643$	11.3028	0.445737	0.222869	+	0.974848i	$0.428458\pi$
$644$	−4.92162	−0.193939
$645$	−1.44628	−0.0569473
$646$	−4.73239	−0.186193
$647$	4.75022	0.186750	0.0933752	−	0.995631i	$-0.470234\pi$
$647$	4.75022	0.186750	0.0933752	+	0.995631i	$0.470234\pi$
$648$	59.9352	2.35448
$649$	−4.77559	−0.187458
$650$	−2.33624	−0.0916349
$651$	17.5056	0.686100
$652$	3.11429	0.121965
$653$	31.3822	1.22808	0.614040	−	0.789275i	$-0.289543\pi$
$653$	31.3822	1.22808	0.614040	+	0.789275i	$0.289543\pi$
$654$	67.4000	2.63555
$655$	0.655274	0.0256037
$656$	−37.8721	−1.47866
$657$	18.6906	0.729189
$658$	25.2763	0.985372
$659$	3.31312	0.129061	0.0645304	−	0.997916i	$-0.479445\pi$
$659$	3.31312	0.129061	0.0645304	+	0.997916i	$0.479445\pi$
$660$	−0.296639	−0.0115467
$661$	−13.3692	−0.520003	−0.260001	−	0.965608i	$-0.583723\pi$
$661$	−13.3692	−0.520003	−0.260001	+	0.965608i	$0.583723\pi$
$662$	−36.3081	−1.41115
$663$	−1.15710	−0.0449381
$664$	21.7754	0.845050
$665$	−5.43287	−0.210678
$666$	32.7431	1.26877
$667$	70.3618	2.72442
$668$	−1.45643	−0.0563510
$669$	9.13731	0.353269
$670$	5.93018	0.229103
$671$	8.94141	0.345179
$672$	12.4305	0.479518
$673$	15.0108	0.578626	0.289313	−	0.957235i	$-0.406573\pi$
$673$	15.0108	0.578626	0.289313	+	0.957235i	$0.406573\pi$
$674$	9.81754	0.378157
$675$	−62.7463	−2.41510
$676$	2.63921	0.101508
$677$	46.2748	1.77848	0.889242	−	0.457437i	$-0.151232\pi$
$677$	46.2748	1.77848	0.889242	+	0.457437i	$0.151232\pi$
$678$	−66.8457	−2.56719
$679$	−14.7887	−0.567539
$680$	1.34339	0.0515165
$681$	34.8765	1.33647
$682$	2.18012	0.0834809
$683$	4.10490	0.157070	0.0785348	−	0.996911i	$-0.474976\pi$
$683$	4.10490	0.157070	0.0785348	+	0.996911i	$0.474976\pi$
$684$	−5.15540	−0.197122
$685$	−2.60624	−0.0995792
$686$	−11.5996	−0.442874
$687$	39.1190	1.49248
$688$	3.54772	0.135256
$689$	3.98405	0.151780
$690$	13.7464	0.523316
$691$	−20.5828	−0.783005	−0.391502	−	0.920177i	$-0.628045\pi$
$691$	−20.5828	−0.783005	−0.391502	+	0.920177i	$0.628045\pi$
$692$	−3.30863	−0.125775
$693$	−24.0682	−0.914275
$694$	−8.02024	−0.304444
$695$	−8.48740	−0.321945
$696$	−93.1903	−3.53237
$697$	10.6751	0.404347
$698$	−44.1355	−1.67055
$699$	−13.6102	−0.514785
$700$	−3.32522	−0.125681
$701$	13.4225	0.506961	0.253481	−	0.967340i	$-0.418425\pi$
$701$	13.4225	0.506961	0.253481	+	0.967340i	$0.418425\pi$
$702$	6.38044	0.240814
$703$	−12.1321	−0.457569
$704$	8.64352	0.325765
$705$	8.06733	0.303833
$706$	1.89950	0.0714886
$707$	34.3846	1.29317
$708$	3.11533	0.117081
$709$	21.4218	0.804511	0.402256	−	0.915527i	$-0.368226\pi$
$709$	21.4218	0.804511	0.402256	+	0.915527i	$0.368226\pi$
$710$	5.82418	0.218578
$711$	−13.8601	−0.519795
$712$	−3.20387	−0.120070
$713$	11.5445	0.432346
$714$	14.4124	0.539372
$715$	−0.165433	−0.00618685
$716$	4.74847	0.177459
$717$	40.5526	1.51446
$718$	37.9967	1.41802
$719$	45.7611	1.70660	0.853300	−	0.521421i	$-0.174598\pi$
$719$	45.7611	1.70660	0.853300	+	0.521421i	$0.174598\pi$
$720$	−11.4796	−0.427821
$721$	58.7074	2.18638
$722$	8.73867	0.325220
$723$	−20.8834	−0.776663
$724$	−1.34180	−0.0498676
$725$	47.5388	1.76555
$726$	−4.26109	−0.158144
$727$	30.3365	1.12512	0.562560	−	0.826757i	$-0.309817\pi$
$727$	30.3365	1.12512	0.562560	+	0.826757i	$0.309817\pi$
$728$	3.63527	0.134732
$729$	19.4586	0.720688
$730$	−1.60018	−0.0592251
$731$	−1.00000	−0.0369863
$732$	−5.83288	−0.215589
$733$	2.34394	0.0865754	0.0432877	−	0.999063i	$-0.486217\pi$
$733$	2.34394	0.0865754	0.0432877	+	0.999063i	$0.486217\pi$
$734$	35.0606	1.29411
$735$	6.42179	0.236871
$736$	8.19762	0.302168
$737$	−9.73412	−0.358561
$738$	−101.769	−3.74618
$739$	−6.81773	−0.250794	−0.125397	−	0.992107i	$-0.540021\pi$
$739$	−6.81773	−0.250794	−0.125397	+	0.992107i	$0.540021\pi$
$740$	0.320333	0.0117757
$741$	−4.08726	−0.150149
$742$	−49.6238	−1.82175
$743$	11.8465	0.434605	0.217302	−	0.976104i	$-0.430274\pi$
$743$	11.8465	0.434605	0.217302	+	0.976104i	$0.430274\pi$
$744$	−15.2901	−0.560562
$745$	8.51963	0.312135
$746$	18.3956	0.673510
$747$	52.4498	1.91904
$748$	−0.205105	−0.00749937
$749$	32.9802	1.20507
$750$	18.9757	0.692895
$751$	5.66980	0.206894	0.103447	−	0.994635i	$-0.467013\pi$
$751$	5.66980	0.206894	0.103447	+	0.994635i	$0.467013\pi$
$752$	−19.7891	−0.721635
$753$	−3.49754	−0.127458
$754$	−4.83405	−0.176046
$755$	−1.07100	−0.0389777
$756$	9.08139	0.330287
$757$	−47.7264	−1.73465	−0.867323	−	0.497746i	$-0.834161\pi$
$757$	−47.7264	−1.73465	−0.867323	+	0.497746i	$0.834161\pi$
$758$	−8.97980	−0.326161
$759$	−22.5641	−0.819024
$760$	4.74528	0.172129
$761$	−34.7006	−1.25790	−0.628948	−	0.777448i	$-0.716514\pi$
$761$	−34.7006	−1.25790	−0.628948	+	0.777448i	$0.716514\pi$
$762$	7.46535	0.270441
$763$	53.5003	1.93684
$764$	−1.33508	−0.0483015
$765$	3.23578	0.116990
$766$	−46.4115	−1.67692
$767$	1.73739	0.0627336
$768$	−15.4860	−0.558802
$769$	42.1319	1.51932	0.759658	−	0.650323i	$-0.225367\pi$
$769$	42.1319	1.51932	0.759658	+	0.650323i	$0.225367\pi$
$770$	2.06058	0.0742580
$771$	66.4878	2.39450
$772$	−3.02639	−0.108922
$773$	28.6118	1.02910	0.514548	−	0.857462i	$-0.327960\pi$
$773$	28.6118	1.02910	0.514548	+	0.857462i	$0.327960\pi$
$774$	9.53336	0.342670
$775$	7.79987	0.280180
$776$	12.9170	0.463694
$777$	36.9481	1.32550
$778$	14.8337	0.531814
$779$	37.7078	1.35102
$780$	0.107919	0.00386413
$781$	−9.56012	−0.342088
$782$	9.50464	0.339885
$783$	−129.832	−4.63981
$784$	−15.7526	−0.562593
$785$	−5.40913	−0.193060
$786$	−6.14033	−0.219018
$787$	13.8565	0.493930	0.246965	−	0.969024i	$-0.420567\pi$
$787$	13.8565	0.493930	0.246965	+	0.969024i	$0.420567\pi$
$788$	−3.30449	−0.117718
$789$	7.49933	0.266983
$790$	1.18662	0.0422181
$791$	−53.0603	−1.88661
$792$	21.0221	0.746987
$793$	−3.25295	−0.115515
$794$	6.89389	0.244655
$795$	−15.8382	−0.561724
$796$	−3.78654	−0.134210
$797$	−19.5986	−0.694218	−0.347109	−	0.937825i	$-0.612837\pi$
$797$	−19.5986	−0.694218	−0.347109	+	0.937825i	$0.612837\pi$
$798$	50.9094	1.80217
$799$	5.57798	0.197335
$800$	5.53859	0.195819
$801$	−7.71707	−0.272669
$802$	43.0245	1.51925
$803$	2.62661	0.0926911
$804$	6.34999	0.223947
$805$	10.9115	0.384580
$806$	−0.793140	−0.0279372
$807$	−3.59846	−0.126672
$808$	−30.0328	−1.05655
$809$	−24.5865	−0.864417	−0.432208	−	0.901774i	$-0.642266\pi$
$809$	−24.5865	−0.864417	−0.432208	+	0.901774i	$0.642266\pi$
$810$	−12.3596	−0.434273
$811$	−17.0850	−0.599934	−0.299967	−	0.953950i	$-0.596976\pi$
$811$	−17.0850	−0.599934	−0.299967	+	0.953950i	$0.596976\pi$
$812$	−6.88039	−0.241454
$813$	−40.3447	−1.41495
$814$	4.60143	0.161280
$815$	−6.90455	−0.241856
$816$	−11.2837	−0.395008
$817$	−3.53233	−0.123580
$818$	13.8885	0.485599
$819$	8.75617	0.305965
$820$	−0.995629	−0.0347689
$821$	−14.7658	−0.515330	−0.257665	−	0.966234i	$-0.582953\pi$
$821$	−14.7658	−0.515330	−0.257665	+	0.966234i	$0.582953\pi$
$822$	24.4221	0.851818
$823$	25.2719	0.880923	0.440461	−	0.897772i	$-0.354815\pi$
$823$	25.2719	0.880923	0.440461	+	0.897772i	$0.354815\pi$
$824$	−51.2772	−1.78633
$825$	−15.2450	−0.530764
$826$	−21.6403	−0.752963
$827$	−44.8351	−1.55907	−0.779535	−	0.626359i	$-0.784544\pi$
$827$	−44.8351	−1.55907	−0.779535	+	0.626359i	$0.784544\pi$
$828$	10.3542	0.359835
$829$	−47.2167	−1.63990	−0.819951	−	0.572433i	$-0.806000\pi$
$829$	−47.2167	−1.63990	−0.819951	+	0.572433i	$0.806000\pi$
$830$	−4.49044	−0.155865
$831$	−61.6393	−2.13824
$832$	−3.14457	−0.109018
$833$	4.44020	0.153844
$834$	79.5322	2.75398
$835$	3.22899	0.111744
$836$	−0.724496	−0.0250572
$837$	−21.3020	−0.736304
$838$	50.5219	1.74525
$839$	−40.8118	−1.40898	−0.704489	−	0.709714i	$-0.748824\pi$
$839$	−40.8118	−1.40898	−0.704489	+	0.709714i	$0.748824\pi$
$840$	−14.4517	−0.498631
$841$	69.3652	2.39190
$842$	−9.84236	−0.339190
$843$	−54.2760	−1.86936
$844$	1.85031	0.0636903
$845$	−5.85128	−0.201290
$846$	−53.1769	−1.82826
$847$	−3.38234	−0.116218
$848$	38.8511	1.33415
$849$	49.1094	1.68543
$850$	6.42166	0.220261
$851$	24.3663	0.835267
$852$	6.23649	0.213658
$853$	−24.8068	−0.849368	−0.424684	−	0.905342i	$-0.639615\pi$
$853$	−24.8068	−0.849368	−0.424684	+	0.905342i	$0.639615\pi$
$854$	40.5175	1.38648
$855$	11.4298	0.390892
$856$	−28.8061	−0.984573
$857$	30.9124	1.05595	0.527975	−	0.849260i	$-0.322952\pi$
$857$	30.9124	1.05595	0.527975	+	0.849260i	$0.322952\pi$
$858$	1.55021	0.0529234
$859$	23.9984	0.818815	0.409408	−	0.912352i	$-0.365735\pi$
$859$	23.9984	0.818815	0.409408	+	0.912352i	$0.365735\pi$
$860$	0.0932669	0.00318037
$861$	−114.839	−3.91369
$862$	46.6258	1.58808
$863$	41.0024	1.39574	0.697868	−	0.716226i	$-0.254132\pi$
$863$	41.0024	1.39574	0.697868	+	0.716226i	$0.254132\pi$
$864$	−15.1263	−0.514606
$865$	7.33541	0.249411
$866$	−8.20887	−0.278949
$867$	3.18054	0.108017
$868$	−1.12889	−0.0383171
$869$	−1.94778	−0.0660740
$870$	19.2173	0.651529
$871$	3.54134	0.119994
$872$	−46.7291	−1.58245
$873$	31.1129	1.05301
$874$	33.5735	1.13564
$875$	15.0624	0.509202
$876$	−1.71345	−0.0578922
$877$	36.8489	1.24430	0.622150	−	0.782898i	$-0.286259\pi$
$877$	36.8489	1.24430	0.622150	+	0.782898i	$0.286259\pi$
$878$	−5.89506	−0.198949
$879$	−29.7426	−1.00319
$880$	−1.61325	−0.0543826
$881$	22.8256	0.769013	0.384507	−	0.923122i	$-0.374372\pi$
$881$	22.8256	0.769013	0.384507	+	0.923122i	$0.374372\pi$
$882$	−42.3301	−1.42533
$883$	0.0988568	0.00332680	0.00166340	−	0.999999i	$-0.499471\pi$
$883$	0.0988568	0.00332680	0.00166340	+	0.999999i	$0.499471\pi$
$884$	0.0746184	0.00250969
$885$	−6.90685	−0.232171
$886$	−38.4326	−1.29117
$887$	−11.3072	−0.379658	−0.189829	−	0.981817i	$-0.560793\pi$
$887$	−11.3072	−0.379658	−0.189829	+	0.981817i	$0.560793\pi$
$888$	−32.2718	−1.08297
$889$	5.92579	0.198745
$890$	0.660690	0.0221464
$891$	20.2877	0.679665
$892$	−0.589241	−0.0197292
$893$	19.7032	0.659344
$894$	−79.8343	−2.67006
$895$	−10.5276	−0.351900
$896$	31.3510	1.04736
$897$	8.20896	0.274089
$898$	19.9545	0.665890
$899$	16.1391	0.538271
$900$	6.99567	0.233189
$901$	−10.9510	−0.364831
$902$	−14.3018	−0.476197
$903$	10.7577	0.357993
$904$	46.3448	1.54141
$905$	2.97484	0.0988871
$906$	10.0360	0.333422
$907$	31.9410	1.06058	0.530292	−	0.847815i	$-0.322082\pi$
$907$	31.9410	1.06058	0.530292	+	0.847815i	$0.322082\pi$
$908$	−2.24909	−0.0746388
$909$	−72.3393	−2.39934
$910$	−0.749651	−0.0248507
$911$	−46.2435	−1.53212	−0.766058	−	0.642771i	$-0.777785\pi$
$911$	−46.2435	−1.53212	−0.766058	+	0.642771i	$0.777785\pi$
$912$	−39.8576	−1.31982
$913$	7.37085	0.243939
$914$	−17.7436	−0.586904
$915$	12.9318	0.427512
$916$	−2.52268	−0.0833517
$917$	−4.87402	−0.160954
$918$	−17.5380	−0.578840
$919$	−41.1409	−1.35711	−0.678556	−	0.734549i	$-0.737394\pi$
$919$	−41.1409	−1.35711	−0.678556	+	0.734549i	$0.737394\pi$
$920$	−9.53053	−0.314212
$921$	2.46596	0.0812560
$922$	42.0787	1.38579
$923$	3.47804	0.114481
$924$	2.20645	0.0725868
$925$	16.4627	0.541291
$926$	8.62265	0.283358
$927$	−123.510	−4.05660
$928$	11.4602	0.376200
$929$	−16.4081	−0.538333	−0.269167	−	0.963094i	$-0.586748\pi$
$929$	−16.4081	−0.538333	−0.269167	+	0.963094i	$0.586748\pi$
$930$	3.15306	0.103393
$931$	15.6842	0.514030
$932$	0.877684	0.0287495
$933$	42.3759	1.38732
$934$	38.3785	1.25578
$935$	0.454728	0.0148712
$936$	−7.64797	−0.249982
$937$	−13.4099	−0.438083	−0.219041	−	0.975716i	$-0.570293\pi$
$937$	−13.4099	−0.438083	−0.219041	+	0.975716i	$0.570293\pi$
$938$	−44.1096	−1.44023
$939$	−92.7738	−3.02756
$940$	−0.520240	−0.0169684
$941$	−13.8502	−0.451503	−0.225751	−	0.974185i	$-0.572484\pi$
$941$	−13.8502	−0.451503	−0.225751	+	0.974185i	$0.572484\pi$
$942$	50.6869	1.65147
$943$	−75.7332	−2.46621
$944$	16.9425	0.551431
$945$	−20.1340	−0.654958
$946$	1.33974	0.0435586
$947$	−26.1153	−0.848634	−0.424317	−	0.905514i	$-0.639486\pi$
$947$	−26.1153	−0.848634	−0.424317	+	0.905514i	$0.639486\pi$
$948$	1.27062	0.0412679
$949$	−0.955579	−0.0310194
$950$	22.6834	0.735946
$951$	−48.6414	−1.57731
$952$	−9.99230	−0.323852
$953$	−31.5319	−1.02142	−0.510709	−	0.859754i	$-0.670617\pi$
$953$	−31.5319	−1.02142	−0.510709	+	0.859754i	$0.670617\pi$
$954$	104.400	3.38007
$955$	2.95995	0.0957817
$956$	−2.61513	−0.0845792
$957$	−31.5444	−1.01968
$958$	41.2006	1.33113
$959$	19.3856	0.625993
$960$	12.5010	0.403467
$961$	−28.3520	−0.914580
$962$	−1.67403	−0.0539730
$963$	−69.3845	−2.23589
$964$	1.34672	0.0433748
$965$	6.70968	0.215992
$966$	−102.248	−3.28977
$967$	3.17175	0.101997	0.0509984	−	0.998699i	$-0.483760\pi$
$967$	3.17175	0.101997	0.0509984	+	0.998699i	$0.483760\pi$
$968$	2.95426	0.0949535
$969$	11.2347	0.360911
$970$	−2.66370	−0.0855262
$971$	−12.9902	−0.416875	−0.208438	−	0.978036i	$-0.566838\pi$
$971$	−12.9902	−0.416875	−0.208438	+	0.978036i	$0.566838\pi$
$972$	−5.17974	−0.166140
$973$	63.1305	2.02387
$974$	40.6285	1.30182
$975$	5.54625	0.177622
$976$	−31.7217	−1.01539
$977$	−41.4730	−1.32684	−0.663419	−	0.748248i	$-0.730895\pi$
$977$	−41.4730	−1.32684	−0.663419	+	0.748248i	$0.730895\pi$
$978$	64.6999	2.06888
$979$	−1.08449	−0.0346605
$980$	−0.414124	−0.0132287
$981$	−112.555	−3.59361
$982$	−28.8688	−0.921242
$983$	−10.9614	−0.349614	−0.174807	−	0.984603i	$-0.555930\pi$
$983$	−10.9614	−0.349614	−0.174807	+	0.984603i	$0.555930\pi$
$984$	100.304	3.19759
$985$	7.32623	0.233433
$986$	13.2874	0.423157
$987$	−60.0060	−1.91001
$988$	0.263577	0.00838549
$989$	7.09441	0.225589
$990$	−4.33509	−0.137778
$991$	−30.9879	−0.984363	−0.492181	−	0.870493i	$-0.663800\pi$
$991$	−30.9879	−0.984363	−0.492181	+	0.870493i	$0.663800\pi$
$992$	1.88032	0.0597002
$993$	86.1955	2.73533
$994$	−43.3211	−1.37406
$995$	8.39496	0.266138
$996$	−4.80833	−0.152358
$997$	31.3196	0.991900	0.495950	−	0.868351i	$-0.334820\pi$
$997$	31.3196	0.991900	0.495950	+	0.868351i	$0.334820\pi$
$998$	24.0653	0.761774
$999$	−44.9608	−1.42250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.c.1.18	✓	60

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.c.1.18	✓	60	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)

\(f(q)\)	\(=\)	\(q-1.33974 q^{2} +3.18054 q^{3} -0.205105 q^{4} +0.454728 q^{5} -4.26109 q^{6} -3.38234 q^{7} +2.95426 q^{8} +7.11585 q^{9} +O(q^{10})\)	\(q-1.33974 q^{2} +3.18054 q^{3} -0.205105 q^{4} +0.454728 q^{5} -4.26109 q^{6} -3.38234 q^{7} +2.95426 q^{8} +7.11585 q^{9} -0.609216 q^{10} +1.00000 q^{11} -0.652344 q^{12} -0.363807 q^{13} +4.53144 q^{14} +1.44628 q^{15} -3.54772 q^{16} +1.00000 q^{17} -9.53336 q^{18} +3.53233 q^{19} -0.0932669 q^{20} -10.7577 q^{21} -1.33974 q^{22} -7.09441 q^{23} +9.39615 q^{24} -4.79322 q^{25} +0.487405 q^{26} +13.0906 q^{27} +0.693733 q^{28} -9.91792 q^{29} -1.93764 q^{30} -1.62727 q^{31} -1.15550 q^{32} +3.18054 q^{33} -1.33974 q^{34} -1.53804 q^{35} -1.45949 q^{36} -3.43458 q^{37} -4.73239 q^{38} -1.15710 q^{39} +1.34339 q^{40} +10.6751 q^{41} +14.4124 q^{42} -1.00000 q^{43} -0.205105 q^{44} +3.23578 q^{45} +9.50464 q^{46} +5.57798 q^{47} -11.2837 q^{48} +4.44020 q^{49} +6.42166 q^{50} +3.18054 q^{51} +0.0746184 q^{52} -10.9510 q^{53} -17.5380 q^{54} +0.454728 q^{55} -9.99230 q^{56} +11.2347 q^{57} +13.2874 q^{58} -4.77559 q^{59} -0.296639 q^{60} +8.94141 q^{61} +2.18012 q^{62} -24.0682 q^{63} +8.64352 q^{64} -0.165433 q^{65} -4.26109 q^{66} -9.73412 q^{67} -0.205105 q^{68} -22.5641 q^{69} +2.06058 q^{70} -9.56012 q^{71} +21.0221 q^{72} +2.62661 q^{73} +4.60143 q^{74} -15.2450 q^{75} -0.724496 q^{76} -3.38234 q^{77} +1.55021 q^{78} -1.94778 q^{79} -1.61325 q^{80} +20.2877 q^{81} -14.3018 q^{82} +7.37085 q^{83} +2.20645 q^{84} +0.454728 q^{85} +1.33974 q^{86} -31.5444 q^{87} +2.95426 q^{88} -1.08449 q^{89} -4.33509 q^{90} +1.23052 q^{91} +1.45510 q^{92} -5.17560 q^{93} -7.47302 q^{94} +1.60625 q^{95} -3.67513 q^{96} +4.37233 q^{97} -5.94870 q^{98} +7.11585 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q - 9 q^{2} - 6 q^{3} + 43 q^{4} - 15 q^{5} - 4 q^{6} - 17 q^{7} - 21 q^{8} + 40 q^{9}+O(q^{10}) \) 60 * q - 9 * q^2 - 6 * q^3 + 43 * q^4 - 15 * q^5 - 4 * q^6 - 17 * q^7 - 21 * q^8 + 40 * q^9	\( 60 q - 9 q^{2} - 6 q^{3} + 43 q^{4} - 15 q^{5} - 4 q^{6} - 17 q^{7} - 21 q^{8} + 40 q^{9} + q^{10} + 60 q^{11} - 13 q^{12} - 2 q^{13} - 15 q^{14} - 32 q^{15} + 21 q^{16} + 60 q^{17} - 41 q^{18} - 24 q^{19} - 33 q^{20} - 12 q^{21} - 9 q^{22} - 20 q^{23} + 9 q^{24} + 25 q^{25} - 40 q^{26} - 18 q^{27} - 3 q^{28} - 54 q^{29} - 18 q^{30} - 50 q^{31} - 51 q^{32} - 6 q^{33} - 9 q^{34} - 30 q^{35} + 27 q^{36} - 63 q^{37} - 38 q^{38} - 55 q^{39} - 5 q^{40} - 53 q^{41} - 47 q^{42} - 60 q^{43} + 43 q^{44} - 36 q^{45} - 27 q^{46} - 47 q^{47} - 38 q^{48} + 5 q^{49} - 4 q^{50} - 6 q^{51} - 13 q^{52} - 24 q^{53} - 58 q^{54} - 15 q^{55} - 45 q^{56} + 23 q^{57} + 16 q^{58} - 57 q^{59} - 4 q^{60} - 8 q^{61} - 3 q^{62} - 57 q^{63} - 5 q^{64} - 27 q^{65} - 4 q^{66} - 49 q^{67} + 43 q^{68} - 57 q^{69} - 7 q^{70} - 151 q^{71} - 29 q^{72} - 12 q^{73} + 18 q^{74} - 23 q^{75} - 38 q^{76} - 17 q^{77} + 37 q^{78} - 11 q^{79} - 21 q^{80} + 28 q^{81} + 44 q^{82} - 42 q^{83} + 16 q^{84} - 15 q^{85} + 9 q^{86} - 56 q^{87} - 21 q^{88} - 88 q^{89} + 47 q^{90} - 60 q^{91} + 26 q^{92} - 16 q^{93} + 37 q^{94} - 57 q^{95} + 108 q^{96} - 22 q^{97} + 8 q^{98} + 40 q^{99}+O(q^{100}) \) 60 * q - 9 * q^2 - 6 * q^3 + 43 * q^4 - 15 * q^5 - 4 * q^6 - 17 * q^7 - 21 * q^8 + 40 * q^9 + q^10 + 60 * q^11 - 13 * q^12 - 2 * q^13 - 15 * q^14 - 32 * q^15 + 21 * q^16 + 60 * q^17 - 41 * q^18 - 24 * q^19 - 33 * q^20 - 12 * q^21 - 9 * q^22 - 20 * q^23 + 9 * q^24 + 25 * q^25 - 40 * q^26 - 18 * q^27 - 3 * q^28 - 54 * q^29 - 18 * q^30 - 50 * q^31 - 51 * q^32 - 6 * q^33 - 9 * q^34 - 30 * q^35 + 27 * q^36 - 63 * q^37 - 38 * q^38 - 55 * q^39 - 5 * q^40 - 53 * q^41 - 47 * q^42 - 60 * q^43 + 43 * q^44 - 36 * q^45 - 27 * q^46 - 47 * q^47 - 38 * q^48 + 5 * q^49 - 4 * q^50 - 6 * q^51 - 13 * q^52 - 24 * q^53 - 58 * q^54 - 15 * q^55 - 45 * q^56 + 23 * q^57 + 16 * q^58 - 57 * q^59 - 4 * q^60 - 8 * q^61 - 3 * q^62 - 57 * q^63 - 5 * q^64 - 27 * q^65 - 4 * q^66 - 49 * q^67 + 43 * q^68 - 57 * q^69 - 7 * q^70 - 151 * q^71 - 29 * q^72 - 12 * q^73 + 18 * q^74 - 23 * q^75 - 38 * q^76 - 17 * q^77 + 37 * q^78 - 11 * q^79 - 21 * q^80 + 28 * q^81 + 44 * q^82 - 42 * q^83 + 16 * q^84 - 15 * q^85 + 9 * q^86 - 56 * q^87 - 21 * q^88 - 88 * q^89 + 47 * q^90 - 60 * q^91 + 26 * q^92 - 16 * q^93 + 37 * q^94 - 57 * q^95 + 108 * q^96 - 22 * q^97 + 8 * q^98 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more