LMFDB - Embedded newform 6041.2.a.e.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6041,2,Mod(1,6041)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6041, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6041 = 7 \cdot 863 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6041.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:	$48.2376278611$
Analytic rank:	$0$
Dimension:	$112$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	6041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.25493 q^{2} +0.275101 q^{3} +3.08473 q^{4} +2.79647 q^{5} -0.620335 q^{6} -1.00000 q^{7} -2.44598 q^{8} -2.92432 q^{9} +O(q^{10})$	\(q-2.25493 q^{2} +0.275101 q^{3} +3.08473 q^{4} +2.79647 q^{5} -0.620335 q^{6} -1.00000 q^{7} -2.44598 q^{8} -2.92432 q^{9} -6.30585 q^{10} -2.15009 q^{11} +0.848612 q^{12} -2.68812 q^{13} +2.25493 q^{14} +0.769312 q^{15} -0.653919 q^{16} -5.18356 q^{17} +6.59415 q^{18} -5.88738 q^{19} +8.62633 q^{20} -0.275101 q^{21} +4.84831 q^{22} +7.71467 q^{23} -0.672894 q^{24} +2.82023 q^{25} +6.06153 q^{26} -1.62979 q^{27} -3.08473 q^{28} +1.52358 q^{29} -1.73475 q^{30} -7.44409 q^{31} +6.36651 q^{32} -0.591493 q^{33} +11.6886 q^{34} -2.79647 q^{35} -9.02072 q^{36} +0.972403 q^{37} +13.2757 q^{38} -0.739505 q^{39} -6.84011 q^{40} -5.46669 q^{41} +0.620335 q^{42} -5.86038 q^{43} -6.63244 q^{44} -8.17776 q^{45} -17.3961 q^{46} +2.35392 q^{47} -0.179894 q^{48} +1.00000 q^{49} -6.35942 q^{50} -1.42600 q^{51} -8.29211 q^{52} +7.46686 q^{53} +3.67506 q^{54} -6.01265 q^{55} +2.44598 q^{56} -1.61963 q^{57} -3.43558 q^{58} +3.93950 q^{59} +2.37312 q^{60} +0.0331323 q^{61} +16.7859 q^{62} +2.92432 q^{63} -13.0482 q^{64} -7.51724 q^{65} +1.33378 q^{66} +4.54895 q^{67} -15.9898 q^{68} +2.12232 q^{69} +6.30585 q^{70} -14.7331 q^{71} +7.15284 q^{72} -2.66416 q^{73} -2.19270 q^{74} +0.775848 q^{75} -18.1610 q^{76} +2.15009 q^{77} +1.66754 q^{78} +5.66291 q^{79} -1.82866 q^{80} +8.32460 q^{81} +12.3270 q^{82} +3.95471 q^{83} -0.848612 q^{84} -14.4956 q^{85} +13.2148 q^{86} +0.419140 q^{87} +5.25909 q^{88} -5.82973 q^{89} +18.4403 q^{90} +2.68812 q^{91} +23.7976 q^{92} -2.04788 q^{93} -5.30793 q^{94} -16.4639 q^{95} +1.75144 q^{96} +9.78109 q^{97} -2.25493 q^{98} +6.28755 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 112 q - 3 q^{2} + 14 q^{3} + 131 q^{4} + 13 q^{5} + 18 q^{6} - 112 q^{7} - 9 q^{8} + 116 q^{9}+O(q^{10}) $ 112 * q - 3 * q^2 + 14 * q^3 + 131 * q^4 + 13 * q^5 + 18 * q^6 - 112 * q^7 - 9 * q^8 + 116 * q^9	\( 112 q - 3 q^{2} + 14 q^{3} + 131 q^{4} + 13 q^{5} + 18 q^{6} - 112 q^{7} - 9 q^{8} + 116 q^{9} + 32 q^{10} + 14 q^{11} + 36 q^{12} + 22 q^{13} + 3 q^{14} + 19 q^{15} + 169 q^{16} + 11 q^{17} - 18 q^{18} + 52 q^{19} + 40 q^{20} - 14 q^{21} + 16 q^{22} + 38 q^{23} + 64 q^{24} + 99 q^{25} + 45 q^{26} + 65 q^{27} - 131 q^{28} + 10 q^{29} + q^{30} + 133 q^{31} - 26 q^{32} + 27 q^{33} + 52 q^{34} - 13 q^{35} + 183 q^{36} - 13 q^{37} + 20 q^{38} + 74 q^{39} + 92 q^{40} + 25 q^{41} - 18 q^{42} - 11 q^{43} + 16 q^{44} + 63 q^{45} + 28 q^{46} + 71 q^{47} + 70 q^{48} + 112 q^{49} + 5 q^{50} + 57 q^{51} + 79 q^{52} - 10 q^{53} + 75 q^{54} + 146 q^{55} + 9 q^{56} - 83 q^{57} - 19 q^{58} + 56 q^{59} - 3 q^{60} + 80 q^{61} + 42 q^{62} - 116 q^{63} + 263 q^{64} - 26 q^{65} + 48 q^{66} + 29 q^{67} + 57 q^{68} + 56 q^{69} - 32 q^{70} + 100 q^{71} - 62 q^{72} + 73 q^{73} + 24 q^{74} + 89 q^{75} + 155 q^{76} - 14 q^{77} + 33 q^{78} + 140 q^{79} + 80 q^{80} + 120 q^{81} + 114 q^{82} + 36 q^{83} - 36 q^{84} - 2 q^{85} + 12 q^{86} + 96 q^{87} + 29 q^{88} + 47 q^{89} + 52 q^{90} - 22 q^{91} + 81 q^{92} - 10 q^{93} + 127 q^{94} + 96 q^{95} + 175 q^{96} + 80 q^{97} - 3 q^{98} + 74 q^{99}+O(q^{100}) \) 112 * q - 3 * q^2 + 14 * q^3 + 131 * q^4 + 13 * q^5 + 18 * q^6 - 112 * q^7 - 9 * q^8 + 116 * q^9 + 32 * q^10 + 14 * q^11 + 36 * q^12 + 22 * q^13 + 3 * q^14 + 19 * q^15 + 169 * q^16 + 11 * q^17 - 18 * q^18 + 52 * q^19 + 40 * q^20 - 14 * q^21 + 16 * q^22 + 38 * q^23 + 64 * q^24 + 99 * q^25 + 45 * q^26 + 65 * q^27 - 131 * q^28 + 10 * q^29 + q^30 + 133 * q^31 - 26 * q^32 + 27 * q^33 + 52 * q^34 - 13 * q^35 + 183 * q^36 - 13 * q^37 + 20 * q^38 + 74 * q^39 + 92 * q^40 + 25 * q^41 - 18 * q^42 - 11 * q^43 + 16 * q^44 + 63 * q^45 + 28 * q^46 + 71 * q^47 + 70 * q^48 + 112 * q^49 + 5 * q^50 + 57 * q^51 + 79 * q^52 - 10 * q^53 + 75 * q^54 + 146 * q^55 + 9 * q^56 - 83 * q^57 - 19 * q^58 + 56 * q^59 - 3 * q^60 + 80 * q^61 + 42 * q^62 - 116 * q^63 + 263 * q^64 - 26 * q^65 + 48 * q^66 + 29 * q^67 + 57 * q^68 + 56 * q^69 - 32 * q^70 + 100 * q^71 - 62 * q^72 + 73 * q^73 + 24 * q^74 + 89 * q^75 + 155 * q^76 - 14 * q^77 + 33 * q^78 + 140 * q^79 + 80 * q^80 + 120 * q^81 + 114 * q^82 + 36 * q^83 - 36 * q^84 - 2 * q^85 + 12 * q^86 + 96 * q^87 + 29 * q^88 + 47 * q^89 + 52 * q^90 - 22 * q^91 + 81 * q^92 - 10 * q^93 + 127 * q^94 + 96 * q^95 + 175 * q^96 + 80 * q^97 - 3 * q^98 + 74 * 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 $

$2$	−2.25493	−1.59448	−0.797239	−	0.603663i	$-0.793707\pi$
$2$	−2.25493	−1.59448	−0.797239	+	0.603663i	$0.793707\pi$
$3$	0.275101	0.158830	0.0794149	−	0.996842i	$-0.474695\pi$
$3$	0.275101	0.158830	0.0794149	+	0.996842i	$0.474695\pi$
$4$	3.08473	1.54236
$5$	2.79647	1.25062	0.625309	−	0.780377i	$-0.284973\pi$
$5$	2.79647	1.25062	0.625309	+	0.780377i	$0.284973\pi$
$6$	−0.620335	−0.253251
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.44598	−0.864786
$9$	−2.92432	−0.974773
$10$	−6.30585	−1.99408
$11$	−2.15009	−0.648277	−0.324138	−	0.946010i	$-0.605074\pi$
$11$	−2.15009	−0.648277	−0.324138	+	0.946010i	$0.605074\pi$
$12$	0.848612	0.244973
$13$	−2.68812	−0.745550	−0.372775	−	0.927922i	$-0.621594\pi$
$13$	−2.68812	−0.745550	−0.372775	+	0.927922i	$0.621594\pi$
$14$	2.25493	0.602656
$15$	0.769312	0.198635
$16$	−0.653919	−0.163480
$17$	−5.18356	−1.25720	−0.628598	−	0.777730i	$-0.716371\pi$
$17$	−5.18356	−1.25720	−0.628598	+	0.777730i	$0.716371\pi$
$18$	6.59415	1.55426
$19$	−5.88738	−1.35066	−0.675329	−	0.737516i	$-0.735998\pi$
$19$	−5.88738	−1.35066	−0.675329	+	0.737516i	$0.735998\pi$
$20$	8.62633	1.92891
$21$	−0.275101	−0.0600320
$22$	4.84831	1.03366
$23$	7.71467	1.60862	0.804310	−	0.594210i	$-0.202535\pi$
$23$	7.71467	1.60862	0.804310	+	0.594210i	$0.202535\pi$
$24$	−0.672894	−0.137354
$25$	2.82023	0.564045
$26$	6.06153	1.18876
$27$	−1.62979	−0.313653
$28$	−3.08473	−0.582958
$29$	1.52358	0.282922	0.141461	−	0.989944i	$-0.454820\pi$
$29$	1.52358	0.282922	0.141461	+	0.989944i	$0.454820\pi$
$30$	−1.73475	−0.316720
$31$	−7.44409	−1.33700	−0.668499	−	0.743713i	$-0.733063\pi$
$31$	−7.44409	−1.33700	−0.668499	+	0.743713i	$0.733063\pi$
$32$	6.36651	1.12545
$33$	−0.591493	−0.102966
$34$	11.6886	2.00457
$35$	−2.79647	−0.472689
$36$	−9.02072	−1.50345
$37$	0.972403	0.159862	0.0799310	−	0.996800i	$-0.474530\pi$
$37$	0.972403	0.159862	0.0799310	+	0.996800i	$0.474530\pi$
$38$	13.2757	2.15360
$39$	−0.739505	−0.118416
$40$	−6.84011	−1.08152
$41$	−5.46669	−0.853754	−0.426877	−	0.904310i	$-0.640386\pi$
$41$	−5.46669	−0.853754	−0.426877	+	0.904310i	$0.640386\pi$
$42$	0.620335	0.0957198
$43$	−5.86038	−0.893700	−0.446850	−	0.894609i	$-0.647454\pi$
$43$	−5.86038	−0.893700	−0.446850	+	0.894609i	$0.647454\pi$
$44$	−6.63244	−0.999878
$45$	−8.17776	−1.21907
$46$	−17.3961	−2.56491
$47$	2.35392	0.343354	0.171677	−	0.985153i	$-0.445081\pi$
$47$	2.35392	0.343354	0.171677	+	0.985153i	$0.445081\pi$
$48$	−0.179894	−0.0259655
$49$	1.00000	0.142857
$50$	−6.35942	−0.899358
$51$	−1.42600	−0.199680
$52$	−8.29211	−1.14991
$53$	7.46686	1.02565	0.512826	−	0.858493i	$-0.328599\pi$
$53$	7.46686	1.02565	0.512826	+	0.858493i	$0.328599\pi$
$54$	3.67506	0.500113
$55$	−6.01265	−0.810746
$56$	2.44598	0.326858
$57$	−1.61963	−0.214525
$58$	−3.43558	−0.451113
$59$	3.93950	0.512880	0.256440	−	0.966560i	$-0.417450\pi$
$59$	3.93950	0.512880	0.256440	+	0.966560i	$0.417450\pi$
$60$	2.37312	0.306368
$61$	0.0331323	0.00424215	0.00212108	−	0.999998i	$-0.499325\pi$
$61$	0.0331323	0.00424215	0.00212108	+	0.999998i	$0.499325\pi$
$62$	16.7859	2.13182
$63$	2.92432	0.368430
$64$	−13.0482	−1.63103
$65$	−7.51724	−0.932398
$66$	1.33378	0.164177
$67$	4.54895	0.555743	0.277872	−	0.960618i	$-0.410371\pi$
$67$	4.54895	0.555743	0.277872	+	0.960618i	$0.410371\pi$
$68$	−15.9898	−1.93905
$69$	2.12232	0.255497
$70$	6.30585	0.753693
$71$	−14.7331	−1.74850	−0.874250	−	0.485475i	$-0.838647\pi$
$71$	−14.7331	−1.74850	−0.874250	+	0.485475i	$0.838647\pi$
$72$	7.15284	0.842970
$73$	−2.66416	−0.311817	−0.155908	−	0.987772i	$-0.549830\pi$
$73$	−2.66416	−0.311817	−0.155908	+	0.987772i	$0.549830\pi$
$74$	−2.19270	−0.254897
$75$	0.775848	0.0895872
$76$	−18.1610	−2.08321
$77$	2.15009	0.245025
$78$	1.66754	0.188811
$79$	5.66291	0.637127	0.318563	−	0.947902i	$-0.396800\pi$
$79$	5.66291	0.637127	0.318563	+	0.947902i	$0.396800\pi$
$80$	−1.82866	−0.204451
$81$	8.32460	0.924956
$82$	12.3270	1.36129
$83$	3.95471	0.434086	0.217043	−	0.976162i	$-0.430359\pi$
$83$	3.95471	0.434086	0.217043	+	0.976162i	$0.430359\pi$
$84$	−0.848612	−0.0925912
$85$	−14.4956	−1.57227
$86$	13.2148	1.42499
$87$	0.419140	0.0449365
$88$	5.25909	0.560620
$89$	−5.82973	−0.617950	−0.308975	−	0.951070i	$-0.599986\pi$
$89$	−5.82973	−0.617950	−0.308975	+	0.951070i	$0.599986\pi$
$90$	18.4403	1.94378
$91$	2.68812	0.281791
$92$	23.7976	2.48108
$93$	−2.04788	−0.212355
$94$	−5.30793	−0.547471
$95$	−16.4639	−1.68916
$96$	1.75144	0.178755
$97$	9.78109	0.993119	0.496559	−	0.868003i	$-0.334596\pi$
$97$	9.78109	0.993119	0.496559	+	0.868003i	$0.334596\pi$
$98$	−2.25493	−0.227783
$99$	6.28755	0.631922
$100$	8.69962	0.869962
$101$	16.6257	1.65432	0.827161	−	0.561965i	$-0.189955\pi$
$101$	16.6257	1.65432	0.827161	+	0.561965i	$0.189955\pi$
$102$	3.21554	0.318386
$103$	9.48253	0.934341	0.467171	−	0.884167i	$-0.345273\pi$
$103$	9.48253	0.934341	0.467171	+	0.884167i	$0.345273\pi$
$104$	6.57510	0.644741
$105$	−0.769312	−0.0750771
$106$	−16.8373	−1.63538
$107$	18.2518	1.76447	0.882234	−	0.470811i	$-0.156039\pi$
$107$	18.2518	1.76447	0.882234	+	0.470811i	$0.156039\pi$
$108$	−5.02745	−0.483767
$109$	6.00203	0.574890	0.287445	−	0.957797i	$-0.407194\pi$
$109$	6.00203	0.574890	0.287445	+	0.957797i	$0.407194\pi$
$110$	13.5581	1.29272
$111$	0.267509	0.0253909
$112$	0.653919	0.0617896
$113$	5.83368	0.548787	0.274393	−	0.961618i	$-0.411523\pi$
$113$	5.83368	0.548787	0.274393	+	0.961618i	$0.411523\pi$
$114$	3.65215	0.342055
$115$	21.5738	2.01177
$116$	4.69983	0.436369
$117$	7.86092	0.726742
$118$	−8.88332	−0.817776
$119$	5.18356	0.475176
$120$	−1.88172	−0.171777
$121$	−6.37711	−0.579738
$122$	−0.0747111	−0.00676402
$123$	−1.50389	−0.135602
$124$	−22.9630	−2.06214
$125$	−6.09567	−0.545213
$126$	−6.59415	−0.587453
$127$	−5.26055	−0.466799	−0.233399	−	0.972381i	$-0.574985\pi$
$127$	−5.26055	−0.466799	−0.233399	+	0.972381i	$0.574985\pi$
$128$	16.6899	1.47519
$129$	−1.61220	−0.141946
$130$	16.9509	1.48669
$131$	7.47647	0.653222	0.326611	−	0.945159i	$-0.394093\pi$
$131$	7.47647	0.653222	0.326611	+	0.945159i	$0.394093\pi$
$132$	−1.82459	−0.158810
$133$	5.88738	0.510501
$134$	−10.2576	−0.886121
$135$	−4.55765	−0.392260
$136$	12.6789	1.08721
$137$	3.07950	0.263100	0.131550	−	0.991310i	$-0.458005\pi$
$137$	3.07950	0.263100	0.131550	+	0.991310i	$0.458005\pi$
$138$	−4.78568	−0.407384
$139$	15.9395	1.35197	0.675987	−	0.736914i	$-0.263718\pi$
$139$	15.9395	1.35197	0.675987	+	0.736914i	$0.263718\pi$
$140$	−8.62633	−0.729058
$141$	0.647566	0.0545349
$142$	33.2222	2.78795
$143$	5.77970	0.483323
$144$	1.91227	0.159356
$145$	4.26065	0.353827
$146$	6.00751	0.497185
$147$	0.275101	0.0226900
$148$	2.99959	0.246565
$149$	−15.7348	−1.28905	−0.644524	−	0.764584i	$-0.722944\pi$
$149$	−15.7348	−1.28905	−0.644524	+	0.764584i	$0.722944\pi$
$150$	−1.74949	−0.142845
$151$	21.3855	1.74033	0.870164	−	0.492762i	$-0.164013\pi$
$151$	21.3855	1.74033	0.870164	+	0.492762i	$0.164013\pi$
$152$	14.4004	1.16803
$153$	15.1584	1.22548
$154$	−4.84831	−0.390688
$155$	−20.8172	−1.67207
$156$	−2.28117	−0.182640
$157$	14.9957	1.19678	0.598392	−	0.801203i	$-0.295806\pi$
$157$	14.9957	1.19678	0.598392	+	0.801203i	$0.295806\pi$
$158$	−12.7695	−1.01589
$159$	2.05414	0.162904
$160$	17.8037	1.40751
$161$	−7.71467	−0.608001
$162$	−18.7714	−1.47482
$163$	−23.2855	−1.82386	−0.911929	−	0.410348i	$-0.865407\pi$
$163$	−23.2855	−1.82386	−0.911929	+	0.410348i	$0.865407\pi$
$164$	−16.8632	−1.31680
$165$	−1.65409	−0.128771
$166$	−8.91761	−0.692141
$167$	−0.879020	−0.0680206	−0.0340103	−	0.999421i	$-0.510828\pi$
$167$	−0.879020	−0.0680206	−0.0340103	+	0.999421i	$0.510828\pi$
$168$	0.672894	0.0519149
$169$	−5.77401	−0.444155
$170$	32.6867	2.50696
$171$	17.2166	1.31659
$172$	−18.0777	−1.37841
$173$	−2.81029	−0.213662	−0.106831	−	0.994277i	$-0.534070\pi$
$173$	−2.81029	−0.213662	−0.106831	+	0.994277i	$0.534070\pi$
$174$	−0.945132	−0.0716503
$175$	−2.82023	−0.213189
$176$	1.40599	0.105980
$177$	1.08376	0.0814606
$178$	13.1457	0.985308
$179$	22.8517	1.70802	0.854008	−	0.520259i	$-0.174165\pi$
$179$	22.8517	1.70802	0.854008	+	0.520259i	$0.174165\pi$
$180$	−25.2261	−1.88025
$181$	24.9047	1.85115	0.925576	−	0.378561i	$-0.123581\pi$
$181$	24.9047	1.85115	0.925576	+	0.378561i	$0.123581\pi$
$182$	−6.06153	−0.449311
$183$	0.00911473	0.000673781 0
$184$	−18.8700	−1.39111
$185$	2.71929	0.199926
$186$	4.61783	0.338596
$187$	11.1451	0.815011
$188$	7.26119	0.529577
$189$	1.62979	0.118550
$190$	37.1249	2.69333
$191$	10.9556	0.792722	0.396361	−	0.918095i	$-0.370273\pi$
$191$	10.9556	0.792722	0.396361	+	0.918095i	$0.370273\pi$
$192$	−3.58958	−0.259056
$193$	−8.60103	−0.619116	−0.309558	−	0.950881i	$-0.600181\pi$
$193$	−8.60103	−0.619116	−0.309558	+	0.950881i	$0.600181\pi$
$194$	−22.0557	−1.58351
$195$	−2.06800	−0.148093
$196$	3.08473	0.220338
$197$	25.7269	1.83297	0.916484	−	0.400072i	$-0.131015\pi$
$197$	25.7269	1.83297	0.916484	+	0.400072i	$0.131015\pi$
$198$	−14.1780	−1.00759
$199$	5.31227	0.376577	0.188288	−	0.982114i	$-0.439706\pi$
$199$	5.31227	0.376577	0.188288	+	0.982114i	$0.439706\pi$
$200$	−6.89823	−0.487778
$201$	1.25142	0.0882686
$202$	−37.4899	−2.63778
$203$	−1.52358	−0.106935
$204$	−4.39883	−0.307980
$205$	−15.2874	−1.06772
$206$	−21.3825	−1.48979
$207$	−22.5602	−1.56804
$208$	1.75781	0.121882
$209$	12.6584	0.875600
$210$	1.73475	0.119709
$211$	−4.60468	−0.316999	−0.158500	−	0.987359i	$-0.550666\pi$
$211$	−4.60468	−0.316999	−0.158500	+	0.987359i	$0.550666\pi$
$212$	23.0332	1.58193
$213$	−4.05310	−0.277714
$214$	−41.1566	−2.81341
$215$	−16.3884	−1.11768
$216$	3.98644	0.271243
$217$	7.44409	0.505338
$218$	−13.5342	−0.916650
$219$	−0.732915	−0.0495258
$220$	−18.5474	−1.25046
$221$	13.9340	0.937303
$222$	−0.603216	−0.0404852
$223$	16.5305	1.10696	0.553482	−	0.832861i	$-0.313299\pi$
$223$	16.5305	1.10696	0.553482	+	0.832861i	$0.313299\pi$
$224$	−6.36651	−0.425381
$225$	−8.24724	−0.549816
$226$	−13.1546	−0.875029
$227$	−5.49253	−0.364552	−0.182276	−	0.983247i	$-0.558346\pi$
$227$	−5.49253	−0.364552	−0.182276	+	0.983247i	$0.558346\pi$
$228$	−4.99611	−0.330875
$229$	5.70232	0.376820	0.188410	−	0.982090i	$-0.439667\pi$
$229$	5.70232	0.376820	0.188410	+	0.982090i	$0.439667\pi$
$230$	−48.6475	−3.20772
$231$	0.591493	0.0389174
$232$	−3.72666	−0.244667
$233$	−28.5455	−1.87008	−0.935039	−	0.354544i	$-0.884636\pi$
$233$	−28.5455	−1.87008	−0.935039	+	0.354544i	$0.884636\pi$
$234$	−17.7259	−1.15878
$235$	6.58265	0.429405
$236$	12.1523	0.791047
$237$	1.55787	0.101195
$238$	−11.6886	−0.757658
$239$	−13.8855	−0.898182	−0.449091	−	0.893486i	$-0.648252\pi$
$239$	−13.8855	−0.898182	−0.449091	+	0.893486i	$0.648252\pi$
$240$	−0.503068	−0.0324729
$241$	−3.10831	−0.200224	−0.100112	−	0.994976i	$-0.531920\pi$
$241$	−3.10831	−0.200224	−0.100112	+	0.994976i	$0.531920\pi$
$242$	14.3800	0.924379
$243$	7.17947	0.460563
$244$	0.102204	0.00654294
$245$	2.79647	0.178660
$246$	3.39118	0.216214
$247$	15.8260	1.00698
$248$	18.2081	1.15622
$249$	1.08795	0.0689458
$250$	13.7453	0.869331
$251$	23.0720	1.45629	0.728147	−	0.685421i	$-0.240382\pi$
$251$	23.0720	1.45629	0.728147	+	0.685421i	$0.240382\pi$
$252$	9.02072	0.568252
$253$	−16.5872	−1.04283
$254$	11.8622	0.744301
$255$	−3.98777	−0.249724
$256$	−11.5381	−0.721129
$257$	−20.0175	−1.24866	−0.624329	−	0.781162i	$-0.714627\pi$
$257$	−20.0175	−1.24866	−0.624329	+	0.781162i	$0.714627\pi$
$258$	3.63540	0.226330
$259$	−0.972403	−0.0604222
$260$	−23.1886	−1.43810
$261$	−4.45544	−0.275785
$262$	−16.8589	−1.04155
$263$	−22.6675	−1.39774	−0.698869	−	0.715250i	$-0.746313\pi$
$263$	−22.6675	−1.39774	−0.698869	+	0.715250i	$0.746313\pi$
$264$	1.44678	0.0890433
$265$	20.8808	1.28270
$266$	−13.2757	−0.813983
$267$	−1.60377	−0.0981489
$268$	14.0323	0.857158
$269$	2.60875	0.159058	0.0795290	−	0.996833i	$-0.474658\pi$
$269$	2.60875	0.159058	0.0795290	+	0.996833i	$0.474658\pi$
$270$	10.2772	0.625450
$271$	9.10048	0.552815	0.276407	−	0.961041i	$-0.410856\pi$
$271$	9.10048	0.552815	0.276407	+	0.961041i	$0.410856\pi$
$272$	3.38963	0.205526
$273$	0.739505	0.0447569
$274$	−6.94408	−0.419507
$275$	−6.06374	−0.365657
$276$	6.54676	0.394069
$277$	−17.3373	−1.04170	−0.520850	−	0.853648i	$-0.674385\pi$
$277$	−17.3373	−1.04170	−0.520850	+	0.853648i	$0.674385\pi$
$278$	−35.9426	−2.15569
$279$	21.7689	1.30327
$280$	6.84011	0.408775
$281$	−11.8537	−0.707132	−0.353566	−	0.935410i	$-0.615031\pi$
$281$	−11.8537	−0.707132	−0.353566	+	0.935410i	$0.615031\pi$
$282$	−1.46022	−0.0869548
$283$	−3.16122	−0.187915	−0.0939576	−	0.995576i	$-0.529952\pi$
$283$	−3.16122	−0.187915	−0.0939576	+	0.995576i	$0.529952\pi$
$284$	−45.4477	−2.69682
$285$	−4.52923	−0.268289
$286$	−13.0328	−0.770648
$287$	5.46669	0.322689
$288$	−18.6177	−1.09706
$289$	9.86924	0.580544
$290$	−9.60748	−0.564170
$291$	2.69079	0.157737
$292$	−8.21821	−0.480934
$293$	−5.81998	−0.340007	−0.170003	−	0.985443i	$-0.554378\pi$
$293$	−5.81998	−0.340007	−0.170003	+	0.985443i	$0.554378\pi$
$294$	−0.620335	−0.0361787
$295$	11.0167	0.641417
$296$	−2.37848	−0.138246
$297$	3.50419	0.203334
$298$	35.4810	2.05536
$299$	−20.7380	−1.19931
$300$	2.39328	0.138176
$301$	5.86038	0.337787
$302$	−48.2229	−2.77492
$303$	4.57376	0.262756
$304$	3.84987	0.220805
$305$	0.0926533	0.00530531
$306$	−34.1811	−1.95400
$307$	−7.85723	−0.448436	−0.224218	−	0.974539i	$-0.571983\pi$
$307$	−7.85723	−0.448436	−0.224218	+	0.974539i	$0.571983\pi$
$308$	6.63244	0.377918
$309$	2.60866	0.148401
$310$	46.9413	2.66609
$311$	12.3178	0.698477	0.349239	−	0.937034i	$-0.386440\pi$
$311$	12.3178	0.698477	0.349239	+	0.937034i	$0.386440\pi$
$312$	1.80882	0.102404
$313$	4.62318	0.261317	0.130659	−	0.991427i	$-0.458291\pi$
$313$	4.62318	0.261317	0.130659	+	0.991427i	$0.458291\pi$
$314$	−33.8142	−1.90825
$315$	8.17776	0.460765
$316$	17.4685	0.982681
$317$	23.1054	1.29773	0.648865	−	0.760904i	$-0.275244\pi$
$317$	23.1054	1.29773	0.648865	+	0.760904i	$0.275244\pi$
$318$	−4.63195	−0.259747
$319$	−3.27584	−0.183412
$320$	−36.4889	−2.03979
$321$	5.02109	0.280250
$322$	17.3961	0.969445
$323$	30.5176	1.69804
$324$	25.6791	1.42662
$325$	−7.58110	−0.420524
$326$	52.5072	2.90810
$327$	1.65117	0.0913097
$328$	13.3714	0.738314
$329$	−2.35392	−0.129776
$330$	3.72986	0.205322
$331$	−5.42307	−0.298079	−0.149039	−	0.988831i	$-0.547618\pi$
$331$	−5.42307	−0.298079	−0.149039	+	0.988831i	$0.547618\pi$
$332$	12.1992	0.669518
$333$	−2.84362	−0.155829
$334$	1.98213	0.108457
$335$	12.7210	0.695022
$336$	0.179894	0.00981403
$337$	4.21709	0.229719	0.114860	−	0.993382i	$-0.463358\pi$
$337$	4.21709	0.229719	0.114860	+	0.993382i	$0.463358\pi$
$338$	13.0200	0.708196
$339$	1.60485	0.0871637
$340$	−44.7151	−2.42502
$341$	16.0055	0.866745
$342$	−38.8223	−2.09927
$343$	−1.00000	−0.0539949
$344$	14.3344	0.772859
$345$	5.93499	0.319529
$346$	6.33702	0.340680
$347$	23.9748	1.28703	0.643517	−	0.765432i	$-0.277475\pi$
$347$	23.9748	1.28703	0.643517	+	0.765432i	$0.277475\pi$
$348$	1.29293	0.0693083
$349$	19.3041	1.03332	0.516662	−	0.856189i	$-0.327174\pi$
$349$	19.3041	1.03332	0.516662	+	0.856189i	$0.327174\pi$
$350$	6.35942	0.339925
$351$	4.38107	0.233844
$352$	−13.6886	−0.729604
$353$	1.31913	0.0702101	0.0351050	−	0.999384i	$-0.488823\pi$
$353$	1.31913	0.0702101	0.0351050	+	0.999384i	$0.488823\pi$
$354$	−2.44381	−0.129887
$355$	−41.2007	−2.18671
$356$	−17.9831	−0.953103
$357$	1.42600	0.0754721
$358$	−51.5291	−2.72340
$359$	−18.7779	−0.991059	−0.495530	−	0.868591i	$-0.665026\pi$
$359$	−18.7779	−0.991059	−0.495530	+	0.868591i	$0.665026\pi$
$360$	20.0027	1.05423
$361$	15.6613	0.824279
$362$	−56.1585	−2.95162
$363$	−1.75435	−0.0920796
$364$	8.29211	0.434625
$365$	−7.45024	−0.389963
$366$	−0.0205531	−0.00107433
$367$	3.89126	0.203122	0.101561	−	0.994829i	$-0.467616\pi$
$367$	3.89126	0.203122	0.101561	+	0.994829i	$0.467616\pi$
$368$	−5.04477	−0.262977
$369$	15.9864	0.832216
$370$	−6.13182	−0.318778
$371$	−7.46686	−0.387660
$372$	−6.31715	−0.327529
$373$	−21.7680	−1.12710	−0.563551	−	0.826081i	$-0.690565\pi$
$373$	−21.7680	−1.12710	−0.563551	+	0.826081i	$0.690565\pi$
$374$	−25.1315	−1.29952
$375$	−1.67693	−0.0865961
$376$	−5.75765	−0.296928
$377$	−4.09557	−0.210933
$378$	−3.67506	−0.189025
$379$	−21.7113	−1.11523	−0.557617	−	0.830098i	$-0.688284\pi$
$379$	−21.7113	−1.11523	−0.557617	+	0.830098i	$0.688284\pi$
$380$	−50.7865	−2.60529
$381$	−1.44719	−0.0741416
$382$	−24.7042	−1.26398
$383$	−23.8196	−1.21713	−0.608563	−	0.793506i	$-0.708254\pi$
$383$	−23.8196	−1.21713	−0.608563	+	0.793506i	$0.708254\pi$
$384$	4.59140	0.234304
$385$	6.01265	0.306433
$386$	19.3948	0.987167
$387$	17.1376	0.871154
$388$	30.1720	1.53175
$389$	0.344702	0.0174771	0.00873855	−	0.999962i	$-0.497218\pi$
$389$	0.344702	0.0174771	0.00873855	+	0.999962i	$0.497218\pi$
$390$	4.66321	0.236131
$391$	−39.9894	−2.02235
$392$	−2.44598	−0.123541
$393$	2.05679	0.103751
$394$	−58.0125	−2.92263
$395$	15.8361	0.796802
$396$	19.3954	0.974654
$397$	−27.3900	−1.37467	−0.687333	−	0.726342i	$-0.741219\pi$
$397$	−27.3900	−1.37467	−0.687333	+	0.726342i	$0.741219\pi$
$398$	−11.9788	−0.600444
$399$	1.61963	0.0810828
$400$	−1.84420	−0.0922100
$401$	−18.8047	−0.939062	−0.469531	−	0.882916i	$-0.655577\pi$
$401$	−18.8047	−0.939062	−0.469531	+	0.882916i	$0.655577\pi$
$402$	−2.82188	−0.140742
$403$	20.0106	0.996799
$404$	51.2858	2.55156
$405$	23.2795	1.15677
$406$	3.43558	0.170505
$407$	−2.09075	−0.103635
$408$	3.48798	0.172681
$409$	−11.0157	−0.544690	−0.272345	−	0.962200i	$-0.587799\pi$
$409$	−11.0157	−0.544690	−0.272345	+	0.962200i	$0.587799\pi$
$410$	34.4721	1.70246
$411$	0.847176	0.0417881
$412$	29.2510	1.44109
$413$	−3.93950	−0.193850
$414$	50.8717	2.50021
$415$	11.0592	0.542876
$416$	−17.1139	−0.839080
$417$	4.38499	0.214734
$418$	−28.5439	−1.39613
$419$	5.79883	0.283291	0.141646	−	0.989917i	$-0.454761\pi$
$419$	5.79883	0.283291	0.141646	+	0.989917i	$0.454761\pi$
$420$	−2.37312	−0.115796
$421$	4.80643	0.234251	0.117126	−	0.993117i	$-0.462632\pi$
$421$	4.80643	0.234251	0.117126	+	0.993117i	$0.462632\pi$
$422$	10.3832	0.505448
$423$	−6.88361	−0.334693
$424$	−18.2638	−0.886969
$425$	−14.6188	−0.709116
$426$	9.13948	0.442809
$427$	−0.0331323	−0.00160338
$428$	56.3018	2.72145
$429$	1.59000	0.0767661
$430$	36.9547	1.78211
$431$	−12.6200	−0.607882	−0.303941	−	0.952691i	$-0.598303\pi$
$431$	−12.6200	−0.607882	−0.303941	+	0.952691i	$0.598303\pi$
$432$	1.06575	0.0512759
$433$	18.1111	0.870365	0.435182	−	0.900342i	$-0.356684\pi$
$433$	18.1111	0.870365	0.435182	+	0.900342i	$0.356684\pi$
$434$	−16.7859	−0.805750
$435$	1.17211	0.0561984
$436$	18.5146	0.886689
$437$	−45.4192	−2.17270
$438$	1.65267	0.0789678
$439$	11.0072	0.525343	0.262671	−	0.964885i	$-0.415397\pi$
$439$	11.0072	0.525343	0.262671	+	0.964885i	$0.415397\pi$
$440$	14.7069	0.701122
$441$	−2.92432	−0.139253
$442$	−31.4203	−1.49451
$443$	−14.8329	−0.704730	−0.352365	−	0.935863i	$-0.614622\pi$
$443$	−14.8329	−0.704730	−0.352365	+	0.935863i	$0.614622\pi$
$444$	0.825193	0.0391619
$445$	−16.3026	−0.772819
$446$	−37.2751	−1.76503
$447$	−4.32868	−0.204739
$448$	13.0482	0.616471
$449$	−0.498037	−0.0235038	−0.0117519	−	0.999931i	$-0.503741\pi$
$449$	−0.498037	−0.0235038	−0.0117519	+	0.999931i	$0.503741\pi$
$450$	18.5970	0.876670
$451$	11.7539	0.553468
$452$	17.9953	0.846428
$453$	5.88318	0.276416
$454$	12.3853	0.581270
$455$	7.51724	0.352413
$456$	3.96158	0.185518
$457$	7.08593	0.331466	0.165733	−	0.986171i	$-0.447001\pi$
$457$	7.08593	0.331466	0.165733	+	0.986171i	$0.447001\pi$
$458$	−12.8583	−0.600831
$459$	8.44810	0.394323
$460$	66.5493	3.10288
$461$	16.5944	0.772880	0.386440	−	0.922315i	$-0.373705\pi$
$461$	16.5944	0.772880	0.386440	+	0.922315i	$0.373705\pi$
$462$	−1.33378	−0.0620529
$463$	31.5599	1.46672	0.733358	−	0.679843i	$-0.237952\pi$
$463$	31.5599	1.46672	0.733358	+	0.679843i	$0.237952\pi$
$464$	−0.996300	−0.0462521
$465$	−5.72683	−0.265575
$466$	64.3682	2.98180
$467$	−26.5820	−1.23007	−0.615034	−	0.788501i	$-0.710858\pi$
$467$	−26.5820	−1.23007	−0.615034	+	0.788501i	$0.710858\pi$
$468$	24.2488	1.12090
$469$	−4.54895	−0.210051
$470$	−14.8434	−0.684677
$471$	4.12533	0.190085
$472$	−9.63596	−0.443531
$473$	12.6003	0.579364
$474$	−3.51290	−0.161353
$475$	−16.6038	−0.761832
$476$	15.9898	0.732893
$477$	−21.8355	−0.999777
$478$	31.3110	1.43213
$479$	−18.1466	−0.829139	−0.414570	−	0.910018i	$-0.636068\pi$
$479$	−18.1466	−0.829139	−0.414570	+	0.910018i	$0.636068\pi$
$480$	4.89783	0.223554
$481$	−2.61393	−0.119185
$482$	7.00904	0.319253
$483$	−2.12232	−0.0965687
$484$	−19.6716	−0.894166
$485$	27.3525	1.24201
$486$	−16.1892	−0.734359
$487$	16.6583	0.754861	0.377431	−	0.926038i	$-0.376808\pi$
$487$	16.6583	0.754861	0.377431	+	0.926038i	$0.376808\pi$
$488$	−0.0810410	−0.00366855
$489$	−6.40586	−0.289683
$490$	−6.30585	−0.284869
$491$	17.4997	0.789751	0.394876	−	0.918735i	$-0.370788\pi$
$491$	17.4997	0.789751	0.394876	+	0.918735i	$0.370788\pi$
$492$	−4.63910	−0.209147
$493$	−7.89757	−0.355689
$494$	−35.6866	−1.60561
$495$	17.5829	0.790294
$496$	4.86783	0.218572
$497$	14.7331	0.660871
$498$	−2.45325	−0.109933
$499$	37.5384	1.68045	0.840225	−	0.542238i	$-0.182423\pi$
$499$	37.5384	1.68045	0.840225	+	0.542238i	$0.182423\pi$
$500$	−18.8035	−0.840916
$501$	−0.241820	−0.0108037
$502$	−52.0259	−2.32203
$503$	−31.0135	−1.38282	−0.691411	−	0.722462i	$-0.743010\pi$
$503$	−31.0135	−1.38282	−0.691411	+	0.722462i	$0.743010\pi$
$504$	−7.15284	−0.318613
$505$	46.4933	2.06892
$506$	37.4031	1.66277
$507$	−1.58844	−0.0705450
$508$	−16.2274	−0.719973
$509$	−20.3674	−0.902771	−0.451385	−	0.892329i	$-0.649070\pi$
$509$	−20.3674	−0.902771	−0.451385	+	0.892329i	$0.649070\pi$
$510$	8.99216	0.398179
$511$	2.66416	0.117856
$512$	−7.36214	−0.325364
$513$	9.59519	0.423638
$514$	45.1381	1.99096
$515$	26.5176	1.16850
$516$	−4.97319	−0.218932
$517$	−5.06114	−0.222589
$518$	2.19270	0.0963418
$519$	−0.773115	−0.0339360
$520$	18.3870	0.806325
$521$	27.1191	1.18811	0.594054	−	0.804425i	$-0.297526\pi$
$521$	27.1191	1.18811	0.594054	+	0.804425i	$0.297526\pi$
$522$	10.0467	0.439733
$523$	−26.8425	−1.17374	−0.586871	−	0.809681i	$-0.699640\pi$
$523$	−26.8425	−1.17374	−0.586871	+	0.809681i	$0.699640\pi$
$524$	23.0628	1.00751
$525$	−0.775848	−0.0338608
$526$	51.1137	2.22866
$527$	38.5869	1.68087
$528$	0.386788	0.0168328
$529$	36.5162	1.58766
$530$	−47.0848	−2.04523
$531$	−11.5204	−0.499941
$532$	18.1610	0.787378
$533$	14.6951	0.636516
$534$	3.61639	0.156496
$535$	51.0405	2.20668
$536$	−11.1267	−0.480599
$537$	6.28653	0.271284
$538$	−5.88255	−0.253615
$539$	−2.15009	−0.0926109
$540$	−14.0591	−0.605007
$541$	−45.2118	−1.94381	−0.971904	−	0.235376i	$-0.924368\pi$
$541$	−45.2118	−1.94381	−0.971904	+	0.235376i	$0.924368\pi$
$542$	−20.5210	−0.881452
$543$	6.85132	0.294018
$544$	−33.0012	−1.41491
$545$	16.7845	0.718968
$546$	−1.66754	−0.0713639
$547$	−33.6083	−1.43699	−0.718493	−	0.695534i	$-0.755168\pi$
$547$	−33.6083	−1.43699	−0.718493	+	0.695534i	$0.755168\pi$
$548$	9.49943	0.405795
$549$	−0.0968894	−0.00413514
$550$	13.6733	0.583033
$551$	−8.96991	−0.382131
$552$	−5.19115	−0.220950
$553$	−5.66291	−0.240811
$554$	39.0946	1.66097
$555$	0.748081	0.0317543
$556$	49.1691	2.08523
$557$	−6.49222	−0.275084	−0.137542	−	0.990496i	$-0.543920\pi$
$557$	−6.49222	−0.275084	−0.137542	+	0.990496i	$0.543920\pi$
$558$	−49.0874	−2.07804
$559$	15.7534	0.666298
$560$	1.82866	0.0772751
$561$	3.06603	0.129448
$562$	26.7293	1.12751
$563$	13.6790	0.576500	0.288250	−	0.957555i	$-0.406927\pi$
$563$	13.6790	0.576500	0.288250	+	0.957555i	$0.406927\pi$
$564$	1.99756	0.0841126
$565$	16.3137	0.686322
$566$	7.12835	0.299627
$567$	−8.32460	−0.349600
$568$	36.0370	1.51208
$569$	−0.753644	−0.0315944	−0.0157972	−	0.999875i	$-0.505029\pi$
$569$	−0.753644	−0.0315944	−0.0157972	+	0.999875i	$0.505029\pi$
$570$	10.2131	0.427781
$571$	−4.35178	−0.182116	−0.0910581	−	0.995846i	$-0.529025\pi$
$571$	−4.35178	−0.182116	−0.0910581	+	0.995846i	$0.529025\pi$
$572$	17.8288	0.745459
$573$	3.01391	0.125908
$574$	−12.3270	−0.514520
$575$	21.7571	0.907334
$576$	38.1572	1.58988
$577$	27.9504	1.16359	0.581795	−	0.813335i	$-0.302350\pi$
$577$	27.9504	1.16359	0.581795	+	0.813335i	$0.302350\pi$
$578$	−22.2545	−0.925665
$579$	−2.36616	−0.0983340
$580$	13.1429	0.545730
$581$	−3.95471	−0.164069
$582$	−6.06755	−0.251508
$583$	−16.0544	−0.664906
$584$	6.51650	0.269655
$585$	21.9828	0.908877
$586$	13.1237	0.542134
$587$	0.406205	0.0167659	0.00838295	−	0.999965i	$-0.497332\pi$
$587$	0.406205	0.0167659	0.00838295	+	0.999965i	$0.497332\pi$
$588$	0.848612	0.0349962
$589$	43.8262	1.80583
$590$	−24.8419	−1.02273
$591$	7.07751	0.291130
$592$	−0.635873	−0.0261342
$593$	43.6038	1.79059	0.895296	−	0.445472i	$-0.146964\pi$
$593$	43.6038	1.79059	0.895296	+	0.445472i	$0.146964\pi$
$594$	−7.90172	−0.324211
$595$	14.4956	0.594263
$596$	−48.5377	−1.98818
$597$	1.46141	0.0598117
$598$	46.7627	1.91227
$599$	−34.1271	−1.39440	−0.697198	−	0.716879i	$-0.745570\pi$
$599$	−34.1271	−1.39440	−0.697198	+	0.716879i	$0.745570\pi$
$600$	−1.89771	−0.0774737
$601$	17.0991	0.697486	0.348743	−	0.937218i	$-0.386609\pi$
$601$	17.0991	0.697486	0.348743	+	0.937218i	$0.386609\pi$
$602$	−13.2148	−0.538594
$603$	−13.3026	−0.541723
$604$	65.9684	2.68422
$605$	−17.8334	−0.725030
$606$	−10.3135	−0.418958
$607$	21.1326	0.857747	0.428873	−	0.903365i	$-0.358911\pi$
$607$	21.1326	0.857747	0.428873	+	0.903365i	$0.358911\pi$
$608$	−37.4821	−1.52010
$609$	−0.419140	−0.0169844
$610$	−0.208927	−0.00845921
$611$	−6.32761	−0.255988
$612$	46.7594	1.89014
$613$	45.6694	1.84457	0.922285	−	0.386510i	$-0.126319\pi$
$613$	45.6694	1.84457	0.922285	+	0.386510i	$0.126319\pi$
$614$	17.7175	0.715021
$615$	−4.20559	−0.169586
$616$	−5.25909	−0.211895
$617$	−21.2722	−0.856386	−0.428193	−	0.903687i	$-0.640850\pi$
$617$	−21.2722	−0.856386	−0.428193	+	0.903687i	$0.640850\pi$
$618$	−5.88235	−0.236623
$619$	12.2730	0.493295	0.246648	−	0.969105i	$-0.420671\pi$
$619$	12.2730	0.493295	0.246648	+	0.969105i	$0.420671\pi$
$620$	−64.2152	−2.57894
$621$	−12.5733	−0.504548
$622$	−27.7758	−1.11371
$623$	5.82973	0.233563
$624$	0.483577	0.0193586
$625$	−31.1475	−1.24590
$626$	−10.4250	−0.416665
$627$	3.48234	0.139071
$628$	46.2575	1.84588
$629$	−5.04050	−0.200978
$630$	−18.4403	−0.734679
$631$	−21.6158	−0.860511	−0.430256	−	0.902707i	$-0.641577\pi$
$631$	−21.6158	−0.860511	−0.430256	+	0.902707i	$0.641577\pi$
$632$	−13.8514	−0.550978
$633$	−1.26675	−0.0503489
$634$	−52.1012	−2.06920
$635$	−14.7110	−0.583787
$636$	6.33646	0.251257
$637$	−2.68812	−0.106507
$638$	7.38680	0.292446
$639$	43.0844	1.70439
$640$	46.6726	1.84490
$641$	−4.15457	−0.164096	−0.0820478	−	0.996628i	$-0.526146\pi$
$641$	−4.15457	−0.164096	−0.0820478	+	0.996628i	$0.526146\pi$
$642$	−11.3222	−0.446853
$643$	15.5987	0.615154	0.307577	−	0.951523i	$-0.400482\pi$
$643$	15.5987	0.615154	0.307577	+	0.951523i	$0.400482\pi$
$644$	−23.7976	−0.937759
$645$	−4.50846	−0.177520
$646$	−68.8151	−2.70749
$647$	−18.2033	−0.715646	−0.357823	−	0.933789i	$-0.616481\pi$
$647$	−18.2033	−0.715646	−0.357823	+	0.933789i	$0.616481\pi$
$648$	−20.3618	−0.799889
$649$	−8.47029	−0.332488
$650$	17.0949	0.670517
$651$	2.04788	0.0802627
$652$	−71.8293	−2.81305
$653$	1.07859	0.0422086	0.0211043	−	0.999777i	$-0.493282\pi$
$653$	1.07859	0.0422086	0.0211043	+	0.999777i	$0.493282\pi$
$654$	−3.72327	−0.145591
$655$	20.9077	0.816931
$656$	3.57477	0.139571
$657$	7.79086	0.303950
$658$	5.30793	0.206925
$659$	24.8985	0.969909	0.484954	−	0.874539i	$-0.338836\pi$
$659$	24.8985	0.969909	0.484954	+	0.874539i	$0.338836\pi$
$660$	−5.10241	−0.198611
$661$	−6.14054	−0.238839	−0.119420	−	0.992844i	$-0.538103\pi$
$661$	−6.14054	−0.238839	−0.119420	+	0.992844i	$0.538103\pi$
$662$	12.2287	0.475280
$663$	3.83327	0.148872
$664$	−9.67316	−0.375392
$665$	16.4639	0.638442
$666$	6.41216	0.248466
$667$	11.7539	0.455114
$668$	−2.71154	−0.104913
$669$	4.54756	0.175819
$670$	−28.6850	−1.10820
$671$	−0.0712374	−0.00275009
$672$	−1.75144	−0.0675631
$673$	16.8207	0.648390	0.324195	−	0.945990i	$-0.394907\pi$
$673$	16.8207	0.648390	0.324195	+	0.945990i	$0.394907\pi$
$674$	−9.50925	−0.366283
$675$	−4.59637	−0.176914
$676$	−17.8112	−0.685048
$677$	34.7896	1.33707	0.668537	−	0.743679i	$-0.266921\pi$
$677$	34.7896	1.33707	0.668537	+	0.743679i	$0.266921\pi$
$678$	−3.61884	−0.138981
$679$	−9.78109	−0.375364
$680$	35.4561	1.35968
$681$	−1.51100	−0.0579017
$682$	−36.0913	−1.38201
$683$	24.4610	0.935972	0.467986	−	0.883736i	$-0.344980\pi$
$683$	24.4610	0.935972	0.467986	+	0.883736i	$0.344980\pi$
$684$	53.1085	2.03065
$685$	8.61173	0.329037
$686$	2.25493	0.0860938
$687$	1.56872	0.0598502
$688$	3.83222	0.146102
$689$	−20.0718	−0.764675
$690$	−13.3830	−0.509482
$691$	33.7025	1.28210	0.641052	−	0.767497i	$-0.278498\pi$
$691$	33.7025	1.28210	0.641052	+	0.767497i	$0.278498\pi$
$692$	−8.66897	−0.329545
$693$	−6.28755	−0.238844
$694$	−54.0615	−2.05215
$695$	44.5744	1.69080
$696$	−1.02521	−0.0388604
$697$	28.3369	1.07334
$698$	−43.5295	−1.64761
$699$	−7.85291	−0.297024
$700$	−8.69962	−0.328815
$701$	39.1725	1.47952	0.739762	−	0.672868i	$-0.234938\pi$
$701$	39.1725	1.47952	0.739762	+	0.672868i	$0.234938\pi$
$702$	−9.87901	−0.372859
$703$	−5.72491	−0.215919
$704$	28.0549	1.05736
$705$	1.81090	0.0682023
$706$	−2.97455	−0.111949
$707$	−16.6257	−0.625275
$708$	3.34311	0.125642
$709$	−10.9982	−0.413045	−0.206523	−	0.978442i	$-0.566215\pi$
$709$	−10.9982	−0.413045	−0.206523	+	0.978442i	$0.566215\pi$
$710$	92.9049	3.48666
$711$	−16.5602	−0.621054
$712$	14.2594	0.534395
$713$	−57.4287	−2.15072
$714$	−3.21554	−0.120339
$715$	16.1627	0.604452
$716$	70.4912	2.63438
$717$	−3.81993	−0.142658
$718$	42.3429	1.58022
$719$	13.2016	0.492338	0.246169	−	0.969227i	$-0.420828\pi$
$719$	13.2016	0.492338	0.246169	+	0.969227i	$0.420828\pi$
$720$	5.34760	0.199293
$721$	−9.48253	−0.353148
$722$	−35.3152	−1.31429
$723$	−0.855101	−0.0318015
$724$	76.8242	2.85515
$725$	4.29685	0.159581
$726$	3.95595	0.146819
$727$	−4.55333	−0.168874	−0.0844368	−	0.996429i	$-0.526909\pi$
$727$	−4.55333	−0.168874	−0.0844368	+	0.996429i	$0.526909\pi$
$728$	−6.57510	−0.243689
$729$	−22.9987	−0.851804
$730$	16.7998	0.621788
$731$	30.3776	1.12356
$732$	0.0281165	0.00103921
$733$	−43.2084	−1.59594	−0.797970	−	0.602698i	$-0.794092\pi$
$733$	−43.2084	−1.59594	−0.797970	+	0.602698i	$0.794092\pi$
$734$	−8.77454	−0.323874
$735$	0.769312	0.0283765
$736$	49.1156	1.81042
$737$	−9.78066	−0.360275
$738$	−36.0482	−1.32695
$739$	30.3605	1.11683	0.558414	−	0.829563i	$-0.311410\pi$
$739$	30.3605	1.11683	0.558414	+	0.829563i	$0.311410\pi$
$740$	8.38827	0.308359
$741$	4.35375	0.159939
$742$	16.8373	0.618115
$743$	20.2042	0.741221	0.370610	−	0.928788i	$-0.379149\pi$
$743$	20.2042	0.741221	0.370610	+	0.928788i	$0.379149\pi$
$744$	5.00908	0.183642
$745$	−44.0020	−1.61211
$746$	49.0853	1.79714
$747$	−11.5648	−0.423135
$748$	34.3796	1.25704
$749$	−18.2518	−0.666906
$750$	3.78136	0.138076
$751$	−16.9359	−0.617998	−0.308999	−	0.951062i	$-0.599994\pi$
$751$	−16.9359	−0.617998	−0.308999	+	0.951062i	$0.599994\pi$
$752$	−1.53927	−0.0561315
$753$	6.34715	0.231303
$754$	9.23524	0.336328
$755$	59.8039	2.17649
$756$	5.02745	0.182847
$757$	12.8297	0.466304	0.233152	−	0.972440i	$-0.425096\pi$
$757$	12.8297	0.466304	0.233152	+	0.972440i	$0.425096\pi$
$758$	48.9575	1.77822
$759$	−4.56317	−0.165633
$760$	40.2704	1.46076
$761$	18.2142	0.660265	0.330133	−	0.943935i	$-0.392907\pi$
$761$	18.2142	0.660265	0.330133	+	0.943935i	$0.392907\pi$
$762$	3.26331	0.118217
$763$	−6.00203	−0.217288
$764$	33.7951	1.22266
$765$	42.3899	1.53261
$766$	53.7117	1.94068
$767$	−10.5899	−0.382378
$768$	−3.17414	−0.114537
$769$	−11.9286	−0.430157	−0.215078	−	0.976597i	$-0.569001\pi$
$769$	−11.9286	−0.430157	−0.215078	+	0.976597i	$0.569001\pi$
$770$	−13.5581	−0.488601
$771$	−5.50684	−0.198324
$772$	−26.5318	−0.954901
$773$	−11.3628	−0.408692	−0.204346	−	0.978899i	$-0.565507\pi$
$773$	−11.3628	−0.408692	−0.204346	+	0.978899i	$0.565507\pi$
$774$	−38.6442	−1.38904
$775$	−20.9940	−0.754127
$776$	−23.9244	−0.858835
$777$	−0.267509	−0.00959684
$778$	−0.777281	−0.0278669
$779$	32.1845	1.15313
$780$	−6.37922	−0.228413
$781$	31.6776	1.13351
$782$	90.1735	3.22460
$783$	−2.48312	−0.0887393
$784$	−0.653919	−0.0233543
$785$	41.9349	1.49672
$786$	−4.63792	−0.165429
$787$	28.9377	1.03152	0.515758	−	0.856734i	$-0.327510\pi$
$787$	28.9377	1.03152	0.515758	+	0.856734i	$0.327510\pi$
$788$	79.3605	2.82710
$789$	−6.23586	−0.222002
$790$	−35.7094	−1.27048
$791$	−5.83368	−0.207422
$792$	−15.3792	−0.546478
$793$	−0.0890635	−0.00316274
$794$	61.7627	2.19188
$795$	5.74434	0.203731
$796$	16.3869	0.580818
$797$	21.6035	0.765235	0.382617	−	0.923907i	$-0.375023\pi$
$797$	21.6035	0.765235	0.382617	+	0.923907i	$0.375023\pi$
$798$	−3.65215	−0.129285
$799$	−12.2017	−0.431664
$800$	17.9550	0.634805
$801$	17.0480	0.602361
$802$	42.4034	1.49732
$803$	5.72819	0.202143
$804$	3.86030	0.136142
$805$	−21.5738	−0.760377
$806$	−45.1226	−1.58938
$807$	0.717669	0.0252632
$808$	−40.6663	−1.43063
$809$	30.1322	1.05939	0.529696	−	0.848188i	$-0.322306\pi$
$809$	30.1322	1.05939	0.529696	+	0.848188i	$0.322306\pi$
$810$	−52.4937	−1.84444
$811$	38.1079	1.33815	0.669076	−	0.743194i	$-0.266690\pi$
$811$	38.1079	1.33815	0.669076	+	0.743194i	$0.266690\pi$
$812$	−4.69983	−0.164932
$813$	2.50356	0.0878035
$814$	4.71451	0.165243
$815$	−65.1170	−2.28095
$816$	0.932491	0.0326437
$817$	34.5023	1.20708
$818$	24.8396	0.868497
$819$	−7.86092	−0.274683
$820$	−47.1575	−1.64681
$821$	−50.1768	−1.75118	−0.875591	−	0.483052i	$-0.839528\pi$
$821$	−50.1768	−1.75118	−0.875591	+	0.483052i	$0.839528\pi$
$822$	−1.91033	−0.0666302
$823$	9.74108	0.339553	0.169776	−	0.985483i	$-0.445695\pi$
$823$	9.74108	0.339553	0.169776	+	0.985483i	$0.445695\pi$
$824$	−23.1941	−0.808005
$825$	−1.66814	−0.0580773
$826$	8.88332	0.309090
$827$	−25.9467	−0.902254	−0.451127	−	0.892460i	$-0.648978\pi$
$827$	−25.9467	−0.902254	−0.451127	+	0.892460i	$0.648978\pi$
$828$	−69.5919	−2.41849
$829$	−3.32563	−0.115504	−0.0577519	−	0.998331i	$-0.518393\pi$
$829$	−3.32563	−0.115504	−0.0577519	+	0.998331i	$0.518393\pi$
$830$	−24.9378	−0.865604
$831$	−4.76953	−0.165453
$832$	35.0752	1.21601
$833$	−5.18356	−0.179600
$834$	−9.88786	−0.342389
$835$	−2.45815	−0.0850678
$836$	39.0477	1.35049
$837$	12.1323	0.419353
$838$	−13.0760	−0.451702
$839$	−5.04595	−0.174205	−0.0871027	−	0.996199i	$-0.527761\pi$
$839$	−5.04595	−0.174205	−0.0871027	+	0.996199i	$0.527761\pi$
$840$	1.88172	0.0649257
$841$	−26.6787	−0.919955
$842$	−10.8382	−0.373509
$843$	−3.26097	−0.112314
$844$	−14.2042	−0.488928
$845$	−16.1468	−0.555468
$846$	15.5221	0.533660
$847$	6.37711	0.219120
$848$	−4.88272	−0.167673
$849$	−0.869657	−0.0298466
$850$	32.9644	1.13067
$851$	7.50177	0.257157
$852$	−12.5027	−0.428336
$853$	−50.5711	−1.73152	−0.865761	−	0.500458i	$-0.833165\pi$
$853$	−50.5711	−1.73152	−0.865761	+	0.500458i	$0.833165\pi$
$854$	0.0747111	0.00255656
$855$	48.1456	1.64655
$856$	−44.6436	−1.52589
$857$	31.3686	1.07153	0.535766	−	0.844367i	$-0.320023\pi$
$857$	31.3686	1.07153	0.535766	+	0.844367i	$0.320023\pi$
$858$	−3.58535	−0.122402
$859$	38.5218	1.31435	0.657174	−	0.753739i	$-0.271752\pi$
$859$	38.5218	1.31435	0.657174	+	0.753739i	$0.271752\pi$
$860$	−50.5536	−1.72386
$861$	1.50389	0.0512526
$862$	28.4572	0.969255
$863$	1.00000	0.0340404
$863$	1.00000	0.0340404
$864$	−10.3761	−0.353001
$865$	−7.85888	−0.267210
$866$	−40.8394	−1.38778
$867$	2.71504	0.0922077
$868$	22.9630	0.779414
$869$	−12.1758	−0.413034
$870$	−2.64303	−0.0896071
$871$	−12.2281	−0.414334
$872$	−14.6809	−0.497157
$873$	−28.6030	−0.968065
$874$	102.417	3.46432
$875$	6.09567	0.206071
$876$	−2.26084	−0.0763867
$877$	−50.6750	−1.71117	−0.855587	−	0.517659i	$-0.826803\pi$
$877$	−50.6750	−1.71117	−0.855587	+	0.517659i	$0.826803\pi$
$878$	−24.8204	−0.837648
$879$	−1.60108	−0.0540033
$880$	3.93179	0.132541
$881$	−33.3529	−1.12369	−0.561844	−	0.827243i	$-0.689908\pi$
$881$	−33.3529	−1.12369	−0.561844	+	0.827243i	$0.689908\pi$
$882$	6.59415	0.222036
$883$	30.5756	1.02895	0.514475	−	0.857505i	$-0.327987\pi$
$883$	30.5756	1.02895	0.514475	+	0.857505i	$0.327987\pi$
$884$	42.9826	1.44566
$885$	3.03071	0.101876
$886$	33.4471	1.12368
$887$	−5.10504	−0.171410	−0.0857052	−	0.996321i	$-0.527314\pi$
$887$	−5.10504	−0.171410	−0.0857052	+	0.996321i	$0.527314\pi$
$888$	−0.654323	−0.0219577
$889$	5.26055	0.176433
$890$	36.7614	1.23224
$891$	−17.8986	−0.599627
$892$	50.9920	1.70734
$893$	−13.8584	−0.463754
$894$	9.76088	0.326453
$895$	63.9040	2.13608
$896$	−16.6899	−0.557569
$897$	−5.70504	−0.190486
$898$	1.12304	0.0374763
$899$	−11.3417	−0.378266
$900$	−25.4405	−0.848016
$901$	−38.7049	−1.28945
$902$	−26.5042	−0.882494
$903$	1.61220	0.0536506
$904$	−14.2691	−0.474583
$905$	69.6452	2.31508
$906$	−13.2662	−0.440740
$907$	36.1876	1.20159	0.600795	−	0.799403i	$-0.294851\pi$
$907$	36.1876	1.20159	0.600795	+	0.799403i	$0.294851\pi$
$908$	−16.9429	−0.562271
$909$	−48.6189	−1.61259
$910$	−16.9509	−0.561916
$911$	−47.9333	−1.58810	−0.794050	−	0.607853i	$-0.792031\pi$
$911$	−47.9333	−1.58810	−0.794050	+	0.607853i	$0.792031\pi$
$912$	1.05911	0.0350705
$913$	−8.50299	−0.281408
$914$	−15.9783	−0.528515
$915$	0.0254890	0.000842642 0
$916$	17.5901	0.581193
$917$	−7.47647	−0.246895
$918$	−19.0499	−0.628740
$919$	−29.7853	−0.982528	−0.491264	−	0.871011i	$-0.663465\pi$
$919$	−29.7853	−0.982528	−0.491264	+	0.871011i	$0.663465\pi$
$920$	−52.7692	−1.73975
$921$	−2.16153	−0.0712250
$922$	−37.4194	−1.23234
$923$	39.6044	1.30360
$924$	1.82459	0.0600247
$925$	2.74239	0.0901694
$926$	−71.1656	−2.33865
$927$	−27.7299	−0.910771
$928$	9.69991	0.318415
$929$	34.5563	1.13376	0.566878	−	0.823802i	$-0.308151\pi$
$929$	34.5563	1.13376	0.566878	+	0.823802i	$0.308151\pi$
$930$	12.9136	0.423454
$931$	−5.88738	−0.192951
$932$	−88.0551	−2.88434
$933$	3.38864	0.110939
$934$	59.9406	1.96132
$935$	31.1669	1.01927
$936$	−19.2277	−0.628477
$937$	2.19655	0.0717582	0.0358791	−	0.999356i	$-0.488577\pi$
$937$	2.19655	0.0717582	0.0358791	+	0.999356i	$0.488577\pi$
$938$	10.2576	0.334922
$939$	1.27184	0.0415050
$940$	20.3057	0.662298
$941$	−25.3365	−0.825946	−0.412973	−	0.910743i	$-0.635510\pi$
$941$	−25.3365	−0.825946	−0.412973	+	0.910743i	$0.635510\pi$
$942$	−9.30234	−0.303087
$943$	−42.1737	−1.37337
$944$	−2.57612	−0.0838455
$945$	4.55765	0.148260
$946$	−28.4129	−0.923784
$947$	7.74188	0.251577	0.125789	−	0.992057i	$-0.459854\pi$
$947$	7.74188	0.251577	0.125789	+	0.992057i	$0.459854\pi$
$948$	4.80561	0.156079
$949$	7.16159	0.232475
$950$	37.4404	1.21473
$951$	6.35633	0.206118
$952$	−12.6789	−0.410925
$953$	−26.6801	−0.864254	−0.432127	−	0.901813i	$-0.642237\pi$
$953$	−26.6801	−0.864254	−0.432127	+	0.901813i	$0.642237\pi$
$954$	49.2375	1.59412
$955$	30.6371	0.991392
$956$	−42.8331	−1.38532
$957$	−0.901188	−0.0291313
$958$	40.9194	1.32204
$959$	−3.07950	−0.0994424
$960$	−10.0382	−0.323980
$961$	24.4145	0.787564
$962$	5.89425	0.190038
$963$	−53.3741	−1.71996
$964$	−9.58829	−0.308818
$965$	−24.0525	−0.774277
$966$	4.78568	0.153977
$967$	3.23380	0.103992	0.0519960	−	0.998647i	$-0.483442\pi$
$967$	3.23380	0.103992	0.0519960	+	0.998647i	$0.483442\pi$
$968$	15.5983	0.501349
$969$	8.39543	0.269700
$970$	−61.6780	−1.98036
$971$	−12.5599	−0.403066	−0.201533	−	0.979482i	$-0.564592\pi$
$971$	−12.5599	−0.403066	−0.201533	+	0.979482i	$0.564592\pi$
$972$	22.1467	0.710356
$973$	−15.9395	−0.510998
$974$	−37.5634	−1.20361
$975$	−2.08557	−0.0667917
$976$	−0.0216658	−0.000693506 0
$977$	9.32316	0.298274	0.149137	−	0.988817i	$-0.452350\pi$
$977$	9.32316	0.298274	0.149137	+	0.988817i	$0.452350\pi$
$978$	14.4448	0.461894
$979$	12.5344	0.400603
$980$	8.62633	0.275558
$981$	−17.5518	−0.560387
$982$	−39.4607	−1.25924
$983$	−36.4941	−1.16398	−0.581990	−	0.813196i	$-0.697726\pi$
$983$	−36.4941	−1.16398	−0.581990	+	0.813196i	$0.697726\pi$
$984$	3.67850	0.117266
$985$	71.9445	2.29234
$986$	17.8085	0.567138
$987$	−0.647566	−0.0206123
$988$	48.8188	1.55313
$989$	−45.2109	−1.43762
$990$	−39.6483	−1.26011
$991$	55.6711	1.76845	0.884226	−	0.467060i	$-0.154687\pi$
$991$	55.6711	1.76845	0.884226	+	0.467060i	$0.154687\pi$
$992$	−47.3929	−1.50473
$993$	−1.49189	−0.0473438
$994$	−33.2222	−1.05375
$995$	14.8556	0.470954
$996$	3.35602	0.106339
$997$	−21.2222	−0.672115	−0.336057	−	0.941842i	$-0.609094\pi$
$997$	−21.2222	−0.672115	−0.336057	+	0.941842i	$0.609094\pi$
$998$	−84.6466	−2.67944
$999$	−1.58481	−0.0501412

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6041.2.a.e.1.16	✓	112

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6041.2.a.e.1.16	✓	112	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:	\( N \)	\(=\)	\( 6041 = 7 \cdot 863 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6041.a (trivial)

\(f(q)\)	\(=\)	\(q-2.25493 q^{2} +0.275101 q^{3} +3.08473 q^{4} +2.79647 q^{5} -0.620335 q^{6} -1.00000 q^{7} -2.44598 q^{8} -2.92432 q^{9} +O(q^{10})\)	\(q-2.25493 q^{2} +0.275101 q^{3} +3.08473 q^{4} +2.79647 q^{5} -0.620335 q^{6} -1.00000 q^{7} -2.44598 q^{8} -2.92432 q^{9} -6.30585 q^{10} -2.15009 q^{11} +0.848612 q^{12} -2.68812 q^{13} +2.25493 q^{14} +0.769312 q^{15} -0.653919 q^{16} -5.18356 q^{17} +6.59415 q^{18} -5.88738 q^{19} +8.62633 q^{20} -0.275101 q^{21} +4.84831 q^{22} +7.71467 q^{23} -0.672894 q^{24} +2.82023 q^{25} +6.06153 q^{26} -1.62979 q^{27} -3.08473 q^{28} +1.52358 q^{29} -1.73475 q^{30} -7.44409 q^{31} +6.36651 q^{32} -0.591493 q^{33} +11.6886 q^{34} -2.79647 q^{35} -9.02072 q^{36} +0.972403 q^{37} +13.2757 q^{38} -0.739505 q^{39} -6.84011 q^{40} -5.46669 q^{41} +0.620335 q^{42} -5.86038 q^{43} -6.63244 q^{44} -8.17776 q^{45} -17.3961 q^{46} +2.35392 q^{47} -0.179894 q^{48} +1.00000 q^{49} -6.35942 q^{50} -1.42600 q^{51} -8.29211 q^{52} +7.46686 q^{53} +3.67506 q^{54} -6.01265 q^{55} +2.44598 q^{56} -1.61963 q^{57} -3.43558 q^{58} +3.93950 q^{59} +2.37312 q^{60} +0.0331323 q^{61} +16.7859 q^{62} +2.92432 q^{63} -13.0482 q^{64} -7.51724 q^{65} +1.33378 q^{66} +4.54895 q^{67} -15.9898 q^{68} +2.12232 q^{69} +6.30585 q^{70} -14.7331 q^{71} +7.15284 q^{72} -2.66416 q^{73} -2.19270 q^{74} +0.775848 q^{75} -18.1610 q^{76} +2.15009 q^{77} +1.66754 q^{78} +5.66291 q^{79} -1.82866 q^{80} +8.32460 q^{81} +12.3270 q^{82} +3.95471 q^{83} -0.848612 q^{84} -14.4956 q^{85} +13.2148 q^{86} +0.419140 q^{87} +5.25909 q^{88} -5.82973 q^{89} +18.4403 q^{90} +2.68812 q^{91} +23.7976 q^{92} -2.04788 q^{93} -5.30793 q^{94} -16.4639 q^{95} +1.75144 q^{96} +9.78109 q^{97} -2.25493 q^{98} +6.28755 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 3 q^{2} + 14 q^{3} + 131 q^{4} + 13 q^{5} + 18 q^{6} - 112 q^{7} - 9 q^{8} + 116 q^{9}+O(q^{10}) \) 112 * q - 3 * q^2 + 14 * q^3 + 131 * q^4 + 13 * q^5 + 18 * q^6 - 112 * q^7 - 9 * q^8 + 116 * q^9	\( 112 q - 3 q^{2} + 14 q^{3} + 131 q^{4} + 13 q^{5} + 18 q^{6} - 112 q^{7} - 9 q^{8} + 116 q^{9} + 32 q^{10} + 14 q^{11} + 36 q^{12} + 22 q^{13} + 3 q^{14} + 19 q^{15} + 169 q^{16} + 11 q^{17} - 18 q^{18} + 52 q^{19} + 40 q^{20} - 14 q^{21} + 16 q^{22} + 38 q^{23} + 64 q^{24} + 99 q^{25} + 45 q^{26} + 65 q^{27} - 131 q^{28} + 10 q^{29} + q^{30} + 133 q^{31} - 26 q^{32} + 27 q^{33} + 52 q^{34} - 13 q^{35} + 183 q^{36} - 13 q^{37} + 20 q^{38} + 74 q^{39} + 92 q^{40} + 25 q^{41} - 18 q^{42} - 11 q^{43} + 16 q^{44} + 63 q^{45} + 28 q^{46} + 71 q^{47} + 70 q^{48} + 112 q^{49} + 5 q^{50} + 57 q^{51} + 79 q^{52} - 10 q^{53} + 75 q^{54} + 146 q^{55} + 9 q^{56} - 83 q^{57} - 19 q^{58} + 56 q^{59} - 3 q^{60} + 80 q^{61} + 42 q^{62} - 116 q^{63} + 263 q^{64} - 26 q^{65} + 48 q^{66} + 29 q^{67} + 57 q^{68} + 56 q^{69} - 32 q^{70} + 100 q^{71} - 62 q^{72} + 73 q^{73} + 24 q^{74} + 89 q^{75} + 155 q^{76} - 14 q^{77} + 33 q^{78} + 140 q^{79} + 80 q^{80} + 120 q^{81} + 114 q^{82} + 36 q^{83} - 36 q^{84} - 2 q^{85} + 12 q^{86} + 96 q^{87} + 29 q^{88} + 47 q^{89} + 52 q^{90} - 22 q^{91} + 81 q^{92} - 10 q^{93} + 127 q^{94} + 96 q^{95} + 175 q^{96} + 80 q^{97} - 3 q^{98} + 74 q^{99}+O(q^{100}) \) 112 * q - 3 * q^2 + 14 * q^3 + 131 * q^4 + 13 * q^5 + 18 * q^6 - 112 * q^7 - 9 * q^8 + 116 * q^9 + 32 * q^10 + 14 * q^11 + 36 * q^12 + 22 * q^13 + 3 * q^14 + 19 * q^15 + 169 * q^16 + 11 * q^17 - 18 * q^18 + 52 * q^19 + 40 * q^20 - 14 * q^21 + 16 * q^22 + 38 * q^23 + 64 * q^24 + 99 * q^25 + 45 * q^26 + 65 * q^27 - 131 * q^28 + 10 * q^29 + q^30 + 133 * q^31 - 26 * q^32 + 27 * q^33 + 52 * q^34 - 13 * q^35 + 183 * q^36 - 13 * q^37 + 20 * q^38 + 74 * q^39 + 92 * q^40 + 25 * q^41 - 18 * q^42 - 11 * q^43 + 16 * q^44 + 63 * q^45 + 28 * q^46 + 71 * q^47 + 70 * q^48 + 112 * q^49 + 5 * q^50 + 57 * q^51 + 79 * q^52 - 10 * q^53 + 75 * q^54 + 146 * q^55 + 9 * q^56 - 83 * q^57 - 19 * q^58 + 56 * q^59 - 3 * q^60 + 80 * q^61 + 42 * q^62 - 116 * q^63 + 263 * q^64 - 26 * q^65 + 48 * q^66 + 29 * q^67 + 57 * q^68 + 56 * q^69 - 32 * q^70 + 100 * q^71 - 62 * q^72 + 73 * q^73 + 24 * q^74 + 89 * q^75 + 155 * q^76 - 14 * q^77 + 33 * q^78 + 140 * q^79 + 80 * q^80 + 120 * q^81 + 114 * q^82 + 36 * q^83 - 36 * q^84 - 2 * q^85 + 12 * q^86 + 96 * q^87 + 29 * q^88 + 47 * q^89 + 52 * q^90 - 22 * q^91 + 81 * q^92 - 10 * q^93 + 127 * q^94 + 96 * q^95 + 175 * q^96 + 80 * q^97 - 3 * q^98 + 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more