LMFDB - Embedded newform 8049.2.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8049,2,Mod(1,8049)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8049, 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("8049.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8049 = 3 \cdot 2683 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8049.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:	$64.2715885869$
Analytic rank:	$1$
Dimension:	$104$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68757 q^{2} -1.00000 q^{3} +5.22302 q^{4} +2.72258 q^{5} +2.68757 q^{6} -1.36352 q^{7} -8.66207 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.68757 q^{2} -1.00000 q^{3} +5.22302 q^{4} +2.72258 q^{5} +2.68757 q^{6} -1.36352 q^{7} -8.66207 q^{8} +1.00000 q^{9} -7.31712 q^{10} -2.97086 q^{11} -5.22302 q^{12} +5.64512 q^{13} +3.66456 q^{14} -2.72258 q^{15} +12.8339 q^{16} +3.24798 q^{17} -2.68757 q^{18} +0.763077 q^{19} +14.2201 q^{20} +1.36352 q^{21} +7.98438 q^{22} -1.93376 q^{23} +8.66207 q^{24} +2.41244 q^{25} -15.1716 q^{26} -1.00000 q^{27} -7.12171 q^{28} +0.0377820 q^{29} +7.31712 q^{30} +6.48532 q^{31} -17.1677 q^{32} +2.97086 q^{33} -8.72917 q^{34} -3.71230 q^{35} +5.22302 q^{36} -1.42045 q^{37} -2.05082 q^{38} -5.64512 q^{39} -23.5832 q^{40} -3.12949 q^{41} -3.66456 q^{42} -11.0489 q^{43} -15.5168 q^{44} +2.72258 q^{45} +5.19710 q^{46} +1.30173 q^{47} -12.8339 q^{48} -5.14080 q^{49} -6.48360 q^{50} -3.24798 q^{51} +29.4846 q^{52} -7.18369 q^{53} +2.68757 q^{54} -8.08840 q^{55} +11.8109 q^{56} -0.763077 q^{57} -0.101542 q^{58} -10.0409 q^{59} -14.2201 q^{60} +0.969751 q^{61} -17.4297 q^{62} -1.36352 q^{63} +20.4717 q^{64} +15.3693 q^{65} -7.98438 q^{66} +7.67453 q^{67} +16.9643 q^{68} +1.93376 q^{69} +9.97707 q^{70} -15.0749 q^{71} -8.66207 q^{72} +10.7120 q^{73} +3.81756 q^{74} -2.41244 q^{75} +3.98556 q^{76} +4.05084 q^{77} +15.1716 q^{78} -11.8498 q^{79} +34.9412 q^{80} +1.00000 q^{81} +8.41071 q^{82} +1.29760 q^{83} +7.12171 q^{84} +8.84289 q^{85} +29.6947 q^{86} -0.0377820 q^{87} +25.7338 q^{88} +15.0291 q^{89} -7.31712 q^{90} -7.69726 q^{91} -10.1000 q^{92} -6.48532 q^{93} -3.49850 q^{94} +2.07754 q^{95} +17.1677 q^{96} -0.766873 q^{97} +13.8162 q^{98} -2.97086 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9}+O(q^{10}) $ 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9} + 8 q^{10} - 52 q^{11} - 87 q^{12} + 35 q^{13} - 23 q^{14} + 15 q^{15} + 53 q^{16} - 19 q^{17} - 9 q^{18} - 22 q^{19} - 35 q^{20} + 10 q^{21} - q^{22} - 70 q^{23} + 27 q^{24} + 79 q^{25} - 39 q^{26} - 104 q^{27} - 9 q^{28} - 37 q^{29} - 8 q^{30} - 47 q^{31} - 53 q^{32} + 52 q^{33} - 17 q^{34} - 54 q^{35} + 87 q^{36} + 65 q^{37} - 33 q^{38} - 35 q^{39} + 14 q^{40} - 47 q^{41} + 23 q^{42} - 30 q^{43} - 122 q^{44} - 15 q^{45} - 6 q^{46} - 101 q^{47} - 53 q^{48} + 78 q^{49} - 64 q^{50} + 19 q^{51} + 41 q^{52} - 48 q^{53} + 9 q^{54} - 29 q^{55} - 71 q^{56} + 22 q^{57} - 2 q^{58} - 86 q^{59} + 35 q^{60} + 34 q^{61} - 36 q^{62} - 10 q^{63} - 15 q^{64} - 64 q^{65} + q^{66} - 38 q^{67} - 33 q^{68} + 70 q^{69} - 29 q^{70} - 176 q^{71} - 27 q^{72} + 69 q^{73} - 86 q^{74} - 79 q^{75} - 54 q^{76} - 45 q^{77} + 39 q^{78} - 101 q^{79} - 76 q^{80} + 104 q^{81} + 38 q^{82} - 67 q^{83} + 9 q^{84} + 3 q^{85} - 90 q^{86} + 37 q^{87} + 7 q^{88} - 91 q^{89} + 8 q^{90} - 47 q^{91} - 136 q^{92} + 47 q^{93} - 20 q^{94} - 130 q^{95} + 53 q^{96} + 86 q^{97} - 44 q^{98} - 52 q^{99}+O(q^{100}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9 + 8 * q^10 - 52 * q^11 - 87 * q^12 + 35 * q^13 - 23 * q^14 + 15 * q^15 + 53 * q^16 - 19 * q^17 - 9 * q^18 - 22 * q^19 - 35 * q^20 + 10 * q^21 - q^22 - 70 * q^23 + 27 * q^24 + 79 * q^25 - 39 * q^26 - 104 * q^27 - 9 * q^28 - 37 * q^29 - 8 * q^30 - 47 * q^31 - 53 * q^32 + 52 * q^33 - 17 * q^34 - 54 * q^35 + 87 * q^36 + 65 * q^37 - 33 * q^38 - 35 * q^39 + 14 * q^40 - 47 * q^41 + 23 * q^42 - 30 * q^43 - 122 * q^44 - 15 * q^45 - 6 * q^46 - 101 * q^47 - 53 * q^48 + 78 * q^49 - 64 * q^50 + 19 * q^51 + 41 * q^52 - 48 * q^53 + 9 * q^54 - 29 * q^55 - 71 * q^56 + 22 * q^57 - 2 * q^58 - 86 * q^59 + 35 * q^60 + 34 * q^61 - 36 * q^62 - 10 * q^63 - 15 * q^64 - 64 * q^65 + q^66 - 38 * q^67 - 33 * q^68 + 70 * q^69 - 29 * q^70 - 176 * q^71 - 27 * q^72 + 69 * q^73 - 86 * q^74 - 79 * q^75 - 54 * q^76 - 45 * q^77 + 39 * q^78 - 101 * q^79 - 76 * q^80 + 104 * q^81 + 38 * q^82 - 67 * q^83 + 9 * q^84 + 3 * q^85 - 90 * q^86 + 37 * q^87 + 7 * q^88 - 91 * q^89 + 8 * q^90 - 47 * q^91 - 136 * q^92 + 47 * q^93 - 20 * q^94 - 130 * q^95 + 53 * q^96 + 86 * q^97 - 44 * q^98 - 52 * 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 $

$2$	−2.68757	−1.90040	−0.950198	−	0.311646i	$-0.899120\pi$
$2$	−2.68757	−1.90040	−0.950198	+	0.311646i	$0.899120\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.22302	2.61151
$5$	2.72258	1.21757	0.608787	−	0.793333i	$-0.291656\pi$
$5$	2.72258	1.21757	0.608787	+	0.793333i	$0.291656\pi$
$6$	2.68757	1.09719
$7$	−1.36352	−0.515364	−0.257682	−	0.966230i	$-0.582959\pi$
$7$	−1.36352	−0.515364	−0.257682	+	0.966230i	$0.582959\pi$
$8$	−8.66207	−3.06250
$9$	1.00000	0.333333
$10$	−7.31712	−2.31388
$11$	−2.97086	−0.895747	−0.447874	−	0.894097i	$-0.647819\pi$
$11$	−2.97086	−0.895747	−0.447874	+	0.894097i	$0.647819\pi$
$12$	−5.22302	−1.50775
$13$	5.64512	1.56568	0.782838	−	0.622226i	$-0.213772\pi$
$13$	5.64512	1.56568	0.782838	+	0.622226i	$0.213772\pi$
$14$	3.66456	0.979396
$15$	−2.72258	−0.702967
$16$	12.8339	3.20846
$17$	3.24798	0.787752	0.393876	−	0.919164i	$-0.371134\pi$
$17$	3.24798	0.787752	0.393876	+	0.919164i	$0.371134\pi$
$18$	−2.68757	−0.633466
$19$	0.763077	0.175062	0.0875309	−	0.996162i	$-0.472102\pi$
$19$	0.763077	0.175062	0.0875309	+	0.996162i	$0.472102\pi$
$20$	14.2201	3.17971
$21$	1.36352	0.297545
$22$	7.98438	1.70227
$23$	−1.93376	−0.403216	−0.201608	−	0.979466i	$-0.564617\pi$
$23$	−1.93376	−0.403216	−0.201608	+	0.979466i	$0.564617\pi$
$24$	8.66207	1.76814
$25$	2.41244	0.482489
$26$	−15.1716	−2.97540
$27$	−1.00000	−0.192450
$28$	−7.12171	−1.34588
$29$	0.0377820	0.00701594	0.00350797	−	0.999994i	$-0.498883\pi$
$29$	0.0377820	0.00701594	0.00350797	+	0.999994i	$0.498883\pi$
$30$	7.31712	1.33592
$31$	6.48532	1.16480	0.582399	−	0.812903i	$-0.302114\pi$
$31$	6.48532	1.16480	0.582399	+	0.812903i	$0.302114\pi$
$32$	−17.1677	−3.03485
$33$	2.97086	0.517160
$34$	−8.72917	−1.49704
$35$	−3.71230	−0.627494
$36$	5.22302	0.870503
$37$	−1.42045	−0.233521	−0.116760	−	0.993160i	$-0.537251\pi$
$37$	−1.42045	−0.233521	−0.116760	+	0.993160i	$0.537251\pi$
$38$	−2.05082	−0.332687
$39$	−5.64512	−0.903943
$40$	−23.5832	−3.72883
$41$	−3.12949	−0.488744	−0.244372	−	0.969682i	$-0.578582\pi$
$41$	−3.12949	−0.488744	−0.244372	+	0.969682i	$0.578582\pi$
$42$	−3.66456	−0.565454
$43$	−11.0489	−1.68494	−0.842471	−	0.538741i	$-0.818900\pi$
$43$	−11.0489	−1.68494	−0.842471	+	0.538741i	$0.818900\pi$
$44$	−15.5168	−2.33925
$45$	2.72258	0.405858
$46$	5.19710	0.766270
$47$	1.30173	0.189877	0.0949387	−	0.995483i	$-0.469734\pi$
$47$	1.30173	0.189877	0.0949387	+	0.995483i	$0.469734\pi$
$48$	−12.8339	−1.85241
$49$	−5.14080	−0.734400
$50$	−6.48360	−0.916920
$51$	−3.24798	−0.454809
$52$	29.4846	4.08877
$53$	−7.18369	−0.986756	−0.493378	−	0.869815i	$-0.664238\pi$
$53$	−7.18369	−0.986756	−0.493378	+	0.869815i	$0.664238\pi$
$54$	2.68757	0.365732
$55$	−8.08840	−1.09064
$56$	11.8109	1.57830
$57$	−0.763077	−0.101072
$58$	−0.101542	−0.0133331
$59$	−10.0409	−1.30721	−0.653606	−	0.756835i	$-0.726745\pi$
$59$	−10.0409	−1.30721	−0.653606	+	0.756835i	$0.726745\pi$
$60$	−14.2201	−1.83580
$61$	0.969751	0.124164	0.0620819	−	0.998071i	$-0.480226\pi$
$61$	0.969751	0.124164	0.0620819	+	0.998071i	$0.480226\pi$
$62$	−17.4297	−2.21358
$63$	−1.36352	−0.171788
$64$	20.4717	2.55896
$65$	15.3693	1.90633
$66$	−7.98438	−0.982809
$67$	7.67453	0.937593	0.468796	−	0.883306i	$-0.344688\pi$
$67$	7.67453	0.937593	0.468796	+	0.883306i	$0.344688\pi$
$68$	16.9643	2.05722
$69$	1.93376	0.232797
$70$	9.97707	1.19249
$71$	−15.0749	−1.78907	−0.894533	−	0.447003i	$-0.852491\pi$
$71$	−15.0749	−1.78907	−0.894533	+	0.447003i	$0.852491\pi$
$72$	−8.66207	−1.02083
$73$	10.7120	1.25375	0.626874	−	0.779121i	$-0.284334\pi$
$73$	10.7120	1.25375	0.626874	+	0.779121i	$0.284334\pi$
$74$	3.81756	0.443782
$75$	−2.41244	−0.278565
$76$	3.98556	0.457175
$77$	4.05084	0.461636
$78$	15.1716	1.71785
$79$	−11.8498	−1.33321	−0.666605	−	0.745411i	$-0.732253\pi$
$79$	−11.8498	−1.33321	−0.666605	+	0.745411i	$0.732253\pi$
$80$	34.9412	3.90655
$81$	1.00000	0.111111
$82$	8.41071	0.928807
$83$	1.29760	0.142430	0.0712152	−	0.997461i	$-0.477312\pi$
$83$	1.29760	0.142430	0.0712152	+	0.997461i	$0.477312\pi$
$84$	7.12171	0.777042
$85$	8.84289	0.959147
$86$	29.6947	3.20206
$87$	−0.0377820	−0.00405065
$88$	25.7338	2.74323
$89$	15.0291	1.59308	0.796541	−	0.604584i	$-0.206661\pi$
$89$	15.0291	1.59308	0.796541	+	0.604584i	$0.206661\pi$
$90$	−7.31712	−0.771292
$91$	−7.69726	−0.806892
$92$	−10.1000	−1.05300
$93$	−6.48532	−0.672496
$94$	−3.49850	−0.360843
$95$	2.07754	0.213151
$96$	17.1677	1.75217
$97$	−0.766873	−0.0778642	−0.0389321	−	0.999242i	$-0.512396\pi$
$97$	−0.766873	−0.0778642	−0.0389321	+	0.999242i	$0.512396\pi$
$98$	13.8162	1.39565
$99$	−2.97086	−0.298582
$100$	12.6002	1.26002
$101$	−5.55508	−0.552751	−0.276375	−	0.961050i	$-0.589133\pi$
$101$	−5.55508	−0.552751	−0.276375	+	0.961050i	$0.589133\pi$
$102$	8.72917	0.864317
$103$	8.57434	0.844855	0.422428	−	0.906397i	$-0.361178\pi$
$103$	8.57434	0.844855	0.422428	+	0.906397i	$0.361178\pi$
$104$	−48.8984	−4.79489
$105$	3.71230	0.362284
$106$	19.3067	1.87523
$107$	−1.66099	−0.160574	−0.0802869	−	0.996772i	$-0.525584\pi$
$107$	−1.66099	−0.160574	−0.0802869	+	0.996772i	$0.525584\pi$
$108$	−5.22302	−0.502585
$109$	−15.9070	−1.52361	−0.761806	−	0.647805i	$-0.775687\pi$
$109$	−15.9070	−1.52361	−0.761806	+	0.647805i	$0.775687\pi$
$110$	21.7381	2.07265
$111$	1.42045	0.134823
$112$	−17.4993	−1.65353
$113$	0.569882	0.0536100	0.0268050	−	0.999641i	$-0.491467\pi$
$113$	0.569882	0.0536100	0.0268050	+	0.999641i	$0.491467\pi$
$114$	2.05082	0.192077
$115$	−5.26480	−0.490946
$116$	0.197336	0.0183222
$117$	5.64512	0.521892
$118$	26.9855	2.48422
$119$	−4.42870	−0.405979
$120$	23.5832	2.15284
$121$	−2.17401	−0.197637
$122$	−2.60627	−0.235961
$123$	3.12949	0.282176
$124$	33.8729	3.04188
$125$	−7.04483	−0.630109
$126$	3.66456	0.326465
$127$	−14.7472	−1.30861	−0.654303	−	0.756232i	$-0.727038\pi$
$127$	−14.7472	−1.30861	−0.654303	+	0.756232i	$0.727038\pi$
$128$	−20.6835	−1.82818
$129$	11.0489	0.972802
$130$	−41.3060	−3.62278
$131$	−10.7855	−0.942330	−0.471165	−	0.882045i	$-0.656166\pi$
$131$	−10.7855	−0.942330	−0.471165	+	0.882045i	$0.656166\pi$
$132$	15.5168	1.35057
$133$	−1.04047	−0.0902206
$134$	−20.6258	−1.78180
$135$	−2.72258	−0.234322
$136$	−28.1343	−2.41249
$137$	1.29674	0.110788	0.0553939	−	0.998465i	$-0.482359\pi$
$137$	1.29674	0.110788	0.0553939	+	0.998465i	$0.482359\pi$
$138$	−5.19710	−0.442406
$139$	22.0796	1.87277	0.936385	−	0.350975i	$-0.114150\pi$
$139$	22.0796	1.87277	0.936385	+	0.350975i	$0.114150\pi$
$140$	−19.3894	−1.63871
$141$	−1.30173	−0.109626
$142$	40.5149	3.39993
$143$	−16.7709	−1.40245
$144$	12.8339	1.06949
$145$	0.102865	0.00854243
$146$	−28.7893	−2.38262
$147$	5.14080	0.424006
$148$	−7.41904	−0.609841
$149$	−17.1802	−1.40746	−0.703728	−	0.710470i	$-0.748483\pi$
$149$	−17.1802	−1.40746	−0.703728	+	0.710470i	$0.748483\pi$
$150$	6.48360	0.529384
$151$	5.59222	0.455088	0.227544	−	0.973768i	$-0.426930\pi$
$151$	5.59222	0.455088	0.227544	+	0.973768i	$0.426930\pi$
$152$	−6.60983	−0.536128
$153$	3.24798	0.262584
$154$	−10.8869	−0.877291
$155$	17.6568	1.41823
$156$	−29.4846	−2.36065
$157$	−14.3750	−1.14725	−0.573623	−	0.819119i	$-0.694463\pi$
$157$	−14.3750	−1.14725	−0.573623	+	0.819119i	$0.694463\pi$
$158$	31.8472	2.53363
$159$	7.18369	0.569704
$160$	−46.7405	−3.69516
$161$	2.63672	0.207803
$162$	−2.68757	−0.211155
$163$	−0.222093	−0.0173956	−0.00869782	−	0.999962i	$-0.502769\pi$
$163$	−0.222093	−0.0173956	−0.00869782	+	0.999962i	$0.502769\pi$
$164$	−16.3454	−1.27636
$165$	8.08840	0.629681
$166$	−3.48740	−0.270674
$167$	−15.0646	−1.16573	−0.582867	−	0.812568i	$-0.698069\pi$
$167$	−15.0646	−1.16573	−0.582867	+	0.812568i	$0.698069\pi$
$168$	−11.8109	−0.911234
$169$	18.8674	1.45134
$170$	−23.7659	−1.82276
$171$	0.763077	0.0583540
$172$	−57.7086	−4.40024
$173$	24.0670	1.82978	0.914889	−	0.403707i	$-0.132278\pi$
$173$	24.0670	1.82978	0.914889	+	0.403707i	$0.132278\pi$
$174$	0.101542	0.00769785
$175$	−3.28943	−0.248657
$176$	−38.1276	−2.87397
$177$	10.0409	0.754719
$178$	−40.3917	−3.02749
$179$	−19.9068	−1.48791	−0.743954	−	0.668231i	$-0.767052\pi$
$179$	−19.9068	−1.48791	−0.743954	+	0.668231i	$0.767052\pi$
$180$	14.2201	1.05990
$181$	−14.8184	−1.10144	−0.550722	−	0.834689i	$-0.685648\pi$
$181$	−14.8184	−1.10144	−0.550722	+	0.834689i	$0.685648\pi$
$182$	20.6869	1.53342
$183$	−0.969751	−0.0716860
$184$	16.7503	1.23485
$185$	−3.86729	−0.284329
$186$	17.4297	1.27801
$187$	−9.64929	−0.705626
$188$	6.79898	0.495866
$189$	1.36352	0.0991818
$190$	−5.58352	−0.405071
$191$	−18.2546	−1.32086	−0.660428	−	0.750889i	$-0.729625\pi$
$191$	−18.2546	−1.32086	−0.660428	+	0.750889i	$0.729625\pi$
$192$	−20.4717	−1.47741
$193$	2.11086	0.151943	0.0759717	−	0.997110i	$-0.475794\pi$
$193$	2.11086	0.151943	0.0759717	+	0.997110i	$0.475794\pi$
$194$	2.06102	0.147973
$195$	−15.3693	−1.10062
$196$	−26.8505	−1.91789
$197$	5.12488	0.365133	0.182566	−	0.983194i	$-0.441560\pi$
$197$	5.12488	0.365133	0.182566	+	0.983194i	$0.441560\pi$
$198$	7.98438	0.567425
$199$	−3.73939	−0.265079	−0.132539	−	0.991178i	$-0.542313\pi$
$199$	−3.73939	−0.265079	−0.132539	+	0.991178i	$0.542313\pi$
$200$	−20.8968	−1.47762
$201$	−7.67453	−0.541320
$202$	14.9296	1.05045
$203$	−0.0515167	−0.00361576
$204$	−16.9643	−1.18774
$205$	−8.52028	−0.595082
$206$	−23.0441	−1.60556
$207$	−1.93376	−0.134405
$208$	72.4487	5.02341
$209$	−2.26699	−0.156811
$210$	−9.97707	−0.688483
$211$	17.9581	1.23629	0.618143	−	0.786066i	$-0.287885\pi$
$211$	17.9581	1.23629	0.618143	+	0.786066i	$0.287885\pi$
$212$	−37.5205	−2.57692
$213$	15.0749	1.03292
$214$	4.46401	0.305154
$215$	−30.0815	−2.05154
$216$	8.66207	0.589379
$217$	−8.84289	−0.600295
$218$	42.7511	2.89547
$219$	−10.7120	−0.723851
$220$	−42.2458	−2.84821
$221$	18.3353	1.23336
$222$	−3.81756	−0.256218
$223$	−10.4387	−0.699026	−0.349513	−	0.936932i	$-0.613653\pi$
$223$	−10.4387	−0.699026	−0.349513	+	0.936932i	$0.613653\pi$
$224$	23.4086	1.56405
$225$	2.41244	0.160830
$226$	−1.53160	−0.101880
$227$	20.8950	1.38685	0.693425	−	0.720528i	$-0.256101\pi$
$227$	20.8950	1.38685	0.693425	+	0.720528i	$0.256101\pi$
$228$	−3.98556	−0.263950
$229$	26.3820	1.74337	0.871685	−	0.490066i	$-0.163027\pi$
$229$	26.3820	1.74337	0.871685	+	0.490066i	$0.163027\pi$
$230$	14.1495	0.932991
$231$	−4.05084	−0.266525
$232$	−0.327270	−0.0214863
$233$	4.44631	0.291287	0.145644	−	0.989337i	$-0.453475\pi$
$233$	4.44631	0.291287	0.145644	+	0.989337i	$0.453475\pi$
$234$	−15.1716	−0.991801
$235$	3.54408	0.231190
$236$	−52.4437	−3.41379
$237$	11.8498	0.769729
$238$	11.9024	0.771520
$239$	−13.2207	−0.855176	−0.427588	−	0.903974i	$-0.640637\pi$
$239$	−13.2207	−0.855176	−0.427588	+	0.903974i	$0.640637\pi$
$240$	−34.9412	−2.25545
$241$	−9.71192	−0.625600	−0.312800	−	0.949819i	$-0.601267\pi$
$241$	−9.71192	−0.625600	−0.312800	+	0.949819i	$0.601267\pi$
$242$	5.84280	0.375589
$243$	−1.00000	−0.0641500
$244$	5.06502	0.324255
$245$	−13.9962	−0.894187
$246$	−8.41071	−0.536247
$247$	4.30766	0.274090
$248$	−56.1763	−3.56720
$249$	−1.29760	−0.0822323
$250$	18.9335	1.19746
$251$	11.3709	0.717725	0.358863	−	0.933390i	$-0.383165\pi$
$251$	11.3709	0.717725	0.358863	+	0.933390i	$0.383165\pi$
$252$	−7.12171	−0.448625
$253$	5.74491	0.361179
$254$	39.6342	2.48687
$255$	−8.84289	−0.553764
$256$	14.6450	0.915314
$257$	−25.0029	−1.55964	−0.779819	−	0.626005i	$-0.784689\pi$
$257$	−25.0029	−1.55964	−0.779819	+	0.626005i	$0.784689\pi$
$258$	−29.6947	−1.84871
$259$	1.93682	0.120348
$260$	80.2741	4.97839
$261$	0.0377820	0.00233865
$262$	28.9866	1.79080
$263$	−6.28785	−0.387725	−0.193863	−	0.981029i	$-0.562102\pi$
$263$	−6.28785	−0.387725	−0.193863	+	0.981029i	$0.562102\pi$
$264$	−25.7338	−1.58380
$265$	−19.5582	−1.20145
$266$	2.79634	0.171455
$267$	−15.0291	−0.919767
$268$	40.0842	2.44853
$269$	4.68367	0.285568	0.142784	−	0.989754i	$-0.454394\pi$
$269$	4.68367	0.285568	0.142784	+	0.989754i	$0.454394\pi$
$270$	7.31712	0.445306
$271$	−12.6140	−0.766244	−0.383122	−	0.923698i	$-0.625151\pi$
$271$	−12.6140	−0.766244	−0.383122	+	0.923698i	$0.625151\pi$
$272$	41.6841	2.52747
$273$	7.69726	0.465860
$274$	−3.48507	−0.210541
$275$	−7.16702	−0.432188
$276$	10.1000	0.607951
$277$	9.61144	0.577495	0.288748	−	0.957405i	$-0.406761\pi$
$277$	9.61144	0.577495	0.288748	+	0.957405i	$0.406761\pi$
$278$	−59.3405	−3.55901
$279$	6.48532	0.388266
$280$	32.1562	1.92170
$281$	−1.96310	−0.117109	−0.0585544	−	0.998284i	$-0.518649\pi$
$281$	−1.96310	−0.117109	−0.0585544	+	0.998284i	$0.518649\pi$
$282$	3.49850	0.208333
$283$	−3.78315	−0.224885	−0.112442	−	0.993658i	$-0.535867\pi$
$283$	−3.78315	−0.224885	−0.112442	+	0.993658i	$0.535867\pi$
$284$	−78.7366	−4.67216
$285$	−2.07754	−0.123063
$286$	45.0728	2.66521
$287$	4.26713	0.251881
$288$	−17.1677	−1.01162
$289$	−6.45061	−0.379447
$290$	−0.276455	−0.0162340
$291$	0.766873	0.0449549
$292$	55.9490	3.27417
$293$	−25.9123	−1.51381	−0.756906	−	0.653524i	$-0.773290\pi$
$293$	−25.9123	−1.51381	−0.756906	+	0.653524i	$0.773290\pi$
$294$	−13.8162	−0.805780
$295$	−27.3371	−1.59163
$296$	12.3040	0.715158
$297$	2.97086	0.172387
$298$	46.1729	2.67472
$299$	−10.9163	−0.631305
$300$	−12.6002	−0.727475
$301$	15.0655	0.868358
$302$	−15.0295	−0.864848
$303$	5.55508	0.319131
$304$	9.79322	0.561680
$305$	2.64022	0.151179
$306$	−8.72917	−0.499014
$307$	17.2804	0.986246	0.493123	−	0.869960i	$-0.335855\pi$
$307$	17.2804	0.986246	0.493123	+	0.869960i	$0.335855\pi$
$308$	21.1576	1.20556
$309$	−8.57434	−0.487777
$310$	−47.4539	−2.69520
$311$	−16.7118	−0.947637	−0.473818	−	0.880623i	$-0.657125\pi$
$311$	−16.7118	−0.947637	−0.473818	+	0.880623i	$0.657125\pi$
$312$	48.8984	2.76833
$313$	15.3623	0.868329	0.434164	−	0.900834i	$-0.357044\pi$
$313$	15.3623	0.868329	0.434164	+	0.900834i	$0.357044\pi$
$314$	38.6337	2.18022
$315$	−3.71230	−0.209165
$316$	−61.8918	−3.48169
$317$	−5.50279	−0.309068	−0.154534	−	0.987988i	$-0.549388\pi$
$317$	−5.50279	−0.309068	−0.154534	+	0.987988i	$0.549388\pi$
$318$	−19.3067	−1.08266
$319$	−0.112245	−0.00628451
$320$	55.7357	3.11572
$321$	1.66099	0.0927073
$322$	−7.08637	−0.394908
$323$	2.47846	0.137905
$324$	5.22302	0.290168
$325$	13.6185	0.755421
$326$	0.596889	0.0330586
$327$	15.9070	0.879658
$328$	27.1078	1.49678
$329$	−1.77495	−0.0978560
$330$	−21.7381	−1.19664
$331$	15.9923	0.879017	0.439508	−	0.898238i	$-0.355153\pi$
$331$	15.9923	0.879017	0.439508	+	0.898238i	$0.355153\pi$
$332$	6.77740	0.371958
$333$	−1.42045	−0.0778403
$334$	40.4871	2.21536
$335$	20.8945	1.14159
$336$	17.4993	0.954664
$337$	34.0179	1.85307	0.926537	−	0.376203i	$-0.122770\pi$
$337$	34.0179	1.85307	0.926537	+	0.376203i	$0.122770\pi$
$338$	−50.7074	−2.75812
$339$	−0.569882	−0.0309518
$340$	46.1866	2.50482
$341$	−19.2670	−1.04336
$342$	−2.05082	−0.110896
$343$	16.5543	0.893847
$344$	95.7064	5.16014
$345$	5.26480	0.283448
$346$	−64.6816	−3.47730
$347$	−5.00807	−0.268847	−0.134424	−	0.990924i	$-0.542918\pi$
$347$	−5.00807	−0.268847	−0.134424	+	0.990924i	$0.542918\pi$
$348$	−0.197336	−0.0105783
$349$	3.40431	0.182229	0.0911143	−	0.995840i	$-0.470957\pi$
$349$	3.40431	0.182229	0.0911143	+	0.995840i	$0.470957\pi$
$350$	8.84055	0.472547
$351$	−5.64512	−0.301314
$352$	51.0028	2.71846
$353$	−9.25610	−0.492653	−0.246326	−	0.969187i	$-0.579224\pi$
$353$	−9.25610	−0.492653	−0.246326	+	0.969187i	$0.579224\pi$
$354$	−26.9855	−1.43427
$355$	−41.0427	−2.17832
$356$	78.4973	4.16035
$357$	4.42870	0.234392
$358$	53.5010	2.82762
$359$	14.3762	0.758746	0.379373	−	0.925244i	$-0.376140\pi$
$359$	14.3762	0.758746	0.379373	+	0.925244i	$0.376140\pi$
$360$	−23.5832	−1.24294
$361$	−18.4177	−0.969353
$362$	39.8255	2.09318
$363$	2.17401	0.114106
$364$	−40.2029	−2.10721
$365$	29.1643	1.52653
$366$	2.60627	0.136232
$367$	11.5172	0.601195	0.300597	−	0.953751i	$-0.402814\pi$
$367$	11.5172	0.601195	0.300597	+	0.953751i	$0.402814\pi$
$368$	−24.8175	−1.29370
$369$	−3.12949	−0.162915
$370$	10.3936	0.540338
$371$	9.79514	0.508538
$372$	−33.8729	−1.75623
$373$	6.64730	0.344184	0.172092	−	0.985081i	$-0.444947\pi$
$373$	6.64730	0.344184	0.172092	+	0.985081i	$0.444947\pi$
$374$	25.9331	1.34097
$375$	7.04483	0.363793
$376$	−11.2757	−0.581500
$377$	0.213284	0.0109847
$378$	−3.66456	−0.188485
$379$	−8.99504	−0.462044	−0.231022	−	0.972949i	$-0.574207\pi$
$379$	−8.99504	−0.462044	−0.231022	+	0.972949i	$0.574207\pi$
$380$	10.8510	0.556645
$381$	14.7472	0.755524
$382$	49.0605	2.51015
$383$	−7.84844	−0.401037	−0.200518	−	0.979690i	$-0.564263\pi$
$383$	−7.84844	−0.401037	−0.200518	+	0.979690i	$0.564263\pi$
$384$	20.6835	1.05550
$385$	11.0287	0.562076
$386$	−5.67309	−0.288753
$387$	−11.0489	−0.561648
$388$	−4.00539	−0.203343
$389$	30.3089	1.53672	0.768361	−	0.640017i	$-0.221073\pi$
$389$	30.3089	1.53672	0.768361	+	0.640017i	$0.221073\pi$
$390$	41.3060	2.09161
$391$	−6.28080	−0.317634
$392$	44.5300	2.24910
$393$	10.7855	0.544054
$394$	−13.7735	−0.693897
$395$	−32.2621	−1.62328
$396$	−15.5168	−0.779750
$397$	−14.8656	−0.746081	−0.373040	−	0.927815i	$-0.621685\pi$
$397$	−14.8656	−0.746081	−0.373040	+	0.927815i	$0.621685\pi$
$398$	10.0499	0.503754
$399$	1.04047	0.0520889
$400$	30.9610	1.54805
$401$	0.0854940	0.00426937	0.00213468	−	0.999998i	$-0.499321\pi$
$401$	0.0854940	0.00426937	0.00213468	+	0.999998i	$0.499321\pi$
$402$	20.6258	1.02872
$403$	36.6104	1.82370
$404$	−29.0143	−1.44351
$405$	2.72258	0.135286
$406$	0.138454	0.00687138
$407$	4.21996	0.209176
$408$	28.1343	1.39285
$409$	18.7804	0.928633	0.464317	−	0.885669i	$-0.346300\pi$
$409$	18.7804	0.928633	0.464317	+	0.885669i	$0.346300\pi$
$410$	22.8988	1.13089
$411$	−1.29674	−0.0639633
$412$	44.7839	2.20635
$413$	13.6910	0.673690
$414$	5.19710	0.255423
$415$	3.53283	0.173420
$416$	−96.9138	−4.75159
$417$	−22.0796	−1.08124
$418$	6.09269	0.298003
$419$	−9.83046	−0.480250	−0.240125	−	0.970742i	$-0.577188\pi$
$419$	−9.83046	−0.480250	−0.240125	+	0.970742i	$0.577188\pi$
$420$	19.3894	0.946107
$421$	−36.4892	−1.77838	−0.889188	−	0.457541i	$-0.848730\pi$
$421$	−36.4892	−1.77838	−0.889188	+	0.457541i	$0.848730\pi$
$422$	−48.2635	−2.34943
$423$	1.30173	0.0632925
$424$	62.2256	3.02194
$425$	7.83558	0.380081
$426$	−40.5149	−1.96295
$427$	−1.32228	−0.0639896
$428$	−8.67536	−0.419339
$429$	16.7709	0.809704
$430$	80.8461	3.89875
$431$	25.3873	1.22286	0.611432	−	0.791297i	$-0.290594\pi$
$431$	25.3873	1.22286	0.611432	+	0.791297i	$0.290594\pi$
$432$	−12.8339	−0.617469
$433$	33.8526	1.62685	0.813426	−	0.581668i	$-0.197600\pi$
$433$	33.8526	1.62685	0.813426	+	0.581668i	$0.197600\pi$
$434$	23.7659	1.14080
$435$	−0.102865	−0.00493198
$436$	−83.0824	−3.97892
$437$	−1.47560	−0.0705877
$438$	28.7893	1.37560
$439$	−18.3592	−0.876237	−0.438118	−	0.898917i	$-0.644355\pi$
$439$	−18.3592	−0.876237	−0.438118	+	0.898917i	$0.644355\pi$
$440$	70.0622	3.34009
$441$	−5.14080	−0.244800
$442$	−49.2772	−2.34388
$443$	23.9909	1.13984	0.569921	−	0.821699i	$-0.306974\pi$
$443$	23.9909	1.13984	0.569921	+	0.821699i	$0.306974\pi$
$444$	7.41904	0.352092
$445$	40.9180	1.93970
$446$	28.0547	1.32843
$447$	17.1802	0.812595
$448$	−27.9136	−1.31879
$449$	−2.22093	−0.104812	−0.0524061	−	0.998626i	$-0.516689\pi$
$449$	−2.22093	−0.104812	−0.0524061	+	0.998626i	$0.516689\pi$
$450$	−6.48360	−0.305640
$451$	9.29726	0.437791
$452$	2.97650	0.140003
$453$	−5.59222	−0.262745
$454$	−56.1567	−2.63557
$455$	−20.9564	−0.982452
$456$	6.60983	0.309533
$457$	−0.785259	−0.0367329	−0.0183664	−	0.999831i	$-0.505847\pi$
$457$	−0.785259	−0.0367329	−0.0183664	+	0.999831i	$0.505847\pi$
$458$	−70.9033	−3.31310
$459$	−3.24798	−0.151603
$460$	−27.4982	−1.28211
$461$	23.5332	1.09605	0.548024	−	0.836462i	$-0.315380\pi$
$461$	23.5332	1.09605	0.548024	+	0.836462i	$0.315380\pi$
$462$	10.8869	0.506504
$463$	−1.60893	−0.0747731	−0.0373866	−	0.999301i	$-0.511903\pi$
$463$	−1.60893	−0.0747731	−0.0373866	+	0.999301i	$0.511903\pi$
$464$	0.484889	0.0225104
$465$	−17.6568	−0.818815
$466$	−11.9498	−0.553562
$467$	1.56889	0.0725994	0.0362997	−	0.999341i	$-0.488443\pi$
$467$	1.56889	0.0725994	0.0362997	+	0.999341i	$0.488443\pi$
$468$	29.4846	1.36292
$469$	−10.4644	−0.483201
$470$	−9.52494	−0.439353
$471$	14.3750	0.662363
$472$	86.9748	4.00334
$473$	32.8247	1.50928
$474$	−31.8472	−1.46279
$475$	1.84088	0.0844654
$476$	−23.1312	−1.06022
$477$	−7.18369	−0.328919
$478$	35.5315	1.62517
$479$	−26.3261	−1.20287	−0.601435	−	0.798922i	$-0.705404\pi$
$479$	−26.3261	−1.20287	−0.601435	+	0.798922i	$0.705404\pi$
$480$	46.7405	2.13340
$481$	−8.01862	−0.365618
$482$	26.1014	1.18889
$483$	−2.63672	−0.119975
$484$	−11.3549	−0.516131
$485$	−2.08787	−0.0948055
$486$	2.68757	0.121911
$487$	5.77982	0.261909	0.130954	−	0.991388i	$-0.458196\pi$
$487$	5.77982	0.261909	0.130954	+	0.991388i	$0.458196\pi$
$488$	−8.40005	−0.380252
$489$	0.222093	0.0100434
$490$	37.6158	1.69931
$491$	38.0989	1.71938	0.859689	−	0.510818i	$-0.170657\pi$
$491$	38.0989	1.71938	0.859689	+	0.510818i	$0.170657\pi$
$492$	16.3454	0.736906
$493$	0.122715	0.00552682
$494$	−11.5771	−0.520880
$495$	−8.08840	−0.363546
$496$	83.2317	3.73721
$497$	20.5550	0.922019
$498$	3.48740	0.156274
$499$	−9.90581	−0.443445	−0.221722	−	0.975110i	$-0.571168\pi$
$499$	−9.90581	−0.443445	−0.221722	+	0.975110i	$0.571168\pi$
$500$	−36.7953	−1.64553
$501$	15.0646	0.673036
$502$	−30.5601	−1.36396
$503$	−36.0165	−1.60590	−0.802948	−	0.596049i	$-0.796737\pi$
$503$	−36.0165	−1.60590	−0.802948	+	0.596049i	$0.796737\pi$
$504$	11.8109	0.526101
$505$	−15.1241	−0.673016
$506$	−15.4398	−0.686384
$507$	−18.8674	−0.837931
$508$	−77.0251	−3.41744
$509$	−30.4860	−1.35127	−0.675635	−	0.737236i	$-0.736130\pi$
$509$	−30.4860	−1.35127	−0.675635	+	0.737236i	$0.736130\pi$
$510$	23.7659	1.05237
$511$	−14.6061	−0.646136
$512$	2.00757	0.0887227
$513$	−0.763077	−0.0336907
$514$	67.1969	2.96393
$515$	23.3443	1.02867
$516$	57.7086	2.54048
$517$	−3.86727	−0.170082
$518$	−5.20533	−0.228709
$519$	−24.0670	−1.05642
$520$	−133.130	−5.83813
$521$	−40.1775	−1.76021	−0.880104	−	0.474780i	$-0.842527\pi$
$521$	−40.1775	−1.76021	−0.880104	+	0.474780i	$0.842527\pi$
$522$	−0.101542	−0.00444436
$523$	12.5957	0.550773	0.275386	−	0.961334i	$-0.411194\pi$
$523$	12.5957	0.550773	0.275386	+	0.961334i	$0.411194\pi$
$524$	−56.3326	−2.46090
$525$	3.28943	0.143562
$526$	16.8990	0.736832
$527$	21.0642	0.917572
$528$	38.1276	1.65929
$529$	−19.2606	−0.837417
$530$	52.5639	2.28323
$531$	−10.0409	−0.435737
$532$	−5.43441	−0.235612
$533$	−17.6663	−0.765214
$534$	40.3917	1.74792
$535$	−4.52217	−0.195511
$536$	−66.4773	−2.87138
$537$	19.9068	0.859044
$538$	−12.5877	−0.542693
$539$	15.2726	0.657837
$540$	−14.2201	−0.611935
$541$	−21.7243	−0.934002	−0.467001	−	0.884257i	$-0.654666\pi$
$541$	−21.7243	−0.934002	−0.467001	+	0.884257i	$0.654666\pi$
$542$	33.9009	1.45617
$543$	14.8184	0.635919
$544$	−55.7604	−2.39071
$545$	−43.3080	−1.85511
$546$	−20.6869	−0.885318
$547$	19.8150	0.847229	0.423615	−	0.905843i	$-0.360761\pi$
$547$	19.8150	0.847229	0.423615	+	0.905843i	$0.360761\pi$
$548$	6.77288	0.289323
$549$	0.969751	0.0413880
$550$	19.2619	0.821328
$551$	0.0288306	0.00122822
$552$	−16.7503	−0.712941
$553$	16.1575	0.687088
$554$	−25.8314	−1.09747
$555$	3.86729	0.164157
$556$	115.322	4.89075
$557$	−2.03072	−0.0860443	−0.0430222	−	0.999074i	$-0.513699\pi$
$557$	−2.03072	−0.0860443	−0.0430222	+	0.999074i	$0.513699\pi$
$558$	−17.4297	−0.737859
$559$	−62.3724	−2.63807
$560$	−47.6432	−2.01329
$561$	9.64929	0.407393
$562$	5.27596	0.222553
$563$	−31.4186	−1.32414	−0.662068	−	0.749444i	$-0.730321\pi$
$563$	−31.4186	−1.32414	−0.662068	+	0.749444i	$0.730321\pi$
$564$	−6.79898	−0.286289
$565$	1.55155	0.0652742
$566$	10.1675	0.427370
$567$	−1.36352	−0.0572626
$568$	130.580	5.47902
$569$	27.0000	1.13190	0.565949	−	0.824440i	$-0.308510\pi$
$569$	27.0000	1.13190	0.565949	+	0.824440i	$0.308510\pi$
$570$	5.58352	0.233868
$571$	2.39397	0.100184	0.0500922	−	0.998745i	$-0.484048\pi$
$571$	2.39397	0.100184	0.0500922	+	0.998745i	$0.484048\pi$
$572$	−87.5944	−3.66251
$573$	18.2546	0.762597
$574$	−11.4682	−0.478674
$575$	−4.66508	−0.194547
$576$	20.4717	0.852985
$577$	15.9742	0.665014	0.332507	−	0.943101i	$-0.392106\pi$
$577$	15.9742	0.665014	0.332507	+	0.943101i	$0.392106\pi$
$578$	17.3364	0.721101
$579$	−2.11086	−0.0877245
$580$	0.537263	0.0223086
$581$	−1.76931	−0.0734035
$582$	−2.06102	−0.0854321
$583$	21.3417	0.883884
$584$	−92.7883	−3.83961
$585$	15.3693	0.635442
$586$	69.6410	2.87684
$587$	−1.96044	−0.0809160	−0.0404580	−	0.999181i	$-0.512882\pi$
$587$	−1.96044	−0.0809160	−0.0404580	+	0.999181i	$0.512882\pi$
$588$	26.8505	1.10730
$589$	4.94880	0.203912
$590$	73.4703	3.02472
$591$	−5.12488	−0.210809
$592$	−18.2299	−0.749243
$593$	−4.43116	−0.181966	−0.0909830	−	0.995852i	$-0.529001\pi$
$593$	−4.43116	−0.181966	−0.0909830	+	0.995852i	$0.529001\pi$
$594$	−7.98438	−0.327603
$595$	−12.0575	−0.494309
$596$	−89.7324	−3.67558
$597$	3.73939	0.153043
$598$	29.3382	1.19973
$599$	45.8707	1.87423	0.937114	−	0.349023i	$-0.113487\pi$
$599$	45.8707	1.87423	0.937114	+	0.349023i	$0.113487\pi$
$600$	20.8968	0.853106
$601$	19.7206	0.804419	0.402210	−	0.915548i	$-0.368242\pi$
$601$	19.7206	0.804419	0.402210	+	0.915548i	$0.368242\pi$
$602$	−40.4894	−1.65023
$603$	7.67453	0.312531
$604$	29.2082	1.18847
$605$	−5.91892	−0.240638
$606$	−14.9296	−0.606475
$607$	−17.7980	−0.722397	−0.361198	−	0.932489i	$-0.617632\pi$
$607$	−17.7980	−0.722397	−0.361198	+	0.932489i	$0.617632\pi$
$608$	−13.1003	−0.531287
$609$	0.0515167	0.00208756
$610$	−7.09578	−0.287300
$611$	7.34845	0.297286
$612$	16.9643	0.685740
$613$	38.7211	1.56393	0.781966	−	0.623321i	$-0.214217\pi$
$613$	38.7211	1.56393	0.781966	+	0.623321i	$0.214217\pi$
$614$	−46.4423	−1.87426
$615$	8.52028	0.343571
$616$	−35.0886	−1.41376
$617$	−37.0004	−1.48958	−0.744790	−	0.667299i	$-0.767450\pi$
$617$	−37.0004	−1.48958	−0.744790	+	0.667299i	$0.767450\pi$
$618$	23.0441	0.926971
$619$	25.6528	1.03107	0.515536	−	0.856868i	$-0.327593\pi$
$619$	25.6528	1.03107	0.515536	+	0.856868i	$0.327593\pi$
$620$	92.2218	3.70372
$621$	1.93376	0.0775989
$622$	44.9140	1.80089
$623$	−20.4926	−0.821017
$624$	−72.4487	−2.90027
$625$	−31.2423	−1.24969
$626$	−41.2872	−1.65017
$627$	2.26699	0.0905350
$628$	−75.0806	−2.99604
$629$	−4.61360	−0.183956
$630$	9.97707	0.397496
$631$	−10.6963	−0.425814	−0.212907	−	0.977073i	$-0.568293\pi$
$631$	−10.6963	−0.425814	−0.212907	+	0.977073i	$0.568293\pi$
$632$	102.644	4.08296
$633$	−17.9581	−0.713770
$634$	14.7891	0.587351
$635$	−40.1506	−1.59333
$636$	37.5205	1.48779
$637$	−29.0205	−1.14983
$638$	0.301666	0.0119431
$639$	−15.0749	−0.596355
$640$	−56.3125	−2.22595
$641$	18.6162	0.735294	0.367647	−	0.929965i	$-0.380164\pi$
$641$	18.6162	0.735294	0.367647	+	0.929965i	$0.380164\pi$
$642$	−4.46401	−0.176181
$643$	−13.6982	−0.540206	−0.270103	−	0.962831i	$-0.587058\pi$
$643$	−13.6982	−0.540206	−0.270103	+	0.962831i	$0.587058\pi$
$644$	13.7716	0.542679
$645$	30.0815	1.18446
$646$	−6.66103	−0.262075
$647$	21.6490	0.851110	0.425555	−	0.904933i	$-0.360079\pi$
$647$	21.6490	0.851110	0.425555	+	0.904933i	$0.360079\pi$
$648$	−8.66207	−0.340278
$649$	29.8300	1.17093
$650$	−36.6007	−1.43560
$651$	8.84289	0.346580
$652$	−1.15999	−0.0454288
$653$	−40.0916	−1.56890	−0.784452	−	0.620189i	$-0.787056\pi$
$653$	−40.0916	−1.56890	−0.784452	+	0.620189i	$0.787056\pi$
$654$	−42.7511	−1.67170
$655$	−29.3643	−1.14736
$656$	−40.1634	−1.56812
$657$	10.7120	0.417916
$658$	4.77029	0.185965
$659$	−36.6132	−1.42625	−0.713124	−	0.701038i	$-0.752720\pi$
$659$	−36.6132	−1.42625	−0.713124	+	0.701038i	$0.752720\pi$
$660$	42.2458	1.64442
$661$	−21.4809	−0.835512	−0.417756	−	0.908559i	$-0.637183\pi$
$661$	−21.4809	−0.835512	−0.417756	+	0.908559i	$0.637183\pi$
$662$	−42.9804	−1.67048
$663$	−18.3353	−0.712083
$664$	−11.2399	−0.436194
$665$	−2.83277	−0.109850
$666$	3.81756	0.147927
$667$	−0.0730611	−0.00282894
$668$	−78.6826	−3.04432
$669$	10.4387	0.403583
$670$	−56.1554	−2.16947
$671$	−2.88099	−0.111219
$672$	−23.4086	−0.903006
$673$	−17.0925	−0.658868	−0.329434	−	0.944179i	$-0.606858\pi$
$673$	−17.0925	−0.658868	−0.329434	+	0.944179i	$0.606858\pi$
$674$	−91.4255	−3.52158
$675$	−2.41244	−0.0928550
$676$	98.5448	3.79018
$677$	31.9156	1.22662	0.613308	−	0.789844i	$-0.289839\pi$
$677$	31.9156	1.22662	0.613308	+	0.789844i	$0.289839\pi$
$678$	1.53160	0.0588206
$679$	1.04565	0.0401284
$680$	−76.5978	−2.93739
$681$	−20.8950	−0.800699
$682$	51.7812	1.98281
$683$	−31.5013	−1.20536	−0.602681	−	0.797982i	$-0.705901\pi$
$683$	−31.5013	−1.20536	−0.602681	+	0.797982i	$0.705901\pi$
$684$	3.98556	0.152392
$685$	3.53047	0.134892
$686$	−44.4907	−1.69866
$687$	−26.3820	−1.00654
$688$	−141.800	−5.40608
$689$	−40.5528	−1.54494
$690$	−14.1495	−0.538663
$691$	−30.8653	−1.17417	−0.587086	−	0.809524i	$-0.699725\pi$
$691$	−30.8653	−1.17417	−0.587086	+	0.809524i	$0.699725\pi$
$692$	125.702	4.77848
$693$	4.05084	0.153879
$694$	13.4595	0.510917
$695$	60.1136	2.28024
$696$	0.327270	0.0124051
$697$	−10.1645	−0.385009
$698$	−9.14932	−0.346307
$699$	−4.44631	−0.168175
$700$	−17.1807	−0.649370
$701$	−18.9833	−0.716988	−0.358494	−	0.933532i	$-0.616710\pi$
$701$	−18.9833	−0.716988	−0.358494	+	0.933532i	$0.616710\pi$
$702$	15.1716	0.572617
$703$	−1.08391	−0.0408806
$704$	−60.8183	−2.29218
$705$	−3.54408	−0.133478
$706$	24.8764	0.936236
$707$	7.57448	0.284868
$708$	52.4437	1.97095
$709$	22.8927	0.859753	0.429876	−	0.902888i	$-0.358557\pi$
$709$	22.8927	0.859753	0.429876	+	0.902888i	$0.358557\pi$
$710$	110.305	4.13967
$711$	−11.8498	−0.444403
$712$	−130.183	−4.87882
$713$	−12.5410	−0.469665
$714$	−11.9024	−0.445438
$715$	−45.6600	−1.70759
$716$	−103.974	−3.88568
$717$	13.2207	0.493736
$718$	−38.6369	−1.44192
$719$	−23.6317	−0.881315	−0.440657	−	0.897675i	$-0.645255\pi$
$719$	−23.6317	−0.881315	−0.440657	+	0.897675i	$0.645255\pi$
$720$	34.9412	1.30218
$721$	−11.6913	−0.435408
$722$	49.4988	1.84216
$723$	9.71192	0.361190
$724$	−77.3968	−2.87643
$725$	0.0911469	0.00338511
$726$	−5.84280	−0.216846
$727$	14.8953	0.552436	0.276218	−	0.961095i	$-0.410919\pi$
$727$	14.8953	0.552436	0.276218	+	0.961095i	$0.410919\pi$
$728$	66.6742	2.47111
$729$	1.00000	0.0370370
$730$	−78.3811	−2.90101
$731$	−35.8867	−1.32732
$732$	−5.06502	−0.187209
$733$	−41.7019	−1.54030	−0.770148	−	0.637865i	$-0.779818\pi$
$733$	−41.7019	−1.54030	−0.770148	+	0.637865i	$0.779818\pi$
$734$	−30.9533	−1.14251
$735$	13.9962	0.516259
$736$	33.1981	1.22370
$737$	−22.7999	−0.839846
$738$	8.41071	0.309602
$739$	−24.1279	−0.887559	−0.443780	−	0.896136i	$-0.646363\pi$
$739$	−24.1279	−0.887559	−0.443780	+	0.896136i	$0.646363\pi$
$740$	−20.1989	−0.742527
$741$	−4.30766	−0.158246
$742$	−26.3251	−0.966425
$743$	−41.4440	−1.52043	−0.760217	−	0.649670i	$-0.774907\pi$
$743$	−41.4440	−1.52043	−0.760217	+	0.649670i	$0.774907\pi$
$744$	56.1763	2.05952
$745$	−46.7744	−1.71368
$746$	−17.8651	−0.654087
$747$	1.29760	0.0474768
$748$	−50.3984	−1.84275
$749$	2.26480	0.0827539
$750$	−18.9335	−0.691352
$751$	−24.2929	−0.886460	−0.443230	−	0.896408i	$-0.646167\pi$
$751$	−24.2929	−0.886460	−0.443230	+	0.896408i	$0.646167\pi$
$752$	16.7063	0.609215
$753$	−11.3709	−0.414379
$754$	−0.573215	−0.0208753
$755$	15.2253	0.554104
$756$	7.12171	0.259014
$757$	−24.6573	−0.896184	−0.448092	−	0.893988i	$-0.647896\pi$
$757$	−24.6573	−0.896184	−0.448092	+	0.893988i	$0.647896\pi$
$758$	24.1748	0.878067
$759$	−5.74491	−0.208527
$760$	−17.9958	−0.652776
$761$	31.9902	1.15965	0.579823	−	0.814743i	$-0.303122\pi$
$761$	31.9902	1.15965	0.579823	+	0.814743i	$0.303122\pi$
$762$	−39.6342	−1.43580
$763$	21.6896	0.785214
$764$	−95.3441	−3.44943
$765$	8.84289	0.319716
$766$	21.0932	0.762129
$767$	−56.6820	−2.04667
$768$	−14.6450	−0.528457
$769$	−3.24694	−0.117087	−0.0585437	−	0.998285i	$-0.518646\pi$
$769$	−3.24694	−0.117087	−0.0585437	+	0.998285i	$0.518646\pi$
$770$	−29.6404	−1.06817
$771$	25.0029	0.900457
$772$	11.0251	0.396801
$773$	50.7432	1.82511	0.912554	−	0.408957i	$-0.134107\pi$
$773$	50.7432	1.82511	0.912554	+	0.408957i	$0.134107\pi$
$774$	29.6947	1.06735
$775$	15.6455	0.562002
$776$	6.64271	0.238459
$777$	−1.93682	−0.0694830
$778$	−81.4572	−2.92038
$779$	−2.38804	−0.0855604
$780$	−80.2741	−2.87427
$781$	44.7855	1.60255
$782$	16.8801	0.603630
$783$	−0.0377820	−0.00135022
$784$	−65.9763	−2.35630
$785$	−39.1370	−1.39686
$786$	−28.9866	−1.03392
$787$	−46.4898	−1.65718	−0.828590	−	0.559855i	$-0.810857\pi$
$787$	−46.4898	−1.65718	−0.828590	+	0.559855i	$0.810857\pi$
$788$	26.7673	0.953547
$789$	6.28785	0.223853
$790$	86.7066	3.08488
$791$	−0.777048	−0.0276287
$792$	25.7338	0.914410
$793$	5.47436	0.194400
$794$	39.9522	1.41785
$795$	19.5582	0.693657
$796$	−19.5309	−0.692255
$797$	40.8337	1.44640	0.723201	−	0.690637i	$-0.242670\pi$
$797$	40.8337	1.44640	0.723201	+	0.690637i	$0.242670\pi$
$798$	−2.79634	−0.0989895
$799$	4.22801	0.149576
$800$	−41.4161	−1.46428
$801$	15.0291	0.531028
$802$	−0.229771	−0.00811349
$803$	−31.8239	−1.12304
$804$	−40.0842	−1.41366
$805$	7.17869	0.253016
$806$	−98.3930	−3.46575
$807$	−4.68367	−0.164873
$808$	48.1185	1.69280
$809$	1.82107	0.0640253	0.0320127	−	0.999487i	$-0.489808\pi$
$809$	1.82107	0.0640253	0.0320127	+	0.999487i	$0.489808\pi$
$810$	−7.31712	−0.257097
$811$	23.5467	0.826838	0.413419	−	0.910541i	$-0.364334\pi$
$811$	23.5467	0.826838	0.413419	+	0.910541i	$0.364334\pi$
$812$	−0.269072	−0.00944259
$813$	12.6140	0.442391
$814$	−11.3414	−0.397517
$815$	−0.604665	−0.0211805
$816$	−41.6841	−1.45924
$817$	−8.43117	−0.294969
$818$	−50.4737	−1.76477
$819$	−7.69726	−0.268964
$820$	−44.5016	−1.55406
$821$	29.8274	1.04098	0.520492	−	0.853866i	$-0.325748\pi$
$821$	29.8274	1.04098	0.520492	+	0.853866i	$0.325748\pi$
$822$	3.48507	0.121556
$823$	7.11756	0.248103	0.124051	−	0.992276i	$-0.460411\pi$
$823$	7.11756	0.248103	0.124051	+	0.992276i	$0.460411\pi$
$824$	−74.2716	−2.58737
$825$	7.16702	0.249524
$826$	−36.7954	−1.28028
$827$	3.08454	0.107260	0.0536300	−	0.998561i	$-0.482921\pi$
$827$	3.08454	0.107260	0.0536300	+	0.998561i	$0.482921\pi$
$828$	−10.1000	−0.351000
$829$	1.47050	0.0510724	0.0255362	−	0.999674i	$-0.491871\pi$
$829$	1.47050	0.0510724	0.0255362	+	0.999674i	$0.491871\pi$
$830$	−9.49472	−0.329566
$831$	−9.61144	−0.333417
$832$	115.565	4.00649
$833$	−16.6972	−0.578525
$834$	59.3405	2.05479
$835$	−41.0146	−1.41937
$836$	−11.8405	−0.409514
$837$	−6.48532	−0.224165
$838$	26.4200	0.912665
$839$	7.55231	0.260735	0.130367	−	0.991466i	$-0.458384\pi$
$839$	7.55231	0.260735	0.130367	+	0.991466i	$0.458384\pi$
$840$	−32.1562	−1.10950
$841$	−28.9986	−0.999951
$842$	98.0673	3.37962
$843$	1.96310	0.0676128
$844$	93.7953	3.22857
$845$	51.3680	1.76711
$846$	−3.49850	−0.120281
$847$	2.96431	0.101855
$848$	−92.1945	−3.16597
$849$	3.78315	0.129837
$850$	−21.0586	−0.722305
$851$	2.74681	0.0941593
$852$	78.7366	2.69747
$853$	13.5517	0.464000	0.232000	−	0.972716i	$-0.425473\pi$
$853$	13.5517	0.464000	0.232000	+	0.972716i	$0.425473\pi$
$854$	3.55371	0.121606
$855$	2.07754	0.0710503
$856$	14.3876	0.491758
$857$	−9.28080	−0.317026	−0.158513	−	0.987357i	$-0.550670\pi$
$857$	−9.28080	−0.317026	−0.158513	+	0.987357i	$0.550670\pi$
$858$	−45.0728	−1.53876
$859$	−45.0609	−1.53746	−0.768729	−	0.639575i	$-0.779110\pi$
$859$	−45.0609	−1.53746	−0.768729	+	0.639575i	$0.779110\pi$
$860$	−157.116	−5.35762
$861$	−4.26713	−0.145424
$862$	−68.2301	−2.32392
$863$	26.6393	0.906812	0.453406	−	0.891304i	$-0.350209\pi$
$863$	26.6393	0.906812	0.453406	+	0.891304i	$0.350209\pi$
$864$	17.1677	0.584057
$865$	65.5242	2.22789
$866$	−90.9812	−3.09167
$867$	6.45061	0.219074
$868$	−46.1866	−1.56767
$869$	35.2041	1.19422
$870$	0.276455	0.00937271
$871$	43.3237	1.46797
$872$	137.787	4.66607
$873$	−0.766873	−0.0259547
$874$	3.96579	0.134145
$875$	9.60580	0.324735
$876$	−55.9490	−1.89034
$877$	25.5246	0.861904	0.430952	−	0.902375i	$-0.358178\pi$
$877$	25.5246	0.861904	0.430952	+	0.902375i	$0.358178\pi$
$878$	49.3416	1.66520
$879$	25.9123	0.874000
$880$	−103.805	−3.49928
$881$	33.3316	1.12297	0.561484	−	0.827487i	$-0.310231\pi$
$881$	33.3316	1.12297	0.561484	+	0.827487i	$0.310231\pi$
$882$	13.8162	0.465217
$883$	−14.6012	−0.491371	−0.245685	−	0.969350i	$-0.579013\pi$
$883$	−14.6012	−0.491371	−0.245685	+	0.969350i	$0.579013\pi$
$884$	95.7654	3.22094
$885$	27.3371	0.918927
$886$	−64.4771	−2.16615
$887$	8.01137	0.268995	0.134498	−	0.990914i	$-0.457058\pi$
$887$	8.01137	0.268995	0.134498	+	0.990914i	$0.457058\pi$
$888$	−12.3040	−0.412897
$889$	20.1082	0.674409
$890$	−109.970	−3.68619
$891$	−2.97086	−0.0995275
$892$	−54.5214	−1.82551
$893$	0.993324	0.0332403
$894$	−46.1729	−1.54425
$895$	−54.1980	−1.81164
$896$	28.2025	0.942179
$897$	10.9163	0.364484
$898$	5.96890	0.199185
$899$	0.245028	0.00817215
$900$	12.6002	0.420008
$901$	−23.3325	−0.777319
$902$	−24.9870	−0.831976
$903$	−15.0655	−0.501347
$904$	−4.93636	−0.164181
$905$	−40.3443	−1.34109
$906$	15.0295	0.499320
$907$	13.0763	0.434191	0.217096	−	0.976150i	$-0.430342\pi$
$907$	13.0763	0.434191	0.217096	+	0.976150i	$0.430342\pi$
$908$	109.135	3.62177
$909$	−5.55508	−0.184250
$910$	56.3218	1.86705
$911$	15.7031	0.520266	0.260133	−	0.965573i	$-0.416234\pi$
$911$	15.7031	0.520266	0.260133	+	0.965573i	$0.416234\pi$
$912$	−9.79322	−0.324286
$913$	−3.85499	−0.127582
$914$	2.11044	0.0698070
$915$	−2.64022	−0.0872831
$916$	137.793	4.55283
$917$	14.7062	0.485643
$918$	8.72917	0.288106
$919$	4.58151	0.151130	0.0755650	−	0.997141i	$-0.475924\pi$
$919$	4.58151	0.151130	0.0755650	+	0.997141i	$0.475924\pi$
$920$	45.6041	1.50352
$921$	−17.2804	−0.569409
$922$	−63.2469	−2.08293
$923$	−85.0998	−2.80110
$924$	−21.1576	−0.696033
$925$	−3.42676	−0.112671
$926$	4.32410	0.142099
$927$	8.57434	0.281618
$928$	−0.648630	−0.0212923
$929$	−1.62337	−0.0532610	−0.0266305	−	0.999645i	$-0.508478\pi$
$929$	−1.62337	−0.0532610	−0.0266305	+	0.999645i	$0.508478\pi$
$930$	47.4539	1.55607
$931$	−3.92283	−0.128565
$932$	23.2231	0.760699
$933$	16.7118	0.547118
$934$	−4.21649	−0.137968
$935$	−26.2710	−0.859153
$936$	−48.8984	−1.59830
$937$	−26.4155	−0.862955	−0.431478	−	0.902124i	$-0.642008\pi$
$937$	−26.4155	−0.862955	−0.431478	+	0.902124i	$0.642008\pi$
$938$	28.1238	0.918275
$939$	−15.3623	−0.501330
$940$	18.5108	0.603755
$941$	−0.0584648	−0.00190590	−0.000952949	−	1.00000i	$-0.500303\pi$
$941$	−0.0584648	−0.00190590	−0.000952949		1.00000i	$0.500303\pi$
$942$	−38.6337	−1.25875
$943$	6.05166	0.197069
$944$	−128.863	−4.19414
$945$	3.71230	0.120761
$946$	−88.2186	−2.86824
$947$	−21.9013	−0.711698	−0.355849	−	0.934543i	$-0.615808\pi$
$947$	−21.9013	−0.711698	−0.355849	+	0.934543i	$0.615808\pi$
$948$	61.8918	2.01015
$949$	60.4707	1.96296
$950$	−4.94749	−0.160518
$951$	5.50279	0.178440
$952$	38.3617	1.24331
$953$	36.8403	1.19337	0.596687	−	0.802474i	$-0.296483\pi$
$953$	36.8403	1.19337	0.596687	+	0.802474i	$0.296483\pi$
$954$	19.3067	0.625076
$955$	−49.6996	−1.60824
$956$	−69.0519	−2.23330
$957$	0.112245	0.00362836
$958$	70.7532	2.28593
$959$	−1.76813	−0.0570960
$960$	−55.7357	−1.79886
$961$	11.0594	0.356755
$962$	21.5506	0.694819
$963$	−1.66099	−0.0535246
$964$	−50.7255	−1.63376
$965$	5.74700	0.185002
$966$	7.08637	0.228000
$967$	−2.06131	−0.0662874	−0.0331437	−	0.999451i	$-0.510552\pi$
$967$	−2.06131	−0.0662874	−0.0331437	+	0.999451i	$0.510552\pi$
$968$	18.8314	0.605265
$969$	−2.47846	−0.0796196
$970$	5.61130	0.180168
$971$	−30.1229	−0.966689	−0.483344	−	0.875430i	$-0.660578\pi$
$971$	−30.1229	−0.966689	−0.483344	+	0.875430i	$0.660578\pi$
$972$	−5.22302	−0.167528
$973$	−30.1061	−0.965158
$974$	−15.5337	−0.497731
$975$	−13.6185	−0.436142
$976$	12.4456	0.398375
$977$	30.1880	0.965800	0.482900	−	0.875675i	$-0.339583\pi$
$977$	30.1880	0.965800	0.482900	+	0.875675i	$0.339583\pi$
$978$	−0.596889	−0.0190864
$979$	−44.6493	−1.42700
$980$	−73.1026	−2.33518
$981$	−15.9070	−0.507871
$982$	−102.393	−3.26750
$983$	22.2427	0.709433	0.354717	−	0.934974i	$-0.384577\pi$
$983$	22.2427	0.709433	0.354717	+	0.934974i	$0.384577\pi$
$984$	−27.1078	−0.864166
$985$	13.9529	0.444576
$986$	−0.329805	−0.0105031
$987$	1.77495	0.0564972
$988$	22.4990	0.715788
$989$	21.3659	0.679396
$990$	21.7381	0.690882
$991$	−27.9897	−0.889123	−0.444561	−	0.895748i	$-0.646640\pi$
$991$	−27.9897	−0.889123	−0.444561	+	0.895748i	$0.646640\pi$
$992$	−111.338	−3.53499
$993$	−15.9923	−0.507501
$994$	−55.2430	−1.75220
$995$	−10.1808	−0.322753
$996$	−6.77740	−0.214750
$997$	−37.6275	−1.19167	−0.595837	−	0.803105i	$-0.703180\pi$
$997$	−37.6275	−1.19167	−0.595837	+	0.803105i	$0.703180\pi$
$998$	26.6225	0.842721
$999$	1.42045	0.0449411

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8049.2.a.b.1.3	✓	104

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8049.2.a.b.1.3	✓	104	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:	\( N \)	\(=\)	\( 8049 = 3 \cdot 2683 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8049.a (trivial)

\(f(q)\)	\(=\)	\(q-2.68757 q^{2} -1.00000 q^{3} +5.22302 q^{4} +2.72258 q^{5} +2.68757 q^{6} -1.36352 q^{7} -8.66207 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.68757 q^{2} -1.00000 q^{3} +5.22302 q^{4} +2.72258 q^{5} +2.68757 q^{6} -1.36352 q^{7} -8.66207 q^{8} +1.00000 q^{9} -7.31712 q^{10} -2.97086 q^{11} -5.22302 q^{12} +5.64512 q^{13} +3.66456 q^{14} -2.72258 q^{15} +12.8339 q^{16} +3.24798 q^{17} -2.68757 q^{18} +0.763077 q^{19} +14.2201 q^{20} +1.36352 q^{21} +7.98438 q^{22} -1.93376 q^{23} +8.66207 q^{24} +2.41244 q^{25} -15.1716 q^{26} -1.00000 q^{27} -7.12171 q^{28} +0.0377820 q^{29} +7.31712 q^{30} +6.48532 q^{31} -17.1677 q^{32} +2.97086 q^{33} -8.72917 q^{34} -3.71230 q^{35} +5.22302 q^{36} -1.42045 q^{37} -2.05082 q^{38} -5.64512 q^{39} -23.5832 q^{40} -3.12949 q^{41} -3.66456 q^{42} -11.0489 q^{43} -15.5168 q^{44} +2.72258 q^{45} +5.19710 q^{46} +1.30173 q^{47} -12.8339 q^{48} -5.14080 q^{49} -6.48360 q^{50} -3.24798 q^{51} +29.4846 q^{52} -7.18369 q^{53} +2.68757 q^{54} -8.08840 q^{55} +11.8109 q^{56} -0.763077 q^{57} -0.101542 q^{58} -10.0409 q^{59} -14.2201 q^{60} +0.969751 q^{61} -17.4297 q^{62} -1.36352 q^{63} +20.4717 q^{64} +15.3693 q^{65} -7.98438 q^{66} +7.67453 q^{67} +16.9643 q^{68} +1.93376 q^{69} +9.97707 q^{70} -15.0749 q^{71} -8.66207 q^{72} +10.7120 q^{73} +3.81756 q^{74} -2.41244 q^{75} +3.98556 q^{76} +4.05084 q^{77} +15.1716 q^{78} -11.8498 q^{79} +34.9412 q^{80} +1.00000 q^{81} +8.41071 q^{82} +1.29760 q^{83} +7.12171 q^{84} +8.84289 q^{85} +29.6947 q^{86} -0.0377820 q^{87} +25.7338 q^{88} +15.0291 q^{89} -7.31712 q^{90} -7.69726 q^{91} -10.1000 q^{92} -6.48532 q^{93} -3.49850 q^{94} +2.07754 q^{95} +17.1677 q^{96} -0.766873 q^{97} +13.8162 q^{98} -2.97086 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9}+O(q^{10}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9	\( 104 q - 9 q^{2} - 104 q^{3} + 87 q^{4} - 15 q^{5} + 9 q^{6} - 10 q^{7} - 27 q^{8} + 104 q^{9} + 8 q^{10} - 52 q^{11} - 87 q^{12} + 35 q^{13} - 23 q^{14} + 15 q^{15} + 53 q^{16} - 19 q^{17} - 9 q^{18} - 22 q^{19} - 35 q^{20} + 10 q^{21} - q^{22} - 70 q^{23} + 27 q^{24} + 79 q^{25} - 39 q^{26} - 104 q^{27} - 9 q^{28} - 37 q^{29} - 8 q^{30} - 47 q^{31} - 53 q^{32} + 52 q^{33} - 17 q^{34} - 54 q^{35} + 87 q^{36} + 65 q^{37} - 33 q^{38} - 35 q^{39} + 14 q^{40} - 47 q^{41} + 23 q^{42} - 30 q^{43} - 122 q^{44} - 15 q^{45} - 6 q^{46} - 101 q^{47} - 53 q^{48} + 78 q^{49} - 64 q^{50} + 19 q^{51} + 41 q^{52} - 48 q^{53} + 9 q^{54} - 29 q^{55} - 71 q^{56} + 22 q^{57} - 2 q^{58} - 86 q^{59} + 35 q^{60} + 34 q^{61} - 36 q^{62} - 10 q^{63} - 15 q^{64} - 64 q^{65} + q^{66} - 38 q^{67} - 33 q^{68} + 70 q^{69} - 29 q^{70} - 176 q^{71} - 27 q^{72} + 69 q^{73} - 86 q^{74} - 79 q^{75} - 54 q^{76} - 45 q^{77} + 39 q^{78} - 101 q^{79} - 76 q^{80} + 104 q^{81} + 38 q^{82} - 67 q^{83} + 9 q^{84} + 3 q^{85} - 90 q^{86} + 37 q^{87} + 7 q^{88} - 91 q^{89} + 8 q^{90} - 47 q^{91} - 136 q^{92} + 47 q^{93} - 20 q^{94} - 130 q^{95} + 53 q^{96} + 86 q^{97} - 44 q^{98} - 52 q^{99}+O(q^{100}) \) 104 * q - 9 * q^2 - 104 * q^3 + 87 * q^4 - 15 * q^5 + 9 * q^6 - 10 * q^7 - 27 * q^8 + 104 * q^9 + 8 * q^10 - 52 * q^11 - 87 * q^12 + 35 * q^13 - 23 * q^14 + 15 * q^15 + 53 * q^16 - 19 * q^17 - 9 * q^18 - 22 * q^19 - 35 * q^20 + 10 * q^21 - q^22 - 70 * q^23 + 27 * q^24 + 79 * q^25 - 39 * q^26 - 104 * q^27 - 9 * q^28 - 37 * q^29 - 8 * q^30 - 47 * q^31 - 53 * q^32 + 52 * q^33 - 17 * q^34 - 54 * q^35 + 87 * q^36 + 65 * q^37 - 33 * q^38 - 35 * q^39 + 14 * q^40 - 47 * q^41 + 23 * q^42 - 30 * q^43 - 122 * q^44 - 15 * q^45 - 6 * q^46 - 101 * q^47 - 53 * q^48 + 78 * q^49 - 64 * q^50 + 19 * q^51 + 41 * q^52 - 48 * q^53 + 9 * q^54 - 29 * q^55 - 71 * q^56 + 22 * q^57 - 2 * q^58 - 86 * q^59 + 35 * q^60 + 34 * q^61 - 36 * q^62 - 10 * q^63 - 15 * q^64 - 64 * q^65 + q^66 - 38 * q^67 - 33 * q^68 + 70 * q^69 - 29 * q^70 - 176 * q^71 - 27 * q^72 + 69 * q^73 - 86 * q^74 - 79 * q^75 - 54 * q^76 - 45 * q^77 + 39 * q^78 - 101 * q^79 - 76 * q^80 + 104 * q^81 + 38 * q^82 - 67 * q^83 + 9 * q^84 + 3 * q^85 - 90 * q^86 + 37 * q^87 + 7 * q^88 - 91 * q^89 + 8 * q^90 - 47 * q^91 - 136 * q^92 + 47 * q^93 - 20 * q^94 - 130 * q^95 + 53 * q^96 + 86 * q^97 - 44 * q^98 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more