LMFDB - Embedded newform 8041.2.a.j.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8041, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8041.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

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

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8041.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.55417 q^{2} -3.23448 q^{3} +4.52378 q^{4} -3.35627 q^{5} +8.26140 q^{6} +1.37818 q^{7} -6.44616 q^{8} +7.46183 q^{9} +O(q^{10})$	\(q-2.55417 q^{2} -3.23448 q^{3} +4.52378 q^{4} -3.35627 q^{5} +8.26140 q^{6} +1.37818 q^{7} -6.44616 q^{8} +7.46183 q^{9} +8.57249 q^{10} +1.00000 q^{11} -14.6321 q^{12} -5.93482 q^{13} -3.52011 q^{14} +10.8558 q^{15} +7.41702 q^{16} +1.00000 q^{17} -19.0588 q^{18} +0.524776 q^{19} -15.1830 q^{20} -4.45770 q^{21} -2.55417 q^{22} +5.85725 q^{23} +20.8499 q^{24} +6.26457 q^{25} +15.1585 q^{26} -14.4317 q^{27} +6.23460 q^{28} -0.806143 q^{29} -27.7275 q^{30} -8.85305 q^{31} -6.05200 q^{32} -3.23448 q^{33} -2.55417 q^{34} -4.62556 q^{35} +33.7557 q^{36} -1.29326 q^{37} -1.34037 q^{38} +19.1960 q^{39} +21.6351 q^{40} +11.2002 q^{41} +11.3857 q^{42} +1.00000 q^{43} +4.52378 q^{44} -25.0440 q^{45} -14.9604 q^{46} -9.73357 q^{47} -23.9902 q^{48} -5.10061 q^{49} -16.0008 q^{50} -3.23448 q^{51} -26.8478 q^{52} +5.69213 q^{53} +36.8610 q^{54} -3.35627 q^{55} -8.88399 q^{56} -1.69738 q^{57} +2.05903 q^{58} +10.4120 q^{59} +49.1092 q^{60} -2.19688 q^{61} +22.6122 q^{62} +10.2838 q^{63} +0.623802 q^{64} +19.9189 q^{65} +8.26140 q^{66} +2.89361 q^{67} +4.52378 q^{68} -18.9451 q^{69} +11.8145 q^{70} +8.86691 q^{71} -48.1002 q^{72} -11.0044 q^{73} +3.30320 q^{74} -20.2626 q^{75} +2.37397 q^{76} +1.37818 q^{77} -49.0299 q^{78} +11.8348 q^{79} -24.8935 q^{80} +24.2935 q^{81} -28.6071 q^{82} -15.7774 q^{83} -20.1657 q^{84} -3.35627 q^{85} -2.55417 q^{86} +2.60745 q^{87} -6.44616 q^{88} -1.57241 q^{89} +63.9665 q^{90} -8.17928 q^{91} +26.4969 q^{92} +28.6350 q^{93} +24.8612 q^{94} -1.76129 q^{95} +19.5751 q^{96} +12.2296 q^{97} +13.0278 q^{98} +7.46183 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9}+O(q^{10}) $ 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9	\( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9} + q^{10} + 82 q^{11} + 3 q^{12} + 26 q^{13} + 17 q^{14} + 66 q^{15} + 122 q^{16} + 82 q^{17} + 18 q^{18} + 12 q^{19} + 9 q^{20} + 22 q^{21} + 8 q^{22} + 50 q^{23} + 15 q^{24} + 117 q^{25} + 36 q^{26} + 30 q^{27} + 11 q^{28} + 33 q^{29} - 26 q^{30} + 40 q^{31} + 58 q^{32} + 6 q^{33} + 8 q^{34} + 16 q^{35} + 160 q^{36} + 31 q^{37} + 18 q^{38} + 41 q^{39} - 29 q^{40} + 42 q^{41} - 51 q^{42} + 82 q^{43} + 98 q^{44} - 2 q^{45} - 19 q^{46} + 84 q^{47} - 46 q^{48} + 136 q^{49} + 59 q^{50} + 6 q^{51} + 45 q^{52} + 83 q^{53} + 24 q^{54} + 11 q^{55} + 21 q^{56} + 23 q^{57} + 14 q^{58} + 96 q^{59} + 184 q^{60} - 6 q^{61} - 23 q^{62} + 8 q^{63} + 148 q^{64} + 5 q^{65} + 10 q^{66} + 78 q^{67} + 98 q^{68} + 61 q^{69} - 3 q^{70} + 155 q^{71} + 50 q^{72} - 23 q^{73} + 10 q^{74} - 19 q^{75} + 44 q^{76} + 8 q^{77} - 27 q^{78} + 31 q^{79} + 19 q^{80} + 150 q^{81} - 12 q^{82} + 54 q^{83} + 8 q^{84} + 11 q^{85} + 8 q^{86} + 20 q^{87} + 30 q^{88} + 25 q^{89} - 81 q^{90} - 14 q^{91} + 60 q^{92} + 36 q^{93} + 19 q^{94} + 111 q^{95} - 6 q^{96} + 2 q^{97} - 5 q^{98} + 108 q^{99}+O(q^{100}) \) 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9 + q^10 + 82 * q^11 + 3 * q^12 + 26 * q^13 + 17 * q^14 + 66 * q^15 + 122 * q^16 + 82 * q^17 + 18 * q^18 + 12 * q^19 + 9 * q^20 + 22 * q^21 + 8 * q^22 + 50 * q^23 + 15 * q^24 + 117 * q^25 + 36 * q^26 + 30 * q^27 + 11 * q^28 + 33 * q^29 - 26 * q^30 + 40 * q^31 + 58 * q^32 + 6 * q^33 + 8 * q^34 + 16 * q^35 + 160 * q^36 + 31 * q^37 + 18 * q^38 + 41 * q^39 - 29 * q^40 + 42 * q^41 - 51 * q^42 + 82 * q^43 + 98 * q^44 - 2 * q^45 - 19 * q^46 + 84 * q^47 - 46 * q^48 + 136 * q^49 + 59 * q^50 + 6 * q^51 + 45 * q^52 + 83 * q^53 + 24 * q^54 + 11 * q^55 + 21 * q^56 + 23 * q^57 + 14 * q^58 + 96 * q^59 + 184 * q^60 - 6 * q^61 - 23 * q^62 + 8 * q^63 + 148 * q^64 + 5 * q^65 + 10 * q^66 + 78 * q^67 + 98 * q^68 + 61 * q^69 - 3 * q^70 + 155 * q^71 + 50 * q^72 - 23 * q^73 + 10 * q^74 - 19 * q^75 + 44 * q^76 + 8 * q^77 - 27 * q^78 + 31 * q^79 + 19 * q^80 + 150 * q^81 - 12 * q^82 + 54 * q^83 + 8 * q^84 + 11 * q^85 + 8 * q^86 + 20 * q^87 + 30 * q^88 + 25 * q^89 - 81 * q^90 - 14 * q^91 + 60 * q^92 + 36 * q^93 + 19 * q^94 + 111 * q^95 - 6 * q^96 + 2 * q^97 - 5 * q^98 + 108 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.55417	−1.80607	−0.903035	−	0.429567i	$-0.858666\pi$
$2$	−2.55417	−1.80607	−0.903035	+	0.429567i	$0.858666\pi$
$3$	−3.23448	−1.86743	−0.933713	−	0.358023i	$-0.883451\pi$
$3$	−3.23448	−1.86743	−0.933713	+	0.358023i	$0.883451\pi$
$4$	4.52378	2.26189
$5$	−3.35627	−1.50097	−0.750486	−	0.660887i	$-0.770180\pi$
$5$	−3.35627	−1.50097	−0.750486	+	0.660887i	$0.770180\pi$
$6$	8.26140	3.37270
$7$	1.37818	0.520905	0.260452	−	0.965487i	$-0.416128\pi$
$7$	1.37818	0.520905	0.260452	+	0.965487i	$0.416128\pi$
$8$	−6.44616	−2.27906
$9$	7.46183	2.48728
$10$	8.57249	2.71086
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−14.6321	−4.22391
$13$	−5.93482	−1.64602	−0.823012	−	0.568024i	$-0.807708\pi$
$13$	−5.93482	−1.64602	−0.823012	+	0.568024i	$0.807708\pi$
$14$	−3.52011	−0.940790
$15$	10.8558	2.80295
$16$	7.41702	1.85425
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−19.0588	−4.49220
$19$	0.524776	0.120392	0.0601960	−	0.998187i	$-0.480827\pi$
$19$	0.524776	0.120392	0.0601960	+	0.998187i	$0.480827\pi$
$20$	−15.1830	−3.39503
$21$	−4.45770	−0.972750
$22$	−2.55417	−0.544551
$23$	5.85725	1.22132	0.610661	−	0.791892i	$-0.290904\pi$
$23$	5.85725	1.22132	0.610661	+	0.791892i	$0.290904\pi$
$24$	20.8499	4.25598
$25$	6.26457	1.25291
$26$	15.1585	2.97283
$27$	−14.4317	−2.77738
$28$	6.23460	1.17823
$29$	−0.806143	−0.149697	−0.0748485	−	0.997195i	$-0.523847\pi$
$29$	−0.806143	−0.149697	−0.0748485	+	0.997195i	$0.523847\pi$
$30$	−27.7275	−5.06233
$31$	−8.85305	−1.59005	−0.795027	−	0.606574i	$-0.792543\pi$
$31$	−8.85305	−1.59005	−0.795027	+	0.606574i	$0.792543\pi$
$32$	−6.05200	−1.06985
$33$	−3.23448	−0.563050
$34$	−2.55417	−0.438036
$35$	−4.62556	−0.781863
$36$	33.7557	5.62595
$37$	−1.29326	−0.212610	−0.106305	−	0.994334i	$-0.533902\pi$
$37$	−1.29326	−0.212610	−0.106305	+	0.994334i	$0.533902\pi$
$38$	−1.34037	−0.217436
$39$	19.1960	3.07383
$40$	21.6351	3.42080
$41$	11.2002	1.74917	0.874587	−	0.484870i	$-0.161133\pi$
$41$	11.2002	1.74917	0.874587	+	0.484870i	$0.161133\pi$
$42$	11.3857	1.75686
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	4.52378	0.681985
$45$	−25.0440	−3.73333
$46$	−14.9604	−2.20579
$47$	−9.73357	−1.41979	−0.709893	−	0.704309i	$-0.751257\pi$
$47$	−9.73357	−1.41979	−0.709893	+	0.704309i	$0.751257\pi$
$48$	−23.9902	−3.46268
$49$	−5.10061	−0.728658
$50$	−16.0008	−2.26285
$51$	−3.23448	−0.452917
$52$	−26.8478	−3.72312
$53$	5.69213	0.781874	0.390937	−	0.920417i	$-0.372151\pi$
$53$	5.69213	0.781874	0.390937	+	0.920417i	$0.372151\pi$
$54$	36.8610	5.01615
$55$	−3.35627	−0.452560
$56$	−8.88399	−1.18717
$57$	−1.69738	−0.224823
$58$	2.05903	0.270363
$59$	10.4120	1.35553	0.677763	−	0.735281i	$-0.262950\pi$
$59$	10.4120	1.35553	0.677763	+	0.735281i	$0.262950\pi$
$60$	49.1092	6.33997
$61$	−2.19688	−0.281282	−0.140641	−	0.990061i	$-0.544916\pi$
$61$	−2.19688	−0.281282	−0.140641	+	0.990061i	$0.544916\pi$
$62$	22.6122	2.87175
$63$	10.2838	1.29563
$64$	0.623802	0.0779752
$65$	19.9189	2.47063
$66$	8.26140	1.01691
$67$	2.89361	0.353511	0.176755	−	0.984255i	$-0.443440\pi$
$67$	2.89361	0.353511	0.176755	+	0.984255i	$0.443440\pi$
$68$	4.52378	0.548589
$69$	−18.9451	−2.28073
$70$	11.8145	1.41210
$71$	8.86691	1.05231	0.526154	−	0.850389i	$-0.323633\pi$
$71$	8.86691	1.05231	0.526154	+	0.850389i	$0.323633\pi$
$72$	−48.1002	−5.66866
$73$	−11.0044	−1.28796	−0.643981	−	0.765041i	$-0.722719\pi$
$73$	−11.0044	−1.28796	−0.643981	+	0.765041i	$0.722719\pi$
$74$	3.30320	0.383989
$75$	−20.2626	−2.33972
$76$	2.37397	0.272313
$77$	1.37818	0.157059
$78$	−49.0299	−5.55155
$79$	11.8348	1.33152	0.665758	−	0.746167i	$-0.268108\pi$
$79$	11.8348	1.33152	0.665758	+	0.746167i	$0.268108\pi$
$80$	−24.8935	−2.78318
$81$	24.2935	2.69927
$82$	−28.6071	−3.15913
$83$	−15.7774	−1.73179	−0.865895	−	0.500226i	$-0.833250\pi$
$83$	−15.7774	−1.73179	−0.865895	+	0.500226i	$0.833250\pi$
$84$	−20.1657	−2.20025
$85$	−3.35627	−0.364039
$86$	−2.55417	−0.275423
$87$	2.60745	0.279548
$88$	−6.44616	−0.687163
$89$	−1.57241	−0.166675	−0.0833376	−	0.996521i	$-0.526558\pi$
$89$	−1.57241	−0.166675	−0.0833376	+	0.996521i	$0.526558\pi$
$90$	63.9665	6.74266
$91$	−8.17928	−0.857421
$92$	26.4969	2.76249
$93$	28.6350	2.96931
$94$	24.8612	2.56423
$95$	−1.76129	−0.180705
$96$	19.5751	1.99787
$97$	12.2296	1.24173	0.620864	−	0.783918i	$-0.286782\pi$
$97$	12.2296	1.24173	0.620864	+	0.783918i	$0.286782\pi$
$98$	13.0278	1.31601
$99$	7.46183	0.749943
$100$	28.3395	2.83395
$101$	13.2908	1.32249	0.661243	−	0.750172i	$-0.270029\pi$
$101$	13.2908	1.32249	0.661243	+	0.750172i	$0.270029\pi$
$102$	8.26140	0.818000
$103$	14.7617	1.45451	0.727257	−	0.686365i	$-0.240795\pi$
$103$	14.7617	1.45451	0.727257	+	0.686365i	$0.240795\pi$
$104$	38.2568	3.75139
$105$	14.9613	1.46007
$106$	−14.5387	−1.41212
$107$	1.55105	0.149946	0.0749730	−	0.997186i	$-0.476113\pi$
$107$	1.55105	0.149946	0.0749730	+	0.997186i	$0.476113\pi$
$108$	−65.2858	−6.28213
$109$	7.67263	0.734904	0.367452	−	0.930042i	$-0.380230\pi$
$109$	7.67263	0.734904	0.367452	+	0.930042i	$0.380230\pi$
$110$	8.57249	0.817355
$111$	4.18301	0.397034
$112$	10.2220	0.965890
$113$	7.41517	0.697561	0.348780	−	0.937205i	$-0.386596\pi$
$113$	7.41517	0.697561	0.348780	+	0.937205i	$0.386596\pi$
$114$	4.33539	0.406046
$115$	−19.6585	−1.83317
$116$	−3.64681	−0.338598
$117$	−44.2847	−4.09412
$118$	−26.5940	−2.44817
$119$	1.37818	0.126338
$120$	−69.9781	−6.38810
$121$	1.00000	0.0909091
$122$	5.61120	0.508014
$123$	−36.2267	−3.26645
$124$	−40.0492	−3.59653
$125$	−4.24425	−0.379617
$126$	−26.2665	−2.34001
$127$	0.552737	0.0490474	0.0245237	−	0.999699i	$-0.492193\pi$
$127$	0.552737	0.0490474	0.0245237	+	0.999699i	$0.492193\pi$
$128$	10.5107	0.929025
$129$	−3.23448	−0.284780
$130$	−50.8762	−4.46214
$131$	1.76387	0.154110	0.0770548	−	0.997027i	$-0.475448\pi$
$131$	1.76387	0.154110	0.0770548	+	0.997027i	$0.475448\pi$
$132$	−14.6321	−1.27356
$133$	0.723239	0.0627127
$134$	−7.39077	−0.638465
$135$	48.4367	4.16877
$136$	−6.44616	−0.552754
$137$	4.44502	0.379764	0.189882	−	0.981807i	$-0.439189\pi$
$137$	4.44502	0.379764	0.189882	+	0.981807i	$0.439189\pi$
$138$	48.3891	4.11915
$139$	−2.58705	−0.219431	−0.109715	−	0.993963i	$-0.534994\pi$
$139$	−2.58705	−0.219431	−0.109715	+	0.993963i	$0.534994\pi$
$140$	−20.9250	−1.76849
$141$	31.4830	2.65135
$142$	−22.6476	−1.90054
$143$	−5.93482	−0.496295
$144$	55.3446	4.61205
$145$	2.70564	0.224691
$146$	28.1070	2.32615
$147$	16.4978	1.36072
$148$	−5.85041	−0.480901
$149$	0.313474	0.0256808	0.0128404	−	0.999918i	$-0.495913\pi$
$149$	0.313474	0.0256808	0.0128404	+	0.999918i	$0.495913\pi$
$150$	51.7541	4.22571
$151$	5.19538	0.422794	0.211397	−	0.977400i	$-0.432199\pi$
$151$	5.19538	0.422794	0.211397	+	0.977400i	$0.432199\pi$
$152$	−3.38279	−0.274381
$153$	7.46183	0.603254
$154$	−3.52011	−0.283659
$155$	29.7132	2.38662
$156$	86.8387	6.95266
$157$	−3.30167	−0.263502	−0.131751	−	0.991283i	$-0.542060\pi$
$157$	−3.30167	−0.263502	−0.131751	+	0.991283i	$0.542060\pi$
$158$	−30.2280	−2.40481
$159$	−18.4111	−1.46009
$160$	20.3122	1.60582
$161$	8.07237	0.636192
$162$	−62.0496	−4.87508
$163$	−22.4799	−1.76076	−0.880380	−	0.474269i	$-0.842712\pi$
$163$	−22.4799	−1.76076	−0.880380	+	0.474269i	$0.842712\pi$
$164$	50.6671	3.95644
$165$	10.8558	0.845122
$166$	40.2980	3.12773
$167$	1.31422	0.101697	0.0508486	−	0.998706i	$-0.483807\pi$
$167$	1.31422	0.101697	0.0508486	+	0.998706i	$0.483807\pi$
$168$	28.7351	2.21696
$169$	22.2221	1.70939
$170$	8.57249	0.657480
$171$	3.91580	0.299448
$172$	4.52378	0.344935
$173$	−21.0326	−1.59908	−0.799540	−	0.600613i	$-0.794923\pi$
$173$	−21.0326	−1.59908	−0.799540	+	0.600613i	$0.794923\pi$
$174$	−6.65987	−0.504883
$175$	8.63373	0.652649
$176$	7.41702	0.559079
$177$	−33.6773	−2.53134
$178$	4.01620	0.301027
$179$	−3.94618	−0.294952	−0.147476	−	0.989066i	$-0.547115\pi$
$179$	−3.94618	−0.294952	−0.147476	+	0.989066i	$0.547115\pi$
$180$	−113.293	−8.44439
$181$	−15.3567	−1.14145	−0.570727	−	0.821140i	$-0.693338\pi$
$181$	−15.3567	−1.14145	−0.570727	+	0.821140i	$0.693338\pi$
$182$	20.8913	1.54856
$183$	7.10575	0.525272
$184$	−37.7568	−2.78347
$185$	4.34052	0.319122
$186$	−73.1385	−5.36278
$187$	1.00000	0.0731272
$188$	−44.0325	−3.21140
$189$	−19.8895	−1.44675
$190$	4.49864	0.326366
$191$	4.06337	0.294015	0.147008	−	0.989135i	$-0.453036\pi$
$191$	4.06337	0.294015	0.147008	+	0.989135i	$0.453036\pi$
$192$	−2.01767	−0.145613
$193$	−13.1024	−0.943133	−0.471566	−	0.881831i	$-0.656311\pi$
$193$	−13.1024	−0.943133	−0.471566	+	0.881831i	$0.656311\pi$
$194$	−31.2365	−2.24265
$195$	−64.4272	−4.61373
$196$	−23.0740	−1.64814
$197$	20.1168	1.43327	0.716633	−	0.697451i	$-0.245682\pi$
$197$	20.1168	1.43327	0.716633	+	0.697451i	$0.245682\pi$
$198$	−19.0588	−1.35445
$199$	−16.5753	−1.17499	−0.587496	−	0.809227i	$-0.699886\pi$
$199$	−16.5753	−1.17499	−0.587496	+	0.809227i	$0.699886\pi$
$200$	−40.3824	−2.85547
$201$	−9.35931	−0.660155
$202$	−33.9470	−2.38850
$203$	−1.11101	−0.0779778
$204$	−14.6321	−1.02445
$205$	−37.5908	−2.62546
$206$	−37.7039	−2.62696
$207$	43.7058	3.03777
$208$	−44.0187	−3.05215
$209$	0.524776	0.0362995
$210$	−38.2136	−2.63699
$211$	−0.334550	−0.0230314	−0.0115157	−	0.999934i	$-0.503666\pi$
$211$	−0.334550	−0.0230314	−0.0115157	+	0.999934i	$0.503666\pi$
$212$	25.7499	1.76851
$213$	−28.6798	−1.96511
$214$	−3.96165	−0.270813
$215$	−3.35627	−0.228896
$216$	93.0290	6.32982
$217$	−12.2011	−0.828266
$218$	−19.5972	−1.32729
$219$	35.5933	2.40517
$220$	−15.1830	−1.02364
$221$	−5.93482	−0.399219
$222$	−10.6841	−0.717071
$223$	−5.22511	−0.349899	−0.174950	−	0.984577i	$-0.555976\pi$
$223$	−5.22511	−0.349899	−0.174950	+	0.984577i	$0.555976\pi$
$224$	−8.34077	−0.557291
$225$	46.7452	3.11635
$226$	−18.9396	−1.25984
$227$	−9.06116	−0.601410	−0.300705	−	0.953717i	$-0.597222\pi$
$227$	−9.06116	−0.601410	−0.300705	+	0.953717i	$0.597222\pi$
$228$	−7.67856	−0.508525
$229$	−2.89718	−0.191451	−0.0957257	−	0.995408i	$-0.530517\pi$
$229$	−2.89718	−0.191451	−0.0957257	+	0.995408i	$0.530517\pi$
$230$	50.2112	3.31083
$231$	−4.45770	−0.293295
$232$	5.19652	0.341169
$233$	−24.5253	−1.60671	−0.803354	−	0.595502i	$-0.796953\pi$
$233$	−24.5253	−1.60671	−0.803354	+	0.595502i	$0.796953\pi$
$234$	113.111	7.39427
$235$	32.6685	2.13106
$236$	47.1015	3.06605
$237$	−38.2793	−2.48651
$238$	−3.52011	−0.228175
$239$	−8.04199	−0.520193	−0.260096	−	0.965583i	$-0.583754\pi$
$239$	−8.04199	−0.520193	−0.260096	+	0.965583i	$0.583754\pi$
$240$	80.5176	5.19739
$241$	−17.3835	−1.11977	−0.559885	−	0.828570i	$-0.689155\pi$
$241$	−17.3835	−1.11977	−0.559885	+	0.828570i	$0.689155\pi$
$242$	−2.55417	−0.164188
$243$	−35.2815	−2.26331
$244$	−9.93819	−0.636228
$245$	17.1190	1.09370
$246$	92.5291	5.89944
$247$	−3.11446	−0.198168
$248$	57.0681	3.62383
$249$	51.0315	3.23399
$250$	10.8405	0.685615
$251$	−23.3782	−1.47562	−0.737810	−	0.675009i	$-0.764140\pi$
$251$	−23.3782	−1.47562	−0.737810	+	0.675009i	$0.764140\pi$
$252$	46.5216	2.93058
$253$	5.85725	0.368242
$254$	−1.41178	−0.0885831
$255$	10.8558	0.679816
$256$	−28.0937	−1.75586
$257$	9.17874	0.572554	0.286277	−	0.958147i	$-0.407582\pi$
$257$	9.17874	0.572554	0.286277	+	0.958147i	$0.407582\pi$
$258$	8.26140	0.514332
$259$	−1.78235	−0.110750
$260$	90.1087	5.58830
$261$	−6.01530	−0.372338
$262$	−4.50521	−0.278333
$263$	−25.7970	−1.59071	−0.795355	−	0.606143i	$-0.792716\pi$
$263$	−25.7970	−1.59071	−0.795355	+	0.606143i	$0.792716\pi$
$264$	20.8499	1.28323
$265$	−19.1044	−1.17357
$266$	−1.84727	−0.113264
$267$	5.08592	0.311254
$268$	13.0901	0.799602
$269$	8.71268	0.531222	0.265611	−	0.964080i	$-0.414426\pi$
$269$	8.71268	0.531222	0.265611	+	0.964080i	$0.414426\pi$
$270$	−123.716	−7.52909
$271$	31.9214	1.93909	0.969543	−	0.244923i	$-0.0787626\pi$
$271$	31.9214	1.93909	0.969543	+	0.244923i	$0.0787626\pi$
$272$	7.41702	0.449723
$273$	26.4557	1.60117
$274$	−11.3533	−0.685880
$275$	6.26457	0.377768
$276$	−85.7036	−5.15875
$277$	−22.4204	−1.34711	−0.673557	−	0.739135i	$-0.735234\pi$
$277$	−22.4204	−1.34711	−0.673557	+	0.739135i	$0.735234\pi$
$278$	6.60776	0.396307
$279$	−66.0600	−3.95491
$280$	29.8171	1.78191
$281$	−6.59654	−0.393516	−0.196758	−	0.980452i	$-0.563041\pi$
$281$	−6.59654	−0.393516	−0.196758	+	0.980452i	$0.563041\pi$
$282$	−80.4129	−4.78852
$283$	1.26338	0.0750999	0.0375500	−	0.999295i	$-0.488045\pi$
$283$	1.26338	0.0750999	0.0375500	+	0.999295i	$0.488045\pi$
$284$	40.1119	2.38021
$285$	5.69686	0.337453
$286$	15.1585	0.896343
$287$	15.4359	0.911152
$288$	−45.1590	−2.66102
$289$	1.00000	0.0588235
$290$	−6.91065	−0.405807
$291$	−39.5564	−2.31884
$292$	−49.7813	−2.91323
$293$	31.0522	1.81409	0.907043	−	0.421037i	$-0.138334\pi$
$293$	31.0522	1.81409	0.907043	+	0.421037i	$0.138334\pi$
$294$	−42.1382	−2.45755
$295$	−34.9455	−2.03460
$296$	8.33654	0.484552
$297$	−14.4317	−0.837412
$298$	−0.800665	−0.0463813
$299$	−34.7617	−2.01032
$300$	−91.6636	−5.29220
$301$	1.37818	0.0794372
$302$	−13.2699	−0.763596
$303$	−42.9888	−2.46964
$304$	3.89228	0.223237
$305$	7.37333	0.422195
$306$	−19.0588	−1.08952
$307$	20.5178	1.17101	0.585506	−	0.810668i	$-0.300896\pi$
$307$	20.5178	1.17101	0.585506	+	0.810668i	$0.300896\pi$
$308$	6.23460	0.355249
$309$	−47.7464	−2.71620
$310$	−75.8926	−4.31041
$311$	−14.8101	−0.839806	−0.419903	−	0.907569i	$-0.637936\pi$
$311$	−14.8101	−0.839806	−0.419903	+	0.907569i	$0.637936\pi$
$312$	−123.741	−7.00544
$313$	32.8935	1.85925	0.929626	−	0.368505i	$-0.120130\pi$
$313$	32.8935	1.85925	0.929626	+	0.368505i	$0.120130\pi$
$314$	8.43303	0.475903
$315$	−34.5152	−1.94471
$316$	53.5379	3.01174
$317$	7.02870	0.394771	0.197386	−	0.980326i	$-0.436755\pi$
$317$	7.02870	0.394771	0.197386	+	0.980326i	$0.436755\pi$
$318$	47.0250	2.63703
$319$	−0.806143	−0.0451353
$320$	−2.09365	−0.117039
$321$	−5.01684	−0.280013
$322$	−20.6182	−1.14901
$323$	0.524776	0.0291993
$324$	109.898	6.10546
$325$	−37.1791	−2.06233
$326$	57.4174	3.18006
$327$	−24.8169	−1.37238
$328$	−72.1981	−3.98647
$329$	−13.4146	−0.739574
$330$	−27.7275	−1.52635
$331$	12.0888	0.664463	0.332231	−	0.943198i	$-0.392199\pi$
$331$	12.0888	0.664463	0.332231	+	0.943198i	$0.392199\pi$
$332$	−71.3733	−3.91712
$333$	−9.65007	−0.528821
$334$	−3.35673	−0.183672
$335$	−9.71175	−0.530610
$336$	−33.0629	−1.80373
$337$	−13.8127	−0.752424	−0.376212	−	0.926534i	$-0.622774\pi$
$337$	−13.8127	−0.752424	−0.376212	+	0.926534i	$0.622774\pi$
$338$	−56.7591	−3.08729
$339$	−23.9842	−1.30264
$340$	−15.1830	−0.823416
$341$	−8.85305	−0.479419
$342$	−10.0016	−0.540825
$343$	−16.6769	−0.900466
$344$	−6.44616	−0.347554
$345$	63.5850	3.42330
$346$	53.7209	2.88805
$347$	−8.94658	−0.480277	−0.240139	−	0.970739i	$-0.577193\pi$
$347$	−8.94658	−0.480277	−0.240139	+	0.970739i	$0.577193\pi$
$348$	11.7955	0.632307
$349$	8.83961	0.473174	0.236587	−	0.971610i	$-0.423971\pi$
$349$	8.83961	0.473174	0.236587	+	0.971610i	$0.423971\pi$
$350$	−22.0520	−1.17873
$351$	85.6496	4.57164
$352$	−6.05200	−0.322573
$353$	−11.6584	−0.620513	−0.310257	−	0.950653i	$-0.600415\pi$
$353$	−11.6584	−0.620513	−0.310257	+	0.950653i	$0.600415\pi$
$354$	86.0176	4.57178
$355$	−29.7598	−1.57948
$356$	−7.11324	−0.377001
$357$	−4.45770	−0.235927
$358$	10.0792	0.532703
$359$	7.79998	0.411667	0.205834	−	0.978587i	$-0.434009\pi$
$359$	7.79998	0.411667	0.205834	+	0.978587i	$0.434009\pi$
$360$	161.437	8.50849
$361$	−18.7246	−0.985506
$362$	39.2236	2.06154
$363$	−3.23448	−0.169766
$364$	−37.0013	−1.93939
$365$	36.9336	1.93319
$366$	−18.1493	−0.948679
$367$	−18.7115	−0.976731	−0.488366	−	0.872639i	$-0.662407\pi$
$367$	−18.7115	−0.976731	−0.488366	+	0.872639i	$0.662407\pi$
$368$	43.4433	2.26464
$369$	83.5738	4.35068
$370$	−11.0864	−0.576356
$371$	7.84481	0.407282
$372$	129.538	6.71624
$373$	−15.9039	−0.823475	−0.411738	−	0.911302i	$-0.635078\pi$
$373$	−15.9039	−0.823475	−0.411738	+	0.911302i	$0.635078\pi$
$374$	−2.55417	−0.132073
$375$	13.7279	0.708907
$376$	62.7441	3.23578
$377$	4.78432	0.246405
$378$	50.8012	2.61293
$379$	11.1411	0.572278	0.286139	−	0.958188i	$-0.407628\pi$
$379$	11.1411	0.572278	0.286139	+	0.958188i	$0.407628\pi$
$380$	−7.96770	−0.408735
$381$	−1.78781	−0.0915925
$382$	−10.3785	−0.531012
$383$	−3.19351	−0.163181	−0.0815905	−	0.996666i	$-0.526000\pi$
$383$	−3.19351	−0.163181	−0.0815905	+	0.996666i	$0.526000\pi$
$384$	−33.9966	−1.73488
$385$	−4.62556	−0.235740
$386$	33.4658	1.70336
$387$	7.46183	0.379306
$388$	55.3240	2.80865
$389$	1.08134	0.0548260	0.0274130	−	0.999624i	$-0.491273\pi$
$389$	1.08134	0.0548260	0.0274130	+	0.999624i	$0.491273\pi$
$390$	164.558	8.33271
$391$	5.85725	0.296214
$392$	32.8793	1.66066
$393$	−5.70518	−0.287788
$394$	−51.3818	−2.58858
$395$	−39.7208	−1.99857
$396$	33.7557	1.69629
$397$	−31.7452	−1.59325	−0.796624	−	0.604475i	$-0.793383\pi$
$397$	−31.7452	−1.59325	−0.796624	+	0.604475i	$0.793383\pi$
$398$	42.3361	2.12212
$399$	−2.33930	−0.117111
$400$	46.4644	2.32322
$401$	−27.3622	−1.36640	−0.683201	−	0.730231i	$-0.739413\pi$
$401$	−27.3622	−1.36640	−0.683201	+	0.730231i	$0.739413\pi$
$402$	23.9053	1.19229
$403$	52.5413	2.61727
$404$	60.1247	2.99132
$405$	−81.5355	−4.05153
$406$	2.83772	0.140833
$407$	−1.29326	−0.0641044
$408$	20.8499	1.03223
$409$	10.0088	0.494903	0.247451	−	0.968900i	$-0.420407\pi$
$409$	10.0088	0.494903	0.247451	+	0.968900i	$0.420407\pi$
$410$	96.0134	4.74176
$411$	−14.3773	−0.709181
$412$	66.7787	3.28995
$413$	14.3496	0.706100
$414$	−111.632	−5.48642
$415$	52.9531	2.59937
$416$	35.9176	1.76100
$417$	8.36775	0.409771
$418$	−1.34037	−0.0655595
$419$	−30.7938	−1.50438	−0.752188	−	0.658948i	$-0.771002\pi$
$419$	−30.7938	−1.50438	−0.752188	+	0.658948i	$0.771002\pi$
$420$	67.6815	3.30252
$421$	−20.8669	−1.01699	−0.508495	−	0.861065i	$-0.669798\pi$
$421$	−20.8669	−1.01699	−0.508495	+	0.861065i	$0.669798\pi$
$422$	0.854497	0.0415963
$423$	−72.6303	−3.53140
$424$	−36.6924	−1.78194
$425$	6.26457	0.303876
$426$	73.2530	3.54912
$427$	−3.02770	−0.146521
$428$	7.01662	0.339161
$429$	19.1960	0.926794
$430$	8.57249	0.413402
$431$	5.79030	0.278909	0.139455	−	0.990228i	$-0.455465\pi$
$431$	5.79030	0.278909	0.139455	+	0.990228i	$0.455465\pi$
$432$	−107.040	−5.14997
$433$	−23.5918	−1.13375	−0.566875	−	0.823804i	$-0.691848\pi$
$433$	−23.5918	−1.13375	−0.566875	+	0.823804i	$0.691848\pi$
$434$	31.1637	1.49591
$435$	−8.75131	−0.419593
$436$	34.7093	1.66227
$437$	3.07375	0.147037
$438$	−90.9114	−4.34391
$439$	−16.9166	−0.807388	−0.403694	−	0.914894i	$-0.632274\pi$
$439$	−16.9166	−0.807388	−0.403694	+	0.914894i	$0.632274\pi$
$440$	21.6351	1.03141
$441$	−38.0599	−1.81238
$442$	15.1585	0.721018
$443$	6.90701	0.328162	0.164081	−	0.986447i	$-0.447534\pi$
$443$	6.90701	0.328162	0.164081	+	0.986447i	$0.447534\pi$
$444$	18.9230	0.898046
$445$	5.27744	0.250175
$446$	13.3458	0.631942
$447$	−1.01392	−0.0479569
$448$	0.859714	0.0406176
$449$	34.7352	1.63926	0.819628	−	0.572897i	$-0.194180\pi$
$449$	34.7352	1.63926	0.819628	+	0.572897i	$0.194180\pi$
$450$	−119.395	−5.62834
$451$	11.2002	0.527396
$452$	33.5446	1.57781
$453$	−16.8043	−0.789537
$454$	23.1437	1.08619
$455$	27.4519	1.28696
$456$	10.9416	0.512385
$457$	32.3147	1.51162	0.755809	−	0.654793i	$-0.227244\pi$
$457$	32.3147	1.51162	0.755809	+	0.654793i	$0.227244\pi$
$458$	7.39990	0.345775
$459$	−14.4317	−0.673614
$460$	−88.9309	−4.14642
$461$	10.8304	0.504422	0.252211	−	0.967672i	$-0.418842\pi$
$461$	10.8304	0.504422	0.252211	+	0.967672i	$0.418842\pi$
$462$	11.3857	0.529712
$463$	24.5364	1.14030	0.570152	−	0.821539i	$-0.306884\pi$
$463$	24.5364	1.14030	0.570152	+	0.821539i	$0.306884\pi$
$464$	−5.97918	−0.277576
$465$	−96.1068	−4.45684
$466$	62.6419	2.90183
$467$	−26.1616	−1.21062	−0.605308	−	0.795991i	$-0.706950\pi$
$467$	−26.1616	−1.21062	−0.605308	+	0.795991i	$0.706950\pi$
$468$	−200.334	−9.26045
$469$	3.98793	0.184145
$470$	−83.4409	−3.84884
$471$	10.6792	0.492071
$472$	−67.1173	−3.08933
$473$	1.00000	0.0459800
$474$	97.7718	4.49081
$475$	3.28750	0.150841
$476$	6.23460	0.285762
$477$	42.4737	1.94474
$478$	20.5406	0.939505
$479$	18.2621	0.834415	0.417208	−	0.908811i	$-0.363009\pi$
$479$	18.2621	0.834415	0.417208	+	0.908811i	$0.363009\pi$
$480$	−65.6993	−2.99875
$481$	7.67525	0.349961
$482$	44.4004	2.02238
$483$	−26.1099	−1.18804
$484$	4.52378	0.205626
$485$	−41.0459	−1.86380
$486$	90.1150	4.08770
$487$	21.4633	0.972597	0.486298	−	0.873793i	$-0.338347\pi$
$487$	21.4633	0.972597	0.486298	+	0.873793i	$0.338347\pi$
$488$	14.1614	0.641058
$489$	72.7106	3.28809
$490$	−43.7249	−1.97529
$491$	−32.7277	−1.47698	−0.738490	−	0.674264i	$-0.764461\pi$
$491$	−32.7277	−1.47698	−0.738490	+	0.674264i	$0.764461\pi$
$492$	−163.882	−7.38835
$493$	−0.806143	−0.0363068
$494$	7.95485	0.357905
$495$	−25.0440	−1.12564
$496$	−65.6632	−2.94836
$497$	12.2202	0.548152
$498$	−130.343	−5.84081
$499$	−35.9455	−1.60914	−0.804571	−	0.593857i	$-0.797605\pi$
$499$	−35.9455	−1.60914	−0.804571	+	0.593857i	$0.797605\pi$
$500$	−19.2000	−0.858652
$501$	−4.25080	−0.189912
$502$	59.7119	2.66507
$503$	21.9715	0.979662	0.489831	−	0.871818i	$-0.337059\pi$
$503$	21.9715	0.979662	0.489831	+	0.871818i	$0.337059\pi$
$504$	−66.2909	−2.95283
$505$	−44.6076	−1.98501
$506$	−14.9604	−0.665071
$507$	−71.8770	−3.19217
$508$	2.50046	0.110940
$509$	20.5308	0.910010	0.455005	−	0.890489i	$-0.349637\pi$
$509$	20.5308	0.910010	0.455005	+	0.890489i	$0.349637\pi$
$510$	−27.7275	−1.22779
$511$	−15.1660	−0.670906
$512$	50.7347	2.24218
$513$	−7.57341	−0.334374
$514$	−23.4441	−1.03407
$515$	−49.5443	−2.18318
$516$	−14.6321	−0.644140
$517$	−9.73357	−0.428082
$518$	4.55241	0.200022
$519$	68.0295	2.98616
$520$	−128.400	−5.63073
$521$	26.5945	1.16512	0.582562	−	0.812786i	$-0.302050\pi$
$521$	26.5945	1.16512	0.582562	+	0.812786i	$0.302050\pi$
$522$	15.3641	0.672469
$523$	−9.86949	−0.431563	−0.215781	−	0.976442i	$-0.569230\pi$
$523$	−9.86949	−0.431563	−0.215781	+	0.976442i	$0.569230\pi$
$524$	7.97934	0.348579
$525$	−27.9256	−1.21877
$526$	65.8899	2.87294
$527$	−8.85305	−0.385645
$528$	−23.9902	−1.04404
$529$	11.3074	0.491625
$530$	48.7957	2.11955
$531$	77.6925	3.37157
$532$	3.27177	0.141849
$533$	−66.4710	−2.87918
$534$	−12.9903	−0.562146
$535$	−5.20576	−0.225065
$536$	−18.6527	−0.805673
$537$	12.7638	0.550800
$538$	−22.2537	−0.959423
$539$	−5.10061	−0.219699
$540$	219.117	9.42929
$541$	0.336012	0.0144463	0.00722315	−	0.999974i	$-0.497701\pi$
$541$	0.336012	0.0144463	0.00722315	+	0.999974i	$0.497701\pi$
$542$	−81.5326	−3.50212
$543$	49.6708	2.13158
$544$	−6.05200	−0.259478
$545$	−25.7514	−1.10307
$546$	−67.5723	−2.89183
$547$	−21.9888	−0.940172	−0.470086	−	0.882621i	$-0.655777\pi$
$547$	−21.9888	−0.940172	−0.470086	+	0.882621i	$0.655777\pi$
$548$	20.1083	0.858984
$549$	−16.3927	−0.699625
$550$	−16.0008	−0.682275
$551$	−0.423045	−0.0180223
$552$	122.123	5.19791
$553$	16.3105	0.693593
$554$	57.2656	2.43298
$555$	−14.0393	−0.595936
$556$	−11.7032	−0.496328
$557$	5.51343	0.233611	0.116806	−	0.993155i	$-0.462735\pi$
$557$	5.51343	0.233611	0.116806	+	0.993155i	$0.462735\pi$
$558$	168.728	7.14284
$559$	−5.93482	−0.251016
$560$	−34.3079	−1.44977
$561$	−3.23448	−0.136560
$562$	16.8487	0.710718
$563$	10.9402	0.461075	0.230537	−	0.973063i	$-0.425952\pi$
$563$	10.9402	0.461075	0.230537	+	0.973063i	$0.425952\pi$
$564$	142.422	5.99705
$565$	−24.8873	−1.04702
$566$	−3.22688	−0.135636
$567$	33.4809	1.40606
$568$	−57.1575	−2.39827
$569$	−46.1074	−1.93292	−0.966462	−	0.256809i	$-0.917329\pi$
$569$	−46.1074	−1.93292	−0.966462	+	0.256809i	$0.917329\pi$
$570$	−14.5507	−0.609464
$571$	−10.8126	−0.452493	−0.226247	−	0.974070i	$-0.572646\pi$
$571$	−10.8126	−0.452493	−0.226247	+	0.974070i	$0.572646\pi$
$572$	−26.8478	−1.12256
$573$	−13.1429	−0.549052
$574$	−39.4259	−1.64560
$575$	36.6932	1.53021
$576$	4.65471	0.193946
$577$	28.4683	1.18515	0.592575	−	0.805515i	$-0.298111\pi$
$577$	28.4683	1.18515	0.592575	+	0.805515i	$0.298111\pi$
$578$	−2.55417	−0.106239
$579$	42.3795	1.76123
$580$	12.2397	0.508226
$581$	−21.7441	−0.902097
$582$	101.034	4.18798
$583$	5.69213	0.235744
$584$	70.9358	2.93535
$585$	148.631	6.14515
$586$	−79.3125	−3.27637
$587$	48.2610	1.99194	0.995971	−	0.0896706i	$-0.0285814\pi$
$587$	48.2610	1.99194	0.995971	+	0.0896706i	$0.0285814\pi$
$588$	74.6324	3.07779
$589$	−4.64587	−0.191430
$590$	89.2567	3.67464
$591$	−65.0674	−2.67652
$592$	−9.59211	−0.394233
$593$	37.0365	1.52091	0.760454	−	0.649392i	$-0.224977\pi$
$593$	37.0365	1.52091	0.760454	+	0.649392i	$0.224977\pi$
$594$	36.8610	1.51242
$595$	−4.62556	−0.189630
$596$	1.41809	0.0580871
$597$	53.6124	2.19421
$598$	88.7874	3.63079
$599$	23.3037	0.952165	0.476082	−	0.879401i	$-0.342056\pi$
$599$	23.3037	0.952165	0.476082	+	0.879401i	$0.342056\pi$
$600$	130.616	5.33237
$601$	36.3416	1.48240	0.741202	−	0.671282i	$-0.234256\pi$
$601$	36.3416	1.48240	0.741202	+	0.671282i	$0.234256\pi$
$602$	−3.52011	−0.143469
$603$	21.5916	0.879280
$604$	23.5028	0.956314
$605$	−3.35627	−0.136452
$606$	109.801	4.46035
$607$	−44.7723	−1.81725	−0.908627	−	0.417609i	$-0.862868\pi$
$607$	−44.7723	−1.81725	−0.908627	+	0.417609i	$0.862868\pi$
$608$	−3.17595	−0.128802
$609$	3.59355	0.145618
$610$	−18.8327	−0.762515
$611$	57.7670	2.33700
$612$	33.7557	1.36449
$613$	22.4024	0.904823	0.452411	−	0.891809i	$-0.350564\pi$
$613$	22.4024	0.904823	0.452411	+	0.891809i	$0.350564\pi$
$614$	−52.4059	−2.11493
$615$	121.587	4.90285
$616$	−8.88399	−0.357946
$617$	−24.7601	−0.996805	−0.498403	−	0.866946i	$-0.666080\pi$
$617$	−24.7601	−0.996805	−0.498403	+	0.866946i	$0.666080\pi$
$618$	121.952	4.90564
$619$	5.72666	0.230174	0.115087	−	0.993355i	$-0.463285\pi$
$619$	5.72666	0.230174	0.115087	+	0.993355i	$0.463285\pi$
$620$	134.416	5.39828
$621$	−84.5300	−3.39207
$622$	37.8276	1.51675
$623$	−2.16707	−0.0868219
$624$	142.377	5.69966
$625$	−17.0780	−0.683120
$626$	−84.0156	−3.35794
$627$	−1.69738	−0.0677867
$628$	−14.9360	−0.596013
$629$	−1.29326	−0.0515655
$630$	88.1576	3.51228
$631$	−17.6119	−0.701118	−0.350559	−	0.936541i	$-0.614008\pi$
$631$	−17.6119	−0.701118	−0.350559	+	0.936541i	$0.614008\pi$
$632$	−76.2889	−3.03461
$633$	1.08209	0.0430094
$634$	−17.9525	−0.712984
$635$	−1.85514	−0.0736188
$636$	−83.2876	−3.30257
$637$	30.2712	1.19939
$638$	2.05903	0.0815176
$639$	66.1634	2.61738
$640$	−35.2768	−1.39444
$641$	−5.36006	−0.211710	−0.105855	−	0.994382i	$-0.533758\pi$
$641$	−5.36006	−0.211710	−0.105855	+	0.994382i	$0.533758\pi$
$642$	12.8139	0.505723
$643$	−23.4496	−0.924760	−0.462380	−	0.886682i	$-0.653004\pi$
$643$	−23.4496	−0.924760	−0.462380	+	0.886682i	$0.653004\pi$
$644$	36.5176	1.43900
$645$	10.8558	0.427446
$646$	−1.34037	−0.0527361
$647$	21.7715	0.855926	0.427963	−	0.903796i	$-0.359231\pi$
$647$	21.7715	0.855926	0.427963	+	0.903796i	$0.359231\pi$
$648$	−156.600	−6.15181
$649$	10.4120	0.408706
$650$	94.9618	3.72471
$651$	39.4642	1.54673
$652$	−101.694	−3.98264
$653$	7.20019	0.281765	0.140883	−	0.990026i	$-0.455006\pi$
$653$	7.20019	0.281765	0.140883	+	0.990026i	$0.455006\pi$
$654$	63.3866	2.47861
$655$	−5.92001	−0.231314
$656$	83.0719	3.24341
$657$	−82.1127	−3.20352
$658$	34.2633	1.33572
$659$	7.27702	0.283472	0.141736	−	0.989904i	$-0.454732\pi$
$659$	7.27702	0.283472	0.141736	+	0.989904i	$0.454732\pi$
$660$	49.1092	1.91157
$661$	−8.85333	−0.344355	−0.172177	−	0.985066i	$-0.555080\pi$
$661$	−8.85333	−0.344355	−0.172177	+	0.985066i	$0.555080\pi$
$662$	−30.8769	−1.20007
$663$	19.1960	0.745513
$664$	101.703	3.94686
$665$	−2.42739	−0.0941300
$666$	24.6479	0.955087
$667$	−4.72178	−0.182828
$668$	5.94523	0.230028
$669$	16.9005	0.653411
$670$	24.8054	0.958318
$671$	−2.19688	−0.0848096
$672$	26.9780	1.04070
$673$	−40.1144	−1.54630	−0.773148	−	0.634225i	$-0.781319\pi$
$673$	−40.1144	−1.54630	−0.773148	+	0.634225i	$0.781319\pi$
$674$	35.2799	1.35893
$675$	−90.4084	−3.47982
$676$	100.528	3.86646
$677$	−38.9840	−1.49828	−0.749139	−	0.662413i	$-0.769532\pi$
$677$	−38.9840	−1.49828	−0.749139	+	0.662413i	$0.769532\pi$
$678$	61.2597	2.35266
$679$	16.8546	0.646822
$680$	21.6351	0.829667
$681$	29.3081	1.12309
$682$	22.6122	0.865865
$683$	−25.9123	−0.991506	−0.495753	−	0.868464i	$-0.665108\pi$
$683$	−25.9123	−0.991506	−0.495753	+	0.868464i	$0.665108\pi$
$684$	17.7142	0.677319
$685$	−14.9187	−0.570015
$686$	42.5955	1.62630
$687$	9.37087	0.357521
$688$	7.41702	0.282771
$689$	−33.7818	−1.28698
$690$	−162.407	−6.18273
$691$	10.3583	0.394049	0.197025	−	0.980399i	$-0.436872\pi$
$691$	10.3583	0.394049	0.197025	+	0.980399i	$0.436872\pi$
$692$	−95.1469	−3.61694
$693$	10.2838	0.390649
$694$	22.8511	0.867415
$695$	8.68285	0.329359
$696$	−16.8080	−0.637107
$697$	11.2002	0.424237
$698$	−22.5779	−0.854585
$699$	79.3266	3.00041
$700$	39.0571	1.47622
$701$	−13.2522	−0.500529	−0.250265	−	0.968178i	$-0.580518\pi$
$701$	−13.2522	−0.500529	−0.250265	+	0.968178i	$0.580518\pi$
$702$	−218.763	−8.25670
$703$	−0.678671	−0.0255966
$704$	0.623802	0.0235104
$705$	−105.666	−3.97959
$706$	29.7775	1.12069
$707$	18.3172	0.688889
$708$	−152.349	−5.72562
$709$	1.18051	0.0443348	0.0221674	−	0.999754i	$-0.492943\pi$
$709$	1.18051	0.0443348	0.0221674	+	0.999754i	$0.492943\pi$
$710$	76.0115	2.85266
$711$	88.3092	3.31185
$712$	10.1360	0.379863
$713$	−51.8545	−1.94197
$714$	11.3857	0.426100
$715$	19.9189	0.744924
$716$	−17.8517	−0.667148
$717$	26.0116	0.971422
$718$	−19.9225	−0.743500
$719$	26.7223	0.996572	0.498286	−	0.867013i	$-0.333963\pi$
$719$	26.7223	0.996572	0.498286	+	0.867013i	$0.333963\pi$
$720$	−185.752	−6.92255
$721$	20.3444	0.757663
$722$	47.8258	1.77989
$723$	56.2266	2.09109
$724$	−69.4702	−2.58184
$725$	−5.05014	−0.187557
$726$	8.26140	0.306609
$727$	22.8693	0.848175	0.424088	−	0.905621i	$-0.360595\pi$
$727$	22.8693	0.848175	0.424088	+	0.905621i	$0.360595\pi$
$728$	52.7249	1.95412
$729$	41.2369	1.52729
$730$	−94.3348	−3.49149
$731$	1.00000	0.0369863
$732$	32.1449	1.18811
$733$	−4.91443	−0.181518	−0.0907592	−	0.995873i	$-0.528929\pi$
$733$	−4.91443	−0.181518	−0.0907592	+	0.995873i	$0.528929\pi$
$734$	47.7923	1.76404
$735$	−55.3711	−2.04239
$736$	−35.4481	−1.30663
$737$	2.89361	0.106588
$738$	−213.462	−7.85763
$739$	5.04742	0.185672	0.0928362	−	0.995681i	$-0.470407\pi$
$739$	5.04742	0.185672	0.0928362	+	0.995681i	$0.470407\pi$
$740$	19.6356	0.721818
$741$	10.0736	0.370064
$742$	−20.0370	−0.735580
$743$	17.9495	0.658504	0.329252	−	0.944242i	$-0.393203\pi$
$743$	17.9495	0.658504	0.329252	+	0.944242i	$0.393203\pi$
$744$	−184.585	−6.76723
$745$	−1.05210	−0.0385461
$746$	40.6214	1.48725
$747$	−117.728	−4.30744
$748$	4.52378	0.165406
$749$	2.13764	0.0781075
$750$	−35.0634	−1.28034
$751$	17.7075	0.646156	0.323078	−	0.946372i	$-0.395282\pi$
$751$	17.7075	0.646156	0.323078	+	0.946372i	$0.395282\pi$
$752$	−72.1941	−2.63265
$753$	75.6162	2.75561
$754$	−12.2200	−0.445024
$755$	−17.4371	−0.634602
$756$	−89.9758	−3.27239
$757$	32.9132	1.19625	0.598126	−	0.801402i	$-0.295912\pi$
$757$	32.9132	1.19625	0.598126	+	0.801402i	$0.295912\pi$
$758$	−28.4561	−1.03357
$759$	−18.9451	−0.687665
$760$	11.3536	0.411837
$761$	−23.0245	−0.834636	−0.417318	−	0.908760i	$-0.637030\pi$
$761$	−23.0245	−0.834636	−0.417318	+	0.908760i	$0.637030\pi$
$762$	4.56638	0.165422
$763$	10.5743	0.382815
$764$	18.3818	0.665030
$765$	−25.0440	−0.905466
$766$	8.15677	0.294716
$767$	−61.7933	−2.23123
$768$	90.8685	3.27894
$769$	−13.7130	−0.494503	−0.247251	−	0.968951i	$-0.579527\pi$
$769$	−13.7130	−0.494503	−0.247251	+	0.968951i	$0.579527\pi$
$770$	11.8145	0.425764
$771$	−29.6884	−1.06920
$772$	−59.2725	−2.13326
$773$	2.20396	0.0792708	0.0396354	−	0.999214i	$-0.487380\pi$
$773$	2.20396	0.0792708	0.0396354	+	0.999214i	$0.487380\pi$
$774$	−19.0588	−0.685054
$775$	−55.4605	−1.99220
$776$	−78.8340	−2.82998
$777$	5.76496	0.206817
$778$	−2.76192	−0.0990196
$779$	5.87759	0.210586
$780$	−291.454	−10.4357
$781$	8.86691	0.317283
$782$	−14.9604	−0.534983
$783$	11.6340	0.415766
$784$	−37.8313	−1.35112
$785$	11.0813	0.395509
$786$	14.5720	0.519766
$787$	17.9522	0.639927	0.319964	−	0.947430i	$-0.396329\pi$
$787$	17.9522	0.639927	0.319964	+	0.947430i	$0.396329\pi$
$788$	91.0042	3.24189
$789$	83.4398	2.97053
$790$	101.454	3.60955
$791$	10.2195	0.363363
$792$	−48.1002	−1.70916
$793$	13.0381	0.462996
$794$	81.0827	2.87752
$795$	61.7926	2.19156
$796$	−74.9830	−2.65770
$797$	−38.7993	−1.37434	−0.687170	−	0.726497i	$-0.741147\pi$
$797$	−38.7993	−1.37434	−0.687170	+	0.726497i	$0.741147\pi$
$798$	5.97496	0.211511
$799$	−9.73357	−0.344349
$800$	−37.9132	−1.34043
$801$	−11.7331	−0.414568
$802$	69.8876	2.46782
$803$	−11.0044	−0.388335
$804$	−42.3395	−1.49320
$805$	−27.0931	−0.954905
$806$	−134.199	−4.72697
$807$	−28.1810	−0.992017
$808$	−85.6747	−3.01402
$809$	44.1907	1.55366	0.776831	−	0.629709i	$-0.216826\pi$
$809$	44.1907	1.55366	0.776831	+	0.629709i	$0.216826\pi$
$810$	208.255	7.31735
$811$	48.8606	1.71573	0.857864	−	0.513877i	$-0.171791\pi$
$811$	48.8606	1.71573	0.857864	+	0.513877i	$0.171791\pi$
$812$	−5.02598	−0.176377
$813$	−103.249	−3.62110
$814$	3.30320	0.115777
$815$	75.4486	2.64285
$816$	−23.9902	−0.839824
$817$	0.524776	0.0183596
$818$	−25.5641	−0.893829
$819$	−61.0324	−2.13265
$820$	−170.053	−5.93850
$821$	−38.4371	−1.34147	−0.670733	−	0.741699i	$-0.734020\pi$
$821$	−38.4371	−1.34147	−0.670733	+	0.741699i	$0.734020\pi$
$822$	36.7221	1.28083
$823$	28.2441	0.984527	0.492263	−	0.870446i	$-0.336170\pi$
$823$	28.2441	0.984527	0.492263	+	0.870446i	$0.336170\pi$
$824$	−95.1563	−3.31493
$825$	−20.2626	−0.705453
$826$	−36.6514	−1.27527
$827$	−18.6964	−0.650137	−0.325069	−	0.945690i	$-0.605387\pi$
$827$	−18.6964	−0.650137	−0.325069	+	0.945690i	$0.605387\pi$
$828$	197.716	6.87109
$829$	17.8475	0.619870	0.309935	−	0.950758i	$-0.399693\pi$
$829$	17.8475	0.619870	0.309935	+	0.950758i	$0.399693\pi$
$830$	−135.251	−4.69464
$831$	72.5184	2.51563
$832$	−3.70215	−0.128349
$833$	−5.10061	−0.176726
$834$	−21.3727	−0.740074
$835$	−4.41087	−0.152645
$836$	2.37397	0.0821056
$837$	127.764	4.41619
$838$	78.6526	2.71701
$839$	49.5585	1.71095	0.855474	−	0.517845i	$-0.173266\pi$
$839$	49.5585	1.71095	0.855474	+	0.517845i	$0.173266\pi$
$840$	−96.4427	−3.32759
$841$	−28.3501	−0.977591
$842$	53.2976	1.83676
$843$	21.3363	0.734863
$844$	−1.51343	−0.0520944
$845$	−74.5836	−2.56575
$846$	185.510	6.37797
$847$	1.37818	0.0473550
$848$	42.2187	1.44979
$849$	−4.08636	−0.140244
$850$	−16.0008	−0.548822
$851$	−7.57493	−0.259665
$852$	−129.741	−4.44486
$853$	18.4565	0.631939	0.315970	−	0.948769i	$-0.397670\pi$
$853$	18.4565	0.631939	0.315970	+	0.948769i	$0.397670\pi$
$854$	7.73327	0.264627
$855$	−13.1425	−0.449463
$856$	−9.99833	−0.341736
$857$	−43.3772	−1.48174	−0.740869	−	0.671650i	$-0.765586\pi$
$857$	−43.3772	−1.48174	−0.740869	+	0.671650i	$0.765586\pi$
$858$	−49.0299	−1.67385
$859$	26.9066	0.918041	0.459020	−	0.888426i	$-0.348201\pi$
$859$	26.9066	0.918041	0.459020	+	0.888426i	$0.348201\pi$
$860$	−15.1830	−0.517737
$861$	−49.9270	−1.70151
$862$	−14.7894	−0.503729
$863$	26.1250	0.889306	0.444653	−	0.895703i	$-0.353327\pi$
$863$	26.1250	0.889306	0.444653	+	0.895703i	$0.353327\pi$
$864$	87.3407	2.97139
$865$	70.5912	2.40017
$866$	60.2575	2.04763
$867$	−3.23448	−0.109849
$868$	−55.1952	−1.87345
$869$	11.8348	0.401467
$870$	22.3523	0.757815
$871$	−17.1731	−0.581887
$872$	−49.4590	−1.67489
$873$	91.2553	3.08852
$874$	−7.85087	−0.265560
$875$	−5.84935	−0.197744
$876$	161.016	5.44024
$877$	4.20254	0.141910	0.0709548	−	0.997480i	$-0.477395\pi$
$877$	4.20254	0.141910	0.0709548	+	0.997480i	$0.477395\pi$
$878$	43.2080	1.45820
$879$	−100.437	−3.38767
$880$	−24.8935	−0.839161
$881$	−38.0143	−1.28073	−0.640367	−	0.768069i	$-0.721218\pi$
$881$	−38.0143	−1.28073	−0.640367	+	0.768069i	$0.721218\pi$
$882$	97.2114	3.27328
$883$	−11.5452	−0.388528	−0.194264	−	0.980949i	$-0.562232\pi$
$883$	−11.5452	−0.388528	−0.194264	+	0.980949i	$0.562232\pi$
$884$	−26.8478	−0.902990
$885$	113.030	3.79947
$886$	−17.6417	−0.592684
$887$	16.4918	0.553741	0.276871	−	0.960907i	$-0.410703\pi$
$887$	16.4918	0.553741	0.276871	+	0.960907i	$0.410703\pi$
$888$	−26.9643	−0.904864
$889$	0.761773	0.0255490
$890$	−13.4795	−0.451833
$891$	24.2935	0.813862
$892$	−23.6372	−0.791433
$893$	−5.10795	−0.170931
$894$	2.58973	0.0866136
$895$	13.2445	0.442714
$896$	14.4857	0.483933
$897$	112.436	3.75413
$898$	−88.7196	−2.96061
$899$	7.13682	0.238026
$900$	211.465	7.04883
$901$	5.69213	0.189632
$902$	−28.6071	−0.952513
$903$	−4.45770	−0.148343
$904$	−47.7994	−1.58978
$905$	51.5412	1.71329
$906$	42.9211	1.42596
$907$	16.3851	0.544059	0.272029	−	0.962289i	$-0.412305\pi$
$907$	16.3851	0.544059	0.272029	+	0.962289i	$0.412305\pi$
$908$	−40.9907	−1.36032
$909$	99.1738	3.28939
$910$	−70.1168	−2.32435
$911$	−52.2144	−1.72994	−0.864970	−	0.501824i	$-0.832662\pi$
$911$	−52.2144	−1.72994	−0.864970	+	0.501824i	$0.832662\pi$
$912$	−12.5895	−0.416879
$913$	−15.7774	−0.522154
$914$	−82.5372	−2.73009
$915$	−23.8488	−0.788419
$916$	−13.1062	−0.433042
$917$	2.43093	0.0802764
$918$	36.8610	1.21659
$919$	29.8832	0.985757	0.492879	−	0.870098i	$-0.335945\pi$
$919$	29.8832	0.985757	0.492879	+	0.870098i	$0.335945\pi$
$920$	126.722	4.17790
$921$	−66.3643	−2.18678
$922$	−27.6627	−0.911022
$923$	−52.6235	−1.73212
$924$	−20.1657	−0.663402
$925$	−8.10170	−0.266382
$926$	−62.6701	−2.05947
$927$	110.149	3.61778
$928$	4.87878	0.160154
$929$	−47.7925	−1.56802	−0.784011	−	0.620747i	$-0.786829\pi$
$929$	−47.7925	−1.56802	−0.784011	+	0.620747i	$0.786829\pi$
$930$	245.473	8.04937
$931$	−2.67668	−0.0877246
$932$	−110.947	−3.63420
$933$	47.9030	1.56827
$934$	66.8212	2.18646
$935$	−3.35627	−0.109762
$936$	285.466	9.33075
$937$	−19.2335	−0.628332	−0.314166	−	0.949368i	$-0.601725\pi$
$937$	−19.2335	−0.628332	−0.314166	+	0.949368i	$0.601725\pi$
$938$	−10.1858	−0.332580
$939$	−106.393	−3.47201
$940$	147.785	4.82022
$941$	−49.3214	−1.60783	−0.803916	−	0.594743i	$-0.797254\pi$
$941$	−49.3214	−1.60783	−0.803916	+	0.594743i	$0.797254\pi$
$942$	−27.2764	−0.888714
$943$	65.6022	2.13630
$944$	77.2259	2.51349
$945$	66.7547	2.17153
$946$	−2.55417	−0.0830432
$947$	30.1197	0.978759	0.489379	−	0.872071i	$-0.337223\pi$
$947$	30.1197	0.978759	0.489379	+	0.872071i	$0.337223\pi$
$948$	−173.167	−5.62421
$949$	65.3089	2.12002
$950$	−8.39683	−0.272429
$951$	−22.7342	−0.737206
$952$	−8.88399	−0.287932
$953$	−32.0301	−1.03756	−0.518778	−	0.854909i	$-0.673613\pi$
$953$	−32.0301	−1.03756	−0.518778	+	0.854909i	$0.673613\pi$
$954$	−108.485	−3.51234
$955$	−13.6378	−0.441308
$956$	−36.3802	−1.17662
$957$	2.60745	0.0842869
$958$	−46.6444	−1.50701
$959$	6.12606	0.197821
$960$	6.77186	0.218561
$961$	47.3764	1.52827
$962$	−19.6039	−0.632055
$963$	11.5737	0.372957
$964$	−78.6392	−2.53280
$965$	43.9753	1.41562
$966$	66.6890	2.14568
$967$	−46.4739	−1.49450	−0.747250	−	0.664543i	$-0.768626\pi$
$967$	−46.4739	−1.49450	−0.747250	+	0.664543i	$0.768626\pi$
$968$	−6.44616	−0.207187
$969$	−1.69738	−0.0545276
$970$	104.838	3.36615
$971$	−21.2545	−0.682090	−0.341045	−	0.940047i	$-0.610781\pi$
$971$	−21.2545	−0.682090	−0.341045	+	0.940047i	$0.610781\pi$
$972$	−159.606	−5.11936
$973$	−3.56543	−0.114302
$974$	−54.8210	−1.75658
$975$	120.255	3.85124
$976$	−16.2943	−0.521568
$977$	33.8197	1.08199	0.540994	−	0.841027i	$-0.318048\pi$
$977$	33.8197	1.08199	0.540994	+	0.841027i	$0.318048\pi$
$978$	−185.715	−5.93852
$979$	−1.57241	−0.0502545
$980$	77.4427	2.47382
$981$	57.2519	1.82791
$982$	83.5921	2.66753
$983$	17.4845	0.557668	0.278834	−	0.960339i	$-0.410052\pi$
$983$	17.4845	0.557668	0.278834	+	0.960339i	$0.410052\pi$
$984$	233.523	7.44444
$985$	−67.5176	−2.15129
$986$	2.05903	0.0655727
$987$	43.3894	1.38110
$988$	−14.0891	−0.448234
$989$	5.85725	0.186250
$990$	63.9665	2.03299
$991$	−19.4549	−0.618005	−0.309003	−	0.951061i	$-0.599995\pi$
$991$	−19.4549	−0.618005	−0.309003	+	0.951061i	$0.599995\pi$
$992$	53.5787	1.70112
$993$	−39.1011	−1.24083
$994$	−31.2125	−0.990001
$995$	55.6312	1.76363
$996$	230.855	7.31493
$997$	4.47879	0.141845	0.0709224	−	0.997482i	$-0.477406\pi$
$997$	4.47879	0.141845	0.0709224	+	0.997482i	$0.477406\pi$
$998$	91.8109	2.90622
$999$	18.6639	0.590499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8041.2.a.j.1.6	✓	82

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8041.2.a.j.1.6	✓	82	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.55417 q^{2} -3.23448 q^{3} +4.52378 q^{4} -3.35627 q^{5} +8.26140 q^{6} +1.37818 q^{7} -6.44616 q^{8} +7.46183 q^{9} +O(q^{10})\)	\(q-2.55417 q^{2} -3.23448 q^{3} +4.52378 q^{4} -3.35627 q^{5} +8.26140 q^{6} +1.37818 q^{7} -6.44616 q^{8} +7.46183 q^{9} +8.57249 q^{10} +1.00000 q^{11} -14.6321 q^{12} -5.93482 q^{13} -3.52011 q^{14} +10.8558 q^{15} +7.41702 q^{16} +1.00000 q^{17} -19.0588 q^{18} +0.524776 q^{19} -15.1830 q^{20} -4.45770 q^{21} -2.55417 q^{22} +5.85725 q^{23} +20.8499 q^{24} +6.26457 q^{25} +15.1585 q^{26} -14.4317 q^{27} +6.23460 q^{28} -0.806143 q^{29} -27.7275 q^{30} -8.85305 q^{31} -6.05200 q^{32} -3.23448 q^{33} -2.55417 q^{34} -4.62556 q^{35} +33.7557 q^{36} -1.29326 q^{37} -1.34037 q^{38} +19.1960 q^{39} +21.6351 q^{40} +11.2002 q^{41} +11.3857 q^{42} +1.00000 q^{43} +4.52378 q^{44} -25.0440 q^{45} -14.9604 q^{46} -9.73357 q^{47} -23.9902 q^{48} -5.10061 q^{49} -16.0008 q^{50} -3.23448 q^{51} -26.8478 q^{52} +5.69213 q^{53} +36.8610 q^{54} -3.35627 q^{55} -8.88399 q^{56} -1.69738 q^{57} +2.05903 q^{58} +10.4120 q^{59} +49.1092 q^{60} -2.19688 q^{61} +22.6122 q^{62} +10.2838 q^{63} +0.623802 q^{64} +19.9189 q^{65} +8.26140 q^{66} +2.89361 q^{67} +4.52378 q^{68} -18.9451 q^{69} +11.8145 q^{70} +8.86691 q^{71} -48.1002 q^{72} -11.0044 q^{73} +3.30320 q^{74} -20.2626 q^{75} +2.37397 q^{76} +1.37818 q^{77} -49.0299 q^{78} +11.8348 q^{79} -24.8935 q^{80} +24.2935 q^{81} -28.6071 q^{82} -15.7774 q^{83} -20.1657 q^{84} -3.35627 q^{85} -2.55417 q^{86} +2.60745 q^{87} -6.44616 q^{88} -1.57241 q^{89} +63.9665 q^{90} -8.17928 q^{91} +26.4969 q^{92} +28.6350 q^{93} +24.8612 q^{94} -1.76129 q^{95} +19.5751 q^{96} +12.2296 q^{97} +13.0278 q^{98} +7.46183 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9}+O(q^{10}) \) 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9	\( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9} + q^{10} + 82 q^{11} + 3 q^{12} + 26 q^{13} + 17 q^{14} + 66 q^{15} + 122 q^{16} + 82 q^{17} + 18 q^{18} + 12 q^{19} + 9 q^{20} + 22 q^{21} + 8 q^{22} + 50 q^{23} + 15 q^{24} + 117 q^{25} + 36 q^{26} + 30 q^{27} + 11 q^{28} + 33 q^{29} - 26 q^{30} + 40 q^{31} + 58 q^{32} + 6 q^{33} + 8 q^{34} + 16 q^{35} + 160 q^{36} + 31 q^{37} + 18 q^{38} + 41 q^{39} - 29 q^{40} + 42 q^{41} - 51 q^{42} + 82 q^{43} + 98 q^{44} - 2 q^{45} - 19 q^{46} + 84 q^{47} - 46 q^{48} + 136 q^{49} + 59 q^{50} + 6 q^{51} + 45 q^{52} + 83 q^{53} + 24 q^{54} + 11 q^{55} + 21 q^{56} + 23 q^{57} + 14 q^{58} + 96 q^{59} + 184 q^{60} - 6 q^{61} - 23 q^{62} + 8 q^{63} + 148 q^{64} + 5 q^{65} + 10 q^{66} + 78 q^{67} + 98 q^{68} + 61 q^{69} - 3 q^{70} + 155 q^{71} + 50 q^{72} - 23 q^{73} + 10 q^{74} - 19 q^{75} + 44 q^{76} + 8 q^{77} - 27 q^{78} + 31 q^{79} + 19 q^{80} + 150 q^{81} - 12 q^{82} + 54 q^{83} + 8 q^{84} + 11 q^{85} + 8 q^{86} + 20 q^{87} + 30 q^{88} + 25 q^{89} - 81 q^{90} - 14 q^{91} + 60 q^{92} + 36 q^{93} + 19 q^{94} + 111 q^{95} - 6 q^{96} + 2 q^{97} - 5 q^{98} + 108 q^{99}+O(q^{100}) \) 82 * q + 8 * q^2 + 6 * q^3 + 98 * q^4 + 11 * q^5 + 10 * q^6 + 8 * q^7 + 30 * q^8 + 108 * q^9 + q^10 + 82 * q^11 + 3 * q^12 + 26 * q^13 + 17 * q^14 + 66 * q^15 + 122 * q^16 + 82 * q^17 + 18 * q^18 + 12 * q^19 + 9 * q^20 + 22 * q^21 + 8 * q^22 + 50 * q^23 + 15 * q^24 + 117 * q^25 + 36 * q^26 + 30 * q^27 + 11 * q^28 + 33 * q^29 - 26 * q^30 + 40 * q^31 + 58 * q^32 + 6 * q^33 + 8 * q^34 + 16 * q^35 + 160 * q^36 + 31 * q^37 + 18 * q^38 + 41 * q^39 - 29 * q^40 + 42 * q^41 - 51 * q^42 + 82 * q^43 + 98 * q^44 - 2 * q^45 - 19 * q^46 + 84 * q^47 - 46 * q^48 + 136 * q^49 + 59 * q^50 + 6 * q^51 + 45 * q^52 + 83 * q^53 + 24 * q^54 + 11 * q^55 + 21 * q^56 + 23 * q^57 + 14 * q^58 + 96 * q^59 + 184 * q^60 - 6 * q^61 - 23 * q^62 + 8 * q^63 + 148 * q^64 + 5 * q^65 + 10 * q^66 + 78 * q^67 + 98 * q^68 + 61 * q^69 - 3 * q^70 + 155 * q^71 + 50 * q^72 - 23 * q^73 + 10 * q^74 - 19 * q^75 + 44 * q^76 + 8 * q^77 - 27 * q^78 + 31 * q^79 + 19 * q^80 + 150 * q^81 - 12 * q^82 + 54 * q^83 + 8 * q^84 + 11 * q^85 + 8 * q^86 + 20 * q^87 + 30 * q^88 + 25 * q^89 - 81 * q^90 - 14 * q^91 + 60 * q^92 + 36 * q^93 + 19 * q^94 + 111 * q^95 - 6 * q^96 + 2 * q^97 - 5 * q^98 + 108 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more