LMFDB - Embedded newform 8015.2.a.n.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8015,2,Mod(1,8015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8015, 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("8015.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8015 = 5 \cdot 7 \cdot 229 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8015.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:	$64.0000972201$
Analytic rank:	$0$
Dimension:	$68$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.64550 q^{2} +3.30855 q^{3} +0.707660 q^{4} -1.00000 q^{5} -5.44420 q^{6} +1.00000 q^{7} +2.12654 q^{8} +7.94648 q^{9} +O(q^{10})$	\(q-1.64550 q^{2} +3.30855 q^{3} +0.707660 q^{4} -1.00000 q^{5} -5.44420 q^{6} +1.00000 q^{7} +2.12654 q^{8} +7.94648 q^{9} +1.64550 q^{10} +2.22946 q^{11} +2.34132 q^{12} +2.85017 q^{13} -1.64550 q^{14} -3.30855 q^{15} -4.91454 q^{16} -0.634986 q^{17} -13.0759 q^{18} +0.555045 q^{19} -0.707660 q^{20} +3.30855 q^{21} -3.66858 q^{22} -7.93359 q^{23} +7.03576 q^{24} +1.00000 q^{25} -4.68995 q^{26} +16.3657 q^{27} +0.707660 q^{28} -5.45385 q^{29} +5.44420 q^{30} +5.20170 q^{31} +3.83377 q^{32} +7.37629 q^{33} +1.04487 q^{34} -1.00000 q^{35} +5.62340 q^{36} +1.50197 q^{37} -0.913325 q^{38} +9.42993 q^{39} -2.12654 q^{40} +0.623475 q^{41} -5.44420 q^{42} +11.2489 q^{43} +1.57770 q^{44} -7.94648 q^{45} +13.0547 q^{46} -0.738828 q^{47} -16.2600 q^{48} +1.00000 q^{49} -1.64550 q^{50} -2.10088 q^{51} +2.01695 q^{52} -1.67894 q^{53} -26.9296 q^{54} -2.22946 q^{55} +2.12654 q^{56} +1.83639 q^{57} +8.97430 q^{58} -1.35236 q^{59} -2.34132 q^{60} +2.67685 q^{61} -8.55938 q^{62} +7.94648 q^{63} +3.52062 q^{64} -2.85017 q^{65} -12.1377 q^{66} +12.7294 q^{67} -0.449354 q^{68} -26.2487 q^{69} +1.64550 q^{70} +10.1580 q^{71} +16.8985 q^{72} -11.0936 q^{73} -2.47148 q^{74} +3.30855 q^{75} +0.392783 q^{76} +2.22946 q^{77} -15.5169 q^{78} -3.98868 q^{79} +4.91454 q^{80} +30.3071 q^{81} -1.02593 q^{82} -2.30652 q^{83} +2.34132 q^{84} +0.634986 q^{85} -18.5101 q^{86} -18.0443 q^{87} +4.74105 q^{88} -8.69038 q^{89} +13.0759 q^{90} +2.85017 q^{91} -5.61428 q^{92} +17.2101 q^{93} +1.21574 q^{94} -0.555045 q^{95} +12.6842 q^{96} -2.47563 q^{97} -1.64550 q^{98} +17.7164 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9}+O(q^{10}) $ 68 * q + 9 * q^2 + 83 * q^4 - 68 * q^5 + 5 * q^6 + 68 * q^7 + 30 * q^8 + 86 * q^9	\( 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9} - 9 q^{10} + 5 q^{11} + 9 q^{12} + 15 q^{13} + 9 q^{14} + 109 q^{16} + 7 q^{17} + 39 q^{18} + 20 q^{19} - 83 q^{20} + 56 q^{22} + 36 q^{23} + q^{24} + 68 q^{25} + q^{26} + 12 q^{27} + 83 q^{28} - 16 q^{29} - 5 q^{30} + 31 q^{31} + 79 q^{32} + 45 q^{33} + 31 q^{34} - 68 q^{35} + 114 q^{36} + 72 q^{37} + 8 q^{38} + 47 q^{39} - 30 q^{40} + 6 q^{41} + 5 q^{42} + 75 q^{43} + 15 q^{44} - 86 q^{45} + 29 q^{46} - 10 q^{47} + 44 q^{48} + 68 q^{49} + 9 q^{50} + 23 q^{51} + 37 q^{52} + 41 q^{53} + 4 q^{54} - 5 q^{55} + 30 q^{56} + 55 q^{57} + 66 q^{58} - 5 q^{59} - 9 q^{60} - 2 q^{61} + 3 q^{62} + 86 q^{63} + 162 q^{64} - 15 q^{65} - 23 q^{66} + 92 q^{67} + 35 q^{68} - 25 q^{69} - 9 q^{70} - 2 q^{71} + 128 q^{72} + 80 q^{73} + 18 q^{74} + 71 q^{76} + 5 q^{77} + 20 q^{78} + 100 q^{79} - 109 q^{80} + 140 q^{81} + 36 q^{82} - 60 q^{83} + 9 q^{84} - 7 q^{85} - 27 q^{86} + 24 q^{87} + 175 q^{88} + 19 q^{89} - 39 q^{90} + 15 q^{91} + 75 q^{92} + 37 q^{93} + 11 q^{94} - 20 q^{95} + 15 q^{96} + 96 q^{97} + 9 q^{98} + 3 q^{99}+O(q^{100}) \) 68 * q + 9 * q^2 + 83 * q^4 - 68 * q^5 + 5 * q^6 + 68 * q^7 + 30 * q^8 + 86 * q^9 - 9 * q^10 + 5 * q^11 + 9 * q^12 + 15 * q^13 + 9 * q^14 + 109 * q^16 + 7 * q^17 + 39 * q^18 + 20 * q^19 - 83 * q^20 + 56 * q^22 + 36 * q^23 + q^24 + 68 * q^25 + q^26 + 12 * q^27 + 83 * q^28 - 16 * q^29 - 5 * q^30 + 31 * q^31 + 79 * q^32 + 45 * q^33 + 31 * q^34 - 68 * q^35 + 114 * q^36 + 72 * q^37 + 8 * q^38 + 47 * q^39 - 30 * q^40 + 6 * q^41 + 5 * q^42 + 75 * q^43 + 15 * q^44 - 86 * q^45 + 29 * q^46 - 10 * q^47 + 44 * q^48 + 68 * q^49 + 9 * q^50 + 23 * q^51 + 37 * q^52 + 41 * q^53 + 4 * q^54 - 5 * q^55 + 30 * q^56 + 55 * q^57 + 66 * q^58 - 5 * q^59 - 9 * q^60 - 2 * q^61 + 3 * q^62 + 86 * q^63 + 162 * q^64 - 15 * q^65 - 23 * q^66 + 92 * q^67 + 35 * q^68 - 25 * q^69 - 9 * q^70 - 2 * q^71 + 128 * q^72 + 80 * q^73 + 18 * q^74 + 71 * q^76 + 5 * q^77 + 20 * q^78 + 100 * q^79 - 109 * q^80 + 140 * q^81 + 36 * q^82 - 60 * q^83 + 9 * q^84 - 7 * q^85 - 27 * q^86 + 24 * q^87 + 175 * q^88 + 19 * q^89 - 39 * q^90 + 15 * q^91 + 75 * q^92 + 37 * q^93 + 11 * q^94 - 20 * q^95 + 15 * q^96 + 96 * q^97 + 9 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.64550	−1.16354	−0.581771	−	0.813353i	$-0.697640\pi$
$2$	−1.64550	−1.16354	−0.581771	+	0.813353i	$0.697640\pi$
$3$	3.30855	1.91019	0.955095	−	0.296300i	$-0.0957527\pi$
$3$	3.30855	1.91019	0.955095	+	0.296300i	$0.0957527\pi$
$4$	0.707660	0.353830
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−5.44420	−2.22259
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.12654	0.751846
$9$	7.94648	2.64883
$10$	1.64550	0.520352
$11$	2.22946	0.672209	0.336104	−	0.941825i	$-0.390890\pi$
$11$	2.22946	0.672209	0.336104	+	0.941825i	$0.390890\pi$
$12$	2.34132	0.675882
$13$	2.85017	0.790496	0.395248	−	0.918574i	$-0.370659\pi$
$13$	2.85017	0.790496	0.395248	+	0.918574i	$0.370659\pi$
$14$	−1.64550	−0.439778
$15$	−3.30855	−0.854263
$16$	−4.91454	−1.22863
$17$	−0.634986	−0.154007	−0.0770034	−	0.997031i	$-0.524535\pi$
$17$	−0.634986	−0.154007	−0.0770034	+	0.997031i	$0.524535\pi$
$18$	−13.0759	−3.08202
$19$	0.555045	0.127336	0.0636680	−	0.997971i	$-0.479720\pi$
$19$	0.555045	0.127336	0.0636680	+	0.997971i	$0.479720\pi$
$20$	−0.707660	−0.158238
$21$	3.30855	0.721984
$22$	−3.66858	−0.782143
$23$	−7.93359	−1.65427	−0.827134	−	0.562004i	$-0.810030\pi$
$23$	−7.93359	−1.65427	−0.827134	+	0.562004i	$0.810030\pi$
$24$	7.03576	1.43617
$25$	1.00000	0.200000
$26$	−4.68995	−0.919775
$27$	16.3657	3.14957
$28$	0.707660	0.133735
$29$	−5.45385	−1.01276	−0.506378	−	0.862312i	$-0.669016\pi$
$29$	−5.45385	−1.01276	−0.506378	+	0.862312i	$0.669016\pi$
$30$	5.44420	0.993971
$31$	5.20170	0.934253	0.467126	−	0.884191i	$-0.345289\pi$
$31$	5.20170	0.934253	0.467126	+	0.884191i	$0.345289\pi$
$32$	3.83377	0.677721
$33$	7.37629	1.28405
$34$	1.04487	0.179193
$35$	−1.00000	−0.169031
$36$	5.62340	0.937234
$37$	1.50197	0.246922	0.123461	−	0.992349i	$-0.460601\pi$
$37$	1.50197	0.246922	0.123461	+	0.992349i	$0.460601\pi$
$38$	−0.913325	−0.148161
$39$	9.42993	1.51000
$40$	−2.12654	−0.336236
$41$	0.623475	0.0973704	0.0486852	−	0.998814i	$-0.484497\pi$
$41$	0.623475	0.0973704	0.0486852	+	0.998814i	$0.484497\pi$
$42$	−5.44420	−0.840059
$43$	11.2489	1.71545	0.857723	−	0.514113i	$-0.171879\pi$
$43$	11.2489	1.71545	0.857723	+	0.514113i	$0.171879\pi$
$44$	1.57770	0.237848
$45$	−7.94648	−1.18459
$46$	13.0547	1.92481
$47$	−0.738828	−0.107769	−0.0538846	−	0.998547i	$-0.517160\pi$
$47$	−0.738828	−0.107769	−0.0538846	+	0.998547i	$0.517160\pi$
$48$	−16.2600	−2.34692
$49$	1.00000	0.142857
$50$	−1.64550	−0.232708
$51$	−2.10088	−0.294182
$52$	2.01695	0.279701
$53$	−1.67894	−0.230621	−0.115310	−	0.993330i	$-0.536786\pi$
$53$	−1.67894	−0.230621	−0.115310	+	0.993330i	$0.536786\pi$
$54$	−26.9296	−3.66466
$55$	−2.22946	−0.300621
$56$	2.12654	0.284171
$57$	1.83639	0.243236
$58$	8.97430	1.17838
$59$	−1.35236	−0.176063	−0.0880315	−	0.996118i	$-0.528058\pi$
$59$	−1.35236	−0.176063	−0.0880315	+	0.996118i	$0.528058\pi$
$60$	−2.34132	−0.302264
$61$	2.67685	0.342735	0.171368	−	0.985207i	$-0.445181\pi$
$61$	2.67685	0.342735	0.171368	+	0.985207i	$0.445181\pi$
$62$	−8.55938	−1.08704
$63$	7.94648	1.00116
$64$	3.52062	0.440077
$65$	−2.85017	−0.353520
$66$	−12.1377	−1.49404
$67$	12.7294	1.55515	0.777573	−	0.628793i	$-0.216451\pi$
$67$	12.7294	1.55515	0.777573	+	0.628793i	$0.216451\pi$
$68$	−0.449354	−0.0544922
$69$	−26.2487	−3.15997
$70$	1.64550	0.196674
$71$	10.1580	1.20553	0.602766	−	0.797918i	$-0.294065\pi$
$71$	10.1580	1.20553	0.602766	+	0.797918i	$0.294065\pi$
$72$	16.8985	1.99151
$73$	−11.0936	−1.29840	−0.649201	−	0.760617i	$-0.724897\pi$
$73$	−11.0936	−1.29840	−0.649201	+	0.760617i	$0.724897\pi$
$74$	−2.47148	−0.287304
$75$	3.30855	0.382038
$76$	0.392783	0.0450553
$77$	2.22946	0.254071
$78$	−15.5169	−1.75695
$79$	−3.98868	−0.448762	−0.224381	−	0.974502i	$-0.572036\pi$
$79$	−3.98868	−0.448762	−0.224381	+	0.974502i	$0.572036\pi$
$80$	4.91454	0.549462
$81$	30.3071	3.36745
$82$	−1.02593	−0.113295
$83$	−2.30652	−0.253174	−0.126587	−	0.991956i	$-0.540402\pi$
$83$	−2.30652	−0.253174	−0.126587	+	0.991956i	$0.540402\pi$
$84$	2.34132	0.255459
$85$	0.634986	0.0688739
$86$	−18.5101	−1.99599
$87$	−18.0443	−1.93456
$88$	4.74105	0.505398
$89$	−8.69038	−0.921178	−0.460589	−	0.887613i	$-0.652362\pi$
$89$	−8.69038	−0.921178	−0.460589	+	0.887613i	$0.652362\pi$
$90$	13.0759	1.37832
$91$	2.85017	0.298779
$92$	−5.61428	−0.585330
$93$	17.2101	1.78460
$94$	1.21574	0.125394
$95$	−0.555045	−0.0569464
$96$	12.6842	1.29458
$97$	−2.47563	−0.251362	−0.125681	−	0.992071i	$-0.540112\pi$
$97$	−2.47563	−0.251362	−0.125681	+	0.992071i	$0.540112\pi$
$98$	−1.64550	−0.166220
$99$	17.7164	1.78056
$100$	0.707660	0.0707660
$101$	14.6200	1.45474	0.727372	−	0.686243i	$-0.240741\pi$
$101$	14.6200	1.45474	0.727372	+	0.686243i	$0.240741\pi$
$102$	3.45699	0.342293
$103$	10.8258	1.06670	0.533348	−	0.845896i	$-0.320934\pi$
$103$	10.8258	1.06670	0.533348	+	0.845896i	$0.320934\pi$
$104$	6.06101	0.594331
$105$	−3.30855	−0.322881
$106$	2.76270	0.268337
$107$	5.65375	0.546569	0.273285	−	0.961933i	$-0.411890\pi$
$107$	5.65375	0.546569	0.273285	+	0.961933i	$0.411890\pi$
$108$	11.5813	1.11441
$109$	−6.84952	−0.656065	−0.328033	−	0.944666i	$-0.606386\pi$
$109$	−6.84952	−0.656065	−0.328033	+	0.944666i	$0.606386\pi$
$110$	3.66858	0.349785
$111$	4.96933	0.471668
$112$	−4.91454	−0.464380
$113$	18.7352	1.76246	0.881232	−	0.472684i	$-0.156715\pi$
$113$	18.7352	1.76246	0.881232	+	0.472684i	$0.156715\pi$
$114$	−3.02178	−0.283015
$115$	7.93359	0.739811
$116$	−3.85947	−0.358343
$117$	22.6488	2.09389
$118$	2.22531	0.204857
$119$	−0.634986	−0.0582091
$120$	−7.03576	−0.642274
$121$	−6.02949	−0.548135
$122$	−4.40474	−0.398787
$123$	2.06279	0.185996
$124$	3.68103	0.330566
$125$	−1.00000	−0.0894427
$126$	−13.0759	−1.16489
$127$	6.20232	0.550367	0.275183	−	0.961392i	$-0.411261\pi$
$127$	6.20232	0.550367	0.275183	+	0.961392i	$0.411261\pi$
$128$	−13.4607	−1.18977
$129$	37.2176	3.27683
$130$	4.68995	0.411336
$131$	15.0265	1.31287	0.656436	−	0.754381i	$-0.272063\pi$
$131$	15.0265	1.31287	0.656436	+	0.754381i	$0.272063\pi$
$132$	5.21990	0.454334
$133$	0.555045	0.0481285
$134$	−20.9462	−1.80948
$135$	−16.3657	−1.40853
$136$	−1.35033	−0.115789
$137$	0.951897	0.0813261	0.0406630	−	0.999173i	$-0.487053\pi$
$137$	0.951897	0.0813261	0.0406630	+	0.999173i	$0.487053\pi$
$138$	43.1921	3.67675
$139$	14.6190	1.23997	0.619986	−	0.784613i	$-0.287138\pi$
$139$	14.6190	1.23997	0.619986	+	0.784613i	$0.287138\pi$
$140$	−0.707660	−0.0598082
$141$	−2.44445	−0.205859
$142$	−16.7149	−1.40269
$143$	6.35436	0.531378
$144$	−39.0533	−3.25444
$145$	5.45385	0.452918
$146$	18.2544	1.51075
$147$	3.30855	0.272884
$148$	1.06288	0.0873684
$149$	−21.9214	−1.79587	−0.897937	−	0.440123i	$-0.854935\pi$
$149$	−21.9214	−1.79587	−0.897937	+	0.440123i	$0.854935\pi$
$150$	−5.44420	−0.444517
$151$	15.1075	1.22943	0.614717	−	0.788747i	$-0.289270\pi$
$151$	15.1075	1.22943	0.614717	+	0.788747i	$0.289270\pi$
$152$	1.18033	0.0957371
$153$	−5.04591	−0.407937
$154$	−3.66858	−0.295622
$155$	−5.20170	−0.417811
$156$	6.67318	0.534282
$157$	−19.5662	−1.56155	−0.780776	−	0.624811i	$-0.785176\pi$
$157$	−19.5662	−1.56155	−0.780776	+	0.624811i	$0.785176\pi$
$158$	6.56336	0.522153
$159$	−5.55486	−0.440529
$160$	−3.83377	−0.303086
$161$	−7.93359	−0.625255
$162$	−49.8702	−3.91817
$163$	17.6203	1.38013	0.690064	−	0.723748i	$-0.257582\pi$
$163$	17.6203	1.38013	0.690064	+	0.723748i	$0.257582\pi$
$164$	0.441208	0.0344525
$165$	−7.37629	−0.574243
$166$	3.79538	0.294578
$167$	−4.37332	−0.338418	−0.169209	−	0.985580i	$-0.554121\pi$
$167$	−4.37332	−0.338418	−0.169209	+	0.985580i	$0.554121\pi$
$168$	7.03576	0.542821
$169$	−4.87651	−0.375116
$170$	−1.04487	−0.0801377
$171$	4.41065	0.337291
$172$	7.96041	0.606976
$173$	−20.8739	−1.58701	−0.793506	−	0.608563i	$-0.791746\pi$
$173$	−20.8739	−1.58701	−0.793506	+	0.608563i	$0.791746\pi$
$174$	29.6919	2.25094
$175$	1.00000	0.0755929
$176$	−10.9568	−0.825899
$177$	−4.47436	−0.336314
$178$	14.3000	1.07183
$179$	−17.0409	−1.27370	−0.636848	−	0.770990i	$-0.719762\pi$
$179$	−17.0409	−1.27370	−0.636848	+	0.770990i	$0.719762\pi$
$180$	−5.62340	−0.419144
$181$	1.66482	0.123745	0.0618726	−	0.998084i	$-0.480293\pi$
$181$	1.66482	0.123745	0.0618726	+	0.998084i	$0.480293\pi$
$182$	−4.68995	−0.347642
$183$	8.85647	0.654689
$184$	−16.8711	−1.24376
$185$	−1.50197	−0.110427
$186$	−28.3191	−2.07646
$187$	−1.41568	−0.103525
$188$	−0.522839	−0.0381319
$189$	16.3657	1.19043
$190$	0.913325	0.0662595
$191$	9.11648	0.659645	0.329823	−	0.944043i	$-0.393011\pi$
$191$	9.11648	0.659645	0.329823	+	0.944043i	$0.393011\pi$
$192$	11.6481	0.840631
$193$	−12.7105	−0.914921	−0.457461	−	0.889230i	$-0.651241\pi$
$193$	−12.7105	−0.914921	−0.457461	+	0.889230i	$0.651241\pi$
$194$	4.07364	0.292470
$195$	−9.42993	−0.675291
$196$	0.707660	0.0505471
$197$	13.4415	0.957664	0.478832	−	0.877907i	$-0.341060\pi$
$197$	13.4415	0.957664	0.478832	+	0.877907i	$0.341060\pi$
$198$	−29.1523	−2.07176
$199$	−5.34654	−0.379006	−0.189503	−	0.981880i	$-0.560688\pi$
$199$	−5.34654	−0.379006	−0.189503	+	0.981880i	$0.560688\pi$
$200$	2.12654	0.150369
$201$	42.1159	2.97062
$202$	−24.0572	−1.69266
$203$	−5.45385	−0.382786
$204$	−1.48671	−0.104090
$205$	−0.623475	−0.0435454
$206$	−17.8138	−1.24114
$207$	−63.0441	−4.38187
$208$	−14.0073	−0.971230
$209$	1.23745	0.0855964
$210$	5.44420	0.375686
$211$	14.5466	1.00143	0.500715	−	0.865612i	$-0.333070\pi$
$211$	14.5466	1.00143	0.500715	+	0.865612i	$0.333070\pi$
$212$	−1.18812	−0.0816005
$213$	33.6082	2.30279
$214$	−9.30323	−0.635956
$215$	−11.2489	−0.767170
$216$	34.8022	2.36799
$217$	5.20170	0.353114
$218$	11.2709	0.763359
$219$	−36.7035	−2.48020
$220$	−1.57770	−0.106369
$221$	−1.80982	−0.121742
$222$	−8.17702	−0.548806
$223$	−19.6969	−1.31900	−0.659501	−	0.751703i	$-0.729233\pi$
$223$	−19.6969	−1.31900	−0.659501	+	0.751703i	$0.729233\pi$
$224$	3.83377	0.256155
$225$	7.94648	0.529765
$226$	−30.8288	−2.05070
$227$	17.7136	1.17569	0.587847	−	0.808972i	$-0.299976\pi$
$227$	17.7136	1.17569	0.587847	+	0.808972i	$0.299976\pi$
$228$	1.29954	0.0860642
$229$	−1.00000	−0.0660819
$229$	−1.00000	−0.0660819
$230$	−13.0547	−0.860802
$231$	7.37629	0.485324
$232$	−11.5978	−0.761436
$233$	−26.6423	−1.74540	−0.872698	−	0.488261i	$-0.837631\pi$
$233$	−26.6423	−1.74540	−0.872698	+	0.488261i	$0.837631\pi$
$234$	−37.2686	−2.43632
$235$	0.738828	0.0481958
$236$	−0.957014	−0.0622963
$237$	−13.1967	−0.857220
$238$	1.04487	0.0677287
$239$	5.50220	0.355908	0.177954	−	0.984039i	$-0.443052\pi$
$239$	5.50220	0.355908	0.177954	+	0.984039i	$0.443052\pi$
$240$	16.2600	1.04958
$241$	−8.37639	−0.539571	−0.269785	−	0.962920i	$-0.586953\pi$
$241$	−8.37639	−0.539571	−0.269785	+	0.962920i	$0.586953\pi$
$242$	9.92150	0.637778
$243$	51.1754	3.28291
$244$	1.89430	0.121270
$245$	−1.00000	−0.0638877
$246$	−3.39432	−0.216414
$247$	1.58197	0.100659
$248$	11.0616	0.702414
$249$	−7.63124	−0.483610
$250$	1.64550	0.104070
$251$	8.26857	0.521908	0.260954	−	0.965351i	$-0.415963\pi$
$251$	8.26857	0.521908	0.260954	+	0.965351i	$0.415963\pi$
$252$	5.62340	0.354241
$253$	−17.6877	−1.11201
$254$	−10.2059	−0.640375
$255$	2.10088	0.131562
$256$	15.1083	0.944270
$257$	1.34680	0.0840108	0.0420054	−	0.999117i	$-0.486625\pi$
$257$	1.34680	0.0840108	0.0420054	+	0.999117i	$0.486625\pi$
$258$	−61.2414	−3.81272
$259$	1.50197	0.0933278
$260$	−2.01695	−0.125086
$261$	−43.3389	−2.68261
$262$	−24.7261	−1.52758
$263$	10.5044	0.647727	0.323864	−	0.946104i	$-0.395018\pi$
$263$	10.5044	0.647727	0.323864	+	0.946104i	$0.395018\pi$
$264$	15.6860	0.965405
$265$	1.67894	0.103137
$266$	−0.913325	−0.0559995
$267$	−28.7525	−1.75963
$268$	9.00810	0.550257
$269$	25.8285	1.57479	0.787395	−	0.616449i	$-0.211429\pi$
$269$	25.8285	1.57479	0.787395	+	0.616449i	$0.211429\pi$
$270$	26.9296	1.63888
$271$	−5.45250	−0.331216	−0.165608	−	0.986192i	$-0.552959\pi$
$271$	−5.45250	−0.331216	−0.165608	+	0.986192i	$0.552959\pi$
$272$	3.12066	0.189218
$273$	9.42993	0.570725
$274$	−1.56634	−0.0946263
$275$	2.22946	0.134442
$276$	−18.5751	−1.11809
$277$	3.94065	0.236771	0.118385	−	0.992968i	$-0.462228\pi$
$277$	3.94065	0.236771	0.118385	+	0.992968i	$0.462228\pi$
$278$	−24.0556	−1.44276
$279$	41.3352	2.47467
$280$	−2.12654	−0.127085
$281$	−14.1918	−0.846609	−0.423305	−	0.905987i	$-0.639130\pi$
$281$	−14.1918	−0.846609	−0.423305	+	0.905987i	$0.639130\pi$
$282$	4.02233	0.239526
$283$	−1.21626	−0.0722990	−0.0361495	−	0.999346i	$-0.511509\pi$
$283$	−1.21626	−0.0722990	−0.0361495	+	0.999346i	$0.511509\pi$
$284$	7.18840	0.426553
$285$	−1.83639	−0.108778
$286$	−10.4561	−0.618281
$287$	0.623475	0.0368025
$288$	30.4650	1.79517
$289$	−16.5968	−0.976282
$290$	−8.97430	−0.526989
$291$	−8.19073	−0.480149
$292$	−7.85046	−0.459413
$293$	−20.3645	−1.18971	−0.594853	−	0.803835i	$-0.702790\pi$
$293$	−20.3645	−1.18971	−0.594853	+	0.803835i	$0.702790\pi$
$294$	−5.44420	−0.317512
$295$	1.35236	0.0787377
$296$	3.19400	0.185647
$297$	36.4866	2.11717
$298$	36.0717	2.08958
$299$	−22.6121	−1.30769
$300$	2.34132	0.135176
$301$	11.2489	0.648377
$302$	−24.8594	−1.43050
$303$	48.3709	2.77884
$304$	−2.72779	−0.156449
$305$	−2.67685	−0.153276
$306$	8.30302	0.474652
$307$	18.2715	1.04281	0.521404	−	0.853310i	$-0.325409\pi$
$307$	18.2715	1.04281	0.521404	+	0.853310i	$0.325409\pi$
$308$	1.57770	0.0898979
$309$	35.8176	2.03759
$310$	8.55938	0.486140
$311$	−16.1755	−0.917227	−0.458613	−	0.888636i	$-0.651654\pi$
$311$	−16.1755	−0.917227	−0.458613	+	0.888636i	$0.651654\pi$
$312$	20.0531	1.13529
$313$	−19.0496	−1.07675	−0.538374	−	0.842706i	$-0.680961\pi$
$313$	−19.0496	−1.07675	−0.538374	+	0.842706i	$0.680961\pi$
$314$	32.1961	1.81693
$315$	−7.94648	−0.447733
$316$	−2.82263	−0.158785
$317$	−28.0296	−1.57430	−0.787150	−	0.616762i	$-0.788444\pi$
$317$	−28.0296	−1.57430	−0.787150	+	0.616762i	$0.788444\pi$
$318$	9.14051	0.512574
$319$	−12.1592	−0.680783
$320$	−3.52062	−0.196808
$321$	18.7057	1.04405
$322$	13.0547	0.727510
$323$	−0.352446	−0.0196106
$324$	21.4471	1.19151
$325$	2.85017	0.158099
$326$	−28.9941	−1.60584
$327$	−22.6620	−1.25321
$328$	1.32585	0.0732075
$329$	−0.738828	−0.0407329
$330$	12.1377	0.668156
$331$	9.01450	0.495482	0.247741	−	0.968826i	$-0.420312\pi$
$331$	9.01450	0.495482	0.247741	+	0.968826i	$0.420312\pi$
$332$	−1.63223	−0.0895804
$333$	11.9354	0.654054
$334$	7.19629	0.393764
$335$	−12.7294	−0.695482
$336$	−16.2600	−0.887054
$337$	9.43948	0.514201	0.257101	−	0.966385i	$-0.417233\pi$
$337$	9.43948	0.514201	0.257101	+	0.966385i	$0.417233\pi$
$338$	8.02429	0.436464
$339$	61.9864	3.36664
$340$	0.449354	0.0243697
$341$	11.5970	0.628013
$342$	−7.25771	−0.392452
$343$	1.00000	0.0539949
$344$	23.9213	1.28975
$345$	26.2487	1.41318
$346$	34.3479	1.84655
$347$	−12.3213	−0.661440	−0.330720	−	0.943729i	$-0.607292\pi$
$347$	−12.3213	−0.661440	−0.330720	+	0.943729i	$0.607292\pi$
$348$	−12.7692	−0.684503
$349$	17.0063	0.910325	0.455163	−	0.890408i	$-0.349581\pi$
$349$	17.0063	0.910325	0.455163	+	0.890408i	$0.349581\pi$
$350$	−1.64550	−0.0879555
$351$	46.6449	2.48972
$352$	8.54726	0.455570
$353$	36.0498	1.91874	0.959369	−	0.282154i	$-0.0910488\pi$
$353$	36.0498	1.91874	0.959369	+	0.282154i	$0.0910488\pi$
$354$	7.36255	0.391315
$355$	−10.1580	−0.539130
$356$	−6.14983	−0.325940
$357$	−2.10088	−0.111190
$358$	28.0407	1.48200
$359$	1.24351	0.0656301	0.0328150	−	0.999461i	$-0.489553\pi$
$359$	1.24351	0.0656301	0.0328150	+	0.999461i	$0.489553\pi$
$360$	−16.8985	−0.890630
$361$	−18.6919	−0.983786
$362$	−2.73946	−0.143983
$363$	−19.9488	−1.04704
$364$	2.01695	0.105717
$365$	11.0936	0.580663
$366$	−14.5733	−0.761759
$367$	4.63254	0.241817	0.120908	−	0.992664i	$-0.461419\pi$
$367$	4.63254	0.241817	0.120908	+	0.992664i	$0.461419\pi$
$368$	38.9899	2.03249
$369$	4.95443	0.257917
$370$	2.47148	0.128486
$371$	−1.67894	−0.0871664
$372$	12.1789	0.631445
$373$	10.2154	0.528931	0.264465	−	0.964395i	$-0.414805\pi$
$373$	10.2154	0.528931	0.264465	+	0.964395i	$0.414805\pi$
$374$	2.32950	0.120455
$375$	−3.30855	−0.170853
$376$	−1.57115	−0.0810258
$377$	−15.5444	−0.800579
$378$	−26.9296	−1.38511
$379$	8.60642	0.442082	0.221041	−	0.975265i	$-0.429055\pi$
$379$	8.60642	0.442082	0.221041	+	0.975265i	$0.429055\pi$
$380$	−0.392783	−0.0201493
$381$	20.5207	1.05130
$382$	−15.0011	−0.767525
$383$	−3.78344	−0.193325	−0.0966623	−	0.995317i	$-0.530817\pi$
$383$	−3.78344	−0.193325	−0.0966623	+	0.995317i	$0.530817\pi$
$384$	−44.5354	−2.27269
$385$	−2.22946	−0.113624
$386$	20.9151	1.06455
$387$	89.3893	4.54392
$388$	−1.75190	−0.0889394
$389$	28.9759	1.46914	0.734568	−	0.678536i	$-0.237385\pi$
$389$	28.9759	1.46914	0.734568	+	0.678536i	$0.237385\pi$
$390$	15.5169	0.785730
$391$	5.03772	0.254769
$392$	2.12654	0.107407
$393$	49.7159	2.50784
$394$	−22.1179	−1.11428
$395$	3.98868	0.200692
$396$	12.5372	0.630017
$397$	−23.3424	−1.17152	−0.585762	−	0.810483i	$-0.699205\pi$
$397$	−23.3424	−1.17152	−0.585762	+	0.810483i	$0.699205\pi$
$398$	8.79772	0.440990
$399$	1.83639	0.0919346
$400$	−4.91454	−0.245727
$401$	36.0262	1.79906	0.899531	−	0.436856i	$-0.143908\pi$
$401$	36.0262	1.79906	0.899531	+	0.436856i	$0.143908\pi$
$402$	−69.3015	−3.45645
$403$	14.8257	0.738523
$404$	10.3460	0.514732
$405$	−30.3071	−1.50597
$406$	8.97430	0.445387
$407$	3.34859	0.165983
$408$	−4.46761	−0.221180
$409$	6.93881	0.343102	0.171551	−	0.985175i	$-0.445122\pi$
$409$	6.93881	0.343102	0.171551	+	0.985175i	$0.445122\pi$
$410$	1.02593	0.0506669
$411$	3.14940	0.155348
$412$	7.66096	0.377429
$413$	−1.35236	−0.0665455
$414$	103.739	5.09849
$415$	2.30652	0.113223
$416$	10.9269	0.535736
$417$	48.3678	2.36858
$418$	−2.03622	−0.0995950
$419$	−22.7246	−1.11017	−0.555084	−	0.831795i	$-0.687314\pi$
$419$	−22.7246	−1.11017	−0.555084	+	0.831795i	$0.687314\pi$
$420$	−2.34132	−0.114245
$421$	9.51831	0.463894	0.231947	−	0.972728i	$-0.425490\pi$
$421$	9.51831	0.463894	0.231947	+	0.972728i	$0.425490\pi$
$422$	−23.9364	−1.16521
$423$	−5.87108	−0.285462
$424$	−3.57034	−0.173391
$425$	−0.634986	−0.0308014
$426$	−55.3021	−2.67940
$427$	2.67685	0.129542
$428$	4.00093	0.193392
$429$	21.0237	1.01503
$430$	18.5101	0.892635
$431$	21.0431	1.01361	0.506806	−	0.862060i	$-0.330826\pi$
$431$	21.0431	1.01361	0.506806	+	0.862060i	$0.330826\pi$
$432$	−80.4296	−3.86967
$433$	8.39542	0.403458	0.201729	−	0.979441i	$-0.435344\pi$
$433$	8.39542	0.403458	0.201729	+	0.979441i	$0.435344\pi$
$434$	−8.55938	−0.410863
$435$	18.0443	0.865159
$436$	−4.84713	−0.232135
$437$	−4.40350	−0.210648
$438$	60.3956	2.88581
$439$	−4.57944	−0.218565	−0.109282	−	0.994011i	$-0.534855\pi$
$439$	−4.57944	−0.218565	−0.109282	+	0.994011i	$0.534855\pi$
$440$	−4.74105	−0.226021
$441$	7.94648	0.378404
$442$	2.97805	0.141652
$443$	−22.1518	−1.05247	−0.526233	−	0.850341i	$-0.676396\pi$
$443$	−22.1518	−1.05247	−0.526233	+	0.850341i	$0.676396\pi$
$444$	3.51660	0.166890
$445$	8.69038	0.411964
$446$	32.4112	1.53472
$447$	−72.5281	−3.43046
$448$	3.52062	0.166333
$449$	19.2350	0.907758	0.453879	−	0.891063i	$-0.350040\pi$
$449$	19.2350	0.907758	0.453879	+	0.891063i	$0.350040\pi$
$450$	−13.0759	−0.616404
$451$	1.39001	0.0654532
$452$	13.2582	0.623612
$453$	49.9840	2.34845
$454$	−29.1477	−1.36797
$455$	−2.85017	−0.133618
$456$	3.90516	0.182876
$457$	−6.11096	−0.285859	−0.142929	−	0.989733i	$-0.545652\pi$
$457$	−6.11096	−0.285859	−0.142929	+	0.989733i	$0.545652\pi$
$458$	1.64550	0.0768890
$459$	−10.3920	−0.485055
$460$	5.61428	0.261767
$461$	6.64863	0.309657	0.154829	−	0.987941i	$-0.450517\pi$
$461$	6.64863	0.309657	0.154829	+	0.987941i	$0.450517\pi$
$462$	−12.1377	−0.564695
$463$	12.6491	0.587856	0.293928	−	0.955828i	$-0.405037\pi$
$463$	12.6491	0.587856	0.293928	+	0.955828i	$0.405037\pi$
$464$	26.8032	1.24431
$465$	−17.2101	−0.798097
$466$	43.8398	2.03084
$467$	24.5704	1.13698	0.568492	−	0.822689i	$-0.307527\pi$
$467$	24.5704	1.13698	0.568492	+	0.822689i	$0.307527\pi$
$468$	16.0277	0.740879
$469$	12.7294	0.587790
$470$	−1.21574	−0.0560779
$471$	−64.7356	−2.98286
$472$	−2.87586	−0.132372
$473$	25.0791	1.15314
$474$	21.7152	0.997412
$475$	0.555045	0.0254672
$476$	−0.449354	−0.0205961
$477$	−13.3417	−0.610874
$478$	−9.05386	−0.414114
$479$	21.9819	1.00438	0.502189	−	0.864758i	$-0.332528\pi$
$479$	21.9819	1.00438	0.502189	+	0.864758i	$0.332528\pi$
$480$	−12.6842	−0.578952
$481$	4.28087	0.195191
$482$	13.7833	0.627813
$483$	−26.2487	−1.19436
$484$	−4.26683	−0.193947
$485$	2.47563	0.112413
$486$	−84.2090	−3.81980
$487$	−41.8100	−1.89459	−0.947296	−	0.320360i	$-0.896196\pi$
$487$	−41.8100	−1.89459	−0.947296	+	0.320360i	$0.896196\pi$
$488$	5.69243	0.257684
$489$	58.2976	2.63631
$490$	1.64550	0.0743360
$491$	16.4321	0.741569	0.370785	−	0.928719i	$-0.379089\pi$
$491$	16.4321	0.741569	0.370785	+	0.928719i	$0.379089\pi$
$492$	1.45976	0.0658109
$493$	3.46312	0.155971
$494$	−2.60313	−0.117121
$495$	−17.7164	−0.796293
$496$	−25.5639	−1.14785
$497$	10.1580	0.455648
$498$	12.5572	0.562701
$499$	−9.00357	−0.403055	−0.201528	−	0.979483i	$-0.564591\pi$
$499$	−9.00357	−0.403055	−0.201528	+	0.979483i	$0.564591\pi$
$500$	−0.707660	−0.0316475
$501$	−14.4693	−0.646443
$502$	−13.6059	−0.607262
$503$	−15.0260	−0.669978	−0.334989	−	0.942222i	$-0.608733\pi$
$503$	−15.0260	−0.669978	−0.334989	+	0.942222i	$0.608733\pi$
$504$	16.8985	0.752720
$505$	−14.6200	−0.650581
$506$	29.1050	1.29387
$507$	−16.1342	−0.716544
$508$	4.38913	0.194736
$509$	−39.5544	−1.75322	−0.876610	−	0.481202i	$-0.840200\pi$
$509$	−39.5544	−1.75322	−0.876610	+	0.481202i	$0.840200\pi$
$510$	−3.45699	−0.153078
$511$	−11.0936	−0.490750
$512$	2.06073	0.0910722
$513$	9.08367	0.401054
$514$	−2.21615	−0.0977501
$515$	−10.8258	−0.477041
$516$	26.3374	1.15944
$517$	−1.64719	−0.0724434
$518$	−2.47148	−0.108591
$519$	−69.0622	−3.03149
$520$	−6.06101	−0.265793
$521$	5.53002	0.242275	0.121137	−	0.992636i	$-0.461346\pi$
$521$	5.53002	0.242275	0.121137	+	0.992636i	$0.461346\pi$
$522$	71.3141	3.12133
$523$	38.8149	1.69726	0.848629	−	0.528988i	$-0.177428\pi$
$523$	38.8149	1.69726	0.848629	+	0.528988i	$0.177428\pi$
$524$	10.6337	0.464534
$525$	3.30855	0.144397
$526$	−17.2849	−0.753658
$527$	−3.30301	−0.143881
$528$	−36.2510	−1.57762
$529$	39.9419	1.73661
$530$	−2.76270	−0.120004
$531$	−10.7465	−0.466360
$532$	0.392783	0.0170293
$533$	1.77701	0.0769709
$534$	47.3122	2.04740
$535$	−5.65375	−0.244433
$536$	27.0696	1.16923
$537$	−56.3805	−2.43300
$538$	−42.5007	−1.83233
$539$	2.22946	0.0960298
$540$	−11.5813	−0.498380
$541$	−27.8937	−1.19924	−0.599622	−	0.800283i	$-0.704683\pi$
$541$	−27.8937	−1.19924	−0.599622	+	0.800283i	$0.704683\pi$
$542$	8.97208	0.385384
$543$	5.50814	0.236377
$544$	−2.43439	−0.104374
$545$	6.84952	0.293401
$546$	−15.5169	−0.664063
$547$	34.2372	1.46388	0.731939	−	0.681371i	$-0.238616\pi$
$547$	34.2372	1.46388	0.731939	+	0.681371i	$0.238616\pi$
$548$	0.673619	0.0287756
$549$	21.2715	0.907846
$550$	−3.66858	−0.156429
$551$	−3.02713	−0.128960
$552$	−55.8189	−2.37581
$553$	−3.98868	−0.169616
$554$	−6.48433	−0.275493
$555$	−4.96933	−0.210936
$556$	10.3453	0.438739
$557$	−18.2466	−0.773131	−0.386566	−	0.922262i	$-0.626339\pi$
$557$	−18.2466	−0.773131	−0.386566	+	0.922262i	$0.626339\pi$
$558$	−68.0169	−2.87939
$559$	32.0614	1.35605
$560$	4.91454	0.207677
$561$	−4.68384	−0.197752
$562$	23.3525	0.985065
$563$	−34.0223	−1.43387	−0.716934	−	0.697141i	$-0.754455\pi$
$563$	−34.0223	−1.43387	−0.716934	+	0.697141i	$0.754455\pi$
$564$	−1.72984	−0.0728392
$565$	−18.7352	−0.788198
$566$	2.00135	0.0841230
$567$	30.3071	1.27278
$568$	21.6014	0.906374
$569$	−36.2421	−1.51935	−0.759674	−	0.650304i	$-0.774641\pi$
$569$	−36.2421	−1.51935	−0.759674	+	0.650304i	$0.774641\pi$
$570$	3.02178	0.126568
$571$	1.54934	0.0648377	0.0324189	−	0.999474i	$-0.489679\pi$
$571$	1.54934	0.0648377	0.0324189	+	0.999474i	$0.489679\pi$
$572$	4.49672	0.188017
$573$	30.1623	1.26005
$574$	−1.02593	−0.0428213
$575$	−7.93359	−0.330854
$576$	27.9765	1.16569
$577$	40.5250	1.68708	0.843538	−	0.537069i	$-0.180468\pi$
$577$	40.5250	1.68708	0.843538	+	0.537069i	$0.180468\pi$
$578$	27.3100	1.13594
$579$	−42.0532	−1.74767
$580$	3.85947	0.160256
$581$	−2.30652	−0.0956907
$582$	13.4778	0.558674
$583$	−3.74315	−0.155025
$584$	−23.5909	−0.976199
$585$	−22.6488	−0.936414
$586$	33.5097	1.38427
$587$	−7.30725	−0.301603	−0.150801	−	0.988564i	$-0.548185\pi$
$587$	−7.30725	−0.301603	−0.150801	+	0.988564i	$0.548185\pi$
$588$	2.34132	0.0965546
$589$	2.88718	0.118964
$590$	−2.22531	−0.0916146
$591$	44.4717	1.82932
$592$	−7.38148	−0.303377
$593$	30.2268	1.24127	0.620634	−	0.784100i	$-0.286875\pi$
$593$	30.2268	1.24127	0.620634	+	0.784100i	$0.286875\pi$
$594$	−60.0386	−2.46342
$595$	0.634986	0.0260319
$596$	−15.5129	−0.635434
$597$	−17.6893	−0.723974
$598$	37.2082	1.52155
$599$	1.41629	0.0578681	0.0289341	−	0.999581i	$-0.490789\pi$
$599$	1.41629	0.0578681	0.0289341	+	0.999581i	$0.490789\pi$
$600$	7.03576	0.287234
$601$	0.0722398	0.00294672	0.00147336	−	0.999999i	$-0.499531\pi$
$601$	0.0722398	0.00294672	0.00147336	+	0.999999i	$0.499531\pi$
$602$	−18.5101	−0.754414
$603$	101.154	4.11931
$604$	10.6910	0.435011
$605$	6.02949	0.245134
$606$	−79.5942	−3.23329
$607$	3.22724	0.130990	0.0654948	−	0.997853i	$-0.479137\pi$
$607$	3.22724	0.130990	0.0654948	+	0.997853i	$0.479137\pi$
$608$	2.12792	0.0862984
$609$	−18.0443	−0.731193
$610$	4.40474	0.178343
$611$	−2.10579	−0.0851910
$612$	−3.57078	−0.144340
$613$	28.4910	1.15074	0.575370	−	0.817893i	$-0.304858\pi$
$613$	28.4910	1.15074	0.575370	+	0.817893i	$0.304858\pi$
$614$	−30.0656	−1.21335
$615$	−2.06279	−0.0831799
$616$	4.74105	0.191022
$617$	−39.5059	−1.59045	−0.795224	−	0.606315i	$-0.792647\pi$
$617$	−39.5059	−1.59045	−0.795224	+	0.606315i	$0.792647\pi$
$618$	−58.9377	−2.37082
$619$	10.6413	0.427711	0.213855	−	0.976865i	$-0.431398\pi$
$619$	10.6413	0.427711	0.213855	+	0.976865i	$0.431398\pi$
$620$	−3.68103	−0.147834
$621$	−129.838	−5.21024
$622$	26.6167	1.06723
$623$	−8.69038	−0.348173
$624$	−46.3437	−1.85523
$625$	1.00000	0.0400000
$626$	31.3461	1.25284
$627$	4.09417	0.163505
$628$	−13.8462	−0.552523
$629$	−0.953729	−0.0380277
$630$	13.0759	0.520957
$631$	−16.3587	−0.651228	−0.325614	−	0.945503i	$-0.605571\pi$
$631$	−16.3587	−0.651228	−0.325614	+	0.945503i	$0.605571\pi$
$632$	−8.48210	−0.337400
$633$	48.1282	1.91292
$634$	46.1226	1.83176
$635$	−6.20232	−0.246131
$636$	−3.93095	−0.155872
$637$	2.85017	0.112928
$638$	20.0079	0.792120
$639$	80.7202	3.19324
$640$	13.4607	0.532081
$641$	−43.2716	−1.70913	−0.854563	−	0.519347i	$-0.826175\pi$
$641$	−43.2716	−1.70913	−0.854563	+	0.519347i	$0.826175\pi$
$642$	−30.7802	−1.21480
$643$	2.26870	0.0894688	0.0447344	−	0.998999i	$-0.485756\pi$
$643$	2.26870	0.0894688	0.0447344	+	0.998999i	$0.485756\pi$
$644$	−5.61428	−0.221234
$645$	−37.2176	−1.46544
$646$	0.579949	0.0228178
$647$	−42.5250	−1.67183	−0.835916	−	0.548858i	$-0.815063\pi$
$647$	−42.5250	−1.67183	−0.835916	+	0.548858i	$0.815063\pi$
$648$	64.4493	2.53181
$649$	−3.01505	−0.118351
$650$	−4.68995	−0.183955
$651$	17.2101	0.674515
$652$	12.4692	0.488331
$653$	−5.90644	−0.231137	−0.115568	−	0.993300i	$-0.536869\pi$
$653$	−5.90644	−0.231137	−0.115568	+	0.993300i	$0.536869\pi$
$654$	37.2902	1.45816
$655$	−15.0265	−0.587135
$656$	−3.06409	−0.119633
$657$	−88.1547	−3.43924
$658$	1.21574	0.0473944
$659$	−39.6997	−1.54648	−0.773241	−	0.634113i	$-0.781365\pi$
$659$	−39.6997	−1.54648	−0.773241	+	0.634113i	$0.781365\pi$
$660$	−5.21990	−0.203184
$661$	12.8157	0.498472	0.249236	−	0.968443i	$-0.419820\pi$
$661$	12.8157	0.498472	0.249236	+	0.968443i	$0.419820\pi$
$662$	−14.8333	−0.576514
$663$	−5.98788	−0.232550
$664$	−4.90492	−0.190348
$665$	−0.555045	−0.0215237
$666$	−19.6396	−0.761019
$667$	43.2687	1.67537
$668$	−3.09483	−0.119742
$669$	−65.1682	−2.51955
$670$	20.9462	0.809223
$671$	5.96794	0.230390
$672$	12.6842	0.489304
$673$	4.35441	0.167850	0.0839250	−	0.996472i	$-0.473254\pi$
$673$	4.35441	0.167850	0.0839250	+	0.996472i	$0.473254\pi$
$674$	−15.5326	−0.598295
$675$	16.3657	0.629914
$676$	−3.45091	−0.132727
$677$	7.04455	0.270744	0.135372	−	0.990795i	$-0.456777\pi$
$677$	7.04455	0.270744	0.135372	+	0.990795i	$0.456777\pi$
$678$	−101.998	−3.91723
$679$	−2.47563	−0.0950059
$680$	1.35033	0.0517826
$681$	58.6063	2.24580
$682$	−19.0828	−0.730719
$683$	7.32200	0.280169	0.140084	−	0.990140i	$-0.455263\pi$
$683$	7.32200	0.280169	0.140084	+	0.990140i	$0.455263\pi$
$684$	3.12124	0.119344
$685$	−0.951897	−0.0363701
$686$	−1.64550	−0.0628254
$687$	−3.30855	−0.126229
$688$	−55.2833	−2.10765
$689$	−4.78528	−0.182305
$690$	−43.1921	−1.64429
$691$	9.40655	0.357842	0.178921	−	0.983863i	$-0.442739\pi$
$691$	9.40655	0.357842	0.178921	+	0.983863i	$0.442739\pi$
$692$	−14.7716	−0.561532
$693$	17.7164	0.672990
$694$	20.2746	0.769614
$695$	−14.6190	−0.554532
$696$	−38.3720	−1.45449
$697$	−0.395898	−0.0149957
$698$	−27.9838	−1.05920
$699$	−88.1473	−3.33404
$700$	0.707660	0.0267470
$701$	−35.7070	−1.34863	−0.674316	−	0.738442i	$-0.735562\pi$
$701$	−35.7070	−1.34863	−0.674316	+	0.738442i	$0.735562\pi$
$702$	−76.7541	−2.89690
$703$	0.833660	0.0314421
$704$	7.84909	0.295824
$705$	2.44445	0.0920632
$706$	−59.3199	−2.23253
$707$	14.6200	0.549842
$708$	−3.16633	−0.118998
$709$	−21.0350	−0.789986	−0.394993	−	0.918684i	$-0.629253\pi$
$709$	−21.0350	−0.789986	−0.394993	+	0.918684i	$0.629253\pi$
$710$	16.7149	0.627300
$711$	−31.6960	−1.18869
$712$	−18.4805	−0.692584
$713$	−41.2682	−1.54551
$714$	3.45699	0.129375
$715$	−6.35436	−0.237640
$716$	−12.0591	−0.450671
$717$	18.2043	0.679852
$718$	−2.04620	−0.0763633
$719$	26.3251	0.981760	0.490880	−	0.871227i	$-0.336675\pi$
$719$	26.3251	0.981760	0.490880	+	0.871227i	$0.336675\pi$
$720$	39.0533	1.45543
$721$	10.8258	0.403173
$722$	30.7575	1.14468
$723$	−27.7137	−1.03068
$724$	1.17813	0.0437848
$725$	−5.45385	−0.202551
$726$	32.8258	1.21828
$727$	−21.5027	−0.797490	−0.398745	−	0.917062i	$-0.630554\pi$
$727$	−21.5027	−0.797490	−0.398745	+	0.917062i	$0.630554\pi$
$728$	6.06101	0.224636
$729$	78.3950	2.90352
$730$	−18.2544	−0.675626
$731$	−7.14292	−0.264190
$732$	6.26737	0.231649
$733$	28.8620	1.06604	0.533022	−	0.846101i	$-0.321056\pi$
$733$	28.8620	1.06604	0.533022	+	0.846101i	$0.321056\pi$
$734$	−7.62283	−0.281364
$735$	−3.30855	−0.122038
$736$	−30.4156	−1.12113
$737$	28.3798	1.04538
$738$	−8.15250	−0.300098
$739$	−16.1170	−0.592872	−0.296436	−	0.955053i	$-0.595798\pi$
$739$	−16.1170	−0.592872	−0.296436	+	0.955053i	$0.595798\pi$
$740$	−1.06288	−0.0390723
$741$	5.23403	0.192277
$742$	2.76270	0.101422
$743$	−31.9078	−1.17058	−0.585292	−	0.810823i	$-0.699020\pi$
$743$	−31.9078	−1.17058	−0.585292	+	0.810823i	$0.699020\pi$
$744$	36.5979	1.34174
$745$	21.9214	0.803140
$746$	−16.8093	−0.615433
$747$	−18.3287	−0.670613
$748$	−1.00182	−0.0366301
$749$	5.65375	0.206584
$750$	5.44420	0.198794
$751$	−11.2908	−0.412007	−0.206004	−	0.978551i	$-0.566046\pi$
$751$	−11.2908	−0.412007	−0.206004	+	0.978551i	$0.566046\pi$
$752$	3.63100	0.132409
$753$	27.3570	0.996943
$754$	25.5783	0.931507
$755$	−15.1075	−0.549820
$756$	11.5813	0.421208
$757$	19.4179	0.705754	0.352877	−	0.935670i	$-0.385203\pi$
$757$	19.4179	0.705754	0.352877	+	0.935670i	$0.385203\pi$
$758$	−14.1618	−0.514381
$759$	−58.5205	−2.12416
$760$	−1.18033	−0.0428149
$761$	−26.0111	−0.942902	−0.471451	−	0.881892i	$-0.656270\pi$
$761$	−26.0111	−0.942902	−0.471451	+	0.881892i	$0.656270\pi$
$762$	−33.7667	−1.22324
$763$	−6.84952	−0.247969
$764$	6.45136	0.233402
$765$	5.04591	0.182435
$766$	6.22563	0.224941
$767$	−3.85447	−0.139177
$768$	49.9866	1.80373
$769$	37.7665	1.36190	0.680948	−	0.732332i	$-0.261568\pi$
$769$	37.7665	1.36190	0.680948	+	0.732332i	$0.261568\pi$
$770$	3.66858	0.132206
$771$	4.45594	0.160477
$772$	−8.99470	−0.323726
$773$	5.57689	0.200587	0.100293	−	0.994958i	$-0.468022\pi$
$773$	5.57689	0.200587	0.100293	+	0.994958i	$0.468022\pi$
$774$	−147.090	−5.28704
$775$	5.20170	0.186851
$776$	−5.26453	−0.188986
$777$	4.96933	0.178274
$778$	−47.6797	−1.70940
$779$	0.346056	0.0123988
$780$	−6.67318	−0.238938
$781$	22.6469	0.810369
$782$	−8.28956	−0.296434
$783$	−89.2559	−3.18975
$784$	−4.91454	−0.175519
$785$	19.5662	0.698347
$786$	−81.8074	−2.91797
$787$	48.9999	1.74666	0.873329	−	0.487131i	$-0.161957\pi$
$787$	48.9999	1.74666	0.873329	+	0.487131i	$0.161957\pi$
$788$	9.51198	0.338850
$789$	34.7542	1.23728
$790$	−6.56336	−0.233514
$791$	18.7352	0.666149
$792$	37.6746	1.33871
$793$	7.62948	0.270931
$794$	38.4099	1.36312
$795$	5.55486	0.197011
$796$	−3.78353	−0.134104
$797$	25.4704	0.902206	0.451103	−	0.892472i	$-0.351031\pi$
$797$	25.4704	0.902206	0.451103	+	0.892472i	$0.351031\pi$
$798$	−3.02178	−0.106970
$799$	0.469146	0.0165972
$800$	3.83377	0.135544
$801$	−69.0579	−2.44004
$802$	−59.2810	−2.09328
$803$	−24.7327	−0.872797
$804$	29.8037	1.05110
$805$	7.93359	0.279622
$806$	−24.3957	−0.859302
$807$	85.4547	3.00815
$808$	31.0900	1.09374
$809$	27.0969	0.952675	0.476338	−	0.879262i	$-0.341964\pi$
$809$	27.0969	0.952675	0.476338	+	0.879262i	$0.341964\pi$
$810$	49.8702	1.75226
$811$	−13.5510	−0.475842	−0.237921	−	0.971285i	$-0.576466\pi$
$811$	−13.5510	−0.475842	−0.237921	+	0.971285i	$0.576466\pi$
$812$	−3.85947	−0.135441
$813$	−18.0399	−0.632685
$814$	−5.51009	−0.193128
$815$	−17.6203	−0.617212
$816$	10.3249	0.361442
$817$	6.24366	0.218438
$818$	−11.4178	−0.399214
$819$	22.6488	0.791414
$820$	−0.441208	−0.0154076
$821$	21.1165	0.736972	0.368486	−	0.929633i	$-0.379876\pi$
$821$	21.1165	0.736972	0.368486	+	0.929633i	$0.379876\pi$
$822$	−5.18232	−0.180754
$823$	−15.7378	−0.548585	−0.274293	−	0.961646i	$-0.588444\pi$
$823$	−15.7378	−0.548585	−0.274293	+	0.961646i	$0.588444\pi$
$824$	23.0215	0.801991
$825$	7.37629	0.256809
$826$	2.22531	0.0774285
$827$	4.68107	0.162777	0.0813883	−	0.996682i	$-0.474065\pi$
$827$	4.68107	0.162777	0.0813883	+	0.996682i	$0.474065\pi$
$828$	−44.6138	−1.55044
$829$	16.7036	0.580139	0.290069	−	0.957006i	$-0.406322\pi$
$829$	16.7036	0.580139	0.290069	+	0.957006i	$0.406322\pi$
$830$	−3.79538	−0.131739
$831$	13.0378	0.452278
$832$	10.0344	0.347879
$833$	−0.634986	−0.0220010
$834$	−79.5890	−2.75594
$835$	4.37332	0.151345
$836$	0.875696	0.0302866
$837$	85.1292	2.94250
$838$	37.3932	1.29173
$839$	−19.4319	−0.670864	−0.335432	−	0.942064i	$-0.608882\pi$
$839$	−19.4319	−0.670864	−0.335432	+	0.942064i	$0.608882\pi$
$840$	−7.03576	−0.242757
$841$	0.744526	0.0256733
$842$	−15.6624	−0.539760
$843$	−46.9541	−1.61718
$844$	10.2941	0.354336
$845$	4.87651	0.167757
$846$	9.66084	0.332147
$847$	−6.02949	−0.207176
$848$	8.25123	0.283348
$849$	−4.02405	−0.138105
$850$	1.04487	0.0358387
$851$	−11.9160	−0.408475
$852$	23.7831	0.814797
$853$	−6.38399	−0.218584	−0.109292	−	0.994010i	$-0.534858\pi$
$853$	−6.38399	−0.218584	−0.109292	+	0.994010i	$0.534858\pi$
$854$	−4.40474	−0.150727
$855$	−4.41065	−0.150841
$856$	12.0229	0.410936
$857$	15.7865	0.539258	0.269629	−	0.962964i	$-0.413099\pi$
$857$	15.7865	0.539258	0.269629	+	0.962964i	$0.413099\pi$
$858$	−34.5944	−1.18103
$859$	−10.4304	−0.355882	−0.177941	−	0.984041i	$-0.556944\pi$
$859$	−10.4304	−0.355882	−0.177941	+	0.984041i	$0.556944\pi$
$860$	−7.96041	−0.271448
$861$	2.06279	0.0702999
$862$	−34.6264	−1.17938
$863$	−7.64897	−0.260374	−0.130187	−	0.991489i	$-0.541558\pi$
$863$	−7.64897	−0.260374	−0.130187	+	0.991489i	$0.541558\pi$
$864$	62.7422	2.13453
$865$	20.8739	0.709733
$866$	−13.8146	−0.469441
$867$	−54.9113	−1.86488
$868$	3.68103	0.124942
$869$	−8.89262	−0.301662
$870$	−29.6919	−1.00665
$871$	36.2810	1.22934
$872$	−14.5658	−0.493260
$873$	−19.6725	−0.665814
$874$	7.24595	0.245098
$875$	−1.00000	−0.0338062
$876$	−25.9736	−0.877567
$877$	43.1913	1.45847	0.729233	−	0.684266i	$-0.239877\pi$
$877$	43.1913	1.45847	0.729233	+	0.684266i	$0.239877\pi$
$878$	7.53546	0.254309
$879$	−67.3768	−2.27256
$880$	10.9568	0.369353
$881$	−31.6402	−1.06598	−0.532992	−	0.846120i	$-0.678933\pi$
$881$	−31.6402	−1.06598	−0.532992	+	0.846120i	$0.678933\pi$
$882$	−13.0759	−0.440289
$883$	25.5330	0.859253	0.429626	−	0.903007i	$-0.358645\pi$
$883$	25.5330	0.859253	0.429626	+	0.903007i	$0.358645\pi$
$884$	−1.28074	−0.0430759
$885$	4.47436	0.150404
$886$	36.4508	1.22459
$887$	−27.1648	−0.912105	−0.456053	−	0.889953i	$-0.650737\pi$
$887$	−27.1648	−0.912105	−0.456053	+	0.889953i	$0.650737\pi$
$888$	10.5675	0.354622
$889$	6.20232	0.208019
$890$	−14.3000	−0.479337
$891$	67.5686	2.26363
$892$	−13.9387	−0.466703
$893$	−0.410083	−0.0137229
$894$	119.345	3.99149
$895$	17.0409	0.569614
$896$	−13.4607	−0.449691
$897$	−74.8132	−2.49794
$898$	−31.6512	−1.05621
$899$	−28.3693	−0.946169
$900$	5.62340	0.187447
$901$	1.06611	0.0355172
$902$	−2.28726	−0.0761576
$903$	37.2176	1.23852
$904$	39.8413	1.32510
$905$	−1.66482	−0.0553405
$906$	−82.2485	−2.73252
$907$	−12.7652	−0.423863	−0.211931	−	0.977285i	$-0.567975\pi$
$907$	−12.7652	−0.423863	−0.211931	+	0.977285i	$0.567975\pi$
$908$	12.5352	0.415996
$909$	116.177	3.85336
$910$	4.68995	0.155470
$911$	3.09851	0.102658	0.0513292	−	0.998682i	$-0.483654\pi$
$911$	3.09851	0.102658	0.0513292	+	0.998682i	$0.483654\pi$
$912$	−9.02502	−0.298848
$913$	−5.14231	−0.170186
$914$	10.0556	0.332609
$915$	−8.85647	−0.292786
$916$	−0.707660	−0.0233817
$917$	15.0265	0.496219
$918$	17.0999	0.564382
$919$	56.5282	1.86469	0.932346	−	0.361568i	$-0.117758\pi$
$919$	56.5282	1.86469	0.932346	+	0.361568i	$0.117758\pi$
$920$	16.8711	0.556224
$921$	60.4520	1.99196
$922$	−10.9403	−0.360299
$923$	28.9520	0.952967
$924$	5.21990	0.171722
$925$	1.50197	0.0493844
$926$	−20.8141	−0.683995
$927$	86.0268	2.82549
$928$	−20.9088	−0.686366
$929$	8.95801	0.293903	0.146951	−	0.989144i	$-0.453054\pi$
$929$	8.95801	0.293903	0.146951	+	0.989144i	$0.453054\pi$
$930$	28.3191	0.928620
$931$	0.555045	0.0181909
$932$	−18.8537	−0.617573
$933$	−53.5173	−1.75208
$934$	−40.4305	−1.32293
$935$	1.41568	0.0462977
$936$	48.1637	1.57428
$937$	0.280288	0.00915661	0.00457831	−	0.999990i	$-0.498543\pi$
$937$	0.280288	0.00915661	0.00457831	+	0.999990i	$0.498543\pi$
$938$	−20.9462	−0.683918
$939$	−63.0266	−2.05679
$940$	0.522839	0.0170531
$941$	2.19279	0.0714830	0.0357415	−	0.999361i	$-0.488621\pi$
$941$	2.19279	0.0714830	0.0357415	+	0.999361i	$0.488621\pi$
$942$	106.522	3.47068
$943$	−4.94640	−0.161077
$944$	6.64625	0.216317
$945$	−16.3657	−0.532375
$946$	−41.2675	−1.34172
$947$	26.4176	0.858456	0.429228	−	0.903196i	$-0.358786\pi$
$947$	26.4176	0.858456	0.429228	+	0.903196i	$0.358786\pi$
$948$	−9.33880	−0.303310
$949$	−31.6185	−1.02638
$950$	−0.913325	−0.0296322
$951$	−92.7373	−3.00721
$952$	−1.35033	−0.0437643
$953$	−30.0610	−0.973772	−0.486886	−	0.873465i	$-0.661867\pi$
$953$	−30.0610	−0.973772	−0.486886	+	0.873465i	$0.661867\pi$
$954$	21.9537	0.710778
$955$	−9.11648	−0.295002
$956$	3.89369	0.125931
$957$	−40.2292	−1.30043
$958$	−36.1711	−1.16863
$959$	0.951897	0.0307384
$960$	−11.6481	−0.375942
$961$	−3.94233	−0.127172
$962$	−7.04416	−0.227113
$963$	44.9274	1.44777
$964$	−5.92763	−0.190916
$965$	12.7105	0.409165
$966$	43.1921	1.38968
$967$	−19.2931	−0.620424	−0.310212	−	0.950667i	$-0.600400\pi$
$967$	−19.2931	−0.620424	−0.310212	+	0.950667i	$0.600400\pi$
$968$	−12.8220	−0.412113
$969$	−1.16608	−0.0374600
$970$	−4.07364	−0.130797
$971$	16.2955	0.522946	0.261473	−	0.965211i	$-0.415792\pi$
$971$	16.2955	0.522946	0.261473	+	0.965211i	$0.415792\pi$
$972$	36.2148	1.16159
$973$	14.6190	0.468665
$974$	68.7982	2.20444
$975$	9.42993	0.301999
$976$	−13.1555	−0.421096
$977$	38.0193	1.21635	0.608173	−	0.793805i	$-0.291903\pi$
$977$	38.0193	1.21635	0.608173	+	0.793805i	$0.291903\pi$
$978$	−95.9285	−3.06745
$979$	−19.3749	−0.619224
$980$	−0.707660	−0.0226054
$981$	−54.4296	−1.73780
$982$	−27.0389	−0.862847
$983$	7.27123	0.231916	0.115958	−	0.993254i	$-0.463006\pi$
$983$	7.27123	0.231916	0.115958	+	0.993254i	$0.463006\pi$
$984$	4.38662	0.139840
$985$	−13.4415	−0.428280
$986$	−5.69856	−0.181479
$987$	−2.44445	−0.0778076
$988$	1.11950	0.0356160
$989$	−89.2444	−2.83781
$990$	29.1523	0.926520
$991$	35.4358	1.12566	0.562828	−	0.826574i	$-0.309713\pi$
$991$	35.4358	1.12566	0.562828	+	0.826574i	$0.309713\pi$
$992$	19.9421	0.633163
$993$	29.8249	0.946465
$994$	−16.7149	−0.530166
$995$	5.34654	0.169497
$996$	−5.40032	−0.171116
$997$	−40.8943	−1.29513	−0.647567	−	0.762008i	$-0.724214\pi$
$997$	−40.8943	−1.29513	−0.647567	+	0.762008i	$0.724214\pi$
$998$	14.8153	0.468972
$999$	24.5807	0.777699

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8015.2.a.n.1.16	✓	68

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8015.2.a.n.1.16	✓	68	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:	\( N \)	\(=\)	\( 8015 = 5 \cdot 7 \cdot 229 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8015.a (trivial)

\(f(q)\)	\(=\)	\(q-1.64550 q^{2} +3.30855 q^{3} +0.707660 q^{4} -1.00000 q^{5} -5.44420 q^{6} +1.00000 q^{7} +2.12654 q^{8} +7.94648 q^{9} +O(q^{10})\)	\(q-1.64550 q^{2} +3.30855 q^{3} +0.707660 q^{4} -1.00000 q^{5} -5.44420 q^{6} +1.00000 q^{7} +2.12654 q^{8} +7.94648 q^{9} +1.64550 q^{10} +2.22946 q^{11} +2.34132 q^{12} +2.85017 q^{13} -1.64550 q^{14} -3.30855 q^{15} -4.91454 q^{16} -0.634986 q^{17} -13.0759 q^{18} +0.555045 q^{19} -0.707660 q^{20} +3.30855 q^{21} -3.66858 q^{22} -7.93359 q^{23} +7.03576 q^{24} +1.00000 q^{25} -4.68995 q^{26} +16.3657 q^{27} +0.707660 q^{28} -5.45385 q^{29} +5.44420 q^{30} +5.20170 q^{31} +3.83377 q^{32} +7.37629 q^{33} +1.04487 q^{34} -1.00000 q^{35} +5.62340 q^{36} +1.50197 q^{37} -0.913325 q^{38} +9.42993 q^{39} -2.12654 q^{40} +0.623475 q^{41} -5.44420 q^{42} +11.2489 q^{43} +1.57770 q^{44} -7.94648 q^{45} +13.0547 q^{46} -0.738828 q^{47} -16.2600 q^{48} +1.00000 q^{49} -1.64550 q^{50} -2.10088 q^{51} +2.01695 q^{52} -1.67894 q^{53} -26.9296 q^{54} -2.22946 q^{55} +2.12654 q^{56} +1.83639 q^{57} +8.97430 q^{58} -1.35236 q^{59} -2.34132 q^{60} +2.67685 q^{61} -8.55938 q^{62} +7.94648 q^{63} +3.52062 q^{64} -2.85017 q^{65} -12.1377 q^{66} +12.7294 q^{67} -0.449354 q^{68} -26.2487 q^{69} +1.64550 q^{70} +10.1580 q^{71} +16.8985 q^{72} -11.0936 q^{73} -2.47148 q^{74} +3.30855 q^{75} +0.392783 q^{76} +2.22946 q^{77} -15.5169 q^{78} -3.98868 q^{79} +4.91454 q^{80} +30.3071 q^{81} -1.02593 q^{82} -2.30652 q^{83} +2.34132 q^{84} +0.634986 q^{85} -18.5101 q^{86} -18.0443 q^{87} +4.74105 q^{88} -8.69038 q^{89} +13.0759 q^{90} +2.85017 q^{91} -5.61428 q^{92} +17.2101 q^{93} +1.21574 q^{94} -0.555045 q^{95} +12.6842 q^{96} -2.47563 q^{97} -1.64550 q^{98} +17.7164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9}+O(q^{10}) \) 68 * q + 9 * q^2 + 83 * q^4 - 68 * q^5 + 5 * q^6 + 68 * q^7 + 30 * q^8 + 86 * q^9	\( 68 q + 9 q^{2} + 83 q^{4} - 68 q^{5} + 5 q^{6} + 68 q^{7} + 30 q^{8} + 86 q^{9} - 9 q^{10} + 5 q^{11} + 9 q^{12} + 15 q^{13} + 9 q^{14} + 109 q^{16} + 7 q^{17} + 39 q^{18} + 20 q^{19} - 83 q^{20} + 56 q^{22} + 36 q^{23} + q^{24} + 68 q^{25} + q^{26} + 12 q^{27} + 83 q^{28} - 16 q^{29} - 5 q^{30} + 31 q^{31} + 79 q^{32} + 45 q^{33} + 31 q^{34} - 68 q^{35} + 114 q^{36} + 72 q^{37} + 8 q^{38} + 47 q^{39} - 30 q^{40} + 6 q^{41} + 5 q^{42} + 75 q^{43} + 15 q^{44} - 86 q^{45} + 29 q^{46} - 10 q^{47} + 44 q^{48} + 68 q^{49} + 9 q^{50} + 23 q^{51} + 37 q^{52} + 41 q^{53} + 4 q^{54} - 5 q^{55} + 30 q^{56} + 55 q^{57} + 66 q^{58} - 5 q^{59} - 9 q^{60} - 2 q^{61} + 3 q^{62} + 86 q^{63} + 162 q^{64} - 15 q^{65} - 23 q^{66} + 92 q^{67} + 35 q^{68} - 25 q^{69} - 9 q^{70} - 2 q^{71} + 128 q^{72} + 80 q^{73} + 18 q^{74} + 71 q^{76} + 5 q^{77} + 20 q^{78} + 100 q^{79} - 109 q^{80} + 140 q^{81} + 36 q^{82} - 60 q^{83} + 9 q^{84} - 7 q^{85} - 27 q^{86} + 24 q^{87} + 175 q^{88} + 19 q^{89} - 39 q^{90} + 15 q^{91} + 75 q^{92} + 37 q^{93} + 11 q^{94} - 20 q^{95} + 15 q^{96} + 96 q^{97} + 9 q^{98} + 3 q^{99}+O(q^{100}) \) 68 * q + 9 * q^2 + 83 * q^4 - 68 * q^5 + 5 * q^6 + 68 * q^7 + 30 * q^8 + 86 * q^9 - 9 * q^10 + 5 * q^11 + 9 * q^12 + 15 * q^13 + 9 * q^14 + 109 * q^16 + 7 * q^17 + 39 * q^18 + 20 * q^19 - 83 * q^20 + 56 * q^22 + 36 * q^23 + q^24 + 68 * q^25 + q^26 + 12 * q^27 + 83 * q^28 - 16 * q^29 - 5 * q^30 + 31 * q^31 + 79 * q^32 + 45 * q^33 + 31 * q^34 - 68 * q^35 + 114 * q^36 + 72 * q^37 + 8 * q^38 + 47 * q^39 - 30 * q^40 + 6 * q^41 + 5 * q^42 + 75 * q^43 + 15 * q^44 - 86 * q^45 + 29 * q^46 - 10 * q^47 + 44 * q^48 + 68 * q^49 + 9 * q^50 + 23 * q^51 + 37 * q^52 + 41 * q^53 + 4 * q^54 - 5 * q^55 + 30 * q^56 + 55 * q^57 + 66 * q^58 - 5 * q^59 - 9 * q^60 - 2 * q^61 + 3 * q^62 + 86 * q^63 + 162 * q^64 - 15 * q^65 - 23 * q^66 + 92 * q^67 + 35 * q^68 - 25 * q^69 - 9 * q^70 - 2 * q^71 + 128 * q^72 + 80 * q^73 + 18 * q^74 + 71 * q^76 + 5 * q^77 + 20 * q^78 + 100 * q^79 - 109 * q^80 + 140 * q^81 + 36 * q^82 - 60 * q^83 + 9 * q^84 - 7 * q^85 - 27 * q^86 + 24 * q^87 + 175 * q^88 + 19 * q^89 - 39 * q^90 + 15 * q^91 + 75 * q^92 + 37 * q^93 + 11 * q^94 - 20 * q^95 + 15 * q^96 + 96 * q^97 + 9 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more