LMFDB - Embedded newform 538.2.a.e.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [538,2,Mod(1,538)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(538, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("538.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$538 = 2 \cdot 269$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	538.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:	$4.29595162874$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48$ x^7 - x^6 - 15x^5 + 16x^4 + 49x^3 - 53x^2 - 44*x + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.14366$ of defining polynomial
Character	$\chi$	$=$	538.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.65255 q^{3} +1.00000 q^{4} +0.725043 q^{5} +1.65255 q^{6} +1.29978 q^{7} +1.00000 q^{8} -0.269065 q^{9} +0.725043 q^{10} -3.71687 q^{11} +1.65255 q^{12} +2.30856 q^{13} +1.29978 q^{14} +1.19817 q^{15} +1.00000 q^{16} +6.32982 q^{17} -0.269065 q^{18} +0.0212539 q^{19} +0.725043 q^{20} +2.14795 q^{21} -3.71687 q^{22} -1.30511 q^{23} +1.65255 q^{24} -4.47431 q^{25} +2.30856 q^{26} -5.40231 q^{27} +1.29978 q^{28} -2.06303 q^{29} +1.19817 q^{30} -5.82316 q^{31} +1.00000 q^{32} -6.14233 q^{33} +6.32982 q^{34} +0.942394 q^{35} -0.269065 q^{36} -7.25272 q^{37} +0.0212539 q^{38} +3.81503 q^{39} +0.725043 q^{40} +8.77454 q^{41} +2.14795 q^{42} +4.52327 q^{43} -3.71687 q^{44} -0.195084 q^{45} -1.30511 q^{46} +3.85502 q^{47} +1.65255 q^{48} -5.31058 q^{49} -4.47431 q^{50} +10.4604 q^{51} +2.30856 q^{52} -6.86923 q^{53} -5.40231 q^{54} -2.69489 q^{55} +1.29978 q^{56} +0.0351232 q^{57} -2.06303 q^{58} +7.84518 q^{59} +1.19817 q^{60} +3.84958 q^{61} -5.82316 q^{62} -0.349724 q^{63} +1.00000 q^{64} +1.67381 q^{65} -6.14233 q^{66} -9.31541 q^{67} +6.32982 q^{68} -2.15676 q^{69} +0.942394 q^{70} -0.643097 q^{71} -0.269065 q^{72} -2.97920 q^{73} -7.25272 q^{74} -7.39404 q^{75} +0.0212539 q^{76} -4.83110 q^{77} +3.81503 q^{78} +0.418245 q^{79} +0.725043 q^{80} -8.12041 q^{81} +8.77454 q^{82} -15.6322 q^{83} +2.14795 q^{84} +4.58939 q^{85} +4.52327 q^{86} -3.40927 q^{87} -3.71687 q^{88} +10.3879 q^{89} -0.195084 q^{90} +3.00062 q^{91} -1.30511 q^{92} -9.62309 q^{93} +3.85502 q^{94} +0.0154100 q^{95} +1.65255 q^{96} +5.07659 q^{97} -5.31058 q^{98} +1.00008 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9} + 7 q^{10} - 3 q^{11} + q^{12} - 9 q^{13} + 6 q^{14} + 8 q^{15} + 7 q^{16} + 8 q^{17} + 12 q^{18} - 11 q^{19} + 7 q^{20}+ \cdots - 37 q^{99}+O(q^{100})$ 7 * q + 7 * q^2 + q^3 + 7 * q^4 + 7 * q^5 + q^6 + 6 * q^7 + 7 * q^8 + 12 * q^9 + 7 * q^10 - 3 * q^11 + q^12 - 9 * q^13 + 6 * q^14 + 8 * q^15 + 7 * q^16 + 8 * q^17 + 12 * q^18 - 11 * q^19 + 7 * q^20 - 6 * q^21 - 3 * q^22 + 12 * q^23 + q^24 + 22 * q^25 - 9 * q^26 - 14 * q^27 + 6 * q^28 - 5 * q^29 + 8 * q^30 + 14 * q^31 + 7 * q^32 - 4 * q^33 + 8 * q^34 - 4 * q^35 + 12 * q^36 + 13 * q^37 - 11 * q^38 - 18 * q^39 + 7 * q^40 + 12 * q^41 - 6 * q^42 - 11 * q^43 - 3 * q^44 + 3 * q^45 + 12 * q^46 + 2 * q^47 + q^48 + 15 * q^49 + 22 * q^50 - 26 * q^51 - 9 * q^52 + 19 * q^53 - 14 * q^54 - 40 * q^55 + 6 * q^56 - 12 * q^57 - 5 * q^58 - 9 * q^59 + 8 * q^60 - 3 * q^61 + 14 * q^62 - 26 * q^63 + 7 * q^64 - 10 * q^65 - 4 * q^66 - 33 * q^67 + 8 * q^68 - 64 * q^69 - 4 * q^70 + 28 * q^71 + 12 * q^72 - 14 * q^73 + 13 * q^74 - 45 * q^75 - 11 * q^76 + 10 * q^77 - 18 * q^78 + 2 * q^79 + 7 * q^80 + 15 * q^81 + 12 * q^82 - 7 * q^83 - 6 * q^84 - 16 * q^85 - 11 * q^86 + 16 * q^87 - 3 * q^88 + 18 * q^89 + 3 * q^90 - 26 * q^91 + 12 * q^92 + 6 * q^93 + 2 * q^94 - 34 * q^95 + q^96 - 4 * q^97 + 15 * q^98 - 37 * q^99

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

)

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.65255	0.954103	0.477051	−	0.878875i	$-0.341706\pi$
$3$	1.65255	0.954103	0.477051	+	0.878875i	$0.341706\pi$
$4$	1.00000	0.500000
$5$	0.725043	0.324249	0.162125	−	0.986770i	$-0.448165\pi$
$5$	0.725043	0.324249	0.162125	+	0.986770i	$0.448165\pi$
$6$	1.65255	0.674652
$7$	1.29978	0.491269	0.245635	−	0.969362i	$-0.421004\pi$
$7$	1.29978	0.491269	0.245635	+	0.969362i	$0.421004\pi$
$8$	1.00000	0.353553
$9$	−0.269065	−0.0896884
$10$	0.725043	0.229279
$11$	−3.71687	−1.12068	−0.560340	−	0.828263i	$-0.689329\pi$
$11$	−3.71687	−1.12068	−0.560340	+	0.828263i	$0.689329\pi$
$12$	1.65255	0.477051
$13$	2.30856	0.640280	0.320140	−	0.947370i	$-0.396270\pi$
$13$	2.30856	0.640280	0.320140	+	0.947370i	$0.396270\pi$
$14$	1.29978	0.347380
$15$	1.19817	0.309367
$16$	1.00000	0.250000
$17$	6.32982	1.53521	0.767603	−	0.640925i	$-0.221449\pi$
$17$	6.32982	1.53521	0.767603	+	0.640925i	$0.221449\pi$
$18$	−0.269065	−0.0634192
$19$	0.0212539	0.00487598	0.00243799	−	0.999997i	$-0.499224\pi$
$19$	0.0212539	0.00487598	0.00243799	+	0.999997i	$0.499224\pi$
$20$	0.725043	0.162125
$21$	2.14795	0.468721
$22$	−3.71687	−0.792440
$23$	−1.30511	−0.272134	−0.136067	−	0.990700i	$-0.543446\pi$
$23$	−1.30511	−0.272134	−0.136067	+	0.990700i	$0.543446\pi$
$24$	1.65255	0.337326
$25$	−4.47431	−0.894863
$26$	2.30856	0.452747
$27$	−5.40231	−1.03967
$28$	1.29978	0.245635
$29$	−2.06303	−0.383095	−0.191548	−	0.981483i	$-0.561351\pi$
$29$	−2.06303	−0.383095	−0.191548	+	0.981483i	$0.561351\pi$
$30$	1.19817	0.218755
$31$	−5.82316	−1.04587	−0.522935	−	0.852372i	$-0.675163\pi$
$31$	−5.82316	−1.04587	−0.522935	+	0.852372i	$0.675163\pi$
$32$	1.00000	0.176777
$33$	−6.14233	−1.06924
$34$	6.32982	1.08555
$35$	0.942394	0.159294
$36$	−0.269065	−0.0448442
$37$	−7.25272	−1.19234	−0.596170	−	0.802858i	$-0.703312\pi$
$37$	−7.25272	−1.19234	−0.596170	+	0.802858i	$0.703312\pi$
$38$	0.0212539	0.00344784
$39$	3.81503	0.610893
$40$	0.725043	0.114639
$41$	8.77454	1.37035	0.685176	−	0.728377i	$-0.259725\pi$
$41$	8.77454	1.37035	0.685176	+	0.728377i	$0.259725\pi$
$42$	2.14795	0.331436
$43$	4.52327	0.689792	0.344896	−	0.938641i	$-0.387914\pi$
$43$	4.52327	0.689792	0.344896	+	0.938641i	$0.387914\pi$
$44$	−3.71687	−0.560340
$45$	−0.195084	−0.0290814
$46$	−1.30511	−0.192428
$47$	3.85502	0.562313	0.281156	−	0.959662i	$-0.409282\pi$
$47$	3.85502	0.562313	0.281156	+	0.959662i	$0.409282\pi$
$48$	1.65255	0.238526
$49$	−5.31058	−0.758654
$50$	−4.47431	−0.632763
$51$	10.4604	1.46474
$52$	2.30856	0.320140
$53$	−6.86923	−0.943562	−0.471781	−	0.881716i	$-0.656389\pi$
$53$	−6.86923	−0.943562	−0.471781	+	0.881716i	$0.656389\pi$
$54$	−5.40231	−0.735161
$55$	−2.69489	−0.363379
$56$	1.29978	0.173690
$57$	0.0351232	0.00465218
$58$	−2.06303	−0.270889
$59$	7.84518	1.02136	0.510678	−	0.859772i	$-0.329395\pi$
$59$	7.84518	1.02136	0.510678	+	0.859772i	$0.329395\pi$
$60$	1.19817	0.154683
$61$	3.84958	0.492888	0.246444	−	0.969157i	$-0.420738\pi$
$61$	3.84958	0.492888	0.246444	+	0.969157i	$0.420738\pi$
$62$	−5.82316	−0.739542
$63$	−0.349724	−0.0440611
$64$	1.00000	0.125000
$65$	1.67381	0.207610
$66$	−6.14233	−0.756069
$67$	−9.31541	−1.13806	−0.569029	−	0.822317i	$-0.692681\pi$
$67$	−9.31541	−1.13806	−0.569029	+	0.822317i	$0.692681\pi$
$68$	6.32982	0.767603
$69$	−2.15676	−0.259644
$70$	0.942394	0.112638
$71$	−0.643097	−0.0763215	−0.0381608	−	0.999272i	$-0.512150\pi$
$71$	−0.643097	−0.0763215	−0.0381608	+	0.999272i	$0.512150\pi$
$72$	−0.269065	−0.0317096
$73$	−2.97920	−0.348689	−0.174345	−	0.984685i	$-0.555781\pi$
$73$	−2.97920	−0.348689	−0.174345	+	0.984685i	$0.555781\pi$
$74$	−7.25272	−0.843112
$75$	−7.39404	−0.853791
$76$	0.0212539	0.00243799
$77$	−4.83110	−0.550555
$78$	3.81503	0.431967
$79$	0.418245	0.0470563	0.0235281	−	0.999723i	$-0.492510\pi$
$79$	0.418245	0.0470563	0.0235281	+	0.999723i	$0.492510\pi$
$80$	0.725043	0.0810623
$81$	−8.12041	−0.902268
$82$	8.77454	0.968985
$83$	−15.6322	−1.71585	−0.857927	−	0.513771i	$-0.828248\pi$
$83$	−15.6322	−1.71585	−0.857927	+	0.513771i	$0.828248\pi$
$84$	2.14795	0.234361
$85$	4.58939	0.497789
$86$	4.52327	0.487757
$87$	−3.40927	−0.365512
$88$	−3.71687	−0.396220
$89$	10.3879	1.10111	0.550556	−	0.834798i	$-0.314416\pi$
$89$	10.3879	1.10111	0.550556	+	0.834798i	$0.314416\pi$
$90$	−0.195084	−0.0205636
$91$	3.00062	0.314550
$92$	−1.30511	−0.136067
$93$	−9.62309	−0.997868
$94$	3.85502	0.397615
$95$	0.0154100	0.00158103
$96$	1.65255	0.168663
$97$	5.07659	0.515450	0.257725	−	0.966218i	$-0.417027\pi$
$97$	5.07659	0.515450	0.257725	+	0.966218i	$0.417027\pi$
$98$	−5.31058	−0.536450
$99$	1.00008	0.100512
$100$	−4.47431	−0.447431
$101$	12.5321	1.24699	0.623495	−	0.781827i	$-0.285712\pi$
$101$	12.5321	1.24699	0.623495	+	0.781827i	$0.285712\pi$
$102$	10.4604	1.03573
$103$	5.43374	0.535403	0.267701	−	0.963502i	$-0.413736\pi$
$103$	5.43374	0.535403	0.267701	+	0.963502i	$0.413736\pi$
$104$	2.30856	0.226373
$105$	1.55736	0.151982
$106$	−6.86923	−0.667199
$107$	−3.45500	−0.334007	−0.167004	−	0.985956i	$-0.553409\pi$
$107$	−3.45500	−0.334007	−0.167004	+	0.985956i	$0.553409\pi$
$108$	−5.40231	−0.519837
$109$	−6.80801	−0.652089	−0.326044	−	0.945354i	$-0.605716\pi$
$109$	−6.80801	−0.652089	−0.326044	+	0.945354i	$0.605716\pi$
$110$	−2.69489	−0.256948
$111$	−11.9855	−1.13762
$112$	1.29978	0.122817
$113$	0.407582	0.0383421	0.0191710	−	0.999816i	$-0.493897\pi$
$113$	0.407582	0.0383421	0.0191710	+	0.999816i	$0.493897\pi$
$114$	0.0351232	0.00328959
$115$	−0.946259	−0.0882391
$116$	−2.06303	−0.191548
$117$	−0.621154	−0.0574257
$118$	7.84518	0.722207
$119$	8.22735	0.754200
$120$	1.19817	0.109378
$121$	2.81514	0.255922
$122$	3.84958	0.348524
$123$	14.5004	1.30746
$124$	−5.82316	−0.522935
$125$	−6.86928	−0.614407
$126$	−0.349724	−0.0311559
$127$	−9.52589	−0.845286	−0.422643	−	0.906296i	$-0.638898\pi$
$127$	−9.52589	−0.845286	−0.422643	+	0.906296i	$0.638898\pi$
$128$	1.00000	0.0883883
$129$	7.47495	0.658132
$130$	1.67381	0.146803
$131$	−21.9325	−1.91625	−0.958127	−	0.286342i	$-0.907561\pi$
$131$	−21.9325	−1.91625	−0.958127	+	0.286342i	$0.907561\pi$
$132$	−6.14233	−0.534621
$133$	0.0276253	0.00239542
$134$	−9.31541	−0.804729
$135$	−3.91690	−0.337113
$136$	6.32982	0.542777
$137$	−2.48849	−0.212606	−0.106303	−	0.994334i	$-0.533901\pi$
$137$	−2.48849	−0.212606	−0.106303	+	0.994334i	$0.533901\pi$
$138$	−2.15676	−0.183596
$139$	21.6244	1.83416	0.917078	−	0.398708i	$-0.130541\pi$
$139$	21.6244	1.83416	0.917078	+	0.398708i	$0.130541\pi$
$140$	0.942394	0.0796468
$141$	6.37063	0.536504
$142$	−0.643097	−0.0539675
$143$	−8.58064	−0.717549
$144$	−0.269065	−0.0224221
$145$	−1.49579	−0.124218
$146$	−2.97920	−0.246561
$147$	−8.77602	−0.723834
$148$	−7.25272	−0.596170
$149$	20.9106	1.71306	0.856532	−	0.516093i	$-0.172614\pi$
$149$	20.9106	1.71306	0.856532	+	0.516093i	$0.172614\pi$
$150$	−7.39404	−0.603721
$151$	−2.07209	−0.168624	−0.0843120	−	0.996439i	$-0.526869\pi$
$151$	−2.07209	−0.168624	−0.0843120	+	0.996439i	$0.526869\pi$
$152$	0.0212539	0.00172392
$153$	−1.70313	−0.137690
$154$	−4.83110	−0.389301
$155$	−4.22204	−0.339123
$156$	3.81503	0.305447
$157$	−2.60975	−0.208281	−0.104140	−	0.994563i	$-0.533209\pi$
$157$	−2.60975	−0.208281	−0.104140	+	0.994563i	$0.533209\pi$
$158$	0.418245	0.0332738
$159$	−11.3518	−0.900255
$160$	0.725043	0.0573197
$161$	−1.69635	−0.133691
$162$	−8.12041	−0.638000
$163$	−6.63147	−0.519417	−0.259708	−	0.965687i	$-0.583626\pi$
$163$	−6.63147	−0.519417	−0.259708	+	0.965687i	$0.583626\pi$
$164$	8.77454	0.685176
$165$	−4.45345	−0.346701
$166$	−15.6322	−1.21329
$167$	22.6369	1.75170	0.875850	−	0.482584i	$-0.160302\pi$
$167$	22.6369	1.75170	0.875850	+	0.482584i	$0.160302\pi$
$168$	2.14795	0.165718
$169$	−7.67053	−0.590041
$170$	4.58939	0.351990
$171$	−0.00571868	−0.000437318 0
$172$	4.52327	0.344896
$173$	12.2456	0.931018	0.465509	−	0.885043i	$-0.345871\pi$
$173$	12.2456	0.931018	0.465509	+	0.885043i	$0.345871\pi$
$174$	−3.40927	−0.258456
$175$	−5.81561	−0.439619
$176$	−3.71687	−0.280170
$177$	12.9646	0.974478
$178$	10.3879	0.778603
$179$	−18.0181	−1.34673	−0.673367	−	0.739308i	$-0.735153\pi$
$179$	−18.0181	−1.34673	−0.673367	+	0.739308i	$0.735153\pi$
$180$	−0.195084	−0.0145407
$181$	7.77690	0.578052	0.289026	−	0.957321i	$-0.406669\pi$
$181$	7.77690	0.578052	0.289026	+	0.957321i	$0.406669\pi$
$182$	3.00062	0.222421
$183$	6.36163	0.470266
$184$	−1.30511	−0.0962138
$185$	−5.25854	−0.386615
$186$	−9.62309	−0.705599
$187$	−23.5271	−1.72047
$188$	3.85502	0.281156
$189$	−7.02179	−0.510760
$190$	0.0154100	0.00111796
$191$	0.312243	0.0225931	0.0112965	−	0.999936i	$-0.496404\pi$
$191$	0.312243	0.0225931	0.0112965	+	0.999936i	$0.496404\pi$
$192$	1.65255	0.119263
$193$	−0.336949	−0.0242541	−0.0121271	−	0.999926i	$-0.503860\pi$
$193$	−0.336949	−0.0242541	−0.0121271	+	0.999926i	$0.503860\pi$
$194$	5.07659	0.364478
$195$	2.76606	0.198082
$196$	−5.31058	−0.379327
$197$	26.8275	1.91138	0.955691	−	0.294370i	$-0.0951099\pi$
$197$	26.8275	1.91138	0.955691	+	0.294370i	$0.0951099\pi$
$198$	1.00008	0.0710726
$199$	4.03560	0.286076	0.143038	−	0.989717i	$-0.454313\pi$
$199$	4.03560	0.286076	0.143038	+	0.989717i	$0.454313\pi$
$200$	−4.47431	−0.316382
$201$	−15.3942	−1.08582
$202$	12.5321	0.881755
$203$	−2.68148	−0.188203
$204$	10.4604	0.732372
$205$	6.36192	0.444335
$206$	5.43374	0.378587
$207$	0.351159	0.0244072
$208$	2.30856	0.160070
$209$	−0.0789980	−0.00546440
$210$	1.55736	0.107468
$211$	9.22688	0.635205	0.317602	−	0.948224i	$-0.397122\pi$
$211$	9.22688	0.635205	0.317602	+	0.948224i	$0.397122\pi$
$212$	−6.86923	−0.471781
$213$	−1.06275	−0.0728186
$214$	−3.45500	−0.236179
$215$	3.27956	0.223664
$216$	−5.40231	−0.367580
$217$	−7.56881	−0.513804
$218$	−6.80801	−0.461096
$219$	−4.92329	−0.332685
$220$	−2.69489	−0.181690
$221$	14.6128	0.982963
$222$	−11.9855	−0.804415
$223$	−9.44842	−0.632713	−0.316357	−	0.948640i	$-0.602460\pi$
$223$	−9.44842	−0.632713	−0.316357	+	0.948640i	$0.602460\pi$
$224$	1.29978	0.0868450
$225$	1.20388	0.0802588
$226$	0.407582	0.0271119
$227$	15.2061	1.00926	0.504632	−	0.863335i	$-0.331628\pi$
$227$	15.2061	1.00926	0.504632	+	0.863335i	$0.331628\pi$
$228$	0.0351232	0.00232609
$229$	28.6411	1.89266	0.946329	−	0.323205i	$-0.104760\pi$
$229$	28.6411	1.89266	0.946329	+	0.323205i	$0.104760\pi$
$230$	−0.946259	−0.0623945
$231$	−7.98366	−0.525286
$232$	−2.06303	−0.135445
$233$	3.14875	0.206281	0.103141	−	0.994667i	$-0.467111\pi$
$233$	3.14875	0.206281	0.103141	+	0.994667i	$0.467111\pi$
$234$	−0.621154	−0.0406061
$235$	2.79506	0.182329
$236$	7.84518	0.510678
$237$	0.691173	0.0448965
$238$	8.22735	0.533300
$239$	12.5059	0.808940	0.404470	−	0.914551i	$-0.367456\pi$
$239$	12.5059	0.808940	0.404470	+	0.914551i	$0.367456\pi$
$240$	1.19817	0.0773417
$241$	3.73027	0.240287	0.120144	−	0.992757i	$-0.461664\pi$
$241$	3.73027	0.240287	0.120144	+	0.992757i	$0.461664\pi$
$242$	2.81514	0.180964
$243$	2.78751	0.178819
$244$	3.84958	0.246444
$245$	−3.85040	−0.245993
$246$	14.5004	0.924511
$247$	0.0490660	0.00312199
$248$	−5.82316	−0.369771
$249$	−25.8330	−1.63710
$250$	−6.86928	−0.434452
$251$	−26.2795	−1.65875	−0.829374	−	0.558694i	$-0.811303\pi$
$251$	−26.2795	−1.65875	−0.829374	+	0.558694i	$0.811303\pi$
$252$	−0.349724	−0.0220306
$253$	4.85092	0.304975
$254$	−9.52589	−0.597708
$255$	7.58421	0.474942
$256$	1.00000	0.0625000
$257$	28.2237	1.76055	0.880273	−	0.474468i	$-0.157359\pi$
$257$	28.2237	1.76055	0.880273	+	0.474468i	$0.157359\pi$
$258$	7.47495	0.465370
$259$	−9.42692	−0.585760
$260$	1.67381	0.103805
$261$	0.555090	0.0343592
$262$	−21.9325	−1.35500
$263$	25.3865	1.56540	0.782701	−	0.622398i	$-0.213842\pi$
$263$	25.3865	1.56540	0.782701	+	0.622398i	$0.213842\pi$
$264$	−6.14233	−0.378034
$265$	−4.98049	−0.305949
$266$	0.0276253	0.00169382
$267$	17.1665	1.05057
$268$	−9.31541	−0.569029
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	−3.91690	−0.238375
$271$	−17.9078	−1.08782	−0.543911	−	0.839143i	$-0.683057\pi$
$271$	−17.9078	−1.08782	−0.543911	+	0.839143i	$0.683057\pi$
$272$	6.32982	0.383802
$273$	4.95868	0.300113
$274$	−2.48849	−0.150335
$275$	16.6304	1.00285
$276$	−2.15676	−0.129822
$277$	−6.51988	−0.391741	−0.195871	−	0.980630i	$-0.562753\pi$
$277$	−6.51988	−0.391741	−0.195871	+	0.980630i	$0.562753\pi$
$278$	21.6244	1.29694
$279$	1.56681	0.0938024
$280$	0.942394	0.0563188
$281$	−2.61394	−0.155934	−0.0779672	−	0.996956i	$-0.524843\pi$
$281$	−2.61394	−0.155934	−0.0779672	+	0.996956i	$0.524843\pi$
$282$	6.37063	0.379366
$283$	−9.65338	−0.573834	−0.286917	−	0.957955i	$-0.592630\pi$
$283$	−9.65338	−0.573834	−0.286917	+	0.957955i	$0.592630\pi$
$284$	−0.643097	−0.0381608
$285$	0.0254658	0.00150847
$286$	−8.58064	−0.507384
$287$	11.4049	0.673212
$288$	−0.269065	−0.0158548
$289$	23.0666	1.35686
$290$	−1.49579	−0.0878356
$291$	8.38934	0.491792
$292$	−2.97920	−0.174345
$293$	2.04724	0.119601	0.0598005	−	0.998210i	$-0.480954\pi$
$293$	2.04724	0.119601	0.0598005	+	0.998210i	$0.480954\pi$
$294$	−8.77602	−0.511828
$295$	5.68809	0.331173
$296$	−7.25272	−0.421556
$297$	20.0797	1.16514
$298$	20.9106	1.21132
$299$	−3.01293	−0.174242
$300$	−7.39404	−0.426895
$301$	5.87924	0.338874
$302$	−2.07209	−0.119235
$303$	20.7100	1.18976
$304$	0.0212539	0.00121899
$305$	2.79111	0.159818
$306$	−1.70313	−0.0973616
$307$	−18.6159	−1.06246	−0.531231	−	0.847227i	$-0.678271\pi$
$307$	−18.6159	−1.06246	−0.531231	+	0.847227i	$0.678271\pi$
$308$	−4.83110	−0.275278
$309$	8.97956	0.510829
$310$	−4.22204	−0.239796
$311$	−9.88503	−0.560529	−0.280264	−	0.959923i	$-0.590422\pi$
$311$	−9.88503	−0.560529	−0.280264	+	0.959923i	$0.590422\pi$
$312$	3.81503	0.215983
$313$	17.4438	0.985983	0.492991	−	0.870034i	$-0.335903\pi$
$313$	17.4438	0.985983	0.492991	+	0.870034i	$0.335903\pi$
$314$	−2.60975	−0.147277
$315$	−0.253565	−0.0142868
$316$	0.418245	0.0235281
$317$	2.92828	0.164468	0.0822342	−	0.996613i	$-0.473794\pi$
$317$	2.92828	0.164468	0.0822342	+	0.996613i	$0.473794\pi$
$318$	−11.3518	−0.636576
$319$	7.66802	0.429327
$320$	0.725043	0.0405311
$321$	−5.70957	−0.318677
$322$	−1.69635	−0.0945338
$323$	0.134533	0.00748563
$324$	−8.12041	−0.451134
$325$	−10.3292	−0.572963
$326$	−6.63147	−0.367283
$327$	−11.2506	−0.622159
$328$	8.77454	0.484493
$329$	5.01067	0.276247
$330$	−4.45345	−0.245155
$331$	5.44140	0.299087	0.149543	−	0.988755i	$-0.452220\pi$
$331$	5.44140	0.299087	0.149543	+	0.988755i	$0.452220\pi$
$332$	−15.6322	−0.857927
$333$	1.95145	0.106939
$334$	22.6369	1.23864
$335$	−6.75407	−0.369014
$336$	2.14795	0.117180
$337$	−3.73840	−0.203644	−0.101822	−	0.994803i	$-0.532467\pi$
$337$	−3.73840	−0.203644	−0.101822	+	0.994803i	$0.532467\pi$
$338$	−7.67053	−0.417222
$339$	0.673551	0.0365823
$340$	4.58939	0.248895
$341$	21.6439	1.17209
$342$	−0.00571868	−0.000309231 0
$343$	−16.0010	−0.863973
$344$	4.52327	0.243878
$345$	−1.56374	−0.0841892
$346$	12.2456	0.658329
$347$	10.8089	0.580252	0.290126	−	0.956988i	$-0.406303\pi$
$347$	10.8089	0.580252	0.290126	+	0.956988i	$0.406303\pi$
$348$	−3.40927	−0.182756
$349$	19.8978	1.06510	0.532552	−	0.846397i	$-0.321233\pi$
$349$	19.8978	1.06510	0.532552	+	0.846397i	$0.321233\pi$
$350$	−5.81561	−0.310857
$351$	−12.4716	−0.665683
$352$	−3.71687	−0.198110
$353$	−11.8505	−0.630736	−0.315368	−	0.948969i	$-0.602128\pi$
$353$	−11.8505	−0.630736	−0.315368	+	0.948969i	$0.602128\pi$
$354$	12.9646	0.689060
$355$	−0.466273	−0.0247472
$356$	10.3879	0.550556
$357$	13.5961	0.719584
$358$	−18.0181	−0.952285
$359$	28.6632	1.51279	0.756393	−	0.654118i	$-0.226960\pi$
$359$	28.6632	1.51279	0.756393	+	0.654118i	$0.226960\pi$
$360$	−0.195084	−0.0102818
$361$	−18.9995	−0.999976
$362$	7.77690	0.408745
$363$	4.65217	0.244176
$364$	3.00062	0.157275
$365$	−2.16005	−0.113062
$366$	6.36163	0.332528
$367$	13.1083	0.684246	0.342123	−	0.939655i	$-0.388854\pi$
$367$	13.1083	0.684246	0.342123	+	0.939655i	$0.388854\pi$
$368$	−1.30511	−0.0680335
$369$	−2.36092	−0.122905
$370$	−5.25854	−0.273378
$371$	−8.92847	−0.463543
$372$	−9.62309	−0.498934
$373$	−17.2738	−0.894406	−0.447203	−	0.894432i	$-0.647580\pi$
$373$	−17.2738	−0.894406	−0.447203	+	0.894432i	$0.647580\pi$
$374$	−23.5271	−1.21656
$375$	−11.3519	−0.586208
$376$	3.85502	0.198808
$377$	−4.76264	−0.245288
$378$	−7.02179	−0.361162
$379$	6.30803	0.324022	0.162011	−	0.986789i	$-0.448202\pi$
$379$	6.30803	0.324022	0.162011	+	0.986789i	$0.448202\pi$
$380$	0.0154100	0.000790515 0
$381$	−15.7421	−0.806490
$382$	0.312243	0.0159757
$383$	−28.2196	−1.44196	−0.720978	−	0.692958i	$-0.756307\pi$
$383$	−28.2196	−1.44196	−0.720978	+	0.692958i	$0.756307\pi$
$384$	1.65255	0.0843315
$385$	−3.50276	−0.178517
$386$	−0.336949	−0.0171502
$387$	−1.21705	−0.0618663
$388$	5.07659	0.257725
$389$	−3.15029	−0.159726	−0.0798629	−	0.996806i	$-0.525448\pi$
$389$	−3.15029	−0.159726	−0.0798629	+	0.996806i	$0.525448\pi$
$390$	2.76606	0.140065
$391$	−8.26110	−0.417782
$392$	−5.31058	−0.268225
$393$	−36.2447	−1.82830
$394$	26.8275	1.35155
$395$	0.303246	0.0152579
$396$	1.00008	0.0502559
$397$	31.2926	1.57053	0.785265	−	0.619159i	$-0.212527\pi$
$397$	31.2926	1.57053	0.785265	+	0.619159i	$0.212527\pi$
$398$	4.03560	0.202286
$399$	0.0456523	0.00228547
$400$	−4.47431	−0.223716
$401$	0.744072	0.0371572	0.0185786	−	0.999827i	$-0.494086\pi$
$401$	0.744072	0.0371572	0.0185786	+	0.999827i	$0.494086\pi$
$402$	−15.3942	−0.767794
$403$	−13.4431	−0.669651
$404$	12.5321	0.623495
$405$	−5.88764	−0.292559
$406$	−2.68148	−0.133080
$407$	26.9574	1.33623
$408$	10.4604	0.517865
$409$	3.60242	0.178128	0.0890641	−	0.996026i	$-0.471612\pi$
$409$	3.60242	0.178128	0.0890641	+	0.996026i	$0.471612\pi$
$410$	6.36192	0.314193
$411$	−4.11237	−0.202848
$412$	5.43374	0.267701
$413$	10.1970	0.501760
$414$	0.351159	0.0172585
$415$	−11.3340	−0.556364
$416$	2.30856	0.113187
$417$	35.7355	1.74997
$418$	−0.0789980	−0.00386392
$419$	−17.1323	−0.836966	−0.418483	−	0.908225i	$-0.637438\pi$
$419$	−17.1323	−0.836966	−0.418483	+	0.908225i	$0.637438\pi$
$420$	1.55736	0.0759912
$421$	24.1651	1.17773	0.588866	−	0.808230i	$-0.299574\pi$
$421$	24.1651	1.17773	0.588866	+	0.808230i	$0.299574\pi$
$422$	9.22688	0.449158
$423$	−1.03725	−0.0504329
$424$	−6.86923	−0.333600
$425$	−28.3216	−1.37380
$426$	−1.06275	−0.0514905
$427$	5.00359	0.242141
$428$	−3.45500	−0.167004
$429$	−14.1800	−0.684615
$430$	3.27956	0.158155
$431$	16.8823	0.813193	0.406596	−	0.913608i	$-0.366716\pi$
$431$	16.8823	0.813193	0.406596	+	0.913608i	$0.366716\pi$
$432$	−5.40231	−0.259919
$433$	−13.0644	−0.627834	−0.313917	−	0.949450i	$-0.601641\pi$
$433$	−13.0644	−0.627834	−0.313917	+	0.949450i	$0.601641\pi$
$434$	−7.56881	−0.363314
$435$	−2.47187	−0.118517
$436$	−6.80801	−0.326044
$437$	−0.0277386	−0.00132692
$438$	−4.92329	−0.235244
$439$	3.91599	0.186900	0.0934499	−	0.995624i	$-0.470211\pi$
$439$	3.91599	0.186900	0.0934499	+	0.995624i	$0.470211\pi$
$440$	−2.69489	−0.128474
$441$	1.42889	0.0680425
$442$	14.6128	0.695060
$443$	25.2673	1.20049	0.600243	−	0.799818i	$-0.295071\pi$
$443$	25.2673	1.20049	0.600243	+	0.799818i	$0.295071\pi$
$444$	−11.9855	−0.568808
$445$	7.53165	0.357034
$446$	−9.44842	−0.447396
$447$	34.5559	1.63444
$448$	1.29978	0.0614087
$449$	27.4706	1.29642	0.648210	−	0.761462i	$-0.275518\pi$
$449$	27.4706	1.29642	0.648210	+	0.761462i	$0.275518\pi$
$450$	1.20388	0.0567515
$451$	−32.6138	−1.53573
$452$	0.407582	0.0191710
$453$	−3.42423	−0.160885
$454$	15.2061	0.713657
$455$	2.17558	0.101993
$456$	0.0351232	0.00164479
$457$	−14.5345	−0.679896	−0.339948	−	0.940444i	$-0.610409\pi$
$457$	−14.5345	−0.679896	−0.339948	+	0.940444i	$0.610409\pi$
$458$	28.6411	1.33831
$459$	−34.1956	−1.59611
$460$	−0.946259	−0.0441196
$461$	−10.4630	−0.487312	−0.243656	−	0.969862i	$-0.578347\pi$
$461$	−10.4630	−0.487312	−0.243656	+	0.969862i	$0.578347\pi$
$462$	−7.98366	−0.371433
$463$	−27.7519	−1.28974	−0.644871	−	0.764292i	$-0.723089\pi$
$463$	−27.7519	−1.28974	−0.644871	+	0.764292i	$0.723089\pi$
$464$	−2.06303	−0.0957738
$465$	−6.97715	−0.323558
$466$	3.14875	0.145863
$467$	−12.3322	−0.570665	−0.285333	−	0.958429i	$-0.592104\pi$
$467$	−12.3322	−0.570665	−0.285333	+	0.958429i	$0.592104\pi$
$468$	−0.621154	−0.0287129
$469$	−12.1080	−0.559093
$470$	2.79506	0.128926
$471$	−4.31276	−0.198721
$472$	7.84518	0.361104
$473$	−16.8124	−0.773036
$474$	0.691173	0.0317466
$475$	−0.0950965	−0.00436333
$476$	8.22735	0.377100
$477$	1.84827	0.0846265
$478$	12.5059	0.572007
$479$	−2.92366	−0.133586	−0.0667928	−	0.997767i	$-0.521277\pi$
$479$	−2.92366	−0.133586	−0.0667928	+	0.997767i	$0.521277\pi$
$480$	1.19817	0.0546888
$481$	−16.7434	−0.763432
$482$	3.73027	0.169909
$483$	−2.80331	−0.127555
$484$	2.81514	0.127961
$485$	3.68075	0.167134
$486$	2.78751	0.126444
$487$	10.1739	0.461025	0.230513	−	0.973069i	$-0.425960\pi$
$487$	10.1739	0.461025	0.230513	+	0.973069i	$0.425960\pi$
$488$	3.84958	0.174262
$489$	−10.9589	−0.495577
$490$	−3.85040	−0.173943
$491$	−16.8181	−0.758989	−0.379494	−	0.925194i	$-0.623902\pi$
$491$	−16.8181	−0.758989	−0.379494	+	0.925194i	$0.623902\pi$
$492$	14.5004	0.653728
$493$	−13.0586	−0.588130
$494$	0.0490660	0.00220758
$495$	0.725101	0.0325909
$496$	−5.82316	−0.261468
$497$	−0.835882	−0.0374944
$498$	−25.8330	−1.15761
$499$	−30.0514	−1.34529	−0.672643	−	0.739967i	$-0.734841\pi$
$499$	−30.0514	−1.34529	−0.672643	+	0.739967i	$0.734841\pi$
$500$	−6.86928	−0.307204
$501$	37.4088	1.67130
$502$	−26.2795	−1.17291
$503$	−6.83648	−0.304823	−0.152412	−	0.988317i	$-0.548704\pi$
$503$	−6.83648	−0.304823	−0.152412	+	0.988317i	$0.548704\pi$
$504$	−0.349724	−0.0155780
$505$	9.08630	0.404335
$506$	4.85092	0.215650
$507$	−12.6760	−0.562960
$508$	−9.52589	−0.422643
$509$	−16.9547	−0.751502	−0.375751	−	0.926721i	$-0.622615\pi$
$509$	−16.9547	−0.751502	−0.375751	+	0.926721i	$0.622615\pi$
$510$	7.58421	0.335835
$511$	−3.87230	−0.171300
$512$	1.00000	0.0441942
$513$	−0.114820	−0.00506943
$514$	28.2237	1.24489
$515$	3.93970	0.173604
$516$	7.47495	0.329066
$517$	−14.3286	−0.630172
$518$	−9.42692	−0.414195
$519$	20.2366	0.888287
$520$	1.67381	0.0734013
$521$	37.1136	1.62597	0.812987	−	0.582282i	$-0.197840\pi$
$521$	37.1136	1.62597	0.812987	+	0.582282i	$0.197840\pi$
$522$	0.555090	0.0242956
$523$	−9.80481	−0.428734	−0.214367	−	0.976753i	$-0.568769\pi$
$523$	−9.80481	−0.428734	−0.214367	+	0.976753i	$0.568769\pi$
$524$	−21.9325	−0.958127
$525$	−9.61060	−0.419441
$526$	25.3865	1.10691
$527$	−36.8596	−1.60563
$528$	−6.14233	−0.267311
$529$	−21.2967	−0.925943
$530$	−4.98049	−0.216339
$531$	−2.11086	−0.0916037
$532$	0.0276253	0.00119771
$533$	20.2566	0.877410
$534$	17.1665	0.742867
$535$	−2.50502	−0.108302
$536$	−9.31541	−0.402365
$537$	−29.7758	−1.28492
$538$	−1.00000	−0.0431131
$539$	19.7388	0.850208
$540$	−3.91690	−0.168557
$541$	−15.4647	−0.664878	−0.332439	−	0.943125i	$-0.607871\pi$
$541$	−15.4647	−0.664878	−0.332439	+	0.943125i	$0.607871\pi$
$542$	−17.9078	−0.769206
$543$	12.8517	0.551521
$544$	6.32982	0.271389
$545$	−4.93610	−0.211439
$546$	4.95868	0.212212
$547$	−31.5080	−1.34719	−0.673593	−	0.739103i	$-0.735250\pi$
$547$	−31.5080	−1.34719	−0.673593	+	0.739103i	$0.735250\pi$
$548$	−2.48849	−0.106303
$549$	−1.03579	−0.0442063
$550$	16.6304	0.709125
$551$	−0.0438474	−0.00186796
$552$	−2.15676	−0.0917979
$553$	0.543625	0.0231173
$554$	−6.51988	−0.277003
$555$	−8.69001	−0.368871
$556$	21.6244	0.917078
$557$	−5.45649	−0.231199	−0.115599	−	0.993296i	$-0.536879\pi$
$557$	−5.45649	−0.231199	−0.115599	+	0.993296i	$0.536879\pi$
$558$	1.56681	0.0663283
$559$	10.4423	0.441660
$560$	0.942394	0.0398234
$561$	−38.8798	−1.64151
$562$	−2.61394	−0.110262
$563$	−5.91106	−0.249121	−0.124561	−	0.992212i	$-0.539752\pi$
$563$	−5.91106	−0.249121	−0.124561	+	0.992212i	$0.539752\pi$
$564$	6.37063	0.268252
$565$	0.295514	0.0124324
$566$	−9.65338	−0.405762
$567$	−10.5547	−0.443256
$568$	−0.643097	−0.0269837
$569$	31.9474	1.33930	0.669652	−	0.742675i	$-0.266443\pi$
$569$	31.9474	1.33930	0.669652	+	0.742675i	$0.266443\pi$
$570$	0.0254658	0.00106665
$571$	−13.7554	−0.575647	−0.287824	−	0.957683i	$-0.592932\pi$
$571$	−13.7554	−0.575647	−0.287824	+	0.957683i	$0.592932\pi$
$572$	−8.58064	−0.358774
$573$	0.515998	0.0215561
$574$	11.4049	0.476033
$575$	5.83946	0.243522
$576$	−0.269065	−0.0112110
$577$	−2.93964	−0.122379	−0.0611893	−	0.998126i	$-0.519489\pi$
$577$	−2.93964	−0.122379	−0.0611893	+	0.998126i	$0.519489\pi$
$578$	23.0666	0.959444
$579$	−0.556826	−0.0231409
$580$	−1.49579	−0.0621091
$581$	−20.3183	−0.842947
$582$	8.38934	0.347749
$583$	25.5321	1.05743
$584$	−2.97920	−0.123280
$585$	−0.450363	−0.0186202
$586$	2.04724	0.0845707
$587$	15.2876	0.630989	0.315494	−	0.948927i	$-0.397830\pi$
$587$	15.2876	0.630989	0.315494	+	0.948927i	$0.397830\pi$
$588$	−8.77602	−0.361917
$589$	−0.123765	−0.00509964
$590$	5.68809	0.234175
$591$	44.3340	1.82366
$592$	−7.25272	−0.298085
$593$	−33.5005	−1.37570	−0.687850	−	0.725853i	$-0.741445\pi$
$593$	−33.5005	−1.37570	−0.687850	+	0.725853i	$0.741445\pi$
$594$	20.0797	0.823879
$595$	5.96518	0.244549
$596$	20.9106	0.856532
$597$	6.66904	0.272946
$598$	−3.01293	−0.123208
$599$	0.150988	0.00616920	0.00308460	−	0.999995i	$-0.499018\pi$
$599$	0.150988	0.00616920	0.00308460	+	0.999995i	$0.499018\pi$
$600$	−7.39404	−0.301861
$601$	−34.9546	−1.42583	−0.712915	−	0.701251i	$-0.752625\pi$
$601$	−34.9546	−1.42583	−0.712915	+	0.701251i	$0.752625\pi$
$602$	5.87924	0.239620
$603$	2.50645	0.102071
$604$	−2.07209	−0.0843120
$605$	2.04110	0.0829824
$606$	20.7100	0.841285
$607$	10.5301	0.427404	0.213702	−	0.976899i	$-0.431448\pi$
$607$	10.5301	0.427404	0.213702	+	0.976899i	$0.431448\pi$
$608$	0.0212539	0.000861959 0
$609$	−4.43129	−0.179565
$610$	2.79111	0.113009
$611$	8.89957	0.360038
$612$	−1.70313	−0.0688451
$613$	−20.2271	−0.816965	−0.408483	−	0.912766i	$-0.633942\pi$
$613$	−20.2271	−0.816965	−0.408483	+	0.912766i	$0.633942\pi$
$614$	−18.6159	−0.751275
$615$	10.5134	0.423942
$616$	−4.83110	−0.194651
$617$	7.82969	0.315211	0.157606	−	0.987502i	$-0.449622\pi$
$617$	7.82969	0.315211	0.157606	+	0.987502i	$0.449622\pi$
$618$	8.97956	0.361211
$619$	−23.6992	−0.952549	−0.476275	−	0.879297i	$-0.658013\pi$
$619$	−23.6992	−0.952549	−0.476275	+	0.879297i	$0.658013\pi$
$620$	−4.22204	−0.169561
$621$	7.05059	0.282931
$622$	−9.88503	−0.396354
$623$	13.5019	0.540942
$624$	3.81503	0.152723
$625$	17.3910	0.695642
$626$	17.4438	0.697195
$627$	−0.130548	−0.00521360
$628$	−2.60975	−0.104140
$629$	−45.9084	−1.83049
$630$	−0.253565	−0.0101023
$631$	−31.8687	−1.26867	−0.634336	−	0.773058i	$-0.718726\pi$
$631$	−31.8687	−1.26867	−0.634336	+	0.773058i	$0.718726\pi$
$632$	0.418245	0.0166369
$633$	15.2479	0.606050
$634$	2.92828	0.116297
$635$	−6.90668	−0.274083
$636$	−11.3518	−0.450127
$637$	−12.2598	−0.485752
$638$	7.66802	0.303580
$639$	0.173035	0.00684515
$640$	0.725043	0.0286598
$641$	10.3578	0.409108	0.204554	−	0.978855i	$-0.434426\pi$
$641$	10.3578	0.409108	0.204554	+	0.978855i	$0.434426\pi$
$642$	−5.70957	−0.225339
$643$	−21.9761	−0.866653	−0.433326	−	0.901237i	$-0.642660\pi$
$643$	−21.9761	−0.866653	−0.433326	+	0.901237i	$0.642660\pi$
$644$	−1.69635	−0.0668455
$645$	5.41966	0.213399
$646$	0.134533	0.00529314
$647$	9.10376	0.357906	0.178953	−	0.983858i	$-0.442729\pi$
$647$	9.10376	0.357906	0.178953	+	0.983858i	$0.442729\pi$
$648$	−8.12041	−0.319000
$649$	−29.1595	−1.14461
$650$	−10.3292	−0.405146
$651$	−12.5079	−0.490222
$652$	−6.63147	−0.259708
$653$	6.84892	0.268019	0.134010	−	0.990980i	$-0.457215\pi$
$653$	6.84892	0.268019	0.134010	+	0.990980i	$0.457215\pi$
$654$	−11.2506	−0.439933
$655$	−15.9020	−0.621344
$656$	8.77454	0.342588
$657$	0.801599	0.0312734
$658$	5.01067	0.195336
$659$	−10.9102	−0.424999	−0.212500	−	0.977161i	$-0.568160\pi$
$659$	−10.9102	−0.424999	−0.212500	+	0.977161i	$0.568160\pi$
$660$	−4.45345	−0.173350
$661$	−9.26687	−0.360439	−0.180220	−	0.983626i	$-0.557681\pi$
$661$	−9.26687	−0.360439	−0.180220	+	0.983626i	$0.557681\pi$
$662$	5.44140	0.211486
$663$	24.1484	0.937847
$664$	−15.6322	−0.606646
$665$	0.0200295	0.000776712 0
$666$	1.95145	0.0756173
$667$	2.69248	0.104253
$668$	22.6369	0.875850
$669$	−15.6140	−0.603673
$670$	−6.75407	−0.260933
$671$	−14.3084	−0.552369
$672$	2.14795	0.0828590
$673$	−15.0272	−0.579255	−0.289627	−	0.957139i	$-0.593531\pi$
$673$	−15.0272	−0.579255	−0.289627	+	0.957139i	$0.593531\pi$
$674$	−3.73840	−0.143998
$675$	24.1716	0.930366
$676$	−7.67053	−0.295020
$677$	−46.3947	−1.78309	−0.891547	−	0.452929i	$-0.850379\pi$
$677$	−46.3947	−1.78309	−0.891547	+	0.452929i	$0.850379\pi$
$678$	0.673551	0.0258676
$679$	6.59843	0.253225
$680$	4.58939	0.175995
$681$	25.1289	0.962941
$682$	21.6439	0.828790
$683$	−34.2506	−1.31056	−0.655281	−	0.755385i	$-0.727450\pi$
$683$	−34.2506	−1.31056	−0.655281	+	0.755385i	$0.727450\pi$
$684$	−0.00571868	−0.000218659 0
$685$	−1.80426	−0.0689373
$686$	−16.0010	−0.610921
$687$	47.3310	1.80579
$688$	4.52327	0.172448
$689$	−15.8581	−0.604144
$690$	−1.56374	−0.0595307
$691$	−1.19208	−0.0453489	−0.0226744	−	0.999743i	$-0.507218\pi$
$691$	−1.19208	−0.0453489	−0.0226744	+	0.999743i	$0.507218\pi$
$692$	12.2456	0.465509
$693$	1.29988	0.0493784
$694$	10.8089	0.410300
$695$	15.6786	0.594723
$696$	−3.40927	−0.129228
$697$	55.5412	2.10377
$698$	19.8978	0.753142
$699$	5.20348	0.196814
$700$	−5.81561	−0.219809
$701$	−24.6698	−0.931767	−0.465883	−	0.884846i	$-0.654263\pi$
$701$	−24.6698	−0.931767	−0.465883	+	0.884846i	$0.654263\pi$
$702$	−12.4716	−0.470709
$703$	−0.154149	−0.00581382
$704$	−3.71687	−0.140085
$705$	4.61898	0.173961
$706$	−11.8505	−0.445998
$707$	16.2889	0.612608
$708$	12.9646	0.487239
$709$	23.3891	0.878398	0.439199	−	0.898390i	$-0.355262\pi$
$709$	23.3891	0.878398	0.439199	+	0.898390i	$0.355262\pi$
$710$	−0.466273	−0.0174989
$711$	−0.112535	−0.00422040
$712$	10.3879	0.389302
$713$	7.59986	0.284617
$714$	13.5961	0.508823
$715$	−6.22133	−0.232665
$716$	−18.0181	−0.673367
$717$	20.6667	0.771812
$718$	28.6632	1.06970
$719$	−1.52136	−0.0567371	−0.0283686	−	0.999598i	$-0.509031\pi$
$719$	−1.52136	−0.0567371	−0.0283686	+	0.999598i	$0.509031\pi$
$720$	−0.195084	−0.00727034
$721$	7.06265	0.263027
$722$	−18.9995	−0.707090
$723$	6.16446	0.229259
$724$	7.77690	0.289026
$725$	9.23065	0.342818
$726$	4.65217	0.172658
$727$	36.3855	1.34946	0.674732	−	0.738063i	$-0.264259\pi$
$727$	36.3855	1.34946	0.674732	+	0.738063i	$0.264259\pi$
$728$	3.00062	0.111210
$729$	28.9677	1.07288
$730$	−2.16005	−0.0799470
$731$	28.6315	1.05897
$732$	6.36163	0.235133
$733$	−18.7869	−0.693908	−0.346954	−	0.937882i	$-0.612784\pi$
$733$	−18.7869	−0.693908	−0.346954	+	0.937882i	$0.612784\pi$
$734$	13.1083	0.483835
$735$	−6.36299	−0.234703
$736$	−1.30511	−0.0481069
$737$	34.6242	1.27540
$738$	−2.36092	−0.0869067
$739$	−38.7776	−1.42646	−0.713228	−	0.700932i	$-0.752767\pi$
$739$	−38.7776	−1.42646	−0.713228	+	0.700932i	$0.752767\pi$
$740$	−5.25854	−0.193308
$741$	0.0810841	0.00297870
$742$	−8.92847	−0.327774
$743$	−6.80190	−0.249537	−0.124769	−	0.992186i	$-0.539819\pi$
$743$	−6.80190	−0.249537	−0.124769	+	0.992186i	$0.539819\pi$
$744$	−9.62309	−0.352800
$745$	15.1611	0.555460
$746$	−17.2738	−0.632441
$747$	4.20607	0.153892
$748$	−23.5271	−0.860237
$749$	−4.49073	−0.164088
$750$	−11.3519	−0.414511
$751$	10.5672	0.385601	0.192801	−	0.981238i	$-0.438243\pi$
$751$	10.5672	0.385601	0.192801	+	0.981238i	$0.438243\pi$
$752$	3.85502	0.140578
$753$	−43.4283	−1.58262
$754$	−4.76264	−0.173445
$755$	−1.50235	−0.0546762
$756$	−7.02179	−0.255380
$757$	−22.6389	−0.822825	−0.411412	−	0.911449i	$-0.634964\pi$
$757$	−22.6389	−0.822825	−0.411412	+	0.911449i	$0.634964\pi$
$758$	6.30803	0.229118
$759$	8.01641	0.290977
$760$	0.0154100	0.000558979 0
$761$	1.83584	0.0665490	0.0332745	−	0.999446i	$-0.489406\pi$
$761$	1.83584	0.0665490	0.0332745	+	0.999446i	$0.489406\pi$
$762$	−15.7421	−0.570274
$763$	−8.84889	−0.320351
$764$	0.312243	0.0112965
$765$	−1.23484	−0.0446459
$766$	−28.2196	−1.01962
$767$	18.1111	0.653954
$768$	1.65255	0.0596314
$769$	50.0137	1.80354	0.901770	−	0.432215i	$-0.142268\pi$
$769$	50.0137	1.80354	0.901770	+	0.432215i	$0.142268\pi$
$770$	−3.50276	−0.126231
$771$	46.6412	1.67974
$772$	−0.336949	−0.0121271
$773$	40.2988	1.44945	0.724724	−	0.689039i	$-0.241967\pi$
$773$	40.2988	1.44945	0.724724	+	0.689039i	$0.241967\pi$
$774$	−1.21705	−0.0437461
$775$	26.0546	0.935911
$776$	5.07659	0.182239
$777$	−15.5785	−0.558875
$778$	−3.15029	−0.112943
$779$	0.186493	0.00668180
$780$	2.76606	0.0990408
$781$	2.39031	0.0855319
$782$	−8.26110	−0.295416
$783$	11.1451	0.398294
$784$	−5.31058	−0.189664
$785$	−1.89218	−0.0675349
$786$	−36.2447	−1.29281
$787$	38.2500	1.36347	0.681733	−	0.731601i	$-0.261227\pi$
$787$	38.2500	1.36347	0.681733	+	0.731601i	$0.261227\pi$
$788$	26.8275	0.955691
$789$	41.9526	1.49355
$790$	0.303246	0.0107890
$791$	0.529765	0.0188363
$792$	1.00008	0.0355363
$793$	8.88700	0.315586
$794$	31.2926	1.11053
$795$	−8.23053	−0.291907
$796$	4.03560	0.143038
$797$	−33.4247	−1.18397	−0.591983	−	0.805951i	$-0.701655\pi$
$797$	−33.4247	−1.18397	−0.591983	+	0.805951i	$0.701655\pi$
$798$	0.0456523	0.00161607
$799$	24.4016	0.863266
$800$	−4.47431	−0.158191
$801$	−2.79501	−0.0987569
$802$	0.744072	0.0262741
$803$	11.0733	0.390769
$804$	−15.3942	−0.542912
$805$	−1.22993	−0.0433492
$806$	−13.4431	−0.473514
$807$	−1.65255	−0.0581727
$808$	12.5321	0.440877
$809$	−39.8841	−1.40225	−0.701125	−	0.713038i	$-0.747319\pi$
$809$	−39.8841	−1.40225	−0.701125	+	0.713038i	$0.747319\pi$
$810$	−5.88764	−0.206871
$811$	34.7237	1.21931	0.609657	−	0.792665i	$-0.291307\pi$
$811$	34.7237	1.21931	0.609657	+	0.792665i	$0.291307\pi$
$812$	−2.68148	−0.0941015
$813$	−29.5936	−1.03789
$814$	26.9574	0.944858
$815$	−4.80810	−0.168420
$816$	10.4604	0.366186
$817$	0.0961370	0.00336341
$818$	3.60242	0.125956
$819$	−0.807361	−0.0282115
$820$	6.36192	0.222168
$821$	−48.5169	−1.69325	−0.846626	−	0.532189i	$-0.821370\pi$
$821$	−48.5169	−1.69325	−0.846626	+	0.532189i	$0.821370\pi$
$822$	−4.11237	−0.143435
$823$	19.5287	0.680728	0.340364	−	0.940294i	$-0.389450\pi$
$823$	19.5287	0.680728	0.340364	+	0.940294i	$0.389450\pi$
$824$	5.43374	0.189293
$825$	27.4827	0.956825
$826$	10.1970	0.354798
$827$	18.5400	0.644698	0.322349	−	0.946621i	$-0.395528\pi$
$827$	18.5400	0.644698	0.322349	+	0.946621i	$0.395528\pi$
$828$	0.351159	0.0122036
$829$	−20.2344	−0.702769	−0.351385	−	0.936231i	$-0.614289\pi$
$829$	−20.2344	−0.702769	−0.351385	+	0.936231i	$0.614289\pi$
$830$	−11.3340	−0.393409
$831$	−10.7744	−0.373762
$832$	2.30856	0.0800351
$833$	−33.6150	−1.16469
$834$	35.7355	1.23742
$835$	16.4128	0.567987
$836$	−0.0789980	−0.00273220
$837$	31.4585	1.08737
$838$	−17.1323	−0.591824
$839$	46.3931	1.60167	0.800833	−	0.598887i	$-0.204390\pi$
$839$	46.3931	1.60167	0.800833	+	0.598887i	$0.204390\pi$
$840$	1.55736	0.0537339
$841$	−24.7439	−0.853238
$842$	24.1651	0.832783
$843$	−4.31967	−0.148777
$844$	9.22688	0.317602
$845$	−5.56146	−0.191320
$846$	−1.03725	−0.0356615
$847$	3.65905	0.125727
$848$	−6.86923	−0.235890
$849$	−15.9527	−0.547496
$850$	−28.3216	−0.971422
$851$	9.46559	0.324476
$852$	−1.06275	−0.0364093
$853$	−0.133230	−0.00456172	−0.00228086	−	0.999997i	$-0.500726\pi$
$853$	−0.133230	−0.00456172	−0.00228086	+	0.999997i	$0.500726\pi$
$854$	5.00359	0.171219
$855$	−0.00414629	−0.000141800 0
$856$	−3.45500	−0.118089
$857$	−10.2942	−0.351642	−0.175821	−	0.984422i	$-0.556258\pi$
$857$	−10.2942	−0.351642	−0.175821	+	0.984422i	$0.556258\pi$
$858$	−14.1800	−0.484096
$859$	35.5814	1.21402	0.607011	−	0.794694i	$-0.292368\pi$
$859$	35.5814	1.21402	0.607011	+	0.794694i	$0.292368\pi$
$860$	3.27956	0.111832
$861$	18.8473	0.642313
$862$	16.8823	0.575014
$863$	17.9896	0.612372	0.306186	−	0.951972i	$-0.400947\pi$
$863$	17.9896	0.612372	0.306186	+	0.951972i	$0.400947\pi$
$864$	−5.40231	−0.183790
$865$	8.87861	0.301882
$866$	−13.0644	−0.443946
$867$	38.1188	1.29458
$868$	−7.56881	−0.256902
$869$	−1.55456	−0.0527350
$870$	−2.47187	−0.0838042
$871$	−21.5052	−0.728677
$872$	−6.80801	−0.230548
$873$	−1.36593	−0.0462298
$874$	−0.0277386	−0.000938273 0
$875$	−8.92853	−0.301839
$876$	−4.92329	−0.166343
$877$	47.6131	1.60778	0.803890	−	0.594778i	$-0.202760\pi$
$877$	47.6131	1.60778	0.803890	+	0.594778i	$0.202760\pi$
$878$	3.91599	0.132158
$879$	3.38318	0.114112
$880$	−2.69489	−0.0908448
$881$	−15.7287	−0.529914	−0.264957	−	0.964260i	$-0.585358\pi$
$881$	−15.7287	−0.529914	−0.264957	+	0.964260i	$0.585358\pi$
$882$	1.42889	0.0481133
$883$	−20.9111	−0.703713	−0.351856	−	0.936054i	$-0.614449\pi$
$883$	−20.9111	−0.703713	−0.351856	+	0.936054i	$0.614449\pi$
$884$	14.6128	0.491481
$885$	9.39988	0.315973
$886$	25.2673	0.848871
$887$	3.47543	0.116694	0.0583468	−	0.998296i	$-0.481417\pi$
$887$	3.47543	0.116694	0.0583468	+	0.998296i	$0.481417\pi$
$888$	−11.9855	−0.402208
$889$	−12.3815	−0.415263
$890$	7.53165	0.252461
$891$	30.1825	1.01115
$892$	−9.44842	−0.316357
$893$	0.0819342	0.00274182
$894$	34.5559	1.15572
$895$	−13.0639	−0.436677
$896$	1.29978	0.0434225
$897$	−4.97902	−0.166245
$898$	27.4706	0.916707
$899$	12.0134	0.400668
$900$	1.20388	0.0401294
$901$	−43.4810	−1.44856
$902$	−32.6138	−1.08592
$903$	9.71576	0.323320
$904$	0.407582	0.0135560
$905$	5.63859	0.187433
$906$	−3.42423	−0.113763
$907$	5.99370	0.199018	0.0995088	−	0.995037i	$-0.468273\pi$
$907$	5.99370	0.199018	0.0995088	+	0.995037i	$0.468273\pi$
$908$	15.2061	0.504632
$909$	−3.37195	−0.111840
$910$	2.17558	0.0721196
$911$	3.66404	0.121395	0.0606976	−	0.998156i	$-0.480667\pi$
$911$	3.66404	0.121395	0.0606976	+	0.998156i	$0.480667\pi$
$912$	0.0351232	0.00116305
$913$	58.1028	1.92292
$914$	−14.5345	−0.480759
$915$	4.61246	0.152483
$916$	28.6411	0.946329
$917$	−28.5074	−0.941397
$918$	−34.1956	−1.12862
$919$	44.8323	1.47888	0.739441	−	0.673222i	$-0.235090\pi$
$919$	44.8323	1.47888	0.739441	+	0.673222i	$0.235090\pi$
$920$	−0.946259	−0.0311972
$921$	−30.7637	−1.01370
$922$	−10.4630	−0.344581
$923$	−1.48463	−0.0488672
$924$	−7.98366	−0.262643
$925$	32.4510	1.06698
$926$	−27.7519	−0.911985
$927$	−1.46203	−0.0480194
$928$	−2.06303	−0.0677223
$929$	−35.1823	−1.15429	−0.577146	−	0.816641i	$-0.695834\pi$
$929$	−35.1823	−1.15429	−0.577146	+	0.816641i	$0.695834\pi$
$930$	−6.97715	−0.228790
$931$	−0.112870	−0.00369918
$932$	3.14875	0.103141
$933$	−16.3355	−0.534802
$934$	−12.3322	−0.403521
$935$	−17.0582	−0.557862
$936$	−0.621154	−0.0203031
$937$	37.7329	1.23268	0.616339	−	0.787481i	$-0.288615\pi$
$937$	37.7329	1.23268	0.616339	+	0.787481i	$0.288615\pi$
$938$	−12.1080	−0.395339
$939$	28.8268	0.940729
$940$	2.79506	0.0911647
$941$	−31.0321	−1.01162	−0.505809	−	0.862646i	$-0.668806\pi$
$941$	−31.0321	−1.01162	−0.505809	+	0.862646i	$0.668806\pi$
$942$	−4.31276	−0.140517
$943$	−11.4517	−0.372919
$944$	7.84518	0.255339
$945$	−5.09110	−0.165613
$946$	−16.8124	−0.546619
$947$	19.5388	0.634924	0.317462	−	0.948271i	$-0.397169\pi$
$947$	19.5388	0.634924	0.317462	+	0.948271i	$0.397169\pi$
$948$	0.691173	0.0224482
$949$	−6.87768	−0.223259
$950$	−0.0950965	−0.00308534
$951$	4.83914	0.156920
$952$	8.22735	0.266650
$953$	15.9894	0.517948	0.258974	−	0.965884i	$-0.416616\pi$
$953$	15.9894	0.517948	0.258974	+	0.965884i	$0.416616\pi$
$954$	1.84827	0.0598400
$955$	0.226389	0.00732579
$956$	12.5059	0.404470
$957$	12.6718	0.409622
$958$	−2.92366	−0.0944592
$959$	−3.23448	−0.104447
$960$	1.19817	0.0386709
$961$	2.90921	0.0938456
$962$	−16.7434	−0.539828
$963$	0.929619	0.0299566
$964$	3.73027	0.120144
$965$	−0.244302	−0.00786437
$966$	−2.80331	−0.0901950
$967$	38.1787	1.22774	0.613872	−	0.789406i	$-0.289611\pi$
$967$	38.1787	1.22774	0.613872	+	0.789406i	$0.289611\pi$
$968$	2.81514	0.0904820
$969$	0.222323	0.00714206
$970$	3.68075	0.118182
$971$	2.16411	0.0694497	0.0347248	−	0.999397i	$-0.488945\pi$
$971$	2.16411	0.0694497	0.0347248	+	0.999397i	$0.488945\pi$
$972$	2.78751	0.0894093
$973$	28.1069	0.901064
$974$	10.1739	0.325994
$975$	−17.0696	−0.546665
$976$	3.84958	0.123222
$977$	−14.7297	−0.471243	−0.235622	−	0.971845i	$-0.575713\pi$
$977$	−14.7297	−0.471243	−0.235622	+	0.971845i	$0.575713\pi$
$978$	−10.9589	−0.350426
$979$	−38.6104	−1.23399
$980$	−3.85040	−0.122996
$981$	1.83180	0.0584848
$982$	−16.8181	−0.536686
$983$	36.4217	1.16167	0.580836	−	0.814020i	$-0.302726\pi$
$983$	36.4217	1.16167	0.580836	+	0.814020i	$0.302726\pi$
$984$	14.5004	0.462256
$985$	19.4511	0.619764
$986$	−13.0586	−0.415871
$987$	8.28040	0.263568
$988$	0.0490660	0.00156100
$989$	−5.90336	−0.187716
$990$	0.725101	0.0230452
$991$	−24.5429	−0.779632	−0.389816	−	0.920893i	$-0.627461\pi$
$991$	−24.5429	−0.779632	−0.389816	+	0.920893i	$0.627461\pi$
$992$	−5.82316	−0.184886
$993$	8.99221	0.285359
$994$	−0.835882	−0.0265126
$995$	2.92598	0.0927598
$996$	−25.8330	−0.818551
$997$	−52.7413	−1.67033	−0.835167	−	0.549996i	$-0.814629\pi$
$997$	−52.7413	−1.67033	−0.835167	+	0.549996i	$0.814629\pi$
$998$	−30.0514	−0.951260
$999$	39.1814	1.23965

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	538.2.a.e.1.5	✓	7
3.2	odd	2		4842.2.a.n.1.4		7
4.3	odd	2		4304.2.a.h.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.e.1.5	✓	7	1.1	even	1	trivial
4304.2.a.h.1.3		7	4.3	odd	2
4842.2.a.n.1.4		7	3.2	odd	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

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Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	538.2.a.e.1.5

Level	$538$
Weight	$2$
Character	538.1
Self dual	yes
Analytic conductor	$4.296$
Analytic rank	$0$
Dimension	$7$
CM	no
Inner twists	$1$