LMFDB - Embedded newform 600.8.f.d.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,8,Mod(49,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("600.49");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$N$	$=$	$600 = 2^{3} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	600.f (of order $2$ , degree $1$ , not minimal)

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:	no
Analytic conductor:	$187.431015290$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 24)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	600.49
Dual form			600.8.f.d.49.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000i q^{3} +504.000i q^{7} -729.000 q^{9} +3812.00 q^{11} -9574.00i q^{13} +26098.0i q^{17} +38308.0 q^{19} +13608.0 q^{21} +71128.0i q^{23} +19683.0i q^{27} -74262.0 q^{29} -275680. q^{31} -102924. i q^{33} -266610. i q^{37} -258498. q^{39} +684762. q^{41} -245956. i q^{43} +478800. i q^{47} +569527. q^{49} +704646. q^{51} +569410. i q^{53} -1.03432e6i q^{57} +1.52532e6 q^{59} -2.64046e6 q^{61} -367416. i q^{63} +1.41624e6i q^{67} +1.92046e6 q^{69} -3.51130e6 q^{71} -4.73862e6i q^{73} +1.92125e6i q^{77} -4.66149e6 q^{79} +531441. q^{81} +5.72925e6i q^{83} +2.00507e6i q^{87} -1.19935e7 q^{89} +4.82530e6 q^{91} +7.44336e6i q^{93} +7.15075e6i q^{97} -2.77895e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 1458 q^{9} + 7624 q^{11} + 76616 q^{19} + 27216 q^{21} - 148524 q^{29} - 551360 q^{31} - 516996 q^{39} + 1369524 q^{41} + 1139054 q^{49} + 1409292 q^{51} + 3050648 q^{59} - 5280916 q^{61} + 3840912 q^{69}+ \cdots - 5557896 q^{99}+O(q^{100})$ 2 * q - 1458 * q^9 + 7624 * q^11 + 76616 * q^19 + 27216 * q^21 - 148524 * q^29 - 551360 * q^31 - 516996 * q^39 + 1369524 * q^41 + 1139054 * q^49 + 1409292 * q^51 + 3050648 * q^59 - 5280916 * q^61 + 3840912 * q^69 - 7022608 * q^71 - 9322976 * q^79 + 1062882 * q^81 - 23987028 * q^89 + 9650592 * q^91 - 5557896 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$ .

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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

)

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

$2$	0	0
$2$	0	0
$3$	− 27.0000i	− 0.577350i
$3$	− 27.0000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	504.000i	0.555376i	0.960671	+	0.277688i	$0.0895682\pi$
$7$	504.000i	0.555376i	−0.960671	+	0.277688i	$0.910432\pi$
$8$	0	0
$9$	−729.000	−0.333333
$10$	0	0
$11$	3812.00	0.863532	0.431766	−	0.901986i	$-0.357891\pi$
$11$	3812.00	0.863532	0.431766	+	0.901986i	$0.357891\pi$
$12$	0	0
$13$	− 9574.00i	− 1.20863i	−0.796747	−	0.604313i	$-0.793448\pi$
$13$	− 9574.00i	− 1.20863i	0.796747	−	0.604313i	$-0.206552\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	26098.0i	1.28836i	0.764875	+	0.644178i	$0.222800\pi$
$17$	26098.0i	1.28836i	−0.764875	+	0.644178i	$0.777200\pi$
$18$	0	0
$19$	38308.0	1.28130	0.640652	−	0.767832i	$-0.278664\pi$
$19$	38308.0	1.28130	0.640652	+	0.767832i	$0.278664\pi$
$20$	0	0
$21$	13608.0	0.320647
$22$	0	0
$23$	71128.0i	1.21897i	0.792797	+	0.609485i	$0.208624\pi$
$23$	71128.0i	1.21897i	−0.792797	+	0.609485i	$0.791376\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	19683.0i	0.192450i
$28$	0	0
$29$	−74262.0	−0.565423	−0.282712	−	0.959205i	$-0.591234\pi$
$29$	−74262.0	−0.565423	−0.282712	+	0.959205i	$0.591234\pi$
$30$	0	0
$31$	−275680.	−1.66203	−0.831016	−	0.556249i	$-0.812240\pi$
$31$	−275680.	−1.66203	−0.831016	+	0.556249i	$0.812240\pi$
$32$	0	0
$33$	− 102924.i	− 0.498560i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 266610.i	− 0.865307i	−0.901560	−	0.432654i	$-0.857577\pi$
$37$	− 266610.i	− 0.865307i	0.901560	−	0.432654i	$-0.142423\pi$
$38$	0	0
$39$	−258498.	−0.697800
$40$	0	0
$41$	684762.	1.55166	0.775829	−	0.630943i	$-0.217332\pi$
$41$	684762.	1.55166	0.775829	+	0.630943i	$0.217332\pi$
$42$	0	0
$43$	− 245956.i	− 0.471756i	−0.971783	−	0.235878i	$-0.924203\pi$
$43$	− 245956.i	− 0.471756i	0.971783	−	0.235878i	$-0.0757966\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	478800.i	0.672685i	0.941740	+	0.336342i	$0.109190\pi$
$47$	478800.i	0.672685i	−0.941740	+	0.336342i	$0.890810\pi$
$48$	0	0
$49$	569527.	0.691557
$50$	0	0
$51$	704646.	0.743833
$52$	0	0
$53$	569410.i	0.525363i	0.964883	+	0.262682i	$0.0846069\pi$
$53$	569410.i	0.525363i	−0.964883	+	0.262682i	$0.915393\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.03432e6i	− 0.739761i
$58$	0	0
$59$	1.52532e6	0.966897	0.483448	−	0.875373i	$-0.339384\pi$
$59$	1.52532e6	0.966897	0.483448	+	0.875373i	$0.339384\pi$
$60$	0	0
$61$	−2.64046e6	−1.48945	−0.744723	−	0.667374i	$-0.767418\pi$
$61$	−2.64046e6	−1.48945	−0.744723	+	0.667374i	$0.767418\pi$
$62$	0	0
$63$	− 367416.i	− 0.185125i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.41624e6i	0.575273i	0.957740	+	0.287636i	$0.0928695\pi$
$67$	1.41624e6i	0.575273i	−0.957740	+	0.287636i	$0.907131\pi$
$68$	0	0
$69$	1.92046e6	0.703773
$70$	0	0
$71$	−3.51130e6	−1.16430	−0.582149	−	0.813082i	$-0.697788\pi$
$71$	−3.51130e6	−1.16430	−0.582149	+	0.813082i	$0.697788\pi$
$72$	0	0
$73$	− 4.73862e6i	− 1.42568i	−0.701327	−	0.712839i	$-0.747409\pi$
$73$	− 4.73862e6i	− 1.42568i	0.701327	−	0.712839i	$-0.252591\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.92125e6i	0.479585i
$78$	0	0
$79$	−4.66149e6	−1.06373	−0.531863	−	0.846830i	$-0.678508\pi$
$79$	−4.66149e6	−1.06373	−0.531863	+	0.846830i	$0.678508\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	5.72925e6i	1.09983i	0.835221	+	0.549914i	$0.185339\pi$
$83$	5.72925e6i	1.09983i	−0.835221	+	0.549914i	$0.814661\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.00507e6i	0.326447i
$88$	0	0
$89$	−1.19935e7	−1.80336	−0.901678	−	0.432408i	$-0.857664\pi$
$89$	−1.19935e7	−1.80336	−0.901678	+	0.432408i	$0.857664\pi$
$90$	0	0
$91$	4.82530e6	0.671242
$92$	0	0
$93$	7.44336e6i	0.959575i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.15075e6i	0.795519i	0.917490	+	0.397760i	$0.130212\pi$
$97$	7.15075e6i	0.795519i	−0.917490	+	0.397760i	$0.869788\pi$
$98$	0	0
$99$	−2.77895e6	−0.287844
$100$	0	0
$101$	−8.78373e6	−0.848309	−0.424155	−	0.905590i	$-0.639429\pi$
$101$	−8.78373e6	−0.848309	−0.424155	+	0.905590i	$0.639429\pi$
$102$	0	0
$103$	8.01610e6i	0.722825i	0.932406	+	0.361412i	$0.117705\pi$
$103$	8.01610e6i	0.722825i	−0.932406	+	0.361412i	$0.882295\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 5.15123e6i	− 0.406507i	−0.979126	−	0.203253i	$-0.934849\pi$
$107$	− 5.15123e6i	− 0.406507i	0.979126	−	0.203253i	$-0.0651515\pi$
$108$	0	0
$109$	2.41280e7	1.78455	0.892274	−	0.451493i	$-0.149109\pi$
$109$	2.41280e7	1.78455	0.892274	+	0.451493i	$0.149109\pi$
$110$	0	0
$111$	−7.19847e6	−0.499585
$112$	0	0
$113$	− 2.04827e6i	− 0.133541i	−0.997768	−	0.0667703i	$-0.978731\pi$
$113$	− 2.04827e6i	− 0.133541i	0.997768	−	0.0667703i	$-0.0212695\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.97945e6i	0.402875i
$118$	0	0
$119$	−1.31534e7	−0.715523
$120$	0	0
$121$	−4.95583e6	−0.254312
$122$	0	0
$123$	− 1.84886e7i	− 0.895850i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.36634e6i	0.0591895i	0.999562	+	0.0295947i	$0.00942167\pi$
$127$	1.36634e6i	0.0591895i	−0.999562	+	0.0295947i	$0.990578\pi$
$128$	0	0
$129$	−6.64081e6	−0.272369
$130$	0	0
$131$	3.84645e7	1.49489	0.747447	−	0.664321i	$-0.231279\pi$
$131$	3.84645e7	1.49489	0.747447	+	0.664321i	$0.231279\pi$
$132$	0	0
$133$	1.93072e7i	0.711605i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.62585e6i	0.253376i	0.991943	+	0.126688i	$0.0404348\pi$
$137$	7.62585e6i	0.253376i	−0.991943	+	0.126688i	$0.959565\pi$
$138$	0	0
$139$	−5.32324e6	−0.168122	−0.0840609	−	0.996461i	$-0.526789\pi$
$139$	−5.32324e6	−0.168122	−0.0840609	+	0.996461i	$0.526789\pi$
$140$	0	0
$141$	1.29276e7	0.388375
$142$	0	0
$143$	− 3.64961e7i	− 1.04369i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.53772e7i	− 0.399271i
$148$	0	0
$149$	−7.61366e6	−0.188557	−0.0942783	−	0.995546i	$-0.530054\pi$
$149$	−7.61366e6	−0.188557	−0.0942783	+	0.995546i	$0.530054\pi$
$150$	0	0
$151$	−2.50221e7	−0.591432	−0.295716	−	0.955276i	$-0.595558\pi$
$151$	−2.50221e7	−0.591432	−0.295716	+	0.955276i	$0.595558\pi$
$152$	0	0
$153$	− 1.90254e7i	− 0.429452i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.93145e7i	0.810782i	0.914143	+	0.405391i	$0.132865\pi$
$157$	3.93145e7i	0.810782i	−0.914143	+	0.405391i	$0.867135\pi$
$158$	0	0
$159$	1.53741e7	0.303319
$160$	0	0
$161$	−3.58485e7	−0.676987
$162$	0	0
$163$	6.28387e7i	1.13650i	0.822855	+	0.568252i	$0.192380\pi$
$163$	6.28387e7i	1.13650i	−0.822855	+	0.568252i	$0.807620\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.04133e7i	0.173014i	0.996251	+	0.0865072i	$0.0275706\pi$
$167$	1.04133e7i	0.173014i	−0.996251	+	0.0865072i	$0.972429\pi$
$168$	0	0
$169$	−2.89130e7	−0.460775
$170$	0	0
$171$	−2.79265e7	−0.427101
$172$	0	0
$173$	8.03551e7i	1.17992i	0.807433	+	0.589959i	$0.200856\pi$
$173$	8.03551e7i	1.17992i	−0.807433	+	0.589959i	$0.799144\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 4.11837e7i	− 0.558238i
$178$	0	0
$179$	8.40084e7	1.09481	0.547403	−	0.836869i	$-0.315617\pi$
$179$	8.40084e7	1.09481	0.547403	+	0.836869i	$0.315617\pi$
$180$	0	0
$181$	1.15469e8	1.44741	0.723703	−	0.690112i	$-0.242439\pi$
$181$	1.15469e8	1.44741	0.723703	+	0.690112i	$0.242439\pi$
$182$	0	0
$183$	7.12924e7i	0.859932i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.94856e7i	1.11254i
$188$	0	0
$189$	−9.92023e6	−0.106882
$190$	0	0
$191$	9.97154e7	1.03549	0.517744	−	0.855535i	$-0.326772\pi$
$191$	9.97154e7	1.03549	0.517744	+	0.855535i	$0.326772\pi$
$192$	0	0
$193$	1.86157e7i	0.186393i	0.995648	+	0.0931965i	$0.0297085\pi$
$193$	1.86157e7i	0.186393i	−0.995648	+	0.0931965i	$0.970292\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.30384e7i	0.867022i	0.901148	+	0.433511i	$0.142726\pi$
$197$	9.30384e7i	0.867022i	−0.901148	+	0.433511i	$0.857274\pi$
$198$	0	0
$199$	−7.39686e7	−0.665367	−0.332684	−	0.943038i	$-0.607954\pi$
$199$	−7.39686e7	−0.665367	−0.332684	+	0.943038i	$0.607954\pi$
$200$	0	0
$201$	3.82384e7	0.332134
$202$	0	0
$203$	− 3.74280e7i	− 0.314023i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 5.18523e7i	− 0.406323i
$208$	0	0
$209$	1.46030e8	1.10645
$210$	0	0
$211$	1.85163e8	1.35695	0.678476	−	0.734623i	$-0.262641\pi$
$211$	1.85163e8	1.35695	0.678476	+	0.734623i	$0.262641\pi$
$212$	0	0
$213$	9.48052e7i	0.672208i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.38943e8i	− 0.923053i
$218$	0	0
$219$	−1.27943e8	−0.823116
$220$	0	0
$221$	2.49862e8	1.55714
$222$	0	0
$223$	1.20862e8i	0.729830i	0.931041	+	0.364915i	$0.118902\pi$
$223$	1.20862e8i	0.729830i	−0.931041	+	0.364915i	$0.881098\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2.82315e8i	− 1.60193i	−0.598710	−	0.800966i	$-0.704320\pi$
$227$	− 2.82315e8i	− 1.60193i	0.598710	−	0.800966i	$-0.295680\pi$
$228$	0	0
$229$	8.91913e7	0.490793	0.245397	−	0.969423i	$-0.421082\pi$
$229$	8.91913e7	0.490793	0.245397	+	0.969423i	$0.421082\pi$
$230$	0	0
$231$	5.18737e7	0.276889
$232$	0	0
$233$	2.32240e8i	1.20279i	0.798951	+	0.601396i	$0.205389\pi$
$233$	2.32240e8i	1.20279i	−0.798951	+	0.601396i	$0.794611\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.25860e8i	0.614142i
$238$	0	0
$239$	−3.21986e8	−1.52561	−0.762807	−	0.646626i	$-0.776179\pi$
$239$	−3.21986e8	−1.52561	−0.762807	+	0.646626i	$0.776179\pi$
$240$	0	0
$241$	−2.00366e8	−0.922072	−0.461036	−	0.887381i	$-0.652522\pi$
$241$	−2.00366e8	−0.922072	−0.461036	+	0.887381i	$0.652522\pi$
$242$	0	0
$243$	− 1.43489e7i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3.66761e8i	− 1.54862i
$248$	0	0
$249$	1.54690e8	0.634986
$250$	0	0
$251$	−8.70560e7	−0.347489	−0.173744	−	0.984791i	$-0.555587\pi$
$251$	−8.70560e7	−0.347489	−0.173744	+	0.984791i	$0.555587\pi$
$252$	0	0
$253$	2.71140e8i	1.05262i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5.22879e8i	1.92148i	0.277457	+	0.960738i	$0.410509\pi$
$257$	5.22879e8i	1.92148i	−0.277457	+	0.960738i	$0.589491\pi$
$258$	0	0
$259$	1.34371e8	0.480571
$260$	0	0
$261$	5.41370e7	0.188474
$262$	0	0
$263$	4.06215e8i	1.37693i	0.725270	+	0.688464i	$0.241715\pi$
$263$	4.06215e8i	1.37693i	−0.725270	+	0.688464i	$0.758285\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.23825e8i	1.04117i
$268$	0	0
$269$	−3.82347e8	−1.19764	−0.598818	−	0.800885i	$-0.704363\pi$
$269$	−3.82347e8	−1.19764	−0.598818	+	0.800885i	$0.704363\pi$
$270$	0	0
$271$	2.84165e8	0.867317	0.433658	−	0.901077i	$-0.357222\pi$
$271$	2.84165e8	0.867317	0.433658	+	0.901077i	$0.357222\pi$
$272$	0	0
$273$	− 1.30283e8i	− 0.387542i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.93752e8i	0.830427i	0.909724	+	0.415213i	$0.136293\pi$
$277$	2.93752e8i	0.830427i	−0.909724	+	0.415213i	$0.863707\pi$
$278$	0	0
$279$	2.00971e8	0.554011
$280$	0	0
$281$	4.15399e8	1.11685	0.558424	−	0.829556i	$-0.311406\pi$
$281$	4.15399e8	1.11685	0.558424	+	0.829556i	$0.311406\pi$
$282$	0	0
$283$	− 5.06429e7i	− 0.132821i	−0.997792	−	0.0664104i	$-0.978845\pi$
$283$	− 5.06429e7i	− 0.132821i	0.997792	−	0.0664104i	$-0.0211547\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.45120e8i	0.861754i
$288$	0	0
$289$	−2.70767e8	−0.659862
$290$	0	0
$291$	1.93070e8	0.459293
$292$	0	0
$293$	− 7.47714e7i	− 0.173660i	−0.996223	−	0.0868298i	$-0.972326\pi$
$293$	− 7.47714e7i	− 0.173660i	0.996223	−	0.0868298i	$-0.0276736\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.50316e7i	0.166187i
$298$	0	0
$299$	6.80979e8	1.47328
$300$	0	0
$301$	1.23962e8	0.262002
$302$	0	0
$303$	2.37161e8i	0.489772i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	8.52577e7i	0.168170i	0.996459	+	0.0840851i	$0.0267968\pi$
$307$	8.52577e7i	0.168170i	−0.996459	+	0.0840851i	$0.973203\pi$
$308$	0	0
$309$	2.16435e8	0.417323
$310$	0	0
$311$	9.39129e8	1.77037	0.885184	−	0.465240i	$-0.154032\pi$
$311$	9.39129e8	1.77037	0.885184	+	0.465240i	$0.154032\pi$
$312$	0	0
$313$	3.43040e8i	0.632323i	0.948705	+	0.316162i	$0.102394\pi$
$313$	3.43040e8i	0.632323i	−0.948705	+	0.316162i	$0.897606\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.03960e9i	− 1.83298i	−0.400054	−	0.916492i	$-0.631009\pi$
$317$	− 1.03960e9i	− 1.83298i	0.400054	−	0.916492i	$-0.368991\pi$
$318$	0	0
$319$	−2.83087e8	−0.488261
$320$	0	0
$321$	−1.39083e8	−0.234697
$322$	0	0
$323$	9.99762e8i	1.65077i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 6.51456e8i	− 1.03031i
$328$	0	0
$329$	−2.41315e8	−0.373593
$330$	0	0
$331$	1.10022e9	1.66756	0.833779	−	0.552098i	$-0.186173\pi$
$331$	1.10022e9	1.66756	0.833779	+	0.552098i	$0.186173\pi$
$332$	0	0
$333$	1.94359e8i	0.288436i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.28272e9i	1.82569i	0.408302	+	0.912847i	$0.366121\pi$
$337$	1.28272e9i	1.82569i	−0.408302	+	0.912847i	$0.633879\pi$
$338$	0	0
$339$	−5.53034e7	−0.0770997
$340$	0	0
$341$	−1.05089e9	−1.43522
$342$	0	0
$343$	7.02107e8i	0.939451i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.25822e9i	1.61660i	0.588770	+	0.808301i	$0.299612\pi$
$347$	1.25822e9i	1.61660i	−0.588770	+	0.808301i	$0.700388\pi$
$348$	0	0
$349$	−1.35371e8	−0.170465	−0.0852327	−	0.996361i	$-0.527163\pi$
$349$	−1.35371e8	−0.170465	−0.0852327	+	0.996361i	$0.527163\pi$
$350$	0	0
$351$	1.88445e8	0.232600
$352$	0	0
$353$	8.49221e8i	1.02756i	0.857921	+	0.513782i	$0.171756\pi$
$353$	8.49221e8i	1.02756i	−0.857921	+	0.513782i	$0.828244\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.55142e8i	0.413107i
$358$	0	0
$359$	−2.20121e8	−0.251091	−0.125546	−	0.992088i	$-0.540068\pi$
$359$	−2.20121e8	−0.251091	−0.125546	+	0.992088i	$0.540068\pi$
$360$	0	0
$361$	5.73631e8	0.641738
$362$	0	0
$363$	1.33807e8i	0.146827i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.66505e8i	0.492635i	0.969189	+	0.246317i	$0.0792205\pi$
$367$	4.66505e8i	0.492635i	−0.969189	+	0.246317i	$0.920779\pi$
$368$	0	0
$369$	−4.99191e8	−0.517220
$370$	0	0
$371$	−2.86983e8	−0.291774
$372$	0	0
$373$	2.98453e8i	0.297780i	0.988854	+	0.148890i	$0.0475700\pi$
$373$	2.98453e8i	0.297780i	−0.988854	+	0.148890i	$0.952430\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.10984e8i	0.683385i
$378$	0	0
$379$	−1.46218e9	−1.37964	−0.689818	−	0.723983i	$-0.742309\pi$
$379$	−1.46218e9	−1.37964	−0.689818	+	0.723983i	$0.742309\pi$
$380$	0	0
$381$	3.68911e7	0.0341731
$382$	0	0
$383$	− 1.58702e9i	− 1.44340i	−0.692206	−	0.721700i	$-0.743361\pi$
$383$	− 1.58702e9i	− 1.44340i	0.692206	−	0.721700i	$-0.256639\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.79302e8i	0.157252i
$388$	0	0
$389$	3.14439e8	0.270840	0.135420	−	0.990788i	$-0.456762\pi$
$389$	3.14439e8	0.270840	0.135420	+	0.990788i	$0.456762\pi$
$390$	0	0
$391$	−1.85630e9	−1.57047
$392$	0	0
$393$	− 1.03854e9i	− 0.863078i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 8.52757e8i	− 0.684004i	−0.939699	−	0.342002i	$-0.888895\pi$
$397$	− 8.52757e8i	− 0.684004i	0.939699	−	0.342002i	$-0.111105\pi$
$398$	0	0
$399$	5.21295e8	0.410846
$400$	0	0
$401$	6.92522e8	0.536325	0.268163	−	0.963374i	$-0.413584\pi$
$401$	6.92522e8	0.536325	0.268163	+	0.963374i	$0.413584\pi$
$402$	0	0
$403$	2.63936e9i	2.00877i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.01632e9i	− 0.747221i
$408$	0	0
$409$	−6.17357e8	−0.446174	−0.223087	−	0.974799i	$-0.571613\pi$
$409$	−6.17357e8	−0.446174	−0.223087	+	0.974799i	$0.571613\pi$
$410$	0	0
$411$	2.05898e8	0.146287
$412$	0	0
$413$	7.68763e8i	0.536992i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.43727e8i	0.0970651i
$418$	0	0
$419$	1.65512e9	1.09921	0.549604	−	0.835425i	$-0.314779\pi$
$419$	1.65512e9	1.09921	0.549604	+	0.835425i	$0.314779\pi$
$420$	0	0
$421$	7.01472e7	0.0458166	0.0229083	−	0.999738i	$-0.492707\pi$
$421$	7.01472e7	0.0458166	0.0229083	+	0.999738i	$0.492707\pi$
$422$	0	0
$423$	− 3.49045e8i	− 0.224228i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.33079e9i	− 0.827203i
$428$	0	0
$429$	−9.85394e8	−0.602573
$430$	0	0
$431$	1.81387e9	1.09128	0.545640	−	0.838020i	$-0.316287\pi$
$431$	1.81387e9	1.09128	0.545640	+	0.838020i	$0.316287\pi$
$432$	0	0
$433$	2.59970e9i	1.53892i	0.638695	+	0.769460i	$0.279475\pi$
$433$	2.59970e9i	1.53892i	−0.638695	+	0.769460i	$0.720525\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.72477e9i	1.56187i
$438$	0	0
$439$	1.67431e9	0.944517	0.472258	−	0.881460i	$-0.343439\pi$
$439$	1.67431e9	0.944517	0.472258	+	0.881460i	$0.343439\pi$
$440$	0	0
$441$	−4.15185e8	−0.230519
$442$	0	0
$443$	2.52711e8i	0.138105i	0.997613	+	0.0690527i	$0.0219977\pi$
$443$	2.52711e8i	0.138105i	−0.997613	+	0.0690527i	$0.978002\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.05569e8i	0.108863i
$448$	0	0
$449$	−7.55311e8	−0.393789	−0.196895	−	0.980425i	$-0.563086\pi$
$449$	−7.55311e8	−0.393789	−0.196895	+	0.980425i	$0.563086\pi$
$450$	0	0
$451$	2.61031e9	1.33991
$452$	0	0
$453$	6.75597e8i	0.341463i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 1.51584e8i	− 0.0742928i	−0.999310	−	0.0371464i	$-0.988173\pi$
$457$	− 1.51584e8i	− 0.0742928i	0.999310	−	0.0371464i	$-0.0118268\pi$
$458$	0	0
$459$	−5.13687e8	−0.247944
$460$	0	0
$461$	−7.78405e8	−0.370043	−0.185022	−	0.982734i	$-0.559236\pi$
$461$	−7.78405e8	−0.370043	−0.185022	+	0.982734i	$0.559236\pi$
$462$	0	0
$463$	− 2.41052e9i	− 1.12870i	−0.825536	−	0.564349i	$-0.809127\pi$
$463$	− 2.41052e9i	− 1.12870i	0.825536	−	0.564349i	$-0.190873\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.76192e9i	− 0.800527i	−0.916400	−	0.400264i	$-0.868919\pi$
$467$	− 1.76192e9i	− 0.800527i	0.916400	−	0.400264i	$-0.131081\pi$
$468$	0	0
$469$	−7.13783e8	−0.319493
$470$	0	0
$471$	1.06149e9	0.468105
$472$	0	0
$473$	− 9.37584e8i	− 0.407377i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 4.15100e8i	− 0.175121i
$478$	0	0
$479$	−6.43811e8	−0.267661	−0.133830	−	0.991004i	$-0.542728\pi$
$479$	−6.43811e8	−0.267661	−0.133830	+	0.991004i	$0.542728\pi$
$480$	0	0
$481$	−2.55252e9	−1.04583
$482$	0	0
$483$	9.67910e8i	0.390859i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.16421e9i	1.24141i	0.784045	+	0.620704i	$0.213153\pi$
$487$	3.16421e9i	1.24141i	−0.784045	+	0.620704i	$0.786847\pi$
$488$	0	0
$489$	1.69665e9	0.656160
$490$	0	0
$491$	3.62406e9	1.38169	0.690844	−	0.723004i	$-0.257239\pi$
$491$	3.62406e9	1.38169	0.690844	+	0.723004i	$0.257239\pi$
$492$	0	0
$493$	− 1.93809e9i	− 0.728467i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.76970e9i	− 0.646624i
$498$	0	0
$499$	1.35483e9	0.488128	0.244064	−	0.969759i	$-0.421519\pi$
$499$	1.35483e9	0.488128	0.244064	+	0.969759i	$0.421519\pi$
$500$	0	0
$501$	2.81160e8	0.0998899
$502$	0	0
$503$	4.66389e9i	1.63403i	0.576616	+	0.817015i	$0.304373\pi$
$503$	4.66389e9i	1.63403i	−0.576616	+	0.817015i	$0.695627\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.80650e8i	0.266029i
$508$	0	0
$509$	1.34292e9	0.451376	0.225688	−	0.974200i	$-0.427537\pi$
$509$	1.34292e9	0.451376	0.225688	+	0.974200i	$0.427537\pi$
$510$	0	0
$511$	2.38826e9	0.791788
$512$	0	0
$513$	7.54016e8i	0.246587i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.82519e9i	0.580885i
$518$	0	0
$519$	2.16959e9	0.681226
$520$	0	0
$521$	−1.45400e9	−0.450435	−0.225217	−	0.974309i	$-0.572309\pi$
$521$	−1.45400e9	−0.450435	−0.225217	+	0.974309i	$0.572309\pi$
$522$	0	0
$523$	− 4.90309e8i	− 0.149870i	−0.997188	−	0.0749349i	$-0.976125\pi$
$523$	− 4.90309e8i	− 0.149870i	0.997188	−	0.0749349i	$-0.0238749\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 7.19470e9i	− 2.14129i
$528$	0	0
$529$	−1.65437e9	−0.485889
$530$	0	0
$531$	−1.11196e9	−0.322299
$532$	0	0
$533$	− 6.55591e9i	− 1.87537i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2.26823e9i	− 0.632086i
$538$	0	0
$539$	2.17104e9	0.597182
$540$	0	0
$541$	−4.82889e9	−1.31116	−0.655582	−	0.755124i	$-0.727577\pi$
$541$	−4.82889e9	−1.31116	−0.655582	+	0.755124i	$0.727577\pi$
$542$	0	0
$543$	− 3.11766e9i	− 0.835660i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 3.08793e9i	− 0.806698i	−0.915046	−	0.403349i	$-0.867846\pi$
$547$	− 3.08793e9i	− 0.806698i	0.915046	−	0.403349i	$-0.132154\pi$
$548$	0	0
$549$	1.92489e9	0.496482
$550$	0	0
$551$	−2.84483e9	−0.724479
$552$	0	0
$553$	− 2.34939e9i	− 0.590768i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.09889e9i	1.00502i	0.864573	+	0.502508i	$0.167589\pi$
$557$	4.09889e9i	1.00502i	−0.864573	+	0.502508i	$0.832411\pi$
$558$	0	0
$559$	−2.35478e9	−0.570177
$560$	0	0
$561$	2.68611e9	0.642324
$562$	0	0
$563$	− 4.97105e9i	− 1.17400i	−0.809587	−	0.587001i	$-0.800309\pi$
$563$	− 4.97105e9i	− 1.17400i	0.809587	−	0.587001i	$-0.199691\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.67846e8i	0.0617085i
$568$	0	0
$569$	−3.71316e9	−0.844988	−0.422494	−	0.906366i	$-0.638845\pi$
$569$	−3.71316e9	−0.844988	−0.422494	+	0.906366i	$0.638845\pi$
$570$	0	0
$571$	−2.36205e9	−0.530961	−0.265481	−	0.964116i	$-0.585531\pi$
$571$	−2.36205e9	−0.530961	−0.265481	+	0.964116i	$0.585531\pi$
$572$	0	0
$573$	− 2.69232e9i	− 0.597840i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.81146e8i	− 0.0392566i	−0.999807	−	0.0196283i	$-0.993752\pi$
$577$	− 1.81146e8i	− 0.0392566i	0.999807	−	0.0196283i	$-0.00624828\pi$
$578$	0	0
$579$	5.02625e8	0.107614
$580$	0	0
$581$	−2.88754e9	−0.610818
$582$	0	0
$583$	2.17059e9i	0.453668i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.31976e9i	0.473380i	0.971585	+	0.236690i	$0.0760626\pi$
$587$	2.31976e9i	0.473380i	−0.971585	+	0.236690i	$0.923937\pi$
$588$	0	0
$589$	−1.05607e10	−2.12957
$590$	0	0
$591$	2.51204e9	0.500576
$592$	0	0
$593$	− 2.27806e9i	− 0.448615i	−0.974518	−	0.224308i	$-0.927988\pi$
$593$	− 2.27806e9i	− 0.448615i	0.974518	−	0.224308i	$-0.0720120\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.99715e9i	0.384150i
$598$	0	0
$599$	4.88253e9	0.928220	0.464110	−	0.885778i	$-0.346374\pi$
$599$	4.88253e9	0.928220	0.464110	+	0.885778i	$0.346374\pi$
$600$	0	0
$601$	−6.74758e9	−1.26791	−0.633954	−	0.773371i	$-0.718569\pi$
$601$	−6.74758e9	−1.26791	−0.633954	+	0.773371i	$0.718569\pi$
$602$	0	0
$603$	− 1.03244e9i	− 0.191758i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 9.05928e9i	− 1.64412i	−0.569401	−	0.822060i	$-0.692825\pi$
$607$	− 9.05928e9i	− 1.64412i	0.569401	−	0.822060i	$-0.307175\pi$
$608$	0	0
$609$	−1.01056e9	−0.181301
$610$	0	0
$611$	4.58403e9	0.813024
$612$	0	0
$613$	8.48777e9i	1.48827i	0.668029	+	0.744135i	$0.267138\pi$
$613$	8.48777e9i	1.48827i	−0.668029	+	0.744135i	$0.732862\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.34407e9i	0.401766i	0.979615	+	0.200883i	$0.0643810\pi$
$617$	2.34407e9i	0.401766i	−0.979615	+	0.200883i	$0.935619\pi$
$618$	0	0
$619$	1.01541e9	0.172077	0.0860384	−	0.996292i	$-0.472579\pi$
$619$	1.01541e9	0.172077	0.0860384	+	0.996292i	$0.472579\pi$
$620$	0	0
$621$	−1.40001e9	−0.234591
$622$	0	0
$623$	− 6.04473e9i	− 1.00154i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 3.94281e9i	− 0.638807i
$628$	0	0
$629$	6.95799e9	1.11482
$630$	0	0
$631$	7.01911e9	1.11219	0.556095	−	0.831119i	$-0.312299\pi$
$631$	7.01911e9	1.11219	0.556095	+	0.831119i	$0.312299\pi$
$632$	0	0
$633$	− 4.99939e9i	− 0.783437i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 5.45265e9i	− 0.835833i
$638$	0	0
$639$	2.55974e9	0.388099
$640$	0	0
$641$	−4.52776e9	−0.679016	−0.339508	−	0.940603i	$-0.610261\pi$
$641$	−4.52776e9	−0.679016	−0.339508	+	0.940603i	$0.610261\pi$
$642$	0	0
$643$	8.63094e9i	1.28032i	0.768240	+	0.640162i	$0.221133\pi$
$643$	8.63094e9i	1.28032i	−0.768240	+	0.640162i	$0.778867\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.57401e9i	0.373632i	0.982395	+	0.186816i	$0.0598169\pi$
$647$	2.57401e9i	0.373632i	−0.982395	+	0.186816i	$0.940183\pi$
$648$	0	0
$649$	5.81454e9	0.834946
$650$	0	0
$651$	−3.75145e9	−0.532925
$652$	0	0
$653$	9.31827e9i	1.30960i	0.755802	+	0.654800i	$0.227247\pi$
$653$	9.31827e9i	1.30960i	−0.755802	+	0.654800i	$0.772753\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.45445e9i	0.475226i
$658$	0	0
$659$	−1.04422e10	−1.42133	−0.710663	−	0.703532i	$-0.751605\pi$
$659$	−1.04422e10	−1.42133	−0.710663	+	0.703532i	$0.751605\pi$
$660$	0	0
$661$	−1.04761e10	−1.41090	−0.705449	−	0.708761i	$-0.749254\pi$
$661$	−1.04761e10	−1.41090	−0.705449	+	0.708761i	$0.749254\pi$
$662$	0	0
$663$	− 6.74628e9i	− 0.899015i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 5.28211e9i	− 0.689234i
$668$	0	0
$669$	3.26327e9	0.421368
$670$	0	0
$671$	−1.00654e10	−1.28618
$672$	0	0
$673$	− 1.38891e10i	− 1.75639i	−0.478299	−	0.878197i	$-0.658746\pi$
$673$	− 1.38891e10i	− 1.75639i	0.478299	−	0.878197i	$-0.341254\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 7.48893e8i	− 0.0927598i	−0.998924	−	0.0463799i	$-0.985232\pi$
$677$	− 7.48893e8i	− 0.0927598i	0.998924	−	0.0463799i	$-0.0147685\pi$
$678$	0	0
$679$	−3.60398e9	−0.441813
$680$	0	0
$681$	−7.62251e9	−0.924875
$682$	0	0
$683$	− 1.15581e10i	− 1.38808i	−0.719938	−	0.694038i	$-0.755830\pi$
$683$	− 1.15581e10i	− 1.38808i	0.719938	−	0.694038i	$-0.244170\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 2.40817e9i	− 0.283360i
$688$	0	0
$689$	5.45153e9	0.634967
$690$	0	0
$691$	3.34337e8	0.0385489	0.0192744	−	0.999814i	$-0.493864\pi$
$691$	3.34337e8	0.0385489	0.0192744	+	0.999814i	$0.493864\pi$
$692$	0	0
$693$	− 1.40059e9i	− 0.159862i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.78709e10i	1.99909i
$698$	0	0
$699$	6.27047e9	0.694433
$700$	0	0
$701$	−5.55383e9	−0.608948	−0.304474	−	0.952521i	$-0.598481\pi$
$701$	−5.55383e9	−0.608948	−0.304474	+	0.952521i	$0.598481\pi$
$702$	0	0
$703$	− 1.02133e10i	− 1.10872i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.42700e9i	− 0.471131i
$708$	0	0
$709$	−1.30817e10	−1.37849	−0.689243	−	0.724530i	$-0.742057\pi$
$709$	−1.30817e10	−1.37849	−0.689243	+	0.724530i	$0.742057\pi$
$710$	0	0
$711$	3.39822e9	0.354575
$712$	0	0
$713$	− 1.96086e10i	− 2.02597i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.69363e9i	0.880814i
$718$	0	0
$719$	1.10847e10	1.11217	0.556085	−	0.831125i	$-0.312303\pi$
$719$	1.10847e10	1.11217	0.556085	+	0.831125i	$0.312303\pi$
$720$	0	0
$721$	−4.04012e9	−0.401440
$722$	0	0
$723$	5.40989e9i	0.532358i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 7.79416e9i	− 0.752314i	−0.926556	−	0.376157i	$-0.877245\pi$
$727$	− 7.79416e9i	− 0.752314i	0.926556	−	0.376157i	$-0.122755\pi$
$728$	0	0
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	6.41896e9	0.607790
$732$	0	0
$733$	− 6.83552e9i	− 0.641073i	−0.947236	−	0.320537i	$-0.896137\pi$
$733$	− 6.83552e9i	− 0.641073i	0.947236	−	0.320537i	$-0.103863\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.39869e9i	0.496767i
$738$	0	0
$739$	−1.73862e10	−1.58471	−0.792356	−	0.610059i	$-0.791146\pi$
$739$	−1.73862e10	−1.58471	−0.792356	+	0.610059i	$0.791146\pi$
$740$	0	0
$741$	−9.90254e9	−0.894094
$742$	0	0
$743$	− 2.25537e9i	− 0.201724i	−0.994900	−	0.100862i	$-0.967840\pi$
$743$	− 2.25537e9i	− 0.201724i	0.994900	−	0.100862i	$-0.0321600\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4.17662e9i	− 0.366609i
$748$	0	0
$749$	2.59622e9	0.225764
$750$	0	0
$751$	−2.05027e10	−1.76632	−0.883162	−	0.469068i	$-0.844590\pi$
$751$	−2.05027e10	−1.76632	−0.883162	+	0.469068i	$0.844590\pi$
$752$	0	0
$753$	2.35051e9i	0.200623i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 2.57872e8i	− 0.0216057i	−0.999942	−	0.0108029i	$-0.996561\pi$
$757$	− 2.57872e8i	− 0.0216057i	0.999942	−	0.0108029i	$-0.00343872\pi$
$758$	0	0
$759$	7.32078e9	0.607731
$760$	0	0
$761$	−1.34452e10	−1.10591	−0.552957	−	0.833210i	$-0.686501\pi$
$761$	−1.34452e10	−1.10591	−0.552957	+	0.833210i	$0.686501\pi$
$762$	0	0
$763$	1.21605e10i	0.991096i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.46035e10i	− 1.16862i
$768$	0	0
$769$	−8.28541e9	−0.657009	−0.328505	−	0.944502i	$-0.606545\pi$
$769$	−8.28541e9	−0.657009	−0.328505	+	0.944502i	$0.606545\pi$
$770$	0	0
$771$	1.41177e10	1.10936
$772$	0	0
$773$	− 1.43430e10i	− 1.11689i	−0.829540	−	0.558447i	$-0.811397\pi$
$773$	− 1.43430e10i	− 1.11689i	0.829540	−	0.558447i	$-0.188603\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3.62803e9i	− 0.277458i
$778$	0	0
$779$	2.62319e10	1.98814
$780$	0	0
$781$	−1.33851e10	−1.00541
$782$	0	0
$783$	− 1.46170e9i	− 0.108816i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 3.83137e9i	− 0.280184i	−0.990139	−	0.140092i	$-0.955260\pi$
$787$	− 3.83137e9i	− 0.280184i	0.990139	−	0.140092i	$-0.0447398\pi$
$788$	0	0
$789$	1.09678e10	0.794970
$790$	0	0
$791$	1.03233e9	0.0741653
$792$	0	0
$793$	2.52797e10i	1.80018i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.95859e9i	− 0.137038i	−0.997650	−	0.0685188i	$-0.978173\pi$
$797$	− 1.95859e9i	− 0.137038i	0.997650	−	0.0685188i	$-0.0218273\pi$
$798$	0	0
$799$	−1.24957e10	−0.866658
$800$	0	0
$801$	8.74327e9	0.601119
$802$	0	0
$803$	− 1.80636e10i	− 1.23112i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.03234e10i	0.691455i
$808$	0	0
$809$	−2.07415e9	−0.137727	−0.0688637	−	0.997626i	$-0.521937\pi$
$809$	−2.07415e9	−0.137727	−0.0688637	+	0.997626i	$0.521937\pi$
$810$	0	0
$811$	−5.71508e9	−0.376227	−0.188113	−	0.982147i	$-0.560237\pi$
$811$	−5.71508e9	−0.376227	−0.188113	+	0.982147i	$0.560237\pi$
$812$	0	0
$813$	− 7.67245e9i	− 0.500746i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.42208e9i	− 0.604463i
$818$	0	0
$819$	−3.51764e9	−0.223747
$820$	0	0
$821$	2.82748e10	1.78319	0.891596	−	0.452832i	$-0.149586\pi$
$821$	2.82748e10	1.78319	0.891596	+	0.452832i	$0.149586\pi$
$822$	0	0
$823$	2.09283e9i	0.130868i	0.997857	+	0.0654342i	$0.0208432\pi$
$823$	2.09283e9i	0.130868i	−0.997857	+	0.0654342i	$0.979157\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 3.71453e9i	− 0.228368i	−0.993460	−	0.114184i	$-0.963575\pi$
$827$	− 3.71453e9i	− 0.228368i	0.993460	−	0.114184i	$-0.0364253\pi$
$828$	0	0
$829$	−3.37924e9	−0.206005	−0.103003	−	0.994681i	$-0.532845\pi$
$829$	−3.37924e9	−0.206005	−0.103003	+	0.994681i	$0.532845\pi$
$830$	0	0
$831$	7.93130e9	0.479447
$832$	0	0
$833$	1.48635e10i	0.890972i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.42621e9i	− 0.319858i
$838$	0	0
$839$	−1.64907e10	−0.963990	−0.481995	−	0.876174i	$-0.660088\pi$
$839$	−1.64907e10	−0.963990	−0.481995	+	0.876174i	$0.660088\pi$
$840$	0	0
$841$	−1.17350e10	−0.680297
$842$	0	0
$843$	− 1.12158e10i	− 0.644812i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.49774e9i	− 0.141239i
$848$	0	0
$849$	−1.36736e9	−0.0766841
$850$	0	0
$851$	1.89634e10	1.05478
$852$	0	0
$853$	4.77028e9i	0.263162i	0.991305	+	0.131581i	$0.0420053\pi$
$853$	4.77028e9i	0.263162i	−0.991305	+	0.131581i	$0.957995\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.61514e10i	0.876554i	0.898840	+	0.438277i	$0.144411\pi$
$857$	1.61514e10i	0.876554i	−0.898840	+	0.438277i	$0.855589\pi$
$858$	0	0
$859$	3.41593e8	0.0183879	0.00919397	−	0.999958i	$-0.497073\pi$
$859$	3.41593e8	0.0183879	0.00919397	+	0.999958i	$0.497073\pi$
$860$	0	0
$861$	9.31824e9	0.497534
$862$	0	0
$863$	− 6.07878e9i	− 0.321943i	−0.986959	−	0.160971i	$-0.948537\pi$
$863$	− 6.07878e9i	− 0.321943i	0.986959	−	0.160971i	$-0.0514627\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.31071e9i	0.380972i
$868$	0	0
$869$	−1.77696e10	−0.918562
$870$	0	0
$871$	1.35590e10	0.695289
$872$	0	0
$873$	− 5.21290e9i	− 0.265173i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.23852e10i	0.620020i	0.950733	+	0.310010i	$0.100332\pi$
$877$	1.23852e10i	0.620020i	−0.950733	+	0.310010i	$0.899668\pi$
$878$	0	0
$879$	−2.01883e9	−0.100262
$880$	0	0
$881$	−1.37801e10	−0.678949	−0.339475	−	0.940615i	$-0.610249\pi$
$881$	−1.37801e10	−0.678949	−0.339475	+	0.940615i	$0.610249\pi$
$882$	0	0
$883$	− 1.89296e9i	− 0.0925292i	−0.998929	−	0.0462646i	$-0.985268\pi$
$883$	− 1.89296e9i	− 0.0925292i	0.998929	−	0.0462646i	$-0.0147317\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.23912e9i	0.203959i	0.994787	+	0.101979i	$0.0325176\pi$
$887$	4.23912e9i	0.203959i	−0.994787	+	0.101979i	$0.967482\pi$
$888$	0	0
$889$	−6.88633e8	−0.0328724
$890$	0	0
$891$	2.02585e9	0.0959480
$892$	0	0
$893$	1.83419e10i	0.861913i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.83864e10i	− 0.850598i
$898$	0	0
$899$	2.04725e10	0.939751
$900$	0	0
$901$	−1.48605e10	−0.676855
$902$	0	0
$903$	− 3.34697e9i	− 0.151267i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 9.51367e9i	− 0.423372i	−0.977338	−	0.211686i	$-0.932105\pi$
$907$	− 9.51367e9i	− 0.423372i	0.977338	−	0.211686i	$-0.0678955\pi$
$908$	0	0
$909$	6.40334e9	0.282770
$910$	0	0
$911$	−1.16235e10	−0.509359	−0.254680	−	0.967025i	$-0.581970\pi$
$911$	−1.16235e10	−0.509359	−0.254680	+	0.967025i	$0.581970\pi$
$912$	0	0
$913$	2.18399e10i	0.949736i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.93861e10i	0.830229i
$918$	0	0
$919$	9.22943e9	0.392257	0.196128	−	0.980578i	$-0.437163\pi$
$919$	9.22943e9	0.392257	0.196128	+	0.980578i	$0.437163\pi$
$920$	0	0
$921$	2.30196e9	0.0970931
$922$	0	0
$923$	3.36172e10i	1.40720i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 5.84374e9i	− 0.240942i
$928$	0	0
$929$	4.09353e10	1.67511	0.837555	−	0.546353i	$-0.183984\pi$
$929$	4.09353e10	1.67511	0.837555	+	0.546353i	$0.183984\pi$
$930$	0	0
$931$	2.18174e10	0.886094
$932$	0	0
$933$	− 2.53565e10i	− 1.02212i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.75085e9i	− 0.0695281i	−0.999396	−	0.0347641i	$-0.988932\pi$
$937$	− 1.75085e9i	− 0.0695281i	0.999396	−	0.0347641i	$-0.0110680\pi$
$938$	0	0
$939$	9.26207e9	0.365072
$940$	0	0
$941$	5.91102e9	0.231259	0.115630	−	0.993292i	$-0.463111\pi$
$941$	5.91102e9	0.231259	0.115630	+	0.993292i	$0.463111\pi$
$942$	0	0
$943$	4.87058e10i	1.89143i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 2.26089e10i	− 0.865077i	−0.901616	−	0.432538i	$-0.857618\pi$
$947$	− 2.26089e10i	− 0.865077i	0.901616	−	0.432538i	$-0.142382\pi$
$948$	0	0
$949$	−4.53675e10	−1.72311
$950$	0	0
$951$	−2.80692e10	−1.05827
$952$	0	0
$953$	− 1.11773e10i	− 0.418322i	−0.977881	−	0.209161i	$-0.932927\pi$
$953$	− 1.11773e10i	− 0.418322i	0.977881	−	0.209161i	$-0.0670733\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7.64334e9i	0.281898i
$958$	0	0
$959$	−3.84343e9	−0.140719
$960$	0	0
$961$	4.84868e10	1.76235
$962$	0	0
$963$	3.75525e9i	0.135502i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.55518e10i	0.553078i	0.961003	+	0.276539i	$0.0891875\pi$
$967$	1.55518e10i	0.553078i	−0.961003	+	0.276539i	$0.910812\pi$
$968$	0	0
$969$	2.69936e10	0.953075
$970$	0	0
$971$	8.34508e9	0.292525	0.146263	−	0.989246i	$-0.453276\pi$
$971$	8.34508e9	0.292525	0.146263	+	0.989246i	$0.453276\pi$
$972$	0	0
$973$	− 2.68291e9i	− 0.0933709i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 9.85180e9i	− 0.337975i	−0.985618	−	0.168988i	$-0.945950\pi$
$977$	− 9.85180e9i	− 0.337975i	0.985618	−	0.168988i	$-0.0540498\pi$
$978$	0	0
$979$	−4.57193e10	−1.55726
$980$	0	0
$981$	−1.75893e10	−0.594850
$982$	0	0
$983$	− 3.70884e10i	− 1.24538i	−0.782470	−	0.622688i	$-0.786041\pi$
$983$	− 3.70884e10i	− 1.24538i	0.782470	−	0.622688i	$-0.213959\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.51551e9i	0.215694i
$988$	0	0
$989$	1.74944e10	0.575057
$990$	0	0
$991$	6.43526e9	0.210043	0.105022	−	0.994470i	$-0.466509\pi$
$991$	6.43526e9	0.210043	0.105022	+	0.994470i	$0.466509\pi$
$992$	0	0
$993$	− 2.97059e10i	− 0.962765i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.23071e10i	− 0.393299i	−0.980474	−	0.196650i	$-0.936994\pi$
$997$	− 1.23071e10i	− 0.393299i	0.980474	−	0.196650i	$-0.0630062\pi$
$998$	0	0
$999$	5.24768e9	0.166528

Display

a_p

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a_n

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a_n

instead) (See

a_n

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a_n

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n

up to: 50 250 1000 (See only

a_p

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a_p

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.8.f.d.49.1		2
5.2	odd	4		600.8.a.a.1.1		1
5.3	odd	4		24.8.a.c.1.1	✓	1
5.4	even	2	inner	600.8.f.d.49.2		2
15.8	even	4		72.8.a.b.1.1		1
20.3	even	4		48.8.a.c.1.1		1
40.3	even	4		192.8.a.k.1.1		1
40.13	odd	4		192.8.a.c.1.1		1
60.23	odd	4		144.8.a.d.1.1		1
120.53	even	4		576.8.a.t.1.1		1
120.83	odd	4		576.8.a.s.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.8.a.c.1.1	✓	1	5.3	odd	4
48.8.a.c.1.1		1	20.3	even	4
72.8.a.b.1.1		1	15.8	even	4
144.8.a.d.1.1		1	60.23	odd	4
192.8.a.c.1.1		1	40.13	odd	4
192.8.a.k.1.1		1	40.3	even	4
576.8.a.s.1.1		1	120.83	odd	4
576.8.a.t.1.1		1	120.53	even	4
600.8.a.a.1.1		1	5.2	odd	4
600.8.f.d.49.1		2	1.1	even	1	trivial
600.8.f.d.49.2		2	5.4	even	2	inner

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

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Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

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	600.8.f.d.49.1

Level	$600$
Weight	$8$
Character	600.49
Analytic conductor	$187.431$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$