LMFDB - Embedded newform 48.24.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,24,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 24, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 24 $
Character orbit:	$[\chi]$	$=$	48.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:	$160.897937926$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-177147. q^{3} -3.54832e7 q^{5} +2.38585e9 q^{7} +3.13811e10 q^{9} +O(q^{10})$

$q-177147. q^{3} -3.54832e7 q^{5} +2.38585e9 q^{7} +3.13811e10 q^{9} -4.27835e11 q^{11} +4.30351e12 q^{13} +6.28575e12 q^{15} -2.11567e14 q^{17} +3.03300e14 q^{19} -4.22646e14 q^{21} +4.08483e15 q^{23} -1.06619e16 q^{25} -5.55906e15 q^{27} -7.67245e16 q^{29} +9.56625e16 q^{31} +7.57897e16 q^{33} -8.46576e16 q^{35} +1.91679e18 q^{37} -7.62354e17 q^{39} -3.82193e18 q^{41} +5.02883e18 q^{43} -1.11350e18 q^{45} +2.05876e19 q^{47} -2.16765e19 q^{49} +3.74784e19 q^{51} -1.72053e19 q^{53} +1.51810e19 q^{55} -5.37286e19 q^{57} -1.09298e20 q^{59} +4.75260e20 q^{61} +7.48704e19 q^{63} -1.52703e20 q^{65} -4.72132e20 q^{67} -7.23615e20 q^{69} +3.01752e21 q^{71} +4.69713e21 q^{73} +1.88872e21 q^{75} -1.02075e21 q^{77} -9.68876e21 q^{79} +9.84771e20 q^{81} -9.02054e21 q^{83} +7.50707e21 q^{85} +1.35915e22 q^{87} -7.81397e21 q^{89} +1.02675e22 q^{91} -1.69463e22 q^{93} -1.07621e22 q^{95} -5.74319e22 q^{97} -1.34259e22 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−177147.	−0.577350
$3$	−177147.	−0.577350
$4$	0	0
$5$	−3.54832e7	−0.324989	−0.162494	−	0.986709i	$-0.551954\pi$
$5$	−3.54832e7	−0.324989	−0.162494	+	0.986709i	$0.551954\pi$
$6$	0	0
$7$	2.38585e9	0.456053	0.228026	−	0.973655i	$-0.426773\pi$
$7$	2.38585e9	0.456053	0.228026	+	0.973655i	$0.426773\pi$
$8$	0	0
$9$	3.13811e10	0.333333
$10$	0	0
$11$	−4.27835e11	−0.452127	−0.226064	−	0.974113i	$-0.572586\pi$
$11$	−4.27835e11	−0.452127	−0.226064	+	0.974113i	$0.572586\pi$
$12$	0	0
$13$	4.30351e12	0.666000	0.333000	−	0.942927i	$-0.391939\pi$
$13$	4.30351e12	0.666000	0.333000	+	0.942927i	$0.391939\pi$
$14$	0	0
$15$	6.28575e12	0.187632
$16$	0	0
$17$	−2.11567e14	−1.49722	−0.748608	−	0.663012i	$-0.769278\pi$
$17$	−2.11567e14	−1.49722	−0.748608	+	0.663012i	$0.769278\pi$
$18$	0	0
$19$	3.03300e14	0.597318	0.298659	−	0.954360i	$-0.403461\pi$
$19$	3.03300e14	0.597318	0.298659	+	0.954360i	$0.403461\pi$
$20$	0	0
$21$	−4.22646e14	−0.263302
$22$	0	0
$23$	4.08483e15	0.893930	0.446965	−	0.894551i	$-0.352505\pi$
$23$	4.08483e15	0.893930	0.446965	+	0.894551i	$0.352505\pi$
$24$	0	0
$25$	−1.06619e16	−0.894382
$26$	0	0
$27$	−5.55906e15	−0.192450
$28$	0	0
$29$	−7.67245e16	−1.16777	−0.583885	−	0.811837i	$-0.698468\pi$
$29$	−7.67245e16	−1.16777	−0.583885	+	0.811837i	$0.698468\pi$
$30$	0	0
$31$	9.56625e16	0.676211	0.338106	−	0.941108i	$-0.390214\pi$
$31$	9.56625e16	0.676211	0.338106	+	0.941108i	$0.390214\pi$
$32$	0	0
$33$	7.57897e16	0.261036
$34$	0	0
$35$	−8.46576e16	−0.148212
$36$	0	0
$37$	1.91679e18	1.77114	0.885572	−	0.464501i	$-0.153766\pi$
$37$	1.91679e18	1.77114	0.885572	+	0.464501i	$0.153766\pi$
$38$	0	0
$39$	−7.62354e17	−0.384515
$40$	0	0
$41$	−3.82193e18	−1.08460	−0.542298	−	0.840186i	$-0.682446\pi$
$41$	−3.82193e18	−1.08460	−0.542298	+	0.840186i	$0.682446\pi$
$42$	0	0
$43$	5.02883e18	0.825240	0.412620	−	0.910903i	$-0.364614\pi$
$43$	5.02883e18	0.825240	0.412620	+	0.910903i	$0.364614\pi$
$44$	0	0
$45$	−1.11350e18	−0.108330
$46$	0	0
$47$	2.05876e19	1.21473	0.607366	−	0.794422i	$-0.292226\pi$
$47$	2.05876e19	1.21473	0.607366	+	0.794422i	$0.292226\pi$
$48$	0	0
$49$	−2.16765e19	−0.792016
$50$	0	0
$51$	3.74784e19	0.864418
$52$	0	0
$53$	−1.72053e19	−0.254971	−0.127486	−	0.991840i	$-0.540691\pi$
$53$	−1.72053e19	−0.254971	−0.127486	+	0.991840i	$0.540691\pi$
$54$	0	0
$55$	1.51810e19	0.146936
$56$	0	0
$57$	−5.37286e19	−0.344862
$58$	0	0
$59$	−1.09298e20	−0.471859	−0.235930	−	0.971770i	$-0.575813\pi$
$59$	−1.09298e20	−0.471859	−0.235930	+	0.971770i	$0.575813\pi$
$60$	0	0
$61$	4.75260e20	1.39842	0.699211	−	0.714915i	$-0.253535\pi$
$61$	4.75260e20	1.39842	0.699211	+	0.714915i	$0.253535\pi$
$62$	0	0
$63$	7.48704e19	0.152018
$64$	0	0
$65$	−1.52703e20	−0.216443
$66$	0	0
$67$	−4.72132e20	−0.472284	−0.236142	−	0.971719i	$-0.575883\pi$
$67$	−4.72132e20	−0.472284	−0.236142	+	0.971719i	$0.575883\pi$
$68$	0	0
$69$	−7.23615e20	−0.516111
$70$	0	0
$71$	3.01752e21	1.54946	0.774728	−	0.632294i	$-0.217887\pi$
$71$	3.01752e21	1.54946	0.774728	+	0.632294i	$0.217887\pi$
$72$	0	0
$73$	4.69713e21	1.75234	0.876172	−	0.481998i	$-0.160089\pi$
$73$	4.69713e21	1.75234	0.876172	+	0.481998i	$0.160089\pi$
$74$	0	0
$75$	1.88872e21	0.516372
$76$	0	0
$77$	−1.02075e21	−0.206194
$78$	0	0
$79$	−9.68876e21	−1.45733	−0.728663	−	0.684873i	$-0.759858\pi$
$79$	−9.68876e21	−1.45733	−0.728663	+	0.684873i	$0.759858\pi$
$80$	0	0
$81$	9.84771e20	0.111111
$82$	0	0
$83$	−9.02054e21	−0.768837	−0.384418	−	0.923159i	$-0.625598\pi$
$83$	−9.02054e21	−0.768837	−0.384418	+	0.923159i	$0.625598\pi$
$84$	0	0
$85$	7.50707e21	0.486579
$86$	0	0
$87$	1.35915e22	0.674212
$88$	0	0
$89$	−7.81397e21	−0.298461	−0.149230	−	0.988802i	$-0.547680\pi$
$89$	−7.81397e21	−0.298461	−0.149230	+	0.988802i	$0.547680\pi$
$90$	0	0
$91$	1.02675e22	0.303731
$92$	0	0
$93$	−1.69463e22	−0.390411
$94$	0	0
$95$	−1.07621e22	−0.194122
$96$	0	0
$97$	−5.74319e22	−0.815226	−0.407613	−	0.913155i	$-0.633639\pi$
$97$	−5.74319e22	−0.815226	−0.407613	+	0.913155i	$0.633639\pi$
$98$	0	0
$99$	−1.34259e22	−0.150709
$100$	0	0
$101$	−1.15980e23	−1.03440	−0.517198	−	0.855866i	$-0.673025\pi$
$101$	−1.15980e23	−1.03440	−0.517198	+	0.855866i	$0.673025\pi$
$102$	0	0
$103$	1.75357e23	1.24823	0.624115	−	0.781332i	$-0.285460\pi$
$103$	1.75357e23	1.24823	0.624115	+	0.781332i	$0.285460\pi$
$104$	0	0
$105$	1.49968e22	0.0855703
$106$	0	0
$107$	−1.42394e23	−0.653999	−0.327000	−	0.945024i	$-0.606038\pi$
$107$	−1.42394e23	−0.653999	−0.327000	+	0.945024i	$0.606038\pi$
$108$	0	0
$109$	−4.29786e22	−0.159532	−0.0797660	−	0.996814i	$-0.525417\pi$
$109$	−4.29786e22	−0.159532	−0.0797660	+	0.996814i	$0.525417\pi$
$110$	0	0
$111$	−3.39553e23	−1.02257
$112$	0	0
$113$	6.13546e23	1.50468	0.752342	−	0.658773i	$-0.228924\pi$
$113$	6.13546e23	1.50468	0.752342	+	0.658773i	$0.228924\pi$
$114$	0	0
$115$	−1.44943e23	−0.290517
$116$	0	0
$117$	1.35049e23	0.222000
$118$	0	0
$119$	−5.04766e23	−0.682810
$120$	0	0
$121$	−7.12387e23	−0.795581
$122$	0	0
$123$	6.77043e23	0.626192
$124$	0	0
$125$	8.01311e23	0.615653
$126$	0	0
$127$	1.95955e24	1.25434	0.627168	−	0.778884i	$-0.284214\pi$
$127$	1.95955e24	1.25434	0.627168	+	0.778884i	$0.284214\pi$
$128$	0	0
$129$	−8.90843e23	−0.476452
$130$	0	0
$131$	−7.09683e23	−0.318013	−0.159006	−	0.987278i	$-0.550829\pi$
$131$	−7.09683e23	−0.318013	−0.159006	+	0.987278i	$0.550829\pi$
$132$	0	0
$133$	7.23627e23	0.272409
$134$	0	0
$135$	1.97254e23	0.0625441
$136$	0	0
$137$	−4.01533e23	−0.107506	−0.0537532	−	0.998554i	$-0.517118\pi$
$137$	−4.01533e23	−0.107506	−0.0537532	+	0.998554i	$0.517118\pi$
$138$	0	0
$139$	−4.99216e24	−1.13140	−0.565702	−	0.824609i	$-0.691395\pi$
$139$	−4.99216e24	−1.13140	−0.565702	+	0.824609i	$0.691395\pi$
$140$	0	0
$141$	−3.64703e24	−0.701326
$142$	0	0
$143$	−1.84119e24	−0.301117
$144$	0	0
$145$	2.72244e24	0.379512
$146$	0	0
$147$	3.83992e24	0.457270
$148$	0	0
$149$	−7.07281e24	−0.721024	−0.360512	−	0.932754i	$-0.617398\pi$
$149$	−7.07281e24	−0.721024	−0.360512	+	0.932754i	$0.617398\pi$
$150$	0	0
$151$	−8.62065e23	−0.0753885	−0.0376943	−	0.999289i	$-0.512001\pi$
$151$	−8.62065e23	−0.0753885	−0.0376943	+	0.999289i	$0.512001\pi$
$152$	0	0
$153$	−6.63919e24	−0.499072
$154$	0	0
$155$	−3.39442e24	−0.219761
$156$	0	0
$157$	−2.09496e25	−1.17039	−0.585193	−	0.810894i	$-0.698981\pi$
$157$	−2.09496e25	−1.17039	−0.585193	+	0.810894i	$0.698981\pi$
$158$	0	0
$159$	3.04788e24	0.147208
$160$	0	0
$161$	9.74577e24	0.407679
$162$	0	0
$163$	−1.02247e25	−0.371102	−0.185551	−	0.982635i	$-0.559407\pi$
$163$	−1.02247e25	−0.371102	−0.185551	+	0.982635i	$0.559407\pi$
$164$	0	0
$165$	−2.68927e24	−0.0848337
$166$	0	0
$167$	−6.50477e25	−1.78646	−0.893228	−	0.449604i	$-0.851565\pi$
$167$	−6.50477e25	−1.78646	−0.893228	+	0.449604i	$0.851565\pi$
$168$	0	0
$169$	−2.32337e25	−0.556444
$170$	0	0
$171$	9.51786e24	0.199106
$172$	0	0
$173$	1.43650e25	0.262892	0.131446	−	0.991323i	$-0.458038\pi$
$173$	1.43650e25	0.262892	0.131446	+	0.991323i	$0.458038\pi$
$174$	0	0
$175$	−2.54376e25	−0.407886
$176$	0	0
$177$	1.93617e25	0.272428
$178$	0	0
$179$	−1.46620e26	−1.81294	−0.906468	−	0.422274i	$-0.861232\pi$
$179$	−1.46620e26	−1.81294	−0.906468	+	0.422274i	$0.861232\pi$
$180$	0	0
$181$	−8.58088e25	−0.933745	−0.466873	−	0.884325i	$-0.654619\pi$
$181$	−8.58088e25	−0.933745	−0.466873	+	0.884325i	$0.654619\pi$
$182$	0	0
$183$	−8.41909e25	−0.807380
$184$	0	0
$185$	−6.80138e25	−0.575602
$186$	0	0
$187$	9.05157e25	0.676933
$188$	0	0
$189$	−1.32631e25	−0.0877674
$190$	0	0
$191$	2.57595e26	1.51027	0.755134	−	0.655571i	$-0.227572\pi$
$191$	2.57595e26	1.51027	0.755134	+	0.655571i	$0.227572\pi$
$192$	0	0
$193$	−2.71962e26	−1.41449	−0.707244	−	0.706970i	$-0.750062\pi$
$193$	−2.71962e26	−1.41449	−0.707244	+	0.706970i	$0.750062\pi$
$194$	0	0
$195$	2.70508e25	0.124963
$196$	0	0
$197$	−2.22519e26	−0.914126	−0.457063	−	0.889434i	$-0.651099\pi$
$197$	−2.22519e26	−0.914126	−0.457063	+	0.889434i	$0.651099\pi$
$198$	0	0
$199$	−3.09111e26	−1.13059	−0.565294	−	0.824890i	$-0.691237\pi$
$199$	−3.09111e26	−1.13059	−0.565294	+	0.824890i	$0.691237\pi$
$200$	0	0
$201$	8.36367e25	0.272673
$202$	0	0
$203$	−1.83053e26	−0.532565
$204$	0	0
$205$	1.35614e26	0.352482
$206$	0	0
$207$	1.28186e26	0.297977
$208$	0	0
$209$	−1.29762e26	−0.270064
$210$	0	0
$211$	−8.30827e26	−1.54975	−0.774876	−	0.632113i	$-0.782188\pi$
$211$	−8.30827e26	−1.54975	−0.774876	+	0.632113i	$0.782188\pi$
$212$	0	0
$213$	−5.34545e26	−0.894579
$214$	0	0
$215$	−1.78439e26	−0.268194
$216$	0	0
$217$	2.28236e26	0.308388
$218$	0	0
$219$	−8.32082e26	−1.01172
$220$	0	0
$221$	−9.10479e26	−0.997147
$222$	0	0
$223$	−5.82216e26	−0.574882	−0.287441	−	0.957798i	$-0.592804\pi$
$223$	−5.82216e26	−0.574882	−0.287441	+	0.957798i	$0.592804\pi$
$224$	0	0
$225$	−3.34581e26	−0.298127
$226$	0	0
$227$	−5.37591e26	−0.432668	−0.216334	−	0.976319i	$-0.569410\pi$
$227$	−5.37591e26	−0.432668	−0.216334	+	0.976319i	$0.569410\pi$
$228$	0	0
$229$	4.88640e26	0.355534	0.177767	−	0.984073i	$-0.443113\pi$
$229$	4.88640e26	0.355534	0.177767	+	0.984073i	$0.443113\pi$
$230$	0	0
$231$	1.80823e26	0.119046
$232$	0	0
$233$	−2.89872e27	−1.72828	−0.864139	−	0.503253i	$-0.832136\pi$
$233$	−2.89872e27	−1.72828	−0.864139	+	0.503253i	$0.832136\pi$
$234$	0	0
$235$	−7.30515e26	−0.394774
$236$	0	0
$237$	1.71633e27	0.841387
$238$	0	0
$239$	7.63733e26	0.339912	0.169956	−	0.985452i	$-0.445638\pi$
$239$	7.63733e26	0.339912	0.169956	+	0.985452i	$0.445638\pi$
$240$	0	0
$241$	−1.87570e27	−0.758522	−0.379261	−	0.925290i	$-0.623822\pi$
$241$	−1.87570e27	−0.758522	−0.379261	+	0.925290i	$0.623822\pi$
$242$	0	0
$243$	−1.74449e26	−0.0641500
$244$	0	0
$245$	7.69152e26	0.257396
$246$	0	0
$247$	1.30525e27	0.397814
$248$	0	0
$249$	1.59796e27	0.443888
$250$	0	0
$251$	−1.47826e27	−0.374543	−0.187271	−	0.982308i	$-0.559964\pi$
$251$	−1.47826e27	−0.374543	−0.187271	+	0.982308i	$0.559964\pi$
$252$	0	0
$253$	−1.74763e27	−0.404170
$254$	0	0
$255$	−1.32986e27	−0.280926
$256$	0	0
$257$	6.12854e27	1.18339	0.591693	−	0.806164i	$-0.298460\pi$
$257$	6.12854e27	1.18339	0.591693	+	0.806164i	$0.298460\pi$
$258$	0	0
$259$	4.57316e27	0.807736
$260$	0	0
$261$	−2.40770e27	−0.389257
$262$	0	0
$263$	1.39751e27	0.206949	0.103474	−	0.994632i	$-0.467004\pi$
$263$	1.39751e27	0.206949	0.103474	+	0.994632i	$0.467004\pi$
$264$	0	0
$265$	6.10502e26	0.0828627
$266$	0	0
$267$	1.38422e27	0.172316
$268$	0	0
$269$	6.49585e26	0.0742138	0.0371069	−	0.999311i	$-0.488186\pi$
$269$	6.49585e26	0.0742138	0.0371069	+	0.999311i	$0.488186\pi$
$270$	0	0
$271$	1.61019e28	1.68939	0.844694	−	0.535250i	$-0.179783\pi$
$271$	1.61019e28	1.68939	0.844694	+	0.535250i	$0.179783\pi$
$272$	0	0
$273$	−1.81886e27	−0.175359
$274$	0	0
$275$	4.56152e27	0.404375
$276$	0	0
$277$	3.21829e27	0.262487	0.131244	−	0.991350i	$-0.458103\pi$
$277$	3.21829e27	0.262487	0.131244	+	0.991350i	$0.458103\pi$
$278$	0	0
$279$	3.00199e27	0.225404
$280$	0	0
$281$	4.95540e27	0.342733	0.171367	−	0.985207i	$-0.445182\pi$
$281$	4.95540e27	0.342733	0.171367	+	0.985207i	$0.445182\pi$
$282$	0	0
$283$	−2.85963e28	−1.82291	−0.911456	−	0.411399i	$-0.865040\pi$
$283$	−2.85963e28	−1.82291	−0.911456	+	0.411399i	$0.865040\pi$
$284$	0	0
$285$	1.90647e27	0.112076
$286$	0	0
$287$	−9.11854e27	−0.494633
$288$	0	0
$289$	2.47929e28	1.24166
$290$	0	0
$291$	1.01739e28	0.470671
$292$	0	0
$293$	3.15243e28	1.34793	0.673967	−	0.738762i	$-0.264589\pi$
$293$	3.15243e28	1.34793	0.673967	+	0.738762i	$0.264589\pi$
$294$	0	0
$295$	3.87823e27	0.153349
$296$	0	0
$297$	2.37836e27	0.0870120
$298$	0	0
$299$	1.75791e28	0.595358
$300$	0	0
$301$	1.19980e28	0.376353
$302$	0	0
$303$	2.05455e28	0.597209
$304$	0	0
$305$	−1.68638e28	−0.454472
$306$	0	0
$307$	2.26454e28	0.566094	0.283047	−	0.959106i	$-0.408655\pi$
$307$	2.26454e28	0.566094	0.283047	+	0.959106i	$0.408655\pi$
$308$	0	0
$309$	−3.10640e28	−0.720666
$310$	0	0
$311$	5.08124e28	1.09452	0.547262	−	0.836962i	$-0.315670\pi$
$311$	5.08124e28	1.09452	0.547262	+	0.836962i	$0.315670\pi$
$312$	0	0
$313$	−3.34545e28	−0.669415	−0.334707	−	0.942322i	$-0.608637\pi$
$313$	−3.34545e28	−0.669415	−0.334707	+	0.942322i	$0.608637\pi$
$314$	0	0
$315$	−2.65665e27	−0.0494040
$316$	0	0
$317$	3.18867e28	0.551352	0.275676	−	0.961251i	$-0.411098\pi$
$317$	3.18867e28	0.551352	0.275676	+	0.961251i	$0.411098\pi$
$318$	0	0
$319$	3.28255e28	0.527981
$320$	0	0
$321$	2.52246e28	0.377587
$322$	0	0
$323$	−6.41681e28	−0.894314
$324$	0	0
$325$	−4.58835e28	−0.595659
$326$	0	0
$327$	7.61354e27	0.0921059
$328$	0	0
$329$	4.91189e28	0.553982
$330$	0	0
$331$	1.66920e29	1.75584	0.877922	−	0.478804i	$-0.158930\pi$
$331$	1.66920e29	1.75584	0.877922	+	0.478804i	$0.158930\pi$
$332$	0	0
$333$	6.01508e28	0.590382
$334$	0	0
$335$	1.67528e28	0.153487
$336$	0	0
$337$	1.98063e29	1.69457	0.847286	−	0.531138i	$-0.178235\pi$
$337$	1.98063e29	1.69457	0.847286	+	0.531138i	$0.178235\pi$
$338$	0	0
$339$	−1.08688e29	−0.868730
$340$	0	0
$341$	−4.09278e28	−0.305734
$342$	0	0
$343$	−1.17014e29	−0.817254
$344$	0	0
$345$	2.56762e28	0.167730
$346$	0	0
$347$	−2.35903e28	−0.144193	−0.0720966	−	0.997398i	$-0.522969\pi$
$347$	−2.35903e28	−0.144193	−0.0720966	+	0.997398i	$0.522969\pi$
$348$	0	0
$349$	−4.57973e28	−0.262028	−0.131014	−	0.991381i	$-0.541823\pi$
$349$	−4.57973e28	−0.262028	−0.131014	+	0.991381i	$0.541823\pi$
$350$	0	0
$351$	−2.39235e28	−0.128172
$352$	0	0
$353$	−2.02926e29	−1.01842	−0.509212	−	0.860641i	$-0.670063\pi$
$353$	−2.02926e29	−1.01842	−0.509212	+	0.860641i	$0.670063\pi$
$354$	0	0
$355$	−1.07071e29	−0.503556
$356$	0	0
$357$	8.94178e28	0.394221
$358$	0	0
$359$	1.48550e28	0.0614168	0.0307084	−	0.999528i	$-0.490224\pi$
$359$	1.48550e28	0.0614168	0.0307084	+	0.999528i	$0.490224\pi$
$360$	0	0
$361$	−1.65839e29	−0.643211
$362$	0	0
$363$	1.26197e29	0.459329
$364$	0	0
$365$	−1.66669e29	−0.569492
$366$	0	0
$367$	−8.72251e28	−0.279886	−0.139943	−	0.990160i	$-0.544692\pi$
$367$	−8.72251e28	−0.279886	−0.139943	+	0.990160i	$0.544692\pi$
$368$	0	0
$369$	−1.19936e29	−0.361532
$370$	0	0
$371$	−4.10493e28	−0.116280
$372$	0	0
$373$	1.18888e29	0.316582	0.158291	−	0.987392i	$-0.449402\pi$
$373$	1.18888e29	0.316582	0.158291	+	0.987392i	$0.449402\pi$
$374$	0	0
$375$	−1.41950e29	−0.355447
$376$	0	0
$377$	−3.30185e29	−0.777735
$378$	0	0
$379$	−3.43993e29	−0.762427	−0.381214	−	0.924487i	$-0.624494\pi$
$379$	−3.43993e29	−0.762427	−0.381214	+	0.924487i	$0.624494\pi$
$380$	0	0
$381$	−3.47129e29	−0.724192
$382$	0	0
$383$	−2.35072e29	−0.461759	−0.230879	−	0.972982i	$-0.574160\pi$
$383$	−2.35072e29	−0.461759	−0.230879	+	0.972982i	$0.574160\pi$
$384$	0	0
$385$	3.62195e28	0.0670107
$386$	0	0
$387$	1.57810e29	0.275080
$388$	0	0
$389$	−4.99930e28	−0.0821276	−0.0410638	−	0.999157i	$-0.513075\pi$
$389$	−4.99930e28	−0.0821276	−0.0410638	+	0.999157i	$0.513075\pi$
$390$	0	0
$391$	−8.64213e29	−1.33841
$392$	0	0
$393$	1.25718e29	0.183605
$394$	0	0
$395$	3.43789e29	0.473614
$396$	0	0
$397$	−3.83154e29	−0.498060	−0.249030	−	0.968496i	$-0.580112\pi$
$397$	−3.83154e29	−0.498060	−0.249030	+	0.968496i	$0.580112\pi$
$398$	0	0
$399$	−1.28188e29	−0.157275
$400$	0	0
$401$	−8.85838e29	−1.02611	−0.513055	−	0.858356i	$-0.671486\pi$
$401$	−8.85838e29	−1.02611	−0.513055	+	0.858356i	$0.671486\pi$
$402$	0	0
$403$	4.11685e29	0.450357
$404$	0	0
$405$	−3.49429e28	−0.0361099
$406$	0	0
$407$	−8.20069e29	−0.800783
$408$	0	0
$409$	−1.13656e30	−1.04900	−0.524498	−	0.851412i	$-0.675747\pi$
$409$	−1.13656e30	−1.04900	−0.524498	+	0.851412i	$0.675747\pi$
$410$	0	0
$411$	7.11303e28	0.0620689
$412$	0	0
$413$	−2.60767e29	−0.215193
$414$	0	0
$415$	3.20078e29	0.249863
$416$	0	0
$417$	8.84346e29	0.653217
$418$	0	0
$419$	−1.22802e30	−0.858511	−0.429255	−	0.903183i	$-0.641224\pi$
$419$	−1.22802e30	−0.858511	−0.429255	+	0.903183i	$0.641224\pi$
$420$	0	0
$421$	2.08955e29	0.138296	0.0691478	−	0.997606i	$-0.477972\pi$
$421$	2.08955e29	0.138296	0.0691478	+	0.997606i	$0.477972\pi$
$422$	0	0
$423$	6.46061e29	0.404910
$424$	0	0
$425$	2.25570e30	1.33908
$426$	0	0
$427$	1.13390e30	0.637755
$428$	0	0
$429$	3.26162e29	0.173850
$430$	0	0
$431$	−2.18709e30	−1.10504	−0.552520	−	0.833500i	$-0.686334\pi$
$431$	−2.18709e30	−1.10504	−0.552520	+	0.833500i	$0.686334\pi$
$432$	0	0
$433$	6.48538e29	0.310688	0.155344	−	0.987860i	$-0.450351\pi$
$433$	6.48538e29	0.310688	0.155344	+	0.987860i	$0.450351\pi$
$434$	0	0
$435$	−4.82271e29	−0.219111
$436$	0	0
$437$	1.23893e30	0.533960
$438$	0	0
$439$	2.39546e30	0.979597	0.489798	−	0.871836i	$-0.337070\pi$
$439$	2.39546e30	0.979597	0.489798	+	0.871836i	$0.337070\pi$
$440$	0	0
$441$	−6.80231e29	−0.264005
$442$	0	0
$443$	−1.12838e30	−0.415732	−0.207866	−	0.978157i	$-0.566652\pi$
$443$	−1.12838e30	−0.415732	−0.207866	+	0.978157i	$0.566652\pi$
$444$	0	0
$445$	2.77265e29	0.0969963
$446$	0	0
$447$	1.25293e30	0.416284
$448$	0	0
$449$	−7.16916e29	−0.226274	−0.113137	−	0.993579i	$-0.536090\pi$
$449$	−7.16916e29	−0.226274	−0.113137	+	0.993579i	$0.536090\pi$
$450$	0	0
$451$	1.63516e30	0.490376
$452$	0	0
$453$	1.52712e29	0.0435256
$454$	0	0
$455$	−3.64325e29	−0.0987093
$456$	0	0
$457$	2.08316e30	0.536645	0.268322	−	0.963329i	$-0.413531\pi$
$457$	2.08316e30	0.536645	0.268322	+	0.963329i	$0.413531\pi$
$458$	0	0
$459$	1.17611e30	0.288139
$460$	0	0
$461$	−5.58501e30	−1.30156	−0.650779	−	0.759267i	$-0.725558\pi$
$461$	−5.58501e30	−1.30156	−0.650779	+	0.759267i	$0.725558\pi$
$462$	0	0
$463$	8.06785e29	0.178886	0.0894431	−	0.995992i	$-0.471491\pi$
$463$	8.06785e29	0.178886	0.0894431	+	0.995992i	$0.471491\pi$
$464$	0	0
$465$	6.01311e29	0.126879
$466$	0	0
$467$	6.09237e30	1.22361	0.611804	−	0.791009i	$-0.290444\pi$
$467$	6.09237e30	1.22361	0.611804	+	0.791009i	$0.290444\pi$
$468$	0	0
$469$	−1.12643e30	−0.215386
$470$	0	0
$471$	3.71116e30	0.675723
$472$	0	0
$473$	−2.15151e30	−0.373113
$474$	0	0
$475$	−3.23374e30	−0.534231
$476$	0	0
$477$	−5.39922e29	−0.0849904
$478$	0	0
$479$	6.95461e30	1.04331	0.521656	−	0.853156i	$-0.325314\pi$
$479$	6.95461e30	1.04331	0.521656	+	0.853156i	$0.325314\pi$
$480$	0	0
$481$	8.24891e30	1.17958
$482$	0	0
$483$	−1.72643e30	−0.235374
$484$	0	0
$485$	2.03787e30	0.264939
$486$	0	0
$487$	−8.03571e30	−0.996418	−0.498209	−	0.867057i	$-0.666009\pi$
$487$	−8.03571e30	−0.996418	−0.498209	+	0.867057i	$0.666009\pi$
$488$	0	0
$489$	1.81127e30	0.214256
$490$	0	0
$491$	−1.60987e31	−1.81700	−0.908498	−	0.417890i	$-0.862770\pi$
$491$	−1.60987e31	−1.81700	−0.908498	+	0.417890i	$0.862770\pi$
$492$	0	0
$493$	1.62324e31	1.74840
$494$	0	0
$495$	4.76396e29	0.0489788
$496$	0	0
$497$	7.19935e30	0.706634
$498$	0	0
$499$	2.00345e31	1.87768	0.938841	−	0.344350i	$-0.111901\pi$
$499$	2.00345e31	1.87768	0.938841	+	0.344350i	$0.111901\pi$
$500$	0	0
$501$	1.15230e31	1.03141
$502$	0	0
$503$	1.05260e31	0.899980	0.449990	−	0.893033i	$-0.351427\pi$
$503$	1.05260e31	0.899980	0.449990	+	0.893033i	$0.351427\pi$
$504$	0	0
$505$	4.11534e30	0.336167
$506$	0	0
$507$	4.11578e30	0.321263
$508$	0	0
$509$	−1.11320e31	−0.830457	−0.415228	−	0.909717i	$-0.636298\pi$
$509$	−1.11320e31	−0.830457	−0.415228	+	0.909717i	$0.636298\pi$
$510$	0	0
$511$	1.12066e31	0.799162
$512$	0	0
$513$	−1.68606e30	−0.114954
$514$	0	0
$515$	−6.22223e30	−0.405661
$516$	0	0
$517$	−8.80810e30	−0.549213
$518$	0	0
$519$	−2.54472e30	−0.151781
$520$	0	0
$521$	−1.51217e31	−0.862914	−0.431457	−	0.902134i	$-0.642000\pi$
$521$	−1.51217e31	−0.862914	−0.431457	+	0.902134i	$0.642000\pi$
$522$	0	0
$523$	1.01336e31	0.553342	0.276671	−	0.960965i	$-0.410769\pi$
$523$	1.01336e31	0.553342	0.276671	+	0.960965i	$0.410769\pi$
$524$	0	0
$525$	4.50619e30	0.235493
$526$	0	0
$527$	−2.02390e31	−1.01243
$528$	0	0
$529$	−4.19466e30	−0.200889
$530$	0	0
$531$	−3.42987e30	−0.157286
$532$	0	0
$533$	−1.64477e31	−0.722341
$534$	0	0
$535$	5.05259e30	0.212542
$536$	0	0
$537$	2.59732e31	1.04670
$538$	0	0
$539$	9.27396e30	0.358092
$540$	0	0
$541$	−3.92790e30	−0.145342	−0.0726712	−	0.997356i	$-0.523152\pi$
$541$	−3.92790e30	−0.145342	−0.0726712	+	0.997356i	$0.523152\pi$
$542$	0	0
$543$	1.52008e31	0.539098
$544$	0	0
$545$	1.52502e30	0.0518461
$546$	0	0
$547$	−5.08022e31	−1.65588	−0.827938	−	0.560819i	$-0.810486\pi$
$547$	−5.08022e31	−1.65588	−0.827938	+	0.560819i	$0.810486\pi$
$548$	0	0
$549$	1.49142e31	0.466141
$550$	0	0
$551$	−2.32705e31	−0.697530
$552$	0	0
$553$	−2.31159e31	−0.664618
$554$	0	0
$555$	1.20484e31	0.332324
$556$	0	0
$557$	−5.32703e31	−1.40978	−0.704889	−	0.709318i	$-0.749003\pi$
$557$	−5.32703e31	−1.40978	−0.704889	+	0.709318i	$0.749003\pi$
$558$	0	0
$559$	2.16416e31	0.549610
$560$	0	0
$561$	−1.60346e31	−0.390827
$562$	0	0
$563$	−6.35019e31	−1.48573	−0.742866	−	0.669441i	$-0.766534\pi$
$563$	−6.35019e31	−1.48573	−0.742866	+	0.669441i	$0.766534\pi$
$564$	0	0
$565$	−2.17706e31	−0.489005
$566$	0	0
$567$	2.34951e30	0.0506726
$568$	0	0
$569$	−2.81648e31	−0.583332	−0.291666	−	0.956520i	$-0.594210\pi$
$569$	−2.81648e31	−0.583332	−0.291666	+	0.956520i	$0.594210\pi$
$570$	0	0
$571$	−1.67583e30	−0.0333361	−0.0166680	−	0.999861i	$-0.505306\pi$
$571$	−1.67583e30	−0.0333361	−0.0166680	+	0.999861i	$0.505306\pi$
$572$	0	0
$573$	−4.56322e31	−0.871954
$574$	0	0
$575$	−4.35519e31	−0.799515
$576$	0	0
$577$	1.07229e32	1.89143	0.945715	−	0.324996i	$-0.105363\pi$
$577$	1.07229e32	1.89143	0.945715	+	0.324996i	$0.105363\pi$
$578$	0	0
$579$	4.81772e31	0.816655
$580$	0	0
$581$	−2.15216e31	−0.350630
$582$	0	0
$583$	7.36106e30	0.115279
$584$	0	0
$585$	−4.79197e30	−0.0721475
$586$	0	0
$587$	−1.23744e32	−1.79137	−0.895684	−	0.444691i	$-0.853313\pi$
$587$	−1.23744e32	−1.79137	−0.895684	+	0.444691i	$0.853313\pi$
$588$	0	0
$589$	2.90144e31	0.403913
$590$	0	0
$591$	3.94186e31	0.527771
$592$	0	0
$593$	−1.30128e32	−1.67587	−0.837935	−	0.545770i	$-0.816237\pi$
$593$	−1.30128e32	−1.67587	−0.837935	+	0.545770i	$0.816237\pi$
$594$	0	0
$595$	1.79107e31	0.221906
$596$	0	0
$597$	5.47581e31	0.652745
$598$	0	0
$599$	−8.58244e31	−0.984469	−0.492235	−	0.870463i	$-0.663820\pi$
$599$	−8.58244e31	−0.984469	−0.492235	+	0.870463i	$0.663820\pi$
$600$	0	0
$601$	−3.54795e31	−0.391670	−0.195835	−	0.980637i	$-0.562742\pi$
$601$	−3.54795e31	−0.391670	−0.195835	+	0.980637i	$0.562742\pi$
$602$	0	0
$603$	−1.48160e31	−0.157428
$604$	0	0
$605$	2.52778e31	0.258555
$606$	0	0
$607$	−1.08336e32	−1.06685	−0.533425	−	0.845847i	$-0.679095\pi$
$607$	−1.08336e32	−1.06685	−0.533425	+	0.845847i	$0.679095\pi$
$608$	0	0
$609$	3.24273e31	0.307476
$610$	0	0
$611$	8.85989e31	0.809011
$612$	0	0
$613$	−9.17001e31	−0.806444	−0.403222	−	0.915102i	$-0.632110\pi$
$613$	−9.17001e31	−0.806444	−0.403222	+	0.915102i	$0.632110\pi$
$614$	0	0
$615$	−2.40237e31	−0.203505
$616$	0	0
$617$	2.40717e32	1.96439	0.982193	−	0.187874i	$-0.0601598\pi$
$617$	2.40717e32	1.96439	0.982193	+	0.187874i	$0.0601598\pi$
$618$	0	0
$619$	1.50899e32	1.18643	0.593216	−	0.805043i	$-0.297858\pi$
$619$	1.50899e32	1.18643	0.593216	+	0.805043i	$0.297858\pi$
$620$	0	0
$621$	−2.27078e31	−0.172037
$622$	0	0
$623$	−1.86430e31	−0.136114
$624$	0	0
$625$	9.86663e31	0.694302
$626$	0	0
$627$	2.29870e31	0.155921
$628$	0	0
$629$	−4.05528e32	−2.65179
$630$	0	0
$631$	1.64482e32	1.03701	0.518503	−	0.855076i	$-0.326489\pi$
$631$	1.64482e32	1.03701	0.518503	+	0.855076i	$0.326489\pi$
$632$	0	0
$633$	1.47178e32	0.894750
$634$	0	0
$635$	−6.95313e31	−0.407645
$636$	0	0
$637$	−9.32850e31	−0.527483
$638$	0	0
$639$	9.46930e31	0.516485
$640$	0	0
$641$	−1.11001e32	−0.584063	−0.292032	−	0.956409i	$-0.594331\pi$
$641$	−1.11001e32	−0.584063	−0.292032	+	0.956409i	$0.594331\pi$
$642$	0	0
$643$	3.32442e32	1.68768	0.843838	−	0.536598i	$-0.180291\pi$
$643$	3.32442e32	1.68768	0.843838	+	0.536598i	$0.180291\pi$
$644$	0	0
$645$	3.16100e31	0.154842
$646$	0	0
$647$	−1.67496e32	−0.791783	−0.395891	−	0.918297i	$-0.629564\pi$
$647$	−1.67496e32	−0.791783	−0.395891	+	0.918297i	$0.629564\pi$
$648$	0	0
$649$	4.67614e31	0.213340
$650$	0	0
$651$	−4.04314e31	−0.178048
$652$	0	0
$653$	7.32573e31	0.311422	0.155711	−	0.987803i	$-0.450233\pi$
$653$	7.32573e31	0.311422	0.155711	+	0.987803i	$0.450233\pi$
$654$	0	0
$655$	2.51819e31	0.103351
$656$	0	0
$657$	1.47401e32	0.584115
$658$	0	0
$659$	−6.03858e31	−0.231075	−0.115537	−	0.993303i	$-0.536859\pi$
$659$	−6.03858e31	−0.231075	−0.115537	+	0.993303i	$0.536859\pi$
$660$	0	0
$661$	3.76425e32	1.39111	0.695555	−	0.718473i	$-0.255158\pi$
$661$	3.76425e32	1.39111	0.695555	+	0.718473i	$0.255158\pi$
$662$	0	0
$663$	1.61289e32	0.575703
$664$	0	0
$665$	−2.56766e31	−0.0885297
$666$	0	0
$667$	−3.13406e32	−1.04390
$668$	0	0
$669$	1.03138e32	0.331908
$670$	0	0
$671$	−2.03333e32	−0.632265
$672$	0	0
$673$	3.92456e32	1.17928	0.589642	−	0.807665i	$-0.299269\pi$
$673$	3.92456e32	1.17928	0.589642	+	0.807665i	$0.299269\pi$
$674$	0	0
$675$	5.92700e31	0.172124
$676$	0	0
$677$	−1.22862e32	−0.344865	−0.172433	−	0.985021i	$-0.555163\pi$
$677$	−1.22862e32	−0.344865	−0.172433	+	0.985021i	$0.555163\pi$
$678$	0	0
$679$	−1.37024e32	−0.371786
$680$	0	0
$681$	9.52327e31	0.249801
$682$	0	0
$683$	5.86596e32	1.48765	0.743825	−	0.668374i	$-0.233010\pi$
$683$	5.86596e32	1.48765	0.743825	+	0.668374i	$0.233010\pi$
$684$	0	0
$685$	1.42477e31	0.0349384
$686$	0	0
$687$	−8.65611e31	−0.205268
$688$	0	0
$689$	−7.40434e31	−0.169811
$690$	0	0
$691$	−1.25069e32	−0.277429	−0.138714	−	0.990332i	$-0.544297\pi$
$691$	−1.25069e32	−0.277429	−0.138714	+	0.990332i	$0.544297\pi$
$692$	0	0
$693$	−3.20322e31	−0.0687313
$694$	0	0
$695$	1.77138e32	0.367694
$696$	0	0
$697$	8.08593e32	1.62388
$698$	0	0
$699$	5.13500e32	0.997822
$700$	0	0
$701$	−8.69124e32	−1.63427	−0.817137	−	0.576444i	$-0.804440\pi$
$701$	−8.69124e32	−1.63427	−0.817137	+	0.576444i	$0.804440\pi$
$702$	0	0
$703$	5.81361e32	1.05794
$704$	0	0
$705$	1.29409e32	0.227923
$706$	0	0
$707$	−2.76710e32	−0.471740
$708$	0	0
$709$	−7.34080e32	−1.21147	−0.605734	−	0.795667i	$-0.707120\pi$
$709$	−7.34080e32	−1.21147	−0.605734	+	0.795667i	$0.707120\pi$
$710$	0	0
$711$	−3.04043e32	−0.485775
$712$	0	0
$713$	3.90765e32	0.604485
$714$	0	0
$715$	6.53315e31	0.0978596
$716$	0	0
$717$	−1.35293e32	−0.196248
$718$	0	0
$719$	1.15251e33	1.61905	0.809526	−	0.587084i	$-0.199724\pi$
$719$	1.15251e33	1.61905	0.809526	+	0.587084i	$0.199724\pi$
$720$	0	0
$721$	4.18375e32	0.569259
$722$	0	0
$723$	3.32275e32	0.437933
$724$	0	0
$725$	8.18027e32	1.04443
$726$	0	0
$727$	3.95293e32	0.488960	0.244480	−	0.969654i	$-0.421383\pi$
$727$	3.95293e32	0.488960	0.244480	+	0.969654i	$0.421383\pi$
$728$	0	0
$729$	3.09032e31	0.0370370
$730$	0	0
$731$	−1.06393e33	−1.23556
$732$	0	0
$733$	−8.03668e32	−0.904443	−0.452221	−	0.891906i	$-0.649368\pi$
$733$	−8.03668e32	−0.904443	−0.452221	+	0.891906i	$0.649368\pi$
$734$	0	0
$735$	−1.36253e32	−0.148608
$736$	0	0
$737$	2.01995e32	0.213532
$738$	0	0
$739$	−1.84582e32	−0.189138	−0.0945689	−	0.995518i	$-0.530147\pi$
$739$	−1.84582e32	−0.189138	−0.0945689	+	0.995518i	$0.530147\pi$
$740$	0	0
$741$	−2.31222e32	−0.229678
$742$	0	0
$743$	−2.05453e33	−1.97853	−0.989263	−	0.146145i	$-0.953313\pi$
$743$	−2.05453e33	−1.97853	−0.989263	+	0.146145i	$0.953313\pi$
$744$	0	0
$745$	2.50966e32	0.234325
$746$	0	0
$747$	−2.83074e32	−0.256279
$748$	0	0
$749$	−3.39730e32	−0.298258
$750$	0	0
$751$	9.96861e32	0.848741	0.424371	−	0.905489i	$-0.360495\pi$
$751$	9.96861e32	0.848741	0.424371	+	0.905489i	$0.360495\pi$
$752$	0	0
$753$	2.61868e32	0.216242
$754$	0	0
$755$	3.05889e31	0.0245004
$756$	0	0
$757$	−1.14407e33	−0.888898	−0.444449	−	0.895804i	$-0.646600\pi$
$757$	−1.14407e33	−0.888898	−0.444449	+	0.895804i	$0.646600\pi$
$758$	0	0
$759$	3.09588e32	0.233348
$760$	0	0
$761$	1.02852e33	0.752126	0.376063	−	0.926594i	$-0.377278\pi$
$761$	1.02852e33	0.752126	0.376063	+	0.926594i	$0.377278\pi$
$762$	0	0
$763$	−1.02541e32	−0.0727551
$764$	0	0
$765$	2.35580e32	0.162193
$766$	0	0
$767$	−4.70363e32	−0.314258
$768$	0	0
$769$	−1.15966e33	−0.751930	−0.375965	−	0.926634i	$-0.622689\pi$
$769$	−1.15966e33	−0.751930	−0.375965	+	0.926634i	$0.622689\pi$
$770$	0	0
$771$	−1.08565e33	−0.683228
$772$	0	0
$773$	−2.37717e33	−1.45210	−0.726049	−	0.687642i	$-0.758646\pi$
$773$	−2.37717e33	−1.45210	−0.726049	+	0.687642i	$0.758646\pi$
$774$	0	0
$775$	−1.01994e33	−0.604791
$776$	0	0
$777$	−8.10122e32	−0.466346
$778$	0	0
$779$	−1.15919e33	−0.647849
$780$	0	0
$781$	−1.29100e33	−0.700552
$782$	0	0
$783$	4.26516e32	0.224737
$784$	0	0
$785$	7.43359e32	0.380363
$786$	0	0
$787$	−4.22154e32	−0.209779	−0.104889	−	0.994484i	$-0.533449\pi$
$787$	−4.22154e32	−0.209779	−0.104889	+	0.994484i	$0.533449\pi$
$788$	0	0
$789$	−2.47564e32	−0.119482
$790$	0	0
$791$	1.46383e33	0.686215
$792$	0	0
$793$	2.04529e33	0.931350
$794$	0	0
$795$	−1.08149e32	−0.0478408
$796$	0	0
$797$	−1.84906e33	−0.794657	−0.397328	−	0.917677i	$-0.630063\pi$
$797$	−1.84906e33	−0.794657	−0.397328	+	0.917677i	$0.630063\pi$
$798$	0	0
$799$	−4.35565e33	−1.81872
$800$	0	0
$801$	−2.45211e32	−0.0994868
$802$	0	0
$803$	−2.00960e33	−0.792283
$804$	0	0
$805$	−3.45812e32	−0.132491
$806$	0	0
$807$	−1.15072e32	−0.0428473
$808$	0	0
$809$	3.42497e33	1.23951	0.619753	−	0.784797i	$-0.287233\pi$
$809$	3.42497e33	1.23951	0.619753	+	0.784797i	$0.287233\pi$
$810$	0	0
$811$	−2.27584e33	−0.800574	−0.400287	−	0.916390i	$-0.631090\pi$
$811$	−2.27584e33	−0.800574	−0.400287	+	0.916390i	$0.631090\pi$
$812$	0	0
$813$	−2.85240e33	−0.975368
$814$	0	0
$815$	3.62805e32	0.120604
$816$	0	0
$817$	1.52524e33	0.492930
$818$	0	0
$819$	3.22206e32	0.101244
$820$	0	0
$821$	−5.01922e33	−1.53352	−0.766760	−	0.641934i	$-0.778132\pi$
$821$	−5.01922e33	−1.53352	−0.766760	+	0.641934i	$0.778132\pi$
$822$	0	0
$823$	3.43402e33	1.02025	0.510123	−	0.860102i	$-0.329600\pi$
$823$	3.43402e33	1.02025	0.510123	+	0.860102i	$0.329600\pi$
$824$	0	0
$825$	−8.08060e32	−0.233466
$826$	0	0
$827$	6.15512e33	1.72951	0.864755	−	0.502194i	$-0.167474\pi$
$827$	6.15512e33	1.72951	0.864755	+	0.502194i	$0.167474\pi$
$828$	0	0
$829$	−4.40716e33	−1.20443	−0.602216	−	0.798333i	$-0.705715\pi$
$829$	−4.40716e33	−1.20443	−0.602216	+	0.798333i	$0.705715\pi$
$830$	0	0
$831$	−5.70111e32	−0.151547
$832$	0	0
$833$	4.58602e33	1.18582
$834$	0	0
$835$	2.30810e33	0.580578
$836$	0	0
$837$	−5.31794e32	−0.130137
$838$	0	0
$839$	3.66089e33	0.871614	0.435807	−	0.900040i	$-0.356463\pi$
$839$	3.66089e33	0.871614	0.435807	+	0.900040i	$0.356463\pi$
$840$	0	0
$841$	1.56993e33	0.363686
$842$	0	0
$843$	−8.77835e32	−0.197877
$844$	0	0
$845$	8.24407e32	0.180838
$846$	0	0
$847$	−1.69965e33	−0.362827
$848$	0	0
$849$	5.06574e33	1.05246
$850$	0	0
$851$	7.82974e33	1.58328
$852$	0	0
$853$	6.76757e32	0.133204	0.0666022	−	0.997780i	$-0.478784\pi$
$853$	6.76757e32	0.133204	0.0666022	+	0.997780i	$0.478784\pi$
$854$	0	0
$855$	−3.37725e32	−0.0647072
$856$	0	0
$857$	3.94570e33	0.735943	0.367972	−	0.929837i	$-0.380052\pi$
$857$	3.94570e33	0.735943	0.367972	+	0.929837i	$0.380052\pi$
$858$	0	0
$859$	−6.90977e33	−1.25471	−0.627354	−	0.778734i	$-0.715862\pi$
$859$	−6.90977e33	−1.25471	−0.627354	+	0.778734i	$0.715862\pi$
$860$	0	0
$861$	1.61532e33	0.285577
$862$	0	0
$863$	−2.53657e33	−0.436639	−0.218319	−	0.975877i	$-0.570057\pi$
$863$	−2.53657e33	−0.436639	−0.218319	+	0.975877i	$0.570057\pi$
$864$	0	0
$865$	−5.09718e32	−0.0854368
$866$	0	0
$867$	−4.39199e33	−0.716872
$868$	0	0
$869$	4.14519e33	0.658897
$870$	0	0
$871$	−2.03182e33	−0.314541
$872$	0	0
$873$	−1.80227e33	−0.271742
$874$	0	0
$875$	1.91181e33	0.280770
$876$	0	0
$877$	−6.49775e33	−0.929539	−0.464770	−	0.885432i	$-0.653863\pi$
$877$	−6.49775e33	−0.929539	−0.464770	+	0.885432i	$0.653863\pi$
$878$	0	0
$879$	−5.58444e33	−0.778230
$880$	0	0
$881$	5.94105e33	0.806567	0.403283	−	0.915075i	$-0.367869\pi$
$881$	5.94105e33	0.806567	0.403283	+	0.915075i	$0.367869\pi$
$882$	0	0
$883$	4.54083e33	0.600602	0.300301	−	0.953844i	$-0.402913\pi$
$883$	4.54083e33	0.600602	0.300301	+	0.953844i	$0.402913\pi$
$884$	0	0
$885$	−6.87017e32	−0.0885360
$886$	0	0
$887$	5.73254e33	0.719822	0.359911	−	0.932987i	$-0.382807\pi$
$887$	5.73254e33	0.719822	0.359911	+	0.932987i	$0.382807\pi$
$888$	0	0
$889$	4.67519e33	0.572044
$890$	0	0
$891$	−4.21320e32	−0.0502364
$892$	0	0
$893$	6.24421e33	0.725581
$894$	0	0
$895$	5.20254e33	0.589184
$896$	0	0
$897$	−3.11408e33	−0.343730
$898$	0	0
$899$	−7.33966e33	−0.789659
$900$	0	0
$901$	3.64008e33	0.381747
$902$	0	0
$903$	−2.12542e33	−0.217287
$904$	0	0
$905$	3.04478e33	0.303457
$906$	0	0
$907$	1.42714e34	1.38670	0.693351	−	0.720600i	$-0.256133\pi$
$907$	1.42714e34	1.38670	0.693351	+	0.720600i	$0.256133\pi$
$908$	0	0
$909$	−3.63957e33	−0.344799
$910$	0	0
$911$	1.46389e34	1.35222	0.676108	−	0.736803i	$-0.263665\pi$
$911$	1.46389e34	1.35222	0.676108	+	0.736803i	$0.263665\pi$
$912$	0	0
$913$	3.85930e33	0.347612
$914$	0	0
$915$	2.98737e33	0.262389
$916$	0	0
$917$	−1.69320e33	−0.145031
$918$	0	0
$919$	−2.69602e33	−0.225214	−0.112607	−	0.993640i	$-0.535920\pi$
$919$	−2.69602e33	−0.225214	−0.112607	+	0.993640i	$0.535920\pi$
$920$	0	0
$921$	−4.01156e33	−0.326835
$922$	0	0
$923$	1.29859e34	1.03194
$924$	0	0
$925$	−2.04365e34	−1.58408
$926$	0	0
$927$	5.50289e33	0.416077
$928$	0	0
$929$	1.49108e34	1.09981	0.549907	−	0.835226i	$-0.314663\pi$
$929$	1.49108e34	1.09981	0.549907	+	0.835226i	$0.314663\pi$
$930$	0	0
$931$	−6.57447e33	−0.473085
$932$	0	0
$933$	−9.00126e33	−0.631923
$934$	0	0
$935$	−3.21179e33	−0.219996
$936$	0	0
$937$	1.00441e34	0.671282	0.335641	−	0.941990i	$-0.391047\pi$
$937$	1.00441e34	0.671282	0.335641	+	0.941990i	$0.391047\pi$
$938$	0	0
$939$	5.92637e33	0.386487
$940$	0	0
$941$	−9.16842e33	−0.583464	−0.291732	−	0.956500i	$-0.594231\pi$
$941$	−9.16842e33	−0.583464	−0.291732	+	0.956500i	$0.594231\pi$
$942$	0	0
$943$	−1.56119e34	−0.969553
$944$	0	0
$945$	4.70617e32	0.0285234
$946$	0	0
$947$	9.71592e33	0.574724	0.287362	−	0.957822i	$-0.407222\pi$
$947$	9.71592e33	0.574724	0.287362	+	0.957822i	$0.407222\pi$
$948$	0	0
$949$	2.02141e34	1.16706
$950$	0	0
$951$	−5.64864e33	−0.318323
$952$	0	0
$953$	2.43208e34	1.33786	0.668929	−	0.743326i	$-0.266753\pi$
$953$	2.43208e34	1.33786	0.668929	+	0.743326i	$0.266753\pi$
$954$	0	0
$955$	−9.14031e33	−0.490820
$956$	0	0
$957$	−5.81493e33	−0.304830
$958$	0	0
$959$	−9.57996e32	−0.0490286
$960$	0	0
$961$	−1.08620e34	−0.542739
$962$	0	0
$963$	−4.46847e33	−0.218000
$964$	0	0
$965$	9.65009e33	0.459693
$966$	0	0
$967$	−6.14332e33	−0.285758	−0.142879	−	0.989740i	$-0.545636\pi$
$967$	−6.14332e33	−0.285758	−0.142879	+	0.989740i	$0.545636\pi$
$968$	0	0
$969$	1.13672e34	0.516333
$970$	0	0
$971$	−1.48663e34	−0.659451	−0.329725	−	0.944077i	$-0.606956\pi$
$971$	−1.48663e34	−0.659451	−0.329725	+	0.944077i	$0.606956\pi$
$972$	0	0
$973$	−1.19105e34	−0.515981
$974$	0	0
$975$	8.12812e33	0.343904
$976$	0	0
$977$	−3.28586e34	−1.35788	−0.678940	−	0.734194i	$-0.737560\pi$
$977$	−3.28586e34	−1.35788	−0.678940	+	0.734194i	$0.737560\pi$
$978$	0	0
$979$	3.34309e33	0.134942
$980$	0	0
$981$	−1.34872e33	−0.0531774
$982$	0	0
$983$	−1.25289e34	−0.482555	−0.241277	−	0.970456i	$-0.577566\pi$
$983$	−1.25289e34	−0.482555	−0.241277	+	0.970456i	$0.577566\pi$
$984$	0	0
$985$	7.89571e33	0.297081
$986$	0	0
$987$	−8.70126e33	−0.319842
$988$	0	0
$989$	2.05419e34	0.737706
$990$	0	0
$991$	−2.95000e33	−0.103508	−0.0517542	−	0.998660i	$-0.516481\pi$
$991$	−2.95000e33	−0.103508	−0.0517542	+	0.998660i	$0.516481\pi$
$992$	0	0
$993$	−2.95693e34	−1.01374
$994$	0	0
$995$	1.09683e34	0.367428
$996$	0	0
$997$	−4.78570e34	−1.56658	−0.783288	−	0.621659i	$-0.786459\pi$
$997$	−4.78570e34	−1.56658	−0.783288	+	0.621659i	$0.786459\pi$
$998$	0	0
$999$	−1.06555e34	−0.340857

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.24.a.b.1.1		1
4.3	odd	2		6.24.a.b.1.1	✓	1
12.11	even	2		18.24.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.24.a.b.1.1	✓	1	4.3	odd	2
18.24.a.c.1.1		1	12.11	even	2
48.24.a.b.1.1		1	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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