LMFDB - Embedded newform 128.5.d.c.63.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,5,Mod(63,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("128.63");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	128.d (of order $2$, degree $1$, 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:	$13.2313552747$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			63.3
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	128.63
Dual form			128.5.d.c.63.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.79796 q^{3} -8.00000i q^{5} -78.3837i q^{7} +15.0000 q^{9} +O(q^{10})$	$q+9.79796 q^{3} -8.00000i q^{5} -78.3837i q^{7} +15.0000 q^{9} -107.778 q^{11} -216.000i q^{13} -78.3837i q^{15} -162.000 q^{17} +440.908 q^{19} -768.000i q^{21} +705.453i q^{23} +561.000 q^{25} -646.665 q^{27} -1304.00i q^{29} +627.069i q^{31} -1056.00 q^{33} -627.069 q^{35} -1512.00i q^{37} -2116.36i q^{39} +1890.00 q^{41} +2909.99 q^{43} -120.000i q^{45} -1410.91i q^{47} -3743.00 q^{49} -1587.27 q^{51} +1976.00i q^{53} +862.220i q^{55} +4320.00 q^{57} -2263.33 q^{59} +2376.00i q^{61} -1175.76i q^{63} -1728.00 q^{65} -1675.45 q^{67} +6912.00i q^{69} +7759.98i q^{71} +2750.00 q^{73} +5496.65 q^{75} +8448.00i q^{77} +7995.13i q^{79} -7551.00 q^{81} +9337.45 q^{83} +1296.00i q^{85} -12776.5i q^{87} +2430.00 q^{89} -16930.9 q^{91} +6144.00i q^{93} -3527.27i q^{95} +7454.00 q^{97} -1616.66 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 60 q^{9}+O(q^{10}) $ 4 * q + 60 * q^9	$ 4 q + 60 q^{9} - 648 q^{17} + 2244 q^{25} - 4224 q^{33} + 7560 q^{41} - 14972 q^{49} + 17280 q^{57} - 6912 q^{65} + 11000 q^{73} - 30204 q^{81} + 9720 q^{89} + 29816 q^{97}+O(q^{100}) $ 4 * q + 60 * q^9 - 648 * q^17 + 2244 * q^25 - 4224 * q^33 + 7560 * q^41 - 14972 * q^49 + 17280 * q^57 - 6912 * q^65 + 11000 * q^73 - 30204 * q^81 + 9720 * q^89 + 29816 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$.

$n$	$5$	$127$
$\chi(n)$	$-1$	$-1$

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
$3$	9.79796	1.08866	0.544331	−	0.838870i	$-0.316784\pi$
$3$	9.79796	1.08866	0.544331	+	0.838870i	$0.316784\pi$
$4$	0	0
$5$	− 8.00000i	− 0.320000i	−0.987117	−	0.160000i	$-0.948851\pi$
$5$	− 8.00000i	− 0.320000i	0.987117	−	0.160000i	$-0.0511494\pi$
$6$	0	0
$7$	− 78.3837i	− 1.59967i	−0.600222	−	0.799833i	$-0.704921\pi$
$7$	− 78.3837i	− 1.59967i	0.600222	−	0.799833i	$-0.295079\pi$
$8$	0	0
$9$	15.0000	0.185185
$10$	0	0
$11$	−107.778	−0.890724	−0.445362	−	0.895351i	$-0.646925\pi$
$11$	−107.778	−0.890724	−0.445362	+	0.895351i	$0.646925\pi$
$12$	0	0
$13$	− 216.000i	− 1.27811i	−0.769162	−	0.639053i	$-0.779326\pi$
$13$	− 216.000i	− 1.27811i	0.769162	−	0.639053i	$-0.220674\pi$
$14$	0	0
$15$	− 78.3837i	− 0.348372i
$16$	0	0
$17$	−162.000	−0.560554	−0.280277	−	0.959919i	$-0.590426\pi$
$17$	−162.000	−0.560554	−0.280277	+	0.959919i	$0.590426\pi$
$18$	0	0
$19$	440.908	1.22135	0.610676	−	0.791880i	$-0.290898\pi$
$19$	440.908	1.22135	0.610676	+	0.791880i	$0.290898\pi$
$20$	0	0
$21$	− 768.000i	− 1.74150i
$22$	0	0
$23$	705.453i	1.33356i	0.745255	+	0.666780i	$0.232328\pi$
$23$	705.453i	1.33356i	−0.745255	+	0.666780i	$0.767672\pi$
$24$	0	0
$25$	561.000	0.897600
$26$	0	0
$27$	−646.665	−0.887058
$28$	0	0
$29$	− 1304.00i	− 1.55054i	−0.631633	−	0.775268i	$-0.717615\pi$
$29$	− 1304.00i	− 1.55054i	0.631633	−	0.775268i	$-0.282385\pi$
$30$	0	0
$31$	627.069i	0.652518i	0.945280	+	0.326259i	$0.105788\pi$
$31$	627.069i	0.652518i	−0.945280	+	0.326259i	$0.894212\pi$
$32$	0	0
$33$	−1056.00	−0.969697
$34$	0	0
$35$	−627.069	−0.511893
$36$	0	0
$37$	− 1512.00i	− 1.10446i	−0.833693	−	0.552228i	$-0.813778\pi$
$37$	− 1512.00i	− 1.10446i	0.833693	−	0.552228i	$-0.186222\pi$
$38$	0	0
$39$	− 2116.36i	− 1.39143i
$40$	0	0
$41$	1890.00	1.12433	0.562165	−	0.827025i	$-0.309968\pi$
$41$	1890.00	1.12433	0.562165	+	0.827025i	$0.309968\pi$
$42$	0	0
$43$	2909.99	1.57382	0.786910	−	0.617068i	$-0.211679\pi$
$43$	2909.99	1.57382	0.786910	+	0.617068i	$0.211679\pi$
$44$	0	0
$45$	− 120.000i	− 0.0592593i
$46$	0	0
$47$	− 1410.91i	− 0.638708i	−0.947635	−	0.319354i	$-0.896534\pi$
$47$	− 1410.91i	− 0.638708i	0.947635	−	0.319354i	$-0.103466\pi$
$48$	0	0
$49$	−3743.00	−1.55893
$50$	0	0
$51$	−1587.27	−0.610253
$52$	0	0
$53$	1976.00i	0.703453i	0.936103	+	0.351727i	$0.114405\pi$
$53$	1976.00i	0.703453i	−0.936103	+	0.351727i	$0.885595\pi$
$54$	0	0
$55$	862.220i	0.285032i
$56$	0	0
$57$	4320.00	1.32964
$58$	0	0
$59$	−2263.33	−0.650195	−0.325097	−	0.945681i	$-0.605397\pi$
$59$	−2263.33	−0.650195	−0.325097	+	0.945681i	$0.605397\pi$
$60$	0	0
$61$	2376.00i	0.638538i	0.947664	+	0.319269i	$0.103437\pi$
$61$	2376.00i	0.638538i	−0.947664	+	0.319269i	$0.896563\pi$
$62$	0	0
$63$	− 1175.76i	− 0.296235i
$64$	0	0
$65$	−1728.00	−0.408994
$66$	0	0
$67$	−1675.45	−0.373235	−0.186617	−	0.982433i	$-0.559752\pi$
$67$	−1675.45	−0.373235	−0.186617	+	0.982433i	$0.559752\pi$
$68$	0	0
$69$	6912.00i	1.45180i
$70$	0	0
$71$	7759.98i	1.53937i	0.638422	+	0.769687i	$0.279588\pi$
$71$	7759.98i	1.53937i	−0.638422	+	0.769687i	$0.720412\pi$
$72$	0	0
$73$	2750.00	0.516044	0.258022	−	0.966139i	$-0.416929\pi$
$73$	2750.00	0.516044	0.258022	+	0.966139i	$0.416929\pi$
$74$	0	0
$75$	5496.65	0.977183
$76$	0	0
$77$	8448.00i	1.42486i
$78$	0	0
$79$	7995.13i	1.28107i	0.767931	+	0.640533i	$0.221287\pi$
$79$	7995.13i	1.28107i	−0.767931	+	0.640533i	$0.778713\pi$
$80$	0	0
$81$	−7551.00	−1.15089
$82$	0	0
$83$	9337.45	1.35542	0.677708	−	0.735332i	$-0.262974\pi$
$83$	9337.45	1.35542	0.677708	+	0.735332i	$0.262974\pi$
$84$	0	0
$85$	1296.00i	0.179377i
$86$	0	0
$87$	− 12776.5i	− 1.68801i
$88$	0	0
$89$	2430.00	0.306779	0.153390	−	0.988166i	$-0.450981\pi$
$89$	2430.00	0.306779	0.153390	+	0.988166i	$0.450981\pi$
$90$	0	0
$91$	−16930.9	−2.04454
$92$	0	0
$93$	6144.00i	0.710371i
$94$	0	0
$95$	− 3527.27i	− 0.390833i
$96$	0	0
$97$	7454.00	0.792220	0.396110	−	0.918203i	$-0.370360\pi$
$97$	7454.00	0.792220	0.396110	+	0.918203i	$0.370360\pi$
$98$	0	0
$99$	−1616.66	−0.164949
$100$	0	0
$101$	1496.00i	0.146652i	0.997308	+	0.0733261i	$0.0233614\pi$
$101$	1496.00i	0.146652i	−0.997308	+	0.0733261i	$0.976639\pi$
$102$	0	0
$103$	− 2586.66i	− 0.243818i	−0.992541	−	0.121909i	$-0.961098\pi$
$103$	− 2586.66i	− 0.243818i	0.992541	−	0.121909i	$-0.0389015\pi$
$104$	0	0
$105$	−6144.00	−0.557279
$106$	0	0
$107$	1067.98	0.0932813	0.0466406	−	0.998912i	$-0.485148\pi$
$107$	1067.98	0.0932813	0.0466406	+	0.998912i	$0.485148\pi$
$108$	0	0
$109$	− 14904.0i	− 1.25444i	−0.778842	−	0.627220i	$-0.784193\pi$
$109$	− 14904.0i	− 1.25444i	0.778842	−	0.627220i	$-0.215807\pi$
$110$	0	0
$111$	− 14814.5i	− 1.20238i
$112$	0	0
$113$	−702.000	−0.0549769	−0.0274884	−	0.999622i	$-0.508751\pi$
$113$	−702.000	−0.0549769	−0.0274884	+	0.999622i	$0.508751\pi$
$114$	0	0
$115$	5643.62	0.426739
$116$	0	0
$117$	− 3240.00i	− 0.236686i
$118$	0	0
$119$	12698.2i	0.896699i
$120$	0	0
$121$	−3025.00	−0.206612
$122$	0	0
$123$	18518.1	1.22402
$124$	0	0
$125$	− 9488.00i	− 0.607232i
$126$	0	0
$127$	3448.88i	0.213831i	0.994268	+	0.106916i	$0.0340974\pi$
$127$	3448.88i	0.213831i	−0.994268	+	0.106916i	$0.965903\pi$
$128$	0	0
$129$	28512.0	1.71336
$130$	0	0
$131$	−9592.20	−0.558954	−0.279477	−	0.960152i	$-0.590161\pi$
$131$	−9592.20	−0.558954	−0.279477	+	0.960152i	$0.590161\pi$
$132$	0	0
$133$	− 34560.0i	− 1.95376i
$134$	0	0
$135$	5173.32i	0.283859i
$136$	0	0
$137$	−32670.0	−1.74064	−0.870318	−	0.492490i	$-0.836087\pi$
$137$	−32670.0	−1.74064	−0.870318	+	0.492490i	$0.836087\pi$
$138$	0	0
$139$	−1322.72	−0.0684605	−0.0342302	−	0.999414i	$-0.510898\pi$
$139$	−1322.72	−0.0684605	−0.0342302	+	0.999414i	$0.510898\pi$
$140$	0	0
$141$	− 13824.0i	− 0.695337i
$142$	0	0
$143$	23280.0i	1.13844i
$144$	0	0
$145$	−10432.0	−0.496171
$146$	0	0
$147$	−36673.8	−1.69715
$148$	0	0
$149$	− 872.000i	− 0.0392775i	−0.999807	−	0.0196388i	$-0.993748\pi$
$149$	− 872.000i	− 0.0392775i	0.999807	−	0.0196388i	$-0.00625161\pi$
$150$	0	0
$151$	9484.42i	0.415965i	0.978133	+	0.207983i	$0.0666898\pi$
$151$	9484.42i	0.415965i	−0.978133	+	0.207983i	$0.933310\pi$
$152$	0	0
$153$	−2430.00	−0.103806
$154$	0	0
$155$	5016.55	0.208806
$156$	0	0
$157$	− 28728.0i	− 1.16548i	−0.812657	−	0.582742i	$-0.801980\pi$
$157$	− 28728.0i	− 1.16548i	0.812657	−	0.582742i	$-0.198020\pi$
$158$	0	0
$159$	19360.8i	0.765823i
$160$	0	0
$161$	55296.0	2.13325
$162$	0	0
$163$	31128.1	1.17160	0.585798	−	0.810457i	$-0.300781\pi$
$163$	31128.1	1.17160	0.585798	+	0.810457i	$0.300781\pi$
$164$	0	0
$165$	8448.00i	0.310303i
$166$	0	0
$167$	− 54319.9i	− 1.94772i	−0.227155	−	0.973859i	$-0.572942\pi$
$167$	− 54319.9i	− 1.94772i	0.227155	−	0.973859i	$-0.427058\pi$
$168$	0	0
$169$	−18095.0	−0.633556
$170$	0	0
$171$	6613.62	0.226176
$172$	0	0
$173$	25256.0i	0.843864i	0.906628	+	0.421932i	$0.138648\pi$
$173$	25256.0i	0.843864i	−0.906628	+	0.421932i	$0.861352\pi$
$174$	0	0
$175$	− 43973.2i	− 1.43586i
$176$	0	0
$177$	−22176.0	−0.707843
$178$	0	0
$179$	8945.54	0.279190	0.139595	−	0.990209i	$-0.455420\pi$
$179$	8945.54	0.279190	0.139595	+	0.990209i	$0.455420\pi$
$180$	0	0
$181$	− 36072.0i	− 1.10107i	−0.834814	−	0.550533i	$-0.814425\pi$
$181$	− 36072.0i	− 1.10107i	0.834814	−	0.550533i	$-0.185575\pi$
$182$	0	0
$183$	23280.0i	0.695152i
$184$	0	0
$185$	−12096.0	−0.353426
$186$	0	0
$187$	17460.0	0.499298
$188$	0	0
$189$	50688.0i	1.41900i
$190$	0	0
$191$	− 31039.9i	− 0.850852i	−0.904993	−	0.425426i	$-0.860124\pi$
$191$	− 31039.9i	− 0.850852i	0.904993	−	0.425426i	$-0.139876\pi$
$192$	0	0
$193$	41374.0	1.11074	0.555371	−	0.831603i	$-0.312576\pi$
$193$	41374.0	1.11074	0.555371	+	0.831603i	$0.312576\pi$
$194$	0	0
$195$	−16930.9	−0.445256
$196$	0	0
$197$	67640.0i	1.74289i	0.490489	+	0.871447i	$0.336818\pi$
$197$	67640.0i	1.74289i	−0.490489	+	0.871447i	$0.663182\pi$
$198$	0	0
$199$	25004.4i	0.631408i	0.948858	+	0.315704i	$0.102241\pi$
$199$	25004.4i	0.631408i	−0.948858	+	0.315704i	$0.897759\pi$
$200$	0	0
$201$	−16416.0	−0.406327
$202$	0	0
$203$	−102212.	−2.48034
$204$	0	0
$205$	− 15120.0i	− 0.359786i
$206$	0	0
$207$	10581.8i	0.246955i
$208$	0	0
$209$	−47520.0	−1.08789
$210$	0	0
$211$	26189.9	0.588260	0.294130	−	0.955765i	$-0.404970\pi$
$211$	26189.9	0.588260	0.294130	+	0.955765i	$0.404970\pi$
$212$	0	0
$213$	76032.0i	1.67586i
$214$	0	0
$215$	− 23280.0i	− 0.503623i
$216$	0	0
$217$	49152.0	1.04381
$218$	0	0
$219$	26944.4	0.561798
$220$	0	0
$221$	34992.0i	0.716447i
$222$	0	0
$223$	20693.3i	0.416121i	0.978116	+	0.208061i	$0.0667151\pi$
$223$	20693.3i	0.416121i	−0.978116	+	0.208061i	$0.933285\pi$
$224$	0	0
$225$	8415.00	0.166222
$226$	0	0
$227$	77923.2	1.51222	0.756110	−	0.654445i	$-0.227098\pi$
$227$	77923.2	1.51222	0.756110	+	0.654445i	$0.227098\pi$
$228$	0	0
$229$	− 14472.0i	− 0.275967i	−0.990435	−	0.137984i	$-0.955938\pi$
$229$	− 14472.0i	− 0.275967i	0.990435	−	0.137984i	$-0.0440621\pi$
$230$	0	0
$231$	82773.2i	1.55119i
$232$	0	0
$233$	−2754.00	−0.0507285	−0.0253643	−	0.999678i	$-0.508075\pi$
$233$	−2754.00	−0.0507285	−0.0253643	+	0.999678i	$0.508075\pi$
$234$	0	0
$235$	−11287.2	−0.204387
$236$	0	0
$237$	78336.0i	1.39465i
$238$	0	0
$239$	91708.9i	1.60552i	0.596302	+	0.802760i	$0.296636\pi$
$239$	91708.9i	1.60552i	−0.596302	+	0.802760i	$0.703364\pi$
$240$	0	0
$241$	−97570.0	−1.67990	−0.839948	−	0.542667i	$-0.817414\pi$
$241$	−97570.0	−1.67990	−0.839948	+	0.542667i	$0.817414\pi$
$242$	0	0
$243$	−21604.5	−0.365874
$244$	0	0
$245$	29944.0i	0.498859i
$246$	0	0
$247$	− 95236.2i	− 1.56102i
$248$	0	0
$249$	91488.0	1.47559
$250$	0	0
$251$	67811.7	1.07636	0.538179	−	0.842830i	$-0.319112\pi$
$251$	67811.7	1.07636	0.538179	+	0.842830i	$0.319112\pi$
$252$	0	0
$253$	− 76032.0i	− 1.18783i
$254$	0	0
$255$	12698.2i	0.195281i
$256$	0	0
$257$	80514.0	1.21900	0.609502	−	0.792785i	$-0.291369\pi$
$257$	80514.0	1.21900	0.609502	+	0.792785i	$0.291369\pi$
$258$	0	0
$259$	−118516.	−1.76676
$260$	0	0
$261$	− 19560.0i	− 0.287136i
$262$	0	0
$263$	− 14814.5i	− 0.214179i	−0.994249	−	0.107089i	$-0.965847\pi$
$263$	− 14814.5i	− 0.214179i	0.994249	−	0.107089i	$-0.0341531\pi$
$264$	0	0
$265$	15808.0	0.225105
$266$	0	0
$267$	23809.0	0.333979
$268$	0	0
$269$	44872.0i	0.620113i	0.950718	+	0.310057i	$0.100348\pi$
$269$	44872.0i	0.620113i	−0.950718	+	0.310057i	$0.899652\pi$
$270$	0	0
$271$	− 91395.4i	− 1.24447i	−0.782829	−	0.622237i	$-0.786224\pi$
$271$	− 91395.4i	− 1.24447i	0.782829	−	0.622237i	$-0.213776\pi$
$272$	0	0
$273$	−165888.	−2.22582
$274$	0	0
$275$	−60463.2	−0.799513
$276$	0	0
$277$	− 90504.0i	− 1.17953i	−0.807576	−	0.589764i	$-0.799221\pi$
$277$	− 90504.0i	− 1.17953i	0.807576	−	0.589764i	$-0.200779\pi$
$278$	0	0
$279$	9406.04i	0.120837i
$280$	0	0
$281$	23166.0	0.293385	0.146693	−	0.989182i	$-0.453137\pi$
$281$	23166.0	0.293385	0.146693	+	0.989182i	$0.453137\pi$
$282$	0	0
$283$	−70809.8	−0.884140	−0.442070	−	0.896981i	$-0.645756\pi$
$283$	−70809.8	−0.884140	−0.442070	+	0.896981i	$0.645756\pi$
$284$	0	0
$285$	− 34560.0i	− 0.425485i
$286$	0	0
$287$	− 148145.i	− 1.79855i
$288$	0	0
$289$	−57277.0	−0.685780
$290$	0	0
$291$	73034.0	0.862460
$292$	0	0
$293$	− 37768.0i	− 0.439935i	−0.975507	−	0.219968i	$-0.929405\pi$
$293$	− 37768.0i	− 0.439935i	0.975507	−	0.219968i	$-0.0705952\pi$
$294$	0	0
$295$	18106.6i	0.208062i
$296$	0	0
$297$	69696.0	0.790123
$298$	0	0
$299$	152378.	1.70443
$300$	0	0
$301$	− 228096.i	− 2.51759i
$302$	0	0
$303$	14657.7i	0.159655i
$304$	0	0
$305$	19008.0	0.204332
$306$	0	0
$307$	38535.4	0.408868	0.204434	−	0.978880i	$-0.434465\pi$
$307$	38535.4	0.408868	0.204434	+	0.978880i	$0.434465\pi$
$308$	0	0
$309$	− 25344.0i	− 0.265435i
$310$	0	0
$311$	85359.8i	0.882537i	0.897375	+	0.441268i	$0.145471\pi$
$311$	85359.8i	0.882537i	−0.897375	+	0.441268i	$0.854529\pi$
$312$	0	0
$313$	93602.0	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$313$	93602.0	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$314$	0	0
$315$	−9406.04	−0.0947951
$316$	0	0
$317$	49192.0i	0.489526i	0.969583	+	0.244763i	$0.0787102\pi$
$317$	49192.0i	0.489526i	−0.969583	+	0.244763i	$0.921290\pi$
$318$	0	0
$319$	140542.i	1.38110i
$320$	0	0
$321$	10464.0	0.101552
$322$	0	0
$323$	−71427.1	−0.684633
$324$	0	0
$325$	− 121176.i	− 1.14723i
$326$	0	0
$327$	− 146029.i	− 1.36566i
$328$	0	0
$329$	−110592.	−1.02172
$330$	0	0
$331$	−16490.0	−0.150509	−0.0752547	−	0.997164i	$-0.523977\pi$
$331$	−16490.0	−0.150509	−0.0752547	+	0.997164i	$0.523977\pi$
$332$	0	0
$333$	− 22680.0i	− 0.204529i
$334$	0	0
$335$	13403.6i	0.119435i
$336$	0	0
$337$	−93758.0	−0.825560	−0.412780	−	0.910831i	$-0.635442\pi$
$337$	−93758.0	−0.825560	−0.412780	+	0.910831i	$0.635442\pi$
$338$	0	0
$339$	−6878.17	−0.0598513
$340$	0	0
$341$	− 67584.0i	− 0.581213i
$342$	0	0
$343$	105191.i	0.894108i
$344$	0	0
$345$	55296.0	0.464575
$346$	0	0
$347$	−220405.	−1.83047	−0.915235	−	0.402920i	$-0.867995\pi$
$347$	−220405.	−1.83047	−0.915235	+	0.402920i	$0.867995\pi$
$348$	0	0
$349$	− 48600.0i	− 0.399012i	−0.979897	−	0.199506i	$-0.936066\pi$
$349$	− 48600.0i	− 0.399012i	0.979897	−	0.199506i	$-0.0639337\pi$
$350$	0	0
$351$	139680.i	1.13375i
$352$	0	0
$353$	108162.	0.868011	0.434006	−	0.900910i	$-0.357100\pi$
$353$	108162.	0.868011	0.434006	+	0.900910i	$0.357100\pi$
$354$	0	0
$355$	62079.9	0.492600
$356$	0	0
$357$	124416.i	0.976202i
$358$	0	0
$359$	50087.2i	0.388631i	0.980939	+	0.194316i	$0.0622486\pi$
$359$	50087.2i	0.388631i	−0.980939	+	0.194316i	$0.937751\pi$
$360$	0	0
$361$	64079.0	0.491701
$362$	0	0
$363$	−29638.8	−0.224930
$364$	0	0
$365$	− 22000.0i	− 0.165134i
$366$	0	0
$367$	61296.0i	0.455093i	0.973767	+	0.227547i	$0.0730704\pi$
$367$	61296.0i	0.455093i	−0.973767	+	0.227547i	$0.926930\pi$
$368$	0	0
$369$	28350.0	0.208209
$370$	0	0
$371$	154886.	1.12529
$372$	0	0
$373$	54648.0i	0.392787i	0.980525	+	0.196393i	$0.0629229\pi$
$373$	54648.0i	0.392787i	−0.980525	+	0.196393i	$0.937077\pi$
$374$	0	0
$375$	− 92963.0i	− 0.661070i
$376$	0	0
$377$	−281664.	−1.98175
$378$	0	0
$379$	103790.	0.722564	0.361282	−	0.932457i	$-0.382339\pi$
$379$	103790.	0.722564	0.361282	+	0.932457i	$0.382339\pi$
$380$	0	0
$381$	33792.0i	0.232790i
$382$	0	0
$383$	− 124160.i	− 0.846415i	−0.906033	−	0.423207i	$-0.860904\pi$
$383$	− 124160.i	− 0.846415i	0.906033	−	0.423207i	$-0.139096\pi$
$384$	0	0
$385$	67584.0	0.455955
$386$	0	0
$387$	43649.9	0.291448
$388$	0	0
$389$	129112.i	0.853233i	0.904433	+	0.426616i	$0.140295\pi$
$389$	129112.i	0.853233i	−0.904433	+	0.426616i	$0.859705\pi$
$390$	0	0
$391$	− 114283.i	− 0.747532i
$392$	0	0
$393$	−93984.0	−0.608512
$394$	0	0
$395$	63961.1	0.409941
$396$	0	0
$397$	232200.i	1.47327i	0.676293	+	0.736633i	$0.263585\pi$
$397$	232200.i	1.47327i	−0.676293	+	0.736633i	$0.736415\pi$
$398$	0	0
$399$	− 338617.i	− 2.12698i
$400$	0	0
$401$	151902.	0.944658	0.472329	−	0.881422i	$-0.343413\pi$
$401$	151902.	0.944658	0.472329	+	0.881422i	$0.343413\pi$
$402$	0	0
$403$	135447.	0.833987
$404$	0	0
$405$	60408.0i	0.368285i
$406$	0	0
$407$	162960.i	0.983765i
$408$	0	0
$409$	28450.0	0.170073	0.0850366	−	0.996378i	$-0.472899\pi$
$409$	28450.0	0.170073	0.0850366	+	0.996378i	$0.472899\pi$
$410$	0	0
$411$	−320099.	−1.89496
$412$	0	0
$413$	177408.i	1.04010i
$414$	0	0
$415$	− 74699.6i	− 0.433733i
$416$	0	0
$417$	−12960.0	−0.0745303
$418$	0	0
$419$	−291382.	−1.65972	−0.829858	−	0.557974i	$-0.811579\pi$
$419$	−291382.	−1.65972	−0.829858	+	0.557974i	$0.811579\pi$
$420$	0	0
$421$	− 119880.i	− 0.676367i	−0.941080	−	0.338184i	$-0.890188\pi$
$421$	− 119880.i	− 0.676367i	0.941080	−	0.338184i	$-0.109812\pi$
$422$	0	0
$423$	− 21163.6i	− 0.118279i
$424$	0	0
$425$	−90882.0	−0.503153
$426$	0	0
$427$	186240.	1.02145
$428$	0	0
$429$	228096.i	1.23938i
$430$	0	0
$431$	170720.i	0.919028i	0.888170	+	0.459514i	$0.151976\pi$
$431$	170720.i	0.919028i	−0.888170	+	0.459514i	$0.848024\pi$
$432$	0	0
$433$	215518.	1.14950	0.574748	−	0.818330i	$-0.305100\pi$
$433$	215518.	1.14950	0.574748	+	0.818330i	$0.305100\pi$
$434$	0	0
$435$	−102212.	−0.540163
$436$	0	0
$437$	311040.i	1.62875i
$438$	0	0
$439$	− 75169.9i	− 0.390045i	−0.980799	−	0.195023i	$-0.937522\pi$
$439$	− 75169.9i	− 0.390045i	0.980799	−	0.195023i	$-0.0624781\pi$
$440$	0	0
$441$	−56145.0	−0.288691
$442$	0	0
$443$	185270.	0.944054	0.472027	−	0.881584i	$-0.343523\pi$
$443$	185270.	0.944054	0.472027	+	0.881584i	$0.343523\pi$
$444$	0	0
$445$	− 19440.0i	− 0.0981694i
$446$	0	0
$447$	− 8543.82i	− 0.0427599i
$448$	0	0
$449$	177822.	0.882049	0.441025	−	0.897495i	$-0.354615\pi$
$449$	177822.	0.882049	0.441025	+	0.897495i	$0.354615\pi$
$450$	0	0
$451$	−203700.	−1.00147
$452$	0	0
$453$	92928.0i	0.452846i
$454$	0	0
$455$	135447.i	0.654254i
$456$	0	0
$457$	−203294.	−0.973402	−0.486701	−	0.873569i	$-0.661800\pi$
$457$	−203294.	−0.973402	−0.486701	+	0.873569i	$0.661800\pi$
$458$	0	0
$459$	104760.	0.497244
$460$	0	0
$461$	− 286648.i	− 1.34880i	−0.738367	−	0.674399i	$-0.764403\pi$
$461$	− 286648.i	− 1.34880i	0.738367	−	0.674399i	$-0.235597\pi$
$462$	0	0
$463$	332817.i	1.55254i	0.630399	+	0.776271i	$0.282891\pi$
$463$	332817.i	1.55254i	−0.630399	+	0.776271i	$0.717109\pi$
$464$	0	0
$465$	49152.0	0.227319
$466$	0	0
$467$	−58738.8	−0.269334	−0.134667	−	0.990891i	$-0.542996\pi$
$467$	−58738.8	−0.269334	−0.134667	+	0.990891i	$0.542996\pi$
$468$	0	0
$469$	131328.i	0.597051i
$470$	0	0
$471$	− 281476.i	− 1.26882i
$472$	0	0
$473$	−313632.	−1.40184
$474$	0	0
$475$	247349.	1.09629
$476$	0	0
$477$	29640.0i	0.130269i
$478$	0	0
$479$	186240.i	0.811710i	0.913937	+	0.405855i	$0.133026\pi$
$479$	186240.i	0.811710i	−0.913937	+	0.405855i	$0.866974\pi$
$480$	0	0
$481$	−326592.	−1.41161
$482$	0	0
$483$	541788.	2.32239
$484$	0	0
$485$	− 59632.0i	− 0.253510i
$486$	0	0
$487$	363308.i	1.53185i	0.642927	+	0.765927i	$0.277720\pi$
$487$	363308.i	1.53185i	−0.642927	+	0.765927i	$0.722280\pi$
$488$	0	0
$489$	304992.	1.27547
$490$	0	0
$491$	−107533.	−0.446043	−0.223022	−	0.974813i	$-0.571592\pi$
$491$	−107533.	−0.446043	−0.223022	+	0.974813i	$0.571592\pi$
$492$	0	0
$493$	211248.i	0.869158i
$494$	0	0
$495$	12933.3i	0.0527836i
$496$	0	0
$497$	608256.	2.46249
$498$	0	0
$499$	−385089.	−1.54654	−0.773268	−	0.634079i	$-0.781379\pi$
$499$	−385089.	−1.54654	−0.773268	+	0.634079i	$0.781379\pi$
$500$	0	0
$501$	− 532224.i	− 2.12041i
$502$	0	0
$503$	− 210930.i	− 0.833688i	−0.908978	−	0.416844i	$-0.863136\pi$
$503$	− 210930.i	− 0.833688i	0.908978	−	0.416844i	$-0.136864\pi$
$504$	0	0
$505$	11968.0	0.0469287
$506$	0	0
$507$	−177294.	−0.689729
$508$	0	0
$509$	− 171512.i	− 0.662001i	−0.943630	−	0.331001i	$-0.892614\pi$
$509$	− 171512.i	− 0.662001i	0.943630	−	0.331001i	$-0.107386\pi$
$510$	0	0
$511$	− 215555.i	− 0.825499i
$512$	0	0
$513$	−285120.	−1.08341
$514$	0	0
$515$	−20693.3	−0.0780216
$516$	0	0
$517$	152064.i	0.568912i
$518$	0	0
$519$	247457.i	0.918683i
$520$	0	0
$521$	−184734.	−0.680568	−0.340284	−	0.940323i	$-0.610523\pi$
$521$	−184734.	−0.680568	−0.340284	+	0.940323i	$0.610523\pi$
$522$	0	0
$523$	−420009.	−1.53552	−0.767760	−	0.640738i	$-0.778629\pi$
$523$	−420009.	−1.53552	−0.767760	+	0.640738i	$0.778629\pi$
$524$	0	0
$525$	− 430848.i	− 1.56317i
$526$	0	0
$527$	− 101585.i	− 0.365771i
$528$	0	0
$529$	−217823.	−0.778381
$530$	0	0
$531$	−33949.9	−0.120406
$532$	0	0
$533$	− 408240.i	− 1.43701i
$534$	0	0
$535$	− 8543.82i	− 0.0298500i
$536$	0	0
$537$	87648.0	0.303944
$538$	0	0
$539$	403411.	1.38858
$540$	0	0
$541$	− 230904.i	− 0.788927i	−0.918912	−	0.394464i	$-0.870930\pi$
$541$	− 230904.i	− 0.788927i	0.918912	−	0.394464i	$-0.129070\pi$
$542$	0	0
$543$	− 353432.i	− 1.19869i
$544$	0	0
$545$	−119232.	−0.401421
$546$	0	0
$547$	−39769.9	−0.132917	−0.0664584	−	0.997789i	$-0.521170\pi$
$547$	−39769.9	−0.132917	−0.0664584	+	0.997789i	$0.521170\pi$
$548$	0	0
$549$	35640.0i	0.118248i
$550$	0	0
$551$	− 574944.i	− 1.89375i
$552$	0	0
$553$	626688.	2.04928
$554$	0	0
$555$	−118516.	−0.384761
$556$	0	0
$557$	23144.0i	0.0745981i	0.999304	+	0.0372991i	$0.0118754\pi$
$557$	23144.0i	0.0745981i	−0.999304	+	0.0372991i	$0.988125\pi$
$558$	0	0
$559$	− 628559.i	− 2.01151i
$560$	0	0
$561$	171072.	0.543567
$562$	0	0
$563$	491123.	1.54943	0.774717	−	0.632308i	$-0.217892\pi$
$563$	491123.	1.54943	0.774717	+	0.632308i	$0.217892\pi$
$564$	0	0
$565$	5616.00i	0.0175926i
$566$	0	0
$567$	591875.i	1.84104i
$568$	0	0
$569$	−127710.	−0.394458	−0.197229	−	0.980357i	$-0.563194\pi$
$569$	−127710.	−0.394458	−0.197229	+	0.980357i	$0.563194\pi$
$570$	0	0
$571$	−39064.5	−0.119815	−0.0599073	−	0.998204i	$-0.519081\pi$
$571$	−39064.5	−0.119815	−0.0599073	+	0.998204i	$0.519081\pi$
$572$	0	0
$573$	− 304128.i	− 0.926290i
$574$	0	0
$575$	395759.i	1.19700i
$576$	0	0
$577$	−270718.	−0.813140	−0.406570	−	0.913620i	$-0.633275\pi$
$577$	−270718.	−0.813140	−0.406570	+	0.913620i	$0.633275\pi$
$578$	0	0
$579$	405381.	1.20922
$580$	0	0
$581$	− 731904.i	− 2.16821i
$582$	0	0
$583$	− 212968.i	− 0.626582i
$584$	0	0
$585$	−25920.0	−0.0757396
$586$	0	0
$587$	−273353.	−0.793319	−0.396660	−	0.917966i	$-0.629831\pi$
$587$	−273353.	−0.793319	−0.396660	+	0.917966i	$0.629831\pi$
$588$	0	0
$589$	276480.i	0.796954i
$590$	0	0
$591$	662734.i	1.89742i
$592$	0	0
$593$	−62910.0	−0.178900	−0.0894500	−	0.995991i	$-0.528511\pi$
$593$	−62910.0	−0.178900	−0.0894500	+	0.995991i	$0.528511\pi$
$594$	0	0
$595$	101585.	0.286944
$596$	0	0
$597$	244992.i	0.687390i
$598$	0	0
$599$	− 261723.i	− 0.729438i	−0.931118	−	0.364719i	$-0.881165\pi$
$599$	− 261723.i	− 0.729438i	0.931118	−	0.364719i	$-0.118835\pi$
$600$	0	0
$601$	203902.	0.564511	0.282256	−	0.959339i	$-0.408917\pi$
$601$	203902.	0.564511	0.282256	+	0.959339i	$0.408917\pi$
$602$	0	0
$603$	−25131.8	−0.0691176
$604$	0	0
$605$	24200.0i	0.0661157i
$606$	0	0
$607$	369657.i	1.00328i	0.865077	+	0.501640i	$0.167270\pi$
$607$	369657.i	1.00328i	−0.865077	+	0.501640i	$0.832730\pi$
$608$	0	0
$609$	−1.00147e6	−2.70025
$610$	0	0
$611$	−304756.	−0.816337
$612$	0	0
$613$	222264.i	0.591491i	0.955267	+	0.295746i	$0.0955680\pi$
$613$	222264.i	0.591491i	−0.955267	+	0.295746i	$0.904432\pi$
$614$	0	0
$615$	− 148145.i	− 0.391685i
$616$	0	0
$617$	363582.	0.955063	0.477532	−	0.878615i	$-0.341532\pi$
$617$	363582.	0.955063	0.477532	+	0.878615i	$0.341532\pi$
$618$	0	0
$619$	−317718.	−0.829203	−0.414602	−	0.910003i	$-0.636079\pi$
$619$	−317718.	−0.829203	−0.414602	+	0.910003i	$0.636079\pi$
$620$	0	0
$621$	− 456192.i	− 1.18294i
$622$	0	0
$623$	− 190472.i	− 0.490745i
$624$	0	0
$625$	274721.	0.703286
$626$	0	0
$627$	−465599.	−1.18434
$628$	0	0
$629$	244944.i	0.619107i
$630$	0	0
$631$	348415.i	0.875062i	0.899203	+	0.437531i	$0.144147\pi$
$631$	348415.i	0.875062i	−0.899203	+	0.437531i	$0.855853\pi$
$632$	0	0
$633$	256608.	0.640417
$634$	0	0
$635$	27591.1	0.0684259
$636$	0	0
$637$	808488.i	1.99248i
$638$	0	0
$639$	116400.i	0.285069i
$640$	0	0
$641$	67230.0	0.163624	0.0818120	−	0.996648i	$-0.473929\pi$
$641$	67230.0	0.163624	0.0818120	+	0.996648i	$0.473929\pi$
$642$	0	0
$643$	74513.5	0.180224	0.0901121	−	0.995932i	$-0.471277\pi$
$643$	74513.5	0.180224	0.0901121	+	0.995932i	$0.471277\pi$
$644$	0	0
$645$	− 228096.i	− 0.548275i
$646$	0	0
$647$	380239.i	0.908340i	0.890915	+	0.454170i	$0.150064\pi$
$647$	380239.i	0.908340i	−0.890915	+	0.454170i	$0.849936\pi$
$648$	0	0
$649$	243936.	0.579144
$650$	0	0
$651$	481589.	1.13636
$652$	0	0
$653$	89288.0i	0.209395i	0.994504	+	0.104698i	$0.0333875\pi$
$653$	89288.0i	0.209395i	−0.994504	+	0.104698i	$0.966613\pi$
$654$	0	0
$655$	76737.6i	0.178865i
$656$	0	0
$657$	41250.0	0.0955638
$658$	0	0
$659$	−223149.	−0.513834	−0.256917	−	0.966433i	$-0.582707\pi$
$659$	−223149.	−0.513834	−0.256917	+	0.966433i	$0.582707\pi$
$660$	0	0
$661$	560088.i	1.28190i	0.767584	+	0.640949i	$0.221459\pi$
$661$	560088.i	1.28190i	−0.767584	+	0.640949i	$0.778541\pi$
$662$	0	0
$663$	342850.i	0.779969i
$664$	0	0
$665$	−276480.	−0.625202
$666$	0	0
$667$	919911.	2.06773
$668$	0	0
$669$	202752.i	0.453015i
$670$	0	0
$671$	− 256079.i	− 0.568761i
$672$	0	0
$673$	−810722.	−1.78995	−0.894977	−	0.446112i	$-0.852808\pi$
$673$	−810722.	−1.78995	−0.894977	+	0.446112i	$0.852808\pi$
$674$	0	0
$675$	−362779.	−0.796223
$676$	0	0
$677$	655768.i	1.43078i	0.698725	+	0.715390i	$0.253751\pi$
$677$	655768.i	1.43078i	−0.698725	+	0.715390i	$0.746249\pi$
$678$	0	0
$679$	− 584272.i	− 1.26729i
$680$	0	0
$681$	763488.	1.64630
$682$	0	0
$683$	−189208.	−0.405601	−0.202800	−	0.979220i	$-0.565004\pi$
$683$	−189208.	−0.405601	−0.202800	+	0.979220i	$0.565004\pi$
$684$	0	0
$685$	261360.i	0.557004i
$686$	0	0
$687$	− 141796.i	− 0.300435i
$688$	0	0
$689$	426816.	0.899088
$690$	0	0
$691$	569389.	1.19248	0.596242	−	0.802804i	$-0.296660\pi$
$691$	569389.	1.19248	0.596242	+	0.802804i	$0.296660\pi$
$692$	0	0
$693$	126720.i	0.263863i
$694$	0	0
$695$	10581.8i	0.0219073i
$696$	0	0
$697$	−306180.	−0.630248
$698$	0	0
$699$	−26983.6	−0.0552262
$700$	0	0
$701$	− 519224.i	− 1.05662i	−0.849052	−	0.528310i	$-0.822826\pi$
$701$	− 519224.i	− 1.05662i	0.849052	−	0.528310i	$-0.177174\pi$
$702$	0	0
$703$	− 666653.i	− 1.34893i
$704$	0	0
$705$	−110592.	−0.222508
$706$	0	0
$707$	117262.	0.234595
$708$	0	0
$709$	919512.i	1.82922i	0.404342	+	0.914608i	$0.367501\pi$
$709$	919512.i	1.82922i	−0.404342	+	0.914608i	$0.632499\pi$
$710$	0	0
$711$	119927.i	0.237234i
$712$	0	0
$713$	−442368.	−0.870171
$714$	0	0
$715$	186240.	0.364301
$716$	0	0
$717$	898560.i	1.74787i
$718$	0	0
$719$	− 656071.i	− 1.26909i	−0.772885	−	0.634546i	$-0.781187\pi$
$719$	− 656071.i	− 1.26909i	0.772885	−	0.634546i	$-0.218813\pi$
$720$	0	0
$721$	−202752.	−0.390027
$722$	0	0
$723$	−955987.	−1.82884
$724$	0	0
$725$	− 731544.i	− 1.39176i
$726$	0	0
$727$	− 488722.i	− 0.924684i	−0.886702	−	0.462342i	$-0.847009\pi$
$727$	− 488722.i	− 0.924684i	0.886702	−	0.462342i	$-0.152991\pi$
$728$	0	0
$729$	399951.	0.752578
$730$	0	0
$731$	−471419.	−0.882211
$732$	0	0
$733$	706536.i	1.31500i	0.753454	+	0.657501i	$0.228386\pi$
$733$	706536.i	1.31500i	−0.753454	+	0.657501i	$0.771614\pi$
$734$	0	0
$735$	293390.i	0.543089i
$736$	0	0
$737$	180576.	0.332449
$738$	0	0
$739$	−172748.	−0.316318	−0.158159	−	0.987414i	$-0.550556\pi$
$739$	−172748.	−0.316318	−0.158159	+	0.987414i	$0.550556\pi$
$740$	0	0
$741$	− 933120.i	− 1.69942i
$742$	0	0
$743$	250436.i	0.453648i	0.973936	+	0.226824i	$0.0728342\pi$
$743$	250436.i	0.453648i	−0.973936	+	0.226824i	$0.927166\pi$
$744$	0	0
$745$	−6976.00	−0.0125688
$746$	0	0
$747$	140062.	0.251003
$748$	0	0
$749$	− 83712.0i	− 0.149219i
$750$	0	0
$751$	536301.i	0.950887i	0.879746	+	0.475443i	$0.157712\pi$
$751$	536301.i	0.950887i	−0.879746	+	0.475443i	$0.842288\pi$
$752$	0	0
$753$	664416.	1.17179
$754$	0	0
$755$	75875.4	0.133109
$756$	0	0
$757$	470232.i	0.820579i	0.911955	+	0.410290i	$0.134572\pi$
$757$	470232.i	0.820579i	−0.911955	+	0.410290i	$0.865428\pi$
$758$	0	0
$759$	− 744958.i	− 1.29315i
$760$	0	0
$761$	−691038.	−1.19325	−0.596627	−	0.802519i	$-0.703493\pi$
$761$	−691038.	−1.19325	−0.596627	+	0.802519i	$0.703493\pi$
$762$	0	0
$763$	−1.16823e6	−2.00669
$764$	0	0
$765$	19440.0i	0.0332180i
$766$	0	0
$767$	488879.i	0.831018i
$768$	0	0
$769$	304030.	0.514119	0.257060	−	0.966396i	$-0.417246\pi$
$769$	304030.	0.514119	0.257060	+	0.966396i	$0.417246\pi$
$770$	0	0
$771$	788873.	1.32708
$772$	0	0
$773$	533720.i	0.893212i	0.894731	+	0.446606i	$0.147367\pi$
$773$	533720.i	0.893212i	−0.894731	+	0.446606i	$0.852633\pi$
$774$	0	0
$775$	351786.i	0.585700i
$776$	0	0
$777$	−1.16122e6	−1.92341
$778$	0	0
$779$	833316.	1.37320
$780$	0	0
$781$	− 836352.i	− 1.37116i
$782$	0	0
$783$	843252.i	1.37541i
$784$	0	0
$785$	−229824.	−0.372955
$786$	0	0
$787$	1.17467e6	1.89656	0.948278	−	0.317442i	$-0.102824\pi$
$787$	1.17467e6	1.89656	0.948278	+	0.317442i	$0.102824\pi$
$788$	0	0
$789$	− 145152.i	− 0.233168i
$790$	0	0
$791$	55025.3i	0.0879447i
$792$	0	0
$793$	513216.	0.816120
$794$	0	0
$795$	154886.	0.245063
$796$	0	0
$797$	− 552376.i	− 0.869597i	−0.900528	−	0.434799i	$-0.856820\pi$
$797$	− 552376.i	− 0.869597i	0.900528	−	0.434799i	$-0.143180\pi$
$798$	0	0
$799$	228567.i	0.358030i
$800$	0	0
$801$	36450.0	0.0568110
$802$	0	0
$803$	−296388.	−0.459653
$804$	0	0
$805$	− 442368.i	− 0.682640i
$806$	0	0
$807$	439654.i	0.675094i
$808$	0	0
$809$	803682.	1.22797	0.613984	−	0.789318i	$-0.289566\pi$
$809$	803682.	1.22797	0.613984	+	0.789318i	$0.289566\pi$
$810$	0	0
$811$	68869.9	0.104710	0.0523549	−	0.998629i	$-0.483327\pi$
$811$	68869.9	0.104710	0.0523549	+	0.998629i	$0.483327\pi$
$812$	0	0
$813$	− 895488.i	− 1.35481i
$814$	0	0
$815$	− 249025.i	− 0.374910i
$816$	0	0
$817$	1.28304e6	1.92219
$818$	0	0
$819$	−253963.	−0.378619
$820$	0	0
$821$	− 867592.i	− 1.28715i	−0.765383	−	0.643575i	$-0.777450\pi$
$821$	− 867592.i	− 1.28715i	0.765383	−	0.643575i	$-0.222550\pi$
$822$	0	0
$823$	30177.7i	0.0445540i	0.999752	+	0.0222770i	$0.00709158\pi$
$823$	30177.7i	0.0445540i	−0.999752	+	0.0222770i	$0.992908\pi$
$824$	0	0
$825$	−592416.	−0.870400
$826$	0	0
$827$	−864680.	−1.26428	−0.632141	−	0.774853i	$-0.717824\pi$
$827$	−864680.	−1.26428	−0.632141	+	0.774853i	$0.717824\pi$
$828$	0	0
$829$	449928.i	0.654687i	0.944905	+	0.327344i	$0.106153\pi$
$829$	449928.i	0.654687i	−0.944905	+	0.327344i	$0.893847\pi$
$830$	0	0
$831$	− 886754.i	− 1.28411i
$832$	0	0
$833$	606366.	0.873866
$834$	0	0
$835$	−434559.	−0.623269
$836$	0	0
$837$	− 405504.i	− 0.578821i
$838$	0	0
$839$	800689.i	1.13747i	0.822521	+	0.568735i	$0.192567\pi$
$839$	800689.i	1.13747i	−0.822521	+	0.568735i	$0.807433\pi$
$840$	0	0
$841$	−993135.	−1.40416
$842$	0	0
$843$	226980.	0.319398
$844$	0	0
$845$	144760.i	0.202738i
$846$	0	0
$847$	237111.i	0.330510i
$848$	0	0
$849$	−693792.	−0.962529
$850$	0	0
$851$	1.06665e6	1.47286
$852$	0	0
$853$	− 121608.i	− 0.167134i	−0.996502	−	0.0835669i	$-0.973369\pi$
$853$	− 121608.i	− 0.167134i	0.996502	−	0.0835669i	$-0.0266312\pi$
$854$	0	0
$855$	− 52909.0i	− 0.0723764i
$856$	0	0
$857$	−1.26473e6	−1.72202	−0.861009	−	0.508590i	$-0.830167\pi$
$857$	−1.26473e6	−1.72202	−0.861009	+	0.508590i	$0.830167\pi$
$858$	0	0
$859$	−967088.	−1.31063	−0.655314	−	0.755356i	$-0.727464\pi$
$859$	−967088.	−1.31063	−0.655314	+	0.755356i	$0.727464\pi$
$860$	0	0
$861$	− 1.45152e6i	− 1.95802i
$862$	0	0
$863$	− 428915.i	− 0.575904i	−0.957645	−	0.287952i	$-0.907026\pi$
$863$	− 428915.i	− 0.575904i	0.957645	−	0.287952i	$-0.0929744\pi$
$864$	0	0
$865$	202048.	0.270036
$866$	0	0
$867$	−561198.	−0.746582
$868$	0	0
$869$	− 861696.i	− 1.14108i
$870$	0	0
$871$	361897.i	0.477034i
$872$	0	0
$873$	111810.	0.146707
$874$	0	0
$875$	−743704.	−0.971369
$876$	0	0
$877$	519048.i	0.674852i	0.941352	+	0.337426i	$0.109556\pi$
$877$	519048.i	0.674852i	−0.941352	+	0.337426i	$0.890444\pi$
$878$	0	0
$879$	− 370049.i	− 0.478941i
$880$	0	0
$881$	434754.	0.560134	0.280067	−	0.959980i	$-0.409643\pi$
$881$	434754.	0.560134	0.280067	+	0.959980i	$0.409643\pi$
$882$	0	0
$883$	−474329.	−0.608357	−0.304178	−	0.952615i	$-0.598382\pi$
$883$	−474329.	−0.608357	−0.304178	+	0.952615i	$0.598382\pi$
$884$	0	0
$885$	177408.i	0.226510i
$886$	0	0
$887$	− 298407.i	− 0.379281i	−0.981854	−	0.189641i	$-0.939268\pi$
$887$	− 298407.i	− 0.379281i	0.981854	−	0.189641i	$-0.0607323\pi$
$888$	0	0
$889$	270336.	0.342058
$890$	0	0
$891$	813828.	1.02513
$892$	0	0
$893$	− 622080.i	− 0.780088i
$894$	0	0
$895$	− 71564.3i	− 0.0893409i
$896$	0	0
$897$	1.49299e6	1.85555
$898$	0	0
$899$	817698.	1.01175
$900$	0	0
$901$	− 320112.i	− 0.394323i
$902$	0	0
$903$	− 2.23488e6i	− 2.74080i
$904$	0	0
$905$	−288576.	−0.352341
$906$	0	0
$907$	693549.	0.843067	0.421534	−	0.906813i	$-0.361492\pi$
$907$	693549.	0.843067	0.421534	+	0.906813i	$0.361492\pi$
$908$	0	0
$909$	22440.0i	0.0271578i
$910$	0	0
$911$	− 1.13296e6i	− 1.36514i	−0.730821	−	0.682570i	$-0.760862\pi$
$911$	− 1.13296e6i	− 1.36514i	0.730821	−	0.682570i	$-0.239138\pi$
$912$	0	0
$913$	−1.00637e6	−1.20730
$914$	0	0
$915$	186240.	0.222449
$916$	0	0
$917$	751872.i	0.894139i
$918$	0	0
$919$	526817.i	0.623776i	0.950119	+	0.311888i	$0.100961\pi$
$919$	526817.i	0.623776i	−0.950119	+	0.311888i	$0.899039\pi$
$920$	0	0
$921$	377568.	0.445119
$922$	0	0
$923$	1.67616e6	1.96748
$924$	0	0
$925$	− 848232.i	− 0.991360i
$926$	0	0
$927$	− 38799.9i	− 0.0451514i
$928$	0	0
$929$	−371682.	−0.430666	−0.215333	−	0.976541i	$-0.569084\pi$
$929$	−371682.	−0.430666	−0.215333	+	0.976541i	$0.569084\pi$
$930$	0	0
$931$	−1.65032e6	−1.90401
$932$	0	0
$933$	836352.i	0.960784i
$934$	0	0
$935$	− 139680.i	− 0.159775i
$936$	0	0
$937$	−532418.	−0.606420	−0.303210	−	0.952924i	$-0.598058\pi$
$937$	−532418.	−0.606420	−0.303210	+	0.952924i	$0.598058\pi$
$938$	0	0
$939$	917109.	1.04013
$940$	0	0
$941$	614600.i	0.694086i	0.937849	+	0.347043i	$0.112814\pi$
$941$	614600.i	0.694086i	−0.937849	+	0.347043i	$0.887186\pi$
$942$	0	0
$943$	1.33331e6i	1.49936i
$944$	0	0
$945$	405504.	0.454079
$946$	0	0
$947$	−583988.	−0.651184	−0.325592	−	0.945510i	$-0.605564\pi$
$947$	−583988.	−0.651184	−0.325592	+	0.945510i	$0.605564\pi$
$948$	0	0
$949$	− 594000.i	− 0.659560i
$950$	0	0
$951$	481981.i	0.532929i
$952$	0	0
$953$	604962.	0.666104	0.333052	−	0.942908i	$-0.391921\pi$
$953$	604962.	0.666104	0.333052	+	0.942908i	$0.391921\pi$
$954$	0	0
$955$	−248319.	−0.272273
$956$	0	0
$957$	1.37702e6i	1.50355i
$958$	0	0
$959$	2.56079e6i	2.78444i
$960$	0	0
$961$	530305.	0.574221
$962$	0	0
$963$	16019.7	0.0172743
$964$	0	0
$965$	− 330992.i	− 0.355437i
$966$	0	0
$967$	− 1.39405e6i	− 1.49082i	−0.666604	−	0.745412i	$-0.732253\pi$
$967$	− 1.39405e6i	− 1.49082i	0.666604	−	0.745412i	$-0.267747\pi$
$968$	0	0
$969$	−699840.	−0.745334
$970$	0	0
$971$	−662175.	−0.702319	−0.351160	−	0.936316i	$-0.614213\pi$
$971$	−662175.	−0.702319	−0.351160	+	0.936316i	$0.614213\pi$
$972$	0	0
$973$	103680.i	0.109514i
$974$	0	0
$975$	− 1.18728e6i	− 1.24894i
$976$	0	0
$977$	−815778.	−0.854639	−0.427320	−	0.904101i	$-0.640542\pi$
$977$	−815778.	−0.854639	−0.427320	+	0.904101i	$0.640542\pi$
$978$	0	0
$979$	−261899.	−0.273256
$980$	0	0
$981$	− 223560.i	− 0.232304i
$982$	0	0
$983$	1.23384e6i	1.27688i	0.769671	+	0.638441i	$0.220420\pi$
$983$	1.23384e6i	1.27688i	−0.769671	+	0.638441i	$0.779580\pi$
$984$	0	0
$985$	541120.	0.557726
$986$	0	0
$987$	−1.08358e6	−1.11231
$988$	0	0
$989$	2.05286e6i	2.09878i
$990$	0	0
$991$	− 417315.i	− 0.424929i	−0.977169	−	0.212464i	$-0.931851\pi$
$991$	− 417315.i	− 0.424929i	0.977169	−	0.212464i	$-0.0681490\pi$
$992$	0	0
$993$	−161568.	−0.163854
$994$	0	0
$995$	200035.	0.202051
$996$	0	0
$997$	1.80252e6i	1.81338i	0.421793	+	0.906692i	$0.361401\pi$
$997$	1.80252e6i	1.81338i	−0.421793	+	0.906692i	$0.638599\pi$
$998$	0	0
$999$	977758.i	0.979716i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.5.d.c.63.3	yes	4
3.2	odd	2		1152.5.b.i.703.3		4
4.3	odd	2	inner	128.5.d.c.63.1	✓	4
8.3	odd	2	inner	128.5.d.c.63.4	yes	4
8.5	even	2	inner	128.5.d.c.63.2	yes	4
12.11	even	2		1152.5.b.i.703.4		4
16.3	odd	4		256.5.c.c.255.1		2
16.5	even	4		256.5.c.f.255.1		2
16.11	odd	4		256.5.c.f.255.2		2
16.13	even	4		256.5.c.c.255.2		2
24.5	odd	2		1152.5.b.i.703.1		4
24.11	even	2		1152.5.b.i.703.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.d.c.63.1	✓	4	4.3	odd	2	inner
128.5.d.c.63.2	yes	4	8.5	even	2	inner
128.5.d.c.63.3	yes	4	1.1	even	1	trivial
128.5.d.c.63.4	yes	4	8.3	odd	2	inner
256.5.c.c.255.1		2	16.3	odd	4
256.5.c.c.255.2		2	16.13	even	4
256.5.c.f.255.1		2	16.5	even	4
256.5.c.f.255.2		2	16.11	odd	4
1152.5.b.i.703.1		4	24.5	odd	2
1152.5.b.i.703.2		4	24.11	even	2
1152.5.b.i.703.3		4	3.2	odd	2
1152.5.b.i.703.4		4	12.11	even	2

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:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	128.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.79796 q^{3} -8.00000i q^{5} -78.3837i q^{7} +15.0000 q^{9} +O(q^{10})\)	\(q+9.79796 q^{3} -8.00000i q^{5} -78.3837i q^{7} +15.0000 q^{9} -107.778 q^{11} -216.000i q^{13} -78.3837i q^{15} -162.000 q^{17} +440.908 q^{19} -768.000i q^{21} +705.453i q^{23} +561.000 q^{25} -646.665 q^{27} -1304.00i q^{29} +627.069i q^{31} -1056.00 q^{33} -627.069 q^{35} -1512.00i q^{37} -2116.36i q^{39} +1890.00 q^{41} +2909.99 q^{43} -120.000i q^{45} -1410.91i q^{47} -3743.00 q^{49} -1587.27 q^{51} +1976.00i q^{53} +862.220i q^{55} +4320.00 q^{57} -2263.33 q^{59} +2376.00i q^{61} -1175.76i q^{63} -1728.00 q^{65} -1675.45 q^{67} +6912.00i q^{69} +7759.98i q^{71} +2750.00 q^{73} +5496.65 q^{75} +8448.00i q^{77} +7995.13i q^{79} -7551.00 q^{81} +9337.45 q^{83} +1296.00i q^{85} -12776.5i q^{87} +2430.00 q^{89} -16930.9 q^{91} +6144.00i q^{93} -3527.27i q^{95} +7454.00 q^{97} -1616.66 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 60 q^{9}+O(q^{10}) \) 4 * q + 60 * q^9	\( 4 q + 60 q^{9} - 648 q^{17} + 2244 q^{25} - 4224 q^{33} + 7560 q^{41} - 14972 q^{49} + 17280 q^{57} - 6912 q^{65} + 11000 q^{73} - 30204 q^{81} + 9720 q^{89} + 29816 q^{97}+O(q^{100}) \) 4 * q + 60 * q^9 - 648 * q^17 + 2244 * q^25 - 4224 * q^33 + 7560 * q^41 - 14972 * q^49 + 17280 * q^57 - 6912 * q^65 + 11000 * q^73 - 30204 * q^81 + 9720 * q^89 + 29816 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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