LMFDB - Embedded newform 1152.5.b.i.703.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,5,Mod(703,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1152.703");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1152.b (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:	$119.082197473$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.2
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1152.703
Dual form			1152.5.b.i.703.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000i q^{5} +78.3837i q^{7} +O(q^{10})$	$q-8.00000i q^{5} +78.3837i q^{7} +107.778 q^{11} +216.000i q^{13} +162.000 q^{17} +440.908 q^{19} +705.453i q^{23} +561.000 q^{25} -1304.00i q^{29} -627.069i q^{31} +627.069 q^{35} +1512.00i q^{37} -1890.00 q^{41} +2909.99 q^{43} -1410.91i q^{47} -3743.00 q^{49} +1976.00i q^{53} -862.220i q^{55} +2263.33 q^{59} -2376.00i q^{61} +1728.00 q^{65} -1675.45 q^{67} +7759.98i q^{71} +2750.00 q^{73} +8448.00i q^{77} -7995.13i q^{79} -9337.45 q^{83} -1296.00i q^{85} -2430.00 q^{89} -16930.9 q^{91} -3527.27i q^{95} +7454.00 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 648 q^{17} + 2244 q^{25} - 7560 q^{41} - 14972 q^{49} + 6912 q^{65} + 11000 q^{73} - 9720 q^{89} + 29816 q^{97}+O(q^{100}) $ 4 * q + 648 * q^17 + 2244 * q^25 - 7560 * q^41 - 14972 * q^49 + 6912 * q^65 + 11000 * q^73 - 9720 * q^89 + 29816 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	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	$0.295079\pi$
$7$	78.3837i	1.59967i	−0.600222	+	0.799833i	$0.704921\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	107.778	0.890724	0.445362	−	0.895351i	$-0.353075\pi$
$11$	107.778	0.890724	0.445362	+	0.895351i	$0.353075\pi$
$12$	0	0
$13$	216.000i	1.27811i	0.769162	+	0.639053i	$0.220674\pi$
$13$	216.000i	1.27811i	−0.769162	+	0.639053i	$0.779326\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	162.000	0.560554	0.280277	−	0.959919i	$-0.409574\pi$
$17$	162.000	0.560554	0.280277	+	0.959919i	$0.409574\pi$
$18$	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
$21$	0	0
$22$	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
$27$	0	0
$28$	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	$-0.894212\pi$
$31$	− 627.069i	− 0.652518i	0.945280	−	0.326259i	$-0.105788\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	627.069	0.511893
$36$	0	0
$37$	1512.00i	1.10446i	0.833693	+	0.552228i	$0.186222\pi$
$37$	1512.00i	1.10446i	−0.833693	+	0.552228i	$0.813778\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1890.00	−1.12433	−0.562165	−	0.827025i	$-0.690032\pi$
$41$	−1890.00	−1.12433	−0.562165	+	0.827025i	$0.690032\pi$
$42$	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
$45$	0	0
$46$	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
$51$	0	0
$52$	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
$57$	0	0
$58$	0	0
$59$	2263.33	0.650195	0.325097	−	0.945681i	$-0.394603\pi$
$59$	2263.33	0.650195	0.325097	+	0.945681i	$0.394603\pi$
$60$	0	0
$61$	− 2376.00i	− 0.638538i	−0.947664	−	0.319269i	$-0.896563\pi$
$61$	− 2376.00i	− 0.638538i	0.947664	−	0.319269i	$-0.103437\pi$
$62$	0	0
$63$	0	0
$64$	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
$69$	0	0
$70$	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
$75$	0	0
$76$	0	0
$77$	8448.00i	1.42486i
$78$	0	0
$79$	− 7995.13i	− 1.28107i	−0.767931	−	0.640533i	$-0.778713\pi$
$79$	− 7995.13i	− 1.28107i	0.767931	−	0.640533i	$-0.221287\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9337.45	−1.35542	−0.677708	−	0.735332i	$-0.737026\pi$
$83$	−9337.45	−1.35542	−0.677708	+	0.735332i	$0.737026\pi$
$84$	0	0
$85$	− 1296.00i	− 0.179377i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2430.00	−0.306779	−0.153390	−	0.988166i	$-0.549019\pi$
$89$	−2430.00	−0.306779	−0.153390	+	0.988166i	$0.549019\pi$
$90$	0	0
$91$	−16930.9	−2.04454
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	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	$0.0389015\pi$
$103$	2586.66i	0.243818i	−0.992541	+	0.121909i	$0.961098\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1067.98	−0.0932813	−0.0466406	−	0.998912i	$-0.514852\pi$
$107$	−1067.98	−0.0932813	−0.0466406	+	0.998912i	$0.514852\pi$
$108$	0	0
$109$	14904.0i	1.25444i	0.778842	+	0.627220i	$0.215807\pi$
$109$	14904.0i	1.25444i	−0.778842	+	0.627220i	$0.784193\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	702.000	0.0549769	0.0274884	−	0.999622i	$-0.491249\pi$
$113$	702.000	0.0549769	0.0274884	+	0.999622i	$0.491249\pi$
$114$	0	0
$115$	5643.62	0.426739
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12698.2i	0.896699i
$120$	0	0
$121$	−3025.00	−0.206612
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9488.00i	− 0.607232i
$126$	0	0
$127$	− 3448.88i	− 0.213831i	−0.994268	−	0.106916i	$-0.965903\pi$
$127$	− 3448.88i	− 0.213831i	0.994268	−	0.106916i	$-0.0340974\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9592.20	0.558954	0.279477	−	0.960152i	$-0.409839\pi$
$131$	9592.20	0.558954	0.279477	+	0.960152i	$0.409839\pi$
$132$	0	0
$133$	34560.0i	1.95376i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	32670.0	1.74064	0.870318	−	0.492490i	$-0.163913\pi$
$137$	32670.0	1.74064	0.870318	+	0.492490i	$0.163913\pi$
$138$	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
$141$	0	0
$142$	0	0
$143$	23280.0i	1.13844i
$144$	0	0
$145$	−10432.0	−0.496171
$146$	0	0
$147$	0	0
$148$	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	$-0.933310\pi$
$151$	− 9484.42i	− 0.415965i	0.978133	−	0.207983i	$-0.0666898\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5016.55	−0.208806
$156$	0	0
$157$	28728.0i	1.16548i	0.812657	+	0.582742i	$0.198020\pi$
$157$	28728.0i	1.16548i	−0.812657	+	0.582742i	$0.801980\pi$
$158$	0	0
$159$	0	0
$160$	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
$165$	0	0
$166$	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
$171$	0	0
$172$	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
$177$	0	0
$178$	0	0
$179$	−8945.54	−0.279190	−0.139595	−	0.990209i	$-0.544580\pi$
$179$	−8945.54	−0.279190	−0.139595	+	0.990209i	$0.544580\pi$
$180$	0	0
$181$	36072.0i	1.10107i	0.834814	+	0.550533i	$0.185575\pi$
$181$	36072.0i	1.10107i	−0.834814	+	0.550533i	$0.814425\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12096.0	0.353426
$186$	0	0
$187$	17460.0	0.499298
$188$	0	0
$189$	0	0
$190$	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
$195$	0	0
$196$	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	$-0.897759\pi$
$199$	− 25004.4i	− 0.631408i	0.948858	−	0.315704i	$-0.102241\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	102212.	2.48034
$204$	0	0
$205$	15120.0i	0.359786i
$206$	0	0
$207$	0	0
$208$	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
$213$	0	0
$214$	0	0
$215$	− 23280.0i	− 0.503623i
$216$	0	0
$217$	49152.0	1.04381
$218$	0	0
$219$	0	0
$220$	0	0
$221$	34992.0i	0.716447i
$222$	0	0
$223$	− 20693.3i	− 0.416121i	−0.978116	−	0.208061i	$-0.933285\pi$
$223$	− 20693.3i	− 0.416121i	0.978116	−	0.208061i	$-0.0667151\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−77923.2	−1.51222	−0.756110	−	0.654445i	$-0.772902\pi$
$227$	−77923.2	−1.51222	−0.756110	+	0.654445i	$0.772902\pi$
$228$	0	0
$229$	14472.0i	0.275967i	0.990435	+	0.137984i	$0.0440621\pi$
$229$	14472.0i	0.275967i	−0.990435	+	0.137984i	$0.955938\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2754.00	0.0507285	0.0253643	−	0.999678i	$-0.491925\pi$
$233$	2754.00	0.0507285	0.0253643	+	0.999678i	$0.491925\pi$
$234$	0	0
$235$	−11287.2	−0.204387
$236$	0	0
$237$	0	0
$238$	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
$243$	0	0
$244$	0	0
$245$	29944.0i	0.498859i
$246$	0	0
$247$	95236.2i	1.56102i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−67811.7	−1.07636	−0.538179	−	0.842830i	$-0.680888\pi$
$251$	−67811.7	−1.07636	−0.538179	+	0.842830i	$0.680888\pi$
$252$	0	0
$253$	76032.0i	1.18783i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−80514.0	−1.21900	−0.609502	−	0.792785i	$-0.708631\pi$
$257$	−80514.0	−1.21900	−0.609502	+	0.792785i	$0.708631\pi$
$258$	0	0
$259$	−118516.	−1.76676
$260$	0	0
$261$	0	0
$262$	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
$267$	0	0
$268$	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	$0.213776\pi$
$271$	91395.4i	1.24447i	−0.782829	+	0.622237i	$0.786224\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	60463.2	0.799513
$276$	0	0
$277$	90504.0i	1.17953i	0.807576	+	0.589764i	$0.200779\pi$
$277$	90504.0i	1.17953i	−0.807576	+	0.589764i	$0.799221\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−23166.0	−0.293385	−0.146693	−	0.989182i	$-0.546863\pi$
$281$	−23166.0	−0.293385	−0.146693	+	0.989182i	$0.546863\pi$
$282$	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
$285$	0	0
$286$	0	0
$287$	− 148145.i	− 1.79855i
$288$	0	0
$289$	−57277.0	−0.685780
$290$	0	0
$291$	0	0
$292$	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
$297$	0	0
$298$	0	0
$299$	−152378.	−1.70443
$300$	0	0
$301$	228096.i	2.51759i
$302$	0	0
$303$	0	0
$304$	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
$309$	0	0
$310$	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
$315$	0	0
$316$	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
$321$	0	0
$322$	0	0
$323$	71427.1	0.684633
$324$	0	0
$325$	121176.i	1.14723i
$326$	0	0
$327$	0	0
$328$	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
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	− 67584.0i	− 0.581213i
$342$	0	0
$343$	− 105191.i	− 0.894108i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	220405.	1.83047	0.915235	−	0.402920i	$-0.132005\pi$
$347$	220405.	1.83047	0.915235	+	0.402920i	$0.132005\pi$
$348$	0	0
$349$	48600.0i	0.399012i	0.979897	+	0.199506i	$0.0639337\pi$
$349$	48600.0i	0.399012i	−0.979897	+	0.199506i	$0.936066\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−108162.	−0.868011	−0.434006	−	0.900910i	$-0.642900\pi$
$353$	−108162.	−0.868011	−0.434006	+	0.900910i	$0.642900\pi$
$354$	0	0
$355$	62079.9	0.492600
$356$	0	0
$357$	0	0
$358$	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
$363$	0	0
$364$	0	0
$365$	− 22000.0i	− 0.165134i
$366$	0	0
$367$	− 61296.0i	− 0.455093i	−0.973767	−	0.227547i	$-0.926930\pi$
$367$	− 61296.0i	− 0.455093i	0.973767	−	0.227547i	$-0.0730704\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−154886.	−1.12529
$372$	0	0
$373$	− 54648.0i	− 0.392787i	−0.980525	−	0.196393i	$-0.937077\pi$
$373$	− 54648.0i	− 0.392787i	0.980525	−	0.196393i	$-0.0629229\pi$
$374$	0	0
$375$	0	0
$376$	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
$381$	0	0
$382$	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
$387$	0	0
$388$	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
$393$	0	0
$394$	0	0
$395$	−63961.1	−0.409941
$396$	0	0
$397$	− 232200.i	− 1.47327i	−0.676293	−	0.736633i	$-0.736415\pi$
$397$	− 232200.i	− 1.47327i	0.676293	−	0.736633i	$-0.263585\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−151902.	−0.944658	−0.472329	−	0.881422i	$-0.656587\pi$
$401$	−151902.	−0.944658	−0.472329	+	0.881422i	$0.656587\pi$
$402$	0	0
$403$	135447.	0.833987
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	177408.i	1.04010i
$414$	0	0
$415$	74699.6i	0.433733i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	291382.	1.65972	0.829858	−	0.557974i	$-0.188421\pi$
$419$	291382.	1.65972	0.829858	+	0.557974i	$0.188421\pi$
$420$	0	0
$421$	119880.i	0.676367i	0.941080	+	0.338184i	$0.109812\pi$
$421$	119880.i	0.676367i	−0.941080	+	0.338184i	$0.890188\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	90882.0	0.503153
$426$	0	0
$427$	186240.	1.02145
$428$	0	0
$429$	0	0
$430$	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
$435$	0	0
$436$	0	0
$437$	311040.i	1.62875i
$438$	0	0
$439$	75169.9i	0.390045i	0.980799	+	0.195023i	$0.0624781\pi$
$439$	75169.9i	0.390045i	−0.980799	+	0.195023i	$0.937522\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−185270.	−0.944054	−0.472027	−	0.881584i	$-0.656477\pi$
$443$	−185270.	−0.944054	−0.472027	+	0.881584i	$0.656477\pi$
$444$	0	0
$445$	19440.0i	0.0981694i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−177822.	−0.882049	−0.441025	−	0.897495i	$-0.645385\pi$
$449$	−177822.	−0.882049	−0.441025	+	0.897495i	$0.645385\pi$
$450$	0	0
$451$	−203700.	−1.00147
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	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	$-0.717109\pi$
$463$	− 332817.i	− 1.55254i	0.630399	−	0.776271i	$-0.282891\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	58738.8	0.269334	0.134667	−	0.990891i	$-0.457004\pi$
$467$	58738.8	0.269334	0.134667	+	0.990891i	$0.457004\pi$
$468$	0	0
$469$	− 131328.i	− 0.597051i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	313632.	1.40184
$474$	0	0
$475$	247349.	1.09629
$476$	0	0
$477$	0	0
$478$	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
$483$	0	0
$484$	0	0
$485$	− 59632.0i	− 0.253510i
$486$	0	0
$487$	− 363308.i	− 1.53185i	−0.642927	−	0.765927i	$-0.722280\pi$
$487$	− 363308.i	− 1.53185i	0.642927	−	0.765927i	$-0.277720\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	107533.	0.446043	0.223022	−	0.974813i	$-0.428408\pi$
$491$	107533.	0.446043	0.223022	+	0.974813i	$0.428408\pi$
$492$	0	0
$493$	− 211248.i	− 0.869158i
$494$	0	0
$495$	0	0
$496$	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
$501$	0	0
$502$	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
$507$	0	0
$508$	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
$513$	0	0
$514$	0	0
$515$	20693.3	0.0780216
$516$	0	0
$517$	− 152064.i	− 0.568912i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	184734.	0.680568	0.340284	−	0.940323i	$-0.389477\pi$
$521$	184734.	0.680568	0.340284	+	0.940323i	$0.389477\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	− 101585.i	− 0.365771i
$528$	0	0
$529$	−217823.	−0.778381
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 408240.i	− 1.43701i
$534$	0	0
$535$	8543.82i	0.0298500i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−403411.	−1.38858
$540$	0	0
$541$	230904.i	0.788927i	0.918912	+	0.394464i	$0.129070\pi$
$541$	230904.i	0.788927i	−0.918912	+	0.394464i	$0.870930\pi$
$542$	0	0
$543$	0	0
$544$	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
$549$	0	0
$550$	0	0
$551$	− 574944.i	− 1.89375i
$552$	0	0
$553$	626688.	2.04928
$554$	0	0
$555$	0	0
$556$	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
$561$	0	0
$562$	0	0
$563$	−491123.	−1.54943	−0.774717	−	0.632308i	$-0.782108\pi$
$563$	−491123.	−1.54943	−0.774717	+	0.632308i	$0.782108\pi$
$564$	0	0
$565$	− 5616.00i	− 0.0175926i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	127710.	0.394458	0.197229	−	0.980357i	$-0.436806\pi$
$569$	127710.	0.394458	0.197229	+	0.980357i	$0.436806\pi$
$570$	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
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	− 731904.i	− 2.16821i
$582$	0	0
$583$	212968.i	0.626582i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	273353.	0.793319	0.396660	−	0.917966i	$-0.370169\pi$
$587$	273353.	0.793319	0.396660	+	0.917966i	$0.370169\pi$
$588$	0	0
$589$	− 276480.i	− 0.796954i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	62910.0	0.178900	0.0894500	−	0.995991i	$-0.471489\pi$
$593$	62910.0	0.178900	0.0894500	+	0.995991i	$0.471489\pi$
$594$	0	0
$595$	101585.	0.286944
$596$	0	0
$597$	0	0
$598$	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
$603$	0	0
$604$	0	0
$605$	24200.0i	0.0661157i
$606$	0	0
$607$	− 369657.i	− 1.00328i	−0.865077	−	0.501640i	$-0.832730\pi$
$607$	− 369657.i	− 1.00328i	0.865077	−	0.501640i	$-0.167270\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	304756.	0.816337
$612$	0	0
$613$	− 222264.i	− 0.591491i	−0.955267	−	0.295746i	$-0.904432\pi$
$613$	− 222264.i	− 0.591491i	0.955267	−	0.295746i	$-0.0955680\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−363582.	−0.955063	−0.477532	−	0.878615i	$-0.658468\pi$
$617$	−363582.	−0.955063	−0.477532	+	0.878615i	$0.658468\pi$
$618$	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
$621$	0	0
$622$	0	0
$623$	− 190472.i	− 0.490745i
$624$	0	0
$625$	274721.	0.703286
$626$	0	0
$627$	0	0
$628$	0	0
$629$	244944.i	0.619107i
$630$	0	0
$631$	− 348415.i	− 0.875062i	−0.899203	−	0.437531i	$-0.855853\pi$
$631$	− 348415.i	− 0.875062i	0.899203	−	0.437531i	$-0.144147\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−27591.1	−0.0684259
$636$	0	0
$637$	− 808488.i	− 1.99248i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−67230.0	−0.163624	−0.0818120	−	0.996648i	$-0.526071\pi$
$641$	−67230.0	−0.163624	−0.0818120	+	0.996648i	$0.526071\pi$
$642$	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
$645$	0	0
$646$	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
$651$	0	0
$652$	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
$657$	0	0
$658$	0	0
$659$	223149.	0.513834	0.256917	−	0.966433i	$-0.417293\pi$
$659$	223149.	0.513834	0.256917	+	0.966433i	$0.417293\pi$
$660$	0	0
$661$	− 560088.i	− 1.28190i	−0.767584	−	0.640949i	$-0.778541\pi$
$661$	− 560088.i	− 1.28190i	0.767584	−	0.640949i	$-0.221459\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	276480.	0.625202
$666$	0	0
$667$	919911.	2.06773
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	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
$681$	0	0
$682$	0	0
$683$	189208.	0.405601	0.202800	−	0.979220i	$-0.434996\pi$
$683$	189208.	0.405601	0.202800	+	0.979220i	$0.434996\pi$
$684$	0	0
$685$	− 261360.i	− 0.557004i
$686$	0	0
$687$	0	0
$688$	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
$693$	0	0
$694$	0	0
$695$	10581.8i	0.0219073i
$696$	0	0
$697$	−306180.	−0.630248
$698$	0	0
$699$	0	0
$700$	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
$705$	0	0
$706$	0	0
$707$	−117262.	−0.234595
$708$	0	0
$709$	− 919512.i	− 1.82922i	−0.404342	−	0.914608i	$-0.632499\pi$
$709$	− 919512.i	− 1.82922i	0.404342	−	0.914608i	$-0.367501\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	442368.	0.870171
$714$	0	0
$715$	186240.	0.364301
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	0	0
$725$	− 731544.i	− 1.39176i
$726$	0	0
$727$	488722.i	0.924684i	0.886702	+	0.462342i	$0.152991\pi$
$727$	488722.i	0.924684i	−0.886702	+	0.462342i	$0.847009\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	471419.	0.882211
$732$	0	0
$733$	− 706536.i	− 1.31500i	−0.753454	−	0.657501i	$-0.771614\pi$
$733$	− 706536.i	− 1.31500i	0.753454	−	0.657501i	$-0.228386\pi$
$734$	0	0
$735$	0	0
$736$	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
$741$	0	0
$742$	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
$747$	0	0
$748$	0	0
$749$	− 83712.0i	− 0.149219i
$750$	0	0
$751$	− 536301.i	− 0.950887i	−0.879746	−	0.475443i	$-0.842288\pi$
$751$	− 536301.i	− 0.950887i	0.879746	−	0.475443i	$-0.157712\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−75875.4	−0.133109
$756$	0	0
$757$	− 470232.i	− 0.820579i	−0.911955	−	0.410290i	$-0.865428\pi$
$757$	− 470232.i	− 0.820579i	0.911955	−	0.410290i	$-0.134572\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	691038.	1.19325	0.596627	−	0.802519i	$-0.296507\pi$
$761$	691038.	1.19325	0.596627	+	0.802519i	$0.296507\pi$
$762$	0	0
$763$	−1.16823e6	−2.00669
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	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
$777$	0	0
$778$	0	0
$779$	−833316.	−1.37320
$780$	0	0
$781$	836352.i	1.37116i
$782$	0	0
$783$	0	0
$784$	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
$789$	0	0
$790$	0	0
$791$	55025.3i	0.0879447i
$792$	0	0
$793$	513216.	0.816120
$794$	0	0
$795$	0	0
$796$	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
$801$	0	0
$802$	0	0
$803$	296388.	0.459653
$804$	0	0
$805$	442368.i	0.682640i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−803682.	−1.22797	−0.613984	−	0.789318i	$-0.710434\pi$
$809$	−803682.	−1.22797	−0.613984	+	0.789318i	$0.710434\pi$
$810$	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
$813$	0	0
$814$	0	0
$815$	− 249025.i	− 0.374910i
$816$	0	0
$817$	1.28304e6	1.92219
$818$	0	0
$819$	0	0
$820$	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	$-0.992908\pi$
$823$	− 30177.7i	− 0.0445540i	0.999752	−	0.0222770i	$-0.00709158\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	864680.	1.26428	0.632141	−	0.774853i	$-0.282176\pi$
$827$	864680.	1.26428	0.632141	+	0.774853i	$0.282176\pi$
$828$	0	0
$829$	− 449928.i	− 0.654687i	−0.944905	−	0.327344i	$-0.893847\pi$
$829$	− 449928.i	− 0.654687i	0.944905	−	0.327344i	$-0.106153\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−606366.	−0.873866
$834$	0	0
$835$	−434559.	−0.623269
$836$	0	0
$837$	0	0
$838$	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
$843$	0	0
$844$	0	0
$845$	144760.i	0.202738i
$846$	0	0
$847$	− 237111.i	− 0.330510i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.06665e6	−1.47286
$852$	0	0
$853$	121608.i	0.167134i	0.996502	+	0.0835669i	$0.0266312\pi$
$853$	121608.i	0.167134i	−0.996502	+	0.0835669i	$0.973369\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.26473e6	1.72202	0.861009	−	0.508590i	$-0.169833\pi$
$857$	1.26473e6	1.72202	0.861009	+	0.508590i	$0.169833\pi$
$858$	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
$861$	0	0
$862$	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
$867$	0	0
$868$	0	0
$869$	− 861696.i	− 1.14108i
$870$	0	0
$871$	− 361897.i	− 0.477034i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	743704.	0.971369
$876$	0	0
$877$	− 519048.i	− 0.674852i	−0.941352	−	0.337426i	$-0.890444\pi$
$877$	− 519048.i	− 0.674852i	0.941352	−	0.337426i	$-0.109556\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−434754.	−0.560134	−0.280067	−	0.959980i	$-0.590357\pi$
$881$	−434754.	−0.560134	−0.280067	+	0.959980i	$0.590357\pi$
$882$	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
$885$	0	0
$886$	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
$891$	0	0
$892$	0	0
$893$	− 622080.i	− 0.780088i
$894$	0	0
$895$	71564.3i	0.0893409i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−817698.	−1.01175
$900$	0	0
$901$	320112.i	0.394323i
$902$	0	0
$903$	0	0
$904$	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
$909$	0	0
$910$	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
$915$	0	0
$916$	0	0
$917$	751872.i	0.894139i
$918$	0	0
$919$	− 526817.i	− 0.623776i	−0.950119	−	0.311888i	$-0.899039\pi$
$919$	− 526817.i	− 0.623776i	0.950119	−	0.311888i	$-0.100961\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.67616e6	−1.96748
$924$	0	0
$925$	848232.i	0.991360i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	371682.	0.430666	0.215333	−	0.976541i	$-0.430916\pi$
$929$	371682.	0.430666	0.215333	+	0.976541i	$0.430916\pi$
$930$	0	0
$931$	−1.65032e6	−1.90401
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	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
$945$	0	0
$946$	0	0
$947$	583988.	0.651184	0.325592	−	0.945510i	$-0.394436\pi$
$947$	583988.	0.651184	0.325592	+	0.945510i	$0.394436\pi$
$948$	0	0
$949$	594000.i	0.659560i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−604962.	−0.666104	−0.333052	−	0.942908i	$-0.608079\pi$
$953$	−604962.	−0.666104	−0.333052	+	0.942908i	$0.608079\pi$
$954$	0	0
$955$	−248319.	−0.272273
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.56079e6i	2.78444i
$960$	0	0
$961$	530305.	0.574221
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 330992.i	− 0.355437i
$966$	0	0
$967$	1.39405e6i	1.49082i	0.666604	+	0.745412i	$0.267747\pi$
$967$	1.39405e6i	1.49082i	−0.666604	+	0.745412i	$0.732253\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	662175.	0.702319	0.351160	−	0.936316i	$-0.385787\pi$
$971$	662175.	0.702319	0.351160	+	0.936316i	$0.385787\pi$
$972$	0	0
$973$	− 103680.i	− 0.109514i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	815778.	0.854639	0.427320	−	0.904101i	$-0.359458\pi$
$977$	815778.	0.854639	0.427320	+	0.904101i	$0.359458\pi$
$978$	0	0
$979$	−261899.	−0.273256
$980$	0	0
$981$	0	0
$982$	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
$987$	0	0
$988$	0	0
$989$	2.05286e6i	2.09878i
$990$	0	0
$991$	417315.i	0.424929i	0.977169	+	0.212464i	$0.0681490\pi$
$991$	417315.i	0.424929i	−0.977169	+	0.212464i	$0.931851\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−200035.	−0.202051
$996$	0	0
$997$	− 1.80252e6i	− 1.81338i	−0.421793	−	0.906692i	$-0.638599\pi$
$997$	− 1.80252e6i	− 1.81338i	0.421793	−	0.906692i	$-0.361401\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.5.d.c.63.1	✓	4	24.5	odd	2
128.5.d.c.63.2	yes	4	12.11	even	2
128.5.d.c.63.3	yes	4	24.11	even	2
128.5.d.c.63.4	yes	4	3.2	odd	2
256.5.c.c.255.1		2	48.5	odd	4
256.5.c.c.255.2		2	48.11	even	4
256.5.c.f.255.1		2	48.35	even	4
256.5.c.f.255.2		2	48.29	odd	4
1152.5.b.i.703.1		4	4.3	odd	2	inner
1152.5.b.i.703.2		4	1.1	even	1	trivial
1152.5.b.i.703.3		4	8.3	odd	2	inner
1152.5.b.i.703.4		4	8.5	even	2	inner

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:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1152.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.00000i q^{5} +78.3837i q^{7} +O(q^{10})\)	\(q-8.00000i q^{5} +78.3837i q^{7} +107.778 q^{11} +216.000i q^{13} +162.000 q^{17} +440.908 q^{19} +705.453i q^{23} +561.000 q^{25} -1304.00i q^{29} -627.069i q^{31} +627.069 q^{35} +1512.00i q^{37} -1890.00 q^{41} +2909.99 q^{43} -1410.91i q^{47} -3743.00 q^{49} +1976.00i q^{53} -862.220i q^{55} +2263.33 q^{59} -2376.00i q^{61} +1728.00 q^{65} -1675.45 q^{67} +7759.98i q^{71} +2750.00 q^{73} +8448.00i q^{77} -7995.13i q^{79} -9337.45 q^{83} -1296.00i q^{85} -2430.00 q^{89} -16930.9 q^{91} -3527.27i q^{95} +7454.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 648 q^{17} + 2244 q^{25} - 7560 q^{41} - 14972 q^{49} + 6912 q^{65} + 11000 q^{73} - 9720 q^{89} + 29816 q^{97}+O(q^{100}) \) 4 * q + 648 * q^17 + 2244 * q^25 - 7560 * q^41 - 14972 * q^49 + 6912 * q^65 + 11000 * q^73 - 9720 * q^89 + 29816 * q^97

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