LMFDB - Embedded newform 1472.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1472,4,Mod(1,1472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1472.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-5,0,6,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$86.8508115285$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1472.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-5.00000 q^{3} +6.00000 q^{5} +8.00000 q^{7} -2.00000 q^{9} +34.0000 q^{11} +57.0000 q^{13} -30.0000 q^{15} -80.0000 q^{17} -70.0000 q^{19} -40.0000 q^{21} -23.0000 q^{23} -89.0000 q^{25} +145.000 q^{27} -245.000 q^{29} -103.000 q^{31} -170.000 q^{33} +48.0000 q^{35} +298.000 q^{37} -285.000 q^{39} +95.0000 q^{41} +88.0000 q^{43} -12.0000 q^{45} +357.000 q^{47} -279.000 q^{49} +400.000 q^{51} +414.000 q^{53} +204.000 q^{55} +350.000 q^{57} -408.000 q^{59} -822.000 q^{61} -16.0000 q^{63} +342.000 q^{65} +926.000 q^{67} +115.000 q^{69} -335.000 q^{71} -899.000 q^{73} +445.000 q^{75} +272.000 q^{77} +1322.00 q^{79} -671.000 q^{81} -36.0000 q^{83} -480.000 q^{85} +1225.00 q^{87} -460.000 q^{89} +456.000 q^{91} +515.000 q^{93} -420.000 q^{95} -964.000 q^{97} -68.0000 q^{99} +O(q^{100})$

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
$3$	−5.00000	−0.962250	−0.481125	−	0.876652i	$-0.659772\pi$
$3$	−5.00000	−0.962250	−0.481125	+	0.876652i	$0.659772\pi$
$4$	0	0
$5$	6.00000	0.536656	0.268328	−	0.963328i	$-0.413529\pi$
$5$	6.00000	0.536656	0.268328	+	0.963328i	$0.413529\pi$
$6$	0	0
$7$	8.00000	0.431959	0.215980	−	0.976398i	$-0.430705\pi$
$7$	8.00000	0.431959	0.215980	+	0.976398i	$0.430705\pi$
$8$	0	0
$9$	−2.00000	−0.0740741
$10$	0	0
$11$	34.0000	0.931944	0.465972	−	0.884799i	$-0.345705\pi$
$11$	34.0000	0.931944	0.465972	+	0.884799i	$0.345705\pi$
$12$	0	0
$13$	57.0000	1.21607	0.608037	−	0.793909i	$-0.291957\pi$
$13$	57.0000	1.21607	0.608037	+	0.793909i	$0.291957\pi$
$14$	0	0
$15$	−30.0000	−0.516398
$16$	0	0
$17$	−80.0000	−1.14134	−0.570672	−	0.821178i	$-0.693317\pi$
$17$	−80.0000	−1.14134	−0.570672	+	0.821178i	$0.693317\pi$
$18$	0	0
$19$	−70.0000	−0.845216	−0.422608	−	0.906313i	$-0.638885\pi$
$19$	−70.0000	−0.845216	−0.422608	+	0.906313i	$0.638885\pi$
$20$	0	0
$21$	−40.0000	−0.415653
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−89.0000	−0.712000
$26$	0	0
$27$	145.000	1.03353
$28$	0	0
$29$	−245.000	−1.56881	−0.784403	−	0.620252i	$-0.787030\pi$
$29$	−245.000	−1.56881	−0.784403	+	0.620252i	$0.787030\pi$
$30$	0	0
$31$	−103.000	−0.596753	−0.298377	−	0.954448i	$-0.596445\pi$
$31$	−103.000	−0.596753	−0.298377	+	0.954448i	$0.596445\pi$
$32$	0	0
$33$	−170.000	−0.896764
$34$	0	0
$35$	48.0000	0.231814
$36$	0	0
$37$	298.000	1.32408	0.662039	−	0.749469i	$-0.269691\pi$
$37$	298.000	1.32408	0.662039	+	0.749469i	$0.269691\pi$
$38$	0	0
$39$	−285.000	−1.17017
$40$	0	0
$41$	95.0000	0.361866	0.180933	−	0.983495i	$-0.442088\pi$
$41$	95.0000	0.361866	0.180933	+	0.983495i	$0.442088\pi$
$42$	0	0
$43$	88.0000	0.312090	0.156045	−	0.987750i	$-0.450125\pi$
$43$	88.0000	0.312090	0.156045	+	0.987750i	$0.450125\pi$
$44$	0	0
$45$	−12.0000	−0.0397523
$46$	0	0
$47$	357.000	1.10795	0.553977	−	0.832532i	$-0.313110\pi$
$47$	357.000	1.10795	0.553977	+	0.832532i	$0.313110\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	400.000	1.09826
$52$	0	0
$53$	414.000	1.07297	0.536484	−	0.843911i	$-0.319752\pi$
$53$	414.000	1.07297	0.536484	+	0.843911i	$0.319752\pi$
$54$	0	0
$55$	204.000	0.500134
$56$	0	0
$57$	350.000	0.813309
$58$	0	0
$59$	−408.000	−0.900289	−0.450145	−	0.892956i	$-0.648628\pi$
$59$	−408.000	−0.900289	−0.450145	+	0.892956i	$0.648628\pi$
$60$	0	0
$61$	−822.000	−1.72535	−0.862675	−	0.505759i	$-0.831212\pi$
$61$	−822.000	−1.72535	−0.862675	+	0.505759i	$0.831212\pi$
$62$	0	0
$63$	−16.0000	−0.0319970
$64$	0	0
$65$	342.000	0.652614
$66$	0	0
$67$	926.000	1.68849	0.844246	−	0.535957i	$-0.180049\pi$
$67$	926.000	1.68849	0.844246	+	0.535957i	$0.180049\pi$
$68$	0	0
$69$	115.000	0.200643
$70$	0	0
$71$	−335.000	−0.559960	−0.279980	−	0.960006i	$-0.590328\pi$
$71$	−335.000	−0.559960	−0.279980	+	0.960006i	$0.590328\pi$
$72$	0	0
$73$	−899.000	−1.44137	−0.720685	−	0.693263i	$-0.756173\pi$
$73$	−899.000	−1.44137	−0.720685	+	0.693263i	$0.756173\pi$
$74$	0	0
$75$	445.000	0.685122
$76$	0	0
$77$	272.000	0.402562
$78$	0	0
$79$	1322.00	1.88274	0.941371	−	0.337373i	$-0.109538\pi$
$79$	1322.00	1.88274	0.941371	+	0.337373i	$0.109538\pi$
$80$	0	0
$81$	−671.000	−0.920439
$82$	0	0
$83$	−36.0000	−0.0476086	−0.0238043	−	0.999717i	$-0.507578\pi$
$83$	−36.0000	−0.0476086	−0.0238043	+	0.999717i	$0.507578\pi$
$84$	0	0
$85$	−480.000	−0.612510
$86$	0	0
$87$	1225.00	1.50958
$88$	0	0
$89$	−460.000	−0.547864	−0.273932	−	0.961749i	$-0.588324\pi$
$89$	−460.000	−0.547864	−0.273932	+	0.961749i	$0.588324\pi$
$90$	0	0
$91$	456.000	0.525294
$92$	0	0
$93$	515.000	0.574226
$94$	0	0
$95$	−420.000	−0.453590
$96$	0	0
$97$	−964.000	−1.00907	−0.504533	−	0.863393i	$-0.668335\pi$
$97$	−964.000	−1.00907	−0.504533	+	0.863393i	$0.668335\pi$
$98$	0	0
$99$	−68.0000	−0.0690329
$100$	0	0
$101$	310.000	0.305407	0.152704	−	0.988272i	$-0.451202\pi$
$101$	310.000	0.305407	0.152704	+	0.988272i	$0.451202\pi$
$102$	0	0
$103$	−1044.00	−0.998722	−0.499361	−	0.866394i	$-0.666432\pi$
$103$	−1044.00	−0.998722	−0.499361	+	0.866394i	$0.666432\pi$
$104$	0	0
$105$	−240.000	−0.223063
$106$	0	0
$107$	414.000	0.374046	0.187023	−	0.982356i	$-0.440116\pi$
$107$	414.000	0.374046	0.187023	+	0.982356i	$0.440116\pi$
$108$	0	0
$109$	−704.000	−0.618633	−0.309316	−	0.950959i	$-0.600100\pi$
$109$	−704.000	−0.618633	−0.309316	+	0.950959i	$0.600100\pi$
$110$	0	0
$111$	−1490.00	−1.27409
$112$	0	0
$113$	952.000	0.792537	0.396268	−	0.918135i	$-0.370305\pi$
$113$	952.000	0.792537	0.396268	+	0.918135i	$0.370305\pi$
$114$	0	0
$115$	−138.000	−0.111901
$116$	0	0
$117$	−114.000	−0.0900795
$118$	0	0
$119$	−640.000	−0.493014
$120$	0	0
$121$	−175.000	−0.131480
$122$	0	0
$123$	−475.000	−0.348206
$124$	0	0
$125$	−1284.00	−0.918756
$126$	0	0
$127$	−261.000	−0.182362	−0.0911811	−	0.995834i	$-0.529064\pi$
$127$	−261.000	−0.182362	−0.0911811	+	0.995834i	$0.529064\pi$
$128$	0	0
$129$	−440.000	−0.300309
$130$	0	0
$131$	−1441.00	−0.961074	−0.480537	−	0.876974i	$-0.659558\pi$
$131$	−1441.00	−0.961074	−0.480537	+	0.876974i	$0.659558\pi$
$132$	0	0
$133$	−560.000	−0.365099
$134$	0	0
$135$	870.000	0.554649
$136$	0	0
$137$	1556.00	0.970351	0.485175	−	0.874417i	$-0.338756\pi$
$137$	1556.00	0.970351	0.485175	+	0.874417i	$0.338756\pi$
$138$	0	0
$139$	25.0000	0.0152552	0.00762760	−	0.999971i	$-0.497572\pi$
$139$	25.0000	0.0152552	0.00762760	+	0.999971i	$0.497572\pi$
$140$	0	0
$141$	−1785.00	−1.06613
$142$	0	0
$143$	1938.00	1.13331
$144$	0	0
$145$	−1470.00	−0.841909
$146$	0	0
$147$	1395.00	0.782705
$148$	0	0
$149$	−822.000	−0.451952	−0.225976	−	0.974133i	$-0.572557\pi$
$149$	−822.000	−0.451952	−0.225976	+	0.974133i	$0.572557\pi$
$150$	0	0
$151$	1489.00	0.802471	0.401235	−	0.915975i	$-0.368581\pi$
$151$	1489.00	0.802471	0.401235	+	0.915975i	$0.368581\pi$
$152$	0	0
$153$	160.000	0.0845440
$154$	0	0
$155$	−618.000	−0.320251
$156$	0	0
$157$	632.000	0.321268	0.160634	−	0.987014i	$-0.448646\pi$
$157$	632.000	0.321268	0.160634	+	0.987014i	$0.448646\pi$
$158$	0	0
$159$	−2070.00	−1.03246
$160$	0	0
$161$	−184.000	−0.0900698
$162$	0	0
$163$	−3043.00	−1.46225	−0.731123	−	0.682245i	$-0.761004\pi$
$163$	−3043.00	−1.46225	−0.731123	+	0.682245i	$0.761004\pi$
$164$	0	0
$165$	−1020.00	−0.481254
$166$	0	0
$167$	2224.00	1.03053	0.515264	−	0.857031i	$-0.327694\pi$
$167$	2224.00	1.03053	0.515264	+	0.857031i	$0.327694\pi$
$168$	0	0
$169$	1052.00	0.478835
$170$	0	0
$171$	140.000	0.0626086
$172$	0	0
$173$	−3230.00	−1.41949	−0.709747	−	0.704457i	$-0.751191\pi$
$173$	−3230.00	−1.41949	−0.709747	+	0.704457i	$0.751191\pi$
$174$	0	0
$175$	−712.000	−0.307555
$176$	0	0
$177$	2040.00	0.866304
$178$	0	0
$179$	369.000	0.154080	0.0770401	−	0.997028i	$-0.475453\pi$
$179$	369.000	0.154080	0.0770401	+	0.997028i	$0.475453\pi$
$180$	0	0
$181$	1370.00	0.562604	0.281302	−	0.959619i	$-0.409234\pi$
$181$	1370.00	0.562604	0.281302	+	0.959619i	$0.409234\pi$
$182$	0	0
$183$	4110.00	1.66022
$184$	0	0
$185$	1788.00	0.710575
$186$	0	0
$187$	−2720.00	−1.06367
$188$	0	0
$189$	1160.00	0.446442
$190$	0	0
$191$	−4410.00	−1.67066	−0.835331	−	0.549747i	$-0.814724\pi$
$191$	−4410.00	−1.67066	−0.835331	+	0.549747i	$0.814724\pi$
$192$	0	0
$193$	−135.000	−0.0503498	−0.0251749	−	0.999683i	$-0.508014\pi$
$193$	−135.000	−0.0503498	−0.0251749	+	0.999683i	$0.508014\pi$
$194$	0	0
$195$	−1710.00	−0.627978
$196$	0	0
$197$	−1221.00	−0.441587	−0.220794	−	0.975321i	$-0.570865\pi$
$197$	−1221.00	−0.441587	−0.220794	+	0.975321i	$0.570865\pi$
$198$	0	0
$199$	1098.00	0.391131	0.195566	−	0.980691i	$-0.437346\pi$
$199$	1098.00	0.391131	0.195566	+	0.980691i	$0.437346\pi$
$200$	0	0
$201$	−4630.00	−1.62475
$202$	0	0
$203$	−1960.00	−0.677660
$204$	0	0
$205$	570.000	0.194198
$206$	0	0
$207$	46.0000	0.0154455
$208$	0	0
$209$	−2380.00	−0.787694
$210$	0	0
$211$	−3676.00	−1.19937	−0.599683	−	0.800238i	$-0.704707\pi$
$211$	−3676.00	−1.19937	−0.599683	+	0.800238i	$0.704707\pi$
$212$	0	0
$213$	1675.00	0.538822
$214$	0	0
$215$	528.000	0.167485
$216$	0	0
$217$	−824.000	−0.257773
$218$	0	0
$219$	4495.00	1.38696
$220$	0	0
$221$	−4560.00	−1.38796
$222$	0	0
$223$	−1656.00	−0.497282	−0.248641	−	0.968596i	$-0.579984\pi$
$223$	−1656.00	−0.497282	−0.248641	+	0.968596i	$0.579984\pi$
$224$	0	0
$225$	178.000	0.0527407
$226$	0	0
$227$	2940.00	0.859624	0.429812	−	0.902918i	$-0.358580\pi$
$227$	2940.00	0.859624	0.429812	+	0.902918i	$0.358580\pi$
$228$	0	0
$229$	−3612.00	−1.04230	−0.521152	−	0.853464i	$-0.674498\pi$
$229$	−3612.00	−1.04230	−0.521152	+	0.853464i	$0.674498\pi$
$230$	0	0
$231$	−1360.00	−0.387366
$232$	0	0
$233$	−4325.00	−1.21605	−0.608026	−	0.793917i	$-0.708038\pi$
$233$	−4325.00	−1.21605	−0.608026	+	0.793917i	$0.708038\pi$
$234$	0	0
$235$	2142.00	0.594590
$236$	0	0
$237$	−6610.00	−1.81167
$238$	0	0
$239$	−2735.00	−0.740219	−0.370110	−	0.928988i	$-0.620680\pi$
$239$	−2735.00	−0.740219	−0.370110	+	0.928988i	$0.620680\pi$
$240$	0	0
$241$	−6710.00	−1.79348	−0.896741	−	0.442556i	$-0.854072\pi$
$241$	−6710.00	−1.79348	−0.896741	+	0.442556i	$0.854072\pi$
$242$	0	0
$243$	−560.000	−0.147835
$244$	0	0
$245$	−1674.00	−0.436522
$246$	0	0
$247$	−3990.00	−1.02784
$248$	0	0
$249$	180.000	0.0458114
$250$	0	0
$251$	−6948.00	−1.74723	−0.873613	−	0.486621i	$-0.838229\pi$
$251$	−6948.00	−1.74723	−0.873613	+	0.486621i	$0.838229\pi$
$252$	0	0
$253$	−782.000	−0.194324
$254$	0	0
$255$	2400.00	0.589388
$256$	0	0
$257$	−4929.00	−1.19635	−0.598176	−	0.801365i	$-0.704108\pi$
$257$	−4929.00	−1.19635	−0.598176	+	0.801365i	$0.704108\pi$
$258$	0	0
$259$	2384.00	0.571948
$260$	0	0
$261$	490.000	0.116208
$262$	0	0
$263$	−6138.00	−1.43911	−0.719554	−	0.694437i	$-0.755654\pi$
$263$	−6138.00	−1.43911	−0.719554	+	0.694437i	$0.755654\pi$
$264$	0	0
$265$	2484.00	0.575815
$266$	0	0
$267$	2300.00	0.527182
$268$	0	0
$269$	2063.00	0.467596	0.233798	−	0.972285i	$-0.424885\pi$
$269$	2063.00	0.467596	0.233798	+	0.972285i	$0.424885\pi$
$270$	0	0
$271$	1064.00	0.238500	0.119250	−	0.992864i	$-0.461951\pi$
$271$	1064.00	0.238500	0.119250	+	0.992864i	$0.461951\pi$
$272$	0	0
$273$	−2280.00	−0.505465
$274$	0	0
$275$	−3026.00	−0.663544
$276$	0	0
$277$	−5729.00	−1.24268	−0.621340	−	0.783541i	$-0.713411\pi$
$277$	−5729.00	−1.24268	−0.621340	+	0.783541i	$0.713411\pi$
$278$	0	0
$279$	206.000	0.0442039
$280$	0	0
$281$	−960.000	−0.203804	−0.101902	−	0.994794i	$-0.532493\pi$
$281$	−960.000	−0.203804	−0.101902	+	0.994794i	$0.532493\pi$
$282$	0	0
$283$	−114.000	−0.0239456	−0.0119728	−	0.999928i	$-0.503811\pi$
$283$	−114.000	−0.0239456	−0.0119728	+	0.999928i	$0.503811\pi$
$284$	0	0
$285$	2100.00	0.436468
$286$	0	0
$287$	760.000	0.156311
$288$	0	0
$289$	1487.00	0.302666
$290$	0	0
$291$	4820.00	0.970974
$292$	0	0
$293$	7048.00	1.40529	0.702643	−	0.711543i	$-0.252003\pi$
$293$	7048.00	1.40529	0.702643	+	0.711543i	$0.252003\pi$
$294$	0	0
$295$	−2448.00	−0.483146
$296$	0	0
$297$	4930.00	0.963191
$298$	0	0
$299$	−1311.00	−0.253569
$300$	0	0
$301$	704.000	0.134810
$302$	0	0
$303$	−1550.00	−0.293878
$304$	0	0
$305$	−4932.00	−0.925920
$306$	0	0
$307$	3872.00	0.719826	0.359913	−	0.932986i	$-0.382806\pi$
$307$	3872.00	0.719826	0.359913	+	0.932986i	$0.382806\pi$
$308$	0	0
$309$	5220.00	0.961021
$310$	0	0
$311$	4977.00	0.907459	0.453730	−	0.891139i	$-0.350093\pi$
$311$	4977.00	0.907459	0.453730	+	0.891139i	$0.350093\pi$
$312$	0	0
$313$	−2536.00	−0.457965	−0.228983	−	0.973430i	$-0.573540\pi$
$313$	−2536.00	−0.457965	−0.228983	+	0.973430i	$0.573540\pi$
$314$	0	0
$315$	−96.0000	−0.0171714
$316$	0	0
$317$	−1434.00	−0.254074	−0.127037	−	0.991898i	$-0.540547\pi$
$317$	−1434.00	−0.254074	−0.127037	+	0.991898i	$0.540547\pi$
$318$	0	0
$319$	−8330.00	−1.46204
$320$	0	0
$321$	−2070.00	−0.359926
$322$	0	0
$323$	5600.00	0.964682
$324$	0	0
$325$	−5073.00	−0.865844
$326$	0	0
$327$	3520.00	0.595280
$328$	0	0
$329$	2856.00	0.478591
$330$	0	0
$331$	5469.00	0.908167	0.454084	−	0.890959i	$-0.349967\pi$
$331$	5469.00	0.908167	0.454084	+	0.890959i	$0.349967\pi$
$332$	0	0
$333$	−596.000	−0.0980799
$334$	0	0
$335$	5556.00	0.906139
$336$	0	0
$337$	−7796.00	−1.26016	−0.630082	−	0.776529i	$-0.716979\pi$
$337$	−7796.00	−1.26016	−0.630082	+	0.776529i	$0.716979\pi$
$338$	0	0
$339$	−4760.00	−0.762619
$340$	0	0
$341$	−3502.00	−0.556141
$342$	0	0
$343$	−4976.00	−0.783320
$344$	0	0
$345$	690.000	0.107676
$346$	0	0
$347$	−10068.0	−1.55758	−0.778788	−	0.627288i	$-0.784165\pi$
$347$	−10068.0	−1.55758	−0.778788	+	0.627288i	$0.784165\pi$
$348$	0	0
$349$	7495.00	1.14956	0.574782	−	0.818306i	$-0.305087\pi$
$349$	7495.00	1.14956	0.574782	+	0.818306i	$0.305087\pi$
$350$	0	0
$351$	8265.00	1.25685
$352$	0	0
$353$	10617.0	1.60081	0.800405	−	0.599460i	$-0.204618\pi$
$353$	10617.0	1.60081	0.800405	+	0.599460i	$0.204618\pi$
$354$	0	0
$355$	−2010.00	−0.300506
$356$	0	0
$357$	3200.00	0.474403
$358$	0	0
$359$	−2522.00	−0.370769	−0.185384	−	0.982666i	$-0.559353\pi$
$359$	−2522.00	−0.370769	−0.185384	+	0.982666i	$0.559353\pi$
$360$	0	0
$361$	−1959.00	−0.285610
$362$	0	0
$363$	875.000	0.126517
$364$	0	0
$365$	−5394.00	−0.773520
$366$	0	0
$367$	−7204.00	−1.02465	−0.512324	−	0.858792i	$-0.671215\pi$
$367$	−7204.00	−1.02465	−0.512324	+	0.858792i	$0.671215\pi$
$368$	0	0
$369$	−190.000	−0.0268049
$370$	0	0
$371$	3312.00	0.463478
$372$	0	0
$373$	13310.0	1.84763	0.923815	−	0.382840i	$-0.125054\pi$
$373$	13310.0	1.84763	0.923815	+	0.382840i	$0.125054\pi$
$374$	0	0
$375$	6420.00	0.884073
$376$	0	0
$377$	−13965.0	−1.90778
$378$	0	0
$379$	12952.0	1.75541	0.877704	−	0.479203i	$-0.159074\pi$
$379$	12952.0	1.75541	0.877704	+	0.479203i	$0.159074\pi$
$380$	0	0
$381$	1305.00	0.175478
$382$	0	0
$383$	2812.00	0.375161	0.187580	−	0.982249i	$-0.439936\pi$
$383$	2812.00	0.375161	0.187580	+	0.982249i	$0.439936\pi$
$384$	0	0
$385$	1632.00	0.216037
$386$	0	0
$387$	−176.000	−0.0231178
$388$	0	0
$389$	−1264.00	−0.164749	−0.0823745	−	0.996601i	$-0.526250\pi$
$389$	−1264.00	−0.164749	−0.0823745	+	0.996601i	$0.526250\pi$
$390$	0	0
$391$	1840.00	0.237987
$392$	0	0
$393$	7205.00	0.924794
$394$	0	0
$395$	7932.00	1.01039
$396$	0	0
$397$	−7119.00	−0.899981	−0.449990	−	0.893033i	$-0.648573\pi$
$397$	−7119.00	−0.899981	−0.449990	+	0.893033i	$0.648573\pi$
$398$	0	0
$399$	2800.00	0.351317
$400$	0	0
$401$	4262.00	0.530758	0.265379	−	0.964144i	$-0.414503\pi$
$401$	4262.00	0.530758	0.265379	+	0.964144i	$0.414503\pi$
$402$	0	0
$403$	−5871.00	−0.725696
$404$	0	0
$405$	−4026.00	−0.493959
$406$	0	0
$407$	10132.0	1.23397
$408$	0	0
$409$	229.000	0.0276854	0.0138427	−	0.999904i	$-0.495594\pi$
$409$	229.000	0.0276854	0.0138427	+	0.999904i	$0.495594\pi$
$410$	0	0
$411$	−7780.00	−0.933720
$412$	0	0
$413$	−3264.00	−0.388888
$414$	0	0
$415$	−216.000	−0.0255495
$416$	0	0
$417$	−125.000	−0.0146793
$418$	0	0
$419$	15776.0	1.83940	0.919699	−	0.392623i	$-0.128432\pi$
$419$	15776.0	1.83940	0.919699	+	0.392623i	$0.128432\pi$
$420$	0	0
$421$	8728.00	1.01040	0.505198	−	0.863003i	$-0.331419\pi$
$421$	8728.00	1.01040	0.505198	+	0.863003i	$0.331419\pi$
$422$	0	0
$423$	−714.000	−0.0820706
$424$	0	0
$425$	7120.00	0.812637
$426$	0	0
$427$	−6576.00	−0.745281
$428$	0	0
$429$	−9690.00	−1.09053
$430$	0	0
$431$	2928.00	0.327232	0.163616	−	0.986524i	$-0.447684\pi$
$431$	2928.00	0.327232	0.163616	+	0.986524i	$0.447684\pi$
$432$	0	0
$433$	−5314.00	−0.589780	−0.294890	−	0.955531i	$-0.595283\pi$
$433$	−5314.00	−0.589780	−0.294890	+	0.955531i	$0.595283\pi$
$434$	0	0
$435$	7350.00	0.810128
$436$	0	0
$437$	1610.00	0.176240
$438$	0	0
$439$	−2585.00	−0.281037	−0.140519	−	0.990078i	$-0.544877\pi$
$439$	−2585.00	−0.281037	−0.140519	+	0.990078i	$0.544877\pi$
$440$	0	0
$441$	558.000	0.0602527
$442$	0	0
$443$	−2997.00	−0.321426	−0.160713	−	0.987001i	$-0.551379\pi$
$443$	−2997.00	−0.321426	−0.160713	+	0.987001i	$0.551379\pi$
$444$	0	0
$445$	−2760.00	−0.294015
$446$	0	0
$447$	4110.00	0.434891
$448$	0	0
$449$	−16562.0	−1.74078	−0.870389	−	0.492365i	$-0.836132\pi$
$449$	−16562.0	−1.74078	−0.870389	+	0.492365i	$0.836132\pi$
$450$	0	0
$451$	3230.00	0.337239
$452$	0	0
$453$	−7445.00	−0.772178
$454$	0	0
$455$	2736.00	0.281903
$456$	0	0
$457$	3924.00	0.401656	0.200828	−	0.979626i	$-0.435637\pi$
$457$	3924.00	0.401656	0.200828	+	0.979626i	$0.435637\pi$
$458$	0	0
$459$	−11600.0	−1.17961
$460$	0	0
$461$	4543.00	0.458977	0.229489	−	0.973311i	$-0.426295\pi$
$461$	4543.00	0.458977	0.229489	+	0.973311i	$0.426295\pi$
$462$	0	0
$463$	−9616.00	−0.965213	−0.482606	−	0.875837i	$-0.660310\pi$
$463$	−9616.00	−0.965213	−0.482606	+	0.875837i	$0.660310\pi$
$464$	0	0
$465$	3090.00	0.308162
$466$	0	0
$467$	7826.00	0.775469	0.387735	−	0.921771i	$-0.373258\pi$
$467$	7826.00	0.775469	0.387735	+	0.921771i	$0.373258\pi$
$468$	0	0
$469$	7408.00	0.729360
$470$	0	0
$471$	−3160.00	−0.309140
$472$	0	0
$473$	2992.00	0.290851
$474$	0	0
$475$	6230.00	0.601794
$476$	0	0
$477$	−828.000	−0.0794791
$478$	0	0
$479$	−11404.0	−1.08781	−0.543906	−	0.839146i	$-0.683055\pi$
$479$	−11404.0	−1.08781	−0.543906	+	0.839146i	$0.683055\pi$
$480$	0	0
$481$	16986.0	1.61018
$482$	0	0
$483$	920.000	0.0866697
$484$	0	0
$485$	−5784.00	−0.541521
$486$	0	0
$487$	9267.00	0.862275	0.431137	−	0.902286i	$-0.358112\pi$
$487$	9267.00	0.862275	0.431137	+	0.902286i	$0.358112\pi$
$488$	0	0
$489$	15215.0	1.40705
$490$	0	0
$491$	−18191.0	−1.67199	−0.835996	−	0.548735i	$-0.815110\pi$
$491$	−18191.0	−1.67199	−0.835996	+	0.548735i	$0.815110\pi$
$492$	0	0
$493$	19600.0	1.79055
$494$	0	0
$495$	−408.000	−0.0370469
$496$	0	0
$497$	−2680.00	−0.241880
$498$	0	0
$499$	19315.0	1.73278	0.866391	−	0.499366i	$-0.166434\pi$
$499$	19315.0	1.73278	0.866391	+	0.499366i	$0.166434\pi$
$500$	0	0
$501$	−11120.0	−0.991627
$502$	0	0
$503$	−8422.00	−0.746557	−0.373279	−	0.927719i	$-0.621766\pi$
$503$	−8422.00	−0.746557	−0.373279	+	0.927719i	$0.621766\pi$
$504$	0	0
$505$	1860.00	0.163899
$506$	0	0
$507$	−5260.00	−0.460759
$508$	0	0
$509$	863.000	0.0751509	0.0375754	−	0.999294i	$-0.488037\pi$
$509$	863.000	0.0751509	0.0375754	+	0.999294i	$0.488037\pi$
$510$	0	0
$511$	−7192.00	−0.622613
$512$	0	0
$513$	−10150.0	−0.873554
$514$	0	0
$515$	−6264.00	−0.535971
$516$	0	0
$517$	12138.0	1.03255
$518$	0	0
$519$	16150.0	1.36591
$520$	0	0
$521$	19260.0	1.61957	0.809785	−	0.586727i	$-0.199584\pi$
$521$	19260.0	1.61957	0.809785	+	0.586727i	$0.199584\pi$
$522$	0	0
$523$	−11740.0	−0.981557	−0.490779	−	0.871284i	$-0.663288\pi$
$523$	−11740.0	−0.981557	−0.490779	+	0.871284i	$0.663288\pi$
$524$	0	0
$525$	3560.00	0.295945
$526$	0	0
$527$	8240.00	0.681101
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	816.000	0.0666881
$532$	0	0
$533$	5415.00	0.440056
$534$	0	0
$535$	2484.00	0.200734
$536$	0	0
$537$	−1845.00	−0.148264
$538$	0	0
$539$	−9486.00	−0.758054
$540$	0	0
$541$	−17741.0	−1.40988	−0.704940	−	0.709267i	$-0.749026\pi$
$541$	−17741.0	−1.40988	−0.704940	+	0.709267i	$0.749026\pi$
$542$	0	0
$543$	−6850.00	−0.541366
$544$	0	0
$545$	−4224.00	−0.331993
$546$	0	0
$547$	−6571.00	−0.513630	−0.256815	−	0.966461i	$-0.582673\pi$
$547$	−6571.00	−0.513630	−0.256815	+	0.966461i	$0.582673\pi$
$548$	0	0
$549$	1644.00	0.127804
$550$	0	0
$551$	17150.0	1.32598
$552$	0	0
$553$	10576.0	0.813268
$554$	0	0
$555$	−8940.00	−0.683751
$556$	0	0
$557$	1372.00	0.104369	0.0521845	−	0.998637i	$-0.483382\pi$
$557$	1372.00	0.104369	0.0521845	+	0.998637i	$0.483382\pi$
$558$	0	0
$559$	5016.00	0.379524
$560$	0	0
$561$	13600.0	1.02352
$562$	0	0
$563$	4332.00	0.324284	0.162142	−	0.986767i	$-0.448160\pi$
$563$	4332.00	0.324284	0.162142	+	0.986767i	$0.448160\pi$
$564$	0	0
$565$	5712.00	0.425320
$566$	0	0
$567$	−5368.00	−0.397592
$568$	0	0
$569$	−3546.00	−0.261258	−0.130629	−	0.991431i	$-0.541700\pi$
$569$	−3546.00	−0.261258	−0.130629	+	0.991431i	$0.541700\pi$
$570$	0	0
$571$	−6160.00	−0.451468	−0.225734	−	0.974189i	$-0.572478\pi$
$571$	−6160.00	−0.451468	−0.225734	+	0.974189i	$0.572478\pi$
$572$	0	0
$573$	22050.0	1.60760
$574$	0	0
$575$	2047.00	0.148462
$576$	0	0
$577$	2953.00	0.213059	0.106529	−	0.994310i	$-0.466026\pi$
$577$	2953.00	0.213059	0.106529	+	0.994310i	$0.466026\pi$
$578$	0	0
$579$	675.000	0.0484491
$580$	0	0
$581$	−288.000	−0.0205650
$582$	0	0
$583$	14076.0	0.999946
$584$	0	0
$585$	−684.000	−0.0483417
$586$	0	0
$587$	−2949.00	−0.207356	−0.103678	−	0.994611i	$-0.533061\pi$
$587$	−2949.00	−0.207356	−0.103678	+	0.994611i	$0.533061\pi$
$588$	0	0
$589$	7210.00	0.504385
$590$	0	0
$591$	6105.00	0.424917
$592$	0	0
$593$	16390.0	1.13500	0.567501	−	0.823372i	$-0.307910\pi$
$593$	16390.0	1.13500	0.567501	+	0.823372i	$0.307910\pi$
$594$	0	0
$595$	−3840.00	−0.264579
$596$	0	0
$597$	−5490.00	−0.376366
$598$	0	0
$599$	12920.0	0.881297	0.440648	−	0.897680i	$-0.354749\pi$
$599$	12920.0	0.881297	0.440648	+	0.897680i	$0.354749\pi$
$600$	0	0
$601$	−13835.0	−0.939004	−0.469502	−	0.882931i	$-0.655567\pi$
$601$	−13835.0	−0.939004	−0.469502	+	0.882931i	$0.655567\pi$
$602$	0	0
$603$	−1852.00	−0.125073
$604$	0	0
$605$	−1050.00	−0.0705596
$606$	0	0
$607$	−6004.00	−0.401474	−0.200737	−	0.979645i	$-0.564334\pi$
$607$	−6004.00	−0.401474	−0.200737	+	0.979645i	$0.564334\pi$
$608$	0	0
$609$	9800.00	0.652079
$610$	0	0
$611$	20349.0	1.34735
$612$	0	0
$613$	16416.0	1.08162	0.540812	−	0.841143i	$-0.318117\pi$
$613$	16416.0	1.08162	0.540812	+	0.841143i	$0.318117\pi$
$614$	0	0
$615$	−2850.00	−0.186867
$616$	0	0
$617$	3786.00	0.247032	0.123516	−	0.992343i	$-0.460583\pi$
$617$	3786.00	0.247032	0.123516	+	0.992343i	$0.460583\pi$
$618$	0	0
$619$	15824.0	1.02750	0.513748	−	0.857941i	$-0.328257\pi$
$619$	15824.0	1.02750	0.513748	+	0.857941i	$0.328257\pi$
$620$	0	0
$621$	−3335.00	−0.215506
$622$	0	0
$623$	−3680.00	−0.236655
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	11900.0	0.757959
$628$	0	0
$629$	−23840.0	−1.51123
$630$	0	0
$631$	−17852.0	−1.12627	−0.563135	−	0.826365i	$-0.690405\pi$
$631$	−17852.0	−1.12627	−0.563135	+	0.826365i	$0.690405\pi$
$632$	0	0
$633$	18380.0	1.15409
$634$	0	0
$635$	−1566.00	−0.0978658
$636$	0	0
$637$	−15903.0	−0.989168
$638$	0	0
$639$	670.000	0.0414785
$640$	0	0
$641$	10324.0	0.636152	0.318076	−	0.948065i	$-0.396963\pi$
$641$	10324.0	0.636152	0.318076	+	0.948065i	$0.396963\pi$
$642$	0	0
$643$	−14702.0	−0.901696	−0.450848	−	0.892601i	$-0.648878\pi$
$643$	−14702.0	−0.901696	−0.450848	+	0.892601i	$0.648878\pi$
$644$	0	0
$645$	−2640.00	−0.161163
$646$	0	0
$647$	−11939.0	−0.725457	−0.362728	−	0.931895i	$-0.618155\pi$
$647$	−11939.0	−0.725457	−0.362728	+	0.931895i	$0.618155\pi$
$648$	0	0
$649$	−13872.0	−0.839019
$650$	0	0
$651$	4120.00	0.248042
$652$	0	0
$653$	−6159.00	−0.369097	−0.184548	−	0.982823i	$-0.559082\pi$
$653$	−6159.00	−0.369097	−0.184548	+	0.982823i	$0.559082\pi$
$654$	0	0
$655$	−8646.00	−0.515767
$656$	0	0
$657$	1798.00	0.106768
$658$	0	0
$659$	−21692.0	−1.28225	−0.641123	−	0.767438i	$-0.721531\pi$
$659$	−21692.0	−1.28225	−0.641123	+	0.767438i	$0.721531\pi$
$660$	0	0
$661$	−16502.0	−0.971034	−0.485517	−	0.874227i	$-0.661369\pi$
$661$	−16502.0	−0.971034	−0.485517	+	0.874227i	$0.661369\pi$
$662$	0	0
$663$	22800.0	1.33556
$664$	0	0
$665$	−3360.00	−0.195933
$666$	0	0
$667$	5635.00	0.327119
$668$	0	0
$669$	8280.00	0.478510
$670$	0	0
$671$	−27948.0	−1.60793
$672$	0	0
$673$	−27733.0	−1.58845	−0.794226	−	0.607622i	$-0.792124\pi$
$673$	−27733.0	−1.58845	−0.794226	+	0.607622i	$0.792124\pi$
$674$	0	0
$675$	−12905.0	−0.735872
$676$	0	0
$677$	8814.00	0.500369	0.250184	−	0.968198i	$-0.419509\pi$
$677$	8814.00	0.500369	0.250184	+	0.968198i	$0.419509\pi$
$678$	0	0
$679$	−7712.00	−0.435875
$680$	0	0
$681$	−14700.0	−0.827174
$682$	0	0
$683$	−22999.0	−1.28848	−0.644240	−	0.764823i	$-0.722826\pi$
$683$	−22999.0	−1.28848	−0.644240	+	0.764823i	$0.722826\pi$
$684$	0	0
$685$	9336.00	0.520745
$686$	0	0
$687$	18060.0	1.00296
$688$	0	0
$689$	23598.0	1.30481
$690$	0	0
$691$	−12140.0	−0.668346	−0.334173	−	0.942512i	$-0.608457\pi$
$691$	−12140.0	−0.668346	−0.334173	+	0.942512i	$0.608457\pi$
$692$	0	0
$693$	−544.000	−0.0298194
$694$	0	0
$695$	150.000	0.00818680
$696$	0	0
$697$	−7600.00	−0.413014
$698$	0	0
$699$	21625.0	1.17015
$700$	0	0
$701$	20024.0	1.07888	0.539441	−	0.842024i	$-0.318636\pi$
$701$	20024.0	1.07888	0.539441	+	0.842024i	$0.318636\pi$
$702$	0	0
$703$	−20860.0	−1.11913
$704$	0	0
$705$	−10710.0	−0.572145
$706$	0	0
$707$	2480.00	0.131924
$708$	0	0
$709$	4956.00	0.262520	0.131260	−	0.991348i	$-0.458098\pi$
$709$	4956.00	0.262520	0.131260	+	0.991348i	$0.458098\pi$
$710$	0	0
$711$	−2644.00	−0.139462
$712$	0	0
$713$	2369.00	0.124432
$714$	0	0
$715$	11628.0	0.608199
$716$	0	0
$717$	13675.0	0.712276
$718$	0	0
$719$	−2760.00	−0.143158	−0.0715790	−	0.997435i	$-0.522804\pi$
$719$	−2760.00	−0.143158	−0.0715790	+	0.997435i	$0.522804\pi$
$720$	0	0
$721$	−8352.00	−0.431407
$722$	0	0
$723$	33550.0	1.72578
$724$	0	0
$725$	21805.0	1.11699
$726$	0	0
$727$	−7746.00	−0.395163	−0.197581	−	0.980287i	$-0.563309\pi$
$727$	−7746.00	−0.395163	−0.197581	+	0.980287i	$0.563309\pi$
$728$	0	0
$729$	20917.0	1.06269
$730$	0	0
$731$	−7040.00	−0.356202
$732$	0	0
$733$	11976.0	0.603470	0.301735	−	0.953392i	$-0.402434\pi$
$733$	11976.0	0.603470	0.301735	+	0.953392i	$0.402434\pi$
$734$	0	0
$735$	8370.00	0.420044
$736$	0	0
$737$	31484.0	1.57358
$738$	0	0
$739$	15057.0	0.749500	0.374750	−	0.927126i	$-0.377728\pi$
$739$	15057.0	0.749500	0.374750	+	0.927126i	$0.377728\pi$
$740$	0	0
$741$	19950.0	0.989044
$742$	0	0
$743$	−18532.0	−0.915038	−0.457519	−	0.889200i	$-0.651262\pi$
$743$	−18532.0	−0.915038	−0.457519	+	0.889200i	$0.651262\pi$
$744$	0	0
$745$	−4932.00	−0.242543
$746$	0	0
$747$	72.0000	0.00352656
$748$	0	0
$749$	3312.00	0.161573
$750$	0	0
$751$	192.000	0.00932913	0.00466457	−	0.999989i	$-0.498515\pi$
$751$	192.000	0.00932913	0.00466457	+	0.999989i	$0.498515\pi$
$752$	0	0
$753$	34740.0	1.68127
$754$	0	0
$755$	8934.00	0.430651
$756$	0	0
$757$	9830.00	0.471965	0.235982	−	0.971757i	$-0.424169\pi$
$757$	9830.00	0.471965	0.235982	+	0.971757i	$0.424169\pi$
$758$	0	0
$759$	3910.00	0.186988
$760$	0	0
$761$	−30219.0	−1.43947	−0.719736	−	0.694248i	$-0.755737\pi$
$761$	−30219.0	−1.43947	−0.719736	+	0.694248i	$0.755737\pi$
$762$	0	0
$763$	−5632.00	−0.267224
$764$	0	0
$765$	960.000	0.0453711
$766$	0	0
$767$	−23256.0	−1.09482
$768$	0	0
$769$	1122.00	0.0526142	0.0263071	−	0.999654i	$-0.491625\pi$
$769$	1122.00	0.0526142	0.0263071	+	0.999654i	$0.491625\pi$
$770$	0	0
$771$	24645.0	1.15119
$772$	0	0
$773$	−19300.0	−0.898024	−0.449012	−	0.893526i	$-0.648224\pi$
$773$	−19300.0	−0.898024	−0.449012	+	0.893526i	$0.648224\pi$
$774$	0	0
$775$	9167.00	0.424888
$776$	0	0
$777$	−11920.0	−0.550357
$778$	0	0
$779$	−6650.00	−0.305855
$780$	0	0
$781$	−11390.0	−0.521852
$782$	0	0
$783$	−35525.0	−1.62140
$784$	0	0
$785$	3792.00	0.172411
$786$	0	0
$787$	−19396.0	−0.878517	−0.439258	−	0.898361i	$-0.644759\pi$
$787$	−19396.0	−0.878517	−0.439258	+	0.898361i	$0.644759\pi$
$788$	0	0
$789$	30690.0	1.38478
$790$	0	0
$791$	7616.00	0.342344
$792$	0	0
$793$	−46854.0	−2.09815
$794$	0	0
$795$	−12420.0	−0.554078
$796$	0	0
$797$	39034.0	1.73482	0.867412	−	0.497590i	$-0.165782\pi$
$797$	39034.0	1.73482	0.867412	+	0.497590i	$0.165782\pi$
$798$	0	0
$799$	−28560.0	−1.26456
$800$	0	0
$801$	920.000	0.0405825
$802$	0	0
$803$	−30566.0	−1.34328
$804$	0	0
$805$	−1104.00	−0.0483365
$806$	0	0
$807$	−10315.0	−0.449944
$808$	0	0
$809$	−10310.0	−0.448060	−0.224030	−	0.974582i	$-0.571921\pi$
$809$	−10310.0	−0.448060	−0.224030	+	0.974582i	$0.571921\pi$
$810$	0	0
$811$	−40693.0	−1.76193	−0.880965	−	0.473182i	$-0.843105\pi$
$811$	−40693.0	−1.76193	−0.880965	+	0.473182i	$0.843105\pi$
$812$	0	0
$813$	−5320.00	−0.229496
$814$	0	0
$815$	−18258.0	−0.784724
$816$	0	0
$817$	−6160.00	−0.263784
$818$	0	0
$819$	−912.000	−0.0389107
$820$	0	0
$821$	13934.0	0.592326	0.296163	−	0.955137i	$-0.404293\pi$
$821$	13934.0	0.592326	0.296163	+	0.955137i	$0.404293\pi$
$822$	0	0
$823$	−6175.00	−0.261539	−0.130770	−	0.991413i	$-0.541745\pi$
$823$	−6175.00	−0.261539	−0.130770	+	0.991413i	$0.541745\pi$
$824$	0	0
$825$	15130.0	0.638496
$826$	0	0
$827$	28664.0	1.20525	0.602627	−	0.798023i	$-0.294121\pi$
$827$	28664.0	1.20525	0.602627	+	0.798023i	$0.294121\pi$
$828$	0	0
$829$	39590.0	1.65865	0.829323	−	0.558770i	$-0.188726\pi$
$829$	39590.0	1.65865	0.829323	+	0.558770i	$0.188726\pi$
$830$	0	0
$831$	28645.0	1.19577
$832$	0	0
$833$	22320.0	0.928382
$834$	0	0
$835$	13344.0	0.553040
$836$	0	0
$837$	−14935.0	−0.616761
$838$	0	0
$839$	14316.0	0.589086	0.294543	−	0.955638i	$-0.404833\pi$
$839$	14316.0	0.589086	0.294543	+	0.955638i	$0.404833\pi$
$840$	0	0
$841$	35636.0	1.46115
$842$	0	0
$843$	4800.00	0.196110
$844$	0	0
$845$	6312.00	0.256970
$846$	0	0
$847$	−1400.00	−0.0567941
$848$	0	0
$849$	570.000	0.0230416
$850$	0	0
$851$	−6854.00	−0.276089
$852$	0	0
$853$	−28366.0	−1.13861	−0.569304	−	0.822127i	$-0.692787\pi$
$853$	−28366.0	−1.13861	−0.569304	+	0.822127i	$0.692787\pi$
$854$	0	0
$855$	840.000	0.0335993
$856$	0	0
$857$	19283.0	0.768605	0.384303	−	0.923207i	$-0.374442\pi$
$857$	19283.0	0.768605	0.384303	+	0.923207i	$0.374442\pi$
$858$	0	0
$859$	−26101.0	−1.03673	−0.518367	−	0.855158i	$-0.673460\pi$
$859$	−26101.0	−1.03673	−0.518367	+	0.855158i	$0.673460\pi$
$860$	0	0
$861$	−3800.00	−0.150411
$862$	0	0
$863$	−973.000	−0.0383793	−0.0191896	−	0.999816i	$-0.506109\pi$
$863$	−973.000	−0.0383793	−0.0191896	+	0.999816i	$0.506109\pi$
$864$	0	0
$865$	−19380.0	−0.761780
$866$	0	0
$867$	−7435.00	−0.291241
$868$	0	0
$869$	44948.0	1.75461
$870$	0	0
$871$	52782.0	2.05333
$872$	0	0
$873$	1928.00	0.0747456
$874$	0	0
$875$	−10272.0	−0.396865
$876$	0	0
$877$	−5694.00	−0.219239	−0.109620	−	0.993974i	$-0.534963\pi$
$877$	−5694.00	−0.219239	−0.109620	+	0.993974i	$0.534963\pi$
$878$	0	0
$879$	−35240.0	−1.35224
$880$	0	0
$881$	45960.0	1.75758	0.878792	−	0.477205i	$-0.158350\pi$
$881$	45960.0	1.75758	0.878792	+	0.477205i	$0.158350\pi$
$882$	0	0
$883$	17188.0	0.655065	0.327532	−	0.944840i	$-0.393783\pi$
$883$	17188.0	0.655065	0.327532	+	0.944840i	$0.393783\pi$
$884$	0	0
$885$	12240.0	0.464907
$886$	0	0
$887$	−8451.00	−0.319906	−0.159953	−	0.987125i	$-0.551134\pi$
$887$	−8451.00	−0.319906	−0.159953	+	0.987125i	$0.551134\pi$
$888$	0	0
$889$	−2088.00	−0.0787731
$890$	0	0
$891$	−22814.0	−0.857798
$892$	0	0
$893$	−24990.0	−0.936460
$894$	0	0
$895$	2214.00	0.0826881
$896$	0	0
$897$	6555.00	0.243997
$898$	0	0
$899$	25235.0	0.936190
$900$	0	0
$901$	−33120.0	−1.22463
$902$	0	0
$903$	−3520.00	−0.129721
$904$	0	0
$905$	8220.00	0.301925
$906$	0	0
$907$	32774.0	1.19983	0.599913	−	0.800065i	$-0.295202\pi$
$907$	32774.0	1.19983	0.599913	+	0.800065i	$0.295202\pi$
$908$	0	0
$909$	−620.000	−0.0226228
$910$	0	0
$911$	23690.0	0.861564	0.430782	−	0.902456i	$-0.358238\pi$
$911$	23690.0	0.861564	0.430782	+	0.902456i	$0.358238\pi$
$912$	0	0
$913$	−1224.00	−0.0443686
$914$	0	0
$915$	24660.0	0.890967
$916$	0	0
$917$	−11528.0	−0.415145
$918$	0	0
$919$	30044.0	1.07841	0.539206	−	0.842174i	$-0.318725\pi$
$919$	30044.0	1.07841	0.539206	+	0.842174i	$0.318725\pi$
$920$	0	0
$921$	−19360.0	−0.692653
$922$	0	0
$923$	−19095.0	−0.680953
$924$	0	0
$925$	−26522.0	−0.942744
$926$	0	0
$927$	2088.00	0.0739794
$928$	0	0
$929$	−39705.0	−1.40224	−0.701119	−	0.713044i	$-0.747316\pi$
$929$	−39705.0	−1.40224	−0.701119	+	0.713044i	$0.747316\pi$
$930$	0	0
$931$	19530.0	0.687508
$932$	0	0
$933$	−24885.0	−0.873203
$934$	0	0
$935$	−16320.0	−0.570825
$936$	0	0
$937$	17422.0	0.607419	0.303710	−	0.952765i	$-0.401775\pi$
$937$	17422.0	0.607419	0.303710	+	0.952765i	$0.401775\pi$
$938$	0	0
$939$	12680.0	0.440677
$940$	0	0
$941$	25292.0	0.876191	0.438095	−	0.898928i	$-0.355653\pi$
$941$	25292.0	0.876191	0.438095	+	0.898928i	$0.355653\pi$
$942$	0	0
$943$	−2185.00	−0.0754543
$944$	0	0
$945$	6960.00	0.239586
$946$	0	0
$947$	33211.0	1.13961	0.569806	−	0.821779i	$-0.307018\pi$
$947$	33211.0	1.13961	0.569806	+	0.821779i	$0.307018\pi$
$948$	0	0
$949$	−51243.0	−1.75281
$950$	0	0
$951$	7170.00	0.244483
$952$	0	0
$953$	−14154.0	−0.481105	−0.240552	−	0.970636i	$-0.577329\pi$
$953$	−14154.0	−0.481105	−0.240552	+	0.970636i	$0.577329\pi$
$954$	0	0
$955$	−26460.0	−0.896571
$956$	0	0
$957$	41650.0	1.40685
$958$	0	0
$959$	12448.0	0.419152
$960$	0	0
$961$	−19182.0	−0.643886
$962$	0	0
$963$	−828.000	−0.0277071
$964$	0	0
$965$	−810.000	−0.0270205
$966$	0	0
$967$	46343.0	1.54115	0.770574	−	0.637350i	$-0.219970\pi$
$967$	46343.0	1.54115	0.770574	+	0.637350i	$0.219970\pi$
$968$	0	0
$969$	−28000.0	−0.928266
$970$	0	0
$971$	11710.0	0.387015	0.193508	−	0.981099i	$-0.438014\pi$
$971$	11710.0	0.387015	0.193508	+	0.981099i	$0.438014\pi$
$972$	0	0
$973$	200.000	0.00658963
$974$	0	0
$975$	25365.0	0.833159
$976$	0	0
$977$	47854.0	1.56703	0.783513	−	0.621375i	$-0.213426\pi$
$977$	47854.0	1.56703	0.783513	+	0.621375i	$0.213426\pi$
$978$	0	0
$979$	−15640.0	−0.510579
$980$	0	0
$981$	1408.00	0.0458246
$982$	0	0
$983$	22078.0	0.716357	0.358178	−	0.933653i	$-0.383398\pi$
$983$	22078.0	0.716357	0.358178	+	0.933653i	$0.383398\pi$
$984$	0	0
$985$	−7326.00	−0.236980
$986$	0	0
$987$	−14280.0	−0.460524
$988$	0	0
$989$	−2024.00	−0.0650753
$990$	0	0
$991$	4288.00	0.137450	0.0687249	−	0.997636i	$-0.478107\pi$
$991$	4288.00	0.137450	0.0687249	+	0.997636i	$0.478107\pi$
$992$	0	0
$993$	−27345.0	−0.873885
$994$	0	0
$995$	6588.00	0.209903
$996$	0	0
$997$	−28966.0	−0.920123	−0.460061	−	0.887887i	$-0.652173\pi$
$997$	−28966.0	−0.920123	−0.460061	+	0.887887i	$0.652173\pi$
$998$	0	0
$999$	43210.0	1.36847

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.c.1.1		1
4.3	odd	2		1472.4.a.h.1.1		1
8.3	odd	2		23.4.a.a.1.1	✓	1
8.5	even	2		368.4.a.d.1.1		1
24.11	even	2		207.4.a.a.1.1		1
40.3	even	4		575.4.b.b.24.2		2
40.19	odd	2		575.4.a.g.1.1		1
40.27	even	4		575.4.b.b.24.1		2
56.27	even	2		1127.4.a.a.1.1		1
184.91	even	2		529.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.4.a.a.1.1	✓	1	8.3	odd	2
207.4.a.a.1.1		1	24.11	even	2
368.4.a.d.1.1		1	8.5	even	2
529.4.a.a.1.1		1	184.91	even	2
575.4.a.g.1.1		1	40.19	odd	2
575.4.b.b.24.1		2	40.27	even	4
575.4.b.b.24.2		2	40.3	even	4
1127.4.a.a.1.1		1	56.27	even	2
1472.4.a.c.1.1		1	1.1	even	1	trivial
1472.4.a.h.1.1		1	4.3	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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