LMFDB - Embedded newform 2240.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,4,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2240.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$132.164278413$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} -11.0000 q^{9} +O(q^{10})$

$q-4.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} -11.0000 q^{9} -36.0000 q^{11} +82.0000 q^{13} +20.0000 q^{15} +6.00000 q^{17} -56.0000 q^{19} +28.0000 q^{21} -16.0000 q^{23} +25.0000 q^{25} +152.000 q^{27} -118.000 q^{29} -104.000 q^{31} +144.000 q^{33} +35.0000 q^{35} +230.000 q^{37} -328.000 q^{39} -30.0000 q^{41} +228.000 q^{43} +55.0000 q^{45} -72.0000 q^{47} +49.0000 q^{49} -24.0000 q^{51} +606.000 q^{53} +180.000 q^{55} +224.000 q^{57} +40.0000 q^{59} -102.000 q^{61} +77.0000 q^{63} -410.000 q^{65} +836.000 q^{67} +64.0000 q^{69} +1020.00 q^{71} -578.000 q^{73} -100.000 q^{75} +252.000 q^{77} -1324.00 q^{79} -311.000 q^{81} +244.000 q^{83} -30.0000 q^{85} +472.000 q^{87} +722.000 q^{89} -574.000 q^{91} +416.000 q^{93} +280.000 q^{95} +166.000 q^{97} +396.000 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0	0
$2$	0	0
$3$	−4.00000	−0.769800	−0.384900	−	0.922958i	$-0.625764\pi$
$3$	−4.00000	−0.769800	−0.384900	+	0.922958i	$0.625764\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−11.0000	−0.407407
$10$	0	0
$11$	−36.0000	−0.986764	−0.493382	−	0.869813i	$-0.664240\pi$
$11$	−36.0000	−0.986764	−0.493382	+	0.869813i	$0.664240\pi$
$12$	0	0
$13$	82.0000	1.74944	0.874720	−	0.484629i	$-0.161046\pi$
$13$	82.0000	1.74944	0.874720	+	0.484629i	$0.161046\pi$
$14$	0	0
$15$	20.0000	0.344265
$16$	0	0
$17$	6.00000	0.0856008	0.0428004	−	0.999084i	$-0.486372\pi$
$17$	6.00000	0.0856008	0.0428004	+	0.999084i	$0.486372\pi$
$18$	0	0
$19$	−56.0000	−0.676173	−0.338086	−	0.941115i	$-0.609780\pi$
$19$	−56.0000	−0.676173	−0.338086	+	0.941115i	$0.609780\pi$
$20$	0	0
$21$	28.0000	0.290957
$22$	0	0
$23$	−16.0000	−0.145054	−0.0725268	−	0.997366i	$-0.523106\pi$
$23$	−16.0000	−0.145054	−0.0725268	+	0.997366i	$0.523106\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	152.000	1.08342
$28$	0	0
$29$	−118.000	−0.755588	−0.377794	−	0.925890i	$-0.623317\pi$
$29$	−118.000	−0.755588	−0.377794	+	0.925890i	$0.623317\pi$
$30$	0	0
$31$	−104.000	−0.602547	−0.301273	−	0.953538i	$-0.597412\pi$
$31$	−104.000	−0.602547	−0.301273	+	0.953538i	$0.597412\pi$
$32$	0	0
$33$	144.000	0.759612
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	230.000	1.02194	0.510970	−	0.859599i	$-0.329286\pi$
$37$	230.000	1.02194	0.510970	+	0.859599i	$0.329286\pi$
$38$	0	0
$39$	−328.000	−1.34672
$40$	0	0
$41$	−30.0000	−0.114273	−0.0571367	−	0.998366i	$-0.518197\pi$
$41$	−30.0000	−0.114273	−0.0571367	+	0.998366i	$0.518197\pi$
$42$	0	0
$43$	228.000	0.808597	0.404299	−	0.914627i	$-0.367516\pi$
$43$	228.000	0.808597	0.404299	+	0.914627i	$0.367516\pi$
$44$	0	0
$45$	55.0000	0.182198
$46$	0	0
$47$	−72.0000	−0.223453	−0.111726	−	0.993739i	$-0.535638\pi$
$47$	−72.0000	−0.223453	−0.111726	+	0.993739i	$0.535638\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−24.0000	−0.0658955
$52$	0	0
$53$	606.000	1.57058	0.785288	−	0.619131i	$-0.212515\pi$
$53$	606.000	1.57058	0.785288	+	0.619131i	$0.212515\pi$
$54$	0	0
$55$	180.000	0.441294
$56$	0	0
$57$	224.000	0.520518
$58$	0	0
$59$	40.0000	0.0882637	0.0441318	−	0.999026i	$-0.485948\pi$
$59$	40.0000	0.0882637	0.0441318	+	0.999026i	$0.485948\pi$
$60$	0	0
$61$	−102.000	−0.214094	−0.107047	−	0.994254i	$-0.534140\pi$
$61$	−102.000	−0.214094	−0.107047	+	0.994254i	$0.534140\pi$
$62$	0	0
$63$	77.0000	0.153986
$64$	0	0
$65$	−410.000	−0.782373
$66$	0	0
$67$	836.000	1.52438	0.762191	−	0.647352i	$-0.224123\pi$
$67$	836.000	1.52438	0.762191	+	0.647352i	$0.224123\pi$
$68$	0	0
$69$	64.0000	0.111662
$70$	0	0
$71$	1020.00	1.70495	0.852477	−	0.522765i	$-0.175099\pi$
$71$	1020.00	1.70495	0.852477	+	0.522765i	$0.175099\pi$
$72$	0	0
$73$	−578.000	−0.926709	−0.463355	−	0.886173i	$-0.653354\pi$
$73$	−578.000	−0.926709	−0.463355	+	0.886173i	$0.653354\pi$
$74$	0	0
$75$	−100.000	−0.153960
$76$	0	0
$77$	252.000	0.372962
$78$	0	0
$79$	−1324.00	−1.88559	−0.942795	−	0.333373i	$-0.891813\pi$
$79$	−1324.00	−1.88559	−0.942795	+	0.333373i	$0.891813\pi$
$80$	0	0
$81$	−311.000	−0.426612
$82$	0	0
$83$	244.000	0.322680	0.161340	−	0.986899i	$-0.448418\pi$
$83$	244.000	0.322680	0.161340	+	0.986899i	$0.448418\pi$
$84$	0	0
$85$	−30.0000	−0.0382818
$86$	0	0
$87$	472.000	0.581652
$88$	0	0
$89$	722.000	0.859908	0.429954	−	0.902851i	$-0.358530\pi$
$89$	722.000	0.859908	0.429954	+	0.902851i	$0.358530\pi$
$90$	0	0
$91$	−574.000	−0.661226
$92$	0	0
$93$	416.000	0.463841
$94$	0	0
$95$	280.000	0.302394
$96$	0	0
$97$	166.000	0.173760	0.0868801	−	0.996219i	$-0.472310\pi$
$97$	166.000	0.173760	0.0868801	+	0.996219i	$0.472310\pi$
$98$	0	0
$99$	396.000	0.402015
$100$	0	0
$101$	−190.000	−0.187185	−0.0935926	−	0.995611i	$-0.529835\pi$
$101$	−190.000	−0.187185	−0.0935926	+	0.995611i	$0.529835\pi$
$102$	0	0
$103$	−1152.00	−1.10204	−0.551019	−	0.834493i	$-0.685761\pi$
$103$	−1152.00	−1.10204	−0.551019	+	0.834493i	$0.685761\pi$
$104$	0	0
$105$	−140.000	−0.130120
$106$	0	0
$107$	−108.000	−0.0975771	−0.0487886	−	0.998809i	$-0.515536\pi$
$107$	−108.000	−0.0975771	−0.0487886	+	0.998809i	$0.515536\pi$
$108$	0	0
$109$	1010.00	0.887527	0.443764	−	0.896144i	$-0.353643\pi$
$109$	1010.00	0.887527	0.443764	+	0.896144i	$0.353643\pi$
$110$	0	0
$111$	−920.000	−0.786690
$112$	0	0
$113$	898.000	0.747582	0.373791	−	0.927513i	$-0.378058\pi$
$113$	898.000	0.747582	0.373791	+	0.927513i	$0.378058\pi$
$114$	0	0
$115$	80.0000	0.0648699
$116$	0	0
$117$	−902.000	−0.712734
$118$	0	0
$119$	−42.0000	−0.0323541
$120$	0	0
$121$	−35.0000	−0.0262960
$122$	0	0
$123$	120.000	0.0879678
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1392.00	−0.972599	−0.486299	−	0.873792i	$-0.661654\pi$
$127$	−1392.00	−0.972599	−0.486299	+	0.873792i	$0.661654\pi$
$128$	0	0
$129$	−912.000	−0.622458
$130$	0	0
$131$	1536.00	1.02443	0.512217	−	0.858856i	$-0.328824\pi$
$131$	1536.00	1.02443	0.512217	+	0.858856i	$0.328824\pi$
$132$	0	0
$133$	392.000	0.255569
$134$	0	0
$135$	−760.000	−0.484521
$136$	0	0
$137$	2202.00	1.37321	0.686604	−	0.727031i	$-0.259101\pi$
$137$	2202.00	1.37321	0.686604	+	0.727031i	$0.259101\pi$
$138$	0	0
$139$	−1600.00	−0.976333	−0.488166	−	0.872751i	$-0.662334\pi$
$139$	−1600.00	−0.976333	−0.488166	+	0.872751i	$0.662334\pi$
$140$	0	0
$141$	288.000	0.172014
$142$	0	0
$143$	−2952.00	−1.72628
$144$	0	0
$145$	590.000	0.337909
$146$	0	0
$147$	−196.000	−0.109971
$148$	0	0
$149$	−1254.00	−0.689474	−0.344737	−	0.938699i	$-0.612032\pi$
$149$	−1254.00	−0.689474	−0.344737	+	0.938699i	$0.612032\pi$
$150$	0	0
$151$	−2916.00	−1.57153	−0.785764	−	0.618526i	$-0.787730\pi$
$151$	−2916.00	−1.57153	−0.785764	+	0.618526i	$0.787730\pi$
$152$	0	0
$153$	−66.0000	−0.0348744
$154$	0	0
$155$	520.000	0.269467
$156$	0	0
$157$	−1934.00	−0.983121	−0.491561	−	0.870843i	$-0.663573\pi$
$157$	−1934.00	−0.983121	−0.491561	+	0.870843i	$0.663573\pi$
$158$	0	0
$159$	−2424.00	−1.20903
$160$	0	0
$161$	112.000	0.0548251
$162$	0	0
$163$	−172.000	−0.0826508	−0.0413254	−	0.999146i	$-0.513158\pi$
$163$	−172.000	−0.0826508	−0.0413254	+	0.999146i	$0.513158\pi$
$164$	0	0
$165$	−720.000	−0.339709
$166$	0	0
$167$	1640.00	0.759922	0.379961	−	0.925002i	$-0.375937\pi$
$167$	1640.00	0.759922	0.379961	+	0.925002i	$0.375937\pi$
$168$	0	0
$169$	4527.00	2.06054
$170$	0	0
$171$	616.000	0.275478
$172$	0	0
$173$	−1614.00	−0.709307	−0.354654	−	0.934998i	$-0.615401\pi$
$173$	−1614.00	−0.709307	−0.354654	+	0.934998i	$0.615401\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	−160.000	−0.0679454
$178$	0	0
$179$	−2420.00	−1.01050	−0.505249	−	0.862973i	$-0.668600\pi$
$179$	−2420.00	−1.01050	−0.505249	+	0.862973i	$0.668600\pi$
$180$	0	0
$181$	3682.00	1.51205	0.756025	−	0.654543i	$-0.227139\pi$
$181$	3682.00	1.51205	0.756025	+	0.654543i	$0.227139\pi$
$182$	0	0
$183$	408.000	0.164810
$184$	0	0
$185$	−1150.00	−0.457025
$186$	0	0
$187$	−216.000	−0.0844678
$188$	0	0
$189$	−1064.00	−0.409495
$190$	0	0
$191$	−1580.00	−0.598559	−0.299280	−	0.954165i	$-0.596746\pi$
$191$	−1580.00	−0.598559	−0.299280	+	0.954165i	$0.596746\pi$
$192$	0	0
$193$	−1958.00	−0.730259	−0.365129	−	0.930957i	$-0.618975\pi$
$193$	−1958.00	−0.730259	−0.365129	+	0.930957i	$0.618975\pi$
$194$	0	0
$195$	1640.00	0.602271
$196$	0	0
$197$	2038.00	0.737063	0.368532	−	0.929615i	$-0.379861\pi$
$197$	2038.00	0.737063	0.368532	+	0.929615i	$0.379861\pi$
$198$	0	0
$199$	−1824.00	−0.649748	−0.324874	−	0.945757i	$-0.605322\pi$
$199$	−1824.00	−0.649748	−0.324874	+	0.945757i	$0.605322\pi$
$200$	0	0
$201$	−3344.00	−1.17347
$202$	0	0
$203$	826.000	0.285585
$204$	0	0
$205$	150.000	0.0511047
$206$	0	0
$207$	176.000	0.0590959
$208$	0	0
$209$	2016.00	0.667223
$210$	0	0
$211$	−468.000	−0.152694	−0.0763470	−	0.997081i	$-0.524326\pi$
$211$	−468.000	−0.152694	−0.0763470	+	0.997081i	$0.524326\pi$
$212$	0	0
$213$	−4080.00	−1.31247
$214$	0	0
$215$	−1140.00	−0.361616
$216$	0	0
$217$	728.000	0.227741
$218$	0	0
$219$	2312.00	0.713381
$220$	0	0
$221$	492.000	0.149753
$222$	0	0
$223$	−4448.00	−1.33570	−0.667848	−	0.744298i	$-0.732784\pi$
$223$	−4448.00	−1.33570	−0.667848	+	0.744298i	$0.732784\pi$
$224$	0	0
$225$	−275.000	−0.0814815
$226$	0	0
$227$	2604.00	0.761381	0.380691	−	0.924702i	$-0.375686\pi$
$227$	2604.00	0.761381	0.380691	+	0.924702i	$0.375686\pi$
$228$	0	0
$229$	4130.00	1.19178	0.595891	−	0.803065i	$-0.296799\pi$
$229$	4130.00	1.19178	0.595891	+	0.803065i	$0.296799\pi$
$230$	0	0
$231$	−1008.00	−0.287106
$232$	0	0
$233$	2162.00	0.607886	0.303943	−	0.952690i	$-0.401697\pi$
$233$	2162.00	0.607886	0.303943	+	0.952690i	$0.401697\pi$
$234$	0	0
$235$	360.000	0.0999311
$236$	0	0
$237$	5296.00	1.45153
$238$	0	0
$239$	−2404.00	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$239$	−2404.00	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$240$	0	0
$241$	554.000	0.148076	0.0740379	−	0.997255i	$-0.476411\pi$
$241$	554.000	0.148076	0.0740379	+	0.997255i	$0.476411\pi$
$242$	0	0
$243$	−2860.00	−0.755017
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	−4592.00	−1.18292
$248$	0	0
$249$	−976.000	−0.248400
$250$	0	0
$251$	−600.000	−0.150883	−0.0754416	−	0.997150i	$-0.524037\pi$
$251$	−600.000	−0.150883	−0.0754416	+	0.997150i	$0.524037\pi$
$252$	0	0
$253$	576.000	0.143134
$254$	0	0
$255$	120.000	0.0294694
$256$	0	0
$257$	2814.00	0.683006	0.341503	−	0.939881i	$-0.389064\pi$
$257$	2814.00	0.683006	0.341503	+	0.939881i	$0.389064\pi$
$258$	0	0
$259$	−1610.00	−0.386257
$260$	0	0
$261$	1298.00	0.307832
$262$	0	0
$263$	112.000	0.0262594	0.0131297	−	0.999914i	$-0.495821\pi$
$263$	112.000	0.0262594	0.0131297	+	0.999914i	$0.495821\pi$
$264$	0	0
$265$	−3030.00	−0.702383
$266$	0	0
$267$	−2888.00	−0.661958
$268$	0	0
$269$	−3782.00	−0.857222	−0.428611	−	0.903489i	$-0.640997\pi$
$269$	−3782.00	−0.857222	−0.428611	+	0.903489i	$0.640997\pi$
$270$	0	0
$271$	544.000	0.121940	0.0609698	−	0.998140i	$-0.480581\pi$
$271$	544.000	0.121940	0.0609698	+	0.998140i	$0.480581\pi$
$272$	0	0
$273$	2296.00	0.509012
$274$	0	0
$275$	−900.000	−0.197353
$276$	0	0
$277$	−8738.00	−1.89536	−0.947681	−	0.319218i	$-0.896580\pi$
$277$	−8738.00	−1.89536	−0.947681	+	0.319218i	$0.896580\pi$
$278$	0	0
$279$	1144.00	0.245482
$280$	0	0
$281$	−4566.00	−0.969341	−0.484670	−	0.874697i	$-0.661060\pi$
$281$	−4566.00	−0.969341	−0.484670	+	0.874697i	$0.661060\pi$
$282$	0	0
$283$	8300.00	1.74341	0.871703	−	0.490035i	$-0.163016\pi$
$283$	8300.00	1.74341	0.871703	+	0.490035i	$0.163016\pi$
$284$	0	0
$285$	−1120.00	−0.232783
$286$	0	0
$287$	210.000	0.0431913
$288$	0	0
$289$	−4877.00	−0.992673
$290$	0	0
$291$	−664.000	−0.133761
$292$	0	0
$293$	1026.00	0.204572	0.102286	−	0.994755i	$-0.467384\pi$
$293$	1026.00	0.204572	0.102286	+	0.994755i	$0.467384\pi$
$294$	0	0
$295$	−200.000	−0.0394727
$296$	0	0
$297$	−5472.00	−1.06908
$298$	0	0
$299$	−1312.00	−0.253762
$300$	0	0
$301$	−1596.00	−0.305621
$302$	0	0
$303$	760.000	0.144095
$304$	0	0
$305$	510.000	0.0957460
$306$	0	0
$307$	−8140.00	−1.51327	−0.756636	−	0.653837i	$-0.773158\pi$
$307$	−8140.00	−1.51327	−0.756636	+	0.653837i	$0.773158\pi$
$308$	0	0
$309$	4608.00	0.848349
$310$	0	0
$311$	−544.000	−0.0991878	−0.0495939	−	0.998769i	$-0.515793\pi$
$311$	−544.000	−0.0991878	−0.0495939	+	0.998769i	$0.515793\pi$
$312$	0	0
$313$	−3418.00	−0.617242	−0.308621	−	0.951185i	$-0.599867\pi$
$313$	−3418.00	−0.617242	−0.308621	+	0.951185i	$0.599867\pi$
$314$	0	0
$315$	−385.000	−0.0688644
$316$	0	0
$317$	−7242.00	−1.28313	−0.641563	−	0.767070i	$-0.721714\pi$
$317$	−7242.00	−1.28313	−0.641563	+	0.767070i	$0.721714\pi$
$318$	0	0
$319$	4248.00	0.745587
$320$	0	0
$321$	432.000	0.0751149
$322$	0	0
$323$	−336.000	−0.0578809
$324$	0	0
$325$	2050.00	0.349888
$326$	0	0
$327$	−4040.00	−0.683219
$328$	0	0
$329$	504.000	0.0844572
$330$	0	0
$331$	−9348.00	−1.55230	−0.776152	−	0.630546i	$-0.782831\pi$
$331$	−9348.00	−1.55230	−0.776152	+	0.630546i	$0.782831\pi$
$332$	0	0
$333$	−2530.00	−0.416346
$334$	0	0
$335$	−4180.00	−0.681725
$336$	0	0
$337$	−1046.00	−0.169078	−0.0845389	−	0.996420i	$-0.526942\pi$
$337$	−1046.00	−0.169078	−0.0845389	+	0.996420i	$0.526942\pi$
$338$	0	0
$339$	−3592.00	−0.575489
$340$	0	0
$341$	3744.00	0.594572
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−320.000	−0.0499369
$346$	0	0
$347$	−6036.00	−0.933802	−0.466901	−	0.884309i	$-0.654630\pi$
$347$	−6036.00	−0.933802	−0.466901	+	0.884309i	$0.654630\pi$
$348$	0	0
$349$	−6630.00	−1.01689	−0.508447	−	0.861093i	$-0.669780\pi$
$349$	−6630.00	−1.01689	−0.508447	+	0.861093i	$0.669780\pi$
$350$	0	0
$351$	12464.0	1.89538
$352$	0	0
$353$	4894.00	0.737907	0.368954	−	0.929448i	$-0.379716\pi$
$353$	4894.00	0.737907	0.368954	+	0.929448i	$0.379716\pi$
$354$	0	0
$355$	−5100.00	−0.762479
$356$	0	0
$357$	168.000	0.0249062
$358$	0	0
$359$	−5924.00	−0.870910	−0.435455	−	0.900210i	$-0.643413\pi$
$359$	−5924.00	−0.870910	−0.435455	+	0.900210i	$0.643413\pi$
$360$	0	0
$361$	−3723.00	−0.542790
$362$	0	0
$363$	140.000	0.0202427
$364$	0	0
$365$	2890.00	0.414437
$366$	0	0
$367$	−9704.00	−1.38023	−0.690115	−	0.723699i	$-0.742440\pi$
$367$	−9704.00	−1.38023	−0.690115	+	0.723699i	$0.742440\pi$
$368$	0	0
$369$	330.000	0.0465559
$370$	0	0
$371$	−4242.00	−0.593622
$372$	0	0
$373$	2182.00	0.302895	0.151447	−	0.988465i	$-0.451607\pi$
$373$	2182.00	0.302895	0.151447	+	0.988465i	$0.451607\pi$
$374$	0	0
$375$	500.000	0.0688530
$376$	0	0
$377$	−9676.00	−1.32186
$378$	0	0
$379$	−7932.00	−1.07504	−0.537519	−	0.843252i	$-0.680638\pi$
$379$	−7932.00	−1.07504	−0.537519	+	0.843252i	$0.680638\pi$
$380$	0	0
$381$	5568.00	0.748707
$382$	0	0
$383$	−12840.0	−1.71304	−0.856519	−	0.516116i	$-0.827377\pi$
$383$	−12840.0	−1.71304	−0.856519	+	0.516116i	$0.827377\pi$
$384$	0	0
$385$	−1260.00	−0.166794
$386$	0	0
$387$	−2508.00	−0.329428
$388$	0	0
$389$	10714.0	1.39646	0.698228	−	0.715875i	$-0.253972\pi$
$389$	10714.0	1.39646	0.698228	+	0.715875i	$0.253972\pi$
$390$	0	0
$391$	−96.0000	−0.0124167
$392$	0	0
$393$	−6144.00	−0.788610
$394$	0	0
$395$	6620.00	0.843262
$396$	0	0
$397$	−13214.0	−1.67051	−0.835254	−	0.549864i	$-0.814679\pi$
$397$	−13214.0	−1.67051	−0.835254	+	0.549864i	$0.814679\pi$
$398$	0	0
$399$	−1568.00	−0.196737
$400$	0	0
$401$	−3902.00	−0.485927	−0.242963	−	0.970035i	$-0.578119\pi$
$401$	−3902.00	−0.485927	−0.242963	+	0.970035i	$0.578119\pi$
$402$	0	0
$403$	−8528.00	−1.05412
$404$	0	0
$405$	1555.00	0.190787
$406$	0	0
$407$	−8280.00	−1.00841
$408$	0	0
$409$	8602.00	1.03995	0.519977	−	0.854180i	$-0.325940\pi$
$409$	8602.00	1.03995	0.519977	+	0.854180i	$0.325940\pi$
$410$	0	0
$411$	−8808.00	−1.05710
$412$	0	0
$413$	−280.000	−0.0333605
$414$	0	0
$415$	−1220.00	−0.144307
$416$	0	0
$417$	6400.00	0.751581
$418$	0	0
$419$	11736.0	1.36836	0.684178	−	0.729315i	$-0.260161\pi$
$419$	11736.0	1.36836	0.684178	+	0.729315i	$0.260161\pi$
$420$	0	0
$421$	−5582.00	−0.646200	−0.323100	−	0.946365i	$-0.604725\pi$
$421$	−5582.00	−0.646200	−0.323100	+	0.946365i	$0.604725\pi$
$422$	0	0
$423$	792.000	0.0910363
$424$	0	0
$425$	150.000	0.0171202
$426$	0	0
$427$	714.000	0.0809201
$428$	0	0
$429$	11808.0	1.32889
$430$	0	0
$431$	15548.0	1.73764	0.868818	−	0.495132i	$-0.164880\pi$
$431$	15548.0	1.73764	0.868818	+	0.495132i	$0.164880\pi$
$432$	0	0
$433$	5726.00	0.635506	0.317753	−	0.948174i	$-0.397072\pi$
$433$	5726.00	0.635506	0.317753	+	0.948174i	$0.397072\pi$
$434$	0	0
$435$	−2360.00	−0.260123
$436$	0	0
$437$	896.000	0.0980812
$438$	0	0
$439$	9536.00	1.03674	0.518370	−	0.855157i	$-0.326539\pi$
$439$	9536.00	1.03674	0.518370	+	0.855157i	$0.326539\pi$
$440$	0	0
$441$	−539.000	−0.0582011
$442$	0	0
$443$	7772.00	0.833541	0.416771	−	0.909012i	$-0.363162\pi$
$443$	7772.00	0.833541	0.416771	+	0.909012i	$0.363162\pi$
$444$	0	0
$445$	−3610.00	−0.384563
$446$	0	0
$447$	5016.00	0.530758
$448$	0	0
$449$	8370.00	0.879743	0.439872	−	0.898061i	$-0.355024\pi$
$449$	8370.00	0.879743	0.439872	+	0.898061i	$0.355024\pi$
$450$	0	0
$451$	1080.00	0.112761
$452$	0	0
$453$	11664.0	1.20976
$454$	0	0
$455$	2870.00	0.295709
$456$	0	0
$457$	−3166.00	−0.324068	−0.162034	−	0.986785i	$-0.551805\pi$
$457$	−3166.00	−0.324068	−0.162034	+	0.986785i	$0.551805\pi$
$458$	0	0
$459$	912.000	0.0927419
$460$	0	0
$461$	762.000	0.0769846	0.0384923	−	0.999259i	$-0.487745\pi$
$461$	762.000	0.0769846	0.0384923	+	0.999259i	$0.487745\pi$
$462$	0	0
$463$	−12408.0	−1.24546	−0.622731	−	0.782436i	$-0.713977\pi$
$463$	−12408.0	−1.24546	−0.622731	+	0.782436i	$0.713977\pi$
$464$	0	0
$465$	−2080.00	−0.207436
$466$	0	0
$467$	−5436.00	−0.538647	−0.269323	−	0.963050i	$-0.586800\pi$
$467$	−5436.00	−0.538647	−0.269323	+	0.963050i	$0.586800\pi$
$468$	0	0
$469$	−5852.00	−0.576163
$470$	0	0
$471$	7736.00	0.756807
$472$	0	0
$473$	−8208.00	−0.797895
$474$	0	0
$475$	−1400.00	−0.135235
$476$	0	0
$477$	−6666.00	−0.639864
$478$	0	0
$479$	−1952.00	−0.186199	−0.0930993	−	0.995657i	$-0.529677\pi$
$479$	−1952.00	−0.186199	−0.0930993	+	0.995657i	$0.529677\pi$
$480$	0	0
$481$	18860.0	1.78782
$482$	0	0
$483$	−448.000	−0.0422044
$484$	0	0
$485$	−830.000	−0.0777080
$486$	0	0
$487$	12248.0	1.13965	0.569825	−	0.821766i	$-0.307011\pi$
$487$	12248.0	1.13965	0.569825	+	0.821766i	$0.307011\pi$
$488$	0	0
$489$	688.000	0.0636246
$490$	0	0
$491$	−9268.00	−0.851851	−0.425926	−	0.904758i	$-0.640051\pi$
$491$	−9268.00	−0.851851	−0.425926	+	0.904758i	$0.640051\pi$
$492$	0	0
$493$	−708.000	−0.0646789
$494$	0	0
$495$	−1980.00	−0.179787
$496$	0	0
$497$	−7140.00	−0.644412
$498$	0	0
$499$	−10388.0	−0.931925	−0.465963	−	0.884804i	$-0.654292\pi$
$499$	−10388.0	−0.931925	−0.465963	+	0.884804i	$0.654292\pi$
$500$	0	0
$501$	−6560.00	−0.584988
$502$	0	0
$503$	20472.0	1.81471	0.907357	−	0.420360i	$-0.138096\pi$
$503$	20472.0	1.81471	0.907357	+	0.420360i	$0.138096\pi$
$504$	0	0
$505$	950.000	0.0837118
$506$	0	0
$507$	−18108.0	−1.58620
$508$	0	0
$509$	−21990.0	−1.91491	−0.957455	−	0.288581i	$-0.906816\pi$
$509$	−21990.0	−1.91491	−0.957455	+	0.288581i	$0.906816\pi$
$510$	0	0
$511$	4046.00	0.350263
$512$	0	0
$513$	−8512.00	−0.732581
$514$	0	0
$515$	5760.00	0.492846
$516$	0	0
$517$	2592.00	0.220495
$518$	0	0
$519$	6456.00	0.546025
$520$	0	0
$521$	14618.0	1.22922	0.614612	−	0.788829i	$-0.289312\pi$
$521$	14618.0	1.22922	0.614612	+	0.788829i	$0.289312\pi$
$522$	0	0
$523$	10492.0	0.877214	0.438607	−	0.898679i	$-0.355472\pi$
$523$	10492.0	0.877214	0.438607	+	0.898679i	$0.355472\pi$
$524$	0	0
$525$	700.000	0.0581914
$526$	0	0
$527$	−624.000	−0.0515785
$528$	0	0
$529$	−11911.0	−0.978959
$530$	0	0
$531$	−440.000	−0.0359593
$532$	0	0
$533$	−2460.00	−0.199914
$534$	0	0
$535$	540.000	0.0436378
$536$	0	0
$537$	9680.00	0.777882
$538$	0	0
$539$	−1764.00	−0.140966
$540$	0	0
$541$	19410.0	1.54252	0.771258	−	0.636523i	$-0.219628\pi$
$541$	19410.0	1.54252	0.771258	+	0.636523i	$0.219628\pi$
$542$	0	0
$543$	−14728.0	−1.16398
$544$	0	0
$545$	−5050.00	−0.396914
$546$	0	0
$547$	−19900.0	−1.55551	−0.777754	−	0.628569i	$-0.783641\pi$
$547$	−19900.0	−1.55551	−0.777754	+	0.628569i	$0.783641\pi$
$548$	0	0
$549$	1122.00	0.0872237
$550$	0	0
$551$	6608.00	0.510908
$552$	0	0
$553$	9268.00	0.712686
$554$	0	0
$555$	4600.00	0.351818
$556$	0	0
$557$	−19810.0	−1.50696	−0.753480	−	0.657471i	$-0.771626\pi$
$557$	−19810.0	−1.50696	−0.753480	+	0.657471i	$0.771626\pi$
$558$	0	0
$559$	18696.0	1.41459
$560$	0	0
$561$	864.000	0.0650234
$562$	0	0
$563$	−5068.00	−0.379380	−0.189690	−	0.981844i	$-0.560748\pi$
$563$	−5068.00	−0.379380	−0.189690	+	0.981844i	$0.560748\pi$
$564$	0	0
$565$	−4490.00	−0.334329
$566$	0	0
$567$	2177.00	0.161244
$568$	0	0
$569$	−8486.00	−0.625223	−0.312611	−	0.949881i	$-0.601204\pi$
$569$	−8486.00	−0.625223	−0.312611	+	0.949881i	$0.601204\pi$
$570$	0	0
$571$	9924.00	0.727332	0.363666	−	0.931529i	$-0.381525\pi$
$571$	9924.00	0.727332	0.363666	+	0.931529i	$0.381525\pi$
$572$	0	0
$573$	6320.00	0.460771
$574$	0	0
$575$	−400.000	−0.0290107
$576$	0	0
$577$	−938.000	−0.0676767	−0.0338383	−	0.999427i	$-0.510773\pi$
$577$	−938.000	−0.0676767	−0.0338383	+	0.999427i	$0.510773\pi$
$578$	0	0
$579$	7832.00	0.562153
$580$	0	0
$581$	−1708.00	−0.121962
$582$	0	0
$583$	−21816.0	−1.54979
$584$	0	0
$585$	4510.00	0.318745
$586$	0	0
$587$	−16084.0	−1.13093	−0.565467	−	0.824771i	$-0.691304\pi$
$587$	−16084.0	−1.13093	−0.565467	+	0.824771i	$0.691304\pi$
$588$	0	0
$589$	5824.00	0.407426
$590$	0	0
$591$	−8152.00	−0.567392
$592$	0	0
$593$	25542.0	1.76878	0.884388	−	0.466752i	$-0.154576\pi$
$593$	25542.0	1.76878	0.884388	+	0.466752i	$0.154576\pi$
$594$	0	0
$595$	210.000	0.0144692
$596$	0	0
$597$	7296.00	0.500177
$598$	0	0
$599$	−2564.00	−0.174895	−0.0874476	−	0.996169i	$-0.527871\pi$
$599$	−2564.00	−0.174895	−0.0874476	+	0.996169i	$0.527871\pi$
$600$	0	0
$601$	10834.0	0.735321	0.367661	−	0.929960i	$-0.380159\pi$
$601$	10834.0	0.735321	0.367661	+	0.929960i	$0.380159\pi$
$602$	0	0
$603$	−9196.00	−0.621045
$604$	0	0
$605$	175.000	0.0117599
$606$	0	0
$607$	11576.0	0.774062	0.387031	−	0.922067i	$-0.373501\pi$
$607$	11576.0	0.774062	0.387031	+	0.922067i	$0.373501\pi$
$608$	0	0
$609$	−3304.00	−0.219844
$610$	0	0
$611$	−5904.00	−0.390917
$612$	0	0
$613$	−19034.0	−1.25412	−0.627060	−	0.778971i	$-0.715742\pi$
$613$	−19034.0	−1.25412	−0.627060	+	0.778971i	$0.715742\pi$
$614$	0	0
$615$	−600.000	−0.0393404
$616$	0	0
$617$	−28606.0	−1.86651	−0.933253	−	0.359220i	$-0.883043\pi$
$617$	−28606.0	−1.86651	−0.933253	+	0.359220i	$0.883043\pi$
$618$	0	0
$619$	−10352.0	−0.672184	−0.336092	−	0.941829i	$-0.609105\pi$
$619$	−10352.0	−0.672184	−0.336092	+	0.941829i	$0.609105\pi$
$620$	0	0
$621$	−2432.00	−0.157154
$622$	0	0
$623$	−5054.00	−0.325015
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−8064.00	−0.513629
$628$	0	0
$629$	1380.00	0.0874789
$630$	0	0
$631$	2204.00	0.139049	0.0695244	−	0.997580i	$-0.477852\pi$
$631$	2204.00	0.139049	0.0695244	+	0.997580i	$0.477852\pi$
$632$	0	0
$633$	1872.00	0.117544
$634$	0	0
$635$	6960.00	0.434959
$636$	0	0
$637$	4018.00	0.249920
$638$	0	0
$639$	−11220.0	−0.694611
$640$	0	0
$641$	−8654.00	−0.533249	−0.266624	−	0.963801i	$-0.585908\pi$
$641$	−8654.00	−0.533249	−0.266624	+	0.963801i	$0.585908\pi$
$642$	0	0
$643$	12212.0	0.748980	0.374490	−	0.927231i	$-0.377818\pi$
$643$	12212.0	0.748980	0.374490	+	0.927231i	$0.377818\pi$
$644$	0	0
$645$	4560.00	0.278372
$646$	0	0
$647$	11096.0	0.674233	0.337117	−	0.941463i	$-0.390548\pi$
$647$	11096.0	0.674233	0.337117	+	0.941463i	$0.390548\pi$
$648$	0	0
$649$	−1440.00	−0.0870954
$650$	0	0
$651$	−2912.00	−0.175315
$652$	0	0
$653$	22134.0	1.32645	0.663224	−	0.748421i	$-0.269188\pi$
$653$	22134.0	1.32645	0.663224	+	0.748421i	$0.269188\pi$
$654$	0	0
$655$	−7680.00	−0.458141
$656$	0	0
$657$	6358.00	0.377548
$658$	0	0
$659$	−12948.0	−0.765376	−0.382688	−	0.923878i	$-0.625001\pi$
$659$	−12948.0	−0.765376	−0.382688	+	0.923878i	$0.625001\pi$
$660$	0	0
$661$	−2894.00	−0.170293	−0.0851464	−	0.996368i	$-0.527136\pi$
$661$	−2894.00	−0.170293	−0.0851464	+	0.996368i	$0.527136\pi$
$662$	0	0
$663$	−1968.00	−0.115280
$664$	0	0
$665$	−1960.00	−0.114294
$666$	0	0
$667$	1888.00	0.109601
$668$	0	0
$669$	17792.0	1.02822
$670$	0	0
$671$	3672.00	0.211261
$672$	0	0
$673$	4482.00	0.256714	0.128357	−	0.991728i	$-0.459030\pi$
$673$	4482.00	0.256714	0.128357	+	0.991728i	$0.459030\pi$
$674$	0	0
$675$	3800.00	0.216685
$676$	0	0
$677$	9498.00	0.539199	0.269600	−	0.962973i	$-0.413109\pi$
$677$	9498.00	0.539199	0.269600	+	0.962973i	$0.413109\pi$
$678$	0	0
$679$	−1162.00	−0.0656752
$680$	0	0
$681$	−10416.0	−0.586112
$682$	0	0
$683$	21972.0	1.23094	0.615472	−	0.788158i	$-0.288965\pi$
$683$	21972.0	1.23094	0.615472	+	0.788158i	$0.288965\pi$
$684$	0	0
$685$	−11010.0	−0.614117
$686$	0	0
$687$	−16520.0	−0.917434
$688$	0	0
$689$	49692.0	2.74763
$690$	0	0
$691$	−18208.0	−1.00241	−0.501205	−	0.865329i	$-0.667110\pi$
$691$	−18208.0	−1.00241	−0.501205	+	0.865329i	$0.667110\pi$
$692$	0	0
$693$	−2772.00	−0.151947
$694$	0	0
$695$	8000.00	0.436629
$696$	0	0
$697$	−180.000	−0.00978190
$698$	0	0
$699$	−8648.00	−0.467951
$700$	0	0
$701$	506.000	0.0272630	0.0136315	−	0.999907i	$-0.495661\pi$
$701$	506.000	0.0272630	0.0136315	+	0.999907i	$0.495661\pi$
$702$	0	0
$703$	−12880.0	−0.691008
$704$	0	0
$705$	−1440.00	−0.0769270
$706$	0	0
$707$	1330.00	0.0707494
$708$	0	0
$709$	10066.0	0.533197	0.266598	−	0.963808i	$-0.414100\pi$
$709$	10066.0	0.533197	0.266598	+	0.963808i	$0.414100\pi$
$710$	0	0
$711$	14564.0	0.768203
$712$	0	0
$713$	1664.00	0.0874015
$714$	0	0
$715$	14760.0	0.772018
$716$	0	0
$717$	9616.00	0.500859
$718$	0	0
$719$	−17984.0	−0.932809	−0.466405	−	0.884571i	$-0.654451\pi$
$719$	−17984.0	−0.932809	−0.466405	+	0.884571i	$0.654451\pi$
$720$	0	0
$721$	8064.00	0.416531
$722$	0	0
$723$	−2216.00	−0.113989
$724$	0	0
$725$	−2950.00	−0.151118
$726$	0	0
$727$	32856.0	1.67615	0.838075	−	0.545554i	$-0.183681\pi$
$727$	32856.0	1.67615	0.838075	+	0.545554i	$0.183681\pi$
$728$	0	0
$729$	19837.0	1.00782
$730$	0	0
$731$	1368.00	0.0692166
$732$	0	0
$733$	30906.0	1.55735	0.778676	−	0.627426i	$-0.215892\pi$
$733$	30906.0	1.55735	0.778676	+	0.627426i	$0.215892\pi$
$734$	0	0
$735$	980.000	0.0491807
$736$	0	0
$737$	−30096.0	−1.50421
$738$	0	0
$739$	−34596.0	−1.72210	−0.861052	−	0.508517i	$-0.830194\pi$
$739$	−34596.0	−1.72210	−0.861052	+	0.508517i	$0.830194\pi$
$740$	0	0
$741$	18368.0	0.910614
$742$	0	0
$743$	12168.0	0.600808	0.300404	−	0.953812i	$-0.402878\pi$
$743$	12168.0	0.600808	0.300404	+	0.953812i	$0.402878\pi$
$744$	0	0
$745$	6270.00	0.308342
$746$	0	0
$747$	−2684.00	−0.131462
$748$	0	0
$749$	756.000	0.0368807
$750$	0	0
$751$	29524.0	1.43455	0.717274	−	0.696791i	$-0.245389\pi$
$751$	29524.0	1.43455	0.717274	+	0.696791i	$0.245389\pi$
$752$	0	0
$753$	2400.00	0.116150
$754$	0	0
$755$	14580.0	0.702809
$756$	0	0
$757$	20310.0	0.975138	0.487569	−	0.873084i	$-0.337884\pi$
$757$	20310.0	0.975138	0.487569	+	0.873084i	$0.337884\pi$
$758$	0	0
$759$	−2304.00	−0.110184
$760$	0	0
$761$	−16950.0	−0.807407	−0.403704	−	0.914890i	$-0.632277\pi$
$761$	−16950.0	−0.807407	−0.403704	+	0.914890i	$0.632277\pi$
$762$	0	0
$763$	−7070.00	−0.335454
$764$	0	0
$765$	330.000	0.0155963
$766$	0	0
$767$	3280.00	0.154412
$768$	0	0
$769$	−17126.0	−0.803094	−0.401547	−	0.915838i	$-0.631527\pi$
$769$	−17126.0	−0.803094	−0.401547	+	0.915838i	$0.631527\pi$
$770$	0	0
$771$	−11256.0	−0.525778
$772$	0	0
$773$	−28718.0	−1.33624	−0.668121	−	0.744053i	$-0.732901\pi$
$773$	−28718.0	−1.33624	−0.668121	+	0.744053i	$0.732901\pi$
$774$	0	0
$775$	−2600.00	−0.120509
$776$	0	0
$777$	6440.00	0.297341
$778$	0	0
$779$	1680.00	0.0772686
$780$	0	0
$781$	−36720.0	−1.68239
$782$	0	0
$783$	−17936.0	−0.818621
$784$	0	0
$785$	9670.00	0.439665
$786$	0	0
$787$	31012.0	1.40465	0.702324	−	0.711857i	$-0.252146\pi$
$787$	31012.0	1.40465	0.702324	+	0.711857i	$0.252146\pi$
$788$	0	0
$789$	−448.000	−0.0202145
$790$	0	0
$791$	−6286.00	−0.282559
$792$	0	0
$793$	−8364.00	−0.374545
$794$	0	0
$795$	12120.0	0.540694
$796$	0	0
$797$	14450.0	0.642215	0.321107	−	0.947043i	$-0.395945\pi$
$797$	14450.0	0.642215	0.321107	+	0.947043i	$0.395945\pi$
$798$	0	0
$799$	−432.000	−0.0191277
$800$	0	0
$801$	−7942.00	−0.350333
$802$	0	0
$803$	20808.0	0.914444
$804$	0	0
$805$	−560.000	−0.0245185
$806$	0	0
$807$	15128.0	0.659889
$808$	0	0
$809$	12026.0	0.522635	0.261317	−	0.965253i	$-0.415843\pi$
$809$	12026.0	0.522635	0.261317	+	0.965253i	$0.415843\pi$
$810$	0	0
$811$	33336.0	1.44339	0.721693	−	0.692214i	$-0.243364\pi$
$811$	33336.0	1.44339	0.721693	+	0.692214i	$0.243364\pi$
$812$	0	0
$813$	−2176.00	−0.0938692
$814$	0	0
$815$	860.000	0.0369626
$816$	0	0
$817$	−12768.0	−0.546751
$818$	0	0
$819$	6314.00	0.269388
$820$	0	0
$821$	−15198.0	−0.646058	−0.323029	−	0.946389i	$-0.604701\pi$
$821$	−15198.0	−0.646058	−0.323029	+	0.946389i	$0.604701\pi$
$822$	0	0
$823$	−30696.0	−1.30012	−0.650058	−	0.759885i	$-0.725255\pi$
$823$	−30696.0	−1.30012	−0.650058	+	0.759885i	$0.725255\pi$
$824$	0	0
$825$	3600.00	0.151922
$826$	0	0
$827$	21900.0	0.920844	0.460422	−	0.887700i	$-0.347698\pi$
$827$	21900.0	0.920844	0.460422	+	0.887700i	$0.347698\pi$
$828$	0	0
$829$	24954.0	1.04546	0.522731	−	0.852498i	$-0.324913\pi$
$829$	24954.0	1.04546	0.522731	+	0.852498i	$0.324913\pi$
$830$	0	0
$831$	34952.0	1.45905
$832$	0	0
$833$	294.000	0.0122287
$834$	0	0
$835$	−8200.00	−0.339848
$836$	0	0
$837$	−15808.0	−0.652813
$838$	0	0
$839$	−15928.0	−0.655418	−0.327709	−	0.944779i	$-0.606277\pi$
$839$	−15928.0	−0.655418	−0.327709	+	0.944779i	$0.606277\pi$
$840$	0	0
$841$	−10465.0	−0.429087
$842$	0	0
$843$	18264.0	0.746199
$844$	0	0
$845$	−22635.0	−0.921500
$846$	0	0
$847$	245.000	0.00993896
$848$	0	0
$849$	−33200.0	−1.34207
$850$	0	0
$851$	−3680.00	−0.148236
$852$	0	0
$853$	35498.0	1.42489	0.712443	−	0.701730i	$-0.247589\pi$
$853$	35498.0	1.42489	0.712443	+	0.701730i	$0.247589\pi$
$854$	0	0
$855$	−3080.00	−0.123197
$856$	0	0
$857$	37414.0	1.49129	0.745646	−	0.666342i	$-0.232141\pi$
$857$	37414.0	1.49129	0.745646	+	0.666342i	$0.232141\pi$
$858$	0	0
$859$	32064.0	1.27359	0.636793	−	0.771035i	$-0.280261\pi$
$859$	32064.0	1.27359	0.636793	+	0.771035i	$0.280261\pi$
$860$	0	0
$861$	−840.000	−0.0332487
$862$	0	0
$863$	−33696.0	−1.32911	−0.664557	−	0.747238i	$-0.731380\pi$
$863$	−33696.0	−1.32911	−0.664557	+	0.747238i	$0.731380\pi$
$864$	0	0
$865$	8070.00	0.317212
$866$	0	0
$867$	19508.0	0.764160
$868$	0	0
$869$	47664.0	1.86063
$870$	0	0
$871$	68552.0	2.66682
$872$	0	0
$873$	−1826.00	−0.0707912
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	9958.00	0.383418	0.191709	−	0.981452i	$-0.438597\pi$
$877$	9958.00	0.383418	0.191709	+	0.981452i	$0.438597\pi$
$878$	0	0
$879$	−4104.00	−0.157480
$880$	0	0
$881$	−13958.0	−0.533776	−0.266888	−	0.963728i	$-0.585995\pi$
$881$	−13958.0	−0.533776	−0.266888	+	0.963728i	$0.585995\pi$
$882$	0	0
$883$	25628.0	0.976728	0.488364	−	0.872640i	$-0.337594\pi$
$883$	25628.0	0.976728	0.488364	+	0.872640i	$0.337594\pi$
$884$	0	0
$885$	800.000	0.0303861
$886$	0	0
$887$	−38544.0	−1.45905	−0.729527	−	0.683952i	$-0.760260\pi$
$887$	−38544.0	−1.45905	−0.729527	+	0.683952i	$0.760260\pi$
$888$	0	0
$889$	9744.00	0.367608
$890$	0	0
$891$	11196.0	0.420965
$892$	0	0
$893$	4032.00	0.151093
$894$	0	0
$895$	12100.0	0.451909
$896$	0	0
$897$	5248.00	0.195346
$898$	0	0
$899$	12272.0	0.455277
$900$	0	0
$901$	3636.00	0.134443
$902$	0	0
$903$	6384.00	0.235267
$904$	0	0
$905$	−18410.0	−0.676209
$906$	0	0
$907$	33676.0	1.23285	0.616424	−	0.787414i	$-0.288581\pi$
$907$	33676.0	1.23285	0.616424	+	0.787414i	$0.288581\pi$
$908$	0	0
$909$	2090.00	0.0762606
$910$	0	0
$911$	1356.00	0.0493154	0.0246577	−	0.999696i	$-0.492150\pi$
$911$	1356.00	0.0493154	0.0246577	+	0.999696i	$0.492150\pi$
$912$	0	0
$913$	−8784.00	−0.318410
$914$	0	0
$915$	−2040.00	−0.0737053
$916$	0	0
$917$	−10752.0	−0.387200
$918$	0	0
$919$	−6100.00	−0.218956	−0.109478	−	0.993989i	$-0.534918\pi$
$919$	−6100.00	−0.218956	−0.109478	+	0.993989i	$0.534918\pi$
$920$	0	0
$921$	32560.0	1.16492
$922$	0	0
$923$	83640.0	2.98271
$924$	0	0
$925$	5750.00	0.204388
$926$	0	0
$927$	12672.0	0.448979
$928$	0	0
$929$	−20142.0	−0.711343	−0.355671	−	0.934611i	$-0.615748\pi$
$929$	−20142.0	−0.711343	−0.355671	+	0.934611i	$0.615748\pi$
$930$	0	0
$931$	−2744.00	−0.0965961
$932$	0	0
$933$	2176.00	0.0763548
$934$	0	0
$935$	1080.00	0.0377752
$936$	0	0
$937$	−39994.0	−1.39439	−0.697197	−	0.716880i	$-0.745570\pi$
$937$	−39994.0	−1.39439	−0.697197	+	0.716880i	$0.745570\pi$
$938$	0	0
$939$	13672.0	0.475153
$940$	0	0
$941$	−51830.0	−1.79555	−0.897773	−	0.440457i	$-0.854816\pi$
$941$	−51830.0	−1.79555	−0.897773	+	0.440457i	$0.854816\pi$
$942$	0	0
$943$	480.000	0.0165758
$944$	0	0
$945$	5320.00	0.183132
$946$	0	0
$947$	−1972.00	−0.0676678	−0.0338339	−	0.999427i	$-0.510772\pi$
$947$	−1972.00	−0.0676678	−0.0338339	+	0.999427i	$0.510772\pi$
$948$	0	0
$949$	−47396.0	−1.62122
$950$	0	0
$951$	28968.0	0.987752
$952$	0	0
$953$	−44526.0	−1.51347	−0.756736	−	0.653721i	$-0.773207\pi$
$953$	−44526.0	−1.51347	−0.756736	+	0.653721i	$0.773207\pi$
$954$	0	0
$955$	7900.00	0.267684
$956$	0	0
$957$	−16992.0	−0.573953
$958$	0	0
$959$	−15414.0	−0.519024
$960$	0	0
$961$	−18975.0	−0.636937
$962$	0	0
$963$	1188.00	0.0397537
$964$	0	0
$965$	9790.00	0.326582
$966$	0	0
$967$	−55056.0	−1.83090	−0.915451	−	0.402430i	$-0.868166\pi$
$967$	−55056.0	−1.83090	−0.915451	+	0.402430i	$0.868166\pi$
$968$	0	0
$969$	1344.00	0.0445568
$970$	0	0
$971$	35288.0	1.16627	0.583134	−	0.812376i	$-0.301826\pi$
$971$	35288.0	1.16627	0.583134	+	0.812376i	$0.301826\pi$
$972$	0	0
$973$	11200.0	0.369019
$974$	0	0
$975$	−8200.00	−0.269344
$976$	0	0
$977$	−30238.0	−0.990173	−0.495087	−	0.868844i	$-0.664864\pi$
$977$	−30238.0	−0.990173	−0.495087	+	0.868844i	$0.664864\pi$
$978$	0	0
$979$	−25992.0	−0.848527
$980$	0	0
$981$	−11110.0	−0.361585
$982$	0	0
$983$	−9768.00	−0.316939	−0.158469	−	0.987364i	$-0.550656\pi$
$983$	−9768.00	−0.316939	−0.158469	+	0.987364i	$0.550656\pi$
$984$	0	0
$985$	−10190.0	−0.329625
$986$	0	0
$987$	−2016.00	−0.0650152
$988$	0	0
$989$	−3648.00	−0.117290
$990$	0	0
$991$	31580.0	1.01228	0.506141	−	0.862451i	$-0.331071\pi$
$991$	31580.0	1.01228	0.506141	+	0.862451i	$0.331071\pi$
$992$	0	0
$993$	37392.0	1.19496
$994$	0	0
$995$	9120.00	0.290576
$996$	0	0
$997$	−34046.0	−1.08149	−0.540746	−	0.841186i	$-0.681858\pi$
$997$	−34046.0	−1.08149	−0.540746	+	0.841186i	$0.681858\pi$
$998$	0	0
$999$	34960.0	1.10719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.k.1.1		1
4.3	odd	2		2240.4.a.ba.1.1		1
8.3	odd	2		1120.4.a.b.1.1	✓	1
8.5	even	2		1120.4.a.d.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.4.a.b.1.1	✓	1	8.3	odd	2
1120.4.a.d.1.1	yes	1	8.5	even	2
2240.4.a.k.1.1		1	1.1	even	1	trivial
2240.4.a.ba.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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