LMFDB - Embedded newform 2352.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -15.0000 q^{5} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -15.0000 q^{5} +9.00000 q^{9} +9.00000 q^{11} -88.0000 q^{13} -45.0000 q^{15} -84.0000 q^{17} -104.000 q^{19} +84.0000 q^{23} +100.000 q^{25} +27.0000 q^{27} +51.0000 q^{29} -185.000 q^{31} +27.0000 q^{33} +44.0000 q^{37} -264.000 q^{39} -168.000 q^{41} -326.000 q^{43} -135.000 q^{45} +138.000 q^{47} -252.000 q^{51} +639.000 q^{53} -135.000 q^{55} -312.000 q^{57} -159.000 q^{59} +722.000 q^{61} +1320.00 q^{65} +166.000 q^{67} +252.000 q^{69} -1086.00 q^{71} +218.000 q^{73} +300.000 q^{75} +583.000 q^{79} +81.0000 q^{81} +597.000 q^{83} +1260.00 q^{85} +153.000 q^{87} -1038.00 q^{89} -555.000 q^{93} +1560.00 q^{95} -169.000 q^{97} +81.0000 q^{99} +O(q^{100})$

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−15.0000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−15.0000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	9.00000	0.246691	0.123346	−	0.992364i	$-0.460638\pi$
$11$	9.00000	0.246691	0.123346	+	0.992364i	$0.460638\pi$
$12$	0	0
$13$	−88.0000	−1.87745	−0.938723	−	0.344671i	$-0.887990\pi$
$13$	−88.0000	−1.87745	−0.938723	+	0.344671i	$0.887990\pi$
$14$	0	0
$15$	−45.0000	−0.774597
$16$	0	0
$17$	−84.0000	−1.19841	−0.599206	−	0.800595i	$-0.704517\pi$
$17$	−84.0000	−1.19841	−0.599206	+	0.800595i	$0.704517\pi$
$18$	0	0
$19$	−104.000	−1.25575	−0.627875	−	0.778314i	$-0.716075\pi$
$19$	−104.000	−1.25575	−0.627875	+	0.778314i	$0.716075\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	84.0000	0.761531	0.380765	−	0.924672i	$-0.375661\pi$
$23$	84.0000	0.761531	0.380765	+	0.924672i	$0.375661\pi$
$24$	0	0
$25$	100.000	0.800000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	51.0000	0.326568	0.163284	−	0.986579i	$-0.447791\pi$
$29$	51.0000	0.326568	0.163284	+	0.986579i	$0.447791\pi$
$30$	0	0
$31$	−185.000	−1.07184	−0.535919	−	0.844269i	$-0.680035\pi$
$31$	−185.000	−1.07184	−0.535919	+	0.844269i	$0.680035\pi$
$32$	0	0
$33$	27.0000	0.142427
$34$	0	0
$35$	0	0
$36$	0	0
$37$	44.0000	0.195501	0.0977507	−	0.995211i	$-0.468835\pi$
$37$	44.0000	0.195501	0.0977507	+	0.995211i	$0.468835\pi$
$38$	0	0
$39$	−264.000	−1.08394
$40$	0	0
$41$	−168.000	−0.639932	−0.319966	−	0.947429i	$-0.603671\pi$
$41$	−168.000	−0.639932	−0.319966	+	0.947429i	$0.603671\pi$
$42$	0	0
$43$	−326.000	−1.15615	−0.578076	−	0.815983i	$-0.696196\pi$
$43$	−326.000	−1.15615	−0.578076	+	0.815983i	$0.696196\pi$
$44$	0	0
$45$	−135.000	−0.447214
$46$	0	0
$47$	138.000	0.428284	0.214142	−	0.976802i	$-0.431304\pi$
$47$	138.000	0.428284	0.214142	+	0.976802i	$0.431304\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−252.000	−0.691903
$52$	0	0
$53$	639.000	1.65610	0.828051	−	0.560653i	$-0.189450\pi$
$53$	639.000	1.65610	0.828051	+	0.560653i	$0.189450\pi$
$54$	0	0
$55$	−135.000	−0.330971
$56$	0	0
$57$	−312.000	−0.725007
$58$	0	0
$59$	−159.000	−0.350848	−0.175424	−	0.984493i	$-0.556130\pi$
$59$	−159.000	−0.350848	−0.175424	+	0.984493i	$0.556130\pi$
$60$	0	0
$61$	722.000	1.51545	0.757726	−	0.652572i	$-0.226310\pi$
$61$	722.000	1.51545	0.757726	+	0.652572i	$0.226310\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1320.00	2.51886
$66$	0	0
$67$	166.000	0.302688	0.151344	−	0.988481i	$-0.451640\pi$
$67$	166.000	0.302688	0.151344	+	0.988481i	$0.451640\pi$
$68$	0	0
$69$	252.000	0.439670
$70$	0	0
$71$	−1086.00	−1.81527	−0.907637	−	0.419755i	$-0.862116\pi$
$71$	−1086.00	−1.81527	−0.907637	+	0.419755i	$0.862116\pi$
$72$	0	0
$73$	218.000	0.349520	0.174760	−	0.984611i	$-0.444085\pi$
$73$	218.000	0.349520	0.174760	+	0.984611i	$0.444085\pi$
$74$	0	0
$75$	300.000	0.461880
$76$	0	0
$77$	0	0
$78$	0	0
$79$	583.000	0.830286	0.415143	−	0.909756i	$-0.363731\pi$
$79$	583.000	0.830286	0.415143	+	0.909756i	$0.363731\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	597.000	0.789509	0.394755	−	0.918787i	$-0.370830\pi$
$83$	597.000	0.789509	0.394755	+	0.918787i	$0.370830\pi$
$84$	0	0
$85$	1260.00	1.60784
$86$	0	0
$87$	153.000	0.188544
$88$	0	0
$89$	−1038.00	−1.23627	−0.618134	−	0.786073i	$-0.712111\pi$
$89$	−1038.00	−1.23627	−0.618134	+	0.786073i	$0.712111\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−555.000	−0.618826
$94$	0	0
$95$	1560.00	1.68476
$96$	0	0
$97$	−169.000	−0.176901	−0.0884503	−	0.996081i	$-0.528191\pi$
$97$	−169.000	−0.176901	−0.0884503	+	0.996081i	$0.528191\pi$
$98$	0	0
$99$	81.0000	0.0822304
$100$	0	0
$101$	642.000	0.632489	0.316244	−	0.948678i	$-0.397578\pi$
$101$	642.000	0.632489	0.316244	+	0.948678i	$0.397578\pi$
$102$	0	0
$103$	−464.000	−0.443876	−0.221938	−	0.975061i	$-0.571238\pi$
$103$	−464.000	−0.443876	−0.221938	+	0.975061i	$0.571238\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−393.000	−0.355072	−0.177536	−	0.984114i	$-0.556813\pi$
$107$	−393.000	−0.355072	−0.177536	+	0.984114i	$0.556813\pi$
$108$	0	0
$109$	14.0000	0.0123024	0.00615118	−	0.999981i	$-0.498042\pi$
$109$	14.0000	0.0123024	0.00615118	+	0.999981i	$0.498042\pi$
$110$	0	0
$111$	132.000	0.112873
$112$	0	0
$113$	−2184.00	−1.81817	−0.909086	−	0.416608i	$-0.863219\pi$
$113$	−2184.00	−1.81817	−0.909086	+	0.416608i	$0.863219\pi$
$114$	0	0
$115$	−1260.00	−1.02170
$116$	0	0
$117$	−792.000	−0.625816
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1250.00	−0.939144
$122$	0	0
$123$	−504.000	−0.369465
$124$	0	0
$125$	375.000	0.268328
$126$	0	0
$127$	373.000	0.260617	0.130309	−	0.991473i	$-0.458403\pi$
$127$	373.000	0.260617	0.130309	+	0.991473i	$0.458403\pi$
$128$	0	0
$129$	−978.000	−0.667505
$130$	0	0
$131$	1173.00	0.782332	0.391166	−	0.920320i	$-0.372072\pi$
$131$	1173.00	0.782332	0.391166	+	0.920320i	$0.372072\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−405.000	−0.258199
$136$	0	0
$137$	30.0000	0.0187086	0.00935428	−	0.999956i	$-0.497022\pi$
$137$	30.0000	0.0187086	0.00935428	+	0.999956i	$0.497022\pi$
$138$	0	0
$139$	82.0000	0.0500370	0.0250185	−	0.999687i	$-0.492036\pi$
$139$	82.0000	0.0500370	0.0250185	+	0.999687i	$0.492036\pi$
$140$	0	0
$141$	414.000	0.247270
$142$	0	0
$143$	−792.000	−0.463149
$144$	0	0
$145$	−765.000	−0.438136
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1434.00	−0.788442	−0.394221	−	0.919016i	$-0.628986\pi$
$149$	−1434.00	−0.788442	−0.394221	+	0.919016i	$0.628986\pi$
$150$	0	0
$151$	2671.00	1.43949	0.719745	−	0.694239i	$-0.244259\pi$
$151$	2671.00	1.43949	0.719745	+	0.694239i	$0.244259\pi$
$152$	0	0
$153$	−756.000	−0.399470
$154$	0	0
$155$	2775.00	1.43802
$156$	0	0
$157$	2252.00	1.14477	0.572386	−	0.819984i	$-0.306018\pi$
$157$	2252.00	1.14477	0.572386	+	0.819984i	$0.306018\pi$
$158$	0	0
$159$	1917.00	0.956151
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1676.00	−0.805365	−0.402682	−	0.915340i	$-0.631922\pi$
$163$	−1676.00	−0.805365	−0.402682	+	0.915340i	$0.631922\pi$
$164$	0	0
$165$	−405.000	−0.191086
$166$	0	0
$167$	−3030.00	−1.40400	−0.702001	−	0.712176i	$-0.747710\pi$
$167$	−3030.00	−1.40400	−0.702001	+	0.712176i	$0.747710\pi$
$168$	0	0
$169$	5547.00	2.52481
$170$	0	0
$171$	−936.000	−0.418583
$172$	0	0
$173$	3438.00	1.51090	0.755452	−	0.655204i	$-0.227417\pi$
$173$	3438.00	1.51090	0.755452	+	0.655204i	$0.227417\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−477.000	−0.202562
$178$	0	0
$179$	1212.00	0.506085	0.253042	−	0.967455i	$-0.418569\pi$
$179$	1212.00	0.506085	0.253042	+	0.967455i	$0.418569\pi$
$180$	0	0
$181$	3032.00	1.24512	0.622560	−	0.782572i	$-0.286093\pi$
$181$	3032.00	1.24512	0.622560	+	0.782572i	$0.286093\pi$
$182$	0	0
$183$	2166.00	0.874947
$184$	0	0
$185$	−660.000	−0.262293
$186$	0	0
$187$	−756.000	−0.295637
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2520.00	−0.954664	−0.477332	−	0.878723i	$-0.658396\pi$
$191$	−2520.00	−0.954664	−0.477332	+	0.878723i	$0.658396\pi$
$192$	0	0
$193$	365.000	0.136131	0.0680655	−	0.997681i	$-0.478317\pi$
$193$	365.000	0.136131	0.0680655	+	0.997681i	$0.478317\pi$
$194$	0	0
$195$	3960.00	1.45426
$196$	0	0
$197$	−1590.00	−0.575040	−0.287520	−	0.957775i	$-0.592831\pi$
$197$	−1590.00	−0.575040	−0.287520	+	0.957775i	$0.592831\pi$
$198$	0	0
$199$	5380.00	1.91647	0.958236	−	0.285977i	$-0.0923182\pi$
$199$	5380.00	1.91647	0.958236	+	0.285977i	$0.0923182\pi$
$200$	0	0
$201$	498.000	0.174757
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2520.00	0.858558
$206$	0	0
$207$	756.000	0.253844
$208$	0	0
$209$	−936.000	−0.309782
$210$	0	0
$211$	5362.00	1.74946	0.874728	−	0.484614i	$-0.161040\pi$
$211$	5362.00	1.74946	0.874728	+	0.484614i	$0.161040\pi$
$212$	0	0
$213$	−3258.00	−1.04805
$214$	0	0
$215$	4890.00	1.55114
$216$	0	0
$217$	0	0
$218$	0	0
$219$	654.000	0.201796
$220$	0	0
$221$	7392.00	2.24995
$222$	0	0
$223$	1573.00	0.472358	0.236179	−	0.971710i	$-0.424105\pi$
$223$	1573.00	0.472358	0.236179	+	0.971710i	$0.424105\pi$
$224$	0	0
$225$	900.000	0.266667
$226$	0	0
$227$	−921.000	−0.269290	−0.134645	−	0.990894i	$-0.542989\pi$
$227$	−921.000	−0.269290	−0.134645	+	0.990894i	$0.542989\pi$
$228$	0	0
$229$	4052.00	1.16927	0.584637	−	0.811295i	$-0.301237\pi$
$229$	4052.00	1.16927	0.584637	+	0.811295i	$0.301237\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−468.000	−0.131587	−0.0657933	−	0.997833i	$-0.520958\pi$
$233$	−468.000	−0.131587	−0.0657933	+	0.997833i	$0.520958\pi$
$234$	0	0
$235$	−2070.00	−0.574604
$236$	0	0
$237$	1749.00	0.479366
$238$	0	0
$239$	−4932.00	−1.33483	−0.667415	−	0.744686i	$-0.732599\pi$
$239$	−4932.00	−1.33483	−0.667415	+	0.744686i	$0.732599\pi$
$240$	0	0
$241$	−1537.00	−0.410817	−0.205408	−	0.978676i	$-0.565852\pi$
$241$	−1537.00	−0.410817	−0.205408	+	0.978676i	$0.565852\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	9152.00	2.35760
$248$	0	0
$249$	1791.00	0.455823
$250$	0	0
$251$	−5319.00	−1.33758	−0.668789	−	0.743452i	$-0.733187\pi$
$251$	−5319.00	−1.33758	−0.668789	+	0.743452i	$0.733187\pi$
$252$	0	0
$253$	756.000	0.187863
$254$	0	0
$255$	3780.00	0.928285
$256$	0	0
$257$	5346.00	1.29757	0.648783	−	0.760974i	$-0.275278\pi$
$257$	5346.00	1.29757	0.648783	+	0.760974i	$0.275278\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	459.000	0.108856
$262$	0	0
$263$	−774.000	−0.181471	−0.0907355	−	0.995875i	$-0.528922\pi$
$263$	−774.000	−0.181471	−0.0907355	+	0.995875i	$0.528922\pi$
$264$	0	0
$265$	−9585.00	−2.22189
$266$	0	0
$267$	−3114.00	−0.713759
$268$	0	0
$269$	2415.00	0.547380	0.273690	−	0.961818i	$-0.411756\pi$
$269$	2415.00	0.547380	0.273690	+	0.961818i	$0.411756\pi$
$270$	0	0
$271$	475.000	0.106473	0.0532365	−	0.998582i	$-0.483046\pi$
$271$	475.000	0.106473	0.0532365	+	0.998582i	$0.483046\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	900.000	0.197353
$276$	0	0
$277$	3728.00	0.808642	0.404321	−	0.914617i	$-0.367508\pi$
$277$	3728.00	0.808642	0.404321	+	0.914617i	$0.367508\pi$
$278$	0	0
$279$	−1665.00	−0.357279
$280$	0	0
$281$	1602.00	0.340097	0.170049	−	0.985436i	$-0.445608\pi$
$281$	1602.00	0.340097	0.170049	+	0.985436i	$0.445608\pi$
$282$	0	0
$283$	−686.000	−0.144094	−0.0720468	−	0.997401i	$-0.522953\pi$
$283$	−686.000	−0.144094	−0.0720468	+	0.997401i	$0.522953\pi$
$284$	0	0
$285$	4680.00	0.972699
$286$	0	0
$287$	0	0
$288$	0	0
$289$	2143.00	0.436190
$290$	0	0
$291$	−507.000	−0.102134
$292$	0	0
$293$	−1101.00	−0.219526	−0.109763	−	0.993958i	$-0.535009\pi$
$293$	−1101.00	−0.219526	−0.109763	+	0.993958i	$0.535009\pi$
$294$	0	0
$295$	2385.00	0.470712
$296$	0	0
$297$	243.000	0.0474757
$298$	0	0
$299$	−7392.00	−1.42973
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1926.00	0.365168
$304$	0	0
$305$	−10830.0	−2.03319
$306$	0	0
$307$	−2780.00	−0.516818	−0.258409	−	0.966036i	$-0.583198\pi$
$307$	−2780.00	−0.516818	−0.258409	+	0.966036i	$0.583198\pi$
$308$	0	0
$309$	−1392.00	−0.256272
$310$	0	0
$311$	−4296.00	−0.783292	−0.391646	−	0.920116i	$-0.628094\pi$
$311$	−4296.00	−0.783292	−0.391646	+	0.920116i	$0.628094\pi$
$312$	0	0
$313$	5489.00	0.991235	0.495618	−	0.868541i	$-0.334942\pi$
$313$	5489.00	0.991235	0.495618	+	0.868541i	$0.334942\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4491.00	0.795709	0.397854	−	0.917449i	$-0.369755\pi$
$317$	4491.00	0.795709	0.397854	+	0.917449i	$0.369755\pi$
$318$	0	0
$319$	459.000	0.0805613
$320$	0	0
$321$	−1179.00	−0.205001
$322$	0	0
$323$	8736.00	1.50490
$324$	0	0
$325$	−8800.00	−1.50196
$326$	0	0
$327$	42.0000	0.00710277
$328$	0	0
$329$	0	0
$330$	0	0
$331$	3964.00	0.658251	0.329126	−	0.944286i	$-0.393246\pi$
$331$	3964.00	0.658251	0.329126	+	0.944286i	$0.393246\pi$
$332$	0	0
$333$	396.000	0.0651672
$334$	0	0
$335$	−2490.00	−0.406099
$336$	0	0
$337$	161.000	0.0260244	0.0130122	−	0.999915i	$-0.495858\pi$
$337$	161.000	0.0260244	0.0130122	+	0.999915i	$0.495858\pi$
$338$	0	0
$339$	−6552.00	−1.04972
$340$	0	0
$341$	−1665.00	−0.264413
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−3780.00	−0.589879
$346$	0	0
$347$	5916.00	0.915238	0.457619	−	0.889148i	$-0.348702\pi$
$347$	5916.00	0.915238	0.457619	+	0.889148i	$0.348702\pi$
$348$	0	0
$349$	−142.000	−0.0217796	−0.0108898	−	0.999941i	$-0.503466\pi$
$349$	−142.000	−0.0217796	−0.0108898	+	0.999941i	$0.503466\pi$
$350$	0	0
$351$	−2376.00	−0.361315
$352$	0	0
$353$	−4440.00	−0.669454	−0.334727	−	0.942315i	$-0.608644\pi$
$353$	−4440.00	−0.669454	−0.334727	+	0.942315i	$0.608644\pi$
$354$	0	0
$355$	16290.0	2.43545
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2286.00	−0.336074	−0.168037	−	0.985781i	$-0.553743\pi$
$359$	−2286.00	−0.336074	−0.168037	+	0.985781i	$0.553743\pi$
$360$	0	0
$361$	3957.00	0.576906
$362$	0	0
$363$	−3750.00	−0.542215
$364$	0	0
$365$	−3270.00	−0.468930
$366$	0	0
$367$	2869.00	0.408067	0.204033	−	0.978964i	$-0.434595\pi$
$367$	2869.00	0.408067	0.204033	+	0.978964i	$0.434595\pi$
$368$	0	0
$369$	−1512.00	−0.213311
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−3064.00	−0.425330	−0.212665	−	0.977125i	$-0.568214\pi$
$373$	−3064.00	−0.425330	−0.212665	+	0.977125i	$0.568214\pi$
$374$	0	0
$375$	1125.00	0.154919
$376$	0	0
$377$	−4488.00	−0.613113
$378$	0	0
$379$	6040.00	0.818612	0.409306	−	0.912397i	$-0.365771\pi$
$379$	6040.00	0.818612	0.409306	+	0.912397i	$0.365771\pi$
$380$	0	0
$381$	1119.00	0.150467
$382$	0	0
$383$	−1842.00	−0.245749	−0.122874	−	0.992422i	$-0.539211\pi$
$383$	−1842.00	−0.245749	−0.122874	+	0.992422i	$0.539211\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2934.00	−0.385384
$388$	0	0
$389$	7830.00	1.02056	0.510279	−	0.860009i	$-0.329542\pi$
$389$	7830.00	1.02056	0.510279	+	0.860009i	$0.329542\pi$
$390$	0	0
$391$	−7056.00	−0.912627
$392$	0	0
$393$	3519.00	0.451680
$394$	0	0
$395$	−8745.00	−1.11395
$396$	0	0
$397$	−14764.0	−1.86646	−0.933229	−	0.359282i	$-0.883022\pi$
$397$	−14764.0	−1.86646	−0.933229	+	0.359282i	$0.883022\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6264.00	0.780073	0.390036	−	0.920799i	$-0.372462\pi$
$401$	6264.00	0.780073	0.390036	+	0.920799i	$0.372462\pi$
$402$	0	0
$403$	16280.0	2.01232
$404$	0	0
$405$	−1215.00	−0.149071
$406$	0	0
$407$	396.000	0.0482285
$408$	0	0
$409$	4751.00	0.574381	0.287191	−	0.957873i	$-0.407279\pi$
$409$	4751.00	0.574381	0.287191	+	0.957873i	$0.407279\pi$
$410$	0	0
$411$	90.0000	0.0108014
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−8955.00	−1.05924
$416$	0	0
$417$	246.000	0.0288889
$418$	0	0
$419$	4704.00	0.548462	0.274231	−	0.961664i	$-0.411577\pi$
$419$	4704.00	0.548462	0.274231	+	0.961664i	$0.411577\pi$
$420$	0	0
$421$	−4474.00	−0.517932	−0.258966	−	0.965886i	$-0.583382\pi$
$421$	−4474.00	−0.517932	−0.258966	+	0.965886i	$0.583382\pi$
$422$	0	0
$423$	1242.00	0.142761
$424$	0	0
$425$	−8400.00	−0.958729
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−2376.00	−0.267399
$430$	0	0
$431$	12804.0	1.43097	0.715484	−	0.698629i	$-0.246206\pi$
$431$	12804.0	1.43097	0.715484	+	0.698629i	$0.246206\pi$
$432$	0	0
$433$	−5074.00	−0.563143	−0.281571	−	0.959540i	$-0.590856\pi$
$433$	−5074.00	−0.563143	−0.281571	+	0.959540i	$0.590856\pi$
$434$	0	0
$435$	−2295.00	−0.252958
$436$	0	0
$437$	−8736.00	−0.956292
$438$	0	0
$439$	1267.00	0.137746	0.0688731	−	0.997625i	$-0.478060\pi$
$439$	1267.00	0.137746	0.0688731	+	0.997625i	$0.478060\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6933.00	0.743559	0.371780	−	0.928321i	$-0.378748\pi$
$443$	6933.00	0.743559	0.371780	+	0.928321i	$0.378748\pi$
$444$	0	0
$445$	15570.0	1.65863
$446$	0	0
$447$	−4302.00	−0.455207
$448$	0	0
$449$	11688.0	1.22849	0.614244	−	0.789116i	$-0.289461\pi$
$449$	11688.0	1.22849	0.614244	+	0.789116i	$0.289461\pi$
$450$	0	0
$451$	−1512.00	−0.157865
$452$	0	0
$453$	8013.00	0.831090
$454$	0	0
$455$	0	0
$456$	0	0
$457$	551.000	0.0563998	0.0281999	−	0.999602i	$-0.491023\pi$
$457$	551.000	0.0563998	0.0281999	+	0.999602i	$0.491023\pi$
$458$	0	0
$459$	−2268.00	−0.230634
$460$	0	0
$461$	13386.0	1.35238	0.676191	−	0.736726i	$-0.263629\pi$
$461$	13386.0	1.35238	0.676191	+	0.736726i	$0.263629\pi$
$462$	0	0
$463$	6376.00	0.639995	0.319998	−	0.947418i	$-0.396318\pi$
$463$	6376.00	0.639995	0.319998	+	0.947418i	$0.396318\pi$
$464$	0	0
$465$	8325.00	0.830242
$466$	0	0
$467$	−5700.00	−0.564806	−0.282403	−	0.959296i	$-0.591132\pi$
$467$	−5700.00	−0.564806	−0.282403	+	0.959296i	$0.591132\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	6756.00	0.660934
$472$	0	0
$473$	−2934.00	−0.285212
$474$	0	0
$475$	−10400.0	−1.00460
$476$	0	0
$477$	5751.00	0.552034
$478$	0	0
$479$	−19794.0	−1.88812	−0.944062	−	0.329769i	$-0.893029\pi$
$479$	−19794.0	−1.88812	−0.944062	+	0.329769i	$0.893029\pi$
$480$	0	0
$481$	−3872.00	−0.367044
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2535.00	0.237337
$486$	0	0
$487$	−15935.0	−1.48272	−0.741359	−	0.671109i	$-0.765818\pi$
$487$	−15935.0	−1.48272	−0.741359	+	0.671109i	$0.765818\pi$
$488$	0	0
$489$	−5028.00	−0.464978
$490$	0	0
$491$	−9963.00	−0.915731	−0.457865	−	0.889021i	$-0.651386\pi$
$491$	−9963.00	−0.915731	−0.457865	+	0.889021i	$0.651386\pi$
$492$	0	0
$493$	−4284.00	−0.391362
$494$	0	0
$495$	−1215.00	−0.110324
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−19142.0	−1.71726	−0.858631	−	0.512594i	$-0.828684\pi$
$499$	−19142.0	−1.71726	−0.858631	+	0.512594i	$0.828684\pi$
$500$	0	0
$501$	−9090.00	−0.810601
$502$	0	0
$503$	12192.0	1.08074	0.540372	−	0.841426i	$-0.318283\pi$
$503$	12192.0	1.08074	0.540372	+	0.841426i	$0.318283\pi$
$504$	0	0
$505$	−9630.00	−0.848573
$506$	0	0
$507$	16641.0	1.45770
$508$	0	0
$509$	−19809.0	−1.72499	−0.862494	−	0.506068i	$-0.831098\pi$
$509$	−19809.0	−1.72499	−0.862494	+	0.506068i	$0.831098\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−2808.00	−0.241669
$514$	0	0
$515$	6960.00	0.595523
$516$	0	0
$517$	1242.00	0.105654
$518$	0	0
$519$	10314.0	0.872321
$520$	0	0
$521$	−1794.00	−0.150857	−0.0754286	−	0.997151i	$-0.524032\pi$
$521$	−1794.00	−0.150857	−0.0754286	+	0.997151i	$0.524032\pi$
$522$	0	0
$523$	6448.00	0.539104	0.269552	−	0.962986i	$-0.413124\pi$
$523$	6448.00	0.539104	0.269552	+	0.962986i	$0.413124\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15540.0	1.28450
$528$	0	0
$529$	−5111.00	−0.420071
$530$	0	0
$531$	−1431.00	−0.116949
$532$	0	0
$533$	14784.0	1.20144
$534$	0	0
$535$	5895.00	0.476380
$536$	0	0
$537$	3636.00	0.292188
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7262.00	0.577112	0.288556	−	0.957463i	$-0.406825\pi$
$541$	7262.00	0.577112	0.288556	+	0.957463i	$0.406825\pi$
$542$	0	0
$543$	9096.00	0.718871
$544$	0	0
$545$	−210.000	−0.0165053
$546$	0	0
$547$	−14204.0	−1.11027	−0.555136	−	0.831759i	$-0.687334\pi$
$547$	−14204.0	−1.11027	−0.555136	+	0.831759i	$0.687334\pi$
$548$	0	0
$549$	6498.00	0.505151
$550$	0	0
$551$	−5304.00	−0.410087
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1980.00	−0.151435
$556$	0	0
$557$	15825.0	1.20382	0.601909	−	0.798565i	$-0.294407\pi$
$557$	15825.0	1.20382	0.601909	+	0.798565i	$0.294407\pi$
$558$	0	0
$559$	28688.0	2.17061
$560$	0	0
$561$	−2268.00	−0.170686
$562$	0	0
$563$	−1059.00	−0.0792745	−0.0396372	−	0.999214i	$-0.512620\pi$
$563$	−1059.00	−0.0792745	−0.0396372	+	0.999214i	$0.512620\pi$
$564$	0	0
$565$	32760.0	2.43933
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3960.00	0.291761	0.145880	−	0.989302i	$-0.453399\pi$
$569$	3960.00	0.291761	0.145880	+	0.989302i	$0.453399\pi$
$570$	0	0
$571$	2530.00	0.185424	0.0927121	−	0.995693i	$-0.470446\pi$
$571$	2530.00	0.185424	0.0927121	+	0.995693i	$0.470446\pi$
$572$	0	0
$573$	−7560.00	−0.551175
$574$	0	0
$575$	8400.00	0.609225
$576$	0	0
$577$	11831.0	0.853607	0.426803	−	0.904344i	$-0.359640\pi$
$577$	11831.0	0.853607	0.426803	+	0.904344i	$0.359640\pi$
$578$	0	0
$579$	1095.00	0.0785952
$580$	0	0
$581$	0	0
$582$	0	0
$583$	5751.00	0.408546
$584$	0	0
$585$	11880.0	0.839620
$586$	0	0
$587$	−4809.00	−0.338141	−0.169070	−	0.985604i	$-0.554077\pi$
$587$	−4809.00	−0.338141	−0.169070	+	0.985604i	$0.554077\pi$
$588$	0	0
$589$	19240.0	1.34596
$590$	0	0
$591$	−4770.00	−0.331999
$592$	0	0
$593$	21804.0	1.50992	0.754960	−	0.655770i	$-0.227656\pi$
$593$	21804.0	1.50992	0.754960	+	0.655770i	$0.227656\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	16140.0	1.10648
$598$	0	0
$599$	−14166.0	−0.966289	−0.483144	−	0.875541i	$-0.660505\pi$
$599$	−14166.0	−0.966289	−0.483144	+	0.875541i	$0.660505\pi$
$600$	0	0
$601$	5891.00	0.399832	0.199916	−	0.979813i	$-0.435933\pi$
$601$	5891.00	0.399832	0.199916	+	0.979813i	$0.435933\pi$
$602$	0	0
$603$	1494.00	0.100896
$604$	0	0
$605$	18750.0	1.25999
$606$	0	0
$607$	2737.00	0.183017	0.0915086	−	0.995804i	$-0.470831\pi$
$607$	2737.00	0.183017	0.0915086	+	0.995804i	$0.470831\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12144.0	−0.804081
$612$	0	0
$613$	−26188.0	−1.72549	−0.862743	−	0.505642i	$-0.831256\pi$
$613$	−26188.0	−1.72549	−0.862743	+	0.505642i	$0.831256\pi$
$614$	0	0
$615$	7560.00	0.495689
$616$	0	0
$617$	−2358.00	−0.153857	−0.0769283	−	0.997037i	$-0.524511\pi$
$617$	−2358.00	−0.153857	−0.0769283	+	0.997037i	$0.524511\pi$
$618$	0	0
$619$	−13766.0	−0.893865	−0.446932	−	0.894568i	$-0.647484\pi$
$619$	−13766.0	−0.893865	−0.446932	+	0.894568i	$0.647484\pi$
$620$	0	0
$621$	2268.00	0.146557
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−18125.0	−1.16000
$626$	0	0
$627$	−2808.00	−0.178853
$628$	0	0
$629$	−3696.00	−0.234291
$630$	0	0
$631$	−21287.0	−1.34298	−0.671491	−	0.741012i	$-0.734346\pi$
$631$	−21287.0	−1.34298	−0.671491	+	0.741012i	$0.734346\pi$
$632$	0	0
$633$	16086.0	1.01005
$634$	0	0
$635$	−5595.00	−0.349655
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−9774.00	−0.605091
$640$	0	0
$641$	21426.0	1.32024	0.660122	−	0.751159i	$-0.270505\pi$
$641$	21426.0	1.32024	0.660122	+	0.751159i	$0.270505\pi$
$642$	0	0
$643$	−9962.00	−0.610984	−0.305492	−	0.952195i	$-0.598821\pi$
$643$	−9962.00	−0.610984	−0.305492	+	0.952195i	$0.598821\pi$
$644$	0	0
$645$	14670.0	0.895551
$646$	0	0
$647$	18174.0	1.10432	0.552159	−	0.833739i	$-0.313804\pi$
$647$	18174.0	1.10432	0.552159	+	0.833739i	$0.313804\pi$
$648$	0	0
$649$	−1431.00	−0.0865511
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−19167.0	−1.14864	−0.574321	−	0.818630i	$-0.694734\pi$
$653$	−19167.0	−1.14864	−0.574321	+	0.818630i	$0.694734\pi$
$654$	0	0
$655$	−17595.0	−1.04961
$656$	0	0
$657$	1962.00	0.116507
$658$	0	0
$659$	−13080.0	−0.773178	−0.386589	−	0.922252i	$-0.626347\pi$
$659$	−13080.0	−0.773178	−0.386589	+	0.922252i	$0.626347\pi$
$660$	0	0
$661$	−15190.0	−0.893831	−0.446916	−	0.894576i	$-0.647478\pi$
$661$	−15190.0	−0.893831	−0.446916	+	0.894576i	$0.647478\pi$
$662$	0	0
$663$	22176.0	1.29901
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4284.00	0.248691
$668$	0	0
$669$	4719.00	0.272716
$670$	0	0
$671$	6498.00	0.373849
$672$	0	0
$673$	4397.00	0.251845	0.125923	−	0.992040i	$-0.459811\pi$
$673$	4397.00	0.251845	0.125923	+	0.992040i	$0.459811\pi$
$674$	0	0
$675$	2700.00	0.153960
$676$	0	0
$677$	4029.00	0.228725	0.114363	−	0.993439i	$-0.463517\pi$
$677$	4029.00	0.228725	0.114363	+	0.993439i	$0.463517\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−2763.00	−0.155475
$682$	0	0
$683$	−15021.0	−0.841526	−0.420763	−	0.907170i	$-0.638238\pi$
$683$	−15021.0	−0.841526	−0.420763	+	0.907170i	$0.638238\pi$
$684$	0	0
$685$	−450.000	−0.0251002
$686$	0	0
$687$	12156.0	0.675081
$688$	0	0
$689$	−56232.0	−3.10924
$690$	0	0
$691$	13984.0	0.769865	0.384932	−	0.922945i	$-0.374225\pi$
$691$	13984.0	0.769865	0.384932	+	0.922945i	$0.374225\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1230.00	−0.0671317
$696$	0	0
$697$	14112.0	0.766901
$698$	0	0
$699$	−1404.00	−0.0759716
$700$	0	0
$701$	−31053.0	−1.67312	−0.836559	−	0.547877i	$-0.815436\pi$
$701$	−31053.0	−1.67312	−0.836559	+	0.547877i	$0.815436\pi$
$702$	0	0
$703$	−4576.00	−0.245501
$704$	0	0
$705$	−6210.00	−0.331748
$706$	0	0
$707$	0	0
$708$	0	0
$709$	15086.0	0.799107	0.399553	−	0.916710i	$-0.369165\pi$
$709$	15086.0	0.799107	0.399553	+	0.916710i	$0.369165\pi$
$710$	0	0
$711$	5247.00	0.276762
$712$	0	0
$713$	−15540.0	−0.816238
$714$	0	0
$715$	11880.0	0.621380
$716$	0	0
$717$	−14796.0	−0.770665
$718$	0	0
$719$	6378.00	0.330820	0.165410	−	0.986225i	$-0.447105\pi$
$719$	6378.00	0.330820	0.165410	+	0.986225i	$0.447105\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−4611.00	−0.237185
$724$	0	0
$725$	5100.00	0.261254
$726$	0	0
$727$	7363.00	0.375624	0.187812	−	0.982205i	$-0.439860\pi$
$727$	7363.00	0.375624	0.187812	+	0.982205i	$0.439860\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	27384.0	1.38555
$732$	0	0
$733$	32810.0	1.65329	0.826647	−	0.562720i	$-0.190245\pi$
$733$	32810.0	1.65329	0.826647	+	0.562720i	$0.190245\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1494.00	0.0746706
$738$	0	0
$739$	24034.0	1.19635	0.598177	−	0.801364i	$-0.295892\pi$
$739$	24034.0	1.19635	0.598177	+	0.801364i	$0.295892\pi$
$740$	0	0
$741$	27456.0	1.36116
$742$	0	0
$743$	8022.00	0.396095	0.198048	−	0.980192i	$-0.436540\pi$
$743$	8022.00	0.396095	0.198048	+	0.980192i	$0.436540\pi$
$744$	0	0
$745$	21510.0	1.05781
$746$	0	0
$747$	5373.00	0.263170
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−29519.0	−1.43431	−0.717153	−	0.696916i	$-0.754555\pi$
$751$	−29519.0	−1.43431	−0.717153	+	0.696916i	$0.754555\pi$
$752$	0	0
$753$	−15957.0	−0.772252
$754$	0	0
$755$	−40065.0	−1.93128
$756$	0	0
$757$	−3742.00	−0.179664	−0.0898318	−	0.995957i	$-0.528633\pi$
$757$	−3742.00	−0.179664	−0.0898318	+	0.995957i	$0.528633\pi$
$758$	0	0
$759$	2268.00	0.108463
$760$	0	0
$761$	−10896.0	−0.519027	−0.259514	−	0.965739i	$-0.583562\pi$
$761$	−10896.0	−0.519027	−0.259514	+	0.965739i	$0.583562\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	11340.0	0.535946
$766$	0	0
$767$	13992.0	0.658699
$768$	0	0
$769$	17285.0	0.810550	0.405275	−	0.914195i	$-0.367176\pi$
$769$	17285.0	0.810550	0.405275	+	0.914195i	$0.367176\pi$
$770$	0	0
$771$	16038.0	0.749150
$772$	0	0
$773$	11826.0	0.550261	0.275130	−	0.961407i	$-0.411279\pi$
$773$	11826.0	0.550261	0.275130	+	0.961407i	$0.411279\pi$
$774$	0	0
$775$	−18500.0	−0.857470
$776$	0	0
$777$	0	0
$778$	0	0
$779$	17472.0	0.803594
$780$	0	0
$781$	−9774.00	−0.447812
$782$	0	0
$783$	1377.00	0.0628480
$784$	0	0
$785$	−33780.0	−1.53587
$786$	0	0
$787$	−17714.0	−0.802333	−0.401166	−	0.916005i	$-0.631395\pi$
$787$	−17714.0	−0.802333	−0.401166	+	0.916005i	$0.631395\pi$
$788$	0	0
$789$	−2322.00	−0.104772
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−63536.0	−2.84518
$794$	0	0
$795$	−28755.0	−1.28281
$796$	0	0
$797$	−24939.0	−1.10839	−0.554194	−	0.832388i	$-0.686973\pi$
$797$	−24939.0	−1.10839	−0.554194	+	0.832388i	$0.686973\pi$
$798$	0	0
$799$	−11592.0	−0.513261
$800$	0	0
$801$	−9342.00	−0.412089
$802$	0	0
$803$	1962.00	0.0862235
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7245.00	0.316030
$808$	0	0
$809$	−29064.0	−1.26309	−0.631543	−	0.775341i	$-0.717578\pi$
$809$	−29064.0	−1.26309	−0.631543	+	0.775341i	$0.717578\pi$
$810$	0	0
$811$	15370.0	0.665492	0.332746	−	0.943017i	$-0.392025\pi$
$811$	15370.0	0.665492	0.332746	+	0.943017i	$0.392025\pi$
$812$	0	0
$813$	1425.00	0.0614722
$814$	0	0
$815$	25140.0	1.08051
$816$	0	0
$817$	33904.0	1.45184
$818$	0	0
$819$	0	0
$820$	0	0
$821$	44031.0	1.87173	0.935866	−	0.352355i	$-0.114619\pi$
$821$	44031.0	1.87173	0.935866	+	0.352355i	$0.114619\pi$
$822$	0	0
$823$	4192.00	0.177550	0.0887752	−	0.996052i	$-0.471705\pi$
$823$	4192.00	0.177550	0.0887752	+	0.996052i	$0.471705\pi$
$824$	0	0
$825$	2700.00	0.113942
$826$	0	0
$827$	−33195.0	−1.39577	−0.697886	−	0.716209i	$-0.745876\pi$
$827$	−33195.0	−1.39577	−0.697886	+	0.716209i	$0.745876\pi$
$828$	0	0
$829$	16448.0	0.689098	0.344549	−	0.938768i	$-0.388032\pi$
$829$	16448.0	0.689098	0.344549	+	0.938768i	$0.388032\pi$
$830$	0	0
$831$	11184.0	0.466870
$832$	0	0
$833$	0	0
$834$	0	0
$835$	45450.0	1.88367
$836$	0	0
$837$	−4995.00	−0.206275
$838$	0	0
$839$	−16860.0	−0.693769	−0.346884	−	0.937908i	$-0.612760\pi$
$839$	−16860.0	−0.693769	−0.346884	+	0.937908i	$0.612760\pi$
$840$	0	0
$841$	−21788.0	−0.893354
$842$	0	0
$843$	4806.00	0.196355
$844$	0	0
$845$	−83205.0	−3.38738
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2058.00	−0.0831924
$850$	0	0
$851$	3696.00	0.148880
$852$	0	0
$853$	29054.0	1.16623	0.583113	−	0.812391i	$-0.301835\pi$
$853$	29054.0	1.16623	0.583113	+	0.812391i	$0.301835\pi$
$854$	0	0
$855$	14040.0	0.561588
$856$	0	0
$857$	41958.0	1.67241	0.836207	−	0.548415i	$-0.184768\pi$
$857$	41958.0	1.67241	0.836207	+	0.548415i	$0.184768\pi$
$858$	0	0
$859$	−5546.00	−0.220288	−0.110144	−	0.993916i	$-0.535131\pi$
$859$	−5546.00	−0.220288	−0.110144	+	0.993916i	$0.535131\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32538.0	1.28344	0.641719	−	0.766940i	$-0.278222\pi$
$863$	32538.0	1.28344	0.641719	+	0.766940i	$0.278222\pi$
$864$	0	0
$865$	−51570.0	−2.02709
$866$	0	0
$867$	6429.00	0.251834
$868$	0	0
$869$	5247.00	0.204824
$870$	0	0
$871$	−14608.0	−0.568282
$872$	0	0
$873$	−1521.00	−0.0589668
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32096.0	1.23581	0.617905	−	0.786253i	$-0.287982\pi$
$877$	32096.0	1.23581	0.617905	+	0.786253i	$0.287982\pi$
$878$	0	0
$879$	−3303.00	−0.126743
$880$	0	0
$881$	−8490.00	−0.324671	−0.162336	−	0.986736i	$-0.551903\pi$
$881$	−8490.00	−0.324671	−0.162336	+	0.986736i	$0.551903\pi$
$882$	0	0
$883$	48352.0	1.84278	0.921390	−	0.388640i	$-0.127055\pi$
$883$	48352.0	1.84278	0.921390	+	0.388640i	$0.127055\pi$
$884$	0	0
$885$	7155.00	0.271766
$886$	0	0
$887$	−15492.0	−0.586438	−0.293219	−	0.956045i	$-0.594726\pi$
$887$	−15492.0	−0.586438	−0.293219	+	0.956045i	$0.594726\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	729.000	0.0274101
$892$	0	0
$893$	−14352.0	−0.537818
$894$	0	0
$895$	−18180.0	−0.678984
$896$	0	0
$897$	−22176.0	−0.825457
$898$	0	0
$899$	−9435.00	−0.350028
$900$	0	0
$901$	−53676.0	−1.98469
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−45480.0	−1.67050
$906$	0	0
$907$	8116.00	0.297119	0.148560	−	0.988903i	$-0.452536\pi$
$907$	8116.00	0.297119	0.148560	+	0.988903i	$0.452536\pi$
$908$	0	0
$909$	5778.00	0.210830
$910$	0	0
$911$	4446.00	0.161693	0.0808466	−	0.996727i	$-0.474238\pi$
$911$	4446.00	0.161693	0.0808466	+	0.996727i	$0.474238\pi$
$912$	0	0
$913$	5373.00	0.194765
$914$	0	0
$915$	−32490.0	−1.17386
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−26504.0	−0.951345	−0.475673	−	0.879622i	$-0.657795\pi$
$919$	−26504.0	−0.951345	−0.475673	+	0.879622i	$0.657795\pi$
$920$	0	0
$921$	−8340.00	−0.298385
$922$	0	0
$923$	95568.0	3.40808
$924$	0	0
$925$	4400.00	0.156401
$926$	0	0
$927$	−4176.00	−0.147959
$928$	0	0
$929$	−5430.00	−0.191768	−0.0958840	−	0.995393i	$-0.530568\pi$
$929$	−5430.00	−0.191768	−0.0958840	+	0.995393i	$0.530568\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−12888.0	−0.452234
$934$	0	0
$935$	11340.0	0.396639
$936$	0	0
$937$	33803.0	1.17854	0.589272	−	0.807935i	$-0.299415\pi$
$937$	33803.0	1.17854	0.589272	+	0.807935i	$0.299415\pi$
$938$	0	0
$939$	16467.0	0.572290
$940$	0	0
$941$	−48483.0	−1.67960	−0.839798	−	0.542898i	$-0.817327\pi$
$941$	−48483.0	−1.67960	−0.839798	+	0.542898i	$0.817327\pi$
$942$	0	0
$943$	−14112.0	−0.487328
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−37296.0	−1.27979	−0.639893	−	0.768464i	$-0.721021\pi$
$947$	−37296.0	−1.27979	−0.639893	+	0.768464i	$0.721021\pi$
$948$	0	0
$949$	−19184.0	−0.656205
$950$	0	0
$951$	13473.0	0.459403
$952$	0	0
$953$	−38478.0	−1.30790	−0.653948	−	0.756540i	$-0.726888\pi$
$953$	−38478.0	−1.30790	−0.653948	+	0.756540i	$0.726888\pi$
$954$	0	0
$955$	37800.0	1.28082
$956$	0	0
$957$	1377.00	0.0465121
$958$	0	0
$959$	0	0
$960$	0	0
$961$	4434.00	0.148837
$962$	0	0
$963$	−3537.00	−0.118357
$964$	0	0
$965$	−5475.00	−0.182639
$966$	0	0
$967$	−27257.0	−0.906438	−0.453219	−	0.891399i	$-0.649725\pi$
$967$	−27257.0	−0.906438	−0.453219	+	0.891399i	$0.649725\pi$
$968$	0	0
$969$	26208.0	0.868857
$970$	0	0
$971$	−34341.0	−1.13497	−0.567485	−	0.823384i	$-0.692083\pi$
$971$	−34341.0	−1.13497	−0.567485	+	0.823384i	$0.692083\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−26400.0	−0.867156
$976$	0	0
$977$	27426.0	0.898092	0.449046	−	0.893509i	$-0.351764\pi$
$977$	27426.0	0.898092	0.449046	+	0.893509i	$0.351764\pi$
$978$	0	0
$979$	−9342.00	−0.304976
$980$	0	0
$981$	126.000	0.00410079
$982$	0	0
$983$	−12324.0	−0.399872	−0.199936	−	0.979809i	$-0.564073\pi$
$983$	−12324.0	−0.399872	−0.199936	+	0.979809i	$0.564073\pi$
$984$	0	0
$985$	23850.0	0.771497
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−27384.0	−0.880445
$990$	0	0
$991$	−47597.0	−1.52570	−0.762850	−	0.646576i	$-0.776200\pi$
$991$	−47597.0	−1.52570	−0.762850	+	0.646576i	$0.776200\pi$
$992$	0	0
$993$	11892.0	0.380042
$994$	0	0
$995$	−80700.0	−2.57122
$996$	0	0
$997$	−11242.0	−0.357109	−0.178555	−	0.983930i	$-0.557142\pi$
$997$	−11242.0	−0.357109	−0.178555	+	0.983930i	$0.557142\pi$
$998$	0	0
$999$	1188.00	0.0376243

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.u.1.1		1
4.3	odd	2		294.4.a.a.1.1		1
7.2	even	3		336.4.q.d.193.1		2
7.4	even	3		336.4.q.d.289.1		2
7.6	odd	2		2352.4.a.q.1.1		1
12.11	even	2		882.4.a.r.1.1		1
28.3	even	6		294.4.e.e.79.1		2
28.11	odd	6		42.4.e.b.37.1	yes	2
28.19	even	6		294.4.e.e.67.1		2
28.23	odd	6		42.4.e.b.25.1	✓	2
28.27	even	2		294.4.a.g.1.1		1
84.11	even	6		126.4.g.a.37.1		2
84.23	even	6		126.4.g.a.109.1		2
84.47	odd	6		882.4.g.l.361.1		2
84.59	odd	6		882.4.g.l.667.1		2
84.83	odd	2		882.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.e.b.25.1	✓	2	28.23	odd	6
42.4.e.b.37.1	yes	2	28.11	odd	6
126.4.g.a.37.1		2	84.11	even	6
126.4.g.a.109.1		2	84.23	even	6
294.4.a.a.1.1		1	4.3	odd	2
294.4.a.g.1.1		1	28.27	even	2
294.4.e.e.67.1		2	28.19	even	6
294.4.e.e.79.1		2	28.3	even	6
336.4.q.d.193.1		2	7.2	even	3
336.4.q.d.289.1		2	7.4	even	3
882.4.a.h.1.1		1	84.83	odd	2
882.4.a.r.1.1		1	12.11	even	2
882.4.g.l.361.1		2	84.47	odd	6
882.4.g.l.667.1		2	84.59	odd	6
2352.4.a.q.1.1		1	7.6	odd	2
2352.4.a.u.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

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