LMFDB - Embedded newform 2352.4.a.x.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 1176)
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)

$f(q)$

$=$

$q+3.00000 q^{3} -12.0000 q^{5} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -12.0000 q^{5} +9.00000 q^{9} +60.0000 q^{11} -44.0000 q^{13} -36.0000 q^{15} +128.000 q^{17} +52.0000 q^{19} +160.000 q^{23} +19.0000 q^{25} +27.0000 q^{27} -230.000 q^{29} +136.000 q^{31} +180.000 q^{33} -318.000 q^{37} -132.000 q^{39} +192.000 q^{41} -220.000 q^{43} -108.000 q^{45} +184.000 q^{47} +384.000 q^{51} -498.000 q^{53} -720.000 q^{55} +156.000 q^{57} -492.000 q^{59} -20.0000 q^{61} +528.000 q^{65} -380.000 q^{67} +480.000 q^{69} +264.000 q^{71} +560.000 q^{73} +57.0000 q^{75} -104.000 q^{79} +81.0000 q^{81} +1508.00 q^{83} -1536.00 q^{85} -690.000 q^{87} -1144.00 q^{89} +408.000 q^{93} -624.000 q^{95} +904.000 q^{97} +540.000 q^{99} +O(q^{100})$

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−12.0000	−1.07331	−0.536656	−	0.843801i	$-0.680313\pi$
$5$	−12.0000	−1.07331	−0.536656	+	0.843801i	$0.680313\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	60.0000	1.64461	0.822304	−	0.569049i	$-0.192689\pi$
$11$	60.0000	1.64461	0.822304	+	0.569049i	$0.192689\pi$
$12$	0	0
$13$	−44.0000	−0.938723	−0.469362	−	0.883006i	$-0.655516\pi$
$13$	−44.0000	−0.938723	−0.469362	+	0.883006i	$0.655516\pi$
$14$	0	0
$15$	−36.0000	−0.619677
$16$	0	0
$17$	128.000	1.82615	0.913075	−	0.407791i	$-0.133701\pi$
$17$	128.000	1.82615	0.913075	+	0.407791i	$0.133701\pi$
$18$	0	0
$19$	52.0000	0.627875	0.313937	−	0.949444i	$-0.398352\pi$
$19$	52.0000	0.627875	0.313937	+	0.949444i	$0.398352\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	160.000	1.45054	0.725268	−	0.688467i	$-0.241716\pi$
$23$	160.000	1.45054	0.725268	+	0.688467i	$0.241716\pi$
$24$	0	0
$25$	19.0000	0.152000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−230.000	−1.47276	−0.736378	−	0.676570i	$-0.763465\pi$
$29$	−230.000	−1.47276	−0.736378	+	0.676570i	$0.763465\pi$
$30$	0	0
$31$	136.000	0.787946	0.393973	−	0.919122i	$-0.371100\pi$
$31$	136.000	0.787946	0.393973	+	0.919122i	$0.371100\pi$
$32$	0	0
$33$	180.000	0.949514
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−318.000	−1.41294	−0.706471	−	0.707742i	$-0.749714\pi$
$37$	−318.000	−1.41294	−0.706471	+	0.707742i	$0.749714\pi$
$38$	0	0
$39$	−132.000	−0.541972
$40$	0	0
$41$	192.000	0.731350	0.365675	−	0.930743i	$-0.380838\pi$
$41$	192.000	0.731350	0.365675	+	0.930743i	$0.380838\pi$
$42$	0	0
$43$	−220.000	−0.780225	−0.390113	−	0.920767i	$-0.627564\pi$
$43$	−220.000	−0.780225	−0.390113	+	0.920767i	$0.627564\pi$
$44$	0	0
$45$	−108.000	−0.357771
$46$	0	0
$47$	184.000	0.571046	0.285523	−	0.958372i	$-0.407833\pi$
$47$	184.000	0.571046	0.285523	+	0.958372i	$0.407833\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	384.000	1.05433
$52$	0	0
$53$	−498.000	−1.29067	−0.645335	−	0.763899i	$-0.723282\pi$
$53$	−498.000	−1.29067	−0.645335	+	0.763899i	$0.723282\pi$
$54$	0	0
$55$	−720.000	−1.76518
$56$	0	0
$57$	156.000	0.362504
$58$	0	0
$59$	−492.000	−1.08564	−0.542822	−	0.839848i	$-0.682644\pi$
$59$	−492.000	−1.08564	−0.542822	+	0.839848i	$0.682644\pi$
$60$	0	0
$61$	−20.0000	−0.0419793	−0.0209897	−	0.999780i	$-0.506682\pi$
$61$	−20.0000	−0.0419793	−0.0209897	+	0.999780i	$0.506682\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	528.000	1.00754
$66$	0	0
$67$	−380.000	−0.692901	−0.346451	−	0.938068i	$-0.612613\pi$
$67$	−380.000	−0.692901	−0.346451	+	0.938068i	$0.612613\pi$
$68$	0	0
$69$	480.000	0.837467
$70$	0	0
$71$	264.000	0.441282	0.220641	−	0.975355i	$-0.429185\pi$
$71$	264.000	0.441282	0.220641	+	0.975355i	$0.429185\pi$
$72$	0	0
$73$	560.000	0.897850	0.448925	−	0.893569i	$-0.351807\pi$
$73$	560.000	0.897850	0.448925	+	0.893569i	$0.351807\pi$
$74$	0	0
$75$	57.0000	0.0877572
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−104.000	−0.148113	−0.0740564	−	0.997254i	$-0.523594\pi$
$79$	−104.000	−0.148113	−0.0740564	+	0.997254i	$0.523594\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1508.00	1.99427	0.997136	−	0.0756351i	$-0.0240984\pi$
$83$	1508.00	1.99427	0.997136	+	0.0756351i	$0.0240984\pi$
$84$	0	0
$85$	−1536.00	−1.96003
$86$	0	0
$87$	−690.000	−0.850296
$88$	0	0
$89$	−1144.00	−1.36251	−0.681257	−	0.732044i	$-0.738566\pi$
$89$	−1144.00	−1.36251	−0.681257	+	0.732044i	$0.738566\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	408.000	0.454921
$94$	0	0
$95$	−624.000	−0.673906
$96$	0	0
$97$	904.000	0.946261	0.473130	−	0.880992i	$-0.343124\pi$
$97$	904.000	0.946261	0.473130	+	0.880992i	$0.343124\pi$
$98$	0	0
$99$	540.000	0.548202
$100$	0	0
$101$	500.000	0.492593	0.246296	−	0.969195i	$-0.420786\pi$
$101$	500.000	0.492593	0.246296	+	0.969195i	$0.420786\pi$
$102$	0	0
$103$	248.000	0.237244	0.118622	−	0.992939i	$-0.462152\pi$
$103$	248.000	0.237244	0.118622	+	0.992939i	$0.462152\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1028.00	0.928790	0.464395	−	0.885628i	$-0.346272\pi$
$107$	1028.00	0.928790	0.464395	+	0.885628i	$0.346272\pi$
$108$	0	0
$109$	1066.00	0.936737	0.468368	−	0.883533i	$-0.344842\pi$
$109$	1066.00	0.936737	0.468368	+	0.883533i	$0.344842\pi$
$110$	0	0
$111$	−954.000	−0.815763
$112$	0	0
$113$	466.000	0.387943	0.193972	−	0.981007i	$-0.437863\pi$
$113$	466.000	0.387943	0.193972	+	0.981007i	$0.437863\pi$
$114$	0	0
$115$	−1920.00	−1.55688
$116$	0	0
$117$	−396.000	−0.312908
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2269.00	1.70473
$122$	0	0
$123$	576.000	0.422245
$124$	0	0
$125$	1272.00	0.910169
$126$	0	0
$127$	−1616.00	−1.12911	−0.564554	−	0.825396i	$-0.690952\pi$
$127$	−1616.00	−1.12911	−0.564554	+	0.825396i	$0.690952\pi$
$128$	0	0
$129$	−660.000	−0.450463
$130$	0	0
$131$	1700.00	1.13381	0.566907	−	0.823782i	$-0.308140\pi$
$131$	1700.00	1.13381	0.566907	+	0.823782i	$0.308140\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−324.000	−0.206559
$136$	0	0
$137$	1958.00	1.22105	0.610523	−	0.791999i	$-0.290959\pi$
$137$	1958.00	1.22105	0.610523	+	0.791999i	$0.290959\pi$
$138$	0	0
$139$	−1284.00	−0.783507	−0.391753	−	0.920070i	$-0.628131\pi$
$139$	−1284.00	−0.783507	−0.391753	+	0.920070i	$0.628131\pi$
$140$	0	0
$141$	552.000	0.329694
$142$	0	0
$143$	−2640.00	−1.54383
$144$	0	0
$145$	2760.00	1.58073
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−306.000	−0.168245	−0.0841225	−	0.996455i	$-0.526809\pi$
$149$	−306.000	−0.168245	−0.0841225	+	0.996455i	$0.526809\pi$
$150$	0	0
$151$	2264.00	1.22014	0.610072	−	0.792346i	$-0.291141\pi$
$151$	2264.00	1.22014	0.610072	+	0.792346i	$0.291141\pi$
$152$	0	0
$153$	1152.00	0.608717
$154$	0	0
$155$	−1632.00	−0.845712
$156$	0	0
$157$	2068.00	1.05124	0.525619	−	0.850720i	$-0.323834\pi$
$157$	2068.00	1.05124	0.525619	+	0.850720i	$0.323834\pi$
$158$	0	0
$159$	−1494.00	−0.745169
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1908.00	−0.916847	−0.458424	−	0.888734i	$-0.651586\pi$
$163$	−1908.00	−0.916847	−0.458424	+	0.888734i	$0.651586\pi$
$164$	0	0
$165$	−2160.00	−1.01913
$166$	0	0
$167$	−3128.00	−1.44941	−0.724706	−	0.689058i	$-0.758025\pi$
$167$	−3128.00	−1.44941	−0.724706	+	0.689058i	$0.758025\pi$
$168$	0	0
$169$	−261.000	−0.118798
$170$	0	0
$171$	468.000	0.209292
$172$	0	0
$173$	−532.000	−0.233799	−0.116899	−	0.993144i	$-0.537296\pi$
$173$	−532.000	−0.233799	−0.116899	+	0.993144i	$0.537296\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1476.00	−0.626796
$178$	0	0
$179$	1068.00	0.445956	0.222978	−	0.974824i	$-0.428422\pi$
$179$	1068.00	0.445956	0.222978	+	0.974824i	$0.428422\pi$
$180$	0	0
$181$	4164.00	1.70999	0.854994	−	0.518639i	$-0.173561\pi$
$181$	4164.00	1.70999	0.854994	+	0.518639i	$0.173561\pi$
$182$	0	0
$183$	−60.0000	−0.0242368
$184$	0	0
$185$	3816.00	1.51653
$186$	0	0
$187$	7680.00	3.00330
$188$	0	0
$189$	0	0
$190$	0	0
$191$	288.000	0.109104	0.0545522	−	0.998511i	$-0.482627\pi$
$191$	288.000	0.109104	0.0545522	+	0.998511i	$0.482627\pi$
$192$	0	0
$193$	1282.00	0.478137	0.239068	−	0.971003i	$-0.423158\pi$
$193$	1282.00	0.478137	0.239068	+	0.971003i	$0.423158\pi$
$194$	0	0
$195$	1584.00	0.581706
$196$	0	0
$197$	2622.00	0.948273	0.474136	−	0.880451i	$-0.342760\pi$
$197$	2622.00	0.948273	0.474136	+	0.880451i	$0.342760\pi$
$198$	0	0
$199$	776.000	0.276428	0.138214	−	0.990402i	$-0.455864\pi$
$199$	776.000	0.276428	0.138214	+	0.990402i	$0.455864\pi$
$200$	0	0
$201$	−1140.00	−0.400047
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2304.00	−0.784968
$206$	0	0
$207$	1440.00	0.483512
$208$	0	0
$209$	3120.00	1.03261
$210$	0	0
$211$	5476.00	1.78665	0.893326	−	0.449410i	$-0.148366\pi$
$211$	5476.00	1.78665	0.893326	+	0.449410i	$0.148366\pi$
$212$	0	0
$213$	792.000	0.254774
$214$	0	0
$215$	2640.00	0.837426
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1680.00	0.518374
$220$	0	0
$221$	−5632.00	−1.71425
$222$	0	0
$223$	5680.00	1.70565	0.852827	−	0.522193i	$-0.174886\pi$
$223$	5680.00	1.70565	0.852827	+	0.522193i	$0.174886\pi$
$224$	0	0
$225$	171.000	0.0506667
$226$	0	0
$227$	−5180.00	−1.51458	−0.757288	−	0.653081i	$-0.773476\pi$
$227$	−5180.00	−1.51458	−0.757288	+	0.653081i	$0.773476\pi$
$228$	0	0
$229$	2996.00	0.864547	0.432273	−	0.901743i	$-0.357712\pi$
$229$	2996.00	0.864547	0.432273	+	0.901743i	$0.357712\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1818.00	−0.511164	−0.255582	−	0.966787i	$-0.582267\pi$
$233$	−1818.00	−0.511164	−0.255582	+	0.966787i	$0.582267\pi$
$234$	0	0
$235$	−2208.00	−0.612911
$236$	0	0
$237$	−312.000	−0.0855130
$238$	0	0
$239$	−2264.00	−0.612745	−0.306372	−	0.951912i	$-0.599115\pi$
$239$	−2264.00	−0.612745	−0.306372	+	0.951912i	$0.599115\pi$
$240$	0	0
$241$	2952.00	0.789025	0.394513	−	0.918891i	$-0.370913\pi$
$241$	2952.00	0.789025	0.394513	+	0.918891i	$0.370913\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2288.00	−0.589401
$248$	0	0
$249$	4524.00	1.15139
$250$	0	0
$251$	2460.00	0.618621	0.309310	−	0.950961i	$-0.399902\pi$
$251$	2460.00	0.618621	0.309310	+	0.950961i	$0.399902\pi$
$252$	0	0
$253$	9600.00	2.38556
$254$	0	0
$255$	−4608.00	−1.13162
$256$	0	0
$257$	2952.00	0.716501	0.358250	−	0.933626i	$-0.383373\pi$
$257$	2952.00	0.716501	0.358250	+	0.933626i	$0.383373\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2070.00	−0.490919
$262$	0	0
$263$	5448.00	1.27733	0.638666	−	0.769484i	$-0.279487\pi$
$263$	5448.00	1.27733	0.638666	+	0.769484i	$0.279487\pi$
$264$	0	0
$265$	5976.00	1.38529
$266$	0	0
$267$	−3432.00	−0.786648
$268$	0	0
$269$	−8052.00	−1.82505	−0.912526	−	0.409018i	$-0.865871\pi$
$269$	−8052.00	−1.82505	−0.912526	+	0.409018i	$0.865871\pi$
$270$	0	0
$271$	3872.00	0.867923	0.433962	−	0.900931i	$-0.357115\pi$
$271$	3872.00	0.867923	0.433962	+	0.900931i	$0.357115\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1140.00	0.249980
$276$	0	0
$277$	6062.00	1.31491	0.657455	−	0.753493i	$-0.271633\pi$
$277$	6062.00	1.31491	0.657455	+	0.753493i	$0.271633\pi$
$278$	0	0
$279$	1224.00	0.262649
$280$	0	0
$281$	−7018.00	−1.48989	−0.744944	−	0.667127i	$-0.767524\pi$
$281$	−7018.00	−1.48989	−0.744944	+	0.667127i	$0.767524\pi$
$282$	0	0
$283$	−3596.00	−0.755336	−0.377668	−	0.925941i	$-0.623274\pi$
$283$	−3596.00	−0.755336	−0.377668	+	0.925941i	$0.623274\pi$
$284$	0	0
$285$	−1872.00	−0.389080
$286$	0	0
$287$	0	0
$288$	0	0
$289$	11471.0	2.33483
$290$	0	0
$291$	2712.00	0.546324
$292$	0	0
$293$	8100.00	1.61504	0.807521	−	0.589839i	$-0.200809\pi$
$293$	8100.00	1.61504	0.807521	+	0.589839i	$0.200809\pi$
$294$	0	0
$295$	5904.00	1.16523
$296$	0	0
$297$	1620.00	0.316505
$298$	0	0
$299$	−7040.00	−1.36165
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1500.00	0.284399
$304$	0	0
$305$	240.000	0.0450569
$306$	0	0
$307$	−7164.00	−1.33183	−0.665914	−	0.746029i	$-0.731958\pi$
$307$	−7164.00	−1.33183	−0.665914	+	0.746029i	$0.731958\pi$
$308$	0	0
$309$	744.000	0.136973
$310$	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
$313$	−2792.00	−0.504195	−0.252098	−	0.967702i	$-0.581120\pi$
$313$	−2792.00	−0.504195	−0.252098	+	0.967702i	$0.581120\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3642.00	−0.645284	−0.322642	−	0.946521i	$-0.604571\pi$
$317$	−3642.00	−0.645284	−0.322642	+	0.946521i	$0.604571\pi$
$318$	0	0
$319$	−13800.0	−2.42211
$320$	0	0
$321$	3084.00	0.536237
$322$	0	0
$323$	6656.00	1.14659
$324$	0	0
$325$	−836.000	−0.142686
$326$	0	0
$327$	3198.00	0.540825
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−2380.00	−0.395216	−0.197608	−	0.980281i	$-0.563317\pi$
$331$	−2380.00	−0.395216	−0.197608	+	0.980281i	$0.563317\pi$
$332$	0	0
$333$	−2862.00	−0.470981
$334$	0	0
$335$	4560.00	0.743700
$336$	0	0
$337$	−12306.0	−1.98917	−0.994585	−	0.103923i	$-0.966861\pi$
$337$	−12306.0	−1.98917	−0.994585	+	0.103923i	$0.966861\pi$
$338$	0	0
$339$	1398.00	0.223979
$340$	0	0
$341$	8160.00	1.29586
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−5760.00	−0.898864
$346$	0	0
$347$	−8244.00	−1.27539	−0.637696	−	0.770288i	$-0.720112\pi$
$347$	−8244.00	−1.27539	−0.637696	+	0.770288i	$0.720112\pi$
$348$	0	0
$349$	−9276.00	−1.42273	−0.711365	−	0.702823i	$-0.751923\pi$
$349$	−9276.00	−1.42273	−0.711365	+	0.702823i	$0.751923\pi$
$350$	0	0
$351$	−1188.00	−0.180657
$352$	0	0
$353$	−10648.0	−1.60548	−0.802742	−	0.596326i	$-0.796626\pi$
$353$	−10648.0	−1.60548	−0.802742	+	0.596326i	$0.796626\pi$
$354$	0	0
$355$	−3168.00	−0.473634
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2000.00	−0.294028	−0.147014	−	0.989134i	$-0.546966\pi$
$359$	−2000.00	−0.294028	−0.147014	+	0.989134i	$0.546966\pi$
$360$	0	0
$361$	−4155.00	−0.605773
$362$	0	0
$363$	6807.00	0.984228
$364$	0	0
$365$	−6720.00	−0.963674
$366$	0	0
$367$	−848.000	−0.120614	−0.0603069	−	0.998180i	$-0.519208\pi$
$367$	−848.000	−0.120614	−0.0603069	+	0.998180i	$0.519208\pi$
$368$	0	0
$369$	1728.00	0.243783
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1810.00	−0.251255	−0.125628	−	0.992077i	$-0.540094\pi$
$373$	−1810.00	−0.251255	−0.125628	+	0.992077i	$0.540094\pi$
$374$	0	0
$375$	3816.00	0.525486
$376$	0	0
$377$	10120.0	1.38251
$378$	0	0
$379$	1668.00	0.226067	0.113034	−	0.993591i	$-0.463943\pi$
$379$	1668.00	0.226067	0.113034	+	0.993591i	$0.463943\pi$
$380$	0	0
$381$	−4848.00	−0.651891
$382$	0	0
$383$	11736.0	1.56575	0.782874	−	0.622180i	$-0.213753\pi$
$383$	11736.0	1.56575	0.782874	+	0.622180i	$0.213753\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1980.00	−0.260075
$388$	0	0
$389$	−10046.0	−1.30939	−0.654695	−	0.755893i	$-0.727203\pi$
$389$	−10046.0	−1.30939	−0.654695	+	0.755893i	$0.727203\pi$
$390$	0	0
$391$	20480.0	2.64890
$392$	0	0
$393$	5100.00	0.654608
$394$	0	0
$395$	1248.00	0.158971
$396$	0	0
$397$	7636.00	0.965340	0.482670	−	0.875802i	$-0.339667\pi$
$397$	7636.00	0.965340	0.482670	+	0.875802i	$0.339667\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1842.00	−0.229389	−0.114695	−	0.993401i	$-0.536589\pi$
$401$	−1842.00	−0.229389	−0.114695	+	0.993401i	$0.536589\pi$
$402$	0	0
$403$	−5984.00	−0.739663
$404$	0	0
$405$	−972.000	−0.119257
$406$	0	0
$407$	−19080.0	−2.32374
$408$	0	0
$409$	−664.000	−0.0802755	−0.0401378	−	0.999194i	$-0.512780\pi$
$409$	−664.000	−0.0802755	−0.0401378	+	0.999194i	$0.512780\pi$
$410$	0	0
$411$	5874.00	0.704971
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−18096.0	−2.14048
$416$	0	0
$417$	−3852.00	−0.452358
$418$	0	0
$419$	6572.00	0.766261	0.383130	−	0.923694i	$-0.374846\pi$
$419$	6572.00	0.766261	0.383130	+	0.923694i	$0.374846\pi$
$420$	0	0
$421$	−2370.00	−0.274363	−0.137181	−	0.990546i	$-0.543804\pi$
$421$	−2370.00	−0.274363	−0.137181	+	0.990546i	$0.543804\pi$
$422$	0	0
$423$	1656.00	0.190349
$424$	0	0
$425$	2432.00	0.277575
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−7920.00	−0.891331
$430$	0	0
$431$	−9048.00	−1.01120	−0.505600	−	0.862768i	$-0.668729\pi$
$431$	−9048.00	−1.01120	−0.505600	+	0.862768i	$0.668729\pi$
$432$	0	0
$433$	−3208.00	−0.356043	−0.178022	−	0.984027i	$-0.556970\pi$
$433$	−3208.00	−0.356043	−0.178022	+	0.984027i	$0.556970\pi$
$434$	0	0
$435$	8280.00	0.912634
$436$	0	0
$437$	8320.00	0.910754
$438$	0	0
$439$	8832.00	0.960201	0.480101	−	0.877213i	$-0.340600\pi$
$439$	8832.00	0.960201	0.480101	+	0.877213i	$0.340600\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16756.0	1.79707	0.898535	−	0.438903i	$-0.144633\pi$
$443$	16756.0	1.79707	0.898535	+	0.438903i	$0.144633\pi$
$444$	0	0
$445$	13728.0	1.46240
$446$	0	0
$447$	−918.000	−0.0971363
$448$	0	0
$449$	16994.0	1.78618	0.893092	−	0.449874i	$-0.148531\pi$
$449$	16994.0	1.78618	0.893092	+	0.449874i	$0.148531\pi$
$450$	0	0
$451$	11520.0	1.20278
$452$	0	0
$453$	6792.00	0.704450
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11194.0	1.14581	0.572903	−	0.819623i	$-0.305817\pi$
$457$	11194.0	1.14581	0.572903	+	0.819623i	$0.305817\pi$
$458$	0	0
$459$	3456.00	0.351443
$460$	0	0
$461$	−4292.00	−0.433619	−0.216810	−	0.976214i	$-0.569565\pi$
$461$	−4292.00	−0.433619	−0.216810	+	0.976214i	$0.569565\pi$
$462$	0	0
$463$	4328.00	0.434426	0.217213	−	0.976124i	$-0.430303\pi$
$463$	4328.00	0.434426	0.217213	+	0.976124i	$0.430303\pi$
$464$	0	0
$465$	−4896.00	−0.488272
$466$	0	0
$467$	3844.00	0.380897	0.190449	−	0.981697i	$-0.439006\pi$
$467$	3844.00	0.380897	0.190449	+	0.981697i	$0.439006\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	6204.00	0.606933
$472$	0	0
$473$	−13200.0	−1.28316
$474$	0	0
$475$	988.000	0.0954369
$476$	0	0
$477$	−4482.00	−0.430224
$478$	0	0
$479$	12712.0	1.21258	0.606290	−	0.795243i	$-0.292657\pi$
$479$	12712.0	1.21258	0.606290	+	0.795243i	$0.292657\pi$
$480$	0	0
$481$	13992.0	1.32636
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10848.0	−1.01563
$486$	0	0
$487$	13912.0	1.29448	0.647241	−	0.762285i	$-0.275923\pi$
$487$	13912.0	1.29448	0.647241	+	0.762285i	$0.275923\pi$
$488$	0	0
$489$	−5724.00	−0.529342
$490$	0	0
$491$	−11340.0	−1.04230	−0.521148	−	0.853467i	$-0.674496\pi$
$491$	−11340.0	−1.04230	−0.521148	+	0.853467i	$0.674496\pi$
$492$	0	0
$493$	−29440.0	−2.68947
$494$	0	0
$495$	−6480.00	−0.588393
$496$	0	0
$497$	0	0
$498$	0	0
$499$	3444.00	0.308967	0.154484	−	0.987995i	$-0.450629\pi$
$499$	3444.00	0.308967	0.154484	+	0.987995i	$0.450629\pi$
$500$	0	0
$501$	−9384.00	−0.836819
$502$	0	0
$503$	7856.00	0.696385	0.348193	−	0.937423i	$-0.386796\pi$
$503$	7856.00	0.696385	0.348193	+	0.937423i	$0.386796\pi$
$504$	0	0
$505$	−6000.00	−0.528706
$506$	0	0
$507$	−783.000	−0.0685883
$508$	0	0
$509$	12460.0	1.08503	0.542515	−	0.840046i	$-0.317472\pi$
$509$	12460.0	1.08503	0.542515	+	0.840046i	$0.317472\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1404.00	0.120835
$514$	0	0
$515$	−2976.00	−0.254637
$516$	0	0
$517$	11040.0	0.939146
$518$	0	0
$519$	−1596.00	−0.134984
$520$	0	0
$521$	3552.00	0.298687	0.149344	−	0.988785i	$-0.452284\pi$
$521$	3552.00	0.298687	0.149344	+	0.988785i	$0.452284\pi$
$522$	0	0
$523$	−14380.0	−1.20228	−0.601141	−	0.799143i	$-0.705287\pi$
$523$	−14380.0	−1.20228	−0.601141	+	0.799143i	$0.705287\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17408.0	1.43891
$528$	0	0
$529$	13433.0	1.10405
$530$	0	0
$531$	−4428.00	−0.361881
$532$	0	0
$533$	−8448.00	−0.686536
$534$	0	0
$535$	−12336.0	−0.996882
$536$	0	0
$537$	3204.00	0.257473
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14118.0	1.12196	0.560980	−	0.827829i	$-0.310424\pi$
$541$	14118.0	1.12196	0.560980	+	0.827829i	$0.310424\pi$
$542$	0	0
$543$	12492.0	0.987262
$544$	0	0
$545$	−12792.0	−1.00541
$546$	0	0
$547$	3052.00	0.238563	0.119282	−	0.992860i	$-0.461941\pi$
$547$	3052.00	0.238563	0.119282	+	0.992860i	$0.461941\pi$
$548$	0	0
$549$	−180.000	−0.0139931
$550$	0	0
$551$	−11960.0	−0.924706
$552$	0	0
$553$	0	0
$554$	0	0
$555$	11448.0	0.875569
$556$	0	0
$557$	−17162.0	−1.30552	−0.652762	−	0.757563i	$-0.726390\pi$
$557$	−17162.0	−1.30552	−0.652762	+	0.757563i	$0.726390\pi$
$558$	0	0
$559$	9680.00	0.732416
$560$	0	0
$561$	23040.0	1.73396
$562$	0	0
$563$	15948.0	1.19383	0.596917	−	0.802303i	$-0.296392\pi$
$563$	15948.0	1.19383	0.596917	+	0.802303i	$0.296392\pi$
$564$	0	0
$565$	−5592.00	−0.416384
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15958.0	1.17574	0.587868	−	0.808957i	$-0.299967\pi$
$569$	15958.0	1.17574	0.587868	+	0.808957i	$0.299967\pi$
$570$	0	0
$571$	4548.00	0.333324	0.166662	−	0.986014i	$-0.446701\pi$
$571$	4548.00	0.333324	0.166662	+	0.986014i	$0.446701\pi$
$572$	0	0
$573$	864.000	0.0629915
$574$	0	0
$575$	3040.00	0.220481
$576$	0	0
$577$	−12304.0	−0.887733	−0.443867	−	0.896093i	$-0.646394\pi$
$577$	−12304.0	−0.887733	−0.443867	+	0.896093i	$0.646394\pi$
$578$	0	0
$579$	3846.00	0.276052
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−29880.0	−2.12265
$584$	0	0
$585$	4752.00	0.335848
$586$	0	0
$587$	28236.0	1.98539	0.992695	−	0.120647i	$-0.0384970\pi$
$587$	28236.0	1.98539	0.992695	+	0.120647i	$0.0384970\pi$
$588$	0	0
$589$	7072.00	0.494731
$590$	0	0
$591$	7866.00	0.547486
$592$	0	0
$593$	−4888.00	−0.338493	−0.169246	−	0.985574i	$-0.554133\pi$
$593$	−4888.00	−0.338493	−0.169246	+	0.985574i	$0.554133\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2328.00	0.159596
$598$	0	0
$599$	5208.00	0.355247	0.177624	−	0.984098i	$-0.443159\pi$
$599$	5208.00	0.355247	0.177624	+	0.984098i	$0.443159\pi$
$600$	0	0
$601$	27040.0	1.83525	0.917624	−	0.397449i	$-0.130104\pi$
$601$	27040.0	1.83525	0.917624	+	0.397449i	$0.130104\pi$
$602$	0	0
$603$	−3420.00	−0.230967
$604$	0	0
$605$	−27228.0	−1.82971
$606$	0	0
$607$	−4032.00	−0.269611	−0.134805	−	0.990872i	$-0.543041\pi$
$607$	−4032.00	−0.269611	−0.134805	+	0.990872i	$0.543041\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8096.00	−0.536054
$612$	0	0
$613$	21602.0	1.42332	0.711661	−	0.702523i	$-0.247943\pi$
$613$	21602.0	1.42332	0.711661	+	0.702523i	$0.247943\pi$
$614$	0	0
$615$	−6912.00	−0.453201
$616$	0	0
$617$	−2554.00	−0.166645	−0.0833227	−	0.996523i	$-0.526553\pi$
$617$	−2554.00	−0.166645	−0.0833227	+	0.996523i	$0.526553\pi$
$618$	0	0
$619$	12580.0	0.816854	0.408427	−	0.912791i	$-0.366077\pi$
$619$	12580.0	0.816854	0.408427	+	0.912791i	$0.366077\pi$
$620$	0	0
$621$	4320.00	0.279156
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−17639.0	−1.12890
$626$	0	0
$627$	9360.00	0.596176
$628$	0	0
$629$	−40704.0	−2.58025
$630$	0	0
$631$	−14784.0	−0.932713	−0.466356	−	0.884597i	$-0.654434\pi$
$631$	−14784.0	−0.932713	−0.466356	+	0.884597i	$0.654434\pi$
$632$	0	0
$633$	16428.0	1.03152
$634$	0	0
$635$	19392.0	1.21189
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2376.00	0.147094
$640$	0	0
$641$	1278.00	0.0787488	0.0393744	−	0.999225i	$-0.487464\pi$
$641$	1278.00	0.0787488	0.0393744	+	0.999225i	$0.487464\pi$
$642$	0	0
$643$	24428.0	1.49821	0.749103	−	0.662454i	$-0.230485\pi$
$643$	24428.0	1.49821	0.749103	+	0.662454i	$0.230485\pi$
$644$	0	0
$645$	7920.00	0.483488
$646$	0	0
$647$	424.000	0.0257638	0.0128819	−	0.999917i	$-0.495899\pi$
$647$	424.000	0.0257638	0.0128819	+	0.999917i	$0.495899\pi$
$648$	0	0
$649$	−29520.0	−1.78546
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15754.0	0.944107	0.472053	−	0.881570i	$-0.343513\pi$
$653$	15754.0	0.944107	0.472053	+	0.881570i	$0.343513\pi$
$654$	0	0
$655$	−20400.0	−1.21694
$656$	0	0
$657$	5040.00	0.299283
$658$	0	0
$659$	5860.00	0.346393	0.173197	−	0.984887i	$-0.444590\pi$
$659$	5860.00	0.346393	0.173197	+	0.984887i	$0.444590\pi$
$660$	0	0
$661$	−10580.0	−0.622563	−0.311282	−	0.950318i	$-0.600758\pi$
$661$	−10580.0	−0.622563	−0.311282	+	0.950318i	$0.600758\pi$
$662$	0	0
$663$	−16896.0	−0.989723
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−36800.0	−2.13628
$668$	0	0
$669$	17040.0	0.984760
$670$	0	0
$671$	−1200.00	−0.0690395
$672$	0	0
$673$	−8290.00	−0.474823	−0.237412	−	0.971409i	$-0.576299\pi$
$673$	−8290.00	−0.474823	−0.237412	+	0.971409i	$0.576299\pi$
$674$	0	0
$675$	513.000	0.0292524
$676$	0	0
$677$	−14500.0	−0.823161	−0.411581	−	0.911373i	$-0.635023\pi$
$677$	−14500.0	−0.823161	−0.411581	+	0.911373i	$0.635023\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−15540.0	−0.874441
$682$	0	0
$683$	7572.00	0.424209	0.212104	−	0.977247i	$-0.431968\pi$
$683$	7572.00	0.424209	0.212104	+	0.977247i	$0.431968\pi$
$684$	0	0
$685$	−23496.0	−1.31056
$686$	0	0
$687$	8988.00	0.499146
$688$	0	0
$689$	21912.0	1.21158
$690$	0	0
$691$	−17020.0	−0.937006	−0.468503	−	0.883462i	$-0.655207\pi$
$691$	−17020.0	−0.937006	−0.468503	+	0.883462i	$0.655207\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15408.0	0.840948
$696$	0	0
$697$	24576.0	1.33556
$698$	0	0
$699$	−5454.00	−0.295120
$700$	0	0
$701$	−12422.0	−0.669290	−0.334645	−	0.942344i	$-0.608616\pi$
$701$	−12422.0	−0.669290	−0.334645	+	0.942344i	$0.608616\pi$
$702$	0	0
$703$	−16536.0	−0.887151
$704$	0	0
$705$	−6624.00	−0.353864
$706$	0	0
$707$	0	0
$708$	0	0
$709$	22370.0	1.18494	0.592470	−	0.805592i	$-0.298153\pi$
$709$	22370.0	1.18494	0.592470	+	0.805592i	$0.298153\pi$
$710$	0	0
$711$	−936.000	−0.0493709
$712$	0	0
$713$	21760.0	1.14294
$714$	0	0
$715$	31680.0	1.65701
$716$	0	0
$717$	−6792.00	−0.353768
$718$	0	0
$719$	−14936.0	−0.774713	−0.387357	−	0.921930i	$-0.626612\pi$
$719$	−14936.0	−0.774713	−0.387357	+	0.921930i	$0.626612\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	8856.00	0.455544
$724$	0	0
$725$	−4370.00	−0.223859
$726$	0	0
$727$	−24456.0	−1.24762	−0.623812	−	0.781574i	$-0.714417\pi$
$727$	−24456.0	−1.24762	−0.623812	+	0.781574i	$0.714417\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−28160.0	−1.42481
$732$	0	0
$733$	34796.0	1.75337	0.876685	−	0.481066i	$-0.159750\pi$
$733$	34796.0	1.75337	0.876685	+	0.481066i	$0.159750\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−22800.0	−1.13955
$738$	0	0
$739$	−14228.0	−0.708235	−0.354117	−	0.935201i	$-0.615219\pi$
$739$	−14228.0	−0.708235	−0.354117	+	0.935201i	$0.615219\pi$
$740$	0	0
$741$	−6864.00	−0.340291
$742$	0	0
$743$	−14368.0	−0.709436	−0.354718	−	0.934973i	$-0.615423\pi$
$743$	−14368.0	−0.709436	−0.354718	+	0.934973i	$0.615423\pi$
$744$	0	0
$745$	3672.00	0.180579
$746$	0	0
$747$	13572.0	0.664757
$748$	0	0
$749$	0	0
$750$	0	0
$751$	7360.00	0.357617	0.178808	−	0.983884i	$-0.442776\pi$
$751$	7360.00	0.357617	0.178808	+	0.983884i	$0.442776\pi$
$752$	0	0
$753$	7380.00	0.357161
$754$	0	0
$755$	−27168.0	−1.30960
$756$	0	0
$757$	−4558.00	−0.218842	−0.109421	−	0.993996i	$-0.534900\pi$
$757$	−4558.00	−0.218842	−0.109421	+	0.993996i	$0.534900\pi$
$758$	0	0
$759$	28800.0	1.37730
$760$	0	0
$761$	13008.0	0.619632	0.309816	−	0.950797i	$-0.399733\pi$
$761$	13008.0	0.619632	0.309816	+	0.950797i	$0.399733\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−13824.0	−0.653343
$766$	0	0
$767$	21648.0	1.01912
$768$	0	0
$769$	−14168.0	−0.664384	−0.332192	−	0.943212i	$-0.607788\pi$
$769$	−14168.0	−0.664384	−0.332192	+	0.943212i	$0.607788\pi$
$770$	0	0
$771$	8856.00	0.413672
$772$	0	0
$773$	−14700.0	−0.683987	−0.341994	−	0.939702i	$-0.611102\pi$
$773$	−14700.0	−0.683987	−0.341994	+	0.939702i	$0.611102\pi$
$774$	0	0
$775$	2584.00	0.119768
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9984.00	0.459196
$780$	0	0
$781$	15840.0	0.725736
$782$	0	0
$783$	−6210.00	−0.283432
$784$	0	0
$785$	−24816.0	−1.12831
$786$	0	0
$787$	8068.00	0.365430	0.182715	−	0.983166i	$-0.441511\pi$
$787$	8068.00	0.365430	0.182715	+	0.983166i	$0.441511\pi$
$788$	0	0
$789$	16344.0	0.737467
$790$	0	0
$791$	0	0
$792$	0	0
$793$	880.000	0.0394070
$794$	0	0
$795$	17928.0	0.799800
$796$	0	0
$797$	−4228.00	−0.187909	−0.0939545	−	0.995576i	$-0.529951\pi$
$797$	−4228.00	−0.187909	−0.0939545	+	0.995576i	$0.529951\pi$
$798$	0	0
$799$	23552.0	1.04282
$800$	0	0
$801$	−10296.0	−0.454171
$802$	0	0
$803$	33600.0	1.47661
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−24156.0	−1.05369
$808$	0	0
$809$	−36934.0	−1.60511	−0.802553	−	0.596581i	$-0.796525\pi$
$809$	−36934.0	−1.60511	−0.802553	+	0.596581i	$0.796525\pi$
$810$	0	0
$811$	−16540.0	−0.716150	−0.358075	−	0.933693i	$-0.616567\pi$
$811$	−16540.0	−0.716150	−0.358075	+	0.933693i	$0.616567\pi$
$812$	0	0
$813$	11616.0	0.501096
$814$	0	0
$815$	22896.0	0.984064
$816$	0	0
$817$	−11440.0	−0.489884
$818$	0	0
$819$	0	0
$820$	0	0
$821$	40718.0	1.73090	0.865449	−	0.500996i	$-0.167033\pi$
$821$	40718.0	1.73090	0.865449	+	0.500996i	$0.167033\pi$
$822$	0	0
$823$	34832.0	1.47529	0.737647	−	0.675186i	$-0.235937\pi$
$823$	34832.0	1.47529	0.737647	+	0.675186i	$0.235937\pi$
$824$	0	0
$825$	3420.00	0.144326
$826$	0	0
$827$	−38300.0	−1.61043	−0.805213	−	0.592986i	$-0.797949\pi$
$827$	−38300.0	−1.61043	−0.805213	+	0.592986i	$0.797949\pi$
$828$	0	0
$829$	20476.0	0.857854	0.428927	−	0.903339i	$-0.358892\pi$
$829$	20476.0	0.857854	0.428927	+	0.903339i	$0.358892\pi$
$830$	0	0
$831$	18186.0	0.759164
$832$	0	0
$833$	0	0
$834$	0	0
$835$	37536.0	1.55567
$836$	0	0
$837$	3672.00	0.151640
$838$	0	0
$839$	−42024.0	−1.72924	−0.864618	−	0.502429i	$-0.832440\pi$
$839$	−42024.0	−1.72924	−0.864618	+	0.502429i	$0.832440\pi$
$840$	0	0
$841$	28511.0	1.16901
$842$	0	0
$843$	−21054.0	−0.860188
$844$	0	0
$845$	3132.00	0.127508
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−10788.0	−0.436093
$850$	0	0
$851$	−50880.0	−2.04952
$852$	0	0
$853$	−412.000	−0.0165376	−0.00826882	−	0.999966i	$-0.502632\pi$
$853$	−412.000	−0.0165376	−0.00826882	+	0.999966i	$0.502632\pi$
$854$	0	0
$855$	−5616.00	−0.224635
$856$	0	0
$857$	42816.0	1.70661	0.853306	−	0.521410i	$-0.174594\pi$
$857$	42816.0	1.70661	0.853306	+	0.521410i	$0.174594\pi$
$858$	0	0
$859$	−43572.0	−1.73068	−0.865342	−	0.501182i	$-0.832899\pi$
$859$	−43572.0	−1.73068	−0.865342	+	0.501182i	$0.832899\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16208.0	0.639313	0.319656	−	0.947534i	$-0.396433\pi$
$863$	16208.0	0.639313	0.319656	+	0.947534i	$0.396433\pi$
$864$	0	0
$865$	6384.00	0.250939
$866$	0	0
$867$	34413.0	1.34801
$868$	0	0
$869$	−6240.00	−0.243587
$870$	0	0
$871$	16720.0	0.650443
$872$	0	0
$873$	8136.00	0.315420
$874$	0	0
$875$	0	0
$876$	0	0
$877$	34378.0	1.32367	0.661837	−	0.749648i	$-0.269777\pi$
$877$	34378.0	1.32367	0.661837	+	0.749648i	$0.269777\pi$
$878$	0	0
$879$	24300.0	0.932444
$880$	0	0
$881$	7416.00	0.283600	0.141800	−	0.989895i	$-0.454711\pi$
$881$	7416.00	0.283600	0.141800	+	0.989895i	$0.454711\pi$
$882$	0	0
$883$	16772.0	0.639210	0.319605	−	0.947551i	$-0.396450\pi$
$883$	16772.0	0.639210	0.319605	+	0.947551i	$0.396450\pi$
$884$	0	0
$885$	17712.0	0.672748
$886$	0	0
$887$	−23336.0	−0.883367	−0.441683	−	0.897171i	$-0.645619\pi$
$887$	−23336.0	−0.883367	−0.441683	+	0.897171i	$0.645619\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	4860.00	0.182734
$892$	0	0
$893$	9568.00	0.358545
$894$	0	0
$895$	−12816.0	−0.478650
$896$	0	0
$897$	−21120.0	−0.786150
$898$	0	0
$899$	−31280.0	−1.16045
$900$	0	0
$901$	−63744.0	−2.35696
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−49968.0	−1.83535
$906$	0	0
$907$	22732.0	0.832198	0.416099	−	0.909319i	$-0.363397\pi$
$907$	22732.0	0.832198	0.416099	+	0.909319i	$0.363397\pi$
$908$	0	0
$909$	4500.00	0.164198
$910$	0	0
$911$	−16584.0	−0.603131	−0.301566	−	0.953445i	$-0.597509\pi$
$911$	−16584.0	−0.603131	−0.301566	+	0.953445i	$0.597509\pi$
$912$	0	0
$913$	90480.0	3.27979
$914$	0	0
$915$	720.000	0.0260136
$916$	0	0
$917$	0	0
$918$	0	0
$919$	41616.0	1.49378	0.746891	−	0.664947i	$-0.231546\pi$
$919$	41616.0	1.49378	0.746891	+	0.664947i	$0.231546\pi$
$920$	0	0
$921$	−21492.0	−0.768931
$922$	0	0
$923$	−11616.0	−0.414242
$924$	0	0
$925$	−6042.00	−0.214767
$926$	0	0
$927$	2232.00	0.0790814
$928$	0	0
$929$	−41680.0	−1.47199	−0.735994	−	0.676988i	$-0.763285\pi$
$929$	−41680.0	−1.47199	−0.735994	+	0.676988i	$0.763285\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1632.00	−0.0572661
$934$	0	0
$935$	−92160.0	−3.22348
$936$	0	0
$937$	−13248.0	−0.461893	−0.230946	−	0.972967i	$-0.574182\pi$
$937$	−13248.0	−0.461893	−0.230946	+	0.972967i	$0.574182\pi$
$938$	0	0
$939$	−8376.00	−0.291097
$940$	0	0
$941$	−6204.00	−0.214925	−0.107463	−	0.994209i	$-0.534273\pi$
$941$	−6204.00	−0.214925	−0.107463	+	0.994209i	$0.534273\pi$
$942$	0	0
$943$	30720.0	1.06085
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44996.0	1.54401	0.772003	−	0.635619i	$-0.219255\pi$
$947$	44996.0	1.54401	0.772003	+	0.635619i	$0.219255\pi$
$948$	0	0
$949$	−24640.0	−0.842833
$950$	0	0
$951$	−10926.0	−0.372555
$952$	0	0
$953$	−24918.0	−0.846981	−0.423491	−	0.905900i	$-0.639195\pi$
$953$	−24918.0	−0.846981	−0.423491	+	0.905900i	$0.639195\pi$
$954$	0	0
$955$	−3456.00	−0.117103
$956$	0	0
$957$	−41400.0	−1.39840
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−11295.0	−0.379141
$962$	0	0
$963$	9252.00	0.309597
$964$	0	0
$965$	−15384.0	−0.513190
$966$	0	0
$967$	8984.00	0.298765	0.149383	−	0.988779i	$-0.452271\pi$
$967$	8984.00	0.298765	0.149383	+	0.988779i	$0.452271\pi$
$968$	0	0
$969$	19968.0	0.661986
$970$	0	0
$971$	−15020.0	−0.496411	−0.248205	−	0.968707i	$-0.579841\pi$
$971$	−15020.0	−0.496411	−0.248205	+	0.968707i	$0.579841\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−2508.00	−0.0823798
$976$	0	0
$977$	−53138.0	−1.74006	−0.870028	−	0.493002i	$-0.835900\pi$
$977$	−53138.0	−1.74006	−0.870028	+	0.493002i	$0.835900\pi$
$978$	0	0
$979$	−68640.0	−2.24080
$980$	0	0
$981$	9594.00	0.312246
$982$	0	0
$983$	−2120.00	−0.0687869	−0.0343934	−	0.999408i	$-0.510950\pi$
$983$	−2120.00	−0.0687869	−0.0343934	+	0.999408i	$0.510950\pi$
$984$	0	0
$985$	−31464.0	−1.01779
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−35200.0	−1.13174
$990$	0	0
$991$	−18520.0	−0.593650	−0.296825	−	0.954932i	$-0.595928\pi$
$991$	−18520.0	−0.593650	−0.296825	+	0.954932i	$0.595928\pi$
$992$	0	0
$993$	−7140.00	−0.228178
$994$	0	0
$995$	−9312.00	−0.296694
$996$	0	0
$997$	−46996.0	−1.49286	−0.746428	−	0.665466i	$-0.768233\pi$
$997$	−46996.0	−1.49286	−0.746428	+	0.665466i	$0.768233\pi$
$998$	0	0
$999$	−8586.00	−0.271921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.x.1.1		1
4.3	odd	2		1176.4.a.b.1.1	✓	1
7.6	odd	2		2352.4.a.p.1.1		1
28.27	even	2		1176.4.a.o.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.4.a.b.1.1	✓	1	4.3	odd	2
1176.4.a.o.1.1	yes	1	28.27	even	2
2352.4.a.p.1.1		1	7.6	odd	2
2352.4.a.x.1.1		1	1.1	even	1	trivial

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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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