LMFDB - Embedded newform 1680.4.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,4,Mod(1,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1680.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1680.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:	$99.1232088096$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 210)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -5.00000 q^{5} -7.00000 q^{7} +9.00000 q^{9} -12.0000 q^{11} +2.00000 q^{13} -15.0000 q^{15} -18.0000 q^{17} -56.0000 q^{19} -21.0000 q^{21} +156.000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -186.000 q^{29} +52.0000 q^{31} -36.0000 q^{33} +35.0000 q^{35} -178.000 q^{37} +6.00000 q^{39} -138.000 q^{41} +412.000 q^{43} -45.0000 q^{45} +456.000 q^{47} +49.0000 q^{49} -54.0000 q^{51} -198.000 q^{53} +60.0000 q^{55} -168.000 q^{57} -348.000 q^{59} +110.000 q^{61} -63.0000 q^{63} -10.0000 q^{65} +196.000 q^{67} +468.000 q^{69} +936.000 q^{71} +542.000 q^{73} +75.0000 q^{75} +84.0000 q^{77} -992.000 q^{79} +81.0000 q^{81} +276.000 q^{83} +90.0000 q^{85} -558.000 q^{87} +630.000 q^{89} -14.0000 q^{91} +156.000 q^{93} +280.000 q^{95} +110.000 q^{97} -108.000 q^{99} +O(q^{100})$

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
$5$	−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
$9$	9.00000	0.333333
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	2.00000	0.0426692	0.0213346	−	0.999772i	$-0.493208\pi$
$13$	2.00000	0.0426692	0.0213346	+	0.999772i	$0.493208\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	−18.0000	−0.256802	−0.128401	−	0.991722i	$-0.540985\pi$
$17$	−18.0000	−0.256802	−0.128401	+	0.991722i	$0.540985\pi$
$18$	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
$21$	−21.0000	−0.218218
$22$	0	0
$23$	156.000	1.41427	0.707136	−	0.707078i	$-0.249987\pi$
$23$	156.000	1.41427	0.707136	+	0.707078i	$0.249987\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−186.000	−1.19101	−0.595506	−	0.803351i	$-0.703048\pi$
$29$	−186.000	−1.19101	−0.595506	+	0.803351i	$0.703048\pi$
$30$	0	0
$31$	52.0000	0.301273	0.150637	−	0.988589i	$-0.451868\pi$
$31$	52.0000	0.301273	0.150637	+	0.988589i	$0.451868\pi$
$32$	0	0
$33$	−36.0000	−0.189903
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	−178.000	−0.790892	−0.395446	−	0.918489i	$-0.629410\pi$
$37$	−178.000	−0.790892	−0.395446	+	0.918489i	$0.629410\pi$
$38$	0	0
$39$	6.00000	0.0246351
$40$	0	0
$41$	−138.000	−0.525658	−0.262829	−	0.964842i	$-0.584656\pi$
$41$	−138.000	−0.525658	−0.262829	+	0.964842i	$0.584656\pi$
$42$	0	0
$43$	412.000	1.46115	0.730575	−	0.682833i	$-0.239252\pi$
$43$	412.000	1.46115	0.730575	+	0.682833i	$0.239252\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	456.000	1.41520	0.707600	−	0.706613i	$-0.249778\pi$
$47$	456.000	1.41520	0.707600	+	0.706613i	$0.249778\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−54.0000	−0.148265
$52$	0	0
$53$	−198.000	−0.513158	−0.256579	−	0.966523i	$-0.582595\pi$
$53$	−198.000	−0.513158	−0.256579	+	0.966523i	$0.582595\pi$
$54$	0	0
$55$	60.0000	0.147098
$56$	0	0
$57$	−168.000	−0.390388
$58$	0	0
$59$	−348.000	−0.767894	−0.383947	−	0.923355i	$-0.625435\pi$
$59$	−348.000	−0.767894	−0.383947	+	0.923355i	$0.625435\pi$
$60$	0	0
$61$	110.000	0.230886	0.115443	−	0.993314i	$-0.463171\pi$
$61$	110.000	0.230886	0.115443	+	0.993314i	$0.463171\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−10.0000	−0.0190823
$66$	0	0
$67$	196.000	0.357391	0.178696	−	0.983904i	$-0.442812\pi$
$67$	196.000	0.357391	0.178696	+	0.983904i	$0.442812\pi$
$68$	0	0
$69$	468.000	0.816530
$70$	0	0
$71$	936.000	1.56455	0.782273	−	0.622936i	$-0.214060\pi$
$71$	936.000	1.56455	0.782273	+	0.622936i	$0.214060\pi$
$72$	0	0
$73$	542.000	0.868990	0.434495	−	0.900674i	$-0.356927\pi$
$73$	542.000	0.868990	0.434495	+	0.900674i	$0.356927\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	84.0000	0.124321
$78$	0	0
$79$	−992.000	−1.41277	−0.706384	−	0.707829i	$-0.749675\pi$
$79$	−992.000	−1.41277	−0.706384	+	0.707829i	$0.749675\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	276.000	0.364999	0.182500	−	0.983206i	$-0.441581\pi$
$83$	276.000	0.364999	0.182500	+	0.983206i	$0.441581\pi$
$84$	0	0
$85$	90.0000	0.114846
$86$	0	0
$87$	−558.000	−0.687631
$88$	0	0
$89$	630.000	0.750336	0.375168	−	0.926957i	$-0.377585\pi$
$89$	630.000	0.750336	0.375168	+	0.926957i	$0.377585\pi$
$90$	0	0
$91$	−14.0000	−0.0161275
$92$	0	0
$93$	156.000	0.173940
$94$	0	0
$95$	280.000	0.302394
$96$	0	0
$97$	110.000	0.115142	0.0575712	−	0.998341i	$-0.481664\pi$
$97$	110.000	0.115142	0.0575712	+	0.998341i	$0.481664\pi$
$98$	0	0
$99$	−108.000	−0.109640
$100$	0	0
$101$	570.000	0.561556	0.280778	−	0.959773i	$-0.409408\pi$
$101$	570.000	0.561556	0.280778	+	0.959773i	$0.409408\pi$
$102$	0	0
$103$	304.000	0.290816	0.145408	−	0.989372i	$-0.453551\pi$
$103$	304.000	0.290816	0.145408	+	0.989372i	$0.453551\pi$
$104$	0	0
$105$	105.000	0.0975900
$106$	0	0
$107$	−216.000	−0.195154	−0.0975771	−	0.995228i	$-0.531109\pi$
$107$	−216.000	−0.195154	−0.0975771	+	0.995228i	$0.531109\pi$
$108$	0	0
$109$	614.000	0.539546	0.269773	−	0.962924i	$-0.413051\pi$
$109$	614.000	0.539546	0.269773	+	0.962924i	$0.413051\pi$
$110$	0	0
$111$	−534.000	−0.456622
$112$	0	0
$113$	−498.000	−0.414583	−0.207292	−	0.978279i	$-0.566465\pi$
$113$	−498.000	−0.414583	−0.207292	+	0.978279i	$0.566465\pi$
$114$	0	0
$115$	−780.000	−0.632482
$116$	0	0
$117$	18.0000	0.0142231
$118$	0	0
$119$	126.000	0.0970622
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	−414.000	−0.303489
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1888.00	1.31916	0.659578	−	0.751636i	$-0.270735\pi$
$127$	1888.00	1.31916	0.659578	+	0.751636i	$0.270735\pi$
$128$	0	0
$129$	1236.00	0.843595
$130$	0	0
$131$	2892.00	1.92882	0.964409	−	0.264414i	$-0.0851786\pi$
$131$	2892.00	1.92882	0.964409	+	0.264414i	$0.0851786\pi$
$132$	0	0
$133$	392.000	0.255569
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	822.000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	822.000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	376.000	0.229438	0.114719	−	0.993398i	$-0.463403\pi$
$139$	376.000	0.229438	0.114719	+	0.993398i	$0.463403\pi$
$140$	0	0
$141$	1368.00	0.817067
$142$	0	0
$143$	−24.0000	−0.0140348
$144$	0	0
$145$	930.000	0.532637
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	3390.00	1.86389	0.931945	−	0.362600i	$-0.118111\pi$
$149$	3390.00	1.86389	0.931945	+	0.362600i	$0.118111\pi$
$150$	0	0
$151$	2968.00	1.59955	0.799776	−	0.600298i	$-0.204951\pi$
$151$	2968.00	1.59955	0.799776	+	0.600298i	$0.204951\pi$
$152$	0	0
$153$	−162.000	−0.0856008
$154$	0	0
$155$	−260.000	−0.134734
$156$	0	0
$157$	1874.00	0.952621	0.476310	−	0.879277i	$-0.341974\pi$
$157$	1874.00	0.952621	0.476310	+	0.879277i	$0.341974\pi$
$158$	0	0
$159$	−594.000	−0.296272
$160$	0	0
$161$	−1092.00	−0.534544
$162$	0	0
$163$	−452.000	−0.217199	−0.108599	−	0.994086i	$-0.534637\pi$
$163$	−452.000	−0.217199	−0.108599	+	0.994086i	$0.534637\pi$
$164$	0	0
$165$	180.000	0.0849272
$166$	0	0
$167$	1416.00	0.656128	0.328064	−	0.944656i	$-0.393604\pi$
$167$	1416.00	0.656128	0.328064	+	0.944656i	$0.393604\pi$
$168$	0	0
$169$	−2193.00	−0.998179
$170$	0	0
$171$	−504.000	−0.225391
$172$	0	0
$173$	426.000	0.187215	0.0936075	−	0.995609i	$-0.470160\pi$
$173$	426.000	0.187215	0.0936075	+	0.995609i	$0.470160\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	−1044.00	−0.443344
$178$	0	0
$179$	−2700.00	−1.12742	−0.563708	−	0.825974i	$-0.690626\pi$
$179$	−2700.00	−1.12742	−0.563708	+	0.825974i	$0.690626\pi$
$180$	0	0
$181$	−1978.00	−0.812285	−0.406142	−	0.913810i	$-0.633126\pi$
$181$	−1978.00	−0.812285	−0.406142	+	0.913810i	$0.633126\pi$
$182$	0	0
$183$	330.000	0.133302
$184$	0	0
$185$	890.000	0.353698
$186$	0	0
$187$	216.000	0.0844678
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	2328.00	0.881928	0.440964	−	0.897525i	$-0.354637\pi$
$191$	2328.00	0.881928	0.440964	+	0.897525i	$0.354637\pi$
$192$	0	0
$193$	−3166.00	−1.18080	−0.590398	−	0.807112i	$-0.701029\pi$
$193$	−3166.00	−1.18080	−0.590398	+	0.807112i	$0.701029\pi$
$194$	0	0
$195$	−30.0000	−0.0110172
$196$	0	0
$197$	−414.000	−0.149727	−0.0748637	−	0.997194i	$-0.523852\pi$
$197$	−414.000	−0.149727	−0.0748637	+	0.997194i	$0.523852\pi$
$198$	0	0
$199$	1636.00	0.582779	0.291389	−	0.956605i	$-0.405882\pi$
$199$	1636.00	0.582779	0.291389	+	0.956605i	$0.405882\pi$
$200$	0	0
$201$	588.000	0.206340
$202$	0	0
$203$	1302.00	0.450160
$204$	0	0
$205$	690.000	0.235081
$206$	0	0
$207$	1404.00	0.471424
$208$	0	0
$209$	672.000	0.222408
$210$	0	0
$211$	2860.00	0.933130	0.466565	−	0.884487i	$-0.345491\pi$
$211$	2860.00	0.933130	0.466565	+	0.884487i	$0.345491\pi$
$212$	0	0
$213$	2808.00	0.903291
$214$	0	0
$215$	−2060.00	−0.653446
$216$	0	0
$217$	−364.000	−0.113871
$218$	0	0
$219$	1626.00	0.501712
$220$	0	0
$221$	−36.0000	−0.0109576
$222$	0	0
$223$	1096.00	0.329119	0.164560	−	0.986367i	$-0.447380\pi$
$223$	1096.00	0.329119	0.164560	+	0.986367i	$0.447380\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−6276.00	−1.83503	−0.917517	−	0.397696i	$-0.869810\pi$
$227$	−6276.00	−1.83503	−0.917517	+	0.397696i	$0.869810\pi$
$228$	0	0
$229$	−754.000	−0.217580	−0.108790	−	0.994065i	$-0.534698\pi$
$229$	−754.000	−0.217580	−0.108790	+	0.994065i	$0.534698\pi$
$230$	0	0
$231$	252.000	0.0717765
$232$	0	0
$233$	3870.00	1.08812	0.544060	−	0.839046i	$-0.316886\pi$
$233$	3870.00	1.08812	0.544060	+	0.839046i	$0.316886\pi$
$234$	0	0
$235$	−2280.00	−0.632897
$236$	0	0
$237$	−2976.00	−0.815662
$238$	0	0
$239$	−744.000	−0.201361	−0.100681	−	0.994919i	$-0.532102\pi$
$239$	−744.000	−0.201361	−0.100681	+	0.994919i	$0.532102\pi$
$240$	0	0
$241$	5474.00	1.46312	0.731559	−	0.681778i	$-0.238793\pi$
$241$	5474.00	1.46312	0.731559	+	0.681778i	$0.238793\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	−112.000	−0.0288518
$248$	0	0
$249$	828.000	0.210732
$250$	0	0
$251$	−1980.00	−0.497914	−0.248957	−	0.968514i	$-0.580088\pi$
$251$	−1980.00	−0.497914	−0.248957	+	0.968514i	$0.580088\pi$
$252$	0	0
$253$	−1872.00	−0.465184
$254$	0	0
$255$	270.000	0.0663061
$256$	0	0
$257$	−210.000	−0.0509706	−0.0254853	−	0.999675i	$-0.508113\pi$
$257$	−210.000	−0.0509706	−0.0254853	+	0.999675i	$0.508113\pi$
$258$	0	0
$259$	1246.00	0.298929
$260$	0	0
$261$	−1674.00	−0.397004
$262$	0	0
$263$	1428.00	0.334807	0.167404	−	0.985888i	$-0.446462\pi$
$263$	1428.00	0.334807	0.167404	+	0.985888i	$0.446462\pi$
$264$	0	0
$265$	990.000	0.229491
$266$	0	0
$267$	1890.00	0.433206
$268$	0	0
$269$	4122.00	0.934285	0.467143	−	0.884182i	$-0.345283\pi$
$269$	4122.00	0.934285	0.467143	+	0.884182i	$0.345283\pi$
$270$	0	0
$271$	−5780.00	−1.29561	−0.647804	−	0.761807i	$-0.724313\pi$
$271$	−5780.00	−1.29561	−0.647804	+	0.761807i	$0.724313\pi$
$272$	0	0
$273$	−42.0000	−0.00931119
$274$	0	0
$275$	−300.000	−0.0657843
$276$	0	0
$277$	4574.00	0.992148	0.496074	−	0.868280i	$-0.334774\pi$
$277$	4574.00	0.992148	0.496074	+	0.868280i	$0.334774\pi$
$278$	0	0
$279$	468.000	0.100424
$280$	0	0
$281$	3450.00	0.732419	0.366210	−	0.930532i	$-0.380655\pi$
$281$	3450.00	0.732419	0.366210	+	0.930532i	$0.380655\pi$
$282$	0	0
$283$	700.000	0.147034	0.0735171	−	0.997294i	$-0.476578\pi$
$283$	700.000	0.147034	0.0735171	+	0.997294i	$0.476578\pi$
$284$	0	0
$285$	840.000	0.174587
$286$	0	0
$287$	966.000	0.198680
$288$	0	0
$289$	−4589.00	−0.934053
$290$	0	0
$291$	330.000	0.0664775
$292$	0	0
$293$	7170.00	1.42961	0.714805	−	0.699324i	$-0.246515\pi$
$293$	7170.00	1.42961	0.714805	+	0.699324i	$0.246515\pi$
$294$	0	0
$295$	1740.00	0.343413
$296$	0	0
$297$	−324.000	−0.0633010
$298$	0	0
$299$	312.000	0.0603459
$300$	0	0
$301$	−2884.00	−0.552262
$302$	0	0
$303$	1710.00	0.324214
$304$	0	0
$305$	−550.000	−0.103255
$306$	0	0
$307$	−6644.00	−1.23516	−0.617578	−	0.786509i	$-0.711886\pi$
$307$	−6644.00	−1.23516	−0.617578	+	0.786509i	$0.711886\pi$
$308$	0	0
$309$	912.000	0.167902
$310$	0	0
$311$	5376.00	0.980209	0.490104	−	0.871664i	$-0.336959\pi$
$311$	5376.00	0.980209	0.490104	+	0.871664i	$0.336959\pi$
$312$	0	0
$313$	2126.00	0.383925	0.191963	−	0.981402i	$-0.438515\pi$
$313$	2126.00	0.383925	0.191963	+	0.981402i	$0.438515\pi$
$314$	0	0
$315$	315.000	0.0563436
$316$	0	0
$317$	1074.00	0.190290	0.0951449	−	0.995463i	$-0.469669\pi$
$317$	1074.00	0.190290	0.0951449	+	0.995463i	$0.469669\pi$
$318$	0	0
$319$	2232.00	0.391749
$320$	0	0
$321$	−648.000	−0.112672
$322$	0	0
$323$	1008.00	0.173643
$324$	0	0
$325$	50.0000	0.00853385
$326$	0	0
$327$	1842.00	0.311507
$328$	0	0
$329$	−3192.00	−0.534896
$330$	0	0
$331$	2788.00	0.462968	0.231484	−	0.972839i	$-0.425642\pi$
$331$	2788.00	0.462968	0.231484	+	0.972839i	$0.425642\pi$
$332$	0	0
$333$	−1602.00	−0.263631
$334$	0	0
$335$	−980.000	−0.159830
$336$	0	0
$337$	−6334.00	−1.02384	−0.511921	−	0.859032i	$-0.671066\pi$
$337$	−6334.00	−1.02384	−0.511921	+	0.859032i	$0.671066\pi$
$338$	0	0
$339$	−1494.00	−0.239360
$340$	0	0
$341$	−624.000	−0.0990953
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−2340.00	−0.365163
$346$	0	0
$347$	7032.00	1.08789	0.543945	−	0.839121i	$-0.316930\pi$
$347$	7032.00	1.08789	0.543945	+	0.839121i	$0.316930\pi$
$348$	0	0
$349$	−1474.00	−0.226079	−0.113039	−	0.993591i	$-0.536059\pi$
$349$	−1474.00	−0.226079	−0.113039	+	0.993591i	$0.536059\pi$
$350$	0	0
$351$	54.0000	0.00821170
$352$	0	0
$353$	7950.00	1.19868	0.599342	−	0.800493i	$-0.295429\pi$
$353$	7950.00	1.19868	0.599342	+	0.800493i	$0.295429\pi$
$354$	0	0
$355$	−4680.00	−0.699686
$356$	0	0
$357$	378.000	0.0560389
$358$	0	0
$359$	−6624.00	−0.973820	−0.486910	−	0.873452i	$-0.661876\pi$
$359$	−6624.00	−0.973820	−0.486910	+	0.873452i	$0.661876\pi$
$360$	0	0
$361$	−3723.00	−0.542790
$362$	0	0
$363$	−3561.00	−0.514887
$364$	0	0
$365$	−2710.00	−0.388624
$366$	0	0
$367$	−1784.00	−0.253744	−0.126872	−	0.991919i	$-0.540494\pi$
$367$	−1784.00	−0.253744	−0.126872	+	0.991919i	$0.540494\pi$
$368$	0	0
$369$	−1242.00	−0.175219
$370$	0	0
$371$	1386.00	0.193956
$372$	0	0
$373$	−1978.00	−0.274576	−0.137288	−	0.990531i	$-0.543839\pi$
$373$	−1978.00	−0.274576	−0.137288	+	0.990531i	$0.543839\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−372.000	−0.0508196
$378$	0	0
$379$	10780.0	1.46103	0.730516	−	0.682895i	$-0.239279\pi$
$379$	10780.0	1.46103	0.730516	+	0.682895i	$0.239279\pi$
$380$	0	0
$381$	5664.00	0.761616
$382$	0	0
$383$	5880.00	0.784475	0.392238	−	0.919864i	$-0.371701\pi$
$383$	5880.00	0.784475	0.392238	+	0.919864i	$0.371701\pi$
$384$	0	0
$385$	−420.000	−0.0555979
$386$	0	0
$387$	3708.00	0.487050
$388$	0	0
$389$	6438.00	0.839125	0.419562	−	0.907726i	$-0.362184\pi$
$389$	6438.00	0.839125	0.419562	+	0.907726i	$0.362184\pi$
$390$	0	0
$391$	−2808.00	−0.363188
$392$	0	0
$393$	8676.00	1.11360
$394$	0	0
$395$	4960.00	0.631809
$396$	0	0
$397$	2954.00	0.373443	0.186722	−	0.982413i	$-0.440214\pi$
$397$	2954.00	0.373443	0.186722	+	0.982413i	$0.440214\pi$
$398$	0	0
$399$	1176.00	0.147553
$400$	0	0
$401$	−5574.00	−0.694145	−0.347073	−	0.937838i	$-0.612824\pi$
$401$	−5574.00	−0.694145	−0.347073	+	0.937838i	$0.612824\pi$
$402$	0	0
$403$	104.000	0.0128551
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	2136.00	0.260141
$408$	0	0
$409$	−15406.0	−1.86254	−0.931269	−	0.364333i	$-0.881297\pi$
$409$	−15406.0	−1.86254	−0.931269	+	0.364333i	$0.881297\pi$
$410$	0	0
$411$	2466.00	0.295958
$412$	0	0
$413$	2436.00	0.290237
$414$	0	0
$415$	−1380.00	−0.163233
$416$	0	0
$417$	1128.00	0.132466
$418$	0	0
$419$	−2940.00	−0.342789	−0.171394	−	0.985203i	$-0.554827\pi$
$419$	−2940.00	−0.342789	−0.171394	+	0.985203i	$0.554827\pi$
$420$	0	0
$421$	254.000	0.0294043	0.0147021	−	0.999892i	$-0.495320\pi$
$421$	254.000	0.0294043	0.0147021	+	0.999892i	$0.495320\pi$
$422$	0	0
$423$	4104.00	0.471734
$424$	0	0
$425$	−450.000	−0.0513605
$426$	0	0
$427$	−770.000	−0.0872668
$428$	0	0
$429$	−72.0000	−0.00810301
$430$	0	0
$431$	13248.0	1.48059	0.740294	−	0.672283i	$-0.234686\pi$
$431$	13248.0	1.48059	0.740294	+	0.672283i	$0.234686\pi$
$432$	0	0
$433$	16598.0	1.84215	0.921073	−	0.389391i	$-0.127314\pi$
$433$	16598.0	1.84215	0.921073	+	0.389391i	$0.127314\pi$
$434$	0	0
$435$	2790.00	0.307518
$436$	0	0
$437$	−8736.00	−0.956292
$438$	0	0
$439$	6532.00	0.710149	0.355074	−	0.934838i	$-0.384456\pi$
$439$	6532.00	0.710149	0.355074	+	0.934838i	$0.384456\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−12216.0	−1.31016	−0.655079	−	0.755561i	$-0.727365\pi$
$443$	−12216.0	−1.31016	−0.655079	+	0.755561i	$0.727365\pi$
$444$	0	0
$445$	−3150.00	−0.335560
$446$	0	0
$447$	10170.0	1.07612
$448$	0	0
$449$	306.000	0.0321627	0.0160813	−	0.999871i	$-0.494881\pi$
$449$	306.000	0.0321627	0.0160813	+	0.999871i	$0.494881\pi$
$450$	0	0
$451$	1656.00	0.172900
$452$	0	0
$453$	8904.00	0.923502
$454$	0	0
$455$	70.0000	0.00721242
$456$	0	0
$457$	−6046.00	−0.618862	−0.309431	−	0.950922i	$-0.600139\pi$
$457$	−6046.00	−0.618862	−0.309431	+	0.950922i	$0.600139\pi$
$458$	0	0
$459$	−486.000	−0.0494217
$460$	0	0
$461$	7122.00	0.719533	0.359766	−	0.933042i	$-0.382856\pi$
$461$	7122.00	0.719533	0.359766	+	0.933042i	$0.382856\pi$
$462$	0	0
$463$	11248.0	1.12903	0.564513	−	0.825424i	$-0.309064\pi$
$463$	11248.0	1.12903	0.564513	+	0.825424i	$0.309064\pi$
$464$	0	0
$465$	−780.000	−0.0777885
$466$	0	0
$467$	18252.0	1.80857	0.904285	−	0.426930i	$-0.140405\pi$
$467$	18252.0	1.80857	0.904285	+	0.426930i	$0.140405\pi$
$468$	0	0
$469$	−1372.00	−0.135081
$470$	0	0
$471$	5622.00	0.549996
$472$	0	0
$473$	−4944.00	−0.480603
$474$	0	0
$475$	−1400.00	−0.135235
$476$	0	0
$477$	−1782.00	−0.171053
$478$	0	0
$479$	18168.0	1.73302	0.866511	−	0.499159i	$-0.166358\pi$
$479$	18168.0	1.73302	0.866511	+	0.499159i	$0.166358\pi$
$480$	0	0
$481$	−356.000	−0.0337468
$482$	0	0
$483$	−3276.00	−0.308619
$484$	0	0
$485$	−550.000	−0.0514932
$486$	0	0
$487$	−19856.0	−1.84756	−0.923780	−	0.382925i	$-0.874917\pi$
$487$	−19856.0	−1.84756	−0.923780	+	0.382925i	$0.874917\pi$
$488$	0	0
$489$	−1356.00	−0.125400
$490$	0	0
$491$	−11220.0	−1.03127	−0.515633	−	0.856810i	$-0.672443\pi$
$491$	−11220.0	−1.03127	−0.515633	+	0.856810i	$0.672443\pi$
$492$	0	0
$493$	3348.00	0.305855
$494$	0	0
$495$	540.000	0.0490327
$496$	0	0
$497$	−6552.00	−0.591343
$498$	0	0
$499$	9268.00	0.831448	0.415724	−	0.909491i	$-0.363528\pi$
$499$	9268.00	0.831448	0.415724	+	0.909491i	$0.363528\pi$
$500$	0	0
$501$	4248.00	0.378816
$502$	0	0
$503$	−18576.0	−1.64665	−0.823323	−	0.567573i	$-0.807882\pi$
$503$	−18576.0	−1.64665	−0.823323	+	0.567573i	$0.807882\pi$
$504$	0	0
$505$	−2850.00	−0.251135
$506$	0	0
$507$	−6579.00	−0.576299
$508$	0	0
$509$	−11190.0	−0.974436	−0.487218	−	0.873280i	$-0.661988\pi$
$509$	−11190.0	−0.974436	−0.487218	+	0.873280i	$0.661988\pi$
$510$	0	0
$511$	−3794.00	−0.328448
$512$	0	0
$513$	−1512.00	−0.130129
$514$	0	0
$515$	−1520.00	−0.130057
$516$	0	0
$517$	−5472.00	−0.465490
$518$	0	0
$519$	1278.00	0.108089
$520$	0	0
$521$	−306.000	−0.0257315	−0.0128657	−	0.999917i	$-0.504095\pi$
$521$	−306.000	−0.0257315	−0.0128657	+	0.999917i	$0.504095\pi$
$522$	0	0
$523$	−17444.0	−1.45846	−0.729228	−	0.684270i	$-0.760121\pi$
$523$	−17444.0	−1.45846	−0.729228	+	0.684270i	$0.760121\pi$
$524$	0	0
$525$	−525.000	−0.0436436
$526$	0	0
$527$	−936.000	−0.0773677
$528$	0	0
$529$	12169.0	1.00016
$530$	0	0
$531$	−3132.00	−0.255965
$532$	0	0
$533$	−276.000	−0.0224294
$534$	0	0
$535$	1080.00	0.0872756
$536$	0	0
$537$	−8100.00	−0.650914
$538$	0	0
$539$	−588.000	−0.0469888
$540$	0	0
$541$	−538.000	−0.0427549	−0.0213775	−	0.999771i	$-0.506805\pi$
$541$	−538.000	−0.0427549	−0.0213775	+	0.999771i	$0.506805\pi$
$542$	0	0
$543$	−5934.00	−0.468973
$544$	0	0
$545$	−3070.00	−0.241292
$546$	0	0
$547$	−19820.0	−1.54925	−0.774627	−	0.632418i	$-0.782062\pi$
$547$	−19820.0	−1.54925	−0.774627	+	0.632418i	$0.782062\pi$
$548$	0	0
$549$	990.000	0.0769621
$550$	0	0
$551$	10416.0	0.805329
$552$	0	0
$553$	6944.00	0.533976
$554$	0	0
$555$	2670.00	0.204208
$556$	0	0
$557$	−16686.0	−1.26932	−0.634658	−	0.772794i	$-0.718859\pi$
$557$	−16686.0	−1.26932	−0.634658	+	0.772794i	$0.718859\pi$
$558$	0	0
$559$	824.000	0.0623461
$560$	0	0
$561$	648.000	0.0487675
$562$	0	0
$563$	16788.0	1.25671	0.628357	−	0.777925i	$-0.283728\pi$
$563$	16788.0	1.25671	0.628357	+	0.777925i	$0.283728\pi$
$564$	0	0
$565$	2490.00	0.185407
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	15906.0	1.17191	0.585953	−	0.810345i	$-0.300720\pi$
$569$	15906.0	1.17191	0.585953	+	0.810345i	$0.300720\pi$
$570$	0	0
$571$	−17084.0	−1.25209	−0.626045	−	0.779787i	$-0.715327\pi$
$571$	−17084.0	−1.25209	−0.626045	+	0.779787i	$0.715327\pi$
$572$	0	0
$573$	6984.00	0.509181
$574$	0	0
$575$	3900.00	0.282854
$576$	0	0
$577$	25382.0	1.83131	0.915656	−	0.401964i	$-0.131672\pi$
$577$	25382.0	1.83131	0.915656	+	0.401964i	$0.131672\pi$
$578$	0	0
$579$	−9498.00	−0.681733
$580$	0	0
$581$	−1932.00	−0.137957
$582$	0	0
$583$	2376.00	0.168789
$584$	0	0
$585$	−90.0000	−0.00636076
$586$	0	0
$587$	−13764.0	−0.967804	−0.483902	−	0.875122i	$-0.660781\pi$
$587$	−13764.0	−0.967804	−0.483902	+	0.875122i	$0.660781\pi$
$588$	0	0
$589$	−2912.00	−0.203713
$590$	0	0
$591$	−1242.00	−0.0864451
$592$	0	0
$593$	−13266.0	−0.918667	−0.459333	−	0.888264i	$-0.651912\pi$
$593$	−13266.0	−0.918667	−0.459333	+	0.888264i	$0.651912\pi$
$594$	0	0
$595$	−630.000	−0.0434075
$596$	0	0
$597$	4908.00	0.336467
$598$	0	0
$599$	−9600.00	−0.654834	−0.327417	−	0.944880i	$-0.606178\pi$
$599$	−9600.00	−0.654834	−0.327417	+	0.944880i	$0.606178\pi$
$600$	0	0
$601$	−19582.0	−1.32906	−0.664531	−	0.747261i	$-0.731369\pi$
$601$	−19582.0	−1.32906	−0.664531	+	0.747261i	$0.731369\pi$
$602$	0	0
$603$	1764.00	0.119130
$604$	0	0
$605$	5935.00	0.398830
$606$	0	0
$607$	−3944.00	−0.263727	−0.131863	−	0.991268i	$-0.542096\pi$
$607$	−3944.00	−0.263727	−0.131863	+	0.991268i	$0.542096\pi$
$608$	0	0
$609$	3906.00	0.259900
$610$	0	0
$611$	912.000	0.0603855
$612$	0	0
$613$	20846.0	1.37351	0.686755	−	0.726889i	$-0.259034\pi$
$613$	20846.0	1.37351	0.686755	+	0.726889i	$0.259034\pi$
$614$	0	0
$615$	2070.00	0.135724
$616$	0	0
$617$	15342.0	1.00105	0.500523	−	0.865723i	$-0.333141\pi$
$617$	15342.0	1.00105	0.500523	+	0.865723i	$0.333141\pi$
$618$	0	0
$619$	5128.00	0.332975	0.166488	−	0.986044i	$-0.446757\pi$
$619$	5128.00	0.332975	0.166488	+	0.986044i	$0.446757\pi$
$620$	0	0
$621$	4212.00	0.272177
$622$	0	0
$623$	−4410.00	−0.283600
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	2016.00	0.128407
$628$	0	0
$629$	3204.00	0.203103
$630$	0	0
$631$	−31016.0	−1.95678	−0.978389	−	0.206771i	$-0.933705\pi$
$631$	−31016.0	−1.95678	−0.978389	+	0.206771i	$0.933705\pi$
$632$	0	0
$633$	8580.00	0.538743
$634$	0	0
$635$	−9440.00	−0.589945
$636$	0	0
$637$	98.0000	0.00609561
$638$	0	0
$639$	8424.00	0.521515
$640$	0	0
$641$	15474.0	0.953489	0.476744	−	0.879042i	$-0.341817\pi$
$641$	15474.0	0.953489	0.476744	+	0.879042i	$0.341817\pi$
$642$	0	0
$643$	−24644.0	−1.51145	−0.755727	−	0.654887i	$-0.772716\pi$
$643$	−24644.0	−1.51145	−0.755727	+	0.654887i	$0.772716\pi$
$644$	0	0
$645$	−6180.00	−0.377267
$646$	0	0
$647$	16632.0	1.01062	0.505310	−	0.862938i	$-0.331378\pi$
$647$	16632.0	1.01062	0.505310	+	0.862938i	$0.331378\pi$
$648$	0	0
$649$	4176.00	0.252577
$650$	0	0
$651$	−1092.00	−0.0657432
$652$	0	0
$653$	−10542.0	−0.631762	−0.315881	−	0.948799i	$-0.602300\pi$
$653$	−10542.0	−0.631762	−0.315881	+	0.948799i	$0.602300\pi$
$654$	0	0
$655$	−14460.0	−0.862594
$656$	0	0
$657$	4878.00	0.289663
$658$	0	0
$659$	−15276.0	−0.902987	−0.451494	−	0.892274i	$-0.649109\pi$
$659$	−15276.0	−0.902987	−0.451494	+	0.892274i	$0.649109\pi$
$660$	0	0
$661$	1478.00	0.0869706	0.0434853	−	0.999054i	$-0.486154\pi$
$661$	1478.00	0.0869706	0.0434853	+	0.999054i	$0.486154\pi$
$662$	0	0
$663$	−108.000	−0.00632635
$664$	0	0
$665$	−1960.00	−0.114294
$666$	0	0
$667$	−29016.0	−1.68441
$668$	0	0
$669$	3288.00	0.190017
$670$	0	0
$671$	−1320.00	−0.0759434
$672$	0	0
$673$	−19366.0	−1.10922	−0.554610	−	0.832111i	$-0.687132\pi$
$673$	−19366.0	−1.10922	−0.554610	+	0.832111i	$0.687132\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	−21390.0	−1.21430	−0.607152	−	0.794585i	$-0.707688\pi$
$677$	−21390.0	−1.21430	−0.607152	+	0.794585i	$0.707688\pi$
$678$	0	0
$679$	−770.000	−0.0435197
$680$	0	0
$681$	−18828.0	−1.05946
$682$	0	0
$683$	21672.0	1.21414	0.607069	−	0.794649i	$-0.292345\pi$
$683$	21672.0	1.21414	0.607069	+	0.794649i	$0.292345\pi$
$684$	0	0
$685$	−4110.00	−0.229248
$686$	0	0
$687$	−2262.00	−0.125620
$688$	0	0
$689$	−396.000	−0.0218961
$690$	0	0
$691$	5992.00	0.329879	0.164940	−	0.986304i	$-0.447257\pi$
$691$	5992.00	0.329879	0.164940	+	0.986304i	$0.447257\pi$
$692$	0	0
$693$	756.000	0.0414402
$694$	0	0
$695$	−1880.00	−0.102608
$696$	0	0
$697$	2484.00	0.134990
$698$	0	0
$699$	11610.0	0.628227
$700$	0	0
$701$	17766.0	0.957222	0.478611	−	0.878027i	$-0.341140\pi$
$701$	17766.0	0.957222	0.478611	+	0.878027i	$0.341140\pi$
$702$	0	0
$703$	9968.00	0.534780
$704$	0	0
$705$	−6840.00	−0.365403
$706$	0	0
$707$	−3990.00	−0.212248
$708$	0	0
$709$	−24514.0	−1.29851	−0.649254	−	0.760571i	$-0.724919\pi$
$709$	−24514.0	−1.29851	−0.649254	+	0.760571i	$0.724919\pi$
$710$	0	0
$711$	−8928.00	−0.470923
$712$	0	0
$713$	8112.00	0.426082
$714$	0	0
$715$	120.000	0.00627657
$716$	0	0
$717$	−2232.00	−0.116256
$718$	0	0
$719$	13176.0	0.683424	0.341712	−	0.939805i	$-0.388993\pi$
$719$	13176.0	0.683424	0.341712	+	0.939805i	$0.388993\pi$
$720$	0	0
$721$	−2128.00	−0.109918
$722$	0	0
$723$	16422.0	0.844731
$724$	0	0
$725$	−4650.00	−0.238202
$726$	0	0
$727$	−20792.0	−1.06071	−0.530353	−	0.847777i	$-0.677940\pi$
$727$	−20792.0	−1.06071	−0.530353	+	0.847777i	$0.677940\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−7416.00	−0.375227
$732$	0	0
$733$	−21742.0	−1.09558	−0.547789	−	0.836616i	$-0.684530\pi$
$733$	−21742.0	−1.09558	−0.547789	+	0.836616i	$0.684530\pi$
$734$	0	0
$735$	−735.000	−0.0368856
$736$	0	0
$737$	−2352.00	−0.117554
$738$	0	0
$739$	−39044.0	−1.94351	−0.971757	−	0.235984i	$-0.924169\pi$
$739$	−39044.0	−1.94351	−0.971757	+	0.235984i	$0.924169\pi$
$740$	0	0
$741$	−336.000	−0.0166576
$742$	0	0
$743$	31116.0	1.53639	0.768193	−	0.640218i	$-0.221156\pi$
$743$	31116.0	1.53639	0.768193	+	0.640218i	$0.221156\pi$
$744$	0	0
$745$	−16950.0	−0.833557
$746$	0	0
$747$	2484.00	0.121666
$748$	0	0
$749$	1512.00	0.0737614
$750$	0	0
$751$	29320.0	1.42464	0.712318	−	0.701857i	$-0.247645\pi$
$751$	29320.0	1.42464	0.712318	+	0.701857i	$0.247645\pi$
$752$	0	0
$753$	−5940.00	−0.287471
$754$	0	0
$755$	−14840.0	−0.715342
$756$	0	0
$757$	−2266.00	−0.108797	−0.0543984	−	0.998519i	$-0.517324\pi$
$757$	−2266.00	−0.108797	−0.0543984	+	0.998519i	$0.517324\pi$
$758$	0	0
$759$	−5616.00	−0.268574
$760$	0	0
$761$	−29946.0	−1.42647	−0.713234	−	0.700926i	$-0.752770\pi$
$761$	−29946.0	−1.42647	−0.713234	+	0.700926i	$0.752770\pi$
$762$	0	0
$763$	−4298.00	−0.203929
$764$	0	0
$765$	810.000	0.0382818
$766$	0	0
$767$	−696.000	−0.0327655
$768$	0	0
$769$	−23110.0	−1.08370	−0.541852	−	0.840474i	$-0.682277\pi$
$769$	−23110.0	−1.08370	−0.541852	+	0.840474i	$0.682277\pi$
$770$	0	0
$771$	−630.000	−0.0294279
$772$	0	0
$773$	−31950.0	−1.48663	−0.743313	−	0.668944i	$-0.766747\pi$
$773$	−31950.0	−1.48663	−0.743313	+	0.668944i	$0.766747\pi$
$774$	0	0
$775$	1300.00	0.0602547
$776$	0	0
$777$	3738.00	0.172587
$778$	0	0
$779$	7728.00	0.355436
$780$	0	0
$781$	−11232.0	−0.514613
$782$	0	0
$783$	−5022.00	−0.229210
$784$	0	0
$785$	−9370.00	−0.426025
$786$	0	0
$787$	−6284.00	−0.284626	−0.142313	−	0.989822i	$-0.545454\pi$
$787$	−6284.00	−0.284626	−0.142313	+	0.989822i	$0.545454\pi$
$788$	0	0
$789$	4284.00	0.193301
$790$	0	0
$791$	3486.00	0.156698
$792$	0	0
$793$	220.000	0.00985174
$794$	0	0
$795$	2970.00	0.132497
$796$	0	0
$797$	5946.00	0.264264	0.132132	−	0.991232i	$-0.457818\pi$
$797$	5946.00	0.264264	0.132132	+	0.991232i	$0.457818\pi$
$798$	0	0
$799$	−8208.00	−0.363427
$800$	0	0
$801$	5670.00	0.250112
$802$	0	0
$803$	−6504.00	−0.285830
$804$	0	0
$805$	5460.00	0.239056
$806$	0	0
$807$	12366.0	0.539410
$808$	0	0
$809$	27090.0	1.17730	0.588649	−	0.808389i	$-0.299660\pi$
$809$	27090.0	1.17730	0.588649	+	0.808389i	$0.299660\pi$
$810$	0	0
$811$	20104.0	0.870465	0.435232	−	0.900318i	$-0.356666\pi$
$811$	20104.0	0.870465	0.435232	+	0.900318i	$0.356666\pi$
$812$	0	0
$813$	−17340.0	−0.748020
$814$	0	0
$815$	2260.00	0.0971342
$816$	0	0
$817$	−23072.0	−0.987989
$818$	0	0
$819$	−126.000	−0.00537582
$820$	0	0
$821$	7302.00	0.310404	0.155202	−	0.987883i	$-0.450397\pi$
$821$	7302.00	0.310404	0.155202	+	0.987883i	$0.450397\pi$
$822$	0	0
$823$	24136.0	1.02227	0.511135	−	0.859500i	$-0.329225\pi$
$823$	24136.0	1.02227	0.511135	+	0.859500i	$0.329225\pi$
$824$	0	0
$825$	−900.000	−0.0379806
$826$	0	0
$827$	−22680.0	−0.953641	−0.476820	−	0.879001i	$-0.658211\pi$
$827$	−22680.0	−0.953641	−0.476820	+	0.879001i	$0.658211\pi$
$828$	0	0
$829$	−20338.0	−0.852072	−0.426036	−	0.904706i	$-0.640090\pi$
$829$	−20338.0	−0.852072	−0.426036	+	0.904706i	$0.640090\pi$
$830$	0	0
$831$	13722.0	0.572817
$832$	0	0
$833$	−882.000	−0.0366861
$834$	0	0
$835$	−7080.00	−0.293429
$836$	0	0
$837$	1404.00	0.0579801
$838$	0	0
$839$	−6600.00	−0.271582	−0.135791	−	0.990738i	$-0.543358\pi$
$839$	−6600.00	−0.271582	−0.135791	+	0.990738i	$0.543358\pi$
$840$	0	0
$841$	10207.0	0.418508
$842$	0	0
$843$	10350.0	0.422862
$844$	0	0
$845$	10965.0	0.446399
$846$	0	0
$847$	8309.00	0.337073
$848$	0	0
$849$	2100.00	0.0848902
$850$	0	0
$851$	−27768.0	−1.11854
$852$	0	0
$853$	−40174.0	−1.61258	−0.806290	−	0.591520i	$-0.798528\pi$
$853$	−40174.0	−1.61258	−0.806290	+	0.591520i	$0.798528\pi$
$854$	0	0
$855$	2520.00	0.100798
$856$	0	0
$857$	−20778.0	−0.828195	−0.414097	−	0.910233i	$-0.635903\pi$
$857$	−20778.0	−0.828195	−0.414097	+	0.910233i	$0.635903\pi$
$858$	0	0
$859$	−7400.00	−0.293929	−0.146964	−	0.989142i	$-0.546950\pi$
$859$	−7400.00	−0.293929	−0.146964	+	0.989142i	$0.546950\pi$
$860$	0	0
$861$	2898.00	0.114708
$862$	0	0
$863$	−684.000	−0.0269799	−0.0134899	−	0.999909i	$-0.504294\pi$
$863$	−684.000	−0.0269799	−0.0134899	+	0.999909i	$0.504294\pi$
$864$	0	0
$865$	−2130.00	−0.0837251
$866$	0	0
$867$	−13767.0	−0.539275
$868$	0	0
$869$	11904.0	0.464690
$870$	0	0
$871$	392.000	0.0152496
$872$	0	0
$873$	990.000	0.0383808
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−9754.00	−0.375563	−0.187782	−	0.982211i	$-0.560130\pi$
$877$	−9754.00	−0.375563	−0.187782	+	0.982211i	$0.560130\pi$
$878$	0	0
$879$	21510.0	0.825386
$880$	0	0
$881$	14310.0	0.547237	0.273619	−	0.961838i	$-0.411779\pi$
$881$	14310.0	0.547237	0.273619	+	0.961838i	$0.411779\pi$
$882$	0	0
$883$	14092.0	0.537071	0.268535	−	0.963270i	$-0.413460\pi$
$883$	14092.0	0.537071	0.268535	+	0.963270i	$0.413460\pi$
$884$	0	0
$885$	5220.00	0.198269
$886$	0	0
$887$	−45600.0	−1.72615	−0.863077	−	0.505073i	$-0.831466\pi$
$887$	−45600.0	−1.72615	−0.863077	+	0.505073i	$0.831466\pi$
$888$	0	0
$889$	−13216.0	−0.498594
$890$	0	0
$891$	−972.000	−0.0365468
$892$	0	0
$893$	−25536.0	−0.956920
$894$	0	0
$895$	13500.0	0.504196
$896$	0	0
$897$	936.000	0.0348407
$898$	0	0
$899$	−9672.00	−0.358820
$900$	0	0
$901$	3564.00	0.131780
$902$	0	0
$903$	−8652.00	−0.318849
$904$	0	0
$905$	9890.00	0.363265
$906$	0	0
$907$	−8012.00	−0.293312	−0.146656	−	0.989188i	$-0.546851\pi$
$907$	−8012.00	−0.293312	−0.146656	+	0.989188i	$0.546851\pi$
$908$	0	0
$909$	5130.00	0.187185
$910$	0	0
$911$	−2136.00	−0.0776826	−0.0388413	−	0.999245i	$-0.512367\pi$
$911$	−2136.00	−0.0776826	−0.0388413	+	0.999245i	$0.512367\pi$
$912$	0	0
$913$	−3312.00	−0.120056
$914$	0	0
$915$	−1650.00	−0.0596146
$916$	0	0
$917$	−20244.0	−0.729025
$918$	0	0
$919$	15280.0	0.548466	0.274233	−	0.961663i	$-0.411576\pi$
$919$	15280.0	0.548466	0.274233	+	0.961663i	$0.411576\pi$
$920$	0	0
$921$	−19932.0	−0.713118
$922$	0	0
$923$	1872.00	0.0667580
$924$	0	0
$925$	−4450.00	−0.158178
$926$	0	0
$927$	2736.00	0.0969385
$928$	0	0
$929$	20910.0	0.738466	0.369233	−	0.929337i	$-0.379620\pi$
$929$	20910.0	0.738466	0.369233	+	0.929337i	$0.379620\pi$
$930$	0	0
$931$	−2744.00	−0.0965961
$932$	0	0
$933$	16128.0	0.565924
$934$	0	0
$935$	−1080.00	−0.0377752
$936$	0	0
$937$	−38122.0	−1.32913	−0.664563	−	0.747232i	$-0.731382\pi$
$937$	−38122.0	−1.32913	−0.664563	+	0.747232i	$0.731382\pi$
$938$	0	0
$939$	6378.00	0.221659
$940$	0	0
$941$	42810.0	1.48307	0.741534	−	0.670916i	$-0.234099\pi$
$941$	42810.0	1.48307	0.741534	+	0.670916i	$0.234099\pi$
$942$	0	0
$943$	−21528.0	−0.743423
$944$	0	0
$945$	945.000	0.0325300
$946$	0	0
$947$	−39864.0	−1.36790	−0.683952	−	0.729527i	$-0.739740\pi$
$947$	−39864.0	−1.36790	−0.683952	+	0.729527i	$0.739740\pi$
$948$	0	0
$949$	1084.00	0.0370792
$950$	0	0
$951$	3222.00	0.109864
$952$	0	0
$953$	−23850.0	−0.810679	−0.405340	−	0.914166i	$-0.632847\pi$
$953$	−23850.0	−0.810679	−0.405340	+	0.914166i	$0.632847\pi$
$954$	0	0
$955$	−11640.0	−0.394410
$956$	0	0
$957$	6696.00	0.226177
$958$	0	0
$959$	−5754.00	−0.193750
$960$	0	0
$961$	−27087.0	−0.909234
$962$	0	0
$963$	−1944.00	−0.0650514
$964$	0	0
$965$	15830.0	0.528068
$966$	0	0
$967$	12832.0	0.426731	0.213366	−	0.976972i	$-0.431557\pi$
$967$	12832.0	0.426731	0.213366	+	0.976972i	$0.431557\pi$
$968$	0	0
$969$	3024.00	0.100253
$970$	0	0
$971$	15804.0	0.522322	0.261161	−	0.965295i	$-0.415895\pi$
$971$	15804.0	0.522322	0.261161	+	0.965295i	$0.415895\pi$
$972$	0	0
$973$	−2632.00	−0.0867195
$974$	0	0
$975$	150.000	0.00492702
$976$	0	0
$977$	−33114.0	−1.08435	−0.542175	−	0.840265i	$-0.682399\pi$
$977$	−33114.0	−1.08435	−0.542175	+	0.840265i	$0.682399\pi$
$978$	0	0
$979$	−7560.00	−0.246801
$980$	0	0
$981$	5526.00	0.179849
$982$	0	0
$983$	58632.0	1.90241	0.951206	−	0.308558i	$-0.0998462\pi$
$983$	58632.0	1.90241	0.951206	+	0.308558i	$0.0998462\pi$
$984$	0	0
$985$	2070.00	0.0669601
$986$	0	0
$987$	−9576.00	−0.308822
$988$	0	0
$989$	64272.0	2.06646
$990$	0	0
$991$	−55784.0	−1.78813	−0.894065	−	0.447937i	$-0.852159\pi$
$991$	−55784.0	−1.78813	−0.894065	+	0.447937i	$0.852159\pi$
$992$	0	0
$993$	8364.00	0.267295
$994$	0	0
$995$	−8180.00	−0.260627
$996$	0	0
$997$	−23326.0	−0.740965	−0.370482	−	0.928840i	$-0.620808\pi$
$997$	−23326.0	−0.740965	−0.370482	+	0.928840i	$0.620808\pi$
$998$	0	0
$999$	−4806.00	−0.152207

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.4.a.n.1.1		1
4.3	odd	2		210.4.a.a.1.1	✓	1
12.11	even	2		630.4.a.v.1.1		1
20.3	even	4		1050.4.g.o.799.2		2
20.7	even	4		1050.4.g.o.799.1		2
20.19	odd	2		1050.4.a.t.1.1		1
28.27	even	2		1470.4.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.4.a.a.1.1	✓	1	4.3	odd	2
630.4.a.v.1.1		1	12.11	even	2
1050.4.a.t.1.1		1	20.19	odd	2
1050.4.g.o.799.1		2	20.7	even	4
1050.4.g.o.799.2		2	20.3	even	4
1470.4.a.n.1.1		1	28.27	even	2
1680.4.a.n.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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1680.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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