LMFDB - Embedded newform 2112.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,4,Mod(1,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, 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("2112.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2112.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:	$124.612033932$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +4.00000 q^{5} -26.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} +4.00000 q^{5} -26.0000 q^{7} +9.00000 q^{9} -11.0000 q^{11} +32.0000 q^{13} +12.0000 q^{15} +74.0000 q^{17} +60.0000 q^{19} -78.0000 q^{21} -182.000 q^{23} -109.000 q^{25} +27.0000 q^{27} +90.0000 q^{29} -8.00000 q^{31} -33.0000 q^{33} -104.000 q^{35} +66.0000 q^{37} +96.0000 q^{39} +422.000 q^{41} -408.000 q^{43} +36.0000 q^{45} -506.000 q^{47} +333.000 q^{49} +222.000 q^{51} -348.000 q^{53} -44.0000 q^{55} +180.000 q^{57} +200.000 q^{59} -132.000 q^{61} -234.000 q^{63} +128.000 q^{65} +1036.00 q^{67} -546.000 q^{69} +762.000 q^{71} -542.000 q^{73} -327.000 q^{75} +286.000 q^{77} -550.000 q^{79} +81.0000 q^{81} +132.000 q^{83} +296.000 q^{85} +270.000 q^{87} +570.000 q^{89} -832.000 q^{91} -24.0000 q^{93} +240.000 q^{95} +14.0000 q^{97} -99.0000 q^{99} +O(q^{100})$

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
$5$	4.00000	0.357771	0.178885	−	0.983870i	$-0.442751\pi$
$5$	4.00000	0.357771	0.178885	+	0.983870i	$0.442751\pi$
$6$	0	0
$7$	−26.0000	−1.40387	−0.701934	−	0.712242i	$-0.747680\pi$
$7$	−26.0000	−1.40387	−0.701934	+	0.712242i	$0.747680\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	32.0000	0.682708	0.341354	−	0.939935i	$-0.389115\pi$
$13$	32.0000	0.682708	0.341354	+	0.939935i	$0.389115\pi$
$14$	0	0
$15$	12.0000	0.206559
$16$	0	0
$17$	74.0000	1.05574	0.527872	−	0.849324i	$-0.322990\pi$
$17$	74.0000	1.05574	0.527872	+	0.849324i	$0.322990\pi$
$18$	0	0
$19$	60.0000	0.724471	0.362235	−	0.932087i	$-0.382014\pi$
$19$	60.0000	0.724471	0.362235	+	0.932087i	$0.382014\pi$
$20$	0	0
$21$	−78.0000	−0.810524
$22$	0	0
$23$	−182.000	−1.64998	−0.824992	−	0.565145i	$-0.808820\pi$
$23$	−182.000	−1.64998	−0.824992	+	0.565145i	$0.808820\pi$
$24$	0	0
$25$	−109.000	−0.872000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	90.0000	0.576296	0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000	0.576296	0.288148	+	0.957586i	$0.406961\pi$
$30$	0	0
$31$	−8.00000	−0.0463498	−0.0231749	−	0.999731i	$-0.507377\pi$
$31$	−8.00000	−0.0463498	−0.0231749	+	0.999731i	$0.507377\pi$
$32$	0	0
$33$	−33.0000	−0.174078
$34$	0	0
$35$	−104.000	−0.502263
$36$	0	0
$37$	66.0000	0.293252	0.146626	−	0.989192i	$-0.453159\pi$
$37$	66.0000	0.293252	0.146626	+	0.989192i	$0.453159\pi$
$38$	0	0
$39$	96.0000	0.394162
$40$	0	0
$41$	422.000	1.60745	0.803724	−	0.595003i	$-0.202849\pi$
$41$	422.000	1.60745	0.803724	+	0.595003i	$0.202849\pi$
$42$	0	0
$43$	−408.000	−1.44696	−0.723482	−	0.690344i	$-0.757459\pi$
$43$	−408.000	−1.44696	−0.723482	+	0.690344i	$0.757459\pi$
$44$	0	0
$45$	36.0000	0.119257
$46$	0	0
$47$	−506.000	−1.57038	−0.785188	−	0.619257i	$-0.787434\pi$
$47$	−506.000	−1.57038	−0.785188	+	0.619257i	$0.787434\pi$
$48$	0	0
$49$	333.000	0.970845
$50$	0	0
$51$	222.000	0.609534
$52$	0	0
$53$	−348.000	−0.901915	−0.450957	−	0.892546i	$-0.648917\pi$
$53$	−348.000	−0.901915	−0.450957	+	0.892546i	$0.648917\pi$
$54$	0	0
$55$	−44.0000	−0.107872
$56$	0	0
$57$	180.000	0.418273
$58$	0	0
$59$	200.000	0.441318	0.220659	−	0.975351i	$-0.429179\pi$
$59$	200.000	0.441318	0.220659	+	0.975351i	$0.429179\pi$
$60$	0	0
$61$	−132.000	−0.277063	−0.138532	−	0.990358i	$-0.544238\pi$
$61$	−132.000	−0.277063	−0.138532	+	0.990358i	$0.544238\pi$
$62$	0	0
$63$	−234.000	−0.467956
$64$	0	0
$65$	128.000	0.244253
$66$	0	0
$67$	1036.00	1.88907	0.944534	−	0.328414i	$-0.106514\pi$
$67$	1036.00	1.88907	0.944534	+	0.328414i	$0.106514\pi$
$68$	0	0
$69$	−546.000	−0.952618
$70$	0	0
$71$	762.000	1.27370	0.636850	−	0.770987i	$-0.280237\pi$
$71$	762.000	1.27370	0.636850	+	0.770987i	$0.280237\pi$
$72$	0	0
$73$	−542.000	−0.868990	−0.434495	−	0.900674i	$-0.643073\pi$
$73$	−542.000	−0.868990	−0.434495	+	0.900674i	$0.643073\pi$
$74$	0	0
$75$	−327.000	−0.503449
$76$	0	0
$77$	286.000	0.423282
$78$	0	0
$79$	−550.000	−0.783289	−0.391645	−	0.920117i	$-0.628094\pi$
$79$	−550.000	−0.783289	−0.391645	+	0.920117i	$0.628094\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	132.000	0.174565	0.0872824	−	0.996184i	$-0.472182\pi$
$83$	132.000	0.174565	0.0872824	+	0.996184i	$0.472182\pi$
$84$	0	0
$85$	296.000	0.377714
$86$	0	0
$87$	270.000	0.332725
$88$	0	0
$89$	570.000	0.678875	0.339438	−	0.940629i	$-0.389763\pi$
$89$	570.000	0.678875	0.339438	+	0.940629i	$0.389763\pi$
$90$	0	0
$91$	−832.000	−0.958432
$92$	0	0
$93$	−24.0000	−0.0267600
$94$	0	0
$95$	240.000	0.259195
$96$	0	0
$97$	14.0000	0.0146545	0.00732724	−	0.999973i	$-0.497668\pi$
$97$	14.0000	0.0146545	0.00732724	+	0.999973i	$0.497668\pi$
$98$	0	0
$99$	−99.0000	−0.100504
$100$	0	0
$101$	−1702.00	−1.67679	−0.838393	−	0.545067i	$-0.816504\pi$
$101$	−1702.00	−1.67679	−0.838393	+	0.545067i	$0.816504\pi$
$102$	0	0
$103$	−1132.00	−1.08291	−0.541453	−	0.840731i	$-0.682126\pi$
$103$	−1132.00	−1.08291	−0.541453	+	0.840731i	$0.682126\pi$
$104$	0	0
$105$	−312.000	−0.289982
$106$	0	0
$107$	−564.000	−0.509570	−0.254785	−	0.966998i	$-0.582005\pi$
$107$	−564.000	−0.509570	−0.254785	+	0.966998i	$0.582005\pi$
$108$	0	0
$109$	320.000	0.281197	0.140598	−	0.990067i	$-0.455097\pi$
$109$	320.000	0.281197	0.140598	+	0.990067i	$0.455097\pi$
$110$	0	0
$111$	198.000	0.169309
$112$	0	0
$113$	−2142.00	−1.78321	−0.891604	−	0.452817i	$-0.850419\pi$
$113$	−2142.00	−1.78321	−0.891604	+	0.452817i	$0.850419\pi$
$114$	0	0
$115$	−728.000	−0.590316
$116$	0	0
$117$	288.000	0.227569
$118$	0	0
$119$	−1924.00	−1.48212
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	1266.00	0.928060
$124$	0	0
$125$	−936.000	−0.669747
$126$	0	0
$127$	−1606.00	−1.12212	−0.561061	−	0.827775i	$-0.689607\pi$
$127$	−1606.00	−1.12212	−0.561061	+	0.827775i	$0.689607\pi$
$128$	0	0
$129$	−1224.00	−0.835405
$130$	0	0
$131$	1908.00	1.27254	0.636270	−	0.771466i	$-0.280476\pi$
$131$	1908.00	1.27254	0.636270	+	0.771466i	$0.280476\pi$
$132$	0	0
$133$	−1560.00	−1.01706
$134$	0	0
$135$	108.000	0.0688530
$136$	0	0
$137$	−2186.00	−1.36323	−0.681615	−	0.731711i	$-0.738722\pi$
$137$	−2186.00	−1.36323	−0.681615	+	0.731711i	$0.738722\pi$
$138$	0	0
$139$	−2740.00	−1.67197	−0.835985	−	0.548753i	$-0.815103\pi$
$139$	−2740.00	−1.67197	−0.835985	+	0.548753i	$0.815103\pi$
$140$	0	0
$141$	−1518.00	−0.906657
$142$	0	0
$143$	−352.000	−0.205844
$144$	0	0
$145$	360.000	0.206182
$146$	0	0
$147$	999.000	0.560518
$148$	0	0
$149$	1310.00	0.720264	0.360132	−	0.932901i	$-0.382732\pi$
$149$	1310.00	0.720264	0.360132	+	0.932901i	$0.382732\pi$
$150$	0	0
$151$	−1198.00	−0.645641	−0.322821	−	0.946460i	$-0.604631\pi$
$151$	−1198.00	−0.645641	−0.322821	+	0.946460i	$0.604631\pi$
$152$	0	0
$153$	666.000	0.351914
$154$	0	0
$155$	−32.0000	−0.0165826
$156$	0	0
$157$	−2114.00	−1.07462	−0.537311	−	0.843384i	$-0.680560\pi$
$157$	−2114.00	−1.07462	−0.537311	+	0.843384i	$0.680560\pi$
$158$	0	0
$159$	−1044.00	−0.520721
$160$	0	0
$161$	4732.00	2.31636
$162$	0	0
$163$	−3868.00	−1.85868	−0.929341	−	0.369223i	$-0.879624\pi$
$163$	−3868.00	−1.85868	−0.929341	+	0.369223i	$0.879624\pi$
$164$	0	0
$165$	−132.000	−0.0622799
$166$	0	0
$167$	2004.00	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	2004.00	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	−1173.00	−0.533910
$170$	0	0
$171$	540.000	0.241490
$172$	0	0
$173$	−678.000	−0.297962	−0.148981	−	0.988840i	$-0.547599\pi$
$173$	−678.000	−0.297962	−0.148981	+	0.988840i	$0.547599\pi$
$174$	0	0
$175$	2834.00	1.22417
$176$	0	0
$177$	600.000	0.254795
$178$	0	0
$179$	1680.00	0.701503	0.350752	−	0.936469i	$-0.385926\pi$
$179$	1680.00	0.701503	0.350752	+	0.936469i	$0.385926\pi$
$180$	0	0
$181$	4358.00	1.78966	0.894828	−	0.446412i	$-0.147298\pi$
$181$	4358.00	1.78966	0.894828	+	0.446412i	$0.147298\pi$
$182$	0	0
$183$	−396.000	−0.159963
$184$	0	0
$185$	264.000	0.104917
$186$	0	0
$187$	−814.000	−0.318319
$188$	0	0
$189$	−702.000	−0.270175
$190$	0	0
$191$	−1778.00	−0.673568	−0.336784	−	0.941582i	$-0.609339\pi$
$191$	−1778.00	−0.673568	−0.336784	+	0.941582i	$0.609339\pi$
$192$	0	0
$193$	−3962.00	−1.47767	−0.738837	−	0.673884i	$-0.764625\pi$
$193$	−3962.00	−1.47767	−0.738837	+	0.673884i	$0.764625\pi$
$194$	0	0
$195$	384.000	0.141020
$196$	0	0
$197$	−374.000	−0.135261	−0.0676304	−	0.997710i	$-0.521544\pi$
$197$	−374.000	−0.135261	−0.0676304	+	0.997710i	$0.521544\pi$
$198$	0	0
$199$	2100.00	0.748066	0.374033	−	0.927415i	$-0.377975\pi$
$199$	2100.00	0.748066	0.374033	+	0.927415i	$0.377975\pi$
$200$	0	0
$201$	3108.00	1.09065
$202$	0	0
$203$	−2340.00	−0.809043
$204$	0	0
$205$	1688.00	0.575098
$206$	0	0
$207$	−1638.00	−0.549995
$208$	0	0
$209$	−660.000	−0.218436
$210$	0	0
$211$	−2232.00	−0.728233	−0.364117	−	0.931353i	$-0.618629\pi$
$211$	−2232.00	−0.728233	−0.364117	+	0.931353i	$0.618629\pi$
$212$	0	0
$213$	2286.00	0.735372
$214$	0	0
$215$	−1632.00	−0.517681
$216$	0	0
$217$	208.000	0.0650689
$218$	0	0
$219$	−1626.00	−0.501712
$220$	0	0
$221$	2368.00	0.720764
$222$	0	0
$223$	2128.00	0.639020	0.319510	−	0.947583i	$-0.396482\pi$
$223$	2128.00	0.639020	0.319510	+	0.947583i	$0.396482\pi$
$224$	0	0
$225$	−981.000	−0.290667
$226$	0	0
$227$	−2964.00	−0.866641	−0.433321	−	0.901240i	$-0.642658\pi$
$227$	−2964.00	−0.866641	−0.433321	+	0.901240i	$0.642658\pi$
$228$	0	0
$229$	2550.00	0.735846	0.367923	−	0.929856i	$-0.380069\pi$
$229$	2550.00	0.735846	0.367923	+	0.929856i	$0.380069\pi$
$230$	0	0
$231$	858.000	0.244382
$232$	0	0
$233$	−3042.00	−0.855314	−0.427657	−	0.903941i	$-0.640661\pi$
$233$	−3042.00	−0.855314	−0.427657	+	0.903941i	$0.640661\pi$
$234$	0	0
$235$	−2024.00	−0.561835
$236$	0	0
$237$	−1650.00	−0.452232
$238$	0	0
$239$	2700.00	0.730747	0.365373	−	0.930861i	$-0.380941\pi$
$239$	2700.00	0.730747	0.365373	+	0.930861i	$0.380941\pi$
$240$	0	0
$241$	−578.000	−0.154491	−0.0772453	−	0.997012i	$-0.524612\pi$
$241$	−578.000	−0.154491	−0.0772453	+	0.997012i	$0.524612\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	1332.00	0.347340
$246$	0	0
$247$	1920.00	0.494602
$248$	0	0
$249$	396.000	0.100785
$250$	0	0
$251$	−3752.00	−0.943522	−0.471761	−	0.881726i	$-0.656382\pi$
$251$	−3752.00	−0.943522	−0.471761	+	0.881726i	$0.656382\pi$
$252$	0	0
$253$	2002.00	0.497489
$254$	0	0
$255$	888.000	0.218073
$256$	0	0
$257$	674.000	0.163591	0.0817957	−	0.996649i	$-0.473935\pi$
$257$	674.000	0.163591	0.0817957	+	0.996649i	$0.473935\pi$
$258$	0	0
$259$	−1716.00	−0.411687
$260$	0	0
$261$	810.000	0.192099
$262$	0	0
$263$	−4352.00	−1.02036	−0.510182	−	0.860066i	$-0.670422\pi$
$263$	−4352.00	−1.02036	−0.510182	+	0.860066i	$0.670422\pi$
$264$	0	0
$265$	−1392.00	−0.322679
$266$	0	0
$267$	1710.00	0.391949
$268$	0	0
$269$	−500.000	−0.113329	−0.0566646	−	0.998393i	$-0.518047\pi$
$269$	−500.000	−0.113329	−0.0566646	+	0.998393i	$0.518047\pi$
$270$	0	0
$271$	−6538.00	−1.46552	−0.732759	−	0.680489i	$-0.761768\pi$
$271$	−6538.00	−1.46552	−0.732759	+	0.680489i	$0.761768\pi$
$272$	0	0
$273$	−2496.00	−0.553351
$274$	0	0
$275$	1199.00	0.262918
$276$	0	0
$277$	−124.000	−0.0268969	−0.0134484	−	0.999910i	$-0.504281\pi$
$277$	−124.000	−0.0268969	−0.0134484	+	0.999910i	$0.504281\pi$
$278$	0	0
$279$	−72.0000	−0.0154499
$280$	0	0
$281$	3642.00	0.773180	0.386590	−	0.922252i	$-0.373653\pi$
$281$	3642.00	0.773180	0.386590	+	0.922252i	$0.373653\pi$
$282$	0	0
$283$	−4648.00	−0.976307	−0.488154	−	0.872758i	$-0.662329\pi$
$283$	−4648.00	−0.976307	−0.488154	+	0.872758i	$0.662329\pi$
$284$	0	0
$285$	720.000	0.149646
$286$	0	0
$287$	−10972.0	−2.25664
$288$	0	0
$289$	563.000	0.114594
$290$	0	0
$291$	42.0000	0.00846077
$292$	0	0
$293$	3102.00	0.618501	0.309250	−	0.950981i	$-0.399922\pi$
$293$	3102.00	0.618501	0.309250	+	0.950981i	$0.399922\pi$
$294$	0	0
$295$	800.000	0.157891
$296$	0	0
$297$	−297.000	−0.0580259
$298$	0	0
$299$	−5824.00	−1.12646
$300$	0	0
$301$	10608.0	2.03135
$302$	0	0
$303$	−5106.00	−0.968093
$304$	0	0
$305$	−528.000	−0.0991252
$306$	0	0
$307$	−1244.00	−0.231267	−0.115633	−	0.993292i	$-0.536890\pi$
$307$	−1244.00	−0.231267	−0.115633	+	0.993292i	$0.536890\pi$
$308$	0	0
$309$	−3396.00	−0.625216
$310$	0	0
$311$	2082.00	0.379612	0.189806	−	0.981822i	$-0.439214\pi$
$311$	2082.00	0.379612	0.189806	+	0.981822i	$0.439214\pi$
$312$	0	0
$313$	2378.00	0.429433	0.214716	−	0.976676i	$-0.431117\pi$
$313$	2378.00	0.429433	0.214716	+	0.976676i	$0.431117\pi$
$314$	0	0
$315$	−936.000	−0.167421
$316$	0	0
$317$	496.000	0.0878806	0.0439403	−	0.999034i	$-0.486009\pi$
$317$	496.000	0.0878806	0.0439403	+	0.999034i	$0.486009\pi$
$318$	0	0
$319$	−990.000	−0.173760
$320$	0	0
$321$	−1692.00	−0.294200
$322$	0	0
$323$	4440.00	0.764855
$324$	0	0
$325$	−3488.00	−0.595321
$326$	0	0
$327$	960.000	0.162349
$328$	0	0
$329$	13156.0	2.20460
$330$	0	0
$331$	2708.00	0.449683	0.224842	−	0.974395i	$-0.427814\pi$
$331$	2708.00	0.449683	0.224842	+	0.974395i	$0.427814\pi$
$332$	0	0
$333$	594.000	0.0977507
$334$	0	0
$335$	4144.00	0.675853
$336$	0	0
$337$	4034.00	0.652065	0.326033	−	0.945359i	$-0.394288\pi$
$337$	4034.00	0.652065	0.326033	+	0.945359i	$0.394288\pi$
$338$	0	0
$339$	−6426.00	−1.02954
$340$	0	0
$341$	88.0000	0.0139750
$342$	0	0
$343$	260.000	0.0409291
$344$	0	0
$345$	−2184.00	−0.340819
$346$	0	0
$347$	−11084.0	−1.71476	−0.857378	−	0.514687i	$-0.827908\pi$
$347$	−11084.0	−1.71476	−0.857378	+	0.514687i	$0.827908\pi$
$348$	0	0
$349$	3120.00	0.478538	0.239269	−	0.970953i	$-0.423092\pi$
$349$	3120.00	0.478538	0.239269	+	0.970953i	$0.423092\pi$
$350$	0	0
$351$	864.000	0.131387
$352$	0	0
$353$	−5622.00	−0.847674	−0.423837	−	0.905739i	$-0.639317\pi$
$353$	−5622.00	−0.847674	−0.423837	+	0.905739i	$0.639317\pi$
$354$	0	0
$355$	3048.00	0.455693
$356$	0	0
$357$	−5772.00	−0.855705
$358$	0	0
$359$	−8500.00	−1.24962	−0.624809	−	0.780778i	$-0.714823\pi$
$359$	−8500.00	−1.24962	−0.624809	+	0.780778i	$0.714823\pi$
$360$	0	0
$361$	−3259.00	−0.475142
$362$	0	0
$363$	363.000	0.0524864
$364$	0	0
$365$	−2168.00	−0.310899
$366$	0	0
$367$	7144.00	1.01611	0.508057	−	0.861324i	$-0.330364\pi$
$367$	7144.00	1.01611	0.508057	+	0.861324i	$0.330364\pi$
$368$	0	0
$369$	3798.00	0.535816
$370$	0	0
$371$	9048.00	1.26617
$372$	0	0
$373$	632.000	0.0877312	0.0438656	−	0.999037i	$-0.486033\pi$
$373$	632.000	0.0877312	0.0438656	+	0.999037i	$0.486033\pi$
$374$	0	0
$375$	−2808.00	−0.386679
$376$	0	0
$377$	2880.00	0.393442
$378$	0	0
$379$	4220.00	0.571944	0.285972	−	0.958238i	$-0.407684\pi$
$379$	4220.00	0.571944	0.285972	+	0.958238i	$0.407684\pi$
$380$	0	0
$381$	−4818.00	−0.647857
$382$	0	0
$383$	8458.00	1.12842	0.564208	−	0.825632i	$-0.309181\pi$
$383$	8458.00	1.12842	0.564208	+	0.825632i	$0.309181\pi$
$384$	0	0
$385$	1144.00	0.151438
$386$	0	0
$387$	−3672.00	−0.482321
$388$	0	0
$389$	−1740.00	−0.226790	−0.113395	−	0.993550i	$-0.536173\pi$
$389$	−1740.00	−0.226790	−0.113395	+	0.993550i	$0.536173\pi$
$390$	0	0
$391$	−13468.0	−1.74196
$392$	0	0
$393$	5724.00	0.734701
$394$	0	0
$395$	−2200.00	−0.280238
$396$	0	0
$397$	5126.00	0.648027	0.324013	−	0.946053i	$-0.394968\pi$
$397$	5126.00	0.648027	0.324013	+	0.946053i	$0.394968\pi$
$398$	0	0
$399$	−4680.00	−0.587201
$400$	0	0
$401$	−3098.00	−0.385802	−0.192901	−	0.981218i	$-0.561790\pi$
$401$	−3098.00	−0.385802	−0.192901	+	0.981218i	$0.561790\pi$
$402$	0	0
$403$	−256.000	−0.0316433
$404$	0	0
$405$	324.000	0.0397523
$406$	0	0
$407$	−726.000	−0.0884189
$408$	0	0
$409$	6390.00	0.772531	0.386265	−	0.922388i	$-0.373765\pi$
$409$	6390.00	0.772531	0.386265	+	0.922388i	$0.373765\pi$
$410$	0	0
$411$	−6558.00	−0.787062
$412$	0	0
$413$	−5200.00	−0.619553
$414$	0	0
$415$	528.000	0.0624542
$416$	0	0
$417$	−8220.00	−0.965312
$418$	0	0
$419$	−9760.00	−1.13796	−0.568982	−	0.822350i	$-0.692663\pi$
$419$	−9760.00	−1.13796	−0.568982	+	0.822350i	$0.692663\pi$
$420$	0	0
$421$	5138.00	0.594800	0.297400	−	0.954753i	$-0.403881\pi$
$421$	5138.00	0.594800	0.297400	+	0.954753i	$0.403881\pi$
$422$	0	0
$423$	−4554.00	−0.523459
$424$	0	0
$425$	−8066.00	−0.920608
$426$	0	0
$427$	3432.00	0.388960
$428$	0	0
$429$	−1056.00	−0.118844
$430$	0	0
$431$	−7008.00	−0.783210	−0.391605	−	0.920133i	$-0.628080\pi$
$431$	−7008.00	−0.783210	−0.391605	+	0.920133i	$0.628080\pi$
$432$	0	0
$433$	5578.00	0.619080	0.309540	−	0.950886i	$-0.399825\pi$
$433$	5578.00	0.619080	0.309540	+	0.950886i	$0.399825\pi$
$434$	0	0
$435$	1080.00	0.119039
$436$	0	0
$437$	−10920.0	−1.19536
$438$	0	0
$439$	−10430.0	−1.13393	−0.566967	−	0.823741i	$-0.691883\pi$
$439$	−10430.0	−1.13393	−0.566967	+	0.823741i	$0.691883\pi$
$440$	0	0
$441$	2997.00	0.323615
$442$	0	0
$443$	4432.00	0.475329	0.237664	−	0.971347i	$-0.423618\pi$
$443$	4432.00	0.475329	0.237664	+	0.971347i	$0.423618\pi$
$444$	0	0
$445$	2280.00	0.242882
$446$	0	0
$447$	3930.00	0.415845
$448$	0	0
$449$	−6290.00	−0.661121	−0.330561	−	0.943785i	$-0.607238\pi$
$449$	−6290.00	−0.661121	−0.330561	+	0.943785i	$0.607238\pi$
$450$	0	0
$451$	−4642.00	−0.484664
$452$	0	0
$453$	−3594.00	−0.372761
$454$	0	0
$455$	−3328.00	−0.342899
$456$	0	0
$457$	3054.00	0.312604	0.156302	−	0.987709i	$-0.450043\pi$
$457$	3054.00	0.312604	0.156302	+	0.987709i	$0.450043\pi$
$458$	0	0
$459$	1998.00	0.203178
$460$	0	0
$461$	−12882.0	−1.30146	−0.650732	−	0.759308i	$-0.725538\pi$
$461$	−12882.0	−1.30146	−0.650732	+	0.759308i	$0.725538\pi$
$462$	0	0
$463$	6148.00	0.617110	0.308555	−	0.951207i	$-0.400155\pi$
$463$	6148.00	0.617110	0.308555	+	0.951207i	$0.400155\pi$
$464$	0	0
$465$	−96.0000	−0.00957396
$466$	0	0
$467$	−5124.00	−0.507731	−0.253866	−	0.967240i	$-0.581702\pi$
$467$	−5124.00	−0.507731	−0.253866	+	0.967240i	$0.581702\pi$
$468$	0	0
$469$	−26936.0	−2.65200
$470$	0	0
$471$	−6342.00	−0.620433
$472$	0	0
$473$	4488.00	0.436276
$474$	0	0
$475$	−6540.00	−0.631738
$476$	0	0
$477$	−3132.00	−0.300638
$478$	0	0
$479$	−16520.0	−1.57582	−0.787910	−	0.615790i	$-0.788837\pi$
$479$	−16520.0	−1.57582	−0.787910	+	0.615790i	$0.788837\pi$
$480$	0	0
$481$	2112.00	0.200206
$482$	0	0
$483$	14196.0	1.33735
$484$	0	0
$485$	56.0000	0.00524295
$486$	0	0
$487$	524.000	0.0487571	0.0243785	−	0.999703i	$-0.492239\pi$
$487$	524.000	0.0487571	0.0243785	+	0.999703i	$0.492239\pi$
$488$	0	0
$489$	−11604.0	−1.07311
$490$	0	0
$491$	15028.0	1.38127	0.690636	−	0.723203i	$-0.257331\pi$
$491$	15028.0	1.38127	0.690636	+	0.723203i	$0.257331\pi$
$492$	0	0
$493$	6660.00	0.608421
$494$	0	0
$495$	−396.000	−0.0359573
$496$	0	0
$497$	−19812.0	−1.78811
$498$	0	0
$499$	−9020.00	−0.809200	−0.404600	−	0.914494i	$-0.632589\pi$
$499$	−9020.00	−0.809200	−0.404600	+	0.914494i	$0.632589\pi$
$500$	0	0
$501$	6012.00	0.536120
$502$	0	0
$503$	−14812.0	−1.31299	−0.656495	−	0.754330i	$-0.727962\pi$
$503$	−14812.0	−1.31299	−0.656495	+	0.754330i	$0.727962\pi$
$504$	0	0
$505$	−6808.00	−0.599905
$506$	0	0
$507$	−3519.00	−0.308253
$508$	0	0
$509$	−12660.0	−1.10245	−0.551223	−	0.834358i	$-0.685839\pi$
$509$	−12660.0	−1.10245	−0.551223	+	0.834358i	$0.685839\pi$
$510$	0	0
$511$	14092.0	1.21995
$512$	0	0
$513$	1620.00	0.139424
$514$	0	0
$515$	−4528.00	−0.387432
$516$	0	0
$517$	5566.00	0.473486
$518$	0	0
$519$	−2034.00	−0.172028
$520$	0	0
$521$	−3738.00	−0.314328	−0.157164	−	0.987573i	$-0.550235\pi$
$521$	−3738.00	−0.314328	−0.157164	+	0.987573i	$0.550235\pi$
$522$	0	0
$523$	6352.00	0.531078	0.265539	−	0.964100i	$-0.414450\pi$
$523$	6352.00	0.531078	0.265539	+	0.964100i	$0.414450\pi$
$524$	0	0
$525$	8502.00	0.706777
$526$	0	0
$527$	−592.000	−0.0489334
$528$	0	0
$529$	20957.0	1.72245
$530$	0	0
$531$	1800.00	0.147106
$532$	0	0
$533$	13504.0	1.09742
$534$	0	0
$535$	−2256.00	−0.182309
$536$	0	0
$537$	5040.00	0.405013
$538$	0	0
$539$	−3663.00	−0.292721
$540$	0	0
$541$	24728.0	1.96514	0.982569	−	0.185898i	$-0.0595193\pi$
$541$	24728.0	1.96514	0.982569	+	0.185898i	$0.0595193\pi$
$542$	0	0
$543$	13074.0	1.03326
$544$	0	0
$545$	1280.00	0.100604
$546$	0	0
$547$	22756.0	1.77875	0.889375	−	0.457178i	$-0.151140\pi$
$547$	22756.0	1.77875	0.889375	+	0.457178i	$0.151140\pi$
$548$	0	0
$549$	−1188.00	−0.0923545
$550$	0	0
$551$	5400.00	0.417509
$552$	0	0
$553$	14300.0	1.09963
$554$	0	0
$555$	792.000	0.0605739
$556$	0	0
$557$	9526.00	0.724649	0.362325	−	0.932052i	$-0.381983\pi$
$557$	9526.00	0.724649	0.362325	+	0.932052i	$0.381983\pi$
$558$	0	0
$559$	−13056.0	−0.987853
$560$	0	0
$561$	−2442.00	−0.183781
$562$	0	0
$563$	−12068.0	−0.903385	−0.451692	−	0.892174i	$-0.649180\pi$
$563$	−12068.0	−0.903385	−0.451692	+	0.892174i	$0.649180\pi$
$564$	0	0
$565$	−8568.00	−0.637980
$566$	0	0
$567$	−2106.00	−0.155985
$568$	0	0
$569$	15090.0	1.11179	0.555893	−	0.831254i	$-0.312377\pi$
$569$	15090.0	1.11179	0.555893	+	0.831254i	$0.312377\pi$
$570$	0	0
$571$	−4412.00	−0.323356	−0.161678	−	0.986844i	$-0.551691\pi$
$571$	−4412.00	−0.323356	−0.161678	+	0.986844i	$0.551691\pi$
$572$	0	0
$573$	−5334.00	−0.388885
$574$	0	0
$575$	19838.0	1.43879
$576$	0	0
$577$	−3906.00	−0.281818	−0.140909	−	0.990023i	$-0.545002\pi$
$577$	−3906.00	−0.281818	−0.140909	+	0.990023i	$0.545002\pi$
$578$	0	0
$579$	−11886.0	−0.853135
$580$	0	0
$581$	−3432.00	−0.245066
$582$	0	0
$583$	3828.00	0.271937
$584$	0	0
$585$	1152.00	0.0814177
$586$	0	0
$587$	12016.0	0.844895	0.422448	−	0.906387i	$-0.361171\pi$
$587$	12016.0	0.844895	0.422448	+	0.906387i	$0.361171\pi$
$588$	0	0
$589$	−480.000	−0.0335790
$590$	0	0
$591$	−1122.00	−0.0780929
$592$	0	0
$593$	−11342.0	−0.785430	−0.392715	−	0.919660i	$-0.628464\pi$
$593$	−11342.0	−0.785430	−0.392715	+	0.919660i	$0.628464\pi$
$594$	0	0
$595$	−7696.00	−0.530261
$596$	0	0
$597$	6300.00	0.431896
$598$	0	0
$599$	20690.0	1.41130	0.705651	−	0.708559i	$-0.250654\pi$
$599$	20690.0	1.41130	0.705651	+	0.708559i	$0.250654\pi$
$600$	0	0
$601$	−598.000	−0.0405872	−0.0202936	−	0.999794i	$-0.506460\pi$
$601$	−598.000	−0.0405872	−0.0202936	+	0.999794i	$0.506460\pi$
$602$	0	0
$603$	9324.00	0.629689
$604$	0	0
$605$	484.000	0.0325246
$606$	0	0
$607$	−166.000	−0.0111001	−0.00555003	−	0.999985i	$-0.501767\pi$
$607$	−166.000	−0.0111001	−0.00555003	+	0.999985i	$0.501767\pi$
$608$	0	0
$609$	−7020.00	−0.467101
$610$	0	0
$611$	−16192.0	−1.07211
$612$	0	0
$613$	−20108.0	−1.32488	−0.662442	−	0.749113i	$-0.730480\pi$
$613$	−20108.0	−1.32488	−0.662442	+	0.749113i	$0.730480\pi$
$614$	0	0
$615$	5064.00	0.332033
$616$	0	0
$617$	−2286.00	−0.149159	−0.0745793	−	0.997215i	$-0.523761\pi$
$617$	−2286.00	−0.149159	−0.0745793	+	0.997215i	$0.523761\pi$
$618$	0	0
$619$	25660.0	1.66618	0.833088	−	0.553141i	$-0.186571\pi$
$619$	25660.0	1.66618	0.833088	+	0.553141i	$0.186571\pi$
$620$	0	0
$621$	−4914.00	−0.317539
$622$	0	0
$623$	−14820.0	−0.953051
$624$	0	0
$625$	9881.00	0.632384
$626$	0	0
$627$	−1980.00	−0.126114
$628$	0	0
$629$	4884.00	0.309599
$630$	0	0
$631$	−11408.0	−0.719723	−0.359862	−	0.933006i	$-0.617176\pi$
$631$	−11408.0	−0.719723	−0.359862	+	0.933006i	$0.617176\pi$
$632$	0	0
$633$	−6696.00	−0.420446
$634$	0	0
$635$	−6424.00	−0.401462
$636$	0	0
$637$	10656.0	0.662804
$638$	0	0
$639$	6858.00	0.424567
$640$	0	0
$641$	−3378.00	−0.208148	−0.104074	−	0.994570i	$-0.533188\pi$
$641$	−3378.00	−0.208148	−0.104074	+	0.994570i	$0.533188\pi$
$642$	0	0
$643$	11212.0	0.687649	0.343824	−	0.939034i	$-0.388278\pi$
$643$	11212.0	0.687649	0.343824	+	0.939034i	$0.388278\pi$
$644$	0	0
$645$	−4896.00	−0.298883
$646$	0	0
$647$	−86.0000	−0.00522567	−0.00261284	−	0.999997i	$-0.500832\pi$
$647$	−86.0000	−0.00522567	−0.00261284	+	0.999997i	$0.500832\pi$
$648$	0	0
$649$	−2200.00	−0.133062
$650$	0	0
$651$	624.000	0.0375676
$652$	0	0
$653$	4432.00	0.265601	0.132801	−	0.991143i	$-0.457603\pi$
$653$	4432.00	0.265601	0.132801	+	0.991143i	$0.457603\pi$
$654$	0	0
$655$	7632.00	0.455278
$656$	0	0
$657$	−4878.00	−0.289663
$658$	0	0
$659$	−4580.00	−0.270731	−0.135365	−	0.990796i	$-0.543221\pi$
$659$	−4580.00	−0.270731	−0.135365	+	0.990796i	$0.543221\pi$
$660$	0	0
$661$	−4282.00	−0.251967	−0.125984	−	0.992032i	$-0.540209\pi$
$661$	−4282.00	−0.251967	−0.125984	+	0.992032i	$0.540209\pi$
$662$	0	0
$663$	7104.00	0.416133
$664$	0	0
$665$	−6240.00	−0.363875
$666$	0	0
$667$	−16380.0	−0.950879
$668$	0	0
$669$	6384.00	0.368938
$670$	0	0
$671$	1452.00	0.0835378
$672$	0	0
$673$	8438.00	0.483300	0.241650	−	0.970363i	$-0.422311\pi$
$673$	8438.00	0.483300	0.241650	+	0.970363i	$0.422311\pi$
$674$	0	0
$675$	−2943.00	−0.167816
$676$	0	0
$677$	−34494.0	−1.95822	−0.979108	−	0.203341i	$-0.934820\pi$
$677$	−34494.0	−1.95822	−0.979108	+	0.203341i	$0.934820\pi$
$678$	0	0
$679$	−364.000	−0.0205730
$680$	0	0
$681$	−8892.00	−0.500356
$682$	0	0
$683$	13712.0	0.768192	0.384096	−	0.923293i	$-0.374513\pi$
$683$	13712.0	0.768192	0.384096	+	0.923293i	$0.374513\pi$
$684$	0	0
$685$	−8744.00	−0.487724
$686$	0	0
$687$	7650.00	0.424841
$688$	0	0
$689$	−11136.0	−0.615744
$690$	0	0
$691$	−11372.0	−0.626066	−0.313033	−	0.949742i	$-0.601345\pi$
$691$	−11372.0	−0.626066	−0.313033	+	0.949742i	$0.601345\pi$
$692$	0	0
$693$	2574.00	0.141094
$694$	0	0
$695$	−10960.0	−0.598182
$696$	0	0
$697$	31228.0	1.69705
$698$	0	0
$699$	−9126.00	−0.493815
$700$	0	0
$701$	6398.00	0.344721	0.172360	−	0.985034i	$-0.444861\pi$
$701$	6398.00	0.344721	0.172360	+	0.985034i	$0.444861\pi$
$702$	0	0
$703$	3960.00	0.212453
$704$	0	0
$705$	−6072.00	−0.324376
$706$	0	0
$707$	44252.0	2.35399
$708$	0	0
$709$	5830.00	0.308816	0.154408	−	0.988007i	$-0.450653\pi$
$709$	5830.00	0.308816	0.154408	+	0.988007i	$0.450653\pi$
$710$	0	0
$711$	−4950.00	−0.261096
$712$	0	0
$713$	1456.00	0.0764763
$714$	0	0
$715$	−1408.00	−0.0736451
$716$	0	0
$717$	8100.00	0.421897
$718$	0	0
$719$	34530.0	1.79103	0.895516	−	0.445030i	$-0.146807\pi$
$719$	34530.0	1.79103	0.895516	+	0.445030i	$0.146807\pi$
$720$	0	0
$721$	29432.0	1.52026
$722$	0	0
$723$	−1734.00	−0.0891952
$724$	0	0
$725$	−9810.00	−0.502530
$726$	0	0
$727$	−17316.0	−0.883377	−0.441688	−	0.897169i	$-0.645620\pi$
$727$	−17316.0	−0.883377	−0.441688	+	0.897169i	$0.645620\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−30192.0	−1.52762
$732$	0	0
$733$	27072.0	1.36416	0.682079	−	0.731279i	$-0.261076\pi$
$733$	27072.0	1.36416	0.682079	+	0.731279i	$0.261076\pi$
$734$	0	0
$735$	3996.00	0.200537
$736$	0	0
$737$	−11396.0	−0.569575
$738$	0	0
$739$	17320.0	0.862147	0.431073	−	0.902317i	$-0.358135\pi$
$739$	17320.0	0.862147	0.431073	+	0.902317i	$0.358135\pi$
$740$	0	0
$741$	5760.00	0.285559
$742$	0	0
$743$	14588.0	0.720299	0.360149	−	0.932895i	$-0.382726\pi$
$743$	14588.0	0.720299	0.360149	+	0.932895i	$0.382726\pi$
$744$	0	0
$745$	5240.00	0.257690
$746$	0	0
$747$	1188.00	0.0581883
$748$	0	0
$749$	14664.0	0.715368
$750$	0	0
$751$	26152.0	1.27071	0.635353	−	0.772222i	$-0.280855\pi$
$751$	26152.0	1.27071	0.635353	+	0.772222i	$0.280855\pi$
$752$	0	0
$753$	−11256.0	−0.544743
$754$	0	0
$755$	−4792.00	−0.230992
$756$	0	0
$757$	1066.00	0.0511815	0.0255908	−	0.999673i	$-0.491853\pi$
$757$	1066.00	0.0511815	0.0255908	+	0.999673i	$0.491853\pi$
$758$	0	0
$759$	6006.00	0.287225
$760$	0	0
$761$	−37518.0	−1.78716	−0.893578	−	0.448907i	$-0.851813\pi$
$761$	−37518.0	−1.78716	−0.893578	+	0.448907i	$0.851813\pi$
$762$	0	0
$763$	−8320.00	−0.394763
$764$	0	0
$765$	2664.00	0.125905
$766$	0	0
$767$	6400.00	0.301292
$768$	0	0
$769$	−17290.0	−0.810785	−0.405392	−	0.914143i	$-0.632865\pi$
$769$	−17290.0	−0.810785	−0.405392	+	0.914143i	$0.632865\pi$
$770$	0	0
$771$	2022.00	0.0944495
$772$	0	0
$773$	17172.0	0.799009	0.399504	−	0.916731i	$-0.369182\pi$
$773$	17172.0	0.799009	0.399504	+	0.916731i	$0.369182\pi$
$774$	0	0
$775$	872.000	0.0404170
$776$	0	0
$777$	−5148.00	−0.237688
$778$	0	0
$779$	25320.0	1.16455
$780$	0	0
$781$	−8382.00	−0.384035
$782$	0	0
$783$	2430.00	0.110908
$784$	0	0
$785$	−8456.00	−0.384468
$786$	0	0
$787$	9536.00	0.431921	0.215960	−	0.976402i	$-0.430712\pi$
$787$	9536.00	0.431921	0.215960	+	0.976402i	$0.430712\pi$
$788$	0	0
$789$	−13056.0	−0.589108
$790$	0	0
$791$	55692.0	2.50339
$792$	0	0
$793$	−4224.00	−0.189153
$794$	0	0
$795$	−4176.00	−0.186299
$796$	0	0
$797$	20516.0	0.911812	0.455906	−	0.890028i	$-0.349315\pi$
$797$	20516.0	0.911812	0.455906	+	0.890028i	$0.349315\pi$
$798$	0	0
$799$	−37444.0	−1.65791
$800$	0	0
$801$	5130.00	0.226292
$802$	0	0
$803$	5962.00	0.262010
$804$	0	0
$805$	18928.0	0.828726
$806$	0	0
$807$	−1500.00	−0.0654306
$808$	0	0
$809$	22470.0	0.976518	0.488259	−	0.872699i	$-0.337632\pi$
$809$	22470.0	0.976518	0.488259	+	0.872699i	$0.337632\pi$
$810$	0	0
$811$	3368.00	0.145828	0.0729140	−	0.997338i	$-0.476770\pi$
$811$	3368.00	0.145828	0.0729140	+	0.997338i	$0.476770\pi$
$812$	0	0
$813$	−19614.0	−0.846117
$814$	0	0
$815$	−15472.0	−0.664982
$816$	0	0
$817$	−24480.0	−1.04828
$818$	0	0
$819$	−7488.00	−0.319477
$820$	0	0
$821$	10738.0	0.456466	0.228233	−	0.973607i	$-0.426705\pi$
$821$	10738.0	0.456466	0.228233	+	0.973607i	$0.426705\pi$
$822$	0	0
$823$	−15912.0	−0.673946	−0.336973	−	0.941514i	$-0.609403\pi$
$823$	−15912.0	−0.673946	−0.336973	+	0.941514i	$0.609403\pi$
$824$	0	0
$825$	3597.00	0.151796
$826$	0	0
$827$	−22924.0	−0.963900	−0.481950	−	0.876199i	$-0.660071\pi$
$827$	−22924.0	−0.963900	−0.481950	+	0.876199i	$0.660071\pi$
$828$	0	0
$829$	41690.0	1.74663	0.873313	−	0.487159i	$-0.161967\pi$
$829$	41690.0	1.74663	0.873313	+	0.487159i	$0.161967\pi$
$830$	0	0
$831$	−372.000	−0.0155289
$832$	0	0
$833$	24642.0	1.02496
$834$	0	0
$835$	8016.00	0.332222
$836$	0	0
$837$	−216.000	−0.00892001
$838$	0	0
$839$	−16450.0	−0.676898	−0.338449	−	0.940985i	$-0.609902\pi$
$839$	−16450.0	−0.676898	−0.338449	+	0.940985i	$0.609902\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	0	0
$843$	10926.0	0.446396
$844$	0	0
$845$	−4692.00	−0.191017
$846$	0	0
$847$	−3146.00	−0.127624
$848$	0	0
$849$	−13944.0	−0.563671
$850$	0	0
$851$	−12012.0	−0.483861
$852$	0	0
$853$	30892.0	1.24000	0.620001	−	0.784601i	$-0.287132\pi$
$853$	30892.0	1.24000	0.620001	+	0.784601i	$0.287132\pi$
$854$	0	0
$855$	2160.00	0.0863982
$856$	0	0
$857$	−38906.0	−1.55076	−0.775381	−	0.631493i	$-0.782442\pi$
$857$	−38906.0	−1.55076	−0.775381	+	0.631493i	$0.782442\pi$
$858$	0	0
$859$	1020.00	0.0405145	0.0202572	−	0.999795i	$-0.493551\pi$
$859$	1020.00	0.0405145	0.0202572	+	0.999795i	$0.493551\pi$
$860$	0	0
$861$	−32916.0	−1.30287
$862$	0	0
$863$	15078.0	0.594741	0.297370	−	0.954762i	$-0.403890\pi$
$863$	15078.0	0.594741	0.297370	+	0.954762i	$0.403890\pi$
$864$	0	0
$865$	−2712.00	−0.106602
$866$	0	0
$867$	1689.00	0.0661608
$868$	0	0
$869$	6050.00	0.236171
$870$	0	0
$871$	33152.0	1.28968
$872$	0	0
$873$	126.000	0.00488483
$874$	0	0
$875$	24336.0	0.940237
$876$	0	0
$877$	−22704.0	−0.874184	−0.437092	−	0.899417i	$-0.643992\pi$
$877$	−22704.0	−0.874184	−0.437092	+	0.899417i	$0.643992\pi$
$878$	0	0
$879$	9306.00	0.357092
$880$	0	0
$881$	−19358.0	−0.740281	−0.370141	−	0.928976i	$-0.620690\pi$
$881$	−19358.0	−0.740281	−0.370141	+	0.928976i	$0.620690\pi$
$882$	0	0
$883$	11252.0	0.428833	0.214417	−	0.976742i	$-0.431215\pi$
$883$	11252.0	0.428833	0.214417	+	0.976742i	$0.431215\pi$
$884$	0	0
$885$	2400.00	0.0911583
$886$	0	0
$887$	43684.0	1.65362	0.826812	−	0.562478i	$-0.190152\pi$
$887$	43684.0	1.65362	0.826812	+	0.562478i	$0.190152\pi$
$888$	0	0
$889$	41756.0	1.57531
$890$	0	0
$891$	−891.000	−0.0335013
$892$	0	0
$893$	−30360.0	−1.13769
$894$	0	0
$895$	6720.00	0.250977
$896$	0	0
$897$	−17472.0	−0.650360
$898$	0	0
$899$	−720.000	−0.0267112
$900$	0	0
$901$	−25752.0	−0.952190
$902$	0	0
$903$	31824.0	1.17280
$904$	0	0
$905$	17432.0	0.640287
$906$	0	0
$907$	−45804.0	−1.67684	−0.838422	−	0.545022i	$-0.816521\pi$
$907$	−45804.0	−1.67684	−0.838422	+	0.545022i	$0.816521\pi$
$908$	0	0
$909$	−15318.0	−0.558928
$910$	0	0
$911$	−15318.0	−0.557089	−0.278544	−	0.960423i	$-0.589852\pi$
$911$	−15318.0	−0.557089	−0.278544	+	0.960423i	$0.589852\pi$
$912$	0	0
$913$	−1452.00	−0.0526333
$914$	0	0
$915$	−1584.00	−0.0572300
$916$	0	0
$917$	−49608.0	−1.78648
$918$	0	0
$919$	11350.0	0.407401	0.203701	−	0.979033i	$-0.434703\pi$
$919$	11350.0	0.407401	0.203701	+	0.979033i	$0.434703\pi$
$920$	0	0
$921$	−3732.00	−0.133522
$922$	0	0
$923$	24384.0	0.869566
$924$	0	0
$925$	−7194.00	−0.255716
$926$	0	0
$927$	−10188.0	−0.360969
$928$	0	0
$929$	33030.0	1.16650	0.583250	−	0.812292i	$-0.301781\pi$
$929$	33030.0	1.16650	0.583250	+	0.812292i	$0.301781\pi$
$930$	0	0
$931$	19980.0	0.703349
$932$	0	0
$933$	6246.00	0.219169
$934$	0	0
$935$	−3256.00	−0.113885
$936$	0	0
$937$	−10006.0	−0.348860	−0.174430	−	0.984670i	$-0.555808\pi$
$937$	−10006.0	−0.348860	−0.174430	+	0.984670i	$0.555808\pi$
$938$	0	0
$939$	7134.00	0.247933
$940$	0	0
$941$	−2622.00	−0.0908340	−0.0454170	−	0.998968i	$-0.514462\pi$
$941$	−2622.00	−0.0908340	−0.0454170	+	0.998968i	$0.514462\pi$
$942$	0	0
$943$	−76804.0	−2.65226
$944$	0	0
$945$	−2808.00	−0.0966606
$946$	0	0
$947$	39876.0	1.36832	0.684158	−	0.729334i	$-0.260170\pi$
$947$	39876.0	1.36832	0.684158	+	0.729334i	$0.260170\pi$
$948$	0	0
$949$	−17344.0	−0.593267
$950$	0	0
$951$	1488.00	0.0507379
$952$	0	0
$953$	38918.0	1.32285	0.661426	−	0.750011i	$-0.269952\pi$
$953$	38918.0	1.32285	0.661426	+	0.750011i	$0.269952\pi$
$954$	0	0
$955$	−7112.00	−0.240983
$956$	0	0
$957$	−2970.00	−0.100320
$958$	0	0
$959$	56836.0	1.91380
$960$	0	0
$961$	−29727.0	−0.997852
$962$	0	0
$963$	−5076.00	−0.169857
$964$	0	0
$965$	−15848.0	−0.528669
$966$	0	0
$967$	1114.00	0.0370464	0.0185232	−	0.999828i	$-0.494104\pi$
$967$	1114.00	0.0370464	0.0185232	+	0.999828i	$0.494104\pi$
$968$	0	0
$969$	13320.0	0.441589
$970$	0	0
$971$	1688.00	0.0557884	0.0278942	−	0.999611i	$-0.491120\pi$
$971$	1688.00	0.0557884	0.0278942	+	0.999611i	$0.491120\pi$
$972$	0	0
$973$	71240.0	2.34722
$974$	0	0
$975$	−10464.0	−0.343709
$976$	0	0
$977$	−41826.0	−1.36963	−0.684817	−	0.728715i	$-0.740118\pi$
$977$	−41826.0	−1.36963	−0.684817	+	0.728715i	$0.740118\pi$
$978$	0	0
$979$	−6270.00	−0.204689
$980$	0	0
$981$	2880.00	0.0937322
$982$	0	0
$983$	978.000	0.0317328	0.0158664	−	0.999874i	$-0.494949\pi$
$983$	978.000	0.0317328	0.0158664	+	0.999874i	$0.494949\pi$
$984$	0	0
$985$	−1496.00	−0.0483924
$986$	0	0
$987$	39468.0	1.27283
$988$	0	0
$989$	74256.0	2.38747
$990$	0	0
$991$	47272.0	1.51528	0.757641	−	0.652671i	$-0.226352\pi$
$991$	47272.0	1.51528	0.757641	+	0.652671i	$0.226352\pi$
$992$	0	0
$993$	8124.00	0.259625
$994$	0	0
$995$	8400.00	0.267636
$996$	0	0
$997$	−51104.0	−1.62335	−0.811675	−	0.584109i	$-0.801444\pi$
$997$	−51104.0	−1.62335	−0.811675	+	0.584109i	$0.801444\pi$
$998$	0	0
$999$	1782.00	0.0564364

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.4.a.u.1.1		1
4.3	odd	2		2112.4.a.h.1.1		1
8.3	odd	2		528.4.a.h.1.1		1
8.5	even	2		33.4.a.b.1.1	✓	1
24.5	odd	2		99.4.a.a.1.1		1
24.11	even	2		1584.4.a.l.1.1		1
40.13	odd	4		825.4.c.f.199.2		2
40.29	even	2		825.4.a.f.1.1		1
40.37	odd	4		825.4.c.f.199.1		2
56.13	odd	2		1617.4.a.d.1.1		1
88.21	odd	2		363.4.a.d.1.1		1
120.29	odd	2		2475.4.a.e.1.1		1
264.197	even	2		1089.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.4.a.b.1.1	✓	1	8.5	even	2
99.4.a.a.1.1		1	24.5	odd	2
363.4.a.d.1.1		1	88.21	odd	2
528.4.a.h.1.1		1	8.3	odd	2
825.4.a.f.1.1		1	40.29	even	2
825.4.c.f.199.1		2	40.37	odd	4
825.4.c.f.199.2		2	40.13	odd	4
1089.4.a.e.1.1		1	264.197	even	2
1584.4.a.l.1.1		1	24.11	even	2
1617.4.a.d.1.1		1	56.13	odd	2
2112.4.a.h.1.1		1	4.3	odd	2
2112.4.a.u.1.1		1	1.1	even	1	trivial
2475.4.a.e.1.1		1	120.29	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2112.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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