LMFDB - Embedded newform 825.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("825.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.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:	$48.6765757547$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -8.00000 q^{4} -2.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -8.00000 q^{4} -2.00000 q^{7} +9.00000 q^{9} -11.0000 q^{11} -24.0000 q^{12} +22.0000 q^{13} +64.0000 q^{16} -72.0000 q^{17} +122.000 q^{19} -6.00000 q^{21} -72.0000 q^{23} +27.0000 q^{27} +16.0000 q^{28} +96.0000 q^{29} -112.000 q^{31} -33.0000 q^{33} -72.0000 q^{36} -266.000 q^{37} +66.0000 q^{39} -96.0000 q^{41} +382.000 q^{43} +88.0000 q^{44} -360.000 q^{47} +192.000 q^{48} -339.000 q^{49} -216.000 q^{51} -176.000 q^{52} -318.000 q^{53} +366.000 q^{57} +660.000 q^{59} -430.000 q^{61} -18.0000 q^{63} -512.000 q^{64} -380.000 q^{67} +576.000 q^{68} -216.000 q^{69} +168.000 q^{71} -218.000 q^{73} -976.000 q^{76} +22.0000 q^{77} -706.000 q^{79} +81.0000 q^{81} -1068.00 q^{83} +48.0000 q^{84} +288.000 q^{87} -6.00000 q^{89} -44.0000 q^{91} +576.000 q^{92} -336.000 q^{93} -686.000 q^{97} -99.0000 q^{99} +O(q^{100})$

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−8.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.00000	−0.107990	−0.0539949	−	0.998541i	$-0.517195\pi$
$7$	−2.00000	−0.107990	−0.0539949	+	0.998541i	$0.517195\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−24.0000	−0.577350
$13$	22.0000	0.469362	0.234681	−	0.972072i	$-0.424595\pi$
$13$	22.0000	0.469362	0.234681	+	0.972072i	$0.424595\pi$
$14$	0	0
$15$	0	0
$16$	64.0000	1.00000
$17$	−72.0000	−1.02721	−0.513605	−	0.858027i	$-0.671690\pi$
$17$	−72.0000	−1.02721	−0.513605	+	0.858027i	$0.671690\pi$
$18$	0	0
$19$	122.000	1.47309	0.736545	−	0.676388i	$-0.236456\pi$
$19$	122.000	1.47309	0.736545	+	0.676388i	$0.236456\pi$
$20$	0	0
$21$	−6.00000	−0.0623480
$22$	0	0
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	16.0000	0.107990
$29$	96.0000	0.614716	0.307358	−	0.951594i	$-0.400555\pi$
$29$	96.0000	0.614716	0.307358	+	0.951594i	$0.400555\pi$
$30$	0	0
$31$	−112.000	−0.648897	−0.324448	−	0.945903i	$-0.605179\pi$
$31$	−112.000	−0.648897	−0.324448	+	0.945903i	$0.605179\pi$
$32$	0	0
$33$	−33.0000	−0.174078
$34$	0	0
$35$	0	0
$36$	−72.0000	−0.333333
$37$	−266.000	−1.18190	−0.590948	−	0.806710i	$-0.701246\pi$
$37$	−266.000	−1.18190	−0.590948	+	0.806710i	$0.701246\pi$
$38$	0	0
$39$	66.0000	0.270986
$40$	0	0
$41$	−96.0000	−0.365675	−0.182838	−	0.983143i	$-0.558528\pi$
$41$	−96.0000	−0.365675	−0.182838	+	0.983143i	$0.558528\pi$
$42$	0	0
$43$	382.000	1.35475	0.677377	−	0.735636i	$-0.263116\pi$
$43$	382.000	1.35475	0.677377	+	0.735636i	$0.263116\pi$
$44$	88.0000	0.301511
$45$	0	0
$46$	0	0
$47$	−360.000	−1.11726	−0.558632	−	0.829416i	$-0.688674\pi$
$47$	−360.000	−1.11726	−0.558632	+	0.829416i	$0.688674\pi$
$48$	192.000	0.577350
$49$	−339.000	−0.988338
$50$	0	0
$51$	−216.000	−0.593060
$52$	−176.000	−0.469362
$53$	−318.000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−318.000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	366.000	0.850489
$58$	0	0
$59$	660.000	1.45635	0.728175	−	0.685391i	$-0.240369\pi$
$59$	660.000	1.45635	0.728175	+	0.685391i	$0.240369\pi$
$60$	0	0
$61$	−430.000	−0.902555	−0.451278	−	0.892384i	$-0.649032\pi$
$61$	−430.000	−0.902555	−0.451278	+	0.892384i	$0.649032\pi$
$62$	0	0
$63$	−18.0000	−0.0359966
$64$	−512.000	−1.00000
$65$	0	0
$66$	0	0
$67$	−380.000	−0.692901	−0.346451	−	0.938068i	$-0.612613\pi$
$67$	−380.000	−0.692901	−0.346451	+	0.938068i	$0.612613\pi$
$68$	576.000	1.02721
$69$	−216.000	−0.376860
$70$	0	0
$71$	168.000	0.280816	0.140408	−	0.990094i	$-0.455159\pi$
$71$	168.000	0.280816	0.140408	+	0.990094i	$0.455159\pi$
$72$	0	0
$73$	−218.000	−0.349520	−0.174760	−	0.984611i	$-0.555915\pi$
$73$	−218.000	−0.349520	−0.174760	+	0.984611i	$0.555915\pi$
$74$	0	0
$75$	0	0
$76$	−976.000	−1.47309
$77$	22.0000	0.0325602
$78$	0	0
$79$	−706.000	−1.00546	−0.502729	−	0.864444i	$-0.667671\pi$
$79$	−706.000	−1.00546	−0.502729	+	0.864444i	$0.667671\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1068.00	−1.41239	−0.706194	−	0.708018i	$-0.749589\pi$
$83$	−1068.00	−1.41239	−0.706194	+	0.708018i	$0.749589\pi$
$84$	48.0000	0.0623480
$85$	0	0
$86$	0	0
$87$	288.000	0.354906
$88$	0	0
$89$	−6.00000	−0.00714605	−0.00357303	−	0.999994i	$-0.501137\pi$
$89$	−6.00000	−0.00714605	−0.00357303	+	0.999994i	$0.501137\pi$
$90$	0	0
$91$	−44.0000	−0.0506863
$92$	576.000	0.652741
$93$	−336.000	−0.374641
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−686.000	−0.718070	−0.359035	−	0.933324i	$-0.616894\pi$
$97$	−686.000	−0.718070	−0.359035	+	0.933324i	$0.616894\pi$
$98$	0	0
$99$	−99.0000	−0.100504
$100$	0	0
$101$	−960.000	−0.945778	−0.472889	−	0.881122i	$-0.656789\pi$
$101$	−960.000	−0.945778	−0.472889	+	0.881122i	$0.656789\pi$
$102$	0	0
$103$	844.000	0.807396	0.403698	−	0.914892i	$-0.367725\pi$
$103$	844.000	0.807396	0.403698	+	0.914892i	$0.367725\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2172.00	−1.96238	−0.981192	−	0.193033i	$-0.938168\pi$
$107$	−2172.00	−1.96238	−0.981192	+	0.193033i	$0.938168\pi$
$108$	−216.000	−0.192450
$109$	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
$111$	−798.000	−0.682368
$112$	−128.000	−0.107990
$113$	−1254.00	−1.04395	−0.521975	−	0.852961i	$-0.674805\pi$
$113$	−1254.00	−1.04395	−0.521975	+	0.852961i	$0.674805\pi$
$114$	0	0
$115$	0	0
$116$	−768.000	−0.614716
$117$	198.000	0.156454
$118$	0	0
$119$	144.000	0.110928
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−288.000	−0.211123
$124$	896.000	0.648897
$125$	0	0
$126$	0	0
$127$	−1394.00	−0.973996	−0.486998	−	0.873403i	$-0.661908\pi$
$127$	−1394.00	−0.973996	−0.486998	+	0.873403i	$0.661908\pi$
$128$	0	0
$129$	1146.00	0.782168
$130$	0	0
$131$	−252.000	−0.168071	−0.0840357	−	0.996463i	$-0.526781\pi$
$131$	−252.000	−0.168071	−0.0840357	+	0.996463i	$0.526781\pi$
$132$	264.000	0.174078
$133$	−244.000	−0.159079
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1050.00	0.654800	0.327400	−	0.944886i	$-0.393828\pi$
$137$	1050.00	0.654800	0.327400	+	0.944886i	$0.393828\pi$
$138$	0	0
$139$	1874.00	1.14353	0.571765	−	0.820418i	$-0.306259\pi$
$139$	1874.00	1.14353	0.571765	+	0.820418i	$0.306259\pi$
$140$	0	0
$141$	−1080.00	−0.645053
$142$	0	0
$143$	−242.000	−0.141518
$144$	576.000	0.333333
$145$	0	0
$146$	0	0
$147$	−1017.00	−0.570617
$148$	2128.00	1.18190
$149$	1476.00	0.811534	0.405767	−	0.913976i	$-0.367004\pi$
$149$	1476.00	0.811534	0.405767	+	0.913976i	$0.367004\pi$
$150$	0	0
$151$	1478.00	0.796543	0.398271	−	0.917268i	$-0.369610\pi$
$151$	1478.00	0.796543	0.398271	+	0.917268i	$0.369610\pi$
$152$	0	0
$153$	−648.000	−0.342403
$154$	0	0
$155$	0	0
$156$	−528.000	−0.270986
$157$	−854.000	−0.434119	−0.217059	−	0.976158i	$-0.569646\pi$
$157$	−854.000	−0.434119	−0.217059	+	0.976158i	$0.569646\pi$
$158$	0	0
$159$	−954.000	−0.475831
$160$	0	0
$161$	144.000	0.0704894
$162$	0	0
$163$	−1544.00	−0.741935	−0.370968	−	0.928646i	$-0.620974\pi$
$163$	−1544.00	−0.741935	−0.370968	+	0.928646i	$0.620974\pi$
$164$	768.000	0.365675
$165$	0	0
$166$	0	0
$167$	−240.000	−0.111208	−0.0556041	−	0.998453i	$-0.517708\pi$
$167$	−240.000	−0.111208	−0.0556041	+	0.998453i	$0.517708\pi$
$168$	0	0
$169$	−1713.00	−0.779700
$170$	0	0
$171$	1098.00	0.491030
$172$	−3056.00	−1.35475
$173$	−2532.00	−1.11274	−0.556371	−	0.830934i	$-0.687807\pi$
$173$	−2532.00	−1.11274	−0.556371	+	0.830934i	$0.687807\pi$
$174$	0	0
$175$	0	0
$176$	−704.000	−0.301511
$177$	1980.00	0.840824
$178$	0	0
$179$	1092.00	0.455977	0.227989	−	0.973664i	$-0.426785\pi$
$179$	1092.00	0.455977	0.227989	+	0.973664i	$0.426785\pi$
$180$	0	0
$181$	−2290.00	−0.940411	−0.470205	−	0.882557i	$-0.655820\pi$
$181$	−2290.00	−0.940411	−0.470205	+	0.882557i	$0.655820\pi$
$182$	0	0
$183$	−1290.00	−0.521090
$184$	0	0
$185$	0	0
$186$	0	0
$187$	792.000	0.309715
$188$	2880.00	1.11726
$189$	−54.0000	−0.0207827
$190$	0	0
$191$	−4392.00	−1.66384	−0.831921	−	0.554894i	$-0.812759\pi$
$191$	−4392.00	−1.66384	−0.831921	+	0.554894i	$0.812759\pi$
$192$	−1536.00	−0.577350
$193$	5074.00	1.89241	0.946203	−	0.323572i	$-0.104884\pi$
$193$	5074.00	1.89241	0.946203	+	0.323572i	$0.104884\pi$
$194$	0	0
$195$	0	0
$196$	2712.00	0.988338
$197$	−1692.00	−0.611929	−0.305964	−	0.952043i	$-0.598979\pi$
$197$	−1692.00	−0.611929	−0.305964	+	0.952043i	$0.598979\pi$
$198$	0	0
$199$	4664.00	1.66142	0.830709	−	0.556707i	$-0.187935\pi$
$199$	4664.00	1.66142	0.830709	+	0.556707i	$0.187935\pi$
$200$	0	0
$201$	−1140.00	−0.400047
$202$	0	0
$203$	−192.000	−0.0663830
$204$	1728.00	0.593060
$205$	0	0
$206$	0	0
$207$	−648.000	−0.217580
$208$	1408.00	0.469362
$209$	−1342.00	−0.444153
$210$	0	0
$211$	−1870.00	−0.610124	−0.305062	−	0.952333i	$-0.598677\pi$
$211$	−1870.00	−0.610124	−0.305062	+	0.952333i	$0.598677\pi$
$212$	2544.00	0.824163
$213$	504.000	0.162129
$214$	0	0
$215$	0	0
$216$	0	0
$217$	224.000	0.0700742
$218$	0	0
$219$	−654.000	−0.201796
$220$	0	0
$221$	−1584.00	−0.482133
$222$	0	0
$223$	−2300.00	−0.690670	−0.345335	−	0.938479i	$-0.612235\pi$
$223$	−2300.00	−0.690670	−0.345335	+	0.938479i	$0.612235\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1332.00	0.389462	0.194731	−	0.980857i	$-0.437617\pi$
$227$	1332.00	0.389462	0.194731	+	0.980857i	$0.437617\pi$
$228$	−2928.00	−0.850489
$229$	−6022.00	−1.73775	−0.868875	−	0.495031i	$-0.835157\pi$
$229$	−6022.00	−1.73775	−0.868875	+	0.495031i	$0.835157\pi$
$230$	0	0
$231$	66.0000	0.0187986
$232$	0	0
$233$	4716.00	1.32599	0.662994	−	0.748624i	$-0.269285\pi$
$233$	4716.00	1.32599	0.662994	+	0.748624i	$0.269285\pi$
$234$	0	0
$235$	0	0
$236$	−5280.00	−1.45635
$237$	−2118.00	−0.580502
$238$	0	0
$239$	−6420.00	−1.73755	−0.868777	−	0.495204i	$-0.835093\pi$
$239$	−6420.00	−1.73755	−0.868777	+	0.495204i	$0.835093\pi$
$240$	0	0
$241$	3302.00	0.882575	0.441287	−	0.897366i	$-0.354522\pi$
$241$	3302.00	0.882575	0.441287	+	0.897366i	$0.354522\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	3440.00	0.902555
$245$	0	0
$246$	0	0
$247$	2684.00	0.691412
$248$	0	0
$249$	−3204.00	−0.815443
$250$	0	0
$251$	732.000	0.184077	0.0920387	−	0.995755i	$-0.470662\pi$
$251$	732.000	0.184077	0.0920387	+	0.995755i	$0.470662\pi$
$252$	144.000	0.0359966
$253$	792.000	0.196809
$254$	0	0
$255$	0	0
$256$	4096.00	1.00000
$257$	3438.00	0.834461	0.417231	−	0.908801i	$-0.363001\pi$
$257$	3438.00	0.834461	0.417231	+	0.908801i	$0.363001\pi$
$258$	0	0
$259$	532.000	0.127633
$260$	0	0
$261$	864.000	0.204905
$262$	0	0
$263$	696.000	0.163183	0.0815916	−	0.996666i	$-0.474000\pi$
$263$	696.000	0.163183	0.0815916	+	0.996666i	$0.474000\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−18.0000	−0.00412578
$268$	3040.00	0.692901
$269$	−7338.00	−1.66322	−0.831609	−	0.555361i	$-0.812580\pi$
$269$	−7338.00	−1.66322	−0.831609	+	0.555361i	$0.812580\pi$
$270$	0	0
$271$	5114.00	1.14632	0.573161	−	0.819443i	$-0.305717\pi$
$271$	5114.00	1.14632	0.573161	+	0.819443i	$0.305717\pi$
$272$	−4608.00	−1.02721
$273$	−132.000	−0.0292637
$274$	0	0
$275$	0	0
$276$	1728.00	0.376860
$277$	−986.000	−0.213874	−0.106937	−	0.994266i	$-0.534104\pi$
$277$	−986.000	−0.213874	−0.106937	+	0.994266i	$0.534104\pi$
$278$	0	0
$279$	−1008.00	−0.216299
$280$	0	0
$281$	3312.00	0.703122	0.351561	−	0.936165i	$-0.385651\pi$
$281$	3312.00	0.703122	0.351561	+	0.936165i	$0.385651\pi$
$282$	0	0
$283$	−4298.00	−0.902790	−0.451395	−	0.892324i	$-0.649073\pi$
$283$	−4298.00	−0.902790	−0.451395	+	0.892324i	$0.649073\pi$
$284$	−1344.00	−0.280816
$285$	0	0
$286$	0	0
$287$	192.000	0.0394892
$288$	0	0
$289$	271.000	0.0551598
$290$	0	0
$291$	−2058.00	−0.414578
$292$	1744.00	0.349520
$293$	−2736.00	−0.545525	−0.272763	−	0.962081i	$-0.587937\pi$
$293$	−2736.00	−0.545525	−0.272763	+	0.962081i	$0.587937\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−297.000	−0.0580259
$298$	0	0
$299$	−1584.00	−0.306372
$300$	0	0
$301$	−764.000	−0.146300
$302$	0	0
$303$	−2880.00	−0.546045
$304$	7808.00	1.47309
$305$	0	0
$306$	0	0
$307$	250.000	0.0464764	0.0232382	−	0.999730i	$-0.492602\pi$
$307$	250.000	0.0464764	0.0232382	+	0.999730i	$0.492602\pi$
$308$	−176.000	−0.0325602
$309$	2532.00	0.466150
$310$	0	0
$311$	7248.00	1.32153	0.660766	−	0.750592i	$-0.270232\pi$
$311$	7248.00	1.32153	0.660766	+	0.750592i	$0.270232\pi$
$312$	0	0
$313$	7786.00	1.40604	0.703020	−	0.711170i	$-0.251834\pi$
$313$	7786.00	1.40604	0.703020	+	0.711170i	$0.251834\pi$
$314$	0	0
$315$	0	0
$316$	5648.00	1.00546
$317$	4230.00	0.749465	0.374733	−	0.927133i	$-0.377735\pi$
$317$	4230.00	0.749465	0.374733	+	0.927133i	$0.377735\pi$
$318$	0	0
$319$	−1056.00	−0.185344
$320$	0	0
$321$	−6516.00	−1.13298
$322$	0	0
$323$	−8784.00	−1.51317
$324$	−648.000	−0.111111
$325$	0	0
$326$	0	0
$327$	1842.00	0.311507
$328$	0	0
$329$	720.000	0.120653
$330$	0	0
$331$	7736.00	1.28462	0.642310	−	0.766445i	$-0.277976\pi$
$331$	7736.00	1.28462	0.642310	+	0.766445i	$0.277976\pi$
$332$	8544.00	1.41239
$333$	−2394.00	−0.393965
$334$	0	0
$335$	0	0
$336$	−384.000	−0.0623480
$337$	2014.00	0.325548	0.162774	−	0.986663i	$-0.447956\pi$
$337$	2014.00	0.325548	0.162774	+	0.986663i	$0.447956\pi$
$338$	0	0
$339$	−3762.00	−0.602725
$340$	0	0
$341$	1232.00	0.195650
$342$	0	0
$343$	1364.00	0.214720
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7692.00	−1.18999	−0.594997	−	0.803728i	$-0.702847\pi$
$347$	−7692.00	−1.18999	−0.594997	+	0.803728i	$0.702847\pi$
$348$	−2304.00	−0.354906
$349$	−1750.00	−0.268411	−0.134205	−	0.990954i	$-0.542848\pi$
$349$	−1750.00	−0.268411	−0.134205	+	0.990954i	$0.542848\pi$
$350$	0	0
$351$	594.000	0.0903287
$352$	0	0
$353$	−8034.00	−1.21135	−0.605675	−	0.795712i	$-0.707097\pi$
$353$	−8034.00	−1.21135	−0.605675	+	0.795712i	$0.707097\pi$
$354$	0	0
$355$	0	0
$356$	48.0000	0.00714605
$357$	432.000	0.0640444
$358$	0	0
$359$	2304.00	0.338720	0.169360	−	0.985554i	$-0.445830\pi$
$359$	2304.00	0.338720	0.169360	+	0.985554i	$0.445830\pi$
$360$	0	0
$361$	8025.00	1.17000
$362$	0	0
$363$	363.000	0.0524864
$364$	352.000	0.0506863
$365$	0	0
$366$	0	0
$367$	2356.00	0.335101	0.167551	−	0.985863i	$-0.446414\pi$
$367$	2356.00	0.335101	0.167551	+	0.985863i	$0.446414\pi$
$368$	−4608.00	−0.652741
$369$	−864.000	−0.121892
$370$	0	0
$371$	636.000	0.0890013
$372$	2688.00	0.374641
$373$	8602.00	1.19409	0.597044	−	0.802209i	$-0.296342\pi$
$373$	8602.00	1.19409	0.597044	+	0.802209i	$0.296342\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2112.00	0.288524
$378$	0	0
$379$	−12016.0	−1.62855	−0.814275	−	0.580479i	$-0.802865\pi$
$379$	−12016.0	−1.62855	−0.814275	+	0.580479i	$0.802865\pi$
$380$	0	0
$381$	−4182.00	−0.562337
$382$	0	0
$383$	−1728.00	−0.230540	−0.115270	−	0.993334i	$-0.536773\pi$
$383$	−1728.00	−0.230540	−0.115270	+	0.993334i	$0.536773\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3438.00	0.451585
$388$	5488.00	0.718070
$389$	8010.00	1.04402	0.522009	−	0.852940i	$-0.325183\pi$
$389$	8010.00	1.04402	0.522009	+	0.852940i	$0.325183\pi$
$390$	0	0
$391$	5184.00	0.670502
$392$	0	0
$393$	−756.000	−0.0970360
$394$	0	0
$395$	0	0
$396$	792.000	0.100504
$397$	10150.0	1.28316	0.641579	−	0.767057i	$-0.278280\pi$
$397$	10150.0	1.28316	0.641579	+	0.767057i	$0.278280\pi$
$398$	0	0
$399$	−732.000	−0.0918442
$400$	0	0
$401$	11862.0	1.47721	0.738604	−	0.674140i	$-0.235486\pi$
$401$	11862.0	1.47721	0.738604	+	0.674140i	$0.235486\pi$
$402$	0	0
$403$	−2464.00	−0.304567
$404$	7680.00	0.945778
$405$	0	0
$406$	0	0
$407$	2926.00	0.356355
$408$	0	0
$409$	−682.000	−0.0824517	−0.0412258	−	0.999150i	$-0.513126\pi$
$409$	−682.000	−0.0824517	−0.0412258	+	0.999150i	$0.513126\pi$
$410$	0	0
$411$	3150.00	0.378049
$412$	−6752.00	−0.807396
$413$	−1320.00	−0.157271
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5622.00	0.660217
$418$	0	0
$419$	10836.0	1.26342	0.631710	−	0.775205i	$-0.282353\pi$
$419$	10836.0	1.26342	0.631710	+	0.775205i	$0.282353\pi$
$420$	0	0
$421$	12350.0	1.42970	0.714848	−	0.699280i	$-0.246496\pi$
$421$	12350.0	1.42970	0.714848	+	0.699280i	$0.246496\pi$
$422$	0	0
$423$	−3240.00	−0.372421
$424$	0	0
$425$	0	0
$426$	0	0
$427$	860.000	0.0974668
$428$	17376.0	1.96238
$429$	−726.000	−0.0817054
$430$	0	0
$431$	−5940.00	−0.663851	−0.331925	−	0.943306i	$-0.607698\pi$
$431$	−5940.00	−0.663851	−0.331925	+	0.943306i	$0.607698\pi$
$432$	1728.00	0.192450
$433$	12898.0	1.43150	0.715749	−	0.698358i	$-0.246086\pi$
$433$	12898.0	1.43150	0.715749	+	0.698358i	$0.246086\pi$
$434$	0	0
$435$	0	0
$436$	−4912.00	−0.539546
$437$	−8784.00	−0.961546
$438$	0	0
$439$	11450.0	1.24483	0.622413	−	0.782689i	$-0.286152\pi$
$439$	11450.0	1.24483	0.622413	+	0.782689i	$0.286152\pi$
$440$	0	0
$441$	−3051.00	−0.329446
$442$	0	0
$443$	−2100.00	−0.225224	−0.112612	−	0.993639i	$-0.535922\pi$
$443$	−2100.00	−0.225224	−0.112612	+	0.993639i	$0.535922\pi$
$444$	6384.00	0.682368
$445$	0	0
$446$	0	0
$447$	4428.00	0.468540
$448$	1024.00	0.107990
$449$	−11934.0	−1.25434	−0.627172	−	0.778881i	$-0.715788\pi$
$449$	−11934.0	−1.25434	−0.627172	+	0.778881i	$0.715788\pi$
$450$	0	0
$451$	1056.00	0.110255
$452$	10032.0	1.04395
$453$	4434.00	0.459884
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−578.000	−0.0591635	−0.0295817	−	0.999562i	$-0.509418\pi$
$457$	−578.000	−0.0591635	−0.0295817	+	0.999562i	$0.509418\pi$
$458$	0	0
$459$	−1944.00	−0.197687
$460$	0	0
$461$	−324.000	−0.0327336	−0.0163668	−	0.999866i	$-0.505210\pi$
$461$	−324.000	−0.0327336	−0.0163668	+	0.999866i	$0.505210\pi$
$462$	0	0
$463$	11788.0	1.18323	0.591614	−	0.806221i	$-0.298491\pi$
$463$	11788.0	1.18323	0.591614	+	0.806221i	$0.298491\pi$
$464$	6144.00	0.614716
$465$	0	0
$466$	0	0
$467$	14484.0	1.43520	0.717601	−	0.696454i	$-0.245240\pi$
$467$	14484.0	1.43520	0.717601	+	0.696454i	$0.245240\pi$
$468$	−1584.00	−0.156454
$469$	760.000	0.0748263
$470$	0	0
$471$	−2562.00	−0.250638
$472$	0	0
$473$	−4202.00	−0.408474
$474$	0	0
$475$	0	0
$476$	−1152.00	−0.110928
$477$	−2862.00	−0.274721
$478$	0	0
$479$	−3084.00	−0.294179	−0.147089	−	0.989123i	$-0.546990\pi$
$479$	−3084.00	−0.294179	−0.147089	+	0.989123i	$0.546990\pi$
$480$	0	0
$481$	−5852.00	−0.554736
$482$	0	0
$483$	432.000	0.0406971
$484$	−968.000	−0.0909091
$485$	0	0
$486$	0	0
$487$	5584.00	0.519579	0.259790	−	0.965665i	$-0.416347\pi$
$487$	5584.00	0.519579	0.259790	+	0.965665i	$0.416347\pi$
$488$	0	0
$489$	−4632.00	−0.428356
$490$	0	0
$491$	−10752.0	−0.988250	−0.494125	−	0.869391i	$-0.664512\pi$
$491$	−10752.0	−0.988250	−0.494125	+	0.869391i	$0.664512\pi$
$492$	2304.00	0.211123
$493$	−6912.00	−0.631442
$494$	0	0
$495$	0	0
$496$	−7168.00	−0.648897
$497$	−336.000	−0.0303253
$498$	0	0
$499$	−13372.0	−1.19963	−0.599813	−	0.800141i	$-0.704758\pi$
$499$	−13372.0	−1.19963	−0.599813	+	0.800141i	$0.704758\pi$
$500$	0	0
$501$	−720.000	−0.0642060
$502$	0	0
$503$	−9072.00	−0.804176	−0.402088	−	0.915601i	$-0.631715\pi$
$503$	−9072.00	−0.804176	−0.402088	+	0.915601i	$0.631715\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−5139.00	−0.450160
$508$	11152.0	0.973996
$509$	−14586.0	−1.27016	−0.635082	−	0.772445i	$-0.719034\pi$
$509$	−14586.0	−1.27016	−0.635082	+	0.772445i	$0.719034\pi$
$510$	0	0
$511$	436.000	0.0377446
$512$	0	0
$513$	3294.00	0.283496
$514$	0	0
$515$	0	0
$516$	−9168.00	−0.782168
$517$	3960.00	0.336868
$518$	0	0
$519$	−7596.00	−0.642442
$520$	0	0
$521$	2718.00	0.228556	0.114278	−	0.993449i	$-0.463545\pi$
$521$	2718.00	0.228556	0.114278	+	0.993449i	$0.463545\pi$
$522$	0	0
$523$	2086.00	0.174406	0.0872031	−	0.996191i	$-0.472207\pi$
$523$	2086.00	0.174406	0.0872031	+	0.996191i	$0.472207\pi$
$524$	2016.00	0.168071
$525$	0	0
$526$	0	0
$527$	8064.00	0.666553
$528$	−2112.00	−0.174078
$529$	−6983.00	−0.573929
$530$	0	0
$531$	5940.00	0.485450
$532$	1952.00	0.159079
$533$	−2112.00	−0.171634
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3276.00	0.263259
$538$	0	0
$539$	3729.00	0.297995
$540$	0	0
$541$	13838.0	1.09971	0.549854	−	0.835261i	$-0.314683\pi$
$541$	13838.0	1.09971	0.549854	+	0.835261i	$0.314683\pi$
$542$	0	0
$543$	−6870.00	−0.542946
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−23546.0	−1.84050	−0.920251	−	0.391329i	$-0.872015\pi$
$547$	−23546.0	−1.84050	−0.920251	+	0.391329i	$0.872015\pi$
$548$	−8400.00	−0.654800
$549$	−3870.00	−0.300852
$550$	0	0
$551$	11712.0	0.905532
$552$	0	0
$553$	1412.00	0.108579
$554$	0	0
$555$	0	0
$556$	−14992.0	−1.14353
$557$	−15624.0	−1.18853	−0.594264	−	0.804270i	$-0.702557\pi$
$557$	−15624.0	−1.18853	−0.594264	+	0.804270i	$0.702557\pi$
$558$	0	0
$559$	8404.00	0.635870
$560$	0	0
$561$	2376.00	0.178814
$562$	0	0
$563$	−2400.00	−0.179659	−0.0898294	−	0.995957i	$-0.528632\pi$
$563$	−2400.00	−0.179659	−0.0898294	+	0.995957i	$0.528632\pi$
$564$	8640.00	0.645053
$565$	0	0
$566$	0	0
$567$	−162.000	−0.0119989
$568$	0	0
$569$	−18300.0	−1.34829	−0.674144	−	0.738600i	$-0.735487\pi$
$569$	−18300.0	−1.34829	−0.674144	+	0.738600i	$0.735487\pi$
$570$	0	0
$571$	25454.0	1.86553	0.932764	−	0.360487i	$-0.117389\pi$
$571$	25454.0	1.86553	0.932764	+	0.360487i	$0.117389\pi$
$572$	1936.00	0.141518
$573$	−13176.0	−0.960620
$574$	0	0
$575$	0	0
$576$	−4608.00	−0.333333
$577$	−19802.0	−1.42871	−0.714357	−	0.699781i	$-0.753281\pi$
$577$	−19802.0	−1.42871	−0.714357	+	0.699781i	$0.753281\pi$
$578$	0	0
$579$	15222.0	1.09258
$580$	0	0
$581$	2136.00	0.152524
$582$	0	0
$583$	3498.00	0.248495
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18396.0	−1.29350	−0.646750	−	0.762702i	$-0.723872\pi$
$587$	−18396.0	−1.29350	−0.646750	+	0.762702i	$0.723872\pi$
$588$	8136.00	0.570617
$589$	−13664.0	−0.955883
$590$	0	0
$591$	−5076.00	−0.353297
$592$	−17024.0	−1.18190
$593$	−15012.0	−1.03958	−0.519788	−	0.854295i	$-0.673989\pi$
$593$	−15012.0	−1.03958	−0.519788	+	0.854295i	$0.673989\pi$
$594$	0	0
$595$	0	0
$596$	−11808.0	−0.811534
$597$	13992.0	0.959220
$598$	0	0
$599$	15408.0	1.05101	0.525504	−	0.850791i	$-0.323877\pi$
$599$	15408.0	1.05101	0.525504	+	0.850791i	$0.323877\pi$
$600$	0	0
$601$	−1558.00	−0.105744	−0.0528720	−	0.998601i	$-0.516838\pi$
$601$	−1558.00	−0.105744	−0.0528720	+	0.998601i	$0.516838\pi$
$602$	0	0
$603$	−3420.00	−0.230967
$604$	−11824.0	−0.796543
$605$	0	0
$606$	0	0
$607$	−22970.0	−1.53595	−0.767977	−	0.640478i	$-0.778736\pi$
$607$	−22970.0	−1.53595	−0.767977	+	0.640478i	$0.778736\pi$
$608$	0	0
$609$	−576.000	−0.0383263
$610$	0	0
$611$	−7920.00	−0.524401
$612$	5184.00	0.342403
$613$	11482.0	0.756531	0.378266	−	0.925697i	$-0.376521\pi$
$613$	11482.0	0.756531	0.378266	+	0.925697i	$0.376521\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−246.000	−0.0160512	−0.00802560	−	0.999968i	$-0.502555\pi$
$617$	−246.000	−0.0160512	−0.00802560	+	0.999968i	$0.502555\pi$
$618$	0	0
$619$	11648.0	0.756337	0.378169	−	0.925737i	$-0.376554\pi$
$619$	11648.0	0.756337	0.378169	+	0.925737i	$0.376554\pi$
$620$	0	0
$621$	−1944.00	−0.125620
$622$	0	0
$623$	12.0000	0.000771701 0
$624$	4224.00	0.270986
$625$	0	0
$626$	0	0
$627$	−4026.00	−0.256432
$628$	6832.00	0.434119
$629$	19152.0	1.21405
$630$	0	0
$631$	−22024.0	−1.38948	−0.694740	−	0.719261i	$-0.744480\pi$
$631$	−22024.0	−1.38948	−0.694740	+	0.719261i	$0.744480\pi$
$632$	0	0
$633$	−5610.00	−0.352255
$634$	0	0
$635$	0	0
$636$	7632.00	0.475831
$637$	−7458.00	−0.463888
$638$	0	0
$639$	1512.00	0.0936053
$640$	0	0
$641$	2322.00	0.143079	0.0715394	−	0.997438i	$-0.477209\pi$
$641$	2322.00	0.143079	0.0715394	+	0.997438i	$0.477209\pi$
$642$	0	0
$643$	−14024.0	−0.860113	−0.430056	−	0.902802i	$-0.641506\pi$
$643$	−14024.0	−0.860113	−0.430056	+	0.902802i	$0.641506\pi$
$644$	−1152.00	−0.0704894
$645$	0	0
$646$	0	0
$647$	7152.00	0.434581	0.217291	−	0.976107i	$-0.430278\pi$
$647$	7152.00	0.434581	0.217291	+	0.976107i	$0.430278\pi$
$648$	0	0
$649$	−7260.00	−0.439106
$650$	0	0
$651$	672.000	0.0404574
$652$	12352.0	0.741935
$653$	3138.00	0.188054	0.0940271	−	0.995570i	$-0.470026\pi$
$653$	3138.00	0.188054	0.0940271	+	0.995570i	$0.470026\pi$
$654$	0	0
$655$	0	0
$656$	−6144.00	−0.365675
$657$	−1962.00	−0.116507
$658$	0	0
$659$	−15876.0	−0.938454	−0.469227	−	0.883078i	$-0.655467\pi$
$659$	−15876.0	−0.938454	−0.469227	+	0.883078i	$0.655467\pi$
$660$	0	0
$661$	−20554.0	−1.20947	−0.604734	−	0.796428i	$-0.706720\pi$
$661$	−20554.0	−1.20947	−0.604734	+	0.796428i	$0.706720\pi$
$662$	0	0
$663$	−4752.00	−0.278360
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6912.00	−0.401250
$668$	1920.00	0.111208
$669$	−6900.00	−0.398758
$670$	0	0
$671$	4730.00	0.272131
$672$	0	0
$673$	−27806.0	−1.59263	−0.796317	−	0.604880i	$-0.793221\pi$
$673$	−27806.0	−1.59263	−0.796317	+	0.604880i	$0.793221\pi$
$674$	0	0
$675$	0	0
$676$	13704.0	0.779700
$677$	−20820.0	−1.18195	−0.590973	−	0.806691i	$-0.701256\pi$
$677$	−20820.0	−1.18195	−0.590973	+	0.806691i	$0.701256\pi$
$678$	0	0
$679$	1372.00	0.0775442
$680$	0	0
$681$	3996.00	0.224856
$682$	0	0
$683$	7020.00	0.393284	0.196642	−	0.980475i	$-0.436996\pi$
$683$	7020.00	0.393284	0.196642	+	0.980475i	$0.436996\pi$
$684$	−8784.00	−0.491030
$685$	0	0
$686$	0	0
$687$	−18066.0	−1.00329
$688$	24448.0	1.35475
$689$	−6996.00	−0.386831
$690$	0	0
$691$	12536.0	0.690147	0.345074	−	0.938576i	$-0.387854\pi$
$691$	12536.0	0.690147	0.345074	+	0.938576i	$0.387854\pi$
$692$	20256.0	1.11274
$693$	198.000	0.0108534
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6912.00	0.375625
$698$	0	0
$699$	14148.0	0.765560
$700$	0	0
$701$	−33276.0	−1.79289	−0.896446	−	0.443153i	$-0.853860\pi$
$701$	−33276.0	−1.79289	−0.896446	+	0.443153i	$0.853860\pi$
$702$	0	0
$703$	−32452.0	−1.74104
$704$	5632.00	0.301511
$705$	0	0
$706$	0	0
$707$	1920.00	0.102134
$708$	−15840.0	−0.840824
$709$	9818.00	0.520060	0.260030	−	0.965601i	$-0.416267\pi$
$709$	9818.00	0.520060	0.260030	+	0.965601i	$0.416267\pi$
$710$	0	0
$711$	−6354.00	−0.335153
$712$	0	0
$713$	8064.00	0.423561
$714$	0	0
$715$	0	0
$716$	−8736.00	−0.455977
$717$	−19260.0	−1.00318
$718$	0	0
$719$	3216.00	0.166810	0.0834051	−	0.996516i	$-0.473420\pi$
$719$	3216.00	0.166810	0.0834051	+	0.996516i	$0.473420\pi$
$720$	0	0
$721$	−1688.00	−0.0871906
$722$	0	0
$723$	9906.00	0.509555
$724$	18320.0	0.940411
$725$	0	0
$726$	0	0
$727$	10960.0	0.559125	0.279563	−	0.960127i	$-0.409811\pi$
$727$	10960.0	0.559125	0.279563	+	0.960127i	$0.409811\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−27504.0	−1.39162
$732$	10320.0	0.521090
$733$	−14618.0	−0.736600	−0.368300	−	0.929707i	$-0.620060\pi$
$733$	−14618.0	−0.736600	−0.368300	+	0.929707i	$0.620060\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4180.00	0.208918
$738$	0	0
$739$	36518.0	1.81778	0.908888	−	0.417040i	$-0.136933\pi$
$739$	36518.0	1.81778	0.908888	+	0.417040i	$0.136933\pi$
$740$	0	0
$741$	8052.00	0.399187
$742$	0	0
$743$	37452.0	1.84923	0.924617	−	0.380899i	$-0.124385\pi$
$743$	37452.0	1.84923	0.924617	+	0.380899i	$0.124385\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9612.00	−0.470796
$748$	−6336.00	−0.309715
$749$	4344.00	0.211918
$750$	0	0
$751$	−10648.0	−0.517378	−0.258689	−	0.965961i	$-0.583291\pi$
$751$	−10648.0	−0.517378	−0.258689	+	0.965961i	$0.583291\pi$
$752$	−23040.0	−1.11726
$753$	2196.00	0.106277
$754$	0	0
$755$	0	0
$756$	432.000	0.0207827
$757$	1258.00	0.0604000	0.0302000	−	0.999544i	$-0.490386\pi$
$757$	1258.00	0.0604000	0.0302000	+	0.999544i	$0.490386\pi$
$758$	0	0
$759$	2376.00	0.113628
$760$	0	0
$761$	1740.00	0.0828843	0.0414421	−	0.999141i	$-0.486805\pi$
$761$	1740.00	0.0828843	0.0414421	+	0.999141i	$0.486805\pi$
$762$	0	0
$763$	−1228.00	−0.0582655
$764$	35136.0	1.66384
$765$	0	0
$766$	0	0
$767$	14520.0	0.683555
$768$	12288.0	0.577350
$769$	−10774.0	−0.505228	−0.252614	−	0.967567i	$-0.581290\pi$
$769$	−10774.0	−0.505228	−0.252614	+	0.967567i	$0.581290\pi$
$770$	0	0
$771$	10314.0	0.481776
$772$	−40592.0	−1.89241
$773$	19146.0	0.890859	0.445429	−	0.895317i	$-0.353051\pi$
$773$	19146.0	0.890859	0.445429	+	0.895317i	$0.353051\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1596.00	0.0736888
$778$	0	0
$779$	−11712.0	−0.538673
$780$	0	0
$781$	−1848.00	−0.0846692
$782$	0	0
$783$	2592.00	0.118302
$784$	−21696.0	−0.988338
$785$	0	0
$786$	0	0
$787$	30670.0	1.38916	0.694579	−	0.719416i	$-0.255591\pi$
$787$	30670.0	1.38916	0.694579	+	0.719416i	$0.255591\pi$
$788$	13536.0	0.611929
$789$	2088.00	0.0942139
$790$	0	0
$791$	2508.00	0.112736
$792$	0	0
$793$	−9460.00	−0.423625
$794$	0	0
$795$	0	0
$796$	−37312.0	−1.66142
$797$	−11970.0	−0.531994	−0.265997	−	0.963974i	$-0.585701\pi$
$797$	−11970.0	−0.531994	−0.265997	+	0.963974i	$0.585701\pi$
$798$	0	0
$799$	25920.0	1.14766
$800$	0	0
$801$	−54.0000	−0.00238202
$802$	0	0
$803$	2398.00	0.105384
$804$	9120.00	0.400047
$805$	0	0
$806$	0	0
$807$	−22014.0	−0.960260
$808$	0	0
$809$	−19932.0	−0.866220	−0.433110	−	0.901341i	$-0.642584\pi$
$809$	−19932.0	−0.866220	−0.433110	+	0.901341i	$0.642584\pi$
$810$	0	0
$811$	−31462.0	−1.36224	−0.681122	−	0.732170i	$-0.738508\pi$
$811$	−31462.0	−1.36224	−0.681122	+	0.732170i	$0.738508\pi$
$812$	1536.00	0.0663830
$813$	15342.0	0.661830
$814$	0	0
$815$	0	0
$816$	−13824.0	−0.593060
$817$	46604.0	1.99568
$818$	0	0
$819$	−396.000	−0.0168954
$820$	0	0
$821$	−39720.0	−1.68847	−0.844237	−	0.535970i	$-0.819946\pi$
$821$	−39720.0	−1.68847	−0.844237	+	0.535970i	$0.819946\pi$
$822$	0	0
$823$	28492.0	1.20677	0.603383	−	0.797451i	$-0.293819\pi$
$823$	28492.0	1.20677	0.603383	+	0.797451i	$0.293819\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	18324.0	0.770481	0.385241	−	0.922816i	$-0.374118\pi$
$827$	18324.0	0.770481	0.385241	+	0.922816i	$0.374118\pi$
$828$	5184.00	0.217580
$829$	21626.0	0.906034	0.453017	−	0.891502i	$-0.350348\pi$
$829$	21626.0	0.906034	0.453017	+	0.891502i	$0.350348\pi$
$830$	0	0
$831$	−2958.00	−0.123480
$832$	−11264.0	−0.469362
$833$	24408.0	1.01523
$834$	0	0
$835$	0	0
$836$	10736.0	0.444153
$837$	−3024.00	−0.124880
$838$	0	0
$839$	−36960.0	−1.52086	−0.760430	−	0.649420i	$-0.775012\pi$
$839$	−36960.0	−1.52086	−0.760430	+	0.649420i	$0.775012\pi$
$840$	0	0
$841$	−15173.0	−0.622125
$842$	0	0
$843$	9936.00	0.405948
$844$	14960.0	0.610124
$845$	0	0
$846$	0	0
$847$	−242.000	−0.00981726
$848$	−20352.0	−0.824163
$849$	−12894.0	−0.521226
$850$	0	0
$851$	19152.0	0.771471
$852$	−4032.00	−0.162129
$853$	−31502.0	−1.26449	−0.632244	−	0.774769i	$-0.717866\pi$
$853$	−31502.0	−1.26449	−0.632244	+	0.774769i	$0.717866\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5640.00	−0.224806	−0.112403	−	0.993663i	$-0.535855\pi$
$857$	−5640.00	−0.224806	−0.112403	+	0.993663i	$0.535855\pi$
$858$	0	0
$859$	−2056.00	−0.0816645	−0.0408323	−	0.999166i	$-0.513001\pi$
$859$	−2056.00	−0.0816645	−0.0408323	+	0.999166i	$0.513001\pi$
$860$	0	0
$861$	576.000	0.0227991
$862$	0	0
$863$	−18336.0	−0.723250	−0.361625	−	0.932324i	$-0.617778\pi$
$863$	−18336.0	−0.723250	−0.361625	+	0.932324i	$0.617778\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	813.000	0.0318465
$868$	−1792.00	−0.0700742
$869$	7766.00	0.303157
$870$	0	0
$871$	−8360.00	−0.325221
$872$	0	0
$873$	−6174.00	−0.239357
$874$	0	0
$875$	0	0
$876$	5232.00	0.201796
$877$	51346.0	1.97700	0.988501	−	0.151213i	$-0.0483178\pi$
$877$	51346.0	1.97700	0.988501	+	0.151213i	$0.0483178\pi$
$878$	0	0
$879$	−8208.00	−0.314959
$880$	0	0
$881$	32910.0	1.25853	0.629266	−	0.777190i	$-0.283356\pi$
$881$	32910.0	1.25853	0.629266	+	0.777190i	$0.283356\pi$
$882$	0	0
$883$	−15356.0	−0.585244	−0.292622	−	0.956228i	$-0.594528\pi$
$883$	−15356.0	−0.585244	−0.292622	+	0.956228i	$0.594528\pi$
$884$	12672.0	0.482133
$885$	0	0
$886$	0	0
$887$	−18372.0	−0.695458	−0.347729	−	0.937595i	$-0.613047\pi$
$887$	−18372.0	−0.695458	−0.347729	+	0.937595i	$0.613047\pi$
$888$	0	0
$889$	2788.00	0.105182
$890$	0	0
$891$	−891.000	−0.0335013
$892$	18400.0	0.690670
$893$	−43920.0	−1.64583
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4752.00	−0.176884
$898$	0	0
$899$	−10752.0	−0.398887
$900$	0	0
$901$	22896.0	0.846589
$902$	0	0
$903$	−2292.00	−0.0844662
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−7640.00	−0.279694	−0.139847	−	0.990173i	$-0.544661\pi$
$907$	−7640.00	−0.279694	−0.139847	+	0.990173i	$0.544661\pi$
$908$	−10656.0	−0.389462
$909$	−8640.00	−0.315259
$910$	0	0
$911$	−53040.0	−1.92897	−0.964486	−	0.264134i	$-0.914914\pi$
$911$	−53040.0	−1.92897	−0.964486	+	0.264134i	$0.914914\pi$
$912$	23424.0	0.850489
$913$	11748.0	0.425851
$914$	0	0
$915$	0	0
$916$	48176.0	1.73775
$917$	504.000	0.0181500
$918$	0	0
$919$	−11302.0	−0.405679	−0.202839	−	0.979212i	$-0.565017\pi$
$919$	−11302.0	−0.405679	−0.202839	+	0.979212i	$0.565017\pi$
$920$	0	0
$921$	750.000	0.0268332
$922$	0	0
$923$	3696.00	0.131804
$924$	−528.000	−0.0187986
$925$	0	0
$926$	0	0
$927$	7596.00	0.269132
$928$	0	0
$929$	19254.0	0.679982	0.339991	−	0.940429i	$-0.389576\pi$
$929$	19254.0	0.679982	0.339991	+	0.940429i	$0.389576\pi$
$930$	0	0
$931$	−41358.0	−1.45591
$932$	−37728.0	−1.32599
$933$	21744.0	0.762987
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−22214.0	−0.774493	−0.387246	−	0.921976i	$-0.626574\pi$
$937$	−22214.0	−0.774493	−0.387246	+	0.921976i	$0.626574\pi$
$938$	0	0
$939$	23358.0	0.811778
$940$	0	0
$941$	41736.0	1.44586	0.722930	−	0.690921i	$-0.242795\pi$
$941$	41736.0	1.44586	0.722930	+	0.690921i	$0.242795\pi$
$942$	0	0
$943$	6912.00	0.238691
$944$	42240.0	1.45635
$945$	0	0
$946$	0	0
$947$	−42732.0	−1.46632	−0.733159	−	0.680057i	$-0.761955\pi$
$947$	−42732.0	−1.46632	−0.733159	+	0.680057i	$0.761955\pi$
$948$	16944.0	0.580502
$949$	−4796.00	−0.164051
$950$	0	0
$951$	12690.0	0.432704
$952$	0	0
$953$	25056.0	0.851672	0.425836	−	0.904800i	$-0.359980\pi$
$953$	25056.0	0.851672	0.425836	+	0.904800i	$0.359980\pi$
$954$	0	0
$955$	0	0
$956$	51360.0	1.73755
$957$	−3168.00	−0.107008
$958$	0	0
$959$	−2100.00	−0.0707117
$960$	0	0
$961$	−17247.0	−0.578933
$962$	0	0
$963$	−19548.0	−0.654128
$964$	−26416.0	−0.882575
$965$	0	0
$966$	0	0
$967$	14326.0	0.476415	0.238207	−	0.971214i	$-0.423440\pi$
$967$	14326.0	0.476415	0.238207	+	0.971214i	$0.423440\pi$
$968$	0	0
$969$	−26352.0	−0.873631
$970$	0	0
$971$	45924.0	1.51779	0.758894	−	0.651215i	$-0.225740\pi$
$971$	45924.0	1.51779	0.758894	+	0.651215i	$0.225740\pi$
$972$	−1944.00	−0.0641500
$973$	−3748.00	−0.123490
$974$	0	0
$975$	0	0
$976$	−27520.0	−0.902555
$977$	38946.0	1.27533	0.637663	−	0.770316i	$-0.279901\pi$
$977$	38946.0	1.27533	0.637663	+	0.770316i	$0.279901\pi$
$978$	0	0
$979$	66.0000	0.00215462
$980$	0	0
$981$	5526.00	0.179849
$982$	0	0
$983$	−21000.0	−0.681379	−0.340690	−	0.940176i	$-0.610661\pi$
$983$	−21000.0	−0.681379	−0.340690	+	0.940176i	$0.610661\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2160.00	0.0696591
$988$	−21472.0	−0.691412
$989$	−27504.0	−0.884304
$990$	0	0
$991$	7760.00	0.248743	0.124372	−	0.992236i	$-0.460309\pi$
$991$	7760.00	0.248743	0.124372	+	0.992236i	$0.460309\pi$
$992$	0	0
$993$	23208.0	0.741675
$994$	0	0
$995$	0	0
$996$	25632.0	0.815443
$997$	−21350.0	−0.678196	−0.339098	−	0.940751i	$-0.610122\pi$
$997$	−21350.0	−0.678196	−0.339098	+	0.940751i	$0.610122\pi$
$998$	0	0
$999$	−7182.00	−0.227456

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.a.e.1.1		1
3.2	odd	2		2475.4.a.f.1.1		1
5.2	odd	4		825.4.c.g.199.1		2
5.3	odd	4		825.4.c.g.199.2		2
5.4	even	2		165.4.a.a.1.1	✓	1
15.14	odd	2		495.4.a.c.1.1		1
55.54	odd	2		1815.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.4.a.a.1.1	✓	1	5.4	even	2
495.4.a.c.1.1		1	15.14	odd	2
825.4.a.e.1.1		1	1.1	even	1	trivial
825.4.c.g.199.1		2	5.2	odd	4
825.4.c.g.199.2		2	5.3	odd	4
1815.4.a.f.1.1		1	55.54	odd	2
2475.4.a.f.1.1		1	3.2	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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