LMFDB - Embedded newform 693.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{2} +1.00000 q^{4} +14.0000 q^{5} -7.00000 q^{7} +21.0000 q^{8} -42.0000 q^{10} +11.0000 q^{11} +2.00000 q^{13} +21.0000 q^{14} -71.0000 q^{16} +74.0000 q^{17} +14.0000 q^{20} -33.0000 q^{22} +148.000 q^{23} +71.0000 q^{25} -6.00000 q^{26} -7.00000 q^{28} -26.0000 q^{29} +112.000 q^{31} +45.0000 q^{32} -222.000 q^{34} -98.0000 q^{35} -98.0000 q^{37} +294.000 q^{40} +10.0000 q^{41} +208.000 q^{43} +11.0000 q^{44} -444.000 q^{46} -460.000 q^{47} +49.0000 q^{49} -213.000 q^{50} +2.00000 q^{52} -258.000 q^{53} +154.000 q^{55} -147.000 q^{56} +78.0000 q^{58} +204.000 q^{59} +178.000 q^{61} -336.000 q^{62} +433.000 q^{64} +28.0000 q^{65} -924.000 q^{67} +74.0000 q^{68} +294.000 q^{70} +748.000 q^{71} -230.000 q^{73} +294.000 q^{74} -77.0000 q^{77} -456.000 q^{79} -994.000 q^{80} -30.0000 q^{82} +228.000 q^{83} +1036.00 q^{85} -624.000 q^{86} +231.000 q^{88} +198.000 q^{89} -14.0000 q^{91} +148.000 q^{92} +1380.00 q^{94} +562.000 q^{97} -147.000 q^{98} +O(q^{100})$

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 $

$2$	−3.00000	−1.06066	−0.530330	−	0.847791i	$-0.677932\pi$
$2$	−3.00000	−1.06066	−0.530330	+	0.847791i	$0.677932\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	14.0000	1.25220	0.626099	−	0.779744i	$-0.284651\pi$
$5$	14.0000	1.25220	0.626099	+	0.779744i	$0.284651\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	21.0000	0.928078
$9$	0	0
$10$	−42.0000	−1.32816
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	2.00000	0.0426692	0.0213346	−	0.999772i	$-0.493208\pi$
$13$	2.00000	0.0426692	0.0213346	+	0.999772i	$0.493208\pi$
$14$	21.0000	0.400892
$15$	0	0
$16$	−71.0000	−1.10938
$17$	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
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	14.0000	0.156525
$21$	0	0
$22$	−33.0000	−0.319801
$23$	148.000	1.34174	0.670872	−	0.741573i	$-0.265920\pi$
$23$	148.000	1.34174	0.670872	+	0.741573i	$0.265920\pi$
$24$	0	0
$25$	71.0000	0.568000
$26$	−6.00000	−0.0452576
$27$	0	0
$28$	−7.00000	−0.0472456
$29$	−26.0000	−0.166485	−0.0832427	−	0.996529i	$-0.526528\pi$
$29$	−26.0000	−0.166485	−0.0832427	+	0.996529i	$0.526528\pi$
$30$	0	0
$31$	112.000	0.648897	0.324448	−	0.945903i	$-0.394821\pi$
$31$	112.000	0.648897	0.324448	+	0.945903i	$0.394821\pi$
$32$	45.0000	0.248592
$33$	0	0
$34$	−222.000	−1.11978
$35$	−98.0000	−0.473286
$36$	0	0
$37$	−98.0000	−0.435435	−0.217718	−	0.976012i	$-0.569861\pi$
$37$	−98.0000	−0.435435	−0.217718	+	0.976012i	$0.569861\pi$
$38$	0	0
$39$	0	0
$40$	294.000	1.16214
$41$	10.0000	0.0380912	0.0190456	−	0.999819i	$-0.493937\pi$
$41$	10.0000	0.0380912	0.0190456	+	0.999819i	$0.493937\pi$
$42$	0	0
$43$	208.000	0.737668	0.368834	−	0.929495i	$-0.379757\pi$
$43$	208.000	0.737668	0.368834	+	0.929495i	$0.379757\pi$
$44$	11.0000	0.0376889
$45$	0	0
$46$	−444.000	−1.42314
$47$	−460.000	−1.42761	−0.713807	−	0.700342i	$-0.753031\pi$
$47$	−460.000	−1.42761	−0.713807	+	0.700342i	$0.753031\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−213.000	−0.602455
$51$	0	0
$52$	2.00000	0.00533366
$53$	−258.000	−0.668661	−0.334330	−	0.942456i	$-0.608510\pi$
$53$	−258.000	−0.668661	−0.334330	+	0.942456i	$0.608510\pi$
$54$	0	0
$55$	154.000	0.377552
$56$	−147.000	−0.350780
$57$	0	0
$58$	78.0000	0.176585
$59$	204.000	0.450145	0.225072	−	0.974342i	$-0.427738\pi$
$59$	204.000	0.450145	0.225072	+	0.974342i	$0.427738\pi$
$60$	0	0
$61$	178.000	0.373616	0.186808	−	0.982396i	$-0.440186\pi$
$61$	178.000	0.373616	0.186808	+	0.982396i	$0.440186\pi$
$62$	−336.000	−0.688259
$63$	0	0
$64$	433.000	0.845703
$65$	28.0000	0.0534303
$66$	0	0
$67$	−924.000	−1.68484	−0.842422	−	0.538818i	$-0.818871\pi$
$67$	−924.000	−1.68484	−0.842422	+	0.538818i	$0.818871\pi$
$68$	74.0000	0.131968
$69$	0	0
$70$	294.000	0.501996
$71$	748.000	1.25030	0.625150	−	0.780505i	$-0.285038\pi$
$71$	748.000	1.25030	0.625150	+	0.780505i	$0.285038\pi$
$72$	0	0
$73$	−230.000	−0.368760	−0.184380	−	0.982855i	$-0.559028\pi$
$73$	−230.000	−0.368760	−0.184380	+	0.982855i	$0.559028\pi$
$74$	294.000	0.461849
$75$	0	0
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−456.000	−0.649418	−0.324709	−	0.945814i	$-0.605266\pi$
$79$	−456.000	−0.649418	−0.324709	+	0.945814i	$0.605266\pi$
$80$	−994.000	−1.38916
$81$	0	0
$82$	−30.0000	−0.0404018
$83$	228.000	0.301521	0.150761	−	0.988570i	$-0.451828\pi$
$83$	228.000	0.301521	0.150761	+	0.988570i	$0.451828\pi$
$84$	0	0
$85$	1036.00	1.32200
$86$	−624.000	−0.782415
$87$	0	0
$88$	231.000	0.279826
$89$	198.000	0.235820	0.117910	−	0.993024i	$-0.462381\pi$
$89$	198.000	0.235820	0.117910	+	0.993024i	$0.462381\pi$
$90$	0	0
$91$	−14.0000	−0.0161275
$92$	148.000	0.167718
$93$	0	0
$94$	1380.00	1.51421
$95$	0	0
$96$	0	0
$97$	562.000	0.588273	0.294136	−	0.955763i	$-0.404968\pi$
$97$	562.000	0.588273	0.294136	+	0.955763i	$0.404968\pi$
$98$	−147.000	−0.151523
$99$	0	0
$100$	71.0000	0.0710000
$101$	414.000	0.407867	0.203933	−	0.978985i	$-0.434627\pi$
$101$	414.000	0.407867	0.203933	+	0.978985i	$0.434627\pi$
$102$	0	0
$103$	984.000	0.941324	0.470662	−	0.882314i	$-0.344015\pi$
$103$	984.000	0.941324	0.470662	+	0.882314i	$0.344015\pi$
$104$	42.0000	0.0396004
$105$	0	0
$106$	774.000	0.709222
$107$	1916.00	1.73109	0.865545	−	0.500831i	$-0.166972\pi$
$107$	1916.00	1.73109	0.865545	+	0.500831i	$0.166972\pi$
$108$	0	0
$109$	−494.000	−0.434097	−0.217049	−	0.976161i	$-0.569643\pi$
$109$	−494.000	−0.434097	−0.217049	+	0.976161i	$0.569643\pi$
$110$	−462.000	−0.400454
$111$	0	0
$112$	497.000	0.419304
$113$	1238.00	1.03063	0.515315	−	0.857001i	$-0.327675\pi$
$113$	1238.00	1.03063	0.515315	+	0.857001i	$0.327675\pi$
$114$	0	0
$115$	2072.00	1.68013
$116$	−26.0000	−0.0208107
$117$	0	0
$118$	−612.000	−0.477451
$119$	−518.000	−0.399033
$120$	0	0
$121$	121.000	0.0909091
$122$	−534.000	−0.396279
$123$	0	0
$124$	112.000	0.0811121
$125$	−756.000	−0.540950
$126$	0	0
$127$	1016.00	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	1016.00	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1659.00	−1.14560
$129$	0	0
$130$	−84.0000	−0.0566714
$131$	2556.00	1.70472	0.852362	−	0.522953i	$-0.175170\pi$
$131$	2556.00	1.70472	0.852362	+	0.522953i	$0.175170\pi$
$132$	0	0
$133$	0	0
$134$	2772.00	1.78705
$135$	0	0
$136$	1554.00	0.979812
$137$	−1626.00	−1.01400	−0.507002	−	0.861945i	$-0.669246\pi$
$137$	−1626.00	−1.01400	−0.507002	+	0.861945i	$0.669246\pi$
$138$	0	0
$139$	536.000	0.327071	0.163536	−	0.986537i	$-0.447710\pi$
$139$	536.000	0.327071	0.163536	+	0.986537i	$0.447710\pi$
$140$	−98.0000	−0.0591608
$141$	0	0
$142$	−2244.00	−1.32614
$143$	22.0000	0.0128653
$144$	0	0
$145$	−364.000	−0.208473
$146$	690.000	0.391129
$147$	0	0
$148$	−98.0000	−0.0544294
$149$	3470.00	1.90788	0.953938	−	0.300004i	$-0.0969881\pi$
$149$	3470.00	1.90788	0.953938	+	0.300004i	$0.0969881\pi$
$150$	0	0
$151$	1392.00	0.750194	0.375097	−	0.926985i	$-0.377609\pi$
$151$	1392.00	0.750194	0.375097	+	0.926985i	$0.377609\pi$
$152$	0	0
$153$	0	0
$154$	231.000	0.120873
$155$	1568.00	0.812547
$156$	0	0
$157$	2758.00	1.40199	0.700995	−	0.713166i	$-0.252740\pi$
$157$	2758.00	1.40199	0.700995	+	0.713166i	$0.252740\pi$
$158$	1368.00	0.688812
$159$	0	0
$160$	630.000	0.311287
$161$	−1036.00	−0.507132
$162$	0	0
$163$	−1388.00	−0.666973	−0.333486	−	0.942755i	$-0.608225\pi$
$163$	−1388.00	−0.666973	−0.333486	+	0.942755i	$0.608225\pi$
$164$	10.0000	0.00476140
$165$	0	0
$166$	−684.000	−0.319811
$167$	−1688.00	−0.782164	−0.391082	−	0.920356i	$-0.627899\pi$
$167$	−1688.00	−0.782164	−0.391082	+	0.920356i	$0.627899\pi$
$168$	0	0
$169$	−2193.00	−0.998179
$170$	−3108.00	−1.40219
$171$	0	0
$172$	208.000	0.0922084
$173$	118.000	0.0518577	0.0259288	−	0.999664i	$-0.491746\pi$
$173$	118.000	0.0518577	0.0259288	+	0.999664i	$0.491746\pi$
$174$	0	0
$175$	−497.000	−0.214684
$176$	−781.000	−0.334489
$177$	0	0
$178$	−594.000	−0.250125
$179$	3004.00	1.25435	0.627177	−	0.778876i	$-0.284210\pi$
$179$	3004.00	1.25435	0.627177	+	0.778876i	$0.284210\pi$
$180$	0	0
$181$	782.000	0.321136	0.160568	−	0.987025i	$-0.448667\pi$
$181$	782.000	0.321136	0.160568	+	0.987025i	$0.448667\pi$
$182$	42.0000	0.0171058
$183$	0	0
$184$	3108.00	1.24524
$185$	−1372.00	−0.545251
$186$	0	0
$187$	814.000	0.318319
$188$	−460.000	−0.178452
$189$	0	0
$190$	0	0
$191$	4940.00	1.87144	0.935722	−	0.352738i	$-0.114749\pi$
$191$	4940.00	1.87144	0.935722	+	0.352738i	$0.114749\pi$
$192$	0	0
$193$	3290.00	1.22704	0.613522	−	0.789678i	$-0.289752\pi$
$193$	3290.00	1.22704	0.613522	+	0.789678i	$0.289752\pi$
$194$	−1686.00	−0.623957
$195$	0	0
$196$	49.0000	0.0178571
$197$	−66.0000	−0.0238696	−0.0119348	−	0.999929i	$-0.503799\pi$
$197$	−66.0000	−0.0238696	−0.0119348	+	0.999929i	$0.503799\pi$
$198$	0	0
$199$	−2672.00	−0.951824	−0.475912	−	0.879493i	$-0.657882\pi$
$199$	−2672.00	−0.951824	−0.475912	+	0.879493i	$0.657882\pi$
$200$	1491.00	0.527148
$201$	0	0
$202$	−1242.00	−0.432608
$203$	182.000	0.0629256
$204$	0	0
$205$	140.000	0.0476977
$206$	−2952.00	−0.998425
$207$	0	0
$208$	−142.000	−0.0473362
$209$	0	0
$210$	0	0
$211$	2992.00	0.976198	0.488099	−	0.872788i	$-0.337690\pi$
$211$	2992.00	0.976198	0.488099	+	0.872788i	$0.337690\pi$
$212$	−258.000	−0.0835826
$213$	0	0
$214$	−5748.00	−1.83610
$215$	2912.00	0.923706
$216$	0	0
$217$	−784.000	−0.245260
$218$	1482.00	0.460430
$219$	0	0
$220$	154.000	0.0471940
$221$	148.000	0.0450478
$222$	0	0
$223$	−2408.00	−0.723101	−0.361551	−	0.932352i	$-0.617753\pi$
$223$	−2408.00	−0.723101	−0.361551	+	0.932352i	$0.617753\pi$
$224$	−315.000	−0.0939590
$225$	0	0
$226$	−3714.00	−1.09315
$227$	2380.00	0.695886	0.347943	−	0.937516i	$-0.386880\pi$
$227$	2380.00	0.695886	0.347943	+	0.937516i	$0.386880\pi$
$228$	0	0
$229$	−4498.00	−1.29797	−0.648987	−	0.760799i	$-0.724807\pi$
$229$	−4498.00	−1.29797	−0.648987	+	0.760799i	$0.724807\pi$
$230$	−6216.00	−1.78205
$231$	0	0
$232$	−546.000	−0.154511
$233$	5010.00	1.40865	0.704326	−	0.709876i	$-0.251249\pi$
$233$	5010.00	1.40865	0.704326	+	0.709876i	$0.251249\pi$
$234$	0	0
$235$	−6440.00	−1.78766
$236$	204.000	0.0562681
$237$	0	0
$238$	1554.00	0.423239
$239$	−704.000	−0.190535	−0.0952677	−	0.995452i	$-0.530371\pi$
$239$	−704.000	−0.190535	−0.0952677	+	0.995452i	$0.530371\pi$
$240$	0	0
$241$	−6102.00	−1.63097	−0.815486	−	0.578776i	$-0.803530\pi$
$241$	−6102.00	−1.63097	−0.815486	+	0.578776i	$0.803530\pi$
$242$	−363.000	−0.0964237
$243$	0	0
$244$	178.000	0.0467020
$245$	686.000	0.178885
$246$	0	0
$247$	0	0
$248$	2352.00	0.602226
$249$	0	0
$250$	2268.00	0.573764
$251$	−6156.00	−1.54806	−0.774030	−	0.633148i	$-0.781762\pi$
$251$	−6156.00	−1.54806	−0.774030	+	0.633148i	$0.781762\pi$
$252$	0	0
$253$	1628.00	0.404551
$254$	−3048.00	−0.752947
$255$	0	0
$256$	1513.00	0.369385
$257$	3054.00	0.741258	0.370629	−	0.928781i	$-0.379142\pi$
$257$	3054.00	0.741258	0.370629	+	0.928781i	$0.379142\pi$
$258$	0	0
$259$	686.000	0.164579
$260$	28.0000	0.00667879
$261$	0	0
$262$	−7668.00	−1.80813
$263$	4944.00	1.15916	0.579582	−	0.814914i	$-0.303216\pi$
$263$	4944.00	1.15916	0.579582	+	0.814914i	$0.303216\pi$
$264$	0	0
$265$	−3612.00	−0.837296
$266$	0	0
$267$	0	0
$268$	−924.000	−0.210606
$269$	−7602.00	−1.72306	−0.861528	−	0.507710i	$-0.830492\pi$
$269$	−7602.00	−1.72306	−0.861528	+	0.507710i	$0.830492\pi$
$270$	0	0
$271$	4552.00	1.02035	0.510174	−	0.860071i	$-0.329581\pi$
$271$	4552.00	1.02035	0.510174	+	0.860071i	$0.329581\pi$
$272$	−5254.00	−1.17122
$273$	0	0
$274$	4878.00	1.07551
$275$	781.000	0.171258
$276$	0	0
$277$	−3294.00	−0.714503	−0.357251	−	0.934008i	$-0.616286\pi$
$277$	−3294.00	−0.714503	−0.357251	+	0.934008i	$0.616286\pi$
$278$	−1608.00	−0.346912
$279$	0	0
$280$	−2058.00	−0.439247
$281$	−3454.00	−0.733268	−0.366634	−	0.930365i	$-0.619490\pi$
$281$	−3454.00	−0.733268	−0.366634	+	0.930365i	$0.619490\pi$
$282$	0	0
$283$	8520.00	1.78962	0.894808	−	0.446451i	$-0.147312\pi$
$283$	8520.00	1.78962	0.894808	+	0.446451i	$0.147312\pi$
$284$	748.000	0.156287
$285$	0	0
$286$	−66.0000	−0.0136457
$287$	−70.0000	−0.0143971
$288$	0	0
$289$	563.000	0.114594
$290$	1092.00	0.221119
$291$	0	0
$292$	−230.000	−0.0460950
$293$	−5682.00	−1.13292	−0.566461	−	0.824089i	$-0.691688\pi$
$293$	−5682.00	−1.13292	−0.566461	+	0.824089i	$0.691688\pi$
$294$	0	0
$295$	2856.00	0.563670
$296$	−2058.00	−0.404118
$297$	0	0
$298$	−10410.0	−2.02361
$299$	296.000	0.0572512
$300$	0	0
$301$	−1456.00	−0.278812
$302$	−4176.00	−0.795701
$303$	0	0
$304$	0	0
$305$	2492.00	0.467841
$306$	0	0
$307$	1040.00	0.193342	0.0966709	−	0.995316i	$-0.469181\pi$
$307$	1040.00	0.193342	0.0966709	+	0.995316i	$0.469181\pi$
$308$	−77.0000	−0.0142451
$309$	0	0
$310$	−4704.00	−0.861836
$311$	−7972.00	−1.45354	−0.726770	−	0.686881i	$-0.758979\pi$
$311$	−7972.00	−1.45354	−0.726770	+	0.686881i	$0.758979\pi$
$312$	0	0
$313$	−3158.00	−0.570290	−0.285145	−	0.958484i	$-0.592042\pi$
$313$	−3158.00	−0.570290	−0.285145	+	0.958484i	$0.592042\pi$
$314$	−8274.00	−1.48703
$315$	0	0
$316$	−456.000	−0.0811772
$317$	6246.00	1.10666	0.553329	−	0.832963i	$-0.313357\pi$
$317$	6246.00	1.10666	0.553329	+	0.832963i	$0.313357\pi$
$318$	0	0
$319$	−286.000	−0.0501973
$320$	6062.00	1.05899
$321$	0	0
$322$	3108.00	0.537895
$323$	0	0
$324$	0	0
$325$	142.000	0.0242361
$326$	4164.00	0.707431
$327$	0	0
$328$	210.000	0.0353516
$329$	3220.00	0.539588
$330$	0	0
$331$	7900.00	1.31185	0.655926	−	0.754825i	$-0.272278\pi$
$331$	7900.00	1.31185	0.655926	+	0.754825i	$0.272278\pi$
$332$	228.000	0.0376901
$333$	0	0
$334$	5064.00	0.829610
$335$	−12936.0	−2.10976
$336$	0	0
$337$	4890.00	0.790431	0.395216	−	0.918588i	$-0.370670\pi$
$337$	4890.00	0.790431	0.395216	+	0.918588i	$0.370670\pi$
$338$	6579.00	1.05873
$339$	0	0
$340$	1036.00	0.165250
$341$	1232.00	0.195650
$342$	0	0
$343$	−343.000	−0.0539949
$344$	4368.00	0.684613
$345$	0	0
$346$	−354.000	−0.0550033
$347$	−10732.0	−1.66030	−0.830150	−	0.557540i	$-0.811745\pi$
$347$	−10732.0	−1.66030	−0.830150	+	0.557540i	$0.811745\pi$
$348$	0	0
$349$	7370.00	1.13039	0.565196	−	0.824956i	$-0.308800\pi$
$349$	7370.00	1.13039	0.565196	+	0.824956i	$0.308800\pi$
$350$	1491.00	0.227707
$351$	0	0
$352$	495.000	0.0749534
$353$	2526.00	0.380865	0.190433	−	0.981700i	$-0.439011\pi$
$353$	2526.00	0.380865	0.190433	+	0.981700i	$0.439011\pi$
$354$	0	0
$355$	10472.0	1.56562
$356$	198.000	0.0294775
$357$	0	0
$358$	−9012.00	−1.33044
$359$	−8080.00	−1.18787	−0.593936	−	0.804512i	$-0.702427\pi$
$359$	−8080.00	−1.18787	−0.593936	+	0.804512i	$0.702427\pi$
$360$	0	0
$361$	−6859.00	−1.00000
$362$	−2346.00	−0.340616
$363$	0	0
$364$	−14.0000	−0.00201593
$365$	−3220.00	−0.461760
$366$	0	0
$367$	−152.000	−0.0216194	−0.0108097	−	0.999942i	$-0.503441\pi$
$367$	−152.000	−0.0216194	−0.0108097	+	0.999942i	$0.503441\pi$
$368$	−10508.0	−1.48850
$369$	0	0
$370$	4116.00	0.578326
$371$	1806.00	0.252730
$372$	0	0
$373$	1106.00	0.153530	0.0767648	−	0.997049i	$-0.475541\pi$
$373$	1106.00	0.153530	0.0767648	+	0.997049i	$0.475541\pi$
$374$	−2442.00	−0.337628
$375$	0	0
$376$	−9660.00	−1.32494
$377$	−52.0000	−0.00710381
$378$	0	0
$379$	−3428.00	−0.464603	−0.232301	−	0.972644i	$-0.574626\pi$
$379$	−3428.00	−0.464603	−0.232301	+	0.972644i	$0.574626\pi$
$380$	0	0
$381$	0	0
$382$	−14820.0	−1.98497
$383$	4052.00	0.540594	0.270297	−	0.962777i	$-0.412878\pi$
$383$	4052.00	0.540594	0.270297	+	0.962777i	$0.412878\pi$
$384$	0	0
$385$	−1078.00	−0.142701
$386$	−9870.00	−1.30148
$387$	0	0
$388$	562.000	0.0735341
$389$	−4554.00	−0.593565	−0.296783	−	0.954945i	$-0.595914\pi$
$389$	−4554.00	−0.593565	−0.296783	+	0.954945i	$0.595914\pi$
$390$	0	0
$391$	10952.0	1.41654
$392$	1029.00	0.132583
$393$	0	0
$394$	198.000	0.0253175
$395$	−6384.00	−0.813200
$396$	0	0
$397$	−6666.00	−0.842713	−0.421356	−	0.906895i	$-0.638446\pi$
$397$	−6666.00	−0.842713	−0.421356	+	0.906895i	$0.638446\pi$
$398$	8016.00	1.00956
$399$	0	0
$400$	−5041.00	−0.630125
$401$	−12210.0	−1.52054	−0.760272	−	0.649605i	$-0.774934\pi$
$401$	−12210.0	−1.52054	−0.760272	+	0.649605i	$0.774934\pi$
$402$	0	0
$403$	224.000	0.0276879
$404$	414.000	0.0509833
$405$	0	0
$406$	−546.000	−0.0667427
$407$	−1078.00	−0.131289
$408$	0	0
$409$	−10190.0	−1.23194	−0.615970	−	0.787770i	$-0.711236\pi$
$409$	−10190.0	−1.23194	−0.615970	+	0.787770i	$0.711236\pi$
$410$	−420.000	−0.0505910
$411$	0	0
$412$	984.000	0.117666
$413$	−1428.00	−0.170139
$414$	0	0
$415$	3192.00	0.377564
$416$	90.0000	0.0106072
$417$	0	0
$418$	0	0
$419$	16780.0	1.95646	0.978230	−	0.207524i	$-0.0665406\pi$
$419$	16780.0	1.95646	0.978230	+	0.207524i	$0.0665406\pi$
$420$	0	0
$421$	9214.00	1.06666	0.533329	−	0.845908i	$-0.320941\pi$
$421$	9214.00	1.06666	0.533329	+	0.845908i	$0.320941\pi$
$422$	−8976.00	−1.03541
$423$	0	0
$424$	−5418.00	−0.620569
$425$	5254.00	0.599662
$426$	0	0
$427$	−1246.00	−0.141214
$428$	1916.00	0.216386
$429$	0	0
$430$	−8736.00	−0.979738
$431$	−9896.00	−1.10597	−0.552986	−	0.833191i	$-0.686512\pi$
$431$	−9896.00	−1.10597	−0.552986	+	0.833191i	$0.686512\pi$
$432$	0	0
$433$	−8878.00	−0.985334	−0.492667	−	0.870218i	$-0.663978\pi$
$433$	−8878.00	−0.985334	−0.492667	+	0.870218i	$0.663978\pi$
$434$	2352.00	0.260137
$435$	0	0
$436$	−494.000	−0.0542622
$437$	0	0
$438$	0	0
$439$	5464.00	0.594038	0.297019	−	0.954872i	$-0.404008\pi$
$439$	5464.00	0.594038	0.297019	+	0.954872i	$0.404008\pi$
$440$	3234.00	0.350398
$441$	0	0
$442$	−444.000	−0.0477804
$443$	−1668.00	−0.178892	−0.0894459	−	0.995992i	$-0.528510\pi$
$443$	−1668.00	−0.178892	−0.0894459	+	0.995992i	$0.528510\pi$
$444$	0	0
$445$	2772.00	0.295293
$446$	7224.00	0.766965
$447$	0	0
$448$	−3031.00	−0.319646
$449$	10374.0	1.09038	0.545189	−	0.838313i	$-0.316458\pi$
$449$	10374.0	1.09038	0.545189	+	0.838313i	$0.316458\pi$
$450$	0	0
$451$	110.000	0.0114849
$452$	1238.00	0.128829
$453$	0	0
$454$	−7140.00	−0.738099
$455$	−196.000	−0.0201948
$456$	0	0
$457$	−3046.00	−0.311785	−0.155893	−	0.987774i	$-0.549825\pi$
$457$	−3046.00	−0.311785	−0.155893	+	0.987774i	$0.549825\pi$
$458$	13494.0	1.37671
$459$	0	0
$460$	2072.00	0.210016
$461$	−1770.00	−0.178822	−0.0894112	−	0.995995i	$-0.528499\pi$
$461$	−1770.00	−0.178822	−0.0894112	+	0.995995i	$0.528499\pi$
$462$	0	0
$463$	12088.0	1.21334	0.606671	−	0.794953i	$-0.292505\pi$
$463$	12088.0	1.21334	0.606671	+	0.794953i	$0.292505\pi$
$464$	1846.00	0.184695
$465$	0	0
$466$	−15030.0	−1.49410
$467$	19836.0	1.96553	0.982763	−	0.184870i	$-0.0591864\pi$
$467$	19836.0	1.96553	0.982763	+	0.184870i	$0.0591864\pi$
$468$	0	0
$469$	6468.00	0.636811
$470$	19320.0	1.89610
$471$	0	0
$472$	4284.00	0.417769
$473$	2288.00	0.222415
$474$	0	0
$475$	0	0
$476$	−518.000	−0.0498792
$477$	0	0
$478$	2112.00	0.202093
$479$	4520.00	0.431157	0.215578	−	0.976487i	$-0.430836\pi$
$479$	4520.00	0.431157	0.215578	+	0.976487i	$0.430836\pi$
$480$	0	0
$481$	−196.000	−0.0185797
$482$	18306.0	1.72991
$483$	0	0
$484$	121.000	0.0113636
$485$	7868.00	0.736634
$486$	0	0
$487$	−1768.00	−0.164509	−0.0822543	−	0.996611i	$-0.526212\pi$
$487$	−1768.00	−0.164509	−0.0822543	+	0.996611i	$0.526212\pi$
$488$	3738.00	0.346744
$489$	0	0
$490$	−2058.00	−0.189737
$491$	−17988.0	−1.65333	−0.826667	−	0.562691i	$-0.809766\pi$
$491$	−17988.0	−1.65333	−0.826667	+	0.562691i	$0.809766\pi$
$492$	0	0
$493$	−1924.00	−0.175766
$494$	0	0
$495$	0	0
$496$	−7952.00	−0.719870
$497$	−5236.00	−0.472569
$498$	0	0
$499$	1916.00	0.171888	0.0859438	−	0.996300i	$-0.472609\pi$
$499$	1916.00	0.171888	0.0859438	+	0.996300i	$0.472609\pi$
$500$	−756.000	−0.0676187
$501$	0	0
$502$	18468.0	1.64197
$503$	18552.0	1.64452	0.822259	−	0.569113i	$-0.192713\pi$
$503$	18552.0	1.64452	0.822259	+	0.569113i	$0.192713\pi$
$504$	0	0
$505$	5796.00	0.510730
$506$	−4884.00	−0.429091
$507$	0	0
$508$	1016.00	0.0887357
$509$	9326.00	0.812117	0.406059	−	0.913847i	$-0.366903\pi$
$509$	9326.00	0.812117	0.406059	+	0.913847i	$0.366903\pi$
$510$	0	0
$511$	1610.00	0.139378
$512$	8733.00	0.753804
$513$	0	0
$514$	−9162.00	−0.786223
$515$	13776.0	1.17872
$516$	0	0
$517$	−5060.00	−0.430442
$518$	−2058.00	−0.174562
$519$	0	0
$520$	588.000	0.0495875
$521$	14038.0	1.18045	0.590226	−	0.807238i	$-0.299038\pi$
$521$	14038.0	1.18045	0.590226	+	0.807238i	$0.299038\pi$
$522$	0	0
$523$	5384.00	0.450145	0.225073	−	0.974342i	$-0.427738\pi$
$523$	5384.00	0.450145	0.225073	+	0.974342i	$0.427738\pi$
$524$	2556.00	0.213090
$525$	0	0
$526$	−14832.0	−1.22948
$527$	8288.00	0.685068
$528$	0	0
$529$	9737.00	0.800279
$530$	10836.0	0.888086
$531$	0	0
$532$	0	0
$533$	20.0000	0.00162532
$534$	0	0
$535$	26824.0	2.16767
$536$	−19404.0	−1.56367
$537$	0	0
$538$	22806.0	1.82758
$539$	539.000	0.0430730
$540$	0	0
$541$	−8398.00	−0.667390	−0.333695	−	0.942681i	$-0.608296\pi$
$541$	−8398.00	−0.667390	−0.333695	+	0.942681i	$0.608296\pi$
$542$	−13656.0	−1.08224
$543$	0	0
$544$	3330.00	0.262450
$545$	−6916.00	−0.543576
$546$	0	0
$547$	21312.0	1.66588	0.832939	−	0.553365i	$-0.186656\pi$
$547$	21312.0	1.66588	0.832939	+	0.553365i	$0.186656\pi$
$548$	−1626.00	−0.126751
$549$	0	0
$550$	−2343.00	−0.181647
$551$	0	0
$552$	0	0
$553$	3192.00	0.245457
$554$	9882.00	0.757845
$555$	0	0
$556$	536.000	0.0408839
$557$	−12778.0	−0.972031	−0.486015	−	0.873950i	$-0.661550\pi$
$557$	−12778.0	−0.972031	−0.486015	+	0.873950i	$0.661550\pi$
$558$	0	0
$559$	416.000	0.0314757
$560$	6958.00	0.525052
$561$	0	0
$562$	10362.0	0.777748
$563$	−22444.0	−1.68011	−0.840055	−	0.542501i	$-0.817477\pi$
$563$	−22444.0	−1.68011	−0.840055	+	0.542501i	$0.817477\pi$
$564$	0	0
$565$	17332.0	1.29055
$566$	−25560.0	−1.89817
$567$	0	0
$568$	15708.0	1.16038
$569$	26874.0	1.97999	0.989997	−	0.141088i	$-0.0450600\pi$
$569$	26874.0	1.97999	0.989997	+	0.141088i	$0.0450600\pi$
$570$	0	0
$571$	12712.0	0.931665	0.465832	−	0.884873i	$-0.345755\pi$
$571$	12712.0	0.931665	0.465832	+	0.884873i	$0.345755\pi$
$572$	22.0000	0.00160816
$573$	0	0
$574$	210.000	0.0152704
$575$	10508.0	0.762111
$576$	0	0
$577$	−17470.0	−1.26046	−0.630230	−	0.776408i	$-0.717039\pi$
$577$	−17470.0	−1.26046	−0.630230	+	0.776408i	$0.717039\pi$
$578$	−1689.00	−0.121545
$579$	0	0
$580$	−364.000	−0.0260591
$581$	−1596.00	−0.113964
$582$	0	0
$583$	−2838.00	−0.201609
$584$	−4830.00	−0.342238
$585$	0	0
$586$	17046.0	1.20164
$587$	−8340.00	−0.586420	−0.293210	−	0.956048i	$-0.594724\pi$
$587$	−8340.00	−0.586420	−0.293210	+	0.956048i	$0.594724\pi$
$588$	0	0
$589$	0	0
$590$	−8568.00	−0.597863
$591$	0	0
$592$	6958.00	0.483061
$593$	10818.0	0.749143	0.374572	−	0.927198i	$-0.377790\pi$
$593$	10818.0	0.749143	0.374572	+	0.927198i	$0.377790\pi$
$594$	0	0
$595$	−7252.00	−0.499669
$596$	3470.00	0.238484
$597$	0	0
$598$	−888.000	−0.0607241
$599$	3348.00	0.228373	0.114187	−	0.993459i	$-0.463574\pi$
$599$	3348.00	0.228373	0.114187	+	0.993459i	$0.463574\pi$
$600$	0	0
$601$	−9934.00	−0.674237	−0.337118	−	0.941462i	$-0.609452\pi$
$601$	−9934.00	−0.674237	−0.337118	+	0.941462i	$0.609452\pi$
$602$	4368.00	0.295725
$603$	0	0
$604$	1392.00	0.0937743
$605$	1694.00	0.113836
$606$	0	0
$607$	19640.0	1.31328	0.656642	−	0.754203i	$-0.271976\pi$
$607$	19640.0	1.31328	0.656642	+	0.754203i	$0.271976\pi$
$608$	0	0
$609$	0	0
$610$	−7476.00	−0.496220
$611$	−920.000	−0.0609152
$612$	0	0
$613$	10042.0	0.661652	0.330826	−	0.943692i	$-0.392673\pi$
$613$	10042.0	0.661652	0.330826	+	0.943692i	$0.392673\pi$
$614$	−3120.00	−0.205070
$615$	0	0
$616$	−1617.00	−0.105764
$617$	−26226.0	−1.71121	−0.855607	−	0.517626i	$-0.826816\pi$
$617$	−26226.0	−1.71121	−0.855607	+	0.517626i	$0.826816\pi$
$618$	0	0
$619$	−29932.0	−1.94357	−0.971784	−	0.235872i	$-0.924205\pi$
$619$	−29932.0	−1.94357	−0.971784	+	0.235872i	$0.924205\pi$
$620$	1568.00	0.101568
$621$	0	0
$622$	23916.0	1.54171
$623$	−1386.00	−0.0891315
$624$	0	0
$625$	−19459.0	−1.24538
$626$	9474.00	0.604884
$627$	0	0
$628$	2758.00	0.175249
$629$	−7252.00	−0.459708
$630$	0	0
$631$	−1920.00	−0.121132	−0.0605658	−	0.998164i	$-0.519290\pi$
$631$	−1920.00	−0.121132	−0.0605658	+	0.998164i	$0.519290\pi$
$632$	−9576.00	−0.602710
$633$	0	0
$634$	−18738.0	−1.17379
$635$	14224.0	0.888917
$636$	0	0
$637$	98.0000	0.00609561
$638$	858.000	0.0532422
$639$	0	0
$640$	−23226.0	−1.43451
$641$	2550.00	0.157128	0.0785639	−	0.996909i	$-0.474967\pi$
$641$	2550.00	0.157128	0.0785639	+	0.996909i	$0.474967\pi$
$642$	0	0
$643$	−24500.0	−1.50262	−0.751311	−	0.659949i	$-0.770578\pi$
$643$	−24500.0	−1.50262	−0.751311	+	0.659949i	$0.770578\pi$
$644$	−1036.00	−0.0633915
$645$	0	0
$646$	0	0
$647$	−20436.0	−1.24177	−0.620883	−	0.783904i	$-0.713226\pi$
$647$	−20436.0	−1.24177	−0.620883	+	0.783904i	$0.713226\pi$
$648$	0	0
$649$	2244.00	0.135724
$650$	−426.000	−0.0257063
$651$	0	0
$652$	−1388.00	−0.0833716
$653$	20062.0	1.20228	0.601138	−	0.799145i	$-0.294714\pi$
$653$	20062.0	1.20228	0.601138	+	0.799145i	$0.294714\pi$
$654$	0	0
$655$	35784.0	2.13465
$656$	−710.000	−0.0422574
$657$	0	0
$658$	−9660.00	−0.572319
$659$	13324.0	0.787601	0.393801	−	0.919196i	$-0.371160\pi$
$659$	13324.0	0.787601	0.393801	+	0.919196i	$0.371160\pi$
$660$	0	0
$661$	4958.00	0.291746	0.145873	−	0.989303i	$-0.453401\pi$
$661$	4958.00	0.291746	0.145873	+	0.989303i	$0.453401\pi$
$662$	−23700.0	−1.39143
$663$	0	0
$664$	4788.00	0.279835
$665$	0	0
$666$	0	0
$667$	−3848.00	−0.223381
$668$	−1688.00	−0.0977705
$669$	0	0
$670$	38808.0	2.23774
$671$	1958.00	0.112649
$672$	0	0
$673$	2458.00	0.140786	0.0703930	−	0.997519i	$-0.477575\pi$
$673$	2458.00	0.140786	0.0703930	+	0.997519i	$0.477575\pi$
$674$	−14670.0	−0.838379
$675$	0	0
$676$	−2193.00	−0.124772
$677$	−18658.0	−1.05921	−0.529605	−	0.848244i	$-0.677660\pi$
$677$	−18658.0	−1.05921	−0.529605	+	0.848244i	$0.677660\pi$
$678$	0	0
$679$	−3934.00	−0.222346
$680$	21756.0	1.22692
$681$	0	0
$682$	−3696.00	−0.207518
$683$	−28092.0	−1.57381	−0.786904	−	0.617076i	$-0.788317\pi$
$683$	−28092.0	−1.57381	−0.786904	+	0.617076i	$0.788317\pi$
$684$	0	0
$685$	−22764.0	−1.26973
$686$	1029.00	0.0572703
$687$	0	0
$688$	−14768.0	−0.818350
$689$	−516.000	−0.0285313
$690$	0	0
$691$	25596.0	1.40914	0.704571	−	0.709633i	$-0.251139\pi$
$691$	25596.0	1.40914	0.704571	+	0.709633i	$0.251139\pi$
$692$	118.000	0.00648221
$693$	0	0
$694$	32196.0	1.76101
$695$	7504.00	0.409558
$696$	0	0
$697$	740.000	0.0402145
$698$	−22110.0	−1.19896
$699$	0	0
$700$	−497.000	−0.0268355
$701$	−21498.0	−1.15830	−0.579150	−	0.815221i	$-0.696615\pi$
$701$	−21498.0	−1.15830	−0.579150	+	0.815221i	$0.696615\pi$
$702$	0	0
$703$	0	0
$704$	4763.00	0.254989
$705$	0	0
$706$	−7578.00	−0.403969
$707$	−2898.00	−0.154159
$708$	0	0
$709$	−17706.0	−0.937888	−0.468944	−	0.883228i	$-0.655365\pi$
$709$	−17706.0	−0.937888	−0.468944	+	0.883228i	$0.655365\pi$
$710$	−31416.0	−1.66059
$711$	0	0
$712$	4158.00	0.218859
$713$	16576.0	0.870654
$714$	0	0
$715$	308.000	0.0161099
$716$	3004.00	0.156794
$717$	0	0
$718$	24240.0	1.25993
$719$	−31796.0	−1.64922	−0.824611	−	0.565700i	$-0.808606\pi$
$719$	−31796.0	−1.64922	−0.824611	+	0.565700i	$0.808606\pi$
$720$	0	0
$721$	−6888.00	−0.355787
$722$	20577.0	1.06066
$723$	0	0
$724$	782.000	0.0401420
$725$	−1846.00	−0.0945638
$726$	0	0
$727$	−9952.00	−0.507702	−0.253851	−	0.967243i	$-0.581697\pi$
$727$	−9952.00	−0.507702	−0.253851	+	0.967243i	$0.581697\pi$
$728$	−294.000	−0.0149675
$729$	0	0
$730$	9660.00	0.489771
$731$	15392.0	0.778788
$732$	0	0
$733$	22850.0	1.15141	0.575705	−	0.817657i	$-0.304728\pi$
$733$	22850.0	1.15141	0.575705	+	0.817657i	$0.304728\pi$
$734$	456.000	0.0229309
$735$	0	0
$736$	6660.00	0.333547
$737$	−10164.0	−0.508000
$738$	0	0
$739$	−33144.0	−1.64983	−0.824913	−	0.565259i	$-0.808776\pi$
$739$	−33144.0	−1.64983	−0.824913	+	0.565259i	$0.808776\pi$
$740$	−1372.00	−0.0681564
$741$	0	0
$742$	−5418.00	−0.268061
$743$	15408.0	0.760787	0.380393	−	0.924825i	$-0.375789\pi$
$743$	15408.0	0.760787	0.380393	+	0.924825i	$0.375789\pi$
$744$	0	0
$745$	48580.0	2.38904
$746$	−3318.00	−0.162843
$747$	0	0
$748$	814.000	0.0397898
$749$	−13412.0	−0.654291
$750$	0	0
$751$	−23288.0	−1.13155	−0.565773	−	0.824561i	$-0.691422\pi$
$751$	−23288.0	−1.13155	−0.565773	+	0.824561i	$0.691422\pi$
$752$	32660.0	1.58376
$753$	0	0
$754$	156.000	0.00753473
$755$	19488.0	0.939392
$756$	0	0
$757$	−34018.0	−1.63330	−0.816648	−	0.577136i	$-0.804170\pi$
$757$	−34018.0	−1.63330	−0.816648	+	0.577136i	$0.804170\pi$
$758$	10284.0	0.492786
$759$	0	0
$760$	0	0
$761$	−17118.0	−0.815410	−0.407705	−	0.913114i	$-0.633671\pi$
$761$	−17118.0	−0.815410	−0.407705	+	0.913114i	$0.633671\pi$
$762$	0	0
$763$	3458.00	0.164073
$764$	4940.00	0.233931
$765$	0	0
$766$	−12156.0	−0.573387
$767$	408.000	0.0192073
$768$	0	0
$769$	−10622.0	−0.498100	−0.249050	−	0.968491i	$-0.580118\pi$
$769$	−10622.0	−0.498100	−0.249050	+	0.968491i	$0.580118\pi$
$770$	3234.00	0.151357
$771$	0	0
$772$	3290.00	0.153380
$773$	12870.0	0.598838	0.299419	−	0.954122i	$-0.403207\pi$
$773$	12870.0	0.598838	0.299419	+	0.954122i	$0.403207\pi$
$774$	0	0
$775$	7952.00	0.368573
$776$	11802.0	0.545963
$777$	0	0
$778$	13662.0	0.629571
$779$	0	0
$780$	0	0
$781$	8228.00	0.376979
$782$	−32856.0	−1.50247
$783$	0	0
$784$	−3479.00	−0.158482
$785$	38612.0	1.75557
$786$	0	0
$787$	20288.0	0.918919	0.459459	−	0.888199i	$-0.348043\pi$
$787$	20288.0	0.918919	0.459459	+	0.888199i	$0.348043\pi$
$788$	−66.0000	−0.00298370
$789$	0	0
$790$	19152.0	0.862529
$791$	−8666.00	−0.389542
$792$	0	0
$793$	356.000	0.0159419
$794$	19998.0	0.893832
$795$	0	0
$796$	−2672.00	−0.118978
$797$	−9754.00	−0.433506	−0.216753	−	0.976226i	$-0.569547\pi$
$797$	−9754.00	−0.433506	−0.216753	+	0.976226i	$0.569547\pi$
$798$	0	0
$799$	−34040.0	−1.50719
$800$	3195.00	0.141200
$801$	0	0
$802$	36630.0	1.61278
$803$	−2530.00	−0.111185
$804$	0	0
$805$	−14504.0	−0.635030
$806$	−672.000	−0.0293675
$807$	0	0
$808$	8694.00	0.378532
$809$	4554.00	0.197911	0.0989556	−	0.995092i	$-0.468450\pi$
$809$	4554.00	0.197911	0.0989556	+	0.995092i	$0.468450\pi$
$810$	0	0
$811$	−5584.00	−0.241777	−0.120888	−	0.992666i	$-0.538574\pi$
$811$	−5584.00	−0.241777	−0.120888	+	0.992666i	$0.538574\pi$
$812$	182.000	0.00786570
$813$	0	0
$814$	3234.00	0.139253
$815$	−19432.0	−0.835182
$816$	0	0
$817$	0	0
$818$	30570.0	1.30667
$819$	0	0
$820$	140.000	0.00596221
$821$	−29490.0	−1.25360	−0.626802	−	0.779179i	$-0.715636\pi$
$821$	−29490.0	−1.25360	−0.626802	+	0.779179i	$0.715636\pi$
$822$	0	0
$823$	−5032.00	−0.213128	−0.106564	−	0.994306i	$-0.533985\pi$
$823$	−5032.00	−0.213128	−0.106564	+	0.994306i	$0.533985\pi$
$824$	20664.0	0.873622
$825$	0	0
$826$	4284.00	0.180459
$827$	−36604.0	−1.53911	−0.769556	−	0.638579i	$-0.779522\pi$
$827$	−36604.0	−1.53911	−0.769556	+	0.638579i	$0.779522\pi$
$828$	0	0
$829$	−37154.0	−1.55659	−0.778294	−	0.627900i	$-0.783915\pi$
$829$	−37154.0	−1.55659	−0.778294	+	0.627900i	$0.783915\pi$
$830$	−9576.00	−0.400467
$831$	0	0
$832$	866.000	0.0360855
$833$	3626.00	0.150820
$834$	0	0
$835$	−23632.0	−0.979424
$836$	0	0
$837$	0	0
$838$	−50340.0	−2.07514
$839$	−3852.00	−0.158505	−0.0792526	−	0.996855i	$-0.525253\pi$
$839$	−3852.00	−0.158505	−0.0792526	+	0.996855i	$0.525253\pi$
$840$	0	0
$841$	−23713.0	−0.972283
$842$	−27642.0	−1.13136
$843$	0	0
$844$	2992.00	0.122025
$845$	−30702.0	−1.24992
$846$	0	0
$847$	−847.000	−0.0343604
$848$	18318.0	0.741796
$849$	0	0
$850$	−15762.0	−0.636038
$851$	−14504.0	−0.584243
$852$	0	0
$853$	−9622.00	−0.386226	−0.193113	−	0.981176i	$-0.561858\pi$
$853$	−9622.00	−0.386226	−0.193113	+	0.981176i	$0.561858\pi$
$854$	3738.00	0.149780
$855$	0	0
$856$	40236.0	1.60659
$857$	25202.0	1.00453	0.502266	−	0.864713i	$-0.332500\pi$
$857$	25202.0	1.00453	0.502266	+	0.864713i	$0.332500\pi$
$858$	0	0
$859$	20188.0	0.801869	0.400935	−	0.916107i	$-0.368685\pi$
$859$	20188.0	0.801869	0.400935	+	0.916107i	$0.368685\pi$
$860$	2912.00	0.115463
$861$	0	0
$862$	29688.0	1.17306
$863$	−26876.0	−1.06010	−0.530052	−	0.847965i	$-0.677828\pi$
$863$	−26876.0	−1.06010	−0.530052	+	0.847965i	$0.677828\pi$
$864$	0	0
$865$	1652.00	0.0649361
$866$	26634.0	1.04510
$867$	0	0
$868$	−784.000	−0.0306575
$869$	−5016.00	−0.195807
$870$	0	0
$871$	−1848.00	−0.0718910
$872$	−10374.0	−0.402876
$873$	0	0
$874$	0	0
$875$	5292.00	0.204460
$876$	0	0
$877$	−38198.0	−1.47076	−0.735379	−	0.677656i	$-0.762996\pi$
$877$	−38198.0	−1.47076	−0.735379	+	0.677656i	$0.762996\pi$
$878$	−16392.0	−0.630072
$879$	0	0
$880$	−10934.0	−0.418847
$881$	27222.0	1.04101	0.520507	−	0.853858i	$-0.325743\pi$
$881$	27222.0	1.04101	0.520507	+	0.853858i	$0.325743\pi$
$882$	0	0
$883$	−34316.0	−1.30784	−0.653921	−	0.756562i	$-0.726877\pi$
$883$	−34316.0	−1.30784	−0.653921	+	0.756562i	$0.726877\pi$
$884$	148.000	0.00563097
$885$	0	0
$886$	5004.00	0.189743
$887$	8944.00	0.338568	0.169284	−	0.985567i	$-0.445854\pi$
$887$	8944.00	0.338568	0.169284	+	0.985567i	$0.445854\pi$
$888$	0	0
$889$	−7112.00	−0.268311
$890$	−8316.00	−0.313206
$891$	0	0
$892$	−2408.00	−0.0903877
$893$	0	0
$894$	0	0
$895$	42056.0	1.57070
$896$	11613.0	0.432995
$897$	0	0
$898$	−31122.0	−1.15652
$899$	−2912.00	−0.108032
$900$	0	0
$901$	−19092.0	−0.705934
$902$	−330.000	−0.0121816
$903$	0	0
$904$	25998.0	0.956505
$905$	10948.0	0.402126
$906$	0	0
$907$	−19948.0	−0.730278	−0.365139	−	0.930953i	$-0.618979\pi$
$907$	−19948.0	−0.730278	−0.365139	+	0.930953i	$0.618979\pi$
$908$	2380.00	0.0869858
$909$	0	0
$910$	588.000	0.0214198
$911$	7812.00	0.284109	0.142054	−	0.989859i	$-0.454629\pi$
$911$	7812.00	0.284109	0.142054	+	0.989859i	$0.454629\pi$
$912$	0	0
$913$	2508.00	0.0909120
$914$	9138.00	0.330698
$915$	0	0
$916$	−4498.00	−0.162247
$917$	−17892.0	−0.644325
$918$	0	0
$919$	−36000.0	−1.29220	−0.646099	−	0.763253i	$-0.723601\pi$
$919$	−36000.0	−1.29220	−0.646099	+	0.763253i	$0.723601\pi$
$920$	43512.0	1.55929
$921$	0	0
$922$	5310.00	0.189670
$923$	1496.00	0.0533493
$924$	0	0
$925$	−6958.00	−0.247327
$926$	−36264.0	−1.28694
$927$	0	0
$928$	−1170.00	−0.0413870
$929$	15246.0	0.538434	0.269217	−	0.963080i	$-0.413235\pi$
$929$	15246.0	0.538434	0.269217	+	0.963080i	$0.413235\pi$
$930$	0	0
$931$	0	0
$932$	5010.00	0.176082
$933$	0	0
$934$	−59508.0	−2.08476
$935$	11396.0	0.398598
$936$	0	0
$937$	32706.0	1.14030	0.570149	−	0.821542i	$-0.306886\pi$
$937$	32706.0	1.14030	0.570149	+	0.821542i	$0.306886\pi$
$938$	−19404.0	−0.675440
$939$	0	0
$940$	−6440.00	−0.223457
$941$	10006.0	0.346638	0.173319	−	0.984866i	$-0.444551\pi$
$941$	10006.0	0.346638	0.173319	+	0.984866i	$0.444551\pi$
$942$	0	0
$943$	1480.00	0.0511086
$944$	−14484.0	−0.499379
$945$	0	0
$946$	−6864.00	−0.235907
$947$	5540.00	0.190101	0.0950506	−	0.995472i	$-0.469699\pi$
$947$	5540.00	0.190101	0.0950506	+	0.995472i	$0.469699\pi$
$948$	0	0
$949$	−460.000	−0.0157347
$950$	0	0
$951$	0	0
$952$	−10878.0	−0.370334
$953$	32522.0	1.10545	0.552723	−	0.833365i	$-0.313589\pi$
$953$	32522.0	1.10545	0.552723	+	0.833365i	$0.313589\pi$
$954$	0	0
$955$	69160.0	2.34342
$956$	−704.000	−0.0238169
$957$	0	0
$958$	−13560.0	−0.457311
$959$	11382.0	0.383258
$960$	0	0
$961$	−17247.0	−0.578933
$962$	588.000	0.0197067
$963$	0	0
$964$	−6102.00	−0.203872
$965$	46060.0	1.53650
$966$	0	0
$967$	−6976.00	−0.231989	−0.115994	−	0.993250i	$-0.537005\pi$
$967$	−6976.00	−0.231989	−0.115994	+	0.993250i	$0.537005\pi$
$968$	2541.00	0.0843707
$969$	0	0
$970$	−23604.0	−0.781318
$971$	−46660.0	−1.54211	−0.771056	−	0.636767i	$-0.780271\pi$
$971$	−46660.0	−1.54211	−0.771056	+	0.636767i	$0.780271\pi$
$972$	0	0
$973$	−3752.00	−0.123621
$974$	5304.00	0.174488
$975$	0	0
$976$	−12638.0	−0.414480
$977$	−11506.0	−0.376775	−0.188388	−	0.982095i	$-0.560326\pi$
$977$	−11506.0	−0.376775	−0.188388	+	0.982095i	$0.560326\pi$
$978$	0	0
$979$	2178.00	0.0711023
$980$	686.000	0.0223607
$981$	0	0
$982$	53964.0	1.75363
$983$	14844.0	0.481638	0.240819	−	0.970570i	$-0.422584\pi$
$983$	14844.0	0.481638	0.240819	+	0.970570i	$0.422584\pi$
$984$	0	0
$985$	−924.000	−0.0298894
$986$	5772.00	0.186428
$987$	0	0
$988$	0	0
$989$	30784.0	0.989762
$990$	0	0
$991$	−4616.00	−0.147964	−0.0739819	−	0.997260i	$-0.523571\pi$
$991$	−4616.00	−0.147964	−0.0739819	+	0.997260i	$0.523571\pi$
$992$	5040.00	0.161311
$993$	0	0
$994$	15708.0	0.501235
$995$	−37408.0	−1.19187
$996$	0	0
$997$	41298.0	1.31186	0.655928	−	0.754823i	$-0.272277\pi$
$997$	41298.0	1.31186	0.655928	+	0.754823i	$0.272277\pi$
$998$	−5748.00	−0.182314
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.c.1.1		1
3.2	odd	2		231.4.a.d.1.1	✓	1
21.20	even	2		1617.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.d.1.1	✓	1	3.2	odd	2
693.4.a.c.1.1		1	1.1	even	1	trivial
1617.4.a.f.1.1		1	21.20	even	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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