LMFDB - Embedded newform 231.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [231,4,Mod(1,231)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(231, 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("231.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 231 = 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	231.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:	$13.6294412113$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	231.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -14.0000 q^{5} +9.00000 q^{6} -7.00000 q^{7} -21.0000 q^{8} +9.00000 q^{9} -42.0000 q^{10} -11.0000 q^{11} +3.00000 q^{12} +2.00000 q^{13} -21.0000 q^{14} -42.0000 q^{15} -71.0000 q^{16} -74.0000 q^{17} +27.0000 q^{18} -14.0000 q^{20} -21.0000 q^{21} -33.0000 q^{22} -148.000 q^{23} -63.0000 q^{24} +71.0000 q^{25} +6.00000 q^{26} +27.0000 q^{27} -7.00000 q^{28} +26.0000 q^{29} -126.000 q^{30} +112.000 q^{31} -45.0000 q^{32} -33.0000 q^{33} -222.000 q^{34} +98.0000 q^{35} +9.00000 q^{36} -98.0000 q^{37} +6.00000 q^{39} +294.000 q^{40} -10.0000 q^{41} -63.0000 q^{42} +208.000 q^{43} -11.0000 q^{44} -126.000 q^{45} -444.000 q^{46} +460.000 q^{47} -213.000 q^{48} +49.0000 q^{49} +213.000 q^{50} -222.000 q^{51} +2.00000 q^{52} +258.000 q^{53} +81.0000 q^{54} +154.000 q^{55} +147.000 q^{56} +78.0000 q^{58} -204.000 q^{59} -42.0000 q^{60} +178.000 q^{61} +336.000 q^{62} -63.0000 q^{63} +433.000 q^{64} -28.0000 q^{65} -99.0000 q^{66} -924.000 q^{67} -74.0000 q^{68} -444.000 q^{69} +294.000 q^{70} -748.000 q^{71} -189.000 q^{72} -230.000 q^{73} -294.000 q^{74} +213.000 q^{75} +77.0000 q^{77} +18.0000 q^{78} -456.000 q^{79} +994.000 q^{80} +81.0000 q^{81} -30.0000 q^{82} -228.000 q^{83} -21.0000 q^{84} +1036.00 q^{85} +624.000 q^{86} +78.0000 q^{87} +231.000 q^{88} -198.000 q^{89} -378.000 q^{90} -14.0000 q^{91} -148.000 q^{92} +336.000 q^{93} +1380.00 q^{94} -135.000 q^{96} +562.000 q^{97} +147.000 q^{98} -99.0000 q^{99} +O(q^{100})\)

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	$-0.322068\pi$
$2$	3.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	1.00000	0.125000
$5$	−14.0000	−1.25220	−0.626099	−	0.779744i	$-0.715349\pi$
$5$	−14.0000	−1.25220	−0.626099	+	0.779744i	$0.715349\pi$
$6$	9.00000	0.612372
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−21.0000	−0.928078
$9$	9.00000	0.333333
$10$	−42.0000	−1.32816
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	3.00000	0.0721688
$13$	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
$15$	−42.0000	−0.722957
$16$	−71.0000	−1.10938
$17$	−74.0000	−1.05574	−0.527872	−	0.849324i	$-0.677010\pi$
$17$	−74.0000	−1.05574	−0.527872	+	0.849324i	$0.677010\pi$
$18$	27.0000	0.353553
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−14.0000	−0.156525
$21$	−21.0000	−0.218218
$22$	−33.0000	−0.319801
$23$	−148.000	−1.34174	−0.670872	−	0.741573i	$-0.734080\pi$
$23$	−148.000	−1.34174	−0.670872	+	0.741573i	$0.734080\pi$
$24$	−63.0000	−0.535826
$25$	71.0000	0.568000
$26$	6.00000	0.0452576
$27$	27.0000	0.192450
$28$	−7.00000	−0.0472456
$29$	26.0000	0.166485	0.0832427	−	0.996529i	$-0.473472\pi$
$29$	26.0000	0.166485	0.0832427	+	0.996529i	$0.473472\pi$
$30$	−126.000	−0.766812
$31$	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
$33$	−33.0000	−0.174078
$34$	−222.000	−1.11978
$35$	98.0000	0.473286
$36$	9.00000	0.0416667
$37$	−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
$39$	6.00000	0.0246351
$40$	294.000	1.16214
$41$	−10.0000	−0.0380912	−0.0190456	−	0.999819i	$-0.506063\pi$
$41$	−10.0000	−0.0380912	−0.0190456	+	0.999819i	$0.506063\pi$
$42$	−63.0000	−0.231455
$43$	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
$45$	−126.000	−0.417399
$46$	−444.000	−1.42314
$47$	460.000	1.42761	0.713807	−	0.700342i	$-0.246969\pi$
$47$	460.000	1.42761	0.713807	+	0.700342i	$0.246969\pi$
$48$	−213.000	−0.640498
$49$	49.0000	0.142857
$50$	213.000	0.602455
$51$	−222.000	−0.609534
$52$	2.00000	0.00533366
$53$	258.000	0.668661	0.334330	−	0.942456i	$-0.391490\pi$
$53$	258.000	0.668661	0.334330	+	0.942456i	$0.391490\pi$
$54$	81.0000	0.204124
$55$	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	$-0.572262\pi$
$59$	−204.000	−0.450145	−0.225072	+	0.974342i	$0.572262\pi$
$60$	−42.0000	−0.0903696
$61$	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
$63$	−63.0000	−0.125988
$64$	433.000	0.845703
$65$	−28.0000	−0.0534303
$66$	−99.0000	−0.184637
$67$	−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
$69$	−444.000	−0.774657
$70$	294.000	0.501996
$71$	−748.000	−1.25030	−0.625150	−	0.780505i	$-0.714962\pi$
$71$	−748.000	−1.25030	−0.625150	+	0.780505i	$0.714962\pi$
$72$	−189.000	−0.309359
$73$	−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
$75$	213.000	0.327935
$76$	0	0
$77$	77.0000	0.113961
$78$	18.0000	0.0261295
$79$	−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
$81$	81.0000	0.111111
$82$	−30.0000	−0.0404018
$83$	−228.000	−0.301521	−0.150761	−	0.988570i	$-0.548172\pi$
$83$	−228.000	−0.301521	−0.150761	+	0.988570i	$0.548172\pi$
$84$	−21.0000	−0.0272772
$85$	1036.00	1.32200
$86$	624.000	0.782415
$87$	78.0000	0.0961204
$88$	231.000	0.279826
$89$	−198.000	−0.235820	−0.117910	−	0.993024i	$-0.537619\pi$
$89$	−198.000	−0.235820	−0.117910	+	0.993024i	$0.537619\pi$
$90$	−378.000	−0.442719
$91$	−14.0000	−0.0161275
$92$	−148.000	−0.167718
$93$	336.000	0.374641
$94$	1380.00	1.51421
$95$	0	0
$96$	−135.000	−0.143525
$97$	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
$99$	−99.0000	−0.100504
$100$	71.0000	0.0710000
$101$	−414.000	−0.407867	−0.203933	−	0.978985i	$-0.565373\pi$
$101$	−414.000	−0.407867	−0.203933	+	0.978985i	$0.565373\pi$
$102$	−666.000	−0.646508
$103$	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
$105$	294.000	0.273252
$106$	774.000	0.709222
$107$	−1916.00	−1.73109	−0.865545	−	0.500831i	$-0.833028\pi$
$107$	−1916.00	−1.73109	−0.865545	+	0.500831i	$0.833028\pi$
$108$	27.0000	0.0240563
$109$	−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
$111$	−294.000	−0.251399
$112$	497.000	0.419304
$113$	−1238.00	−1.03063	−0.515315	−	0.857001i	$-0.672325\pi$
$113$	−1238.00	−1.03063	−0.515315	+	0.857001i	$0.672325\pi$
$114$	0	0
$115$	2072.00	1.68013
$116$	26.0000	0.0208107
$117$	18.0000	0.0142231
$118$	−612.000	−0.477451
$119$	518.000	0.399033
$120$	882.000	0.670960
$121$	121.000	0.0909091
$122$	534.000	0.396279
$123$	−30.0000	−0.0219919
$124$	112.000	0.0811121
$125$	756.000	0.540950
$126$	−189.000	−0.133631
$127$	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
$129$	624.000	0.425893
$130$	−84.0000	−0.0566714
$131$	−2556.00	−1.70472	−0.852362	−	0.522953i	$-0.824830\pi$
$131$	−2556.00	−1.70472	−0.852362	+	0.522953i	$0.824830\pi$
$132$	−33.0000	−0.0217597
$133$	0	0
$134$	−2772.00	−1.78705
$135$	−378.000	−0.240986
$136$	1554.00	0.979812
$137$	1626.00	1.01400	0.507002	−	0.861945i	$-0.330754\pi$
$137$	1626.00	1.01400	0.507002	+	0.861945i	$0.330754\pi$
$138$	−1332.00	−0.821648
$139$	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
$141$	1380.00	0.824234
$142$	−2244.00	−1.32614
$143$	−22.0000	−0.0128653
$144$	−639.000	−0.369792
$145$	−364.000	−0.208473
$146$	−690.000	−0.391129
$147$	147.000	0.0824786
$148$	−98.0000	−0.0544294
$149$	−3470.00	−1.90788	−0.953938	−	0.300004i	$-0.903012\pi$
$149$	−3470.00	−1.90788	−0.953938	+	0.300004i	$0.903012\pi$
$150$	639.000	0.347828
$151$	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
$153$	−666.000	−0.351914
$154$	231.000	0.120873
$155$	−1568.00	−0.812547
$156$	6.00000	0.00307939
$157$	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
$159$	774.000	0.386052
$160$	630.000	0.311287
$161$	1036.00	0.507132
$162$	243.000	0.117851
$163$	−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
$165$	462.000	0.217980
$166$	−684.000	−0.319811
$167$	1688.00	0.782164	0.391082	−	0.920356i	$-0.372101\pi$
$167$	1688.00	0.782164	0.391082	+	0.920356i	$0.372101\pi$
$168$	441.000	0.202523
$169$	−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	$-0.508254\pi$
$173$	−118.000	−0.0518577	−0.0259288	+	0.999664i	$0.508254\pi$
$174$	234.000	0.101951
$175$	−497.000	−0.214684
$176$	781.000	0.334489
$177$	−612.000	−0.259891
$178$	−594.000	−0.250125
$179$	−3004.00	−1.25435	−0.627177	−	0.778876i	$-0.715790\pi$
$179$	−3004.00	−1.25435	−0.627177	+	0.778876i	$0.715790\pi$
$180$	−126.000	−0.0521749
$181$	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
$183$	534.000	0.215707
$184$	3108.00	1.24524
$185$	1372.00	0.545251
$186$	1008.00	0.397366
$187$	814.000	0.318319
$188$	460.000	0.178452
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−4940.00	−1.87144	−0.935722	−	0.352738i	$-0.885251\pi$
$191$	−4940.00	−1.87144	−0.935722	+	0.352738i	$0.885251\pi$
$192$	1299.00	0.488267
$193$	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
$195$	−84.0000	−0.0308480
$196$	49.0000	0.0178571
$197$	66.0000	0.0238696	0.0119348	−	0.999929i	$-0.496201\pi$
$197$	66.0000	0.0238696	0.0119348	+	0.999929i	$0.496201\pi$
$198$	−297.000	−0.106600
$199$	−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
$201$	−2772.00	−0.972745
$202$	−1242.00	−0.432608
$203$	−182.000	−0.0629256
$204$	−222.000	−0.0761917
$205$	140.000	0.0476977
$206$	2952.00	0.998425
$207$	−1332.00	−0.447248
$208$	−142.000	−0.0473362
$209$	0	0
$210$	882.000	0.289828
$211$	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
$213$	−2244.00	−0.721861
$214$	−5748.00	−1.83610
$215$	−2912.00	−0.923706
$216$	−567.000	−0.178609
$217$	−784.000	−0.245260
$218$	−1482.00	−0.460430
$219$	−690.000	−0.212904
$220$	154.000	0.0471940
$221$	−148.000	−0.0450478
$222$	−882.000	−0.266648
$223$	−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
$225$	639.000	0.189333
$226$	−3714.00	−1.09315
$227$	−2380.00	−0.695886	−0.347943	−	0.937516i	$-0.613120\pi$
$227$	−2380.00	−0.695886	−0.347943	+	0.937516i	$0.613120\pi$
$228$	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
$231$	231.000	0.0657952
$232$	−546.000	−0.154511
$233$	−5010.00	−1.40865	−0.704326	−	0.709876i	$-0.748751\pi$
$233$	−5010.00	−1.40865	−0.704326	+	0.709876i	$0.748751\pi$
$234$	54.0000	0.0150859
$235$	−6440.00	−1.78766
$236$	−204.000	−0.0562681
$237$	−1368.00	−0.374942
$238$	1554.00	0.423239
$239$	704.000	0.190535	0.0952677	−	0.995452i	$-0.469629\pi$
$239$	704.000	0.190535	0.0952677	+	0.995452i	$0.469629\pi$
$240$	2982.00	0.802030
$241$	−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
$243$	243.000	0.0641500
$244$	178.000	0.0467020
$245$	−686.000	−0.178885
$246$	−90.0000	−0.0233260
$247$	0	0
$248$	−2352.00	−0.602226
$249$	−684.000	−0.174083
$250$	2268.00	0.573764
$251$	6156.00	1.54806	0.774030	−	0.633148i	$-0.218238\pi$
$251$	6156.00	1.54806	0.774030	+	0.633148i	$0.218238\pi$
$252$	−63.0000	−0.0157485
$253$	1628.00	0.404551
$254$	3048.00	0.752947
$255$	3108.00	0.763257
$256$	1513.00	0.369385
$257$	−3054.00	−0.741258	−0.370629	−	0.928781i	$-0.620858\pi$
$257$	−3054.00	−0.741258	−0.370629	+	0.928781i	$0.620858\pi$
$258$	1872.00	0.451727
$259$	686.000	0.164579
$260$	−28.0000	−0.00667879
$261$	234.000	0.0554952
$262$	−7668.00	−1.80813
$263$	−4944.00	−1.15916	−0.579582	−	0.814914i	$-0.696784\pi$
$263$	−4944.00	−1.15916	−0.579582	+	0.814914i	$0.696784\pi$
$264$	693.000	0.161558
$265$	−3612.00	−0.837296
$266$	0	0
$267$	−594.000	−0.136151
$268$	−924.000	−0.210606
$269$	7602.00	1.72306	0.861528	−	0.507710i	$-0.169508\pi$
$269$	7602.00	1.72306	0.861528	+	0.507710i	$0.169508\pi$
$270$	−1134.00	−0.255604
$271$	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
$273$	−42.0000	−0.00931119
$274$	4878.00	1.07551
$275$	−781.000	−0.171258
$276$	−444.000	−0.0968321
$277$	−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
$279$	1008.00	0.216299
$280$	−2058.00	−0.439247
$281$	3454.00	0.733268	0.366634	−	0.930365i	$-0.380510\pi$
$281$	3454.00	0.733268	0.366634	+	0.930365i	$0.380510\pi$
$282$	4140.00	0.874232
$283$	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
$288$	−405.000	−0.0828641
$289$	563.000	0.114594
$290$	−1092.00	−0.221119
$291$	1686.00	0.339639
$292$	−230.000	−0.0460950
$293$	5682.00	1.13292	0.566461	−	0.824089i	$-0.308312\pi$
$293$	5682.00	1.13292	0.566461	+	0.824089i	$0.308312\pi$
$294$	441.000	0.0874818
$295$	2856.00	0.563670
$296$	2058.00	0.404118
$297$	−297.000	−0.0580259
$298$	−10410.0	−2.02361
$299$	−296.000	−0.0572512
$300$	213.000	0.0409919
$301$	−1456.00	−0.278812
$302$	4176.00	0.795701
$303$	−1242.00	−0.235482
$304$	0	0
$305$	−2492.00	−0.467841
$306$	−1998.00	−0.373262
$307$	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
$309$	2952.00	0.543474
$310$	−4704.00	−0.861836
$311$	7972.00	1.45354	0.726770	−	0.686881i	$-0.241021\pi$
$311$	7972.00	1.45354	0.726770	+	0.686881i	$0.241021\pi$
$312$	−126.000	−0.0228633
$313$	−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
$315$	882.000	0.157762
$316$	−456.000	−0.0811772
$317$	−6246.00	−1.10666	−0.553329	−	0.832963i	$-0.686643\pi$
$317$	−6246.00	−1.10666	−0.553329	+	0.832963i	$0.686643\pi$
$318$	2322.00	0.409469
$319$	−286.000	−0.0501973
$320$	−6062.00	−1.05899
$321$	−5748.00	−0.999446
$322$	3108.00	0.537895
$323$	0	0
$324$	81.0000	0.0138889
$325$	142.000	0.0242361
$326$	−4164.00	−0.707431
$327$	−1482.00	−0.250626
$328$	210.000	0.0353516
$329$	−3220.00	−0.539588
$330$	1386.00	0.231202
$331$	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
$333$	−882.000	−0.145145
$334$	5064.00	0.829610
$335$	12936.0	2.10976
$336$	1491.00	0.242085
$337$	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
$339$	−3714.00	−0.595035
$340$	1036.00	0.165250
$341$	−1232.00	−0.195650
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−4368.00	−0.684613
$345$	6216.00	0.970024
$346$	−354.000	−0.0550033
$347$	10732.0	1.66030	0.830150	−	0.557540i	$-0.188255\pi$
$347$	10732.0	1.66030	0.830150	+	0.557540i	$0.188255\pi$
$348$	78.0000	0.0120151
$349$	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
$351$	54.0000	0.00821170
$352$	495.000	0.0749534
$353$	−2526.00	−0.380865	−0.190433	−	0.981700i	$-0.560989\pi$
$353$	−2526.00	−0.380865	−0.190433	+	0.981700i	$0.560989\pi$
$354$	−1836.00	−0.275656
$355$	10472.0	1.56562
$356$	−198.000	−0.0294775
$357$	1554.00	0.230382
$358$	−9012.00	−1.33044
$359$	8080.00	1.18787	0.593936	−	0.804512i	$-0.297573\pi$
$359$	8080.00	1.18787	0.593936	+	0.804512i	$0.297573\pi$
$360$	2646.00	0.387379
$361$	−6859.00	−1.00000
$362$	2346.00	0.340616
$363$	363.000	0.0524864
$364$	−14.0000	−0.00201593
$365$	3220.00	0.461760
$366$	1602.00	0.228792
$367$	−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
$369$	−90.0000	−0.0126971
$370$	4116.00	0.578326
$371$	−1806.00	−0.252730
$372$	336.000	0.0468301
$373$	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
$375$	2268.00	0.312317
$376$	−9660.00	−1.32494
$377$	52.0000	0.00710381
$378$	−567.000	−0.0771517
$379$	−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
$381$	3048.00	0.409852
$382$	−14820.0	−1.98497
$383$	−4052.00	−0.540594	−0.270297	−	0.962777i	$-0.587122\pi$
$383$	−4052.00	−0.540594	−0.270297	+	0.962777i	$0.587122\pi$
$384$	4977.00	0.661410
$385$	−1078.00	−0.142701
$386$	9870.00	1.30148
$387$	1872.00	0.245889
$388$	562.000	0.0735341
$389$	4554.00	0.593565	0.296783	−	0.954945i	$-0.404086\pi$
$389$	4554.00	0.593565	0.296783	+	0.954945i	$0.404086\pi$
$390$	−252.000	−0.0327193
$391$	10952.0	1.41654
$392$	−1029.00	−0.132583
$393$	−7668.00	−0.984222
$394$	198.000	0.0253175
$395$	6384.00	0.813200
$396$	−99.0000	−0.0125630
$397$	−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	$-0.225066\pi$
$401$	12210.0	1.52054	0.760272	+	0.649605i	$0.225066\pi$
$402$	−8316.00	−1.03175
$403$	224.000	0.0276879
$404$	−414.000	−0.0509833
$405$	−1134.00	−0.139133
$406$	−546.000	−0.0667427
$407$	1078.00	0.131289
$408$	4662.00	0.565695
$409$	−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
$411$	4878.00	0.585436
$412$	984.000	0.117666
$413$	1428.00	0.170139
$414$	−3996.00	−0.474378
$415$	3192.00	0.377564
$416$	−90.0000	−0.0106072
$417$	1608.00	0.188835
$418$	0	0
$419$	−16780.0	−1.95646	−0.978230	−	0.207524i	$-0.933459\pi$
$419$	−16780.0	−1.95646	−0.978230	+	0.207524i	$0.933459\pi$
$420$	294.000	0.0341565
$421$	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
$423$	4140.00	0.475872
$424$	−5418.00	−0.620569
$425$	−5254.00	−0.599662
$426$	−6732.00	−0.765649
$427$	−1246.00	−0.141214
$428$	−1916.00	−0.216386
$429$	−66.0000	−0.00742776
$430$	−8736.00	−0.979738
$431$	9896.00	1.10597	0.552986	−	0.833191i	$-0.313488\pi$
$431$	9896.00	1.10597	0.552986	+	0.833191i	$0.313488\pi$
$432$	−1917.00	−0.213499
$433$	−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
$435$	−1092.00	−0.120362
$436$	−494.000	−0.0542622
$437$	0	0
$438$	−2070.00	−0.225818
$439$	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
$441$	441.000	0.0476190
$442$	−444.000	−0.0477804
$443$	1668.00	0.178892	0.0894459	−	0.995992i	$-0.471490\pi$
$443$	1668.00	0.178892	0.0894459	+	0.995992i	$0.471490\pi$
$444$	−294.000	−0.0314248
$445$	2772.00	0.295293
$446$	−7224.00	−0.766965
$447$	−10410.0	−1.10151
$448$	−3031.00	−0.319646
$449$	−10374.0	−1.09038	−0.545189	−	0.838313i	$-0.683542\pi$
$449$	−10374.0	−1.09038	−0.545189	+	0.838313i	$0.683542\pi$
$450$	1917.00	0.200818
$451$	110.000	0.0114849
$452$	−1238.00	−0.128829
$453$	4176.00	0.433125
$454$	−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
$459$	−1998.00	−0.203178
$460$	2072.00	0.210016
$461$	1770.00	0.178822	0.0894112	−	0.995995i	$-0.471501\pi$
$461$	1770.00	0.178822	0.0894112	+	0.995995i	$0.471501\pi$
$462$	693.000	0.0697863
$463$	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
$465$	−4704.00	−0.469124
$466$	−15030.0	−1.49410
$467$	−19836.0	−1.96553	−0.982763	−	0.184870i	$-0.940814\pi$
$467$	−19836.0	−1.96553	−0.982763	+	0.184870i	$0.940814\pi$
$468$	18.0000	0.00177789
$469$	6468.00	0.636811
$470$	−19320.0	−1.89610
$471$	8274.00	0.809439
$472$	4284.00	0.417769
$473$	−2288.00	−0.222415
$474$	−4104.00	−0.397686
$475$	0	0
$476$	518.000	0.0498792
$477$	2322.00	0.222887
$478$	2112.00	0.202093
$479$	−4520.00	−0.431157	−0.215578	−	0.976487i	$-0.569164\pi$
$479$	−4520.00	−0.431157	−0.215578	+	0.976487i	$0.569164\pi$
$480$	1890.00	0.179721
$481$	−196.000	−0.0185797
$482$	−18306.0	−1.72991
$483$	3108.00	0.292793
$484$	121.000	0.0113636
$485$	−7868.00	−0.736634
$486$	729.000	0.0680414
$487$	−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
$489$	−4164.00	−0.385077
$490$	−2058.00	−0.189737
$491$	17988.0	1.65333	0.826667	−	0.562691i	$-0.190234\pi$
$491$	17988.0	1.65333	0.826667	+	0.562691i	$0.190234\pi$
$492$	−30.0000	−0.00274899
$493$	−1924.00	−0.175766
$494$	0	0
$495$	1386.00	0.125851
$496$	−7952.00	−0.719870
$497$	5236.00	0.472569
$498$	−2052.00	−0.184643
$499$	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
$501$	5064.00	0.451583
$502$	18468.0	1.64197
$503$	−18552.0	−1.64452	−0.822259	−	0.569113i	$-0.807287\pi$
$503$	−18552.0	−1.64452	−0.822259	+	0.569113i	$0.807287\pi$
$504$	1323.00	0.116927
$505$	5796.00	0.510730
$506$	4884.00	0.429091
$507$	−6579.00	−0.576299
$508$	1016.00	0.0887357
$509$	−9326.00	−0.812117	−0.406059	−	0.913847i	$-0.633097\pi$
$509$	−9326.00	−0.812117	−0.406059	+	0.913847i	$0.633097\pi$
$510$	9324.00	0.809556
$511$	1610.00	0.139378
$512$	−8733.00	−0.753804
$513$	0	0
$514$	−9162.00	−0.786223
$515$	−13776.0	−1.17872
$516$	624.000	0.0532366
$517$	−5060.00	−0.430442
$518$	2058.00	0.174562
$519$	−354.000	−0.0299400
$520$	588.000	0.0495875
$521$	−14038.0	−1.18045	−0.590226	−	0.807238i	$-0.700962\pi$
$521$	−14038.0	−1.18045	−0.590226	+	0.807238i	$0.700962\pi$
$522$	702.000	0.0588615
$523$	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
$525$	−1491.00	−0.123948
$526$	−14832.0	−1.22948
$527$	−8288.00	−0.685068
$528$	2343.00	0.193117
$529$	9737.00	0.800279
$530$	−10836.0	−0.888086
$531$	−1836.00	−0.150048
$532$	0	0
$533$	−20.0000	−0.00162532
$534$	−1782.00	−0.144410
$535$	26824.0	2.16767
$536$	19404.0	1.56367
$537$	−9012.00	−0.724202
$538$	22806.0	1.82758
$539$	−539.000	−0.0430730
$540$	−378.000	−0.0301232
$541$	−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
$543$	2346.00	0.185408
$544$	3330.00	0.262450
$545$	6916.00	0.543576
$546$	−126.000	−0.00987601
$547$	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
$549$	1602.00	0.124539
$550$	−2343.00	−0.181647
$551$	0	0
$552$	9324.00	0.718942
$553$	3192.00	0.245457
$554$	−9882.00	−0.757845
$555$	4116.00	0.314801
$556$	536.000	0.0408839
$557$	12778.0	0.972031	0.486015	−	0.873950i	$-0.338450\pi$
$557$	12778.0	0.972031	0.486015	+	0.873950i	$0.338450\pi$
$558$	3024.00	0.229420
$559$	416.000	0.0314757
$560$	−6958.00	−0.525052
$561$	2442.00	0.183781
$562$	10362.0	0.777748
$563$	22444.0	1.68011	0.840055	−	0.542501i	$-0.182523\pi$
$563$	22444.0	1.68011	0.840055	+	0.542501i	$0.182523\pi$
$564$	1380.00	0.103029
$565$	17332.0	1.29055
$566$	25560.0	1.89817
$567$	−567.000	−0.0419961
$568$	15708.0	1.16038
$569$	−26874.0	−1.97999	−0.989997	−	0.141088i	$-0.954940\pi$
$569$	−26874.0	−1.97999	−0.989997	+	0.141088i	$0.954940\pi$
$570$	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
$573$	−14820.0	−1.08048
$574$	210.000	0.0152704
$575$	−10508.0	−0.762111
$576$	3897.00	0.281901
$577$	−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
$579$	9870.00	0.708434
$580$	−364.000	−0.0260591
$581$	1596.00	0.113964
$582$	5058.00	0.360242
$583$	−2838.00	−0.201609
$584$	4830.00	0.342238
$585$	−252.000	−0.0178101
$586$	17046.0	1.20164
$587$	8340.00	0.586420	0.293210	−	0.956048i	$-0.405276\pi$
$587$	8340.00	0.586420	0.293210	+	0.956048i	$0.405276\pi$
$588$	147.000	0.0103098
$589$	0	0
$590$	8568.00	0.597863
$591$	198.000	0.0137811
$592$	6958.00	0.483061
$593$	−10818.0	−0.749143	−0.374572	−	0.927198i	$-0.622210\pi$
$593$	−10818.0	−0.749143	−0.374572	+	0.927198i	$0.622210\pi$
$594$	−891.000	−0.0615457
$595$	−7252.00	−0.499669
$596$	−3470.00	−0.238484
$597$	−8016.00	−0.549536
$598$	−888.000	−0.0607241
$599$	−3348.00	−0.228373	−0.114187	−	0.993459i	$-0.536426\pi$
$599$	−3348.00	−0.228373	−0.114187	+	0.993459i	$0.536426\pi$
$600$	−4473.00	−0.304349
$601$	−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
$603$	−8316.00	−0.561615
$604$	1392.00	0.0937743
$605$	−1694.00	−0.113836
$606$	−3726.00	−0.249766
$607$	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
$609$	−546.000	−0.0363301
$610$	−7476.00	−0.496220
$611$	920.000	0.0609152
$612$	−666.000	−0.0439893
$613$	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
$615$	420.000	0.0275383
$616$	−1617.00	−0.105764
$617$	26226.0	1.71121	0.855607	−	0.517626i	$-0.173184\pi$
$617$	26226.0	1.71121	0.855607	+	0.517626i	$0.173184\pi$
$618$	8856.00	0.576441
$619$	−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
$621$	−3996.00	−0.258219
$622$	23916.0	1.54171
$623$	1386.00	0.0891315
$624$	−426.000	−0.0273296
$625$	−19459.0	−1.24538
$626$	−9474.00	−0.604884
$627$	0	0
$628$	2758.00	0.175249
$629$	7252.00	0.459708
$630$	2646.00	0.167332
$631$	−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
$633$	8976.00	0.563608
$634$	−18738.0	−1.17379
$635$	−14224.0	−0.888917
$636$	774.000	0.0482564
$637$	98.0000	0.00609561
$638$	−858.000	−0.0532422
$639$	−6732.00	−0.416767
$640$	−23226.0	−1.43451
$641$	−2550.00	−0.157128	−0.0785639	−	0.996909i	$-0.525033\pi$
$641$	−2550.00	−0.157128	−0.0785639	+	0.996909i	$0.525033\pi$
$642$	−17244.0	−1.06007
$643$	−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
$645$	−8736.00	−0.533302
$646$	0	0
$647$	20436.0	1.24177	0.620883	−	0.783904i	$-0.286774\pi$
$647$	20436.0	1.24177	0.620883	+	0.783904i	$0.286774\pi$
$648$	−1701.00	−0.103120
$649$	2244.00	0.135724
$650$	426.000	0.0257063
$651$	−2352.00	−0.141601
$652$	−1388.00	−0.0833716
$653$	−20062.0	−1.20228	−0.601138	−	0.799145i	$-0.705286\pi$
$653$	−20062.0	−1.20228	−0.601138	+	0.799145i	$0.705286\pi$
$654$	−4446.00	−0.265829
$655$	35784.0	2.13465
$656$	710.000	0.0422574
$657$	−2070.00	−0.122920
$658$	−9660.00	−0.572319
$659$	−13324.0	−0.787601	−0.393801	−	0.919196i	$-0.628840\pi$
$659$	−13324.0	−0.787601	−0.393801	+	0.919196i	$0.628840\pi$
$660$	462.000	0.0272475
$661$	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
$663$	−444.000	−0.0260083
$664$	4788.00	0.279835
$665$	0	0
$666$	−2646.00	−0.153950
$667$	−3848.00	−0.223381
$668$	1688.00	0.0977705
$669$	−7224.00	−0.417483
$670$	38808.0	2.23774
$671$	−1958.00	−0.112649
$672$	945.000	0.0542473
$673$	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
$675$	1917.00	0.109312
$676$	−2193.00	−0.124772
$677$	18658.0	1.05921	0.529605	−	0.848244i	$-0.322340\pi$
$677$	18658.0	1.05921	0.529605	+	0.848244i	$0.322340\pi$
$678$	−11142.0	−0.631130
$679$	−3934.00	−0.222346
$680$	−21756.0	−1.22692
$681$	−7140.00	−0.401770
$682$	−3696.00	−0.207518
$683$	28092.0	1.57381	0.786904	−	0.617076i	$-0.211683\pi$
$683$	28092.0	1.57381	0.786904	+	0.617076i	$0.211683\pi$
$684$	0	0
$685$	−22764.0	−1.26973
$686$	−1029.00	−0.0572703
$687$	−13494.0	−0.749386
$688$	−14768.0	−0.818350
$689$	516.000	0.0285313
$690$	18648.0	1.02887
$691$	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
$693$	693.000	0.0379869
$694$	32196.0	1.76101
$695$	−7504.00	−0.409558
$696$	−1638.00	−0.0892072
$697$	740.000	0.0402145
$698$	22110.0	1.19896
$699$	−15030.0	−0.813286
$700$	−497.000	−0.0268355
$701$	21498.0	1.15830	0.579150	−	0.815221i	$-0.303385\pi$
$701$	21498.0	1.15830	0.579150	+	0.815221i	$0.303385\pi$
$702$	162.000	0.00870982
$703$	0	0
$704$	−4763.00	−0.254989
$705$	−19320.0	−1.03210
$706$	−7578.00	−0.403969
$707$	2898.00	0.154159
$708$	−612.000	−0.0324864
$709$	−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
$711$	−4104.00	−0.216473
$712$	4158.00	0.218859
$713$	−16576.0	−0.870654
$714$	4662.00	0.244357
$715$	308.000	0.0161099
$716$	−3004.00	−0.156794
$717$	2112.00	0.110006
$718$	24240.0	1.25993
$719$	31796.0	1.64922	0.824611	−	0.565700i	$-0.191394\pi$
$719$	31796.0	1.64922	0.824611	+	0.565700i	$0.191394\pi$
$720$	8946.00	0.463052
$721$	−6888.00	−0.355787
$722$	−20577.0	−1.06066
$723$	−18306.0	−0.941642
$724$	782.000	0.0401420
$725$	1846.00	0.0945638
$726$	1089.00	0.0556702
$727$	−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
$729$	729.000	0.0370370
$730$	9660.00	0.489771
$731$	−15392.0	−0.778788
$732$	534.000	0.0269634
$733$	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
$735$	−2058.00	−0.103280
$736$	6660.00	0.333547
$737$	10164.0	0.508000
$738$	−270.000	−0.0134673
$739$	−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	$-0.624211\pi$
$743$	−15408.0	−0.760787	−0.380393	+	0.924825i	$0.624211\pi$
$744$	−7056.00	−0.347696
$745$	48580.0	2.38904
$746$	3318.00	0.162843
$747$	−2052.00	−0.100507
$748$	814.000	0.0397898
$749$	13412.0	0.654291
$750$	6804.00	0.331263
$751$	−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
$753$	18468.0	0.893773
$754$	156.000	0.00753473
$755$	−19488.0	−0.939392
$756$	−189.000	−0.00909241
$757$	−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
$759$	4884.00	0.233568
$760$	0	0
$761$	17118.0	0.815410	0.407705	−	0.913114i	$-0.366329\pi$
$761$	17118.0	0.815410	0.407705	+	0.913114i	$0.366329\pi$
$762$	9144.00	0.434714
$763$	3458.00	0.164073
$764$	−4940.00	−0.233931
$765$	9324.00	0.440667
$766$	−12156.0	−0.573387
$767$	−408.000	−0.0192073
$768$	4539.00	0.213264
$769$	−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
$771$	−9162.00	−0.427965
$772$	3290.00	0.153380
$773$	−12870.0	−0.598838	−0.299419	−	0.954122i	$-0.596793\pi$
$773$	−12870.0	−0.598838	−0.299419	+	0.954122i	$0.596793\pi$
$774$	5616.00	0.260805
$775$	7952.00	0.368573
$776$	−11802.0	−0.545963
$777$	2058.00	0.0950197
$778$	13662.0	0.629571
$779$	0	0
$780$	−84.0000	−0.00385600
$781$	8228.00	0.376979
$782$	32856.0	1.50247
$783$	702.000	0.0320401
$784$	−3479.00	−0.158482
$785$	−38612.0	−1.75557
$786$	−23004.0	−1.04393
$787$	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
$789$	−14832.0	−0.669244
$790$	19152.0	0.862529
$791$	8666.00	0.389542
$792$	2079.00	0.0932753
$793$	356.000	0.0159419
$794$	−19998.0	−0.893832
$795$	−10836.0	−0.483413
$796$	−2672.00	−0.118978
$797$	9754.00	0.433506	0.216753	−	0.976226i	$-0.430453\pi$
$797$	9754.00	0.433506	0.216753	+	0.976226i	$0.430453\pi$
$798$	0	0
$799$	−34040.0	−1.50719
$800$	−3195.00	−0.141200
$801$	−1782.00	−0.0786066
$802$	36630.0	1.61278
$803$	2530.00	0.111185
$804$	−2772.00	−0.121593
$805$	−14504.0	−0.635030
$806$	672.000	0.0293675
$807$	22806.0	0.994807
$808$	8694.00	0.378532
$809$	−4554.00	−0.197911	−0.0989556	−	0.995092i	$-0.531550\pi$
$809$	−4554.00	−0.197911	−0.0989556	+	0.995092i	$0.531550\pi$
$810$	−3402.00	−0.147573
$811$	−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
$813$	13656.0	0.589098
$814$	3234.00	0.139253
$815$	19432.0	0.835182
$816$	15762.0	0.676201
$817$	0	0
$818$	−30570.0	−1.30667
$819$	−126.000	−0.00537582
$820$	140.000	0.00596221
$821$	29490.0	1.25360	0.626802	−	0.779179i	$-0.284364\pi$
$821$	29490.0	1.25360	0.626802	+	0.779179i	$0.284364\pi$
$822$	14634.0	0.620948
$823$	−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
$825$	−2343.00	−0.0988761
$826$	4284.00	0.180459
$827$	36604.0	1.53911	0.769556	−	0.638579i	$-0.220478\pi$
$827$	36604.0	1.53911	0.769556	+	0.638579i	$0.220478\pi$
$828$	−1332.00	−0.0559060
$829$	−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
$831$	−9882.00	−0.412518
$832$	866.000	0.0360855
$833$	−3626.00	−0.150820
$834$	4824.00	0.200290
$835$	−23632.0	−0.979424
$836$	0	0
$837$	3024.00	0.124880
$838$	−50340.0	−2.07514
$839$	3852.00	0.158505	0.0792526	−	0.996855i	$-0.474747\pi$
$839$	3852.00	0.158505	0.0792526	+	0.996855i	$0.474747\pi$
$840$	−6174.00	−0.253599
$841$	−23713.0	−0.972283
$842$	27642.0	1.13136
$843$	10362.0	0.423353
$844$	2992.00	0.122025
$845$	30702.0	1.24992
$846$	12420.0	0.504738
$847$	−847.000	−0.0343604
$848$	−18318.0	−0.741796
$849$	25560.0	1.03324
$850$	−15762.0	−0.636038
$851$	14504.0	0.584243
$852$	−2244.00	−0.0902326
$853$	−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	$-0.667500\pi$
$857$	−25202.0	−1.00453	−0.502266	+	0.864713i	$0.667500\pi$
$858$	−198.000	−0.00787833
$859$	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
$861$	210.000	0.00831217
$862$	29688.0	1.17306
$863$	26876.0	1.06010	0.530052	−	0.847965i	$-0.322172\pi$
$863$	26876.0	1.06010	0.530052	+	0.847965i	$0.322172\pi$
$864$	−1215.00	−0.0478416
$865$	1652.00	0.0649361
$866$	−26634.0	−1.04510
$867$	1689.00	0.0661608
$868$	−784.000	−0.0306575
$869$	5016.00	0.195807
$870$	−3276.00	−0.127663
$871$	−1848.00	−0.0718910
$872$	10374.0	0.402876
$873$	5058.00	0.196091
$874$	0	0
$875$	−5292.00	−0.204460
$876$	−690.000	−0.0266129
$877$	−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
$879$	17046.0	0.654093
$880$	−10934.0	−0.418847
$881$	−27222.0	−1.04101	−0.520507	−	0.853858i	$-0.674257\pi$
$881$	−27222.0	−1.04101	−0.520507	+	0.853858i	$0.674257\pi$
$882$	1323.00	0.0505076
$883$	−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
$885$	8568.00	0.325435
$886$	5004.00	0.189743
$887$	−8944.00	−0.338568	−0.169284	−	0.985567i	$-0.554146\pi$
$887$	−8944.00	−0.338568	−0.169284	+	0.985567i	$0.554146\pi$
$888$	6174.00	0.233317
$889$	−7112.00	−0.268311
$890$	8316.00	0.313206
$891$	−891.000	−0.0335013
$892$	−2408.00	−0.0903877
$893$	0	0
$894$	−31230.0	−1.16833
$895$	42056.0	1.57070
$896$	−11613.0	−0.432995
$897$	−888.000	−0.0330540
$898$	−31122.0	−1.15652
$899$	2912.00	0.108032
$900$	639.000	0.0236667
$901$	−19092.0	−0.705934
$902$	330.000	0.0121816
$903$	−4368.00	−0.160972
$904$	25998.0	0.956505
$905$	−10948.0	−0.402126
$906$	12528.0	0.459398
$907$	−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
$909$	−3726.00	−0.135956
$910$	588.000	0.0214198
$911$	−7812.00	−0.284109	−0.142054	−	0.989859i	$-0.545371\pi$
$911$	−7812.00	−0.284109	−0.142054	+	0.989859i	$0.545371\pi$
$912$	0	0
$913$	2508.00	0.0909120
$914$	−9138.00	−0.330698
$915$	−7476.00	−0.270108
$916$	−4498.00	−0.162247
$917$	17892.0	0.644325
$918$	−5994.00	−0.215503
$919$	−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
$921$	3120.00	0.111626
$922$	5310.00	0.189670
$923$	−1496.00	−0.0533493
$924$	231.000	0.00822440
$925$	−6958.00	−0.247327
$926$	36264.0	1.28694
$927$	8856.00	0.313775
$928$	−1170.00	−0.0413870
$929$	−15246.0	−0.538434	−0.269217	−	0.963080i	$-0.586765\pi$
$929$	−15246.0	−0.538434	−0.269217	+	0.963080i	$0.586765\pi$
$930$	−14112.0	−0.497581
$931$	0	0
$932$	−5010.00	−0.176082
$933$	23916.0	0.839201
$934$	−59508.0	−2.08476
$935$	−11396.0	−0.398598
$936$	−378.000	−0.0132001
$937$	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
$939$	−9474.00	−0.329257
$940$	−6440.00	−0.223457
$941$	−10006.0	−0.346638	−0.173319	−	0.984866i	$-0.555449\pi$
$941$	−10006.0	−0.346638	−0.173319	+	0.984866i	$0.555449\pi$
$942$	24822.0	0.858540
$943$	1480.00	0.0511086
$944$	14484.0	0.499379
$945$	2646.00	0.0910840
$946$	−6864.00	−0.235907
$947$	−5540.00	−0.190101	−0.0950506	−	0.995472i	$-0.530301\pi$
$947$	−5540.00	−0.190101	−0.0950506	+	0.995472i	$0.530301\pi$
$948$	−1368.00	−0.0468677
$949$	−460.000	−0.0157347
$950$	0	0
$951$	−18738.0	−0.638929
$952$	−10878.0	−0.370334
$953$	−32522.0	−1.10545	−0.552723	−	0.833365i	$-0.686411\pi$
$953$	−32522.0	−1.10545	−0.552723	+	0.833365i	$0.686411\pi$
$954$	6966.00	0.236407
$955$	69160.0	2.34342
$956$	704.000	0.0238169
$957$	−858.000	−0.0289814
$958$	−13560.0	−0.457311
$959$	−11382.0	−0.383258
$960$	−18186.0	−0.611407
$961$	−17247.0	−0.578933
$962$	−588.000	−0.0197067
$963$	−17244.0	−0.577030
$964$	−6102.00	−0.203872
$965$	−46060.0	−1.53650
$966$	9324.00	0.310554
$967$	−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	$-0.219729\pi$
$971$	46660.0	1.54211	0.771056	+	0.636767i	$0.219729\pi$
$972$	243.000	0.00801875
$973$	−3752.00	−0.123621
$974$	−5304.00	−0.174488
$975$	426.000	0.0139927
$976$	−12638.0	−0.414480
$977$	11506.0	0.376775	0.188388	−	0.982095i	$-0.439674\pi$
$977$	11506.0	0.376775	0.188388	+	0.982095i	$0.439674\pi$
$978$	−12492.0	−0.408436
$979$	2178.00	0.0711023
$980$	−686.000	−0.0223607
$981$	−4446.00	−0.144699
$982$	53964.0	1.75363
$983$	−14844.0	−0.481638	−0.240819	−	0.970570i	$-0.577416\pi$
$983$	−14844.0	−0.481638	−0.240819	+	0.970570i	$0.577416\pi$
$984$	630.000	0.0204102
$985$	−924.000	−0.0298894
$986$	−5772.00	−0.186428
$987$	−9660.00	−0.311531
$988$	0	0
$989$	−30784.0	−0.989762
$990$	4158.00	0.133485
$991$	−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
$993$	23700.0	0.757399
$994$	15708.0	0.501235
$995$	37408.0	1.19187
$996$	−684.000	−0.0217604
$997$	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
$999$	−2646.00	−0.0837995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.4.a.d.1.1	✓	1
3.2	odd	2		693.4.a.c.1.1		1
7.6	odd	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	1.1	even	1	trivial
693.4.a.c.1.1		1	3.2	odd	2
1617.4.a.f.1.1		1	7.6	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:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	231.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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