LMFDB - Embedded newform 128.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,4,Mod(1,128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(128, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("128.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$128 = 2^{7}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	128.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:	$7.55224448073$
Analytic rank:	$0$
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	$\chi$	$=$	128.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+2.00000 q^{3} +6.00000 q^{5} +20.0000 q^{7} -23.0000 q^{9} +14.0000 q^{11} +54.0000 q^{13} +12.0000 q^{15} -66.0000 q^{17} +162.000 q^{19} +40.0000 q^{21} +172.000 q^{23} -89.0000 q^{25} -100.000 q^{27} -2.00000 q^{29} -128.000 q^{31} +28.0000 q^{33} +120.000 q^{35} +158.000 q^{37} +108.000 q^{39} +202.000 q^{41} -298.000 q^{43} -138.000 q^{45} -408.000 q^{47} +57.0000 q^{49} -132.000 q^{51} -690.000 q^{53} +84.0000 q^{55} +324.000 q^{57} -322.000 q^{59} -298.000 q^{61} -460.000 q^{63} +324.000 q^{65} +202.000 q^{67} +344.000 q^{69} -700.000 q^{71} -418.000 q^{73} -178.000 q^{75} +280.000 q^{77} +744.000 q^{79} +421.000 q^{81} -678.000 q^{83} -396.000 q^{85} -4.00000 q^{87} -82.0000 q^{89} +1080.00 q^{91} -256.000 q^{93} +972.000 q^{95} -1122.00 q^{97} -322.000 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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	0	0
$2$	0	0
$3$	2.00000	0.384900	0.192450	−	0.981307i	$-0.438357\pi$
$3$	2.00000	0.384900	0.192450	+	0.981307i	$0.438357\pi$
$4$	0	0
$5$	6.00000	0.536656	0.268328	−	0.963328i	$-0.413529\pi$
$5$	6.00000	0.536656	0.268328	+	0.963328i	$0.413529\pi$
$6$	0	0
$7$	20.0000	1.07990	0.539949	−	0.841698i	$-0.318443\pi$
$7$	20.0000	1.07990	0.539949	+	0.841698i	$0.318443\pi$
$8$	0	0
$9$	−23.0000	−0.851852
$10$	0	0
$11$	14.0000	0.383742	0.191871	−	0.981420i	$-0.438545\pi$
$11$	14.0000	0.383742	0.191871	+	0.981420i	$0.438545\pi$
$12$	0	0
$13$	54.0000	1.15207	0.576035	−	0.817425i	$-0.304599\pi$
$13$	54.0000	1.15207	0.576035	+	0.817425i	$0.304599\pi$
$14$	0	0
$15$	12.0000	0.206559
$16$	0	0
$17$	−66.0000	−0.941609	−0.470804	−	0.882238i	$-0.656036\pi$
$17$	−66.0000	−0.941609	−0.470804	+	0.882238i	$0.656036\pi$
$18$	0	0
$19$	162.000	1.95607	0.978035	−	0.208438i	$-0.0668381\pi$
$19$	162.000	1.95607	0.978035	+	0.208438i	$0.0668381\pi$
$20$	0	0
$21$	40.0000	0.415653
$22$	0	0
$23$	172.000	1.55933	0.779663	−	0.626200i	$-0.215391\pi$
$23$	172.000	1.55933	0.779663	+	0.626200i	$0.215391\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	0	0
$27$	−100.000	−0.712778
$28$	0	0
$29$	−2.00000	−0.0128066	−0.00640329	−	0.999979i	$-0.502038\pi$
$29$	−2.00000	−0.0128066	−0.00640329	+	0.999979i	$0.502038\pi$
$30$	0	0
$31$	−128.000	−0.741596	−0.370798	−	0.928714i	$-0.620916\pi$
$31$	−128.000	−0.741596	−0.370798	+	0.928714i	$0.620916\pi$
$32$	0	0
$33$	28.0000	0.147702
$34$	0	0
$35$	120.000	0.579534
$36$	0	0
$37$	158.000	0.702028	0.351014	−	0.936370i	$-0.385837\pi$
$37$	158.000	0.702028	0.351014	+	0.936370i	$0.385837\pi$
$38$	0	0
$39$	108.000	0.443432
$40$	0	0
$41$	202.000	0.769441	0.384721	−	0.923033i	$-0.374298\pi$
$41$	202.000	0.769441	0.384721	+	0.923033i	$0.374298\pi$
$42$	0	0
$43$	−298.000	−1.05685	−0.528425	−	0.848980i	$-0.677217\pi$
$43$	−298.000	−1.05685	−0.528425	+	0.848980i	$0.677217\pi$
$44$	0	0
$45$	−138.000	−0.457152
$46$	0	0
$47$	−408.000	−1.26623	−0.633116	−	0.774057i	$-0.718224\pi$
$47$	−408.000	−1.26623	−0.633116	+	0.774057i	$0.718224\pi$
$48$	0	0
$49$	57.0000	0.166181
$50$	0	0
$51$	−132.000	−0.362425
$52$	0	0
$53$	−690.000	−1.78828	−0.894140	−	0.447788i	$-0.852212\pi$
$53$	−690.000	−1.78828	−0.894140	+	0.447788i	$0.852212\pi$
$54$	0	0
$55$	84.0000	0.205937
$56$	0	0
$57$	324.000	0.752892
$58$	0	0
$59$	−322.000	−0.710523	−0.355261	−	0.934767i	$-0.615608\pi$
$59$	−322.000	−0.710523	−0.355261	+	0.934767i	$0.615608\pi$
$60$	0	0
$61$	−298.000	−0.625492	−0.312746	−	0.949837i	$-0.601249\pi$
$61$	−298.000	−0.625492	−0.312746	+	0.949837i	$0.601249\pi$
$62$	0	0
$63$	−460.000	−0.919914
$64$	0	0
$65$	324.000	0.618265
$66$	0	0
$67$	202.000	0.368332	0.184166	−	0.982895i	$-0.441042\pi$
$67$	202.000	0.368332	0.184166	+	0.982895i	$0.441042\pi$
$68$	0	0
$69$	344.000	0.600185
$70$	0	0
$71$	−700.000	−1.17007	−0.585033	−	0.811009i	$-0.698919\pi$
$71$	−700.000	−1.17007	−0.585033	+	0.811009i	$0.698919\pi$
$72$	0	0
$73$	−418.000	−0.670181	−0.335090	−	0.942186i	$-0.608767\pi$
$73$	−418.000	−0.670181	−0.335090	+	0.942186i	$0.608767\pi$
$74$	0	0
$75$	−178.000	−0.274049
$76$	0	0
$77$	280.000	0.414402
$78$	0	0
$79$	744.000	1.05958	0.529788	−	0.848130i	$-0.322271\pi$
$79$	744.000	1.05958	0.529788	+	0.848130i	$0.322271\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	0	0
$83$	−678.000	−0.896629	−0.448314	−	0.893876i	$-0.647975\pi$
$83$	−678.000	−0.896629	−0.448314	+	0.893876i	$0.647975\pi$
$84$	0	0
$85$	−396.000	−0.505320
$86$	0	0
$87$	−4.00000	−0.00492925
$88$	0	0
$89$	−82.0000	−0.0976627	−0.0488314	−	0.998807i	$-0.515550\pi$
$89$	−82.0000	−0.0976627	−0.0488314	+	0.998807i	$0.515550\pi$
$90$	0	0
$91$	1080.00	1.24412
$92$	0	0
$93$	−256.000	−0.285440
$94$	0	0
$95$	972.000	1.04974
$96$	0	0
$97$	−1122.00	−1.17445	−0.587226	−	0.809423i	$-0.699780\pi$
$97$	−1122.00	−1.17445	−0.587226	+	0.809423i	$0.699780\pi$
$98$	0	0
$99$	−322.000	−0.326891
$100$	0	0
$101$	1390.00	1.36941	0.684704	−	0.728821i	$-0.259932\pi$
$101$	1390.00	1.36941	0.684704	+	0.728821i	$0.259932\pi$
$102$	0	0
$103$	788.000	0.753825	0.376912	−	0.926249i	$-0.376986\pi$
$103$	788.000	0.753825	0.376912	+	0.926249i	$0.376986\pi$
$104$	0	0
$105$	240.000	0.223063
$106$	0	0
$107$	1614.00	1.45824	0.729118	−	0.684388i	$-0.239930\pi$
$107$	1614.00	1.45824	0.729118	+	0.684388i	$0.239930\pi$
$108$	0	0
$109$	2014.00	1.76978	0.884891	−	0.465798i	$-0.154233\pi$
$109$	2014.00	1.76978	0.884891	+	0.465798i	$0.154233\pi$
$110$	0	0
$111$	316.000	0.270211
$112$	0	0
$113$	−542.000	−0.451213	−0.225607	−	0.974219i	$-0.572436\pi$
$113$	−542.000	−0.451213	−0.225607	+	0.974219i	$0.572436\pi$
$114$	0	0
$115$	1032.00	0.836822
$116$	0	0
$117$	−1242.00	−0.981393
$118$	0	0
$119$	−1320.00	−1.01684
$120$	0	0
$121$	−1135.00	−0.852742
$122$	0	0
$123$	404.000	0.296158
$124$	0	0
$125$	−1284.00	−0.918756
$126$	0	0
$127$	1712.00	1.19618	0.598092	−	0.801427i	$-0.295926\pi$
$127$	1712.00	1.19618	0.598092	+	0.801427i	$0.295926\pi$
$128$	0	0
$129$	−596.000	−0.406782
$130$	0	0
$131$	−2118.00	−1.41260	−0.706300	−	0.707913i	$-0.749637\pi$
$131$	−2118.00	−1.41260	−0.706300	+	0.707913i	$0.749637\pi$
$132$	0	0
$133$	3240.00	2.11236
$134$	0	0
$135$	−600.000	−0.382517
$136$	0	0
$137$	−486.000	−0.303079	−0.151539	−	0.988451i	$-0.548423\pi$
$137$	−486.000	−0.303079	−0.151539	+	0.988451i	$0.548423\pi$
$138$	0	0
$139$	1286.00	0.784727	0.392364	−	0.919810i	$-0.371657\pi$
$139$	1286.00	0.784727	0.392364	+	0.919810i	$0.371657\pi$
$140$	0	0
$141$	−816.000	−0.487373
$142$	0	0
$143$	756.000	0.442097
$144$	0	0
$145$	−12.0000	−0.00687273
$146$	0	0
$147$	114.000	0.0639630
$148$	0	0
$149$	−2666.00	−1.46582	−0.732910	−	0.680325i	$-0.761838\pi$
$149$	−2666.00	−1.46582	−0.732910	+	0.680325i	$0.761838\pi$
$150$	0	0
$151$	172.000	0.0926964	0.0463482	−	0.998925i	$-0.485242\pi$
$151$	172.000	0.0926964	0.0463482	+	0.998925i	$0.485242\pi$
$152$	0	0
$153$	1518.00	0.802111
$154$	0	0
$155$	−768.000	−0.397982
$156$	0	0
$157$	838.000	0.425985	0.212993	−	0.977054i	$-0.431679\pi$
$157$	838.000	0.425985	0.212993	+	0.977054i	$0.431679\pi$
$158$	0	0
$159$	−1380.00	−0.688309
$160$	0	0
$161$	3440.00	1.68391
$162$	0	0
$163$	1346.00	0.646791	0.323395	−	0.946264i	$-0.395176\pi$
$163$	1346.00	0.646791	0.323395	+	0.946264i	$0.395176\pi$
$164$	0	0
$165$	168.000	0.0792653
$166$	0	0
$167$	−1052.00	−0.487462	−0.243731	−	0.969843i	$-0.578371\pi$
$167$	−1052.00	−0.487462	−0.243731	+	0.969843i	$0.578371\pi$
$168$	0	0
$169$	719.000	0.327264
$170$	0	0
$171$	−3726.00	−1.66628
$172$	0	0
$173$	38.0000	0.0166999	0.00834996	−	0.999965i	$-0.497342\pi$
$173$	38.0000	0.0166999	0.00834996	+	0.999965i	$0.497342\pi$
$174$	0	0
$175$	−1780.00	−0.768888
$176$	0	0
$177$	−644.000	−0.273480
$178$	0	0
$179$	−2790.00	−1.16500	−0.582498	−	0.812832i	$-0.697925\pi$
$179$	−2790.00	−1.16500	−0.582498	+	0.812832i	$0.697925\pi$
$180$	0	0
$181$	−3418.00	−1.40364	−0.701818	−	0.712357i	$-0.747628\pi$
$181$	−3418.00	−1.40364	−0.701818	+	0.712357i	$0.747628\pi$
$182$	0	0
$183$	−596.000	−0.240752
$184$	0	0
$185$	948.000	0.376748
$186$	0	0
$187$	−924.000	−0.361335
$188$	0	0
$189$	−2000.00	−0.769728
$190$	0	0
$191$	−1968.00	−0.745547	−0.372774	−	0.927922i	$-0.621593\pi$
$191$	−1968.00	−0.745547	−0.372774	+	0.927922i	$0.621593\pi$
$192$	0	0
$193$	−1058.00	−0.394593	−0.197297	−	0.980344i	$-0.563216\pi$
$193$	−1058.00	−0.394593	−0.197297	+	0.980344i	$0.563216\pi$
$194$	0	0
$195$	648.000	0.237970
$196$	0	0
$197$	726.000	0.262565	0.131283	−	0.991345i	$-0.458090\pi$
$197$	726.000	0.262565	0.131283	+	0.991345i	$0.458090\pi$
$198$	0	0
$199$	4116.00	1.46621	0.733104	−	0.680116i	$-0.238071\pi$
$199$	4116.00	1.46621	0.733104	+	0.680116i	$0.238071\pi$
$200$	0	0
$201$	404.000	0.141771
$202$	0	0
$203$	−40.0000	−0.0138298
$204$	0	0
$205$	1212.00	0.412926
$206$	0	0
$207$	−3956.00	−1.32831
$208$	0	0
$209$	2268.00	0.750626
$210$	0	0
$211$	1482.00	0.483531	0.241766	−	0.970335i	$-0.422273\pi$
$211$	1482.00	0.483531	0.241766	+	0.970335i	$0.422273\pi$
$212$	0	0
$213$	−1400.00	−0.450359
$214$	0	0
$215$	−1788.00	−0.567166
$216$	0	0
$217$	−2560.00	−0.800848
$218$	0	0
$219$	−836.000	−0.257953
$220$	0	0
$221$	−3564.00	−1.08480
$222$	0	0
$223$	−896.000	−0.269061	−0.134530	−	0.990909i	$-0.542953\pi$
$223$	−896.000	−0.269061	−0.134530	+	0.990909i	$0.542953\pi$
$224$	0	0
$225$	2047.00	0.606519
$226$	0	0
$227$	3410.00	0.997047	0.498523	−	0.866876i	$-0.333876\pi$
$227$	3410.00	0.997047	0.498523	+	0.866876i	$0.333876\pi$
$228$	0	0
$229$	4502.00	1.29913	0.649564	−	0.760307i	$-0.274951\pi$
$229$	4502.00	1.29913	0.649564	+	0.760307i	$0.274951\pi$
$230$	0	0
$231$	560.000	0.159503
$232$	0	0
$233$	2302.00	0.647249	0.323625	−	0.946186i	$-0.395099\pi$
$233$	2302.00	0.647249	0.323625	+	0.946186i	$0.395099\pi$
$234$	0	0
$235$	−2448.00	−0.679532
$236$	0	0
$237$	1488.00	0.407831
$238$	0	0
$239$	4024.00	1.08908	0.544542	−	0.838734i	$-0.316704\pi$
$239$	4024.00	1.08908	0.544542	+	0.838734i	$0.316704\pi$
$240$	0	0
$241$	−3586.00	−0.958484	−0.479242	−	0.877683i	$-0.659088\pi$
$241$	−3586.00	−0.958484	−0.479242	+	0.877683i	$0.659088\pi$
$242$	0	0
$243$	3542.00	0.935059
$244$	0	0
$245$	342.000	0.0891820
$246$	0	0
$247$	8748.00	2.25353
$248$	0	0
$249$	−1356.00	−0.345112
$250$	0	0
$251$	−1250.00	−0.314340	−0.157170	−	0.987572i	$-0.550237\pi$
$251$	−1250.00	−0.314340	−0.157170	+	0.987572i	$0.550237\pi$
$252$	0	0
$253$	2408.00	0.598378
$254$	0	0
$255$	−792.000	−0.194498
$256$	0	0
$257$	−6638.00	−1.61116	−0.805578	−	0.592490i	$-0.798145\pi$
$257$	−6638.00	−1.61116	−0.805578	+	0.592490i	$0.798145\pi$
$258$	0	0
$259$	3160.00	0.758119
$260$	0	0
$261$	46.0000	0.0109093
$262$	0	0
$263$	−1724.00	−0.404207	−0.202103	−	0.979364i	$-0.564778\pi$
$263$	−1724.00	−0.404207	−0.202103	+	0.979364i	$0.564778\pi$
$264$	0	0
$265$	−4140.00	−0.959691
$266$	0	0
$267$	−164.000	−0.0375904
$268$	0	0
$269$	4814.00	1.09113	0.545566	−	0.838068i	$-0.316315\pi$
$269$	4814.00	1.09113	0.545566	+	0.838068i	$0.316315\pi$
$270$	0	0
$271$	−1640.00	−0.367612	−0.183806	−	0.982963i	$-0.558842\pi$
$271$	−1640.00	−0.367612	−0.183806	+	0.982963i	$0.558842\pi$
$272$	0	0
$273$	2160.00	0.478861
$274$	0	0
$275$	−1246.00	−0.273224
$276$	0	0
$277$	3982.00	0.863737	0.431869	−	0.901937i	$-0.357854\pi$
$277$	3982.00	0.863737	0.431869	+	0.901937i	$0.357854\pi$
$278$	0	0
$279$	2944.00	0.631730
$280$	0	0
$281$	4126.00	0.875931	0.437965	−	0.898992i	$-0.355699\pi$
$281$	4126.00	0.875931	0.437965	+	0.898992i	$0.355699\pi$
$282$	0	0
$283$	3446.00	0.723828	0.361914	−	0.932211i	$-0.382123\pi$
$283$	3446.00	0.723828	0.361914	+	0.932211i	$0.382123\pi$
$284$	0	0
$285$	1944.00	0.404044
$286$	0	0
$287$	4040.00	0.830919
$288$	0	0
$289$	−557.000	−0.113373
$290$	0	0
$291$	−2244.00	−0.452047
$292$	0	0
$293$	−1514.00	−0.301873	−0.150937	−	0.988543i	$-0.548229\pi$
$293$	−1514.00	−0.301873	−0.150937	+	0.988543i	$0.548229\pi$
$294$	0	0
$295$	−1932.00	−0.381306
$296$	0	0
$297$	−1400.00	−0.273523
$298$	0	0
$299$	9288.00	1.79645
$300$	0	0
$301$	−5960.00	−1.14129
$302$	0	0
$303$	2780.00	0.527085
$304$	0	0
$305$	−1788.00	−0.335674
$306$	0	0
$307$	5490.00	1.02062	0.510311	−	0.859990i	$-0.329530\pi$
$307$	5490.00	1.02062	0.510311	+	0.859990i	$0.329530\pi$
$308$	0	0
$309$	1576.00	0.290147
$310$	0	0
$311$	−5556.00	−1.01303	−0.506514	−	0.862232i	$-0.669066\pi$
$311$	−5556.00	−1.01303	−0.506514	+	0.862232i	$0.669066\pi$
$312$	0	0
$313$	−2054.00	−0.370923	−0.185462	−	0.982652i	$-0.559378\pi$
$313$	−2054.00	−0.370923	−0.185462	+	0.982652i	$0.559378\pi$
$314$	0	0
$315$	−2760.00	−0.493677
$316$	0	0
$317$	2494.00	0.441883	0.220942	−	0.975287i	$-0.429087\pi$
$317$	2494.00	0.441883	0.220942	+	0.975287i	$0.429087\pi$
$318$	0	0
$319$	−28.0000	−0.00491442
$320$	0	0
$321$	3228.00	0.561275
$322$	0	0
$323$	−10692.0	−1.84185
$324$	0	0
$325$	−4806.00	−0.820274
$326$	0	0
$327$	4028.00	0.681189
$328$	0	0
$329$	−8160.00	−1.36740
$330$	0	0
$331$	−2914.00	−0.483891	−0.241946	−	0.970290i	$-0.577786\pi$
$331$	−2914.00	−0.483891	−0.241946	+	0.970290i	$0.577786\pi$
$332$	0	0
$333$	−3634.00	−0.598024
$334$	0	0
$335$	1212.00	0.197668
$336$	0	0
$337$	7186.00	1.16156	0.580781	−	0.814060i	$-0.302747\pi$
$337$	7186.00	1.16156	0.580781	+	0.814060i	$0.302747\pi$
$338$	0	0
$339$	−1084.00	−0.173672
$340$	0	0
$341$	−1792.00	−0.284581
$342$	0	0
$343$	−5720.00	−0.900440
$344$	0	0
$345$	2064.00	0.322093
$346$	0	0
$347$	4446.00	0.687821	0.343910	−	0.939002i	$-0.388248\pi$
$347$	4446.00	0.687821	0.343910	+	0.939002i	$0.388248\pi$
$348$	0	0
$349$	430.000	0.0659524	0.0329762	−	0.999456i	$-0.489501\pi$
$349$	430.000	0.0659524	0.0329762	+	0.999456i	$0.489501\pi$
$350$	0	0
$351$	−5400.00	−0.821170
$352$	0	0
$353$	850.000	0.128161	0.0640806	−	0.997945i	$-0.479589\pi$
$353$	850.000	0.128161	0.0640806	+	0.997945i	$0.479589\pi$
$354$	0	0
$355$	−4200.00	−0.627924
$356$	0	0
$357$	−2640.00	−0.391383
$358$	0	0
$359$	−10988.0	−1.61539	−0.807694	−	0.589602i	$-0.799285\pi$
$359$	−10988.0	−1.61539	−0.807694	+	0.589602i	$0.799285\pi$
$360$	0	0
$361$	19385.0	2.82621
$362$	0	0
$363$	−2270.00	−0.328221
$364$	0	0
$365$	−2508.00	−0.359657
$366$	0	0
$367$	872.000	0.124027	0.0620137	−	0.998075i	$-0.480248\pi$
$367$	872.000	0.124027	0.0620137	+	0.998075i	$0.480248\pi$
$368$	0	0
$369$	−4646.00	−0.655450
$370$	0	0
$371$	−13800.0	−1.93116
$372$	0	0
$373$	3454.00	0.479467	0.239734	−	0.970839i	$-0.422940\pi$
$373$	3454.00	0.479467	0.239734	+	0.970839i	$0.422940\pi$
$374$	0	0
$375$	−2568.00	−0.353629
$376$	0	0
$377$	−108.000	−0.0147541
$378$	0	0
$379$	−1490.00	−0.201942	−0.100971	−	0.994889i	$-0.532195\pi$
$379$	−1490.00	−0.201942	−0.100971	+	0.994889i	$0.532195\pi$
$380$	0	0
$381$	3424.00	0.460412
$382$	0	0
$383$	10240.0	1.36616	0.683080	−	0.730343i	$-0.260640\pi$
$383$	10240.0	1.36616	0.683080	+	0.730343i	$0.260640\pi$
$384$	0	0
$385$	1680.00	0.222392
$386$	0	0
$387$	6854.00	0.900280
$388$	0	0
$389$	−7458.00	−0.972071	−0.486035	−	0.873939i	$-0.661557\pi$
$389$	−7458.00	−0.972071	−0.486035	+	0.873939i	$0.661557\pi$
$390$	0	0
$391$	−11352.0	−1.46827
$392$	0	0
$393$	−4236.00	−0.543710
$394$	0	0
$395$	4464.00	0.568628
$396$	0	0
$397$	−4706.00	−0.594930	−0.297465	−	0.954733i	$-0.596141\pi$
$397$	−4706.00	−0.594930	−0.297465	+	0.954733i	$0.596141\pi$
$398$	0	0
$399$	6480.00	0.813047
$400$	0	0
$401$	5598.00	0.697134	0.348567	−	0.937284i	$-0.386668\pi$
$401$	5598.00	0.697134	0.348567	+	0.937284i	$0.386668\pi$
$402$	0	0
$403$	−6912.00	−0.854370
$404$	0	0
$405$	2526.00	0.309921
$406$	0	0
$407$	2212.00	0.269397
$408$	0	0
$409$	−838.000	−0.101312	−0.0506558	−	0.998716i	$-0.516131\pi$
$409$	−838.000	−0.101312	−0.0506558	+	0.998716i	$0.516131\pi$
$410$	0	0
$411$	−972.000	−0.116655
$412$	0	0
$413$	−6440.00	−0.767292
$414$	0	0
$415$	−4068.00	−0.481181
$416$	0	0
$417$	2572.00	0.302042
$418$	0	0
$419$	26.0000	0.00303146	0.00151573	−	0.999999i	$-0.499518\pi$
$419$	26.0000	0.00303146	0.00151573	+	0.999999i	$0.499518\pi$
$420$	0	0
$421$	−15578.0	−1.80339	−0.901693	−	0.432377i	$-0.857675\pi$
$421$	−15578.0	−1.80339	−0.901693	+	0.432377i	$0.857675\pi$
$422$	0	0
$423$	9384.00	1.07864
$424$	0	0
$425$	5874.00	0.670426
$426$	0	0
$427$	−5960.00	−0.675467
$428$	0	0
$429$	1512.00	0.170163
$430$	0	0
$431$	−6792.00	−0.759070	−0.379535	−	0.925177i	$-0.623916\pi$
$431$	−6792.00	−0.759070	−0.379535	+	0.925177i	$0.623916\pi$
$432$	0	0
$433$	−9314.00	−1.03372	−0.516862	−	0.856069i	$-0.672900\pi$
$433$	−9314.00	−1.03372	−0.516862	+	0.856069i	$0.672900\pi$
$434$	0	0
$435$	−24.0000	−0.00264531
$436$	0	0
$437$	27864.0	3.05015
$438$	0	0
$439$	−3828.00	−0.416174	−0.208087	−	0.978110i	$-0.566724\pi$
$439$	−3828.00	−0.416174	−0.208087	+	0.978110i	$0.566724\pi$
$440$	0	0
$441$	−1311.00	−0.141561
$442$	0	0
$443$	15414.0	1.65314	0.826570	−	0.562834i	$-0.190289\pi$
$443$	15414.0	1.65314	0.826570	+	0.562834i	$0.190289\pi$
$444$	0	0
$445$	−492.000	−0.0524113
$446$	0	0
$447$	−5332.00	−0.564195
$448$	0	0
$449$	−3650.00	−0.383640	−0.191820	−	0.981430i	$-0.561439\pi$
$449$	−3650.00	−0.383640	−0.191820	+	0.981430i	$0.561439\pi$
$450$	0	0
$451$	2828.00	0.295267
$452$	0	0
$453$	344.000	0.0356789
$454$	0	0
$455$	6480.00	0.667664
$456$	0	0
$457$	−15862.0	−1.62362	−0.811809	−	0.583924i	$-0.801517\pi$
$457$	−15862.0	−1.62362	−0.811809	+	0.583924i	$0.801517\pi$
$458$	0	0
$459$	6600.00	0.671158
$460$	0	0
$461$	78.0000	0.00788031	0.00394015	−	0.999992i	$-0.498746\pi$
$461$	78.0000	0.00788031	0.00394015	+	0.999992i	$0.498746\pi$
$462$	0	0
$463$	−4376.00	−0.439244	−0.219622	−	0.975585i	$-0.570482\pi$
$463$	−4376.00	−0.439244	−0.219622	+	0.975585i	$0.570482\pi$
$464$	0	0
$465$	−1536.00	−0.153183
$466$	0	0
$467$	13714.0	1.35890	0.679452	−	0.733720i	$-0.262218\pi$
$467$	13714.0	1.35890	0.679452	+	0.733720i	$0.262218\pi$
$468$	0	0
$469$	4040.00	0.397761
$470$	0	0
$471$	1676.00	0.163962
$472$	0	0
$473$	−4172.00	−0.405558
$474$	0	0
$475$	−14418.0	−1.39272
$476$	0	0
$477$	15870.0	1.52335
$478$	0	0
$479$	19072.0	1.81925	0.909626	−	0.415428i	$-0.136368\pi$
$479$	19072.0	1.81925	0.909626	+	0.415428i	$0.136368\pi$
$480$	0	0
$481$	8532.00	0.808785
$482$	0	0
$483$	6880.00	0.648138
$484$	0	0
$485$	−6732.00	−0.630277
$486$	0	0
$487$	19428.0	1.80773	0.903867	−	0.427813i	$-0.140716\pi$
$487$	19428.0	1.80773	0.903867	+	0.427813i	$0.140716\pi$
$488$	0	0
$489$	2692.00	0.248950
$490$	0	0
$491$	−2490.00	−0.228864	−0.114432	−	0.993431i	$-0.536505\pi$
$491$	−2490.00	−0.228864	−0.114432	+	0.993431i	$0.536505\pi$
$492$	0	0
$493$	132.000	0.0120588
$494$	0	0
$495$	−1932.00	−0.175428
$496$	0	0
$497$	−14000.0	−1.26355
$498$	0	0
$499$	2826.00	0.253525	0.126763	−	0.991933i	$-0.459541\pi$
$499$	2826.00	0.253525	0.126763	+	0.991933i	$0.459541\pi$
$500$	0	0
$501$	−2104.00	−0.187624
$502$	0	0
$503$	2268.00	0.201044	0.100522	−	0.994935i	$-0.467949\pi$
$503$	2268.00	0.201044	0.100522	+	0.994935i	$0.467949\pi$
$504$	0	0
$505$	8340.00	0.734901
$506$	0	0
$507$	1438.00	0.125964
$508$	0	0
$509$	10534.0	0.917311	0.458656	−	0.888614i	$-0.348331\pi$
$509$	10534.0	0.917311	0.458656	+	0.888614i	$0.348331\pi$
$510$	0	0
$511$	−8360.00	−0.723727
$512$	0	0
$513$	−16200.0	−1.39424
$514$	0	0
$515$	4728.00	0.404545
$516$	0	0
$517$	−5712.00	−0.485906
$518$	0	0
$519$	76.0000	0.00642780
$520$	0	0
$521$	−9478.00	−0.797003	−0.398502	−	0.917168i	$-0.630470\pi$
$521$	−9478.00	−0.797003	−0.398502	+	0.917168i	$0.630470\pi$
$522$	0	0
$523$	−5858.00	−0.489775	−0.244888	−	0.969551i	$-0.578751\pi$
$523$	−5858.00	−0.489775	−0.244888	+	0.969551i	$0.578751\pi$
$524$	0	0
$525$	−3560.00	−0.295945
$526$	0	0
$527$	8448.00	0.698293
$528$	0	0
$529$	17417.0	1.43150
$530$	0	0
$531$	7406.00	0.605260
$532$	0	0
$533$	10908.0	0.886450
$534$	0	0
$535$	9684.00	0.782572
$536$	0	0
$537$	−5580.00	−0.448407
$538$	0	0
$539$	798.000	0.0637705
$540$	0	0
$541$	1910.00	0.151788	0.0758940	−	0.997116i	$-0.475819\pi$
$541$	1910.00	0.151788	0.0758940	+	0.997116i	$0.475819\pi$
$542$	0	0
$543$	−6836.00	−0.540259
$544$	0	0
$545$	12084.0	0.949765
$546$	0	0
$547$	1754.00	0.137104	0.0685518	−	0.997648i	$-0.478162\pi$
$547$	1754.00	0.137104	0.0685518	+	0.997648i	$0.478162\pi$
$548$	0	0
$549$	6854.00	0.532826
$550$	0	0
$551$	−324.000	−0.0250506
$552$	0	0
$553$	14880.0	1.14424
$554$	0	0
$555$	1896.00	0.145010
$556$	0	0
$557$	2294.00	0.174506	0.0872531	−	0.996186i	$-0.472191\pi$
$557$	2294.00	0.174506	0.0872531	+	0.996186i	$0.472191\pi$
$558$	0	0
$559$	−16092.0	−1.21757
$560$	0	0
$561$	−1848.00	−0.139078
$562$	0	0
$563$	16242.0	1.21584	0.607921	−	0.793998i	$-0.292004\pi$
$563$	16242.0	1.21584	0.607921	+	0.793998i	$0.292004\pi$
$564$	0	0
$565$	−3252.00	−0.242146
$566$	0	0
$567$	8420.00	0.623645
$568$	0	0
$569$	−15990.0	−1.17809	−0.589047	−	0.808099i	$-0.700497\pi$
$569$	−15990.0	−1.17809	−0.589047	+	0.808099i	$0.700497\pi$
$570$	0	0
$571$	−21674.0	−1.58849	−0.794246	−	0.607597i	$-0.792134\pi$
$571$	−21674.0	−1.58849	−0.794246	+	0.607597i	$0.792134\pi$
$572$	0	0
$573$	−3936.00	−0.286961
$574$	0	0
$575$	−15308.0	−1.11024
$576$	0	0
$577$	−12542.0	−0.904905	−0.452453	−	0.891788i	$-0.649451\pi$
$577$	−12542.0	−0.904905	−0.452453	+	0.891788i	$0.649451\pi$
$578$	0	0
$579$	−2116.00	−0.151879
$580$	0	0
$581$	−13560.0	−0.968268
$582$	0	0
$583$	−9660.00	−0.686237
$584$	0	0
$585$	−7452.00	−0.526671
$586$	0	0
$587$	−18578.0	−1.30630	−0.653148	−	0.757230i	$-0.726552\pi$
$587$	−18578.0	−1.30630	−0.653148	+	0.757230i	$0.726552\pi$
$588$	0	0
$589$	−20736.0	−1.45061
$590$	0	0
$591$	1452.00	0.101061
$592$	0	0
$593$	14514.0	1.00509	0.502545	−	0.864551i	$-0.332397\pi$
$593$	14514.0	1.00509	0.502545	+	0.864551i	$0.332397\pi$
$594$	0	0
$595$	−7920.00	−0.545695
$596$	0	0
$597$	8232.00	0.564344
$598$	0	0
$599$	−3460.00	−0.236013	−0.118006	−	0.993013i	$-0.537650\pi$
$599$	−3460.00	−0.236013	−0.118006	+	0.993013i	$0.537650\pi$
$600$	0	0
$601$	20686.0	1.40399	0.701996	−	0.712181i	$-0.252292\pi$
$601$	20686.0	1.40399	0.701996	+	0.712181i	$0.252292\pi$
$602$	0	0
$603$	−4646.00	−0.313764
$604$	0	0
$605$	−6810.00	−0.457630
$606$	0	0
$607$	−9776.00	−0.653700	−0.326850	−	0.945076i	$-0.605987\pi$
$607$	−9776.00	−0.653700	−0.326850	+	0.945076i	$0.605987\pi$
$608$	0	0
$609$	−80.0000	−0.00532309
$610$	0	0
$611$	−22032.0	−1.45879
$612$	0	0
$613$	−8794.00	−0.579423	−0.289712	−	0.957114i	$-0.593559\pi$
$613$	−8794.00	−0.579423	−0.289712	+	0.957114i	$0.593559\pi$
$614$	0	0
$615$	2424.00	0.158935
$616$	0	0
$617$	16398.0	1.06995	0.534975	−	0.844868i	$-0.320321\pi$
$617$	16398.0	1.06995	0.534975	+	0.844868i	$0.320321\pi$
$618$	0	0
$619$	21374.0	1.38787	0.693937	−	0.720036i	$-0.255875\pi$
$619$	21374.0	1.38787	0.693937	+	0.720036i	$0.255875\pi$
$620$	0	0
$621$	−17200.0	−1.11145
$622$	0	0
$623$	−1640.00	−0.105466
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	4536.00	0.288916
$628$	0	0
$629$	−10428.0	−0.661036
$630$	0	0
$631$	−18916.0	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$631$	−18916.0	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$632$	0	0
$633$	2964.00	0.186111
$634$	0	0
$635$	10272.0	0.641940
$636$	0	0
$637$	3078.00	0.191452
$638$	0	0
$639$	16100.0	0.996723
$640$	0	0
$641$	−5250.00	−0.323498	−0.161749	−	0.986832i	$-0.551714\pi$
$641$	−5250.00	−0.323498	−0.161749	+	0.986832i	$0.551714\pi$
$642$	0	0
$643$	−4502.00	−0.276114	−0.138057	−	0.990424i	$-0.544086\pi$
$643$	−4502.00	−0.276114	−0.138057	+	0.990424i	$0.544086\pi$
$644$	0	0
$645$	−3576.00	−0.218302
$646$	0	0
$647$	11076.0	0.673018	0.336509	−	0.941680i	$-0.390754\pi$
$647$	11076.0	0.673018	0.336509	+	0.941680i	$0.390754\pi$
$648$	0	0
$649$	−4508.00	−0.272657
$650$	0	0
$651$	−5120.00	−0.308247
$652$	0	0
$653$	30766.0	1.84375	0.921873	−	0.387491i	$-0.126658\pi$
$653$	30766.0	1.84375	0.921873	+	0.387491i	$0.126658\pi$
$654$	0	0
$655$	−12708.0	−0.758080
$656$	0	0
$657$	9614.00	0.570895
$658$	0	0
$659$	−20518.0	−1.21285	−0.606425	−	0.795141i	$-0.707397\pi$
$659$	−20518.0	−1.21285	−0.606425	+	0.795141i	$0.707397\pi$
$660$	0	0
$661$	70.0000	0.00411904	0.00205952	−	0.999998i	$-0.499344\pi$
$661$	70.0000	0.00411904	0.00205952	+	0.999998i	$0.499344\pi$
$662$	0	0
$663$	−7128.00	−0.417539
$664$	0	0
$665$	19440.0	1.13361
$666$	0	0
$667$	−344.000	−0.0199696
$668$	0	0
$669$	−1792.00	−0.103562
$670$	0	0
$671$	−4172.00	−0.240027
$672$	0	0
$673$	23070.0	1.32137	0.660686	−	0.750663i	$-0.270266\pi$
$673$	23070.0	1.32137	0.660686	+	0.750663i	$0.270266\pi$
$674$	0	0
$675$	8900.00	0.507498
$676$	0	0
$677$	2622.00	0.148850	0.0744251	−	0.997227i	$-0.476288\pi$
$677$	2622.00	0.148850	0.0744251	+	0.997227i	$0.476288\pi$
$678$	0	0
$679$	−22440.0	−1.26829
$680$	0	0
$681$	6820.00	0.383764
$682$	0	0
$683$	−14682.0	−0.822535	−0.411267	−	0.911515i	$-0.634914\pi$
$683$	−14682.0	−0.822535	−0.411267	+	0.911515i	$0.634914\pi$
$684$	0	0
$685$	−2916.00	−0.162649
$686$	0	0
$687$	9004.00	0.500035
$688$	0	0
$689$	−37260.0	−2.06022
$690$	0	0
$691$	−23270.0	−1.28109	−0.640545	−	0.767921i	$-0.721291\pi$
$691$	−23270.0	−1.28109	−0.640545	+	0.767921i	$0.721291\pi$
$692$	0	0
$693$	−6440.00	−0.353009
$694$	0	0
$695$	7716.00	0.421129
$696$	0	0
$697$	−13332.0	−0.724513
$698$	0	0
$699$	4604.00	0.249126
$700$	0	0
$701$	−31530.0	−1.69882	−0.849409	−	0.527735i	$-0.823041\pi$
$701$	−31530.0	−1.69882	−0.849409	+	0.527735i	$0.823041\pi$
$702$	0	0
$703$	25596.0	1.37322
$704$	0	0
$705$	−4896.00	−0.261552
$706$	0	0
$707$	27800.0	1.47882
$708$	0	0
$709$	22014.0	1.16608	0.583042	−	0.812442i	$-0.301862\pi$
$709$	22014.0	1.16608	0.583042	+	0.812442i	$0.301862\pi$
$710$	0	0
$711$	−17112.0	−0.902602
$712$	0	0
$713$	−22016.0	−1.15639
$714$	0	0
$715$	4536.00	0.237254
$716$	0	0
$717$	8048.00	0.419188
$718$	0	0
$719$	19016.0	0.986338	0.493169	−	0.869933i	$-0.335838\pi$
$719$	19016.0	0.986338	0.493169	+	0.869933i	$0.335838\pi$
$720$	0	0
$721$	15760.0	0.814054
$722$	0	0
$723$	−7172.00	−0.368921
$724$	0	0
$725$	178.000	0.00911828
$726$	0	0
$727$	15996.0	0.816037	0.408018	−	0.912974i	$-0.366220\pi$
$727$	15996.0	0.816037	0.408018	+	0.912974i	$0.366220\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	0	0
$731$	19668.0	0.995140
$732$	0	0
$733$	10622.0	0.535242	0.267621	−	0.963524i	$-0.413762\pi$
$733$	10622.0	0.535242	0.267621	+	0.963524i	$0.413762\pi$
$734$	0	0
$735$	684.000	0.0343261
$736$	0	0
$737$	2828.00	0.141344
$738$	0	0
$739$	−8878.00	−0.441925	−0.220962	−	0.975282i	$-0.570920\pi$
$739$	−8878.00	−0.441925	−0.220962	+	0.975282i	$0.570920\pi$
$740$	0	0
$741$	17496.0	0.867384
$742$	0	0
$743$	−21852.0	−1.07897	−0.539483	−	0.841996i	$-0.681380\pi$
$743$	−21852.0	−1.07897	−0.539483	+	0.841996i	$0.681380\pi$
$744$	0	0
$745$	−15996.0	−0.786642
$746$	0	0
$747$	15594.0	0.763795
$748$	0	0
$749$	32280.0	1.57475
$750$	0	0
$751$	32024.0	1.55602	0.778011	−	0.628251i	$-0.216229\pi$
$751$	32024.0	1.55602	0.778011	+	0.628251i	$0.216229\pi$
$752$	0	0
$753$	−2500.00	−0.120989
$754$	0	0
$755$	1032.00	0.0497461
$756$	0	0
$757$	−26602.0	−1.27723	−0.638617	−	0.769525i	$-0.720493\pi$
$757$	−26602.0	−1.27723	−0.638617	+	0.769525i	$0.720493\pi$
$758$	0	0
$759$	4816.00	0.230316
$760$	0	0
$761$	−13958.0	−0.664885	−0.332442	−	0.943124i	$-0.607873\pi$
$761$	−13958.0	−0.664885	−0.332442	+	0.943124i	$0.607873\pi$
$762$	0	0
$763$	40280.0	1.91118
$764$	0	0
$765$	9108.00	0.430458
$766$	0	0
$767$	−17388.0	−0.818571
$768$	0	0
$769$	−11970.0	−0.561312	−0.280656	−	0.959808i	$-0.590552\pi$
$769$	−11970.0	−0.561312	−0.280656	+	0.959808i	$0.590552\pi$
$770$	0	0
$771$	−13276.0	−0.620134
$772$	0	0
$773$	12318.0	0.573154	0.286577	−	0.958057i	$-0.407483\pi$
$773$	12318.0	0.573154	0.286577	+	0.958057i	$0.407483\pi$
$774$	0	0
$775$	11392.0	0.528016
$776$	0	0
$777$	6320.00	0.291800
$778$	0	0
$779$	32724.0	1.50508
$780$	0	0
$781$	−9800.00	−0.449003
$782$	0	0
$783$	200.000	0.00912825
$784$	0	0
$785$	5028.00	0.228608
$786$	0	0
$787$	26698.0	1.20925	0.604626	−	0.796510i	$-0.293323\pi$
$787$	26698.0	1.20925	0.604626	+	0.796510i	$0.293323\pi$
$788$	0	0
$789$	−3448.00	−0.155579
$790$	0	0
$791$	−10840.0	−0.487264
$792$	0	0
$793$	−16092.0	−0.720610
$794$	0	0
$795$	−8280.00	−0.369385
$796$	0	0
$797$	−4794.00	−0.213064	−0.106532	−	0.994309i	$-0.533975\pi$
$797$	−4794.00	−0.213064	−0.106532	+	0.994309i	$0.533975\pi$
$798$	0	0
$799$	26928.0	1.19230
$800$	0	0
$801$	1886.00	0.0831942
$802$	0	0
$803$	−5852.00	−0.257176
$804$	0	0
$805$	20640.0	0.903683
$806$	0	0
$807$	9628.00	0.419977
$808$	0	0
$809$	39114.0	1.69985	0.849923	−	0.526907i	$-0.176649\pi$
$809$	39114.0	1.69985	0.849923	+	0.526907i	$0.176649\pi$
$810$	0	0
$811$	−1090.00	−0.0471949	−0.0235975	−	0.999722i	$-0.507512\pi$
$811$	−1090.00	−0.0471949	−0.0235975	+	0.999722i	$0.507512\pi$
$812$	0	0
$813$	−3280.00	−0.141494
$814$	0	0
$815$	8076.00	0.347104
$816$	0	0
$817$	−48276.0	−2.06727
$818$	0	0
$819$	−24840.0	−1.05980
$820$	0	0
$821$	−9730.00	−0.413617	−0.206808	−	0.978381i	$-0.566308\pi$
$821$	−9730.00	−0.413617	−0.206808	+	0.978381i	$0.566308\pi$
$822$	0	0
$823$	1100.00	0.0465900	0.0232950	−	0.999729i	$-0.492584\pi$
$823$	1100.00	0.0465900	0.0232950	+	0.999729i	$0.492584\pi$
$824$	0	0
$825$	−2492.00	−0.105164
$826$	0	0
$827$	−38074.0	−1.60092	−0.800461	−	0.599385i	$-0.795412\pi$
$827$	−38074.0	−1.60092	−0.800461	+	0.599385i	$0.795412\pi$
$828$	0	0
$829$	24230.0	1.01513	0.507565	−	0.861613i	$-0.330546\pi$
$829$	24230.0	1.01513	0.507565	+	0.861613i	$0.330546\pi$
$830$	0	0
$831$	7964.00	0.332453
$832$	0	0
$833$	−3762.00	−0.156477
$834$	0	0
$835$	−6312.00	−0.261600
$836$	0	0
$837$	12800.0	0.528593
$838$	0	0
$839$	16820.0	0.692123	0.346061	−	0.938212i	$-0.387519\pi$
$839$	16820.0	0.692123	0.346061	+	0.938212i	$0.387519\pi$
$840$	0	0
$841$	−24385.0	−0.999836
$842$	0	0
$843$	8252.00	0.337146
$844$	0	0
$845$	4314.00	0.175629
$846$	0	0
$847$	−22700.0	−0.920875
$848$	0	0
$849$	6892.00	0.278602
$850$	0	0
$851$	27176.0	1.09469
$852$	0	0
$853$	−22162.0	−0.889581	−0.444790	−	0.895635i	$-0.646722\pi$
$853$	−22162.0	−0.889581	−0.444790	+	0.895635i	$0.646722\pi$
$854$	0	0
$855$	−22356.0	−0.894221
$856$	0	0
$857$	−8790.00	−0.350363	−0.175181	−	0.984536i	$-0.556051\pi$
$857$	−8790.00	−0.350363	−0.175181	+	0.984536i	$0.556051\pi$
$858$	0	0
$859$	10558.0	0.419365	0.209682	−	0.977770i	$-0.432757\pi$
$859$	10558.0	0.419365	0.209682	+	0.977770i	$0.432757\pi$
$860$	0	0
$861$	8080.00	0.319821
$862$	0	0
$863$	−7392.00	−0.291572	−0.145786	−	0.989316i	$-0.546571\pi$
$863$	−7392.00	−0.291572	−0.145786	+	0.989316i	$0.546571\pi$
$864$	0	0
$865$	228.000	0.00896212
$866$	0	0
$867$	−1114.00	−0.0436372
$868$	0	0
$869$	10416.0	0.406604
$870$	0	0
$871$	10908.0	0.424344
$872$	0	0
$873$	25806.0	1.00046
$874$	0	0
$875$	−25680.0	−0.992163
$876$	0	0
$877$	6574.00	0.253122	0.126561	−	0.991959i	$-0.459606\pi$
$877$	6574.00	0.253122	0.126561	+	0.991959i	$0.459606\pi$
$878$	0	0
$879$	−3028.00	−0.116191
$880$	0	0
$881$	47154.0	1.80324	0.901622	−	0.432524i	$-0.142377\pi$
$881$	47154.0	1.80324	0.901622	+	0.432524i	$0.142377\pi$
$882$	0	0
$883$	642.000	0.0244677	0.0122339	−	0.999925i	$-0.496106\pi$
$883$	642.000	0.0244677	0.0122339	+	0.999925i	$0.496106\pi$
$884$	0	0
$885$	−3864.00	−0.146765
$886$	0	0
$887$	23308.0	0.882307	0.441153	−	0.897432i	$-0.354569\pi$
$887$	23308.0	0.882307	0.441153	+	0.897432i	$0.354569\pi$
$888$	0	0
$889$	34240.0	1.29176
$890$	0	0
$891$	5894.00	0.221612
$892$	0	0
$893$	−66096.0	−2.47684
$894$	0	0
$895$	−16740.0	−0.625203
$896$	0	0
$897$	18576.0	0.691454
$898$	0	0
$899$	256.000	0.00949731
$900$	0	0
$901$	45540.0	1.68386
$902$	0	0
$903$	−11920.0	−0.439283
$904$	0	0
$905$	−20508.0	−0.753270
$906$	0	0
$907$	−21450.0	−0.785265	−0.392633	−	0.919695i	$-0.628436\pi$
$907$	−21450.0	−0.785265	−0.392633	+	0.919695i	$0.628436\pi$
$908$	0	0
$909$	−31970.0	−1.16653
$910$	0	0
$911$	−40904.0	−1.48761	−0.743804	−	0.668398i	$-0.766980\pi$
$911$	−40904.0	−1.48761	−0.743804	+	0.668398i	$0.766980\pi$
$912$	0	0
$913$	−9492.00	−0.344074
$914$	0	0
$915$	−3576.00	−0.129201
$916$	0	0
$917$	−42360.0	−1.52546
$918$	0	0
$919$	−27380.0	−0.982789	−0.491394	−	0.870937i	$-0.663513\pi$
$919$	−27380.0	−0.982789	−0.491394	+	0.870937i	$0.663513\pi$
$920$	0	0
$921$	10980.0	0.392837
$922$	0	0
$923$	−37800.0	−1.34800
$924$	0	0
$925$	−14062.0	−0.499844
$926$	0	0
$927$	−18124.0	−0.642147
$928$	0	0
$929$	10302.0	0.363830	0.181915	−	0.983314i	$-0.441770\pi$
$929$	10302.0	0.363830	0.181915	+	0.983314i	$0.441770\pi$
$930$	0	0
$931$	9234.00	0.325061
$932$	0	0
$933$	−11112.0	−0.389915
$934$	0	0
$935$	−5544.00	−0.193912
$936$	0	0
$937$	5054.00	0.176208	0.0881040	−	0.996111i	$-0.471919\pi$
$937$	5054.00	0.176208	0.0881040	+	0.996111i	$0.471919\pi$
$938$	0	0
$939$	−4108.00	−0.142768
$940$	0	0
$941$	30462.0	1.05530	0.527648	−	0.849463i	$-0.323074\pi$
$941$	30462.0	1.05530	0.527648	+	0.849463i	$0.323074\pi$
$942$	0	0
$943$	34744.0	1.19981
$944$	0	0
$945$	−12000.0	−0.413079
$946$	0	0
$947$	32082.0	1.10087	0.550436	−	0.834878i	$-0.314462\pi$
$947$	32082.0	1.10087	0.550436	+	0.834878i	$0.314462\pi$
$948$	0	0
$949$	−22572.0	−0.772095
$950$	0	0
$951$	4988.00	0.170081
$952$	0	0
$953$	12970.0	0.440860	0.220430	−	0.975403i	$-0.429254\pi$
$953$	12970.0	0.440860	0.220430	+	0.975403i	$0.429254\pi$
$954$	0	0
$955$	−11808.0	−0.400103
$956$	0	0
$957$	−56.0000	−0.00189156
$958$	0	0
$959$	−9720.00	−0.327294
$960$	0	0
$961$	−13407.0	−0.450035
$962$	0	0
$963$	−37122.0	−1.24220
$964$	0	0
$965$	−6348.00	−0.211761
$966$	0	0
$967$	1652.00	0.0549377	0.0274688	−	0.999623i	$-0.491255\pi$
$967$	1652.00	0.0549377	0.0274688	+	0.999623i	$0.491255\pi$
$968$	0	0
$969$	−21384.0	−0.708930
$970$	0	0
$971$	−24650.0	−0.814682	−0.407341	−	0.913276i	$-0.633544\pi$
$971$	−24650.0	−0.814682	−0.407341	+	0.913276i	$0.633544\pi$
$972$	0	0
$973$	25720.0	0.847426
$974$	0	0
$975$	−9612.00	−0.315723
$976$	0	0
$977$	23646.0	0.774312	0.387156	−	0.922014i	$-0.373458\pi$
$977$	23646.0	0.774312	0.387156	+	0.922014i	$0.373458\pi$
$978$	0	0
$979$	−1148.00	−0.0374773
$980$	0	0
$981$	−46322.0	−1.50759
$982$	0	0
$983$	38108.0	1.23648	0.618238	−	0.785991i	$-0.287847\pi$
$983$	38108.0	1.23648	0.618238	+	0.785991i	$0.287847\pi$
$984$	0	0
$985$	4356.00	0.140907
$986$	0	0
$987$	−16320.0	−0.526313
$988$	0	0
$989$	−51256.0	−1.64797
$990$	0	0
$991$	−18640.0	−0.597497	−0.298748	−	0.954332i	$-0.596569\pi$
$991$	−18640.0	−0.597497	−0.298748	+	0.954332i	$0.596569\pi$
$992$	0	0
$993$	−5828.00	−0.186250
$994$	0	0
$995$	24696.0	0.786850
$996$	0	0
$997$	29022.0	0.921902	0.460951	−	0.887426i	$-0.347508\pi$
$997$	29022.0	0.921902	0.460951	+	0.887426i	$0.347508\pi$
$998$	0	0
$999$	−15800.0	−0.500390

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.4.a.d.1.1	yes	1
3.2	odd	2		1152.4.a.d.1.1		1
4.3	odd	2		128.4.a.b.1.1	yes	1
8.3	odd	2		128.4.a.c.1.1	yes	1
8.5	even	2		128.4.a.a.1.1	✓	1
12.11	even	2		1152.4.a.c.1.1		1
16.3	odd	4		256.4.b.f.129.1		2
16.5	even	4		256.4.b.b.129.1		2
16.11	odd	4		256.4.b.f.129.2		2
16.13	even	4		256.4.b.b.129.2		2
24.5	odd	2		1152.4.a.j.1.1		1
24.11	even	2		1152.4.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.a.a.1.1	✓	1	8.5	even	2
128.4.a.b.1.1	yes	1	4.3	odd	2
128.4.a.c.1.1	yes	1	8.3	odd	2
128.4.a.d.1.1	yes	1	1.1	even	1	trivial
256.4.b.b.129.1		2	16.5	even	4
256.4.b.b.129.2		2	16.13	even	4
256.4.b.f.129.1		2	16.3	odd	4
256.4.b.f.129.2		2	16.11	odd	4
1152.4.a.c.1.1		1	12.11	even	2
1152.4.a.d.1.1		1	3.2	odd	2
1152.4.a.i.1.1		1	24.11	even	2
1152.4.a.j.1.1		1	24.5	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	128.4.a.d.1.1

Level	$128$
Weight	$4$
Character	128.1
Self dual	yes
Analytic conductor	$7.552$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$