LMFDB - Embedded newform 128.4.b.a.65.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [128,4,Mod(65,128)]

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, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("128.65");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 128 = 2^{7} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	128.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$7.55224448073$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			65.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	128.65
Dual form			128.4.b.a.65.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000i q^{3} +12.0000i q^{5} -32.0000 q^{7} -37.0000 q^{9} +O(q^{10})$	$q-8.00000i q^{3} +12.0000i q^{5} -32.0000 q^{7} -37.0000 q^{9} -8.00000i q^{11} +20.0000i q^{13} +96.0000 q^{15} -98.0000 q^{17} +88.0000i q^{19} +256.000i q^{21} +32.0000 q^{23} -19.0000 q^{25} +80.0000i q^{27} -172.000i q^{29} -256.000 q^{31} -64.0000 q^{33} -384.000i q^{35} +92.0000i q^{37} +160.000 q^{39} -102.000 q^{41} -296.000i q^{43} -444.000i q^{45} -320.000 q^{47} +681.000 q^{49} +784.000i q^{51} +76.0000i q^{53} +96.0000 q^{55} +704.000 q^{57} +408.000i q^{59} -636.000i q^{61} +1184.00 q^{63} -240.000 q^{65} -552.000i q^{67} -256.000i q^{69} -416.000 q^{71} -138.000 q^{73} +152.000i q^{75} +256.000i q^{77} -64.0000 q^{79} -359.000 q^{81} -392.000i q^{83} -1176.00i q^{85} -1376.00 q^{87} +582.000 q^{89} -640.000i q^{91} +2048.00i q^{93} -1056.00 q^{95} +238.000 q^{97} +296.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 64 q^{7} - 74 q^{9}+O(q^{10}) $ 2 * q - 64 * q^7 - 74 * q^9	$ 2 q - 64 q^{7} - 74 q^{9} + 192 q^{15} - 196 q^{17} + 64 q^{23} - 38 q^{25} - 512 q^{31} - 128 q^{33} + 320 q^{39} - 204 q^{41} - 640 q^{47} + 1362 q^{49} + 192 q^{55} + 1408 q^{57} + 2368 q^{63} - 480 q^{65} - 832 q^{71} - 276 q^{73} - 128 q^{79} - 718 q^{81} - 2752 q^{87} + 1164 q^{89} - 2112 q^{95} + 476 q^{97}+O(q^{100}) $ 2 * q - 64 * q^7 - 74 * q^9 + 192 * q^15 - 196 * q^17 + 64 * q^23 - 38 * q^25 - 512 * q^31 - 128 * q^33 + 320 * q^39 - 204 * q^41 - 640 * q^47 + 1362 * q^49 + 192 * q^55 + 1408 * q^57 + 2368 * q^63 - 480 * q^65 - 832 * q^71 - 276 * q^73 - 128 * q^79 - 718 * q^81 - 2752 * q^87 + 1164 * q^89 - 2112 * q^95 + 476 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/128\mathbb{Z}\right)^\times$.

$n$	$5$	$127$
$\chi(n)$	$-1$	$1$

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
$3$	− 8.00000i	− 1.53960i	−0.638285	−	0.769800i	$-0.720356\pi$
$3$	− 8.00000i	− 1.53960i	0.638285	−	0.769800i	$-0.279644\pi$
$4$	0	0
$5$	12.0000i	1.07331i	0.843801	+	0.536656i	$0.180313\pi$
$5$	12.0000i	1.07331i	−0.843801	+	0.536656i	$0.819687\pi$
$6$	0	0
$7$	−32.0000	−1.72784	−0.863919	−	0.503631i	$-0.831997\pi$
$7$	−32.0000	−1.72784	−0.863919	+	0.503631i	$0.831997\pi$
$8$	0	0
$9$	−37.0000	−1.37037
$10$	0	0
$11$	− 8.00000i	− 0.219281i	−0.993971	−	0.109640i	$-0.965030\pi$
$11$	− 8.00000i	− 0.219281i	0.993971	−	0.109640i	$-0.0349700\pi$
$12$	0	0
$13$	20.0000i	0.426692i	0.976977	+	0.213346i	$0.0684362\pi$
$13$	20.0000i	0.426692i	−0.976977	+	0.213346i	$0.931564\pi$
$14$	0	0
$15$	96.0000	1.65247
$16$	0	0
$17$	−98.0000	−1.39815	−0.699073	−	0.715050i	$-0.746404\pi$
$17$	−98.0000	−1.39815	−0.699073	+	0.715050i	$0.746404\pi$
$18$	0	0
$19$	88.0000i	1.06256i	0.847197	+	0.531279i	$0.178288\pi$
$19$	88.0000i	1.06256i	−0.847197	+	0.531279i	$0.821712\pi$
$20$	0	0
$21$	256.000i	2.66018i
$22$	0	0
$23$	32.0000	0.290107	0.145054	−	0.989424i	$-0.453665\pi$
$23$	32.0000	0.290107	0.145054	+	0.989424i	$0.453665\pi$
$24$	0	0
$25$	−19.0000	−0.152000
$26$	0	0
$27$	80.0000i	0.570222i
$28$	0	0
$29$	− 172.000i	− 1.10137i	−0.834715	−	0.550683i	$-0.814367\pi$
$29$	− 172.000i	− 1.10137i	0.834715	−	0.550683i	$-0.185633\pi$
$30$	0	0
$31$	−256.000	−1.48319	−0.741596	−	0.670847i	$-0.765931\pi$
$31$	−256.000	−1.48319	−0.741596	+	0.670847i	$0.765931\pi$
$32$	0	0
$33$	−64.0000	−0.337605
$34$	0	0
$35$	− 384.000i	− 1.85451i
$36$	0	0
$37$	92.0000i	0.408776i	0.978890	+	0.204388i	$0.0655204\pi$
$37$	92.0000i	0.408776i	−0.978890	+	0.204388i	$0.934480\pi$
$38$	0	0
$39$	160.000	0.656936
$40$	0	0
$41$	−102.000	−0.388530	−0.194265	−	0.980949i	$-0.562232\pi$
$41$	−102.000	−0.388530	−0.194265	+	0.980949i	$0.562232\pi$
$42$	0	0
$43$	− 296.000i	− 1.04976i	−0.851177	−	0.524879i	$-0.824111\pi$
$43$	− 296.000i	− 1.04976i	0.851177	−	0.524879i	$-0.175889\pi$
$44$	0	0
$45$	− 444.000i	− 1.47084i
$46$	0	0
$47$	−320.000	−0.993123	−0.496562	−	0.868001i	$-0.665404\pi$
$47$	−320.000	−0.993123	−0.496562	+	0.868001i	$0.665404\pi$
$48$	0	0
$49$	681.000	1.98542
$50$	0	0
$51$	784.000i	2.15259i
$52$	0	0
$53$	76.0000i	0.196970i	0.995139	+	0.0984849i	$0.0313996\pi$
$53$	76.0000i	0.196970i	−0.995139	+	0.0984849i	$0.968600\pi$
$54$	0	0
$55$	96.0000	0.235357
$56$	0	0
$57$	704.000	1.63591
$58$	0	0
$59$	408.000i	0.900289i	0.892956	+	0.450145i	$0.148628\pi$
$59$	408.000i	0.900289i	−0.892956	+	0.450145i	$0.851372\pi$
$60$	0	0
$61$	− 636.000i	− 1.33494i	−0.744636	−	0.667471i	$-0.767377\pi$
$61$	− 636.000i	− 1.33494i	0.744636	−	0.667471i	$-0.232623\pi$
$62$	0	0
$63$	1184.00	2.36778
$64$	0	0
$65$	−240.000	−0.457974
$66$	0	0
$67$	− 552.000i	− 1.00653i	−0.864132	−	0.503265i	$-0.832132\pi$
$67$	− 552.000i	− 1.00653i	0.864132	−	0.503265i	$-0.167868\pi$
$68$	0	0
$69$	− 256.000i	− 0.446649i
$70$	0	0
$71$	−416.000	−0.695354	−0.347677	−	0.937614i	$-0.613029\pi$
$71$	−416.000	−0.695354	−0.347677	+	0.937614i	$0.613029\pi$
$72$	0	0
$73$	−138.000	−0.221256	−0.110628	−	0.993862i	$-0.535286\pi$
$73$	−138.000	−0.221256	−0.110628	+	0.993862i	$0.535286\pi$
$74$	0	0
$75$	152.000i	0.234019i
$76$	0	0
$77$	256.000i	0.378882i
$78$	0	0
$79$	−64.0000	−0.0911464	−0.0455732	−	0.998961i	$-0.514511\pi$
$79$	−64.0000	−0.0911464	−0.0455732	+	0.998961i	$0.514511\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	− 392.000i	− 0.518405i	−0.965823	−	0.259202i	$-0.916540\pi$
$83$	− 392.000i	− 0.518405i	0.965823	−	0.259202i	$-0.0834597\pi$
$84$	0	0
$85$	− 1176.00i	− 1.50065i
$86$	0	0
$87$	−1376.00	−1.69566
$88$	0	0
$89$	582.000	0.693167	0.346584	−	0.938019i	$-0.387342\pi$
$89$	582.000	0.693167	0.346584	+	0.938019i	$0.387342\pi$
$90$	0	0
$91$	− 640.000i	− 0.737255i
$92$	0	0
$93$	2048.00i	2.28352i
$94$	0	0
$95$	−1056.00	−1.14046
$96$	0	0
$97$	238.000	0.249126	0.124563	−	0.992212i	$-0.460247\pi$
$97$	238.000	0.249126	0.124563	+	0.992212i	$0.460247\pi$
$98$	0	0
$99$	296.000i	0.300496i
$100$	0	0
$101$	1468.00i	1.44625i	0.690716	+	0.723126i	$0.257295\pi$
$101$	1468.00i	1.44625i	−0.690716	+	0.723126i	$0.742705\pi$
$102$	0	0
$103$	992.000	0.948977	0.474489	−	0.880262i	$-0.342633\pi$
$103$	992.000	0.948977	0.474489	+	0.880262i	$0.342633\pi$
$104$	0	0
$105$	−3072.00	−2.85520
$106$	0	0
$107$	− 584.000i	− 0.527639i	−0.964572	−	0.263820i	$-0.915018\pi$
$107$	− 584.000i	− 0.527639i	0.964572	−	0.263820i	$-0.0849824\pi$
$108$	0	0
$109$	740.000i	0.650267i	0.945668	+	0.325134i	$0.105409\pi$
$109$	740.000i	0.650267i	−0.945668	+	0.325134i	$0.894591\pi$
$110$	0	0
$111$	736.000	0.629352
$112$	0	0
$113$	−302.000	−0.251414	−0.125707	−	0.992067i	$-0.540120\pi$
$113$	−302.000	−0.251414	−0.125707	+	0.992067i	$0.540120\pi$
$114$	0	0
$115$	384.000i	0.311376i
$116$	0	0
$117$	− 740.000i	− 0.584727i
$118$	0	0
$119$	3136.00	2.41577
$120$	0	0
$121$	1267.00	0.951916
$122$	0	0
$123$	816.000i	0.598181i
$124$	0	0
$125$	1272.00i	0.910169i
$126$	0	0
$127$	−1664.00	−1.16265	−0.581323	−	0.813673i	$-0.697465\pi$
$127$	−1664.00	−1.16265	−0.581323	+	0.813673i	$0.697465\pi$
$128$	0	0
$129$	−2368.00	−1.61621
$130$	0	0
$131$	2328.00i	1.55266i	0.630327	+	0.776329i	$0.282921\pi$
$131$	2328.00i	1.55266i	−0.630327	+	0.776329i	$0.717079\pi$
$132$	0	0
$133$	− 2816.00i	− 1.83593i
$134$	0	0
$135$	−960.000	−0.612027
$136$	0	0
$137$	−1734.00	−1.08135	−0.540677	−	0.841230i	$-0.681832\pi$
$137$	−1734.00	−1.08135	−0.540677	+	0.841230i	$0.681832\pi$
$138$	0	0
$139$	3032.00i	1.85015i	0.379784	+	0.925075i	$0.375998\pi$
$139$	3032.00i	1.85015i	−0.379784	+	0.925075i	$0.624002\pi$
$140$	0	0
$141$	2560.00i	1.52901i
$142$	0	0
$143$	160.000	0.0935655
$144$	0	0
$145$	2064.00	1.18211
$146$	0	0
$147$	− 5448.00i	− 3.05676i
$148$	0	0
$149$	1788.00i	0.983078i	0.870855	+	0.491539i	$0.163566\pi$
$149$	1788.00i	0.983078i	−0.870855	+	0.491539i	$0.836434\pi$
$150$	0	0
$151$	−480.000	−0.258688	−0.129344	−	0.991600i	$-0.541287\pi$
$151$	−480.000	−0.258688	−0.129344	+	0.991600i	$0.541287\pi$
$152$	0	0
$153$	3626.00	1.91598
$154$	0	0
$155$	− 3072.00i	− 1.59193i
$156$	0	0
$157$	− 2300.00i	− 1.16917i	−0.811332	−	0.584586i	$-0.801257\pi$
$157$	− 2300.00i	− 1.16917i	0.811332	−	0.584586i	$-0.198743\pi$
$158$	0	0
$159$	608.000	0.303255
$160$	0	0
$161$	−1024.00	−0.501258
$162$	0	0
$163$	1592.00i	0.765000i	0.923955	+	0.382500i	$0.124937\pi$
$163$	1592.00i	0.765000i	−0.923955	+	0.382500i	$0.875063\pi$
$164$	0	0
$165$	− 768.000i	− 0.362356i
$166$	0	0
$167$	−2208.00	−1.02311	−0.511557	−	0.859249i	$-0.670931\pi$
$167$	−2208.00	−1.02311	−0.511557	+	0.859249i	$0.670931\pi$
$168$	0	0
$169$	1797.00	0.817934
$170$	0	0
$171$	− 3256.00i	− 1.45610i
$172$	0	0
$173$	− 3948.00i	− 1.73503i	−0.497408	−	0.867517i	$-0.665715\pi$
$173$	− 3948.00i	− 1.73503i	0.497408	−	0.867517i	$-0.334285\pi$
$174$	0	0
$175$	608.000	0.262631
$176$	0	0
$177$	3264.00	1.38609
$178$	0	0
$179$	2104.00i	0.878549i	0.898353	+	0.439275i	$0.144765\pi$
$179$	2104.00i	0.878549i	−0.898353	+	0.439275i	$0.855235\pi$
$180$	0	0
$181$	− 1412.00i	− 0.579852i	−0.957049	−	0.289926i	$-0.906369\pi$
$181$	− 1412.00i	− 0.579852i	0.957049	−	0.289926i	$-0.0936306\pi$
$182$	0	0
$183$	−5088.00	−2.05528
$184$	0	0
$185$	−1104.00	−0.438744
$186$	0	0
$187$	784.000i	0.306587i
$188$	0	0
$189$	− 2560.00i	− 0.985252i
$190$	0	0
$191$	3712.00	1.40624	0.703118	−	0.711074i	$-0.251791\pi$
$191$	3712.00	1.40624	0.703118	+	0.711074i	$0.251791\pi$
$192$	0	0
$193$	1614.00	0.601960	0.300980	−	0.953630i	$-0.402686\pi$
$193$	1614.00	0.601960	0.300980	+	0.953630i	$0.402686\pi$
$194$	0	0
$195$	1920.00i	0.705098i
$196$	0	0
$197$	684.000i	0.247376i	0.992321	+	0.123688i	$0.0394721\pi$
$197$	684.000i	0.247376i	−0.992321	+	0.123688i	$0.960528\pi$
$198$	0	0
$199$	4064.00	1.44769	0.723843	−	0.689965i	$-0.242374\pi$
$199$	4064.00	1.44769	0.723843	+	0.689965i	$0.242374\pi$
$200$	0	0
$201$	−4416.00	−1.54965
$202$	0	0
$203$	5504.00i	1.90298i
$204$	0	0
$205$	− 1224.00i	− 0.417014i
$206$	0	0
$207$	−1184.00	−0.397554
$208$	0	0
$209$	704.000	0.232999
$210$	0	0
$211$	− 2120.00i	− 0.691691i	−0.938291	−	0.345846i	$-0.887592\pi$
$211$	− 2120.00i	− 0.691691i	0.938291	−	0.345846i	$-0.112408\pi$
$212$	0	0
$213$	3328.00i	1.07057i
$214$	0	0
$215$	3552.00	1.12672
$216$	0	0
$217$	8192.00	2.56272
$218$	0	0
$219$	1104.00i	0.340646i
$220$	0	0
$221$	− 1960.00i	− 0.596579i
$222$	0	0
$223$	2816.00	0.845620	0.422810	−	0.906218i	$-0.361044\pi$
$223$	2816.00	0.845620	0.422810	+	0.906218i	$0.361044\pi$
$224$	0	0
$225$	703.000	0.208296
$226$	0	0
$227$	− 3848.00i	− 1.12511i	−0.826759	−	0.562557i	$-0.809818\pi$
$227$	− 3848.00i	− 1.12511i	0.826759	−	0.562557i	$-0.190182\pi$
$228$	0	0
$229$	652.000i	0.188146i	0.995565	+	0.0940729i	$0.0299887\pi$
$229$	652.000i	0.188146i	−0.995565	+	0.0940729i	$0.970011\pi$
$230$	0	0
$231$	2048.00	0.583327
$232$	0	0
$233$	−3050.00	−0.857563	−0.428781	−	0.903408i	$-0.641057\pi$
$233$	−3050.00	−0.857563	−0.428781	+	0.903408i	$0.641057\pi$
$234$	0	0
$235$	− 3840.00i	− 1.06593i
$236$	0	0
$237$	512.000i	0.140329i
$238$	0	0
$239$	−6336.00	−1.71482	−0.857410	−	0.514635i	$-0.827928\pi$
$239$	−6336.00	−1.71482	−0.857410	+	0.514635i	$0.827928\pi$
$240$	0	0
$241$	−4610.00	−1.23218	−0.616092	−	0.787674i	$-0.711285\pi$
$241$	−4610.00	−1.23218	−0.616092	+	0.787674i	$0.711285\pi$
$242$	0	0
$243$	5032.00i	1.32841i
$244$	0	0
$245$	8172.00i	2.13098i
$246$	0	0
$247$	−1760.00	−0.453385
$248$	0	0
$249$	−3136.00	−0.798136
$250$	0	0
$251$	792.000i	0.199166i	0.995029	+	0.0995829i	$0.0317508\pi$
$251$	792.000i	0.199166i	−0.995029	+	0.0995829i	$0.968249\pi$
$252$	0	0
$253$	− 256.000i	− 0.0636149i
$254$	0	0
$255$	−9408.00	−2.31040
$256$	0	0
$257$	−5374.00	−1.30436	−0.652181	−	0.758063i	$-0.726146\pi$
$257$	−5374.00	−1.30436	−0.652181	+	0.758063i	$0.726146\pi$
$258$	0	0
$259$	− 2944.00i	− 0.706298i
$260$	0	0
$261$	6364.00i	1.50928i
$262$	0	0
$263$	−3488.00	−0.817792	−0.408896	−	0.912581i	$-0.634086\pi$
$263$	−3488.00	−0.817792	−0.408896	+	0.912581i	$0.634086\pi$
$264$	0	0
$265$	−912.000	−0.211410
$266$	0	0
$267$	− 4656.00i	− 1.06720i
$268$	0	0
$269$	− 4764.00i	− 1.07980i	−0.841729	−	0.539900i	$-0.818462\pi$
$269$	− 4764.00i	− 1.07980i	0.841729	−	0.539900i	$-0.181538\pi$
$270$	0	0
$271$	1344.00	0.301263	0.150631	−	0.988590i	$-0.451869\pi$
$271$	1344.00	0.301263	0.150631	+	0.988590i	$0.451869\pi$
$272$	0	0
$273$	−5120.00	−1.13508
$274$	0	0
$275$	152.000i	0.0333307i
$276$	0	0
$277$	− 8596.00i	− 1.86456i	−0.361735	−	0.932281i	$-0.617816\pi$
$277$	− 8596.00i	− 1.86456i	0.361735	−	0.932281i	$-0.382184\pi$
$278$	0	0
$279$	9472.00	2.03252
$280$	0	0
$281$	−2874.00	−0.610137	−0.305068	−	0.952330i	$-0.598679\pi$
$281$	−2874.00	−0.610137	−0.305068	+	0.952330i	$0.598679\pi$
$282$	0	0
$283$	− 2888.00i	− 0.606621i	−0.952892	−	0.303311i	$-0.901908\pi$
$283$	− 2888.00i	− 0.606621i	0.952892	−	0.303311i	$-0.0980920\pi$
$284$	0	0
$285$	8448.00i	1.75585i
$286$	0	0
$287$	3264.00	0.671316
$288$	0	0
$289$	4691.00	0.954814
$290$	0	0
$291$	− 1904.00i	− 0.383555i
$292$	0	0
$293$	6540.00i	1.30400i	0.758221	+	0.651998i	$0.226069\pi$
$293$	6540.00i	1.30400i	−0.758221	+	0.651998i	$0.773931\pi$
$294$	0	0
$295$	−4896.00	−0.966292
$296$	0	0
$297$	640.000	0.125039
$298$	0	0
$299$	640.000i	0.123786i
$300$	0	0
$301$	9472.00i	1.81381i
$302$	0	0
$303$	11744.0	2.22665
$304$	0	0
$305$	7632.00	1.43281
$306$	0	0
$307$	10584.0i	1.96762i	0.179202	+	0.983812i	$0.442649\pi$
$307$	10584.0i	1.96762i	−0.179202	+	0.983812i	$0.557351\pi$
$308$	0	0
$309$	− 7936.00i	− 1.46105i
$310$	0	0
$311$	−6368.00	−1.16108	−0.580540	−	0.814231i	$-0.697159\pi$
$311$	−6368.00	−1.16108	−0.580540	+	0.814231i	$0.697159\pi$
$312$	0	0
$313$	−8758.00	−1.58157	−0.790785	−	0.612094i	$-0.790327\pi$
$313$	−8758.00	−1.58157	−0.790785	+	0.612094i	$0.790327\pi$
$314$	0	0
$315$	14208.0i	2.54137i
$316$	0	0
$317$	− 716.000i	− 0.126860i	−0.997986	−	0.0634299i	$-0.979796\pi$
$317$	− 716.000i	− 0.126860i	0.997986	−	0.0634299i	$-0.0202039\pi$
$318$	0	0
$319$	−1376.00	−0.241508
$320$	0	0
$321$	−4672.00	−0.812354
$322$	0	0
$323$	− 8624.00i	− 1.48561i
$324$	0	0
$325$	− 380.000i	− 0.0648573i
$326$	0	0
$327$	5920.00	1.00115
$328$	0	0
$329$	10240.0	1.71596
$330$	0	0
$331$	4408.00i	0.731981i	0.930619	+	0.365990i	$0.119270\pi$
$331$	4408.00i	0.731981i	−0.930619	+	0.365990i	$0.880730\pi$
$332$	0	0
$333$	− 3404.00i	− 0.560174i
$334$	0	0
$335$	6624.00	1.08032
$336$	0	0
$337$	1202.00	0.194294	0.0971471	−	0.995270i	$-0.469028\pi$
$337$	1202.00	0.194294	0.0971471	+	0.995270i	$0.469028\pi$
$338$	0	0
$339$	2416.00i	0.387077i
$340$	0	0
$341$	2048.00i	0.325236i
$342$	0	0
$343$	−10816.0	−1.70265
$344$	0	0
$345$	3072.00	0.479394
$346$	0	0
$347$	− 5160.00i	− 0.798280i	−0.916890	−	0.399140i	$-0.869309\pi$
$347$	− 5160.00i	− 0.798280i	0.916890	−	0.399140i	$-0.130691\pi$
$348$	0	0
$349$	− 4876.00i	− 0.747869i	−0.927455	−	0.373935i	$-0.878008\pi$
$349$	− 4876.00i	− 0.747869i	0.927455	−	0.373935i	$-0.121992\pi$
$350$	0	0
$351$	−1600.00	−0.243310
$352$	0	0
$353$	4834.00	0.728861	0.364430	−	0.931231i	$-0.381264\pi$
$353$	4834.00	0.728861	0.364430	+	0.931231i	$0.381264\pi$
$354$	0	0
$355$	− 4992.00i	− 0.746332i
$356$	0	0
$357$	− 25088.0i	− 3.71932i
$358$	0	0
$359$	−4128.00	−0.606873	−0.303437	−	0.952852i	$-0.598134\pi$
$359$	−4128.00	−0.606873	−0.303437	+	0.952852i	$0.598134\pi$
$360$	0	0
$361$	−885.000	−0.129028
$362$	0	0
$363$	− 10136.0i	− 1.46557i
$364$	0	0
$365$	− 1656.00i	− 0.237477i
$366$	0	0
$367$	−4416.00	−0.628102	−0.314051	−	0.949406i	$-0.601686\pi$
$367$	−4416.00	−0.628102	−0.314051	+	0.949406i	$0.601686\pi$
$368$	0	0
$369$	3774.00	0.532430
$370$	0	0
$371$	− 2432.00i	− 0.340332i
$372$	0	0
$373$	− 4180.00i	− 0.580247i	−0.956989	−	0.290124i	$-0.906304\pi$
$373$	− 4180.00i	− 0.580247i	0.956989	−	0.290124i	$-0.0936964\pi$
$374$	0	0
$375$	10176.0	1.40130
$376$	0	0
$377$	3440.00	0.469944
$378$	0	0
$379$	− 13736.0i	− 1.86166i	−0.365446	−	0.930832i	$-0.619084\pi$
$379$	− 13736.0i	− 1.86166i	0.365446	−	0.930832i	$-0.380916\pi$
$380$	0	0
$381$	13312.0i	1.79001i
$382$	0	0
$383$	512.000	0.0683080	0.0341540	−	0.999417i	$-0.489126\pi$
$383$	512.000	0.0683080	0.0341540	+	0.999417i	$0.489126\pi$
$384$	0	0
$385$	−3072.00	−0.406659
$386$	0	0
$387$	10952.0i	1.43856i
$388$	0	0
$389$	− 1732.00i	− 0.225748i	−0.993609	−	0.112874i	$-0.963994\pi$
$389$	− 1732.00i	− 0.225748i	0.993609	−	0.112874i	$-0.0360056\pi$
$390$	0	0
$391$	−3136.00	−0.405612
$392$	0	0
$393$	18624.0	2.39047
$394$	0	0
$395$	− 768.000i	− 0.0978285i
$396$	0	0
$397$	10436.0i	1.31931i	0.751567	+	0.659657i	$0.229299\pi$
$397$	10436.0i	1.31931i	−0.751567	+	0.659657i	$0.770701\pi$
$398$	0	0
$399$	−22528.0	−2.82659
$400$	0	0
$401$	−12130.0	−1.51058	−0.755291	−	0.655390i	$-0.772504\pi$
$401$	−12130.0	−1.51058	−0.755291	+	0.655390i	$0.772504\pi$
$402$	0	0
$403$	− 5120.00i	− 0.632867i
$404$	0	0
$405$	− 4308.00i	− 0.528559i
$406$	0	0
$407$	736.000	0.0896368
$408$	0	0
$409$	−5014.00	−0.606177	−0.303088	−	0.952962i	$-0.598018\pi$
$409$	−5014.00	−0.606177	−0.303088	+	0.952962i	$0.598018\pi$
$410$	0	0
$411$	13872.0i	1.66485i
$412$	0	0
$413$	− 13056.0i	− 1.55555i
$414$	0	0
$415$	4704.00	0.556410
$416$	0	0
$417$	24256.0	2.84849
$418$	0	0
$419$	8024.00i	0.935556i	0.883846	+	0.467778i	$0.154945\pi$
$419$	8024.00i	0.935556i	−0.883846	+	0.467778i	$0.845055\pi$
$420$	0	0
$421$	2348.00i	0.271816i	0.990721	+	0.135908i	$0.0433952\pi$
$421$	2348.00i	0.271816i	−0.990721	+	0.135908i	$0.956605\pi$
$422$	0	0
$423$	11840.0	1.36095
$424$	0	0
$425$	1862.00	0.212518
$426$	0	0
$427$	20352.0i	2.30656i
$428$	0	0
$429$	− 1280.00i	− 0.144054i
$430$	0	0
$431$	−1728.00	−0.193120	−0.0965601	−	0.995327i	$-0.530784\pi$
$431$	−1728.00	−0.193120	−0.0965601	+	0.995327i	$0.530784\pi$
$432$	0	0
$433$	62.0000	0.00688113	0.00344057	−	0.999994i	$-0.498905\pi$
$433$	62.0000	0.00688113	0.00344057	+	0.999994i	$0.498905\pi$
$434$	0	0
$435$	− 16512.0i	− 1.81998i
$436$	0	0
$437$	2816.00i	0.308255i
$438$	0	0
$439$	14112.0	1.53423	0.767117	−	0.641507i	$-0.221690\pi$
$439$	14112.0	1.53423	0.767117	+	0.641507i	$0.221690\pi$
$440$	0	0
$441$	−25197.0	−2.72076
$442$	0	0
$443$	− 4488.00i	− 0.481335i	−0.970608	−	0.240667i	$-0.922634\pi$
$443$	− 4488.00i	− 0.481335i	0.970608	−	0.240667i	$-0.0773663\pi$
$444$	0	0
$445$	6984.00i	0.743985i
$446$	0	0
$447$	14304.0	1.51355
$448$	0	0
$449$	−2482.00	−0.260875	−0.130437	−	0.991457i	$-0.541638\pi$
$449$	−2482.00	−0.260875	−0.130437	+	0.991457i	$0.541638\pi$
$450$	0	0
$451$	816.000i	0.0851972i
$452$	0	0
$453$	3840.00i	0.398276i
$454$	0	0
$455$	7680.00	0.791305
$456$	0	0
$457$	−5894.00	−0.603303	−0.301652	−	0.953418i	$-0.597538\pi$
$457$	−5894.00	−0.603303	−0.301652	+	0.953418i	$0.597538\pi$
$458$	0	0
$459$	− 7840.00i	− 0.797255i
$460$	0	0
$461$	− 7068.00i	− 0.714077i	−0.934090	−	0.357039i	$-0.883786\pi$
$461$	− 7068.00i	− 0.714077i	0.934090	−	0.357039i	$-0.116214\pi$
$462$	0	0
$463$	7616.00	0.764461	0.382231	−	0.924067i	$-0.375156\pi$
$463$	7616.00	0.764461	0.382231	+	0.924067i	$0.375156\pi$
$464$	0	0
$465$	−24576.0	−2.45093
$466$	0	0
$467$	13080.0i	1.29608i	0.761606	+	0.648041i	$0.224411\pi$
$467$	13080.0i	1.29608i	−0.761606	+	0.648041i	$0.775589\pi$
$468$	0	0
$469$	17664.0i	1.73912i
$470$	0	0
$471$	−18400.0	−1.80006
$472$	0	0
$473$	−2368.00	−0.230192
$474$	0	0
$475$	− 1672.00i	− 0.161509i
$476$	0	0
$477$	− 2812.00i	− 0.269922i
$478$	0	0
$479$	−13568.0	−1.29423	−0.647117	−	0.762391i	$-0.724025\pi$
$479$	−13568.0	−1.29423	−0.647117	+	0.762391i	$0.724025\pi$
$480$	0	0
$481$	−1840.00	−0.174422
$482$	0	0
$483$	8192.00i	0.771737i
$484$	0	0
$485$	2856.00i	0.267390i
$486$	0	0
$487$	−1696.00	−0.157809	−0.0789046	−	0.996882i	$-0.525142\pi$
$487$	−1696.00	−0.157809	−0.0789046	+	0.996882i	$0.525142\pi$
$488$	0	0
$489$	12736.0	1.17780
$490$	0	0
$491$	3096.00i	0.284563i	0.989826	+	0.142282i	$0.0454439\pi$
$491$	3096.00i	0.284563i	−0.989826	+	0.142282i	$0.954556\pi$
$492$	0	0
$493$	16856.0i	1.53987i
$494$	0	0
$495$	−3552.00	−0.322526
$496$	0	0
$497$	13312.0	1.20146
$498$	0	0
$499$	− 19208.0i	− 1.72318i	−0.507603	−	0.861591i	$-0.669468\pi$
$499$	− 19208.0i	− 1.72318i	0.507603	−	0.861591i	$-0.330532\pi$
$500$	0	0
$501$	17664.0i	1.57519i
$502$	0	0
$503$	−16224.0	−1.43816	−0.719078	−	0.694929i	$-0.755436\pi$
$503$	−16224.0	−1.43816	−0.719078	+	0.694929i	$0.755436\pi$
$504$	0	0
$505$	−17616.0	−1.55228
$506$	0	0
$507$	− 14376.0i	− 1.25929i
$508$	0	0
$509$	− 11292.0i	− 0.983318i	−0.870788	−	0.491659i	$-0.836391\pi$
$509$	− 11292.0i	− 0.983318i	0.870788	−	0.491659i	$-0.163609\pi$
$510$	0	0
$511$	4416.00	0.382294
$512$	0	0
$513$	−7040.00	−0.605894
$514$	0	0
$515$	11904.0i	1.01855i
$516$	0	0
$517$	2560.00i	0.217773i
$518$	0	0
$519$	−31584.0	−2.67126
$520$	0	0
$521$	5178.00	0.435417	0.217709	−	0.976014i	$-0.430142\pi$
$521$	5178.00	0.435417	0.217709	+	0.976014i	$0.430142\pi$
$522$	0	0
$523$	− 6856.00i	− 0.573216i	−0.958048	−	0.286608i	$-0.907472\pi$
$523$	− 6856.00i	− 0.573216i	0.958048	−	0.286608i	$-0.0925277\pi$
$524$	0	0
$525$	− 4864.00i	− 0.404347i
$526$	0	0
$527$	25088.0	2.07372
$528$	0	0
$529$	−11143.0	−0.915838
$530$	0	0
$531$	− 15096.0i	− 1.23373i
$532$	0	0
$533$	− 2040.00i	− 0.165783i
$534$	0	0
$535$	7008.00	0.566322
$536$	0	0
$537$	16832.0	1.35262
$538$	0	0
$539$	− 5448.00i	− 0.435365i
$540$	0	0
$541$	13732.0i	1.09128i	0.838018	+	0.545642i	$0.183714\pi$
$541$	13732.0i	1.09128i	−0.838018	+	0.545642i	$0.816286\pi$
$542$	0	0
$543$	−11296.0	−0.892740
$544$	0	0
$545$	−8880.00	−0.697940
$546$	0	0
$547$	10968.0i	0.857327i	0.903464	+	0.428663i	$0.141015\pi$
$547$	10968.0i	0.857327i	−0.903464	+	0.428663i	$0.858985\pi$
$548$	0	0
$549$	23532.0i	1.82936i
$550$	0	0
$551$	15136.0	1.17026
$552$	0	0
$553$	2048.00	0.157486
$554$	0	0
$555$	8832.00i	0.675491i
$556$	0	0
$557$	− 25612.0i	− 1.94832i	−0.225855	−	0.974161i	$-0.572518\pi$
$557$	− 25612.0i	− 1.94832i	0.225855	−	0.974161i	$-0.427482\pi$
$558$	0	0
$559$	5920.00	0.447924
$560$	0	0
$561$	6272.00	0.472021
$562$	0	0
$563$	− 9768.00i	− 0.731212i	−0.930770	−	0.365606i	$-0.880862\pi$
$563$	− 9768.00i	− 0.731212i	0.930770	−	0.365606i	$-0.119138\pi$
$564$	0	0
$565$	− 3624.00i	− 0.269846i
$566$	0	0
$567$	11488.0	0.850883
$568$	0	0
$569$	−22838.0	−1.68263	−0.841317	−	0.540542i	$-0.818219\pi$
$569$	−22838.0	−1.68263	−0.841317	+	0.540542i	$0.818219\pi$
$570$	0	0
$571$	9208.00i	0.674856i	0.941351	+	0.337428i	$0.109557\pi$
$571$	9208.00i	0.674856i	−0.941351	+	0.337428i	$0.890443\pi$
$572$	0	0
$573$	− 29696.0i	− 2.16504i
$574$	0	0
$575$	−608.000	−0.0440963
$576$	0	0
$577$	−10878.0	−0.784848	−0.392424	−	0.919785i	$-0.628363\pi$
$577$	−10878.0	−0.784848	−0.392424	+	0.919785i	$0.628363\pi$
$578$	0	0
$579$	− 12912.0i	− 0.926778i
$580$	0	0
$581$	12544.0i	0.895719i
$582$	0	0
$583$	608.000	0.0431917
$584$	0	0
$585$	8880.00	0.627595
$586$	0	0
$587$	18040.0i	1.26847i	0.773141	+	0.634234i	$0.218684\pi$
$587$	18040.0i	1.26847i	−0.773141	+	0.634234i	$0.781316\pi$
$588$	0	0
$589$	− 22528.0i	− 1.57598i
$590$	0	0
$591$	5472.00	0.380860
$592$	0	0
$593$	26994.0	1.86933	0.934663	−	0.355534i	$-0.115701\pi$
$593$	26994.0	1.86933	0.934663	+	0.355534i	$0.115701\pi$
$594$	0	0
$595$	37632.0i	2.59288i
$596$	0	0
$597$	− 32512.0i	− 2.22886i
$598$	0	0
$599$	18336.0	1.25073	0.625366	−	0.780331i	$-0.284950\pi$
$599$	18336.0	1.25073	0.625366	+	0.780331i	$0.284950\pi$
$600$	0	0
$601$	9286.00	0.630256	0.315128	−	0.949049i	$-0.397953\pi$
$601$	9286.00	0.630256	0.315128	+	0.949049i	$0.397953\pi$
$602$	0	0
$603$	20424.0i	1.37932i
$604$	0	0
$605$	15204.0i	1.02170i
$606$	0	0
$607$	−17536.0	−1.17259	−0.586297	−	0.810096i	$-0.699415\pi$
$607$	−17536.0	−1.17259	−0.586297	+	0.810096i	$0.699415\pi$
$608$	0	0
$609$	44032.0	2.92983
$610$	0	0
$611$	− 6400.00i	− 0.423758i
$612$	0	0
$613$	5868.00i	0.386633i	0.981136	+	0.193317i	$0.0619245\pi$
$613$	5868.00i	0.386633i	−0.981136	+	0.193317i	$0.938076\pi$
$614$	0	0
$615$	−9792.00	−0.642035
$616$	0	0
$617$	19286.0	1.25839	0.629194	−	0.777248i	$-0.283385\pi$
$617$	19286.0	1.25839	0.629194	+	0.777248i	$0.283385\pi$
$618$	0	0
$619$	5240.00i	0.340248i	0.985423	+	0.170124i	$0.0544168\pi$
$619$	5240.00i	0.340248i	−0.985423	+	0.170124i	$0.945583\pi$
$620$	0	0
$621$	2560.00i	0.165426i
$622$	0	0
$623$	−18624.0	−1.19768
$624$	0	0
$625$	−17639.0	−1.12890
$626$	0	0
$627$	− 5632.00i	− 0.358725i
$628$	0	0
$629$	− 9016.00i	− 0.571529i
$630$	0	0
$631$	15520.0	0.979147	0.489573	−	0.871962i	$-0.337153\pi$
$631$	15520.0	0.979147	0.489573	+	0.871962i	$0.337153\pi$
$632$	0	0
$633$	−16960.0	−1.06493
$634$	0	0
$635$	− 19968.0i	− 1.24788i
$636$	0	0
$637$	13620.0i	0.847165i
$638$	0	0
$639$	15392.0	0.952892
$640$	0	0
$641$	654.000	0.0402987	0.0201493	−	0.999797i	$-0.493586\pi$
$641$	654.000	0.0402987	0.0201493	+	0.999797i	$0.493586\pi$
$642$	0	0
$643$	− 8232.00i	− 0.504881i	−0.967612	−	0.252440i	$-0.918767\pi$
$643$	− 8232.00i	− 0.504881i	0.967612	−	0.252440i	$-0.0812332\pi$
$644$	0	0
$645$	− 28416.0i	− 1.73470i
$646$	0	0
$647$	24672.0	1.49916	0.749580	−	0.661914i	$-0.230256\pi$
$647$	24672.0	1.49916	0.749580	+	0.661914i	$0.230256\pi$
$648$	0	0
$649$	3264.00	0.197416
$650$	0	0
$651$	− 65536.0i	− 3.94556i
$652$	0	0
$653$	22052.0i	1.32153i	0.750591	+	0.660767i	$0.229769\pi$
$653$	22052.0i	1.32153i	−0.750591	+	0.660767i	$0.770231\pi$
$654$	0	0
$655$	−27936.0	−1.66649
$656$	0	0
$657$	5106.00	0.303202
$658$	0	0
$659$	12024.0i	0.710757i	0.934723	+	0.355378i	$0.115648\pi$
$659$	12024.0i	0.710757i	−0.934723	+	0.355378i	$0.884352\pi$
$660$	0	0
$661$	19100.0i	1.12391i	0.827168	+	0.561955i	$0.189950\pi$
$661$	19100.0i	1.12391i	−0.827168	+	0.561955i	$0.810050\pi$
$662$	0	0
$663$	−15680.0	−0.918493
$664$	0	0
$665$	33792.0	1.97052
$666$	0	0
$667$	− 5504.00i	− 0.319514i
$668$	0	0
$669$	− 22528.0i	− 1.30192i
$670$	0	0
$671$	−5088.00	−0.292727
$672$	0	0
$673$	9902.00	0.567153	0.283577	−	0.958950i	$-0.408479\pi$
$673$	9902.00	0.567153	0.283577	+	0.958950i	$0.408479\pi$
$674$	0	0
$675$	− 1520.00i	− 0.0866738i
$676$	0	0
$677$	− 19684.0i	− 1.11746i	−0.829351	−	0.558728i	$-0.811290\pi$
$677$	− 19684.0i	− 1.11746i	0.829351	−	0.558728i	$-0.188710\pi$
$678$	0	0
$679$	−7616.00	−0.430450
$680$	0	0
$681$	−30784.0	−1.73223
$682$	0	0
$683$	19864.0i	1.11285i	0.830899	+	0.556424i	$0.187827\pi$
$683$	19864.0i	1.11285i	−0.830899	+	0.556424i	$0.812173\pi$
$684$	0	0
$685$	− 20808.0i	− 1.16063i
$686$	0	0
$687$	5216.00	0.289669
$688$	0	0
$689$	−1520.00	−0.0840456
$690$	0	0
$691$	3256.00i	0.179253i	0.995975	+	0.0896267i	$0.0285674\pi$
$691$	3256.00i	0.179253i	−0.995975	+	0.0896267i	$0.971433\pi$
$692$	0	0
$693$	− 9472.00i	− 0.519209i
$694$	0	0
$695$	−36384.0	−1.98579
$696$	0	0
$697$	9996.00	0.543222
$698$	0	0
$699$	24400.0i	1.32030i
$700$	0	0
$701$	− 2876.00i	− 0.154957i	−0.996994	−	0.0774786i	$-0.975313\pi$
$701$	− 2876.00i	− 0.154957i	0.996994	−	0.0774786i	$-0.0246870\pi$
$702$	0	0
$703$	−8096.00	−0.434348
$704$	0	0
$705$	−30720.0	−1.64111
$706$	0	0
$707$	− 46976.0i	− 2.49889i
$708$	0	0
$709$	− 7300.00i	− 0.386682i	−0.981132	−	0.193341i	$-0.938068\pi$
$709$	− 7300.00i	− 0.386682i	0.981132	−	0.193341i	$-0.0619323\pi$
$710$	0	0
$711$	2368.00	0.124904
$712$	0	0
$713$	−8192.00	−0.430284
$714$	0	0
$715$	1920.00i	0.100425i
$716$	0	0
$717$	50688.0i	2.64014i
$718$	0	0
$719$	−2880.00	−0.149382	−0.0746912	−	0.997207i	$-0.523797\pi$
$719$	−2880.00	−0.149382	−0.0746912	+	0.997207i	$0.523797\pi$
$720$	0	0
$721$	−31744.0	−1.63968
$722$	0	0
$723$	36880.0i	1.89707i
$724$	0	0
$725$	3268.00i	0.167408i
$726$	0	0
$727$	−8800.00	−0.448933	−0.224466	−	0.974482i	$-0.572064\pi$
$727$	−8800.00	−0.448933	−0.224466	+	0.974482i	$0.572064\pi$
$728$	0	0
$729$	30563.0	1.55276
$730$	0	0
$731$	29008.0i	1.46771i
$732$	0	0
$733$	21076.0i	1.06202i	0.847366	+	0.531009i	$0.178187\pi$
$733$	21076.0i	1.06202i	−0.847366	+	0.531009i	$0.821813\pi$
$734$	0	0
$735$	65376.0	3.28086
$736$	0	0
$737$	−4416.00	−0.220713
$738$	0	0
$739$	− 19336.0i	− 0.962498i	−0.876584	−	0.481249i	$-0.840183\pi$
$739$	− 19336.0i	− 0.962498i	0.876584	−	0.481249i	$-0.159817\pi$
$740$	0	0
$741$	14080.0i	0.698032i
$742$	0	0
$743$	13664.0	0.674675	0.337338	−	0.941384i	$-0.390474\pi$
$743$	13664.0	0.674675	0.337338	+	0.941384i	$0.390474\pi$
$744$	0	0
$745$	−21456.0	−1.05515
$746$	0	0
$747$	14504.0i	0.710406i
$748$	0	0
$749$	18688.0i	0.911675i
$750$	0	0
$751$	19520.0	0.948462	0.474231	−	0.880400i	$-0.342726\pi$
$751$	19520.0	0.948462	0.474231	+	0.880400i	$0.342726\pi$
$752$	0	0
$753$	6336.00	0.306636
$754$	0	0
$755$	− 5760.00i	− 0.277653i
$756$	0	0
$757$	− 20004.0i	− 0.960446i	−0.877146	−	0.480223i	$-0.840556\pi$
$757$	− 20004.0i	− 0.960446i	0.877146	−	0.480223i	$-0.159444\pi$
$758$	0	0
$759$	−2048.00	−0.0979416
$760$	0	0
$761$	−31478.0	−1.49944	−0.749722	−	0.661753i	$-0.769813\pi$
$761$	−31478.0	−1.49944	−0.749722	+	0.661753i	$0.769813\pi$
$762$	0	0
$763$	− 23680.0i	− 1.12356i
$764$	0	0
$765$	43512.0i	2.05644i
$766$	0	0
$767$	−8160.00	−0.384147
$768$	0	0
$769$	7054.00	0.330785	0.165393	−	0.986228i	$-0.447111\pi$
$769$	7054.00	0.330785	0.165393	+	0.986228i	$0.447111\pi$
$770$	0	0
$771$	42992.0i	2.00820i
$772$	0	0
$773$	− 9604.00i	− 0.446872i	−0.974719	−	0.223436i	$-0.928273\pi$
$773$	− 9604.00i	− 0.446872i	0.974719	−	0.223436i	$-0.0717273\pi$
$774$	0	0
$775$	4864.00	0.225445
$776$	0	0
$777$	−23552.0	−1.08742
$778$	0	0
$779$	− 8976.00i	− 0.412835i
$780$	0	0
$781$	3328.00i	0.152478i
$782$	0	0
$783$	13760.0	0.628023
$784$	0	0
$785$	27600.0	1.25489
$786$	0	0
$787$	− 3144.00i	− 0.142403i	−0.997462	−	0.0712017i	$-0.977317\pi$
$787$	− 3144.00i	− 0.142403i	0.997462	−	0.0712017i	$-0.0226834\pi$
$788$	0	0
$789$	27904.0i	1.25907i
$790$	0	0
$791$	9664.00	0.434402
$792$	0	0
$793$	12720.0	0.569610
$794$	0	0
$795$	7296.00i	0.325487i
$796$	0	0
$797$	22084.0i	0.981500i	0.871300	+	0.490750i	$0.163277\pi$
$797$	22084.0i	0.981500i	−0.871300	+	0.490750i	$0.836723\pi$
$798$	0	0
$799$	31360.0	1.38853
$800$	0	0
$801$	−21534.0	−0.949896
$802$	0	0
$803$	1104.00i	0.0485172i
$804$	0	0
$805$	− 12288.0i	− 0.538006i
$806$	0	0
$807$	−38112.0	−1.66246
$808$	0	0
$809$	−14950.0	−0.649708	−0.324854	−	0.945764i	$-0.605315\pi$
$809$	−14950.0	−0.649708	−0.324854	+	0.945764i	$0.605315\pi$
$810$	0	0
$811$	− 23432.0i	− 1.01456i	−0.861781	−	0.507280i	$-0.830651\pi$
$811$	− 23432.0i	− 1.01456i	0.861781	−	0.507280i	$-0.169349\pi$
$812$	0	0
$813$	− 10752.0i	− 0.463824i
$814$	0	0
$815$	−19104.0	−0.821085
$816$	0	0
$817$	26048.0	1.11543
$818$	0	0
$819$	23680.0i	1.01031i
$820$	0	0
$821$	− 1044.00i	− 0.0443798i	−0.999754	−	0.0221899i	$-0.992936\pi$
$821$	− 1044.00i	− 0.0443798i	0.999754	−	0.0221899i	$-0.00706385\pi$
$822$	0	0
$823$	18208.0	0.771192	0.385596	−	0.922668i	$-0.373996\pi$
$823$	18208.0	0.771192	0.385596	+	0.922668i	$0.373996\pi$
$824$	0	0
$825$	1216.00	0.0513160
$826$	0	0
$827$	− 12488.0i	− 0.525091i	−0.964920	−	0.262546i	$-0.915438\pi$
$827$	− 12488.0i	− 0.525091i	0.964920	−	0.262546i	$-0.0845620\pi$
$828$	0	0
$829$	− 30172.0i	− 1.26407i	−0.774938	−	0.632037i	$-0.782219\pi$
$829$	− 30172.0i	− 1.26407i	0.774938	−	0.632037i	$-0.217781\pi$
$830$	0	0
$831$	−68768.0	−2.87068
$832$	0	0
$833$	−66738.0	−2.77591
$834$	0	0
$835$	− 26496.0i	− 1.09812i
$836$	0	0
$837$	− 20480.0i	− 0.845750i
$838$	0	0
$839$	−32544.0	−1.33915	−0.669573	−	0.742746i	$-0.733523\pi$
$839$	−32544.0	−1.33915	−0.669573	+	0.742746i	$0.733523\pi$
$840$	0	0
$841$	−5195.00	−0.213006
$842$	0	0
$843$	22992.0i	0.939367i
$844$	0	0
$845$	21564.0i	0.877898i
$846$	0	0
$847$	−40544.0	−1.64476
$848$	0	0
$849$	−23104.0	−0.933954
$850$	0	0
$851$	2944.00i	0.118589i
$852$	0	0
$853$	26156.0i	1.04990i	0.851133	+	0.524950i	$0.175916\pi$
$853$	26156.0i	1.04990i	−0.851133	+	0.524950i	$0.824084\pi$
$854$	0	0
$855$	39072.0	1.56285
$856$	0	0
$857$	−18646.0	−0.743215	−0.371607	−	0.928390i	$-0.621193\pi$
$857$	−18646.0	−0.743215	−0.371607	+	0.928390i	$0.621193\pi$
$858$	0	0
$859$	− 5800.00i	− 0.230377i	−0.993344	−	0.115188i	$-0.963253\pi$
$859$	− 5800.00i	− 0.230377i	0.993344	−	0.115188i	$-0.0367471\pi$
$860$	0	0
$861$	− 26112.0i	− 1.03356i
$862$	0	0
$863$	−25088.0	−0.989578	−0.494789	−	0.869013i	$-0.664755\pi$
$863$	−25088.0	−0.989578	−0.494789	+	0.869013i	$0.664755\pi$
$864$	0	0
$865$	47376.0	1.86223
$866$	0	0
$867$	− 37528.0i	− 1.47003i
$868$	0	0
$869$	512.000i	0.0199867i
$870$	0	0
$871$	11040.0	0.429479
$872$	0	0
$873$	−8806.00	−0.341395
$874$	0	0
$875$	− 40704.0i	− 1.57262i
$876$	0	0
$877$	− 3004.00i	− 0.115665i	−0.998326	−	0.0578323i	$-0.981581\pi$
$877$	− 3004.00i	− 0.115665i	0.998326	−	0.0578323i	$-0.0184189\pi$
$878$	0	0
$879$	52320.0	2.00763
$880$	0	0
$881$	43282.0	1.65517	0.827587	−	0.561338i	$-0.189713\pi$
$881$	43282.0	1.65517	0.827587	+	0.561338i	$0.189713\pi$
$882$	0	0
$883$	− 27880.0i	− 1.06256i	−0.847198	−	0.531278i	$-0.821712\pi$
$883$	− 27880.0i	− 1.06256i	0.847198	−	0.531278i	$-0.178288\pi$
$884$	0	0
$885$	39168.0i	1.48770i
$886$	0	0
$887$	−7392.00	−0.279819	−0.139909	−	0.990164i	$-0.544681\pi$
$887$	−7392.00	−0.279819	−0.139909	+	0.990164i	$0.544681\pi$
$888$	0	0
$889$	53248.0	2.00886
$890$	0	0
$891$	2872.00i	0.107986i
$892$	0	0
$893$	− 28160.0i	− 1.05525i
$894$	0	0
$895$	−25248.0	−0.942958
$896$	0	0
$897$	5120.00	0.190582
$898$	0	0
$899$	44032.0i	1.63354i
$900$	0	0
$901$	− 7448.00i	− 0.275393i
$902$	0	0
$903$	75776.0	2.79254
$904$	0	0
$905$	16944.0	0.622362
$906$	0	0
$907$	29080.0i	1.06459i	0.846558	+	0.532296i	$0.178671\pi$
$907$	29080.0i	1.06459i	−0.846558	+	0.532296i	$0.821329\pi$
$908$	0	0
$909$	− 54316.0i	− 1.98190i
$910$	0	0
$911$	26688.0	0.970596	0.485298	−	0.874349i	$-0.338711\pi$
$911$	26688.0	0.970596	0.485298	+	0.874349i	$0.338711\pi$
$912$	0	0
$913$	−3136.00	−0.113676
$914$	0	0
$915$	− 61056.0i	− 2.20596i
$916$	0	0
$917$	− 74496.0i	− 2.68274i
$918$	0	0
$919$	−19680.0	−0.706402	−0.353201	−	0.935547i	$-0.614907\pi$
$919$	−19680.0	−0.706402	−0.353201	+	0.935547i	$0.614907\pi$
$920$	0	0
$921$	84672.0	3.02936
$922$	0	0
$923$	− 8320.00i	− 0.296702i
$924$	0	0
$925$	− 1748.00i	− 0.0621339i
$926$	0	0
$927$	−36704.0	−1.30045
$928$	0	0
$929$	−48466.0	−1.71164	−0.855822	−	0.517270i	$-0.826948\pi$
$929$	−48466.0	−1.71164	−0.855822	+	0.517270i	$0.826948\pi$
$930$	0	0
$931$	59928.0i	2.10962i
$932$	0	0
$933$	50944.0i	1.78760i
$934$	0	0
$935$	−9408.00	−0.329064
$936$	0	0
$937$	−13610.0	−0.474514	−0.237257	−	0.971447i	$-0.576248\pi$
$937$	−13610.0	−0.474514	−0.237257	+	0.971447i	$0.576248\pi$
$938$	0	0
$939$	70064.0i	2.43499i
$940$	0	0
$941$	30692.0i	1.06326i	0.846976	+	0.531632i	$0.178421\pi$
$941$	30692.0i	1.06326i	−0.846976	+	0.531632i	$0.821579\pi$
$942$	0	0
$943$	−3264.00	−0.112715
$944$	0	0
$945$	30720.0	1.05748
$946$	0	0
$947$	4824.00i	0.165532i	0.996569	+	0.0827661i	$0.0263754\pi$
$947$	4824.00i	0.165532i	−0.996569	+	0.0827661i	$0.973625\pi$
$948$	0	0
$949$	− 2760.00i	− 0.0944082i
$950$	0	0
$951$	−5728.00	−0.195313
$952$	0	0
$953$	22986.0	0.781311	0.390656	−	0.920537i	$-0.372248\pi$
$953$	22986.0	0.781311	0.390656	+	0.920537i	$0.372248\pi$
$954$	0	0
$955$	44544.0i	1.50933i
$956$	0	0
$957$	11008.0i	0.371827i
$958$	0	0
$959$	55488.0	1.86841
$960$	0	0
$961$	35745.0	1.19986
$962$	0	0
$963$	21608.0i	0.723061i
$964$	0	0
$965$	19368.0i	0.646091i
$966$	0	0
$967$	−17184.0	−0.571458	−0.285729	−	0.958310i	$-0.592236\pi$
$967$	−17184.0	−0.571458	−0.285729	+	0.958310i	$0.592236\pi$
$968$	0	0
$969$	−68992.0	−2.28725
$970$	0	0
$971$	− 2920.00i	− 0.0965059i	−0.998835	−	0.0482530i	$-0.984635\pi$
$971$	− 2920.00i	− 0.0965059i	0.998835	−	0.0482530i	$-0.0153654\pi$
$972$	0	0
$973$	− 97024.0i	− 3.19676i
$974$	0	0
$975$	−3040.00	−0.0998543
$976$	0	0
$977$	−27042.0	−0.885517	−0.442759	−	0.896641i	$-0.646000\pi$
$977$	−27042.0	−0.885517	−0.442759	+	0.896641i	$0.646000\pi$
$978$	0	0
$979$	− 4656.00i	− 0.151998i
$980$	0	0
$981$	− 27380.0i	− 0.891107i
$982$	0	0
$983$	44192.0	1.43388	0.716941	−	0.697134i	$-0.245542\pi$
$983$	44192.0	1.43388	0.716941	+	0.697134i	$0.245542\pi$
$984$	0	0
$985$	−8208.00	−0.265511
$986$	0	0
$987$	− 81920.0i	− 2.64189i
$988$	0	0
$989$	− 9472.00i	− 0.304542i
$990$	0	0
$991$	29824.0	0.955995	0.477997	−	0.878361i	$-0.341363\pi$
$991$	29824.0	0.955995	0.477997	+	0.878361i	$0.341363\pi$
$992$	0	0
$993$	35264.0	1.12696
$994$	0	0
$995$	48768.0i	1.55382i
$996$	0	0
$997$	11612.0i	0.368862i	0.982845	+	0.184431i	$0.0590443\pi$
$997$	11612.0i	0.368862i	−0.982845	+	0.184431i	$0.940956\pi$
$998$	0	0
$999$	−7360.00	−0.233093

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	128.4.b.a.65.1	✓	2
3.2	odd	2		1152.4.d.a.577.1		2
4.3	odd	2		128.4.b.d.65.2	yes	2
8.3	odd	2		128.4.b.d.65.1	yes	2
8.5	even	2	inner	128.4.b.a.65.2	yes	2
12.11	even	2		1152.4.d.h.577.1		2
16.3	odd	4		256.4.a.c.1.1		1
16.5	even	4		256.4.a.b.1.1		1
16.11	odd	4		256.4.a.f.1.1		1
16.13	even	4		256.4.a.g.1.1		1
24.5	odd	2		1152.4.d.a.577.2		2
24.11	even	2		1152.4.d.h.577.2		2
48.5	odd	4		2304.4.a.n.1.1		1
48.11	even	4		2304.4.a.m.1.1		1
48.29	odd	4		2304.4.a.d.1.1		1
48.35	even	4		2304.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.4.b.a.65.1	✓	2	1.1	even	1	trivial
128.4.b.a.65.2	yes	2	8.5	even	2	inner
128.4.b.d.65.1	yes	2	8.3	odd	2
128.4.b.d.65.2	yes	2	4.3	odd	2
256.4.a.b.1.1		1	16.5	even	4
256.4.a.c.1.1		1	16.3	odd	4
256.4.a.f.1.1		1	16.11	odd	4
256.4.a.g.1.1		1	16.13	even	4
1152.4.d.a.577.1		2	3.2	odd	2
1152.4.d.a.577.2		2	24.5	odd	2
1152.4.d.h.577.1		2	12.11	even	2
1152.4.d.h.577.2		2	24.11	even	2
2304.4.a.c.1.1		1	48.35	even	4
2304.4.a.d.1.1		1	48.29	odd	4
2304.4.a.m.1.1		1	48.11	even	4
2304.4.a.n.1.1		1	48.5	odd	4

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:	\( N \)	\(=\)	\( 128 = 2^{7} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	128.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.00000i q^{3} +12.0000i q^{5} -32.0000 q^{7} -37.0000 q^{9} +O(q^{10})\)	\(q-8.00000i q^{3} +12.0000i q^{5} -32.0000 q^{7} -37.0000 q^{9} -8.00000i q^{11} +20.0000i q^{13} +96.0000 q^{15} -98.0000 q^{17} +88.0000i q^{19} +256.000i q^{21} +32.0000 q^{23} -19.0000 q^{25} +80.0000i q^{27} -172.000i q^{29} -256.000 q^{31} -64.0000 q^{33} -384.000i q^{35} +92.0000i q^{37} +160.000 q^{39} -102.000 q^{41} -296.000i q^{43} -444.000i q^{45} -320.000 q^{47} +681.000 q^{49} +784.000i q^{51} +76.0000i q^{53} +96.0000 q^{55} +704.000 q^{57} +408.000i q^{59} -636.000i q^{61} +1184.00 q^{63} -240.000 q^{65} -552.000i q^{67} -256.000i q^{69} -416.000 q^{71} -138.000 q^{73} +152.000i q^{75} +256.000i q^{77} -64.0000 q^{79} -359.000 q^{81} -392.000i q^{83} -1176.00i q^{85} -1376.00 q^{87} +582.000 q^{89} -640.000i q^{91} +2048.00i q^{93} -1056.00 q^{95} +238.000 q^{97} +296.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 64 q^{7} - 74 q^{9}+O(q^{10}) \) 2 * q - 64 * q^7 - 74 * q^9	\( 2 q - 64 q^{7} - 74 q^{9} + 192 q^{15} - 196 q^{17} + 64 q^{23} - 38 q^{25} - 512 q^{31} - 128 q^{33} + 320 q^{39} - 204 q^{41} - 640 q^{47} + 1362 q^{49} + 192 q^{55} + 1408 q^{57} + 2368 q^{63} - 480 q^{65} - 832 q^{71} - 276 q^{73} - 128 q^{79} - 718 q^{81} - 2752 q^{87} + 1164 q^{89} - 2112 q^{95} + 476 q^{97}+O(q^{100}) \) 2 * q - 64 * q^7 - 74 * q^9 + 192 * q^15 - 196 * q^17 + 64 * q^23 - 38 * q^25 - 512 * q^31 - 128 * q^33 + 320 * q^39 - 204 * q^41 - 640 * q^47 + 1362 * q^49 + 192 * q^55 + 1408 * q^57 + 2368 * q^63 - 480 * q^65 - 832 * q^71 - 276 * q^73 - 128 * q^79 - 718 * q^81 - 2752 * q^87 + 1164 * q^89 - 2112 * q^95 + 476 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more