LMFDB - Embedded newform 1764.4.f.a.881.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(881,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("1764.881");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.f (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:	$104.079369250$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + \cdots + 7375227456 $ x^16 - 4x^15 - 290x^14 + 1728x^13 + 29275x^12 - 246984x^11 - 955194x^10 + 14344616x^9 - 18123280x^8 - 273588032x^7 + 1239640536x^6 - 1407381792x^5 - 1961185792x^4 + 4297169408x^3 + 2991779296x^2 - 11217342336*x + 7375227456
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{18}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.4
Root			$-10.4548 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1764.881
Dual form			1764.4.f.a.881.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.73625 q^{5} +O(q^{10})$	$q-8.73625 q^{5} +8.78630i q^{11} +11.8322i q^{13} +44.5839 q^{17} +11.6458i q^{19} -142.630i q^{23} -48.6779 q^{25} +234.018i q^{29} -291.919i q^{31} -88.9030 q^{37} -145.961 q^{41} +144.633 q^{43} -240.367 q^{47} +304.308i q^{53} -76.7594i q^{55} +7.08392 q^{59} -172.985i q^{61} -103.369i q^{65} -486.560 q^{67} +653.710i q^{71} +114.359i q^{73} +294.615 q^{79} +877.193 q^{83} -389.496 q^{85} +1420.76 q^{89} -101.741i q^{95} -738.981i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 424 q^{25} - 152 q^{37} + 1408 q^{43} + 3056 q^{67} + 728 q^{79} + 7392 q^{85}+O(q^{100}) $ 16 * q + 424 * q^25 - 152 * q^37 + 1408 * q^43 + 3056 * q^67 + 728 * q^79 + 7392 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−8.73625	−0.781394	−0.390697	−	0.920519i	$-0.627766\pi$
$5$	−8.73625	−0.781394	−0.390697	+	0.920519i	$0.627766\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	8.78630i	0.240834i	0.992723	+	0.120417i	$0.0384231\pi$
$11$	8.78630i	0.240834i	−0.992723	+	0.120417i	$0.961577\pi$
$12$	0	0
$13$	11.8322i	0.252435i	0.992003	+	0.126217i	$0.0402837\pi$
$13$	11.8322i	0.252435i	−0.992003	+	0.126217i	$0.959716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	44.5839	0.636070	0.318035	−	0.948079i	$-0.396977\pi$
$17$	44.5839	0.636070	0.318035	+	0.948079i	$0.396977\pi$
$18$	0	0
$19$	11.6458i	0.140618i	0.997525	+	0.0703088i	$0.0223985\pi$
$19$	11.6458i	0.140618i	−0.997525	+	0.0703088i	$0.977602\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 142.630i	− 1.29306i	−0.762888	−	0.646531i	$-0.776219\pi$
$23$	− 142.630i	− 1.29306i	0.762888	−	0.646531i	$-0.223781\pi$
$24$	0	0
$25$	−48.6779	−0.389423
$26$	0	0
$27$	0	0
$28$	0	0
$29$	234.018i	1.49848i	0.662298	+	0.749241i	$0.269581\pi$
$29$	234.018i	1.49848i	−0.662298	+	0.749241i	$0.730419\pi$
$30$	0	0
$31$	− 291.919i	− 1.69130i	−0.533739	−	0.845649i	$-0.679214\pi$
$31$	− 291.919i	− 1.69130i	0.533739	−	0.845649i	$-0.320786\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−88.9030	−0.395015	−0.197508	−	0.980301i	$-0.563285\pi$
$37$	−88.9030	−0.395015	−0.197508	+	0.980301i	$0.563285\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−145.961	−0.555984	−0.277992	−	0.960583i	$-0.589669\pi$
$41$	−145.961	−0.555984	−0.277992	+	0.960583i	$0.589669\pi$
$42$	0	0
$43$	144.633	0.512938	0.256469	−	0.966552i	$-0.417441\pi$
$43$	144.633	0.512938	0.256469	+	0.966552i	$0.417441\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−240.367	−0.745981	−0.372991	−	0.927835i	$-0.621668\pi$
$47$	−240.367	−0.745981	−0.372991	+	0.927835i	$0.621668\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	304.308i	0.788678i	0.918965	+	0.394339i	$0.129026\pi$
$53$	304.308i	0.788678i	−0.918965	+	0.394339i	$0.870974\pi$
$54$	0	0
$55$	− 76.7594i	− 0.188186i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.08392	0.0156313	0.00781566	−	0.999969i	$-0.497512\pi$
$59$	7.08392	0.0156313	0.00781566	+	0.999969i	$0.497512\pi$
$60$	0	0
$61$	− 172.985i	− 0.363090i	−0.983383	−	0.181545i	$-0.941890\pi$
$61$	− 172.985i	− 0.363090i	0.983383	−	0.181545i	$-0.0581098\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 103.369i	− 0.197251i
$66$	0	0
$67$	−486.560	−0.887205	−0.443603	−	0.896224i	$-0.646300\pi$
$67$	−486.560	−0.887205	−0.443603	+	0.896224i	$0.646300\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	653.710i	1.09269i	0.837560	+	0.546345i	$0.183981\pi$
$71$	653.710i	1.09269i	−0.837560	+	0.546345i	$0.816019\pi$
$72$	0	0
$73$	114.359i	0.183352i	0.995789	+	0.0916761i	$0.0292224\pi$
$73$	114.359i	0.183352i	−0.995789	+	0.0916761i	$0.970778\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	294.615	0.419579	0.209789	−	0.977747i	$-0.432722\pi$
$79$	294.615	0.419579	0.209789	+	0.977747i	$0.432722\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	877.193	1.16005	0.580027	−	0.814597i	$-0.303042\pi$
$83$	877.193	1.16005	0.580027	+	0.814597i	$0.303042\pi$
$84$	0	0
$85$	−389.496	−0.497021
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1420.76	1.69214	0.846068	−	0.533075i	$-0.178964\pi$
$89$	1420.76	1.69214	0.846068	+	0.533075i	$0.178964\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 101.741i	− 0.109878i
$96$	0	0
$97$	− 738.981i	− 0.773527i	−0.922179	−	0.386764i	$-0.873593\pi$
$97$	− 738.981i	− 0.773527i	0.922179	−	0.386764i	$-0.126407\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1662.88	−1.63825	−0.819125	−	0.573615i	$-0.805540\pi$
$101$	−1662.88	−1.63825	−0.819125	+	0.573615i	$0.805540\pi$
$102$	0	0
$103$	− 455.883i	− 0.436112i	−0.975936	−	0.218056i	$-0.930028\pi$
$103$	− 455.883i	− 0.436112i	0.975936	−	0.218056i	$-0.0699715\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1476.95i	1.33441i	0.744874	+	0.667205i	$0.232510\pi$
$107$	1476.95i	1.33441i	−0.744874	+	0.667205i	$0.767490\pi$
$108$	0	0
$109$	1568.40	1.37822	0.689110	−	0.724657i	$-0.258002\pi$
$109$	1568.40	1.37822	0.689110	+	0.724657i	$0.258002\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1077.27i	− 0.896820i	−0.893828	−	0.448410i	$-0.851990\pi$
$113$	− 1077.27i	− 0.896820i	0.893828	−	0.448410i	$-0.148010\pi$
$114$	0	0
$115$	1246.05i	1.01039i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1253.80	0.941999
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1517.29	1.08569
$126$	0	0
$127$	1518.00	1.06063	0.530317	−	0.847799i	$-0.322073\pi$
$127$	1518.00	1.06063	0.530317	+	0.847799i	$0.322073\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	403.343	0.269010	0.134505	−	0.990913i	$-0.457056\pi$
$131$	403.343	0.269010	0.134505	+	0.990913i	$0.457056\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	539.996i	0.336752i	0.985723	+	0.168376i	$0.0538522\pi$
$137$	539.996i	0.336752i	−0.985723	+	0.168376i	$0.946148\pi$
$138$	0	0
$139$	− 2042.22i	− 1.24618i	−0.782151	−	0.623089i	$-0.785877\pi$
$139$	− 2042.22i	− 1.24618i	0.782151	−	0.623089i	$-0.214123\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−103.961	−0.0607947
$144$	0	0
$145$	− 2044.44i	− 1.17090i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	208.443i	0.114606i	0.998357	+	0.0573032i	$0.0182502\pi$
$149$	208.443i	0.114606i	−0.998357	+	0.0573032i	$0.981750\pi$
$150$	0	0
$151$	−460.089	−0.247957	−0.123979	−	0.992285i	$-0.539565\pi$
$151$	−460.089	−0.247957	−0.123979	+	0.992285i	$0.539565\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2550.28i	1.32157i
$156$	0	0
$157$	2690.62i	1.36774i	0.729605	+	0.683869i	$0.239704\pi$
$157$	2690.62i	1.36774i	−0.729605	+	0.683869i	$0.760296\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−2274.88	−1.09314	−0.546571	−	0.837413i	$-0.684067\pi$
$163$	−2274.88	−1.09314	−0.546571	+	0.837413i	$0.684067\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3214.68	1.48958	0.744789	−	0.667300i	$-0.232550\pi$
$167$	3214.68	1.48958	0.744789	+	0.667300i	$0.232550\pi$
$168$	0	0
$169$	2057.00	0.936277
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3564.22	1.56637	0.783187	−	0.621786i	$-0.213593\pi$
$173$	3564.22	1.56637	0.783187	+	0.621786i	$0.213593\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 349.703i	− 0.146023i	−0.997331	−	0.0730113i	$-0.976739\pi$
$179$	− 349.703i	− 0.146023i	0.997331	−	0.0730113i	$-0.0232609\pi$
$180$	0	0
$181$	1664.25i	0.683439i	0.939802	+	0.341720i	$0.111009\pi$
$181$	1664.25i	0.683439i	−0.939802	+	0.341720i	$0.888991\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	776.679	0.308663
$186$	0	0
$187$	391.728i	0.153187i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	246.784i	0.0934903i	0.998907	+	0.0467452i	$0.0148849\pi$
$191$	246.784i	0.0934903i	−0.998907	+	0.0467452i	$0.985115\pi$
$192$	0	0
$193$	2548.44	0.950469	0.475235	−	0.879859i	$-0.342363\pi$
$193$	2548.44	0.950469	0.475235	+	0.879859i	$0.342363\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4579.00i	1.65604i	0.560697	+	0.828021i	$0.310533\pi$
$197$	4579.00i	1.65604i	−0.560697	+	0.828021i	$0.689467\pi$
$198$	0	0
$199$	− 2484.31i	− 0.884965i	−0.896777	−	0.442482i	$-0.854098\pi$
$199$	− 2484.31i	− 0.884965i	0.896777	−	0.442482i	$-0.145902\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1275.16	0.434443
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−102.324	−0.0338655
$210$	0	0
$211$	5736.87	1.87177	0.935883	−	0.352311i	$-0.114604\pi$
$211$	5736.87	1.87177	0.935883	+	0.352311i	$0.114604\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1263.55	−0.400807
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	527.524i	0.160566i
$222$	0	0
$223$	− 5391.46i	− 1.61901i	−0.587115	−	0.809504i	$-0.699736\pi$
$223$	− 5391.46i	− 1.61901i	0.587115	−	0.809504i	$-0.300264\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1494.89	0.437091	0.218545	−	0.975827i	$-0.429869\pi$
$227$	1494.89	0.437091	0.218545	+	0.975827i	$0.429869\pi$
$228$	0	0
$229$	800.761i	0.231073i	0.993303	+	0.115537i	$0.0368588\pi$
$229$	800.761i	0.231073i	−0.993303	+	0.115537i	$0.963141\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2625.50i	− 0.738206i	−0.929389	−	0.369103i	$-0.879665\pi$
$233$	− 2625.50i	− 0.738206i	0.929389	−	0.369103i	$-0.120335\pi$
$234$	0	0
$235$	2099.91	0.582906
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6283.85i	1.70070i	0.526214	+	0.850352i	$0.323611\pi$
$239$	6283.85i	1.70070i	−0.526214	+	0.850352i	$0.676389\pi$
$240$	0	0
$241$	936.722i	0.250372i	0.992133	+	0.125186i	$0.0399527\pi$
$241$	936.722i	0.250372i	−0.992133	+	0.125186i	$0.960047\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−137.795	−0.0354968
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2831.45	0.712031	0.356015	−	0.934480i	$-0.384135\pi$
$251$	2831.45	0.712031	0.356015	+	0.934480i	$0.384135\pi$
$252$	0	0
$253$	1253.19	0.311413
$254$	0	0
$255$	0	0
$256$	0	0
$257$	5689.30	1.38089	0.690445	−	0.723384i	$-0.257415\pi$
$257$	5689.30	1.38089	0.690445	+	0.723384i	$0.257415\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4017.90i	0.942031i	0.882125	+	0.471016i	$0.156112\pi$
$263$	4017.90i	0.942031i	−0.882125	+	0.471016i	$0.843888\pi$
$264$	0	0
$265$	− 2658.51i	− 0.616268i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3817.62	−0.865295	−0.432648	−	0.901563i	$-0.642421\pi$
$269$	−3817.62	−0.865295	−0.432648	+	0.901563i	$0.642421\pi$
$270$	0	0
$271$	− 7151.24i	− 1.60298i	−0.598009	−	0.801489i	$-0.704041\pi$
$271$	− 7151.24i	− 1.60298i	0.598009	−	0.801489i	$-0.295959\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 427.698i	− 0.0937861i
$276$	0	0
$277$	−5483.81	−1.18950	−0.594748	−	0.803912i	$-0.702748\pi$
$277$	−5483.81	−1.18950	−0.594748	+	0.803912i	$0.702748\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5525.72i	1.17308i	0.809919	+	0.586542i	$0.199511\pi$
$281$	5525.72i	1.17308i	−0.809919	+	0.586542i	$0.800489\pi$
$282$	0	0
$283$	− 251.526i	− 0.0528328i	−0.999651	−	0.0264164i	$-0.991590\pi$
$283$	− 251.526i	− 0.0528328i	0.999651	−	0.0264164i	$-0.00840957\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2925.27	−0.595415
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9296.44	1.85360	0.926798	−	0.375560i	$-0.122550\pi$
$293$	9296.44	1.85360	0.926798	+	0.375560i	$0.122550\pi$
$294$	0	0
$295$	−61.8869	−0.0122142
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1687.62	0.326414
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1511.24i	0.283717i
$306$	0	0
$307$	1498.96i	0.278666i	0.990246	+	0.139333i	$0.0444958\pi$
$307$	1498.96i	0.278666i	−0.990246	+	0.139333i	$0.955504\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6258.21	−1.14106	−0.570531	−	0.821276i	$-0.693263\pi$
$311$	−6258.21	−1.14106	−0.570531	+	0.821276i	$0.693263\pi$
$312$	0	0
$313$	− 2128.72i	− 0.384417i	−0.981354	−	0.192208i	$-0.938435\pi$
$313$	− 2128.72i	− 0.384417i	0.981354	−	0.192208i	$-0.0615650\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	5250.94i	0.930353i	0.885218	+	0.465177i	$0.154009\pi$
$317$	5250.94i	0.930353i	−0.885218	+	0.465177i	$0.845991\pi$
$318$	0	0
$319$	−2056.15	−0.360885
$320$	0	0
$321$	0	0
$322$	0	0
$323$	519.216i	0.0894427i
$324$	0	0
$325$	− 575.964i	− 0.0983038i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	8803.98	1.46197	0.730983	−	0.682396i	$-0.239062\pi$
$331$	8803.98	1.46197	0.730983	+	0.682396i	$0.239062\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4250.71	0.693257
$336$	0	0
$337$	−1428.63	−0.230927	−0.115463	−	0.993312i	$-0.536835\pi$
$337$	−1428.63	−0.230927	−0.115463	+	0.993312i	$0.536835\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2564.89	0.407321
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9118.34i	1.41066i	0.708880	+	0.705329i	$0.249201\pi$
$347$	9118.34i	1.41066i	−0.708880	+	0.705329i	$0.750799\pi$
$348$	0	0
$349$	− 11899.1i	− 1.82506i	−0.409011	−	0.912530i	$-0.634126\pi$
$349$	− 11899.1i	− 1.82506i	0.409011	−	0.912530i	$-0.365874\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9157.64	−1.38077	−0.690385	−	0.723442i	$-0.742559\pi$
$353$	−9157.64	−1.38077	−0.690385	+	0.723442i	$0.742559\pi$
$354$	0	0
$355$	− 5710.97i	− 0.853822i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7770.94i	1.14244i	0.820798	+	0.571218i	$0.193529\pi$
$359$	7770.94i	1.14244i	−0.820798	+	0.571218i	$0.806471\pi$
$360$	0	0
$361$	6723.37	0.980227
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 999.070i	− 0.143270i
$366$	0	0
$367$	6300.78i	0.896180i	0.893989	+	0.448090i	$0.147896\pi$
$367$	6300.78i	0.896180i	−0.893989	+	0.448090i	$0.852104\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10103.5	1.40252	0.701260	−	0.712905i	$-0.252621\pi$
$373$	10103.5	1.40252	0.701260	+	0.712905i	$0.252621\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2768.93	−0.378269
$378$	0	0
$379$	1122.50	0.152134	0.0760671	−	0.997103i	$-0.475764\pi$
$379$	1122.50	0.152134	0.0760671	+	0.997103i	$0.475764\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10628.0	1.41792	0.708960	−	0.705249i	$-0.249165\pi$
$383$	10628.0	1.41792	0.708960	+	0.705249i	$0.249165\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 13361.9i	− 1.74158i	−0.491652	−	0.870792i	$-0.663607\pi$
$389$	− 13361.9i	− 1.74158i	0.491652	−	0.870792i	$-0.336393\pi$
$390$	0	0
$391$	− 6359.01i	− 0.822478i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2573.83	−0.327857
$396$	0	0
$397$	− 2660.79i	− 0.336376i	−0.985755	−	0.168188i	$-0.946208\pi$
$397$	− 2660.79i	− 0.336376i	0.985755	−	0.168188i	$-0.0537916\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 6106.75i	− 0.760490i	−0.924886	−	0.380245i	$-0.875840\pi$
$401$	− 6106.75i	− 0.760490i	0.924886	−	0.380245i	$-0.124160\pi$
$402$	0	0
$403$	3454.03	0.426942
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 781.129i	− 0.0951330i
$408$	0	0
$409$	2307.13i	0.278925i	0.990227	+	0.139462i	$0.0445374\pi$
$409$	2307.13i	0.278925i	−0.990227	+	0.139462i	$0.955463\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−7663.38	−0.906460
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11116.9	1.29617	0.648083	−	0.761570i	$-0.275571\pi$
$419$	11116.9	1.29617	0.648083	+	0.761570i	$0.275571\pi$
$420$	0	0
$421$	4188.14	0.484840	0.242420	−	0.970171i	$-0.422059\pi$
$421$	4188.14	0.484840	0.242420	+	0.970171i	$0.422059\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2170.25	−0.247700
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4911.11i	0.548863i	0.961607	+	0.274432i	$0.0884897\pi$
$431$	4911.11i	0.548863i	−0.961607	+	0.274432i	$0.911510\pi$
$432$	0	0
$433$	14412.6i	1.59960i	0.600268	+	0.799799i	$0.295061\pi$
$433$	14412.6i	1.59960i	−0.600268	+	0.799799i	$0.704939\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1661.04	0.181827
$438$	0	0
$439$	8916.56i	0.969394i	0.874682	+	0.484697i	$0.161070\pi$
$439$	8916.56i	0.969394i	−0.874682	+	0.484697i	$0.838930\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1180.27i	0.126583i	0.997995	+	0.0632913i	$0.0201597\pi$
$443$	1180.27i	0.126583i	−0.997995	+	0.0632913i	$0.979840\pi$
$444$	0	0
$445$	−12412.1	−1.32223
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12536.1i	1.31763i	0.752307	+	0.658813i	$0.228941\pi$
$449$	12536.1i	1.31763i	−0.752307	+	0.658813i	$0.771059\pi$
$450$	0	0
$451$	− 1282.46i	− 0.133900i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10103.8	−1.03422	−0.517109	−	0.855920i	$-0.672992\pi$
$457$	−10103.8	−1.03422	−0.517109	+	0.855920i	$0.672992\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	418.675	0.0422985	0.0211493	−	0.999776i	$-0.493267\pi$
$461$	418.675	0.0422985	0.0211493	+	0.999776i	$0.493267\pi$
$462$	0	0
$463$	−2178.42	−0.218661	−0.109330	−	0.994005i	$-0.534871\pi$
$463$	−2178.42	−0.218661	−0.109330	+	0.994005i	$0.534871\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3859.69	0.382452	0.191226	−	0.981546i	$-0.438754\pi$
$467$	3859.69	0.382452	0.191226	+	0.981546i	$0.438754\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1270.79i	0.123533i
$474$	0	0
$475$	− 566.894i	− 0.0547597i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	16213.7	1.54660	0.773302	−	0.634038i	$-0.218604\pi$
$479$	16213.7	1.54660	0.773302	+	0.634038i	$0.218604\pi$
$480$	0	0
$481$	− 1051.91i	− 0.0997155i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6455.93i	0.604430i
$486$	0	0
$487$	19805.7	1.84287	0.921437	−	0.388527i	$-0.127016\pi$
$487$	19805.7	1.84287	0.921437	+	0.388527i	$0.127016\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 11104.5i	− 1.02065i	−0.859983	−	0.510323i	$-0.829526\pi$
$491$	− 11104.5i	− 1.02065i	0.859983	−	0.510323i	$-0.170474\pi$
$492$	0	0
$493$	10433.4i	0.953139i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	870.047	0.0780535	0.0390267	−	0.999238i	$-0.487574\pi$
$499$	870.047	0.0780535	0.0390267	+	0.999238i	$0.487574\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−12435.5	−1.10233	−0.551166	−	0.834396i	$-0.685817\pi$
$503$	−12435.5	−1.10233	−0.551166	+	0.834396i	$0.685817\pi$
$504$	0	0
$505$	14527.4	1.28012
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−19700.1	−1.71550	−0.857751	−	0.514065i	$-0.828139\pi$
$509$	−19700.1	−1.71550	−0.857751	+	0.514065i	$0.828139\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3982.71i	0.340775i
$516$	0	0
$517$	− 2111.94i	− 0.179657i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9626.53	0.809493	0.404747	−	0.914429i	$-0.367360\pi$
$521$	9626.53	0.809493	0.404747	+	0.914429i	$0.367360\pi$
$522$	0	0
$523$	1675.95i	0.140123i	0.997543	+	0.0700615i	$0.0223196\pi$
$523$	1675.95i	0.140123i	−0.997543	+	0.0700615i	$0.977680\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 13014.9i	− 1.07578i
$528$	0	0
$529$	−8176.35	−0.672010
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1727.04i	− 0.140350i
$534$	0	0
$535$	− 12903.0i	− 1.04270i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8646.12	−0.687109	−0.343554	−	0.939133i	$-0.611631\pi$
$541$	−8646.12	−0.687109	−0.343554	+	0.939133i	$0.611631\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−13702.0	−1.07693
$546$	0	0
$547$	183.297	0.0143276	0.00716382	−	0.999974i	$-0.497720\pi$
$547$	183.297	0.0143276	0.00716382	+	0.999974i	$0.497720\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2725.33	−0.210713
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 21850.0i	− 1.66214i	−0.556165	−	0.831072i	$-0.687728\pi$
$557$	− 21850.0i	− 1.66214i	0.556165	−	0.831072i	$-0.312272\pi$
$558$	0	0
$559$	1711.32i	0.129483i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15248.2	1.14145	0.570724	−	0.821142i	$-0.306663\pi$
$563$	15248.2	1.14145	0.570724	+	0.821142i	$0.306663\pi$
$564$	0	0
$565$	9411.27i	0.700770i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 12542.2i	− 0.924071i	−0.886862	−	0.462035i	$-0.847119\pi$
$569$	− 12542.2i	− 0.924071i	0.886862	−	0.462035i	$-0.152881\pi$
$570$	0	0
$571$	11442.8	0.838645	0.419323	−	0.907837i	$-0.362268\pi$
$571$	11442.8	0.838645	0.419323	+	0.907837i	$0.362268\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6942.93i	0.503548i
$576$	0	0
$577$	− 11084.4i	− 0.799737i	−0.916572	−	0.399869i	$-0.869056\pi$
$577$	− 11084.4i	− 0.799737i	0.916572	−	0.399869i	$-0.130944\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2673.74	−0.189940
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1493.49	−0.105014	−0.0525068	−	0.998621i	$-0.516721\pi$
$587$	−1493.49	−0.105014	−0.0525068	+	0.998621i	$0.516721\pi$
$588$	0	0
$589$	3399.64	0.237826
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5665.45	0.392331	0.196165	−	0.980571i	$-0.437151\pi$
$593$	5665.45	0.392331	0.196165	+	0.980571i	$0.437151\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 5181.35i	− 0.353430i	−0.984262	−	0.176715i	$-0.943453\pi$
$599$	− 5181.35i	− 0.353430i	0.984262	−	0.176715i	$-0.0565470\pi$
$600$	0	0
$601$	13911.4i	0.944186i	0.881549	+	0.472093i	$0.156501\pi$
$601$	13911.4i	0.944186i	−0.881549	+	0.472093i	$0.843499\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10953.5	−0.736073
$606$	0	0
$607$	7549.41i	0.504812i	0.967621	+	0.252406i	$0.0812219\pi$
$607$	7549.41i	0.504812i	−0.967621	+	0.252406i	$0.918778\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 2844.06i	− 0.188311i
$612$	0	0
$613$	−12887.5	−0.849140	−0.424570	−	0.905395i	$-0.639575\pi$
$613$	−12887.5	−0.849140	−0.424570	+	0.905395i	$0.639575\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 22388.7i	− 1.46084i	−0.683000	−	0.730419i	$-0.739325\pi$
$617$	− 22388.7i	− 1.46084i	0.683000	−	0.730419i	$-0.260675\pi$
$618$	0	0
$619$	− 5532.37i	− 0.359232i	−0.983737	−	0.179616i	$-0.942514\pi$
$619$	− 5532.37i	− 0.359232i	0.983737	−	0.179616i	$-0.0574855\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−7170.73	−0.458927
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3963.65	−0.251257
$630$	0	0
$631$	−24188.4	−1.52603	−0.763015	−	0.646381i	$-0.776282\pi$
$631$	−24188.4	−1.52603	−0.763015	+	0.646381i	$0.776282\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13261.6	−0.828774
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9186.15i	0.566039i	0.959114	+	0.283020i	$0.0913362\pi$
$641$	9186.15i	0.566039i	−0.959114	+	0.283020i	$0.908664\pi$
$642$	0	0
$643$	21051.2i	1.29110i	0.763718	+	0.645550i	$0.223372\pi$
$643$	21051.2i	1.29110i	−0.763718	+	0.645550i	$0.776628\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3668.31	−0.222900	−0.111450	−	0.993770i	$-0.535549\pi$
$647$	−3668.31	−0.222900	−0.111450	+	0.993770i	$0.535549\pi$
$648$	0	0
$649$	62.2415i	0.00376455i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	7682.88i	0.460420i	0.973141	+	0.230210i	$0.0739414\pi$
$653$	7682.88i	0.460420i	−0.973141	+	0.230210i	$0.926059\pi$
$654$	0	0
$655$	−3523.71	−0.210203
$656$	0	0
$657$	0	0
$658$	0	0
$659$	529.107i	0.0312763i	0.999878	+	0.0156382i	$0.00497798\pi$
$659$	529.107i	0.0312763i	−0.999878	+	0.0156382i	$0.995022\pi$
$660$	0	0
$661$	13761.7i	0.809785i	0.914364	+	0.404892i	$0.132691\pi$
$661$	13761.7i	0.809785i	−0.914364	+	0.404892i	$0.867309\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	33377.9	1.93763
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1519.90	0.0874443
$672$	0	0
$673$	−31045.7	−1.77819	−0.889095	−	0.457722i	$-0.848666\pi$
$673$	−31045.7	−1.77819	−0.889095	+	0.457722i	$0.848666\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−5848.48	−0.332017	−0.166009	−	0.986124i	$-0.553088\pi$
$677$	−5848.48	−0.332017	−0.166009	+	0.986124i	$0.553088\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	13641.6i	0.764246i	0.924111	+	0.382123i	$0.124807\pi$
$683$	13641.6i	0.764246i	−0.924111	+	0.382123i	$0.875193\pi$
$684$	0	0
$685$	− 4717.54i	− 0.263136i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3600.62	−0.199089
$690$	0	0
$691$	13896.5i	0.765050i	0.923945	+	0.382525i	$0.124945\pi$
$691$	13896.5i	0.765050i	−0.923945	+	0.382525i	$0.875055\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17841.4i	0.973757i
$696$	0	0
$697$	−6507.53	−0.353645
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 30902.8i	− 1.66503i	−0.554005	−	0.832513i	$-0.686901\pi$
$701$	− 30902.8i	− 1.66503i	0.554005	−	0.832513i	$-0.313099\pi$
$702$	0	0
$703$	− 1035.35i	− 0.0555461i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−4321.29	−0.228899	−0.114450	−	0.993429i	$-0.536510\pi$
$709$	−4321.29	−0.228899	−0.114450	+	0.993429i	$0.536510\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−41636.5	−2.18695
$714$	0	0
$715$	908.229	0.0475047
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32262.0	−1.67339	−0.836696	−	0.547667i	$-0.815516\pi$
$719$	−32262.0	−1.67339	−0.836696	+	0.547667i	$0.815516\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 11391.5i	− 0.583543i
$726$	0	0
$727$	− 12056.4i	− 0.615056i	−0.951539	−	0.307528i	$-0.900498\pi$
$727$	− 12056.4i	− 0.615056i	0.951539	−	0.307528i	$-0.0995017\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6448.31	0.326264
$732$	0	0
$733$	− 25545.3i	− 1.28723i	−0.765350	−	0.643615i	$-0.777434\pi$
$733$	− 25545.3i	− 1.28723i	0.765350	−	0.643615i	$-0.222566\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4275.06i	− 0.213669i
$738$	0	0
$739$	−24805.7	−1.23477	−0.617384	−	0.786662i	$-0.711808\pi$
$739$	−24805.7	−1.23477	−0.617384	+	0.786662i	$0.711808\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 23318.4i	− 1.15137i	−0.817671	−	0.575686i	$-0.804735\pi$
$743$	− 23318.4i	− 1.15137i	0.817671	−	0.575686i	$-0.195265\pi$
$744$	0	0
$745$	− 1821.02i	− 0.0895528i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−311.841	−0.0151521	−0.00757607	−	0.999971i	$-0.502412\pi$
$751$	−311.841	−0.0151521	−0.00757607	+	0.999971i	$0.502412\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4019.46	0.193752
$756$	0	0
$757$	−26444.1	−1.26965	−0.634825	−	0.772656i	$-0.718928\pi$
$757$	−26444.1	−1.26965	−0.634825	+	0.772656i	$0.718928\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	20531.2	0.977995	0.488997	−	0.872285i	$-0.337363\pi$
$761$	20531.2	0.977995	0.488997	+	0.872285i	$0.337363\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	83.8181i	0.00394589i
$768$	0	0
$769$	19950.3i	0.935533i	0.883852	+	0.467767i	$0.154941\pi$
$769$	19950.3i	0.935533i	−0.883852	+	0.467767i	$0.845059\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−39619.3	−1.84347	−0.921737	−	0.387815i	$-0.873230\pi$
$773$	−39619.3	−1.84347	−0.921737	+	0.387815i	$0.873230\pi$
$774$	0	0
$775$	14210.0i	0.658630i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1699.84i	− 0.0781811i
$780$	0	0
$781$	−5743.69	−0.263157
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 23506.0i	− 1.06874i
$786$	0	0
$787$	− 33345.2i	− 1.51033i	−0.655537	−	0.755163i	$-0.727558\pi$
$787$	− 33345.2i	− 1.51033i	0.655537	−	0.755163i	$-0.272442\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	2046.79	0.0916565
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15138.0	−0.672793	−0.336397	−	0.941720i	$-0.609208\pi$
$797$	−15138.0	−0.672793	−0.336397	+	0.941720i	$0.609208\pi$
$798$	0	0
$799$	−10716.5	−0.474496
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1004.79	−0.0441574
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 19585.8i	− 0.851175i	−0.904917	−	0.425588i	$-0.860067\pi$
$809$	− 19585.8i	− 0.851175i	0.904917	−	0.425588i	$-0.139933\pi$
$810$	0	0
$811$	− 11787.4i	− 0.510371i	−0.966892	−	0.255186i	$-0.917863\pi$
$811$	− 11787.4i	− 0.510371i	0.966892	−	0.255186i	$-0.0821365\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	19873.9	0.854175
$816$	0	0
$817$	1684.37i	0.0721281i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 922.533i	− 0.0392163i	−0.999808	−	0.0196082i	$-0.993758\pi$
$821$	− 922.533i	− 0.0392163i	0.999808	−	0.0196082i	$-0.00624187\pi$
$822$	0	0
$823$	23576.0	0.998550	0.499275	−	0.866444i	$-0.333600\pi$
$823$	23576.0	0.998550	0.499275	+	0.866444i	$0.333600\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 9010.27i	− 0.378861i	−0.981894	−	0.189430i	$-0.939336\pi$
$827$	− 9010.27i	− 0.378861i	0.981894	−	0.189430i	$-0.0606641\pi$
$828$	0	0
$829$	1399.19i	0.0586197i	0.999570	+	0.0293098i	$0.00933095\pi$
$829$	1399.19i	0.0586197i	−0.999570	+	0.0293098i	$0.990669\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−28084.3	−1.16395
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10339.8	0.425472	0.212736	−	0.977110i	$-0.431763\pi$
$839$	10339.8	0.425472	0.212736	+	0.977110i	$0.431763\pi$
$840$	0	0
$841$	−30375.2	−1.24545
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−17970.5	−0.731601
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	12680.2i	0.510779i
$852$	0	0
$853$	39316.9i	1.57818i	0.614280	+	0.789088i	$0.289447\pi$
$853$	39316.9i	1.57818i	−0.614280	+	0.789088i	$0.710553\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	359.729	0.0143385	0.00716926	−	0.999974i	$-0.497718\pi$
$857$	359.729	0.0143385	0.00716926	+	0.999974i	$0.497718\pi$
$858$	0	0
$859$	37191.8i	1.47726i	0.674111	+	0.738630i	$0.264527\pi$
$859$	37191.8i	1.47726i	−0.674111	+	0.738630i	$0.735473\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 12709.0i	− 0.501297i	−0.968078	−	0.250649i	$-0.919356\pi$
$863$	− 12709.0i	− 0.501297i	0.968078	−	0.250649i	$-0.0806439\pi$
$864$	0	0
$865$	−31137.9	−1.22396
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2588.57i	0.101049i
$870$	0	0
$871$	− 5757.05i	− 0.223961i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14962.8	0.576121	0.288061	−	0.957612i	$-0.406990\pi$
$877$	14962.8	0.576121	0.288061	+	0.957612i	$0.406990\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−48954.8	−1.87211	−0.936056	−	0.351851i	$-0.885552\pi$
$881$	−48954.8	−1.87211	−0.936056	+	0.351851i	$0.885552\pi$
$882$	0	0
$883$	4761.13	0.181455	0.0907276	−	0.995876i	$-0.471081\pi$
$883$	4761.13	0.181455	0.0907276	+	0.995876i	$0.471081\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38955.1	−1.47462	−0.737308	−	0.675557i	$-0.763903\pi$
$887$	−38955.1	−1.47462	−0.737308	+	0.675557i	$0.763903\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 2799.27i	− 0.104898i
$894$	0	0
$895$	3055.10i	0.114101i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	68314.2	2.53438
$900$	0	0
$901$	13567.2i	0.501654i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 14539.3i	− 0.534035i
$906$	0	0
$907$	15221.5	0.557245	0.278622	−	0.960401i	$-0.410122\pi$
$907$	15221.5	0.557245	0.278622	+	0.960401i	$0.410122\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	6441.70i	0.234273i	0.993116	+	0.117137i	$0.0373716\pi$
$911$	6441.70i	0.234273i	−0.993116	+	0.117137i	$0.962628\pi$
$912$	0	0
$913$	7707.29i	0.279380i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	18668.2	0.670085	0.335042	−	0.942203i	$-0.391249\pi$
$919$	18668.2	0.670085	0.335042	+	0.942203i	$0.391249\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7734.79	−0.275833
$924$	0	0
$925$	4327.61	0.153828
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−22548.9	−0.796347	−0.398173	−	0.917310i	$-0.630356\pi$
$929$	−22548.9	−0.796347	−0.398173	+	0.917310i	$0.630356\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 3422.23i	− 0.119699i
$936$	0	0
$937$	22939.2i	0.799779i	0.916563	+	0.399889i	$0.130951\pi$
$937$	22939.2i	0.799779i	−0.916563	+	0.399889i	$0.869049\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3616.90	0.125300	0.0626501	−	0.998036i	$-0.480045\pi$
$941$	3616.90	0.125300	0.0626501	+	0.998036i	$0.480045\pi$
$942$	0	0
$943$	20818.5i	0.718922i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 14511.5i	− 0.497952i	−0.968510	−	0.248976i	$-0.919906\pi$
$947$	− 14511.5i	− 0.497952i	0.968510	−	0.248976i	$-0.0800940\pi$
$948$	0	0
$949$	−1353.11	−0.0462844
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 31649.8i	− 1.07580i	−0.843008	−	0.537901i	$-0.819218\pi$
$953$	− 31649.8i	− 1.07580i	0.843008	−	0.537901i	$-0.180782\pi$
$954$	0	0
$955$	− 2155.97i	− 0.0730528i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−55425.8	−1.86049
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−22263.8	−0.742691
$966$	0	0
$967$	26409.0	0.878238	0.439119	−	0.898429i	$-0.355291\pi$
$967$	26409.0	0.878238	0.439119	+	0.898429i	$0.355291\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−9050.09	−0.299105	−0.149553	−	0.988754i	$-0.547783\pi$
$971$	−9050.09	−0.299105	−0.149553	+	0.988754i	$0.547783\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 20093.7i	− 0.657988i	−0.944332	−	0.328994i	$-0.893290\pi$
$977$	− 20093.7i	− 0.657988i	0.944332	−	0.328994i	$-0.106710\pi$
$978$	0	0
$979$	12483.2i	0.407523i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	25677.3	0.833142	0.416571	−	0.909103i	$-0.363232\pi$
$983$	25677.3	0.833142	0.416571	+	0.909103i	$0.363232\pi$
$984$	0	0
$985$	− 40003.3i	− 1.29402i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 20629.0i	− 0.663260i
$990$	0	0
$991$	5703.10	0.182810	0.0914051	−	0.995814i	$-0.470864\pi$
$991$	5703.10	0.182810	0.0914051	+	0.995814i	$0.470864\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	21703.6i	0.691507i
$996$	0	0
$997$	9407.98i	0.298850i	0.988773	+	0.149425i	$0.0477423\pi$
$997$	9407.98i	0.298850i	−0.988773	+	0.149425i	$0.952258\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.f.a.881.4		16
3.2	odd	2	inner	1764.4.f.a.881.13		16
7.2	even	3		252.4.t.a.17.7	yes	16
7.3	odd	6		252.4.t.a.89.2	yes	16
7.4	even	3		1764.4.t.b.1097.7		16
7.5	odd	6		1764.4.t.b.521.2		16
7.6	odd	2	inner	1764.4.f.a.881.14		16
21.2	odd	6		252.4.t.a.17.2	✓	16
21.5	even	6		1764.4.t.b.521.7		16
21.11	odd	6		1764.4.t.b.1097.2		16
21.17	even	6		252.4.t.a.89.7	yes	16
21.20	even	2	inner	1764.4.f.a.881.3		16
28.3	even	6		1008.4.bt.b.593.2		16
28.23	odd	6		1008.4.bt.b.17.7		16
84.23	even	6		1008.4.bt.b.17.2		16
84.59	odd	6		1008.4.bt.b.593.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.t.a.17.2	✓	16	21.2	odd	6
252.4.t.a.17.7	yes	16	7.2	even	3
252.4.t.a.89.2	yes	16	7.3	odd	6
252.4.t.a.89.7	yes	16	21.17	even	6
1008.4.bt.b.17.2		16	84.23	even	6
1008.4.bt.b.17.7		16	28.23	odd	6
1008.4.bt.b.593.2		16	28.3	even	6
1008.4.bt.b.593.7		16	84.59	odd	6
1764.4.f.a.881.3		16	21.20	even	2	inner
1764.4.f.a.881.4		16	1.1	even	1	trivial
1764.4.f.a.881.13		16	3.2	odd	2	inner
1764.4.f.a.881.14		16	7.6	odd	2	inner
1764.4.t.b.521.2		16	7.5	odd	6
1764.4.t.b.521.7		16	21.5	even	6
1764.4.t.b.1097.2		16	21.11	odd	6
1764.4.t.b.1097.7		16	7.4	even	3

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-8.73625 q^{5} +O(q^{10})\)	\(q-8.73625 q^{5} +8.78630i q^{11} +11.8322i q^{13} +44.5839 q^{17} +11.6458i q^{19} -142.630i q^{23} -48.6779 q^{25} +234.018i q^{29} -291.919i q^{31} -88.9030 q^{37} -145.961 q^{41} +144.633 q^{43} -240.367 q^{47} +304.308i q^{53} -76.7594i q^{55} +7.08392 q^{59} -172.985i q^{61} -103.369i q^{65} -486.560 q^{67} +653.710i q^{71} +114.359i q^{73} +294.615 q^{79} +877.193 q^{83} -389.496 q^{85} +1420.76 q^{89} -101.741i q^{95} -738.981i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 424 q^{25} - 152 q^{37} + 1408 q^{43} + 3056 q^{67} + 728 q^{79} + 7392 q^{85}+O(q^{100}) \) 16 * q + 424 * q^25 - 152 * q^37 + 1408 * q^43 + 3056 * q^67 + 728 * q^79 + 7392 * q^85

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