LMFDB - Embedded newform 1764.3.c.h.197.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(197,1764)]

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, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1764.197");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.c (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:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.224054542336.12
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 199x^{4} + 14641 $ x^8 + 199*x^4 + 14641
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.4
Root			$1.96485 + 2.67196i$ of defining polynomial
Character	$\chi$	$=$	1764.197
Dual form			1764.3.c.h.197.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.55744i q^{5} +O(q^{10})$	$q-3.55744i q^{5} +9.27362i q^{11} +4.24264 q^{13} +2.44256i q^{17} -20.7498 q^{19} -0.788337i q^{23} +12.3446 q^{25} +7.69694i q^{29} -23.5782 q^{31} -23.3446 q^{37} -51.5574i q^{41} -49.3446 q^{43} +15.7702i q^{47} +75.2782i q^{53} +32.9903 q^{55} +15.7702i q^{59} +8.02187 q^{61} -15.0929i q^{65} -87.3446 q^{67} +40.0614i q^{71} -34.4045 q^{73} +8.68926 q^{79} -115.115i q^{83} +8.68926 q^{85} +94.2467i q^{89} +73.8161i q^{95} +154.173 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 216 q^{25} + 128 q^{37} - 80 q^{43} - 384 q^{67} - 560 q^{79} - 560 q^{85}+O(q^{100}) $ 8 * q - 216 * q^25 + 128 * q^37 - 80 * q^43 - 384 * q^67 - 560 * q^79 - 560 * 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
$5$	− 3.55744i	− 0.711488i	−0.934583	−	0.355744i	$-0.884228\pi$
$5$	− 3.55744i	− 0.711488i	0.934583	−	0.355744i	$-0.115772\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	9.27362i	0.843056i	0.906815	+	0.421528i	$0.138506\pi$
$11$	9.27362i	0.843056i	−0.906815	+	0.421528i	$0.861494\pi$
$12$	0	0
$13$	4.24264	0.326357	0.163178	−	0.986597i	$-0.447825\pi$
$13$	4.24264	0.326357	0.163178	+	0.986597i	$0.447825\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.44256i	0.143680i	0.997416	+	0.0718400i	$0.0228871\pi$
$17$	2.44256i	0.143680i	−0.997416	+	0.0718400i	$0.977113\pi$
$18$	0	0
$19$	−20.7498	−1.09209	−0.546047	−	0.837755i	$-0.683868\pi$
$19$	−20.7498	−1.09209	−0.546047	+	0.837755i	$0.683868\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 0.788337i	− 0.0342755i	−0.999853	−	0.0171378i	$-0.994545\pi$
$23$	− 0.788337i	− 0.0342755i	0.999853	−	0.0171378i	$-0.00545539\pi$
$24$	0	0
$25$	12.3446	0.493785
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.69694i	0.265412i	0.991155	+	0.132706i	$0.0423666\pi$
$29$	7.69694i	0.265412i	−0.991155	+	0.132706i	$0.957633\pi$
$30$	0	0
$31$	−23.5782	−0.760588	−0.380294	−	0.924866i	$-0.624177\pi$
$31$	−23.5782	−0.760588	−0.380294	+	0.924866i	$0.624177\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−23.3446	−0.630936	−0.315468	−	0.948936i	$-0.602161\pi$
$37$	−23.3446	−0.630936	−0.315468	+	0.948936i	$0.602161\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 51.5574i	− 1.25750i	−0.777608	−	0.628749i	$-0.783567\pi$
$41$	− 51.5574i	− 1.25750i	0.777608	−	0.628749i	$-0.216433\pi$
$42$	0	0
$43$	−49.3446	−1.14755	−0.573775	−	0.819013i	$-0.694522\pi$
$43$	−49.3446	−1.14755	−0.573775	+	0.819013i	$0.694522\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	15.7702i	0.335537i	0.985826	+	0.167769i	$0.0536561\pi$
$47$	15.7702i	0.335537i	−0.985826	+	0.167769i	$0.946344\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	75.2782i	1.42034i	0.704028	+	0.710172i	$0.251383\pi$
$53$	75.2782i	1.42034i	−0.704028	+	0.710172i	$0.748617\pi$
$54$	0	0
$55$	32.9903	0.599824
$56$	0	0
$57$	0	0
$58$	0	0
$59$	15.7702i	0.267292i	0.991029	+	0.133646i	$0.0426686\pi$
$59$	15.7702i	0.267292i	−0.991029	+	0.133646i	$0.957331\pi$
$60$	0	0
$61$	8.02187	0.131506	0.0657530	−	0.997836i	$-0.479055\pi$
$61$	8.02187	0.131506	0.0657530	+	0.997836i	$0.479055\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 15.0929i	− 0.232199i
$66$	0	0
$67$	−87.3446	−1.30365	−0.651826	−	0.758369i	$-0.725997\pi$
$67$	−87.3446	−1.30365	−0.651826	+	0.758369i	$0.725997\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	40.0614i	0.564245i	0.959378	+	0.282123i	$0.0910385\pi$
$71$	40.0614i	0.564245i	−0.959378	+	0.282123i	$0.908962\pi$
$72$	0	0
$73$	−34.4045	−0.471295	−0.235648	−	0.971839i	$-0.575721\pi$
$73$	−34.4045	−0.471295	−0.235648	+	0.971839i	$0.575721\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.68926	0.109991	0.0549953	−	0.998487i	$-0.482486\pi$
$79$	8.68926	0.109991	0.0549953	+	0.998487i	$0.482486\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 115.115i	− 1.38693i	−0.720492	−	0.693463i	$-0.756084\pi$
$83$	− 115.115i	− 1.38693i	0.720492	−	0.693463i	$-0.243916\pi$
$84$	0	0
$85$	8.68926	0.102227
$86$	0	0
$87$	0	0
$88$	0	0
$89$	94.2467i	1.05895i	0.848325	+	0.529476i	$0.177611\pi$
$89$	94.2467i	1.05895i	−0.848325	+	0.529476i	$0.822389\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	73.8161i	0.777012i
$96$	0	0
$97$	154.173	1.58941	0.794707	−	0.606993i	$-0.207624\pi$
$97$	154.173	1.58941	0.794707	+	0.606993i	$0.207624\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	191.362i	1.89467i	0.320246	+	0.947335i	$0.396235\pi$
$101$	191.362i	1.89467i	−0.320246	+	0.947335i	$0.603765\pi$
$102$	0	0
$103$	−81.0736	−0.787122	−0.393561	−	0.919298i	$-0.628757\pi$
$103$	−81.0736	−0.787122	−0.393561	+	0.919298i	$0.628757\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	166.137i	1.55268i	0.630314	+	0.776340i	$0.282926\pi$
$107$	166.137i	1.55268i	−0.630314	+	0.776340i	$0.717074\pi$
$108$	0	0
$109$	−128.689	−1.18064	−0.590318	−	0.807171i	$-0.700998\pi$
$109$	−128.689	−1.18064	−0.590318	+	0.807171i	$0.700998\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	188.138i	1.66494i	0.554069	+	0.832471i	$0.313074\pi$
$113$	188.138i	1.66494i	−0.554069	+	0.832471i	$0.686926\pi$
$114$	0	0
$115$	−2.80446	−0.0243866
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	35.0000	0.289256
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 132.851i	− 1.06281i
$126$	0	0
$127$	−40.6554	−0.320121	−0.160061	−	0.987107i	$-0.551169\pi$
$127$	−40.6554	−0.320121	−0.160061	+	0.987107i	$0.551169\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	96.0000i	0.732824i	0.930453	+	0.366412i	$0.119414\pi$
$131$	96.0000i	0.732824i	−0.930453	+	0.366412i	$0.880586\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	116.429i	0.849846i	0.905229	+	0.424923i	$0.139699\pi$
$137$	116.429i	0.849846i	−0.905229	+	0.424923i	$0.860301\pi$
$138$	0	0
$139$	99.9218	0.718862	0.359431	−	0.933172i	$-0.382971\pi$
$139$	99.9218	0.718862	0.359431	+	0.933172i	$0.382971\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	39.3446i	0.275137i
$144$	0	0
$145$	27.3814	0.188837
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 108.617i	− 0.728976i	−0.931208	−	0.364488i	$-0.881244\pi$
$149$	− 108.617i	− 0.728976i	0.931208	−	0.364488i	$-0.118756\pi$
$150$	0	0
$151$	−224.034	−1.48367	−0.741834	−	0.670583i	$-0.766044\pi$
$151$	−224.034	−1.48367	−0.741834	+	0.670583i	$0.766044\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	83.8780i	0.541149i
$156$	0	0
$157$	−105.115	−0.669524	−0.334762	−	0.942303i	$-0.608656\pi$
$157$	−105.115	−0.669524	−0.334762	+	0.942303i	$0.608656\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−21.9661	−0.134761	−0.0673807	−	0.997727i	$-0.521464\pi$
$163$	−21.9661	−0.134761	−0.0673807	+	0.997727i	$0.521464\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 311.838i	− 1.86729i	−0.358195	−	0.933647i	$-0.616608\pi$
$167$	− 311.838i	− 1.86729i	0.358195	−	0.933647i	$-0.383392\pi$
$168$	0	0
$169$	−151.000	−0.893491
$170$	0	0
$171$	0	0
$172$	0	0
$173$	64.6723i	0.373828i	0.982376	+	0.186914i	$0.0598486\pi$
$173$	64.6723i	0.373828i	−0.982376	+	0.186914i	$0.940151\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	334.266i	1.86741i	0.358048	+	0.933703i	$0.383442\pi$
$179$	334.266i	1.86741i	−0.358048	+	0.933703i	$0.616558\pi$
$180$	0	0
$181$	229.590	1.26845	0.634226	−	0.773147i	$-0.281319\pi$
$181$	229.590	1.26845	0.634226	+	0.773147i	$0.281319\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	83.0471i	0.448903i
$186$	0	0
$187$	−22.6514	−0.121130
$188$	0	0
$189$	0	0
$190$	0	0
$191$	75.5792i	0.395703i	0.980232	+	0.197851i	$0.0633963\pi$
$191$	75.5792i	0.395703i	−0.980232	+	0.197851i	$0.936604\pi$
$192$	0	0
$193$	9.37852	0.0485934	0.0242967	−	0.999705i	$-0.492265\pi$
$193$	9.37852	0.0485934	0.0242967	+	0.999705i	$0.492265\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 38.5566i	− 0.195719i	−0.995200	−	0.0978594i	$-0.968800\pi$
$197$	− 38.5566i	− 0.195719i	0.995200	−	0.0978594i	$-0.0311996\pi$
$198$	0	0
$199$	−65.1018	−0.327145	−0.163572	−	0.986531i	$-0.552302\pi$
$199$	−65.1018	−0.327145	−0.163572	+	0.986531i	$0.552302\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−183.412	−0.894695
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 192.426i	− 0.920697i
$210$	0	0
$211$	−176.068	−0.834444	−0.417222	−	0.908804i	$-0.636996\pi$
$211$	−176.068	−0.834444	−0.417222	+	0.908804i	$0.636996\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	175.540i	0.816467i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.3629i	0.0468910i
$222$	0	0
$223$	−146.199	−0.655602	−0.327801	−	0.944747i	$-0.606308\pi$
$223$	−146.199	−0.655602	−0.327801	+	0.944747i	$0.606308\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.6893i	0.108763i	0.998520	+	0.0543816i	$0.0173188\pi$
$227$	24.6893i	0.108763i	−0.998520	+	0.0543816i	$0.982681\pi$
$228$	0	0
$229$	−209.839	−0.916327	−0.458164	−	0.888868i	$-0.651493\pi$
$229$	−209.839	−0.916327	−0.458164	+	0.888868i	$0.651493\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 40.0614i	− 0.171937i	−0.996298	−	0.0859687i	$-0.972602\pi$
$233$	− 40.0614i	− 0.171937i	0.996298	−	0.0859687i	$-0.0273985\pi$
$234$	0	0
$235$	56.1017	0.238731
$236$	0	0
$237$	0	0
$238$	0	0
$239$	229.662i	0.960928i	0.877014	+	0.480464i	$0.159532\pi$
$239$	229.662i	0.960928i	−0.877014	+	0.480464i	$0.840468\pi$
$240$	0	0
$241$	−381.422	−1.58266	−0.791332	−	0.611386i	$-0.790612\pi$
$241$	−381.422	−1.58266	−0.791332	+	0.611386i	$0.790612\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−88.0339	−0.356413
$248$	0	0
$249$	0	0
$250$	0	0
$251$	72.2636i	0.287903i	0.989585	+	0.143951i	$0.0459809\pi$
$251$	72.2636i	0.287903i	−0.989585	+	0.143951i	$0.954019\pi$
$252$	0	0
$253$	7.31074	0.0288962
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 222.740i	− 0.866693i	−0.901227	−	0.433347i	$-0.857333\pi$
$257$	− 222.740i	− 0.866693i	0.901227	−	0.433347i	$-0.142667\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	322.025i	1.22443i	0.790691	+	0.612215i	$0.209721\pi$
$263$	322.025i	1.22443i	−0.790691	+	0.612215i	$0.790279\pi$
$264$	0	0
$265$	267.798	1.01056
$266$	0	0
$267$	0	0
$268$	0	0
$269$	307.591i	1.14346i	0.820441	+	0.571731i	$0.193728\pi$
$269$	307.591i	1.14346i	−0.820441	+	0.571731i	$0.806272\pi$
$270$	0	0
$271$	−362.111	−1.33620	−0.668101	−	0.744071i	$-0.732892\pi$
$271$	−362.111	−1.33620	−0.668101	+	0.744071i	$0.732892\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	114.479i	0.416289i
$276$	0	0
$277$	28.6893	0.103571	0.0517857	−	0.998658i	$-0.483509\pi$
$277$	28.6893	0.103571	0.0517857	+	0.998658i	$0.483509\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 89.7693i	− 0.319464i	−0.987160	−	0.159732i	$-0.948937\pi$
$281$	− 89.7693i	− 0.319464i	0.987160	−	0.159732i	$-0.0510629\pi$
$282$	0	0
$283$	301.739	1.06621	0.533107	−	0.846048i	$-0.321024\pi$
$283$	301.739	1.06621	0.533107	+	0.846048i	$0.321024\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	283.034	0.979356
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 235.855i	− 0.804966i	−0.915428	−	0.402483i	$-0.868147\pi$
$293$	− 235.855i	− 0.804966i	0.915428	−	0.402483i	$-0.131853\pi$
$294$	0	0
$295$	56.1017	0.190175
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 3.34463i	− 0.0111861i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 28.5373i	− 0.0935649i
$306$	0	0
$307$	68.8810	0.224368	0.112184	−	0.993687i	$-0.464215\pi$
$307$	68.8810	0.224368	0.112184	+	0.993687i	$0.464215\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 499.906i	− 1.60741i	−0.595025	−	0.803707i	$-0.702858\pi$
$311$	− 499.906i	− 1.60741i	0.595025	−	0.803707i	$-0.297142\pi$
$312$	0	0
$313$	288.012	0.920167	0.460083	−	0.887876i	$-0.347820\pi$
$313$	288.012	0.920167	0.460083	+	0.887876i	$0.347820\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 165.090i	− 0.520789i	−0.965502	−	0.260395i	$-0.916147\pi$
$317$	− 165.090i	− 0.520789i	0.965502	−	0.260395i	$-0.0838526\pi$
$318$	0	0
$319$	−71.3785	−0.223757
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 50.6826i	− 0.156912i
$324$	0	0
$325$	52.3738	0.161150
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−265.311	−0.801543	−0.400772	−	0.916178i	$-0.631258\pi$
$331$	−265.311	−0.801543	−0.400772	+	0.916178i	$0.631258\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	310.723i	0.927532i
$336$	0	0
$337$	220.655	0.654764	0.327382	−	0.944892i	$-0.393834\pi$
$337$	220.655	0.654764	0.327382	+	0.944892i	$0.393834\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 218.655i	− 0.641218i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 187.235i	− 0.539583i	−0.962919	−	0.269792i	$-0.913045\pi$
$347$	− 187.235i	− 0.539583i	0.962919	−	0.269792i	$-0.0869549\pi$
$348$	0	0
$349$	−318.270	−0.911948	−0.455974	−	0.889993i	$-0.650709\pi$
$349$	−318.270	−0.911948	−0.455974	+	0.889993i	$0.650709\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	169.166i	0.479223i	0.970869	+	0.239611i	$0.0770201\pi$
$353$	169.166i	0.479223i	−0.970869	+	0.239611i	$0.922980\pi$
$354$	0	0
$355$	142.516	0.401453
$356$	0	0
$357$	0	0
$358$	0	0
$359$	111.097i	0.309462i	0.987957	+	0.154731i	$0.0494511\pi$
$359$	111.097i	0.309462i	−0.987957	+	0.154731i	$0.950549\pi$
$360$	0	0
$361$	69.5537	0.192669
$362$	0	0
$363$	0	0
$364$	0	0
$365$	122.392i	0.335321i
$366$	0	0
$367$	81.0496	0.220844	0.110422	−	0.993885i	$-0.464780\pi$
$367$	81.0496	0.220844	0.110422	+	0.993885i	$0.464780\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	439.446	1.17814	0.589070	−	0.808082i	$-0.299494\pi$
$373$	439.446	1.17814	0.589070	+	0.808082i	$0.299494\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	32.6554i	0.0866190i
$378$	0	0
$379$	−81.3785	−0.214719	−0.107360	−	0.994220i	$-0.534240\pi$
$379$	−81.3785	−0.214719	−0.107360	+	0.994220i	$0.534240\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	460.987i	1.20362i	0.798639	+	0.601810i	$0.205554\pi$
$383$	460.987i	1.20362i	−0.798639	+	0.601810i	$0.794446\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 555.256i	− 1.42739i	−0.700455	−	0.713697i	$-0.747020\pi$
$389$	− 555.256i	− 1.42739i	0.700455	−	0.713697i	$-0.252980\pi$
$390$	0	0
$391$	1.92556	0.00492471
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 30.9115i	− 0.0782570i
$396$	0	0
$397$	304.128	0.766065	0.383033	−	0.923735i	$-0.374880\pi$
$397$	304.128	0.766065	0.383033	+	0.923735i	$0.374880\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 246.489i	− 0.614685i	−0.951599	−	0.307342i	$-0.900560\pi$
$401$	− 246.489i	− 0.614685i	0.951599	−	0.307342i	$-0.0994397\pi$
$402$	0	0
$403$	−100.034	−0.248223
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 216.489i	− 0.531915i
$408$	0	0
$409$	−14.6056	−0.0357104	−0.0178552	−	0.999841i	$-0.505684\pi$
$409$	−14.6056	−0.0357104	−0.0178552	+	0.999841i	$0.505684\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−409.514	−0.986781
$416$	0	0
$417$	0	0
$418$	0	0
$419$	619.804i	1.47925i	0.673021	+	0.739623i	$0.264996\pi$
$419$	619.804i	1.47925i	−0.673021	+	0.739623i	$0.735004\pi$
$420$	0	0
$421$	−307.379	−0.730115	−0.365058	−	0.930985i	$-0.618951\pi$
$421$	−307.379	−0.730115	−0.365058	+	0.930985i	$0.618951\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	30.1525i	0.0709471i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 522.891i	− 1.21321i	−0.795005	−	0.606603i	$-0.792532\pi$
$431$	− 522.891i	− 1.21321i	0.795005	−	0.606603i	$-0.207468\pi$
$432$	0	0
$433$	−547.444	−1.26431	−0.632153	−	0.774844i	$-0.717829\pi$
$433$	−547.444	−1.26431	−0.632153	+	0.774844i	$0.717829\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.3578i	0.0374321i
$438$	0	0
$439$	516.771	1.17716	0.588578	−	0.808441i	$-0.299688\pi$
$439$	516.771	1.17716	0.588578	+	0.808441i	$0.299688\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	547.974i	1.23696i	0.785799	+	0.618481i	$0.212252\pi$
$443$	547.974i	1.23696i	−0.785799	+	0.618481i	$0.787748\pi$
$444$	0	0
$445$	335.277	0.753431
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 575.681i	− 1.28214i	−0.767482	−	0.641070i	$-0.778491\pi$
$449$	− 575.681i	− 1.28214i	0.767482	−	0.641070i	$-0.221509\pi$
$450$	0	0
$451$	478.124	1.06014
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	406.825	0.890208	0.445104	−	0.895479i	$-0.353167\pi$
$457$	406.825	0.890208	0.445104	+	0.895479i	$0.353167\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	209.524i	0.454498i	0.973837	+	0.227249i	$0.0729731\pi$
$461$	209.524i	0.454498i	−0.973837	+	0.227249i	$0.927027\pi$
$462$	0	0
$463$	870.169	1.87942	0.939708	−	0.341978i	$-0.111097\pi$
$463$	870.169	1.87942	0.939708	+	0.341978i	$0.111097\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 232.196i	− 0.497207i	−0.968605	−	0.248604i	$-0.920028\pi$
$467$	− 232.196i	− 0.497207i	0.968605	−	0.248604i	$-0.0799717\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 457.603i	− 0.967449i
$474$	0	0
$475$	−256.148	−0.539260
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 889.642i	− 1.85729i	−0.370969	−	0.928645i	$-0.620974\pi$
$479$	− 889.642i	− 1.85729i	0.370969	−	0.928645i	$-0.379026\pi$
$480$	0	0
$481$	−99.0429	−0.205910
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 548.462i	− 1.13085i
$486$	0	0
$487$	−65.4124	−0.134317	−0.0671585	−	0.997742i	$-0.521393\pi$
$487$	−65.4124	−0.134317	−0.0671585	+	0.997742i	$0.521393\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 82.4878i	− 0.168000i	−0.996466	−	0.0839998i	$-0.973230\pi$
$491$	− 82.4878i	− 0.168000i	0.996466	−	0.0839998i	$-0.0267695\pi$
$492$	0	0
$493$	−18.8003	−0.0381344
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−66.6554	−0.133578	−0.0667889	−	0.997767i	$-0.521275\pi$
$499$	−66.6554	−0.133578	−0.0667889	+	0.997767i	$0.521275\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	592.885i	1.17870i	0.807879	+	0.589349i	$0.200616\pi$
$503$	592.885i	1.17870i	−0.807879	+	0.589349i	$0.799384\pi$
$504$	0	0
$505$	680.757	1.34803
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 777.923i	− 1.52834i	−0.645018	−	0.764168i	$-0.723150\pi$
$509$	− 777.923i	− 1.52834i	0.645018	−	0.764168i	$-0.276850\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	288.414i	0.560028i
$516$	0	0
$517$	−146.247	−0.282877
$518$	0	0
$519$	0	0
$520$	0	0
$521$	433.166i	0.831412i	0.909499	+	0.415706i	$0.136465\pi$
$521$	433.166i	0.831412i	−0.909499	+	0.415706i	$0.863535\pi$
$522$	0	0
$523$	−277.234	−0.530084	−0.265042	−	0.964237i	$-0.585386\pi$
$523$	−277.234	−0.530084	−0.265042	+	0.964237i	$0.585386\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 57.5912i	− 0.109281i
$528$	0	0
$529$	528.379	0.998825
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 218.740i	− 0.410393i
$534$	0	0
$535$	591.021	1.10471
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	720.757	1.33227	0.666134	−	0.745832i	$-0.267948\pi$
$541$	720.757	1.33227	0.666134	+	0.745832i	$0.267948\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	457.804i	0.840008i
$546$	0	0
$547$	−744.825	−1.36165	−0.680827	−	0.732444i	$-0.738379\pi$
$547$	−744.825	−1.36165	−0.680827	+	0.732444i	$0.738379\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 159.710i	− 0.289855i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 716.032i	− 1.28551i	−0.766070	−	0.642757i	$-0.777790\pi$
$557$	− 716.032i	− 1.28551i	0.766070	−	0.642757i	$-0.222210\pi$
$558$	0	0
$559$	−209.352	−0.374511
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 750.851i	− 1.33366i	−0.745209	−	0.666831i	$-0.767650\pi$
$563$	− 750.851i	− 1.33366i	0.745209	−	0.666831i	$-0.232350\pi$
$564$	0	0
$565$	669.291	1.18459
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 169.748i	− 0.298327i	−0.988813	−	0.149164i	$-0.952342\pi$
$569$	− 169.748i	− 0.298327i	0.988813	−	0.149164i	$-0.0476581\pi$
$570$	0	0
$571$	627.514	1.09897	0.549487	−	0.835502i	$-0.314823\pi$
$571$	627.514	1.09897	0.549487	+	0.835502i	$0.314823\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 9.73173i	− 0.0169247i
$576$	0	0
$577$	−164.632	−0.285324	−0.142662	−	0.989771i	$-0.545566\pi$
$577$	−164.632	−0.285324	−0.142662	+	0.989771i	$0.545566\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−698.102	−1.19743
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1040.76i	1.77301i	0.462719	+	0.886505i	$0.346874\pi$
$587$	1040.76i	1.77301i	−0.462719	+	0.886505i	$0.653126\pi$
$588$	0	0
$589$	489.243	0.830633
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 627.923i	− 1.05889i	−0.848344	−	0.529446i	$-0.822400\pi$
$593$	− 627.923i	− 1.05889i	0.848344	−	0.529446i	$-0.177600\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 263.832i	− 0.440454i	−0.975449	−	0.220227i	$-0.929320\pi$
$599$	− 263.832i	− 0.440454i	0.975449	−	0.220227i	$-0.0706798\pi$
$600$	0	0
$601$	−502.094	−0.835431	−0.417715	−	0.908578i	$-0.637169\pi$
$601$	−502.094	−0.835431	−0.417715	+	0.908578i	$0.637169\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 124.510i	− 0.205802i
$606$	0	0
$607$	−999.601	−1.64679	−0.823395	−	0.567469i	$-0.807923\pi$
$607$	−999.601	−1.64679	−0.823395	+	0.567469i	$0.807923\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	66.9075i	0.109505i
$612$	0	0
$613$	920.169	1.50109	0.750546	−	0.660818i	$-0.229791\pi$
$613$	920.169	1.50109	0.750546	+	0.660818i	$0.229791\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 520.713i	− 0.843943i	−0.906609	−	0.421972i	$-0.861338\pi$
$617$	− 520.713i	− 0.843943i	0.906609	−	0.421972i	$-0.138662\pi$
$618$	0	0
$619$	238.611	0.385478	0.192739	−	0.981250i	$-0.438263\pi$
$619$	238.611	0.385478	0.192739	+	0.981250i	$0.438263\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−163.994	−0.262391
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 57.0207i	− 0.0906529i
$630$	0	0
$631$	534.034	0.846329	0.423165	−	0.906053i	$-0.360919\pi$
$631$	534.034	0.846329	0.423165	+	0.906053i	$0.360919\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	144.629i	0.227762i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1003.03i	− 1.56478i	−0.622787	−	0.782392i	$-0.713999\pi$
$641$	− 1003.03i	− 1.56478i	0.622787	−	0.782392i	$-0.286001\pi$
$642$	0	0
$643$	−744.923	−1.15851	−0.579256	−	0.815146i	$-0.696657\pi$
$643$	−744.923	−1.15851	−0.579256	+	0.815146i	$0.696657\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 891.446i	− 1.37782i	−0.724849	−	0.688908i	$-0.758091\pi$
$647$	− 891.446i	− 1.37782i	0.724849	−	0.688908i	$-0.241909\pi$
$648$	0	0
$649$	−146.247	−0.225342
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 81.5130i	− 0.124829i	−0.998050	−	0.0624143i	$-0.980120\pi$
$653$	− 81.5130i	− 0.124829i	0.998050	−	0.0624143i	$-0.0198800\pi$
$654$	0	0
$655$	341.514	0.521396
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 185.430i	− 0.281380i	−0.990054	−	0.140690i	$-0.955068\pi$
$659$	− 185.430i	− 0.281380i	0.990054	−	0.140690i	$-0.0449322\pi$
$660$	0	0
$661$	−545.567	−0.825366	−0.412683	−	0.910875i	$-0.635408\pi$
$661$	−545.567	−0.825366	−0.412683	+	0.910875i	$0.635408\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.06779	0.00909713
$668$	0	0
$669$	0	0
$670$	0	0
$671$	74.3917i	0.110867i
$672$	0	0
$673$	−994.136	−1.47717	−0.738585	−	0.674160i	$-0.764506\pi$
$673$	−994.136	−1.47717	−0.738585	+	0.674160i	$0.764506\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 209.949i	− 0.310117i	−0.987905	−	0.155058i	$-0.950443\pi$
$677$	− 209.949i	− 0.310117i	0.987905	−	0.155058i	$-0.0495566\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	410.633i	0.601220i	0.953747	+	0.300610i	$0.0971903\pi$
$683$	410.633i	0.601220i	−0.953747	+	0.300610i	$0.902810\pi$
$684$	0	0
$685$	414.189	0.604655
$686$	0	0
$687$	0	0
$688$	0	0
$689$	319.379i	0.463539i
$690$	0	0
$691$	1324.68	1.91705	0.958523	−	0.285016i	$-0.0919988\pi$
$691$	1324.68	1.91705	0.958523	+	0.285016i	$0.0919988\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 355.466i	− 0.511461i
$696$	0	0
$697$	125.932	0.180677
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 580.339i	− 0.827873i	−0.910306	−	0.413936i	$-0.864154\pi$
$701$	− 580.339i	− 0.827873i	0.910306	−	0.413936i	$-0.135846\pi$
$702$	0	0
$703$	484.396	0.689041
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−327.379	−0.461747	−0.230873	−	0.972984i	$-0.574158\pi$
$709$	−327.379	−0.461747	−0.230873	+	0.972984i	$0.574158\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	18.5876i	0.0260695i
$714$	0	0
$715$	139.966	0.195757
$716$	0	0
$717$	0	0
$718$	0	0
$719$	392.392i	0.545746i	0.962050	+	0.272873i	$0.0879740\pi$
$719$	392.392i	0.545746i	−0.962050	+	0.272873i	$0.912026\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	95.0159i	0.131056i
$726$	0	0
$727$	−1340.99	−1.84455	−0.922274	−	0.386537i	$-0.873671\pi$
$727$	−1340.99	−1.84455	−0.922274	+	0.386537i	$0.873671\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 120.527i	− 0.164880i
$732$	0	0
$733$	−130.403	−0.177904	−0.0889518	−	0.996036i	$-0.528352\pi$
$733$	−130.403	−0.177904	−0.0889518	+	0.996036i	$0.528352\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 810.001i	− 1.09905i
$738$	0	0
$739$	−202.034	−0.273388	−0.136694	−	0.990613i	$-0.543648\pi$
$739$	−202.034	−0.273388	−0.136694	+	0.990613i	$0.543648\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 821.453i	− 1.10559i	−0.833317	−	0.552795i	$-0.813561\pi$
$743$	− 821.453i	− 1.10559i	0.833317	−	0.552795i	$-0.186439\pi$
$744$	0	0
$745$	−386.400	−0.518658
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	804.136	1.07075	0.535377	−	0.844613i	$-0.320170\pi$
$751$	804.136	1.07075	0.535377	+	0.844613i	$0.320170\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	796.987i	1.05561i
$756$	0	0
$757$	518.000	0.684280	0.342140	−	0.939649i	$-0.388848\pi$
$757$	518.000	0.684280	0.342140	+	0.939649i	$0.388848\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 673.855i	− 0.885486i	−0.896649	−	0.442743i	$-0.854005\pi$
$761$	− 673.855i	− 0.885486i	0.896649	−	0.442743i	$-0.145995\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	66.9075i	0.0872327i
$768$	0	0
$769$	653.023	0.849185	0.424592	−	0.905385i	$-0.360417\pi$
$769$	653.023	0.849185	0.424592	+	0.905385i	$0.360417\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 461.301i	− 0.596768i	−0.954446	−	0.298384i	$-0.903552\pi$
$773$	− 461.301i	− 0.596768i	0.954446	−	0.298384i	$-0.0964475\pi$
$774$	0	0
$775$	−291.064	−0.375567
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1069.81i	1.37331i
$780$	0	0
$781$	−371.514	−0.475690
$782$	0	0
$783$	0	0
$784$	0	0
$785$	373.941i	0.476358i
$786$	0	0
$787$	793.126	1.00778	0.503892	−	0.863767i	$-0.331901\pi$
$787$	793.126	1.00778	0.503892	+	0.863767i	$0.331901\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	34.0339	0.0429179
$794$	0	0
$795$	0	0
$796$	0	0
$797$	435.599i	0.546548i	0.961936	+	0.273274i	$0.0881066\pi$
$797$	435.599i	0.546548i	−0.961936	+	0.273274i	$0.911893\pi$
$798$	0	0
$799$	−38.5198	−0.0482100
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 319.055i	− 0.397328i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 577.028i	− 0.713261i	−0.934245	−	0.356631i	$-0.883925\pi$
$809$	− 577.028i	− 0.713261i	0.934245	−	0.356631i	$-0.116075\pi$
$810$	0	0
$811$	38.5274	0.0475060	0.0237530	−	0.999718i	$-0.492438\pi$
$811$	38.5274	0.0475060	0.0237530	+	0.999718i	$0.492438\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	78.1431i	0.0958811i
$816$	0	0
$817$	1023.89	1.25323
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 455.768i	− 0.555138i	−0.960706	−	0.277569i	$-0.910471\pi$
$821$	− 455.768i	− 0.555138i	0.960706	−	0.277569i	$-0.0895287\pi$
$822$	0	0
$823$	−399.582	−0.485519	−0.242759	−	0.970087i	$-0.578053\pi$
$823$	−399.582	−0.485519	−0.242759	+	0.970087i	$0.578053\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1182.94i	1.43040i	0.698922	+	0.715198i	$0.253664\pi$
$827$	1182.94i	1.43040i	−0.698922	+	0.715198i	$0.746336\pi$
$828$	0	0
$829$	−163.561	−0.197300	−0.0986498	−	0.995122i	$-0.531452\pi$
$829$	−163.561	−0.197300	−0.0986498	+	0.995122i	$0.531452\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−1109.34	−1.32856
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 980.535i	− 1.16869i	−0.811504	−	0.584347i	$-0.801351\pi$
$839$	− 980.535i	− 1.16869i	0.811504	−	0.584347i	$-0.198649\pi$
$840$	0	0
$841$	781.757	0.929557
$842$	0	0
$843$	0	0
$844$	0	0
$845$	537.173i	0.635708i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	18.4034i	0.0216257i
$852$	0	0
$853$	184.455	0.216243	0.108121	−	0.994138i	$-0.465517\pi$
$853$	184.455	0.216243	0.108121	+	0.994138i	$0.465517\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1146.42i	1.33771i	0.743394	+	0.668854i	$0.233215\pi$
$857$	1146.42i	1.33771i	−0.743394	+	0.668854i	$0.766785\pi$
$858$	0	0
$859$	−891.074	−1.03734	−0.518670	−	0.854975i	$-0.673573\pi$
$859$	−891.074	−1.03734	−0.518670	+	0.854975i	$0.673573\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1430.73i	1.65786i	0.559354	+	0.828929i	$0.311049\pi$
$863$	1430.73i	1.65786i	−0.559354	+	0.828929i	$0.688951\pi$
$864$	0	0
$865$	230.068	0.265974
$866$	0	0
$867$	0	0
$868$	0	0
$869$	80.5809i	0.0927283i
$870$	0	0
$871$	−370.572	−0.425456
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1414.86	1.61329	0.806647	−	0.591034i	$-0.201280\pi$
$877$	1414.86	1.61329	0.806647	+	0.591034i	$0.201280\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 407.260i	− 0.462270i	−0.972922	−	0.231135i	$-0.925756\pi$
$881$	− 407.260i	− 0.462270i	0.972922	−	0.231135i	$-0.0742439\pi$
$882$	0	0
$883$	−386.723	−0.437965	−0.218983	−	0.975729i	$-0.570274\pi$
$883$	−386.723	−0.437965	−0.218983	+	0.975729i	$0.570274\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1153.74i	1.30073i	0.759624	+	0.650363i	$0.225383\pi$
$887$	1153.74i	1.30073i	−0.759624	+	0.650363i	$0.774617\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 327.229i	− 0.366438i
$894$	0	0
$895$	1189.13	1.32864
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 181.480i	− 0.201869i
$900$	0	0
$901$	−183.872	−0.204075
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 816.752i	− 0.902489i
$906$	0	0
$907$	−568.169	−0.626427	−0.313214	−	0.949683i	$-0.601406\pi$
$907$	−568.169	−0.626427	−0.313214	+	0.949683i	$0.601406\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	78.5035i	0.0861729i	0.999071	+	0.0430864i	$0.0137191\pi$
$911$	78.5035i	0.0861729i	−0.999071	+	0.0430864i	$0.986281\pi$
$912$	0	0
$913$	1067.53	1.16926
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	542.927	0.590780	0.295390	−	0.955377i	$-0.404550\pi$
$919$	542.927	0.590780	0.295390	+	0.955377i	$0.404550\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	169.966i	0.184145i
$924$	0	0
$925$	−288.181	−0.311547
$926$	0	0
$927$	0	0
$928$	0	0
$929$	183.335i	0.197347i	0.995120	+	0.0986734i	$0.0314599\pi$
$929$	183.335i	0.197347i	−0.995120	+	0.0986734i	$0.968540\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	80.5809i	0.0861828i
$936$	0	0
$937$	532.184	0.567966	0.283983	−	0.958829i	$-0.408344\pi$
$937$	532.184	0.567966	0.283983	+	0.958829i	$0.408344\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1666.13i	1.77059i	0.465029	+	0.885295i	$0.346044\pi$
$941$	1666.13i	1.77059i	−0.465029	+	0.885295i	$0.653956\pi$
$942$	0	0
$943$	−40.6446	−0.0431014
$944$	0	0
$945$	0	0
$946$	0	0
$947$	538.887i	0.569047i	0.958669	+	0.284523i	$0.0918353\pi$
$947$	538.887i	0.569047i	−0.958669	+	0.284523i	$0.908165\pi$
$948$	0	0
$949$	−145.966	−0.153810
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1047.62i	1.09928i	0.835400	+	0.549642i	$0.185236\pi$
$953$	1047.62i	1.09928i	−0.835400	+	0.549642i	$0.814764\pi$
$954$	0	0
$955$	268.868	0.281538
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−405.068	−0.421507
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 33.3635i	− 0.0345736i
$966$	0	0
$967$	347.209	0.359058	0.179529	−	0.983753i	$-0.442543\pi$
$967$	347.209	0.359058	0.179529	+	0.983753i	$0.442543\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 448.885i	− 0.462292i	−0.972919	−	0.231146i	$-0.925753\pi$
$971$	− 448.885i	− 0.462292i	0.972919	−	0.231146i	$-0.0742474\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	945.494i	0.967752i	0.875137	+	0.483876i	$0.160771\pi$
$977$	945.494i	0.967752i	−0.875137	+	0.483876i	$0.839229\pi$
$978$	0	0
$979$	−874.008	−0.892756
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1079.25i	− 1.09792i	−0.835850	−	0.548958i	$-0.815025\pi$
$983$	− 1079.25i	− 1.09792i	0.835850	−	0.548958i	$-0.184975\pi$
$984$	0	0
$985$	−137.163	−0.139252
$986$	0	0
$987$	0	0
$988$	0	0
$989$	38.9002i	0.0393329i
$990$	0	0
$991$	−786.859	−0.794005	−0.397002	−	0.917818i	$-0.629950\pi$
$991$	−786.859	−0.794005	−0.397002	+	0.917818i	$0.629950\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	231.595i	0.232759i
$996$	0	0
$997$	−1876.00	−1.88164	−0.940822	−	0.338902i	$-0.889944\pi$
$997$	−1876.00	−1.88164	−0.940822	+	0.338902i	$0.889944\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.c.h.197.4	yes	8
3.2	odd	2	inner	1764.3.c.h.197.5	yes	8
7.2	even	3		1764.3.bk.g.557.3		16
7.3	odd	6		1764.3.bk.g.1745.4		16
7.4	even	3		1764.3.bk.g.1745.6		16
7.5	odd	6		1764.3.bk.g.557.5		16
7.6	odd	2	inner	1764.3.c.h.197.6	yes	8
21.2	odd	6		1764.3.bk.g.557.6		16
21.5	even	6		1764.3.bk.g.557.4		16
21.11	odd	6		1764.3.bk.g.1745.3		16
21.17	even	6		1764.3.bk.g.1745.5		16
21.20	even	2	inner	1764.3.c.h.197.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.3.c.h.197.3	✓	8	21.20	even	2	inner
1764.3.c.h.197.4	yes	8	1.1	even	1	trivial
1764.3.c.h.197.5	yes	8	3.2	odd	2	inner
1764.3.c.h.197.6	yes	8	7.6	odd	2	inner
1764.3.bk.g.557.3		16	7.2	even	3
1764.3.bk.g.557.4		16	21.5	even	6
1764.3.bk.g.557.5		16	7.5	odd	6
1764.3.bk.g.557.6		16	21.2	odd	6
1764.3.bk.g.1745.3		16	21.11	odd	6
1764.3.bk.g.1745.4		16	7.3	odd	6
1764.3.bk.g.1745.5		16	21.17	even	6
1764.3.bk.g.1745.6		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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.55744i q^{5} +O(q^{10})\)	\(q-3.55744i q^{5} +9.27362i q^{11} +4.24264 q^{13} +2.44256i q^{17} -20.7498 q^{19} -0.788337i q^{23} +12.3446 q^{25} +7.69694i q^{29} -23.5782 q^{31} -23.3446 q^{37} -51.5574i q^{41} -49.3446 q^{43} +15.7702i q^{47} +75.2782i q^{53} +32.9903 q^{55} +15.7702i q^{59} +8.02187 q^{61} -15.0929i q^{65} -87.3446 q^{67} +40.0614i q^{71} -34.4045 q^{73} +8.68926 q^{79} -115.115i q^{83} +8.68926 q^{85} +94.2467i q^{89} +73.8161i q^{95} +154.173 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 216 q^{25} + 128 q^{37} - 80 q^{43} - 384 q^{67} - 560 q^{79} - 560 q^{85}+O(q^{100}) \) 8 * q - 216 * q^25 + 128 * q^37 - 80 * q^43 - 384 * q^67 - 560 * q^79 - 560 * q^85

Introduction

L-functions

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