LMFDB - Embedded newform 1764.3.d.h.685.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,3,Mod(685,1764)]

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, 0, 1]))

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

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

chi := DirichletCharacter("1764.685");

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.d (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:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 588)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.1
Root			$-0.662827 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	1764.685
Dual form			1764.3.d.h.685.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.82798i q^{5} +O(q^{10})$	$q-5.82798i q^{5} +6.86137 q^{11} -3.62063i q^{13} -9.71521i q^{17} +30.2903i q^{19} +18.1558 q^{23} -8.96538 q^{25} +40.4570 q^{29} -55.2124i q^{31} +54.9745 q^{37} +56.3322i q^{41} -66.0512 q^{43} -49.4215i q^{47} +81.0162 q^{53} -39.9879i q^{55} -34.7914i q^{59} -0.0382805i q^{61} -21.1010 q^{65} -64.0898 q^{67} -50.2730 q^{71} +21.3464i q^{73} -47.5044 q^{79} +33.6039i q^{83} -56.6201 q^{85} -156.092i q^{89} +176.531 q^{95} -43.7452i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 16 q^{23} + 72 q^{25} - 80 q^{29} + 128 q^{37} - 112 q^{43} + 144 q^{53} - 240 q^{65} - 64 q^{67} - 224 q^{71} - 432 q^{79} - 96 q^{85} + 272 q^{95}+O(q^{100}) $ 8 * q + 16 * q^23 + 72 * q^25 - 80 * q^29 + 128 * q^37 - 112 * q^43 + 144 * q^53 - 240 * q^65 - 64 * q^67 - 224 * q^71 - 432 * q^79 - 96 * q^85 + 272 * q^95

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
$5$	− 5.82798i	− 1.16560i	−0.812617	−	0.582798i	$-0.801958\pi$
$5$	− 5.82798i	− 1.16560i	0.812617	−	0.582798i	$-0.198042\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.86137	0.623761	0.311880	−	0.950121i	$-0.399041\pi$
$11$	6.86137	0.623761	0.311880	+	0.950121i	$0.399041\pi$
$12$	0	0
$13$	− 3.62063i	− 0.278510i	−0.990257	−	0.139255i	$-0.955529\pi$
$13$	− 3.62063i	− 0.278510i	0.990257	−	0.139255i	$-0.0444708\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 9.71521i	− 0.571483i	−0.958307	−	0.285742i	$-0.907760\pi$
$17$	− 9.71521i	− 0.571483i	0.958307	−	0.285742i	$-0.0922399\pi$
$18$	0	0
$19$	30.2903i	1.59423i	0.603831	+	0.797113i	$0.293640\pi$
$19$	30.2903i	1.59423i	−0.603831	+	0.797113i	$0.706360\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	18.1558	0.789382	0.394691	−	0.918814i	$-0.370852\pi$
$23$	18.1558	0.789382	0.394691	+	0.918814i	$0.370852\pi$
$24$	0	0
$25$	−8.96538	−0.358615
$26$	0	0
$27$	0	0
$28$	0	0
$29$	40.4570	1.39507	0.697534	−	0.716552i	$-0.254281\pi$
$29$	40.4570	1.39507	0.697534	+	0.716552i	$0.254281\pi$
$30$	0	0
$31$	− 55.2124i	− 1.78104i	−0.454940	−	0.890522i	$-0.650339\pi$
$31$	− 55.2124i	− 1.78104i	0.454940	−	0.890522i	$-0.349661\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	54.9745	1.48580	0.742899	−	0.669403i	$-0.233450\pi$
$37$	54.9745	1.48580	0.742899	+	0.669403i	$0.233450\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	56.3322i	1.37396i	0.726678	+	0.686978i	$0.241063\pi$
$41$	56.3322i	1.37396i	−0.726678	+	0.686978i	$0.758937\pi$
$42$	0	0
$43$	−66.0512	−1.53608	−0.768038	−	0.640405i	$-0.778767\pi$
$43$	−66.0512	−1.53608	−0.768038	+	0.640405i	$0.778767\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 49.4215i	− 1.05152i	−0.850632	−	0.525761i	$-0.823781\pi$
$47$	− 49.4215i	− 1.05152i	0.850632	−	0.525761i	$-0.176219\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	81.0162	1.52861	0.764303	−	0.644857i	$-0.223083\pi$
$53$	81.0162	1.52861	0.764303	+	0.644857i	$0.223083\pi$
$54$	0	0
$55$	− 39.9879i	− 0.727054i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 34.7914i	− 0.589685i	−0.955546	−	0.294842i	$-0.904733\pi$
$59$	− 34.7914i	− 0.589685i	0.955546	−	0.294842i	$-0.0952671\pi$
$60$	0	0
$61$	− 0.0382805i	0 0.000627550i	−1.00000		0.000313775i	$-0.999900\pi$
$61$	− 0.0382805i	0 0.000627550i	1.00000		0.000313775i	$-9.98776e-5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−21.1010	−0.324630
$66$	0	0
$67$	−64.0898	−0.956565	−0.478282	−	0.878206i	$-0.658740\pi$
$67$	−64.0898	−0.956565	−0.478282	+	0.878206i	$0.658740\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−50.2730	−0.708070	−0.354035	−	0.935232i	$-0.615191\pi$
$71$	−50.2730	−0.708070	−0.354035	+	0.935232i	$0.615191\pi$
$72$	0	0
$73$	21.3464i	0.292416i	0.989254	+	0.146208i	$0.0467069\pi$
$73$	21.3464i	0.292416i	−0.989254	+	0.146208i	$0.953293\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−47.5044	−0.601321	−0.300660	−	0.953731i	$-0.597207\pi$
$79$	−47.5044	−0.601321	−0.300660	+	0.953731i	$0.597207\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	33.6039i	0.404866i	0.979296	+	0.202433i	$0.0648849\pi$
$83$	33.6039i	0.404866i	−0.979296	+	0.202433i	$0.935115\pi$
$84$	0	0
$85$	−56.6201	−0.666119
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 156.092i	− 1.75384i	−0.480636	−	0.876920i	$-0.659594\pi$
$89$	− 156.092i	− 1.75384i	0.480636	−	0.876920i	$-0.340406\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	176.531	1.85822
$96$	0	0
$97$	− 43.7452i	− 0.450981i	−0.974245	−	0.225491i	$-0.927601\pi$
$97$	− 43.7452i	− 0.450981i	0.974245	−	0.225491i	$-0.0723985\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 169.746i	− 1.68066i	−0.542077	−	0.840329i	$-0.682362\pi$
$101$	− 169.746i	− 1.68066i	0.542077	−	0.840329i	$-0.317638\pi$
$102$	0	0
$103$	− 59.7216i	− 0.579822i	−0.957054	−	0.289911i	$-0.906374\pi$
$103$	− 59.7216i	− 0.579822i	0.957054	−	0.289911i	$-0.0936257\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	57.7678	0.539886	0.269943	−	0.962876i	$-0.412995\pi$
$107$	57.7678	0.539886	0.269943	+	0.962876i	$0.412995\pi$
$108$	0	0
$109$	179.846	1.64996	0.824980	−	0.565162i	$-0.191186\pi$
$109$	179.846	1.64996	0.824980	+	0.565162i	$0.191186\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−96.6900	−0.855664	−0.427832	−	0.903858i	$-0.640722\pi$
$113$	−96.6900	−0.855664	−0.427832	+	0.903858i	$0.640722\pi$
$114$	0	0
$115$	− 105.812i	− 0.920101i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−73.9216	−0.610922
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 93.4495i	− 0.747596i
$126$	0	0
$127$	−136.454	−1.07444	−0.537221	−	0.843441i	$-0.680526\pi$
$127$	−136.454	−1.07444	−0.537221	+	0.843441i	$0.680526\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	26.5364i	0.202568i	0.994858	+	0.101284i	$0.0322950\pi$
$131$	26.5364i	0.202568i	−0.994858	+	0.101284i	$0.967705\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−210.866	−1.53917	−0.769583	−	0.638546i	$-0.779536\pi$
$137$	−210.866	−1.53917	−0.769583	+	0.638546i	$0.779536\pi$
$138$	0	0
$139$	83.7490i	0.602511i	0.953543	+	0.301256i	$0.0974057\pi$
$139$	83.7490i	0.602511i	−0.953543	+	0.301256i	$0.902594\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 24.8425i	− 0.173724i
$144$	0	0
$145$	− 235.783i	− 1.62609i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.99481	−0.0536564	−0.0268282	−	0.999640i	$-0.508541\pi$
$149$	−7.99481	−0.0536564	−0.0268282	+	0.999640i	$0.508541\pi$
$150$	0	0
$151$	60.8831	0.403199	0.201600	−	0.979468i	$-0.435386\pi$
$151$	60.8831	0.403199	0.201600	+	0.979468i	$0.435386\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−321.777	−2.07598
$156$	0	0
$157$	29.1758i	0.185833i	0.995674	+	0.0929166i	$0.0296190\pi$
$157$	29.1758i	0.185833i	−0.995674	+	0.0929166i	$0.970381\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−62.4556	−0.383163	−0.191582	−	0.981477i	$-0.561362\pi$
$163$	−62.4556	−0.383163	−0.191582	+	0.981477i	$0.561362\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 141.404i	− 0.846730i	−0.905959	−	0.423365i	$-0.860849\pi$
$167$	− 141.404i	− 0.846730i	0.905959	−	0.423365i	$-0.139151\pi$
$168$	0	0
$169$	155.891	0.922432
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 243.848i	− 1.40953i	−0.709443	−	0.704763i	$-0.751054\pi$
$173$	− 243.848i	− 1.40953i	0.709443	−	0.704763i	$-0.248946\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	286.642	1.60135	0.800676	−	0.599098i	$-0.204474\pi$
$179$	286.642	1.60135	0.800676	+	0.599098i	$0.204474\pi$
$180$	0	0
$181$	214.838i	1.18695i	0.804852	+	0.593475i	$0.202245\pi$
$181$	214.838i	1.18695i	−0.804852	+	0.593475i	$0.797755\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 320.391i	− 1.73184i
$186$	0	0
$187$	− 66.6597i	− 0.356469i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	45.3249	0.237303	0.118652	−	0.992936i	$-0.462143\pi$
$191$	45.3249	0.237303	0.118652	+	0.992936i	$0.462143\pi$
$192$	0	0
$193$	314.031	1.62710	0.813551	−	0.581494i	$-0.197532\pi$
$193$	314.031	1.62710	0.813551	+	0.581494i	$0.197532\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−259.231	−1.31590	−0.657948	−	0.753063i	$-0.728575\pi$
$197$	−259.231	−1.31590	−0.657948	+	0.753063i	$0.728575\pi$
$198$	0	0
$199$	− 73.5255i	− 0.369475i	−0.982788	−	0.184737i	$-0.940857\pi$
$199$	− 73.5255i	− 0.369475i	0.982788	−	0.184737i	$-0.0591435\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	328.303	1.60148
$206$	0	0
$207$	0	0
$208$	0	0
$209$	207.833i	0.994415i
$210$	0	0
$211$	263.537	1.24899	0.624496	−	0.781028i	$-0.285304\pi$
$211$	263.537	1.24899	0.624496	+	0.781028i	$0.285304\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	384.945i	1.79044i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−35.1752	−0.159164
$222$	0	0
$223$	191.042i	0.856689i	0.903616	+	0.428344i	$0.140903\pi$
$223$	191.042i	0.856689i	−0.903616	+	0.428344i	$0.859097\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 431.405i	− 1.90046i	−0.311547	−	0.950231i	$-0.600847\pi$
$227$	− 431.405i	− 1.90046i	0.311547	−	0.950231i	$-0.399153\pi$
$228$	0	0
$229$	− 34.8689i	− 0.152266i	−0.997098	−	0.0761329i	$-0.975743\pi$
$229$	− 34.8689i	− 0.152266i	0.997098	−	0.0761329i	$-0.0242573\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	350.771	1.50546	0.752728	−	0.658332i	$-0.228738\pi$
$233$	350.771	1.50546	0.752728	+	0.658332i	$0.228738\pi$
$234$	0	0
$235$	−288.028	−1.22565
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−167.400	−0.700419	−0.350209	−	0.936671i	$-0.613890\pi$
$239$	−167.400	−0.700419	−0.350209	+	0.936671i	$0.613890\pi$
$240$	0	0
$241$	− 82.0658i	− 0.340522i	−0.985399	−	0.170261i	$-0.945539\pi$
$241$	− 82.0658i	− 0.340522i	0.985399	−	0.170261i	$-0.0544611\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	109.670	0.444008
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 232.918i	− 0.927959i	−0.885846	−	0.463979i	$-0.846421\pi$
$251$	− 232.918i	− 0.927959i	0.885846	−	0.463979i	$-0.153579\pi$
$252$	0	0
$253$	124.574	0.492385
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 37.3589i	− 0.145365i	−0.997355	−	0.0726827i	$-0.976844\pi$
$257$	− 37.3589i	− 0.145365i	0.997355	−	0.0726827i	$-0.0231560\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.37261	−0.0128236	−0.00641181	−	0.999979i	$-0.502041\pi$
$263$	−3.37261	−0.0128236	−0.00641181	+	0.999979i	$0.502041\pi$
$264$	0	0
$265$	− 472.161i	− 1.78174i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	87.2686i	0.324419i	0.986756	+	0.162209i	$0.0518620\pi$
$269$	87.2686i	0.324419i	−0.986756	+	0.162209i	$0.948138\pi$
$270$	0	0
$271$	92.0075i	0.339511i	0.985486	+	0.169756i	$0.0542978\pi$
$271$	92.0075i	0.339511i	−0.985486	+	0.169756i	$0.945702\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−61.5148	−0.223690
$276$	0	0
$277$	152.481	0.550474	0.275237	−	0.961376i	$-0.411244\pi$
$277$	152.481	0.550474	0.275237	+	0.961376i	$0.411244\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	219.880	0.782491	0.391245	−	0.920286i	$-0.372044\pi$
$281$	219.880	0.782491	0.391245	+	0.920286i	$0.372044\pi$
$282$	0	0
$283$	− 387.389i	− 1.36887i	−0.729076	−	0.684433i	$-0.760050\pi$
$283$	− 387.389i	− 1.36887i	0.729076	−	0.684433i	$-0.239950\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	194.615	0.673407
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 14.8794i	− 0.0507828i	−0.999678	−	0.0253914i	$-0.991917\pi$
$293$	− 14.8794i	− 0.0507828i	0.999678	−	0.0253914i	$-0.00808320\pi$
$294$	0	0
$295$	−202.764	−0.687335
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 65.7354i	− 0.219851i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−0.223098	−0.000731470 0
$306$	0	0
$307$	453.211i	1.47626i	0.674660	+	0.738128i	$0.264290\pi$
$307$	453.211i	1.47626i	−0.674660	+	0.738128i	$0.735710\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	435.246i	1.39950i	0.714386	+	0.699752i	$0.246706\pi$
$311$	435.246i	1.39950i	−0.714386	+	0.699752i	$0.753294\pi$
$312$	0	0
$313$	− 54.9405i	− 0.175529i	−0.996141	−	0.0877644i	$-0.972028\pi$
$313$	− 54.9405i	− 0.175529i	0.996141	−	0.0877644i	$-0.0279723\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−433.691	−1.36811	−0.684055	−	0.729430i	$-0.739785\pi$
$317$	−433.691	−1.36811	−0.684055	+	0.729430i	$0.739785\pi$
$318$	0	0
$319$	277.590	0.870189
$320$	0	0
$321$	0	0
$322$	0	0
$323$	294.277	0.911073
$324$	0	0
$325$	32.4603i	0.0998780i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−576.548	−1.74184	−0.870919	−	0.491427i	$-0.836475\pi$
$331$	−576.548	−1.74184	−0.870919	+	0.491427i	$0.836475\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	373.514i	1.11497i
$336$	0	0
$337$	−301.108	−0.893495	−0.446747	−	0.894660i	$-0.647418\pi$
$337$	−301.108	−0.893495	−0.446747	+	0.894660i	$0.647418\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 378.832i	− 1.11095i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	252.112	0.726547	0.363273	−	0.931683i	$-0.381659\pi$
$347$	252.112	0.726547	0.363273	+	0.931683i	$0.381659\pi$
$348$	0	0
$349$	− 406.452i	− 1.16462i	−0.812967	−	0.582309i	$-0.802149\pi$
$349$	− 406.452i	− 1.16462i	0.812967	−	0.582309i	$-0.197851\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 134.429i	− 0.380819i	−0.981705	−	0.190410i	$-0.939018\pi$
$353$	− 134.429i	− 0.380819i	0.981705	−	0.190410i	$-0.0609816\pi$
$354$	0	0
$355$	292.990i	0.825324i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	116.676	0.325003	0.162501	−	0.986708i	$-0.448044\pi$
$359$	116.676	0.325003	0.162501	+	0.986708i	$0.448044\pi$
$360$	0	0
$361$	−556.501	−1.54155
$362$	0	0
$363$	0	0
$364$	0	0
$365$	124.406	0.340839
$366$	0	0
$367$	419.148i	1.14209i	0.820918	+	0.571046i	$0.193462\pi$
$367$	419.148i	1.14209i	−0.820918	+	0.571046i	$0.806538\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−126.992	−0.340462	−0.170231	−	0.985404i	$-0.554451\pi$
$373$	−126.992	−0.340462	−0.170231	+	0.985404i	$0.554451\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 146.480i	− 0.388541i
$378$	0	0
$379$	−366.675	−0.967479	−0.483740	−	0.875212i	$-0.660722\pi$
$379$	−366.675	−0.967479	−0.483740	+	0.875212i	$0.660722\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 258.364i	− 0.674579i	−0.941401	−	0.337290i	$-0.890490\pi$
$383$	− 258.364i	− 0.674579i	0.941401	−	0.337290i	$-0.109510\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−542.395	−1.39433	−0.697166	−	0.716910i	$-0.745556\pi$
$389$	−542.395	−1.39433	−0.697166	+	0.716910i	$0.745556\pi$
$390$	0	0
$391$	− 176.387i	− 0.451118i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	276.855i	0.700898i
$396$	0	0
$397$	404.764i	1.01956i	0.860306	+	0.509778i	$0.170273\pi$
$397$	404.764i	1.01956i	−0.860306	+	0.509778i	$0.829727\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	473.436	1.18064	0.590319	−	0.807170i	$-0.299002\pi$
$401$	473.436	1.18064	0.590319	+	0.807170i	$0.299002\pi$
$402$	0	0
$403$	−199.904	−0.496039
$404$	0	0
$405$	0	0
$406$	0	0
$407$	377.201	0.926783
$408$	0	0
$409$	403.658i	0.986938i	0.869763	+	0.493469i	$0.164271\pi$
$409$	403.658i	0.986938i	−0.869763	+	0.493469i	$0.835729\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	195.843	0.471911
$416$	0	0
$417$	0	0
$418$	0	0
$419$	454.684i	1.08517i	0.840003	+	0.542583i	$0.182553\pi$
$419$	454.684i	1.08517i	−0.840003	+	0.542583i	$0.817447\pi$
$420$	0	0
$421$	−180.928	−0.429758	−0.214879	−	0.976641i	$-0.568936\pi$
$421$	−180.928	−0.429758	−0.214879	+	0.976641i	$0.568936\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	87.1006i	0.204943i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	807.892	1.87446	0.937229	−	0.348714i	$-0.113382\pi$
$431$	807.892	1.87446	0.937229	+	0.348714i	$0.113382\pi$
$432$	0	0
$433$	− 166.000i	− 0.383372i	−0.981456	−	0.191686i	$-0.938604\pi$
$433$	− 166.000i	− 0.383372i	0.981456	−	0.191686i	$-0.0613955\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	549.944i	1.25845i
$438$	0	0
$439$	29.2895i	0.0667186i	0.999443	+	0.0333593i	$0.0106206\pi$
$439$	29.2895i	0.0667186i	−0.999443	+	0.0333593i	$0.989379\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−822.751	−1.85723	−0.928613	−	0.371049i	$-0.878998\pi$
$443$	−822.751	−1.85723	−0.928613	+	0.371049i	$0.878998\pi$
$444$	0	0
$445$	−909.701	−2.04427
$446$	0	0
$447$	0	0
$448$	0	0
$449$	428.131	0.953522	0.476761	−	0.879033i	$-0.341811\pi$
$449$	428.131	0.953522	0.476761	+	0.879033i	$0.341811\pi$
$450$	0	0
$451$	386.516i	0.857020i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−187.311	−0.409871	−0.204935	−	0.978775i	$-0.565698\pi$
$457$	−187.311	−0.409871	−0.204935	+	0.978775i	$0.565698\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 182.821i	− 0.396576i	−0.980144	−	0.198288i	$-0.936462\pi$
$461$	− 182.821i	− 0.396576i	0.980144	−	0.198288i	$-0.0635381\pi$
$462$	0	0
$463$	232.389	0.501920	0.250960	−	0.967997i	$-0.419254\pi$
$463$	232.389	0.501920	0.250960	+	0.967997i	$0.419254\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	448.927i	0.961299i	0.876913	+	0.480650i	$0.159599\pi$
$467$	448.927i	0.961299i	−0.876913	+	0.480650i	$0.840401\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−453.202	−0.958144
$474$	0	0
$475$	− 271.564i	− 0.571713i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	631.644i	1.31867i	0.751848	+	0.659336i	$0.229163\pi$
$479$	631.644i	1.31867i	−0.751848	+	0.659336i	$0.770837\pi$
$480$	0	0
$481$	− 199.043i	− 0.413810i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−254.946	−0.525662
$486$	0	0
$487$	701.535	1.44052	0.720262	−	0.693702i	$-0.244022\pi$
$487$	701.535	1.44052	0.720262	+	0.693702i	$0.244022\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	414.789	0.844785	0.422392	−	0.906413i	$-0.361190\pi$
$491$	414.789	0.844785	0.422392	+	0.906413i	$0.361190\pi$
$492$	0	0
$493$	− 393.048i	− 0.797258i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−612.340	−1.22713	−0.613567	−	0.789643i	$-0.710266\pi$
$499$	−612.340	−1.22713	−0.613567	+	0.789643i	$0.710266\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	467.449i	0.929323i	0.885488	+	0.464661i	$0.153824\pi$
$503$	467.449i	0.929323i	−0.885488	+	0.464661i	$0.846176\pi$
$504$	0	0
$505$	−989.279	−1.95897
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 62.5666i	− 0.122921i	−0.998110	−	0.0614603i	$-0.980424\pi$
$509$	− 62.5666i	− 0.122921i	0.998110	−	0.0614603i	$-0.0195758\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−348.057	−0.675838
$516$	0	0
$517$	− 339.099i	− 0.655898i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	712.575i	1.36771i	0.729620	+	0.683853i	$0.239697\pi$
$521$	712.575i	1.36771i	−0.729620	+	0.683853i	$0.760303\pi$
$522$	0	0
$523$	− 280.114i	− 0.535590i	−0.963476	−	0.267795i	$-0.913705\pi$
$523$	− 280.114i	− 0.535590i	0.963476	−	0.267795i	$-0.0862950\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−536.400	−1.01784
$528$	0	0
$529$	−199.368	−0.376876
$530$	0	0
$531$	0	0
$532$	0	0
$533$	203.958	0.382660
$534$	0	0
$535$	− 336.670i	− 0.629289i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	633.002	1.17006	0.585030	−	0.811012i	$-0.301083\pi$
$541$	633.002	1.17006	0.585030	+	0.811012i	$0.301083\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1048.14i	− 1.92319i
$546$	0	0
$547$	1047.16	1.91438	0.957189	−	0.289465i	$-0.0934773\pi$
$547$	1047.16	1.91438	0.957189	+	0.289465i	$0.0934773\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1225.45i	2.22405i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−681.732	−1.22393	−0.611967	−	0.790883i	$-0.709622\pi$
$557$	−681.732	−1.22393	−0.611967	+	0.790883i	$0.709622\pi$
$558$	0	0
$559$	239.147i	0.427812i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 201.588i	− 0.358061i	−0.983844	−	0.179031i	$-0.942704\pi$
$563$	− 201.588i	− 0.358061i	0.983844	−	0.179031i	$-0.0572961\pi$
$564$	0	0
$565$	563.508i	0.997359i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	684.800	1.20352	0.601758	−	0.798679i	$-0.294467\pi$
$569$	684.800	1.20352	0.601758	+	0.798679i	$0.294467\pi$
$570$	0	0
$571$	−25.0602	−0.0438884	−0.0219442	−	0.999759i	$-0.506986\pi$
$571$	−25.0602	−0.0438884	−0.0219442	+	0.999759i	$0.506986\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−162.773	−0.283084
$576$	0	0
$577$	1019.96i	1.76770i	0.467773	+	0.883849i	$0.345056\pi$
$577$	1019.96i	1.76770i	−0.467773	+	0.883849i	$0.654944\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	555.882	0.953485
$584$	0	0
$585$	0	0
$586$	0	0
$587$	581.810i	0.991158i	0.868563	+	0.495579i	$0.165044\pi$
$587$	581.810i	0.991158i	−0.868563	+	0.495579i	$0.834956\pi$
$588$	0	0
$589$	1672.40	2.83939
$590$	0	0
$591$	0	0
$592$	0	0
$593$	143.863i	0.242602i	0.992616	+	0.121301i	$0.0387066\pi$
$593$	143.863i	0.242602i	−0.992616	+	0.121301i	$0.961293\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−710.736	−1.18654	−0.593269	−	0.805005i	$-0.702163\pi$
$599$	−710.736	−1.18654	−0.593269	+	0.805005i	$0.702163\pi$
$600$	0	0
$601$	914.930i	1.52235i	0.648549	+	0.761173i	$0.275376\pi$
$601$	914.930i	1.52235i	−0.648549	+	0.761173i	$0.724624\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	430.814i	0.712089i
$606$	0	0
$607$	622.481i	1.02550i	0.858537	+	0.512752i	$0.171374\pi$
$607$	622.481i	1.02550i	−0.858537	+	0.512752i	$0.828626\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−178.937	−0.292859
$612$	0	0
$613$	−345.019	−0.562837	−0.281419	−	0.959585i	$-0.590805\pi$
$613$	−345.019	−0.562837	−0.281419	+	0.959585i	$0.590805\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	854.311	1.38462	0.692310	−	0.721600i	$-0.256593\pi$
$617$	854.311	1.38462	0.692310	+	0.721600i	$0.256593\pi$
$618$	0	0
$619$	49.2180i	0.0795121i	0.999209	+	0.0397561i	$0.0126581\pi$
$619$	49.2180i	0.0795121i	−0.999209	+	0.0397561i	$0.987342\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−768.756	−1.23001
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 534.089i	− 0.849109i
$630$	0	0
$631$	457.699	0.725355	0.362677	−	0.931915i	$-0.381863\pi$
$631$	457.699	0.725355	0.362677	+	0.931915i	$0.381863\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	795.252i	1.25237i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	323.931	0.505353	0.252677	−	0.967551i	$-0.418689\pi$
$641$	323.931	0.505353	0.252677	+	0.967551i	$0.418689\pi$
$642$	0	0
$643$	− 117.018i	− 0.181987i	−0.995851	−	0.0909935i	$-0.970996\pi$
$643$	− 117.018i	− 0.181987i	0.995851	−	0.0909935i	$-0.0290043\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 539.247i	− 0.833458i	−0.909031	−	0.416729i	$-0.863176\pi$
$647$	− 539.247i	− 0.833458i	0.909031	−	0.416729i	$-0.136824\pi$
$648$	0	0
$649$	− 238.717i	− 0.367822i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	797.906	1.22191	0.610954	−	0.791666i	$-0.290786\pi$
$653$	797.906	1.22191	0.610954	+	0.791666i	$0.290786\pi$
$654$	0	0
$655$	154.654	0.236112
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−295.186	−0.447929	−0.223965	−	0.974597i	$-0.571900\pi$
$659$	−295.186	−0.447929	−0.223965	+	0.974597i	$0.571900\pi$
$660$	0	0
$661$	− 1032.24i	− 1.56163i	−0.624759	−	0.780817i	$-0.714803\pi$
$661$	− 1032.24i	− 1.56163i	0.624759	−	0.780817i	$-0.285197\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	734.528	1.10124
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 0.262657i	0 0.000391441i
$672$	0	0
$673$	418.188	0.621379	0.310689	−	0.950512i	$-0.399440\pi$
$673$	418.188	0.621379	0.310689	+	0.950512i	$0.399440\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 465.460i	− 0.687534i	−0.939055	−	0.343767i	$-0.888297\pi$
$677$	− 465.460i	− 0.687534i	0.939055	−	0.343767i	$-0.111703\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1259.41	−1.84394	−0.921969	−	0.387263i	$-0.873420\pi$
$683$	−1259.41	−1.84394	−0.921969	+	0.387263i	$0.873420\pi$
$684$	0	0
$685$	1228.92i	1.79405i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 293.330i	− 0.425732i
$690$	0	0
$691$	458.874i	0.664073i	0.943267	+	0.332036i	$0.107736\pi$
$691$	458.874i	0.664073i	−0.943267	+	0.332036i	$0.892264\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	488.088	0.702285
$696$	0	0
$697$	547.279	0.785192
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−822.907	−1.17390	−0.586952	−	0.809622i	$-0.699672\pi$
$701$	−822.907	−1.17390	−0.586952	+	0.809622i	$0.699672\pi$
$702$	0	0
$703$	1665.19i	2.36870i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	285.908	0.403255	0.201627	−	0.979462i	$-0.435377\pi$
$709$	285.908	0.403255	0.201627	+	0.979462i	$0.435377\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1002.42i	− 1.40592i
$714$	0	0
$715$	−144.782	−0.202492
$716$	0	0
$717$	0	0
$718$	0	0
$719$	699.242i	0.972520i	0.873814	+	0.486260i	$0.161639\pi$
$719$	699.242i	0.972520i	−0.873814	+	0.486260i	$0.838361\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−362.712	−0.500293
$726$	0	0
$727$	− 216.138i	− 0.297302i	−0.988890	−	0.148651i	$-0.952507\pi$
$727$	− 216.138i	− 0.297302i	0.988890	−	0.148651i	$-0.0474930\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	641.702i	0.877841i
$732$	0	0
$733$	− 727.188i	− 0.992071i	−0.868302	−	0.496035i	$-0.834789\pi$
$733$	− 727.188i	− 0.992071i	0.868302	−	0.496035i	$-0.165211\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−439.744	−0.596668
$738$	0	0
$739$	396.413	0.536418	0.268209	−	0.963361i	$-0.413568\pi$
$739$	396.413	0.536418	0.268209	+	0.963361i	$0.413568\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−739.481	−0.995263	−0.497632	−	0.867388i	$-0.665797\pi$
$743$	−739.481	−0.995263	−0.497632	+	0.867388i	$0.665797\pi$
$744$	0	0
$745$	46.5936i	0.0625417i
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−748.332	−0.996448	−0.498224	−	0.867048i	$-0.666014\pi$
$751$	−748.332	−0.996448	−0.498224	+	0.867048i	$0.666014\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 354.825i	− 0.469967i
$756$	0	0
$757$	199.145	0.263071	0.131536	−	0.991311i	$-0.458009\pi$
$757$	199.145	0.263071	0.131536	+	0.991311i	$0.458009\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	981.456i	1.28969i	0.764312	+	0.644846i	$0.223079\pi$
$761$	981.456i	1.28969i	−0.764312	+	0.644846i	$0.776921\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−125.967	−0.164233
$768$	0	0
$769$	− 724.214i	− 0.941760i	−0.882197	−	0.470880i	$-0.843936\pi$
$769$	− 724.214i	− 0.941760i	0.882197	−	0.470880i	$-0.156064\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1404.90i	1.81746i	0.417383	+	0.908731i	$0.362947\pi$
$773$	1404.90i	1.81746i	−0.417383	+	0.908731i	$0.637053\pi$
$774$	0	0
$775$	495.000i	0.638709i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1706.32	−2.19039
$780$	0	0
$781$	−344.942	−0.441667
$782$	0	0
$783$	0	0
$784$	0	0
$785$	170.036	0.216607
$786$	0	0
$787$	− 255.814i	− 0.325049i	−0.986704	−	0.162525i	$-0.948036\pi$
$787$	− 255.814i	− 0.325049i	0.986704	−	0.162525i	$-0.0519637\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.138600	−0.000174779 0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	474.641i	0.595534i	0.954639	+	0.297767i	$0.0962418\pi$
$797$	474.641i	0.595534i	−0.954639	+	0.297767i	$0.903758\pi$
$798$	0	0
$799$	−480.141	−0.600927
$800$	0	0
$801$	0	0
$802$	0	0
$803$	146.465i	0.182398i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−467.727	−0.578155	−0.289077	−	0.957306i	$-0.593349\pi$
$809$	−467.727	−0.578155	−0.289077	+	0.957306i	$0.593349\pi$
$810$	0	0
$811$	1567.70i	1.93305i	0.256571	+	0.966525i	$0.417407\pi$
$811$	1567.70i	1.93305i	−0.256571	+	0.966525i	$0.582593\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	363.990i	0.446614i
$816$	0	0
$817$	− 2000.71i	− 2.44885i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	438.250	0.533801	0.266900	−	0.963724i	$-0.414001\pi$
$821$	438.250	0.533801	0.266900	+	0.963724i	$0.414001\pi$
$822$	0	0
$823$	937.213	1.13878	0.569388	−	0.822069i	$-0.307180\pi$
$823$	937.213	1.13878	0.569388	+	0.822069i	$0.307180\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1160.31	1.40303	0.701516	−	0.712654i	$-0.252507\pi$
$827$	1160.31	1.40303	0.701516	+	0.712654i	$0.252507\pi$
$828$	0	0
$829$	− 254.599i	− 0.307116i	−0.988140	−	0.153558i	$-0.950927\pi$
$829$	− 254.599i	− 0.307116i	0.988140	−	0.153558i	$-0.0490732\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−824.100	−0.986946
$836$	0	0
$837$	0	0
$838$	0	0
$839$	129.902i	0.154829i	0.996999	+	0.0774146i	$0.0246665\pi$
$839$	129.902i	0.154829i	−0.996999	+	0.0774146i	$0.975333\pi$
$840$	0	0
$841$	795.767	0.946215
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 908.530i	− 1.07518i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	998.106	1.17286
$852$	0	0
$853$	− 1229.45i	− 1.44133i	−0.693284	−	0.720665i	$-0.743837\pi$
$853$	− 1229.45i	− 1.44133i	0.693284	−	0.720665i	$-0.256163\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 668.190i	− 0.779685i	−0.920881	−	0.389843i	$-0.872529\pi$
$857$	− 668.190i	− 0.779685i	0.920881	−	0.389843i	$-0.127471\pi$
$858$	0	0
$859$	− 19.3907i	− 0.0225736i	−0.999936	−	0.0112868i	$-0.996407\pi$
$859$	− 19.3907i	− 0.0225736i	0.999936	−	0.0112868i	$-0.00359278\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−92.9711	−0.107730	−0.0538651	−	0.998548i	$-0.517154\pi$
$863$	−92.9711	−0.107730	−0.0538651	+	0.998548i	$0.517154\pi$
$864$	0	0
$865$	−1421.14	−1.64294
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−325.945	−0.375080
$870$	0	0
$871$	232.046i	0.266413i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	312.308	0.356110	0.178055	−	0.984021i	$-0.443020\pi$
$877$	312.308	0.356110	0.178055	+	0.984021i	$0.443020\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	772.018i	0.876298i	0.898902	+	0.438149i	$0.144366\pi$
$881$	772.018i	0.876298i	−0.898902	+	0.438149i	$0.855634\pi$
$882$	0	0
$883$	−959.154	−1.08624	−0.543122	−	0.839654i	$-0.682758\pi$
$883$	−959.154	−1.08624	−0.543122	+	0.839654i	$0.682758\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 554.964i	− 0.625664i	−0.949808	−	0.312832i	$-0.898722\pi$
$887$	− 554.964i	− 0.625664i	0.949808	−	0.312832i	$-0.101278\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1496.99	1.67636
$894$	0	0
$895$	− 1670.54i	− 1.86653i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 2233.73i	− 2.48468i
$900$	0	0
$901$	− 787.089i	− 0.873573i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1252.07	1.38351
$906$	0	0
$907$	573.116	0.631881	0.315940	−	0.948779i	$-0.397680\pi$
$907$	573.116	0.631881	0.315940	+	0.948779i	$0.397680\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−726.088	−0.797023	−0.398511	−	0.917163i	$-0.630473\pi$
$911$	−726.088	−0.797023	−0.398511	+	0.917163i	$0.630473\pi$
$912$	0	0
$913$	230.569i	0.252540i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	784.533	0.853681	0.426840	−	0.904327i	$-0.359627\pi$
$919$	784.533	0.853681	0.426840	+	0.904327i	$0.359627\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	182.020i	0.197205i
$924$	0	0
$925$	−492.868	−0.532830
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 159.109i	− 0.171269i	−0.996327	−	0.0856347i	$-0.972708\pi$
$929$	− 159.109i	− 0.171269i	0.996327	−	0.0856347i	$-0.0272918\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−388.491	−0.415499
$936$	0	0
$937$	728.876i	0.777883i	0.921262	+	0.388941i	$0.127159\pi$
$937$	728.876i	0.777883i	−0.921262	+	0.388941i	$0.872841\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	323.245i	0.343513i	0.985139	+	0.171756i	$0.0549442\pi$
$941$	323.245i	0.343513i	−0.985139	+	0.171756i	$0.945056\pi$
$942$	0	0
$943$	1022.75i	1.08458i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−673.049	−0.710717	−0.355359	−	0.934730i	$-0.615641\pi$
$947$	−673.049	−0.710717	−0.355359	+	0.934730i	$0.615641\pi$
$948$	0	0
$949$	77.2873	0.0814408
$950$	0	0
$951$	0	0
$952$	0	0
$953$	195.034	0.204653	0.102326	−	0.994751i	$-0.467371\pi$
$953$	195.034	0.204653	0.102326	+	0.994751i	$0.467371\pi$
$954$	0	0
$955$	− 264.153i	− 0.276600i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2087.40	−2.17212
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1830.16i	− 1.89654i
$966$	0	0
$967$	−731.573	−0.756538	−0.378269	−	0.925696i	$-0.623481\pi$
$967$	−731.573	−0.756538	−0.378269	+	0.925696i	$0.623481\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 498.822i	− 0.513720i	−0.966449	−	0.256860i	$-0.917312\pi$
$971$	− 498.822i	− 0.513720i	0.966449	−	0.256860i	$-0.0826879\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	180.705	0.184959	0.0924794	−	0.995715i	$-0.470521\pi$
$977$	180.705	0.184959	0.0924794	+	0.995715i	$0.470521\pi$
$978$	0	0
$979$	− 1071.00i	− 1.09398i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	326.658i	0.332307i	0.986100	+	0.166153i	$0.0531347\pi$
$983$	326.658i	0.332307i	−0.986100	+	0.166153i	$0.946865\pi$
$984$	0	0
$985$	1510.80i	1.53380i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1199.21	−1.21255
$990$	0	0
$991$	−1646.75	−1.66171	−0.830854	−	0.556491i	$-0.812147\pi$
$991$	−1646.75	−1.66171	−0.830854	+	0.556491i	$0.812147\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−428.505	−0.430659
$996$	0	0
$997$	140.158i	0.140580i	0.997527	+	0.0702900i	$0.0223925\pi$
$997$	140.158i	0.140580i	−0.997527	+	0.0702900i	$0.977608\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.d.h.685.1		8
3.2	odd	2		588.3.d.c.97.8	yes	8
7.2	even	3		1764.3.z.m.325.1		8
7.3	odd	6		1764.3.z.m.901.1		8
7.4	even	3		1764.3.z.l.901.4		8
7.5	odd	6		1764.3.z.l.325.4		8
7.6	odd	2	inner	1764.3.d.h.685.8		8
12.11	even	2		2352.3.f.j.97.4		8
21.2	odd	6		588.3.m.e.325.4		8
21.5	even	6		588.3.m.f.325.1		8
21.11	odd	6		588.3.m.f.313.1		8
21.17	even	6		588.3.m.e.313.4		8
21.20	even	2		588.3.d.c.97.1	✓	8
84.83	odd	2		2352.3.f.j.97.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.3.d.c.97.1	✓	8	21.20	even	2
588.3.d.c.97.8	yes	8	3.2	odd	2
588.3.m.e.313.4		8	21.17	even	6
588.3.m.e.325.4		8	21.2	odd	6
588.3.m.f.313.1		8	21.11	odd	6
588.3.m.f.325.1		8	21.5	even	6
1764.3.d.h.685.1		8	1.1	even	1	trivial
1764.3.d.h.685.8		8	7.6	odd	2	inner
1764.3.z.l.325.4		8	7.5	odd	6
1764.3.z.l.901.4		8	7.4	even	3
1764.3.z.m.325.1		8	7.2	even	3
1764.3.z.m.901.1		8	7.3	odd	6
2352.3.f.j.97.4		8	12.11	even	2
2352.3.f.j.97.5		8	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-5.82798i q^{5} +O(q^{10})\)	\(q-5.82798i q^{5} +6.86137 q^{11} -3.62063i q^{13} -9.71521i q^{17} +30.2903i q^{19} +18.1558 q^{23} -8.96538 q^{25} +40.4570 q^{29} -55.2124i q^{31} +54.9745 q^{37} +56.3322i q^{41} -66.0512 q^{43} -49.4215i q^{47} +81.0162 q^{53} -39.9879i q^{55} -34.7914i q^{59} -0.0382805i q^{61} -21.1010 q^{65} -64.0898 q^{67} -50.2730 q^{71} +21.3464i q^{73} -47.5044 q^{79} +33.6039i q^{83} -56.6201 q^{85} -156.092i q^{89} +176.531 q^{95} -43.7452i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 16 q^{23} + 72 q^{25} - 80 q^{29} + 128 q^{37} - 112 q^{43} + 144 q^{53} - 240 q^{65} - 64 q^{67} - 224 q^{71} - 432 q^{79} - 96 q^{85} + 272 q^{95}+O(q^{100}) \) 8 * q + 16 * q^23 + 72 * q^25 - 80 * q^29 + 128 * q^37 - 112 * q^43 + 144 * q^53 - 240 * q^65 - 64 * q^67 - 224 * q^71 - 432 * q^79 - 96 * q^85 + 272 * q^95

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