LMFDB - Embedded newform 1764.3.bk.e.1745.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,3,Mod(557,1764)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1764.557"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1764.bk (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$48.0655186332$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.12745506816.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 $ x^8 - 8x^6 + 55x^4 - 72*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			1745.1
Root			$-2.23256 - 1.28897i$ of defining polynomial
Character	$\chi$	$=$	1764.1745
Dual form			1764.3.bk.e.557.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.46512 - 2.57794i) q^{5} +(-5.25600 + 3.03455i) q^{11} +12.5830 q^{13} +(14.9771 - 8.64704i) q^{17} +(-12.9373 + 22.4080i) q^{19} +(-2.09247 - 1.20809i) q^{23} +(0.791503 + 1.37092i) q^{25} -55.8014i q^{29} +(7.64575 + 13.2428i) q^{31} +(11.8745 - 20.5673i) q^{37} +15.4676i q^{41} +27.7490 q^{43} +(-24.6486 - 14.2309i) q^{47} +(40.4166 - 23.3345i) q^{53} +31.2915 q^{55} +(-69.8601 + 40.3337i) q^{59} +(-31.3542 + 54.3072i) q^{61} +(-56.1846 - 32.4382i) q^{65} +(-58.0405 - 100.529i) q^{67} +49.7896i q^{71} +(15.3542 + 26.5943i) q^{73} +(-71.3320 + 123.551i) q^{79} -28.5763i q^{83} -89.1660 q^{85} +(-145.668 - 84.1013i) q^{89} +(115.533 - 66.7028i) q^{95} -113.956 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{13} - 40 q^{19} - 36 q^{25} + 40 q^{31} - 32 q^{37} - 32 q^{43} + 208 q^{55} - 272 q^{61} - 168 q^{67} + 144 q^{73} - 232 q^{79} - 544 q^{85} - 192 q^{97}+O(q^{100}) $ 8 * q + 16 * q^13 - 40 * q^19 - 36 * q^25 + 40 * q^31 - 32 * q^37 - 32 * q^43 + 208 * q^55 - 272 * q^61 - 168 * q^67 + 144 * q^73 - 232 * q^79 - 544 * q^85 - 192 * q^97

Character values

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

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−4.46512	−	2.57794i	−0.893023	−	0.515587i	−0.0180929	−	0.999836i	$-0.505759\pi$
$5$	−4.46512	−	2.57794i	−0.893023	−	0.515587i	−0.874930	+	0.484249i	$0.839093\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.25600	+	3.03455i	−0.477818	+	0.275868i	−0.719507	−	0.694486i	$-0.755632\pi$
$11$	−5.25600	+	3.03455i	−0.477818	+	0.275868i	0.241689	+	0.970354i	$0.422299\pi$
$12$		0			0
$13$	12.5830			0.967923			0.483962	−	0.875089i	$-0.339197\pi$
$13$	12.5830			0.967923			0.483962	+	0.875089i	$0.339197\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14.9771	−	8.64704i	0.881006	−	0.508649i	0.0100162	−	0.999950i	$-0.496812\pi$
$17$	14.9771	−	8.64704i	0.881006	−	0.508649i	0.870990	+	0.491301i	$0.163478\pi$
$18$		0			0
$19$	−12.9373	+	22.4080i	−0.680908	+	1.17937i	0.293796	+	0.955868i	$0.405081\pi$
$19$	−12.9373	+	22.4080i	−0.680908	+	1.17937i	−0.974704	+	0.223499i	$0.928252\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.09247	−	1.20809i	−0.0909771	−	0.0525257i	0.453821	−	0.891093i	$-0.350060\pi$
$23$	−2.09247	−	1.20809i	−0.0909771	−	0.0525257i	−0.544798	+	0.838567i	$0.683394\pi$
$24$		0			0
$25$	0.791503	+	1.37092i	0.0316601	+	0.0548369i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	55.8014i		−	1.92418i	−0.272724	−	0.962092i	$-0.587925\pi$
$29$		−	55.8014i		−	1.92418i	0.272724	−	0.962092i	$-0.412075\pi$
$30$		0			0
$31$	7.64575	+	13.2428i	0.246637	+	0.427188i	0.962591	−	0.270960i	$-0.0873411\pi$
$31$	7.64575	+	13.2428i	0.246637	+	0.427188i	−0.715953	+	0.698148i	$0.754008\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	11.8745	−	20.5673i	0.320933	−	0.555872i	−0.659748	−	0.751487i	$-0.729337\pi$
$37$	11.8745	−	20.5673i	0.320933	−	0.555872i	0.980681	+	0.195615i	$0.0626704\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			15.4676i			0.377259i	0.982048	+	0.188629i	$0.0604045\pi$
$41$			15.4676i			0.377259i	−0.982048	+	0.188629i	$0.939596\pi$
$42$		0			0
$43$	27.7490			0.645326			0.322663	−	0.946514i	$-0.395422\pi$
$43$	27.7490			0.645326			0.322663	+	0.946514i	$0.395422\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−24.6486	−	14.2309i	−0.524438	−	0.302785i	0.214310	−	0.976766i	$-0.431250\pi$
$47$	−24.6486	−	14.2309i	−0.524438	−	0.302785i	−0.738749	+	0.673981i	$0.764583\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	40.4166	−	23.3345i	0.762577	−	0.440274i	−0.0676432	−	0.997710i	$-0.521548\pi$
$53$	40.4166	−	23.3345i	0.762577	−	0.440274i	0.830220	+	0.557436i	$0.188215\pi$
$54$		0			0
$55$	31.2915			0.568936
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−69.8601	+	40.3337i	−1.18407	+	0.683623i	−0.956952	−	0.290245i	$-0.906263\pi$
$59$	−69.8601	+	40.3337i	−1.18407	+	0.683623i	−0.227117	+	0.973868i	$0.572930\pi$
$60$		0			0
$61$	−31.3542	+	54.3072i	−0.514004	+	0.890281i	0.485864	+	0.874034i	$0.338505\pi$
$61$	−31.3542	+	54.3072i	−0.514004	+	0.890281i	−0.999868	+	0.0162467i	$0.994828\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−56.1846	−	32.4382i	−0.864378	−	0.499049i
$66$		0			0
$67$	−58.0405	−	100.529i	−0.866276	−	1.50043i	−0.865774	−	0.500435i	$-0.833173\pi$
$67$	−58.0405	−	100.529i	−0.866276	−	1.50043i	−0.000502198	−	1.00000i	$-0.500160\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			49.7896i			0.701261i	0.936514	+	0.350631i	$0.114033\pi$
$71$			49.7896i			0.701261i	−0.936514	+	0.350631i	$0.885967\pi$
$72$		0			0
$73$	15.3542	+	26.5943i	0.210332	+	0.364306i	0.951818	−	0.306662i	$-0.0992121\pi$
$73$	15.3542	+	26.5943i	0.210332	+	0.364306i	−0.741486	+	0.670968i	$0.765879\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−71.3320	+	123.551i	−0.902937	+	1.56393i	−0.0792742	+	0.996853i	$0.525260\pi$
$79$	−71.3320	+	123.551i	−0.902937	+	1.56393i	−0.823663	+	0.567080i	$0.808073\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	28.5763i		−	0.344293i	−0.985071	−	0.172147i	$-0.944930\pi$
$83$		−	28.5763i		−	0.344293i	0.985071	−	0.172147i	$-0.0550703\pi$
$84$		0			0
$85$	−89.1660			−1.04901
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−145.668	−	84.1013i	−1.63672	−	0.944959i	−0.981953	−	0.189125i	$-0.939435\pi$
$89$	−145.668	−	84.1013i	−1.63672	−	0.944959i	−0.654764	−	0.755834i	$-0.727232\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	115.533	−	66.7028i	1.21613	−	0.702135i
$96$		0			0
$97$	−113.956			−1.17480			−0.587400	−	0.809297i	$-0.699848\pi$
$97$	−113.956			−1.17480			−0.587400	+	0.809297i	$0.699848\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−34.8804	+	20.1382i	−0.345351	+	0.199388i	−0.662636	−	0.748942i	$-0.730562\pi$
$101$	−34.8804	+	20.1382i	−0.345351	+	0.199388i	0.317285	+	0.948330i	$0.397229\pi$
$102$		0			0
$103$	14.5608	−	25.2200i	0.141367	−	0.244854i	−0.786645	−	0.617406i	$-0.788184\pi$
$103$	14.5608	−	25.2200i	0.141367	−	0.244854i	0.928012	+	0.372551i	$0.121517\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−157.531	−	90.9506i	−1.47225	−	0.850005i	−0.472739	−	0.881202i	$-0.656735\pi$
$107$	−157.531	−	90.9506i	−1.47225	−	0.850005i	−0.999513	+	0.0311972i	$0.990068\pi$
$108$		0			0
$109$	44.0810	+	76.3506i	0.404413	+	0.700464i	0.994253	−	0.107056i	$-0.0341424\pi$
$109$	44.0810	+	76.3506i	0.404413	+	0.700464i	−0.589840	+	0.807520i	$0.700809\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	64.8190i		−	0.573620i	−0.957988	−	0.286810i	$-0.907405\pi$
$113$		−	64.8190i		−	0.573620i	0.957988	−	0.286810i	$-0.0925948\pi$
$114$		0			0
$115$	6.22876	+	10.7885i	0.0541631	+	0.0938132i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−42.0830	+	72.8899i	−0.347793	+	0.602396i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			120.735i			0.965880i
$126$		0			0
$127$	−223.579			−1.76047			−0.880233	−	0.474543i	$-0.842613\pi$
$127$	−223.579			−1.76047			−0.880233	+	0.474543i	$0.842613\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−179.527	−	103.650i	−1.37043	−	0.791220i	−0.379451	−	0.925212i	$-0.623887\pi$
$131$	−179.527	−	103.650i	−1.37043	−	0.791220i	−0.990982	+	0.133992i	$0.957220\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	152.225	−	87.8874i	1.11113	−	0.641514i	0.172011	−	0.985095i	$-0.444973\pi$
$137$	152.225	−	87.8874i	1.11113	−	0.641514i	0.939123	+	0.343581i	$0.111640\pi$
$138$		0			0
$139$	117.166			0.842921			0.421460	−	0.906847i	$-0.361518\pi$
$139$	117.166			0.842921			0.421460	+	0.906847i	$0.361518\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−66.1362	+	38.1838i	−0.462491	+	0.267019i
$144$		0			0
$145$	−143.852	+	249.159i	−0.992085	+	1.71834i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	161.354	+	93.1578i	1.08291	+	0.625220i	0.931681	−	0.363278i	$-0.118342\pi$
$149$	161.354	+	93.1578i	1.08291	+	0.625220i	0.151233	+	0.988498i	$0.451676\pi$
$150$		0			0
$151$	−91.0405	−	157.687i	−0.602917	−	1.04428i	−0.992377	−	0.123240i	$-0.960672\pi$
$151$	−91.0405	−	157.687i	−0.602917	−	1.04428i	0.389460	−	0.921044i	$-0.372662\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	78.8410i		−	0.508652i
$156$		0			0
$157$	34.1882	+	59.2158i	0.217759	+	0.377170i	0.954123	−	0.299416i	$-0.0967918\pi$
$157$	34.1882	+	59.2158i	0.217759	+	0.377170i	−0.736363	+	0.676586i	$0.763458\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	18.3725	−	31.8222i	0.112715	−	0.195228i	−0.804149	−	0.594428i	$-0.797379\pi$
$163$	18.3725	−	31.8222i	0.112715	−	0.195228i	0.916864	+	0.399200i	$0.130712\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			240.708i			1.44137i	0.693264	+	0.720684i	$0.256172\pi$
$167$			240.708i			1.44137i	−0.693264	+	0.720684i	$0.743828\pi$
$168$		0			0
$169$	−10.6680			−0.0631241
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−295.340	−	170.514i	−1.70717	−	0.985632i	−0.938035	−	0.346540i	$-0.887356\pi$
$173$	−295.340	−	170.514i	−1.70717	−	0.985632i	−0.769130	−	0.639092i	$-0.779310\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−119.866	+	69.2049i	−0.669645	+	0.386620i	−0.795942	−	0.605373i	$-0.793024\pi$
$179$	−119.866	+	69.2049i	−0.669645	+	0.386620i	0.126297	+	0.991992i	$0.459691\pi$
$180$		0			0
$181$	135.328			0.747669			0.373834	−	0.927495i	$-0.378043\pi$
$181$	135.328			0.747669			0.373834	+	0.927495i	$0.378043\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−106.042	+	61.2234i	−0.573200	+	0.330937i
$186$		0			0
$187$	−52.4797	+	90.8976i	−0.280640	+	0.486083i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−145.898	−	84.2344i	−0.763866	−	0.441018i	0.0668163	−	0.997765i	$-0.478716\pi$
$191$	−145.898	−	84.2344i	−0.763866	−	0.441018i	−0.830682	+	0.556747i	$0.812049\pi$
$192$		0			0
$193$	−71.7490	−	124.273i	−0.371757	−	0.643901i	0.618079	−	0.786116i	$-0.287911\pi$
$193$	−71.7490	−	124.273i	−0.371757	−	0.643901i	−0.989836	+	0.142215i	$0.954578\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	296.510i		−	1.50513i	−0.658521	−	0.752563i	$-0.728817\pi$
$197$		−	296.510i		−	1.50513i	0.658521	−	0.752563i	$-0.271183\pi$
$198$		0			0
$199$	−71.1216	−	123.186i	−0.357395	−	0.619026i	0.630130	−	0.776490i	$-0.283002\pi$
$199$	−71.1216	−	123.186i	−0.357395	−	0.619026i	−0.987525	+	0.157464i	$0.949668\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	39.8745	−	69.0647i	0.194510	−	0.336901i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	157.035i		−	0.751364i
$210$		0			0
$211$	110.996			0.526048			0.263024	−	0.964789i	$-0.415280\pi$
$211$	110.996			0.526048			0.263024	+	0.964789i	$0.415280\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−123.903	−	71.5352i	−0.576291	−	0.332722i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	188.457	−	108.806i	0.852747	−	0.492334i
$222$		0			0
$223$	104.251			0.467493			0.233747	−	0.972298i	$-0.424901\pi$
$223$	104.251			0.467493			0.233747	+	0.972298i	$0.424901\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	141.302	−	81.5807i	0.622475	−	0.359386i	−0.155357	−	0.987858i	$-0.549653\pi$
$227$	141.302	−	81.5807i	0.622475	−	0.359386i	0.777832	+	0.628472i	$0.216319\pi$
$228$		0			0
$229$	−139.207	+	241.113i	−0.607889	+	1.05289i	0.383699	+	0.923458i	$0.374650\pi$
$229$	−139.207	+	241.113i	−0.607889	+	1.05289i	−0.991588	+	0.129436i	$0.958683\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−349.910	−	202.021i	−1.50176	−	0.867042i	−0.999998	−	0.00203732i	$-0.999352\pi$
$233$	−349.910	−	202.021i	−1.50176	−	0.867042i	−0.501763	−	0.865005i	$-0.667315\pi$
$234$		0			0
$235$	73.3725	+	127.085i	0.312224	+	0.540787i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	257.204i		−	1.07617i	−0.842892	−	0.538083i	$-0.819149\pi$
$239$		−	257.204i		−	1.07617i	0.842892	−	0.538083i	$-0.180851\pi$
$240$		0			0
$241$	124.808	+	216.173i	0.517875	+	0.896985i	0.999784	+	0.0207645i	$0.00661002\pi$
$241$	124.808	+	216.173i	0.517875	+	0.896985i	−0.481910	+	0.876221i	$0.660057\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−162.790	+	281.960i	−0.659067	+	1.14154i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	69.2909i		−	0.276059i	−0.990428	−	0.138030i	$-0.955923\pi$
$251$		−	69.2909i		−	0.276059i	0.990428	−	0.138030i	$-0.0440769\pi$
$252$		0			0
$253$	14.6640			0.0579606
$254$		0			0
$255$		0			0
$256$		0			0
$257$	221.592	+	127.936i	0.862227	+	0.497807i	0.864757	−	0.502190i	$-0.167472\pi$
$257$	221.592	+	127.936i	0.862227	+	0.497807i	−0.00253044	+	0.999997i	$0.500805\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.7037	−	7.33449i	0.0483031	−	0.0278878i	−0.475654	−	0.879632i	$-0.657789\pi$
$263$	12.7037	−	7.33449i	0.0483031	−	0.0278878i	0.523957	+	0.851745i	$0.324455\pi$
$264$		0			0
$265$	−240.620			−0.907998
$266$		0			0
$267$		0			0
$268$		0			0
$269$	64.9339	−	37.4896i	0.241390	−	0.139367i	−0.374425	−	0.927257i	$-0.622160\pi$
$269$	64.9339	−	37.4896i	0.241390	−	0.139367i	0.615815	+	0.787890i	$0.288827\pi$
$270$		0			0
$271$	−105.601	+	182.907i	−0.389673	+	0.674933i	−0.992405	−	0.123010i	$-0.960745\pi$
$271$	−105.601	+	182.907i	−0.389673	+	0.674933i	0.602733	+	0.797943i	$0.294079\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.32027	−	4.80371i	−0.0302555	−	0.0174680i
$276$		0			0
$277$	−204.247	−	353.766i	−0.737354	−	1.27713i	−0.953683	−	0.300814i	$-0.902742\pi$
$277$	−204.247	−	353.766i	−0.737354	−	1.27713i	0.216329	−	0.976321i	$-0.430592\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			200.395i			0.713149i	0.934267	+	0.356575i	$0.116055\pi$
$281$			200.395i			0.713149i	−0.934267	+	0.356575i	$0.883945\pi$
$282$		0			0
$283$	179.225	+	310.426i	0.633303	+	1.09691i	0.986872	+	0.161505i	$0.0516347\pi$
$283$	179.225	+	310.426i	0.633303	+	1.09691i	−0.353569	+	0.935409i	$0.615032\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	5.04249	−	8.73384i	0.0174481	−	0.0302209i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	325.352i		−	1.11042i	−0.831711	−	0.555209i	$-0.812638\pi$
$293$		−	325.352i		−	1.11042i	0.831711	−	0.555209i	$-0.187362\pi$
$294$		0			0
$295$	415.911			1.40987
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−26.3296	−	15.2014i	−0.0880589	−	0.0508408i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	280.001	−	161.658i	0.918035	−	0.530028i
$306$		0			0
$307$	338.037			1.10110			0.550548	−	0.834803i	$-0.314419\pi$
$307$	338.037			1.10110			0.550548	+	0.834803i	$0.314419\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−269.489	+	155.589i	−0.866523	+	0.500287i	−0.866191	−	0.499713i	$-0.833439\pi$
$311$	−269.489	+	155.589i	−0.866523	+	0.500287i	−0.000331809		1.00000i	$0.500106\pi$
$312$		0			0
$313$	200.535	−	347.336i	0.640686	−	1.10970i	−0.344594	−	0.938752i	$-0.611984\pi$
$313$	200.535	−	347.336i	0.640686	−	1.10970i	0.985280	−	0.170948i	$-0.0546831\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	72.8747	+	42.0742i	0.229889	+	0.132726i	0.610521	−	0.792000i	$-0.290960\pi$
$317$	72.8747	+	42.0742i	0.229889	+	0.132726i	−0.380632	+	0.924727i	$0.624294\pi$
$318$		0			0
$319$	169.332	+	293.292i	0.530821	+	0.919410i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			447.476i			1.38537i
$324$		0			0
$325$	9.95948	+	17.2503i	0.0306446	+	0.0530779i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	111.749	−	193.555i	0.337610	−	0.584758i	−0.646372	−	0.763022i	$-0.723715\pi$
$331$	111.749	−	193.555i	0.337610	−	0.584758i	0.983983	+	0.178264i	$0.0570481\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			598.499i			1.78656i
$336$		0			0
$337$	−234.583			−0.696092			−0.348046	−	0.937477i	$-0.613155\pi$
$337$	−234.583			−0.696092			−0.348046	+	0.937477i	$0.613155\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−80.3721	−	46.4028i	−0.235695	−	0.136079i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−72.4136	+	41.8080i	−0.208685	+	0.120484i	−0.600700	−	0.799474i	$-0.705111\pi$
$347$	−72.4136	+	41.8080i	−0.208685	+	0.120484i	0.392015	+	0.919959i	$0.371778\pi$
$348$		0			0
$349$	−298.959			−0.856617			−0.428309	−	0.903632i	$-0.640890\pi$
$349$	−298.959			−0.856617			−0.428309	+	0.903632i	$0.640890\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−370.704	+	214.026i	−1.05015	+	0.606306i	−0.922692	−	0.385538i	$-0.874016\pi$
$353$	−370.704	+	214.026i	−1.05015	+	0.606306i	−0.127460	+	0.991844i	$0.540683\pi$
$354$		0			0
$355$	128.354	−	222.316i	0.361561	−	0.626242i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	125.172	+	72.2681i	0.348669	+	0.201304i	0.664099	−	0.747645i	$-0.268815\pi$
$359$	125.172	+	72.2681i	0.348669	+	0.201304i	−0.315430	+	0.948949i	$0.602149\pi$
$360$		0			0
$361$	−154.245	−	267.160i	−0.427272	−	0.740056i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	158.329i		−	0.433778i
$366$		0			0
$367$	−305.122	−	528.486i	−0.831394	−	1.44002i	−0.896933	−	0.442166i	$-0.854210\pi$
$367$	−305.122	−	528.486i	−0.831394	−	1.44002i	0.0655391	−	0.997850i	$-0.479123\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.166010	+	0.287539i	−0.000445068	+	0.000770881i	−0.866248	−	0.499615i	$-0.833475\pi$
$373$	−0.166010	+	0.287539i	−0.000445068	+	0.000770881i	0.865803	+	0.500385i	$0.166808\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	702.149i		−	1.86246i
$378$		0			0
$379$	−393.652			−1.03866			−0.519330	−	0.854574i	$-0.673818\pi$
$379$	−393.652			−1.03866			−0.519330	+	0.854574i	$0.673818\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	468.259	+	270.350i	1.22261	+	0.705874i	0.965473	−	0.260502i	$-0.0838880\pi$
$383$	468.259	+	270.350i	1.22261	+	0.705874i	0.257135	+	0.966375i	$0.417221\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−410.840	+	237.199i	−1.05614	+	0.609765i	−0.924363	−	0.381513i	$-0.875403\pi$
$389$	−410.840	+	237.199i	−1.05614	+	0.609765i	−0.131781	+	0.991279i	$0.542070\pi$
$390$		0			0
$391$	−41.7856			−0.106869
$392$		0			0
$393$		0			0
$394$		0			0
$395$	637.011	−	367.779i	1.61269	−	0.931085i
$396$		0			0
$397$	200.144	−	346.659i	0.504141	−	0.873197i	−0.495848	−	0.868409i	$-0.665143\pi$
$397$	200.144	−	346.659i	0.504141	−	0.873197i	0.999989	−	0.00478767i	$-0.00152397\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	563.839	+	325.533i	1.40608	+	0.811802i	0.995008	−	0.0997999i	$-0.0318203\pi$
$401$	563.839	+	325.533i	1.40608	+	0.811802i	0.411075	+	0.911602i	$0.365154\pi$
$402$		0			0
$403$	96.2065	+	166.635i	0.238726	+	0.413485i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			144.135i			0.354140i
$408$		0			0
$409$	−94.5568	−	163.777i	−0.231190	−	0.400433i	0.726968	−	0.686671i	$-0.240929\pi$
$409$	−94.5568	−	163.777i	−0.231190	−	0.400433i	−0.958159	+	0.286238i	$0.907595\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−73.6680	+	127.597i	−0.177513	+	0.307462i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			829.772i			1.98036i	0.139791	+	0.990181i	$0.455357\pi$
$419$			829.772i			1.98036i	−0.139791	+	0.990181i	$0.544643\pi$
$420$		0			0
$421$	−226.324			−0.537587			−0.268794	−	0.963198i	$-0.586625\pi$
$421$	−226.324			−0.537587			−0.268794	+	0.963198i	$0.586625\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	23.7088	+	13.6883i	0.0557855	+	0.0322078i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−377.985	+	218.230i	−0.876996	+	0.506334i	−0.869667	−	0.493640i	$-0.835666\pi$
$431$	−377.985	+	218.230i	−0.876996	+	0.506334i	−0.00732893	+	0.999973i	$0.502333\pi$
$432$		0			0
$433$	−404.243			−0.933587			−0.466793	−	0.884366i	$-0.654591\pi$
$433$	−404.243			−0.933587			−0.466793	+	0.884366i	$0.654591\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	54.1417	−	31.2587i	0.123894	−	0.0715303i
$438$		0			0
$439$	−91.4091	+	158.325i	−0.208221	+	0.360650i	−0.951154	−	0.308716i	$-0.900101\pi$
$439$	−91.4091	+	158.325i	−0.208221	+	0.360650i	0.742933	+	0.669366i	$0.233434\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−286.838	−	165.606i	−0.647491	−	0.373829i	0.140003	−	0.990151i	$-0.455289\pi$
$443$	−286.838	−	165.606i	−0.647491	−	0.373829i	−0.787494	+	0.616322i	$0.788622\pi$
$444$		0			0
$445$	433.616	+	751.044i	0.974417	+	1.68774i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	12.8425i		−	0.0286025i	−0.999898	−	0.0143013i	$-0.995448\pi$
$449$		−	12.8425i		−	0.0286025i	0.999898	−	0.0143013i	$-0.00455239\pi$
$450$		0			0
$451$	−46.9373	−	81.2977i	−0.104074	−	0.180261i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	155.336	−	269.050i	0.339904	−	0.588730i	−0.644511	−	0.764595i	$-0.722939\pi$
$457$	155.336	−	269.050i	0.339904	−	0.588730i	0.984414	+	0.175865i	$0.0562722\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	389.770i		−	0.845489i	−0.906249	−	0.422744i	$-0.861067\pi$
$461$		−	389.770i		−	0.845489i	0.906249	−	0.422744i	$-0.138933\pi$
$462$		0			0
$463$	152.081			0.328469			0.164234	−	0.986421i	$-0.447485\pi$
$463$	152.081			0.328469			0.164234	+	0.986421i	$0.447485\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	75.2298	+	43.4339i	0.161092	+	0.0930063i	0.578379	−	0.815768i	$-0.303686\pi$
$467$	75.2298	+	43.4339i	0.161092	+	0.0930063i	−0.417287	+	0.908775i	$0.637019\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−145.849	+	84.2058i	−0.308348	+	0.178025i
$474$		0			0
$475$	−40.9595			−0.0862305
$476$		0			0
$477$		0			0
$478$		0			0
$479$	476.005	−	274.821i	0.993747	−	0.573740i	0.0873545	−	0.996177i	$-0.472159\pi$
$479$	476.005	−	274.821i	0.993747	−	0.573740i	0.906392	+	0.422437i	$0.138825\pi$
$480$		0			0
$481$	149.417	−	258.798i	0.310638	−	0.538041i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	508.825	+	293.770i	1.04912	+	0.605711i
$486$		0			0
$487$	345.373	+	598.203i	0.709184	+	1.22834i	0.965160	+	0.261659i	$0.0842697\pi$
$487$	345.373	+	598.203i	0.709184	+	1.22834i	−0.255976	+	0.966683i	$0.582397\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			732.380i			1.49161i	0.666165	+	0.745804i	$0.267934\pi$
$491$			732.380i			1.49161i	−0.666165	+	0.745804i	$0.732066\pi$
$492$		0			0
$493$	−482.516	−	835.743i	−0.978735	−	1.69522i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	227.203	−	393.526i	0.455316	−	0.788630i	−0.543390	−	0.839480i	$-0.682860\pi$
$499$	227.203	−	393.526i	0.455316	−	0.788630i	0.998706	+	0.0508499i	$0.0161930\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			304.594i			0.605554i	0.953061	+	0.302777i	$0.0979138\pi$
$503$			304.594i			0.605554i	−0.953061	+	0.302777i	$0.902086\pi$
$504$		0			0
$505$	207.660			0.411208
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−64.2102	−	37.0718i	−0.126150	−	0.0728326i	0.435597	−	0.900142i	$-0.356537\pi$
$509$	−64.2102	−	37.0718i	−0.126150	−	0.0728326i	−0.561747	+	0.827309i	$0.689871\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−130.031	+	75.0735i	−0.252488	+	0.145774i
$516$		0			0
$517$	172.737			0.334115
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−185.772	+	107.256i	−0.356568	+	0.205865i	−0.667574	−	0.744543i	$-0.732667\pi$
$521$	−185.772	+	107.256i	−0.356568	+	0.205865i	0.311006	+	0.950408i	$0.399334\pi$
$522$		0			0
$523$	466.243	−	807.557i	0.891478	−	1.54409i	0.0533746	−	0.998575i	$-0.483002\pi$
$523$	466.243	−	807.557i	0.891478	−	1.54409i	0.838104	−	0.545511i	$-0.183664\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	229.022	+	132.226i	0.434578	+	0.250904i
$528$		0			0
$529$	−261.581	−	453.072i	−0.494482	−	0.856468i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			194.629i			0.365158i
$534$		0			0
$535$	468.929	+	812.210i	0.876504	+	1.51815i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	93.1699	−	161.375i	0.172218	−	0.298290i	−0.766977	−	0.641675i	$-0.778240\pi$
$541$	93.1699	−	161.375i	0.172218	−	0.298290i	0.939195	+	0.343384i	$0.111573\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	454.552i		−	0.834041i
$546$		0			0
$547$	792.170			1.44821			0.724104	−	0.689691i	$-0.242253\pi$
$547$	792.170			1.44821			0.724104	+	0.689691i	$0.242253\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1250.40	+	721.916i	2.26932	+	1.31019i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−346.087	+	199.814i	−0.621342	+	0.358732i	−0.777391	−	0.629017i	$-0.783457\pi$
$557$	−346.087	+	199.814i	−0.621342	+	0.358732i	0.156050	+	0.987749i	$0.450124\pi$
$558$		0			0
$559$	349.166			0.624626
$560$		0			0
$561$		0			0
$562$		0			0
$563$	302.571	−	174.690i	0.537427	−	0.310283i	−0.206609	−	0.978424i	$-0.566243\pi$
$563$	302.571	−	174.690i	0.537427	−	0.310283i	0.744035	+	0.668140i	$0.232909\pi$
$564$		0			0
$565$	−167.099	+	289.425i	−0.295751	+	0.512256i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	654.163	+	377.681i	1.14967	+	0.663763i	0.948807	−	0.315856i	$-0.102292\pi$
$569$	654.163	+	377.681i	1.14967	+	0.663763i	0.200864	+	0.979619i	$0.435625\pi$
$570$		0			0
$571$	−47.5751	−	82.4025i	−0.0833190	−	0.144313i	0.821355	−	0.570418i	$-0.193219\pi$
$571$	−47.5751	−	82.4025i	−0.0833190	−	0.144313i	−0.904674	+	0.426105i	$0.859885\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	3.82483i		−	0.00665187i
$576$		0			0
$577$	378.033	+	654.772i	0.655169	+	1.13479i	0.981851	+	0.189652i	$0.0607360\pi$
$577$	378.033	+	654.772i	0.655169	+	1.13479i	−0.326682	+	0.945134i	$0.605931\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−141.620	+	245.292i	−0.242915	+	0.420742i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	688.642i		−	1.17316i	−0.809893	−	0.586578i	$-0.800475\pi$
$587$		−	688.642i		−	1.17316i	0.809893	−	0.586578i	$-0.199525\pi$
$588$		0			0
$589$	−395.660			−0.671749
$590$		0			0
$591$		0			0
$592$		0			0
$593$	879.890	+	508.005i	1.48379	+	0.856669i	0.999830	−	0.0184184i	$-0.00586309\pi$
$593$	879.890	+	508.005i	1.48379	+	0.856669i	0.483964	+	0.875088i	$0.339196\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−511.180	+	295.130i	−0.853388	+	0.492704i	−0.861793	−	0.507261i	$-0.830658\pi$
$599$	−511.180	+	295.130i	−0.853388	+	0.492704i	0.00840423	+	0.999965i	$0.497325\pi$
$600$		0			0
$601$	−300.761			−0.500434			−0.250217	−	0.968190i	$-0.580502\pi$
$601$	−300.761			−0.500434			−0.250217	+	0.968190i	$0.580502\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	375.811	−	216.975i	0.621175	−	0.358636i
$606$		0			0
$607$	348.207	−	603.111i	0.573652	−	0.993594i	−0.422535	−	0.906347i	$-0.638860\pi$
$607$	348.207	−	603.111i	0.573652	−	0.993594i	0.996187	−	0.0872472i	$-0.0278070\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−310.153	−	179.067i	−0.507616	−	0.293072i
$612$		0			0
$613$	−278.697	−	482.717i	−0.454644	−	0.787466i	0.544024	−	0.839070i	$-0.316900\pi$
$613$	−278.697	−	482.717i	−0.454644	−	0.787466i	−0.998668	+	0.0516035i	$0.983567\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	91.9868i		−	0.149087i	−0.997218	−	0.0745436i	$-0.976250\pi$
$617$		−	91.9868i		−	0.149087i	0.997218	−	0.0745436i	$-0.0237500\pi$
$618$		0			0
$619$	−558.708	−	967.712i	−0.902599	−	1.56335i	−0.824109	−	0.566432i	$-0.808324\pi$
$619$	−558.708	−	967.712i	−0.902599	−	1.56335i	−0.0784898	−	0.996915i	$-0.525010\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	331.035	−	573.369i	0.529655	−	0.917390i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	410.717i		−	0.652969i
$630$		0			0
$631$	962.219			1.52491			0.762456	−	0.647040i	$-0.223993\pi$
$631$	962.219			1.52491			0.762456	+	0.647040i	$0.223993\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	998.306	+	576.372i	1.57214	+	0.907673i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	194.373	−	112.221i	0.303233	−	0.175072i	−0.340661	−	0.940186i	$-0.610651\pi$
$641$	194.373	−	112.221i	0.303233	−	0.175072i	0.643895	+	0.765114i	$0.277318\pi$
$642$		0			0
$643$	332.384			0.516927			0.258464	−	0.966021i	$-0.416784\pi$
$643$	332.384			0.516927			0.258464	+	0.966021i	$0.416784\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−788.563	+	455.277i	−1.21880	+	0.703674i	−0.964661	−	0.263494i	$-0.915125\pi$
$647$	−788.563	+	455.277i	−1.21880	+	0.703674i	−0.254138	+	0.967168i	$0.581792\pi$
$648$		0			0
$649$	244.790	−	423.988i	0.377180	−	0.653294i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−128.435	−	74.1519i	−0.196684	−	0.113556i	0.398424	−	0.917202i	$-0.369557\pi$
$653$	−128.435	−	74.1519i	−0.196684	−	0.113556i	−0.595108	+	0.803646i	$0.702891\pi$
$654$		0			0
$655$	534.405	+	925.617i	0.815886	+	1.41316i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			45.5636i			0.0691405i	0.999402	+	0.0345703i	$0.0110063\pi$
$659$			45.5636i			0.0691405i	−0.999402	+	0.0345703i	$0.988994\pi$
$660$		0			0
$661$	202.107	+	350.060i	0.305760	+	0.529591i	0.977430	−	0.211259i	$-0.0677562\pi$
$661$	202.107	+	350.060i	0.305760	+	0.529591i	−0.671670	+	0.740850i	$0.734423\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−67.4131	+	116.763i	−0.101069	+	0.175057i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	380.584i		−	0.567190i
$672$		0			0
$673$	1016.49			1.51038			0.755190	−	0.655506i	$-0.227544\pi$
$673$	1016.49			1.51038			0.755190	+	0.655506i	$0.227544\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	201.391	+	116.273i	0.297476	+	0.171748i	0.641308	−	0.767283i	$-0.278392\pi$
$677$	201.391	+	116.273i	0.297476	+	0.171748i	−0.343832	+	0.939031i	$0.611725\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	102.403	−	59.1224i	0.149931	−	0.0865628i	−0.423158	−	0.906056i	$-0.639078\pi$
$683$	102.403	−	59.1224i	0.149931	−	0.0865628i	0.573089	+	0.819493i	$0.305745\pi$
$684$		0			0
$685$	−906.272			−1.32302
$686$		0			0
$687$		0			0
$688$		0			0
$689$	508.562	−	293.618i	0.738116	−	0.426152i
$690$		0			0
$691$	24.1255	−	41.7866i	0.0349139	−	0.0604726i	−0.848040	−	0.529931i	$-0.822218\pi$
$691$	24.1255	−	41.7866i	0.0349139	−	0.0604726i	0.882954	+	0.469459i	$0.155551\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−523.160	−	302.046i	−0.752748	−	0.434599i
$696$		0			0
$697$	133.749	+	231.660i	0.191892	+	0.332367i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	236.466i		−	0.337326i	−0.985674	−	0.168663i	$-0.946055\pi$
$701$		−	236.466i		−	0.337326i	0.985674	−	0.168663i	$-0.0539450\pi$
$702$		0			0
$703$	307.247	+	532.167i	0.437051	+	0.756995i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	670.988	−	1162.19i	0.946387	−	1.63919i	0.193436	−	0.981113i	$-0.438037\pi$
$709$	670.988	−	1162.19i	0.946387	−	1.63919i	0.752951	−	0.658077i	$-0.228630\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	36.9470i		−	0.0518191i
$714$		0			0
$715$	393.741			0.550687
$716$		0			0
$717$		0			0
$718$		0			0
$719$	246.486	+	142.309i	0.342818	+	0.197926i	0.661517	−	0.749930i	$-0.269913\pi$
$719$	246.486	+	142.309i	0.342818	+	0.197926i	−0.318700	+	0.947856i	$0.603246\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	76.4993	−	44.1669i	0.105516	−	0.0609199i
$726$		0			0
$727$	−430.450			−0.592090			−0.296045	−	0.955174i	$-0.595668\pi$
$727$	−430.450			−0.592090			−0.296045	+	0.955174i	$0.595668\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	415.600	−	239.947i	0.568536	−	0.328245i
$732$		0			0
$733$	−193.907	+	335.857i	−0.264539	+	0.458195i	−0.967443	−	0.253090i	$-0.918553\pi$
$733$	−193.907	+	335.857i	−0.264539	+	0.458195i	0.702904	+	0.711285i	$0.251886\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	610.121	+	352.254i	0.827845	+	0.477956i
$738$		0			0
$739$	−176.454	−	305.627i	−0.238773	−	0.413568i	0.721589	−	0.692322i	$-0.243412\pi$
$739$	−176.454	−	305.627i	−0.238773	−	0.413568i	−0.960363	+	0.278754i	$0.910079\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1068.60i			1.43822i	0.694897	+	0.719109i	$0.255450\pi$
$743$			1068.60i			1.43822i	−0.694897	+	0.719109i	$0.744550\pi$
$744$		0			0
$745$	−480.310	−	831.921i	−0.644711	−	1.11667i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−612.073	+	1060.14i	−0.815011	+	1.41164i	0.0943092	+	0.995543i	$0.469936\pi$
$751$	−612.073	+	1060.14i	−0.815011	+	1.41164i	−0.909320	+	0.416097i	$0.863398\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			938.786i			1.24343i
$756$		0			0
$757$	−96.3399			−0.127265			−0.0636327	−	0.997973i	$-0.520269\pi$
$757$	−96.3399			−0.127265			−0.0636327	+	0.997973i	$0.520269\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−821.762	−	474.445i	−1.07985	−	0.623449i	−0.148990	−	0.988839i	$-0.547602\pi$
$761$	−821.762	−	474.445i	−1.07985	−	0.623449i	−0.930855	+	0.365390i	$0.880936\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−879.050	+	507.520i	−1.14609	+	0.661694i
$768$		0			0
$769$	998.633			1.29861			0.649306	−	0.760527i	$-0.275059\pi$
$769$	998.633			1.29861			0.649306	+	0.760527i	$0.275059\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	74.8855	−	43.2352i	0.0968765	−	0.0559317i	−0.450779	−	0.892636i	$-0.648854\pi$
$773$	74.8855	−	43.2352i	0.0968765	−	0.0559317i	0.547656	+	0.836704i	$0.315520\pi$
$774$		0			0
$775$	−12.1033	+	20.9635i	−0.0156171	+	0.0270496i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−346.598	−	200.108i	−0.444927	−	0.256879i
$780$		0			0
$781$	−151.089	−	261.694i	−0.193456	−	0.335075i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	352.540i		−	0.449096i
$786$		0			0
$787$	−244.907	−	424.192i	−0.311191	−	0.538998i	0.667430	−	0.744673i	$-0.267394\pi$
$787$	−244.907	−	424.192i	−0.311191	−	0.538998i	−0.978620	+	0.205675i	$0.934061\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−394.531	+	683.347i	−0.497517	+	0.861724i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	500.193i		−	0.627595i	−0.949490	−	0.313798i	$-0.898399\pi$
$797$		−	500.193i		−	0.627595i	0.949490	−	0.313798i	$-0.101601\pi$
$798$		0			0
$799$	−492.219			−0.616044
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−161.404	−	93.1865i	−0.201001	−	0.116048i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−268.914	+	155.257i	−0.332403	+	0.191913i	−0.656907	−	0.753971i	$-0.728136\pi$
$809$	−268.914	+	155.257i	−0.332403	+	0.191913i	0.324505	+	0.945884i	$0.394802\pi$
$810$		0			0
$811$	45.9190			0.0566202			0.0283101	−	0.999599i	$-0.490987\pi$
$811$	45.9190			0.0566202			0.0283101	+	0.999599i	$0.490987\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−164.071	+	94.7264i	−0.201314	+	0.116229i
$816$		0			0
$817$	−358.996	+	621.799i	−0.439408	+	0.761076i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1046.73	+	604.330i	1.27495	+	0.736091i	0.975915	−	0.218153i	$-0.0700030\pi$
$821$	1046.73	+	604.330i	1.27495	+	0.736091i	0.299032	+	0.954243i	$0.403336\pi$
$822$		0			0
$823$	59.5909	+	103.214i	0.0724069	+	0.125412i	0.899956	−	0.435981i	$-0.143599\pi$
$823$	59.5909	+	103.214i	0.0724069	+	0.125412i	−0.827549	+	0.561394i	$0.810265\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			872.026i			1.05445i	0.849727	+	0.527223i	$0.176767\pi$
$827$			872.026i			1.05445i	−0.849727	+	0.527223i	$0.823233\pi$
$828$		0			0
$829$	368.911	+	638.973i	0.445007	+	0.770775i	0.998053	−	0.0623758i	$-0.0198677\pi$
$829$	368.911	+	638.973i	0.445007	+	0.770775i	−0.553045	+	0.833151i	$0.686534\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	620.531	−	1074.79i	0.743151	−	1.28717i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			896.131i			1.06809i	0.845455	+	0.534047i	$0.179329\pi$
$839$			896.131i			1.06809i	−0.845455	+	0.534047i	$0.820671\pi$
$840$		0			0
$841$	−2272.79			−2.70249
$842$		0			0
$843$		0			0
$844$		0			0
$845$	47.6338	+	27.5014i	0.0563713	+	0.0325460i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−49.6942	+	28.6910i	−0.0583950	+	0.0337144i
$852$		0			0
$853$	−311.875			−0.365621			−0.182810	−	0.983148i	$-0.558519\pi$
$853$	−311.875			−0.365621			−0.182810	+	0.983148i	$0.558519\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−312.540	+	180.445i	−0.364691	+	0.210555i	−0.671137	−	0.741334i	$-0.734194\pi$
$857$	−312.540	+	180.445i	−0.364691	+	0.210555i	0.306445	+	0.951888i	$0.400860\pi$
$858$		0			0
$859$	605.549	−	1048.84i	0.704946	−	1.22100i	−0.261764	−	0.965132i	$-0.584304\pi$
$859$	605.549	−	1048.84i	0.704946	−	1.22100i	0.966711	−	0.255871i	$-0.0823623\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	442.972	+	255.750i	0.513293	+	0.296350i	0.734186	−	0.678948i	$-0.237564\pi$
$863$	442.972	+	255.750i	0.513293	+	0.296350i	−0.220893	+	0.975298i	$0.570897\pi$
$864$		0			0
$865$	879.150	+	1522.73i	1.01636	+	1.76038i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	865.843i		−	0.996367i
$870$		0			0
$871$	−730.324	−	1264.96i	−0.838489	−	1.45231i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−67.2954	+	116.559i	−0.0767337	+	0.132907i	−0.901839	−	0.432072i	$-0.857782\pi$
$877$	−67.2954	+	116.559i	−0.0767337	+	0.132907i	0.825105	+	0.564979i	$0.191116\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	405.447i		−	0.460212i	−0.973166	−	0.230106i	$-0.926093\pi$
$881$		−	405.447i		−	0.460212i	0.973166	−	0.230106i	$-0.0739073\pi$
$882$		0			0
$883$	−552.235			−0.625408			−0.312704	−	0.949851i	$-0.601235\pi$
$883$	−552.235			−0.625408			−0.312704	+	0.949851i	$0.601235\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1042.69	−	602.000i	−1.17553	−	0.678692i	−0.220554	−	0.975375i	$-0.570786\pi$
$887$	−1042.69	−	602.000i	−1.17553	−	0.678692i	−0.954976	+	0.296682i	$0.904120\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	637.770	−	368.217i	0.714188	−	0.412337i
$894$		0			0
$895$	713.624			0.797345
$896$		0			0
$897$		0			0
$898$		0			0
$899$	738.968	−	426.643i	0.821989	−	0.474575i
$900$		0			0
$901$	403.549	−	698.967i	0.447890	−	0.775768i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−604.255	−	348.867i	−0.667686	−	0.385488i
$906$		0			0
$907$	−660.207	−	1143.51i	−0.727901	−	1.26076i	−0.957769	−	0.287540i	$-0.907163\pi$
$907$	−660.207	−	1143.51i	−0.727901	−	1.26076i	0.229867	−	0.973222i	$-0.426171\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	965.020i		−	1.05930i	−0.848217	−	0.529649i	$-0.822324\pi$
$911$		−	965.020i		−	1.05930i	0.848217	−	0.529649i	$-0.177676\pi$
$912$		0			0
$913$	86.7164	+	150.197i	0.0949796	+	0.164509i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−360.044	+	623.615i	−0.391779	+	0.678580i	−0.992684	−	0.120740i	$-0.961473\pi$
$919$	−360.044	+	623.615i	−0.391779	+	0.678580i	0.600906	+	0.799320i	$0.294807\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			626.502i			0.678767i
$924$		0			0
$925$	37.5948			0.0406430
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−806.867	−	465.845i	−0.868532	−	0.501447i	−0.00167220	−	0.999999i	$-0.500532\pi$
$929$	−806.867	−	465.845i	−0.868532	−	0.501447i	−0.866860	+	0.498551i	$0.833866\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	468.656	−	270.579i	0.501237	−	0.289389i
$936$		0			0
$937$	−1370.97			−1.46315			−0.731575	−	0.681760i	$-0.761215\pi$
$937$	−1370.97			−1.46315			−0.731575	+	0.681760i	$0.761215\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−197.008	+	113.743i	−0.209360	+	0.120874i	−0.601014	−	0.799239i	$-0.705236\pi$
$941$	−197.008	+	113.743i	−0.209360	+	0.120874i	0.391654	+	0.920113i	$0.371903\pi$
$942$		0			0
$943$	18.6863	−	32.3656i	0.0198158	−	0.0343219i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	570.662	+	329.472i	0.602600	+	0.347911i	0.770064	−	0.637967i	$-0.220224\pi$
$947$	570.662	+	329.472i	0.602600	+	0.347911i	−0.167464	+	0.985878i	$0.553558\pi$
$948$		0			0
$949$	193.203	+	334.637i	0.203585	+	0.352620i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	217.669i		−	0.228404i	−0.993458	−	0.114202i	$-0.963569\pi$
$953$		−	217.669i		−	0.228404i	0.993458	−	0.114202i	$-0.0364311\pi$
$954$		0			0
$955$	434.302	+	752.233i	0.454766	+	0.787679i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	363.585	−	629.748i	0.378340	−	0.655305i
$962$		0			0
$963$		0			0
$964$		0			0
$965$			739.857i			0.766692i
$966$		0			0
$967$	−1018.75			−1.05352			−0.526760	−	0.850014i	$-0.676593\pi$
$967$	−1018.75			−1.05352			−0.526760	+	0.850014i	$0.676593\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−353.812	−	204.273i	−0.364379	−	0.210374i	0.306621	−	0.951832i	$-0.400802\pi$
$971$	−353.812	−	204.273i	−0.364379	−	0.210374i	−0.671000	+	0.741457i	$0.734135\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	391.760	−	226.183i	0.400982	−	0.231507i	−0.285925	−	0.958252i	$-0.592301\pi$
$977$	391.760	−	226.183i	0.400982	−	0.231507i	0.686908	+	0.726745i	$0.258968\pi$
$978$		0			0
$979$	1020.84			1.04274
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1500.44	+	866.281i	−1.52639	+	0.881262i	−0.526881	+	0.849939i	$0.676639\pi$
$983$	−1500.44	+	866.281i	−1.52639	+	0.881262i	−0.999509	+	0.0313233i	$0.990028\pi$
$984$		0			0
$985$	−764.383	+	1323.95i	−0.776023	+	1.34411i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−58.0641	−	33.5233i	−0.0587099	−	0.0338962i
$990$		0			0
$991$	−197.373	−	341.859i	−0.199165	−	0.344964i	0.749093	−	0.662465i	$-0.230490\pi$
$991$	−197.373	−	341.859i	−0.199165	−	0.344964i	−0.948258	+	0.317501i	$0.897156\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			733.387i			0.737072i
$996$		0			0
$997$	82.6745	+	143.196i	0.0829232	+	0.143627i	0.904504	−	0.426464i	$-0.140241\pi$
$997$	82.6745	+	143.196i	0.0829232	+	0.143627i	−0.821581	+	0.570092i	$0.806908\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.3.bk.e.1745.1		8
3.2	odd	2	inner	1764.3.bk.e.1745.4		8
7.2	even	3		1764.3.c.f.197.4		4
7.3	odd	6		1764.3.bk.d.557.1		8
7.4	even	3	inner	1764.3.bk.e.557.4		8
7.5	odd	6		252.3.c.a.197.1	✓	4
7.6	odd	2		1764.3.bk.d.1745.4		8
21.2	odd	6		1764.3.c.f.197.1		4
21.5	even	6		252.3.c.a.197.4	yes	4
21.11	odd	6	inner	1764.3.bk.e.557.1		8
21.17	even	6		1764.3.bk.d.557.4		8
21.20	even	2		1764.3.bk.d.1745.1		8
28.19	even	6		1008.3.d.c.449.1		4
56.5	odd	6		4032.3.d.h.449.4		4
56.19	even	6		4032.3.d.e.449.4		4
63.5	even	6		2268.3.bg.c.2213.4		8
63.40	odd	6		2268.3.bg.c.2213.1		8
63.47	even	6		2268.3.bg.c.701.1		8
63.61	odd	6		2268.3.bg.c.701.4		8
84.47	odd	6		1008.3.d.c.449.4		4
168.5	even	6		4032.3.d.h.449.1		4
168.131	odd	6		4032.3.d.e.449.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.3.c.a.197.1	✓	4	7.5	odd	6
252.3.c.a.197.4	yes	4	21.5	even	6
1008.3.d.c.449.1		4	28.19	even	6
1008.3.d.c.449.4		4	84.47	odd	6
1764.3.c.f.197.1		4	21.2	odd	6
1764.3.c.f.197.4		4	7.2	even	3
1764.3.bk.d.557.1		8	7.3	odd	6
1764.3.bk.d.557.4		8	21.17	even	6
1764.3.bk.d.1745.1		8	21.20	even	2
1764.3.bk.d.1745.4		8	7.6	odd	2
1764.3.bk.e.557.1		8	21.11	odd	6	inner
1764.3.bk.e.557.4		8	7.4	even	3	inner
1764.3.bk.e.1745.1		8	1.1	even	1	trivial
1764.3.bk.e.1745.4		8	3.2	odd	2	inner
2268.3.bg.c.701.1		8	63.47	even	6
2268.3.bg.c.701.4		8	63.61	odd	6
2268.3.bg.c.2213.1		8	63.40	odd	6
2268.3.bg.c.2213.4		8	63.5	even	6
4032.3.d.e.449.1		4	168.131	odd	6
4032.3.d.e.449.4		4	56.19	even	6
4032.3.d.h.449.1		4	168.5	even	6
4032.3.d.h.449.4		4	56.5	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1764.bk (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.46512 - 2.57794i) q^{5} +(-5.25600 + 3.03455i) q^{11} +12.5830 q^{13} +(14.9771 - 8.64704i) q^{17} +(-12.9373 + 22.4080i) q^{19} +(-2.09247 - 1.20809i) q^{23} +(0.791503 + 1.37092i) q^{25} -55.8014i q^{29} +(7.64575 + 13.2428i) q^{31} +(11.8745 - 20.5673i) q^{37} +15.4676i q^{41} +27.7490 q^{43} +(-24.6486 - 14.2309i) q^{47} +(40.4166 - 23.3345i) q^{53} +31.2915 q^{55} +(-69.8601 + 40.3337i) q^{59} +(-31.3542 + 54.3072i) q^{61} +(-56.1846 - 32.4382i) q^{65} +(-58.0405 - 100.529i) q^{67} +49.7896i q^{71} +(15.3542 + 26.5943i) q^{73} +(-71.3320 + 123.551i) q^{79} -28.5763i q^{83} -89.1660 q^{85} +(-145.668 - 84.1013i) q^{89} +(115.533 - 66.7028i) q^{95} -113.956 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{13} - 40 q^{19} - 36 q^{25} + 40 q^{31} - 32 q^{37} - 32 q^{43} + 208 q^{55} - 272 q^{61} - 168 q^{67} + 144 q^{73} - 232 q^{79} - 544 q^{85} - 192 q^{97}+O(q^{100}) \) 8 * q + 16 * q^13 - 40 * q^19 - 36 * q^25 + 40 * q^31 - 32 * q^37 - 32 * q^43 + 208 * q^55 - 272 * q^61 - 168 * q^67 + 144 * q^73 - 232 * q^79 - 544 * q^85 - 192 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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