LMFDB - Embedded newform 5292.2.x.b.4409.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5292,2,Mod(881,5292)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5292, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 5, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5292.881"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$5292 = 2^{2} \cdot 3^{3} \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5292.x (of order $6$ , degree $2$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,-6,0,3] 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:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - 2 x^{15} + 5 x^{14} - 17 x^{13} + 22 x^{12} - 31 x^{11} + 62 x^{10} - 52 x^{9} + 52 x^{8} + \cdots + 6561$ x^16 - 2x^15 + 5x^14 - 17x^13 + 22x^12 - 31x^11 + 62x^10 - 52x^9 + 52x^8 - 156x^7 + 558x^6 - 837x^5 + 1782x^4 - 4131x^3 + 3645x^2 - 4374*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$3^{6}$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4409.2
Root			$1.68042 - 0.419752i$ of defining polynomial
Character	$\chi$	$=$	5292.4409
Dual form			5292.2.x.b.881.2

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.48494 - 2.57199i) q^{5} +(-4.09466 - 2.36406i) q^{11} +(3.54045 - 2.04408i) q^{13} +1.67056 q^{17} +4.91183i q^{19} +(4.25297 - 2.45545i) q^{23} +(-1.91009 + 3.30837i) q^{25} +(-0.238557 - 0.137731i) q^{29} +(1.38847 - 0.801636i) q^{31} +3.39362 q^{37} +(-3.55632 - 6.15972i) q^{41} +(5.22930 - 9.05742i) q^{43} +(5.49885 - 9.52430i) q^{47} +0.816814i q^{53} +14.0419i q^{55} +(-1.37428 - 2.38032i) q^{59} +(6.23807 + 3.60155i) q^{61} +(-10.5147 - 6.07067i) q^{65} +(-5.80513 - 10.0548i) q^{67} -10.4406i q^{71} +15.7608i q^{73} +(6.15163 - 10.6549i) q^{79} +(-4.03981 + 6.99715i) q^{83} +(-2.48067 - 4.29665i) q^{85} -9.21744 q^{89} +(12.6332 - 7.29377i) q^{95} +(-7.00772 - 4.04591i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 6 q^{11} + 3 q^{13} - 18 q^{17} + 21 q^{23} - 8 q^{25} - 6 q^{29} - 6 q^{31} - 2 q^{37} - 6 q^{41} - 2 q^{43} + 18 q^{47} + 15 q^{59} - 3 q^{61} - 39 q^{65} - 7 q^{67} - q^{79} + 6 q^{85} - 42 q^{89}+ \cdots + 3 q^{97}+O(q^{100})$ 16 * q - 6 * q^11 + 3 * q^13 - 18 * q^17 + 21 * q^23 - 8 * q^25 - 6 * q^29 - 6 * q^31 - 2 * q^37 - 6 * q^41 - 2 * q^43 + 18 * q^47 + 15 * q^59 - 3 * q^61 - 39 * q^65 - 7 * q^67 - q^79 + 6 * q^85 - 42 * q^89 + 6 * q^95 + 3 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$ .

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$	$1$

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
$5$	−1.48494	−	2.57199i	−0.664085	−	1.15023i	−0.979532	−	0.201286i	$-0.935488\pi$
$5$	−1.48494	−	2.57199i	−0.664085	−	1.15023i	0.315447	−	0.948943i	$-0.397845\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−4.09466	−	2.36406i	−1.23459	−	0.712790i	−0.266605	−	0.963806i	$-0.585902\pi$
$11$	−4.09466	−	2.36406i	−1.23459	−	0.712790i	−0.967983	+	0.251016i	$0.919235\pi$
$12$		0			0
$13$	3.54045	−	2.04408i	0.981945	−	0.566926i	0.0790880	−	0.996868i	$-0.474799\pi$
$13$	3.54045	−	2.04408i	0.981945	−	0.566926i	0.902857	+	0.429942i	$0.141466\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.67056			0.405169			0.202585	−	0.979265i	$-0.435066\pi$
$17$	1.67056			0.405169			0.202585	+	0.979265i	$0.435066\pi$
$18$		0			0
$19$			4.91183i			1.12685i	0.826167	+	0.563426i	$0.190517\pi$
$19$			4.91183i			1.12685i	−0.826167	+	0.563426i	$0.809483\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.25297	−	2.45545i	0.886805	−	0.511997i	0.0139086	−	0.999903i	$-0.495573\pi$
$23$	4.25297	−	2.45545i	0.886805	−	0.511997i	0.872896	+	0.487906i	$0.162239\pi$
$24$		0			0
$25$	−1.91009	+	3.30837i	−0.382018	+	0.661675i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−0.238557	−	0.137731i	−0.0442989	−	0.0255760i	0.477687	−	0.878530i	$-0.341475\pi$
$29$	−0.238557	−	0.137731i	−0.0442989	−	0.0255760i	−0.521986	+	0.852954i	$0.674809\pi$
$30$		0			0
$31$	1.38847	−	0.801636i	0.249377	−	0.143978i	−0.370102	−	0.928991i	$-0.620677\pi$
$31$	1.38847	−	0.801636i	0.249377	−	0.143978i	0.619479	+	0.785013i	$0.287344\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.39362			0.557907			0.278954	−	0.960305i	$-0.410012\pi$
$37$	3.39362			0.557907			0.278954	+	0.960305i	$0.410012\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.55632	−	6.15972i	−0.555404	−	0.961987i	−0.997872	−	0.0652031i	$-0.979230\pi$
$41$	−3.55632	−	6.15972i	−0.555404	−	0.961987i	0.442468	−	0.896784i	$-0.354103\pi$
$42$		0			0
$43$	5.22930	−	9.05742i	0.797461	−	1.38124i	−0.123804	−	0.992307i	$-0.539509\pi$
$43$	5.22930	−	9.05742i	0.797461	−	1.38124i	0.921265	−	0.388936i	$-0.127157\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.49885	−	9.52430i	0.802090	−	1.38926i	−0.116148	−	0.993232i	$-0.537055\pi$
$47$	5.49885	−	9.52430i	0.802090	−	1.38926i	0.918238	−	0.396029i	$-0.129612\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$			0.816814i			0.112198i	0.998425	+	0.0560990i	$0.0178662\pi$
$53$			0.816814i			0.112198i	−0.998425	+	0.0560990i	$0.982134\pi$
$54$		0			0
$55$			14.0419i			1.89341i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.37428	−	2.38032i	−0.178916	−	0.309891i	0.762594	−	0.646878i	$-0.223926\pi$
$59$	−1.37428	−	2.38032i	−0.178916	−	0.309891i	−0.941509	+	0.336986i	$0.890592\pi$
$60$		0			0
$61$	6.23807	+	3.60155i	0.798703	+	0.461131i	0.843017	−	0.537886i	$-0.180777\pi$
$61$	6.23807	+	3.60155i	0.798703	+	0.461131i	−0.0443147	+	0.999018i	$0.514110\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−10.5147	−	6.07067i	−1.30419	−	0.752974i
$66$		0			0
$67$	−5.80513	−	10.0548i	−0.709210	−	1.22839i	−0.965151	−	0.261695i	$-0.915719\pi$
$67$	−5.80513	−	10.0548i	−0.709210	−	1.22839i	0.255941	−	0.966692i	$-0.417615\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	10.4406i		−	1.23907i	−0.784968	−	0.619537i	$-0.787320\pi$
$71$		−	10.4406i		−	1.23907i	0.784968	−	0.619537i	$-0.212680\pi$
$72$		0			0
$73$			15.7608i			1.84467i	0.386395	+	0.922334i	$0.373720\pi$
$73$			15.7608i			1.84467i	−0.386395	+	0.922334i	$0.626280\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	6.15163	−	10.6549i	0.692112	−	1.19877i	−0.279032	−	0.960282i	$-0.590014\pi$
$79$	6.15163	−	10.6549i	0.692112	−	1.19877i	0.971145	−	0.238492i	$-0.0766530\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.03981	+	6.99715i	−0.443426	+	0.768037i	−0.997941	−	0.0641368i	$-0.979571\pi$
$83$	−4.03981	+	6.99715i	−0.443426	+	0.768037i	0.554515	+	0.832174i	$0.312904\pi$
$84$		0			0
$85$	−2.48067	−	4.29665i	−0.269067	−	0.466037i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.21744			−0.977047			−0.488523	−	0.872551i	$-0.662464\pi$
$89$	−9.21744			−0.977047			−0.488523	+	0.872551i	$0.662464\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	12.6332	−	7.29377i	1.29614	−	0.748325i
$96$		0			0
$97$	−7.00772	−	4.04591i	−0.711527	−	0.410800i	0.100099	−	0.994977i	$-0.468084\pi$
$97$	−7.00772	−	4.04591i	−0.711527	−	0.410800i	−0.811626	+	0.584177i	$0.801417\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.65365	+	6.32831i	−0.363552	+	0.629690i	−0.988543	−	0.150942i	$-0.951769\pi$
$101$	−3.65365	+	6.32831i	−0.363552	+	0.629690i	0.624991	+	0.780632i	$0.285103\pi$
$102$		0			0
$103$	−6.08409	+	3.51265i	−0.599483	+	0.346112i	−0.768838	−	0.639443i	$-0.779165\pi$
$103$	−6.08409	+	3.51265i	−0.599483	+	0.346112i	0.169355	+	0.985555i	$0.445832\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			14.1588i			1.36878i	0.729117	+	0.684389i	$0.239931\pi$
$107$			14.1588i			1.36878i	−0.729117	+	0.684389i	$0.760069\pi$
$108$		0			0
$109$	5.64405			0.540602			0.270301	−	0.962776i	$-0.412877\pi$
$109$	5.64405			0.540602			0.270301	+	0.962776i	$0.412877\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.6411	+	6.72099i	−1.09510	+	0.632258i	−0.934930	−	0.354831i	$-0.884538\pi$
$113$	−11.6411	+	6.72099i	−1.09510	+	0.632258i	−0.160172	+	0.987089i	$0.551205\pi$
$114$		0			0
$115$	−12.6308	−	7.29239i	−1.17783	−	0.680019i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.67752	+	9.83375i	0.516138	+	0.893977i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.50392			−0.313400
$126$		0			0
$127$	−12.7730			−1.13342			−0.566712	−	0.823916i	$-0.691785\pi$
$127$	−12.7730			−1.13342			−0.566712	+	0.823916i	$0.691785\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	6.70890	+	11.6202i	0.586159	+	1.01526i	0.994730	+	0.102531i	$0.0326941\pi$
$131$	6.70890	+	11.6202i	0.586159	+	1.01526i	−0.408570	+	0.912727i	$0.633973\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.79449	−	4.50015i	−0.665928	−	0.384474i	0.128604	−	0.991696i	$-0.458950\pi$
$137$	−7.79449	−	4.50015i	−0.665928	−	0.384474i	−0.794532	+	0.607222i	$0.792284\pi$
$138$		0			0
$139$	−1.54902	+	0.894326i	−0.131386	+	0.0758557i	−0.564252	−	0.825602i	$-0.690836\pi$
$139$	−1.54902	+	0.894326i	−0.131386	+	0.0758557i	0.432866	+	0.901458i	$0.357502\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−19.3293			−1.61640
$144$		0			0
$145$			0.818088i			0.0679385i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.1779	+	6.45358i	−0.915732	+	0.528698i	−0.882271	−	0.470742i	$-0.843986\pi$
$149$	−11.1779	+	6.45358i	−0.915732	+	0.528698i	−0.0334609	+	0.999440i	$0.510653\pi$
$150$		0			0
$151$	6.48364	−	11.2300i	0.527631	−	0.913884i	−0.471850	−	0.881679i	$-0.656414\pi$
$151$	6.48364	−	11.2300i	0.527631	−	0.913884i	0.999481	−	0.0322054i	$-0.0102531\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.12360	−	2.38076i	−0.331216	−	0.191227i
$156$		0			0
$157$	−14.8720	+	8.58638i	−1.18692	+	0.685268i	−0.957605	−	0.288085i	$-0.906981\pi$
$157$	−14.8720	+	8.58638i	−1.18692	+	0.685268i	−0.229314	+	0.973353i	$0.573648\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−5.06214			−0.396497			−0.198249	−	0.980152i	$-0.563525\pi$
$163$	−5.06214			−0.396497			−0.198249	+	0.980152i	$0.563525\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−5.79673	−	10.0402i	−0.448564	−	0.776936i	0.549729	−	0.835343i	$-0.314731\pi$
$167$	−5.79673	−	10.0402i	−0.448564	−	0.776936i	−0.998293	+	0.0584072i	$0.981398\pi$
$168$		0			0
$169$	1.85653	−	3.21561i	0.142810	−	0.247354i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.13346	+	5.42730i	−0.238232	+	0.412630i	−0.960207	−	0.279289i	$-0.909901\pi$
$173$	−3.13346	+	5.42730i	−0.238232	+	0.412630i	0.721975	+	0.691919i	$0.243235\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	14.7418i		−	1.10185i	−0.834554	−	0.550927i	$-0.814274\pi$
$179$		−	14.7418i		−	1.10185i	0.834554	−	0.550927i	$-0.185726\pi$
$180$		0			0
$181$		−	0.0833642i		−	0.00619641i	−0.999995	−	0.00309821i	$-0.999014\pi$
$181$		−	0.0833642i		−	0.00619641i	0.999995	−	0.00309821i	$-0.000986191\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.03932	−	8.72835i	−0.370498	−	0.641721i
$186$		0			0
$187$	−6.84036	−	3.94929i	−0.500217	−	0.288800i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.3672	+	7.71754i	0.967214	+	0.558421i	0.898386	−	0.439207i	$-0.144741\pi$
$191$	13.3672	+	7.71754i	0.967214	+	0.558421i	0.0688282	+	0.997629i	$0.478074\pi$
$192$		0			0
$193$	−10.7779	−	18.6678i	−0.775808	−	1.34374i	−0.934339	−	0.356385i	$-0.884009\pi$
$193$	−10.7779	−	18.6678i	−0.775808	−	1.34374i	0.158532	−	0.987354i	$-0.449324\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	9.88306i		−	0.704139i	−0.935974	−	0.352069i	$-0.885478\pi$
$197$		−	9.88306i		−	0.704139i	0.935974	−	0.352069i	$-0.114522\pi$
$198$		0			0
$199$		−	10.5612i		−	0.748660i	−0.927295	−	0.374330i	$-0.877873\pi$
$199$		−	10.5612i		−	0.748660i	0.927295	−	0.374330i	$-0.122127\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−10.5618	+	18.2936i	−0.737670	+	1.27768i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	11.6118	−	20.1123i	0.803208	−	1.39120i
$210$		0			0
$211$	6.08453	+	10.5387i	0.418876	+	0.725514i	0.995827	−	0.0912645i	$-0.0290909\pi$
$211$	6.08453	+	10.5387i	0.418876	+	0.725514i	−0.576951	+	0.816779i	$0.695758\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−31.0608			−2.11833
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.91452	−	3.41475i	0.397854	−	0.229701i
$222$		0			0
$223$	−0.714485	−	0.412508i	−0.0478455	−	0.0276236i	0.475886	−	0.879507i	$-0.342127\pi$
$223$	−0.714485	−	0.412508i	−0.0478455	−	0.0276236i	−0.523732	+	0.851883i	$0.675461\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	0.166778	−	0.288869i	0.0110695	−	0.0191729i	−0.860438	−	0.509556i	$-0.829810\pi$
$227$	0.166778	−	0.288869i	0.0110695	−	0.0191729i	0.871507	+	0.490383i	$0.163143\pi$
$228$		0			0
$229$	12.4893	−	7.21072i	0.825319	−	0.476498i	−0.0269285	−	0.999637i	$-0.508573\pi$
$229$	12.4893	−	7.21072i	0.825319	−	0.476498i	0.852247	+	0.523139i	$0.175239\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			14.7748i			0.967927i	0.875088	+	0.483964i	$0.160803\pi$
$233$			14.7748i			0.967927i	−0.875088	+	0.483964i	$0.839197\pi$
$234$		0			0
$235$	−32.6619			−2.13063
$236$		0			0
$237$		0			0
$238$		0			0
$239$	22.5339	−	13.0100i	1.45760	−	0.841545i	0.458707	−	0.888588i	$-0.348313\pi$
$239$	22.5339	−	13.0100i	1.45760	−	0.841545i	0.998893	+	0.0470423i	$0.0149795\pi$
$240$		0			0
$241$	1.66295	+	0.960105i	0.107120	+	0.0618458i	0.552603	−	0.833445i	$-0.313635\pi$
$241$	1.66295	+	0.960105i	0.107120	+	0.0618458i	−0.445483	+	0.895290i	$0.646968\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	10.0402	+	17.3901i	0.638841	+	1.10651i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.97663			0.629719			0.314860	−	0.949138i	$-0.398043\pi$
$251$	9.97663			0.629719			0.314860	+	0.949138i	$0.398043\pi$
$252$		0			0
$253$	−23.2193			−1.45978
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.50364	+	12.9967i	0.468064	+	0.810711i	0.999334	−	0.0364915i	$-0.0116182\pi$
$257$	7.50364	+	12.9967i	0.468064	+	0.810711i	−0.531270	+	0.847203i	$0.678285\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	6.11010	+	3.52767i	0.376765	+	0.217525i	0.676410	−	0.736525i	$-0.263535\pi$
$263$	6.11010	+	3.52767i	0.376765	+	0.217525i	−0.299645	+	0.954051i	$0.596868\pi$
$264$		0			0
$265$	2.10084	−	1.21292i	0.129053	−	0.0745090i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−29.7795			−1.81569			−0.907844	−	0.419308i	$-0.862273\pi$
$269$	−29.7795			−1.81569			−0.907844	+	0.419308i	$0.862273\pi$
$270$		0			0
$271$			2.78816i			0.169369i	0.996408	+	0.0846843i	$0.0269882\pi$
$271$			2.78816i			0.169369i	−0.996408	+	0.0846843i	$0.973012\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	15.6424	−	9.03112i	0.943270	−	0.544597i
$276$		0			0
$277$	−6.79074	+	11.7619i	−0.408016	+	0.706705i	−0.994667	−	0.103135i	$-0.967113\pi$
$277$	−6.79074	+	11.7619i	−0.408016	+	0.706705i	0.586651	+	0.809840i	$0.300446\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	3.95777	+	2.28502i	0.236101	+	0.136313i	0.613383	−	0.789785i	$-0.289808\pi$
$281$	3.95777	+	2.28502i	0.236101	+	0.136313i	−0.377283	+	0.926098i	$0.623141\pi$
$282$		0			0
$283$	−17.6685	+	10.2009i	−1.05029	+	0.606383i	−0.922729	−	0.385449i	$-0.874047\pi$
$283$	−17.6685	+	10.2009i	−1.05029	+	0.606383i	−0.127556	+	0.991831i	$0.540713\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−14.2092			−0.835838
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6.41037	+	11.1031i	0.374498	+	0.648649i	0.990252	−	0.139289i	$-0.0444818\pi$
$293$	6.41037	+	11.1031i	0.374498	+	0.648649i	−0.615754	+	0.787939i	$0.711148\pi$
$294$		0			0
$295$	−4.08144	+	7.06926i	−0.237631	+	0.411589i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10.0383	−	17.3868i	0.580529	−	1.00551i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	21.3923i		−	1.22492i
$306$		0			0
$307$		−	1.93411i		−	0.110386i	−0.998476	−	0.0551928i	$-0.982423\pi$
$307$		−	1.93411i		−	0.110386i	0.998476	−	0.0551928i	$-0.0175773\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.04458	−	1.80926i	−0.0592326	−	0.102594i	0.834889	−	0.550419i	$-0.185532\pi$
$311$	−1.04458	−	1.80926i	−0.0592326	−	0.102594i	−0.894121	+	0.447825i	$0.852199\pi$
$312$		0			0
$313$	−19.4066	−	11.2044i	−1.09692	−	0.633309i	−0.161512	−	0.986871i	$-0.551637\pi$
$313$	−19.4066	−	11.2044i	−1.09692	−	0.633309i	−0.935411	+	0.353562i	$0.884970\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.01788	−	1.74237i	−0.169501	−	0.0978614i	0.412850	−	0.910799i	$-0.364534\pi$
$317$	−3.01788	−	1.74237i	−0.169501	−	0.0978614i	−0.582350	+	0.812938i	$0.697867\pi$
$318$		0			0
$319$	0.651207	+	1.12792i	0.0364606	+	0.0631516i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			8.20549i			0.456565i
$324$		0			0
$325$			15.6175i			0.866304i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.28857	−	3.96392i	0.125791	−	0.217877i	−0.796251	−	0.604967i	$-0.793186\pi$
$331$	2.28857	−	3.96392i	0.125791	−	0.217877i	0.922042	+	0.387090i	$0.126520\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−17.2405	+	29.8615i	−0.941951	+	1.63151i
$336$		0			0
$337$	−14.7062	−	25.4720i	−0.801100	−	1.38755i	−0.918893	−	0.394508i	$-0.870915\pi$
$337$	−14.7062	−	25.4720i	−0.801100	−	1.38755i	0.117793	−	0.993038i	$-0.462418\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7.58045			−0.410504
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	17.0245	−	9.82911i	0.913924	−	0.527654i	0.0322323	−	0.999480i	$-0.489738\pi$
$347$	17.0245	−	9.82911i	0.913924	−	0.527654i	0.881692	+	0.471826i	$0.156405\pi$
$348$		0			0
$349$	8.47286	+	4.89181i	0.453542	+	0.261852i	0.709325	−	0.704882i	$-0.249000\pi$
$349$	8.47286	+	4.89181i	0.453542	+	0.261852i	−0.255783	+	0.966734i	$0.582333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	12.5322	−	21.7065i	0.667023	−	1.15532i	−0.311709	−	0.950178i	$-0.600901\pi$
$353$	12.5322	−	21.7065i	0.667023	−	1.15532i	0.978733	−	0.205141i	$-0.0657652\pi$
$354$		0			0
$355$	−26.8532	+	15.5037i	−1.42522	+	0.822850i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			9.35147i			0.493552i	0.969073	+	0.246776i	$0.0793712\pi$
$359$			9.35147i			0.493552i	−0.969073	+	0.246776i	$0.920629\pi$
$360$		0			0
$361$	−5.12609			−0.269794
$362$		0			0
$363$		0			0
$364$		0			0
$365$	40.5367	−	23.4039i	2.12179	−	1.22502i
$366$		0			0
$367$	18.9530	+	10.9425i	0.989337	+	0.571194i	0.905076	−	0.425250i	$-0.139814\pi$
$367$	18.9530	+	10.9425i	0.989337	+	0.571194i	0.0842608	+	0.996444i	$0.473147\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−2.30822	−	3.99795i	−0.119515	−	0.207006i	0.800061	−	0.599919i	$-0.204801\pi$
$373$	−2.30822	−	3.99795i	−0.119515	−	0.207006i	−0.919576	+	0.392913i	$0.871467\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.12613			−0.0579988
$378$		0			0
$379$	−6.22396			−0.319703			−0.159852	−	0.987141i	$-0.551102\pi$
$379$	−6.22396			−0.319703			−0.159852	+	0.987141i	$0.551102\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.9989	−	19.0506i	−0.562015	−	0.973439i	−0.997321	−	0.0731560i	$-0.976693\pi$
$383$	−10.9989	−	19.0506i	−0.562015	−	0.973439i	0.435305	−	0.900283i	$-0.356640\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−8.51109	−	4.91388i	−0.431529	−	0.249144i	0.268469	−	0.963288i	$-0.413482\pi$
$389$	−8.51109	−	4.91388i	−0.431529	−	0.249144i	−0.699998	+	0.714145i	$0.746816\pi$
$390$		0			0
$391$	7.10481	−	4.10197i	0.359306	−	0.207445i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−36.5392			−1.83849
$396$		0			0
$397$		−	5.25762i		−	0.263873i	−0.991258	−	0.131936i	$-0.957881\pi$
$397$		−	5.25762i		−	0.263873i	0.991258	−	0.131936i	$-0.0421194\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.7847	−	8.53594i	0.738312	−	0.426265i	−0.0831432	−	0.996538i	$-0.526496\pi$
$401$	14.7847	−	8.53594i	0.738312	−	0.426265i	0.821455	+	0.570273i	$0.193163\pi$
$402$		0			0
$403$	3.27722	−	5.67631i	0.163250	−	0.282757i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−13.8957	−	8.02270i	−0.688786	−	0.397671i
$408$		0			0
$409$	−16.9484	+	9.78516i	−0.838044	+	0.483845i	−0.856599	−	0.515983i	$-0.827427\pi$
$409$	−16.9484	+	9.78516i	−0.838044	+	0.483845i	0.0185546	+	0.999828i	$0.494094\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	23.9955			1.17789
$416$		0			0
$417$		0			0
$418$		0			0
$419$	10.3073	+	17.8529i	0.503547	+	0.872169i	0.999992	+	0.00410056i	$0.00130525\pi$
$419$	10.3073	+	17.8529i	0.503547	+	0.872169i	−0.496445	+	0.868068i	$0.665361\pi$
$420$		0			0
$421$	0.704748	−	1.22066i	0.0343473	−	0.0594913i	−0.848341	−	0.529451i	$-0.822398\pi$
$421$	0.704748	−	1.22066i	0.0343473	−	0.0594913i	0.882688	+	0.469959i	$0.155731\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.19091	+	5.52682i	−0.154782	+	0.268090i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$			13.4714i			0.648894i	0.945904	+	0.324447i	$0.105178\pi$
$431$			13.4714i			0.648894i	−0.945904	+	0.324447i	$0.894822\pi$
$432$		0			0
$433$		−	12.9356i		−	0.621646i	−0.950468	−	0.310823i	$-0.899395\pi$
$433$		−	12.9356i		−	0.621646i	0.950468	−	0.310823i	$-0.100605\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	12.0608	+	20.8899i	0.576944	+	0.999297i
$438$		0			0
$439$	−8.75023	−	5.05195i	−0.417626	−	0.241116i	0.276435	−	0.961033i	$-0.410847\pi$
$439$	−8.75023	−	5.05195i	−0.417626	−	0.241116i	−0.694061	+	0.719916i	$0.744180\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−25.1220	−	14.5042i	−1.19358	−	0.689115i	−0.234466	−	0.972124i	$-0.575334\pi$
$443$	−25.1220	−	14.5042i	−1.19358	−	0.689115i	−0.959117	+	0.283009i	$0.908667\pi$
$444$		0			0
$445$	13.6873	+	23.7072i	0.648842	+	1.12383i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	7.94881i		−	0.375127i	−0.982252	−	0.187564i	$-0.939941\pi$
$449$		−	7.94881i		−	0.375127i	0.982252	−	0.187564i	$-0.0600591\pi$
$450$		0			0
$451$			33.6293i			1.58354i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6.98084	−	12.0912i	0.326550	−	0.565601i	−0.655275	−	0.755391i	$-0.727447\pi$
$457$	6.98084	−	12.0912i	0.326550	−	0.565601i	0.981825	+	0.189789i	$0.0607805\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−16.4030	+	28.4108i	−0.763964	+	1.32322i	0.176829	+	0.984242i	$0.443416\pi$
$461$	−16.4030	+	28.4108i	−0.763964	+	1.32322i	−0.940793	+	0.338983i	$0.889917\pi$
$462$		0			0
$463$	−13.8812	−	24.0429i	−0.645112	−	1.11737i	−0.984276	−	0.176640i	$-0.943477\pi$
$463$	−13.8812	−	24.0429i	−0.645112	−	1.11737i	0.339163	−	0.940727i	$-0.389856\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	22.8621			1.05793			0.528966	−	0.848643i	$-0.322580\pi$
$467$	22.8621			1.05793			0.528966	+	0.848643i	$0.322580\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−42.8245	+	24.7247i	−1.96907	+	1.13684i
$474$		0			0
$475$	−16.2502	−	9.38204i	−0.745609	−	0.430478i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.21212	−	2.09946i	0.0553834	−	0.0959269i	−0.837004	−	0.547196i	$-0.815695\pi$
$479$	1.21212	−	2.09946i	0.0553834	−	0.0959269i	0.892388	+	0.451269i	$0.149029\pi$
$480$		0			0
$481$	12.0149	−	6.93683i	0.547834	−	0.316292i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			24.0317i			1.09122i
$486$		0			0
$487$	−10.3930			−0.470952			−0.235476	−	0.971880i	$-0.575665\pi$
$487$	−10.3930			−0.470952			−0.235476	+	0.971880i	$0.575665\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.93014	−	1.69172i	0.132235	−	0.0763462i	−0.432423	−	0.901671i	$-0.642341\pi$
$491$	2.93014	−	1.69172i	0.132235	−	0.0763462i	0.564658	+	0.825325i	$0.309008\pi$
$492$		0			0
$493$	−0.398522	−	0.230087i	−0.0179485	−	0.0103626i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−19.7801	−	34.2602i	−0.885481	−	1.53370i	−0.845162	−	0.534511i	$-0.820496\pi$
$499$	−19.7801	−	34.2602i	−0.885481	−	1.53370i	−0.0403188	−	0.999187i	$-0.512837\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	14.5476			0.648645			0.324323	−	0.945947i	$-0.394864\pi$
$503$	14.5476			0.648645			0.324323	+	0.945947i	$0.394864\pi$
$504$		0			0
$505$	21.7018			0.965717
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10.1958	+	17.6596i	0.451921	+	0.782750i	0.998505	−	0.0546542i	$-0.0174057\pi$
$509$	10.1958	+	17.6596i	0.451921	+	0.782750i	−0.546585	+	0.837404i	$0.684072\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	18.0690	+	10.4322i	0.796216	+	0.459696i
$516$		0			0
$517$	−45.0319	+	25.9992i	−1.98050	+	1.14344i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	15.5024			0.679175			0.339587	−	0.940575i	$-0.389713\pi$
$521$	15.5024			0.679175			0.339587	+	0.940575i	$0.389713\pi$
$522$		0			0
$523$		−	10.8079i		−	0.472595i	−0.971681	−	0.236298i	$-0.924066\pi$
$523$		−	10.8079i		−	0.472595i	0.971681	−	0.236298i	$-0.0759340\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.31952	−	1.33918i	0.101040	−	0.0583355i
$528$		0			0
$529$	0.558476	−	0.967309i	0.0242816	−	0.0420569i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−25.1819	−	14.5388i	−1.09075	−	0.629745i
$534$		0			0
$535$	36.4162	−	21.0249i	1.57441	−	0.908986i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	17.5871			0.756130			0.378065	−	0.925779i	$-0.376590\pi$
$541$	17.5871			0.756130			0.378065	+	0.925779i	$0.376590\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.38108	−	14.5165i	−0.359006	−	0.621817i
$546$		0			0
$547$	−5.72451	+	9.91513i	−0.244762	+	0.423940i	−0.962065	−	0.272821i	$-0.912043\pi$
$547$	−5.72451	+	9.91513i	−0.244762	+	0.423940i	0.717303	+	0.696762i	$0.245377\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	0.676511	−	1.17175i	0.0288203	−	0.0499183i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	38.0080i		−	1.61045i	−0.592968	−	0.805226i	$-0.702044\pi$
$557$		−	38.0080i		−	1.61045i	0.592968	−	0.805226i	$-0.297956\pi$
$558$		0			0
$559$		−	42.7565i		−	1.80841i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8.88438	+	15.3882i	0.374432	+	0.648535i	0.990242	−	0.139360i	$-0.0445044\pi$
$563$	8.88438	+	15.3882i	0.374432	+	0.648535i	−0.615810	+	0.787895i	$0.711171\pi$
$564$		0			0
$565$	34.5727	+	19.9605i	1.45448	+	0.839746i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−33.7404	−	19.4801i	−1.41447	−	0.816646i	−0.418667	−	0.908140i	$-0.637503\pi$
$569$	−33.7404	−	19.4801i	−1.41447	−	0.816646i	−0.995806	+	0.0914936i	$0.970836\pi$
$570$		0			0
$571$	−8.45245	−	14.6401i	−0.353724	−	0.612668i	0.633175	−	0.774009i	$-0.281752\pi$
$571$	−8.45245	−	14.6401i	−0.353724	−	0.612668i	−0.986899	+	0.161341i	$0.948418\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			18.7605i			0.782368i
$576$		0			0
$577$		−	47.2653i		−	1.96768i	−0.179050	−	0.983840i	$-0.557302\pi$
$577$		−	47.2653i		−	1.96768i	0.179050	−	0.983840i	$-0.442698\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.93099	−	3.34458i	0.0799736	−	0.138518i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	11.6343	−	20.1513i	0.480200	−	0.831731i	−0.519542	−	0.854445i	$-0.673897\pi$
$587$	11.6343	−	20.1513i	0.480200	−	0.831731i	0.999742	+	0.0227138i	$0.00723065\pi$
$588$		0			0
$589$	3.93750	+	6.81995i	0.162242	+	0.281011i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	37.1924			1.52731			0.763654	−	0.645626i	$-0.223404\pi$
$593$	37.1924			1.52731			0.763654	+	0.645626i	$0.223404\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	27.9591	−	16.1422i	1.14238	−	0.659552i	0.195359	−	0.980732i	$-0.437413\pi$
$599$	27.9591	−	16.1422i	1.14238	−	0.659552i	0.947018	+	0.321180i	$0.104079\pi$
$600$		0			0
$601$	14.7559	+	8.51933i	0.601906	+	0.347511i	0.769791	−	0.638296i	$-0.220360\pi$
$601$	14.7559	+	8.51933i	0.601906	+	0.347511i	−0.167885	+	0.985807i	$0.553694\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	16.8615	−	29.2051i	0.685519	−	1.18735i
$606$		0			0
$607$	8.44393	−	4.87510i	0.342728	−	0.197874i	−0.318749	−	0.947839i	$-0.603263\pi$
$607$	8.44393	−	4.87510i	0.342728	−	0.197874i	0.661478	+	0.749965i	$0.269930\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	44.9604i		−	1.81890i
$612$		0			0
$613$	13.7266			0.554414			0.277207	−	0.960810i	$-0.410591\pi$
$613$	13.7266			0.554414			0.277207	+	0.960810i	$0.410591\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	2.84301	−	1.64141i	0.114455	−	0.0660807i	−0.441680	−	0.897173i	$-0.645617\pi$
$617$	2.84301	−	1.64141i	0.114455	−	0.0660807i	0.556135	+	0.831092i	$0.312284\pi$
$618$		0			0
$619$	−14.9907	−	8.65490i	−0.602528	−	0.347870i	0.167507	−	0.985871i	$-0.446428\pi$
$619$	−14.9907	−	8.65490i	−0.602528	−	0.347870i	−0.770036	+	0.638001i	$0.779762\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	14.7536	+	25.5539i	0.590142	+	1.02216i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.66923			0.226047
$630$		0			0
$631$	−6.27821			−0.249932			−0.124966	−	0.992161i	$-0.539882\pi$
$631$	−6.27821			−0.249932			−0.124966	+	0.992161i	$0.539882\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.9672	+	32.8522i	0.752691	+	1.30370i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−17.9788	−	10.3801i	−0.710120	−	0.409988i	0.100986	−	0.994888i	$-0.467800\pi$
$641$	−17.9788	−	10.3801i	−0.710120	−	0.409988i	−0.811105	+	0.584900i	$0.801134\pi$
$642$		0			0
$643$	17.2553	−	9.96236i	0.680483	−	0.392877i	−0.119554	−	0.992828i	$-0.538146\pi$
$643$	17.2553	−	9.96236i	0.680483	−	0.392877i	0.800037	+	0.599950i	$0.204813\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	29.5340			1.16110			0.580551	−	0.814224i	$-0.302837\pi$
$647$	29.5340			1.16110			0.580551	+	0.814224i	$0.302837\pi$
$648$		0			0
$649$			12.9955i			0.510118i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13.7914	−	7.96249i	0.539701	−	0.311596i	−0.205257	−	0.978708i	$-0.565803\pi$
$653$	13.7914	−	7.96249i	0.539701	−	0.311596i	0.744958	+	0.667112i	$0.232470\pi$
$654$		0			0
$655$	19.9246	−	34.5105i	0.778520	−	1.34844i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−2.80283	−	1.61822i	−0.109183	−	0.0630368i	0.444414	−	0.895821i	$-0.353412\pi$
$659$	−2.80283	−	1.61822i	−0.109183	−	0.0630368i	−0.553597	+	0.832785i	$0.686745\pi$
$660$		0			0
$661$	−7.71194	+	4.45249i	−0.299960	+	0.173182i	−0.642425	−	0.766349i	$-0.722072\pi$
$661$	−7.71194	+	4.45249i	−0.299960	+	0.173182i	0.342465	+	0.939531i	$0.388738\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.35277			−0.0523793
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−17.0285	−	29.4943i	−0.657379	−	1.13861i
$672$		0			0
$673$	−13.2311	+	22.9169i	−0.510021	+	0.883382i	0.489912	+	0.871772i	$0.337029\pi$
$673$	−13.2311	+	22.9169i	−0.510021	+	0.883382i	−0.999933	+	0.0116101i	$0.996304\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.46424	+	7.73229i	−0.171575	+	0.297176i	−0.938971	−	0.343997i	$-0.888219\pi$
$677$	−4.46424	+	7.73229i	−0.171575	+	0.297176i	0.767396	+	0.641174i	$0.221552\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			37.8628i			1.44878i	0.689390	+	0.724390i	$0.257879\pi$
$683$			37.8628i			1.44878i	−0.689390	+	0.724390i	$0.742121\pi$
$684$		0			0
$685$			26.7298i			1.02129i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.66963	+	2.89189i	0.0636080	+	0.110172i
$690$		0			0
$691$	4.94211	+	2.85333i	0.188007	+	0.108546i	0.591049	−	0.806636i	$-0.298714\pi$
$691$	4.94211	+	2.85333i	0.188007	+	0.108546i	−0.403042	+	0.915181i	$0.632047\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4.60039	+	2.65604i	0.174503	+	0.100749i
$696$		0			0
$697$	−5.94103	−	10.2902i	−0.225032	−	0.389768i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			8.19949i			0.309690i	0.987939	+	0.154845i	$0.0494879\pi$
$701$			8.19949i			0.309690i	−0.987939	+	0.154845i	$0.950512\pi$
$702$		0			0
$703$			16.6689i			0.628679i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−10.0757	+	17.4517i	−0.378402	+	0.655412i	−0.990830	−	0.135115i	$-0.956860\pi$
$709$	−10.0757	+	17.4517i	−0.378402	+	0.655412i	0.612428	+	0.790527i	$0.290193\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.93676	−	6.81866i	0.147433	−	0.255361i
$714$		0			0
$715$	28.7028	+	49.7147i	1.07342	+	1.85923i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−51.1991			−1.90940			−0.954702	−	0.297563i	$-0.903826\pi$
$719$	−51.1991			−1.90940			−0.954702	+	0.297563i	$0.903826\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.911330	−	0.526157i	0.0338460	−	0.0195410i
$726$		0			0
$727$	−13.7848	−	7.95865i	−0.511249	−	0.295170i	0.222098	−	0.975024i	$-0.428710\pi$
$727$	−13.7848	−	7.95865i	−0.511249	−	0.295170i	−0.733347	+	0.679854i	$0.762043\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8.73584	−	15.1309i	0.323107	−	0.559637i
$732$		0			0
$733$	3.67216	−	2.12012i	0.135634	−	0.0783086i	−0.430647	−	0.902520i	$-0.641715\pi$
$733$	3.67216	−	2.12012i	0.135634	−	0.0783086i	0.566282	+	0.824212i	$0.308381\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			54.8946i			2.02207i
$738$		0			0
$739$	28.3669			1.04349			0.521747	−	0.853100i	$-0.325280\pi$
$739$	28.3669			1.04349			0.521747	+	0.853100i	$0.325280\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−21.8850	+	12.6353i	−0.802884	+	0.463545i	−0.844479	−	0.535589i	$-0.820090\pi$
$743$	−21.8850	+	12.6353i	−0.802884	+	0.463545i	0.0415945	+	0.999135i	$0.486756\pi$
$744$		0			0
$745$	33.1971	+	19.1664i	1.21625	+	0.702201i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−23.7730	−	41.1761i	−0.867490	−	1.50254i	−0.864554	−	0.502540i	$-0.832399\pi$
$751$	−23.7730	−	41.1761i	−0.867490	−	1.50254i	−0.00293597	−	0.999996i	$-0.500935\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−38.5113			−1.40157
$756$		0			0
$757$	37.3922			1.35904			0.679521	−	0.733656i	$-0.262188\pi$
$757$	37.3922			1.35904			0.679521	+	0.733656i	$0.262188\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4.12142	−	7.13850i	−0.149401	−	0.258770i	0.781605	−	0.623774i	$-0.214401\pi$
$761$	−4.12142	−	7.13850i	−0.149401	−	0.258770i	−0.931006	+	0.365003i	$0.881068\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.73114	−	5.61827i	−0.351371	−	0.202864i
$768$		0			0
$769$	20.2182	−	11.6730i	0.729086	−	0.420938i	−0.0890020	−	0.996031i	$-0.528368\pi$
$769$	20.2182	−	11.6730i	0.729086	−	0.420938i	0.818088	+	0.575094i	$0.195034\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−34.4402			−1.23873			−0.619364	−	0.785104i	$-0.712609\pi$
$773$	−34.4402			−1.23873			−0.619364	+	0.785104i	$0.712609\pi$
$774$		0			0
$775$			6.12479i			0.220009i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	30.2555	−	17.4680i	1.08402	−	0.625857i
$780$		0			0
$781$	−24.6822	+	42.7508i	−0.883199	+	1.52975i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	44.1682	+	25.5005i	1.57643	+	0.910152i
$786$		0			0
$787$	7.19975	−	4.15678i	0.256643	−	0.148173i	−0.366159	−	0.930552i	$-0.619327\pi$
$787$	7.19975	−	4.15678i	0.256643	−	0.148173i	0.622802	+	0.782379i	$0.285994\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	29.4474			1.04571
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.426036	−	0.737916i	−0.0150910	−	0.0261383i	0.858381	−	0.513012i	$-0.171470\pi$
$797$	−0.426036	−	0.737916i	−0.0150910	−	0.0261383i	−0.873472	+	0.486874i	$0.838137\pi$
$798$		0			0
$799$	9.18614	−	15.9109i	0.324982	−	0.562886i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	37.2595	−	64.5354i	1.31486	−	2.27740i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			36.4400i			1.28116i	0.767891	+	0.640581i	$0.221306\pi$
$809$			36.4400i			1.28116i	−0.767891	+	0.640581i	$0.778694\pi$
$810$		0			0
$811$			1.08986i			0.0382702i	0.999817	+	0.0191351i	$0.00609126\pi$
$811$			1.08986i			0.0382702i	−0.999817	+	0.0191351i	$0.993909\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	7.51697	+	13.0198i	0.263308	+	0.456063i
$816$		0			0
$817$	44.4885	+	25.6855i	1.55646	+	0.898620i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−20.9748	−	12.1098i	−0.732025	−	0.422635i	0.0871374	−	0.996196i	$-0.472228\pi$
$821$	−20.9748	−	12.1098i	−0.732025	−	0.422635i	−0.819163	+	0.573561i	$0.805561\pi$
$822$		0			0
$823$	−2.85592	−	4.94660i	−0.0995512	−	0.172428i	0.811948	−	0.583730i	$-0.198407\pi$
$823$	−2.85592	−	4.94660i	−0.0995512	−	0.172428i	−0.911499	+	0.411302i	$0.865074\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			36.4579i			1.26777i	0.773429	+	0.633883i	$0.218540\pi$
$827$			36.4579i			1.26777i	−0.773429	+	0.633883i	$0.781460\pi$
$828$		0			0
$829$		−	0.575352i		−	0.0199828i	−0.999950	−	0.00999140i	$-0.996820\pi$
$829$		−	0.575352i		−	0.0199828i	0.999950	−	0.00999140i	$-0.00318041\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−17.2156	+	29.8183i	−0.595770	+	1.03190i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	23.9341	−	41.4550i	0.826295	−	1.43119i	−0.0746300	−	0.997211i	$-0.523778\pi$
$839$	23.9341	−	41.4550i	0.826295	−	1.43119i	0.900925	−	0.433974i	$-0.142889\pi$
$840$		0			0
$841$	−14.4621	−	25.0490i	−0.498692	−	0.863759i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−11.0274			−0.379352
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	14.4329	−	8.33286i	0.494755	−	0.285647i
$852$		0			0
$853$	40.5393	+	23.4054i	1.38804	+	0.801385i	0.993094	−	0.117320i	$-0.0374303\pi$
$853$	40.5393	+	23.4054i	1.38804	+	0.801385i	0.394945	+	0.918705i	$0.370764\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.78220	−	8.28302i	0.163357	−	0.282943i	−0.772714	−	0.634755i	$-0.781101\pi$
$857$	4.78220	−	8.28302i	0.163357	−	0.282943i	0.936071	+	0.351812i	$0.114434\pi$
$858$		0			0
$859$	−4.68311	+	2.70379i	−0.159786	+	0.0922523i	−0.577761	−	0.816206i	$-0.696073\pi$
$859$	−4.68311	+	2.70379i	−0.159786	+	0.0922523i	0.417975	+	0.908459i	$0.362740\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			41.0383i			1.39696i	0.715630	+	0.698480i	$0.246140\pi$
$863$			41.0383i			1.39696i	−0.715630	+	0.698480i	$0.753860\pi$
$864$		0			0
$865$	18.6120			0.632826
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−50.3777	+	29.0856i	−1.70895	+	0.986661i
$870$		0			0
$871$	−41.1056	−	23.7323i	−1.39281	−	0.804139i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	7.32509	+	12.6874i	0.247351	+	0.428424i	0.962790	−	0.270251i	$-0.0871067\pi$
$877$	7.32509	+	12.6874i	0.247351	+	0.428424i	−0.715439	+	0.698675i	$0.753773\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	44.8295			1.51034			0.755172	−	0.655527i	$-0.227553\pi$
$881$	44.8295			1.51034			0.755172	+	0.655527i	$0.227553\pi$
$882$		0			0
$883$	33.8527			1.13923			0.569617	−	0.821910i	$-0.307091\pi$
$883$	33.8527			1.13923			0.569617	+	0.821910i	$0.307091\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−13.3422	−	23.1093i	−0.447987	−	0.775936i	0.550268	−	0.834988i	$-0.314525\pi$
$887$	−13.3422	−	23.1093i	−0.447987	−	0.775936i	−0.998255	+	0.0590523i	$0.981192\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	46.7817	+	27.0094i	1.56549	+	0.903837i
$894$		0			0
$895$	−37.9157	+	21.8907i	−1.26738	+	0.731724i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.441640			−0.0147295
$900$		0			0
$901$			1.36453i			0.0454592i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.214412	+	0.123791i	−0.00712729	+	0.00411494i
$906$		0			0
$907$	7.97211	−	13.8081i	0.264710	−	0.458490i	−0.702778	−	0.711409i	$-0.748057\pi$
$907$	7.97211	−	13.8081i	0.264710	−	0.458490i	0.967487	+	0.252919i	$0.0813906\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−40.9207	−	23.6256i	−1.35576	−	0.782750i	−0.366713	−	0.930334i	$-0.619517\pi$
$911$	−40.9207	−	23.6256i	−1.35576	−	0.782750i	−0.989050	+	0.147584i	$0.952850\pi$
$912$		0			0
$913$	33.0833	−	19.1007i	1.09490	−	0.632140i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−29.6325			−0.977486			−0.488743	−	0.872428i	$-0.662544\pi$
$919$	−29.6325			−0.977486			−0.488743	+	0.872428i	$0.662544\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−21.3415	−	36.9645i	−0.702463	−	1.21670i
$924$		0			0
$925$	−6.48212	+	11.2274i	−0.213131	+	0.369153i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−16.6186	+	28.7842i	−0.545238	+	0.944380i	0.453354	+	0.891331i	$0.350227\pi$
$929$	−16.6186	+	28.7842i	−0.545238	+	0.944380i	−0.998592	+	0.0530496i	$0.983106\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$			23.4578i			0.767152i
$936$		0			0
$937$			23.8190i			0.778134i	0.921209	+	0.389067i	$0.127203\pi$
$937$			23.8190i			0.778134i	−0.921209	+	0.389067i	$0.872797\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	27.1201	+	46.9734i	0.884091	+	1.53129i	0.846752	+	0.531988i	$0.178555\pi$
$941$	27.1201	+	46.9734i	0.884091	+	1.53129i	0.0373389	+	0.999303i	$0.488112\pi$
$942$		0			0
$943$	−30.2498	−	17.4647i	−0.985069	−	0.568730i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18.2427	+	10.5324i	0.592807	+	0.342257i	0.766207	−	0.642594i	$-0.222142\pi$
$947$	18.2427	+	10.5324i	0.592807	+	0.342257i	−0.173399	+	0.984852i	$0.555475\pi$
$948$		0			0
$949$	32.2164	+	55.8005i	1.04579	+	1.81136i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			4.50028i			0.145778i	0.997340	+	0.0728892i	$0.0232219\pi$
$953$			4.50028i			0.145778i	−0.997340	+	0.0728892i	$0.976778\pi$
$954$		0			0
$955$		−	45.8403i		−	1.48336i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−14.2148	+	24.6207i	−0.458541	+	0.794216i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−32.0090	+	55.4411i	−1.03040	+	1.78471i
$966$		0			0
$967$	−10.8811	−	18.8466i	−0.349912	−	0.606065i	0.636322	−	0.771424i	$-0.280455\pi$
$967$	−10.8811	−	18.8466i	−0.349912	−	0.606065i	−0.986233	+	0.165359i	$0.947122\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−47.0443			−1.50972			−0.754862	−	0.655884i	$-0.772296\pi$
$971$	−47.0443			−1.50972			−0.754862	+	0.655884i	$0.772296\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−21.7766	+	12.5727i	−0.696695	+	0.402237i	−0.806115	−	0.591758i	$-0.798434\pi$
$977$	−21.7766	+	12.5727i	−0.696695	+	0.402237i	0.109420	+	0.993996i	$0.465101\pi$
$978$		0			0
$979$	37.7423	+	21.7905i	1.20625	+	0.696429i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18.1071	+	31.3624i	−0.577527	+	1.00031i	0.418235	+	0.908339i	$0.362649\pi$
$983$	−18.1071	+	31.3624i	−0.577527	+	1.00031i	−0.995762	+	0.0919674i	$0.970684\pi$
$984$		0			0
$985$	−25.4191	+	14.6757i	−0.809921	+	0.467608i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	51.3612i		−	1.63319i
$990$		0			0
$991$	18.6554			0.592608			0.296304	−	0.955094i	$-0.404246\pi$
$991$	18.6554			0.592608			0.296304	+	0.955094i	$0.404246\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−27.1632	+	15.6827i	−0.861131	+	0.497174i
$996$		0			0
$997$	−15.1413	−	8.74181i	−0.479528	−	0.276856i	0.240691	−	0.970602i	$-0.422626\pi$
$997$	−15.1413	−	8.74181i	−0.479528	−	0.276856i	−0.720220	+	0.693746i	$0.755959\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.x.b.4409.2		16
3.2	odd	2		1764.2.x.b.1469.5		16
7.2	even	3		5292.2.w.b.521.2		16
7.3	odd	6		5292.2.bm.a.4625.2		16
7.4	even	3		756.2.bm.a.89.7		16
7.5	odd	6		756.2.w.a.521.7		16
7.6	odd	2		5292.2.x.a.4409.7		16
9.4	even	3		1764.2.x.a.293.4		16
9.5	odd	6		5292.2.x.a.881.7		16
21.2	odd	6		1764.2.w.b.1109.6		16
21.5	even	6		252.2.w.a.101.3	yes	16
21.11	odd	6		252.2.bm.a.173.1	yes	16
21.17	even	6		1764.2.bm.a.1685.8		16
21.20	even	2		1764.2.x.a.1469.4		16
28.11	odd	6		3024.2.df.d.1601.7		16
28.19	even	6		3024.2.ca.d.2033.7		16
63.4	even	3		252.2.w.a.5.3	✓	16
63.5	even	6		756.2.bm.a.17.7		16
63.11	odd	6		2268.2.t.a.2105.7		16
63.13	odd	6		1764.2.x.b.293.5		16
63.23	odd	6		5292.2.bm.a.2285.2		16
63.25	even	3		2268.2.t.b.2105.2		16
63.31	odd	6		1764.2.w.b.509.6		16
63.32	odd	6		756.2.w.a.341.7		16
63.40	odd	6		252.2.bm.a.185.1	yes	16
63.41	even	6	inner	5292.2.x.b.881.2		16
63.47	even	6		2268.2.t.b.1781.2		16
63.58	even	3		1764.2.bm.a.1697.8		16
63.59	even	6		5292.2.w.b.1097.2		16
63.61	odd	6		2268.2.t.a.1781.7		16
84.11	even	6		1008.2.df.d.929.8		16
84.47	odd	6		1008.2.ca.d.353.6		16
252.67	odd	6		1008.2.ca.d.257.6		16
252.95	even	6		3024.2.ca.d.2609.7		16
252.103	even	6		1008.2.df.d.689.8		16
252.131	odd	6		3024.2.df.d.17.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.w.a.5.3	✓	16	63.4	even	3
252.2.w.a.101.3	yes	16	21.5	even	6
252.2.bm.a.173.1	yes	16	21.11	odd	6
252.2.bm.a.185.1	yes	16	63.40	odd	6
756.2.w.a.341.7		16	63.32	odd	6
756.2.w.a.521.7		16	7.5	odd	6
756.2.bm.a.17.7		16	63.5	even	6
756.2.bm.a.89.7		16	7.4	even	3
1008.2.ca.d.257.6		16	252.67	odd	6
1008.2.ca.d.353.6		16	84.47	odd	6
1008.2.df.d.689.8		16	252.103	even	6
1008.2.df.d.929.8		16	84.11	even	6
1764.2.w.b.509.6		16	63.31	odd	6
1764.2.w.b.1109.6		16	21.2	odd	6
1764.2.x.a.293.4		16	9.4	even	3
1764.2.x.a.1469.4		16	21.20	even	2
1764.2.x.b.293.5		16	63.13	odd	6
1764.2.x.b.1469.5		16	3.2	odd	2
1764.2.bm.a.1685.8		16	21.17	even	6
1764.2.bm.a.1697.8		16	63.58	even	3
2268.2.t.a.1781.7		16	63.61	odd	6
2268.2.t.a.2105.7		16	63.11	odd	6
2268.2.t.b.1781.2		16	63.47	even	6
2268.2.t.b.2105.2		16	63.25	even	3
3024.2.ca.d.2033.7		16	28.19	even	6
3024.2.ca.d.2609.7		16	252.95	even	6
3024.2.df.d.17.7		16	252.131	odd	6
3024.2.df.d.1601.7		16	28.11	odd	6
5292.2.w.b.521.2		16	7.2	even	3
5292.2.w.b.1097.2		16	63.59	even	6
5292.2.x.a.881.7		16	9.5	odd	6
5292.2.x.a.4409.7		16	7.6	odd	2
5292.2.x.b.881.2		16	63.41	even	6	inner
5292.2.x.b.4409.2		16	1.1	even	1	trivial
5292.2.bm.a.2285.2		16	63.23	odd	6
5292.2.bm.a.4625.2		16	7.3	odd	6

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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

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Newspace parameters

Newform invariants

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$q$ -expansion

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	5292.2.x.b.4409.2

Level	$5292$
Weight	$2$
Character	5292.4409
Analytic conductor	$42.257$
Analytic rank	$0$
Dimension	$16$
Inner twists	$2$