Embedded newform 403.2.ba.a.6.16

• Label: 403.2.ba.a.6.16
• Level: $403$
• Weight: $2$
• Character: 403.6
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.ba (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(35\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			6.16
Character	\(\chi\)	\(=\)	403.6
Dual form			403.2.ba.a.336.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.135765 - 0.506680i) q^{2} -2.82155i q^{3} +(1.49376 - 0.862422i) q^{4} +(0.266274 + 0.993746i) q^{5} +(-1.42962 + 0.383067i) q^{6} +(-2.70478 + 0.724745i) q^{7} +(-1.38160 - 1.38160i) q^{8} -4.96115 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 12 q^{4} - 2 q^{5} + 6 q^{6} + 12 q^{7} - 10 q^{8} - 124 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.135765	−	0.506680i	−0.0960000	−	0.358277i	0.901169	−	0.433468i	\(-0.142710\pi\)
							−0.997169	+	0.0751906i	\(0.976043\pi\)
\(3\)		−	2.82155i		−	1.62902i	−0.580147	−	0.814511i	\(-0.697005\pi\)
							0.580147	−	0.814511i	\(-0.302995\pi\)
\(5\)	0.266274	+	0.993746i	0.119081	+	0.444417i	0.999560	−	0.0296690i	\(-0.00944531\pi\)
							−0.880479	+	0.474086i	\(0.842779\pi\)
\(7\)	−2.70478	+	0.724745i	−1.02231	+	0.273928i	−0.730765	−	0.682629i	\(-0.760837\pi\)
							−0.291547	+	0.956557i	\(0.594170\pi\)
\(11\)	−1.71234	−	0.458819i	−0.516289	−	0.138339i	−0.00873958	−	0.999962i	\(-0.502782\pi\)
							−0.507550	+	0.861623i	\(0.669449\pi\)
\(13\)	−3.47369	−	0.966163i	−0.963429	−	0.267965i
							\(17\)	1.52040	−	2.63341i	0.368751	−	0.638696i	−0.620619	−	0.784112i	\(-0.713119\pi\)
0.989371	+	0.145416i	\(0.0464521\pi\)
\(19\)	−0.0648041	−	0.241852i	−0.0148671	−	0.0554847i	0.958094	−	0.286455i	\(-0.0924768\pi\)
							−0.972961	+	0.230970i	\(0.925810\pi\)
\(23\)	2.63588	−	4.56549i	0.549620	−	0.951970i	−0.448680	−	0.893692i	\(-0.648106\pi\)
							0.998300	−	0.0582775i	\(-0.0185608\pi\)
\(29\)	7.26207	+	4.19276i	1.34853	+	0.778575i	0.988042	−	0.154188i	\(-0.0492761\pi\)
							0.360490	+	0.932763i	\(0.382609\pi\)
\(31\)	−5.43271	+	1.21888i	−0.975743	+	0.218918i
							\(37\)	6.63337	−	6.63337i	1.09052	−	1.09052i	0.0950461	−	0.995473i	\(-0.469700\pi\)
0.995473	−	0.0950461i	\(-0.0302998\pi\)
\(41\)	5.39542	+	1.44570i	0.842623	+	0.225780i	0.654213	−	0.756310i	\(-0.273000\pi\)
							0.188410	+	0.982090i	\(0.439667\pi\)
\(43\)	−0.862836	+	1.49448i	−0.131581	+	0.227906i	−0.924286	−	0.381700i	\(-0.875339\pi\)
							0.792705	+	0.609605i	\(0.208672\pi\)
\(47\)	−1.73502	−	1.73502i	−0.253079	−	0.253079i	0.569153	−	0.822232i	\(-0.307271\pi\)
							−0.822232	+	0.569153i	\(0.807271\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ba.a.6.16	✓	140
13.11	odd	12		403.2.bf.a.37.16	yes	140
31.26	odd	6		403.2.bf.a.305.16	yes	140
403.336	even	12	inner	403.2.ba.a.336.16	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.16	✓	140	1.1	even	1	trivial
403.2.ba.a.336.16	yes	140	403.336	even	12	inner
403.2.bf.a.37.16	yes	140	13.11	odd	12
403.2.bf.a.305.16	yes	140	31.26	odd	6