Embedded newform 403.2.r.a.342.8

• Label: 403.2.r.a.342.8
• Level: $403$
• Weight: $2$
• Character: 403.342
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $68$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			342.8
Character	\(\chi\)	\(=\)	403.342
Dual form			403.2.r.a.218.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.56202 - 0.901835i) q^{2} +(0.553992 - 0.959542i) q^{3} +(0.626612 + 1.08532i) q^{4} -0.771735i q^{5} +(-1.73070 + 0.999218i) q^{6} +(3.80672 - 2.19781i) q^{7} +1.34694i q^{8} +(0.886187 + 1.53492i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 32 q^{4} - 34 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(1\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.56202	−	0.901835i	−1.10452	−	0.637694i	−0.167114	−	0.985938i	\(-0.553445\pi\)
							−0.937404	+	0.348244i	\(0.886778\pi\)
\(3\)	0.553992	−	0.959542i	0.319847	−	0.553992i	−0.660609	−	0.750730i	\(-0.729702\pi\)
							0.980456	+	0.196739i	\(0.0630351\pi\)
\(5\)		−	0.771735i		−	0.345130i	−0.984998	−	0.172565i	\(-0.944794\pi\)
							0.984998	−	0.172565i	\(-0.0552055\pi\)
\(7\)	3.80672	−	2.19781i	1.43881	−	0.830695i	0.441040	−	0.897488i	\(-0.354610\pi\)
							0.997767	+	0.0667923i	\(0.0212765\pi\)
\(11\)	4.70330	+	2.71545i	1.41810	+	0.818740i	0.996132	−	0.0878717i	\(-0.0280066\pi\)
							0.421967	+	0.906611i	\(0.361340\pi\)
\(13\)	3.40753	+	1.17844i	0.945080	+	0.326840i
							\(17\)	−0.585633	−	1.01435i	−0.142037	−	0.246015i	0.786227	−	0.617938i	\(-0.212032\pi\)
−0.928263	+	0.371923i	\(0.878698\pi\)
\(19\)	−5.24197	+	3.02646i	−1.20259	+	0.694316i	−0.961131	−	0.276094i	\(-0.910960\pi\)
							−0.241461	+	0.970411i	\(0.577626\pi\)
\(23\)	−0.575237	+	0.996341i	−0.119945	+	0.207751i	−0.919746	−	0.392515i	\(-0.871605\pi\)
							0.799800	+	0.600266i	\(0.204939\pi\)
\(29\)	−2.41217	+	4.17801i	−0.447929	+	0.775836i	−0.998251	−	0.0591165i	\(-0.981172\pi\)
							0.550322	+	0.834953i	\(0.314505\pi\)
\(31\)		−	1.00000i		−	0.179605i
							\(37\)	−5.52259	−	3.18847i	−0.907909	−	0.524181i	−0.0281510	−	0.999604i	\(-0.508962\pi\)
−0.879758	+	0.475422i	\(0.842295\pi\)
\(41\)	−10.1628	−	5.86751i	−1.58717	−	0.916350i	−0.993771	−	0.111438i	\(-0.964454\pi\)
							−0.593394	−	0.804912i	\(-0.702212\pi\)
\(43\)	−2.82424	−	4.89172i	−0.430692	−	0.745981i	0.566241	−	0.824240i	\(-0.308397\pi\)
							−0.996933	+	0.0782590i	\(0.975064\pi\)
\(47\)			4.06481i			0.592914i	0.955046	+	0.296457i	\(0.0958052\pi\)
							−0.955046	+	0.296457i	\(0.904195\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.r.a.342.8	yes	68
13.6	odd	12		5239.2.a.r.1.25		34
13.7	odd	12		5239.2.a.q.1.10		34
13.10	even	6	inner	403.2.r.a.218.8	✓	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.r.a.218.8	✓	68	13.10	even	6	inner
403.2.r.a.342.8	yes	68	1.1	even	1	trivial
5239.2.a.q.1.10		34	13.7	odd	12
5239.2.a.r.1.25		34	13.6	odd	12