Embedded newform 403.2.r.a.342.21

• Label: 403.2.r.a.342.21
• 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			342.21
Character	\(\chi\)	\(=\)	403.342
Dual form			403.2.r.a.218.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.519051 + 0.299674i) q^{2} +(0.801804 - 1.38877i) q^{3} +(-0.820391 - 1.42096i) q^{4} +0.195832i q^{5} +(0.832355 - 0.480560i) q^{6} +(2.33812 - 1.34992i) q^{7} -2.18210i q^{8} +(0.214221 + 0.371041i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 32 q^{4} - 34 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.519051	+	0.299674i	0.367025	+	0.211902i	0.672158	−	0.740408i	\(-0.265368\pi\)
							−0.305133	+	0.952310i	\(0.598701\pi\)
\(3\)	0.801804	−	1.38877i	0.462922	−	0.801804i	−0.536183	−	0.844102i	\(-0.680134\pi\)
							0.999105	+	0.0422975i	\(0.0134677\pi\)
\(5\)			0.195832i			0.0875787i	0.999041	+	0.0437893i	\(0.0139430\pi\)
							−0.999041	+	0.0437893i	\(0.986057\pi\)
\(7\)	2.33812	−	1.34992i	0.883727	−	0.510220i	0.0118418	−	0.999930i	\(-0.496231\pi\)
							0.871886	+	0.489710i	\(0.162897\pi\)
\(11\)	0.499279	+	0.288259i	0.150538	+	0.0869133i	0.573377	−	0.819292i	\(-0.305633\pi\)
							−0.422839	+	0.906205i	\(0.638966\pi\)
\(13\)	−3.39340	−	1.21853i	−0.941161	−	0.337960i
							\(17\)	−1.81374	−	3.14150i	−0.439898	−	0.761925i	0.557783	−	0.829986i	\(-0.311652\pi\)
−0.997681	+	0.0680614i	\(0.978319\pi\)
\(19\)	−1.74940	+	1.01002i	−0.401341	+	0.231714i	−0.687062	−	0.726599i	\(-0.741100\pi\)
							0.285722	+	0.958313i	\(0.407767\pi\)
\(23\)	1.32064	−	2.28742i	0.275372	−	0.476959i	−0.694857	−	0.719148i	\(-0.744532\pi\)
							0.970229	+	0.242189i	\(0.0778655\pi\)
\(29\)	1.15635	−	2.00286i	0.214729	−	0.371921i	−0.738460	−	0.674297i	\(-0.764447\pi\)
							0.953189	+	0.302376i	\(0.0977799\pi\)
\(31\)			1.00000i			0.179605i
							\(37\)	6.65790	+	3.84394i	1.09455	+	0.631940i	0.934785	−	0.355215i	\(-0.115592\pi\)
0.159767	+	0.987155i	\(0.448926\pi\)
\(41\)	−7.10424	−	4.10164i	−1.10950	−	0.640568i	−0.170797	−	0.985306i	\(-0.554634\pi\)
							−0.938699	+	0.344738i	\(0.887968\pi\)
\(43\)	2.73090	+	4.73006i	0.416459	+	0.721328i	0.995580	−	0.0939136i	\(-0.0299377\pi\)
							−0.579122	+	0.815241i	\(0.696604\pi\)
\(47\)			9.19538i			1.34128i	0.741781	+	0.670642i	\(0.233981\pi\)
							−0.741781	+	0.670642i	\(0.766019\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.r.a.342.21	yes	68
13.6	odd	12		5239.2.a.q.1.16		34
13.7	odd	12		5239.2.a.r.1.19		34
13.10	even	6	inner	403.2.r.a.218.21	✓	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.r.a.218.21	✓	68	13.10	even	6	inner
403.2.r.a.342.21	yes	68	1.1	even	1	trivial
5239.2.a.q.1.16		34	13.6	odd	12
5239.2.a.r.1.19		34	13.7	odd	12