Embedded newform 403.2.k.e.66.14

• Label: 403.2.k.e.66.14
• Level: $403$
• Weight: $2$
• Character: 403.66
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.k (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(17\) over \(\Q(\zeta_{5})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label			66.14
Character	\(\chi\)	\(=\)	403.66
Dual form			403.2.k.e.287.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.614130 + 1.89010i) q^{2} +(-0.776144 + 2.38873i) q^{3} +(-1.57728 + 1.14596i) q^{4} +0.0289380 q^{5} -4.99158 q^{6} +(-3.33658 + 2.42417i) q^{7} +(0.0810008 + 0.0588505i) q^{8} +(-2.67656 - 1.94463i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q - 3 q^{2} - 2 q^{3} - 23 q^{4} + 12 q^{5} + 4 q^{6} + 2 q^{7} - 3 q^{8} - 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.614130	+	1.89010i	0.434255	+	1.33650i	0.893848	+	0.448370i	\(0.147995\pi\)
							−0.459593	+	0.888130i	\(0.652005\pi\)
\(3\)	−0.776144	+	2.38873i	−0.448107	+	1.37913i	0.430933	+	0.902384i	\(0.358184\pi\)
							−0.879040	+	0.476748i	\(0.841816\pi\)
\(5\)	0.0289380			0.0129415			0.00647074	−	0.999979i	\(-0.497940\pi\)
							0.00647074	+	0.999979i	\(0.497940\pi\)
\(7\)	−3.33658	+	2.42417i	−1.26111	+	0.916249i	−0.998812	−	0.0487387i	\(-0.984480\pi\)
							−0.262296	+	0.964987i	\(0.584480\pi\)
\(11\)	2.87936	−	2.09197i	0.868158	−	0.630754i	−0.0619338	−	0.998080i	\(-0.519727\pi\)
							0.930092	+	0.367326i	\(0.119727\pi\)
\(13\)	0.309017	−	0.951057i	0.0857059	−	0.263776i
							\(17\)	1.48973	+	1.08235i	0.361312	+	0.262508i	0.753599	−	0.657334i	\(-0.228316\pi\)
−0.392287	+	0.919843i	\(0.628316\pi\)
\(19\)	−1.05495	−	3.24679i	−0.242021	−	0.744865i	−0.996112	−	0.0880948i	\(-0.971922\pi\)
							0.754091	−	0.656770i	\(-0.228078\pi\)
\(23\)	5.02249	+	3.64906i	1.04726	+	0.760881i	0.971690	−	0.236260i	\(-0.0759216\pi\)
							0.0755726	+	0.997140i	\(0.475922\pi\)
\(29\)	−1.55148	−	4.77496i	−0.288102	−	0.886687i	−0.985452	−	0.169956i	\(-0.945637\pi\)
							0.697349	−	0.716731i	\(-0.254363\pi\)
\(31\)	4.33742	+	3.49096i	0.779024	+	0.626994i
							\(37\)	0.217652			0.0357817			0.0178909	−	0.999840i	\(-0.494305\pi\)
0.0178909	+	0.999840i	\(0.494305\pi\)
\(41\)	2.99954	+	9.23163i	0.468449	+	1.44174i	0.854592	+	0.519300i	\(0.173807\pi\)
							−0.386143	+	0.922439i	\(0.626193\pi\)
\(43\)	1.46154	+	4.49817i	0.222883	+	0.685964i	0.998500	+	0.0547593i	\(0.0174392\pi\)
							−0.775616	+	0.631205i	\(0.782561\pi\)
\(47\)	1.88949	−	5.81524i	0.275610	−	0.848240i	−0.713448	−	0.700709i	\(-0.752867\pi\)
							0.989057	−	0.147531i	\(-0.0471327\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.k.e.66.14	✓	68
31.8	even	5	inner	403.2.k.e.287.14	yes	68

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.k.e.66.14	✓	68	1.1	even	1	trivial
403.2.k.e.287.14	yes	68	31.8	even	5	inner