Embedded newform 403.2.bi.b.14.9

• Label: 403.2.bi.b.14.9
• Level: $403$
• Weight: $2$
• Character: 403.14
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bi (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(17\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			14.9
Character	\(\chi\)	\(=\)	403.14
Dual form			403.2.bi.b.144.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.124680 - 0.383725i) q^{2} +(-0.653315 + 0.725580i) q^{3} +(1.48633 + 1.07989i) q^{4} +(-0.174215 + 0.301749i) q^{5} +(0.196968 + 0.341158i) q^{6} +(0.256927 - 2.44450i) q^{7} +(1.25253 - 0.910013i) q^{8} +(0.213940 + 2.03550i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 6 q^{2} + 2 q^{3} - 34 q^{4} - 15 q^{5} - 10 q^{6} - 8 q^{7} - 6 q^{8} + 23 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{11}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.124680	−	0.383725i	0.0881619	−	0.271334i	−0.897249	−	0.441524i	\(-0.854438\pi\)
							0.985411	+	0.170190i	\(0.0544380\pi\)
\(3\)	−0.653315	+	0.725580i	−0.377191	+	0.418914i	−0.901612	−	0.432546i	\(-0.857615\pi\)
							0.524420	+	0.851459i	\(0.324282\pi\)
\(5\)	−0.174215	+	0.301749i	−0.0779113	+	0.134946i	−0.902349	−	0.431007i	\(-0.858158\pi\)
							0.824437	+	0.565953i	\(0.191492\pi\)
\(7\)	0.256927	−	2.44450i	0.0971094	−	0.923934i	−0.832161	−	0.554534i	\(-0.812896\pi\)
							0.929270	−	0.369400i	\(-0.120437\pi\)
\(11\)	3.19304	−	1.42163i	0.962738	−	0.428639i	0.135679	−	0.990753i	\(-0.456678\pi\)
							0.827060	+	0.562114i	\(0.190012\pi\)
\(13\)	−0.978148	−	0.207912i	−0.271289	−	0.0576643i
							\(17\)	5.10275	+	2.27189i	1.23760	+	0.551014i	0.918015	−	0.396545i	\(-0.129791\pi\)
0.319582	+	0.947559i	\(0.396457\pi\)
\(19\)	−0.729275	+	0.155012i	−0.167307	+	0.0355622i	−0.290804	−	0.956783i	\(-0.593923\pi\)
							0.123496	+	0.992345i	\(0.460589\pi\)
\(23\)	−5.94448	+	4.31892i	−1.23951	+	0.900556i	−0.997566	−	0.0697337i	\(-0.977785\pi\)
							−0.241944	+	0.970290i	\(0.577785\pi\)
\(29\)	1.19524	−	3.67857i	0.221950	−	0.683092i	−0.776637	−	0.629949i	\(-0.783076\pi\)
							0.998587	−	0.0531437i	\(-0.0169241\pi\)
\(31\)	4.33911	+	3.48886i	0.779327	+	0.626618i
							\(37\)	0.0308864	+	0.0534968i	0.00507769	+	0.00879482i	0.868553	−	0.495596i	\(-0.165050\pi\)
−0.863475	+	0.504391i	\(0.831717\pi\)
\(41\)	−3.53526	−	3.92631i	−0.552115	−	0.613186i	0.400895	−	0.916124i	\(-0.368699\pi\)
							−0.953010	+	0.302938i	\(0.902033\pi\)
\(43\)	−2.62585	+	0.558142i	−0.400439	+	0.0851159i	−0.403729	−	0.914879i	\(-0.632286\pi\)
							0.00328995	+	0.999995i	\(0.498953\pi\)
\(47\)	−3.23333	−	9.95117i	−0.471630	−	1.45153i	−0.850449	−	0.526057i	\(-0.823670\pi\)
							0.378819	−	0.925471i	\(-0.376330\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.b.14.9	✓	136
31.20	even	15	inner	403.2.bi.b.144.9	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.b.14.9	✓	136	1.1	even	1	trivial
403.2.bi.b.144.9	yes	136	31.20	even	15	inner