Embedded newform 403.2.bi.a.14.9

• Label: 403.2.bi.a.14.9
• Level: $403$
• Weight: $2$
• Character: 403.14
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $120$
• 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:	\(120\)
Relative dimension:	\(15\) 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.a.144.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0727036 + 0.223759i) q^{2} +(0.337206 - 0.374505i) q^{3} +(1.57325 + 1.14303i) q^{4} +(0.254802 - 0.441330i) q^{5} +(0.0592827 + 0.102681i) q^{6} +(-0.0906697 + 0.862665i) q^{7} +(-0.750826 + 0.545507i) q^{8} +(0.287039 + 2.73099i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 13 q^{5} - 10 q^{6} - 16 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0727036	+	0.223759i	−0.0514092	+	0.158221i	−0.973465	−	0.228835i	\(-0.926508\pi\)
							0.922056	+	0.387057i	\(0.126508\pi\)
\(3\)	0.337206	−	0.374505i	0.194686	−	0.216221i	−0.637896	−	0.770123i	\(-0.720195\pi\)
							0.832582	+	0.553902i	\(0.186862\pi\)
\(5\)	0.254802	−	0.441330i	0.113951	−	0.197369i	−0.803409	−	0.595427i	\(-0.796983\pi\)
							0.917360	+	0.398059i	\(0.130316\pi\)
\(7\)	−0.0906697	+	0.862665i	−0.0342699	+	0.326057i	0.963934	+	0.266143i	\(0.0857493\pi\)
							−0.998204	+	0.0599138i	\(0.980917\pi\)
\(11\)	1.65022	−	0.734724i	0.497559	−	0.221528i	−0.142585	−	0.989783i	\(-0.545541\pi\)
							0.640144	+	0.768255i	\(0.278875\pi\)
\(13\)	0.978148	+	0.207912i	0.271289	+	0.0576643i
							\(17\)	−2.17444	−	0.968123i	−0.527379	−	0.234804i	0.125735	−	0.992064i	\(-0.459871\pi\)
−0.653114	+	0.757260i	\(0.726538\pi\)
\(19\)	−1.61598	+	0.343488i	−0.370732	+	0.0788014i	−0.389510	−	0.921022i	\(-0.627356\pi\)
							0.0187784	+	0.999824i	\(0.494022\pi\)
\(23\)	3.94538	−	2.86649i	0.822670	−	0.597704i	−0.0948064	−	0.995496i	\(-0.530223\pi\)
							0.917476	+	0.397791i	\(0.130223\pi\)
\(29\)	2.25562	−	6.94209i	0.418858	−	1.28911i	−0.489896	−	0.871781i	\(-0.662965\pi\)
							0.908754	−	0.417332i	\(-0.137035\pi\)
\(31\)	−5.12122	−	2.18473i	−0.919799	−	0.392390i
							\(37\)	−2.83218	−	4.90549i	−0.465608	−	0.806457i	0.533620	−	0.845724i	\(-0.320831\pi\)
−0.999229	+	0.0392668i	\(0.987498\pi\)
\(41\)	5.71914	+	6.35175i	0.893180	+	0.991977i	0.999997	−	0.00230190i	\(-0.000732720\pi\)
							−0.106817	+	0.994279i	\(0.534066\pi\)
\(43\)	−4.25431	+	0.904281i	−0.648776	+	0.137902i	−0.520533	−	0.853841i	\(-0.674267\pi\)
							−0.128242	+	0.991743i	\(0.540934\pi\)
\(47\)	0.681023	+	2.09597i	0.0993374	+	0.305729i	0.988360	−	0.152135i	\(-0.0486148\pi\)
							−0.889022	+	0.457864i	\(0.848615\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.a.14.9	✓	120
31.20	even	15	inner	403.2.bi.a.144.9	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.a.14.9	✓	120	1.1	even	1	trivial
403.2.bi.a.144.9	yes	120	31.20	even	15	inner