Embedded newform 403.2.be.c.57.9

• Label: 403.2.be.c.57.9
• Level: $403$
• Weight: $2$
• Character: 403.57
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.be (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(34\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			57.9
Character	\(\chi\)	\(=\)	403.57
Dual form			403.2.be.c.99.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.06684 + 1.06684i) q^{2} +(-1.24662 - 0.719735i) q^{3} -0.276298i q^{4} +(3.17907 + 0.851829i) q^{5} +(2.09779 - 0.562100i) q^{6} +(4.69129 - 1.25703i) q^{7} +(-1.83892 - 1.83892i) q^{8} +(-0.463962 - 0.803606i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 4 q^{2} - 24 q^{3} + 2 q^{5} - 36 q^{6} - 2 q^{7} - 8 q^{8} + 60 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.06684	+	1.06684i	−0.754370	+	0.754370i	−0.975292	−	0.220921i	\(-0.929094\pi\)
							0.220921	+	0.975292i	\(0.429094\pi\)
\(3\)	−1.24662	−	0.719735i	−0.719735	−	0.415539i	0.0949200	−	0.995485i	\(-0.469740\pi\)
							−0.814655	+	0.579946i	\(0.803074\pi\)
\(5\)	3.17907	+	0.851829i	1.42172	+	0.380949i	0.886095	−	0.463504i	\(-0.153408\pi\)
							0.535628	+	0.844454i	\(0.320075\pi\)
\(7\)	4.69129	−	1.25703i	1.77314	−	0.475112i	0.783836	−	0.620968i	\(-0.213260\pi\)
							0.989306	+	0.145857i	\(0.0465938\pi\)
\(11\)	−0.954007	−	0.255625i	−0.287644	−	0.0770740i	0.112112	−	0.993696i	\(-0.464238\pi\)
							−0.399756	+	0.916622i	\(0.630905\pi\)
\(13\)	−2.54693	−	2.55209i	−0.706390	−	0.707823i
							\(17\)	2.18818	−	3.79004i	0.530712	−	0.919220i	−0.468646	−	0.883386i	\(-0.655258\pi\)
0.999358	−	0.0358341i	\(-0.0114088\pi\)
\(19\)	−1.32542	−	4.94653i	−0.304072	−	1.13481i	−0.933741	−	0.357949i	\(-0.883476\pi\)
							0.629669	−	0.776864i	\(-0.283191\pi\)
\(23\)	2.63177			0.548763			0.274381	−	0.961621i	\(-0.411527\pi\)
							0.274381	+	0.961621i	\(0.411527\pi\)
\(29\)			6.85337i			1.27264i	0.771425	+	0.636320i	\(0.219544\pi\)
							−0.771425	+	0.636320i	\(0.780456\pi\)
\(31\)	3.59222	−	4.25393i	0.645182	−	0.764029i
							\(37\)	1.11523	+	4.16208i	0.183342	+	0.684242i	0.994979	+	0.100080i	\(0.0319100\pi\)
−0.811637	+	0.584162i	\(0.801423\pi\)
\(41\)	−8.75767	−	2.34661i	−1.36772	−	0.366479i	−0.501073	−	0.865405i	\(-0.667061\pi\)
							−0.866645	+	0.498926i	\(0.833728\pi\)
\(43\)	1.84166	−	3.18985i	0.280851	−	0.486448i	−0.690744	−	0.723100i	\(-0.742717\pi\)
							0.971595	+	0.236652i	\(0.0760500\pi\)
\(47\)	4.87646	+	4.87646i	0.711305	+	0.711305i	0.966808	−	0.255503i	\(-0.0822412\pi\)
							−0.255503	+	0.966808i	\(0.582241\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.be.c.57.9	✓	136
13.8	odd	4	inner	403.2.be.c.398.9	yes	136
31.6	odd	6	inner	403.2.be.c.161.9	yes	136
403.99	even	12	inner	403.2.be.c.99.9	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.be.c.57.9	✓	136	1.1	even	1	trivial
403.2.be.c.99.9	yes	136	403.99	even	12	inner
403.2.be.c.161.9	yes	136	31.6	odd	6	inner
403.2.be.c.398.9	yes	136	13.8	odd	4	inner