Embedded newform 403.2.ba.a.6.2

• Label: 403.2.ba.a.6.2
• Level: $403$
• Weight: $2$
• Character: 403.6
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(35\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			6.2
Character	\(\chi\)	\(=\)	403.6
Dual form			403.2.ba.a.336.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.648505 - 2.42025i) q^{2} +0.854246i q^{3} +(-3.70501 + 2.13909i) q^{4} +(-0.846300 - 3.15843i) q^{5} +(2.06749 - 0.553983i) q^{6} +(4.17770 - 1.11941i) q^{7} +(4.03636 + 4.03636i) q^{8} +2.27026 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 12 q^{4} - 2 q^{5} + 6 q^{6} + 12 q^{7} - 10 q^{8} - 124 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.648505	−	2.42025i	−0.458562	−	1.71138i	−0.677400	−	0.735615i	\(-0.736893\pi\)
							0.218838	−	0.975761i	\(-0.429774\pi\)
\(3\)			0.854246i			0.493199i	0.969117	+	0.246600i	\(0.0793133\pi\)
							−0.969117	+	0.246600i	\(0.920687\pi\)
\(5\)	−0.846300	−	3.15843i	−0.378477	−	1.41249i	−0.848198	−	0.529680i	\(-0.822312\pi\)
							0.469721	−	0.882815i	\(-0.344354\pi\)
\(7\)	4.17770	−	1.11941i	1.57902	−	0.423097i	0.640398	−	0.768043i	\(-0.278769\pi\)
							0.938623	+	0.344946i	\(0.112103\pi\)
\(11\)	−4.86400	−	1.30331i	−1.46655	−	0.392961i	−0.564805	−	0.825225i	\(-0.691049\pi\)
							−0.901748	+	0.432263i	\(0.857715\pi\)
\(13\)	2.39253	−	2.69737i	0.663569	−	0.748115i
							\(17\)	−1.96727	+	3.40742i	−0.477134	+	0.826420i	−0.999657	−	0.0262051i	\(-0.991658\pi\)
0.522523	+	0.852625i	\(0.324991\pi\)
\(19\)	−1.67744	−	6.26029i	−0.384831	−	1.43621i	−0.838433	−	0.545005i	\(-0.816528\pi\)
							0.453602	−	0.891204i	\(-0.350139\pi\)
\(23\)	−0.344515	+	0.596718i	−0.0718364	+	0.124424i	−0.899706	−	0.436496i	\(-0.856219\pi\)
							0.827870	+	0.560920i	\(0.189553\pi\)
\(29\)	−5.24115	−	3.02598i	−0.973257	−	0.561910i	−0.0730294	−	0.997330i	\(-0.523267\pi\)
							−0.900228	+	0.435420i	\(0.856600\pi\)
\(31\)	1.72909	+	5.29247i	0.310554	+	0.950556i
							\(37\)	−2.76478	+	2.76478i	−0.454527	+	0.454527i	−0.896854	−	0.442327i	\(-0.854153\pi\)
0.442327	+	0.896854i	\(0.354153\pi\)
\(41\)	−1.22652	−	0.328645i	−0.191550	−	0.0513257i	0.161769	−	0.986829i	\(-0.448280\pi\)
							−0.353319	+	0.935503i	\(0.614947\pi\)
\(43\)	2.63637	−	4.56633i	0.402043	−	0.696359i	−0.591929	−	0.805990i	\(-0.701633\pi\)
							0.993972	+	0.109631i	\(0.0349668\pi\)
\(47\)	−1.44871	−	1.44871i	−0.211316	−	0.211316i	0.593510	−	0.804826i	\(-0.297742\pi\)
							−0.804826	+	0.593510i	\(0.797742\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ba.a.6.2	✓	140
13.11	odd	12		403.2.bf.a.37.2	yes	140
31.26	odd	6		403.2.bf.a.305.2	yes	140
403.336	even	12	inner	403.2.ba.a.336.2	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.2	✓	140	1.1	even	1	trivial
403.2.ba.a.336.2	yes	140	403.336	even	12	inner
403.2.bf.a.37.2	yes	140	13.11	odd	12
403.2.bf.a.305.2	yes	140	31.26	odd	6