Embedded newform 403.2.ba.a.6.3

• Label: 403.2.ba.a.6.3
• 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			6.3
Character	\(\chi\)	\(=\)	403.6
Dual form			403.2.ba.a.336.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.626284 - 2.33732i) q^{2} +1.39283i q^{3} +(-3.33879 + 1.92765i) q^{4} +(-0.0315189 - 0.117630i) q^{5} +(3.25550 - 0.872308i) q^{6} +(-1.74827 + 0.468447i) q^{7} +(3.17450 + 3.17450i) q^{8} +1.06002 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 12 q^{4} - 2 q^{5} + 6 q^{6} + 12 q^{7} - 10 q^{8} - 124 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.626284	−	2.33732i	−0.442849	−	1.65274i	−0.721553	−	0.692359i	\(-0.756571\pi\)
							0.278704	−	0.960377i	\(-0.410095\pi\)
\(3\)			1.39283i			0.804152i	0.915606	+	0.402076i	\(0.131711\pi\)
							−0.915606	+	0.402076i	\(0.868289\pi\)
\(5\)	−0.0315189	−	0.117630i	−0.0140957	−	0.0526057i	0.958520	−	0.285026i	\(-0.0920022\pi\)
							−0.972615	+	0.232420i	\(0.925336\pi\)
\(7\)	−1.74827	+	0.468447i	−0.660783	+	0.177056i	−0.573599	−	0.819136i	\(-0.694453\pi\)
							−0.0871835	+	0.996192i	\(0.527787\pi\)
\(11\)	1.33954	+	0.358930i	0.403887	+	0.108221i	0.455043	−	0.890469i	\(-0.349624\pi\)
							−0.0511557	+	0.998691i	\(0.516290\pi\)
\(13\)	3.37853	−	1.25919i	0.937035	−	0.349236i
							\(17\)	3.20275	−	5.54733i	0.776782	−	1.34543i	−0.157006	−	0.987598i	\(-0.550184\pi\)
0.933788	−	0.357828i	\(-0.116482\pi\)
\(19\)	2.21110	+	8.25193i	0.507261	+	1.89312i	0.446075	+	0.894996i	\(0.352822\pi\)
							0.0611859	+	0.998126i	\(0.480512\pi\)
\(23\)	3.44035	−	5.95887i	0.717363	−	1.24251i	−0.244678	−	0.969604i	\(-0.578682\pi\)
							0.962041	−	0.272905i	\(-0.0879845\pi\)
\(29\)	−2.28168	−	1.31733i	−0.423698	−	0.244622i	0.272960	−	0.962025i	\(-0.411997\pi\)
							−0.696658	+	0.717403i	\(0.745331\pi\)
\(31\)	1.53500	−	5.35199i	0.275694	−	0.961245i
							\(37\)	−1.23350	+	1.23350i	−0.202786	+	0.202786i	−0.801193	−	0.598406i	\(-0.795801\pi\)
0.598406	+	0.801193i	\(0.295801\pi\)
\(41\)	−1.03573	−	0.277522i	−0.161753	−	0.0433417i	0.177034	−	0.984205i	\(-0.443350\pi\)
							−0.338787	+	0.940863i	\(0.610017\pi\)
\(43\)	−2.03840	+	3.53062i	−0.310854	+	0.538414i	−0.978547	−	0.206022i	\(-0.933948\pi\)
							0.667694	+	0.744436i	\(0.267282\pi\)
\(47\)	6.84577	+	6.84577i	0.998558	+	0.998558i	0.999999	−	0.00144096i	\(-0.000458671\pi\)
							−0.00144096	+	0.999999i	\(0.500459\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ba.a.6.3	✓	140
13.11	odd	12		403.2.bf.a.37.3	yes	140
31.26	odd	6		403.2.bf.a.305.3	yes	140
403.336	even	12	inner	403.2.ba.a.336.3	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.3	✓	140	1.1	even	1	trivial
403.2.ba.a.336.3	yes	140	403.336	even	12	inner
403.2.bf.a.37.3	yes	140	13.11	odd	12
403.2.bf.a.305.3	yes	140	31.26	odd	6