Embedded newform 403.2.ba.a.336.3

• Label: 403.2.ba.a.336.3
• Level: $403$
• Weight: $2$
• Character: 403.336
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $140$
• 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			336.3
Character	\(\chi\)	\(=\)	403.336
Dual form			403.2.ba.a.6.3

$q$-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 \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.3	✓	140	403.6	even	12	inner
403.2.ba.a.336.3	yes	140	1.1	even	1	trivial
403.2.bf.a.37.3	yes	140	31.6	odd	6
403.2.bf.a.305.3	yes	140	13.6	odd	12