Embedded newform 403.2.bf.a.305.3

• Label: 403.2.bf.a.305.3
• Level: $403$
• Weight: $2$
• Character: 403.305
• 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.bf (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			305.3
Character	\(\chi\)	\(=\)	403.305
Dual form			403.2.bf.a.37.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.626284 - 2.33732i) q^{2} +(-1.20623 + 0.696416i) q^{3} +(-3.33879 + 1.92765i) q^{4} +(0.117630 + 0.0315189i) q^{5} +(2.38319 + 2.38319i) q^{6} +(1.27982 + 1.27982i) q^{7} +(3.17450 + 3.17450i) q^{8} +(-0.530010 + 0.918004i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 6 q^{3} - 12 q^{4} - 2 q^{5} + 12 q^{6} - 12 q^{7} - 10 q^{8} + 62 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)

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	−0.721553	−	0.692359i	\(-0.756571\pi\)
							0.278704	−	0.960377i	\(-0.410095\pi\)
\(3\)	−1.20623	+	0.696416i	−0.696416	+	0.402076i	−0.806011	−	0.591900i	\(-0.798378\pi\)
							0.109595	+	0.993976i	\(0.465045\pi\)
\(5\)	0.117630	+	0.0315189i	0.0526057	+	0.0140957i	0.285026	−	0.958520i	\(-0.407998\pi\)
							−0.232420	+	0.972615i	\(0.574664\pi\)
\(7\)	1.27982	+	1.27982i	0.483726	+	0.483726i	0.906320	−	0.422593i	\(-0.138880\pi\)
							−0.422593	+	0.906320i	\(0.638880\pi\)
\(11\)	0.980614	−	0.980614i	0.295666	−	0.295666i	−0.543647	−	0.839314i	\(-0.682957\pi\)
							0.839314	+	0.543647i	\(0.182957\pi\)
\(13\)	0.598774	−	3.55548i	0.166070	−	0.986114i
							\(17\)	6.40551			1.55356			0.776782	−	0.629770i	\(-0.216851\pi\)
0.776782	+	0.629770i	\(0.216851\pi\)
\(19\)	6.04083	−	6.04083i	1.38586	−	1.38586i	0.552052	−	0.833810i	\(-0.313845\pi\)
							0.833810	−	0.552052i	\(-0.186155\pi\)
\(23\)	−3.44035	+	5.95887i	−0.717363	+	1.24251i	0.244678	+	0.969604i	\(0.421318\pi\)
							−0.962041	+	0.272905i	\(0.912015\pi\)
\(29\)	2.28168	+	1.31733i	0.423698	+	0.244622i	0.696658	−	0.717403i	\(-0.254669\pi\)
							−0.272960	+	0.962025i	\(0.588003\pi\)
\(31\)	5.35199	−	1.53500i	0.961245	−	0.275694i
							\(37\)	−1.68499	−	0.451493i	−0.277011	−	0.0742250i	0.117639	−	0.993056i	\(-0.462468\pi\)
−0.394650	+	0.918831i	\(0.629134\pi\)
\(41\)	0.758205	−	0.758205i	0.118412	−	0.118412i	−0.645418	−	0.763830i	\(-0.723317\pi\)
							0.763830	+	0.645418i	\(0.223317\pi\)
\(43\)	−4.07681			−0.621707			−0.310854	−	0.950458i	\(-0.600615\pi\)
							−0.310854	+	0.950458i	\(0.600615\pi\)
\(47\)	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.bf.a.305.3	yes	140
13.11	odd	12		403.2.ba.a.336.3	yes	140
31.6	odd	6		403.2.ba.a.6.3	✓	140
403.37	even	12	inner	403.2.bf.a.37.3	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.3	✓	140	31.6	odd	6
403.2.ba.a.336.3	yes	140	13.11	odd	12
403.2.bf.a.37.3	yes	140	403.37	even	12	inner
403.2.bf.a.305.3	yes	140	1.1	even	1	trivial