Embedded newform 403.2.v.a.36.15

• Label: 403.2.v.a.36.15
• Level: $403$
• Weight: $2$
• Character: 403.36
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.v (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			36.15
Character	\(\chi\)	\(=\)	403.36
Dual form			403.2.v.a.56.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.700842 + 0.404632i) q^{2} +1.07124 q^{3} +(-0.672547 + 1.16488i) q^{4} +(-3.37908 + 1.95091i) q^{5} +(-0.750767 + 0.433456i) q^{6} +(2.06539 - 1.19246i) q^{7} -2.70706i q^{8} -1.85245 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} + 4 q^{3} + 30 q^{4} + 6 q^{6} - 12 q^{7} + 58 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.700842	+	0.404632i	−0.495570	+	0.286118i	−0.726882	−	0.686762i	\(-0.759032\pi\)
							0.231312	+	0.972880i	\(0.425698\pi\)
\(3\)	1.07124			0.618478			0.309239	−	0.950984i	\(-0.399926\pi\)
							0.309239	+	0.950984i	\(0.399926\pi\)
\(5\)	−3.37908	+	1.95091i	−1.51117	+	0.872474i	−0.511254	+	0.859430i	\(0.670819\pi\)
							−0.999915	+	0.0130443i	\(0.995848\pi\)
\(7\)	2.06539	−	1.19246i	0.780645	−	0.450706i	−0.0560136	−	0.998430i	\(-0.517839\pi\)
							0.836659	+	0.547724i	\(0.184506\pi\)
\(11\)	0.548190	+	0.316498i	0.165285	+	0.0954276i	0.580361	−	0.814359i	\(-0.302911\pi\)
							−0.415075	+	0.909787i	\(0.636245\pi\)
\(13\)	−2.35258	+	2.73228i	−0.652488	+	0.757799i
							\(17\)	−1.34141	−	2.32339i	−0.325339	−	0.563504i	0.656242	−	0.754551i	\(-0.272145\pi\)
−0.981581	+	0.191047i	\(0.938812\pi\)
\(19\)	−3.14186	+	1.81395i	−0.720792	+	0.416150i	−0.815044	−	0.579399i	\(-0.803287\pi\)
							0.0942519	+	0.995548i	\(0.469954\pi\)
\(23\)	−2.45825	−	4.25782i	−0.512581	−	0.887817i	−0.999894	−	0.0145892i	\(-0.995356\pi\)
							0.487312	−	0.873228i	\(-0.337977\pi\)
\(29\)	−3.82172	−	6.61941i	−0.709675	−	1.22919i	−0.964978	−	0.262332i	\(-0.915508\pi\)
							0.255303	−	0.966861i	\(-0.417825\pi\)
\(31\)	−0.0308301	+	5.56768i	−0.00553725	+	0.999985i
							\(37\)			6.64926i			1.09313i	0.837416	+	0.546566i	\(0.184065\pi\)
−0.837416	+	0.546566i	\(0.815935\pi\)
\(41\)	−5.47362	−	3.16019i	−0.854835	−	0.493539i	0.00744400	−	0.999972i	\(-0.497630\pi\)
							−0.862279	+	0.506433i	\(0.830964\pi\)
\(43\)	1.47396	+	2.55297i	0.224777	+	0.389325i	0.956252	−	0.292543i	\(-0.0945014\pi\)
							−0.731476	+	0.681868i	\(0.761168\pi\)
\(47\)			2.05460i			0.299695i	0.988709	+	0.149847i	\(0.0478782\pi\)
							−0.988709	+	0.149847i	\(0.952122\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.v.a.36.15	yes	70
13.4	even	6		403.2.s.a.160.15	✓	70
31.25	even	3		403.2.s.a.335.15	yes	70
403.56	even	6	inner	403.2.v.a.56.15	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.15	✓	70	13.4	even	6
403.2.s.a.335.15	yes	70	31.25	even	3
403.2.v.a.36.15	yes	70	1.1	even	1	trivial
403.2.v.a.56.15	yes	70	403.56	even	6	inner