Embedded newform 403.2.bi.a.40.1

• Label: 403.2.bi.a.40.1
• Level: $403$
• Weight: $2$
• Character: 403.40
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bi (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(15\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			40.1
Character	\(\chi\)	\(=\)	403.40
Dual form			403.2.bi.a.131.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.844756 + 2.59989i) q^{2} +(1.33170 + 0.283062i) q^{3} +(-4.42779 - 3.21698i) q^{4} +(1.99341 + 3.45269i) q^{5} +(-1.86090 + 3.22317i) q^{6} +(3.32958 - 1.48242i) q^{7} +(7.68099 - 5.58056i) q^{8} +(-1.04733 - 0.466300i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 13 q^{5} - 10 q^{6} - 16 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.844756	+	2.59989i	−0.597333	+	1.83840i	−0.0545774	+	0.998510i	\(0.517381\pi\)
							−0.542755	+	0.839891i	\(0.682619\pi\)
\(3\)	1.33170	+	0.283062i	0.768859	+	0.163426i	0.575611	−	0.817724i	\(-0.304764\pi\)
							0.193248	+	0.981150i	\(0.438098\pi\)
\(5\)	1.99341	+	3.45269i	0.891480	+	1.54409i	0.838102	+	0.545514i	\(0.183665\pi\)
							0.0533777	+	0.998574i	\(0.483001\pi\)
\(7\)	3.32958	−	1.48242i	1.25846	−	0.560304i	0.334359	−	0.942446i	\(-0.391480\pi\)
							0.924104	+	0.382142i	\(0.124813\pi\)
\(11\)	−0.349508	+	3.32535i	−0.105381	+	1.00263i	0.806238	+	0.591592i	\(0.201500\pi\)
							−0.911618	+	0.411038i	\(0.865166\pi\)
\(13\)	−0.669131	+	0.743145i	−0.185583	+	0.206111i
							\(17\)	0.400223	+	3.80787i	0.0970683	+	0.923543i	0.929352	+	0.369194i	\(0.120366\pi\)
−0.832284	+	0.554349i	\(0.812967\pi\)
\(19\)	−1.76247	−	1.95742i	−0.404339	−	0.449064i	0.506244	−	0.862390i	\(-0.331034\pi\)
							−0.910583	+	0.413326i	\(0.864367\pi\)
\(23\)	2.33886	−	1.69928i	0.487686	−	0.354324i	−0.316608	−	0.948556i	\(-0.602544\pi\)
							0.804294	+	0.594232i	\(0.202544\pi\)
\(29\)	1.70409	−	5.24467i	0.316442	−	0.973910i	−0.658714	−	0.752393i	\(-0.728899\pi\)
							0.975157	−	0.221517i	\(-0.0711007\pi\)
\(31\)	4.07400	−	3.79507i	0.731712	−	0.681614i
							\(37\)	−0.0895119	+	0.155039i	−0.0147157	+	0.0254883i	−0.873289	−	0.487202i	\(-0.838018\pi\)
0.858574	+	0.512690i	\(0.171351\pi\)
\(41\)	−3.95885	+	0.841479i	−0.618268	+	0.131417i	−0.506389	−	0.862305i	\(-0.669020\pi\)
							−0.111879	+	0.993722i	\(0.535687\pi\)
\(43\)	−6.43222	−	7.14371i	−0.980905	−	1.08941i	−0.995985	−	0.0895188i	\(-0.971467\pi\)
							0.0150804	−	0.999886i	\(-0.495200\pi\)
\(47\)	−1.13041	−	3.47904i	−0.164887	−	0.507470i	0.834141	−	0.551552i	\(-0.185964\pi\)
							−0.999028	+	0.0440811i	\(0.985964\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.a.40.1	✓	120
31.7	even	15	inner	403.2.bi.a.131.1	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.a.40.1	✓	120	1.1	even	1	trivial
403.2.bi.a.131.1	yes	120	31.7	even	15	inner