Embedded newform 91.10.w.a.24.9

• Label: 91.10.w.a.24.9
• Level: $91$
• Weight: $10$
• Character: 91.24
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $328$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.w (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(328\)
Relative dimension:	\(82\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			24.9
Character	\(\chi\)	\(=\)	91.24
Dual form			91.10.w.a.19.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-36.3027 + 9.72728i) q^{2} -195.464i q^{3} +(779.863 - 450.254i) q^{4} +(-142.800 - 38.2632i) q^{5} +(1901.33 + 7095.88i) q^{6} +(5520.27 - 3143.27i) q^{7} +(-10324.8 + 10324.8i) q^{8} -18523.2 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 328 q - 2 q^{2} - 6 q^{4} - 6 q^{5} + 3072 q^{6} + 1136 q^{7} + 1016 q^{8} - 2047036 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−36.3027	+	9.72728i	−1.60437	+	0.429889i	−0.946358	−	0.323119i	\(-0.895268\pi\)
							−0.658011	+	0.753009i	\(0.728602\pi\)
\(3\)		−	195.464i		−	1.39323i	−0.717447	−	0.696613i	\(-0.754690\pi\)
							0.717447	−	0.696613i	\(-0.245310\pi\)
\(5\)	−142.800	−	38.2632i	−0.102180	−	0.0273789i	0.207367	−	0.978263i	\(-0.433511\pi\)
							−0.309547	+	0.950884i	\(0.600177\pi\)
\(7\)	5520.27	−	3143.27i	0.868999	−	0.494813i
							\(11\)	−37600.8	+	37600.8i	−0.774336	+	0.774336i	−0.978861	−	0.204525i	\(-0.934435\pi\)
0.204525	+	0.978861i	\(0.434435\pi\)
\(13\)	48978.4	−	90584.9i	0.475619	−	0.879651i
							\(17\)	−158102.	−	273840.i	−0.459109	−	0.795200i	0.539805	−	0.841790i	\(-0.318498\pi\)
−0.998914	+	0.0465897i	\(0.985165\pi\)
\(19\)	713276.	−	713276.i	1.25564	−	1.25564i	0.302492	−	0.953152i	\(-0.402182\pi\)
							0.953152	−	0.302492i	\(-0.0978184\pi\)
\(23\)	1.84283e6	+	1.06396e6i	1.37312	+	0.792774i	0.991320	−	0.131470i	\(-0.0419696\pi\)
							0.381804	+	0.924243i	\(0.375303\pi\)
\(29\)	2.72703e6	+	4.72335e6i	0.715976	+	1.24011i	0.962582	+	0.270991i	\(0.0873513\pi\)
							−0.246606	+	0.969116i	\(0.579315\pi\)
\(31\)	128343.	+	478981.i	0.0249599	+	0.0931517i	0.977282	−	0.211942i	\(-0.0679788\pi\)
							−0.952322	+	0.305094i	\(0.901312\pi\)
\(37\)	−1.95133e6	−	7.28247e6i	−0.171168	−	0.638809i	−0.997173	−	0.0751449i	\(-0.976058\pi\)
							0.826004	−	0.563664i	\(-0.190609\pi\)
\(41\)	2.34388e7	+	6.28042e6i	1.29541	+	0.347105i	0.839714	−	0.543028i	\(-0.182722\pi\)
							0.455700	+	0.890133i	\(0.349389\pi\)
\(43\)	1.99603e7	+	1.15241e7i	0.890347	+	0.514042i	0.874056	−	0.485825i	\(-0.161481\pi\)
							0.0162909	+	0.999867i	\(0.494814\pi\)
\(47\)	7.18888e6	−	2.68293e7i	0.214892	−	0.801989i	−0.771312	−	0.636457i	\(-0.780399\pi\)
							0.986204	−	0.165532i	\(-0.0529341\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.w.a.24.9	yes	328
7.5	odd	6		91.10.ba.a.89.74	yes	328
13.6	odd	12		91.10.ba.a.45.74	yes	328
91.19	even	12	inner	91.10.w.a.19.9	✓	328

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.w.a.19.9	✓	328	91.19	even	12	inner
91.10.w.a.24.9	yes	328	1.1	even	1	trivial
91.10.ba.a.45.74	yes	328	13.6	odd	12
91.10.ba.a.89.74	yes	328	7.5	odd	6