Embedded newform 91.10.w.a.19.9

• Label: 91.10.w.a.19.9
• Level: $91$
• Weight: $10$
• Character: 91.19
• 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			19.9
Character	\(\chi\)	\(=\)	91.19
Dual form			91.10.w.a.24.9

$q$-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 \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{5}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.w.a.19.9	✓	328	1.1	even	1	trivial
91.10.w.a.24.9	yes	328	91.24	even	12	inner
91.10.ba.a.45.74	yes	328	7.3	odd	6
91.10.ba.a.89.74	yes	328	13.11	odd	12