Embedded newform 91.10.u.a.30.15

• Label: 91.10.u.a.30.15
• Level: $91$
• Weight: $10$
• Character: 91.30
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			30.15
Character	\(\chi\)	\(=\)	91.30
Dual form			91.10.u.a.88.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-27.3854 + 15.8110i) q^{2} +258.288 q^{3} +(243.973 - 422.574i) q^{4} +(2120.68 + 1224.37i) q^{5} +(-7073.31 + 4083.78i) q^{6} +(-4845.78 - 4107.55i) q^{7} -760.607i q^{8} +47029.5 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 3 q^{2} + 322 q^{3} + 20481 q^{4} - 6 q^{6} + 1023 q^{7} + 1021818 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−27.3854	+	15.8110i	−1.21028	+	0.698753i	−0.962819	−	0.270146i	\(-0.912928\pi\)
							−0.247456	+	0.968899i	\(0.579595\pi\)
\(3\)	258.288			1.84102			0.920509	−	0.390722i	\(-0.127775\pi\)
							0.920509	+	0.390722i	\(0.127775\pi\)
\(5\)	2120.68	+	1224.37i	1.51743	+	0.876091i	0.999790	+	0.0204951i	\(0.00652424\pi\)
							0.517644	+	0.855596i	\(0.326809\pi\)
\(7\)	−4845.78	−	4107.55i	−0.762821	−	0.646609i
							\(11\)		−	33756.5i		−	0.695169i	−0.937649	−	0.347585i	\(-0.887002\pi\)
0.937649	−	0.347585i	\(-0.112998\pi\)
\(13\)	−82390.5	−	61776.3i	−0.800077	−	0.599897i
							\(17\)	171205.	−	296536.i	0.497160	−	0.861107i	−0.502834	−	0.864383i	\(-0.667709\pi\)
0.999995	+	0.00327599i	\(0.00104278\pi\)
\(19\)			744528.i			1.31066i	0.755343	+	0.655330i	\(0.227470\pi\)
							−0.755343	+	0.655330i	\(0.772530\pi\)
\(23\)	821488.	+	1.42286e6i	0.612105	+	1.06020i	0.990885	+	0.134711i	\(0.0430105\pi\)
							−0.378780	+	0.925487i	\(0.623656\pi\)
\(29\)	1.85385e6	−	3.21095e6i	0.486724	−	0.843030i	−0.513160	−	0.858293i	\(-0.671525\pi\)
							0.999884	+	0.0152629i	\(0.00485852\pi\)
\(31\)	3.70113e6	−	2.13685e6i	0.719792	−	0.415572i	−0.0948840	−	0.995488i	\(-0.530248\pi\)
							0.814676	+	0.579916i	\(0.196915\pi\)
\(37\)	1.01833e7	−	5.87932e6i	0.893265	−	0.515727i	0.0182558	−	0.999833i	\(-0.494189\pi\)
							0.875009	+	0.484107i	\(0.160855\pi\)
\(41\)	456047.	+	263299.i	0.0252048	+	0.0145520i	0.512549	−	0.858658i	\(-0.328701\pi\)
							−0.487345	+	0.873210i	\(0.662034\pi\)
\(43\)	3.24058e6	+	5.61286e6i	0.144549	+	0.250366i	0.929205	−	0.369566i	\(-0.120494\pi\)
							−0.784656	+	0.619932i	\(0.787160\pi\)
\(47\)	−4.45286e6	−	2.57086e6i	−0.133106	−	0.0768490i	0.431968	−	0.901889i	\(-0.357819\pi\)
							−0.565075	+	0.825040i	\(0.691153\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.u.a.30.15	yes	164
7.4	even	3		91.10.k.a.4.15	✓	164
13.10	even	6		91.10.k.a.23.68	yes	164
91.88	even	6	inner	91.10.u.a.88.15	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.k.a.4.15	✓	164	7.4	even	3
91.10.k.a.23.68	yes	164	13.10	even	6
91.10.u.a.30.15	yes	164	1.1	even	1	trivial
91.10.u.a.88.15	yes	164	91.88	even	6	inner