Embedded newform 91.10.i.a.34.16

• Label: 91.10.i.a.34.16
• Level: $91$
• Weight: $10$
• Character: 91.34
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			34.16
Character	\(\chi\)	\(=\)	91.34
Dual form			91.10.i.a.83.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-21.7082 - 21.7082i) q^{2} +61.2207i q^{3} +430.495i q^{4} +(1460.95 - 1460.95i) q^{5} +(1328.99 - 1328.99i) q^{6} +(-2376.03 + 5891.36i) q^{7} +(-1769.32 + 1769.32i) q^{8} +15935.0 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 4 q^{2} - 1142 q^{7} + 2044 q^{8} - 1023524 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−21.7082	−	21.7082i	−0.959378	−	0.959378i	0.0398288	−	0.999207i	\(-0.487319\pi\)
							−0.999207	+	0.0398288i	\(0.987319\pi\)
\(3\)			61.2207i			0.436368i	0.975908	+	0.218184i	\(0.0700132\pi\)
							−0.975908	+	0.218184i	\(0.929987\pi\)
\(5\)	1460.95	−	1460.95i	1.04537	−	1.04537i	0.0464491	−	0.998921i	\(-0.485209\pi\)
							0.998921	−	0.0464491i	\(-0.0147905\pi\)
\(7\)	−2376.03	+	5891.36i	−0.374034	+	0.927415i
							\(11\)	51114.1	−	51114.1i	1.05262	−	1.05262i	0.0540884	−	0.998536i	\(-0.482775\pi\)
0.998536	−	0.0540884i	\(-0.0172253\pi\)
\(13\)	56516.7	+	86083.5i	0.548822	+	0.835939i
							\(17\)	−244293.			−0.709400			−0.354700	−	0.934980i	\(-0.615417\pi\)
−0.354700	+	0.934980i	\(0.615417\pi\)
\(19\)	31172.6	−	31172.6i	0.0548758	−	0.0548758i	−0.679136	−	0.734012i	\(-0.737645\pi\)
							0.734012	+	0.679136i	\(0.237645\pi\)
\(23\)			1.79997e6i			1.34119i	0.741823	+	0.670595i	\(0.233961\pi\)
							−0.741823	+	0.670595i	\(0.766039\pi\)
\(29\)	823404.			0.216183			0.108092	−	0.994141i	\(-0.465526\pi\)
							0.108092	+	0.994141i	\(0.465526\pi\)
\(31\)	−5.70880e6	+	5.70880e6i	−1.11024	+	1.11024i	−0.117122	+	0.993117i	\(0.537367\pi\)
							−0.993117	+	0.117122i	\(0.962633\pi\)
\(37\)	−1.28889e7	+	1.28889e7i	−1.13060	+	1.13060i	−0.140520	+	0.990078i	\(0.544877\pi\)
							−0.990078	+	0.140520i	\(0.955123\pi\)
\(41\)	−670979.	+	670979.i	−0.0370836	+	0.0370836i	−0.725405	−	0.688322i	\(-0.758348\pi\)
							0.688322	+	0.725405i	\(0.258348\pi\)
\(43\)			2.01846e7i			0.900351i	0.892940	+	0.450176i	\(0.148639\pi\)
							−0.892940	+	0.450176i	\(0.851361\pi\)
\(47\)	2.39226e7	+	2.39226e7i	0.715101	+	0.715101i	0.967598	−	0.252497i	\(-0.0812516\pi\)
							−0.252497	+	0.967598i	\(0.581252\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.i.a.34.16	yes	164
7.6	odd	2	inner	91.10.i.a.34.15	✓	164
13.5	odd	4	inner	91.10.i.a.83.16	yes	164
91.83	even	4	inner	91.10.i.a.83.15	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.i.a.34.15	✓	164	7.6	odd	2	inner
91.10.i.a.34.16	yes	164	1.1	even	1	trivial
91.10.i.a.83.15	yes	164	91.83	even	4	inner
91.10.i.a.83.16	yes	164	13.5	odd	4	inner