Embedded newform 91.8.r.a.25.12

• Label: 91.8.r.a.25.12
• Level: $91$
• Weight: $8$
• Character: 91.25
• Analytic conductor: $28.427$
• Analytic rank: $0$
• Dimension: $128$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	91.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.4270373191\)
Analytic rank:	\(0\)
Dimension:	\(128\)
Relative dimension:	\(64\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			25.12
Character	\(\chi\)	\(=\)	91.25
Dual form			91.8.r.a.51.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.5528 + 7.82474i) q^{2} +(-9.72553 + 16.8451i) q^{3} +(58.4531 - 101.244i) q^{4} +(-434.311 + 250.749i) q^{5} -304.399i q^{6} +(-299.078 - 856.794i) q^{7} -173.614i q^{8} +(904.328 + 1566.34i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 52 q^{3} + 4094 q^{4} - 44052 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.5528	+	7.82474i	−1.19791	+	0.691616i	−0.960090	−	0.279693i	\(-0.909767\pi\)
							−0.237824	+	0.971308i	\(0.576434\pi\)
\(3\)	−9.72553	+	16.8451i	−0.207964	+	0.360205i	−0.951073	−	0.308966i	\(-0.900017\pi\)
							0.743109	+	0.669171i	\(0.233350\pi\)
\(5\)	−434.311	+	250.749i	−1.55384	+	0.897109i	−0.556014	+	0.831173i	\(0.687670\pi\)
							−0.997824	+	0.0659356i	\(0.978997\pi\)
\(7\)	−299.078	−	856.794i	−0.329566	−	0.944133i
							\(11\)	−1042.99	−	602.172i	−0.236269	−	0.136410i	0.377192	−	0.926135i	\(-0.376890\pi\)
−0.613461	+	0.789725i	\(0.710223\pi\)
\(13\)	−6401.54	−	4665.71i	−0.808132	−	0.589001i
							\(17\)	6928.07	−	11999.8i	0.342012	−	0.592382i	−0.642794	−	0.766039i	\(-0.722225\pi\)
0.984806	+	0.173657i	\(0.0555584\pi\)
\(19\)	11556.9	−	6672.35i	0.386547	−	0.223173i	−0.294116	−	0.955770i	\(-0.595025\pi\)
							0.680663	+	0.732597i	\(0.261692\pi\)
\(23\)	−20897.2	−	36194.9i	−0.358129	−	0.620298i	0.629519	−	0.776985i	\(-0.283252\pi\)
							−0.987648	+	0.156687i	\(0.949919\pi\)
\(29\)	176050.			1.34043			0.670215	−	0.742167i	\(-0.266202\pi\)
							0.670215	+	0.742167i	\(0.266202\pi\)
\(31\)	−201847.	−	116536.i	−1.21690	−	0.702579i	−0.252649	−	0.967558i	\(-0.581302\pi\)
							−0.964254	+	0.264979i	\(0.914635\pi\)
\(37\)	−41851.8	+	24163.1i	−0.135834	+	0.0784237i	−0.566377	−	0.824146i	\(-0.691655\pi\)
							0.430543	+	0.902570i	\(0.358322\pi\)
\(41\)			619520.i			1.40382i	0.712265	+	0.701910i	\(0.247669\pi\)
							−0.712265	+	0.701910i	\(0.752331\pi\)
\(43\)	−657462.			−1.26105			−0.630523	−	0.776171i	\(-0.717159\pi\)
							−0.630523	+	0.776171i	\(0.717159\pi\)
\(47\)	−799638.	+	461671.i	−1.12344	+	0.648620i	−0.942278	−	0.334832i	\(-0.891320\pi\)
							−0.181165	+	0.983453i	\(0.557987\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.r.a.25.12	✓	128
7.2	even	3	inner	91.8.r.a.51.53	yes	128
13.12	even	2	inner	91.8.r.a.25.53	yes	128
91.51	even	6	inner	91.8.r.a.51.12	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.r.a.25.12	✓	128	1.1	even	1	trivial
91.8.r.a.25.53	yes	128	13.12	even	2	inner
91.8.r.a.51.12	yes	128	91.51	even	6	inner
91.8.r.a.51.53	yes	128	7.2	even	3	inner