Embedded newform 91.8.r.a.51.48

• Label: 91.8.r.a.51.48
• Level: $91$
• Weight: $8$
• Character: 91.51
• 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			51.48
Character	\(\chi\)	\(=\)	91.51
Dual form			91.8.r.a.25.48

$q$-expansion

\(f(q)\)	\(=\)	\(q+(10.9207 + 6.30508i) q^{2} +(-7.65371 - 13.2566i) q^{3} +(15.5082 + 26.8609i) q^{4} +(-37.8419 - 21.8480i) q^{5} -193.029i q^{6} +(534.194 + 733.608i) q^{7} -1222.98i q^{8} +(976.341 - 1691.07i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 52 q^{3} + 4094 q^{4} - 44052 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	10.9207	+	6.30508i	0.965265	+	0.557296i	0.897789	−	0.440425i	\(-0.145172\pi\)
							0.0674754	+	0.997721i	\(0.478506\pi\)
\(3\)	−7.65371	−	13.2566i	−0.163662	−	0.283471i	0.772517	−	0.634994i	\(-0.218997\pi\)
							−0.936179	+	0.351523i	\(0.885664\pi\)
\(5\)	−37.8419	−	21.8480i	−0.135387	−	0.0781658i	0.430777	−	0.902459i	\(-0.358240\pi\)
							−0.566164	+	0.824293i	\(0.691573\pi\)
\(7\)	534.194	+	733.608i	0.588648	+	0.808389i
							\(11\)	−2780.19	+	1605.14i	−0.629797	+	0.363613i	−0.780673	−	0.624939i	\(-0.785124\pi\)
0.150877	+	0.988553i	\(0.451790\pi\)
\(13\)	−7921.40	−	3.49801i	−1.00000	−	0.000441591i
							\(17\)	−11334.8	−	19632.5i	−0.559556	−	0.969180i	−0.997533	−	0.0701939i	\(-0.977638\pi\)
0.437977	−	0.898986i	\(-0.355695\pi\)
\(19\)	8795.81	+	5078.26i	0.294197	+	0.169855i	0.639833	−	0.768514i	\(-0.279004\pi\)
							−0.345636	+	0.938369i	\(0.612337\pi\)
\(23\)	46168.0	−	79965.4i	0.791214	−	1.37042i	−0.134001	−	0.990981i	\(-0.542783\pi\)
							0.925215	−	0.379442i	\(-0.123884\pi\)
\(29\)	14743.7			0.112257			0.0561285	−	0.998424i	\(-0.482124\pi\)
							0.0561285	+	0.998424i	\(0.482124\pi\)
\(31\)	205488.	−	118639.i	1.23886	−	0.715254i	0.269996	−	0.962862i	\(-0.412978\pi\)
							0.968860	+	0.247608i	\(0.0796444\pi\)
\(37\)	−408067.	−	235597.i	−1.32442	−	0.764653i	−0.339988	−	0.940430i	\(-0.610423\pi\)
							−0.984430	+	0.175777i	\(0.943756\pi\)
\(41\)		−	280765.i		−	0.636209i	−0.948056	−	0.318104i	\(-0.896954\pi\)
							0.948056	−	0.318104i	\(-0.103046\pi\)
\(43\)	−16564.7			−0.0317720			−0.0158860	−	0.999874i	\(-0.505057\pi\)
							−0.0158860	+	0.999874i	\(0.505057\pi\)
\(47\)	−93465.3	−	53962.2i	−0.131313	−	0.0758136i	0.432904	−	0.901440i	\(-0.357489\pi\)
							−0.564218	+	0.825626i	\(0.690822\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.r.a.51.48	yes	128
7.4	even	3	inner	91.8.r.a.25.17	✓	128
13.12	even	2	inner	91.8.r.a.51.17	yes	128
91.25	even	6	inner	91.8.r.a.25.48	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.r.a.25.17	✓	128	7.4	even	3	inner
91.8.r.a.25.48	yes	128	91.25	even	6	inner
91.8.r.a.51.17	yes	128	13.12	even	2	inner
91.8.r.a.51.48	yes	128	1.1	even	1	trivial