Embedded newform 91.8.r.a.25.13

• Label: 91.8.r.a.25.13
• 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			25.13
Character	\(\chi\)	\(=\)	91.25
Dual form			91.8.r.a.51.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-13.0889 + 7.55687i) q^{2} +(9.02507 - 15.6319i) q^{3} +(50.2126 - 86.9708i) q^{4} +(23.7868 - 13.7333i) q^{5} +272.805i q^{6} +(635.750 - 647.584i) q^{7} -416.758i q^{8} +(930.596 + 1611.84i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 128 q + 52 q^{3} + 4094 q^{4} - 44052 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−13.0889	+	7.55687i	−1.15691	+	0.667939i	−0.950560	−	0.310540i	\(-0.899490\pi\)
							−0.206345	+	0.978479i	\(0.566157\pi\)
\(3\)	9.02507	−	15.6319i	0.192986	−	0.334262i	−0.753252	−	0.657732i	\(-0.771516\pi\)
							0.946239	+	0.323470i	\(0.104849\pi\)
\(5\)	23.7868	−	13.7333i	0.0851021	−	0.0491337i	−0.456845	−	0.889546i	\(-0.651021\pi\)
							0.541947	+	0.840413i	\(0.317687\pi\)
\(7\)	635.750	−	647.584i	0.700556	−	0.713597i
							\(11\)	−1170.38	−	675.718i	−0.265126	−	0.153070i	0.361545	−	0.932355i	\(-0.382249\pi\)
−0.626671	+	0.779284i	\(0.715583\pi\)
\(13\)	5143.23	−	6024.59i	0.649284	−	0.760546i
							\(17\)	586.797	−	1016.36i	0.0289679	−	0.0501738i	−0.851178	−	0.524877i	\(-0.824111\pi\)
0.880146	+	0.474703i	\(0.157445\pi\)
\(19\)	−23179.1	+	13382.5i	−0.775282	+	0.447609i	−0.834756	−	0.550621i	\(-0.814391\pi\)
							0.0594736	+	0.998230i	\(0.481058\pi\)
\(23\)	−30707.2	−	53186.5i	−0.526252	−	0.911495i	−0.999532	−	0.0305828i	\(-0.990264\pi\)
							0.473281	−	0.880912i	\(-0.343070\pi\)
\(29\)	−103086.			−0.784883			−0.392441	−	0.919777i	\(-0.628369\pi\)
							−0.392441	+	0.919777i	\(0.628369\pi\)
\(31\)	−93473.9	−	53967.2i	−0.563540	−	0.325360i	0.191025	−	0.981585i	\(-0.438819\pi\)
							−0.754565	+	0.656225i	\(0.772152\pi\)
\(37\)	203403.	−	117435.i	0.660164	−	0.381146i	−0.132175	−	0.991226i	\(-0.542196\pi\)
							0.792339	+	0.610080i	\(0.208863\pi\)
\(41\)		−	241410.i		−	0.547030i	−0.961868	−	0.273515i	\(-0.911814\pi\)
							0.961868	−	0.273515i	\(-0.0881863\pi\)
\(43\)	852537.			1.63521			0.817606	−	0.575779i	\(-0.195301\pi\)
							0.817606	+	0.575779i	\(0.195301\pi\)
\(47\)	1.10928e6	−	640441.i	1.55846	−	0.899780i	0.561061	−	0.827775i	\(-0.310393\pi\)
							0.997404	−	0.0720054i	\(-0.0229399\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.r.a.25.13	✓	128
7.2	even	3	inner	91.8.r.a.51.52	yes	128
13.12	even	2	inner	91.8.r.a.25.52	yes	128
91.51	even	6	inner	91.8.r.a.51.13	yes	128

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.r.a.25.13	✓	128	1.1	even	1	trivial
91.8.r.a.25.52	yes	128	13.12	even	2	inner
91.8.r.a.51.13	yes	128	91.51	even	6	inner
91.8.r.a.51.52	yes	128	7.2	even	3	inner