Embedded newform 91.10.i.a.34.3

• Label: 91.10.i.a.34.3
• 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			34.3
Character	\(\chi\)	\(=\)	91.34
Dual form			91.10.i.a.83.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-30.1481 - 30.1481i) q^{2} -41.9129i q^{3} +1305.82i q^{4} +(47.1446 - 47.1446i) q^{5} +(-1263.60 + 1263.60i) q^{6} +(6068.45 + 1878.17i) q^{7} +(23932.2 - 23932.2i) q^{8} +17926.3 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 4 q^{2} - 1142 q^{7} + 2044 q^{8} - 1023524 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−30.1481	−	30.1481i	−1.33237	−	1.33237i	−0.903243	−	0.429129i	\(-0.858820\pi\)
							−0.429129	−	0.903243i	\(-0.641180\pi\)
\(3\)		−	41.9129i		−	0.298746i	−0.988781	−	0.149373i	\(-0.952274\pi\)
							0.988781	−	0.149373i	\(-0.0477256\pi\)
\(5\)	47.1446	−	47.1446i	0.0337339	−	0.0337339i	−0.690039	−	0.723773i	\(-0.742406\pi\)
							0.723773	+	0.690039i	\(0.242406\pi\)
\(7\)	6068.45	+	1878.17i	0.955293	+	0.295661i
							\(11\)	6097.56	−	6097.56i	0.125571	−	0.125571i	−0.641528	−	0.767099i	\(-0.721699\pi\)
0.767099	+	0.641528i	\(0.221699\pi\)
\(13\)	−99897.3	+	25000.4i	−0.970083	+	0.242774i
							\(17\)	287963.			0.836213			0.418106	−	0.908398i	\(-0.362694\pi\)
0.418106	+	0.908398i	\(0.362694\pi\)
\(19\)	254439.	−	254439.i	0.447912	−	0.447912i	−0.446748	−	0.894660i	\(-0.647418\pi\)
							0.894660	+	0.446748i	\(0.147418\pi\)
\(23\)			847990.i			0.631852i	0.948784	+	0.315926i	\(0.102315\pi\)
							−0.948784	+	0.315926i	\(0.897685\pi\)
\(29\)	2.72388e6			0.715148			0.357574	−	0.933885i	\(-0.383604\pi\)
							0.357574	+	0.933885i	\(0.383604\pi\)
\(31\)	−3.46378e6	+	3.46378e6i	−0.673632	+	0.673632i	−0.958552	−	0.284919i	\(-0.908033\pi\)
							0.284919	+	0.958552i	\(0.408033\pi\)
\(37\)	−7.21559e6	+	7.21559e6i	−0.632942	+	0.632942i	−0.948805	−	0.315863i	\(-0.897706\pi\)
							0.315863	+	0.948805i	\(0.397706\pi\)
\(41\)	−1.29794e7	+	1.29794e7i	−0.717342	+	0.717342i	−0.968060	−	0.250718i	\(-0.919333\pi\)
							0.250718	+	0.968060i	\(0.419333\pi\)
\(43\)			6.76198e6i			0.301624i	0.988562	+	0.150812i	\(0.0481888\pi\)
							−0.988562	+	0.150812i	\(0.951811\pi\)
\(47\)	2.96519e7	+	2.96519e7i	0.886364	+	0.886364i	0.994172	−	0.107807i	\(-0.0343830\pi\)
							−0.107807	+	0.994172i	\(0.534383\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.i.a.34.3	✓	164
7.6	odd	2	inner	91.10.i.a.34.4	yes	164
13.5	odd	4	inner	91.10.i.a.83.3	yes	164
91.83	even	4	inner	91.10.i.a.83.4	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.i.a.34.3	✓	164	1.1	even	1	trivial
91.10.i.a.34.4	yes	164	7.6	odd	2	inner
91.10.i.a.83.3	yes	164	13.5	odd	4	inner
91.10.i.a.83.4	yes	164	91.83	even	4	inner