Embedded newform 91.10.i.a.34.1

• Label: 91.10.i.a.34.1
• 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			34.1
Character	\(\chi\)	\(=\)	91.34
Dual form			91.10.i.a.83.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-30.6699 - 30.6699i) q^{2} -210.836i q^{3} +1369.29i q^{4} +(1324.26 - 1324.26i) q^{5} +(-6466.33 + 6466.33i) q^{6} +(-5803.54 + 2583.12i) q^{7} +(26293.0 - 26293.0i) q^{8} -24768.9 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 4 q^{2} - 1142 q^{7} + 2044 q^{8} - 1023524 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−30.6699	−	30.6699i	−1.35543	−	1.35543i	−0.879461	−	0.475971i	\(-0.842097\pi\)
							−0.475971	−	0.879461i	\(-0.657903\pi\)
\(3\)		−	210.836i		−	1.50279i	−0.659850	−	0.751397i	\(-0.729380\pi\)
							0.659850	−	0.751397i	\(-0.270620\pi\)
\(5\)	1324.26	−	1324.26i	0.947562	−	0.947562i	−0.0511304	−	0.998692i	\(-0.516282\pi\)
							0.998692	+	0.0511304i	\(0.0162824\pi\)
\(7\)	−5803.54	+	2583.12i	−0.913591	+	0.406634i
							\(11\)	53904.3	−	53904.3i	1.11008	−	1.11008i	0.116946	−	0.993138i	\(-0.462689\pi\)
0.993138	−	0.116946i	\(-0.0373106\pi\)
\(13\)	31012.0	−	98197.5i	0.301151	−	0.953576i
							\(17\)	129197.			0.375173			0.187586	−	0.982248i	\(-0.439934\pi\)
0.187586	+	0.982248i	\(0.439934\pi\)
\(19\)	553281.	−	553281.i	0.973990	−	0.973990i	−0.0256799	−	0.999670i	\(-0.508175\pi\)
							0.999670	+	0.0256799i	\(0.00817506\pi\)
\(23\)		−	2.04492e6i		−	1.52371i	−0.647749	−	0.761854i	\(-0.724289\pi\)
							0.647749	−	0.761854i	\(-0.275711\pi\)
\(29\)	−2.66791e6			−0.700454			−0.350227	−	0.936665i	\(-0.613895\pi\)
							−0.350227	+	0.936665i	\(0.613895\pi\)
\(31\)	−1.49435e6	+	1.49435e6i	−0.290619	+	0.290619i	−0.837325	−	0.546706i	\(-0.815882\pi\)
							0.546706	+	0.837325i	\(0.315882\pi\)
\(37\)	−771628.	+	771628.i	−0.0676862	+	0.0676862i	−0.740140	−	0.672453i	\(-0.765241\pi\)
							0.672453	+	0.740140i	\(0.265241\pi\)
\(41\)	−2.72584e6	+	2.72584e6i	−0.150651	+	0.150651i	−0.778409	−	0.627758i	\(-0.783973\pi\)
							0.627758	+	0.778409i	\(0.283973\pi\)
\(43\)		−	3.23111e7i		−	1.44127i	−0.693316	−	0.720633i	\(-0.743851\pi\)
							0.693316	−	0.720633i	\(-0.256149\pi\)
\(47\)	1.40620e7	+	1.40620e7i	0.420347	+	0.420347i	0.885323	−	0.464976i	\(-0.153937\pi\)
							−0.464976	+	0.885323i	\(0.653937\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.i.a.34.1	✓	164
7.6	odd	2	inner	91.10.i.a.34.2	yes	164
13.5	odd	4	inner	91.10.i.a.83.1	yes	164
91.83	even	4	inner	91.10.i.a.83.2	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.i.a.34.1	✓	164	1.1	even	1	trivial
91.10.i.a.34.2	yes	164	7.6	odd	2	inner
91.10.i.a.83.1	yes	164	13.5	odd	4	inner
91.10.i.a.83.2	yes	164	91.83	even	4	inner