Embedded newform 91.10.i.a.34.9

• Label: 91.10.i.a.34.9
• 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			34.9
Character	\(\chi\)	\(=\)	91.34
Dual form			91.10.i.a.83.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-26.0147 - 26.0147i) q^{2} -172.872i q^{3} +841.530i q^{4} +(-924.961 + 924.961i) q^{5} +(-4497.22 + 4497.22i) q^{6} +(3450.17 + 5333.85i) q^{7} +(8572.61 - 8572.61i) q^{8} -10201.8 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 4 q^{2} - 1142 q^{7} + 2044 q^{8} - 1023524 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−26.0147	−	26.0147i	−1.14970	−	1.14970i	−0.986612	−	0.163087i	\(-0.947855\pi\)
							−0.163087	−	0.986612i	\(-0.552145\pi\)
\(3\)		−	172.872i		−	1.23220i	−0.787670	−	0.616098i	\(-0.788713\pi\)
							0.787670	−	0.616098i	\(-0.211287\pi\)
\(5\)	−924.961	+	924.961i	−0.661848	+	0.661848i	−0.955815	−	0.293967i	\(-0.905024\pi\)
							0.293967	+	0.955815i	\(0.405024\pi\)
\(7\)	3450.17	+	5333.85i	0.543124	+	0.839652i
							\(11\)	11181.3	−	11181.3i	0.230264	−	0.230264i	−0.582539	−	0.812803i	\(-0.697941\pi\)
0.812803	+	0.582539i	\(0.197941\pi\)
\(13\)	100662.	−	21719.3i	0.977505	−	0.210911i
							\(17\)	−265306.			−0.770419			−0.385209	−	0.922829i	\(-0.625871\pi\)
−0.385209	+	0.922829i	\(0.625871\pi\)
\(19\)	−317865.	+	317865.i	−0.559565	+	0.559565i	−0.929184	−	0.369618i	\(-0.879488\pi\)
							0.369618	+	0.929184i	\(0.379488\pi\)
\(23\)			439495.i			0.327475i	0.986504	+	0.163738i	\(0.0523550\pi\)
							−0.986504	+	0.163738i	\(0.947645\pi\)
\(29\)	−3.55792e6			−0.934127			−0.467063	−	0.884224i	\(-0.654688\pi\)
							−0.467063	+	0.884224i	\(0.654688\pi\)
\(31\)	−1.61453e6	+	1.61453e6i	−0.313992	+	0.313992i	−0.846454	−	0.532462i	\(-0.821267\pi\)
							0.532462	+	0.846454i	\(0.321267\pi\)
\(37\)	1.22514e7	−	1.22514e7i	1.07468	−	1.07468i	0.0777034	−	0.996977i	\(-0.475241\pi\)
							0.996977	−	0.0777034i	\(-0.0247587\pi\)
\(41\)	5.54493e6	−	5.54493e6i	0.306456	−	0.306456i	−0.537077	−	0.843533i	\(-0.680471\pi\)
							0.843533	+	0.537077i	\(0.180471\pi\)
\(43\)		−	1.30339e7i		−	0.581387i	−0.956816	−	0.290694i	\(-0.906114\pi\)
							0.956816	−	0.290694i	\(-0.0938861\pi\)
\(47\)	−4.76843e6	−	4.76843e6i	−0.142540	−	0.142540i	0.632236	−	0.774776i	\(-0.282137\pi\)
							−0.774776	+	0.632236i	\(0.782137\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.i.a.34.9	✓	164
7.6	odd	2	inner	91.10.i.a.34.10	yes	164
13.5	odd	4	inner	91.10.i.a.83.9	yes	164
91.83	even	4	inner	91.10.i.a.83.10	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.i.a.34.9	✓	164	1.1	even	1	trivial
91.10.i.a.34.10	yes	164	7.6	odd	2	inner
91.10.i.a.83.9	yes	164	13.5	odd	4	inner
91.10.i.a.83.10	yes	164	91.83	even	4	inner