Embedded newform 91.10.u.a.30.12

• Label: 91.10.u.a.30.12
• Level: $91$
• Weight: $10$
• Character: 91.30
• Analytic conductor: $46.868$
• Analytic rank: $0$
• Dimension: $164$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	91.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(46.8682610909\)
Analytic rank:	\(0\)
Dimension:	\(164\)
Relative dimension:	\(82\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			30.12
Character	\(\chi\)	\(=\)	91.30
Dual form			91.10.u.a.88.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-29.1216 + 16.8134i) q^{2} -81.0362 q^{3} +(309.379 - 535.861i) q^{4} +(1658.21 + 957.369i) q^{5} +(2359.91 - 1362.49i) q^{6} +(4828.80 + 4127.50i) q^{7} +3589.95i q^{8} -13116.1 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 3 q^{2} + 322 q^{3} + 20481 q^{4} - 6 q^{6} + 1023 q^{7} + 1021818 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(15\)	\(66\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−29.1216	+	16.8134i	−1.28701	+	0.743053i	−0.978119	−	0.208045i	\(-0.933290\pi\)
							−0.308887	+	0.951099i	\(0.599956\pi\)
\(3\)	−81.0362			−0.577608			−0.288804	−	0.957388i	\(-0.593258\pi\)
							−0.288804	+	0.957388i	\(0.593258\pi\)
\(5\)	1658.21	+	957.369i	1.18652	+	0.685037i	0.957514	−	0.288388i	\(-0.0931192\pi\)
							0.229006	+	0.973425i	\(0.426452\pi\)
\(7\)	4828.80	+	4127.50i	0.760148	+	0.649750i
							\(11\)		−	22677.8i		−	0.467018i	−0.972355	−	0.233509i	\(-0.924979\pi\)
0.972355	−	0.233509i	\(-0.0750208\pi\)
\(13\)	−34236.5	−	97120.3i	−0.332464	−	0.943116i
							\(17\)	−7081.97	+	12266.3i	−0.0205653	+	0.0356201i	−0.876125	−	0.482084i	\(-0.839880\pi\)
0.855560	+	0.517704i	\(0.173213\pi\)
\(19\)		−	67371.7i		−	0.118600i	−0.998240	−	0.0593002i	\(-0.981113\pi\)
							0.998240	−	0.0593002i	\(-0.0188869\pi\)
\(23\)	429748.	+	744345.i	0.320213	+	0.554624i	0.980532	−	0.196361i	\(-0.0629123\pi\)
							−0.660319	+	0.750985i	\(0.729579\pi\)
\(29\)	−1.73194e6	+	2.99981e6i	−0.454718	+	0.787594i	−0.998672	−	0.0515208i	\(-0.983593\pi\)
							0.543954	+	0.839115i	\(0.316926\pi\)
\(31\)	−3.87359e6	+	2.23642e6i	−0.753330	+	0.434936i	−0.826896	−	0.562355i	\(-0.809896\pi\)
							0.0735656	+	0.997290i	\(0.476562\pi\)
\(37\)	4.23670e6	−	2.44606e6i	0.371638	−	0.214565i	−0.302536	−	0.953138i	\(-0.597833\pi\)
							0.674174	+	0.738573i	\(0.264500\pi\)
\(41\)	1.94601e7	+	1.12353e7i	1.07552	+	0.620951i	0.929684	−	0.368358i	\(-0.120080\pi\)
							0.145834	+	0.989309i	\(0.453413\pi\)
\(43\)	−1.57427e6	−	2.72672e6i	−0.0702217	−	0.121627i	0.828777	−	0.559579i	\(-0.189037\pi\)
							−0.898998	+	0.437952i	\(0.855704\pi\)
\(47\)	2.99674e7	+	1.73017e7i	0.895794	+	0.517187i	0.875833	−	0.482614i	\(-0.160313\pi\)
							0.0199609	+	0.999801i	\(0.493646\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.u.a.30.12	yes	164
7.4	even	3		91.10.k.a.4.12	✓	164
13.10	even	6		91.10.k.a.23.71	yes	164
91.88	even	6	inner	91.10.u.a.88.12	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.k.a.4.12	✓	164	7.4	even	3
91.10.k.a.23.71	yes	164	13.10	even	6
91.10.u.a.30.12	yes	164	1.1	even	1	trivial
91.10.u.a.88.12	yes	164	91.88	even	6	inner