Embedded newform 91.10.u.a.30.14

• Label: 91.10.u.a.30.14
• 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			30.14
Character	\(\chi\)	\(=\)	91.30
Dual form			91.10.u.a.88.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-28.2880 + 16.3321i) q^{2} +21.2897 q^{3} +(277.473 - 480.598i) q^{4} +(611.718 + 353.175i) q^{5} +(-602.244 + 347.706i) q^{6} +(-6100.90 - 1769.93i) q^{7} +1402.82i q^{8} -19229.7 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 164 q - 3 q^{2} + 322 q^{3} + 20481 q^{4} - 6 q^{6} + 1023 q^{7} + 1021818 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−28.2880	+	16.3321i	−1.25016	+	0.721783i	−0.971142	−	0.238502i	\(-0.923344\pi\)
							−0.279022	+	0.960285i	\(0.590010\pi\)
\(3\)	21.2897			0.151749			0.0758743	−	0.997117i	\(-0.475825\pi\)
							0.0758743	+	0.997117i	\(0.475825\pi\)
\(5\)	611.718	+	353.175i	0.437709	+	0.252712i	0.702626	−	0.711560i	\(-0.252011\pi\)
							−0.264916	+	0.964271i	\(0.585344\pi\)
\(7\)	−6100.90	−	1769.93i	−0.960401	−	0.278622i
							\(11\)			51484.3i			1.06025i	0.847920	+	0.530124i	\(0.177855\pi\)
−0.847920	+	0.530124i	\(0.822145\pi\)
\(13\)	32769.4	−	97625.1i	0.318217	−	0.948018i
							\(17\)	−298943.	+	517784.i	−0.868096	+	1.50359i	−0.00415534	+	0.999991i	\(0.501323\pi\)
−0.863940	+	0.503594i	\(0.832011\pi\)
\(19\)			622576.i			1.09598i	0.836486	+	0.547988i	\(0.184606\pi\)
							−0.836486	+	0.547988i	\(0.815394\pi\)
\(23\)	−769746.	−	1.33324e6i	−0.573551	−	0.993420i	−0.996197	−	0.0871249i	\(-0.972232\pi\)
							0.422646	−	0.906295i	\(-0.361101\pi\)
\(29\)	−1.23260e6	+	2.13492e6i	−0.323617	+	0.560521i	−0.981231	−	0.192834i	\(-0.938232\pi\)
							0.657615	+	0.753354i	\(0.271565\pi\)
\(31\)	4.65040e6	−	2.68491e6i	0.904405	−	0.522158i	0.0257781	−	0.999668i	\(-0.491794\pi\)
							0.878627	+	0.477509i	\(0.158460\pi\)
\(37\)	−3.87908e6	+	2.23959e6i	−0.340268	+	0.196454i	−0.660391	−	0.750922i	\(-0.729609\pi\)
							0.320122	+	0.947376i	\(0.396276\pi\)
\(41\)	−1.10201e7	−	6.36245e6i	−0.609056	−	0.351639i	0.163540	−	0.986537i	\(-0.447709\pi\)
							−0.772596	+	0.634898i	\(0.781042\pi\)
\(43\)	−1.11051e7	−	1.92346e7i	−0.495353	−	0.857976i	0.504633	−	0.863334i	\(-0.331628\pi\)
							−0.999986	+	0.00535808i	\(0.998294\pi\)
\(47\)	−1.42729e7	−	8.24045e6i	−0.426649	−	0.246326i	0.271269	−	0.962504i	\(-0.412557\pi\)
							−0.697918	+	0.716178i	\(0.745890\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.10.u.a.30.14	yes	164
7.4	even	3		91.10.k.a.4.14	✓	164
13.10	even	6		91.10.k.a.23.69	yes	164
91.88	even	6	inner	91.10.u.a.88.14	yes	164

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.10.k.a.4.14	✓	164	7.4	even	3
91.10.k.a.23.69	yes	164	13.10	even	6
91.10.u.a.30.14	yes	164	1.1	even	1	trivial
91.10.u.a.88.14	yes	164	91.88	even	6	inner