Embedded newform 91.8.h.a.16.4

• Label: 91.8.h.a.16.4
• Level: $91$
• Weight: $8$
• Character: 91.16
• Analytic conductor: $28.427$
• Analytic rank: $0$
• Dimension: $126$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 91 = 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	91.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.4270373191\)
Analytic rank:	\(0\)
Dimension:	\(126\)
Relative dimension:	\(63\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			16.4
Character	\(\chi\)	\(=\)	91.16
Dual form			91.8.h.a.74.4

$q$-expansion

\(f(q)\)	\(=\)	\(q-20.9336 q^{2} +(26.1734 - 45.3336i) q^{3} +310.215 q^{4} +(-90.5560 + 156.848i) q^{5} +(-547.902 + 948.995i) q^{6} +(-623.340 + 659.538i) q^{7} -3814.40 q^{8} +(-276.592 - 479.071i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q - 2 q^{2} - 80 q^{3} + 7678 q^{4} - 2 q^{5} - 258 q^{6} + 953 q^{7} - 264 q^{8} - 41707 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−20.9336			−1.85028			−0.925142	−	0.379621i	\(-0.876054\pi\)
							−0.925142	+	0.379621i	\(0.876054\pi\)
\(3\)	26.1734	−	45.3336i	0.559674	−	0.969384i	−0.437849	−	0.899049i	\(-0.644260\pi\)
							0.997523	−	0.0703359i	\(-0.0224071\pi\)
\(5\)	−90.5560	+	156.848i	−0.323983	+	0.561155i	−0.981306	−	0.192454i	\(-0.938355\pi\)
							0.657323	+	0.753609i	\(0.271689\pi\)
\(7\)	−623.340	+	659.538i	−0.686882	+	0.726769i
							\(11\)	−1138.25	+	1971.50i	−0.257847	+	0.446604i	−0.965665	−	0.259791i	\(-0.916346\pi\)
0.707818	+	0.706395i	\(0.249680\pi\)
\(13\)	−3529.51	−	7091.62i	−0.445566	−	0.895249i
							\(17\)	−20366.3			−1.00540			−0.502702	−	0.864460i	\(-0.667661\pi\)
−0.502702	+	0.864460i	\(0.667661\pi\)
\(19\)	8628.14	+	14944.4i	0.288589	+	0.499851i	0.973473	−	0.228801i	\(-0.0734806\pi\)
							−0.684884	+	0.728652i	\(0.740147\pi\)
\(23\)	1416.24			0.0242711			0.0121356	−	0.999926i	\(-0.496137\pi\)
							0.0121356	+	0.999926i	\(0.496137\pi\)
\(29\)	−82149.8	−	142288.i	−0.625480	−	1.08336i	−0.988448	−	0.151562i	\(-0.951570\pi\)
							0.362968	−	0.931802i	\(-0.381764\pi\)
\(31\)	−36618.3	−	63424.8i	−0.220766	−	0.382378i	0.734275	−	0.678853i	\(-0.237522\pi\)
							−0.955041	+	0.296474i	\(0.904189\pi\)
\(37\)	449902.			1.46020			0.730100	−	0.683340i	\(-0.239473\pi\)
							0.730100	+	0.683340i	\(0.239473\pi\)
\(41\)	269436.	+	466677.i	0.610537	+	1.05748i	0.991150	+	0.132747i	\(0.0423797\pi\)
							−0.380613	+	0.924734i	\(0.624287\pi\)
\(43\)	361131.	−	625497.i	0.692668	−	1.19974i	−0.278293	−	0.960496i	\(-0.589769\pi\)
							0.970961	−	0.239239i	\(-0.0768981\pi\)
\(47\)	478301.	−	828441.i	0.671983	−	1.16391i	−0.305357	−	0.952238i	\(-0.598776\pi\)
							0.977341	−	0.211672i	\(-0.0678907\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	91.8.h.a.16.4	yes	126
7.4	even	3		91.8.g.a.81.60	yes	126
13.9	even	3		91.8.g.a.9.60	✓	126
91.74	even	3	inner	91.8.h.a.74.4	yes	126

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.8.g.a.9.60	✓	126	13.9	even	3
91.8.g.a.81.60	yes	126	7.4	even	3
91.8.h.a.16.4	yes	126	1.1	even	1	trivial
91.8.h.a.74.4	yes	126	91.74	even	3	inner