Embedded newform 92.4.h.a.7.9

• Label: 92.4.h.a.7.9
• Level: $92$
• Weight: $4$
• Character: 92.7
• Analytic conductor: $5.428$
• Analytic rank: $0$
• Dimension: $340$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 92 = 2^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	92.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.42817572053\)
Analytic rank:	\(0\)
Dimension:	\(340\)
Relative dimension:	\(34\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			7.9
Character	\(\chi\)	\(=\)	92.7
Dual form			92.4.h.a.79.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.95779 - 2.04134i) q^{2} +(-2.24967 - 3.50055i) q^{3} +(-0.334121 + 7.99302i) q^{4} +(15.6896 - 7.16519i) q^{5} +(-2.74143 + 11.4457i) q^{6} +(8.54748 - 2.50977i) q^{7} +(16.9706 - 14.9666i) q^{8} +(4.02336 - 8.80993i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 340 q - 11 q^{2} + 13 q^{4} - 22 q^{5} + 2 q^{6} + 22 q^{8} + 252 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/92\mathbb{Z}\right)^\times\).

\(n\)	\(5\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{19}{22}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.95779	−	2.04134i	−0.692183	−	0.721722i
							\(3\)	−2.24967	−	3.50055i	−0.432948	−	0.673681i	0.554399	−	0.832251i	\(-0.312948\pi\)
−0.987348	+	0.158570i	\(0.949312\pi\)
\(5\)	15.6896	−	7.16519i	1.40332	−	0.640874i	0.437292	−	0.899320i	\(-0.355938\pi\)
							0.966026	+	0.258446i	\(0.0832103\pi\)
\(7\)	8.54748	−	2.50977i	0.461520	−	0.135515i	−0.0427031	−	0.999088i	\(-0.513597\pi\)
							0.504224	+	0.863573i	\(0.331779\pi\)
\(11\)	9.60234	+	11.0817i	0.263201	+	0.303750i	0.871933	−	0.489626i	\(-0.162867\pi\)
							−0.608731	+	0.793376i	\(0.708321\pi\)
\(13\)	−30.9800	−	9.09654i	−0.660946	−	0.194071i	−0.0659783	−	0.997821i	\(-0.521017\pi\)
							−0.594967	+	0.803750i	\(0.702835\pi\)
\(17\)	−45.3388	−	6.51873i	−0.646839	−	0.0930014i	−0.188916	−	0.981993i	\(-0.560497\pi\)
							−0.457923	+	0.888992i	\(0.651406\pi\)
\(19\)	−13.6286	−	94.7890i	−0.164559	−	1.14453i	−0.889905	−	0.456147i	\(-0.849229\pi\)
							0.725346	−	0.688385i	\(-0.241680\pi\)
\(23\)	55.9300	+	95.0728i	0.507053	+	0.861915i
							\(29\)	28.4308	−	197.741i	0.182051	−	1.26619i	−0.669854	−	0.742493i	\(-0.733643\pi\)
0.851905	−	0.523697i	\(-0.175448\pi\)
\(31\)	21.9905	−	34.2179i	0.127407	−	0.198249i	−0.771690	−	0.635999i	\(-0.780588\pi\)
							0.899097	+	0.437750i	\(0.144225\pi\)
\(37\)	296.672	+	135.486i	1.31818	+	0.601991i	0.945395	−	0.325927i	\(-0.105676\pi\)
							0.372783	+	0.927919i	\(0.378404\pi\)
\(41\)	−104.334	−	228.459i	−0.397420	−	0.870228i	−0.997525	−	0.0703074i	\(-0.977602\pi\)
							0.600105	−	0.799921i	\(-0.295125\pi\)
\(43\)	16.8020	−	10.7980i	0.0595880	−	0.0382949i	−0.510508	−	0.859873i	\(-0.670542\pi\)
							0.570096	+	0.821578i	\(0.306906\pi\)
\(47\)			385.676i			1.19695i	0.801141	+	0.598475i	\(0.204226\pi\)
							−0.801141	+	0.598475i	\(0.795774\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	92.4.h.a.7.9	✓	340
4.3	odd	2	inner	92.4.h.a.7.28	yes	340
23.10	odd	22	inner	92.4.h.a.79.28	yes	340
92.79	even	22	inner	92.4.h.a.79.9	yes	340

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
92.4.h.a.7.9	✓	340	1.1	even	1	trivial
92.4.h.a.7.28	yes	340	4.3	odd	2	inner
92.4.h.a.79.9	yes	340	92.79	even	22	inner
92.4.h.a.79.28	yes	340	23.10	odd	22	inner