Embedded newform 117.4.t.a.103.30

• Label: 117.4.t.a.103.30
• Level: $117$
• Weight: $4$
• Character: 117.103
• Analytic conductor: $6.903$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.t (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.90322347067\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			103.30
Character	\(\chi\)	\(=\)	117.103
Dual form			117.4.t.a.25.30

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.63673 - 1.52232i) q^{2} +(0.490546 + 5.17295i) q^{3} +(0.634892 - 1.09966i) q^{4} +(4.80701 + 2.77533i) q^{5} +(9.16829 + 12.8929i) q^{6} +(8.73788 - 5.04482i) q^{7} +20.4910i q^{8} +(-26.5187 + 5.07513i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 150 q^{4} - 68 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(92\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.63673	−	1.52232i	0.932224	−	0.538220i	0.0447100	−	0.999000i	\(-0.485764\pi\)
							0.887514	+	0.460780i	\(0.152430\pi\)
\(3\)	0.490546	+	5.17295i	0.0944055	+	0.995534i
							\(5\)	4.80701	+	2.77533i	0.429952	+	0.248233i	0.699326	−	0.714802i	\(-0.253483\pi\)
−0.269374	+	0.963036i	\(0.586817\pi\)
\(7\)	8.73788	−	5.04482i	0.471801	−	0.272395i	−0.245192	−	0.969474i	\(-0.578851\pi\)
							0.716993	+	0.697080i	\(0.245518\pi\)
\(11\)	−13.9366	+	8.04628i	−0.382003	+	0.220549i	−0.678689	−	0.734425i	\(-0.737452\pi\)
							0.296686	+	0.954975i	\(0.404118\pi\)
\(13\)	24.3800	+	40.0327i	0.520137	+	0.854083i
							\(17\)	68.7556			0.980923			0.490461	−	0.871463i	\(-0.336828\pi\)
0.490461	+	0.871463i	\(0.336828\pi\)
\(19\)		−	7.26478i		−	0.0877187i	−0.999038	−	0.0438593i	\(-0.986035\pi\)
							0.999038	−	0.0438593i	\(-0.0139653\pi\)
\(23\)	59.2055	−	102.547i	0.536748	−	0.929674i	−0.462329	−	0.886709i	\(-0.652986\pi\)
							0.999077	−	0.0429658i	\(-0.0136806\pi\)
\(29\)	−69.2119	−	119.879i	−0.443184	−	0.767617i	0.554740	−	0.832024i	\(-0.312818\pi\)
							−0.997924	+	0.0644070i	\(0.979484\pi\)
\(31\)	5.73632	+	3.31187i	0.0332346	+	0.0191880i	0.516525	−	0.856272i	\(-0.327225\pi\)
							−0.483291	+	0.875460i	\(0.660559\pi\)
\(37\)		−	228.565i		−	1.01556i	−0.861485	−	0.507782i	\(-0.830465\pi\)
							0.861485	−	0.507782i	\(-0.169535\pi\)
\(41\)	213.771	+	123.421i	0.814280	+	0.470125i	0.848440	−	0.529292i	\(-0.177542\pi\)
							−0.0341602	+	0.999416i	\(0.510876\pi\)
\(43\)	−48.2284	−	83.5341i	−0.171041	−	0.296252i	0.767743	−	0.640758i	\(-0.221380\pi\)
							−0.938784	+	0.344506i	\(0.888046\pi\)
\(47\)	245.615	−	141.806i	0.762268	−	0.440096i	−0.0678412	−	0.997696i	\(-0.521611\pi\)
							0.830110	+	0.557600i	\(0.188278\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.t.a.103.30	yes	80
3.2	odd	2		351.4.t.a.64.11		80
9.2	odd	6		351.4.t.a.181.30		80
9.7	even	3	inner	117.4.t.a.25.11	✓	80
13.12	even	2	inner	117.4.t.a.103.11	yes	80
39.38	odd	2		351.4.t.a.64.30		80
117.25	even	6	inner	117.4.t.a.25.30	yes	80
117.38	odd	6		351.4.t.a.181.11		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.t.a.25.11	✓	80	9.7	even	3	inner
117.4.t.a.25.30	yes	80	117.25	even	6	inner
117.4.t.a.103.11	yes	80	13.12	even	2	inner
117.4.t.a.103.30	yes	80	1.1	even	1	trivial
351.4.t.a.64.11		80	3.2	odd	2
351.4.t.a.64.30		80	39.38	odd	2
351.4.t.a.181.11		80	117.38	odd	6
351.4.t.a.181.30		80	9.2	odd	6