Embedded newform 117.4.t.a.103.33

• Label: 117.4.t.a.103.33
• 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			103.33
Character	\(\chi\)	\(=\)	117.103
Dual form			117.4.t.a.25.33

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.17659 - 1.83400i) q^{2} +(4.26570 + 2.96712i) q^{3} +(2.72714 - 4.72354i) q^{4} +(-1.60561 - 0.926997i) q^{5} +(18.9921 + 1.60199i) q^{6} +(12.1556 - 7.01804i) q^{7} +9.33774i q^{8} +(9.39244 + 25.3137i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 150 q^{4} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.17659	−	1.83400i	1.12309	−	0.648418i	0.180904	−	0.983501i	\(-0.442098\pi\)
							0.942189	+	0.335083i	\(0.108764\pi\)
\(3\)	4.26570	+	2.96712i	0.820935	+	0.571022i
							\(5\)	−1.60561	−	0.926997i	−0.143610	−	0.0829131i	0.426474	−	0.904500i	\(-0.359756\pi\)
−0.570083	+	0.821587i	\(0.693089\pi\)
\(7\)	12.1556	−	7.01804i	0.656341	−	0.378939i	−0.134541	−	0.990908i	\(-0.542956\pi\)
							0.790881	+	0.611970i	\(0.209623\pi\)
\(11\)	52.3521	−	30.2255i	1.43498	−	0.828484i	0.437482	−	0.899227i	\(-0.355871\pi\)
							0.997495	+	0.0707435i	\(0.0225372\pi\)
\(13\)	−31.9328	−	34.3117i	−0.681275	−	0.732028i
							\(17\)	−82.6381			−1.17898			−0.589491	−	0.807775i	\(-0.700671\pi\)
−0.589491	+	0.807775i	\(0.700671\pi\)
\(19\)			66.3365i			0.800981i	0.916301	+	0.400490i	\(0.131160\pi\)
							−0.916301	+	0.400490i	\(0.868840\pi\)
\(23\)	−69.7305	+	120.777i	−0.632166	+	1.09494i	0.354942	+	0.934888i	\(0.384501\pi\)
							−0.987108	+	0.160056i	\(0.948833\pi\)
\(29\)	−148.936	−	257.964i	−0.953678	−	1.65182i	−0.737366	−	0.675494i	\(-0.763931\pi\)
							−0.216312	−	0.976324i	\(-0.569403\pi\)
\(31\)	−19.3931	−	11.1966i	−0.112358	−	0.0648702i	0.442768	−	0.896636i	\(-0.353997\pi\)
							−0.555126	+	0.831766i	\(0.687330\pi\)
\(37\)		−	113.509i		−	0.504344i	−0.967682	−	0.252172i	\(-0.918855\pi\)
							0.967682	−	0.252172i	\(-0.0811449\pi\)
\(41\)	−402.401	−	232.326i	−1.53279	−	0.884958i	−0.999232	−	0.0391958i	\(-0.987520\pi\)
							−0.533560	−	0.845762i	\(-0.679146\pi\)
\(43\)	149.438	+	258.834i	0.529979	+	0.917951i	0.999388	+	0.0349702i	\(0.0111336\pi\)
							−0.469409	+	0.882981i	\(0.655533\pi\)
\(47\)	58.4696	−	33.7575i	0.181461	−	0.104767i	−0.406518	−	0.913643i	\(-0.633257\pi\)
							0.587979	+	0.808876i	\(0.299924\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.t.a.103.33	yes	80
3.2	odd	2		351.4.t.a.64.8		80
9.2	odd	6		351.4.t.a.181.33		80
9.7	even	3	inner	117.4.t.a.25.8	✓	80
13.12	even	2	inner	117.4.t.a.103.8	yes	80
39.38	odd	2		351.4.t.a.64.33		80
117.25	even	6	inner	117.4.t.a.25.33	yes	80
117.38	odd	6		351.4.t.a.181.8		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.t.a.25.8	✓	80	9.7	even	3	inner
117.4.t.a.25.33	yes	80	117.25	even	6	inner
117.4.t.a.103.8	yes	80	13.12	even	2	inner
117.4.t.a.103.33	yes	80	1.1	even	1	trivial
351.4.t.a.64.8		80	3.2	odd	2
351.4.t.a.64.33		80	39.38	odd	2
351.4.t.a.181.8		80	117.38	odd	6
351.4.t.a.181.33		80	9.2	odd	6