Embedded newform 117.4.t.a.103.17

• Label: 117.4.t.a.103.17
• 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			103.17
Character	\(\chi\)	\(=\)	117.103
Dual form			117.4.t.a.25.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00243 + 0.578751i) q^{2} +(4.54931 + 2.51073i) q^{3} +(-3.33009 + 5.76789i) q^{4} +(9.98169 + 5.76293i) q^{5} +(-6.01344 + 0.116103i) q^{6} +(23.3870 - 13.5025i) q^{7} -16.9692i q^{8} +(14.3925 + 22.8442i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 150 q^{4} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00243	+	0.578751i	−0.354411	+	0.204619i	−0.666626	−	0.745392i	\(-0.732262\pi\)
							0.312215	+	0.950011i	\(0.398929\pi\)
\(3\)	4.54931	+	2.51073i	0.875516	+	0.483189i
							\(5\)	9.98169	+	5.76293i	0.892789	+	0.515452i	0.874854	−	0.484387i	\(-0.160957\pi\)
0.0179354	+	0.999839i	\(0.494291\pi\)
\(7\)	23.3870	−	13.5025i	1.26278	−	0.729066i	0.289168	−	0.957278i	\(-0.406621\pi\)
							0.973611	+	0.228212i	\(0.0732880\pi\)
\(11\)	−7.49643	+	4.32807i	−0.205478	+	0.118633i	−0.599208	−	0.800593i	\(-0.704518\pi\)
							0.393730	+	0.919226i	\(0.371184\pi\)
\(13\)	−35.5404	+	30.5595i	−0.758241	+	0.651975i
							\(17\)	30.4896			0.434989			0.217494	−	0.976062i	\(-0.430212\pi\)
0.217494	+	0.976062i	\(0.430212\pi\)
\(19\)			30.3459i			0.366412i	0.983075	+	0.183206i	\(0.0586476\pi\)
							−0.983075	+	0.183206i	\(0.941352\pi\)
\(23\)	3.48285	−	6.03248i	0.0315750	−	0.0546895i	−0.849806	−	0.527095i	\(-0.823281\pi\)
							0.881381	+	0.472406i	\(0.156614\pi\)
\(29\)	88.4796	+	153.251i	0.566560	+	0.981311i	0.996903	+	0.0786451i	\(0.0250594\pi\)
							−0.430343	+	0.902666i	\(0.641607\pi\)
\(31\)	−272.984	−	157.608i	−1.58160	−	0.913134i	−0.994627	−	0.103525i	\(-0.966988\pi\)
							−0.586968	−	0.809610i	\(-0.699679\pi\)
\(37\)		−	122.460i		−	0.544116i	−0.962281	−	0.272058i	\(-0.912296\pi\)
							0.962281	−	0.272058i	\(-0.0877043\pi\)
\(41\)	92.3863	+	53.3393i	0.351910	+	0.203175i	0.665526	−	0.746374i	\(-0.268207\pi\)
							−0.313616	+	0.949550i	\(0.601540\pi\)
\(43\)	−185.168	−	320.721i	−0.656696	−	1.13743i	−0.981466	−	0.191637i	\(-0.938620\pi\)
							0.324770	−	0.945793i	\(-0.394713\pi\)
\(47\)	246.673	−	142.417i	0.765552	−	0.441991i	−0.0657338	−	0.997837i	\(-0.520939\pi\)
							0.831285	+	0.555846i	\(0.187605\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.t.a.103.17	yes	80
3.2	odd	2		351.4.t.a.64.24		80
9.2	odd	6		351.4.t.a.181.17		80
9.7	even	3	inner	117.4.t.a.25.24	yes	80
13.12	even	2	inner	117.4.t.a.103.24	yes	80
39.38	odd	2		351.4.t.a.64.17		80
117.25	even	6	inner	117.4.t.a.25.17	✓	80
117.38	odd	6		351.4.t.a.181.24		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.t.a.25.17	✓	80	117.25	even	6	inner
117.4.t.a.25.24	yes	80	9.7	even	3	inner
117.4.t.a.103.17	yes	80	1.1	even	1	trivial
117.4.t.a.103.24	yes	80	13.12	even	2	inner
351.4.t.a.64.17		80	39.38	odd	2
351.4.t.a.64.24		80	3.2	odd	2
351.4.t.a.181.17		80	9.2	odd	6
351.4.t.a.181.24		80	117.38	odd	6