Embedded newform 117.4.t.a.103.29

• Label: 117.4.t.a.103.29
• 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			103.29
Character	\(\chi\)	\(=\)	117.103
Dual form			117.4.t.a.25.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.61982 - 1.51255i) q^{2} +(1.89916 - 4.83665i) q^{3} +(0.575620 - 0.997004i) q^{4} +(-18.8187 - 10.8650i) q^{5} +(-2.34023 - 15.5437i) q^{6} +(-0.934009 + 0.539251i) q^{7} +20.7182i q^{8} +(-19.7864 - 18.3712i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 150 q^{4} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.61982	−	1.51255i	0.926244	−	0.534767i	0.0406226	−	0.999175i	\(-0.487066\pi\)
							0.885622	+	0.464407i	\(0.153733\pi\)
\(3\)	1.89916	−	4.83665i	0.365494	−	0.930814i
							\(5\)	−18.8187	−	10.8650i	−1.68320	−	0.971793i	−0.959515	−	0.281659i	\(-0.909115\pi\)
−0.723681	−	0.690135i	\(-0.757551\pi\)
\(7\)	−0.934009	+	0.539251i	−0.0504318	+	0.0291168i	−0.525004	−	0.851100i	\(-0.675936\pi\)
							0.474572	+	0.880217i	\(0.342603\pi\)
\(11\)	25.6854	−	14.8295i	0.704039	−	0.406477i	−0.104811	−	0.994492i	\(-0.533424\pi\)
							0.808850	+	0.588015i	\(0.200090\pi\)
\(13\)	−35.7767	−	30.2825i	−0.763282	−	0.646066i
							\(17\)	85.5254			1.22017			0.610087	−	0.792334i	\(-0.291134\pi\)
0.610087	+	0.792334i	\(0.291134\pi\)
\(19\)		−	93.0601i		−	1.12365i	−0.827254	−	0.561827i	\(-0.810099\pi\)
							0.827254	−	0.561827i	\(-0.189901\pi\)
\(23\)	79.6585	−	137.973i	0.722172	−	1.25084i	−0.237956	−	0.971276i	\(-0.576477\pi\)
							0.960128	−	0.279562i	\(-0.0901894\pi\)
\(29\)	−35.2280	−	61.0167i	−0.225575	−	0.390708i	0.730917	−	0.682467i	\(-0.239093\pi\)
							−0.956492	+	0.291759i	\(0.905759\pi\)
\(31\)	−181.969	−	105.060i	−1.05428	−	0.608688i	−0.130435	−	0.991457i	\(-0.541637\pi\)
							−0.923844	+	0.382769i	\(0.874971\pi\)
\(37\)			112.336i			0.499135i	0.968357	+	0.249568i	\(0.0802885\pi\)
							−0.968357	+	0.249568i	\(0.919712\pi\)
\(41\)	73.5944	+	42.4898i	0.280330	+	0.161848i	0.633573	−	0.773683i	\(-0.281588\pi\)
							−0.353243	+	0.935532i	\(0.614921\pi\)
\(43\)	−36.5216	−	63.2572i	−0.129523	−	0.224340i	0.793969	−	0.607958i	\(-0.208011\pi\)
							−0.923492	+	0.383618i	\(0.874678\pi\)
\(47\)	194.809	−	112.473i	0.604593	−	0.349062i	−0.166253	−	0.986083i	\(-0.553167\pi\)
							0.770846	+	0.637021i	\(0.219834\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.t.a.103.29	yes	80
3.2	odd	2		351.4.t.a.64.12		80
9.2	odd	6		351.4.t.a.181.29		80
9.7	even	3	inner	117.4.t.a.25.12	✓	80
13.12	even	2	inner	117.4.t.a.103.12	yes	80
39.38	odd	2		351.4.t.a.64.29		80
117.25	even	6	inner	117.4.t.a.25.29	yes	80
117.38	odd	6		351.4.t.a.181.12		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.t.a.25.12	✓	80	9.7	even	3	inner
117.4.t.a.25.29	yes	80	117.25	even	6	inner
117.4.t.a.103.12	yes	80	13.12	even	2	inner
117.4.t.a.103.29	yes	80	1.1	even	1	trivial
351.4.t.a.64.12		80	3.2	odd	2
351.4.t.a.64.29		80	39.38	odd	2
351.4.t.a.181.12		80	117.38	odd	6
351.4.t.a.181.29		80	9.2	odd	6