Embedded newform 117.4.t.a.103.13

• Label: 117.4.t.a.103.13
• 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			103.13
Character	\(\chi\)	\(=\)	117.103
Dual form			117.4.t.a.25.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.88513 + 1.08838i) q^{2} +(5.16808 - 0.539440i) q^{3} +(-1.63085 + 2.82471i) q^{4} +(-12.6071 - 7.27874i) q^{5} +(-9.15540 + 6.64176i) q^{6} +(9.24346 - 5.33672i) q^{7} -24.5141i q^{8} +(26.4180 - 5.57573i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 150 q^{4} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.88513	+	1.08838i	−0.666496	+	0.384801i	−0.794748	−	0.606940i	\(-0.792397\pi\)
							0.128252	+	0.991742i	\(0.459063\pi\)
\(3\)	5.16808	−	0.539440i	0.994597	−	0.103815i
							\(5\)	−12.6071	−	7.27874i	−1.12762	−	0.651030i	−0.184283	−	0.982873i	\(-0.558996\pi\)
−0.943335	+	0.331843i	\(0.892330\pi\)
\(7\)	9.24346	−	5.33672i	0.499100	−	0.288156i	−0.229242	−	0.973370i	\(-0.573625\pi\)
							0.728342	+	0.685214i	\(0.240291\pi\)
\(11\)	23.1403	−	13.3600i	0.634277	−	0.366200i	−0.148129	−	0.988968i	\(-0.547325\pi\)
							0.782407	+	0.622768i	\(0.213992\pi\)
\(13\)	36.9761	−	28.8057i	0.788871	−	0.614558i
							\(17\)	−10.9453			−0.156155			−0.0780774	−	0.996947i	\(-0.524878\pi\)
−0.0780774	+	0.996947i	\(0.524878\pi\)
\(19\)			62.0323i			0.749010i	0.927225	+	0.374505i	\(0.122187\pi\)
							−0.927225	+	0.374505i	\(0.877813\pi\)
\(23\)	97.2817	−	168.497i	0.881941	−	1.52757i	0.0327611	−	0.999463i	\(-0.489570\pi\)
							0.849180	−	0.528104i	\(-0.177097\pi\)
\(29\)	−34.2711	−	59.3593i	−0.219448	−	0.380095i	0.735191	−	0.677860i	\(-0.237092\pi\)
							−0.954639	+	0.297765i	\(0.903759\pi\)
\(31\)	83.0915	+	47.9729i	0.481409	+	0.277941i	0.721003	−	0.692932i	\(-0.243681\pi\)
							−0.239595	+	0.970873i	\(0.577015\pi\)
\(37\)		−	184.787i		−	0.821051i	−0.911849	−	0.410525i	\(-0.865345\pi\)
							0.911849	−	0.410525i	\(-0.134655\pi\)
\(41\)	−271.060	−	156.497i	−1.03250	−	0.596114i	−0.114800	−	0.993389i	\(-0.536623\pi\)
							−0.917700	+	0.397274i	\(0.869956\pi\)
\(43\)	0.207090	+	0.358690i	0.000734440	+	0.00127209i	0.866392	−	0.499364i	\(-0.166433\pi\)
							−0.865658	+	0.500636i	\(0.833100\pi\)
\(47\)	−30.0741	+	17.3633i	−0.0933353	+	0.0538872i	−0.545941	−	0.837824i	\(-0.683828\pi\)
							0.452606	+	0.891711i	\(0.350494\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.t.a.103.13	yes	80
3.2	odd	2		351.4.t.a.64.28		80
9.2	odd	6		351.4.t.a.181.13		80
9.7	even	3	inner	117.4.t.a.25.28	yes	80
13.12	even	2	inner	117.4.t.a.103.28	yes	80
39.38	odd	2		351.4.t.a.64.13		80
117.25	even	6	inner	117.4.t.a.25.13	✓	80
117.38	odd	6		351.4.t.a.181.28		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.t.a.25.13	✓	80	117.25	even	6	inner
117.4.t.a.25.28	yes	80	9.7	even	3	inner
117.4.t.a.103.13	yes	80	1.1	even	1	trivial
117.4.t.a.103.28	yes	80	13.12	even	2	inner
351.4.t.a.64.13		80	39.38	odd	2
351.4.t.a.64.28		80	3.2	odd	2
351.4.t.a.181.13		80	9.2	odd	6
351.4.t.a.181.28		80	117.38	odd	6