Embedded newform 117.4.t.a.103.10

• Label: 117.4.t.a.103.10
• 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			103.10
Character	\(\chi\)	\(=\)	117.103
Dual form			117.4.t.a.25.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.81478 + 1.62512i) q^{2} +(-5.02430 - 1.32530i) q^{3} +(1.28200 - 2.22049i) q^{4} +(8.08964 + 4.67056i) q^{5} +(16.2961 - 4.43465i) q^{6} +(5.60310 - 3.23495i) q^{7} -17.6683i q^{8} +(23.4872 + 13.3174i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 150 q^{4} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.81478	+	1.62512i	−0.995176	+	0.574565i	−0.906817	−	0.421524i	\(-0.861495\pi\)
							−0.0883585	+	0.996089i	\(0.528162\pi\)
\(3\)	−5.02430	−	1.32530i	−0.966927	−	0.255053i
							\(5\)	8.08964	+	4.67056i	0.723560	+	0.417747i	0.816061	−	0.577965i	\(-0.196153\pi\)
−0.0925018	+	0.995713i	\(0.529486\pi\)
\(7\)	5.60310	−	3.23495i	0.302539	−	0.174671i	−0.341044	−	0.940047i	\(-0.610780\pi\)
							0.643583	+	0.765376i	\(0.277447\pi\)
\(11\)	28.4864	−	16.4466i	0.780816	−	0.450804i	−0.0559037	−	0.998436i	\(-0.517804\pi\)
							0.836719	+	0.547632i	\(0.184471\pi\)
\(13\)	−17.9757	+	43.2883i	−0.383505	+	0.923539i
							\(17\)	−99.5008			−1.41956			−0.709779	−	0.704425i	\(-0.751205\pi\)
−0.709779	+	0.704425i	\(0.751205\pi\)
\(19\)			136.315i			1.64594i	0.568087	+	0.822968i	\(0.307684\pi\)
							−0.568087	+	0.822968i	\(0.692316\pi\)
\(23\)	−7.91924	+	13.7165i	−0.0717946	+	0.124352i	−0.899688	−	0.436534i	\(-0.856206\pi\)
							0.827893	+	0.560886i	\(0.189539\pi\)
\(29\)	29.1948	+	50.5668i	0.186943	+	0.323794i	0.944229	−	0.329288i	\(-0.106809\pi\)
							−0.757287	+	0.653082i	\(0.773476\pi\)
\(31\)	−80.0415	−	46.2120i	−0.463738	−	0.267739i	0.249877	−	0.968278i	\(-0.419610\pi\)
							−0.713615	+	0.700539i	\(0.752943\pi\)
\(37\)			384.369i			1.70783i	0.520410	+	0.853916i	\(0.325779\pi\)
							−0.520410	+	0.853916i	\(0.674221\pi\)
\(41\)	−33.4192	−	19.2946i	−0.127298	−	0.0734953i	0.434999	−	0.900431i	\(-0.356749\pi\)
							−0.562297	+	0.826936i	\(0.690082\pi\)
\(43\)	240.935	+	417.312i	0.854472	+	1.47999i	0.877134	+	0.480245i	\(0.159452\pi\)
							−0.0226626	+	0.999743i	\(0.507214\pi\)
\(47\)	232.759	−	134.383i	0.722369	−	0.417060i	−0.0932547	−	0.995642i	\(-0.529727\pi\)
							0.815624	+	0.578582i	\(0.196394\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.t.a.103.10	yes	80
3.2	odd	2		351.4.t.a.64.31		80
9.2	odd	6		351.4.t.a.181.10		80
9.7	even	3	inner	117.4.t.a.25.31	yes	80
13.12	even	2	inner	117.4.t.a.103.31	yes	80
39.38	odd	2		351.4.t.a.64.10		80
117.25	even	6	inner	117.4.t.a.25.10	✓	80
117.38	odd	6		351.4.t.a.181.31		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.t.a.25.10	✓	80	117.25	even	6	inner
117.4.t.a.25.31	yes	80	9.7	even	3	inner
117.4.t.a.103.10	yes	80	1.1	even	1	trivial
117.4.t.a.103.31	yes	80	13.12	even	2	inner
351.4.t.a.64.10		80	39.38	odd	2
351.4.t.a.64.31		80	3.2	odd	2
351.4.t.a.181.10		80	9.2	odd	6
351.4.t.a.181.31		80	117.38	odd	6