Embedded newform 117.4.t.a.103.36

• Label: 117.4.t.a.103.36
• 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			103.36
Character	\(\chi\)	\(=\)	117.103
Dual form			117.4.t.a.25.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.99826 - 2.30840i) q^{2} +(3.49652 - 3.84374i) q^{3} +(6.65738 - 11.5309i) q^{4} +(5.26914 + 3.04214i) q^{5} +(5.10714 - 23.4396i) q^{6} +(-14.5470 + 8.39873i) q^{7} -24.5372i q^{8} +(-2.54865 - 26.8794i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q - 4 q^{3} + 150 q^{4} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.99826	−	2.30840i	1.41360	−	0.816141i	0.417873	−	0.908506i	\(-0.362776\pi\)
							0.995725	+	0.0923644i	\(0.0294425\pi\)
\(3\)	3.49652	−	3.84374i	0.672906	−	0.739728i
							\(5\)	5.26914	+	3.04214i	0.471286	+	0.272097i	0.716778	−	0.697302i	\(-0.245616\pi\)
−0.245492	+	0.969399i	\(0.578949\pi\)
\(7\)	−14.5470	+	8.39873i	−0.785465	+	0.453489i	−0.838364	−	0.545111i	\(-0.816487\pi\)
							0.0528984	+	0.998600i	\(0.483154\pi\)
\(11\)	31.9191	−	18.4285i	0.874908	−	0.505128i	0.00593166	−	0.999982i	\(-0.498112\pi\)
							0.868976	+	0.494854i	\(0.164779\pi\)
\(13\)	−12.5798	+	45.1525i	−0.268385	+	0.963312i
							\(17\)	−83.7613			−1.19501			−0.597503	−	0.801867i	\(-0.703840\pi\)
−0.597503	+	0.801867i	\(0.703840\pi\)
\(19\)			113.228i			1.36717i	0.729870	+	0.683586i	\(0.239580\pi\)
							−0.729870	+	0.683586i	\(0.760420\pi\)
\(23\)	49.5385	−	85.8033i	0.449109	−	0.777879i	−0.549220	−	0.835678i	\(-0.685075\pi\)
							0.998328	+	0.0577990i	\(0.0184083\pi\)
\(29\)	55.2727	+	95.7351i	0.353927	+	0.613019i	0.986934	−	0.161127i	\(-0.0515129\pi\)
							−0.633007	+	0.774146i	\(0.718180\pi\)
\(31\)	67.0340	+	38.7021i	0.388376	+	0.224229i	0.681456	−	0.731859i	\(-0.261347\pi\)
							−0.293080	+	0.956088i	\(0.594680\pi\)
\(37\)		−	142.423i		−	0.632814i	−0.948623	−	0.316407i	\(-0.897523\pi\)
							0.948623	−	0.316407i	\(-0.102477\pi\)
\(41\)	286.330	+	165.313i	1.09066	+	0.629695i	0.933753	−	0.357918i	\(-0.116513\pi\)
							0.156910	+	0.987613i	\(0.449847\pi\)
\(43\)	−144.446	−	250.188i	−0.512276	−	0.887287i	−0.999899	−	0.0142332i	\(-0.995469\pi\)
							0.487623	−	0.873054i	\(-0.337864\pi\)
\(47\)	−131.767	+	76.0756i	−0.408939	+	0.236101i	−0.690334	−	0.723491i	\(-0.742536\pi\)
							0.281394	+	0.959592i	\(0.409203\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.t.a.103.36	yes	80
3.2	odd	2		351.4.t.a.64.5		80
9.2	odd	6		351.4.t.a.181.36		80
9.7	even	3	inner	117.4.t.a.25.5	✓	80
13.12	even	2	inner	117.4.t.a.103.5	yes	80
39.38	odd	2		351.4.t.a.64.36		80
117.25	even	6	inner	117.4.t.a.25.36	yes	80
117.38	odd	6		351.4.t.a.181.5		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.t.a.25.5	✓	80	9.7	even	3	inner
117.4.t.a.25.36	yes	80	117.25	even	6	inner
117.4.t.a.103.5	yes	80	13.12	even	2	inner
117.4.t.a.103.36	yes	80	1.1	even	1	trivial
351.4.t.a.64.5		80	3.2	odd	2
351.4.t.a.64.36		80	39.38	odd	2
351.4.t.a.181.5		80	117.38	odd	6
351.4.t.a.181.36		80	9.2	odd	6