Embedded newform 117.2.h.a.22.5

• Label: 117.2.h.a.22.5
• Level: $117$
• Weight: $2$
• Character: 117.22
• Analytic conductor: $0.934$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	117.h (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.934249703649\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			22.5
Character	\(\chi\)	\(=\)	117.22
Dual form			117.2.h.a.16.5

$q$-expansion

\(f(q)\)	\(=\)	\(q-0.867378 q^{2} +(0.973656 - 1.43248i) q^{3} -1.24766 q^{4} +(-0.0324057 + 0.0561283i) q^{5} +(-0.844528 + 1.24250i) q^{6} +(1.96209 - 3.39845i) q^{7} +2.81694 q^{8} +(-1.10399 - 2.78948i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{2} - q^{3} + 18 q^{4} - 2 q^{5} - 12 q^{6} + 3 q^{7} - 18 q^{8} - 3 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(92\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.867378			−0.613329			−0.306664	−	0.951818i	\(-0.599213\pi\)
							−0.306664	+	0.951818i	\(0.599213\pi\)
\(3\)	0.973656	−	1.43248i	0.562141	−	0.827042i
							\(5\)	−0.0324057	+	0.0561283i	−0.0144923	+	0.0251013i	−0.873181	−	0.487397i	\(-0.837947\pi\)
0.858688	+	0.512498i	\(0.171280\pi\)
\(7\)	1.96209	−	3.39845i	0.741602	−	1.28449i	−0.210164	−	0.977666i	\(-0.567400\pi\)
							0.951766	−	0.306826i	\(-0.0992669\pi\)
\(11\)	−5.29315			−1.59595			−0.797973	−	0.602694i	\(-0.794094\pi\)
							−0.797973	+	0.602694i	\(0.794094\pi\)
\(13\)	3.21261	−	1.63680i	0.891018	−	0.453967i
							\(17\)	2.28144	+	3.95157i	0.553330	+	0.958395i	0.998031	+	0.0627166i	\(0.0199764\pi\)
−0.444702	+	0.895679i	\(0.646690\pi\)
\(19\)	0.281137	+	0.486944i	0.0644973	+	0.111713i	0.896471	−	0.443103i	\(-0.146122\pi\)
							−0.831974	+	0.554815i	\(0.812789\pi\)
\(23\)	1.42839	+	2.47404i	0.297840	+	0.515873i	0.975641	−	0.219371i	\(-0.0704006\pi\)
							−0.677802	+	0.735245i	\(0.737067\pi\)
\(29\)	6.00595			1.11528			0.557639	−	0.830084i	\(-0.311708\pi\)
							0.557639	+	0.830084i	\(0.311708\pi\)
\(31\)	−4.23254	+	7.33098i	−0.760187	+	1.31668i	0.182567	+	0.983193i	\(0.441559\pi\)
							−0.942754	+	0.333489i	\(0.891774\pi\)
\(37\)	−0.506751	+	0.877718i	−0.0833094	+	0.144296i	−0.904670	−	0.426114i	\(-0.859882\pi\)
							0.821360	+	0.570410i	\(0.193216\pi\)
\(41\)	−0.674907	−	1.16897i	−0.105403	−	0.182563i	0.808500	−	0.588496i	\(-0.200280\pi\)
							−0.913903	+	0.405933i	\(0.866947\pi\)
\(43\)	3.45051	−	5.97645i	0.526197	−	0.911400i	−0.473337	−	0.880881i	\(-0.656951\pi\)
							0.999534	−	0.0305189i	\(-0.00971599\pi\)
\(47\)	−2.22815	−	3.85927i	−0.325009	−	0.562933i	0.656505	−	0.754322i	\(-0.272034\pi\)
							−0.981514	+	0.191389i	\(0.938701\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.2.h.a.22.5	yes	24
3.2	odd	2		351.2.h.a.334.8		24
9.2	odd	6		351.2.f.a.100.5		24
9.7	even	3		117.2.f.a.61.8	✓	24
13.3	even	3		117.2.f.a.94.8	yes	24
39.29	odd	6		351.2.f.a.172.5		24
117.16	even	3	inner	117.2.h.a.16.5	yes	24
117.29	odd	6		351.2.h.a.289.8		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.f.a.61.8	✓	24	9.7	even	3
117.2.f.a.94.8	yes	24	13.3	even	3
117.2.h.a.16.5	yes	24	117.16	even	3	inner
117.2.h.a.22.5	yes	24	1.1	even	1	trivial
351.2.f.a.100.5		24	9.2	odd	6
351.2.f.a.172.5		24	39.29	odd	6
351.2.h.a.289.8		24	117.29	odd	6
351.2.h.a.334.8		24	3.2	odd	2