Embedded newform 54.4.e.a.31.3

• Label: 54.4.e.a.31.3
• Level: $54$
• Weight: $4$
• Character: 54.31
• Analytic conductor: $3.186$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	54.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.18610314031\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			31.3
Character	\(\chi\)	\(=\)	54.31
Dual form			54.4.e.a.7.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.87939 - 0.684040i) q^{2} +(2.43378 - 4.59094i) q^{3} +(3.06418 + 2.57115i) q^{4} +(1.31542 - 7.46011i) q^{5} +(-7.71439 + 6.96335i) q^{6} +(-2.90083 + 2.43409i) q^{7} +(-4.00000 - 6.92820i) q^{8} +(-15.1535 - 22.3466i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{5} - 18 q^{6} - 33 q^{7} - 96 q^{8} + 54 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.87939	−	0.684040i	−0.664463	−	0.241845i
							\(3\)	2.43378	−	4.59094i	0.468380	−	0.883527i
\(5\)	1.31542	−	7.46011i	0.117655	−	0.667252i								−0.867747	−	0.497006i	\(-0.834433\pi\)
							0.985402	−	0.170246i	\(-0.0544563\pi\)
\(7\)	−2.90083	+	2.43409i	−0.156630	+	0.131428i	−0.717735	−	0.696317i	\(-0.754821\pi\)
							0.561105	+	0.827745i	\(0.310377\pi\)
\(11\)	−8.73232	−	49.5234i	−0.239354	−	1.35744i	−0.833248	−	0.552900i	\(-0.813521\pi\)
							0.593894	−	0.804544i	\(-0.297590\pi\)
\(13\)	−7.25902	+	2.64207i	−0.154869	+	0.0563675i	−0.418291	−	0.908313i	\(-0.637371\pi\)
							0.263423	+	0.964681i	\(0.415149\pi\)
\(17\)	17.4793	−	30.2751i	0.249374	−	0.431928i	−0.713978	−	0.700168i	\(-0.753108\pi\)
							0.963352	+	0.268239i	\(0.0864418\pi\)
\(19\)	36.2933	+	62.8618i	0.438224	+	0.759026i	0.997553	−	0.0699200i	\(-0.0222744\pi\)
							−0.559329	+	0.828946i	\(0.688941\pi\)
\(23\)	116.403	+	97.6734i	1.05529	+	0.885492i	0.993640	−	0.112606i	\(-0.0359199\pi\)
							0.0616481	+	0.998098i	\(0.480364\pi\)
\(29\)	−60.4031	−	21.9849i	−0.386778	−	0.140776i	0.141310	−	0.989965i	\(-0.454869\pi\)
							−0.528088	+	0.849190i	\(0.677091\pi\)
\(31\)	175.978	+	147.663i	1.01957	+	0.855518i	0.989573	−	0.144032i	\(-0.0460067\pi\)
							0.0299941	+	0.999550i	\(0.490451\pi\)
\(37\)	121.156	−	209.848i	0.538322	−	0.932402i	−0.460672	−	0.887570i	\(-0.652392\pi\)
							0.998995	−	0.0448313i	\(-0.0142750\pi\)
\(41\)	382.412	−	139.187i	1.45665	−	0.530178i	0.512212	−	0.858859i	\(-0.328826\pi\)
							0.944441	+	0.328681i	\(0.106604\pi\)
\(43\)	−62.2961	−	353.299i	−0.220932	−	1.25297i	−0.870311	−	0.492502i	\(-0.836082\pi\)
							0.649380	−	0.760464i	\(-0.275029\pi\)
\(47\)	−385.943	+	323.845i	−1.19778	+	1.00506i	−0.198087	+	0.980184i	\(0.563473\pi\)
							−0.999691	+	0.0248708i	\(0.992083\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.4.e.a.31.3	yes	24
3.2	odd	2		162.4.e.a.145.1		24
27.7	even	9	inner	54.4.e.a.7.3	✓	24
27.13	even	9		1458.4.a.h.1.11		12
27.14	odd	18		1458.4.a.e.1.2		12
27.20	odd	18		162.4.e.a.19.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.4.e.a.7.3	✓	24	27.7	even	9	inner
54.4.e.a.31.3	yes	24	1.1	even	1	trivial
162.4.e.a.19.1		24	27.20	odd	18
162.4.e.a.145.1		24	3.2	odd	2
1458.4.a.e.1.2		12	27.14	odd	18
1458.4.a.h.1.11		12	27.13	even	9