Embedded newform 54.8.e.b.13.6

• Label: 54.8.e.b.13.6
• Level: $54$
• Weight: $8$
• Character: 54.13
• Analytic conductor: $16.869$
• Analytic rank: $0$
• Dimension: $66$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	54.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(16.8687913761\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			13.6
Character	\(\chi\)	\(=\)	54.13
Dual form			54.8.e.b.25.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38919 - 7.87846i) q^{2} +(-2.46752 - 46.7002i) q^{3} +(-60.1403 + 21.8893i) q^{4} +(138.474 + 116.193i) q^{5} +(-364.498 + 84.3155i) q^{6} +(95.8895 + 34.9009i) q^{7} +(256.000 + 443.405i) q^{8} +(-2174.82 + 230.467i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 213 q^{5} - 1320 q^{6} - 1677 q^{7} + 16896 q^{8} + 6468 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(29\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38919	−	7.87846i	−0.122788	−	0.696364i
							\(3\)	−2.46752	−	46.7002i	−0.0527638	−	0.998607i
\(5\)	138.474	+	116.193i	0.495418	+	0.415705i								0.855963	−	0.517036i	\(-0.172965\pi\)
							−0.360545	+	0.932742i	\(0.617409\pi\)
\(7\)	95.8895	+	34.9009i	0.105664	+	0.0384586i	0.394311	−	0.918977i	\(-0.370983\pi\)
							−0.288647	+	0.957436i	\(0.593205\pi\)
\(11\)	−5496.12	+	4611.79i	−1.24504	+	1.04471i	−0.247923	+	0.968780i	\(0.579748\pi\)
							−0.997113	+	0.0759295i	\(0.975808\pi\)
\(13\)	−1895.76	+	10751.4i	−0.239321	+	1.35726i	0.593998	+	0.804466i	\(0.297549\pi\)
							−0.833320	+	0.552792i	\(0.813563\pi\)
\(17\)	4324.68	−	7490.57i	0.213493	−	0.369780i	−0.739312	−	0.673362i	\(-0.764849\pi\)
							0.952805	+	0.303582i	\(0.0981827\pi\)
\(19\)	10374.4	+	17968.9i	0.346996	+	0.601014i	0.985714	−	0.168426i	\(-0.0538685\pi\)
							−0.638718	+	0.769441i	\(0.720535\pi\)
\(23\)	45284.0	−	16482.0i	0.776064	−	0.282464i	0.0765336	−	0.997067i	\(-0.475615\pi\)
							0.699530	+	0.714603i	\(0.253393\pi\)
\(29\)	15047.3	+	85337.5i	0.114569	+	0.649751i	0.986963	+	0.160948i	\(0.0514552\pi\)
							−0.872394	+	0.488803i	\(0.837434\pi\)
\(31\)	−46178.4	+	16807.6i	−0.278403	+	0.101330i	−0.477448	−	0.878660i	\(-0.658438\pi\)
							0.199045	+	0.979990i	\(0.436216\pi\)
\(37\)	−62533.7	+	108312.i	−0.202959	+	0.351535i	−0.949481	−	0.313826i	\(-0.898389\pi\)
							0.746522	+	0.665361i	\(0.231722\pi\)
\(41\)	−20899.8	+	118529.i	−0.0473586	+	0.268584i	−0.999288	−	0.0377342i	\(-0.987986\pi\)
							0.951929	+	0.306318i	\(0.0990971\pi\)
\(43\)	−554105.	+	464949.i	−1.06280	+	0.891797i	−0.994381	−	0.105859i	\(-0.966241\pi\)
							−0.0684216	+	0.997656i	\(0.521796\pi\)
\(47\)	985182.	+	358577.i	1.38412	+	0.503779i	0.923425	−	0.383780i	\(-0.125378\pi\)
							0.460696	+	0.887558i	\(0.347600\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.8.e.b.13.6	✓	66
27.25	even	9	inner	54.8.e.b.25.6	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.8.e.b.13.6	✓	66	1.1	even	1	trivial
54.8.e.b.25.6	yes	66	27.25	even	9	inner