Embedded newform 54.8.e.b.13.9

• Label: 54.8.e.b.13.9
• Level: $54$
• Weight: $8$
• Character: 54.13
• Analytic conductor: $16.869$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• 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			13.9
Character	\(\chi\)	\(=\)	54.13
Dual form			54.8.e.b.25.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38919 - 7.87846i) q^{2} +(38.4590 - 26.6064i) q^{3} +(-60.1403 + 21.8893i) q^{4} +(337.131 + 282.886i) q^{5} +(-263.044 - 266.037i) q^{6} +(-1612.97 - 587.074i) q^{7} +(256.000 + 443.405i) q^{8} +(771.197 - 2046.52i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 213 q^{5} - 1320 q^{6} - 1677 q^{7} + 16896 q^{8} + 6468 q^{9}+O(q^{10}) \)

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
							\(3\)	38.4590	−	26.6064i	0.822383	−	0.568934i
\(5\)	337.131	+	282.886i	1.20616	+	1.01209i								0.999432	+	0.0336856i	\(0.0107245\pi\)
							0.206723	+	0.978399i	\(0.433720\pi\)
\(7\)	−1612.97	−	587.074i	−1.77739	−	0.646919i	−0.999836	−	0.0181076i	\(-0.994236\pi\)
							−0.777558	−	0.628811i	\(-0.783542\pi\)
\(11\)	1482.01	−	1243.55i	0.335719	−	0.281702i	−0.459306	−	0.888278i	\(-0.651902\pi\)
							0.795025	+	0.606576i	\(0.207457\pi\)
\(13\)	2422.20	−	13737.0i	0.305779	−	1.73416i	−0.314034	−	0.949412i	\(-0.601681\pi\)
							0.619813	−	0.784749i	\(-0.287208\pi\)
\(17\)	2639.88	−	4572.41i	0.130321	−	0.225722i	−0.793479	−	0.608597i	\(-0.791733\pi\)
							0.923800	+	0.382875i	\(0.125066\pi\)
\(19\)	−20832.2	−	36082.5i	−0.696783	−	1.20686i	−0.969576	−	0.244791i	\(-0.921281\pi\)
							0.272792	−	0.962073i	\(-0.412053\pi\)
\(23\)	60627.6	−	22066.7i	1.03902	−	0.378172i	0.234510	−	0.972114i	\(-0.424651\pi\)
							0.804508	+	0.593942i	\(0.202429\pi\)
\(29\)	820.737	+	4654.63i	0.00624901	+	0.0354399i	0.987773	−	0.155900i	\(-0.0498278\pi\)
							−0.981524	+	0.191340i	\(0.938717\pi\)
\(31\)	−116546.	+	42419.3i	−0.702638	+	0.255739i	−0.668537	−	0.743679i	\(-0.733079\pi\)
							−0.0341012	+	0.999418i	\(0.510857\pi\)
\(37\)	−14744.9	+	25539.0i	−0.0478560	+	0.0828891i	−0.888961	−	0.457983i	\(-0.848572\pi\)
							0.841105	+	0.540872i	\(0.181906\pi\)
\(41\)	−57433.1	+	325720.i	−0.130142	+	0.738075i	0.847977	+	0.530032i	\(0.177820\pi\)
							−0.978120	+	0.208042i	\(0.933291\pi\)
\(43\)	−221637.	+	185975.i	−0.425110	+	0.356710i	−0.830103	−	0.557610i	\(-0.811718\pi\)
							0.404993	+	0.914320i	\(0.367274\pi\)
\(47\)	1.01664e6	+	370026.i	1.42831	+	0.519863i	0.936447	−	0.350808i	\(-0.114093\pi\)
							0.491865	+	0.870671i	\(0.336315\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.8.e.b.13.9	✓	66
27.25	even	9	inner	54.8.e.b.25.9	yes	66

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.8.e.b.13.9	✓	66	1.1	even	1	trivial
54.8.e.b.25.9	yes	66	27.25	even	9	inner