Embedded newform 648.2.bd.a.133.76

• Label: 648.2.bd.a.133.76
• Level: $648$
• Weight: $2$
• Character: 648.133
• Analytic conductor: $5.174$
• Analytic rank: $0$
• Dimension: $1908$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	648.bd (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.17430605098\)
Analytic rank:	\(0\)
Dimension:	\(1908\)
Relative dimension:	\(106\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			133.76
Character	\(\chi\)	\(=\)	648.133
Dual form			648.2.bd.a.229.76

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.876064 + 1.11019i) q^{2} +(-0.817491 + 1.52699i) q^{3} +(-0.465024 + 1.94519i) q^{4} +(0.00150451 - 0.00141943i) q^{5} +(-2.41142 + 0.430176i) q^{6} +(-0.227742 + 0.0266192i) q^{7} +(-2.56691 + 1.18785i) q^{8} +(-1.66342 - 2.49661i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1908 q - 18 q^{2} - 18 q^{4} - 18 q^{6} - 36 q^{7} - 18 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{5}{27}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.876064	+	1.11019i	0.619471	+	0.785020i
							\(3\)	−0.817491	+	1.52699i	−0.471979	+	0.881610i
\(5\)	0.00150451	−	0.00141943i	0.000672836	−	0.000634788i								−0.685905	−	0.727691i	\(-0.740594\pi\)
							0.686578	+	0.727056i	\(0.259112\pi\)
\(7\)	−0.227742	+	0.0266192i	−0.0860782	+	0.0100611i	−0.159023	−	0.987275i	\(-0.550834\pi\)
							0.0729448	+	0.997336i	\(0.476760\pi\)
\(11\)	−1.22997	+	0.368229i	−0.370850	+	0.111025i	−0.466804	−	0.884361i	\(-0.654595\pi\)
							0.0959547	+	0.995386i	\(0.469410\pi\)
\(13\)	−4.36193	+	0.254054i	−1.20978	+	0.0704618i	−0.651147	−	0.758951i	\(-0.725712\pi\)
							−0.558635	+	0.829413i	\(0.688675\pi\)
\(17\)	−3.40788	−	1.24037i	−0.826533	−	0.300833i	−0.106098	−	0.994356i	\(-0.533836\pi\)
							−0.720435	+	0.693522i	\(0.756058\pi\)
\(19\)	0.999151	+	2.74515i	0.229221	+	0.629779i	0.999973	−	0.00735584i	\(-0.00234146\pi\)
							−0.770752	+	0.637135i	\(0.780119\pi\)
\(23\)	3.60470	+	0.421329i	0.751632	+	0.0878532i	0.483278	−	0.875467i	\(-0.339446\pi\)
							0.268354	+	0.963320i	\(0.413520\pi\)
\(29\)	1.10532	+	2.20087i	0.205253	+	0.408692i	0.972582	−	0.232562i	\(-0.0747109\pi\)
							−0.767329	+	0.641254i	\(0.778415\pi\)
\(31\)	−0.0468457	+	0.0629248i	−0.00841374	+	0.0113016i	−0.806310	−	0.591493i	\(-0.798539\pi\)
							0.797896	+	0.602795i	\(0.205946\pi\)
\(37\)	−0.538535	−	0.641801i	−0.0885346	−	0.105511i	0.719958	−	0.694017i	\(-0.244161\pi\)
							−0.808493	+	0.588506i	\(0.799716\pi\)
\(41\)	−5.74457	+	3.77826i	−0.897151	+	0.590065i	−0.912166	−	0.409822i	\(-0.865591\pi\)
							0.0150148	+	0.999887i	\(0.495220\pi\)
\(43\)	0.284470	−	1.20027i	0.0433813	−	0.183040i	−0.947041	−	0.321114i	\(-0.895943\pi\)
							0.990422	+	0.138074i	\(0.0440910\pi\)
\(47\)	−0.816271	−	1.09644i	−0.119065	−	0.159933i	0.738594	−	0.674150i	\(-0.235490\pi\)
							−0.857660	+	0.514217i	\(0.828082\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.2.bd.a.133.76	✓	1908
8.5	even	2	inner	648.2.bd.a.133.98	yes	1908
81.67	even	27	inner	648.2.bd.a.229.98	yes	1908
648.229	even	54	inner	648.2.bd.a.229.76	yes	1908

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.133.76	✓	1908	1.1	even	1	trivial
648.2.bd.a.133.98	yes	1908	8.5	even	2	inner
648.2.bd.a.229.76	yes	1908	648.229	even	54	inner
648.2.bd.a.229.98	yes	1908	81.67	even	27	inner