Embedded newform 648.2.bd.a.229.76

• Label: 648.2.bd.a.229.76
• Level: $648$
• Weight: $2$
• Character: 648.229
• 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			229.76
Character	\(\chi\)	\(=\)	648.229
Dual form			648.2.bd.a.133.76

$q$-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 \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(487\)	\(569\)
\(\chi(n)\)	\(-1\)	\(1\)	\(e\left(\frac{22}{27}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
648.2.bd.a.133.76	✓	1908	648.133	even	54	inner
648.2.bd.a.133.98	yes	1908	81.52	even	27	inner
648.2.bd.a.229.76	yes	1908	1.1	even	1	trivial
648.2.bd.a.229.98	yes	1908	8.5	even	2	inner