Embedded newform 567.2.bm.a.104.65

• Label: 567.2.bm.a.104.65
• Level: $567$
• Weight: $2$
• Character: 567.104
• Analytic conductor: $4.528$
• Analytic rank: $0$
• Dimension: $1260$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 567 = 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	567.bm (of order \(54\), degree \(18\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.52751779461\)
Analytic rank:	\(0\)
Dimension:	\(1260\)
Relative dimension:	\(70\) over \(\Q(\zeta_{54})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{54}]$

Embedding invariants

Embedding label			104.65
Character	\(\chi\)	\(=\)	567.104
Dual form			567.2.bm.a.398.65

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.04558 + 1.52288i) q^{2} +(-0.0916956 + 1.72962i) q^{3} +(1.29164 + 4.31438i) q^{4} +(-3.13084 - 2.05918i) q^{5} +(-2.82158 + 3.39844i) q^{6} +(-2.42808 + 1.05092i) q^{7} +(-2.18367 + 5.99960i) q^{8} +(-2.98318 - 0.317197i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1260 q - 36 q^{2} - 36 q^{4} - 18 q^{7} - 36 q^{8} - 36 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\)	\(325\)	\(407\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{11}{54}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.04558	+	1.52288i	1.44645	+	1.07684i	0.981492	+	0.191504i	\(0.0613364\pi\)
							0.464954	+	0.885335i	\(0.346071\pi\)
\(3\)	−0.0916956	+	1.72962i	−0.0529405	+	0.998598i
							\(5\)	−3.13084	−	2.05918i	−1.40015	−	0.920895i	−0.999990	−	0.00445014i	\(-0.998583\pi\)
−0.400162	−	0.916444i	\(-0.631046\pi\)
\(7\)	−2.42808	+	1.05092i	−0.917727	+	0.397212i
							\(11\)	−0.382687	+	0.761992i	−0.115384	+	0.229749i	−0.943911	−	0.330201i	\(-0.892883\pi\)
0.828526	+	0.559950i	\(0.189180\pi\)
\(13\)	4.42071	+	1.90691i	1.22609	+	0.528882i	0.907916	−	0.419152i	\(-0.137673\pi\)
							0.318169	+	0.948034i	\(0.396932\pi\)
\(17\)	−0.405617	−	0.340353i	−0.0983765	−	0.0825477i	0.592273	−	0.805737i	\(-0.298231\pi\)
							−0.690649	+	0.723190i	\(0.742675\pi\)
\(19\)	3.99854	+	4.76528i	0.917328	+	1.09323i	0.995354	+	0.0962780i	\(0.0306938\pi\)
							−0.0780261	+	0.996951i	\(0.524862\pi\)
\(23\)	−2.49899	−	2.35768i	−0.521076	−	0.491610i	0.380170	−	0.924916i	\(-0.375865\pi\)
							−0.901247	+	0.433306i	\(0.857347\pi\)
\(29\)	−1.14496	+	9.79576i	−0.212614	+	1.81903i	0.292826	+	0.956166i	\(0.405404\pi\)
							−0.505440	+	0.862862i	\(0.668670\pi\)
\(31\)	−0.613511	+	2.58860i	−0.110190	+	0.464927i	0.889774	+	0.456402i	\(0.150862\pi\)
							−0.999964	+	0.00852554i	\(0.997286\pi\)
\(37\)	−0.172054	+	0.975769i	−0.0282856	+	0.160415i	−0.995679	−	0.0928636i	\(-0.970398\pi\)
							0.967393	+	0.253279i	\(0.0815091\pi\)
\(41\)	−1.29219	−	1.73571i	−0.201806	−	0.271072i	0.689687	−	0.724108i	\(-0.257748\pi\)
							−0.891492	+	0.453036i	\(0.850341\pi\)
\(43\)	−0.239881	−	4.11859i	−0.0365814	−	0.628079i	−0.965918	−	0.258846i	\(-0.916658\pi\)
							0.929337	−	0.369233i	\(-0.120379\pi\)
\(47\)	−5.36763	+	1.27215i	−0.782949	+	0.185562i	−0.602603	−	0.798041i	\(-0.705870\pi\)
							−0.180346	+	0.983603i	\(0.557722\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	567.2.bm.a.104.65	✓	1260
7.6	odd	2	inner	567.2.bm.a.104.66	yes	1260
81.74	odd	54	inner	567.2.bm.a.398.66	yes	1260
567.398	even	54	inner	567.2.bm.a.398.65	yes	1260

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.bm.a.104.65	✓	1260	1.1	even	1	trivial
567.2.bm.a.104.66	yes	1260	7.6	odd	2	inner
567.2.bm.a.398.65	yes	1260	567.398	even	54	inner
567.2.bm.a.398.66	yes	1260	81.74	odd	54	inner