Embedded newform 804.2.bf.b.7.18

• Label: 804.2.bf.b.7.18
• Level: $804$
• Weight: $2$
• Character: 804.7
• Analytic conductor: $6.420$
• Analytic rank: $0$
• Dimension: $680$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.bf (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41997232251\)
Analytic rank:	\(0\)
Dimension:	\(680\)
Relative dimension:	\(34\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			7.18
Character	\(\chi\)	\(=\)	804.7
Dual form			804.2.bf.b.115.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.112466 + 1.40973i) q^{2} +(0.959493 + 0.281733i) q^{3} +(-1.97470 + 0.317094i) q^{4} +(1.50481 + 1.30392i) q^{5} +(-0.289258 + 1.38432i) q^{6} +(0.198031 - 4.15718i) q^{7} +(-0.669104 - 2.74814i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q + 68 q^{3} + 2 q^{4} + 11 q^{6} - 4 q^{7} - 27 q^{8} - 68 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\)	\(269\)	\(337\)	\(403\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{23}{66}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.112466	+	1.40973i	0.0795253	+	0.996833i
							\(3\)	0.959493	+	0.281733i	0.553964	+	0.162658i
\(5\)	1.50481	+	1.30392i	0.672969	+	0.583131i								0.922857	−	0.385144i	\(-0.125848\pi\)
							−0.249887	+	0.968275i	\(0.580394\pi\)
\(7\)	0.198031	−	4.15718i	0.0748486	−	1.57127i	−0.579436	−	0.815017i	\(-0.696727\pi\)
							0.654285	−	0.756248i	\(-0.272970\pi\)
\(11\)	3.68590	+	0.710398i	1.11134	+	0.214193i	0.711703	−	0.702481i	\(-0.247924\pi\)
							0.399637	+	0.916674i	\(0.369136\pi\)
\(13\)	1.05947	−	2.64643i	0.293844	−	0.733987i	−0.705828	−	0.708383i	\(-0.749425\pi\)
							0.999672	−	0.0256036i	\(-0.00815078\pi\)
\(17\)	0.613140	+	0.861034i	0.148708	+	0.208832i	0.882210	−	0.470856i	\(-0.156055\pi\)
							−0.733502	+	0.679687i	\(0.762115\pi\)
\(19\)	2.44867	−	0.116645i	0.561764	−	0.0267601i	0.235220	−	0.971942i	\(-0.424419\pi\)
							0.326544	+	0.945182i	\(0.394116\pi\)
\(23\)	1.25188	+	1.31293i	0.261034	+	0.273765i	0.841189	−	0.540742i	\(-0.181856\pi\)
							−0.580155	+	0.814506i	\(0.697008\pi\)
\(29\)	3.37481	+	5.84535i	0.626687	+	1.08545i	0.988212	+	0.153091i	\(0.0489229\pi\)
							−0.361525	+	0.932362i	\(0.617744\pi\)
\(31\)	−8.64523	+	3.46103i	−1.55273	+	0.621619i	−0.980155	−	0.198234i	\(-0.936480\pi\)
							−0.572575	+	0.819853i	\(0.694055\pi\)
\(37\)	−0.469928	+	0.813939i	−0.0772557	+	0.133811i	−0.902065	−	0.431600i	\(-0.857949\pi\)
							0.824809	+	0.565411i	\(0.191282\pi\)
\(41\)	−0.212240	−	2.22267i	−0.0331463	−	0.347123i	−0.996798	−	0.0799615i	\(-0.974520\pi\)
							0.963652	−	0.267162i	\(-0.0860858\pi\)
\(43\)	3.93893	−	8.62506i	0.600682	−	1.31531i	−0.328086	−	0.944648i	\(-0.606404\pi\)
							0.928768	−	0.370662i	\(-0.120869\pi\)
\(47\)	−1.24559	+	0.302177i	−0.181688	+	0.0440771i	−0.325571	−	0.945518i	\(-0.605556\pi\)
							0.143883	+	0.989595i	\(0.454041\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.b.7.18	yes	680
4.3	odd	2		804.2.bf.a.7.28	✓	680
67.48	odd	66		804.2.bf.a.115.28	yes	680
268.115	even	66	inner	804.2.bf.b.115.18	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.28	✓	680	4.3	odd	2
804.2.bf.a.115.28	yes	680	67.48	odd	66
804.2.bf.b.7.18	yes	680	1.1	even	1	trivial
804.2.bf.b.115.18	yes	680	268.115	even	66	inner