Embedded newform 804.2.bf.a.7.5

• Label: 804.2.bf.a.7.5
• 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			7.5
Character	\(\chi\)	\(=\)	804.7
Dual form			804.2.bf.a.115.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.32744 + 0.487752i) q^{2} +(-0.959493 - 0.281733i) q^{3} +(1.52420 - 1.29492i) q^{4} +(2.75137 + 2.38407i) q^{5} +(1.41109 - 0.0940111i) q^{6} +(0.196492 - 4.12487i) q^{7} +(-1.39168 + 2.46236i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q - 68 q^{3} + 2 q^{4} - 11 q^{6} + 4 q^{7} - 39 q^{8} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.32744	+	0.487752i	−0.938642	+	0.344893i
							\(3\)	−0.959493	−	0.281733i	−0.553964	−	0.162658i
\(5\)	2.75137	+	2.38407i	1.23045	+	1.06619i								0.995556	+	0.0941678i	\(0.0300190\pi\)
							0.234892	+	0.972022i	\(0.424526\pi\)
\(7\)	0.196492	−	4.12487i	0.0742669	−	1.55905i	−0.587425	−	0.809279i	\(-0.699858\pi\)
							0.661692	−	0.749776i	\(-0.269839\pi\)
\(11\)	3.62694	+	0.699036i	1.09356	+	0.210767i	0.703977	−	0.710223i	\(-0.251406\pi\)
							0.389587	+	0.920990i	\(0.372618\pi\)
\(13\)	1.76455	−	4.40764i	0.489398	−	1.22246i	−0.454061	−	0.890970i	\(-0.650025\pi\)
							0.943460	−	0.331488i	\(-0.107550\pi\)
\(17\)	−1.15602	−	1.62341i	−0.280376	−	0.393734i	0.650240	−	0.759728i	\(-0.274668\pi\)
							−0.930617	+	0.365995i	\(0.880729\pi\)
\(19\)	0.322878	−	0.0153806i	0.0740733	−	0.00352855i	−0.0105125	−	0.999945i	\(-0.503346\pi\)
							0.0845859	+	0.996416i	\(0.473043\pi\)
\(23\)	−4.61821	−	4.84344i	−0.962963	−	1.00993i	−0.999944	−	0.0106107i	\(-0.996622\pi\)
							0.0369806	−	0.999316i	\(-0.488226\pi\)
\(29\)	0.327728	+	0.567642i	0.0608576	+	0.105408i	0.894849	−	0.446369i	\(-0.147283\pi\)
							−0.833991	+	0.551777i	\(0.813950\pi\)
\(31\)	−0.376844	+	0.150866i	−0.0676833	+	0.0270963i	−0.405257	−	0.914203i	\(-0.632818\pi\)
							0.337574	+	0.941299i	\(0.390394\pi\)
\(37\)	−1.74966	+	3.03049i	−0.287642	+	0.498210i	−0.973246	−	0.229764i	\(-0.926205\pi\)
							0.685605	+	0.727974i	\(0.259538\pi\)
\(41\)	−0.922286	−	9.65862i	−0.144037	−	1.50842i	−0.721734	−	0.692171i	\(-0.756654\pi\)
							0.577697	−	0.816252i	\(-0.303952\pi\)
\(43\)	−0.938352	+	2.05470i	−0.143097	+	0.313339i	−0.967587	−	0.252537i	\(-0.918735\pi\)
							0.824490	+	0.565877i	\(0.191462\pi\)
\(47\)	−9.17484	+	2.22579i	−1.33829	+	0.324665i	−0.840113	−	0.542411i	\(-0.817512\pi\)
							−0.498175	+	0.867076i	\(0.665996\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.a.7.5	✓	680
4.3	odd	2		804.2.bf.b.7.29	yes	680
67.48	odd	66		804.2.bf.b.115.29	yes	680
268.115	even	66	inner	804.2.bf.a.115.5	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.5	✓	680	1.1	even	1	trivial
804.2.bf.a.115.5	yes	680	268.115	even	66	inner
804.2.bf.b.7.29	yes	680	4.3	odd	2
804.2.bf.b.115.29	yes	680	67.48	odd	66