Embedded newform 804.2.bf.a.115.7

• Label: 804.2.bf.a.115.7
• Level: $804$
• Weight: $2$
• Character: 804.115
• 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			115.7
Character	\(\chi\)	\(=\)	804.115
Dual form			804.2.bf.a.7.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.16345 - 0.803974i) q^{2} +(-0.959493 + 0.281733i) q^{3} +(0.707250 + 1.87077i) q^{4} +(0.406963 - 0.352635i) q^{5} +(1.34283 + 0.443625i) q^{6} +(0.0731812 + 1.53626i) q^{7} +(0.681202 - 2.74517i) q^{8} +(0.841254 - 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q - 68 q^{3} + 2 q^{4} - 11 q^{6} + 4 q^{7} - 39 q^{8} - 68 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(269\)	\(337\)	\(403\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{43}{66}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.16345	−	0.803974i	−0.822686	−	0.568496i
							\(3\)	−0.959493	+	0.281733i	−0.553964	+	0.162658i
\(5\)	0.406963	−	0.352635i	0.181999	−	0.157703i								−0.559096	−	0.829103i	\(-0.688852\pi\)
							0.741096	+	0.671399i	\(0.234306\pi\)
\(7\)	0.0731812	+	1.53626i	0.0276599	+	0.580653i	0.969574	+	0.244799i	\(0.0787219\pi\)
							−0.941914	+	0.335854i	\(0.890975\pi\)
\(11\)	−3.09397	+	0.596315i	−0.932868	+	0.179796i	−0.632971	−	0.774176i	\(-0.718165\pi\)
							−0.299898	+	0.953971i	\(0.596952\pi\)
\(13\)	−0.850492	−	2.12443i	−0.235884	−	0.589210i	0.762441	−	0.647057i	\(-0.224001\pi\)
							−0.998325	+	0.0578473i	\(0.981576\pi\)
\(17\)	3.37866	−	4.74466i	0.819445	−	1.15075i	−0.166886	−	0.985976i	\(-0.553371\pi\)
							0.986331	−	0.164774i	\(-0.0526894\pi\)
\(19\)	5.09896	+	0.242893i	1.16978	+	0.0557235i	0.623462	−	0.781854i	\(-0.285726\pi\)
							0.546319	+	0.837577i	\(0.316029\pi\)
\(23\)	3.11215	−	3.26393i	0.648928	−	0.680576i	−0.314281	−	0.949330i	\(-0.601763\pi\)
							0.963209	+	0.268754i	\(0.0866118\pi\)
\(29\)	−3.97439	+	6.88384i	−0.738025	+	1.27830i	0.215358	+	0.976535i	\(0.430908\pi\)
							−0.953383	+	0.301762i	\(0.902425\pi\)
\(31\)	−3.18764	−	1.27614i	−0.572516	−	0.229201i	0.0672947	−	0.997733i	\(-0.478563\pi\)
							−0.639811	+	0.768532i	\(0.720987\pi\)
\(37\)	4.68447	+	8.11374i	0.770122	+	1.33389i	0.937495	+	0.347997i	\(0.113138\pi\)
							−0.167373	+	0.985894i	\(0.553528\pi\)
\(41\)	0.725666	−	7.59951i	0.113330	−	1.18684i	−0.742594	−	0.669742i	\(-0.766405\pi\)
							0.855924	−	0.517102i	\(-0.172989\pi\)
\(43\)	1.20679	+	2.64250i	0.184034	+	0.402978i	0.979053	−	0.203607i	\(-0.0652664\pi\)
							−0.795019	+	0.606585i	\(0.792539\pi\)
\(47\)	9.11198	+	2.21054i	1.32912	+	0.322441i	0.836578	−	0.547848i	\(-0.184553\pi\)
							0.492541	+	0.870289i	\(0.336068\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.a.115.7	yes	680
4.3	odd	2		804.2.bf.b.115.30	yes	680
67.7	odd	66		804.2.bf.b.7.30	yes	680
268.7	even	66	inner	804.2.bf.a.7.7	✓	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.7	✓	680	268.7	even	66	inner
804.2.bf.a.115.7	yes	680	1.1	even	1	trivial
804.2.bf.b.7.30	yes	680	67.7	odd	66
804.2.bf.b.115.30	yes	680	4.3	odd	2