Embedded newform 804.2.bf.b.7.20

• Label: 804.2.bf.b.7.20
• 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			7.20
Character	\(\chi\)	\(=\)	804.7
Dual form			804.2.bf.b.115.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.319658 - 1.37761i) q^{2} +(0.959493 + 0.281733i) q^{3} +(-1.79564 - 0.880730i) q^{4} +(2.42058 + 2.09744i) q^{5} +(0.694828 - 1.23175i) q^{6} +(0.217948 - 4.57528i) q^{7} +(-1.78730 + 2.19216i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q + 68 q^{3} + 2 q^{4} + 11 q^{6} - 4 q^{7} - 27 q^{8} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.319658	−	1.37761i	0.226032	−	0.974120i
							\(3\)	0.959493	+	0.281733i	0.553964	+	0.162658i
\(5\)	2.42058	+	2.09744i	1.08251	+	0.938004i								0.998291	−	0.0584429i	\(-0.0186136\pi\)
							0.0842236	+	0.996447i	\(0.473159\pi\)
\(7\)	0.217948	−	4.57528i	0.0823765	−	1.72929i	−0.458932	−	0.888471i	\(-0.651768\pi\)
							0.541309	−	0.840824i	\(-0.317929\pi\)
\(11\)	−4.38859	−	0.845832i	−1.32321	−	0.255028i	−0.521762	−	0.853091i	\(-0.674725\pi\)
							−0.801449	+	0.598063i	\(0.795937\pi\)
\(13\)	2.32683	−	5.81213i	0.645346	−	1.61200i	−0.138016	−	0.990430i	\(-0.544073\pi\)
							0.783362	−	0.621566i	\(-0.213503\pi\)
\(17\)	1.76820	+	2.48309i	0.428852	+	0.602238i	0.971302	−	0.237850i	\(-0.0764427\pi\)
							−0.542450	+	0.840088i	\(0.682503\pi\)
\(19\)	1.08054	−	0.0514725i	0.247893	−	0.0118086i	0.0767324	−	0.997052i	\(-0.475551\pi\)
							0.171161	+	0.985243i	\(0.445248\pi\)
\(23\)	−1.85636	−	1.94689i	−0.387077	−	0.405955i	0.501149	−	0.865361i	\(-0.332911\pi\)
							−0.888226	+	0.459406i	\(0.848062\pi\)
\(29\)	−0.373248	−	0.646484i	−0.0693104	−	0.120049i	0.829288	−	0.558822i	\(-0.188747\pi\)
							−0.898598	+	0.438773i	\(0.855413\pi\)
\(31\)	8.46111	−	3.38732i	1.51966	−	0.608380i	0.546285	−	0.837599i	\(-0.316042\pi\)
							0.973375	+	0.229219i	\(0.0736173\pi\)
\(37\)	4.72660	−	8.18672i	0.777049	−	1.34589i	−0.156587	−	0.987664i	\(-0.550049\pi\)
							0.933636	−	0.358224i	\(-0.116617\pi\)
\(41\)	0.787394	+	8.24596i	0.122970	+	1.28780i	0.820666	+	0.571408i	\(0.193603\pi\)
							−0.697696	+	0.716394i	\(0.745791\pi\)
\(43\)	−2.54826	+	5.57990i	−0.388605	+	0.850927i	0.609694	+	0.792637i	\(0.291292\pi\)
							−0.998300	+	0.0582905i	\(0.981435\pi\)
\(47\)	−2.93540	+	0.712121i	−0.428172	+	0.103874i	−0.444053	−	0.896000i	\(-0.646460\pi\)
							0.0158809	+	0.999874i	\(0.494945\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.b.7.20	yes	680
4.3	odd	2		804.2.bf.a.7.6	✓	680
67.48	odd	66		804.2.bf.a.115.6	yes	680
268.115	even	66	inner	804.2.bf.b.115.20	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.6	✓	680	4.3	odd	2
804.2.bf.a.115.6	yes	680	67.48	odd	66
804.2.bf.b.7.20	yes	680	1.1	even	1	trivial
804.2.bf.b.115.20	yes	680	268.115	even	66	inner