Embedded newform 804.2.bf.a.7.20

• Label: 804.2.bf.a.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.a.115.20

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.335515 + 1.37384i) q^{2} +(-0.959493 - 0.281733i) q^{3} +(-1.77486 + 0.921885i) q^{4} +(-2.14113 - 1.85530i) q^{5} +(0.0651310 - 1.41271i) q^{6} +(0.0976058 - 2.04900i) q^{7} +(-1.86201 - 2.12906i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q - 68 q^{3} + 2 q^{4} - 11 q^{6} + 4 q^{7} - 39 q^{8} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.335515	+	1.37384i	0.237245	+	0.971450i
							\(3\)	−0.959493	−	0.281733i	−0.553964	−	0.162658i
\(5\)	−2.14113	−	1.85530i	−0.957541	−	0.829714i								0.0279211	−	0.999610i	\(-0.491111\pi\)
							−0.985462	+	0.169896i	\(0.945657\pi\)
\(7\)	0.0976058	−	2.04900i	0.0368915	−	0.774449i	−0.902311	−	0.431085i	\(-0.858131\pi\)
							0.939203	−	0.343363i	\(-0.111566\pi\)
\(11\)	−2.10284	−	0.405288i	−0.634029	−	0.122199i	−0.137897	−	0.990447i	\(-0.544034\pi\)
							−0.496131	+	0.868247i	\(0.665247\pi\)
\(13\)	−1.84520	+	4.60909i	−0.511767	+	1.27833i	0.417418	+	0.908715i	\(0.362935\pi\)
							−0.929184	+	0.369617i	\(0.879489\pi\)
\(17\)	3.89931	+	5.47582i	0.945723	+	1.32808i	0.944649	+	0.328082i	\(0.106402\pi\)
							0.00107349	+	0.999999i	\(0.499658\pi\)
\(19\)	2.58362	−	0.123073i	0.592723	−	0.0282349i	0.250922	−	0.968007i	\(-0.419266\pi\)
							0.341801	+	0.939772i	\(0.388963\pi\)
\(23\)	3.62060	+	3.79717i	0.754946	+	0.791765i	0.983335	−	0.181805i	\(-0.0581940\pi\)
							−0.228388	+	0.973570i	\(0.573346\pi\)
\(29\)	3.56641	+	6.17720i	0.662266	+	1.14708i	0.980019	+	0.198904i	\(0.0637383\pi\)
							−0.317753	+	0.948173i	\(0.602928\pi\)
\(31\)	8.54097	−	3.41929i	1.53400	−	0.614122i	0.557607	−	0.830105i	\(-0.311720\pi\)
							0.976397	+	0.215983i	\(0.0692955\pi\)
\(37\)	1.80622	−	3.12847i	0.296941	−	0.514317i	−0.678494	−	0.734606i	\(-0.737367\pi\)
							0.975434	+	0.220290i	\(0.0707003\pi\)
\(41\)	−0.415609	−	4.35245i	−0.0649072	−	0.679739i	−0.968110	−	0.250525i	\(-0.919397\pi\)
							0.903203	−	0.429214i	\(-0.141209\pi\)
\(43\)	0.342508	−	0.749988i	0.0522320	−	0.114372i	−0.881717	−	0.471779i	\(-0.843612\pi\)
							0.933949	+	0.357407i	\(0.116339\pi\)
\(47\)	−8.38941	+	2.03525i	−1.22372	+	0.296872i	−0.795025	−	0.606577i	\(-0.792542\pi\)
							−0.428697	+	0.903448i	\(0.641027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.a.7.20	✓	680
4.3	odd	2		804.2.bf.b.7.26	yes	680
67.48	odd	66		804.2.bf.b.115.26	yes	680
268.115	even	66	inner	804.2.bf.a.115.20	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.20	✓	680	1.1	even	1	trivial
804.2.bf.a.115.20	yes	680	268.115	even	66	inner
804.2.bf.b.7.26	yes	680	4.3	odd	2
804.2.bf.b.115.26	yes	680	67.48	odd	66