Embedded newform 804.2.bf.a.7.15

• Label: 804.2.bf.a.7.15
• 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			7.15
Character	\(\chi\)	\(=\)	804.7
Dual form			804.2.bf.a.115.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.518960 + 1.31555i) q^{2} +(-0.959493 - 0.281733i) q^{3} +(-1.46136 - 1.36544i) q^{4} +(-0.507778 - 0.439992i) q^{5} +(0.868573 - 1.11606i) q^{6} +(0.160858 - 3.37682i) q^{7} +(2.55470 - 1.21389i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q - 68 q^{3} + 2 q^{4} - 11 q^{6} + 4 q^{7} - 39 q^{8} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.518960	+	1.31555i	−0.366960	+	0.930237i
							\(3\)	−0.959493	−	0.281733i	−0.553964	−	0.162658i
\(5\)	−0.507778	−	0.439992i	−0.227085	−	0.196770i								0.533885	−	0.845557i	\(-0.320731\pi\)
							−0.760971	+	0.648786i	\(0.775277\pi\)
\(7\)	0.160858	−	3.37682i	0.0607985	−	1.27632i	−0.738033	−	0.674764i	\(-0.764245\pi\)
							0.798832	−	0.601554i	\(-0.205452\pi\)
\(11\)	2.24657	+	0.432990i	0.677365	+	0.130551i	0.516320	−	0.856396i	\(-0.327301\pi\)
							0.161045	+	0.986947i	\(0.448514\pi\)
\(13\)	−1.35977	+	3.39654i	−0.377132	+	0.942031i	0.611441	+	0.791290i	\(0.290590\pi\)
							−0.988573	+	0.150741i	\(0.951834\pi\)
\(17\)	−0.955723	−	1.34212i	−0.231797	−	0.325513i	0.682252	−	0.731117i	\(-0.261001\pi\)
							−0.914049	+	0.405604i	\(0.867061\pi\)
\(19\)	−5.29406	+	0.252187i	−1.21454	+	0.0578557i	−0.645049	−	0.764141i	\(-0.723163\pi\)
							−0.569492	+	0.821997i	\(0.692860\pi\)
\(23\)	−0.106791	−	0.111999i	−0.0222675	−	0.0233535i	0.712506	−	0.701666i	\(-0.247560\pi\)
							−0.734774	+	0.678312i	\(0.762712\pi\)
\(29\)	−1.04972	−	1.81817i	−0.194929	−	0.337626i	0.751948	−	0.659222i	\(-0.229114\pi\)
							−0.946877	+	0.321595i	\(0.895781\pi\)
\(31\)	−5.90632	+	2.36453i	−1.06081	+	0.424683i	−0.835434	−	0.549590i	\(-0.814784\pi\)
							−0.225372	+	0.974273i	\(0.572360\pi\)
\(37\)	−0.889661	+	1.54094i	−0.146259	+	0.253329i	−0.929842	−	0.367959i	\(-0.880057\pi\)
							0.783583	+	0.621287i	\(0.213390\pi\)
\(41\)	−0.866395	−	9.07330i	−0.135308	−	1.41701i	−0.766928	−	0.641733i	\(-0.778216\pi\)
							0.631620	−	0.775278i	\(-0.282390\pi\)
\(43\)	−1.39524	+	3.05516i	−0.212773	+	0.465907i	−0.985683	−	0.168608i	\(-0.946073\pi\)
							0.772911	+	0.634515i	\(0.218800\pi\)
\(47\)	−4.26648	+	1.03504i	−0.622330	+	0.150976i	−0.534512	−	0.845161i	\(-0.679505\pi\)
							−0.0878180	+	0.996137i	\(0.527989\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.a.7.15	✓	680
4.3	odd	2		804.2.bf.b.7.32	yes	680
67.48	odd	66		804.2.bf.b.115.32	yes	680
268.115	even	66	inner	804.2.bf.a.115.15	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.15	✓	680	1.1	even	1	trivial
804.2.bf.a.115.15	yes	680	268.115	even	66	inner
804.2.bf.b.7.32	yes	680	4.3	odd	2
804.2.bf.b.115.32	yes	680	67.48	odd	66