Embedded newform 804.2.bf.a.7.11

• Label: 804.2.bf.a.7.11
• 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			7.11
Character	\(\chi\)	\(=\)	804.7
Dual form			804.2.bf.a.115.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.774785 + 1.18309i) q^{2} +(-0.959493 - 0.281733i) q^{3} +(-0.799417 - 1.83328i) q^{4} +(2.32348 + 2.01331i) q^{5} +(1.07672 - 0.916887i) q^{6} +(-0.144795 + 3.03962i) q^{7} +(2.78832 + 0.474616i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q - 68 q^{3} + 2 q^{4} - 11 q^{6} + 4 q^{7} - 39 q^{8} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.774785	+	1.18309i	−0.547856	+	0.836573i
							\(3\)	−0.959493	−	0.281733i	−0.553964	−	0.162658i
\(5\)	2.32348	+	2.01331i	1.03909	+	0.900379i								0.995122	−	0.0986541i	\(-0.0314537\pi\)
							0.0439706	+	0.999033i	\(0.485999\pi\)
\(7\)	−0.144795	+	3.03962i	−0.0547273	+	1.14887i	0.789804	+	0.613360i	\(0.210182\pi\)
							−0.844531	+	0.535507i	\(0.820121\pi\)
\(11\)	1.56145	+	0.300944i	0.470794	+	0.0907381i	0.419130	−	0.907926i	\(-0.362335\pi\)
							0.0516644	+	0.998665i	\(0.483547\pi\)
\(13\)	−0.787476	+	1.96702i	−0.218406	+	0.545553i	−0.996593	−	0.0824806i	\(-0.973716\pi\)
							0.778186	+	0.628034i	\(0.216140\pi\)
\(17\)	−2.30601	−	3.23834i	−0.559291	−	0.785414i	0.434018	−	0.900904i	\(-0.357095\pi\)
							−0.993309	+	0.115490i	\(0.963156\pi\)
\(19\)	4.99883	−	0.238123i	1.14681	−	0.0546293i	0.534444	−	0.845204i	\(-0.320521\pi\)
							0.612365	+	0.790575i	\(0.290218\pi\)
\(23\)	2.74857	+	2.88262i	0.573116	+	0.601067i	0.944943	−	0.327235i	\(-0.106117\pi\)
							−0.371827	+	0.928302i	\(0.621268\pi\)
\(29\)	−0.465554	−	0.806363i	−0.0864512	−	0.149738i	0.819557	−	0.572997i	\(-0.194219\pi\)
							−0.906009	+	0.423259i	\(0.860886\pi\)
\(31\)	4.60569	−	1.84384i	0.827207	−	0.331164i	0.0808637	−	0.996725i	\(-0.474232\pi\)
							0.746343	+	0.665562i	\(0.231808\pi\)
\(37\)	−3.99128	+	6.91311i	−0.656163	+	1.13651i	0.325438	+	0.945563i	\(0.394488\pi\)
							−0.981601	+	0.190944i	\(0.938845\pi\)
\(41\)	−0.00406842	−	0.0426065i	−0.000635381	−	0.00665401i	0.995149	−	0.0983825i	\(-0.0313669\pi\)
							−0.995784	+	0.0917285i	\(0.970761\pi\)
\(43\)	−1.23022	+	2.69380i	−0.187606	+	0.410800i	−0.979941	−	0.199286i	\(-0.936138\pi\)
							0.792335	+	0.610086i	\(0.208865\pi\)
\(47\)	−6.45799	+	1.56669i	−0.941995	+	0.228525i	−0.677216	−	0.735784i	\(-0.736814\pi\)
							−0.264778	+	0.964309i	\(0.585299\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.a.7.11	✓	680
4.3	odd	2		804.2.bf.b.7.34	yes	680
67.48	odd	66		804.2.bf.b.115.34	yes	680
268.115	even	66	inner	804.2.bf.a.115.11	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.11	✓	680	1.1	even	1	trivial
804.2.bf.a.115.11	yes	680	268.115	even	66	inner
804.2.bf.b.7.34	yes	680	4.3	odd	2
804.2.bf.b.115.34	yes	680	67.48	odd	66