Embedded newform 804.2.bf.b.7.6

• Label: 804.2.bf.b.7.6
• 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			7.6
Character	\(\chi\)	\(=\)	804.7
Dual form			804.2.bf.b.115.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28701 - 0.586177i) q^{2} +(0.959493 + 0.281733i) q^{3} +(1.31279 + 1.50883i) q^{4} +(1.87962 + 1.62870i) q^{5} +(-1.06973 - 0.925026i) q^{6} +(-0.129508 + 2.71872i) q^{7} +(-0.805133 - 2.71141i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q + 68 q^{3} + 2 q^{4} + 11 q^{6} - 4 q^{7} - 27 q^{8} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28701	−	0.586177i	−0.910054	−	0.414490i
							\(3\)	0.959493	+	0.281733i	0.553964	+	0.162658i
\(5\)	1.87962	+	1.62870i	0.840594	+	0.728379i								0.964547	−	0.263909i	\(-0.0850120\pi\)
							−0.123954	+	0.992288i	\(0.539557\pi\)
\(7\)	−0.129508	+	2.71872i	−0.0489496	+	1.02758i	0.832121	+	0.554594i	\(0.187126\pi\)
							−0.881071	+	0.472985i	\(0.843177\pi\)
\(11\)	−5.61675	−	1.08254i	−1.69351	−	0.326398i	−0.750934	−	0.660378i	\(-0.770396\pi\)
							−0.942580	+	0.333980i	\(0.891608\pi\)
\(13\)	−1.03267	+	2.57947i	−0.286410	+	0.715417i	0.713481	+	0.700674i	\(0.247117\pi\)
							−0.999891	+	0.0147431i	\(0.995307\pi\)
\(17\)	3.68975	+	5.18153i	0.894897	+	1.25671i	0.965817	+	0.259224i	\(0.0834669\pi\)
							−0.0709206	+	0.997482i	\(0.522594\pi\)
\(19\)	3.53866	−	0.168567i	0.811825	−	0.0386720i	0.362437	−	0.932008i	\(-0.381945\pi\)
							0.449388	+	0.893336i	\(0.351642\pi\)
\(23\)	−1.65357	−	1.73422i	−0.344793	−	0.361609i	0.528389	−	0.849002i	\(-0.322796\pi\)
							−0.873182	+	0.487393i	\(0.837948\pi\)
\(29\)	−1.54025	−	2.66779i	−0.286017	−	0.495396i	0.686838	−	0.726810i	\(-0.258998\pi\)
							−0.972855	+	0.231414i	\(0.925665\pi\)
\(31\)	−6.86269	+	2.74740i	−1.23257	+	0.493448i	−0.894244	−	0.447579i	\(-0.852286\pi\)
							−0.338330	+	0.941027i	\(0.609862\pi\)
\(37\)	−0.230601	+	0.399413i	−0.0379106	+	0.0656630i	−0.884358	−	0.466809i	\(-0.845404\pi\)
							0.846448	+	0.532472i	\(0.178737\pi\)
\(41\)	−0.779554	−	8.16386i	−0.121746	−	1.27498i	−0.825445	−	0.564483i	\(-0.809076\pi\)
							0.703699	−	0.710498i	\(-0.251530\pi\)
\(43\)	−3.59951	+	7.88183i	−0.548920	+	1.20197i	0.408364	+	0.912819i	\(0.366099\pi\)
							−0.957284	+	0.289149i	\(0.906628\pi\)
\(47\)	−5.65467	+	1.37181i	−0.824818	+	0.200099i	−0.625886	−	0.779915i	\(-0.715262\pi\)
							−0.198932	+	0.980013i	\(0.563747\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.b.7.6	yes	680
4.3	odd	2		804.2.bf.a.7.19	✓	680
67.48	odd	66		804.2.bf.a.115.19	yes	680
268.115	even	66	inner	804.2.bf.b.115.6	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.19	✓	680	4.3	odd	2
804.2.bf.a.115.19	yes	680	67.48	odd	66
804.2.bf.b.7.6	yes	680	1.1	even	1	trivial
804.2.bf.b.115.6	yes	680	268.115	even	66	inner