Embedded newform 804.2.bf.b.7.17

• Label: 804.2.bf.b.7.17
• 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			7.17
Character	\(\chi\)	\(=\)	804.7
Dual form			804.2.bf.b.115.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.50135e-5 - 1.41421i) q^{2} +(0.959493 + 0.281733i) q^{3} +(-2.00000 + 0.000212170i) q^{4} +(-1.53250 - 1.32792i) q^{5} +(0.398358 - 1.35695i) q^{6} +(0.182174 - 3.82430i) q^{7} +(0.000450081 + 2.82843i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q + 68 q^{3} + 2 q^{4} + 11 q^{6} - 4 q^{7} - 27 q^{8} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.50135e−5	−	1.41421i	−5.30425e−5	−	1.00000i
							\(3\)	0.959493	+	0.281733i	0.553964	+	0.162658i
\(5\)	−1.53250	−	1.32792i	−0.685355	−	0.593864i								0.240995	−	0.970526i	\(-0.422526\pi\)
							−0.926350	+	0.376663i	\(0.877072\pi\)
\(7\)	0.182174	−	3.82430i	0.0688553	−	1.44545i	−0.655068	−	0.755570i	\(-0.727360\pi\)
							0.723923	−	0.689881i	\(-0.242337\pi\)
\(11\)	5.58439	+	1.07630i	1.68376	+	0.324517i	0.939285	−	0.343138i	\(-0.111490\pi\)
							0.744471	+	0.667655i	\(0.232702\pi\)
\(13\)	0.0445192	−	0.111204i	0.0123474	−	0.0308423i	−0.922066	−	0.387032i	\(-0.873500\pi\)
							0.934414	+	0.356190i	\(0.115924\pi\)
\(17\)	−2.90732	−	4.08276i	−0.705128	−	0.990214i	−0.999405	−	0.0344940i	\(-0.989018\pi\)
							0.294276	−	0.955720i	\(-0.404921\pi\)
\(19\)	−4.27871	+	0.203820i	−0.981604	+	0.0467595i	−0.532244	−	0.846591i	\(-0.678651\pi\)
							−0.449360	+	0.893351i	\(0.648348\pi\)
\(23\)	−6.21510	−	6.51821i	−1.29594	−	1.35914i	−0.896917	−	0.442198i	\(-0.854199\pi\)
							−0.399020	−	0.916942i	\(-0.630650\pi\)
\(29\)	2.20079	+	3.81188i	0.408677	+	0.707849i	0.994742	−	0.102415i	\(-0.0326570\pi\)
							−0.586065	+	0.810264i	\(0.699324\pi\)
\(31\)	−5.59606	+	2.24032i	−1.00508	+	0.402374i	−0.815041	−	0.579403i	\(-0.803286\pi\)
							−0.190040	+	0.981776i	\(0.560862\pi\)
\(37\)	−0.475519	+	0.823624i	−0.0781749	+	0.135403i	−0.902463	−	0.430768i	\(-0.858243\pi\)
							0.824288	+	0.566171i	\(0.191576\pi\)
\(41\)	−0.233102	−	2.44115i	−0.0364044	−	0.381244i	−0.995253	−	0.0973248i	\(-0.968971\pi\)
							0.958848	−	0.283919i	\(-0.0916346\pi\)
\(43\)	−4.57968	+	10.0281i	−0.698394	+	1.52927i	0.143514	+	0.989648i	\(0.454160\pi\)
							−0.841908	+	0.539621i	\(0.818568\pi\)
\(47\)	9.25257	−	2.24465i	1.34963	−	0.327416i	0.505160	−	0.863025i	\(-0.331433\pi\)
							0.844466	+	0.535610i	\(0.179918\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.b.7.17	yes	680
4.3	odd	2		804.2.bf.a.7.8	✓	680
67.48	odd	66		804.2.bf.a.115.8	yes	680
268.115	even	66	inner	804.2.bf.b.115.17	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.8	✓	680	4.3	odd	2
804.2.bf.a.115.8	yes	680	67.48	odd	66
804.2.bf.b.7.17	yes	680	1.1	even	1	trivial
804.2.bf.b.115.17	yes	680	268.115	even	66	inner