Embedded newform 804.2.bf.b.7.2

• Label: 804.2.bf.b.7.2
• 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			7.2
Character	\(\chi\)	\(=\)	804.7
Dual form			804.2.bf.b.115.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41249 + 0.0698158i) q^{2} +(0.959493 + 0.281733i) q^{3} +(1.99025 - 0.197228i) q^{4} +(-2.97198 - 2.57523i) q^{5} +(-1.37494 - 0.330956i) q^{6} +(-0.164722 + 3.45794i) q^{7} +(-2.79744 + 0.417533i) q^{8} +(0.841254 + 0.540641i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 680 q + 68 q^{3} + 2 q^{4} + 11 q^{6} - 4 q^{7} - 27 q^{8} - 68 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41249	+	0.0698158i	−0.998781	+	0.0493672i
							\(3\)	0.959493	+	0.281733i	0.553964	+	0.162658i
\(5\)	−2.97198	−	2.57523i	−1.32911	−	1.15168i								−0.976406	−	0.215943i	\(-0.930717\pi\)
							−0.352702	−	0.935736i	\(-0.614737\pi\)
\(7\)	−0.164722	+	3.45794i	−0.0622591	+	1.30698i	0.724309	+	0.689476i	\(0.242159\pi\)
							−0.786568	+	0.617504i	\(0.788144\pi\)
\(11\)	−2.56840	−	0.495018i	−0.774400	−	0.149253i	−0.213284	−	0.976990i	\(-0.568416\pi\)
							−0.561116	+	0.827737i	\(0.689628\pi\)
\(13\)	1.32013	−	3.29752i	0.366137	−	0.914566i	−0.624867	−	0.780732i	\(-0.714847\pi\)
							0.991004	−	0.133835i	\(-0.0427291\pi\)
\(17\)	2.57721	+	3.61919i	0.625066	+	0.877783i	0.998761	−	0.0497691i	\(-0.0158485\pi\)
							−0.373694	+	0.927552i	\(0.621909\pi\)
\(19\)	2.29567	−	0.109356i	0.526664	−	0.0250881i	0.217433	−	0.976075i	\(-0.430232\pi\)
							0.309231	+	0.950987i	\(0.399929\pi\)
\(23\)	2.69458	+	2.82600i	0.561859	+	0.589261i	0.941977	−	0.335677i	\(-0.108965\pi\)
							−0.380118	+	0.924938i	\(0.624117\pi\)
\(29\)	2.52951	+	4.38125i	0.469719	+	0.813577i	0.999401	−	0.0346194i	\(-0.0110219\pi\)
							−0.529682	+	0.848197i	\(0.677689\pi\)
\(31\)	2.68829	−	1.07623i	0.482832	−	0.193297i	−0.117460	−	0.993078i	\(-0.537475\pi\)
							0.600292	+	0.799781i	\(0.295051\pi\)
\(37\)	4.38998	−	7.60367i	0.721708	−	1.25004i	−0.238606	−	0.971116i	\(-0.576690\pi\)
							0.960315	−	0.278919i	\(-0.0899762\pi\)
\(41\)	0.388194	+	4.06535i	0.0606258	+	0.634902i	0.973811	+	0.227358i	\(0.0730087\pi\)
							−0.913186	+	0.407544i	\(0.866385\pi\)
\(43\)	−2.60673	+	5.70795i	−0.397523	+	0.870454i	0.599993	+	0.800006i	\(0.295170\pi\)
							−0.997515	+	0.0704481i	\(0.977557\pi\)
\(47\)	12.8665	−	3.12137i	1.87676	−	0.455298i	0.877079	−	0.480347i	\(-0.159489\pi\)
							0.999686	+	0.0250483i	\(0.00797397\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bf.b.7.2	yes	680
4.3	odd	2		804.2.bf.a.7.25	✓	680
67.48	odd	66		804.2.bf.a.115.25	yes	680
268.115	even	66	inner	804.2.bf.b.115.2	yes	680

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bf.a.7.25	✓	680	4.3	odd	2
804.2.bf.a.115.25	yes	680	67.48	odd	66
804.2.bf.b.7.2	yes	680	1.1	even	1	trivial
804.2.bf.b.115.2	yes	680	268.115	even	66	inner