Embedded newform 804.2.bd.a.35.1

• Label: 804.2.bd.a.35.1
• Level: $804$
• Weight: $2$
• Character: 804.35
• Analytic conductor: $6.420$
• Analytic rank: $0$
• Dimension: $2640$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	804.bd (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41997232251\)
Analytic rank:	\(0\)
Dimension:	\(2640\)
Relative dimension:	\(132\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label			35.1
Character	\(\chi\)	\(=\)	804.35
Dual form			804.2.bd.a.23.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41230 - 0.0734794i) q^{2} +(-0.651270 + 1.60494i) q^{3} +(1.98920 + 0.207550i) q^{4} +(1.42405 + 0.650344i) q^{5} +(1.03772 - 2.21881i) q^{6} +(-2.49089 + 2.61237i) q^{7} +(-2.79411 - 0.439290i) q^{8} +(-2.15170 - 2.09050i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2640 q - 42 q^{4} - 19 q^{6} - 32 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\)	\(269\)	\(337\)	\(403\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{19}{33}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41230	−	0.0734794i	−0.998649	−	0.0519578i
							\(3\)	−0.651270	+	1.60494i	−0.376011	+	0.926615i
\(5\)	1.42405	+	0.650344i	0.636856	+	0.290843i								0.707558	−	0.706655i	\(-0.249797\pi\)
							−0.0707020	+	0.997497i	\(0.522524\pi\)
\(7\)	−2.49089	+	2.61237i	−0.941468	+	0.987383i	−0.999941	−	0.0109056i	\(-0.996529\pi\)
							0.0584723	+	0.998289i	\(0.481377\pi\)
\(11\)	3.33341	−	0.318302i	1.00506	−	0.0959717i	0.420476	−	0.907304i	\(-0.361863\pi\)
							0.584586	+	0.811332i	\(0.301257\pi\)
\(13\)	−5.81919	+	1.12156i	−1.61395	+	0.311064i	−0.914799	−	0.403909i	\(-0.867651\pi\)
							−0.699153	+	0.714972i	\(0.746439\pi\)
\(17\)	−3.01536	−	5.84899i	−0.731333	−	1.41859i	−0.901326	−	0.433142i	\(-0.857405\pi\)
							0.169993	−	0.985445i	\(-0.445626\pi\)
\(19\)	−2.35023	−	2.46485i	−0.539180	−	0.565475i	0.396631	−	0.917978i	\(-0.370179\pi\)
							−0.935811	+	0.352503i	\(0.885331\pi\)
\(23\)	0.843917	+	0.337853i	0.175969	+	0.0704473i	0.457974	−	0.888966i	\(-0.348575\pi\)
							−0.282005	+	0.959413i	\(0.591000\pi\)
\(29\)	−3.86686	+	2.23253i	−0.718058	+	0.414571i	−0.814038	−	0.580812i	\(-0.802735\pi\)
							0.0959793	+	0.995383i	\(0.469402\pi\)
\(31\)	−1.50080	+	7.78687i	−0.269551	+	1.39856i	0.554564	+	0.832141i	\(0.312885\pi\)
							−0.824115	+	0.566422i	\(0.808327\pi\)
\(37\)	3.98212	−	6.89723i	0.654656	−	1.13390i	−0.327324	−	0.944912i	\(-0.606147\pi\)
							0.981980	−	0.188985i	\(-0.0605198\pi\)
\(41\)	11.3009	+	0.538328i	1.76490	+	0.0840727i	0.904744	−	0.425955i	\(-0.140062\pi\)
							0.860158	+	0.510028i	\(0.170365\pi\)
\(43\)	−1.72186	+	2.67927i	−0.262582	+	0.408585i	−0.947362	−	0.320164i	\(-0.896262\pi\)
							0.684780	+	0.728750i	\(0.259898\pi\)
\(47\)	−4.66470	+	3.66836i	−0.680417	+	0.535086i	−0.897471	−	0.441074i	\(-0.854598\pi\)
							0.217054	+	0.976160i	\(0.430355\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	804.2.bd.a.35.1	yes	2640
3.2	odd	2	inner	804.2.bd.a.35.132	yes	2640
4.3	odd	2	inner	804.2.bd.a.35.19	yes	2640
12.11	even	2	inner	804.2.bd.a.35.114	yes	2640
67.23	even	33	inner	804.2.bd.a.23.114	yes	2640
201.23	odd	66	inner	804.2.bd.a.23.19	yes	2640
268.23	odd	66	inner	804.2.bd.a.23.132	yes	2640
804.23	even	66	inner	804.2.bd.a.23.1	✓	2640

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
804.2.bd.a.23.1	✓	2640	804.23	even	66	inner
804.2.bd.a.23.19	yes	2640	201.23	odd	66	inner
804.2.bd.a.23.114	yes	2640	67.23	even	33	inner
804.2.bd.a.23.132	yes	2640	268.23	odd	66	inner
804.2.bd.a.35.1	yes	2640	1.1	even	1	trivial
804.2.bd.a.35.19	yes	2640	4.3	odd	2	inner
804.2.bd.a.35.114	yes	2640	12.11	even	2	inner
804.2.bd.a.35.132	yes	2640	3.2	odd	2	inner