Embedded newform 803.2.bk.a.3.18

• Label: 803.2.bk.a.3.18
• Level: $803$
• Weight: $2$
• Character: 803.3
• Analytic conductor: $6.412$
• Analytic rank: $0$
• Dimension: $1152$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 803 = 11 \cdot 73 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	803.bk (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.41198728231\)
Analytic rank:	\(0\)
Dimension:	\(1152\)
Relative dimension:	\(72\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

Embedding invariants

Embedding label			3.18
Character	\(\chi\)	\(=\)	803.3
Dual form			803.2.bk.a.268.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.153529 - 1.46073i) q^{2} +(1.53227 - 0.497863i) q^{3} +(-0.153864 + 0.0327049i) q^{4} +(-0.726055 - 0.896603i) q^{5} +(-0.962490 - 2.16179i) q^{6} +(0.389717 + 0.764864i) q^{7} +(-0.836358 - 2.57404i) q^{8} +(-0.327082 + 0.237639i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 2 q^{2} + 134 q^{4} - 8 q^{5} - 42 q^{6} - 8 q^{7} - 64 q^{8} + 272 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/803\mathbb{Z}\right)^\times\).

\(n\)	\(78\)	\(585\)
\(\chi(n)\)	\(e\left(\frac{1}{12}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.153529	−	1.46073i	−0.108561	−	1.03289i	−0.904197	−	0.427116i	\(-0.859530\pi\)
							0.795635	−	0.605776i	\(-0.207137\pi\)
\(3\)	1.53227	−	0.497863i	0.884654	−	0.287441i	0.168765	−	0.985656i	\(-0.446022\pi\)
							0.715888	+	0.698215i	\(0.246022\pi\)
\(5\)	−0.726055	−	0.896603i	−0.324701	−	0.400973i	0.588489	−	0.808505i	\(-0.299723\pi\)
							−0.913191	+	0.407532i	\(0.866390\pi\)
\(7\)	0.389717	+	0.764864i	0.147299	+	0.289091i	0.952853	−	0.303431i	\(-0.0981322\pi\)
							−0.805554	+	0.592522i	\(0.798132\pi\)
\(11\)	−1.45542	−	2.98023i	−0.438825	−	0.898572i
							\(13\)	3.97455	−	4.90817i	1.10234	−	1.36128i	0.176092	−	0.984374i	\(-0.443654\pi\)
0.926251	−	0.376907i	\(-0.123012\pi\)
\(17\)	−0.302460	+	0.0479050i	−0.0733574	+	0.0116187i	−0.193005	−	0.981198i	\(-0.561823\pi\)
							0.119648	+	0.992816i	\(0.461823\pi\)
\(19\)	−1.38101	+	6.49712i	−0.316824	+	1.49054i	0.475105	+	0.879929i	\(0.342410\pi\)
							−0.791929	+	0.610613i	\(0.790923\pi\)
\(23\)	2.54900	−	1.47166i	0.531502	−	0.306863i	−0.210126	−	0.977674i	\(-0.567387\pi\)
							0.741628	+	0.670811i	\(0.234054\pi\)
\(29\)	0.185633	−	3.54209i	0.0344712	−	0.657749i	−0.925687	−	0.378291i	\(-0.876512\pi\)
							0.960158	−	0.279458i	\(-0.0901548\pi\)
\(31\)	1.81672	−	2.24346i	0.326292	−	0.402937i	−0.587431	−	0.809274i	\(-0.699861\pi\)
							0.913723	+	0.406337i	\(0.133194\pi\)
\(37\)	−10.8606	+	2.30850i	−1.78548	+	0.379515i	−0.977706	−	0.209981i	\(-0.932660\pi\)
							−0.807772	+	0.589496i	\(0.799327\pi\)
\(41\)	2.88197	−	3.20076i	0.450089	−	0.499874i	−0.474810	−	0.880089i	\(-0.657483\pi\)
							0.924898	+	0.380214i	\(0.124150\pi\)
\(43\)	7.61745	+	7.61745i	1.16165	+	1.16165i	0.984114	+	0.177535i	\(0.0568124\pi\)
							0.177535	+	0.984114i	\(0.443188\pi\)
\(47\)	−3.92672	+	0.205791i	−0.572771	+	0.0300176i	−0.336523	−	0.941675i	\(-0.609251\pi\)
							−0.236248	+	0.971693i	\(0.575918\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	803.2.bk.a.3.18	✓	1152
11.4	even	5	inner	803.2.bk.a.587.55	yes	1152
73.49	even	12	inner	803.2.bk.a.487.55	yes	1152
803.268	even	60	inner	803.2.bk.a.268.18	yes	1152

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
803.2.bk.a.3.18	✓	1152	1.1	even	1	trivial
803.2.bk.a.268.18	yes	1152	803.268	even	60	inner
803.2.bk.a.487.55	yes	1152	73.49	even	12	inner
803.2.bk.a.587.55	yes	1152	11.4	even	5	inner