Embedded newform 803.2.bk.a.487.55

• Label: 803.2.bk.a.487.55
• Level: $803$
• Weight: $2$
• Character: 803.487
• Analytic conductor: $6.412$
• Analytic rank: $0$
• Dimension: $1152$
• 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			487.55
Character	\(\chi\)	\(=\)	803.487
Dual form			803.2.bk.a.587.55

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.34179 + 0.597405i) q^{2} +(-1.53227 + 0.497863i) q^{3} +(0.105255 + 0.116898i) q^{4} +(-1.07708 - 0.413453i) q^{5} +(-2.35341 - 0.247353i) q^{6} +(-0.764864 + 0.389717i) q^{7} +(-0.836358 - 2.57404i) q^{8} +(-0.327082 + 0.237639i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1152 q - 2 q^{2} + 134 q^{4} - 8 q^{5} - 42 q^{6} - 8 q^{7} - 64 q^{8} + 272 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(78\)	\(585\)
\(\chi(n)\)	\(e\left(\frac{11}{12}\right)\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.34179	+	0.597405i	0.948791	+	0.422429i	0.821992	−	0.569499i	\(-0.192863\pi\)
							0.126799	+	0.991928i	\(0.459530\pi\)
\(3\)	−1.53227	+	0.497863i	−0.884654	+	0.287441i	−0.715888	−	0.698215i	\(-0.753978\pi\)
							−0.168765	+	0.985656i	\(0.553978\pi\)
\(5\)	−1.07708	−	0.413453i	−0.481686	−	0.184902i	0.105394	−	0.994431i	\(-0.466390\pi\)
							−0.587081	+	0.809528i	\(0.699723\pi\)
\(7\)	−0.764864	+	0.389717i	−0.289091	+	0.147299i	−0.592522	−	0.805554i	\(-0.701868\pi\)
							0.303431	+	0.952853i	\(0.401868\pi\)
\(11\)	2.75054	+	1.85324i	0.829320	+	0.558774i
							\(13\)	5.89615	−	2.26332i	1.63530	−	0.627732i	0.644687	−	0.764447i	\(-0.276988\pi\)
0.990610	+	0.136715i	\(0.0436544\pi\)
\(17\)	−0.0479050	−	0.302460i	−0.0116187	−	0.0733574i	0.981198	−	0.193005i	\(-0.0618234\pi\)
							−0.992816	+	0.119648i	\(0.961823\pi\)
\(19\)	4.93617	+	4.44455i	1.13243	+	1.01965i	0.999593	+	0.0285115i	\(0.00907673\pi\)
							0.132841	+	0.991137i	\(0.457590\pi\)
\(23\)	2.54900	+	1.47166i	0.531502	+	0.306863i	0.741628	−	0.670811i	\(-0.234054\pi\)
							−0.210126	+	0.977674i	\(0.567387\pi\)
\(29\)	1.93181	−	2.97472i	0.358727	−	0.552392i	−0.612523	−	0.790453i	\(-0.709845\pi\)
							0.971250	+	0.238061i	\(0.0765119\pi\)
\(31\)	2.69505	−	1.03453i	0.484045	−	0.185808i	−0.104094	−	0.994568i	\(-0.533194\pi\)
							0.588139	+	0.808760i	\(0.299861\pi\)
\(37\)	7.42954	+	8.25134i	1.22141	+	1.35651i	0.914404	+	0.404803i	\(0.132660\pi\)
							0.307004	+	0.951708i	\(0.400673\pi\)
\(41\)	−4.21292	−	0.895485i	−0.657948	−	0.139851i	−0.133174	−	0.991093i	\(-0.542517\pi\)
							−0.524774	+	0.851241i	\(0.675850\pi\)
\(43\)	7.61745	−	7.61745i	1.16165	−	1.16165i	0.177535	−	0.984114i	\(-0.443188\pi\)
							0.984114	−	0.177535i	\(-0.0568124\pi\)
\(47\)	3.29774	−	2.14158i	0.481025	−	0.312381i	−0.281250	−	0.959635i	\(-0.590749\pi\)
							0.762275	+	0.647253i	\(0.224082\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
803.2.bk.a.3.18	✓	1152	73.3	even	12	inner
803.2.bk.a.268.18	yes	1152	11.4	even	5	inner
803.2.bk.a.487.55	yes	1152	1.1	even	1	trivial
803.2.bk.a.587.55	yes	1152	803.587	even	60	inner