Embedded newform 637.2.bc.b.411.6

• Label: 637.2.bc.b.411.6
• Level: $637$
• Weight: $2$
• Character: 637.411
• Analytic conductor: $5.086$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.bc (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.08647060876\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(8\) over \(\Q(\zeta_{12})\)
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			411.6
Character	\(\chi\)	\(=\)	637.411
Dual form			637.2.bc.b.31.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.788958 - 0.211401i) q^{2} +(2.60482 - 1.50389i) q^{3} +(-1.15429 + 0.666428i) q^{4} +(-0.814012 - 3.03793i) q^{5} +(1.73717 - 1.73717i) q^{6} +(-1.92491 + 1.92491i) q^{8} +(3.02338 - 5.23665i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 2 q^{2} + 12 q^{3} + 6 q^{5} - 16 q^{8} + 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\)	\(197\)	\(248\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.788958	−	0.211401i	0.557877	−	0.149483i	0.0311464	−	0.999515i	\(-0.490084\pi\)
							0.526731	+	0.850032i	\(0.323418\pi\)
\(3\)	2.60482	−	1.50389i	1.50389	−	0.868272i	0.503902	−	0.863761i	\(-0.331897\pi\)
							0.999990	−	0.00451177i	\(-0.00143615\pi\)
\(5\)	−0.814012	−	3.03793i	−0.364037	−	1.35860i	−0.868721	−	0.495301i	\(-0.835058\pi\)
							0.504684	−	0.863304i	\(-0.331609\pi\)
\(7\)		0			0
							\(11\)	0.491212	+	0.131620i	0.148106	+	0.0396849i	0.332110	−	0.943241i	\(-0.392239\pi\)
−0.184004	+	0.982925i	\(0.558906\pi\)
\(13\)	−1.73717	−	3.15947i	−0.481804	−	0.876279i
							\(17\)	−0.606654	−	1.05076i	−0.147135	−	0.254846i	0.783032	−	0.621981i	\(-0.213672\pi\)
−0.930168	+	0.367135i	\(0.880339\pi\)
\(19\)	0.461325	+	1.72169i	0.105835	+	0.394982i	0.998439	−	0.0558606i	\(-0.0177902\pi\)
							−0.892603	+	0.450843i	\(0.851124\pi\)
\(23\)	4.51168	+	2.60482i	0.940750	+	0.543142i	0.890195	−	0.455579i	\(-0.150568\pi\)
							0.0505543	+	0.998721i	\(0.483901\pi\)
\(29\)	1.64443			0.305362			0.152681	−	0.988276i	\(-0.451209\pi\)
							0.152681	+	0.988276i	\(0.451209\pi\)
\(31\)	3.64327	+	0.976210i	0.654350	+	0.175333i	0.570695	−	0.821162i	\(-0.306674\pi\)
							0.0836552	+	0.996495i	\(0.473341\pi\)
\(37\)	0.715128	+	2.66889i	0.117566	+	0.438764i	0.999466	−	0.0326734i	\(-0.0104021\pi\)
							−0.881900	+	0.471437i	\(0.843735\pi\)
\(41\)	−5.55629	+	5.55629i	−0.867747	+	0.867747i	−0.992223	−	0.124476i	\(-0.960275\pi\)
							0.124476	+	0.992223i	\(0.460275\pi\)
\(43\)		−	7.46499i		−	1.13840i	−0.822199	−	0.569200i	\(-0.807253\pi\)
							0.822199	−	0.569200i	\(-0.192747\pi\)
\(47\)	4.73504	−	1.26875i	0.690677	−	0.185066i	0.103626	−	0.994616i	\(-0.466956\pi\)
							0.587051	+	0.809550i	\(0.300289\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.2.bc.b.411.6		32
7.2	even	3		637.2.i.a.489.6		32
7.3	odd	6	inner	637.2.bc.b.619.3		32
7.4	even	3		91.2.bb.a.73.3	yes	32
7.5	odd	6		637.2.i.a.489.5		32
7.6	odd	2		91.2.bb.a.47.6	yes	32
13.5	odd	4	inner	637.2.bc.b.460.3		32
21.11	odd	6		819.2.fn.e.73.6		32
21.20	even	2		819.2.fn.e.775.3		32
91.5	even	12		637.2.i.a.538.5		32
91.18	odd	12		91.2.bb.a.31.6	yes	32
91.31	even	12	inner	637.2.bc.b.31.6		32
91.44	odd	12		637.2.i.a.538.6		32
91.83	even	4		91.2.bb.a.5.3	✓	32
273.83	odd	4		819.2.fn.e.460.6		32
273.200	even	12		819.2.fn.e.577.3		32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.bb.a.5.3	✓	32	91.83	even	4
91.2.bb.a.31.6	yes	32	91.18	odd	12
91.2.bb.a.47.6	yes	32	7.6	odd	2
91.2.bb.a.73.3	yes	32	7.4	even	3
637.2.i.a.489.5		32	7.5	odd	6
637.2.i.a.489.6		32	7.2	even	3
637.2.i.a.538.5		32	91.5	even	12
637.2.i.a.538.6		32	91.44	odd	12
637.2.bc.b.31.6		32	91.31	even	12	inner
637.2.bc.b.411.6		32	1.1	even	1	trivial
637.2.bc.b.460.3		32	13.5	odd	4	inner
637.2.bc.b.619.3		32	7.3	odd	6	inner
819.2.fn.e.73.6		32	21.11	odd	6
819.2.fn.e.460.6		32	273.83	odd	4
819.2.fn.e.577.3		32	273.200	even	12
819.2.fn.e.775.3		32	21.20	even	2