Newform orbit 504.2.p.a

• Label: 504.2.p.a
• Level: $504$
• Weight: $2$
• Character orbit: 504.p
• Analytic conductor: $4.024$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -7
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.02446026187\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-7}) \)
Defining polynomial:	\( x^{2} - x + 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-7})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b - 1) * q^2 + (-b - 1) * q^4 + (2*b - 1) * q^7 + (-b + 3) * q^8 + 4 * q^11 + (-b - 3) * q^14 + (3*b - 1) * q^16 + (4*b - 4) * q^22 + (4*b - 2) * q^23 - 5 * q^25 + (-3*b + 5) * q^28 + (8*b - 4) * q^29 + (-b - 5) * q^32 + (8*b - 4) * q^37 + 12 * q^43 + (-4*b - 4) * q^44 + (-2*b - 6) * q^46 - 7 * q^49 + (-5*b + 5) * q^50 + (-8*b + 4) * q^53 + (5*b + 1) * q^56 + (-4*b - 12) * q^58 + (-5*b + 7) * q^64 + 4 * q^67 + (-4*b + 2) * q^71 + (-4*b - 12) * q^74 + (8*b - 4) * q^77 + (-12*b + 6) * q^79 + (12*b - 12) * q^86 + (-4*b + 12) * q^88 + (-6*b + 10) * q^92 + (-7*b + 7) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - q^2 - 3 * q^4 + 5 * q^8 + 8 * q^11 - 7 * q^14 + q^16 - 4 * q^22 - 10 * q^25 + 7 * q^28 - 11 * q^32 + 24 * q^43 - 12 * q^44 - 14 * q^46 - 14 * q^49 + 5 * q^50 + 7 * q^56 - 28 * q^58 + 9 * q^64 + 8 * q^67 - 28 * q^74 - 12 * q^86 + 20 * q^88 + 14 * q^92 + 7 * q^98

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\)	\(73\)	\(127\)	\(253\)	\(281\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

307.1

0.500000	−	1.32288i
0.500000	+	1.32288i

−0.500000

−

1.32288i

0

−1.50000

+

1.32288i

0

0

−

2.64575i

2.50000

+

1.32288i

0

0

307.2

−0.500000

+

1.32288i

0

−1.50000

−

1.32288i

0

0

2.64575i

2.50000

−

1.32288i

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
8.d	odd	2	1	inner
56.e	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	504.2.p.a		2
3.b	odd	2	1		56.2.e.a	✓	2
4.b	odd	2	1		2016.2.p.a		2
7.b	odd	2	1	CM	504.2.p.a		2
8.b	even	2	1		2016.2.p.a		2
8.d	odd	2	1	inner	504.2.p.a		2
12.b	even	2	1		224.2.e.a		2
21.c	even	2	1		56.2.e.a	✓	2
21.g	even	6	2		392.2.m.a		4
21.h	odd	6	2		392.2.m.a		4
24.f	even	2	1		56.2.e.a	✓	2
24.h	odd	2	1		224.2.e.a		2
28.d	even	2	1		2016.2.p.a		2
48.i	odd	4	2		1792.2.f.d		4
48.k	even	4	2		1792.2.f.d		4
56.e	even	2	1	inner	504.2.p.a		2
56.h	odd	2	1		2016.2.p.a		2
84.h	odd	2	1		224.2.e.a		2
84.j	odd	6	2		1568.2.q.a		4
84.n	even	6	2		1568.2.q.a		4
168.e	odd	2	1		56.2.e.a	✓	2
168.i	even	2	1		224.2.e.a		2
168.s	odd	6	2		1568.2.q.a		4
168.v	even	6	2		392.2.m.a		4
168.ba	even	6	2		1568.2.q.a		4
168.be	odd	6	2		392.2.m.a		4
336.v	odd	4	2		1792.2.f.d		4
336.y	even	4	2		1792.2.f.d		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.2.e.a	✓	2	3.b	odd	2	1
56.2.e.a	✓	2	21.c	even	2	1
56.2.e.a	✓	2	24.f	even	2	1
56.2.e.a	✓	2	168.e	odd	2	1
224.2.e.a		2	12.b	even	2	1
224.2.e.a		2	24.h	odd	2	1
224.2.e.a		2	84.h	odd	2	1
224.2.e.a		2	168.i	even	2	1
392.2.m.a		4	21.g	even	6	2
392.2.m.a		4	21.h	odd	6	2
392.2.m.a		4	168.v	even	6	2
392.2.m.a		4	168.be	odd	6	2
504.2.p.a		2	1.a	even	1	1	trivial
504.2.p.a		2	7.b	odd	2	1	CM
504.2.p.a		2	8.d	odd	2	1	inner
504.2.p.a		2	56.e	even	2	1	inner
1568.2.q.a		4	84.j	odd	6	2
1568.2.q.a		4	84.n	even	6	2
1568.2.q.a		4	168.s	odd	6	2
1568.2.q.a		4	168.ba	even	6	2
1792.2.f.d		4	48.i	odd	4	2
1792.2.f.d		4	48.k	even	4	2
1792.2.f.d		4	336.v	odd	4	2
1792.2.f.d		4	336.y	even	4	2
2016.2.p.a		2	4.b	odd	2	1
2016.2.p.a		2	8.b	even	2	1
2016.2.p.a		2	28.d	even	2	1
2016.2.p.a		2	56.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\):

\( T_{5} \)

\( T_{11} - 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + T + 2 \)
$3$	\( T^{2} \)
$5$	\( T^{2} \)
$7$	\( T^{2} + 7 \)
$11$	\( (T - 4)^{2} \)