Newform orbit 1056.3.p.c

• Label: 1056.3.p.c
• Level: $1056$
• Weight: $3$
• Character orbit: 1056.p
• Analytic conductor: $28.774$
• Analytic rank: $0$
• Dimension: $2$
• CM discriminant: -8
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1056 = 2^{5} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1056.p (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(28.7739159164\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 264)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

\(f(q)\)	\(=\)	q + (-b - 1) * q^3 + (2*b - 7) * q^9 + (-3*b - 7) * q^11 + 2 * q^17 + 6*b * q^19 - 25 * q^25 + (5*b + 23) * q^27 + (10*b - 17) * q^33 + 46 * q^41 + 30*b * q^43 - 49 * q^49 + (-2*b - 2) * q^51 + (-6*b + 48) * q^57 + 30*b * q^59 - 62 * q^67 - 12*b * q^73 + (25*b + 25) * q^75 + (-28*b + 17) * q^81 + 158 * q^83 + 36*b * q^89 + 94 * q^97 + (7*b + 97) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 14 q^{9} - 14 q^{11} + 4 q^{17} - 50 q^{25} + 46 q^{27} - 34 q^{33} + 92 q^{41} - 98 q^{49} - 4 q^{51} + 96 q^{57} - 124 q^{67} + 50 q^{75} + 34 q^{81} + 316 q^{83} + 188 q^{97} + 194 q^{99}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1056\mathbb{Z}\right)^\times\).

\(n\)	\(133\)	\(353\)	\(673\)	\(991\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(-1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

527.1

		1.41421i
	−	1.41421i

0

−1.00000

−

2.82843i

0

0

0

0

0

−7.00000

+

5.65685i

0

527.2

0

−1.00000

+

2.82843i

0

0

0

0

0

−7.00000

−

5.65685i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
33.d	even	2	1	inner
264.p	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1056.3.p.c		2
3.b	odd	2	1		1056.3.p.d		2
4.b	odd	2	1		264.3.p.e	yes	2
8.b	even	2	1		264.3.p.e	yes	2
8.d	odd	2	1	CM	1056.3.p.c		2
11.b	odd	2	1		1056.3.p.d		2
12.b	even	2	1		264.3.p.b	✓	2
24.f	even	2	1		1056.3.p.d		2
24.h	odd	2	1		264.3.p.b	✓	2
33.d	even	2	1	inner	1056.3.p.c		2
44.c	even	2	1		264.3.p.b	✓	2
88.b	odd	2	1		264.3.p.b	✓	2
88.g	even	2	1		1056.3.p.d		2
132.d	odd	2	1		264.3.p.e	yes	2
264.m	even	2	1		264.3.p.e	yes	2
264.p	odd	2	1	inner	1056.3.p.c		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
264.3.p.b	✓	2	12.b	even	2	1
264.3.p.b	✓	2	24.h	odd	2	1
264.3.p.b	✓	2	44.c	even	2	1
264.3.p.b	✓	2	88.b	odd	2	1
264.3.p.e	yes	2	4.b	odd	2	1
264.3.p.e	yes	2	8.b	even	2	1
264.3.p.e	yes	2	132.d	odd	2	1
264.3.p.e	yes	2	264.m	even	2	1
1056.3.p.c		2	1.a	even	1	1	trivial
1056.3.p.c		2	8.d	odd	2	1	CM
1056.3.p.c		2	33.d	even	2	1	inner
1056.3.p.c		2	264.p	odd	2	1	inner
1056.3.p.d		2	3.b	odd	2	1
1056.3.p.d		2	11.b	odd	2	1
1056.3.p.d		2	24.f	even	2	1
1056.3.p.d		2	88.g	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1056, [\chi])\):

\( T_{5} \)

\( T_{17} - 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + 2T + 9 \)
$5$	\( T^{2} \)
$7$	\( T^{2} \)
$11$	\( T^{2} + 14T + 121 \)