Newform orbit 1008.3.m.a

• Label: 1008.3.m.a
• Level: $1008$
• Weight: $3$
• Character orbit: 1008.m
• Analytic conductor: $27.466$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1008.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(27.4660106475\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-7}) \)
Defining polynomial:	\( x^{2} - x + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\)	\(=\)	q - 8 * q^5 - b * q^7 - 4*b * q^11 - 4 * q^13 + 2 * q^17 - 10*b * q^19 + 8*b * q^23 + 39 * q^25 - 14 * q^29 + 12*b * q^31 + 8*b * q^35 + 14 * q^37 - 46 * q^41 - 4*b * q^43 + 12*b * q^47 - 7 * q^49 + 22 * q^53 + 32*b * q^55 + 34*b * q^59 + 48 * q^61 + 32 * q^65 + 24*b * q^67 + 32*b * q^71 - 110 * q^73 - 28 * q^77 + 48*b * q^79 - 14*b * q^83 - 16 * q^85 + 134 * q^89 + 4*b * q^91 + 80*b * q^95 - 178 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 16 q^{5} - 8 q^{13} + 4 q^{17} + 78 q^{25} - 28 q^{29} + 28 q^{37} - 92 q^{41} - 14 q^{49} + 44 q^{53} + 96 q^{61} + 64 q^{65} - 220 q^{73} - 56 q^{77} - 32 q^{85} + 268 q^{89} - 356 q^{97}+O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(577\)	\(757\)	\(785\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

127.1

0.500000	+	1.32288i
0.500000	−	1.32288i

0

0

0

−8.00000

0

−

2.64575i

0

0

0

127.2

0

0

0

−8.00000

0

2.64575i

0

0

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1008.3.m.a		2
3.b	odd	2	1		112.3.d.a	✓	2
4.b	odd	2	1	inner	1008.3.m.a		2
12.b	even	2	1		112.3.d.a	✓	2
21.c	even	2	1		784.3.d.d		2
21.g	even	6	2		784.3.r.m		4
21.h	odd	6	2		784.3.r.k		4
24.f	even	2	1		448.3.d.a		2
24.h	odd	2	1		448.3.d.a		2
48.i	odd	4	2		1792.3.g.b		4
48.k	even	4	2		1792.3.g.b		4
84.h	odd	2	1		784.3.d.d		2
84.j	odd	6	2		784.3.r.m		4
84.n	even	6	2		784.3.r.k		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
112.3.d.a	✓	2	3.b	odd	2	1
112.3.d.a	✓	2	12.b	even	2	1
448.3.d.a		2	24.f	even	2	1
448.3.d.a		2	24.h	odd	2	1
784.3.d.d		2	21.c	even	2	1
784.3.d.d		2	84.h	odd	2	1
784.3.r.k		4	21.h	odd	6	2
784.3.r.k		4	84.n	even	6	2
784.3.r.m		4	21.g	even	6	2
784.3.r.m		4	84.j	odd	6	2
1008.3.m.a		2	1.a	even	1	1	trivial
1008.3.m.a		2	4.b	odd	2	1	inner
1792.3.g.b		4	48.i	odd	4	2
1792.3.g.b		4	48.k	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 8 \) acting on \(S_{3}^{\mathrm{new}}(1008, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( (T + 8)^{2} \)
$7$	\( T^{2} + 7 \)
$11$	\( T^{2} + 112 \)