Newform orbit 1008.4.a.y

• Label: 1008.4.a.y
• Level: $1008$
• Weight: $4$
• Character orbit: 1008.a
• Self dual: yes
• Analytic conductor: $59.474$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(59.4739252858\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{177}) \)
Defining polynomial:	\( x^{2} - x - 44 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2 \)
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{177}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (-b - 7) * q^5 - 7 * q^7 + (-3*b + 9) * q^11 + (2*b + 24) * q^13 + (9*b - 17) * q^17 + (4*b + 8) * q^19 + (7*b + 55) * q^23 + (14*b + 101) * q^25 + (-4*b - 106) * q^29 + (4*b + 68) * q^31 + (7*b + 49) * q^35 + (-26*b - 12) * q^37 + (-9*b - 347) * q^41 + (-8*b + 292) * q^43 + (2*b - 158) * q^47 + 49 * q^49 + (6*b - 280) * q^53 + (12*b + 468) * q^55 + (-26*b - 246) * q^59 + (28*b - 302) * q^61 + (-38*b - 522) * q^65 + (10*b + 510) * q^67 + (25*b - 855) * q^71 + (6*b - 656) * q^73 + (21*b - 63) * q^77 + (-42*b + 278) * q^79 + (-32*b - 132) * q^83 + (-46*b - 1474) * q^85 + (95*b - 35) * q^89 + (-14*b - 168) * q^91 + (-36*b - 764) * q^95 + (10*b - 68) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 14 * q^5 - 14 * q^7 + 18 * q^11 + 48 * q^13 - 34 * q^17 + 16 * q^19 + 110 * q^23 + 202 * q^25 - 212 * q^29 + 136 * q^31 + 98 * q^35 - 24 * q^37 - 694 * q^41 + 584 * q^43 - 316 * q^47 + 98 * q^49 - 560 * q^53 + 936 * q^55 - 492 * q^59 - 604 * q^61 - 1044 * q^65 + 1020 * q^67 - 1710 * q^71 - 1312 * q^73 - 126 * q^77 + 556 * q^79 - 264 * q^83 - 2948 * q^85 - 70 * q^89 - 336 * q^91 - 1528 * q^95 - 136 * q^97

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

7.15207
−6.15207

0

0

0

−20.3041

0

−7.00000

0

0

0

1.2

0

0

0

6.30413

0

−7.00000

0

0

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(3\)	\( -1 \)
\(7\)	\( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1008.4.a.y		2
3.b	odd	2	1		336.4.a.n		2
4.b	odd	2	1		504.4.a.j		2
12.b	even	2	1		168.4.a.h	✓	2
21.c	even	2	1		2352.4.a.bv		2
24.f	even	2	1		1344.4.a.bd		2
24.h	odd	2	1		1344.4.a.bl		2
84.h	odd	2	1		1176.4.a.p		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.4.a.h	✓	2	12.b	even	2	1
336.4.a.n		2	3.b	odd	2	1
504.4.a.j		2	4.b	odd	2	1
1008.4.a.y		2	1.a	even	1	1	trivial
1176.4.a.p		2	84.h	odd	2	1
1344.4.a.bd		2	24.f	even	2	1
1344.4.a.bl		2	24.h	odd	2	1
2352.4.a.bv		2	21.c	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5}^{2} + 14T_{5} - 128 \)

\( T_{11}^{2} - 18T_{11} - 1512 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 14T - 128 \)
$7$	\( (T + 7)^{2} \)
$11$	\( T^{2} - 18T - 1512 \)