Newform orbit 576.7.e.a

• Label: 576.7.e.a
• Level: $576$
• Weight: $7$
• Character orbit: 576.e
• Analytic conductor: $132.511$
• Analytic rank: $0$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	576.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(132.511152165\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-2}) \)
Defining polynomial:	\( x^{2} + 2 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	no (minimal twist has level 9)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 9\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + 5*b * q^5 - 524 * q^7 + 68*b * q^11 - 344 * q^13 + 561*b * q^17 - 2320 * q^19 + 452*b * q^23 + 11575 * q^25 + 1819*b * q^29 + 10564 * q^31 - 2620*b * q^35 + 24082 * q^37 + 8551*b * q^41 - 90952 * q^43 + 10132*b * q^47 + 156927 * q^49 - 15453*b * q^53 - 55080 * q^55 - 3128*b * q^59 - 251138 * q^61 - 1720*b * q^65 - 216088 * q^67 + 4236*b * q^71 - 308176 * q^73 - 35632*b * q^77 + 540124 * q^79 - 73252*b * q^83 - 454410 * q^85 - 17553*b * q^89 + 180256 * q^91 - 11600*b * q^95 - 37168 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q - 1048 * q^7 - 688 * q^13 - 4640 * q^19 + 23150 * q^25 + 21128 * q^31 + 48164 * q^37 - 181904 * q^43 + 313854 * q^49 - 110160 * q^55 - 502276 * q^61 - 432176 * q^67 - 616352 * q^73 + 1080248 * q^79 - 908820 * q^85 + 360512 * q^91 - 74336 * q^97

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(-1\)	\(1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

449.1

	−	1.41421i
		1.41421i

0

0

0

−

63.6396i

0

−524.000

0

0

0

449.2

0

0

0

63.6396i

0

−524.000

0

0

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	576.7.e.a		2
3.b	odd	2	1	inner	576.7.e.a		2
4.b	odd	2	1		576.7.e.l		2
8.b	even	2	1		144.7.e.a		2
8.d	odd	2	1		9.7.b.a	✓	2
12.b	even	2	1		576.7.e.l		2
24.f	even	2	1		9.7.b.a	✓	2
24.h	odd	2	1		144.7.e.a		2
40.e	odd	2	1		225.7.c.a		2
40.k	even	4	2		225.7.d.a		4
56.e	even	2	1		441.7.b.a		2
72.l	even	6	2		81.7.d.d		4
72.p	odd	6	2		81.7.d.d		4
120.m	even	2	1		225.7.c.a		2
120.q	odd	4	2		225.7.d.a		4
168.e	odd	2	1		441.7.b.a		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
9.7.b.a	✓	2	8.d	odd	2	1
9.7.b.a	✓	2	24.f	even	2	1
81.7.d.d		4	72.l	even	6	2
81.7.d.d		4	72.p	odd	6	2
144.7.e.a		2	8.b	even	2	1
144.7.e.a		2	24.h	odd	2	1
225.7.c.a		2	40.e	odd	2	1
225.7.c.a		2	120.m	even	2	1
225.7.d.a		4	40.k	even	4	2
225.7.d.a		4	120.q	odd	4	2
441.7.b.a		2	56.e	even	2	1
441.7.b.a		2	168.e	odd	2	1
576.7.e.a		2	1.a	even	1	1	trivial
576.7.e.a		2	3.b	odd	2	1	inner
576.7.e.l		2	4.b	odd	2	1
576.7.e.l		2	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(576, [\chi])\):

\( T_{5}^{2} + 4050 \)

\( T_{7} + 524 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} \)
$5$	\( T^{2} + 4050 \)
$7$	\( (T + 524)^{2} \)
$11$	\( T^{2} + 749088 \)