Newform orbit 2304.4.a.bs

• Label: 2304.4.a.bs
• Level: $2304$
• Weight: $4$
• Character orbit: 2304.a
• Self dual: yes
• Analytic conductor: $135.940$
• Analytic rank: $1$
• Dimension: $2$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$2304 = 2^{8} \cdot 3^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	2304.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$135.940400653$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7})$
Defining polynomial:	$x^{2} - 7$
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	no (minimal twist has level 1152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $q$-expansion are expressed in terms of $\beta = 8\sqrt{7}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	q + 6 * q^5 - b * q^7 + 2*b * q^11 + 20 * q^13 - 8 * q^17 + 4*b * q^19 - 8*b * q^23 - 89 * q^25 - 46 * q^29 - b * q^31 - 6*b * q^35 - 164 * q^37 - 312 * q^41 + 20*b * q^43 - 8*b * q^47 + 105 * q^49 + 266 * q^53 + 12*b * q^55 - 12*b * q^59 - 132 * q^61 + 120 * q^65 + 24*b * q^67 + 32*b * q^71 + 246 * q^73 - 896 * q^77 + 11*b * q^79 + 46*b * q^83 - 48 * q^85 - 1392 * q^89 - 20*b * q^91 + 24*b * q^95 - 302 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 12 q^{5} + 40 q^{13} - 16 q^{17} - 178 q^{25} - 92 q^{29} - 328 q^{37} - 624 q^{41} + 210 q^{49} + 532 q^{53} - 264 q^{61} + 240 q^{65} + 492 q^{73} - 1792 q^{77} - 96 q^{85} - 2784 q^{89} - 604 q^{97}+O(q^{100})$

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

2.64575
−2.64575

0

0

0

6.00000

0

−21.1660

0

0

0

1.2

0

0

0

6.00000

0

21.1660

0

0

0

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$3$	$+1$

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	2304.4.a.bs		2
3.b	odd	2	1		2304.4.a.r		2
4.b	odd	2	1	inner	2304.4.a.bs		2
8.b	even	2	1		2304.4.a.q		2
8.d	odd	2	1		2304.4.a.q		2
12.b	even	2	1		2304.4.a.r		2
16.e	even	4	2		1152.4.d.k	✓	4
16.f	odd	4	2		1152.4.d.k	✓	4
24.f	even	2	1		2304.4.a.br		2
24.h	odd	2	1		2304.4.a.br		2
48.i	odd	4	2		1152.4.d.m	yes	4
48.k	even	4	2		1152.4.d.m	yes	4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1152.4.d.k	✓	4	16.e	even	4	2
1152.4.d.k	✓	4	16.f	odd	4	2
1152.4.d.m	yes	4	48.i	odd	4	2
1152.4.d.m	yes	4	48.k	even	4	2
2304.4.a.q		2	8.b	even	2	1
2304.4.a.q		2	8.d	odd	2	1
2304.4.a.r		2	3.b	odd	2	1
2304.4.a.r		2	12.b	even	2	1
2304.4.a.br		2	24.f	even	2	1
2304.4.a.br		2	24.h	odd	2	1
2304.4.a.bs		2	1.a	even	1	1	trivial
2304.4.a.bs		2	4.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(2304))$:

$T_{5} - 6$

$T_{7}^{2} - 448$

$T_{11}^{2} - 1792$

$T_{13} - 20$

$T_{17} + 8$

$T_{19}^{2} - 7168$

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$
$3$	$T^{2}$
$5$	$(T - 6)^{2}$
$7$	$T^{2} - 448$
$11$	$T^{2} - 1792$