LMFDB - Newform orbit 2304.4.a.bs

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2304,4,Mod(1,2304)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2304.1"); S:= CuspForms(chi, 4); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,12,0,0,0,0,0,0,0,40,0,0,0,-16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$135.940400653$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 7$ 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

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

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.

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

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.

comment:embeddings in the coefficient field

gp:mfembed(f)

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

6.00000

−21.1660

1.2

6.00000

21.1660

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

T5 - 6

T_{7}^{2} - 448

T7^2 - 448

T_{11}^{2} - 1792

T11^2 - 1792

T_{13} - 20

T13 - 20

T_{17} + 8

T17 + 8

T_{19}^{2} - 7168

T19^2 - 7168

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2}$ T^2
$5$	$(T - 6)^{2}$ (T - 6)^2
$7$	$T^{2} - 448$ T^2 - 448
$11$	$T^{2} - 1792$ T^2 - 1792
$13$	$(T - 20)^{2}$ (T - 20)^2
$17$	$(T + 8)^{2}$ (T + 8)^2
$19$	$T^{2} - 7168$ T^2 - 7168
$23$	$T^{2} - 28672$ T^2 - 28672
$29$	$(T + 46)^{2}$ (T + 46)^2
$31$	$T^{2} - 448$ T^2 - 448
$37$	$(T + 164)^{2}$ (T + 164)^2
$41$	$(T + 312)^{2}$ (T + 312)^2
$43$	$T^{2} - 179200$ T^2 - 179200
$47$	$T^{2} - 28672$ T^2 - 28672
$53$	$(T - 266)^{2}$ (T - 266)^2
$59$	$T^{2} - 64512$ T^2 - 64512
$61$	$(T + 132)^{2}$ (T + 132)^2
$67$	$T^{2} - 258048$ T^2 - 258048
$71$	$T^{2} - 458752$ T^2 - 458752
$73$	$(T - 246)^{2}$ (T - 246)^2
$79$	$T^{2} - 54208$ T^2 - 54208
$83$	$T^{2} - 947968$ T^2 - 947968
$89$	$(T + 1392)^{2}$ (T + 1392)^2
$97$	$(T + 302)^{2}$ (T + 302)^2
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