LMFDB - Newform orbit 5376.2.a.r

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5376,2,Mod(1,5376)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5376.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5376, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$5376 = 2^{8} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5376.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,0,0,2,0,2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$42.9275761266$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 6$ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 2688)
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 $\beta = \sqrt{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - q^{3} + \beta q^{5} + q^{7} + q^{9} + ( - \beta - 2) q^{11} + (2 \beta - 2) q^{13} - \beta q^{15} + (\beta + 4) q^{17} + ( - 2 \beta + 2) q^{19} - q^{21} + (\beta - 2) q^{23} + q^{25} - q^{27} + \cdots + ( - \beta - 2) q^{99} +O(q^{100})$ q - q^3 + b * q^5 + q^7 + q^9 + (-b - 2) * q^11 + (2b - 2) q^13 - b * q^15 + (b + 4) * q^17 + (-2b + 2) q^19 - q^21 + (b - 2) * q^23 + q^25 - q^27 + (2b - 2) q^29 + (2b + 2) q^31 + (b + 2) * q^33 + b * q^35 + 2 * q^37 + (-2b + 2) q^39 - 3b q^41 + (2b - 8) q^43 + b * q^45 + (-2b + 4) q^47 + q^49 + (-b - 4) * q^51 + 10 * q^53 + (-2b - 6) q^55 + (2b - 2) q^57 - 2b q^59 - 6 * q^61 + q^63 + (-2b + 12) q^65 + 4b q^67 + (-b + 2) * q^69 + (3b + 2) q^71 + (-2b - 6) q^73 - q^75 + (-b - 2) * q^77 + (2b + 8) q^79 + q^81 + (4b - 4) q^83 + (4b + 6) q^85 + (-2b + 2) q^87 + 3b q^89 + (2b - 2) q^91 + (-2b - 2) q^93 + (2b - 12) q^95 + (-2b + 10) q^97 + (-b - 2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 4 q^{11} - 4 q^{13} + 8 q^{17} + 4 q^{19} - 2 q^{21} - 4 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 4 q^{31} + 4 q^{33} + 4 q^{37} + 4 q^{39} - 16 q^{43} + 8 q^{47} + 2 q^{49}+ \cdots - 4 q^{99}+O(q^{100})$ 2 * q - 2 * q^3 + 2 * q^7 + 2 * q^9 - 4 * q^11 - 4 * q^13 + 8 * q^17 + 4 * q^19 - 2 * q^21 - 4 * q^23 + 2 * q^25 - 2 * q^27 - 4 * q^29 + 4 * q^31 + 4 * q^33 + 4 * q^37 + 4 * q^39 - 16 * q^43 + 8 * q^47 + 2 * q^49 - 8 * q^51 + 20 * q^53 - 12 * q^55 - 4 * q^57 - 12 * q^61 + 2 * q^63 + 24 * q^65 + 4 * q^69 + 4 * q^71 - 12 * q^73 - 2 * q^75 - 4 * q^77 + 16 * q^79 + 2 * q^81 - 8 * q^83 + 12 * q^85 + 4 * q^87 - 4 * q^91 - 4 * q^93 - 24 * q^95 + 20 * q^97 - 4 * q^99

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

−2.44949
2.44949

−1.00000

−2.44949

1.00000

1.2

−1.00000

2.44949

1.00000

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	5376.2.a.r		2
4.b	odd	2	1		5376.2.a.ba		2
8.b	even	2	1		5376.2.a.bb		2
8.d	odd	2	1		5376.2.a.q		2
16.e	even	4	2		2688.2.c.d	✓	4
16.f	odd	4	2		2688.2.c.e	yes	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2688.2.c.d	✓	4	16.e	even	4	2
2688.2.c.e	yes	4	16.f	odd	4	2
5376.2.a.q		2	8.d	odd	2	1
5376.2.a.r		2	1.a	even	1	1	trivial
5376.2.a.ba		2	4.b	odd	2	1
5376.2.a.bb		2	8.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 $S_{2}^{\mathrm{new}}(\Gamma_0(5376))$ :

T_{5}^{2} - 6

T5^2 - 6

T_{11}^{2} + 4T_{11} - 2

T11^2 + 4*T11 - 2

T_{13}^{2} + 4T_{13} - 20

T13^2 + 4*T13 - 20

T_{19}^{2} - 4T_{19} - 20

T19^2 - 4*T19 - 20

T_{29}^{2} + 4T_{29} - 20

T29^2 + 4*T29 - 20

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$(T + 1)^{2}$ (T + 1)^2
$5$	$T^{2} - 6$ T^2 - 6
$7$	$(T - 1)^{2}$ (T - 1)^2
$11$	$T^{2} + 4T - 2$ T^2 + 4*T - 2
$13$	$T^{2} + 4T - 20$ T^2 + 4*T - 20
$17$	$T^{2} - 8T + 10$ T^2 - 8*T + 10
$19$	$T^{2} - 4T - 20$ T^2 - 4*T - 20
$23$	$T^{2} + 4T - 2$ T^2 + 4*T - 2
$29$	$T^{2} + 4T - 20$ T^2 + 4*T - 20
$31$	$T^{2} - 4T - 20$ T^2 - 4*T - 20
$37$	$(T - 2)^{2}$ (T - 2)^2
$41$	$T^{2} - 54$ T^2 - 54
$43$	$T^{2} + 16T + 40$ T^2 + 16*T + 40
$47$	$T^{2} - 8T - 8$ T^2 - 8*T - 8
$53$	$(T - 10)^{2}$ (T - 10)^2
$59$	$T^{2} - 24$ T^2 - 24
$61$	$(T + 6)^{2}$ (T + 6)^2
$67$	$T^{2} - 96$ T^2 - 96
$71$	$T^{2} - 4T - 50$ T^2 - 4*T - 50
$73$	$T^{2} + 12T + 12$ T^2 + 12*T + 12
$79$	$T^{2} - 16T + 40$ T^2 - 16*T + 40
$83$	$T^{2} + 8T - 80$ T^2 + 8*T - 80
$89$	$T^{2} - 54$ T^2 - 54
$97$	$T^{2} - 20T + 76$ T^2 - 20*T + 76
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