LMFDB - Newform orbit 6080.2.a.bg

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6080,2,Mod(1,6080)] 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("6080.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$6080 = 2^{6} \cdot 5 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	6080.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2,0,0,0,-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:	$48.5490444289$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 2$ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 760)
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 $\beta = \sqrt{2}$ . We also show the integral $q$ -expansion of the trace form.

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

−1.41421
1.41421

−1.41421

1.00000

2.82843

−1.00000

1.2

1.41421

1.00000

−2.82843

−1.00000

Atkin-Lehner signs

$p$	Sign
$2$	$-1$
$5$	$-1$
$19$	$+1$

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	6080.2.a.bg		2
4.b	odd	2	1		6080.2.a.bf		2
8.b	even	2	1		1520.2.a.m		2
8.d	odd	2	1		760.2.a.f	✓	2
24.f	even	2	1		6840.2.a.z		2
40.e	odd	2	1		3800.2.a.n		2
40.f	even	2	1		7600.2.a.ba		2
40.k	even	4	2		3800.2.d.i		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
760.2.a.f	✓	2	8.d	odd	2	1
1520.2.a.m		2	8.b	even	2	1
3800.2.a.n		2	40.e	odd	2	1
3800.2.d.i		4	40.k	even	4	2
6080.2.a.bf		2	4.b	odd	2	1
6080.2.a.bg		2	1.a	even	1	1	trivial
6840.2.a.z		2	24.f	even	2	1
7600.2.a.ba		2	40.f	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(6080))$ :

T_{3}^{2} - 2

T3^2 - 2

T_{7}^{2} - 8

T7^2 - 8

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

T11^2 - 4*T11 - 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} - 2$ T^2 - 2
$5$	$(T - 1)^{2}$ (T - 1)^2
$7$	$T^{2} - 8$ T^2 - 8
$11$	$T^{2} - 4T - 4$ T^2 - 4*T - 4
$13$	$T^{2} + 4T + 2$ T^2 + 4*T + 2
$17$	$T^{2} - 4T - 4$ T^2 - 4*T - 4
$19$	$(T + 1)^{2}$ (T + 1)^2
$23$	$(T + 4)^{2}$ (T + 4)^2
$29$	$T^{2} + 4T - 4$ T^2 + 4*T - 4
$31$	$T^{2}$ T^2
$37$	$T^{2} + 12T + 18$ T^2 + 12*T + 18
$41$	$T^{2} - 4T - 4$ T^2 - 4*T - 4
$43$	$T^{2} - 8$ T^2 - 8
$47$	$T^{2} - 72$ T^2 - 72
$53$	$T^{2} + 12T - 14$ T^2 + 12*T - 14
$59$	$T^{2} - 8$ T^2 - 8
$61$	$T^{2} + 8T - 16$ T^2 + 8*T - 16
$67$	$T^{2} + 16T + 62$ T^2 + 16*T + 62
$71$	$T^{2} + 8T - 112$ T^2 + 8*T - 112
$73$	$T^{2} + 4T - 196$ T^2 + 4*T - 196
$79$	$T^{2} - 24T + 136$ T^2 - 24*T + 136
$83$	$(T - 8)^{2}$ (T - 8)^2
$89$	$T^{2} + 12T + 28$ T^2 + 12*T + 28
$97$	$T^{2} - 4T - 14$ T^2 - 4*T - 14
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