LMFDB - Newform orbit 1920.2.bh.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(703,1920)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 0, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1920.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1920 = 2^{7} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1920.bh (of order $4$ , degree $2$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 1$ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$ -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 a primitive root of unity $\zeta_{8}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \zeta_{8}^{3} q^{3} + (\zeta_{8}^{3} - 2 \zeta_{8}) q^{5} + ( - \zeta_{8}^{2} + 1) q^{7} - \zeta_{8}^{2} q^{9} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{11} + 2 \zeta_{8} q^{13} + ( - \zeta_{8}^{2} + 2) q^{15} + \cdots + ( - \zeta_{8}^{3} - \zeta_{8}) q^{99} +O(q^{100})$ q + z^3 * q^3 + (z^3 - 2z) q^5 + (-z^2 + 1) * q^7 - z^2 * q^9 + (-z^3 + z) * q^11 + 2z q^13 + (-z^2 + 2) * q^15 + (-2z^2 - 2) q^17 + (4z^3 + 4z) * q^19 + (z^3 + z) * q^21 + (-2z^2 - 2) q^23 + (3z^2 + 4) q^25 + z * q^27 + (z^3 - z) * q^29 - 10z^2 q^31 + (z^2 - 1) * q^33 + (3z^3 - z) q^35 + 6z^3 q^37 - 2 * q^39 - 2 * q^41 - 4z^3 q^43 + (2z^3 + z) q^45 + (-8z^2 + 8) q^47 + 5z^2 q^49 + (-2z^3 + 2z) * q^51 - 6z q^53 + (-z^2 - 3) * q^55 + (-4z^2 - 4) q^57 + (-3z^3 - 3z) * q^59 + (-2z^3 - 2z) * q^61 + (-z^2 - 1) * q^63 + (-4z^2 - 2) q^65 - 12z q^67 + (-2z^3 + 2z) * q^69 - 8z^2 q^71 + (-3z^2 + 3) q^73 + (4z^3 - 3z) * q^75 - 2z^3 q^77 + 10 * q^79 - q^81 - 10z^3 q^83 + (2z^3 + 6z) * q^85 + (-z^2 + 1) * q^87 + 10z^2 q^89 + (-2z^3 + 2z) * q^91 + 10z q^93 + (-12z^2 + 4) q^95 + (3z^2 + 3) q^97 + (-z^3 - z) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 4 q^{7} + 8 q^{15} - 8 q^{17} - 8 q^{23} + 16 q^{25} - 4 q^{33} - 8 q^{39} - 8 q^{41} + 32 q^{47} - 12 q^{55} - 16 q^{57} - 4 q^{63} - 8 q^{65} + 12 q^{73} + 40 q^{79} - 4 q^{81} + 4 q^{87} + 16 q^{95}+ \cdots + 12 q^{97}+O(q^{100})$ 4 * q + 4 * q^7 + 8 * q^15 - 8 * q^17 - 8 * q^23 + 16 * q^25 - 4 * q^33 - 8 * q^39 - 8 * q^41 + 32 * q^47 - 12 * q^55 - 16 * q^57 - 4 * q^63 - 8 * q^65 + 12 * q^73 + 40 * q^79 - 4 * q^81 + 4 * q^87 + 16 * q^95 + 12 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$ .

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$-1$	$1$	$-1$	$\zeta_{8}^{2}$

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}

703.1

0.707107	−	0.707107i
−0.707107	+	0.707107i
0.707107	+	0.707107i
−0.707107	−	0.707107i

−0.707107

−

0.707107i

−2.12132

0.707107i

1.00000

1.00000i

703.2

0.707107

0.707107i

2.12132

−

0.707107i

1.00000

1.00000i

1087.1

−0.707107

0.707107i

−2.12132

−

0.707107i

1.00000

−

1.00000i

−

1.00000i

1087.2

0.707107

−

0.707107i

2.12132

0.707107i

1.00000

−

1.00000i

−

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
20.e	even	4	1	inner
40.k	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1920.2.bh.f	yes	4
4.b	odd	2	1		1920.2.bh.c	✓	4
5.c	odd	4	1		1920.2.bh.c	✓	4
8.b	even	2	1	inner	1920.2.bh.f	yes	4
8.d	odd	2	1		1920.2.bh.c	✓	4
20.e	even	4	1	inner	1920.2.bh.f	yes	4
40.i	odd	4	1		1920.2.bh.c	✓	4
40.k	even	4	1	inner	1920.2.bh.f	yes	4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1920.2.bh.c	✓	4	4.b	odd	2	1
1920.2.bh.c	✓	4	5.c	odd	4	1
1920.2.bh.c	✓	4	8.d	odd	2	1
1920.2.bh.c	✓	4	40.i	odd	4	1
1920.2.bh.f	yes	4	1.a	even	1	1	trivial
1920.2.bh.f	yes	4	8.b	even	2	1	inner
1920.2.bh.f	yes	4	20.e	even	4	1	inner
1920.2.bh.f	yes	4	40.k	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(1920, [\chi])$ :

T_{7}^{2} - 2T_{7} + 2

T_{11}^{2} - 2

T_{13}^{4} + 16

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4}$ T^4
$3$	$T^{4} + 1$ T^4 + 1
$5$	$T^{4} - 8T^{2} + 25$ T^4 - 8*T^2 + 25
$7$	$(T^{2} - 2 T + 2)^{2}$ (T^2 - 2*T + 2)^2
$11$	$(T^{2} - 2)^{2}$ (T^2 - 2)^2
$13$	$T^{4} + 16$ T^4 + 16
$17$	$(T^{2} + 4 T + 8)^{2}$ (T^2 + 4*T + 8)^2
$19$	$(T^{2} + 32)^{2}$ (T^2 + 32)^2
$23$	$(T^{2} + 4 T + 8)^{2}$ (T^2 + 4*T + 8)^2
$29$	$(T^{2} - 2)^{2}$ (T^2 - 2)^2
$31$	$(T^{2} + 100)^{2}$ (T^2 + 100)^2
$37$	$T^{4} + 1296$ T^4 + 1296
$41$	$(T + 2)^{4}$ (T + 2)^4
$43$	$T^{4} + 256$ T^4 + 256
$47$	$(T^{2} - 16 T + 128)^{2}$ (T^2 - 16*T + 128)^2
$53$	$T^{4} + 1296$ T^4 + 1296
$59$	$(T^{2} + 18)^{2}$ (T^2 + 18)^2
$61$	$(T^{2} + 8)^{2}$ (T^2 + 8)^2
$67$	$T^{4} + 20736$ T^4 + 20736
$71$	$(T^{2} + 64)^{2}$ (T^2 + 64)^2
$73$	$(T^{2} - 6 T + 18)^{2}$ (T^2 - 6*T + 18)^2
$79$	$(T - 10)^{4}$ (T - 10)^4
$83$	$T^{4} + 10000$ T^4 + 10000
$89$	$(T^{2} + 100)^{2}$ (T^2 + 100)^2
$97$	$(T^{2} - 6 T + 18)^{2}$ (T^2 - 6*T + 18)^2
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Newspace parameters

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Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1920.2.bh.f

Level	$1920$
Weight	$2$
Character orbit	1920.bh
Analytic conductor	$15.331$
Analytic rank	$0$
Dimension	$4$
Inner twists	$4$