LMFDB - Newform orbit 4800.2.d.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4800,2,Mod(1249,4800)] 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("4800.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4800 = 2^{6} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4800.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,0,0,0,0,2,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,0,0, 0,-16,0,0,0,0,0,12,0,-4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)

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

Self dual:	no
Analytic conductor:	$38.3281929702$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 960)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 $\beta = 2i$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - q^{3} + 2 \beta q^{7} + q^{9} + 2 \beta q^{11} + 2 q^{13} + 3 \beta q^{19} - 2 \beta q^{21} - \beta q^{23} - q^{27} + 5 \beta q^{29} - 8 q^{31} - 2 \beta q^{33} + 6 q^{37} - 2 q^{39} - 2 q^{41} + \cdots + 2 \beta q^{99} +O(q^{100}) $ q - q^3 + 2b q^7 + q^9 + 2b q^11 + 2 * q^13 + 3b q^19 - 2b q^21 - b * q^23 - q^27 + 5b q^29 - 8 * q^31 - 2b q^33 + 6 * q^37 - 2 * q^39 - 2 * q^41 + 5b q^47 - 9 * q^49 + 6 * q^53 - 3b q^57 - 2b q^61 + 2b q^63 + 12 * q^67 + b * q^69 + 12 * q^71 - 7b q^73 - 16 * q^77 + 8 * q^79 + q^81 - 4 * q^83 - 5b q^87 - 18 * q^89 + 4b q^91 + 8 * q^93 + 3b q^97 + 2b q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{9} + 4 q^{13} - 2 q^{27} - 16 q^{31} + 12 q^{37} - 4 q^{39} - 4 q^{41} - 18 q^{49} + 12 q^{53} + 24 q^{67} + 24 q^{71} - 32 q^{77} + 16 q^{79} + 2 q^{81} - 8 q^{83} - 36 q^{89} + 16 q^{93}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^13 - 2 * q^27 - 16 * q^31 + 12 * q^37 - 4 * q^39 - 4 * q^41 - 18 * q^49 + 12 * q^53 + 24 * q^67 + 24 * q^71 - 32 * q^77 + 16 * q^79 + 2 * q^81 - 8 * q^83 - 36 * q^89 + 16 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$4351$
$\chi(n)$	$-1$	$-1$	$1$	$1$

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} $

1249.1

	−	1.00000i
		1.00000i

−1.00000

−

4.00000i

1.00000

1249.2

−1.00000

4.00000i

1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
40.f	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	4800.2.d.c		2
4.b	odd	2	1		4800.2.d.g		2
5.b	even	2	1		4800.2.d.f		2
5.c	odd	4	1		960.2.k.a	✓	2
5.c	odd	4	1		4800.2.k.h		2
8.b	even	2	1		4800.2.d.f		2
8.d	odd	2	1		4800.2.d.b		2
15.e	even	4	1		2880.2.k.a		2
20.d	odd	2	1		4800.2.d.b		2
20.e	even	4	1		960.2.k.d	yes	2
20.e	even	4	1		4800.2.k.a		2
40.e	odd	2	1		4800.2.d.g		2
40.f	even	2	1	inner	4800.2.d.c		2
40.i	odd	4	1		960.2.k.a	✓	2
40.i	odd	4	1		4800.2.k.h		2
40.k	even	4	1		960.2.k.d	yes	2
40.k	even	4	1		4800.2.k.a		2
60.l	odd	4	1		2880.2.k.d		2
80.i	odd	4	1		3840.2.a.u		1
80.j	even	4	1		3840.2.a.v		1
80.s	even	4	1		3840.2.a.a		1
80.t	odd	4	1		3840.2.a.n		1
120.q	odd	4	1		2880.2.k.d		2
120.w	even	4	1		2880.2.k.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
960.2.k.a	✓	2	5.c	odd	4	1
960.2.k.a	✓	2	40.i	odd	4	1
960.2.k.d	yes	2	20.e	even	4	1
960.2.k.d	yes	2	40.k	even	4	1
2880.2.k.a		2	15.e	even	4	1
2880.2.k.a		2	120.w	even	4	1
2880.2.k.d		2	60.l	odd	4	1
2880.2.k.d		2	120.q	odd	4	1
3840.2.a.a		1	80.s	even	4	1
3840.2.a.n		1	80.t	odd	4	1
3840.2.a.u		1	80.i	odd	4	1
3840.2.a.v		1	80.j	even	4	1
4800.2.d.b		2	8.d	odd	2	1
4800.2.d.b		2	20.d	odd	2	1
4800.2.d.c		2	1.a	even	1	1	trivial
4800.2.d.c		2	40.f	even	2	1	inner
4800.2.d.f		2	5.b	even	2	1
4800.2.d.f		2	8.b	even	2	1
4800.2.d.g		2	4.b	odd	2	1
4800.2.d.g		2	40.e	odd	2	1
4800.2.k.a		2	20.e	even	4	1
4800.2.k.a		2	40.k	even	4	1
4800.2.k.h		2	5.c	odd	4	1
4800.2.k.h		2	40.i	odd	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(4800, [\chi])$:

$ T_{7}^{2} + 16 $

T7^2 + 16

$ T_{13} - 2 $

T13 - 2

$ T_{31} + 8 $

T31 + 8

$ T_{43} $

T43

$ T_{83} + 4 $

T83 + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ (T + 1)^{2} $ (T + 1)^2
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 16 $ T^2 + 16
$11$	$ T^{2} + 16 $ T^2 + 16
$13$	$ (T - 2)^{2} $ (T - 2)^2
$17$	$ T^{2} $ T^2
$19$	$ T^{2} + 36 $ T^2 + 36
$23$	$ T^{2} + 4 $ T^2 + 4
$29$	$ T^{2} + 100 $ T^2 + 100
$31$	$ (T + 8)^{2} $ (T + 8)^2
$37$	$ (T - 6)^{2} $ (T - 6)^2
$41$	$ (T + 2)^{2} $ (T + 2)^2
$43$	$ T^{2} $ T^2
$47$	$ T^{2} + 100 $ T^2 + 100
$53$	$ (T - 6)^{2} $ (T - 6)^2
$59$	$ T^{2} $ T^2
$61$	$ T^{2} + 16 $ T^2 + 16
$67$	$ (T - 12)^{2} $ (T - 12)^2
$71$	$ (T - 12)^{2} $ (T - 12)^2
$73$	$ T^{2} + 196 $ T^2 + 196
$79$	$ (T - 8)^{2} $ (T - 8)^2
$83$	$ (T + 4)^{2} $ (T + 4)^2
$89$	$ (T + 18)^{2} $ (T + 18)^2
$97$	$ T^{2} + 36 $ T^2 + 36
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Level:	\( N \)	\(=\)	\( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4800.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - q^{3} + 2 \beta q^{7} + q^{9} + 2 \beta q^{11} + 2 q^{13} + 3 \beta q^{19} - 2 \beta q^{21} - \beta q^{23} - q^{27} + 5 \beta q^{29} - 8 q^{31} - 2 \beta q^{33} + 6 q^{37} - 2 q^{39} - 2 q^{41} + \cdots + 2 \beta q^{99} +O(q^{100}) \) q - q^3 + 2b q^7 + q^9 + 2b q^11 + 2 * q^13 + 3b q^19 - 2b q^21 - b * q^23 - q^27 + 5b q^29 - 8 * q^31 - 2b q^33 + 6 * q^37 - 2 * q^39 - 2 * q^41 + 5b q^47 - 9 * q^49 + 6 * q^53 - 3b q^57 - 2b q^61 + 2b q^63 + 12 * q^67 + b * q^69 + 12 * q^71 - 7b q^73 - 16 * q^77 + 8 * q^79 + q^81 - 4 * q^83 - 5b q^87 - 18 * q^89 + 4b q^91 + 8 * q^93 + 3b q^97 + 2b q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{9} + 4 q^{13} - 2 q^{27} - 16 q^{31} + 12 q^{37} - 4 q^{39} - 4 q^{41} - 18 q^{49} + 12 q^{53} + 24 q^{67} + 24 q^{71} - 32 q^{77} + 16 q^{79} + 2 q^{81} - 8 q^{83} - 36 q^{89} + 16 q^{93}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^9 + 4 * q^13 - 2 * q^27 - 16 * q^31 + 12 * q^37 - 4 * q^39 - 4 * q^41 - 18 * q^49 + 12 * q^53 + 24 * q^67 + 24 * q^71 - 32 * q^77 + 16 * q^79 + 2 * q^81 - 8 * q^83 - 36 * q^89 + 16 * q^93

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