LMFDB - Newform orbit 1920.4.a.p

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1920,4,Mod(1,1920)] 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("1920.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,5,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$113.283667211$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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)

$f(q)$

$=$

$ q + 3 q^{3} + 5 q^{5} + 10 q^{7} + 9 q^{9} - 22 q^{11} + 26 q^{13} + 15 q^{15} + 14 q^{17} - 34 q^{19} + 30 q^{21} - 190 q^{23} + 25 q^{25} + 27 q^{27} - 162 q^{29} - 268 q^{31} - 66 q^{33} + 50 q^{35}+ \cdots - 198 q^{99}+O(q^{100}) $

q + 3 * q^3 + 5 * q^5 + 10 * q^7 + 9 * q^9 - 22 * q^11 + 26 * q^13 + 15 * q^15 + 14 * q^17 - 34 * q^19 + 30 * q^21 - 190 * q^23 + 25 * q^25 + 27 * q^27 - 162 * q^29 - 268 * q^31 - 66 * q^33 + 50 * q^35 - 362 * q^37 + 78 * q^39 - 170 * q^41 + 16 * q^43 + 45 * q^45 + 434 * q^47 - 243 * q^49 + 42 * q^51 + 594 * q^53 - 110 * q^55 - 102 * q^57 - 170 * q^59 + 130 * q^61 + 90 * q^63 + 130 * q^65 - 1024 * q^67 - 570 * q^69 + 280 * q^71 + 282 * q^73 + 75 * q^75 - 220 * q^77 - 160 * q^79 + 81 * q^81 - 732 * q^83 + 70 * q^85 - 486 * q^87 - 746 * q^89 + 260 * q^91 - 804 * q^93 - 170 * q^95 - 534 * q^97 - 198 * 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

3.00000

5.00000

10.0000

9.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -1 $
$5$	$ -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	1920.4.a.p	yes	1
4.b	odd	2	1		1920.4.a.e	yes	1
8.b	even	2	1		1920.4.a.d	✓	1
8.d	odd	2	1		1920.4.a.i	yes	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1920.4.a.d	✓	1	8.b	even	2	1
1920.4.a.e	yes	1	4.b	odd	2	1
1920.4.a.i	yes	1	8.d	odd	2	1
1920.4.a.p	yes	1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1920))$:

$ T_{7} - 10 $

T7 - 10

$ T_{11} + 22 $

T11 + 22

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 3 $ T - 3
$5$	$ T - 5 $ T - 5
$7$	$ T - 10 $ T - 10
$11$	$ T + 22 $ T + 22
$13$	$ T - 26 $ T - 26
$17$	$ T - 14 $ T - 14
$19$	$ T + 34 $ T + 34
$23$	$ T + 190 $ T + 190
$29$	$ T + 162 $ T + 162
$31$	$ T + 268 $ T + 268
$37$	$ T + 362 $ T + 362
$41$	$ T + 170 $ T + 170
$43$	$ T - 16 $ T - 16
$47$	$ T - 434 $ T - 434
$53$	$ T - 594 $ T - 594
$59$	$ T + 170 $ T + 170
$61$	$ T - 130 $ T - 130
$67$	$ T + 1024 $ T + 1024
$71$	$ T - 280 $ T - 280
$73$	$ T - 282 $ T - 282
$79$	$ T + 160 $ T + 160
$83$	$ T + 732 $ T + 732
$89$	$ T + 746 $ T + 746
$97$	$ T + 534 $ T + 534
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Level:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1920.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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