Properties

Label 1960.4.a.a
Level $1960$
Weight $4$
Character orbit 1960.a
Self dual yes
Analytic conductor $115.644$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1960,4,Mod(1,1960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1960.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1960 = 2^{3} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1960.a (trivial)

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: yes
Analytic conductor: \(115.643743611\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
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)
 
\(f(q)\) \(=\) \( q - 10 q^{3} + 5 q^{5} + 73 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 10 q^{3} + 5 q^{5} + 73 q^{9} - 16 q^{11} + 6 q^{13} - 50 q^{15} + 6 q^{17} + 124 q^{19} + 42 q^{23} + 25 q^{25} - 460 q^{27} + 142 q^{29} + 188 q^{31} + 160 q^{33} + 202 q^{37} - 60 q^{39} - 54 q^{41} + 66 q^{43} + 365 q^{45} - 38 q^{47} - 60 q^{51} + 738 q^{53} - 80 q^{55} - 1240 q^{57} - 564 q^{59} + 262 q^{61} + 30 q^{65} - 554 q^{67} - 420 q^{69} + 140 q^{71} - 882 q^{73} - 250 q^{75} - 1160 q^{79} + 2629 q^{81} - 642 q^{83} + 30 q^{85} - 1420 q^{87} + 854 q^{89} - 1880 q^{93} + 620 q^{95} + 478 q^{97} - 1168 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
0
0 −10.0000 0 5.00000 0 0 0 73.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 1960.4.a.a 1
7.b odd 2 1 40.4.a.c 1
21.c even 2 1 360.4.a.i 1
28.d even 2 1 80.4.a.a 1
35.c odd 2 1 200.4.a.a 1
35.f even 4 2 200.4.c.a 2
56.e even 2 1 320.4.a.n 1
56.h odd 2 1 320.4.a.a 1
84.h odd 2 1 720.4.a.ba 1
105.g even 2 1 1800.4.a.bd 1
105.k odd 4 2 1800.4.f.n 2
112.j even 4 2 1280.4.d.b 2
112.l odd 4 2 1280.4.d.o 2
140.c even 2 1 400.4.a.u 1
140.j odd 4 2 400.4.c.a 2
280.c odd 2 1 1600.4.a.ca 1
280.n even 2 1 1600.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.c 1 7.b odd 2 1
80.4.a.a 1 28.d even 2 1
200.4.a.a 1 35.c odd 2 1
200.4.c.a 2 35.f even 4 2
320.4.a.a 1 56.h odd 2 1
320.4.a.n 1 56.e even 2 1
360.4.a.i 1 21.c even 2 1
400.4.a.u 1 140.c even 2 1
400.4.c.a 2 140.j odd 4 2
720.4.a.ba 1 84.h odd 2 1
1280.4.d.b 2 112.j even 4 2
1280.4.d.o 2 112.l odd 4 2
1600.4.a.a 1 280.n even 2 1
1600.4.a.ca 1 280.c odd 2 1
1800.4.a.bd 1 105.g even 2 1
1800.4.f.n 2 105.k odd 4 2
1960.4.a.a 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1960))\):

\( T_{3} + 10 \) Copy content Toggle raw display
\( T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 10 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 16 \) Copy content Toggle raw display
$13$ \( T - 6 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T - 124 \) Copy content Toggle raw display
$23$ \( T - 42 \) Copy content Toggle raw display
$29$ \( T - 142 \) Copy content Toggle raw display
$31$ \( T - 188 \) Copy content Toggle raw display
$37$ \( T - 202 \) Copy content Toggle raw display
$41$ \( T + 54 \) Copy content Toggle raw display
$43$ \( T - 66 \) Copy content Toggle raw display
$47$ \( T + 38 \) Copy content Toggle raw display
$53$ \( T - 738 \) Copy content Toggle raw display
$59$ \( T + 564 \) Copy content Toggle raw display
$61$ \( T - 262 \) Copy content Toggle raw display
$67$ \( T + 554 \) Copy content Toggle raw display
$71$ \( T - 140 \) Copy content Toggle raw display
$73$ \( T + 882 \) Copy content Toggle raw display
$79$ \( T + 1160 \) Copy content Toggle raw display
$83$ \( T + 642 \) Copy content Toggle raw display
$89$ \( T - 854 \) Copy content Toggle raw display
$97$ \( T - 478 \) Copy content Toggle raw display
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