Properties

Label 3520.2.a.r
Level $3520$
Weight $2$
Character orbit 3520.a
Self dual yes
Analytic conductor $28.107$
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] = [3520,2,Mod(1,3520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3520.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3520.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: \(28.1073415115\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
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)
 
\(f(q)\) \(=\) \( q - q^{5} + 4 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + 4 q^{7} - 3 q^{9} + q^{11} - 6 q^{13} - 6 q^{17} - 4 q^{19} + 4 q^{23} + q^{25} + 2 q^{29} + 8 q^{31} - 4 q^{35} + 10 q^{37} + 10 q^{41} + 3 q^{45} + 4 q^{47} + 9 q^{49} + 10 q^{53} - q^{55} + 4 q^{59} + 2 q^{61} - 12 q^{63} + 6 q^{65} + 8 q^{67} - 14 q^{73} + 4 q^{77} - 16 q^{79} + 9 q^{81} + 8 q^{83} + 6 q^{85} - 6 q^{89} - 24 q^{91} + 4 q^{95} + 2 q^{97} - 3 q^{99}+O(q^{100}) \) 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 0 0 −1.00000 0 4.00000 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(1\)
\(11\) \(-1\)

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 3520.2.a.r 1
4.b odd 2 1 3520.2.a.m 1
8.b even 2 1 440.2.a.c 1
8.d odd 2 1 880.2.a.g 1
24.f even 2 1 7920.2.a.b 1
24.h odd 2 1 3960.2.a.i 1
40.e odd 2 1 4400.2.a.r 1
40.f even 2 1 2200.2.a.e 1
40.i odd 4 2 2200.2.b.c 2
40.k even 4 2 4400.2.b.o 2
88.b odd 2 1 4840.2.a.e 1
88.g even 2 1 9680.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.c 1 8.b even 2 1
880.2.a.g 1 8.d odd 2 1
2200.2.a.e 1 40.f even 2 1
2200.2.b.c 2 40.i odd 4 2
3520.2.a.m 1 4.b odd 2 1
3520.2.a.r 1 1.a even 1 1 trivial
3960.2.a.i 1 24.h odd 2 1
4400.2.a.r 1 40.e odd 2 1
4400.2.b.o 2 40.k even 4 2
4840.2.a.e 1 88.b odd 2 1
7920.2.a.b 1 24.f even 2 1
9680.2.a.s 1 88.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3520))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display
\( T_{19} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T - 4 \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T + 6 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T - 10 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 4 \) Copy content Toggle raw display
$53$ \( T - 10 \) Copy content Toggle raw display
$59$ \( T - 4 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T - 8 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 14 \) Copy content Toggle raw display
$79$ \( T + 16 \) Copy content Toggle raw display
$83$ \( T - 8 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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