Properties

Label 2240.4.a.u
Level $2240$
Weight $4$
Character orbit 2240.a
Self dual yes
Analytic conductor $132.164$
Analytic rank $1$
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] = [2240,4,Mod(1,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, 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("2240.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2240.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: \(132.164278413\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
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)
 
\(f(q)\) \(=\) \( q + q^{3} + 5 q^{5} + 7 q^{7} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 5 q^{5} + 7 q^{7} - 26 q^{9} - 7 q^{11} + 23 q^{13} + 5 q^{15} - 25 q^{17} - 62 q^{19} + 7 q^{21} + 86 q^{23} + 25 q^{25} - 53 q^{27} + 29 q^{29} + 12 q^{31} - 7 q^{33} + 35 q^{35} + 150 q^{37} + 23 q^{39} + 204 q^{41} - 178 q^{43} - 130 q^{45} - 33 q^{47} + 49 q^{49} - 25 q^{51} - 452 q^{53} - 35 q^{55} - 62 q^{57} + 120 q^{59} - 920 q^{61} - 182 q^{63} + 115 q^{65} - 300 q^{67} + 86 q^{69} - 520 q^{71} + 370 q^{73} + 25 q^{75} - 49 q^{77} + 1013 q^{79} + 649 q^{81} - 636 q^{83} - 125 q^{85} + 29 q^{87} + 292 q^{89} + 161 q^{91} + 12 q^{93} - 310 q^{95} - 1381 q^{97} + 182 q^{99}+O(q^{100}) \) 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 1.00000 0 5.00000 0 7.00000 0 −26.0000 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 2240.4.a.u 1
4.b odd 2 1 2240.4.a.s 1
8.b even 2 1 560.4.a.h 1
8.d odd 2 1 140.4.a.d 1
24.f even 2 1 1260.4.a.i 1
40.e odd 2 1 700.4.a.g 1
40.k even 4 2 700.4.e.i 2
56.e even 2 1 980.4.a.g 1
56.k odd 6 2 980.4.i.i 2
56.m even 6 2 980.4.i.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.4.a.d 1 8.d odd 2 1
560.4.a.h 1 8.b even 2 1
700.4.a.g 1 40.e odd 2 1
700.4.e.i 2 40.k even 4 2
980.4.a.g 1 56.e even 2 1
980.4.i.i 2 56.k odd 6 2
980.4.i.k 2 56.m even 6 2
1260.4.a.i 1 24.f even 2 1
2240.4.a.s 1 4.b odd 2 1
2240.4.a.u 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2240))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{11} + 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T + 7 \) Copy content Toggle raw display
$13$ \( T - 23 \) Copy content Toggle raw display
$17$ \( T + 25 \) Copy content Toggle raw display
$19$ \( T + 62 \) Copy content Toggle raw display
$23$ \( T - 86 \) Copy content Toggle raw display
$29$ \( T - 29 \) Copy content Toggle raw display
$31$ \( T - 12 \) Copy content Toggle raw display
$37$ \( T - 150 \) Copy content Toggle raw display
$41$ \( T - 204 \) Copy content Toggle raw display
$43$ \( T + 178 \) Copy content Toggle raw display
$47$ \( T + 33 \) Copy content Toggle raw display
$53$ \( T + 452 \) Copy content Toggle raw display
$59$ \( T - 120 \) Copy content Toggle raw display
$61$ \( T + 920 \) Copy content Toggle raw display
$67$ \( T + 300 \) Copy content Toggle raw display
$71$ \( T + 520 \) Copy content Toggle raw display
$73$ \( T - 370 \) Copy content Toggle raw display
$79$ \( T - 1013 \) Copy content Toggle raw display
$83$ \( T + 636 \) Copy content Toggle raw display
$89$ \( T - 292 \) Copy content Toggle raw display
$97$ \( T + 1381 \) Copy content Toggle raw display
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