Properties

Label 2240.4.a.f
Level $2240$
Weight $4$
Character orbit 2240.a
Self dual yes
Analytic conductor $132.164$
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] = [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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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)
 
\(f(q)\) \(=\) \( q - 7 q^{3} + 5 q^{5} + 7 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 7 q^{3} + 5 q^{5} + 7 q^{7} + 22 q^{9} + 33 q^{11} + 43 q^{13} - 35 q^{15} + 111 q^{17} + 70 q^{19} - 49 q^{21} + 42 q^{23} + 25 q^{25} + 35 q^{27} + 225 q^{29} - 88 q^{31} - 231 q^{33} + 35 q^{35} + 34 q^{37} - 301 q^{39} + 432 q^{41} + 178 q^{43} + 110 q^{45} + 411 q^{47} + 49 q^{49} - 777 q^{51} + 708 q^{53} + 165 q^{55} - 490 q^{57} - 480 q^{59} - 812 q^{61} + 154 q^{63} + 215 q^{65} - 596 q^{67} - 294 q^{69} + 432 q^{71} - 358 q^{73} - 175 q^{75} + 231 q^{77} + 425 q^{79} - 839 q^{81} - 972 q^{83} + 555 q^{85} - 1575 q^{87} + 960 q^{89} + 301 q^{91} + 616 q^{93} + 350 q^{95} - 709 q^{97} + 726 q^{99}+O(q^{100}) \) 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 −7.00000 0 5.00000 0 7.00000 0 22.0000 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.f 1
4.b odd 2 1 2240.4.a.bh 1
8.b even 2 1 70.4.a.f 1
8.d odd 2 1 560.4.a.c 1
24.h odd 2 1 630.4.a.j 1
40.f even 2 1 350.4.a.b 1
40.i odd 4 2 350.4.c.l 2
56.h odd 2 1 490.4.a.i 1
56.j odd 6 2 490.4.e.h 2
56.p even 6 2 490.4.e.b 2
280.c odd 2 1 2450.4.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.a.f 1 8.b even 2 1
350.4.a.b 1 40.f even 2 1
350.4.c.l 2 40.i odd 4 2
490.4.a.i 1 56.h odd 2 1
490.4.e.b 2 56.p even 6 2
490.4.e.h 2 56.j odd 6 2
560.4.a.c 1 8.d odd 2 1
630.4.a.j 1 24.h odd 2 1
2240.4.a.f 1 1.a even 1 1 trivial
2240.4.a.bh 1 4.b odd 2 1
2450.4.a.s 1 280.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2240))\):

\( T_{3} + 7 \) Copy content Toggle raw display
\( T_{11} - 33 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 7 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T - 33 \) Copy content Toggle raw display
$13$ \( T - 43 \) Copy content Toggle raw display
$17$ \( T - 111 \) Copy content Toggle raw display
$19$ \( T - 70 \) Copy content Toggle raw display
$23$ \( T - 42 \) Copy content Toggle raw display
$29$ \( T - 225 \) Copy content Toggle raw display
$31$ \( T + 88 \) Copy content Toggle raw display
$37$ \( T - 34 \) Copy content Toggle raw display
$41$ \( T - 432 \) Copy content Toggle raw display
$43$ \( T - 178 \) Copy content Toggle raw display
$47$ \( T - 411 \) Copy content Toggle raw display
$53$ \( T - 708 \) Copy content Toggle raw display
$59$ \( T + 480 \) Copy content Toggle raw display
$61$ \( T + 812 \) Copy content Toggle raw display
$67$ \( T + 596 \) Copy content Toggle raw display
$71$ \( T - 432 \) Copy content Toggle raw display
$73$ \( T + 358 \) Copy content Toggle raw display
$79$ \( T - 425 \) Copy content Toggle raw display
$83$ \( T + 972 \) Copy content Toggle raw display
$89$ \( T - 960 \) Copy content Toggle raw display
$97$ \( T + 709 \) Copy content Toggle raw display
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