Properties

Label 2240.4.a.p
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)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - 5 q^{5} + 7 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 5 q^{5} + 7 q^{7} - 18 q^{9} - 17 q^{11} + 81 q^{13} + 15 q^{15} - 91 q^{17} + 102 q^{19} - 21 q^{21} + 90 q^{23} + 25 q^{25} + 135 q^{27} + 129 q^{29} - 116 q^{31} + 51 q^{33} - 35 q^{35} - 314 q^{37} - 243 q^{39} - 124 q^{41} - 434 q^{43} + 90 q^{45} - 497 q^{47} + 49 q^{49} + 273 q^{51} + 584 q^{53} + 85 q^{55} - 306 q^{57} - 332 q^{59} - 220 q^{61} - 126 q^{63} - 405 q^{65} + 384 q^{67} - 270 q^{69} + 664 q^{71} + 230 q^{73} - 75 q^{75} - 119 q^{77} - 361 q^{79} + 81 q^{81} + 1172 q^{83} + 455 q^{85} - 387 q^{87} + 40 q^{89} + 567 q^{91} + 348 q^{93} - 510 q^{95} - 175 q^{97} + 306 q^{99}+O(q^{100}) \) 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 −3.00000 0 −5.00000 0 7.00000 0 −18.0000 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.p 1
4.b odd 2 1 2240.4.a.w 1
8.b even 2 1 560.4.a.k 1
8.d odd 2 1 70.4.a.b 1
24.f even 2 1 630.4.a.m 1
40.e odd 2 1 350.4.a.t 1
40.k even 4 2 350.4.c.j 2
56.e even 2 1 490.4.a.f 1
56.k odd 6 2 490.4.e.p 2
56.m even 6 2 490.4.e.l 2
280.n even 2 1 2450.4.a.ba 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.a.b 1 8.d odd 2 1
350.4.a.t 1 40.e odd 2 1
350.4.c.j 2 40.k even 4 2
490.4.a.f 1 56.e even 2 1
490.4.e.l 2 56.m even 6 2
490.4.e.p 2 56.k odd 6 2
560.4.a.k 1 8.b even 2 1
630.4.a.m 1 24.f even 2 1
2240.4.a.p 1 1.a even 1 1 trivial
2240.4.a.w 1 4.b odd 2 1
2450.4.a.ba 1 280.n even 2 1

Hecke kernels

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

\( T_{3} + 3 \) Copy content Toggle raw display
\( T_{11} + 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T + 17 \) Copy content Toggle raw display
$13$ \( T - 81 \) Copy content Toggle raw display
$17$ \( T + 91 \) Copy content Toggle raw display
$19$ \( T - 102 \) Copy content Toggle raw display
$23$ \( T - 90 \) Copy content Toggle raw display
$29$ \( T - 129 \) Copy content Toggle raw display
$31$ \( T + 116 \) Copy content Toggle raw display
$37$ \( T + 314 \) Copy content Toggle raw display
$41$ \( T + 124 \) Copy content Toggle raw display
$43$ \( T + 434 \) Copy content Toggle raw display
$47$ \( T + 497 \) Copy content Toggle raw display
$53$ \( T - 584 \) Copy content Toggle raw display
$59$ \( T + 332 \) Copy content Toggle raw display
$61$ \( T + 220 \) Copy content Toggle raw display
$67$ \( T - 384 \) Copy content Toggle raw display
$71$ \( T - 664 \) Copy content Toggle raw display
$73$ \( T - 230 \) Copy content Toggle raw display
$79$ \( T + 361 \) Copy content Toggle raw display
$83$ \( T - 1172 \) Copy content Toggle raw display
$89$ \( T - 40 \) Copy content Toggle raw display
$97$ \( T + 175 \) Copy content Toggle raw display
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