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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [600,4,Mod(1,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 600.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,-3,0,0,0,-20,0,9,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.4011460034\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 3 q^{3} - 20 q^{7} + 9 q^{9} + 16 q^{11} - 58 q^{13} - 38 q^{17} + 4 q^{19} + 60 q^{21} + 80 q^{23} - 27 q^{27} + 82 q^{29} - 8 q^{31} - 48 q^{33} - 426 q^{37} + 174 q^{39} - 246 q^{41} + 524 q^{43}+ \cdots + 144 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 0 0 −20.0000 0 9.00000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)
\(5\) \( +1 \)

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 600.4.a.a 1
3.b odd 2 1 1800.4.a.e 1
4.b odd 2 1 1200.4.a.bj 1
5.b even 2 1 120.4.a.e ✓ 1
5.c odd 4 2 600.4.f.f 2
15.d odd 2 1 360.4.a.m 1
15.e even 4 2 1800.4.f.k 2
20.d odd 2 1 240.4.a.a 1
20.e even 4 2 1200.4.f.h 2
40.e odd 2 1 960.4.a.bd 1
40.f even 2 1 960.4.a.q 1
60.h even 2 1 720.4.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.e ✓ 1 5.b even 2 1
240.4.a.a 1 20.d odd 2 1
360.4.a.m 1 15.d odd 2 1
600.4.a.a 1 1.a even 1 1 trivial
600.4.f.f 2 5.c odd 4 2
720.4.a.s 1 60.h even 2 1
960.4.a.q 1 40.f even 2 1
960.4.a.bd 1 40.e odd 2 1
1200.4.a.bj 1 4.b odd 2 1
1200.4.f.h 2 20.e even 4 2
1800.4.a.e 1 3.b odd 2 1
1800.4.f.k 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(600))\):

\( T_{7} + 20 \) 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
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 20 \) Copy content Toggle raw display
$11$ \( T - 16 \) Copy content Toggle raw display
$13$ \( T + 58 \) Copy content Toggle raw display
$17$ \( T + 38 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T - 80 \) Copy content Toggle raw display
$29$ \( T - 82 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T + 426 \) Copy content Toggle raw display
$41$ \( T + 246 \) Copy content Toggle raw display
$43$ \( T - 524 \) Copy content Toggle raw display
$47$ \( T - 464 \) Copy content Toggle raw display
$53$ \( T - 702 \) Copy content Toggle raw display
$59$ \( T + 592 \) Copy content Toggle raw display
$61$ \( T - 574 \) Copy content Toggle raw display
$67$ \( T - 172 \) Copy content Toggle raw display
$71$ \( T - 768 \) Copy content Toggle raw display
$73$ \( T - 558 \) Copy content Toggle raw display
$79$ \( T - 408 \) Copy content Toggle raw display
$83$ \( T + 164 \) Copy content Toggle raw display
$89$ \( T + 510 \) Copy content Toggle raw display
$97$ \( T + 514 \) Copy content Toggle raw display
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