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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1584,4,Mod(1,1584)] 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("1584.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1584.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-11,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(93.4590254491\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{97})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 5) q^{5} + ( - 6 \beta - 2) q^{7} + 11 q^{11} + (2 \beta - 12) q^{13} + ( - 4 \beta + 82) q^{17} + (20 \beta - 16) q^{19} + (23 \beta - 71) q^{23} + (11 \beta - 76) q^{25} + (42 \beta + 48) q^{29}+ \cdots + (217 \beta + 625) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 11 q^{5} - 10 q^{7} + 22 q^{11} - 22 q^{13} + 160 q^{17} - 12 q^{19} - 119 q^{23} - 141 q^{25} + 138 q^{29} - 401 q^{31} + 346 q^{35} + 341 q^{37} + 318 q^{41} - 450 q^{43} - 312 q^{47} + 1110 q^{49}+ \cdots + 1467 q^{97}+O(q^{100}) \) 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
5.42443
−4.42443
0 0 0 −10.4244 0 −34.5466 0 0 0
1.2 0 0 0 −0.575571 0 24.5466 0 0 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(11\) \( -1 \)

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 1584.4.a.z 2
3.b odd 2 1 176.4.a.g 2
4.b odd 2 1 396.4.a.i 2
12.b even 2 1 44.4.a.b ✓ 2
24.f even 2 1 704.4.a.m 2
24.h odd 2 1 704.4.a.r 2
33.d even 2 1 1936.4.a.o 2
60.h even 2 1 1100.4.a.e 2
60.l odd 4 2 1100.4.b.e 4
84.h odd 2 1 2156.4.a.d 2
132.d odd 2 1 484.4.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.4.a.b ✓ 2 12.b even 2 1
176.4.a.g 2 3.b odd 2 1
396.4.a.i 2 4.b odd 2 1
484.4.a.e 2 132.d odd 2 1
704.4.a.m 2 24.f even 2 1
704.4.a.r 2 24.h odd 2 1
1100.4.a.e 2 60.h even 2 1
1100.4.b.e 4 60.l odd 4 2
1584.4.a.z 2 1.a even 1 1 trivial
1936.4.a.o 2 33.d even 2 1
2156.4.a.d 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1584))\):

\( T_{5}^{2} + 11T_{5} + 6 \) Copy content Toggle raw display
\( T_{7}^{2} + 10T_{7} - 848 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 11T + 6 \) Copy content Toggle raw display
$7$ \( T^{2} + 10T - 848 \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 22T + 24 \) Copy content Toggle raw display
$17$ \( T^{2} - 160T + 6012 \) Copy content Toggle raw display
$19$ \( T^{2} + 12T - 9664 \) Copy content Toggle raw display
$23$ \( T^{2} + 119T - 9288 \) Copy content Toggle raw display
$29$ \( T^{2} - 138T - 38016 \) Copy content Toggle raw display
$31$ \( T^{2} + 401T + 40176 \) Copy content Toggle raw display
$37$ \( T^{2} - 341T + 16242 \) Copy content Toggle raw display
$41$ \( T^{2} - 318T + 17424 \) Copy content Toggle raw display
$43$ \( T^{2} + 450T + 34232 \) Copy content Toggle raw display
$47$ \( T^{2} + 312T - 101376 \) Copy content Toggle raw display
$53$ \( T^{2} - 400T + 30300 \) Copy content Toggle raw display
$59$ \( T^{2} + 21T - 62964 \) Copy content Toggle raw display
$61$ \( T^{2} + 462T + 18344 \) Copy content Toggle raw display
$67$ \( T^{2} + 327T - 192124 \) Copy content Toggle raw display
$71$ \( T^{2} - 603T - 24552 \) Copy content Toggle raw display
$73$ \( T^{2} - 206T - 70968 \) Copy content Toggle raw display
$79$ \( T^{2} + 2026 T + 983392 \) Copy content Toggle raw display
$83$ \( T^{2} + 1046 T + 110472 \) Copy content Toggle raw display
$89$ \( T^{2} + 721T - 647034 \) Copy content Toggle raw display
$97$ \( T^{2} - 1467 T - 603886 \) Copy content Toggle raw display
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