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Properties

Label 864.5.h.a
Level $864$
Weight $5$
Character orbit 864.h
Self dual yes
Analytic conductor $89.312$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [864,5,Mod(593,864)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 5, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("864.593"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 864.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-46] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(89.3116481044\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 12\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 23) q^{5} + ( - 5 \beta + 1) q^{7} + ( - 10 \beta + 71) q^{11} + ( - 46 \beta + 192) q^{25} - 818 q^{29} + ( - 95 \beta - 239) q^{31} + (116 \beta - 1463) q^{35} + ( - 10 \beta + 4800) q^{49}+ \cdots + (380 \beta - 8641) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 46 q^{5} + 2 q^{7} + 142 q^{11} + 384 q^{25} - 1636 q^{29} - 478 q^{31} - 2926 q^{35} + 9600 q^{49} + 3218 q^{53} - 9026 q^{55} - 13724 q^{59} + 8158 q^{73} + 28942 q^{77} + 18236 q^{79} - 4178 q^{83}+ \cdots - 17282 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
−1.41421
1.41421
0 0 0 −39.9706 0 85.8528 0 0 0
593.2 0 0 0 −6.02944 0 −83.8528 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.5.h.a 2
3.b odd 2 1 864.5.h.b 2
4.b odd 2 1 216.5.h.a ✓ 2
8.b even 2 1 864.5.h.b 2
8.d odd 2 1 216.5.h.b yes 2
12.b even 2 1 216.5.h.b yes 2
24.f even 2 1 216.5.h.a ✓ 2
24.h odd 2 1 CM 864.5.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.5.h.a ✓ 2 4.b odd 2 1
216.5.h.a ✓ 2 24.f even 2 1
216.5.h.b yes 2 8.d odd 2 1
216.5.h.b yes 2 12.b even 2 1
864.5.h.a 2 1.a even 1 1 trivial
864.5.h.a 2 24.h odd 2 1 CM
864.5.h.b 2 3.b odd 2 1
864.5.h.b 2 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 46T_{5} + 241 \) acting on \(S_{5}^{\mathrm{new}}(864, [\chi])\). 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
$5$ \( T^{2} + 46T + 241 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 7199 \) Copy content Toggle raw display
$11$ \( T^{2} - 142T - 23759 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 818)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 478 T - 2542079 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 3218 T - 13315919 \) Copy content Toggle raw display
$59$ \( (T + 6862)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 8158 T - 18641759 \) Copy content Toggle raw display
$79$ \( (T - 9118)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4178 T - 124919279 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 17282 T + 33079681 \) Copy content Toggle raw display
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