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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,8,Mod(361,588)] 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("588.361"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 588.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,27,0,-100] 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: no
Analytic conductor: \(183.682394985\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 27 \zeta_{6} + 27) q^{3} - 100 \zeta_{6} q^{5} - 729 \zeta_{6} q^{9} + (2774 \zeta_{6} - 2774) q^{11} - 3294 q^{13} - 2700 q^{15} + (5900 \zeta_{6} - 5900) q^{17} - 6644 \zeta_{6} q^{19} - 1982 \zeta_{6} q^{23} + \cdots + 2022246 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 27 q^{3} - 100 q^{5} - 729 q^{9} - 2774 q^{11} - 6588 q^{13} - 5400 q^{15} - 5900 q^{17} - 6644 q^{19} - 1982 q^{23} + 68125 q^{25} - 39366 q^{27} - 416212 q^{29} + 117792 q^{31} + 74898 q^{33} + 335686 q^{37}+ \cdots + 4044492 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 13.5000 − 23.3827i 0 −50.0000 − 86.6025i 0 0 0 −364.500 − 631.333i 0
373.1 0 13.5000 + 23.3827i 0 −50.0000 + 86.6025i 0 0 0 −364.500 + 631.333i 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.8.i.f 2
7.b odd 2 1 588.8.i.c 2
7.c even 3 1 84.8.a.a ✓ 1
7.c even 3 1 inner 588.8.i.f 2
7.d odd 6 1 588.8.a.c 1
7.d odd 6 1 588.8.i.c 2
21.h odd 6 1 252.8.a.a 1
28.g odd 6 1 336.8.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.8.a.a ✓ 1 7.c even 3 1
252.8.a.a 1 21.h odd 6 1
336.8.a.j 1 28.g odd 6 1
588.8.a.c 1 7.d odd 6 1
588.8.i.c 2 7.b odd 2 1
588.8.i.c 2 7.d odd 6 1
588.8.i.f 2 1.a even 1 1 trivial
588.8.i.f 2 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 100T_{5} + 10000 \) acting on \(S_{8}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 27T + 729 \) Copy content Toggle raw display
$5$ \( T^{2} + 100T + 10000 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 2774 T + 7695076 \) Copy content Toggle raw display
$13$ \( (T + 3294)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 5900 T + 34810000 \) Copy content Toggle raw display
$19$ \( T^{2} + 6644 T + 44142736 \) Copy content Toggle raw display
$23$ \( T^{2} + 1982 T + 3928324 \) Copy content Toggle raw display
$29$ \( (T + 208106)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 13874955264 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 112685090596 \) Copy content Toggle raw display
$41$ \( (T + 265488)^{2} \) Copy content Toggle raw display
$43$ \( (T + 93292)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 432327290256 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 370537603524 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 287424654400 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 3229532468100 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 4507876326976 \) Copy content Toggle raw display
$71$ \( (T + 1191214)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 1116044344900 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 996970298256 \) Copy content Toggle raw display
$83$ \( (T - 3898004)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 21366138011904 \) Copy content Toggle raw display
$97$ \( (T - 15287710)^{2} \) Copy content Toggle raw display
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