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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 588.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,18,0,47,0,0,0,-162,0,407] 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: no
Analytic conductor: \(94.3056860500\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7081})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 1771x^{2} + 1770x + 3132900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 9 \beta_{2} + 9) q^{3} + (24 \beta_{2} - \beta_1) q^{5} - 81 \beta_{2} q^{9} + (\beta_{3} - 204 \beta_{2} + \cdots + 203) q^{11} + ( - 5 \beta_{3} - 222) q^{13} + (9 \beta_{3} + 207) q^{15} + ( - 4 \beta_{3} + 936 \beta_{2} + \cdots - 932) q^{17}+ \cdots + ( - 81 \beta_{3} - 16443) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 18 q^{3} + 47 q^{5} - 162 q^{9} + 407 q^{11} - 898 q^{13} + 846 q^{15} - 1868 q^{17} - 1463 q^{19} + 44 q^{23} + 1605 q^{25} - 2916 q^{27} + 1534 q^{29} - 11170 q^{31} - 3663 q^{33} + 3113 q^{37}+ \cdots - 65934 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 1771x^{2} + 1770x + 3132900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 1771\nu^{2} - 1771\nu + 3132900 ) / 3134670 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 3541 ) / 1771 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 1770\beta_{2} + \beta _1 - 1771 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 1771\beta_{3} - 3541 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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
21.2872 + 36.8705i
−20.7872 − 36.0044i
21.2872 − 36.8705i
−20.7872 + 36.0044i
0 4.50000 − 7.79423i 0 −9.28717 − 16.0858i 0 0 0 −40.5000 − 70.1481i 0
361.2 0 4.50000 − 7.79423i 0 32.7872 + 56.7890i 0 0 0 −40.5000 − 70.1481i 0
373.1 0 4.50000 + 7.79423i 0 −9.28717 + 16.0858i 0 0 0 −40.5000 + 70.1481i 0
373.2 0 4.50000 + 7.79423i 0 32.7872 − 56.7890i 0 0 0 −40.5000 + 70.1481i 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 588.6.i.m 4
7.b odd 2 1 84.6.i.b ✓ 4
7.c even 3 1 588.6.a.h 2
7.c even 3 1 inner 588.6.i.m 4
7.d odd 6 1 84.6.i.b ✓ 4
7.d odd 6 1 588.6.a.l 2
21.c even 2 1 252.6.k.e 4
21.g even 6 1 252.6.k.e 4
28.d even 2 1 336.6.q.g 4
28.f even 6 1 336.6.q.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.i.b ✓ 4 7.b odd 2 1
84.6.i.b ✓ 4 7.d odd 6 1
252.6.k.e 4 21.c even 2 1
252.6.k.e 4 21.g even 6 1
336.6.q.g 4 28.d even 2 1
336.6.q.g 4 28.f even 6 1
588.6.a.h 2 7.c even 3 1
588.6.a.l 2 7.d odd 6 1
588.6.i.m 4 1.a even 1 1 trivial
588.6.i.m 4 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 47T_{5}^{3} + 3427T_{5}^{2} + 57246T_{5} + 1483524 \) acting on \(S_{6}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 47 T^{3} + \cdots + 1483524 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 1571488164 \) Copy content Toggle raw display
$13$ \( (T^{2} + 449 T + 6144)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 712390017024 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 16559430371584 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 313361370464256 \) Copy content Toggle raw display
$29$ \( (T^{2} - 767 T - 20885268)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 439626033842121 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 2696203408144 \) Copy content Toggle raw display
$41$ \( (T^{2} - 7842 T + 15310512)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 12629 T + 39092230)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 425625782490000 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 44\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{2} + 125840 T + 3516363900)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 21\!\cdots\!29 \) Copy content Toggle raw display
$83$ \( (T^{2} + 39325 T - 6214225254)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + 184785 T + 6255893750)^{2} \) Copy content Toggle raw display
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