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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 = [2,0,-9,0,34,0,0,0,-81,0,332] 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: \(2\)
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.

\(f(q)\) \(=\) \( q + (9 \zeta_{6} - 9) q^{3} + 34 \zeta_{6} q^{5} - 81 \zeta_{6} q^{9} + ( - 332 \zeta_{6} + 332) q^{11} - 1026 q^{13} - 306 q^{15} + (922 \zeta_{6} - 922) q^{17} - 452 \zeta_{6} q^{19} + 3776 \zeta_{6} q^{23} + \cdots - 26892 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{3} + 34 q^{5} - 81 q^{9} + 332 q^{11} - 2052 q^{13} - 612 q^{15} - 922 q^{17} - 452 q^{19} + 3776 q^{23} + 1969 q^{25} + 1458 q^{27} + 2332 q^{29} + 9792 q^{31} + 2988 q^{33} - 2390 q^{37} + 9234 q^{39}+ \cdots - 53784 q^{99}+O(q^{100}) \) 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 −4.50000 + 7.79423i 0 17.0000 + 29.4449i 0 0 0 −40.5000 − 70.1481i 0
373.1 0 −4.50000 − 7.79423i 0 17.0000 − 29.4449i 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.c 2
7.b odd 2 1 588.6.i.e 2
7.c even 3 1 84.6.a.b ✓ 1
7.c even 3 1 inner 588.6.i.c 2
7.d odd 6 1 588.6.a.b 1
7.d odd 6 1 588.6.i.e 2
21.h odd 6 1 252.6.a.c 1
28.g odd 6 1 336.6.a.e 1
84.n even 6 1 1008.6.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.b ✓ 1 7.c even 3 1
252.6.a.c 1 21.h odd 6 1
336.6.a.e 1 28.g odd 6 1
588.6.a.b 1 7.d odd 6 1
588.6.i.c 2 1.a even 1 1 trivial
588.6.i.c 2 7.c even 3 1 inner
588.6.i.e 2 7.b odd 2 1
588.6.i.e 2 7.d odd 6 1
1008.6.a.u 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 34T_{5} + 1156 \) acting on \(S_{6}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} - 34T + 1156 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 332T + 110224 \) Copy content Toggle raw display
$13$ \( (T + 1026)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 922T + 850084 \) Copy content Toggle raw display
$19$ \( T^{2} + 452T + 204304 \) Copy content Toggle raw display
$23$ \( T^{2} - 3776 T + 14258176 \) Copy content Toggle raw display
$29$ \( (T - 1166)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 9792 T + 95883264 \) Copy content Toggle raw display
$37$ \( T^{2} + 2390 T + 5712100 \) Copy content Toggle raw display
$41$ \( (T + 7230)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4652)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 24672 T + 608707584 \) Copy content Toggle raw display
$53$ \( T^{2} + 1110 T + 1232100 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 2198859664 \) Copy content Toggle raw display
$61$ \( T^{2} - 9762 T + 95296644 \) Copy content Toggle raw display
$67$ \( T^{2} - 26252 T + 689167504 \) Copy content Toggle raw display
$71$ \( (T - 65440)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 5606 T + 31427236 \) Copy content Toggle raw display
$79$ \( T^{2} - 9840 T + 96825600 \) Copy content Toggle raw display
$83$ \( (T - 61108)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 3963709764 \) Copy content Toggle raw display
$97$ \( (T + 37838)^{2} \) Copy content Toggle raw display
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