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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,-18,0,90,0,0,0,-162,0,-566] 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: \(\Q(\sqrt{-3}, \sqrt{2641})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 661x^{2} + 660x + 435600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (9 \beta_1 - 9) q^{3} + (\beta_{2} + 45 \beta_1) q^{5} - 81 \beta_1 q^{9} + ( - 9 \beta_{3} - 9 \beta_{2} + \cdots - 283) q^{11} + (8 \beta_{3} + 468) q^{13} + (9 \beta_{3} - 405) q^{15} + ( - 33 \beta_{3} - 33 \beta_{2} + \cdots - 279) q^{17}+ \cdots + (729 \beta_{3} + 22923) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{3} + 90 q^{5} - 162 q^{9} - 566 q^{11} + 1872 q^{13} - 1620 q^{15} - 558 q^{17} - 324 q^{19} + 2862 q^{23} - 3082 q^{25} + 2916 q^{27} - 8912 q^{29} - 1116 q^{31} - 5094 q^{33} + 23712 q^{37}+ \cdots + 91692 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 661x^{2} + 660x + 435600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 661\nu^{2} - 661\nu + 435600 ) / 436260 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 661\nu^{2} + 873181\nu - 435600 ) / 436260 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + 1981 ) / 661 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 1321\beta _1 - 1321 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 661\beta_{3} - 1981 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−12.5977 − 21.8198i
13.0977 + 22.6858i
−12.5977 + 21.8198i
13.0977 − 22.6858i
0 −4.50000 + 7.79423i 0 −3.19533 − 5.53447i 0 0 0 −40.5000 − 70.1481i 0
361.2 0 −4.50000 + 7.79423i 0 48.1953 + 83.4768i 0 0 0 −40.5000 − 70.1481i 0
373.1 0 −4.50000 − 7.79423i 0 −3.19533 + 5.53447i 0 0 0 −40.5000 + 70.1481i 0
373.2 0 −4.50000 − 7.79423i 0 48.1953 − 83.4768i 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.j 4
7.b odd 2 1 588.6.i.k 4
7.c even 3 1 588.6.a.j yes 2
7.c even 3 1 inner 588.6.i.j 4
7.d odd 6 1 588.6.a.i ✓ 2
7.d odd 6 1 588.6.i.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.6.a.i ✓ 2 7.d odd 6 1
588.6.a.j yes 2 7.c even 3 1
588.6.i.j 4 1.a even 1 1 trivial
588.6.i.j 4 7.c even 3 1 inner
588.6.i.k 4 7.b odd 2 1
588.6.i.k 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 90T_{5}^{3} + 8716T_{5}^{2} + 55440T_{5} + 379456 \) 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
$5$ \( T^{4} - 90 T^{3} + \cdots + 379456 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 17911004224 \) Copy content Toggle raw display
$13$ \( (T^{2} - 936 T + 50000)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 7829968011264 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 1567504000000 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 71138563559424 \) Copy content Toggle raw display
$29$ \( (T^{2} + 4456 T - 2737172)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{2} + 21582 T + 99118080)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 3752 T - 215535728)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 265256727183616 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 67\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 477270959104 \) Copy content Toggle raw display
$71$ \( (T^{2} - 59846 T + 499846000)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 60\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( (T^{2} + 17640 T - 1893703536)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + 4500 T - 19276360864)^{2} \) Copy content Toggle raw display
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