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Properties

Label 588.8.i.j
Level $588$
Weight $8$
Character orbit 588.i
Analytic conductor $183.682$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,8,Mod(361,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(183.682394985\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{3649})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 913x^{2} + 912x + 831744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - 27 \beta_1 q^{3} + ( - \beta_{3} - \beta_{2} - 132 \beta_1 + 132) q^{5} + (729 \beta_1 - 729) q^{9} + ( - 7 \beta_{2} + 2490 \beta_1) q^{11} + (24 \beta_{3} + 5074) q^{13} + (27 \beta_{3} - 3564) q^{15}+ \cdots + ( - 5103 \beta_{3} - 1815210) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 54 q^{3} + 264 q^{5} - 1458 q^{9} + 4980 q^{11} + 20296 q^{13} - 14256 q^{15} + 17832 q^{17} + 6256 q^{19} - 14052 q^{23} - 141326 q^{25} + 78732 q^{27} + 487176 q^{29} + 470824 q^{31} + 134460 q^{33}+ \cdots - 7260840 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 913x^{2} + 912x + 831744 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 913\nu^{2} - 913\nu + 831744 ) / 832656 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 913\nu^{2} + 1666225\nu - 831744 ) / 138776 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\nu^{3} + 16422 ) / 913 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 6\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 10950\beta _1 - 10950 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 913\beta_{3} - 16422 ) / 12 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
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
−14.8517 + 25.7240i
15.3517 − 26.5900i
−14.8517 − 25.7240i
15.3517 + 26.5900i
0 −13.5000 + 23.3827i 0 −115.221 − 199.568i 0 0 0 −364.500 − 631.333i 0
361.2 0 −13.5000 + 23.3827i 0 247.221 + 428.199i 0 0 0 −364.500 − 631.333i 0
373.1 0 −13.5000 − 23.3827i 0 −115.221 + 199.568i 0 0 0 −364.500 + 631.333i 0
373.2 0 −13.5000 − 23.3827i 0 247.221 − 428.199i 0 0 0 −364.500 + 631.333i 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
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.j 4
7.b odd 2 1 588.8.i.k 4
7.c even 3 1 588.8.a.f 2
7.c even 3 1 inner 588.8.i.j 4
7.d odd 6 1 84.8.a.c ✓ 2
7.d odd 6 1 588.8.i.k 4
21.g even 6 1 252.8.a.c 2
28.f even 6 1 336.8.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.8.a.c ✓ 2 7.d odd 6 1
252.8.a.c 2 21.g even 6 1
336.8.a.q 2 28.f even 6 1
588.8.a.f 2 7.c even 3 1
588.8.i.j 4 1.a even 1 1 trivial
588.8.i.j 4 7.c even 3 1 inner
588.8.i.k 4 7.b odd 2 1
588.8.i.k 4 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 264T_{5}^{3} + 183636T_{5}^{2} + 30080160T_{5} + 12982323600 \) acting on \(S_{8}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 27 T + 729)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 12982323600 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 56043933696 \) Copy content Toggle raw display
$13$ \( (T^{2} - 10148 T - 49920188)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 10\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 33\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{2} - 243588 T + 14505368436)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{2} + 919248 T + 189493906140)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 112616 T - 246999010736)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 30\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 18\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 82\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots - 3623141894400)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{2} + \cdots - 4261335046704)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots - 89368449155180)^{2} \) Copy content Toggle raw display
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