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Properties

Label 588.3.p.d
Level $588$
Weight $3$
Character orbit 588.p
Analytic conductor $16.022$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,3,Mod(557,588)]
 
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, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.557");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 588.p (of order \(6\), 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: \(16.0218395444\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-35})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} - 9x + 81 \) 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{2} + 2 \beta_1 - 1) q^{5} + ( - 9 \beta_{2} - \beta_1 + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + (\beta_{2} + 2 \beta_1 - 1) q^{5} + ( - 9 \beta_{2} - \beta_1 + 9) q^{9} + ( - 2 \beta_{3} - \beta_{2}) q^{11} + 6 q^{13} + (\beta_{3} - \beta_1 + 18) q^{15} + (2 \beta_{3} + \beta_{2}) q^{17} + ( - 23 \beta_{2} + 23) q^{19} + (7 \beta_{2} + 14 \beta_1 - 7) q^{23} + 10 \beta_{2} q^{25} + (8 \beta_{3} - 8 \beta_1 - 9) q^{27} + (16 \beta_{3} - 16 \beta_1 + 8) q^{29} + 39 \beta_{2} q^{31} + (18 \beta_{2} + \beta_1 - 18) q^{33} + (47 \beta_{2} - 47) q^{37} + 6 \beta_{3} q^{39} - 22 q^{43} + (17 \beta_{3} - 9 \beta_{2}) q^{45} + (9 \beta_{2} + 18 \beta_1 - 9) q^{47} + ( - 18 \beta_{2} - \beta_1 + 18) q^{51} + ( - 18 \beta_{3} - 9 \beta_{2}) q^{53} - 35 q^{55} + (23 \beta_{3} - 23 \beta_1) q^{57} + (34 \beta_{3} + 17 \beta_{2}) q^{59} + (81 \beta_{2} - 81) q^{61} + (6 \beta_{2} + 12 \beta_1 - 6) q^{65} - 31 \beta_{2} q^{67} + (7 \beta_{3} - 7 \beta_1 + 126) q^{69} + (32 \beta_{3} - 32 \beta_1 + 16) q^{71} - 17 \beta_{2} q^{73} + 10 \beta_1 q^{75} + ( - 9 \beta_{2} + 9) q^{79} + ( - 17 \beta_{3} - 72 \beta_{2}) q^{81} + (16 \beta_{3} - 16 \beta_1 + 8) q^{83} + 35 q^{85} + ( - 8 \beta_{3} - 144 \beta_{2}) q^{87} + ( - 15 \beta_{2} - 30 \beta_1 + 15) q^{89} + 39 \beta_1 q^{93} + (46 \beta_{3} + 23 \beta_{2}) q^{95} - 82 q^{97} + ( - 17 \beta_{3} + 17 \beta_1 + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} + 17 q^{9} + 24 q^{13} + 70 q^{15} + 46 q^{19} + 20 q^{25} - 52 q^{27} + 78 q^{31} - 35 q^{33} - 94 q^{37} - 6 q^{39} - 88 q^{43} - 35 q^{45} + 35 q^{51} - 140 q^{55} - 46 q^{57} - 162 q^{61} - 62 q^{67} + 490 q^{69} - 34 q^{73} + 10 q^{75} + 18 q^{79} - 127 q^{81} + 140 q^{85} - 280 q^{87} + 39 q^{93} - 328 q^{97} + 70 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 8x^{2} - 9x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 8\nu^{2} - 8\nu - 9 ) / 72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 8\nu + 9 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 9\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -8\beta_{3} + 8\beta _1 + 9 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
−2.31174 + 1.91203i
2.81174 − 1.04601i
−2.31174 − 1.91203i
2.81174 + 1.04601i
0 −2.81174 − 1.04601i 0 −5.12348 + 2.95804i 0 0 0 6.81174 + 5.88220i 0
557.2 0 2.31174 + 1.91203i 0 5.12348 − 2.95804i 0 0 0 1.68826 + 8.84024i 0
569.1 0 −2.81174 + 1.04601i 0 −5.12348 − 2.95804i 0 0 0 6.81174 − 5.88220i 0
569.2 0 2.31174 − 1.91203i 0 5.12348 + 2.95804i 0 0 0 1.68826 − 8.84024i 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
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.3.p.d 4
3.b odd 2 1 inner 588.3.p.d 4
7.b odd 2 1 84.3.p.c ✓ 4
7.c even 3 1 588.3.c.f 2
7.c even 3 1 inner 588.3.p.d 4
7.d odd 6 1 84.3.p.c ✓ 4
7.d odd 6 1 588.3.c.e 2
21.c even 2 1 84.3.p.c ✓ 4
21.g even 6 1 84.3.p.c ✓ 4
21.g even 6 1 588.3.c.e 2
21.h odd 6 1 588.3.c.f 2
21.h odd 6 1 inner 588.3.p.d 4
28.d even 2 1 336.3.bn.d 4
28.f even 6 1 336.3.bn.d 4
84.h odd 2 1 336.3.bn.d 4
84.j odd 6 1 336.3.bn.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.3.p.c ✓ 4 7.b odd 2 1
84.3.p.c ✓ 4 7.d odd 6 1
84.3.p.c ✓ 4 21.c even 2 1
84.3.p.c ✓ 4 21.g even 6 1
336.3.bn.d 4 28.d even 2 1
336.3.bn.d 4 28.f even 6 1
336.3.bn.d 4 84.h odd 2 1
336.3.bn.d 4 84.j odd 6 1
588.3.c.e 2 7.d odd 6 1
588.3.c.e 2 21.g even 6 1
588.3.c.f 2 7.c even 3 1
588.3.c.f 2 21.h odd 6 1
588.3.p.d 4 1.a even 1 1 trivial
588.3.p.d 4 3.b odd 2 1 inner
588.3.p.d 4 7.c even 3 1 inner
588.3.p.d 4 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(588, [\chi])\):

\( T_{5}^{4} - 35T_{5}^{2} + 1225 \) Copy content Toggle raw display
\( T_{13} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{4} - 35T^{2} + 1225 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 35T^{2} + 1225 \) Copy content Toggle raw display
$13$ \( (T - 6)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 35T^{2} + 1225 \) Copy content Toggle raw display
$19$ \( (T^{2} - 23 T + 529)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 1715 T^{2} + 2941225 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2240)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 39 T + 1521)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 47 T + 2209)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T + 22)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 2835 T^{2} + 8037225 \) Copy content Toggle raw display
$53$ \( T^{4} - 2835 T^{2} + 8037225 \) Copy content Toggle raw display
$59$ \( T^{4} - 10115 T^{2} + 102313225 \) Copy content Toggle raw display
$61$ \( (T^{2} + 81 T + 6561)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 31 T + 961)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8960)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 17 T + 289)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 9 T + 81)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2240)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 7875 T^{2} + 62015625 \) Copy content Toggle raw display
$97$ \( (T + 82)^{4} \) Copy content Toggle raw display
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