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Properties

Label 588.6.i.i
Level $588$
Weight $6$
Character orbit 588.i
Analytic conductor $94.306$
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] = [588,6,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, 6, names="a")
 
//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);
 
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

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: \(94.3056860500\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5569})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 1393x^{2} + 1392x + 1937664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - 9 \beta_1 q^{3} + (\beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{5} + (81 \beta_1 - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 \beta_1 q^{3} + (\beta_{3} + \beta_{2} + 3 \beta_1 - 3) q^{5} + (81 \beta_1 - 81) q^{9} + ( - 7 \beta_{2} + 45 \beta_1) q^{11} + ( - 6 \beta_{3} - 384) q^{13} + ( - 9 \beta_{3} + 27) q^{15} + (\beta_{2} + 963 \beta_1) q^{17} + (24 \beta_{3} + 24 \beta_{2} + \cdots + 1124) q^{19}+ \cdots + ( - 567 \beta_{3} - 3645) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{3} - 6 q^{5} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 18 q^{3} - 6 q^{5} - 162 q^{9} + 90 q^{11} - 1536 q^{13} + 108 q^{15} + 1926 q^{17} + 2248 q^{19} - 6354 q^{23} - 4906 q^{25} + 2916 q^{27} + 21144 q^{29} - 3312 q^{31} + 810 q^{33} - 2104 q^{37} + 6912 q^{39} - 2532 q^{41} - 11536 q^{43} - 486 q^{45} + 15612 q^{47} + 17334 q^{51} - 16512 q^{53} + 155392 q^{55} - 40464 q^{57} - 13140 q^{59} - 5796 q^{61} - 64524 q^{65} + 56116 q^{67} + 114372 q^{69} + 22044 q^{71} - 85384 q^{73} - 44154 q^{75} + 19620 q^{79} - 13122 q^{81} + 88848 q^{83} - 33832 q^{85} - 95148 q^{87} + 211218 q^{89} - 29808 q^{93} - 260568 q^{95} - 89728 q^{97} - 14580 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 1393x^{2} + 1392x + 1937664 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 1393\nu^{2} - 1393\nu + 1937664 ) / 1939056 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 1393\nu^{2} + 3879505\nu - 1937664 ) / 1939056 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + 4177 ) / 1393 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 2785\beta _1 - 2785 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1393\beta_{3} - 4177 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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
18.9064 − 32.7469i
−18.4064 + 31.8809i
18.9064 + 32.7469i
−18.4064 − 31.8809i
0 −4.50000 + 7.79423i 0 −38.8129 − 67.2259i 0 0 0 −40.5000 − 70.1481i 0
361.2 0 −4.50000 + 7.79423i 0 35.8129 + 62.0297i 0 0 0 −40.5000 − 70.1481i 0
373.1 0 −4.50000 − 7.79423i 0 −38.8129 + 67.2259i 0 0 0 −40.5000 + 70.1481i 0
373.2 0 −4.50000 − 7.79423i 0 35.8129 − 62.0297i 0 0 0 −40.5000 + 70.1481i 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 588.6.i.i 4
7.b odd 2 1 588.6.i.l 4
7.c even 3 1 588.6.a.k 2
7.c even 3 1 inner 588.6.i.i 4
7.d odd 6 1 84.6.a.c ✓ 2
7.d odd 6 1 588.6.i.l 4
21.g even 6 1 252.6.a.h 2
28.f even 6 1 336.6.a.x 2
84.j odd 6 1 1008.6.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.c ✓ 2 7.d odd 6 1
252.6.a.h 2 21.g even 6 1
336.6.a.x 2 28.f even 6 1
588.6.a.k 2 7.c even 3 1
588.6.i.i 4 1.a even 1 1 trivial
588.6.i.i 4 7.c even 3 1 inner
588.6.i.l 4 7.b odd 2 1
588.6.i.l 4 7.d odd 6 1
1008.6.a.bo 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 6T_{5}^{3} + 5596T_{5}^{2} - 33360T_{5} + 30913600 \) 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
$5$ \( T^{4} + 6 T^{3} + \cdots + 30913600 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 73362972736 \) Copy content Toggle raw display
$13$ \( (T^{2} + 768 T - 53028)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 849715240000 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 3780566919424 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 101762901817600 \) Copy content Toggle raw display
$29$ \( (T^{2} - 10572 T + 19031396)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 682638940569600 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{2} + 1266 T - 5663952)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 5768 T - 312456944)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 56\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 855117957760000 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{2} - 11022 T - 4430537104)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 38\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( (T^{2} - 44424 T - 76892656)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{2} + 44864 T - 14004028100)^{2} \) Copy content Toggle raw display
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