Properties

Label 588.6.i.n
Level $588$
Weight $6$
Character orbit 588.i
Analytic conductor $94.306$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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: \(\Q(\sqrt{-3}, \sqrt{505})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - 9 \beta_1 + 9) q^{3} + (\beta_{2} + 39 \beta_1) q^{5} - 81 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 9 \beta_1 + 9) q^{3} + (\beta_{2} + 39 \beta_1) q^{5} - 81 \beta_1 q^{9} + ( - 5 \beta_{3} - 5 \beta_{2} + \cdots - 87) q^{11}+ \cdots + (405 \beta_{3} + 7047) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 18 q^{3} + 78 q^{5} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 18 q^{3} + 78 q^{5} - 162 q^{9} - 174 q^{11} - 416 q^{13} + 1404 q^{15} + 1482 q^{17} + 352 q^{19} - 3354 q^{23} - 5882 q^{25} - 2916 q^{27} + 552 q^{29} + 6520 q^{31} + 1566 q^{33} - 13864 q^{37} - 1872 q^{39} + 25860 q^{41} + 25424 q^{43} + 6318 q^{45} - 28116 q^{47} - 13338 q^{51} + 46992 q^{53} + 77328 q^{55} + 6336 q^{57} - 65556 q^{59} - 13148 q^{61} + 46428 q^{65} + 75236 q^{67} - 60372 q^{69} - 132084 q^{71} + 60496 q^{73} + 52938 q^{75} + 34916 q^{79} - 13122 q^{81} + 164976 q^{83} + 461016 q^{85} + 2484 q^{87} + 42510 q^{89} - 58680 q^{93} + 313512 q^{95} - 426512 q^{97} + 28188 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 127\nu^{2} - 127\nu + 15876 ) / 16002 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 127\nu^{2} + 32131\nu - 15876 ) / 5334 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{3} + 1137 ) / 127 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 3\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 759\beta _1 - 759 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 127\beta_{3} - 1137 ) / 6 \) Copy content Toggle raw display

Character values

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

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

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
−5.36805 9.29774i
5.86805 + 10.1638i
−5.36805 + 9.29774i
5.86805 10.1638i
0 4.50000 7.79423i 0 −14.2083 24.6095i 0 0 0 −40.5000 70.1481i 0
361.2 0 4.50000 7.79423i 0 53.2083 + 92.1595i 0 0 0 −40.5000 70.1481i 0
373.1 0 4.50000 + 7.79423i 0 −14.2083 + 24.6095i 0 0 0 −40.5000 + 70.1481i 0
373.2 0 4.50000 + 7.79423i 0 53.2083 92.1595i 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.n 4
7.b odd 2 1 588.6.i.h 4
7.c even 3 1 588.6.a.g 2
7.c even 3 1 inner 588.6.i.n 4
7.d odd 6 1 84.6.a.d 2
7.d odd 6 1 588.6.i.h 4
21.g even 6 1 252.6.a.e 2
28.f even 6 1 336.6.a.u 2
84.j odd 6 1 1008.6.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.d 2 7.d odd 6 1
252.6.a.e 2 21.g even 6 1
336.6.a.u 2 28.f even 6 1
588.6.a.g 2 7.c even 3 1
588.6.i.h 4 7.b odd 2 1
588.6.i.h 4 7.d odd 6 1
588.6.i.n 4 1.a even 1 1 trivial
588.6.i.n 4 7.c even 3 1 inner
1008.6.a.bf 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 78T_{5}^{3} + 9108T_{5}^{2} + 235872T_{5} + 9144576 \) 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
$5$ \( T^{4} - 78 T^{3} + \cdots + 9144576 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 11247875136 \) Copy content Toggle raw display
$13$ \( (T^{2} + 208 T - 152804)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 1191730288896 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 34331912110336 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 652838144256 \) Copy content Toggle raw display
$29$ \( (T^{2} - 276 T - 45430956)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 22441821798400 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{2} - 12930 T + 41246280)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 12712 T - 25049264)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{2} + 66042 T + 931452336)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 71\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( (T^{2} - 82488 T + 1671979536)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + 213256 T + 11297373964)^{2} \) Copy content Toggle raw display
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