Properties

Label 588.6.i.f
Level $588$
Weight $6$
Character orbit 588.i
Analytic conductor $94.306$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 9 \zeta_{6} + 9) q^{3} - 6 \zeta_{6} q^{5} - 81 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 9 \zeta_{6} + 9) q^{3} - 6 \zeta_{6} q^{5} - 81 \zeta_{6} q^{9} + ( - 108 \zeta_{6} + 108) q^{11} - 346 q^{13} - 54 q^{15} + ( - 1398 \zeta_{6} + 1398) q^{17} + 1012 \zeta_{6} q^{19} + 1536 \zeta_{6} q^{23} + ( - 3089 \zeta_{6} + 3089) q^{25} - 729 q^{27} - 3762 q^{29} + ( - 736 \zeta_{6} + 736) q^{31} - 972 \zeta_{6} q^{33} - 2054 \zeta_{6} q^{37} + (3114 \zeta_{6} - 3114) q^{39} - 15534 q^{41} + 11036 q^{43} + (486 \zeta_{6} - 486) q^{45} - 4560 \zeta_{6} q^{47} - 12582 \zeta_{6} q^{51} + ( - 7962 \zeta_{6} + 7962) q^{53} - 648 q^{55} + 9108 q^{57} + ( - 7020 \zeta_{6} + 7020) q^{59} - 26870 \zeta_{6} q^{61} + 2076 \zeta_{6} q^{65} + (52148 \zeta_{6} - 52148) q^{67} + 13824 q^{69} - 2544 q^{71} + ( - 9766 \zeta_{6} + 9766) q^{73} - 27801 \zeta_{6} q^{75} - 68672 \zeta_{6} q^{79} + (6561 \zeta_{6} - 6561) q^{81} - 61668 q^{83} - 8388 q^{85} + (33858 \zeta_{6} - 33858) q^{87} + 41454 \zeta_{6} q^{89} - 6624 \zeta_{6} q^{93} + ( - 6072 \zeta_{6} + 6072) q^{95} - 111262 q^{97} - 8748 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{3} - 6 q^{5} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 9 q^{3} - 6 q^{5} - 81 q^{9} + 108 q^{11} - 692 q^{13} - 108 q^{15} + 1398 q^{17} + 1012 q^{19} + 1536 q^{23} + 3089 q^{25} - 1458 q^{27} - 7524 q^{29} + 736 q^{31} - 972 q^{33} - 2054 q^{37} - 3114 q^{39} - 31068 q^{41} + 22072 q^{43} - 486 q^{45} - 4560 q^{47} - 12582 q^{51} + 7962 q^{53} - 1296 q^{55} + 18216 q^{57} + 7020 q^{59} - 26870 q^{61} + 2076 q^{65} - 52148 q^{67} + 27648 q^{69} - 5088 q^{71} + 9766 q^{73} - 27801 q^{75} - 68672 q^{79} - 6561 q^{81} - 123336 q^{83} - 16776 q^{85} - 33858 q^{87} + 41454 q^{89} - 6624 q^{93} + 6072 q^{95} - 222524 q^{97} - 17496 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0 4.50000 7.79423i 0 −3.00000 5.19615i 0 0 0 −40.5000 70.1481i 0
373.1 0 4.50000 + 7.79423i 0 −3.00000 + 5.19615i 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.f 2
7.b odd 2 1 588.6.i.b 2
7.c even 3 1 84.6.a.a 1
7.c even 3 1 inner 588.6.i.f 2
7.d odd 6 1 588.6.a.e 1
7.d odd 6 1 588.6.i.b 2
21.h odd 6 1 252.6.a.b 1
28.g odd 6 1 336.6.a.n 1
84.n even 6 1 1008.6.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.a 1 7.c even 3 1
252.6.a.b 1 21.h odd 6 1
336.6.a.n 1 28.g odd 6 1
588.6.a.e 1 7.d odd 6 1
588.6.i.b 2 7.b odd 2 1
588.6.i.b 2 7.d odd 6 1
588.6.i.f 2 1.a even 1 1 trivial
588.6.i.f 2 7.c even 3 1 inner
1008.6.a.o 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 6T_{5} + 36 \) acting on \(S_{6}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 108T + 11664 \) Copy content Toggle raw display
$13$ \( (T + 346)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1398 T + 1954404 \) Copy content Toggle raw display
$19$ \( T^{2} - 1012 T + 1024144 \) Copy content Toggle raw display
$23$ \( T^{2} - 1536 T + 2359296 \) Copy content Toggle raw display
$29$ \( (T + 3762)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 736T + 541696 \) Copy content Toggle raw display
$37$ \( T^{2} + 2054 T + 4218916 \) Copy content Toggle raw display
$41$ \( (T + 15534)^{2} \) Copy content Toggle raw display
$43$ \( (T - 11036)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4560 T + 20793600 \) Copy content Toggle raw display
$53$ \( T^{2} - 7962 T + 63393444 \) Copy content Toggle raw display
$59$ \( T^{2} - 7020 T + 49280400 \) Copy content Toggle raw display
$61$ \( T^{2} + 26870 T + 721996900 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 2719413904 \) Copy content Toggle raw display
$71$ \( (T + 2544)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 9766 T + 95374756 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 4715843584 \) Copy content Toggle raw display
$83$ \( (T + 61668)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 1718434116 \) Copy content Toggle raw display
$97$ \( (T + 111262)^{2} \) Copy content Toggle raw display
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