Properties

Label 588.4.i.c
Level $588$
Weight $4$
Character orbit 588.i
Analytic conductor $34.693$
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] = [588,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(34.6931230834\)
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.

\(f(q)\) \(=\) \( q + (3 \zeta_{6} - 3) q^{3} + 6 \zeta_{6} q^{5} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (3 \zeta_{6} - 3) q^{3} + 6 \zeta_{6} q^{5} - 9 \zeta_{6} q^{9} + (36 \zeta_{6} - 36) q^{11} - 62 q^{13} - 18 q^{15} + ( - 114 \zeta_{6} + 114) q^{17} - 76 \zeta_{6} q^{19} + 24 \zeta_{6} q^{23} + ( - 89 \zeta_{6} + 89) q^{25} + 27 q^{27} + 54 q^{29} + (112 \zeta_{6} - 112) q^{31} - 108 \zeta_{6} q^{33} + 178 \zeta_{6} q^{37} + ( - 186 \zeta_{6} + 186) q^{39} - 378 q^{41} - 172 q^{43} + ( - 54 \zeta_{6} + 54) q^{45} - 192 \zeta_{6} q^{47} + 342 \zeta_{6} q^{51} + ( - 402 \zeta_{6} + 402) q^{53} - 216 q^{55} + 228 q^{57} + ( - 396 \zeta_{6} + 396) q^{59} + 254 \zeta_{6} q^{61} - 372 \zeta_{6} q^{65} + ( - 1012 \zeta_{6} + 1012) q^{67} - 72 q^{69} + 840 q^{71} + ( - 890 \zeta_{6} + 890) q^{73} + 267 \zeta_{6} q^{75} - 80 \zeta_{6} q^{79} + (81 \zeta_{6} - 81) q^{81} + 108 q^{83} + 684 q^{85} + (162 \zeta_{6} - 162) q^{87} - 1638 \zeta_{6} q^{89} - 336 \zeta_{6} q^{93} + ( - 456 \zeta_{6} + 456) q^{95} - 1010 q^{97} + 324 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 6 q^{5} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} + 6 q^{5} - 9 q^{9} - 36 q^{11} - 124 q^{13} - 36 q^{15} + 114 q^{17} - 76 q^{19} + 24 q^{23} + 89 q^{25} + 54 q^{27} + 108 q^{29} - 112 q^{31} - 108 q^{33} + 178 q^{37} + 186 q^{39} - 756 q^{41} - 344 q^{43} + 54 q^{45} - 192 q^{47} + 342 q^{51} + 402 q^{53} - 432 q^{55} + 456 q^{57} + 396 q^{59} + 254 q^{61} - 372 q^{65} + 1012 q^{67} - 144 q^{69} + 1680 q^{71} + 890 q^{73} + 267 q^{75} - 80 q^{79} - 81 q^{81} + 216 q^{83} + 1368 q^{85} - 162 q^{87} - 1638 q^{89} - 336 q^{93} + 456 q^{95} - 2020 q^{97} + 648 q^{99}+O(q^{100}) \) 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 −1.50000 + 2.59808i 0 3.00000 + 5.19615i 0 0 0 −4.50000 7.79423i 0
373.1 0 −1.50000 2.59808i 0 3.00000 5.19615i 0 0 0 −4.50000 + 7.79423i 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 588.4.i.c 2
3.b odd 2 1 1764.4.k.f 2
7.b odd 2 1 588.4.i.f 2
7.c even 3 1 588.4.a.d 1
7.c even 3 1 inner 588.4.i.c 2
7.d odd 6 1 84.4.a.a 1
7.d odd 6 1 588.4.i.f 2
21.c even 2 1 1764.4.k.l 2
21.g even 6 1 252.4.a.b 1
21.g even 6 1 1764.4.k.l 2
21.h odd 6 1 1764.4.a.j 1
21.h odd 6 1 1764.4.k.f 2
28.f even 6 1 336.4.a.k 1
28.g odd 6 1 2352.4.a.d 1
35.i odd 6 1 2100.4.a.l 1
35.k even 12 2 2100.4.k.j 2
56.j odd 6 1 1344.4.a.q 1
56.m even 6 1 1344.4.a.d 1
84.j odd 6 1 1008.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.a 1 7.d odd 6 1
252.4.a.b 1 21.g even 6 1
336.4.a.k 1 28.f even 6 1
588.4.a.d 1 7.c even 3 1
588.4.i.c 2 1.a even 1 1 trivial
588.4.i.c 2 7.c even 3 1 inner
588.4.i.f 2 7.b odd 2 1
588.4.i.f 2 7.d odd 6 1
1008.4.a.h 1 84.j odd 6 1
1344.4.a.d 1 56.m even 6 1
1344.4.a.q 1 56.j odd 6 1
1764.4.a.j 1 21.h odd 6 1
1764.4.k.f 2 3.b odd 2 1
1764.4.k.f 2 21.h odd 6 1
1764.4.k.l 2 21.c even 2 1
1764.4.k.l 2 21.g even 6 1
2100.4.a.l 1 35.i odd 6 1
2100.4.k.j 2 35.k even 12 2
2352.4.a.d 1 28.g odd 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 \(S_{4}^{\mathrm{new}}(588, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 36T + 1296 \) Copy content Toggle raw display
$13$ \( (T + 62)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 114T + 12996 \) Copy content Toggle raw display
$19$ \( T^{2} + 76T + 5776 \) Copy content Toggle raw display
$23$ \( T^{2} - 24T + 576 \) Copy content Toggle raw display
$29$ \( (T - 54)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 112T + 12544 \) Copy content Toggle raw display
$37$ \( T^{2} - 178T + 31684 \) Copy content Toggle raw display
$41$ \( (T + 378)^{2} \) Copy content Toggle raw display
$43$ \( (T + 172)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 192T + 36864 \) Copy content Toggle raw display
$53$ \( T^{2} - 402T + 161604 \) Copy content Toggle raw display
$59$ \( T^{2} - 396T + 156816 \) Copy content Toggle raw display
$61$ \( T^{2} - 254T + 64516 \) Copy content Toggle raw display
$67$ \( T^{2} - 1012 T + 1024144 \) Copy content Toggle raw display
$71$ \( (T - 840)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 890T + 792100 \) Copy content Toggle raw display
$79$ \( T^{2} + 80T + 6400 \) Copy content Toggle raw display
$83$ \( (T - 108)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1638 T + 2683044 \) Copy content Toggle raw display
$97$ \( (T + 1010)^{2} \) Copy content Toggle raw display
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