Properties

Label 588.2.o.d
Level $588$
Weight $2$
Character orbit 588.o
Analytic conductor $4.695$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,2,Mod(19,588)]
 
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([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.o (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: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.562828176.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + x^{6} + 2x^{5} - 6x^{4} + 4x^{3} + 4x^{2} - 16x + 16 \) 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{3} + 1) q^{3} + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{6} + \beta_1) q^{5} + (\beta_{6} - \beta_{4}) q^{6} + ( - \beta_{7} - \beta_{5} - 2 \beta_{3} - \beta_1 + 2) q^{8} - \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_{3} + 1) q^{3} + ( - \beta_{7} + \beta_{5} + \beta_{2}) q^{4} + ( - \beta_{7} + \beta_{6} + \beta_1) q^{5} + (\beta_{6} - \beta_{4}) q^{6} + ( - \beta_{7} - \beta_{5} - 2 \beta_{3} - \beta_1 + 2) q^{8} - \beta_{3} q^{9} + (\beta_{7} - \beta_{5} + 2 \beta_{3} - \beta_{2} - 2 \beta_1) q^{10} + (\beta_{6} - \beta_{5} - \beta_{4} + \beta_{2}) q^{11} + (\beta_{2} - \beta_1) q^{12} + ( - \beta_{7} - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{7} + \beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{15} + (2 \beta_{6} - 2 \beta_{4} - \beta_{2} - \beta_1 - 2) q^{16} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_1) q^{17} + \beta_{6} q^{18} + ( - \beta_{7} + \beta_{6} + \beta_{3} + 2 \beta_{2} - \beta_1) q^{19} + (\beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{3} + \beta_1 - 2) q^{20} + ( - \beta_{7} - \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{22} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{23} + ( - \beta_{7} - \beta_{5} - 2 \beta_{3} + \beta_{2}) q^{24} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{25} + ( - 2 \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{2} - \beta_1 - 2) q^{26} - q^{27} + ( - \beta_{7} + 3 \beta_{5} - 3 \beta_{4} - \beta_{2} + 3 \beta_1 - 2) q^{29} + ( - 2 \beta_{5} - \beta_{2} - \beta_1 + 2) q^{30} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 4 \beta_1 - 3) q^{31} + (\beta_{7} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{32} + (\beta_{7} + \beta_{6} + \beta_1) q^{33} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_1 + 4) q^{34} + (\beta_{7} - \beta_{5} - \beta_1) q^{36} + (\beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} + \beta_1) q^{37} + ( - \beta_{7} - \beta_{6} - 3 \beta_{5} - 2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{38} + (\beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 1) q^{39} + ( - 2 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \beta_{2} + 3 \beta_1 + 2) q^{40} + (4 \beta_{7} + 4 \beta_{3} - 4 \beta_{2} - 2) q^{41} + ( - \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{43} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1 - 2) q^{44} + ( - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2}) q^{45} + ( - 4 \beta_1 + 4) q^{46} + ( - 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3}) q^{47} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{3} - \beta_1 - 2) q^{48} + (\beta_{7} + \beta_{6} - 3 \beta_{5} - \beta_{4} + 6 \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{50} + (2 \beta_{5} - 2 \beta_{4}) q^{51} + ( - 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 4 \beta_1) q^{52} + (2 \beta_{7} - \beta_{6} - \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{53} + \beta_{4} q^{54} + ( - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{2} - \beta_1 + 2) q^{55} + (\beta_{7} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 + 1) q^{57} + ( - 2 \beta_{7} + 4 \beta_{5} + 2 \beta_{4} + \beta_{2} - \beta_1 + 6) q^{58} + (2 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} + 2) q^{59} + (\beta_{7} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 4) q^{60} + ( - 4 \beta_{6} - 4 \beta_{5} + 4 \beta_{4} - 4 \beta_1) q^{61} + (4 \beta_{7} - 3 \beta_{6} - 4 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + \cdots + 4) q^{62}+ \cdots + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 4 q^{3} - q^{4} + 2 q^{6} + 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 4 q^{3} - q^{4} + 2 q^{6} + 4 q^{8} - 4 q^{9} + 5 q^{10} + 6 q^{11} + q^{12} - 17 q^{16} + q^{18} + 6 q^{19} - 22 q^{20} - 6 q^{22} - 7 q^{24} + 2 q^{25} - 18 q^{26} - 8 q^{27} - 16 q^{29} + 13 q^{30} - 6 q^{31} - 9 q^{32} + 6 q^{33} + 28 q^{34} + 2 q^{36} + 6 q^{37} - 10 q^{38} - 6 q^{39} + 17 q^{40} - 23 q^{44} + 24 q^{46} - 4 q^{47} - 10 q^{48} + 2 q^{50} - 16 q^{52} - 4 q^{53} - q^{54} + 8 q^{55} + 12 q^{57} + 37 q^{58} + 14 q^{59} - 23 q^{60} - 12 q^{61} + 48 q^{62} + 2 q^{64} + 4 q^{65} + 15 q^{66} + 42 q^{67} + 26 q^{68} - 11 q^{72} + 18 q^{73} - 10 q^{74} - 2 q^{75} - 44 q^{76} - 6 q^{78} - 6 q^{79} + 39 q^{80} - 4 q^{81} + 10 q^{82} - 4 q^{83} - 32 q^{85} + 36 q^{86} - 8 q^{87} - 37 q^{88} + 8 q^{90} - 28 q^{92} + 6 q^{93} - 16 q^{94} - 24 q^{95} - 21 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + x^{6} + 2x^{5} - 6x^{4} + 4x^{3} + 4x^{2} - 16x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + \nu^{5} + 4\nu^{4} + 2\nu^{3} + 8\nu^{2} + 4\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + \nu^{5} + 4\nu^{4} - 6\nu^{3} + 4\nu - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - \nu^{5} - 4\nu^{4} - 2\nu^{3} + 8\nu^{2} - 4\nu + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - \nu^{5} + 2\nu^{3} - 4\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 4\nu^{6} - 3\nu^{5} - 8\nu^{4} + 10\nu^{3} - 12\nu + 40 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 2\nu^{6} + \nu^{5} - 6\nu^{4} + 10\nu^{3} + 4\nu^{2} - 12\nu + 32 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} - 2\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} - 2\beta_{6} - 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{6} - 2\beta_{5} - \beta_{4} + 6\beta_{3} + \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{7} + 2\beta_{6} - 6\beta_{5} - \beta_{4} - 6\beta_{3} + 3\beta_{2} - 4\beta _1 + 8 \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−1.33790 + 0.458297i
0.856419 + 1.12541i
1.40376 0.171630i
0.0777157 1.41208i
−1.33790 0.458297i
0.856419 1.12541i
1.40376 + 0.171630i
0.0777157 + 1.41208i
−1.06584 + 0.929502i 0.500000 + 0.866025i 0.272050 1.98141i −2.12403 1.22631i −1.33790 0.458297i 0 1.55176 + 2.36475i −0.500000 + 0.866025i 3.40374 0.667235i
19.2 −0.546424 1.30439i 0.500000 + 0.866025i −1.40284 + 1.42549i 3.33878 + 1.92764i 0.856419 1.12541i 0 2.62594 + 1.05092i −0.500000 + 0.866025i 0.690004 5.40836i
19.3 0.850516 1.12988i 0.500000 + 0.866025i −0.553244 1.92196i −0.834598 0.481855i 1.40376 + 0.171630i 0 −2.64212 1.00956i −0.500000 + 0.866025i −1.25428 + 0.533167i
19.4 1.26175 + 0.638735i 0.500000 + 0.866025i 1.18404 + 1.61185i −0.380152 0.219481i 0.0777157 + 1.41208i 0 0.464416 + 2.79004i −0.500000 + 0.866025i −0.339468 0.519747i
31.1 −1.06584 0.929502i 0.500000 0.866025i 0.272050 + 1.98141i −2.12403 + 1.22631i −1.33790 + 0.458297i 0 1.55176 2.36475i −0.500000 0.866025i 3.40374 + 0.667235i
31.2 −0.546424 + 1.30439i 0.500000 0.866025i −1.40284 1.42549i 3.33878 1.92764i 0.856419 + 1.12541i 0 2.62594 1.05092i −0.500000 0.866025i 0.690004 + 5.40836i
31.3 0.850516 + 1.12988i 0.500000 0.866025i −0.553244 + 1.92196i −0.834598 + 0.481855i 1.40376 0.171630i 0 −2.64212 + 1.00956i −0.500000 0.866025i −1.25428 0.533167i
31.4 1.26175 0.638735i 0.500000 0.866025i 1.18404 1.61185i −0.380152 + 0.219481i 0.0777157 1.41208i 0 0.464416 2.79004i −0.500000 0.866025i −0.339468 + 0.519747i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.o.d 8
4.b odd 2 1 588.2.o.b 8
7.b odd 2 1 84.2.o.a 8
7.c even 3 1 84.2.o.b yes 8
7.c even 3 1 588.2.b.a 8
7.d odd 6 1 588.2.b.b 8
7.d odd 6 1 588.2.o.b 8
21.c even 2 1 252.2.bf.g 8
21.g even 6 1 1764.2.b.i 8
21.h odd 6 1 252.2.bf.f 8
21.h odd 6 1 1764.2.b.j 8
28.d even 2 1 84.2.o.b yes 8
28.f even 6 1 588.2.b.a 8
28.f even 6 1 inner 588.2.o.d 8
28.g odd 6 1 84.2.o.a 8
28.g odd 6 1 588.2.b.b 8
56.e even 2 1 1344.2.bl.i 8
56.h odd 2 1 1344.2.bl.j 8
56.k odd 6 1 1344.2.bl.j 8
56.p even 6 1 1344.2.bl.i 8
84.h odd 2 1 252.2.bf.f 8
84.j odd 6 1 1764.2.b.j 8
84.n even 6 1 252.2.bf.g 8
84.n even 6 1 1764.2.b.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.o.a 8 7.b odd 2 1
84.2.o.a 8 28.g odd 6 1
84.2.o.b yes 8 7.c even 3 1
84.2.o.b yes 8 28.d even 2 1
252.2.bf.f 8 21.h odd 6 1
252.2.bf.f 8 84.h odd 2 1
252.2.bf.g 8 21.c even 2 1
252.2.bf.g 8 84.n even 6 1
588.2.b.a 8 7.c even 3 1
588.2.b.a 8 28.f even 6 1
588.2.b.b 8 7.d odd 6 1
588.2.b.b 8 28.g odd 6 1
588.2.o.b 8 4.b odd 2 1
588.2.o.b 8 7.d odd 6 1
588.2.o.d 8 1.a even 1 1 trivial
588.2.o.d 8 28.f even 6 1 inner
1344.2.bl.i 8 56.e even 2 1
1344.2.bl.i 8 56.p even 6 1
1344.2.bl.j 8 56.h odd 2 1
1344.2.bl.j 8 56.k odd 6 1
1764.2.b.i 8 21.g even 6 1
1764.2.b.i 8 84.n even 6 1
1764.2.b.j 8 21.h odd 6 1
1764.2.b.j 8 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(588, [\chi])\):

\( T_{5}^{8} - 11T_{5}^{6} + 125T_{5}^{4} + 264T_{5}^{3} + 236T_{5}^{2} + 96T_{5} + 16 \) Copy content Toggle raw display
\( T_{11}^{8} - 6T_{11}^{7} - T_{11}^{6} + 78T_{11}^{5} + 125T_{11}^{4} - 156T_{11}^{3} - 212T_{11}^{2} + 240T_{11} + 400 \) Copy content Toggle raw display
\( T_{19}^{8} - 6T_{19}^{7} + 43T_{19}^{6} - 78T_{19}^{5} + 405T_{19}^{4} - 372T_{19}^{3} + 3628T_{19}^{2} + 240T_{19} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + T^{6} - 2 T^{5} + 6 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 11 T^{6} + 125 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} - T^{6} + 78 T^{5} + \cdots + 400 \) Copy content Toggle raw display
$13$ \( T^{8} + 38 T^{6} + 473 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} - 28 T^{6} + 752 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + 43 T^{6} - 78 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{8} - 40 T^{6} + 1472 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$29$ \( (T^{4} + 8 T^{3} - 45 T^{2} - 352 T - 512)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 6 T^{7} + 120 T^{6} + \cdots + 4173849 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + 79 T^{6} + \cdots + 355216 \) Copy content Toggle raw display
$41$ \( T^{8} + 208 T^{6} + 14048 T^{4} + \cdots + 350464 \) Copy content Toggle raw display
$43$ \( T^{8} + 134 T^{6} + 6593 T^{4} + \cdots + 1073296 \) Copy content Toggle raw display
$47$ \( T^{8} + 4 T^{7} + 44 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$53$ \( T^{8} + 4 T^{7} + 77 T^{6} - 476 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( T^{8} - 14 T^{7} + 223 T^{6} + \cdots + 1420864 \) Copy content Toggle raw display
$61$ \( T^{8} + 12 T^{7} - 64 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$67$ \( T^{8} - 42 T^{7} + 755 T^{6} + \cdots + 4129024 \) Copy content Toggle raw display
$71$ \( T^{8} + 280 T^{6} + 19856 T^{4} + \cdots + 200704 \) Copy content Toggle raw display
$73$ \( T^{8} - 18 T^{7} + 55 T^{6} + \cdots + 952576 \) Copy content Toggle raw display
$79$ \( T^{8} + 6 T^{7} - 184 T^{6} + \cdots + 241081 \) Copy content Toggle raw display
$83$ \( (T^{4} + 2 T^{3} - 103 T^{2} + 304 T + 196)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 92 T^{6} + 8528 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$97$ \( T^{8} + 182 T^{6} + 8249 T^{4} + \cdots + 246016 \) Copy content Toggle raw display
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