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Properties

Label 588.2.b.c
Level $588$
Weight $2$
Character orbit 588.b
Analytic conductor $4.695$
Analytic rank $0$
Dimension $12$
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(391,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.391");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.b (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: 12.0.15911316233388032.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 10 x^{10} - 20 x^{9} + 35 x^{8} - 56 x^{7} + 84 x^{6} - 112 x^{5} + 140 x^{4} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{6} + \beta_{5}) q^{4} + ( - \beta_{11} - \beta_{7} - \beta_{6} + \cdots - 1) q^{5}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{6} + \beta_{5}) q^{4} + ( - \beta_{11} - \beta_{7} - \beta_{6} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{11} + \beta_{9} - \beta_{8} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} - 12 q^{3} - 4 q^{4} - 4 q^{6} + 4 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} - 12 q^{3} - 4 q^{4} - 4 q^{6} + 4 q^{8} + 12 q^{9} + 4 q^{12} - 4 q^{16} + 4 q^{18} + 24 q^{20} - 4 q^{24} - 12 q^{25} - 24 q^{26} - 12 q^{27} + 32 q^{29} - 16 q^{31} + 4 q^{32} - 32 q^{34} - 4 q^{36} + 32 q^{37} + 24 q^{38} + 32 q^{40} - 24 q^{44} + 24 q^{46} + 4 q^{48} - 28 q^{50} - 32 q^{52} - 32 q^{53} - 4 q^{54} - 16 q^{55} + 16 q^{58} + 16 q^{59} - 24 q^{60} - 8 q^{62} - 4 q^{64} + 8 q^{68} + 4 q^{72} - 32 q^{74} + 12 q^{75} + 32 q^{76} + 24 q^{78} + 16 q^{80} + 12 q^{81} - 32 q^{82} + 16 q^{83} + 16 q^{85} - 24 q^{86} - 32 q^{87} + 24 q^{88} + 16 q^{93} - 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 10 x^{10} - 20 x^{9} + 35 x^{8} - 56 x^{7} + 84 x^{6} - 112 x^{5} + 140 x^{4} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{11} + 2\nu^{9} + 5\nu^{7} - 4\nu^{6} + 16\nu^{5} - 20\nu^{4} + 20\nu^{3} - 24\nu^{2} + 48\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2 \nu^{11} - 3 \nu^{10} + 8 \nu^{9} - 8 \nu^{8} + 18 \nu^{7} - 25 \nu^{6} + 32 \nu^{5} - 34 \nu^{4} + \cdots + 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} - 4 \nu^{9} + 12 \nu^{8} - 19 \nu^{7} + 26 \nu^{6} - 42 \nu^{5} + 52 \nu^{4} + \cdots + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} + 8 \nu^{9} - 16 \nu^{8} + 23 \nu^{7} - 37 \nu^{6} + 58 \nu^{5} - 70 \nu^{4} + \cdots - 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{11} + 3 \nu^{10} - 8 \nu^{9} + 16 \nu^{8} - 23 \nu^{7} + 37 \nu^{6} - 58 \nu^{5} + 70 \nu^{4} + \cdots + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{11} - 5 \nu^{10} + 10 \nu^{9} - 20 \nu^{8} + 31 \nu^{7} - 51 \nu^{6} + 72 \nu^{5} - 102 \nu^{4} + \cdots - 96 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2 \nu^{11} + 5 \nu^{10} - 12 \nu^{9} + 24 \nu^{8} - 42 \nu^{7} + 63 \nu^{6} - 84 \nu^{5} + 110 \nu^{4} + \cdots + 64 ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} - 6 \nu^{9} + 12 \nu^{8} - 19 \nu^{7} + 34 \nu^{6} - 48 \nu^{5} + 60 \nu^{4} + \cdots + 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - \nu^{11} + 3 \nu^{10} - 8 \nu^{9} + 14 \nu^{8} - 23 \nu^{7} + 37 \nu^{6} - 50 \nu^{5} + 64 \nu^{4} + \cdots + 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5 \nu^{11} - 15 \nu^{10} + 32 \nu^{9} - 60 \nu^{8} + 99 \nu^{7} - 153 \nu^{6} + 218 \nu^{5} + \cdots - 192 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{8} + \beta_{6} - \beta_{4} - \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{8} - 2\beta_{7} - \beta_{5} - \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + 3\beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - \beta_{8} + \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{10} + 2\beta_{9} - 3\beta_{8} - 2\beta_{7} + \beta_{6} - 2\beta_{5} - 3\beta_{4} + 2\beta_{2} + \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{11} + 3 \beta_{10} - 4 \beta_{9} + \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - 7 \beta_{5} + 3 \beta_{3} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 3 \beta_{11} - 7 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} - \beta_{7} - 7 \beta_{6} + \beta_{5} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7 \beta_{11} - 8 \beta_{10} - 2 \beta_{9} - 7 \beta_{8} - 11 \beta_{7} - 7 \beta_{6} + 4 \beta_{5} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 6 \beta_{11} + 21 \beta_{10} - 8 \beta_{9} - 7 \beta_{8} + 8 \beta_{7} - 17 \beta_{6} - 4 \beta_{5} + \cdots - 13 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
391.1
−0.988865 − 1.01101i
−0.988865 + 1.01101i
−0.476589 − 1.33149i
−0.476589 + 1.33149i
0.250649 − 1.39182i
0.250649 + 1.39182i
0.639847 − 1.26119i
0.639847 + 1.26119i
1.22594 − 0.705031i
1.22594 + 0.705031i
1.34902 − 0.424442i
1.34902 + 0.424442i
−0.988865 − 1.01101i −1.00000 −0.0442929 + 1.99951i − 1.12886i 0.988865 + 1.01101i 0 2.06533 − 1.93246i 1.00000 −1.14129 + 1.11629i
391.2 −0.988865 + 1.01101i −1.00000 −0.0442929 − 1.99951i 1.12886i 0.988865 − 1.01101i 0 2.06533 + 1.93246i 1.00000 −1.14129 − 1.11629i
391.3 −0.476589 − 1.33149i −1.00000 −1.54572 + 1.26915i − 0.509876i 0.476589 + 1.33149i 0 2.42653 + 1.45325i 1.00000 −0.678894 + 0.243001i
391.4 −0.476589 + 1.33149i −1.00000 −1.54572 − 1.26915i 0.509876i 0.476589 − 1.33149i 0 2.42653 − 1.45325i 1.00000 −0.678894 − 0.243001i
391.5 0.250649 − 1.39182i −1.00000 −1.87435 − 0.697718i − 3.39209i −0.250649 + 1.39182i 0 −1.44090 + 2.43388i 1.00000 −4.72119 − 0.850222i
391.6 0.250649 + 1.39182i −1.00000 −1.87435 + 0.697718i 3.39209i −0.250649 − 1.39182i 0 −1.44090 − 2.43388i 1.00000 −4.72119 + 0.850222i
391.7 0.639847 − 1.26119i −1.00000 −1.18119 − 1.61393i 3.10455i −0.639847 + 1.26119i 0 −2.79126 + 0.457034i 1.00000 3.91542 + 1.98644i
391.8 0.639847 + 1.26119i −1.00000 −1.18119 + 1.61393i − 3.10455i −0.639847 − 1.26119i 0 −2.79126 − 0.457034i 1.00000 3.91542 − 1.98644i
391.9 1.22594 − 0.705031i −1.00000 1.00586 − 1.72865i 3.64758i −1.22594 + 0.705031i 0 0.0143727 − 2.82839i 1.00000 2.57166 + 4.47172i
391.10 1.22594 + 0.705031i −1.00000 1.00586 + 1.72865i − 3.64758i −1.22594 − 0.705031i 0 0.0143727 + 2.82839i 1.00000 2.57166 − 4.47172i
391.11 1.34902 − 0.424442i −1.00000 1.63970 − 1.14516i 0.127929i −1.34902 + 0.424442i 0 1.72593 − 2.24080i 1.00000 0.0542984 + 0.172579i
391.12 1.34902 + 0.424442i −1.00000 1.63970 + 1.14516i − 0.127929i −1.34902 − 0.424442i 0 1.72593 + 2.24080i 1.00000 0.0542984 − 0.172579i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 391.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 588.2.b.c ✓ 12
3.b odd 2 1 1764.2.b.l 12
4.b odd 2 1 588.2.b.d yes 12
7.b odd 2 1 588.2.b.d yes 12
7.c even 3 2 588.2.o.f 24
7.d odd 6 2 588.2.o.e 24
12.b even 2 1 1764.2.b.m 12
21.c even 2 1 1764.2.b.m 12
28.d even 2 1 inner 588.2.b.c ✓ 12
28.f even 6 2 588.2.o.f 24
28.g odd 6 2 588.2.o.e 24
84.h odd 2 1 1764.2.b.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
588.2.b.c ✓ 12 1.a even 1 1 trivial
588.2.b.c ✓ 12 28.d even 2 1 inner
588.2.b.d yes 12 4.b odd 2 1
588.2.b.d yes 12 7.b odd 2 1
588.2.o.e 24 7.d odd 6 2
588.2.o.e 24 28.g odd 6 2
588.2.o.f 24 7.c even 3 2
588.2.o.f 24 28.f even 6 2
1764.2.b.l 12 3.b odd 2 1
1764.2.b.l 12 84.h odd 2 1
1764.2.b.m 12 12.b even 2 1
1764.2.b.m 12 21.c even 2 1

Hecke kernels

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

\( T_{5}^{12} + 36T_{5}^{10} + 446T_{5}^{8} + 2096T_{5}^{6} + 2428T_{5}^{4} + 528T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{19}^{6} - 72T_{19}^{4} + 32T_{19}^{3} + 1288T_{19}^{2} - 1280T_{19} - 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 4 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T + 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 36 T^{10} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 104 T^{10} + \cdots + 1968128 \) Copy content Toggle raw display
$13$ \( T^{12} + 92 T^{10} + \cdots + 40328 \) Copy content Toggle raw display
$17$ \( T^{12} + 116 T^{10} + \cdots + 8405000 \) Copy content Toggle raw display
$19$ \( (T^{6} - 72 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 518162432 \) Copy content Toggle raw display
$29$ \( (T^{6} - 16 T^{5} + \cdots + 9544)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 8 T^{5} + \cdots - 6272)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 16 T^{5} + \cdots + 10424)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 228 T^{10} + \cdots + 84872 \) Copy content Toggle raw display
$43$ \( T^{12} + 240 T^{10} + \cdots + 6422528 \) Copy content Toggle raw display
$47$ \( (T^{6} - 192 T^{4} + \cdots - 62336)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 16 T^{5} + \cdots - 256)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 8 T^{5} + \cdots + 256)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 188 T^{10} + \cdots + 7688 \) Copy content Toggle raw display
$67$ \( T^{12} + 448 T^{10} + \cdots + 8388608 \) Copy content Toggle raw display
$71$ \( T^{12} + 296 T^{10} + \cdots + 4917248 \) Copy content Toggle raw display
$73$ \( T^{12} + 396 T^{10} + \cdots + 19046792 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 1038221312 \) Copy content Toggle raw display
$83$ \( (T^{6} - 8 T^{5} + \cdots + 82432)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 146068232 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 1057448072 \) Copy content Toggle raw display
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