Properties

Label 588.2.o.f
Level $588$
Weight $2$
Character orbit 588.o
Analytic conductor $4.695$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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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: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} + 12 q^{3} + 4 q^{4} - 8 q^{6} + 8 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} + 12 q^{3} + 4 q^{4} - 8 q^{6} + 8 q^{8} - 12 q^{9} - 4 q^{12} + 4 q^{16} - 4 q^{18} + 48 q^{20} + 4 q^{24} + 12 q^{25} + 24 q^{26} - 24 q^{27} + 64 q^{29} + 16 q^{31} - 4 q^{32} - 64 q^{34} - 8 q^{36} - 32 q^{37} - 24 q^{38} - 32 q^{40} + 24 q^{44} - 24 q^{46} + 8 q^{48} - 56 q^{50} + 32 q^{52} + 32 q^{53} + 4 q^{54} - 32 q^{55} - 16 q^{58} - 16 q^{59} + 24 q^{60} - 16 q^{62} - 8 q^{64} - 8 q^{68} - 4 q^{72} + 32 q^{74} - 12 q^{75} + 64 q^{76} + 48 q^{78} - 16 q^{80} - 12 q^{81} + 32 q^{82} + 32 q^{83} + 32 q^{85} + 24 q^{86} + 32 q^{87} - 24 q^{88} - 16 q^{93} + 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.41214 0.0764704i 0.500000 + 0.866025i 1.98830 + 0.215975i 2.68862 + 1.55228i −0.639847 1.26119i 0 −2.79126 0.457034i −0.500000 + 0.866025i −3.67802 2.39764i
19.2 −1.33068 0.478844i 0.500000 + 0.866025i 1.54142 + 1.27438i −2.93763 1.69604i −0.250649 1.39182i 0 −1.44090 2.43388i −0.500000 + 0.866025i 3.09691 + 3.66356i
19.3 −1.22355 + 0.709180i 0.500000 + 0.866025i 0.994127 1.73543i 3.15890 + 1.82379i −1.22594 0.705031i 0 0.0143727 + 2.82839i −0.500000 + 0.866025i −5.15845 + 0.00873760i
19.4 −1.04209 + 0.956063i 0.500000 + 0.866025i 0.171888 1.99260i 0.110790 + 0.0639645i −1.34902 0.424442i 0 1.72593 + 2.24080i −0.500000 + 0.866025i −0.176607 + 0.0392655i
19.5 −0.914808 1.07848i 0.500000 + 0.866025i −0.326251 + 1.97321i −0.441565 0.254938i 0.476589 1.33149i 0 2.42653 1.45325i −0.500000 + 0.866025i 0.129001 + 0.709440i
19.6 −0.381130 1.36189i 0.500000 + 0.866025i −1.70948 + 1.03811i −0.977624 0.564431i 0.988865 1.01101i 0 2.06533 + 1.93246i −0.500000 + 0.866025i −0.396091 + 1.54654i
19.7 −0.306932 + 1.38050i 0.500000 + 0.866025i −1.81159 0.847441i −0.110790 0.0639645i −1.34902 + 0.424442i 0 1.72593 2.24080i −0.500000 + 0.866025i 0.122308 0.133313i
19.8 −0.00239545 + 1.41421i 0.500000 + 0.866025i −1.99999 0.00677535i −3.15890 1.82379i −1.22594 + 0.705031i 0 0.0143727 2.82839i −0.500000 + 0.866025i 2.58679 4.46298i
19.9 0.772298 + 1.18472i 0.500000 + 0.866025i −0.807113 + 1.82991i −2.68862 1.55228i −0.639847 + 1.26119i 0 −2.79126 + 0.457034i −0.500000 + 0.866025i −0.237406 4.38407i
19.10 1.08003 + 0.912980i 0.500000 + 0.866025i 0.332934 + 1.97209i 2.93763 + 1.69604i −0.250649 + 1.39182i 0 −1.44090 + 2.43388i −0.500000 + 0.866025i 1.62428 + 4.51378i
19.11 1.36999 0.350876i 0.500000 + 0.866025i 1.75377 0.961396i 0.977624 + 0.564431i 0.988865 + 1.01101i 0 2.06533 1.93246i −0.500000 + 0.866025i 1.53738 + 0.430244i
19.12 1.39140 + 0.253006i 0.500000 + 0.866025i 1.87198 + 0.704063i 0.441565 + 0.254938i 0.476589 + 1.33149i 0 2.42653 + 1.45325i −0.500000 + 0.866025i 0.549892 + 0.466438i
31.1 −1.41214 + 0.0764704i 0.500000 0.866025i 1.98830 0.215975i 2.68862 1.55228i −0.639847 + 1.26119i 0 −2.79126 + 0.457034i −0.500000 0.866025i −3.67802 + 2.39764i
31.2 −1.33068 + 0.478844i 0.500000 0.866025i 1.54142 1.27438i −2.93763 + 1.69604i −0.250649 + 1.39182i 0 −1.44090 + 2.43388i −0.500000 0.866025i 3.09691 3.66356i
31.3 −1.22355 0.709180i 0.500000 0.866025i 0.994127 + 1.73543i 3.15890 1.82379i −1.22594 + 0.705031i 0 0.0143727 2.82839i −0.500000 0.866025i −5.15845 0.00873760i
31.4 −1.04209 0.956063i 0.500000 0.866025i 0.171888 + 1.99260i 0.110790 0.0639645i −1.34902 + 0.424442i 0 1.72593 2.24080i −0.500000 0.866025i −0.176607 0.0392655i
31.5 −0.914808 + 1.07848i 0.500000 0.866025i −0.326251 1.97321i −0.441565 + 0.254938i 0.476589 + 1.33149i 0 2.42653 + 1.45325i −0.500000 0.866025i 0.129001 0.709440i
31.6 −0.381130 + 1.36189i 0.500000 0.866025i −1.70948 1.03811i −0.977624 + 0.564431i 0.988865 + 1.01101i 0 2.06533 1.93246i −0.500000 0.866025i −0.396091 1.54654i
31.7 −0.306932 1.38050i 0.500000 0.866025i −1.81159 + 0.847441i −0.110790 + 0.0639645i −1.34902 0.424442i 0 1.72593 + 2.24080i −0.500000 0.866025i 0.122308 + 0.133313i
31.8 −0.00239545 1.41421i 0.500000 0.866025i −1.99999 + 0.00677535i −3.15890 + 1.82379i −1.22594 0.705031i 0 0.0143727 + 2.82839i −0.500000 0.866025i 2.58679 + 4.46298i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{5}^{24} - 36 T_{5}^{22} + 850 T_{5}^{20} - 11864 T_{5}^{18} + 121032 T_{5}^{16} - 760528 T_{5}^{14} + 3291336 T_{5}^{12} - 4618400 T_{5}^{10} + 4784928 T_{5}^{8} - 1248448 T_{5}^{6} + 259360 T_{5}^{4} + \cdots + 64 \) Copy content Toggle raw display
\( T_{11}^{24} - 104 T_{11}^{22} + 6776 T_{11}^{20} - 274816 T_{11}^{18} + 8159792 T_{11}^{16} - 169942016 T_{11}^{14} + 2641219456 T_{11}^{12} - 28235791872 T_{11}^{10} + 214149619968 T_{11}^{8} + \cdots + 3873527824384 \) Copy content Toggle raw display
\( T_{19}^{12} + 72 T_{19}^{10} + 64 T_{19}^{9} + 3896 T_{19}^{8} + 3584 T_{19}^{7} + 93248 T_{19}^{6} + 143104 T_{19}^{5} + 1681472 T_{19}^{4} + 1632256 T_{19}^{3} + 1968128 T_{19}^{2} - 327680 T_{19} + 65536 \) Copy content Toggle raw display