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Properties

Label 588.2.n.d
Level $588$
Weight $2$
Character orbit 588.n
Analytic conductor $4.695$
Analytic rank $0$
Dimension $24$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [588,2,Mod(263,588)]
 
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, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.263");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.n (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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{4} - 36 q^{16} - 12 q^{18} + 12 q^{25} + 12 q^{30} + 12 q^{36} + 96 q^{39} - 96 q^{46} - 12 q^{51} - 24 q^{57} - 120 q^{58} - 84 q^{60} - 48 q^{64} + 48 q^{67} - 72 q^{72} - 24 q^{78} + 144 q^{79}+ \cdots - 24 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
263.1 −1.13414 − 0.844822i −1.36816 + 1.06214i 0.572551 + 1.91629i 3.44057 − 1.98641i 2.44900 − 0.0487576i 0 0.969574 − 2.65705i 0.743736 − 2.90635i −5.58025 − 0.653796i
263.2 −1.13414 − 0.844822i 1.36816 − 1.06214i 0.572551 + 1.91629i −3.44057 + 1.98641i −2.44900 + 0.0487576i 0 0.969574 − 2.65705i 0.743736 − 2.90635i 5.58025 + 0.653796i
263.3 −0.938613 + 1.05783i −0.299892 − 1.70589i −0.238012 − 1.98579i −0.695526 + 0.401562i 2.08603 + 1.28394i 0 2.32403 + 1.61211i −2.82013 + 1.02317i 0.228045 − 1.11266i
263.4 −0.938613 + 1.05783i 0.299892 + 1.70589i −0.238012 − 1.98579i 0.695526 − 0.401562i −2.08603 − 1.28394i 0 2.32403 + 1.61211i −2.82013 + 1.02317i −0.228045 + 1.11266i
263.5 −0.287629 + 1.38466i −1.66143 − 0.489533i −1.83454 − 0.796533i −1.08570 + 0.626827i 1.15571 − 2.15971i 0 1.63059 − 2.31110i 2.52072 + 1.62665i −0.555662 − 1.68361i
263.6 −0.287629 + 1.38466i 1.66143 + 0.489533i −1.83454 − 0.796533i 1.08570 − 0.626827i −1.15571 + 2.15971i 0 1.63059 − 2.31110i 2.52072 + 1.62665i 0.555662 + 1.68361i
263.7 0.287629 − 1.38466i −1.25466 − 1.19408i −1.83454 − 0.796533i 1.08570 − 0.626827i −2.01426 + 1.39383i 0 −1.63059 + 2.31110i 0.148364 + 2.99633i −0.555662 − 1.68361i
263.8 0.287629 − 1.38466i 1.25466 + 1.19408i −1.83454 − 0.796533i −1.08570 + 0.626827i 2.01426 − 1.39383i 0 −1.63059 + 2.31110i 0.148364 + 2.99633i 0.555662 + 1.68361i
263.9 0.938613 − 1.05783i −1.62729 + 0.593231i −0.238012 − 1.98579i 0.695526 − 0.401562i −0.899858 + 2.27821i 0 −2.32403 − 1.61211i 2.29615 − 1.93072i 0.228045 − 1.11266i
263.10 0.938613 − 1.05783i 1.62729 − 0.593231i −0.238012 − 1.98579i −0.695526 + 0.401562i 0.899858 − 2.27821i 0 −2.32403 − 1.61211i 2.29615 − 1.93072i −0.228045 + 1.11266i
263.11 1.13414 + 0.844822i −0.235755 + 1.71593i 0.572551 + 1.91629i 3.44057 − 1.98641i −1.71704 + 1.74694i 0 −0.969574 + 2.65705i −2.88884 − 0.809079i 5.58025 + 0.653796i
263.12 1.13414 + 0.844822i 0.235755 − 1.71593i 0.572551 + 1.91629i −3.44057 + 1.98641i 1.71704 − 1.74694i 0 −0.969574 + 2.65705i −2.88884 − 0.809079i −5.58025 − 0.653796i
275.1 −1.13414 + 0.844822i −1.36816 − 1.06214i 0.572551 − 1.91629i 3.44057 + 1.98641i 2.44900 + 0.0487576i 0 0.969574 + 2.65705i 0.743736 + 2.90635i −5.58025 + 0.653796i
275.2 −1.13414 + 0.844822i 1.36816 + 1.06214i 0.572551 − 1.91629i −3.44057 − 1.98641i −2.44900 − 0.0487576i 0 0.969574 + 2.65705i 0.743736 + 2.90635i 5.58025 − 0.653796i
275.3 −0.938613 − 1.05783i −0.299892 + 1.70589i −0.238012 + 1.98579i −0.695526 − 0.401562i 2.08603 − 1.28394i 0 2.32403 − 1.61211i −2.82013 − 1.02317i 0.228045 + 1.11266i
275.4 −0.938613 − 1.05783i 0.299892 − 1.70589i −0.238012 + 1.98579i 0.695526 + 0.401562i −2.08603 + 1.28394i 0 2.32403 − 1.61211i −2.82013 − 1.02317i −0.228045 − 1.11266i
275.5 −0.287629 − 1.38466i −1.66143 + 0.489533i −1.83454 + 0.796533i −1.08570 − 0.626827i 1.15571 + 2.15971i 0 1.63059 + 2.31110i 2.52072 − 1.62665i −0.555662 + 1.68361i
275.6 −0.287629 − 1.38466i 1.66143 − 0.489533i −1.83454 + 0.796533i 1.08570 + 0.626827i −1.15571 − 2.15971i 0 1.63059 + 2.31110i 2.52072 − 1.62665i 0.555662 − 1.68361i
275.7 0.287629 + 1.38466i −1.25466 + 1.19408i −1.83454 + 0.796533i 1.08570 + 0.626827i −2.01426 − 1.39383i 0 −1.63059 − 2.31110i 0.148364 − 2.99633i −0.555662 + 1.68361i
275.8 0.287629 + 1.38466i 1.25466 − 1.19408i −1.83454 + 0.796533i −1.08570 − 0.626827i 2.01426 + 1.39383i 0 −1.63059 − 2.31110i 0.148364 − 2.99633i 0.555662 − 1.68361i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 263.12
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{5}^{12} - 18T_{5}^{10} + 288T_{5}^{8} - 616T_{5}^{6} + 1008T_{5}^{4} - 576T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{12} + 42T_{11}^{10} + 1284T_{11}^{8} + 17088T_{11}^{6} + 165888T_{11}^{4} + 737280T_{11}^{2} + 2359296 \) Copy content Toggle raw display
\( T_{13}^{6} - 48T_{13}^{4} + 576T_{13}^{2} - 392 \) Copy content Toggle raw display
\( T_{19}^{12} - 66T_{19}^{10} + 3552T_{19}^{8} - 49992T_{19}^{6} + 545040T_{19}^{4} - 1234944T_{19}^{2} + 2359296 \) Copy content Toggle raw display
\( T_{67}^{6} - 12T_{67}^{5} - 60T_{67}^{4} + 1296T_{67}^{3} + 6096T_{67}^{2} - 150336T_{67} + 645888 \) Copy content Toggle raw display