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Properties

Label 588.2.n.f
Level $588$
Weight $2$
Character orbit 588.n
Analytic conductor $4.695$
Analytic rank $0$
Dimension $24$
CM no
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: no (minimal twist has level 84)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{4} - 12 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{4} - 12 q^{6} + 4 q^{9} - 4 q^{10} + 6 q^{12} - 4 q^{16} + 8 q^{18} - 32 q^{22} - 2 q^{24} + 12 q^{25} - 20 q^{30} + 16 q^{33} + 64 q^{34} - 40 q^{36} + 16 q^{37} - 20 q^{40} - 24 q^{45} + 92 q^{48} + 28 q^{52} - 10 q^{54} + 32 q^{57} + 32 q^{58} - 28 q^{60} + 16 q^{61} + 40 q^{64} + 12 q^{66} - 48 q^{69} + 32 q^{72} - 24 q^{73} - 120 q^{76} + 40 q^{78} - 28 q^{81} - 8 q^{82} + 80 q^{85} + 56 q^{88} - 160 q^{90} - 24 q^{93} + 34 q^{96} - 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 −1.41332 − 0.0503882i 1.22461 + 1.22488i 1.99492 + 0.142429i 3.08528 − 1.78129i −1.66903 − 1.79285i 0 −2.81228 − 0.301817i −0.000681912 3.00000i −4.45023 + 2.36206i
263.2 −1.36159 + 0.382199i 1.03615 − 1.38794i 1.70785 − 1.04079i −1.97228 + 1.13870i −0.880346 + 2.28582i 0 −1.92760 + 2.06987i −0.852770 − 2.87624i 2.25023 − 2.30424i
263.3 −1.04635 − 0.951394i −0.814407 − 1.52864i 0.189699 + 1.99098i −0.301907 + 0.174306i −0.602184 + 2.37432i 0 1.69572 − 2.26374i −1.67348 + 2.48987i 0.481734 + 0.104847i
263.4 −1.01179 + 0.988071i −1.03615 + 1.38794i 0.0474302 − 1.99944i −1.97228 + 1.13870i −0.323018 − 2.42810i 0 1.92760 + 2.06987i −0.852770 − 2.87624i 0.870418 − 3.10088i
263.5 −0.663020 + 1.24916i −1.22461 − 1.22488i −1.12081 − 1.65644i 3.08528 − 1.78129i 2.34202 − 0.717607i 0 2.81228 − 0.301817i −0.000681912 3.00000i 0.179512 + 5.03504i
263.6 −0.300756 − 1.38186i 1.73104 − 0.0590228i −1.81909 + 0.831208i 0.301907 − 0.174306i −0.602184 − 2.37432i 0 1.69572 + 2.26374i 2.99303 − 0.204342i −0.331667 − 0.364770i
263.7 0.300756 + 1.38186i 0.814407 + 1.52864i −1.81909 + 0.831208i −0.301907 + 0.174306i −1.86743 + 1.58515i 0 −1.69572 − 2.26374i −1.67348 + 2.48987i −0.331667 − 0.364770i
263.8 0.663020 − 1.24916i −1.67308 − 0.448098i −1.12081 − 1.65644i −3.08528 + 1.78129i −1.66903 + 1.79285i 0 −2.81228 + 0.301817i 2.59842 + 1.49941i 0.179512 + 5.03504i
263.9 1.01179 − 0.988071i 0.683917 − 1.59131i 0.0474302 − 1.99944i 1.97228 − 1.13870i −0.880346 − 2.28582i 0 −1.92760 − 2.06987i −2.06452 − 2.17664i 0.870418 − 3.10088i
263.10 1.04635 + 0.951394i −1.73104 + 0.0590228i 0.189699 + 1.99098i 0.301907 − 0.174306i −1.86743 − 1.58515i 0 −1.69572 + 2.26374i 2.99303 − 0.204342i 0.481734 + 0.104847i
263.11 1.36159 − 0.382199i −0.683917 + 1.59131i 1.70785 − 1.04079i 1.97228 − 1.13870i −0.323018 + 2.42810i 0 1.92760 − 2.06987i −2.06452 − 2.17664i 2.25023 − 2.30424i
263.12 1.41332 + 0.0503882i 1.67308 + 0.448098i 1.99492 + 0.142429i −3.08528 + 1.78129i 2.34202 + 0.717607i 0 2.81228 + 0.301817i 2.59842 + 1.49941i −4.45023 + 2.36206i
275.1 −1.41332 + 0.0503882i 1.22461 − 1.22488i 1.99492 − 0.142429i 3.08528 + 1.78129i −1.66903 + 1.79285i 0 −2.81228 + 0.301817i −0.000681912 − 3.00000i −4.45023 − 2.36206i
275.2 −1.36159 − 0.382199i 1.03615 + 1.38794i 1.70785 + 1.04079i −1.97228 − 1.13870i −0.880346 − 2.28582i 0 −1.92760 − 2.06987i −0.852770 + 2.87624i 2.25023 + 2.30424i
275.3 −1.04635 + 0.951394i −0.814407 + 1.52864i 0.189699 − 1.99098i −0.301907 − 0.174306i −0.602184 − 2.37432i 0 1.69572 + 2.26374i −1.67348 − 2.48987i 0.481734 − 0.104847i
275.4 −1.01179 − 0.988071i −1.03615 − 1.38794i 0.0474302 + 1.99944i −1.97228 − 1.13870i −0.323018 + 2.42810i 0 1.92760 − 2.06987i −0.852770 + 2.87624i 0.870418 + 3.10088i
275.5 −0.663020 − 1.24916i −1.22461 + 1.22488i −1.12081 + 1.65644i 3.08528 + 1.78129i 2.34202 + 0.717607i 0 2.81228 + 0.301817i −0.000681912 − 3.00000i 0.179512 − 5.03504i
275.6 −0.300756 + 1.38186i 1.73104 + 0.0590228i −1.81909 − 0.831208i 0.301907 + 0.174306i −0.602184 + 2.37432i 0 1.69572 − 2.26374i 2.99303 + 0.204342i −0.331667 + 0.364770i
275.7 0.300756 − 1.38186i 0.814407 − 1.52864i −1.81909 − 0.831208i −0.301907 − 0.174306i −1.86743 − 1.58515i 0 −1.69572 + 2.26374i −1.67348 − 2.48987i −0.331667 + 0.364770i
275.8 0.663020 + 1.24916i −1.67308 + 0.448098i −1.12081 + 1.65644i −3.08528 − 1.78129i −1.66903 − 1.79285i 0 −2.81228 − 0.301817i 2.59842 − 1.49941i 0.179512 − 5.03504i
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
4.b odd 2 1 inner
7.c even 3 1 inner
12.b even 2 1 inner
21.h odd 6 1 inner
28.g 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.f 24
3.b odd 2 1 inner 588.2.n.f 24
4.b odd 2 1 inner 588.2.n.f 24
7.b odd 2 1 588.2.n.g 24
7.c even 3 1 84.2.e.a ✓ 12
7.c even 3 1 inner 588.2.n.f 24
7.d odd 6 1 588.2.e.c 12
7.d odd 6 1 588.2.n.g 24
12.b even 2 1 inner 588.2.n.f 24
21.c even 2 1 588.2.n.g 24
21.g even 6 1 588.2.e.c 12
21.g even 6 1 588.2.n.g 24
21.h odd 6 1 84.2.e.a ✓ 12
21.h odd 6 1 inner 588.2.n.f 24
28.d even 2 1 588.2.n.g 24
28.f even 6 1 588.2.e.c 12
28.f even 6 1 588.2.n.g 24
28.g odd 6 1 84.2.e.a ✓ 12
28.g odd 6 1 inner 588.2.n.f 24
56.k odd 6 1 1344.2.h.h 12
56.p even 6 1 1344.2.h.h 12
84.h odd 2 1 588.2.n.g 24
84.j odd 6 1 588.2.e.c 12
84.j odd 6 1 588.2.n.g 24
84.n even 6 1 84.2.e.a ✓ 12
84.n even 6 1 inner 588.2.n.f 24
168.s odd 6 1 1344.2.h.h 12
168.v even 6 1 1344.2.h.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.e.a ✓ 12 7.c even 3 1
84.2.e.a ✓ 12 21.h odd 6 1
84.2.e.a ✓ 12 28.g odd 6 1
84.2.e.a ✓ 12 84.n even 6 1
588.2.e.c 12 7.d odd 6 1
588.2.e.c 12 21.g even 6 1
588.2.e.c 12 28.f even 6 1
588.2.e.c 12 84.j odd 6 1
588.2.n.f 24 1.a even 1 1 trivial
588.2.n.f 24 3.b odd 2 1 inner
588.2.n.f 24 4.b odd 2 1 inner
588.2.n.f 24 7.c even 3 1 inner
588.2.n.f 24 12.b even 2 1 inner
588.2.n.f 24 21.h odd 6 1 inner
588.2.n.f 24 28.g odd 6 1 inner
588.2.n.f 24 84.n even 6 1 inner
588.2.n.g 24 7.b odd 2 1
588.2.n.g 24 7.d odd 6 1
588.2.n.g 24 21.c even 2 1
588.2.n.g 24 21.g even 6 1
588.2.n.g 24 28.d even 2 1
588.2.n.g 24 28.f even 6 1
588.2.n.g 24 84.h odd 2 1
588.2.n.g 24 84.j odd 6 1
1344.2.h.h 12 56.k odd 6 1
1344.2.h.h 12 56.p even 6 1
1344.2.h.h 12 168.s odd 6 1
1344.2.h.h 12 168.v even 6 1

Hecke kernels

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

\( T_{5}^{12} - 18T_{5}^{10} + 256T_{5}^{8} - 1208T_{5}^{6} + 4480T_{5}^{4} - 544T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{12} + 34T_{11}^{10} + 868T_{11}^{8} + 9728T_{11}^{6} + 81856T_{11}^{4} + 9216T_{11}^{2} + 1024 \) Copy content Toggle raw display
\( T_{13}^{3} - 10T_{13} - 4 \) Copy content Toggle raw display
\( T_{19}^{12} - 64T_{19}^{10} + 2972T_{19}^{8} - 65664T_{19}^{6} + 1062672T_{19}^{4} - 3524864T_{19}^{2} + 9834496 \) Copy content Toggle raw display
\( T_{67}^{12} - 72T_{67}^{10} + 4528T_{67}^{8} - 46720T_{67}^{6} + 411904T_{67}^{4} - 167936T_{67}^{2} + 65536 \) Copy content Toggle raw display