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Properties

Label 810.3.b.c
Level $810$
Weight $3$
Character orbit 810.b
Analytic conductor $22.071$
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] = [810,3,Mod(809,810)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(810, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.809");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 810.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: \(22.0709014132\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 48 q^{4} - 12 q^{10} + 96 q^{16} - 48 q^{25} - 120 q^{34} - 24 q^{40} + 72 q^{49} + 216 q^{55} + 120 q^{61} + 192 q^{64} + 192 q^{70} + 480 q^{79} + 444 q^{85} + 48 q^{91} + 96 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
809.1 −1.41421 0 2.00000 −4.81687 − 1.34080i 0 − 0.364778i −2.82843 0 6.81209 + 1.89618i
809.2 −1.41421 0 2.00000 −4.81687 + 1.34080i 0 0.364778i −2.82843 0 6.81209 − 1.89618i
809.3 −1.41421 0 2.00000 −1.68174 − 4.70869i 0 9.41831i −2.82843 0 2.37833 + 6.65909i
809.4 −1.41421 0 2.00000 −1.68174 + 4.70869i 0 − 9.41831i −2.82843 0 2.37833 − 6.65909i
809.5 −1.41421 0 2.00000 −0.420464 − 4.98229i 0 − 1.96530i −2.82843 0 0.594626 + 7.04602i
809.6 −1.41421 0 2.00000 −0.420464 + 4.98229i 0 1.96530i −2.82843 0 0.594626 − 7.04602i
809.7 −1.41421 0 2.00000 −0.174155 − 4.99697i 0 − 9.47560i −2.82843 0 0.246293 + 7.06678i
809.8 −1.41421 0 2.00000 −0.174155 + 4.99697i 0 9.47560i −2.82843 0 0.246293 − 7.06678i
809.9 −1.41421 0 2.00000 4.21457 − 2.69024i 0 − 7.64672i −2.82843 0 −5.96030 + 3.80458i
809.10 −1.41421 0 2.00000 4.21457 + 2.69024i 0 7.64672i −2.82843 0 −5.96030 − 3.80458i
809.11 −1.41421 0 2.00000 4.99998 − 0.0147805i 0 − 5.91951i −2.82843 0 −7.07104 + 0.0209028i
809.12 −1.41421 0 2.00000 4.99998 + 0.0147805i 0 5.91951i −2.82843 0 −7.07104 − 0.0209028i
809.13 1.41421 0 2.00000 −4.99998 − 0.0147805i 0 5.91951i 2.82843 0 −7.07104 − 0.0209028i
809.14 1.41421 0 2.00000 −4.99998 + 0.0147805i 0 − 5.91951i 2.82843 0 −7.07104 + 0.0209028i
809.15 1.41421 0 2.00000 −4.21457 − 2.69024i 0 7.64672i 2.82843 0 −5.96030 − 3.80458i
809.16 1.41421 0 2.00000 −4.21457 + 2.69024i 0 − 7.64672i 2.82843 0 −5.96030 + 3.80458i
809.17 1.41421 0 2.00000 0.174155 − 4.99697i 0 9.47560i 2.82843 0 0.246293 − 7.06678i
809.18 1.41421 0 2.00000 0.174155 + 4.99697i 0 − 9.47560i 2.82843 0 0.246293 + 7.06678i
809.19 1.41421 0 2.00000 0.420464 − 4.98229i 0 1.96530i 2.82843 0 0.594626 − 7.04602i
809.20 1.41421 0 2.00000 0.420464 + 4.98229i 0 − 1.96530i 2.82843 0 0.594626 + 7.04602i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 809.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.3.b.c ✓ 24
3.b odd 2 1 inner 810.3.b.c ✓ 24
5.b even 2 1 inner 810.3.b.c ✓ 24
9.c even 3 1 810.3.j.g 24
9.c even 3 1 810.3.j.h 24
9.d odd 6 1 810.3.j.g 24
9.d odd 6 1 810.3.j.h 24
15.d odd 2 1 inner 810.3.b.c ✓ 24
45.h odd 6 1 810.3.j.g 24
45.h odd 6 1 810.3.j.h 24
45.j even 6 1 810.3.j.g 24
45.j even 6 1 810.3.j.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
810.3.b.c ✓ 24 1.a even 1 1 trivial
810.3.b.c ✓ 24 3.b odd 2 1 inner
810.3.b.c ✓ 24 5.b even 2 1 inner
810.3.b.c ✓ 24 15.d odd 2 1 inner
810.3.j.g 24 9.c even 3 1
810.3.j.g 24 9.d odd 6 1
810.3.j.g 24 45.h odd 6 1
810.3.j.g 24 45.j even 6 1
810.3.j.h 24 9.c even 3 1
810.3.j.h 24 9.d odd 6 1
810.3.j.h 24 45.h odd 6 1
810.3.j.h 24 45.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 276T_{7}^{10} + 27792T_{7}^{8} + 1217336T_{7}^{6} + 20769252T_{7}^{4} + 65771136T_{7}^{2} + 8386816 \) acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\). Copy content Toggle raw display