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Properties

Label 810.5.d.c
Level $810$
Weight $5$
Character orbit 810.d
Analytic conductor $83.730$
Analytic rank $0$
Dimension $32$
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] = [810,5,Mod(161,810)]
 
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, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("810.161");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 810.d (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: \(83.7296700979\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 90)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 256 q^{4} - 104 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 256 q^{4} - 104 q^{7} + 40 q^{13} + 2048 q^{16} - 200 q^{19} + 1344 q^{22} - 4000 q^{25} + 832 q^{28} - 1472 q^{31} - 384 q^{34} - 4136 q^{37} - 272 q^{43} - 2112 q^{46} + 22296 q^{49} - 320 q^{52} - 12344 q^{61} - 16384 q^{64} + 40936 q^{67} + 4800 q^{70} - 41432 q^{73} + 1600 q^{76} - 14048 q^{79} - 17664 q^{82} - 17400 q^{85} - 10752 q^{88} + 69392 q^{91} + 1344 q^{94} + 36664 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1 − 2.82843i 0 −8.00000 11.1803i 0 47.7167 22.6274i 0 31.6228
161.2 2.82843i 0 −8.00000 − 11.1803i 0 47.7167 − 22.6274i 0 31.6228
161.3 − 2.82843i 0 −8.00000 11.1803i 0 −91.1350 22.6274i 0 31.6228
161.4 2.82843i 0 −8.00000 − 11.1803i 0 −91.1350 − 22.6274i 0 31.6228
161.5 − 2.82843i 0 −8.00000 − 11.1803i 0 90.5214 22.6274i 0 −31.6228
161.6 2.82843i 0 −8.00000 11.1803i 0 90.5214 − 22.6274i 0 −31.6228
161.7 − 2.82843i 0 −8.00000 − 11.1803i 0 49.1951 22.6274i 0 −31.6228
161.8 2.82843i 0 −8.00000 11.1803i 0 49.1951 − 22.6274i 0 −31.6228
161.9 − 2.82843i 0 −8.00000 11.1803i 0 −26.3454 22.6274i 0 31.6228
161.10 2.82843i 0 −8.00000 − 11.1803i 0 −26.3454 − 22.6274i 0 31.6228
161.11 − 2.82843i 0 −8.00000 11.1803i 0 21.7826 22.6274i 0 31.6228
161.12 2.82843i 0 −8.00000 − 11.1803i 0 21.7826 − 22.6274i 0 31.6228
161.13 − 2.82843i 0 −8.00000 − 11.1803i 0 21.4683 22.6274i 0 −31.6228
161.14 2.82843i 0 −8.00000 11.1803i 0 21.4683 − 22.6274i 0 −31.6228
161.15 − 2.82843i 0 −8.00000 − 11.1803i 0 −61.8752 22.6274i 0 −31.6228
161.16 2.82843i 0 −8.00000 11.1803i 0 −61.8752 − 22.6274i 0 −31.6228
161.17 − 2.82843i 0 −8.00000 − 11.1803i 0 −19.6152 22.6274i 0 −31.6228
161.18 2.82843i 0 −8.00000 11.1803i 0 −19.6152 − 22.6274i 0 −31.6228
161.19 − 2.82843i 0 −8.00000 − 11.1803i 0 −55.7017 22.6274i 0 −31.6228
161.20 2.82843i 0 −8.00000 11.1803i 0 −55.7017 − 22.6274i 0 −31.6228
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.32
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 810.5.d.c 32
3.b odd 2 1 inner 810.5.d.c 32
9.c even 3 1 90.5.h.a ✓ 32
9.c even 3 1 270.5.h.a 32
9.d odd 6 1 90.5.h.a ✓ 32
9.d odd 6 1 270.5.h.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.5.h.a ✓ 32 9.c even 3 1
90.5.h.a ✓ 32 9.d odd 6 1
270.5.h.a 32 9.c even 3 1
270.5.h.a 32 9.d odd 6 1
810.5.d.c 32 1.a even 1 1 trivial
810.5.d.c 32 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 52 T_{7}^{15} - 23430 T_{7}^{14} - 1195696 T_{7}^{13} + 210696827 T_{7}^{12} + \cdots - 68\!\cdots\!00 \) acting on \(S_{5}^{\mathrm{new}}(810, [\chi])\). Copy content Toggle raw display