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Properties

Label 880.2.cd.e
Level $880$
Weight $2$
Character orbit 880.cd
Analytic conductor $7.027$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [880,2,Mod(49,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.cd (of order \(10\), degree \(4\), 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: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 440)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 18 q^{9} - 2 q^{11} + 12 q^{15} + 16 q^{19} + 32 q^{21} + 12 q^{29} - 20 q^{31} + 18 q^{35} + 44 q^{39} + 2 q^{41} - 20 q^{45} - 24 q^{49} - 84 q^{51} + 38 q^{55} - 32 q^{59} - 12 q^{61} + 12 q^{69} + 40 q^{71} + 18 q^{75} - 44 q^{79} - 2 q^{81} - 40 q^{85} + 4 q^{89} + 2 q^{91} + 28 q^{95} + 90 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −1.97047 + 2.71211i 0 −1.53430 − 1.62663i 0 −2.38609 − 3.28417i 0 −2.54578 − 7.83509i 0
49.2 0 −1.51545 + 2.08584i 0 2.17840 − 0.504557i 0 2.00676 + 2.76207i 0 −1.12708 − 3.46880i 0
49.3 0 −1.35923 + 1.87082i 0 0.432742 − 2.19379i 0 0.618827 + 0.851742i 0 −0.725409 − 2.23258i 0
49.4 0 −1.28339 + 1.76644i 0 −1.10872 + 1.94184i 0 −2.55269 − 3.51348i 0 −0.546162 − 1.68091i 0
49.5 0 −1.27634 + 1.75674i 0 −2.09377 + 0.784938i 0 2.56608 + 3.53191i 0 −0.530019 − 1.63123i 0
49.6 0 −0.824263 + 1.13450i 0 −0.443981 + 2.19155i 0 0.697744 + 0.960362i 0 0.319368 + 0.982913i 0
49.7 0 −0.662675 + 0.912094i 0 2.15810 + 0.585337i 0 −1.59515 − 2.19553i 0 0.534274 + 1.64433i 0
49.8 0 −0.126982 + 0.174776i 0 −2.23589 − 0.0280054i 0 −0.564009 − 0.776292i 0 0.912629 + 2.80878i 0
49.9 0 −0.00529914 + 0.00729365i 0 1.69294 − 1.46081i 0 −1.52189 − 2.09471i 0 0.927026 + 2.85309i 0
49.10 0 0.00529914 − 0.00729365i 0 −0.510973 − 2.17690i 0 1.52189 + 2.09471i 0 0.927026 + 2.85309i 0
49.11 0 0.126982 − 0.174776i 0 1.82534 + 1.29157i 0 0.564009 + 0.776292i 0 0.912629 + 2.80878i 0
49.12 0 0.662675 − 0.912094i 0 −2.08999 − 0.794950i 0 1.59515 + 2.19553i 0 0.534274 + 1.64433i 0
49.13 0 0.824263 − 1.13450i 0 −0.928972 + 2.03396i 0 −0.697744 − 0.960362i 0 0.319368 + 0.982913i 0
49.14 0 1.27634 − 1.75674i 0 1.23252 + 1.86572i 0 −2.56608 − 3.53191i 0 −0.530019 − 1.63123i 0
49.15 0 1.28339 − 1.76644i 0 −0.244412 + 2.22267i 0 2.55269 + 3.51348i 0 −0.546162 − 1.68091i 0
49.16 0 1.35923 − 1.87082i 0 0.939384 − 2.02918i 0 −0.618827 − 0.851742i 0 −0.725409 − 2.23258i 0
49.17 0 1.51545 − 2.08584i 0 −1.46579 − 1.68863i 0 −2.00676 − 2.76207i 0 −1.12708 − 3.46880i 0
49.18 0 1.97047 − 2.71211i 0 2.19738 − 0.414130i 0 2.38609 + 3.28417i 0 −2.54578 − 7.83509i 0
289.1 0 −3.04168 + 0.988301i 0 2.10808 + 0.745657i 0 2.77138 + 0.900475i 0 5.84802 − 4.24883i 0
289.2 0 −2.67406 + 0.868854i 0 −2.19265 + 0.438479i 0 −3.10880 − 1.01011i 0 3.96863 − 2.88338i 0
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.cd.e 72
4.b odd 2 1 440.2.bn.a ✓ 72
5.b even 2 1 inner 880.2.cd.e 72
11.c even 5 1 inner 880.2.cd.e 72
20.d odd 2 1 440.2.bn.a ✓ 72
44.h odd 10 1 440.2.bn.a ✓ 72
55.j even 10 1 inner 880.2.cd.e 72
220.n odd 10 1 440.2.bn.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.bn.a ✓ 72 4.b odd 2 1
440.2.bn.a ✓ 72 20.d odd 2 1
440.2.bn.a ✓ 72 44.h odd 10 1
440.2.bn.a ✓ 72 220.n odd 10 1
880.2.cd.e 72 1.a even 1 1 trivial
880.2.cd.e 72 5.b even 2 1 inner
880.2.cd.e 72 11.c even 5 1 inner
880.2.cd.e 72 55.j even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{72} - 36 T_{3}^{70} + 770 T_{3}^{68} - 12856 T_{3}^{66} + 185757 T_{3}^{64} - 2263828 T_{3}^{62} + \cdots + 256 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display