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Properties

Label 880.2.bi.d
Level $880$
Weight $2$
Character orbit 880.bi
Analytic conductor $7.027$
Analytic rank $0$
Dimension $256$
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] = [880,2,Mod(219,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.219");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.bi (of order \(4\), degree \(2\), 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: \(256\)
Relative dimension: \(128\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 256 q - 8 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 256 q - 8 q^{4} + 4 q^{5} + 12 q^{11} - 32 q^{14} + 24 q^{16} - 28 q^{20} - 80 q^{26} - 32 q^{34} + 24 q^{36} - 96 q^{44} + 4 q^{45} + 88 q^{49} - 4 q^{55} + 32 q^{56} + 96 q^{59} + 84 q^{60} - 104 q^{64} - 16 q^{66} - 8 q^{69} + 160 q^{70} - 216 q^{71} - 120 q^{80} - 160 q^{81} - 152 q^{86} - 80 q^{91} + 76 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
219.1 −1.41417 + 0.0106423i −1.85941 + 1.85941i 1.99977 − 0.0301002i −0.310397 + 2.21442i 2.60974 − 2.64931i 3.64924 −2.82771 + 0.0638491i − 3.91479i 0.415389 − 3.13488i
219.2 −1.41417 + 0.0106423i 1.85941 − 1.85941i 1.99977 − 0.0301002i 2.21442 − 0.310397i −2.60974 + 2.64931i 3.64924 −2.82771 + 0.0638491i − 3.91479i −3.12827 + 0.462522i
219.3 −1.41046 − 0.103015i −0.151274 + 0.151274i 1.97878 + 0.290595i −1.59081 + 1.57141i 0.228949 − 0.197782i −1.36640 −2.76104 − 0.613715i 2.95423i 2.40564 − 2.05253i
219.4 −1.41046 − 0.103015i 0.151274 − 0.151274i 1.97878 + 0.290595i 1.57141 − 1.59081i −0.228949 + 0.197782i −1.36640 −2.76104 − 0.613715i 2.95423i −2.38028 + 2.08189i
219.5 −1.39809 + 0.212916i −1.56533 + 1.56533i 1.90933 − 0.595354i −2.06921 − 0.847567i 1.85519 − 2.52176i 3.42027 −2.54267 + 1.23889i − 1.90049i 3.07341 + 0.744410i
219.6 −1.39809 + 0.212916i 1.56533 − 1.56533i 1.90933 − 0.595354i −0.847567 − 2.06921i −1.85519 + 2.52176i 3.42027 −2.54267 + 1.23889i − 1.90049i 1.62555 + 2.71249i
219.7 −1.39090 − 0.255703i −1.13891 + 1.13891i 1.86923 + 0.711318i −1.86700 − 1.23057i 1.87533 − 1.29289i −1.28295 −2.41804 − 1.46734i 0.405790i 2.28216 + 2.18901i
219.8 −1.39090 − 0.255703i 1.13891 − 1.13891i 1.86923 + 0.711318i −1.23057 − 1.86700i −1.87533 + 1.29289i −1.28295 −2.41804 − 1.46734i 0.405790i 1.23421 + 2.91148i
219.9 −1.38290 − 0.295964i −2.08446 + 2.08446i 1.82481 + 0.818575i 1.73936 − 1.40521i 3.49952 − 2.26567i −1.54550 −2.28126 − 1.67208i − 5.68993i −2.82125 + 1.42848i
219.10 −1.38290 − 0.295964i 2.08446 − 2.08446i 1.82481 + 0.818575i −1.40521 + 1.73936i −3.49952 + 2.26567i −1.54550 −2.28126 − 1.67208i − 5.68993i 2.45805 − 1.98947i
219.11 −1.36322 + 0.376332i −0.892427 + 0.892427i 1.71675 − 1.02605i 0.0993035 − 2.23386i 0.880728 − 1.55243i −3.79028 −1.95417 + 2.04480i 1.40715i 0.705301 + 3.08262i
219.12 −1.36322 + 0.376332i 0.892427 − 0.892427i 1.71675 − 1.02605i −2.23386 + 0.0993035i −0.880728 + 1.55243i −3.79028 −1.95417 + 2.04480i 1.40715i 3.00788 − 0.976046i
219.13 −1.31487 + 0.520698i −0.367903 + 0.367903i 1.45775 − 1.36930i 2.17281 − 0.528100i 0.292177 − 0.675310i 1.12175 −1.20375 + 2.55949i 2.72929i −2.58198 + 1.82576i
219.14 −1.31487 + 0.520698i 0.367903 − 0.367903i 1.45775 − 1.36930i −0.528100 + 2.17281i −0.292177 + 0.675310i 1.12175 −1.20375 + 2.55949i 2.72929i −0.436997 − 3.13194i
219.15 −1.30149 − 0.553288i −0.929906 + 0.929906i 1.38774 + 1.44020i 0.790029 − 2.09185i 1.72477 − 0.695756i 4.78468 −1.00929 − 2.64222i 1.27055i −2.18561 + 2.28541i
219.16 −1.30149 − 0.553288i 0.929906 − 0.929906i 1.38774 + 1.44020i −2.09185 + 0.790029i −1.72477 + 0.695756i 4.78468 −1.00929 − 2.64222i 1.27055i 3.15964 + 0.129184i
219.17 −1.28439 + 0.591906i −1.49814 + 1.49814i 1.29929 − 1.52047i 1.65473 + 1.50395i 1.03743 − 2.81094i −3.30239 −0.768819 + 2.72193i − 1.48884i −3.01551 − 0.952214i
219.18 −1.28439 + 0.591906i 1.49814 − 1.49814i 1.29929 − 1.52047i 1.50395 + 1.65473i −1.03743 + 2.81094i −3.30239 −0.768819 + 2.72193i − 1.48884i −2.91110 − 1.23511i
219.19 −1.23763 + 0.684311i −2.17945 + 2.17945i 1.06344 − 1.69384i −0.702012 − 2.12301i 1.20592 − 4.18878i −0.566910 −0.157019 + 2.82407i − 6.50005i 2.32163 + 2.14710i
219.20 −1.23763 + 0.684311i 2.17945 − 2.17945i 1.06344 − 1.69384i −2.12301 − 0.702012i −1.20592 + 4.18878i −0.566910 −0.157019 + 2.82407i − 6.50005i 3.10789 − 0.583973i
See next 80 embeddings (of 256 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 219.128
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
16.f odd 4 1 inner
55.d odd 2 1 inner
80.k odd 4 1 inner
176.i even 4 1 inner
880.bi even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.bi.d ✓ 256
5.b even 2 1 inner 880.2.bi.d ✓ 256
11.b odd 2 1 inner 880.2.bi.d ✓ 256
16.f odd 4 1 inner 880.2.bi.d ✓ 256
55.d odd 2 1 inner 880.2.bi.d ✓ 256
80.k odd 4 1 inner 880.2.bi.d ✓ 256
176.i even 4 1 inner 880.2.bi.d ✓ 256
880.bi even 4 1 inner 880.2.bi.d ✓ 256
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.bi.d ✓ 256 1.a even 1 1 trivial
880.2.bi.d ✓ 256 5.b even 2 1 inner
880.2.bi.d ✓ 256 11.b odd 2 1 inner
880.2.bi.d ✓ 256 16.f odd 4 1 inner
880.2.bi.d ✓ 256 55.d odd 2 1 inner
880.2.bi.d ✓ 256 80.k odd 4 1 inner
880.2.bi.d ✓ 256 176.i even 4 1 inner
880.2.bi.d ✓ 256 880.bi even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\):

\( T_{3}^{128} + 947 T_{3}^{124} + 413101 T_{3}^{120} + 110293763 T_{3}^{116} + 20201548449 T_{3}^{112} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{64} - 235 T_{7}^{62} + 25963 T_{7}^{60} - 1793973 T_{7}^{58} + 87007629 T_{7}^{56} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display