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Properties

Label 880.2.bl.b
Level $880$
Weight $2$
Character orbit 880.bl
Analytic conductor $7.027$
Analytic rank $0$
Dimension $272$
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(373,880)]
 
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([0, 1, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.373");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.bl (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: \(272\)
Relative dimension: \(136\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 272 q - 16 q^{3} + 16 q^{4} - 4 q^{5} + 280 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 272 q - 16 q^{3} + 16 q^{4} - 4 q^{5} + 280 q^{9} - 4 q^{11} - 16 q^{12} - 8 q^{15} - 40 q^{16} - 4 q^{20} - 8 q^{22} + 24 q^{23} + 40 q^{25} + 72 q^{26} + 8 q^{27} + 16 q^{31} - 4 q^{33} + 32 q^{34} - 72 q^{36} - 24 q^{37} - 16 q^{38} - 32 q^{42} + 40 q^{44} - 36 q^{45} - 8 q^{47} - 48 q^{48} - 40 q^{55} - 56 q^{56} - 48 q^{58} - 8 q^{59} - 20 q^{60} + 64 q^{64} + 12 q^{66} - 16 q^{70} + 56 q^{75} + 40 q^{78} + 48 q^{80} + 192 q^{81} + 24 q^{82} - 88 q^{88} - 24 q^{89} - 24 q^{91} - 32 q^{92} - 40 q^{97} + 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1 −1.41416 + 0.0126155i −1.07050 1.99968 − 0.0356805i −1.17083 + 1.90504i 1.51385 − 0.0135048i 1.86340 − 1.86340i −2.82741 + 0.0756848i −1.85403 1.63170 − 2.70879i
373.2 −1.41310 − 0.0561765i −0.674924 1.99369 + 0.158766i 1.75433 + 1.38649i 0.953733 + 0.0379149i 0.608318 − 0.608318i −2.80836 − 0.336350i −2.54448 −2.40115 − 2.05779i
373.3 −1.40976 − 0.112085i 2.23985 1.97487 + 0.316026i −2.06123 + 0.866789i −3.15766 − 0.251053i −1.42896 + 1.42896i −2.74869 − 0.666876i 2.01692 3.00301 − 0.990936i
373.4 −1.40869 + 0.124829i −2.90556 1.96884 − 0.351692i 2.22179 − 0.252307i 4.09304 − 0.362698i −1.92690 + 1.92690i −2.72958 + 0.741194i 5.44225 −3.09832 + 0.632767i
373.5 −1.40750 + 0.137681i −2.51387 1.96209 − 0.387571i −2.07972 − 0.821436i 3.53826 − 0.346112i 1.84550 − 1.84550i −2.70827 + 0.815646i 3.31953 3.04030 + 0.869829i
373.6 −1.40709 + 0.141779i 1.21446 1.95980 − 0.398991i 2.00886 − 0.982090i −1.70885 + 0.172184i 0.290836 − 0.290836i −2.70104 + 0.839275i −1.52510 −2.68740 + 1.66670i
373.7 −1.39296 − 0.244255i 3.14227 1.88068 + 0.680475i −0.608141 − 2.15178i −4.37706 − 0.767515i 1.49653 − 1.49653i −2.45350 − 1.40724i 6.87388 0.321534 + 3.14589i
373.8 −1.38952 − 0.263134i −1.93078 1.86152 + 0.731258i 0.545557 + 2.16849i 2.68285 + 0.508053i −1.70562 + 1.70562i −2.39420 − 1.50593i 0.727913 −0.187457 − 3.15672i
373.9 −1.38872 + 0.267297i 1.30603 1.85710 − 0.742403i 0.912935 + 2.04121i −1.81372 + 0.349099i −2.09764 + 2.09764i −2.38056 + 1.52739i −1.29427 −1.81342 − 2.59065i
373.10 −1.38654 + 0.278384i 0.462443 1.84500 − 0.771983i 0.0764625 − 2.23476i −0.641197 + 0.128737i −0.0535265 + 0.0535265i −2.34327 + 1.58401i −2.78615 0.516103 + 3.11988i
373.11 −1.37686 − 0.322900i 0.220858 1.79147 + 0.889174i −1.72614 − 1.42143i −0.304090 − 0.0713151i 2.99005 − 2.99005i −2.17949 − 1.80273i −2.95122 1.91766 + 2.51447i
373.12 −1.36747 − 0.360598i −0.414112 1.73994 + 0.986212i −0.822406 − 2.07934i 0.566285 + 0.149328i −2.56460 + 2.56460i −2.02368 − 1.97603i −2.82851 0.374809 + 3.13999i
373.13 −1.36677 + 0.363222i 2.65632 1.73614 − 0.992884i −1.03050 + 1.98446i −3.63059 + 0.964835i 3.60504 − 3.60504i −2.01227 + 1.98765i 4.05605 0.687666 − 3.08660i
373.14 −1.35322 + 0.410831i −2.21300 1.66244 − 1.11189i −0.390688 − 2.20167i 2.99469 − 0.909169i 0.871779 − 0.871779i −1.79285 + 2.18762i 1.89737 1.43320 + 2.81885i
373.15 −1.33887 − 0.455443i 1.62510 1.58514 + 1.21956i 2.17787 − 0.506846i −2.17579 − 0.740138i 1.44642 − 1.44642i −1.56686 − 2.35477i −0.359065 −3.14672 − 0.313293i
373.16 −1.29984 + 0.557156i −2.75660 1.37916 − 1.44842i −1.57568 + 1.58658i 3.58313 − 1.53585i −0.888257 + 0.888257i −0.985681 + 2.65112i 4.59884 1.16415 − 2.94020i
373.17 −1.29600 − 0.566036i −1.46363 1.35921 + 1.46716i 1.00355 − 1.99822i 1.89686 + 0.828469i −1.46033 + 1.46033i −0.931059 − 2.67079i −0.857776 −2.43166 + 2.02164i
373.18 −1.28387 − 0.593029i 0.882124 1.29663 + 1.52274i −1.67383 + 1.48266i −1.13253 − 0.523125i −0.753990 + 0.753990i −0.761678 − 2.72394i −2.22186 3.02824 − 0.910911i
373.19 −1.28184 + 0.597392i 2.27765 1.28625 − 1.53153i −1.60411 − 1.55783i −2.91959 + 1.36065i −1.99404 + 1.99404i −0.733843 + 2.73157i 2.18768 2.98685 + 1.03862i
373.20 −1.27771 − 0.606179i −2.77877 1.26509 + 1.54904i −2.23553 + 0.0492729i 3.55047 + 1.68443i −2.14400 + 2.14400i −0.677428 − 2.74610i 4.72158 2.88622 + 1.29217i
See next 80 embeddings (of 272 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.136
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
80.t odd 4 1 inner
880.bl 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.bl.b yes 272
5.c odd 4 1 880.2.t.b ✓ 272
11.b odd 2 1 inner 880.2.bl.b yes 272
16.e even 4 1 880.2.t.b ✓ 272
55.e even 4 1 880.2.t.b ✓ 272
80.t odd 4 1 inner 880.2.bl.b yes 272
176.l odd 4 1 880.2.t.b ✓ 272
880.bl even 4 1 inner 880.2.bl.b yes 272
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.t.b ✓ 272 5.c odd 4 1
880.2.t.b ✓ 272 16.e even 4 1
880.2.t.b ✓ 272 55.e even 4 1
880.2.t.b ✓ 272 176.l odd 4 1
880.2.bl.b yes 272 1.a even 1 1 trivial
880.2.bl.b yes 272 11.b odd 2 1 inner
880.2.bl.b yes 272 80.t odd 4 1 inner
880.2.bl.b yes 272 880.bl even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{68} + 4 T_{3}^{67} - 129 T_{3}^{66} - 530 T_{3}^{65} + 7853 T_{3}^{64} + 33292 T_{3}^{63} + \cdots - 398131200 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display