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Properties

Label 880.2.t.b
Level $880$
Weight $2$
Character orbit 880.t
Analytic conductor $7.027$
Analytic rank $0$
Dimension $272$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [880,2,Mod(197,880)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("880.197"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.t (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [272] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content 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.

\(\operatorname{Tr}(f)(q) = \) \( 272 q - 16 q^{4} - 4 q^{5} - 280 q^{9} - 4 q^{11} - 8 q^{12} - 8 q^{15} - 40 q^{16} - 4 q^{20} - 12 q^{22} - 24 q^{23} - 40 q^{25} + 72 q^{26} + 16 q^{31} - 4 q^{33} - 32 q^{34} - 72 q^{36} - 24 q^{42}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 −1.41343 − 0.0470882i 1.97860i 1.99557 + 0.133112i −1.31699 + 1.80708i 0.0931686 − 2.79661i −1.68608 − 1.68608i −2.81432 − 0.282112i −0.914843 1.94656 − 2.49217i
197.2 −1.41063 + 0.100546i 3.18943i 1.97978 − 0.283668i 1.45766 − 1.69565i −0.320686 − 4.49912i −1.16971 − 1.16971i −2.76423 + 0.599212i −7.17247 −1.88573 + 2.53851i
197.3 −1.40790 − 0.133454i 1.34191i 1.96438 + 0.375780i −0.819461 − 2.08050i 0.179083 − 1.88928i 0.494450 + 0.494450i −2.71551 − 0.791216i 1.19928 0.876071 + 3.03850i
197.4 −1.40694 + 0.143216i − 1.51143i 1.95898 − 0.402994i 0.315092 − 2.21376i 0.216460 + 2.12649i −0.540984 − 0.540984i −2.69846 + 0.847546i 0.715594 −0.126271 + 3.15976i
197.5 −1.40011 − 0.199242i − 2.95867i 1.92060 + 0.557922i 1.22252 + 1.87228i −0.589492 + 4.14245i −2.14958 − 2.14958i −2.57789 − 1.16382i −5.75372 −1.33862 − 2.86498i
197.6 −1.39893 − 0.207327i − 0.723444i 1.91403 + 0.580073i 0.561818 + 2.16434i −0.149989 + 1.01205i 2.33007 + 2.33007i −2.55734 − 1.20831i 2.47663 −0.337220 − 3.14425i
197.7 −1.39434 − 0.236245i 1.93211i 1.88838 + 0.658812i −2.15477 + 0.597459i 0.456452 − 2.69403i 2.32575 + 2.32575i −2.47740 − 1.36473i −0.733060 3.14563 − 0.324008i
197.8 −1.38316 + 0.294724i − 2.92808i 1.82628 − 0.815301i −1.26567 + 1.84339i 0.862975 + 4.05001i 2.56078 + 2.56078i −2.28575 + 1.66594i −5.57366 1.20733 − 2.92273i
197.9 −1.36620 + 0.365375i 2.54864i 1.73300 − 0.998351i 0.718923 + 2.11734i −0.931208 − 3.48194i −0.0911983 − 0.0911983i −2.00285 + 1.99714i −3.49554 −1.75582 − 2.63004i
197.10 −1.35914 + 0.390807i 0.351635i 1.69454 − 1.06232i 1.79685 + 1.33091i −0.137421 − 0.477922i −1.79868 − 1.79868i −1.88796 + 2.10609i 2.87635 −2.96231 − 1.10668i
197.11 −1.34873 − 0.425368i − 0.164610i 1.63812 + 1.14741i 1.03203 − 1.98366i −0.0700198 + 0.222014i −3.03309 − 3.03309i −1.72131 − 2.24435i 2.97290 −2.23571 + 2.23642i
197.12 −1.32787 + 0.486582i − 0.206834i 1.52648 − 1.29224i −1.72338 + 1.42477i 0.100642 + 0.274649i 0.242373 + 0.242373i −1.39818 + 2.45868i 2.95722 1.59516 − 2.73047i
197.13 −1.32559 + 0.492757i 0.550992i 1.51438 − 1.30639i −0.0313544 − 2.23585i −0.271505 − 0.730389i 2.90951 + 2.90951i −1.36372 + 2.47796i 2.69641 1.14329 + 2.94837i
197.14 −1.31356 − 0.523987i − 3.37787i 1.45088 + 1.37658i −0.219739 − 2.22524i −1.76996 + 4.43703i −0.0174711 − 0.0174711i −1.18450 − 2.56845i −8.41001 −0.877360 + 3.03813i
197.15 −1.31339 − 0.524410i 1.92611i 1.44999 + 1.37751i 2.10484 − 0.754755i 1.01007 − 2.52973i 1.16754 + 1.16754i −1.18202 − 2.56960i −0.709896 −3.16028 − 0.112510i
197.16 −1.30784 + 0.538105i − 2.91705i 1.42089 − 1.40751i 2.17349 − 0.525313i 1.56968 + 3.81504i 1.11629 + 1.11629i −1.10090 + 2.60538i −5.50920 −2.55990 + 1.85659i
197.17 −1.29761 + 0.562331i 2.09978i 1.36757 − 1.45937i 2.23540 − 0.0545997i −1.18077 − 2.72469i 3.59060 + 3.59060i −0.953917 + 2.66271i −1.40907 −2.86997 + 1.32788i
197.18 −1.28720 − 0.585763i − 1.88479i 1.31376 + 1.50799i 1.78899 − 1.34146i −1.10404 + 2.42609i 2.18859 + 2.18859i −0.807749 − 2.71063i −0.552419 −3.08856 + 0.678800i
197.19 −1.27080 − 0.620547i − 1.93723i 1.22984 + 1.57718i −2.22302 + 0.241218i −1.20214 + 2.46182i −3.02388 − 3.02388i −0.584165 − 2.76744i −0.752857 2.97469 + 1.07295i
197.20 −1.24724 − 0.666634i 1.18128i 1.11120 + 1.66290i −1.55569 − 1.60618i 0.787483 − 1.47334i −0.718948 − 0.718948i −0.277382 − 2.81479i 1.60457 0.869577 + 3.04037i
See next 80 embeddings (of 272 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.136
Significant digits:
Format:

Inner twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{136} + 274 T_{3}^{134} + 36587 T_{3}^{132} + 3173080 T_{3}^{130} + 200989453 T_{3}^{128} + \cdots + 15\!\cdots\!00 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display