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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [100,11,Mod(51,100)] 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("100.51"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 100.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(63.5357252674\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 608 q^{4} - 19584 q^{6} - 597192 q^{9} + 1706016 q^{14} - 4733376 q^{16} - 13030368 q^{21} - 10190784 q^{24} - 9454368 q^{26} - 121656816 q^{29} + 335231168 q^{34} - 276632160 q^{36} + 892843248 q^{41}+ \cdots - 5617046784 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1 −31.9986 − 0.302723i 375.794i 1023.82 + 19.3734i 0 113.762 − 12024.9i − 19635.7i −32754.8 − 929.854i −82172.5 0
51.2 −31.9986 + 0.302723i − 375.794i 1023.82 − 19.3734i 0 113.762 + 12024.9i 19635.7i −32754.8 + 929.854i −82172.5 0
51.3 −26.9619 − 17.2353i 146.443i 429.889 + 929.393i 0 2523.99 − 3948.38i 13979.7i 4427.76 − 32467.5i 37603.5 0
51.4 −26.9619 + 17.2353i − 146.443i 429.889 − 929.393i 0 2523.99 + 3948.38i − 13979.7i 4427.76 + 32467.5i 37603.5 0
51.5 −23.3997 − 21.8278i − 416.738i 71.0958 + 1021.53i 0 −9096.47 + 9751.56i − 9550.48i 20634.1 − 25455.4i −114622. 0
51.6 −23.3997 + 21.8278i 416.738i 71.0958 − 1021.53i 0 −9096.47 − 9751.56i 9550.48i 20634.1 + 25455.4i −114622. 0
51.7 −21.2538 − 23.9223i − 94.4407i −120.549 + 1016.88i 0 −2259.24 + 2007.23i − 20081.9i 26888.2 − 18728.8i 50129.9 0
51.8 −21.2538 + 23.9223i 94.4407i −120.549 − 1016.88i 0 −2259.24 − 2007.23i 20081.9i 26888.2 + 18728.8i 50129.9 0
51.9 −15.1426 − 28.1904i 349.582i −565.401 + 853.755i 0 9854.87 − 5293.59i 15402.9i 32629.4 + 3010.79i −63158.6 0
51.10 −15.1426 + 28.1904i − 349.582i −565.401 − 853.755i 0 9854.87 + 5293.59i − 15402.9i 32629.4 − 3010.79i −63158.6 0
51.11 −4.07116 − 31.7400i − 190.073i −990.851 + 258.437i 0 −6032.92 + 773.819i 14056.7i 12236.7 + 30397.4i 22921.2 0
51.12 −4.07116 + 31.7400i 190.073i −990.851 − 258.437i 0 −6032.92 − 773.819i − 14056.7i 12236.7 − 30397.4i 22921.2 0
51.13 4.07116 − 31.7400i − 190.073i −990.851 − 258.437i 0 −6032.92 − 773.819i 14056.7i −12236.7 + 30397.4i 22921.2 0
51.14 4.07116 + 31.7400i 190.073i −990.851 + 258.437i 0 −6032.92 + 773.819i − 14056.7i −12236.7 − 30397.4i 22921.2 0
51.15 15.1426 − 28.1904i 349.582i −565.401 − 853.755i 0 9854.87 + 5293.59i 15402.9i −32629.4 + 3010.79i −63158.6 0
51.16 15.1426 + 28.1904i − 349.582i −565.401 + 853.755i 0 9854.87 − 5293.59i − 15402.9i −32629.4 − 3010.79i −63158.6 0
51.17 21.2538 − 23.9223i − 94.4407i −120.549 − 1016.88i 0 −2259.24 − 2007.23i − 20081.9i −26888.2 − 18728.8i 50129.9 0
51.18 21.2538 + 23.9223i 94.4407i −120.549 + 1016.88i 0 −2259.24 + 2007.23i 20081.9i −26888.2 + 18728.8i 50129.9 0
51.19 23.3997 − 21.8278i − 416.738i 71.0958 − 1021.53i 0 −9096.47 − 9751.56i − 9550.48i −20634.1 − 25455.4i −114622. 0
51.20 23.3997 + 21.8278i 416.738i 71.0958 + 1021.53i 0 −9096.47 + 9751.56i 9550.48i −20634.1 + 25455.4i −114622. 0
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 51.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.11.b.h 24
4.b odd 2 1 inner 100.11.b.h 24
5.b even 2 1 inner 100.11.b.h 24
5.c odd 4 2 20.11.d.d ✓ 24
20.d odd 2 1 inner 100.11.b.h 24
20.e even 4 2 20.11.d.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.11.d.d ✓ 24 5.c odd 4 2
20.11.d.d ✓ 24 20.e even 4 2
100.11.b.h 24 1.a even 1 1 trivial
100.11.b.h 24 4.b odd 2 1 inner
100.11.b.h 24 5.b even 2 1 inner
100.11.b.h 24 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{11}^{\mathrm{new}}(100, [\chi])\):

\( T_{3}^{12} + 503592 T_{3}^{10} + 93360281616 T_{3}^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
\( T_{13}^{12} - 854958307584 T_{13}^{10} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display