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Properties

Label 56.4.p.a
Level $56$
Weight $4$
Character orbit 56.p
Analytic conductor $3.304$
Analytic rank $0$
Dimension $44$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [56,4,Mod(37,56)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(56, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 56.p (of order \(6\), 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: \(3.30410696032\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 4 q^{4} - 20 q^{6} - 4 q^{7} - 12 q^{8} + 160 q^{9} - 14 q^{10} - 110 q^{12} + 102 q^{14} - 116 q^{15} - 104 q^{16} - 2 q^{17} - 38 q^{18} + 256 q^{20} - 360 q^{22} + 162 q^{23} + 186 q^{24} + 348 q^{25}+ \cdots - 2670 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −2.82711 + 0.0861459i −0.982205 + 0.567076i 7.98516 − 0.487088i −5.55772 − 3.20875i 2.72796 − 1.68780i 10.9028 + 14.9709i −22.5330 + 2.06494i −12.8568 + 22.2687i 15.9887 + 8.59273i
37.2 −2.72280 + 0.765755i 5.66005 − 3.26783i 6.82724 − 4.16999i −3.76893 − 2.17599i −12.9088 + 13.2318i −3.65934 − 18.1551i −15.3960 + 16.5820i 7.85742 − 13.6095i 11.9283 + 3.03871i
37.3 −2.70712 − 0.819467i −8.54801 + 4.93520i 6.65695 + 4.43678i 3.19579 + 1.84509i 27.1847 − 6.35534i −18.3785 − 2.28733i −14.3853 − 17.4660i 35.2124 − 60.9896i −7.13937 − 7.61370i
37.4 −2.39920 − 1.49795i 6.82792 − 3.94210i 3.51228 + 7.18776i 13.5811 + 7.84104i −22.2866 − 0.770025i −1.13912 + 18.4852i 2.34028 − 22.5061i 17.5803 − 30.4500i −20.8382 − 39.1560i
37.5 −2.15037 − 1.83737i 1.18522 − 0.684286i 1.24817 + 7.90203i −10.2713 − 5.93013i −3.80594 − 0.706211i −12.2158 − 13.9203i 11.8349 − 19.2856i −12.5635 + 21.7606i 11.1912 + 31.6241i
37.6 −2.11891 + 1.87356i −1.06569 + 0.615275i 0.979578 − 7.93980i 15.9258 + 9.19477i 1.10535 − 3.30034i −17.4761 + 6.13062i 12.8000 + 18.6590i −12.7429 + 22.0713i −50.9723 + 10.3550i
37.7 −1.96852 + 2.03099i −5.91297 + 3.41385i −0.249876 − 7.99610i −6.49634 − 3.75066i 4.70627 − 18.7294i 17.5560 − 5.89813i 16.7319 + 15.2330i 9.80880 − 16.9893i 20.4057 − 5.81078i
37.8 −1.48377 − 2.40799i −2.65541 + 1.53310i −3.59684 + 7.14582i 7.57927 + 4.37589i 7.63171 + 4.11942i 17.7736 − 5.20552i 22.5440 − 1.94162i −8.79921 + 15.2407i −0.708800 − 24.7436i
37.9 −0.774634 + 2.72028i 5.91297 − 3.41385i −6.79988 − 4.21445i 6.49634 + 3.75066i 4.70627 + 18.7294i 17.5560 − 5.89813i 16.7319 − 15.2330i 9.80880 − 16.9893i −15.2351 + 14.7665i
37.10 −0.563091 + 2.77181i 1.06569 − 0.615275i −7.36586 − 3.12156i −15.9258 − 9.19477i 1.10535 + 3.30034i −17.4761 + 6.13062i 12.8000 − 18.6590i −12.7429 + 22.0713i 34.4538 − 38.9658i
37.11 −0.181977 − 2.82257i 7.07996 − 4.08762i −7.93377 + 1.02729i −17.0228 − 9.82813i −12.8260 − 19.2398i 15.6931 + 9.83491i 4.34335 + 22.2066i 19.9172 − 34.4977i −24.6428 + 49.8365i
37.12 0.0844787 − 2.82717i −3.95946 + 2.28599i −7.98573 − 0.477670i −0.420105 − 0.242548i 6.12839 + 11.3872i −11.5119 + 14.5078i −2.02508 + 22.5366i −3.04846 + 5.28008i −0.721213 + 1.16722i
37.13 0.698235 + 2.74089i −5.66005 + 3.26783i −7.02494 + 3.82757i 3.76893 + 2.17599i −12.9088 − 13.2318i −3.65934 − 18.1551i −15.3960 − 16.5820i 7.85742 − 13.6095i −3.33255 + 11.8496i
37.14 0.941296 − 2.66720i 4.43871 − 2.56269i −6.22792 − 5.02125i 13.1631 + 7.59971i −2.65707 − 14.2512i 1.45519 − 18.4630i −19.2550 + 11.8846i −0.365251 + 0.632633i 32.6603 − 27.9550i
37.15 1.33895 + 2.49143i 0.982205 − 0.567076i −4.41441 + 6.67181i 5.55772 + 3.20875i 2.72796 + 1.68780i 10.9028 + 14.9709i −22.5330 − 2.06494i −12.8568 + 22.2687i −0.552841 + 18.1430i
37.16 1.83922 − 2.14879i −4.43871 + 2.56269i −1.23457 − 7.90417i −13.1631 − 7.59971i −2.65707 + 14.2512i 1.45519 − 18.4630i −19.2550 − 11.8846i −0.365251 + 0.632633i −40.5399 + 14.3072i
37.17 2.06324 + 1.93470i 8.54801 − 4.93520i 0.513891 + 7.98348i −3.19579 − 1.84509i 27.1847 + 6.35534i −18.3785 − 2.28733i −14.3853 + 17.4660i 35.2124 − 60.9896i −3.02397 − 9.98973i
37.18 2.40616 − 1.48674i 3.95946 − 2.28599i 3.57919 − 7.15468i 0.420105 + 0.242548i 6.12839 − 11.3872i −11.5119 + 14.5078i −2.02508 − 22.5366i −3.04846 + 5.28008i 1.37145 − 0.0409803i
37.19 2.49686 + 1.32879i −6.82792 + 3.94210i 4.46864 + 6.63560i −13.5811 − 7.84104i −22.2866 + 0.770025i −1.13912 + 18.4852i 2.34028 + 22.5061i 17.5803 − 30.4500i −23.4910 − 37.6244i
37.20 2.53540 − 1.25369i −7.07996 + 4.08762i 4.85654 − 6.35720i 17.0228 + 9.82813i −12.8260 + 19.2398i 15.6931 + 9.83491i 4.34335 − 22.2066i 19.9172 − 34.4977i 55.4811 + 3.57700i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.22
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
8.b even 2 1 inner
56.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.4.p.a ✓ 44
4.b odd 2 1 224.4.t.a 44
7.c even 3 1 inner 56.4.p.a ✓ 44
8.b even 2 1 inner 56.4.p.a ✓ 44
8.d odd 2 1 224.4.t.a 44
28.g odd 6 1 224.4.t.a 44
56.k odd 6 1 224.4.t.a 44
56.p even 6 1 inner 56.4.p.a ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.p.a ✓ 44 1.a even 1 1 trivial
56.4.p.a ✓ 44 7.c even 3 1 inner
56.4.p.a ✓ 44 8.b even 2 1 inner
56.4.p.a ✓ 44 56.p even 6 1 inner
224.4.t.a 44 4.b odd 2 1
224.4.t.a 44 8.d odd 2 1
224.4.t.a 44 28.g odd 6 1
224.4.t.a 44 56.k odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(56, [\chi])\).