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Properties

Label 56.5.k.a
Level $56$
Weight $5$
Character orbit 56.k
Analytic conductor $5.789$
Analytic rank $0$
Dimension $60$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [56,5,Mod(11,56)]
 
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([3, 3, 4]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.11");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 56.k (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: \(5.78871793270\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 4 q^{2} - 2 q^{3} - 4 q^{4} + 28 q^{6} - 4 q^{8} - 704 q^{9} + 66 q^{10} + 94 q^{11} + 46 q^{12} + 6 q^{14} + 120 q^{16} - 2 q^{17} - 730 q^{18} - 2 q^{19} - 1872 q^{20} + 2160 q^{22} + 550 q^{24}+ \cdots + 640 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −3.99270 + 0.241514i 2.90447 − 5.03069i 15.8833 − 1.92859i −35.7453 + 20.6376i −10.3817 + 20.7875i 33.5978 − 35.6678i −62.9517 + 11.5363i 23.6281 + 40.9250i 137.736 − 91.0326i
11.2 −3.91997 − 0.796124i 4.42688 − 7.66758i 14.7324 + 6.24157i 19.7872 − 11.4241i −23.4576 + 26.5324i 24.1747 + 42.6214i −52.7814 − 36.1956i 1.30543 + 2.26108i −86.6602 + 29.0292i
11.3 −3.73587 − 1.42944i −2.68606 + 4.65239i 11.9134 + 10.6804i 15.6318 − 9.02500i 16.6851 − 13.5412i −44.4242 − 20.6759i −29.2399 − 56.9301i 26.0702 + 45.1548i −71.2989 + 11.3715i
11.4 −3.72154 + 1.46634i −5.61505 + 9.72556i 11.6997 − 10.9141i 29.2132 − 16.8663i 6.63571 − 44.4276i 48.9331 + 2.56001i −27.5373 + 57.7728i −22.5576 − 39.0709i −83.9866 + 105.605i
11.5 −3.66880 + 1.59370i −4.03038 + 6.98083i 10.9202 − 11.6939i −15.7223 + 9.07726i 3.66135 − 32.0345i −48.8060 + 4.35649i −21.4276 + 60.3064i 8.01201 + 13.8772i 43.2155 − 58.3593i
11.6 −3.49436 + 1.94665i 6.97904 − 12.0881i 8.42110 − 13.6046i 12.6763 − 7.31864i −0.856068 + 55.8258i −37.6711 − 31.3351i −2.94298 + 63.9323i −56.9141 − 98.5781i −30.0486 + 50.2502i
11.7 −3.42567 − 2.06513i −8.62600 + 14.9407i 7.47046 + 14.1489i −27.4474 + 15.8467i 60.4043 − 33.3680i 24.0616 + 42.6853i 3.62804 − 63.8971i −108.316 − 187.609i 126.751 + 2.39667i
11.8 −2.66196 − 2.98562i 6.96678 − 12.0668i −1.82790 + 15.8952i −33.4946 + 19.3381i −54.5723 + 11.3212i −35.7007 + 33.5627i 52.3230 − 36.8552i −56.5720 − 97.9856i 146.898 + 48.5249i
11.9 −2.66151 − 2.98603i −0.962732 + 1.66750i −1.83276 + 15.8947i −3.89042 + 2.24614i 7.54153 − 1.56332i 31.0044 − 37.9437i 52.3399 − 36.8312i 38.6463 + 66.9373i 17.0614 + 5.63881i
11.10 −2.31118 + 3.26473i 2.16063 − 3.74231i −5.31692 − 15.0907i −4.23878 + 2.44726i 7.22405 + 15.7030i 7.11583 + 48.4806i 61.5555 + 17.5191i 31.1634 + 53.9766i 1.80693 − 19.4945i
11.11 −1.61448 + 3.65971i −5.08078 + 8.80018i −10.7869 − 11.8170i −25.7945 + 14.8925i −24.0033 − 32.8019i 24.6862 − 42.3272i 60.6622 − 20.3986i −11.1287 − 19.2755i −12.8574 − 118.444i
11.12 −1.25522 − 3.79795i −0.962732 + 1.66750i −12.8488 + 9.53456i 3.89042 − 2.24614i 7.54153 + 1.56332i −31.0044 + 37.9437i 52.3399 + 36.8312i 38.6463 + 66.9373i −13.4141 − 11.9562i
11.13 −1.25464 − 3.79814i 6.96678 − 12.0668i −12.8517 + 9.53063i 33.4946 − 19.3381i −54.5723 − 11.3212i 35.7007 − 33.5627i 52.3230 + 36.8552i −56.5720 − 97.9856i −115.473 − 102.955i
11.14 −0.907680 + 3.89565i −0.285840 + 0.495090i −14.3522 − 7.07202i 40.2341 − 23.2292i −1.66925 − 1.56292i −22.8899 − 43.3249i 40.5774 − 49.4922i 40.3366 + 69.8650i 53.9731 + 177.823i
11.15 −0.229998 + 3.99338i 7.74459 − 13.4140i −15.8942 − 1.83694i −13.5276 + 7.81018i 51.7861 + 34.0123i 43.5544 − 22.4503i 10.9912 − 63.0491i −79.4574 − 137.624i −28.0777 − 55.8173i
11.16 −0.0756202 − 3.99929i −8.62600 + 14.9407i −15.9886 + 0.604854i 27.4474 − 15.8467i 60.4043 + 33.3680i −24.0616 − 42.6853i 3.62804 + 63.8971i −108.316 − 187.609i −65.4512 − 108.571i
11.17 0.630001 − 3.95008i −2.68606 + 4.65239i −15.2062 − 4.97710i −15.6318 + 9.02500i 16.6851 + 13.5412i 44.4242 + 20.6759i −29.2399 + 56.9301i 26.0702 + 45.1548i 25.8014 + 67.4324i
11.18 0.635272 + 3.94923i −6.43733 + 11.1498i −15.1929 + 5.01767i 8.48321 − 4.89778i −48.1225 − 18.3394i −4.25339 + 48.8150i −29.4675 − 56.8125i −42.3785 − 73.4017i 24.7316 + 30.3907i
11.19 1.27052 − 3.79286i 4.42688 − 7.66758i −12.7715 − 9.63782i −19.7872 + 11.4241i −23.4576 − 26.5324i −24.1747 − 42.6214i −52.7814 + 36.1956i 1.30543 + 2.26108i 18.1900 + 89.5645i
11.20 1.42092 + 3.73911i 2.04179 − 3.53649i −11.9620 + 10.6260i −20.9617 + 12.1023i 16.1246 + 2.60942i −48.9779 − 1.47008i −56.7288 − 29.6284i 32.1621 + 55.7065i −75.0368 − 61.1819i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.30
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.d odd 2 1 inner
56.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 56.5.k.a ✓ 60
4.b odd 2 1 224.5.o.a 60
7.c even 3 1 inner 56.5.k.a ✓ 60
8.b even 2 1 224.5.o.a 60
8.d odd 2 1 inner 56.5.k.a ✓ 60
28.g odd 6 1 224.5.o.a 60
56.k odd 6 1 inner 56.5.k.a ✓ 60
56.p even 6 1 224.5.o.a 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.5.k.a ✓ 60 1.a even 1 1 trivial
56.5.k.a ✓ 60 7.c even 3 1 inner
56.5.k.a ✓ 60 8.d odd 2 1 inner
56.5.k.a ✓ 60 56.k odd 6 1 inner
224.5.o.a 60 4.b odd 2 1
224.5.o.a 60 8.b even 2 1
224.5.o.a 60 28.g odd 6 1
224.5.o.a 60 56.p even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(56, [\chi])\).