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Properties

Label 110.4.j.a
Level $110$
Weight $4$
Character orbit 110.j
Analytic conductor $6.490$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [110,4,Mod(9,110)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(110, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 6]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("110.9");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 110.j (of order \(10\), degree \(4\), 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: \(6.49021010063\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 72 q^{4} + 24 q^{5} - 24 q^{6} + 194 q^{9} - 24 q^{10} + 78 q^{11} - 12 q^{14} + 144 q^{15} - 288 q^{16} + 48 q^{19} - 96 q^{20} + 448 q^{21} + 96 q^{24} - 560 q^{25} - 152 q^{26} + 764 q^{29} + 420 q^{30}+ \cdots + 8194 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −1.90211 − 0.618034i −4.82478 − 6.64074i 3.23607 + 2.35114i −11.1132 + 1.22313i 5.07307 + 15.6133i −8.39040 + 11.5484i −4.70228 − 6.47214i −12.4774 + 38.4016i 21.8946 + 4.54183i
9.2 −1.90211 − 0.618034i −4.49475 − 6.18649i 3.23607 + 2.35114i 6.00836 − 9.42866i 4.72606 + 14.5453i 18.5677 − 25.5562i −4.70228 − 6.47214i −9.72643 + 29.9349i −17.2558 + 14.2210i
9.3 −1.90211 − 0.618034i −3.68663 − 5.07421i 3.23607 + 2.35114i 9.59889 + 5.73247i 3.87635 + 11.9302i −10.5049 + 14.4588i −4.70228 − 6.47214i −3.81291 + 11.7349i −14.7153 − 16.8363i
9.4 −1.90211 − 0.618034i −1.72273 − 2.37114i 3.23607 + 2.35114i −2.73555 + 10.8405i 1.81139 + 5.57488i 10.7547 − 14.8025i −4.70228 − 6.47214i 5.68897 − 17.5089i 11.9031 − 18.9292i
9.5 −1.90211 − 0.618034i −0.306528 − 0.421899i 3.23607 + 2.35114i −2.18646 − 10.9645i 0.322302 + 0.991945i −15.6145 + 21.4916i −4.70228 − 6.47214i 8.25942 − 25.4199i −2.61751 + 22.2070i
9.6 −1.90211 − 0.618034i 1.23002 + 1.69297i 3.23607 + 2.35114i −11.1788 − 0.186140i −1.29332 − 3.98041i 2.97886 − 4.10005i −4.70228 − 6.47214i 6.99025 − 21.5138i 21.1483 + 7.26293i
9.7 −1.90211 − 0.618034i 2.80775 + 3.86454i 3.23607 + 2.35114i 3.39411 − 10.6527i −2.95225 − 9.08608i 3.93988 − 5.42279i −4.70228 − 6.47214i 1.29226 − 3.97716i −13.0397 + 18.1650i
9.8 −1.90211 − 0.618034i 3.61498 + 4.97560i 3.23607 + 2.35114i 2.97826 + 10.7764i −3.80102 − 11.6983i −16.5460 + 22.7736i −4.70228 − 6.47214i −3.34501 + 10.2949i 0.995167 − 22.3385i
9.9 −1.90211 − 0.618034i 3.85595 + 5.30726i 3.23607 + 2.35114i 10.6149 + 3.51062i −4.05438 − 12.4781i 15.7658 − 21.6998i −4.70228 − 6.47214i −4.95519 + 15.2505i −18.0210 − 13.2380i
9.10 1.90211 + 0.618034i −3.85595 − 5.30726i 3.23607 + 2.35114i −6.52412 + 9.07942i −4.05438 − 12.4781i −15.7658 + 21.6998i 4.70228 + 6.47214i −4.95519 + 15.2505i −18.0210 + 13.2380i
9.11 1.90211 + 0.618034i −3.61498 − 4.97560i 3.23607 + 2.35114i 3.92472 + 10.4688i −3.80102 − 11.6983i 16.5460 − 22.7736i 4.70228 + 6.47214i −3.34501 + 10.2949i 0.995167 + 22.3385i
9.12 1.90211 + 0.618034i −2.80775 − 3.86454i 3.23607 + 2.35114i −9.00739 − 6.62321i −2.95225 − 9.08608i −3.93988 + 5.42279i 4.70228 + 6.47214i 1.29226 − 3.97716i −13.0397 − 18.1650i
9.13 1.90211 + 0.618034i −1.23002 − 1.69297i 3.23607 + 2.35114i 8.93442 − 6.72132i −1.29332 − 3.98041i −2.97886 + 4.10005i 4.70228 + 6.47214i 6.99025 − 21.5138i 21.1483 − 7.26293i
9.14 1.90211 + 0.618034i 0.306528 + 0.421899i 3.23607 + 2.35114i −4.67586 − 10.1556i 0.322302 + 0.991945i 15.6145 − 21.4916i 4.70228 + 6.47214i 8.25942 − 25.4199i −2.61751 − 22.2070i
9.15 1.90211 + 0.618034i 1.72273 + 2.37114i 3.23607 + 2.35114i 8.58500 + 7.16224i 1.81139 + 5.57488i −10.7547 + 14.8025i 4.70228 + 6.47214i 5.68897 − 17.5089i 11.9031 + 18.9292i
9.16 1.90211 + 0.618034i 3.68663 + 5.07421i 3.23607 + 2.35114i −4.39620 + 10.2798i 3.87635 + 11.9302i 10.5049 − 14.4588i 4.70228 + 6.47214i −3.81291 + 11.7349i −14.7153 + 16.8363i
9.17 1.90211 + 0.618034i 4.49475 + 6.18649i 3.23607 + 2.35114i −10.4029 − 4.09632i 4.72606 + 14.5453i −18.5677 + 25.5562i 4.70228 + 6.47214i −9.72643 + 29.9349i −17.2558 − 14.2210i
9.18 1.90211 + 0.618034i 4.82478 + 6.64074i 3.23607 + 2.35114i 9.70973 − 5.54266i 5.07307 + 15.6133i 8.39040 − 11.5484i 4.70228 + 6.47214i −12.4774 + 38.4016i 21.8946 − 4.54183i
49.1 −1.90211 + 0.618034i −4.82478 + 6.64074i 3.23607 − 2.35114i −11.1132 − 1.22313i 5.07307 − 15.6133i −8.39040 − 11.5484i −4.70228 + 6.47214i −12.4774 − 38.4016i 21.8946 − 4.54183i
49.2 −1.90211 + 0.618034i −4.49475 + 6.18649i 3.23607 − 2.35114i 6.00836 + 9.42866i 4.72606 − 14.5453i 18.5677 + 25.5562i −4.70228 + 6.47214i −9.72643 − 29.9349i −17.2558 − 14.2210i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.c even 5 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 110.4.j.a ✓ 72
5.b even 2 1 inner 110.4.j.a ✓ 72
11.c even 5 1 inner 110.4.j.a ✓ 72
55.j even 10 1 inner 110.4.j.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.4.j.a ✓ 72 1.a even 1 1 trivial
110.4.j.a ✓ 72 5.b even 2 1 inner
110.4.j.a ✓ 72 11.c even 5 1 inner
110.4.j.a ✓ 72 55.j even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(110, [\chi])\).