↑

Properties

Label 110.4.k.a
Level $110$
Weight $4$
Character orbit 110.k
Analytic conductor $6.490$
Analytic rank $0$
Dimension $144$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [110,4,Mod(7,110)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(110, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([5, 14])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("110.7"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 110 = 2 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 110.k (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.49021010063\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 8 q^{3} + 32 q^{5} - 40 q^{7} - 44 q^{11} + 128 q^{12} - 256 q^{15} + 576 q^{16} - 640 q^{17} - 64 q^{20} + 312 q^{22} - 984 q^{23} - 520 q^{25} + 304 q^{26} - 364 q^{27} + 240 q^{28} - 432 q^{31}+ \cdots + 7044 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −1.97538 + 0.312869i −3.48140 − 6.83262i 3.80423 − 1.23607i 6.72660 + 8.93045i 9.01478 + 12.4078i 14.5104 + 7.39343i −7.12805 + 3.63192i −18.6944 + 25.7307i −16.0816 − 15.5365i
7.2 −1.97538 + 0.312869i −2.81267 − 5.52017i 3.80423 − 1.23607i −9.79201 − 5.39597i 7.28317 + 10.0244i −13.7656 − 7.01394i −7.12805 + 3.63192i −6.69100 + 9.20938i 21.0311 + 7.59546i
7.3 −1.97538 + 0.312869i −2.35230 − 4.61664i 3.80423 − 1.23607i 3.06446 − 10.7522i 6.09107 + 8.38364i 26.5017 + 13.5033i −7.12805 + 3.63192i 0.0901319 − 0.124056i −2.68945 + 22.1984i
7.4 −1.97538 + 0.312869i −0.941282 − 1.84737i 3.80423 − 1.23607i 2.15025 + 10.9716i 2.43737 + 3.35475i −9.71350 − 4.94928i −7.12805 + 3.63192i 13.3434 − 18.3657i −7.68023 − 21.0003i
7.5 −1.97538 + 0.312869i 0.305523 + 0.599623i 3.80423 − 1.23607i −11.0972 + 1.36087i −0.791127 − 1.08889i 6.44885 + 3.28585i −7.12805 + 3.63192i 15.6040 − 21.4771i 21.4954 − 6.16021i
7.6 −1.97538 + 0.312869i 0.636330 + 1.24887i 3.80423 − 1.23607i 9.71128 − 5.53994i −1.64772 − 2.26790i −23.2491 − 11.8460i −7.12805 + 3.63192i 14.7154 − 20.2541i −17.4502 + 13.9818i
7.7 −1.97538 + 0.312869i 2.35028 + 4.61268i 3.80423 − 1.23607i 11.1644 + 0.595895i −6.08585 − 8.37645i 11.8045 + 6.01470i −7.12805 + 3.63192i 0.117179 − 0.161283i −22.2404 + 2.31589i
7.8 −1.97538 + 0.312869i 3.43959 + 6.75057i 3.80423 − 1.23607i −6.05483 + 9.39889i −8.90653 − 12.2588i 16.3583 + 8.33497i −7.12805 + 3.63192i −17.8693 + 24.5949i 9.01995 − 20.4607i
7.9 −1.97538 + 0.312869i 4.14000 + 8.12521i 3.80423 − 1.23607i −7.11680 − 8.62271i −10.7202 − 14.7551i −29.7959 − 15.1817i −7.12805 + 3.63192i −33.0092 + 45.4333i 16.7561 + 14.8065i
7.10 1.97538 − 0.312869i −4.48373 − 8.79981i 3.80423 − 1.23607i −0.827166 − 11.1497i −11.6102 − 15.9801i 0.659738 + 0.336154i 7.12805 − 3.63192i −41.4626 + 57.0684i −5.12236 − 21.7661i
7.11 1.97538 − 0.312869i −2.36332 − 4.63829i 3.80423 − 1.23607i 11.0203 + 1.88505i −6.11963 − 8.42295i 7.23245 + 3.68512i 7.12805 − 3.63192i −0.0581894 + 0.0800909i 22.3590 + 0.275778i
7.12 1.97538 − 0.312869i −1.67333 − 3.28409i 3.80423 − 1.23607i −10.5883 + 3.59002i −4.33294 − 5.96378i −23.0882 − 11.7640i 7.12805 − 3.63192i 7.88499 − 10.8528i −19.7926 + 10.4044i
7.13 1.97538 − 0.312869i −0.134038 − 0.263065i 3.80423 − 1.23607i 0.491989 + 11.1695i −0.347081 − 0.477716i 9.77114 + 4.97864i 7.12805 − 3.63192i 15.8190 − 21.7729i 4.46646 + 21.9101i
7.14 1.97538 − 0.312869i 0.0345940 + 0.0678945i 3.80423 − 1.23607i −9.05578 − 6.55690i 0.0895782 + 0.123294i 26.9536 + 13.7335i 7.12805 − 3.63192i 15.8668 − 21.8388i −19.9400 − 10.1191i
7.15 1.97538 − 0.312869i 0.114151 + 0.224033i 3.80423 − 1.23607i 1.74009 − 11.0441i 0.295583 + 0.406836i −27.7439 − 14.1362i 7.12805 − 3.63192i 15.8330 − 21.7923i −0.0180186 − 22.3607i
7.16 1.97538 − 0.312869i 2.68986 + 5.27915i 3.80423 − 1.23607i 6.17250 − 9.32203i 6.96517 + 9.58673i 12.2499 + 6.24166i 7.12805 − 3.63192i −4.76385 + 6.55687i 9.27645 − 20.3457i
7.17 1.97538 − 0.312869i 3.05899 + 6.00362i 3.80423 − 1.23607i 10.3687 + 4.18211i 7.92101 + 10.9023i −6.69380 − 3.41066i 7.12805 − 3.63192i −10.8157 + 14.8866i 21.7905 + 5.01720i
7.18 1.97538 − 0.312869i 4.04090 + 7.93071i 3.80423 − 1.23607i −8.78081 + 6.92079i 10.4636 + 14.4019i 0.860054 + 0.438219i 7.12805 − 3.63192i −30.6971 + 42.2509i −15.1801 + 16.4184i
13.1 −1.78201 + 0.907981i −9.15200 − 1.44953i 2.35114 − 3.23607i 10.9659 + 2.17925i 17.6251 − 5.72675i −4.53048 − 28.6043i −1.25148 + 7.90151i 55.9793 + 18.1888i −21.5201 + 6.07338i
13.2 −1.78201 + 0.907981i −6.39172 − 1.01235i 2.35114 − 3.23607i −3.68265 + 10.5564i 12.3093 − 3.99954i 3.66890 + 23.1645i −1.25148 + 7.90151i 14.1507 + 4.59786i −3.02250 − 22.1555i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.d odd 10 1 inner
55.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 110.4.k.a ✓ 144
5.c odd 4 1 inner 110.4.k.a ✓ 144
11.d odd 10 1 inner 110.4.k.a ✓ 144
55.l even 20 1 inner 110.4.k.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.4.k.a ✓ 144 1.a even 1 1 trivial
110.4.k.a ✓ 144 5.c odd 4 1 inner
110.4.k.a ✓ 144 11.d odd 10 1 inner
110.4.k.a ✓ 144 55.l even 20 1 inner

Hecke kernels

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