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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [138,4,Mod(13,138)] 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("138.13"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(138, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 14])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 138.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [30,-6,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.14226358079\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(3\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 6 q^{2} - 9 q^{3} - 12 q^{4} - 20 q^{5} - 18 q^{6} - 10 q^{7} - 24 q^{8} - 27 q^{9} + 4 q^{10} + 35 q^{11} - 36 q^{12} + 13 q^{13} - 64 q^{14} + 72 q^{15} - 48 q^{16} + 45 q^{17} - 54 q^{18} - 7 q^{19}+ \cdots - 675 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −1.30972 − 1.51150i 2.52376 + 1.62192i −0.569259 + 3.95929i −2.73603 + 5.99106i −0.853889 − 5.93893i −4.46365 − 1.31065i 6.73003 − 4.32513i 3.73874 + 8.18669i 12.6389 − 3.71112i
13.2 −1.30972 − 1.51150i 2.52376 + 1.62192i −0.569259 + 3.95929i 0.452038 − 0.989824i −0.853889 − 5.93893i 6.64594 + 1.95142i 6.73003 − 4.32513i 3.73874 + 8.18669i −2.08816 + 0.613140i
13.3 −1.30972 − 1.51150i 2.52376 + 1.62192i −0.569259 + 3.95929i 7.75778 − 16.9872i −0.853889 − 5.93893i −16.8919 − 4.95990i 6.73003 − 4.32513i 3.73874 + 8.18669i −35.8366 + 10.5226i
25.1 1.68251 + 1.08128i −0.426945 − 2.96946i 1.66166 + 3.63853i −10.4597 − 3.07126i 2.49249 − 5.45779i 11.4417 − 13.2044i −1.13852 + 7.91857i −8.63544 + 2.53559i −14.2777 − 16.4773i
25.2 1.68251 + 1.08128i −0.426945 − 2.96946i 1.66166 + 3.63853i −10.3127 − 3.02807i 2.49249 − 5.45779i −14.8488 + 17.1364i −1.13852 + 7.91857i −8.63544 + 2.53559i −14.0769 − 16.2456i
25.3 1.68251 + 1.08128i −0.426945 − 2.96946i 1.66166 + 3.63853i 15.5804 + 4.57481i 2.49249 − 5.45779i −6.98605 + 8.06233i −1.13852 + 7.91857i −8.63544 + 2.53559i 21.2674 + 24.5439i
31.1 −0.284630 + 1.97964i 1.24625 + 2.72890i −3.83797 − 1.12693i −13.9794 − 16.1331i −5.75696 + 1.69040i 18.5923 + 11.9485i 3.32332 − 7.27706i −5.89375 + 6.80175i 35.9168 − 23.0823i
31.2 −0.284630 + 1.97964i 1.24625 + 2.72890i −3.83797 − 1.12693i −1.58076 − 1.82430i −5.75696 + 1.69040i −19.6617 − 12.6358i 3.32332 − 7.27706i −5.89375 + 6.80175i 4.06139 − 2.61010i
31.3 −0.284630 + 1.97964i 1.24625 + 2.72890i −3.83797 − 1.12693i 3.37498 + 3.89494i −5.75696 + 1.69040i 15.9890 + 10.2755i 3.32332 − 7.27706i −5.89375 + 6.80175i −8.67120 + 5.57264i
49.1 −0.284630 − 1.97964i 1.24625 − 2.72890i −3.83797 + 1.12693i −13.9794 + 16.1331i −5.75696 − 1.69040i 18.5923 − 11.9485i 3.32332 + 7.27706i −5.89375 − 6.80175i 35.9168 + 23.0823i
49.2 −0.284630 − 1.97964i 1.24625 − 2.72890i −3.83797 + 1.12693i −1.58076 + 1.82430i −5.75696 − 1.69040i −19.6617 + 12.6358i 3.32332 + 7.27706i −5.89375 − 6.80175i 4.06139 + 2.61010i
49.3 −0.284630 − 1.97964i 1.24625 − 2.72890i −3.83797 + 1.12693i 3.37498 − 3.89494i −5.75696 − 1.69040i 15.9890 − 10.2755i 3.32332 + 7.27706i −5.89375 − 6.80175i −8.67120 − 5.57264i
55.1 −1.91899 + 0.563465i −1.96458 − 2.26725i 3.36501 − 2.16256i −2.21950 − 15.4369i 5.04752 + 3.24384i 4.21769 − 9.23545i −5.23889 + 6.04600i −1.28083 + 8.90839i 12.9574 + 28.3727i
55.2 −1.91899 + 0.563465i −1.96458 − 2.26725i 3.36501 − 2.16256i −0.141588 − 0.984766i 5.04752 + 3.24384i −6.60477 + 14.4624i −5.23889 + 6.04600i −1.28083 + 8.90839i 0.826586 + 1.80997i
55.3 −1.91899 + 0.563465i −1.96458 − 2.26725i 3.36501 − 2.16256i 1.68648 + 11.7297i 5.04752 + 3.24384i 8.69247 − 19.0339i −5.23889 + 6.04600i −1.28083 + 8.90839i −9.84560 − 21.5588i
73.1 0.830830 + 1.81926i −2.87848 + 0.845198i −2.61944 + 3.02300i −10.0675 − 6.46998i −3.92916 − 4.53450i 2.57211 + 17.8894i −7.67594 − 2.25386i 7.57128 − 4.86577i 3.40623 − 23.6909i
73.2 0.830830 + 1.81926i −2.87848 + 0.845198i −2.61944 + 3.02300i 0.656703 + 0.422037i −3.92916 − 4.53450i −4.63598 − 32.2439i −7.67594 − 2.25386i 7.57128 − 4.86577i −0.222189 + 1.54536i
73.3 0.830830 + 1.81926i −2.87848 + 0.845198i −2.61944 + 3.02300i 11.9888 + 7.70474i −3.92916 − 4.53450i 0.941694 + 6.54963i −7.67594 − 2.25386i 7.57128 − 4.86577i −4.05629 + 28.2122i
85.1 −1.30972 + 1.51150i 2.52376 − 1.62192i −0.569259 − 3.95929i −2.73603 − 5.99106i −0.853889 + 5.93893i −4.46365 + 1.31065i 6.73003 + 4.32513i 3.73874 − 8.18669i 12.6389 + 3.71112i
85.2 −1.30972 + 1.51150i 2.52376 − 1.62192i −0.569259 − 3.95929i 0.452038 + 0.989824i −0.853889 + 5.93893i 6.64594 − 1.95142i 6.73003 + 4.32513i 3.73874 − 8.18669i −2.08816 − 0.613140i
See all 30 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 13.3
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.4.e.a ✓ 30
23.c even 11 1 inner 138.4.e.a ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.4.e.a ✓ 30 1.a even 1 1 trivial
138.4.e.a ✓ 30 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{30} + 20 T_{5}^{29} + 333 T_{5}^{28} + 1351 T_{5}^{27} + 64930 T_{5}^{26} - 217335 T_{5}^{25} + \cdots + 27\!\cdots\!41 \) acting on \(S_{4}^{\mathrm{new}}(138, [\chi])\). Copy content Toggle raw display