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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.

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9} - 56 q^{10} - 105 q^{11} + 36 q^{12} - 21 q^{13} - 114 q^{15} - 48 q^{16} + 41 q^{17} - 54 q^{18} - 149 q^{19} + 152 q^{20}+ \cdots - 945 q^{99}+O(q^{100}) \) 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 −3.80011 + 8.32109i 0.853889 + 5.93893i 2.38181 + 0.699363i 6.73003 − 4.32513i 3.73874 + 8.18669i 17.5544 − 5.15444i
13.2 −1.30972 − 1.51150i −2.52376 − 1.62192i −0.569259 + 3.95929i 2.02822 − 4.44117i 0.853889 + 5.93893i −16.2703 − 4.77739i 6.73003 − 4.32513i 3.73874 + 8.18669i −9.36923 + 2.75105i
13.3 −1.30972 − 1.51150i −2.52376 − 1.62192i −0.569259 + 3.95929i 6.04376 − 13.2340i 0.853889 + 5.93893i 29.2734 + 8.59546i 6.73003 − 4.32513i 3.73874 + 8.18669i −27.9188 + 8.19771i
25.1 1.68251 + 1.08128i 0.426945 + 2.96946i 1.66166 + 3.63853i −20.3614 − 5.97866i −2.49249 + 5.45779i 5.13322 − 5.92406i −1.13852 + 7.91857i −8.63544 + 2.53559i −27.7936 − 32.0756i
25.2 1.68251 + 1.08128i 0.426945 + 2.96946i 1.66166 + 3.63853i 2.07213 + 0.608431i −2.49249 + 5.45779i −14.7194 + 16.9870i −1.13852 + 7.91857i −8.63544 + 2.53559i 2.82848 + 3.26424i
25.3 1.68251 + 1.08128i 0.426945 + 2.96946i 1.66166 + 3.63853i 16.0712 + 4.71893i −2.49249 + 5.45779i 17.1563 − 19.7994i −1.13852 + 7.91857i −8.63544 + 2.53559i 21.9374 + 25.3171i
31.1 −0.284630 + 1.97964i −1.24625 − 2.72890i −3.83797 − 1.12693i −11.4557 − 13.2206i 5.75696 − 1.69040i 16.3172 + 10.4864i 3.32332 − 7.27706i −5.89375 + 6.80175i 29.4328 − 18.9153i
31.2 −0.284630 + 1.97964i −1.24625 − 2.72890i −3.83797 − 1.12693i 1.48009 + 1.70811i 5.75696 − 1.69040i 3.78550 + 2.43279i 3.32332 − 7.27706i −5.89375 + 6.80175i −3.80273 + 2.44386i
31.3 −0.284630 + 1.97964i −1.24625 − 2.72890i −3.83797 − 1.12693i 11.2596 + 12.9943i 5.75696 − 1.69040i −30.1361 − 19.3673i 3.32332 − 7.27706i −5.89375 + 6.80175i −28.9289 + 18.5915i
49.1 −0.284630 − 1.97964i −1.24625 + 2.72890i −3.83797 + 1.12693i −11.4557 + 13.2206i 5.75696 + 1.69040i 16.3172 − 10.4864i 3.32332 + 7.27706i −5.89375 − 6.80175i 29.4328 + 18.9153i
49.2 −0.284630 − 1.97964i −1.24625 + 2.72890i −3.83797 + 1.12693i 1.48009 − 1.70811i 5.75696 + 1.69040i 3.78550 − 2.43279i 3.32332 + 7.27706i −5.89375 − 6.80175i −3.80273 − 2.44386i
49.3 −0.284630 − 1.97964i −1.24625 + 2.72890i −3.83797 + 1.12693i 11.2596 − 12.9943i 5.75696 + 1.69040i −30.1361 + 19.3673i 3.32332 + 7.27706i −5.89375 − 6.80175i −28.9289 − 18.5915i
55.1 −1.91899 + 0.563465i 1.96458 + 2.26725i 3.36501 − 2.16256i −1.40880 − 9.79842i −5.04752 − 3.24384i −9.01039 + 19.7300i −5.23889 + 6.04600i −1.28083 + 8.90839i 8.22454 + 18.0092i
55.2 −1.91899 + 0.563465i 1.96458 + 2.26725i 3.36501 − 2.16256i −0.966187 − 6.71998i −5.04752 − 3.24384i 5.61851 − 12.3028i −5.23889 + 6.04600i −1.28083 + 8.90839i 5.64057 + 12.3511i
55.3 −1.91899 + 0.563465i 1.96458 + 2.26725i 3.36501 − 2.16256i 3.11759 + 21.6833i −5.04752 − 3.24384i −0.846192 + 1.85290i −5.23889 + 6.04600i −1.28083 + 8.90839i −18.2004 − 39.8533i
73.1 0.830830 + 1.81926i 2.87848 − 0.845198i −2.61944 + 3.02300i −11.8567 − 7.61986i 3.92916 + 4.53450i −2.71532 − 18.8855i −7.67594 − 2.25386i 7.57128 − 4.86577i 4.01160 − 27.9013i
73.2 0.830830 + 1.81926i 2.87848 − 0.845198i −2.61944 + 3.02300i −8.49581 − 5.45993i 3.92916 + 4.53450i 4.18194 + 29.0860i −7.67594 − 2.25386i 7.57128 − 4.86577i 2.87447 − 19.9924i
73.3 0.830830 + 1.81926i 2.87848 − 0.845198i −2.61944 + 3.02300i 13.2722 + 8.52953i 3.92916 + 4.53450i 0.849820 + 5.91063i −7.67594 − 2.25386i 7.57128 − 4.86577i −4.49052 + 31.2322i
85.1 −1.30972 + 1.51150i −2.52376 + 1.62192i −0.569259 − 3.95929i −3.80011 − 8.32109i 0.853889 − 5.93893i 2.38181 − 0.699363i 6.73003 + 4.32513i 3.73874 − 8.18669i 17.5544 + 5.15444i
85.2 −1.30972 + 1.51150i −2.52376 + 1.62192i −0.569259 − 3.95929i 2.02822 + 4.44117i 0.853889 − 5.93893i −16.2703 + 4.77739i 6.73003 + 4.32513i 3.73874 − 8.18669i −9.36923 − 2.75105i
See all 30 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 13.3
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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.b ✓ 30
23.c even 11 1 inner 138.4.e.b ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.4.e.b ✓ 30 1.a even 1 1 trivial
138.4.e.b ✓ 30 23.c even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{30} + 6 T_{5}^{29} + 65 T_{5}^{28} + 6409 T_{5}^{27} + 5482 T_{5}^{26} - 1016815 T_{5}^{25} + \cdots + 12\!\cdots\!69 \) acting on \(S_{4}^{\mathrm{new}}(138, [\chi])\). Copy content Toggle raw display