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Properties

Label 415.4.a.c
Level $415$
Weight $4$
Character orbit 415.a
Self dual yes
Analytic conductor $24.486$
Analytic rank $1$
Dimension $21$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [415,4,Mod(1,415)] 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("415.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(415, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 415 = 5 \cdot 83 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 415.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [21] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(24.4857926524\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 21 q - 5 q^{2} - 12 q^{3} + 87 q^{4} - 105 q^{5} - 7 q^{6} - 11 q^{7} - 84 q^{8} + 153 q^{9} + 25 q^{10} - 30 q^{11} - 244 q^{12} - 89 q^{13} - 191 q^{14} + 60 q^{15} + 583 q^{16} - 357 q^{17} - 281 q^{18}+ \cdots - 5369 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.53348 −2.19798 22.6194 −5.00000 12.1625 33.9918 −80.8964 −22.1689 27.6674
1.2 −5.22755 5.95879 19.3272 −5.00000 −31.1498 −21.5721 −59.2137 8.50716 26.1377
1.3 −5.13790 −8.68410 18.3981 −5.00000 44.6181 −3.62891 −53.4243 48.4135 25.6895
1.4 −4.39847 −5.91845 11.3466 −5.00000 26.0321 −33.6660 −14.7198 8.02804 21.9924
1.5 −4.08773 3.96482 8.70951 −5.00000 −16.2071 32.6977 −2.90029 −11.2802 20.4386
1.6 −2.92996 7.50768 0.584655 −5.00000 −21.9972 −14.0550 21.7267 29.3653 14.6498
1.7 −2.70189 −4.13473 −0.699769 −5.00000 11.1716 4.29122 23.5059 −9.90398 13.5095
1.8 −1.52351 2.27996 −5.67892 −5.00000 −3.47354 9.53696 20.8400 −21.8018 7.61755
1.9 −1.06933 −4.93076 −6.85653 −5.00000 5.27261 −16.4681 15.8865 −2.68759 5.34665
1.10 −0.612344 9.34361 −7.62503 −5.00000 −5.72151 9.19593 9.56790 60.3030 3.06172
1.11 −0.505655 −2.37524 −7.74431 −5.00000 1.20105 18.6532 7.96119 −21.3582 2.52828
1.12 −0.483861 −9.34705 −7.76588 −5.00000 4.52267 −16.2706 7.62849 60.3674 2.41930
1.13 0.486234 4.96720 −7.76358 −5.00000 2.41522 6.56959 −7.66478 −2.32690 −2.43117
1.14 1.63541 −9.34891 −5.32544 −5.00000 −15.2893 24.9057 −21.7925 60.4022 −8.17703
1.15 2.30385 6.02335 −2.69226 −5.00000 13.8769 −6.96885 −24.6334 9.28076 −11.5193
1.16 3.03570 1.75697 1.21547 −5.00000 5.33363 19.4847 −20.5958 −23.9131 −15.1785
1.17 3.38049 5.94262 3.42769 −5.00000 20.0889 −25.0025 −15.4566 8.31472 −16.9024
1.18 4.19650 −7.79463 9.61062 −5.00000 −32.7102 14.8766 6.75897 33.7563 −20.9825
1.19 4.28686 −0.965562 10.3772 −5.00000 −4.13923 6.97800 10.1906 −26.0677 −21.4343
1.20 4.36783 0.581825 11.0779 −5.00000 2.54131 −27.7923 13.4438 −26.6615 −21.8391
See all 21 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.21
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(83\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 415.4.a.c ✓ 21
5.b even 2 1 2075.4.a.g 21
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
415.4.a.c ✓ 21 1.a even 1 1 trivial
2075.4.a.g 21 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{21} + 5 T_{2}^{20} - 115 T_{2}^{19} - 562 T_{2}^{18} + 5494 T_{2}^{17} + 26144 T_{2}^{16} + \cdots - 42094592 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(415))\). Copy content Toggle raw display