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Properties

Label 13.14.e.a
Level $13$
Weight $14$
Character orbit 13.e
Analytic conductor $13.940$
Analytic rank $0$
Dimension $30$
Inner twists $2$

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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] = [13,14,Mod(4,13)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 14, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("13.4"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 13.e (of order \(6\), degree \(2\), 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: \(13.9400207637\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(15\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q - 3 q^{2} - 730 q^{3} + 65535 q^{4} - 382584 q^{6} + 438984 q^{7} - 9117071 q^{9} + 3893475 q^{10} - 5364528 q^{11} + 27694876 q^{12} - 1794135 q^{13} + 61125588 q^{14} - 113380170 q^{15} - 350182521 q^{16}+ \cdots + 86959564160763 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −148.000 − 85.4479i 1092.19 − 1891.73i 10506.7 + 18198.1i − 22940.5i −323289. + 186651.i 137378. − 79315.3i − 2.19112e6i −1.58860e6 − 2.75154e6i −1.96022e6 + 3.39519e6i
4.2 −139.455 − 80.5145i −619.056 + 1072.24i 8869.17 + 15361.9i 40204.7i 172661. − 99686.1i −4340.50 + 2505.99i − 1.53724e6i 30699.8 + 53173.6i 3.23706e6 − 5.60676e6i
4.3 −103.966 − 60.0249i −41.2753 + 71.4909i 3109.97 + 5386.62i − 28055.4i 8582.46 − 4955.09i −389915. + 225118.i 236746.i 793754. + 1.37482e6i −1.68402e6 + 2.91681e6i
4.4 −87.3849 − 50.4517i 197.507 − 342.093i 994.753 + 1722.96i 4300.66i −34518.3 + 19929.2i 411054. − 237322.i 625853.i 719143. + 1.24559e6i 216976. − 375813.i
4.5 −77.2063 − 44.5751i −1229.77 + 2130.03i −122.125 − 211.526i − 52376.5i 189892. − 109634.i 240349. − 138765.i 752093.i −2.22751e6 − 3.85817e6i −2.33469e6 + 4.04379e6i
4.6 −55.2505 − 31.8989i 924.809 − 1601.82i −2060.92 − 3569.63i 57759.0i −102192. + 59000.7i −148669. + 85833.9i 785596.i −913381. − 1.58202e6i 1.84245e6 − 3.19121e6i
4.7 −11.3819 − 6.57136i −864.242 + 1496.91i −4009.63 − 6944.89i 50287.2i 19673.5 − 11358.5i −255242. + 147364.i 213060.i −696665. − 1.20666e6i 330455. − 572365.i
4.8 −5.65944 − 3.26748i 865.700 − 1499.44i −4074.65 − 7057.50i − 60806.7i −9798.77 + 5657.32i −175141. + 101118.i 106790.i −701713. − 1.21540e6i −198685. + 344132.i
4.9 −1.87656 − 1.08343i −123.317 + 213.592i −4093.65 − 7090.41i − 3830.61i 462.824 − 267.212i 38591.2 − 22280.6i 35491.7i 766747. + 1.32805e6i −4150.20 + 7188.36i
4.10 57.2913 + 33.0771i −469.367 + 812.968i −1907.80 − 3304.41i − 11059.1i −53781.3 + 31050.6i 299288. − 172794.i − 794355.i 356551. + 617564.i 365805. − 633593.i
4.11 73.3921 + 42.3729i 812.878 − 1407.95i −505.068 − 874.804i 11734.8i 119318. − 68888.1i 138253. − 79820.2i − 779843.i −524381. − 908254.i −497238. + 861241.i
4.12 97.8824 + 56.5124i −748.252 + 1296.01i 2291.31 + 3968.66i − 55447.0i −146481. + 84571.1i −434658. + 250950.i − 407950.i −322601. − 558762.i 3.13344e6 − 5.42728e6i
4.13 113.733 + 65.6639i 197.560 − 342.184i 4527.50 + 7841.87i 34433.6i 44938.3 − 25945.1i −324633. + 187427.i 113337.i 719102. + 1.24552e6i −2.26105e6 + 3.91624e6i
4.14 134.717 + 77.7787i −960.268 + 1663.23i 8003.06 + 13861.7i 38811.6i −258728. + 149377.i 442407. − 255424.i 1.21554e6i −1.04707e6 − 1.81358e6i −3.01871e6 + 5.22857e6i
4.15 151.665 + 87.5640i 599.903 − 1039.06i 11238.9 + 19466.4i − 54580.7i 181969. − 105060.i 244771. − 141318.i 2.50185e6i 77394.3 + 134051.i 4.77930e6 − 8.27800e6i
10.1 −148.000 + 85.4479i 1092.19 + 1891.73i 10506.7 − 18198.1i 22940.5i −323289. − 186651.i 137378. + 79315.3i 2.19112e6i −1.58860e6 + 2.75154e6i −1.96022e6 − 3.39519e6i
10.2 −139.455 + 80.5145i −619.056 − 1072.24i 8869.17 − 15361.9i − 40204.7i 172661. + 99686.1i −4340.50 − 2505.99i 1.53724e6i 30699.8 − 53173.6i 3.23706e6 + 5.60676e6i
10.3 −103.966 + 60.0249i −41.2753 − 71.4909i 3109.97 − 5386.62i 28055.4i 8582.46 + 4955.09i −389915. − 225118.i − 236746.i 793754. − 1.37482e6i −1.68402e6 − 2.91681e6i
10.4 −87.3849 + 50.4517i 197.507 + 342.093i 994.753 − 1722.96i − 4300.66i −34518.3 − 19929.2i 411054. + 237322.i − 625853.i 719143. − 1.24559e6i 216976. + 375813.i
10.5 −77.2063 + 44.5751i −1229.77 − 2130.03i −122.125 + 211.526i 52376.5i 189892. + 109634.i 240349. + 138765.i − 752093.i −2.22751e6 + 3.85817e6i −2.33469e6 − 4.04379e6i
See all 30 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.15
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 13.14.e.a ✓ 30
13.e even 6 1 inner 13.14.e.a ✓ 30
13.f odd 12 2 169.14.a.f 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.14.e.a ✓ 30 1.a even 1 1 trivial
13.14.e.a ✓ 30 13.e even 6 1 inner
169.14.a.f 30 13.f odd 12 2

Hecke kernels

This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(13, [\chi])\).