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Properties

Label 6039.2.a.m
Level $6039$
Weight $2$
Character orbit 6039.a
Self dual yes
Analytic conductor $48.222$
Analytic rank $1$
Dimension $25$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6039,2,Mod(1,6039)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6039, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6039.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6039.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.2216577807\)
Analytic rank: \(1\)
Dimension: \(25\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.81809 0 5.94161 −0.621459 0 3.29277 −11.1078 0 1.75133
1.2 −2.53522 0 4.42736 3.72510 0 −0.951731 −6.15390 0 −9.44395
1.3 −2.52609 0 4.38115 −1.93230 0 0.113912 −6.01501 0 4.88117
1.4 −2.43689 0 3.93843 −3.08470 0 4.69684 −4.72373 0 7.51707
1.5 −2.03124 0 2.12592 −3.40634 0 −3.47017 −0.255767 0 6.91908
1.6 −1.98740 0 1.94976 1.69625 0 1.47319 0.0998540 0 −3.37113
1.7 −1.84847 0 1.41684 3.30338 0 −3.95713 1.07795 0 −6.10619
1.8 −1.38389 0 −0.0848502 −4.17750 0 3.01863 2.88520 0 5.78120
1.9 −1.18061 0 −0.606158 0.892999 0 −4.86985 3.07686 0 −1.05428
1.10 −1.12582 0 −0.732534 0.691275 0 −1.12590 3.07634 0 −0.778249
1.11 −0.800654 0 −1.35895 −0.757549 0 0.484724 2.68936 0 0.606535
1.12 −0.649968 0 −1.57754 3.31582 0 1.03899 2.32529 0 −2.15517
1.13 −0.536820 0 −1.71182 −1.82707 0 2.96010 1.99258 0 0.980806
1.14 0.228656 0 −1.94772 0.872276 0 −1.27506 −0.902667 0 0.199451
1.15 0.325798 0 −1.89386 −3.19527 0 −2.43974 −1.26861 0 −1.04101
1.16 0.608354 0 −1.62991 1.59846 0 −1.08777 −2.20827 0 0.972431
1.17 0.870092 0 −1.24294 −3.50217 0 −4.57282 −2.82166 0 −3.04721
1.18 1.04320 0 −0.911728 1.64856 0 5.01087 −3.03752 0 1.71978
1.19 1.23685 0 −0.470198 −0.403292 0 −0.303505 −3.05527 0 −0.498813
1.20 1.53628 0 0.360147 −2.16725 0 2.96367 −2.51927 0 −3.32950
See all 25 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.25
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)
\(61\) \(-1\)

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 6039.2.a.m ✓ 25
3.b odd 2 1 6039.2.a.p yes 25
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6039.2.a.m ✓ 25 1.a even 1 1 trivial
6039.2.a.p yes 25 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{25} + 5 T_{2}^{24} - 25 T_{2}^{23} - 155 T_{2}^{22} + 226 T_{2}^{21} + 2056 T_{2}^{20} + \cdots + 657 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6039))\). Copy content Toggle raw display