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Properties

Label 6038.2.a.d
Level $6038$
Weight $2$
Character orbit 6038.a
Self dual yes
Analytic conductor $48.214$
Analytic rank $0$
Dimension $69$
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] = [6038,2,Mod(1,6038)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6038, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6038.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6038 = 2 \cdot 3019 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6038.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: \(48.2136727404\)
Analytic rank: \(0\)
Dimension: \(69\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 69 q - 69 q^{2} + 8 q^{3} + 69 q^{4} + 18 q^{5} - 8 q^{6} + 32 q^{7} - 69 q^{8} + 79 q^{9} - 18 q^{10} - 4 q^{11} + 8 q^{12} + 40 q^{13} - 32 q^{14} + 6 q^{15} + 69 q^{16} + 6 q^{17} - 79 q^{18} + 13 q^{19} + 18 q^{20} + 23 q^{21} + 4 q^{22} + 11 q^{23} - 8 q^{24} + 111 q^{25} - 40 q^{26} + 32 q^{27} + 32 q^{28} + 23 q^{29} - 6 q^{30} + 30 q^{31} - 69 q^{32} + 37 q^{33} - 6 q^{34} - 12 q^{35} + 79 q^{36} + 81 q^{37} - 13 q^{38} - 8 q^{39} - 18 q^{40} - 13 q^{41} - 23 q^{42} + 42 q^{43} - 4 q^{44} + 89 q^{45} - 11 q^{46} + 35 q^{47} + 8 q^{48} + 115 q^{49} - 111 q^{50} - 21 q^{51} + 40 q^{52} + 41 q^{53} - 32 q^{54} + 28 q^{55} - 32 q^{56} + 39 q^{57} - 23 q^{58} - 24 q^{59} + 6 q^{60} + 69 q^{61} - 30 q^{62} + 79 q^{63} + 69 q^{64} + 15 q^{65} - 37 q^{66} + 87 q^{67} + 6 q^{68} + 51 q^{69} + 12 q^{70} - 22 q^{71} - 79 q^{72} + 104 q^{73} - 81 q^{74} + 39 q^{75} + 13 q^{76} + 47 q^{77} + 8 q^{78} + 16 q^{79} + 18 q^{80} + 105 q^{81} + 13 q^{82} + 30 q^{83} + 23 q^{84} + 74 q^{85} - 42 q^{86} + 47 q^{87} + 4 q^{88} - 4 q^{89} - 89 q^{90} + 70 q^{91} + 11 q^{92} + 104 q^{93} - 35 q^{94} - 36 q^{95} - 8 q^{96} + 158 q^{97} - 115 q^{98} + 34 q^{99}+O(q^{100}) \) 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 −1.00000 −3.38264 1.00000 3.55013 3.38264 −0.929098 −1.00000 8.44223 −3.55013
1.2 −1.00000 −3.22022 1.00000 2.70863 3.22022 4.47236 −1.00000 7.36980 −2.70863
1.3 −1.00000 −3.17982 1.00000 −0.0288660 3.17982 0.302400 −1.00000 7.11126 0.0288660
1.4 −1.00000 −2.96065 1.00000 0.357077 2.96065 −2.81647 −1.00000 5.76543 −0.357077
1.5 −1.00000 −2.91924 1.00000 −2.24533 2.91924 −3.88903 −1.00000 5.52197 2.24533
1.6 −1.00000 −2.88111 1.00000 2.68970 2.88111 4.10918 −1.00000 5.30078 −2.68970
1.7 −1.00000 −2.79695 1.00000 −3.22250 2.79695 3.03239 −1.00000 4.82293 3.22250
1.8 −1.00000 −2.63337 1.00000 −0.783693 2.63337 4.82371 −1.00000 3.93463 0.783693
1.9 −1.00000 −2.40625 1.00000 −0.733460 2.40625 −2.00296 −1.00000 2.79002 0.733460
1.10 −1.00000 −2.29973 1.00000 4.30527 2.29973 −2.14109 −1.00000 2.28878 −4.30527
1.11 −1.00000 −2.23277 1.00000 1.71292 2.23277 0.0975688 −1.00000 1.98527 −1.71292
1.12 −1.00000 −2.22571 1.00000 1.16705 2.22571 1.13637 −1.00000 1.95376 −1.16705
1.13 −1.00000 −2.18407 1.00000 2.82880 2.18407 −4.34635 −1.00000 1.77018 −2.82880
1.14 −1.00000 −2.15476 1.00000 −2.42408 2.15476 −0.124964 −1.00000 1.64299 2.42408
1.15 −1.00000 −2.10359 1.00000 2.00437 2.10359 2.39594 −1.00000 1.42511 −2.00437
1.16 −1.00000 −2.10090 1.00000 −3.50554 2.10090 3.00259 −1.00000 1.41376 3.50554
1.17 −1.00000 −1.78840 1.00000 −4.39275 1.78840 −0.240516 −1.00000 0.198361 4.39275
1.18 −1.00000 −1.59881 1.00000 4.26429 1.59881 −0.810088 −1.00000 −0.443805 −4.26429
1.19 −1.00000 −1.58255 1.00000 −3.72234 1.58255 −1.18949 −1.00000 −0.495522 3.72234
1.20 −1.00000 −1.49405 1.00000 −0.329476 1.49405 1.19581 −1.00000 −0.767828 0.329476
See all 69 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.69
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3019\) \(-1\)

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 6038.2.a.d ✓ 69
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6038.2.a.d ✓ 69 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{69} - 8 T_{3}^{68} - 111 T_{3}^{67} + 1032 T_{3}^{66} + 5518 T_{3}^{65} - 62959 T_{3}^{64} + \cdots + 82462107721 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6038))\). Copy content Toggle raw display