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Properties

Label 6038.2.a.c
Level $6038$
Weight $2$
Character orbit 6038.a
Self dual yes
Analytic conductor $48.214$
Analytic rank $1$
Dimension $57$
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: \(1\)
Dimension: \(57\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 57 q - 57 q^{2} - 5 q^{3} + 57 q^{4} - 15 q^{5} + 5 q^{6} - 28 q^{7} - 57 q^{8} + 50 q^{9} + 15 q^{10} + 13 q^{11} - 5 q^{12} - 43 q^{13} + 28 q^{14} - 10 q^{15} + 57 q^{16} - 50 q^{18} - 6 q^{19} - 15 q^{20} - 23 q^{21} - 13 q^{22} - q^{23} + 5 q^{24} + 20 q^{25} + 43 q^{26} - 20 q^{27} - 28 q^{28} - 4 q^{29} + 10 q^{30} - 34 q^{31} - 57 q^{32} - 43 q^{33} + 26 q^{35} + 50 q^{36} - 64 q^{37} + 6 q^{38} + 8 q^{39} + 15 q^{40} + 27 q^{41} + 23 q^{42} - 29 q^{43} + 13 q^{44} - 76 q^{45} + q^{46} - 25 q^{47} - 5 q^{48} + 7 q^{49} - 20 q^{50} + 27 q^{51} - 43 q^{52} - 34 q^{53} + 20 q^{54} - 36 q^{55} + 28 q^{56} - 33 q^{57} + 4 q^{58} + 19 q^{59} - 10 q^{60} - 58 q^{61} + 34 q^{62} - 65 q^{63} + 57 q^{64} + 17 q^{65} + 43 q^{66} - 84 q^{67} - 33 q^{69} - 26 q^{70} + 22 q^{71} - 50 q^{72} - 82 q^{73} + 64 q^{74} + 8 q^{75} - 6 q^{76} - 41 q^{77} - 8 q^{78} + 8 q^{79} - 15 q^{80} + 25 q^{81} - 27 q^{82} - 23 q^{83} - 23 q^{84} - 58 q^{85} + 29 q^{86} - 17 q^{87} - 13 q^{88} + 18 q^{89} + 76 q^{90} - 4 q^{91} - q^{92} - 60 q^{93} + 25 q^{94} + 36 q^{95} + 5 q^{96} - 156 q^{97} - 7 q^{98} + 25 q^{99}+O(q^{100}) \) 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 −3.40132 1.00000 −0.259244 3.40132 0.796720 −1.00000 8.56895 0.259244
1.2 −1.00000 −3.09911 1.00000 −3.06696 3.09911 −3.37062 −1.00000 6.60450 3.06696
1.3 −1.00000 −3.09194 1.00000 −2.28798 3.09194 0.749696 −1.00000 6.56008 2.28798
1.4 −1.00000 −2.99307 1.00000 2.62099 2.99307 −1.28357 −1.00000 5.95844 −2.62099
1.5 −1.00000 −2.94445 1.00000 −1.42608 2.94445 1.57265 −1.00000 5.66979 1.42608
1.6 −1.00000 −2.85836 1.00000 −3.68447 2.85836 2.06392 −1.00000 5.17024 3.68447
1.7 −1.00000 −2.75036 1.00000 0.139827 2.75036 0.205180 −1.00000 4.56449 −0.139827
1.8 −1.00000 −2.68115 1.00000 1.78050 2.68115 −4.24119 −1.00000 4.18856 −1.78050
1.9 −1.00000 −2.46647 1.00000 3.27492 2.46647 3.14216 −1.00000 3.08346 −3.27492
1.10 −1.00000 −2.39662 1.00000 −1.22776 2.39662 −4.40908 −1.00000 2.74377 1.22776
1.11 −1.00000 −2.25784 1.00000 0.481380 2.25784 −0.819021 −1.00000 2.09784 −0.481380
1.12 −1.00000 −2.15416 1.00000 1.67498 2.15416 2.16308 −1.00000 1.64039 −1.67498
1.13 −1.00000 −2.12089 1.00000 −1.22072 2.12089 −2.75912 −1.00000 1.49816 1.22072
1.14 −1.00000 −1.79555 1.00000 3.22751 1.79555 −1.51507 −1.00000 0.223993 −3.22751
1.15 −1.00000 −1.71541 1.00000 1.06645 1.71541 4.85447 −1.00000 −0.0573586 −1.06645
1.16 −1.00000 −1.68023 1.00000 −2.51181 1.68023 3.66839 −1.00000 −0.176828 2.51181
1.17 −1.00000 −1.57568 1.00000 1.43358 1.57568 2.19919 −1.00000 −0.517233 −1.43358
1.18 −1.00000 −1.49224 1.00000 −4.20825 1.49224 −4.82054 −1.00000 −0.773233 4.20825
1.19 −1.00000 −1.42937 1.00000 −1.03254 1.42937 −0.605000 −1.00000 −0.956891 1.03254
1.20 −1.00000 −1.21264 1.00000 1.98344 1.21264 −0.999136 −1.00000 −1.52951 −1.98344
See all 57 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.57
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.c ✓ 57
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6038.2.a.c ✓ 57 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{57} + 5 T_{3}^{56} - 98 T_{3}^{55} - 515 T_{3}^{54} + 4458 T_{3}^{53} + 24847 T_{3}^{52} + \cdots - 33684056 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6038))\). Copy content Toggle raw display