Properties

Label 6038.2.a.e
Level $6038$
Weight $2$
Character orbit 6038.a
Self dual yes
Analytic conductor $48.214$
Analytic rank $0$
Dimension $70$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(70\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 70 q + 70 q^{2} + 25 q^{3} + 70 q^{4} + 18 q^{5} + 25 q^{6} + 50 q^{7} + 70 q^{8} + 89 q^{9} + 18 q^{10} + 41 q^{11} + 25 q^{12} + 41 q^{13} + 50 q^{14} + 13 q^{15} + 70 q^{16} + 40 q^{17} + 89 q^{18} + 55 q^{19} + 18 q^{20} + 2 q^{21} + 41 q^{22} + 41 q^{23} + 25 q^{24} + 104 q^{25} + 41 q^{26} + 82 q^{27} + 50 q^{28} + 11 q^{29} + 13 q^{30} + 78 q^{31} + 70 q^{32} + 45 q^{33} + 40 q^{34} + 25 q^{35} + 89 q^{36} + 46 q^{37} + 55 q^{38} + 19 q^{39} + 18 q^{40} + 51 q^{41} + 2 q^{42} + 68 q^{43} + 41 q^{44} + 37 q^{45} + 41 q^{46} + 69 q^{47} + 25 q^{48} + 126 q^{49} + 104 q^{50} + 36 q^{51} + 41 q^{52} + 23 q^{53} + 82 q^{54} + 42 q^{55} + 50 q^{56} + 14 q^{57} + 11 q^{58} + 89 q^{59} + 13 q^{60} + 32 q^{61} + 78 q^{62} + 106 q^{63} + 70 q^{64} + 18 q^{65} + 45 q^{66} + 90 q^{67} + 40 q^{68} - 12 q^{69} + 25 q^{70} + 54 q^{71} + 89 q^{72} + 94 q^{73} + 46 q^{74} + 72 q^{75} + 55 q^{76} - 16 q^{77} + 19 q^{78} + 54 q^{79} + 18 q^{80} + 102 q^{81} + 51 q^{82} + 60 q^{83} + 2 q^{84} - 5 q^{85} + 68 q^{86} + 9 q^{87} + 41 q^{88} + 77 q^{89} + 37 q^{90} + 54 q^{91} + 41 q^{92} - 2 q^{93} + 69 q^{94} + 39 q^{95} + 25 q^{96} + 139 q^{97} + 126 q^{98} + 60 q^{99}+O(q^{100}) \) 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.

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 −3.07733 1.00000 0.607795 −3.07733 4.61887 1.00000 6.46998 0.607795
1.2 1.00000 −3.00085 1.00000 −0.777425 −3.00085 3.31557 1.00000 6.00511 −0.777425
1.3 1.00000 −2.94455 1.00000 3.92217 −2.94455 4.11176 1.00000 5.67038 3.92217
1.4 1.00000 −2.93737 1.00000 −3.45627 −2.93737 2.34218 1.00000 5.62815 −3.45627
1.5 1.00000 −2.81205 1.00000 2.74866 −2.81205 −2.36582 1.00000 4.90764 2.74866
1.6 1.00000 −2.80845 1.00000 1.85496 −2.80845 −0.963784 1.00000 4.88737 1.85496
1.7 1.00000 −2.79976 1.00000 −2.91282 −2.79976 −0.947335 1.00000 4.83868 −2.91282
1.8 1.00000 −2.73176 1.00000 1.22157 −2.73176 −0.448839 1.00000 4.46253 1.22157
1.9 1.00000 −2.44970 1.00000 −2.67633 −2.44970 2.27602 1.00000 3.00102 −2.67633
1.10 1.00000 −2.34935 1.00000 −0.768821 −2.34935 −1.35127 1.00000 2.51947 −0.768821
1.11 1.00000 −2.23514 1.00000 3.00438 −2.23514 1.56958 1.00000 1.99587 3.00438
1.12 1.00000 −2.08835 1.00000 −0.884909 −2.08835 −2.93795 1.00000 1.36120 −0.884909
1.13 1.00000 −2.05123 1.00000 3.94879 −2.05123 5.22685 1.00000 1.20754 3.94879
1.14 1.00000 −1.76800 1.00000 1.61299 −1.76800 3.26382 1.00000 0.125835 1.61299
1.15 1.00000 −1.60124 1.00000 −1.76472 −1.60124 −0.621570 1.00000 −0.436015 −1.76472
1.16 1.00000 −1.50199 1.00000 −3.07076 −1.50199 4.02705 1.00000 −0.744026 −3.07076
1.17 1.00000 −1.49174 1.00000 1.96904 −1.49174 1.25123 1.00000 −0.774726 1.96904
1.18 1.00000 −1.19115 1.00000 −1.23162 −1.19115 −2.44516 1.00000 −1.58116 −1.23162
1.19 1.00000 −1.12952 1.00000 −0.938247 −1.12952 0.310012 1.00000 −1.72418 −0.938247
1.20 1.00000 −1.00890 1.00000 1.23789 −1.00890 −4.49878 1.00000 −1.98213 1.23789
See all 70 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.70
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.e 70
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6038.2.a.e 70 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{70} - 25 T_{3}^{69} + 163 T_{3}^{68} + 1056 T_{3}^{67} - 17875 T_{3}^{66} + 29034 T_{3}^{65} + 676447 T_{3}^{64} - 3380437 T_{3}^{63} - 11337576 T_{3}^{62} + 121690816 T_{3}^{61} + \cdots + 4147175912 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6038))\). Copy content Toggle raw display