Properties

Label 6044.2.a.a
Level $6044$
Weight $2$
Character orbit 6044.a
Self dual yes
Analytic conductor $48.262$
Analytic rank $1$
Dimension $63$
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] = [6044,2,Mod(1,6044)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6044, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6044.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6044 = 2^{2} \cdot 1511 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6044.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: \(48.2615829817\)
Analytic rank: \(1\)
Dimension: \(63\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) \) 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 0 −3.43683 0 1.31396 0 2.97589 0 8.81178 0
1.2 0 −3.31051 0 0.0245297 0 −4.08379 0 7.95946 0
1.3 0 −3.15665 0 −3.22660 0 1.21507 0 6.96442 0
1.4 0 −3.10897 0 4.27127 0 −2.96851 0 6.66567 0
1.5 0 −3.06921 0 2.30904 0 −0.616615 0 6.42006 0
1.6 0 −2.97134 0 −2.23756 0 4.75272 0 5.82886 0
1.7 0 −2.95257 0 −0.800433 0 −2.68929 0 5.71766 0
1.8 0 −2.59590 0 −0.0339717 0 0.331169 0 3.73871 0
1.9 0 −2.53520 0 −2.05681 0 1.80178 0 3.42722 0
1.10 0 −2.53490 0 4.14677 0 1.26365 0 3.42570 0
1.11 0 −2.46300 0 −1.99757 0 3.35645 0 3.06637 0
1.12 0 −2.40035 0 2.30377 0 −2.94550 0 2.76170 0
1.13 0 −2.33939 0 2.09632 0 1.25063 0 2.47276 0
1.14 0 −1.98161 0 −3.75516 0 −4.03693 0 0.926797 0
1.15 0 −1.75230 0 −3.12629 0 −3.74178 0 0.0705422 0
1.16 0 −1.69210 0 −3.14690 0 −4.59556 0 −0.136814 0
1.17 0 −1.68302 0 3.28650 0 −4.68087 0 −0.167449 0
1.18 0 −1.59751 0 1.94100 0 −3.58958 0 −0.447960 0
1.19 0 −1.58770 0 −0.675402 0 −1.82211 0 −0.479212 0
1.20 0 −1.57704 0 3.78585 0 1.51605 0 −0.512930 0
See all 63 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.63
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(1511\) \(-1\)

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 6044.2.a.a 63
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6044.2.a.a 63 1.a even 1 1 trivial