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Properties

Label 5239.2.a.q
Level $5239$
Weight $2$
Character orbit 5239.a
Self dual yes
Analytic conductor $41.834$
Analytic rank $1$
Dimension $34$
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] = [5239,2,Mod(1,5239)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5239, 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("5239.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5239 = 13^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5239.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: \(41.8336256189\)
Analytic rank: \(1\)
Dimension: \(34\)
Twist minimal: no (minimal twist has level 403)
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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q - 8 q^{2} + 32 q^{4} - 16 q^{5} - 12 q^{6} - 8 q^{7} - 24 q^{8} + 34 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q - 8 q^{2} + 32 q^{4} - 16 q^{5} - 12 q^{6} - 8 q^{7} - 24 q^{8} + 34 q^{9} + 8 q^{10} - 26 q^{11} + 8 q^{12} - 4 q^{14} - 16 q^{15} + 36 q^{16} - 6 q^{17} - 64 q^{18} - 4 q^{19} - 40 q^{20} - 32 q^{21} + 20 q^{22} - 8 q^{23} + 16 q^{24} + 36 q^{25} - 6 q^{27} - 24 q^{28} + 32 q^{30} + 34 q^{31} - 36 q^{32} - 40 q^{33} - 16 q^{34} - 30 q^{35} + 40 q^{36} - 2 q^{37} + 18 q^{38} + 4 q^{40} - 80 q^{41} + 16 q^{42} + 12 q^{43} - 108 q^{44} - 12 q^{45} - 48 q^{46} - 24 q^{47} + 46 q^{48} + 22 q^{49} + 44 q^{50} - 28 q^{51} + 10 q^{53} - 48 q^{54} + 6 q^{55} - 2 q^{56} - 66 q^{57} + 44 q^{58} - 64 q^{59} - 48 q^{60} - 6 q^{61} - 8 q^{62} + 52 q^{63} - 12 q^{64} + 4 q^{66} - 16 q^{67} - 58 q^{68} - 28 q^{69} - 72 q^{70} - 52 q^{71} - 152 q^{72} - 42 q^{73} - 8 q^{74} - 4 q^{75} + 48 q^{76} + 10 q^{77} + 8 q^{79} - 48 q^{80} + 58 q^{81} - 42 q^{82} - 44 q^{83} + 8 q^{84} - 96 q^{85} - 16 q^{86} + 20 q^{87} + 64 q^{88} - 74 q^{89} - 26 q^{90} + 24 q^{92} - 8 q^{94} - 32 q^{95} - 50 q^{96} - 40 q^{97} - 72 q^{98} - 74 q^{99}+O(q^{100}) \) 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 −2.70002 −2.95194 5.29011 0.483354 7.97031 0.876420 −8.88337 5.71396 −1.30506
1.2 −2.67393 2.09266 5.14990 −2.17311 −5.59562 2.88696 −8.42260 1.37922 5.81073
1.3 −2.61388 0.389596 4.83236 −3.31493 −1.01836 −2.18405 −7.40344 −2.84822 8.66481
1.4 −2.53476 3.38270 4.42503 −0.452856 −8.57433 1.99898 −6.14687 8.44263 1.14788
1.5 −2.36247 2.70874 3.58124 2.43529 −6.39930 −3.91769 −3.73563 4.33727 −5.75329
1.6 −2.28834 −1.65573 3.23650 −1.16564 3.78886 −2.98977 −2.82952 −0.258573 2.66737
1.7 −2.15340 −2.65889 2.63714 3.07578 5.72566 −0.505620 −1.37202 4.06970 −6.62339
1.8 −2.12811 0.449534 2.52886 −3.56888 −0.956659 −3.15924 −1.12547 −2.79792 7.59498
1.9 −1.80539 −2.61742 1.25945 −2.29293 4.72548 3.15704 1.33699 3.85091 4.13963
1.10 −1.80367 −1.10798 1.25322 0.771735 1.99844 4.39563 1.34694 −1.77237 −1.39195
1.11 −1.64709 2.61371 0.712902 −2.48483 −4.30501 −4.71248 2.11996 3.83146 4.09274
1.12 −1.07687 0.369892 −0.840351 1.90084 −0.398325 −1.51757 3.05869 −2.86318 −2.04695
1.13 −0.831936 3.06444 −1.30788 −1.49771 −2.54941 −2.02107 2.75195 6.39077 1.24600
1.14 −0.720723 2.26158 −1.48056 −4.09116 −1.62997 4.35864 2.50852 2.11475 2.94859
1.15 −0.653108 0.962267 −1.57345 −0.172354 −0.628465 2.47185 2.33385 −2.07404 0.112566
1.16 −0.599349 −1.60361 −1.64078 0.195832 0.961120 −2.69983 2.18210 −0.428441 −0.117372
1.17 −0.594114 1.69775 −1.64703 2.73915 −1.00866 2.94642 2.16675 −0.117635 −1.62737
1.18 −0.282682 −3.22372 −1.92009 −0.0582096 0.911287 2.82624 1.10814 7.39237 0.0164548
1.19 −0.135932 −0.0702798 −1.98152 −1.52124 0.00955329 −2.32668 0.541217 −2.99506 0.206785
1.20 0.264008 0.508006 −1.93030 2.92638 0.134118 −4.76393 −1.03763 −2.74193 0.772588
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.34
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(31\) \(-1\)

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 5239.2.a.q 34
13.b even 2 1 5239.2.a.r 34
13.f odd 12 2 403.2.r.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
403.2.r.a ✓ 68 13.f odd 12 2
5239.2.a.q 34 1.a even 1 1 trivial
5239.2.a.r 34 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5239))\):

\( T_{2}^{34} + 8 T_{2}^{33} - 18 T_{2}^{32} - 296 T_{2}^{31} - 155 T_{2}^{30} + 4748 T_{2}^{29} + \cdots - 543 \) Copy content Toggle raw display
\( T_{5}^{34} + 16 T_{5}^{33} + 25 T_{5}^{32} - 892 T_{5}^{31} - 4564 T_{5}^{30} + 16488 T_{5}^{29} + \cdots + 915088 \) Copy content Toggle raw display