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Properties

Label 5239.2.a.r
Level $5239$
Weight $2$
Character orbit 5239.a
Self dual yes
Analytic conductor $41.834$
Analytic rank $0$
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: \(0\)
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 −2.51897 0.871779 4.34522 3.48615 −2.19599 −2.34079 −5.90753 −2.24000 −8.78152
1.2 −2.48206 0.663833 4.16060 −0.685218 −1.64767 4.46792 −5.36274 −2.55933 1.70075
1.3 −2.46213 −0.495389 4.06206 −0.570785 1.21971 0.485018 −5.07705 −2.75459 1.40534
1.4 −2.10967 −0.521252 2.45069 3.77709 1.09967 −1.46991 −0.950815 −2.72830 −7.96840
1.5 −1.90555 2.34815 1.63114 1.14932 −4.47453 2.28537 0.702890 2.51380 −2.19008
1.6 −1.85841 −2.54430 1.45371 −3.22618 4.72837 2.78583 1.01524 3.47347 5.99558
1.7 −1.55751 −3.11259 0.425844 3.93166 4.84789 −0.386283 2.45177 6.68820 −6.12361
1.8 −1.44544 0.0252265 0.0893109 1.27994 −0.0364636 −2.11070 2.76180 −2.99936 −1.85008
1.9 −1.32498 −1.54906 −0.244422 −1.40687 2.05248 0.535818 2.97382 −0.600418 1.86407
1.10 −1.20974 1.92888 −0.536537 −3.63568 −2.33344 1.11880 3.06854 0.720590 4.39822
1.11 −0.805226 −1.45456 −1.35161 −3.57617 1.17125 −1.18499 2.69880 −0.884247 2.87963
1.12 −0.628504 −2.72152 −1.60498 1.23905 1.71048 −4.08330 2.26575 4.40665 −0.778745
1.13 −0.524150 −0.819346 −1.72527 4.38147 0.429461 2.72437 1.95260 −2.32867 −2.29655
1.14 −0.509421 2.76885 −1.74049 1.59074 −1.41051 0.293111 1.90549 4.66653 −0.810359
1.15 −0.264008 0.508006 −1.93030 −2.92638 −0.134118 4.76393 1.03763 −2.74193 0.772588
1.16 0.135932 −0.0702798 −1.98152 1.52124 −0.00955329 2.32668 −0.541217 −2.99506 0.206785
1.17 0.282682 −3.22372 −1.92009 0.0582096 −0.911287 −2.82624 −1.10814 7.39237 0.0164548
1.18 0.594114 1.69775 −1.64703 −2.73915 1.00866 −2.94642 −2.16675 −0.117635 −1.62737
1.19 0.599349 −1.60361 −1.64078 −0.195832 −0.961120 2.69983 −2.18210 −0.428441 −0.117372
1.20 0.653108 0.962267 −1.57345 0.172354 0.628465 −2.47185 −2.33385 −2.07404 0.112566
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.r 34
13.b even 2 1 5239.2.a.q 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 13.b even 2 1
5239.2.a.r 34 1.a even 1 1 trivial

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