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Properties

Label 520.2.g.b
Level $520$
Weight $2$
Character orbit 520.g
Analytic conductor $4.152$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [520,2,Mod(261,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("520.261");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.g (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(4.15222090511\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 2 q^{6} + 24 q^{7} + 2 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} + 2 q^{6} + 24 q^{7} + 2 q^{8} - 24 q^{9} - 2 q^{10} + 10 q^{12} + 8 q^{14} - 4 q^{15} - 12 q^{16} - 10 q^{18} - 2 q^{22} + 12 q^{23} - 38 q^{24} - 24 q^{25} + 2 q^{26} + 28 q^{28} - 10 q^{30} - 44 q^{31} - 38 q^{32} - 8 q^{33} + 24 q^{34} + 40 q^{36} - 18 q^{38} + 4 q^{39} + 2 q^{40} + 8 q^{41} + 48 q^{42} + 14 q^{44} - 18 q^{46} - 56 q^{47} - 22 q^{48} + 24 q^{49} - 2 q^{50} - 72 q^{54} + 20 q^{55} - 56 q^{56} + 8 q^{57} + 8 q^{58} - 2 q^{60} - 38 q^{62} - 40 q^{63} - 24 q^{64} + 24 q^{65} + 64 q^{66} + 68 q^{68} + 12 q^{70} + 44 q^{71} - 58 q^{72} + 68 q^{74} + 46 q^{76} + 10 q^{78} - 24 q^{79} - 16 q^{80} + 40 q^{81} + 52 q^{82} + 32 q^{84} - 78 q^{86} + 56 q^{87} - 2 q^{88} - 8 q^{89} + 30 q^{90} + 22 q^{92} - 64 q^{94} - 12 q^{95} - 70 q^{96} - 8 q^{97} + 34 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
261.1 −1.41298 − 0.0590782i − 0.566638i 1.99302 + 0.166952i − 1.00000i −0.0334759 + 0.800647i 3.79026 −2.80623 − 0.353644i 2.67892 −0.0590782 + 1.41298i
261.2 −1.41298 + 0.0590782i 0.566638i 1.99302 − 0.166952i 1.00000i −0.0334759 − 0.800647i 3.79026 −2.80623 + 0.353644i 2.67892 −0.0590782 − 1.41298i
261.3 −1.25921 − 0.643738i − 1.77634i 1.17120 + 1.62120i 1.00000i −1.14350 + 2.23678i 0.311706 −0.431160 − 2.79537i −0.155378 0.643738 − 1.25921i
261.4 −1.25921 + 0.643738i 1.77634i 1.17120 − 1.62120i − 1.00000i −1.14350 − 2.23678i 0.311706 −0.431160 + 2.79537i −0.155378 0.643738 + 1.25921i
261.5 −0.952203 − 1.04561i − 1.78588i −0.186619 + 1.99127i − 1.00000i −1.86734 + 1.70052i 2.12592 2.25980 − 1.70097i −0.189352 −1.04561 + 0.952203i
261.6 −0.952203 + 1.04561i 1.78588i −0.186619 − 1.99127i 1.00000i −1.86734 − 1.70052i 2.12592 2.25980 + 1.70097i −0.189352 −1.04561 − 0.952203i
261.7 −0.699924 − 1.22886i − 1.18763i −1.02021 + 1.72022i 1.00000i −1.45944 + 0.831254i −1.83473 2.82799 + 0.0496750i 1.58953 1.22886 − 0.699924i
261.8 −0.699924 + 1.22886i 1.18763i −1.02021 − 1.72022i − 1.00000i −1.45944 − 0.831254i −1.83473 2.82799 − 0.0496750i 1.58953 1.22886 + 0.699924i
261.9 −0.556931 − 1.29993i 3.15202i −1.37966 + 1.44795i − 1.00000i 4.09742 − 1.75546i −0.934859 2.65061 + 0.987055i −6.93524 −1.29993 + 0.556931i
261.10 −0.556931 + 1.29993i − 3.15202i −1.37966 − 1.44795i 1.00000i 4.09742 + 1.75546i −0.934859 2.65061 − 0.987055i −6.93524 −1.29993 − 0.556931i
261.11 −0.0177740 − 1.41410i − 2.22897i −1.99937 + 0.0502684i − 1.00000i −3.15199 + 0.0396175i −3.77941 0.106621 + 2.82642i −1.96829 −1.41410 + 0.0177740i
261.12 −0.0177740 + 1.41410i 2.22897i −1.99937 − 0.0502684i 1.00000i −3.15199 − 0.0396175i −3.77941 0.106621 − 2.82642i −1.96829 −1.41410 − 0.0177740i
261.13 0.179690 − 1.40275i 2.75242i −1.93542 − 0.504121i 1.00000i 3.86097 + 0.494583i 3.52133 −1.05493 + 2.62433i −4.57584 1.40275 + 0.179690i
261.14 0.179690 + 1.40275i − 2.75242i −1.93542 + 0.504121i − 1.00000i 3.86097 − 0.494583i 3.52133 −1.05493 − 2.62433i −4.57584 1.40275 − 0.179690i
261.15 0.866990 − 1.11729i 2.31565i −0.496657 − 1.93735i − 1.00000i 2.58724 + 2.00764i 4.14961 −2.59517 − 1.12476i −2.36221 −1.11729 − 0.866990i
261.16 0.866990 + 1.11729i − 2.31565i −0.496657 + 1.93735i 1.00000i 2.58724 − 2.00764i 4.14961 −2.59517 + 1.12476i −2.36221 −1.11729 + 0.866990i
261.17 1.09076 − 0.900136i 0.561263i 0.379509 − 1.96366i 1.00000i 0.505213 + 0.612203i −0.350098 −1.35361 − 2.48349i 2.68498 0.900136 + 1.09076i
261.18 1.09076 + 0.900136i − 0.561263i 0.379509 + 1.96366i − 1.00000i 0.505213 − 0.612203i −0.350098 −1.35361 + 2.48349i 2.68498 0.900136 − 1.09076i
261.19 1.15314 − 0.818700i − 1.05646i 0.659461 − 1.88815i 1.00000i −0.864921 − 1.21824i 4.50891 −0.785377 − 2.71720i 1.88390 0.818700 + 1.15314i
261.20 1.15314 + 0.818700i 1.05646i 0.659461 + 1.88815i − 1.00000i −0.864921 + 1.21824i 4.50891 −0.785377 + 2.71720i 1.88390 0.818700 − 1.15314i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 261.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 520.2.g.b ✓ 24
4.b odd 2 1 2080.2.g.a 24
8.b even 2 1 inner 520.2.g.b ✓ 24
8.d odd 2 1 2080.2.g.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.g.b ✓ 24 1.a even 1 1 trivial
520.2.g.b ✓ 24 8.b even 2 1 inner
2080.2.g.a 24 4.b odd 2 1
2080.2.g.a 24 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 48 T_{3}^{22} + 980 T_{3}^{20} + 11152 T_{3}^{18} + 77936 T_{3}^{16} + 347712 T_{3}^{14} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\). Copy content Toggle raw display