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Properties

Label 520.2.j.b
Level $520$
Weight $2$
Character orbit 520.j
Analytic conductor $4.152$
Analytic rank $0$
Dimension $36$
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(469,520)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("520.469");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.j (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: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 8 q^{3} + 2 q^{4} - 2 q^{6} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 8 q^{3} + 2 q^{4} - 2 q^{6} + 36 q^{9} + 3 q^{10} - 10 q^{12} - 36 q^{13} - 6 q^{14} + 6 q^{16} + 7 q^{20} + 20 q^{22} - 16 q^{24} + 32 q^{27} - 11 q^{30} - 20 q^{31} + 10 q^{32} + 22 q^{34} - 12 q^{36} + 10 q^{38} - 8 q^{39} - 29 q^{40} - 26 q^{42} - 32 q^{43} + 26 q^{44} + 12 q^{45} - 6 q^{46} + 20 q^{48} - 36 q^{49} + 25 q^{50} - 2 q^{52} - 22 q^{54} - 8 q^{55} - 2 q^{56} - 10 q^{58} + 33 q^{60} + 18 q^{62} - 10 q^{64} - 28 q^{66} - 44 q^{67} - 26 q^{68} - 13 q^{70} + 28 q^{71} + 60 q^{72} + 10 q^{74} - 36 q^{76} + 2 q^{78} - 8 q^{79} - 29 q^{80} + 28 q^{81} - 34 q^{82} + 68 q^{83} - 18 q^{84} - 12 q^{86} + 2 q^{88} - 8 q^{89} + 48 q^{90} - 34 q^{92} + 16 q^{93} - 10 q^{94} + 12 q^{95} + 2 q^{96} - 38 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
469.1 −1.40912 − 0.119867i −0.190712 1.97126 + 0.337816i −1.77411 − 1.36108i 0.268737 + 0.0228602i − 4.09016i −2.73726 − 0.712315i −2.96363 2.33679 + 2.13059i
469.2 −1.40912 + 0.119867i −0.190712 1.97126 − 0.337816i −1.77411 + 1.36108i 0.268737 − 0.0228602i 4.09016i −2.73726 + 0.712315i −2.96363 2.33679 − 2.13059i
469.3 −1.35623 − 0.400797i 3.00223 1.67872 + 1.08715i 1.65971 − 1.49846i −4.07172 − 1.20328i 1.46837i −1.84101 − 2.14725i 6.01339 −2.85152 + 1.36705i
469.4 −1.35623 + 0.400797i 3.00223 1.67872 − 1.08715i 1.65971 + 1.49846i −4.07172 + 1.20328i − 1.46837i −1.84101 + 2.14725i 6.01339 −2.85152 − 1.36705i
469.5 −1.27176 − 0.618572i −0.137253 1.23474 + 1.57335i 2.07691 + 0.828531i 0.174553 + 0.0849011i − 3.21234i −0.597058 − 2.76469i −2.98116 −2.12882 − 2.33841i
469.6 −1.27176 + 0.618572i −0.137253 1.23474 − 1.57335i 2.07691 − 0.828531i 0.174553 − 0.0849011i 3.21234i −0.597058 + 2.76469i −2.98116 −2.12882 + 2.33841i
469.7 −1.26322 − 0.635825i −1.97518 1.19145 + 1.60637i 0.0576388 − 2.23532i 2.49509 + 1.25587i 2.57704i −0.483697 − 2.78676i 0.901344 −1.49409 + 2.78706i
469.8 −1.26322 + 0.635825i −1.97518 1.19145 − 1.60637i 0.0576388 + 2.23532i 2.49509 − 1.25587i − 2.57704i −0.483697 + 2.78676i 0.901344 −1.49409 − 2.78706i
469.9 −0.939649 − 1.05691i 0.918092 −0.234120 + 1.98625i −1.18780 + 1.89450i −0.862684 − 0.970341i − 0.287745i 2.31928 − 1.61893i −2.15711 3.11843 − 0.524764i
469.10 −0.939649 + 1.05691i 0.918092 −0.234120 − 1.98625i −1.18780 − 1.89450i −0.862684 + 0.970341i 0.287745i 2.31928 + 1.61893i −2.15711 3.11843 + 0.524764i
469.11 −0.864514 − 1.11920i −2.89320 −0.505233 + 1.93513i 0.761512 + 2.10240i 2.50121 + 3.23807i 0.661218i 2.60259 − 1.10749i 5.37058 1.69468 − 2.66984i
469.12 −0.864514 + 1.11920i −2.89320 −0.505233 − 1.93513i 0.761512 − 2.10240i 2.50121 − 3.23807i − 0.661218i 2.60259 + 1.10749i 5.37058 1.69468 + 2.66984i
469.13 −0.739040 − 1.20574i 2.15299 −0.907640 + 1.78219i 1.09748 − 1.94822i −1.59114 − 2.59595i − 3.34911i 2.81964 − 0.222725i 1.63535 −3.16013 + 0.116534i
469.14 −0.739040 + 1.20574i 2.15299 −0.907640 − 1.78219i 1.09748 + 1.94822i −1.59114 + 2.59595i 3.34911i 2.81964 + 0.222725i 1.63535 −3.16013 − 0.116534i
469.15 −0.235661 − 1.39444i 3.32382 −1.88893 + 0.657231i −1.37647 + 1.76219i −0.783296 − 4.63487i 3.24635i 1.36162 + 2.47911i 8.04780 2.78165 + 1.50413i
469.16 −0.235661 + 1.39444i 3.32382 −1.88893 − 0.657231i −1.37647 − 1.76219i −0.783296 + 4.63487i − 3.24635i 1.36162 − 2.47911i 8.04780 2.78165 − 1.50413i
469.17 −0.0685086 − 1.41255i −2.36486 −1.99061 + 0.193544i −2.14667 + 0.625954i 0.162014 + 3.34049i 0.617984i 0.409766 + 2.79859i 2.59258 1.03126 + 2.98940i
469.18 −0.0685086 + 1.41255i −2.36486 −1.99061 − 0.193544i −2.14667 − 0.625954i 0.162014 − 3.34049i − 0.617984i 0.409766 − 2.79859i 2.59258 1.03126 − 2.98940i
469.19 0.110689 − 1.40988i 0.434449 −1.97550 − 0.312116i 1.54684 + 1.61471i 0.0480888 − 0.612519i 2.08355i −0.658710 + 2.75065i −2.81125 2.44776 − 2.00211i
469.20 0.110689 + 1.40988i 0.434449 −1.97550 + 0.312116i 1.54684 − 1.61471i 0.0480888 + 0.612519i − 2.08355i −0.658710 − 2.75065i −2.81125 2.44776 + 2.00211i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 469.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f 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.j.b yes 36
4.b odd 2 1 2080.2.j.a 36
5.b even 2 1 520.2.j.a ✓ 36
8.b even 2 1 520.2.j.a ✓ 36
8.d odd 2 1 2080.2.j.b 36
20.d odd 2 1 2080.2.j.b 36
40.e odd 2 1 2080.2.j.a 36
40.f even 2 1 inner 520.2.j.b yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.j.a ✓ 36 5.b even 2 1
520.2.j.a ✓ 36 8.b even 2 1
520.2.j.b yes 36 1.a even 1 1 trivial
520.2.j.b yes 36 40.f even 2 1 inner
2080.2.j.a 36 4.b odd 2 1
2080.2.j.a 36 40.e odd 2 1
2080.2.j.b 36 8.d odd 2 1
2080.2.j.b 36 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 4 T_{3}^{17} - 28 T_{3}^{16} + 120 T_{3}^{15} + 299 T_{3}^{14} - 1432 T_{3}^{13} + \cdots - 64 \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\). Copy content Toggle raw display