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Properties

Label 520.2.j.a
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
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{3} + 2 q^{4} - 2 q^{6} + 36 q^{9} - 13 q^{10} + 10 q^{12} + 36 q^{13} - 6 q^{14} + 6 q^{16} - 15 q^{20} - 20 q^{22} - 16 q^{24} - 32 q^{27} + 37 q^{30} - 20 q^{31} - 10 q^{32} + 22 q^{34} - 12 q^{36} - 10 q^{38} - 8 q^{39} + 23 q^{40} + 26 q^{42} + 32 q^{43} + 26 q^{44} - 12 q^{45} - 6 q^{46} - 20 q^{48} - 36 q^{49} - 37 q^{50} + 2 q^{52} - 22 q^{54} - 8 q^{55} - 2 q^{56} + 10 q^{58} - 57 q^{60} - 18 q^{62} - 10 q^{64} - 28 q^{66} + 44 q^{67} + 26 q^{68} + 81 q^{70} + 28 q^{71} - 60 q^{72} + 10 q^{74} - 36 q^{76} - 2 q^{78} - 8 q^{79} + 65 q^{80} + 28 q^{81} + 34 q^{82} - 68 q^{83} - 18 q^{84} - 12 q^{86} - 2 q^{88} - 8 q^{89} - 88 q^{90} + 34 q^{92} - 16 q^{93} - 10 q^{94} + 12 q^{95} + 2 q^{96} + 38 q^{98}+O(q^{100}) \) 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 −1.41338 − 0.0486040i 1.28444 1.99528 + 0.137392i 0.102281 + 2.23373i −1.81540 − 0.0624289i − 0.878987i −2.81340 − 0.291165i −1.35022 −0.0359940 − 3.16207i
469.2 −1.41338 + 0.0486040i 1.28444 1.99528 − 0.137392i 0.102281 − 2.23373i −1.81540 + 0.0624289i 0.878987i −2.81340 + 0.291165i −1.35022 −0.0359940 + 3.16207i
469.3 −1.40724 − 0.140267i −2.42022 1.96065 + 0.394779i 1.90368 + 1.17304i 3.40583 + 0.339476i 3.11812i −2.70373 − 0.830563i 2.85745 −2.51439 − 1.91777i
469.4 −1.40724 + 0.140267i −2.42022 1.96065 − 0.394779i 1.90368 − 1.17304i 3.40583 − 0.339476i − 3.11812i −2.70373 + 0.830563i 2.85745 −2.51439 + 1.91777i
469.5 −1.25753 − 0.647009i 2.75252 1.16276 + 1.62727i −2.22751 − 0.195492i −3.46138 − 1.78091i − 2.88285i −0.409346 − 2.79865i 4.57638 2.67467 + 1.68705i
469.6 −1.25753 + 0.647009i 2.75252 1.16276 − 1.62727i −2.22751 + 0.195492i −3.46138 + 1.78091i 2.88285i −0.409346 + 2.79865i 4.57638 2.67467 − 1.68705i
469.7 −1.14972 − 0.823497i −1.34955 0.643705 + 1.89358i −2.19960 + 0.402172i 1.55161 + 1.11135i 1.03254i 0.819277 − 2.70717i −1.17871 2.86011 + 1.34898i
469.8 −1.14972 + 0.823497i −1.34955 0.643705 − 1.89358i −2.19960 − 0.402172i 1.55161 − 1.11135i − 1.03254i 0.819277 + 2.70717i −1.17871 2.86011 − 1.34898i
469.9 −1.11757 − 0.866629i 1.60954 0.497910 + 1.93703i 1.87746 + 1.21455i −1.79876 − 1.39487i 5.19937i 1.12224 − 2.59626i −0.409396 −1.04562 − 2.98441i
469.10 −1.11757 + 0.866629i 1.60954 0.497910 − 1.93703i 1.87746 − 1.21455i −1.79876 + 1.39487i − 5.19937i 1.12224 + 2.59626i −0.409396 −1.04562 + 2.98441i
469.11 −0.769904 − 1.18628i −2.67002 −0.814497 + 1.82663i −0.868389 − 2.06056i 2.05566 + 3.16738i − 4.78032i 2.79398 − 0.440115i 4.12901 −1.77581 + 2.61658i
469.12 −0.769904 + 1.18628i −2.67002 −0.814497 − 1.82663i −0.868389 + 2.06056i 2.05566 − 3.16738i 4.78032i 2.79398 + 0.440115i 4.12901 −1.77581 − 2.61658i
469.13 −0.641501 − 1.26035i −1.27105 −1.17695 + 1.61703i 2.22857 − 0.182959i 0.815378 + 1.60196i − 0.963418i 2.79303 + 0.446045i −1.38444 −1.66022 − 2.69141i
469.14 −0.641501 + 1.26035i −1.27105 −1.17695 − 1.61703i 2.22857 + 0.182959i 0.815378 − 1.60196i 0.963418i 2.79303 − 0.446045i −1.38444 −1.66022 + 2.69141i
469.15 −0.280181 − 1.38618i 0.334714 −1.84300 + 0.776762i −0.101467 + 2.23376i −0.0937803 − 0.463974i − 3.23581i 1.59311 + 2.33710i −2.88797 3.12483 − 0.485206i
469.16 −0.280181 + 1.38618i 0.334714 −1.84300 − 0.776762i −0.101467 − 2.23376i −0.0937803 + 0.463974i 3.23581i 1.59311 − 2.33710i −2.88797 3.12483 + 0.485206i
469.17 −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 −2.10533 + 2.35958i
469.18 −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 −2.10533 − 2.35958i
469.19 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 −0.737128 − 3.07517i
469.20 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 −0.737128 + 3.07517i
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.a ✓ 36
4.b odd 2 1 2080.2.j.b 36
5.b even 2 1 520.2.j.b yes 36
8.b even 2 1 520.2.j.b yes 36
8.d odd 2 1 2080.2.j.a 36
20.d odd 2 1 2080.2.j.a 36
40.e odd 2 1 2080.2.j.b 36
40.f even 2 1 inner 520.2.j.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.j.a ✓ 36 1.a even 1 1 trivial
520.2.j.a ✓ 36 40.f even 2 1 inner
520.2.j.b yes 36 5.b even 2 1
520.2.j.b yes 36 8.b even 2 1
2080.2.j.a 36 8.d odd 2 1
2080.2.j.a 36 20.d odd 2 1
2080.2.j.b 36 4.b odd 2 1
2080.2.j.b 36 40.e 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