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Properties

Label 720.2.bm.h
Level $720$
Weight $2$
Character orbit 720.bm
Analytic conductor $5.749$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [720,2,Mod(109,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.bm (of order \(4\), degree \(2\), 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: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{10} + 16 q^{14} - 4 q^{16} + 8 q^{19} + 40 q^{26} - 48 q^{31} - 28 q^{34} - 24 q^{35} - 16 q^{40} + 40 q^{44} - 4 q^{46} + 48 q^{49} + 32 q^{50} - 48 q^{56} + 32 q^{59} + 16 q^{61} + 48 q^{64} - 16 q^{65} - 40 q^{74} + 60 q^{76} - 96 q^{79} - 72 q^{80} - 16 q^{86} - 32 q^{91} + 44 q^{94} + 48 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −1.40976 − 0.112110i 0 1.97486 + 0.316097i 0.466917 + 2.18678i 0 −1.00010 −2.74865 − 0.667024i 0 −0.413083 − 3.13518i
109.2 −1.39033 − 0.258835i 0 1.86601 + 0.719729i −2.19925 + 0.404088i 0 −1.81567 −2.40807 − 1.48364i 0 3.16227 + 0.00742837i
109.3 −1.36038 + 0.386472i 0 1.70128 − 1.05150i 1.03097 − 1.98421i 0 3.91927 −1.90801 + 2.08794i 0 −0.635669 + 3.09773i
109.4 −1.22294 − 0.710230i 0 0.991146 + 1.73713i −0.607542 − 2.15195i 0 −2.25286 0.0216568 − 2.82834i 0 −0.785395 + 3.06319i
109.5 −1.20386 + 0.742113i 0 0.898535 − 1.78679i −0.860885 − 2.06370i 0 −0.707398 0.244298 + 2.81786i 0 2.56788 + 1.84553i
109.6 −1.06224 − 0.933621i 0 0.256702 + 1.98346i 1.24079 − 1.86022i 0 −1.58988 1.57912 − 2.34657i 0 −3.05476 + 0.817572i
109.7 −0.903247 + 1.08818i 0 −0.368289 − 1.96580i 2.09919 + 0.770325i 0 3.05002 2.47181 + 1.37484i 0 −2.73434 + 1.58851i
109.8 −0.750333 − 1.19875i 0 −0.874002 + 1.79892i −1.95942 + 1.07735i 0 1.22137 2.81225 − 0.302081i 0 2.76169 + 1.54047i
109.9 −0.550383 + 1.30272i 0 −1.39416 − 1.43399i 2.23019 + 0.162008i 0 −2.93661 2.63541 − 1.02695i 0 −1.43851 + 2.81615i
109.10 −0.456856 + 1.33839i 0 −1.58257 − 1.22290i −1.65754 − 1.50085i 0 2.58977 2.35972 − 1.55940i 0 2.76598 − 1.53277i
109.11 −0.382275 + 1.36157i 0 −1.70773 − 1.04099i −1.75308 + 1.38805i 0 −4.66030 2.07020 − 1.92725i 0 −1.21977 − 2.91756i
109.12 −0.345118 − 1.37146i 0 −1.76179 + 0.946629i 0.561697 + 2.16437i 0 −4.51614 1.90629 + 2.08952i 0 2.77449 − 1.51731i
109.13 0.345118 + 1.37146i 0 −1.76179 + 0.946629i −2.16437 − 0.561697i 0 4.51614 −1.90629 − 2.08952i 0 0.0233792 − 3.16219i
109.14 0.382275 − 1.36157i 0 −1.70773 − 1.04099i −1.38805 + 1.75308i 0 4.66030 −2.07020 + 1.92725i 0 1.85633 + 2.56009i
109.15 0.456856 − 1.33839i 0 −1.58257 − 1.22290i 1.50085 + 1.65754i 0 −2.58977 −2.35972 + 1.55940i 0 2.90411 − 1.25146i
109.16 0.550383 − 1.30272i 0 −1.39416 − 1.43399i −0.162008 − 2.23019i 0 2.93661 −2.63541 + 1.02695i 0 −2.99448 − 1.01641i
109.17 0.750333 + 1.19875i 0 −0.874002 + 1.79892i −1.07735 + 1.95942i 0 −1.22137 −2.81225 + 0.302081i 0 −3.15722 + 0.178735i
109.18 0.903247 − 1.08818i 0 −0.368289 − 1.96580i −0.770325 − 2.09919i 0 −3.05002 −2.47181 − 1.37484i 0 −2.98010 − 1.05783i
109.19 1.06224 + 0.933621i 0 0.256702 + 1.98346i 1.86022 − 1.24079i 0 1.58988 −1.57912 + 2.34657i 0 3.13443 + 0.418728i
109.20 1.20386 − 0.742113i 0 0.898535 − 1.78679i 2.06370 + 0.860885i 0 0.707398 −0.244298 − 2.81786i 0 3.12328 − 0.495122i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.e even 4 1 inner
80.q even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.bm.h 48
3.b odd 2 1 240.2.bl.a ✓ 48
5.b even 2 1 inner 720.2.bm.h 48
12.b even 2 1 960.2.bl.a 48
15.d odd 2 1 240.2.bl.a ✓ 48
16.e even 4 1 inner 720.2.bm.h 48
24.f even 2 1 1920.2.bl.b 48
24.h odd 2 1 1920.2.bl.a 48
48.i odd 4 1 240.2.bl.a ✓ 48
48.i odd 4 1 1920.2.bl.a 48
48.k even 4 1 960.2.bl.a 48
48.k even 4 1 1920.2.bl.b 48
60.h even 2 1 960.2.bl.a 48
80.q even 4 1 inner 720.2.bm.h 48
120.i odd 2 1 1920.2.bl.a 48
120.m even 2 1 1920.2.bl.b 48
240.t even 4 1 960.2.bl.a 48
240.t even 4 1 1920.2.bl.b 48
240.bm odd 4 1 240.2.bl.a ✓ 48
240.bm odd 4 1 1920.2.bl.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.bl.a ✓ 48 3.b odd 2 1
240.2.bl.a ✓ 48 15.d odd 2 1
240.2.bl.a ✓ 48 48.i odd 4 1
240.2.bl.a ✓ 48 240.bm odd 4 1
720.2.bm.h 48 1.a even 1 1 trivial
720.2.bm.h 48 5.b even 2 1 inner
720.2.bm.h 48 16.e even 4 1 inner
720.2.bm.h 48 80.q even 4 1 inner
960.2.bl.a 48 12.b even 2 1
960.2.bl.a 48 48.k even 4 1
960.2.bl.a 48 60.h even 2 1
960.2.bl.a 48 240.t even 4 1
1920.2.bl.a 48 24.h odd 2 1
1920.2.bl.a 48 48.i odd 4 1
1920.2.bl.a 48 120.i odd 2 1
1920.2.bl.a 48 240.bm odd 4 1
1920.2.bl.b 48 24.f even 2 1
1920.2.bl.b 48 48.k even 4 1
1920.2.bl.b 48 120.m even 2 1
1920.2.bl.b 48 240.t even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(720, [\chi])\):

\( T_{7}^{24} - 96 T_{7}^{22} + 3920 T_{7}^{20} - 89504 T_{7}^{18} + 1265760 T_{7}^{16} - 11615360 T_{7}^{14} + \cdots + 115605504 \) Copy content Toggle raw display
\( T_{11}^{24} - 96 T_{11}^{21} + 1744 T_{11}^{20} - 2944 T_{11}^{19} + 4608 T_{11}^{18} + \cdots + 1849688064 \) Copy content Toggle raw display
\( T_{13}^{48} + 4928 T_{13}^{44} + 8830400 T_{13}^{40} + 7352414208 T_{13}^{36} + 3017663342080 T_{13}^{32} + \cdots + 4294967296 \) Copy content Toggle raw display