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Properties

Label 1080.2.bm.b
Level $1080$
Weight $2$
Character orbit 1080.bm
Analytic conductor $8.624$
Analytic rank $0$
Dimension $48$
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] = [1080,2,Mod(251,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 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("1080.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.bm (of order \(6\), degree \(2\), not 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: \(8.62384341830\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{5} + 6 q^{8} - 15 q^{14} + 12 q^{16} + 21 q^{22} - 24 q^{25} + 33 q^{34} + 33 q^{38} - 6 q^{40} - 12 q^{41} + 24 q^{44} - 6 q^{46} - 12 q^{47} + 24 q^{49} - 36 q^{52} - 21 q^{56} - 51 q^{58} + 36 q^{59} + 12 q^{61} - 42 q^{62} - 12 q^{64} - 57 q^{68} - 15 q^{70} - 30 q^{74} + 57 q^{76} - 18 q^{82} + 60 q^{83} - 27 q^{86} + 57 q^{88} + 51 q^{92} + 57 q^{94} - 42 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1 −1.39113 + 0.254469i 0 1.87049 − 0.708000i 0.500000 + 0.866025i 0 2.40441 + 1.38819i −2.42193 + 1.46090i 0 −0.915942 − 1.07752i
251.2 −1.38197 − 0.300285i 0 1.81966 + 0.829967i 0.500000 + 0.866025i 0 −1.13105 − 0.653010i −2.26548 − 1.69340i 0 −0.430929 − 1.34696i
251.3 −1.37962 − 0.310881i 0 1.80671 + 0.857795i 0.500000 + 0.866025i 0 1.21691 + 0.702581i −2.22590 − 1.74510i 0 −0.420580 − 1.35023i
251.4 −1.26042 + 0.641367i 0 1.17730 − 1.61678i 0.500000 + 0.866025i 0 −1.88846 − 1.09030i −0.446938 + 2.79289i 0 −1.18565 − 0.770869i
251.5 −1.20329 + 0.743030i 0 0.895814 − 1.78816i 0.500000 + 0.866025i 0 −0.947055 − 0.546782i 0.250732 + 2.81729i 0 −1.24513 − 0.670565i
251.6 −0.962641 − 1.03601i 0 −0.146644 + 1.99462i 0.500000 + 0.866025i 0 1.68164 + 0.970893i 2.20761 − 1.76818i 0 0.415893 − 1.35168i
251.7 −0.893891 − 1.09588i 0 −0.401917 + 1.95920i 0.500000 + 0.866025i 0 2.53202 + 1.46186i 2.50632 − 1.31086i 0 0.502116 − 1.32207i
251.8 −0.767212 + 1.18802i 0 −0.822770 − 1.82292i 0.500000 + 0.866025i 0 4.07138 + 2.35061i 2.79690 + 0.421104i 0 −1.41246 − 0.0704168i
251.9 −0.652259 + 1.25481i 0 −1.14912 − 1.63693i 0.500000 + 0.866025i 0 −1.06767 − 0.616420i 2.80356 − 0.374228i 0 −1.41283 + 0.0625343i
251.10 −0.524388 − 1.31340i 0 −1.45003 + 1.37746i 0.500000 + 0.866025i 0 −3.45090 − 1.99238i 2.56954 + 1.18215i 0 0.875243 − 1.11083i
251.11 −0.158141 − 1.40534i 0 −1.94998 + 0.444484i 0.500000 + 0.866025i 0 −2.24682 − 1.29720i 0.933024 + 2.67011i 0 1.13799 − 0.839626i
251.12 −0.102357 + 1.41050i 0 −1.97905 − 0.288749i 0.500000 + 0.866025i 0 −3.97204 − 2.29326i 0.609850 − 2.76190i 0 −1.27271 + 0.616609i
251.13 −0.0218455 + 1.41404i 0 −1.99905 − 0.0617811i 0.500000 + 0.866025i 0 0.550736 + 0.317967i 0.131031 − 2.82539i 0 −1.23552 + 0.688104i
251.14 0.325988 − 1.37613i 0 −1.78746 − 0.897203i 0.500000 + 0.866025i 0 0.518944 + 0.299612i −1.81736 + 2.16730i 0 1.35476 − 0.405751i
251.15 0.477617 − 1.33112i 0 −1.54376 − 1.27153i 0.500000 + 0.866025i 0 −1.98473 − 1.14588i −2.42989 + 1.44763i 0 1.39159 − 0.251932i
251.16 0.513744 − 1.31760i 0 −1.47213 − 1.35382i 0.500000 + 0.866025i 0 2.20775 + 1.27465i −2.54009 + 1.24417i 0 1.39795 − 0.213885i
251.17 0.537760 + 1.30798i 0 −1.42163 + 1.40676i 0.500000 + 0.866025i 0 3.22730 + 1.86328i −2.60451 − 1.10297i 0 −0.863865 + 1.11970i
251.18 1.06985 + 0.924887i 0 0.289169 + 1.97898i 0.500000 + 0.866025i 0 3.14000 + 1.81288i −1.52097 + 2.38467i 0 −0.266049 + 1.38896i
251.19 1.10684 + 0.880287i 0 0.450191 + 1.94867i 0.500000 + 0.866025i 0 −3.88456 − 2.24275i −1.21710 + 2.55317i 0 −0.208931 + 1.39870i
251.20 1.27893 − 0.603598i 0 1.27134 − 1.54392i 0.500000 + 0.866025i 0 −4.17077 − 2.40800i 0.694050 − 2.74195i 0 1.16220 + 0.805790i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
72.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.2.bm.b 48
3.b odd 2 1 360.2.bm.a ✓ 48
4.b odd 2 1 4320.2.cc.b 48
8.b even 2 1 4320.2.cc.a 48
8.d odd 2 1 1080.2.bm.a 48
9.c even 3 1 360.2.bm.b yes 48
9.d odd 6 1 1080.2.bm.a 48
12.b even 2 1 1440.2.cc.a 48
24.f even 2 1 360.2.bm.b yes 48
24.h odd 2 1 1440.2.cc.b 48
36.f odd 6 1 1440.2.cc.b 48
36.h even 6 1 4320.2.cc.a 48
72.j odd 6 1 4320.2.cc.b 48
72.l even 6 1 inner 1080.2.bm.b 48
72.n even 6 1 1440.2.cc.a 48
72.p odd 6 1 360.2.bm.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bm.a ✓ 48 3.b odd 2 1
360.2.bm.a ✓ 48 72.p odd 6 1
360.2.bm.b yes 48 9.c even 3 1
360.2.bm.b yes 48 24.f even 2 1
1080.2.bm.a 48 8.d odd 2 1
1080.2.bm.a 48 9.d odd 6 1
1080.2.bm.b 48 1.a even 1 1 trivial
1080.2.bm.b 48 72.l even 6 1 inner
1440.2.cc.a 48 12.b even 2 1
1440.2.cc.a 48 72.n even 6 1
1440.2.cc.b 48 24.h odd 2 1
1440.2.cc.b 48 36.f odd 6 1
4320.2.cc.a 48 8.b even 2 1
4320.2.cc.a 48 36.h even 6 1
4320.2.cc.b 48 4.b odd 2 1
4320.2.cc.b 48 72.j odd 6 1