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Properties

Label 1440.2.cc.a
Level $1440$
Weight $2$
Character orbit 1440.cc
Analytic conductor $11.498$
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] = [1440,2,Mod(911,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, 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("1440.911");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.cc (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: \(11.4984578911\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 24 q^{5} - 4 q^{21} - 24 q^{25} - 12 q^{27} - 8 q^{33} - 16 q^{39} + 12 q^{41} - 12 q^{47} + 24 q^{49} + 20 q^{51} + 4 q^{57} + 36 q^{59} + 12 q^{61} - 56 q^{63} - 40 q^{69} - 8 q^{81} + 60 q^{83} - 36 q^{87} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
911.1 0 −1.72054 + 0.199393i 0 −0.500000 − 0.866025i 0 3.97204 + 2.29326i 0 2.92049 − 0.686124i 0
911.2 0 −1.67034 + 0.458221i 0 −0.500000 − 0.866025i 0 −3.22730 − 1.86328i 0 2.58007 − 1.53077i 0
911.3 0 −1.62132 + 0.609351i 0 −0.500000 − 0.866025i 0 −0.518944 − 0.299612i 0 2.25738 − 1.97591i 0
911.4 0 −1.59734 − 0.669699i 0 −0.500000 − 0.866025i 0 −2.53202 − 1.46186i 0 2.10301 + 2.13948i 0
911.5 0 −1.46420 − 0.925262i 0 −0.500000 − 0.866025i 0 0.947055 + 0.546782i 0 1.28778 + 2.70954i 0
911.6 0 −1.36390 − 1.06761i 0 −0.500000 − 0.866025i 0 −1.02179 − 0.589933i 0 0.720424 + 2.91221i 0
911.7 0 −1.28580 + 1.16048i 0 −0.500000 − 0.866025i 0 1.13105 + 0.653010i 0 0.306582 − 2.98429i 0
911.8 0 −0.875353 + 1.49458i 0 −0.500000 − 0.866025i 0 1.05351 + 0.608247i 0 −1.46751 − 2.61656i 0
911.9 0 −0.801968 + 1.53520i 0 −0.500000 − 0.866025i 0 −4.07138 − 2.35061i 0 −1.71369 − 2.46237i 0
911.10 0 −0.424591 − 1.67920i 0 −0.500000 − 0.866025i 0 3.88456 + 2.24275i 0 −2.63944 + 1.42595i 0
911.11 0 −0.284002 − 1.70861i 0 −0.500000 − 0.866025i 0 2.24682 + 1.29720i 0 −2.83869 + 0.970497i 0
911.12 0 −0.0478668 − 1.73139i 0 −0.500000 − 0.866025i 0 −1.21691 − 0.702581i 0 −2.99542 + 0.165752i 0
911.13 0 0.0300586 + 1.73179i 0 −0.500000 − 0.866025i 0 3.45090 + 1.99238i 0 −2.99819 + 0.104110i 0
911.14 0 0.452164 − 1.67199i 0 −0.500000 − 0.866025i 0 −0.550736 − 0.317967i 0 −2.59109 − 1.51203i 0
911.15 0 0.478797 + 1.66456i 0 −0.500000 − 0.866025i 0 1.88846 + 1.09030i 0 −2.54151 + 1.59397i 0
911.16 0 0.857820 − 1.50471i 0 −0.500000 − 0.866025i 0 −0.661806 − 0.382094i 0 −1.52829 − 2.58154i 0
911.17 0 0.952268 + 1.44678i 0 −0.500000 − 0.866025i 0 −2.56287 − 1.47968i 0 −1.18637 + 2.75545i 0
911.18 0 1.18636 + 1.26196i 0 −0.500000 − 0.866025i 0 −2.20775 − 1.27465i 0 −0.185091 + 2.99428i 0
911.19 0 1.29784 − 1.14700i 0 −0.500000 − 0.866025i 0 −3.14000 − 1.81288i 0 0.368796 − 2.97725i 0
911.20 0 1.41981 + 0.992033i 0 −0.500000 − 0.866025i 0 4.17077 + 2.40800i 0 1.03174 + 2.81700i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 911.24
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 1440.2.cc.a 48
3.b odd 2 1 4320.2.cc.b 48
4.b odd 2 1 360.2.bm.a ✓ 48
8.b even 2 1 360.2.bm.b yes 48
8.d odd 2 1 1440.2.cc.b 48
9.c even 3 1 4320.2.cc.a 48
9.d odd 6 1 1440.2.cc.b 48
12.b even 2 1 1080.2.bm.b 48
24.f even 2 1 4320.2.cc.a 48
24.h odd 2 1 1080.2.bm.a 48
36.f odd 6 1 1080.2.bm.a 48
36.h even 6 1 360.2.bm.b yes 48
72.j odd 6 1 360.2.bm.a ✓ 48
72.l even 6 1 inner 1440.2.cc.a 48
72.n even 6 1 1080.2.bm.b 48
72.p odd 6 1 4320.2.cc.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.bm.a ✓ 48 4.b odd 2 1
360.2.bm.a ✓ 48 72.j odd 6 1
360.2.bm.b yes 48 8.b even 2 1
360.2.bm.b yes 48 36.h even 6 1
1080.2.bm.a 48 24.h odd 2 1
1080.2.bm.a 48 36.f odd 6 1
1080.2.bm.b 48 12.b even 2 1
1080.2.bm.b 48 72.n even 6 1
1440.2.cc.a 48 1.a even 1 1 trivial
1440.2.cc.a 48 72.l even 6 1 inner
1440.2.cc.b 48 8.d odd 2 1
1440.2.cc.b 48 9.d odd 6 1
4320.2.cc.a 48 9.c even 3 1
4320.2.cc.a 48 24.f even 2 1
4320.2.cc.b 48 3.b odd 2 1
4320.2.cc.b 48 72.p odd 6 1