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Properties

Label 270.4.f.b
Level $270$
Weight $4$
Character orbit 270.f
Analytic conductor $15.931$
Analytic rank $0$
Dimension $24$
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] = [270,4,Mod(53,270)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(270, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("270.53");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 270.f (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: \(15.9305157015\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{7} + 48 q^{10} - 168 q^{13} - 384 q^{16} + 192 q^{22} + 72 q^{25} + 96 q^{28} + 504 q^{31} - 480 q^{37} + 192 q^{40} + 96 q^{43} + 144 q^{46} + 672 q^{52} - 2616 q^{55} - 1776 q^{58} - 1344 q^{61} - 768 q^{67} + 384 q^{70} + 4032 q^{73} + 1152 q^{76} - 336 q^{82} + 6264 q^{85} + 768 q^{88} + 1680 q^{91} - 144 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −1.41421 − 1.41421i 0 4.00000i −10.9770 + 2.12276i 0 17.9871 − 17.9871i 5.65685 − 5.65685i 0 18.5258 + 12.5217i
53.2 −1.41421 − 1.41421i 0 4.00000i 11.0071 + 1.96063i 0 14.6053 − 14.6053i 5.65685 − 5.65685i 0 −12.7936 − 18.3391i
53.3 −1.41421 − 1.41421i 0 4.00000i 5.83234 − 9.53854i 0 −3.50204 + 3.50204i 5.65685 − 5.65685i 0 −21.7377 + 5.24136i
53.4 −1.41421 − 1.41421i 0 4.00000i 5.14463 + 9.92637i 0 −1.68712 + 1.68712i 5.65685 − 5.65685i 0 6.76240 − 21.3136i
53.5 −1.41421 − 1.41421i 0 4.00000i −2.24097 + 10.9535i 0 −23.8734 + 23.8734i 5.65685 − 5.65685i 0 18.6597 − 12.3213i
53.6 −1.41421 − 1.41421i 0 4.00000i −8.76612 − 6.93939i 0 2.47012 − 2.47012i 5.65685 − 5.65685i 0 2.58338 + 22.2109i
53.7 1.41421 + 1.41421i 0 4.00000i 10.9770 − 2.12276i 0 17.9871 − 17.9871i −5.65685 + 5.65685i 0 18.5258 + 12.5217i
53.8 1.41421 + 1.41421i 0 4.00000i 8.76612 + 6.93939i 0 2.47012 − 2.47012i −5.65685 + 5.65685i 0 2.58338 + 22.2109i
53.9 1.41421 + 1.41421i 0 4.00000i −11.0071 − 1.96063i 0 14.6053 − 14.6053i −5.65685 + 5.65685i 0 −12.7936 − 18.3391i
53.10 1.41421 + 1.41421i 0 4.00000i −5.83234 + 9.53854i 0 −3.50204 + 3.50204i −5.65685 + 5.65685i 0 −21.7377 + 5.24136i
53.11 1.41421 + 1.41421i 0 4.00000i −5.14463 − 9.92637i 0 −1.68712 + 1.68712i −5.65685 + 5.65685i 0 6.76240 − 21.3136i
53.12 1.41421 + 1.41421i 0 4.00000i 2.24097 − 10.9535i 0 −23.8734 + 23.8734i −5.65685 + 5.65685i 0 18.6597 − 12.3213i
107.1 −1.41421 + 1.41421i 0 − 4.00000i −10.9770 − 2.12276i 0 17.9871 + 17.9871i 5.65685 + 5.65685i 0 18.5258 − 12.5217i
107.2 −1.41421 + 1.41421i 0 − 4.00000i 11.0071 − 1.96063i 0 14.6053 + 14.6053i 5.65685 + 5.65685i 0 −12.7936 + 18.3391i
107.3 −1.41421 + 1.41421i 0 − 4.00000i 5.83234 + 9.53854i 0 −3.50204 − 3.50204i 5.65685 + 5.65685i 0 −21.7377 − 5.24136i
107.4 −1.41421 + 1.41421i 0 − 4.00000i 5.14463 − 9.92637i 0 −1.68712 − 1.68712i 5.65685 + 5.65685i 0 6.76240 + 21.3136i
107.5 −1.41421 + 1.41421i 0 − 4.00000i −2.24097 − 10.9535i 0 −23.8734 − 23.8734i 5.65685 + 5.65685i 0 18.6597 + 12.3213i
107.6 −1.41421 + 1.41421i 0 − 4.00000i −8.76612 + 6.93939i 0 2.47012 + 2.47012i 5.65685 + 5.65685i 0 2.58338 − 22.2109i
107.7 1.41421 − 1.41421i 0 − 4.00000i 10.9770 + 2.12276i 0 17.9871 + 17.9871i −5.65685 − 5.65685i 0 18.5258 − 12.5217i
107.8 1.41421 − 1.41421i 0 − 4.00000i −5.83234 − 9.53854i 0 −3.50204 − 3.50204i −5.65685 − 5.65685i 0 −21.7377 − 5.24136i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 270.4.f.b ✓ 24
3.b odd 2 1 inner 270.4.f.b ✓ 24
5.c odd 4 1 inner 270.4.f.b ✓ 24
15.e even 4 1 inner 270.4.f.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.f.b ✓ 24 1.a even 1 1 trivial
270.4.f.b ✓ 24 3.b odd 2 1 inner
270.4.f.b ✓ 24 5.c odd 4 1 inner
270.4.f.b ✓ 24 15.e even 4 1 inner