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Properties

Label 224.4.q.a
Level $224$
Weight $4$
Character orbit 224.q
Analytic conductor $13.216$
Analytic rank $0$
Dimension $44$
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] = [224,4,Mod(47,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.47");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 224.q (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: \(13.2164278413\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 56)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q + 6 q^{3} + 160 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q + 6 q^{3} + 160 q^{9} + 22 q^{11} - 6 q^{17} + 6 q^{19} - 352 q^{25} - 6 q^{33} - 18 q^{35} - 800 q^{43} + 356 q^{49} + 810 q^{51} + 220 q^{57} + 2070 q^{59} - 252 q^{65} + 98 q^{67} + 642 q^{73} - 744 q^{75} + 182 q^{81} - 1278 q^{89} + 424 q^{91} - 1000 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −8.09715 + 4.67489i 0 −1.43573 + 2.48675i 0 −16.2515 − 8.88188i 0 30.2092 − 52.3239i 0
47.2 0 −8.09715 + 4.67489i 0 1.43573 − 2.48675i 0 16.2515 + 8.88188i 0 30.2092 − 52.3239i 0
47.3 0 −5.56780 + 3.21457i 0 −7.75010 + 13.4236i 0 7.54552 + 16.9135i 0 7.16690 − 12.4134i 0
47.4 0 −5.56780 + 3.21457i 0 7.75010 − 13.4236i 0 −7.54552 − 16.9135i 0 7.16690 − 12.4134i 0
47.5 0 −4.91139 + 2.83559i 0 −0.988157 + 1.71154i 0 7.56347 − 16.9054i 0 2.58118 − 4.47073i 0
47.6 0 −4.91139 + 2.83559i 0 0.988157 − 1.71154i 0 −7.56347 + 16.9054i 0 2.58118 − 4.47073i 0
47.7 0 −3.30655 + 1.90904i 0 −10.3944 + 18.0036i 0 −18.4727 + 1.32673i 0 −6.21115 + 10.7580i 0
47.8 0 −3.30655 + 1.90904i 0 10.3944 − 18.0036i 0 18.4727 − 1.32673i 0 −6.21115 + 10.7580i 0
47.9 0 −2.16602 + 1.25055i 0 −7.11229 + 12.3188i 0 11.6310 − 14.4125i 0 −10.3722 + 17.9652i 0
47.10 0 −2.16602 + 1.25055i 0 7.11229 − 12.3188i 0 −11.6310 + 14.4125i 0 −10.3722 + 17.9652i 0
47.11 0 1.25815 − 0.726393i 0 −3.13408 + 5.42838i 0 17.9566 − 4.53443i 0 −12.4447 + 21.5549i 0
47.12 0 1.25815 − 0.726393i 0 3.13408 − 5.42838i 0 −17.9566 + 4.53443i 0 −12.4447 + 21.5549i 0
47.13 0 1.97959 − 1.14292i 0 −1.53618 + 2.66074i 0 −12.8356 − 13.3509i 0 −10.8875 + 18.8577i 0
47.14 0 1.97959 − 1.14292i 0 1.53618 − 2.66074i 0 12.8356 + 13.3509i 0 −10.8875 + 18.8577i 0
47.15 0 2.41498 − 1.39429i 0 −4.21456 + 7.29983i 0 4.99534 + 17.8339i 0 −9.61190 + 16.6483i 0
47.16 0 2.41498 − 1.39429i 0 4.21456 − 7.29983i 0 −4.99534 − 17.8339i 0 −9.61190 + 16.6483i 0
47.17 0 5.18267 − 2.99222i 0 −8.14672 + 14.1105i 0 −15.0041 − 10.8571i 0 4.40672 − 7.63267i 0
47.18 0 5.18267 − 2.99222i 0 8.14672 − 14.1105i 0 15.0041 + 10.8571i 0 4.40672 − 7.63267i 0
47.19 0 7.33084 − 4.23246i 0 −6.81740 + 11.8081i 0 −6.02295 + 17.5135i 0 22.3275 − 38.6723i 0
47.20 0 7.33084 − 4.23246i 0 6.81740 − 11.8081i 0 6.02295 − 17.5135i 0 22.3275 − 38.6723i 0
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
8.d odd 2 1 inner
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 224.4.q.a 44
4.b odd 2 1 56.4.m.a ✓ 44
7.d odd 6 1 inner 224.4.q.a 44
8.b even 2 1 56.4.m.a ✓ 44
8.d odd 2 1 inner 224.4.q.a 44
28.f even 6 1 56.4.m.a ✓ 44
56.j odd 6 1 56.4.m.a ✓ 44
56.m even 6 1 inner 224.4.q.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.m.a ✓ 44 4.b odd 2 1
56.4.m.a ✓ 44 8.b even 2 1
56.4.m.a ✓ 44 28.f even 6 1
56.4.m.a ✓ 44 56.j odd 6 1
224.4.q.a 44 1.a even 1 1 trivial
224.4.q.a 44 7.d odd 6 1 inner
224.4.q.a 44 8.d odd 2 1 inner
224.4.q.a 44 56.m even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(224, [\chi])\).