Properties

Label 448.4.p.i
Level $448$
Weight $4$
Character orbit 448.p
Analytic conductor $26.433$
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] = [448,4,Mod(255,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.255");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.p (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: \(26.4328556826\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 224)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 216 q^{9} + 104 q^{21} + 432 q^{25} - 112 q^{29} + 72 q^{33} + 504 q^{37} + 1320 q^{45} + 160 q^{49} - 392 q^{53} + 1360 q^{57} - 600 q^{61} - 744 q^{65} - 648 q^{73} + 2880 q^{77} - 400 q^{81} + 240 q^{85} + 3816 q^{89} - 2872 q^{93}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1 0 −4.91971 + 8.52120i 0 4.22368 2.43854i 0 18.4862 + 1.12186i 0 −34.9072 60.4610i 0
255.2 0 −4.28593 + 7.42345i 0 −13.1506 + 7.59249i 0 −18.3478 + 2.52164i 0 −23.2384 40.2500i 0
255.3 0 −4.15093 + 7.18962i 0 −10.5034 + 6.06415i 0 0.898453 18.4985i 0 −20.9604 36.3045i 0
255.4 0 −3.65183 + 6.32516i 0 12.0402 6.95142i 0 −1.67411 + 18.4444i 0 −13.1717 22.8141i 0
255.5 0 −3.12400 + 5.41092i 0 0.225941 0.130447i 0 −14.3061 + 11.7616i 0 −6.01870 10.4247i 0
255.6 0 −3.02960 + 5.24743i 0 12.5389 7.23935i 0 −0.978923 18.4944i 0 −4.85698 8.41254i 0
255.7 0 −2.69336 + 4.66504i 0 −12.5403 + 7.24016i 0 18.3374 + 2.59641i 0 −1.00837 1.74655i 0
255.8 0 −1.86026 + 3.22206i 0 4.38985 2.53448i 0 −18.5198 + 0.132997i 0 6.57889 + 11.3950i 0
255.9 0 −1.62864 + 2.82088i 0 16.4571 9.50153i 0 15.1537 10.6473i 0 8.19508 + 14.1943i 0
255.10 0 −1.35521 + 2.34729i 0 −3.53108 + 2.03867i 0 13.7196 + 12.4408i 0 9.82683 + 17.0206i 0
255.11 0 −0.692224 + 1.19897i 0 4.20674 2.42876i 0 −6.91230 17.1820i 0 12.5417 + 21.7228i 0
255.12 0 −0.490265 + 0.849164i 0 −14.3571 + 8.28906i 0 6.74534 17.2482i 0 13.0193 + 22.5501i 0
255.13 0 0.490265 0.849164i 0 −14.3571 + 8.28906i 0 −6.74534 + 17.2482i 0 13.0193 + 22.5501i 0
255.14 0 0.692224 1.19897i 0 4.20674 2.42876i 0 6.91230 + 17.1820i 0 12.5417 + 21.7228i 0
255.15 0 1.35521 2.34729i 0 −3.53108 + 2.03867i 0 −13.7196 12.4408i 0 9.82683 + 17.0206i 0
255.16 0 1.62864 2.82088i 0 16.4571 9.50153i 0 −15.1537 + 10.6473i 0 8.19508 + 14.1943i 0
255.17 0 1.86026 3.22206i 0 4.38985 2.53448i 0 18.5198 0.132997i 0 6.57889 + 11.3950i 0
255.18 0 2.69336 4.66504i 0 −12.5403 + 7.24016i 0 −18.3374 2.59641i 0 −1.00837 1.74655i 0
255.19 0 3.02960 5.24743i 0 12.5389 7.23935i 0 0.978923 + 18.4944i 0 −4.85698 8.41254i 0
255.20 0 3.12400 5.41092i 0 0.225941 0.130447i 0 14.3061 11.7616i 0 −6.01870 10.4247i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 255.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.4.p.i 48
4.b odd 2 1 inner 448.4.p.i 48
7.d odd 6 1 inner 448.4.p.i 48
8.b even 2 1 224.4.p.a 48
8.d odd 2 1 224.4.p.a 48
28.f even 6 1 inner 448.4.p.i 48
56.j odd 6 1 224.4.p.a 48
56.m even 6 1 224.4.p.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.4.p.a 48 8.b even 2 1
224.4.p.a 48 8.d odd 2 1
224.4.p.a 48 56.j odd 6 1
224.4.p.a 48 56.m even 6 1
448.4.p.i 48 1.a even 1 1 trivial
448.4.p.i 48 4.b odd 2 1 inner
448.4.p.i 48 7.d odd 6 1 inner
448.4.p.i 48 28.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + 432 T_{3}^{46} + 106534 T_{3}^{44} + 17868192 T_{3}^{42} + 2257274037 T_{3}^{40} + 223414613312 T_{3}^{38} + 17851857344466 T_{3}^{36} + \cdots + 46\!\cdots\!21 \) acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display