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Properties

Label 448.4.j.b
Level $448$
Weight $4$
Character orbit 448.j
Analytic conductor $26.433$
Analytic rank $0$
Dimension $88$
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(111,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.111");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.j (of order \(4\), 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: \(88\)
Relative dimension: \(44\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{7} - 112 q^{11} + 52 q^{21} - 160 q^{23} + 528 q^{29} - 476 q^{35} - 896 q^{37} + 8 q^{39} - 40 q^{43} - 1376 q^{49} - 1504 q^{51} - 1560 q^{53} - 8 q^{65} - 648 q^{67} + 456 q^{71} + 292 q^{77}+ \cdots - 7592 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1 0 −7.12471 − 7.12471i 0 5.44380 + 5.44380i 0 −16.6758 + 8.05711i 0 74.5229i 0
111.2 0 −6.75145 − 6.75145i 0 −10.7426 − 10.7426i 0 −7.90955 − 16.7463i 0 64.1641i 0
111.3 0 −6.00789 − 6.00789i 0 8.45808 + 8.45808i 0 17.6848 − 5.49968i 0 45.1894i 0
111.4 0 −5.85052 − 5.85052i 0 −6.82314 − 6.82314i 0 18.5177 − 0.310856i 0 41.4573i 0
111.5 0 −5.75197 − 5.75197i 0 6.05552 + 6.05552i 0 4.50707 + 17.9635i 0 39.1703i 0
111.6 0 −5.70883 − 5.70883i 0 −14.7783 − 14.7783i 0 2.64946 + 18.3298i 0 38.1815i 0
111.7 0 −5.68355 − 5.68355i 0 −2.84203 − 2.84203i 0 2.19712 − 18.3895i 0 37.6054i 0
111.8 0 −4.79702 − 4.79702i 0 −2.04624 − 2.04624i 0 −13.3740 + 12.8115i 0 19.0229i 0
111.9 0 −4.73059 − 4.73059i 0 5.14976 + 5.14976i 0 −10.9791 − 14.9150i 0 17.7570i 0
111.10 0 −4.69197 − 4.69197i 0 9.81242 + 9.81242i 0 17.7338 − 5.33982i 0 17.0292i 0
111.11 0 −3.57396 − 3.57396i 0 −2.98764 − 2.98764i 0 −17.0858 + 7.14665i 0 − 1.45369i 0
111.12 0 −3.51698 − 3.51698i 0 14.7032 + 14.7032i 0 −16.3514 − 8.69658i 0 − 2.26172i 0
111.13 0 −3.36528 − 3.36528i 0 −8.35027 − 8.35027i 0 17.6921 + 5.47632i 0 − 4.34978i 0
111.14 0 −2.83962 − 2.83962i 0 −7.83669 − 7.83669i 0 12.7115 − 13.4692i 0 − 10.8731i 0
111.15 0 −2.77244 − 2.77244i 0 13.8731 + 13.8731i 0 7.54098 + 16.9155i 0 − 11.6272i 0
111.16 0 −2.68963 − 2.68963i 0 −1.79622 − 1.79622i 0 −8.50315 + 16.4529i 0 − 12.5318i 0
111.17 0 −2.09861 − 2.09861i 0 4.88560 + 4.88560i 0 −4.27896 − 18.0192i 0 − 18.1917i 0
111.18 0 −2.04007 − 2.04007i 0 −1.81821 − 1.81821i 0 10.2362 + 15.4344i 0 − 18.6762i 0
111.19 0 −1.83086 − 1.83086i 0 −11.2147 − 11.2147i 0 −18.0262 − 4.24923i 0 − 20.2959i 0
111.20 0 −1.20488 − 1.20488i 0 −11.4522 − 11.4522i 0 −13.8299 − 12.3180i 0 − 24.0965i 0
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
16.f odd 4 1 inner
112.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.4.j.b 88
4.b odd 2 1 112.4.j.b ✓ 88
7.b odd 2 1 inner 448.4.j.b 88
16.e even 4 1 112.4.j.b ✓ 88
16.f odd 4 1 inner 448.4.j.b 88
28.d even 2 1 112.4.j.b ✓ 88
112.j even 4 1 inner 448.4.j.b 88
112.l odd 4 1 112.4.j.b ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.j.b ✓ 88 4.b odd 2 1
112.4.j.b ✓ 88 16.e even 4 1
112.4.j.b ✓ 88 28.d even 2 1
112.4.j.b ✓ 88 112.l odd 4 1
448.4.j.b 88 1.a even 1 1 trivial
448.4.j.b 88 7.b odd 2 1 inner
448.4.j.b 88 16.f odd 4 1 inner
448.4.j.b 88 112.j even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{88} + 50060 T_{3}^{84} + 1106718216 T_{3}^{80} + 14309556615488 T_{3}^{76} + \cdots + 20\!\cdots\!36 \) acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display