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Properties

Label 1472.2.i.b
Level $1472$
Weight $2$
Character orbit 1472.i
Analytic conductor $11.754$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1472,2,Mod(367,1472)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1472, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1472.367");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1472.i (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: \(11.7539791775\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 368)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{3} - 4 q^{13} - 4 q^{23} - 44 q^{27} - 20 q^{29} - 16 q^{35} + 128 q^{39} - 160 q^{49} + 8 q^{55} - 80 q^{59} - 24 q^{69} + 8 q^{71} + 12 q^{75} + 40 q^{77} + 40 q^{81} - 16 q^{85} + 8 q^{87}+ \cdots - 28 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
367.1 0 −2.02263 + 2.02263i 0 2.65107 + 2.65107i 0 3.33656i 0 − 5.18204i 0
367.2 0 −2.02263 + 2.02263i 0 −2.65107 − 2.65107i 0 − 3.33656i 0 − 5.18204i 0
367.3 0 −1.81358 + 1.81358i 0 1.82931 + 1.82931i 0 − 3.62156i 0 − 3.57815i 0
367.4 0 −1.81358 + 1.81358i 0 −1.82931 − 1.82931i 0 3.62156i 0 − 3.57815i 0
367.5 0 −1.79623 + 1.79623i 0 2.02942 + 2.02942i 0 − 3.36009i 0 − 3.45290i 0
367.6 0 −1.79623 + 1.79623i 0 −2.02942 − 2.02942i 0 3.36009i 0 − 3.45290i 0
367.7 0 −1.60868 + 1.60868i 0 −1.03338 − 1.03338i 0 − 1.93918i 0 − 2.17568i 0
367.8 0 −1.60868 + 1.60868i 0 1.03338 + 1.03338i 0 1.93918i 0 − 2.17568i 0
367.9 0 −1.07531 + 1.07531i 0 2.00611 + 2.00611i 0 1.45118i 0 0.687411i 0
367.10 0 −1.07531 + 1.07531i 0 −2.00611 − 2.00611i 0 − 1.45118i 0 0.687411i 0
367.11 0 −0.860099 + 0.860099i 0 0.0592628 + 0.0592628i 0 2.65917i 0 1.52046i 0
367.12 0 −0.860099 + 0.860099i 0 −0.0592628 − 0.0592628i 0 − 2.65917i 0 1.52046i 0
367.13 0 −0.802883 + 0.802883i 0 2.06308 + 2.06308i 0 4.69475i 0 1.71076i 0
367.14 0 −0.802883 + 0.802883i 0 −2.06308 − 2.06308i 0 − 4.69475i 0 1.71076i 0
367.15 0 −0.670749 + 0.670749i 0 1.44943 + 1.44943i 0 − 2.27607i 0 2.10019i 0
367.16 0 −0.670749 + 0.670749i 0 −1.44943 − 1.44943i 0 2.27607i 0 2.10019i 0
367.17 0 −0.290642 + 0.290642i 0 −0.682402 − 0.682402i 0 1.60196i 0 2.83105i 0
367.18 0 −0.290642 + 0.290642i 0 0.682402 + 0.682402i 0 − 1.60196i 0 2.83105i 0
367.19 0 −0.112079 + 0.112079i 0 −0.729430 − 0.729430i 0 4.66292i 0 2.97488i 0
367.20 0 −0.112079 + 0.112079i 0 0.729430 + 0.729430i 0 − 4.66292i 0 2.97488i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 367.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner
23.b odd 2 1 inner
368.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1472.2.i.b 80
4.b odd 2 1 368.2.i.b ✓ 80
16.e even 4 1 368.2.i.b ✓ 80
16.f odd 4 1 inner 1472.2.i.b 80
23.b odd 2 1 inner 1472.2.i.b 80
92.b even 2 1 368.2.i.b ✓ 80
368.i even 4 1 inner 1472.2.i.b 80
368.k odd 4 1 368.2.i.b ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.2.i.b ✓ 80 4.b odd 2 1
368.2.i.b ✓ 80 16.e even 4 1
368.2.i.b ✓ 80 92.b even 2 1
368.2.i.b ✓ 80 368.k odd 4 1
1472.2.i.b 80 1.a even 1 1 trivial
1472.2.i.b 80 16.f odd 4 1 inner
1472.2.i.b 80 23.b odd 2 1 inner
1472.2.i.b 80 368.i even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 2 T_{3}^{39} + 2 T_{3}^{38} + 10 T_{3}^{37} + 207 T_{3}^{36} - 360 T_{3}^{35} + 356 T_{3}^{34} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(1472, [\chi])\). Copy content Toggle raw display