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Properties

Label 440.2.p.a
Level $440$
Weight $2$
Character orbit 440.p
Analytic conductor $3.513$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [440,2,Mod(131,440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.131");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.p (of order \(2\), degree \(1\), 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: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{4} + 48 q^{9} - 8 q^{11} + 16 q^{16} - 4 q^{20} - 4 q^{22} - 48 q^{25} - 20 q^{26} - 8 q^{33} - 8 q^{34} + 24 q^{36} - 44 q^{44} - 96 q^{48} + 48 q^{49} + 8 q^{58} + 16 q^{59} - 28 q^{60} + 4 q^{64}+ \cdots - 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
131.1 −1.41231 − 0.0734309i 2.21821 1.98922 + 0.207414i − 1.00000i −3.13279 − 0.162885i 2.12688 −2.79415 − 0.439002i 1.92044 −0.0734309 + 1.41231i
131.2 −1.41231 + 0.0734309i 2.21821 1.98922 − 0.207414i 1.00000i −3.13279 + 0.162885i 2.12688 −2.79415 + 0.439002i 1.92044 −0.0734309 − 1.41231i
131.3 −1.40209 − 0.184781i −1.29072 1.93171 + 0.518159i 1.00000i 1.80970 + 0.238500i −1.50456 −2.61269 − 1.08345i −1.33405 0.184781 − 1.40209i
131.4 −1.40209 + 0.184781i −1.29072 1.93171 − 0.518159i − 1.00000i 1.80970 − 0.238500i −1.50456 −2.61269 + 1.08345i −1.33405 0.184781 + 1.40209i
131.5 −1.31268 − 0.526174i −1.79872 1.44628 + 1.38140i 1.00000i 2.36115 + 0.946439i 4.67916 −1.17165 − 2.57434i 0.235384 0.526174 − 1.31268i
131.6 −1.31268 + 0.526174i −1.79872 1.44628 − 1.38140i − 1.00000i 2.36115 − 0.946439i 4.67916 −1.17165 + 2.57434i 0.235384 0.526174 + 1.31268i
131.7 −1.30050 − 0.555606i −3.23763 1.38260 + 1.44513i − 1.00000i 4.21054 + 1.79885i −1.25292 −0.995156 − 2.64758i 7.48225 −0.555606 + 1.30050i
131.8 −1.30050 + 0.555606i −3.23763 1.38260 − 1.44513i 1.00000i 4.21054 − 1.79885i −1.25292 −0.995156 + 2.64758i 7.48225 −0.555606 − 1.30050i
131.9 −1.29763 − 0.562281i 1.04788 1.36768 + 1.45926i − 1.00000i −1.35976 − 0.589201i −2.10014 −0.954227 − 2.66260i −1.90195 −0.562281 + 1.29763i
131.10 −1.29763 + 0.562281i 1.04788 1.36768 − 1.45926i 1.00000i −1.35976 + 0.589201i −2.10014 −0.954227 + 2.66260i −1.90195 −0.562281 − 1.29763i
131.11 −1.14906 − 0.824415i 1.21401 0.640680 + 1.89461i 1.00000i −1.39497 − 1.00084i −4.53805 0.825760 − 2.70520i −1.52619 0.824415 − 1.14906i
131.12 −1.14906 + 0.824415i 1.21401 0.640680 − 1.89461i − 1.00000i −1.39497 + 1.00084i −4.53805 0.825760 + 2.70520i −1.52619 0.824415 + 1.14906i
131.13 −0.913040 − 1.07998i 3.16023 −0.332715 + 1.97213i 1.00000i −2.88542 − 3.41299i 3.07151 2.43364 − 1.44131i 6.98705 1.07998 − 0.913040i
131.14 −0.913040 + 1.07998i 3.16023 −0.332715 − 1.97213i − 1.00000i −2.88542 + 3.41299i 3.07151 2.43364 + 1.44131i 6.98705 1.07998 + 0.913040i
131.15 −0.719724 − 1.21737i −0.178950 −0.963995 + 1.75235i 1.00000i 0.128795 + 0.217849i −0.405414 2.82707 − 0.0876636i −2.96798 1.21737 − 0.719724i
131.16 −0.719724 + 1.21737i −0.178950 −0.963995 − 1.75235i − 1.00000i 0.128795 − 0.217849i −0.405414 2.82707 + 0.0876636i −2.96798 1.21737 + 0.719724i
131.17 −0.697721 − 1.23012i 2.53719 −1.02637 + 1.71656i − 1.00000i −1.77025 − 3.12103i −0.924276 2.82768 + 0.0648795i 3.43732 −1.23012 + 0.697721i
131.18 −0.697721 + 1.23012i 2.53719 −1.02637 − 1.71656i 1.00000i −1.77025 + 3.12103i −0.924276 2.82768 − 0.0648795i 3.43732 −1.23012 − 0.697721i
131.19 −0.458142 − 1.33795i −1.38393 −1.58021 + 1.22594i − 1.00000i 0.634037 + 1.85163i −1.48233 2.36421 + 1.55258i −1.08474 −1.33795 + 0.458142i
131.20 −0.458142 + 1.33795i −1.38393 −1.58021 − 1.22594i 1.00000i 0.634037 − 1.85163i −1.48233 2.36421 − 1.55258i −1.08474 −1.33795 − 0.458142i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 131.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
11.b odd 2 1 inner
88.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 440.2.p.a ✓ 48
4.b odd 2 1 1760.2.p.a 48
8.b even 2 1 1760.2.p.a 48
8.d odd 2 1 inner 440.2.p.a ✓ 48
11.b odd 2 1 inner 440.2.p.a ✓ 48
44.c even 2 1 1760.2.p.a 48
88.b odd 2 1 1760.2.p.a 48
88.g even 2 1 inner 440.2.p.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.p.a ✓ 48 1.a even 1 1 trivial
440.2.p.a ✓ 48 8.d odd 2 1 inner
440.2.p.a ✓ 48 11.b odd 2 1 inner
440.2.p.a ✓ 48 88.g even 2 1 inner
1760.2.p.a 48 4.b odd 2 1
1760.2.p.a 48 8.b even 2 1
1760.2.p.a 48 44.c even 2 1
1760.2.p.a 48 88.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(440, [\chi])\).