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Properties

Label 440.2.c.c
Level $440$
Weight $2$
Character orbit 440.c
Analytic conductor $3.513$
Analytic rank $0$
Dimension $56$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [440,2,Mod(219,440)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("440.219");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.c (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: \(56\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 4 q^{4} - 104 q^{9} - 4 q^{14} - 12 q^{16} - 24 q^{20} + 16 q^{25} - 52 q^{34} + 32 q^{36} - 8 q^{44} + 8 q^{49} + 108 q^{56} + 96 q^{59} + 44 q^{60} + 4 q^{64} - 60 q^{66} - 8 q^{70} - 24 q^{75}+ \cdots + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
219.1 −1.37885 − 0.314264i − 1.93054i 1.80248 + 0.866649i 0.651439 + 2.13907i −0.606701 + 2.66194i 0.116559i −2.21299 − 1.76144i −0.727003 −0.226005 − 3.15419i
219.2 −1.37885 − 0.314264i 1.93054i 1.80248 + 0.866649i −0.651439 + 2.13907i 0.606701 − 2.66194i 0.116559i −2.21299 − 1.76144i −0.727003 1.57047 − 2.74474i
219.3 −1.37885 + 0.314264i − 1.93054i 1.80248 − 0.866649i −0.651439 − 2.13907i 0.606701 + 2.66194i − 0.116559i −2.21299 + 1.76144i −0.727003 1.57047 + 2.74474i
219.4 −1.37885 + 0.314264i 1.93054i 1.80248 − 0.866649i 0.651439 − 2.13907i −0.606701 − 2.66194i − 0.116559i −2.21299 + 1.76144i −0.727003 −0.226005 + 3.15419i
219.5 −1.35227 − 0.413972i − 3.03912i 1.65725 + 1.11960i 2.09783 − 0.774024i −1.25811 + 4.10971i 1.67292i −1.77757 − 2.20006i −6.23628 −3.15725 + 0.178244i
219.6 −1.35227 − 0.413972i 3.03912i 1.65725 + 1.11960i −2.09783 − 0.774024i 1.25811 − 4.10971i 1.67292i −1.77757 − 2.20006i −6.23628 2.51640 + 1.91513i
219.7 −1.35227 + 0.413972i − 3.03912i 1.65725 − 1.11960i −2.09783 + 0.774024i 1.25811 + 4.10971i − 1.67292i −1.77757 + 2.20006i −6.23628 2.51640 − 1.91513i
219.8 −1.35227 + 0.413972i 3.03912i 1.65725 − 1.11960i 2.09783 + 0.774024i −1.25811 − 4.10971i − 1.67292i −1.77757 + 2.20006i −6.23628 −3.15725 − 0.178244i
219.9 −1.24688 − 0.667297i − 1.20551i 1.10943 + 1.66408i −1.79732 − 1.33028i −0.804436 + 1.50313i 3.49920i −0.272891 − 2.81523i 1.54674 1.35335 + 2.85805i
219.10 −1.24688 − 0.667297i 1.20551i 1.10943 + 1.66408i 1.79732 − 1.33028i 0.804436 − 1.50313i 3.49920i −0.272891 − 2.81523i 1.54674 −3.12874 + 0.459355i
219.11 −1.24688 + 0.667297i − 1.20551i 1.10943 − 1.66408i 1.79732 + 1.33028i 0.804436 + 1.50313i − 3.49920i −0.272891 + 2.81523i 1.54674 −3.12874 − 0.459355i
219.12 −1.24688 + 0.667297i 1.20551i 1.10943 − 1.66408i −1.79732 + 1.33028i −0.804436 − 1.50313i − 3.49920i −0.272891 + 2.81523i 1.54674 1.35335 − 2.85805i
219.13 −0.944607 − 1.05248i − 0.907092i −0.215435 + 1.98836i −1.66478 + 1.49282i −0.954697 + 0.856845i − 0.330142i 2.29622 − 1.65148i 2.17718 3.14373 + 0.342028i
219.14 −0.944607 − 1.05248i 0.907092i −0.215435 + 1.98836i 1.66478 + 1.49282i 0.954697 − 0.856845i − 0.330142i 2.29622 − 1.65148i 2.17718 −0.00140387 − 3.16228i
219.15 −0.944607 + 1.05248i − 0.907092i −0.215435 − 1.98836i 1.66478 − 1.49282i 0.954697 + 0.856845i 0.330142i 2.29622 + 1.65148i 2.17718 −0.00140387 + 3.16228i
219.16 −0.944607 + 1.05248i 0.907092i −0.215435 − 1.98836i −1.66478 − 1.49282i −0.954697 − 0.856845i 0.330142i 2.29622 + 1.65148i 2.17718 3.14373 − 0.342028i
219.17 −0.861423 − 1.12158i − 1.93862i −0.515899 + 1.93232i 2.16613 − 0.554857i −2.17432 + 1.66997i − 3.35344i 2.61166 − 1.08592i −0.758244 −2.48828 − 1.95153i
219.18 −0.861423 − 1.12158i 1.93862i −0.515899 + 1.93232i −2.16613 − 0.554857i 2.17432 − 1.66997i − 3.35344i 2.61166 − 1.08592i −0.758244 1.24364 + 2.90747i
219.19 −0.861423 + 1.12158i − 1.93862i −0.515899 − 1.93232i −2.16613 + 0.554857i 2.17432 + 1.66997i 3.35344i 2.61166 + 1.08592i −0.758244 1.24364 − 2.90747i
219.20 −0.861423 + 1.12158i 1.93862i −0.515899 − 1.93232i 2.16613 + 0.554857i −2.17432 − 1.66997i 3.35344i 2.61166 + 1.08592i −0.758244 −2.48828 + 1.95153i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 219.56
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 2 1 inner
11.b odd 2 1 inner
40.e odd 2 1 inner
55.d odd 2 1 inner
88.g even 2 1 inner
440.c 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.c.c ✓ 56
4.b odd 2 1 1760.2.c.c 56
5.b even 2 1 inner 440.2.c.c ✓ 56
8.b even 2 1 1760.2.c.c 56
8.d odd 2 1 inner 440.2.c.c ✓ 56
11.b odd 2 1 inner 440.2.c.c ✓ 56
20.d odd 2 1 1760.2.c.c 56
40.e odd 2 1 inner 440.2.c.c ✓ 56
40.f even 2 1 1760.2.c.c 56
44.c even 2 1 1760.2.c.c 56
55.d odd 2 1 inner 440.2.c.c ✓ 56
88.b odd 2 1 1760.2.c.c 56
88.g even 2 1 inner 440.2.c.c ✓ 56
220.g even 2 1 1760.2.c.c 56
440.c even 2 1 inner 440.2.c.c ✓ 56
440.o odd 2 1 1760.2.c.c 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.c.c ✓ 56 1.a even 1 1 trivial
440.2.c.c ✓ 56 5.b even 2 1 inner
440.2.c.c ✓ 56 8.d odd 2 1 inner
440.2.c.c ✓ 56 11.b odd 2 1 inner
440.2.c.c ✓ 56 40.e odd 2 1 inner
440.2.c.c ✓ 56 55.d odd 2 1 inner
440.2.c.c ✓ 56 88.g even 2 1 inner
440.2.c.c ✓ 56 440.c even 2 1 inner
1760.2.c.c 56 4.b odd 2 1
1760.2.c.c 56 8.b even 2 1
1760.2.c.c 56 20.d odd 2 1
1760.2.c.c 56 40.f even 2 1
1760.2.c.c 56 44.c even 2 1
1760.2.c.c 56 88.b odd 2 1
1760.2.c.c 56 220.g even 2 1
1760.2.c.c 56 440.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(440, [\chi])\):

\( T_{3}^{14} + 34T_{3}^{12} + 460T_{3}^{10} + 3174T_{3}^{8} + 11911T_{3}^{6} + 23872T_{3}^{4} + 23036T_{3}^{2} + 8136 \) Copy content Toggle raw display
\( T_{7}^{14} + 48T_{7}^{12} + 889T_{7}^{10} + 7834T_{7}^{8} + 32084T_{7}^{6} + 47048T_{7}^{4} + 5344T_{7}^{2} + 64 \) Copy content Toggle raw display