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Properties

Label 880.2.co.b
Level $880$
Weight $2$
Character orbit 880.co
Analytic conductor $7.027$
Analytic rank $0$
Dimension $192$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [880,2,Mod(47,880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(880, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 0, 5, 16]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("880.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.co (of order \(20\), degree \(8\), 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: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(24\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 8 q^{5} - 16 q^{13} - 12 q^{17} - 16 q^{21} + 4 q^{25} + 72 q^{33} - 12 q^{37} + 48 q^{41} + 48 q^{45} + 92 q^{53} + 48 q^{57} + 48 q^{61} + 128 q^{65} + 16 q^{73} - 24 q^{77} + 32 q^{81} - 16 q^{85} - 180 q^{93} + 52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −1.41714 + 2.78130i 0 2.23427 + 0.0896165i 0 1.90544 + 3.73963i 0 −3.96398 − 5.45595i 0
47.2 0 −1.33045 + 2.61115i 0 −1.58357 + 1.57870i 0 −0.322438 − 0.632820i 0 −3.28468 − 4.52097i 0
47.3 0 −1.18245 + 2.32068i 0 0.659089 + 2.13673i 0 −1.96468 − 3.85590i 0 −2.22403 − 3.06111i 0
47.4 0 −1.04038 + 2.04187i 0 2.04850 − 0.896456i 0 −0.197283 − 0.387191i 0 −1.32347 − 1.82160i 0
47.5 0 −0.990507 + 1.94398i 0 −1.08955 − 1.95266i 0 0.631670 + 1.23972i 0 −1.03460 − 1.42400i 0
47.6 0 −0.801582 + 1.57319i 0 −1.67845 − 1.47743i 0 −1.96067 − 3.84803i 0 −0.0690469 − 0.0950349i 0
47.7 0 −0.783847 + 1.53839i 0 −1.35150 + 1.78141i 0 0.875451 + 1.71817i 0 0.0111394 + 0.0153321i 0
47.8 0 −0.619906 + 1.21663i 0 0.711828 − 2.11974i 0 −1.12045 − 2.19901i 0 0.667441 + 0.918653i 0
47.9 0 −0.462768 + 0.908233i 0 −1.40636 + 1.73844i 0 2.34769 + 4.60760i 0 1.15262 + 1.58645i 0
47.10 0 −0.328292 + 0.644308i 0 1.53607 + 1.62496i 0 −0.679425 − 1.33345i 0 1.45600 + 2.00401i 0
47.11 0 −0.139851 + 0.274472i 0 −2.20029 − 0.398397i 0 0.338043 + 0.663447i 0 1.70758 + 2.35028i 0
47.12 0 −0.0581806 + 0.114186i 0 2.03218 − 0.932875i 0 0.790318 + 1.55109i 0 1.75370 + 2.41376i 0
47.13 0 0.0581806 − 0.114186i 0 2.03218 − 0.932875i 0 −0.790318 − 1.55109i 0 1.75370 + 2.41376i 0
47.14 0 0.139851 − 0.274472i 0 −2.20029 − 0.398397i 0 −0.338043 − 0.663447i 0 1.70758 + 2.35028i 0
47.15 0 0.328292 − 0.644308i 0 1.53607 + 1.62496i 0 0.679425 + 1.33345i 0 1.45600 + 2.00401i 0
47.16 0 0.462768 − 0.908233i 0 −1.40636 + 1.73844i 0 −2.34769 − 4.60760i 0 1.15262 + 1.58645i 0
47.17 0 0.619906 − 1.21663i 0 0.711828 − 2.11974i 0 1.12045 + 2.19901i 0 0.667441 + 0.918653i 0
47.18 0 0.783847 − 1.53839i 0 −1.35150 + 1.78141i 0 −0.875451 − 1.71817i 0 0.0111394 + 0.0153321i 0
47.19 0 0.801582 − 1.57319i 0 −1.67845 − 1.47743i 0 1.96067 + 3.84803i 0 −0.0690469 − 0.0950349i 0
47.20 0 0.990507 − 1.94398i 0 −1.08955 − 1.95266i 0 −0.631670 − 1.23972i 0 −1.03460 − 1.42400i 0
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
11.c even 5 1 inner
20.e even 4 1 inner
44.h odd 10 1 inner
55.k odd 20 1 inner
220.v even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 880.2.co.b ✓ 192
4.b odd 2 1 inner 880.2.co.b ✓ 192
5.c odd 4 1 inner 880.2.co.b ✓ 192
11.c even 5 1 inner 880.2.co.b ✓ 192
20.e even 4 1 inner 880.2.co.b ✓ 192
44.h odd 10 1 inner 880.2.co.b ✓ 192
55.k odd 20 1 inner 880.2.co.b ✓ 192
220.v even 20 1 inner 880.2.co.b ✓ 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
880.2.co.b ✓ 192 1.a even 1 1 trivial
880.2.co.b ✓ 192 4.b odd 2 1 inner
880.2.co.b ✓ 192 5.c odd 4 1 inner
880.2.co.b ✓ 192 11.c even 5 1 inner
880.2.co.b ✓ 192 20.e even 4 1 inner
880.2.co.b ✓ 192 44.h odd 10 1 inner
880.2.co.b ✓ 192 55.k odd 20 1 inner
880.2.co.b ✓ 192 220.v even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{192} - 368 T_{3}^{188} + 75898 T_{3}^{184} - 11772098 T_{3}^{180} + 1543949087 T_{3}^{176} + \cdots + 70\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(880, [\chi])\). Copy content Toggle raw display