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Properties

Label 1122.3.d.a
Level $1122$
Weight $3$
Character orbit 1122.d
Analytic conductor $30.572$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1122,3,Mod(373,1122)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1122, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1122.373");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1122.d (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: \(30.5722856612\)
Analytic rank: \(0\)
Dimension: \(72\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 144 q^{4} - 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q - 144 q^{4} - 216 q^{9} - 24 q^{15} + 288 q^{16} - 208 q^{25} - 12 q^{33} + 24 q^{34} + 432 q^{36} - 160 q^{38} - 48 q^{42} + 280 q^{47} + 400 q^{49} + 136 q^{53} - 260 q^{55} - 72 q^{59} + 48 q^{60} - 576 q^{64} - 96 q^{66} - 104 q^{67} - 504 q^{69} + 288 q^{70} + 336 q^{77} + 648 q^{81} + 32 q^{86} + 296 q^{89} - 96 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1 − 1.41421i − 1.73205i −2.00000 9.39230i −2.44949 10.2466 2.82843i −3.00000 13.2827
373.2 − 1.41421i 1.73205i −2.00000 − 9.39230i 2.44949 −10.2466 2.82843i −3.00000 −13.2827
373.3 1.41421i 1.73205i −2.00000 − 9.39230i −2.44949 10.2466 − 2.82843i −3.00000 13.2827
373.4 1.41421i − 1.73205i −2.00000 9.39230i 2.44949 −10.2466 − 2.82843i −3.00000 −13.2827
373.5 − 1.41421i 1.73205i −2.00000 3.82664i 2.44949 13.2588 2.82843i −3.00000 5.41169
373.6 − 1.41421i − 1.73205i −2.00000 − 3.82664i −2.44949 −13.2588 2.82843i −3.00000 −5.41169
373.7 1.41421i − 1.73205i −2.00000 − 3.82664i 2.44949 13.2588 − 2.82843i −3.00000 5.41169
373.8 1.41421i 1.73205i −2.00000 3.82664i −2.44949 −13.2588 − 2.82843i −3.00000 −5.41169
373.9 − 1.41421i − 1.73205i −2.00000 − 8.87653i −2.44949 7.87926 2.82843i −3.00000 −12.5533
373.10 − 1.41421i 1.73205i −2.00000 8.87653i 2.44949 −7.87926 2.82843i −3.00000 12.5533
373.11 1.41421i 1.73205i −2.00000 8.87653i −2.44949 7.87926 − 2.82843i −3.00000 −12.5533
373.12 1.41421i − 1.73205i −2.00000 − 8.87653i 2.44949 −7.87926 − 2.82843i −3.00000 12.5533
373.13 − 1.41421i 1.73205i −2.00000 7.88226i 2.44949 8.44553 2.82843i −3.00000 11.1472
373.14 − 1.41421i − 1.73205i −2.00000 − 7.88226i −2.44949 −8.44553 2.82843i −3.00000 −11.1472
373.15 1.41421i − 1.73205i −2.00000 − 7.88226i 2.44949 8.44553 − 2.82843i −3.00000 11.1472
373.16 1.41421i 1.73205i −2.00000 7.88226i −2.44949 −8.44553 − 2.82843i −3.00000 −11.1472
373.17 − 1.41421i − 1.73205i −2.00000 1.13418i −2.44949 10.9244 2.82843i −3.00000 1.60397
373.18 − 1.41421i 1.73205i −2.00000 − 1.13418i 2.44949 −10.9244 2.82843i −3.00000 −1.60397
373.19 1.41421i 1.73205i −2.00000 − 1.13418i −2.44949 10.9244 − 2.82843i −3.00000 1.60397
373.20 1.41421i − 1.73205i −2.00000 1.13418i 2.44949 −10.9244 − 2.82843i −3.00000 −1.60397
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.72
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
17.b even 2 1 inner
187.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1122.3.d.a ✓ 72
11.b odd 2 1 inner 1122.3.d.a ✓ 72
17.b even 2 1 inner 1122.3.d.a ✓ 72
187.b odd 2 1 inner 1122.3.d.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1122.3.d.a ✓ 72 1.a even 1 1 trivial
1122.3.d.a ✓ 72 11.b odd 2 1 inner
1122.3.d.a ✓ 72 17.b even 2 1 inner
1122.3.d.a ✓ 72 187.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(1122, [\chi])\).