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Properties

Label 473.2.d.e
Level $473$
Weight $2$
Character orbit 473.d
Analytic conductor $3.777$
Analytic rank $0$
Dimension $28$
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] = [473,2,Mod(472,473)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(473, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("473.472");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 473 = 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 473.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: \(3.77692401561\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 32 q^{4} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 32 q^{4} - 20 q^{9} + 10 q^{11} - 12 q^{14} - 28 q^{15} + 8 q^{16} + 36 q^{23} - 68 q^{25} - 36 q^{31} - 64 q^{36} - 36 q^{38} + 76 q^{44} - 4 q^{47} + 64 q^{49} + 4 q^{53} - 136 q^{56} - 72 q^{58} + 4 q^{59} + 104 q^{60} + 28 q^{64} + 128 q^{66} + 56 q^{67} - 84 q^{78} + 20 q^{81} - 76 q^{86} + 64 q^{92} - 110 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
472.1 −2.59909 − 2.97823i 4.75526 0.117207i 7.74068i 2.68373 −7.16115 −5.86986 − 0.304632i
472.2 −2.59909 2.97823i 4.75526 − 0.117207i − 7.74068i 2.68373 −7.16115 −5.86986 0.304632i
472.3 −2.38709 − 0.708405i 3.69819 3.86342i 1.69102i 2.03056 −4.05372 2.49816 − 9.22232i
472.4 −2.38709 0.708405i 3.69819 − 3.86342i − 1.69102i 2.03056 −4.05372 2.49816 9.22232i
472.5 −2.09675 − 1.56270i 2.39634 1.80196i 3.27659i −3.07814 −0.831030 0.557964 − 3.77825i
472.6 −2.09675 1.56270i 2.39634 − 1.80196i − 3.27659i −3.07814 −0.831030 0.557964 3.77825i
472.7 −1.53912 − 1.28951i 0.368904 − 1.69510i 1.98472i 2.28809 2.51046 1.33715 2.60897i
472.8 −1.53912 1.28951i 0.368904 1.69510i − 1.98472i 2.28809 2.51046 1.33715 − 2.60897i
472.9 −1.16998 − 3.00312i −0.631139 − 3.57799i 3.51361i −4.51575 3.07839 −6.01876 4.18619i
472.10 −1.16998 3.00312i −0.631139 3.57799i − 3.51361i −4.51575 3.07839 −6.01876 − 4.18619i
472.11 −0.956037 − 1.04613i −1.08599 3.54053i 1.00014i −2.95263 2.95032 1.90561 − 3.38488i
472.12 −0.956037 1.04613i −1.08599 − 3.54053i − 1.00014i −2.95263 2.95032 1.90561 3.38488i
472.13 −0.706003 − 1.55250i −1.50156 − 2.36695i 1.09607i 3.13911 2.47211 0.589732 1.67107i
472.14 −0.706003 1.55250i −1.50156 2.36695i − 1.09607i 3.13911 2.47211 0.589732 − 1.67107i
472.15 0.706003 − 1.55250i −1.50156 − 2.36695i − 1.09607i −3.13911 −2.47211 0.589732 − 1.67107i
472.16 0.706003 1.55250i −1.50156 2.36695i 1.09607i −3.13911 −2.47211 0.589732 1.67107i
472.17 0.956037 − 1.04613i −1.08599 3.54053i − 1.00014i 2.95263 −2.95032 1.90561 3.38488i
472.18 0.956037 1.04613i −1.08599 − 3.54053i 1.00014i 2.95263 −2.95032 1.90561 − 3.38488i
472.19 1.16998 − 3.00312i −0.631139 − 3.57799i − 3.51361i 4.51575 −3.07839 −6.01876 − 4.18619i
472.20 1.16998 3.00312i −0.631139 3.57799i 3.51361i 4.51575 −3.07839 −6.01876 4.18619i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 472.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
43.b odd 2 1 inner
473.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 473.2.d.e ✓ 28
11.b odd 2 1 inner 473.2.d.e ✓ 28
43.b odd 2 1 inner 473.2.d.e ✓ 28
473.d even 2 1 inner 473.2.d.e ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
473.2.d.e ✓ 28 1.a even 1 1 trivial
473.2.d.e ✓ 28 11.b odd 2 1 inner
473.2.d.e ✓ 28 43.b odd 2 1 inner
473.2.d.e ✓ 28 473.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(473, [\chi])\):

\( T_{2}^{14} - 22T_{2}^{12} + 189T_{2}^{10} - 807T_{2}^{8} + 1816T_{2}^{6} - 2130T_{2}^{4} + 1201T_{2}^{2} - 250 \) Copy content Toggle raw display
\( T_{3}^{14} + 26T_{3}^{12} + 250T_{3}^{10} + 1130T_{3}^{8} + 2653T_{3}^{6} + 3273T_{3}^{4} + 1959T_{3}^{2} + 430 \) Copy content Toggle raw display