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Properties

Label 483.2.e.a
Level $483$
Weight $2$
Character orbit 483.e
Analytic conductor $3.857$
Analytic rank $0$
Dimension $48$
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] = [483,2,Mod(344,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("483.344");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.e (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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{3} - 40 q^{4} + 6 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{3} - 40 q^{4} + 6 q^{6} + 4 q^{9} + 22 q^{12} - 8 q^{13} + 24 q^{16} - 14 q^{18} - 12 q^{24} + 88 q^{25} - 16 q^{27} - 8 q^{31} - 10 q^{36} + 8 q^{46} - 98 q^{48} - 48 q^{49} + 28 q^{52} - 28 q^{54} - 92 q^{58} + 92 q^{64} + 8 q^{69} + 8 q^{70} + 60 q^{72} + 40 q^{73} - 80 q^{75} + 114 q^{78} - 28 q^{81} - 20 q^{82} - 40 q^{85} - 28 q^{87} - 76 q^{93} - 12 q^{94} - 6 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
344.1 − 2.64103i −1.65834 + 0.499912i −4.97503 −2.97721 1.32028 + 4.37972i − 1.00000i 7.85715i 2.50018 − 1.65805i 7.86291i
344.2 − 2.64103i −1.65834 + 0.499912i −4.97503 2.97721 1.32028 + 4.37972i 1.00000i 7.85715i 2.50018 − 1.65805i − 7.86291i
344.3 − 2.35218i 1.12077 − 1.32055i −3.53277 −2.32116 −3.10619 − 2.63627i 1.00000i 3.60535i −0.487727 − 2.96009i 5.45980i
344.4 − 2.35218i 1.12077 − 1.32055i −3.53277 2.32116 −3.10619 − 2.63627i − 1.00000i 3.60535i −0.487727 − 2.96009i − 5.45980i
344.5 − 2.27845i −0.307505 + 1.70454i −3.19131 −0.401391 3.88369 + 0.700634i 1.00000i 2.71435i −2.81088 − 1.04831i 0.914548i
344.6 − 2.27845i −0.307505 + 1.70454i −3.19131 0.401391 3.88369 + 0.700634i − 1.00000i 2.71435i −2.81088 − 1.04831i − 0.914548i
344.7 − 2.26372i −1.42471 − 0.984983i −3.12443 −3.16242 −2.22973 + 3.22515i 1.00000i 2.54539i 1.05962 + 2.80664i 7.15884i
344.8 − 2.26372i −1.42471 − 0.984983i −3.12443 3.16242 −2.22973 + 3.22515i − 1.00000i 2.54539i 1.05962 + 2.80664i − 7.15884i
344.9 − 1.98227i 0.949942 + 1.44831i −1.92939 −4.16373 2.87095 − 1.88304i − 1.00000i − 0.139961i −1.19522 + 2.75163i 8.25364i
344.10 − 1.98227i 0.949942 + 1.44831i −1.92939 4.16373 2.87095 − 1.88304i 1.00000i − 0.139961i −1.19522 + 2.75163i − 8.25364i
344.11 − 1.67931i 1.72585 + 0.146449i −0.820088 −1.07520 0.245933 − 2.89824i 1.00000i − 1.98144i 2.95711 + 0.505497i 1.80559i
344.12 − 1.67931i 1.72585 + 0.146449i −0.820088 1.07520 0.245933 − 2.89824i − 1.00000i − 1.98144i 2.95711 + 0.505497i − 1.80559i
344.13 − 1.44207i 0.0348577 − 1.73170i −0.0795775 −2.43097 −2.49724 − 0.0502674i − 1.00000i − 2.76939i −2.99757 − 0.120726i 3.50564i
344.14 − 1.44207i 0.0348577 − 1.73170i −0.0795775 2.43097 −2.49724 − 0.0502674i 1.00000i − 2.76939i −2.99757 − 0.120726i − 3.50564i
344.15 − 1.06181i −1.10086 + 1.33720i 0.872554 −4.21218 1.41986 + 1.16891i 1.00000i − 3.05011i −0.576205 − 2.94414i 4.47255i
344.16 − 1.06181i −1.10086 + 1.33720i 0.872554 4.21218 1.41986 + 1.16891i − 1.00000i − 3.05011i −0.576205 − 2.94414i − 4.47255i
344.17 − 0.800018i −1.73165 − 0.0370806i 1.35997 −1.73737 −0.0296651 + 1.38535i − 1.00000i − 2.68804i 2.99725 + 0.128422i 1.38993i
344.18 − 0.800018i −1.73165 − 0.0370806i 1.35997 1.73737 −0.0296651 + 1.38535i 1.00000i − 2.68804i 2.99725 + 0.128422i − 1.38993i
344.19 − 0.548382i 1.56357 − 0.745160i 1.69928 −2.39464 −0.408632 − 0.857431i − 1.00000i − 2.02862i 1.88947 − 2.33021i 1.31318i
344.20 − 0.548382i 1.56357 − 0.745160i 1.69928 2.39464 −0.408632 − 0.857431i 1.00000i − 2.02862i 1.88947 − 2.33021i − 1.31318i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 344.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.b odd 2 1 inner
69.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.e.a ✓ 48
3.b odd 2 1 inner 483.2.e.a ✓ 48
23.b odd 2 1 inner 483.2.e.a ✓ 48
69.c even 2 1 inner 483.2.e.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.e.a ✓ 48 1.a even 1 1 trivial
483.2.e.a ✓ 48 3.b odd 2 1 inner
483.2.e.a ✓ 48 23.b odd 2 1 inner
483.2.e.a ✓ 48 69.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(483, [\chi])\).