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Properties

Label 285.2.h.a
Level $285$
Weight $2$
Character orbit 285.h
Analytic conductor $2.276$
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] = [285,2,Mod(56,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, 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("285.56");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.h (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: \(2.27573645761\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{4} - 4 q^{6} - 16 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 28 q^{4} - 4 q^{6} - 16 q^{7} + 12 q^{9} + 36 q^{16} - 24 q^{19} - 20 q^{24} - 28 q^{25} - 56 q^{28} + 8 q^{30} - 20 q^{36} - 8 q^{39} - 8 q^{42} + 24 q^{43} + 8 q^{45} + 60 q^{49} - 28 q^{54} - 8 q^{58} - 24 q^{61} + 48 q^{63} - 4 q^{64} - 16 q^{66} - 72 q^{76} + 12 q^{81} - 40 q^{82} - 16 q^{85} - 24 q^{87} + 8 q^{93} - 68 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1 −2.66824 0.143711 − 1.72608i 5.11951 − 1.00000i −0.383456 + 4.60559i −3.27641 −8.32359 −2.95869 − 0.496114i 2.66824i
56.2 −2.66824 0.143711 + 1.72608i 5.11951 1.00000i −0.383456 − 4.60559i −3.27641 −8.32359 −2.95869 + 0.496114i − 2.66824i
56.3 −2.36049 1.61464 − 0.626851i 3.57192 1.00000i −3.81134 + 1.47968i 2.15365 −3.71051 2.21412 − 2.02427i − 2.36049i
56.4 −2.36049 1.61464 + 0.626851i 3.57192 − 1.00000i −3.81134 − 1.47968i 2.15365 −3.71051 2.21412 + 2.02427i 2.36049i
56.5 −2.02062 −1.58918 − 0.688844i 2.08292 1.00000i 3.21114 + 1.39189i −2.69011 −0.167558 2.05099 + 2.18939i − 2.02062i
56.6 −2.02062 −1.58918 + 0.688844i 2.08292 − 1.00000i 3.21114 − 1.39189i −2.69011 −0.167558 2.05099 − 2.18939i 2.02062i
56.7 −1.57778 −1.18476 − 1.26346i 0.489381 − 1.00000i 1.86929 + 1.99346i 3.13519 2.38342 −0.192677 + 2.99381i 1.57778i
56.8 −1.57778 −1.18476 + 1.26346i 0.489381 1.00000i 1.86929 − 1.99346i 3.13519 2.38342 −0.192677 − 2.99381i − 1.57778i
56.9 −1.24076 1.48157 − 0.897192i −0.460525 1.00000i −1.83827 + 1.11320i −4.78567 3.05291 1.39009 − 2.65850i − 1.24076i
56.10 −1.24076 1.48157 + 0.897192i −0.460525 − 1.00000i −1.83827 − 1.11320i −4.78567 3.05291 1.39009 + 2.65850i 1.24076i
56.11 −0.409913 −0.589151 − 1.62877i −1.83197 1.00000i 0.241501 + 0.667656i −1.21606 1.57078 −2.30580 + 1.91919i − 0.409913i
56.12 −0.409913 −0.589151 + 1.62877i −1.83197 − 1.00000i 0.241501 − 0.667656i −1.21606 1.57078 −2.30580 − 1.91919i 0.409913i
56.13 −0.169599 1.70323 − 0.314664i −1.97124 − 1.00000i −0.288866 + 0.0533667i 2.67942 0.673518 2.80197 − 1.07189i 0.169599i
56.14 −0.169599 1.70323 + 0.314664i −1.97124 1.00000i −0.288866 − 0.0533667i 2.67942 0.673518 2.80197 + 1.07189i − 0.169599i
56.15 0.169599 −1.70323 − 0.314664i −1.97124 1.00000i −0.288866 − 0.0533667i 2.67942 −0.673518 2.80197 + 1.07189i 0.169599i
56.16 0.169599 −1.70323 + 0.314664i −1.97124 − 1.00000i −0.288866 + 0.0533667i 2.67942 −0.673518 2.80197 − 1.07189i − 0.169599i
56.17 0.409913 0.589151 − 1.62877i −1.83197 − 1.00000i 0.241501 − 0.667656i −1.21606 −1.57078 −2.30580 − 1.91919i − 0.409913i
56.18 0.409913 0.589151 + 1.62877i −1.83197 1.00000i 0.241501 + 0.667656i −1.21606 −1.57078 −2.30580 + 1.91919i 0.409913i
56.19 1.24076 −1.48157 − 0.897192i −0.460525 − 1.00000i −1.83827 − 1.11320i −4.78567 −3.05291 1.39009 + 2.65850i − 1.24076i
56.20 1.24076 −1.48157 + 0.897192i −0.460525 1.00000i −1.83827 + 1.11320i −4.78567 −3.05291 1.39009 − 2.65850i 1.24076i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 56.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.b odd 2 1 inner
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 285.2.h.a ✓ 28
3.b odd 2 1 inner 285.2.h.a ✓ 28
19.b odd 2 1 inner 285.2.h.a ✓ 28
57.d even 2 1 inner 285.2.h.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.h.a ✓ 28 1.a even 1 1 trivial
285.2.h.a ✓ 28 3.b odd 2 1 inner
285.2.h.a ✓ 28 19.b odd 2 1 inner
285.2.h.a ✓ 28 57.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(285, [\chi])\).