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Properties

Label 696.2.bs.a
Level $696$
Weight $2$
Character orbit 696.bs
Analytic conductor $5.558$
Analytic rank $0$
Dimension $24$
CM discriminant -24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [696,2,Mod(77,696)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(696, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([0, 14, 14, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("696.77");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 696 = 2^{3} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 696.bs (of order \(28\), degree \(12\), 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: \(5.55758798068\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{28}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} + 28 q^{5} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{2} + 28 q^{5} + 8 q^{8} + 8 q^{10} - 56 q^{11} - 28 q^{14} - 12 q^{15} + 16 q^{16} + 12 q^{18} + 16 q^{20} + 24 q^{21} + 20 q^{25} + 4 q^{29} - 24 q^{30} - 20 q^{31} + 16 q^{32} + 24 q^{36} + 16 q^{40} + 24 q^{45} - 68 q^{49} + 20 q^{50} - 40 q^{53} - 52 q^{55} + 56 q^{56} + 8 q^{58} + 32 q^{59} - 24 q^{60} + 84 q^{63} - 60 q^{66} - 176 q^{70} + 24 q^{72} - 28 q^{73} - 48 q^{75} + 20 q^{77} - 20 q^{79} + 36 q^{81} - 16 q^{83} - 48 q^{84} + 24 q^{90} - 4 q^{97} - 68 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1 0.752407 + 1.19745i −0.572060 + 1.63485i −0.867767 + 1.80194i 4.32668 + 0.987537i −2.38808 + 0.545063i −2.73893 + 1.31900i −2.81064 + 0.316683i −2.34549 − 1.87047i 2.07290 + 5.92401i
77.2 0.752407 + 1.19745i 0.572060 − 1.63485i −0.867767 + 1.80194i −1.39251 − 0.317832i 2.38808 − 0.545063i −0.774590 + 0.373023i −2.81064 + 0.316683i −2.34549 − 1.87047i −0.667149 − 1.90660i
101.1 −0.158342 − 1.40532i −0.921507 − 1.46657i −1.94986 + 0.445042i −2.44372 − 1.94880i −1.91509 + 1.52723i −0.331734 + 1.45342i 0.934170 + 2.66971i −1.30165 + 2.70291i −2.35175 + 3.74279i
101.2 −0.158342 − 1.40532i 0.921507 + 1.46657i −1.94986 + 0.445042i 2.93891 + 2.34370i 1.91509 − 1.52723i 1.02763 − 4.50234i 0.934170 + 2.66971i −1.30165 + 2.70291i 2.82830 − 4.50121i
221.1 −0.467085 − 1.33485i −1.72116 + 0.193928i −1.56366 + 1.24698i −1.82399 + 3.78756i 1.06279 + 2.20691i −2.21093 + 2.77242i 2.39490 + 1.50481i 2.92478 − 0.667563i 5.90780 + 0.665649i
221.2 −0.467085 − 1.33485i 1.72116 − 0.193928i −1.56366 + 1.24698i 1.01335 − 2.10425i −1.06279 − 2.20691i 3.29302 − 4.12932i 2.39490 + 1.50481i 2.92478 − 0.667563i −3.28218 − 0.369813i
269.1 −1.33485 − 0.467085i −0.193928 + 1.72116i 1.56366 + 1.24698i 0.661927 + 1.37450i 1.06279 − 2.20691i 2.21093 + 2.77242i −1.50481 − 2.39490i −2.92478 − 0.667563i −0.241564 − 2.14394i
269.2 −1.33485 − 0.467085i 0.193928 − 1.72116i 1.56366 + 1.24698i 1.65476 + 3.43614i −1.06279 + 2.20691i −3.29302 − 4.12932i −1.50481 − 2.39490i −2.92478 − 0.667563i −0.603888 − 5.35965i
293.1 −1.19745 − 0.752407i −1.63485 + 0.572060i 0.867767 + 1.80194i 4.13166 − 0.943024i 2.38808 + 0.545063i 0.774590 + 0.373023i 0.316683 − 2.81064i 2.34549 − 1.87047i −5.65699 − 1.97947i
293.2 −1.19745 − 0.752407i 1.63485 − 0.572060i 0.867767 + 1.80194i 0.538048 − 0.122806i −2.38808 − 0.545063i 2.73893 + 1.31900i 0.316683 − 2.81064i 2.34549 − 1.87047i −0.736684 − 0.257777i
317.1 −0.158342 + 1.40532i −0.921507 + 1.46657i −1.94986 − 0.445042i −2.44372 + 1.94880i −1.91509 − 1.52723i −0.331734 − 1.45342i 0.934170 − 2.66971i −1.30165 − 2.70291i −2.35175 − 3.74279i
317.2 −0.158342 + 1.40532i 0.921507 − 1.46657i −1.94986 − 0.445042i 2.93891 − 2.34370i 1.91509 + 1.52723i 1.02763 + 4.50234i 0.934170 − 2.66971i −1.30165 − 2.70291i 2.82830 + 4.50121i
437.1 1.40532 + 0.158342i −1.46657 − 0.921507i 1.94986 + 0.445042i 2.50069 − 1.99423i −1.91509 − 1.52723i 0.331734 + 1.45342i 2.66971 + 0.934170i 1.30165 + 2.70291i 3.83004 − 2.40657i
437.2 1.40532 + 0.158342i 1.46657 + 0.921507i 1.94986 + 0.445042i 1.89421 − 1.51058i 1.91509 + 1.52723i −1.02763 − 4.50234i 2.66971 + 0.934170i 1.30165 + 2.70291i 2.90116 − 1.82292i
461.1 0.752407 − 1.19745i −0.572060 − 1.63485i −0.867767 − 1.80194i 4.32668 − 0.987537i −2.38808 − 0.545063i −2.73893 − 1.31900i −2.81064 − 0.316683i −2.34549 + 1.87047i 2.07290 − 5.92401i
461.2 0.752407 − 1.19745i 0.572060 + 1.63485i −0.867767 − 1.80194i −1.39251 + 0.317832i 2.38808 + 0.545063i −0.774590 − 0.373023i −2.81064 − 0.316683i −2.34549 + 1.87047i −0.667149 + 1.90660i
485.1 −0.467085 + 1.33485i −1.72116 − 0.193928i −1.56366 − 1.24698i −1.82399 − 3.78756i 1.06279 − 2.20691i −2.21093 − 2.77242i 2.39490 − 1.50481i 2.92478 + 0.667563i 5.90780 − 0.665649i
485.2 −0.467085 + 1.33485i 1.72116 + 0.193928i −1.56366 − 1.24698i 1.01335 + 2.10425i −1.06279 + 2.20691i 3.29302 + 4.12932i 2.39490 − 1.50481i 2.92478 + 0.667563i −3.28218 + 0.369813i
533.1 −1.33485 + 0.467085i −0.193928 − 1.72116i 1.56366 − 1.24698i 0.661927 − 1.37450i 1.06279 + 2.20691i 2.21093 − 2.77242i −1.50481 + 2.39490i −2.92478 + 0.667563i −0.241564 + 2.14394i
533.2 −1.33485 + 0.467085i 0.193928 + 1.72116i 1.56366 − 1.24698i 1.65476 − 3.43614i −1.06279 − 2.20691i −3.29302 + 4.12932i −1.50481 + 2.39490i −2.92478 + 0.667563i −0.603888 + 5.35965i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
29.f odd 28 1 inner
696.bs even 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 696.2.bs.a ✓ 24
3.b odd 2 1 696.2.bs.b yes 24
8.b even 2 1 696.2.bs.b yes 24
24.h odd 2 1 CM 696.2.bs.a ✓ 24
29.f odd 28 1 inner 696.2.bs.a ✓ 24
87.k even 28 1 696.2.bs.b yes 24
232.u odd 28 1 696.2.bs.b yes 24
696.bs even 28 1 inner 696.2.bs.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
696.2.bs.a ✓ 24 1.a even 1 1 trivial
696.2.bs.a ✓ 24 24.h odd 2 1 CM
696.2.bs.a ✓ 24 29.f odd 28 1 inner
696.2.bs.a ✓ 24 696.bs even 28 1 inner
696.2.bs.b yes 24 3.b odd 2 1
696.2.bs.b yes 24 8.b even 2 1
696.2.bs.b yes 24 87.k even 28 1
696.2.bs.b yes 24 232.u odd 28 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 28 T_{5}^{23} + 372 T_{5}^{22} - 3164 T_{5}^{21} + 19835 T_{5}^{20} - 101192 T_{5}^{19} + \cdots + 5946106321 \) acting on \(S_{2}^{\mathrm{new}}(696, [\chi])\). Copy content Toggle raw display