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Properties

Label 714.2.bf.a
Level $714$
Weight $2$
Character orbit 714.bf
Analytic conductor $5.701$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [714,2,Mod(29,714)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(714, base_ring=CyclotomicField(16))
 
chi = DirichletCharacter(H, H._module([8, 0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("714.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 714.bf (of order \(16\), degree \(8\), 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: \(5.70131870432\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 16 q^{11} + 32 q^{17} + 16 q^{24} + 32 q^{25} + 16 q^{29} + 32 q^{31} + 48 q^{38} - 16 q^{39} + 16 q^{43} - 64 q^{45} - 16 q^{46} + 32 q^{47} + 176 q^{50} - 64 q^{51} - 32 q^{52} - 128 q^{53} - 64 q^{54} - 16 q^{57} - 16 q^{58} - 16 q^{60} - 32 q^{61} - 32 q^{62} + 16 q^{70} - 16 q^{71} - 16 q^{72} - 16 q^{74} + 112 q^{75} + 16 q^{78} + 64 q^{79} + 32 q^{80} + 128 q^{82} - 144 q^{83} - 16 q^{85} + 112 q^{87} + 16 q^{88} + 96 q^{91} - 16 q^{92} - 32 q^{93} + 32 q^{97} + 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −0.923880 + 0.382683i −1.72339 − 0.172963i 0.707107 − 0.707107i −0.579269 + 2.91218i 1.65840 − 0.499717i −0.980785 + 0.195090i −0.382683 + 0.923880i 2.94017 + 0.596166i −0.579269 − 2.91218i
29.2 −0.923880 + 0.382683i −1.60893 − 0.641372i 0.707107 − 0.707107i −0.114246 + 0.574355i 1.73190 − 0.0231583i −0.980785 + 0.195090i −0.382683 + 0.923880i 2.17728 + 2.06384i −0.114246 − 0.574355i
29.3 −0.923880 + 0.382683i −1.56989 + 0.731736i 0.707107 − 0.707107i −0.163150 + 0.820210i 1.17037 − 1.27681i 0.980785 − 0.195090i −0.382683 + 0.923880i 1.92913 − 2.29749i −0.163150 − 0.820210i
29.4 −0.923880 + 0.382683i −1.47593 − 0.906434i 0.707107 − 0.707107i 0.605443 − 3.04377i 1.71046 + 0.272621i 0.980785 − 0.195090i −0.382683 + 0.923880i 1.35676 + 2.67567i 0.605443 + 3.04377i
29.5 −0.923880 + 0.382683i −1.10904 − 1.33042i 0.707107 − 0.707107i −0.799212 + 4.01791i 1.53375 + 0.804739i 0.980785 − 0.195090i −0.382683 + 0.923880i −0.540054 + 2.95099i −0.799212 − 4.01791i
29.6 −0.923880 + 0.382683i −1.10851 + 1.33086i 0.707107 − 0.707107i 0.207463 − 1.04299i 0.514835 − 1.65377i −0.980785 + 0.195090i −0.382683 + 0.923880i −0.542389 − 2.95056i 0.207463 + 1.04299i
29.7 −0.923880 + 0.382683i −0.892446 + 1.48443i 0.707107 − 0.707107i −0.773360 + 3.88794i 0.256445 − 1.71296i −0.980785 + 0.195090i −0.382683 + 0.923880i −1.40708 − 2.64955i −0.773360 − 3.88794i
29.8 −0.923880 + 0.382683i −0.510946 − 1.65497i 0.707107 − 0.707107i 0.195499 − 0.982838i 1.10538 + 1.33346i 0.980785 − 0.195090i −0.382683 + 0.923880i −2.47787 + 1.69120i 0.195499 + 0.982838i
29.9 −0.923880 + 0.382683i −0.0543633 − 1.73120i 0.707107 − 0.707107i 0.725066 − 3.64515i 0.712726 + 1.57861i −0.980785 + 0.195090i −0.382683 + 0.923880i −2.99409 + 0.188227i 0.725066 + 3.64515i
29.10 −0.923880 + 0.382683i −0.0182428 + 1.73195i 0.707107 − 0.707107i 0.494369 − 2.48536i −0.645936 − 1.60710i −0.980785 + 0.195090i −0.382683 + 0.923880i −2.99933 − 0.0631916i 0.494369 + 2.48536i
29.11 −0.923880 + 0.382683i 0.201124 + 1.72033i 0.707107 − 0.707107i 0.277065 − 1.39290i −0.844158 − 1.51241i 0.980785 − 0.195090i −0.382683 + 0.923880i −2.91910 + 0.692001i 0.277065 + 1.39290i
29.12 −0.923880 + 0.382683i 0.285068 − 1.70843i 0.707107 − 0.707107i −0.344420 + 1.73152i 0.390419 + 1.68748i −0.980785 + 0.195090i −0.382683 + 0.923880i −2.83747 − 0.974039i −0.344420 − 1.73152i
29.13 −0.923880 + 0.382683i 0.963319 − 1.43945i 0.707107 − 0.707107i −0.316357 + 1.59043i −0.339137 + 1.69852i 0.980785 − 0.195090i −0.382683 + 0.923880i −1.14403 − 2.77330i −0.316357 − 1.59043i
29.14 −0.923880 + 0.382683i 1.38369 + 1.04183i 0.707107 − 0.707107i 0.00481222 − 0.0241926i −1.67705 − 0.433010i 0.980785 − 0.195090i −0.382683 + 0.923880i 0.829183 + 2.88313i 0.00481222 + 0.0241926i
29.15 −0.923880 + 0.382683i 1.56201 − 0.748417i 0.707107 − 0.707107i −0.152300 + 0.765662i −1.15670 + 1.28920i −0.980785 + 0.195090i −0.382683 + 0.923880i 1.87974 − 2.33807i −0.152300 − 0.765662i
29.16 −0.923880 + 0.382683i 1.61610 − 0.623081i 0.707107 − 0.707107i 0.385439 − 1.93773i −1.25464 + 1.19411i 0.980785 − 0.195090i −0.382683 + 0.923880i 2.22354 − 2.01392i 0.385439 + 1.93773i
29.17 −0.923880 + 0.382683i 1.63977 + 0.557812i 0.707107 − 0.707107i −0.377986 + 1.90027i −1.72842 + 0.112162i −0.980785 + 0.195090i −0.382683 + 0.923880i 2.37769 + 1.82937i −0.377986 − 1.90027i
29.18 −0.923880 + 0.382683i 1.65526 + 0.510023i 0.707107 − 0.707107i −0.805590 + 4.04997i −1.72444 + 0.162240i 0.980785 − 0.195090i −0.382683 + 0.923880i 2.47975 + 1.68844i −0.805590 − 4.04997i
71.1 0.382683 + 0.923880i −1.70795 + 0.287914i −0.707107 + 0.707107i −0.734221 + 0.490591i −0.919603 − 1.46776i 0.555570 − 0.831470i −0.923880 − 0.382683i 2.83421 − 0.983487i −0.734221 − 0.490591i
71.2 0.382683 + 0.923880i −1.61638 + 0.622352i −0.707107 + 0.707107i 3.02182 − 2.01912i −1.19354 − 1.25517i −0.555570 + 0.831470i −0.923880 − 0.382683i 2.22536 − 2.01191i 3.02182 + 2.01912i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
51.i even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 714.2.bf.a ✓ 144
3.b odd 2 1 714.2.bf.b yes 144
17.e odd 16 1 714.2.bf.b yes 144
51.i even 16 1 inner 714.2.bf.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
714.2.bf.a ✓ 144 1.a even 1 1 trivial
714.2.bf.a ✓ 144 51.i even 16 1 inner
714.2.bf.b yes 144 3.b odd 2 1
714.2.bf.b yes 144 17.e odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{144} - 16 T_{5}^{142} + 32 T_{5}^{141} + 88 T_{5}^{140} + 160 T_{5}^{139} - 256 T_{5}^{138} + \cdots + 37\!\cdots\!76 \) acting on \(S_{2}^{\mathrm{new}}(714, [\chi])\). Copy content Toggle raw display