Properties

Label 784.2.bg.e
Level $784$
Weight $2$
Character orbit 784.bg
Analytic conductor $6.260$
Analytic rank $0$
Dimension $84$
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] = [784,2,Mod(65,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 0, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.bg (of order \(21\), degree \(12\), not 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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(7\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 392)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 10 q^{3} - q^{5} + 4 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 10 q^{3} - q^{5} + 4 q^{7} - 11 q^{9} - q^{11} + 10 q^{13} - 9 q^{15} + 3 q^{17} + 23 q^{19} - 10 q^{21} + q^{23} + 6 q^{25} + 32 q^{27} - 13 q^{29} + 29 q^{31} + 11 q^{33} - 30 q^{35} + 48 q^{37} - 51 q^{39} + 20 q^{41} + 14 q^{43} + 22 q^{45} + 34 q^{47} - 2 q^{49} - 3 q^{51} - 40 q^{53} + 9 q^{55} - 10 q^{57} - 38 q^{59} + 10 q^{61} - 9 q^{63} - 2 q^{65} + 37 q^{67} - 87 q^{69} - 5 q^{71} + 21 q^{73} + 12 q^{75} + 71 q^{77} - 19 q^{79} - 70 q^{81} + 55 q^{83} + 29 q^{85} + 165 q^{87} - 65 q^{89} + 15 q^{91} - 120 q^{93} - 7 q^{95} + 170 q^{97} - 110 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0 −0.243123 3.24426i 0 −2.30951 + 1.57459i 0 −2.57062 0.626010i 0 −7.49960 + 1.13038i 0
65.2 0 −0.145655 1.94364i 0 1.90959 1.30194i 0 −0.0445909 + 2.64538i 0 −0.790013 + 0.119075i 0
65.3 0 −0.0762273 1.01718i 0 −1.98810 + 1.35546i 0 2.53769 + 0.748408i 0 1.93764 0.292053i 0
65.4 0 −0.0603171 0.804876i 0 1.77912 1.21299i 0 −0.287907 2.63004i 0 2.32231 0.350031i 0
65.5 0 0.138470 + 1.84775i 0 −3.02338 + 2.06131i 0 −0.184852 2.63929i 0 −0.428529 + 0.0645904i 0
65.6 0 0.167965 + 2.24134i 0 1.72106 1.17340i 0 −2.19081 + 1.48335i 0 −2.02891 + 0.305809i 0
65.7 0 0.192917 + 2.57430i 0 0.277188 0.188984i 0 2.63686 + 0.216760i 0 −3.62334 + 0.546130i 0
81.1 0 −2.47210 0.762543i 0 −2.35987 + 2.18964i 0 0.219770 2.63661i 0 3.05111 + 2.08021i 0
81.2 0 −2.43830 0.752115i 0 2.77023 2.57040i 0 1.75630 1.97874i 0 2.90091 + 1.97780i 0
81.3 0 −1.73701 0.535797i 0 −0.482840 + 0.448010i 0 −1.73126 + 2.00068i 0 0.251414 + 0.171411i 0
81.4 0 −0.291667 0.0899675i 0 −1.69575 + 1.57343i 0 1.90711 + 1.83383i 0 −2.40174 1.63748i 0
81.5 0 −0.157271 0.0485115i 0 2.54579 2.36215i 0 −2.50741 + 0.844343i 0 −2.45634 1.67470i 0
81.6 0 1.70610 + 0.526262i 0 0.444592 0.412521i 0 0.436186 2.60955i 0 0.155111 + 0.105753i 0
81.7 0 2.07814 + 0.641021i 0 −2.43350 + 2.25796i 0 −2.64362 0.106260i 0 1.42903 + 0.974298i 0
193.1 0 −0.243123 + 3.24426i 0 −2.30951 1.57459i 0 −2.57062 + 0.626010i 0 −7.49960 1.13038i 0
193.2 0 −0.145655 + 1.94364i 0 1.90959 + 1.30194i 0 −0.0445909 2.64538i 0 −0.790013 0.119075i 0
193.3 0 −0.0762273 + 1.01718i 0 −1.98810 1.35546i 0 2.53769 0.748408i 0 1.93764 + 0.292053i 0
193.4 0 −0.0603171 + 0.804876i 0 1.77912 + 1.21299i 0 −0.287907 + 2.63004i 0 2.32231 + 0.350031i 0
193.5 0 0.138470 1.84775i 0 −3.02338 2.06131i 0 −0.184852 + 2.63929i 0 −0.428529 0.0645904i 0
193.6 0 0.167965 2.24134i 0 1.72106 + 1.17340i 0 −2.19081 1.48335i 0 −2.02891 0.305809i 0
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.g even 21 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.bg.e 84
4.b odd 2 1 392.2.y.b 84
49.g even 21 1 inner 784.2.bg.e 84
196.o odd 42 1 392.2.y.b 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.y.b 84 4.b odd 2 1
392.2.y.b 84 196.o odd 42 1
784.2.bg.e 84 1.a even 1 1 trivial
784.2.bg.e 84 49.g even 21 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{84} + 10 T_{3}^{83} + 45 T_{3}^{82} + 94 T_{3}^{81} - 84 T_{3}^{80} - 1196 T_{3}^{79} - 2795 T_{3}^{78} + 3055 T_{3}^{77} + 30672 T_{3}^{76} + 46074 T_{3}^{75} - 188603 T_{3}^{74} - 1063528 T_{3}^{73} + \cdots + 7127918329 \) acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display