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Properties

Label 784.2.bb.b
Level $784$
Weight $2$
Character orbit 784.bb
Analytic conductor $6.260$
Analytic rank $0$
Dimension $120$
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] = [784,2,Mod(111,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 0, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.111");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.bb (of order \(14\), degree \(6\), 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: \(120\)
Relative dimension: \(20\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 24 q^{9} - 14 q^{17} + 16 q^{21} + 40 q^{25} + 32 q^{29} - 62 q^{37} - 28 q^{41} - 60 q^{49} + 14 q^{53} - 34 q^{57} - 112 q^{61} - 32 q^{65} + 112 q^{69} + 42 q^{73} + 66 q^{77} - 44 q^{81} - 12 q^{85} + 28 q^{89} - 58 q^{93}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
111.1 0 −2.07723 − 2.60477i 0 −0.165391 + 0.131895i 0 2.53455 − 0.758996i 0 −1.80236 + 7.89664i 0
111.2 0 −1.94065 − 2.43350i 0 −2.10945 + 1.68223i 0 −1.58011 − 2.12208i 0 −1.48823 + 6.52037i 0
111.3 0 −1.44547 − 1.81256i 0 2.77297 − 2.21137i 0 −1.94347 − 1.79525i 0 −0.528427 + 2.31519i 0
111.4 0 −1.36747 − 1.71475i 0 1.64860 − 1.31472i 0 −1.20605 + 2.35488i 0 −0.402843 + 1.76497i 0
111.5 0 −1.31050 − 1.64332i 0 −0.883194 + 0.704324i 0 2.18004 + 1.49914i 0 −0.315513 + 1.38235i 0
111.6 0 −0.977383 − 1.22560i 0 −0.470082 + 0.374878i 0 1.48039 + 2.19281i 0 0.120747 − 0.529025i 0
111.7 0 −0.912783 − 1.14459i 0 −0.583523 + 0.465344i 0 −2.34056 − 1.23360i 0 0.190641 − 0.835251i 0
111.8 0 −0.881327 − 1.10515i 0 2.63324 − 2.09994i 0 0.910280 − 2.48423i 0 0.222946 − 0.976789i 0
111.9 0 −0.0556550 − 0.0697892i 0 −3.27604 + 2.61256i 0 1.51276 − 2.17061i 0 0.665790 − 2.91702i 0
111.10 0 −0.0314318 − 0.0394142i 0 −0.245577 + 0.195841i 0 −1.27396 + 2.31884i 0 0.666997 − 2.92231i 0
111.11 0 0.0314318 + 0.0394142i 0 −0.245577 + 0.195841i 0 1.27396 − 2.31884i 0 0.666997 − 2.92231i 0
111.12 0 0.0556550 + 0.0697892i 0 −3.27604 + 2.61256i 0 −1.51276 + 2.17061i 0 0.665790 − 2.91702i 0
111.13 0 0.881327 + 1.10515i 0 2.63324 − 2.09994i 0 −0.910280 + 2.48423i 0 0.222946 − 0.976789i 0
111.14 0 0.912783 + 1.14459i 0 −0.583523 + 0.465344i 0 2.34056 + 1.23360i 0 0.190641 − 0.835251i 0
111.15 0 0.977383 + 1.22560i 0 −0.470082 + 0.374878i 0 −1.48039 − 2.19281i 0 0.120747 − 0.529025i 0
111.16 0 1.31050 + 1.64332i 0 −0.883194 + 0.704324i 0 −2.18004 − 1.49914i 0 −0.315513 + 1.38235i 0
111.17 0 1.36747 + 1.71475i 0 1.64860 − 1.31472i 0 1.20605 − 2.35488i 0 −0.402843 + 1.76497i 0
111.18 0 1.44547 + 1.81256i 0 2.77297 − 2.21137i 0 1.94347 + 1.79525i 0 −0.528427 + 2.31519i 0
111.19 0 1.94065 + 2.43350i 0 −2.10945 + 1.68223i 0 1.58011 + 2.12208i 0 −1.48823 + 6.52037i 0
111.20 0 2.07723 + 2.60477i 0 −0.165391 + 0.131895i 0 −2.53455 + 0.758996i 0 −1.80236 + 7.89664i 0
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
49.f odd 14 1 inner
196.j even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.bb.b ✓ 120
4.b odd 2 1 inner 784.2.bb.b ✓ 120
49.f odd 14 1 inner 784.2.bb.b ✓ 120
196.j even 14 1 inner 784.2.bb.b ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.2.bb.b ✓ 120 1.a even 1 1 trivial
784.2.bb.b ✓ 120 4.b odd 2 1 inner
784.2.bb.b ✓ 120 49.f odd 14 1 inner
784.2.bb.b ✓ 120 196.j even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{120} + 42 T_{3}^{118} + 1037 T_{3}^{116} + 19853 T_{3}^{114} + 327058 T_{3}^{112} + \cdots + 3418801 \) acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display