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Properties

Label 784.2.bb.a
Level $784$
Weight $2$
Character orbit 784.bb
Analytic conductor $6.260$
Analytic rank $0$
Dimension $48$
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: \(48\)
Relative dimension: \(8\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 4 q^{9} + 14 q^{17} + 12 q^{25} + 28 q^{29} + 42 q^{37} + 28 q^{41} + 56 q^{49} - 38 q^{53} + 42 q^{57} + 84 q^{61} + 8 q^{65} + 56 q^{69} - 42 q^{73} - 42 q^{77} - 44 q^{81} - 12 q^{85} - 28 q^{89} + 98 q^{93}+O(q^{100}) \) 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 −1.92908 − 2.41899i 0 1.36103 − 1.08539i 0 −2.17477 + 1.50678i 0 −1.46260 + 6.40807i 0
111.2 0 −0.969349 − 1.21553i 0 −1.39014 + 1.10860i 0 0.267553 − 2.63219i 0 0.129699 − 0.568247i 0
111.3 0 −0.683449 − 0.857018i 0 −1.48324 + 1.18285i 0 −2.57717 + 0.598513i 0 0.400186 − 1.75333i 0
111.4 0 −0.560590 − 0.702957i 0 2.19080 − 1.74711i 0 2.31112 + 1.28791i 0 0.487675 − 2.13664i 0
111.5 0 0.560590 + 0.702957i 0 2.19080 − 1.74711i 0 −2.31112 − 1.28791i 0 0.487675 − 2.13664i 0
111.6 0 0.683449 + 0.857018i 0 −1.48324 + 1.18285i 0 2.57717 − 0.598513i 0 0.400186 − 1.75333i 0
111.7 0 0.969349 + 1.21553i 0 −1.39014 + 1.10860i 0 −0.267553 + 2.63219i 0 0.129699 − 0.568247i 0
111.8 0 1.92908 + 2.41899i 0 1.36103 − 1.08539i 0 2.17477 − 1.50678i 0 −1.46260 + 6.40807i 0
223.1 0 −0.569888 + 2.49684i 0 −0.751141 − 0.171443i 0 −0.485208 + 2.60088i 0 −3.20653 − 1.54419i 0
223.2 0 −0.539159 + 2.36221i 0 0.323546 + 0.0738472i 0 −1.69316 − 2.03303i 0 −2.58644 − 1.24556i 0
223.3 0 −0.278678 + 1.22097i 0 −3.68584 − 0.841270i 0 2.64465 − 0.0764139i 0 1.28980 + 0.621136i 0
223.4 0 −0.00959582 + 0.0420420i 0 2.58898 + 0.590918i 0 2.63502 + 0.238031i 0 2.70123 + 1.30084i 0
223.5 0 0.00959582 − 0.0420420i 0 2.58898 + 0.590918i 0 −2.63502 − 0.238031i 0 2.70123 + 1.30084i 0
223.6 0 0.278678 − 1.22097i 0 −3.68584 − 0.841270i 0 −2.64465 + 0.0764139i 0 1.28980 + 0.621136i 0
223.7 0 0.539159 − 2.36221i 0 0.323546 + 0.0738472i 0 1.69316 + 2.03303i 0 −2.58644 − 1.24556i 0
223.8 0 0.569888 − 2.49684i 0 −0.751141 − 0.171443i 0 0.485208 − 2.60088i 0 −3.20653 − 1.54419i 0
335.1 0 −2.37272 + 1.14264i 0 −0.850196 − 1.76545i 0 2.39479 + 1.12472i 0 2.45369 − 3.07683i 0
335.2 0 −1.83590 + 0.884123i 0 1.40671 + 2.92107i 0 1.59819 − 2.10851i 0 0.718387 − 0.900829i 0
335.3 0 −1.46912 + 0.707493i 0 −0.362567 − 0.752877i 0 −2.62064 + 0.363622i 0 −0.212691 + 0.266706i 0
335.4 0 −0.453641 + 0.218462i 0 0.652059 + 1.35401i 0 0.138020 + 2.64215i 0 −1.71241 + 2.14729i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 111.8
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.a ✓ 48
4.b odd 2 1 inner 784.2.bb.a ✓ 48
49.f odd 14 1 inner 784.2.bb.a ✓ 48
196.j even 14 1 inner 784.2.bb.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.2.bb.a ✓ 48 1.a even 1 1 trivial
784.2.bb.a ✓ 48 4.b odd 2 1 inner
784.2.bb.a ✓ 48 49.f odd 14 1 inner
784.2.bb.a ✓ 48 196.j even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} + 14 T_{3}^{46} + 154 T_{3}^{44} + 1197 T_{3}^{42} + 6944 T_{3}^{40} + 55713 T_{3}^{38} + \cdots + 2401 \) acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display