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Properties

Label 784.2.u.g
Level $784$
Weight $2$
Character orbit 784.u
Analytic conductor $6.260$
Analytic rank $0$
Dimension $42$
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(113,784)]
 
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([0, 0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.113");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.u (of order \(7\), degree \(6\), 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: \(42\)
Relative dimension: \(7\) over \(\Q(\zeta_{7})\)
Twist minimal: no (minimal twist has level 392)
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 7 q^{3} - 2 q^{5} - q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 42 q + 7 q^{3} - 2 q^{5} - q^{7} - 22 q^{9} - 2 q^{11} - 16 q^{13} + 9 q^{15} + 6 q^{17} - 62 q^{19} + 7 q^{21} + 2 q^{23} - 9 q^{25} + 7 q^{27} + 13 q^{29} - 20 q^{31} - 14 q^{33} + 33 q^{35} + 12 q^{37} - 39 q^{39} - 2 q^{41} - 14 q^{43} - 22 q^{45} + 41 q^{47} - 43 q^{49} - 6 q^{51} + 25 q^{53} - 15 q^{55} + 10 q^{57} - 7 q^{59} - q^{61} - 18 q^{63} - 4 q^{65} + 74 q^{67} - 21 q^{69} + 5 q^{71} - 24 q^{73} - 21 q^{75} + 31 q^{77} - 38 q^{79} + 28 q^{81} - 13 q^{83} - 29 q^{85} + 84 q^{87} - 22 q^{89} + 33 q^{91} - 9 q^{93} + 7 q^{95} - 8 q^{97} + 110 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
113.1 0 −2.08664 + 2.61656i 0 1.04068 − 1.30497i 0 1.76173 − 1.97391i 0 −1.82477 − 7.99484i 0
113.2 0 −1.22307 + 1.53368i 0 −1.68803 + 2.11672i 0 −1.16466 + 2.37562i 0 −0.188711 − 0.826797i 0
113.3 0 −0.869265 + 1.09002i 0 −1.43583 + 1.80047i 0 −1.29356 − 2.30796i 0 0.235032 + 1.02974i 0
113.4 0 0.344955 − 0.432560i 0 2.00421 − 2.51319i 0 −0.288138 − 2.63001i 0 0.599449 + 2.62636i 0
113.5 0 0.417314 − 0.523295i 0 −0.142507 + 0.178698i 0 2.28538 + 1.33305i 0 0.567876 + 2.48803i 0
113.6 0 1.18898 − 1.49094i 0 −1.29751 + 1.62703i 0 −2.61471 − 0.404112i 0 −0.141653 − 0.620621i 0
113.7 0 1.75822 − 2.20473i 0 1.24152 − 1.55681i 0 −0.265459 + 2.63240i 0 −1.10196 − 4.82801i 0
225.1 0 −0.667279 − 2.92354i 0 0.0973281 + 0.426422i 0 1.39946 − 2.24533i 0 −5.39891 + 2.59998i 0
225.2 0 −0.296281 − 1.29809i 0 0.886136 + 3.88242i 0 −2.34019 + 1.23430i 0 1.10565 − 0.532453i 0
225.3 0 −0.249488 − 1.09308i 0 −0.565224 − 2.47641i 0 1.39149 + 2.25028i 0 1.57033 − 0.756230i 0
225.4 0 0.112665 + 0.493620i 0 0.367834 + 1.61159i 0 2.63261 + 0.263327i 0 2.47194 − 1.19042i 0
225.5 0 0.243915 + 1.06866i 0 −0.0904045 − 0.396088i 0 −2.10610 − 1.60136i 0 1.62036 − 0.780324i 0
225.6 0 0.646066 + 2.83060i 0 −0.857236 − 3.75579i 0 −0.477544 + 2.60230i 0 −4.89199 + 2.35586i 0
225.7 0 0.754474 + 3.30557i 0 0.562535 + 2.46463i 0 1.64822 − 2.06963i 0 −7.65462 + 3.68627i 0
337.1 0 −1.86848 − 0.899813i 0 0.757173 + 0.364635i 0 −0.700840 − 2.55124i 0 0.811089 + 1.01707i 0
337.2 0 −1.08347 − 0.521774i 0 −3.09666 − 1.49127i 0 −1.82858 + 1.91214i 0 −0.968801 − 1.21484i 0
337.3 0 −0.221227 − 0.106537i 0 2.79736 + 1.34714i 0 −0.700359 + 2.55137i 0 −1.83288 − 2.29836i 0
337.4 0 0.0885962 + 0.0426657i 0 −0.834433 − 0.401842i 0 2.57690 + 0.599657i 0 −1.86444 − 2.33793i 0
337.5 0 1.57698 + 0.759435i 0 1.03378 + 0.497843i 0 2.35755 − 1.20081i 0 0.0396651 + 0.0497384i 0
337.6 0 2.06904 + 0.996398i 0 −3.59189 − 1.72977i 0 −0.967089 − 2.46267i 0 1.41766 + 1.77768i 0
See all 42 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.u.g 42
4.b odd 2 1 392.2.q.a ✓ 42
49.e even 7 1 inner 784.2.u.g 42
196.k odd 14 1 392.2.q.a ✓ 42
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.q.a ✓ 42 4.b odd 2 1
392.2.q.a ✓ 42 196.k odd 14 1
784.2.u.g 42 1.a even 1 1 trivial
784.2.u.g 42 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{42} - 7 T_{3}^{41} + 46 T_{3}^{40} - 210 T_{3}^{39} + 901 T_{3}^{38} - 3360 T_{3}^{37} + \cdots + 117649 \) acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display