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Properties

Label 784.3.r.a
Level $784$
Weight $3$
Character orbit 784.r
Analytic conductor $21.362$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [784,3,Mod(79,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 2])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.79"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 784.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-6,0,-2,0,0,0,3,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.3624527258\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 4) q^{3} + (2 \zeta_{6} - 2) q^{5} + ( - 3 \zeta_{6} + 3) q^{9} + ( - 4 \zeta_{6} + 8) q^{11} - 18 q^{13} + ( - 8 \zeta_{6} + 4) q^{15} + 28 \zeta_{6} q^{17} + ( - 10 \zeta_{6} - 10) q^{19}+ \cdots + ( - 24 \zeta_{6} + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 2 q^{5} + 3 q^{9} + 12 q^{11} - 36 q^{13} + 28 q^{17} - 30 q^{19} - 60 q^{23} + 21 q^{25} + 12 q^{29} - 60 q^{31} - 24 q^{33} + 50 q^{37} + 108 q^{39} - 40 q^{41} + 6 q^{45} + 84 q^{47}+ \cdots + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(-1 + \zeta_{6}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 −3.00000 + 1.73205i 0 −1.00000 + 1.73205i 0 0 0 1.50000 − 2.59808i 0
655.1 0 −3.00000 − 1.73205i 0 −1.00000 − 1.73205i 0 0 0 1.50000 + 2.59808i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.3.r.a 2
4.b odd 2 1 784.3.r.g 2
7.b odd 2 1 784.3.r.h 2
7.c even 3 1 784.3.d.h yes 2
7.c even 3 1 784.3.r.g 2
7.d odd 6 1 784.3.d.e ✓ 2
7.d odd 6 1 784.3.r.b 2
28.d even 2 1 784.3.r.b 2
28.f even 6 1 784.3.d.e ✓ 2
28.f even 6 1 784.3.r.h 2
28.g odd 6 1 784.3.d.h yes 2
28.g odd 6 1 inner 784.3.r.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.3.d.e ✓ 2 7.d odd 6 1
784.3.d.e ✓ 2 28.f even 6 1
784.3.d.h yes 2 7.c even 3 1
784.3.d.h yes 2 28.g odd 6 1
784.3.r.a 2 1.a even 1 1 trivial
784.3.r.a 2 28.g odd 6 1 inner
784.3.r.b 2 7.d odd 6 1
784.3.r.b 2 28.d even 2 1
784.3.r.g 2 4.b odd 2 1
784.3.r.g 2 7.c even 3 1
784.3.r.h 2 7.b odd 2 1
784.3.r.h 2 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{2} + 6T_{3} + 12 \) Copy content Toggle raw display
\( T_{5}^{2} + 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 12T_{11} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 12T + 48 \) Copy content Toggle raw display
$13$ \( (T + 18)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 28T + 784 \) Copy content Toggle raw display
$19$ \( T^{2} + 30T + 300 \) Copy content Toggle raw display
$23$ \( T^{2} + 60T + 1200 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 60T + 1200 \) Copy content Toggle raw display
$37$ \( T^{2} - 50T + 2500 \) Copy content Toggle raw display
$41$ \( (T + 20)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1200 \) Copy content Toggle raw display
$47$ \( T^{2} - 84T + 2352 \) Copy content Toggle raw display
$53$ \( T^{2} - 30T + 900 \) Copy content Toggle raw display
$59$ \( T^{2} - 150T + 7500 \) Copy content Toggle raw display
$61$ \( T^{2} - 30T + 900 \) Copy content Toggle raw display
$67$ \( T^{2} - 180T + 10800 \) Copy content Toggle raw display
$71$ \( T^{2} + 4800 \) Copy content Toggle raw display
$73$ \( T^{2} + 96T + 9216 \) Copy content Toggle raw display
$79$ \( T^{2} + 192T + 12288 \) Copy content Toggle raw display
$83$ \( T^{2} + 300 \) Copy content Toggle raw display
$89$ \( T^{2} - 80T + 6400 \) Copy content Toggle raw display
$97$ \( (T - 12)^{2} \) Copy content Toggle raw display
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