↑

Properties

Label 784.3.s.f
Level $784$
Weight $3$
Character orbit 784.s
Analytic conductor $21.362$
Analytic rank $0$
Dimension $8$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [784,3,Mod(129,784)] 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([0, 0, 1])) 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("784.129"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 784.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,-4,0,8] 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 56)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{6} + \beta_{3}) q^{9} + ( - 2 \beta_{7} + 2 \beta_{3} + 2) q^{11} + ( - 5 \beta_{5} - \beta_{4} + \cdots - \beta_1) q^{13} + (\beta_{7} - \beta_{6} + 8) q^{15}+ \cdots + 62 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{9} + 8 q^{11} + 64 q^{15} - 24 q^{23} + 60 q^{25} + 208 q^{29} - 40 q^{37} + 160 q^{39} + 368 q^{43} + 152 q^{53} + 448 q^{57} - 32 q^{65} + 8 q^{67} + 496 q^{71} - 248 q^{79} + 156 q^{81} + 512 q^{85}+ \cdots + 496 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 34\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{6} - 7\nu^{4} + 28\nu^{2} - 16 ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4\nu^{7} - 14\nu^{5} + 56\nu^{3} - 32\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{7} - 21\nu^{5} + 70\nu^{3} - 40\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -6\nu^{6} + 28\nu^{4} - 84\nu^{2} + 48 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -8\nu^{6} + 28\nu^{4} - 84\nu^{2} + 8 ) / 7 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 8\beta_{3} + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{5} + 3\beta_{4} - 2\beta_{2} + 3\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 6\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{5} + 5\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -7\beta_{7} + 7\beta_{6} - 40 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 14\beta_{2} - 17\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
0.662827 − 0.382683i
−1.60021 + 0.923880i
1.60021 − 0.923880i
−0.662827 + 0.382683i
0.662827 + 0.382683i
−1.60021 − 0.923880i
1.60021 + 0.923880i
−0.662827 − 0.382683i
0 −3.20041 − 1.84776i 0 −0.549104 + 0.317025i 0 0 0 2.32843 + 4.03295i 0
129.2 0 −1.32565 − 0.765367i 0 −7.72648 + 4.46088i 0 0 0 −3.32843 − 5.76500i 0
129.3 0 1.32565 + 0.765367i 0 7.72648 − 4.46088i 0 0 0 −3.32843 − 5.76500i 0
129.4 0 3.20041 + 1.84776i 0 0.549104 − 0.317025i 0 0 0 2.32843 + 4.03295i 0
705.1 0 −3.20041 + 1.84776i 0 −0.549104 − 0.317025i 0 0 0 2.32843 − 4.03295i 0
705.2 0 −1.32565 + 0.765367i 0 −7.72648 − 4.46088i 0 0 0 −3.32843 + 5.76500i 0
705.3 0 1.32565 − 0.765367i 0 7.72648 + 4.46088i 0 0 0 −3.32843 + 5.76500i 0
705.4 0 3.20041 − 1.84776i 0 0.549104 + 0.317025i 0 0 0 2.32843 − 4.03295i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d 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.s.f 8
4.b odd 2 1 392.3.o.b 8
7.b odd 2 1 inner 784.3.s.f 8
7.c even 3 1 112.3.c.c 4
7.c even 3 1 inner 784.3.s.f 8
7.d odd 6 1 112.3.c.c 4
7.d odd 6 1 inner 784.3.s.f 8
21.g even 6 1 1008.3.f.h 4
21.h odd 6 1 1008.3.f.h 4
28.d even 2 1 392.3.o.b 8
28.f even 6 1 56.3.c.a ✓ 4
28.f even 6 1 392.3.o.b 8
28.g odd 6 1 56.3.c.a ✓ 4
28.g odd 6 1 392.3.o.b 8
56.j odd 6 1 448.3.c.e 4
56.k odd 6 1 448.3.c.f 4
56.m even 6 1 448.3.c.f 4
56.p even 6 1 448.3.c.e 4
84.j odd 6 1 504.3.f.a 4
84.n even 6 1 504.3.f.a 4
140.p odd 6 1 1400.3.f.a 4
140.s even 6 1 1400.3.f.a 4
140.w even 12 2 1400.3.p.a 8
140.x odd 12 2 1400.3.p.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.c.a ✓ 4 28.f even 6 1
56.3.c.a ✓ 4 28.g odd 6 1
112.3.c.c 4 7.c even 3 1
112.3.c.c 4 7.d odd 6 1
392.3.o.b 8 4.b odd 2 1
392.3.o.b 8 28.d even 2 1
392.3.o.b 8 28.f even 6 1
392.3.o.b 8 28.g odd 6 1
448.3.c.e 4 56.j odd 6 1
448.3.c.e 4 56.p even 6 1
448.3.c.f 4 56.k odd 6 1
448.3.c.f 4 56.m even 6 1
504.3.f.a 4 84.j odd 6 1
504.3.f.a 4 84.n even 6 1
784.3.s.f 8 1.a even 1 1 trivial
784.3.s.f 8 7.b odd 2 1 inner
784.3.s.f 8 7.c even 3 1 inner
784.3.s.f 8 7.d odd 6 1 inner
1008.3.f.h 4 21.g even 6 1
1008.3.f.h 4 21.h odd 6 1
1400.3.f.a 4 140.p odd 6 1
1400.3.f.a 4 140.s even 6 1
1400.3.p.a 8 140.w even 12 2
1400.3.p.a 8 140.x odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 16T_{3}^{6} + 224T_{3}^{4} - 512T_{3}^{2} + 1024 \) acting on \(S_{3}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 16 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$5$ \( T^{8} - 80 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{3} + \cdots + 15376)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 464 T^{2} + 32)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 256 T^{6} + \cdots + 67108864 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 64248054784 \) Copy content Toggle raw display
$23$ \( (T^{4} + 12 T^{3} + \cdots + 226576)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 52 T + 548)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 1073741824 \) Copy content Toggle raw display
$37$ \( (T^{4} + 20 T^{3} + \cdots + 1106704)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 5440 T^{2} + 2580992)^{2} \) Copy content Toggle raw display
$43$ \( (T - 46)^{8} \) Copy content Toggle raw display
$47$ \( T^{8} - 7424 T^{6} + \cdots + 67108864 \) Copy content Toggle raw display
$53$ \( (T^{4} - 76 T^{3} + \cdots + 868624)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 561132831744 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 561132831744 \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{3} + \cdots + 258064)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 124 T + 3556)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 1511207993344 \) Copy content Toggle raw display
$79$ \( (T^{4} + 124 T^{3} + \cdots + 9265936)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 11152 T^{2} + 30921248)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 6208 T^{6} + \cdots + 629407744 \) Copy content Toggle raw display
$97$ \( (T^{4} + 33280 T^{2} + 204505088)^{2} \) Copy content Toggle raw display
show more
show less