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Properties

Label 784.2.i.k
Level $784$
Weight $2$
Character orbit 784.i
Analytic conductor $6.260$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [784,2,Mod(177,784)] 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, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.177"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-10,0,-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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 392)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} + \beta_1) q^{5} + 5 \beta_{2} q^{9} + ( - 4 \beta_{2} - 4) q^{11} + \beta_{3} q^{13} - 8 q^{15} + 2 \beta_1 q^{17} + (\beta_{3} + \beta_1) q^{19} + ( - 3 \beta_{2} - 3) q^{25}+ \cdots + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{9} - 8 q^{11} - 32 q^{15} - 6 q^{25} + 8 q^{29} - 20 q^{37} - 16 q^{39} + 16 q^{43} - 32 q^{51} - 12 q^{53} - 32 q^{57} + 16 q^{65} + 24 q^{67} + 16 q^{79} - 2 q^{81} - 64 q^{85} + 32 q^{93}+ \cdots + 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
177.1
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 − 1.22474i
0 −1.41421 − 2.44949i 0 1.41421 − 2.44949i 0 0 0 −2.50000 + 4.33013i 0
177.2 0 1.41421 + 2.44949i 0 −1.41421 + 2.44949i 0 0 0 −2.50000 + 4.33013i 0
753.1 0 −1.41421 + 2.44949i 0 1.41421 + 2.44949i 0 0 0 −2.50000 − 4.33013i 0
753.2 0 1.41421 − 2.44949i 0 −1.41421 − 2.44949i 0 0 0 −2.50000 − 4.33013i 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
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 784.2.i.k 4
4.b odd 2 1 392.2.i.g 4
7.b odd 2 1 inner 784.2.i.k 4
7.c even 3 1 784.2.a.n 2
7.c even 3 1 inner 784.2.i.k 4
7.d odd 6 1 784.2.a.n 2
7.d odd 6 1 inner 784.2.i.k 4
12.b even 2 1 3528.2.s.be 4
21.g even 6 1 7056.2.a.cj 2
21.h odd 6 1 7056.2.a.cj 2
28.d even 2 1 392.2.i.g 4
28.f even 6 1 392.2.a.h ✓ 2
28.f even 6 1 392.2.i.g 4
28.g odd 6 1 392.2.a.h ✓ 2
28.g odd 6 1 392.2.i.g 4
56.j odd 6 1 3136.2.a.bq 2
56.k odd 6 1 3136.2.a.bt 2
56.m even 6 1 3136.2.a.bt 2
56.p even 6 1 3136.2.a.bq 2
84.h odd 2 1 3528.2.s.be 4
84.j odd 6 1 3528.2.a.bj 2
84.j odd 6 1 3528.2.s.be 4
84.n even 6 1 3528.2.a.bj 2
84.n even 6 1 3528.2.s.be 4
140.p odd 6 1 9800.2.a.bw 2
140.s even 6 1 9800.2.a.bw 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.h ✓ 2 28.f even 6 1
392.2.a.h ✓ 2 28.g odd 6 1
392.2.i.g 4 4.b odd 2 1
392.2.i.g 4 28.d even 2 1
392.2.i.g 4 28.f even 6 1
392.2.i.g 4 28.g odd 6 1
784.2.a.n 2 7.c even 3 1
784.2.a.n 2 7.d odd 6 1
784.2.i.k 4 1.a even 1 1 trivial
784.2.i.k 4 7.b odd 2 1 inner
784.2.i.k 4 7.c even 3 1 inner
784.2.i.k 4 7.d odd 6 1 inner
3136.2.a.bq 2 56.j odd 6 1
3136.2.a.bq 2 56.p even 6 1
3136.2.a.bt 2 56.k odd 6 1
3136.2.a.bt 2 56.m even 6 1
3528.2.a.bj 2 84.j odd 6 1
3528.2.a.bj 2 84.n even 6 1
3528.2.s.be 4 12.b even 2 1
3528.2.s.be 4 84.h odd 2 1
3528.2.s.be 4 84.j odd 6 1
3528.2.s.be 4 84.n even 6 1
7056.2.a.cj 2 21.g even 6 1
7056.2.a.cj 2 21.h odd 6 1
9800.2.a.bw 2 140.p odd 6 1
9800.2.a.bw 2 140.s even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\):

\( T_{3}^{4} + 8T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{5}^{4} + 8T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T - 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$37$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 8T^{2} + 64 \) Copy content Toggle raw display
$61$ \( T^{4} + 200 T^{2} + 40000 \) Copy content Toggle raw display
$67$ \( (T^{2} - 12 T + 144)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 200)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
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