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Properties

Label 784.3.r.n
Level $784$
Weight $3$
Character orbit 784.r
Analytic conductor $21.362$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,3,Mod(79,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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

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: \(21.3624527258\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) 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 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2} + 2) q^{3} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{5} + ( - 4 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2} + 2) q^{3} + (2 \beta_{3} + \beta_{2} - \beta_1 - 1) q^{5} + ( - 4 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{9} + (2 \beta_{3} - 2 \beta_{2} + 4) q^{11} + ( - 3 \beta_{3} - 3 \beta_1 + 7) q^{13} + (4 \beta_{3} + 16 \beta_{2} - 4 \beta_1 - 8) q^{15} + (2 \beta_{3} + 16 \beta_{2} - 4 \beta_1) q^{17} + ( - 7 \beta_{2} + 7 \beta_1 - 7) q^{19} + (8 \beta_{2} + 4 \beta_1 + 8) q^{23} + ( - 2 \beta_{3} + 3 \beta_{2} + 4 \beta_1) q^{25} + (2 \beta_{3} - 12 \beta_{2} - 2 \beta_1 + 6) q^{27} + ( - 10 \beta_{3} - 10 \beta_1 + 8) q^{29} + (6 \beta_{3} - 26 \beta_{2} + 52) q^{31} + (8 \beta_{2} - 8) q^{33} + (4 \beta_{3} + 8 \beta_{2} - 2 \beta_1 - 8) q^{37} + ( - 16 \beta_{3} - 28 \beta_{2} + 56) q^{39} + (6 \beta_{3} + 6 \beta_1 + 20) q^{41} + (2 \beta_{3} + 60 \beta_{2} + \cdots - 30) q^{43}+ \cdots + ( - 10 \beta_{3} + 52 \beta_{2} + \cdots - 26) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} - 2 q^{5} + 2 q^{9} + 12 q^{11} + 28 q^{13} + 32 q^{17} - 42 q^{19} + 48 q^{23} + 6 q^{25} + 32 q^{29} + 156 q^{31} - 16 q^{33} - 16 q^{37} + 168 q^{39} + 80 q^{41} + 86 q^{45} - 36 q^{47} + 180 q^{51} + 92 q^{53} - 280 q^{57} + 306 q^{59} + 102 q^{61} - 140 q^{65} + 48 q^{67} - 16 q^{69} + 92 q^{73} - 66 q^{75} + 72 q^{79} + 10 q^{81} - 232 q^{85} + 468 q^{87} - 28 q^{89} + 72 q^{93} + 336 q^{95} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - x^{2} - 2x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu^{2} + 3\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} - \nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{3} + \nu^{2} + 3\nu + 6 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + \beta _1 + 5 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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.

comment: embeddings in the coefficient field
 
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
1.39564 + 0.228425i
−0.895644 − 1.09445i
1.39564 − 0.228425i
−0.895644 + 1.09445i
0 −0.791288 + 0.456850i 0 1.79129 − 3.10260i 0 0 0 −4.08258 + 7.07123i 0
79.2 0 3.79129 − 2.18890i 0 −2.79129 + 4.83465i 0 0 0 5.08258 − 8.80328i 0
655.1 0 −0.791288 − 0.456850i 0 1.79129 + 3.10260i 0 0 0 −4.08258 − 7.07123i 0
655.2 0 3.79129 + 2.18890i 0 −2.79129 − 4.83465i 0 0 0 5.08258 + 8.80328i 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.n 4
4.b odd 2 1 784.3.r.i 4
7.b odd 2 1 784.3.r.j 4
7.c even 3 1 784.3.d.j 4
7.c even 3 1 784.3.r.i 4
7.d odd 6 1 112.3.d.b ✓ 4
7.d odd 6 1 784.3.r.o 4
21.g even 6 1 1008.3.m.d 4
28.d even 2 1 784.3.r.o 4
28.f even 6 1 112.3.d.b ✓ 4
28.f even 6 1 784.3.r.j 4
28.g odd 6 1 784.3.d.j 4
28.g odd 6 1 inner 784.3.r.n 4
56.j odd 6 1 448.3.d.b 4
56.m even 6 1 448.3.d.b 4
84.j odd 6 1 1008.3.m.d 4
112.v even 12 2 1792.3.g.e 8
112.x odd 12 2 1792.3.g.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.3.d.b ✓ 4 7.d odd 6 1
112.3.d.b ✓ 4 28.f even 6 1
448.3.d.b 4 56.j odd 6 1
448.3.d.b 4 56.m even 6 1
784.3.d.j 4 7.c even 3 1
784.3.d.j 4 28.g odd 6 1
784.3.r.i 4 4.b odd 2 1
784.3.r.i 4 7.c even 3 1
784.3.r.j 4 7.b odd 2 1
784.3.r.j 4 28.f even 6 1
784.3.r.n 4 1.a even 1 1 trivial
784.3.r.n 4 28.g odd 6 1 inner
784.3.r.o 4 7.d odd 6 1
784.3.r.o 4 28.d even 2 1
1008.3.m.d 4 21.g even 6 1
1008.3.m.d 4 84.j odd 6 1
1792.3.g.e 8 112.v even 12 2
1792.3.g.e 8 112.x odd 12 2

Hecke kernels

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

\( T_{3}^{4} - 6T_{3}^{3} + 8T_{3}^{2} + 24T_{3} + 16 \) Copy content Toggle raw display
\( T_{5}^{4} + 2T_{5}^{3} + 24T_{5}^{2} - 40T_{5} + 400 \) Copy content Toggle raw display
\( T_{11}^{4} - 12T_{11}^{3} + 32T_{11}^{2} + 192T_{11} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 6 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 12 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( (T^{2} - 14 T - 140)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 32 T^{3} + \cdots + 29584 \) Copy content Toggle raw display
$19$ \( T^{4} + 42 T^{3} + \cdots + 38416 \) Copy content Toggle raw display
$23$ \( T^{4} - 48 T^{3} + \cdots + 6400 \) Copy content Toggle raw display
$29$ \( (T^{2} - 16 T - 2036)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 156 T^{3} + \cdots + 3154176 \) Copy content Toggle raw display
$37$ \( T^{4} + 16 T^{3} + \cdots + 400 \) Copy content Toggle raw display
$41$ \( (T^{2} - 40 T - 356)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 5456 T^{2} + 7139584 \) Copy content Toggle raw display
$47$ \( T^{4} + 36 T^{3} + \cdots + 4665600 \) Copy content Toggle raw display
$53$ \( (T^{2} - 46 T + 2116)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 306 T^{3} + \cdots + 58186384 \) Copy content Toggle raw display
$61$ \( T^{4} - 102 T^{3} + \cdots + 4309776 \) Copy content Toggle raw display
$67$ \( T^{4} - 48 T^{3} + \cdots + 78854400 \) Copy content Toggle raw display
$71$ \( T^{4} + 6464 T^{2} + 5607424 \) Copy content Toggle raw display
$73$ \( T^{4} - 92 T^{3} + \cdots + 39488656 \) Copy content Toggle raw display
$79$ \( T^{4} - 72 T^{3} + \cdots + 331776 \) Copy content Toggle raw display
$83$ \( T^{4} + 23540 T^{2} + 59660176 \) Copy content Toggle raw display
$89$ \( T^{4} + 28 T^{3} + \cdots + 26832400 \) Copy content Toggle raw display
$97$ \( (T^{2} + 16 T - 20)^{2} \) Copy content Toggle raw display
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