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Properties

Label 784.3.r.k
Level $784$
Weight $3$
Character orbit 784.r
Analytic conductor $21.362$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( - 8 \beta_1 - 8) q^{5} + (19 \beta_1 + 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - 8 \beta_1 - 8) q^{5} + (19 \beta_1 + 19) q^{9} + 2 \beta_{2} q^{11} - 4 q^{13} + (8 \beta_{3} - 8 \beta_{2}) q^{15} - 2 \beta_1 q^{17} + 5 \beta_{3} q^{19} + 4 \beta_{3} q^{23} + 39 \beta_1 q^{25} + ( - 10 \beta_{3} + 10 \beta_{2}) q^{27} + 14 q^{29} + 6 \beta_{2} q^{31} + (56 \beta_1 + 56) q^{33} + ( - 14 \beta_1 - 14) q^{37} - 4 \beta_{2} q^{39} + 46 q^{41} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{43} - 152 \beta_1 q^{45} + 6 \beta_{3} q^{47} + 2 \beta_{3} q^{51} - 22 \beta_1 q^{53} + (16 \beta_{3} - 16 \beta_{2}) q^{55} + 140 q^{57} - 17 \beta_{2} q^{59} + ( - 48 \beta_1 - 48) q^{61} + (32 \beta_1 + 32) q^{65} + 12 \beta_{2} q^{67} + 112 q^{69} + ( - 16 \beta_{3} + 16 \beta_{2}) q^{71} - 110 \beta_1 q^{73} - 39 \beta_{3} q^{75} - 24 \beta_{3} q^{79} + 109 \beta_1 q^{81} + (7 \beta_{3} - 7 \beta_{2}) q^{83} - 16 q^{85} + 14 \beta_{2} q^{87} + (134 \beta_1 + 134) q^{89} + (168 \beta_1 + 168) q^{93} - 40 \beta_{2} q^{95} - 178 q^{97} + ( - 38 \beta_{3} + 38 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{5} + 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{5} + 38 q^{9} - 16 q^{13} + 4 q^{17} - 78 q^{25} + 56 q^{29} + 112 q^{33} - 28 q^{37} + 184 q^{41} + 304 q^{45} + 44 q^{53} + 560 q^{57} - 96 q^{61} + 64 q^{65} + 448 q^{69} + 220 q^{73} - 218 q^{81} - 64 q^{85} + 268 q^{89} + 336 q^{93} - 712 q^{97}+O(q^{100}) \) 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

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

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

\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 6\beta _1 + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} + 10 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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
−0.895644 + 1.09445i
1.39564 − 0.228425i
−0.895644 − 1.09445i
1.39564 + 0.228425i
0 −4.58258 + 2.64575i 0 −4.00000 + 6.92820i 0 0 0 9.50000 − 16.4545i 0
79.2 0 4.58258 − 2.64575i 0 −4.00000 + 6.92820i 0 0 0 9.50000 − 16.4545i 0
655.1 0 −4.58258 − 2.64575i 0 −4.00000 − 6.92820i 0 0 0 9.50000 + 16.4545i 0
655.2 0 4.58258 + 2.64575i 0 −4.00000 − 6.92820i 0 0 0 9.50000 + 16.4545i 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
4.b odd 2 1 inner
7.c even 3 1 inner
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.k 4
4.b odd 2 1 inner 784.3.r.k 4
7.b odd 2 1 784.3.r.m 4
7.c even 3 1 112.3.d.a ✓ 2
7.c even 3 1 inner 784.3.r.k 4
7.d odd 6 1 784.3.d.d 2
7.d odd 6 1 784.3.r.m 4
21.h odd 6 1 1008.3.m.a 2
28.d even 2 1 784.3.r.m 4
28.f even 6 1 784.3.d.d 2
28.f even 6 1 784.3.r.m 4
28.g odd 6 1 112.3.d.a ✓ 2
28.g odd 6 1 inner 784.3.r.k 4
56.k odd 6 1 448.3.d.a 2
56.p even 6 1 448.3.d.a 2
84.n even 6 1 1008.3.m.a 2
112.u odd 12 2 1792.3.g.b 4
112.w even 12 2 1792.3.g.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.3.d.a ✓ 2 7.c even 3 1
112.3.d.a ✓ 2 28.g odd 6 1
448.3.d.a 2 56.k odd 6 1
448.3.d.a 2 56.p even 6 1
784.3.d.d 2 7.d odd 6 1
784.3.d.d 2 28.f even 6 1
784.3.r.k 4 1.a even 1 1 trivial
784.3.r.k 4 4.b odd 2 1 inner
784.3.r.k 4 7.c even 3 1 inner
784.3.r.k 4 28.g odd 6 1 inner
784.3.r.m 4 7.b odd 2 1
784.3.r.m 4 7.d odd 6 1
784.3.r.m 4 28.d even 2 1
784.3.r.m 4 28.f even 6 1
1008.3.m.a 2 21.h odd 6 1
1008.3.m.a 2 84.n even 6 1
1792.3.g.b 4 112.u odd 12 2
1792.3.g.b 4 112.w even 12 2

Hecke kernels

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

\( T_{3}^{4} - 28T_{3}^{2} + 784 \) Copy content Toggle raw display
\( T_{5}^{2} + 8T_{5} + 64 \) Copy content Toggle raw display
\( T_{11}^{4} - 112T_{11}^{2} + 12544 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 28T^{2} + 784 \) Copy content Toggle raw display
$5$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 112 T^{2} + 12544 \) Copy content Toggle raw display
$13$ \( (T + 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 700 T^{2} + 490000 \) Copy content Toggle raw display
$23$ \( T^{4} - 448 T^{2} + 200704 \) Copy content Toggle raw display
$29$ \( (T - 14)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 1008 T^{2} + 1016064 \) Copy content Toggle raw display
$37$ \( (T^{2} + 14 T + 196)^{2} \) Copy content Toggle raw display
$41$ \( (T - 46)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 112)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 1008 T^{2} + 1016064 \) Copy content Toggle raw display
$53$ \( (T^{2} - 22 T + 484)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 8092 T^{2} + 65480464 \) Copy content Toggle raw display
$61$ \( (T^{2} + 48 T + 2304)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4032 T^{2} + 16257024 \) Copy content Toggle raw display
$71$ \( (T^{2} + 7168)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 110 T + 12100)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 16128 T^{2} + 260112384 \) Copy content Toggle raw display
$83$ \( (T^{2} + 1372)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 134 T + 17956)^{2} \) Copy content Toggle raw display
$97$ \( (T + 178)^{4} \) Copy content Toggle raw display
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