Properties

Label 784.2.p.e
Level $784$
Weight $2$
Character orbit 784.p
Analytic conductor $6.260$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [784,2,Mod(31,784)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.p (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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
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: 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{3} + (2 \zeta_{6} - 4) q^{5} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{3} + (2 \zeta_{6} - 4) q^{5} - \zeta_{6} q^{9} + (2 \zeta_{6} + 2) q^{11} + (4 \zeta_{6} - 2) q^{13} + (8 \zeta_{6} - 4) q^{15} + 2 \zeta_{6} q^{19} + ( - 2 \zeta_{6} + 4) q^{23} + ( - 7 \zeta_{6} + 7) q^{25} + 4 q^{27} + 6 q^{29} + (8 \zeta_{6} - 8) q^{31} + ( - 4 \zeta_{6} + 8) q^{33} + 2 \zeta_{6} q^{37} + (4 \zeta_{6} + 4) q^{39} + ( - 8 \zeta_{6} + 4) q^{41} + (12 \zeta_{6} - 6) q^{43} + (2 \zeta_{6} + 2) q^{45} + (6 \zeta_{6} - 6) q^{53} - 12 q^{55} + 4 q^{57} + ( - 6 \zeta_{6} + 6) q^{59} + ( - 2 \zeta_{6} + 4) q^{61} - 12 \zeta_{6} q^{65} + (2 \zeta_{6} + 2) q^{67} + ( - 8 \zeta_{6} + 4) q^{69} + ( - 4 \zeta_{6} + 2) q^{71} + ( - 4 \zeta_{6} - 4) q^{73} - 14 \zeta_{6} q^{75} + (2 \zeta_{6} - 4) q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 6 q^{83} + ( - 12 \zeta_{6} + 12) q^{87} + (4 \zeta_{6} - 8) q^{89} + 16 \zeta_{6} q^{93} + ( - 4 \zeta_{6} - 4) q^{95} + (16 \zeta_{6} - 8) q^{97} + ( - 4 \zeta_{6} + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 6 q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 6 q^{5} - q^{9} + 6 q^{11} + 2 q^{19} + 6 q^{23} + 7 q^{25} + 8 q^{27} + 12 q^{29} - 8 q^{31} + 12 q^{33} + 2 q^{37} + 12 q^{39} + 6 q^{45} - 6 q^{53} - 24 q^{55} + 8 q^{57} + 6 q^{59} + 6 q^{61} - 12 q^{65} + 6 q^{67} - 12 q^{73} - 14 q^{75} - 6 q^{79} + 11 q^{81} + 12 q^{83} + 12 q^{87} - 12 q^{89} + 16 q^{93} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.00000 1.73205i 0 −3.00000 + 1.73205i 0 0 0 −0.500000 0.866025i 0
607.1 0 1.00000 + 1.73205i 0 −3.00000 1.73205i 0 0 0 −0.500000 + 0.866025i 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.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.p.e 2
4.b odd 2 1 784.2.p.a 2
7.b odd 2 1 784.2.p.b 2
7.c even 3 1 112.2.f.a 2
7.c even 3 1 784.2.p.f 2
7.d odd 6 1 112.2.f.b yes 2
7.d odd 6 1 784.2.p.a 2
21.g even 6 1 1008.2.b.b 2
21.h odd 6 1 1008.2.b.g 2
28.d even 2 1 784.2.p.f 2
28.f even 6 1 112.2.f.a 2
28.f even 6 1 inner 784.2.p.e 2
28.g odd 6 1 112.2.f.b yes 2
28.g odd 6 1 784.2.p.b 2
35.i odd 6 1 2800.2.k.b 2
35.j even 6 1 2800.2.k.e 2
35.k even 12 2 2800.2.e.b 4
35.l odd 12 2 2800.2.e.c 4
56.j odd 6 1 448.2.f.a 2
56.k odd 6 1 448.2.f.a 2
56.m even 6 1 448.2.f.c 2
56.p even 6 1 448.2.f.c 2
84.j odd 6 1 1008.2.b.g 2
84.n even 6 1 1008.2.b.b 2
112.u odd 12 2 1792.2.e.c 4
112.v even 12 2 1792.2.e.a 4
112.w even 12 2 1792.2.e.a 4
112.x odd 12 2 1792.2.e.c 4
140.p odd 6 1 2800.2.k.b 2
140.s even 6 1 2800.2.k.e 2
140.w even 12 2 2800.2.e.b 4
140.x odd 12 2 2800.2.e.c 4
168.s odd 6 1 4032.2.b.h 2
168.v even 6 1 4032.2.b.b 2
168.ba even 6 1 4032.2.b.b 2
168.be odd 6 1 4032.2.b.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.f.a 2 7.c even 3 1
112.2.f.a 2 28.f even 6 1
112.2.f.b yes 2 7.d odd 6 1
112.2.f.b yes 2 28.g odd 6 1
448.2.f.a 2 56.j odd 6 1
448.2.f.a 2 56.k odd 6 1
448.2.f.c 2 56.m even 6 1
448.2.f.c 2 56.p even 6 1
784.2.p.a 2 4.b odd 2 1
784.2.p.a 2 7.d odd 6 1
784.2.p.b 2 7.b odd 2 1
784.2.p.b 2 28.g odd 6 1
784.2.p.e 2 1.a even 1 1 trivial
784.2.p.e 2 28.f even 6 1 inner
784.2.p.f 2 7.c even 3 1
784.2.p.f 2 28.d even 2 1
1008.2.b.b 2 21.g even 6 1
1008.2.b.b 2 84.n even 6 1
1008.2.b.g 2 21.h odd 6 1
1008.2.b.g 2 84.j odd 6 1
1792.2.e.a 4 112.v even 12 2
1792.2.e.a 4 112.w even 12 2
1792.2.e.c 4 112.u odd 12 2
1792.2.e.c 4 112.x odd 12 2
2800.2.e.b 4 35.k even 12 2
2800.2.e.b 4 140.w even 12 2
2800.2.e.c 4 35.l odd 12 2
2800.2.e.c 4 140.x odd 12 2
2800.2.k.b 2 35.i odd 6 1
2800.2.k.b 2 140.p odd 6 1
2800.2.k.e 2 35.j even 6 1
2800.2.k.e 2 140.s even 6 1
4032.2.b.b 2 168.v even 6 1
4032.2.b.b 2 168.ba even 6 1
4032.2.b.h 2 168.s odd 6 1
4032.2.b.h 2 168.be odd 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 12 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 48 \) Copy content Toggle raw display
$43$ \( T^{2} + 108 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$67$ \( T^{2} - 6T + 12 \) Copy content Toggle raw display
$71$ \( T^{2} + 12 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$79$ \( T^{2} + 6T + 12 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} + 192 \) Copy content Toggle raw display
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