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Properties

Label 784.2.x.b
Level $784$
Weight $2$
Character orbit 784.x
Analytic conductor $6.260$
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] = [784,2,Mod(165,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.x (of order \(12\), degree \(4\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{2} + (2 \zeta_{12}^{3} + \cdots - 2 \zeta_{12}) q^{3}+ \cdots + 5 \zeta_{12} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{2} - \zeta_{12} + 1) q^{2} + (2 \zeta_{12}^{3} + \cdots - 2 \zeta_{12}) q^{3}+ \cdots + (15 \zeta_{12}^{3} - 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 4 q^{3} - 4 q^{5} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 4 q^{3} - 4 q^{5} + 8 q^{8} + 8 q^{10} + 6 q^{11} + 8 q^{12} + 32 q^{15} + 8 q^{16} - 12 q^{17} - 10 q^{18} - 8 q^{19} + 16 q^{20} + 24 q^{22} - 16 q^{24} - 16 q^{27} - 4 q^{29} + 16 q^{30} + 8 q^{31} - 8 q^{32} + 24 q^{33} - 24 q^{34} - 40 q^{36} - 6 q^{37} - 20 q^{43} + 12 q^{44} - 20 q^{45} + 4 q^{46} - 16 q^{47} - 32 q^{48} - 12 q^{50} - 24 q^{51} - 14 q^{53} - 16 q^{54} + 4 q^{59} + 12 q^{61} + 16 q^{62} - 24 q^{66} + 10 q^{67} + 16 q^{69} - 20 q^{72} + 12 q^{74} - 12 q^{75} - 32 q^{76} + 28 q^{79} + 16 q^{80} + 2 q^{81} - 4 q^{82} + 48 q^{85} - 20 q^{86} + 16 q^{92} + 16 q^{93} + 16 q^{94} - 32 q^{95} + 24 q^{97} - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(1\) \(-1 + \zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
1.36603 + 0.366025i 0.732051 + 2.73205i 1.73205 + 1.00000i 0.732051 − 2.73205i 4.00000i 0 2.00000 + 2.00000i −4.33013 + 2.50000i 2.00000 − 3.46410i
373.1 −0.366025 − 1.36603i −2.73205 − 0.732051i −1.73205 + 1.00000i −2.73205 + 0.732051i 4.00000i 0 2.00000 + 2.00000i 4.33013 + 2.50000i 2.00000 + 3.46410i
557.1 −0.366025 + 1.36603i −2.73205 + 0.732051i −1.73205 − 1.00000i −2.73205 − 0.732051i − 4.00000i 0 2.00000 − 2.00000i 4.33013 − 2.50000i 2.00000 − 3.46410i
765.1 1.36603 − 0.366025i 0.732051 − 2.73205i 1.73205 − 1.00000i 0.732051 + 2.73205i − 4.00000i 0 2.00000 − 2.00000i −4.33013 − 2.50000i 2.00000 + 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
16.e even 4 1 inner
112.w even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.x.b 4
7.b odd 2 1 784.2.x.g 4
7.c even 3 1 784.2.m.c 2
7.c even 3 1 inner 784.2.x.b 4
7.d odd 6 1 112.2.m.a ✓ 2
7.d odd 6 1 784.2.x.g 4
16.e even 4 1 inner 784.2.x.b 4
28.f even 6 1 448.2.m.b 2
56.j odd 6 1 896.2.m.d 2
56.m even 6 1 896.2.m.a 2
112.l odd 4 1 784.2.x.g 4
112.v even 12 1 448.2.m.b 2
112.v even 12 1 896.2.m.a 2
112.w even 12 1 784.2.m.c 2
112.w even 12 1 inner 784.2.x.b 4
112.x odd 12 1 112.2.m.a ✓ 2
112.x odd 12 1 784.2.x.g 4
112.x odd 12 1 896.2.m.d 2
224.bc odd 24 2 7168.2.a.q 2
224.be even 24 2 7168.2.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.m.a ✓ 2 7.d odd 6 1
112.2.m.a ✓ 2 112.x odd 12 1
448.2.m.b 2 28.f even 6 1
448.2.m.b 2 112.v even 12 1
784.2.m.c 2 7.c even 3 1
784.2.m.c 2 112.w even 12 1
784.2.x.b 4 1.a even 1 1 trivial
784.2.x.b 4 7.c even 3 1 inner
784.2.x.b 4 16.e even 4 1 inner
784.2.x.b 4 112.w even 12 1 inner
784.2.x.g 4 7.b odd 2 1
784.2.x.g 4 7.d odd 6 1
784.2.x.g 4 112.l odd 4 1
784.2.x.g 4 112.x odd 12 1
896.2.m.a 2 56.m even 6 1
896.2.m.a 2 112.v even 12 1
896.2.m.d 2 56.j odd 6 1
896.2.m.d 2 112.x odd 12 1
7168.2.a.i 2 224.be even 24 2
7168.2.a.q 2 224.bc odd 24 2

Hecke kernels

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

\( T_{3}^{4} + 4T_{3}^{3} + 8T_{3}^{2} + 32T_{3} + 64 \) Copy content Toggle raw display
\( T_{5}^{4} + 4T_{5}^{3} + 8T_{5}^{2} + 32T_{5} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{4} + 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 8 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10 T + 50)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 14 T^{3} + \cdots + 9604 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + \cdots + 5184 \) Copy content Toggle raw display
$67$ \( T^{4} - 10 T^{3} + \cdots + 2500 \) Copy content Toggle raw display
$71$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$97$ \( (T - 6)^{4} \) Copy content Toggle raw display
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