Properties

Label 3136.2.f.e
Level $3136$
Weight $2$
Character orbit 3136.f
Analytic conductor $25.041$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3136,2,Mod(3135,3136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3136, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3136.3135");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.f (of order \(2\), degree \(1\), 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: \(25.0410860739\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - \beta_{2} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - \beta_{2} q^{5} + \beta_1 q^{11} - 2 \beta_{2} q^{13} + 3 \beta_1 q^{15} - \beta_{2} q^{17} + 3 \beta_{3} q^{19} + \beta_1 q^{23} + 2 q^{25} + 3 \beta_{3} q^{27} - 4 q^{29} - \beta_{3} q^{31} - \beta_{2} q^{33} - 3 q^{37} + 6 \beta_1 q^{39} - 2 \beta_{2} q^{41} - 2 \beta_1 q^{43} + 5 \beta_{3} q^{47} + 3 \beta_1 q^{51} + q^{53} + \beta_{3} q^{55} - 9 q^{57} + 3 \beta_{3} q^{59} - 3 \beta_{2} q^{61} - 6 q^{65} + 3 \beta_1 q^{67} - \beta_{2} q^{69} - 14 \beta_1 q^{71} + 5 \beta_{2} q^{73} - 2 \beta_{3} q^{75} + 9 \beta_1 q^{79} - 9 q^{81} - 8 \beta_{3} q^{83} - 3 q^{85} + 4 \beta_{3} q^{87} - 9 \beta_{2} q^{89} + 3 q^{93} - 9 \beta_1 q^{95} - 10 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{25} - 16 q^{29} - 12 q^{37} + 4 q^{53} - 36 q^{57} - 24 q^{65} - 36 q^{81} - 12 q^{85} + 12 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3135.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0 −1.73205 0 1.73205i 0 0 0 0 0
3135.2 0 −1.73205 0 1.73205i 0 0 0 0 0
3135.3 0 1.73205 0 1.73205i 0 0 0 0 0
3135.4 0 1.73205 0 1.73205i 0 0 0 0 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.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.f.e 4
4.b odd 2 1 inner 3136.2.f.e 4
7.b odd 2 1 inner 3136.2.f.e 4
7.c even 3 1 448.2.p.d 4
7.d odd 6 1 448.2.p.d 4
8.b even 2 1 196.2.d.b 4
8.d odd 2 1 196.2.d.b 4
24.f even 2 1 1764.2.b.a 4
24.h odd 2 1 1764.2.b.a 4
28.d even 2 1 inner 3136.2.f.e 4
28.f even 6 1 448.2.p.d 4
28.g odd 6 1 448.2.p.d 4
56.e even 2 1 196.2.d.b 4
56.h odd 2 1 196.2.d.b 4
56.j odd 6 1 28.2.f.a 4
56.j odd 6 1 196.2.f.a 4
56.k odd 6 1 28.2.f.a 4
56.k odd 6 1 196.2.f.a 4
56.m even 6 1 28.2.f.a 4
56.m even 6 1 196.2.f.a 4
56.p even 6 1 28.2.f.a 4
56.p even 6 1 196.2.f.a 4
168.e odd 2 1 1764.2.b.a 4
168.i even 2 1 1764.2.b.a 4
168.s odd 6 1 252.2.bf.e 4
168.v even 6 1 252.2.bf.e 4
168.ba even 6 1 252.2.bf.e 4
168.be odd 6 1 252.2.bf.e 4
280.ba even 6 1 700.2.p.a 4
280.bf even 6 1 700.2.p.a 4
280.bi odd 6 1 700.2.p.a 4
280.bk odd 6 1 700.2.p.a 4
280.bp odd 12 1 700.2.t.a 4
280.bp odd 12 1 700.2.t.b 4
280.br even 12 1 700.2.t.a 4
280.br even 12 1 700.2.t.b 4
280.bt odd 12 1 700.2.t.a 4
280.bt odd 12 1 700.2.t.b 4
280.bv even 12 1 700.2.t.a 4
280.bv even 12 1 700.2.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 56.j odd 6 1
28.2.f.a 4 56.k odd 6 1
28.2.f.a 4 56.m even 6 1
28.2.f.a 4 56.p even 6 1
196.2.d.b 4 8.b even 2 1
196.2.d.b 4 8.d odd 2 1
196.2.d.b 4 56.e even 2 1
196.2.d.b 4 56.h odd 2 1
196.2.f.a 4 56.j odd 6 1
196.2.f.a 4 56.k odd 6 1
196.2.f.a 4 56.m even 6 1
196.2.f.a 4 56.p even 6 1
252.2.bf.e 4 168.s odd 6 1
252.2.bf.e 4 168.v even 6 1
252.2.bf.e 4 168.ba even 6 1
252.2.bf.e 4 168.be odd 6 1
448.2.p.d 4 7.c even 3 1
448.2.p.d 4 7.d odd 6 1
448.2.p.d 4 28.f even 6 1
448.2.p.d 4 28.g odd 6 1
700.2.p.a 4 280.ba even 6 1
700.2.p.a 4 280.bf even 6 1
700.2.p.a 4 280.bi odd 6 1
700.2.p.a 4 280.bk odd 6 1
700.2.t.a 4 280.bp odd 12 1
700.2.t.a 4 280.br even 12 1
700.2.t.a 4 280.bt odd 12 1
700.2.t.a 4 280.bv even 12 1
700.2.t.b 4 280.bp odd 12 1
700.2.t.b 4 280.br even 12 1
700.2.t.b 4 280.bt odd 12 1
700.2.t.b 4 280.bv even 12 1
1764.2.b.a 4 24.f even 2 1
1764.2.b.a 4 24.h odd 2 1
1764.2.b.a 4 168.e odd 2 1
1764.2.b.a 4 168.i even 2 1
3136.2.f.e 4 1.a even 1 1 trivial
3136.2.f.e 4 4.b odd 2 1 inner
3136.2.f.e 4 7.b odd 2 1 inner
3136.2.f.e 4 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3136, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T + 4)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$37$ \( (T + 3)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 75)^{2} \) Copy content Toggle raw display
$53$ \( (T - 1)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 75)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 243)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 300)^{2} \) Copy content Toggle raw display
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