Properties

Label 3136.1.d.b
Level $3136$
Weight $1$
Character orbit 3136.d
Analytic conductor $1.565$
Analytic rank $0$
Dimension $2$
Projective image $A_{4}$
CM/RM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3136,1,Mod(1471,3136)]
 
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, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3136.1471");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3136.d (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: \(1.56506787962\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
Projective image: \(A_{4}\)
Projective field: Galois closure of 4.0.3136.1
Artin image: $\SL(2,3):C_2$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + i q^{3} - q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + i q^{3} - q^{5} - i q^{11} - i q^{15} + q^{17} + i q^{19} + i q^{23} + i q^{27} + i q^{31} + q^{33} + q^{37} - i q^{47} + i q^{51} + q^{53} + i q^{55} - q^{57} + i q^{59} + q^{61} + i q^{67} - q^{69} - q^{73} + i q^{79} - q^{81} - q^{85} - q^{89} - q^{93} - i q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 2 q^{17} + 2 q^{33} + 2 q^{37} + 2 q^{53} - 2 q^{57} + 2 q^{61} - 2 q^{69} - 2 q^{73} - 2 q^{81} - 2 q^{85} - 2 q^{89} - 2 q^{93}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
1.00000i
1.00000i
0 1.00000i 0 −1.00000 0 0 0 0 0
1471.2 0 1.00000i 0 −1.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.1.d.b 2
4.b odd 2 1 inner 3136.1.d.b 2
7.b odd 2 1 3136.1.d.d 2
7.c even 3 2 448.1.r.a 4
7.d odd 6 2 3136.1.r.b 4
8.b even 2 1 1568.1.d.b 2
8.d odd 2 1 1568.1.d.b 2
28.d even 2 1 3136.1.d.d 2
28.f even 6 2 3136.1.r.b 4
28.g odd 6 2 448.1.r.a 4
56.e even 2 1 1568.1.d.a 2
56.h odd 2 1 1568.1.d.a 2
56.j odd 6 2 1568.1.r.a 4
56.k odd 6 2 224.1.r.a 4
56.m even 6 2 1568.1.r.a 4
56.p even 6 2 224.1.r.a 4
112.u odd 12 2 1792.1.o.a 4
112.u odd 12 2 1792.1.o.b 4
112.w even 12 2 1792.1.o.a 4
112.w even 12 2 1792.1.o.b 4
168.s odd 6 2 2016.1.cd.a 4
168.v even 6 2 2016.1.cd.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.r.a 4 56.k odd 6 2
224.1.r.a 4 56.p even 6 2
448.1.r.a 4 7.c even 3 2
448.1.r.a 4 28.g odd 6 2
1568.1.d.a 2 56.e even 2 1
1568.1.d.a 2 56.h odd 2 1
1568.1.d.b 2 8.b even 2 1
1568.1.d.b 2 8.d odd 2 1
1568.1.r.a 4 56.j odd 6 2
1568.1.r.a 4 56.m even 6 2
1792.1.o.a 4 112.u odd 12 2
1792.1.o.a 4 112.w even 12 2
1792.1.o.b 4 112.u odd 12 2
1792.1.o.b 4 112.w even 12 2
2016.1.cd.a 4 168.s odd 6 2
2016.1.cd.a 4 168.v even 6 2
3136.1.d.b 2 1.a even 1 1 trivial
3136.1.d.b 2 4.b odd 2 1 inner
3136.1.d.d 2 7.b odd 2 1
3136.1.d.d 2 28.d even 2 1
3136.1.r.b 4 7.d odd 6 2
3136.1.r.b 4 28.f even 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3136, [\chi])\):

\( T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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