Properties

Label 3136.2.b.b
Level $3136$
Weight $2$
Character orbit 3136.b
Analytic conductor $25.041$
Analytic rank $0$
Dimension $2$
CM discriminant -8
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(1569,3136)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3136.1569");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.b (of order \(2\), degree \(1\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
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: \( 2 \)
Twist minimal: no (minimal twist has level 64)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - q^{9} + 3 \beta q^{11} + 6 q^{17} + \beta q^{19} + 5 q^{25} + 2 \beta q^{27} - 12 q^{33} + 6 q^{41} - 5 \beta q^{43} + 6 \beta q^{51} - 4 q^{57} - 3 \beta q^{59} + 7 \beta q^{67} - 2 q^{73} + 5 \beta q^{75} - 11 q^{81} + 9 \beta q^{83} - 18 q^{89} - 10 q^{97} - 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{9} + 12 q^{17} + 10 q^{25} - 24 q^{33} + 12 q^{41} - 8 q^{57} - 4 q^{73} - 22 q^{81} - 36 q^{89} - 20 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1569.1
1.00000i
1.00000i
0 2.00000i 0 0 0 0 0 −1.00000 0
1569.2 0 2.00000i 0 0 0 0 0 −1.00000 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b 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.b.b 2
4.b odd 2 1 inner 3136.2.b.b 2
7.b odd 2 1 64.2.b.a 2
8.b even 2 1 inner 3136.2.b.b 2
8.d odd 2 1 CM 3136.2.b.b 2
21.c even 2 1 576.2.d.a 2
28.d even 2 1 64.2.b.a 2
35.c odd 2 1 1600.2.d.a 2
35.f even 4 1 1600.2.f.a 2
35.f even 4 1 1600.2.f.b 2
56.e even 2 1 64.2.b.a 2
56.h odd 2 1 64.2.b.a 2
84.h odd 2 1 576.2.d.a 2
112.j even 4 1 256.2.a.a 1
112.j even 4 1 256.2.a.d 1
112.l odd 4 1 256.2.a.a 1
112.l odd 4 1 256.2.a.d 1
140.c even 2 1 1600.2.d.a 2
140.j odd 4 1 1600.2.f.a 2
140.j odd 4 1 1600.2.f.b 2
168.e odd 2 1 576.2.d.a 2
168.i even 2 1 576.2.d.a 2
224.v odd 8 4 1024.2.e.l 4
224.x even 8 4 1024.2.e.l 4
280.c odd 2 1 1600.2.d.a 2
280.n even 2 1 1600.2.d.a 2
280.s even 4 1 1600.2.f.a 2
280.s even 4 1 1600.2.f.b 2
280.y odd 4 1 1600.2.f.a 2
280.y odd 4 1 1600.2.f.b 2
336.v odd 4 1 2304.2.a.h 1
336.v odd 4 1 2304.2.a.i 1
336.y even 4 1 2304.2.a.h 1
336.y even 4 1 2304.2.a.i 1
560.be even 4 1 6400.2.a.a 1
560.be even 4 1 6400.2.a.x 1
560.bf odd 4 1 6400.2.a.a 1
560.bf odd 4 1 6400.2.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a 2 7.b odd 2 1
64.2.b.a 2 28.d even 2 1
64.2.b.a 2 56.e even 2 1
64.2.b.a 2 56.h odd 2 1
256.2.a.a 1 112.j even 4 1
256.2.a.a 1 112.l odd 4 1
256.2.a.d 1 112.j even 4 1
256.2.a.d 1 112.l odd 4 1
576.2.d.a 2 21.c even 2 1
576.2.d.a 2 84.h odd 2 1
576.2.d.a 2 168.e odd 2 1
576.2.d.a 2 168.i even 2 1
1024.2.e.l 4 224.v odd 8 4
1024.2.e.l 4 224.x even 8 4
1600.2.d.a 2 35.c odd 2 1
1600.2.d.a 2 140.c even 2 1
1600.2.d.a 2 280.c odd 2 1
1600.2.d.a 2 280.n even 2 1
1600.2.f.a 2 35.f even 4 1
1600.2.f.a 2 140.j odd 4 1
1600.2.f.a 2 280.s even 4 1
1600.2.f.a 2 280.y odd 4 1
1600.2.f.b 2 35.f even 4 1
1600.2.f.b 2 140.j odd 4 1
1600.2.f.b 2 280.s even 4 1
1600.2.f.b 2 280.y odd 4 1
2304.2.a.h 1 336.v odd 4 1
2304.2.a.h 1 336.y even 4 1
2304.2.a.i 1 336.v odd 4 1
2304.2.a.i 1 336.y even 4 1
3136.2.b.b 2 1.a even 1 1 trivial
3136.2.b.b 2 4.b odd 2 1 inner
3136.2.b.b 2 8.b even 2 1 inner
3136.2.b.b 2 8.d odd 2 1 CM
6400.2.a.a 1 560.be even 4 1
6400.2.a.a 1 560.bf odd 4 1
6400.2.a.x 1 560.be even 4 1
6400.2.a.x 1 560.bf odd 4 1

Hecke kernels

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

\( T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{17} - 6 \) Copy content Toggle raw display
\( T_{31} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T - 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 100 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 196 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 324 \) Copy content Toggle raw display
$89$ \( (T + 18)^{2} \) Copy content Toggle raw display
$97$ \( (T + 10)^{2} \) Copy content Toggle raw display
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