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Properties

Label 3136.2.a.m
Level $3136$
Weight $2$
Character orbit 3136.a
Self dual yes
Analytic conductor $25.041$
Analytic rank $1$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3136,2,Mod(1,3136)]
 
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, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3136.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.a (trivial)

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: yes
Analytic conductor: \(25.0410860739\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(+1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 2 q^{5} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{5} - 3 q^{9} + 6 q^{13} - 2 q^{17} - q^{25} + 10 q^{29} + 2 q^{37} - 10 q^{41} + 6 q^{45} - 14 q^{53} - 10 q^{61} - 12 q^{65} + 6 q^{73} + 9 q^{81} + 4 q^{85} - 10 q^{89} - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 0 −2.00000 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3136.2.a.m 1
4.b odd 2 1 CM 3136.2.a.m 1
7.b odd 2 1 64.2.a.a 1
8.b even 2 1 1568.2.a.e 1
8.d odd 2 1 1568.2.a.e 1
21.c even 2 1 576.2.a.c 1
28.d even 2 1 64.2.a.a 1
35.c odd 2 1 1600.2.a.n 1
35.f even 4 2 1600.2.c.l 2
56.e even 2 1 32.2.a.a ✓ 1
56.h odd 2 1 32.2.a.a ✓ 1
56.j odd 6 2 1568.2.i.g 2
56.k odd 6 2 1568.2.i.f 2
56.m even 6 2 1568.2.i.g 2
56.p even 6 2 1568.2.i.f 2
77.b even 2 1 7744.2.a.v 1
84.h odd 2 1 576.2.a.c 1
112.j even 4 2 256.2.b.b 2
112.l odd 4 2 256.2.b.b 2
140.c even 2 1 1600.2.a.n 1
140.j odd 4 2 1600.2.c.l 2
168.e odd 2 1 288.2.a.d 1
168.i even 2 1 288.2.a.d 1
224.v odd 8 4 1024.2.e.j 4
224.x even 8 4 1024.2.e.j 4
280.c odd 2 1 800.2.a.d 1
280.n even 2 1 800.2.a.d 1
280.s even 4 2 800.2.c.e 2
280.y odd 4 2 800.2.c.e 2
308.g odd 2 1 7744.2.a.v 1
336.v odd 4 2 2304.2.d.j 2
336.y even 4 2 2304.2.d.j 2
504.be even 6 2 2592.2.i.t 2
504.bn odd 6 2 2592.2.i.t 2
504.cc even 6 2 2592.2.i.e 2
504.co odd 6 2 2592.2.i.e 2
616.g odd 2 1 3872.2.a.f 1
616.o even 2 1 3872.2.a.f 1
728.b even 2 1 5408.2.a.g 1
728.l odd 2 1 5408.2.a.g 1
840.b odd 2 1 7200.2.a.v 1
840.u even 2 1 7200.2.a.v 1
840.bm even 4 2 7200.2.f.m 2
840.bp odd 4 2 7200.2.f.m 2
952.e odd 2 1 9248.2.a.f 1
952.k even 2 1 9248.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a ✓ 1 56.e even 2 1
32.2.a.a ✓ 1 56.h odd 2 1
64.2.a.a 1 7.b odd 2 1
64.2.a.a 1 28.d even 2 1
256.2.b.b 2 112.j even 4 2
256.2.b.b 2 112.l odd 4 2
288.2.a.d 1 168.e odd 2 1
288.2.a.d 1 168.i even 2 1
576.2.a.c 1 21.c even 2 1
576.2.a.c 1 84.h odd 2 1
800.2.a.d 1 280.c odd 2 1
800.2.a.d 1 280.n even 2 1
800.2.c.e 2 280.s even 4 2
800.2.c.e 2 280.y odd 4 2
1024.2.e.j 4 224.v odd 8 4
1024.2.e.j 4 224.x even 8 4
1568.2.a.e 1 8.b even 2 1
1568.2.a.e 1 8.d odd 2 1
1568.2.i.f 2 56.k odd 6 2
1568.2.i.f 2 56.p even 6 2
1568.2.i.g 2 56.j odd 6 2
1568.2.i.g 2 56.m even 6 2
1600.2.a.n 1 35.c odd 2 1
1600.2.a.n 1 140.c even 2 1
1600.2.c.l 2 35.f even 4 2
1600.2.c.l 2 140.j odd 4 2
2304.2.d.j 2 336.v odd 4 2
2304.2.d.j 2 336.y even 4 2
2592.2.i.e 2 504.cc even 6 2
2592.2.i.e 2 504.co odd 6 2
2592.2.i.t 2 504.be even 6 2
2592.2.i.t 2 504.bn odd 6 2
3136.2.a.m 1 1.a even 1 1 trivial
3136.2.a.m 1 4.b odd 2 1 CM
3872.2.a.f 1 616.g odd 2 1
3872.2.a.f 1 616.o even 2 1
5408.2.a.g 1 728.b even 2 1
5408.2.a.g 1 728.l odd 2 1
7200.2.a.v 1 840.b odd 2 1
7200.2.a.v 1 840.u even 2 1
7200.2.f.m 2 840.bm even 4 2
7200.2.f.m 2 840.bp odd 4 2
7744.2.a.v 1 77.b even 2 1
7744.2.a.v 1 308.g odd 2 1
9248.2.a.f 1 952.e odd 2 1
9248.2.a.f 1 952.k even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3136))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 6 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 10 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T + 10 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 14 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 10 \) Copy content Toggle raw display
$97$ \( T + 18 \) Copy content Toggle raw display
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