Properties

Label 4900.2.a.e
Level $4900$
Weight $2$
Character orbit 4900.a
Self dual yes
Analytic conductor $39.127$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4900,2,Mod(1,4900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4900, 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("4900.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4900.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: \(39.1266969904\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 2 q^{3} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} + q^{9} + 2 q^{13} - 6 q^{17} + 4 q^{19} - 6 q^{23} + 4 q^{27} + 6 q^{29} + 4 q^{31} - 2 q^{37} - 4 q^{39} - 6 q^{41} + 10 q^{43} - 6 q^{47} + 12 q^{51} + 6 q^{53} - 8 q^{57} - 12 q^{59} - 2 q^{61} - 2 q^{67} + 12 q^{69} - 12 q^{71} + 2 q^{73} + 8 q^{79} - 11 q^{81} + 6 q^{83} - 12 q^{87} + 6 q^{89} - 8 q^{93} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 −2.00000 0 0 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(7\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4900.2.a.e 1
5.b even 2 1 980.2.a.h 1
5.c odd 4 2 4900.2.e.f 2
7.b odd 2 1 100.2.a.a 1
15.d odd 2 1 8820.2.a.g 1
20.d odd 2 1 3920.2.a.h 1
21.c even 2 1 900.2.a.b 1
28.d even 2 1 400.2.a.c 1
35.c odd 2 1 20.2.a.a 1
35.f even 4 2 100.2.c.a 2
35.i odd 6 2 980.2.i.i 2
35.j even 6 2 980.2.i.c 2
56.e even 2 1 1600.2.a.w 1
56.h odd 2 1 1600.2.a.c 1
84.h odd 2 1 3600.2.a.be 1
105.g even 2 1 180.2.a.a 1
105.k odd 4 2 900.2.d.c 2
140.c even 2 1 80.2.a.b 1
140.j odd 4 2 400.2.c.b 2
280.c odd 2 1 320.2.a.f 1
280.n even 2 1 320.2.a.a 1
280.s even 4 2 1600.2.c.d 2
280.y odd 4 2 1600.2.c.e 2
315.z even 6 2 1620.2.i.b 2
315.bg odd 6 2 1620.2.i.h 2
385.h even 2 1 2420.2.a.a 1
420.o odd 2 1 720.2.a.h 1
420.w even 4 2 3600.2.f.j 2
455.h odd 2 1 3380.2.a.c 1
455.u even 4 2 3380.2.f.b 2
560.be even 4 2 1280.2.d.g 2
560.bf odd 4 2 1280.2.d.c 2
595.b odd 2 1 5780.2.a.f 1
595.u odd 4 2 5780.2.c.a 2
665.g even 2 1 7220.2.a.f 1
840.b odd 2 1 2880.2.a.f 1
840.u even 2 1 2880.2.a.m 1
1540.b odd 2 1 9680.2.a.ba 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 35.c odd 2 1
80.2.a.b 1 140.c even 2 1
100.2.a.a 1 7.b odd 2 1
100.2.c.a 2 35.f even 4 2
180.2.a.a 1 105.g even 2 1
320.2.a.a 1 280.n even 2 1
320.2.a.f 1 280.c odd 2 1
400.2.a.c 1 28.d even 2 1
400.2.c.b 2 140.j odd 4 2
720.2.a.h 1 420.o odd 2 1
900.2.a.b 1 21.c even 2 1
900.2.d.c 2 105.k odd 4 2
980.2.a.h 1 5.b even 2 1
980.2.i.c 2 35.j even 6 2
980.2.i.i 2 35.i odd 6 2
1280.2.d.c 2 560.bf odd 4 2
1280.2.d.g 2 560.be even 4 2
1600.2.a.c 1 56.h odd 2 1
1600.2.a.w 1 56.e even 2 1
1600.2.c.d 2 280.s even 4 2
1600.2.c.e 2 280.y odd 4 2
1620.2.i.b 2 315.z even 6 2
1620.2.i.h 2 315.bg odd 6 2
2420.2.a.a 1 385.h even 2 1
2880.2.a.f 1 840.b odd 2 1
2880.2.a.m 1 840.u even 2 1
3380.2.a.c 1 455.h odd 2 1
3380.2.f.b 2 455.u even 4 2
3600.2.a.be 1 84.h odd 2 1
3600.2.f.j 2 420.w even 4 2
3920.2.a.h 1 20.d odd 2 1
4900.2.a.e 1 1.a even 1 1 trivial
4900.2.e.f 2 5.c odd 4 2
5780.2.a.f 1 595.b odd 2 1
5780.2.c.a 2 595.u odd 4 2
7220.2.a.f 1 665.g even 2 1
8820.2.a.g 1 15.d odd 2 1
9680.2.a.ba 1 1540.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4900))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 2 \) Copy content Toggle raw display
\( T_{19} - 4 \) Copy content Toggle raw display
\( T_{23} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T - 10 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T - 2 \) Copy content Toggle raw display
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