Properties

Label 4900.2.a.bf
Level $4900$
Weight $2$
Character orbit 4900.a
Self dual yes
Analytic conductor $39.127$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 11x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
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)
 

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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + ( - \beta_{3} - 2) q^{11} + ( - 2 \beta_{2} + 2 \beta_1) q^{13} + (2 \beta_{2} - \beta_1) q^{17} - \beta_{3} q^{19} + (\beta_{2} + 2 \beta_1) q^{23} - 3 \beta_{2} q^{27} + (\beta_{3} + 1) q^{29} - \beta_{3} q^{31} - 3 \beta_1 q^{33} + (4 \beta_{2} + \beta_1) q^{37} + (2 \beta_{3} - 2) q^{39} + ( - \beta_{3} + 7) q^{41} + (3 \beta_{2} - \beta_1) q^{43} + (2 \beta_{2} + \beta_1) q^{47} + ( - \beta_{3} + 4) q^{51} + ( - 2 \beta_{2} + 3 \beta_1) q^{53} + (2 \beta_{2} - 3 \beta_1) q^{57} + \beta_{3} q^{59} + (2 \beta_{3} + 7) q^{61} + (5 \beta_{2} - 4 \beta_1) q^{67} + (2 \beta_{3} + 7) q^{69} + ( - 2 \beta_{3} + 2) q^{71} + (2 \beta_{2} + \beta_1) q^{73} + (\beta_{3} + 4) q^{79} - 9 q^{81} + (\beta_{2} - 3 \beta_1) q^{83} + ( - \beta_{2} + 3 \beta_1) q^{87} + 7 q^{89} + (2 \beta_{2} - 3 \beta_1) q^{93} - 4 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{11} + 2 q^{19} + 2 q^{29} + 2 q^{31} - 12 q^{39} + 30 q^{41} + 18 q^{51} - 2 q^{59} + 24 q^{61} + 24 q^{69} + 12 q^{71} + 14 q^{79} - 36 q^{81} + 28 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 11x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 7\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} + 7\beta_1 \) 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
−3.04547
1.31342
3.04547
−1.31342
0 −1.73205 0 0 0 0 0 0 0
1.2 0 −1.73205 0 0 0 0 0 0 0
1.3 0 1.73205 0 0 0 0 0 0 0
1.4 0 1.73205 0 0 0 0 0 0 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4900.2.a.bf 4
5.b even 2 1 inner 4900.2.a.bf 4
5.c odd 4 2 980.2.e.c 4
7.b odd 2 1 4900.2.a.be 4
7.d odd 6 2 700.2.i.f 8
35.c odd 2 1 4900.2.a.be 4
35.f even 4 2 980.2.e.f 4
35.i odd 6 2 700.2.i.f 8
35.k even 12 2 140.2.q.a 4
35.k even 12 2 140.2.q.b yes 4
35.l odd 12 2 980.2.q.b 4
35.l odd 12 2 980.2.q.g 4
105.w odd 12 2 1260.2.bm.a 4
105.w odd 12 2 1260.2.bm.b 4
140.x odd 12 2 560.2.bw.a 4
140.x odd 12 2 560.2.bw.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a 4 35.k even 12 2
140.2.q.b yes 4 35.k even 12 2
560.2.bw.a 4 140.x odd 12 2
560.2.bw.e 4 140.x odd 12 2
700.2.i.f 8 7.d odd 6 2
700.2.i.f 8 35.i odd 6 2
980.2.e.c 4 5.c odd 4 2
980.2.e.f 4 35.f even 4 2
980.2.q.b 4 35.l odd 12 2
980.2.q.g 4 35.l odd 12 2
1260.2.bm.a 4 105.w odd 12 2
1260.2.bm.b 4 105.w odd 12 2
4900.2.a.be 4 7.b odd 2 1
4900.2.a.be 4 35.c odd 2 1
4900.2.a.bf 4 1.a even 1 1 trivial
4900.2.a.bf 4 5.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4900))\):

\( T_{3}^{2} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} - 12 \) Copy content Toggle raw display
\( T_{13}^{4} - 44T_{13}^{2} + 256 \) Copy content Toggle raw display
\( T_{19}^{2} - T_{19} - 14 \) Copy content Toggle raw display
\( T_{23}^{4} - 62T_{23}^{2} + 49 \) 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
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3 T - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 44T^{2} + 256 \) Copy content Toggle raw display
$17$ \( T^{4} - 23T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T^{2} - T - 14)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 62T^{2} + 49 \) Copy content Toggle raw display
$29$ \( (T^{2} - T - 14)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - T - 14)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 131T^{2} + 3136 \) Copy content Toggle raw display
$41$ \( (T^{2} - 15 T + 42)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 47T^{2} + 196 \) Copy content Toggle raw display
$47$ \( T^{4} - 47T^{2} + 196 \) Copy content Toggle raw display
$53$ \( T^{4} - 87T^{2} + 1764 \) Copy content Toggle raw display
$59$ \( (T^{2} + T - 14)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 12 T - 21)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 206T^{2} + 2401 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T - 48)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 47T^{2} + 196 \) Copy content Toggle raw display
$79$ \( (T^{2} - 7 T - 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 87T^{2} + 1764 \) Copy content Toggle raw display
$89$ \( (T - 7)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
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