Properties

Label 6272.2.a.bu
Level $6272$
Weight $2$
Character orbit 6272.a
Self dual yes
Analytic conductor $50.082$
Analytic rank $0$
Dimension $4$
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] = [6272,2,Mod(1,6272)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6272, 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("6272.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6272 = 2^{7} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6272.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: \(50.0821721477\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11344.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) 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 896)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + (\beta_{3} + 1) q^{5} + ( - \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + (\beta_{3} + 1) q^{5} + ( - \beta_{2} + \beta_1) q^{9} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{11} + ( - \beta_{2} - \beta_1 + 2) q^{13} + (2 \beta_{3} - \beta_1 + 2) q^{15} + ( - \beta_{2} + 3 \beta_1 - 1) q^{17} + (\beta_{3} - \beta_{2} + 2 \beta_1) q^{19} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{23} + (2 \beta_{3} - 2 \beta_{2}) q^{25} + ( - \beta_{3} + \beta_{2} + 2) q^{27} + (2 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{29} + ( - \beta_1 - 2) q^{31} + ( - 4 \beta_{3} - 2 \beta_{2} - 1) q^{33} + ( - \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 3) q^{37} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{39} + (2 \beta_{3} + \beta_{2} + \beta_1 + 4) q^{41} + 2 \beta_{3} q^{43} + (2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{45} + ( - 2 \beta_{2} + 3 \beta_1 - 6) q^{47} + ( - 3 \beta_{3} - 3 \beta_{2} - 2 \beta_1) q^{51} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 - 1) q^{53} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 - 4) q^{55} + ( - 2 \beta_{2} - 2 \beta_1 + 3) q^{57} + ( - 2 \beta_{3} - \beta_{2}) q^{59} + ( - 3 \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{61} + (4 \beta_{3} - \beta_{2} - \beta_1 + 4) q^{65} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 6) q^{67} + ( - \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 5) q^{69} - 6 q^{71} + ( - 2 \beta_{2} - 4 \beta_1 + 5) q^{73} + (4 \beta_{3} + 10) q^{75} + (\beta_{3} - 4 \beta_{2} + \beta_1 - 4) q^{79} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 - 5) q^{81} + ( - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{83} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{85} + (3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 6) q^{87} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 5) q^{89} + (\beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{93} + (3 \beta_{3} - \beta_1 + 6) q^{95} + (2 \beta_{3} + 5 \beta_{2} - \beta_1 + 8) q^{97} + ( - 5 \beta_{3} - 2 \beta_{2} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 2 q^{5} + 4 q^{9} + 8 q^{13} + 2 q^{15} + 4 q^{17} + 4 q^{19} + 4 q^{23} + 8 q^{27} + 8 q^{29} - 10 q^{31} + 8 q^{33} + 2 q^{37} + 22 q^{39} + 12 q^{41} - 4 q^{43} - 14 q^{47} + 8 q^{51} - 10 q^{53} - 24 q^{55} + 12 q^{57} + 6 q^{59} + 6 q^{61} + 8 q^{65} - 22 q^{67} + 34 q^{69} - 24 q^{71} + 16 q^{73} + 32 q^{75} - 8 q^{79} - 24 q^{81} + 16 q^{83} - 6 q^{85} + 18 q^{87} + 8 q^{89} - 2 q^{93} + 16 q^{95} + 16 q^{97} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 4x^{2} + 4x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 3\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 2\beta_{2} + 5\beta _1 + 4 \) 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
2.78165
−1.51658
−0.552409
1.28734
0 −1.95594 0 0.703158 0 0 0 0.825711 0
1.2 0 −0.816594 0 −0.538445 0 0 0 −2.33317 0
1.3 0 2.14243 0 3.87834 0 0 0 1.59002 0
1.4 0 2.63010 0 −2.04306 0 0 0 3.91744 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 6272.2.a.bu 4
4.b odd 2 1 6272.2.a.be 4
7.b odd 2 1 6272.2.a.z 4
7.d odd 6 2 896.2.i.g yes 8
8.b even 2 1 6272.2.a.ba 4
8.d odd 2 1 6272.2.a.bq 4
28.d even 2 1 6272.2.a.bp 4
28.f even 6 2 896.2.i.d yes 8
56.e even 2 1 6272.2.a.bd 4
56.h odd 2 1 6272.2.a.bt 4
56.j odd 6 2 896.2.i.b 8
56.m even 6 2 896.2.i.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
896.2.i.b 8 56.j odd 6 2
896.2.i.d yes 8 28.f even 6 2
896.2.i.e yes 8 56.m even 6 2
896.2.i.g yes 8 7.d odd 6 2
6272.2.a.z 4 7.b odd 2 1
6272.2.a.ba 4 8.b even 2 1
6272.2.a.bd 4 56.e even 2 1
6272.2.a.be 4 4.b odd 2 1
6272.2.a.bp 4 28.d even 2 1
6272.2.a.bq 4 8.d odd 2 1
6272.2.a.bt 4 56.h odd 2 1
6272.2.a.bu 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6272))\):

\( T_{3}^{4} - 2T_{3}^{3} - 6T_{3}^{2} + 8T_{3} + 9 \) Copy content Toggle raw display
\( T_{5}^{4} - 2T_{5}^{3} - 8T_{5}^{2} + 2T_{5} + 3 \) Copy content Toggle raw display
\( T_{11}^{4} - 38T_{11}^{2} - 38T_{11} + 197 \) Copy content Toggle raw display
\( T_{13}^{4} - 8T_{13}^{3} + 8T_{13}^{2} + 60T_{13} - 116 \) Copy content Toggle raw display
\( T_{23}^{4} - 4T_{23}^{3} - 40T_{23}^{2} + 150T_{23} - 125 \) Copy content Toggle raw display
\( T_{29}^{4} - 8T_{29}^{3} - 20T_{29}^{2} + 68T_{29} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} - 6 T^{2} + 8 T + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - 2 T^{3} - 8 T^{2} + 2 T + 3 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 38 T^{2} - 38 T + 197 \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} + 8 T^{2} + 60 T - 116 \) Copy content Toggle raw display
$17$ \( T^{4} - 4 T^{3} - 42 T^{2} + 168 T + 97 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} - 22 T^{2} + 126 T - 151 \) Copy content Toggle raw display
$23$ \( T^{4} - 4 T^{3} - 40 T^{2} + 150 T - 125 \) Copy content Toggle raw display
$29$ \( T^{4} - 8 T^{3} - 20 T^{2} + 68 T + 12 \) Copy content Toggle raw display
$31$ \( T^{4} + 10 T^{3} + 32 T^{2} + 36 T + 11 \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} - 92 T^{2} - 230 T + 251 \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + 8 T^{2} + 196 T - 44 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} - 32 T^{2} - 128 T - 64 \) Copy content Toggle raw display
$47$ \( T^{4} + 14 T^{3} + 12 T^{2} + \cdots - 201 \) Copy content Toggle raw display
$53$ \( T^{4} + 10 T^{3} - 156 T^{2} + \cdots - 605 \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} - 30 T^{2} + 44 T + 97 \) Copy content Toggle raw display
$61$ \( T^{4} - 6 T^{3} - 172 T^{2} + \cdots - 3557 \) Copy content Toggle raw display
$67$ \( T^{4} + 22 T^{3} + 122 T^{2} + \cdots - 11 \) Copy content Toggle raw display
$71$ \( (T + 6)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 16 T^{3} - 46 T^{2} + \cdots - 5563 \) Copy content Toggle raw display
$79$ \( T^{4} + 8 T^{3} - 100 T^{2} + \cdots + 3167 \) Copy content Toggle raw display
$83$ \( T^{4} - 16 T^{3} - 152 T^{2} + \cdots - 14640 \) Copy content Toggle raw display
$89$ \( T^{4} - 8 T^{3} - 62 T^{2} + \cdots - 1227 \) Copy content Toggle raw display
$97$ \( T^{4} - 16 T^{3} - 116 T^{2} + \cdots - 6828 \) Copy content Toggle raw display
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