Properties

Label 8281.2.a.bo
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.105456.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 13x^{2} + 39 \) 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 637)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} + 1) q^{4} - \beta_1 q^{5} + \beta_{3} q^{6} - 3 q^{8} + (\beta_{2} + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_1 q^{3} + ( - \beta_{2} + 1) q^{4} - \beta_1 q^{5} + \beta_{3} q^{6} - 3 q^{8} + (\beta_{2} + 4) q^{9} - \beta_{3} q^{10} + (3 \beta_{2} + 2) q^{11} + ( - \beta_{3} + \beta_1) q^{12} + ( - \beta_{2} - 7) q^{15} + ( - \beta_{2} - 2) q^{16} + ( - \beta_{3} + \beta_1) q^{17} + (3 \beta_{2} + 3) q^{18} - \beta_1 q^{19} + (\beta_{3} - \beta_1) q^{20} + ( - \beta_{2} + 9) q^{22} + (2 \beta_{2} + 4) q^{23} - 3 \beta_1 q^{24} + (\beta_{2} + 2) q^{25} + (\beta_{3} + \beta_1) q^{27} + \beta_{2} q^{29} + ( - 6 \beta_{2} - 3) q^{30} + ( - \beta_{3} + \beta_1) q^{31} + ( - \beta_{2} + 3) q^{32} + (3 \beta_{3} + 2 \beta_1) q^{33} + (2 \beta_{3} - 3 \beta_1) q^{34} + ( - 2 \beta_{2} + 1) q^{36} + (2 \beta_{2} - 4) q^{37} - \beta_{3} q^{38} + 3 \beta_1 q^{40} - 2 \beta_{3} q^{41} + (5 \beta_{2} - 1) q^{43} + (4 \beta_{2} - 7) q^{44} + ( - \beta_{3} - 4 \beta_1) q^{45} + (2 \beta_{2} + 6) q^{46} + (\beta_{3} + 3 \beta_1) q^{47} + ( - \beta_{3} - 2 \beta_1) q^{48} + (\beta_{2} + 3) q^{50} + ( - 5 \beta_{2} + 4) q^{51} + (2 \beta_{2} + 7) q^{53} + 3 \beta_1 q^{54} + ( - 3 \beta_{3} - 2 \beta_1) q^{55} + ( - \beta_{2} - 7) q^{57} + ( - \beta_{2} + 3) q^{58} + ( - \beta_{3} - \beta_1) q^{59} + (5 \beta_{2} - 4) q^{60} + 2 \beta_1 q^{61} + (2 \beta_{3} - 3 \beta_1) q^{62} + (6 \beta_{2} + 1) q^{64} + ( - \beta_{3} + 9 \beta_1) q^{66} - q^{67} + ( - 3 \beta_{3} + 4 \beta_1) q^{68} + (2 \beta_{3} + 4 \beta_1) q^{69} - 4 q^{71} + ( - 3 \beta_{2} - 12) q^{72} + 2 \beta_1 q^{73} + ( - 6 \beta_{2} + 6) q^{74} + (\beta_{3} + 2 \beta_1) q^{75} + (\beta_{3} - \beta_1) q^{76} + (2 \beta_{2} - 2) q^{79} + (\beta_{3} + 2 \beta_1) q^{80} + (4 \beta_{2} - 2) q^{81} + (2 \beta_{3} - 6 \beta_1) q^{82} + (\beta_{3} + \beta_1) q^{83} + (5 \beta_{2} - 4) q^{85} + ( - 6 \beta_{2} + 15) q^{86} + \beta_{3} q^{87} + ( - 9 \beta_{2} - 6) q^{88} - 3 \beta_1 q^{89} + ( - 3 \beta_{3} - 3 \beta_1) q^{90} - 2 q^{92} + ( - 5 \beta_{2} + 4) q^{93} + (2 \beta_{3} + 3 \beta_1) q^{94} + (\beta_{2} + 7) q^{95} + ( - \beta_{3} + 3 \beta_1) q^{96} + ( - \beta_{3} + 4 \beta_1) q^{97} + (11 \beta_{2} + 17) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 6 q^{4} - 12 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 6 q^{4} - 12 q^{8} + 14 q^{9} + 2 q^{11} - 26 q^{15} - 6 q^{16} + 6 q^{18} + 38 q^{22} + 12 q^{23} + 6 q^{25} - 2 q^{29} + 14 q^{32} + 8 q^{36} - 20 q^{37} - 14 q^{43} - 36 q^{44} + 20 q^{46} + 10 q^{50} + 26 q^{51} + 24 q^{53} - 26 q^{57} + 14 q^{58} - 26 q^{60} - 8 q^{64} - 4 q^{67} - 16 q^{71} - 42 q^{72} + 36 q^{74} - 12 q^{79} - 16 q^{81} - 26 q^{85} + 72 q^{86} - 6 q^{88} - 8 q^{92} + 26 q^{93} + 26 q^{95} + 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 13x^{2} + 39 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 7\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 7\beta_1 \) 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
−2.16731
2.16731
−2.88145
2.88145
−2.30278 −2.16731 3.30278 2.16731 4.99082 0 −3.00000 1.69722 −4.99082
1.2 −2.30278 2.16731 3.30278 −2.16731 −4.99082 0 −3.00000 1.69722 4.99082
1.3 1.30278 −2.88145 −0.302776 2.88145 −3.75389 0 −3.00000 5.30278 3.75389
1.4 1.30278 2.88145 −0.302776 −2.88145 3.75389 0 −3.00000 5.30278 −3.75389
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( -1 \)
\(13\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8281.2.a.bo 4
7.b odd 2 1 inner 8281.2.a.bo 4
13.b even 2 1 8281.2.a.bu 4
13.e even 6 2 637.2.f.h 8
91.b odd 2 1 8281.2.a.bu 4
91.k even 6 2 637.2.h.j 8
91.l odd 6 2 637.2.h.j 8
91.p odd 6 2 637.2.g.i 8
91.t odd 6 2 637.2.f.h 8
91.u even 6 2 637.2.g.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.h 8 13.e even 6 2
637.2.f.h 8 91.t odd 6 2
637.2.g.i 8 91.p odd 6 2
637.2.g.i 8 91.u even 6 2
637.2.h.j 8 91.k even 6 2
637.2.h.j 8 91.l odd 6 2
8281.2.a.bo 4 1.a even 1 1 trivial
8281.2.a.bo 4 7.b odd 2 1 inner
8281.2.a.bu 4 13.b even 2 1
8281.2.a.bu 4 91.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{2} + T_{2} - 3 \) Copy content Toggle raw display
\( T_{3}^{4} - 13T_{3}^{2} + 39 \) Copy content Toggle raw display
\( T_{5}^{4} - 13T_{5}^{2} + 39 \) Copy content Toggle raw display
\( T_{11}^{2} - T_{11} - 29 \) Copy content Toggle raw display
\( T_{17}^{4} - 52T_{17}^{2} + 39 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T - 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 13T^{2} + 39 \) Copy content Toggle raw display
$5$ \( T^{4} - 13T^{2} + 39 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - T - 29)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 52T^{2} + 39 \) Copy content Toggle raw display
$19$ \( T^{4} - 13T^{2} + 39 \) Copy content Toggle raw display
$23$ \( (T^{2} - 6 T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 52T^{2} + 39 \) Copy content Toggle raw display
$37$ \( (T^{2} + 10 T + 12)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 156T^{2} + 5616 \) Copy content Toggle raw display
$43$ \( (T^{2} + 7 T - 69)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 156T^{2} + 351 \) Copy content Toggle raw display
$53$ \( (T^{2} - 12 T + 23)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 52T^{2} + 351 \) Copy content Toggle raw display
$61$ \( T^{4} - 52T^{2} + 624 \) Copy content Toggle raw display
$67$ \( (T + 1)^{4} \) Copy content Toggle raw display
$71$ \( (T + 4)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 52T^{2} + 624 \) Copy content Toggle raw display
$79$ \( (T^{2} + 6 T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 52T^{2} + 351 \) Copy content Toggle raw display
$89$ \( T^{4} - 117T^{2} + 3159 \) Copy content Toggle raw display
$97$ \( T^{4} - 247 T^{2} + 11271 \) Copy content Toggle raw display
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