Properties

Label 8281.2.a.s
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $2$
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] = [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: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{3} + q^{4} - \beta q^{5} - \beta q^{6} - \beta q^{8} - 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - q^{3} + q^{4} - \beta q^{5} - \beta q^{6} - \beta q^{8} - 2 q^{9} - 3 q^{10} - 3 \beta q^{11} - q^{12} + \beta q^{15} - 5 q^{16} - 6 q^{17} - 2 \beta q^{18} - \beta q^{19} - \beta q^{20} - 9 q^{22} + \beta q^{24} - 2 q^{25} + 5 q^{27} + 3 q^{29} + 3 q^{30} - \beta q^{31} - 3 \beta q^{32} + 3 \beta q^{33} - 6 \beta q^{34} - 2 q^{36} - 3 q^{38} + 3 q^{40} - 3 \beta q^{41} - 11 q^{43} - 3 \beta q^{44} + 2 \beta q^{45} + 5 \beta q^{47} + 5 q^{48} - 2 \beta q^{50} + 6 q^{51} - 9 q^{53} + 5 \beta q^{54} + 9 q^{55} + \beta q^{57} + 3 \beta q^{58} + 2 \beta q^{59} + \beta q^{60} + 7 q^{61} - 3 q^{62} + q^{64} + 9 q^{66} - 5 \beta q^{67} - 6 q^{68} - \beta q^{71} + 2 \beta q^{72} - 5 \beta q^{73} + 2 q^{75} - \beta q^{76} - 5 q^{79} + 5 \beta q^{80} + q^{81} - 9 q^{82} + 2 \beta q^{83} + 6 \beta q^{85} - 11 \beta q^{86} - 3 q^{87} + 9 q^{88} - 4 \beta q^{89} + 6 q^{90} + \beta q^{93} + 15 q^{94} + 3 q^{95} + 3 \beta q^{96} + 3 \beta q^{97} + 6 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{4} - 4 q^{9} - 6 q^{10} - 2 q^{12} - 10 q^{16} - 12 q^{17} - 18 q^{22} - 4 q^{25} + 10 q^{27} + 6 q^{29} + 6 q^{30} - 4 q^{36} - 6 q^{38} + 6 q^{40} - 22 q^{43} + 10 q^{48} + 12 q^{51} - 18 q^{53} + 18 q^{55} + 14 q^{61} - 6 q^{62} + 2 q^{64} + 18 q^{66} - 12 q^{68} + 4 q^{75} - 10 q^{79} + 2 q^{81} - 18 q^{82} - 6 q^{87} + 18 q^{88} + 12 q^{90} + 30 q^{94} + 6 q^{95}+O(q^{100}) \) 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
−1.73205
1.73205
−1.73205 −1.00000 1.00000 1.73205 1.73205 0 1.73205 −2.00000 −3.00000
1.2 1.73205 −1.00000 1.00000 −1.73205 −1.73205 0 −1.73205 −2.00000 −3.00000
\(n\): 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
13.b even 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.s 2
7.b odd 2 1 8281.2.a.w 2
7.c even 3 2 1183.2.e.e 4
13.b even 2 1 inner 8281.2.a.s 2
13.f odd 12 2 637.2.q.c 2
91.b odd 2 1 8281.2.a.w 2
91.r even 6 2 1183.2.e.e 4
91.w even 12 2 637.2.u.a 2
91.x odd 12 2 91.2.k.a 2
91.ba even 12 2 637.2.k.b 2
91.bc even 12 2 637.2.q.b 2
91.bd odd 12 2 91.2.u.a yes 2
273.bv even 12 2 819.2.bm.a 2
273.bw even 12 2 819.2.do.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.k.a 2 91.x odd 12 2
91.2.u.a yes 2 91.bd odd 12 2
637.2.k.b 2 91.ba even 12 2
637.2.q.b 2 91.bc even 12 2
637.2.q.c 2 13.f odd 12 2
637.2.u.a 2 91.w even 12 2
819.2.bm.a 2 273.bv even 12 2
819.2.do.c 2 273.bw even 12 2
1183.2.e.e 4 7.c even 3 2
1183.2.e.e 4 91.r even 6 2
8281.2.a.s 2 1.a even 1 1 trivial
8281.2.a.s 2 13.b even 2 1 inner
8281.2.a.w 2 7.b odd 2 1
8281.2.a.w 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{2} - 3 \) Copy content Toggle raw display
\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 27 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 27 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 3 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 3 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 27 \) Copy content Toggle raw display
$43$ \( (T + 11)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 75 \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 12 \) Copy content Toggle raw display
$61$ \( (T - 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 75 \) Copy content Toggle raw display
$71$ \( T^{2} - 3 \) Copy content Toggle raw display
$73$ \( T^{2} - 75 \) Copy content Toggle raw display
$79$ \( (T + 5)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 12 \) Copy content Toggle raw display
$89$ \( T^{2} - 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 27 \) Copy content Toggle raw display
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