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Properties

Label 8281.2.a.ck
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 3 \) 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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{4} + 1) q^{3} + (\beta_{2} + 1) q^{4} + \beta_{5} q^{5} + (\beta_{5} + \beta_{3} + \beta_1) q^{6} + \beta_{3} q^{8} + (\beta_{6} + \beta_{4} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{4} + 1) q^{3} + (\beta_{2} + 1) q^{4} + \beta_{5} q^{5} + (\beta_{5} + \beta_{3} + \beta_1) q^{6} + \beta_{3} q^{8} + (\beta_{6} + \beta_{4} + 2) q^{9} + (\beta_{6} + 2 \beta_{4} - \beta_{2}) q^{10} + (\beta_{7} - \beta_1) q^{11} + (\beta_{6} + \beta_{4} + \beta_{2} + 3) q^{12} + (\beta_{5} + 2 \beta_1) q^{15} + (\beta_{4} - \beta_{2}) q^{16} + ( - \beta_{6} + 1) q^{17} + (\beta_{7} + 3 \beta_{5} + 3 \beta_1) q^{18} + ( - \beta_{3} + 2 \beta_1) q^{19} + (\beta_{7} + 2 \beta_{5}) q^{20} + (\beta_{6} + \beta_{4} - \beta_{2} - 2) q^{22} + (\beta_{4} - 1) q^{23} + (\beta_{7} + \beta_{5} + \cdots + 3 \beta_1) q^{24}+ \cdots + ( - \beta_{7} - \beta_{5} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9} - 6 q^{10} + 18 q^{12} - 2 q^{16} + 8 q^{17} - 18 q^{22} - 12 q^{23} + 16 q^{27} - 8 q^{29} + 38 q^{30} + 28 q^{36} + 34 q^{38} + 4 q^{40} - 8 q^{43} + 18 q^{48} + 16 q^{51} + 20 q^{53} + 12 q^{55} + 12 q^{61} - 22 q^{62} - 44 q^{64} - 2 q^{66} + 2 q^{68} + 28 q^{69} - 42 q^{74} + 8 q^{75} - 20 q^{79} + 24 q^{81} - 16 q^{82} + 68 q^{87} + 4 q^{88} + 108 q^{90} + 6 q^{92} + 26 q^{94} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 5\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 6\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 8\nu^{4} + 16\nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 10\nu^{5} + 29\nu^{3} - 20\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 6\beta_{3} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 8\beta_{4} + 24\beta_{2} + 69 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} + 10\beta_{5} + 31\beta_{3} + 94\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−2.28481
−2.12549
−1.07305
−0.332375
0.332375
1.07305
2.12549
2.28481
−2.28481 3.15042 3.22037 −2.12499 −7.19813 0 −2.78832 6.92516 4.85521
1.2 −2.12549 −0.178854 2.51771 3.60603 0.380153 0 −1.10038 −2.96801 −7.66457
1.3 −1.07305 −2.43140 −0.848553 0.625432 2.60903 0 3.05665 2.91173 −0.671123
1.4 −0.332375 1.45984 −1.88953 −1.44562 −0.485214 0 1.29278 −0.868875 0.480489
1.5 0.332375 1.45984 −1.88953 1.44562 0.485214 0 −1.29278 −0.868875 0.480489
1.6 1.07305 −2.43140 −0.848553 −0.625432 −2.60903 0 −3.05665 2.91173 −0.671123
1.7 2.12549 −0.178854 2.51771 −3.60603 −0.380153 0 1.10038 −2.96801 −7.66457
1.8 2.28481 3.15042 3.22037 2.12499 7.19813 0 2.78832 6.92516 4.85521
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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.ck 8
7.b odd 2 1 8281.2.a.cj 8
7.c even 3 2 1183.2.e.i 16
13.b even 2 1 inner 8281.2.a.ck 8
13.d odd 4 2 637.2.c.f 8
91.b odd 2 1 8281.2.a.cj 8
91.i even 4 2 637.2.c.e 8
91.r even 6 2 1183.2.e.i 16
91.z odd 12 4 91.2.r.a ✓ 16
91.bb even 12 4 637.2.r.f 16
273.cd even 12 4 819.2.dl.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.r.a ✓ 16 91.z odd 12 4
637.2.c.e 8 91.i even 4 2
637.2.c.f 8 13.d odd 4 2
637.2.r.f 16 91.bb even 12 4
819.2.dl.e 16 273.cd even 12 4
1183.2.e.i 16 7.c even 3 2
1183.2.e.i 16 91.r even 6 2
8281.2.a.cj 8 7.b odd 2 1
8281.2.a.cj 8 91.b odd 2 1
8281.2.a.ck 8 1.a even 1 1 trivial
8281.2.a.ck 8 13.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{8} - 11T_{2}^{6} + 36T_{2}^{4} - 31T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{3}^{4} - 2T_{3}^{3} - 7T_{3}^{2} + 10T_{3} + 2 \) Copy content Toggle raw display
\( T_{5}^{8} - 20T_{5}^{6} + 103T_{5}^{4} - 160T_{5}^{2} + 48 \) Copy content Toggle raw display
\( T_{11}^{8} - 52T_{11}^{6} + 596T_{11}^{4} - 340T_{11}^{2} + 27 \) Copy content Toggle raw display
\( T_{17}^{4} - 4T_{17}^{3} - 20T_{17}^{2} + 52T_{17} + 123 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 11 T^{6} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{3} - 7 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 20 T^{6} + \cdots + 48 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 52 T^{6} + \cdots + 27 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 4 T^{3} + \cdots + 123)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} - 44 T^{6} + \cdots + 3267 \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} + 5 T^{2} + \cdots - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + \cdots + 624)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 80 T^{6} + \cdots + 33708 \) Copy content Toggle raw display
$37$ \( T^{8} - 120 T^{6} + \cdots + 8748 \) Copy content Toggle raw display
$41$ \( T^{8} - 132 T^{6} + \cdots + 292032 \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} + \cdots - 104)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 196 T^{6} + \cdots + 240267 \) Copy content Toggle raw display
$53$ \( (T^{4} - 10 T^{3} + \cdots + 87)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 188 T^{6} + \cdots + 111747 \) Copy content Toggle raw display
$61$ \( (T^{4} - 6 T^{3} + \cdots + 223)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 284 T^{6} + \cdots + 257547 \) Copy content Toggle raw display
$71$ \( T^{8} - 292 T^{6} + \cdots + 397488 \) Copy content Toggle raw display
$73$ \( T^{8} - 260 T^{6} + \cdots + 2904768 \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} + 9 T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 296 T^{6} + \cdots + 5483712 \) Copy content Toggle raw display
$89$ \( T^{8} - 440 T^{6} + \cdots + 7622508 \) Copy content Toggle raw display
$97$ \( T^{8} - 104 T^{6} + \cdots + 192 \) Copy content Toggle raw display
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