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Properties

Label 8281.2.a.cl
Level $8281$
Weight $2$
Character orbit 8281.a
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
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: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.8446345216.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 19x^{4} - 14x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} + 1) q^{2} + ( - \beta_{4} - \beta_1) q^{3} + ( - \beta_{6} - \beta_{5} - \beta_{2} + 1) q^{4} + (\beta_{7} - \beta_{3}) q^{5} + ( - \beta_{4} + \beta_{3} - 2 \beta_1) q^{6} + ( - \beta_{6} + \beta_{5} - 1) q^{8} + (\beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + 1) q^{2} + ( - \beta_{4} - \beta_1) q^{3} + ( - \beta_{6} - \beta_{5} - \beta_{2} + 1) q^{4} + (\beta_{7} - \beta_{3}) q^{5} + ( - \beta_{4} + \beta_{3} - 2 \beta_1) q^{6} + ( - \beta_{6} + \beta_{5} - 1) q^{8} + (\beta_{5} + 1) q^{9} + (2 \beta_{7} - 2 \beta_{3} + \beta_1) q^{10} + ( - \beta_{5} - 1) q^{11} + ( - 2 \beta_{7} - \beta_{4} + 2 \beta_{3}) q^{12} + (2 \beta_{6} + \beta_{2} - 1) q^{15} + ( - \beta_{6} - \beta_{5} + \beta_{2}) q^{16} + (2 \beta_{4} - \beta_{3} + \beta_1) q^{17} + ( - \beta_{6} - \beta_{5} - \beta_{2} + 3) q^{18} + ( - 2 \beta_{7} - 2 \beta_{4} + \beta_{3}) q^{19} + (3 \beta_{7} + \beta_{4} + \cdots + 2 \beta_1) q^{20}+ \cdots + (\beta_{6} + \beta_{5} + \beta_{2} - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} + 12 q^{4} - 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} + 12 q^{4} - 12 q^{8} + 4 q^{9} - 4 q^{11} - 8 q^{15} + 4 q^{16} + 28 q^{18} - 28 q^{22} - 12 q^{23} - 12 q^{25} - 8 q^{29} - 28 q^{30} - 4 q^{36} - 8 q^{37} - 32 q^{43} + 4 q^{44} - 4 q^{46} + 36 q^{50} - 44 q^{51} - 4 q^{53} + 48 q^{57} - 48 q^{58} - 64 q^{60} - 32 q^{64} + 20 q^{67} + 8 q^{71} + 28 q^{72} - 76 q^{74} - 4 q^{79} - 56 q^{81} + 36 q^{85} - 4 q^{86} - 28 q^{88} - 80 q^{92} + 8 q^{93} - 52 q^{95} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 19x^{4} - 14x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 6\nu^{3} + 7\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - 6\nu^{4} + 7\nu^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{6} + 7\nu^{4} - 12\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 7\nu^{5} + 13\nu^{3} - 6\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} + 5\beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{4} + 6\beta_{3} + 11\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 6\beta_{6} + 7\beta_{5} + 23\beta_{2} + 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} + 7\beta_{4} + 29\beta_{3} + 44\beta_1 \) 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
−1.51373
1.51373
2.09282
−2.09282
0.282452
−0.282452
1.11758
−1.11758
−2.43210 −0.753592 3.91511 0.341537 1.83281 0 −4.65773 −2.43210 −0.830652
1.2 −2.43210 0.753592 3.91511 −0.341537 −1.83281 0 −4.65773 −2.43210 0.830652
1.3 0.579810 −1.89204 −1.66382 −1.47362 −1.09702 0 −2.12432 0.579810 −0.854419
1.4 0.579810 1.89204 −1.66382 1.47362 1.09702 0 −2.12432 0.579810 0.854419
1.5 1.52077 −2.12621 0.312752 −0.589391 −3.23349 0 −2.56592 1.52077 −0.896331
1.6 1.52077 2.12621 0.312752 0.589391 3.23349 0 −2.56592 1.52077 0.896331
1.7 2.33152 −2.30901 3.43596 3.37112 −5.38349 0 3.34797 2.33152 7.85981
1.8 2.33152 2.30901 3.43596 −3.37112 5.38349 0 3.34797 2.33152 −7.85981
\(n\): 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
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.cl 8
7.b odd 2 1 inner 8281.2.a.cl 8
13.b even 2 1 8281.2.a.ci 8
13.e even 6 2 637.2.f.l ✓ 16
91.b odd 2 1 8281.2.a.ci 8
91.k even 6 2 637.2.h.m 16
91.l odd 6 2 637.2.h.m 16
91.p odd 6 2 637.2.g.m 16
91.t odd 6 2 637.2.f.l ✓ 16
91.u even 6 2 637.2.g.m 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.f.l ✓ 16 13.e even 6 2
637.2.f.l ✓ 16 91.t odd 6 2
637.2.g.m 16 91.p odd 6 2
637.2.g.m 16 91.u even 6 2
637.2.h.m 16 91.k even 6 2
637.2.h.m 16 91.l odd 6 2
8281.2.a.ci 8 13.b even 2 1
8281.2.a.ci 8 91.b odd 2 1
8281.2.a.cl 8 1.a even 1 1 trivial
8281.2.a.cl 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):

\( T_{2}^{4} - 2T_{2}^{3} - 5T_{2}^{2} + 12T_{2} - 5 \) Copy content Toggle raw display
\( T_{3}^{8} - 14T_{3}^{6} + 67T_{3}^{4} - 120T_{3}^{2} + 49 \) Copy content Toggle raw display
\( T_{5}^{8} - 14T_{5}^{6} + 31T_{5}^{4} - 12T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{4} + 2T_{11}^{3} - 5T_{11}^{2} - 12T_{11} - 5 \) Copy content Toggle raw display
\( T_{17}^{8} - 58T_{17}^{6} + 966T_{17}^{4} - 3866T_{17}^{2} + 3721 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} - 5 T^{2} + \cdots - 5)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 14 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{8} - 14 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2 T^{3} - 5 T^{2} + \cdots - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} - 58 T^{6} + \cdots + 3721 \) Copy content Toggle raw display
$19$ \( T^{8} - 94 T^{6} + \cdots + 1225 \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + \cdots + 1781)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 70 T^{6} + \cdots + 26569 \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + \cdots - 380)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 176 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$43$ \( (T^{4} + 16 T^{3} + \cdots - 3205)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 226 T^{6} + \cdots + 27889 \) Copy content Toggle raw display
$53$ \( (T^{4} + 2 T^{3} + \cdots + 271)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 194 T^{6} + \cdots + 169 \) Copy content Toggle raw display
$61$ \( T^{8} - 108 T^{6} + \cdots + 19600 \) Copy content Toggle raw display
$67$ \( (T^{4} - 10 T^{3} + \cdots - 283)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} + \cdots + 5956)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 428 T^{6} + \cdots + 2226064 \) Copy content Toggle raw display
$79$ \( (T^{4} + 2 T^{3} + \cdots + 8164)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 350 T^{6} + \cdots + 405769 \) Copy content Toggle raw display
$89$ \( T^{8} - 606 T^{6} + \cdots + 45225625 \) Copy content Toggle raw display
$97$ \( T^{8} - 364 T^{6} + \cdots + 310249 \) Copy content Toggle raw display
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