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Properties

Label 8016.2.a.x
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $8$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
Analytic rank: \(1\)
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: \( x^{8} - 3x^{7} - 8x^{6} + 28x^{5} + 9x^{4} - 64x^{3} + 17x^{2} + 23x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 501)
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.

\(f(q)\) \(=\) \( q - q^{3} + (\beta_{2} + 1) q^{5} + (\beta_{5} + \beta_{3} - \beta_1 + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + (\beta_{2} + 1) q^{5} + (\beta_{5} + \beta_{3} - \beta_1 + 1) q^{7} + q^{9} + (\beta_{4} + \beta_1 - 2) q^{11} + ( - \beta_{7} - \beta_{5}) q^{13} + ( - \beta_{2} - 1) q^{15} + (\beta_{7} - \beta_{6} + \beta_{3} + \cdots + 1) q^{17}+ \cdots + (\beta_{4} + \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} + 7 q^{5} + 4 q^{7} + 8 q^{9} - 13 q^{11} - 7 q^{15} + 11 q^{17} - 12 q^{19} - 4 q^{21} - 7 q^{23} - 5 q^{25} - 8 q^{27} + q^{29} + 2 q^{31} + 13 q^{33} + 4 q^{35} - 9 q^{37} + 4 q^{41} - 2 q^{43} + 7 q^{45} - 17 q^{47} - 2 q^{49} - 11 q^{51} + 9 q^{53} - 7 q^{55} + 12 q^{57} - 29 q^{59} - 12 q^{61} + 4 q^{63} + 8 q^{65} + 7 q^{69} - 13 q^{71} - 20 q^{73} + 5 q^{75} - 22 q^{77} - 8 q^{79} + 8 q^{81} - 33 q^{83} - 31 q^{85} - q^{87} + 4 q^{89} - q^{91} - 2 q^{93} - 3 q^{95} - 31 q^{97} - 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 8x^{6} + 28x^{5} + 9x^{4} - 64x^{3} + 17x^{2} + 23x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} + 38\nu^{5} + 2\nu^{4} - 147\nu^{3} - 11\nu^{2} + 168\nu + 15 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 15\nu^{5} + 3\nu^{4} - 70\nu^{3} - 20\nu^{2} + 98\nu + 19 ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - 7\nu^{6} - 16\nu^{5} + 64\nu^{4} + 21\nu^{3} - 135\nu^{2} + 21\nu + 25 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} + 24\nu^{5} - 61\nu^{4} - 28\nu^{3} + 122\nu^{2} - 42\nu - 27 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + 10\nu^{5} - 18\nu^{4} - 25\nu^{3} + 37\nu^{2} + 8\nu - 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} - 7\nu^{6} - 39\nu^{5} + 58\nu^{4} + 98\nu^{3} - 95\nu^{2} - 56\nu - 13 ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{5} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{4} - \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{7} + 7\beta_{5} + \beta_{4} + 8\beta_{3} - 2\beta_{2} - \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 9\beta_{6} + 8\beta_{4} + 2\beta_{3} - 9\beta_{2} + 28\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37\beta_{7} + \beta_{6} + 45\beta_{5} + 8\beta_{4} + 57\beta_{3} - 21\beta_{2} - 8\beta _1 + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{7} + 65\beta_{6} + \beta_{5} + 53\beta_{4} + 27\beta_{3} - 71\beta_{2} + 165\beta _1 + 4 \) 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
−0.459587
2.63734
1.60389
−2.45154
−0.0452510
−1.71120
2.18779
1.23856
0 −1.00000 0 −2.63865 0 0.604857 0 1.00000 0
1.2 0 −1.00000 0 −1.29727 0 2.24503 0 1.00000 0
1.3 0 −1.00000 0 −0.122001 0 −3.47013 0 1.00000 0
1.4 0 −1.00000 0 1.48532 0 5.10210 0 1.00000 0
1.5 0 −1.00000 0 1.52778 0 −0.428515 0 1.00000 0
1.6 0 −1.00000 0 2.45529 0 −2.43610 0 1.00000 0
1.7 0 −1.00000 0 2.52133 0 0.307143 0 1.00000 0
1.8 0 −1.00000 0 3.06821 0 2.07562 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(167\) \(-1\)

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 8016.2.a.x 8
4.b odd 2 1 501.2.a.e ✓ 8
12.b even 2 1 1503.2.a.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.a.e ✓ 8 4.b odd 2 1
1503.2.a.e 8 12.b even 2 1
8016.2.a.x 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{8} - 7T_{5}^{7} + 7T_{5}^{6} + 50T_{5}^{5} - 125T_{5}^{4} + 9T_{5}^{3} + 196T_{5}^{2} - 124T_{5} - 18 \) Copy content Toggle raw display
\( T_{7}^{8} - 4T_{7}^{7} - 19T_{7}^{6} + 65T_{7}^{5} + 63T_{7}^{4} - 255T_{7}^{3} + 84T_{7}^{2} + 48T_{7} - 16 \) Copy content Toggle raw display
\( T_{11}^{8} + 13T_{11}^{7} + 41T_{11}^{6} - 104T_{11}^{5} - 691T_{11}^{4} - 435T_{11}^{3} + 1804T_{11}^{2} + 1316T_{11} - 1596 \) Copy content Toggle raw display
\( T_{13}^{8} - 37T_{13}^{6} - 52T_{13}^{5} + 320T_{13}^{4} + 973T_{13}^{3} + 994T_{13}^{2} + 412T_{13} + 56 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 7 T^{7} + \cdots - 18 \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( T^{8} + 13 T^{7} + \cdots - 1596 \) Copy content Toggle raw display
$13$ \( T^{8} - 37 T^{6} + \cdots + 56 \) Copy content Toggle raw display
$17$ \( T^{8} - 11 T^{7} + \cdots + 7822 \) Copy content Toggle raw display
$19$ \( T^{8} + 12 T^{7} + \cdots + 4384 \) Copy content Toggle raw display
$23$ \( T^{8} + 7 T^{7} + \cdots + 384 \) Copy content Toggle raw display
$29$ \( T^{8} - T^{7} + \cdots + 88 \) Copy content Toggle raw display
$31$ \( T^{8} - 2 T^{7} + \cdots - 1017696 \) Copy content Toggle raw display
$37$ \( T^{8} + 9 T^{7} + \cdots - 42872 \) Copy content Toggle raw display
$41$ \( T^{8} - 4 T^{7} + \cdots - 859446 \) Copy content Toggle raw display
$43$ \( T^{8} + 2 T^{7} + \cdots + 114 \) Copy content Toggle raw display
$47$ \( T^{8} + 17 T^{7} + \cdots - 68732 \) Copy content Toggle raw display
$53$ \( T^{8} - 9 T^{7} + \cdots + 20326 \) Copy content Toggle raw display
$59$ \( T^{8} + 29 T^{7} + \cdots - 21056 \) Copy content Toggle raw display
$61$ \( T^{8} + 12 T^{7} + \cdots + 51244 \) Copy content Toggle raw display
$67$ \( T^{8} - 177 T^{6} + \cdots - 1226366 \) Copy content Toggle raw display
$71$ \( T^{8} + 13 T^{7} + \cdots + 32816 \) Copy content Toggle raw display
$73$ \( T^{8} + 20 T^{7} + \cdots - 9386184 \) Copy content Toggle raw display
$79$ \( T^{8} + 8 T^{7} + \cdots + 39988958 \) Copy content Toggle raw display
$83$ \( T^{8} + 33 T^{7} + \cdots - 2562224 \) Copy content Toggle raw display
$89$ \( T^{8} - 4 T^{7} + \cdots - 216312 \) Copy content Toggle raw display
$97$ \( T^{8} + 31 T^{7} + \cdots - 24584 \) Copy content Toggle raw display
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