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Properties

Label 8040.2.a.w
Level $8040$
Weight $2$
Character orbit 8040.a
Self dual yes
Analytic conductor $64.200$
Analytic rank $0$
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] = [8040,2,Mod(1,8040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8040 = 2^{3} \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8040.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: \(64.1997232251\)
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: \( x^{8} - 3x^{7} - 24x^{6} + 30x^{5} + 182x^{4} + 115x^{3} - 99x^{2} - 90x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + \beta_{4} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{5} + \beta_{4} q^{7} + q^{9} + ( - \beta_{3} + 1) q^{11} + ( - \beta_{6} - \beta_{3} + \beta_{2}) q^{13} - q^{15} + (\beta_{7} + 1) q^{17} + ( - \beta_{6} + \beta_{5} + \beta_{2} + 2) q^{19} + \beta_{4} q^{21} + ( - \beta_{7} + \beta_{6} + 2) q^{23} + q^{25} + q^{27} + (\beta_{4} - \beta_{2} - \beta_1) q^{29} + ( - \beta_{6} + \beta_{4} + \beta_{2} + 1) q^{31} + ( - \beta_{3} + 1) q^{33} - \beta_{4} q^{35} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots + 1) q^{37}+ \cdots + ( - \beta_{3} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} - 8 q^{5} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} - 8 q^{5} + 8 q^{9} + 12 q^{11} + q^{13} - 8 q^{15} + 11 q^{17} + 10 q^{19} + 13 q^{23} + 8 q^{25} + 8 q^{27} + 5 q^{31} + 12 q^{33} + 14 q^{37} + q^{39} - 5 q^{41} + 25 q^{43} - 8 q^{45} - 6 q^{47} + 14 q^{49} + 11 q^{51} - 25 q^{53} - 12 q^{55} + 10 q^{57} + 10 q^{59} - 8 q^{61} - q^{65} - 8 q^{67} + 13 q^{69} - 13 q^{71} + 13 q^{73} + 8 q^{75} - 4 q^{77} - 7 q^{79} + 8 q^{81} + 17 q^{83} - 11 q^{85} + 28 q^{89} + 27 q^{91} + 5 q^{93} - 10 q^{95} + 14 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 24x^{6} + 30x^{5} + 182x^{4} + 115x^{3} - 99x^{2} - 90x - 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -83\nu^{7} + 907\nu^{6} - 998\nu^{5} - 13686\nu^{4} + 22150\nu^{3} + 51563\nu^{2} - 11237\nu - 10928 ) / 3418 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 577\nu^{7} - 1796\nu^{6} - 13776\nu^{5} + 19164\nu^{4} + 106362\nu^{3} + 50345\nu^{2} - 87532\nu - 44196 ) / 3418 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -839\nu^{7} + 3506\nu^{6} + 15938\nu^{5} - 43546\nu^{4} - 101220\nu^{3} + 18281\nu^{2} + 65280\nu + 7106 ) / 3418 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -1085\nu^{7} + 4094\nu^{6} + 22534\nu^{5} - 48488\nu^{4} - 153924\nu^{3} - 23555\nu^{2} + 92552\nu + 28952 ) / 3418 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1118\nu^{7} - 3631\nu^{6} - 26420\nu^{5} + 42440\nu^{4} + 200176\nu^{3} + 49300\nu^{2} - 153191\nu - 36426 ) / 3418 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 756\nu^{7} - 2599\nu^{6} - 16936\nu^{5} + 29860\nu^{4} + 123370\nu^{3} + 34991\nu^{2} - 85062\nu - 29997 ) / 1709 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2203\nu^{7} - 7725\nu^{6} - 48954\nu^{5} + 90928\nu^{4} + 354100\nu^{3} + 76273\nu^{2} - 249161\nu - 85886 ) / 3418 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 2\beta_{3} + 2\beta _1 - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{7} - \beta_{6} - 5\beta_{5} + 5\beta_{4} - 2\beta_{3} + 2\beta _1 + 23 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 17\beta_{7} - \beta_{6} - 21\beta_{5} + 33\beta_{4} - 30\beta_{3} + 22\beta _1 + 39 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 97\beta_{7} + 35\beta_{6} - 113\beta_{5} + 157\beta_{4} - 62\beta_{3} - 32\beta_{2} + 42\beta _1 + 367 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 365\beta_{7} + 279\beta_{6} - 529\beta_{5} + 865\beta_{4} - 494\beta_{3} - 148\beta_{2} + 302\beta _1 + 1183 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1913 \beta_{7} + 1983 \beta_{6} - 2705 \beta_{5} + 4297 \beta_{4} - 1474 \beta_{3} - 1216 \beta_{2} + \cdots + 7351 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8029 \beta_{7} + 11791 \beta_{6} - 13141 \beta_{5} + 22445 \beta_{4} - 8922 \beta_{3} - 6232 \beta_{2} + \cdots + 29895 ) / 4 \) 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
0.838130
−1.63802
−0.498005
−3.19279
−0.281360
4.06099
−1.28420
4.99526
0 1.00000 0 −1.00000 0 −4.14001 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −3.14300 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 −2.21535 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 −1.60891 0 1.00000 0
1.5 0 1.00000 0 −1.00000 0 1.20944 0 1.00000 0
1.6 0 1.00000 0 −1.00000 0 2.47311 0 1.00000 0
1.7 0 1.00000 0 −1.00000 0 3.29804 0 1.00000 0
1.8 0 1.00000 0 −1.00000 0 4.12668 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
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(67\) \(1\)

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 8040.2.a.w ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8040.2.a.w ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):

\( T_{7}^{8} - 35T_{7}^{6} - 2T_{7}^{5} + 395T_{7}^{4} + 51T_{7}^{3} - 1630T_{7}^{2} - 272T_{7} + 1888 \) Copy content Toggle raw display
\( T_{11}^{8} - 12T_{11}^{7} + 19T_{11}^{6} + 272T_{11}^{5} - 1117T_{11}^{4} - 345T_{11}^{3} + 7826T_{11}^{2} - 11800T_{11} + 4672 \) Copy content Toggle raw display
\( T_{13}^{8} - T_{13}^{7} - 77T_{13}^{6} - T_{13}^{5} + 1407T_{13}^{4} + 2172T_{13}^{3} - 144T_{13}^{2} - 928T_{13} + 144 \) Copy content Toggle raw display
\( T_{17}^{8} - 11 T_{17}^{7} - 48 T_{17}^{6} + 697 T_{17}^{5} + 1022 T_{17}^{4} - 15751 T_{17}^{3} + \cdots + 211192 \) 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
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 35 T^{6} + \cdots + 1888 \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{7} + \cdots + 4672 \) Copy content Toggle raw display
$13$ \( T^{8} - T^{7} + \cdots + 144 \) Copy content Toggle raw display
$17$ \( T^{8} - 11 T^{7} + \cdots + 211192 \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{7} + \cdots - 1696 \) Copy content Toggle raw display
$23$ \( T^{8} - 13 T^{7} + \cdots - 13824 \) Copy content Toggle raw display
$29$ \( T^{8} - 92 T^{6} + \cdots - 57488 \) Copy content Toggle raw display
$31$ \( T^{8} - 5 T^{7} + \cdots + 512 \) Copy content Toggle raw display
$37$ \( T^{8} - 14 T^{7} + \cdots - 68004 \) Copy content Toggle raw display
$41$ \( T^{8} + 5 T^{7} + \cdots - 82608 \) Copy content Toggle raw display
$43$ \( T^{8} - 25 T^{7} + \cdots + 31424 \) Copy content Toggle raw display
$47$ \( T^{8} + 6 T^{7} + \cdots + 380928 \) Copy content Toggle raw display
$53$ \( T^{8} + 25 T^{7} + \cdots + 195904 \) Copy content Toggle raw display
$59$ \( T^{8} - 10 T^{7} + \cdots + 2352128 \) Copy content Toggle raw display
$61$ \( T^{8} + 8 T^{7} + \cdots - 12496 \) Copy content Toggle raw display
$67$ \( (T + 1)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + 13 T^{7} + \cdots + 67584 \) Copy content Toggle raw display
$73$ \( T^{8} - 13 T^{7} + \cdots - 24368 \) Copy content Toggle raw display
$79$ \( T^{8} + 7 T^{7} + \cdots + 1987072 \) Copy content Toggle raw display
$83$ \( T^{8} - 17 T^{7} + \cdots - 480768 \) Copy content Toggle raw display
$89$ \( T^{8} - 28 T^{7} + \cdots + 1743332 \) Copy content Toggle raw display
$97$ \( T^{8} - 14 T^{7} + \cdots - 9266216 \) Copy content Toggle raw display
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