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Properties

Label 8040.2.a.y
Level $8040$
Weight $2$
Character orbit 8040.a
Self dual yes
Analytic conductor $64.200$
Analytic rank $1$
Dimension $9$
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 2x^{8} - 30x^{7} + 32x^{6} + 248x^{5} - 38x^{4} - 705x^{3} - 450x^{2} + 168x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} - q^{5} + ( - \beta_{6} - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - q^{5} + ( - \beta_{6} - 1) q^{7} + q^{9} + ( - \beta_{5} - \beta_{3}) q^{11} + ( - \beta_{7} + \beta_{5} + \beta_{3} - 1) q^{13} + q^{15} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{17} + ( - \beta_{5} + \beta_{4} + 1) q^{19} + (\beta_{6} + 1) q^{21} + (\beta_{7} - \beta_{3} - 1) q^{23} + q^{25} - q^{27} + ( - \beta_{8} + \beta_{7} + \beta_{5} + \cdots + 1) q^{29}+ \cdots + ( - \beta_{5} - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{3} - 9 q^{5} - 6 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{3} - 9 q^{5} - 6 q^{7} + 9 q^{9} - 4 q^{11} - q^{13} + 9 q^{15} - 2 q^{17} + 5 q^{19} + 6 q^{21} - 14 q^{23} + 9 q^{25} - 9 q^{27} + 13 q^{29} - q^{31} + 4 q^{33} + 6 q^{35} + 9 q^{37} + q^{39} + 5 q^{41} + 7 q^{43} - 9 q^{45} - 15 q^{47} + 15 q^{49} + 2 q^{51} + 3 q^{53} + 4 q^{55} - 5 q^{57} - q^{59} + 12 q^{61} - 6 q^{63} + q^{65} + 9 q^{67} + 14 q^{69} - 15 q^{71} + 4 q^{73} - 9 q^{75} + 6 q^{77} + 9 q^{79} + 9 q^{81} - 17 q^{83} + 2 q^{85} - 13 q^{87} + 15 q^{89} + 11 q^{91} + q^{93} - 5 q^{95} - 6 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 30x^{7} + 32x^{6} + 248x^{5} - 38x^{4} - 705x^{3} - 450x^{2} + 168x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 33 \nu^{8} + 212 \nu^{7} + 968 \nu^{6} - 5254 \nu^{5} - 10924 \nu^{4} + 25516 \nu^{3} + \cdots - 16836 ) / 508 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 107 \nu^{8} - 37 \nu^{7} - 3054 \nu^{6} - 1514 \nu^{5} + 17875 \nu^{4} + 17096 \nu^{3} - 16575 \nu^{2} + \cdots + 6468 ) / 762 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 235 \nu^{8} + 740 \nu^{7} + 6216 \nu^{6} - 14678 \nu^{5} - 42032 \nu^{4} + 57368 \nu^{3} + \cdots - 37920 ) / 1524 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 128 \nu^{8} + 322 \nu^{7} + 3543 \nu^{6} - 5905 \nu^{5} - 25046 \nu^{4} + 20489 \nu^{3} + \cdots - 19514 ) / 254 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 436 \nu^{8} - 1073 \nu^{7} - 11985 \nu^{6} + 19340 \nu^{5} + 82511 \nu^{4} - 65429 \nu^{3} + \cdots + 62604 ) / 762 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 917 \nu^{8} + 2920 \nu^{7} + 25290 \nu^{6} - 59422 \nu^{5} - 191764 \nu^{4} + 235618 \nu^{3} + \cdots - 184188 ) / 1524 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1021 \nu^{8} + 3134 \nu^{7} + 28002 \nu^{6} - 62672 \nu^{5} - 206510 \nu^{4} + 243542 \nu^{3} + \cdots - 181644 ) / 1524 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 533 \nu^{8} - 1954 \nu^{7} - 14661 \nu^{6} + 41707 \nu^{5} + 116626 \nu^{4} - 174151 \nu^{3} + \cdots + 120312 ) / 762 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{8} - 3\beta_{7} + \beta_{6} - 3\beta_{5} - 2\beta_{4} + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{8} - 21\beta_{7} - 7\beta_{6} - 15\beta_{5} + 4\beta_{4} - 2\beta_{3} + 6\beta_{2} - 4\beta _1 + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 23 \beta_{8} - 71 \beta_{7} + 23 \beta_{6} - 77 \beta_{5} - 40 \beta_{4} - 20 \beta_{3} + 18 \beta_{2} + \cdots + 220 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 253 \beta_{8} - 483 \beta_{7} - 55 \beta_{6} - 343 \beta_{5} + 62 \beta_{4} - 96 \beta_{3} + \cdots + 318 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 637 \beta_{8} - 1861 \beta_{7} + 513 \beta_{6} - 1891 \beta_{5} - 748 \beta_{4} - 634 \beta_{3} + \cdots + 4378 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 5571 \beta_{8} - 11607 \beta_{7} - 185 \beta_{6} - 8693 \beta_{5} + 604 \beta_{4} - 2888 \beta_{3} + \cdots + 10028 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 18041 \beta_{8} - 49331 \beta_{7} + 11113 \beta_{6} - 47155 \beta_{5} - 14530 \beta_{4} + \cdots + 96622 ) / 2 \) 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
−4.03122
−2.04086
−0.843391
5.05973
2.70246
−1.30489
0.541204
−0.793032
2.71000
0 −1.00000 0 −1.00000 0 −4.52964 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −4.23309 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 −3.31312 0 1.00000 0
1.4 0 −1.00000 0 −1.00000 0 −1.99685 0 1.00000 0
1.5 0 −1.00000 0 −1.00000 0 −0.596451 0 1.00000 0
1.6 0 −1.00000 0 −1.00000 0 0.168719 0 1.00000 0
1.7 0 −1.00000 0 −1.00000 0 2.54469 0 1.00000 0
1.8 0 −1.00000 0 −1.00000 0 2.93320 0 1.00000 0
1.9 0 −1.00000 0 −1.00000 0 3.02255 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.y ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8040.2.a.y ✓ 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):

\( T_{7}^{9} + 6T_{7}^{8} - 21T_{7}^{7} - 150T_{7}^{6} + 131T_{7}^{5} + 1223T_{7}^{4} - 94T_{7}^{3} - 3214T_{7}^{2} - 1168T_{7} + 288 \) Copy content Toggle raw display
\( T_{11}^{9} + 4 T_{11}^{8} - 49 T_{11}^{7} - 210 T_{11}^{6} + 419 T_{11}^{5} + 1951 T_{11}^{4} + \cdots + 384 \) Copy content Toggle raw display
\( T_{13}^{9} + T_{13}^{8} - 65 T_{13}^{7} + 13 T_{13}^{6} + 1207 T_{13}^{5} - 1248 T_{13}^{4} - 5370 T_{13}^{3} + \cdots + 96 \) Copy content Toggle raw display
\( T_{17}^{9} + 2 T_{17}^{8} - 92 T_{17}^{7} - 245 T_{17}^{6} + 1819 T_{17}^{5} + 7946 T_{17}^{4} + \cdots - 384 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( (T + 1)^{9} \) Copy content Toggle raw display
$5$ \( (T + 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} + 6 T^{8} + \cdots + 288 \) Copy content Toggle raw display
$11$ \( T^{9} + 4 T^{8} + \cdots + 384 \) Copy content Toggle raw display
$13$ \( T^{9} + T^{8} + \cdots + 96 \) Copy content Toggle raw display
$17$ \( T^{9} + 2 T^{8} + \cdots - 384 \) Copy content Toggle raw display
$19$ \( T^{9} - 5 T^{8} + \cdots - 384 \) Copy content Toggle raw display
$23$ \( T^{9} + 14 T^{8} + \cdots + 14848 \) Copy content Toggle raw display
$29$ \( T^{9} - 13 T^{8} + \cdots - 1500768 \) Copy content Toggle raw display
$31$ \( T^{9} + T^{8} + \cdots + 713472 \) Copy content Toggle raw display
$37$ \( T^{9} - 9 T^{8} + \cdots + 18992 \) Copy content Toggle raw display
$41$ \( T^{9} - 5 T^{8} + \cdots - 2996352 \) Copy content Toggle raw display
$43$ \( T^{9} - 7 T^{8} + \cdots - 59392 \) Copy content Toggle raw display
$47$ \( T^{9} + 15 T^{8} + \cdots - 712512 \) Copy content Toggle raw display
$53$ \( T^{9} - 3 T^{8} + \cdots - 14602752 \) Copy content Toggle raw display
$59$ \( T^{9} + T^{8} + \cdots + 72736 \) Copy content Toggle raw display
$61$ \( T^{9} - 12 T^{8} + \cdots - 1128 \) Copy content Toggle raw display
$67$ \( (T - 1)^{9} \) Copy content Toggle raw display
$71$ \( T^{9} + 15 T^{8} + \cdots - 217704 \) Copy content Toggle raw display
$73$ \( T^{9} - 4 T^{8} + \cdots + 36286912 \) Copy content Toggle raw display
$79$ \( T^{9} - 9 T^{8} + \cdots - 200784 \) Copy content Toggle raw display
$83$ \( T^{9} + 17 T^{8} + \cdots - 94582784 \) Copy content Toggle raw display
$89$ \( T^{9} - 15 T^{8} + \cdots - 309744 \) Copy content Toggle raw display
$97$ \( T^{9} + 6 T^{8} + \cdots - 110362688 \) Copy content Toggle raw display
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