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Properties

Label 8040.2.a.bb
Level $8040$
Weight $2$
Character orbit 8040.a
Self dual yes
Analytic conductor $64.200$
Analytic rank $0$
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: \(0\)
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: \( x^{9} - 4x^{8} - 34x^{7} + 123x^{6} + 375x^{5} - 1146x^{4} - 1662x^{3} + 3086x^{2} + 3372x - 64 \) 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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + q^{5} + ( - \beta_1 + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{5} + ( - \beta_1 + 1) q^{7} + q^{9} + (\beta_{5} - \beta_{2} + \beta_1 - 1) q^{11} - \beta_{2} q^{13} + q^{15} + ( - \beta_{8} - \beta_{6} + 1) q^{17} + ( - \beta_{3} + 1) q^{19} + ( - \beta_1 + 1) q^{21} + ( - \beta_{6} + 2) q^{23} + q^{25} + q^{27} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots + 1) q^{29}+ \cdots + (\beta_{5} - \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{3} + 9 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{3} + 9 q^{5} + 5 q^{7} + 9 q^{9} - 7 q^{11} - 3 q^{13} + 9 q^{15} + 7 q^{17} + 8 q^{19} + 5 q^{21} + 19 q^{23} + 9 q^{25} + 9 q^{27} + 14 q^{29} + 27 q^{31} - 7 q^{33} + 5 q^{35} + 15 q^{37} - 3 q^{39} - 5 q^{41} + 11 q^{43} + 9 q^{45} + 6 q^{47} + 30 q^{49} + 7 q^{51} - 11 q^{53} - 7 q^{55} + 8 q^{57} + 14 q^{59} - 7 q^{61} + 5 q^{63} - 3 q^{65} - 9 q^{67} + 19 q^{69} - 2 q^{71} - 23 q^{73} + 9 q^{75} - 9 q^{77} + 13 q^{79} + 9 q^{81} + 48 q^{83} + 7 q^{85} + 14 q^{87} + 5 q^{89} + 41 q^{91} + 27 q^{93} + 8 q^{95} + 19 q^{97} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 34x^{7} + 123x^{6} + 375x^{5} - 1146x^{4} - 1662x^{3} + 3086x^{2} + 3372x - 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 60429 \nu^{8} + 54882 \nu^{7} + 2230564 \nu^{6} - 586265 \nu^{5} - 24550545 \nu^{4} + \cdots - 3019562 ) / 1184894 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 48698 \nu^{8} - 55375 \nu^{7} - 1801381 \nu^{6} + 782418 \nu^{5} + 20127849 \nu^{4} + \cdots - 1839938 ) / 592447 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 261557 \nu^{8} + 206018 \nu^{7} + 9819906 \nu^{6} - 1328383 \nu^{5} - 110744457 \nu^{4} + \cdots + 868204 ) / 2369788 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 157825 \nu^{8} - 165632 \nu^{7} - 5833326 \nu^{6} + 2151101 \nu^{5} + 64806243 \nu^{4} + \cdots - 1845208 ) / 1184894 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 193129 \nu^{8} - 195460 \nu^{7} - 7087766 \nu^{6} + 2388727 \nu^{5} + 77645393 \nu^{4} + \cdots - 708176 ) / 1184894 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 529981 \nu^{8} + 507450 \nu^{7} + 19528990 \nu^{6} - 5598063 \nu^{5} - 215175981 \nu^{4} + \cdots - 12210920 ) / 2369788 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 589495 \nu^{8} - 544442 \nu^{7} - 21728250 \nu^{6} + 5811565 \nu^{5} + 238666899 \nu^{4} + \cdots + 15388292 ) / 2369788 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 157825 \nu^{8} - 165632 \nu^{7} - 5833326 \nu^{6} + 2151101 \nu^{5} + 64806243 \nu^{4} + \cdots + 4671709 ) / 592447 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} + \beta_{4} + \beta_{2} - \beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} + \beta_{6} + 3\beta_{5} + 8\beta_{4} - \beta_{3} - 9\beta_{2} + 4\beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -19\beta_{8} - 9\beta_{7} - 5\beta_{6} + \beta_{5} + 26\beta_{4} + 10\beta_{2} - 34\beta _1 + 168 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{8} - 59\beta_{7} + 26\beta_{6} + 75\beta_{5} + 168\beta_{4} - 23\beta_{3} - 166\beta_{2} - 9\beta _1 + 396 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 384 \beta_{8} - 311 \beta_{7} - 132 \beta_{6} + 105 \beta_{5} + 661 \beta_{4} - 15 \beta_{3} + \cdots + 3365 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 618 \beta_{8} - 1558 \beta_{7} + 488 \beta_{6} + 1701 \beta_{5} + 3882 \beta_{4} - 508 \beta_{3} + \cdots + 10628 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 8260 \beta_{8} - 8484 \beta_{7} - 2741 \beta_{6} + 4014 \beta_{5} + 16901 \beta_{4} - 701 \beta_{3} + \cdots + 74577 \) 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
3.11607
2.94981
3.03344
−1.81912
−1.02526
−4.02322
0.0186643
5.01663
−3.26701
0 1.00000 0 1.00000 0 −5.24081 0 1.00000 0
1.2 0 1.00000 0 1.00000 0 −2.42687 0 1.00000 0
1.3 0 1.00000 0 1.00000 0 −1.95262 0 1.00000 0
1.4 0 1.00000 0 1.00000 0 0.227701 0 1.00000 0
1.5 0 1.00000 0 1.00000 0 0.487682 0 1.00000 0
1.6 0 1.00000 0 1.00000 0 1.81240 0 1.00000 0
1.7 0 1.00000 0 1.00000 0 2.52099 0 1.00000 0
1.8 0 1.00000 0 1.00000 0 4.56825 0 1.00000 0
1.9 0 1.00000 0 1.00000 0 5.00327 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.bb ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8040.2.a.bb ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):

\( T_{7}^{9} - 5T_{7}^{8} - 34T_{7}^{7} + 186T_{7}^{6} + 175T_{7}^{5} - 1430T_{7}^{4} + 264T_{7}^{3} + 2924T_{7}^{2} - 1928T_{7} + 288 \) Copy content Toggle raw display
\( T_{11}^{9} + 7 T_{11}^{8} - 48 T_{11}^{7} - 352 T_{11}^{6} + 553 T_{11}^{5} + 3848 T_{11}^{4} + \cdots + 2752 \) Copy content Toggle raw display
\( T_{13}^{9} + 3 T_{13}^{8} - 65 T_{13}^{7} - 107 T_{13}^{6} + 1471 T_{13}^{5} + 560 T_{13}^{4} + \cdots - 1296 \) Copy content Toggle raw display
\( T_{17}^{9} - 7 T_{17}^{8} - 82 T_{17}^{7} + 519 T_{17}^{6} + 1990 T_{17}^{5} - 10273 T_{17}^{4} + \cdots + 18240 \) 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
$7$ \( T^{9} - 5 T^{8} + \cdots + 288 \) Copy content Toggle raw display
$11$ \( T^{9} + 7 T^{8} + \cdots + 2752 \) Copy content Toggle raw display
$13$ \( T^{9} + 3 T^{8} + \cdots - 1296 \) Copy content Toggle raw display
$17$ \( T^{9} - 7 T^{8} + \cdots + 18240 \) Copy content Toggle raw display
$19$ \( T^{9} - 8 T^{8} + \cdots - 163136 \) Copy content Toggle raw display
$23$ \( T^{9} - 19 T^{8} + \cdots - 8640 \) Copy content Toggle raw display
$29$ \( T^{9} - 14 T^{8} + \cdots - 4648552 \) Copy content Toggle raw display
$31$ \( T^{9} - 27 T^{8} + \cdots - 2255200 \) Copy content Toggle raw display
$37$ \( T^{9} - 15 T^{8} + \cdots - 1611936 \) Copy content Toggle raw display
$41$ \( T^{9} + 5 T^{8} + \cdots - 40896 \) Copy content Toggle raw display
$43$ \( T^{9} - 11 T^{8} + \cdots - 1108992 \) Copy content Toggle raw display
$47$ \( T^{9} - 6 T^{8} + \cdots - 2160432 \) Copy content Toggle raw display
$53$ \( T^{9} + 11 T^{8} + \cdots - 1057920 \) Copy content Toggle raw display
$59$ \( T^{9} - 14 T^{8} + \cdots + 742272 \) Copy content Toggle raw display
$61$ \( T^{9} + 7 T^{8} + \cdots + 314122240 \) Copy content Toggle raw display
$67$ \( (T + 1)^{9} \) Copy content Toggle raw display
$71$ \( T^{9} + 2 T^{8} + \cdots - 1509888 \) Copy content Toggle raw display
$73$ \( T^{9} + 23 T^{8} + \cdots + 94324848 \) Copy content Toggle raw display
$79$ \( T^{9} + \cdots + 2154480320 \) Copy content Toggle raw display
$83$ \( T^{9} - 48 T^{8} + \cdots - 1350144 \) Copy content Toggle raw display
$89$ \( T^{9} - 5 T^{8} + \cdots - 51857200 \) Copy content Toggle raw display
$97$ \( T^{9} - 19 T^{8} + \cdots - 886976 \) Copy content Toggle raw display
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