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Properties

Label 8040.2.a.ba
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: \( x^{9} - x^{8} - 20x^{7} - 11x^{6} + 84x^{5} + 92x^{4} - 62x^{3} - 99x^{2} - 20x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 20x^{7} - 11x^{6} + 84x^{5} + 92x^{4} - 62x^{3} - 99x^{2} - 20x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 73 \nu^{8} - 575 \nu^{7} - 178 \nu^{6} + 6993 \nu^{5} - 160 \nu^{4} - 21270 \nu^{3} - 4448 \nu^{2} + \cdots + 2878 ) / 1318 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 149 \nu^{8} + 307 \nu^{7} + 2566 \nu^{6} - 967 \nu^{5} - 9748 \nu^{4} - 2752 \nu^{3} + 4204 \nu^{2} + \cdots + 4760 ) / 1318 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 405 \nu^{8} + 861 \nu^{7} + 7072 \nu^{6} - 3283 \nu^{5} - 29318 \nu^{4} - 6808 \nu^{3} + \cdots + 174 ) / 1318 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 485 \nu^{8} - 787 \nu^{7} - 8892 \nu^{6} - 771 \nu^{5} + 36834 \nu^{4} + 27516 \nu^{3} - 32260 \nu^{2} + \cdots - 1624 ) / 1318 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 563 \nu^{8} - 1275 \nu^{7} - 9678 \nu^{6} + 6051 \nu^{5} + 40274 \nu^{4} + 1774 \nu^{3} - 39432 \nu^{2} + \cdots + 422 ) / 1318 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 303 \nu^{8} + 527 \nu^{7} + 5740 \nu^{6} - 1104 \nu^{5} - 25732 \nu^{4} - 7622 \nu^{3} + \cdots - 4146 ) / 659 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 907 \nu^{8} + 2143 \nu^{7} + 14868 \nu^{6} - 9575 \nu^{5} - 57304 \nu^{4} - 6730 \nu^{3} + \cdots - 118 ) / 1318 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 933 \nu^{8} + 1427 \nu^{7} + 17766 \nu^{6} + 1331 \nu^{5} - 77342 \nu^{4} - 47794 \nu^{3} + \cdots - 6072 ) / 1318 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 3\beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{8} + 6\beta_{7} + 15\beta_{6} + 31\beta_{5} + \beta_{4} + 17\beta_{3} + 6\beta_{2} - 4\beta _1 + 48 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{8} + 36\beta_{7} + 51\beta_{6} + 147\beta_{5} + 37\beta_{4} + 95\beta_{3} + 4\beta_{2} - 12\beta _1 + 308 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 18 \beta_{8} + 75 \beta_{7} + 139 \beta_{6} + 359 \beta_{5} + 61 \beta_{4} + 208 \beta_{3} + \cdots + 630 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 73 \beta_{8} + 750 \beta_{7} + 1207 \beta_{6} + 3391 \beta_{5} + 765 \beta_{4} + 2105 \beta_{3} + \cdots + 6400 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 467 \beta_{8} + 3460 \beta_{7} + 5963 \beta_{6} + 16279 \beta_{5} + 3293 \beta_{4} + 9735 \beta_{3} + \cdots + 29412 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 856 \beta_{8} + 8331 \beta_{7} + 13863 \beta_{6} + 38637 \beta_{5} + 8385 \beta_{4} + 23604 \beta_{3} + \cdots + 71148 ) / 2 \) 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
0.121924
−0.508296
−0.844624
4.76225
−1.76328
−1.54537
−2.46691
1.12789
2.11643
0 1.00000 0 −1.00000 0 −4.85861 0 1.00000 0
1.2 0 1.00000 0 −1.00000 0 −4.41706 0 1.00000 0
1.3 0 1.00000 0 −1.00000 0 −3.88188 0 1.00000 0
1.4 0 1.00000 0 −1.00000 0 −0.697968 0 1.00000 0
1.5 0 1.00000 0 −1.00000 0 0.374236 0 1.00000 0
1.6 0 1.00000 0 −1.00000 0 0.442587 0 1.00000 0
1.7 0 1.00000 0 −1.00000 0 1.43483 0 1.00000 0
1.8 0 1.00000 0 −1.00000 0 1.48494 0 1.00000 0
1.9 0 1.00000 0 −1.00000 0 3.11892 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.ba ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8040.2.a.ba ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):

\( T_{7}^{9} + 7T_{7}^{8} - 12T_{7}^{7} - 124T_{7}^{6} + 75T_{7}^{5} + 596T_{7}^{4} - 656T_{7}^{3} - 152T_{7}^{2} + 288T_{7} - 64 \) Copy content Toggle raw display
\( T_{11}^{9} - 3 T_{11}^{8} - 48 T_{11}^{7} + 108 T_{11}^{6} + 799 T_{11}^{5} - 1026 T_{11}^{4} + \cdots + 7168 \) Copy content Toggle raw display
\( T_{13}^{9} + 3 T_{13}^{8} - 53 T_{13}^{7} - 65 T_{13}^{6} + 971 T_{13}^{5} - 244 T_{13}^{4} - 5936 T_{13}^{3} + \cdots - 128 \) Copy content Toggle raw display
\( T_{17}^{9} - T_{17}^{8} - 86 T_{17}^{7} + 101 T_{17}^{6} + 2108 T_{17}^{5} - 2307 T_{17}^{4} + \cdots + 512 \) 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
$7$ \( T^{9} + 7 T^{8} + \cdots - 64 \) Copy content Toggle raw display
$11$ \( T^{9} - 3 T^{8} + \cdots + 7168 \) Copy content Toggle raw display
$13$ \( T^{9} + 3 T^{8} + \cdots - 128 \) Copy content Toggle raw display
$17$ \( T^{9} - T^{8} + \cdots + 512 \) Copy content Toggle raw display
$19$ \( T^{9} + 8 T^{8} + \cdots + 2304 \) Copy content Toggle raw display
$23$ \( T^{9} + 5 T^{8} + \cdots - 5344 \) Copy content Toggle raw display
$29$ \( T^{9} + 6 T^{8} + \cdots + 2062768 \) Copy content Toggle raw display
$31$ \( T^{9} + 17 T^{8} + \cdots - 9373504 \) Copy content Toggle raw display
$37$ \( T^{9} + 11 T^{8} + \cdots + 26089136 \) Copy content Toggle raw display
$41$ \( T^{9} - T^{8} + \cdots - 500832 \) Copy content Toggle raw display
$43$ \( T^{9} + 13 T^{8} + \cdots - 1691648 \) Copy content Toggle raw display
$47$ \( T^{9} + 14 T^{8} + \cdots + 381088 \) Copy content Toggle raw display
$53$ \( T^{9} + 17 T^{8} + \cdots - 726208 \) Copy content Toggle raw display
$59$ \( T^{9} + 8 T^{8} + \cdots + 13410864 \) Copy content Toggle raw display
$61$ \( T^{9} + T^{8} + \cdots - 11855808 \) Copy content Toggle raw display
$67$ \( (T + 1)^{9} \) Copy content Toggle raw display
$71$ \( T^{9} + 8 T^{8} + \cdots + 1030592 \) Copy content Toggle raw display
$73$ \( T^{9} + 27 T^{8} + \cdots + 4618384 \) Copy content Toggle raw display
$79$ \( T^{9} + 35 T^{8} + \cdots + 1744128 \) Copy content Toggle raw display
$83$ \( T^{9} + 2 T^{8} + \cdots + 19039232 \) Copy content Toggle raw display
$89$ \( T^{9} + 17 T^{8} + \cdots + 436208 \) Copy content Toggle raw display
$97$ \( T^{9} + 11 T^{8} + \cdots - 109568 \) Copy content Toggle raw display
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