↑

Properties

Label 8040.2.a.bc
Level $8040$
Weight $2$
Character orbit 8040.a
Self dual yes
Analytic conductor $64.200$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3x^{9} - 37x^{8} + 132x^{7} + 358x^{6} - 1708x^{5} - 92x^{4} + 5969x^{3} - 3864x^{2} - 4752x + 3524 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{5} - \beta_{9} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} - \beta_{9} q^{7} + q^{9} + ( - \beta_{7} + 1) q^{11} + \beta_{8} q^{13} - q^{15} + \beta_{6} q^{17} - \beta_{3} q^{19} + \beta_{9} q^{21} + ( - \beta_{5} - \beta_1 - 1) q^{23} + q^{25} - q^{27} + ( - \beta_{9} + \beta_{5} - \beta_{4} + \cdots + 1) q^{29}+ \cdots + ( - \beta_{7} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 10 q^{3} + 10 q^{5} + q^{7} + 10 q^{9} + 7 q^{11} - q^{13} - 10 q^{15} + 3 q^{17} + 4 q^{19} - q^{21} - 13 q^{23} + 10 q^{25} - 10 q^{27} + 18 q^{29} - 9 q^{31} - 7 q^{33} + q^{35} - 3 q^{37} + q^{39} + 19 q^{41} + 5 q^{43} + 10 q^{45} - 8 q^{47} + 43 q^{49} - 3 q^{51} + 17 q^{53} + 7 q^{55} - 4 q^{57} + 24 q^{59} + 21 q^{61} + q^{63} - q^{65} - 10 q^{67} + 13 q^{69} + 2 q^{71} + 25 q^{73} - 10 q^{75} + 15 q^{77} - q^{79} + 10 q^{81} - 6 q^{83} + 3 q^{85} - 18 q^{87} + 23 q^{89} + 29 q^{91} + 9 q^{93} + 4 q^{95} + 21 q^{97} + 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 37x^{8} + 132x^{7} + 358x^{6} - 1708x^{5} - 92x^{4} + 5969x^{3} - 3864x^{2} - 4752x + 3524 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1820505 \nu^{9} + 648324 \nu^{8} - 58839604 \nu^{7} + 40103260 \nu^{6} + 547588947 \nu^{5} + \cdots - 6029040447 ) / 712963777 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 9643245 \nu^{9} + 13049510 \nu^{8} + 405600855 \nu^{7} - 622469771 \nu^{6} + \cdots + 24517354298 ) / 1425927554 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26086723 \nu^{9} - 252840 \nu^{8} + 996434659 \nu^{7} - 442082665 \nu^{6} + \cdots + 28946015068 ) / 1425927554 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18725497 \nu^{9} + 12691238 \nu^{8} + 723832368 \nu^{7} - 798514962 \nu^{6} + \cdots + 28921908461 ) / 712963777 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 39985935 \nu^{9} + 21312224 \nu^{8} + 1524928207 \nu^{7} - 1501477001 \nu^{6} + \cdots + 50050926048 ) / 1425927554 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 53420509 \nu^{9} + 49948754 \nu^{8} + 2092831555 \nu^{7} - 2752544585 \nu^{6} + \cdots + 97540858780 ) / 1425927554 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27977725 \nu^{9} + 22939251 \nu^{8} + 1085047513 \nu^{7} - 1328495831 \nu^{6} + \cdots + 44873983953 ) / 712963777 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 29521266 \nu^{9} + 37943916 \nu^{8} + 1166227892 \nu^{7} - 1880005231 \nu^{6} + \cdots + 60525448260 ) / 712963777 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 45139551 \nu^{9} - 37028569 \nu^{8} - 1746534425 \nu^{7} + 2162038384 \nu^{6} + \cdots - 75217331604 ) / 712963777 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{9} - \beta_{8} - 3\beta_{6} - \beta_{4} - \beta_{3} + 2\beta_{2} + 2\beta _1 + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{9} + \beta_{8} - 8\beta_{7} + 15\beta_{6} + 14\beta_{5} - 15\beta_{4} + \beta_{3} - 4\beta _1 - 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 57 \beta_{9} - 13 \beta_{8} + 4 \beta_{7} - 67 \beta_{6} - 12 \beta_{5} - 13 \beta_{4} - 11 \beta_{3} + \cdots + 245 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 119 \beta_{9} + 17 \beta_{8} - 96 \beta_{7} + 287 \beta_{6} + 222 \beta_{5} - 265 \beta_{4} + 27 \beta_{3} + \cdots - 469 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1003 \beta_{9} - 139 \beta_{8} + 110 \beta_{7} - 1365 \beta_{6} - 326 \beta_{5} - 91 \beta_{4} + \cdots + 4161 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2423 \beta_{9} + 145 \beta_{8} - 1406 \beta_{7} + 5743 \beta_{6} + 3766 \beta_{5} - 4859 \beta_{4} + \cdots - 10369 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 17727 \beta_{9} - 927 \beta_{8} + 2254 \beta_{7} - 27377 \beta_{6} - 7094 \beta_{5} + 1069 \beta_{4} + \cdots + 75113 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 47525 \beta_{9} - 1463 \beta_{8} - 22344 \beta_{7} + 115779 \beta_{6} + 66128 \beta_{5} - 90593 \beta_{4} + \cdots - 215295 ) / 2 \) 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
−1.93365
1.90235
2.61555
4.04435
−1.02133
−4.09581
0.731185
3.07629
−4.47145
2.15252
0 −1.00000 0 1.00000 0 −4.46032 0 1.00000 0
1.2 0 −1.00000 0 1.00000 0 −4.35860 0 1.00000 0
1.3 0 −1.00000 0 1.00000 0 −3.08369 0 1.00000 0
1.4 0 −1.00000 0 1.00000 0 −1.71989 0 1.00000 0
1.5 0 −1.00000 0 1.00000 0 −1.55723 0 1.00000 0
1.6 0 −1.00000 0 1.00000 0 1.83837 0 1.00000 0
1.7 0 −1.00000 0 1.00000 0 1.87892 0 1.00000 0
1.8 0 −1.00000 0 1.00000 0 3.78235 0 1.00000 0
1.9 0 −1.00000 0 1.00000 0 3.93512 0 1.00000 0
1.10 0 −1.00000 0 1.00000 0 4.74497 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.bc ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8040.2.a.bc ✓ 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):

\( T_{7}^{10} - T_{7}^{9} - 56 T_{7}^{8} + 48 T_{7}^{7} + 1117 T_{7}^{6} - 766 T_{7}^{5} - 9504 T_{7}^{4} + \cdots - 39168 \) Copy content Toggle raw display
\( T_{11}^{10} - 7 T_{11}^{9} - 46 T_{11}^{8} + 384 T_{11}^{7} + 399 T_{11}^{6} - 5740 T_{11}^{5} + \cdots - 32768 \) Copy content Toggle raw display
\( T_{13}^{10} + T_{13}^{9} - 79 T_{13}^{8} + 19 T_{13}^{7} + 2135 T_{13}^{6} - 2590 T_{13}^{5} + \cdots + 73856 \) Copy content Toggle raw display
\( T_{17}^{10} - 3 T_{17}^{9} - 86 T_{17}^{8} + 247 T_{17}^{7} + 2100 T_{17}^{6} - 5747 T_{17}^{5} + \cdots - 3264 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - T^{9} + \cdots - 39168 \) Copy content Toggle raw display
$11$ \( T^{10} - 7 T^{9} + \cdots - 32768 \) Copy content Toggle raw display
$13$ \( T^{10} + T^{9} + \cdots + 73856 \) Copy content Toggle raw display
$17$ \( T^{10} - 3 T^{9} + \cdots - 3264 \) Copy content Toggle raw display
$19$ \( T^{10} - 4 T^{9} + \cdots - 1378048 \) Copy content Toggle raw display
$23$ \( T^{10} + 13 T^{9} + \cdots + 24576 \) Copy content Toggle raw display
$29$ \( T^{10} - 18 T^{9} + \cdots - 501312 \) Copy content Toggle raw display
$31$ \( T^{10} + 9 T^{9} + \cdots + 459072 \) Copy content Toggle raw display
$37$ \( T^{10} + 3 T^{9} + \cdots + 2000 \) Copy content Toggle raw display
$41$ \( T^{10} - 19 T^{9} + \cdots - 3379104 \) Copy content Toggle raw display
$43$ \( T^{10} - 5 T^{9} + \cdots - 25090048 \) Copy content Toggle raw display
$47$ \( T^{10} + 8 T^{9} + \cdots - 1438592 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 113774592 \) Copy content Toggle raw display
$59$ \( T^{10} - 24 T^{9} + \cdots - 5059008 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 135811584 \) Copy content Toggle raw display
$67$ \( (T + 1)^{10} \) Copy content Toggle raw display
$71$ \( T^{10} - 2 T^{9} + \cdots + 704384 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 369762464 \) Copy content Toggle raw display
$79$ \( T^{10} + T^{9} + \cdots + 4942848 \) Copy content Toggle raw display
$83$ \( T^{10} + 6 T^{9} + \cdots + 56870912 \) Copy content Toggle raw display
$89$ \( T^{10} - 23 T^{9} + \cdots + 2304288 \) Copy content Toggle raw display
$97$ \( T^{10} - 21 T^{9} + \cdots - 87472384 \) Copy content Toggle raw display
show more
show less