↑

Properties

Label 8040.2.a.u
Level $8040$
Weight $2$
Character orbit 8040.a
Self dual yes
Analytic conductor $64.200$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 29x^{6} - 16x^{5} + 228x^{4} + 209x^{3} - 280x^{2} - 121x + 88 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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 \( q - q^{3} - q^{5} - \beta_{4} q^{7} + q^{9} + ( - \beta_{6} + \beta_1 + 1) q^{11} + ( - \beta_{6} + \beta_{5} + \beta_1) q^{13} + q^{15} + (\beta_{7} - \beta_{6} - \beta_{2} + 1) q^{17} + ( - \beta_{6} + \beta_{5}) q^{19} + \beta_{4} q^{21} + (\beta_{7} - \beta_{6} + \beta_{5} + 2) q^{23} + q^{25} - q^{27} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 2) q^{29}+ \cdots + ( - \beta_{6} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} - 8 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} - 8 q^{5} + q^{7} + 8 q^{9} + 3 q^{11} - q^{13} + 8 q^{15} + 7 q^{17} - q^{21} + 15 q^{23} + 8 q^{25} - 8 q^{27} - 14 q^{29} + 5 q^{31} - 3 q^{33} - q^{35} + 5 q^{37} + q^{39} - q^{41} - 15 q^{43} - 8 q^{45} + 36 q^{47} + 5 q^{49} - 7 q^{51} - 5 q^{53} - 3 q^{55} + 10 q^{59} - 19 q^{61} + q^{63} + q^{65} - 8 q^{67} - 15 q^{69} + 8 q^{71} - 11 q^{73} - 8 q^{75} + 7 q^{77} + q^{79} + 8 q^{81} + 14 q^{83} - 7 q^{85} + 14 q^{87} - 3 q^{89} - 15 q^{91} - 5 q^{93} - 9 q^{97} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 29x^{6} - 16x^{5} + 228x^{4} + 209x^{3} - 280x^{2} - 121x + 88 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -15\nu^{7} - 3463\nu^{6} + 3924\nu^{5} + 91372\nu^{4} - 10520\nu^{3} - 631051\nu^{2} - 486715\nu + 259432 ) / 88564 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 277\nu^{7} - 3949\nu^{6} - 1612\nu^{5} + 89848\nu^{4} - 12380\nu^{3} - 556615\nu^{2} - 148849\nu + 440336 ) / 88564 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1671 \nu^{7} - 4953 \nu^{6} + 47452 \nu^{5} + 153396 \nu^{4} - 286288 \nu^{3} - 1166023 \nu^{2} + \cdots + 631096 ) / 177128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 863 \nu^{7} + 1445 \nu^{6} - 30920 \nu^{5} - 37564 \nu^{4} + 280516 \nu^{3} + 249111 \nu^{2} + \cdots + 35424 ) / 88564 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3951 \nu^{7} - 55 \nu^{6} - 112516 \nu^{5} - 48828 \nu^{4} + 822560 \nu^{3} + 551015 \nu^{2} + \cdots + 497552 ) / 177128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7429 \nu^{7} - 81 \nu^{6} - 207572 \nu^{5} - 133100 \nu^{4} + 1549560 \nu^{3} + 1672981 \nu^{2} + \cdots - 471712 ) / 177128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2991 \nu^{7} - 277 \nu^{6} - 82790 \nu^{5} - 46244 \nu^{4} + 592100 \nu^{3} + 637499 \nu^{2} + \cdots - 257344 ) / 44282 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 2\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - 3\beta_{2} + 2\beta _1 + 15 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 4\beta_{6} + 11\beta_{5} - 13\beta_{4} + 11\beta_{3} - 15\beta_{2} + 4\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 28\beta_{7} - 34\beta_{6} + 5\beta_{5} - 25\beta_{4} + 15\beta_{3} - 59\beta_{2} + 42\beta _1 + 185 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -30\beta_{7} + 56\beta_{6} + 169\beta_{5} - 221\beta_{4} + 121\beta_{3} - 249\beta_{2} + 108\beta _1 + 79 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 348\beta_{7} - 484\beta_{6} + 319\beta_{5} - 531\beta_{4} + 171\beta_{3} - 1087\beta_{2} + 792\beta _1 + 2525 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 366 \beta_{7} + 614 \beta_{6} + 2929 \beta_{5} - 3875 \beta_{4} + 1315 \beta_{3} - 4405 \beta_{2} + \cdots + 2741 ) / 2 \) 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
−0.764001
−3.52792
−1.80149
0.801322
3.52958
0.479122
4.35461
−3.07123
0 −1.00000 0 −1.00000 0 −4.13113 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −3.01317 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 −1.50458 0 1.00000 0
1.4 0 −1.00000 0 −1.00000 0 0.175251 0 1.00000 0
1.5 0 −1.00000 0 −1.00000 0 0.732722 0 1.00000 0
1.6 0 −1.00000 0 −1.00000 0 0.865456 0 1.00000 0
1.7 0 −1.00000 0 −1.00000 0 3.57523 0 1.00000 0
1.8 0 −1.00000 0 −1.00000 0 4.30022 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.u ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8040.2.a.u ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8040))\):

\( T_{7}^{8} - T_{7}^{7} - 30T_{7}^{6} + 22T_{7}^{5} + 237T_{7}^{4} - 108T_{7}^{3} - 316T_{7}^{2} + 240T_{7} - 32 \) Copy content Toggle raw display
\( T_{11}^{8} - 3T_{11}^{7} - 40T_{11}^{6} + 124T_{11}^{5} + 265T_{11}^{4} - 764T_{11}^{3} - 562T_{11}^{2} + 916T_{11} + 496 \) Copy content Toggle raw display
\( T_{13}^{8} + T_{13}^{7} - 47T_{13}^{6} - 23T_{13}^{5} + 667T_{13}^{4} + 54T_{13}^{3} - 3012T_{13}^{2} - 296T_{13} + 3616 \) Copy content Toggle raw display
\( T_{17}^{8} - 7T_{17}^{7} - 48T_{17}^{6} + 293T_{17}^{5} + 904T_{17}^{4} - 3629T_{17}^{3} - 7472T_{17}^{2} + 13702T_{17} + 21388 \) 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
$7$ \( T^{8} - T^{7} + \cdots - 32 \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} + \cdots + 496 \) Copy content Toggle raw display
$13$ \( T^{8} + T^{7} + \cdots + 3616 \) Copy content Toggle raw display
$17$ \( T^{8} - 7 T^{7} + \cdots + 21388 \) Copy content Toggle raw display
$19$ \( T^{8} - 54 T^{6} + \cdots - 1072 \) Copy content Toggle raw display
$23$ \( T^{8} - 15 T^{7} + \cdots - 1472 \) Copy content Toggle raw display
$29$ \( T^{8} + 14 T^{7} + \cdots + 6592 \) Copy content Toggle raw display
$31$ \( T^{8} - 5 T^{7} + \cdots + 104656 \) Copy content Toggle raw display
$37$ \( T^{8} - 5 T^{7} + \cdots + 79292 \) Copy content Toggle raw display
$41$ \( T^{8} + T^{7} + \cdots + 1228424 \) Copy content Toggle raw display
$43$ \( T^{8} + 15 T^{7} + \cdots + 204448 \) Copy content Toggle raw display
$47$ \( T^{8} - 36 T^{7} + \cdots + 572720 \) Copy content Toggle raw display
$53$ \( T^{8} + 5 T^{7} + \cdots - 61888 \) Copy content Toggle raw display
$59$ \( T^{8} - 10 T^{7} + \cdots - 10912 \) Copy content Toggle raw display
$61$ \( T^{8} + 19 T^{7} + \cdots - 65036672 \) Copy content Toggle raw display
$67$ \( (T + 1)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} - 8 T^{7} + \cdots + 1060064 \) Copy content Toggle raw display
$73$ \( T^{8} + 11 T^{7} + \cdots + 91768 \) Copy content Toggle raw display
$79$ \( T^{8} - T^{7} + \cdots - 36593920 \) Copy content Toggle raw display
$83$ \( T^{8} - 14 T^{7} + \cdots + 113824 \) Copy content Toggle raw display
$89$ \( T^{8} + 3 T^{7} + \cdots - 200632 \) Copy content Toggle raw display
$97$ \( T^{8} + 9 T^{7} + \cdots - 3533104 \) Copy content Toggle raw display
show more
show less