Properties

Label 7569.2.a.be
Level $7569$
Weight $2$
Character orbit 7569.a
Self dual yes
Analytic conductor $60.439$
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] = [7569,2,Mod(1,7569)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7569, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7569.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7569 = 3^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7569.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: \(60.4387692899\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.5878828125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 13x^{6} + 7x^{5} + 55x^{4} + 3x^{3} - 78x^{2} - 54x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2523)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - \beta_{7} - \beta_{3} + \beta_{2} + \cdots + 2) q^{4}+ \cdots + (\beta_{7} + \beta_{6} + 2 \beta_{5} + \cdots - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + ( - \beta_{7} - \beta_{3} + \beta_{2} + \cdots + 2) q^{4}+ \cdots + (5 \beta_{7} + 4 \beta_{6} + 10 \beta_{5} + \cdots + 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + 11 q^{4} + 9 q^{5} - 12 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{2} + 11 q^{4} + 9 q^{5} - 12 q^{7} - 15 q^{8} + 2 q^{10} + 12 q^{11} - 15 q^{13} - 11 q^{14} + 13 q^{16} + 9 q^{17} + 9 q^{19} + 8 q^{20} - 24 q^{22} + 15 q^{23} + 7 q^{25} + 5 q^{26} - 39 q^{28} + 20 q^{31} + 9 q^{32} - 23 q^{34} - 6 q^{35} + 17 q^{37} + 2 q^{38} + 5 q^{40} - 20 q^{41} + 20 q^{43} + 24 q^{44} - 20 q^{46} - 9 q^{47} + 2 q^{49} + 51 q^{50} - 35 q^{52} + 39 q^{53} + 36 q^{55} + 15 q^{56} + 34 q^{61} - 15 q^{62} - 9 q^{64} + 25 q^{65} - 21 q^{67} + 28 q^{68} - 8 q^{70} - 15 q^{71} - 2 q^{73} - 9 q^{74} + 18 q^{76} - 3 q^{77} + 6 q^{79} + 54 q^{80} - 60 q^{82} - 17 q^{83} + 22 q^{85} + 45 q^{86} - 60 q^{88} + 20 q^{89} + 30 q^{91} + 70 q^{92} - 2 q^{94} + 12 q^{95} - 16 q^{97} + 51 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 13x^{6} + 7x^{5} + 55x^{4} + 3x^{3} - 78x^{2} - 54x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - \nu^{5} - 10\nu^{4} + 7\nu^{3} + 25\nu^{2} - 6\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 11\nu^{5} - 3\nu^{4} + 32\nu^{3} + 19\nu^{2} - 18\nu - 12 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 2\nu^{6} + 12\nu^{5} - 17\nu^{4} - 48\nu^{3} + 25\nu^{2} + 72\nu + 27 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} - 4\nu^{6} - 24\nu^{5} + 37\nu^{4} + 93\nu^{3} - 71\nu^{2} - 129\nu - 33 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} - 5\nu^{6} - 23\nu^{5} + 47\nu^{4} + 89\nu^{3} - 96\nu^{2} - 138\nu - 27 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 5\nu^{6} - 34\nu^{5} + 44\nu^{4} + 118\nu^{3} - 80\nu^{2} - 138\nu - 24 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{5} + \beta_{3} - \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{7} + \beta_{6} + 7\beta_{5} + 2\beta_{4} + 7\beta_{3} - 6\beta_{2} + 7\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} + 10\beta_{6} - 8\beta_{5} + \beta_{4} + 3\beta_{3} + 6\beta_{2} + 30\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -47\beta_{7} + 13\beta_{6} + 44\beta_{5} + 21\beta_{4} + 48\beta_{3} - 33\beta_{2} + 46\beta _1 + 129 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -24\beta_{7} + 81\beta_{6} - 54\beta_{5} + 17\beta_{4} + 38\beta_{3} + 35\beta_{2} + 190\beta _1 + 47 \) Copy content Toggle raw display

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
−0.851203
−2.46784
−0.758720
−1.69344
2.09698
2.67829
−0.262852
2.25879
−2.67829 0 5.17326 0.923940 0 −3.83323 −8.49892 0 −2.47458
1.2 −2.25879 0 3.10211 −0.0968284 0 1.11297 −2.48944 0 0.218715
1.3 −2.09698 0 2.39733 1.13206 0 −0.0912936 −0.833198 0 −2.37390
1.4 0.262852 0 −1.93091 4.31701 0 −0.159034 −1.03325 0 1.13473
1.5 0.758720 0 −1.42434 2.69503 0 0.874680 −2.59812 0 2.04478
1.6 0.851203 0 −1.27545 −0.880235 0 −0.914011 −2.78808 0 −0.749259
1.7 1.69344 0 0.867750 −2.52606 0 −4.38828 −1.91740 0 −4.27775
1.8 2.46784 0 4.09025 3.43509 0 −4.60180 5.15841 0 8.47726
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(29\) \( +1 \)

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 7569.2.a.be 8
3.b odd 2 1 2523.2.a.n yes 8
29.b even 2 1 7569.2.a.bh 8
87.d odd 2 1 2523.2.a.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2523.2.a.m 8 87.d odd 2 1
2523.2.a.n yes 8 3.b odd 2 1
7569.2.a.be 8 1.a even 1 1 trivial
7569.2.a.bh 8 29.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):

\( T_{2}^{8} + T_{2}^{7} - 13T_{2}^{6} - 7T_{2}^{5} + 55T_{2}^{4} - 3T_{2}^{3} - 78T_{2}^{2} + 54T_{2} - 9 \) Copy content Toggle raw display
\( T_{5}^{8} - 9T_{5}^{7} + 17T_{5}^{6} + 48T_{5}^{5} - 170T_{5}^{4} + 72T_{5}^{3} + 132T_{5}^{2} - 81T_{5} - 9 \) Copy content Toggle raw display
\( T_{7}^{8} + 12T_{7}^{7} + 43T_{7}^{6} + 19T_{7}^{5} - 115T_{7}^{4} - 46T_{7}^{3} + 63T_{7}^{2} + 17T_{7} + 1 \) Copy content Toggle raw display
\( T_{19}^{8} - 9T_{19}^{7} - 53T_{19}^{6} + 683T_{19}^{5} - 565T_{19}^{4} - 10613T_{19}^{3} + 33402T_{19}^{2} - 30176T_{19} + 3541 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} - 13 T^{6} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 9 T^{7} + \cdots - 9 \) Copy content Toggle raw display
$7$ \( T^{8} + 12 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{8} - 12 T^{7} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{8} + 15 T^{7} + \cdots - 5 \) Copy content Toggle raw display
$17$ \( T^{8} - 9 T^{7} + \cdots + 531 \) Copy content Toggle raw display
$19$ \( T^{8} - 9 T^{7} + \cdots + 3541 \) Copy content Toggle raw display
$23$ \( T^{8} - 15 T^{7} + \cdots - 2655 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} - 20 T^{7} + \cdots - 8055 \) Copy content Toggle raw display
$37$ \( T^{8} - 17 T^{7} + \cdots + 865611 \) Copy content Toggle raw display
$41$ \( T^{8} + 20 T^{7} + \cdots + 249525 \) Copy content Toggle raw display
$43$ \( T^{8} - 20 T^{7} + \cdots + 6525 \) Copy content Toggle raw display
$47$ \( T^{8} + 9 T^{7} + \cdots - 72909 \) Copy content Toggle raw display
$53$ \( T^{8} - 39 T^{7} + \cdots - 306099 \) Copy content Toggle raw display
$59$ \( T^{8} - 250 T^{6} + \cdots - 573975 \) Copy content Toggle raw display
$61$ \( T^{8} - 34 T^{7} + \cdots - 464039 \) Copy content Toggle raw display
$67$ \( T^{8} + 21 T^{7} + \cdots - 169679 \) Copy content Toggle raw display
$71$ \( T^{8} + 15 T^{7} + \cdots + 3645 \) Copy content Toggle raw display
$73$ \( T^{8} + 2 T^{7} + \cdots + 2369671 \) Copy content Toggle raw display
$79$ \( T^{8} - 6 T^{7} + \cdots - 2579879 \) Copy content Toggle raw display
$83$ \( T^{8} + 17 T^{7} + \cdots - 32427819 \) Copy content Toggle raw display
$89$ \( T^{8} - 20 T^{7} + \cdots + 291645 \) Copy content Toggle raw display
$97$ \( T^{8} + 16 T^{7} + \cdots + 40453821 \) Copy content Toggle raw display
show more
show less