Properties

Label 7569.2.a.bd
Level $7569$
Weight $2$
Character orbit 7569.a
Self dual yes
Analytic conductor $60.439$
Analytic rank $1$
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: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.2841328125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 3x^{6} + 23x^{5} - 43x^{3} + 2x^{2} + 24x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 841)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{6} + \beta_{3}) q^{5} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_1) q^{7} + ( - \beta_{3} - \beta_{2} - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{6} + \beta_{3}) q^{5} + (\beta_{7} + \beta_{6} - \beta_{5} + \beta_1) q^{7} + ( - \beta_{3} - \beta_{2} - 2) q^{8} + (\beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{10}+ \cdots + ( - 4 \beta_{7} - 7 \beta_{6} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 6 q^{4} + q^{5} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 6 q^{4} + q^{5} - 15 q^{8} - 3 q^{10} - 5 q^{11} - 4 q^{13} - 15 q^{14} - 2 q^{16} - 9 q^{17} + 17 q^{19} + 7 q^{20} + 25 q^{22} + 7 q^{23} - 13 q^{25} - 3 q^{26} + 25 q^{28} + q^{31} + 6 q^{32} + 12 q^{34} + 5 q^{35} + 6 q^{37} + 14 q^{38} - 25 q^{40} - 6 q^{41} - 5 q^{43} - 20 q^{44} + 14 q^{46} - 31 q^{47} + 4 q^{49} - 11 q^{50} + 2 q^{52} - 19 q^{53} + 20 q^{55} - 20 q^{56} + q^{59} + 21 q^{61} + 12 q^{62} + q^{64} + 17 q^{65} + 2 q^{67} - 8 q^{68} - 5 q^{70} + 12 q^{71} - 34 q^{73} + 27 q^{74} - 16 q^{76} + 10 q^{77} + 29 q^{79} + 41 q^{80} - 22 q^{82} + 3 q^{83} + 2 q^{85} + 5 q^{86} - 15 q^{88} - 36 q^{89} + 4 q^{92} + 13 q^{94} - 11 q^{95} - 18 q^{97} - 27 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 3x^{6} + 23x^{5} - 43x^{3} + 2x^{2} + 24x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 2\nu^{3} - 3\nu^{2} + 4\nu + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 3\nu^{4} - 2\nu^{3} + 8\nu^{2} + \nu - 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 3\nu^{5} - 3\nu^{4} + 11\nu^{3} + 3\nu^{2} - 9\nu - 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 4\nu^{6} + 14\nu^{4} - 8\nu^{3} - 12\nu^{2} + 7\nu + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 2\beta_{3} + 5\beta_{2} + 7\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 3\beta_{4} + 8\beta_{3} + 9\beta_{2} + 20\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 3\beta_{5} + 12\beta_{4} + 19\beta_{3} + 28\beta_{2} + 43\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} + 4\beta_{6} + 12\beta_{5} + 34\beta_{4} + 56\beta_{3} + 62\beta_{2} + 111\beta _1 + 86 \) 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
2.51918
2.39427
1.92862
1.04195
−0.0419454
−0.928616
−1.39427
−1.51918
−2.51918 0 4.34627 1.46556 0 4.12665 −5.91068 0 −3.69200
1.2 −2.39427 0 3.73253 2.42792 0 −0.378370 −4.14815 0 −5.81310
1.3 −1.92862 0 1.71956 −2.94984 0 2.60541 0.540863 0 5.68911
1.4 −1.04195 0 −0.914350 −1.46226 0 −3.80089 3.03659 0 1.52360
1.5 0.0419454 0 −1.99824 1.74204 0 2.05366 −0.167708 0 0.0730704
1.6 0.928616 0 −1.13767 0.504715 0 −4.15272 −2.91369 0 0.468687
1.7 1.39427 0 −0.0560094 1.14641 0 −0.110460 −2.86663 0 1.59840
1.8 1.51918 0 0.307910 −1.87453 0 −0.343265 −2.57059 0 −2.84776
\(n\): 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.bd 8
3.b odd 2 1 841.2.a.j yes 8
29.b even 2 1 7569.2.a.bi 8
87.d odd 2 1 841.2.a.i 8
87.f even 4 2 841.2.b.f 16
87.h odd 14 6 841.2.d.q 48
87.j odd 14 6 841.2.d.p 48
87.k even 28 12 841.2.e.m 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
841.2.a.i 8 87.d odd 2 1
841.2.a.j yes 8 3.b odd 2 1
841.2.b.f 16 87.f even 4 2
841.2.d.p 48 87.j odd 14 6
841.2.d.q 48 87.h odd 14 6
841.2.e.m 96 87.k even 28 12
7569.2.a.bd 8 1.a even 1 1 trivial
7569.2.a.bi 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):

\( T_{2}^{8} + 4T_{2}^{7} - 3T_{2}^{6} - 23T_{2}^{5} + 43T_{2}^{3} + 2T_{2}^{2} - 24T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} - T_{5}^{7} - 13T_{5}^{6} + 17T_{5}^{5} + 45T_{5}^{4} - 72T_{5}^{3} - 33T_{5}^{2} + 86T_{5} - 29 \) Copy content Toggle raw display
\( T_{7}^{8} - 30T_{7}^{6} + 10T_{7}^{5} + 235T_{7}^{4} - 165T_{7}^{3} - 245T_{7}^{2} - 70T_{7} - 5 \) Copy content Toggle raw display
\( T_{19}^{8} - 17T_{19}^{7} + 83T_{19}^{6} - 44T_{19}^{5} - 335T_{19}^{4} - 34T_{19}^{3} + 203T_{19}^{2} + 23T_{19} - 29 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{7} + \cdots - 29 \) Copy content Toggle raw display
$7$ \( T^{8} - 30 T^{6} + \cdots - 5 \) Copy content Toggle raw display
$11$ \( T^{8} + 5 T^{7} + \cdots + 295 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} + \cdots - 1769 \) Copy content Toggle raw display
$17$ \( T^{8} + 9 T^{7} + \cdots - 4799 \) Copy content Toggle raw display
$19$ \( T^{8} - 17 T^{7} + \cdots - 29 \) Copy content Toggle raw display
$23$ \( T^{8} - 7 T^{7} + \cdots + 25681 \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} - T^{7} + \cdots + 631 \) Copy content Toggle raw display
$37$ \( T^{8} - 6 T^{7} + \cdots - 229709 \) Copy content Toggle raw display
$41$ \( T^{8} + 6 T^{7} + \cdots + 210811 \) Copy content Toggle raw display
$43$ \( T^{8} + 5 T^{7} + \cdots - 305 \) Copy content Toggle raw display
$47$ \( T^{8} + 31 T^{7} + \cdots + 25111 \) Copy content Toggle raw display
$53$ \( T^{8} + 19 T^{7} + \cdots - 322349 \) Copy content Toggle raw display
$59$ \( T^{8} - T^{7} + \cdots + 179131 \) Copy content Toggle raw display
$61$ \( T^{8} - 21 T^{7} + \cdots - 365879 \) Copy content Toggle raw display
$67$ \( T^{8} - 2 T^{7} + \cdots + 14622931 \) Copy content Toggle raw display
$71$ \( T^{8} - 12 T^{7} + \cdots + 2574301 \) Copy content Toggle raw display
$73$ \( T^{8} + 34 T^{7} + \cdots + 1223191 \) Copy content Toggle raw display
$79$ \( T^{8} - 29 T^{7} + \cdots + 878851 \) Copy content Toggle raw display
$83$ \( T^{8} - 3 T^{7} + \cdots - 343139 \) Copy content Toggle raw display
$89$ \( T^{8} + 36 T^{7} + \cdots - 108359 \) Copy content Toggle raw display
$97$ \( T^{8} + 18 T^{7} + \cdots + 3862591 \) Copy content Toggle raw display
show more
show less