Properties

Label 7569.2.a.bs
Level $7569$
Weight $2$
Character orbit 7569.a
Self dual yes
Analytic conductor $60.439$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 17 x^{10} + 13 x^{9} + 105 x^{8} - 49 x^{7} - 292 x^{6} + 44 x^{5} + 355 x^{4} + \cdots + 16 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{9} - \beta_{6} + \beta_{3} + 1) q^{4} + (\beta_{11} - \beta_{7} + \beta_{3} + \cdots - 1) q^{5}+ \cdots + (\beta_{11} + \beta_{9} - 2 \beta_{8} + \cdots + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{9} - \beta_{6} + \beta_{3} + 1) q^{4} + (\beta_{11} - \beta_{7} + \beta_{3} + \cdots - 1) q^{5}+ \cdots + (2 \beta_{11} + \beta_{9} - 4 \beta_{8} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{2} + 11 q^{4} - 7 q^{5} + 9 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + q^{2} + 11 q^{4} - 7 q^{5} + 9 q^{7} + 9 q^{8} + 6 q^{10} + 12 q^{11} + 19 q^{13} + 23 q^{14} + 13 q^{16} - 7 q^{17} + 16 q^{19} - 6 q^{20} + 12 q^{22} - 29 q^{23} + 27 q^{25} + 13 q^{26} - q^{28} + 8 q^{31} - 11 q^{32} + 17 q^{34} - 26 q^{35} - 4 q^{37} + 17 q^{40} - 3 q^{43} + 38 q^{44} + 60 q^{46} + 11 q^{47} + 35 q^{49} - 7 q^{50} + 53 q^{52} - 41 q^{53} + 36 q^{55} + 9 q^{56} + 16 q^{59} + q^{61} + 15 q^{62} + 7 q^{64} - 25 q^{65} + 24 q^{67} + 56 q^{68} - 44 q^{70} - 51 q^{71} - 17 q^{74} + 16 q^{76} - 25 q^{77} + 69 q^{79} + 44 q^{80} + 16 q^{82} - 25 q^{83} - 10 q^{85} - q^{86} + 40 q^{88} - 32 q^{91} - 42 q^{92} + 6 q^{94} + 86 q^{95} + 14 q^{97} + 37 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 17 x^{10} + 13 x^{9} + 105 x^{8} - 49 x^{7} - 292 x^{6} + 44 x^{5} + 355 x^{4} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 485 \nu^{11} + 834 \nu^{10} + 7634 \nu^{9} - 12224 \nu^{8} - 41950 \nu^{7} + 59596 \nu^{6} + \cdots - 15412 ) / 1916 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 292 \nu^{11} + 352 \nu^{10} + 4767 \nu^{9} - 4933 \nu^{8} - 27691 \nu^{7} + 22347 \nu^{6} + \cdots - 6004 ) / 958 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1381 \nu^{11} + 2275 \nu^{10} + 21835 \nu^{9} - 32643 \nu^{8} - 121795 \nu^{7} + 154447 \nu^{6} + \cdots - 45200 ) / 3832 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1561 \nu^{11} + 2387 \nu^{10} + 24767 \nu^{9} - 33407 \nu^{8} - 138727 \nu^{7} + 150867 \nu^{6} + \cdots - 32392 ) / 3832 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 921 \nu^{11} - 1563 \nu^{10} - 14555 \nu^{9} + 22175 \nu^{8} + 81079 \nu^{7} - 102071 \nu^{6} + \cdots + 19600 ) / 1916 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1031 \nu^{11} + 1525 \nu^{10} + 16879 \nu^{9} - 21471 \nu^{8} - 98771 \nu^{7} + 97435 \nu^{6} + \cdots - 23056 ) / 1916 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1225 \nu^{11} + 2146 \nu^{10} + 19262 \nu^{9} - 30480 \nu^{8} - 106450 \nu^{7} + 141104 \nu^{6} + \cdots - 29604 ) / 1916 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1505 \nu^{11} - 2267 \nu^{10} - 24089 \nu^{9} + 32041 \nu^{8} + 136461 \nu^{7} - 146765 \nu^{6} + \cdots + 25860 ) / 1916 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4049 \nu^{11} - 7171 \nu^{10} - 64059 \nu^{9} + 102171 \nu^{8} + 358331 \nu^{7} - 475855 \nu^{6} + \cdots + 101712 ) / 3832 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3807 \nu^{11} + 6584 \nu^{10} + 60000 \nu^{9} - 93086 \nu^{8} - 333108 \nu^{7} + 426754 \nu^{6} + \cdots - 81080 ) / 1916 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{6} + \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{9} - 2\beta_{8} + \beta_{6} + \beta_{3} + \beta_{2} + 5\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + 7\beta_{9} - \beta_{7} - 7\beta_{6} - \beta_{5} - 2\beta_{4} + 9\beta_{3} - \beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{11} + \beta_{10} + 7 \beta_{9} - 16 \beta_{8} + 10 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + \cdots + 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{11} - 11 \beta_{10} + 47 \beta_{9} + 3 \beta_{8} - 10 \beta_{7} - 44 \beta_{6} - 8 \beta_{5} + \cdots + 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 68 \beta_{11} + 13 \beta_{10} + 44 \beta_{9} - 107 \beta_{8} + \beta_{7} + 85 \beta_{6} - 35 \beta_{5} + \cdots + 97 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12 \beta_{11} - 91 \beta_{10} + 316 \beta_{9} + 43 \beta_{8} - 79 \beta_{7} - 274 \beta_{6} + \cdots + 601 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 488 \beta_{11} + 117 \beta_{10} + 275 \beta_{9} - 693 \beta_{8} + 17 \beta_{7} + 672 \beta_{6} + \cdots + 611 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 106 \beta_{11} - 687 \beta_{10} + 2136 \beta_{9} + 430 \beta_{8} - 577 \beta_{7} - 1726 \beta_{6} + \cdots + 3857 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3430 \beta_{11} + 917 \beta_{10} + 1743 \beta_{9} - 4490 \beta_{8} + 188 \beta_{7} + 5084 \beta_{6} + \cdots + 3844 \) 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
−2.63048
−2.01690
−1.46888
−1.06848
−0.726454
−0.668708
0.282500
0.742506
1.40064
2.21330
2.32830
2.61265
−2.63048 0 4.91944 −0.886923 0 −3.51834 −7.67955 0 2.33304
1.2 −2.01690 0 2.06787 −3.13689 0 3.05231 −0.136879 0 6.32677
1.3 −1.46888 0 0.157605 3.40129 0 −1.63441 2.70625 0 −4.99608
1.4 −1.06848 0 −0.858356 −3.93653 0 2.58610 3.05409 0 4.20609
1.5 −0.726454 0 −1.47226 1.17772 0 0.775671 2.52244 0 −0.855558
1.6 −0.668708 0 −1.55283 −3.36024 0 −4.86588 2.37581 0 2.24702
1.7 0.282500 0 −1.92019 2.99747 0 2.93733 −1.10745 0 0.846784
1.8 0.742506 0 −1.44869 0.0547619 0 1.46183 −2.56067 0 0.0406610
1.9 1.40064 0 −0.0382014 −2.49985 0 2.29755 −2.85479 0 −3.50140
1.10 2.21330 0 2.89869 −3.59353 0 4.20833 1.98908 0 −7.95355
1.11 2.32830 0 3.42098 −0.126501 0 4.62264 3.30846 0 −0.294531
1.12 2.61265 0 4.82594 2.90921 0 −2.92312 7.38321 0 7.60076
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.bs 12
3.b odd 2 1 2523.2.a.t 12
29.b even 2 1 7569.2.a.bo 12
87.d odd 2 1 2523.2.a.u yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2523.2.a.t 12 3.b odd 2 1
2523.2.a.u yes 12 87.d odd 2 1
7569.2.a.bo 12 29.b even 2 1
7569.2.a.bs 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):

\( T_{2}^{12} - T_{2}^{11} - 17 T_{2}^{10} + 13 T_{2}^{9} + 105 T_{2}^{8} - 49 T_{2}^{7} - 292 T_{2}^{6} + \cdots + 16 \) Copy content Toggle raw display
\( T_{5}^{12} + 7 T_{5}^{11} - 19 T_{5}^{10} - 214 T_{5}^{9} - 26 T_{5}^{8} + 2258 T_{5}^{7} + 2294 T_{5}^{6} + \cdots - 80 \) Copy content Toggle raw display
\( T_{7}^{12} - 9 T_{7}^{11} - 19 T_{7}^{10} + 390 T_{7}^{9} - 601 T_{7}^{8} - 4803 T_{7}^{7} + \cdots + 96111 \) Copy content Toggle raw display
\( T_{19}^{12} - 16 T_{19}^{11} - 12 T_{19}^{10} + 1258 T_{19}^{9} - 3375 T_{19}^{8} - 30829 T_{19}^{7} + \cdots + 605191 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{11} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 7 T^{11} + \cdots - 80 \) Copy content Toggle raw display
$7$ \( T^{12} - 9 T^{11} + \cdots + 96111 \) Copy content Toggle raw display
$11$ \( T^{12} - 12 T^{11} + \cdots - 3098864 \) Copy content Toggle raw display
$13$ \( T^{12} - 19 T^{11} + \cdots - 1355625 \) Copy content Toggle raw display
$17$ \( T^{12} + 7 T^{11} + \cdots - 3600 \) Copy content Toggle raw display
$19$ \( T^{12} - 16 T^{11} + \cdots + 605191 \) Copy content Toggle raw display
$23$ \( T^{12} + 29 T^{11} + \cdots - 1265904 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} - 8 T^{11} + \cdots - 412619 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots - 107348609 \) Copy content Toggle raw display
$41$ \( T^{12} - 193 T^{10} + \cdots + 919120 \) Copy content Toggle raw display
$43$ \( T^{12} + 3 T^{11} + \cdots + 418491 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 386568976 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 400419584 \) Copy content Toggle raw display
$59$ \( T^{12} - 16 T^{11} + \cdots - 2116080 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 54907178661 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 825555631 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 1261879664 \) Copy content Toggle raw display
$73$ \( T^{12} - 216 T^{10} + \cdots + 106111 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 8602913331 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 203096304 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 201310544 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 13491351961 \) Copy content Toggle raw display
show more
show less