Properties

Label 7569.2.a.bq
Level $7569$
Weight $2$
Character orbit 7569.a
Self dual yes
Analytic conductor $60.439$
Analytic rank $1$
Dimension $12$
Inner twists $2$

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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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{10} + 93x^{8} - 241x^{6} + 282x^{4} - 149x^{2} + 29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 261)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{11} q^{5} + ( - \beta_{9} + \beta_{8} - 1) q^{7} + (\beta_{6} - \beta_{4} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} - \beta_{11} q^{5} + ( - \beta_{9} + \beta_{8} - 1) q^{7} + (\beta_{6} - \beta_{4} + \beta_1) q^{8} + ( - \beta_{9} + \beta_{8}) q^{10} + (\beta_{11} - \beta_{7}) q^{11} + (\beta_{8} + \beta_{5} - \beta_{3}) q^{13} + ( - \beta_{11} + \beta_{7} - \beta_1) q^{14} + ( - \beta_{5} + \beta_{3} - 1) q^{16} + (\beta_{7} - \beta_{6} + \beta_{4} - \beta_1) q^{17} + (\beta_{9} - \beta_{3} - \beta_{2} - 1) q^{19} + (\beta_{11} + \beta_{7}) q^{20} + (2 \beta_{9} + \beta_{8} + 2 \beta_{5} + 2) q^{22} + (\beta_{11} - \beta_{7} + \beta_{6} + \cdots + \beta_1) q^{23}+ \cdots + (3 \beta_{11} - 2 \beta_{7} + \cdots - 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 8 q^{4} - 14 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 8 q^{4} - 14 q^{7} - 2 q^{10} - 12 q^{13} - 4 q^{16} - 14 q^{19} + 8 q^{22} + 2 q^{25} - 12 q^{28} + 28 q^{31} - 18 q^{34} + 36 q^{37} - 20 q^{43} + 20 q^{46} - 10 q^{49} - 12 q^{52} - 58 q^{55} + 4 q^{61} - 18 q^{64} - 54 q^{67} + 60 q^{70} + 32 q^{73} - 56 q^{76} - 2 q^{79} - 96 q^{82} - 4 q^{85} - 10 q^{88} + 6 q^{91} - 32 q^{94} + 26 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{10} + 93x^{8} - 241x^{6} + 282x^{4} - 149x^{2} + 29 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{10} - 15\nu^{8} + 78\nu^{6} - 162\nu^{4} + 112\nu^{2} - 21 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{11} - 15\nu^{9} + 78\nu^{7} - 163\nu^{5} + 118\nu^{3} - 25\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{10} - 15\nu^{8} + 78\nu^{6} - 163\nu^{4} + 118\nu^{2} - 26 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{11} - 15\nu^{9} + 78\nu^{7} - 163\nu^{5} + 119\nu^{3} - 30\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 3\nu^{11} - 46\nu^{9} + 248\nu^{7} - 554\nu^{5} + 465\nu^{3} - 127\nu \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -4\nu^{10} + 61\nu^{8} - 326\nu^{6} + 717\nu^{4} - 583\nu^{2} + 152 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 4\nu^{10} - 61\nu^{8} + 327\nu^{6} - 727\nu^{4} + 610\nu^{2} - 167 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -6\nu^{11} + 91\nu^{9} - 482\nu^{7} + 1042\nu^{5} - 813\nu^{3} + 199\nu \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( 11\nu^{11} - 168\nu^{9} + 901\nu^{7} - 1998\nu^{5} + 1658\nu^{3} - 446\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{4} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{5} + \beta_{3} + 6\beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{10} - \beta_{7} + 6\beta_{6} - 9\beta_{4} + 27\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} + \beta_{8} - 10\beta_{5} + 10\beta_{3} + 33\beta_{2} + 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{11} - 10\beta_{10} - 13\beta_{7} + 33\beta_{6} - 65\beta_{4} + 150\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 14\beta_{9} + 15\beta_{8} - 71\beta_{5} + 75\beta_{3} + 183\beta_{2} + 336 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 14\beta_{11} - 75\beta_{10} - 118\beta_{7} + 183\beta_{6} - 433\beta_{4} + 848\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 132\beta_{9} + 147\beta_{8} - 447\beta_{5} + 508\beta_{3} + 1031\beta_{2} + 1839 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 132\beta_{11} - 508\beta_{10} - 919\beta_{7} + 1031\beta_{6} - 2773\beta_{4} + 4856\beta_1 \) 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
−2.44987
−2.17124
−1.79061
−0.894469
−0.854850
−0.739423
0.739423
0.854850
0.894469
1.79061
2.17124
2.44987
−2.44987 0 4.00186 0.217978 0 −1.53402 −4.90430 0 −0.534017
1.2 −2.17124 0 2.71427 −0.728942 0 0.582706 −1.55086 0 1.58271
1.3 −1.79061 0 1.20627 1.46245 0 −3.61867 1.42125 0 −2.61867
1.4 −0.894469 0 −1.19993 −1.99112 0 0.780997 2.86223 0 1.78100
1.5 −0.854850 0 −1.26923 3.95934 0 −4.38464 2.79470 0 −3.38464
1.6 −0.739423 0 −1.45325 −2.93963 0 1.17363 2.55342 0 2.17363
1.7 0.739423 0 −1.45325 2.93963 0 1.17363 −2.55342 0 2.17363
1.8 0.854850 0 −1.26923 −3.95934 0 −4.38464 −2.79470 0 −3.38464
1.9 0.894469 0 −1.19993 1.99112 0 0.780997 −2.86223 0 1.78100
1.10 1.79061 0 1.20627 −1.46245 0 −3.61867 −1.42125 0 −2.61867
1.11 2.17124 0 2.71427 0.728942 0 0.582706 1.55086 0 1.58271
1.12 2.44987 0 4.00186 −0.217978 0 −1.53402 4.90430 0 −0.534017
\(n\): 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7569.2.a.bq 12
3.b odd 2 1 inner 7569.2.a.bq 12
29.b even 2 1 7569.2.a.br 12
29.d even 7 2 261.2.k.d 24
87.d odd 2 1 7569.2.a.br 12
87.j odd 14 2 261.2.k.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
261.2.k.d 24 29.d even 7 2
261.2.k.d 24 87.j odd 14 2
7569.2.a.bq 12 1.a even 1 1 trivial
7569.2.a.bq 12 3.b odd 2 1 inner
7569.2.a.br 12 29.b even 2 1
7569.2.a.br 12 87.d odd 2 1

Hecke kernels

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

\( T_{2}^{12} - 16T_{2}^{10} + 93T_{2}^{8} - 241T_{2}^{6} + 282T_{2}^{4} - 149T_{2}^{2} + 29 \) Copy content Toggle raw display
\( T_{5}^{12} - 31T_{5}^{10} + 310T_{5}^{8} - 1203T_{5}^{6} + 1754T_{5}^{4} - 691T_{5}^{2} + 29 \) Copy content Toggle raw display
\( T_{7}^{6} + 7T_{7}^{5} + 6T_{7}^{4} - 28T_{7}^{3} - 9T_{7}^{2} + 35T_{7} - 13 \) Copy content Toggle raw display
\( T_{19}^{6} + 7T_{19}^{5} - 21T_{19}^{4} - 168T_{19}^{3} + 196T_{19}^{2} + 1029T_{19} - 1421 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 16 T^{10} + \cdots + 29 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 31 T^{10} + \cdots + 29 \) Copy content Toggle raw display
$7$ \( (T^{6} + 7 T^{5} + 6 T^{4} + \cdots - 13)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 49 T^{10} + \cdots + 24389 \) Copy content Toggle raw display
$13$ \( (T^{6} + 6 T^{5} - 3 T^{4} + \cdots + 29)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} - 63 T^{10} + \cdots + 21141 \) Copy content Toggle raw display
$19$ \( (T^{6} + 7 T^{5} + \cdots - 1421)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 90 T^{10} + \cdots + 4901 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( (T^{6} - 14 T^{5} + \cdots - 377)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 18 T^{5} + \cdots - 10933)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 29887475661 \) Copy content Toggle raw display
$43$ \( (T^{6} + 10 T^{5} + \cdots + 12263)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 238 T^{10} + \cdots + 146189 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 52773918149 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 1992518109 \) Copy content Toggle raw display
$61$ \( (T^{6} - 2 T^{5} + \cdots - 10723)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 27 T^{5} + \cdots + 229193)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 473 T^{10} + \cdots + 17779581 \) Copy content Toggle raw display
$73$ \( (T^{6} - 16 T^{5} + \cdots + 10291)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + T^{5} + \cdots - 70713)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 59886921221 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 2865880949 \) Copy content Toggle raw display
$97$ \( (T^{6} - 13 T^{5} + \cdots + 115333)^{2} \) Copy content Toggle raw display
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