Properties

Label 7569.2.a.g
Level $7569$
Weight $2$
Character orbit 7569.a
Self dual yes
Analytic conductor $60.439$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + (\beta - 1) q^{4} + (\beta + 1) q^{5} + (3 \beta - 1) q^{7} + (2 \beta - 1) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + (\beta - 1) q^{4} + (\beta + 1) q^{5} + (3 \beta - 1) q^{7} + (2 \beta - 1) q^{8} + ( - 2 \beta - 1) q^{10} + 3 q^{11} + ( - 4 \beta + 1) q^{13} + ( - 2 \beta - 3) q^{14} - 3 \beta q^{16} + ( - 2 \beta - 2) q^{17} + ( - 3 \beta - 1) q^{19} + \beta q^{20} - 3 \beta q^{22} + (5 \beta - 4) q^{23} + (3 \beta - 3) q^{25} + (3 \beta + 4) q^{26} + ( - \beta + 4) q^{28} - 3 q^{31} + ( - \beta + 5) q^{32} + (4 \beta + 2) q^{34} + (5 \beta + 2) q^{35} + ( - 6 \beta + 6) q^{37} + (4 \beta + 3) q^{38} + (3 \beta + 1) q^{40} + ( - 2 \beta + 9) q^{41} + 3 \beta q^{43} + (3 \beta - 3) q^{44} + ( - \beta - 5) q^{46} + (\beta - 11) q^{47} + (3 \beta + 3) q^{49} - 3 q^{50} + (\beta - 5) q^{52} + ( - 4 \beta + 8) q^{53} + (3 \beta + 3) q^{55} + (\beta + 7) q^{56} + ( - 5 \beta + 10) q^{59} + ( - 2 \beta - 7) q^{61} + 3 \beta q^{62} + (2 \beta + 1) q^{64} + ( - 7 \beta - 3) q^{65} - 7 q^{67} - 2 \beta q^{68} + ( - 7 \beta - 5) q^{70} + (3 \beta - 6) q^{71} + ( - 9 \beta + 1) q^{73} + 6 q^{74} + ( - \beta - 2) q^{76} + (9 \beta - 3) q^{77} + (2 \beta - 11) q^{79} + ( - 6 \beta - 3) q^{80} + ( - 7 \beta + 2) q^{82} + ( - 8 \beta + 5) q^{83} + ( - 6 \beta - 4) q^{85} + ( - 3 \beta - 3) q^{86} + (6 \beta - 3) q^{88} + ( - 8 \beta - 1) q^{89} + ( - 5 \beta - 13) q^{91} + ( - 4 \beta + 9) q^{92} + (10 \beta - 1) q^{94} + ( - 7 \beta - 4) q^{95} + ( - 6 \beta + 6) q^{97} + ( - 6 \beta - 3) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 3 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} + 3 q^{5} + q^{7} - 4 q^{10} + 6 q^{11} - 2 q^{13} - 8 q^{14} - 3 q^{16} - 6 q^{17} - 5 q^{19} + q^{20} - 3 q^{22} - 3 q^{23} - 3 q^{25} + 11 q^{26} + 7 q^{28} - 6 q^{31} + 9 q^{32} + 8 q^{34} + 9 q^{35} + 6 q^{37} + 10 q^{38} + 5 q^{40} + 16 q^{41} + 3 q^{43} - 3 q^{44} - 11 q^{46} - 21 q^{47} + 9 q^{49} - 6 q^{50} - 9 q^{52} + 12 q^{53} + 9 q^{55} + 15 q^{56} + 15 q^{59} - 16 q^{61} + 3 q^{62} + 4 q^{64} - 13 q^{65} - 14 q^{67} - 2 q^{68} - 17 q^{70} - 9 q^{71} - 7 q^{73} + 12 q^{74} - 5 q^{76} + 3 q^{77} - 20 q^{79} - 12 q^{80} - 3 q^{82} + 2 q^{83} - 14 q^{85} - 9 q^{86} - 10 q^{89} - 31 q^{91} + 14 q^{92} + 8 q^{94} - 15 q^{95} + 6 q^{97} - 12 q^{98}+O(q^{100}) \) 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
1.61803
−0.618034
−1.61803 0 0.618034 2.61803 0 3.85410 2.23607 0 −4.23607
1.2 0.618034 0 −1.61803 0.381966 0 −2.85410 −2.23607 0 0.236068
\(n\): e.g. 2-40 or 990-1000
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.g 2
3.b odd 2 1 2523.2.a.f yes 2
29.b even 2 1 7569.2.a.o 2
87.d odd 2 1 2523.2.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2523.2.a.a 2 87.d odd 2 1
2523.2.a.f yes 2 3.b odd 2 1
7569.2.a.g 2 1.a even 1 1 trivial
7569.2.a.o 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7569))\):

\( T_{2}^{2} + T_{2} - 1 \) Copy content Toggle raw display
\( T_{5}^{2} - 3T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - T_{7} - 11 \) Copy content Toggle raw display
\( T_{19}^{2} + 5T_{19} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$11$ \( (T - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 19 \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$23$ \( T^{2} + 3T - 29 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$41$ \( T^{2} - 16T + 59 \) Copy content Toggle raw display
$43$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$47$ \( T^{2} + 21T + 109 \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$59$ \( T^{2} - 15T + 25 \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 59 \) Copy content Toggle raw display
$67$ \( (T + 7)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 9T + 9 \) Copy content Toggle raw display
$73$ \( T^{2} + 7T - 89 \) Copy content Toggle raw display
$79$ \( T^{2} + 20T + 95 \) Copy content Toggle raw display
$83$ \( T^{2} - 2T - 79 \) Copy content Toggle raw display
$89$ \( T^{2} + 10T - 55 \) Copy content Toggle raw display
$97$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
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