Properties

Label 6525.2.a.bf
Level $6525$
Weight $2$
Character orbit 6525.a
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $3$
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] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.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: \(52.1023873189\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.469.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{2} - \beta_1 + 2) q^{7} + (\beta_{2} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + (\beta_{2} - \beta_1 + 2) q^{7} + (\beta_{2} + \beta_1) q^{8} - 3 q^{11} + ( - \beta_{2} - \beta_1 - 2) q^{13} + (3 \beta_1 - 4) q^{14} + \beta_1 q^{16} + ( - \beta_{2} + \beta_1) q^{17} + ( - 2 \beta_{2} - 2) q^{19} - 3 \beta_1 q^{22} + ( - \beta_{2} - 3 \beta_1 + 1) q^{23} + ( - 2 \beta_{2} - 3 \beta_1 - 4) q^{26} + (\beta_{2} - 2 \beta_1 + 8) q^{28} + q^{29} + ( - \beta_{2} - 2 \beta_1 + 4) q^{32} + ( - \beta_1 + 4) q^{34} + (\beta_{2} + 3 \beta_1 - 5) q^{37} + ( - 2 \beta_{2} - 4 \beta_1) q^{38} + ( - \beta_{2} + \beta_1 - 5) q^{41} + ( - \beta_{2} + 3 \beta_1 + 3) q^{43} + ( - 3 \beta_{2} - 6) q^{44} + ( - 4 \beta_{2} - 12) q^{46} + (3 \beta_{2} - \beta_1 + 2) q^{47} + (\beta_{2} - 5 \beta_1 + 5) q^{49} + ( - 3 \beta_{2} - 4 \beta_1 - 8) q^{52} + ( - 3 \beta_{2} - 3 \beta_1 - 1) q^{53} + ( - \beta_{2} + 3 \beta_1) q^{56} + \beta_1 q^{58} + (2 \beta_1 - 8) q^{59} + (2 \beta_{2} + 4) q^{61} + ( - 3 \beta_{2} + \beta_1 - 8) q^{64} + ( - \beta_{2} - 3 \beta_1 + 10) q^{67} + (\beta_{2} + 2 \beta_1 - 4) q^{68} + ( - 6 \beta_1 + 2) q^{71} + ( - 5 \beta_{2} + \beta_1 - 3) q^{73} + (4 \beta_{2} - 4 \beta_1 + 12) q^{74} + ( - 2 \beta_{2} - 2 \beta_1 - 12) q^{76} + ( - 3 \beta_{2} + 3 \beta_1 - 6) q^{77} + (4 \beta_{2} + 2 \beta_1) q^{79} + ( - 6 \beta_1 + 4) q^{82} + (\beta_{2} + \beta_1 - 5) q^{83} + (2 \beta_{2} + 2 \beta_1 + 12) q^{86} + ( - 3 \beta_{2} - 3 \beta_1) q^{88} + (3 \beta_{2} + 3 \beta_1 - 10) q^{89} + ( - \beta_{2} - \beta_1 - 4) q^{91} + ( - 2 \beta_{2} - 10 \beta_1 - 2) q^{92} + (2 \beta_{2} + 5 \beta_1 - 4) q^{94} + (\beta_{2} + 5 \beta_1 - 1) q^{97} + ( - 4 \beta_{2} + 6 \beta_1 - 20) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 5 q^{4} + 4 q^{7} - 9 q^{11} - 6 q^{13} - 9 q^{14} + q^{16} + 2 q^{17} - 4 q^{19} - 3 q^{22} + q^{23} - 13 q^{26} + 21 q^{28} + 3 q^{29} + 11 q^{32} + 11 q^{34} - 13 q^{37} - 2 q^{38} - 13 q^{41} + 13 q^{43} - 15 q^{44} - 32 q^{46} + 2 q^{47} + 9 q^{49} - 25 q^{52} - 3 q^{53} + 4 q^{56} + q^{58} - 22 q^{59} + 10 q^{61} - 20 q^{64} + 28 q^{67} - 11 q^{68} - 3 q^{73} + 28 q^{74} - 36 q^{76} - 12 q^{77} - 2 q^{79} + 6 q^{82} - 15 q^{83} + 36 q^{86} - 30 q^{89} - 12 q^{91} - 14 q^{92} - 9 q^{94} + q^{97} - 50 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) 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
−2.16425
0.772866
2.39138
−2.16425 0 2.68397 0 0 4.84822 −1.48028 0 0
1.2 0.772866 0 −1.40268 0 0 −2.17554 −2.62981 0 0
1.3 2.39138 0 3.71871 0 0 1.32733 4.11009 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(29\) \(-1\)

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 6525.2.a.bf 3
3.b odd 2 1 2175.2.a.u 3
5.b even 2 1 1305.2.a.q 3
15.d odd 2 1 435.2.a.i 3
15.e even 4 2 2175.2.c.m 6
60.h even 2 1 6960.2.a.cl 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.a.i 3 15.d odd 2 1
1305.2.a.q 3 5.b even 2 1
2175.2.a.u 3 3.b odd 2 1
2175.2.c.m 6 15.e even 4 2
6525.2.a.bf 3 1.a even 1 1 trivial
6960.2.a.cl 3 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6525))\):

\( T_{2}^{3} - T_{2}^{2} - 5T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{3} - 4T_{7}^{2} - 7T_{7} + 14 \) Copy content Toggle raw display
\( T_{11} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 5T + 4 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 4 T^{2} + \cdots + 14 \) Copy content Toggle raw display
$11$ \( (T + 3)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + 6T^{2} - T - 2 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$19$ \( T^{3} + 4 T^{2} + \cdots - 88 \) Copy content Toggle raw display
$23$ \( T^{3} - T^{2} + \cdots + 112 \) Copy content Toggle raw display
$29$ \( (T - 1)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} \) Copy content Toggle raw display
$37$ \( T^{3} + 13T^{2} - 256 \) Copy content Toggle raw display
$41$ \( T^{3} + 13 T^{2} + \cdots + 28 \) Copy content Toggle raw display
$43$ \( T^{3} - 13 T^{2} + \cdots + 308 \) Copy content Toggle raw display
$47$ \( T^{3} - 2 T^{2} + \cdots + 266 \) Copy content Toggle raw display
$53$ \( T^{3} + 3 T^{2} + \cdots + 316 \) Copy content Toggle raw display
$59$ \( T^{3} + 22 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$61$ \( T^{3} - 10 T^{2} + \cdots + 112 \) Copy content Toggle raw display
$67$ \( T^{3} - 28 T^{2} + \cdots - 194 \) Copy content Toggle raw display
$71$ \( T^{3} - 192T - 488 \) Copy content Toggle raw display
$73$ \( T^{3} + 3 T^{2} + \cdots - 1168 \) Copy content Toggle raw display
$79$ \( T^{3} + 2 T^{2} + \cdots - 224 \) Copy content Toggle raw display
$83$ \( T^{3} + 15 T^{2} + \cdots + 44 \) Copy content Toggle raw display
$89$ \( T^{3} + 30 T^{2} + \cdots - 602 \) Copy content Toggle raw display
$97$ \( T^{3} - T^{2} + \cdots - 76 \) Copy content Toggle raw display
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