Properties

Label 6525.2.a.be
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: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{7} + (\beta_{2} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{4} + ( - \beta_{2} - \beta_1 - 1) q^{7} + (\beta_{2} + 1) q^{8} + ( - \beta_{2} + \beta_1 - 1) q^{11} + 2 \beta_1 q^{13} + ( - \beta_{2} - 3 \beta_1 - 1) q^{14} + ( - 2 \beta_{2} - 1) q^{16} + (3 \beta_{2} - \beta_1 - 1) q^{17} + ( - \beta_{2} - \beta_1 - 3) q^{19} + (\beta_{2} - \beta_1 + 3) q^{22} + ( - \beta_{2} + \beta_1 + 5) q^{23} + (2 \beta_{2} + 2 \beta_1 + 4) q^{26} + ( - \beta_{2} - 3 \beta_1 - 3) q^{28} + q^{29} + (\beta_{2} + \beta_1 - 5) q^{31} + ( - 2 \beta_{2} - 3 \beta_1) q^{32} + ( - \beta_{2} + \beta_1 - 5) q^{34} + (3 \beta_{2} - \beta_1 + 3) q^{37} + ( - \beta_{2} - 5 \beta_1 - 1) q^{38} + (2 \beta_{2} - 4 \beta_1 + 2) q^{41} + ( - 5 \beta_{2} - 5 \beta_1 + 1) q^{43} + (\beta_{2} + \beta_1 - 1) q^{44} + (\beta_{2} + 5 \beta_1 + 3) q^{46} + ( - \beta_{2} - \beta_1 + 5) q^{47} + (2 \beta_{2} + 4 \beta_1 - 3) q^{49} + (2 \beta_{2} + 4 \beta_1 + 2) q^{52} + ( - 2 \beta_{2} + 2) q^{53} + ( - \beta_{2} - \beta_1 - 3) q^{56} + \beta_1 q^{58} + ( - 2 \beta_{2} + 4 \beta_1 - 4) q^{59} - 6 \beta_1 q^{61} + (\beta_{2} - 3 \beta_1 + 1) q^{62} + (\beta_{2} - 5 \beta_1 - 2) q^{64} + (\beta_{2} - \beta_1 - 9) q^{67} + ( - 5 \beta_{2} - 3 \beta_1 + 5) q^{68} + (2 \beta_{2} - 4 \beta_1 - 8) q^{71} + (7 \beta_{2} + \beta_1 + 5) q^{73} + ( - \beta_{2} + 5 \beta_1 - 5) q^{74} + ( - 3 \beta_{2} - 5 \beta_1 - 3) q^{76} + ( - 2 \beta_1 + 2) q^{77} + ( - 5 \beta_{2} - 3 \beta_1 - 1) q^{79} + ( - 4 \beta_{2} - 10) q^{82} + ( - 3 \beta_{2} - 3 \beta_1 + 5) q^{83} + ( - 5 \beta_{2} - 9 \beta_1 - 5) q^{86} + ( - \beta_{2} + 3 \beta_1 - 5) q^{88} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{89} + ( - 2 \beta_{2} - 6 \beta_1 - 2) q^{91} + (7 \beta_{2} + 7 \beta_1 - 1) q^{92} + ( - \beta_{2} + 3 \beta_1 - 1) q^{94} + ( - 3 \beta_{2} - 5 \beta_1 - 1) q^{97} + (4 \beta_{2} + 3 \beta_1 + 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + q^{4} - 4 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + q^{4} - 4 q^{7} + 3 q^{8} - 2 q^{11} + 2 q^{13} - 6 q^{14} - 3 q^{16} - 4 q^{17} - 10 q^{19} + 8 q^{22} + 16 q^{23} + 14 q^{26} - 12 q^{28} + 3 q^{29} - 14 q^{31} - 3 q^{32} - 14 q^{34} + 8 q^{37} - 8 q^{38} + 2 q^{41} - 2 q^{43} - 2 q^{44} + 14 q^{46} + 14 q^{47} - 5 q^{49} + 10 q^{52} + 6 q^{53} - 10 q^{56} + q^{58} - 8 q^{59} - 6 q^{61} - 11 q^{64} - 28 q^{67} + 12 q^{68} - 28 q^{71} + 16 q^{73} - 10 q^{74} - 14 q^{76} + 4 q^{77} - 6 q^{79} - 30 q^{82} + 12 q^{83} - 24 q^{86} - 12 q^{88} + 10 q^{89} - 12 q^{91} + 4 q^{92} - 8 q^{97} + 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
−1.48119
0.311108
2.17009
−1.48119 0 0.193937 0 0 −1.19394 2.67513 0 0
1.2 0.311108 0 −1.90321 0 0 0.903212 −1.21432 0 0
1.3 2.17009 0 2.70928 0 0 −3.70928 1.53919 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.be 3
3.b odd 2 1 725.2.a.e 3
5.b even 2 1 1305.2.a.p 3
15.d odd 2 1 145.2.a.c 3
15.e even 4 2 725.2.b.e 6
60.h even 2 1 2320.2.a.n 3
105.g even 2 1 7105.2.a.o 3
120.i odd 2 1 9280.2.a.bj 3
120.m even 2 1 9280.2.a.br 3
435.b odd 2 1 4205.2.a.f 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.c 3 15.d odd 2 1
725.2.a.e 3 3.b odd 2 1
725.2.b.e 6 15.e even 4 2
1305.2.a.p 3 5.b even 2 1
2320.2.a.n 3 60.h even 2 1
4205.2.a.f 3 435.b odd 2 1
6525.2.a.be 3 1.a even 1 1 trivial
7105.2.a.o 3 105.g even 2 1
9280.2.a.bj 3 120.i odd 2 1
9280.2.a.br 3 120.m even 2 1

Hecke kernels

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

\( T_{2}^{3} - T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{3} + 4T_{7}^{2} - 4 \) Copy content Toggle raw display
\( T_{11}^{3} + 2T_{11}^{2} - 8T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 3T + 1 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 4T^{2} - 4 \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{3} - 2 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$17$ \( T^{3} + 4 T^{2} + \cdots - 68 \) Copy content Toggle raw display
$19$ \( T^{3} + 10 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$23$ \( T^{3} - 16 T^{2} + \cdots - 92 \) Copy content Toggle raw display
$29$ \( (T - 1)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} + 14 T^{2} + \cdots + 76 \) Copy content Toggle raw display
$37$ \( T^{3} - 8 T^{2} + \cdots + 92 \) Copy content Toggle raw display
$41$ \( T^{3} - 2 T^{2} + \cdots - 232 \) Copy content Toggle raw display
$43$ \( T^{3} + 2 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$47$ \( T^{3} - 14 T^{2} + \cdots - 76 \) Copy content Toggle raw display
$53$ \( T^{3} - 6 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$59$ \( T^{3} + 8 T^{2} + \cdots + 80 \) Copy content Toggle raw display
$61$ \( T^{3} + 6 T^{2} + \cdots - 216 \) Copy content Toggle raw display
$67$ \( T^{3} + 28 T^{2} + \cdots + 716 \) Copy content Toggle raw display
$71$ \( T^{3} + 28 T^{2} + \cdots - 272 \) Copy content Toggle raw display
$73$ \( T^{3} - 16 T^{2} + \cdots + 1700 \) Copy content Toggle raw display
$79$ \( T^{3} + 6 T^{2} + \cdots - 460 \) Copy content Toggle raw display
$83$ \( T^{3} - 12T^{2} + 148 \) Copy content Toggle raw display
$89$ \( T^{3} - 10 T^{2} + \cdots + 40 \) Copy content Toggle raw display
$97$ \( T^{3} + 8 T^{2} + \cdots + 76 \) Copy content Toggle raw display
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