Properties

Label 6525.2.a.bh
Level $6525$
Weight $2$
Character orbit 6525.a
Self dual yes
Analytic conductor $52.102$
Analytic rank $0$
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: \(0\)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{2} - \beta_1 + 1) q^{7} + ( - 3 \beta_1 + 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{2} - \beta_1 + 1) q^{7} + ( - 3 \beta_1 + 4) q^{8} + (\beta_{2} + \beta_1 - 3) q^{11} + ( - 2 \beta_{2} + 2) q^{13} + (\beta_{2} - 3 \beta_1 + 5) q^{14} + (2 \beta_{2} - 4 \beta_1 + 3) q^{16} + ( - \beta_{2} + 3 \beta_1 - 1) q^{17} + ( - \beta_{2} - 3 \beta_1 + 1) q^{19} + ( - 3 \beta_{2} + \beta_1 - 1) q^{22} + ( - \beta_{2} - \beta_1 + 5) q^{23} + (2 \beta_{2} + 2 \beta_1 - 4) q^{26} + (3 \beta_{2} - 5 \beta_1 + 9) q^{28} - q^{29} + (\beta_{2} + 3 \beta_1 + 3) q^{31} + (3 \beta_{2} - 4 \beta_1 + 5) q^{32} + ( - \beta_{2} + 7 \beta_1 - 7) q^{34} + (3 \beta_{2} - \beta_1 - 1) q^{37} + (\beta_{2} - 5 \beta_1 + 1) q^{38} + ( - 2 \beta_1 + 4) q^{41} + (\beta_{2} + \beta_1 + 3) q^{43} + ( - 3 \beta_{2} + 3 \beta_1 - 5) q^{44} + (5 \beta_{2} - \beta_1 + 3) q^{46} + ( - 3 \beta_{2} - 3 \beta_1 + 7) q^{47} + (2 \beta_{2} - 4 \beta_1 + 1) q^{49} + (2 \beta_1 - 4) q^{52} + ( - 2 \beta_1 + 4) q^{53} + (7 \beta_{2} - 7 \beta_1 + 13) q^{56} + ( - \beta_{2} - 1) q^{58} + ( - 2 \beta_{2} - 4 \beta_1) q^{59} + (2 \beta_{2} + 2) q^{61} + (3 \beta_{2} + 5 \beta_1 + 3) q^{62} + (\beta_{2} - 3 \beta_1 + 12) q^{64} + (\beta_{2} + \beta_1 + 3) q^{67} + ( - 5 \beta_{2} + 9 \beta_1 - 15) q^{68} + ( - 2 \beta_{2} - 8) q^{71} + (7 \beta_{2} + \beta_1 + 1) q^{73} + ( - \beta_{2} - 5 \beta_1 + 9) q^{74} + (3 \beta_{2} - 5 \beta_1 + 7) q^{76} + ( - 4 \beta_{2} + 2 \beta_1 - 2) q^{77} + (\beta_{2} + 5 \beta_1 + 1) q^{79} + (4 \beta_{2} - 4 \beta_1 + 6) q^{82} + (3 \beta_{2} + \beta_1 - 1) q^{83} + (3 \beta_{2} + \beta_1 + 5) q^{86} + (\beta_{2} + 7 \beta_1 - 15) q^{88} + ( - 2 \beta_{2} + 2 \beta_1 - 8) q^{89} + (2 \beta_{2} + 2 \beta_1 - 6) q^{91} + (5 \beta_{2} - 5 \beta_1 + 9) q^{92} + (7 \beta_{2} - 3 \beta_1 + 1) q^{94} + ( - 3 \beta_{2} + 3 \beta_1 + 11) q^{97} + (\beta_{2} - 10 \beta_1 + 11) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 5 q^{4} + 2 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 5 q^{4} + 2 q^{7} + 9 q^{8} - 8 q^{11} + 6 q^{13} + 12 q^{14} + 5 q^{16} - 2 q^{22} + 14 q^{23} - 10 q^{26} + 22 q^{28} - 3 q^{29} + 12 q^{31} + 11 q^{32} - 14 q^{34} - 4 q^{37} - 2 q^{38} + 10 q^{41} + 10 q^{43} - 12 q^{44} + 8 q^{46} + 18 q^{47} - q^{49} - 10 q^{52} + 10 q^{53} + 32 q^{56} - 3 q^{58} - 4 q^{59} + 6 q^{61} + 14 q^{62} + 33 q^{64} + 10 q^{67} - 36 q^{68} - 24 q^{71} + 4 q^{73} + 22 q^{74} + 16 q^{76} - 4 q^{77} + 8 q^{79} + 14 q^{82} - 2 q^{83} + 16 q^{86} - 38 q^{88} - 22 q^{89} - 16 q^{91} + 22 q^{92} + 36 q^{97} + 23 q^{98}+O(q^{100}) \) 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
0.311108
2.17009
−1.48119
−1.21432 0 −0.525428 0 0 −1.52543 3.06668 0 0
1.2 1.53919 0 0.369102 0 0 −0.630898 −2.51026 0 0
1.3 2.67513 0 5.15633 0 0 4.15633 8.44358 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.bh 3
3.b odd 2 1 725.2.a.d 3
5.b even 2 1 1305.2.a.o 3
15.d odd 2 1 145.2.a.d 3
15.e even 4 2 725.2.b.d 6
60.h even 2 1 2320.2.a.s 3
105.g even 2 1 7105.2.a.p 3
120.i odd 2 1 9280.2.a.bu 3
120.m even 2 1 9280.2.a.bm 3
435.b odd 2 1 4205.2.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.d 3 15.d odd 2 1
725.2.a.d 3 3.b odd 2 1
725.2.b.d 6 15.e even 4 2
1305.2.a.o 3 5.b even 2 1
2320.2.a.s 3 60.h even 2 1
4205.2.a.e 3 435.b odd 2 1
6525.2.a.bh 3 1.a even 1 1 trivial
7105.2.a.p 3 105.g even 2 1
9280.2.a.bm 3 120.m even 2 1
9280.2.a.bu 3 120.i odd 2 1

Hecke kernels

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

\( T_{2}^{3} - 3T_{2}^{2} - T_{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{3} - 2T_{7}^{2} - 8T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{3} + 8T_{11}^{2} + 16T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 3T^{2} - T + 5 \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 2 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$11$ \( T^{3} + 8 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{3} - 6 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$17$ \( T^{3} - 40T + 76 \) Copy content Toggle raw display
$19$ \( T^{3} - 28T + 52 \) Copy content Toggle raw display
$23$ \( T^{3} - 14 T^{2} + \cdots - 76 \) Copy content Toggle raw display
$29$ \( (T + 1)^{3} \) Copy content Toggle raw display
$31$ \( T^{3} - 12 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$37$ \( T^{3} + 4 T^{2} + \cdots - 68 \) Copy content Toggle raw display
$41$ \( T^{3} - 10 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$43$ \( T^{3} - 10 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$47$ \( T^{3} - 18 T^{2} + \cdots + 92 \) Copy content Toggle raw display
$53$ \( T^{3} - 10 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$59$ \( T^{3} + 4 T^{2} + \cdots + 80 \) Copy content Toggle raw display
$61$ \( T^{3} - 6 T^{2} + \cdots + 40 \) Copy content Toggle raw display
$67$ \( T^{3} - 10 T^{2} + \cdots - 20 \) Copy content Toggle raw display
$71$ \( T^{3} + 24 T^{2} + \cdots + 368 \) Copy content Toggle raw display
$73$ \( T^{3} - 4 T^{2} + \cdots + 1108 \) Copy content Toggle raw display
$79$ \( T^{3} - 8 T^{2} + \cdots + 20 \) Copy content Toggle raw display
$83$ \( T^{3} + 2 T^{2} + \cdots + 52 \) Copy content Toggle raw display
$89$ \( T^{3} + 22 T^{2} + \cdots + 200 \) Copy content Toggle raw display
$97$ \( T^{3} - 36 T^{2} + \cdots - 452 \) Copy content Toggle raw display
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