Properties

Label 6525.2.a.d
Level $6525$
Weight $2$
Character orbit 6525.a
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $1$
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{4} + 2 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{4} + 2 q^{7} + 3 q^{8} + 6 q^{11} - 2 q^{13} - 2 q^{14} - q^{16} - 2 q^{17} - 2 q^{19} - 6 q^{22} + 2 q^{23} + 2 q^{26} - 2 q^{28} + q^{29} + 2 q^{31} - 5 q^{32} + 2 q^{34} - 10 q^{37} + 2 q^{38} - 2 q^{41} - 8 q^{43} - 6 q^{44} - 2 q^{46} - 12 q^{47} - 3 q^{49} + 2 q^{52} - 6 q^{53} + 6 q^{56} - q^{58} + 8 q^{59} - 6 q^{61} - 2 q^{62} + 7 q^{64} - 2 q^{67} + 2 q^{68} + 12 q^{71} + 6 q^{73} + 10 q^{74} + 2 q^{76} + 12 q^{77} - 10 q^{79} + 2 q^{82} - 14 q^{83} + 8 q^{86} + 18 q^{88} - 18 q^{89} - 4 q^{91} - 2 q^{92} + 12 q^{94} - 2 q^{97} + 3 q^{98}+O(q^{100}) \) 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
0
−1.00000 0 −1.00000 0 0 2.00000 3.00000 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.d 1
3.b odd 2 1 725.2.a.a 1
5.b even 2 1 1305.2.a.f 1
15.d odd 2 1 145.2.a.a 1
15.e even 4 2 725.2.b.a 2
60.h even 2 1 2320.2.a.e 1
105.g even 2 1 7105.2.a.b 1
120.i odd 2 1 9280.2.a.l 1
120.m even 2 1 9280.2.a.o 1
435.b odd 2 1 4205.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.a 1 15.d odd 2 1
725.2.a.a 1 3.b odd 2 1
725.2.b.a 2 15.e even 4 2
1305.2.a.f 1 5.b even 2 1
2320.2.a.e 1 60.h even 2 1
4205.2.a.a 1 435.b odd 2 1
6525.2.a.d 1 1.a even 1 1 trivial
7105.2.a.b 1 105.g even 2 1
9280.2.a.l 1 120.i odd 2 1
9280.2.a.o 1 120.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6525))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{11} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T - 2 \) Copy content Toggle raw display
$29$ \( T - 1 \) Copy content Toggle raw display
$31$ \( T - 2 \) Copy content Toggle raw display
$37$ \( T + 10 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T + 8 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T - 8 \) Copy content Toggle raw display
$61$ \( T + 6 \) Copy content Toggle raw display
$67$ \( T + 2 \) Copy content Toggle raw display
$71$ \( T - 12 \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T + 14 \) Copy content Toggle raw display
$89$ \( T + 18 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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