Properties

Label 2535.2.a.l
Level $2535$
Weight $2$
Character orbit 2535.a
Self dual yes
Analytic conductor $20.242$
Analytic rank $0$
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] = [2535,2,Mod(1,2535)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2535, 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("2535.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2535 = 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2535.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: \(20.2420769124\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
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)
 
\(f(q)\) \(=\) \( q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} + 2 q^{7} - 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} + 2 q^{7} - 3 q^{8} + q^{9} + q^{10} - q^{12} + 2 q^{14} + q^{15} - q^{16} - 2 q^{17} + q^{18} + 2 q^{19} - q^{20} + 2 q^{21} + 8 q^{23} - 3 q^{24} + q^{25} + q^{27} - 2 q^{28} + 2 q^{29} + q^{30} - 2 q^{31} + 5 q^{32} - 2 q^{34} + 2 q^{35} - q^{36} + 8 q^{37} + 2 q^{38} - 3 q^{40} - 2 q^{41} + 2 q^{42} + 4 q^{43} + q^{45} + 8 q^{46} + 4 q^{47} - q^{48} - 3 q^{49} + q^{50} - 2 q^{51} - 6 q^{53} + q^{54} - 6 q^{56} + 2 q^{57} + 2 q^{58} + 12 q^{59} - q^{60} + 10 q^{61} - 2 q^{62} + 2 q^{63} + 7 q^{64} - 6 q^{67} + 2 q^{68} + 8 q^{69} + 2 q^{70} + 8 q^{71} - 3 q^{72} - 16 q^{73} + 8 q^{74} + q^{75} - 2 q^{76} - 8 q^{79} - q^{80} + q^{81} - 2 q^{82} + 12 q^{83} - 2 q^{84} - 2 q^{85} + 4 q^{86} + 2 q^{87} - 6 q^{89} + q^{90} - 8 q^{92} - 2 q^{93} + 4 q^{94} + 2 q^{95} + 5 q^{96} + 16 q^{97} - 3 q^{98}+O(q^{100}) \) 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 1.00000 −1.00000 1.00000 1.00000 2.00000 −3.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(-1\)

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 2535.2.a.l 1
3.b odd 2 1 7605.2.a.d 1
13.b even 2 1 2535.2.a.e 1
13.d odd 4 2 195.2.b.b 2
39.d odd 2 1 7605.2.a.p 1
39.f even 4 2 585.2.b.a 2
52.f even 4 2 3120.2.g.a 2
65.f even 4 2 975.2.h.a 2
65.g odd 4 2 975.2.b.b 2
65.k even 4 2 975.2.h.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.b.b 2 13.d odd 4 2
585.2.b.a 2 39.f even 4 2
975.2.b.b 2 65.g odd 4 2
975.2.h.a 2 65.f even 4 2
975.2.h.d 2 65.k even 4 2
2535.2.a.e 1 13.b even 2 1
2535.2.a.l 1 1.a even 1 1 trivial
3120.2.g.a 2 52.f even 4 2
7605.2.a.d 1 3.b odd 2 1
7605.2.a.p 1 39.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2535))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 1 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T - 2 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T - 8 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T + 2 \) Copy content Toggle raw display
$37$ \( T - 8 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 4 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T - 10 \) Copy content Toggle raw display
$67$ \( T + 6 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T + 16 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 16 \) Copy content Toggle raw display
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