Properties

Label 1764.4.a.l
Level $1764$
Weight $4$
Character orbit 1764.a
Self dual yes
Analytic conductor $104.079$
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] = [1764,4,Mod(1,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1764.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: \(104.079369250\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
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)
 
\(f(q)\) \(=\) \( q + 14 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + 14 q^{5} - 4 q^{11} - 54 q^{13} - 14 q^{17} - 92 q^{19} + 152 q^{23} + 71 q^{25} + 106 q^{29} + 144 q^{31} + 158 q^{37} - 390 q^{41} - 508 q^{43} - 528 q^{47} - 606 q^{53} - 56 q^{55} - 364 q^{59} - 678 q^{61} - 756 q^{65} + 844 q^{67} + 8 q^{71} + 422 q^{73} + 384 q^{79} - 548 q^{83} - 196 q^{85} + 1194 q^{89} - 1288 q^{95} + 1502 q^{97}+O(q^{100}) \) 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
0 0 0 14.0000 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)

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 1764.4.a.l 1
3.b odd 2 1 588.4.a.a 1
7.b odd 2 1 252.4.a.a 1
7.c even 3 2 1764.4.k.c 2
7.d odd 6 2 1764.4.k.n 2
12.b even 2 1 2352.4.a.v 1
21.c even 2 1 84.4.a.b 1
21.g even 6 2 588.4.i.a 2
21.h odd 6 2 588.4.i.h 2
28.d even 2 1 1008.4.a.d 1
84.h odd 2 1 336.4.a.e 1
105.g even 2 1 2100.4.a.g 1
105.k odd 4 2 2100.4.k.g 2
168.e odd 2 1 1344.4.a.p 1
168.i even 2 1 1344.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 21.c even 2 1
252.4.a.a 1 7.b odd 2 1
336.4.a.e 1 84.h odd 2 1
588.4.a.a 1 3.b odd 2 1
588.4.i.a 2 21.g even 6 2
588.4.i.h 2 21.h odd 6 2
1008.4.a.d 1 28.d even 2 1
1344.4.a.b 1 168.i even 2 1
1344.4.a.p 1 168.e odd 2 1
1764.4.a.l 1 1.a even 1 1 trivial
1764.4.k.c 2 7.c even 3 2
1764.4.k.n 2 7.d odd 6 2
2100.4.a.g 1 105.g even 2 1
2100.4.k.g 2 105.k odd 4 2
2352.4.a.v 1 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1764))\):

\( T_{5} - 14 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13} + 54 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 14 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T + 54 \) Copy content Toggle raw display
$17$ \( T + 14 \) Copy content Toggle raw display
$19$ \( T + 92 \) Copy content Toggle raw display
$23$ \( T - 152 \) Copy content Toggle raw display
$29$ \( T - 106 \) Copy content Toggle raw display
$31$ \( T - 144 \) Copy content Toggle raw display
$37$ \( T - 158 \) Copy content Toggle raw display
$41$ \( T + 390 \) Copy content Toggle raw display
$43$ \( T + 508 \) Copy content Toggle raw display
$47$ \( T + 528 \) Copy content Toggle raw display
$53$ \( T + 606 \) Copy content Toggle raw display
$59$ \( T + 364 \) Copy content Toggle raw display
$61$ \( T + 678 \) Copy content Toggle raw display
$67$ \( T - 844 \) Copy content Toggle raw display
$71$ \( T - 8 \) Copy content Toggle raw display
$73$ \( T - 422 \) Copy content Toggle raw display
$79$ \( T - 384 \) Copy content Toggle raw display
$83$ \( T + 548 \) Copy content Toggle raw display
$89$ \( T - 1194 \) Copy content Toggle raw display
$97$ \( T - 1502 \) Copy content Toggle raw display
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