Properties

Label 1764.4.k.d
Level $1764$
Weight $4$
Character orbit 1764.k
Analytic conductor $104.079$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,4,Mod(361,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.361");
 
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.k (of order \(3\), degree \(2\), not minimal)

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: no
Analytic conductor: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 \zeta_{6} q^{5} +O(q^{10}) \) Copy content Toggle raw display \( q - 8 \zeta_{6} q^{5} + (40 \zeta_{6} - 40) q^{11} - 12 q^{13} + (58 \zeta_{6} - 58) q^{17} - 26 \zeta_{6} q^{19} - 64 \zeta_{6} q^{23} + ( - 61 \zeta_{6} + 61) q^{25} + 62 q^{29} + (252 \zeta_{6} - 252) q^{31} - 26 \zeta_{6} q^{37} - 6 q^{41} + 416 q^{43} - 396 \zeta_{6} q^{47} + (450 \zeta_{6} - 450) q^{53} + 320 q^{55} + ( - 274 \zeta_{6} + 274) q^{59} + 576 \zeta_{6} q^{61} + 96 \zeta_{6} q^{65} + ( - 476 \zeta_{6} + 476) q^{67} + 448 q^{71} + ( - 158 \zeta_{6} + 158) q^{73} + 936 \zeta_{6} q^{79} - 530 q^{83} + 464 q^{85} - 390 \zeta_{6} q^{89} + (208 \zeta_{6} - 208) q^{95} + 214 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{5} - 40 q^{11} - 24 q^{13} - 58 q^{17} - 26 q^{19} - 64 q^{23} + 61 q^{25} + 124 q^{29} - 252 q^{31} - 26 q^{37} - 12 q^{41} + 832 q^{43} - 396 q^{47} - 450 q^{53} + 640 q^{55} + 274 q^{59} + 576 q^{61} + 96 q^{65} + 476 q^{67} + 896 q^{71} + 158 q^{73} + 936 q^{79} - 1060 q^{83} + 928 q^{85} - 390 q^{89} - 208 q^{95} + 428 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 −4.00000 6.92820i 0 0 0 0 0
1549.1 0 0 0 −4.00000 + 6.92820i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.k.d 2
3.b odd 2 1 196.4.e.f 2
7.b odd 2 1 1764.4.k.m 2
7.c even 3 1 252.4.a.d 1
7.c even 3 1 inner 1764.4.k.d 2
7.d odd 6 1 1764.4.a.c 1
7.d odd 6 1 1764.4.k.m 2
21.c even 2 1 196.4.e.a 2
21.g even 6 1 196.4.a.d 1
21.g even 6 1 196.4.e.a 2
21.h odd 6 1 28.4.a.a 1
21.h odd 6 1 196.4.e.f 2
28.g odd 6 1 1008.4.a.o 1
84.j odd 6 1 784.4.a.a 1
84.n even 6 1 112.4.a.g 1
105.o odd 6 1 700.4.a.n 1
105.x even 12 2 700.4.e.a 2
168.s odd 6 1 448.4.a.p 1
168.v even 6 1 448.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 21.h odd 6 1
112.4.a.g 1 84.n even 6 1
196.4.a.d 1 21.g even 6 1
196.4.e.a 2 21.c even 2 1
196.4.e.a 2 21.g even 6 1
196.4.e.f 2 3.b odd 2 1
196.4.e.f 2 21.h odd 6 1
252.4.a.d 1 7.c even 3 1
448.4.a.a 1 168.v even 6 1
448.4.a.p 1 168.s odd 6 1
700.4.a.n 1 105.o odd 6 1
700.4.e.a 2 105.x even 12 2
784.4.a.a 1 84.j odd 6 1
1008.4.a.o 1 28.g odd 6 1
1764.4.a.c 1 7.d odd 6 1
1764.4.k.d 2 1.a even 1 1 trivial
1764.4.k.d 2 7.c even 3 1 inner
1764.4.k.m 2 7.b odd 2 1
1764.4.k.m 2 7.d odd 6 1

Hecke kernels

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

\( T_{5}^{2} + 8T_{5} + 64 \) Copy content Toggle raw display
\( T_{11}^{2} + 40T_{11} + 1600 \) Copy content Toggle raw display
\( T_{13} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 40T + 1600 \) Copy content Toggle raw display
$13$ \( (T + 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 58T + 3364 \) Copy content Toggle raw display
$19$ \( T^{2} + 26T + 676 \) Copy content Toggle raw display
$23$ \( T^{2} + 64T + 4096 \) Copy content Toggle raw display
$29$ \( (T - 62)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 252T + 63504 \) Copy content Toggle raw display
$37$ \( T^{2} + 26T + 676 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( (T - 416)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 396T + 156816 \) Copy content Toggle raw display
$53$ \( T^{2} + 450T + 202500 \) Copy content Toggle raw display
$59$ \( T^{2} - 274T + 75076 \) Copy content Toggle raw display
$61$ \( T^{2} - 576T + 331776 \) Copy content Toggle raw display
$67$ \( T^{2} - 476T + 226576 \) Copy content Toggle raw display
$71$ \( (T - 448)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 158T + 24964 \) Copy content Toggle raw display
$79$ \( T^{2} - 936T + 876096 \) Copy content Toggle raw display
$83$ \( (T + 530)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 390T + 152100 \) Copy content Toggle raw display
$97$ \( (T - 214)^{2} \) Copy content Toggle raw display
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