Properties

Label 1764.4.k.l
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 84)
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.

\(f(q)\) \(=\) \( q + 6 \zeta_{6} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + 6 \zeta_{6} q^{5} + ( - 36 \zeta_{6} + 36) q^{11} + 62 q^{13} + ( - 114 \zeta_{6} + 114) q^{17} + 76 \zeta_{6} q^{19} - 24 \zeta_{6} q^{23} + ( - 89 \zeta_{6} + 89) q^{25} - 54 q^{29} + ( - 112 \zeta_{6} + 112) q^{31} + 178 \zeta_{6} q^{37} - 378 q^{41} - 172 q^{43} - 192 \zeta_{6} q^{47} + (402 \zeta_{6} - 402) q^{53} + 216 q^{55} + ( - 396 \zeta_{6} + 396) q^{59} - 254 \zeta_{6} q^{61} + 372 \zeta_{6} q^{65} + ( - 1012 \zeta_{6} + 1012) q^{67} - 840 q^{71} + (890 \zeta_{6} - 890) q^{73} - 80 \zeta_{6} q^{79} + 108 q^{83} + 684 q^{85} - 1638 \zeta_{6} q^{89} + (456 \zeta_{6} - 456) q^{95} + 1010 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{5} + 36 q^{11} + 124 q^{13} + 114 q^{17} + 76 q^{19} - 24 q^{23} + 89 q^{25} - 108 q^{29} + 112 q^{31} + 178 q^{37} - 756 q^{41} - 344 q^{43} - 192 q^{47} - 402 q^{53} + 432 q^{55} + 396 q^{59} - 254 q^{61} + 372 q^{65} + 1012 q^{67} - 1680 q^{71} - 890 q^{73} - 80 q^{79} + 216 q^{83} + 1368 q^{85} - 1638 q^{89} - 456 q^{95} + 2020 q^{97}+O(q^{100}) \) 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 3.00000 + 5.19615i 0 0 0 0 0
1549.1 0 0 0 3.00000 5.19615i 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.l 2
3.b odd 2 1 588.4.i.f 2
7.b odd 2 1 1764.4.k.f 2
7.c even 3 1 252.4.a.b 1
7.c even 3 1 inner 1764.4.k.l 2
7.d odd 6 1 1764.4.a.j 1
7.d odd 6 1 1764.4.k.f 2
21.c even 2 1 588.4.i.c 2
21.g even 6 1 588.4.a.d 1
21.g even 6 1 588.4.i.c 2
21.h odd 6 1 84.4.a.a 1
21.h odd 6 1 588.4.i.f 2
28.g odd 6 1 1008.4.a.h 1
84.j odd 6 1 2352.4.a.d 1
84.n even 6 1 336.4.a.k 1
105.o odd 6 1 2100.4.a.l 1
105.x even 12 2 2100.4.k.j 2
168.s odd 6 1 1344.4.a.q 1
168.v even 6 1 1344.4.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.a 1 21.h odd 6 1
252.4.a.b 1 7.c even 3 1
336.4.a.k 1 84.n even 6 1
588.4.a.d 1 21.g even 6 1
588.4.i.c 2 21.c even 2 1
588.4.i.c 2 21.g even 6 1
588.4.i.f 2 3.b odd 2 1
588.4.i.f 2 21.h odd 6 1
1008.4.a.h 1 28.g odd 6 1
1344.4.a.d 1 168.v even 6 1
1344.4.a.q 1 168.s odd 6 1
1764.4.a.j 1 7.d odd 6 1
1764.4.k.f 2 7.b odd 2 1
1764.4.k.f 2 7.d odd 6 1
1764.4.k.l 2 1.a even 1 1 trivial
1764.4.k.l 2 7.c even 3 1 inner
2100.4.a.l 1 105.o odd 6 1
2100.4.k.j 2 105.x even 12 2
2352.4.a.d 1 84.j odd 6 1

Hecke kernels

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

\( T_{5}^{2} - 6T_{5} + 36 \) Copy content Toggle raw display
\( T_{11}^{2} - 36T_{11} + 1296 \) Copy content Toggle raw display
\( T_{13} - 62 \) 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
$5$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 36T + 1296 \) Copy content Toggle raw display
$13$ \( (T - 62)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 114T + 12996 \) Copy content Toggle raw display
$19$ \( T^{2} - 76T + 5776 \) Copy content Toggle raw display
$23$ \( T^{2} + 24T + 576 \) Copy content Toggle raw display
$29$ \( (T + 54)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 112T + 12544 \) Copy content Toggle raw display
$37$ \( T^{2} - 178T + 31684 \) Copy content Toggle raw display
$41$ \( (T + 378)^{2} \) Copy content Toggle raw display
$43$ \( (T + 172)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 192T + 36864 \) Copy content Toggle raw display
$53$ \( T^{2} + 402T + 161604 \) Copy content Toggle raw display
$59$ \( T^{2} - 396T + 156816 \) Copy content Toggle raw display
$61$ \( T^{2} + 254T + 64516 \) Copy content Toggle raw display
$67$ \( T^{2} - 1012 T + 1024144 \) Copy content Toggle raw display
$71$ \( (T + 840)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 890T + 792100 \) Copy content Toggle raw display
$79$ \( T^{2} + 80T + 6400 \) Copy content Toggle raw display
$83$ \( (T - 108)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1638 T + 2683044 \) Copy content Toggle raw display
$97$ \( (T - 1010)^{2} \) Copy content Toggle raw display
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