Properties

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

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Show commands: Magma / PariGP / SageMath

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.

\(f(q)\) \(=\) \( q - 14 \zeta_{6} q^{5} +O(q^{10}) \) Copy content Toggle raw display \( q - 14 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{11} - 54 q^{13} + ( - 14 \zeta_{6} + 14) q^{17} + 92 \zeta_{6} q^{19} - 152 \zeta_{6} q^{23} + (71 \zeta_{6} - 71) q^{25} + 106 q^{29} + (144 \zeta_{6} - 144) q^{31} - 158 \zeta_{6} q^{37} - 390 q^{41} - 508 q^{43} + 528 \zeta_{6} q^{47} + ( - 606 \zeta_{6} + 606) q^{53} - 56 q^{55} + ( - 364 \zeta_{6} + 364) q^{59} + 678 \zeta_{6} q^{61} + 756 \zeta_{6} q^{65} + (844 \zeta_{6} - 844) q^{67} + 8 q^{71} + (422 \zeta_{6} - 422) q^{73} - 384 \zeta_{6} q^{79} - 548 q^{83} - 196 q^{85} - 1194 \zeta_{6} q^{89} + ( - 1288 \zeta_{6} + 1288) q^{95} + 1502 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{5} + 4 q^{11} - 108 q^{13} + 14 q^{17} + 92 q^{19} - 152 q^{23} - 71 q^{25} + 212 q^{29} - 144 q^{31} - 158 q^{37} - 780 q^{41} - 1016 q^{43} + 528 q^{47} + 606 q^{53} - 112 q^{55} + 364 q^{59} + 678 q^{61} + 756 q^{65} - 844 q^{67} + 16 q^{71} - 422 q^{73} - 384 q^{79} - 1096 q^{83} - 392 q^{85} - 1194 q^{89} + 1288 q^{95} + 3004 q^{97}+O(q^{100}) \) 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 −7.00000 12.1244i 0 0 0 0 0
1549.1 0 0 0 −7.00000 + 12.1244i 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.c 2
3.b odd 2 1 588.4.i.h 2
7.b odd 2 1 1764.4.k.n 2
7.c even 3 1 1764.4.a.l 1
7.c even 3 1 inner 1764.4.k.c 2
7.d odd 6 1 252.4.a.a 1
7.d odd 6 1 1764.4.k.n 2
21.c even 2 1 588.4.i.a 2
21.g even 6 1 84.4.a.b 1
21.g even 6 1 588.4.i.a 2
21.h odd 6 1 588.4.a.a 1
21.h odd 6 1 588.4.i.h 2
28.f even 6 1 1008.4.a.d 1
84.j odd 6 1 336.4.a.e 1
84.n even 6 1 2352.4.a.v 1
105.p even 6 1 2100.4.a.g 1
105.w odd 12 2 2100.4.k.g 2
168.ba even 6 1 1344.4.a.b 1
168.be odd 6 1 1344.4.a.p 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.b 1 21.g even 6 1
252.4.a.a 1 7.d odd 6 1
336.4.a.e 1 84.j odd 6 1
588.4.a.a 1 21.h odd 6 1
588.4.i.a 2 21.c even 2 1
588.4.i.a 2 21.g even 6 1
588.4.i.h 2 3.b odd 2 1
588.4.i.h 2 21.h odd 6 1
1008.4.a.d 1 28.f even 6 1
1344.4.a.b 1 168.ba even 6 1
1344.4.a.p 1 168.be odd 6 1
1764.4.a.l 1 7.c even 3 1
1764.4.k.c 2 1.a even 1 1 trivial
1764.4.k.c 2 7.c even 3 1 inner
1764.4.k.n 2 7.b odd 2 1
1764.4.k.n 2 7.d odd 6 1
2100.4.a.g 1 105.p even 6 1
2100.4.k.g 2 105.w odd 12 2
2352.4.a.v 1 84.n even 6 1

Hecke kernels

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

\( T_{5}^{2} + 14T_{5} + 196 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} + 16 \) Copy content Toggle raw display
\( T_{13} + 54 \) 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
$5$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$13$ \( (T + 54)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$19$ \( T^{2} - 92T + 8464 \) Copy content Toggle raw display
$23$ \( T^{2} + 152T + 23104 \) Copy content Toggle raw display
$29$ \( (T - 106)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 144T + 20736 \) Copy content Toggle raw display
$37$ \( T^{2} + 158T + 24964 \) Copy content Toggle raw display
$41$ \( (T + 390)^{2} \) Copy content Toggle raw display
$43$ \( (T + 508)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 528T + 278784 \) Copy content Toggle raw display
$53$ \( T^{2} - 606T + 367236 \) Copy content Toggle raw display
$59$ \( T^{2} - 364T + 132496 \) Copy content Toggle raw display
$61$ \( T^{2} - 678T + 459684 \) Copy content Toggle raw display
$67$ \( T^{2} + 844T + 712336 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 422T + 178084 \) Copy content Toggle raw display
$79$ \( T^{2} + 384T + 147456 \) Copy content Toggle raw display
$83$ \( (T + 548)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 1194 T + 1425636 \) Copy content Toggle raw display
$97$ \( (T - 1502)^{2} \) Copy content Toggle raw display
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