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Properties

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

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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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 252)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{3} - \beta_1) q^{11} + 26 q^{13} + (5 \beta_{3} + 5 \beta_1) q^{17} + ( - 68 \beta_{2} - 68) q^{19} + 3 \beta_1 q^{23} + 127 \beta_{2} q^{25} - 16 \beta_{3} q^{29} + 212 \beta_{2} q^{31}+ \cdots - 1330 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 104 q^{13} - 136 q^{19} - 254 q^{25} - 424 q^{31} - 436 q^{37} + 1040 q^{43} + 1008 q^{55} + 644 q^{61} - 712 q^{67} + 452 q^{73} - 880 q^{79} - 5040 q^{85} - 5320 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{3} ) / 7 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 7\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{3} ) / 6 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{2}\)

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
−1.32288 − 2.29129i
1.32288 + 2.29129i
−1.32288 + 2.29129i
1.32288 − 2.29129i
0 0 0 −7.93725 − 13.7477i 0 0 0 0 0
361.2 0 0 0 7.93725 + 13.7477i 0 0 0 0 0
1549.1 0 0 0 −7.93725 + 13.7477i 0 0 0 0 0
1549.2 0 0 0 7.93725 − 13.7477i 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
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.k.v 4
3.b odd 2 1 inner 1764.4.k.v 4
7.b odd 2 1 1764.4.k.u 4
7.c even 3 1 252.4.a.f ✓ 2
7.c even 3 1 inner 1764.4.k.v 4
7.d odd 6 1 1764.4.a.s 2
7.d odd 6 1 1764.4.k.u 4
21.c even 2 1 1764.4.k.u 4
21.g even 6 1 1764.4.a.s 2
21.g even 6 1 1764.4.k.u 4
21.h odd 6 1 252.4.a.f ✓ 2
21.h odd 6 1 inner 1764.4.k.v 4
28.g odd 6 1 1008.4.a.bc 2
84.n even 6 1 1008.4.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.a.f ✓ 2 7.c even 3 1
252.4.a.f ✓ 2 21.h odd 6 1
1008.4.a.bc 2 28.g odd 6 1
1008.4.a.bc 2 84.n even 6 1
1764.4.a.s 2 7.d odd 6 1
1764.4.a.s 2 21.g even 6 1
1764.4.k.u 4 7.b odd 2 1
1764.4.k.u 4 7.d odd 6 1
1764.4.k.u 4 21.c even 2 1
1764.4.k.u 4 21.g even 6 1
1764.4.k.v 4 1.a even 1 1 trivial
1764.4.k.v 4 3.b odd 2 1 inner
1764.4.k.v 4 7.c even 3 1 inner
1764.4.k.v 4 21.h odd 6 1 inner

Hecke kernels

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

\( T_{5}^{4} + 252T_{5}^{2} + 63504 \) Copy content Toggle raw display
\( T_{11}^{4} + 252T_{11}^{2} + 63504 \) Copy content Toggle raw display
\( T_{13} - 26 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 252 T^{2} + 63504 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 252 T^{2} + 63504 \) Copy content Toggle raw display
$13$ \( (T - 26)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 6300 T^{2} + 39690000 \) Copy content Toggle raw display
$19$ \( (T^{2} + 68 T + 4624)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 2268 T^{2} + 5143824 \) Copy content Toggle raw display
$29$ \( (T^{2} - 64512)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 212 T + 44944)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 218 T + 47524)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 157500)^{2} \) Copy content Toggle raw display
$43$ \( (T - 260)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 29019803904 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 51438240000 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 6666395904 \) Copy content Toggle raw display
$61$ \( (T^{2} - 322 T + 103684)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 356 T + 126736)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 1270332)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 226 T + 51076)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 440 T + 193600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 64512)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 1813737744 \) Copy content Toggle raw display
$97$ \( (T + 1330)^{4} \) Copy content Toggle raw display
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