Properties

Label 1764.3.d.e
Level $1764$
Weight $3$
Character orbit 1764.d
Analytic conductor $48.066$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,3,Mod(685,1764)]
 
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, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.685");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.d (of order \(2\), degree \(1\), 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: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: no (minimal twist has level 196)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - \beta_1) q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - \beta_1) q^{5} + ( - \beta_{3} + 6) q^{11} + ( - 4 \beta_{2} + 3 \beta_1) q^{13} - 3 \beta_{2} q^{17} + ( - \beta_{2} + 5 \beta_1) q^{19} - 26 q^{23} + ( - 2 \beta_{3} - 27) q^{25} + (2 \beta_{3} - 8) q^{29} + (8 \beta_{2} - 2 \beta_1) q^{31} - 32 q^{37} + ( - 5 \beta_{2} - 3 \beta_1) q^{41} + (7 \beta_{3} + 10) q^{43} + ( - 2 \beta_{2} + 12 \beta_1) q^{47} + (2 \beta_{3} - 6) q^{53} + (6 \beta_{2} - 14 \beta_1) q^{55} + (3 \beta_{2} + 11 \beta_1) q^{59} + ( - 8 \beta_{2} - 9 \beta_1) q^{61} + ( - 14 \beta_{3} - 24) q^{65} + ( - 6 \beta_{3} + 28) q^{67} + ( - 4 \beta_{3} - 56) q^{71} + ( - 3 \beta_{2} + 11 \beta_1) q^{73} + (2 \beta_{3} + 60) q^{79} + (33 \beta_{2} + 13 \beta_1) q^{83} + ( - 6 \beta_{3} - 54) q^{85} + ( - 25 \beta_{2} + 11 \beta_1) q^{89} + ( - 12 \beta_{3} + 62) q^{95} + (33 \beta_{2} + 22 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{11} - 104 q^{23} - 108 q^{25} - 32 q^{29} - 128 q^{37} + 40 q^{43} - 24 q^{53} - 96 q^{65} + 112 q^{67} - 224 q^{71} + 240 q^{79} - 216 q^{85} + 248 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 7\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 7\nu^{2} + 14 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 14 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
685.1
0.765367i
1.84776i
1.84776i
0.765367i
0 0 0 8.47343i 0 0 0 0 0
685.2 0 0 0 5.67459i 0 0 0 0 0
685.3 0 0 0 5.67459i 0 0 0 0 0
685.4 0 0 0 8.47343i 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.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.3.d.e 4
3.b odd 2 1 196.3.b.b 4
7.b odd 2 1 inner 1764.3.d.e 4
7.c even 3 2 1764.3.z.k 8
7.d odd 6 2 1764.3.z.k 8
12.b even 2 1 784.3.c.d 4
21.c even 2 1 196.3.b.b 4
21.g even 6 2 196.3.h.c 8
21.h odd 6 2 196.3.h.c 8
84.h odd 2 1 784.3.c.d 4
84.j odd 6 2 784.3.s.g 8
84.n even 6 2 784.3.s.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.3.b.b 4 3.b odd 2 1
196.3.b.b 4 21.c even 2 1
196.3.h.c 8 21.g even 6 2
196.3.h.c 8 21.h odd 6 2
784.3.c.d 4 12.b even 2 1
784.3.c.d 4 84.h odd 2 1
784.3.s.g 8 84.j odd 6 2
784.3.s.g 8 84.n even 6 2
1764.3.d.e 4 1.a even 1 1 trivial
1764.3.d.e 4 7.b odd 2 1 inner
1764.3.z.k 8 7.c even 3 2
1764.3.z.k 8 7.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{4} + 104T_{5}^{2} + 2312 \) Copy content Toggle raw display
\( T_{11}^{2} - 12T_{11} - 62 \) 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
$5$ \( T^{4} + 104T^{2} + 2312 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 12 T - 62)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 776 T^{2} + 150152 \) Copy content Toggle raw display
$17$ \( T^{4} + 180T^{2} + 162 \) Copy content Toggle raw display
$19$ \( T^{4} + 1060 T^{2} + 45602 \) Copy content Toggle raw display
$23$ \( (T + 26)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 16 T - 328)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 1568 T^{2} + 307328 \) Copy content Toggle raw display
$37$ \( (T + 32)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 740 T^{2} + 132098 \) Copy content Toggle raw display
$43$ \( (T^{2} - 20 T - 4702)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 6032 T^{2} + \cdots + 1191968 \) Copy content Toggle raw display
$53$ \( (T^{2} + 12 T - 356)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 4756 T^{2} + 334562 \) Copy content Toggle raw display
$61$ \( T^{4} + 3944 T^{2} + \cdots + 2947592 \) Copy content Toggle raw display
$67$ \( (T^{2} - 56 T - 2744)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 112 T + 1568)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 5284 T^{2} + \cdots + 1659842 \) Copy content Toggle raw display
$79$ \( (T^{2} - 120 T + 3208)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 25108 T^{2} + \cdots + 102330818 \) Copy content Toggle raw display
$89$ \( T^{4} + 19540 T^{2} + \cdots + 81077378 \) Copy content Toggle raw display
$97$ \( T^{4} + 35332 T^{2} + \cdots + 310652738 \) Copy content Toggle raw display
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