Properties

Label 1764.2.b.k
Level $1764$
Weight $2$
Character orbit 1764.b
Analytic conductor $14.086$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(1567,1764)]
 
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([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.b (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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1212153856.10
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 10x^{4} - 16x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 1) q^{2} + ( - \beta_{6} + \beta_{5} + \beta_{3} + 1) q^{4} + (\beta_{7} + \beta_{2}) q^{5} + ( - 2 \beta_{6} - \beta_{3} + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 1) q^{2} + ( - \beta_{6} + \beta_{5} + \beta_{3} + 1) q^{4} + (\beta_{7} + \beta_{2}) q^{5} + ( - 2 \beta_{6} - \beta_{3} + 1) q^{8} + ( - \beta_{4} + \beta_{2} + \beta_1) q^{10} + ( - 2 \beta_{6} + \beta_{5} - \beta_{3} + 1) q^{11} + ( - \beta_{7} + \beta_{2}) q^{13} + ( - \beta_{6} - 3 \beta_{5} - \beta_{3} - 1) q^{16} + (\beta_{7} - 2 \beta_{2}) q^{17} + (2 \beta_{4} - \beta_{2}) q^{19} + ( - 2 \beta_{7} + \beta_{4} - \beta_{2} + \beta_1) q^{20} + ( - \beta_{6} - 3 \beta_{5} - 2 \beta_{3}) q^{22} + ( - 4 \beta_{6} + 2) q^{23} + (2 \beta_{5} + 2 \beta_{3} + 3) q^{25} + (\beta_{4} + \beta_{2} + \beta_1) q^{26} + ( - 2 \beta_{5} - 2 \beta_{3} + 2) q^{29} + (2 \beta_{7} + 4 \beta_{4} - 2 \beta_{2} + 4 \beta_1) q^{31} + ( - 2 \beta_{5} + \beta_{3} - 5) q^{32} + ( - \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{34} - 4 q^{37} + (4 \beta_{7} + \beta_{2} + \beta_1) q^{38} + (2 \beta_{7} + 3 \beta_{4} - \beta_{2} + \beta_1) q^{40} - 3 \beta_{2} q^{41} + ( - 2 \beta_{6} - 3 \beta_{5} + 3 \beta_{3} + 1) q^{43} + (\beta_{6} - 3 \beta_{5} + 2 \beta_{3} - 4) q^{44} + ( - 4 \beta_{6} - 4 \beta_{5} - 2 \beta_{3} - 2) q^{46} + ( - 2 \beta_{7} - 4 \beta_1) q^{47} + ( - 2 \beta_{6} + 2 \beta_{5} + \beta_{3} + 5) q^{50} + (2 \beta_{7} + \beta_{4} + \beta_{2} + 3 \beta_1) q^{52} + (6 \beta_{5} + 6 \beta_{3} + 4) q^{53} + (4 \beta_{4} - 2 \beta_{2}) q^{55} + (2 \beta_{6} - 2 \beta_{5} + 4 \beta_{3}) q^{58} + (2 \beta_{7} + 6 \beta_{4} - 3 \beta_{2} + 4 \beta_1) q^{59} + ( - 5 \beta_{7} + 5 \beta_{2}) q^{61} + (8 \beta_{7} + 2 \beta_{4} - 2 \beta_{2} + 6 \beta_1) q^{62} + ( - \beta_{6} + \beta_{5} - 3 \beta_{3} - 7) q^{64} + ( - 2 \beta_{5} - 2 \beta_{3} - 2) q^{65} + (4 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} - 2) q^{67} + ( - 2 \beta_{7} - 2 \beta_{4} - \beta_{2} - 5 \beta_1) q^{68} + (8 \beta_{7} - \beta_{2}) q^{73} + ( - 4 \beta_{3} - 4) q^{74} + ( - 3 \beta_{4} + 2 \beta_1) q^{76} + ( - 4 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + 2) q^{79} + (6 \beta_{7} - \beta_{4} + \beta_{2} + 3 \beta_1) q^{80} + ( - 3 \beta_{2} - 3 \beta_1) q^{82} + ( - 2 \beta_{7} + 2 \beta_{4} - \beta_{2} - 4 \beta_1) q^{83} + (2 \beta_{5} + 2 \beta_{3} + 4) q^{85} + ( - 5 \beta_{6} + \beta_{5} + 2 \beta_{3} - 4) q^{86} + ( - \beta_{6} + 3 \beta_{5} - 6) q^{88} + (2 \beta_{7} + 3 \beta_{2}) q^{89} + ( - 2 \beta_{6} - 6 \beta_{5} - 2 \beta_{3} - 10) q^{92} + ( - 2 \beta_{4} + 4 \beta_{2} - 4 \beta_1) q^{94} + (4 \beta_{6} - 4 \beta_{5} + 4 \beta_{3} - 2) q^{95} - 5 \beta_{7} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{4} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 4 q^{4} + 4 q^{8} + 4 q^{16} + 16 q^{22} + 8 q^{25} + 32 q^{29} - 36 q^{32} - 32 q^{37} - 24 q^{44} - 8 q^{46} + 20 q^{50} - 16 q^{53} - 52 q^{64} - 16 q^{74} + 16 q^{85} - 64 q^{86} - 64 q^{88} - 56 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 10x^{4} - 16x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 2\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 2\nu^{4} - 2\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} - 2\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} - 2\nu^{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 2\nu^{4} + 6\nu^{2} - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} - 10\nu^{3} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{6} + 2\beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{4} + 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{6} + 4\beta_{5} - 2\beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{7} - 2\beta_{4} + 8\beta_{2} - 2\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
−1.07072 + 0.923880i
1.07072 0.923880i
1.07072 + 0.923880i
−1.07072 0.923880i
−1.36145 0.382683i
1.36145 + 0.382683i
1.36145 0.382683i
−1.36145 + 0.382683i
−0.207107 1.39897i 0 −1.91421 + 0.579471i 2.61313i 0 0 1.20711 + 2.55791i 0 −3.65568 + 0.541196i
1567.2 −0.207107 1.39897i 0 −1.91421 + 0.579471i 2.61313i 0 0 1.20711 + 2.55791i 0 3.65568 0.541196i
1567.3 −0.207107 + 1.39897i 0 −1.91421 0.579471i 2.61313i 0 0 1.20711 2.55791i 0 3.65568 + 0.541196i
1567.4 −0.207107 + 1.39897i 0 −1.91421 0.579471i 2.61313i 0 0 1.20711 2.55791i 0 −3.65568 0.541196i
1567.5 1.20711 0.736813i 0 0.914214 1.77882i 1.08239i 0 0 −0.207107 2.82083i 0 −0.797521 1.30656i
1567.6 1.20711 0.736813i 0 0.914214 1.77882i 1.08239i 0 0 −0.207107 2.82083i 0 0.797521 + 1.30656i
1567.7 1.20711 + 0.736813i 0 0.914214 + 1.77882i 1.08239i 0 0 −0.207107 + 2.82083i 0 0.797521 1.30656i
1567.8 1.20711 + 0.736813i 0 0.914214 + 1.77882i 1.08239i 0 0 −0.207107 + 2.82083i 0 −0.797521 + 1.30656i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.b.k 8
3.b odd 2 1 196.2.d.c 8
4.b odd 2 1 inner 1764.2.b.k 8
7.b odd 2 1 inner 1764.2.b.k 8
12.b even 2 1 196.2.d.c 8
21.c even 2 1 196.2.d.c 8
21.g even 6 2 196.2.f.d 16
21.h odd 6 2 196.2.f.d 16
24.f even 2 1 3136.2.f.i 8
24.h odd 2 1 3136.2.f.i 8
28.d even 2 1 inner 1764.2.b.k 8
84.h odd 2 1 196.2.d.c 8
84.j odd 6 2 196.2.f.d 16
84.n even 6 2 196.2.f.d 16
168.e odd 2 1 3136.2.f.i 8
168.i even 2 1 3136.2.f.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.d.c 8 3.b odd 2 1
196.2.d.c 8 12.b even 2 1
196.2.d.c 8 21.c even 2 1
196.2.d.c 8 84.h odd 2 1
196.2.f.d 16 21.g even 6 2
196.2.f.d 16 21.h odd 6 2
196.2.f.d 16 84.j odd 6 2
196.2.f.d 16 84.n even 6 2
1764.2.b.k 8 1.a even 1 1 trivial
1764.2.b.k 8 4.b odd 2 1 inner
1764.2.b.k 8 7.b odd 2 1 inner
1764.2.b.k 8 28.d even 2 1 inner
3136.2.f.i 8 24.f even 2 1
3136.2.f.i 8 24.h odd 2 1
3136.2.f.i 8 168.e odd 2 1
3136.2.f.i 8 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\):

\( T_{5}^{4} + 8T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{11}^{4} + 20T_{11}^{2} + 68 \) Copy content Toggle raw display
\( T_{19}^{4} - 28T_{19}^{2} + 34 \) Copy content Toggle raw display
\( T_{29}^{2} - 8T_{29} + 8 \) Copy content Toggle raw display
\( T_{53}^{2} + 4T_{53} - 68 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 3 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 8 T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 20 T^{2} + 68)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 8 T^{2} + 8)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 20 T^{2} + 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 28 T^{2} + 34)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 56 T^{2} + 272)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 8 T + 8)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 96 T^{2} + 2176)^{2} \) Copy content Toggle raw display
$37$ \( (T + 4)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 36 T^{2} + 162)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 116 T^{2} + 3332)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 48 T^{2} + 544)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4 T - 68)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 204 T^{2} + 9826)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 200 T^{2} + 5000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 112 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{4} + 260 T^{2} + 12482)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 80 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 108 T^{2} + 1666)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 52 T^{2} + 578)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 100 T^{2} + 1250)^{2} \) Copy content Toggle raw display
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