Properties

Label 1764.2.t.b
Level $1764$
Weight $2$
Character orbit 1764.t
Analytic conductor $14.086$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(521,1764)]
 
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, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.521");
 
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.t (of order \(6\), 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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + ( - \beta_{7} - \beta_{3}) q^{11} + (\beta_{6} - \beta_{5}) q^{13} + ( - \beta_{4} + \beta_{2}) q^{17} - \beta_{5} q^{19} - \beta_{7} q^{23} + (7 \beta_1 - 7) q^{25} + \beta_{3} q^{29}+ \cdots + (\beta_{6} - \beta_{5}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 28 q^{25} + 32 q^{37} - 16 q^{43} - 32 q^{67} + 16 q^{79} + 96 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\zeta_{24}^{5} - 3\zeta_{24}^{3} + 3\zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{6} + 4\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{24}^{7} - 4\zeta_{24}^{5} + 4\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\zeta_{24}^{7} - 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 4\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 3\zeta_{24}^{7} - 3\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( -2\beta_{7} + 2\beta_{6} - \beta_{5} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{6} + \beta_{5} - 2\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -2\beta_{7} + \beta_{6} - 2\beta_{5} - 2\beta_{3} ) / 12 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + 2\beta_{2} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( 2\beta_{7} + 2\beta_{6} - \beta_{5} ) / 12 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
521.1
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0 0 0 −1.73205 3.00000i 0 0 0 0 0
521.2 0 0 0 −1.73205 3.00000i 0 0 0 0 0
521.3 0 0 0 1.73205 + 3.00000i 0 0 0 0 0
521.4 0 0 0 1.73205 + 3.00000i 0 0 0 0 0
1097.1 0 0 0 −1.73205 + 3.00000i 0 0 0 0 0
1097.2 0 0 0 −1.73205 + 3.00000i 0 0 0 0 0
1097.3 0 0 0 1.73205 3.00000i 0 0 0 0 0
1097.4 0 0 0 1.73205 3.00000i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 521.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 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 1764.2.t.b 8
3.b odd 2 1 inner 1764.2.t.b 8
7.b odd 2 1 inner 1764.2.t.b 8
7.c even 3 1 252.2.f.a 4
7.c even 3 1 inner 1764.2.t.b 8
7.d odd 6 1 252.2.f.a 4
7.d odd 6 1 inner 1764.2.t.b 8
21.c even 2 1 inner 1764.2.t.b 8
21.g even 6 1 252.2.f.a 4
21.g even 6 1 inner 1764.2.t.b 8
21.h odd 6 1 252.2.f.a 4
21.h odd 6 1 inner 1764.2.t.b 8
28.f even 6 1 1008.2.k.b 4
28.g odd 6 1 1008.2.k.b 4
35.i odd 6 1 6300.2.d.c 4
35.j even 6 1 6300.2.d.c 4
35.k even 12 2 6300.2.f.b 8
35.l odd 12 2 6300.2.f.b 8
56.j odd 6 1 4032.2.k.a 4
56.k odd 6 1 4032.2.k.d 4
56.m even 6 1 4032.2.k.d 4
56.p even 6 1 4032.2.k.a 4
63.g even 3 1 2268.2.x.i 8
63.h even 3 1 2268.2.x.i 8
63.i even 6 1 2268.2.x.i 8
63.j odd 6 1 2268.2.x.i 8
63.k odd 6 1 2268.2.x.i 8
63.n odd 6 1 2268.2.x.i 8
63.s even 6 1 2268.2.x.i 8
63.t odd 6 1 2268.2.x.i 8
84.j odd 6 1 1008.2.k.b 4
84.n even 6 1 1008.2.k.b 4
105.o odd 6 1 6300.2.d.c 4
105.p even 6 1 6300.2.d.c 4
105.w odd 12 2 6300.2.f.b 8
105.x even 12 2 6300.2.f.b 8
168.s odd 6 1 4032.2.k.a 4
168.v even 6 1 4032.2.k.d 4
168.ba even 6 1 4032.2.k.a 4
168.be odd 6 1 4032.2.k.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.f.a 4 7.c even 3 1
252.2.f.a 4 7.d odd 6 1
252.2.f.a 4 21.g even 6 1
252.2.f.a 4 21.h odd 6 1
1008.2.k.b 4 28.f even 6 1
1008.2.k.b 4 28.g odd 6 1
1008.2.k.b 4 84.j odd 6 1
1008.2.k.b 4 84.n even 6 1
1764.2.t.b 8 1.a even 1 1 trivial
1764.2.t.b 8 3.b odd 2 1 inner
1764.2.t.b 8 7.b odd 2 1 inner
1764.2.t.b 8 7.c even 3 1 inner
1764.2.t.b 8 7.d odd 6 1 inner
1764.2.t.b 8 21.c even 2 1 inner
1764.2.t.b 8 21.g even 6 1 inner
1764.2.t.b 8 21.h odd 6 1 inner
2268.2.x.i 8 63.g even 3 1
2268.2.x.i 8 63.h even 3 1
2268.2.x.i 8 63.i even 6 1
2268.2.x.i 8 63.j odd 6 1
2268.2.x.i 8 63.k odd 6 1
2268.2.x.i 8 63.n odd 6 1
2268.2.x.i 8 63.s even 6 1
2268.2.x.i 8 63.t odd 6 1
4032.2.k.a 4 56.j odd 6 1
4032.2.k.a 4 56.p even 6 1
4032.2.k.a 4 168.s odd 6 1
4032.2.k.a 4 168.ba even 6 1
4032.2.k.d 4 56.k odd 6 1
4032.2.k.d 4 56.m even 6 1
4032.2.k.d 4 168.v even 6 1
4032.2.k.d 4 168.be odd 6 1
6300.2.d.c 4 35.i odd 6 1
6300.2.d.c 4 35.j even 6 1
6300.2.d.c 4 105.o odd 6 1
6300.2.d.c 4 105.p even 6 1
6300.2.f.b 8 35.k even 12 2
6300.2.f.b 8 35.l odd 12 2
6300.2.f.b 8 105.w odd 12 2
6300.2.f.b 8 105.x even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 12T_{5}^{2} + 144 \) acting on \(S_{2}^{\mathrm{new}}(1764, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T + 64)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$43$ \( (T + 2)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 48 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 162 T^{2} + 26244)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 192 T^{2} + 36864)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 96 T^{2} + 9216)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 8 T + 64)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 18)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 24 T^{2} + 576)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 108 T^{2} + 11664)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
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