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Properties

Label 1764.3.bk.e
Level $1764$
Weight $3$
Character orbit 1764.bk
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,3,Mod(557,1764)]
 
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, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.557");
 
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.bk (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: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.12745506816.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + (\beta_{7} - 2 \beta_{5}) q^{11} + (2 \beta_{3} + 2) q^{13} + ( - 2 \beta_{7} + 5 \beta_{5}) q^{17} + (3 \beta_{6} - 10 \beta_{2}) q^{19} + (3 \beta_{4} - 2 \beta_1) q^{23} + (2 \beta_{6} + 2 \beta_{3} + 9 \beta_{2} - 9) q^{25} + (\beta_{7} + 10 \beta_{5} + \cdots + 10 \beta_1) q^{29}+ \cdots + ( - 17 \beta_{3} - 24) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{13} - 40 q^{19} - 36 q^{25} + 40 q^{31} - 32 q^{37} - 32 q^{43} + 208 q^{55} - 272 q^{61} - 168 q^{67} + 144 q^{73} - 232 q^{79} - 544 q^{85} - 192 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 55x^{4} - 72x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{6} + 55\nu^{4} - 440\nu^{2} + 576 ) / 495 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} - 296 ) / 55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 203\nu ) / 55 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{7} - 110\nu^{5} + 880\nu^{3} - 1152\nu ) / 495 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -46\nu^{6} + 440\nu^{4} - 2530\nu^{2} + 3312 ) / 495 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -31\nu^{7} + 275\nu^{5} - 1705\nu^{3} + 2232\nu ) / 495 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{3} - 8\beta_{2} + 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{7} + 5\beta_{5} + 2\beta_{4} + 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{6} - 23\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16\beta_{7} + 31\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -55\beta_{3} - 296 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -110\beta_{4} - 203\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
−2.23256 + 1.28897i
−1.00781 + 0.581861i
1.00781 − 0.581861i
2.23256 − 1.28897i
−2.23256 − 1.28897i
−1.00781 − 0.581861i
1.00781 + 0.581861i
2.23256 + 1.28897i
0 0 0 −4.46512 + 2.57794i 0 0 0 0 0
557.2 0 0 0 −2.01563 + 1.16372i 0 0 0 0 0
557.3 0 0 0 2.01563 − 1.16372i 0 0 0 0 0
557.4 0 0 0 4.46512 − 2.57794i 0 0 0 0 0
1745.1 0 0 0 −4.46512 − 2.57794i 0 0 0 0 0
1745.2 0 0 0 −2.01563 − 1.16372i 0 0 0 0 0
1745.3 0 0 0 2.01563 + 1.16372i 0 0 0 0 0
1745.4 0 0 0 4.46512 + 2.57794i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 557.4
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 1764.3.bk.e 8
3.b odd 2 1 inner 1764.3.bk.e 8
7.b odd 2 1 1764.3.bk.d 8
7.c even 3 1 1764.3.c.f 4
7.c even 3 1 inner 1764.3.bk.e 8
7.d odd 6 1 252.3.c.a ✓ 4
7.d odd 6 1 1764.3.bk.d 8
21.c even 2 1 1764.3.bk.d 8
21.g even 6 1 252.3.c.a ✓ 4
21.g even 6 1 1764.3.bk.d 8
21.h odd 6 1 1764.3.c.f 4
21.h odd 6 1 inner 1764.3.bk.e 8
28.f even 6 1 1008.3.d.c 4
56.j odd 6 1 4032.3.d.h 4
56.m even 6 1 4032.3.d.e 4
63.i even 6 1 2268.3.bg.c 8
63.k odd 6 1 2268.3.bg.c 8
63.s even 6 1 2268.3.bg.c 8
63.t odd 6 1 2268.3.bg.c 8
84.j odd 6 1 1008.3.d.c 4
168.ba even 6 1 4032.3.d.h 4
168.be odd 6 1 4032.3.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.3.c.a ✓ 4 7.d odd 6 1
252.3.c.a ✓ 4 21.g even 6 1
1008.3.d.c 4 28.f even 6 1
1008.3.d.c 4 84.j odd 6 1
1764.3.c.f 4 7.c even 3 1
1764.3.c.f 4 21.h odd 6 1
1764.3.bk.d 8 7.b odd 2 1
1764.3.bk.d 8 7.d odd 6 1
1764.3.bk.d 8 21.c even 2 1
1764.3.bk.d 8 21.g even 6 1
1764.3.bk.e 8 1.a even 1 1 trivial
1764.3.bk.e 8 3.b odd 2 1 inner
1764.3.bk.e 8 7.c even 3 1 inner
1764.3.bk.e 8 21.h odd 6 1 inner
2268.3.bg.c 8 63.i even 6 1
2268.3.bg.c 8 63.k odd 6 1
2268.3.bg.c 8 63.s even 6 1
2268.3.bg.c 8 63.t odd 6 1
4032.3.d.e 4 56.m even 6 1
4032.3.d.e 4 168.be odd 6 1
4032.3.d.h 4 56.j odd 6 1
4032.3.d.h 4 168.ba even 6 1

Hecke kernels

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

\( T_{5}^{8} - 32T_{5}^{6} + 880T_{5}^{4} - 4608T_{5}^{2} + 20736 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} - 108 \) 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
$5$ \( T^{8} - 32 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 116 T^{6} + \cdots + 8503056 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T - 108)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 14666178816 \) Copy content Toggle raw display
$19$ \( (T^{4} + 20 T^{3} + \cdots + 23104)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 308 T^{6} + \cdots + 3111696 \) Copy content Toggle raw display
$29$ \( (T^{4} + 3476 T^{2} + 1127844)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 20 T^{3} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 16 T^{3} + \cdots + 891136)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 288 T^{2} + 11664)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 8 T - 992)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 19007367745536 \) Copy content Toggle raw display
$53$ \( (T^{4} - 2178 T^{2} + 4743684)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 14281868906496 \) Copy content Toggle raw display
$61$ \( (T^{4} + 136 T^{3} + \cdots + 21123216)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 84 T^{3} + \cdots + 13868176)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 15668 T^{2} + 32695524)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 72 T^{3} + \cdots + 1607824)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 116 T^{3} + \cdots + 14470416)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 19328 T^{2} + 15116544)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 36\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{2} + 48 T - 7516)^{4} \) Copy content Toggle raw display
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