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Properties

Label 1776.2.bz.g
Level $1776$
Weight $2$
Character orbit 1776.bz
Analytic conductor $14.181$
Analytic rank $0$
Dimension $6$
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] = [1776,2,Mod(529,1776)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1776, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1776.529");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1776.bz (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: \(14.1814313990\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.16553403.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 111)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{3} - \beta_{2} + 2) q^{5} + \beta_{2} q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + (\beta_{3} - \beta_{2} + 2) q^{5} + \beta_{2} q^{7} + (\beta_{2} - 1) q^{9} + ( - \beta_{4} + 1) q^{11} + \beta_{3} q^{13} + (\beta_{3} + \beta_{2} - \beta_1 + 1) q^{15} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_1) q^{17} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \cdots - 2) q^{19}+ \cdots + ( - \beta_{5} + \beta_{4} + \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 9 q^{5} + 3 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} + 9 q^{5} + 3 q^{7} - 3 q^{9} + 6 q^{11} + 9 q^{15} - 9 q^{19} - 3 q^{21} + 6 q^{25} - 6 q^{27} + 3 q^{33} + 9 q^{35} + 12 q^{37} + 15 q^{41} - 24 q^{47} + 18 q^{49} - 12 q^{53} + 18 q^{55} - 9 q^{57} + 9 q^{59} - 6 q^{63} + 12 q^{65} - 6 q^{67} + 9 q^{69} + 6 q^{71} - 54 q^{73} + 12 q^{75} + 3 q^{77} - 3 q^{81} + 12 q^{83} + 6 q^{85} - 18 q^{87} - 18 q^{89} - 9 q^{93} + 6 q^{95} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 12x^{4} + 36x^{2} + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 6\nu + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 6\nu^{2} + \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} + 10\nu^{3} + \nu^{2} + 24\nu + 4 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 6\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{4} + 2\beta_{3} - \beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - \beta_{4} - 20\beta_{2} + 36\beta _1 + 10 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(593\) \(1297\) \(1333\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
529.1
− 2.28989i
− 0.292861i
2.58275i
2.28989i
0.292861i
− 2.58275i
0 0.500000 − 0.866025i 0 −0.483106 − 0.278921i 0 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0
529.2 0 0.500000 − 0.866025i 0 1.24637 + 0.719595i 0 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0
529.3 0 0.500000 − 0.866025i 0 3.73673 + 2.15740i 0 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0
1729.1 0 0.500000 + 0.866025i 0 −0.483106 + 0.278921i 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0
1729.2 0 0.500000 + 0.866025i 0 1.24637 − 0.719595i 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0
1729.3 0 0.500000 + 0.866025i 0 3.73673 − 2.15740i 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1776.2.bz.g 6
4.b odd 2 1 111.2.j.c ✓ 6
12.b even 2 1 333.2.s.d 6
37.e even 6 1 inner 1776.2.bz.g 6
148.j odd 6 1 111.2.j.c ✓ 6
148.l even 12 2 4107.2.a.l 6
444.p even 6 1 333.2.s.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.2.j.c ✓ 6 4.b odd 2 1
111.2.j.c ✓ 6 148.j odd 6 1
333.2.s.d 6 12.b even 2 1
333.2.s.d 6 444.p even 6 1
1776.2.bz.g 6 1.a even 1 1 trivial
1776.2.bz.g 6 37.e even 6 1 inner
4107.2.a.l 6 148.l even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1776, [\chi])\):

\( T_{5}^{6} - 9T_{5}^{5} + 30T_{5}^{4} - 27T_{5}^{3} - 9T_{5}^{2} + 18T_{5} + 12 \) Copy content Toggle raw display
\( T_{11}^{3} - 3T_{11}^{2} - 9T_{11} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} - 9 T^{5} + \cdots + 12 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T^{3} - 3 T^{2} - 9 T + 24)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 6 T^{4} + \cdots + 3 \) Copy content Toggle raw display
$17$ \( T^{6} - 33 T^{4} + \cdots + 3468 \) Copy content Toggle raw display
$19$ \( T^{6} + 9 T^{5} + \cdots + 2028 \) Copy content Toggle raw display
$23$ \( T^{6} + 21 T^{4} + \cdots + 48 \) Copy content Toggle raw display
$29$ \( T^{6} + 102 T^{4} + \cdots + 17328 \) Copy content Toggle raw display
$31$ \( T^{6} + 111 T^{4} + \cdots + 8112 \) Copy content Toggle raw display
$37$ \( T^{6} - 12 T^{5} + \cdots + 50653 \) Copy content Toggle raw display
$41$ \( T^{6} - 15 T^{5} + \cdots + 6084 \) Copy content Toggle raw display
$43$ \( T^{6} + 12 T^{4} + \cdots + 3 \) Copy content Toggle raw display
$47$ \( (T^{3} + 12 T^{2} + 9 T - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 12 T^{5} + \cdots + 144 \) Copy content Toggle raw display
$59$ \( T^{6} - 9 T^{5} + \cdots + 92928 \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} + 6 T^{5} + \cdots + 16129 \) Copy content Toggle raw display
$71$ \( T^{6} - 6 T^{5} + \cdots + 9216 \) Copy content Toggle raw display
$73$ \( (T^{3} + 27 T^{2} + \cdots + 572)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 42 T^{4} + \cdots + 507 \) Copy content Toggle raw display
$83$ \( T^{6} - 12 T^{5} + \cdots + 5419584 \) Copy content Toggle raw display
$89$ \( T^{6} + 18 T^{5} + \cdots + 5947392 \) Copy content Toggle raw display
$97$ \( T^{6} + 228 T^{4} + \cdots + 363 \) Copy content Toggle raw display
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