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Properties

Label 1764.1.q.b
Level $1764$
Weight $1$
Character orbit 1764.q
Analytic conductor $0.880$
Analytic rank $0$
Dimension $16$
Projective image $D_{8}$
CM discriminant -4
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,1,Mod(215,1764)]
 
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([3, 3, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.215");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1764.q (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: \(0.880350682285\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.38423222208.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{48}^{20} q^{2} - \zeta_{48}^{16} q^{4} + ( - \zeta_{48}^{11} - \zeta_{48}^{5}) q^{5} - \zeta_{48}^{12} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{48}^{20} q^{2} - \zeta_{48}^{16} q^{4} + ( - \zeta_{48}^{11} - \zeta_{48}^{5}) q^{5} - \zeta_{48}^{12} q^{8} + ( - \zeta_{48}^{7} - \zeta_{48}) q^{10} + (\zeta_{48}^{21} + \zeta_{48}^{3}) q^{13} - \zeta_{48}^{8} q^{16} + ( - \zeta_{48}^{7} + \zeta_{48}) q^{17} + (\zeta_{48}^{21} - \zeta_{48}^{3}) q^{20} + (\zeta_{48}^{22} + \zeta_{48}^{16} + \zeta_{48}^{10}) q^{25} + ( - \zeta_{48}^{23} + \zeta_{48}^{17}) q^{26} - \zeta_{48}^{4} q^{32} + ( - \zeta_{48}^{21} - \zeta_{48}^{3}) q^{34} + ( - \zeta_{48}^{14} - \zeta_{48}^{2}) q^{37} + (\zeta_{48}^{23} + \zeta_{48}^{17}) q^{40} + (\zeta_{48}^{15} - \zeta_{48}^{9}) q^{41} + (\zeta_{48}^{18} + \zeta_{48}^{12} + \zeta_{48}^{6}) q^{50} + ( - \zeta_{48}^{19} + \zeta_{48}^{13}) q^{52} + ( - \zeta_{48}^{22} + \zeta_{48}^{10}) q^{53} + ( - \zeta_{48}^{23} - \zeta_{48}^{17}) q^{61} - q^{64} + ( - \zeta_{48}^{14} + \zeta_{48}^{8} + \zeta_{48}^{2}) q^{65} + (\zeta_{48}^{23} - \zeta_{48}^{17}) q^{68} + (\zeta_{48}^{7} + \zeta_{48}) q^{73} + (\zeta_{48}^{22} - \zeta_{48}^{10}) q^{74} + (\zeta_{48}^{19} + \zeta_{48}^{13}) q^{80} + (\zeta_{48}^{11} - \zeta_{48}^{5}) q^{82} + (\zeta_{48}^{18} - \zeta_{48}^{6}) q^{85} + (\zeta_{48}^{11} + \zeta_{48}^{5}) q^{89} + (\zeta_{48}^{15} + \zeta_{48}^{9}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 8 q^{16} - 8 q^{25} - 16 q^{64}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{48}^{8}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
−0.608761 + 0.793353i
−0.793353 − 0.608761i
0.793353 + 0.608761i
0.608761 − 0.793353i
0.991445 − 0.130526i
−0.130526 − 0.991445i
0.130526 + 0.991445i
−0.991445 + 0.130526i
−0.608761 − 0.793353i
−0.793353 + 0.608761i
0.793353 − 0.608761i
0.608761 + 0.793353i
0.991445 + 0.130526i
−0.130526 + 0.991445i
0.130526 − 0.991445i
−0.991445 − 0.130526i
−0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.923880 + 1.60021i 0 0 − 1.00000i 0 1.60021 − 0.923880i
215.2 −0.866025 − 0.500000i 0 0.500000 + 0.866025i −0.382683 + 0.662827i 0 0 − 1.00000i 0 0.662827 − 0.382683i
215.3 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.382683 − 0.662827i 0 0 − 1.00000i 0 −0.662827 + 0.382683i
215.4 −0.866025 − 0.500000i 0 0.500000 + 0.866025i 0.923880 − 1.60021i 0 0 − 1.00000i 0 −1.60021 + 0.923880i
215.5 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.923880 + 1.60021i 0 0 1.00000i 0 −1.60021 + 0.923880i
215.6 0.866025 + 0.500000i 0 0.500000 + 0.866025i −0.382683 + 0.662827i 0 0 1.00000i 0 −0.662827 + 0.382683i
215.7 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.382683 − 0.662827i 0 0 1.00000i 0 0.662827 − 0.382683i
215.8 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.923880 − 1.60021i 0 0 1.00000i 0 1.60021 − 0.923880i
1403.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.923880 − 1.60021i 0 0 1.00000i 0 1.60021 + 0.923880i
1403.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.382683 − 0.662827i 0 0 1.00000i 0 0.662827 + 0.382683i
1403.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.382683 + 0.662827i 0 0 1.00000i 0 −0.662827 − 0.382683i
1403.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.923880 + 1.60021i 0 0 1.00000i 0 −1.60021 − 0.923880i
1403.5 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.923880 − 1.60021i 0 0 − 1.00000i 0 −1.60021 − 0.923880i
1403.6 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.382683 − 0.662827i 0 0 − 1.00000i 0 −0.662827 − 0.382683i
1403.7 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.382683 + 0.662827i 0 0 − 1.00000i 0 0.662827 + 0.382683i
1403.8 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.923880 + 1.60021i 0 0 − 1.00000i 0 1.60021 + 0.923880i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 215.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
84.h odd 2 1 inner
84.j odd 6 1 inner
84.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.q.b 16
3.b odd 2 1 inner 1764.1.q.b 16
4.b odd 2 1 CM 1764.1.q.b 16
7.b odd 2 1 inner 1764.1.q.b 16
7.c even 3 1 1764.1.h.a ✓ 8
7.c even 3 1 inner 1764.1.q.b 16
7.d odd 6 1 1764.1.h.a ✓ 8
7.d odd 6 1 inner 1764.1.q.b 16
12.b even 2 1 inner 1764.1.q.b 16
21.c even 2 1 inner 1764.1.q.b 16
21.g even 6 1 1764.1.h.a ✓ 8
21.g even 6 1 inner 1764.1.q.b 16
21.h odd 6 1 1764.1.h.a ✓ 8
21.h odd 6 1 inner 1764.1.q.b 16
28.d even 2 1 inner 1764.1.q.b 16
28.f even 6 1 1764.1.h.a ✓ 8
28.f even 6 1 inner 1764.1.q.b 16
28.g odd 6 1 1764.1.h.a ✓ 8
28.g odd 6 1 inner 1764.1.q.b 16
84.h odd 2 1 inner 1764.1.q.b 16
84.j odd 6 1 1764.1.h.a ✓ 8
84.j odd 6 1 inner 1764.1.q.b 16
84.n even 6 1 1764.1.h.a ✓ 8
84.n even 6 1 inner 1764.1.q.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.1.h.a ✓ 8 7.c even 3 1
1764.1.h.a ✓ 8 7.d odd 6 1
1764.1.h.a ✓ 8 21.g even 6 1
1764.1.h.a ✓ 8 21.h odd 6 1
1764.1.h.a ✓ 8 28.f even 6 1
1764.1.h.a ✓ 8 28.g odd 6 1
1764.1.h.a ✓ 8 84.j odd 6 1
1764.1.h.a ✓ 8 84.n even 6 1
1764.1.q.b 16 1.a even 1 1 trivial
1764.1.q.b 16 3.b odd 2 1 inner
1764.1.q.b 16 4.b odd 2 1 CM
1764.1.q.b 16 7.b odd 2 1 inner
1764.1.q.b 16 7.c even 3 1 inner
1764.1.q.b 16 7.d odd 6 1 inner
1764.1.q.b 16 12.b even 2 1 inner
1764.1.q.b 16 21.c even 2 1 inner
1764.1.q.b 16 21.g even 6 1 inner
1764.1.q.b 16 21.h odd 6 1 inner
1764.1.q.b 16 28.d even 2 1 inner
1764.1.q.b 16 28.f even 6 1 inner
1764.1.q.b 16 28.g odd 6 1 inner
1764.1.q.b 16 84.h odd 2 1 inner
1764.1.q.b 16 84.j odd 6 1 inner
1764.1.q.b 16 84.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 4T_{5}^{6} + 14T_{5}^{4} + 8T_{5}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1764, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 4 T^{6} + 14 T^{4} + 8 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 4 T^{6} + 14 T^{4} + 8 T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} - 4 T^{6} + 14 T^{4} - 8 T^{2} + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{8} - 4 T^{6} + 14 T^{4} - 8 T^{2} + 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( (T^{8} + 4 T^{6} + 14 T^{4} + 8 T^{2} + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 4 T^{2} + 2)^{4} \) Copy content Toggle raw display
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