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Properties

Label 3528.1.db.a
Level $3528$
Weight $1$
Character orbit 3528.db
Analytic conductor $1.761$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -56
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3528,1,Mod(1733,3528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3528.1733");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.db (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: \(1.76070136457\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{2} q^{2} - \zeta_{24}^{5} q^{3} + \zeta_{24}^{4} q^{4} + ( - \zeta_{24}^{7} + \zeta_{24}^{5}) q^{5} + \zeta_{24}^{7} q^{6} - \zeta_{24}^{6} q^{8} + \zeta_{24}^{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{2} q^{2} - \zeta_{24}^{5} q^{3} + \zeta_{24}^{4} q^{4} + ( - \zeta_{24}^{7} + \zeta_{24}^{5}) q^{5} + \zeta_{24}^{7} q^{6} - \zeta_{24}^{6} q^{8} + \zeta_{24}^{10} q^{9} + (\zeta_{24}^{9} - \zeta_{24}^{7}) q^{10} - \zeta_{24}^{9} q^{12} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{13} + ( - \zeta_{24}^{10} - 1) q^{15} + \zeta_{24}^{8} q^{16} + q^{18} + (\zeta_{24}^{11} + \zeta_{24}^{9}) q^{19} + ( - \zeta_{24}^{11} + \zeta_{24}^{9}) q^{20} + (\zeta_{24}^{8} + \zeta_{24}^{4}) q^{23} + \zeta_{24}^{11} q^{24} + (\zeta_{24}^{10} - \zeta_{24}^{2} + 1) q^{25} + (\zeta_{24}^{7} + \zeta_{24}) q^{26} + \zeta_{24}^{3} q^{27} + (\zeta_{24}^{2} - 1) q^{30} - \zeta_{24}^{10} q^{32} - \zeta_{24}^{2} q^{36} + ( - \zeta_{24}^{11} + \zeta_{24}) q^{38} + (\zeta_{24}^{10} + \zeta_{24}^{4}) q^{39} + ( - \zeta_{24}^{11} - \zeta_{24}) q^{40} + (\zeta_{24}^{5} - \zeta_{24}^{3}) q^{45} + ( - \zeta_{24}^{10} - \zeta_{24}^{6}) q^{46} + \zeta_{24} q^{48} + (\zeta_{24}^{4} - \zeta_{24}^{2} + 1) q^{50} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{52} - \zeta_{24}^{5} q^{54} + (\zeta_{24}^{4} + \zeta_{24}^{2}) q^{57} + (\zeta_{24}^{7} + \zeta_{24}) q^{59} + ( - \zeta_{24}^{4} + \zeta_{24}^{2}) q^{60} + ( - \zeta_{24}^{3} - \zeta_{24}) q^{61} - q^{64} + ( - \zeta_{24}^{10} + \zeta_{24}^{6} + \cdots - 1) q^{65} + \cdots - \zeta_{24}^{3} q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 8 q^{15} - 4 q^{16} + 8 q^{18} + 8 q^{25} - 8 q^{30} + 4 q^{39} + 12 q^{50} + 4 q^{57} - 4 q^{60} - 8 q^{64} - 12 q^{65} + 4 q^{72} + 8 q^{78} + 4 q^{81} - 12 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(-\zeta_{24}^{8}\) \(-\zeta_{24}^{4}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1733.1
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
−0.866025 − 0.500000i −0.258819 − 0.965926i 0.500000 + 0.866025i 0.517638 −0.258819 + 0.965926i 0 − 1.00000i −0.866025 + 0.500000i −0.448288 − 0.258819i
1733.2 −0.866025 − 0.500000i 0.258819 + 0.965926i 0.500000 + 0.866025i −0.517638 0.258819 − 0.965926i 0 − 1.00000i −0.866025 + 0.500000i 0.448288 + 0.258819i
1733.3 0.866025 + 0.500000i −0.965926 + 0.258819i 0.500000 + 0.866025i 1.93185 −0.965926 − 0.258819i 0 1.00000i 0.866025 − 0.500000i 1.67303 + 0.965926i
1733.4 0.866025 + 0.500000i 0.965926 − 0.258819i 0.500000 + 0.866025i −1.93185 0.965926 + 0.258819i 0 1.00000i 0.866025 − 0.500000i −1.67303 − 0.965926i
2333.1 −0.866025 + 0.500000i −0.258819 + 0.965926i 0.500000 − 0.866025i 0.517638 −0.258819 − 0.965926i 0 1.00000i −0.866025 − 0.500000i −0.448288 + 0.258819i
2333.2 −0.866025 + 0.500000i 0.258819 − 0.965926i 0.500000 − 0.866025i −0.517638 0.258819 + 0.965926i 0 1.00000i −0.866025 − 0.500000i 0.448288 − 0.258819i
2333.3 0.866025 − 0.500000i −0.965926 − 0.258819i 0.500000 − 0.866025i 1.93185 −0.965926 + 0.258819i 0 − 1.00000i 0.866025 + 0.500000i 1.67303 − 0.965926i
2333.4 0.866025 − 0.500000i 0.965926 + 0.258819i 0.500000 − 0.866025i −1.93185 0.965926 − 0.258819i 0 − 1.00000i 0.866025 + 0.500000i −1.67303 + 0.965926i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1733.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
8.b even 2 1 inner
63.n odd 6 1 inner
63.s even 6 1 inner
504.y even 6 1 inner
504.db odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.db.a 8
7.b odd 2 1 inner 3528.1.db.a 8
7.c even 3 1 3528.1.bg.a ✓ 8
7.c even 3 1 3528.1.bi.a 8
7.d odd 6 1 3528.1.bg.a ✓ 8
7.d odd 6 1 3528.1.bi.a 8
8.b even 2 1 inner 3528.1.db.a 8
9.d odd 6 1 3528.1.bi.a 8
56.h odd 2 1 CM 3528.1.db.a 8
56.j odd 6 1 3528.1.bg.a ✓ 8
56.j odd 6 1 3528.1.bi.a 8
56.p even 6 1 3528.1.bg.a ✓ 8
56.p even 6 1 3528.1.bi.a 8
63.i even 6 1 3528.1.bg.a ✓ 8
63.j odd 6 1 3528.1.bg.a ✓ 8
63.n odd 6 1 inner 3528.1.db.a 8
63.o even 6 1 3528.1.bi.a 8
63.s even 6 1 inner 3528.1.db.a 8
72.j odd 6 1 3528.1.bi.a 8
504.y even 6 1 inner 3528.1.db.a 8
504.bi odd 6 1 3528.1.bg.a ✓ 8
504.ca even 6 1 3528.1.bg.a ✓ 8
504.cc even 6 1 3528.1.bi.a 8
504.db odd 6 1 inner 3528.1.db.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.bg.a ✓ 8 7.c even 3 1
3528.1.bg.a ✓ 8 7.d odd 6 1
3528.1.bg.a ✓ 8 56.j odd 6 1
3528.1.bg.a ✓ 8 56.p even 6 1
3528.1.bg.a ✓ 8 63.i even 6 1
3528.1.bg.a ✓ 8 63.j odd 6 1
3528.1.bg.a ✓ 8 504.bi odd 6 1
3528.1.bg.a ✓ 8 504.ca even 6 1
3528.1.bi.a 8 7.c even 3 1
3528.1.bi.a 8 7.d odd 6 1
3528.1.bi.a 8 9.d odd 6 1
3528.1.bi.a 8 56.j odd 6 1
3528.1.bi.a 8 56.p even 6 1
3528.1.bi.a 8 63.o even 6 1
3528.1.bi.a 8 72.j odd 6 1
3528.1.bi.a 8 504.cc even 6 1
3528.1.db.a 8 1.a even 1 1 trivial
3528.1.db.a 8 7.b odd 2 1 inner
3528.1.db.a 8 8.b even 2 1 inner
3528.1.db.a 8 56.h odd 2 1 CM
3528.1.db.a 8 63.n odd 6 1 inner
3528.1.db.a 8 63.s even 6 1 inner
3528.1.db.a 8 504.y even 6 1 inner
3528.1.db.a 8 504.db odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3528, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} - 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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