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Properties

Label 3528.1.cm.a
Level $3528$
Weight $1$
Character orbit 3528.cm
Analytic conductor $1.761$
Analytic rank $0$
Dimension $16$
Projective image $D_{24}$
CM discriminant -8
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(227,3528)]
 
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([3, 3, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3528.227");
 
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.cm (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: \(16\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{24}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{24} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{48}^{12} q^{2} + \zeta_{48}^{5} q^{3} - q^{4} + \zeta_{48}^{17} q^{6} - \zeta_{48}^{12} q^{8} + \zeta_{48}^{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{48}^{12} q^{2} + \zeta_{48}^{5} q^{3} - q^{4} + \zeta_{48}^{17} q^{6} - \zeta_{48}^{12} q^{8} + \zeta_{48}^{10} q^{9} + ( - \zeta_{48}^{22} - \zeta_{48}^{18}) q^{11} - \zeta_{48}^{5} q^{12} + q^{16} + ( - \zeta_{48}^{23} - \zeta_{48}^{9}) q^{17} + \zeta_{48}^{22} q^{18} + ( - \zeta_{48}^{9} + \zeta_{48}^{7}) q^{19} + (\zeta_{48}^{10} + \zeta_{48}^{6}) q^{22} - \zeta_{48}^{17} q^{24} - \zeta_{48}^{8} q^{25} + \zeta_{48}^{15} q^{27} + \zeta_{48}^{12} q^{32} + ( - \zeta_{48}^{23} + \zeta_{48}^{3}) q^{33} + ( - \zeta_{48}^{21} + \zeta_{48}^{11}) q^{34} - \zeta_{48}^{10} q^{36} + ( - \zeta_{48}^{21} + \zeta_{48}^{19}) q^{38} + (\zeta_{48}^{13} + \zeta_{48}^{3}) q^{41} + ( - \zeta_{48}^{6} + \zeta_{48}^{2}) q^{43} + (\zeta_{48}^{22} + \zeta_{48}^{18}) q^{44} + \zeta_{48}^{5} q^{48} - \zeta_{48}^{20} q^{50} + ( - \zeta_{48}^{14} + \zeta_{48}^{4}) q^{51} - \zeta_{48}^{3} q^{54} + ( - \zeta_{48}^{14} + \zeta_{48}^{12}) q^{57} + (\zeta_{48}^{23} - \zeta_{48}) q^{59} - q^{64} + (\zeta_{48}^{15} + \zeta_{48}^{11}) q^{66} + ( - \zeta_{48}^{20} + \zeta_{48}^{4}) q^{67} + (\zeta_{48}^{23} + \zeta_{48}^{9}) q^{68} - \zeta_{48}^{22} q^{72} + ( - \zeta_{48}^{5} - \zeta_{48}^{3}) q^{73} - \zeta_{48}^{13} q^{75} + (\zeta_{48}^{9} - \zeta_{48}^{7}) q^{76} + \zeta_{48}^{20} q^{81} + (\zeta_{48}^{15} - \zeta_{48}) q^{82} + (\zeta_{48}^{7} - \zeta_{48}) q^{83} + ( - \zeta_{48}^{18} + \zeta_{48}^{14}) q^{86} + ( - \zeta_{48}^{10} - \zeta_{48}^{6}) q^{88} + (\zeta_{48}^{11} + \zeta_{48}^{5}) q^{89} + \zeta_{48}^{17} q^{96} + (\zeta_{48}^{23} - \zeta_{48}^{9}) q^{97} + (\zeta_{48}^{8} + \zeta_{48}^{4}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 16 q^{16} - 8 q^{25} - 16 q^{64} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(\zeta_{48}^{8}\) \(\zeta_{48}^{8}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
227.1
0.793353 − 0.608761i
0.608761 + 0.793353i
−0.608761 − 0.793353i
−0.793353 + 0.608761i
−0.991445 − 0.130526i
−0.130526 + 0.991445i
0.130526 − 0.991445i
0.991445 + 0.130526i
−0.991445 + 0.130526i
−0.130526 − 0.991445i
0.130526 + 0.991445i
0.991445 − 0.130526i
0.793353 + 0.608761i
0.608761 − 0.793353i
−0.608761 + 0.793353i
−0.793353 − 0.608761i
− 1.00000i −0.991445 + 0.130526i −1.00000 0 0.130526 + 0.991445i 0 1.00000i 0.965926 − 0.258819i 0
227.2 − 1.00000i −0.130526 − 0.991445i −1.00000 0 −0.991445 + 0.130526i 0 1.00000i −0.965926 + 0.258819i 0
227.3 − 1.00000i 0.130526 + 0.991445i −1.00000 0 0.991445 − 0.130526i 0 1.00000i −0.965926 + 0.258819i 0
227.4 − 1.00000i 0.991445 − 0.130526i −1.00000 0 −0.130526 − 0.991445i 0 1.00000i 0.965926 − 0.258819i 0
227.5 1.00000i −0.793353 − 0.608761i −1.00000 0 0.608761 − 0.793353i 0 − 1.00000i 0.258819 + 0.965926i 0
227.6 1.00000i −0.608761 + 0.793353i −1.00000 0 −0.793353 − 0.608761i 0 − 1.00000i −0.258819 − 0.965926i 0
227.7 1.00000i 0.608761 − 0.793353i −1.00000 0 0.793353 + 0.608761i 0 − 1.00000i −0.258819 − 0.965926i 0
227.8 1.00000i 0.793353 + 0.608761i −1.00000 0 −0.608761 + 0.793353i 0 − 1.00000i 0.258819 + 0.965926i 0
3155.1 − 1.00000i −0.793353 + 0.608761i −1.00000 0 0.608761 + 0.793353i 0 1.00000i 0.258819 − 0.965926i 0
3155.2 − 1.00000i −0.608761 − 0.793353i −1.00000 0 −0.793353 + 0.608761i 0 1.00000i −0.258819 + 0.965926i 0
3155.3 − 1.00000i 0.608761 + 0.793353i −1.00000 0 0.793353 − 0.608761i 0 1.00000i −0.258819 + 0.965926i 0
3155.4 − 1.00000i 0.793353 − 0.608761i −1.00000 0 −0.608761 − 0.793353i 0 1.00000i 0.258819 − 0.965926i 0
3155.5 1.00000i −0.991445 − 0.130526i −1.00000 0 0.130526 − 0.991445i 0 − 1.00000i 0.965926 + 0.258819i 0
3155.6 1.00000i −0.130526 + 0.991445i −1.00000 0 −0.991445 − 0.130526i 0 − 1.00000i −0.965926 − 0.258819i 0
3155.7 1.00000i 0.130526 − 0.991445i −1.00000 0 0.991445 + 0.130526i 0 − 1.00000i −0.965926 − 0.258819i 0
3155.8 1.00000i 0.991445 + 0.130526i −1.00000 0 −0.130526 + 0.991445i 0 − 1.00000i 0.965926 + 0.258819i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 227.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
7.b odd 2 1 inner
56.e even 2 1 inner
63.i even 6 1 inner
63.j odd 6 1 inner
504.bt even 6 1 inner
504.cm 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.cm.a 16
7.b odd 2 1 inner 3528.1.cm.a 16
7.c even 3 1 3528.1.u.a 16
7.c even 3 1 3528.1.co.a ✓ 16
7.d odd 6 1 3528.1.u.a 16
7.d odd 6 1 3528.1.co.a ✓ 16
8.d odd 2 1 CM 3528.1.cm.a 16
9.d odd 6 1 3528.1.u.a 16
56.e even 2 1 inner 3528.1.cm.a 16
56.k odd 6 1 3528.1.u.a 16
56.k odd 6 1 3528.1.co.a ✓ 16
56.m even 6 1 3528.1.u.a 16
56.m even 6 1 3528.1.co.a ✓ 16
63.i even 6 1 inner 3528.1.cm.a 16
63.j odd 6 1 inner 3528.1.cm.a 16
63.n odd 6 1 3528.1.co.a ✓ 16
63.o even 6 1 3528.1.u.a 16
63.s even 6 1 3528.1.co.a ✓ 16
72.l even 6 1 3528.1.u.a 16
504.u odd 6 1 3528.1.co.a ✓ 16
504.bt even 6 1 inner 3528.1.cm.a 16
504.cm odd 6 1 inner 3528.1.cm.a 16
504.co odd 6 1 3528.1.u.a 16
504.cy even 6 1 3528.1.co.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.u.a 16 7.c even 3 1
3528.1.u.a 16 7.d odd 6 1
3528.1.u.a 16 9.d odd 6 1
3528.1.u.a 16 56.k odd 6 1
3528.1.u.a 16 56.m even 6 1
3528.1.u.a 16 63.o even 6 1
3528.1.u.a 16 72.l even 6 1
3528.1.u.a 16 504.co odd 6 1
3528.1.cm.a 16 1.a even 1 1 trivial
3528.1.cm.a 16 7.b odd 2 1 inner
3528.1.cm.a 16 8.d odd 2 1 CM
3528.1.cm.a 16 56.e even 2 1 inner
3528.1.cm.a 16 63.i even 6 1 inner
3528.1.cm.a 16 63.j odd 6 1 inner
3528.1.cm.a 16 504.bt even 6 1 inner
3528.1.cm.a 16 504.cm odd 6 1 inner
3528.1.co.a ✓ 16 7.c even 3 1
3528.1.co.a ✓ 16 7.d odd 6 1
3528.1.co.a ✓ 16 56.k odd 6 1
3528.1.co.a ✓ 16 56.m even 6 1
3528.1.co.a ✓ 16 63.n odd 6 1
3528.1.co.a ✓ 16 63.s even 6 1
3528.1.co.a ✓ 16 504.u odd 6 1
3528.1.co.a ✓ 16 504.cy even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{8} + 1 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 4 T^{6} + 15 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} + 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{16} - 8 T^{14} + \cdots + 1 \) 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
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} + 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( (T^{8} + 4 T^{6} + 15 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{8} - 8 T^{6} + 20 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( (T^{2} - 3)^{8} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} - 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{16} \) Copy content Toggle raw display
$83$ \( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 4 T^{6} + 14 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} - 8 T^{14} + \cdots + 1 \) Copy content Toggle raw display
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