Properties

Label 3528.1.dh.a
Level $3528$
Weight $1$
Character orbit 3528.dh
Analytic conductor $1.761$
Analytic rank $0$
Dimension $12$
Projective image $D_{14}$
CM discriminant -24
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(181,3528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 7, 0, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3528.181");
 
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.dh (of order \(14\), degree \(6\), 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: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{28} q^{2} + \zeta_{28}^{2} q^{4} + (\zeta_{28}^{11} + \zeta_{28}^{9}) q^{5} + \zeta_{28}^{4} q^{7} + \zeta_{28}^{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{28} q^{2} + \zeta_{28}^{2} q^{4} + (\zeta_{28}^{11} + \zeta_{28}^{9}) q^{5} + \zeta_{28}^{4} q^{7} + \zeta_{28}^{3} q^{8} + (\zeta_{28}^{12} + \zeta_{28}^{10}) q^{10} + (\zeta_{28}^{13} - \zeta_{28}^{3}) q^{11} + \zeta_{28}^{5} q^{14} + \zeta_{28}^{4} q^{16} + (\zeta_{28}^{13} + \zeta_{28}^{11}) q^{20} + ( - \zeta_{28}^{4} - 1) q^{22} + ( - \zeta_{28}^{8} - \zeta_{28}^{6} - \zeta_{28}^{4}) q^{25} + \zeta_{28}^{6} q^{28} + ( - \zeta_{28}^{13} + \zeta_{28}^{11}) q^{29} + (\zeta_{28}^{10} + \zeta_{28}^{4}) q^{31} + \zeta_{28}^{5} q^{32} + (\zeta_{28}^{13} - \zeta_{28}) q^{35} + (\zeta_{28}^{12} - 1) q^{40} + ( - \zeta_{28}^{5} - \zeta_{28}) q^{44} + \zeta_{28}^{8} q^{49} + ( - \zeta_{28}^{9} - \zeta_{28}^{7} - \zeta_{28}^{5}) q^{50} - \zeta_{28}^{9} q^{53} + ( - \zeta_{28}^{12} - \zeta_{28}^{10} - \zeta_{28}^{8} + 1) q^{55} + \zeta_{28}^{7} q^{56} + (\zeta_{28}^{12} + 1) q^{58} + (\zeta_{28}^{11} + \zeta_{28}^{5}) q^{62} + \zeta_{28}^{6} q^{64} + ( - \zeta_{28}^{2} - 1) q^{70} + (\zeta_{28}^{10} - \zeta_{28}^{2}) q^{73} + ( - \zeta_{28}^{7} - \zeta_{28}^{3}) q^{77} + (\zeta_{28}^{10} - \zeta_{28}^{4}) q^{79} + (\zeta_{28}^{13} - \zeta_{28}) q^{80} + (\zeta_{28}^{7} + \zeta_{28}^{5}) q^{83} + ( - \zeta_{28}^{6} - \zeta_{28}^{2}) q^{88} + ( - \zeta_{28}^{12} - \zeta_{28}^{2}) q^{97} + \zeta_{28}^{9} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{4} - 2 q^{7} - 2 q^{16} - 10 q^{22} + 2 q^{25} + 2 q^{28} - 14 q^{40} - 2 q^{49} + 14 q^{55} + 10 q^{58} + 2 q^{64} - 14 q^{70} + 4 q^{79} - 4 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(\zeta_{28}^{6}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
−0.974928 0.222521i
0.974928 + 0.222521i
−0.974928 + 0.222521i
0.974928 0.222521i
−0.433884 0.900969i
0.433884 + 0.900969i
−0.781831 + 0.623490i
0.781831 0.623490i
−0.781831 0.623490i
0.781831 + 0.623490i
−0.433884 + 0.900969i
0.433884 0.900969i
−0.974928 0.222521i 0 0.900969 + 0.433884i 1.21572 1.52446i 0 0.623490 + 0.781831i −0.781831 0.623490i 0 −1.52446 + 1.21572i
181.2 0.974928 + 0.222521i 0 0.900969 + 0.433884i −1.21572 + 1.52446i 0 0.623490 + 0.781831i 0.781831 + 0.623490i 0 −1.52446 + 1.21572i
1189.1 −0.974928 + 0.222521i 0 0.900969 0.433884i 1.21572 + 1.52446i 0 0.623490 0.781831i −0.781831 + 0.623490i 0 −1.52446 1.21572i
1189.2 0.974928 0.222521i 0 0.900969 0.433884i −1.21572 1.52446i 0 0.623490 0.781831i 0.781831 0.623490i 0 −1.52446 1.21572i
1693.1 −0.433884 0.900969i 0 −0.623490 + 0.781831i −0.193096 + 0.846011i 0 −0.222521 0.974928i 0.974928 + 0.222521i 0 0.846011 0.193096i
1693.2 0.433884 + 0.900969i 0 −0.623490 + 0.781831i 0.193096 0.846011i 0 −0.222521 0.974928i −0.974928 0.222521i 0 0.846011 0.193096i
2197.1 −0.781831 + 0.623490i 0 0.222521 0.974928i −1.40881 + 0.678448i 0 −0.900969 0.433884i 0.433884 + 0.900969i 0 0.678448 1.40881i
2197.2 0.781831 0.623490i 0 0.222521 0.974928i 1.40881 0.678448i 0 −0.900969 0.433884i −0.433884 0.900969i 0 0.678448 1.40881i
2701.1 −0.781831 0.623490i 0 0.222521 + 0.974928i −1.40881 0.678448i 0 −0.900969 + 0.433884i 0.433884 0.900969i 0 0.678448 + 1.40881i
2701.2 0.781831 + 0.623490i 0 0.222521 + 0.974928i 1.40881 + 0.678448i 0 −0.900969 + 0.433884i −0.433884 + 0.900969i 0 0.678448 + 1.40881i
3205.1 −0.433884 + 0.900969i 0 −0.623490 0.781831i −0.193096 0.846011i 0 −0.222521 + 0.974928i 0.974928 0.222521i 0 0.846011 + 0.193096i
3205.2 0.433884 0.900969i 0 −0.623490 0.781831i 0.193096 + 0.846011i 0 −0.222521 + 0.974928i −0.974928 + 0.222521i 0 0.846011 + 0.193096i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
49.f odd 14 1 inner
147.k even 14 1 inner
392.r odd 14 1 inner
1176.bo even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3528.1.dh.a 12
3.b odd 2 1 inner 3528.1.dh.a 12
8.b even 2 1 inner 3528.1.dh.a 12
24.h odd 2 1 CM 3528.1.dh.a 12
49.f odd 14 1 inner 3528.1.dh.a 12
147.k even 14 1 inner 3528.1.dh.a 12
392.r odd 14 1 inner 3528.1.dh.a 12
1176.bo even 14 1 inner 3528.1.dh.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3528.1.dh.a 12 1.a even 1 1 trivial
3528.1.dh.a 12 3.b odd 2 1 inner
3528.1.dh.a 12 8.b even 2 1 inner
3528.1.dh.a 12 24.h odd 2 1 CM
3528.1.dh.a 12 49.f odd 14 1 inner
3528.1.dh.a 12 147.k even 14 1 inner
3528.1.dh.a 12 392.r odd 14 1 inner
3528.1.dh.a 12 1176.bo even 14 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{10} + T^{8} - T^{6} + T^{4} - T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 14 T^{8} - 14 T^{6} + 49 T^{4} + \cdots + 49 \) Copy content Toggle raw display
$7$ \( (T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{10} + 2 T^{8} + 6 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} - 4 T^{10} + 16 T^{8} - 29 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( (T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} \) Copy content Toggle raw display
$53$ \( T^{12} - 4 T^{10} + 16 T^{8} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( T^{12} \) Copy content Toggle raw display
$67$ \( T^{12} \) Copy content Toggle raw display
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( (T^{6} + 7 T^{3} - 7 T + 7)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - T^{2} - 2 T + 1)^{4} \) Copy content Toggle raw display
$83$ \( T^{12} + 7 T^{10} + 21 T^{8} + 35 T^{6} + \cdots + 49 \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( (T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2} \) Copy content Toggle raw display
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