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Properties

Label 3332.1.be.c
Level $3332$
Weight $1$
Character orbit 3332.be
Analytic conductor $1.663$
Analytic rank $0$
Dimension $12$
Projective image $D_{14}$
CM discriminant -68
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3332,1,Mod(407,3332)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 10, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3332.407");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3332.be (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: \(1.66288462209\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q - \zeta_{28}^{8} q^{2} + (\zeta_{28}^{13} + \zeta_{28}^{7}) q^{3} - \zeta_{28}^{2} q^{4} + (\zeta_{28}^{7} + \zeta_{28}) q^{6} + \zeta_{28}^{3} q^{7} + \zeta_{28}^{10} q^{8} + ( - \zeta_{28}^{12} - \zeta_{28}^{6} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{28}^{8} q^{2} + (\zeta_{28}^{13} + \zeta_{28}^{7}) q^{3} - \zeta_{28}^{2} q^{4} + (\zeta_{28}^{7} + \zeta_{28}) q^{6} + \zeta_{28}^{3} q^{7} + \zeta_{28}^{10} q^{8} + ( - \zeta_{28}^{12} - \zeta_{28}^{6} - 1) q^{9} + (\zeta_{28}^{9} + \zeta_{28}^{7}) q^{11} + ( - \zeta_{28}^{9} + \zeta_{28}) q^{12} + (\zeta_{28}^{2} - 1) q^{13} - \zeta_{28}^{11} q^{14} + \zeta_{28}^{4} q^{16} + \zeta_{28}^{12} q^{17} + (\zeta_{28}^{8} - \zeta_{28}^{6} - 1) q^{18} + (\zeta_{28}^{10} - \zeta_{28}^{2}) q^{21} + (\zeta_{28}^{3} + \zeta_{28}) q^{22} + ( - \zeta_{28}^{3} - \zeta_{28}) q^{23} + ( - \zeta_{28}^{9} - \zeta_{28}^{3}) q^{24} - \zeta_{28}^{6} q^{25} + ( - \zeta_{28}^{10} + \zeta_{28}^{8}) q^{26} + ( - \zeta_{28}^{13} + \cdots - \zeta_{28}^{5}) q^{27} + \cdots + ( - \zeta_{28}^{13} + \cdots + \zeta_{28}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{2} - 2 q^{4} + 2 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{2} - 2 q^{4} + 2 q^{8} - 12 q^{9} - 10 q^{13} - 2 q^{16} - 2 q^{17} - 16 q^{18} - 2 q^{25} - 4 q^{26} + 2 q^{32} - 14 q^{33} + 2 q^{34} - 12 q^{36} + 2 q^{49} - 12 q^{50} + 4 q^{52} - 4 q^{53} - 2 q^{64} - 14 q^{66} + 12 q^{68} + 14 q^{69} - 2 q^{72} + 12 q^{81} + 4 q^{89} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(-\zeta_{28}^{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
407.1
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
0.222521 − 0.974928i −0.974928 + 1.22252i −0.900969 − 0.433884i 0 0.974928 + 1.22252i 0.781831 + 0.623490i −0.623490 + 0.781831i −0.321552 − 1.40881i 0
407.2 0.222521 − 0.974928i 0.974928 − 1.22252i −0.900969 − 0.433884i 0 −0.974928 − 1.22252i −0.781831 − 0.623490i −0.623490 + 0.781831i −0.321552 − 1.40881i 0
1359.1 0.222521 + 0.974928i −0.974928 − 1.22252i −0.900969 + 0.433884i 0 0.974928 − 1.22252i 0.781831 − 0.623490i −0.623490 − 0.781831i −0.321552 + 1.40881i 0
1359.2 0.222521 + 0.974928i 0.974928 + 1.22252i −0.900969 + 0.433884i 0 −0.974928 + 1.22252i −0.781831 + 0.623490i −0.623490 − 0.781831i −0.321552 + 1.40881i 0
1835.1 0.900969 − 0.433884i −0.433884 + 1.90097i 0.623490 − 0.781831i 0 0.433884 + 1.90097i −0.974928 − 0.222521i 0.222521 − 0.974928i −2.52446 − 1.21572i 0
1835.2 0.900969 − 0.433884i 0.433884 − 1.90097i 0.623490 − 0.781831i 0 −0.433884 − 1.90097i 0.974928 + 0.222521i 0.222521 − 0.974928i −2.52446 − 1.21572i 0
2311.1 −0.623490 − 0.781831i −0.781831 + 0.376510i −0.222521 + 0.974928i 0 0.781831 + 0.376510i −0.433884 − 0.900969i 0.900969 − 0.433884i −0.153989 + 0.193096i 0
2311.2 −0.623490 − 0.781831i 0.781831 − 0.376510i −0.222521 + 0.974928i 0 −0.781831 − 0.376510i 0.433884 + 0.900969i 0.900969 − 0.433884i −0.153989 + 0.193096i 0
2787.1 −0.623490 + 0.781831i −0.781831 − 0.376510i −0.222521 − 0.974928i 0 0.781831 − 0.376510i −0.433884 + 0.900969i 0.900969 + 0.433884i −0.153989 − 0.193096i 0
2787.2 −0.623490 + 0.781831i 0.781831 + 0.376510i −0.222521 − 0.974928i 0 −0.781831 + 0.376510i 0.433884 − 0.900969i 0.900969 + 0.433884i −0.153989 − 0.193096i 0
3263.1 0.900969 + 0.433884i −0.433884 − 1.90097i 0.623490 + 0.781831i 0 0.433884 − 1.90097i −0.974928 + 0.222521i 0.222521 + 0.974928i −2.52446 + 1.21572i 0
3263.2 0.900969 + 0.433884i 0.433884 + 1.90097i 0.623490 + 0.781831i 0 −0.433884 + 1.90097i 0.974928 − 0.222521i 0.222521 + 0.974928i −2.52446 + 1.21572i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 407.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
4.b odd 2 1 inner
17.b even 2 1 inner
49.e even 7 1 inner
196.k odd 14 1 inner
833.r even 14 1 inner
3332.be odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3332.1.be.c ✓ 12
4.b odd 2 1 inner 3332.1.be.c ✓ 12
17.b even 2 1 inner 3332.1.be.c ✓ 12
49.e even 7 1 inner 3332.1.be.c ✓ 12
68.d odd 2 1 CM 3332.1.be.c ✓ 12
196.k odd 14 1 inner 3332.1.be.c ✓ 12
833.r even 14 1 inner 3332.1.be.c ✓ 12
3332.be odd 14 1 inner 3332.1.be.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.be.c ✓ 12 1.a even 1 1 trivial
3332.1.be.c ✓ 12 4.b odd 2 1 inner
3332.1.be.c ✓ 12 17.b even 2 1 inner
3332.1.be.c ✓ 12 49.e even 7 1 inner
3332.1.be.c ✓ 12 68.d odd 2 1 CM
3332.1.be.c ✓ 12 196.k odd 14 1 inner
3332.1.be.c ✓ 12 833.r even 14 1 inner
3332.1.be.c ✓ 12 3332.be odd 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 7T_{3}^{10} + 21T_{3}^{8} + 35T_{3}^{6} + 49T_{3}^{4} - 49T_{3}^{2} + 49 \) acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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