Properties

Label 3332.1.be.b
Level $3332$
Weight $1$
Character orbit 3332.be
Analytic conductor $1.663$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -68
Inner twists $4$

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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: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{7} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{14}^{4} q^{2} + ( - \zeta_{14}^{3} + 1) q^{3} - \zeta_{14} q^{4} + (\zeta_{14}^{4} + 1) q^{6} - \zeta_{14}^{5} q^{7} - \zeta_{14}^{5} q^{8} + (\zeta_{14}^{6} - \zeta_{14}^{3} + 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{14}^{4} q^{2} + ( - \zeta_{14}^{3} + 1) q^{3} - \zeta_{14} q^{4} + (\zeta_{14}^{4} + 1) q^{6} - \zeta_{14}^{5} q^{7} - \zeta_{14}^{5} q^{8} + (\zeta_{14}^{6} - \zeta_{14}^{3} + 1) q^{9} + ( - \zeta_{14} + 1) q^{11} + (\zeta_{14}^{4} - \zeta_{14}) q^{12} + ( - \zeta_{14} + 1) q^{13} + \zeta_{14}^{2} q^{14} + \zeta_{14}^{2} q^{16} + \zeta_{14}^{6} q^{17} + (\zeta_{14}^{4} - \zeta_{14}^{3} + 1) q^{18} + ( - \zeta_{14}^{5} - \zeta_{14}) q^{21} + ( - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{22} + ( - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{23} + ( - \zeta_{14}^{5} - \zeta_{14}) q^{24} - \zeta_{14}^{3} q^{25} + ( - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{26} + (\zeta_{14}^{6} - \zeta_{14}^{3} + \cdots + 1) q^{27} + \cdots + (\zeta_{14}^{6} + \zeta_{14}^{4} + \cdots + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} + 5 q^{3} - q^{4} + 5 q^{6} - q^{7} - q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} + 5 q^{3} - q^{4} + 5 q^{6} - q^{7} - q^{8} + 4 q^{9} + 5 q^{11} - 2 q^{12} + 5 q^{13} - q^{14} - q^{16} - q^{17} + 4 q^{18} - 2 q^{21} - 2 q^{22} - 2 q^{23} - 2 q^{24} - q^{25} - 2 q^{26} + 3 q^{27} - q^{28} - 2 q^{31} - q^{32} + 3 q^{33} - q^{34} + 4 q^{36} + 3 q^{39} - 2 q^{42} - 2 q^{44} - 2 q^{46} - 2 q^{48} - q^{49} + 6 q^{50} - 2 q^{51} - 2 q^{52} - 2 q^{53} + 3 q^{54} - q^{56} - 2 q^{62} - 3 q^{63} - q^{64} + 3 q^{66} + 6 q^{68} + 3 q^{69} - 2 q^{71} - 3 q^{72} - 2 q^{75} - 2 q^{77} + 3 q^{78} - 2 q^{79} + 2 q^{81} - 2 q^{84} - 2 q^{88} - 2 q^{89} - 2 q^{91} - 2 q^{92} - 4 q^{93} - 2 q^{96} + 6 q^{98} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(885\) \(1667\)
\(\chi(n)\) \(-1\) \(-\zeta_{14}^{3}\) \(-1\)

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
0.900969 + 0.433884i
0.900969 0.433884i
−0.623490 + 0.781831i
0.222521 0.974928i
0.222521 + 0.974928i
−0.623490 0.781831i
−0.222521 + 0.974928i 0.777479 0.974928i −0.900969 0.433884i 0 0.777479 + 0.974928i 0.623490 0.781831i 0.623490 0.781831i −0.123490 0.541044i 0
1359.1 −0.222521 0.974928i 0.777479 + 0.974928i −0.900969 + 0.433884i 0 0.777479 0.974928i 0.623490 + 0.781831i 0.623490 + 0.781831i −0.123490 + 0.541044i 0
1835.1 −0.900969 + 0.433884i 0.0990311 0.433884i 0.623490 0.781831i 0 0.0990311 + 0.433884i −0.222521 + 0.974928i −0.222521 + 0.974928i 0.722521 + 0.347948i 0
2311.1 0.623490 + 0.781831i 1.62349 0.781831i −0.222521 + 0.974928i 0 1.62349 + 0.781831i −0.900969 + 0.433884i −0.900969 + 0.433884i 1.40097 1.75676i 0
2787.1 0.623490 0.781831i 1.62349 + 0.781831i −0.222521 0.974928i 0 1.62349 0.781831i −0.900969 0.433884i −0.900969 0.433884i 1.40097 + 1.75676i 0
3263.1 −0.900969 0.433884i 0.0990311 + 0.433884i 0.623490 + 0.781831i 0 0.0990311 0.433884i −0.222521 0.974928i −0.222521 0.974928i 0.722521 0.347948i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 407.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
68.d odd 2 1 CM by \(\Q(\sqrt{-17}) \)
49.e even 7 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.b yes 6
4.b odd 2 1 3332.1.be.a 6
17.b even 2 1 3332.1.be.a 6
49.e even 7 1 inner 3332.1.be.b yes 6
68.d odd 2 1 CM 3332.1.be.b yes 6
196.k odd 14 1 3332.1.be.a 6
833.r even 14 1 3332.1.be.a 6
3332.be odd 14 1 inner 3332.1.be.b yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3332.1.be.a 6 4.b odd 2 1
3332.1.be.a 6 17.b even 2 1
3332.1.be.a 6 196.k odd 14 1
3332.1.be.a 6 833.r even 14 1
3332.1.be.b yes 6 1.a even 1 1 trivial
3332.1.be.b yes 6 49.e even 7 1 inner
3332.1.be.b yes 6 68.d odd 2 1 CM
3332.1.be.b yes 6 3332.be odd 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 5T_{3}^{5} + 11T_{3}^{4} - 13T_{3}^{3} + 9T_{3}^{2} - 3T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3332, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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