Properties

Label 3311.1.bx.a
Level $3311$
Weight $1$
Character orbit 3311.bx
Analytic conductor $1.652$
Analytic rank $0$
Dimension $6$
Projective image $D_{14}$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3311,1,Mod(538,3311)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3311, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 7, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3311.538");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3311 = 7 \cdot 11 \cdot 43 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3311.bx (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: \(1.65240425683\)
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: \(\Z[a_1, \ldots, a_{4}]\)
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.

\(f(q)\) \(=\) \( q + ( - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{2} + ( - \zeta_{14}^{3} + \zeta_{14}^{2} - \zeta_{14}) q^{4} + q^{7} + (\zeta_{14}^{6} - \zeta_{14}^{5} - \zeta_{14} - 1) q^{8} + \zeta_{14}^{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{2} + ( - \zeta_{14}^{3} + \zeta_{14}^{2} - \zeta_{14}) q^{4} + q^{7} + (\zeta_{14}^{6} - \zeta_{14}^{5} - \zeta_{14} - 1) q^{8} + \zeta_{14}^{5} q^{9} - \zeta_{14}^{4} q^{11} + ( - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{14} + (\zeta_{14}^{6} - \zeta_{14}^{5} + \zeta_{14}^{4} - \zeta_{14}^{3} + \zeta_{14}^{2}) q^{16} + (\zeta_{14}^{3} - \zeta_{14}^{2}) q^{18} + ( - \zeta_{14}^{2} + \zeta_{14}) q^{22} + ( - \zeta_{14}^{2} + \zeta_{14}) q^{23} - \zeta_{14}^{6} q^{25} + ( - \zeta_{14}^{3} + \zeta_{14}^{2} - \zeta_{14}) q^{28} + ( - \zeta_{14}^{6} + \zeta_{14}^{3}) q^{29} + (\zeta_{14}^{6} + \zeta_{14}^{4} - \zeta_{14}^{3} + \zeta_{14}^{2} + \zeta_{14} - 1) q^{32} + ( - \zeta_{14}^{6} + \zeta_{14} - 1) q^{36} + ( - \zeta_{14}^{4} - \zeta_{14}^{3}) q^{37} - \zeta_{14}^{2} q^{43} + ( - \zeta_{14}^{6} + \zeta_{14}^{5} - 1) q^{44} + ( - 2 \zeta_{14}^{6} + \zeta_{14}^{5} - 1) q^{46} + q^{49} + ( - \zeta_{14}^{4} + \zeta_{14}^{3}) q^{50} + ( - \zeta_{14} + 1) q^{53} + (\zeta_{14}^{6} - \zeta_{14}^{5} - \zeta_{14} + 1) q^{56} + ( - \zeta_{14}^{4} + \zeta_{14}^{3} + \zeta_{14} - 1) q^{58} + \zeta_{14}^{5} q^{63} + (\zeta_{14}^{6} - \zeta_{14}^{5} + \zeta_{14}^{4} - \zeta_{14}^{3} + \zeta_{14}^{2} + \zeta_{14} - 1) q^{64} + ( - \zeta_{14}^{3} - \zeta_{14}) q^{67} + ( - \zeta_{14}^{6} + \zeta_{14}^{5} - \zeta_{14}^{4} + \zeta_{14}^{3}) q^{72} + ( - \zeta_{14}^{2} + 1) q^{74} - \zeta_{14}^{4} q^{77} + (\zeta_{14}^{5} + \zeta_{14}^{2}) q^{79} - \zeta_{14}^{3} q^{81} + ( - \zeta_{14}^{6} - 1) q^{86} + (\zeta_{14}^{5} - \zeta_{14}^{4} + \zeta_{14}^{3} - \zeta_{14}^{2}) q^{88} + (\zeta_{14}^{5} - 2 \zeta_{14}^{4} + 2 \zeta_{14}^{3} - \zeta_{14}^{2}) q^{92} + ( - \zeta_{14}^{5} + \zeta_{14}^{4}) q^{98} + \zeta_{14}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} - 3 q^{4} + 6 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} - 3 q^{4} + 6 q^{7} + 3 q^{8} + q^{9} + q^{11} - 2 q^{14} - 5 q^{16} + 2 q^{18} + 2 q^{22} + 2 q^{23} + q^{25} - 3 q^{28} + 2 q^{29} + q^{32} - 4 q^{36} + q^{43} - 4 q^{44} - 3 q^{46} + 6 q^{49} + 2 q^{50} + 5 q^{53} + 3 q^{56} - 3 q^{58} + q^{63} - 2 q^{67} + 4 q^{72} + 7 q^{74} + q^{77} - q^{81} - 5 q^{86} + 4 q^{88} + 6 q^{92} - 2 q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3311\mathbb{Z}\right)^\times\).

\(n\) \(904\) \(1893\) \(2927\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{14}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
538.1
0.900969 + 0.433884i
0.900969 0.433884i
0.222521 + 0.974928i
−0.623490 0.781831i
−0.623490 + 0.781831i
0.222521 0.974928i
0.400969 + 0.193096i 0 −0.500000 0.626980i 0 0 1.00000 −0.178448 0.781831i −0.623490 + 0.781831i 0
1077.1 0.400969 0.193096i 0 −0.500000 + 0.626980i 0 0 1.00000 −0.178448 + 0.781831i −0.623490 0.781831i 0
1231.1 −0.277479 1.21572i 0 −0.500000 + 0.240787i 0 0 1.00000 −0.346011 0.433884i 0.900969 + 0.433884i 0
1924.1 −1.12349 1.40881i 0 −0.500000 + 2.19064i 0 0 1.00000 2.02446 0.974928i 0.222521 + 0.974928i 0
2232.1 −1.12349 + 1.40881i 0 −0.500000 2.19064i 0 0 1.00000 2.02446 + 0.974928i 0.222521 0.974928i 0
3233.1 −0.277479 + 1.21572i 0 −0.500000 0.240787i 0 0 1.00000 −0.346011 + 0.433884i 0.900969 0.433884i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 538.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
473.n even 14 1 inner
3311.bx odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3311.1.bx.a 6
7.b odd 2 1 CM 3311.1.bx.a 6
11.b odd 2 1 3311.1.bx.b yes 6
43.f odd 14 1 3311.1.bx.b yes 6
77.b even 2 1 3311.1.bx.b yes 6
301.w even 14 1 3311.1.bx.b yes 6
473.n even 14 1 inner 3311.1.bx.a 6
3311.bx odd 14 1 inner 3311.1.bx.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3311.1.bx.a 6 1.a even 1 1 trivial
3311.1.bx.a 6 7.b odd 2 1 CM
3311.1.bx.a 6 473.n even 14 1 inner
3311.1.bx.a 6 3311.bx odd 14 1 inner
3311.1.bx.b yes 6 11.b odd 2 1
3311.1.bx.b yes 6 43.f odd 14 1
3311.1.bx.b yes 6 77.b even 2 1
3311.1.bx.b yes 6 301.w even 14 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 2T_{2}^{5} + 4T_{2}^{4} + T_{2}^{3} + 2T_{2}^{2} - 3T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3311, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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