Properties

Label 3364.1.h.b
Level $3364$
Weight $1$
Character orbit 3364.h
Analytic conductor $1.679$
Analytic rank $0$
Dimension $6$
Projective image $D_{3}$
CM discriminant -116
Inner twists $12$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3364,1,Mod(63,3364)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3364, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 11]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3364.63");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3364 = 2^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3364.h (of order \(14\), degree \(6\), 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: \(1.67885470250\)
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: no (minimal twist has level 116)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.116.1
Artin image: $S_3\times C_{14}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{42} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{14}^{2} q^{2} - \zeta_{14}^{3} q^{3} + \zeta_{14}^{4} q^{4} - \zeta_{14}^{2} q^{5} + \zeta_{14}^{5} q^{6} - \zeta_{14}^{6} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{14}^{2} q^{2} - \zeta_{14}^{3} q^{3} + \zeta_{14}^{4} q^{4} - \zeta_{14}^{2} q^{5} + \zeta_{14}^{5} q^{6} - \zeta_{14}^{6} q^{8} + \zeta_{14}^{4} q^{10} - \zeta_{14} q^{11} + q^{12} + \zeta_{14} q^{13} + \zeta_{14}^{5} q^{15} - \zeta_{14} q^{16} - \zeta_{14}^{4} q^{19} - \zeta_{14}^{6} q^{20} + \zeta_{14}^{3} q^{22} - \zeta_{14}^{2} q^{24} - \zeta_{14}^{3} q^{26} - \zeta_{14}^{2} q^{27} + q^{30} + \zeta_{14}^{2} q^{31} + \zeta_{14}^{3} q^{32} + \zeta_{14}^{4} q^{33} + 2 \zeta_{14}^{6} q^{38} - \zeta_{14}^{4} q^{39} - \zeta_{14} q^{40} - \zeta_{14}^{5} q^{43} - \zeta_{14}^{5} q^{44} - \zeta_{14} q^{47} + \zeta_{14}^{4} q^{48} + \zeta_{14}^{6} q^{49} + \zeta_{14}^{5} q^{52} - \zeta_{14}^{2} q^{53} + \zeta_{14}^{4} q^{54} + \zeta_{14}^{3} q^{55} - 2 q^{57} - \zeta_{14}^{2} q^{60} - \zeta_{14}^{4} q^{62} - \zeta_{14}^{5} q^{64} - \zeta_{14}^{3} q^{65} - \zeta_{14}^{6} q^{66} + 2 \zeta_{14} q^{76} + \zeta_{14}^{6} q^{78} + \zeta_{14}^{6} q^{79} + \zeta_{14}^{3} q^{80} + \zeta_{14}^{5} q^{81} - q^{86} - q^{88} - \zeta_{14}^{5} q^{93} + \zeta_{14}^{3} q^{94} + 2 \zeta_{14}^{6} q^{95} - \zeta_{14}^{6} q^{96} + \zeta_{14} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + q^{8} - q^{10} - q^{11} + 6 q^{12} + q^{13} + q^{15} - q^{16} + 2 q^{19} + q^{20} + q^{22} + q^{24} - q^{26} + q^{27} + 6 q^{30} - q^{31} + q^{32} - q^{33} - 2 q^{38} + q^{39} - q^{40} - q^{43} - q^{44} - q^{47} - q^{48} - q^{49} + q^{52} + q^{53} - q^{54} + q^{55} - 12 q^{57} + q^{60} + q^{62} - q^{64} - q^{65} + q^{66} + 2 q^{76} - q^{78} - q^{79} + q^{80} + q^{81} - 6 q^{86} - 6 q^{88} - q^{93} + q^{94} - 2 q^{95} + q^{96} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3364\mathbb{Z}\right)^\times\).

\(n\) \(1683\) \(2525\)
\(\chi(n)\) \(-1\) \(-\zeta_{14}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1
0.900969 + 0.433884i
0.900969 0.433884i
0.222521 0.974928i
0.222521 + 0.974928i
−0.623490 + 0.781831i
−0.623490 0.781831i
−0.623490 0.781831i −0.222521 0.974928i −0.222521 + 0.974928i −0.623490 0.781831i −0.623490 + 0.781831i 0 0.900969 0.433884i 0 −0.222521 + 0.974928i
267.1 −0.623490 + 0.781831i −0.222521 + 0.974928i −0.222521 0.974928i −0.623490 + 0.781831i −0.623490 0.781831i 0 0.900969 + 0.433884i 0 −0.222521 0.974928i
651.1 0.900969 + 0.433884i 0.623490 0.781831i 0.623490 + 0.781831i 0.900969 + 0.433884i 0.900969 0.433884i 0 0.222521 + 0.974928i 0 0.623490 + 0.781831i
1111.1 0.900969 0.433884i 0.623490 + 0.781831i 0.623490 0.781831i 0.900969 0.433884i 0.900969 + 0.433884i 0 0.222521 0.974928i 0 0.623490 0.781831i
2719.1 0.222521 + 0.974928i −0.900969 0.433884i −0.900969 + 0.433884i 0.222521 + 0.974928i 0.222521 0.974928i 0 −0.623490 0.781831i 0 −0.900969 + 0.433884i
2759.1 0.222521 0.974928i −0.900969 + 0.433884i −0.900969 0.433884i 0.222521 0.974928i 0.222521 + 0.974928i 0 −0.623490 + 0.781831i 0 −0.900969 0.433884i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
116.d odd 2 1 CM by \(\Q(\sqrt{-29}) \)
29.d even 7 5 inner
116.h odd 14 5 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3364.1.h.b 6
4.b odd 2 1 3364.1.h.a 6
29.b even 2 1 3364.1.h.a 6
29.c odd 4 2 3364.1.j.f 12
29.d even 7 1 116.1.d.a 1
29.d even 7 5 inner 3364.1.h.b 6
29.e even 14 1 116.1.d.b yes 1
29.e even 14 5 3364.1.h.a 6
29.f odd 28 2 3364.1.b.a 2
29.f odd 28 10 3364.1.j.f 12
87.h odd 14 1 1044.1.g.a 1
87.j odd 14 1 1044.1.g.b 1
116.d odd 2 1 CM 3364.1.h.b 6
116.e even 4 2 3364.1.j.f 12
116.h odd 14 1 116.1.d.a 1
116.h odd 14 5 inner 3364.1.h.b 6
116.j odd 14 1 116.1.d.b yes 1
116.j odd 14 5 3364.1.h.a 6
116.l even 28 2 3364.1.b.a 2
116.l even 28 10 3364.1.j.f 12
145.l even 14 1 2900.1.g.a 1
145.n even 14 1 2900.1.g.d 1
145.p odd 28 2 2900.1.e.b 2
145.q odd 28 2 2900.1.e.a 2
232.o even 14 1 1856.1.h.c 1
232.p odd 14 1 1856.1.h.c 1
232.s even 14 1 1856.1.h.a 1
232.t odd 14 1 1856.1.h.a 1
348.s even 14 1 1044.1.g.a 1
348.t even 14 1 1044.1.g.b 1
580.v odd 14 1 2900.1.g.a 1
580.y odd 14 1 2900.1.g.d 1
580.bh even 28 2 2900.1.e.b 2
580.bi even 28 2 2900.1.e.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.1.d.a 1 29.d even 7 1
116.1.d.a 1 116.h odd 14 1
116.1.d.b yes 1 29.e even 14 1
116.1.d.b yes 1 116.j odd 14 1
1044.1.g.a 1 87.h odd 14 1
1044.1.g.a 1 348.s even 14 1
1044.1.g.b 1 87.j odd 14 1
1044.1.g.b 1 348.t even 14 1
1856.1.h.a 1 232.s even 14 1
1856.1.h.a 1 232.t odd 14 1
1856.1.h.c 1 232.o even 14 1
1856.1.h.c 1 232.p odd 14 1
2900.1.e.a 2 145.q odd 28 2
2900.1.e.a 2 580.bi even 28 2
2900.1.e.b 2 145.p odd 28 2
2900.1.e.b 2 580.bh even 28 2
2900.1.g.a 1 145.l even 14 1
2900.1.g.a 1 580.v odd 14 1
2900.1.g.d 1 145.n even 14 1
2900.1.g.d 1 580.y odd 14 1
3364.1.b.a 2 29.f odd 28 2
3364.1.b.a 2 116.l even 28 2
3364.1.h.a 6 4.b odd 2 1
3364.1.h.a 6 29.b even 2 1
3364.1.h.a 6 29.e even 14 5
3364.1.h.a 6 116.j odd 14 5
3364.1.h.b 6 1.a even 1 1 trivial
3364.1.h.b 6 29.d even 7 5 inner
3364.1.h.b 6 116.d odd 2 1 CM
3364.1.h.b 6 116.h odd 14 5 inner
3364.1.j.f 12 29.c odd 4 2
3364.1.j.f 12 29.f odd 28 10
3364.1.j.f 12 116.e even 4 2
3364.1.j.f 12 116.l even 28 10

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3364, [\chi])\):

\( T_{3}^{6} + T_{3}^{5} + T_{3}^{4} + T_{3}^{3} + T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} - T_{5}^{5} + T_{5}^{4} - T_{5}^{3} + T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$13$ \( T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{6} \) Copy content Toggle raw display
$19$ \( T^{6} - 2 T^{5} + 4 T^{4} - 8 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$37$ \( T^{6} \) 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
$47$ \( T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$53$ \( T^{6} - T^{5} + T^{4} - T^{3} + T^{2} - T + 1 \) 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
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} \) Copy content Toggle raw display
$79$ \( T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1 \) 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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