Properties

Label 3364.1.j.e
Level $3364$
Weight $1$
Character orbit 3364.j
Analytic conductor $1.679$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3364,1,Mod(571,3364)]
 
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, 12]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3364.571");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3364 = 2^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3364.j (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: \(D_{7}\)
Projective field: Galois closure of 7.1.38068692544.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{14}^{6} q^{2} - \zeta_{14}^{5} q^{4} + ( - \zeta_{14}^{5} + 1) q^{5} - \zeta_{14}^{4} q^{8} + \zeta_{14}^{4} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{14}^{6} q^{2} - \zeta_{14}^{5} q^{4} + ( - \zeta_{14}^{5} + 1) q^{5} - \zeta_{14}^{4} q^{8} + \zeta_{14}^{4} q^{9} + ( - \zeta_{14}^{6} - \zeta_{14}^{4}) q^{10} + (\zeta_{14}^{6} + 1) q^{13} - \zeta_{14}^{3} q^{16} + (\zeta_{14}^{5} - \zeta_{14}^{2}) q^{17} + \zeta_{14}^{3} q^{18} + ( - \zeta_{14}^{5} - \zeta_{14}^{3}) q^{20} + ( - \zeta_{14}^{5} - \zeta_{14}^{3} + 1) q^{25} + ( - \zeta_{14}^{6} + \zeta_{14}^{5}) q^{26} - \zeta_{14}^{2} q^{32} + (\zeta_{14}^{4} - \zeta_{14}) q^{34} + \zeta_{14}^{2} q^{36} + (\zeta_{14} - 1) q^{37} + ( - \zeta_{14}^{4} - \zeta_{14}^{2}) q^{40} + ( - \zeta_{14}^{4} + \zeta_{14}^{3}) q^{41} + (\zeta_{14}^{4} + \zeta_{14}^{2}) q^{45} + \zeta_{14}^{4} q^{49} + ( - \zeta_{14}^{6} + \cdots - \zeta_{14}^{2}) q^{50} + \cdots + \zeta_{14}^{3} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - q^{4} + 5 q^{5} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - q^{4} + 5 q^{5} + q^{8} - q^{9} + 2 q^{10} + 5 q^{13} - q^{16} + 2 q^{17} + q^{18} - 2 q^{20} + 4 q^{25} + 2 q^{26} + q^{32} - 2 q^{34} - q^{36} - 5 q^{37} + 2 q^{40} + 2 q^{41} - 2 q^{45} - q^{49} + 3 q^{50} - 2 q^{52} - 2 q^{53} - 5 q^{61} - q^{64} + 3 q^{65} - 5 q^{68} + q^{72} + 2 q^{73} + 5 q^{74} - 2 q^{80} - q^{81} - 2 q^{82} - 3 q^{85} - 5 q^{89} + 2 q^{90} + 2 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1683\) \(2525\)
\(\chi(n)\) \(-1\) \(-\zeta_{14}^{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
571.1
0.900969 + 0.433884i
0.900969 0.433884i
−0.623490 0.781831i
−0.623490 + 0.781831i
0.222521 + 0.974928i
0.222521 0.974928i
0.900969 0.433884i 0 0.623490 0.781831i 1.62349 0.781831i 0 0 0.222521 0.974928i −0.222521 + 0.974928i 1.12349 1.40881i
1031.1 0.900969 + 0.433884i 0 0.623490 + 0.781831i 1.62349 + 0.781831i 0 0 0.222521 + 0.974928i −0.222521 0.974928i 1.12349 + 1.40881i
1415.1 −0.623490 + 0.781831i 0 −0.222521 0.974928i 0.777479 0.974928i 0 0 0.900969 + 0.433884i −0.900969 0.433884i 0.277479 + 1.21572i
1619.1 −0.623490 0.781831i 0 −0.222521 + 0.974928i 0.777479 + 0.974928i 0 0 0.900969 0.433884i −0.900969 + 0.433884i 0.277479 1.21572i
2287.1 0.222521 0.974928i 0 −0.900969 0.433884i 0.0990311 0.433884i 0 0 −0.623490 + 0.781831i 0.623490 0.781831i −0.400969 0.193096i
2327.1 0.222521 + 0.974928i 0 −0.900969 + 0.433884i 0.0990311 + 0.433884i 0 0 −0.623490 0.781831i 0.623490 + 0.781831i −0.400969 + 0.193096i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 571.1
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3364.1.j.e 6
4.b odd 2 1 CM 3364.1.j.e 6
29.b even 2 1 3364.1.j.b 6
29.c odd 4 2 3364.1.h.c 12
29.d even 7 1 3364.1.b.b 3
29.d even 7 2 3364.1.j.c 6
29.d even 7 2 3364.1.j.d 6
29.d even 7 1 inner 3364.1.j.e 6
29.e even 14 2 116.1.j.a 6
29.e even 14 1 3364.1.b.c 3
29.e even 14 2 3364.1.j.a 6
29.e even 14 1 3364.1.j.b 6
29.f odd 28 2 3364.1.d.a 6
29.f odd 28 2 3364.1.h.c 12
29.f odd 28 4 3364.1.h.d 12
29.f odd 28 4 3364.1.h.e 12
87.h odd 14 2 1044.1.bb.a 6
116.d odd 2 1 3364.1.j.b 6
116.e even 4 2 3364.1.h.c 12
116.h odd 14 2 116.1.j.a 6
116.h odd 14 1 3364.1.b.c 3
116.h odd 14 2 3364.1.j.a 6
116.h odd 14 1 3364.1.j.b 6
116.j odd 14 1 3364.1.b.b 3
116.j odd 14 2 3364.1.j.c 6
116.j odd 14 2 3364.1.j.d 6
116.j odd 14 1 inner 3364.1.j.e 6
116.l even 28 2 3364.1.d.a 6
116.l even 28 2 3364.1.h.c 12
116.l even 28 4 3364.1.h.d 12
116.l even 28 4 3364.1.h.e 12
145.l even 14 2 2900.1.bj.a 6
145.q odd 28 4 2900.1.bd.a 12
232.o even 14 2 1856.1.bh.a 6
232.t odd 14 2 1856.1.bh.a 6
348.t even 14 2 1044.1.bb.a 6
580.y odd 14 2 2900.1.bj.a 6
580.bh even 28 4 2900.1.bd.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.1.j.a 6 29.e even 14 2
116.1.j.a 6 116.h odd 14 2
1044.1.bb.a 6 87.h odd 14 2
1044.1.bb.a 6 348.t even 14 2
1856.1.bh.a 6 232.o even 14 2
1856.1.bh.a 6 232.t odd 14 2
2900.1.bd.a 12 145.q odd 28 4
2900.1.bd.a 12 580.bh even 28 4
2900.1.bj.a 6 145.l even 14 2
2900.1.bj.a 6 580.y odd 14 2
3364.1.b.b 3 29.d even 7 1
3364.1.b.b 3 116.j odd 14 1
3364.1.b.c 3 29.e even 14 1
3364.1.b.c 3 116.h odd 14 1
3364.1.d.a 6 29.f odd 28 2
3364.1.d.a 6 116.l even 28 2
3364.1.h.c 12 29.c odd 4 2
3364.1.h.c 12 29.f odd 28 2
3364.1.h.c 12 116.e even 4 2
3364.1.h.c 12 116.l even 28 2
3364.1.h.d 12 29.f odd 28 4
3364.1.h.d 12 116.l even 28 4
3364.1.h.e 12 29.f odd 28 4
3364.1.h.e 12 116.l even 28 4
3364.1.j.a 6 29.e even 14 2
3364.1.j.a 6 116.h odd 14 2
3364.1.j.b 6 29.b even 2 1
3364.1.j.b 6 29.e even 14 1
3364.1.j.b 6 116.d odd 2 1
3364.1.j.b 6 116.h odd 14 1
3364.1.j.c 6 29.d even 7 2
3364.1.j.c 6 116.j odd 14 2
3364.1.j.d 6 29.d even 7 2
3364.1.j.d 6 116.j odd 14 2
3364.1.j.e 6 1.a even 1 1 trivial
3364.1.j.e 6 4.b odd 2 1 CM
3364.1.j.e 6 29.d even 7 1 inner
3364.1.j.e 6 116.j odd 14 1 inner

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{6} - 5T_{5}^{5} + 11T_{5}^{4} - 13T_{5}^{3} + 9T_{5}^{2} - 3T_{5} + 1 \) Copy content Toggle raw display
\( T_{17}^{3} - T_{17}^{2} - 2T_{17} + 1 \) 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
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( (T^{3} - T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} + 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{3} - T^{2} - 2 T + 1)^{2} \) 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
$61$ \( T^{6} + 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{6} \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} - 2 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} \) Copy content Toggle raw display
$89$ \( T^{6} + 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$97$ \( T^{6} - 2 T^{5} + \cdots + 1 \) Copy content Toggle raw display
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