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Properties

Label 3364.1.j.f
Level $3364$
Weight $1$
Character orbit 3364.j
Analytic conductor $1.679$
Analytic rank $0$
Dimension $12$
Projective image $D_{3}$
CM discriminant -116
Inner twists $24$

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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: \(12\)
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: \(\Z[a_1, a_2]\)
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

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{28}^{5} q^{2} + \zeta_{28}^{11} q^{3} + \zeta_{28}^{10} q^{4} + \zeta_{28}^{12} q^{5} - \zeta_{28}^{2} q^{6} - \zeta_{28} q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{28}^{5} q^{2} + \zeta_{28}^{11} q^{3} + \zeta_{28}^{10} q^{4} + \zeta_{28}^{12} q^{5} - \zeta_{28}^{2} q^{6} - \zeta_{28} q^{8} - \zeta_{28}^{3} q^{10} - \zeta_{28}^{13} q^{11} - \zeta_{28}^{7} q^{12} - \zeta_{28}^{6} q^{13} - \zeta_{28}^{9} q^{15} - \zeta_{28}^{6} q^{16} - \zeta_{28}^{3} q^{19} - \zeta_{28}^{8} q^{20} + \zeta_{28}^{4} q^{22} - \zeta_{28}^{12} q^{24} - \zeta_{28}^{11} q^{26} - \zeta_{28}^{5} q^{27} + q^{30} - \zeta_{28}^{5} q^{31} - \zeta_{28}^{11} q^{32} + \zeta_{28}^{10} q^{33} - 2 \zeta_{28}^{8} q^{38} + \zeta_{28}^{3} q^{39} - \zeta_{28}^{13} q^{40} - \zeta_{28}^{9} q^{43} + \zeta_{28}^{9} q^{44} + \zeta_{28}^{13} q^{47} + \zeta_{28}^{3} q^{48} + \zeta_{28}^{8} q^{49} + \zeta_{28}^{2} q^{52} - \zeta_{28}^{12} q^{53} - \zeta_{28}^{10} q^{54} + \zeta_{28}^{11} q^{55} + 2 q^{57} + \zeta_{28}^{5} q^{60} - \zeta_{28}^{10} q^{62} + \zeta_{28}^{2} q^{64} + \zeta_{28}^{4} q^{65} - \zeta_{28} q^{66} - 2 \zeta_{28}^{13} q^{76} + \zeta_{28}^{8} q^{78} - \zeta_{28} q^{79} + \zeta_{28}^{4} q^{80} + \zeta_{28}^{2} q^{81} + q^{86} - q^{88} + \zeta_{28}^{2} q^{93} - \zeta_{28}^{4} q^{94} + 2 \zeta_{28} q^{95} + \zeta_{28}^{8} q^{96} + \zeta_{28}^{13} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} - 2 q^{5} - 2 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{13} - 2 q^{16} + 2 q^{20} - 2 q^{22} + 2 q^{24} + 12 q^{30} + 2 q^{33} + 4 q^{38} - 2 q^{49} + 2 q^{52} + 2 q^{53} - 2 q^{54} + 24 q^{57} - 2 q^{62} + 2 q^{64} - 2 q^{65} - 2 q^{78} - 2 q^{80} + 2 q^{81} + 12 q^{86} - 12 q^{88} + 2 q^{93} + 2 q^{94} - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(1683\) \(2525\)
\(\chi(n)\) \(-1\) \(-\zeta_{28}^{10}\)

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
−0.974928 − 0.222521i
0.974928 + 0.222521i
−0.974928 + 0.222521i
0.974928 − 0.222521i
−0.433884 + 0.900969i
0.433884 − 0.900969i
−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 0.781831 − 0.623490i −0.623490 + 0.781831i −0.900969 + 0.433884i −0.900969 − 0.433884i 0 0.974928 + 0.222521i 0 0.781831 + 0.623490i
571.2 0.433884 + 0.900969i −0.781831 + 0.623490i −0.623490 + 0.781831i −0.900969 + 0.433884i −0.900969 − 0.433884i 0 −0.974928 − 0.222521i 0 −0.781831 − 0.623490i
1031.1 −0.433884 + 0.900969i 0.781831 + 0.623490i −0.623490 − 0.781831i −0.900969 − 0.433884i −0.900969 + 0.433884i 0 0.974928 − 0.222521i 0 0.781831 − 0.623490i
1031.2 0.433884 − 0.900969i −0.781831 − 0.623490i −0.623490 − 0.781831i −0.900969 − 0.433884i −0.900969 + 0.433884i 0 −0.974928 + 0.222521i 0 −0.781831 + 0.623490i
1415.1 −0.781831 − 0.623490i −0.974928 − 0.222521i 0.222521 + 0.974928i 0.623490 − 0.781831i 0.623490 + 0.781831i 0 0.433884 − 0.900969i 0 −0.974928 + 0.222521i
1415.2 0.781831 + 0.623490i 0.974928 + 0.222521i 0.222521 + 0.974928i 0.623490 − 0.781831i 0.623490 + 0.781831i 0 −0.433884 + 0.900969i 0 0.974928 − 0.222521i
1619.1 −0.781831 + 0.623490i −0.974928 + 0.222521i 0.222521 − 0.974928i 0.623490 + 0.781831i 0.623490 − 0.781831i 0 0.433884 + 0.900969i 0 −0.974928 − 0.222521i
1619.2 0.781831 − 0.623490i 0.974928 − 0.222521i 0.222521 − 0.974928i 0.623490 + 0.781831i 0.623490 − 0.781831i 0 −0.433884 − 0.900969i 0 0.974928 + 0.222521i
2287.1 −0.974928 − 0.222521i 0.433884 + 0.900969i 0.900969 + 0.433884i −0.222521 + 0.974928i −0.222521 − 0.974928i 0 −0.781831 − 0.623490i 0 0.433884 − 0.900969i
2287.2 0.974928 + 0.222521i −0.433884 − 0.900969i 0.900969 + 0.433884i −0.222521 + 0.974928i −0.222521 − 0.974928i 0 0.781831 + 0.623490i 0 −0.433884 + 0.900969i
2327.1 −0.974928 + 0.222521i 0.433884 − 0.900969i 0.900969 − 0.433884i −0.222521 − 0.974928i −0.222521 + 0.974928i 0 −0.781831 + 0.623490i 0 0.433884 + 0.900969i
2327.2 0.974928 − 0.222521i −0.433884 + 0.900969i 0.900969 − 0.433884i −0.222521 − 0.974928i −0.222521 + 0.974928i 0 0.781831 − 0.623490i 0 −0.433884 − 0.900969i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 571.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
116.d odd 2 1 CM by \(\Q(\sqrt{-29}) \)
4.b odd 2 1 inner
29.b even 2 1 inner
29.d even 7 5 inner
29.e even 14 5 inner
116.h odd 14 5 inner
116.j 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.j.f 12
4.b odd 2 1 inner 3364.1.j.f 12
29.b even 2 1 inner 3364.1.j.f 12
29.c odd 4 1 3364.1.h.a 6
29.c odd 4 1 3364.1.h.b 6
29.d even 7 1 3364.1.b.a 2
29.d even 7 5 inner 3364.1.j.f 12
29.e even 14 1 3364.1.b.a 2
29.e even 14 5 inner 3364.1.j.f 12
29.f odd 28 1 116.1.d.a ✓ 1
29.f odd 28 1 116.1.d.b yes 1
29.f odd 28 5 3364.1.h.a 6
29.f odd 28 5 3364.1.h.b 6
87.k even 28 1 1044.1.g.a 1
87.k even 28 1 1044.1.g.b 1
116.d odd 2 1 CM 3364.1.j.f 12
116.e even 4 1 3364.1.h.a 6
116.e even 4 1 3364.1.h.b 6
116.h odd 14 1 3364.1.b.a 2
116.h odd 14 5 inner 3364.1.j.f 12
116.j odd 14 1 3364.1.b.a 2
116.j odd 14 5 inner 3364.1.j.f 12
116.l even 28 1 116.1.d.a ✓ 1
116.l even 28 1 116.1.d.b yes 1
116.l even 28 5 3364.1.h.a 6
116.l even 28 5 3364.1.h.b 6
145.o even 28 1 2900.1.e.a 2
145.o even 28 1 2900.1.e.b 2
145.s odd 28 1 2900.1.g.a 1
145.s odd 28 1 2900.1.g.d 1
145.t even 28 1 2900.1.e.a 2
145.t even 28 1 2900.1.e.b 2
232.u odd 28 1 1856.1.h.a 1
232.u odd 28 1 1856.1.h.c 1
232.v even 28 1 1856.1.h.a 1
232.v even 28 1 1856.1.h.c 1
348.v odd 28 1 1044.1.g.a 1
348.v odd 28 1 1044.1.g.b 1
580.bd odd 28 1 2900.1.e.a 2
580.bd odd 28 1 2900.1.e.b 2
580.be even 28 1 2900.1.g.a 1
580.be even 28 1 2900.1.g.d 1
580.bm odd 28 1 2900.1.e.a 2
580.bm odd 28 1 2900.1.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
116.1.d.a ✓ 1 29.f odd 28 1
116.1.d.a ✓ 1 116.l even 28 1
116.1.d.b yes 1 29.f odd 28 1
116.1.d.b yes 1 116.l even 28 1
1044.1.g.a 1 87.k even 28 1
1044.1.g.a 1 348.v odd 28 1
1044.1.g.b 1 87.k even 28 1
1044.1.g.b 1 348.v odd 28 1
1856.1.h.a 1 232.u odd 28 1
1856.1.h.a 1 232.v even 28 1
1856.1.h.c 1 232.u odd 28 1
1856.1.h.c 1 232.v even 28 1
2900.1.e.a 2 145.o even 28 1
2900.1.e.a 2 145.t even 28 1
2900.1.e.a 2 580.bd odd 28 1
2900.1.e.a 2 580.bm odd 28 1
2900.1.e.b 2 145.o even 28 1
2900.1.e.b 2 145.t even 28 1
2900.1.e.b 2 580.bd odd 28 1
2900.1.e.b 2 580.bm odd 28 1
2900.1.g.a 1 145.s odd 28 1
2900.1.g.a 1 580.be even 28 1
2900.1.g.d 1 145.s odd 28 1
2900.1.g.d 1 580.be even 28 1
3364.1.b.a 2 29.d even 7 1
3364.1.b.a 2 29.e even 14 1
3364.1.b.a 2 116.h odd 14 1
3364.1.b.a 2 116.j odd 14 1
3364.1.h.a 6 29.c odd 4 1
3364.1.h.a 6 29.f odd 28 5
3364.1.h.a 6 116.e even 4 1
3364.1.h.a 6 116.l even 28 5
3364.1.h.b 6 29.c odd 4 1
3364.1.h.b 6 29.f odd 28 5
3364.1.h.b 6 116.e even 4 1
3364.1.h.b 6 116.l even 28 5
3364.1.j.f 12 1.a even 1 1 trivial
3364.1.j.f 12 4.b odd 2 1 inner
3364.1.j.f 12 29.b even 2 1 inner
3364.1.j.f 12 29.d even 7 5 inner
3364.1.j.f 12 29.e even 14 5 inner
3364.1.j.f 12 116.d odd 2 1 CM
3364.1.j.f 12 116.h odd 14 5 inner
3364.1.j.f 12 116.j odd 14 5 inner

Hecke kernels

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

\( T_{3}^{12} - T_{3}^{10} + T_{3}^{8} - T_{3}^{6} + T_{3}^{4} - T_{3}^{2} + 1 \) 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
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} \) Copy content Toggle raw display
$19$ \( T^{12} - 4 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$53$ \( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \) 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
$71$ \( T^{12} \) Copy content Toggle raw display
$73$ \( T^{12} \) Copy content Toggle raw display
$79$ \( T^{12} - T^{10} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{12} \) Copy content Toggle raw display
$89$ \( T^{12} \) Copy content Toggle raw display
$97$ \( T^{12} \) Copy content Toggle raw display
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