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Properties

Label 3420.1.fk.a
Level $3420$
Weight $1$
Character orbit 3420.fk
Analytic conductor $1.707$
Analytic rank $0$
Dimension $12$
Projective image $D_{18}$
CM discriminant -15
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3420,1,Mod(199,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 0, 9, 8]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3420.199");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3420.fk (of order \(18\), 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: \(1.70680234320\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{18}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{18} + \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{36}^{5} q^{2} + \zeta_{36}^{10} q^{4} + \zeta_{36} q^{5} + \zeta_{36}^{15} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{36}^{5} q^{2} + \zeta_{36}^{10} q^{4} + \zeta_{36} q^{5} + \zeta_{36}^{15} q^{8} + \zeta_{36}^{6} q^{10} - \zeta_{36}^{2} q^{16} + ( - \zeta_{36}^{17} - \zeta_{36}^{9}) q^{17} - \zeta_{36}^{16} q^{19} + \zeta_{36}^{11} q^{20} + (\zeta_{36}^{11} + \zeta_{36}^{5}) q^{23} + \zeta_{36}^{2} q^{25} + (\zeta_{36}^{8} - \zeta_{36}^{4}) q^{31} - \zeta_{36}^{7} q^{32} + ( - \zeta_{36}^{14} + \zeta_{36}^{4}) q^{34} + \zeta_{36}^{3} q^{38} + \zeta_{36}^{16} q^{40} + (\zeta_{36}^{16} + \zeta_{36}^{10}) q^{46} + (\zeta_{36}^{15} + \zeta_{36}^{13}) q^{47} - \zeta_{36}^{12} q^{49} + \zeta_{36}^{7} q^{50} + (\zeta_{36}^{13} - \zeta_{36}^{3}) q^{53} + \zeta_{36}^{8} q^{61} + (\zeta_{36}^{13} - \zeta_{36}^{9}) q^{62} - \zeta_{36}^{12} q^{64} + (\zeta_{36}^{9} + \zeta_{36}) q^{68} + \zeta_{36}^{8} q^{76} + ( - \zeta_{36}^{10} - \zeta_{36}^{4}) q^{79} - \zeta_{36}^{3} q^{80} + (\zeta_{36}^{17} - \zeta_{36}^{13}) q^{83} + ( - \zeta_{36}^{10} + 1) q^{85} + (\zeta_{36}^{15} - \zeta_{36}^{3}) q^{92} + ( - \zeta_{36}^{2} - 1) q^{94} - \zeta_{36}^{17} q^{95} - \zeta_{36}^{17} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{10} + 6 q^{49} + 6 q^{64} + 12 q^{85} - 12 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(\zeta_{36}^{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
−0.342020 + 0.939693i
0.342020 − 0.939693i
−0.342020 − 0.939693i
0.342020 + 0.939693i
−0.984808 − 0.173648i
0.984808 + 0.173648i
−0.984808 + 0.173648i
0.984808 − 0.173648i
0.642788 − 0.766044i
−0.642788 + 0.766044i
0.642788 + 0.766044i
−0.642788 − 0.766044i
−0.984808 − 0.173648i 0 0.939693 + 0.342020i −0.342020 + 0.939693i 0 0 −0.866025 − 0.500000i 0 0.500000 − 0.866025i
199.2 0.984808 + 0.173648i 0 0.939693 + 0.342020i 0.342020 − 0.939693i 0 0 0.866025 + 0.500000i 0 0.500000 − 0.866025i
739.1 −0.984808 + 0.173648i 0 0.939693 − 0.342020i −0.342020 − 0.939693i 0 0 −0.866025 + 0.500000i 0 0.500000 + 0.866025i
739.2 0.984808 − 0.173648i 0 0.939693 − 0.342020i 0.342020 + 0.939693i 0 0 0.866025 − 0.500000i 0 0.500000 + 0.866025i
1099.1 −0.642788 − 0.766044i 0 −0.173648 + 0.984808i −0.984808 − 0.173648i 0 0 0.866025 − 0.500000i 0 0.500000 + 0.866025i
1099.2 0.642788 + 0.766044i 0 −0.173648 + 0.984808i 0.984808 + 0.173648i 0 0 −0.866025 + 0.500000i 0 0.500000 + 0.866025i
1279.1 −0.642788 + 0.766044i 0 −0.173648 − 0.984808i −0.984808 + 0.173648i 0 0 0.866025 + 0.500000i 0 0.500000 − 0.866025i
1279.2 0.642788 − 0.766044i 0 −0.173648 − 0.984808i 0.984808 − 0.173648i 0 0 −0.866025 − 0.500000i 0 0.500000 − 0.866025i
1639.1 −0.342020 + 0.939693i 0 −0.766044 − 0.642788i 0.642788 − 0.766044i 0 0 0.866025 − 0.500000i 0 0.500000 + 0.866025i
1639.2 0.342020 − 0.939693i 0 −0.766044 − 0.642788i −0.642788 + 0.766044i 0 0 −0.866025 + 0.500000i 0 0.500000 + 0.866025i
1999.1 −0.342020 − 0.939693i 0 −0.766044 + 0.642788i 0.642788 + 0.766044i 0 0 0.866025 + 0.500000i 0 0.500000 − 0.866025i
1999.2 0.342020 + 0.939693i 0 −0.766044 + 0.642788i −0.642788 − 0.766044i 0 0 −0.866025 − 0.500000i 0 0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 199.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner
76.l odd 18 1 inner
228.v even 18 1 inner
380.ba odd 18 1 inner
1140.bz even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3420.1.fk.a ✓ 12
3.b odd 2 1 inner 3420.1.fk.a ✓ 12
4.b odd 2 1 3420.1.fk.b yes 12
5.b even 2 1 inner 3420.1.fk.a ✓ 12
12.b even 2 1 3420.1.fk.b yes 12
15.d odd 2 1 CM 3420.1.fk.a ✓ 12
19.e even 9 1 3420.1.fk.b yes 12
20.d odd 2 1 3420.1.fk.b yes 12
57.l odd 18 1 3420.1.fk.b yes 12
60.h even 2 1 3420.1.fk.b yes 12
76.l odd 18 1 inner 3420.1.fk.a ✓ 12
95.p even 18 1 3420.1.fk.b yes 12
228.v even 18 1 inner 3420.1.fk.a ✓ 12
285.bd odd 18 1 3420.1.fk.b yes 12
380.ba odd 18 1 inner 3420.1.fk.a ✓ 12
1140.bz even 18 1 inner 3420.1.fk.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3420.1.fk.a ✓ 12 1.a even 1 1 trivial
3420.1.fk.a ✓ 12 3.b odd 2 1 inner
3420.1.fk.a ✓ 12 5.b even 2 1 inner
3420.1.fk.a ✓ 12 15.d odd 2 1 CM
3420.1.fk.a ✓ 12 76.l odd 18 1 inner
3420.1.fk.a ✓ 12 228.v even 18 1 inner
3420.1.fk.a ✓ 12 380.ba odd 18 1 inner
3420.1.fk.a ✓ 12 1140.bz even 18 1 inner
3420.1.fk.b yes 12 4.b odd 2 1
3420.1.fk.b yes 12 12.b even 2 1
3420.1.fk.b yes 12 19.e even 9 1
3420.1.fk.b yes 12 20.d odd 2 1
3420.1.fk.b yes 12 57.l odd 18 1
3420.1.fk.b yes 12 60.h even 2 1
3420.1.fk.b yes 12 95.p even 18 1
3420.1.fk.b yes 12 285.bd odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{31}^{6} - 3T_{31}^{4} + 9T_{31}^{2} + 9T_{31} + 3 \) acting on \(S_{1}^{\mathrm{new}}(3420, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + 6 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( (T^{6} - T^{3} + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 27T^{6} + 729 \) Copy content Toggle raw display
$29$ \( T^{12} \) Copy content Toggle raw display
$31$ \( (T^{6} - 3 T^{4} + 9 T^{2} + \cdots + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} \) Copy content Toggle raw display
$41$ \( T^{12} \) Copy content Toggle raw display
$43$ \( T^{12} \) Copy content Toggle raw display
$47$ \( T^{12} - 3 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$53$ \( T^{12} - 3 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{12} \) Copy content Toggle raw display
$61$ \( (T^{6} + T^{3} + 1)^{2} \) 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
$79$ \( (T^{6} - 9 T^{3} + 27)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 6 T^{10} + \cdots + 9 \) 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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