Properties

Label 3380.1.v.c
Level $3380$
Weight $1$
Character orbit 3380.v
Analytic conductor $1.687$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -4
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3380,1,Mod(2219,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3380.2219");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3380.v (of order \(6\), degree \(2\), 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: \(1.68683974270\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.2970344000.2

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{5} - \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{2} + \zeta_{12}^{2} q^{4} + \zeta_{12}^{5} q^{5} - \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + q^{10} + \zeta_{12}^{4} q^{16} + (\zeta_{12}^{4} - 1) q^{17} - \zeta_{12}^{3} q^{18} - \zeta_{12} q^{20} - \zeta_{12}^{4} q^{25} + \zeta_{12}^{4} q^{29} - \zeta_{12}^{5} q^{32} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{34} + \zeta_{12}^{4} q^{36} - \zeta_{12} q^{37} + \zeta_{12}^{2} q^{40} + (\zeta_{12}^{5} + \zeta_{12}^{3}) q^{41} - \zeta_{12} q^{45} - \zeta_{12}^{4} q^{49} + \zeta_{12}^{5} q^{50} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{53} - \zeta_{12}^{5} q^{58} - \zeta_{12}^{2} q^{61} - q^{64} + ( - \zeta_{12}^{2} - 1) q^{68} - \zeta_{12}^{5} q^{72} + \zeta_{12}^{3} q^{73} + \zeta_{12}^{2} q^{74} - \zeta_{12}^{3} q^{80} + \zeta_{12}^{4} q^{81} + ( - \zeta_{12}^{4} + 1) q^{82} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{85} + \zeta_{12}^{2} q^{90} + \zeta_{12}^{5} q^{97} + \zeta_{12}^{5} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{9} + 4 q^{10} - 2 q^{16} - 6 q^{17} + 2 q^{25} - 2 q^{29} - 2 q^{36} + 2 q^{40} + 2 q^{49} - 2 q^{61} - 4 q^{64} - 6 q^{68} + 2 q^{74} - 2 q^{81} + 6 q^{82} + 2 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2219.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i 0 0.500000 + 0.866025i −0.866025 + 0.500000i 0 0 1.00000i 0.500000 + 0.866025i 1.00000
2219.2 0.866025 + 0.500000i 0 0.500000 + 0.866025i 0.866025 0.500000i 0 0 1.00000i 0.500000 + 0.866025i 1.00000
3019.1 −0.866025 + 0.500000i 0 0.500000 0.866025i −0.866025 0.500000i 0 0 1.00000i 0.500000 0.866025i 1.00000
3019.2 0.866025 0.500000i 0 0.500000 0.866025i 0.866025 + 0.500000i 0 0 1.00000i 0.500000 0.866025i 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
13.b even 2 1 inner
52.b odd 2 1 inner
65.l even 6 1 inner
65.n even 6 1 inner
260.v odd 6 1 inner
260.w odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.1.v.c 4
4.b odd 2 1 CM 3380.1.v.c 4
5.b even 2 1 3380.1.v.b 4
13.b even 2 1 inner 3380.1.v.c 4
13.c even 3 1 3380.1.h.f 4
13.c even 3 1 3380.1.v.b 4
13.d odd 4 1 260.1.w.b yes 2
13.d odd 4 1 3380.1.w.b 2
13.e even 6 1 3380.1.h.f 4
13.e even 6 1 3380.1.v.b 4
13.f odd 12 1 260.1.w.a 2
13.f odd 12 1 3380.1.g.a 2
13.f odd 12 1 3380.1.g.b 2
13.f odd 12 1 3380.1.w.c 2
20.d odd 2 1 3380.1.v.b 4
39.f even 4 1 2340.1.dd.a 2
39.k even 12 1 2340.1.dd.b 2
52.b odd 2 1 inner 3380.1.v.c 4
52.f even 4 1 260.1.w.b yes 2
52.f even 4 1 3380.1.w.b 2
52.i odd 6 1 3380.1.h.f 4
52.i odd 6 1 3380.1.v.b 4
52.j odd 6 1 3380.1.h.f 4
52.j odd 6 1 3380.1.v.b 4
52.l even 12 1 260.1.w.a 2
52.l even 12 1 3380.1.g.a 2
52.l even 12 1 3380.1.g.b 2
52.l even 12 1 3380.1.w.c 2
65.d even 2 1 3380.1.v.b 4
65.f even 4 1 1300.1.z.a 4
65.g odd 4 1 260.1.w.a 2
65.g odd 4 1 3380.1.w.c 2
65.k even 4 1 1300.1.z.a 4
65.l even 6 1 3380.1.h.f 4
65.l even 6 1 inner 3380.1.v.c 4
65.n even 6 1 3380.1.h.f 4
65.n even 6 1 inner 3380.1.v.c 4
65.o even 12 1 1300.1.z.a 4
65.s odd 12 1 260.1.w.b yes 2
65.s odd 12 1 3380.1.g.a 2
65.s odd 12 1 3380.1.g.b 2
65.s odd 12 1 3380.1.w.b 2
65.t even 12 1 1300.1.z.a 4
156.l odd 4 1 2340.1.dd.a 2
156.v odd 12 1 2340.1.dd.b 2
195.n even 4 1 2340.1.dd.b 2
195.bh even 12 1 2340.1.dd.a 2
260.g odd 2 1 3380.1.v.b 4
260.l odd 4 1 1300.1.z.a 4
260.s odd 4 1 1300.1.z.a 4
260.u even 4 1 260.1.w.a 2
260.u even 4 1 3380.1.w.c 2
260.v odd 6 1 3380.1.h.f 4
260.v odd 6 1 inner 3380.1.v.c 4
260.w odd 6 1 3380.1.h.f 4
260.w odd 6 1 inner 3380.1.v.c 4
260.bc even 12 1 260.1.w.b yes 2
260.bc even 12 1 3380.1.g.a 2
260.bc even 12 1 3380.1.g.b 2
260.bc even 12 1 3380.1.w.b 2
260.be odd 12 1 1300.1.z.a 4
260.bl odd 12 1 1300.1.z.a 4
780.bb odd 4 1 2340.1.dd.b 2
780.cr odd 12 1 2340.1.dd.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.1.w.a 2 13.f odd 12 1
260.1.w.a 2 52.l even 12 1
260.1.w.a 2 65.g odd 4 1
260.1.w.a 2 260.u even 4 1
260.1.w.b yes 2 13.d odd 4 1
260.1.w.b yes 2 52.f even 4 1
260.1.w.b yes 2 65.s odd 12 1
260.1.w.b yes 2 260.bc even 12 1
1300.1.z.a 4 65.f even 4 1
1300.1.z.a 4 65.k even 4 1
1300.1.z.a 4 65.o even 12 1
1300.1.z.a 4 65.t even 12 1
1300.1.z.a 4 260.l odd 4 1
1300.1.z.a 4 260.s odd 4 1
1300.1.z.a 4 260.be odd 12 1
1300.1.z.a 4 260.bl odd 12 1
2340.1.dd.a 2 39.f even 4 1
2340.1.dd.a 2 156.l odd 4 1
2340.1.dd.a 2 195.bh even 12 1
2340.1.dd.a 2 780.cr odd 12 1
2340.1.dd.b 2 39.k even 12 1
2340.1.dd.b 2 156.v odd 12 1
2340.1.dd.b 2 195.n even 4 1
2340.1.dd.b 2 780.bb odd 4 1
3380.1.g.a 2 13.f odd 12 1
3380.1.g.a 2 52.l even 12 1
3380.1.g.a 2 65.s odd 12 1
3380.1.g.a 2 260.bc even 12 1
3380.1.g.b 2 13.f odd 12 1
3380.1.g.b 2 52.l even 12 1
3380.1.g.b 2 65.s odd 12 1
3380.1.g.b 2 260.bc even 12 1
3380.1.h.f 4 13.c even 3 1
3380.1.h.f 4 13.e even 6 1
3380.1.h.f 4 52.i odd 6 1
3380.1.h.f 4 52.j odd 6 1
3380.1.h.f 4 65.l even 6 1
3380.1.h.f 4 65.n even 6 1
3380.1.h.f 4 260.v odd 6 1
3380.1.h.f 4 260.w odd 6 1
3380.1.v.b 4 5.b even 2 1
3380.1.v.b 4 13.c even 3 1
3380.1.v.b 4 13.e even 6 1
3380.1.v.b 4 20.d odd 2 1
3380.1.v.b 4 52.i odd 6 1
3380.1.v.b 4 52.j odd 6 1
3380.1.v.b 4 65.d even 2 1
3380.1.v.b 4 260.g odd 2 1
3380.1.v.c 4 1.a even 1 1 trivial
3380.1.v.c 4 4.b odd 2 1 CM
3380.1.v.c 4 13.b even 2 1 inner
3380.1.v.c 4 52.b odd 2 1 inner
3380.1.v.c 4 65.l even 6 1 inner
3380.1.v.c 4 65.n even 6 1 inner
3380.1.v.c 4 260.v odd 6 1 inner
3380.1.v.c 4 260.w odd 6 1 inner
3380.1.w.b 2 13.d odd 4 1
3380.1.w.b 2 52.f even 4 1
3380.1.w.b 2 65.s odd 12 1
3380.1.w.b 2 260.bc even 12 1
3380.1.w.c 2 13.f odd 12 1
3380.1.w.c 2 52.l even 12 1
3380.1.w.c 2 65.g odd 4 1
3380.1.w.c 2 260.u even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3380, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{17}^{2} + 3T_{17} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
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