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Properties

Label 3380.1.g.b
Level $3380$
Weight $1$
Character orbit 3380.g
Analytic conductor $1.687$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3380,1,Mod(3379,3380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3380.3379"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3380.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.68683974270\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - \zeta_{6}^{2} q^{5} + q^{8} - q^{9} - \zeta_{6}^{2} q^{10} + q^{16} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{17} - q^{18} - \zeta_{6}^{2} q^{20} - \zeta_{6} q^{25} + q^{29} + q^{32} + \cdots + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{8} - 2 q^{9} + q^{10} + 2 q^{16} - 2 q^{18} + q^{20} - q^{25} + 2 q^{29} + 2 q^{32} - 2 q^{36} - 2 q^{37} + q^{40} - q^{45} + 2 q^{49} - q^{50} + 2 q^{58} + 2 q^{61}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3379.1
0.500000 + 0.866025i
0.500000 − 0.866025i
1.00000 0 1.00000 0.500000 − 0.866025i 0 0 1.00000 −1.00000 0.500000 − 0.866025i
3379.2 1.00000 0 1.00000 0.500000 + 0.866025i 0 0 1.00000 −1.00000 0.500000 + 0.866025i
\(n\): 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 \(\Q(\sqrt{-1}) \)
65.d even 2 1 inner
260.g odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.1.g.b 2
4.b odd 2 1 CM 3380.1.g.b 2
5.b even 2 1 3380.1.g.a 2
13.b even 2 1 3380.1.g.a 2
13.c even 3 1 260.1.w.a ✓ 2
13.c even 3 1 3380.1.w.b 2
13.d odd 4 2 3380.1.h.f 4
13.e even 6 1 260.1.w.b yes 2
13.e even 6 1 3380.1.w.c 2
13.f odd 12 2 3380.1.v.b 4
13.f odd 12 2 3380.1.v.c 4
20.d odd 2 1 3380.1.g.a 2
39.h odd 6 1 2340.1.dd.a 2
39.i odd 6 1 2340.1.dd.b 2
52.b odd 2 1 3380.1.g.a 2
52.f even 4 2 3380.1.h.f 4
52.i odd 6 1 260.1.w.b yes 2
52.i odd 6 1 3380.1.w.c 2
52.j odd 6 1 260.1.w.a ✓ 2
52.j odd 6 1 3380.1.w.b 2
52.l even 12 2 3380.1.v.b 4
52.l even 12 2 3380.1.v.c 4
65.d even 2 1 inner 3380.1.g.b 2
65.g odd 4 2 3380.1.h.f 4
65.l even 6 1 260.1.w.a ✓ 2
65.l even 6 1 3380.1.w.b 2
65.n even 6 1 260.1.w.b yes 2
65.n even 6 1 3380.1.w.c 2
65.q odd 12 2 1300.1.z.a 4
65.r odd 12 2 1300.1.z.a 4
65.s odd 12 2 3380.1.v.b 4
65.s odd 12 2 3380.1.v.c 4
156.p even 6 1 2340.1.dd.b 2
156.r even 6 1 2340.1.dd.a 2
195.x odd 6 1 2340.1.dd.a 2
195.y odd 6 1 2340.1.dd.b 2
260.g odd 2 1 inner 3380.1.g.b 2
260.u even 4 2 3380.1.h.f 4
260.v odd 6 1 260.1.w.b yes 2
260.v odd 6 1 3380.1.w.c 2
260.w odd 6 1 260.1.w.a ✓ 2
260.w odd 6 1 3380.1.w.b 2
260.bc even 12 2 3380.1.v.b 4
260.bc even 12 2 3380.1.v.c 4
260.bg even 12 2 1300.1.z.a 4
260.bj even 12 2 1300.1.z.a 4
780.br even 6 1 2340.1.dd.a 2
780.cb even 6 1 2340.1.dd.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.1.w.a ✓ 2 13.c even 3 1
260.1.w.a ✓ 2 52.j odd 6 1
260.1.w.a ✓ 2 65.l even 6 1
260.1.w.a ✓ 2 260.w odd 6 1
260.1.w.b yes 2 13.e even 6 1
260.1.w.b yes 2 52.i odd 6 1
260.1.w.b yes 2 65.n even 6 1
260.1.w.b yes 2 260.v odd 6 1
1300.1.z.a 4 65.q odd 12 2
1300.1.z.a 4 65.r odd 12 2
1300.1.z.a 4 260.bg even 12 2
1300.1.z.a 4 260.bj even 12 2
2340.1.dd.a 2 39.h odd 6 1
2340.1.dd.a 2 156.r even 6 1
2340.1.dd.a 2 195.x odd 6 1
2340.1.dd.a 2 780.br even 6 1
2340.1.dd.b 2 39.i odd 6 1
2340.1.dd.b 2 156.p even 6 1
2340.1.dd.b 2 195.y odd 6 1
2340.1.dd.b 2 780.cb even 6 1
3380.1.g.a 2 5.b even 2 1
3380.1.g.a 2 13.b even 2 1
3380.1.g.a 2 20.d odd 2 1
3380.1.g.a 2 52.b odd 2 1
3380.1.g.b 2 1.a even 1 1 trivial
3380.1.g.b 2 4.b odd 2 1 CM
3380.1.g.b 2 65.d even 2 1 inner
3380.1.g.b 2 260.g odd 2 1 inner
3380.1.h.f 4 13.d odd 4 2
3380.1.h.f 4 52.f even 4 2
3380.1.h.f 4 65.g odd 4 2
3380.1.h.f 4 260.u even 4 2
3380.1.v.b 4 13.f odd 12 2
3380.1.v.b 4 52.l even 12 2
3380.1.v.b 4 65.s odd 12 2
3380.1.v.b 4 260.bc even 12 2
3380.1.v.c 4 13.f odd 12 2
3380.1.v.c 4 52.l even 12 2
3380.1.v.c 4 65.s odd 12 2
3380.1.v.c 4 260.bc even 12 2
3380.1.w.b 2 13.c even 3 1
3380.1.w.b 2 52.j odd 6 1
3380.1.w.b 2 65.l even 6 1
3380.1.w.b 2 260.w odd 6 1
3380.1.w.c 2 13.e even 6 1
3380.1.w.c 2 52.i odd 6 1
3380.1.w.c 2 65.n even 6 1
3380.1.w.c 2 260.v odd 6 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
\( T_{37} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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