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Properties

Label 380.1.g.a
Level $380$
Weight $1$
Character orbit 380.g
Analytic conductor $0.190$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,1,Mod(189,380)] 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("380.189"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 380.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.189644704801\)
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: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.722000.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{5} + (\zeta_{6}^{2} + \zeta_{6}) q^{7} - q^{9} + q^{11} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{17} + q^{19} - \zeta_{6} q^{25} + ( - \zeta_{6} - 1) q^{35} + (\zeta_{6}^{2} + \zeta_{6}) q^{43} + \cdots - q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - 2 q^{9} + 2 q^{11} + 2 q^{19} - q^{25} - 3 q^{35} + q^{45} - 4 q^{49} - q^{55} + 2 q^{61} + 2 q^{81} + 3 q^{85} - q^{95} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
189.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 0 −0.500000 − 0.866025i 0 − 1.73205i 0 −1.00000 0
189.2 0 0 0 −0.500000 + 0.866025i 0 1.73205i 0 −1.00000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
5.b even 2 1 inner
95.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.1.g.a ✓ 2
3.b odd 2 1 3420.1.h.a 2
4.b odd 2 1 1520.1.m.c 2
5.b even 2 1 inner 380.1.g.a ✓ 2
5.c odd 4 2 1900.1.e.b 2
15.d odd 2 1 3420.1.h.a 2
19.b odd 2 1 CM 380.1.g.a ✓ 2
20.d odd 2 1 1520.1.m.c 2
57.d even 2 1 3420.1.h.a 2
76.d even 2 1 1520.1.m.c 2
95.d odd 2 1 inner 380.1.g.a ✓ 2
95.g even 4 2 1900.1.e.b 2
285.b even 2 1 3420.1.h.a 2
380.d even 2 1 1520.1.m.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.1.g.a ✓ 2 1.a even 1 1 trivial
380.1.g.a ✓ 2 5.b even 2 1 inner
380.1.g.a ✓ 2 19.b odd 2 1 CM
380.1.g.a ✓ 2 95.d odd 2 1 inner
1520.1.m.c 2 4.b odd 2 1
1520.1.m.c 2 20.d odd 2 1
1520.1.m.c 2 76.d even 2 1
1520.1.m.c 2 380.d even 2 1
1900.1.e.b 2 5.c odd 4 2
1900.1.e.b 2 95.g even 4 2
3420.1.h.a 2 3.b odd 2 1
3420.1.h.a 2 15.d odd 2 1
3420.1.h.a 2 57.d even 2 1
3420.1.h.a 2 285.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(380, [\chi])\).

Hecke characteristic polynomials

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