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Properties

Label 390.2.y.a
Level $390$
Weight $2$
Character orbit 390.y
Analytic conductor $3.114$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [390,2,Mod(139,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.139");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.y (of order \(6\), degree \(2\), 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: \(3.11416567883\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} - \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{3} - 1) q^{5} - \zeta_{12}^{2} q^{6} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{12} q^{2} - \zeta_{12} q^{3} + \zeta_{12}^{2} q^{4} + (2 \zeta_{12}^{3} - 1) q^{5} - \zeta_{12}^{2} q^{6} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{7} + \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + (2 \zeta_{12}^{2} - \zeta_{12} - 2) q^{10} + (5 \zeta_{12}^{2} - 5) q^{11} - \zeta_{12}^{3} q^{12} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}) q^{13} + 4 q^{14} + ( - 2 \zeta_{12}^{2} + \zeta_{12} + 2) q^{15} + (\zeta_{12}^{2} - 1) q^{16} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{17} + \zeta_{12}^{3} q^{18} + 8 \zeta_{12}^{2} q^{19} + (2 \zeta_{12}^{3} + \cdots - 2 \zeta_{12}) q^{20} + \cdots - 5 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{9} - 4 q^{10} - 10 q^{11} + 16 q^{14} + 4 q^{15} - 2 q^{16} + 16 q^{19} - 2 q^{20} - 16 q^{21} + 2 q^{24} - 12 q^{25} + 14 q^{26} + 14 q^{29} + 2 q^{30} - 28 q^{31} - 8 q^{34} + 16 q^{35} - 2 q^{36} - 14 q^{39} - 8 q^{40} - 20 q^{44} - 2 q^{45} + 6 q^{46} + 18 q^{49} + 8 q^{50} + 8 q^{51} + 2 q^{54} + 10 q^{55} + 8 q^{56} - 10 q^{59} + 8 q^{60} - 8 q^{61} - 4 q^{64} + 8 q^{65} + 20 q^{66} - 6 q^{69} - 16 q^{70} - 16 q^{71} + 6 q^{74} - 8 q^{75} - 16 q^{76} - 4 q^{79} + 2 q^{80} - 2 q^{81} - 8 q^{84} - 8 q^{85} + 4 q^{86} - 8 q^{90} + 40 q^{91} + 14 q^{94} - 16 q^{95} + 4 q^{96} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/390\mathbb{Z}\right)^\times\).

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(-1\) \(-\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.00000 − 2.00000i −0.500000 − 0.866025i −3.46410 + 2.00000i − 1.00000i 0.500000 + 0.866025i −0.133975 + 2.23205i
139.2 0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 + 0.866025i −1.00000 + 2.00000i −0.500000 − 0.866025i 3.46410 − 2.00000i 1.00000i 0.500000 + 0.866025i −1.86603 + 1.23205i
289.1 −0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 − 0.866025i −1.00000 + 2.00000i −0.500000 + 0.866025i −3.46410 − 2.00000i 1.00000i 0.500000 − 0.866025i −0.133975 − 2.23205i
289.2 0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 − 0.866025i −1.00000 − 2.00000i −0.500000 + 0.866025i 3.46410 + 2.00000i − 1.00000i 0.500000 − 0.866025i −1.86603 − 1.23205i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.y.a ✓ 4
3.b odd 2 1 1170.2.bp.f 4
5.b even 2 1 inner 390.2.y.a ✓ 4
5.c odd 4 1 1950.2.i.g 2
5.c odd 4 1 1950.2.i.t 2
13.c even 3 1 inner 390.2.y.a ✓ 4
15.d odd 2 1 1170.2.bp.f 4
39.i odd 6 1 1170.2.bp.f 4
65.n even 6 1 inner 390.2.y.a ✓ 4
65.q odd 12 1 1950.2.i.g 2
65.q odd 12 1 1950.2.i.t 2
195.x odd 6 1 1170.2.bp.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.y.a ✓ 4 1.a even 1 1 trivial
390.2.y.a ✓ 4 5.b even 2 1 inner
390.2.y.a ✓ 4 13.c even 3 1 inner
390.2.y.a ✓ 4 65.n even 6 1 inner
1170.2.bp.f 4 3.b odd 2 1
1170.2.bp.f 4 15.d odd 2 1
1170.2.bp.f 4 39.i odd 6 1
1170.2.bp.f 4 195.x odd 6 1
1950.2.i.g 2 5.c odd 4 1
1950.2.i.g 2 65.q odd 12 1
1950.2.i.t 2 5.c odd 4 1
1950.2.i.t 2 65.q odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 16T_{7}^{2} + 256 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + 2 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$11$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 22T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$29$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$31$ \( (T + 7)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$47$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$71$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T + 1)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 144 T^{2} + 20736 \) Copy content Toggle raw display
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