Properties

Label 390.2.f.a
Level $390$
Weight $2$
Character orbit 390.f
Analytic conductor $3.114$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [390,2,Mod(259,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.259");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.f (of order \(2\), degree \(1\), 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: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_{4} q^{3} + q^{4} + (\beta_{3} - \beta_1) q^{5} + \beta_{4} q^{6} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{7} - q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_{4} q^{3} + q^{4} + (\beta_{3} - \beta_1) q^{5} + \beta_{4} q^{6} + (\beta_{5} + \beta_{3} - \beta_{2} + 1) q^{7} - q^{8} - q^{9} + ( - \beta_{3} + \beta_1) q^{10} + ( - \beta_{5} + 2 \beta_{4} + \cdots - 2 \beta_1) q^{11}+ \cdots + (\beta_{5} - 2 \beta_{4} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} + 6 q^{4} - 2 q^{5} + 4 q^{7} - 6 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} + 6 q^{4} - 2 q^{5} + 4 q^{7} - 6 q^{8} - 6 q^{9} + 2 q^{10} - 8 q^{13} - 4 q^{14} + 6 q^{16} + 6 q^{18} - 2 q^{20} - 2 q^{25} + 8 q^{26} + 4 q^{28} + 4 q^{29} - 6 q^{32} + 12 q^{33} + 16 q^{35} - 6 q^{36} + 24 q^{37} + 2 q^{39} + 2 q^{40} + 2 q^{45} - 32 q^{47} - 2 q^{49} + 2 q^{50} + 8 q^{51} - 8 q^{52} - 32 q^{55} - 4 q^{56} + 16 q^{57} - 4 q^{58} - 4 q^{61} - 4 q^{63} + 6 q^{64} - 16 q^{65} - 12 q^{66} + 4 q^{67} - 4 q^{69} - 16 q^{70} + 6 q^{72} + 40 q^{73} - 24 q^{74} + 8 q^{75} - 2 q^{78} - 16 q^{79} - 2 q^{80} + 6 q^{81} - 12 q^{85} - 2 q^{90} - 20 q^{91} - 4 q^{93} + 32 q^{94} + 12 q^{95} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 6\nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu^{2} + 80\nu + 2 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 2\beta_{3} - 5\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -9\beta_{4} + 5\beta_{3} - 8\beta_{2} - 8\beta _1 - 9 \) Copy content Toggle raw display

Character values

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

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
259.1
−0.854638 0.854638i
1.45161 + 1.45161i
0.403032 + 0.403032i
−0.854638 + 0.854638i
1.45161 1.45161i
0.403032 0.403032i
−1.00000 1.00000i 1.00000 −2.17009 + 0.539189i 1.00000i −0.630898 −1.00000 −1.00000 2.17009 0.539189i
259.2 −1.00000 1.00000i 1.00000 −0.311108 2.21432i 1.00000i −1.52543 −1.00000 −1.00000 0.311108 + 2.21432i
259.3 −1.00000 1.00000i 1.00000 1.48119 + 1.67513i 1.00000i 4.15633 −1.00000 −1.00000 −1.48119 1.67513i
259.4 −1.00000 1.00000i 1.00000 −2.17009 0.539189i 1.00000i −0.630898 −1.00000 −1.00000 2.17009 + 0.539189i
259.5 −1.00000 1.00000i 1.00000 −0.311108 + 2.21432i 1.00000i −1.52543 −1.00000 −1.00000 0.311108 2.21432i
259.6 −1.00000 1.00000i 1.00000 1.48119 1.67513i 1.00000i 4.15633 −1.00000 −1.00000 −1.48119 + 1.67513i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 259.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.f.a 6
3.b odd 2 1 1170.2.f.d 6
4.b odd 2 1 3120.2.r.g 6
5.b even 2 1 390.2.f.b yes 6
5.c odd 4 1 1950.2.b.l 6
5.c odd 4 1 1950.2.b.m 6
13.b even 2 1 390.2.f.b yes 6
15.d odd 2 1 1170.2.f.c 6
20.d odd 2 1 3120.2.r.h 6
39.d odd 2 1 1170.2.f.c 6
52.b odd 2 1 3120.2.r.h 6
65.d even 2 1 inner 390.2.f.a 6
65.h odd 4 1 1950.2.b.l 6
65.h odd 4 1 1950.2.b.m 6
195.e odd 2 1 1170.2.f.d 6
260.g odd 2 1 3120.2.r.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.f.a 6 1.a even 1 1 trivial
390.2.f.a 6 65.d even 2 1 inner
390.2.f.b yes 6 5.b even 2 1
390.2.f.b yes 6 13.b even 2 1
1170.2.f.c 6 15.d odd 2 1
1170.2.f.c 6 39.d odd 2 1
1170.2.f.d 6 3.b odd 2 1
1170.2.f.d 6 195.e odd 2 1
1950.2.b.l 6 5.c odd 4 1
1950.2.b.l 6 65.h odd 4 1
1950.2.b.m 6 5.c odd 4 1
1950.2.b.m 6 65.h odd 4 1
3120.2.r.g 6 4.b odd 2 1
3120.2.r.g 6 260.g odd 2 1
3120.2.r.h 6 20.d odd 2 1
3120.2.r.h 6 52.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{3} - 2T_{7}^{2} - 8T_{7} - 4 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( (T^{3} - 2 T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 44 T^{4} + \cdots + 1600 \) Copy content Toggle raw display
$13$ \( T^{6} + 8 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} + 16 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( T^{6} + 32 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{6} + 12 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} - 44 T + 20)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 108 T^{4} + \cdots + 1600 \) Copy content Toggle raw display
$37$ \( (T^{3} - 12 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 76 T^{4} + \cdots + 1600 \) Copy content Toggle raw display
$43$ \( T^{6} + 128 T^{4} + \cdots + 73984 \) Copy content Toggle raw display
$47$ \( (T^{3} + 16 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 236 T^{4} + \cdots + 40000 \) Copy content Toggle raw display
$59$ \( T^{6} + 252 T^{4} + \cdots + 61504 \) Copy content Toggle raw display
$61$ \( (T^{3} + 2 T^{2} + \cdots - 104)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 2 T^{2} + \cdots + 184)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 112 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$73$ \( (T^{3} - 20 T^{2} + \cdots + 548)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 8 T^{2} + \cdots - 2000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 144 T - 432)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 252 T^{4} + \cdots + 287296 \) Copy content Toggle raw display
$97$ \( (T^{3} - 100 T - 268)^{2} \) Copy content Toggle raw display
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