Properties

Label 390.2.b.d
Level $390$
Weight $2$
Character orbit 390.b
Analytic conductor $3.114$
Analytic rank $0$
Dimension $4$
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(181,390)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.b (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: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} - q^{4} + \beta_1 q^{5} - \beta_1 q^{6} + (\beta_{3} + \beta_{2} - \beta_1) q^{7} + \beta_1 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + q^{3} - q^{4} + \beta_1 q^{5} - \beta_1 q^{6} + (\beta_{3} + \beta_{2} - \beta_1) q^{7} + \beta_1 q^{8} + q^{9} + q^{10} + (\beta_{3} + \beta_{2} + \beta_1) q^{11} - q^{12} + ( - \beta_{3} + 2 \beta_1 + 1) q^{13} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{14} + \beta_1 q^{15} + q^{16} + 2 q^{17} - \beta_1 q^{18} - 6 \beta_1 q^{19} - \beta_1 q^{20} + (\beta_{3} + \beta_{2} - \beta_1) q^{21} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{22} + (\beta_{3} - \beta_{2} - \beta_1) q^{23} + \beta_1 q^{24} - q^{25} + ( - \beta_{2} - 2 \beta_1 + 2) q^{26} + q^{27} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{28} + 2 q^{29} + q^{30} + (\beta_{3} + \beta_{2} - \beta_1) q^{31} - \beta_1 q^{32} + (\beta_{3} + \beta_{2} + \beta_1) q^{33} - 2 \beta_1 q^{34} + (\beta_{3} - \beta_{2} - \beta_1 + 2) q^{35} - q^{36} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{37} - 6 q^{38} + ( - \beta_{3} + 2 \beta_1 + 1) q^{39} - q^{40} + ( - \beta_{3} - \beta_{2} + 5 \beta_1) q^{41} + ( - \beta_{3} + \beta_{2} + \beta_1 - 2) q^{42} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{43}+ \cdots + (\beta_{3} + \beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{4} + 4 q^{9} + 4 q^{10} - 4 q^{12} + 6 q^{13} - 4 q^{14} + 4 q^{16} + 8 q^{17} + 4 q^{22} - 4 q^{23} - 4 q^{25} + 6 q^{26} + 4 q^{27} + 8 q^{29} + 4 q^{30} + 4 q^{35} - 4 q^{36} - 24 q^{38} + 6 q^{39} - 4 q^{40} - 4 q^{42} - 8 q^{43} + 4 q^{48} - 44 q^{49} + 8 q^{51} - 6 q^{52} - 4 q^{53} - 4 q^{55} + 4 q^{56} + 40 q^{61} - 4 q^{62} - 4 q^{64} - 6 q^{65} + 4 q^{66} - 8 q^{68} - 4 q^{69} - 4 q^{74} - 4 q^{75} - 64 q^{77} + 6 q^{78} + 32 q^{79} + 4 q^{81} + 20 q^{82} + 8 q^{87} - 4 q^{88} + 4 q^{90} + 40 q^{91} + 4 q^{92} + 8 q^{94} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 4\nu^{2} + 9\nu - 20 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} + \beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{3} - 5\beta_{2} + 13\beta_1 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
2.56155i
1.56155i
1.56155i
2.56155i
1.00000i 1.00000 −1.00000 1.00000i 1.00000i 5.12311i 1.00000i 1.00000 1.00000
181.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.12311i 1.00000i 1.00000 1.00000
181.3 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 3.12311i 1.00000i 1.00000 1.00000
181.4 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 5.12311i 1.00000i 1.00000 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
13.b 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.b.d 4
3.b odd 2 1 1170.2.b.f 4
4.b odd 2 1 3120.2.g.o 4
5.b even 2 1 1950.2.b.h 4
5.c odd 4 1 1950.2.f.l 4
5.c odd 4 1 1950.2.f.o 4
13.b even 2 1 inner 390.2.b.d 4
13.d odd 4 1 5070.2.a.bd 2
13.d odd 4 1 5070.2.a.bh 2
39.d odd 2 1 1170.2.b.f 4
52.b odd 2 1 3120.2.g.o 4
65.d even 2 1 1950.2.b.h 4
65.h odd 4 1 1950.2.f.l 4
65.h odd 4 1 1950.2.f.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.b.d 4 1.a even 1 1 trivial
390.2.b.d 4 13.b even 2 1 inner
1170.2.b.f 4 3.b odd 2 1
1170.2.b.f 4 39.d odd 2 1
1950.2.b.h 4 5.b even 2 1
1950.2.b.h 4 65.d even 2 1
1950.2.f.l 4 5.c odd 4 1
1950.2.f.l 4 65.h odd 4 1
1950.2.f.o 4 5.c odd 4 1
1950.2.f.o 4 65.h odd 4 1
3120.2.g.o 4 4.b odd 2 1
3120.2.g.o 4 52.b odd 2 1
5070.2.a.bd 2 13.d odd 4 1
5070.2.a.bh 2 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\):

\( T_{7}^{4} + 36T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{4} + 36T_{11}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$11$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( (T - 2)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( (T - 2)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$37$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$41$ \( T^{4} + 84T^{2} + 64 \) Copy content Toggle raw display
$43$ \( (T^{2} + 4 T - 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 144T^{2} + 4096 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T - 152)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 52T^{2} + 64 \) Copy content Toggle raw display
$61$ \( (T - 10)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 196T^{2} + 4096 \) Copy content Toggle raw display
$71$ \( T^{4} + 144T^{2} + 4096 \) Copy content Toggle raw display
$73$ \( T^{4} + 196T^{2} + 4096 \) Copy content Toggle raw display
$79$ \( (T - 8)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 144T^{2} + 4096 \) Copy content Toggle raw display
$89$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$97$ \( T^{4} + 196T^{2} + 4096 \) Copy content Toggle raw display
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