Properties

Label 390.2.s.a
Level $390$
Weight $2$
Character orbit 390.s
Analytic conductor $3.114$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [390,2,Mod(77,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.77");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.s (of order \(4\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{3} + 1) q^{3} + \beta_{2} q^{4} + (\beta_{7} - 2 \beta_1) q^{5} + ( - \beta_{4} - \beta_1) q^{6} + ( - 2 \beta_{7} - 2 \beta_{6} + \cdots + 2 \beta_1) q^{7}+ \cdots + (\beta_{5} + \beta_{3} + 2 \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{3} + 1) q^{3} + \beta_{2} q^{4} + (\beta_{7} - 2 \beta_1) q^{5} + ( - \beta_{4} - \beta_1) q^{6} + ( - 2 \beta_{7} - 2 \beta_{6} + \cdots + 2 \beta_1) q^{7}+ \cdots + ( - 4 \beta_{7} + 8 \beta_{6} + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 8 q^{10} + 4 q^{12} + 16 q^{13} - 8 q^{16} - 32 q^{22} - 32 q^{25} + 28 q^{27} + 4 q^{30} - 16 q^{36} - 16 q^{40} + 40 q^{42} - 40 q^{43} - 4 q^{48} + 40 q^{51} + 16 q^{52} - 32 q^{55} + 80 q^{61} - 32 q^{66} - 4 q^{75} + 20 q^{78} + 8 q^{81} + 32 q^{82} - 40 q^{87} + 32 q^{88} - 32 q^{90} - 80 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 8\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + \nu^{4} + 5\nu^{2} + 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 8\nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + \nu^{4} - 5\nu^{2} + 2 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + \nu^{5} - 8\nu^{3} + 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} - 13\nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{3} + 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{5} + 3\beta_{3} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{6} - 5\beta_{4} + 11\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4\beta_{5} + 4\beta_{3} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -16\beta_{7} + 13\beta_{6} - 13\beta_{4} - 13\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(-1\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1
−1.14412 1.14412i
0.437016 + 0.437016i
1.14412 + 1.14412i
−0.437016 0.437016i
−1.14412 + 1.14412i
0.437016 0.437016i
1.14412 1.14412i
−0.437016 + 0.437016i
−0.707107 0.707107i −0.618034 1.61803i 1.00000i −0.707107 2.12132i −0.707107 + 1.58114i 3.16228 3.16228i 0.707107 0.707107i −2.23607 + 2.00000i −1.00000 + 2.00000i
77.2 −0.707107 0.707107i 1.61803 + 0.618034i 1.00000i −0.707107 2.12132i −0.707107 1.58114i −3.16228 + 3.16228i 0.707107 0.707107i 2.23607 + 2.00000i −1.00000 + 2.00000i
77.3 0.707107 + 0.707107i −0.618034 1.61803i 1.00000i 0.707107 + 2.12132i 0.707107 1.58114i −3.16228 + 3.16228i −0.707107 + 0.707107i −2.23607 + 2.00000i −1.00000 + 2.00000i
77.4 0.707107 + 0.707107i 1.61803 + 0.618034i 1.00000i 0.707107 + 2.12132i 0.707107 + 1.58114i 3.16228 3.16228i −0.707107 + 0.707107i 2.23607 + 2.00000i −1.00000 + 2.00000i
233.1 −0.707107 + 0.707107i −0.618034 + 1.61803i 1.00000i −0.707107 + 2.12132i −0.707107 1.58114i 3.16228 + 3.16228i 0.707107 + 0.707107i −2.23607 2.00000i −1.00000 2.00000i
233.2 −0.707107 + 0.707107i 1.61803 0.618034i 1.00000i −0.707107 + 2.12132i −0.707107 + 1.58114i −3.16228 3.16228i 0.707107 + 0.707107i 2.23607 2.00000i −1.00000 2.00000i
233.3 0.707107 0.707107i −0.618034 + 1.61803i 1.00000i 0.707107 2.12132i 0.707107 + 1.58114i −3.16228 3.16228i −0.707107 0.707107i −2.23607 2.00000i −1.00000 2.00000i
233.4 0.707107 0.707107i 1.61803 0.618034i 1.00000i 0.707107 2.12132i 0.707107 1.58114i 3.16228 + 3.16228i −0.707107 0.707107i 2.23607 2.00000i −1.00000 2.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
13.b even 2 1 inner
15.e even 4 1 inner
39.d odd 2 1 inner
65.h odd 4 1 inner
195.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.s.a 8
3.b odd 2 1 inner 390.2.s.a 8
5.c odd 4 1 inner 390.2.s.a 8
13.b even 2 1 inner 390.2.s.a 8
15.e even 4 1 inner 390.2.s.a 8
39.d odd 2 1 inner 390.2.s.a 8
65.h odd 4 1 inner 390.2.s.a 8
195.s even 4 1 inner 390.2.s.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.s.a 8 1.a even 1 1 trivial
390.2.s.a 8 3.b odd 2 1 inner
390.2.s.a 8 5.c odd 4 1 inner
390.2.s.a 8 13.b even 2 1 inner
390.2.s.a 8 15.e even 4 1 inner
390.2.s.a 8 39.d odd 2 1 inner
390.2.s.a 8 65.h odd 4 1 inner
390.2.s.a 8 195.s even 4 1 inner

Hecke kernels

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

\( T_{7}^{4} + 400 \) Copy content Toggle raw display
\( T_{17}^{4} + 100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 9)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 8 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 8 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 100)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 40)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{2} - 20)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10 T + 50)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$61$ \( (T - 10)^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 32400)^{2} \) Copy content Toggle raw display
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