Properties

Label 390.2.e.e
Level $390$
Weight $2$
Character orbit 390.e
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(79,390)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.e (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: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_1 q^{3} - q^{4} - \beta_{2} q^{5} - q^{6} + ( - \beta_{2} - \beta_1) q^{7} - \beta_1 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_1 q^{3} - q^{4} - \beta_{2} q^{5} - q^{6} + ( - \beta_{2} - \beta_1) q^{7} - \beta_1 q^{8} - q^{9} + \beta_{3} q^{10} + 2 \beta_{3} q^{11} - \beta_1 q^{12} + \beta_1 q^{13} + (\beta_{3} + 1) q^{14} + \beta_{3} q^{15} + q^{16} + ( - \beta_{2} - 5 \beta_1) q^{17} - \beta_1 q^{18} + ( - \beta_{3} + 5) q^{19} + \beta_{2} q^{20} + (\beta_{3} + 1) q^{21} + 2 \beta_{2} q^{22} + ( - \beta_{2} + 5 \beta_1) q^{23} + q^{24} - 5 q^{25} - q^{26} - \beta_1 q^{27} + (\beta_{2} + \beta_1) q^{28} + (3 \beta_{3} - 3) q^{29} + \beta_{2} q^{30} - 4 q^{31} + \beta_1 q^{32} + 2 \beta_{2} q^{33} + (\beta_{3} + 5) q^{34} + ( - \beta_{3} - 5) q^{35} + q^{36} + (4 \beta_{2} + 2 \beta_1) q^{37} + ( - \beta_{2} + 5 \beta_1) q^{38} - q^{39} - \beta_{3} q^{40} + ( - 2 \beta_{3} + 8) q^{41} + (\beta_{2} + \beta_1) q^{42} + ( - 2 \beta_{2} + 2 \beta_1) q^{43} - 2 \beta_{3} q^{44} + \beta_{2} q^{45} + (\beta_{3} - 5) q^{46} + ( - 4 \beta_{2} - 4 \beta_1) q^{47} + \beta_1 q^{48} + ( - 2 \beta_{3} + 1) q^{49} - 5 \beta_1 q^{50} + (\beta_{3} + 5) q^{51} - \beta_1 q^{52} + ( - 2 \beta_{2} + 4 \beta_1) q^{53} + q^{54} - 10 \beta_1 q^{55} + ( - \beta_{3} - 1) q^{56} + ( - \beta_{2} + 5 \beta_1) q^{57} + (3 \beta_{2} - 3 \beta_1) q^{58} + (2 \beta_{3} + 4) q^{59} - \beta_{3} q^{60} + ( - 4 \beta_{3} - 2) q^{61} - 4 \beta_1 q^{62} + (\beta_{2} + \beta_1) q^{63} - q^{64} + \beta_{3} q^{65} - 2 \beta_{3} q^{66} + (\beta_{2} + 5 \beta_1) q^{68} + (\beta_{3} - 5) q^{69} + ( - \beta_{2} - 5 \beta_1) q^{70} + ( - 2 \beta_{3} + 2) q^{71} + \beta_1 q^{72} + (\beta_{2} + 11 \beta_1) q^{73} + ( - 4 \beta_{3} - 2) q^{74} - 5 \beta_1 q^{75} + (\beta_{3} - 5) q^{76} + ( - 2 \beta_{2} - 10 \beta_1) q^{77} - \beta_1 q^{78} + 4 q^{79} - \beta_{2} q^{80} + q^{81} + ( - 2 \beta_{2} + 8 \beta_1) q^{82} + ( - 4 \beta_{2} + 4 \beta_1) q^{83} + ( - \beta_{3} - 1) q^{84} + ( - 5 \beta_{3} - 5) q^{85} + (2 \beta_{3} - 2) q^{86} + (3 \beta_{2} - 3 \beta_1) q^{87} - 2 \beta_{2} q^{88} + ( - 2 \beta_{3} + 4) q^{89} - \beta_{3} q^{90} + (\beta_{3} + 1) q^{91} + (\beta_{2} - 5 \beta_1) q^{92} - 4 \beta_1 q^{93} + (4 \beta_{3} + 4) q^{94} + ( - 5 \beta_{2} + 5 \beta_1) q^{95} - q^{96} + (3 \beta_{2} - 3 \beta_1) q^{97} + ( - 2 \beta_{2} + \beta_1) q^{98} - 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 4 q^{14} + 4 q^{16} + 20 q^{19} + 4 q^{21} + 4 q^{24} - 20 q^{25} - 4 q^{26} - 12 q^{29} - 16 q^{31} + 20 q^{34} - 20 q^{35} + 4 q^{36} - 4 q^{39} + 32 q^{41} - 20 q^{46} + 4 q^{49} + 20 q^{51} + 4 q^{54} - 4 q^{56} + 16 q^{59} - 8 q^{61} - 4 q^{64} - 20 q^{69} + 8 q^{71} - 8 q^{74} - 20 q^{76} + 16 q^{79} + 4 q^{81} - 4 q^{84} - 20 q^{85} - 8 q^{86} + 16 q^{89} + 4 q^{91} + 16 q^{94} - 4 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 2\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
1.61803i
0.618034i
0.618034i
1.61803i
1.00000i 1.00000i −1.00000 2.23607i −1.00000 1.23607i 1.00000i −1.00000 −2.23607
79.2 1.00000i 1.00000i −1.00000 2.23607i −1.00000 3.23607i 1.00000i −1.00000 2.23607
79.3 1.00000i 1.00000i −1.00000 2.23607i −1.00000 3.23607i 1.00000i −1.00000 2.23607
79.4 1.00000i 1.00000i −1.00000 2.23607i −1.00000 1.23607i 1.00000i −1.00000 −2.23607
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.e.e 4
3.b odd 2 1 1170.2.e.e 4
4.b odd 2 1 3120.2.l.k 4
5.b even 2 1 inner 390.2.e.e 4
5.c odd 4 1 1950.2.a.be 2
5.c odd 4 1 1950.2.a.bf 2
15.d odd 2 1 1170.2.e.e 4
15.e even 4 1 5850.2.a.cf 2
15.e even 4 1 5850.2.a.cm 2
20.d odd 2 1 3120.2.l.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.e.e 4 1.a even 1 1 trivial
390.2.e.e 4 5.b even 2 1 inner
1170.2.e.e 4 3.b odd 2 1
1170.2.e.e 4 15.d odd 2 1
1950.2.a.be 2 5.c odd 4 1
1950.2.a.bf 2 5.c odd 4 1
3120.2.l.k 4 4.b odd 2 1
3120.2.l.k 4 20.d odd 2 1
5850.2.a.cf 2 15.e even 4 1
5850.2.a.cm 2 15.e even 4 1

Hecke kernels

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

\( T_{7}^{4} + 12T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} - 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$19$ \( (T^{2} - 10 T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 36)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 168T^{2} + 5776 \) Copy content Toggle raw display
$41$ \( (T^{2} - 16 T + 44)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 48T^{2} + 256 \) Copy content Toggle raw display
$47$ \( T^{4} + 192T^{2} + 4096 \) Copy content Toggle raw display
$53$ \( T^{4} + 72T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 76)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 252 T^{2} + 13456 \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 192T^{2} + 4096 \) Copy content Toggle raw display
$89$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 108T^{2} + 1296 \) Copy content Toggle raw display
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