Properties

Label 390.2.y.g
Level $390$
Weight $2$
Character orbit 390.y
Analytic conductor $3.114$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.303595776.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} - \beta_{5} q^{3} + (\beta_{4} + 1) q^{4} + ( - \beta_{6} - 2) q^{5} + (\beta_{4} + 1) q^{6} + ( - 2 \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 1) q^{7} + ( - \beta_{5} - \beta_{3}) q^{8} + (\beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} - \beta_{5} q^{3} + (\beta_{4} + 1) q^{4} + ( - \beta_{6} - 2) q^{5} + (\beta_{4} + 1) q^{6} + ( - 2 \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 1) q^{7} + ( - \beta_{5} - \beta_{3}) q^{8} + (\beta_{4} + 1) q^{9} + (\beta_{5} + \beta_1) q^{10} + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_1) q^{11} + ( - \beta_{5} - \beta_{3}) q^{12} + ( - 3 \beta_{5} - 4 \beta_{3}) q^{13} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3} + 2 \beta_1 + 1) q^{14} + (\beta_{5} + \beta_1) q^{15} + \beta_{4} q^{16} + (4 \beta_{6} + 2 \beta_{4} - \beta_{3} - 4 \beta_{2} + 2) q^{17} + ( - \beta_{5} - \beta_{3}) q^{18} + ( - 2 \beta_{7} + \beta_{4} - \beta_{3} + 1) q^{19} + ( - \beta_{6} - 2 \beta_{4} + \beta_{2} - 2) q^{20} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3} + 2 \beta_1 + 1) q^{21} + ( - 2 \beta_{6} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 1) q^{22} + ( - 5 \beta_{5} + \beta_{4} - 2 \beta_{2}) q^{23} + \beta_{4} q^{24} + (3 \beta_{6} + 1) q^{25} + (3 \beta_{4} - 1) q^{26} + ( - \beta_{5} - \beta_{3}) q^{27} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{28} - 3 \beta_{4} q^{29} + ( - \beta_{6} - 2 \beta_{4} + \beta_{2} - 2) q^{30} - 4 q^{31} - \beta_{3} q^{32} + ( - 2 \beta_{6} - \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 1) q^{33} + (4 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} - 4 \beta_1 - 1) q^{34} + (\beta_{7} + 3 \beta_{6} - 4 \beta_{4} - \beta_{3} - 3 \beta_{2} - 4) q^{35} + \beta_{4} q^{36} + (\beta_{5} + 2 \beta_{4} - 4 \beta_{2}) q^{37} + (2 \beta_{6} - \beta_{5} - \beta_{3} + 1) q^{38} + (3 \beta_{4} - 1) q^{39} + ( - \beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{40} + (\beta_{5} - 4 \beta_{4} - 2 \beta_1) q^{41} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{42} + (2 \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 1) q^{43} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3} + 2 \beta_1 + 3) q^{44} + ( - \beta_{6} - 2 \beta_{4} + \beta_{2} - 2) q^{45} + (2 \beta_{7} + 5 \beta_{4} + \beta_{3} + 5) q^{46} + (2 \beta_{6} + 5 \beta_{5} + 5 \beta_{3} + 1) q^{47} - \beta_{3} q^{48} + ( - 2 \beta_{5} - 5 \beta_{4} + 4 \beta_1) q^{49} + (2 \beta_{5} - 3 \beta_1) q^{50} + (4 \beta_{7} + 2 \beta_{5} + 2 \beta_{3} - 4 \beta_1 - 1) q^{51} + (\beta_{5} - 3 \beta_{3}) q^{52} + ( - 2 \beta_{6} - 8 \beta_{5} - 8 \beta_{3} - 1) q^{53} + \beta_{4} q^{54} + (7 \beta_{5} + 6 \beta_{4} - 3 \beta_{2} - 3 \beta_1) q^{55} + ( - 2 \beta_{7} + \beta_{4} - \beta_{3} + 1) q^{56} + (2 \beta_{6} - \beta_{5} - \beta_{3} + 1) q^{57} + 3 \beta_{3} q^{58} + (4 \beta_{7} + 2 \beta_{4} + 2 \beta_{3} + 2) q^{59} + ( - \beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{60} + ( - 2 \beta_{7} - 4 \beta_{4} - \beta_{3} - 4) q^{61} + 4 \beta_{5} q^{62} + ( - \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{63} - q^{64} + ( - 4 \beta_{7} + 3 \beta_{5} + 4 \beta_{3} + 3 \beta_1) q^{65} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3} + 2 \beta_1 + 3) q^{66} + (3 \beta_{5} - 3 \beta_{4} + 6 \beta_{2}) q^{67} + (\beta_{5} + 2 \beta_{4} - 4 \beta_{2}) q^{68} + (2 \beta_{7} + 5 \beta_{4} + \beta_{3} + 5) q^{69} + (3 \beta_{7} - \beta_{6} + 7 \beta_{5} + 7 \beta_{3} - 3 \beta_1 - 2) q^{70} + (2 \beta_{7} - 5 \beta_{4} + \beta_{3} - 5) q^{71} - \beta_{3} q^{72} + ( - 2 \beta_{6} - 4 \beta_{5} - 4 \beta_{3} - 1) q^{73} + (4 \beta_{7} - \beta_{4} + 2 \beta_{3} - 1) q^{74} + (2 \beta_{5} - 3 \beta_1) q^{75} + (\beta_{5} + \beta_{4} - 2 \beta_1) q^{76} + ( - 8 \beta_{6} + 14 \beta_{5} + 14 \beta_{3} - 4) q^{77} + (\beta_{5} - 3 \beta_{3}) q^{78} + 4 q^{79} + ( - 2 \beta_{4} + \beta_{2}) q^{80} + \beta_{4} q^{81} + (2 \beta_{6} + \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 1) q^{82} + (2 \beta_{6} - 5 \beta_{5} - 5 \beta_{3} + 1) q^{83} + ( - 2 \beta_{7} + \beta_{4} - \beta_{3} + 1) q^{84} + ( - \beta_{7} - 6 \beta_{6} + 8 \beta_{4} + \beta_{3} + 6 \beta_{2} + 8) q^{85} + (2 \beta_{7} + \beta_{5} + \beta_{3} - 2 \beta_1 + 1) q^{86} + 3 \beta_{3} q^{87} + ( - 3 \beta_{5} - \beta_{4} + 2 \beta_{2}) q^{88} + (4 \beta_{5} - 2 \beta_{4} - 8 \beta_1) q^{89} + ( - \beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{90} + ( - 6 \beta_{7} + \beta_{5} + 4 \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 3) q^{91} + ( - 2 \beta_{6} - 5 \beta_{5} - 5 \beta_{3} - 1) q^{92} + 4 \beta_{5} q^{93} + (\beta_{5} - 5 \beta_{4} - 2 \beta_1) q^{94} + (3 \beta_{7} - \beta_{6} - 2 \beta_{4} + 7 \beta_{3} + \beta_{2} - 2) q^{95} - q^{96} - 6 \beta_{3} q^{97} + ( - 4 \beta_{6} - 2 \beta_{4} + 5 \beta_{3} + 4 \beta_{2} - 2) q^{98} + ( - 2 \beta_{7} - \beta_{5} - \beta_{3} + 2 \beta_1 + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} - 12 q^{5} + 4 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 12 q^{5} + 4 q^{6} + 4 q^{9} + 12 q^{11} + 8 q^{14} - 4 q^{16} + 4 q^{19} - 6 q^{20} + 8 q^{21} - 4 q^{24} - 4 q^{25} - 20 q^{26} + 12 q^{29} - 6 q^{30} - 32 q^{31} - 8 q^{34} - 22 q^{35} - 4 q^{36} - 20 q^{39} + 16 q^{41} + 24 q^{44} - 6 q^{45} + 20 q^{46} + 20 q^{49} - 8 q^{51} - 4 q^{54} - 18 q^{55} + 4 q^{56} + 8 q^{59} - 16 q^{61} - 8 q^{64} + 24 q^{66} + 20 q^{69} - 12 q^{70} - 20 q^{71} - 4 q^{74} - 4 q^{76} + 32 q^{79} + 6 q^{80} - 4 q^{81} + 4 q^{84} + 44 q^{85} + 8 q^{86} + 8 q^{89} + 8 q^{91} + 20 q^{94} - 6 q^{95} - 8 q^{96} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 32\nu^{4} + 16\nu^{2} + 45 ) / 144 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 13\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} - 13 ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 16\nu^{5} + 80\nu^{3} + 225\nu ) / 144 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 2\beta_{4} - \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} - 3\beta_{5} - 3\beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 15\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} - 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 48\beta_{5} - 13\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(-1\) \(-1 - \beta_{4}\)

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
−0.396143 + 1.68614i
1.26217 1.18614i
0.396143 1.68614i
−1.26217 + 1.18614i
1.26217 + 1.18614i
−0.396143 1.68614i
−1.26217 1.18614i
0.396143 + 1.68614i
−0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −1.50000 1.65831i 0.500000 + 0.866025i 2.00626 1.15831i 1.00000i 0.500000 + 0.866025i 0.469882 + 2.18614i
139.2 −0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i −1.50000 + 1.65831i 0.500000 + 0.866025i −3.73831 + 2.15831i 1.00000i 0.500000 + 0.866025i 2.12819 0.686141i
139.3 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.50000 1.65831i 0.500000 + 0.866025i 3.73831 2.15831i 1.00000i 0.500000 + 0.866025i −0.469882 2.18614i
139.4 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i −1.50000 + 1.65831i 0.500000 + 0.866025i −2.00626 + 1.15831i 1.00000i 0.500000 + 0.866025i −2.12819 + 0.686141i
289.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −1.50000 1.65831i 0.500000 0.866025i −3.73831 2.15831i 1.00000i 0.500000 0.866025i 2.12819 + 0.686141i
289.2 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i −1.50000 + 1.65831i 0.500000 0.866025i 2.00626 + 1.15831i 1.00000i 0.500000 0.866025i 0.469882 2.18614i
289.3 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i −1.50000 1.65831i 0.500000 0.866025i −2.00626 1.15831i 1.00000i 0.500000 0.866025i −2.12819 0.686141i
289.4 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i −1.50000 + 1.65831i 0.500000 0.866025i 3.73831 + 2.15831i 1.00000i 0.500000 0.866025i −0.469882 + 2.18614i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.4
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.g 8
3.b odd 2 1 1170.2.bp.g 8
5.b even 2 1 inner 390.2.y.g 8
5.c odd 4 1 1950.2.i.bb 4
5.c odd 4 1 1950.2.i.be 4
13.c even 3 1 inner 390.2.y.g 8
15.d odd 2 1 1170.2.bp.g 8
39.i odd 6 1 1170.2.bp.g 8
65.n even 6 1 inner 390.2.y.g 8
65.q odd 12 1 1950.2.i.bb 4
65.q odd 12 1 1950.2.i.be 4
195.x odd 6 1 1170.2.bp.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.y.g 8 1.a even 1 1 trivial
390.2.y.g 8 5.b even 2 1 inner
390.2.y.g 8 13.c even 3 1 inner
390.2.y.g 8 65.n even 6 1 inner
1170.2.bp.g 8 3.b odd 2 1
1170.2.bp.g 8 15.d odd 2 1
1170.2.bp.g 8 39.i odd 6 1
1170.2.bp.g 8 195.x odd 6 1
1950.2.i.bb 4 5.c odd 4 1
1950.2.i.bb 4 65.q odd 12 1
1950.2.i.be 4 5.c odd 4 1
1950.2.i.be 4 65.q odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 24T_{7}^{6} + 476T_{7}^{4} - 2400T_{7}^{2} + 10000 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 3 T + 5)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 24 T^{6} + 476 T^{4} + \cdots + 10000 \) Copy content Toggle raw display
$11$ \( (T^{4} - 6 T^{3} + 38 T^{2} + 12 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 23 T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 90 T^{6} + 6251 T^{4} + \cdots + 3418801 \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{3} + 14 T^{2} + 20 T + 100)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 72 T^{6} + 4988 T^{4} + \cdots + 38416 \) Copy content Toggle raw display
$29$ \( (T^{2} - 3 T + 9)^{4} \) Copy content Toggle raw display
$31$ \( (T + 4)^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 90 T^{6} + 6251 T^{4} + \cdots + 3418801 \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} + 59 T^{2} - 40 T + 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 24 T^{6} + 476 T^{4} + \cdots + 10000 \) Copy content Toggle raw display
$47$ \( (T^{4} + 72 T^{2} + 196)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 150 T^{2} + 2809)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4 T^{3} + 56 T^{2} + 160 T + 1600)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8 T^{3} + 59 T^{2} + 40 T + 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 216 T^{6} + \cdots + 65610000 \) Copy content Toggle raw display
$71$ \( (T^{4} + 10 T^{3} + 86 T^{2} + 140 T + 196)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 54 T^{2} + 25)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 72 T^{2} + 196)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 4 T^{3} + 188 T^{2} + 688 T + 29584)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
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