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Properties

Label 390.2.i.g
Level $390$
Weight $2$
Character orbit 390.i
Analytic conductor $3.114$
Analytic rank $0$
Dimension $4$
CM no
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(61,390)]
 
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, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.61");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.i (of order \(3\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1
1.28078 + 2.21837i
−0.780776 − 1.35234i
1.28078 − 2.21837i
−0.780776 + 1.35234i
−0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i −1.78078 − 3.08440i 1.00000 −0.500000 − 0.866025i −0.500000 + 0.866025i
61.2 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i 0.280776 + 0.486319i 1.00000 −0.500000 − 0.866025i −0.500000 + 0.866025i
211.1 −0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −1.78078 + 3.08440i 1.00000 −0.500000 + 0.866025i −0.500000 − 0.866025i
211.2 −0.500000 − 0.866025i −0.500000 − 0.866025i −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0.280776 − 0.486319i 1.00000 −0.500000 + 0.866025i −0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.i.g ✓ 4
3.b odd 2 1 1170.2.i.o 4
5.b even 2 1 1950.2.i.bi 4
5.c odd 4 2 1950.2.z.n 8
13.c even 3 1 inner 390.2.i.g ✓ 4
13.c even 3 1 5070.2.a.bi 2
13.e even 6 1 5070.2.a.bb 2
13.f odd 12 2 5070.2.b.r 4
39.i odd 6 1 1170.2.i.o 4
65.n even 6 1 1950.2.i.bi 4
65.q odd 12 2 1950.2.z.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.g ✓ 4 1.a even 1 1 trivial
390.2.i.g ✓ 4 13.c even 3 1 inner
1170.2.i.o 4 3.b odd 2 1
1170.2.i.o 4 39.i odd 6 1
1950.2.i.bi 4 5.b even 2 1
1950.2.i.bi 4 65.n even 6 1
1950.2.z.n 8 5.c odd 4 2
1950.2.z.n 8 65.q odd 12 2
5070.2.a.bb 2 13.e even 6 1
5070.2.a.bi 2 13.c even 3 1
5070.2.b.r 4 13.f odd 12 2

Hecke kernels

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

\( T_{7}^{4} + 3T_{7}^{3} + 11T_{7}^{2} - 6T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 17T_{11}^{2} + 289 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 3 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 17T^{2} + 289 \) Copy content Toggle raw display
$13$ \( T^{4} + T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( T^{4} + 2 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{4} - 3 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$23$ \( T^{4} - 3 T^{3} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( T^{4} + 9 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$31$ \( (T^{2} + T - 38)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 17T^{2} + 289 \) Copy content Toggle raw display
$41$ \( T^{4} + 8 T^{3} + \cdots + 2704 \) Copy content Toggle raw display
$43$ \( T^{4} + 5 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$47$ \( (T - 7)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 13 T + 38)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 17 T^{3} + \cdots + 4624 \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( T^{4} - 18 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T - 144)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 19 T + 86)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 17 T^{3} + \cdots + 1156 \) Copy content Toggle raw display
$97$ \( T^{4} - 6 T^{3} + \cdots + 64 \) Copy content Toggle raw display
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