Properties

Label 390.2.i.e
Level $390$
Weight $2$
Character orbit 390.i
Analytic conductor $3.114$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i −1.00000 −0.500000 0.866025i −1.50000 2.59808i −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 −1.50000 + 2.59808i −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.e 2
3.b odd 2 1 1170.2.i.c 2
5.b even 2 1 1950.2.i.f 2
5.c odd 4 2 1950.2.z.d 4
13.c even 3 1 inner 390.2.i.e 2
13.c even 3 1 5070.2.a.b 1
13.e even 6 1 5070.2.a.r 1
13.f odd 12 2 5070.2.b.b 2
39.i odd 6 1 1170.2.i.c 2
65.n even 6 1 1950.2.i.f 2
65.q odd 12 2 1950.2.z.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.i.e 2 1.a even 1 1 trivial
390.2.i.e 2 13.c even 3 1 inner
1170.2.i.c 2 3.b odd 2 1
1170.2.i.c 2 39.i odd 6 1
1950.2.i.f 2 5.b even 2 1
1950.2.i.f 2 65.n even 6 1
1950.2.z.d 4 5.c odd 4 2
1950.2.z.d 4 65.q odd 12 2
5070.2.a.b 1 13.c even 3 1
5070.2.a.r 1 13.e even 6 1
5070.2.b.b 2 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 \(S_{2}^{\mathrm{new}}(390, [\chi])\):

\( T_{7}^{2} + 3T_{7} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$11$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$23$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$29$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$31$ \( (T - 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$41$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$43$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$47$ \( (T + 3)^{2} \) Copy content Toggle raw display
$53$ \( (T - 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$71$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$73$ \( (T + 8)^{2} \) Copy content Toggle raw display
$79$ \( (T - 6)^{2} \) Copy content Toggle raw display
$83$ \( (T - 16)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$97$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
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