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Properties

Label 390.2.bb.a
Level $390$
Weight $2$
Character orbit 390.bb
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(121,390)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.bb (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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i − 1.00000i −0.866025 + 0.500000i −2.59808 + 1.50000i − 1.00000i −0.500000 − 0.866025i −0.500000 + 0.866025i
121.2 0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i 1.00000i 0.866025 − 0.500000i 2.59808 − 1.50000i 1.00000i −0.500000 − 0.866025i −0.500000 + 0.866025i
361.1 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i 1.00000i −0.866025 − 0.500000i −2.59808 − 1.50000i 1.00000i −0.500000 + 0.866025i −0.500000 − 0.866025i
361.2 0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i − 1.00000i 0.866025 + 0.500000i 2.59808 + 1.50000i − 1.00000i −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.e 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.bb.a ✓ 4
3.b odd 2 1 1170.2.bs.d 4
5.b even 2 1 1950.2.bc.a 4
5.c odd 4 1 1950.2.y.d 4
5.c odd 4 1 1950.2.y.e 4
13.c even 3 1 5070.2.b.p 4
13.e even 6 1 inner 390.2.bb.a ✓ 4
13.e even 6 1 5070.2.b.p 4
13.f odd 12 1 5070.2.a.ba 2
13.f odd 12 1 5070.2.a.be 2
39.h odd 6 1 1170.2.bs.d 4
65.l even 6 1 1950.2.bc.a 4
65.r odd 12 1 1950.2.y.d 4
65.r odd 12 1 1950.2.y.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.bb.a ✓ 4 1.a even 1 1 trivial
390.2.bb.a ✓ 4 13.e even 6 1 inner
1170.2.bs.d 4 3.b odd 2 1
1170.2.bs.d 4 39.h odd 6 1
1950.2.y.d 4 5.c odd 4 1
1950.2.y.d 4 65.r odd 12 1
1950.2.y.e 4 5.c odd 4 1
1950.2.y.e 4 65.r odd 12 1
1950.2.bc.a 4 5.b even 2 1
1950.2.bc.a 4 65.l even 6 1
5070.2.a.ba 2 13.f odd 12 1
5070.2.a.be 2 13.f odd 12 1
5070.2.b.p 4 13.c even 3 1
5070.2.b.p 4 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 9T_{7}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(390, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$11$ \( T^{4} + 6 T^{3} + 11 T^{2} - 6 T + 1 \) Copy content Toggle raw display
$13$ \( T^{4} + 23T^{2} + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} - T^{2} + 78 T + 169 \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + 24 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$31$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$37$ \( T^{4} + 24 T^{3} + 239 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$41$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$43$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 42T^{2} + 9 \) Copy content Toggle raw display
$53$ \( (T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 12 T^{3} - 4 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + 60 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$67$ \( T^{4} - 12 T^{3} + 56 T^{2} - 96 T + 64 \) Copy content Toggle raw display
$71$ \( T^{4} + 24 T^{3} + 204 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$73$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 20 T + 52)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$89$ \( T^{4} + 42 T^{3} + 731 T^{2} + \cdots + 20449 \) Copy content Toggle raw display
$97$ \( T^{4} - 36 T^{3} + 536 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
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