Properties

Label 390.2.l.b
Level $390$
Weight $2$
Character orbit 390.l
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(53,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.l (of order \(4\), 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 \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + ( - \zeta_{8}^{3} - \zeta_{8} + 1) q^{3} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{5} + ( - \zeta_{8}^{2} + \zeta_{8} + 1) q^{6} + (2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{7} + \zeta_{8}^{3} q^{8} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8} q^{2} + ( - \zeta_{8}^{3} - \zeta_{8} + 1) q^{3} + \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} - 2 \zeta_{8}) q^{5} + ( - \zeta_{8}^{2} + \zeta_{8} + 1) q^{6} + (2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{7} + \zeta_{8}^{3} q^{8} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} - 1) q^{9} + ( - 2 \zeta_{8}^{2} + 1) q^{10} - 4 \zeta_{8}^{2} q^{11} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{12} - \zeta_{8} q^{13} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} - 2) q^{14} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \cdots - 3) q^{15} + \cdots + (8 \zeta_{8}^{3} + 4 \zeta_{8}^{2} - 8 \zeta_{8}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{6} + 8 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{6} + 8 q^{7} - 4 q^{9} + 4 q^{10} - 8 q^{14} - 12 q^{15} - 4 q^{16} + 12 q^{17} + 8 q^{18} + 16 q^{21} + 4 q^{24} - 16 q^{25} - 20 q^{27} + 8 q^{28} + 24 q^{29} + 4 q^{30} + 8 q^{31} + 16 q^{35} - 16 q^{37} - 8 q^{38} - 4 q^{39} + 8 q^{40} - 8 q^{42} - 12 q^{43} + 16 q^{44} - 24 q^{45} + 16 q^{46} - 8 q^{47} - 4 q^{48} + 12 q^{51} - 8 q^{53} + 4 q^{54} + 16 q^{57} + 16 q^{59} - 4 q^{60} + 8 q^{61} + 4 q^{62} + 8 q^{63} - 4 q^{65} - 16 q^{66} + 8 q^{67} - 12 q^{68} - 16 q^{69} - 8 q^{70} + 8 q^{72} - 16 q^{73} + 8 q^{74} - 16 q^{75} - 16 q^{76} - 32 q^{77} - 28 q^{81} + 16 q^{83} + 24 q^{87} - 56 q^{89} - 4 q^{90} + 8 q^{91} + 8 q^{93} + 24 q^{95} - 4 q^{96} + 8 q^{97} - 32 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(-1\) \(-\zeta_{8}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i 1.00000 + 1.41421i 1.00000i 0.707107 + 2.12132i 0.292893 1.70711i 3.41421 3.41421i 0.707107 0.707107i −1.00000 + 2.82843i 1.00000 2.00000i
53.2 0.707107 + 0.707107i 1.00000 1.41421i 1.00000i −0.707107 2.12132i 1.70711 0.292893i 0.585786 0.585786i −0.707107 + 0.707107i −1.00000 2.82843i 1.00000 2.00000i
287.1 −0.707107 + 0.707107i 1.00000 1.41421i 1.00000i 0.707107 2.12132i 0.292893 + 1.70711i 3.41421 + 3.41421i 0.707107 + 0.707107i −1.00000 2.82843i 1.00000 + 2.00000i
287.2 0.707107 0.707107i 1.00000 + 1.41421i 1.00000i −0.707107 + 2.12132i 1.70711 + 0.292893i 0.585786 + 0.585786i −0.707107 0.707107i −1.00000 + 2.82843i 1.00000 + 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.l.b yes 4
3.b odd 2 1 390.2.l.a 4
5.c odd 4 1 390.2.l.a 4
15.e even 4 1 inner 390.2.l.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.l.a 4 3.b odd 2 1
390.2.l.a 4 5.c odd 4 1
390.2.l.b yes 4 1.a even 1 1 trivial
390.2.l.b yes 4 15.e even 4 1 inner

Hecke kernels

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

\( T_{7}^{4} - 8T_{7}^{3} + 32T_{7}^{2} - 32T_{7} + 16 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} + 18 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} - 2 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 8T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} - 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 1 \) Copy content Toggle raw display
$17$ \( (T^{2} - 6 T + 18)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$23$ \( T^{4} + 256 \) Copy content Toggle raw display
$29$ \( (T - 6)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 16 T^{3} + \cdots + 784 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 12 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$47$ \( T^{4} + 8 T^{3} + \cdots + 784 \) Copy content Toggle raw display
$53$ \( T^{4} + 8 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 8 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( T^{4} + 236T^{2} + 6724 \) Copy content Toggle raw display
$73$ \( T^{4} + 16 T^{3} + \cdots + 4624 \) Copy content Toggle raw display
$79$ \( T^{4} + 176T^{2} + 3136 \) Copy content Toggle raw display
$83$ \( T^{4} - 16 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( (T^{2} + 28 T + 188)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 8 T^{3} + \cdots + 8464 \) Copy content Toggle raw display
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