Properties

Label 390.2.p.f
Level $390$
Weight $2$
Character orbit 390.p
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(161,390)]
 
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, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.p (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.

\(f(q)\) \(=\) \( q + \zeta_{8} q^{2} + ( - \zeta_{8}^{2} + \zeta_{8} + 1) q^{3} + \zeta_{8}^{2} q^{4} - \zeta_{8} q^{5} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{6} + (2 \zeta_{8}^{2} + 2) q^{7} + \zeta_{8}^{3} q^{8} + ( - 2 \zeta_{8}^{3} + \cdots + 2 \zeta_{8}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{8} q^{2} + ( - \zeta_{8}^{2} + \zeta_{8} + 1) q^{3} + \zeta_{8}^{2} q^{4} - \zeta_{8} q^{5} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}) q^{6} + (2 \zeta_{8}^{2} + 2) q^{7} + \zeta_{8}^{3} q^{8} + ( - 2 \zeta_{8}^{3} + \cdots + 2 \zeta_{8}) q^{9} + \cdots + ( - 8 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 8 q^{7} + 8 q^{11} + 4 q^{12} - 4 q^{16} + 8 q^{17} + 8 q^{18} + 16 q^{21} - 16 q^{23} - 4 q^{24} - 12 q^{26} + 4 q^{27} - 8 q^{28} - 4 q^{30} + 12 q^{31} - 8 q^{34} + 4 q^{36} - 16 q^{37} - 12 q^{39} + 4 q^{40} - 20 q^{41} - 8 q^{42} + 8 q^{44} - 8 q^{45} + 8 q^{46} - 8 q^{47} - 4 q^{48} + 20 q^{54} - 16 q^{58} - 16 q^{59} + 4 q^{60} + 40 q^{61} - 8 q^{62} + 8 q^{63} + 12 q^{65} + 8 q^{66} - 8 q^{67} - 8 q^{69} + 8 q^{70} - 36 q^{71} - 8 q^{72} + 4 q^{75} + 32 q^{77} - 20 q^{78} - 24 q^{79} + 28 q^{81} + 16 q^{83} + 8 q^{85} + 20 q^{86} - 8 q^{87} + 12 q^{89} - 4 q^{90} - 8 q^{93} - 40 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i 0.292893 1.70711i 1.00000i 0.707107 + 0.707107i −1.41421 + 1.00000i 2.00000 + 2.00000i 0.707107 0.707107i −2.82843 1.00000i 1.00000i
161.2 0.707107 + 0.707107i 1.70711 0.292893i 1.00000i −0.707107 0.707107i 1.41421 + 1.00000i 2.00000 + 2.00000i −0.707107 + 0.707107i 2.82843 1.00000i 1.00000i
281.1 −0.707107 + 0.707107i 0.292893 + 1.70711i 1.00000i 0.707107 0.707107i −1.41421 1.00000i 2.00000 2.00000i 0.707107 + 0.707107i −2.82843 + 1.00000i 1.00000i
281.2 0.707107 0.707107i 1.70711 + 0.292893i 1.00000i −0.707107 + 0.707107i 1.41421 1.00000i 2.00000 2.00000i −0.707107 0.707107i 2.82843 + 1.00000i 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.f 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.p.f yes 4
3.b odd 2 1 390.2.p.e 4
13.d odd 4 1 390.2.p.e 4
39.f even 4 1 inner 390.2.p.f yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.p.e 4 3.b odd 2 1
390.2.p.e 4 13.d odd 4 1
390.2.p.f yes 4 1.a even 1 1 trivial
390.2.p.f yes 4 39.f 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 \(S_{2}^{\mathrm{new}}(390, [\chi])\):

\( T_{7}^{2} - 4T_{7} + 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} + 8 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T + 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 24T^{2} + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T - 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 8 T + 8)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$31$ \( T^{4} - 12 T^{3} + \cdots + 196 \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T + 32)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 20 T^{3} + \cdots + 1156 \) Copy content Toggle raw display
$43$ \( T^{4} + 108T^{2} + 2116 \) Copy content Toggle raw display
$47$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 132T^{2} + 3844 \) Copy content Toggle raw display
$59$ \( (T^{2} + 8 T + 32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 20 T + 92)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 8 T^{3} + \cdots + 784 \) Copy content Toggle raw display
$71$ \( T^{4} + 36 T^{3} + \cdots + 24964 \) Copy content Toggle raw display
$73$ \( T^{4} + 38416 \) Copy content Toggle raw display
$79$ \( (T^{2} + 12 T + 28)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 16 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + \cdots + 2116 \) Copy content Toggle raw display
$97$ \( T^{4} + 40 T^{3} + \cdots + 38416 \) Copy content Toggle raw display
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