Properties

Label 390.2.b.a
Level $390$
Weight $2$
Character orbit 390.b
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(181,390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(390, base_ring=CyclotomicField(2))
 
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("390.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.b (of order \(2\), degree \(1\), 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: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(131\) \(157\) \(301\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
1.00000i
1.00000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 0 1.00000i 1.00000 1.00000
181.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 0 1.00000i 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.b.a 2
3.b odd 2 1 1170.2.b.b 2
4.b odd 2 1 3120.2.g.f 2
5.b even 2 1 1950.2.b.g 2
5.c odd 4 1 1950.2.f.b 2
5.c odd 4 1 1950.2.f.i 2
13.b even 2 1 inner 390.2.b.a 2
13.d odd 4 1 5070.2.a.g 1
13.d odd 4 1 5070.2.a.m 1
39.d odd 2 1 1170.2.b.b 2
52.b odd 2 1 3120.2.g.f 2
65.d even 2 1 1950.2.b.g 2
65.h odd 4 1 1950.2.f.b 2
65.h odd 4 1 1950.2.f.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.b.a 2 1.a even 1 1 trivial
390.2.b.a 2 13.b even 2 1 inner
1170.2.b.b 2 3.b odd 2 1
1170.2.b.b 2 39.d odd 2 1
1950.2.b.g 2 5.b even 2 1
1950.2.b.g 2 65.d even 2 1
1950.2.f.b 2 5.c odd 4 1
1950.2.f.b 2 65.h odd 4 1
1950.2.f.i 2 5.c odd 4 1
1950.2.f.i 2 65.h odd 4 1
3120.2.g.f 2 4.b odd 2 1
3120.2.g.f 2 52.b odd 2 1
5070.2.a.g 1 13.d odd 4 1
5070.2.a.m 1 13.d odd 4 1

Hecke kernels

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

\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

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