Properties

Label 390.2.a.h
Level $390$
Weight $2$
Character orbit 390.a
Self dual yes
Analytic conductor $3.114$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [390,2,Mod(1,390)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("390.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 390.a (trivial)

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: yes
Analytic conductor: \(3.11416567883\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
1.00000 1.00000 1.00000 1.00000 1.00000 −2.82843 1.00000 1.00000 1.00000
1.2 1.00000 1.00000 1.00000 1.00000 1.00000 2.82843 1.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)
\(13\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 390.2.a.h 2
3.b odd 2 1 1170.2.a.o 2
4.b odd 2 1 3120.2.a.bc 2
5.b even 2 1 1950.2.a.bd 2
5.c odd 4 2 1950.2.e.o 4
12.b even 2 1 9360.2.a.ch 2
13.b even 2 1 5070.2.a.bc 2
13.d odd 4 2 5070.2.b.q 4
15.d odd 2 1 5850.2.a.cl 2
15.e even 4 2 5850.2.e.bk 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.h 2 1.a even 1 1 trivial
1170.2.a.o 2 3.b odd 2 1
1950.2.a.bd 2 5.b even 2 1
1950.2.e.o 4 5.c odd 4 2
3120.2.a.bc 2 4.b odd 2 1
5070.2.a.bc 2 13.b even 2 1
5070.2.b.q 4 13.d odd 4 2
5850.2.a.cl 2 15.d odd 2 1
5850.2.e.bk 4 15.e even 4 2
9360.2.a.ch 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(390))\):

\( T_{7}^{2} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 32 \) Copy content Toggle raw display
\( T_{31} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 8 \) Copy content Toggle raw display
$11$ \( T^{2} - 32 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 8 \) Copy content Toggle raw display
$23$ \( T^{2} - 72 \) Copy content Toggle raw display
$29$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$41$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 4T - 124 \) Copy content Toggle raw display
$59$ \( T^{2} + 16T + 32 \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 32 \) Copy content Toggle raw display
$71$ \( T^{2} - 32 \) Copy content Toggle raw display
$73$ \( T^{2} + 12T - 36 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 32 \) Copy content Toggle raw display
$83$ \( T^{2} - 24T + 112 \) Copy content Toggle raw display
$89$ \( T^{2} + 20T + 68 \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 28 \) Copy content Toggle raw display
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