Properties

Label 130.4.a.a
Level $130$
Weight $4$
Character orbit 130.a
Self dual yes
Analytic conductor $7.670$
Analytic rank $0$
Dimension $1$
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] = [130,4,Mod(1,130)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(130, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("130.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 130.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: \(7.67024830075\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
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)
 
\(f(q)\) \(=\) \( q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{5} + 4 q^{6} + 8 q^{7} - 8 q^{8} - 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{5} + 4 q^{6} + 8 q^{7} - 8 q^{8} - 23 q^{9} - 10 q^{10} + 6 q^{11} - 8 q^{12} + 13 q^{13} - 16 q^{14} - 10 q^{15} + 16 q^{16} + 114 q^{17} + 46 q^{18} + 38 q^{19} + 20 q^{20} - 16 q^{21} - 12 q^{22} + 150 q^{23} + 16 q^{24} + 25 q^{25} - 26 q^{26} + 100 q^{27} + 32 q^{28} + 114 q^{29} + 20 q^{30} - 34 q^{31} - 32 q^{32} - 12 q^{33} - 228 q^{34} + 40 q^{35} - 92 q^{36} + 146 q^{37} - 76 q^{38} - 26 q^{39} - 40 q^{40} - 30 q^{41} + 32 q^{42} + 122 q^{43} + 24 q^{44} - 115 q^{45} - 300 q^{46} + 336 q^{47} - 32 q^{48} - 279 q^{49} - 50 q^{50} - 228 q^{51} + 52 q^{52} - 570 q^{53} - 200 q^{54} + 30 q^{55} - 64 q^{56} - 76 q^{57} - 228 q^{58} + 66 q^{59} - 40 q^{60} - 502 q^{61} + 68 q^{62} - 184 q^{63} + 64 q^{64} + 65 q^{65} + 24 q^{66} + 728 q^{67} + 456 q^{68} - 300 q^{69} - 80 q^{70} + 582 q^{71} + 184 q^{72} - 994 q^{73} - 292 q^{74} - 50 q^{75} + 152 q^{76} + 48 q^{77} + 52 q^{78} - 988 q^{79} + 80 q^{80} + 421 q^{81} + 60 q^{82} - 84 q^{83} - 64 q^{84} + 570 q^{85} - 244 q^{86} - 228 q^{87} - 48 q^{88} + 906 q^{89} + 230 q^{90} + 104 q^{91} + 600 q^{92} + 68 q^{93} - 672 q^{94} + 190 q^{95} + 64 q^{96} + 290 q^{97} + 558 q^{98} - 138 q^{99}+O(q^{100}) \) 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
0
−2.00000 −2.00000 4.00000 5.00000 4.00000 8.00000 −8.00000 −23.0000 −10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 130.4.a.a 1
3.b odd 2 1 1170.4.a.k 1
4.b odd 2 1 1040.4.a.c 1
5.b even 2 1 650.4.a.h 1
5.c odd 4 2 650.4.b.e 2
13.b even 2 1 1690.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.4.a.a 1 1.a even 1 1 trivial
650.4.a.h 1 5.b even 2 1
650.4.b.e 2 5.c odd 4 2
1040.4.a.c 1 4.b odd 2 1
1170.4.a.k 1 3.b odd 2 1
1690.4.a.i 1 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 2 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(130))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 2 \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 8 \) Copy content Toggle raw display
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T - 114 \) Copy content Toggle raw display
$19$ \( T - 38 \) Copy content Toggle raw display
$23$ \( T - 150 \) Copy content Toggle raw display
$29$ \( T - 114 \) Copy content Toggle raw display
$31$ \( T + 34 \) Copy content Toggle raw display
$37$ \( T - 146 \) Copy content Toggle raw display
$41$ \( T + 30 \) Copy content Toggle raw display
$43$ \( T - 122 \) Copy content Toggle raw display
$47$ \( T - 336 \) Copy content Toggle raw display
$53$ \( T + 570 \) Copy content Toggle raw display
$59$ \( T - 66 \) Copy content Toggle raw display
$61$ \( T + 502 \) Copy content Toggle raw display
$67$ \( T - 728 \) Copy content Toggle raw display
$71$ \( T - 582 \) Copy content Toggle raw display
$73$ \( T + 994 \) Copy content Toggle raw display
$79$ \( T + 988 \) Copy content Toggle raw display
$83$ \( T + 84 \) Copy content Toggle raw display
$89$ \( T - 906 \) Copy content Toggle raw display
$97$ \( T - 290 \) Copy content Toggle raw display
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