Properties

Label 130.2.g.d
Level $130$
Weight $2$
Character orbit 130.g
Analytic conductor $1.038$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [130,2,Mod(57,130)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(130, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("130.57");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 130.g (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: \(1.03805522628\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{3} + q^{4} + (\beta_{3} + 2) q^{5} + (\beta_{3} - \beta_{2} + \beta_1) q^{6} - 3 \beta_{2} q^{7} - q^{8} + ( - 2 \beta_{3} + 3 \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{3} + q^{4} + (\beta_{3} + 2) q^{5} + (\beta_{3} - \beta_{2} + \beta_1) q^{6} - 3 \beta_{2} q^{7} - q^{8} + ( - 2 \beta_{3} + 3 \beta_{2} - 1) q^{9} + ( - \beta_{3} - 2) q^{10} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{11} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{12} + ( - 3 \beta_{2} - 2) q^{13} + 3 \beta_{2} q^{14} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{15} + q^{16} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{17} + (2 \beta_{3} - 3 \beta_{2} + 1) q^{18} + (2 \beta_{2} - 2) q^{19} + (\beta_{3} + 2) q^{20} + (3 \beta_{3} - 3 \beta_1 + 3) q^{21} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{22} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{23} + (\beta_{3} - \beta_{2} + \beta_1) q^{24} + (3 \beta_{3} + 1) q^{25} + (3 \beta_{2} + 2) q^{26} + ( - \beta_{3} + 5 \beta_{2} + \beta_1 - 6) q^{27} - 3 \beta_{2} q^{28} + (2 \beta_{3} + 3 \beta_{2} + 1) q^{29} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 3) q^{30} + (\beta_{2} + 1) q^{31} - q^{32} + ( - \beta_{2} + 2 \beta_1 - 11) q^{33} + ( - \beta_{3} - \beta_{2} - \beta_1 - 2) q^{34} + ( - 6 \beta_{2} + 3 \beta_1) q^{35} + ( - 2 \beta_{3} + 3 \beta_{2} - 1) q^{36} - 3 \beta_{2} q^{37} + ( - 2 \beta_{2} + 2) q^{38} + (5 \beta_{3} - 2 \beta_{2} - \beta_1 + 3) q^{39} + ( - \beta_{3} - 2) q^{40} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{41} + ( - 3 \beta_{3} + 3 \beta_1 - 3) q^{42} + (3 \beta_{3} - \beta_{2} - 3 \beta_1 + 4) q^{43} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{44} + ( - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1 + 4) q^{45} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{46} + (2 \beta_{3} + 6 \beta_{2} + 1) q^{47} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{48} - 2 q^{49} + ( - 3 \beta_{3} - 1) q^{50} + ( - 2 \beta_{3} - 4 \beta_{2} - 1) q^{51} + ( - 3 \beta_{2} - 2) q^{52} + (4 \beta_{3} + \beta_{2} + 4 \beta_1 + 5) q^{53} + (\beta_{3} - 5 \beta_{2} - \beta_1 + 6) q^{54} + ( - 3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 4) q^{55} + 3 \beta_{2} q^{56} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{57} + ( - 2 \beta_{3} - 3 \beta_{2} - 1) q^{58} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 2) q^{59} + ( - \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{60} + 2 q^{61} + ( - \beta_{2} - 1) q^{62} + (3 \beta_{2} - 6 \beta_1 + 9) q^{63} + q^{64} + ( - 2 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 4) q^{65} + (\beta_{2} - 2 \beta_1 + 11) q^{66} + (3 \beta_{2} - 6 \beta_1 + 5) q^{67} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{68} + ( - \beta_{2} + 2 \beta_1 - 11) q^{69} + (6 \beta_{2} - 3 \beta_1) q^{70} + (\beta_{3} - 5 \beta_{2} + \beta_1 - 4) q^{71} + (2 \beta_{3} - 3 \beta_{2} + 1) q^{72} - 4 q^{73} + 3 \beta_{2} q^{74} + (2 \beta_{3} - 8 \beta_{2} - \beta_1 + 9) q^{75} + (2 \beta_{2} - 2) q^{76} + ( - 6 \beta_{3} + 3 \beta_{2} - 6 \beta_1 - 3) q^{77} + ( - 5 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{78} + ( - 6 \beta_{3} - 3 \beta_{2} - 3) q^{79} + (\beta_{3} + 2) q^{80} + ( - 3 \beta_{2} + 6 \beta_1 - 2) q^{81} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 4) q^{82} + (2 \beta_{3} + 3 \beta_{2} + 1) q^{83} + (3 \beta_{3} - 3 \beta_1 + 3) q^{84} + (3 \beta_{3} + 5 \beta_{2} + 1) q^{85} + ( - 3 \beta_{3} + \beta_{2} + 3 \beta_1 - 4) q^{86} + ( - 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1 + 3) q^{87} + (2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{88} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 + 2) q^{89} + (3 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 4) q^{90} + (6 \beta_{2} - 9) q^{91} + ( - 2 \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{92} + ( - 2 \beta_{3} + \beta_{2} - 1) q^{93} + ( - 2 \beta_{3} - 6 \beta_{2} - 1) q^{94} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 4) q^{95} + (\beta_{3} - \beta_{2} + \beta_1) q^{96} + ( - 3 \beta_{2} + 6 \beta_1 - 1) q^{97} + 2 q^{98} + (6 \beta_{3} - 14 \beta_{2} + 6 \beta_1 - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{6} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 2 q^{3} + 4 q^{4} + 6 q^{5} - 2 q^{6} - 4 q^{8} - 6 q^{10} + 2 q^{12} - 8 q^{13} + 14 q^{15} + 4 q^{16} + 6 q^{17} - 8 q^{19} + 6 q^{20} + 6 q^{21} - 2 q^{24} - 2 q^{25} + 8 q^{26} - 22 q^{27} - 14 q^{30} + 4 q^{31} - 4 q^{32} - 44 q^{33} - 6 q^{34} + 8 q^{38} + 2 q^{39} - 6 q^{40} - 12 q^{41} - 6 q^{42} + 10 q^{43} + 22 q^{45} + 2 q^{48} - 8 q^{49} + 2 q^{50} - 8 q^{52} + 12 q^{53} + 22 q^{54} + 22 q^{55} - 8 q^{57} + 12 q^{59} + 14 q^{60} + 8 q^{61} - 4 q^{62} + 36 q^{63} + 4 q^{64} - 12 q^{65} + 44 q^{66} + 20 q^{67} + 6 q^{68} - 44 q^{69} - 18 q^{71} - 16 q^{73} + 32 q^{75} - 8 q^{76} - 2 q^{78} + 6 q^{80} - 8 q^{81} + 12 q^{82} + 6 q^{84} - 2 q^{85} - 10 q^{86} + 16 q^{87} + 12 q^{89} - 22 q^{90} - 36 q^{91} - 12 q^{95} - 2 q^{96} - 4 q^{97} + 8 q^{98} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 2\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{2} + 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-\beta_{2}\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1
1.65831 0.500000i
−1.65831 0.500000i
1.65831 + 0.500000i
−1.65831 + 0.500000i
−1.00000 −1.15831 + 1.15831i 1.00000 1.50000 1.65831i 1.15831 1.15831i 3.00000i −1.00000 0.316625i −1.50000 + 1.65831i
57.2 −1.00000 2.15831 2.15831i 1.00000 1.50000 + 1.65831i −2.15831 + 2.15831i 3.00000i −1.00000 6.31662i −1.50000 1.65831i
73.1 −1.00000 −1.15831 1.15831i 1.00000 1.50000 + 1.65831i 1.15831 + 1.15831i 3.00000i −1.00000 0.316625i −1.50000 1.65831i
73.2 −1.00000 2.15831 + 2.15831i 1.00000 1.50000 1.65831i −2.15831 2.15831i 3.00000i −1.00000 6.31662i −1.50000 + 1.65831i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.2.g.d 4
3.b odd 2 1 1170.2.m.e 4
4.b odd 2 1 1040.2.bg.k 4
5.b even 2 1 650.2.g.g 4
5.c odd 4 1 130.2.j.d yes 4
5.c odd 4 1 650.2.j.f 4
13.d odd 4 1 130.2.j.d yes 4
15.e even 4 1 1170.2.w.e 4
20.e even 4 1 1040.2.cd.i 4
39.f even 4 1 1170.2.w.e 4
52.f even 4 1 1040.2.cd.i 4
65.f even 4 1 650.2.g.g 4
65.g odd 4 1 650.2.j.f 4
65.k even 4 1 inner 130.2.g.d 4
195.j odd 4 1 1170.2.m.e 4
260.s odd 4 1 1040.2.bg.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.g.d 4 1.a even 1 1 trivial
130.2.g.d 4 65.k even 4 1 inner
130.2.j.d yes 4 5.c odd 4 1
130.2.j.d yes 4 13.d odd 4 1
650.2.g.g 4 5.b even 2 1
650.2.g.g 4 65.f even 4 1
650.2.j.f 4 5.c odd 4 1
650.2.j.f 4 65.g odd 4 1
1040.2.bg.k 4 4.b odd 2 1
1040.2.bg.k 4 260.s odd 4 1
1040.2.cd.i 4 20.e even 4 1
1040.2.cd.i 4 52.f even 4 1
1170.2.m.e 4 3.b odd 2 1
1170.2.m.e 4 195.j odd 4 1
1170.2.w.e 4 15.e even 4 1
1170.2.w.e 4 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2T_{3}^{3} + 2T_{3}^{2} + 10T_{3} + 25 \) acting on \(S_{2}^{\mathrm{new}}(130, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + 2 T^{2} + 10 T + 25 \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 484 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 18 T^{2} + 6 T + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} + 4 T + 8)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 484 \) Copy content Toggle raw display
$29$ \( T^{4} + 40T^{2} + 4 \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + 72 T^{2} - 48 T + 16 \) Copy content Toggle raw display
$43$ \( T^{4} - 10 T^{3} + 50 T^{2} + \cdots + 1369 \) Copy content Toggle raw display
$47$ \( T^{4} + 94T^{2} + 625 \) Copy content Toggle raw display
$53$ \( T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 4900 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + 72 T^{2} + 48 T + 16 \) Copy content Toggle raw display
$61$ \( (T - 2)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 10 T - 74)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 18 T^{3} + 162 T^{2} + \cdots + 1225 \) Copy content Toggle raw display
$73$ \( (T + 4)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 216T^{2} + 8100 \) Copy content Toggle raw display
$83$ \( T^{4} + 40T^{2} + 4 \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + 72 T^{2} + 48 T + 16 \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T - 98)^{2} \) Copy content Toggle raw display
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