Properties

Label 130.2.b.a
Level $130$
Weight $2$
Character orbit 130.b
Analytic conductor $1.038$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [130,2,Mod(79,130)]
 
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([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("130.79");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 130.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: \(1.03805522628\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.3534400.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( - \beta_{4} + \beta_{3}) q^{3} - q^{4} + (\beta_{5} + \beta_{3}) q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{6} + (\beta_{4} - \beta_{2} - \beta_1) q^{7} - \beta_{5} q^{8}+ \cdots + ( - 2 \beta_{4} + 2 \beta_{3} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} - 10 q^{9} - 4 q^{10} + 12 q^{11} + 4 q^{14} - 16 q^{15} + 6 q^{16} - 4 q^{19} + 24 q^{21} - 16 q^{25} + 6 q^{26} + 4 q^{29} - 14 q^{30} + 24 q^{31} - 8 q^{34} - 6 q^{35} + 10 q^{36} + 4 q^{40}+ \cdots - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{5} - 70\nu^{4} + 183\nu^{3} + 120\nu^{2} - 966\nu + 240 ) / 445 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{5} - 5\nu^{4} - 184\nu^{3} + 390\nu^{2} + 643\nu - 1000 ) / 445 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{5} + 8\nu^{4} - 26\nu^{3} - \nu^{2} + 57\nu - 180 ) / 89 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{5} + 24\nu^{4} + 11\nu^{3} - 92\nu^{2} - 7\nu - 6 ) / 178 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{5} - 4\beta_{4} + 2\beta_{3} + \beta_{2} + 2\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 14\beta_{5} - 14\beta_{4} + 9\beta_{3} - 3\beta_{2} - 10\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} - 32\beta_{4} + 6\beta_{3} - 17\beta_{2} - 25\beta _1 - 42 \) Copy content Toggle raw display

Character values

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

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
2.19082 1.44755i
−1.81837 + 0.301352i
0.627553 + 1.14620i
0.627553 1.14620i
−1.81837 0.301352i
2.19082 + 1.44755i
1.00000i 2.89511i −1.00000 1.44755 1.70429i −2.89511 4.38164i 1.00000i −5.38164 −1.70429 1.44755i
79.2 1.00000i 0.602705i −1.00000 −0.301352 2.21567i 0.602705 3.63675i 1.00000i 2.63675 −2.21567 + 0.301352i
79.3 1.00000i 2.29240i −1.00000 −1.14620 + 1.91995i 2.29240 1.25511i 1.00000i −2.25511 1.91995 + 1.14620i
79.4 1.00000i 2.29240i −1.00000 −1.14620 1.91995i 2.29240 1.25511i 1.00000i −2.25511 1.91995 1.14620i
79.5 1.00000i 0.602705i −1.00000 −0.301352 + 2.21567i 0.602705 3.63675i 1.00000i 2.63675 −2.21567 0.301352i
79.6 1.00000i 2.89511i −1.00000 1.44755 + 1.70429i −2.89511 4.38164i 1.00000i −5.38164 −1.70429 + 1.44755i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.2.b.a 6
3.b odd 2 1 1170.2.e.f 6
4.b odd 2 1 1040.2.d.b 6
5.b even 2 1 inner 130.2.b.a 6
5.c odd 4 1 650.2.a.n 3
5.c odd 4 1 650.2.a.o 3
13.b even 2 1 1690.2.b.a 6
13.d odd 4 1 1690.2.c.a 6
13.d odd 4 1 1690.2.c.d 6
15.d odd 2 1 1170.2.e.f 6
15.e even 4 1 5850.2.a.cp 3
15.e even 4 1 5850.2.a.cs 3
20.d odd 2 1 1040.2.d.b 6
20.e even 4 1 5200.2.a.ce 3
20.e even 4 1 5200.2.a.cf 3
65.d even 2 1 1690.2.b.a 6
65.g odd 4 1 1690.2.c.a 6
65.g odd 4 1 1690.2.c.d 6
65.h odd 4 1 8450.2.a.bs 3
65.h odd 4 1 8450.2.a.cc 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.b.a 6 1.a even 1 1 trivial
130.2.b.a 6 5.b even 2 1 inner
650.2.a.n 3 5.c odd 4 1
650.2.a.o 3 5.c odd 4 1
1040.2.d.b 6 4.b odd 2 1
1040.2.d.b 6 20.d odd 2 1
1170.2.e.f 6 3.b odd 2 1
1170.2.e.f 6 15.d odd 2 1
1690.2.b.a 6 13.b even 2 1
1690.2.b.a 6 65.d even 2 1
1690.2.c.a 6 13.d odd 4 1
1690.2.c.a 6 65.g odd 4 1
1690.2.c.d 6 13.d odd 4 1
1690.2.c.d 6 65.g odd 4 1
5200.2.a.ce 3 20.e even 4 1
5200.2.a.cf 3 20.e even 4 1
5850.2.a.cp 3 15.e even 4 1
5850.2.a.cs 3 15.e even 4 1
8450.2.a.bs 3 65.h odd 4 1
8450.2.a.cc 3 65.h odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(130, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 14 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{6} + 8 T^{4} + \cdots + 125 \) Copy content Toggle raw display
$7$ \( T^{6} + 34 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( (T - 2)^{6} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 102 T^{4} + \cdots + 35344 \) Copy content Toggle raw display
$19$ \( (T^{3} + 2 T^{2} - 44 T + 40)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 68 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{3} - 2 T^{2} - 44 T - 40)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 12 T^{2} + \cdots + 80)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 62 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$41$ \( (T^{3} - 2 T^{2} + \cdots + 320)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 174 T^{4} + \cdots + 87616 \) Copy content Toggle raw display
$47$ \( T^{6} + 66 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$53$ \( T^{6} + 68 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( (T^{3} + 2 T^{2} - 44 T + 40)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 10 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 104 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$71$ \( (T^{3} + 8 T^{2} + \cdots - 200)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{3} \) Copy content Toggle raw display
$79$ \( (T^{3} + 28 T^{2} + \cdots + 320)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 176 T^{4} + \cdots + 25600 \) Copy content Toggle raw display
$89$ \( (T^{3} - 2 T^{2} - 44 T - 40)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 572 T^{4} + \cdots + 2534464 \) Copy content Toggle raw display
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