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Properties

Label 1470.2.d.b
Level $1470$
Weight $2$
Character orbit 1470.d
Analytic conductor $11.738$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1470,2,Mod(1469,1470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1470, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1470.1469");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.d (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: \(11.7380090971\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + (\beta_{2} - \beta_1 - 1) q^{5} - \beta_1 q^{6} - q^{8} + (\beta_{3} + 3 \beta_{2}) q^{9} + ( - \beta_{2} + \beta_1 + 1) q^{10} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 + 1) q^{11}+ \cdots + ( - \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + q^{3} + 4 q^{4} - 3 q^{5} - q^{6} - 4 q^{8} + 5 q^{9} + 3 q^{10} + q^{12} - 8 q^{13} - 4 q^{15} + 4 q^{16} - 5 q^{18} - 3 q^{20} - 6 q^{23} - q^{24} + q^{25} + 8 q^{26} + 16 q^{27}+ \cdots + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1469.1
−1.18614 − 1.26217i
−1.18614 + 1.26217i
1.68614 − 0.396143i
1.68614 + 0.396143i
−1.00000 −1.18614 − 1.26217i 1.00000 0.686141 + 2.12819i 1.18614 + 1.26217i 0 −1.00000 −0.186141 + 2.99422i −0.686141 − 2.12819i
1469.2 −1.00000 −1.18614 + 1.26217i 1.00000 0.686141 − 2.12819i 1.18614 − 1.26217i 0 −1.00000 −0.186141 − 2.99422i −0.686141 + 2.12819i
1469.3 −1.00000 1.68614 − 0.396143i 1.00000 −2.18614 − 0.469882i −1.68614 + 0.396143i 0 −1.00000 2.68614 − 1.33591i 2.18614 + 0.469882i
1469.4 −1.00000 1.68614 + 0.396143i 1.00000 −2.18614 + 0.469882i −1.68614 − 0.396143i 0 −1.00000 2.68614 + 1.33591i 2.18614 − 0.469882i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.d.b 4
3.b odd 2 1 1470.2.d.d 4
5.b even 2 1 1470.2.d.c 4
7.b odd 2 1 1470.2.d.a 4
7.c even 3 1 210.2.t.c yes 4
7.d odd 6 1 210.2.t.d yes 4
15.d odd 2 1 1470.2.d.a 4
21.c even 2 1 1470.2.d.c 4
21.g even 6 1 210.2.t.b yes 4
21.h odd 6 1 210.2.t.a ✓ 4
35.c odd 2 1 1470.2.d.d 4
35.i odd 6 1 210.2.t.a ✓ 4
35.j even 6 1 210.2.t.b yes 4
35.k even 12 2 1050.2.s.e 8
35.l odd 12 2 1050.2.s.d 8
105.g even 2 1 inner 1470.2.d.b 4
105.o odd 6 1 210.2.t.d yes 4
105.p even 6 1 210.2.t.c yes 4
105.w odd 12 2 1050.2.s.d 8
105.x even 12 2 1050.2.s.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.t.a ✓ 4 21.h odd 6 1
210.2.t.a ✓ 4 35.i odd 6 1
210.2.t.b yes 4 21.g even 6 1
210.2.t.b yes 4 35.j even 6 1
210.2.t.c yes 4 7.c even 3 1
210.2.t.c yes 4 105.p even 6 1
210.2.t.d yes 4 7.d odd 6 1
210.2.t.d yes 4 105.o odd 6 1
1050.2.s.d 8 35.l odd 12 2
1050.2.s.d 8 105.w odd 12 2
1050.2.s.e 8 35.k even 12 2
1050.2.s.e 8 105.x even 12 2
1470.2.d.a 4 7.b odd 2 1
1470.2.d.a 4 15.d odd 2 1
1470.2.d.b 4 1.a even 1 1 trivial
1470.2.d.b 4 105.g even 2 1 inner
1470.2.d.c 4 5.b even 2 1
1470.2.d.c 4 21.c even 2 1
1470.2.d.d 4 3.b odd 2 1
1470.2.d.d 4 35.c odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\( T_{11}^{4} + 19T_{11}^{2} + 16 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{23}^{2} + 3T_{23} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} - 2 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{4} + 3 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 19T^{2} + 16 \) Copy content Toggle raw display
$13$ \( (T + 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 44)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 3 T - 6)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 11)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 63T^{2} + 324 \) Copy content Toggle raw display
$37$ \( T^{4} + 204T^{2} + 9216 \) Copy content Toggle raw display
$41$ \( (T^{2} - 9 T + 12)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 123T^{2} + 144 \) Copy content Toggle raw display
$47$ \( T^{4} + 76T^{2} + 256 \) Copy content Toggle raw display
$53$ \( (T^{2} - 3 T - 6)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 9 T - 54)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 171T^{2} + 1296 \) Copy content Toggle raw display
$67$ \( T^{4} + 63T^{2} + 324 \) Copy content Toggle raw display
$71$ \( T^{4} + 76T^{2} + 256 \) Copy content Toggle raw display
$73$ \( (T + 2)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - T - 74)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 142T^{2} + 289 \) Copy content Toggle raw display
$89$ \( (T^{2} + 3 T - 6)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 13 T - 32)^{2} \) Copy content Toggle raw display
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