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Properties

Label 1470.2.g.k
Level $1470$
Weight $2$
Character orbit 1470.g
Analytic conductor $11.738$
Analytic rank $0$
Dimension $8$
CM no
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(589,1470)]
 
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([0, 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("1470.589");
 
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.g (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: \(8\)
Coefficient field: 8.0.1698758656.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + \beta_{5} q^{3} - q^{4} + ( - \beta_{7} + 1) q^{5} + q^{6} + \beta_{5} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + \beta_{5} q^{3} - q^{4} + ( - \beta_{7} + 1) q^{5} + q^{6} + \beta_{5} q^{8} - q^{9} - \beta_{4} q^{10} + ( - \beta_{4} + \beta_{3} + \beta_1) q^{11} - \beta_{5} q^{12} + (\beta_{7} - \beta_{6} - 2 \beta_{2}) q^{13} + \beta_{4} q^{15} + q^{16} + (\beta_{7} - \beta_{6} - \beta_{4} + \cdots - \beta_1) q^{17}+ \cdots + (\beta_{4} - \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} + 4 q^{5} + 8 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 4 q^{5} + 8 q^{6} - 8 q^{9} + 8 q^{16} - 24 q^{19} - 4 q^{20} - 8 q^{24} + 4 q^{25} + 16 q^{29} + 4 q^{30} + 32 q^{31} - 8 q^{34} + 8 q^{36} + 24 q^{41} - 4 q^{45} + 8 q^{46} - 4 q^{50} + 8 q^{51} - 8 q^{54} - 40 q^{59} + 24 q^{61} - 8 q^{64} + 28 q^{65} - 8 q^{69} - 40 q^{71} + 16 q^{74} + 4 q^{75} + 24 q^{76} + 16 q^{79} + 4 q^{80} + 8 q^{81} + 28 q^{85} + 8 q^{86} - 88 q^{89} - 24 q^{94} + 24 q^{95} + 8 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 10\nu^{5} - 20\nu^{4} - 15\nu^{3} - 2\nu^{2} - 184\nu + 80 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 18\nu^{5} - 89\nu^{3} - 104\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 37\nu^{2} - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 2\nu^{6} + 10\nu^{5} + 20\nu^{4} - 15\nu^{3} + 2\nu^{2} - 184\nu - 80 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} + 46\nu^{5} + 179\nu^{3} + 168\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 6\nu^{6} - 10\nu^{5} + 92\nu^{4} + 15\nu^{3} + 358\nu^{2} + 120\nu + 336 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 6\nu^{6} + 10\nu^{5} + 92\nu^{4} - 15\nu^{3} + 358\nu^{2} - 120\nu + 336 ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{4} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{4} - 4\beta_{3} - \beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{7} + 9\beta_{6} + 8\beta_{5} + 5\beta_{4} + 8\beta_{2} + 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -9\beta_{7} - 9\beta_{6} - 17\beta_{4} + 44\beta_{3} + 17\beta _1 + 74 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 81\beta_{7} - 81\beta_{6} - 104\beta_{5} - 37\beta_{4} - 112\beta_{2} - 37\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 89\beta_{7} + 89\beta_{6} + 201\beta_{4} - 436\beta_{3} - 201\beta _1 - 650 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -761\beta_{7} + 761\beta_{6} + 1160\beta_{5} + 325\beta_{4} + 1240\beta_{2} + 325\beta_1 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1
0.692297i
2.16053i
− 3.16053i
− 1.69230i
− 0.692297i
− 2.16053i
3.16053i
1.69230i
− 1.00000i 1.00000i −1.00000 −1.88893 + 1.19663i 1.00000 0 1.00000i −1.00000 1.19663 + 1.88893i
589.2 − 1.00000i 1.00000i −1.00000 0.0743018 − 2.23483i 1.00000 0 1.00000i −1.00000 −2.23483 − 0.0743018i
589.3 − 1.00000i 1.00000i −1.00000 1.63280 + 1.52773i 1.00000 0 1.00000i −1.00000 1.52773 − 1.63280i
589.4 − 1.00000i 1.00000i −1.00000 2.18183 − 0.489528i 1.00000 0 1.00000i −1.00000 −0.489528 − 2.18183i
589.5 1.00000i − 1.00000i −1.00000 −1.88893 − 1.19663i 1.00000 0 − 1.00000i −1.00000 1.19663 − 1.88893i
589.6 1.00000i − 1.00000i −1.00000 0.0743018 + 2.23483i 1.00000 0 − 1.00000i −1.00000 −2.23483 + 0.0743018i
589.7 1.00000i − 1.00000i −1.00000 1.63280 − 1.52773i 1.00000 0 − 1.00000i −1.00000 1.52773 + 1.63280i
589.8 1.00000i − 1.00000i −1.00000 2.18183 + 0.489528i 1.00000 0 − 1.00000i −1.00000 −0.489528 + 2.18183i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 589.8
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 1470.2.g.k yes 8
5.b even 2 1 inner 1470.2.g.k yes 8
5.c odd 4 1 7350.2.a.dr 4
5.c odd 4 1 7350.2.a.du 4
7.b odd 2 1 1470.2.g.j ✓ 8
7.c even 3 2 1470.2.n.k 16
7.d odd 6 2 1470.2.n.l 16
35.c odd 2 1 1470.2.g.j ✓ 8
35.f even 4 1 7350.2.a.ds 4
35.f even 4 1 7350.2.a.dt 4
35.i odd 6 2 1470.2.n.l 16
35.j even 6 2 1470.2.n.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.g.j ✓ 8 7.b odd 2 1
1470.2.g.j ✓ 8 35.c odd 2 1
1470.2.g.k yes 8 1.a even 1 1 trivial
1470.2.g.k yes 8 5.b even 2 1 inner
1470.2.n.k 16 7.c even 3 2
1470.2.n.k 16 35.j even 6 2
1470.2.n.l 16 7.d odd 6 2
1470.2.n.l 16 35.i odd 6 2
7350.2.a.dr 4 5.c odd 4 1
7350.2.a.ds 4 35.f even 4 1
7350.2.a.dt 4 35.f even 4 1
7350.2.a.du 4 5.c odd 4 1

Hecke kernels

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

\( T_{11}^{4} - 18T_{11}^{2} - 16T_{11} + 32 \) Copy content Toggle raw display
\( T_{17}^{8} + 80T_{17}^{6} + 1336T_{17}^{4} + 8000T_{17}^{2} + 15376 \) Copy content Toggle raw display
\( T_{19}^{4} + 12T_{19}^{3} + 12T_{19}^{2} - 224T_{19} - 448 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} - 4 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 18 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 52 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} + 80 T^{6} + \cdots + 15376 \) Copy content Toggle raw display
$19$ \( (T^{4} + 12 T^{3} + \cdots - 448)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 88 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + \cdots + 1568)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 16 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 148 T^{6} + \cdots + 40000 \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} + \cdots - 376)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 236 T^{6} + \cdots + 541696 \) Copy content Toggle raw display
$47$ \( T^{8} + 204 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$53$ \( T^{8} + 184 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( (T^{4} + 20 T^{3} + \cdots - 3008)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 12 T^{3} + \cdots - 648)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 44 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( (T^{4} + 20 T^{3} + \cdots + 2944)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 316 T^{6} + \cdots + 61504 \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + \cdots - 3584)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 240 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$89$ \( (T^{4} + 44 T^{3} + \cdots + 5048)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 428 T^{6} + \cdots + 107661376 \) Copy content Toggle raw display
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