Properties

Label 1470.2.i.r
Level $1470$
Weight $2$
Character orbit 1470.i
Analytic conductor $11.738$
Analytic rank $0$
Dimension $2$
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(361,1470)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1470, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1470.361");
 
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.i (of order \(3\), degree \(2\), not 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + q^{6} - q^{8} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{3} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + q^{6} - q^{8} - \zeta_{6} q^{9} + (\zeta_{6} - 1) q^{10} + (6 \zeta_{6} - 6) q^{11} + \zeta_{6} q^{12} - 6 q^{13} + q^{15} - \zeta_{6} q^{16} + ( - \zeta_{6} + 1) q^{18} - 4 \zeta_{6} q^{19} - q^{20} - 6 q^{22} + (\zeta_{6} - 1) q^{24} + (\zeta_{6} - 1) q^{25} - 6 \zeta_{6} q^{26} - q^{27} - 8 q^{29} + \zeta_{6} q^{30} + ( - 2 \zeta_{6} + 2) q^{31} + ( - \zeta_{6} + 1) q^{32} + 6 \zeta_{6} q^{33} + q^{36} - 4 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{38} + (6 \zeta_{6} - 6) q^{39} - \zeta_{6} q^{40} + 10 q^{41} - 6 q^{43} - 6 \zeta_{6} q^{44} + ( - \zeta_{6} + 1) q^{45} - 2 \zeta_{6} q^{47} - q^{48} - q^{50} + ( - 6 \zeta_{6} + 6) q^{52} + (10 \zeta_{6} - 10) q^{53} - \zeta_{6} q^{54} - 6 q^{55} - 4 q^{57} - 8 \zeta_{6} q^{58} + (4 \zeta_{6} - 4) q^{59} + (\zeta_{6} - 1) q^{60} + 14 \zeta_{6} q^{61} + 2 q^{62} + q^{64} - 6 \zeta_{6} q^{65} + (6 \zeta_{6} - 6) q^{66} + (14 \zeta_{6} - 14) q^{67} + 8 q^{71} + \zeta_{6} q^{72} + ( - 6 \zeta_{6} + 6) q^{73} + ( - 4 \zeta_{6} + 4) q^{74} + \zeta_{6} q^{75} + 4 q^{76} - 6 q^{78} + 8 \zeta_{6} q^{79} + ( - \zeta_{6} + 1) q^{80} + (\zeta_{6} - 1) q^{81} + 10 \zeta_{6} q^{82} + 8 q^{83} - 6 \zeta_{6} q^{86} + (8 \zeta_{6} - 8) q^{87} + ( - 6 \zeta_{6} + 6) q^{88} - 18 \zeta_{6} q^{89} + q^{90} - 2 \zeta_{6} q^{93} + ( - 2 \zeta_{6} + 2) q^{94} + ( - 4 \zeta_{6} + 4) q^{95} - \zeta_{6} q^{96} - 2 q^{97} + 6 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} - q^{4} + q^{5} + 2 q^{6} - 2 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + q^{3} - q^{4} + q^{5} + 2 q^{6} - 2 q^{8} - q^{9} - q^{10} - 6 q^{11} + q^{12} - 12 q^{13} + 2 q^{15} - q^{16} + q^{18} - 4 q^{19} - 2 q^{20} - 12 q^{22} - q^{24} - q^{25} - 6 q^{26} - 2 q^{27} - 16 q^{29} + q^{30} + 2 q^{31} + q^{32} + 6 q^{33} + 2 q^{36} - 4 q^{37} + 4 q^{38} - 6 q^{39} - q^{40} + 20 q^{41} - 12 q^{43} - 6 q^{44} + q^{45} - 2 q^{47} - 2 q^{48} - 2 q^{50} + 6 q^{52} - 10 q^{53} - q^{54} - 12 q^{55} - 8 q^{57} - 8 q^{58} - 4 q^{59} - q^{60} + 14 q^{61} + 4 q^{62} + 2 q^{64} - 6 q^{65} - 6 q^{66} - 14 q^{67} + 16 q^{71} + q^{72} + 6 q^{73} + 4 q^{74} + q^{75} + 8 q^{76} - 12 q^{78} + 8 q^{79} + q^{80} - q^{81} + 10 q^{82} + 16 q^{83} - 6 q^{86} - 8 q^{87} + 6 q^{88} - 18 q^{89} + 2 q^{90} - 2 q^{93} + 2 q^{94} + 4 q^{95} - q^{96} - 4 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 0 −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
961.1 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.500000 0.866025i 1.00000 0 −1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.i.r 2
7.b odd 2 1 1470.2.i.k 2
7.c even 3 1 1470.2.a.c 1
7.c even 3 1 inner 1470.2.i.r 2
7.d odd 6 1 1470.2.a.i yes 1
7.d odd 6 1 1470.2.i.k 2
21.g even 6 1 4410.2.a.v 1
21.h odd 6 1 4410.2.a.be 1
35.i odd 6 1 7350.2.a.cf 1
35.j even 6 1 7350.2.a.da 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.c 1 7.c even 3 1
1470.2.a.i yes 1 7.d odd 6 1
1470.2.i.k 2 7.b odd 2 1
1470.2.i.k 2 7.d odd 6 1
1470.2.i.r 2 1.a even 1 1 trivial
1470.2.i.r 2 7.c even 3 1 inner
4410.2.a.v 1 21.g even 6 1
4410.2.a.be 1 21.h odd 6 1
7350.2.a.cf 1 35.i odd 6 1
7350.2.a.da 1 35.j even 6 1

Hecke kernels

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

\( T_{11}^{2} + 6T_{11} + 36 \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{19}^{2} + 4T_{19} + 16 \) Copy content Toggle raw display
\( T_{31}^{2} - 2T_{31} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( (T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 8)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( (T + 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$67$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$79$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$83$ \( (T - 8)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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