Properties

Label 1470.2.i.e
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: no (minimal twist has level 210)
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 \( q - \zeta_{6} q^{2} + (\zeta_{6} - 1) q^{3} + (\zeta_{6} - 1) q^{4} + \zeta_{6} q^{5} + q^{6} + q^{8} - \zeta_{6} q^{9} + ( - \zeta_{6} + 1) q^{10} + ( - 5 \zeta_{6} + 5) q^{11} - \zeta_{6} q^{12} + 5 q^{13} - q^{15} - \zeta_{6} q^{16} + (4 \zeta_{6} - 4) q^{17} + (\zeta_{6} - 1) q^{18} - 7 \zeta_{6} q^{19} - q^{20} - 5 q^{22} - \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{24} + (\zeta_{6} - 1) q^{25} - 5 \zeta_{6} q^{26} + q^{27} + \zeta_{6} q^{30} + (2 \zeta_{6} - 2) q^{31} + (\zeta_{6} - 1) q^{32} + 5 \zeta_{6} q^{33} + 4 q^{34} + q^{36} - \zeta_{6} q^{37} + (7 \zeta_{6} - 7) q^{38} + (5 \zeta_{6} - 5) q^{39} + \zeta_{6} q^{40} - 5 q^{41} + 12 q^{43} + 5 \zeta_{6} q^{44} + ( - \zeta_{6} + 1) q^{45} + (\zeta_{6} - 1) q^{46} - 11 \zeta_{6} q^{47} + q^{48} + q^{50} - 4 \zeta_{6} q^{51} + (5 \zeta_{6} - 5) q^{52} + ( - 9 \zeta_{6} + 9) q^{53} - \zeta_{6} q^{54} + 5 q^{55} + 7 q^{57} + ( - 4 \zeta_{6} + 4) q^{59} + ( - \zeta_{6} + 1) q^{60} + 4 \zeta_{6} q^{61} + 2 q^{62} + q^{64} + 5 \zeta_{6} q^{65} + ( - 5 \zeta_{6} + 5) q^{66} + ( - 12 \zeta_{6} + 12) q^{67} - 4 \zeta_{6} q^{68} + q^{69} + 2 q^{71} - \zeta_{6} q^{72} + ( - 10 \zeta_{6} + 10) q^{73} + (\zeta_{6} - 1) q^{74} - \zeta_{6} q^{75} + 7 q^{76} + 5 q^{78} + 12 \zeta_{6} q^{79} + ( - \zeta_{6} + 1) q^{80} + (\zeta_{6} - 1) q^{81} + 5 \zeta_{6} q^{82} + 12 q^{83} - 4 q^{85} - 12 \zeta_{6} q^{86} + ( - 5 \zeta_{6} + 5) q^{88} + 14 \zeta_{6} q^{89} - q^{90} + q^{92} - 2 \zeta_{6} q^{93} + (11 \zeta_{6} - 11) q^{94} + ( - 7 \zeta_{6} + 7) q^{95} - \zeta_{6} q^{96} + 8 q^{97} - 5 q^{99} +O(q^{100}) \) 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 \( 2 q - q^{2} - q^{3} - q^{4} + q^{5} + 2 q^{6} + 2 q^{8} - q^{9} + q^{10} + 5 q^{11} - q^{12} + 10 q^{13} - 2 q^{15} - q^{16} - 4 q^{17} - q^{18} - 7 q^{19} - 2 q^{20} - 10 q^{22} - q^{23} - q^{24} - q^{25} - 5 q^{26} + 2 q^{27} + q^{30} - 2 q^{31} - q^{32} + 5 q^{33} + 8 q^{34} + 2 q^{36} - q^{37} - 7 q^{38} - 5 q^{39} + q^{40} - 10 q^{41} + 24 q^{43} + 5 q^{44} + q^{45} - q^{46} - 11 q^{47} + 2 q^{48} + 2 q^{50} - 4 q^{51} - 5 q^{52} + 9 q^{53} - q^{54} + 10 q^{55} + 14 q^{57} + 4 q^{59} + q^{60} + 4 q^{61} + 4 q^{62} + 2 q^{64} + 5 q^{65} + 5 q^{66} + 12 q^{67} - 4 q^{68} + 2 q^{69} + 4 q^{71} - q^{72} + 10 q^{73} - q^{74} - q^{75} + 14 q^{76} + 10 q^{78} + 12 q^{79} + q^{80} - q^{81} + 5 q^{82} + 24 q^{83} - 8 q^{85} - 12 q^{86} + 5 q^{88} + 14 q^{89} - 2 q^{90} + 2 q^{92} - 2 q^{93} - 11 q^{94} + 7 q^{95} - q^{96} + 16 q^{97} - 10 q^{99}+O(q^{100}) \) 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.e 2
7.b odd 2 1 210.2.i.b 2
7.c even 3 1 1470.2.a.o 1
7.c even 3 1 inner 1470.2.i.e 2
7.d odd 6 1 210.2.i.b 2
7.d odd 6 1 1470.2.a.l 1
21.c even 2 1 630.2.k.g 2
21.g even 6 1 630.2.k.g 2
21.g even 6 1 4410.2.a.j 1
21.h odd 6 1 4410.2.a.u 1
28.d even 2 1 1680.2.bg.d 2
28.f even 6 1 1680.2.bg.d 2
35.c odd 2 1 1050.2.i.p 2
35.f even 4 2 1050.2.o.g 4
35.i odd 6 1 1050.2.i.p 2
35.i odd 6 1 7350.2.a.u 1
35.j even 6 1 7350.2.a.a 1
35.k even 12 2 1050.2.o.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.b 2 7.b odd 2 1
210.2.i.b 2 7.d odd 6 1
630.2.k.g 2 21.c even 2 1
630.2.k.g 2 21.g even 6 1
1050.2.i.p 2 35.c odd 2 1
1050.2.i.p 2 35.i odd 6 1
1050.2.o.g 4 35.f even 4 2
1050.2.o.g 4 35.k even 12 2
1470.2.a.l 1 7.d odd 6 1
1470.2.a.o 1 7.c even 3 1
1470.2.i.e 2 1.a even 1 1 trivial
1470.2.i.e 2 7.c even 3 1 inner
1680.2.bg.d 2 28.d even 2 1
1680.2.bg.d 2 28.f even 6 1
4410.2.a.j 1 21.g even 6 1
4410.2.a.u 1 21.h odd 6 1
7350.2.a.a 1 35.j even 6 1
7350.2.a.u 1 35.i odd 6 1

Hecke kernels

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

\( T_{11}^{2} - 5T_{11} + 25 \) Copy content Toggle raw display
\( T_{13} - 5 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} + 16 \) Copy content Toggle raw display
\( T_{19}^{2} + 7T_{19} + 49 \) 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
$11$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$13$ \( (T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$41$ \( (T + 5)^{2} \) Copy content Toggle raw display
$43$ \( (T - 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$67$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$71$ \( (T - 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$79$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$83$ \( (T - 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
$97$ \( (T - 8)^{2} \) Copy content Toggle raw display
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