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Properties

Label 1470.2.n.i
Level $1470$
Weight $2$
Character orbit 1470.n
Analytic conductor $11.738$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1470,2,Mod(79,1470)]
 
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, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1470.79");
 
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.n (of order \(6\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{2} + \zeta_{12} q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{5} + q^{6} - \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} + \zeta_{12}) q^{2} + \zeta_{12} q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{5} + q^{6} - \zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + ( - \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{10} + ( - 5 \zeta_{12}^{2} + 5) q^{11} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{12} - \zeta_{12}^{3} q^{13} + (2 \zeta_{12}^{3} + 1) q^{15} - \zeta_{12}^{2} q^{16} + 2 \zeta_{12} q^{17} + \zeta_{12} q^{18} - 7 \zeta_{12}^{2} q^{19} + ( - \zeta_{12}^{3} + 2) q^{20} - 5 \zeta_{12}^{3} q^{22} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{23} + ( - \zeta_{12}^{2} + 1) q^{24} + (3 \zeta_{12}^{2} + 4 \zeta_{12} - 3) q^{25} - \zeta_{12}^{2} q^{26} + \zeta_{12}^{3} q^{27} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12}) q^{30} + (6 \zeta_{12}^{2} - 6) q^{31} - \zeta_{12} q^{32} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{33} + 2 q^{34} + q^{36} + ( - 5 \zeta_{12}^{3} + 5 \zeta_{12}) q^{37} - 7 \zeta_{12} q^{38} + ( - \zeta_{12}^{2} + 1) q^{39} + ( - 2 \zeta_{12}^{3} + \cdots + 2 \zeta_{12}) q^{40} + \cdots + 5 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 4 q^{5} + 4 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 4 q^{5} + 4 q^{6} + 2 q^{9} + 2 q^{10} + 10 q^{11} + 4 q^{15} - 2 q^{16} - 14 q^{19} + 8 q^{20} + 2 q^{24} - 6 q^{25} - 2 q^{26} + 4 q^{30} - 12 q^{31} + 8 q^{34} + 4 q^{36} + 2 q^{39} - 2 q^{40} + 36 q^{41} - 10 q^{44} - 4 q^{45} + 6 q^{46} + 16 q^{50} + 4 q^{51} + 2 q^{54} + 40 q^{55} - 8 q^{59} + 2 q^{60} - 4 q^{61} - 4 q^{64} - 2 q^{65} + 10 q^{66} + 12 q^{69} - 8 q^{71} + 10 q^{74} + 8 q^{75} - 28 q^{76} - 28 q^{79} + 4 q^{80} - 2 q^{81} + 8 q^{85} + 20 q^{86} - 20 q^{89} + 4 q^{90} - 26 q^{94} + 28 q^{95} - 2 q^{96} + 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(-\zeta_{12}^{2}\) \(-1\)

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
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i 0.133975 − 2.23205i 1.00000 0 − 1.00000i 0.500000 − 0.866025i −1.23205 + 1.86603i
79.2 0.866025 + 0.500000i 0.866025 − 0.500000i 0.500000 + 0.866025i 1.86603 − 1.23205i 1.00000 0 1.00000i 0.500000 − 0.866025i 2.23205 − 0.133975i
949.1 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.500000 − 0.866025i 0.133975 + 2.23205i 1.00000 0 1.00000i 0.500000 + 0.866025i −1.23205 − 1.86603i
949.2 0.866025 − 0.500000i 0.866025 + 0.500000i 0.500000 − 0.866025i 1.86603 + 1.23205i 1.00000 0 − 1.00000i 0.500000 + 0.866025i 2.23205 + 0.133975i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.n.i 4
5.b even 2 1 inner 1470.2.n.i 4
7.b odd 2 1 210.2.n.a ✓ 4
7.c even 3 1 1470.2.g.a 2
7.c even 3 1 inner 1470.2.n.i 4
7.d odd 6 1 210.2.n.a ✓ 4
7.d odd 6 1 1470.2.g.f 2
21.c even 2 1 630.2.u.c 4
21.g even 6 1 630.2.u.c 4
28.d even 2 1 1680.2.di.a 4
28.f even 6 1 1680.2.di.a 4
35.c odd 2 1 210.2.n.a ✓ 4
35.f even 4 1 1050.2.i.f 2
35.f even 4 1 1050.2.i.o 2
35.i odd 6 1 210.2.n.a ✓ 4
35.i odd 6 1 1470.2.g.f 2
35.j even 6 1 1470.2.g.a 2
35.j even 6 1 inner 1470.2.n.i 4
35.k even 12 1 1050.2.i.f 2
35.k even 12 1 1050.2.i.o 2
35.k even 12 1 7350.2.a.t 1
35.k even 12 1 7350.2.a.bn 1
35.l odd 12 1 7350.2.a.b 1
35.l odd 12 1 7350.2.a.ch 1
105.g even 2 1 630.2.u.c 4
105.p even 6 1 630.2.u.c 4
140.c even 2 1 1680.2.di.a 4
140.s even 6 1 1680.2.di.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.a ✓ 4 7.b odd 2 1
210.2.n.a ✓ 4 7.d odd 6 1
210.2.n.a ✓ 4 35.c odd 2 1
210.2.n.a ✓ 4 35.i odd 6 1
630.2.u.c 4 21.c even 2 1
630.2.u.c 4 21.g even 6 1
630.2.u.c 4 105.g even 2 1
630.2.u.c 4 105.p even 6 1
1050.2.i.f 2 35.f even 4 1
1050.2.i.f 2 35.k even 12 1
1050.2.i.o 2 35.f even 4 1
1050.2.i.o 2 35.k even 12 1
1470.2.g.a 2 7.c even 3 1
1470.2.g.a 2 35.j even 6 1
1470.2.g.f 2 7.d odd 6 1
1470.2.g.f 2 35.i odd 6 1
1470.2.n.i 4 1.a even 1 1 trivial
1470.2.n.i 4 5.b even 2 1 inner
1470.2.n.i 4 7.c even 3 1 inner
1470.2.n.i 4 35.j even 6 1 inner
1680.2.di.a 4 28.d even 2 1
1680.2.di.a 4 28.f even 6 1
1680.2.di.a 4 140.c even 2 1
1680.2.di.a 4 140.s even 6 1
7350.2.a.b 1 35.l odd 12 1
7350.2.a.t 1 35.k even 12 1
7350.2.a.bn 1 35.k even 12 1
7350.2.a.ch 1 35.l odd 12 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
\( T_{17}^{4} - 4T_{17}^{2} + 16 \) Copy content Toggle raw display
\( T_{19}^{2} + 7T_{19} + 49 \) Copy content Toggle raw display
\( T_{31}^{2} + 6T_{31} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} - 4 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$41$ \( (T - 9)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 169 T^{2} + 28561 \) Copy content Toggle raw display
$53$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$59$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$71$ \( (T + 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$79$ \( (T^{2} + 14 T + 196)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
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