Properties

Label 1470.2.i.x
Level $1470$
Weight $2$
Character orbit 1470.i
Analytic conductor $11.738$
Analytic rank $0$
Dimension $4$
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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + \beta_{2} q^{5} + q^{6} - q^{8} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{2} + 1) q^{3} + ( - \beta_{2} - 1) q^{4} + \beta_{2} q^{5} + q^{6} - q^{8} + \beta_{2} q^{9} + (\beta_{2} + 1) q^{10} + (2 \beta_{2} + 3 \beta_1 + 2) q^{11} - \beta_{2} q^{12} + 4 \beta_{3} q^{13} - q^{15} + \beta_{2} q^{16} + ( - 4 \beta_{2} - \beta_1 - 4) q^{17} + (\beta_{2} + 1) q^{18} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1) q^{19} + q^{20} + ( - 3 \beta_{3} + 2) q^{22} + (2 \beta_{3} + 6 \beta_{2} + 2 \beta_1) q^{23} + ( - \beta_{2} - 1) q^{24} + ( - \beta_{2} - 1) q^{25} + (4 \beta_{3} + 4 \beta_1) q^{26} - q^{27} + ( - \beta_{3} + 4) q^{29} + \beta_{2} q^{30} + (6 \beta_{2} - 3 \beta_1 + 6) q^{31} + (\beta_{2} + 1) q^{32} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{33} + (\beta_{3} - 4) q^{34} + q^{36} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1) q^{37} + ( - 4 \beta_{2} + 2 \beta_1 - 4) q^{38} - 4 \beta_1 q^{39} - \beta_{2} q^{40} + (6 \beta_{3} + 2) q^{41} + (5 \beta_{3} + 2) q^{43} + ( - 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1) q^{44} + ( - \beta_{2} - 1) q^{45} + (6 \beta_{2} + 2 \beta_1 + 6) q^{46} + (3 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{47} - q^{48} - q^{50} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{51} + 4 \beta_1 q^{52} + ( - 6 \beta_{2} - 4 \beta_1 - 6) q^{53} + \beta_{2} q^{54} + (3 \beta_{3} - 2) q^{55} + (2 \beta_{3} + 4) q^{57} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{58} + (6 \beta_{2} + 2 \beta_1 + 6) q^{59} + (\beta_{2} + 1) q^{60} + (4 \beta_{3} - 6 \beta_{2} + 4 \beta_1) q^{61} + (3 \beta_{3} + 6) q^{62} + q^{64} + ( - 4 \beta_{3} - 4 \beta_1) q^{65} + (2 \beta_{2} + 3 \beta_1 + 2) q^{66} + ( - 2 \beta_{2} + 5 \beta_1 - 2) q^{67} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{68} + (2 \beta_{3} - 6) q^{69} + ( - 6 \beta_{3} + 4) q^{71} - \beta_{2} q^{72} + (2 \beta_{2} + 2) q^{73} + (4 \beta_{2} + 3 \beta_1 + 4) q^{74} - \beta_{2} q^{75} + ( - 2 \beta_{3} - 4) q^{76} + 4 \beta_{3} q^{78} + 8 \beta_{2} q^{79} + ( - \beta_{2} - 1) q^{80} + ( - \beta_{2} - 1) q^{81} + (6 \beta_{3} - 2 \beta_{2} + 6 \beta_1) q^{82} + ( - 2 \beta_{3} + 8) q^{83} + ( - \beta_{3} + 4) q^{85} + (5 \beta_{3} - 2 \beta_{2} + 5 \beta_1) q^{86} + (4 \beta_{2} + \beta_1 + 4) q^{87} + ( - 2 \beta_{2} - 3 \beta_1 - 2) q^{88} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{89} - q^{90} + ( - 2 \beta_{3} + 6) q^{92} + ( - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1) q^{93} + (2 \beta_{2} + 3 \beta_1 + 2) q^{94} + (4 \beta_{2} - 2 \beta_1 + 4) q^{95} + \beta_{2} q^{96} + ( - 6 \beta_{3} + 2) q^{97} + (3 \beta_{3} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} + 4 q^{6} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} + 4 q^{6} - 4 q^{8} - 2 q^{9} + 2 q^{10} + 4 q^{11} + 2 q^{12} - 4 q^{15} - 2 q^{16} - 8 q^{17} + 2 q^{18} + 8 q^{19} + 4 q^{20} + 8 q^{22} - 12 q^{23} - 2 q^{24} - 2 q^{25} - 4 q^{27} + 16 q^{29} - 2 q^{30} + 12 q^{31} + 2 q^{32} - 4 q^{33} - 16 q^{34} + 4 q^{36} - 8 q^{37} - 8 q^{38} + 2 q^{40} + 8 q^{41} + 8 q^{43} + 4 q^{44} - 2 q^{45} + 12 q^{46} - 4 q^{47} - 4 q^{48} - 4 q^{50} + 8 q^{51} - 12 q^{53} - 2 q^{54} - 8 q^{55} + 16 q^{57} + 8 q^{58} + 12 q^{59} + 2 q^{60} + 12 q^{61} + 24 q^{62} + 4 q^{64} + 4 q^{66} - 4 q^{67} - 8 q^{68} - 24 q^{69} + 16 q^{71} + 2 q^{72} + 4 q^{73} + 8 q^{74} + 2 q^{75} - 16 q^{76} - 16 q^{79} - 2 q^{80} - 2 q^{81} + 4 q^{82} + 32 q^{83} + 16 q^{85} + 4 q^{86} + 8 q^{87} - 4 q^{88} - 4 q^{89} - 4 q^{90} + 24 q^{92} - 12 q^{93} + 4 q^{94} + 8 q^{95} - 2 q^{96} + 8 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(\beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
0.707107 + 1.22474i
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
361.2 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
961.2 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.x 4
7.b odd 2 1 1470.2.i.w 4
7.c even 3 1 1470.2.a.s 2
7.c even 3 1 inner 1470.2.i.x 4
7.d odd 6 1 1470.2.a.t yes 2
7.d odd 6 1 1470.2.i.w 4
21.g even 6 1 4410.2.a.bz 2
21.h odd 6 1 4410.2.a.bw 2
35.i odd 6 1 7350.2.a.dh 2
35.j even 6 1 7350.2.a.dl 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.s 2 7.c even 3 1
1470.2.a.t yes 2 7.d odd 6 1
1470.2.i.w 4 7.b odd 2 1
1470.2.i.w 4 7.d odd 6 1
1470.2.i.x 4 1.a even 1 1 trivial
1470.2.i.x 4 7.c even 3 1 inner
4410.2.a.bw 2 21.h odd 6 1
4410.2.a.bz 2 21.g even 6 1
7350.2.a.dh 2 35.i odd 6 1
7350.2.a.dl 2 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 \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\( T_{11}^{4} - 4T_{11}^{3} + 30T_{11}^{2} + 56T_{11} + 196 \) Copy content Toggle raw display
\( T_{13}^{2} - 32 \) Copy content Toggle raw display
\( T_{17}^{4} + 8T_{17}^{3} + 50T_{17}^{2} + 112T_{17} + 196 \) Copy content Toggle raw display
\( T_{19}^{4} - 8T_{19}^{3} + 56T_{19}^{2} - 64T_{19} + 64 \) Copy content Toggle raw display
\( T_{31}^{4} - 12T_{31}^{3} + 126T_{31}^{2} - 216T_{31} + 324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + 30 T^{2} + 56 T + 196 \) Copy content Toggle raw display
$13$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 8 T^{3} + 50 T^{2} + 112 T + 196 \) Copy content Toggle raw display
$19$ \( T^{4} - 8 T^{3} + 56 T^{2} - 64 T + 64 \) Copy content Toggle raw display
$23$ \( T^{4} + 12 T^{3} + 116 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$29$ \( (T^{2} - 8 T + 14)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 12 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$37$ \( T^{4} + 8 T^{3} + 66 T^{2} - 16 T + 4 \) Copy content Toggle raw display
$41$ \( (T^{2} - 4 T - 68)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T - 46)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 4 T^{3} + 30 T^{2} - 56 T + 196 \) Copy content Toggle raw display
$53$ \( T^{4} + 12 T^{3} + 140 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + 116 T^{2} + \cdots + 784 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + 140 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$67$ \( T^{4} + 4 T^{3} + 62 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$71$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 16 T + 56)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 4 T^{3} + 20 T^{2} - 16 T + 16 \) Copy content Toggle raw display
$97$ \( (T^{2} - 4 T - 68)^{2} \) Copy content Toggle raw display
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