Properties

Label 1470.2.g.a
Level $1470$
Weight $2$
Character orbit 1470.g
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(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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - i q^{2} + i q^{3} - q^{4} + ( - i - 2) q^{5} + q^{6} + i q^{8} - q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - i q^{2} + i q^{3} - q^{4} + ( - i - 2) q^{5} + q^{6} + i q^{8} - q^{9} + (2 i - 1) q^{10} - 5 q^{11} - i q^{12} + i q^{13} + ( - 2 i + 1) q^{15} + q^{16} + 2 i q^{17} + i q^{18} + 7 q^{19} + (i + 2) q^{20} + 5 i q^{22} - 3 i q^{23} - q^{24} + (4 i + 3) q^{25} + q^{26} - i q^{27} + ( - i - 2) q^{30} + 6 q^{31} - i q^{32} - 5 i q^{33} + 2 q^{34} + q^{36} - 5 i q^{37} - 7 i q^{38} - q^{39} + ( - 2 i + 1) q^{40} + 9 q^{41} - 10 i q^{43} + 5 q^{44} + (i + 2) q^{45} - 3 q^{46} + 13 i q^{47} + i q^{48} + ( - 3 i + 4) q^{50} - 2 q^{51} - i q^{52} + i q^{53} - q^{54} + (5 i + 10) q^{55} + 7 i q^{57} + 4 q^{59} + (2 i - 1) q^{60} + 2 q^{61} - 6 i q^{62} - q^{64} + ( - 2 i + 1) q^{65} - 5 q^{66} + 6 i q^{67} - 2 i q^{68} + 3 q^{69} - 2 q^{71} - i q^{72} + 4 i q^{73} - 5 q^{74} + (3 i - 4) q^{75} - 7 q^{76} + i q^{78} + 14 q^{79} + ( - i - 2) q^{80} + q^{81} - 9 i q^{82} - 10 i q^{83} + ( - 4 i + 2) q^{85} - 10 q^{86} - 5 i q^{88} + 10 q^{89} + ( - 2 i + 1) q^{90} + 3 i q^{92} + 6 i q^{93} + 13 q^{94} + ( - 7 i - 14) q^{95} + q^{96} - 8 i q^{97} + 5 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 4 q^{5} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 4 q^{5} + 2 q^{6} - 2 q^{9} - 2 q^{10} - 10 q^{11} + 2 q^{15} + 2 q^{16} + 14 q^{19} + 4 q^{20} - 2 q^{24} + 6 q^{25} + 2 q^{26} - 4 q^{30} + 12 q^{31} + 4 q^{34} + 2 q^{36} - 2 q^{39} + 2 q^{40} + 18 q^{41} + 10 q^{44} + 4 q^{45} - 6 q^{46} + 8 q^{50} - 4 q^{51} - 2 q^{54} + 20 q^{55} + 8 q^{59} - 2 q^{60} + 4 q^{61} - 2 q^{64} + 2 q^{65} - 10 q^{66} + 6 q^{69} - 4 q^{71} - 10 q^{74} - 8 q^{75} - 14 q^{76} + 28 q^{79} - 4 q^{80} + 2 q^{81} + 4 q^{85} - 20 q^{86} + 20 q^{89} + 2 q^{90} + 26 q^{94} - 28 q^{95} + 2 q^{96} + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(491\) \(1081\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
1.00000i
1.00000i
1.00000i 1.00000i −1.00000 −2.00000 1.00000i 1.00000 0 1.00000i −1.00000 −1.00000 + 2.00000i
589.2 1.00000i 1.00000i −1.00000 −2.00000 + 1.00000i 1.00000 0 1.00000i −1.00000 −1.00000 2.00000i
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.g.a 2
5.b even 2 1 inner 1470.2.g.a 2
5.c odd 4 1 7350.2.a.b 1
5.c odd 4 1 7350.2.a.ch 1
7.b odd 2 1 1470.2.g.f 2
7.c even 3 2 1470.2.n.i 4
7.d odd 6 2 210.2.n.a 4
21.g even 6 2 630.2.u.c 4
28.f even 6 2 1680.2.di.a 4
35.c odd 2 1 1470.2.g.f 2
35.f even 4 1 7350.2.a.t 1
35.f even 4 1 7350.2.a.bn 1
35.i odd 6 2 210.2.n.a 4
35.j even 6 2 1470.2.n.i 4
35.k even 12 2 1050.2.i.f 2
35.k even 12 2 1050.2.i.o 2
105.p even 6 2 630.2.u.c 4
140.s even 6 2 1680.2.di.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.a 4 7.d odd 6 2
210.2.n.a 4 35.i odd 6 2
630.2.u.c 4 21.g even 6 2
630.2.u.c 4 105.p even 6 2
1050.2.i.f 2 35.k even 12 2
1050.2.i.o 2 35.k even 12 2
1470.2.g.a 2 1.a even 1 1 trivial
1470.2.g.a 2 5.b even 2 1 inner
1470.2.g.f 2 7.b odd 2 1
1470.2.g.f 2 35.c odd 2 1
1470.2.n.i 4 7.c even 3 2
1470.2.n.i 4 35.j even 6 2
1680.2.di.a 4 28.f even 6 2
1680.2.di.a 4 140.s even 6 2
7350.2.a.b 1 5.c odd 4 1
7350.2.a.t 1 35.f even 4 1
7350.2.a.bn 1 35.f even 4 1
7350.2.a.ch 1 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 \(S_{2}^{\mathrm{new}}(1470, [\chi])\):

\( T_{11} + 5 \) Copy content Toggle raw display
\( T_{17}^{2} + 4 \) Copy content Toggle raw display
\( T_{19} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

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