Properties

Label 1470.2.a.e
Level $1470$
Weight $2$
Character orbit 1470.a
Self dual yes
Analytic conductor $11.738$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.7380090971\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1.00000 −1.00000 1.00000 1.00000 1.00000 0 −1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1470.2.a.e 1
3.b odd 2 1 4410.2.a.w 1
5.b even 2 1 7350.2.a.cx 1
7.b odd 2 1 1470.2.a.f 1
7.c even 3 2 1470.2.i.p 2
7.d odd 6 2 210.2.i.c 2
21.c even 2 1 4410.2.a.bh 1
21.g even 6 2 630.2.k.a 2
28.f even 6 2 1680.2.bg.n 2
35.c odd 2 1 7350.2.a.cd 1
35.i odd 6 2 1050.2.i.i 2
35.k even 12 4 1050.2.o.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.c 2 7.d odd 6 2
630.2.k.a 2 21.g even 6 2
1050.2.i.i 2 35.i odd 6 2
1050.2.o.c 4 35.k even 12 4
1470.2.a.e 1 1.a even 1 1 trivial
1470.2.a.f 1 7.b odd 2 1
1470.2.i.p 2 7.c even 3 2
1680.2.bg.n 2 28.f even 6 2
4410.2.a.w 1 3.b odd 2 1
4410.2.a.bh 1 21.c even 2 1
7350.2.a.cd 1 35.c odd 2 1
7350.2.a.cx 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1470))\):

\( T_{11} - 3 \)
\( T_{13} + 5 \)
\( T_{17} \)
\( T_{19} + 5 \)
\( T_{31} - 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( -3 + T \)
$13$ \( 5 + T \)
$17$ \( T \)
$19$ \( 5 + T \)
$23$ \( 9 + T \)
$29$ \( T \)
$31$ \( -10 + T \)
$37$ \( 1 + T \)
$41$ \( 9 + T \)
$43$ \( -8 + T \)
$47$ \( 3 + T \)
$53$ \( 3 + T \)
$59$ \( 12 + T \)
$61$ \( 8 + T \)
$67$ \( -8 + T \)
$71$ \( 6 + T \)
$73$ \( 2 + T \)
$79$ \( -8 + T \)
$83$ \( T \)
$89$ \( 6 + T \)
$97$ \( 8 + T \)
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