Properties

Label 1470.2.a.h
Level $1470$
Weight $2$
Character orbit 1470.a
Self dual yes
Analytic conductor $11.738$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} - q^{10} - q^{11} + q^{12} - q^{13} + q^{15} + q^{16} - q^{18} + 3q^{19} + q^{20} + q^{22} + 7q^{23} - q^{24} + q^{25} + q^{26} + q^{27} - 8q^{29} - q^{30} + 2q^{31} - q^{32} - q^{33} + q^{36} + 11q^{37} - 3q^{38} - q^{39} - q^{40} + 11q^{41} + 8q^{43} - q^{44} + q^{45} - 7q^{46} + 5q^{47} + q^{48} - q^{50} - q^{52} - 11q^{53} - q^{54} - q^{55} + 3q^{57} + 8q^{58} - 4q^{59} + q^{60} - 2q^{62} + q^{64} - q^{65} + q^{66} + 7q^{69} - 6q^{71} - q^{72} + 6q^{73} - 11q^{74} + q^{75} + 3q^{76} + q^{78} - 8q^{79} + q^{80} + q^{81} - 11q^{82} - 8q^{83} - 8q^{86} - 8q^{87} + q^{88} + 10q^{89} - q^{90} + 7q^{92} + 2q^{93} - 5q^{94} + 3q^{95} - q^{96} + 16q^{97} - q^{99} + O(q^{100}) \)

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.h 1
3.b odd 2 1 4410.2.a.ba 1
5.b even 2 1 7350.2.a.bu 1
7.b odd 2 1 1470.2.a.a 1
7.c even 3 2 1470.2.i.m 2
7.d odd 6 2 210.2.i.d 2
21.c even 2 1 4410.2.a.bj 1
21.g even 6 2 630.2.k.c 2
28.f even 6 2 1680.2.bg.g 2
35.c odd 2 1 7350.2.a.cp 1
35.i odd 6 2 1050.2.i.b 2
35.k even 12 4 1050.2.o.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.d 2 7.d odd 6 2
630.2.k.c 2 21.g even 6 2
1050.2.i.b 2 35.i odd 6 2
1050.2.o.i 4 35.k even 12 4
1470.2.a.a 1 7.b odd 2 1
1470.2.a.h 1 1.a even 1 1 trivial
1470.2.i.m 2 7.c even 3 2
1680.2.bg.g 2 28.f even 6 2
4410.2.a.ba 1 3.b odd 2 1
4410.2.a.bj 1 21.c even 2 1
7350.2.a.bu 1 5.b even 2 1
7350.2.a.cp 1 35.c odd 2 1

Hecke kernels

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

\( T_{11} + 1 \)
\( T_{13} + 1 \)
\( T_{17} \)
\( T_{19} - 3 \)
\( T_{31} - 2 \)