Properties

Label 1470.2.a.a
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

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

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:

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.a 1
3.b odd 2 1 4410.2.a.bj 1
5.b even 2 1 7350.2.a.cp 1
7.b odd 2 1 1470.2.a.h 1
7.c even 3 2 210.2.i.d 2
7.d odd 6 2 1470.2.i.m 2
21.c even 2 1 4410.2.a.ba 1
21.h odd 6 2 630.2.k.c 2
28.g odd 6 2 1680.2.bg.g 2
35.c odd 2 1 7350.2.a.bu 1
35.j even 6 2 1050.2.i.b 2
35.l odd 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.c even 3 2
630.2.k.c 2 21.h odd 6 2
1050.2.i.b 2 35.j even 6 2
1050.2.o.i 4 35.l odd 12 4
1470.2.a.a 1 1.a even 1 1 trivial
1470.2.a.h 1 7.b odd 2 1
1470.2.i.m 2 7.d odd 6 2
1680.2.bg.g 2 28.g odd 6 2
4410.2.a.ba 1 21.c even 2 1
4410.2.a.bj 1 3.b odd 2 1
7350.2.a.bu 1 35.c odd 2 1
7350.2.a.cp 1 5.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( 1 + T \)
$7$ 1
$11$ \( 1 + T + 11 T^{2} \)
$13$ \( 1 - T + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 + 3 T + 19 T^{2} \)
$23$ \( 1 - 7 T + 23 T^{2} \)
$29$ \( 1 + 8 T + 29 T^{2} \)
$31$ \( 1 + 2 T + 31 T^{2} \)
$37$ \( 1 - 11 T + 37 T^{2} \)
$41$ \( 1 + 11 T + 41 T^{2} \)
$43$ \( 1 - 8 T + 43 T^{2} \)
$47$ \( 1 + 5 T + 47 T^{2} \)
$53$ \( 1 + 11 T + 53 T^{2} \)
$59$ \( 1 - 4 T + 59 T^{2} \)
$61$ \( 1 + 61 T^{2} \)
$67$ \( 1 + 67 T^{2} \)
$71$ \( 1 + 6 T + 71 T^{2} \)
$73$ \( 1 + 6 T + 73 T^{2} \)
$79$ \( 1 + 8 T + 79 T^{2} \)
$83$ \( 1 - 8 T + 83 T^{2} \)
$89$ \( 1 + 10 T + 89 T^{2} \)
$97$ \( 1 + 16 T + 97 T^{2} \)
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