Properties

Label 1470.2.a.o
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

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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

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

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.o 1
3.b odd 2 1 4410.2.a.u 1
5.b even 2 1 7350.2.a.a 1
7.b odd 2 1 1470.2.a.l 1
7.c even 3 2 1470.2.i.e 2
7.d odd 6 2 210.2.i.b 2
21.c even 2 1 4410.2.a.j 1
21.g even 6 2 630.2.k.g 2
28.f even 6 2 1680.2.bg.d 2
35.c odd 2 1 7350.2.a.u 1
35.i odd 6 2 1050.2.i.p 2
35.k even 12 4 1050.2.o.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.b 2 7.d odd 6 2
630.2.k.g 2 21.g even 6 2
1050.2.i.p 2 35.i odd 6 2
1050.2.o.g 4 35.k even 12 4
1470.2.a.l 1 7.b odd 2 1
1470.2.a.o 1 1.a even 1 1 trivial
1470.2.i.e 2 7.c even 3 2
1680.2.bg.d 2 28.f even 6 2
4410.2.a.j 1 21.c even 2 1
4410.2.a.u 1 3.b odd 2 1
7350.2.a.a 1 5.b even 2 1
7350.2.a.u 1 35.c odd 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1470))\):

\( T_{11} + 5 \)
\( T_{13} - 5 \)
\( T_{17} - 4 \)
\( T_{19} - 7 \)
\( T_{31} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 - T \)
$5$ \( 1 + T \)
$7$ 1
$11$ \( 1 + 5 T + 11 T^{2} \)
$13$ \( 1 - 5 T + 13 T^{2} \)
$17$ \( 1 - 4 T + 17 T^{2} \)
$19$ \( 1 - 7 T + 19 T^{2} \)
$23$ \( 1 - T + 23 T^{2} \)
$29$ \( 1 + 29 T^{2} \)
$31$ \( 1 - 2 T + 31 T^{2} \)
$37$ \( 1 - T + 37 T^{2} \)
$41$ \( 1 + 5 T + 41 T^{2} \)
$43$ \( 1 - 12 T + 43 T^{2} \)
$47$ \( 1 - 11 T + 47 T^{2} \)
$53$ \( 1 + 9 T + 53 T^{2} \)
$59$ \( 1 + 4 T + 59 T^{2} \)
$61$ \( 1 + 4 T + 61 T^{2} \)
$67$ \( 1 + 12 T + 67 T^{2} \)
$71$ \( 1 - 2 T + 71 T^{2} \)
$73$ \( 1 + 10 T + 73 T^{2} \)
$79$ \( 1 + 12 T + 79 T^{2} \)
$83$ \( 1 - 12 T + 83 T^{2} \)
$89$ \( 1 + 14 T + 89 T^{2} \)
$97$ \( 1 - 8 T + 97 T^{2} \)
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