Properties

Label 1470.2.a.g
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} + 2q^{13} + q^{15} + q^{16} + 6q^{17} - q^{18} + q^{20} + 4q^{22} - 8q^{23} - q^{24} + q^{25} - 2q^{26} + q^{27} + 10q^{29} - q^{30} + 8q^{31} - q^{32} - 4q^{33} - 6q^{34} + q^{36} + 2q^{37} + 2q^{39} - q^{40} + 2q^{41} + 8q^{43} - 4q^{44} + q^{45} + 8q^{46} - 4q^{47} + q^{48} - q^{50} + 6q^{51} + 2q^{52} + 10q^{53} - q^{54} - 4q^{55} - 10q^{58} - 4q^{59} + q^{60} + 6q^{61} - 8q^{62} + q^{64} + 2q^{65} + 4q^{66} + 6q^{68} - 8q^{69} - 12q^{71} - q^{72} + 6q^{73} - 2q^{74} + q^{75} - 2q^{78} - 8q^{79} + q^{80} + q^{81} - 2q^{82} + 4q^{83} + 6q^{85} - 8q^{86} + 10q^{87} + 4q^{88} - 14q^{89} - q^{90} - 8q^{92} + 8q^{93} + 4q^{94} - q^{96} - 2q^{97} - 4q^{99} + O(q^{100}) \)

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.g 1
3.b odd 2 1 4410.2.a.bc 1
5.b even 2 1 7350.2.a.bo 1
7.b odd 2 1 210.2.a.a 1
7.c even 3 2 1470.2.i.n 2
7.d odd 6 2 1470.2.i.t 2
21.c even 2 1 630.2.a.i 1
28.d even 2 1 1680.2.a.o 1
35.c odd 2 1 1050.2.a.q 1
35.f even 4 2 1050.2.g.f 2
56.e even 2 1 6720.2.a.z 1
56.h odd 2 1 6720.2.a.cg 1
84.h odd 2 1 5040.2.a.bg 1
105.g even 2 1 3150.2.a.t 1
105.k odd 4 2 3150.2.g.t 2
140.c even 2 1 8400.2.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.a 1 7.b odd 2 1
630.2.a.i 1 21.c even 2 1
1050.2.a.q 1 35.c odd 2 1
1050.2.g.f 2 35.f even 4 2
1470.2.a.g 1 1.a even 1 1 trivial
1470.2.i.n 2 7.c even 3 2
1470.2.i.t 2 7.d odd 6 2
1680.2.a.o 1 28.d even 2 1
3150.2.a.t 1 105.g even 2 1
3150.2.g.t 2 105.k odd 4 2
4410.2.a.bc 1 3.b odd 2 1
5040.2.a.bg 1 84.h odd 2 1
6720.2.a.z 1 56.e even 2 1
6720.2.a.cg 1 56.h odd 2 1
7350.2.a.bo 1 5.b even 2 1
8400.2.a.m 1 140.c 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1470))\):

\( T_{11} + 4 \)
\( T_{13} - 2 \)
\( T_{17} - 6 \)
\( T_{19} \)
\( T_{31} - 8 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 - T \)
$5$ \( 1 - T \)
$7$ 1
$11$ \( 1 + 4 T + 11 T^{2} \)
$13$ \( 1 - 2 T + 13 T^{2} \)
$17$ \( 1 - 6 T + 17 T^{2} \)
$19$ \( 1 + 19 T^{2} \)
$23$ \( 1 + 8 T + 23 T^{2} \)
$29$ \( 1 - 10 T + 29 T^{2} \)
$31$ \( 1 - 8 T + 31 T^{2} \)
$37$ \( 1 - 2 T + 37 T^{2} \)
$41$ \( 1 - 2 T + 41 T^{2} \)
$43$ \( 1 - 8 T + 43 T^{2} \)
$47$ \( 1 + 4 T + 47 T^{2} \)
$53$ \( 1 - 10 T + 53 T^{2} \)
$59$ \( 1 + 4 T + 59 T^{2} \)
$61$ \( 1 - 6 T + 61 T^{2} \)
$67$ \( 1 + 67 T^{2} \)
$71$ \( 1 + 12 T + 71 T^{2} \)
$73$ \( 1 - 6 T + 73 T^{2} \)
$79$ \( 1 + 8 T + 79 T^{2} \)
$83$ \( 1 - 4 T + 83 T^{2} \)
$89$ \( 1 + 14 T + 89 T^{2} \)
$97$ \( 1 + 2 T + 97 T^{2} \)
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