Properties

Label 1470.4.a.n
Level 1470
Weight 4
Character orbit 1470.a
Self dual yes
Analytic conductor 86.733
Analytic rank 1
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: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(86.7328077084\)
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

\(f(q)\) \(=\) \( q - 2q^{2} + 3q^{3} + 4q^{4} + 5q^{5} - 6q^{6} - 8q^{8} + 9q^{9} + O(q^{10}) \) \( q - 2q^{2} + 3q^{3} + 4q^{4} + 5q^{5} - 6q^{6} - 8q^{8} + 9q^{9} - 10q^{10} + 12q^{11} + 12q^{12} - 2q^{13} + 15q^{15} + 16q^{16} + 18q^{17} - 18q^{18} - 56q^{19} + 20q^{20} - 24q^{22} - 156q^{23} - 24q^{24} + 25q^{25} + 4q^{26} + 27q^{27} - 186q^{29} - 30q^{30} + 52q^{31} - 32q^{32} + 36q^{33} - 36q^{34} + 36q^{36} - 178q^{37} + 112q^{38} - 6q^{39} - 40q^{40} + 138q^{41} - 412q^{43} + 48q^{44} + 45q^{45} + 312q^{46} + 456q^{47} + 48q^{48} - 50q^{50} + 54q^{51} - 8q^{52} - 198q^{53} - 54q^{54} + 60q^{55} - 168q^{57} + 372q^{58} - 348q^{59} + 60q^{60} - 110q^{61} - 104q^{62} + 64q^{64} - 10q^{65} - 72q^{66} - 196q^{67} + 72q^{68} - 468q^{69} - 936q^{71} - 72q^{72} - 542q^{73} + 356q^{74} + 75q^{75} - 224q^{76} + 12q^{78} + 992q^{79} + 80q^{80} + 81q^{81} - 276q^{82} + 276q^{83} + 90q^{85} + 824q^{86} - 558q^{87} - 96q^{88} - 630q^{89} - 90q^{90} - 624q^{92} + 156q^{93} - 912q^{94} - 280q^{95} - 96q^{96} - 110q^{97} + 108q^{99} + O(q^{100}) \)

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
−2.00000 3.00000 4.00000 5.00000 −6.00000 0 −8.00000 9.00000 −10.0000
\(n\): 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 1470.4.a.n 1
7.b odd 2 1 210.4.a.a 1
21.c even 2 1 630.4.a.v 1
28.d even 2 1 1680.4.a.n 1
35.c odd 2 1 1050.4.a.t 1
35.f even 4 2 1050.4.g.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.4.a.a 1 7.b odd 2 1
630.4.a.v 1 21.c even 2 1
1050.4.a.t 1 35.c odd 2 1
1050.4.g.o 2 35.f even 4 2
1470.4.a.n 1 1.a even 1 1 trivial
1680.4.a.n 1 28.d 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1470))\):

\( T_{11} - 12 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T \)
$3$ \( 1 - 3 T \)
$5$ \( 1 - 5 T \)
$7$ 1
$11$ \( 1 - 12 T + 1331 T^{2} \)
$13$ \( 1 + 2 T + 2197 T^{2} \)
$17$ \( 1 - 18 T + 4913 T^{2} \)
$19$ \( 1 + 56 T + 6859 T^{2} \)
$23$ \( 1 + 156 T + 12167 T^{2} \)
$29$ \( 1 + 186 T + 24389 T^{2} \)
$31$ \( 1 - 52 T + 29791 T^{2} \)
$37$ \( 1 + 178 T + 50653 T^{2} \)
$41$ \( 1 - 138 T + 68921 T^{2} \)
$43$ \( 1 + 412 T + 79507 T^{2} \)
$47$ \( 1 - 456 T + 103823 T^{2} \)
$53$ \( 1 + 198 T + 148877 T^{2} \)
$59$ \( 1 + 348 T + 205379 T^{2} \)
$61$ \( 1 + 110 T + 226981 T^{2} \)
$67$ \( 1 + 196 T + 300763 T^{2} \)
$71$ \( 1 + 936 T + 357911 T^{2} \)
$73$ \( 1 + 542 T + 389017 T^{2} \)
$79$ \( 1 - 992 T + 493039 T^{2} \)
$83$ \( 1 - 276 T + 571787 T^{2} \)
$89$ \( 1 + 630 T + 704969 T^{2} \)
$97$ \( 1 + 110 T + 912673 T^{2} \)
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