Properties

Label 1470.2.a.s
Level $1470$
Weight $2$
Character orbit 1470.a
Self dual yes
Analytic conductor $11.738$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} + ( -2 + 3 \beta ) q^{11} - q^{12} + 4 \beta q^{13} - q^{15} + q^{16} + ( 4 - \beta ) q^{17} - q^{18} + ( -4 - 2 \beta ) q^{19} + q^{20} + ( 2 - 3 \beta ) q^{22} + ( 6 - 2 \beta ) q^{23} + q^{24} + q^{25} -4 \beta q^{26} - q^{27} + ( 4 - \beta ) q^{29} + q^{30} + ( -6 - 3 \beta ) q^{31} - q^{32} + ( 2 - 3 \beta ) q^{33} + ( -4 + \beta ) q^{34} + q^{36} + ( 4 - 3 \beta ) q^{37} + ( 4 + 2 \beta ) q^{38} -4 \beta q^{39} - q^{40} + ( 2 + 6 \beta ) q^{41} + ( 2 + 5 \beta ) q^{43} + ( -2 + 3 \beta ) q^{44} + q^{45} + ( -6 + 2 \beta ) q^{46} + ( 2 - 3 \beta ) q^{47} - q^{48} - q^{50} + ( -4 + \beta ) q^{51} + 4 \beta q^{52} + ( 6 - 4 \beta ) q^{53} + q^{54} + ( -2 + 3 \beta ) q^{55} + ( 4 + 2 \beta ) q^{57} + ( -4 + \beta ) q^{58} + ( -6 + 2 \beta ) q^{59} - q^{60} + ( -6 - 4 \beta ) q^{61} + ( 6 + 3 \beta ) q^{62} + q^{64} + 4 \beta q^{65} + ( -2 + 3 \beta ) q^{66} + ( 2 + 5 \beta ) q^{67} + ( 4 - \beta ) q^{68} + ( -6 + 2 \beta ) q^{69} + ( 4 - 6 \beta ) q^{71} - q^{72} -2 q^{73} + ( -4 + 3 \beta ) q^{74} - q^{75} + ( -4 - 2 \beta ) q^{76} + 4 \beta q^{78} + 8 q^{79} + q^{80} + q^{81} + ( -2 - 6 \beta ) q^{82} + ( 8 - 2 \beta ) q^{83} + ( 4 - \beta ) q^{85} + ( -2 - 5 \beta ) q^{86} + ( -4 + \beta ) q^{87} + ( 2 - 3 \beta ) q^{88} + ( 2 - 2 \beta ) q^{89} - q^{90} + ( 6 - 2 \beta ) q^{92} + ( 6 + 3 \beta ) q^{93} + ( -2 + 3 \beta ) q^{94} + ( -4 - 2 \beta ) q^{95} + q^{96} + ( 2 - 6 \beta ) q^{97} + ( -2 + 3 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - 2q^{8} + 2q^{9} - 2q^{10} - 4q^{11} - 2q^{12} - 2q^{15} + 2q^{16} + 8q^{17} - 2q^{18} - 8q^{19} + 2q^{20} + 4q^{22} + 12q^{23} + 2q^{24} + 2q^{25} - 2q^{27} + 8q^{29} + 2q^{30} - 12q^{31} - 2q^{32} + 4q^{33} - 8q^{34} + 2q^{36} + 8q^{37} + 8q^{38} - 2q^{40} + 4q^{41} + 4q^{43} - 4q^{44} + 2q^{45} - 12q^{46} + 4q^{47} - 2q^{48} - 2q^{50} - 8q^{51} + 12q^{53} + 2q^{54} - 4q^{55} + 8q^{57} - 8q^{58} - 12q^{59} - 2q^{60} - 12q^{61} + 12q^{62} + 2q^{64} - 4q^{66} + 4q^{67} + 8q^{68} - 12q^{69} + 8q^{71} - 2q^{72} - 4q^{73} - 8q^{74} - 2q^{75} - 8q^{76} + 16q^{79} + 2q^{80} + 2q^{81} - 4q^{82} + 16q^{83} + 8q^{85} - 4q^{86} - 8q^{87} + 4q^{88} + 4q^{89} - 2q^{90} + 12q^{92} + 12q^{93} - 4q^{94} - 8q^{95} + 2q^{96} + 4q^{97} - 4q^{99} + O(q^{100}) \)

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
−1.41421
1.41421
−1.00000 −1.00000 1.00000 1.00000 1.00000 0 −1.00000 1.00000 −1.00000
1.2 −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.s 2
3.b odd 2 1 4410.2.a.bw 2
5.b even 2 1 7350.2.a.dl 2
7.b odd 2 1 1470.2.a.t yes 2
7.c even 3 2 1470.2.i.x 4
7.d odd 6 2 1470.2.i.w 4
21.c even 2 1 4410.2.a.bz 2
35.c odd 2 1 7350.2.a.dh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1470.2.a.s 2 1.a even 1 1 trivial
1470.2.a.t yes 2 7.b odd 2 1
1470.2.i.w 4 7.d odd 6 2
1470.2.i.x 4 7.c even 3 2
4410.2.a.bw 2 3.b odd 2 1
4410.2.a.bz 2 21.c even 2 1
7350.2.a.dh 2 35.c odd 2 1
7350.2.a.dl 2 5.b even 2 1

Hecke kernels

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

\( T_{11}^{2} + 4 T_{11} - 14 \)
\( T_{13}^{2} - 32 \)
\( T_{17}^{2} - 8 T_{17} + 14 \)
\( T_{19}^{2} + 8 T_{19} + 8 \)
\( T_{31}^{2} + 12 T_{31} + 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -14 + 4 T + T^{2} \)
$13$ \( -32 + T^{2} \)
$17$ \( 14 - 8 T + T^{2} \)
$19$ \( 8 + 8 T + T^{2} \)
$23$ \( 28 - 12 T + T^{2} \)
$29$ \( 14 - 8 T + T^{2} \)
$31$ \( 18 + 12 T + T^{2} \)
$37$ \( -2 - 8 T + T^{2} \)
$41$ \( -68 - 4 T + T^{2} \)
$43$ \( -46 - 4 T + T^{2} \)
$47$ \( -14 - 4 T + T^{2} \)
$53$ \( 4 - 12 T + T^{2} \)
$59$ \( 28 + 12 T + T^{2} \)
$61$ \( 4 + 12 T + T^{2} \)
$67$ \( -46 - 4 T + T^{2} \)
$71$ \( -56 - 8 T + T^{2} \)
$73$ \( ( 2 + T )^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 56 - 16 T + T^{2} \)
$89$ \( -4 - 4 T + T^{2} \)
$97$ \( -68 - 4 T + T^{2} \)
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