Properties

Label 9.10.a.b
Level 9
Weight 10
Character orbit 9.a
Self dual yes
Analytic conductor 4.635
Analytic rank 1
Dimension 1
CM discriminant -3
Inner twists 2

Related objects

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Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.63532252547\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

\(f(q)\) \(=\) \( q - 512q^{4} - 12580q^{7} + O(q^{10}) \) \( q - 512q^{4} - 12580q^{7} + 118370q^{13} + 262144q^{16} - 976696q^{19} - 1953125q^{25} + 6440960q^{28} + 1691228q^{31} - 15384490q^{37} - 16577080q^{43} + 117902793q^{49} - 60605440q^{52} - 117903058q^{61} - 134217728q^{64} + 112542320q^{67} + 296368310q^{73} + 500068352q^{76} - 616732324q^{79} - 1489094600q^{91} + 1288928270q^{97} + O(q^{100}) \)

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
0 0 −512.000 0 0 −12580.0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.10.a.b 1
3.b odd 2 1 CM 9.10.a.b 1
4.b odd 2 1 144.10.a.h 1
5.b even 2 1 225.10.a.d 1
5.c odd 4 2 225.10.b.f 2
9.c even 3 2 81.10.c.c 2
9.d odd 6 2 81.10.c.c 2
12.b even 2 1 144.10.a.h 1
15.d odd 2 1 225.10.a.d 1
15.e even 4 2 225.10.b.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.10.a.b 1 1.a even 1 1 trivial
9.10.a.b 1 3.b odd 2 1 CM
81.10.c.c 2 9.c even 3 2
81.10.c.c 2 9.d odd 6 2
144.10.a.h 1 4.b odd 2 1
144.10.a.h 1 12.b even 2 1
225.10.a.d 1 5.b even 2 1
225.10.a.d 1 15.d odd 2 1
225.10.b.f 2 5.c odd 4 2
225.10.b.f 2 15.e even 4 2

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 512 T^{2} \)
$3$ \( \)
$5$ \( 1 + 1953125 T^{2} \)
$7$ \( 1 + 12580 T + 40353607 T^{2} \)
$11$ \( 1 + 2357947691 T^{2} \)
$13$ \( 1 - 118370 T + 10604499373 T^{2} \)
$17$ \( 1 + 118587876497 T^{2} \)
$19$ \( 1 + 976696 T + 322687697779 T^{2} \)
$23$ \( 1 + 1801152661463 T^{2} \)
$29$ \( 1 + 14507145975869 T^{2} \)
$31$ \( 1 - 1691228 T + 26439622160671 T^{2} \)
$37$ \( 1 + 15384490 T + 129961739795077 T^{2} \)
$41$ \( 1 + 327381934393961 T^{2} \)
$43$ \( 1 + 16577080 T + 502592611936843 T^{2} \)
$47$ \( 1 + 1119130473102767 T^{2} \)
$53$ \( 1 + 3299763591802133 T^{2} \)
$59$ \( 1 + 8662995818654939 T^{2} \)
$61$ \( 1 + 117903058 T + 11694146092834141 T^{2} \)
$67$ \( 1 - 112542320 T + 27206534396294947 T^{2} \)
$71$ \( 1 + 45848500718449031 T^{2} \)
$73$ \( 1 - 296368310 T + 58871586708267913 T^{2} \)
$79$ \( 1 + 616732324 T + 119851595982618319 T^{2} \)
$83$ \( 1 + 186940255267540403 T^{2} \)
$89$ \( 1 + 350356403707485209 T^{2} \)
$97$ \( 1 - 1288928270 T + 760231058654565217 T^{2} \)
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