Properties

Label 126.10.a.e
Level $126$
Weight $10$
Character orbit 126.a
Self dual yes
Analytic conductor $64.895$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 126.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.8945153566\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 16q^{2} + 256q^{4} - 560q^{5} - 2401q^{7} + 4096q^{8} + O(q^{10}) \) \( q + 16q^{2} + 256q^{4} - 560q^{5} - 2401q^{7} + 4096q^{8} - 8960q^{10} + 54152q^{11} - 113172q^{13} - 38416q^{14} + 65536q^{16} - 6262q^{17} + 257078q^{19} - 143360q^{20} + 866432q^{22} + 266000q^{23} - 1639525q^{25} - 1810752q^{26} - 614656q^{28} - 1574714q^{29} - 4637484q^{31} + 1048576q^{32} - 100192q^{34} + 1344560q^{35} - 11946238q^{37} + 4113248q^{38} - 2293760q^{40} - 21909126q^{41} + 27520592q^{43} + 13862912q^{44} + 4256000q^{46} - 52927836q^{47} + 5764801q^{49} - 26232400q^{50} - 28972032q^{52} - 16221222q^{53} - 30325120q^{55} - 9834496q^{56} - 25195424q^{58} + 140509618q^{59} - 202963560q^{61} - 74199744q^{62} + 16777216q^{64} + 63376320q^{65} + 153734572q^{67} - 1603072q^{68} + 21512960q^{70} - 279655936q^{71} - 404022830q^{73} - 191139808q^{74} + 65811968q^{76} - 130018952q^{77} - 130689816q^{79} - 36700160q^{80} - 350546016q^{82} - 420134014q^{83} + 3506720q^{85} + 440329472q^{86} + 221806592q^{88} + 469542390q^{89} + 271725972q^{91} + 68096000q^{92} - 846845376q^{94} - 143963680q^{95} - 872501690q^{97} + 92236816q^{98} + O(q^{100}) \)

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
16.0000 0 256.000 −560.000 0 −2401.00 4096.00 0 −8960.00
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)

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 126.10.a.e 1
3.b odd 2 1 14.10.a.a 1
12.b even 2 1 112.10.a.b 1
15.d odd 2 1 350.10.a.c 1
15.e even 4 2 350.10.c.b 2
21.c even 2 1 98.10.a.a 1
21.g even 6 2 98.10.c.e 2
21.h odd 6 2 98.10.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.10.a.a 1 3.b odd 2 1
98.10.a.a 1 21.c even 2 1
98.10.c.e 2 21.g even 6 2
98.10.c.f 2 21.h odd 6 2
112.10.a.b 1 12.b even 2 1
126.10.a.e 1 1.a even 1 1 trivial
350.10.a.c 1 15.d odd 2 1
350.10.c.b 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 560 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(126))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -16 + T \)
$3$ \( T \)
$5$ \( 560 + T \)
$7$ \( 2401 + T \)
$11$ \( -54152 + T \)
$13$ \( 113172 + T \)
$17$ \( 6262 + T \)
$19$ \( -257078 + T \)
$23$ \( -266000 + T \)
$29$ \( 1574714 + T \)
$31$ \( 4637484 + T \)
$37$ \( 11946238 + T \)
$41$ \( 21909126 + T \)
$43$ \( -27520592 + T \)
$47$ \( 52927836 + T \)
$53$ \( 16221222 + T \)
$59$ \( -140509618 + T \)
$61$ \( 202963560 + T \)
$67$ \( -153734572 + T \)
$71$ \( 279655936 + T \)
$73$ \( 404022830 + T \)
$79$ \( 130689816 + T \)
$83$ \( 420134014 + T \)
$89$ \( -469542390 + T \)
$97$ \( 872501690 + T \)
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