Properties

Label 150.6.a.h
Level $150$
Weight $6$
Character orbit 150.a
Self dual yes
Analytic conductor $24.058$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.0575729719\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{2} - 9q^{3} + 16q^{4} - 36q^{6} - 164q^{7} + 64q^{8} + 81q^{9} + O(q^{10}) \) \( q + 4q^{2} - 9q^{3} + 16q^{4} - 36q^{6} - 164q^{7} + 64q^{8} + 81q^{9} + 720q^{11} - 144q^{12} - 698q^{13} - 656q^{14} + 256q^{16} + 2226q^{17} + 324q^{18} + 356q^{19} + 1476q^{21} + 2880q^{22} + 1800q^{23} - 576q^{24} - 2792q^{26} - 729q^{27} - 2624q^{28} + 714q^{29} + 848q^{31} + 1024q^{32} - 6480q^{33} + 8904q^{34} + 1296q^{36} + 11302q^{37} + 1424q^{38} + 6282q^{39} + 9354q^{41} + 5904q^{42} + 5956q^{43} + 11520q^{44} + 7200q^{46} + 11160q^{47} - 2304q^{48} + 10089q^{49} - 20034q^{51} - 11168q^{52} - 14106q^{53} - 2916q^{54} - 10496q^{56} - 3204q^{57} + 2856q^{58} + 7920q^{59} - 13450q^{61} + 3392q^{62} - 13284q^{63} + 4096q^{64} - 25920q^{66} + 65476q^{67} + 35616q^{68} - 16200q^{69} + 34560q^{71} + 5184q^{72} - 86258q^{73} + 45208q^{74} + 5696q^{76} - 118080q^{77} + 25128q^{78} - 108832q^{79} + 6561q^{81} + 37416q^{82} - 10668q^{83} + 23616q^{84} + 23824q^{86} - 6426q^{87} + 46080q^{88} + 10818q^{89} + 114472q^{91} + 28800q^{92} - 7632q^{93} + 44640q^{94} - 9216q^{96} - 4418q^{97} + 40356q^{98} + 58320q^{99} + O(q^{100}) \)

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
4.00000 −9.00000 16.0000 0 −36.0000 −164.000 64.0000 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)

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 150.6.a.h 1
3.b odd 2 1 450.6.a.b 1
5.b even 2 1 30.6.a.a 1
5.c odd 4 2 150.6.c.d 2
15.d odd 2 1 90.6.a.g 1
15.e even 4 2 450.6.c.b 2
20.d odd 2 1 240.6.a.a 1
40.e odd 2 1 960.6.a.u 1
40.f even 2 1 960.6.a.n 1
60.h even 2 1 720.6.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.a.a 1 5.b even 2 1
90.6.a.g 1 15.d odd 2 1
150.6.a.h 1 1.a even 1 1 trivial
150.6.c.d 2 5.c odd 4 2
240.6.a.a 1 20.d odd 2 1
450.6.a.b 1 3.b odd 2 1
450.6.c.b 2 15.e even 4 2
720.6.a.m 1 60.h even 2 1
960.6.a.n 1 40.f even 2 1
960.6.a.u 1 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 164 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(150))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( 9 + T \)
$5$ \( T \)
$7$ \( 164 + T \)
$11$ \( -720 + T \)
$13$ \( 698 + T \)
$17$ \( -2226 + T \)
$19$ \( -356 + T \)
$23$ \( -1800 + T \)
$29$ \( -714 + T \)
$31$ \( -848 + T \)
$37$ \( -11302 + T \)
$41$ \( -9354 + T \)
$43$ \( -5956 + T \)
$47$ \( -11160 + T \)
$53$ \( 14106 + T \)
$59$ \( -7920 + T \)
$61$ \( 13450 + T \)
$67$ \( -65476 + T \)
$71$ \( -34560 + T \)
$73$ \( 86258 + T \)
$79$ \( 108832 + T \)
$83$ \( 10668 + T \)
$89$ \( -10818 + T \)
$97$ \( 4418 + T \)
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