Properties

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

Related objects

Downloads

Learn more

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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} + 9q^{3} + 16q^{4} - 36q^{6} - q^{7} - 64q^{8} + 81q^{9} + O(q^{10}) \) \( q - 4q^{2} + 9q^{3} + 16q^{4} - 36q^{6} - q^{7} - 64q^{8} + 81q^{9} - 210q^{11} + 144q^{12} - 667q^{13} + 4q^{14} + 256q^{16} + 114q^{17} - 324q^{18} + 581q^{19} - 9q^{21} + 840q^{22} - 4350q^{23} - 576q^{24} + 2668q^{26} + 729q^{27} - 16q^{28} - 126q^{29} + 7583q^{31} - 1024q^{32} - 1890q^{33} - 456q^{34} + 1296q^{36} - 3742q^{37} - 2324q^{38} - 6003q^{39} - 2856q^{41} + 36q^{42} - 18241q^{43} - 3360q^{44} + 17400q^{46} - 23370q^{47} + 2304q^{48} - 16806q^{49} + 1026q^{51} - 10672q^{52} - 21684q^{53} - 2916q^{54} + 64q^{56} + 5229q^{57} + 504q^{58} - 32310q^{59} - 7165q^{61} - 30332q^{62} - 81q^{63} + 4096q^{64} + 7560q^{66} + 59579q^{67} + 1824q^{68} - 39150q^{69} - 43080q^{71} - 5184q^{72} - 28942q^{73} + 14968q^{74} + 9296q^{76} + 210q^{77} + 24012q^{78} + 27608q^{79} + 6561q^{81} + 11424q^{82} - 1782q^{83} - 144q^{84} + 72964q^{86} - 1134q^{87} + 13440q^{88} + 50208q^{89} + 667q^{91} - 69600q^{92} + 68247q^{93} + 93480q^{94} - 9216q^{96} + 142793q^{97} + 67224q^{98} - 17010q^{99} + O(q^{100}) \)

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 −1.00000 −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.e 1
3.b odd 2 1 450.6.a.r 1
5.b even 2 1 150.6.a.k yes 1
5.c odd 4 2 150.6.c.a 2
15.d odd 2 1 450.6.a.g 1
15.e even 4 2 450.6.c.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.e 1 1.a even 1 1 trivial
150.6.a.k yes 1 5.b even 2 1
150.6.c.a 2 5.c odd 4 2
450.6.a.g 1 15.d odd 2 1
450.6.a.r 1 3.b odd 2 1
450.6.c.k 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 1 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(150))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( -9 + T \)
$5$ \( T \)
$7$ \( 1 + T \)
$11$ \( 210 + T \)
$13$ \( 667 + T \)
$17$ \( -114 + T \)
$19$ \( -581 + T \)
$23$ \( 4350 + T \)
$29$ \( 126 + T \)
$31$ \( -7583 + T \)
$37$ \( 3742 + T \)
$41$ \( 2856 + T \)
$43$ \( 18241 + T \)
$47$ \( 23370 + T \)
$53$ \( 21684 + T \)
$59$ \( 32310 + T \)
$61$ \( 7165 + T \)
$67$ \( -59579 + T \)
$71$ \( 43080 + T \)
$73$ \( 28942 + T \)
$79$ \( -27608 + T \)
$83$ \( 1782 + T \)
$89$ \( -50208 + T \)
$97$ \( -142793 + T \)
show more
show less