Properties

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

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: \(0\)
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

\(f(q)\) \(=\) \( q + 4q^{2} + 9q^{3} + 16q^{4} + 36q^{6} + 47q^{7} + 64q^{8} + 81q^{9} + O(q^{10}) \) \( q + 4q^{2} + 9q^{3} + 16q^{4} + 36q^{6} + 47q^{7} + 64q^{8} + 81q^{9} + 222q^{11} + 144q^{12} + 101q^{13} + 188q^{14} + 256q^{16} + 162q^{17} + 324q^{18} + 1685q^{19} + 423q^{21} + 888q^{22} + 306q^{23} + 576q^{24} + 404q^{26} + 729q^{27} + 752q^{28} + 7890q^{29} - 8593q^{31} + 1024q^{32} + 1998q^{33} + 648q^{34} + 1296q^{36} + 8642q^{37} + 6740q^{38} + 909q^{39} - 18168q^{41} + 1692q^{42} + 14351q^{43} + 3552q^{44} + 1224q^{46} - 1098q^{47} + 2304q^{48} - 14598q^{49} + 1458q^{51} + 1616q^{52} + 17916q^{53} + 2916q^{54} + 3008q^{56} + 15165q^{57} + 31560q^{58} + 17610q^{59} - 21853q^{61} - 34372q^{62} + 3807q^{63} + 4096q^{64} + 7992q^{66} + 107q^{67} + 2592q^{68} + 2754q^{69} - 40728q^{71} + 5184q^{72} + 34706q^{73} + 34568q^{74} + 26960q^{76} + 10434q^{77} + 3636q^{78} - 69160q^{79} + 6561q^{81} - 72672q^{82} - 108534q^{83} + 6768q^{84} + 57404q^{86} + 71010q^{87} + 14208q^{88} + 35040q^{89} + 4747q^{91} + 4896q^{92} - 77337q^{93} - 4392q^{94} + 9216q^{96} - 823q^{97} - 58392q^{98} + 17982q^{99} + O(q^{100}) \)

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 47.0000 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.m yes 1
3.b odd 2 1 450.6.a.i 1
5.b even 2 1 150.6.a.a 1
5.c odd 4 2 150.6.c.g 2
15.d odd 2 1 450.6.a.p 1
15.e even 4 2 450.6.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.a 1 5.b even 2 1
150.6.a.m yes 1 1.a even 1 1 trivial
150.6.c.g 2 5.c odd 4 2
450.6.a.i 1 3.b odd 2 1
450.6.a.p 1 15.d odd 2 1
450.6.c.e 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 47 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(150))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( -9 + T \)
$5$ \( T \)
$7$ \( -47 + T \)
$11$ \( -222 + T \)
$13$ \( -101 + T \)
$17$ \( -162 + T \)
$19$ \( -1685 + T \)
$23$ \( -306 + T \)
$29$ \( -7890 + T \)
$31$ \( 8593 + T \)
$37$ \( -8642 + T \)
$41$ \( 18168 + T \)
$43$ \( -14351 + T \)
$47$ \( 1098 + T \)
$53$ \( -17916 + T \)
$59$ \( -17610 + T \)
$61$ \( 21853 + T \)
$67$ \( -107 + T \)
$71$ \( 40728 + T \)
$73$ \( -34706 + T \)
$79$ \( 69160 + T \)
$83$ \( 108534 + T \)
$89$ \( -35040 + T \)
$97$ \( 823 + T \)
show more
show less