Properties

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

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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: \(1\)
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

\(f(q)\) \(=\) \( q - 4q^{2} + 9q^{3} + 16q^{4} - 36q^{6} + 4q^{7} - 64q^{8} + 81q^{9} + O(q^{10}) \) \( q - 4q^{2} + 9q^{3} + 16q^{4} - 36q^{6} + 4q^{7} - 64q^{8} + 81q^{9} - 500q^{11} + 144q^{12} + 288q^{13} - 16q^{14} + 256q^{16} - 1516q^{17} - 324q^{18} - 1344q^{19} + 36q^{21} + 2000q^{22} + 4100q^{23} - 576q^{24} - 1152q^{26} + 729q^{27} + 64q^{28} - 2646q^{29} - 5612q^{31} - 1024q^{32} - 4500q^{33} + 6064q^{34} + 1296q^{36} + 7288q^{37} + 5376q^{38} + 2592q^{39} - 18986q^{41} - 144q^{42} + 2404q^{43} - 8000q^{44} - 16400q^{46} - 8900q^{47} + 2304q^{48} - 16791q^{49} - 13644q^{51} + 4608q^{52} - 39804q^{53} - 2916q^{54} - 256q^{56} - 12096q^{57} + 10584q^{58} - 28300q^{59} + 18290q^{61} + 22448q^{62} + 324q^{63} + 4096q^{64} + 18000q^{66} - 65956q^{67} - 24256q^{68} + 36900q^{69} - 28800q^{71} - 5184q^{72} + 30808q^{73} - 29152q^{74} - 21504q^{76} - 2000q^{77} - 10368q^{78} + 60228q^{79} + 6561q^{81} + 75944q^{82} + 2468q^{83} + 576q^{84} - 9616q^{86} - 23814q^{87} + 32000q^{88} + 22678q^{89} + 1152q^{91} + 65600q^{92} - 50508q^{93} + 35600q^{94} - 9216q^{96} + 36968q^{97} + 67164q^{98} - 40500q^{99} + O(q^{100}) \)

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 4.00000 −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.f 1
3.b odd 2 1 450.6.a.s 1
5.b even 2 1 150.6.a.j 1
5.c odd 4 2 30.6.c.a 2
15.d odd 2 1 450.6.a.f 1
15.e even 4 2 90.6.c.b 2
20.e even 4 2 240.6.f.a 2
60.l odd 4 2 720.6.f.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.a 2 5.c odd 4 2
90.6.c.b 2 15.e even 4 2
150.6.a.f 1 1.a even 1 1 trivial
150.6.a.j 1 5.b even 2 1
240.6.f.a 2 20.e even 4 2
450.6.a.f 1 15.d odd 2 1
450.6.a.s 1 3.b odd 2 1
720.6.f.g 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 4 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(150))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( -9 + T \)
$5$ \( T \)
$7$ \( -4 + T \)
$11$ \( 500 + T \)
$13$ \( -288 + T \)
$17$ \( 1516 + T \)
$19$ \( 1344 + T \)
$23$ \( -4100 + T \)
$29$ \( 2646 + T \)
$31$ \( 5612 + T \)
$37$ \( -7288 + T \)
$41$ \( 18986 + T \)
$43$ \( -2404 + T \)
$47$ \( 8900 + T \)
$53$ \( 39804 + T \)
$59$ \( 28300 + T \)
$61$ \( -18290 + T \)
$67$ \( 65956 + T \)
$71$ \( 28800 + T \)
$73$ \( -30808 + T \)
$79$ \( -60228 + T \)
$83$ \( -2468 + T \)
$89$ \( -22678 + T \)
$97$ \( -36968 + T \)
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