Properties

Label 150.6.a.c
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} - 9q^{3} + 16q^{4} + 36q^{6} + 233q^{7} - 64q^{8} + 81q^{9} + O(q^{10}) \) \( q - 4q^{2} - 9q^{3} + 16q^{4} + 36q^{6} + 233q^{7} - 64q^{8} + 81q^{9} - 498q^{11} - 144q^{12} + 809q^{13} - 932q^{14} + 256q^{16} - 1002q^{17} - 324q^{18} - 1705q^{19} - 2097q^{21} + 1992q^{22} + 1554q^{23} + 576q^{24} - 3236q^{26} - 729q^{27} + 3728q^{28} + 7830q^{29} + 977q^{31} - 1024q^{32} + 4482q^{33} + 4008q^{34} + 1296q^{36} - 4822q^{37} + 6820q^{38} - 7281q^{39} - 8148q^{41} + 8388q^{42} + 19469q^{43} - 7968q^{44} - 6216q^{46} + 8418q^{47} - 2304q^{48} + 37482q^{49} + 9018q^{51} + 12944q^{52} + 17664q^{53} + 2916q^{54} - 14912q^{56} + 15345q^{57} - 31320q^{58} + 35910q^{59} + 3527q^{61} - 3908q^{62} + 18873q^{63} + 4096q^{64} - 17928q^{66} + 57473q^{67} - 16032q^{68} - 13986q^{69} - 7548q^{71} - 5184q^{72} - 646q^{73} + 19288q^{74} - 27280q^{76} - 116034q^{77} + 29124q^{78} - 22720q^{79} + 6561q^{81} + 32592q^{82} + 11574q^{83} - 33552q^{84} - 77876q^{86} - 70470q^{87} + 31872q^{88} - 78960q^{89} + 188497q^{91} + 24864q^{92} - 8793q^{93} - 33672q^{94} + 9216q^{96} + 54593q^{97} - 149928q^{98} - 40338q^{99} + O(q^{100}) \)

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 233.000 −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.c 1
3.b odd 2 1 450.6.a.x 1
5.b even 2 1 150.6.a.l yes 1
5.c odd 4 2 150.6.c.e 2
15.d odd 2 1 450.6.a.a 1
15.e even 4 2 450.6.c.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.c 1 1.a even 1 1 trivial
150.6.a.l yes 1 5.b even 2 1
150.6.c.e 2 5.c odd 4 2
450.6.a.a 1 15.d odd 2 1
450.6.a.x 1 3.b odd 2 1
450.6.c.n 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 233 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(150))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( 9 + T \)
$5$ \( T \)
$7$ \( -233 + T \)
$11$ \( 498 + T \)
$13$ \( -809 + T \)
$17$ \( 1002 + T \)
$19$ \( 1705 + T \)
$23$ \( -1554 + T \)
$29$ \( -7830 + T \)
$31$ \( -977 + T \)
$37$ \( 4822 + T \)
$41$ \( 8148 + T \)
$43$ \( -19469 + T \)
$47$ \( -8418 + T \)
$53$ \( -17664 + T \)
$59$ \( -35910 + T \)
$61$ \( -3527 + T \)
$67$ \( -57473 + T \)
$71$ \( 7548 + T \)
$73$ \( 646 + T \)
$79$ \( 22720 + T \)
$83$ \( -11574 + T \)
$89$ \( 78960 + T \)
$97$ \( -54593 + T \)
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