Properties

Label 150.6.a.g
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

\(f(q)\) \(=\) \( q - 4q^{2} + 9q^{3} + 16q^{4} - 36q^{6} + 79q^{7} - 64q^{8} + 81q^{9} + O(q^{10}) \) \( q - 4q^{2} + 9q^{3} + 16q^{4} - 36q^{6} + 79q^{7} - 64q^{8} + 81q^{9} + 150q^{11} + 144q^{12} - 137q^{13} - 316q^{14} + 256q^{16} + 2034q^{17} - 324q^{18} - 1969q^{19} + 711q^{21} - 600q^{22} + 1350q^{23} - 576q^{24} + 548q^{26} + 729q^{27} + 1264q^{28} - 2946q^{29} + 713q^{31} - 1024q^{32} + 1350q^{33} - 8136q^{34} + 1296q^{36} + 3238q^{37} + 7876q^{38} - 1233q^{39} + 6564q^{41} - 2844q^{42} + 19579q^{43} + 2400q^{44} - 5400q^{46} + 21150q^{47} + 2304q^{48} - 10566q^{49} + 18306q^{51} - 2192q^{52} + 25896q^{53} - 2916q^{54} - 5056q^{56} - 17721q^{57} + 11784q^{58} + 25350q^{59} + 50615q^{61} - 2852q^{62} + 6399q^{63} + 4096q^{64} - 5400q^{66} + 22519q^{67} + 32544q^{68} + 12150q^{69} + 33900q^{71} - 5184q^{72} - 82442q^{73} - 12952q^{74} - 31504q^{76} + 11850q^{77} + 4932q^{78} - 81472q^{79} + 6561q^{81} - 26256q^{82} - 25782q^{83} + 11376q^{84} - 78316q^{86} - 26514q^{87} - 9600q^{88} + 103728q^{89} - 10823q^{91} + 21600q^{92} + 6417q^{93} - 84600q^{94} - 9216q^{96} + 57343q^{97} + 42264q^{98} + 12150q^{99} + O(q^{100}) \)

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 79.0000 −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.g 1
3.b odd 2 1 450.6.a.t 1
5.b even 2 1 150.6.a.i yes 1
5.c odd 4 2 150.6.c.c 2
15.d odd 2 1 450.6.a.e 1
15.e even 4 2 450.6.c.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.6.a.g 1 1.a even 1 1 trivial
150.6.a.i yes 1 5.b even 2 1
150.6.c.c 2 5.c odd 4 2
450.6.a.e 1 15.d odd 2 1
450.6.a.t 1 3.b odd 2 1
450.6.c.g 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 79 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(150))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4 + T \)
$3$ \( -9 + T \)
$5$ \( T \)
$7$ \( -79 + T \)
$11$ \( -150 + T \)
$13$ \( 137 + T \)
$17$ \( -2034 + T \)
$19$ \( 1969 + T \)
$23$ \( -1350 + T \)
$29$ \( 2946 + T \)
$31$ \( -713 + T \)
$37$ \( -3238 + T \)
$41$ \( -6564 + T \)
$43$ \( -19579 + T \)
$47$ \( -21150 + T \)
$53$ \( -25896 + T \)
$59$ \( -25350 + T \)
$61$ \( -50615 + T \)
$67$ \( -22519 + T \)
$71$ \( -33900 + T \)
$73$ \( 82442 + T \)
$79$ \( 81472 + T \)
$83$ \( 25782 + T \)
$89$ \( -103728 + T \)
$97$ \( -57343 + T \)
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