Properties

Label 150.16.a.f
Level $150$
Weight $16$
Character orbit 150.a
Self dual yes
Analytic conductor $214.040$
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: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(214.040257650\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 128 q^{2} + 2187 q^{3} + 16384 q^{4} - 279936 q^{6} + 3034528 q^{7} - 2097152 q^{8} + 4782969 q^{9} + O(q^{10}) \) \( q - 128 q^{2} + 2187 q^{3} + 16384 q^{4} - 279936 q^{6} + 3034528 q^{7} - 2097152 q^{8} + 4782969 q^{9} - 103451700 q^{11} + 35831808 q^{12} + 104365834 q^{13} - 388419584 q^{14} + 268435456 q^{16} - 997689762 q^{17} - 612220032 q^{18} + 4934015444 q^{19} + 6636512736 q^{21} + 13241817600 q^{22} - 8324920200 q^{23} - 4586471424 q^{24} - 13358826752 q^{26} + 10460353203 q^{27} + 49717706752 q^{28} + 104128242846 q^{29} - 296696681512 q^{31} - 34359738368 q^{32} - 226248867900 q^{33} + 127704289536 q^{34} + 78364164096 q^{36} + 178337455666 q^{37} - 631553976832 q^{38} + 228248078958 q^{39} - 1790882416086 q^{41} - 849473630208 q^{42} + 2863459422772 q^{43} - 1694952652800 q^{44} + 1065589785600 q^{46} - 4332907521600 q^{47} + 587068342272 q^{48} + 4460798672841 q^{49} - 2181947509494 q^{51} + 1709929824256 q^{52} - 9732317104422 q^{53} - 1338925209984 q^{54} - 6363866464256 q^{56} + 10790691776028 q^{57} - 13328415084288 q^{58} - 13514837176500 q^{59} + 5352663511190 q^{61} + 37977175233536 q^{62} + 14514053353632 q^{63} + 4398046511104 q^{64} + 28959855091200 q^{66} + 53233909720108 q^{67} - 16346149060608 q^{68} - 18206600477400 q^{69} - 20229661643400 q^{71} - 10030613004288 q^{72} - 26264166466106 q^{73} - 22827194325248 q^{74} + 80838909034496 q^{76} - 313927080297600 q^{77} - 29215754106624 q^{78} - 339031361615128 q^{79} + 22876792454961 q^{81} + 229232949259008 q^{82} - 131684771045076 q^{83} + 108732624666624 q^{84} - 366522806114816 q^{86} + 227728467104202 q^{87} + 216953939558400 q^{88} - 39352148322678 q^{89} + 316701045516352 q^{91} - 136395492556800 q^{92} - 648875642466744 q^{93} + 554612162764800 q^{94} - 75144747810816 q^{96} - 1128750908801474 q^{97} - 570982230123648 q^{98} - 494806274097300 q^{99} + O(q^{100}) \)

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
−128.000 2187.00 16384.0 0 −279936. 3.03453e6 −2.09715e6 4.78297e6 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 150.16.a.f 1
5.b even 2 1 6.16.a.b 1
5.c odd 4 2 150.16.c.a 2
15.d odd 2 1 18.16.a.b 1
20.d odd 2 1 48.16.a.d 1
60.h even 2 1 144.16.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.16.a.b 1 5.b even 2 1
18.16.a.b 1 15.d odd 2 1
48.16.a.d 1 20.d odd 2 1
144.16.a.j 1 60.h even 2 1
150.16.a.f 1 1.a even 1 1 trivial
150.16.c.a 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} - 3034528 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(150))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 128 + T \)
$3$ \( -2187 + T \)
$5$ \( T \)
$7$ \( -3034528 + T \)
$11$ \( 103451700 + T \)
$13$ \( -104365834 + T \)
$17$ \( 997689762 + T \)
$19$ \( -4934015444 + T \)
$23$ \( 8324920200 + T \)
$29$ \( -104128242846 + T \)
$31$ \( 296696681512 + T \)
$37$ \( -178337455666 + T \)
$41$ \( 1790882416086 + T \)
$43$ \( -2863459422772 + T \)
$47$ \( 4332907521600 + T \)
$53$ \( 9732317104422 + T \)
$59$ \( 13514837176500 + T \)
$61$ \( -5352663511190 + T \)
$67$ \( -53233909720108 + T \)
$71$ \( 20229661643400 + T \)
$73$ \( 26264166466106 + T \)
$79$ \( 339031361615128 + T \)
$83$ \( 131684771045076 + T \)
$89$ \( 39352148322678 + T \)
$97$ \( 1128750908801474 + T \)
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