Properties

 Label 150.6.a.h 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

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: no (minimal twist has level 30) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{2} - 9q^{3} + 16q^{4} - 36q^{6} - 164q^{7} + 64q^{8} + 81q^{9} + O(q^{10})$$ $$q + 4q^{2} - 9q^{3} + 16q^{4} - 36q^{6} - 164q^{7} + 64q^{8} + 81q^{9} + 720q^{11} - 144q^{12} - 698q^{13} - 656q^{14} + 256q^{16} + 2226q^{17} + 324q^{18} + 356q^{19} + 1476q^{21} + 2880q^{22} + 1800q^{23} - 576q^{24} - 2792q^{26} - 729q^{27} - 2624q^{28} + 714q^{29} + 848q^{31} + 1024q^{32} - 6480q^{33} + 8904q^{34} + 1296q^{36} + 11302q^{37} + 1424q^{38} + 6282q^{39} + 9354q^{41} + 5904q^{42} + 5956q^{43} + 11520q^{44} + 7200q^{46} + 11160q^{47} - 2304q^{48} + 10089q^{49} - 20034q^{51} - 11168q^{52} - 14106q^{53} - 2916q^{54} - 10496q^{56} - 3204q^{57} + 2856q^{58} + 7920q^{59} - 13450q^{61} + 3392q^{62} - 13284q^{63} + 4096q^{64} - 25920q^{66} + 65476q^{67} + 35616q^{68} - 16200q^{69} + 34560q^{71} + 5184q^{72} - 86258q^{73} + 45208q^{74} + 5696q^{76} - 118080q^{77} + 25128q^{78} - 108832q^{79} + 6561q^{81} + 37416q^{82} - 10668q^{83} + 23616q^{84} + 23824q^{86} - 6426q^{87} + 46080q^{88} + 10818q^{89} + 114472q^{91} + 28800q^{92} - 7632q^{93} + 44640q^{94} - 9216q^{96} - 4418q^{97} + 40356q^{98} + 58320q^{99} + O(q^{100})$$

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 −164.000 64.0000 81.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

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.h 1
3.b odd 2 1 450.6.a.b 1
5.b even 2 1 30.6.a.a 1
5.c odd 4 2 150.6.c.d 2
15.d odd 2 1 90.6.a.g 1
15.e even 4 2 450.6.c.b 2
20.d odd 2 1 240.6.a.a 1
40.e odd 2 1 960.6.a.u 1
40.f even 2 1 960.6.a.n 1
60.h even 2 1 720.6.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.a.a 1 5.b even 2 1
90.6.a.g 1 15.d odd 2 1
150.6.a.h 1 1.a even 1 1 trivial
150.6.c.d 2 5.c odd 4 2
240.6.a.a 1 20.d odd 2 1
450.6.a.b 1 3.b odd 2 1
450.6.c.b 2 15.e even 4 2
720.6.a.m 1 60.h even 2 1
960.6.a.n 1 40.f even 2 1
960.6.a.u 1 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} + 164$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(150))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + T$$
$3$ $$9 + T$$
$5$ $$T$$
$7$ $$164 + T$$
$11$ $$-720 + T$$
$13$ $$698 + T$$
$17$ $$-2226 + T$$
$19$ $$-356 + T$$
$23$ $$-1800 + T$$
$29$ $$-714 + T$$
$31$ $$-848 + T$$
$37$ $$-11302 + T$$
$41$ $$-9354 + T$$
$43$ $$-5956 + T$$
$47$ $$-11160 + T$$
$53$ $$14106 + T$$
$59$ $$-7920 + T$$
$61$ $$13450 + T$$
$67$ $$-65476 + T$$
$71$ $$-34560 + T$$
$73$ $$86258 + T$$
$79$ $$108832 + T$$
$83$ $$10668 + T$$
$89$ $$-10818 + T$$
$97$ $$4418 + T$$