Properties

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

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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - 4q^{2} + 9q^{3} + 16q^{4} - 36q^{6} - q^{7} - 64q^{8} + 81q^{9} + O(q^{10})$$ $$q - 4q^{2} + 9q^{3} + 16q^{4} - 36q^{6} - q^{7} - 64q^{8} + 81q^{9} - 210q^{11} + 144q^{12} - 667q^{13} + 4q^{14} + 256q^{16} + 114q^{17} - 324q^{18} + 581q^{19} - 9q^{21} + 840q^{22} - 4350q^{23} - 576q^{24} + 2668q^{26} + 729q^{27} - 16q^{28} - 126q^{29} + 7583q^{31} - 1024q^{32} - 1890q^{33} - 456q^{34} + 1296q^{36} - 3742q^{37} - 2324q^{38} - 6003q^{39} - 2856q^{41} + 36q^{42} - 18241q^{43} - 3360q^{44} + 17400q^{46} - 23370q^{47} + 2304q^{48} - 16806q^{49} + 1026q^{51} - 10672q^{52} - 21684q^{53} - 2916q^{54} + 64q^{56} + 5229q^{57} + 504q^{58} - 32310q^{59} - 7165q^{61} - 30332q^{62} - 81q^{63} + 4096q^{64} + 7560q^{66} + 59579q^{67} + 1824q^{68} - 39150q^{69} - 43080q^{71} - 5184q^{72} - 28942q^{73} + 14968q^{74} + 9296q^{76} + 210q^{77} + 24012q^{78} + 27608q^{79} + 6561q^{81} + 11424q^{82} - 1782q^{83} - 144q^{84} + 72964q^{86} - 1134q^{87} + 13440q^{88} + 50208q^{89} + 667q^{91} - 69600q^{92} + 68247q^{93} + 93480q^{94} - 9216q^{96} + 142793q^{97} + 67224q^{98} - 17010q^{99} + O(q^{100})$$

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 −1.00000 −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.e 1
3.b odd 2 1 450.6.a.r 1
5.b even 2 1 150.6.a.k yes 1
5.c odd 4 2 150.6.c.a 2
15.d odd 2 1 450.6.a.g 1
15.e even 4 2 450.6.c.k 2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} + 1$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(150))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + T$$
$3$ $$-9 + T$$
$5$ $$T$$
$7$ $$1 + T$$
$11$ $$210 + T$$
$13$ $$667 + T$$
$17$ $$-114 + T$$
$19$ $$-581 + T$$
$23$ $$4350 + T$$
$29$ $$126 + T$$
$31$ $$-7583 + T$$
$37$ $$3742 + T$$
$41$ $$2856 + T$$
$43$ $$18241 + T$$
$47$ $$23370 + T$$
$53$ $$21684 + T$$
$59$ $$32310 + T$$
$61$ $$7165 + T$$
$67$ $$-59579 + T$$
$71$ $$43080 + T$$
$73$ $$28942 + T$$
$79$ $$-27608 + T$$
$83$ $$1782 + T$$
$89$ $$-50208 + T$$
$97$ $$-142793 + T$$