Properties

 Label 72.8.a.e Level $72$ Weight $8$ Character orbit 72.a Self dual yes Analytic conductor $22.492$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 72.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$22.4917218349$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 24) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 530 q^{5} + 120 q^{7} + O(q^{10})$$ $$q + 530 q^{5} + 120 q^{7} + 7196 q^{11} - 9626 q^{13} - 18674 q^{17} + 7004 q^{19} + 63704 q^{23} + 202775 q^{25} - 29334 q^{29} + 87968 q^{31} + 63600 q^{35} + 227982 q^{37} + 160806 q^{41} + 136132 q^{43} + 1206960 q^{47} - 809143 q^{49} + 398786 q^{53} + 3813880 q^{55} - 1152436 q^{59} - 2070602 q^{61} - 5101780 q^{65} - 4073428 q^{67} + 383752 q^{71} + 3006010 q^{73} + 863520 q^{77} - 4948112 q^{79} + 9163492 q^{83} - 9897220 q^{85} - 7304106 q^{89} - 1155120 q^{91} + 3712120 q^{95} - 690526 q^{97} + O(q^{100})$$

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
0 0 0 530.000 0 120.000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$

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 72.8.a.e 1
3.b odd 2 1 24.8.a.b 1
4.b odd 2 1 144.8.a.k 1
8.b even 2 1 576.8.a.c 1
8.d odd 2 1 576.8.a.b 1
12.b even 2 1 48.8.a.a 1
15.d odd 2 1 600.8.a.b 1
15.e even 4 2 600.8.f.a 2
24.f even 2 1 192.8.a.p 1
24.h odd 2 1 192.8.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.8.a.b 1 3.b odd 2 1
48.8.a.a 1 12.b even 2 1
72.8.a.e 1 1.a even 1 1 trivial
144.8.a.k 1 4.b odd 2 1
192.8.a.h 1 24.h odd 2 1
192.8.a.p 1 24.f even 2 1
576.8.a.b 1 8.d odd 2 1
576.8.a.c 1 8.b even 2 1
600.8.a.b 1 15.d odd 2 1
600.8.f.a 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 530$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(72))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-530 + T$$
$7$ $$-120 + T$$
$11$ $$-7196 + T$$
$13$ $$9626 + T$$
$17$ $$18674 + T$$
$19$ $$-7004 + T$$
$23$ $$-63704 + T$$
$29$ $$29334 + T$$
$31$ $$-87968 + T$$
$37$ $$-227982 + T$$
$41$ $$-160806 + T$$
$43$ $$-136132 + T$$
$47$ $$-1206960 + T$$
$53$ $$-398786 + T$$
$59$ $$1152436 + T$$
$61$ $$2070602 + T$$
$67$ $$4073428 + T$$
$71$ $$-383752 + T$$
$73$ $$-3006010 + T$$
$79$ $$4948112 + T$$
$83$ $$-9163492 + T$$
$89$ $$7304106 + T$$
$97$ $$690526 + T$$