Properties

 Label 180.4.a.b Level $180$ Weight $4$ Character orbit 180.a Self dual yes Analytic conductor $10.620$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 180.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$10.6203438010$$ 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

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + 2 q^{7}+O(q^{10})$$ q - 5 * q^5 + 2 * q^7 $$q - 5 q^{5} + 2 q^{7} - 30 q^{11} - 4 q^{13} - 90 q^{17} - 28 q^{19} - 120 q^{23} + 25 q^{25} - 210 q^{29} - 4 q^{31} - 10 q^{35} + 200 q^{37} - 240 q^{41} - 136 q^{43} + 120 q^{47} - 339 q^{49} + 30 q^{53} + 150 q^{55} + 450 q^{59} - 166 q^{61} + 20 q^{65} + 908 q^{67} + 1020 q^{71} - 250 q^{73} - 60 q^{77} - 916 q^{79} + 1140 q^{83} + 450 q^{85} + 420 q^{89} - 8 q^{91} + 140 q^{95} + 1538 q^{97}+O(q^{100})$$ q - 5 * q^5 + 2 * q^7 - 30 * q^11 - 4 * q^13 - 90 * q^17 - 28 * q^19 - 120 * q^23 + 25 * q^25 - 210 * q^29 - 4 * q^31 - 10 * q^35 + 200 * q^37 - 240 * q^41 - 136 * q^43 + 120 * q^47 - 339 * q^49 + 30 * q^53 + 150 * q^55 + 450 * q^59 - 166 * q^61 + 20 * q^65 + 908 * q^67 + 1020 * q^71 - 250 * q^73 - 60 * q^77 - 916 * q^79 + 1140 * q^83 + 450 * q^85 + 420 * q^89 - 8 * q^91 + 140 * q^95 + 1538 * q^97

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 −5.00000 0 2.00000 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$1$$

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 180.4.a.b 1
3.b odd 2 1 180.4.a.e yes 1
4.b odd 2 1 720.4.a.h 1
5.b even 2 1 900.4.a.i 1
5.c odd 4 2 900.4.d.d 2
9.c even 3 2 1620.4.i.i 2
9.d odd 6 2 1620.4.i.c 2
12.b even 2 1 720.4.a.w 1
15.d odd 2 1 900.4.a.j 1
15.e even 4 2 900.4.d.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.4.a.b 1 1.a even 1 1 trivial
180.4.a.e yes 1 3.b odd 2 1
720.4.a.h 1 4.b odd 2 1
720.4.a.w 1 12.b even 2 1
900.4.a.i 1 5.b even 2 1
900.4.a.j 1 15.d odd 2 1
900.4.d.d 2 5.c odd 4 2
900.4.d.i 2 15.e even 4 2
1620.4.i.c 2 9.d odd 6 2
1620.4.i.i 2 9.c even 3 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(180))$$:

 $$T_{7} - 2$$ T7 - 2 $$T_{11} + 30$$ T11 + 30

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T - 2$$
$11$ $$T + 30$$
$13$ $$T + 4$$
$17$ $$T + 90$$
$19$ $$T + 28$$
$23$ $$T + 120$$
$29$ $$T + 210$$
$31$ $$T + 4$$
$37$ $$T - 200$$
$41$ $$T + 240$$
$43$ $$T + 136$$
$47$ $$T - 120$$
$53$ $$T - 30$$
$59$ $$T - 450$$
$61$ $$T + 166$$
$67$ $$T - 908$$
$71$ $$T - 1020$$
$73$ $$T + 250$$
$79$ $$T + 916$$
$83$ $$T - 1140$$
$89$ $$T - 420$$
$97$ $$T - 1538$$