Properties

 Label 220.2.a.a Level $220$ Weight $2$ Character orbit 220.a Self dual yes Analytic conductor $1.757$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$220 = 2^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 220.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$1.75670884447$$ 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

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + q^{5} - 4 q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^3 + q^5 - 4 * q^7 + q^9 $$q - 2 q^{3} + q^{5} - 4 q^{7} + q^{9} - q^{11} - 4 q^{13} - 2 q^{15} - 4 q^{19} + 8 q^{21} - 6 q^{23} + q^{25} + 4 q^{27} - 6 q^{29} + 8 q^{31} + 2 q^{33} - 4 q^{35} + 2 q^{37} + 8 q^{39} + 6 q^{41} + 8 q^{43} + q^{45} + 6 q^{47} + 9 q^{49} - 6 q^{53} - q^{55} + 8 q^{57} - 12 q^{59} + 2 q^{61} - 4 q^{63} - 4 q^{65} - 10 q^{67} + 12 q^{69} - 12 q^{71} - 16 q^{73} - 2 q^{75} + 4 q^{77} + 8 q^{79} - 11 q^{81} + 12 q^{87} + 6 q^{89} + 16 q^{91} - 16 q^{93} - 4 q^{95} + 14 q^{97} - q^{99}+O(q^{100})$$ q - 2 * q^3 + q^5 - 4 * q^7 + q^9 - q^11 - 4 * q^13 - 2 * q^15 - 4 * q^19 + 8 * q^21 - 6 * q^23 + q^25 + 4 * q^27 - 6 * q^29 + 8 * q^31 + 2 * q^33 - 4 * q^35 + 2 * q^37 + 8 * q^39 + 6 * q^41 + 8 * q^43 + q^45 + 6 * q^47 + 9 * q^49 - 6 * q^53 - q^55 + 8 * q^57 - 12 * q^59 + 2 * q^61 - 4 * q^63 - 4 * q^65 - 10 * q^67 + 12 * q^69 - 12 * q^71 - 16 * q^73 - 2 * q^75 + 4 * q^77 + 8 * q^79 - 11 * q^81 + 12 * q^87 + 6 * q^89 + 16 * q^91 - 16 * q^93 - 4 * q^95 + 14 * q^97 - q^99

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 −2.00000 0 1.00000 0 −4.00000 0 1.00000 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$1$$

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 220.2.a.a 1
3.b odd 2 1 1980.2.a.a 1
4.b odd 2 1 880.2.a.j 1
5.b even 2 1 1100.2.a.e 1
5.c odd 4 2 1100.2.b.a 2
8.b even 2 1 3520.2.a.bd 1
8.d odd 2 1 3520.2.a.d 1
11.b odd 2 1 2420.2.a.b 1
12.b even 2 1 7920.2.a.o 1
15.d odd 2 1 9900.2.a.bd 1
15.e even 4 2 9900.2.c.m 2
20.d odd 2 1 4400.2.a.e 1
20.e even 4 2 4400.2.b.f 2
44.c even 2 1 9680.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.a.a 1 1.a even 1 1 trivial
880.2.a.j 1 4.b odd 2 1
1100.2.a.e 1 5.b even 2 1
1100.2.b.a 2 5.c odd 4 2
1980.2.a.a 1 3.b odd 2 1
2420.2.a.b 1 11.b odd 2 1
3520.2.a.d 1 8.d odd 2 1
3520.2.a.bd 1 8.b even 2 1
4400.2.a.e 1 20.d odd 2 1
4400.2.b.f 2 20.e even 4 2
7920.2.a.o 1 12.b even 2 1
9680.2.a.bb 1 44.c even 2 1
9900.2.a.bd 1 15.d odd 2 1
9900.2.c.m 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(220))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 2$$
$5$ $$T - 1$$
$7$ $$T + 4$$
$11$ $$T + 1$$
$13$ $$T + 4$$
$17$ $$T$$
$19$ $$T + 4$$
$23$ $$T + 6$$
$29$ $$T + 6$$
$31$ $$T - 8$$
$37$ $$T - 2$$
$41$ $$T - 6$$
$43$ $$T - 8$$
$47$ $$T - 6$$
$53$ $$T + 6$$
$59$ $$T + 12$$
$61$ $$T - 2$$
$67$ $$T + 10$$
$71$ $$T + 12$$
$73$ $$T + 16$$
$79$ $$T - 8$$
$83$ $$T$$
$89$ $$T - 6$$
$97$ $$T - 14$$