Properties

 Label 2205.2.a.l Level $2205$ Weight $2$ Character orbit 2205.a Self dual yes Analytic conductor $17.607$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2205.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$17.6070136457$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 245) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 2q^{2} + 2q^{4} + q^{5} + O(q^{10})$$ $$q + 2q^{2} + 2q^{4} + q^{5} + 2q^{10} - q^{11} + 3q^{13} - 4q^{16} + 3q^{17} + 6q^{19} + 2q^{20} - 2q^{22} + 4q^{23} + q^{25} + 6q^{26} + q^{29} + 6q^{31} - 8q^{32} + 6q^{34} + 12q^{38} - 6q^{41} - 6q^{43} - 2q^{44} + 8q^{46} + 9q^{47} + 2q^{50} + 6q^{52} + 10q^{53} - q^{55} + 2q^{58} + 6q^{59} + 12q^{62} - 8q^{64} + 3q^{65} - 14q^{67} + 6q^{68} + 8q^{71} + 6q^{73} + 12q^{76} - q^{79} - 4q^{80} - 12q^{82} - 12q^{83} + 3q^{85} - 12q^{86} - 12q^{89} + 8q^{92} + 18q^{94} + 6q^{95} - 15q^{97} + O(q^{100})$$

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
2.00000 0 2.00000 1.00000 0 0 0 0 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

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 2205.2.a.l 1
3.b odd 2 1 245.2.a.b yes 1
7.b odd 2 1 2205.2.a.j 1
12.b even 2 1 3920.2.a.a 1
15.d odd 2 1 1225.2.a.h 1
15.e even 4 2 1225.2.b.a 2
21.c even 2 1 245.2.a.a 1
21.g even 6 2 245.2.e.d 2
21.h odd 6 2 245.2.e.c 2
84.h odd 2 1 3920.2.a.bj 1
105.g even 2 1 1225.2.a.j 1
105.k odd 4 2 1225.2.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.a 1 21.c even 2 1
245.2.a.b yes 1 3.b odd 2 1
245.2.e.c 2 21.h odd 6 2
245.2.e.d 2 21.g even 6 2
1225.2.a.h 1 15.d odd 2 1
1225.2.a.j 1 105.g even 2 1
1225.2.b.a 2 15.e even 4 2
1225.2.b.b 2 105.k odd 4 2
2205.2.a.j 1 7.b odd 2 1
2205.2.a.l 1 1.a even 1 1 trivial
3920.2.a.a 1 12.b even 2 1
3920.2.a.bj 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2205))$$:

 $$T_{2} - 2$$ $$T_{11} + 1$$ $$T_{13} - 3$$ $$T_{17} - 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$1 + T$$
$13$ $$-3 + T$$
$17$ $$-3 + T$$
$19$ $$-6 + T$$
$23$ $$-4 + T$$
$29$ $$-1 + T$$
$31$ $$-6 + T$$
$37$ $$T$$
$41$ $$6 + T$$
$43$ $$6 + T$$
$47$ $$-9 + T$$
$53$ $$-10 + T$$
$59$ $$-6 + T$$
$61$ $$T$$
$67$ $$14 + T$$
$71$ $$-8 + T$$
$73$ $$-6 + T$$
$79$ $$1 + T$$
$83$ $$12 + T$$
$89$ $$12 + T$$
$97$ $$15 + T$$