Properties

 Label 245.2.a.a Level $245$ Weight $2$ Character orbit 245.a Self dual yes Analytic conductor $1.956$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 245.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$1.95633484952$$ 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

 $$f(q)$$ $$=$$ $$q - 2q^{2} - 3q^{3} + 2q^{4} + q^{5} + 6q^{6} + 6q^{9} + O(q^{10})$$ $$q - 2q^{2} - 3q^{3} + 2q^{4} + q^{5} + 6q^{6} + 6q^{9} - 2q^{10} + q^{11} - 6q^{12} - 3q^{13} - 3q^{15} - 4q^{16} + 3q^{17} - 12q^{18} - 6q^{19} + 2q^{20} - 2q^{22} - 4q^{23} + q^{25} + 6q^{26} - 9q^{27} - q^{29} + 6q^{30} - 6q^{31} + 8q^{32} - 3q^{33} - 6q^{34} + 12q^{36} + 12q^{38} + 9q^{39} - 6q^{41} - 6q^{43} + 2q^{44} + 6q^{45} + 8q^{46} + 9q^{47} + 12q^{48} - 2q^{50} - 9q^{51} - 6q^{52} - 10q^{53} + 18q^{54} + q^{55} + 18q^{57} + 2q^{58} + 6q^{59} - 6q^{60} + 12q^{62} - 8q^{64} - 3q^{65} + 6q^{66} - 14q^{67} + 6q^{68} + 12q^{69} - 8q^{71} - 6q^{73} - 3q^{75} - 12q^{76} - 18q^{78} - q^{79} - 4q^{80} + 9q^{81} + 12q^{82} - 12q^{83} + 3q^{85} + 12q^{86} + 3q^{87} - 12q^{89} - 12q^{90} - 8q^{92} + 18q^{93} - 18q^{94} - 6q^{95} - 24q^{96} + 15q^{97} + 6q^{99} + O(q^{100})$$

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
−2.00000 −3.00000 2.00000 1.00000 6.00000 0 0 6.00000 −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
$$5$$ $$-1$$
$$7$$ $$-1$$

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 245.2.a.a 1
3.b odd 2 1 2205.2.a.j 1
4.b odd 2 1 3920.2.a.bj 1
5.b even 2 1 1225.2.a.j 1
5.c odd 4 2 1225.2.b.b 2
7.b odd 2 1 245.2.a.b yes 1
7.c even 3 2 245.2.e.d 2
7.d odd 6 2 245.2.e.c 2
21.c even 2 1 2205.2.a.l 1
28.d even 2 1 3920.2.a.a 1
35.c odd 2 1 1225.2.a.h 1
35.f even 4 2 1225.2.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.2.a.a 1 1.a even 1 1 trivial
245.2.a.b yes 1 7.b odd 2 1
245.2.e.c 2 7.d odd 6 2
245.2.e.d 2 7.c even 3 2
1225.2.a.h 1 35.c odd 2 1
1225.2.a.j 1 5.b even 2 1
1225.2.b.a 2 35.f even 4 2
1225.2.b.b 2 5.c odd 4 2
2205.2.a.j 1 3.b odd 2 1
2205.2.a.l 1 21.c even 2 1
3920.2.a.a 1 28.d even 2 1
3920.2.a.bj 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2} + 2$$ $$T_{3} + 3$$

Hecke characteristic polynomials

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