# Properties

 Label 2450.2.a.bg Level $2450$ Weight $2$ Character orbit 2450.a Self dual yes Analytic conductor $19.563$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2450 = 2 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.5633484952$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 350) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + 3q^{3} + q^{4} + 3q^{6} + q^{8} + 6q^{9} + O(q^{10})$$ $$q + q^{2} + 3q^{3} + q^{4} + 3q^{6} + q^{8} + 6q^{9} - 5q^{11} + 3q^{12} + 6q^{13} + q^{16} + q^{17} + 6q^{18} + 3q^{19} - 5q^{22} + 3q^{24} + 6q^{26} + 9q^{27} - 6q^{29} + 4q^{31} + q^{32} - 15q^{33} + q^{34} + 6q^{36} + 8q^{37} + 3q^{38} + 18q^{39} - 11q^{41} - 8q^{43} - 5q^{44} - 2q^{47} + 3q^{48} + 3q^{51} + 6q^{52} + 4q^{53} + 9q^{54} + 9q^{57} - 6q^{58} - 4q^{59} + 2q^{61} + 4q^{62} + q^{64} - 15q^{66} + 9q^{67} + q^{68} - 10q^{71} + 6q^{72} + 7q^{73} + 8q^{74} + 3q^{76} + 18q^{78} - 2q^{79} + 9q^{81} - 11q^{82} - 11q^{83} - 8q^{86} - 18q^{87} - 5q^{88} + 11q^{89} + 12q^{93} - 2q^{94} + 3q^{96} + 10q^{97} - 30q^{99} + O(q^{100})$$

## 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
1.00000 3.00000 1.00000 0 3.00000 0 1.00000 6.00000 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 2450.2.a.bg 1
5.b even 2 1 2450.2.a.a 1
5.c odd 4 2 2450.2.c.r 2
7.b odd 2 1 350.2.a.d yes 1
21.c even 2 1 3150.2.a.j 1
28.d even 2 1 2800.2.a.bg 1
35.c odd 2 1 350.2.a.c 1
35.f even 4 2 350.2.c.a 2
105.g even 2 1 3150.2.a.bq 1
105.k odd 4 2 3150.2.g.v 2
140.c even 2 1 2800.2.a.b 1
140.j odd 4 2 2800.2.g.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.c 1 35.c odd 2 1
350.2.a.d yes 1 7.b odd 2 1
350.2.c.a 2 35.f even 4 2
2450.2.a.a 1 5.b even 2 1
2450.2.a.bg 1 1.a even 1 1 trivial
2450.2.c.r 2 5.c odd 4 2
2800.2.a.b 1 140.c even 2 1
2800.2.a.bg 1 28.d even 2 1
2800.2.g.a 2 140.j odd 4 2
3150.2.a.j 1 21.c even 2 1
3150.2.a.bq 1 105.g even 2 1
3150.2.g.v 2 105.k odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2450))$$:

 $$T_{3} - 3$$ $$T_{11} + 5$$ $$T_{13} - 6$$ $$T_{17} - 1$$ $$T_{19} - 3$$ $$T_{23}$$ $$T_{37} - 8$$

## Hecke characteristic polynomials

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