# Properties

 Label 2450.2.a.bf 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 70) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + q^{8} + q^{9} + 3q^{11} + 2q^{12} + q^{13} + q^{16} + 6q^{17} + q^{18} - q^{19} + 3q^{22} - 9q^{23} + 2q^{24} + q^{26} - 4q^{27} + 6q^{29} + 8q^{31} + q^{32} + 6q^{33} + 6q^{34} + q^{36} + 7q^{37} - q^{38} + 2q^{39} + 3q^{41} - 2q^{43} + 3q^{44} - 9q^{46} - 9q^{47} + 2q^{48} + 12q^{51} + q^{52} - 9q^{53} - 4q^{54} - 2q^{57} + 6q^{58} + 8q^{61} + 8q^{62} + q^{64} + 6q^{66} - 8q^{67} + 6q^{68} - 18q^{69} + q^{72} + 4q^{73} + 7q^{74} - q^{76} + 2q^{78} - 10q^{79} - 11q^{81} + 3q^{82} - 2q^{86} + 12q^{87} + 3q^{88} + 6q^{89} - 9q^{92} + 16q^{93} - 9q^{94} + 2q^{96} + 10q^{97} + 3q^{99} + O(q^{100})$$

## 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 2.00000 1.00000 0 2.00000 0 1.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$7$$ $$1$$

## 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.bf 1
5.b even 2 1 490.2.a.a 1
5.c odd 4 2 2450.2.c.q 2
7.b odd 2 1 2450.2.a.v 1
7.c even 3 2 350.2.e.b 2
15.d odd 2 1 4410.2.a.x 1
20.d odd 2 1 3920.2.a.bh 1
35.c odd 2 1 490.2.a.d 1
35.f even 4 2 2450.2.c.e 2
35.i odd 6 2 490.2.e.g 2
35.j even 6 2 70.2.e.d 2
35.l odd 12 4 350.2.j.d 4
105.g even 2 1 4410.2.a.bg 1
105.o odd 6 2 630.2.k.d 2
140.c even 2 1 3920.2.a.e 1
140.p odd 6 2 560.2.q.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.d 2 35.j even 6 2
350.2.e.b 2 7.c even 3 2
350.2.j.d 4 35.l odd 12 4
490.2.a.a 1 5.b even 2 1
490.2.a.d 1 35.c odd 2 1
490.2.e.g 2 35.i odd 6 2
560.2.q.b 2 140.p odd 6 2
630.2.k.d 2 105.o odd 6 2
2450.2.a.v 1 7.b odd 2 1
2450.2.a.bf 1 1.a even 1 1 trivial
2450.2.c.e 2 35.f even 4 2
2450.2.c.q 2 5.c odd 4 2
3920.2.a.e 1 140.c even 2 1
3920.2.a.bh 1 20.d odd 2 1
4410.2.a.x 1 15.d odd 2 1
4410.2.a.bg 1 105.g even 2 1

## Hecke kernels

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

 $$T_{3} - 2$$ $$T_{11} - 3$$ $$T_{13} - 1$$ $$T_{17} - 6$$ $$T_{19} + 1$$ $$T_{23} + 9$$ $$T_{37} - 7$$

## Hecke characteristic polynomials

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