# Properties

 Label 2450.2.a.q 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

 $$f(q)$$ $$=$$ $$q - q^{2} + 3q^{3} + q^{4} - 3q^{6} - q^{8} + 6q^{9} + O(q^{10})$$ $$q - q^{2} + 3q^{3} + q^{4} - 3q^{6} - q^{8} + 6q^{9} - 2q^{11} + 3q^{12} + q^{16} - 4q^{17} - 6q^{18} + 6q^{19} + 2q^{22} - 3q^{23} - 3q^{24} + 9q^{27} + 9q^{29} + 4q^{31} - q^{32} - 6q^{33} + 4q^{34} + 6q^{36} + 4q^{37} - 6q^{38} + 7q^{41} + 5q^{43} - 2q^{44} + 3q^{46} + 8q^{47} + 3q^{48} - 12q^{51} + 2q^{53} - 9q^{54} + 18q^{57} - 9q^{58} - 10q^{59} - q^{61} - 4q^{62} + q^{64} + 6q^{66} + 9q^{67} - 4q^{68} - 9q^{69} + 2q^{71} - 6q^{72} - 4q^{73} - 4q^{74} + 6q^{76} + 10q^{79} + 9q^{81} - 7q^{82} - 7q^{83} - 5q^{86} + 27q^{87} + 2q^{88} - q^{89} - 3q^{92} + 12q^{93} - 8q^{94} - 3q^{96} + 14q^{97} - 12q^{99} + O(q^{100})$$

## 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.q 1
5.b even 2 1 490.2.a.e 1
5.c odd 4 2 2450.2.c.a 2
7.b odd 2 1 2450.2.a.b 1
7.d odd 6 2 350.2.e.l 2
15.d odd 2 1 4410.2.a.h 1
20.d odd 2 1 3920.2.a.bk 1
35.c odd 2 1 490.2.a.k 1
35.f even 4 2 2450.2.c.s 2
35.i odd 6 2 70.2.e.a 2
35.j even 6 2 490.2.e.f 2
35.k even 12 4 350.2.j.f 4
105.g even 2 1 4410.2.a.r 1
105.p even 6 2 630.2.k.f 2
140.c even 2 1 3920.2.a.b 1
140.s even 6 2 560.2.q.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.a 2 35.i odd 6 2
350.2.e.l 2 7.d odd 6 2
350.2.j.f 4 35.k even 12 4
490.2.a.e 1 5.b even 2 1
490.2.a.k 1 35.c odd 2 1
490.2.e.f 2 35.j even 6 2
560.2.q.i 2 140.s even 6 2
630.2.k.f 2 105.p even 6 2
2450.2.a.b 1 7.b odd 2 1
2450.2.a.q 1 1.a even 1 1 trivial
2450.2.c.a 2 5.c odd 4 2
2450.2.c.s 2 35.f even 4 2
3920.2.a.b 1 140.c even 2 1
3920.2.a.bk 1 20.d odd 2 1
4410.2.a.h 1 15.d odd 2 1
4410.2.a.r 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2450))$$:

 $$T_{3} - 3$$ $$T_{11} + 2$$ $$T_{13}$$ $$T_{17} + 4$$ $$T_{19} - 6$$ $$T_{23} + 3$$ $$T_{37} - 4$$

## Hecke characteristic polynomials

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