# Properties

 Label 2450.2.a.ba Level $2450$ Weight $2$ Character orbit 2450.a Self dual yes Analytic conductor $19.563$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{8} - 3q^{9} + O(q^{10})$$ $$q + q^{2} + q^{4} + q^{8} - 3q^{9} + 3q^{11} - 5q^{13} + q^{16} - 2q^{17} - 3q^{18} - 5q^{19} + 3q^{22} - 7q^{23} - 5q^{26} - 4q^{29} - 2q^{31} + q^{32} - 2q^{34} - 3q^{36} + q^{37} - 5q^{38} + 3q^{41} + 2q^{43} + 3q^{44} - 7q^{46} - 7q^{47} - 5q^{52} + 9q^{53} - 4q^{58} - 4q^{59} + 6q^{61} - 2q^{62} + q^{64} + 2q^{67} - 2q^{68} - 6q^{71} - 3q^{72} - 16q^{73} + q^{74} - 5q^{76} + 14q^{79} + 9q^{81} + 3q^{82} - 6q^{83} + 2q^{86} + 3q^{88} + 2q^{89} - 7q^{92} - 7q^{94} - 12q^{97} - 9q^{99} + O(q^{100})$$

## 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 0 1.00000 0 0 0 1.00000 −3.00000 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.ba 1
5.b even 2 1 2450.2.a.k 1
5.c odd 4 2 490.2.c.a 2
7.b odd 2 1 2450.2.a.bb 1
7.c even 3 2 350.2.e.c 2
35.c odd 2 1 2450.2.a.j 1
35.f even 4 2 490.2.c.d 2
35.j even 6 2 350.2.e.j 2
35.k even 12 4 490.2.i.a 4
35.l odd 12 4 70.2.i.b 4
105.x even 12 4 630.2.u.a 4
140.w even 12 4 560.2.bw.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.b 4 35.l odd 12 4
350.2.e.c 2 7.c even 3 2
350.2.e.j 2 35.j even 6 2
490.2.c.a 2 5.c odd 4 2
490.2.c.d 2 35.f even 4 2
490.2.i.a 4 35.k even 12 4
560.2.bw.d 4 140.w even 12 4
630.2.u.a 4 105.x even 12 4
2450.2.a.j 1 35.c odd 2 1
2450.2.a.k 1 5.b even 2 1
2450.2.a.ba 1 1.a even 1 1 trivial
2450.2.a.bb 1 7.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(2450))$$:

 $$T_{3}$$ $$T_{11} - 3$$ $$T_{13} + 5$$ $$T_{17} + 2$$ $$T_{19} + 5$$ $$T_{23} + 7$$ $$T_{37} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$-3 + T$$
$13$ $$5 + T$$
$17$ $$2 + T$$
$19$ $$5 + T$$
$23$ $$7 + T$$
$29$ $$4 + T$$
$31$ $$2 + T$$
$37$ $$-1 + T$$
$41$ $$-3 + T$$
$43$ $$-2 + T$$
$47$ $$7 + T$$
$53$ $$-9 + T$$
$59$ $$4 + T$$
$61$ $$-6 + T$$
$67$ $$-2 + T$$
$71$ $$6 + T$$
$73$ $$16 + T$$
$79$ $$-14 + T$$
$83$ $$6 + T$$
$89$ $$-2 + T$$
$97$ $$12 + T$$