# Properties

 Label 4900.2.a.w Level $4900$ Weight $2$ Character orbit 4900.a Self dual yes Analytic conductor $39.127$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4900 = 2^{2} \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4900.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.1266969904$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3q^{3} + 6q^{9} + O(q^{10})$$ $$q + 3q^{3} + 6q^{9} + 3q^{11} + q^{13} - 5q^{17} + 8q^{19} - 2q^{23} + 9q^{27} - q^{29} + 2q^{31} + 9q^{33} - 10q^{37} + 3q^{39} + 6q^{41} + 4q^{43} + 11q^{47} - 15q^{51} - 6q^{53} + 24q^{57} + 10q^{59} + 10q^{67} - 6q^{69} - 10q^{73} - 7q^{79} + 9q^{81} + 12q^{83} - 3q^{87} - 8q^{89} + 6q^{93} + 3q^{97} + 18q^{99} + O(q^{100})$$

## 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
0 3.00000 0 0 0 0 0 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 4900.2.a.w 1
5.b even 2 1 4900.2.a.b 1
5.c odd 4 2 980.2.e.b 2
7.b odd 2 1 700.2.a.a 1
21.c even 2 1 6300.2.a.c 1
28.d even 2 1 2800.2.a.bf 1
35.c odd 2 1 700.2.a.j 1
35.f even 4 2 140.2.e.a 2
35.k even 12 4 980.2.q.f 4
35.l odd 12 4 980.2.q.c 4
105.g even 2 1 6300.2.a.t 1
105.k odd 4 2 1260.2.k.c 2
140.c even 2 1 2800.2.a.a 1
140.j odd 4 2 560.2.g.a 2
280.s even 4 2 2240.2.g.e 2
280.y odd 4 2 2240.2.g.f 2
420.w even 4 2 5040.2.t.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.e.a 2 35.f even 4 2
560.2.g.a 2 140.j odd 4 2
700.2.a.a 1 7.b odd 2 1
700.2.a.j 1 35.c odd 2 1
980.2.e.b 2 5.c odd 4 2
980.2.q.c 4 35.l odd 12 4
980.2.q.f 4 35.k even 12 4
1260.2.k.c 2 105.k odd 4 2
2240.2.g.e 2 280.s even 4 2
2240.2.g.f 2 280.y odd 4 2
2800.2.a.a 1 140.c even 2 1
2800.2.a.bf 1 28.d even 2 1
4900.2.a.b 1 5.b even 2 1
4900.2.a.w 1 1.a even 1 1 trivial
5040.2.t.s 2 420.w even 4 2
6300.2.a.c 1 21.c even 2 1
6300.2.a.t 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4900))$$:

 $$T_{3} - 3$$ $$T_{11} - 3$$ $$T_{13} - 1$$ $$T_{19} - 8$$ $$T_{23} + 2$$

## Hecke characteristic polynomials

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