Properties

 Label 4900.2.a.k Level $4900$ Weight $2$ Character orbit 4900.a Self dual yes Analytic conductor $39.127$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 700) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 3q^{9} + O(q^{10})$$ $$q - 3q^{9} - 5q^{11} + 6q^{13} - 4q^{17} + 6q^{19} + 3q^{23} - 3q^{29} - 2q^{31} + 7q^{37} + 4q^{41} - 7q^{43} + 2q^{47} - 10q^{53} + 14q^{59} - 4q^{61} + 3q^{67} - 13q^{71} - 16q^{73} + q^{79} + 9q^{81} - 10q^{83} - 10q^{89} + 2q^{97} + 15q^{99} + O(q^{100})$$

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 0 0 0 0 0 0 −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 4900.2.a.k 1
5.b even 2 1 4900.2.a.j 1
5.c odd 4 2 4900.2.e.o 2
7.b odd 2 1 700.2.a.g yes 1
21.c even 2 1 6300.2.a.be 1
28.d even 2 1 2800.2.a.q 1
35.c odd 2 1 700.2.a.e 1
35.f even 4 2 700.2.e.d 2
105.g even 2 1 6300.2.a.o 1
105.k odd 4 2 6300.2.k.q 2
140.c even 2 1 2800.2.a.t 1
140.j odd 4 2 2800.2.g.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.a.e 1 35.c odd 2 1
700.2.a.g yes 1 7.b odd 2 1
700.2.e.d 2 35.f even 4 2
2800.2.a.q 1 28.d even 2 1
2800.2.a.t 1 140.c even 2 1
2800.2.g.q 2 140.j odd 4 2
4900.2.a.j 1 5.b even 2 1
4900.2.a.k 1 1.a even 1 1 trivial
4900.2.e.o 2 5.c odd 4 2
6300.2.a.o 1 105.g even 2 1
6300.2.a.be 1 21.c even 2 1
6300.2.k.q 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4900))$$:

 $$T_{3}$$ $$T_{11} + 5$$ $$T_{13} - 6$$ $$T_{19} - 6$$ $$T_{23} - 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$5 + T$$
$13$ $$-6 + T$$
$17$ $$4 + T$$
$19$ $$-6 + T$$
$23$ $$-3 + T$$
$29$ $$3 + T$$
$31$ $$2 + T$$
$37$ $$-7 + T$$
$41$ $$-4 + T$$
$43$ $$7 + T$$
$47$ $$-2 + T$$
$53$ $$10 + T$$
$59$ $$-14 + T$$
$61$ $$4 + T$$
$67$ $$-3 + T$$
$71$ $$13 + T$$
$73$ $$16 + T$$
$79$ $$-1 + T$$
$83$ $$10 + T$$
$89$ $$10 + T$$
$97$ $$-2 + T$$