# Properties

 Label 4900.2.a.be Level $4900$ Weight $2$ Character orbit 4900.a Self dual yes Analytic conductor $39.127$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# 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: $$4$$ Coefficient field: $$\Q(\sqrt{3}, \sqrt{19})$$ Defining polynomial: $$x^{4} - 11 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( -2 - \beta_{3} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{13} + ( -\beta_{1} + 2 \beta_{2} ) q^{17} + \beta_{3} q^{19} + ( -2 \beta_{1} - \beta_{2} ) q^{23} -3 \beta_{2} q^{27} + ( 1 + \beta_{3} ) q^{29} + \beta_{3} q^{31} -3 \beta_{1} q^{33} + ( -\beta_{1} - 4 \beta_{2} ) q^{37} + ( -2 + 2 \beta_{3} ) q^{39} + ( -7 + \beta_{3} ) q^{41} + ( \beta_{1} - 3 \beta_{2} ) q^{43} + ( \beta_{1} + 2 \beta_{2} ) q^{47} + ( 4 - \beta_{3} ) q^{51} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{57} -\beta_{3} q^{59} + ( -7 - 2 \beta_{3} ) q^{61} + ( 4 \beta_{1} - 5 \beta_{2} ) q^{67} + ( -7 - 2 \beta_{3} ) q^{69} + ( 2 - 2 \beta_{3} ) q^{71} + ( \beta_{1} + 2 \beta_{2} ) q^{73} + ( 4 + \beta_{3} ) q^{79} -9 q^{81} + ( -3 \beta_{1} + \beta_{2} ) q^{83} + ( 3 \beta_{1} - \beta_{2} ) q^{87} -7 q^{89} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{93} -4 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q - 6 q^{11} - 2 q^{19} + 2 q^{29} - 2 q^{31} - 12 q^{39} - 30 q^{41} + 18 q^{51} + 2 q^{59} - 24 q^{61} - 24 q^{69} + 12 q^{71} + 14 q^{79} - 36 q^{81} - 28 q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 11 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 7 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 6$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} + 7 \beta_{1}$$

## 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
 −3.04547 1.31342 3.04547 −1.31342
0 −1.73205 0 0 0 0 0 0 0
1.2 0 −1.73205 0 0 0 0 0 0 0
1.3 0 1.73205 0 0 0 0 0 0 0
1.4 0 1.73205 0 0 0 0 0 0 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4900.2.a.be 4
5.b even 2 1 inner 4900.2.a.be 4
5.c odd 4 2 980.2.e.f 4
7.b odd 2 1 4900.2.a.bf 4
7.c even 3 2 700.2.i.f 8
35.c odd 2 1 4900.2.a.bf 4
35.f even 4 2 980.2.e.c 4
35.j even 6 2 700.2.i.f 8
35.k even 12 2 980.2.q.b 4
35.k even 12 2 980.2.q.g 4
35.l odd 12 2 140.2.q.a 4
35.l odd 12 2 140.2.q.b yes 4
105.x even 12 2 1260.2.bm.a 4
105.x even 12 2 1260.2.bm.b 4
140.w even 12 2 560.2.bw.a 4
140.w even 12 2 560.2.bw.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a 4 35.l odd 12 2
140.2.q.b yes 4 35.l odd 12 2
560.2.bw.a 4 140.w even 12 2
560.2.bw.e 4 140.w even 12 2
700.2.i.f 8 7.c even 3 2
700.2.i.f 8 35.j even 6 2
980.2.e.c 4 35.f even 4 2
980.2.e.f 4 5.c odd 4 2
980.2.q.b 4 35.k even 12 2
980.2.q.g 4 35.k even 12 2
1260.2.bm.a 4 105.x even 12 2
1260.2.bm.b 4 105.x even 12 2
4900.2.a.be 4 1.a even 1 1 trivial
4900.2.a.be 4 5.b even 2 1 inner
4900.2.a.bf 4 7.b odd 2 1
4900.2.a.bf 4 35.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4900))$$:

 $$T_{3}^{2} - 3$$ $$T_{11}^{2} + 3 T_{11} - 12$$ $$T_{13}^{4} - 44 T_{13}^{2} + 256$$ $$T_{19}^{2} + T_{19} - 14$$ $$T_{23}^{4} - 62 T_{23}^{2} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$( -12 + 3 T + T^{2} )^{2}$$
$13$ $$256 - 44 T^{2} + T^{4}$$
$17$ $$4 - 23 T^{2} + T^{4}$$
$19$ $$( -14 + T + T^{2} )^{2}$$
$23$ $$49 - 62 T^{2} + T^{4}$$
$29$ $$( -14 - T + T^{2} )^{2}$$
$31$ $$( -14 + T + T^{2} )^{2}$$
$37$ $$3136 - 131 T^{2} + T^{4}$$
$41$ $$( 42 + 15 T + T^{2} )^{2}$$
$43$ $$196 - 47 T^{2} + T^{4}$$
$47$ $$196 - 47 T^{2} + T^{4}$$
$53$ $$1764 - 87 T^{2} + T^{4}$$
$59$ $$( -14 - T + T^{2} )^{2}$$
$61$ $$( -21 + 12 T + T^{2} )^{2}$$
$67$ $$2401 - 206 T^{2} + T^{4}$$
$71$ $$( -48 - 6 T + T^{2} )^{2}$$
$73$ $$196 - 47 T^{2} + T^{4}$$
$79$ $$( -2 - 7 T + T^{2} )^{2}$$
$83$ $$1764 - 87 T^{2} + T^{4}$$
$89$ $$( 7 + T )^{4}$$
$97$ $$( -48 + T^{2} )^{2}$$