# Properties

 Label 4900.2.a.bb Level $4900$ Weight $2$ Character orbit 4900.a Self dual yes Analytic conductor $39.127$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.257.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 700) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( 2 - \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( 2 - \beta_{1} - \beta_{2} ) q^{9} + ( 1 - \beta_{2} ) q^{11} + ( -3 - \beta_{2} ) q^{13} + ( -3 + \beta_{1} ) q^{17} + ( 1 + \beta_{1} ) q^{19} + ( -1 - \beta_{1} + \beta_{2} ) q^{23} + ( -2 + \beta_{1} + 3 \beta_{2} ) q^{27} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{29} + ( -\beta_{1} - 2 \beta_{2} ) q^{31} + ( -5 + \beta_{1} + 2 \beta_{2} ) q^{33} + ( 1 - 3 \beta_{2} ) q^{37} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{39} + ( -3 - \beta_{1} + 3 \beta_{2} ) q^{41} + ( -3 + \beta_{1} + \beta_{2} ) q^{43} + ( -2 - \beta_{1} - \beta_{2} ) q^{47} + ( -3 - 6 \beta_{2} ) q^{51} + ( 6 + \beta_{1} + 3 \beta_{2} ) q^{53} + ( -3 - 2 \beta_{2} ) q^{57} + ( 2 + \beta_{2} ) q^{59} + ( -6 + \beta_{1} - 2 \beta_{2} ) q^{61} + ( 6 - 2 \beta_{2} ) q^{67} + ( 8 - \beta_{1} + \beta_{2} ) q^{69} + ( -7 - 2 \beta_{2} ) q^{71} -4 q^{73} + ( -1 + \beta_{1} - 3 \beta_{2} ) q^{79} + ( 6 - 5 \beta_{2} ) q^{81} + ( -11 - \beta_{1} - 4 \beta_{2} ) q^{83} + ( 7 - 2 \beta_{1} - 4 \beta_{2} ) q^{87} + ( 4 - \beta_{1} - \beta_{2} ) q^{89} + ( -7 + 2 \beta_{1} + 5 \beta_{2} ) q^{93} + ( -6 - 2 \beta_{1} - \beta_{2} ) q^{97} + ( 4 - 2 \beta_{1} - 7 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - q^{3} + 8 q^{9} + O(q^{10})$$ $$3 q - q^{3} + 8 q^{9} + 4 q^{11} - 8 q^{13} - 10 q^{17} + 2 q^{19} - 3 q^{23} - 10 q^{27} + 3 q^{31} - 18 q^{33} + 6 q^{37} - 14 q^{39} - 11 q^{41} - 11 q^{43} - 4 q^{47} - 3 q^{51} + 14 q^{53} - 7 q^{57} + 5 q^{59} - 17 q^{61} + 20 q^{67} + 24 q^{69} - 19 q^{71} - 12 q^{73} - q^{79} + 23 q^{81} - 28 q^{83} + 27 q^{87} + 14 q^{89} - 28 q^{93} - 15 q^{97} + 21 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + 2 \nu - 4$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1} + 1$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{2} + \beta_{1} + 10$$$$)/3$$

## 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
 0.713538 2.19869 −1.91223
0 −3.20440 0 0 0 0 0 7.26819 0
1.2 0 −0.364448 0 0 0 0 0 −2.86718 0
1.3 0 2.56885 0 0 0 0 0 3.59899 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.bb 3
5.b even 2 1 4900.2.a.bd 3
5.c odd 4 2 4900.2.e.s 6
7.b odd 2 1 4900.2.a.bc 3
7.d odd 6 2 700.2.i.d 6
35.c odd 2 1 4900.2.a.ba 3
35.f even 4 2 4900.2.e.t 6
35.i odd 6 2 700.2.i.e yes 6
35.k even 12 4 700.2.r.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.i.d 6 7.d odd 6 2
700.2.i.e yes 6 35.i odd 6 2
700.2.r.d 12 35.k even 12 4
4900.2.a.ba 3 35.c odd 2 1
4900.2.a.bb 3 1.a even 1 1 trivial
4900.2.a.bc 3 7.b odd 2 1
4900.2.a.bd 3 5.b even 2 1
4900.2.e.s 6 5.c odd 4 2
4900.2.e.t 6 35.f even 4 2

## Hecke kernels

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

 $$T_{3}^{3} + T_{3}^{2} - 8 T_{3} - 3$$ $$T_{11}^{3} - 4 T_{11}^{2} - 3 T_{11} + 9$$ $$T_{13}^{3} + 8 T_{13}^{2} + 13 T_{13} - 3$$ $$T_{19}^{3} - 2 T_{19}^{2} - 23 T_{19} - 21$$ $$T_{23}^{3} + 3 T_{23}^{2} - 36 T_{23} - 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-3 - 8 T + T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$T^{3}$$
$11$ $$9 - 3 T - 4 T^{2} + T^{3}$$
$13$ $$-3 + 13 T + 8 T^{2} + T^{3}$$
$17$ $$-81 + 9 T + 10 T^{2} + T^{3}$$
$19$ $$-21 - 23 T - 2 T^{2} + T^{3}$$
$23$ $$-81 - 36 T + 3 T^{2} + T^{3}$$
$29$ $$81 - 45 T + T^{3}$$
$31$ $$-37 - 42 T - 3 T^{2} + T^{3}$$
$37$ $$149 - 63 T - 6 T^{2} + T^{3}$$
$41$ $$-873 - 78 T + 11 T^{2} + T^{3}$$
$43$ $$-71 + 14 T + 11 T^{2} + T^{3}$$
$47$ $$-9 - 21 T + 4 T^{2} + T^{3}$$
$53$ $$549 - 15 T - 14 T^{2} + T^{3}$$
$59$ $$9 - 5 T^{2} + T^{3}$$
$61$ $$1 + 26 T + 17 T^{2} + T^{3}$$
$67$ $$-72 + 100 T - 20 T^{2} + T^{3}$$
$71$ $$45 + 87 T + 19 T^{2} + T^{3}$$
$73$ $$( 4 + T )^{3}$$
$79$ $$449 - 118 T + T^{2} + T^{3}$$
$83$ $$-981 + 129 T + 28 T^{2} + T^{3}$$
$89$ $$45 + 39 T - 14 T^{2} + T^{3}$$
$97$ $$-5 - 18 T + 15 T^{2} + T^{3}$$