# Properties

 Label 1900.2.a.g Level $1900$ Weight $2$ Character orbit 1900.a Self dual yes Analytic conductor $15.172$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.1715763840$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.321.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{7} + ( 2 - \beta_{1} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{2} ) q^{3} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{7} + ( 2 - \beta_{1} ) q^{9} + \beta_{2} q^{11} + ( -1 - \beta_{1} - \beta_{2} ) q^{13} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{17} + q^{19} + ( -1 - 3 \beta_{1} + 2 \beta_{2} ) q^{21} + ( -5 - \beta_{1} ) q^{23} + ( 2 \beta_{1} + \beta_{2} ) q^{27} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{29} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{31} + ( -4 + \beta_{1} + \beta_{2} ) q^{33} + ( 4 - \beta_{2} ) q^{37} + ( 4 + \beta_{1} ) q^{39} -3 \beta_{1} q^{41} + ( -5 - 2 \beta_{2} ) q^{43} + ( -1 - 5 \beta_{1} + \beta_{2} ) q^{47} + ( 6 + 3 \beta_{1} ) q^{49} + ( -5 + 5 \beta_{1} + 2 \beta_{2} ) q^{51} + ( -1 + 4 \beta_{1} ) q^{53} + ( -1 - \beta_{2} ) q^{57} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -3 + 5 \beta_{1} + \beta_{2} ) q^{61} + ( -7 + 2 \beta_{1} ) q^{63} + ( -4 - 4 \beta_{1} + 3 \beta_{2} ) q^{67} + ( 4 + 2 \beta_{1} + 5 \beta_{2} ) q^{69} + ( -1 - 2 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -4 - \beta_{1} + 2 \beta_{2} ) q^{73} + ( 2 + \beta_{1} - 3 \beta_{2} ) q^{77} + ( -1 - 5 \beta_{2} ) q^{79} + ( -8 + \beta_{2} ) q^{81} + ( -10 + \beta_{1} ) q^{83} + ( -8 + 5 \beta_{1} - \beta_{2} ) q^{87} + ( 3 + 3 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -6 - 5 \beta_{1} ) q^{91} + ( -5 \beta_{1} + 2 \beta_{2} ) q^{93} + ( -1 - \beta_{1} - 5 \beta_{2} ) q^{97} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 2q^{3} - 2q^{7} + 5q^{9} + O(q^{10})$$ $$3q - 2q^{3} - 2q^{7} + 5q^{9} - q^{11} - 3q^{13} - 6q^{17} + 3q^{19} - 8q^{21} - 16q^{23} + q^{27} + 3q^{29} - q^{31} - 12q^{33} + 13q^{37} + 13q^{39} - 3q^{41} - 13q^{43} - 9q^{47} + 21q^{49} - 12q^{51} + q^{53} - 2q^{57} - 6q^{59} - 5q^{61} - 19q^{63} - 19q^{67} + 9q^{69} - 3q^{71} - 15q^{73} + 10q^{77} + 2q^{79} - 25q^{81} - 29q^{83} - 18q^{87} + 14q^{89} - 23q^{91} - 7q^{93} + q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$

## 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
 −1.69963 2.46050 0.239123
0 −2.58836 0 0 0 −2.81089 0 3.69963 0
1.2 0 −1.59358 0 0 0 4.51459 0 −0.460505 0
1.3 0 2.18194 0 0 0 −3.70370 0 1.76088 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
$$2$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$-1$$

## 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 1900.2.a.g 3
4.b odd 2 1 7600.2.a.ca 3
5.b even 2 1 1900.2.a.i yes 3
5.c odd 4 2 1900.2.c.f 6
20.d odd 2 1 7600.2.a.bl 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.a.g 3 1.a even 1 1 trivial
1900.2.a.i yes 3 5.b even 2 1
1900.2.c.f 6 5.c odd 4 2
7600.2.a.bl 3 20.d odd 2 1
7600.2.a.ca 3 4.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1900))$$:

 $$T_{3}^{3} + 2 T_{3}^{2} - 5 T_{3} - 9$$ $$T_{7}^{3} + 2 T_{7}^{2} - 19 T_{7} - 47$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-9 - 5 T + 2 T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$-47 - 19 T + 2 T^{2} + T^{3}$$
$11$ $$3 - 6 T + T^{2} + T^{3}$$
$13$ $$-7 - 6 T + 3 T^{2} + T^{3}$$
$17$ $$-99 - 15 T + 6 T^{2} + T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$129 + 81 T + 16 T^{2} + T^{3}$$
$29$ $$-27 - 24 T - 3 T^{2} + T^{3}$$
$31$ $$-109 - 40 T + T^{2} + T^{3}$$
$37$ $$-59 + 50 T - 13 T^{2} + T^{3}$$
$41$ $$-27 - 36 T + 3 T^{2} + T^{3}$$
$43$ $$-69 + 31 T + 13 T^{2} + T^{3}$$
$47$ $$-621 - 96 T + 9 T^{2} + T^{3}$$
$53$ $$-3 - 69 T - T^{2} + T^{3}$$
$59$ $$-72 - 24 T + 6 T^{2} + T^{3}$$
$61$ $$-489 - 98 T + 5 T^{2} + T^{3}$$
$67$ $$-1323 - 26 T + 19 T^{2} + T^{3}$$
$71$ $$-27 - 33 T + 3 T^{2} + T^{3}$$
$73$ $$-49 + 42 T + 15 T^{2} + T^{3}$$
$79$ $$-529 - 157 T - 2 T^{2} + T^{3}$$
$83$ $$861 + 276 T + 29 T^{2} + T^{3}$$
$89$ $$489 - 9 T - 14 T^{2} + T^{3}$$
$97$ $$-683 - 154 T - T^{2} + T^{3}$$