# Properties

 Label 1900.2.a.h Level $1900$ Weight $2$ Character orbit 1900.a Self dual yes Analytic conductor $15.172$ Analytic rank $0$ 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: $$0$$ 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: $$1$$ 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.

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

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

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

## 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
 2.19869 0.713538 −1.91223
0 −1.19869 0 0 0 −1.19869 0 −1.56314 0
1.2 0 0.286462 0 0 0 0.286462 0 −2.91794 0
1.3 0 2.91223 0 0 0 2.91223 0 5.48108 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.h yes 3
4.b odd 2 1 7600.2.a.bj 3
5.b even 2 1 1900.2.a.f 3
5.c odd 4 2 1900.2.c.g 6
20.d odd 2 1 7600.2.a.by 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.a.f 3 5.b even 2 1
1900.2.a.h yes 3 1.a even 1 1 trivial
1900.2.c.g 6 5.c odd 4 2
7600.2.a.bj 3 4.b odd 2 1
7600.2.a.by 3 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1900))$$:

 $$T_{3}^{3} - 2 T_{3}^{2} - 3 T_{3} + 1$$ $$T_{7}^{3} - 2 T_{7}^{2} - 3 T_{7} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$1 - 3 T - 2 T^{2} + T^{3}$$
$5$ $$T^{3}$$
$7$ $$1 - 3 T - 2 T^{2} + T^{3}$$
$11$ $$15 - 26 T + T^{2} + T^{3}$$
$13$ $$3 - 8 T - T^{2} + T^{3}$$
$17$ $$1 - 3 T - 2 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$9 - 3 T - 8 T^{2} + T^{3}$$
$29$ $$-25 - 10 T + 7 T^{2} + T^{3}$$
$31$ $$241 - 6 T - 11 T^{2} + T^{3}$$
$37$ $$-405 - 72 T + 5 T^{2} + T^{3}$$
$41$ $$-25 + 36 T - 13 T^{2} + T^{3}$$
$43$ $$27 + 23 T - 11 T^{2} + T^{3}$$
$47$ $$63 + 68 T - 19 T^{2} + T^{3}$$
$53$ $$27 + 23 T - 11 T^{2} + T^{3}$$
$59$ $$-856 - 176 T + 6 T^{2} + T^{3}$$
$61$ $$-25 - 30 T - 7 T^{2} + T^{3}$$
$67$ $$599 - 128 T - 3 T^{2} + T^{3}$$
$71$ $$123 - 73 T + 5 T^{2} + T^{3}$$
$73$ $$27 - 18 T - 9 T^{2} + T^{3}$$
$79$ $$-607 + T + 18 T^{2} + T^{3}$$
$83$ $$749 - 248 T - 3 T^{2} + T^{3}$$
$89$ $$-857 - 183 T + T^{3}$$
$97$ $$-81 - 72 T + 3 T^{2} + T^{3}$$