# Properties

 Label 1425.2.a.v Level $1425$ Weight $2$ Character orbit 1425.a Self dual yes Analytic conductor $11.379$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.3786822880$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 Defining polynomial: $$x^{3} - 6 x - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{1} + \beta_{2} ) q^{4} -\beta_{1} q^{6} + \beta_{2} q^{7} + ( -3 - 2 \beta_{1} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + q^{3} + ( 2 + \beta_{1} + \beta_{2} ) q^{4} -\beta_{1} q^{6} + \beta_{2} q^{7} + ( -3 - 2 \beta_{1} ) q^{8} + q^{9} + ( -1 - \beta_{2} ) q^{11} + ( 2 + \beta_{1} + \beta_{2} ) q^{12} + ( 2 - 2 \beta_{1} ) q^{13} + ( 1 - \beta_{1} + \beta_{2} ) q^{14} + ( 4 + 3 \beta_{1} ) q^{16} -\beta_{1} q^{18} + q^{19} + \beta_{2} q^{21} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{22} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{23} + ( -3 - 2 \beta_{1} ) q^{24} + ( 8 + 2 \beta_{2} ) q^{26} + q^{27} + ( 5 - \beta_{1} ) q^{28} + 3 q^{29} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{31} + ( -6 - 3 \beta_{1} - 3 \beta_{2} ) q^{32} + ( -1 - \beta_{2} ) q^{33} + ( 2 + \beta_{1} + \beta_{2} ) q^{36} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{37} -\beta_{1} q^{38} + ( 2 - 2 \beta_{1} ) q^{39} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{41} + ( 1 - \beta_{1} + \beta_{2} ) q^{42} + 8 q^{43} + ( -7 - \beta_{2} ) q^{44} + ( 7 + 4 \beta_{1} + \beta_{2} ) q^{46} + ( -2 + 2 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 4 + 3 \beta_{1} ) q^{48} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{49} + ( -2 - 6 \beta_{1} + 2 \beta_{2} ) q^{52} + ( 3 + 4 \beta_{1} ) q^{53} -\beta_{1} q^{54} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{56} + q^{57} -3 \beta_{1} q^{58} + ( 8 - \beta_{2} ) q^{59} + ( -3 - 2 \beta_{2} ) q^{61} + ( -7 - 6 \beta_{1} - \beta_{2} ) q^{62} + \beta_{2} q^{63} + ( 1 + 6 \beta_{1} ) q^{64} + ( -1 + 2 \beta_{1} - \beta_{2} ) q^{66} + ( 3 + 4 \beta_{1} + \beta_{2} ) q^{67} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{69} + ( 2 - 4 \beta_{1} - \beta_{2} ) q^{71} + ( -3 - 2 \beta_{1} ) q^{72} + ( -1 - 4 \beta_{1} ) q^{73} + ( -6 - 8 \beta_{1} ) q^{74} + ( 2 + \beta_{1} + \beta_{2} ) q^{76} + ( -6 + 2 \beta_{1} ) q^{77} + ( 8 + 2 \beta_{2} ) q^{78} + ( 5 + 2 \beta_{1} - 3 \beta_{2} ) q^{79} + q^{81} + ( -11 + \beta_{1} - 5 \beta_{2} ) q^{82} + ( -5 + \beta_{2} ) q^{83} + ( 5 - \beta_{1} ) q^{84} -8 \beta_{1} q^{86} + 3 q^{87} + ( 1 + 4 \beta_{1} + \beta_{2} ) q^{88} + ( -1 + 2 \beta_{2} ) q^{89} + ( 2 - 2 \beta_{1} + 4 \beta_{2} ) q^{91} + ( -13 - 8 \beta_{1} - \beta_{2} ) q^{92} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{93} + ( -10 + 2 \beta_{1} - 4 \beta_{2} ) q^{94} + ( -6 - 3 \beta_{1} - 3 \beta_{2} ) q^{96} + ( -4 + 6 \beta_{1} ) q^{97} + ( 7 + 4 \beta_{1} + \beta_{2} ) q^{98} + ( -1 - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} + 6q^{4} - 9q^{8} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{3} + 6q^{4} - 9q^{8} + 3q^{9} - 3q^{11} + 6q^{12} + 6q^{13} + 3q^{14} + 12q^{16} + 3q^{19} - 3q^{22} - 3q^{23} - 9q^{24} + 24q^{26} + 3q^{27} + 15q^{28} + 9q^{29} + 9q^{31} - 18q^{32} - 3q^{33} + 6q^{36} + 12q^{37} + 6q^{39} + 3q^{42} + 24q^{43} - 21q^{44} + 21q^{46} - 6q^{47} + 12q^{48} - 3q^{49} - 6q^{52} + 9q^{53} + 6q^{56} + 3q^{57} + 24q^{59} - 9q^{61} - 21q^{62} + 3q^{64} - 3q^{66} + 9q^{67} - 3q^{69} + 6q^{71} - 9q^{72} - 3q^{73} - 18q^{74} + 6q^{76} - 18q^{77} + 24q^{78} + 15q^{79} + 3q^{81} - 33q^{82} - 15q^{83} + 15q^{84} + 9q^{87} + 3q^{88} - 3q^{89} + 6q^{91} - 39q^{92} + 9q^{93} - 30q^{94} - 18q^{96} - 12q^{97} + 21q^{98} - 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6 x - 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 4$$

## 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
 2.66908 −0.523976 −2.14510
−2.66908 1.00000 5.12398 0 −2.66908 0.454904 −8.33816 1.00000 0
1.2 0.523976 1.00000 −1.72545 0 0.523976 −3.20147 −1.95205 1.00000 0
1.3 2.14510 1.00000 2.60147 0 2.14510 2.74657 1.29021 1.00000 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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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 1425.2.a.v yes 3
3.b odd 2 1 4275.2.a.bh 3
5.b even 2 1 1425.2.a.u 3
5.c odd 4 2 1425.2.c.p 6
15.d odd 2 1 4275.2.a.be 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1425.2.a.u 3 5.b even 2 1
1425.2.a.v yes 3 1.a even 1 1 trivial
1425.2.c.p 6 5.c odd 4 2
4275.2.a.be 3 15.d odd 2 1
4275.2.a.bh 3 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1425))$$:

 $$T_{2}^{3} - 6 T_{2} + 3$$ $$T_{7}^{3} - 9 T_{7} + 4$$ $$T_{11}^{3} + 3 T_{11}^{2} - 6 T_{11} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 - 6 T + T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$4 - 9 T + T^{3}$$
$11$ $$-12 - 6 T + 3 T^{2} + T^{3}$$
$13$ $$64 - 12 T - 6 T^{2} + T^{3}$$
$17$ $$T^{3}$$
$19$ $$( -1 + T )^{3}$$
$23$ $$12 - 24 T + 3 T^{2} + T^{3}$$
$29$ $$( -3 + T )^{3}$$
$31$ $$16 - 9 T^{2} + T^{3}$$
$37$ $$184 - 12 T^{2} + T^{3}$$
$41$ $$426 - 123 T + T^{3}$$
$43$ $$( -8 + T )^{3}$$
$47$ $$96 - 60 T + 6 T^{2} + T^{3}$$
$53$ $$69 - 69 T - 9 T^{2} + T^{3}$$
$59$ $$-444 + 183 T - 24 T^{2} + T^{3}$$
$61$ $$-113 - 9 T + 9 T^{2} + T^{3}$$
$67$ $$-92 - 66 T - 9 T^{2} + T^{3}$$
$71$ $$522 - 81 T - 6 T^{2} + T^{3}$$
$73$ $$97 - 93 T + 3 T^{2} + T^{3}$$
$79$ $$916 - 48 T - 15 T^{2} + T^{3}$$
$83$ $$84 + 66 T + 15 T^{2} + T^{3}$$
$89$ $$-3 - 33 T + 3 T^{2} + T^{3}$$
$97$ $$-1448 - 168 T + 12 T^{2} + T^{3}$$