# Properties

 Label 1425.2.a.t 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: 3.3.837.1 Defining polynomial: $$x^{3} - 6 x - 1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + ( \beta_{1} + \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} - q^{3} + ( 2 + \beta_{2} ) q^{4} -\beta_{1} q^{6} + ( \beta_{1} + \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} ) q^{8} + q^{9} + ( -1 - \beta_{1} + \beta_{2} ) q^{11} + ( -2 - \beta_{2} ) q^{12} -4 q^{13} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{14} + ( 4 + \beta_{1} ) q^{16} + ( 2 + 2 \beta_{2} ) q^{17} + \beta_{1} q^{18} - q^{19} + ( -\beta_{1} - \beta_{2} ) q^{21} + ( -3 + \beta_{1} - \beta_{2} ) q^{22} + ( 3 - \beta_{1} - \beta_{2} ) q^{23} + ( -1 - 2 \beta_{1} ) q^{24} -4 \beta_{1} q^{26} - q^{27} + ( 9 + 5 \beta_{1} ) q^{28} -5 q^{29} + ( 5 + \beta_{1} - \beta_{2} ) q^{31} + ( 2 + \beta_{2} ) q^{32} + ( 1 + \beta_{1} - \beta_{2} ) q^{33} + ( 2 + 6 \beta_{1} ) q^{34} + ( 2 + \beta_{2} ) q^{36} + ( -4 - 2 \beta_{1} ) q^{37} -\beta_{1} q^{38} + 4 q^{39} + ( 4 - \beta_{1} + \beta_{2} ) q^{41} + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{42} -4 q^{43} + ( 5 - 3 \beta_{1} - \beta_{2} ) q^{44} + ( -5 + \beta_{1} - \beta_{2} ) q^{46} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{47} + ( -4 - \beta_{1} ) q^{48} + ( 7 + 5 \beta_{1} - \beta_{2} ) q^{49} + ( -2 - 2 \beta_{2} ) q^{51} + ( -8 - 4 \beta_{2} ) q^{52} + ( 3 + 4 \beta_{1} ) q^{53} -\beta_{1} q^{54} + ( 10 + 5 \beta_{1} + 3 \beta_{2} ) q^{56} + q^{57} -5 \beta_{1} q^{58} + ( 4 + 3 \beta_{1} - \beta_{2} ) q^{59} + ( 1 - 2 \beta_{1} + 2 \beta_{2} ) q^{61} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{62} + ( \beta_{1} + \beta_{2} ) q^{63} + ( -7 + 2 \beta_{1} ) q^{64} + ( 3 - \beta_{1} + \beta_{2} ) q^{66} + ( -5 + \beta_{1} - \beta_{2} ) q^{67} + ( 20 + 2 \beta_{1} + 2 \beta_{2} ) q^{68} + ( -3 + \beta_{1} + \beta_{2} ) q^{69} + ( 6 + 3 \beta_{1} - \beta_{2} ) q^{71} + ( 1 + 2 \beta_{1} ) q^{72} + ( -1 - 4 \beta_{1} ) q^{73} + ( -8 - 4 \beta_{1} - 2 \beta_{2} ) q^{74} + ( -2 - \beta_{2} ) q^{76} + ( 4 - 4 \beta_{2} ) q^{77} + 4 \beta_{1} q^{78} + ( 1 - 3 \beta_{1} - 3 \beta_{2} ) q^{79} + q^{81} + ( -3 + 6 \beta_{1} - \beta_{2} ) q^{82} + ( -1 + \beta_{1} - \beta_{2} ) q^{83} + ( -9 - 5 \beta_{1} ) q^{84} -4 \beta_{1} q^{86} + 5 q^{87} + ( -7 + \beta_{1} - \beta_{2} ) q^{88} + ( -1 - 2 \beta_{1} - 2 \beta_{2} ) q^{89} + ( -4 \beta_{1} - 4 \beta_{2} ) q^{91} + ( -3 - 5 \beta_{1} + 3 \beta_{2} ) q^{92} + ( -5 - \beta_{1} + \beta_{2} ) q^{93} + ( -6 - 2 \beta_{2} ) q^{94} + ( -2 - \beta_{2} ) q^{96} + ( 4 + 4 \beta_{1} - 2 \beta_{2} ) q^{97} + ( 19 + 5 \beta_{1} + 5 \beta_{2} ) q^{98} + ( -1 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 6 q^{4} + 3 q^{8} + 3 q^{9} + O(q^{10})$$ $$3 q - 3 q^{3} + 6 q^{4} + 3 q^{8} + 3 q^{9} - 3 q^{11} - 6 q^{12} - 12 q^{13} + 15 q^{14} + 12 q^{16} + 6 q^{17} - 3 q^{19} - 9 q^{22} + 9 q^{23} - 3 q^{24} - 3 q^{27} + 27 q^{28} - 15 q^{29} + 15 q^{31} + 6 q^{32} + 3 q^{33} + 6 q^{34} + 6 q^{36} - 12 q^{37} + 12 q^{39} + 12 q^{41} - 15 q^{42} - 12 q^{43} + 15 q^{44} - 15 q^{46} - 12 q^{47} - 12 q^{48} + 21 q^{49} - 6 q^{51} - 24 q^{52} + 9 q^{53} + 30 q^{56} + 3 q^{57} + 12 q^{59} + 3 q^{61} + 9 q^{62} - 21 q^{64} + 9 q^{66} - 15 q^{67} + 60 q^{68} - 9 q^{69} + 18 q^{71} + 3 q^{72} - 3 q^{73} - 24 q^{74} - 6 q^{76} + 12 q^{77} + 3 q^{79} + 3 q^{81} - 9 q^{82} - 3 q^{83} - 27 q^{84} + 15 q^{87} - 21 q^{88} - 3 q^{89} - 9 q^{92} - 15 q^{93} - 18 q^{94} - 6 q^{96} + 12 q^{97} + 57 q^{98} - 3 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6 x - 1$$:

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

## 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
 −2.36147 −0.167449 2.52892
−2.36147 −1.00000 3.57653 0 2.36147 −0.784934 −3.72294 1.00000 0
1.2 −0.167449 −1.00000 −1.97196 0 0.167449 −4.13941 0.665102 1.00000 0
1.3 2.52892 −1.00000 4.39543 0 −2.52892 4.92434 6.05784 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.t 3
3.b odd 2 1 4275.2.a.bf 3
5.b even 2 1 1425.2.a.w yes 3
5.c odd 4 2 1425.2.c.o 6
15.d odd 2 1 4275.2.a.bg 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1425.2.a.t 3 1.a even 1 1 trivial
1425.2.a.w yes 3 5.b even 2 1
1425.2.c.o 6 5.c odd 4 2
4275.2.a.bf 3 3.b odd 2 1
4275.2.a.bg 3 15.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1425))$$:

 $$T_{2}^{3} - 6 T_{2} - 1$$ $$T_{7}^{3} - 21 T_{7} - 16$$ $$T_{11}^{3} + 3 T_{11}^{2} - 12 T_{11} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 6 T + T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$T^{3}$$
$7$ $$-16 - 21 T + T^{3}$$
$11$ $$-16 - 12 T + 3 T^{2} + T^{3}$$
$13$ $$( 4 + T )^{3}$$
$17$ $$208 - 36 T - 6 T^{2} + T^{3}$$
$19$ $$( 1 + T )^{3}$$
$23$ $$52 + 6 T - 9 T^{2} + T^{3}$$
$29$ $$( 5 + T )^{3}$$
$31$ $$-48 + 60 T - 15 T^{2} + T^{3}$$
$37$ $$-24 + 24 T + 12 T^{2} + T^{3}$$
$41$ $$-6 + 33 T - 12 T^{2} + T^{3}$$
$43$ $$( 4 + T )^{3}$$
$47$ $$-192 - 12 T + 12 T^{2} + T^{3}$$
$53$ $$197 - 69 T - 9 T^{2} + T^{3}$$
$59$ $$320 - 9 T - 12 T^{2} + T^{3}$$
$61$ $$43 - 57 T - 3 T^{2} + T^{3}$$
$67$ $$52 + 60 T + 15 T^{2} + T^{3}$$
$71$ $$282 + 51 T - 18 T^{2} + T^{3}$$
$73$ $$-31 - 93 T + 3 T^{2} + T^{3}$$
$79$ $$620 - 186 T - 3 T^{2} + T^{3}$$
$83$ $$-12 - 12 T + 3 T^{2} + T^{3}$$
$89$ $$45 - 81 T + 3 T^{2} + T^{3}$$
$97$ $$904 - 72 T - 12 T^{2} + T^{3}$$