# Properties

 Label 1425.2.a.q Level $1425$ Weight $2$ Character orbit 1425.a Self dual yes Analytic conductor $11.379$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −0.618034 1.61803
−0.618034 −1.00000 −1.61803 0 0.618034 −2.23607 2.23607 1.00000 0
1.2 1.61803 −1.00000 0.618034 0 −1.61803 2.23607 −2.23607 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.q yes 2
3.b odd 2 1 4275.2.a.s 2
5.b even 2 1 1425.2.a.n 2
5.c odd 4 2 1425.2.c.m 4
15.d odd 2 1 4275.2.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1425.2.a.n 2 5.b even 2 1
1425.2.a.q yes 2 1.a even 1 1 trivial
1425.2.c.m 4 5.c odd 4 2
4275.2.a.s 2 3.b odd 2 1
4275.2.a.v 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1425))$$:

 $$T_{2}^{2} - T_{2} - 1$$ $$T_{7}^{2} - 5$$ $$T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - T + T^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-5 + T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$-16 - 4 T + T^{2}$$
$17$ $$-4 + 2 T + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$-4 - 2 T + T^{2}$$
$29$ $$-11 - 6 T + T^{2}$$
$31$ $$16 + 12 T + T^{2}$$
$37$ $$20 - 10 T + T^{2}$$
$41$ $$( 5 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$-4 + 8 T + T^{2}$$
$53$ $$( 5 + T )^{2}$$
$59$ $$19 + 16 T + T^{2}$$
$61$ $$-79 + 2 T + T^{2}$$
$67$ $$-76 + 4 T + T^{2}$$
$71$ $$-121 + 4 T + T^{2}$$
$73$ $$-19 + 2 T + T^{2}$$
$79$ $$36 + 18 T + T^{2}$$
$83$ $$-76 - 4 T + T^{2}$$
$89$ $$-79 + 2 T + T^{2}$$
$97$ $$-20 - 10 T + T^{2}$$