# Properties

 Label 1425.2.a.r 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.

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

## 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 0.236068 2.23607 1.00000 0
1.2 1.61803 1.00000 0.618034 0 1.61803 −4.23607 −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.r yes 2
3.b odd 2 1 4275.2.a.r 2
5.b even 2 1 1425.2.a.m 2
5.c odd 4 2 1425.2.c.n 4
15.d odd 2 1 4275.2.a.w 2

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

 $$T_{2}^{2} - T_{2} - 1$$ $$T_{7}^{2} + 4 T_{7} - 1$$ $$T_{11}^{2} + 6 T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$T^{2}$$
$7$ $$-1 + 4 T + T^{2}$$
$11$ $$4 + 6 T + T^{2}$$
$13$ $$-4 + 2 T + T^{2}$$
$17$ $$-16 + 4 T + T^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$-4 - 8 T + T^{2}$$
$29$ $$5 + 10 T + T^{2}$$
$31$ $$( 8 + T )^{2}$$
$37$ $$44 + 14 T + T^{2}$$
$41$ $$-11 + 6 T + T^{2}$$
$43$ $$-64 - 8 T + T^{2}$$
$47$ $$4 - 6 T + T^{2}$$
$53$ $$( 1 + T )^{2}$$
$59$ $$-45 + T^{2}$$
$61$ $$-11 + 6 T + T^{2}$$
$67$ $$-36 - 6 T + T^{2}$$
$71$ $$19 + 16 T + T^{2}$$
$73$ $$-179 + 2 T + T^{2}$$
$79$ $$-20 + T^{2}$$
$83$ $$-44 + 2 T + T^{2}$$
$89$ $$( 5 + T )^{2}$$
$97$ $$-36 - 6 T + T^{2}$$