# Properties

 Label 5625.2.a.f Level $5625$ Weight $2$ Character orbit 5625.a Self dual yes Analytic conductor $44.916$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5625 = 3^{2} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5625.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$44.9158511370$$ Analytic rank: $$0$$ 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: no (minimal twist has level 25) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( -1 + \beta ) q^{4} + ( -1 + \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} +O(q^{10})$$ $$q + \beta q^{2} + ( -1 + \beta ) q^{4} + ( -1 + \beta ) q^{7} + ( 1 - 2 \beta ) q^{8} + ( 2 + 2 \beta ) q^{11} + ( -3 + 3 \beta ) q^{13} + q^{14} -3 \beta q^{16} + ( 2 + 2 \beta ) q^{17} + ( -4 + 3 \beta ) q^{19} + ( 2 + 4 \beta ) q^{22} + ( -7 + 2 \beta ) q^{23} + 3 q^{26} + ( 2 - \beta ) q^{28} + ( 2 + \beta ) q^{29} -3 q^{31} + ( -5 + \beta ) q^{32} + ( 2 + 4 \beta ) q^{34} + ( 3 - 2 \beta ) q^{37} + ( 3 - \beta ) q^{38} + ( 4 - 2 \beta ) q^{41} -3 \beta q^{43} + 2 \beta q^{44} + ( 2 - 5 \beta ) q^{46} + ( 1 - \beta ) q^{47} + ( -5 - \beta ) q^{49} + ( 6 - 3 \beta ) q^{52} + ( -3 + 4 \beta ) q^{53} + ( -3 + \beta ) q^{56} + ( 1 + 3 \beta ) q^{58} + ( 6 + 3 \beta ) q^{59} + ( -1 + 6 \beta ) q^{61} -3 \beta q^{62} + ( 1 + 2 \beta ) q^{64} + ( 8 - 2 \beta ) q^{67} + 2 \beta q^{68} + ( 5 + \beta ) q^{71} -9 q^{73} + ( -2 + \beta ) q^{74} + ( 7 - 4 \beta ) q^{76} + 2 \beta q^{77} -5 \beta q^{79} + ( -2 + 2 \beta ) q^{82} + ( 3 + 2 \beta ) q^{83} + ( -3 - 3 \beta ) q^{86} + ( -2 - 6 \beta ) q^{88} + ( -4 + 8 \beta ) q^{89} + ( 6 - 3 \beta ) q^{91} + ( 9 - 7 \beta ) q^{92} - q^{94} + ( 1 - 3 \beta ) q^{97} + ( -1 - 6 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} - q^{4} - q^{7} + O(q^{10})$$ $$2 q + q^{2} - q^{4} - q^{7} + 6 q^{11} - 3 q^{13} + 2 q^{14} - 3 q^{16} + 6 q^{17} - 5 q^{19} + 8 q^{22} - 12 q^{23} + 6 q^{26} + 3 q^{28} + 5 q^{29} - 6 q^{31} - 9 q^{32} + 8 q^{34} + 4 q^{37} + 5 q^{38} + 6 q^{41} - 3 q^{43} + 2 q^{44} - q^{46} + q^{47} - 11 q^{49} + 9 q^{52} - 2 q^{53} - 5 q^{56} + 5 q^{58} + 15 q^{59} + 4 q^{61} - 3 q^{62} + 4 q^{64} + 14 q^{67} + 2 q^{68} + 11 q^{71} - 18 q^{73} - 3 q^{74} + 10 q^{76} + 2 q^{77} - 5 q^{79} - 2 q^{82} + 8 q^{83} - 9 q^{86} - 10 q^{88} + 9 q^{91} + 11 q^{92} - 2 q^{94} - q^{97} - 8 q^{98} + O(q^{100})$$

## 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 0 −1.61803 0 0 −1.61803 2.23607 0 0
1.2 1.61803 0 0.618034 0 0 0.618034 −2.23607 0 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
$$3$$ $$-1$$
$$5$$ $$1$$

## 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 5625.2.a.f 2
3.b odd 2 1 625.2.a.b 2
5.b even 2 1 5625.2.a.d 2
12.b even 2 1 10000.2.a.c 2
15.d odd 2 1 625.2.a.c 2
15.e even 4 2 625.2.b.a 4
25.d even 5 2 225.2.h.b 4
60.h even 2 1 10000.2.a.l 2
75.h odd 10 2 125.2.d.a 4
75.h odd 10 2 625.2.d.b 4
75.j odd 10 2 25.2.d.a 4
75.j odd 10 2 625.2.d.h 4
75.l even 20 4 125.2.e.a 8
75.l even 20 4 625.2.e.c 8
300.n even 10 2 400.2.u.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 75.j odd 10 2
125.2.d.a 4 75.h odd 10 2
125.2.e.a 8 75.l even 20 4
225.2.h.b 4 25.d even 5 2
400.2.u.b 4 300.n even 10 2
625.2.a.b 2 3.b odd 2 1
625.2.a.c 2 15.d odd 2 1
625.2.b.a 4 15.e even 4 2
625.2.d.b 4 75.h odd 10 2
625.2.d.h 4 75.j odd 10 2
625.2.e.c 8 75.l even 20 4
5625.2.a.d 2 5.b even 2 1
5625.2.a.f 2 1.a even 1 1 trivial
10000.2.a.c 2 12.b even 2 1
10000.2.a.l 2 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5625))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$-1 + T + T^{2}$$
$11$ $$4 - 6 T + T^{2}$$
$13$ $$-9 + 3 T + T^{2}$$
$17$ $$4 - 6 T + T^{2}$$
$19$ $$-5 + 5 T + T^{2}$$
$23$ $$31 + 12 T + T^{2}$$
$29$ $$5 - 5 T + T^{2}$$
$31$ $$( 3 + T )^{2}$$
$37$ $$-1 - 4 T + T^{2}$$
$41$ $$4 - 6 T + T^{2}$$
$43$ $$-9 + 3 T + T^{2}$$
$47$ $$-1 - T + T^{2}$$
$53$ $$-19 + 2 T + T^{2}$$
$59$ $$45 - 15 T + T^{2}$$
$61$ $$-41 - 4 T + T^{2}$$
$67$ $$44 - 14 T + T^{2}$$
$71$ $$29 - 11 T + T^{2}$$
$73$ $$( 9 + T )^{2}$$
$79$ $$-25 + 5 T + T^{2}$$
$83$ $$11 - 8 T + T^{2}$$
$89$ $$-80 + T^{2}$$
$97$ $$-11 + T + T^{2}$$