# Properties

 Label 1225.2.a.u Level $1225$ Weight $2$ Character orbit 1225.a Self dual yes Analytic conductor $9.782$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.78167424761$$ 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 175) 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.

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

## 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 3.23607 −1.61803 0 −2.00000 0 2.23607 7.47214 0
1.2 1.61803 −1.23607 0.618034 0 −2.00000 0 −2.23607 −1.47214 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
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 1225.2.a.u 2
5.b even 2 1 1225.2.a.n 2
5.c odd 4 2 1225.2.b.k 4
7.b odd 2 1 175.2.a.e yes 2
21.c even 2 1 1575.2.a.n 2
28.d even 2 1 2800.2.a.bp 2
35.c odd 2 1 175.2.a.d 2
35.f even 4 2 175.2.b.c 4
105.g even 2 1 1575.2.a.s 2
105.k odd 4 2 1575.2.d.k 4
140.c even 2 1 2800.2.a.bh 2
140.j odd 4 2 2800.2.g.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.2.a.d 2 35.c odd 2 1
175.2.a.e yes 2 7.b odd 2 1
175.2.b.c 4 35.f even 4 2
1225.2.a.n 2 5.b even 2 1
1225.2.a.u 2 1.a even 1 1 trivial
1225.2.b.k 4 5.c odd 4 2
1575.2.a.n 2 21.c even 2 1
1575.2.a.s 2 105.g even 2 1
1575.2.d.k 4 105.k odd 4 2
2800.2.a.bh 2 140.c even 2 1
2800.2.a.bp 2 28.d even 2 1
2800.2.g.s 4 140.j odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2}^{2} - T_{2} - 1$$ $$T_{3}^{2} - 2 T_{3} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - T + T^{2}$$
$3$ $$-4 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$-1 - 4 T + T^{2}$$
$13$ $$-4 - 2 T + T^{2}$$
$17$ $$-16 - 4 T + T^{2}$$
$19$ $$-20 + T^{2}$$
$23$ $$11 - 8 T + T^{2}$$
$29$ $$( -5 + T )^{2}$$
$31$ $$-36 - 6 T + T^{2}$$
$37$ $$( -3 + T )^{2}$$
$41$ $$44 + 14 T + T^{2}$$
$43$ $$11 - 8 T + T^{2}$$
$47$ $$( -2 + T )^{2}$$
$53$ $$-4 - 8 T + T^{2}$$
$59$ $$-20 + 10 T + T^{2}$$
$61$ $$-36 - 6 T + T^{2}$$
$67$ $$-1 + 4 T + T^{2}$$
$71$ $$-41 - 4 T + T^{2}$$
$73$ $$116 - 22 T + T^{2}$$
$79$ $$-125 + T^{2}$$
$83$ $$-44 - 2 T + T^{2}$$
$89$ $$220 + 30 T + T^{2}$$
$97$ $$4 + 6 T + T^{2}$$