# Properties

 Label 1250.2.a.a Level $1250$ Weight $2$ Character orbit 1250.a Self dual yes Analytic conductor $9.981$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1250 = 2 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1250.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.98130025266$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 50) 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.

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

## 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
 1.61803 −0.618034
−1.00000 −2.61803 1.00000 0 2.61803 −3.00000 −1.00000 3.85410 0
1.2 −1.00000 −0.381966 1.00000 0 0.381966 −3.00000 −1.00000 −2.85410 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
$$2$$ $$1$$
$$5$$ $$1$$

## 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 1250.2.a.a 2
4.b odd 2 1 10000.2.a.n 2
5.b even 2 1 1250.2.a.d 2
5.c odd 4 2 1250.2.b.b 4
20.d odd 2 1 10000.2.a.a 2
25.d even 5 2 50.2.d.a 4
25.e even 10 2 250.2.d.a 4
25.f odd 20 4 250.2.e.b 8
75.j odd 10 2 450.2.h.a 4
100.j odd 10 2 400.2.u.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.a 4 25.d even 5 2
250.2.d.a 4 25.e even 10 2
250.2.e.b 8 25.f odd 20 4
400.2.u.c 4 100.j odd 10 2
450.2.h.a 4 75.j odd 10 2
1250.2.a.a 2 1.a even 1 1 trivial
1250.2.a.d 2 5.b even 2 1
1250.2.b.b 4 5.c odd 4 2
10000.2.a.a 2 20.d odd 2 1
10000.2.a.n 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 3 T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1250))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$1 + 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$( 3 + T )^{2}$$
$11$ $$-1 - 4 T + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$9 - 9 T + T^{2}$$
$19$ $$-5 - 5 T + T^{2}$$
$23$ $$11 + 8 T + T^{2}$$
$29$ $$5 + 10 T + T^{2}$$
$31$ $$-1 - 4 T + T^{2}$$
$37$ $$-61 + T + T^{2}$$
$41$ $$-11 + 6 T + T^{2}$$
$43$ $$11 + 8 T + T^{2}$$
$47$ $$-71 + 6 T + T^{2}$$
$53$ $$16 - 12 T + T^{2}$$
$59$ $$-20 + T^{2}$$
$61$ $$19 + 11 T + T^{2}$$
$67$ $$59 + 16 T + T^{2}$$
$71$ $$( 3 + T )^{2}$$
$73$ $$-44 - 2 T + T^{2}$$
$79$ $$20 + 10 T + T^{2}$$
$83$ $$61 + 18 T + T^{2}$$
$89$ $$-20 + T^{2}$$
$97$ $$-101 + T + T^{2}$$
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