# Properties

 Label 125.2.a.a Level $125$ Weight $2$ Character orbit 125.a Self dual yes Analytic conductor $0.998$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$125 = 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 125.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.998130025266$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + (\beta - 2) q^{3} + (\beta - 1) q^{4} + (\beta - 1) q^{6} - 3 q^{7} + (2 \beta - 1) q^{8} + ( - 3 \beta + 2) q^{9} +O(q^{10})$$ q - b * q^2 + (b - 2) * q^3 + (b - 1) * q^4 + (b - 1) * q^6 - 3 * q^7 + (2*b - 1) * q^8 + (-3*b + 2) * q^9 $$q - \beta q^{2} + (\beta - 2) q^{3} + (\beta - 1) q^{4} + (\beta - 1) q^{6} - 3 q^{7} + (2 \beta - 1) q^{8} + ( - 3 \beta + 2) q^{9} - 3 q^{11} + ( - 2 \beta + 3) q^{12} - 3 \beta q^{13} + 3 \beta q^{14} - 3 \beta q^{16} + (2 \beta + 1) q^{17} + (\beta + 3) q^{18} + ( - \beta - 2) q^{19} + ( - 3 \beta + 6) q^{21} + 3 \beta q^{22} + ( - 2 \beta + 2) q^{23} + ( - 3 \beta + 4) q^{24} + (3 \beta + 3) q^{26} + (2 \beta - 1) q^{27} + ( - 3 \beta + 3) q^{28} + (6 \beta - 3) q^{29} + (5 \beta - 3) q^{31} + ( - \beta + 5) q^{32} + ( - 3 \beta + 6) q^{33} + ( - 3 \beta - 2) q^{34} + (2 \beta - 5) q^{36} + (6 \beta - 6) q^{37} + (3 \beta + 1) q^{38} + (3 \beta - 3) q^{39} - 3 q^{41} + ( - 3 \beta + 3) q^{42} - 9 q^{43} + ( - 3 \beta + 3) q^{44} + 2 q^{46} + ( - 7 \beta + 3) q^{47} + (3 \beta - 3) q^{48} + 2 q^{49} - \beta q^{51} - 3 q^{52} + (\beta + 3) q^{53} + ( - \beta - 2) q^{54} + ( - 6 \beta + 3) q^{56} + ( - \beta + 3) q^{57} + ( - 3 \beta - 6) q^{58} + ( - 3 \beta + 9) q^{59} + ( - 5 \beta + 2) q^{61} + ( - 2 \beta - 5) q^{62} + (9 \beta - 6) q^{63} + (2 \beta + 1) q^{64} + ( - 3 \beta + 3) q^{66} + ( - 3 \beta - 9) q^{67} + (\beta + 1) q^{68} + (4 \beta - 6) q^{69} - 3 q^{71} + (\beta - 8) q^{72} + (3 \beta - 3) q^{73} - 6 q^{74} + ( - 2 \beta + 1) q^{76} + 9 q^{77} - 3 q^{78} + ( - 4 \beta + 7) q^{79} + (6 \beta - 2) q^{81} + 3 \beta q^{82} + (4 \beta - 6) q^{83} + (6 \beta - 9) q^{84} + 9 \beta q^{86} + ( - 9 \beta + 12) q^{87} + ( - 6 \beta + 3) q^{88} + ( - 12 \beta + 6) q^{89} + 9 \beta q^{91} + (2 \beta - 4) q^{92} + ( - 8 \beta + 11) q^{93} + (4 \beta + 7) q^{94} + (6 \beta - 11) q^{96} + (3 \beta + 3) q^{97} - 2 \beta q^{98} + (9 \beta - 6) q^{99} +O(q^{100})$$ q - b * q^2 + (b - 2) * q^3 + (b - 1) * q^4 + (b - 1) * q^6 - 3 * q^7 + (2*b - 1) * q^8 + (-3*b + 2) * q^9 - 3 * q^11 + (-2*b + 3) * q^12 - 3*b * q^13 + 3*b * q^14 - 3*b * q^16 + (2*b + 1) * q^17 + (b + 3) * q^18 + (-b - 2) * q^19 + (-3*b + 6) * q^21 + 3*b * q^22 + (-2*b + 2) * q^23 + (-3*b + 4) * q^24 + (3*b + 3) * q^26 + (2*b - 1) * q^27 + (-3*b + 3) * q^28 + (6*b - 3) * q^29 + (5*b - 3) * q^31 + (-b + 5) * q^32 + (-3*b + 6) * q^33 + (-3*b - 2) * q^34 + (2*b - 5) * q^36 + (6*b - 6) * q^37 + (3*b + 1) * q^38 + (3*b - 3) * q^39 - 3 * q^41 + (-3*b + 3) * q^42 - 9 * q^43 + (-3*b + 3) * q^44 + 2 * q^46 + (-7*b + 3) * q^47 + (3*b - 3) * q^48 + 2 * q^49 - b * q^51 - 3 * q^52 + (b + 3) * q^53 + (-b - 2) * q^54 + (-6*b + 3) * q^56 + (-b + 3) * q^57 + (-3*b - 6) * q^58 + (-3*b + 9) * q^59 + (-5*b + 2) * q^61 + (-2*b - 5) * q^62 + (9*b - 6) * q^63 + (2*b + 1) * q^64 + (-3*b + 3) * q^66 + (-3*b - 9) * q^67 + (b + 1) * q^68 + (4*b - 6) * q^69 - 3 * q^71 + (b - 8) * q^72 + (3*b - 3) * q^73 - 6 * q^74 + (-2*b + 1) * q^76 + 9 * q^77 - 3 * q^78 + (-4*b + 7) * q^79 + (6*b - 2) * q^81 + 3*b * q^82 + (4*b - 6) * q^83 + (6*b - 9) * q^84 + 9*b * q^86 + (-9*b + 12) * q^87 + (-6*b + 3) * q^88 + (-12*b + 6) * q^89 + 9*b * q^91 + (2*b - 4) * q^92 + (-8*b + 11) * q^93 + (4*b + 7) * q^94 + (6*b - 11) * q^96 + (3*b + 3) * q^97 - 2*b * q^98 + (9*b - 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9}+O(q^{10})$$ 2 * q - q^2 - 3 * q^3 - q^4 - q^6 - 6 * q^7 + q^9 $$2 q - q^{2} - 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9} - 6 q^{11} + 4 q^{12} - 3 q^{13} + 3 q^{14} - 3 q^{16} + 4 q^{17} + 7 q^{18} - 5 q^{19} + 9 q^{21} + 3 q^{22} + 2 q^{23} + 5 q^{24} + 9 q^{26} + 3 q^{28} - q^{31} + 9 q^{32} + 9 q^{33} - 7 q^{34} - 8 q^{36} - 6 q^{37} + 5 q^{38} - 3 q^{39} - 6 q^{41} + 3 q^{42} - 18 q^{43} + 3 q^{44} + 4 q^{46} - q^{47} - 3 q^{48} + 4 q^{49} - q^{51} - 6 q^{52} + 7 q^{53} - 5 q^{54} + 5 q^{57} - 15 q^{58} + 15 q^{59} - q^{61} - 12 q^{62} - 3 q^{63} + 4 q^{64} + 3 q^{66} - 21 q^{67} + 3 q^{68} - 8 q^{69} - 6 q^{71} - 15 q^{72} - 3 q^{73} - 12 q^{74} + 18 q^{77} - 6 q^{78} + 10 q^{79} + 2 q^{81} + 3 q^{82} - 8 q^{83} - 12 q^{84} + 9 q^{86} + 15 q^{87} + 9 q^{91} - 6 q^{92} + 14 q^{93} + 18 q^{94} - 16 q^{96} + 9 q^{97} - 2 q^{98} - 3 q^{99}+O(q^{100})$$ 2 * q - q^2 - 3 * q^3 - q^4 - q^6 - 6 * q^7 + q^9 - 6 * q^11 + 4 * q^12 - 3 * q^13 + 3 * q^14 - 3 * q^16 + 4 * q^17 + 7 * q^18 - 5 * q^19 + 9 * q^21 + 3 * q^22 + 2 * q^23 + 5 * q^24 + 9 * q^26 + 3 * q^28 - q^31 + 9 * q^32 + 9 * q^33 - 7 * q^34 - 8 * q^36 - 6 * q^37 + 5 * q^38 - 3 * q^39 - 6 * q^41 + 3 * q^42 - 18 * q^43 + 3 * q^44 + 4 * q^46 - q^47 - 3 * q^48 + 4 * q^49 - q^51 - 6 * q^52 + 7 * q^53 - 5 * q^54 + 5 * q^57 - 15 * q^58 + 15 * q^59 - q^61 - 12 * q^62 - 3 * q^63 + 4 * q^64 + 3 * q^66 - 21 * q^67 + 3 * q^68 - 8 * q^69 - 6 * q^71 - 15 * q^72 - 3 * q^73 - 12 * q^74 + 18 * q^77 - 6 * q^78 + 10 * q^79 + 2 * q^81 + 3 * q^82 - 8 * q^83 - 12 * q^84 + 9 * q^86 + 15 * q^87 + 9 * q^91 - 6 * q^92 + 14 * q^93 + 18 * q^94 - 16 * q^96 + 9 * q^97 - 2 * q^98 - 3 * q^99

## 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
−1.61803 −0.381966 0.618034 0 0.618034 −3.00000 2.23607 −2.85410 0
1.2 0.618034 −2.61803 −1.61803 0 −1.61803 −3.00000 −2.23607 3.85410 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
$$5$$ $$1$$

## 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 125.2.a.a 2
3.b odd 2 1 1125.2.a.d 2
4.b odd 2 1 2000.2.a.l 2
5.b even 2 1 125.2.a.b yes 2
5.c odd 4 2 125.2.b.b 4
7.b odd 2 1 6125.2.a.d 2
8.b even 2 1 8000.2.a.v 2
8.d odd 2 1 8000.2.a.c 2
15.d odd 2 1 1125.2.a.c 2
15.e even 4 2 1125.2.b.f 4
20.d odd 2 1 2000.2.a.a 2
20.e even 4 2 2000.2.c.e 4
25.d even 5 2 625.2.d.d 4
25.d even 5 2 625.2.d.j 4
25.e even 10 2 625.2.d.a 4
25.e even 10 2 625.2.d.g 4
25.f odd 20 4 625.2.e.d 8
25.f odd 20 4 625.2.e.g 8
35.c odd 2 1 6125.2.a.g 2
40.e odd 2 1 8000.2.a.u 2
40.f even 2 1 8000.2.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
125.2.a.a 2 1.a even 1 1 trivial
125.2.a.b yes 2 5.b even 2 1
125.2.b.b 4 5.c odd 4 2
625.2.d.a 4 25.e even 10 2
625.2.d.d 4 25.d even 5 2
625.2.d.g 4 25.e even 10 2
625.2.d.j 4 25.d even 5 2
625.2.e.d 8 25.f odd 20 4
625.2.e.g 8 25.f odd 20 4
1125.2.a.c 2 15.d odd 2 1
1125.2.a.d 2 3.b odd 2 1
1125.2.b.f 4 15.e even 4 2
2000.2.a.a 2 20.d odd 2 1
2000.2.a.l 2 4.b odd 2 1
2000.2.c.e 4 20.e even 4 2
6125.2.a.d 2 7.b odd 2 1
6125.2.a.g 2 35.c odd 2 1
8000.2.a.c 2 8.d odd 2 1
8000.2.a.d 2 40.f even 2 1
8000.2.a.u 2 40.e odd 2 1
8000.2.a.v 2 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(125))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$T^{2} + 3T + 1$$
$5$ $$T^{2}$$
$7$ $$(T + 3)^{2}$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 3T - 9$$
$17$ $$T^{2} - 4T - 1$$
$19$ $$T^{2} + 5T + 5$$
$23$ $$T^{2} - 2T - 4$$
$29$ $$T^{2} - 45$$
$31$ $$T^{2} + T - 31$$
$37$ $$T^{2} + 6T - 36$$
$41$ $$(T + 3)^{2}$$
$43$ $$(T + 9)^{2}$$
$47$ $$T^{2} + T - 61$$
$53$ $$T^{2} - 7T + 11$$
$59$ $$T^{2} - 15T + 45$$
$61$ $$T^{2} + T - 31$$
$67$ $$T^{2} + 21T + 99$$
$71$ $$(T + 3)^{2}$$
$73$ $$T^{2} + 3T - 9$$
$79$ $$T^{2} - 10T + 5$$
$83$ $$T^{2} + 8T - 4$$
$89$ $$T^{2} - 180$$
$97$ $$T^{2} - 9T + 9$$