# Properties

 Label 625.2.a.a Level $625$ Weight $2$ Character orbit 625.a Self dual yes Analytic conductor $4.991$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.99065012633$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} + ( - \beta - 1) q^{3} + (\beta - 1) q^{4} + (2 \beta + 1) q^{6} + ( - 3 \beta + 1) q^{7} + (2 \beta - 1) q^{8} + (3 \beta - 1) q^{9} +O(q^{10})$$ q - b * q^2 + (-b - 1) * q^3 + (b - 1) * q^4 + (2*b + 1) * q^6 + (-3*b + 1) * q^7 + (2*b - 1) * q^8 + (3*b - 1) * q^9 $$q - \beta q^{2} + ( - \beta - 1) q^{3} + (\beta - 1) q^{4} + (2 \beta + 1) q^{6} + ( - 3 \beta + 1) q^{7} + (2 \beta - 1) q^{8} + (3 \beta - 1) q^{9} + (\beta - 1) q^{11} - \beta q^{12} + (4 \beta - 1) q^{13} + (2 \beta + 3) q^{14} - 3 \beta q^{16} + (4 \beta - 5) q^{17} + ( - 2 \beta - 3) q^{18} + ( - 3 \beta + 4) q^{19} + (5 \beta + 2) q^{21} - q^{22} + (3 \beta - 3) q^{23} + ( - 3 \beta - 1) q^{24} + ( - 3 \beta - 4) q^{26} + ( - 2 \beta + 1) q^{27} + (\beta - 4) q^{28} + (2 \beta - 6) q^{29} + 2 q^{31} + ( - \beta + 5) q^{32} - \beta q^{33} + (\beta - 4) q^{34} + ( - \beta + 4) q^{36} - 3 q^{37} + ( - \beta + 3) q^{38} + ( - 7 \beta - 3) q^{39} + 4 \beta q^{41} + ( - 7 \beta - 5) q^{42} + (4 \beta - 6) q^{43} + ( - \beta + 2) q^{44} - 3 q^{46} + ( - \beta - 10) q^{47} + (6 \beta + 3) q^{48} + (3 \beta + 3) q^{49} + ( - 3 \beta + 1) q^{51} + ( - \beta + 5) q^{52} + ( - 4 \beta - 2) q^{53} + (\beta + 2) q^{54} + ( - \beta - 7) q^{56} + (2 \beta - 1) q^{57} + (4 \beta - 2) q^{58} + ( - 8 \beta - 1) q^{59} + ( - 3 \beta - 4) q^{61} - 2 \beta q^{62} + ( - 3 \beta - 10) q^{63} + (2 \beta + 1) q^{64} + (\beta + 1) q^{66} + (4 \beta + 5) q^{67} + ( - 5 \beta + 9) q^{68} - 3 \beta q^{69} + ( - 7 \beta + 8) q^{71} + (\beta + 7) q^{72} + ( - 3 \beta + 5) q^{73} + 3 \beta q^{74} + (4 \beta - 7) q^{76} + (\beta - 4) q^{77} + (10 \beta + 7) q^{78} + ( - 6 \beta + 3) q^{79} + ( - 6 \beta + 4) q^{81} + ( - 4 \beta - 4) q^{82} - 8 \beta q^{83} + (2 \beta + 3) q^{84} + (2 \beta - 4) q^{86} + (2 \beta + 4) q^{87} + ( - \beta + 3) q^{88} + (\beta + 2) q^{89} + ( - 5 \beta - 13) q^{91} + ( - 3 \beta + 6) q^{92} + ( - 2 \beta - 2) q^{93} + (11 \beta + 1) q^{94} + ( - 3 \beta - 4) q^{96} + (\beta + 9) q^{97} + ( - 6 \beta - 3) q^{98} + ( - \beta + 4) q^{99} +O(q^{100})$$ q - b * q^2 + (-b - 1) * q^3 + (b - 1) * q^4 + (2*b + 1) * q^6 + (-3*b + 1) * q^7 + (2*b - 1) * q^8 + (3*b - 1) * q^9 + (b - 1) * q^11 - b * q^12 + (4*b - 1) * q^13 + (2*b + 3) * q^14 - 3*b * q^16 + (4*b - 5) * q^17 + (-2*b - 3) * q^18 + (-3*b + 4) * q^19 + (5*b + 2) * q^21 - q^22 + (3*b - 3) * q^23 + (-3*b - 1) * q^24 + (-3*b - 4) * q^26 + (-2*b + 1) * q^27 + (b - 4) * q^28 + (2*b - 6) * q^29 + 2 * q^31 + (-b + 5) * q^32 - b * q^33 + (b - 4) * q^34 + (-b + 4) * q^36 - 3 * q^37 + (-b + 3) * q^38 + (-7*b - 3) * q^39 + 4*b * q^41 + (-7*b - 5) * q^42 + (4*b - 6) * q^43 + (-b + 2) * q^44 - 3 * q^46 + (-b - 10) * q^47 + (6*b + 3) * q^48 + (3*b + 3) * q^49 + (-3*b + 1) * q^51 + (-b + 5) * q^52 + (-4*b - 2) * q^53 + (b + 2) * q^54 + (-b - 7) * q^56 + (2*b - 1) * q^57 + (4*b - 2) * q^58 + (-8*b - 1) * q^59 + (-3*b - 4) * q^61 - 2*b * q^62 + (-3*b - 10) * q^63 + (2*b + 1) * q^64 + (b + 1) * q^66 + (4*b + 5) * q^67 + (-5*b + 9) * q^68 - 3*b * q^69 + (-7*b + 8) * q^71 + (b + 7) * q^72 + (-3*b + 5) * q^73 + 3*b * q^74 + (4*b - 7) * q^76 + (b - 4) * q^77 + (10*b + 7) * q^78 + (-6*b + 3) * q^79 + (-6*b + 4) * q^81 + (-4*b - 4) * q^82 - 8*b * q^83 + (2*b + 3) * q^84 + (2*b - 4) * q^86 + (2*b + 4) * q^87 + (-b + 3) * q^88 + (b + 2) * q^89 + (-5*b - 13) * q^91 + (-3*b + 6) * q^92 + (-2*b - 2) * q^93 + (11*b + 1) * q^94 + (-3*b - 4) * q^96 + (b + 9) * q^97 + (-6*b - 3) * q^98 + (-b + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - 3 q^{3} - q^{4} + 4 q^{6} - q^{7} + q^{9}+O(q^{10})$$ 2 * q - q^2 - 3 * q^3 - q^4 + 4 * q^6 - q^7 + q^9 $$2 q - q^{2} - 3 q^{3} - q^{4} + 4 q^{6} - q^{7} + q^{9} - q^{11} - q^{12} + 2 q^{13} + 8 q^{14} - 3 q^{16} - 6 q^{17} - 8 q^{18} + 5 q^{19} + 9 q^{21} - 2 q^{22} - 3 q^{23} - 5 q^{24} - 11 q^{26} - 7 q^{28} - 10 q^{29} + 4 q^{31} + 9 q^{32} - q^{33} - 7 q^{34} + 7 q^{36} - 6 q^{37} + 5 q^{38} - 13 q^{39} + 4 q^{41} - 17 q^{42} - 8 q^{43} + 3 q^{44} - 6 q^{46} - 21 q^{47} + 12 q^{48} + 9 q^{49} - q^{51} + 9 q^{52} - 8 q^{53} + 5 q^{54} - 15 q^{56} - 10 q^{59} - 11 q^{61} - 2 q^{62} - 23 q^{63} + 4 q^{64} + 3 q^{66} + 14 q^{67} + 13 q^{68} - 3 q^{69} + 9 q^{71} + 15 q^{72} + 7 q^{73} + 3 q^{74} - 10 q^{76} - 7 q^{77} + 24 q^{78} + 2 q^{81} - 12 q^{82} - 8 q^{83} + 8 q^{84} - 6 q^{86} + 10 q^{87} + 5 q^{88} + 5 q^{89} - 31 q^{91} + 9 q^{92} - 6 q^{93} + 13 q^{94} - 11 q^{96} + 19 q^{97} - 12 q^{98} + 7 q^{99}+O(q^{100})$$ 2 * q - q^2 - 3 * q^3 - q^4 + 4 * q^6 - q^7 + q^9 - q^11 - q^12 + 2 * q^13 + 8 * q^14 - 3 * q^16 - 6 * q^17 - 8 * q^18 + 5 * q^19 + 9 * q^21 - 2 * q^22 - 3 * q^23 - 5 * q^24 - 11 * q^26 - 7 * q^28 - 10 * q^29 + 4 * q^31 + 9 * q^32 - q^33 - 7 * q^34 + 7 * q^36 - 6 * q^37 + 5 * q^38 - 13 * q^39 + 4 * q^41 - 17 * q^42 - 8 * q^43 + 3 * q^44 - 6 * q^46 - 21 * q^47 + 12 * q^48 + 9 * q^49 - q^51 + 9 * q^52 - 8 * q^53 + 5 * q^54 - 15 * q^56 - 10 * q^59 - 11 * q^61 - 2 * q^62 - 23 * q^63 + 4 * q^64 + 3 * q^66 + 14 * q^67 + 13 * q^68 - 3 * q^69 + 9 * q^71 + 15 * q^72 + 7 * q^73 + 3 * q^74 - 10 * q^76 - 7 * q^77 + 24 * q^78 + 2 * q^81 - 12 * q^82 - 8 * q^83 + 8 * q^84 - 6 * q^86 + 10 * q^87 + 5 * q^88 + 5 * q^89 - 31 * q^91 + 9 * q^92 - 6 * q^93 + 13 * q^94 - 11 * q^96 + 19 * q^97 - 12 * q^98 + 7 * 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 −2.61803 0.618034 0 4.23607 −3.85410 2.23607 3.85410 0
1.2 0.618034 −0.381966 −1.61803 0 −0.236068 2.85410 −2.23607 −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
$$5$$ $$1$$

## 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 625.2.a.a 2
3.b odd 2 1 5625.2.a.e 2
4.b odd 2 1 10000.2.a.m 2
5.b even 2 1 625.2.a.d yes 2
5.c odd 4 2 625.2.b.b 4
15.d odd 2 1 5625.2.a.c 2
20.d odd 2 1 10000.2.a.b 2
25.d even 5 2 625.2.d.e 4
25.d even 5 2 625.2.d.i 4
25.e even 10 2 625.2.d.c 4
25.e even 10 2 625.2.d.f 4
25.f odd 20 4 625.2.e.e 8
25.f odd 20 4 625.2.e.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.a 2 1.a even 1 1 trivial
625.2.a.d yes 2 5.b even 2 1
625.2.b.b 4 5.c odd 4 2
625.2.d.c 4 25.e even 10 2
625.2.d.e 4 25.d even 5 2
625.2.d.f 4 25.e even 10 2
625.2.d.i 4 25.d even 5 2
625.2.e.e 8 25.f odd 20 4
625.2.e.f 8 25.f odd 20 4
5625.2.a.c 2 15.d odd 2 1
5625.2.a.e 2 3.b odd 2 1
10000.2.a.b 2 20.d odd 2 1
10000.2.a.m 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(625))$$:

 $$T_{2}^{2} + T_{2} - 1$$ T2^2 + T2 - 1 $$T_{3}^{2} + 3T_{3} + 1$$ T3^2 + 3*T3 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T - 1$$
$3$ $$T^{2} + 3T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T - 11$$
$11$ $$T^{2} + T - 1$$
$13$ $$T^{2} - 2T - 19$$
$17$ $$T^{2} + 6T - 11$$
$19$ $$T^{2} - 5T - 5$$
$23$ $$T^{2} + 3T - 9$$
$29$ $$T^{2} + 10T + 20$$
$31$ $$(T - 2)^{2}$$
$37$ $$(T + 3)^{2}$$
$41$ $$T^{2} - 4T - 16$$
$43$ $$T^{2} + 8T - 4$$
$47$ $$T^{2} + 21T + 109$$
$53$ $$T^{2} + 8T - 4$$
$59$ $$T^{2} + 10T - 55$$
$61$ $$T^{2} + 11T + 19$$
$67$ $$T^{2} - 14T + 29$$
$71$ $$T^{2} - 9T - 41$$
$73$ $$T^{2} - 7T + 1$$
$79$ $$T^{2} - 45$$
$83$ $$T^{2} + 8T - 64$$
$89$ $$T^{2} - 5T + 5$$
$97$ $$T^{2} - 19T + 89$$