# Properties

 Label 625.2.a.b 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(1,625)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$4.99065012633$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

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

comment: embeddings in the coefficient field

gp: mfembed(f)

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 1.00000 0.618034 0 −1.61803 0.618034 2.23607 −2.00000 0
1.2 0.618034 1.00000 −1.61803 0 0.618034 −1.61803 −2.23607 −2.00000 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.b 2
3.b odd 2 1 5625.2.a.f 2
4.b odd 2 1 10000.2.a.c 2
5.b even 2 1 625.2.a.c 2
5.c odd 4 2 625.2.b.a 4
15.d odd 2 1 5625.2.a.d 2
20.d odd 2 1 10000.2.a.l 2
25.d even 5 2 25.2.d.a 4
25.d even 5 2 625.2.d.h 4
25.e even 10 2 125.2.d.a 4
25.e even 10 2 625.2.d.b 4
25.f odd 20 4 125.2.e.a 8
25.f odd 20 4 625.2.e.c 8
75.j odd 10 2 225.2.h.b 4
100.j odd 10 2 400.2.u.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.2.d.a 4 25.d even 5 2
125.2.d.a 4 25.e even 10 2
125.2.e.a 8 25.f odd 20 4
225.2.h.b 4 75.j odd 10 2
400.2.u.b 4 100.j odd 10 2
625.2.a.b 2 1.a even 1 1 trivial
625.2.a.c 2 5.b even 2 1
625.2.b.a 4 5.c odd 4 2
625.2.d.b 4 25.e even 10 2
625.2.d.h 4 25.d even 5 2
625.2.e.c 8 25.f odd 20 4
5625.2.a.d 2 15.d odd 2 1
5625.2.a.f 2 3.b odd 2 1
10000.2.a.c 2 4.b odd 2 1
10000.2.a.l 2 20.d 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 $$T_{3} - 1$$ T3 - 1

## Hecke characteristic polynomials

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