# Properties

 Label 1875.2.a.a Level $1875$ Weight $2$ Character orbit 1875.a Self dual yes Analytic conductor $14.972$ 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] = [1875,2,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1875.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: $$14.9719503790$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) 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.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} - q^{4} + q^{6} + ( - 4 \beta + 2) q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^3 - q^4 + q^6 + (-4*b + 2) * q^7 + 3 * q^8 + q^9 $$q - q^{2} - q^{3} - q^{4} + q^{6} + ( - 4 \beta + 2) q^{7} + 3 q^{8} + q^{9} + 2 \beta q^{11} + q^{12} + (\beta - 5) q^{13} + (4 \beta - 2) q^{14} - q^{16} + (3 \beta - 2) q^{17} - q^{18} - 2 \beta q^{19} + (4 \beta - 2) q^{21} - 2 \beta q^{22} + (4 \beta - 2) q^{23} - 3 q^{24} + ( - \beta + 5) q^{26} - q^{27} + (4 \beta - 2) q^{28} + ( - \beta + 6) q^{29} + (2 \beta + 4) q^{31} - 5 q^{32} - 2 \beta q^{33} + ( - 3 \beta + 2) q^{34} - q^{36} - 5 \beta q^{37} + 2 \beta q^{38} + ( - \beta + 5) q^{39} + (\beta - 3) q^{41} + ( - 4 \beta + 2) q^{42} + (6 \beta - 4) q^{43} - 2 \beta q^{44} + ( - 4 \beta + 2) q^{46} + (2 \beta + 2) q^{47} + q^{48} + 13 q^{49} + ( - 3 \beta + 2) q^{51} + ( - \beta + 5) q^{52} + (\beta - 3) q^{53} + q^{54} + ( - 12 \beta + 6) q^{56} + 2 \beta q^{57} + (\beta - 6) q^{58} - 4 q^{59} + ( - \beta + 1) q^{61} + ( - 2 \beta - 4) q^{62} + ( - 4 \beta + 2) q^{63} + 7 q^{64} + 2 \beta q^{66} + ( - 2 \beta - 2) q^{67} + ( - 3 \beta + 2) q^{68} + ( - 4 \beta + 2) q^{69} + (2 \beta - 4) q^{71} + 3 q^{72} + ( - 5 \beta + 5) q^{73} + 5 \beta q^{74} + 2 \beta q^{76} + ( - 4 \beta - 8) q^{77} + (\beta - 5) q^{78} + q^{81} + ( - \beta + 3) q^{82} + (4 \beta - 10) q^{83} + ( - 4 \beta + 2) q^{84} + ( - 6 \beta + 4) q^{86} + (\beta - 6) q^{87} + 6 \beta q^{88} + (\beta + 6) q^{89} + (18 \beta - 14) q^{91} + ( - 4 \beta + 2) q^{92} + ( - 2 \beta - 4) q^{93} + ( - 2 \beta - 2) q^{94} + 5 q^{96} + ( - 3 \beta - 4) q^{97} - 13 q^{98} + 2 \beta q^{99} +O(q^{100})$$ q - q^2 - q^3 - q^4 + q^6 + (-4*b + 2) * q^7 + 3 * q^8 + q^9 + 2*b * q^11 + q^12 + (b - 5) * q^13 + (4*b - 2) * q^14 - q^16 + (3*b - 2) * q^17 - q^18 - 2*b * q^19 + (4*b - 2) * q^21 - 2*b * q^22 + (4*b - 2) * q^23 - 3 * q^24 + (-b + 5) * q^26 - q^27 + (4*b - 2) * q^28 + (-b + 6) * q^29 + (2*b + 4) * q^31 - 5 * q^32 - 2*b * q^33 + (-3*b + 2) * q^34 - q^36 - 5*b * q^37 + 2*b * q^38 + (-b + 5) * q^39 + (b - 3) * q^41 + (-4*b + 2) * q^42 + (6*b - 4) * q^43 - 2*b * q^44 + (-4*b + 2) * q^46 + (2*b + 2) * q^47 + q^48 + 13 * q^49 + (-3*b + 2) * q^51 + (-b + 5) * q^52 + (b - 3) * q^53 + q^54 + (-12*b + 6) * q^56 + 2*b * q^57 + (b - 6) * q^58 - 4 * q^59 + (-b + 1) * q^61 + (-2*b - 4) * q^62 + (-4*b + 2) * q^63 + 7 * q^64 + 2*b * q^66 + (-2*b - 2) * q^67 + (-3*b + 2) * q^68 + (-4*b + 2) * q^69 + (2*b - 4) * q^71 + 3 * q^72 + (-5*b + 5) * q^73 + 5*b * q^74 + 2*b * q^76 + (-4*b - 8) * q^77 + (b - 5) * q^78 + q^81 + (-b + 3) * q^82 + (4*b - 10) * q^83 + (-4*b + 2) * q^84 + (-6*b + 4) * q^86 + (b - 6) * q^87 + 6*b * q^88 + (b + 6) * q^89 + (18*b - 14) * q^91 + (-4*b + 2) * q^92 + (-2*b - 4) * q^93 + (-2*b - 2) * q^94 + 5 * q^96 + (-3*b - 4) * q^97 - 13 * q^98 + 2*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 6 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 6 q^{8} + 2 q^{9} + 2 q^{11} + 2 q^{12} - 9 q^{13} - 2 q^{16} - q^{17} - 2 q^{18} - 2 q^{19} - 2 q^{22} - 6 q^{24} + 9 q^{26} - 2 q^{27} + 11 q^{29} + 10 q^{31} - 10 q^{32} - 2 q^{33} + q^{34} - 2 q^{36} - 5 q^{37} + 2 q^{38} + 9 q^{39} - 5 q^{41} - 2 q^{43} - 2 q^{44} + 6 q^{47} + 2 q^{48} + 26 q^{49} + q^{51} + 9 q^{52} - 5 q^{53} + 2 q^{54} + 2 q^{57} - 11 q^{58} - 8 q^{59} + q^{61} - 10 q^{62} + 14 q^{64} + 2 q^{66} - 6 q^{67} + q^{68} - 6 q^{71} + 6 q^{72} + 5 q^{73} + 5 q^{74} + 2 q^{76} - 20 q^{77} - 9 q^{78} + 2 q^{81} + 5 q^{82} - 16 q^{83} + 2 q^{86} - 11 q^{87} + 6 q^{88} + 13 q^{89} - 10 q^{91} - 10 q^{93} - 6 q^{94} + 10 q^{96} - 11 q^{97} - 26 q^{98} + 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 6 * q^8 + 2 * q^9 + 2 * q^11 + 2 * q^12 - 9 * q^13 - 2 * q^16 - q^17 - 2 * q^18 - 2 * q^19 - 2 * q^22 - 6 * q^24 + 9 * q^26 - 2 * q^27 + 11 * q^29 + 10 * q^31 - 10 * q^32 - 2 * q^33 + q^34 - 2 * q^36 - 5 * q^37 + 2 * q^38 + 9 * q^39 - 5 * q^41 - 2 * q^43 - 2 * q^44 + 6 * q^47 + 2 * q^48 + 26 * q^49 + q^51 + 9 * q^52 - 5 * q^53 + 2 * q^54 + 2 * q^57 - 11 * q^58 - 8 * q^59 + q^61 - 10 * q^62 + 14 * q^64 + 2 * q^66 - 6 * q^67 + q^68 - 6 * q^71 + 6 * q^72 + 5 * q^73 + 5 * q^74 + 2 * q^76 - 20 * q^77 - 9 * q^78 + 2 * q^81 + 5 * q^82 - 16 * q^83 + 2 * q^86 - 11 * q^87 + 6 * q^88 + 13 * q^89 - 10 * q^91 - 10 * q^93 - 6 * q^94 + 10 * q^96 - 11 * q^97 - 26 * q^98 + 2 * 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
−1.00000 −1.00000 −1.00000 0 1.00000 −4.47214 3.00000 1.00000 0
1.2 −1.00000 −1.00000 −1.00000 0 1.00000 4.47214 3.00000 1.00000 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
$$3$$ $$+1$$
$$5$$ $$+1$$

## 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 1875.2.a.a 2
3.b odd 2 1 5625.2.a.h 2
5.b even 2 1 1875.2.a.d 2
5.c odd 4 2 1875.2.b.b 4
15.d odd 2 1 5625.2.a.a 2
25.d even 5 2 375.2.g.a 4
25.e even 10 2 75.2.g.a 4
25.f odd 20 4 375.2.i.a 8
75.h odd 10 2 225.2.h.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.a 4 25.e even 10 2
225.2.h.a 4 75.h odd 10 2
375.2.g.a 4 25.d even 5 2
375.2.i.a 8 25.f odd 20 4
1875.2.a.a 2 1.a even 1 1 trivial
1875.2.a.d 2 5.b even 2 1
1875.2.b.b 4 5.c odd 4 2
5625.2.a.a 2 15.d odd 2 1
5625.2.a.h 2 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1875))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 20$$
$11$ $$T^{2} - 2T - 4$$
$13$ $$T^{2} + 9T + 19$$
$17$ $$T^{2} + T - 11$$
$19$ $$T^{2} + 2T - 4$$
$23$ $$T^{2} - 20$$
$29$ $$T^{2} - 11T + 29$$
$31$ $$T^{2} - 10T + 20$$
$37$ $$T^{2} + 5T - 25$$
$41$ $$T^{2} + 5T + 5$$
$43$ $$T^{2} + 2T - 44$$
$47$ $$T^{2} - 6T + 4$$
$53$ $$T^{2} + 5T + 5$$
$59$ $$(T + 4)^{2}$$
$61$ $$T^{2} - T - 1$$
$67$ $$T^{2} + 6T + 4$$
$71$ $$T^{2} + 6T + 4$$
$73$ $$T^{2} - 5T - 25$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16T + 44$$
$89$ $$T^{2} - 13T + 41$$
$97$ $$T^{2} + 11T + 19$$