# Properties

 Label 1875.2.a.c 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{3} + (\beta - 1) q^{4} + \beta q^{6} - 2 q^{7} + ( - 2 \beta + 1) q^{8} + q^{9}+O(q^{10})$$ q + b * q^2 + q^3 + (b - 1) * q^4 + b * q^6 - 2 * q^7 + (-2*b + 1) * q^8 + q^9 $$q + \beta q^{2} + q^{3} + (\beta - 1) q^{4} + \beta q^{6} - 2 q^{7} + ( - 2 \beta + 1) q^{8} + q^{9} - 3 q^{11} + (\beta - 1) q^{12} - q^{13} - 2 \beta q^{14} - 3 \beta q^{16} + ( - 2 \beta - 1) q^{17} + \beta q^{18} + ( - 6 \beta + 3) q^{19} - 2 q^{21} - 3 \beta q^{22} + ( - \beta + 7) q^{23} + ( - 2 \beta + 1) q^{24} - \beta q^{26} + q^{27} + ( - 2 \beta + 2) q^{28} + ( - \beta - 2) q^{29} + (6 \beta - 1) q^{31} + (\beta - 5) q^{32} - 3 q^{33} + ( - 3 \beta - 2) q^{34} + (\beta - 1) q^{36} - 2 q^{37} + ( - 3 \beta - 6) q^{38} - q^{39} + (\beta - 11) q^{41} - 2 \beta q^{42} + (\beta - 9) q^{43} + ( - 3 \beta + 3) q^{44} + (6 \beta - 1) q^{46} + (2 \beta - 8) q^{47} - 3 \beta q^{48} - 3 q^{49} + ( - 2 \beta - 1) q^{51} + ( - \beta + 1) q^{52} + (2 \beta + 8) q^{53} + \beta q^{54} + (4 \beta - 2) q^{56} + ( - 6 \beta + 3) q^{57} + ( - 3 \beta - 1) q^{58} + (8 \beta - 9) q^{59} + (6 \beta - 1) q^{61} + (5 \beta + 6) q^{62} - 2 q^{63} + (2 \beta + 1) q^{64} - 3 \beta q^{66} + ( - 10 \beta + 3) q^{67} + ( - \beta - 1) q^{68} + ( - \beta + 7) q^{69} + (5 \beta + 2) q^{71} + ( - 2 \beta + 1) q^{72} + (6 \beta + 6) q^{73} - 2 \beta q^{74} + (3 \beta - 9) q^{76} + 6 q^{77} - \beta q^{78} + (3 \beta - 14) q^{79} + q^{81} + ( - 10 \beta + 1) q^{82} + 9 q^{83} + ( - 2 \beta + 2) q^{84} + ( - 8 \beta + 1) q^{86} + ( - \beta - 2) q^{87} + (6 \beta - 3) q^{88} + ( - 10 \beta + 5) q^{89} + 2 q^{91} + (7 \beta - 8) q^{92} + (6 \beta - 1) q^{93} + ( - 6 \beta + 2) q^{94} + (\beta - 5) q^{96} + (3 \beta - 1) q^{97} - 3 \beta q^{98} - 3 q^{99} +O(q^{100})$$ q + b * q^2 + q^3 + (b - 1) * q^4 + b * q^6 - 2 * q^7 + (-2*b + 1) * q^8 + q^9 - 3 * q^11 + (b - 1) * q^12 - q^13 - 2*b * q^14 - 3*b * q^16 + (-2*b - 1) * q^17 + b * q^18 + (-6*b + 3) * q^19 - 2 * q^21 - 3*b * q^22 + (-b + 7) * q^23 + (-2*b + 1) * q^24 - b * q^26 + q^27 + (-2*b + 2) * q^28 + (-b - 2) * q^29 + (6*b - 1) * q^31 + (b - 5) * q^32 - 3 * q^33 + (-3*b - 2) * q^34 + (b - 1) * q^36 - 2 * q^37 + (-3*b - 6) * q^38 - q^39 + (b - 11) * q^41 - 2*b * q^42 + (b - 9) * q^43 + (-3*b + 3) * q^44 + (6*b - 1) * q^46 + (2*b - 8) * q^47 - 3*b * q^48 - 3 * q^49 + (-2*b - 1) * q^51 + (-b + 1) * q^52 + (2*b + 8) * q^53 + b * q^54 + (4*b - 2) * q^56 + (-6*b + 3) * q^57 + (-3*b - 1) * q^58 + (8*b - 9) * q^59 + (6*b - 1) * q^61 + (5*b + 6) * q^62 - 2 * q^63 + (2*b + 1) * q^64 - 3*b * q^66 + (-10*b + 3) * q^67 + (-b - 1) * q^68 + (-b + 7) * q^69 + (5*b + 2) * q^71 + (-2*b + 1) * q^72 + (6*b + 6) * q^73 - 2*b * q^74 + (3*b - 9) * q^76 + 6 * q^77 - b * q^78 + (3*b - 14) * q^79 + q^81 + (-10*b + 1) * q^82 + 9 * q^83 + (-2*b + 2) * q^84 + (-8*b + 1) * q^86 + (-b - 2) * q^87 + (6*b - 3) * q^88 + (-10*b + 5) * q^89 + 2 * q^91 + (7*b - 8) * q^92 + (6*b - 1) * q^93 + (-6*b + 2) * q^94 + (b - 5) * q^96 + (3*b - 1) * q^97 - 3*b * q^98 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} - 4 q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 - 4 * q^7 + 2 * q^9 $$2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} - 4 q^{7} + 2 q^{9} - 6 q^{11} - q^{12} - 2 q^{13} - 2 q^{14} - 3 q^{16} - 4 q^{17} + q^{18} - 4 q^{21} - 3 q^{22} + 13 q^{23} - q^{26} + 2 q^{27} + 2 q^{28} - 5 q^{29} + 4 q^{31} - 9 q^{32} - 6 q^{33} - 7 q^{34} - q^{36} - 4 q^{37} - 15 q^{38} - 2 q^{39} - 21 q^{41} - 2 q^{42} - 17 q^{43} + 3 q^{44} + 4 q^{46} - 14 q^{47} - 3 q^{48} - 6 q^{49} - 4 q^{51} + q^{52} + 18 q^{53} + q^{54} - 5 q^{58} - 10 q^{59} + 4 q^{61} + 17 q^{62} - 4 q^{63} + 4 q^{64} - 3 q^{66} - 4 q^{67} - 3 q^{68} + 13 q^{69} + 9 q^{71} + 18 q^{73} - 2 q^{74} - 15 q^{76} + 12 q^{77} - q^{78} - 25 q^{79} + 2 q^{81} - 8 q^{82} + 18 q^{83} + 2 q^{84} - 6 q^{86} - 5 q^{87} + 4 q^{91} - 9 q^{92} + 4 q^{93} - 2 q^{94} - 9 q^{96} + q^{97} - 3 q^{98} - 6 q^{99}+O(q^{100})$$ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 - 4 * q^7 + 2 * q^9 - 6 * q^11 - q^12 - 2 * q^13 - 2 * q^14 - 3 * q^16 - 4 * q^17 + q^18 - 4 * q^21 - 3 * q^22 + 13 * q^23 - q^26 + 2 * q^27 + 2 * q^28 - 5 * q^29 + 4 * q^31 - 9 * q^32 - 6 * q^33 - 7 * q^34 - q^36 - 4 * q^37 - 15 * q^38 - 2 * q^39 - 21 * q^41 - 2 * q^42 - 17 * q^43 + 3 * q^44 + 4 * q^46 - 14 * q^47 - 3 * q^48 - 6 * q^49 - 4 * q^51 + q^52 + 18 * q^53 + q^54 - 5 * q^58 - 10 * q^59 + 4 * q^61 + 17 * q^62 - 4 * q^63 + 4 * q^64 - 3 * q^66 - 4 * q^67 - 3 * q^68 + 13 * q^69 + 9 * q^71 + 18 * q^73 - 2 * q^74 - 15 * q^76 + 12 * q^77 - q^78 - 25 * q^79 + 2 * q^81 - 8 * q^82 + 18 * q^83 + 2 * q^84 - 6 * q^86 - 5 * q^87 + 4 * q^91 - 9 * q^92 + 4 * q^93 - 2 * q^94 - 9 * q^96 + q^97 - 3 * q^98 - 6 * 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
 −0.618034 1.61803
−0.618034 1.00000 −1.61803 0 −0.618034 −2.00000 2.23607 1.00000 0
1.2 1.61803 1.00000 0.618034 0 1.61803 −2.00000 −2.23607 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.c yes 2
3.b odd 2 1 5625.2.a.b 2
5.b even 2 1 1875.2.a.b 2
5.c odd 4 2 1875.2.b.a 4
15.d odd 2 1 5625.2.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.b 2 5.b even 2 1
1875.2.a.c yes 2 1.a even 1 1 trivial
1875.2.b.a 4 5.c odd 4 2
5625.2.a.b 2 3.b odd 2 1
5625.2.a.g 2 15.d odd 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1875))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T - 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 2)^{2}$$
$11$ $$(T + 3)^{2}$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 4T - 1$$
$19$ $$T^{2} - 45$$
$23$ $$T^{2} - 13T + 41$$
$29$ $$T^{2} + 5T + 5$$
$31$ $$T^{2} - 4T - 41$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} + 21T + 109$$
$43$ $$T^{2} + 17T + 71$$
$47$ $$T^{2} + 14T + 44$$
$53$ $$T^{2} - 18T + 76$$
$59$ $$T^{2} + 10T - 55$$
$61$ $$T^{2} - 4T - 41$$
$67$ $$T^{2} + 4T - 121$$
$71$ $$T^{2} - 9T - 11$$
$73$ $$T^{2} - 18T + 36$$
$79$ $$T^{2} + 25T + 145$$
$83$ $$(T - 9)^{2}$$
$89$ $$T^{2} - 125$$
$97$ $$T^{2} - T - 11$$