# Properties

 Label 1875.2.a.h Level $1875$ Weight $2$ Character orbit 1875.a Self dual yes Analytic conductor $14.972$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.5125.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 6x^{2} + 7x + 11$$ x^4 - 2*x^3 - 6*x^2 + 7*x + 11 Coefficient ring: $$\Z[a_1, a_2]$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + \beta_1 q^{6} + ( - 2 \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 4) q^{8} + q^{9}+O(q^{10})$$ q + b1 * q^2 + q^3 + (b2 + b1 + 2) * q^4 + b1 * q^6 + (-2*b2 + b1 - 1) * q^7 + (b3 + b2 + b1 + 4) * q^8 + q^9 $$q + \beta_1 q^{2} + q^{3} + (\beta_{2} + \beta_1 + 2) q^{4} + \beta_1 q^{6} + ( - 2 \beta_{2} + \beta_1 - 1) q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 4) q^{8} + q^{9} + (\beta_{3} - 3 \beta_{2} - 3) q^{11} + (\beta_{2} + \beta_1 + 2) q^{12} + ( - \beta_{3} + 2 \beta_{2} + 1) q^{13} + ( - 2 \beta_{3} + \beta_{2} + 4) q^{14} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 1) q^{16} + ( - \beta_1 + 1) q^{17} + \beta_1 q^{18} + ( - \beta_{3} + 2 \beta_1) q^{19} + ( - 2 \beta_{2} + \beta_1 - 1) q^{21} + ( - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 1) q^{22} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{23} + (\beta_{3} + \beta_{2} + \beta_1 + 4) q^{24} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{26} + q^{27} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{28} + ( - \beta_{2} - \beta_1 - 5) q^{29} + (\beta_{3} - 2 \beta_{2} + 5) q^{31} + (2 \beta_{3} + 7 \beta_{2} + 2 \beta_1 + 6) q^{32} + (\beta_{3} - 3 \beta_{2} - 3) q^{33} + ( - \beta_{2} - 4) q^{34} + (\beta_{2} + \beta_1 + 2) q^{36} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{37} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 + 7) q^{38} + ( - \beta_{3} + 2 \beta_{2} + 1) q^{39} + (4 \beta_{2} - 1) q^{41} + ( - 2 \beta_{3} + \beta_{2} + 4) q^{42} + (2 \beta_{3} - \beta_{2} + 4) q^{43} + ( - \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 8) q^{44} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{46} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{47} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 1) q^{48} + ( - 4 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{49} + ( - \beta_1 + 1) q^{51} + 3 q^{52} + ( - 2 \beta_{3} - \beta_1 - 1) q^{53} + \beta_1 q^{54} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 - 1) q^{56} + ( - \beta_{3} + 2 \beta_1) q^{57} + ( - \beta_{3} - \beta_{2} - 6 \beta_1 - 4) q^{58} + ( - 3 \beta_{3} - 3 \beta_{2} - 6) q^{59} + ( - 2 \beta_{3} + 4 \beta_{2} + 1) q^{61} + ( - \beta_{3} + 3 \beta_{2} + 5 \beta_1 + 1) q^{62} + ( - 2 \beta_{2} + \beta_1 - 1) q^{63} + (5 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 8) q^{64} + ( - 2 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 1) q^{66} + (\beta_{2} - 2 \beta_1 + 2) q^{67} + ( - \beta_{3} - 2 \beta_1 - 2) q^{68} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{69} + (4 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{71} + (\beta_{3} + \beta_{2} + \beta_1 + 4) q^{72} + ( - 2 \beta_{3} - \beta_{2} - 4 \beta_1 + 5) q^{73} + (\beta_{3} + 5 \beta_{2} - 2) q^{74} + ( - \beta_{2} + 5 \beta_1 + 7) q^{76} + ( - \beta_{3} + 6 \beta_{2} - 5 \beta_1 + 10) q^{77} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{78} + ( - 3 \beta_{3} - \beta_{2} - 3 \beta_1 + 9) q^{79} + q^{81} + (4 \beta_{3} - \beta_1) q^{82} + (4 \beta_{3} + 5) q^{83} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{84} + (\beta_{3} + 6 \beta_{2} + 4 \beta_1 + 2) q^{86} + ( - \beta_{2} - \beta_1 - 5) q^{87} + ( - 11 \beta_{2} - 4 \beta_1 - 11) q^{88} + ( - \beta_{3} + 4 \beta_{2} - 7) q^{89} + ( - 3 \beta_{2} + 3 \beta_1 - 6) q^{91} + ( - \beta_{3} - 2 \beta_1 - 5) q^{92} + (\beta_{3} - 2 \beta_{2} + 5) q^{93} + ( - \beta_{3} - 7 \beta_{2} - 6) q^{94} + (2 \beta_{3} + 7 \beta_{2} + 2 \beta_1 + 6) q^{96} + (4 \beta_{3} - 2 \beta_{2} + 3) q^{97} + ( - 3 \beta_{3} - 13 \beta_{2} + \beta_1 - 8) q^{98} + (\beta_{3} - 3 \beta_{2} - 3) q^{99}+O(q^{100})$$ q + b1 * q^2 + q^3 + (b2 + b1 + 2) * q^4 + b1 * q^6 + (-2*b2 + b1 - 1) * q^7 + (b3 + b2 + b1 + 4) * q^8 + q^9 + (b3 - 3*b2 - 3) * q^11 + (b2 + b1 + 2) * q^12 + (-b3 + 2*b2 + 1) * q^13 + (-2*b3 + b2 + 4) * q^14 + (2*b3 + 2*b2 + 3*b1 + 1) * q^16 + (-b1 + 1) * q^17 + b1 * q^18 + (-b3 + 2*b1) * q^19 + (-2*b2 + b1 - 1) * q^21 + (-2*b3 + 3*b2 - 3*b1 + 1) * q^22 + (b3 - 2*b2 - b1) * q^23 + (b3 + b2 + b1 + 4) * q^24 + (b3 - 3*b2 + b1 - 1) * q^26 + q^27 + (-b3 - 2*b2 + 2*b1) * q^28 + (-b2 - b1 - 5) * q^29 + (b3 - 2*b2 + 5) * q^31 + (2*b3 + 7*b2 + 2*b1 + 6) * q^32 + (b3 - 3*b2 - 3) * q^33 + (-b2 - 4) * q^34 + (b2 + b1 + 2) * q^36 + (2*b3 - b2 - b1 + 1) * q^37 + (-b3 - b2 + 2*b1 + 7) * q^38 + (-b3 + 2*b2 + 1) * q^39 + (4*b2 - 1) * q^41 + (-2*b3 + b2 + 4) * q^42 + (2*b3 - b2 + 4) * q^43 + (-b3 - 3*b2 - 2*b1 - 8) * q^44 + (-b3 + 2*b2 - b1 - 3) * q^46 + (-2*b3 + b2 - b1 + 1) * q^47 + (2*b3 + 2*b2 + 3*b1 + 1) * q^48 + (-4*b3 + b2 - b1 + 2) * q^49 + (-b1 + 1) * q^51 + 3 * q^52 + (-2*b3 - b1 - 1) * q^53 + b1 * q^54 + (b3 - 3*b2 + 2*b1 - 1) * q^56 + (-b3 + 2*b1) * q^57 + (-b3 - b2 - 6*b1 - 4) * q^58 + (-3*b3 - 3*b2 - 6) * q^59 + (-2*b3 + 4*b2 + 1) * q^61 + (-b3 + 3*b2 + 5*b1 + 1) * q^62 + (-2*b2 + b1 - 1) * q^63 + (5*b3 + 4*b2 + 2*b1 + 8) * q^64 + (-2*b3 + 3*b2 - 3*b1 + 1) * q^66 + (b2 - 2*b1 + 2) * q^67 + (-b3 - 2*b1 - 2) * q^68 + (b3 - 2*b2 - b1) * q^69 + (4*b3 + 2*b2 + 3*b1) * q^71 + (b3 + b2 + b1 + 4) * q^72 + (-2*b3 - b2 - 4*b1 + 5) * q^73 + (b3 + 5*b2 - 2) * q^74 + (-b2 + 5*b1 + 7) * q^76 + (-b3 + 6*b2 - 5*b1 + 10) * q^77 + (b3 - 3*b2 + b1 - 1) * q^78 + (-3*b3 - b2 - 3*b1 + 9) * q^79 + q^81 + (4*b3 - b1) * q^82 + (4*b3 + 5) * q^83 + (-b3 - 2*b2 + 2*b1) * q^84 + (b3 + 6*b2 + 4*b1 + 2) * q^86 + (-b2 - b1 - 5) * q^87 + (-11*b2 - 4*b1 - 11) * q^88 + (-b3 + 4*b2 - 7) * q^89 + (-3*b2 + 3*b1 - 6) * q^91 + (-b3 - 2*b1 - 5) * q^92 + (b3 - 2*b2 + 5) * q^93 + (-b3 - 7*b2 - 6) * q^94 + (2*b3 + 7*b2 + 2*b1 + 6) * q^96 + (4*b3 - 2*b2 + 3) * q^97 + (-3*b3 - 13*b2 + b1 - 8) * q^98 + (b3 - 3*b2 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 4 q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{7} + 15 q^{8} + 4 q^{9}+O(q^{10})$$ 4 * q + 2 * q^2 + 4 * q^3 + 8 * q^4 + 2 * q^6 + 2 * q^7 + 15 * q^8 + 4 * q^9 $$4 q + 2 q^{2} + 4 q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{7} + 15 q^{8} + 4 q^{9} - 7 q^{11} + 8 q^{12} + q^{13} + 16 q^{14} + 4 q^{16} + 2 q^{17} + 2 q^{18} + 5 q^{19} + 2 q^{21} - 6 q^{22} + q^{23} + 15 q^{24} + 3 q^{26} + 4 q^{27} + 9 q^{28} - 20 q^{29} + 23 q^{31} + 12 q^{32} - 7 q^{33} - 14 q^{34} + 8 q^{36} + 2 q^{37} + 35 q^{38} + q^{39} - 12 q^{41} + 16 q^{42} + 16 q^{43} - 29 q^{44} - 17 q^{46} + 2 q^{47} + 4 q^{48} + 8 q^{49} + 2 q^{51} + 12 q^{52} - 4 q^{53} + 2 q^{54} + 5 q^{56} + 5 q^{57} - 25 q^{58} - 15 q^{59} - 2 q^{61} + 9 q^{62} + 2 q^{63} + 23 q^{64} - 6 q^{66} + 2 q^{67} - 11 q^{68} + q^{69} - 2 q^{71} + 15 q^{72} + 16 q^{73} - 19 q^{74} + 40 q^{76} + 19 q^{77} + 3 q^{78} + 35 q^{79} + 4 q^{81} - 6 q^{82} + 16 q^{83} + 9 q^{84} + 3 q^{86} - 20 q^{87} - 30 q^{88} - 35 q^{89} - 12 q^{91} - 23 q^{92} + 23 q^{93} - 9 q^{94} + 12 q^{96} + 12 q^{97} - q^{98} - 7 q^{99}+O(q^{100})$$ 4 * q + 2 * q^2 + 4 * q^3 + 8 * q^4 + 2 * q^6 + 2 * q^7 + 15 * q^8 + 4 * q^9 - 7 * q^11 + 8 * q^12 + q^13 + 16 * q^14 + 4 * q^16 + 2 * q^17 + 2 * q^18 + 5 * q^19 + 2 * q^21 - 6 * q^22 + q^23 + 15 * q^24 + 3 * q^26 + 4 * q^27 + 9 * q^28 - 20 * q^29 + 23 * q^31 + 12 * q^32 - 7 * q^33 - 14 * q^34 + 8 * q^36 + 2 * q^37 + 35 * q^38 + q^39 - 12 * q^41 + 16 * q^42 + 16 * q^43 - 29 * q^44 - 17 * q^46 + 2 * q^47 + 4 * q^48 + 8 * q^49 + 2 * q^51 + 12 * q^52 - 4 * q^53 + 2 * q^54 + 5 * q^56 + 5 * q^57 - 25 * q^58 - 15 * q^59 - 2 * q^61 + 9 * q^62 + 2 * q^63 + 23 * q^64 - 6 * q^66 + 2 * q^67 - 11 * q^68 + q^69 - 2 * q^71 + 15 * q^72 + 16 * q^73 - 19 * q^74 + 40 * q^76 + 19 * q^77 + 3 * q^78 + 35 * q^79 + 4 * q^81 - 6 * q^82 + 16 * q^83 + 9 * q^84 + 3 * q^86 - 20 * q^87 - 30 * q^88 - 35 * q^89 - 12 * q^91 - 23 * q^92 + 23 * q^93 - 9 * q^94 + 12 * q^96 + 12 * q^97 - q^98 - 7 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 6x^{2} + 7x + 11$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4\nu$$ v^3 - v^2 - 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 4$$ b2 + b1 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5\beta _1 + 4$$ b3 + b2 + 5*b1 + 4

## 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
 −1.70636 −1.12233 2.12233 2.70636
−1.70636 1.00000 0.911672 0 −1.70636 −3.94243 1.85708 1.00000 0
1.2 −1.12233 1.00000 −0.740367 0 −1.12233 1.11373 3.07561 1.00000 0
1.3 2.12233 1.00000 2.50430 0 2.12233 4.35840 1.07029 1.00000 0
1.4 2.70636 1.00000 5.32440 0 2.70636 0.470294 8.99702 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.h 4
3.b odd 2 1 5625.2.a.i 4
5.b even 2 1 1875.2.a.e 4
5.c odd 4 2 1875.2.b.c 8
15.d odd 2 1 5625.2.a.n 4
25.d even 5 2 75.2.g.b 8
25.e even 10 2 375.2.g.b 8
25.f odd 20 4 375.2.i.b 16
75.j odd 10 2 225.2.h.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.b 8 25.d even 5 2
225.2.h.c 8 75.j odd 10 2
375.2.g.b 8 25.e even 10 2
375.2.i.b 16 25.f odd 20 4
1875.2.a.e 4 5.b even 2 1
1875.2.a.h 4 1.a even 1 1 trivial
1875.2.b.c 8 5.c odd 4 2
5625.2.a.i 4 3.b odd 2 1
5625.2.a.n 4 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 2T_{2}^{3} - 6T_{2}^{2} + 7T_{2} + 11$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1875))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} - 6 T^{2} + 7 T + 11$$
$3$ $$(T - 1)^{4}$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 2 T^{3} - 16 T^{2} + 27 T - 9$$
$11$ $$T^{4} + 7 T^{3} - 6 T^{2} - 92 T - 109$$
$13$ $$T^{4} - T^{3} - 14 T^{2} + 24 T - 9$$
$17$ $$T^{4} - 2 T^{3} - 6 T^{2} + 7 T + 11$$
$19$ $$T^{4} - 5 T^{3} - 35 T^{2} + 75 T + 275$$
$23$ $$T^{4} - T^{3} - 24 T^{2} - 46 T - 19$$
$29$ $$T^{4} + 20 T^{3} + 140 T^{2} + \cdots + 405$$
$31$ $$T^{4} - 23 T^{3} + 184 T^{2} + \cdots + 711$$
$37$ $$T^{4} - 2 T^{3} - 46 T^{2} + 47 T + 1$$
$41$ $$(T^{2} + 6 T - 11)^{2}$$
$43$ $$T^{4} - 16 T^{3} + 61 T^{2} + 24 T - 99$$
$47$ $$T^{4} - 2 T^{3} - 36 T^{2} + 37 T + 311$$
$53$ $$T^{4} + 4 T^{3} - 34 T^{2} - 51 T + 261$$
$59$ $$T^{4} + 15 T^{3} - 45 T^{2} + \cdots - 3645$$
$61$ $$T^{4} + 2 T^{3} - 56 T^{2} + 78 T - 9$$
$67$ $$T^{4} - 2 T^{3} - 31 T^{2} + 12 T + 171$$
$71$ $$T^{4} + 2 T^{3} - 216 T^{2} + \cdots + 911$$
$73$ $$T^{4} - 16 T^{3} - 49 T^{2} + \cdots - 4869$$
$79$ $$T^{4} - 35 T^{3} + 320 T^{2} + \cdots - 9845$$
$83$ $$T^{4} - 16 T^{3} - 54 T^{2} + \cdots - 979$$
$89$ $$T^{4} + 35 T^{3} + 420 T^{2} + \cdots + 3305$$
$97$ $$T^{4} - 12 T^{3} - 86 T^{2} + \cdots + 2101$$