# Properties

 Label 1875.2.b.b Level $1875$ Weight $2$ Character orbit 1875.b Analytic conductor $14.972$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

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, 1]))

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

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

chi := DirichletCharacter("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1875.b (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$14.9719503790$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$$.

 $$n$$ $$626$$ $$1252$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1249.1
 1.61803i − 0.618034i 0.618034i − 1.61803i
1.00000i 1.00000i 1.00000 0 1.00000 4.47214i 3.00000i −1.00000 0
1249.2 1.00000i 1.00000i 1.00000 0 1.00000 4.47214i 3.00000i −1.00000 0
1249.3 1.00000i 1.00000i 1.00000 0 1.00000 4.47214i 3.00000i −1.00000 0
1249.4 1.00000i 1.00000i 1.00000 0 1.00000 4.47214i 3.00000i −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

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

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 20)^{2}$$
$11$ $$(T^{2} - 2 T - 4)^{2}$$
$13$ $$T^{4} + 43T^{2} + 361$$
$17$ $$T^{4} + 23T^{2} + 121$$
$19$ $$(T^{2} - 2 T - 4)^{2}$$
$23$ $$(T^{2} + 20)^{2}$$
$29$ $$(T^{2} + 11 T + 29)^{2}$$
$31$ $$(T^{2} - 10 T + 20)^{2}$$
$37$ $$T^{4} + 75T^{2} + 625$$
$41$ $$(T^{2} + 5 T + 5)^{2}$$
$43$ $$T^{4} + 92T^{2} + 1936$$
$47$ $$T^{4} + 28T^{2} + 16$$
$53$ $$T^{4} + 15T^{2} + 25$$
$59$ $$(T - 4)^{4}$$
$61$ $$(T^{2} - T - 1)^{2}$$
$67$ $$T^{4} + 28T^{2} + 16$$
$71$ $$(T^{2} + 6 T + 4)^{2}$$
$73$ $$T^{4} + 75T^{2} + 625$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 168T^{2} + 1936$$
$89$ $$(T^{2} + 13 T + 41)^{2}$$
$97$ $$T^{4} + 83T^{2} + 361$$