# Properties

 Label 1725.2.b.j Level $1725$ Weight $2$ Character orbit 1725.b Analytic conductor $13.774$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1725,2,Mod(1174,1725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1725, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("1725.1174");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1725 = 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1725.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: $$13.7741943487$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 345) 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 2 q^{4} + 3 i q^{7} - q^{9}+O(q^{10})$$ q + i * q^3 + 2 * q^4 + 3*i * q^7 - q^9 $$q + i q^{3} + 2 q^{4} + 3 i q^{7} - q^{9} - 4 q^{11} + 2 i q^{12} + 4 q^{16} + 3 i q^{17} + 8 q^{19} - 3 q^{21} + i q^{23} - i q^{27} + 6 i q^{28} - 9 q^{29} - 5 q^{31} - 4 i q^{33} - 2 q^{36} + 9 i q^{37} + 7 q^{41} + 4 i q^{43} - 8 q^{44} + 2 i q^{47} + 4 i q^{48} - 2 q^{49} - 3 q^{51} + 13 i q^{53} + 8 i q^{57} + 3 q^{59} - 14 q^{61} - 3 i q^{63} + 8 q^{64} - 13 i q^{67} + 6 i q^{68} - q^{69} - 13 q^{71} - 4 i q^{73} + 16 q^{76} - 12 i q^{77} + q^{81} - i q^{83} - 6 q^{84} - 9 i q^{87} + 8 q^{89} + 2 i q^{92} - 5 i q^{93} - 10 i q^{97} + 4 q^{99} +O(q^{100})$$ q + i * q^3 + 2 * q^4 + 3*i * q^7 - q^9 - 4 * q^11 + 2*i * q^12 + 4 * q^16 + 3*i * q^17 + 8 * q^19 - 3 * q^21 + i * q^23 - i * q^27 + 6*i * q^28 - 9 * q^29 - 5 * q^31 - 4*i * q^33 - 2 * q^36 + 9*i * q^37 + 7 * q^41 + 4*i * q^43 - 8 * q^44 + 2*i * q^47 + 4*i * q^48 - 2 * q^49 - 3 * q^51 + 13*i * q^53 + 8*i * q^57 + 3 * q^59 - 14 * q^61 - 3*i * q^63 + 8 * q^64 - 13*i * q^67 + 6*i * q^68 - q^69 - 13 * q^71 - 4*i * q^73 + 16 * q^76 - 12*i * q^77 + q^81 - i * q^83 - 6 * q^84 - 9*i * q^87 + 8 * q^89 + 2*i * q^92 - 5*i * q^93 - 10*i * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} - 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^4 - 2 * q^9 $$2 q + 4 q^{4} - 2 q^{9} - 8 q^{11} + 8 q^{16} + 16 q^{19} - 6 q^{21} - 18 q^{29} - 10 q^{31} - 4 q^{36} + 14 q^{41} - 16 q^{44} - 4 q^{49} - 6 q^{51} + 6 q^{59} - 28 q^{61} + 16 q^{64} - 2 q^{69} - 26 q^{71} + 32 q^{76} + 2 q^{81} - 12 q^{84} + 16 q^{89} + 8 q^{99}+O(q^{100})$$ 2 * q + 4 * q^4 - 2 * q^9 - 8 * q^11 + 8 * q^16 + 16 * q^19 - 6 * q^21 - 18 * q^29 - 10 * q^31 - 4 * q^36 + 14 * q^41 - 16 * q^44 - 4 * q^49 - 6 * q^51 + 6 * q^59 - 28 * q^61 + 16 * q^64 - 2 * q^69 - 26 * q^71 + 32 * q^76 + 2 * q^81 - 12 * q^84 + 16 * q^89 + 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1174.1
 − 1.00000i 1.00000i
0 1.00000i 2.00000 0 0 3.00000i 0 −1.00000 0
1174.2 0 1.00000i 2.00000 0 0 3.00000i 0 −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 1725.2.b.j 2
5.b even 2 1 inner 1725.2.b.j 2
5.c odd 4 1 345.2.a.d 1
5.c odd 4 1 1725.2.a.k 1
15.e even 4 1 1035.2.a.d 1
15.e even 4 1 5175.2.a.o 1
20.e even 4 1 5520.2.a.h 1
115.e even 4 1 7935.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.a.d 1 5.c odd 4 1
1035.2.a.d 1 15.e even 4 1
1725.2.a.k 1 5.c odd 4 1
1725.2.b.j 2 1.a even 1 1 trivial
1725.2.b.j 2 5.b even 2 1 inner
5175.2.a.o 1 15.e even 4 1
5520.2.a.h 1 20.e even 4 1
7935.2.a.i 1 115.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1725, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7}^{2} + 9$$ T7^2 + 9 $$T_{11} + 4$$ T11 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T + 4)^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 9$$
$19$ $$(T - 8)^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T + 9)^{2}$$
$31$ $$(T + 5)^{2}$$
$37$ $$T^{2} + 81$$
$41$ $$(T - 7)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 169$$
$59$ $$(T - 3)^{2}$$
$61$ $$(T + 14)^{2}$$
$67$ $$T^{2} + 169$$
$71$ $$(T + 13)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 1$$
$89$ $$(T - 8)^{2}$$
$97$ $$T^{2} + 100$$