# Properties

 Label 2523.1.b.b Level $2523$ Weight $1$ Character orbit 2523.b Analytic conductor $1.259$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -87 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2523,1,Mod(842,2523)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2523, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2523.842");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2523 = 3 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2523.b (of order $$2$$, degree $$1$$, 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: $$1.25914102687$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 87) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.87.1 Artin image: $C_4\times S_3$ Artin field: Galois closure of 12.0.1175078824045389.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - i q^{2} + i q^{3} + q^{6} - q^{7} - i q^{8} - q^{9} +O(q^{10})$$ q - z * q^2 + z * q^3 + q^6 - q^7 - z * q^8 - q^9 $$q - i q^{2} + i q^{3} + q^{6} - q^{7} - i q^{8} - q^{9} - i q^{11} + q^{13} + i q^{14} - q^{16} - i q^{17} + i q^{18} - i q^{21} - q^{22} + q^{24} + q^{25} - i q^{26} - i q^{27} + q^{33} - q^{34} + i q^{39} - 2 i q^{41} - q^{42} + i q^{47} - i q^{48} - i q^{50} + q^{51} - q^{54} + i q^{56} + q^{63} - q^{64} - i q^{66} + q^{67} + i q^{72} + i q^{75} + i q^{77} + q^{78} + q^{81} - 2 q^{82} - q^{88} - i q^{89} - q^{91} + q^{94} + i q^{99} +O(q^{100})$$ q - z * q^2 + z * q^3 + q^6 - q^7 - z * q^8 - q^9 - z * q^11 + q^13 + z * q^14 - q^16 - z * q^17 + z * q^18 - z * q^21 - q^22 + q^24 + q^25 - z * q^26 - z * q^27 + q^33 - q^34 + z * q^39 - 2*z * q^41 - q^42 + z * q^47 - z * q^48 - z * q^50 + q^51 - q^54 + z * q^56 + q^63 - q^64 - z * q^66 + q^67 + z * q^72 + z * q^75 + z * q^77 + q^78 + q^81 - 2 * q^82 - q^88 - z * q^89 - q^91 + q^94 + z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{6} - 2 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^6 - 2 * q^7 - 2 * q^9 $$2 q + 2 q^{6} - 2 q^{7} - 2 q^{9} + 2 q^{13} - 2 q^{16} - 2 q^{22} + 2 q^{24} + 2 q^{25} + 2 q^{33} - 2 q^{34} - 2 q^{42} + 2 q^{51} - 2 q^{54} + 2 q^{63} - 2 q^{64} + 2 q^{67} + 2 q^{78} + 2 q^{81} - 4 q^{82} - 2 q^{88} - 2 q^{91} + 2 q^{94}+O(q^{100})$$ 2 * q + 2 * q^6 - 2 * q^7 - 2 * q^9 + 2 * q^13 - 2 * q^16 - 2 * q^22 + 2 * q^24 + 2 * q^25 + 2 * q^33 - 2 * q^34 - 2 * q^42 + 2 * q^51 - 2 * q^54 + 2 * q^63 - 2 * q^64 + 2 * q^67 + 2 * q^78 + 2 * q^81 - 4 * q^82 - 2 * q^88 - 2 * q^91 + 2 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2523\mathbb{Z}\right)^\times$$.

 $$n$$ $$842$$ $$1684$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
842.1
 1.00000i − 1.00000i
1.00000i 1.00000i 0 0 1.00000 −1.00000 1.00000i −1.00000 0
842.2 1.00000i 1.00000i 0 0 1.00000 −1.00000 1.00000i −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
87.d odd 2 1 CM by $$\Q(\sqrt{-87})$$
3.b odd 2 1 inner
29.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2523.1.b.b 2
3.b odd 2 1 inner 2523.1.b.b 2
29.b even 2 1 inner 2523.1.b.b 2
29.c odd 4 1 87.1.d.a 1
29.c odd 4 1 87.1.d.b yes 1
29.d even 7 6 2523.1.j.b 12
29.e even 14 6 2523.1.j.b 12
29.f odd 28 6 2523.1.h.a 6
29.f odd 28 6 2523.1.h.b 6
87.d odd 2 1 CM 2523.1.b.b 2
87.f even 4 1 87.1.d.a 1
87.f even 4 1 87.1.d.b yes 1
87.h odd 14 6 2523.1.j.b 12
87.j odd 14 6 2523.1.j.b 12
87.k even 28 6 2523.1.h.a 6
87.k even 28 6 2523.1.h.b 6
116.e even 4 1 1392.1.i.a 1
116.e even 4 1 1392.1.i.b 1
145.e even 4 1 2175.1.b.a 2
145.e even 4 1 2175.1.b.b 2
145.f odd 4 1 2175.1.h.a 1
145.f odd 4 1 2175.1.h.b 1
145.j even 4 1 2175.1.b.a 2
145.j even 4 1 2175.1.b.b 2
261.l even 12 2 2349.1.h.a 2
261.l even 12 2 2349.1.h.b 2
261.m odd 12 2 2349.1.h.a 2
261.m odd 12 2 2349.1.h.b 2
348.k odd 4 1 1392.1.i.a 1
348.k odd 4 1 1392.1.i.b 1
435.i odd 4 1 2175.1.b.a 2
435.i odd 4 1 2175.1.b.b 2
435.l even 4 1 2175.1.h.a 1
435.l even 4 1 2175.1.h.b 1
435.t odd 4 1 2175.1.b.a 2
435.t odd 4 1 2175.1.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 29.c odd 4 1
87.1.d.a 1 87.f even 4 1
87.1.d.b yes 1 29.c odd 4 1
87.1.d.b yes 1 87.f even 4 1
1392.1.i.a 1 116.e even 4 1
1392.1.i.a 1 348.k odd 4 1
1392.1.i.b 1 116.e even 4 1
1392.1.i.b 1 348.k odd 4 1
2175.1.b.a 2 145.e even 4 1
2175.1.b.a 2 145.j even 4 1
2175.1.b.a 2 435.i odd 4 1
2175.1.b.a 2 435.t odd 4 1
2175.1.b.b 2 145.e even 4 1
2175.1.b.b 2 145.j even 4 1
2175.1.b.b 2 435.i odd 4 1
2175.1.b.b 2 435.t odd 4 1
2175.1.h.a 1 145.f odd 4 1
2175.1.h.a 1 435.l even 4 1
2175.1.h.b 1 145.f odd 4 1
2175.1.h.b 1 435.l even 4 1
2349.1.h.a 2 261.l even 12 2
2349.1.h.a 2 261.m odd 12 2
2349.1.h.b 2 261.l even 12 2
2349.1.h.b 2 261.m odd 12 2
2523.1.b.b 2 1.a even 1 1 trivial
2523.1.b.b 2 3.b odd 2 1 inner
2523.1.b.b 2 29.b even 2 1 inner
2523.1.b.b 2 87.d odd 2 1 CM
2523.1.h.a 6 29.f odd 28 6
2523.1.h.a 6 87.k even 28 6
2523.1.h.b 6 29.f odd 28 6
2523.1.h.b 6 87.k even 28 6
2523.1.j.b 12 29.d even 7 6
2523.1.j.b 12 29.e even 14 6
2523.1.j.b 12 87.h odd 14 6
2523.1.j.b 12 87.j odd 14 6

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2523, [\chi])$$:

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{19}$$ T19

## Hecke characteristic polynomials

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