# Properties

 Label 2523.1.h.b Level $2523$ Weight $1$ Character orbit 2523.h Analytic conductor $1.259$ Analytic rank $0$ Dimension $6$ Projective image $D_{3}$ CM discriminant -87 Inner twists $12$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2523,1,Mod(236,2523)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2523, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([7, 1]))

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

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

chi := DirichletCharacter("2523.236");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2523 = 3 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2523.h (of order $$14$$, degree $$6$$, 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: $$1.25914102687$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{14})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ x^6 - x^5 + x^4 - x^3 + x^2 - x + 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: $S_3\times C_7$ Artin field: Galois closure of 21.7.13347832346292311387708944226103.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{14}^{6} q^{2} + \zeta_{14}^{2} q^{3} + \zeta_{14} q^{6} - \zeta_{14}^{2} q^{7} + \zeta_{14}^{4} q^{8} + \zeta_{14}^{4} q^{9} +O(q^{10})$$ q - z^6 * q^2 + z^2 * q^3 + z * q^6 - z^2 * q^7 + z^4 * q^8 + z^4 * q^9 $$q - \zeta_{14}^{6} q^{2} + \zeta_{14}^{2} q^{3} + \zeta_{14} q^{6} - \zeta_{14}^{2} q^{7} + \zeta_{14}^{4} q^{8} + \zeta_{14}^{4} q^{9} + \zeta_{14}^{3} q^{11} + \zeta_{14}^{3} q^{13} - \zeta_{14} q^{14} + \zeta_{14}^{3} q^{16} - q^{17} + \zeta_{14}^{3} q^{18} - \zeta_{14}^{4} q^{21} + \zeta_{14}^{2} q^{22} + \zeta_{14}^{6} q^{24} - \zeta_{14}^{5} q^{25} + \zeta_{14}^{2} q^{26} + \zeta_{14}^{6} q^{27} + \zeta_{14}^{5} q^{33} + \zeta_{14}^{6} q^{34} + \zeta_{14}^{5} q^{39} + 2 q^{41} - \zeta_{14}^{3} q^{42} + \zeta_{14}^{3} q^{47} + \zeta_{14}^{5} q^{48} - \zeta_{14}^{4} q^{50} - \zeta_{14}^{2} q^{51} + \zeta_{14}^{5} q^{54} - \zeta_{14}^{6} q^{56} - \zeta_{14}^{6} q^{63} - \zeta_{14} q^{64} + \zeta_{14}^{4} q^{66} - \zeta_{14}^{4} q^{67} - \zeta_{14} q^{72} + q^{75} - \zeta_{14}^{5} q^{77} + \zeta_{14}^{4} q^{78} - \zeta_{14} q^{81} - 2 \zeta_{14}^{6} q^{82} - q^{88} - \zeta_{14}^{6} q^{89} - \zeta_{14}^{5} q^{91} + \zeta_{14}^{2} q^{94} - q^{99} +O(q^{100})$$ q - z^6 * q^2 + z^2 * q^3 + z * q^6 - z^2 * q^7 + z^4 * q^8 + z^4 * q^9 + z^3 * q^11 + z^3 * q^13 - z * q^14 + z^3 * q^16 - q^17 + z^3 * q^18 - z^4 * q^21 + z^2 * q^22 + z^6 * q^24 - z^5 * q^25 + z^2 * q^26 + z^6 * q^27 + z^5 * q^33 + z^6 * q^34 + z^5 * q^39 + 2 * q^41 - z^3 * q^42 + z^3 * q^47 + z^5 * q^48 - z^4 * q^50 - z^2 * q^51 + z^5 * q^54 - z^6 * q^56 - z^6 * q^63 - z * q^64 + z^4 * q^66 - z^4 * q^67 - z * q^72 + q^75 - z^5 * q^77 + z^4 * q^78 - z * q^81 - 2*z^6 * q^82 - q^88 - z^6 * q^89 - z^5 * q^91 + z^2 * q^94 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + q^{2} - q^{3} + q^{6} + q^{7} - q^{8} - q^{9}+O(q^{10})$$ 6 * q + q^2 - q^3 + q^6 + q^7 - q^8 - q^9 $$6 q + q^{2} - q^{3} + q^{6} + q^{7} - q^{8} - q^{9} + q^{11} + q^{13} - q^{14} + q^{16} - 6 q^{17} + q^{18} + q^{21} - q^{22} - q^{24} - q^{25} - q^{26} - q^{27} + q^{33} - q^{34} + q^{39} + 12 q^{41} - q^{42} + q^{47} + q^{48} + q^{50} + q^{51} + q^{54} + q^{56} + q^{63} - q^{64} - q^{66} + q^{67} - q^{72} + 6 q^{75} - q^{77} - q^{78} - q^{81} + 2 q^{82} - 6 q^{88} + q^{89} - q^{91} - q^{94} - 6 q^{99}+O(q^{100})$$ 6 * q + q^2 - q^3 + q^6 + q^7 - q^8 - q^9 + q^11 + q^13 - q^14 + q^16 - 6 * q^17 + q^18 + q^21 - q^22 - q^24 - q^25 - q^26 - q^27 + q^33 - q^34 + q^39 + 12 * q^41 - q^42 + q^47 + q^48 + q^50 + q^51 + q^54 + q^56 + q^63 - q^64 - q^66 + q^67 - q^72 + 6 * q^75 - q^77 - q^78 - q^81 + 2 * q^82 - 6 * q^88 + q^89 - q^91 - q^94 - 6 * q^99

## Character values

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

 $$n$$ $$842$$ $$1684$$ $$\chi(n)$$ $$-1$$ $$\zeta_{14}^{5}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
236.1
 0.222521 + 0.974928i 0.222521 − 0.974928i −0.623490 + 0.781831i −0.623490 − 0.781831i 0.900969 + 0.433884i 0.900969 − 0.433884i
0.222521 0.974928i −0.900969 + 0.433884i 0 0 0.222521 + 0.974928i 0.900969 0.433884i 0.623490 0.781831i 0.623490 0.781831i 0
1037.1 0.222521 + 0.974928i −0.900969 0.433884i 0 0 0.222521 0.974928i 0.900969 + 0.433884i 0.623490 + 0.781831i 0.623490 + 0.781831i 0
1745.1 −0.623490 0.781831i −0.222521 0.974928i 0 0 −0.623490 + 0.781831i 0.222521 + 0.974928i −0.900969 + 0.433884i −0.900969 + 0.433884i 0
1949.1 −0.623490 + 0.781831i −0.222521 + 0.974928i 0 0 −0.623490 0.781831i 0.222521 0.974928i −0.900969 0.433884i −0.900969 0.433884i 0
1952.1 0.900969 0.433884i 0.623490 + 0.781831i 0 0 0.900969 + 0.433884i −0.623490 0.781831i −0.222521 + 0.974928i −0.222521 + 0.974928i 0
2333.1 0.900969 + 0.433884i 0.623490 0.781831i 0 0 0.900969 0.433884i −0.623490 + 0.781831i −0.222521 0.974928i −0.222521 0.974928i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 236.1 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 $$\Q(\sqrt{-87})$$
29.d even 7 5 inner
87.h odd 14 5 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2523.1.h.b 6
3.b odd 2 1 2523.1.h.a 6
29.b even 2 1 2523.1.h.a 6
29.c odd 4 2 2523.1.j.b 12
29.d even 7 1 87.1.d.a 1
29.d even 7 5 inner 2523.1.h.b 6
29.e even 14 1 87.1.d.b yes 1
29.e even 14 5 2523.1.h.a 6
29.f odd 28 2 2523.1.b.b 2
29.f odd 28 10 2523.1.j.b 12
87.d odd 2 1 CM 2523.1.h.b 6
87.f even 4 2 2523.1.j.b 12
87.h odd 14 1 87.1.d.a 1
87.h odd 14 5 inner 2523.1.h.b 6
87.j odd 14 1 87.1.d.b yes 1
87.j odd 14 5 2523.1.h.a 6
87.k even 28 2 2523.1.b.b 2
87.k even 28 10 2523.1.j.b 12
116.h odd 14 1 1392.1.i.b 1
116.j odd 14 1 1392.1.i.a 1
145.l even 14 1 2175.1.h.a 1
145.n even 14 1 2175.1.h.b 1
145.p odd 28 2 2175.1.b.a 2
145.q odd 28 2 2175.1.b.b 2
261.q even 21 2 2349.1.h.b 2
261.t odd 42 2 2349.1.h.a 2
261.u even 42 2 2349.1.h.a 2
261.v odd 42 2 2349.1.h.b 2
348.s even 14 1 1392.1.i.b 1
348.t even 14 1 1392.1.i.a 1
435.w odd 14 1 2175.1.h.a 1
435.bb odd 14 1 2175.1.h.b 1
435.bg even 28 2 2175.1.b.a 2
435.bj even 28 2 2175.1.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 29.d even 7 1
87.1.d.a 1 87.h odd 14 1
87.1.d.b yes 1 29.e even 14 1
87.1.d.b yes 1 87.j odd 14 1
1392.1.i.a 1 116.j odd 14 1
1392.1.i.a 1 348.t even 14 1
1392.1.i.b 1 116.h odd 14 1
1392.1.i.b 1 348.s even 14 1
2175.1.b.a 2 145.p odd 28 2
2175.1.b.a 2 435.bg even 28 2
2175.1.b.b 2 145.q odd 28 2
2175.1.b.b 2 435.bj even 28 2
2175.1.h.a 1 145.l even 14 1
2175.1.h.a 1 435.w odd 14 1
2175.1.h.b 1 145.n even 14 1
2175.1.h.b 1 435.bb odd 14 1
2349.1.h.a 2 261.t odd 42 2
2349.1.h.a 2 261.u even 42 2
2349.1.h.b 2 261.q even 21 2
2349.1.h.b 2 261.v odd 42 2
2523.1.b.b 2 29.f odd 28 2
2523.1.b.b 2 87.k even 28 2
2523.1.h.a 6 3.b odd 2 1
2523.1.h.a 6 29.b even 2 1
2523.1.h.a 6 29.e even 14 5
2523.1.h.a 6 87.j odd 14 5
2523.1.h.b 6 1.a even 1 1 trivial
2523.1.h.b 6 29.d even 7 5 inner
2523.1.h.b 6 87.d odd 2 1 CM
2523.1.h.b 6 87.h odd 14 5 inner
2523.1.j.b 12 29.c odd 4 2
2523.1.j.b 12 29.f odd 28 10
2523.1.j.b 12 87.f even 4 2
2523.1.j.b 12 87.k even 28 10

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - T_{2}^{5} + T_{2}^{4} - T_{2}^{3} + T_{2}^{2} - T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2523, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{5} + T^{4} + \cdots + 1$$
$3$ $$T^{6} + T^{5} + T^{4} + \cdots + 1$$
$5$ $$T^{6}$$
$7$ $$T^{6} - T^{5} + T^{4} + \cdots + 1$$
$11$ $$T^{6} - T^{5} + T^{4} + \cdots + 1$$
$13$ $$T^{6} - T^{5} + T^{4} + \cdots + 1$$
$17$ $$(T + 1)^{6}$$
$19$ $$T^{6}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$T^{6}$$
$37$ $$T^{6}$$
$41$ $$(T - 2)^{6}$$
$43$ $$T^{6}$$
$47$ $$T^{6} - T^{5} + T^{4} + \cdots + 1$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$T^{6}$$
$67$ $$T^{6} - T^{5} + T^{4} + \cdots + 1$$
$71$ $$T^{6}$$
$73$ $$T^{6}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6} - T^{5} + T^{4} + \cdots + 1$$
$97$ $$T^{6}$$