# Properties

 Label 2523.1.j.b Level $2523$ Weight $1$ Character orbit 2523.j Analytic conductor $1.259$ Analytic rank $0$ Dimension $12$ Projective image $D_{3}$ CM discriminant -87 Inner twists $24$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2523,1,Mod(605,2523)]

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, 8]))

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

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

chi := DirichletCharacter("2523.605");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2523 = 3 \cdot 29^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2523.j (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: $$12$$ Relative dimension: $$2$$ over $$\Q(\zeta_{14})$$ Coefficient field: $$\Q(\zeta_{28})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^12 - x^10 + x^8 - x^6 + x^4 - 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: $S_3\times C_{28}$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{84} - \cdots)$$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{28}^{5} q^{2} + \zeta_{28}^{11} q^{3} - \zeta_{28}^{2} q^{6} - \zeta_{28}^{4} q^{7} + \zeta_{28} q^{8} - \zeta_{28}^{8} q^{9} +O(q^{10})$$ q + z^5 * q^2 + z^11 * q^3 - z^2 * q^6 - z^4 * q^7 + z * q^8 - z^8 * q^9 $$q + \zeta_{28}^{5} q^{2} + \zeta_{28}^{11} q^{3} - \zeta_{28}^{2} q^{6} - \zeta_{28}^{4} q^{7} + \zeta_{28} q^{8} - \zeta_{28}^{8} q^{9} + \zeta_{28}^{13} q^{11} - \zeta_{28}^{6} q^{13} - \zeta_{28}^{9} q^{14} + \zeta_{28}^{6} q^{16} - \zeta_{28}^{7} q^{17} - \zeta_{28}^{13} q^{18} + \zeta_{28} q^{21} - \zeta_{28}^{4} q^{22} + \zeta_{28}^{12} q^{24} - \zeta_{28}^{10} q^{25} - \zeta_{28}^{11} q^{26} + \zeta_{28}^{5} q^{27} - \zeta_{28}^{10} q^{33} - \zeta_{28}^{12} q^{34} + \zeta_{28}^{3} q^{39} - 2 \zeta_{28}^{7} q^{41} + \zeta_{28}^{6} q^{42} - \zeta_{28}^{13} q^{47} - \zeta_{28}^{3} q^{48} + \zeta_{28} q^{50} + \zeta_{28}^{4} q^{51} + \zeta_{28}^{10} q^{54} - \zeta_{28}^{5} q^{56} + \zeta_{28}^{12} q^{63} + \zeta_{28}^{2} q^{64} + \zeta_{28} q^{66} + \zeta_{28}^{8} q^{67} - \zeta_{28}^{9} q^{72} + \zeta_{28}^{7} q^{75} + \zeta_{28}^{3} q^{77} + \zeta_{28}^{8} q^{78} - \zeta_{28}^{2} q^{81} - 2 \zeta_{28}^{12} q^{82} - q^{88} + \zeta_{28}^{5} q^{89} + \zeta_{28}^{10} q^{91} + \zeta_{28}^{4} q^{94} + \zeta_{28}^{7} q^{99} +O(q^{100})$$ q + z^5 * q^2 + z^11 * q^3 - z^2 * q^6 - z^4 * q^7 + z * q^8 - z^8 * q^9 + z^13 * q^11 - z^6 * q^13 - z^9 * q^14 + z^6 * q^16 - z^7 * q^17 - z^13 * q^18 + z * q^21 - z^4 * q^22 + z^12 * q^24 - z^10 * q^25 - z^11 * q^26 + z^5 * q^27 - z^10 * q^33 - z^12 * q^34 + z^3 * q^39 - 2*z^7 * q^41 + z^6 * q^42 - z^13 * q^47 - z^3 * q^48 + z * q^50 + z^4 * q^51 + z^10 * q^54 - z^5 * q^56 + z^12 * q^63 + z^2 * q^64 + z * q^66 + z^8 * q^67 - z^9 * q^72 + z^7 * q^75 + z^3 * q^77 + z^8 * q^78 - z^2 * q^81 - 2*z^12 * q^82 - q^88 + z^5 * q^89 + z^10 * q^91 + z^4 * q^94 + z^7 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 2 q^{6} + 2 q^{7} + 2 q^{9}+O(q^{10})$$ 12 * q - 2 * q^6 + 2 * q^7 + 2 * q^9 $$12 q - 2 q^{6} + 2 q^{7} + 2 q^{9} - 2 q^{13} + 2 q^{16} + 2 q^{22} - 2 q^{24} - 2 q^{25} - 2 q^{33} + 2 q^{34} + 2 q^{42} - 2 q^{51} + 2 q^{54} - 2 q^{63} + 2 q^{64} - 2 q^{67} - 2 q^{78} - 2 q^{81} + 4 q^{82} - 12 q^{88} + 2 q^{91} - 2 q^{94}+O(q^{100})$$ 12 * 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 - 12 * q^88 + 2 * q^91 - 2 * q^94

## Character values

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

 $$n$$ $$842$$ $$1684$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{28}^{10}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
605.1
 0.781831 + 0.623490i −0.781831 − 0.623490i −0.974928 + 0.222521i 0.974928 − 0.222521i −0.974928 − 0.222521i 0.974928 + 0.222521i −0.433884 + 0.900969i 0.433884 − 0.900969i −0.433884 − 0.900969i 0.433884 + 0.900969i 0.781831 − 0.623490i −0.781831 + 0.623490i
−0.974928 0.222521i 0.433884 + 0.900969i 0 0 −0.222521 0.974928i 0.900969 0.433884i 0.781831 + 0.623490i −0.623490 + 0.781831i 0
605.2 0.974928 + 0.222521i −0.433884 0.900969i 0 0 −0.222521 0.974928i 0.900969 0.433884i −0.781831 0.623490i −0.623490 + 0.781831i 0
1031.1 −0.433884 + 0.900969i 0.781831 + 0.623490i 0 0 −0.900969 + 0.433884i −0.623490 + 0.781831i −0.974928 + 0.222521i 0.222521 + 0.974928i 0
1031.2 0.433884 0.900969i −0.781831 0.623490i 0 0 −0.900969 + 0.433884i −0.623490 + 0.781831i 0.974928 0.222521i 0.222521 + 0.974928i 0
1412.1 −0.433884 0.900969i 0.781831 0.623490i 0 0 −0.900969 0.433884i −0.623490 0.781831i −0.974928 0.222521i 0.222521 0.974928i 0
1412.2 0.433884 + 0.900969i −0.781831 + 0.623490i 0 0 −0.900969 0.433884i −0.623490 0.781831i 0.974928 + 0.222521i 0.222521 0.974928i 0
1415.1 −0.781831 0.623490i −0.974928 0.222521i 0 0 0.623490 + 0.781831i 0.222521 0.974928i −0.433884 + 0.900969i 0.900969 + 0.433884i 0
1415.2 0.781831 + 0.623490i 0.974928 + 0.222521i 0 0 0.623490 + 0.781831i 0.222521 0.974928i 0.433884 0.900969i 0.900969 + 0.433884i 0
1619.1 −0.781831 + 0.623490i −0.974928 + 0.222521i 0 0 0.623490 0.781831i 0.222521 + 0.974928i −0.433884 0.900969i 0.900969 0.433884i 0
1619.2 0.781831 0.623490i 0.974928 0.222521i 0 0 0.623490 0.781831i 0.222521 + 0.974928i 0.433884 + 0.900969i 0.900969 0.433884i 0
2327.1 −0.974928 + 0.222521i 0.433884 0.900969i 0 0 −0.222521 + 0.974928i 0.900969 + 0.433884i 0.781831 0.623490i −0.623490 0.781831i 0
2327.2 0.974928 0.222521i −0.433884 + 0.900969i 0 0 −0.222521 + 0.974928i 0.900969 + 0.433884i −0.781831 + 0.623490i −0.623490 0.781831i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 605.2 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
29.d even 7 5 inner
29.e even 14 5 inner
87.h odd 14 5 inner
87.j 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.j.b 12
3.b odd 2 1 inner 2523.1.j.b 12
29.b even 2 1 inner 2523.1.j.b 12
29.c odd 4 1 2523.1.h.a 6
29.c odd 4 1 2523.1.h.b 6
29.d even 7 1 2523.1.b.b 2
29.d even 7 5 inner 2523.1.j.b 12
29.e even 14 1 2523.1.b.b 2
29.e even 14 5 inner 2523.1.j.b 12
29.f odd 28 1 87.1.d.a 1
29.f odd 28 1 87.1.d.b yes 1
29.f odd 28 5 2523.1.h.a 6
29.f odd 28 5 2523.1.h.b 6
87.d odd 2 1 CM 2523.1.j.b 12
87.f even 4 1 2523.1.h.a 6
87.f even 4 1 2523.1.h.b 6
87.h odd 14 1 2523.1.b.b 2
87.h odd 14 5 inner 2523.1.j.b 12
87.j odd 14 1 2523.1.b.b 2
87.j odd 14 5 inner 2523.1.j.b 12
87.k even 28 1 87.1.d.a 1
87.k even 28 1 87.1.d.b yes 1
87.k even 28 5 2523.1.h.a 6
87.k even 28 5 2523.1.h.b 6
116.l even 28 1 1392.1.i.a 1
116.l even 28 1 1392.1.i.b 1
145.o even 28 1 2175.1.b.a 2
145.o even 28 1 2175.1.b.b 2
145.s odd 28 1 2175.1.h.a 1
145.s odd 28 1 2175.1.h.b 1
145.t even 28 1 2175.1.b.a 2
145.t even 28 1 2175.1.b.b 2
261.w odd 84 2 2349.1.h.a 2
261.w odd 84 2 2349.1.h.b 2
261.x even 84 2 2349.1.h.a 2
261.x even 84 2 2349.1.h.b 2
348.v odd 28 1 1392.1.i.a 1
348.v odd 28 1 1392.1.i.b 1
435.bc odd 28 1 2175.1.b.a 2
435.bc odd 28 1 2175.1.b.b 2
435.bk even 28 1 2175.1.h.a 1
435.bk even 28 1 2175.1.h.b 1
435.bn odd 28 1 2175.1.b.a 2
435.bn odd 28 1 2175.1.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 29.f odd 28 1
87.1.d.a 1 87.k even 28 1
87.1.d.b yes 1 29.f odd 28 1
87.1.d.b yes 1 87.k even 28 1
1392.1.i.a 1 116.l even 28 1
1392.1.i.a 1 348.v odd 28 1
1392.1.i.b 1 116.l even 28 1
1392.1.i.b 1 348.v odd 28 1
2175.1.b.a 2 145.o even 28 1
2175.1.b.a 2 145.t even 28 1
2175.1.b.a 2 435.bc odd 28 1
2175.1.b.a 2 435.bn odd 28 1
2175.1.b.b 2 145.o even 28 1
2175.1.b.b 2 145.t even 28 1
2175.1.b.b 2 435.bc odd 28 1
2175.1.b.b 2 435.bn odd 28 1
2175.1.h.a 1 145.s odd 28 1
2175.1.h.a 1 435.bk even 28 1
2175.1.h.b 1 145.s odd 28 1
2175.1.h.b 1 435.bk even 28 1
2349.1.h.a 2 261.w odd 84 2
2349.1.h.a 2 261.x even 84 2
2349.1.h.b 2 261.w odd 84 2
2349.1.h.b 2 261.x even 84 2
2523.1.b.b 2 29.d even 7 1
2523.1.b.b 2 29.e even 14 1
2523.1.b.b 2 87.h odd 14 1
2523.1.b.b 2 87.j odd 14 1
2523.1.h.a 6 29.c odd 4 1
2523.1.h.a 6 29.f odd 28 5
2523.1.h.a 6 87.f even 4 1
2523.1.h.a 6 87.k even 28 5
2523.1.h.b 6 29.c odd 4 1
2523.1.h.b 6 29.f odd 28 5
2523.1.h.b 6 87.f even 4 1
2523.1.h.b 6 87.k even 28 5
2523.1.j.b 12 1.a even 1 1 trivial
2523.1.j.b 12 3.b odd 2 1 inner
2523.1.j.b 12 29.b even 2 1 inner
2523.1.j.b 12 29.d even 7 5 inner
2523.1.j.b 12 29.e even 14 5 inner
2523.1.j.b 12 87.d odd 2 1 CM
2523.1.j.b 12 87.h odd 14 5 inner
2523.1.j.b 12 87.j odd 14 5 inner

## Hecke kernels

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

 $$T_{2}^{12} - T_{2}^{10} + T_{2}^{8} - T_{2}^{6} + T_{2}^{4} - T_{2}^{2} + 1$$ T2^12 - T2^10 + T2^8 - T2^6 + T2^4 - T2^2 + 1 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - T^{10} + \cdots + 1$$
$3$ $$T^{12} - T^{10} + \cdots + 1$$
$5$ $$T^{12}$$
$7$ $$(T^{6} - T^{5} + T^{4} + \cdots + 1)^{2}$$
$11$ $$T^{12} - T^{10} + \cdots + 1$$
$13$ $$(T^{6} + T^{5} + T^{4} + \cdots + 1)^{2}$$
$17$ $$(T^{2} + 1)^{6}$$
$19$ $$T^{12}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$T^{12}$$
$37$ $$T^{12}$$
$41$ $$(T^{2} + 4)^{6}$$
$43$ $$T^{12}$$
$47$ $$T^{12} - T^{10} + \cdots + 1$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$T^{12}$$
$67$ $$(T^{6} + T^{5} + T^{4} + \cdots + 1)^{2}$$
$71$ $$T^{12}$$
$73$ $$T^{12}$$
$79$ $$T^{12}$$
$83$ $$T^{12}$$
$89$ $$T^{12} - T^{10} + \cdots + 1$$
$97$ $$T^{12}$$