# Properties

 Label 2349.1.h.b Level $2349$ Weight $1$ Character orbit 2349.h Analytic conductor $1.172$ 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] = [2349,1,Mod(782,2349)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2349, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 3]))

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

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

chi := DirichletCharacter("2349.782");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2349 = 3^{4} \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2349.h (of order $$6$$, degree $$2$$, 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: $$1.17230371467$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ 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: $C_3\times S_3$ Artin field: Galois closure of 6.0.480048687.2

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + \zeta_{6} q^{7} + q^{8}+O(q^{10})$$ q + z * q^2 + z * q^7 + q^8 $$q + \zeta_{6} q^{2} + \zeta_{6} q^{7} + q^{8} + \zeta_{6} q^{11} - \zeta_{6}^{2} q^{13} + \zeta_{6}^{2} q^{14} + \zeta_{6} q^{16} - q^{17} + \zeta_{6}^{2} q^{22} - \zeta_{6} q^{25} + q^{26} - \zeta_{6} q^{29} - \zeta_{6} q^{34} + 2 \zeta_{6}^{2} q^{41} + \zeta_{6} q^{47} - \zeta_{6}^{2} q^{50} + \zeta_{6} q^{56} - \zeta_{6}^{2} q^{58} + q^{64} - \zeta_{6}^{2} q^{67} + \zeta_{6}^{2} q^{77} - 2 q^{82} + \zeta_{6} q^{88} - q^{89} + q^{91} + \zeta_{6}^{2} q^{94} +O(q^{100})$$ q + z * q^2 + z * q^7 + q^8 + z * q^11 - z^2 * q^13 + z^2 * q^14 + z * q^16 - q^17 + z^2 * q^22 - z * q^25 + q^26 - z * q^29 - z * q^34 + 2*z^2 * q^41 + z * q^47 - z^2 * q^50 + z * q^56 - z^2 * q^58 + q^64 - z^2 * q^67 + z^2 * q^77 - 2 * q^82 + z * q^88 - q^89 + q^91 + z^2 * q^94 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{7} + 2 q^{8}+O(q^{10})$$ 2 * q + q^2 + q^7 + 2 * q^8 $$2 q + q^{2} + q^{7} + 2 q^{8} + q^{11} + q^{13} - q^{14} + q^{16} - 2 q^{17} - q^{22} - q^{25} + 2 q^{26} - q^{29} - q^{34} - 2 q^{41} + q^{47} + q^{50} + q^{56} + q^{58} + 2 q^{64} + q^{67} - q^{77} - 4 q^{82} + q^{88} - 2 q^{89} + 2 q^{91} - q^{94}+O(q^{100})$$ 2 * q + q^2 + q^7 + 2 * q^8 + q^11 + q^13 - q^14 + q^16 - 2 * q^17 - q^22 - q^25 + 2 * q^26 - q^29 - q^34 - 2 * q^41 + q^47 + q^50 + q^56 + q^58 + 2 * q^64 + q^67 - q^77 - 4 * q^82 + q^88 - 2 * q^89 + 2 * q^91 - q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2349\mathbb{Z}\right)^\times$$.

 $$n$$ $$407$$ $$1945$$ $$\chi(n)$$ $$-\zeta_{6}^{2}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
782.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i 0 0 0 0 0.500000 + 0.866025i 1.00000 0 0
1565.1 0.500000 0.866025i 0 0 0 0 0.500000 0.866025i 1.00000 0 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 $$\Q(\sqrt{-87})$$
9.c even 3 1 inner
261.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2349.1.h.b 2
3.b odd 2 1 2349.1.h.a 2
9.c even 3 1 87.1.d.a 1
9.c even 3 1 inner 2349.1.h.b 2
9.d odd 6 1 87.1.d.b yes 1
9.d odd 6 1 2349.1.h.a 2
29.b even 2 1 2349.1.h.a 2
36.f odd 6 1 1392.1.i.a 1
36.h even 6 1 1392.1.i.b 1
45.h odd 6 1 2175.1.h.a 1
45.j even 6 1 2175.1.h.b 1
45.k odd 12 2 2175.1.b.a 2
45.l even 12 2 2175.1.b.b 2
87.d odd 2 1 CM 2349.1.h.b 2
261.h odd 6 1 87.1.d.a 1
261.h odd 6 1 inner 2349.1.h.b 2
261.i even 6 1 87.1.d.b yes 1
261.i even 6 1 2349.1.h.a 2
261.l even 12 2 2523.1.b.b 2
261.m odd 12 2 2523.1.b.b 2
261.q even 21 6 2523.1.h.b 6
261.t odd 42 6 2523.1.h.a 6
261.u even 42 6 2523.1.h.a 6
261.v odd 42 6 2523.1.h.b 6
261.w odd 84 12 2523.1.j.b 12
261.x even 84 12 2523.1.j.b 12
1044.o odd 6 1 1392.1.i.b 1
1044.t even 6 1 1392.1.i.a 1
1305.w even 6 1 2175.1.h.a 1
1305.ba odd 6 1 2175.1.h.b 1
1305.bi even 12 2 2175.1.b.a 2
1305.bk odd 12 2 2175.1.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 9.c even 3 1
87.1.d.a 1 261.h odd 6 1
87.1.d.b yes 1 9.d odd 6 1
87.1.d.b yes 1 261.i even 6 1
1392.1.i.a 1 36.f odd 6 1
1392.1.i.a 1 1044.t even 6 1
1392.1.i.b 1 36.h even 6 1
1392.1.i.b 1 1044.o odd 6 1
2175.1.b.a 2 45.k odd 12 2
2175.1.b.a 2 1305.bi even 12 2
2175.1.b.b 2 45.l even 12 2
2175.1.b.b 2 1305.bk odd 12 2
2175.1.h.a 1 45.h odd 6 1
2175.1.h.a 1 1305.w even 6 1
2175.1.h.b 1 45.j even 6 1
2175.1.h.b 1 1305.ba odd 6 1
2349.1.h.a 2 3.b odd 2 1
2349.1.h.a 2 9.d odd 6 1
2349.1.h.a 2 29.b even 2 1
2349.1.h.a 2 261.i even 6 1
2349.1.h.b 2 1.a even 1 1 trivial
2349.1.h.b 2 9.c even 3 1 inner
2349.1.h.b 2 87.d odd 2 1 CM
2349.1.h.b 2 261.h odd 6 1 inner
2523.1.b.b 2 261.l even 12 2
2523.1.b.b 2 261.m odd 12 2
2523.1.h.a 6 261.t odd 42 6
2523.1.h.a 6 261.u even 42 6
2523.1.h.b 6 261.q even 21 6
2523.1.h.b 6 261.v odd 42 6
2523.1.j.b 12 261.w odd 84 12
2523.1.j.b 12 261.x even 84 12

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2349, [\chi])$$.

## Hecke characteristic polynomials

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