# Properties

 Label 3456.2.i.b Level $3456$ Weight $2$ Character orbit 3456.i Analytic conductor $27.596$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3456,2,Mod(1153,3456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3456, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("3456.1153");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3456 = 2^{7} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3456.i (of order $$3$$, 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: $$27.5962989386$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1152) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 2 \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 2) q^{7} +O(q^{10})$$ q - 2*z * q^5 + (-2*z + 2) * q^7 $$q - 2 \zeta_{6} q^{5} + ( - 2 \zeta_{6} + 2) q^{7} + (5 \zeta_{6} - 5) q^{11} - 4 \zeta_{6} q^{13} - q^{17} + 5 q^{19} - 4 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} + (6 \zeta_{6} - 6) q^{29} - 4 q^{35} - 10 q^{37} - 3 \zeta_{6} q^{41} + (9 \zeta_{6} - 9) q^{43} + ( - 8 \zeta_{6} + 8) q^{47} + 3 \zeta_{6} q^{49} + 12 q^{53} + 10 q^{55} - 7 \zeta_{6} q^{59} + (4 \zeta_{6} - 4) q^{61} + (8 \zeta_{6} - 8) q^{65} - 7 \zeta_{6} q^{67} - 6 q^{71} - 13 q^{73} + 10 \zeta_{6} q^{77} + (2 \zeta_{6} - 2) q^{79} + (12 \zeta_{6} - 12) q^{83} + 2 \zeta_{6} q^{85} - 10 q^{89} - 8 q^{91} - 10 \zeta_{6} q^{95} + (13 \zeta_{6} - 13) q^{97} +O(q^{100})$$ q - 2*z * q^5 + (-2*z + 2) * q^7 + (5*z - 5) * q^11 - 4*z * q^13 - q^17 + 5 * q^19 - 4*z * q^23 + (-z + 1) * q^25 + (6*z - 6) * q^29 - 4 * q^35 - 10 * q^37 - 3*z * q^41 + (9*z - 9) * q^43 + (-8*z + 8) * q^47 + 3*z * q^49 + 12 * q^53 + 10 * q^55 - 7*z * q^59 + (4*z - 4) * q^61 + (8*z - 8) * q^65 - 7*z * q^67 - 6 * q^71 - 13 * q^73 + 10*z * q^77 + (2*z - 2) * q^79 + (12*z - 12) * q^83 + 2*z * q^85 - 10 * q^89 - 8 * q^91 - 10*z * q^95 + (13*z - 13) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^5 + 2 * q^7 $$2 q - 2 q^{5} + 2 q^{7} - 5 q^{11} - 4 q^{13} - 2 q^{17} + 10 q^{19} - 4 q^{23} + q^{25} - 6 q^{29} - 8 q^{35} - 20 q^{37} - 3 q^{41} - 9 q^{43} + 8 q^{47} + 3 q^{49} + 24 q^{53} + 20 q^{55} - 7 q^{59} - 4 q^{61} - 8 q^{65} - 7 q^{67} - 12 q^{71} - 26 q^{73} + 10 q^{77} - 2 q^{79} - 12 q^{83} + 2 q^{85} - 20 q^{89} - 16 q^{91} - 10 q^{95} - 13 q^{97}+O(q^{100})$$ 2 * q - 2 * q^5 + 2 * q^7 - 5 * q^11 - 4 * q^13 - 2 * q^17 + 10 * q^19 - 4 * q^23 + q^25 - 6 * q^29 - 8 * q^35 - 20 * q^37 - 3 * q^41 - 9 * q^43 + 8 * q^47 + 3 * q^49 + 24 * q^53 + 20 * q^55 - 7 * q^59 - 4 * q^61 - 8 * q^65 - 7 * q^67 - 12 * q^71 - 26 * q^73 + 10 * q^77 - 2 * q^79 - 12 * q^83 + 2 * q^85 - 20 * q^89 - 16 * q^91 - 10 * q^95 - 13 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times$$.

 $$n$$ $$2053$$ $$2431$$ $$2945$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1153.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 −1.00000 1.73205i 0 1.00000 1.73205i 0 0 0
2305.1 0 0 0 −1.00000 + 1.73205i 0 1.00000 + 1.73205i 0 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3456.2.i.b 2
3.b odd 2 1 1152.2.i.d yes 2
4.b odd 2 1 3456.2.i.a 2
8.b even 2 1 3456.2.i.d 2
8.d odd 2 1 3456.2.i.c 2
9.c even 3 1 inner 3456.2.i.b 2
9.d odd 6 1 1152.2.i.d yes 2
12.b even 2 1 1152.2.i.b yes 2
24.f even 2 1 1152.2.i.c yes 2
24.h odd 2 1 1152.2.i.a 2
36.f odd 6 1 3456.2.i.a 2
36.h even 6 1 1152.2.i.b yes 2
72.j odd 6 1 1152.2.i.a 2
72.l even 6 1 1152.2.i.c yes 2
72.n even 6 1 3456.2.i.d 2
72.p odd 6 1 3456.2.i.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.i.a 2 24.h odd 2 1
1152.2.i.a 2 72.j odd 6 1
1152.2.i.b yes 2 12.b even 2 1
1152.2.i.b yes 2 36.h even 6 1
1152.2.i.c yes 2 24.f even 2 1
1152.2.i.c yes 2 72.l even 6 1
1152.2.i.d yes 2 3.b odd 2 1
1152.2.i.d yes 2 9.d odd 6 1
3456.2.i.a 2 4.b odd 2 1
3456.2.i.a 2 36.f odd 6 1
3456.2.i.b 2 1.a even 1 1 trivial
3456.2.i.b 2 9.c even 3 1 inner
3456.2.i.c 2 8.d odd 2 1
3456.2.i.c 2 72.p odd 6 1
3456.2.i.d 2 8.b even 2 1
3456.2.i.d 2 72.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3456, [\chi])$$:

 $$T_{5}^{2} + 2T_{5} + 4$$ T5^2 + 2*T5 + 4 $$T_{7}^{2} - 2T_{7} + 4$$ T7^2 - 2*T7 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2} - 2T + 4$$
$11$ $$T^{2} + 5T + 25$$
$13$ $$T^{2} + 4T + 16$$
$17$ $$(T + 1)^{2}$$
$19$ $$(T - 5)^{2}$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$T^{2} + 6T + 36$$
$31$ $$T^{2}$$
$37$ $$(T + 10)^{2}$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} + 9T + 81$$
$47$ $$T^{2} - 8T + 64$$
$53$ $$(T - 12)^{2}$$
$59$ $$T^{2} + 7T + 49$$
$61$ $$T^{2} + 4T + 16$$
$67$ $$T^{2} + 7T + 49$$
$71$ $$(T + 6)^{2}$$
$73$ $$(T + 13)^{2}$$
$79$ $$T^{2} + 2T + 4$$
$83$ $$T^{2} + 12T + 144$$
$89$ $$(T + 10)^{2}$$
$97$ $$T^{2} + 13T + 169$$