# Properties

 Label 5376.2.c.s Level $5376$ Weight $2$ Character orbit 5376.c Analytic conductor $42.928$ Analytic rank $0$ 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] = [5376,2,Mod(2689,5376)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5376, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5376.2689");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5376 = 2^{8} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5376.c (of order $$2$$, degree $$1$$, 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: $$42.9275761266$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 672) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 2 i q^{5} + q^{7} - q^{9}+O(q^{10})$$ q + i * q^3 + 2*i * q^5 + q^7 - q^9 $$q + i q^{3} + 2 i q^{5} + q^{7} - q^{9} + 4 i q^{11} - 6 i q^{13} - 2 q^{15} - 2 q^{17} + 4 i q^{19} + i q^{21} + 4 q^{23} + q^{25} - i q^{27} - 2 i q^{29} + 8 q^{31} - 4 q^{33} + 2 i q^{35} + 10 i q^{37} + 6 q^{39} + 2 q^{41} - 8 i q^{43} - 2 i q^{45} + q^{49} - 2 i q^{51} + 10 i q^{53} - 8 q^{55} - 4 q^{57} + 12 i q^{59} + 10 i q^{61} - q^{63} + 12 q^{65} - 8 i q^{67} + 4 i q^{69} - 12 q^{71} - 2 q^{73} + i q^{75} + 4 i q^{77} + q^{81} + 12 i q^{83} - 4 i q^{85} + 2 q^{87} - 6 q^{89} - 6 i q^{91} + 8 i q^{93} - 8 q^{95} + 2 q^{97} - 4 i q^{99} +O(q^{100})$$ q + i * q^3 + 2*i * q^5 + q^7 - q^9 + 4*i * q^11 - 6*i * q^13 - 2 * q^15 - 2 * q^17 + 4*i * q^19 + i * q^21 + 4 * q^23 + q^25 - i * q^27 - 2*i * q^29 + 8 * q^31 - 4 * q^33 + 2*i * q^35 + 10*i * q^37 + 6 * q^39 + 2 * q^41 - 8*i * q^43 - 2*i * q^45 + q^49 - 2*i * q^51 + 10*i * q^53 - 8 * q^55 - 4 * q^57 + 12*i * q^59 + 10*i * q^61 - q^63 + 12 * q^65 - 8*i * q^67 + 4*i * q^69 - 12 * q^71 - 2 * q^73 + i * q^75 + 4*i * q^77 + q^81 + 12*i * q^83 - 4*i * q^85 + 2 * q^87 - 6 * q^89 - 6*i * q^91 + 8*i * q^93 - 8 * q^95 + 2 * q^97 - 4*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{7} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^7 - 2 * q^9 $$2 q + 2 q^{7} - 2 q^{9} - 4 q^{15} - 4 q^{17} + 8 q^{23} + 2 q^{25} + 16 q^{31} - 8 q^{33} + 12 q^{39} + 4 q^{41} + 2 q^{49} - 16 q^{55} - 8 q^{57} - 2 q^{63} + 24 q^{65} - 24 q^{71} - 4 q^{73} + 2 q^{81} + 4 q^{87} - 12 q^{89} - 16 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^7 - 2 * q^9 - 4 * q^15 - 4 * q^17 + 8 * q^23 + 2 * q^25 + 16 * q^31 - 8 * q^33 + 12 * q^39 + 4 * q^41 + 2 * q^49 - 16 * q^55 - 8 * q^57 - 2 * q^63 + 24 * q^65 - 24 * q^71 - 4 * q^73 + 2 * q^81 + 4 * q^87 - 12 * q^89 - 16 * q^95 + 4 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times$$.

 $$n$$ $$1793$$ $$2815$$ $$4609$$ $$5125$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2689.1
 − 1.00000i 1.00000i
0 1.00000i 0 2.00000i 0 1.00000 0 −1.00000 0
2689.2 0 1.00000i 0 2.00000i 0 1.00000 0 −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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5376.2.c.s 2
4.b odd 2 1 5376.2.c.m 2
8.b even 2 1 inner 5376.2.c.s 2
8.d odd 2 1 5376.2.c.m 2
16.e even 4 1 672.2.a.f yes 1
16.e even 4 1 1344.2.a.h 1
16.f odd 4 1 672.2.a.b 1
16.f odd 4 1 1344.2.a.r 1
48.i odd 4 1 2016.2.a.k 1
48.i odd 4 1 4032.2.a.f 1
48.k even 4 1 2016.2.a.l 1
48.k even 4 1 4032.2.a.n 1
112.j even 4 1 4704.2.a.be 1
112.j even 4 1 9408.2.a.g 1
112.l odd 4 1 4704.2.a.m 1
112.l odd 4 1 9408.2.a.cb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
672.2.a.b 1 16.f odd 4 1
672.2.a.f yes 1 16.e even 4 1
1344.2.a.h 1 16.e even 4 1
1344.2.a.r 1 16.f odd 4 1
2016.2.a.k 1 48.i odd 4 1
2016.2.a.l 1 48.k even 4 1
4032.2.a.f 1 48.i odd 4 1
4032.2.a.n 1 48.k even 4 1
4704.2.a.m 1 112.l odd 4 1
4704.2.a.be 1 112.j even 4 1
5376.2.c.m 2 4.b odd 2 1
5376.2.c.m 2 8.d odd 2 1
5376.2.c.s 2 1.a even 1 1 trivial
5376.2.c.s 2 8.b even 2 1 inner
9408.2.a.g 1 112.j even 4 1
9408.2.a.cb 1 112.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(5376, [\chi])$$:

 $$T_{5}^{2} + 4$$ T5^2 + 4 $$T_{11}^{2} + 16$$ T11^2 + 16 $$T_{13}^{2} + 36$$ T13^2 + 36 $$T_{17} + 2$$ T17 + 2 $$T_{23} - 4$$ T23 - 4 $$T_{31} - 8$$ T31 - 8 $$T_{47}$$ T47 $$T_{71} + 12$$ T71 + 12 $$T_{79}$$ T79

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 4$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} + 16$$
$13$ $$T^{2} + 36$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2} + 16$$
$23$ $$(T - 4)^{2}$$
$29$ $$T^{2} + 4$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 100$$
$59$ $$T^{2} + 144$$
$61$ $$T^{2} + 100$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 12)^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$(T + 6)^{2}$$
$97$ $$(T - 2)^{2}$$