# Properties

 Label 1568.4.a.o Level $1568$ Weight $4$ Character orbit 1568.a Self dual yes Analytic conductor $92.515$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,4,Mod(1,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1568.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1568.a (trivial)

## 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: yes Analytic conductor: $$92.5149948890$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 32) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 8 q^{3} + 10 q^{5} + 37 q^{9}+O(q^{10})$$ q + 8 * q^3 + 10 * q^5 + 37 * q^9 $$q + 8 q^{3} + 10 q^{5} + 37 q^{9} + 40 q^{11} + 50 q^{13} + 80 q^{15} + 30 q^{17} + 40 q^{19} - 48 q^{23} - 25 q^{25} + 80 q^{27} - 34 q^{29} + 320 q^{31} + 320 q^{33} + 310 q^{37} + 400 q^{39} - 410 q^{41} - 152 q^{43} + 370 q^{45} - 416 q^{47} + 240 q^{51} - 410 q^{53} + 400 q^{55} + 320 q^{57} - 200 q^{59} - 30 q^{61} + 500 q^{65} - 776 q^{67} - 384 q^{69} - 400 q^{71} + 630 q^{73} - 200 q^{75} + 1120 q^{79} - 359 q^{81} + 552 q^{83} + 300 q^{85} - 272 q^{87} + 326 q^{89} + 2560 q^{93} + 400 q^{95} + 110 q^{97} + 1480 q^{99}+O(q^{100})$$ q + 8 * q^3 + 10 * q^5 + 37 * q^9 + 40 * q^11 + 50 * q^13 + 80 * q^15 + 30 * q^17 + 40 * q^19 - 48 * q^23 - 25 * q^25 + 80 * q^27 - 34 * q^29 + 320 * q^31 + 320 * q^33 + 310 * q^37 + 400 * q^39 - 410 * q^41 - 152 * q^43 + 370 * q^45 - 416 * q^47 + 240 * q^51 - 410 * q^53 + 400 * q^55 + 320 * q^57 - 200 * q^59 - 30 * q^61 + 500 * q^65 - 776 * q^67 - 384 * q^69 - 400 * q^71 + 630 * q^73 - 200 * q^75 + 1120 * q^79 - 359 * q^81 + 552 * q^83 + 300 * q^85 - 272 * q^87 + 326 * q^89 + 2560 * q^93 + 400 * q^95 + 110 * q^97 + 1480 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 8.00000 0 10.0000 0 0 0 37.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.4.a.o 1
4.b odd 2 1 1568.4.a.c 1
7.b odd 2 1 32.4.a.a 1
21.c even 2 1 288.4.a.h 1
28.d even 2 1 32.4.a.c yes 1
35.c odd 2 1 800.4.a.k 1
35.f even 4 2 800.4.c.b 2
56.e even 2 1 64.4.a.a 1
56.h odd 2 1 64.4.a.e 1
84.h odd 2 1 288.4.a.i 1
112.j even 4 2 256.4.b.c 2
112.l odd 4 2 256.4.b.e 2
140.c even 2 1 800.4.a.a 1
140.j odd 4 2 800.4.c.a 2
168.e odd 2 1 576.4.a.h 1
168.i even 2 1 576.4.a.g 1
280.c odd 2 1 1600.4.a.e 1
280.n even 2 1 1600.4.a.bw 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.a.a 1 7.b odd 2 1
32.4.a.c yes 1 28.d even 2 1
64.4.a.a 1 56.e even 2 1
64.4.a.e 1 56.h odd 2 1
256.4.b.c 2 112.j even 4 2
256.4.b.e 2 112.l odd 4 2
288.4.a.h 1 21.c even 2 1
288.4.a.i 1 84.h odd 2 1
576.4.a.g 1 168.i even 2 1
576.4.a.h 1 168.e odd 2 1
800.4.a.a 1 140.c even 2 1
800.4.a.k 1 35.c odd 2 1
800.4.c.a 2 140.j odd 4 2
800.4.c.b 2 35.f even 4 2
1568.4.a.c 1 4.b odd 2 1
1568.4.a.o 1 1.a even 1 1 trivial
1600.4.a.e 1 280.c odd 2 1
1600.4.a.bw 1 280.n even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1568))$$:

 $$T_{3} - 8$$ T3 - 8 $$T_{5} - 10$$ T5 - 10 $$T_{11} - 40$$ T11 - 40

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 8$$
$5$ $$T - 10$$
$7$ $$T$$
$11$ $$T - 40$$
$13$ $$T - 50$$
$17$ $$T - 30$$
$19$ $$T - 40$$
$23$ $$T + 48$$
$29$ $$T + 34$$
$31$ $$T - 320$$
$37$ $$T - 310$$
$41$ $$T + 410$$
$43$ $$T + 152$$
$47$ $$T + 416$$
$53$ $$T + 410$$
$59$ $$T + 200$$
$61$ $$T + 30$$
$67$ $$T + 776$$
$71$ $$T + 400$$
$73$ $$T - 630$$
$79$ $$T - 1120$$
$83$ $$T - 552$$
$89$ $$T - 326$$
$97$ $$T - 110$$