# Properties

 Label 1568.2.i.m Level $1568$ Weight $2$ Character orbit 1568.i Analytic conductor $12.521$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(961,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("1568.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1568.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: $$12.5205430369$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 2x^{2} + x + 1$$ x^4 - x^3 + 2*x^2 + x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 224) 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - \beta_1 - 1) q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{5} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{9}+O(q^{10})$$ q + (-b2 - b1 - 1) * q^3 + (-b3 - b2 + b1) * q^5 + (2*b3 + 2*b2 + 3*b1) * q^9 $$q + ( - \beta_{2} - \beta_1 - 1) q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{5} + (2 \beta_{3} + 2 \beta_{2} + 3 \beta_1) q^{9} + (2 \beta_{2} - 2 \beta_1 - 2) q^{11} + ( - \beta_{3} + 3) q^{13} - 4 q^{15} + 2 \beta_{2} q^{17} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{19} - 4 \beta_1 q^{23} + (2 \beta_{2} - \beta_1 - 1) q^{25} + ( - 2 \beta_{3} + 10) q^{27} - 2 \beta_{3} q^{29} + (2 \beta_{2} + 2 \beta_1 + 2) q^{31} - 8 \beta_1 q^{33} + (2 \beta_{3} + 2 \beta_{2}) q^{37} + ( - 4 \beta_{2} - 8 \beta_1 - 8) q^{39} + ( - 2 \beta_{3} - 4) q^{41} + ( - 2 \beta_{3} - 2) q^{43} + (\beta_{2} + 7 \beta_1 + 7) q^{45} + (2 \beta_{3} + 2 \beta_{2} - 6 \beta_1) q^{47} + ( - 2 \beta_{3} - 2 \beta_{2} - 10 \beta_1) q^{51} + (10 \beta_1 + 10) q^{53} + (4 \beta_{3} + 12) q^{55} + (2 \beta_{3} - 6) q^{57} + (\beta_{2} - 7 \beta_1 - 7) q^{59} + ( - \beta_{3} - \beta_{2} + 9 \beta_1) q^{61} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{65} + (4 \beta_1 + 4) q^{67} + (4 \beta_{3} - 4) q^{69} + (4 \beta_{3} - 4) q^{71} + ( - 4 \beta_{2} - 6 \beta_1 - 6) q^{73} + ( - \beta_{3} - \beta_{2} - 9 \beta_1) q^{75} + (4 \beta_{3} + 4 \beta_{2} - 4 \beta_1) q^{79} + ( - 6 \beta_{2} - 11 \beta_1 - 11) q^{81} + (\beta_{3} + 7) q^{83} + (2 \beta_{3} + 10) q^{85} + ( - 2 \beta_{2} - 10 \beta_1 - 10) q^{87} - 6 \beta_1 q^{89} + ( - 4 \beta_{3} - 4 \beta_{2} - 12 \beta_1) q^{93} + ( - 4 \beta_1 - 4) q^{95} + (2 \beta_{3} + 8) q^{97} + (2 \beta_{3} - 14) q^{99}+O(q^{100})$$ q + (-b2 - b1 - 1) * q^3 + (-b3 - b2 + b1) * q^5 + (2*b3 + 2*b2 + 3*b1) * q^9 + (2*b2 - 2*b1 - 2) * q^11 + (-b3 + 3) * q^13 - 4 * q^15 + 2*b2 * q^17 + (-b3 - b2 - b1) * q^19 - 4*b1 * q^23 + (2*b2 - b1 - 1) * q^25 + (-2*b3 + 10) * q^27 - 2*b3 * q^29 + (2*b2 + 2*b1 + 2) * q^31 - 8*b1 * q^33 + (2*b3 + 2*b2) * q^37 + (-4*b2 - 8*b1 - 8) * q^39 + (-2*b3 - 4) * q^41 + (-2*b3 - 2) * q^43 + (b2 + 7*b1 + 7) * q^45 + (2*b3 + 2*b2 - 6*b1) * q^47 + (-2*b3 - 2*b2 - 10*b1) * q^51 + (10*b1 + 10) * q^53 + (4*b3 + 12) * q^55 + (2*b3 - 6) * q^57 + (b2 - 7*b1 - 7) * q^59 + (-b3 - b2 + 9*b1) * q^61 + (-2*b3 - 2*b2 - 2*b1) * q^65 + (4*b1 + 4) * q^67 + (4*b3 - 4) * q^69 + (4*b3 - 4) * q^71 + (-4*b2 - 6*b1 - 6) * q^73 + (-b3 - b2 - 9*b1) * q^75 + (4*b3 + 4*b2 - 4*b1) * q^79 + (-6*b2 - 11*b1 - 11) * q^81 + (b3 + 7) * q^83 + (2*b3 + 10) * q^85 + (-2*b2 - 10*b1 - 10) * q^87 - 6*b1 * q^89 + (-4*b3 - 4*b2 - 12*b1) * q^93 + (-4*b1 - 4) * q^95 + (2*b3 + 8) * q^97 + (2*b3 - 14) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 2 q^{5} - 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - 2 * q^5 - 6 * q^9 $$4 q - 2 q^{3} - 2 q^{5} - 6 q^{9} - 4 q^{11} + 12 q^{13} - 16 q^{15} + 2 q^{19} + 8 q^{23} - 2 q^{25} + 40 q^{27} + 4 q^{31} + 16 q^{33} - 16 q^{39} - 16 q^{41} - 8 q^{43} + 14 q^{45} + 12 q^{47} + 20 q^{51} + 20 q^{53} + 48 q^{55} - 24 q^{57} - 14 q^{59} - 18 q^{61} + 4 q^{65} + 8 q^{67} - 16 q^{69} - 16 q^{71} - 12 q^{73} + 18 q^{75} + 8 q^{79} - 22 q^{81} + 28 q^{83} + 40 q^{85} - 20 q^{87} + 12 q^{89} + 24 q^{93} - 8 q^{95} + 32 q^{97} - 56 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 2 * q^5 - 6 * q^9 - 4 * q^11 + 12 * q^13 - 16 * q^15 + 2 * q^19 + 8 * q^23 - 2 * q^25 + 40 * q^27 + 4 * q^31 + 16 * q^33 - 16 * q^39 - 16 * q^41 - 8 * q^43 + 14 * q^45 + 12 * q^47 + 20 * q^51 + 20 * q^53 + 48 * q^55 - 24 * q^57 - 14 * q^59 - 18 * q^61 + 4 * q^65 + 8 * q^67 - 16 * q^69 - 16 * q^71 - 12 * q^73 + 18 * q^75 + 8 * q^79 - 22 * q^81 + 28 * q^83 + 40 * q^85 - 20 * q^87 + 12 * q^89 + 24 * q^93 - 8 * q^95 + 32 * q^97 - 56 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2x^{2} + x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 2\nu^{2} - 2\nu - 1 ) / 2$$ (-v^3 + 2*v^2 - 2*v - 1) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} - 2\nu^{2} + 6\nu - 1 ) / 2$$ (v^3 - 2*v^2 + 6*v - 1) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2$$ v^3 + 2
 $$\nu$$ $$=$$ $$( \beta_{2} + \beta _1 + 1 ) / 2$$ (b2 + b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 3\beta_1 ) / 2$$ (b3 + b2 + 3*b1) / 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2$$ b3 - 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$1471$$ $$1473$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 0.809017 + 1.40126i −0.309017 − 0.535233i 0.809017 − 1.40126i −0.309017 + 0.535233i
0 −1.61803 2.80252i 0 0.618034 1.07047i 0 0 0 −3.73607 + 6.47106i 0
961.2 0 0.618034 + 1.07047i 0 −1.61803 + 2.80252i 0 0 0 0.736068 1.27491i 0
1537.1 0 −1.61803 + 2.80252i 0 0.618034 + 1.07047i 0 0 0 −3.73607 6.47106i 0
1537.2 0 0.618034 1.07047i 0 −1.61803 2.80252i 0 0 0 0.736068 + 1.27491i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.i.m 4
4.b odd 2 1 1568.2.i.v 4
7.b odd 2 1 1568.2.i.w 4
7.c even 3 1 224.2.a.d yes 2
7.c even 3 1 inner 1568.2.i.m 4
7.d odd 6 1 1568.2.a.k 2
7.d odd 6 1 1568.2.i.w 4
21.h odd 6 1 2016.2.a.o 2
28.d even 2 1 1568.2.i.n 4
28.f even 6 1 1568.2.a.v 2
28.f even 6 1 1568.2.i.n 4
28.g odd 6 1 224.2.a.c 2
28.g odd 6 1 1568.2.i.v 4
35.j even 6 1 5600.2.a.z 2
56.j odd 6 1 3136.2.a.by 2
56.k odd 6 1 448.2.a.j 2
56.m even 6 1 3136.2.a.bf 2
56.p even 6 1 448.2.a.i 2
84.n even 6 1 2016.2.a.r 2
112.u odd 12 2 1792.2.b.k 4
112.w even 12 2 1792.2.b.m 4
140.p odd 6 1 5600.2.a.bk 2
168.s odd 6 1 4032.2.a.bv 2
168.v even 6 1 4032.2.a.bw 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.c 2 28.g odd 6 1
224.2.a.d yes 2 7.c even 3 1
448.2.a.i 2 56.p even 6 1
448.2.a.j 2 56.k odd 6 1
1568.2.a.k 2 7.d odd 6 1
1568.2.a.v 2 28.f even 6 1
1568.2.i.m 4 1.a even 1 1 trivial
1568.2.i.m 4 7.c even 3 1 inner
1568.2.i.n 4 28.d even 2 1
1568.2.i.n 4 28.f even 6 1
1568.2.i.v 4 4.b odd 2 1
1568.2.i.v 4 28.g odd 6 1
1568.2.i.w 4 7.b odd 2 1
1568.2.i.w 4 7.d odd 6 1
1792.2.b.k 4 112.u odd 12 2
1792.2.b.m 4 112.w even 12 2
2016.2.a.o 2 21.h odd 6 1
2016.2.a.r 2 84.n even 6 1
3136.2.a.bf 2 56.m even 6 1
3136.2.a.by 2 56.j odd 6 1
4032.2.a.bv 2 168.s odd 6 1
4032.2.a.bw 2 168.v even 6 1
5600.2.a.z 2 35.j even 6 1
5600.2.a.bk 2 140.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1568, [\chi])$$:

 $$T_{3}^{4} + 2T_{3}^{3} + 8T_{3}^{2} - 8T_{3} + 16$$ T3^4 + 2*T3^3 + 8*T3^2 - 8*T3 + 16 $$T_{5}^{4} + 2T_{5}^{3} + 8T_{5}^{2} - 8T_{5} + 16$$ T5^4 + 2*T5^3 + 8*T5^2 - 8*T5 + 16 $$T_{11}^{4} + 4T_{11}^{3} + 32T_{11}^{2} - 64T_{11} + 256$$ T11^4 + 4*T11^3 + 32*T11^2 - 64*T11 + 256

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} + 8 T^{2} - 8 T + 16$$
$5$ $$T^{4} + 2 T^{3} + 8 T^{2} - 8 T + 16$$
$7$ $$T^{4}$$
$11$ $$T^{4} + 4 T^{3} + 32 T^{2} - 64 T + 256$$
$13$ $$(T^{2} - 6 T + 4)^{2}$$
$17$ $$T^{4} + 20T^{2} + 400$$
$19$ $$T^{4} - 2 T^{3} + 8 T^{2} + 8 T + 16$$
$23$ $$(T^{2} - 4 T + 16)^{2}$$
$29$ $$(T^{2} - 20)^{2}$$
$31$ $$T^{4} - 4 T^{3} + 32 T^{2} + 64 T + 256$$
$37$ $$T^{4} + 20T^{2} + 400$$
$41$ $$(T^{2} + 8 T - 4)^{2}$$
$43$ $$(T^{2} + 4 T - 16)^{2}$$
$47$ $$T^{4} - 12 T^{3} + 128 T^{2} + \cdots + 256$$
$53$ $$(T^{2} - 10 T + 100)^{2}$$
$59$ $$T^{4} + 14 T^{3} + 152 T^{2} + \cdots + 1936$$
$61$ $$T^{4} + 18 T^{3} + 248 T^{2} + \cdots + 5776$$
$67$ $$(T^{2} - 4 T + 16)^{2}$$
$71$ $$(T^{2} + 8 T - 64)^{2}$$
$73$ $$T^{4} + 12 T^{3} + 188 T^{2} + \cdots + 1936$$
$79$ $$T^{4} - 8 T^{3} + 128 T^{2} + \cdots + 4096$$
$83$ $$(T^{2} - 14 T + 44)^{2}$$
$89$ $$(T^{2} - 6 T + 36)^{2}$$
$97$ $$(T^{2} - 16 T + 44)^{2}$$