# Properties

 Label 560.3.f.b Level $560$ Weight $3$ Character orbit 560.f Analytic conductor $15.259$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,3,Mod(321,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("560.321");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 560.f (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: $$15.2588948042$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 5$$ x^2 + 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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 $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + \beta q^{5} + ( - 3 \beta + 2) q^{7} + 4 q^{9}+O(q^{10})$$ q + b * q^3 + b * q^5 + (-3*b + 2) * q^7 + 4 * q^9 $$q + \beta q^{3} + \beta q^{5} + ( - 3 \beta + 2) q^{7} + 4 q^{9} + q^{11} - 9 \beta q^{13} - 5 q^{15} - 3 \beta q^{17} - 6 \beta q^{19} + (2 \beta + 15) q^{21} - 8 q^{23} - 5 q^{25} + 13 \beta q^{27} + 41 q^{29} - 18 \beta q^{31} + \beta q^{33} + (2 \beta + 15) q^{35} - 28 q^{37} + 45 q^{39} + 6 \beta q^{41} + 82 q^{43} + 4 \beta q^{45} - 9 \beta q^{47} + ( - 12 \beta - 41) q^{49} + 15 q^{51} + 74 q^{53} + \beta q^{55} + 30 q^{57} + 42 \beta q^{59} + 36 \beta q^{61} + ( - 12 \beta + 8) q^{63} + 45 q^{65} - 2 q^{67} - 8 \beta q^{69} - 14 q^{71} - 30 \beta q^{73} - 5 \beta q^{75} + ( - 3 \beta + 2) q^{77} + 19 q^{79} - 29 q^{81} - 42 \beta q^{83} + 15 q^{85} + 41 \beta q^{87} + 48 \beta q^{89} + ( - 18 \beta - 135) q^{91} + 90 q^{93} + 30 q^{95} - 27 \beta q^{97} + 4 q^{99} +O(q^{100})$$ q + b * q^3 + b * q^5 + (-3*b + 2) * q^7 + 4 * q^9 + q^11 - 9*b * q^13 - 5 * q^15 - 3*b * q^17 - 6*b * q^19 + (2*b + 15) * q^21 - 8 * q^23 - 5 * q^25 + 13*b * q^27 + 41 * q^29 - 18*b * q^31 + b * q^33 + (2*b + 15) * q^35 - 28 * q^37 + 45 * q^39 + 6*b * q^41 + 82 * q^43 + 4*b * q^45 - 9*b * q^47 + (-12*b - 41) * q^49 + 15 * q^51 + 74 * q^53 + b * q^55 + 30 * q^57 + 42*b * q^59 + 36*b * q^61 + (-12*b + 8) * q^63 + 45 * q^65 - 2 * q^67 - 8*b * q^69 - 14 * q^71 - 30*b * q^73 - 5*b * q^75 + (-3*b + 2) * q^77 + 19 * q^79 - 29 * q^81 - 42*b * q^83 + 15 * q^85 + 41*b * q^87 + 48*b * q^89 + (-18*b - 135) * q^91 + 90 * q^93 + 30 * q^95 - 27*b * q^97 + 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{7} + 8 q^{9}+O(q^{10})$$ 2 * q + 4 * q^7 + 8 * q^9 $$2 q + 4 q^{7} + 8 q^{9} + 2 q^{11} - 10 q^{15} + 30 q^{21} - 16 q^{23} - 10 q^{25} + 82 q^{29} + 30 q^{35} - 56 q^{37} + 90 q^{39} + 164 q^{43} - 82 q^{49} + 30 q^{51} + 148 q^{53} + 60 q^{57} + 16 q^{63} + 90 q^{65} - 4 q^{67} - 28 q^{71} + 4 q^{77} + 38 q^{79} - 58 q^{81} + 30 q^{85} - 270 q^{91} + 180 q^{93} + 60 q^{95} + 8 q^{99}+O(q^{100})$$ 2 * q + 4 * q^7 + 8 * q^9 + 2 * q^11 - 10 * q^15 + 30 * q^21 - 16 * q^23 - 10 * q^25 + 82 * q^29 + 30 * q^35 - 56 * q^37 + 90 * q^39 + 164 * q^43 - 82 * q^49 + 30 * q^51 + 148 * q^53 + 60 * q^57 + 16 * q^63 + 90 * q^65 - 4 * q^67 - 28 * q^71 + 4 * q^77 + 38 * q^79 - 58 * q^81 + 30 * q^85 - 270 * q^91 + 180 * q^93 + 60 * q^95 + 8 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
321.1
 − 2.23607i 2.23607i
0 2.23607i 0 2.23607i 0 2.00000 + 6.70820i 0 4.00000 0
321.2 0 2.23607i 0 2.23607i 0 2.00000 6.70820i 0 4.00000 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
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.3.f.b 2
4.b odd 2 1 35.3.d.b 2
7.b odd 2 1 inner 560.3.f.b 2
12.b even 2 1 315.3.h.a 2
20.d odd 2 1 175.3.d.c 2
20.e even 4 2 175.3.c.c 4
28.d even 2 1 35.3.d.b 2
28.f even 6 2 245.3.h.a 4
28.g odd 6 2 245.3.h.a 4
84.h odd 2 1 315.3.h.a 2
140.c even 2 1 175.3.d.c 2
140.j odd 4 2 175.3.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.b 2 4.b odd 2 1
35.3.d.b 2 28.d even 2 1
175.3.c.c 4 20.e even 4 2
175.3.c.c 4 140.j odd 4 2
175.3.d.c 2 20.d odd 2 1
175.3.d.c 2 140.c even 2 1
245.3.h.a 4 28.f even 6 2
245.3.h.a 4 28.g odd 6 2
315.3.h.a 2 12.b even 2 1
315.3.h.a 2 84.h odd 2 1
560.3.f.b 2 1.a even 1 1 trivial
560.3.f.b 2 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 5$$ acting on $$S_{3}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 5$$
$5$ $$T^{2} + 5$$
$7$ $$T^{2} - 4T + 49$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 405$$
$17$ $$T^{2} + 45$$
$19$ $$T^{2} + 180$$
$23$ $$(T + 8)^{2}$$
$29$ $$(T - 41)^{2}$$
$31$ $$T^{2} + 1620$$
$37$ $$(T + 28)^{2}$$
$41$ $$T^{2} + 180$$
$43$ $$(T - 82)^{2}$$
$47$ $$T^{2} + 405$$
$53$ $$(T - 74)^{2}$$
$59$ $$T^{2} + 8820$$
$61$ $$T^{2} + 6480$$
$67$ $$(T + 2)^{2}$$
$71$ $$(T + 14)^{2}$$
$73$ $$T^{2} + 4500$$
$79$ $$(T - 19)^{2}$$
$83$ $$T^{2} + 8820$$
$89$ $$T^{2} + 11520$$
$97$ $$T^{2} + 3645$$