# Properties

 Label 560.4.q.b Level $560$ Weight $4$ Character orbit 560.q Analytic conductor $33.041$ 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] = [560,4,Mod(81,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("560.81");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 560.q (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: $$33.0410696032$$ Analytic rank: $$0$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{3} - 5 \zeta_{6} q^{5} + (14 \zeta_{6} + 7) q^{7} + 23 \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z - 2) * q^3 - 5*z * q^5 + (14*z + 7) * q^7 + 23*z * q^9 $$q + (2 \zeta_{6} - 2) q^{3} - 5 \zeta_{6} q^{5} + (14 \zeta_{6} + 7) q^{7} + 23 \zeta_{6} q^{9} + (45 \zeta_{6} - 45) q^{11} + 59 q^{13} + 10 q^{15} + ( - 54 \zeta_{6} + 54) q^{17} - 121 \zeta_{6} q^{19} + (14 \zeta_{6} - 42) q^{21} + 69 \zeta_{6} q^{23} + (25 \zeta_{6} - 25) q^{25} - 100 q^{27} - 162 q^{29} + (88 \zeta_{6} - 88) q^{31} - 90 \zeta_{6} q^{33} + ( - 105 \zeta_{6} + 70) q^{35} + 259 \zeta_{6} q^{37} + (118 \zeta_{6} - 118) q^{39} + 195 q^{41} + 286 q^{43} + ( - 115 \zeta_{6} + 115) q^{45} + 45 \zeta_{6} q^{47} + (392 \zeta_{6} - 147) q^{49} + 108 \zeta_{6} q^{51} + (597 \zeta_{6} - 597) q^{53} + 225 q^{55} + 242 q^{57} + (360 \zeta_{6} - 360) q^{59} - 392 \zeta_{6} q^{61} + (483 \zeta_{6} - 322) q^{63} - 295 \zeta_{6} q^{65} + (280 \zeta_{6} - 280) q^{67} - 138 q^{69} - 48 q^{71} + (668 \zeta_{6} - 668) q^{73} - 50 \zeta_{6} q^{75} + (315 \zeta_{6} - 945) q^{77} + 782 \zeta_{6} q^{79} + (421 \zeta_{6} - 421) q^{81} - 768 q^{83} - 270 q^{85} + ( - 324 \zeta_{6} + 324) q^{87} + 1194 \zeta_{6} q^{89} + (826 \zeta_{6} + 413) q^{91} - 176 \zeta_{6} q^{93} + (605 \zeta_{6} - 605) q^{95} + 902 q^{97} - 1035 q^{99} +O(q^{100})$$ q + (2*z - 2) * q^3 - 5*z * q^5 + (14*z + 7) * q^7 + 23*z * q^9 + (45*z - 45) * q^11 + 59 * q^13 + 10 * q^15 + (-54*z + 54) * q^17 - 121*z * q^19 + (14*z - 42) * q^21 + 69*z * q^23 + (25*z - 25) * q^25 - 100 * q^27 - 162 * q^29 + (88*z - 88) * q^31 - 90*z * q^33 + (-105*z + 70) * q^35 + 259*z * q^37 + (118*z - 118) * q^39 + 195 * q^41 + 286 * q^43 + (-115*z + 115) * q^45 + 45*z * q^47 + (392*z - 147) * q^49 + 108*z * q^51 + (597*z - 597) * q^53 + 225 * q^55 + 242 * q^57 + (360*z - 360) * q^59 - 392*z * q^61 + (483*z - 322) * q^63 - 295*z * q^65 + (280*z - 280) * q^67 - 138 * q^69 - 48 * q^71 + (668*z - 668) * q^73 - 50*z * q^75 + (315*z - 945) * q^77 + 782*z * q^79 + (421*z - 421) * q^81 - 768 * q^83 - 270 * q^85 + (-324*z + 324) * q^87 + 1194*z * q^89 + (826*z + 413) * q^91 - 176*z * q^93 + (605*z - 605) * q^95 + 902 * q^97 - 1035 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 5 q^{5} + 28 q^{7} + 23 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 5 * q^5 + 28 * q^7 + 23 * q^9 $$2 q - 2 q^{3} - 5 q^{5} + 28 q^{7} + 23 q^{9} - 45 q^{11} + 118 q^{13} + 20 q^{15} + 54 q^{17} - 121 q^{19} - 70 q^{21} + 69 q^{23} - 25 q^{25} - 200 q^{27} - 324 q^{29} - 88 q^{31} - 90 q^{33} + 35 q^{35} + 259 q^{37} - 118 q^{39} + 390 q^{41} + 572 q^{43} + 115 q^{45} + 45 q^{47} + 98 q^{49} + 108 q^{51} - 597 q^{53} + 450 q^{55} + 484 q^{57} - 360 q^{59} - 392 q^{61} - 161 q^{63} - 295 q^{65} - 280 q^{67} - 276 q^{69} - 96 q^{71} - 668 q^{73} - 50 q^{75} - 1575 q^{77} + 782 q^{79} - 421 q^{81} - 1536 q^{83} - 540 q^{85} + 324 q^{87} + 1194 q^{89} + 1652 q^{91} - 176 q^{93} - 605 q^{95} + 1804 q^{97} - 2070 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 5 * q^5 + 28 * q^7 + 23 * q^9 - 45 * q^11 + 118 * q^13 + 20 * q^15 + 54 * q^17 - 121 * q^19 - 70 * q^21 + 69 * q^23 - 25 * q^25 - 200 * q^27 - 324 * q^29 - 88 * q^31 - 90 * q^33 + 35 * q^35 + 259 * q^37 - 118 * q^39 + 390 * q^41 + 572 * q^43 + 115 * q^45 + 45 * q^47 + 98 * q^49 + 108 * q^51 - 597 * q^53 + 450 * q^55 + 484 * q^57 - 360 * q^59 - 392 * q^61 - 161 * q^63 - 295 * q^65 - 280 * q^67 - 276 * q^69 - 96 * q^71 - 668 * q^73 - 50 * q^75 - 1575 * q^77 + 782 * q^79 - 421 * q^81 - 1536 * q^83 - 540 * q^85 + 324 * q^87 + 1194 * q^89 + 1652 * q^91 - 176 * q^93 - 605 * q^95 + 1804 * q^97 - 2070 * q^99

## Character values

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

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
81.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.00000 + 1.73205i 0 −2.50000 4.33013i 0 14.0000 + 12.1244i 0 11.5000 + 19.9186i 0
401.1 0 −1.00000 1.73205i 0 −2.50000 + 4.33013i 0 14.0000 12.1244i 0 11.5000 19.9186i 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 560.4.q.b 2
4.b odd 2 1 35.4.e.a 2
7.c even 3 1 inner 560.4.q.b 2
12.b even 2 1 315.4.j.b 2
20.d odd 2 1 175.4.e.b 2
20.e even 4 2 175.4.k.b 4
28.d even 2 1 245.4.e.a 2
28.f even 6 1 245.4.a.f 1
28.f even 6 1 245.4.e.a 2
28.g odd 6 1 35.4.e.a 2
28.g odd 6 1 245.4.a.e 1
84.j odd 6 1 2205.4.a.g 1
84.n even 6 1 315.4.j.b 2
84.n even 6 1 2205.4.a.e 1
140.p odd 6 1 175.4.e.b 2
140.p odd 6 1 1225.4.a.b 1
140.s even 6 1 1225.4.a.a 1
140.w even 12 2 175.4.k.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.a 2 4.b odd 2 1
35.4.e.a 2 28.g odd 6 1
175.4.e.b 2 20.d odd 2 1
175.4.e.b 2 140.p odd 6 1
175.4.k.b 4 20.e even 4 2
175.4.k.b 4 140.w even 12 2
245.4.a.e 1 28.g odd 6 1
245.4.a.f 1 28.f even 6 1
245.4.e.a 2 28.d even 2 1
245.4.e.a 2 28.f even 6 1
315.4.j.b 2 12.b even 2 1
315.4.j.b 2 84.n even 6 1
560.4.q.b 2 1.a even 1 1 trivial
560.4.q.b 2 7.c even 3 1 inner
1225.4.a.a 1 140.s even 6 1
1225.4.a.b 1 140.p odd 6 1
2205.4.a.e 1 84.n even 6 1
2205.4.a.g 1 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(560, [\chi])$$:

 $$T_{3}^{2} + 2T_{3} + 4$$ T3^2 + 2*T3 + 4 $$T_{11}^{2} + 45T_{11} + 2025$$ T11^2 + 45*T11 + 2025

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} + 5T + 25$$
$7$ $$T^{2} - 28T + 343$$
$11$ $$T^{2} + 45T + 2025$$
$13$ $$(T - 59)^{2}$$
$17$ $$T^{2} - 54T + 2916$$
$19$ $$T^{2} + 121T + 14641$$
$23$ $$T^{2} - 69T + 4761$$
$29$ $$(T + 162)^{2}$$
$31$ $$T^{2} + 88T + 7744$$
$37$ $$T^{2} - 259T + 67081$$
$41$ $$(T - 195)^{2}$$
$43$ $$(T - 286)^{2}$$
$47$ $$T^{2} - 45T + 2025$$
$53$ $$T^{2} + 597T + 356409$$
$59$ $$T^{2} + 360T + 129600$$
$61$ $$T^{2} + 392T + 153664$$
$67$ $$T^{2} + 280T + 78400$$
$71$ $$(T + 48)^{2}$$
$73$ $$T^{2} + 668T + 446224$$
$79$ $$T^{2} - 782T + 611524$$
$83$ $$(T + 768)^{2}$$
$89$ $$T^{2} - 1194 T + 1425636$$
$97$ $$(T - 902)^{2}$$