# Properties

 Label 560.5.p.d Level $560$ Weight $5$ Character orbit 560.p Analytic conductor $57.887$ 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] = [560,5,Mod(209,560)]

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, 1, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("560.209");

S:= CuspForms(chi, 5);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 560.p (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: $$57.8871793270$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 6$$ x^2 + 6 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ 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 $$\beta = 2\sqrt{-6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 5 q^{3} + (5 \beta + 5) q^{5} + (7 \beta - 35) q^{7} - 56 q^{9}+O(q^{10})$$ q - 5 * q^3 + (5*b + 5) * q^5 + (7*b - 35) * q^7 - 56 * q^9 $$q - 5 q^{3} + (5 \beta + 5) q^{5} + (7 \beta - 35) q^{7} - 56 q^{9} - 89 q^{11} - 5 q^{13} + ( - 25 \beta - 25) q^{15} - 485 q^{17} - 45 \beta q^{19} + ( - 35 \beta + 175) q^{21} - 143 \beta q^{23} + (50 \beta - 575) q^{25} + 685 q^{27} + 191 q^{29} + 215 \beta q^{31} + 445 q^{33} + ( - 140 \beta - 1015) q^{35} - 333 \beta q^{37} + 25 q^{39} + 595 \beta q^{41} + 77 \beta q^{43} + ( - 280 \beta - 280) q^{45} + 2195 q^{47} + ( - 490 \beta + 49) q^{49} + 2425 q^{51} + 324 \beta q^{53} + ( - 445 \beta - 445) q^{55} + 225 \beta q^{57} + 740 \beta q^{59} + 395 \beta q^{61} + ( - 392 \beta + 1960) q^{63} + ( - 25 \beta - 25) q^{65} + 418 \beta q^{67} + 715 \beta q^{69} - 4454 q^{71} + 8650 q^{73} + ( - 250 \beta + 2875) q^{75} + ( - 623 \beta + 3115) q^{77} - 5561 q^{79} + 1111 q^{81} - 1990 q^{83} + ( - 2425 \beta - 2425) q^{85} - 955 q^{87} - 165 \beta q^{89} + ( - 35 \beta + 175) q^{91} - 1075 \beta q^{93} + ( - 225 \beta + 5400) q^{95} + 9235 q^{97} + 4984 q^{99} +O(q^{100})$$ q - 5 * q^3 + (5*b + 5) * q^5 + (7*b - 35) * q^7 - 56 * q^9 - 89 * q^11 - 5 * q^13 + (-25*b - 25) * q^15 - 485 * q^17 - 45*b * q^19 + (-35*b + 175) * q^21 - 143*b * q^23 + (50*b - 575) * q^25 + 685 * q^27 + 191 * q^29 + 215*b * q^31 + 445 * q^33 + (-140*b - 1015) * q^35 - 333*b * q^37 + 25 * q^39 + 595*b * q^41 + 77*b * q^43 + (-280*b - 280) * q^45 + 2195 * q^47 + (-490*b + 49) * q^49 + 2425 * q^51 + 324*b * q^53 + (-445*b - 445) * q^55 + 225*b * q^57 + 740*b * q^59 + 395*b * q^61 + (-392*b + 1960) * q^63 + (-25*b - 25) * q^65 + 418*b * q^67 + 715*b * q^69 - 4454 * q^71 + 8650 * q^73 + (-250*b + 2875) * q^75 + (-623*b + 3115) * q^77 - 5561 * q^79 + 1111 * q^81 - 1990 * q^83 + (-2425*b - 2425) * q^85 - 955 * q^87 - 165*b * q^89 + (-35*b + 175) * q^91 - 1075*b * q^93 + (-225*b + 5400) * q^95 + 9235 * q^97 + 4984 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10 q^{3} + 10 q^{5} - 70 q^{7} - 112 q^{9}+O(q^{10})$$ 2 * q - 10 * q^3 + 10 * q^5 - 70 * q^7 - 112 * q^9 $$2 q - 10 q^{3} + 10 q^{5} - 70 q^{7} - 112 q^{9} - 178 q^{11} - 10 q^{13} - 50 q^{15} - 970 q^{17} + 350 q^{21} - 1150 q^{25} + 1370 q^{27} + 382 q^{29} + 890 q^{33} - 2030 q^{35} + 50 q^{39} - 560 q^{45} + 4390 q^{47} + 98 q^{49} + 4850 q^{51} - 890 q^{55} + 3920 q^{63} - 50 q^{65} - 8908 q^{71} + 17300 q^{73} + 5750 q^{75} + 6230 q^{77} - 11122 q^{79} + 2222 q^{81} - 3980 q^{83} - 4850 q^{85} - 1910 q^{87} + 350 q^{91} + 10800 q^{95} + 18470 q^{97} + 9968 q^{99}+O(q^{100})$$ 2 * q - 10 * q^3 + 10 * q^5 - 70 * q^7 - 112 * q^9 - 178 * q^11 - 10 * q^13 - 50 * q^15 - 970 * q^17 + 350 * q^21 - 1150 * q^25 + 1370 * q^27 + 382 * q^29 + 890 * q^33 - 2030 * q^35 + 50 * q^39 - 560 * q^45 + 4390 * q^47 + 98 * q^49 + 4850 * q^51 - 890 * q^55 + 3920 * q^63 - 50 * q^65 - 8908 * q^71 + 17300 * q^73 + 5750 * q^75 + 6230 * q^77 - 11122 * q^79 + 2222 * q^81 - 3980 * q^83 - 4850 * q^85 - 1910 * q^87 + 350 * q^91 + 10800 * q^95 + 18470 * q^97 + 9968 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 − 2.44949i 2.44949i
0 −5.00000 0 5.00000 24.4949i 0 −35.0000 34.2929i 0 −56.0000 0
209.2 0 −5.00000 0 5.00000 + 24.4949i 0 −35.0000 + 34.2929i 0 −56.0000 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
35.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.5.p.d 2
4.b odd 2 1 35.5.c.d yes 2
5.b even 2 1 560.5.p.e 2
7.b odd 2 1 560.5.p.e 2
12.b even 2 1 315.5.e.c 2
20.d odd 2 1 35.5.c.c 2
20.e even 4 2 175.5.d.e 4
28.d even 2 1 35.5.c.c 2
35.c odd 2 1 inner 560.5.p.d 2
60.h even 2 1 315.5.e.d 2
84.h odd 2 1 315.5.e.d 2
140.c even 2 1 35.5.c.d yes 2
140.j odd 4 2 175.5.d.e 4
420.o odd 2 1 315.5.e.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.c 2 20.d odd 2 1
35.5.c.c 2 28.d even 2 1
35.5.c.d yes 2 4.b odd 2 1
35.5.c.d yes 2 140.c even 2 1
175.5.d.e 4 20.e even 4 2
175.5.d.e 4 140.j odd 4 2
315.5.e.c 2 12.b even 2 1
315.5.e.c 2 420.o odd 2 1
315.5.e.d 2 60.h even 2 1
315.5.e.d 2 84.h odd 2 1
560.5.p.d 2 1.a even 1 1 trivial
560.5.p.d 2 35.c odd 2 1 inner
560.5.p.e 2 5.b even 2 1
560.5.p.e 2 7.b odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 5)^{2}$$
$5$ $$T^{2} - 10T + 625$$
$7$ $$T^{2} + 70T + 2401$$
$11$ $$(T + 89)^{2}$$
$13$ $$(T + 5)^{2}$$
$17$ $$(T + 485)^{2}$$
$19$ $$T^{2} + 48600$$
$23$ $$T^{2} + 490776$$
$29$ $$(T - 191)^{2}$$
$31$ $$T^{2} + 1109400$$
$37$ $$T^{2} + 2661336$$
$41$ $$T^{2} + 8496600$$
$43$ $$T^{2} + 142296$$
$47$ $$(T - 2195)^{2}$$
$53$ $$T^{2} + 2519424$$
$59$ $$T^{2} + 13142400$$
$61$ $$T^{2} + 3744600$$
$67$ $$T^{2} + 4193376$$
$71$ $$(T + 4454)^{2}$$
$73$ $$(T - 8650)^{2}$$
$79$ $$(T + 5561)^{2}$$
$83$ $$(T + 1990)^{2}$$
$89$ $$T^{2} + 653400$$
$97$ $$(T - 9235)^{2}$$