# Properties

 Label 320.4.a.p Level $320$ Weight $4$ Character orbit 320.a Self dual yes Analytic conductor $18.881$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,4,Mod(1,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, 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("320.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 320.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: $$18.8806112018$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{10})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 10$$ x^2 - 10 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 160) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - 5 q^{5} - 3 \beta q^{7} + 13 q^{9} +O(q^{10})$$ q + b * q^3 - 5 * q^5 - 3*b * q^7 + 13 * q^9 $$q + \beta q^{3} - 5 q^{5} - 3 \beta q^{7} + 13 q^{9} - 2 \beta q^{11} - 38 q^{13} - 5 \beta q^{15} + 34 q^{17} - 16 \beta q^{19} - 120 q^{21} - 13 \beta q^{23} + 25 q^{25} - 14 \beta q^{27} - 270 q^{29} + 54 \beta q^{31} - 80 q^{33} + 15 \beta q^{35} - 206 q^{37} - 38 \beta q^{39} - 270 q^{41} + 85 \beta q^{43} - 65 q^{45} + 21 \beta q^{47} + 17 q^{49} + 34 \beta q^{51} + 258 q^{53} + 10 \beta q^{55} - 640 q^{57} - 12 \beta q^{59} + 250 q^{61} - 39 \beta q^{63} + 190 q^{65} - 129 \beta q^{67} - 520 q^{69} + 102 \beta q^{71} - 1078 q^{73} + 25 \beta q^{75} + 240 q^{77} + 44 \beta q^{79} - 911 q^{81} - 175 \beta q^{83} - 170 q^{85} - 270 \beta q^{87} + 890 q^{89} + 114 \beta q^{91} + 2160 q^{93} + 80 \beta q^{95} - 254 q^{97} - 26 \beta q^{99} +O(q^{100})$$ q + b * q^3 - 5 * q^5 - 3*b * q^7 + 13 * q^9 - 2*b * q^11 - 38 * q^13 - 5*b * q^15 + 34 * q^17 - 16*b * q^19 - 120 * q^21 - 13*b * q^23 + 25 * q^25 - 14*b * q^27 - 270 * q^29 + 54*b * q^31 - 80 * q^33 + 15*b * q^35 - 206 * q^37 - 38*b * q^39 - 270 * q^41 + 85*b * q^43 - 65 * q^45 + 21*b * q^47 + 17 * q^49 + 34*b * q^51 + 258 * q^53 + 10*b * q^55 - 640 * q^57 - 12*b * q^59 + 250 * q^61 - 39*b * q^63 + 190 * q^65 - 129*b * q^67 - 520 * q^69 + 102*b * q^71 - 1078 * q^73 + 25*b * q^75 + 240 * q^77 + 44*b * q^79 - 911 * q^81 - 175*b * q^83 - 170 * q^85 - 270*b * q^87 + 890 * q^89 + 114*b * q^91 + 2160 * q^93 + 80*b * q^95 - 254 * q^97 - 26*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 10 q^{5} + 26 q^{9}+O(q^{10})$$ 2 * q - 10 * q^5 + 26 * q^9 $$2 q - 10 q^{5} + 26 q^{9} - 76 q^{13} + 68 q^{17} - 240 q^{21} + 50 q^{25} - 540 q^{29} - 160 q^{33} - 412 q^{37} - 540 q^{41} - 130 q^{45} + 34 q^{49} + 516 q^{53} - 1280 q^{57} + 500 q^{61} + 380 q^{65} - 1040 q^{69} - 2156 q^{73} + 480 q^{77} - 1822 q^{81} - 340 q^{85} + 1780 q^{89} + 4320 q^{93} - 508 q^{97}+O(q^{100})$$ 2 * q - 10 * q^5 + 26 * q^9 - 76 * q^13 + 68 * q^17 - 240 * q^21 + 50 * q^25 - 540 * q^29 - 160 * q^33 - 412 * q^37 - 540 * q^41 - 130 * q^45 + 34 * q^49 + 516 * q^53 - 1280 * q^57 + 500 * q^61 + 380 * q^65 - 1040 * q^69 - 2156 * q^73 + 480 * q^77 - 1822 * q^81 - 340 * q^85 + 1780 * q^89 + 4320 * q^93 - 508 * q^97

## 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
 −3.16228 3.16228
0 −6.32456 0 −5.00000 0 18.9737 0 13.0000 0
1.2 0 6.32456 0 −5.00000 0 −18.9737 0 13.0000 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
$$2$$ $$-1$$
$$5$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.4.a.p 2
4.b odd 2 1 inner 320.4.a.p 2
5.b even 2 1 1600.4.a.ch 2
8.b even 2 1 160.4.a.f 2
8.d odd 2 1 160.4.a.f 2
16.e even 4 2 1280.4.d.u 4
16.f odd 4 2 1280.4.d.u 4
20.d odd 2 1 1600.4.a.ch 2
24.f even 2 1 1440.4.a.v 2
24.h odd 2 1 1440.4.a.v 2
40.e odd 2 1 800.4.a.p 2
40.f even 2 1 800.4.a.p 2
40.i odd 4 2 800.4.c.j 4
40.k even 4 2 800.4.c.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.f 2 8.b even 2 1
160.4.a.f 2 8.d odd 2 1
320.4.a.p 2 1.a even 1 1 trivial
320.4.a.p 2 4.b odd 2 1 inner
800.4.a.p 2 40.e odd 2 1
800.4.a.p 2 40.f even 2 1
800.4.c.j 4 40.i odd 4 2
800.4.c.j 4 40.k even 4 2
1280.4.d.u 4 16.e even 4 2
1280.4.d.u 4 16.f odd 4 2
1440.4.a.v 2 24.f even 2 1
1440.4.a.v 2 24.h odd 2 1
1600.4.a.ch 2 5.b even 2 1
1600.4.a.ch 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(320))$$:

 $$T_{3}^{2} - 40$$ T3^2 - 40 $$T_{7}^{2} - 360$$ T7^2 - 360

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 40$$
$5$ $$(T + 5)^{2}$$
$7$ $$T^{2} - 360$$
$11$ $$T^{2} - 160$$
$13$ $$(T + 38)^{2}$$
$17$ $$(T - 34)^{2}$$
$19$ $$T^{2} - 10240$$
$23$ $$T^{2} - 6760$$
$29$ $$(T + 270)^{2}$$
$31$ $$T^{2} - 116640$$
$37$ $$(T + 206)^{2}$$
$41$ $$(T + 270)^{2}$$
$43$ $$T^{2} - 289000$$
$47$ $$T^{2} - 17640$$
$53$ $$(T - 258)^{2}$$
$59$ $$T^{2} - 5760$$
$61$ $$(T - 250)^{2}$$
$67$ $$T^{2} - 665640$$
$71$ $$T^{2} - 416160$$
$73$ $$(T + 1078)^{2}$$
$79$ $$T^{2} - 77440$$
$83$ $$T^{2} - 1225000$$
$89$ $$(T - 890)^{2}$$
$97$ $$(T + 254)^{2}$$