Properties

 Label 320.4.a.n Level $320$ Weight $4$ Character orbit 320.a Self dual yes Analytic conductor $18.881$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) 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)

 $$f(q)$$ $$=$$ $$q + 10 q^{3} + 5 q^{5} + 18 q^{7} + 73 q^{9}+O(q^{10})$$ q + 10 * q^3 + 5 * q^5 + 18 * q^7 + 73 * q^9 $$q + 10 q^{3} + 5 q^{5} + 18 q^{7} + 73 q^{9} - 16 q^{11} + 6 q^{13} + 50 q^{15} - 6 q^{17} - 124 q^{19} + 180 q^{21} - 42 q^{23} + 25 q^{25} + 460 q^{27} - 142 q^{29} + 188 q^{31} - 160 q^{33} + 90 q^{35} - 202 q^{37} + 60 q^{39} + 54 q^{41} + 66 q^{43} + 365 q^{45} - 38 q^{47} - 19 q^{49} - 60 q^{51} - 738 q^{53} - 80 q^{55} - 1240 q^{57} + 564 q^{59} + 262 q^{61} + 1314 q^{63} + 30 q^{65} - 554 q^{67} - 420 q^{69} - 140 q^{71} + 882 q^{73} + 250 q^{75} - 288 q^{77} + 1160 q^{79} + 2629 q^{81} + 642 q^{83} - 30 q^{85} - 1420 q^{87} - 854 q^{89} + 108 q^{91} + 1880 q^{93} - 620 q^{95} - 478 q^{97} - 1168 q^{99}+O(q^{100})$$ q + 10 * q^3 + 5 * q^5 + 18 * q^7 + 73 * q^9 - 16 * q^11 + 6 * q^13 + 50 * q^15 - 6 * q^17 - 124 * q^19 + 180 * q^21 - 42 * q^23 + 25 * q^25 + 460 * q^27 - 142 * q^29 + 188 * q^31 - 160 * q^33 + 90 * q^35 - 202 * q^37 + 60 * q^39 + 54 * q^41 + 66 * q^43 + 365 * q^45 - 38 * q^47 - 19 * q^49 - 60 * q^51 - 738 * q^53 - 80 * q^55 - 1240 * q^57 + 564 * q^59 + 262 * q^61 + 1314 * q^63 + 30 * q^65 - 554 * q^67 - 420 * q^69 - 140 * q^71 + 882 * q^73 + 250 * q^75 - 288 * q^77 + 1160 * q^79 + 2629 * q^81 + 642 * q^83 - 30 * q^85 - 1420 * q^87 - 854 * q^89 + 108 * q^91 + 1880 * q^93 - 620 * q^95 - 478 * q^97 - 1168 * q^99

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
 0
0 10.0000 0 5.00000 0 18.0000 0 73.0000 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

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.4.a.n 1
4.b odd 2 1 320.4.a.a 1
5.b even 2 1 1600.4.a.a 1
8.b even 2 1 80.4.a.a 1
8.d odd 2 1 40.4.a.c 1
16.e even 4 2 1280.4.d.b 2
16.f odd 4 2 1280.4.d.o 2
20.d odd 2 1 1600.4.a.ca 1
24.f even 2 1 360.4.a.i 1
24.h odd 2 1 720.4.a.ba 1
40.e odd 2 1 200.4.a.a 1
40.f even 2 1 400.4.a.u 1
40.i odd 4 2 400.4.c.a 2
40.k even 4 2 200.4.c.a 2
56.e even 2 1 1960.4.a.a 1
120.m even 2 1 1800.4.a.bd 1
120.q odd 4 2 1800.4.f.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.c 1 8.d odd 2 1
80.4.a.a 1 8.b even 2 1
200.4.a.a 1 40.e odd 2 1
200.4.c.a 2 40.k even 4 2
320.4.a.a 1 4.b odd 2 1
320.4.a.n 1 1.a even 1 1 trivial
360.4.a.i 1 24.f even 2 1
400.4.a.u 1 40.f even 2 1
400.4.c.a 2 40.i odd 4 2
720.4.a.ba 1 24.h odd 2 1
1280.4.d.b 2 16.e even 4 2
1280.4.d.o 2 16.f odd 4 2
1600.4.a.a 1 5.b even 2 1
1600.4.a.ca 1 20.d odd 2 1
1800.4.a.bd 1 120.m even 2 1
1800.4.f.n 2 120.q odd 4 2
1960.4.a.a 1 56.e even 2 1

Hecke kernels

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

 $$T_{3} - 10$$ T3 - 10 $$T_{7} - 18$$ T7 - 18

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 10$$
$5$ $$T - 5$$
$7$ $$T - 18$$
$11$ $$T + 16$$
$13$ $$T - 6$$
$17$ $$T + 6$$
$19$ $$T + 124$$
$23$ $$T + 42$$
$29$ $$T + 142$$
$31$ $$T - 188$$
$37$ $$T + 202$$
$41$ $$T - 54$$
$43$ $$T - 66$$
$47$ $$T + 38$$
$53$ $$T + 738$$
$59$ $$T - 564$$
$61$ $$T - 262$$
$67$ $$T + 554$$
$71$ $$T + 140$$
$73$ $$T - 882$$
$79$ $$T - 1160$$
$83$ $$T - 642$$
$89$ $$T + 854$$
$97$ $$T + 478$$