# Properties

 Label 960.4.a.w Level $960$ Weight $4$ Character orbit 960.a Self dual yes Analytic conductor $56.642$ Analytic rank $1$ 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] = [960,4,Mod(1,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("960.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 960.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: $$56.6418336055$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 480) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 9 * q^9 $$q + 3 q^{3} - 5 q^{5} - 4 q^{7} + 9 q^{9} - 40 q^{11} + 90 q^{13} - 15 q^{15} - 70 q^{17} - 40 q^{19} - 12 q^{21} + 108 q^{23} + 25 q^{25} + 27 q^{27} - 166 q^{29} - 40 q^{31} - 120 q^{33} + 20 q^{35} + 130 q^{37} + 270 q^{39} - 310 q^{41} + 268 q^{43} - 45 q^{45} - 556 q^{47} - 327 q^{49} - 210 q^{51} + 370 q^{53} + 200 q^{55} - 120 q^{57} - 240 q^{59} + 130 q^{61} - 36 q^{63} - 450 q^{65} - 876 q^{67} + 324 q^{69} - 840 q^{71} + 250 q^{73} + 75 q^{75} + 160 q^{77} - 880 q^{79} + 81 q^{81} + 188 q^{83} + 350 q^{85} - 498 q^{87} - 726 q^{89} - 360 q^{91} - 120 q^{93} + 200 q^{95} - 1550 q^{97} - 360 q^{99}+O(q^{100})$$ q + 3 * q^3 - 5 * q^5 - 4 * q^7 + 9 * q^9 - 40 * q^11 + 90 * q^13 - 15 * q^15 - 70 * q^17 - 40 * q^19 - 12 * q^21 + 108 * q^23 + 25 * q^25 + 27 * q^27 - 166 * q^29 - 40 * q^31 - 120 * q^33 + 20 * q^35 + 130 * q^37 + 270 * q^39 - 310 * q^41 + 268 * q^43 - 45 * q^45 - 556 * q^47 - 327 * q^49 - 210 * q^51 + 370 * q^53 + 200 * q^55 - 120 * q^57 - 240 * q^59 + 130 * q^61 - 36 * q^63 - 450 * q^65 - 876 * q^67 + 324 * q^69 - 840 * q^71 + 250 * q^73 + 75 * q^75 + 160 * q^77 - 880 * q^79 + 81 * q^81 + 188 * q^83 + 350 * q^85 - 498 * q^87 - 726 * q^89 - 360 * q^91 - 120 * q^93 + 200 * q^95 - 1550 * q^97 - 360 * 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 3.00000 0 −5.00000 0 −4.00000 0 9.00000 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
$$2$$ $$+1$$
$$3$$ $$-1$$
$$5$$ $$+1$$

## 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 960.4.a.w 1
4.b odd 2 1 960.4.a.f 1
8.b even 2 1 480.4.a.d 1
8.d odd 2 1 480.4.a.k yes 1
24.f even 2 1 1440.4.a.f 1
24.h odd 2 1 1440.4.a.e 1
40.e odd 2 1 2400.4.a.d 1
40.f even 2 1 2400.4.a.s 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.4.a.d 1 8.b even 2 1
480.4.a.k yes 1 8.d odd 2 1
960.4.a.f 1 4.b odd 2 1
960.4.a.w 1 1.a even 1 1 trivial
1440.4.a.e 1 24.h odd 2 1
1440.4.a.f 1 24.f even 2 1
2400.4.a.d 1 40.e odd 2 1
2400.4.a.s 1 40.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(960))$$:

 $$T_{7} + 4$$ T7 + 4 $$T_{11} + 40$$ T11 + 40

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 5$$
$7$ $$T + 4$$
$11$ $$T + 40$$
$13$ $$T - 90$$
$17$ $$T + 70$$
$19$ $$T + 40$$
$23$ $$T - 108$$
$29$ $$T + 166$$
$31$ $$T + 40$$
$37$ $$T - 130$$
$41$ $$T + 310$$
$43$ $$T - 268$$
$47$ $$T + 556$$
$53$ $$T - 370$$
$59$ $$T + 240$$
$61$ $$T - 130$$
$67$ $$T + 876$$
$71$ $$T + 840$$
$73$ $$T - 250$$
$79$ $$T + 880$$
$83$ $$T - 188$$
$89$ $$T + 726$$
$97$ $$T + 1550$$