# Properties

 Label 960.4.a.p Level $960$ Weight $4$ Character orbit 960.a Self dual yes Analytic conductor $56.642$ 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] = [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: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 5 q^{5} + 12 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 5 * q^5 + 12 * q^7 + 9 * q^9 $$q - 3 q^{3} + 5 q^{5} + 12 q^{7} + 9 q^{9} + 20 q^{11} + 58 q^{13} - 15 q^{15} - 70 q^{17} + 92 q^{19} - 36 q^{21} + 112 q^{23} + 25 q^{25} - 27 q^{27} - 66 q^{29} - 108 q^{31} - 60 q^{33} + 60 q^{35} + 58 q^{37} - 174 q^{39} + 66 q^{41} + 388 q^{43} + 45 q^{45} - 408 q^{47} - 199 q^{49} + 210 q^{51} - 474 q^{53} + 100 q^{55} - 276 q^{57} + 540 q^{59} - 14 q^{61} + 108 q^{63} + 290 q^{65} + 276 q^{67} - 336 q^{69} - 96 q^{71} - 790 q^{73} - 75 q^{75} + 240 q^{77} + 308 q^{79} + 81 q^{81} + 1036 q^{83} - 350 q^{85} + 198 q^{87} + 1210 q^{89} + 696 q^{91} + 324 q^{93} + 460 q^{95} + 1426 q^{97} + 180 q^{99}+O(q^{100})$$ q - 3 * q^3 + 5 * q^5 + 12 * q^7 + 9 * q^9 + 20 * q^11 + 58 * q^13 - 15 * q^15 - 70 * q^17 + 92 * q^19 - 36 * q^21 + 112 * q^23 + 25 * q^25 - 27 * q^27 - 66 * q^29 - 108 * q^31 - 60 * q^33 + 60 * q^35 + 58 * q^37 - 174 * q^39 + 66 * q^41 + 388 * q^43 + 45 * q^45 - 408 * q^47 - 199 * q^49 + 210 * q^51 - 474 * q^53 + 100 * q^55 - 276 * q^57 + 540 * q^59 - 14 * q^61 + 108 * q^63 + 290 * q^65 + 276 * q^67 - 336 * q^69 - 96 * q^71 - 790 * q^73 - 75 * q^75 + 240 * q^77 + 308 * q^79 + 81 * q^81 + 1036 * q^83 - 350 * q^85 + 198 * q^87 + 1210 * q^89 + 696 * q^91 + 324 * q^93 + 460 * q^95 + 1426 * q^97 + 180 * 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 12.0000 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.p 1
4.b odd 2 1 960.4.a.be 1
8.b even 2 1 480.4.a.h yes 1
8.d odd 2 1 480.4.a.a 1
24.f even 2 1 1440.4.a.l 1
24.h odd 2 1 1440.4.a.q 1
40.e odd 2 1 2400.4.a.u 1
40.f even 2 1 2400.4.a.b 1

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

## Hecke kernels

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

 $$T_{7} - 12$$ T7 - 12 $$T_{11} - 20$$ T11 - 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T - 5$$
$7$ $$T - 12$$
$11$ $$T - 20$$
$13$ $$T - 58$$
$17$ $$T + 70$$
$19$ $$T - 92$$
$23$ $$T - 112$$
$29$ $$T + 66$$
$31$ $$T + 108$$
$37$ $$T - 58$$
$41$ $$T - 66$$
$43$ $$T - 388$$
$47$ $$T + 408$$
$53$ $$T + 474$$
$59$ $$T - 540$$
$61$ $$T + 14$$
$67$ $$T - 276$$
$71$ $$T + 96$$
$73$ $$T + 790$$
$79$ $$T - 308$$
$83$ $$T - 1036$$
$89$ $$T - 1210$$
$97$ $$T - 1426$$