# Properties

 Label 960.4.a.t 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 5 q^{5} - 32 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 5 * q^5 - 32 * q^7 + 9 * q^9 $$q + 3 q^{3} - 5 q^{5} - 32 q^{7} + 9 q^{9} + 64 q^{11} + 6 q^{13} - 15 q^{15} + 38 q^{17} - 116 q^{19} - 96 q^{21} + 120 q^{23} + 25 q^{25} + 27 q^{27} + 122 q^{29} - 164 q^{31} + 192 q^{33} + 160 q^{35} - 146 q^{37} + 18 q^{39} - 238 q^{41} - 148 q^{43} - 45 q^{45} + 184 q^{47} + 681 q^{49} + 114 q^{51} - 470 q^{53} - 320 q^{55} - 348 q^{57} - 216 q^{59} - 806 q^{61} - 288 q^{63} - 30 q^{65} - 732 q^{67} + 360 q^{69} - 264 q^{71} - 638 q^{73} + 75 q^{75} - 2048 q^{77} - 596 q^{79} + 81 q^{81} - 884 q^{83} - 190 q^{85} + 366 q^{87} + 930 q^{89} - 192 q^{91} - 492 q^{93} + 580 q^{95} + 322 q^{97} + 576 q^{99}+O(q^{100})$$ q + 3 * q^3 - 5 * q^5 - 32 * q^7 + 9 * q^9 + 64 * q^11 + 6 * q^13 - 15 * q^15 + 38 * q^17 - 116 * q^19 - 96 * q^21 + 120 * q^23 + 25 * q^25 + 27 * q^27 + 122 * q^29 - 164 * q^31 + 192 * q^33 + 160 * q^35 - 146 * q^37 + 18 * q^39 - 238 * q^41 - 148 * q^43 - 45 * q^45 + 184 * q^47 + 681 * q^49 + 114 * q^51 - 470 * q^53 - 320 * q^55 - 348 * q^57 - 216 * q^59 - 806 * q^61 - 288 * q^63 - 30 * q^65 - 732 * q^67 + 360 * q^69 - 264 * q^71 - 638 * q^73 + 75 * q^75 - 2048 * q^77 - 596 * q^79 + 81 * q^81 - 884 * q^83 - 190 * q^85 + 366 * q^87 + 930 * q^89 - 192 * q^91 - 492 * q^93 + 580 * q^95 + 322 * q^97 + 576 * 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 −32.0000 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.t 1
4.b odd 2 1 960.4.a.i 1
8.b even 2 1 480.4.a.c 1
8.d odd 2 1 480.4.a.l yes 1
24.f even 2 1 1440.4.a.j 1
24.h odd 2 1 1440.4.a.a 1
40.e odd 2 1 2400.4.a.a 1
40.f even 2 1 2400.4.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.4.a.c 1 8.b even 2 1
480.4.a.l yes 1 8.d odd 2 1
960.4.a.i 1 4.b odd 2 1
960.4.a.t 1 1.a even 1 1 trivial
1440.4.a.a 1 24.h odd 2 1
1440.4.a.j 1 24.f even 2 1
2400.4.a.a 1 40.e odd 2 1
2400.4.a.v 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(960))$$:

 $$T_{7} + 32$$ T7 + 32 $$T_{11} - 64$$ T11 - 64

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 5$$
$7$ $$T + 32$$
$11$ $$T - 64$$
$13$ $$T - 6$$
$17$ $$T - 38$$
$19$ $$T + 116$$
$23$ $$T - 120$$
$29$ $$T - 122$$
$31$ $$T + 164$$
$37$ $$T + 146$$
$41$ $$T + 238$$
$43$ $$T + 148$$
$47$ $$T - 184$$
$53$ $$T + 470$$
$59$ $$T + 216$$
$61$ $$T + 806$$
$67$ $$T + 732$$
$71$ $$T + 264$$
$73$ $$T + 638$$
$79$ $$T + 596$$
$83$ $$T + 884$$
$89$ $$T - 930$$
$97$ $$T - 322$$