# Properties

 Label 960.4.a.z Level $960$ Weight $4$ Character orbit 960.a Self dual yes Analytic conductor $56.642$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Learn more

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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 5 q^{5} + 16 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 5 * q^5 + 16 * q^7 + 9 * q^9 $$q + 3 q^{3} - 5 q^{5} + 16 q^{7} + 9 q^{9} - 24 q^{11} + 14 q^{13} - 15 q^{15} - 18 q^{17} - 36 q^{19} + 48 q^{21} + 104 q^{23} + 25 q^{25} + 27 q^{27} + 250 q^{29} - 28 q^{31} - 72 q^{33} - 80 q^{35} + 54 q^{37} + 42 q^{39} + 354 q^{41} - 228 q^{43} - 45 q^{45} + 408 q^{47} - 87 q^{49} - 54 q^{51} - 262 q^{53} + 120 q^{55} - 108 q^{57} + 64 q^{59} - 374 q^{61} + 144 q^{63} - 70 q^{65} - 300 q^{67} + 312 q^{69} + 1016 q^{71} + 274 q^{73} + 75 q^{75} - 384 q^{77} + 788 q^{79} + 81 q^{81} + 396 q^{83} + 90 q^{85} + 750 q^{87} + 786 q^{89} + 224 q^{91} - 84 q^{93} + 180 q^{95} - 1086 q^{97} - 216 q^{99}+O(q^{100})$$ q + 3 * q^3 - 5 * q^5 + 16 * q^7 + 9 * q^9 - 24 * q^11 + 14 * q^13 - 15 * q^15 - 18 * q^17 - 36 * q^19 + 48 * q^21 + 104 * q^23 + 25 * q^25 + 27 * q^27 + 250 * q^29 - 28 * q^31 - 72 * q^33 - 80 * q^35 + 54 * q^37 + 42 * q^39 + 354 * q^41 - 228 * q^43 - 45 * q^45 + 408 * q^47 - 87 * q^49 - 54 * q^51 - 262 * q^53 + 120 * q^55 - 108 * q^57 + 64 * q^59 - 374 * q^61 + 144 * q^63 - 70 * q^65 - 300 * q^67 + 312 * q^69 + 1016 * q^71 + 274 * q^73 + 75 * q^75 - 384 * q^77 + 788 * q^79 + 81 * q^81 + 396 * q^83 + 90 * q^85 + 750 * q^87 + 786 * q^89 + 224 * q^91 - 84 * q^93 + 180 * q^95 - 1086 * q^97 - 216 * 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 16.0000 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.z 1
4.b odd 2 1 960.4.a.c 1
8.b even 2 1 480.4.a.f 1
8.d odd 2 1 480.4.a.i yes 1
24.f even 2 1 1440.4.a.c 1
24.h odd 2 1 1440.4.a.h 1
40.e odd 2 1 2400.4.a.h 1
40.f even 2 1 2400.4.a.o 1

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

 $$T_{7} - 16$$ T7 - 16 $$T_{11} + 24$$ T11 + 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 5$$
$7$ $$T - 16$$
$11$ $$T + 24$$
$13$ $$T - 14$$
$17$ $$T + 18$$
$19$ $$T + 36$$
$23$ $$T - 104$$
$29$ $$T - 250$$
$31$ $$T + 28$$
$37$ $$T - 54$$
$41$ $$T - 354$$
$43$ $$T + 228$$
$47$ $$T - 408$$
$53$ $$T + 262$$
$59$ $$T - 64$$
$61$ $$T + 374$$
$67$ $$T + 300$$
$71$ $$T - 1016$$
$73$ $$T - 274$$
$79$ $$T - 788$$
$83$ $$T - 396$$
$89$ $$T - 786$$
$97$ $$T + 1086$$
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