# Properties

 Label 960.4.a.bc 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 60) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 5 q^{5} - 28 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 5 * q^5 - 28 * q^7 + 9 * q^9 $$q + 3 q^{3} + 5 q^{5} - 28 q^{7} + 9 q^{9} + 24 q^{11} + 70 q^{13} + 15 q^{15} + 102 q^{17} - 20 q^{19} - 84 q^{21} - 72 q^{23} + 25 q^{25} + 27 q^{27} - 306 q^{29} - 136 q^{31} + 72 q^{33} - 140 q^{35} + 214 q^{37} + 210 q^{39} - 150 q^{41} + 292 q^{43} + 45 q^{45} - 72 q^{47} + 441 q^{49} + 306 q^{51} + 414 q^{53} + 120 q^{55} - 60 q^{57} + 744 q^{59} + 418 q^{61} - 252 q^{63} + 350 q^{65} - 188 q^{67} - 216 q^{69} + 480 q^{71} + 434 q^{73} + 75 q^{75} - 672 q^{77} + 1352 q^{79} + 81 q^{81} + 612 q^{83} + 510 q^{85} - 918 q^{87} - 30 q^{89} - 1960 q^{91} - 408 q^{93} - 100 q^{95} - 286 q^{97} + 216 q^{99}+O(q^{100})$$ q + 3 * q^3 + 5 * q^5 - 28 * q^7 + 9 * q^9 + 24 * q^11 + 70 * q^13 + 15 * q^15 + 102 * q^17 - 20 * q^19 - 84 * q^21 - 72 * q^23 + 25 * q^25 + 27 * q^27 - 306 * q^29 - 136 * q^31 + 72 * q^33 - 140 * q^35 + 214 * q^37 + 210 * q^39 - 150 * q^41 + 292 * q^43 + 45 * q^45 - 72 * q^47 + 441 * q^49 + 306 * q^51 + 414 * q^53 + 120 * q^55 - 60 * q^57 + 744 * q^59 + 418 * q^61 - 252 * q^63 + 350 * q^65 - 188 * q^67 - 216 * q^69 + 480 * q^71 + 434 * q^73 + 75 * q^75 - 672 * q^77 + 1352 * q^79 + 81 * q^81 + 612 * q^83 + 510 * q^85 - 918 * q^87 - 30 * q^89 - 1960 * q^91 - 408 * q^93 - 100 * q^95 - 286 * 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 −28.0000 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.bc 1
4.b odd 2 1 960.4.a.r 1
8.b even 2 1 60.4.a.a 1
8.d odd 2 1 240.4.a.i 1
24.f even 2 1 720.4.a.bb 1
24.h odd 2 1 180.4.a.d 1
40.e odd 2 1 1200.4.a.a 1
40.f even 2 1 300.4.a.i 1
40.i odd 4 2 300.4.d.b 2
40.k even 4 2 1200.4.f.n 2
72.j odd 6 2 1620.4.i.f 2
72.n even 6 2 1620.4.i.l 2
120.i odd 2 1 900.4.a.q 1
120.w even 4 2 900.4.d.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.a.a 1 8.b even 2 1
180.4.a.d 1 24.h odd 2 1
240.4.a.i 1 8.d odd 2 1
300.4.a.i 1 40.f even 2 1
300.4.d.b 2 40.i odd 4 2
720.4.a.bb 1 24.f even 2 1
900.4.a.q 1 120.i odd 2 1
900.4.d.h 2 120.w even 4 2
960.4.a.r 1 4.b odd 2 1
960.4.a.bc 1 1.a even 1 1 trivial
1200.4.a.a 1 40.e odd 2 1
1200.4.f.n 2 40.k even 4 2
1620.4.i.f 2 72.j odd 6 2
1620.4.i.l 2 72.n even 6 2

## Hecke kernels

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

 $$T_{7} + 28$$ T7 + 28 $$T_{11} - 24$$ T11 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T - 5$$
$7$ $$T + 28$$
$11$ $$T - 24$$
$13$ $$T - 70$$
$17$ $$T - 102$$
$19$ $$T + 20$$
$23$ $$T + 72$$
$29$ $$T + 306$$
$31$ $$T + 136$$
$37$ $$T - 214$$
$41$ $$T + 150$$
$43$ $$T - 292$$
$47$ $$T + 72$$
$53$ $$T - 414$$
$59$ $$T - 744$$
$61$ $$T - 418$$
$67$ $$T + 188$$
$71$ $$T - 480$$
$73$ $$T - 434$$
$79$ $$T - 1352$$
$83$ $$T - 612$$
$89$ $$T + 30$$
$97$ $$T + 286$$