# Properties

 Label 960.4.a.n 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 30) 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 $$q - 3 q^{3} + 5 q^{5} - 4 q^{7} + 9 q^{9} + 48 q^{11} - 2 q^{13} - 15 q^{15} - 114 q^{17} - 140 q^{19} + 12 q^{21} + 72 q^{23} + 25 q^{25} - 27 q^{27} - 210 q^{29} + 272 q^{31} - 144 q^{33} - 20 q^{35} + 334 q^{37} + 6 q^{39} - 198 q^{41} + 268 q^{43} + 45 q^{45} + 216 q^{47} - 327 q^{49} + 342 q^{51} + 78 q^{53} + 240 q^{55} + 420 q^{57} - 240 q^{59} - 302 q^{61} - 36 q^{63} - 10 q^{65} - 596 q^{67} - 216 q^{69} - 768 q^{71} - 478 q^{73} - 75 q^{75} - 192 q^{77} - 640 q^{79} + 81 q^{81} + 348 q^{83} - 570 q^{85} + 630 q^{87} + 210 q^{89} + 8 q^{91} - 816 q^{93} - 700 q^{95} - 1534 q^{97} + 432 q^{99}+O(q^{100})$$ q - 3 * q^3 + 5 * q^5 - 4 * q^7 + 9 * q^9 + 48 * q^11 - 2 * q^13 - 15 * q^15 - 114 * q^17 - 140 * q^19 + 12 * q^21 + 72 * q^23 + 25 * q^25 - 27 * q^27 - 210 * q^29 + 272 * q^31 - 144 * q^33 - 20 * q^35 + 334 * q^37 + 6 * q^39 - 198 * q^41 + 268 * q^43 + 45 * q^45 + 216 * q^47 - 327 * q^49 + 342 * q^51 + 78 * q^53 + 240 * q^55 + 420 * q^57 - 240 * q^59 - 302 * q^61 - 36 * q^63 - 10 * q^65 - 596 * q^67 - 216 * q^69 - 768 * q^71 - 478 * q^73 - 75 * q^75 - 192 * q^77 - 640 * q^79 + 81 * q^81 + 348 * q^83 - 570 * q^85 + 630 * q^87 + 210 * q^89 + 8 * q^91 - 816 * q^93 - 700 * q^95 - 1534 * q^97 + 432 * 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.n 1
4.b odd 2 1 960.4.a.bg 1
8.b even 2 1 30.4.a.b 1
8.d odd 2 1 240.4.a.b 1
24.f even 2 1 720.4.a.y 1
24.h odd 2 1 90.4.a.c 1
40.e odd 2 1 1200.4.a.ba 1
40.f even 2 1 150.4.a.b 1
40.i odd 4 2 150.4.c.c 2
40.k even 4 2 1200.4.f.r 2
56.h odd 2 1 1470.4.a.r 1
72.j odd 6 2 810.4.e.p 2
72.n even 6 2 810.4.e.i 2
120.i odd 2 1 450.4.a.r 1
120.w even 4 2 450.4.c.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.4.a.b 1 8.b even 2 1
90.4.a.c 1 24.h odd 2 1
150.4.a.b 1 40.f even 2 1
150.4.c.c 2 40.i odd 4 2
240.4.a.b 1 8.d odd 2 1
450.4.a.r 1 120.i odd 2 1
450.4.c.j 2 120.w even 4 2
720.4.a.y 1 24.f even 2 1
810.4.e.i 2 72.n even 6 2
810.4.e.p 2 72.j odd 6 2
960.4.a.n 1 1.a even 1 1 trivial
960.4.a.bg 1 4.b odd 2 1
1200.4.a.ba 1 40.e odd 2 1
1200.4.f.r 2 40.k even 4 2
1470.4.a.r 1 56.h odd 2 1

## Hecke kernels

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

 $$T_{7} + 4$$ T7 + 4 $$T_{11} - 48$$ T11 - 48

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T - 5$$
$7$ $$T + 4$$
$11$ $$T - 48$$
$13$ $$T + 2$$
$17$ $$T + 114$$
$19$ $$T + 140$$
$23$ $$T - 72$$
$29$ $$T + 210$$
$31$ $$T - 272$$
$37$ $$T - 334$$
$41$ $$T + 198$$
$43$ $$T - 268$$
$47$ $$T - 216$$
$53$ $$T - 78$$
$59$ $$T + 240$$
$61$ $$T + 302$$
$67$ $$T + 596$$
$71$ $$T + 768$$
$73$ $$T + 478$$
$79$ $$T + 640$$
$83$ $$T - 348$$
$89$ $$T - 210$$
$97$ $$T + 1534$$