Properties

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

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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 120) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 5 q^{5} - 8 q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 - 5 * q^5 - 8 * q^7 + 9 * q^9 $$q + 3 q^{3} - 5 q^{5} - 8 q^{7} + 9 q^{9} + 20 q^{11} - 22 q^{13} - 15 q^{15} - 14 q^{17} + 76 q^{19} - 24 q^{21} - 56 q^{23} + 25 q^{25} + 27 q^{27} + 154 q^{29} - 160 q^{31} + 60 q^{33} + 40 q^{35} + 162 q^{37} - 66 q^{39} - 390 q^{41} + 388 q^{43} - 45 q^{45} + 544 q^{47} - 279 q^{49} - 42 q^{51} + 210 q^{53} - 100 q^{55} + 228 q^{57} - 380 q^{59} + 794 q^{61} - 72 q^{63} + 110 q^{65} - 148 q^{67} - 168 q^{69} + 840 q^{71} + 858 q^{73} + 75 q^{75} - 160 q^{77} - 144 q^{79} + 81 q^{81} + 316 q^{83} + 70 q^{85} + 462 q^{87} + 1098 q^{89} + 176 q^{91} - 480 q^{93} - 380 q^{95} + 994 q^{97} + 180 q^{99}+O(q^{100})$$ q + 3 * q^3 - 5 * q^5 - 8 * q^7 + 9 * q^9 + 20 * q^11 - 22 * q^13 - 15 * q^15 - 14 * q^17 + 76 * q^19 - 24 * q^21 - 56 * q^23 + 25 * q^25 + 27 * q^27 + 154 * q^29 - 160 * q^31 + 60 * q^33 + 40 * q^35 + 162 * q^37 - 66 * q^39 - 390 * q^41 + 388 * q^43 - 45 * q^45 + 544 * q^47 - 279 * q^49 - 42 * q^51 + 210 * q^53 - 100 * q^55 + 228 * q^57 - 380 * q^59 + 794 * q^61 - 72 * q^63 + 110 * q^65 - 148 * q^67 - 168 * q^69 + 840 * q^71 + 858 * q^73 + 75 * q^75 - 160 * q^77 - 144 * q^79 + 81 * q^81 + 316 * q^83 + 70 * q^85 + 462 * q^87 + 1098 * q^89 + 176 * q^91 - 480 * q^93 - 380 * q^95 + 994 * 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 −8.00000 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.v 1
4.b odd 2 1 960.4.a.g 1
8.b even 2 1 240.4.a.d 1
8.d odd 2 1 120.4.a.f 1
24.f even 2 1 360.4.a.e 1
24.h odd 2 1 720.4.a.f 1
40.e odd 2 1 600.4.a.d 1
40.f even 2 1 1200.4.a.bf 1
40.i odd 4 2 1200.4.f.g 2
40.k even 4 2 600.4.f.g 2
120.m even 2 1 1800.4.a.k 1
120.q odd 4 2 1800.4.f.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.f 1 8.d odd 2 1
240.4.a.d 1 8.b even 2 1
360.4.a.e 1 24.f even 2 1
600.4.a.d 1 40.e odd 2 1
600.4.f.g 2 40.k even 4 2
720.4.a.f 1 24.h odd 2 1
960.4.a.g 1 4.b odd 2 1
960.4.a.v 1 1.a even 1 1 trivial
1200.4.a.bf 1 40.f even 2 1
1200.4.f.g 2 40.i odd 4 2
1800.4.a.k 1 120.m even 2 1
1800.4.f.h 2 120.q odd 4 2

Hecke kernels

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

 $$T_{7} + 8$$ T7 + 8 $$T_{11} - 20$$ T11 - 20

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 3$$
$5$ $$T + 5$$
$7$ $$T + 8$$
$11$ $$T - 20$$
$13$ $$T + 22$$
$17$ $$T + 14$$
$19$ $$T - 76$$
$23$ $$T + 56$$
$29$ $$T - 154$$
$31$ $$T + 160$$
$37$ $$T - 162$$
$41$ $$T + 390$$
$43$ $$T - 388$$
$47$ $$T - 544$$
$53$ $$T - 210$$
$59$ $$T + 380$$
$61$ $$T - 794$$
$67$ $$T + 148$$
$71$ $$T - 840$$
$73$ $$T - 858$$
$79$ $$T + 144$$
$83$ $$T - 316$$
$89$ $$T - 1098$$
$97$ $$T - 994$$
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