# Properties

 Label 960.4.a.h 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 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)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} - 5 q^{5} + 16 q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 - 5 * q^5 + 16 * q^7 + 9 * q^9 $$q - 3 q^{3} - 5 q^{5} + 16 q^{7} + 9 q^{9} - 28 q^{11} + 26 q^{13} + 15 q^{15} - 62 q^{17} - 68 q^{19} - 48 q^{21} + 208 q^{23} + 25 q^{25} - 27 q^{27} + 58 q^{29} - 160 q^{31} + 84 q^{33} - 80 q^{35} - 270 q^{37} - 78 q^{39} + 282 q^{41} + 76 q^{43} - 45 q^{45} + 280 q^{47} - 87 q^{49} + 186 q^{51} + 210 q^{53} + 140 q^{55} + 204 q^{57} + 196 q^{59} - 742 q^{61} + 144 q^{63} - 130 q^{65} + 836 q^{67} - 624 q^{69} + 504 q^{71} - 1062 q^{73} - 75 q^{75} - 448 q^{77} - 768 q^{79} + 81 q^{81} - 1052 q^{83} + 310 q^{85} - 174 q^{87} - 726 q^{89} + 416 q^{91} + 480 q^{93} + 340 q^{95} - 1406 q^{97} - 252 q^{99}+O(q^{100})$$ q - 3 * q^3 - 5 * q^5 + 16 * q^7 + 9 * q^9 - 28 * q^11 + 26 * q^13 + 15 * q^15 - 62 * q^17 - 68 * q^19 - 48 * q^21 + 208 * q^23 + 25 * q^25 - 27 * q^27 + 58 * q^29 - 160 * q^31 + 84 * q^33 - 80 * q^35 - 270 * q^37 - 78 * q^39 + 282 * q^41 + 76 * q^43 - 45 * q^45 + 280 * q^47 - 87 * q^49 + 186 * q^51 + 210 * q^53 + 140 * q^55 + 204 * q^57 + 196 * q^59 - 742 * q^61 + 144 * q^63 - 130 * q^65 + 836 * q^67 - 624 * q^69 + 504 * q^71 - 1062 * q^73 - 75 * q^75 - 448 * q^77 - 768 * q^79 + 81 * q^81 - 1052 * q^83 + 310 * q^85 - 174 * q^87 - 726 * q^89 + 416 * q^91 + 480 * q^93 + 340 * q^95 - 1406 * q^97 - 252 * 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.h 1
4.b odd 2 1 960.4.a.u 1
8.b even 2 1 240.4.a.l 1
8.d odd 2 1 120.4.a.c 1
24.f even 2 1 360.4.a.b 1
24.h odd 2 1 720.4.a.l 1
40.e odd 2 1 600.4.a.q 1
40.f even 2 1 1200.4.a.c 1
40.i odd 4 2 1200.4.f.o 2
40.k even 4 2 600.4.f.c 2
120.m even 2 1 1800.4.a.bb 1
120.q odd 4 2 1800.4.f.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.c 1 8.d odd 2 1
240.4.a.l 1 8.b even 2 1
360.4.a.b 1 24.f even 2 1
600.4.a.q 1 40.e odd 2 1
600.4.f.c 2 40.k even 4 2
720.4.a.l 1 24.h odd 2 1
960.4.a.h 1 1.a even 1 1 trivial
960.4.a.u 1 4.b odd 2 1
1200.4.a.c 1 40.f even 2 1
1200.4.f.o 2 40.i odd 4 2
1800.4.a.bb 1 120.m even 2 1
1800.4.f.r 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(960))$$:

 $$T_{7} - 16$$ T7 - 16 $$T_{11} + 28$$ T11 + 28

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 5$$
$7$ $$T - 16$$
$11$ $$T + 28$$
$13$ $$T - 26$$
$17$ $$T + 62$$
$19$ $$T + 68$$
$23$ $$T - 208$$
$29$ $$T - 58$$
$31$ $$T + 160$$
$37$ $$T + 270$$
$41$ $$T - 282$$
$43$ $$T - 76$$
$47$ $$T - 280$$
$53$ $$T - 210$$
$59$ $$T - 196$$
$61$ $$T + 742$$
$67$ $$T - 836$$
$71$ $$T - 504$$
$73$ $$T + 1062$$
$79$ $$T + 768$$
$83$ $$T + 1052$$
$89$ $$T + 726$$
$97$ $$T + 1406$$