# Properties

 Label 960.4.a.b 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

## Newspace parameters

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 960.a (trivial)

## Newform invariants

 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 15) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 3q^{3} - 5q^{5} - 24q^{7} + 9q^{9} + O(q^{10})$$ $$q - 3q^{3} - 5q^{5} - 24q^{7} + 9q^{9} - 52q^{11} - 22q^{13} + 15q^{15} - 14q^{17} + 20q^{19} + 72q^{21} - 168q^{23} + 25q^{25} - 27q^{27} - 230q^{29} - 288q^{31} + 156q^{33} + 120q^{35} + 34q^{37} + 66q^{39} + 122q^{41} + 188q^{43} - 45q^{45} + 256q^{47} + 233q^{49} + 42q^{51} + 338q^{53} + 260q^{55} - 60q^{57} - 100q^{59} - 742q^{61} - 216q^{63} + 110q^{65} + 84q^{67} + 504q^{69} - 328q^{71} - 38q^{73} - 75q^{75} + 1248q^{77} - 240q^{79} + 81q^{81} - 1212q^{83} + 70q^{85} + 690q^{87} + 330q^{89} + 528q^{91} + 864q^{93} - 100q^{95} + 866q^{97} - 468q^{99} + O(q^{100})$$

## 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.

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 −24.0000 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.b 1
4.b odd 2 1 960.4.a.ba 1
8.b even 2 1 15.4.a.a 1
8.d odd 2 1 240.4.a.e 1
24.f even 2 1 720.4.a.n 1
24.h odd 2 1 45.4.a.c 1
40.e odd 2 1 1200.4.a.t 1
40.f even 2 1 75.4.a.b 1
40.i odd 4 2 75.4.b.b 2
40.k even 4 2 1200.4.f.b 2
56.h odd 2 1 735.4.a.e 1
72.j odd 6 2 405.4.e.i 2
72.n even 6 2 405.4.e.g 2
88.b odd 2 1 1815.4.a.e 1
120.i odd 2 1 225.4.a.f 1
120.w even 4 2 225.4.b.e 2
168.i even 2 1 2205.4.a.l 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.a 1 8.b even 2 1
45.4.a.c 1 24.h odd 2 1
75.4.a.b 1 40.f even 2 1
75.4.b.b 2 40.i odd 4 2
225.4.a.f 1 120.i odd 2 1
225.4.b.e 2 120.w even 4 2
240.4.a.e 1 8.d odd 2 1
405.4.e.g 2 72.n even 6 2
405.4.e.i 2 72.j odd 6 2
720.4.a.n 1 24.f even 2 1
735.4.a.e 1 56.h odd 2 1
960.4.a.b 1 1.a even 1 1 trivial
960.4.a.ba 1 4.b odd 2 1
1200.4.a.t 1 40.e odd 2 1
1200.4.f.b 2 40.k even 4 2
1815.4.a.e 1 88.b odd 2 1
2205.4.a.l 1 168.i even 2 1

## Hecke kernels

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

 $$T_{7} + 24$$ $$T_{11} + 52$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$3 + T$$
$5$ $$5 + T$$
$7$ $$24 + T$$
$11$ $$52 + T$$
$13$ $$22 + T$$
$17$ $$14 + T$$
$19$ $$-20 + T$$
$23$ $$168 + T$$
$29$ $$230 + T$$
$31$ $$288 + T$$
$37$ $$-34 + T$$
$41$ $$-122 + T$$
$43$ $$-188 + T$$
$47$ $$-256 + T$$
$53$ $$-338 + T$$
$59$ $$100 + T$$
$61$ $$742 + T$$
$67$ $$-84 + T$$
$71$ $$328 + T$$
$73$ $$38 + T$$
$79$ $$240 + T$$
$83$ $$1212 + T$$
$89$ $$-330 + T$$
$97$ $$-866 + T$$