# Properties

 Label 960.2.a.f Level $960$ Weight $2$ Character orbit 960.a Self dual yes Analytic conductor $7.666$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$7.66563859404$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 480) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} + q^{9}+O(q^{10})$$ q - q^3 + q^5 + q^9 $$q - q^{3} + q^{5} + q^{9} - 4 q^{11} - 2 q^{13} - q^{15} - 2 q^{17} - 8 q^{19} + 4 q^{23} + q^{25} - q^{27} + 6 q^{29} + 4 q^{33} - 2 q^{37} + 2 q^{39} - 6 q^{41} - 4 q^{43} + q^{45} - 12 q^{47} - 7 q^{49} + 2 q^{51} + 6 q^{53} - 4 q^{55} + 8 q^{57} - 12 q^{59} - 14 q^{61} - 2 q^{65} + 12 q^{67} - 4 q^{69} + 2 q^{73} - q^{75} - 8 q^{79} + q^{81} + 4 q^{83} - 2 q^{85} - 6 q^{87} + 2 q^{89} - 8 q^{95} - 14 q^{97} - 4 q^{99}+O(q^{100})$$ q - q^3 + q^5 + q^9 - 4 * q^11 - 2 * q^13 - q^15 - 2 * q^17 - 8 * q^19 + 4 * q^23 + q^25 - q^27 + 6 * q^29 + 4 * q^33 - 2 * q^37 + 2 * q^39 - 6 * q^41 - 4 * q^43 + q^45 - 12 * q^47 - 7 * q^49 + 2 * q^51 + 6 * q^53 - 4 * q^55 + 8 * q^57 - 12 * q^59 - 14 * q^61 - 2 * q^65 + 12 * q^67 - 4 * q^69 + 2 * q^73 - q^75 - 8 * q^79 + q^81 + 4 * q^83 - 2 * q^85 - 6 * q^87 + 2 * q^89 - 8 * q^95 - 14 * q^97 - 4 * 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.

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 −1.00000 0 1.00000 0 0 0 1.00000 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 960.2.a.f 1
3.b odd 2 1 2880.2.a.j 1
4.b odd 2 1 960.2.a.o 1
5.b even 2 1 4800.2.a.ca 1
5.c odd 4 2 4800.2.f.j 2
8.b even 2 1 480.2.a.e yes 1
8.d odd 2 1 480.2.a.b 1
12.b even 2 1 2880.2.a.i 1
16.e even 4 2 3840.2.k.k 2
16.f odd 4 2 3840.2.k.p 2
20.d odd 2 1 4800.2.a.u 1
20.e even 4 2 4800.2.f.ba 2
24.f even 2 1 1440.2.a.k 1
24.h odd 2 1 1440.2.a.j 1
40.e odd 2 1 2400.2.a.y 1
40.f even 2 1 2400.2.a.j 1
40.i odd 4 2 2400.2.f.n 2
40.k even 4 2 2400.2.f.e 2
120.i odd 2 1 7200.2.a.u 1
120.m even 2 1 7200.2.a.bg 1
120.q odd 4 2 7200.2.f.bb 2
120.w even 4 2 7200.2.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.b 1 8.d odd 2 1
480.2.a.e yes 1 8.b even 2 1
960.2.a.f 1 1.a even 1 1 trivial
960.2.a.o 1 4.b odd 2 1
1440.2.a.j 1 24.h odd 2 1
1440.2.a.k 1 24.f even 2 1
2400.2.a.j 1 40.f even 2 1
2400.2.a.y 1 40.e odd 2 1
2400.2.f.e 2 40.k even 4 2
2400.2.f.n 2 40.i odd 4 2
2880.2.a.i 1 12.b even 2 1
2880.2.a.j 1 3.b odd 2 1
3840.2.k.k 2 16.e even 4 2
3840.2.k.p 2 16.f odd 4 2
4800.2.a.u 1 20.d odd 2 1
4800.2.a.ca 1 5.b even 2 1
4800.2.f.j 2 5.c odd 4 2
4800.2.f.ba 2 20.e even 4 2
7200.2.a.u 1 120.i odd 2 1
7200.2.a.bg 1 120.m even 2 1
7200.2.f.b 2 120.w even 4 2
7200.2.f.bb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(960))$$:

 $$T_{7}$$ T7 $$T_{11} + 4$$ T11 + 4 $$T_{13} + 2$$ T13 + 2 $$T_{19} + 8$$ T19 + 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T + 2$$
$17$ $$T + 2$$
$19$ $$T + 8$$
$23$ $$T - 4$$
$29$ $$T - 6$$
$31$ $$T$$
$37$ $$T + 2$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T + 12$$
$53$ $$T - 6$$
$59$ $$T + 12$$
$61$ $$T + 14$$
$67$ $$T - 12$$
$71$ $$T$$
$73$ $$T - 2$$
$79$ $$T + 8$$
$83$ $$T - 4$$
$89$ $$T - 2$$
$97$ $$T + 14$$