# Properties

 Label 960.2.f.a Level $960$ Weight $2$ Character orbit 960.f Analytic conductor $7.666$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$960 = 2^{6} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 960.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.66563859404$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - i q^{3} + (i - 2) q^{5} + 2 i q^{7} - q^{9} +O(q^{10})$$ q - i * q^3 + (i - 2) * q^5 + 2*i * q^7 - q^9 $$q - i q^{3} + (i - 2) q^{5} + 2 i q^{7} - q^{9} - 2 q^{11} - 2 i q^{13} + (2 i + 1) q^{15} - 6 i q^{17} + 8 q^{19} + 2 q^{21} - 4 i q^{23} + ( - 4 i + 3) q^{25} + i q^{27} + 8 q^{29} + 2 i q^{33} + ( - 4 i - 2) q^{35} - 10 i q^{37} - 2 q^{39} + 2 q^{41} + 12 i q^{43} + ( - i + 2) q^{45} + 3 q^{49} - 6 q^{51} - 10 i q^{53} + ( - 2 i + 4) q^{55} - 8 i q^{57} - 6 q^{59} - 2 q^{61} - 2 i q^{63} + (4 i + 2) q^{65} - 8 i q^{67} - 4 q^{69} - 4 q^{71} + 4 i q^{73} + ( - 3 i - 4) q^{75} - 4 i q^{77} + 8 q^{79} + q^{81} - 4 i q^{83} + (12 i + 6) q^{85} - 8 i q^{87} - 6 q^{89} + 4 q^{91} + (8 i - 16) q^{95} - 8 i q^{97} + 2 q^{99} +O(q^{100})$$ q - i * q^3 + (i - 2) * q^5 + 2*i * q^7 - q^9 - 2 * q^11 - 2*i * q^13 + (2*i + 1) * q^15 - 6*i * q^17 + 8 * q^19 + 2 * q^21 - 4*i * q^23 + (-4*i + 3) * q^25 + i * q^27 + 8 * q^29 + 2*i * q^33 + (-4*i - 2) * q^35 - 10*i * q^37 - 2 * q^39 + 2 * q^41 + 12*i * q^43 + (-i + 2) * q^45 + 3 * q^49 - 6 * q^51 - 10*i * q^53 + (-2*i + 4) * q^55 - 8*i * q^57 - 6 * q^59 - 2 * q^61 - 2*i * q^63 + (4*i + 2) * q^65 - 8*i * q^67 - 4 * q^69 - 4 * q^71 + 4*i * q^73 + (-3*i - 4) * q^75 - 4*i * q^77 + 8 * q^79 + q^81 - 4*i * q^83 + (12*i + 6) * q^85 - 8*i * q^87 - 6 * q^89 + 4 * q^91 + (8*i - 16) * q^95 - 8*i * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^5 - 2 * q^9 $$2 q - 4 q^{5} - 2 q^{9} - 4 q^{11} + 2 q^{15} + 16 q^{19} + 4 q^{21} + 6 q^{25} + 16 q^{29} - 4 q^{35} - 4 q^{39} + 4 q^{41} + 4 q^{45} + 6 q^{49} - 12 q^{51} + 8 q^{55} - 12 q^{59} - 4 q^{61} + 4 q^{65} - 8 q^{69} - 8 q^{71} - 8 q^{75} + 16 q^{79} + 2 q^{81} + 12 q^{85} - 12 q^{89} + 8 q^{91} - 32 q^{95} + 4 q^{99}+O(q^{100})$$ 2 * q - 4 * q^5 - 2 * q^9 - 4 * q^11 + 2 * q^15 + 16 * q^19 + 4 * q^21 + 6 * q^25 + 16 * q^29 - 4 * q^35 - 4 * q^39 + 4 * q^41 + 4 * q^45 + 6 * q^49 - 12 * q^51 + 8 * q^55 - 12 * q^59 - 4 * q^61 + 4 * q^65 - 8 * q^69 - 8 * q^71 - 8 * q^75 + 16 * q^79 + 2 * q^81 + 12 * q^85 - 12 * q^89 + 8 * q^91 - 32 * q^95 + 4 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$$.

 $$n$$ $$511$$ $$577$$ $$641$$ $$901$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
769.1
 1.00000i − 1.00000i
0 1.00000i 0 −2.00000 + 1.00000i 0 2.00000i 0 −1.00000 0
769.2 0 1.00000i 0 −2.00000 1.00000i 0 2.00000i 0 −1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.f.a 2
3.b odd 2 1 2880.2.f.t 2
4.b odd 2 1 960.2.f.b 2
5.b even 2 1 inner 960.2.f.a 2
5.c odd 4 1 4800.2.a.k 1
5.c odd 4 1 4800.2.a.ch 1
8.b even 2 1 120.2.f.a 2
8.d odd 2 1 240.2.f.c 2
12.b even 2 1 2880.2.f.r 2
15.d odd 2 1 2880.2.f.t 2
16.e even 4 1 3840.2.d.d 2
16.e even 4 1 3840.2.d.ba 2
16.f odd 4 1 3840.2.d.m 2
16.f odd 4 1 3840.2.d.v 2
20.d odd 2 1 960.2.f.b 2
20.e even 4 1 4800.2.a.n 1
20.e even 4 1 4800.2.a.ci 1
24.f even 2 1 720.2.f.b 2
24.h odd 2 1 360.2.f.a 2
40.e odd 2 1 240.2.f.c 2
40.f even 2 1 120.2.f.a 2
40.i odd 4 1 600.2.a.d 1
40.i odd 4 1 600.2.a.g 1
40.k even 4 1 1200.2.a.h 1
40.k even 4 1 1200.2.a.l 1
60.h even 2 1 2880.2.f.r 2
80.k odd 4 1 3840.2.d.m 2
80.k odd 4 1 3840.2.d.v 2
80.q even 4 1 3840.2.d.d 2
80.q even 4 1 3840.2.d.ba 2
120.i odd 2 1 360.2.f.a 2
120.m even 2 1 720.2.f.b 2
120.q odd 4 1 3600.2.a.n 1
120.q odd 4 1 3600.2.a.bi 1
120.w even 4 1 1800.2.a.g 1
120.w even 4 1 1800.2.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.f.a 2 8.b even 2 1
120.2.f.a 2 40.f even 2 1
240.2.f.c 2 8.d odd 2 1
240.2.f.c 2 40.e odd 2 1
360.2.f.a 2 24.h odd 2 1
360.2.f.a 2 120.i odd 2 1
600.2.a.d 1 40.i odd 4 1
600.2.a.g 1 40.i odd 4 1
720.2.f.b 2 24.f even 2 1
720.2.f.b 2 120.m even 2 1
960.2.f.a 2 1.a even 1 1 trivial
960.2.f.a 2 5.b even 2 1 inner
960.2.f.b 2 4.b odd 2 1
960.2.f.b 2 20.d odd 2 1
1200.2.a.h 1 40.k even 4 1
1200.2.a.l 1 40.k even 4 1
1800.2.a.g 1 120.w even 4 1
1800.2.a.q 1 120.w even 4 1
2880.2.f.r 2 12.b even 2 1
2880.2.f.r 2 60.h even 2 1
2880.2.f.t 2 3.b odd 2 1
2880.2.f.t 2 15.d odd 2 1
3600.2.a.n 1 120.q odd 4 1
3600.2.a.bi 1 120.q odd 4 1
3840.2.d.d 2 16.e even 4 1
3840.2.d.d 2 80.q even 4 1
3840.2.d.m 2 16.f odd 4 1
3840.2.d.m 2 80.k odd 4 1
3840.2.d.v 2 16.f odd 4 1
3840.2.d.v 2 80.k odd 4 1
3840.2.d.ba 2 16.e even 4 1
3840.2.d.ba 2 80.q even 4 1
4800.2.a.k 1 5.c odd 4 1
4800.2.a.n 1 20.e even 4 1
4800.2.a.ch 1 5.c odd 4 1
4800.2.a.ci 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(960, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 8)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 8)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 144$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 100$$
$59$ $$(T + 6)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 4)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 64$$