# Properties

 Label 960.2.f.i 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 30) 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 $$q - i q^{3} + ( - i + 2) q^{5} + 2 i q^{7} - q^{9} + 2 q^{11} - 6 i q^{13} + ( - 2 i - 1) q^{15} - 2 i q^{17} + 2 q^{21} + 4 i q^{23} + ( - 4 i + 3) q^{25} + i q^{27} + 8 q^{31} - 2 i q^{33} + (4 i + 2) q^{35} + 2 i q^{37} - 6 q^{39} + 2 q^{41} - 4 i q^{43} + (i - 2) q^{45} - 8 i q^{47} + 3 q^{49} - 2 q^{51} - 6 i q^{53} + ( - 2 i + 4) q^{55} - 10 q^{59} - 2 q^{61} - 2 i q^{63} + ( - 12 i - 6) q^{65} + 8 i q^{67} + 4 q^{69} - 12 q^{71} - 4 i q^{73} + ( - 3 i - 4) q^{75} + 4 i q^{77} + q^{81} - 4 i q^{83} + ( - 4 i - 2) q^{85} + 10 q^{89} + 12 q^{91} - 8 i q^{93} + 8 i q^{97} - 2 q^{99} +O(q^{100})$$ q - i * q^3 + (-i + 2) * q^5 + 2*i * q^7 - q^9 + 2 * q^11 - 6*i * q^13 + (-2*i - 1) * q^15 - 2*i * q^17 + 2 * q^21 + 4*i * q^23 + (-4*i + 3) * q^25 + i * q^27 + 8 * q^31 - 2*i * q^33 + (4*i + 2) * q^35 + 2*i * q^37 - 6 * q^39 + 2 * q^41 - 4*i * q^43 + (i - 2) * q^45 - 8*i * q^47 + 3 * q^49 - 2 * q^51 - 6*i * q^53 + (-2*i + 4) * q^55 - 10 * q^59 - 2 * q^61 - 2*i * q^63 + (-12*i - 6) * q^65 + 8*i * q^67 + 4 * q^69 - 12 * q^71 - 4*i * q^73 + (-3*i - 4) * q^75 + 4*i * q^77 + q^81 - 4*i * q^83 + (-4*i - 2) * q^85 + 10 * q^89 + 12 * q^91 - 8*i * q^93 + 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 $$2 q + 4 q^{5} - 2 q^{9} + 4 q^{11} - 2 q^{15} + 4 q^{21} + 6 q^{25} + 16 q^{31} + 4 q^{35} - 12 q^{39} + 4 q^{41} - 4 q^{45} + 6 q^{49} - 4 q^{51} + 8 q^{55} - 20 q^{59} - 4 q^{61} - 12 q^{65} + 8 q^{69} - 24 q^{71} - 8 q^{75} + 2 q^{81} - 4 q^{85} + 20 q^{89} + 24 q^{91} - 4 q^{99}+O(q^{100})$$ 2 * q + 4 * q^5 - 2 * q^9 + 4 * q^11 - 2 * q^15 + 4 * q^21 + 6 * q^25 + 16 * q^31 + 4 * q^35 - 12 * q^39 + 4 * q^41 - 4 * q^45 + 6 * q^49 - 4 * q^51 + 8 * q^55 - 20 * q^59 - 4 * q^61 - 12 * q^65 + 8 * q^69 - 24 * q^71 - 8 * q^75 + 2 * q^81 - 4 * q^85 + 20 * q^89 + 24 * q^91 - 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.i 2
3.b odd 2 1 2880.2.f.c 2
4.b odd 2 1 960.2.f.h 2
5.b even 2 1 inner 960.2.f.i 2
5.c odd 4 1 4800.2.a.m 1
5.c odd 4 1 4800.2.a.cj 1
8.b even 2 1 240.2.f.a 2
8.d odd 2 1 30.2.c.a 2
12.b even 2 1 2880.2.f.e 2
15.d odd 2 1 2880.2.f.c 2
16.e even 4 1 3840.2.d.j 2
16.e even 4 1 3840.2.d.x 2
16.f odd 4 1 3840.2.d.g 2
16.f odd 4 1 3840.2.d.y 2
20.d odd 2 1 960.2.f.h 2
20.e even 4 1 4800.2.a.l 1
20.e even 4 1 4800.2.a.cg 1
24.f even 2 1 90.2.c.a 2
24.h odd 2 1 720.2.f.f 2
40.e odd 2 1 30.2.c.a 2
40.f even 2 1 240.2.f.a 2
40.i odd 4 1 1200.2.a.g 1
40.i odd 4 1 1200.2.a.m 1
40.k even 4 1 150.2.a.a 1
40.k even 4 1 150.2.a.c 1
56.e even 2 1 1470.2.g.g 2
56.k odd 6 2 1470.2.n.h 4
56.m even 6 2 1470.2.n.a 4
60.h even 2 1 2880.2.f.e 2
72.l even 6 2 810.2.i.b 4
72.p odd 6 2 810.2.i.e 4
80.k odd 4 1 3840.2.d.g 2
80.k odd 4 1 3840.2.d.y 2
80.q even 4 1 3840.2.d.j 2
80.q even 4 1 3840.2.d.x 2
120.i odd 2 1 720.2.f.f 2
120.m even 2 1 90.2.c.a 2
120.q odd 4 1 450.2.a.b 1
120.q odd 4 1 450.2.a.f 1
120.w even 4 1 3600.2.a.o 1
120.w even 4 1 3600.2.a.bg 1
280.n even 2 1 1470.2.g.g 2
280.y odd 4 1 7350.2.a.bg 1
280.y odd 4 1 7350.2.a.cc 1
280.ba even 6 2 1470.2.n.a 4
280.bi odd 6 2 1470.2.n.h 4
360.z odd 6 2 810.2.i.e 4
360.bd even 6 2 810.2.i.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.c.a 2 8.d odd 2 1
30.2.c.a 2 40.e odd 2 1
90.2.c.a 2 24.f even 2 1
90.2.c.a 2 120.m even 2 1
150.2.a.a 1 40.k even 4 1
150.2.a.c 1 40.k even 4 1
240.2.f.a 2 8.b even 2 1
240.2.f.a 2 40.f even 2 1
450.2.a.b 1 120.q odd 4 1
450.2.a.f 1 120.q odd 4 1
720.2.f.f 2 24.h odd 2 1
720.2.f.f 2 120.i odd 2 1
810.2.i.b 4 72.l even 6 2
810.2.i.b 4 360.bd even 6 2
810.2.i.e 4 72.p odd 6 2
810.2.i.e 4 360.z odd 6 2
960.2.f.h 2 4.b odd 2 1
960.2.f.h 2 20.d odd 2 1
960.2.f.i 2 1.a even 1 1 trivial
960.2.f.i 2 5.b even 2 1 inner
1200.2.a.g 1 40.i odd 4 1
1200.2.a.m 1 40.i odd 4 1
1470.2.g.g 2 56.e even 2 1
1470.2.g.g 2 280.n even 2 1
1470.2.n.a 4 56.m even 6 2
1470.2.n.a 4 280.ba even 6 2
1470.2.n.h 4 56.k odd 6 2
1470.2.n.h 4 280.bi odd 6 2
2880.2.f.c 2 3.b odd 2 1
2880.2.f.c 2 15.d odd 2 1
2880.2.f.e 2 12.b even 2 1
2880.2.f.e 2 60.h even 2 1
3600.2.a.o 1 120.w even 4 1
3600.2.a.bg 1 120.w even 4 1
3840.2.d.g 2 16.f odd 4 1
3840.2.d.g 2 80.k odd 4 1
3840.2.d.j 2 16.e even 4 1
3840.2.d.j 2 80.q even 4 1
3840.2.d.x 2 16.e even 4 1
3840.2.d.x 2 80.q even 4 1
3840.2.d.y 2 16.f odd 4 1
3840.2.d.y 2 80.k odd 4 1
4800.2.a.l 1 20.e even 4 1
4800.2.a.m 1 5.c odd 4 1
4800.2.a.cg 1 20.e even 4 1
4800.2.a.cj 1 5.c odd 4 1
7350.2.a.bg 1 280.y odd 4 1
7350.2.a.cc 1 280.y odd 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 $$T_{13}^{2} + 36$$ T13^2 + 36 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 2)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$T^{2}$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 64$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 64$$