# Properties

 Label 960.2.d.b Level $960$ Weight $2$ Character orbit 960.d 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.d (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( - \beta + 1) q^{5} + \beta q^{7} + q^{9}+O(q^{10})$$ q - q^3 + (-b + 1) * q^5 + b * q^7 + q^9 $$q - q^{3} + ( - \beta + 1) q^{5} + \beta q^{7} + q^{9} + \beta q^{11} + 2 q^{13} + (\beta - 1) q^{15} + 2 \beta q^{17} - \beta q^{19} - \beta q^{21} + 3 \beta q^{23} + ( - 2 \beta - 3) q^{25} - q^{27} - 2 \beta q^{29} + 8 q^{31} - \beta q^{33} + (\beta + 4) q^{35} + 10 q^{37} - 2 q^{39} - 6 q^{41} + 4 q^{43} + ( - \beta + 1) q^{45} + 3 \beta q^{47} + 3 q^{49} - 2 \beta q^{51} + 10 q^{53} + (\beta + 4) q^{55} + \beta q^{57} - 3 \beta q^{59} + 4 \beta q^{61} + \beta q^{63} + ( - 2 \beta + 2) q^{65} - 12 q^{67} - 3 \beta q^{69} + (2 \beta + 3) q^{75} - 4 q^{77} + 16 q^{79} + q^{81} + 4 q^{83} + (2 \beta + 8) q^{85} + 2 \beta q^{87} + 10 q^{89} + 2 \beta q^{91} - 8 q^{93} + ( - \beta - 4) q^{95} + 8 \beta q^{97} + \beta q^{99} +O(q^{100})$$ q - q^3 + (-b + 1) * q^5 + b * q^7 + q^9 + b * q^11 + 2 * q^13 + (b - 1) * q^15 + 2*b * q^17 - b * q^19 - b * q^21 + 3*b * q^23 + (-2*b - 3) * q^25 - q^27 - 2*b * q^29 + 8 * q^31 - b * q^33 + (b + 4) * q^35 + 10 * q^37 - 2 * q^39 - 6 * q^41 + 4 * q^43 + (-b + 1) * q^45 + 3*b * q^47 + 3 * q^49 - 2*b * q^51 + 10 * q^53 + (b + 4) * q^55 + b * q^57 - 3*b * q^59 + 4*b * q^61 + b * q^63 + (-2*b + 2) * q^65 - 12 * q^67 - 3*b * q^69 + (2*b + 3) * q^75 - 4 * q^77 + 16 * q^79 + q^81 + 4 * q^83 + (2*b + 8) * q^85 + 2*b * q^87 + 10 * q^89 + 2*b * q^91 - 8 * q^93 + (-b - 4) * q^95 + 8*b * q^97 + b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{5} + 2 q^{9} + 4 q^{13} - 2 q^{15} - 6 q^{25} - 2 q^{27} + 16 q^{31} + 8 q^{35} + 20 q^{37} - 4 q^{39} - 12 q^{41} + 8 q^{43} + 2 q^{45} + 6 q^{49} + 20 q^{53} + 8 q^{55} + 4 q^{65} - 24 q^{67} + 6 q^{75} - 8 q^{77} + 32 q^{79} + 2 q^{81} + 8 q^{83} + 16 q^{85} + 20 q^{89} - 16 q^{93} - 8 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^9 + 4 * q^13 - 2 * q^15 - 6 * q^25 - 2 * q^27 + 16 * q^31 + 8 * q^35 + 20 * q^37 - 4 * q^39 - 12 * q^41 + 8 * q^43 + 2 * q^45 + 6 * q^49 + 20 * q^53 + 8 * q^55 + 4 * q^65 - 24 * q^67 + 6 * q^75 - 8 * q^77 + 32 * q^79 + 2 * q^81 + 8 * q^83 + 16 * q^85 + 20 * q^89 - 16 * q^93 - 8 * q^95

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 1.00000i − 1.00000i
0 −1.00000 0 1.00000 2.00000i 0 2.00000i 0 1.00000 0
289.2 0 −1.00000 0 1.00000 + 2.00000i 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
40.f 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.d.b yes 2
3.b odd 2 1 2880.2.d.b 2
4.b odd 2 1 960.2.d.d yes 2
5.b even 2 1 960.2.d.c yes 2
5.c odd 4 1 4800.2.k.b 2
5.c odd 4 1 4800.2.k.g 2
8.b even 2 1 960.2.d.c yes 2
8.d odd 2 1 960.2.d.a 2
12.b even 2 1 2880.2.d.a 2
15.d odd 2 1 2880.2.d.d 2
16.e even 4 1 3840.2.f.b 2
16.e even 4 1 3840.2.f.c 2
16.f odd 4 1 3840.2.f.a 2
16.f odd 4 1 3840.2.f.d 2
20.d odd 2 1 960.2.d.a 2
20.e even 4 1 4800.2.k.c 2
20.e even 4 1 4800.2.k.f 2
24.f even 2 1 2880.2.d.c 2
24.h odd 2 1 2880.2.d.d 2
40.e odd 2 1 960.2.d.d yes 2
40.f even 2 1 inner 960.2.d.b yes 2
40.i odd 4 1 4800.2.k.b 2
40.i odd 4 1 4800.2.k.g 2
40.k even 4 1 4800.2.k.c 2
40.k even 4 1 4800.2.k.f 2
60.h even 2 1 2880.2.d.c 2
80.k odd 4 1 3840.2.f.a 2
80.k odd 4 1 3840.2.f.d 2
80.q even 4 1 3840.2.f.b 2
80.q even 4 1 3840.2.f.c 2
120.i odd 2 1 2880.2.d.b 2
120.m even 2 1 2880.2.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.d.a 2 8.d odd 2 1
960.2.d.a 2 20.d odd 2 1
960.2.d.b yes 2 1.a even 1 1 trivial
960.2.d.b yes 2 40.f even 2 1 inner
960.2.d.c yes 2 5.b even 2 1
960.2.d.c yes 2 8.b even 2 1
960.2.d.d yes 2 4.b odd 2 1
960.2.d.d yes 2 40.e odd 2 1
2880.2.d.a 2 12.b even 2 1
2880.2.d.a 2 120.m even 2 1
2880.2.d.b 2 3.b odd 2 1
2880.2.d.b 2 120.i odd 2 1
2880.2.d.c 2 24.f even 2 1
2880.2.d.c 2 60.h even 2 1
2880.2.d.d 2 15.d odd 2 1
2880.2.d.d 2 24.h odd 2 1
3840.2.f.a 2 16.f odd 4 1
3840.2.f.a 2 80.k odd 4 1
3840.2.f.b 2 16.e even 4 1
3840.2.f.b 2 80.q even 4 1
3840.2.f.c 2 16.e even 4 1
3840.2.f.c 2 80.q even 4 1
3840.2.f.d 2 16.f odd 4 1
3840.2.f.d 2 80.k odd 4 1
4800.2.k.b 2 5.c odd 4 1
4800.2.k.b 2 40.i odd 4 1
4800.2.k.c 2 20.e even 4 1
4800.2.k.c 2 40.k even 4 1
4800.2.k.f 2 20.e even 4 1
4800.2.k.f 2 40.k even 4 1
4800.2.k.g 2 5.c odd 4 1
4800.2.k.g 2 40.i 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 $$T_{13} - 2$$ T13 - 2 $$T_{31} - 8$$ T31 - 8 $$T_{43} - 4$$ T43 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} - 2T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2} + 4$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} + 16$$
$19$ $$T^{2} + 4$$
$23$ $$T^{2} + 36$$
$29$ $$T^{2} + 16$$
$31$ $$(T - 8)^{2}$$
$37$ $$(T - 10)^{2}$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T - 4)^{2}$$
$47$ $$T^{2} + 36$$
$53$ $$(T - 10)^{2}$$
$59$ $$T^{2} + 36$$
$61$ $$T^{2} + 64$$
$67$ $$(T + 12)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$(T - 16)^{2}$$
$83$ $$(T - 4)^{2}$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 256$$