# Properties

 Label 960.2.f.k Level $960$ Weight $2$ Character orbit 960.f Analytic conductor $7.666$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 480) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - \beta_{2} q^{5} + 2 \beta_1 q^{7} - q^{9}+O(q^{10})$$ q + b1 * q^3 - b2 * q^5 + 2*b1 * q^7 - q^9 $$q + \beta_1 q^{3} - \beta_{2} q^{5} + 2 \beta_1 q^{7} - q^{9} + 2 \beta_{3} q^{11} - 2 \beta_{2} q^{13} + \beta_{3} q^{15} + 2 \beta_{2} q^{17} - 2 q^{21} - 4 \beta_1 q^{23} - 5 q^{25} - \beta_1 q^{27} + 4 q^{29} + 4 \beta_{3} q^{31} + 2 \beta_{2} q^{33} + 2 \beta_{3} q^{35} - 2 \beta_{2} q^{37} + 2 \beta_{3} q^{39} + 10 q^{41} - 4 \beta_1 q^{43} + \beta_{2} q^{45} + 8 \beta_1 q^{47} + 3 q^{49} - 2 \beta_{3} q^{51} + 2 \beta_{2} q^{53} - 10 \beta_1 q^{55} + 6 \beta_{3} q^{59} - 10 q^{61} - 2 \beta_1 q^{63} - 10 q^{65} + 8 \beta_1 q^{67} + 4 q^{69} - 4 \beta_{3} q^{71} - 4 \beta_{2} q^{73} - 5 \beta_1 q^{75} + 4 \beta_{2} q^{77} + 4 \beta_{3} q^{79} + q^{81} - 4 \beta_1 q^{83} + 10 q^{85} + 4 \beta_1 q^{87} - 6 q^{89} + 4 \beta_{3} q^{91} + 4 \beta_{2} q^{93} - 8 \beta_{2} q^{97} - 2 \beta_{3} q^{99}+O(q^{100})$$ q + b1 * q^3 - b2 * q^5 + 2*b1 * q^7 - q^9 + 2*b3 * q^11 - 2*b2 * q^13 + b3 * q^15 + 2*b2 * q^17 - 2 * q^21 - 4*b1 * q^23 - 5 * q^25 - b1 * q^27 + 4 * q^29 + 4*b3 * q^31 + 2*b2 * q^33 + 2*b3 * q^35 - 2*b2 * q^37 + 2*b3 * q^39 + 10 * q^41 - 4*b1 * q^43 + b2 * q^45 + 8*b1 * q^47 + 3 * q^49 - 2*b3 * q^51 + 2*b2 * q^53 - 10*b1 * q^55 + 6*b3 * q^59 - 10 * q^61 - 2*b1 * q^63 - 10 * q^65 + 8*b1 * q^67 + 4 * q^69 - 4*b3 * q^71 - 4*b2 * q^73 - 5*b1 * q^75 + 4*b2 * q^77 + 4*b3 * q^79 + q^81 - 4*b1 * q^83 + 10 * q^85 + 4*b1 * q^87 - 6 * q^89 + 4*b3 * q^91 + 4*b2 * q^93 - 8*b2 * q^97 - 2*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 8 q^{21} - 20 q^{25} + 16 q^{29} + 40 q^{41} + 12 q^{49} - 40 q^{61} - 40 q^{65} + 16 q^{69} + 4 q^{81} + 40 q^{85} - 24 q^{89}+O(q^{100})$$ 4 * q - 4 * q^9 - 8 * q^21 - 20 * q^25 + 16 * q^29 + 40 * q^41 + 12 * q^49 - 40 * q^61 - 40 * q^65 + 16 * q^69 + 4 * q^81 + 40 * q^85 - 24 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v $$\beta_{3}$$ $$=$$ $$2\nu^{2} + 3$$ 2*v^2 + 3
 $$\nu$$ $$=$$ $$( \beta_{2} - \beta_1 ) / 2$$ (b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 3 ) / 2$$ (b3 - 3) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2\beta_1$$ -b2 + 2*b1

## 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
 1.61803i − 0.618034i 0.618034i − 1.61803i
0 1.00000i 0 2.23607i 0 2.00000i 0 −1.00000 0
769.2 0 1.00000i 0 2.23607i 0 2.00000i 0 −1.00000 0
769.3 0 1.00000i 0 2.23607i 0 2.00000i 0 −1.00000 0
769.4 0 1.00000i 0 2.23607i 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
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 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.k 4
3.b odd 2 1 2880.2.f.v 4
4.b odd 2 1 inner 960.2.f.k 4
5.b even 2 1 inner 960.2.f.k 4
5.c odd 4 1 4800.2.a.cu 2
5.c odd 4 1 4800.2.a.cv 2
8.b even 2 1 480.2.f.e 4
8.d odd 2 1 480.2.f.e 4
12.b even 2 1 2880.2.f.v 4
15.d odd 2 1 2880.2.f.v 4
16.e even 4 1 3840.2.d.bg 4
16.e even 4 1 3840.2.d.bh 4
16.f odd 4 1 3840.2.d.bg 4
16.f odd 4 1 3840.2.d.bh 4
20.d odd 2 1 inner 960.2.f.k 4
20.e even 4 1 4800.2.a.cu 2
20.e even 4 1 4800.2.a.cv 2
24.f even 2 1 1440.2.f.h 4
24.h odd 2 1 1440.2.f.h 4
40.e odd 2 1 480.2.f.e 4
40.f even 2 1 480.2.f.e 4
40.i odd 4 1 2400.2.a.bi 2
40.i odd 4 1 2400.2.a.bj 2
40.k even 4 1 2400.2.a.bi 2
40.k even 4 1 2400.2.a.bj 2
60.h even 2 1 2880.2.f.v 4
80.k odd 4 1 3840.2.d.bg 4
80.k odd 4 1 3840.2.d.bh 4
80.q even 4 1 3840.2.d.bg 4
80.q even 4 1 3840.2.d.bh 4
120.i odd 2 1 1440.2.f.h 4
120.m even 2 1 1440.2.f.h 4
120.q odd 4 1 7200.2.a.cc 2
120.q odd 4 1 7200.2.a.cq 2
120.w even 4 1 7200.2.a.cc 2
120.w even 4 1 7200.2.a.cq 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.f.e 4 8.b even 2 1
480.2.f.e 4 8.d odd 2 1
480.2.f.e 4 40.e odd 2 1
480.2.f.e 4 40.f even 2 1
960.2.f.k 4 1.a even 1 1 trivial
960.2.f.k 4 4.b odd 2 1 inner
960.2.f.k 4 5.b even 2 1 inner
960.2.f.k 4 20.d odd 2 1 inner
1440.2.f.h 4 24.f even 2 1
1440.2.f.h 4 24.h odd 2 1
1440.2.f.h 4 120.i odd 2 1
1440.2.f.h 4 120.m even 2 1
2400.2.a.bi 2 40.i odd 4 1
2400.2.a.bi 2 40.k even 4 1
2400.2.a.bj 2 40.i odd 4 1
2400.2.a.bj 2 40.k even 4 1
2880.2.f.v 4 3.b odd 2 1
2880.2.f.v 4 12.b even 2 1
2880.2.f.v 4 15.d odd 2 1
2880.2.f.v 4 60.h even 2 1
3840.2.d.bg 4 16.e even 4 1
3840.2.d.bg 4 16.f odd 4 1
3840.2.d.bg 4 80.k odd 4 1
3840.2.d.bg 4 80.q even 4 1
3840.2.d.bh 4 16.e even 4 1
3840.2.d.bh 4 16.f odd 4 1
3840.2.d.bh 4 80.k odd 4 1
3840.2.d.bh 4 80.q even 4 1
4800.2.a.cu 2 5.c odd 4 1
4800.2.a.cu 2 20.e even 4 1
4800.2.a.cv 2 5.c odd 4 1
4800.2.a.cv 2 20.e even 4 1
7200.2.a.cc 2 120.q odd 4 1
7200.2.a.cc 2 120.w even 4 1
7200.2.a.cq 2 120.q odd 4 1
7200.2.a.cq 2 120.w 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 $$T_{11}^{2} - 20$$ T11^2 - 20 $$T_{13}^{2} + 20$$ T13^2 + 20 $$T_{19}$$ T19

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 1)^{2}$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} - 20)^{2}$$
$13$ $$(T^{2} + 20)^{2}$$
$17$ $$(T^{2} + 20)^{2}$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 16)^{2}$$
$29$ $$(T - 4)^{4}$$
$31$ $$(T^{2} - 80)^{2}$$
$37$ $$(T^{2} + 20)^{2}$$
$41$ $$(T - 10)^{4}$$
$43$ $$(T^{2} + 16)^{2}$$
$47$ $$(T^{2} + 64)^{2}$$
$53$ $$(T^{2} + 20)^{2}$$
$59$ $$(T^{2} - 180)^{2}$$
$61$ $$(T + 10)^{4}$$
$67$ $$(T^{2} + 64)^{2}$$
$71$ $$(T^{2} - 80)^{2}$$
$73$ $$(T^{2} + 80)^{2}$$
$79$ $$(T^{2} - 80)^{2}$$
$83$ $$(T^{2} + 16)^{2}$$
$89$ $$(T + 6)^{4}$$
$97$ $$(T^{2} + 320)^{2}$$