# Properties

 Label 960.2.v.f Level $960$ Weight $2$ Character orbit 960.v Analytic conductor $7.666$ Analytic rank $0$ Dimension $4$ 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.v (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.66563859404$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} + \beta_1) q^{3} + (2 \beta_1 + 1) q^{5} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{7} + ( - 2 \beta_{2} + 1) q^{9}+O(q^{10})$$ q + (-b3 + b1) * q^3 + (2*b1 + 1) * q^5 + (b3 + b2 - b1 - 1) * q^7 + (-2*b2 + 1) * q^9 $$q + ( - \beta_{3} + \beta_1) q^{3} + (2 \beta_1 + 1) q^{5} + (\beta_{3} + \beta_{2} - \beta_1 - 1) q^{7} + ( - 2 \beta_{2} + 1) q^{9} + ( - 2 \beta_{2} - 2 \beta_1) q^{11} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{13} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{15} + (2 \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{17} + ( - 2 \beta_{2} - 2 \beta_1) q^{19} + (2 \beta_{2} - 3 \beta_1 - 1) q^{21} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 + 3) q^{23} + (4 \beta_1 - 3) q^{25} + (\beta_{3} + 5 \beta_1) q^{27} + (4 \beta_{3} + 2) q^{29} - 4 \beta_{3} q^{31} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{33} + ( - \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 1) q^{35} + ( - 2 \beta_{3} - 2 \beta_{2} + 3 \beta_1 + 3) q^{37} + (3 \beta_{3} + \beta_{2} + 3 \beta_1 - 5) q^{39} + 4 \beta_{2} q^{41} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{43} + (4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{45} + (\beta_{3} - \beta_{2} - 5 \beta_1 + 5) q^{47} + ( - 4 \beta_{2} - \beta_1) q^{49} + (\beta_{3} + 3 \beta_{2} + 5 \beta_1 - 3) q^{51} + (3 \beta_1 + 3) q^{53} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{55} + (2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{57} + 4 q^{59} + ( - 4 \beta_{3} - 6) q^{61} + ( - \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 3) q^{63} + (6 \beta_{3} + 2 \beta_{2} - \beta_1 - 3) q^{65} + (5 \beta_{3} + 5 \beta_{2} - 3 \beta_1 - 3) q^{67} + ( - 2 \beta_{3} - 4 \beta_{2} + 5 \beta_1 - 1) q^{69} + (6 \beta_{2} - 2 \beta_1) q^{71} + (4 \beta_{3} - 4 \beta_{2} - \beta_1 + 1) q^{73} + (3 \beta_{3} - 4 \beta_{2} - 3 \beta_1 - 4) q^{75} + ( - 2 \beta_1 + 2) q^{77} + ( - 2 \beta_{2} - 2 \beta_1) q^{79} + ( - 4 \beta_{2} - 7) q^{81} + (3 \beta_{3} + 3 \beta_{2} + \beta_1 + 1) q^{83} + (6 \beta_{3} + 2 \beta_{2} + \beta_1 + 3) q^{85} + ( - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 8) q^{87} + ( - 4 \beta_{3} + 10) q^{89} + ( - 6 \beta_{3} + 10) q^{91} + ( - 4 \beta_{2} + 8) q^{93} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 4) q^{95} + (\beta_1 + 1) q^{97} + ( - 4 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 8) q^{99}+O(q^{100})$$ q + (-b3 + b1) * q^3 + (2*b1 + 1) * q^5 + (b3 + b2 - b1 - 1) * q^7 + (-2*b2 + 1) * q^9 + (-2*b2 - 2*b1) * q^11 + (2*b3 - 2*b2 + b1 - 1) * q^13 + (-b3 - 2*b2 + b1 - 2) * q^15 + (2*b3 - 2*b2 - b1 + 1) * q^17 + (-2*b2 - 2*b1) * q^19 + (2*b2 - 3*b1 - 1) * q^21 + (-b3 - b2 + 3*b1 + 3) * q^23 + (4*b1 - 3) * q^25 + (b3 + 5*b1) * q^27 + (4*b3 + 2) * q^29 - 4*b3 * q^31 + (2*b3 + 2*b2 + 4*b1 + 2) * q^33 + (-b3 + 3*b2 - 3*b1 + 1) * q^35 + (-2*b3 - 2*b2 + 3*b1 + 3) * q^37 + (3*b3 + b2 + 3*b1 - 5) * q^39 + 4*b2 * q^41 + (-b3 + b2 + b1 - 1) * q^43 + (4*b3 - 2*b2 + 2*b1 + 1) * q^45 + (b3 - b2 - 5*b1 + 5) * q^47 + (-4*b2 - b1) * q^49 + (b3 + 3*b2 + 5*b1 - 3) * q^51 + (3*b1 + 3) * q^53 + (4*b3 - 2*b2 - 2*b1 + 4) * q^55 + (2*b3 + 2*b2 + 4*b1 + 2) * q^57 + 4 * q^59 + (-4*b3 - 6) * q^61 + (-b3 + 3*b2 - 5*b1 + 3) * q^63 + (6*b3 + 2*b2 - b1 - 3) * q^65 + (5*b3 + 5*b2 - 3*b1 - 3) * q^67 + (-2*b3 - 4*b2 + 5*b1 - 1) * q^69 + (6*b2 - 2*b1) * q^71 + (4*b3 - 4*b2 - b1 + 1) * q^73 + (3*b3 - 4*b2 - 3*b1 - 4) * q^75 + (-2*b1 + 2) * q^77 + (-2*b2 - 2*b1) * q^79 + (-4*b2 - 7) * q^81 + (3*b3 + 3*b2 + b1 + 1) * q^83 + (6*b3 + 2*b2 + b1 + 3) * q^85 + (-2*b3 + 4*b2 + 2*b1 - 8) * q^87 + (-4*b3 + 10) * q^89 + (-6*b3 + 10) * q^91 + (-4*b2 + 8) * q^93 + (4*b3 - 2*b2 - 2*b1 + 4) * q^95 + (b1 + 1) * q^97 + (-4*b3 - 2*b2 - 2*b1 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q + 4 * q^5 - 4 * q^7 + 4 * q^9 $$4 q + 4 q^{5} - 4 q^{7} + 4 q^{9} - 4 q^{13} - 8 q^{15} + 4 q^{17} - 4 q^{21} + 12 q^{23} - 12 q^{25} + 8 q^{29} + 8 q^{33} + 4 q^{35} + 12 q^{37} - 20 q^{39} - 4 q^{43} + 4 q^{45} + 20 q^{47} - 12 q^{51} + 12 q^{53} + 16 q^{55} + 8 q^{57} + 16 q^{59} - 24 q^{61} + 12 q^{63} - 12 q^{65} - 12 q^{67} - 4 q^{69} + 4 q^{73} - 16 q^{75} + 8 q^{77} - 28 q^{81} + 4 q^{83} + 12 q^{85} - 32 q^{87} + 40 q^{89} + 40 q^{91} + 32 q^{93} + 16 q^{95} + 4 q^{97} - 32 q^{99}+O(q^{100})$$ 4 * q + 4 * q^5 - 4 * q^7 + 4 * q^9 - 4 * q^13 - 8 * q^15 + 4 * q^17 - 4 * q^21 + 12 * q^23 - 12 * q^25 + 8 * q^29 + 8 * q^33 + 4 * q^35 + 12 * q^37 - 20 * q^39 - 4 * q^43 + 4 * q^45 + 20 * q^47 - 12 * q^51 + 12 * q^53 + 16 * q^55 + 8 * q^57 + 16 * q^59 - 24 * q^61 + 12 * q^63 - 12 * q^65 - 12 * q^67 - 4 * q^69 + 4 * q^73 - 16 * q^75 + 8 * q^77 - 28 * q^81 + 4 * q^83 + 12 * q^85 - 32 * q^87 + 40 * q^89 + 40 * q^91 + 32 * q^93 + 16 * q^95 + 4 * q^97 - 32 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
0 −1.41421 + 1.00000i 0 1.00000 + 2.00000i 0 0.414214 + 0.414214i 0 1.00000 2.82843i 0
257.2 0 1.41421 + 1.00000i 0 1.00000 + 2.00000i 0 −2.41421 2.41421i 0 1.00000 + 2.82843i 0
833.1 0 −1.41421 1.00000i 0 1.00000 2.00000i 0 0.414214 0.414214i 0 1.00000 + 2.82843i 0
833.2 0 1.41421 1.00000i 0 1.00000 2.00000i 0 −2.41421 + 2.41421i 0 1.00000 2.82843i 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
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.v.f 4
3.b odd 2 1 960.2.v.a 4
4.b odd 2 1 960.2.v.g 4
5.c odd 4 1 960.2.v.a 4
8.b even 2 1 120.2.r.b 4
8.d odd 2 1 240.2.v.c 4
12.b even 2 1 960.2.v.i 4
15.e even 4 1 inner 960.2.v.f 4
20.e even 4 1 960.2.v.i 4
24.f even 2 1 240.2.v.a 4
24.h odd 2 1 120.2.r.c yes 4
40.e odd 2 1 1200.2.v.d 4
40.f even 2 1 600.2.r.c 4
40.i odd 4 1 120.2.r.c yes 4
40.i odd 4 1 600.2.r.b 4
40.k even 4 1 240.2.v.a 4
40.k even 4 1 1200.2.v.j 4
60.l odd 4 1 960.2.v.g 4
120.i odd 2 1 600.2.r.b 4
120.m even 2 1 1200.2.v.j 4
120.q odd 4 1 240.2.v.c 4
120.q odd 4 1 1200.2.v.d 4
120.w even 4 1 120.2.r.b 4
120.w even 4 1 600.2.r.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.r.b 4 8.b even 2 1
120.2.r.b 4 120.w even 4 1
120.2.r.c yes 4 24.h odd 2 1
120.2.r.c yes 4 40.i odd 4 1
240.2.v.a 4 24.f even 2 1
240.2.v.a 4 40.k even 4 1
240.2.v.c 4 8.d odd 2 1
240.2.v.c 4 120.q odd 4 1
600.2.r.b 4 40.i odd 4 1
600.2.r.b 4 120.i odd 2 1
600.2.r.c 4 40.f even 2 1
600.2.r.c 4 120.w even 4 1
960.2.v.a 4 3.b odd 2 1
960.2.v.a 4 5.c odd 4 1
960.2.v.f 4 1.a even 1 1 trivial
960.2.v.f 4 15.e even 4 1 inner
960.2.v.g 4 4.b odd 2 1
960.2.v.g 4 60.l odd 4 1
960.2.v.i 4 12.b even 2 1
960.2.v.i 4 20.e even 4 1
1200.2.v.d 4 40.e odd 2 1
1200.2.v.d 4 120.q odd 4 1
1200.2.v.j 4 40.k even 4 1
1200.2.v.j 4 120.m even 2 1

## Hecke kernels

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

 $$T_{7}^{4} + 4T_{7}^{3} + 8T_{7}^{2} - 8T_{7} + 4$$ T7^4 + 4*T7^3 + 8*T7^2 - 8*T7 + 4 $$T_{11}^{4} + 24T_{11}^{2} + 16$$ T11^4 + 24*T11^2 + 16 $$T_{17}^{4} - 4T_{17}^{3} + 8T_{17}^{2} + 56T_{17} + 196$$ T17^4 - 4*T17^3 + 8*T17^2 + 56*T17 + 196

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 2T^{2} + 9$$
$5$ $$(T^{2} - 2 T + 5)^{2}$$
$7$ $$T^{4} + 4 T^{3} + 8 T^{2} - 8 T + 4$$
$11$ $$T^{4} + 24T^{2} + 16$$
$13$ $$T^{4} + 4 T^{3} + 8 T^{2} - 56 T + 196$$
$17$ $$T^{4} - 4 T^{3} + 8 T^{2} + 56 T + 196$$
$19$ $$T^{4} + 24T^{2} + 16$$
$23$ $$T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 196$$
$29$ $$(T^{2} - 4 T - 28)^{2}$$
$31$ $$(T^{2} - 32)^{2}$$
$37$ $$T^{4} - 12 T^{3} + 72 T^{2} - 24 T + 4$$
$41$ $$(T^{2} + 32)^{2}$$
$43$ $$T^{4} + 4 T^{3} + 8 T^{2} - 8 T + 4$$
$47$ $$T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 2116$$
$53$ $$(T^{2} - 6 T + 18)^{2}$$
$59$ $$(T - 4)^{4}$$
$61$ $$(T^{2} + 12 T + 4)^{2}$$
$67$ $$T^{4} + 12 T^{3} + 72 T^{2} + \cdots + 6724$$
$71$ $$T^{4} + 152T^{2} + 4624$$
$73$ $$T^{4} - 4 T^{3} + 8 T^{2} + 248 T + 3844$$
$79$ $$T^{4} + 24T^{2} + 16$$
$83$ $$T^{4} - 4 T^{3} + 8 T^{2} + 136 T + 1156$$
$89$ $$(T^{2} - 20 T + 68)^{2}$$
$97$ $$(T^{2} - 2 T + 2)^{2}$$