# Properties

 Label 960.2.o.d Level $960$ Weight $2$ Character orbit 960.o Analytic conductor $7.666$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$7.66563859404$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}\cdot 3$$ Twist minimal: no (minimal twist has level 240) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_1 q^{5} + (\beta_{4} - \beta_1 + 1) q^{9}+O(q^{10})$$ q + b2 * q^3 + b1 * q^5 + (b4 - b1 + 1) * q^9 $$q + \beta_{2} q^{3} + \beta_1 q^{5} + (\beta_{4} - \beta_1 + 1) q^{9} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{11} - \beta_{6} q^{13} + (\beta_{7} + \beta_{3}) q^{15} + ( - \beta_{4} - \beta_1) q^{17} - \beta_{3} q^{19} + ( - \beta_{7} - \beta_{5} + 2 \beta_{2}) q^{23} + ( - \beta_{6} + 1) q^{25} + ( - \beta_{7} - \beta_{5} + \beta_{2}) q^{27} + (\beta_{4} - \beta_1) q^{29} - \beta_{3} q^{31} + (\beta_{6} + 2 \beta_{4} + 2 \beta_1) q^{33} + \beta_{6} q^{37} + (\beta_{7} - \beta_{5} - \beta_{3}) q^{39} + ( - 2 \beta_{4} + 2 \beta_1) q^{41} + (\beta_{7} + \beta_{5} + 4 \beta_{2}) q^{43} + (\beta_{6} + \beta_1 + 4) q^{45} + ( - \beta_{7} - \beta_{5} + 2 \beta_{2}) q^{47} - 7 q^{49} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3}) q^{51} + ( - 3 \beta_{4} - 3 \beta_1) q^{53} + (\beta_{7} + \beta_{5} - 2 \beta_{3} + 4 \beta_{2}) q^{55} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{57} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{59} + 2 q^{61} + ( - 5 \beta_{4} + \beta_1) q^{65} + ( - \beta_{7} - \beta_{5} - 4 \beta_{2}) q^{67} + (3 \beta_{4} - 3 \beta_1 - 6) q^{69} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3}) q^{71} + 2 \beta_{6} q^{73} + (\beta_{7} - \beta_{5} - \beta_{3} + \beta_{2}) q^{75} + 3 \beta_{3} q^{79} + (2 \beta_{4} - 2 \beta_1 - 7) q^{81} + (\beta_{7} + \beta_{5} - 2 \beta_{2}) q^{83} + (\beta_{6} - 6) q^{85} + ( - \beta_{7} - \beta_{5}) q^{87} + (2 \beta_{4} - 2 \beta_1) q^{89} + ( - \beta_{6} + \beta_{4} + \beta_1) q^{93} + (2 \beta_{5} - \beta_{3} - 2 \beta_{2}) q^{95} + (\beta_{7} - \beta_{5} + 5 \beta_{3}) q^{99}+O(q^{100})$$ q + b2 * q^3 + b1 * q^5 + (b4 - b1 + 1) * q^9 + (b7 - b5 + b3) * q^11 - b6 * q^13 + (b7 + b3) * q^15 + (-b4 - b1) * q^17 - b3 * q^19 + (-b7 - b5 + 2*b2) * q^23 + (-b6 + 1) * q^25 + (-b7 - b5 + b2) * q^27 + (b4 - b1) * q^29 - b3 * q^31 + (b6 + 2*b4 + 2*b1) * q^33 + b6 * q^37 + (b7 - b5 - b3) * q^39 + (-2*b4 + 2*b1) * q^41 + (b7 + b5 + 4*b2) * q^43 + (b6 + b1 + 4) * q^45 + (-b7 - b5 + 2*b2) * q^47 - 7 * q^49 + (-b7 + b5 - 2*b3) * q^51 + (-3*b4 - 3*b1) * q^53 + (b7 + b5 - 2*b3 + 4*b2) * q^55 + (-b6 + b4 + b1) * q^57 + (b7 - b5 + b3) * q^59 + 2 * q^61 + (-5*b4 + b1) * q^65 + (-b7 - b5 - 4*b2) * q^67 + (3*b4 - 3*b1 - 6) * q^69 + (2*b7 - 2*b5 + 2*b3) * q^71 + 2*b6 * q^73 + (b7 - b5 - b3 + b2) * q^75 + 3*b3 * q^79 + (2*b4 - 2*b1 - 7) * q^81 + (b7 + b5 - 2*b2) * q^83 + (b6 - 6) * q^85 + (-b7 - b5) * q^87 + (2*b4 - 2*b1) * q^89 + (-b6 + b4 + b1) * q^93 + (2*b5 - b3 - 2*b2) * q^95 + (b7 - b5 + 5*b3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 8 q^{9}+O(q^{10})$$ 8 * q + 8 * q^9 $$8 q + 8 q^{9} + 8 q^{25} + 32 q^{45} - 56 q^{49} + 16 q^{61} - 48 q^{69} - 56 q^{81} - 48 q^{85}+O(q^{100})$$ 8 * q + 8 * q^9 + 8 * q^25 + 32 * q^45 - 56 * q^49 + 16 * q^61 - 48 * q^69 - 56 * q^81 - 48 * q^85

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$-\zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + 2\zeta_{24}^{2} + \zeta_{24}$$ -v^6 - v^5 - v^3 + 2*v^2 + v $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ v^6 - v^5 + v^3 + v $$\beta_{3}$$ $$=$$ $$4\zeta_{24}^{4} - 2$$ 4*v^4 - 2 $$\beta_{4}$$ $$=$$ $$-\zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}^{2} - \zeta_{24}$$ -v^6 + v^5 + v^3 + 2*v^2 - v $$\beta_{5}$$ $$=$$ $$2\zeta_{24}^{7} - 2\zeta_{24}^{6} - 2\zeta_{24}^{5} + 2\zeta_{24}^{4} - 1$$ 2*v^7 - 2*v^6 - 2*v^5 + 2*v^4 - 1 $$\beta_{6}$$ $$=$$ $$4\zeta_{24}^{7} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24}$$ 4*v^7 + 2*v^5 - 2*v^3 + 2*v $$\beta_{7}$$ $$=$$ $$-2\zeta_{24}^{7} - 2\zeta_{24}^{6} - 2\zeta_{24}^{4} + 2\zeta_{24}^{3} + 2\zeta_{24} + 1$$ -2*v^7 - 2*v^6 - 2*v^4 + 2*v^3 + 2*v + 1
 $$\zeta_{24}$$ $$=$$ $$( 4\beta_{7} + 3\beta_{6} - 2\beta_{5} - 3\beta_{4} + 3\beta_{3} + 4\beta_{2} + 3\beta_1 ) / 24$$ (4*b7 + 3*b6 - 2*b5 - 3*b4 + 3*b3 + 4*b2 + 3*b1) / 24 $$\zeta_{24}^{2}$$ $$=$$ $$( -\beta_{7} - \beta_{5} + 3\beta_{4} + 2\beta_{2} + 3\beta_1 ) / 12$$ (-b7 - b5 + 3*b4 + 2*b2 + 3*b1) / 12 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{5} + 3\beta_{4} + 4\beta_{2} - 3\beta_1 ) / 12$$ (b7 + b5 + 3*b4 + 4*b2 - 3*b1) / 12 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{3} + 2 ) / 4$$ (b3 + 2) / 4 $$\zeta_{24}^{5}$$ $$=$$ $$( 2\beta_{7} + 3\beta_{6} - 4\beta_{5} + 3\beta_{4} + 3\beta_{3} - 4\beta_{2} - 3\beta_1 ) / 24$$ (2*b7 + 3*b6 - 4*b5 + 3*b4 + 3*b3 - 4*b2 - 3*b1) / 24 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_{7} - \beta_{5} + 2\beta_{2} ) / 6$$ (-b7 - b5 + 2*b2) / 6 $$\zeta_{24}^{7}$$ $$=$$ $$( -2\beta_{7} + 3\beta_{6} + 4\beta_{5} + 3\beta_{4} - 3\beta_{3} + 4\beta_{2} - 3\beta_1 ) / 24$$ (-2*b7 + 3*b6 + 4*b5 + 3*b4 - 3*b3 + 4*b2 - 3*b1) / 24

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
959.1
 0.258819 − 0.965926i −0.965926 + 0.258819i 0.258819 + 0.965926i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.965926 − 0.258819i −0.258819 − 0.965926i 0.965926 + 0.258819i
0 −1.41421 1.00000i 0 −1.73205 1.41421i 0 0 0 1.00000 + 2.82843i 0
959.2 0 −1.41421 1.00000i 0 1.73205 1.41421i 0 0 0 1.00000 + 2.82843i 0
959.3 0 −1.41421 + 1.00000i 0 −1.73205 + 1.41421i 0 0 0 1.00000 2.82843i 0
959.4 0 −1.41421 + 1.00000i 0 1.73205 + 1.41421i 0 0 0 1.00000 2.82843i 0
959.5 0 1.41421 1.00000i 0 −1.73205 + 1.41421i 0 0 0 1.00000 2.82843i 0
959.6 0 1.41421 1.00000i 0 1.73205 + 1.41421i 0 0 0 1.00000 2.82843i 0
959.7 0 1.41421 + 1.00000i 0 −1.73205 1.41421i 0 0 0 1.00000 + 2.82843i 0
959.8 0 1.41421 + 1.00000i 0 1.73205 1.41421i 0 0 0 1.00000 + 2.82843i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 959.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h 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.o.d 8
3.b odd 2 1 inner 960.2.o.d 8
4.b odd 2 1 inner 960.2.o.d 8
5.b even 2 1 inner 960.2.o.d 8
8.b even 2 1 240.2.o.b 8
8.d odd 2 1 240.2.o.b 8
12.b even 2 1 inner 960.2.o.d 8
15.d odd 2 1 inner 960.2.o.d 8
20.d odd 2 1 inner 960.2.o.d 8
24.f even 2 1 240.2.o.b 8
24.h odd 2 1 240.2.o.b 8
40.e odd 2 1 240.2.o.b 8
40.f even 2 1 240.2.o.b 8
40.i odd 4 1 1200.2.h.j 4
40.i odd 4 1 1200.2.h.n 4
40.k even 4 1 1200.2.h.j 4
40.k even 4 1 1200.2.h.n 4
60.h even 2 1 inner 960.2.o.d 8
120.i odd 2 1 240.2.o.b 8
120.m even 2 1 240.2.o.b 8
120.q odd 4 1 1200.2.h.j 4
120.q odd 4 1 1200.2.h.n 4
120.w even 4 1 1200.2.h.j 4
120.w even 4 1 1200.2.h.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.2.o.b 8 8.b even 2 1
240.2.o.b 8 8.d odd 2 1
240.2.o.b 8 24.f even 2 1
240.2.o.b 8 24.h odd 2 1
240.2.o.b 8 40.e odd 2 1
240.2.o.b 8 40.f even 2 1
240.2.o.b 8 120.i odd 2 1
240.2.o.b 8 120.m even 2 1
960.2.o.d 8 1.a even 1 1 trivial
960.2.o.d 8 3.b odd 2 1 inner
960.2.o.d 8 4.b odd 2 1 inner
960.2.o.d 8 5.b even 2 1 inner
960.2.o.d 8 12.b even 2 1 inner
960.2.o.d 8 15.d odd 2 1 inner
960.2.o.d 8 20.d odd 2 1 inner
960.2.o.d 8 60.h even 2 1 inner
1200.2.h.j 4 40.i odd 4 1
1200.2.h.j 4 40.k even 4 1
1200.2.h.j 4 120.q odd 4 1
1200.2.h.j 4 120.w even 4 1
1200.2.h.n 4 40.i odd 4 1
1200.2.h.n 4 40.k even 4 1
1200.2.h.n 4 120.q odd 4 1
1200.2.h.n 4 120.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 $$S_{2}^{\mathrm{new}}(960, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}^{2} - 24$$ T11^2 - 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} - 2 T^{2} + 9)^{2}$$
$5$ $$(T^{4} - 2 T^{2} + 25)^{2}$$
$7$ $$T^{8}$$
$11$ $$(T^{2} - 24)^{4}$$
$13$ $$(T^{2} + 24)^{4}$$
$17$ $$(T^{2} - 12)^{4}$$
$19$ $$(T^{2} + 12)^{4}$$
$23$ $$(T^{2} + 36)^{4}$$
$29$ $$(T^{2} + 8)^{4}$$
$31$ $$(T^{2} + 12)^{4}$$
$37$ $$(T^{2} + 24)^{4}$$
$41$ $$(T^{2} + 32)^{4}$$
$43$ $$(T^{2} - 72)^{4}$$
$47$ $$(T^{2} + 36)^{4}$$
$53$ $$(T^{2} - 108)^{4}$$
$59$ $$(T^{2} - 24)^{4}$$
$61$ $$(T - 2)^{8}$$
$67$ $$(T^{2} - 72)^{4}$$
$71$ $$(T^{2} - 96)^{4}$$
$73$ $$(T^{2} + 96)^{4}$$
$79$ $$(T^{2} + 108)^{4}$$
$83$ $$(T^{2} + 36)^{4}$$
$89$ $$(T^{2} + 32)^{4}$$
$97$ $$T^{8}$$