Properties

 Label 960.2.d.e Level $960$ Weight $2$ Character orbit 960.d Analytic conductor $7.666$ Analytic rank $0$ Dimension $8$ 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.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$7.66563859404$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16$$ x^8 + 3*x^6 + 5*x^4 + 12*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - q^{3} - \beta_{4} q^{5} + (\beta_{5} - \beta_{4}) q^{7} + q^{9}+O(q^{10})$$ q - q^3 - b4 * q^5 + (b5 - b4) * q^7 + q^9 $$q - q^{3} - \beta_{4} q^{5} + (\beta_{5} - \beta_{4}) q^{7} + q^{9} + \beta_1 q^{11} + \beta_{2} q^{13} + \beta_{4} q^{15} - \beta_1 q^{17} - \beta_{3} q^{19} + ( - \beta_{5} + \beta_{4}) q^{21} + \beta_{6} q^{25} - q^{27} + \beta_{7} q^{29} + (\beta_{5} + \beta_{4} - \beta_{2}) q^{31} - \beta_1 q^{33} + (\beta_{6} - 5) q^{35} + (2 \beta_{5} + 2 \beta_{4} + \beta_{2}) q^{37} - \beta_{2} q^{39} + 6 q^{41} + (2 \beta_{6} + \beta_{3} - 2) q^{43} - \beta_{4} q^{45} + ( - \beta_{7} + \beta_{5} - \beta_{4}) q^{47} + (2 \beta_{6} + \beta_{3} - 3) q^{49} + \beta_1 q^{51} + ( - \beta_{5} - \beta_{4} - 2 \beta_{2}) q^{53} + (\beta_{7} + \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{55} + \beta_{3} q^{57} + ( - \beta_{3} + \beta_1) q^{59} + (\beta_{7} + 3 \beta_{5} - 3 \beta_{4}) q^{61} + (\beta_{5} - \beta_{4}) q^{63} + ( - \beta_{3} - \beta_1 + 2) q^{65} + (2 \beta_{6} + \beta_{3} + 6) q^{67} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2}) q^{71} - 3 \beta_{3} q^{73} - \beta_{6} q^{75} + (2 \beta_{5} + 2 \beta_{4} + 6 \beta_{2}) q^{77} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{79} + q^{81} + ( - 2 \beta_{6} - \beta_{3} - 2) q^{83} + ( - \beta_{7} - \beta_{5} - \beta_{4} - 3 \beta_{2}) q^{85} - \beta_{7} q^{87} - 2 q^{89} + ( - 2 \beta_{3} - 2 \beta_1) q^{91} + ( - \beta_{5} - \beta_{4} + \beta_{2}) q^{93} + (\beta_{7} + \beta_{5} - \beta_{4} - 2 \beta_{2}) q^{95} + 2 \beta_1 q^{97} + \beta_1 q^{99}+O(q^{100})$$ q - q^3 - b4 * q^5 + (b5 - b4) * q^7 + q^9 + b1 * q^11 + b2 * q^13 + b4 * q^15 - b1 * q^17 - b3 * q^19 + (-b5 + b4) * q^21 + b6 * q^25 - q^27 + b7 * q^29 + (b5 + b4 - b2) * q^31 - b1 * q^33 + (b6 - 5) * q^35 + (2*b5 + 2*b4 + b2) * q^37 - b2 * q^39 + 6 * q^41 + (2*b6 + b3 - 2) * q^43 - b4 * q^45 + (-b7 + b5 - b4) * q^47 + (2*b6 + b3 - 3) * q^49 + b1 * q^51 + (-b5 - b4 - 2*b2) * q^53 + (b7 + b5 + b4 + 3*b2) * q^55 + b3 * q^57 + (-b3 + b1) * q^59 + (b7 + 3*b5 - 3*b4) * q^61 + (b5 - b4) * q^63 + (-b3 - b1 + 2) * q^65 + (2*b6 + b3 + 6) * q^67 + (2*b5 + 2*b4 + 2*b2) * q^71 - 3*b3 * q^73 - b6 * q^75 + (2*b5 + 2*b4 + 6*b2) * q^77 + (-b5 - b4 + b2) * q^79 + q^81 + (-2*b6 - b3 - 2) * q^83 + (-b7 - b5 - b4 - 3*b2) * q^85 - b7 * q^87 - 2 * q^89 + (-2*b3 - 2*b1) * q^91 + (-b5 - b4 + b2) * q^93 + (b7 + b5 - b4 - 2*b2) * q^95 + 2*b1 * q^97 + b1 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 8 q^{3} + 8 q^{9}+O(q^{10})$$ 8 * q - 8 * q^3 + 8 * q^9 $$8 q - 8 q^{3} + 8 q^{9} - 8 q^{27} - 40 q^{35} + 48 q^{41} - 16 q^{43} - 24 q^{49} + 16 q^{65} + 48 q^{67} + 8 q^{81} - 16 q^{83} - 16 q^{89}+O(q^{100})$$ 8 * q - 8 * q^3 + 8 * q^9 - 8 * q^27 - 40 * q^35 + 48 * q^41 - 16 * q^43 - 24 * q^49 + 16 * q^65 + 48 * q^67 + 8 * q^81 - 16 * q^83 - 16 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} + 5\nu^{5} + 15\nu^{3} + 42\nu ) / 10$$ (v^7 + 5*v^5 + 15*v^3 + 42*v) / 10 $$\beta_{2}$$ $$=$$ $$( -\nu^{7} - 3\nu^{5} - 5\nu^{3} - 4\nu ) / 4$$ (-v^7 - 3*v^5 - 5*v^3 - 4*v) / 4 $$\beta_{3}$$ $$=$$ $$( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 10$$ (-3*v^7 - 5*v^5 + 5*v^3 - 16*v) / 10 $$\beta_{4}$$ $$=$$ $$( 5\nu^{7} - \nu^{6} + 5\nu^{5} + 5\nu^{4} + 15\nu^{3} + 15\nu^{2} + 30\nu + 8 ) / 20$$ (5*v^7 - v^6 + 5*v^5 + 5*v^4 + 15*v^3 + 15*v^2 + 30*v + 8) / 20 $$\beta_{5}$$ $$=$$ $$( 5\nu^{7} + \nu^{6} + 5\nu^{5} - 5\nu^{4} + 15\nu^{3} - 15\nu^{2} + 30\nu - 8 ) / 20$$ (5*v^7 + v^6 + 5*v^5 - 5*v^4 + 15*v^3 - 15*v^2 + 30*v - 8) / 20 $$\beta_{6}$$ $$=$$ $$( 3\nu^{7} + 10\nu^{6} + 5\nu^{5} + 30\nu^{4} - 5\nu^{3} + 10\nu^{2} + 16\nu + 60 ) / 20$$ (3*v^7 + 10*v^6 + 5*v^5 + 30*v^4 - 5*v^3 + 10*v^2 + 16*v + 60) / 20 $$\beta_{7}$$ $$=$$ $$( 13\nu^{6} + 15\nu^{4} + 45\nu^{2} + 96 ) / 10$$ (13*v^6 + 15*v^4 + 45*v^2 + 96) / 10
 $$\nu$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 2\beta_1 ) / 8$$ (-b3 + 2*b2 + 2*b1) / 8 $$\nu^{2}$$ $$=$$ $$( \beta_{7} - 2\beta_{6} - 3\beta_{5} + 3\beta_{4} - \beta_{3} - 6 ) / 8$$ (b7 - 2*b6 - 3*b5 + 3*b4 - b3 - 6) / 8 $$\nu^{3}$$ $$=$$ $$( 2\beta_{5} + 2\beta_{4} + 5\beta_{3} - 2\beta_{2} ) / 8$$ (2*b5 + 2*b4 + 5*b3 - 2*b2) / 8 $$\nu^{4}$$ $$=$$ $$( -2\beta_{7} + 6\beta_{6} - 4\beta_{5} + 4\beta_{4} + 3\beta_{3} - 2 ) / 8$$ (-2*b7 + 6*b6 - 4*b5 + 4*b4 + 3*b3 - 2) / 8 $$\nu^{5}$$ $$=$$ $$( -5\beta_{5} - 5\beta_{4} - 3\beta_{3} - 6\beta_{2} + \beta_1 ) / 4$$ (-5*b5 - 5*b4 - 3*b3 - 6*b2 + b1) / 4 $$\nu^{6}$$ $$=$$ $$( 5\beta_{7} + 15\beta_{5} - 15\beta_{4} - 36 ) / 8$$ (5*b7 + 15*b5 - 15*b4 - 36) / 8 $$\nu^{7}$$ $$=$$ $$( 20\beta_{5} + 20\beta_{4} - 3\beta_{3} + 6\beta_{2} - 14\beta_1 ) / 8$$ (20*b5 + 20*b4 - 3*b3 + 6*b2 - 14*b1) / 8

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
 −0.228425 + 1.39564i −0.228425 − 1.39564i −1.09445 − 0.895644i −1.09445 + 0.895644i 1.09445 + 0.895644i 1.09445 − 0.895644i 0.228425 − 1.39564i 0.228425 + 1.39564i
0 −1.00000 0 −2.18890 0.456850i 0 0.913701i 0 1.00000 0
289.2 0 −1.00000 0 −2.18890 + 0.456850i 0 0.913701i 0 1.00000 0
289.3 0 −1.00000 0 −0.456850 2.18890i 0 4.37780i 0 1.00000 0
289.4 0 −1.00000 0 −0.456850 + 2.18890i 0 4.37780i 0 1.00000 0
289.5 0 −1.00000 0 0.456850 2.18890i 0 4.37780i 0 1.00000 0
289.6 0 −1.00000 0 0.456850 + 2.18890i 0 4.37780i 0 1.00000 0
289.7 0 −1.00000 0 2.18890 0.456850i 0 0.913701i 0 1.00000 0
289.8 0 −1.00000 0 2.18890 + 0.456850i 0 0.913701i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
20.d odd 2 1 inner
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.e 8
3.b odd 2 1 2880.2.d.j 8
4.b odd 2 1 960.2.d.f yes 8
5.b even 2 1 960.2.d.f yes 8
5.c odd 4 1 4800.2.k.q 8
5.c odd 4 1 4800.2.k.r 8
8.b even 2 1 960.2.d.f yes 8
8.d odd 2 1 inner 960.2.d.e 8
12.b even 2 1 2880.2.d.i 8
15.d odd 2 1 2880.2.d.i 8
16.e even 4 1 3840.2.f.i 8
16.e even 4 1 3840.2.f.k 8
16.f odd 4 1 3840.2.f.i 8
16.f odd 4 1 3840.2.f.k 8
20.d odd 2 1 inner 960.2.d.e 8
20.e even 4 1 4800.2.k.q 8
20.e even 4 1 4800.2.k.r 8
24.f even 2 1 2880.2.d.j 8
24.h odd 2 1 2880.2.d.i 8
40.e odd 2 1 960.2.d.f yes 8
40.f even 2 1 inner 960.2.d.e 8
40.i odd 4 1 4800.2.k.q 8
40.i odd 4 1 4800.2.k.r 8
40.k even 4 1 4800.2.k.q 8
40.k even 4 1 4800.2.k.r 8
60.h even 2 1 2880.2.d.j 8
80.k odd 4 1 3840.2.f.i 8
80.k odd 4 1 3840.2.f.k 8
80.q even 4 1 3840.2.f.i 8
80.q even 4 1 3840.2.f.k 8
120.i odd 2 1 2880.2.d.j 8
120.m even 2 1 2880.2.d.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
960.2.d.e 8 1.a even 1 1 trivial
960.2.d.e 8 8.d odd 2 1 inner
960.2.d.e 8 20.d odd 2 1 inner
960.2.d.e 8 40.f even 2 1 inner
960.2.d.f yes 8 4.b odd 2 1
960.2.d.f yes 8 5.b even 2 1
960.2.d.f yes 8 8.b even 2 1
960.2.d.f yes 8 40.e odd 2 1
2880.2.d.i 8 12.b even 2 1
2880.2.d.i 8 15.d odd 2 1
2880.2.d.i 8 24.h odd 2 1
2880.2.d.i 8 120.m even 2 1
2880.2.d.j 8 3.b odd 2 1
2880.2.d.j 8 24.f even 2 1
2880.2.d.j 8 60.h even 2 1
2880.2.d.j 8 120.i odd 2 1
3840.2.f.i 8 16.e even 4 1
3840.2.f.i 8 16.f odd 4 1
3840.2.f.i 8 80.k odd 4 1
3840.2.f.i 8 80.q even 4 1
3840.2.f.k 8 16.e even 4 1
3840.2.f.k 8 16.f odd 4 1
3840.2.f.k 8 80.k odd 4 1
3840.2.f.k 8 80.q even 4 1
4800.2.k.q 8 5.c odd 4 1
4800.2.k.q 8 20.e even 4 1
4800.2.k.q 8 40.i odd 4 1
4800.2.k.q 8 40.k even 4 1
4800.2.k.r 8 5.c odd 4 1
4800.2.k.r 8 20.e even 4 1
4800.2.k.r 8 40.i odd 4 1
4800.2.k.r 8 40.k even 4 1

Hecke kernels

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

 $$T_{7}^{4} + 20T_{7}^{2} + 16$$ T7^4 + 20*T7^2 + 16 $$T_{13}^{4} - 20T_{13}^{2} + 16$$ T13^4 - 20*T13^2 + 16 $$T_{31}^{2} - 28$$ T31^2 - 28 $$T_{43}^{2} + 4T_{43} - 80$$ T43^2 + 4*T43 - 80

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T + 1)^{8}$$
$5$ $$T^{8} - 34T^{4} + 625$$
$7$ $$(T^{4} + 20 T^{2} + 16)^{2}$$
$11$ $$(T^{4} + 44 T^{2} + 400)^{2}$$
$13$ $$(T^{4} - 20 T^{2} + 16)^{2}$$
$17$ $$(T^{4} + 44 T^{2} + 400)^{2}$$
$19$ $$(T^{2} + 16)^{4}$$
$23$ $$T^{8}$$
$29$ $$(T^{4} + 68 T^{2} + 400)^{2}$$
$31$ $$(T^{2} - 28)^{4}$$
$37$ $$(T^{4} - 68 T^{2} + 400)^{2}$$
$41$ $$(T - 6)^{8}$$
$43$ $$(T^{2} + 4 T - 80)^{4}$$
$47$ $$(T^{2} + 48)^{4}$$
$53$ $$(T^{4} - 68 T^{2} + 400)^{2}$$
$59$ $$(T^{4} + 60 T^{2} + 144)^{2}$$
$61$ $$(T^{2} + 112)^{4}$$
$67$ $$(T^{2} - 12 T - 48)^{4}$$
$71$ $$(T^{2} - 48)^{4}$$
$73$ $$(T^{2} + 144)^{4}$$
$79$ $$(T^{2} - 28)^{4}$$
$83$ $$(T^{2} + 4 T - 80)^{4}$$
$89$ $$(T + 2)^{8}$$
$97$ $$(T^{4} + 176 T^{2} + 6400)^{2}$$