# Properties

 Label 960.3.j.e Level $960$ Weight $3$ Character orbit 960.j Analytic conductor $26.158$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$26.1581053786$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.389136420864.4 Defining polynomial: $$x^{8} + 5 x^{6} + 24 x^{4} + 80 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{11}$$ Twist minimal: no (minimal twist has level 60) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -1 + \beta_{3} ) q^{5} + ( -3 \beta_{1} - \beta_{4} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -1 + \beta_{3} ) q^{5} + ( -3 \beta_{1} - \beta_{4} ) q^{7} + 3 q^{9} + \beta_{6} q^{11} + ( -\beta_{3} - \beta_{5} ) q^{13} + ( \beta_{2} + \beta_{4} ) q^{15} + \beta_{7} q^{17} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} ) q^{19} + ( -8 - \beta_{3} + \beta_{5} ) q^{21} + ( 6 \beta_{1} - 2 \beta_{4} ) q^{23} + ( 5 + 2 \beta_{5} - \beta_{7} ) q^{25} + 3 \beta_{1} q^{27} + ( 26 - 3 \beta_{3} + 3 \beta_{5} ) q^{29} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{31} -\beta_{7} q^{33} + ( -10 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} - \beta_{6} ) q^{35} + ( 3 \beta_{3} + 3 \beta_{5} ) q^{37} + ( -\beta_{1} - 2 \beta_{2} + \beta_{4} ) q^{39} + ( -26 - 6 \beta_{3} + 6 \beta_{5} ) q^{41} + ( -4 \beta_{1} - 8 \beta_{4} ) q^{43} + ( -3 + 3 \beta_{3} ) q^{45} + ( 12 \beta_{1} - 8 \beta_{4} ) q^{47} + ( -9 + 6 \beta_{3} - 6 \beta_{5} ) q^{49} -3 \beta_{6} q^{51} + ( 3 \beta_{3} + 3 \beta_{5} ) q^{53} + ( 5 \beta_{1} + 8 \beta_{2} - 13 \beta_{4} - 2 \beta_{6} ) q^{55} + ( 6 \beta_{3} + 6 \beta_{5} ) q^{57} + ( 8 \beta_{1} + 16 \beta_{2} - 8 \beta_{4} - \beta_{6} ) q^{59} -38 q^{61} + ( -9 \beta_{1} - 3 \beta_{4} ) q^{63} + ( 20 - 2 \beta_{5} + \beta_{7} ) q^{65} + ( 18 \beta_{1} - 14 \beta_{4} ) q^{67} + ( 20 - 2 \beta_{3} + 2 \beta_{5} ) q^{69} + ( 4 \beta_{1} + 8 \beta_{2} - 4 \beta_{4} - 6 \beta_{6} ) q^{71} + 2 \beta_{7} q^{73} + ( 5 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} + 3 \beta_{6} ) q^{75} + ( -8 \beta_{3} - 8 \beta_{5} + 2 \beta_{7} ) q^{77} + ( -2 \beta_{1} - 4 \beta_{2} + 2 \beta_{4} - 2 \beta_{6} ) q^{79} + 9 q^{81} + ( 24 \beta_{1} - 4 \beta_{4} ) q^{83} + ( -12 - 3 \beta_{3} - 21 \beta_{5} - 2 \beta_{7} ) q^{85} + ( 23 \beta_{1} - 9 \beta_{4} ) q^{87} + ( 66 + 4 \beta_{3} - 4 \beta_{5} ) q^{89} + ( 4 \beta_{1} + 8 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} ) q^{91} + ( 6 \beta_{3} + 6 \beta_{5} - 2 \beta_{7} ) q^{93} + ( -40 \beta_{1} + 4 \beta_{2} - 8 \beta_{4} + 6 \beta_{6} ) q^{95} + ( -22 \beta_{3} - 22 \beta_{5} - 2 \beta_{7} ) q^{97} + 3 \beta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{5} + 24q^{9} + O(q^{10})$$ $$8q - 4q^{5} + 24q^{9} - 72q^{21} + 32q^{25} + 184q^{29} - 256q^{41} - 12q^{45} - 24q^{49} - 304q^{61} + 168q^{65} + 144q^{69} + 72q^{81} - 24q^{85} + 560q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 24 x^{4} + 80 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + \nu^{3} + 4 \nu$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{6} - 5 \nu^{5} - 12 \nu^{4} - 24 \nu^{3} - 16 \nu - 64$$$$)/64$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} + 4 \nu^{6} + 5 \nu^{5} + 20 \nu^{4} + 24 \nu^{3} + 32 \nu^{2} + 144 \nu + 192$$$$)/64$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 3 \nu^{5} - 22 \nu^{3} - 8 \nu$$$$)/32$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - 4 \nu^{6} + 5 \nu^{5} - 20 \nu^{4} + 24 \nu^{3} - 32 \nu^{2} + 144 \nu - 192$$$$)/64$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{4} + 40 \nu^{2} + 80$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} - 9 \nu^{5} + 24 \nu^{3} - 16 \nu$$$$)/32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{3} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} - \beta_{3} - 4$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} - 5 \beta_{4} - 3 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{6} - 5 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} - 8 \beta_{2} - 4 \beta_{1} - 28$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{7} - 4 \beta_{5} + \beta_{4} - 4 \beta_{3} + 71 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-3 \beta_{6} - 15 \beta_{5} - 20 \beta_{4} + 15 \beta_{3} + 40 \beta_{2} + 20 \beta_{1} - 20$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-19 \beta_{7} + 4 \beta_{5} - 29 \beta_{4} + 4 \beta_{3} - 139 \beta_{1}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
319.1
 1.52274 − 1.29664i 1.52274 + 1.29664i −0.656712 − 1.88911i −0.656712 + 1.88911i −1.52274 − 1.29664i −1.52274 + 1.29664i 0.656712 − 1.88911i 0.656712 + 1.88911i
0 −1.73205 0 −4.27492 2.59328i 0 0.837253 0 3.00000 0
319.2 0 −1.73205 0 −4.27492 + 2.59328i 0 0.837253 0 3.00000 0
319.3 0 −1.73205 0 3.27492 3.77822i 0 9.55505 0 3.00000 0
319.4 0 −1.73205 0 3.27492 + 3.77822i 0 9.55505 0 3.00000 0
319.5 0 1.73205 0 −4.27492 2.59328i 0 −0.837253 0 3.00000 0
319.6 0 1.73205 0 −4.27492 + 2.59328i 0 −0.837253 0 3.00000 0
319.7 0 1.73205 0 3.27492 3.77822i 0 −9.55505 0 3.00000 0
319.8 0 1.73205 0 3.27492 + 3.77822i 0 −9.55505 0 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 319.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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 960.3.j.e 8
4.b odd 2 1 inner 960.3.j.e 8
5.b even 2 1 inner 960.3.j.e 8
8.b even 2 1 60.3.f.b 8
8.d odd 2 1 60.3.f.b 8
20.d odd 2 1 inner 960.3.j.e 8
24.f even 2 1 180.3.f.h 8
24.h odd 2 1 180.3.f.h 8
40.e odd 2 1 60.3.f.b 8
40.f even 2 1 60.3.f.b 8
40.i odd 4 2 300.3.c.f 8
40.k even 4 2 300.3.c.f 8
120.i odd 2 1 180.3.f.h 8
120.m even 2 1 180.3.f.h 8
120.q odd 4 2 900.3.c.r 8
120.w even 4 2 900.3.c.r 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.b 8 8.b even 2 1
60.3.f.b 8 8.d odd 2 1
60.3.f.b 8 40.e odd 2 1
60.3.f.b 8 40.f even 2 1
180.3.f.h 8 24.f even 2 1
180.3.f.h 8 24.h odd 2 1
180.3.f.h 8 120.i odd 2 1
180.3.f.h 8 120.m even 2 1
300.3.c.f 8 40.i odd 4 2
300.3.c.f 8 40.k even 4 2
900.3.c.r 8 120.q odd 4 2
900.3.c.r 8 120.w even 4 2
960.3.j.e 8 1.a even 1 1 trivial
960.3.j.e 8 4.b odd 2 1 inner
960.3.j.e 8 5.b even 2 1 inner
960.3.j.e 8 20.d odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(960, [\chi])$$:

 $$T_{7}^{4} - 92 T_{7}^{2} + 64$$ $$T_{11}^{4} + 348 T_{11}^{2} + 24576$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( -3 + T^{2} )^{4}$$
$5$ $$( 625 + 50 T - 6 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$7$ $$( 64 - 92 T^{2} + T^{4} )^{2}$$
$11$ $$( 24576 + 348 T^{2} + T^{4} )^{2}$$
$13$ $$( 1536 + 84 T^{2} + T^{4} )^{2}$$
$17$ $$( 221184 + 1044 T^{2} + T^{4} )^{2}$$
$19$ $$( 221184 + 1008 T^{2} + T^{4} )^{2}$$
$23$ $$( 1024 - 368 T^{2} + T^{4} )^{2}$$
$29$ $$( 16 - 46 T + T^{2} )^{4}$$
$31$ $$( 393216 + 2304 T^{2} + T^{4} )^{2}$$
$37$ $$( 124416 + 756 T^{2} + T^{4} )^{2}$$
$41$ $$( -1028 + 64 T + T^{2} )^{4}$$
$43$ $$( 1364224 - 2528 T^{2} + T^{4} )^{2}$$
$47$ $$( 614656 - 3296 T^{2} + T^{4} )^{2}$$
$53$ $$( 124416 + 756 T^{2} + T^{4} )^{2}$$
$59$ $$( 69033984 + 16668 T^{2} + T^{4} )^{2}$$
$61$ $$( 38 + T )^{8}$$
$67$ $$( 7573504 - 9392 T^{2} + T^{4} )^{2}$$
$71$ $$( 884736 + 17136 T^{2} + T^{4} )^{2}$$
$73$ $$( 3538944 + 4176 T^{2} + T^{4} )^{2}$$
$79$ $$( 393216 + 2304 T^{2} + T^{4} )^{2}$$
$83$ $$( 2027776 - 4064 T^{2} + T^{4} )^{2}$$
$89$ $$( 3988 - 140 T + T^{2} )^{4}$$
$97$ $$( 495550464 + 45888 T^{2} + T^{4} )^{2}$$