# Properties

 Label 960.3.e.c Level $960$ Weight $3$ Character orbit 960.e Analytic conductor $26.158$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$26.1581053786$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.85100625.1 Defining polynomial: $$x^{8} - x^{7} - 2 x^{6} + x^{5} + 3 x^{4} + 2 x^{3} - 8 x^{2} - 8 x + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{16}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + \beta_{1} q^{5} + ( 2 \beta_{2} + \beta_{5} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + \beta_{1} q^{5} + ( 2 \beta_{2} + \beta_{5} ) q^{7} -3 q^{9} + ( -\beta_{5} - \beta_{7} ) q^{11} + ( -2 - 2 \beta_{1} - \beta_{6} ) q^{13} + \beta_{4} q^{15} + ( -4 \beta_{1} - \beta_{3} ) q^{17} + ( 6 \beta_{2} - 4 \beta_{4} + \beta_{7} ) q^{19} + ( 6 - \beta_{3} ) q^{21} + ( 4 \beta_{2} - 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} ) q^{23} + 5 q^{25} + 3 \beta_{2} q^{27} + ( -8 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{6} ) q^{29} + ( -6 \beta_{2} - 8 \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{31} + ( \beta_{3} + \beta_{6} ) q^{33} + ( -2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{35} + ( 14 - 10 \beta_{1} - 2 \beta_{3} + \beta_{6} ) q^{37} + ( 2 \beta_{2} - 2 \beta_{4} - 3 \beta_{7} ) q^{39} + ( -2 - 2 \beta_{6} ) q^{41} + ( -16 \beta_{2} - 4 \beta_{4} + 6 \beta_{5} + 2 \beta_{7} ) q^{43} -3 \beta_{1} q^{45} + ( -8 \beta_{2} - 10 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{47} + ( -7 - 16 \beta_{1} + 4 \beta_{3} ) q^{49} + ( -4 \beta_{4} - 3 \beta_{5} ) q^{51} + ( -44 - 2 \beta_{1} - 2 \beta_{6} ) q^{53} + ( -3 \beta_{5} + \beta_{7} ) q^{55} + ( 18 + 12 \beta_{1} - \beta_{6} ) q^{57} + ( 4 \beta_{2} - 4 \beta_{4} - 3 \beta_{5} - 3 \beta_{7} ) q^{59} + ( 22 - 4 \beta_{1} + 4 \beta_{3} + 2 \beta_{6} ) q^{61} + ( -6 \beta_{2} - 3 \beta_{5} ) q^{63} + ( -10 - 2 \beta_{1} - \beta_{3} + 2 \beta_{6} ) q^{65} + ( 8 \beta_{2} - 12 \beta_{4} - 4 \beta_{5} - 4 \beta_{7} ) q^{67} + ( 12 + 6 \beta_{1} + 3 \beta_{3} - 2 \beta_{6} ) q^{69} + ( -12 \beta_{2} - 4 \beta_{4} + 8 \beta_{5} ) q^{71} + ( -30 - 4 \beta_{1} - 4 \beta_{3} + 2 \beta_{6} ) q^{73} -5 \beta_{2} q^{75} + ( 36 + 28 \beta_{1} - 2 \beta_{3} - 2 \beta_{6} ) q^{77} + ( 14 \beta_{2} - 16 \beta_{4} + 2 \beta_{5} + 9 \beta_{7} ) q^{79} + 9 q^{81} + ( -4 \beta_{2} - 24 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{83} + ( -20 - 2 \beta_{3} - \beta_{6} ) q^{85} + ( 8 \beta_{2} - 2 \beta_{4} + 6 \beta_{5} + 6 \beta_{7} ) q^{87} + ( 10 + 44 \beta_{1} + 2 \beta_{6} ) q^{89} + ( -12 \beta_{2} - 8 \beta_{4} - 6 \beta_{5} + 4 \beta_{7} ) q^{91} + ( -18 + 24 \beta_{1} - 2 \beta_{3} + \beta_{6} ) q^{93} + ( 20 \beta_{2} - 6 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{95} + ( 54 - 12 \beta_{1} - 6 \beta_{3} - 2 \beta_{6} ) q^{97} + ( 3 \beta_{5} + 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 24q^{9} + O(q^{10})$$ $$8q - 24q^{9} - 16q^{13} + 48q^{21} + 40q^{25} - 64q^{29} + 112q^{37} - 16q^{41} - 56q^{49} - 352q^{53} + 144q^{57} + 176q^{61} - 80q^{65} + 96q^{69} - 240q^{73} + 288q^{77} + 72q^{81} - 160q^{85} + 80q^{89} - 144q^{93} + 432q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 2 x^{6} + x^{5} + 3 x^{4} + 2 x^{3} - 8 x^{2} - 8 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$3 \nu^{7} - \nu^{6} - 4 \nu^{5} - 5 \nu^{4} - 5 \nu^{3} + 16 \nu^{2} - 8 \nu - 24$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$5 \nu^{7} + \nu^{6} - 4 \nu^{5} - 3 \nu^{4} + 5 \nu^{3} + 8 \nu^{2} - 8 \nu - 40$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$3 \nu^{7} + 7 \nu^{6} - 12 \nu^{5} - 21 \nu^{4} + 3 \nu^{3} + 8 \nu^{2} + 40 \nu - 56$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{7} + \nu^{6} - 6 \nu^{5} - \nu^{4} + 5 \nu^{3} + 14 \nu^{2} - 4 \nu - 36$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$3 \nu^{7} - \nu^{6} - 4 \nu^{5} - 5 \nu^{4} + 11 \nu^{3} + 16 \nu^{2} - 8 \nu - 32$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$-\nu^{7} + 3 \nu^{5} + \nu^{4} - 4 \nu^{3} - \nu^{2} + 10 \nu + 10$$ $$\beta_{7}$$ $$=$$ $$($$$$-15 \nu^{7} - 11 \nu^{6} + 20 \nu^{5} + 25 \nu^{4} - 23 \nu^{3} - 80 \nu^{2} + 40 \nu + 184$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_{1} + 2$$$$)/16$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - 10 \beta_{2} + 2 \beta_{1} + 10$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{5} - 4 \beta_{1} + 2$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - 3 \beta_{5} + 10 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 10 \beta_{1} + 2$$$$)/16$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} - 10 \beta_{4} - 3 \beta_{3} + 18 \beta_{2} - 10 \beta_{1} - 18$$$$)/16$$ $$\nu^{6}$$ $$=$$ $$($$$$-2 \beta_{7} - 3 \beta_{5} - 8 \beta_{1} + 10$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$3 \beta_{7} + 3 \beta_{6} - \beta_{5} - 2 \beta_{4} - \beta_{3} + 86 \beta_{2} + 2 \beta_{1} + 86$$$$)/16$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
511.1
 −0.600040 + 1.28061i 1.40906 + 0.120653i 1.04064 − 0.957636i −1.34966 + 0.422403i 1.40906 − 0.120653i −0.600040 − 1.28061i −1.34966 − 0.422403i 1.04064 + 0.957636i
0 1.73205i 0 −2.23607 0 0.596540i 0 −3.00000 0
511.2 0 1.73205i 0 −2.23607 0 6.33166i 0 −3.00000 0
511.3 0 1.73205i 0 2.23607 0 5.46770i 0 −3.00000 0
511.4 0 1.73205i 0 2.23607 0 12.3959i 0 −3.00000 0
511.5 0 1.73205i 0 −2.23607 0 6.33166i 0 −3.00000 0
511.6 0 1.73205i 0 −2.23607 0 0.596540i 0 −3.00000 0
511.7 0 1.73205i 0 2.23607 0 12.3959i 0 −3.00000 0
511.8 0 1.73205i 0 2.23607 0 5.46770i 0 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 511.8 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.3.e.c 8
3.b odd 2 1 2880.3.e.j 8
4.b odd 2 1 inner 960.3.e.c 8
8.b even 2 1 60.3.c.a 8
8.d odd 2 1 60.3.c.a 8
12.b even 2 1 2880.3.e.j 8
24.f even 2 1 180.3.c.b 8
24.h odd 2 1 180.3.c.b 8
40.e odd 2 1 300.3.c.d 8
40.f even 2 1 300.3.c.d 8
40.i odd 4 2 300.3.f.b 16
40.k even 4 2 300.3.f.b 16
120.i odd 2 1 900.3.c.u 8
120.m even 2 1 900.3.c.u 8
120.q odd 4 2 900.3.f.f 16
120.w even 4 2 900.3.f.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.c.a 8 8.b even 2 1
60.3.c.a 8 8.d odd 2 1
180.3.c.b 8 24.f even 2 1
180.3.c.b 8 24.h odd 2 1
300.3.c.d 8 40.e odd 2 1
300.3.c.d 8 40.f even 2 1
300.3.f.b 16 40.i odd 4 2
300.3.f.b 16 40.k even 4 2
900.3.c.u 8 120.i odd 2 1
900.3.c.u 8 120.m even 2 1
900.3.f.f 16 120.q odd 4 2
900.3.f.f 16 120.w even 4 2
960.3.e.c 8 1.a even 1 1 trivial
960.3.e.c 8 4.b odd 2 1 inner
2880.3.e.j 8 3.b odd 2 1
2880.3.e.j 8 12.b even 2 1

## Hecke kernels

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

 $$T_{7}^{8} + 224 T_{7}^{6} + 12032 T_{7}^{4} + 188416 T_{7}^{2} + 65536$$ $$T_{13}^{4} + 8 T_{13}^{3} - 472 T_{13}^{2} - 5792 T_{13} - 12464$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 3 + T^{2} )^{4}$$
$5$ $$( -5 + T^{2} )^{4}$$
$7$ $$65536 + 188416 T^{2} + 12032 T^{4} + 224 T^{6} + T^{8}$$
$11$ $$( 10496 + 208 T^{2} + T^{4} )^{2}$$
$13$ $$( -12464 - 5792 T - 472 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$17$ $$( -8816 + 3840 T - 424 T^{2} + T^{4} )^{2}$$
$19$ $$6544162816 + 173686784 T^{2} + 925952 T^{4} + 1696 T^{6} + T^{8}$$
$23$ $$101419319296 + 1884176384 T^{2} + 4397312 T^{4} + 3616 T^{6} + T^{8}$$
$29$ $$( 1334416 - 34688 T - 2152 T^{2} + 32 T^{3} + T^{4} )^{2}$$
$31$ $$59895709696 + 2731491328 T^{2} + 7432448 T^{4} + 5408 T^{6} + T^{8}$$
$37$ $$( -244784 + 55136 T - 1528 T^{2} - 56 T^{3} + T^{4} )^{2}$$
$41$ $$( 87184 - 7264 T - 1800 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$43$ $$33624411406336 + 62108155904 T^{2} + 40259072 T^{4} + 10816 T^{6} + T^{8}$$
$47$ $$1056981385216 + 9701752832 T^{2} + 15726848 T^{4} + 8032 T^{6} + T^{8}$$
$53$ $$( -478064 + 161344 T + 9752 T^{2} + 176 T^{3} + T^{4} )^{2}$$
$59$ $$173909016576 + 2174459904 T^{2} + 6273792 T^{4} + 4896 T^{6} + T^{8}$$
$61$ $$( -2142704 + 273568 T - 2536 T^{2} - 88 T^{3} + T^{4} )^{2}$$
$67$ $$281086590976 + 15044755456 T^{2} + 57554432 T^{4} + 16064 T^{6} + T^{8}$$
$71$ $$16079971680256 + 101402017792 T^{2} + 64237568 T^{4} + 13952 T^{6} + T^{8}$$
$73$ $$( 4962064 - 325920 T - 1576 T^{2} + 120 T^{3} + T^{4} )^{2}$$
$79$ $$3198642669223936 + 2420601929728 T^{2} + 550899968 T^{4} + 41888 T^{6} + T^{8}$$
$83$ $$4284940379815936 + 2381453017088 T^{2} + 464465408 T^{4} + 36928 T^{6} + T^{8}$$
$89$ $$( 70652944 + 757600 T - 20584 T^{2} - 40 T^{3} + T^{4} )^{2}$$
$97$ $$( -59281776 + 1154592 T + 5880 T^{2} - 216 T^{3} + T^{4} )^{2}$$