# Properties

 Label 960.3.l.e Level $960$ Weight $3$ Character orbit 960.l Analytic conductor $26.158$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$26.1581053786$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 30) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} + \beta_{2} ) q^{3} -\beta_{2} q^{5} + ( 2 - 5 \beta_{1} + \beta_{3} ) q^{7} + ( -2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{1} + \beta_{2} ) q^{3} -\beta_{2} q^{5} + ( 2 - 5 \beta_{1} + \beta_{3} ) q^{7} + ( -2 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{9} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{11} + 10 q^{13} + ( 5 + \beta_{2} + \beta_{3} ) q^{15} + ( 4 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} ) q^{17} + ( -8 - 10 \beta_{1} + 2 \beta_{3} ) q^{19} + ( 13 - 2 \beta_{1} + 5 \beta_{2} - 6 \beta_{3} ) q^{21} + ( \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{23} -5 q^{25} + ( -7 + 10 \beta_{1} + \beta_{2} + 5 \beta_{3} ) q^{27} -12 \beta_{2} q^{29} + 8 q^{31} + ( 6 - 12 \beta_{1} - 6 \beta_{2} ) q^{33} + ( 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{35} + ( 22 - 20 \beta_{1} + 4 \beta_{3} ) q^{37} + ( -10 - 10 \beta_{1} + 10 \beta_{2} ) q^{39} + ( -8 \beta_{1} - 6 \beta_{2} - 8 \beta_{3} ) q^{41} + ( -14 + 15 \beta_{1} - 3 \beta_{3} ) q^{43} + ( -5 - 10 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{45} + ( 5 \beta_{1} - 30 \beta_{2} + 5 \beta_{3} ) q^{47} + ( 45 - 20 \beta_{1} + 4 \beta_{3} ) q^{49} + ( -18 - 24 \beta_{1} - 18 \beta_{2} - 6 \beta_{3} ) q^{51} + ( -4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{53} + ( 10 \beta_{1} - 2 \beta_{3} ) q^{55} + ( 38 + 8 \beta_{1} - 2 \beta_{2} - 12 \beta_{3} ) q^{57} + ( -12 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{59} + ( 16 + 20 \beta_{1} - 4 \beta_{3} ) q^{61} + ( -64 + 19 \beta_{1} + 22 \beta_{2} - \beta_{3} ) q^{63} -10 \beta_{2} q^{65} + ( 82 + 15 \beta_{1} - 3 \beta_{3} ) q^{67} + ( 33 - 6 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{69} + ( -10 \beta_{1} + 12 \beta_{2} - 10 \beta_{3} ) q^{71} + ( 50 + 20 \beta_{1} - 4 \beta_{3} ) q^{73} + ( 5 + 5 \beta_{1} - 5 \beta_{2} ) q^{75} + ( 4 \beta_{1} - 36 \beta_{2} + 4 \beta_{3} ) q^{77} + ( -28 - 10 \beta_{1} + 2 \beta_{3} ) q^{79} + ( 7 - 28 \beta_{1} - 28 \beta_{2} + 4 \beta_{3} ) q^{81} + ( 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} ) q^{83} + ( 30 + 20 \beta_{1} - 4 \beta_{3} ) q^{85} + ( 60 + 12 \beta_{2} + 12 \beta_{3} ) q^{87} + ( -8 \beta_{1} - 12 \beta_{2} - 8 \beta_{3} ) q^{89} + ( 20 - 50 \beta_{1} + 10 \beta_{3} ) q^{91} + ( -8 - 8 \beta_{1} + 8 \beta_{2} ) q^{93} + ( 10 \beta_{1} + 8 \beta_{2} + 10 \beta_{3} ) q^{95} + ( 74 - 20 \beta_{1} + 4 \beta_{3} ) q^{97} + ( 48 + 6 \beta_{1} + 24 \beta_{2} - 6 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} + 8q^{7} - 8q^{9} + O(q^{10})$$ $$4q - 4q^{3} + 8q^{7} - 8q^{9} + 40q^{13} + 20q^{15} - 32q^{19} + 52q^{21} - 20q^{25} - 28q^{27} + 32q^{31} + 24q^{33} + 88q^{37} - 40q^{39} - 56q^{43} - 20q^{45} + 180q^{49} - 72q^{51} + 152q^{57} + 64q^{61} - 256q^{63} + 328q^{67} + 132q^{69} + 200q^{73} + 20q^{75} - 112q^{79} + 28q^{81} + 120q^{85} + 240q^{87} + 80q^{91} - 32q^{93} + 296q^{97} + 192q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
641.1
 1.58114 − 0.707107i 1.58114 + 0.707107i −1.58114 + 0.707107i −1.58114 − 0.707107i
0 −2.58114 1.52896i 0 2.23607i 0 −7.48683 0 4.32456 + 7.89292i 0
641.2 0 −2.58114 + 1.52896i 0 2.23607i 0 −7.48683 0 4.32456 7.89292i 0
641.3 0 0.581139 2.94317i 0 2.23607i 0 11.4868 0 −8.32456 3.42079i 0
641.4 0 0.581139 + 2.94317i 0 2.23607i 0 11.4868 0 −8.32456 + 3.42079i 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
3.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.l.e 4
3.b odd 2 1 inner 960.3.l.e 4
4.b odd 2 1 960.3.l.f 4
8.b even 2 1 30.3.d.a 4
8.d odd 2 1 240.3.l.c 4
12.b even 2 1 960.3.l.f 4
24.f even 2 1 240.3.l.c 4
24.h odd 2 1 30.3.d.a 4
40.e odd 2 1 1200.3.l.u 4
40.f even 2 1 150.3.d.c 4
40.i odd 4 2 150.3.b.b 8
40.k even 4 2 1200.3.c.k 8
72.j odd 6 2 810.3.h.a 8
72.n even 6 2 810.3.h.a 8
120.i odd 2 1 150.3.d.c 4
120.m even 2 1 1200.3.l.u 4
120.q odd 4 2 1200.3.c.k 8
120.w even 4 2 150.3.b.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.d.a 4 8.b even 2 1
30.3.d.a 4 24.h odd 2 1
150.3.b.b 8 40.i odd 4 2
150.3.b.b 8 120.w even 4 2
150.3.d.c 4 40.f even 2 1
150.3.d.c 4 120.i odd 2 1
240.3.l.c 4 8.d odd 2 1
240.3.l.c 4 24.f even 2 1
810.3.h.a 8 72.j odd 6 2
810.3.h.a 8 72.n even 6 2
960.3.l.e 4 1.a even 1 1 trivial
960.3.l.e 4 3.b odd 2 1 inner
960.3.l.f 4 4.b odd 2 1
960.3.l.f 4 12.b even 2 1
1200.3.c.k 8 40.k even 4 2
1200.3.c.k 8 120.q odd 4 2
1200.3.l.u 4 40.e odd 2 1
1200.3.l.u 4 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} - 4 T_{7} - 86$$ acting on $$S_{3}^{\mathrm{new}}(960, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$81 + 36 T + 12 T^{2} + 4 T^{3} + T^{4}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$( -86 - 4 T + T^{2} )^{2}$$
$11$ $$( 72 + T^{2} )^{2}$$
$13$ $$( -10 + T )^{4}$$
$17$ $$11664 + 936 T^{2} + T^{4}$$
$19$ $$( -296 + 16 T + T^{2} )^{2}$$
$23$ $$26244 + 396 T^{2} + T^{4}$$
$29$ $$( 720 + T^{2} )^{2}$$
$31$ $$( -8 + T )^{4}$$
$37$ $$( -956 - 44 T + T^{2} )^{2}$$
$41$ $$944784 + 2664 T^{2} + T^{4}$$
$43$ $$( -614 + 28 T + T^{2} )^{2}$$
$47$ $$16402500 + 9900 T^{2} + T^{4}$$
$53$ $$11664 + 936 T^{2} + T^{4}$$
$59$ $$3504384 + 6624 T^{2} + T^{4}$$
$61$ $$( -1184 - 32 T + T^{2} )^{2}$$
$67$ $$( 5914 - 164 T + T^{2} )^{2}$$
$71$ $$1166400 + 5040 T^{2} + T^{4}$$
$73$ $$( 1060 - 100 T + T^{2} )^{2}$$
$79$ $$( 424 + 56 T + T^{2} )^{2}$$
$83$ $$324 + 684 T^{2} + T^{4}$$
$89$ $$186624 + 3744 T^{2} + T^{4}$$
$97$ $$( 4036 - 148 T + T^{2} )^{2}$$