# Properties

 Label 960.3.l.d Level $960$ Weight $3$ Character orbit 960.l Analytic conductor $26.158$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 + \beta ) q^{3} -\beta q^{5} -2 q^{7} + ( -1 + 4 \beta ) q^{9} +O(q^{10})$$ $$q + ( 2 + \beta ) q^{3} -\beta q^{5} -2 q^{7} + ( -1 + 4 \beta ) q^{9} -6 \beta q^{11} -8 q^{13} + ( 5 - 2 \beta ) q^{15} -6 \beta q^{17} -34 q^{19} + ( -4 - 2 \beta ) q^{21} + 18 \beta q^{23} -5 q^{25} + ( -22 + 7 \beta ) q^{27} -18 \beta q^{29} -14 q^{31} + ( 30 - 12 \beta ) q^{33} + 2 \beta q^{35} -56 q^{37} + ( -16 - 8 \beta ) q^{39} + 12 \beta q^{41} + 8 q^{43} + ( 20 + \beta ) q^{45} -18 \beta q^{47} -45 q^{49} + ( 30 - 12 \beta ) q^{51} -18 \beta q^{53} -30 q^{55} + ( -68 - 34 \beta ) q^{57} -6 \beta q^{59} + 46 q^{61} + ( 2 - 8 \beta ) q^{63} + 8 \beta q^{65} + 32 q^{67} + ( -90 + 36 \beta ) q^{69} + 24 \beta q^{71} -106 q^{73} + ( -10 - 5 \beta ) q^{75} + 12 \beta q^{77} + 22 q^{79} + ( -79 - 8 \beta ) q^{81} + 54 \beta q^{83} -30 q^{85} + ( 90 - 36 \beta ) q^{87} -48 \beta q^{89} + 16 q^{91} + ( -28 - 14 \beta ) q^{93} + 34 \beta q^{95} + 122 q^{97} + ( 120 + 6 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{3} - 4 q^{7} - 2 q^{9} + O(q^{10})$$ $$2 q + 4 q^{3} - 4 q^{7} - 2 q^{9} - 16 q^{13} + 10 q^{15} - 68 q^{19} - 8 q^{21} - 10 q^{25} - 44 q^{27} - 28 q^{31} + 60 q^{33} - 112 q^{37} - 32 q^{39} + 16 q^{43} + 40 q^{45} - 90 q^{49} + 60 q^{51} - 60 q^{55} - 136 q^{57} + 92 q^{61} + 4 q^{63} + 64 q^{67} - 180 q^{69} - 212 q^{73} - 20 q^{75} + 44 q^{79} - 158 q^{81} - 60 q^{85} + 180 q^{87} + 32 q^{91} - 56 q^{93} + 244 q^{97} + 240 q^{99} + O(q^{100})$$

## 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
 − 2.23607i 2.23607i
0 2.00000 2.23607i 0 2.23607i 0 −2.00000 0 −1.00000 8.94427i 0
641.2 0 2.00000 + 2.23607i 0 2.23607i 0 −2.00000 0 −1.00000 + 8.94427i 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.d 2
3.b odd 2 1 inner 960.3.l.d 2
4.b odd 2 1 960.3.l.a 2
8.b even 2 1 240.3.l.a 2
8.d odd 2 1 60.3.g.a 2
12.b even 2 1 960.3.l.a 2
24.f even 2 1 60.3.g.a 2
24.h odd 2 1 240.3.l.a 2
40.e odd 2 1 300.3.g.d 2
40.f even 2 1 1200.3.l.r 2
40.i odd 4 2 1200.3.c.e 4
40.k even 4 2 300.3.b.c 4
72.l even 6 2 1620.3.o.b 4
72.p odd 6 2 1620.3.o.b 4
120.i odd 2 1 1200.3.l.r 2
120.m even 2 1 300.3.g.d 2
120.q odd 4 2 300.3.b.c 4
120.w even 4 2 1200.3.c.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.g.a 2 8.d odd 2 1
60.3.g.a 2 24.f even 2 1
240.3.l.a 2 8.b even 2 1
240.3.l.a 2 24.h odd 2 1
300.3.b.c 4 40.k even 4 2
300.3.b.c 4 120.q odd 4 2
300.3.g.d 2 40.e odd 2 1
300.3.g.d 2 120.m even 2 1
960.3.l.a 2 4.b odd 2 1
960.3.l.a 2 12.b even 2 1
960.3.l.d 2 1.a even 1 1 trivial
960.3.l.d 2 3.b odd 2 1 inner
1200.3.c.e 4 40.i odd 4 2
1200.3.c.e 4 120.w even 4 2
1200.3.l.r 2 40.f even 2 1
1200.3.l.r 2 120.i odd 2 1
1620.3.o.b 4 72.l even 6 2
1620.3.o.b 4 72.p odd 6 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 - 4 T + T^{2}$$
$5$ $$5 + T^{2}$$
$7$ $$( 2 + T )^{2}$$
$11$ $$180 + T^{2}$$
$13$ $$( 8 + T )^{2}$$
$17$ $$180 + T^{2}$$
$19$ $$( 34 + T )^{2}$$
$23$ $$1620 + T^{2}$$
$29$ $$1620 + T^{2}$$
$31$ $$( 14 + T )^{2}$$
$37$ $$( 56 + T )^{2}$$
$41$ $$720 + T^{2}$$
$43$ $$( -8 + T )^{2}$$
$47$ $$1620 + T^{2}$$
$53$ $$1620 + T^{2}$$
$59$ $$180 + T^{2}$$
$61$ $$( -46 + T )^{2}$$
$67$ $$( -32 + T )^{2}$$
$71$ $$2880 + T^{2}$$
$73$ $$( 106 + T )^{2}$$
$79$ $$( -22 + T )^{2}$$
$83$ $$14580 + T^{2}$$
$89$ $$11520 + T^{2}$$
$97$ $$( -122 + T )^{2}$$