# Properties

 Label 960.3.j.b Level $960$ Weight $3$ Character orbit 960.j Analytic conductor $26.158$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ 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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} -5 \zeta_{12}^{3} q^{5} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{7} + 3 q^{9} +O(q^{10})$$ $$q + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{3} -5 \zeta_{12}^{3} q^{5} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{7} + 3 q^{9} + ( -6 + 12 \zeta_{12}^{2} ) q^{11} -18 \zeta_{12}^{3} q^{13} + ( -5 + 10 \zeta_{12}^{2} ) q^{15} -10 \zeta_{12}^{3} q^{17} + ( 8 - 16 \zeta_{12}^{2} ) q^{19} + 18 q^{21} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{23} -25 q^{25} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} -36 q^{29} + ( -4 + 8 \zeta_{12}^{2} ) q^{31} -18 \zeta_{12}^{3} q^{33} + ( -30 + 60 \zeta_{12}^{2} ) q^{35} + 54 \zeta_{12}^{3} q^{37} + ( -18 + 36 \zeta_{12}^{2} ) q^{39} + 18 q^{41} + ( -24 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{43} -15 \zeta_{12}^{3} q^{45} + 59 q^{49} + ( -10 + 20 \zeta_{12}^{2} ) q^{51} + 26 \zeta_{12}^{3} q^{53} + ( 60 \zeta_{12} - 30 \zeta_{12}^{3} ) q^{55} + 24 \zeta_{12}^{3} q^{57} + ( -18 + 36 \zeta_{12}^{2} ) q^{59} + 74 q^{61} + ( -36 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{63} -90 q^{65} + ( 48 \zeta_{12} - 24 \zeta_{12}^{3} ) q^{67} -12 q^{69} + ( -60 + 120 \zeta_{12}^{2} ) q^{71} + 36 \zeta_{12}^{3} q^{73} + ( 50 \zeta_{12} - 25 \zeta_{12}^{3} ) q^{75} -108 \zeta_{12}^{3} q^{77} + ( -52 + 104 \zeta_{12}^{2} ) q^{79} + 9 q^{81} + ( 104 \zeta_{12} - 52 \zeta_{12}^{3} ) q^{83} -50 q^{85} + ( 72 \zeta_{12} - 36 \zeta_{12}^{3} ) q^{87} + 18 q^{89} + ( -108 + 216 \zeta_{12}^{2} ) q^{91} -12 \zeta_{12}^{3} q^{93} + ( -80 \zeta_{12} + 40 \zeta_{12}^{3} ) q^{95} + 72 \zeta_{12}^{3} q^{97} + ( -18 + 36 \zeta_{12}^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} + 72q^{21} - 100q^{25} - 144q^{29} + 72q^{41} + 236q^{49} + 296q^{61} - 360q^{65} - 48q^{69} + 36q^{81} - 200q^{85} + 72q^{89} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
319.1
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i
0 −1.73205 0 5.00000i 0 −10.3923 0 3.00000 0
319.2 0 −1.73205 0 5.00000i 0 −10.3923 0 3.00000 0
319.3 0 1.73205 0 5.00000i 0 10.3923 0 3.00000 0
319.4 0 1.73205 0 5.00000i 0 10.3923 0 3.00000 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
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.b 4
4.b odd 2 1 inner 960.3.j.b 4
5.b even 2 1 inner 960.3.j.b 4
8.b even 2 1 60.3.f.a 4
8.d odd 2 1 60.3.f.a 4
20.d odd 2 1 inner 960.3.j.b 4
24.f even 2 1 180.3.f.e 4
24.h odd 2 1 180.3.f.e 4
40.e odd 2 1 60.3.f.a 4
40.f even 2 1 60.3.f.a 4
40.i odd 4 1 300.3.c.a 2
40.i odd 4 1 300.3.c.c 2
40.k even 4 1 300.3.c.a 2
40.k even 4 1 300.3.c.c 2
120.i odd 2 1 180.3.f.e 4
120.m even 2 1 180.3.f.e 4
120.q odd 4 1 900.3.c.f 2
120.q odd 4 1 900.3.c.j 2
120.w even 4 1 900.3.c.f 2
120.w even 4 1 900.3.c.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.a 4 8.b even 2 1
60.3.f.a 4 8.d odd 2 1
60.3.f.a 4 40.e odd 2 1
60.3.f.a 4 40.f even 2 1
180.3.f.e 4 24.f even 2 1
180.3.f.e 4 24.h odd 2 1
180.3.f.e 4 120.i odd 2 1
180.3.f.e 4 120.m even 2 1
300.3.c.a 2 40.i odd 4 1
300.3.c.a 2 40.k even 4 1
300.3.c.c 2 40.i odd 4 1
300.3.c.c 2 40.k even 4 1
900.3.c.f 2 120.q odd 4 1
900.3.c.f 2 120.w even 4 1
900.3.c.j 2 120.q odd 4 1
900.3.c.j 2 120.w even 4 1
960.3.j.b 4 1.a even 1 1 trivial
960.3.j.b 4 4.b odd 2 1 inner
960.3.j.b 4 5.b even 2 1 inner
960.3.j.b 4 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 $$S_{3}^{\mathrm{new}}(960, [\chi])$$:

 $$T_{7}^{2} - 108$$ $$T_{11}^{2} + 108$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -3 + T^{2} )^{2}$$
$5$ $$( 25 + T^{2} )^{2}$$
$7$ $$( -108 + T^{2} )^{2}$$
$11$ $$( 108 + T^{2} )^{2}$$
$13$ $$( 324 + T^{2} )^{2}$$
$17$ $$( 100 + T^{2} )^{2}$$
$19$ $$( 192 + T^{2} )^{2}$$
$23$ $$( -48 + T^{2} )^{2}$$
$29$ $$( 36 + T )^{4}$$
$31$ $$( 48 + T^{2} )^{2}$$
$37$ $$( 2916 + T^{2} )^{2}$$
$41$ $$( -18 + T )^{4}$$
$43$ $$( -432 + T^{2} )^{2}$$
$47$ $$T^{4}$$
$53$ $$( 676 + T^{2} )^{2}$$
$59$ $$( 972 + T^{2} )^{2}$$
$61$ $$( -74 + T )^{4}$$
$67$ $$( -1728 + T^{2} )^{2}$$
$71$ $$( 10800 + T^{2} )^{2}$$
$73$ $$( 1296 + T^{2} )^{2}$$
$79$ $$( 8112 + T^{2} )^{2}$$
$83$ $$( -8112 + T^{2} )^{2}$$
$89$ $$( -18 + T )^{4}$$
$97$ $$( 5184 + T^{2} )^{2}$$