# Properties

 Label 960.2.o.a Level $960$ Weight $2$ Character orbit 960.o Analytic conductor $7.666$ Analytic rank $0$ Dimension $4$ CM discriminant -15 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$7.66563859404$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} -\beta_{1} q^{5} -3 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} -\beta_{1} q^{5} -3 q^{9} + \beta_{3} q^{15} -2 \beta_{1} q^{17} + 2 \beta_{3} q^{19} + 2 \beta_{2} q^{23} + 5 q^{25} -3 \beta_{2} q^{27} + 2 \beta_{3} q^{31} + 3 \beta_{1} q^{45} -6 \beta_{2} q^{47} -7 q^{49} + 2 \beta_{3} q^{51} -2 \beta_{1} q^{53} + 6 \beta_{1} q^{57} + 2 q^{61} -6 q^{69} + 5 \beta_{2} q^{75} + 2 \beta_{3} q^{79} + 9 q^{81} + 2 \beta_{2} q^{83} + 10 q^{85} + 6 \beta_{1} q^{93} + 10 \beta_{2} q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{9} + O(q^{10})$$ $$4q - 12q^{9} + 20q^{25} - 28q^{49} + 8q^{61} - 24q^{69} + 36q^{81} + 40q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2$$ $$\beta_{2}$$ $$=$$ $$-\nu^{3} + 2 \nu^{2} - 2 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{3} - 2 \nu^{2} + 6 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{2} + \beta_{1} - 3$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$\beta_{1} - 2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
959.1
 −0.309017 + 0.535233i 0.809017 − 1.40126i −0.309017 − 0.535233i 0.809017 + 1.40126i
0 1.73205i 0 −2.23607 0 0 0 −3.00000 0
959.2 0 1.73205i 0 2.23607 0 0 0 −3.00000 0
959.3 0 1.73205i 0 −2.23607 0 0 0 −3.00000 0
959.4 0 1.73205i 0 2.23607 0 0 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 960.2.o.a 4
3.b odd 2 1 inner 960.2.o.a 4
4.b odd 2 1 inner 960.2.o.a 4
5.b even 2 1 inner 960.2.o.a 4
8.b even 2 1 60.2.h.b 4
8.d odd 2 1 60.2.h.b 4
12.b even 2 1 inner 960.2.o.a 4
15.d odd 2 1 CM 960.2.o.a 4
20.d odd 2 1 inner 960.2.o.a 4
24.f even 2 1 60.2.h.b 4
24.h odd 2 1 60.2.h.b 4
40.e odd 2 1 60.2.h.b 4
40.f even 2 1 60.2.h.b 4
40.i odd 4 2 300.2.e.a 4
40.k even 4 2 300.2.e.a 4
60.h even 2 1 inner 960.2.o.a 4
120.i odd 2 1 60.2.h.b 4
120.m even 2 1 60.2.h.b 4
120.q odd 4 2 300.2.e.a 4
120.w even 4 2 300.2.e.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.h.b 4 8.b even 2 1
60.2.h.b 4 8.d odd 2 1
60.2.h.b 4 24.f even 2 1
60.2.h.b 4 24.h odd 2 1
60.2.h.b 4 40.e odd 2 1
60.2.h.b 4 40.f even 2 1
60.2.h.b 4 120.i odd 2 1
60.2.h.b 4 120.m even 2 1
300.2.e.a 4 40.i odd 4 2
300.2.e.a 4 40.k even 4 2
300.2.e.a 4 120.q odd 4 2
300.2.e.a 4 120.w even 4 2
960.2.o.a 4 1.a even 1 1 trivial
960.2.o.a 4 3.b odd 2 1 inner
960.2.o.a 4 4.b odd 2 1 inner
960.2.o.a 4 5.b even 2 1 inner
960.2.o.a 4 12.b even 2 1 inner
960.2.o.a 4 15.d odd 2 1 CM
960.2.o.a 4 20.d odd 2 1 inner
960.2.o.a 4 60.h even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(960, [\chi])$$:

 $$T_{7}$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 3 + T^{2} )^{2}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -20 + T^{2} )^{2}$$
$19$ $$( 60 + T^{2} )^{2}$$
$23$ $$( 12 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 60 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$( 108 + T^{2} )^{2}$$
$53$ $$( -20 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( -2 + T )^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$( 60 + T^{2} )^{2}$$
$83$ $$( 12 + T^{2} )^{2}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$