# Properties

 Label 960.2.o.c Level $960$ Weight $2$ Character orbit 960.o Analytic conductor $7.666$ Analytic rank $0$ Dimension $4$ CM discriminant -20 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: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{3} q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{7} + ( 2 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{3} q^{5} + ( 2 \beta_{1} - \beta_{2} ) q^{7} + ( 2 + \beta_{3} ) q^{9} + ( -\beta_{1} + 3 \beta_{2} ) q^{15} + ( 5 + \beta_{3} ) q^{21} + \beta_{2} q^{23} -5 q^{25} + ( \beta_{1} + 3 \beta_{2} ) q^{27} -4 \beta_{3} q^{29} + 5 \beta_{2} q^{35} -2 \beta_{3} q^{41} + ( -2 \beta_{1} + \beta_{2} ) q^{43} + ( -5 + 2 \beta_{3} ) q^{45} + 7 \beta_{2} q^{47} + 3 q^{49} -8 q^{61} + ( 4 \beta_{1} + 3 \beta_{2} ) q^{63} + ( 10 \beta_{1} - 5 \beta_{2} ) q^{67} + ( -1 + \beta_{3} ) q^{69} -5 \beta_{1} q^{75} + ( -1 + 4 \beta_{3} ) q^{81} + 11 \beta_{2} q^{83} + ( 4 \beta_{1} - 12 \beta_{2} ) q^{87} -8 \beta_{3} q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{9} + O(q^{10})$$ $$4q + 8q^{9} + 20q^{21} - 20q^{25} - 20q^{45} + 12q^{49} - 32q^{61} - 4q^{69} - 4q^{81} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
959.1
 −1.58114 − 0.707107i −1.58114 + 0.707107i 1.58114 − 0.707107i 1.58114 + 0.707107i
0 −1.58114 0.707107i 0 2.23607i 0 −3.16228 0 2.00000 + 2.23607i 0
959.2 0 −1.58114 + 0.707107i 0 2.23607i 0 −3.16228 0 2.00000 2.23607i 0
959.3 0 1.58114 0.707107i 0 2.23607i 0 3.16228 0 2.00000 2.23607i 0
959.4 0 1.58114 + 0.707107i 0 2.23607i 0 3.16228 0 2.00000 + 2.23607i 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.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.c 4
3.b odd 2 1 inner 960.2.o.c 4
4.b odd 2 1 inner 960.2.o.c 4
5.b even 2 1 inner 960.2.o.c 4
8.b even 2 1 60.2.h.a 4
8.d odd 2 1 60.2.h.a 4
12.b even 2 1 inner 960.2.o.c 4
15.d odd 2 1 inner 960.2.o.c 4
20.d odd 2 1 CM 960.2.o.c 4
24.f even 2 1 60.2.h.a 4
24.h odd 2 1 60.2.h.a 4
40.e odd 2 1 60.2.h.a 4
40.f even 2 1 60.2.h.a 4
40.i odd 4 2 300.2.e.b 4
40.k even 4 2 300.2.e.b 4
60.h even 2 1 inner 960.2.o.c 4
120.i odd 2 1 60.2.h.a 4
120.m even 2 1 60.2.h.a 4
120.q odd 4 2 300.2.e.b 4
120.w even 4 2 300.2.e.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.2.h.a 4 8.b even 2 1
60.2.h.a 4 8.d odd 2 1
60.2.h.a 4 24.f even 2 1
60.2.h.a 4 24.h odd 2 1
60.2.h.a 4 40.e odd 2 1
60.2.h.a 4 40.f even 2 1
60.2.h.a 4 120.i odd 2 1
60.2.h.a 4 120.m even 2 1
300.2.e.b 4 40.i odd 4 2
300.2.e.b 4 40.k even 4 2
300.2.e.b 4 120.q odd 4 2
300.2.e.b 4 120.w even 4 2
960.2.o.c 4 1.a even 1 1 trivial
960.2.o.c 4 3.b odd 2 1 inner
960.2.o.c 4 4.b odd 2 1 inner
960.2.o.c 4 5.b even 2 1 inner
960.2.o.c 4 12.b even 2 1 inner
960.2.o.c 4 15.d odd 2 1 inner
960.2.o.c 4 20.d odd 2 1 CM
960.2.o.c 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 $$S_{2}^{\mathrm{new}}(960, [\chi])$$:

 $$T_{7}^{2} - 10$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 - 4 T^{2} + T^{4}$$
$5$ $$( 5 + T^{2} )^{2}$$
$7$ $$( -10 + T^{2} )^{2}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 2 + T^{2} )^{2}$$
$29$ $$( 80 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( 20 + T^{2} )^{2}$$
$43$ $$( -10 + T^{2} )^{2}$$
$47$ $$( 98 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( 8 + T )^{4}$$
$67$ $$( -250 + T^{2} )^{2}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$( 242 + T^{2} )^{2}$$
$89$ $$( 320 + T^{2} )^{2}$$
$97$ $$T^{4}$$