# Properties

 Label 1960.2.q.b Level $1960$ Weight $2$ Character orbit 1960.q Analytic conductor $15.651$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1960 = 2^{3} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1960.q (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.6506787962$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{5} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{5} -\zeta_{6} q^{9} + ( -4 + 4 \zeta_{6} ) q^{11} + 2 q^{13} + 2 q^{15} -2 \zeta_{6} q^{19} + 4 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -4 q^{27} + 10 q^{29} + ( -4 + 4 \zeta_{6} ) q^{31} -8 \zeta_{6} q^{33} + 2 \zeta_{6} q^{37} + ( -4 + 4 \zeta_{6} ) q^{39} -12 q^{41} -4 q^{43} + ( -1 + \zeta_{6} ) q^{45} -4 \zeta_{6} q^{47} + ( -2 + 2 \zeta_{6} ) q^{53} + 4 q^{55} + 4 q^{57} + ( -10 + 10 \zeta_{6} ) q^{59} -6 \zeta_{6} q^{61} -2 \zeta_{6} q^{65} + ( -4 + 4 \zeta_{6} ) q^{67} -8 q^{69} -12 q^{71} + ( 4 - 4 \zeta_{6} ) q^{73} -2 \zeta_{6} q^{75} + 4 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} + 14 q^{83} + ( -20 + 20 \zeta_{6} ) q^{87} -8 \zeta_{6} q^{89} -8 \zeta_{6} q^{93} + ( -2 + 2 \zeta_{6} ) q^{95} -8 q^{97} + 4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - q^{5} - q^{9} + O(q^{10})$$ $$2q - 2q^{3} - q^{5} - q^{9} - 4q^{11} + 4q^{13} + 4q^{15} - 2q^{19} + 4q^{23} - q^{25} - 8q^{27} + 20q^{29} - 4q^{31} - 8q^{33} + 2q^{37} - 4q^{39} - 24q^{41} - 8q^{43} - q^{45} - 4q^{47} - 2q^{53} + 8q^{55} + 8q^{57} - 10q^{59} - 6q^{61} - 2q^{65} - 4q^{67} - 16q^{69} - 24q^{71} + 4q^{73} - 2q^{75} + 4q^{79} + 11q^{81} + 28q^{83} - 20q^{87} - 8q^{89} - 8q^{93} - 2q^{95} - 16q^{97} + 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1960\mathbb{Z}\right)^\times$$.

 $$n$$ $$981$$ $$1081$$ $$1177$$ $$1471$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.00000 + 1.73205i 0 −0.500000 0.866025i 0 0 0 −0.500000 0.866025i 0
961.1 0 −1.00000 1.73205i 0 −0.500000 + 0.866025i 0 0 0 −0.500000 + 0.866025i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.2.q.b 2
7.b odd 2 1 1960.2.q.n 2
7.c even 3 1 1960.2.a.n yes 1
7.c even 3 1 inner 1960.2.q.b 2
7.d odd 6 1 1960.2.a.b 1
7.d odd 6 1 1960.2.q.n 2
28.f even 6 1 3920.2.a.bd 1
28.g odd 6 1 3920.2.a.f 1
35.i odd 6 1 9800.2.a.bn 1
35.j even 6 1 9800.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.b 1 7.d odd 6 1
1960.2.a.n yes 1 7.c even 3 1
1960.2.q.b 2 1.a even 1 1 trivial
1960.2.q.b 2 7.c even 3 1 inner
1960.2.q.n 2 7.b odd 2 1
1960.2.q.n 2 7.d odd 6 1
3920.2.a.f 1 28.g odd 6 1
3920.2.a.bd 1 28.f even 6 1
9800.2.a.i 1 35.j even 6 1
9800.2.a.bn 1 35.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1960, [\chi])$$:

 $$T_{3}^{2} + 2 T_{3} + 4$$ $$T_{11}^{2} + 4 T_{11} + 16$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 + 2 T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$16 + 4 T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$16 - 4 T + T^{2}$$
$29$ $$( -10 + T )^{2}$$
$31$ $$16 + 4 T + T^{2}$$
$37$ $$4 - 2 T + T^{2}$$
$41$ $$( 12 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$16 + 4 T + T^{2}$$
$53$ $$4 + 2 T + T^{2}$$
$59$ $$100 + 10 T + T^{2}$$
$61$ $$36 + 6 T + T^{2}$$
$67$ $$16 + 4 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$16 - 4 T + T^{2}$$
$79$ $$16 - 4 T + T^{2}$$
$83$ $$( -14 + T )^{2}$$
$89$ $$64 + 8 T + T^{2}$$
$97$ $$( 8 + T )^{2}$$