# Properties

 Label 1960.2.q.l 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( 5 - 5 \zeta_{6} ) q^{11} -7 q^{13} - q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} -2 \zeta_{6} q^{19} -8 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + 5 q^{27} -5 q^{29} + ( -10 + 10 \zeta_{6} ) q^{31} -5 \zeta_{6} q^{33} -4 \zeta_{6} q^{37} + ( -7 + 7 \zeta_{6} ) q^{39} + 6 q^{41} + 2 q^{43} + ( 2 - 2 \zeta_{6} ) q^{45} -7 \zeta_{6} q^{47} + 3 \zeta_{6} q^{51} + ( 10 - 10 \zeta_{6} ) q^{53} -5 q^{55} -2 q^{57} + ( -10 + 10 \zeta_{6} ) q^{59} -12 \zeta_{6} q^{61} + 7 \zeta_{6} q^{65} + ( 2 - 2 \zeta_{6} ) q^{67} -8 q^{69} + ( -2 + 2 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + 7 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -4 q^{83} + 3 q^{85} + ( -5 + 5 \zeta_{6} ) q^{87} -8 \zeta_{6} q^{89} + 10 \zeta_{6} q^{93} + ( -2 + 2 \zeta_{6} ) q^{95} -17 q^{97} + 10 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - q^{5} + 2q^{9} + O(q^{10})$$ $$2q + q^{3} - q^{5} + 2q^{9} + 5q^{11} - 14q^{13} - 2q^{15} - 3q^{17} - 2q^{19} - 8q^{23} - q^{25} + 10q^{27} - 10q^{29} - 10q^{31} - 5q^{33} - 4q^{37} - 7q^{39} + 12q^{41} + 4q^{43} + 2q^{45} - 7q^{47} + 3q^{51} + 10q^{53} - 10q^{55} - 4q^{57} - 10q^{59} - 12q^{61} + 7q^{65} + 2q^{67} - 16q^{69} - 2q^{73} + q^{75} + 7q^{79} - q^{81} - 8q^{83} + 6q^{85} - 5q^{87} - 8q^{89} + 10q^{93} - 2q^{95} - 34q^{97} + 20q^{99} + O(q^{100})$$

## 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 0.500000 0.866025i 0 −0.500000 0.866025i 0 0 0 1.00000 + 1.73205i 0
961.1 0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 0 0 1.00000 1.73205i 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.l 2
7.b odd 2 1 1960.2.q.g 2
7.c even 3 1 1960.2.a.d 1
7.c even 3 1 inner 1960.2.q.l 2
7.d odd 6 1 1960.2.a.h yes 1
7.d odd 6 1 1960.2.q.g 2
28.f even 6 1 3920.2.a.o 1
28.g odd 6 1 3920.2.a.bb 1
35.i odd 6 1 9800.2.a.m 1
35.j even 6 1 9800.2.a.y 1

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

## Hecke kernels

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

 $$T_{3}^{2} - T_{3} + 1$$ $$T_{11}^{2} - 5 T_{11} + 25$$ $$T_{13} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$( 7 + T )^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$4 + 2 T + T^{2}$$
$23$ $$64 + 8 T + T^{2}$$
$29$ $$( 5 + T )^{2}$$
$31$ $$100 + 10 T + T^{2}$$
$37$ $$16 + 4 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( -2 + T )^{2}$$
$47$ $$49 + 7 T + T^{2}$$
$53$ $$100 - 10 T + T^{2}$$
$59$ $$100 + 10 T + T^{2}$$
$61$ $$144 + 12 T + T^{2}$$
$67$ $$4 - 2 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$4 + 2 T + T^{2}$$
$79$ $$49 - 7 T + T^{2}$$
$83$ $$( 4 + T )^{2}$$
$89$ $$64 + 8 T + T^{2}$$
$97$ $$( 17 + T )^{2}$$