# Properties

 Label 1960.2.q.e 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: no (minimal twist has level 280) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( 5 - 5 \zeta_{6} ) q^{11} - q^{13} + q^{15} + ( 3 - 3 \zeta_{6} ) q^{17} -6 \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} -5 q^{27} -9 q^{29} + 5 \zeta_{6} q^{33} -6 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{39} -8 q^{41} + 6 q^{43} + ( 2 - 2 \zeta_{6} ) q^{45} + 3 \zeta_{6} q^{47} + 3 \zeta_{6} q^{51} + ( 12 - 12 \zeta_{6} ) q^{53} -5 q^{55} + 6 q^{57} + ( 8 - 8 \zeta_{6} ) q^{59} -4 \zeta_{6} q^{61} + \zeta_{6} q^{65} + ( 4 - 4 \zeta_{6} ) q^{67} -6 q^{69} + 8 q^{71} + ( 10 - 10 \zeta_{6} ) q^{73} -\zeta_{6} q^{75} + 3 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} + 12 q^{83} -3 q^{85} + ( 9 - 9 \zeta_{6} ) q^{87} -16 \zeta_{6} q^{89} + ( -6 + 6 \zeta_{6} ) q^{95} -7 q^{97} + 10 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} - q^{5} + 2q^{9} + O(q^{10})$$ $$2q - q^{3} - q^{5} + 2q^{9} + 5q^{11} - 2q^{13} + 2q^{15} + 3q^{17} - 6q^{19} + 6q^{23} - q^{25} - 10q^{27} - 18q^{29} + 5q^{33} - 6q^{37} + q^{39} - 16q^{41} + 12q^{43} + 2q^{45} + 3q^{47} + 3q^{51} + 12q^{53} - 10q^{55} + 12q^{57} + 8q^{59} - 4q^{61} + q^{65} + 4q^{67} - 12q^{69} + 16q^{71} + 10q^{73} - q^{75} + 3q^{79} - q^{81} + 24q^{83} - 6q^{85} + 9q^{87} - 16q^{89} - 6q^{95} - 14q^{97} + 20q^{99} + O(q^{100})$$

## 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.e 2
7.b odd 2 1 1960.2.q.m 2
7.c even 3 1 1960.2.a.k 1
7.c even 3 1 inner 1960.2.q.e 2
7.d odd 6 1 280.2.a.b 1
7.d odd 6 1 1960.2.q.m 2
21.g even 6 1 2520.2.a.p 1
28.f even 6 1 560.2.a.e 1
28.g odd 6 1 3920.2.a.r 1
35.i odd 6 1 1400.2.a.k 1
35.j even 6 1 9800.2.a.n 1
35.k even 12 2 1400.2.g.e 2
56.j odd 6 1 2240.2.a.v 1
56.m even 6 1 2240.2.a.j 1
84.j odd 6 1 5040.2.a.be 1
140.s even 6 1 2800.2.a.i 1
140.x odd 12 2 2800.2.g.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.b 1 7.d odd 6 1
560.2.a.e 1 28.f even 6 1
1400.2.a.k 1 35.i odd 6 1
1400.2.g.e 2 35.k even 12 2
1960.2.a.k 1 7.c even 3 1
1960.2.q.e 2 1.a even 1 1 trivial
1960.2.q.e 2 7.c even 3 1 inner
1960.2.q.m 2 7.b odd 2 1
1960.2.q.m 2 7.d odd 6 1
2240.2.a.j 1 56.m even 6 1
2240.2.a.v 1 56.j odd 6 1
2520.2.a.p 1 21.g even 6 1
2800.2.a.i 1 140.s even 6 1
2800.2.g.m 2 140.x odd 12 2
3920.2.a.r 1 28.g odd 6 1
5040.2.a.be 1 84.j odd 6 1
9800.2.a.n 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 $$S_{2}^{\mathrm{new}}(1960, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$9 - 3 T + T^{2}$$
$19$ $$36 + 6 T + T^{2}$$
$23$ $$36 - 6 T + T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$36 + 6 T + T^{2}$$
$41$ $$( 8 + T )^{2}$$
$43$ $$( -6 + T )^{2}$$
$47$ $$9 - 3 T + T^{2}$$
$53$ $$144 - 12 T + T^{2}$$
$59$ $$64 - 8 T + T^{2}$$
$61$ $$16 + 4 T + T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$100 - 10 T + T^{2}$$
$79$ $$9 - 3 T + T^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$256 + 16 T + T^{2}$$
$97$ $$( 7 + T )^{2}$$