# Properties

 Label 1960.2.q.k 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: $$x^{2} - x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^3 - z * q^5 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{11} - q^{15} + ( - 4 \zeta_{6} + 4) q^{17} - 2 \zeta_{6} q^{19} - \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 5 q^{27} + 9 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} + 2 \zeta_{6} q^{33} - 4 \zeta_{6} q^{37} - q^{41} + 9 q^{43} + ( - 2 \zeta_{6} + 2) q^{45} - 4 \zeta_{6} q^{51} + ( - 10 \zeta_{6} + 10) q^{53} + 2 q^{55} - 2 q^{57} + (10 \zeta_{6} - 10) q^{59} + 9 \zeta_{6} q^{61} + (5 \zeta_{6} - 5) q^{67} - q^{69} + 14 q^{71} + ( - 12 \zeta_{6} + 12) q^{73} + \zeta_{6} q^{75} - 14 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} - 11 q^{83} - 4 q^{85} + ( - 9 \zeta_{6} + 9) q^{87} - 15 \zeta_{6} q^{89} - 4 \zeta_{6} q^{93} + (2 \zeta_{6} - 2) q^{95} + 18 q^{97} - 4 q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 - z * q^5 + 2*z * q^9 + (2*z - 2) * q^11 - q^15 + (-4*z + 4) * q^17 - 2*z * q^19 - z * q^23 + (z - 1) * q^25 + 5 * q^27 + 9 * q^29 + (-4*z + 4) * q^31 + 2*z * q^33 - 4*z * q^37 - q^41 + 9 * q^43 + (-2*z + 2) * q^45 - 4*z * q^51 + (-10*z + 10) * q^53 + 2 * q^55 - 2 * q^57 + (10*z - 10) * q^59 + 9*z * q^61 + (5*z - 5) * q^67 - q^69 + 14 * q^71 + (-12*z + 12) * q^73 + z * q^75 - 14*z * q^79 + (z - 1) * q^81 - 11 * q^83 - 4 * q^85 + (-9*z + 9) * q^87 - 15*z * q^89 - 4*z * q^93 + (2*z - 2) * q^95 + 18 * q^97 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 - q^5 + 2 * q^9 $$2 q + q^{3} - q^{5} + 2 q^{9} - 2 q^{11} - 2 q^{15} + 4 q^{17} - 2 q^{19} - q^{23} - q^{25} + 10 q^{27} + 18 q^{29} + 4 q^{31} + 2 q^{33} - 4 q^{37} - 2 q^{41} + 18 q^{43} + 2 q^{45} - 4 q^{51} + 10 q^{53} + 4 q^{55} - 4 q^{57} - 10 q^{59} + 9 q^{61} - 5 q^{67} - 2 q^{69} + 28 q^{71} + 12 q^{73} + q^{75} - 14 q^{79} - q^{81} - 22 q^{83} - 8 q^{85} + 9 q^{87} - 15 q^{89} - 4 q^{93} - 2 q^{95} + 36 q^{97} - 8 q^{99}+O(q^{100})$$ 2 * q + q^3 - q^5 + 2 * q^9 - 2 * q^11 - 2 * q^15 + 4 * q^17 - 2 * q^19 - q^23 - q^25 + 10 * q^27 + 18 * q^29 + 4 * q^31 + 2 * q^33 - 4 * q^37 - 2 * q^41 + 18 * q^43 + 2 * q^45 - 4 * q^51 + 10 * q^53 + 4 * q^55 - 4 * q^57 - 10 * q^59 + 9 * q^61 - 5 * q^67 - 2 * q^69 + 28 * q^71 + 12 * q^73 + q^75 - 14 * q^79 - q^81 - 22 * q^83 - 8 * q^85 + 9 * q^87 - 15 * q^89 - 4 * q^93 - 2 * q^95 + 36 * q^97 - 8 * q^99

## 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.k 2
7.b odd 2 1 280.2.q.a 2
7.c even 3 1 1960.2.a.e 1
7.c even 3 1 inner 1960.2.q.k 2
7.d odd 6 1 280.2.q.a 2
7.d odd 6 1 1960.2.a.i 1
21.c even 2 1 2520.2.bi.a 2
21.g even 6 1 2520.2.bi.a 2
28.d even 2 1 560.2.q.h 2
28.f even 6 1 560.2.q.h 2
28.f even 6 1 3920.2.a.m 1
28.g odd 6 1 3920.2.a.y 1
35.c odd 2 1 1400.2.q.e 2
35.f even 4 2 1400.2.bh.b 4
35.i odd 6 1 1400.2.q.e 2
35.i odd 6 1 9800.2.a.r 1
35.j even 6 1 9800.2.a.bc 1
35.k even 12 2 1400.2.bh.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.a 2 7.b odd 2 1
280.2.q.a 2 7.d odd 6 1
560.2.q.h 2 28.d even 2 1
560.2.q.h 2 28.f even 6 1
1400.2.q.e 2 35.c odd 2 1
1400.2.q.e 2 35.i odd 6 1
1400.2.bh.b 4 35.f even 4 2
1400.2.bh.b 4 35.k even 12 2
1960.2.a.e 1 7.c even 3 1
1960.2.a.i 1 7.d odd 6 1
1960.2.q.k 2 1.a even 1 1 trivial
1960.2.q.k 2 7.c even 3 1 inner
2520.2.bi.a 2 21.c even 2 1
2520.2.bi.a 2 21.g even 6 1
3920.2.a.m 1 28.f even 6 1
3920.2.a.y 1 28.g odd 6 1
9800.2.a.r 1 35.i odd 6 1
9800.2.a.bc 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 $$S_{2}^{\mathrm{new}}(1960, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1 $$T_{11}^{2} + 2T_{11} + 4$$ T11^2 + 2*T11 + 4 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$T^{2}$$
$17$ $$T^{2} - 4T + 16$$
$19$ $$T^{2} + 2T + 4$$
$23$ $$T^{2} + T + 1$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2} - 4T + 16$$
$37$ $$T^{2} + 4T + 16$$
$41$ $$(T + 1)^{2}$$
$43$ $$(T - 9)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 10T + 100$$
$59$ $$T^{2} + 10T + 100$$
$61$ $$T^{2} - 9T + 81$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$(T - 14)^{2}$$
$73$ $$T^{2} - 12T + 144$$
$79$ $$T^{2} + 14T + 196$$
$83$ $$(T + 11)^{2}$$
$89$ $$T^{2} + 15T + 225$$
$97$ $$(T - 18)^{2}$$