# Properties

 Label 1960.2.q.d 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 $$q + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} - 4 q^{13} + q^{15} + 6 \zeta_{6} q^{19} - 3 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 5 q^{27} - 3 q^{29} + 2 \zeta_{6} q^{33} + 12 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{39} + 7 q^{41} - 9 q^{43} + ( - 2 \zeta_{6} + 2) q^{45} + ( - 6 \zeta_{6} + 6) q^{53} - 2 q^{55} - 6 q^{57} + (10 \zeta_{6} - 10) q^{59} + 5 \zeta_{6} q^{61} + 4 \zeta_{6} q^{65} + (11 \zeta_{6} - 11) q^{67} + 3 q^{69} - 10 q^{71} + (8 \zeta_{6} - 8) q^{73} - \zeta_{6} q^{75} - 6 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 3 q^{83} + ( - 3 \zeta_{6} + 3) q^{87} + 17 \zeta_{6} q^{89} + ( - 6 \zeta_{6} + 6) q^{95} + 2 q^{97} + 4 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 - z * q^5 + 2*z * q^9 + (-2*z + 2) * q^11 - 4 * q^13 + q^15 + 6*z * q^19 - 3*z * q^23 + (z - 1) * q^25 - 5 * q^27 - 3 * q^29 + 2*z * q^33 + 12*z * q^37 + (-4*z + 4) * q^39 + 7 * q^41 - 9 * q^43 + (-2*z + 2) * q^45 + (-6*z + 6) * q^53 - 2 * q^55 - 6 * q^57 + (10*z - 10) * q^59 + 5*z * q^61 + 4*z * q^65 + (11*z - 11) * q^67 + 3 * q^69 - 10 * q^71 + (8*z - 8) * q^73 - z * q^75 - 6*z * q^79 + (z - 1) * q^81 + 3 * q^83 + (-3*z + 3) * q^87 + 17*z * q^89 + (-6*z + 6) * q^95 + 2 * 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 $$2 q - q^{3} - q^{5} + 2 q^{9} + 2 q^{11} - 8 q^{13} + 2 q^{15} + 6 q^{19} - 3 q^{23} - q^{25} - 10 q^{27} - 6 q^{29} + 2 q^{33} + 12 q^{37} + 4 q^{39} + 14 q^{41} - 18 q^{43} + 2 q^{45} + 6 q^{53} - 4 q^{55} - 12 q^{57} - 10 q^{59} + 5 q^{61} + 4 q^{65} - 11 q^{67} + 6 q^{69} - 20 q^{71} - 8 q^{73} - q^{75} - 6 q^{79} - q^{81} + 6 q^{83} + 3 q^{87} + 17 q^{89} + 6 q^{95} + 4 q^{97} + 8 q^{99}+O(q^{100})$$ 2 * q - q^3 - q^5 + 2 * q^9 + 2 * q^11 - 8 * q^13 + 2 * q^15 + 6 * q^19 - 3 * q^23 - q^25 - 10 * q^27 - 6 * q^29 + 2 * q^33 + 12 * q^37 + 4 * q^39 + 14 * q^41 - 18 * q^43 + 2 * q^45 + 6 * q^53 - 4 * q^55 - 12 * q^57 - 10 * q^59 + 5 * q^61 + 4 * q^65 - 11 * q^67 + 6 * q^69 - 20 * q^71 - 8 * q^73 - q^75 - 6 * q^79 - q^81 + 6 * q^83 + 3 * q^87 + 17 * q^89 + 6 * q^95 + 4 * 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.d 2
7.b odd 2 1 280.2.q.b 2
7.c even 3 1 1960.2.a.l 1
7.c even 3 1 inner 1960.2.q.d 2
7.d odd 6 1 280.2.q.b 2
7.d odd 6 1 1960.2.a.c 1
21.c even 2 1 2520.2.bi.d 2
21.g even 6 1 2520.2.bi.d 2
28.d even 2 1 560.2.q.e 2
28.f even 6 1 560.2.q.e 2
28.f even 6 1 3920.2.a.v 1
28.g odd 6 1 3920.2.a.q 1
35.c odd 2 1 1400.2.q.c 2
35.f even 4 2 1400.2.bh.c 4
35.i odd 6 1 1400.2.q.c 2
35.i odd 6 1 9800.2.a.z 1
35.j even 6 1 9800.2.a.o 1
35.k even 12 2 1400.2.bh.c 4

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 6T + 36$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$(T + 3)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} - 12T + 144$$
$41$ $$(T - 7)^{2}$$
$43$ $$(T + 9)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} + 10T + 100$$
$61$ $$T^{2} - 5T + 25$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$(T + 10)^{2}$$
$73$ $$T^{2} + 8T + 64$$
$79$ $$T^{2} + 6T + 36$$
$83$ $$(T - 3)^{2}$$
$89$ $$T^{2} - 17T + 289$$
$97$ $$(T - 2)^{2}$$