# 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: $$x^{2} - x + 1$$ 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 $$q + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{5} + 2 \zeta_{6} q^{9} + ( - 5 \zeta_{6} + 5) q^{11} - q^{13} + q^{15} + ( - 3 \zeta_{6} + 3) q^{17} - 6 \zeta_{6} q^{19} + 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 5 q^{27} - 9 q^{29} + 5 \zeta_{6} q^{33} - 6 \zeta_{6} q^{37} + ( - \zeta_{6} + 1) q^{39} - 8 q^{41} + 6 q^{43} + ( - 2 \zeta_{6} + 2) q^{45} + 3 \zeta_{6} q^{47} + 3 \zeta_{6} q^{51} + ( - 12 \zeta_{6} + 12) q^{53} - 5 q^{55} + 6 q^{57} + ( - 8 \zeta_{6} + 8) q^{59} - 4 \zeta_{6} q^{61} + \zeta_{6} q^{65} + ( - 4 \zeta_{6} + 4) q^{67} - 6 q^{69} + 8 q^{71} + ( - 10 \zeta_{6} + 10) q^{73} - \zeta_{6} q^{75} + 3 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{81} + 12 q^{83} - 3 q^{85} + ( - 9 \zeta_{6} + 9) q^{87} - 16 \zeta_{6} q^{89} + (6 \zeta_{6} - 6) q^{95} - 7 q^{97} + 10 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 - z * q^5 + 2*z * q^9 + (-5*z + 5) * q^11 - q^13 + q^15 + (-3*z + 3) * q^17 - 6*z * q^19 + 6*z * q^23 + (z - 1) * q^25 - 5 * q^27 - 9 * q^29 + 5*z * q^33 - 6*z * q^37 + (-z + 1) * q^39 - 8 * q^41 + 6 * q^43 + (-2*z + 2) * q^45 + 3*z * q^47 + 3*z * q^51 + (-12*z + 12) * q^53 - 5 * q^55 + 6 * q^57 + (-8*z + 8) * q^59 - 4*z * q^61 + z * q^65 + (-4*z + 4) * q^67 - 6 * q^69 + 8 * q^71 + (-10*z + 10) * q^73 - z * q^75 + 3*z * q^79 + (z - 1) * q^81 + 12 * q^83 - 3 * q^85 + (-9*z + 9) * q^87 - 16*z * q^89 + (6*z - 6) * q^95 - 7 * q^97 + 10 * 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 $$2 q - q^{3} - q^{5} + 2 q^{9} + 5 q^{11} - 2 q^{13} + 2 q^{15} + 3 q^{17} - 6 q^{19} + 6 q^{23} - q^{25} - 10 q^{27} - 18 q^{29} + 5 q^{33} - 6 q^{37} + q^{39} - 16 q^{41} + 12 q^{43} + 2 q^{45} + 3 q^{47} + 3 q^{51} + 12 q^{53} - 10 q^{55} + 12 q^{57} + 8 q^{59} - 4 q^{61} + q^{65} + 4 q^{67} - 12 q^{69} + 16 q^{71} + 10 q^{73} - q^{75} + 3 q^{79} - q^{81} + 24 q^{83} - 6 q^{85} + 9 q^{87} - 16 q^{89} - 6 q^{95} - 14 q^{97} + 20 q^{99}+O(q^{100})$$ 2 * q - q^3 - q^5 + 2 * q^9 + 5 * q^11 - 2 * q^13 + 2 * q^15 + 3 * q^17 - 6 * q^19 + 6 * q^23 - q^25 - 10 * q^27 - 18 * q^29 + 5 * q^33 - 6 * q^37 + q^39 - 16 * q^41 + 12 * q^43 + 2 * q^45 + 3 * q^47 + 3 * q^51 + 12 * q^53 - 10 * q^55 + 12 * q^57 + 8 * q^59 - 4 * q^61 + q^65 + 4 * q^67 - 12 * q^69 + 16 * q^71 + 10 * q^73 - q^75 + 3 * q^79 - q^81 + 24 * q^83 - 6 * q^85 + 9 * q^87 - 16 * q^89 - 6 * q^95 - 14 * q^97 + 20 * 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.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 $$S_{2}^{\mathrm{new}}(1960, [\chi])$$:

 $$T_{3}^{2} + T_{3} + 1$$ T3^2 + T3 + 1 $$T_{11}^{2} - 5T_{11} + 25$$ T11^2 - 5*T11 + 25 $$T_{13} + 1$$ T13 + 1

## Hecke characteristic polynomials

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