# Properties

 Label 1960.2.a.v Level $1960$ Weight $2$ Character orbit 1960.a Self dual yes Analytic conductor $15.651$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1960 = 2^{3} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1960.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.6506787962$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1944.1 Defining polynomial: $$x^{3} - 9x - 6$$ x^3 - 9*x - 6 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - q^{5} + (\beta_{2} + \beta_1 + 3) q^{9}+O(q^{10})$$ q + b1 * q^3 - q^5 + (b2 + b1 + 3) * q^9 $$q + \beta_1 q^{3} - q^{5} + (\beta_{2} + \beta_1 + 3) q^{9} + (\beta_{2} + \beta_1 + 1) q^{11} + (\beta_{2} - \beta_1 - 1) q^{13} - \beta_1 q^{15} - 2 q^{17} + ( - \beta_{2} + \beta_1 - 1) q^{19} + ( - \beta_{2} + 1) q^{23} + q^{25} + (3 \beta_1 + 6) q^{27} + ( - \beta_{2} - \beta_1 + 4) q^{29} + (2 \beta_1 - 4) q^{31} + (4 \beta_1 + 6) q^{33} + ( - \beta_{2} + \beta_1 + 3) q^{37} + ( - 2 \beta_{2} - 6) q^{39} + ( - 2 \beta_{2} - 3) q^{41} + ( - \beta_1 + 4) q^{43} + ( - \beta_{2} - \beta_1 - 3) q^{45} + (3 \beta_{2} + \beta_1 + 5) q^{47} - 2 \beta_1 q^{51} + ( - \beta_{2} - 3 \beta_1 + 3) q^{53} + ( - \beta_{2} - \beta_1 - 1) q^{55} + (2 \beta_{2} - 2 \beta_1 + 6) q^{57} - 8 q^{59} + ( - \beta_{2} - 3 \beta_1 + 2) q^{61} + ( - \beta_{2} + \beta_1 + 1) q^{65} + (3 \beta_1 + 2) q^{67} + (\beta_{2} - \beta_1) q^{69} + (2 \beta_{2} - 2 \beta_1) q^{73} + \beta_1 q^{75} + (2 \beta_{2} - 2 \beta_1 - 6) q^{79} + (6 \beta_1 + 9) q^{81} + ( - \beta_1 + 10) q^{83} + 2 q^{85} + (\beta_1 - 6) q^{87} + (\beta_{2} - \beta_1) q^{89} + (2 \beta_{2} - 2 \beta_1 + 12) q^{93} + (\beta_{2} - \beta_1 + 1) q^{95} + 2 q^{97} + (\beta_{2} + 7 \beta_1 + 21) q^{99}+O(q^{100})$$ q + b1 * q^3 - q^5 + (b2 + b1 + 3) * q^9 + (b2 + b1 + 1) * q^11 + (b2 - b1 - 1) * q^13 - b1 * q^15 - 2 * q^17 + (-b2 + b1 - 1) * q^19 + (-b2 + 1) * q^23 + q^25 + (3*b1 + 6) * q^27 + (-b2 - b1 + 4) * q^29 + (2*b1 - 4) * q^31 + (4*b1 + 6) * q^33 + (-b2 + b1 + 3) * q^37 + (-2*b2 - 6) * q^39 + (-2*b2 - 3) * q^41 + (-b1 + 4) * q^43 + (-b2 - b1 - 3) * q^45 + (3*b2 + b1 + 5) * q^47 - 2*b1 * q^51 + (-b2 - 3*b1 + 3) * q^53 + (-b2 - b1 - 1) * q^55 + (2*b2 - 2*b1 + 6) * q^57 - 8 * q^59 + (-b2 - 3*b1 + 2) * q^61 + (-b2 + b1 + 1) * q^65 + (3*b1 + 2) * q^67 + (b2 - b1) * q^69 + (2*b2 - 2*b1) * q^73 + b1 * q^75 + (2*b2 - 2*b1 - 6) * q^79 + (6*b1 + 9) * q^81 + (-b1 + 10) * q^83 + 2 * q^85 + (b1 - 6) * q^87 + (b2 - b1) * q^89 + (2*b2 - 2*b1 + 12) * q^93 + (b2 - b1 + 1) * q^95 + 2 * q^97 + (b2 + 7*b1 + 21) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} + 9 q^{9}+O(q^{10})$$ 3 * q - 3 * q^5 + 9 * q^9 $$3 q - 3 q^{5} + 9 q^{9} + 3 q^{11} - 3 q^{13} - 6 q^{17} - 3 q^{19} + 3 q^{23} + 3 q^{25} + 18 q^{27} + 12 q^{29} - 12 q^{31} + 18 q^{33} + 9 q^{37} - 18 q^{39} - 9 q^{41} + 12 q^{43} - 9 q^{45} + 15 q^{47} + 9 q^{53} - 3 q^{55} + 18 q^{57} - 24 q^{59} + 6 q^{61} + 3 q^{65} + 6 q^{67} - 18 q^{79} + 27 q^{81} + 30 q^{83} + 6 q^{85} - 18 q^{87} + 36 q^{93} + 3 q^{95} + 6 q^{97} + 63 q^{99}+O(q^{100})$$ 3 * q - 3 * q^5 + 9 * q^9 + 3 * q^11 - 3 * q^13 - 6 * q^17 - 3 * q^19 + 3 * q^23 + 3 * q^25 + 18 * q^27 + 12 * q^29 - 12 * q^31 + 18 * q^33 + 9 * q^37 - 18 * q^39 - 9 * q^41 + 12 * q^43 - 9 * q^45 + 15 * q^47 + 9 * q^53 - 3 * q^55 + 18 * q^57 - 24 * q^59 + 6 * q^61 + 3 * q^65 + 6 * q^67 - 18 * q^79 + 27 * q^81 + 30 * q^83 + 6 * q^85 - 18 * q^87 + 36 * q^93 + 3 * q^95 + 6 * q^97 + 63 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 9x - 6$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 6$$ v^2 - v - 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 6$$ b2 + b1 + 6

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.58423 −0.705720 3.28995
0 −2.58423 0 −1.00000 0 0 0 3.67822 0
1.2 0 −0.705720 0 −1.00000 0 0 0 −2.50196 0
1.3 0 3.28995 0 −1.00000 0 0 0 7.82374 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1960.2.a.v 3
4.b odd 2 1 3920.2.a.cb 3
5.b even 2 1 9800.2.a.cf 3
7.b odd 2 1 1960.2.a.w 3
7.c even 3 2 1960.2.q.w 6
7.d odd 6 2 280.2.q.e 6
21.g even 6 2 2520.2.bi.q 6
28.d even 2 1 3920.2.a.cc 3
28.f even 6 2 560.2.q.l 6
35.c odd 2 1 9800.2.a.ce 3
35.i odd 6 2 1400.2.q.j 6
35.k even 12 4 1400.2.bh.i 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.e 6 7.d odd 6 2
560.2.q.l 6 28.f even 6 2
1400.2.q.j 6 35.i odd 6 2
1400.2.bh.i 12 35.k even 12 4
1960.2.a.v 3 1.a even 1 1 trivial
1960.2.a.w 3 7.b odd 2 1
1960.2.q.w 6 7.c even 3 2
2520.2.bi.q 6 21.g even 6 2
3920.2.a.cb 3 4.b odd 2 1
3920.2.a.cc 3 28.d even 2 1
9800.2.a.ce 3 35.c odd 2 1
9800.2.a.cf 3 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1960))$$:

 $$T_{3}^{3} - 9T_{3} - 6$$ T3^3 - 9*T3 - 6 $$T_{11}^{3} - 3T_{11}^{2} - 24T_{11} + 44$$ T11^3 - 3*T11^2 - 24*T11 + 44 $$T_{13}^{3} + 3T_{13}^{2} - 24T_{13} - 68$$ T13^3 + 3*T13^2 - 24*T13 - 68

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 9T - 6$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3}$$
$11$ $$T^{3} - 3 T^{2} - 24 T + 44$$
$13$ $$T^{3} + 3 T^{2} - 24 T - 68$$
$17$ $$(T + 2)^{3}$$
$19$ $$T^{3} + 3 T^{2} - 24 T + 16$$
$23$ $$T^{3} - 3 T^{2} - 15 T - 7$$
$29$ $$T^{3} - 12 T^{2} + 21 T + 26$$
$31$ $$T^{3} + 12 T^{2} + 12 T - 128$$
$37$ $$T^{3} - 9T^{2} + 96$$
$41$ $$T^{3} + 9 T^{2} - 45 T - 381$$
$43$ $$T^{3} - 12 T^{2} + 39 T - 22$$
$47$ $$T^{3} - 15 T^{2} - 96 T + 1588$$
$53$ $$T^{3} - 9 T^{2} - 72 T + 624$$
$59$ $$(T + 8)^{3}$$
$61$ $$T^{3} - 6 T^{2} - 87 T + 544$$
$67$ $$T^{3} - 6 T^{2} - 69 T - 8$$
$71$ $$T^{3}$$
$73$ $$T^{3} - 108T - 336$$
$79$ $$T^{3} + 18T^{2} - 768$$
$83$ $$T^{3} - 30 T^{2} + 291 T - 904$$
$89$ $$T^{3} - 27T - 42$$
$97$ $$(T - 2)^{3}$$