# Properties

 Label 1960.2.a.s Level $1960$ Weight $2$ Character orbit 1960.a Self dual yes Analytic conductor $15.651$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + q^{5} + (\beta + 5) q^{9}+O(q^{10})$$ q + b * q^3 + q^5 + (b + 5) * q^9 $$q + \beta q^{3} + q^{5} + (\beta + 5) q^{9} + ( - \beta + 4) q^{11} + (\beta - 2) q^{13} + \beta q^{15} + ( - \beta - 2) q^{17} - 2 \beta q^{19} + 2 \beta q^{23} + q^{25} + (3 \beta + 8) q^{27} + (\beta - 2) q^{29} + 8 q^{31} + (3 \beta - 8) q^{33} - 2 q^{37} + ( - \beta + 8) q^{39} + (2 \beta - 2) q^{41} + (2 \beta - 4) q^{43} + (\beta + 5) q^{45} - 3 \beta q^{47} + ( - 3 \beta - 8) q^{51} + ( - 2 \beta + 6) q^{53} + ( - \beta + 4) q^{55} + ( - 2 \beta - 16) q^{57} - 8 q^{59} + ( - 2 \beta - 2) q^{61} + (\beta - 2) q^{65} - 4 q^{67} + (2 \beta + 16) q^{69} + 8 q^{71} + 6 q^{73} + \beta q^{75} + ( - 3 \beta + 8) q^{79} + (8 \beta + 9) q^{81} - 4 \beta q^{83} + ( - \beta - 2) q^{85} + ( - \beta + 8) q^{87} + (2 \beta - 10) q^{89} + 8 \beta q^{93} - 2 \beta q^{95} + ( - 5 \beta - 2) q^{97} + ( - 2 \beta + 12) q^{99} +O(q^{100})$$ q + b * q^3 + q^5 + (b + 5) * q^9 + (-b + 4) * q^11 + (b - 2) * q^13 + b * q^15 + (-b - 2) * q^17 - 2*b * q^19 + 2*b * q^23 + q^25 + (3*b + 8) * q^27 + (b - 2) * q^29 + 8 * q^31 + (3*b - 8) * q^33 - 2 * q^37 + (-b + 8) * q^39 + (2*b - 2) * q^41 + (2*b - 4) * q^43 + (b + 5) * q^45 - 3*b * q^47 + (-3*b - 8) * q^51 + (-2*b + 6) * q^53 + (-b + 4) * q^55 + (-2*b - 16) * q^57 - 8 * q^59 + (-2*b - 2) * q^61 + (b - 2) * q^65 - 4 * q^67 + (2*b + 16) * q^69 + 8 * q^71 + 6 * q^73 + b * q^75 + (-3*b + 8) * q^79 + (8*b + 9) * q^81 - 4*b * q^83 + (-b - 2) * q^85 + (-b + 8) * q^87 + (2*b - 10) * q^89 + 8*b * q^93 - 2*b * q^95 + (-5*b - 2) * q^97 + (-2*b + 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 2 q^{5} + 11 q^{9}+O(q^{10})$$ 2 * q + q^3 + 2 * q^5 + 11 * q^9 $$2 q + q^{3} + 2 q^{5} + 11 q^{9} + 7 q^{11} - 3 q^{13} + q^{15} - 5 q^{17} - 2 q^{19} + 2 q^{23} + 2 q^{25} + 19 q^{27} - 3 q^{29} + 16 q^{31} - 13 q^{33} - 4 q^{37} + 15 q^{39} - 2 q^{41} - 6 q^{43} + 11 q^{45} - 3 q^{47} - 19 q^{51} + 10 q^{53} + 7 q^{55} - 34 q^{57} - 16 q^{59} - 6 q^{61} - 3 q^{65} - 8 q^{67} + 34 q^{69} + 16 q^{71} + 12 q^{73} + q^{75} + 13 q^{79} + 26 q^{81} - 4 q^{83} - 5 q^{85} + 15 q^{87} - 18 q^{89} + 8 q^{93} - 2 q^{95} - 9 q^{97} + 22 q^{99}+O(q^{100})$$ 2 * q + q^3 + 2 * q^5 + 11 * q^9 + 7 * q^11 - 3 * q^13 + q^15 - 5 * q^17 - 2 * q^19 + 2 * q^23 + 2 * q^25 + 19 * q^27 - 3 * q^29 + 16 * q^31 - 13 * q^33 - 4 * q^37 + 15 * q^39 - 2 * q^41 - 6 * q^43 + 11 * q^45 - 3 * q^47 - 19 * q^51 + 10 * q^53 + 7 * q^55 - 34 * q^57 - 16 * q^59 - 6 * q^61 - 3 * q^65 - 8 * q^67 + 34 * q^69 + 16 * q^71 + 12 * q^73 + q^75 + 13 * q^79 + 26 * q^81 - 4 * q^83 - 5 * q^85 + 15 * q^87 - 18 * q^89 + 8 * q^93 - 2 * q^95 - 9 * q^97 + 22 * q^99

## 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
 −2.37228 3.37228
0 −2.37228 0 1.00000 0 0 0 2.62772 0
1.2 0 3.37228 0 1.00000 0 0 0 8.37228 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.s 2
4.b odd 2 1 3920.2.a.bt 2
5.b even 2 1 9800.2.a.bu 2
7.b odd 2 1 280.2.a.c 2
7.c even 3 2 1960.2.q.r 4
7.d odd 6 2 1960.2.q.t 4
21.c even 2 1 2520.2.a.x 2
28.d even 2 1 560.2.a.h 2
35.c odd 2 1 1400.2.a.r 2
35.f even 4 2 1400.2.g.i 4
56.e even 2 1 2240.2.a.bg 2
56.h odd 2 1 2240.2.a.bk 2
84.h odd 2 1 5040.2.a.by 2
140.c even 2 1 2800.2.a.bk 2
140.j odd 4 2 2800.2.g.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.c 2 7.b odd 2 1
560.2.a.h 2 28.d even 2 1
1400.2.a.r 2 35.c odd 2 1
1400.2.g.i 4 35.f even 4 2
1960.2.a.s 2 1.a even 1 1 trivial
1960.2.q.r 4 7.c even 3 2
1960.2.q.t 4 7.d odd 6 2
2240.2.a.bg 2 56.e even 2 1
2240.2.a.bk 2 56.h odd 2 1
2520.2.a.x 2 21.c even 2 1
2800.2.a.bk 2 140.c even 2 1
2800.2.g.r 4 140.j odd 4 2
3920.2.a.bt 2 4.b odd 2 1
5040.2.a.by 2 84.h odd 2 1
9800.2.a.bu 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1960))$$:

 $$T_{3}^{2} - T_{3} - 8$$ T3^2 - T3 - 8 $$T_{11}^{2} - 7T_{11} + 4$$ T11^2 - 7*T11 + 4 $$T_{13}^{2} + 3T_{13} - 6$$ T13^2 + 3*T13 - 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T - 8$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 7T + 4$$
$13$ $$T^{2} + 3T - 6$$
$17$ $$T^{2} + 5T - 2$$
$19$ $$T^{2} + 2T - 32$$
$23$ $$T^{2} - 2T - 32$$
$29$ $$T^{2} + 3T - 6$$
$31$ $$(T - 8)^{2}$$
$37$ $$(T + 2)^{2}$$
$41$ $$T^{2} + 2T - 32$$
$43$ $$T^{2} + 6T - 24$$
$47$ $$T^{2} + 3T - 72$$
$53$ $$T^{2} - 10T - 8$$
$59$ $$(T + 8)^{2}$$
$61$ $$T^{2} + 6T - 24$$
$67$ $$(T + 4)^{2}$$
$71$ $$(T - 8)^{2}$$
$73$ $$(T - 6)^{2}$$
$79$ $$T^{2} - 13T - 32$$
$83$ $$T^{2} + 4T - 128$$
$89$ $$T^{2} + 18T + 48$$
$97$ $$T^{2} + 9T - 186$$