# Properties

 Label 1960.2.a.x Level $1960$ Weight $2$ Character orbit 1960.a Self dual yes Analytic conductor $15.651$ Analytic rank $1$ Dimension $4$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.16448.2 Defining polynomial: $$x^{4} - 2x^{3} - 7x^{2} + 8x + 14$$ x^4 - 2*x^3 - 7*x^2 + 8*x + 14 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - q^{5} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10})$$ q - b1 * q^3 - q^5 + (b2 + b1 + 1) * q^9 $$q - \beta_1 q^{3} - q^{5} + (\beta_{2} + \beta_1 + 1) q^{9} + ( - \beta_{2} + \beta_1) q^{11} + (\beta_{3} - \beta_{2} - \beta_1 - 2) q^{13} + \beta_1 q^{15} + ( - 2 \beta_{3} + \beta_1 - 2) q^{17} + (\beta_{3} + \beta_{2}) q^{19} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{23} + q^{25} + ( - \beta_{3} - \beta_{2} + \beta_1 - 4) q^{27} + ( - 3 \beta_{2} - \beta_1) q^{29} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{31} + (\beta_{3} - \beta_{2} - \beta_1 - 4) q^{33} + (\beta_{3} + 2 \beta_{2}) q^{37} + ( - 3 \beta_{2} + 3 \beta_1 + 2) q^{39} + (2 \beta_{3} + \beta_{2} + 2 \beta_1 - 4) q^{41} + (2 \beta_{3} - 2 \beta_{2} - 2) q^{43} + ( - \beta_{2} - \beta_1 - 1) q^{45} + (\beta_{3} + \beta_{2} + \beta_1) q^{47} + (2 \beta_{3} + 7 \beta_{2} + \beta_1) q^{51} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{53} + (\beta_{2} - \beta_1) q^{55} + ( - 2 \beta_{3} - 4 \beta_{2} - 2) q^{57} + ( - 5 \beta_{2} - 2) q^{59} + ( - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 6) q^{61} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{65} + (\beta_{3} - 6 \beta_{2} - 2) q^{67} + ( - \beta_{3} + 2 \beta_{2} - 6) q^{69} + ( - 4 \beta_{3} + 2 \beta_1) q^{71} + ( - \beta_{3} + 5 \beta_{2} - 4 \beta_1 - 2) q^{73} - \beta_1 q^{75} + (5 \beta_{2} - \beta_1 + 6) q^{79} + (2 \beta_{3} - 5) q^{81} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 - 8) q^{83} + (2 \beta_{3} - \beta_1 + 2) q^{85} + (3 \beta_{3} + \beta_{2} + \beta_1 + 4) q^{87} + ( - \beta_{3} + 3 \beta_{2} - 10) q^{89} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 6) q^{93} + ( - \beta_{3} - \beta_{2}) q^{95} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 6) q^{97} + (2 \beta_1 + 2) q^{99}+O(q^{100})$$ q - b1 * q^3 - q^5 + (b2 + b1 + 1) * q^9 + (-b2 + b1) * q^11 + (b3 - b2 - b1 - 2) * q^13 + b1 * q^15 + (-2*b3 + b1 - 2) * q^17 + (b3 + b2) * q^19 + (-b3 + 2*b2 + 2*b1 - 2) * q^23 + q^25 + (-b3 - b2 + b1 - 4) * q^27 + (-3*b2 - b1) * q^29 + (-b3 + 2*b2 + 2*b1 + 2) * q^31 + (b3 - b2 - b1 - 4) * q^33 + (b3 + 2*b2) * q^37 + (-3*b2 + 3*b1 + 2) * q^39 + (2*b3 + b2 + 2*b1 - 4) * q^41 + (2*b3 - 2*b2 - 2) * q^43 + (-b2 - b1 - 1) * q^45 + (b3 + b2 + b1) * q^47 + (2*b3 + 7*b2 + b1) * q^51 + (-b3 + 2*b2 - 2*b1) * q^53 + (b2 - b1) * q^55 + (-2*b3 - 4*b2 - 2) * q^57 + (-5*b2 - 2) * q^59 + (-2*b3 - 2*b2 + 2*b1 - 6) * q^61 + (-b3 + b2 + b1 + 2) * q^65 + (b3 - 6*b2 - 2) * q^67 + (-b3 + 2*b2 - 6) * q^69 + (-4*b3 + 2*b1) * q^71 + (-b3 + 5*b2 - 4*b1 - 2) * q^73 - b1 * q^75 + (5*b2 - b1 + 6) * q^79 + (2*b3 - 5) * q^81 + (-b3 + b2 - 2*b1 - 8) * q^83 + (2*b3 - b1 + 2) * q^85 + (3*b3 + b2 + b1 + 4) * q^87 + (-b3 + 3*b2 - 10) * q^89 + (-b3 + 2*b2 - 4*b1 - 6) * q^93 + (-b3 - b2) * q^95 + (2*b3 + 2*b2 - b1 - 6) * q^97 + (2*b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 4 q^{5} + 6 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - 4 * q^5 + 6 * q^9 $$4 q - 2 q^{3} - 4 q^{5} + 6 q^{9} + 2 q^{11} - 10 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{23} + 4 q^{25} - 14 q^{27} - 2 q^{29} + 12 q^{31} - 18 q^{33} + 14 q^{39} - 12 q^{41} - 8 q^{43} - 6 q^{45} + 2 q^{47} + 2 q^{51} - 4 q^{53} - 2 q^{55} - 8 q^{57} - 8 q^{59} - 20 q^{61} + 10 q^{65} - 8 q^{67} - 24 q^{69} + 4 q^{71} - 16 q^{73} - 2 q^{75} + 22 q^{79} - 20 q^{81} - 36 q^{83} + 6 q^{85} + 18 q^{87} - 40 q^{89} - 32 q^{93} - 26 q^{97} + 12 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 4 * q^5 + 6 * q^9 + 2 * q^11 - 10 * q^13 + 2 * q^15 - 6 * q^17 - 4 * q^23 + 4 * q^25 - 14 * q^27 - 2 * q^29 + 12 * q^31 - 18 * q^33 + 14 * q^39 - 12 * q^41 - 8 * q^43 - 6 * q^45 + 2 * q^47 + 2 * q^51 - 4 * q^53 - 2 * q^55 - 8 * q^57 - 8 * q^59 - 20 * q^61 + 10 * q^65 - 8 * q^67 - 24 * q^69 + 4 * q^71 - 16 * q^73 - 2 * q^75 + 22 * q^79 - 20 * q^81 - 36 * q^83 + 6 * q^85 + 18 * q^87 - 40 * q^89 - 32 * q^93 - 26 * q^97 + 12 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 7x^{2} + 8x + 14$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4\nu$$ v^3 - v^2 - 4*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 4$$ b2 + b1 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 5\beta _1 + 4$$ b3 + b2 + 5*b1 + 4

## 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
 2.87996 2.18398 −1.18398 −1.87996
0 −2.87996 0 −1.00000 0 0 0 5.29417 0
1.2 0 −2.18398 0 −1.00000 0 0 0 1.76977 0
1.3 0 1.18398 0 −1.00000 0 0 0 −1.59819 0
1.4 0 1.87996 0 −1.00000 0 0 0 0.534253 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.x 4
4.b odd 2 1 3920.2.a.ce 4
5.b even 2 1 9800.2.a.cs 4
7.b odd 2 1 1960.2.a.y yes 4
7.c even 3 2 1960.2.q.y 8
7.d odd 6 2 1960.2.q.x 8
28.d even 2 1 3920.2.a.cd 4
35.c odd 2 1 9800.2.a.cl 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1960.2.a.x 4 1.a even 1 1 trivial
1960.2.a.y yes 4 7.b odd 2 1
1960.2.q.x 8 7.d odd 6 2
1960.2.q.y 8 7.c even 3 2
3920.2.a.cd 4 28.d even 2 1
3920.2.a.ce 4 4.b odd 2 1
9800.2.a.cl 4 35.c odd 2 1
9800.2.a.cs 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1960))$$:

 $$T_{3}^{4} + 2T_{3}^{3} - 7T_{3}^{2} - 8T_{3} + 14$$ T3^4 + 2*T3^3 - 7*T3^2 - 8*T3 + 14 $$T_{11}^{4} - 2T_{11}^{3} - 11T_{11}^{2} + 20T_{11} - 4$$ T11^4 - 2*T11^3 - 11*T11^2 + 20*T11 - 4 $$T_{13}^{4} + 10T_{13}^{3} + 19T_{13}^{2} - 52T_{13} - 124$$ T13^4 + 10*T13^3 + 19*T13^2 - 52*T13 - 124

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} - 7 T^{2} - 8 T + 14$$
$5$ $$(T + 1)^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 2 T^{3} - 11 T^{2} + 20 T - 4$$
$13$ $$T^{4} + 10 T^{3} + 19 T^{2} + \cdots - 124$$
$17$ $$T^{4} + 6 T^{3} - 51 T^{2} - 244 T + 434$$
$19$ $$T^{4} - 26 T^{2} - 24 T + 8$$
$23$ $$T^{4} + 4 T^{3} - 38 T^{2} + 48 T + 16$$
$29$ $$T^{4} + 2 T^{3} - 43 T^{2} - 20 T + 188$$
$31$ $$T^{4} - 12 T^{3} + 10 T^{2} + \cdots - 784$$
$37$ $$T^{4} - 42 T^{2} - 40 T + 8$$
$41$ $$T^{4} + 12 T^{3} - 96 T^{2} + \cdots - 4228$$
$43$ $$T^{4} + 8 T^{3} - 48 T^{2} - 192 T + 752$$
$47$ $$T^{4} - 2 T^{3} - 41 T^{2} - 96 T - 56$$
$53$ $$T^{4} + 4 T^{3} - 70 T^{2} - 424 T - 568$$
$59$ $$(T^{2} + 4 T - 46)^{2}$$
$61$ $$T^{4} + 20 T^{3} + 44 T^{2} + \cdots - 5344$$
$67$ $$T^{4} + 8 T^{3} - 114 T^{2} + \cdots + 2336$$
$71$ $$T^{4} - 4 T^{3} - 252 T^{2} + \cdots + 10976$$
$73$ $$T^{4} + 16 T^{3} - 170 T^{2} + \cdots - 14192$$
$79$ $$T^{4} - 22 T^{3} + 73 T^{2} + 488 T - 56$$
$83$ $$T^{4} + 36 T^{3} + 418 T^{2} + \cdots + 256$$
$89$ $$T^{4} + 40 T^{3} + 558 T^{2} + \cdots + 5408$$
$97$ $$T^{4} + 26 T^{3} + 157 T^{2} + \cdots - 1022$$