# Properties

 Label 1560.4.a.n Level $1560$ Weight $4$ Character orbit 1560.a Self dual yes Analytic conductor $92.043$ Analytic rank $1$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1560.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$92.0429796090$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - x^{3} - 41x^{2} - 30x - 4$$ x^4 - x^3 - 41*x^2 - 30*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$2^{6}$$ 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.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + 5 q^{5} + ( - \beta_{3} - 8) q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + 5 * q^5 + (-b3 - 8) * q^7 + 9 * q^9 $$q + 3 q^{3} + 5 q^{5} + ( - \beta_{3} - 8) q^{7} + 9 q^{9} + (\beta_{2} - 10) q^{11} + 13 q^{13} + 15 q^{15} + (3 \beta_{3} - 2 \beta_1 - 14) q^{17} + (3 \beta_{3} - \beta_{2} - 46) q^{19} + ( - 3 \beta_{3} - 24) q^{21} + (2 \beta_{3} - 3 \beta_{2} + 5 \beta_1 - 42) q^{23} + 25 q^{25} + 27 q^{27} + ( - 6 \beta_{2} - \beta_1 - 18) q^{29} + (\beta_{3} - \beta_{2} + 3 \beta_1 - 102) q^{31} + (3 \beta_{2} - 30) q^{33} + ( - 5 \beta_{3} - 40) q^{35} + (7 \beta_{3} - 6 \beta_{2} + 5 \beta_1 - 10) q^{37} + 39 q^{39} + (9 \beta_{3} + 6 \beta_{2} - \beta_1 - 30) q^{41} + (9 \beta_{3} + 5 \beta_{2} - 13 \beta_1 - 190) q^{43} + 45 q^{45} + ( - 5 \beta_{3} + 5 \beta_{2} + \beta_1 - 130) q^{47} + (24 \beta_{3} + 5 \beta_{2} - 8 \beta_1 + 43) q^{49} + (9 \beta_{3} - 6 \beta_1 - 42) q^{51} + ( - 10 \beta_{3} + \beta_{2} - 10 \beta_1 - 80) q^{53} + (5 \beta_{2} - 50) q^{55} + (9 \beta_{3} - 3 \beta_{2} - 138) q^{57} + ( - 12 \beta_{3} - 52) q^{59} + ( - 14 \beta_{3} + \beta_{2} + 14 \beta_1 + 88) q^{61} + ( - 9 \beta_{3} - 72) q^{63} + 65 q^{65} + ( - 4 \beta_{3} + 4 \beta_{2} + 8 \beta_1 - 260) q^{67} + (6 \beta_{3} - 9 \beta_{2} + 15 \beta_1 - 126) q^{69} + ( - 20 \beta_{3} + \beta_{2} + 16 \beta_1 - 166) q^{71} + ( - 7 \beta_{3} + 15 \beta_{2} + 2 \beta_1 - 96) q^{73} + 75 q^{75} + ( - 2 \beta_{3} - 11 \beta_{2} + 18 \beta_1 + 10) q^{77} + ( - 31 \beta_{3} + 4 \beta_{2} - 21 \beta_1 - 156) q^{79} + 81 q^{81} + ( - 26 \beta_{3} + 10 \beta_{2} - 26 \beta_1 - 464) q^{83} + (15 \beta_{3} - 10 \beta_1 - 70) q^{85} + ( - 18 \beta_{2} - 3 \beta_1 - 54) q^{87} + (15 \beta_{3} - 30 \beta_{2} + 13 \beta_1 - 510) q^{89} + ( - 13 \beta_{3} - 104) q^{91} + (3 \beta_{3} - 3 \beta_{2} + 9 \beta_1 - 306) q^{93} + (15 \beta_{3} - 5 \beta_{2} - 230) q^{95} + ( - 24 \beta_{3} + 19 \beta_{2} + \beta_1 + 284) q^{97} + (9 \beta_{2} - 90) q^{99}+O(q^{100})$$ q + 3 * q^3 + 5 * q^5 + (-b3 - 8) * q^7 + 9 * q^9 + (b2 - 10) * q^11 + 13 * q^13 + 15 * q^15 + (3*b3 - 2*b1 - 14) * q^17 + (3*b3 - b2 - 46) * q^19 + (-3*b3 - 24) * q^21 + (2*b3 - 3*b2 + 5*b1 - 42) * q^23 + 25 * q^25 + 27 * q^27 + (-6*b2 - b1 - 18) * q^29 + (b3 - b2 + 3*b1 - 102) * q^31 + (3*b2 - 30) * q^33 + (-5*b3 - 40) * q^35 + (7*b3 - 6*b2 + 5*b1 - 10) * q^37 + 39 * q^39 + (9*b3 + 6*b2 - b1 - 30) * q^41 + (9*b3 + 5*b2 - 13*b1 - 190) * q^43 + 45 * q^45 + (-5*b3 + 5*b2 + b1 - 130) * q^47 + (24*b3 + 5*b2 - 8*b1 + 43) * q^49 + (9*b3 - 6*b1 - 42) * q^51 + (-10*b3 + b2 - 10*b1 - 80) * q^53 + (5*b2 - 50) * q^55 + (9*b3 - 3*b2 - 138) * q^57 + (-12*b3 - 52) * q^59 + (-14*b3 + b2 + 14*b1 + 88) * q^61 + (-9*b3 - 72) * q^63 + 65 * q^65 + (-4*b3 + 4*b2 + 8*b1 - 260) * q^67 + (6*b3 - 9*b2 + 15*b1 - 126) * q^69 + (-20*b3 + b2 + 16*b1 - 166) * q^71 + (-7*b3 + 15*b2 + 2*b1 - 96) * q^73 + 75 * q^75 + (-2*b3 - 11*b2 + 18*b1 + 10) * q^77 + (-31*b3 + 4*b2 - 21*b1 - 156) * q^79 + 81 * q^81 + (-26*b3 + 10*b2 - 26*b1 - 464) * q^83 + (15*b3 - 10*b1 - 70) * q^85 + (-18*b2 - 3*b1 - 54) * q^87 + (15*b3 - 30*b2 + 13*b1 - 510) * q^89 + (-13*b3 - 104) * q^91 + (3*b3 - 3*b2 + 9*b1 - 306) * q^93 + (15*b3 - 5*b2 - 230) * q^95 + (-24*b3 + 19*b2 + b1 + 284) * q^97 + (9*b2 - 90) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9}+O(q^{10})$$ 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9 $$4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9} - 38 q^{11} + 52 q^{13} + 60 q^{15} - 50 q^{17} - 180 q^{19} - 102 q^{21} - 170 q^{23} + 100 q^{25} + 108 q^{27} - 84 q^{29} - 408 q^{31} - 114 q^{33} - 170 q^{35} - 38 q^{37} + 156 q^{39} - 90 q^{41} - 732 q^{43} + 180 q^{45} - 520 q^{47} + 230 q^{49} - 150 q^{51} - 338 q^{53} - 190 q^{55} - 540 q^{57} - 232 q^{59} + 326 q^{61} - 306 q^{63} + 260 q^{65} - 1040 q^{67} - 510 q^{69} - 702 q^{71} - 368 q^{73} + 300 q^{75} + 14 q^{77} - 678 q^{79} + 324 q^{81} - 1888 q^{83} - 250 q^{85} - 252 q^{87} - 2070 q^{89} - 442 q^{91} - 1224 q^{93} - 900 q^{95} + 1126 q^{97} - 342 q^{99}+O(q^{100})$$ 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9 - 38 * q^11 + 52 * q^13 + 60 * q^15 - 50 * q^17 - 180 * q^19 - 102 * q^21 - 170 * q^23 + 100 * q^25 + 108 * q^27 - 84 * q^29 - 408 * q^31 - 114 * q^33 - 170 * q^35 - 38 * q^37 + 156 * q^39 - 90 * q^41 - 732 * q^43 + 180 * q^45 - 520 * q^47 + 230 * q^49 - 150 * q^51 - 338 * q^53 - 190 * q^55 - 540 * q^57 - 232 * q^59 + 326 * q^61 - 306 * q^63 + 260 * q^65 - 1040 * q^67 - 510 * q^69 - 702 * q^71 - 368 * q^73 + 300 * q^75 + 14 * q^77 - 678 * q^79 + 324 * q^81 - 1888 * q^83 - 250 * q^85 - 252 * q^87 - 2070 * q^89 - 442 * q^91 - 1224 * q^93 - 900 * q^95 + 1126 * q^97 - 342 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 41x^{2} - 30x - 4$$ :

 $$\beta_{1}$$ $$=$$ $$-2\nu^{3} + 4\nu^{2} + 80\nu + 4$$ -2*v^3 + 4*v^2 + 80*v + 4 $$\beta_{2}$$ $$=$$ $$-3\nu^{3} + 4\nu^{2} + 128\nu + 46$$ -3*v^3 + 4*v^2 + 128*v + 46 $$\beta_{3}$$ $$=$$ $$-3\nu^{3} + 4\nu^{2} + 120\nu + 48$$ -3*v^3 + 4*v^2 + 120*v + 48
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} + 2 ) / 8$$ (-b3 + b2 + 2) / 8 $$\nu^{2}$$ $$=$$ $$( -2\beta_{3} + 3\beta _1 + 84 ) / 4$$ (-2*b3 + 3*b1 + 84) / 4 $$\nu^{3}$$ $$=$$ $$-6\beta_{3} + 5\beta_{2} + \beta _1 + 54$$ -6*b3 + 5*b2 + b1 + 54

## 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
 −0.174957 −5.49345 7.24301 −0.574597
0 3.00000 0 5.00000 0 −35.1436 0 9.00000 0
1.2 0 3.00000 0 5.00000 0 −14.8428 0 9.00000 0
1.3 0 3.00000 0 5.00000 0 4.92462 0 9.00000 0
1.4 0 3.00000 0 5.00000 0 11.0618 0 9.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$-1$$

## 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 1560.4.a.n 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.4.a.n 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} + 34T_{7}^{3} - 223T_{7}^{2} - 5616T_{7} + 28416$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1560))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T - 3)^{4}$$
$5$ $$(T - 5)^{4}$$
$7$ $$T^{4} + 34 T^{3} - 223 T^{2} + \cdots + 28416$$
$11$ $$T^{4} + 38 T^{3} - 1759 T^{2} + \cdots + 794640$$
$13$ $$(T - 13)^{4}$$
$17$ $$T^{4} + 50 T^{3} - 10091 T^{2} + \cdots + 2634756$$
$19$ $$T^{4} + 180 T^{3} + 4868 T^{2} + \cdots + 1578560$$
$23$ $$T^{4} + 170 T^{3} + \cdots - 115845696$$
$29$ $$T^{4} + 84 T^{3} + \cdots + 1641072688$$
$31$ $$T^{4} + 408 T^{3} + 49948 T^{2} + \cdots + 9081600$$
$37$ $$T^{4} + 38 T^{3} + \cdots - 153758476$$
$41$ $$T^{4} + 90 T^{3} + \cdots + 1993594900$$
$43$ $$T^{4} + 732 T^{3} + \cdots + 2881893632$$
$47$ $$T^{4} + 520 T^{3} + \cdots - 2001440000$$
$53$ $$T^{4} + 338 T^{3} + \cdots - 9572486780$$
$59$ $$T^{4} + 232 T^{3} + \cdots + 923054080$$
$61$ $$T^{4} - 326 T^{3} + \cdots + 13972040964$$
$67$ $$T^{4} + 1040 T^{3} + \cdots - 15293380352$$
$71$ $$T^{4} + 702 T^{3} + \cdots - 18758062080$$
$73$ $$T^{4} + 368 T^{3} + \cdots + 10624585104$$
$79$ $$T^{4} + 678 T^{3} + \cdots - 203520914688$$
$83$ $$T^{4} + 1888 T^{3} + \cdots - 460820907008$$
$89$ $$T^{4} + 2070 T^{3} + \cdots - 796693724300$$
$97$ $$T^{4} - 1126 T^{3} + \cdots - 13823634476$$