# Properties

 Label 1872.4.a.bo Level $1872$ Weight $4$ Character orbit 1872.a Self dual yes Analytic conductor $110.452$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1872 = 2^{4} \cdot 3^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1872.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$110.451575531$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.1520092.1 Defining polynomial: $$x^{4} - x^{3} - 40x^{2} - 9x + 81$$ x^4 - x^3 - 40*x^2 - 9*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 117) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} + ( - \beta_1 - 9) q^{7}+O(q^{10})$$ q + b3 * q^5 + (-b1 - 9) * q^7 $$q + \beta_{3} q^{5} + ( - \beta_1 - 9) q^{7} + ( - \beta_{3} - \beta_{2}) q^{11} - 13 q^{13} - 4 \beta_{2} q^{17} + (7 \beta_1 - 21) q^{19} + ( - 2 \beta_{3} + 16 \beta_{2}) q^{23} + ( - 6 \beta_1 + 165) q^{25} + ( - 10 \beta_{3} - 10 \beta_{2}) q^{29} + ( - \beta_1 + 151) q^{31} + ( - 4 \beta_{3} + 26 \beta_{2}) q^{35} + (16 \beta_1 + 46) q^{37} + ( - 9 \beta_{3} + 2 \beta_{2}) q^{41} + ( - 4 \beta_1 - 220) q^{43} + ( - 5 \beta_{3} - 27 \beta_{2}) q^{47} + (18 \beta_1 - 29) q^{49} + ( - 8 \beta_{3} - 26 \beta_{2}) q^{53} + (16 \beta_1 - 288) q^{55} + ( - 15 \beta_{3} - 23 \beta_{2}) q^{59} + (22 \beta_1 + 164) q^{61} - 13 \beta_{3} q^{65} + (17 \beta_1 - 763) q^{67} + ( - 29 \beta_{3} + 49 \beta_{2}) q^{71} + ( - 68 \beta_1 - 78) q^{73} + ( - 4 \beta_{3} - 12 \beta_{2}) q^{77} + ( - 56 \beta_1 + 180) q^{79} + ( - 5 \beta_{3} - \beta_{2}) q^{83} + (40 \beta_1 + 8) q^{85} + ( - 7 \beta_{3} + 102 \beta_{2}) q^{89} + (13 \beta_1 + 117) q^{91} + ( - 56 \beta_{3} - 182 \beta_{2}) q^{95} + ( - 52 \beta_1 - 86) q^{97}+O(q^{100})$$ q + b3 * q^5 + (-b1 - 9) * q^7 + (-b3 - b2) * q^11 - 13 * q^13 - 4*b2 * q^17 + (7*b1 - 21) * q^19 + (-2*b3 + 16*b2) * q^23 + (-6*b1 + 165) * q^25 + (-10*b3 - 10*b2) * q^29 + (-b1 + 151) * q^31 + (-4*b3 + 26*b2) * q^35 + (16*b1 + 46) * q^37 + (-9*b3 + 2*b2) * q^41 + (-4*b1 - 220) * q^43 + (-5*b3 - 27*b2) * q^47 + (18*b1 - 29) * q^49 + (-8*b3 - 26*b2) * q^53 + (16*b1 - 288) * q^55 + (-15*b3 - 23*b2) * q^59 + (22*b1 + 164) * q^61 - 13*b3 * q^65 + (17*b1 - 763) * q^67 + (-29*b3 + 49*b2) * q^71 + (-68*b1 - 78) * q^73 + (-4*b3 - 12*b2) * q^77 + (-56*b1 + 180) * q^79 + (-5*b3 - b2) * q^83 + (40*b1 + 8) * q^85 + (-7*b3 + 102*b2) * q^89 + (13*b1 + 117) * q^91 + (-56*b3 - 182*b2) * q^95 + (-52*b1 - 86) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 36 q^{7}+O(q^{10})$$ 4 * q - 36 * q^7 $$4 q - 36 q^{7} - 52 q^{13} - 84 q^{19} + 660 q^{25} + 604 q^{31} + 184 q^{37} - 880 q^{43} - 116 q^{49} - 1152 q^{55} + 656 q^{61} - 3052 q^{67} - 312 q^{73} + 720 q^{79} + 32 q^{85} + 468 q^{91} - 344 q^{97}+O(q^{100})$$ 4 * q - 36 * q^7 - 52 * q^13 - 84 * q^19 + 660 * q^25 + 604 * q^31 + 184 * q^37 - 880 * q^43 - 116 * q^49 - 1152 * q^55 + 656 * q^61 - 3052 * q^67 - 312 * q^73 + 720 * q^79 + 32 * q^85 + 468 * q^91 - 344 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 40x^{2} - 9x + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{3} + 2\nu^{2} + 98\nu + 9 ) / 9$$ (-2*v^3 + 2*v^2 + 98*v + 9) / 9 $$\beta_{2}$$ $$=$$ $$( -2\nu^{3} + 2\nu^{2} + 62\nu + 18 ) / 9$$ (-2*v^3 + 2*v^2 + 62*v + 18) / 9 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 10\nu^{2} + 22\nu - 171 ) / 9$$ (-v^3 + 10*v^2 + 22*v - 171) / 9
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta _1 + 1 ) / 4$$ (-b2 + b1 + 1) / 4 $$\nu^{2}$$ $$=$$ $$( 4\beta_{3} - 3\beta_{2} + \beta _1 + 81 ) / 4$$ (4*b3 - 3*b2 + b1 + 81) / 4 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 13\beta_{2} + 8\beta _1 + 37$$ b3 - 13*b2 + 8*b1 + 37

## 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
 −1.63814 1.32145 6.81072 −5.49403
0 0 0 −19.5342 0 6.26434 0 0 0
1.2 0 0 0 −14.0859 0 −24.2643 0 0 0
1.3 0 0 0 14.0859 0 −24.2643 0 0 0
1.4 0 0 0 19.5342 0 6.26434 0 0 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
$$2$$ $$-1$$
$$3$$ $$1$$
$$13$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1872.4.a.bo 4
3.b odd 2 1 inner 1872.4.a.bo 4
4.b odd 2 1 117.4.a.g 4
12.b even 2 1 117.4.a.g 4
52.b odd 2 1 1521.4.a.ba 4
156.h even 2 1 1521.4.a.ba 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.4.a.g 4 4.b odd 2 1
117.4.a.g 4 12.b even 2 1
1521.4.a.ba 4 52.b odd 2 1
1521.4.a.ba 4 156.h even 2 1
1872.4.a.bo 4 1.a even 1 1 trivial
1872.4.a.bo 4 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1872))$$:

 $$T_{5}^{4} - 580T_{5}^{2} + 75712$$ T5^4 - 580*T5^2 + 75712 $$T_{7}^{2} + 18T_{7} - 152$$ T7^2 + 18*T7 - 152

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} - 580 T^{2} + 75712$$
$7$ $$(T^{2} + 18 T - 152)^{2}$$
$11$ $$T^{4} - 752T^{2} + 7168$$
$13$ $$(T + 13)^{4}$$
$17$ $$T^{4} - 2880 T^{2} + \cdots + 1835008$$
$19$ $$(T^{2} + 42 T - 10976)^{2}$$
$23$ $$T^{4} - 48656 T^{2} + \cdots + 295386112$$
$29$ $$T^{4} - 75200 T^{2} + \cdots + 71680000$$
$31$ $$(T^{2} - 302 T + 22568)^{2}$$
$37$ $$(T^{2} - 92 T - 57532)^{2}$$
$41$ $$T^{4} - 47844 T^{2} + \cdots + 569017792$$
$43$ $$(T^{2} + 440 T + 44672)^{2}$$
$47$ $$T^{4} - 144640 T^{2} + \cdots + 4778706688$$
$53$ $$T^{4} - 157136 T^{2} + \cdots + 3798925312$$
$59$ $$T^{4} - 222960 T^{2} + \cdots + 375897088$$
$61$ $$(T^{2} - 328 T - 85876)^{2}$$
$67$ $$(T^{2} + 1526 T + 514832)^{2}$$
$71$ $$T^{4} - 931328 T^{2} + \cdots + 31867295488$$
$73$ $$(T^{2} + 156 T - 1071308)^{2}$$
$79$ $$(T^{2} - 360 T - 698288)^{2}$$
$83$ $$T^{4} - 14640 T^{2} + \cdots + 39251968$$
$89$ $$T^{4} - 1906852 T^{2} + \cdots + 626946109888$$
$97$ $$(T^{2} + 172 T - 622636)^{2}$$