# Properties

 Label 1840.4.a.j Level $1840$ Weight $4$ Character orbit 1840.a Self dual yes Analytic conductor $108.564$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1840.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$108.563514411$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.318165.1 Defining polynomial: $$x^{3} - x^{2} - 45x + 60$$ x^3 - x^2 - 45*x + 60 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + 5 q^{5} + (\beta_{2} - 3 \beta_1 - 1) q^{7} + (\beta_{2} - \beta_1 + 4) q^{9}+O(q^{10})$$ q + b1 * q^3 + 5 * q^5 + (b2 - 3*b1 - 1) * q^7 + (b2 - b1 + 4) * q^9 $$q + \beta_1 q^{3} + 5 q^{5} + (\beta_{2} - 3 \beta_1 - 1) q^{7} + (\beta_{2} - \beta_1 + 4) q^{9} + (2 \beta_{2} + 5 \beta_1 - 10) q^{11} + (3 \beta_{2} - 3 \beta_1 + 27) q^{13} + 5 \beta_1 q^{15} + ( - 11 \beta_1 + 46) q^{17} + (\beta_{2} + 9 \beta_1 + 59) q^{19} + ( - \beta_{2} + 14 \beta_1 - 91) q^{21} - 23 q^{23} + 25 q^{25} + (\beta_{2} - 10 \beta_1 - 29) q^{27} + (4 \beta_{2} - 18 \beta_1 + 122) q^{29} + (5 \beta_{2} - 17 \beta_1 - 125) q^{31} + (9 \beta_{2} + 9 \beta_1 + 159) q^{33} + (5 \beta_{2} - 15 \beta_1 - 5) q^{35} + (2 \beta_{2} + 8 \beta_1 + 324) q^{37} + (3 \beta_{2} + 66 \beta_1 - 87) q^{39} + (10 \beta_{2} + 47 \beta_1 - 204) q^{41} + (4 \beta_{2} - 40 \beta_1 - 256) q^{43} + (5 \beta_{2} - 5 \beta_1 + 20) q^{45} + (6 \beta_{2} - 42 \beta_1 + 106) q^{47} + ( - 18 \beta_{2} - 49 \beta_1 + 303) q^{49} + ( - 11 \beta_{2} + 57 \beta_1 - 341) q^{51} + ( - 12 \beta_{2} - 60 \beta_1 + 186) q^{53} + (10 \beta_{2} + 25 \beta_1 - 50) q^{55} + (11 \beta_{2} + 62 \beta_1 + 281) q^{57} + ( - 12 \beta_{2} - 34 \beta_1 - 40) q^{59} + (8 \beta_{2} + 49 \beta_1 - 30) q^{61} + ( - 15 \beta_{2} - 36 \beta_1 + 459) q^{63} + (15 \beta_{2} - 15 \beta_1 + 135) q^{65} + (12 \beta_{2} - 20 \beta_1 - 528) q^{67} - 23 \beta_1 q^{69} + ( - 8 \beta_{2} + 67 \beta_1 + 132) q^{71} + ( - 42 \beta_{2} + 30 \beta_1 - 284) q^{73} + 25 \beta_1 q^{75} + ( - 55 \beta_{2} + 80 \beta_1 + 299) q^{77} + ( - 16 \beta_{2} + 28 \beta_1 + 272) q^{79} + ( - 35 \beta_{2} + 20 \beta_1 - 416) q^{81} + ( - 22 \beta_{2} - 52 \beta_1 + 106) q^{83} + ( - 55 \beta_1 + 230) q^{85} + ( - 10 \beta_{2} + 188 \beta_1 - 550) q^{87} + (26 \beta_{2} + 200 \beta_1 - 88) q^{89} + ( - 30 \beta_{2} - 153 \beta_1 + 1362) q^{91} + ( - 7 \beta_{2} - 48 \beta_1 - 517) q^{93} + (5 \beta_{2} + 45 \beta_1 + 295) q^{95} + (40 \beta_{2} - 25 \beta_1 + 462) q^{97} + ( - 27 \beta_{2} + 123 \beta_1 + 567) q^{99}+O(q^{100})$$ q + b1 * q^3 + 5 * q^5 + (b2 - 3*b1 - 1) * q^7 + (b2 - b1 + 4) * q^9 + (2*b2 + 5*b1 - 10) * q^11 + (3*b2 - 3*b1 + 27) * q^13 + 5*b1 * q^15 + (-11*b1 + 46) * q^17 + (b2 + 9*b1 + 59) * q^19 + (-b2 + 14*b1 - 91) * q^21 - 23 * q^23 + 25 * q^25 + (b2 - 10*b1 - 29) * q^27 + (4*b2 - 18*b1 + 122) * q^29 + (5*b2 - 17*b1 - 125) * q^31 + (9*b2 + 9*b1 + 159) * q^33 + (5*b2 - 15*b1 - 5) * q^35 + (2*b2 + 8*b1 + 324) * q^37 + (3*b2 + 66*b1 - 87) * q^39 + (10*b2 + 47*b1 - 204) * q^41 + (4*b2 - 40*b1 - 256) * q^43 + (5*b2 - 5*b1 + 20) * q^45 + (6*b2 - 42*b1 + 106) * q^47 + (-18*b2 - 49*b1 + 303) * q^49 + (-11*b2 + 57*b1 - 341) * q^51 + (-12*b2 - 60*b1 + 186) * q^53 + (10*b2 + 25*b1 - 50) * q^55 + (11*b2 + 62*b1 + 281) * q^57 + (-12*b2 - 34*b1 - 40) * q^59 + (8*b2 + 49*b1 - 30) * q^61 + (-15*b2 - 36*b1 + 459) * q^63 + (15*b2 - 15*b1 + 135) * q^65 + (12*b2 - 20*b1 - 528) * q^67 - 23*b1 * q^69 + (-8*b2 + 67*b1 + 132) * q^71 + (-42*b2 + 30*b1 - 284) * q^73 + 25*b1 * q^75 + (-55*b2 + 80*b1 + 299) * q^77 + (-16*b2 + 28*b1 + 272) * q^79 + (-35*b2 + 20*b1 - 416) * q^81 + (-22*b2 - 52*b1 + 106) * q^83 + (-55*b1 + 230) * q^85 + (-10*b2 + 188*b1 - 550) * q^87 + (26*b2 + 200*b1 - 88) * q^89 + (-30*b2 - 153*b1 + 1362) * q^91 + (-7*b2 - 48*b1 - 517) * q^93 + (5*b2 + 45*b1 + 295) * q^95 + (40*b2 - 25*b1 + 462) * q^97 + (-27*b2 + 123*b1 + 567) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{3} + 15 q^{5} - 7 q^{7} + 10 q^{9}+O(q^{10})$$ 3 * q + q^3 + 15 * q^5 - 7 * q^7 + 10 * q^9 $$3 q + q^{3} + 15 q^{5} - 7 q^{7} + 10 q^{9} - 27 q^{11} + 75 q^{13} + 5 q^{15} + 127 q^{17} + 185 q^{19} - 258 q^{21} - 69 q^{23} + 75 q^{25} - 98 q^{27} + 344 q^{29} - 397 q^{31} + 477 q^{33} - 35 q^{35} + 978 q^{37} - 198 q^{39} - 575 q^{41} - 812 q^{43} + 50 q^{45} + 270 q^{47} + 878 q^{49} - 955 q^{51} + 510 q^{53} - 135 q^{55} + 894 q^{57} - 142 q^{59} - 49 q^{61} + 1356 q^{63} + 375 q^{65} - 1616 q^{67} - 23 q^{69} + 471 q^{71} - 780 q^{73} + 25 q^{75} + 1032 q^{77} + 860 q^{79} - 1193 q^{81} + 288 q^{83} + 635 q^{85} - 1452 q^{87} - 90 q^{89} + 3963 q^{91} - 1592 q^{93} + 925 q^{95} + 1321 q^{97} + 1851 q^{99}+O(q^{100})$$ 3 * q + q^3 + 15 * q^5 - 7 * q^7 + 10 * q^9 - 27 * q^11 + 75 * q^13 + 5 * q^15 + 127 * q^17 + 185 * q^19 - 258 * q^21 - 69 * q^23 + 75 * q^25 - 98 * q^27 + 344 * q^29 - 397 * q^31 + 477 * q^33 - 35 * q^35 + 978 * q^37 - 198 * q^39 - 575 * q^41 - 812 * q^43 + 50 * q^45 + 270 * q^47 + 878 * q^49 - 955 * q^51 + 510 * q^53 - 135 * q^55 + 894 * q^57 - 142 * q^59 - 49 * q^61 + 1356 * q^63 + 375 * q^65 - 1616 * q^67 - 23 * q^69 + 471 * q^71 - 780 * q^73 + 25 * q^75 + 1032 * q^77 + 860 * q^79 - 1193 * q^81 + 288 * q^83 + 635 * q^85 - 1452 * q^87 - 90 * q^89 + 3963 * q^91 - 1592 * q^93 + 925 * q^95 + 1321 * q^97 + 1851 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 45x + 60$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 31$$ v^2 + v - 31
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta _1 + 31$$ b2 - b1 + 31

## 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
 −6.84916 1.34735 6.50182
0 −6.84916 0 5.00000 0 28.6094 0 19.9110 0
1.2 0 1.34735 0 5.00000 0 −32.8794 0 −25.1847 0
1.3 0 6.50182 0 5.00000 0 −2.73001 0 15.2736 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
$$2$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$1$$

## 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 1840.4.a.j 3
4.b odd 2 1 230.4.a.g 3
12.b even 2 1 2070.4.a.ba 3
20.d odd 2 1 1150.4.a.m 3
20.e even 4 2 1150.4.b.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.g 3 4.b odd 2 1
1150.4.a.m 3 20.d odd 2 1
1150.4.b.l 6 20.e even 4 2
1840.4.a.j 3 1.a even 1 1 trivial
2070.4.a.ba 3 12.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{3} - T_{3}^{2} - 45T_{3} + 60$$ T3^3 - T3^2 - 45*T3 + 60 $$T_{7}^{3} + 7T_{7}^{2} - 929T_{7} - 2568$$ T7^3 + 7*T7^2 - 929*T7 - 2568

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - T^{2} - 45 T + 60$$
$5$ $$(T - 5)^{3}$$
$7$ $$T^{3} + 7 T^{2} - 929 T - 2568$$
$11$ $$T^{3} + 27 T^{2} - 3399 T - 89388$$
$13$ $$T^{3} - 75 T^{2} - 3663 T + 275238$$
$17$ $$T^{3} - 127 T^{2} - 109 T + 96550$$
$19$ $$T^{3} - 185 T^{2} + 7003 T - 37596$$
$23$ $$(T + 23)^{3}$$
$29$ $$T^{3} - 344 T^{2} + 16552 T + 291288$$
$31$ $$T^{3} + 397 T^{2} + 26165 T - 1547080$$
$37$ $$T^{3} - 978 T^{2} + \cdots - 33005536$$
$41$ $$T^{3} + 575 T^{2} + \cdots - 50953878$$
$43$ $$T^{3} + 812 T^{2} + \cdots - 10161920$$
$47$ $$T^{3} - 270 T^{2} - 72660 T - 3184000$$
$53$ $$T^{3} - 510 T^{2} + \cdots + 89503704$$
$59$ $$T^{3} + 142 T^{2} - 136780 T + 9906704$$
$61$ $$T^{3} + 49 T^{2} - 151973 T - 23572158$$
$67$ $$T^{3} + 1616 T^{2} + \cdots + 111600960$$
$71$ $$T^{3} - 471 T^{2} + \cdots + 75603760$$
$73$ $$T^{3} + 780 T^{2} + \cdots - 672863896$$
$79$ $$T^{3} - 860 T^{2} + 68208 T + 8296704$$
$83$ $$T^{3} - 288 T^{2} + \cdots + 106176592$$
$89$ $$T^{3} + 90 T^{2} + \cdots - 1109897568$$
$97$ $$T^{3} - 1321 T^{2} + \cdots + 689377182$$