# Properties

 Label 1840.4.a.p Level $1840$ Weight $4$ Character orbit 1840.a Self dual yes Analytic conductor $108.564$ Analytic rank $1$ Dimension $5$ 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: $$1$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Defining polynomial: $$x^{5} - 34x^{3} - 9x^{2} + 260x + 60$$ x^5 - 34*x^3 - 9*x^2 + 260*x + 60 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 115) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_1 + 1) q^{3} - 5 q^{5} + ( - \beta_{2} + 3 \beta_1 + 2) q^{7} + (\beta_{4} + \beta_{3} - 4 \beta_{2} - \beta_1 + 14) q^{9}+O(q^{10})$$ q + (b3 - b1 + 1) * q^3 - 5 * q^5 + (-b2 + 3*b1 + 2) * q^7 + (b4 + b3 - 4*b2 - b1 + 14) * q^9 $$q + (\beta_{3} - \beta_1 + 1) q^{3} - 5 q^{5} + ( - \beta_{2} + 3 \beta_1 + 2) q^{7} + (\beta_{4} + \beta_{3} - 4 \beta_{2} - \beta_1 + 14) q^{9} + (\beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 32) q^{11} + ( - 3 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + \beta_1 + 5) q^{13} + ( - 5 \beta_{3} + 5 \beta_1 - 5) q^{15} + (3 \beta_{4} + 5 \beta_{3} + 4 \beta_{2} - 3 \beta_1 - 68) q^{17} + ( - \beta_{4} - 10 \beta_{3} - 3 \beta_1 + 4) q^{19} + ( - \beta_{4} + 9 \beta_{3} + 13 \beta_{2} - 14 \beta_1 - 38) q^{21} - 23 q^{23} + 25 q^{25} + ( - \beta_{4} + 2 \beta_{3} - 5 \beta_{2} - 26 \beta_1 + 51) q^{27} + ( - 4 \beta_{4} + 8 \beta_{3} + 9 \beta_{2} + 14 \beta_1 - 119) q^{29} + ( - 9 \beta_{4} - 6 \beta_{3} + 11 \beta_{2} + 17 \beta_1 - 129) q^{31} + (3 \beta_{4} + 42 \beta_{3} + 26 \beta_{2} - 9 \beta_1 - 96) q^{33} + (5 \beta_{2} - 15 \beta_1 - 10) q^{35} + ( - 6 \beta_{4} - 11 \beta_{3} - 5 \beta_{2} - 40 \beta_1 - 24) q^{37} + ( - 11 \beta_{4} - 27 \beta_{3} - 24 \beta_{2} - 22 \beta_1 - 11) q^{39} + (10 \beta_{4} - 17 \beta_{3} + 18 \beta_{2} + 19 \beta_1 + 79) q^{41} + (9 \beta_{4} - 9 \beta_{3} - 12 \beta_{2} - 46 \beta_1 + 54) q^{43} + ( - 5 \beta_{4} - 5 \beta_{3} + 20 \beta_{2} + 5 \beta_1 - 70) q^{45} + (7 \beta_{4} + 14 \beta_{3} + 12 \beta_{2} + 68 \beta_1 + 53) q^{47} + (6 \beta_{4} - 61 \beta_{3} - 13 \beta_{2} + 47 \beta_1 + 9) q^{49} + (15 \beta_{4} - 35 \beta_{3} + 19 \beta_{2} + 93 \beta_1 + 44) q^{51} + ( - 2 \beta_{4} - 44 \beta_{3} + 22 \beta_{2} + 20 \beta_1 - 166) q^{53} + ( - 5 \beta_{4} + 10 \beta_{3} - 15 \beta_{2} - 15 \beta_1 - 160) q^{55} + ( - 12 \beta_{4} - 33 \beta_{3} - 32 \beta_{2} + 12 \beta_1 - 180) q^{57} + (11 \beta_{4} - 36 \beta_{3} + 9 \beta_{2} - 4 \beta_1 + 252) q^{59} + ( - 20 \beta_{4} - 34 \beta_{3} + 12 \beta_{2} + 11 \beta_1 + 116) q^{61} + (20 \beta_{4} - 72 \beta_{3} - 15 \beta_{2} + 82 \beta_1 + 252) q^{63} + (15 \beta_{4} + 10 \beta_{3} + 15 \beta_{2} - 5 \beta_1 - 25) q^{65} + ( - 35 \beta_{4} + 31 \beta_{3} - 14 \beta_{2} - 38 \beta_1 + 184) q^{67} + ( - 23 \beta_{3} + 23 \beta_1 - 23) q^{69} + (9 \beta_{4} + 38 \beta_{3} + 34 \beta_{2} - 49 \beta_1 + 11) q^{71} + (23 \beta_{4} + 29 \beta_{3} - 41 \beta_{2} - 64 \beta_1 - 61) q^{73} + (25 \beta_{3} - 25 \beta_1 + 25) q^{75} + (3 \beta_{4} + 20 \beta_{3} - 86 \beta_{2} + 120 \beta_1 - 4) q^{77} + ( - 8 \beta_{4} - 3 \beta_{3} - 95 \beta_{2} - 36 \beta_1 - 12) q^{79} + ( - 32 \beta_{4} - 30 \beta_{3} - 37 \beta_{2} - 42 \beta_1 + 185) q^{81} + ( - 34 \beta_{4} + 26 \beta_{3} + 30 \beta_{2} - 40 \beta_1 + 24) q^{83} + ( - 15 \beta_{4} - 25 \beta_{3} - 20 \beta_{2} + 15 \beta_1 + 340) q^{85} + (9 \beta_{4} - 128 \beta_{3} + 68 \beta_{2} + 190 \beta_1 - 191) q^{87} + (61 \beta_{4} + 49 \beta_{3} + 18 \beta_{2} + 58 \beta_1 - 42) q^{89} + ( - 17 \beta_{4} - 46 \beta_{3} + 31 \beta_{2} + 33 \beta_1 + 286) q^{91} + ( - 13 \beta_{4} - 217 \beta_{3} + 38 \beta_{2} + 244 \beta_1 - 561) q^{93} + (5 \beta_{4} + 50 \beta_{3} + 15 \beta_1 - 20) q^{95} + ( - 17 \beta_{4} - 32 \beta_{3} - 31 \beta_{2} + 27 \beta_1 - 358) q^{97} + (47 \beta_{4} + 31 \beta_{3} - 11 \beta_{2} + 207 \beta_1 - 40) q^{99}+O(q^{100})$$ q + (b3 - b1 + 1) * q^3 - 5 * q^5 + (-b2 + 3*b1 + 2) * q^7 + (b4 + b3 - 4*b2 - b1 + 14) * q^9 + (b4 - 2*b3 + 3*b2 + 3*b1 + 32) * q^11 + (-3*b4 - 2*b3 - 3*b2 + b1 + 5) * q^13 + (-5*b3 + 5*b1 - 5) * q^15 + (3*b4 + 5*b3 + 4*b2 - 3*b1 - 68) * q^17 + (-b4 - 10*b3 - 3*b1 + 4) * q^19 + (-b4 + 9*b3 + 13*b2 - 14*b1 - 38) * q^21 - 23 * q^23 + 25 * q^25 + (-b4 + 2*b3 - 5*b2 - 26*b1 + 51) * q^27 + (-4*b4 + 8*b3 + 9*b2 + 14*b1 - 119) * q^29 + (-9*b4 - 6*b3 + 11*b2 + 17*b1 - 129) * q^31 + (3*b4 + 42*b3 + 26*b2 - 9*b1 - 96) * q^33 + (5*b2 - 15*b1 - 10) * q^35 + (-6*b4 - 11*b3 - 5*b2 - 40*b1 - 24) * q^37 + (-11*b4 - 27*b3 - 24*b2 - 22*b1 - 11) * q^39 + (10*b4 - 17*b3 + 18*b2 + 19*b1 + 79) * q^41 + (9*b4 - 9*b3 - 12*b2 - 46*b1 + 54) * q^43 + (-5*b4 - 5*b3 + 20*b2 + 5*b1 - 70) * q^45 + (7*b4 + 14*b3 + 12*b2 + 68*b1 + 53) * q^47 + (6*b4 - 61*b3 - 13*b2 + 47*b1 + 9) * q^49 + (15*b4 - 35*b3 + 19*b2 + 93*b1 + 44) * q^51 + (-2*b4 - 44*b3 + 22*b2 + 20*b1 - 166) * q^53 + (-5*b4 + 10*b3 - 15*b2 - 15*b1 - 160) * q^55 + (-12*b4 - 33*b3 - 32*b2 + 12*b1 - 180) * q^57 + (11*b4 - 36*b3 + 9*b2 - 4*b1 + 252) * q^59 + (-20*b4 - 34*b3 + 12*b2 + 11*b1 + 116) * q^61 + (20*b4 - 72*b3 - 15*b2 + 82*b1 + 252) * q^63 + (15*b4 + 10*b3 + 15*b2 - 5*b1 - 25) * q^65 + (-35*b4 + 31*b3 - 14*b2 - 38*b1 + 184) * q^67 + (-23*b3 + 23*b1 - 23) * q^69 + (9*b4 + 38*b3 + 34*b2 - 49*b1 + 11) * q^71 + (23*b4 + 29*b3 - 41*b2 - 64*b1 - 61) * q^73 + (25*b3 - 25*b1 + 25) * q^75 + (3*b4 + 20*b3 - 86*b2 + 120*b1 - 4) * q^77 + (-8*b4 - 3*b3 - 95*b2 - 36*b1 - 12) * q^79 + (-32*b4 - 30*b3 - 37*b2 - 42*b1 + 185) * q^81 + (-34*b4 + 26*b3 + 30*b2 - 40*b1 + 24) * q^83 + (-15*b4 - 25*b3 - 20*b2 + 15*b1 + 340) * q^85 + (9*b4 - 128*b3 + 68*b2 + 190*b1 - 191) * q^87 + (61*b4 + 49*b3 + 18*b2 + 58*b1 - 42) * q^89 + (-17*b4 - 46*b3 + 31*b2 + 33*b1 + 286) * q^91 + (-13*b4 - 217*b3 + 38*b2 + 244*b1 - 561) * q^93 + (5*b4 + 50*b3 + 15*b1 - 20) * q^95 + (-17*b4 - 32*b3 - 31*b2 + 27*b1 - 358) * q^97 + (47*b4 + 31*b3 - 11*b2 + 207*b1 - 40) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 6 q^{3} - 25 q^{5} + 15 q^{7} + 79 q^{9}+O(q^{10})$$ 5 * q + 6 * q^3 - 25 * q^5 + 15 * q^7 + 79 * q^9 $$5 q + 6 q^{3} - 25 q^{5} + 15 q^{7} + 79 q^{9} + 153 q^{11} + 28 q^{13} - 30 q^{15} - 341 q^{17} - 3 q^{19} - 212 q^{21} - 115 q^{23} + 125 q^{25} + 243 q^{27} - 583 q^{29} - 662 q^{31} - 457 q^{33} - 75 q^{35} - 172 q^{37} - 83 q^{39} + 344 q^{41} + 230 q^{43} - 395 q^{45} + 337 q^{47} - 4 q^{49} + 205 q^{51} - 942 q^{53} - 765 q^{55} - 890 q^{57} + 1166 q^{59} + 499 q^{61} + 1228 q^{63} - 140 q^{65} + 972 q^{67} - 138 q^{69} + 14 q^{71} - 229 q^{73} + 150 q^{75} + 312 q^{77} + 88 q^{79} + 897 q^{81} + 72 q^{83} + 1705 q^{85} - 1157 q^{87} - 90 q^{89} + 1309 q^{91} - 3071 q^{93} + 15 q^{95} - 1765 q^{97} + 91 q^{99}+O(q^{100})$$ 5 * q + 6 * q^3 - 25 * q^5 + 15 * q^7 + 79 * q^9 + 153 * q^11 + 28 * q^13 - 30 * q^15 - 341 * q^17 - 3 * q^19 - 212 * q^21 - 115 * q^23 + 125 * q^25 + 243 * q^27 - 583 * q^29 - 662 * q^31 - 457 * q^33 - 75 * q^35 - 172 * q^37 - 83 * q^39 + 344 * q^41 + 230 * q^43 - 395 * q^45 + 337 * q^47 - 4 * q^49 + 205 * q^51 - 942 * q^53 - 765 * q^55 - 890 * q^57 + 1166 * q^59 + 499 * q^61 + 1228 * q^63 - 140 * q^65 + 972 * q^67 - 138 * q^69 + 14 * q^71 - 229 * q^73 + 150 * q^75 + 312 * q^77 + 88 * q^79 + 897 * q^81 + 72 * q^83 + 1705 * q^85 - 1157 * q^87 - 90 * q^89 + 1309 * q^91 - 3071 * q^93 + 15 * q^95 - 1765 * q^97 + 91 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 34x^{3} - 9x^{2} + 260x + 60$$ :

 $$\beta_{1}$$ $$=$$ $$( 3\nu^{4} - 22\nu^{3} - 26\nu^{2} + 377\nu - 278 ) / 64$$ (3*v^4 - 22*v^3 - 26*v^2 + 377*v - 278) / 64 $$\beta_{2}$$ $$=$$ $$( \nu^{4} - 2\nu^{3} - 30\nu^{2} + 51\nu + 158 ) / 16$$ (v^4 - 2*v^3 - 30*v^2 + 51*v + 158) / 16 $$\beta_{3}$$ $$=$$ $$( -\nu^{4} + 2\nu^{3} + 30\nu^{2} - 19\nu - 158 ) / 16$$ (-v^4 + 2*v^3 + 30*v^2 - 19*v - 158) / 16 $$\beta_{4}$$ $$=$$ $$( 7\nu^{4} + 2\nu^{3} - 146\nu^{2} - 123\nu + 194 ) / 32$$ (7*v^4 + 2*v^3 - 146*v^2 - 123*v + 194) / 32
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{4} + 4\beta_{3} - \beta_{2} + 2\beta _1 + 52 ) / 4$$ (b4 + 4*b3 - b2 + 2*b1 + 52) / 4 $$\nu^{3}$$ $$=$$ $$\beta_{4} + 11\beta_{3} + 9\beta_{2} - 2\beta _1 + 5$$ b4 + 11*b3 + 9*b2 - 2*b1 + 5 $$\nu^{4}$$ $$=$$ $$( 19\beta_{4} + 53\beta_{3} + 2\beta_{2} + 22\beta _1 + 484 ) / 2$$ (19*b4 + 53*b3 + 2*b2 + 22*b1 + 484) / 2

## 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
 −4.34884 −0.230529 3.26689 4.98640 −3.67392
0 −8.28008 0 −5.00000 0 30.9947 0 41.5597 0
1.2 0 −2.78435 0 −5.00000 0 −24.1991 0 −19.2474 0
1.3 0 0.577397 0 −5.00000 0 10.7178 0 −26.6666 0
1.4 0 7.39409 0 −5.00000 0 3.57544 0 27.6726 0
1.5 0 9.09294 0 −5.00000 0 −6.08885 0 55.6816 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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.p 5
4.b odd 2 1 115.4.a.d 5
12.b even 2 1 1035.4.a.m 5
20.d odd 2 1 575.4.a.k 5
20.e even 4 2 575.4.b.h 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.d 5 4.b odd 2 1
575.4.a.k 5 20.d odd 2 1
575.4.b.h 10 20.e even 4 2
1035.4.a.m 5 12.b even 2 1
1840.4.a.p 5 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1840))$$:

 $$T_{3}^{5} - 6T_{3}^{4} - 89T_{3}^{3} + 417T_{3}^{2} + 1340T_{3} - 895$$ T3^5 - 6*T3^4 - 89*T3^3 + 417*T3^2 + 1340*T3 - 895 $$T_{7}^{5} - 15T_{7}^{4} - 743T_{7}^{3} + 6718T_{7}^{2} + 34948T_{7} - 175008$$ T7^5 - 15*T7^4 - 743*T7^3 + 6718*T7^2 + 34948*T7 - 175008

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - 6 T^{4} - 89 T^{3} + 417 T^{2} + \cdots - 895$$
$5$ $$(T + 5)^{5}$$
$7$ $$T^{5} - 15 T^{4} - 743 T^{3} + \cdots - 175008$$
$11$ $$T^{5} - 153 T^{4} + \cdots + 58105432$$
$13$ $$T^{5} - 28 T^{4} - 5855 T^{3} + \cdots - 42528287$$
$17$ $$T^{5} + 341 T^{4} + \cdots - 1269566848$$
$19$ $$T^{5} + 3 T^{4} - 10481 T^{3} + \cdots + 5566328$$
$23$ $$(T + 23)^{5}$$
$29$ $$T^{5} + 583 T^{4} + \cdots - 63213004636$$
$31$ $$T^{5} + 662 T^{4} + \cdots - 341100199935$$
$37$ $$T^{5} + 172 T^{4} + \cdots + 337199293312$$
$41$ $$T^{5} - 344 T^{4} + \cdots - 554461833173$$
$43$ $$T^{5} - 230 T^{4} + \cdots - 278531891200$$
$47$ $$T^{5} - 337 T^{4} + \cdots - 6082562728660$$
$53$ $$T^{5} + 942 T^{4} + \cdots - 1069522603168$$
$59$ $$T^{5} - 1166 T^{4} + \cdots + 446741827072$$
$61$ $$T^{5} - 499 T^{4} + \cdots + 53636443112$$
$67$ $$T^{5} - 972 T^{4} + \cdots + 19766839800960$$
$71$ $$T^{5} - 14 T^{4} + \cdots - 256345940645$$
$73$ $$T^{5} + 229 T^{4} + \cdots + 3947399121116$$
$79$ $$T^{5} - 88 T^{4} + \cdots + 22913376438144$$
$83$ $$T^{5} - 72 T^{4} + \cdots - 2501024408832$$
$89$ $$T^{5} + \cdots + 279901843479552$$
$97$ $$T^{5} + 1765 T^{4} + \cdots - 113500838416$$