# Properties

 Label 1840.4.a.q Level $1840$ Weight $4$ Character orbit 1840.a Self dual yes Analytic conductor $108.564$ Analytic rank $0$ 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: $$0$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ Defining polynomial: $$x^{5} - 2x^{4} - 55x^{3} + 104x^{2} + 255x + 72$$ x^5 - 2*x^4 - 55*x^3 + 104*x^2 + 255*x + 72 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 460) 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + 1) q^{3} + 5 q^{5} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 5) q^{7} + ( - 2 \beta_{4} + 3 \beta_{2} + \beta_1 + 6) q^{9}+O(q^{10})$$ q + (b3 + 1) * q^3 + 5 * q^5 + (-b3 + 2*b2 + b1 + 5) * q^7 + (-2*b4 + 3*b2 + b1 + 6) * q^9 $$q + (\beta_{3} + 1) q^{3} + 5 q^{5} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 5) q^{7} + ( - 2 \beta_{4} + 3 \beta_{2} + \beta_1 + 6) q^{9} + ( - 3 \beta_{4} - 2 \beta_{3} - \beta_1 + 12) q^{11} + (2 \beta_{4} + 8 \beta_{3} + \beta_{2} - 22) q^{13} + (5 \beta_{3} + 5) q^{15} + ( - 2 \beta_{4} + 6 \beta_{3} + 7 \beta_{2} + 3 \beta_1 - 10) q^{17} + (5 \beta_{4} - 3 \beta_{3} + 7 \beta_{2} - 2 \beta_1 + 44) q^{19} + (6 \beta_{4} + 17 \beta_{3} - \beta_{2} - 4 \beta_1 - 19) q^{21} - 23 q^{23} + 25 q^{25} + ( - 3 \beta_{4} - 3 \beta_{2} + \beta_1 + 4) q^{27} + ( - 6 \beta_{4} + 3 \beta_{3} + 13 \beta_{2} - 11 \beta_1 + 43) q^{29} + ( - 3 \beta_{4} - 4 \beta_{3} + 10 \beta_{2} + 5 \beta_1 + 107) q^{31} + ( - 9 \beta_{4} + 21 \beta_{3} - 15 \beta_{2} + 6 \beta_1 - 36) q^{33} + ( - 5 \beta_{3} + 10 \beta_{2} + 5 \beta_1 + 25) q^{35} + ( - 21 \beta_{4} - 24 \beta_{3} + \beta_{2} + 26 \beta_1 - 16) q^{37} + ( - 3 \beta_{4} - 28 \beta_{3} + 31 \beta_{2} + 211) q^{39} + (9 \beta_{4} - 23 \beta_{3} - \beta_{2} + \beta_1 - 22) q^{41} + ( - 16 \beta_{4} - 12 \beta_{3} - 6 \beta_{2} + 31 \beta_1 + 63) q^{43} + ( - 10 \beta_{4} + 15 \beta_{2} + 5 \beta_1 + 30) q^{45} + ( - 9 \beta_{4} + 25 \beta_{3} - 52 \beta_{2} + 48) q^{47} + (2 \beta_{4} - 14 \beta_{3} + 7 \beta_{2} - 21 \beta_1 - 29) q^{49} + ( - 7 \beta_{4} + 27 \beta_{3} + 19 \beta_{2} + \beta_1 + 223) q^{51} + ( - 5 \beta_{4} - 46 \beta_{3} + \beta_{2} + 45 \beta_1 + 69) q^{53} + ( - 15 \beta_{4} - 10 \beta_{3} - 5 \beta_1 + 60) q^{55} + (56 \beta_{4} + 81 \beta_{3} + 13 \beta_{2} - 34 \beta_1 - 151) q^{57} + (3 \beta_{4} + 38 \beta_{3} + 15 \beta_{2} - 21 \beta_1 + 295) q^{59} + ( - 15 \beta_{4} - 42 \beta_{3} - 30 \beta_{2} - 7 \beta_1 - 144) q^{61} + (\beta_{4} - 23 \beta_{3} + 14 \beta_{2} - 30 \beta_1 + 277) q^{63} + (10 \beta_{4} + 40 \beta_{3} + 5 \beta_{2} - 110) q^{65} + (63 \beta_{4} + 92 \beta_{3} - 31 \beta_{2} - 30 \beta_1 + 84) q^{67} + ( - 23 \beta_{3} - 23) q^{69} + (35 \beta_{4} + 95 \beta_{3} - 45 \beta_{2} - 26 \beta_1 + 309) q^{71} + (\beta_{4} + 79 \beta_{3} + 26 \beta_{2} - 57 \beta_1 - 163) q^{73} + (25 \beta_{3} + 25) q^{75} + ( - 30 \beta_{4} - 95 \beta_{3} + 51 \beta_{2} + 41 \beta_1 - 132) q^{77} + ( - 6 \beta_{4} - 6 \beta_{3} - 6 \beta_{2} + 47 \beta_1 + 599) q^{79} + (28 \beta_{4} - 9 \beta_{3} - 93 \beta_{2} - 11 \beta_1 - 105) q^{81} + (9 \beta_{4} - 84 \beta_{3} + \beta_{2} + 95 \beta_1 + 87) q^{83} + ( - 10 \beta_{4} + 30 \beta_{3} + 35 \beta_{2} + 15 \beta_1 - 50) q^{85} + (25 \beta_{4} + 141 \beta_{3} + 4 \beta_{2} - 16 \beta_1 + 6) q^{87} + (32 \beta_{4} - 112 \beta_{3} - 34 \beta_{2} + 89 \beta_1 - 243) q^{89} + (57 \beta_{4} + 196 \beta_{3} - 64 \beta_{2} - 75 \beta_1 - 76) q^{91} + (13 \beta_{4} + 172 \beta_{3} - 11 \beta_{2} - 10 \beta_1 + 49) q^{93} + (25 \beta_{4} - 15 \beta_{3} + 35 \beta_{2} - 10 \beta_1 + 220) q^{95} + (25 \beta_{4} + 16 \beta_{3} - 148 \beta_{2} - 35 \beta_1 - 308) q^{97} + ( - 63 \beta_{4} - 81 \beta_{3} + 21 \beta_{2} + 111 \beta_1 + 531) q^{99}+O(q^{100})$$ q + (b3 + 1) * q^3 + 5 * q^5 + (-b3 + 2*b2 + b1 + 5) * q^7 + (-2*b4 + 3*b2 + b1 + 6) * q^9 + (-3*b4 - 2*b3 - b1 + 12) * q^11 + (2*b4 + 8*b3 + b2 - 22) * q^13 + (5*b3 + 5) * q^15 + (-2*b4 + 6*b3 + 7*b2 + 3*b1 - 10) * q^17 + (5*b4 - 3*b3 + 7*b2 - 2*b1 + 44) * q^19 + (6*b4 + 17*b3 - b2 - 4*b1 - 19) * q^21 - 23 * q^23 + 25 * q^25 + (-3*b4 - 3*b2 + b1 + 4) * q^27 + (-6*b4 + 3*b3 + 13*b2 - 11*b1 + 43) * q^29 + (-3*b4 - 4*b3 + 10*b2 + 5*b1 + 107) * q^31 + (-9*b4 + 21*b3 - 15*b2 + 6*b1 - 36) * q^33 + (-5*b3 + 10*b2 + 5*b1 + 25) * q^35 + (-21*b4 - 24*b3 + b2 + 26*b1 - 16) * q^37 + (-3*b4 - 28*b3 + 31*b2 + 211) * q^39 + (9*b4 - 23*b3 - b2 + b1 - 22) * q^41 + (-16*b4 - 12*b3 - 6*b2 + 31*b1 + 63) * q^43 + (-10*b4 + 15*b2 + 5*b1 + 30) * q^45 + (-9*b4 + 25*b3 - 52*b2 + 48) * q^47 + (2*b4 - 14*b3 + 7*b2 - 21*b1 - 29) * q^49 + (-7*b4 + 27*b3 + 19*b2 + b1 + 223) * q^51 + (-5*b4 - 46*b3 + b2 + 45*b1 + 69) * q^53 + (-15*b4 - 10*b3 - 5*b1 + 60) * q^55 + (56*b4 + 81*b3 + 13*b2 - 34*b1 - 151) * q^57 + (3*b4 + 38*b3 + 15*b2 - 21*b1 + 295) * q^59 + (-15*b4 - 42*b3 - 30*b2 - 7*b1 - 144) * q^61 + (b4 - 23*b3 + 14*b2 - 30*b1 + 277) * q^63 + (10*b4 + 40*b3 + 5*b2 - 110) * q^65 + (63*b4 + 92*b3 - 31*b2 - 30*b1 + 84) * q^67 + (-23*b3 - 23) * q^69 + (35*b4 + 95*b3 - 45*b2 - 26*b1 + 309) * q^71 + (b4 + 79*b3 + 26*b2 - 57*b1 - 163) * q^73 + (25*b3 + 25) * q^75 + (-30*b4 - 95*b3 + 51*b2 + 41*b1 - 132) * q^77 + (-6*b4 - 6*b3 - 6*b2 + 47*b1 + 599) * q^79 + (28*b4 - 9*b3 - 93*b2 - 11*b1 - 105) * q^81 + (9*b4 - 84*b3 + b2 + 95*b1 + 87) * q^83 + (-10*b4 + 30*b3 + 35*b2 + 15*b1 - 50) * q^85 + (25*b4 + 141*b3 + 4*b2 - 16*b1 + 6) * q^87 + (32*b4 - 112*b3 - 34*b2 + 89*b1 - 243) * q^89 + (57*b4 + 196*b3 - 64*b2 - 75*b1 - 76) * q^91 + (13*b4 + 172*b3 - 11*b2 - 10*b1 + 49) * q^93 + (25*b4 - 15*b3 + 35*b2 - 10*b1 + 220) * q^95 + (25*b4 + 16*b3 - 148*b2 - 35*b1 - 308) * q^97 + (-63*b4 - 81*b3 + 21*b2 + 111*b1 + 531) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 7 q^{3} + 25 q^{5} + 20 q^{7} + 30 q^{9}+O(q^{10})$$ 5 * q + 7 * q^3 + 25 * q^5 + 20 * q^7 + 30 * q^9 $$5 q + 7 q^{3} + 25 q^{5} + 20 q^{7} + 30 q^{9} + 63 q^{11} - 99 q^{13} + 35 q^{15} - 44 q^{17} + 199 q^{19} - 68 q^{21} - 115 q^{23} + 125 q^{25} + 28 q^{27} + 231 q^{29} + 518 q^{31} - 111 q^{33} + 100 q^{35} - 113 q^{37} + 974 q^{39} - 174 q^{41} + 298 q^{43} + 150 q^{45} + 360 q^{47} - 163 q^{49} + 1163 q^{51} + 217 q^{53} + 315 q^{55} - 684 q^{57} + 1551 q^{59} - 737 q^{61} + 1353 q^{63} - 495 q^{65} + 539 q^{67} - 161 q^{69} + 1736 q^{71} - 628 q^{73} + 175 q^{75} - 882 q^{77} + 2954 q^{79} - 495 q^{81} + 153 q^{83} - 220 q^{85} + 274 q^{87} - 1558 q^{89} + 37 q^{91} + 584 q^{93} + 995 q^{95} - 1375 q^{97} + 2487 q^{99}+O(q^{100})$$ 5 * q + 7 * q^3 + 25 * q^5 + 20 * q^7 + 30 * q^9 + 63 * q^11 - 99 * q^13 + 35 * q^15 - 44 * q^17 + 199 * q^19 - 68 * q^21 - 115 * q^23 + 125 * q^25 + 28 * q^27 + 231 * q^29 + 518 * q^31 - 111 * q^33 + 100 * q^35 - 113 * q^37 + 974 * q^39 - 174 * q^41 + 298 * q^43 + 150 * q^45 + 360 * q^47 - 163 * q^49 + 1163 * q^51 + 217 * q^53 + 315 * q^55 - 684 * q^57 + 1551 * q^59 - 737 * q^61 + 1353 * q^63 - 495 * q^65 + 539 * q^67 - 161 * q^69 + 1736 * q^71 - 628 * q^73 + 175 * q^75 - 882 * q^77 + 2954 * q^79 - 495 * q^81 + 153 * q^83 - 220 * q^85 + 274 * q^87 - 1558 * q^89 + 37 * q^91 + 584 * q^93 + 995 * q^95 - 1375 * q^97 + 2487 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2x^{4} - 55x^{3} + 104x^{2} + 255x + 72$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu - 1$$ 2*v - 1 $$\beta_{2}$$ $$=$$ $$( \nu^{4} - \nu^{3} - 61\nu^{2} + 58\nu + 348 ) / 30$$ (v^4 - v^3 - 61*v^2 + 58*v + 348) / 30 $$\beta_{3}$$ $$=$$ $$( 2\nu^{4} - 7\nu^{3} - 107\nu^{2} + 361\nu + 306 ) / 30$$ (2*v^4 - 7*v^3 - 107*v^2 + 361*v + 306) / 30 $$\beta_{4}$$ $$=$$ $$( \nu^{4} - \nu^{3} - 51\nu^{2} + 68\nu + 118 ) / 10$$ (v^4 - v^3 - 51*v^2 + 68*v + 118) / 10
 $$\nu$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{4} - 6\beta_{2} - \beta _1 + 45 ) / 2$$ (2*b4 - 6*b2 - b1 + 45) / 2 $$\nu^{3}$$ $$=$$ $$3\beta_{4} - 6\beta_{3} + 3\beta_{2} + 23\beta _1 + 14$$ 3*b4 - 6*b3 + 3*b2 + 23*b1 + 14 $$\nu^{4}$$ $$=$$ $$( 128\beta_{4} - 12\beta_{3} - 300\beta_{2} - 73\beta _1 + 2019 ) / 2$$ (128*b4 - 12*b3 - 300*b2 - 73*b1 + 2019) / 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
 −1.17944 −7.11066 6.96712 −0.336706 3.65968
0 −7.44221 0 5.00000 0 23.3042 0 28.3866 0
1.2 0 −0.380607 0 5.00000 0 −24.3526 0 −26.8551 0
1.3 0 0.0785019 0 5.00000 0 6.13060 0 −26.9938 0
1.4 0 6.75371 0 5.00000 0 19.0133 0 18.6126 0
1.5 0 7.99061 0 5.00000 0 −4.09549 0 36.8498 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.q 5
4.b odd 2 1 460.4.a.a 5
20.d odd 2 1 2300.4.a.d 5
20.e even 4 2 2300.4.c.c 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
460.4.a.a 5 4.b odd 2 1
1840.4.a.q 5 1.a even 1 1 trivial
2300.4.a.d 5 20.d odd 2 1
2300.4.c.c 10 20.e even 4 2

## Hecke kernels

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

 $$T_{3}^{5} - 7T_{3}^{4} - 58T_{3}^{3} + 385T_{3}^{2} + 123T_{3} - 12$$ T3^5 - 7*T3^4 - 58*T3^3 + 385*T3^2 + 123*T3 - 12 $$T_{7}^{5} - 20T_{7}^{4} - 576T_{7}^{3} + 12437T_{7}^{2} - 7210T_{7} - 270924$$ T7^5 - 20*T7^4 - 576*T7^3 + 12437*T7^2 - 7210*T7 - 270924

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5} - 7 T^{4} - 58 T^{3} + 385 T^{2} + \cdots - 12$$
$5$ $$(T - 5)^{5}$$
$7$ $$T^{5} - 20 T^{4} - 576 T^{3} + \cdots - 270924$$
$11$ $$T^{5} - 63 T^{4} - 1803 T^{3} + \cdots - 72719208$$
$13$ $$T^{5} + 99 T^{4} - 1576 T^{3} + \cdots - 14392890$$
$17$ $$T^{5} + 44 T^{4} + \cdots + 451144080$$
$19$ $$T^{5} - 199 T^{4} + \cdots + 800499864$$
$23$ $$(T + 23)^{5}$$
$29$ $$T^{5} - 231 T^{4} + \cdots + 43256062404$$
$31$ $$T^{5} - 518 T^{4} + \cdots + 9011869285$$
$37$ $$T^{5} + 113 T^{4} + \cdots - 409221703936$$
$41$ $$T^{5} + 174 T^{4} + \cdots + 12760146261$$
$43$ $$T^{5} - 298 T^{4} + \cdots - 803496865792$$
$47$ $$T^{5} - 360 T^{4} + \cdots - 13556115821808$$
$53$ $$T^{5} - 217 T^{4} + \cdots - 4075860543408$$
$59$ $$T^{5} - 1551 T^{4} + \cdots + 897857553600$$
$61$ $$T^{5} + 737 T^{4} + \cdots + 310836562128$$
$67$ $$T^{5} + \cdots - 137714931264960$$
$71$ $$T^{5} + \cdots + 108175489939041$$
$73$ $$T^{5} + 628 T^{4} + \cdots + 13803900176376$$
$79$ $$T^{5} - 2954 T^{4} + \cdots + 8954177264640$$
$83$ $$T^{5} - 153 T^{4} + \cdots + 29112932977344$$
$89$ $$T^{5} + \cdots + 180704830832640$$
$97$ $$T^{5} + \cdots + 347840565067560$$