# Properties

 Label 1856.4.a.u Level $1856$ Weight $4$ Character orbit 1856.a Self dual yes Analytic conductor $109.508$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1856 = 2^{6} \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1856.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$109.507544971$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 232) 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.

 $$f(q)$$ $$=$$ $$q + ( - 3 \beta_{2} + 1) q^{3} + (4 \beta_{2} - 3 \beta_1 - 1) q^{5} + (12 \beta_{2} - 2 \beta_1 - 2) q^{7} + ( - 6 \beta_{2} - 9 \beta_1 + 10) q^{9}+O(q^{10})$$ q + (-3*b2 + 1) * q^3 + (4*b2 - 3*b1 - 1) * q^5 + (12*b2 - 2*b1 - 2) * q^7 + (-6*b2 - 9*b1 + 10) * q^9 $$q + ( - 3 \beta_{2} + 1) q^{3} + (4 \beta_{2} - 3 \beta_1 - 1) q^{5} + (12 \beta_{2} - 2 \beta_1 - 2) q^{7} + ( - 6 \beta_{2} - 9 \beta_1 + 10) q^{9} + ( - 5 \beta_{2} + 4 \beta_1 - 1) q^{11} + ( - 16 \beta_{2} - 13 \beta_1 - 19) q^{13} + ( - 11 \beta_{2} - 49) q^{15} + (10 \beta_{2} + 8 \beta_1 - 16) q^{17} + ( - 16 \beta_{2} - 26 \beta_1 - 36) q^{19} + (6 \beta_{2} + 28 \beta_1 - 146) q^{21} + (10 \beta_{2} + 10 \beta_1 - 52) q^{23} + (22 \beta_{2} + 23 \beta_1 + 12) q^{25} + ( - 9 \beta_{2} - 54 \beta_1 + 55) q^{27} + 29 q^{29} + (7 \beta_{2} - 74 \beta_1 + 41) q^{31} + (22 \beta_{2} + \beta_1 + 59) q^{33} + (56 \beta_{2} + 10 \beta_1 + 242) q^{35} + ( - 76 \beta_{2} - 34 \beta_1 - 184) q^{37} + ( - 37 \beta_{2} - 100 \beta_1 + 173) q^{39} + (42 \beta_{2} + 2 \beta_1 + 402) q^{41} + ( - 77 \beta_{2} - 52 \beta_1 - 33) q^{43} + (28 \beta_{2} + 48 \beta_1 + 110) q^{45} + (59 \beta_{2} - 14 \beta_1 + 261) q^{47} + (40 \beta_{2} - 84 \beta_1 + 269) q^{49} + (106 \beta_{2} + 62 \beta_1 - 136) q^{51} + (112 \beta_{2} + 53 \beta_1 - 49) q^{53} + ( - 37 \beta_{2} - 24 \beta_1 - 175) q^{55} + ( - 64 \beta_{2} - 152 \beta_1 + 156) q^{57} + (62 \beta_{2} - 10 \beta_1 + 32) q^{59} + ( - 126 \beta_{2} - 180 \beta_1 - 120) q^{61} + (288 \beta_{2} + 184 \beta_1 - 164) q^{63} + ( - 130 \beta_{2} + 177 \beta_1 + 75) q^{65} + (12 \beta_{2} - 124 \beta_1 - 76) q^{67} + (226 \beta_{2} + 70 \beta_1 - 172) q^{69} + ( - 82 \beta_{2} + 16 \beta_1 + 378) q^{71} + ( - 172 \beta_{2} + 102 \beta_1 - 396) q^{73} + (124 \beta_{2} + 158 \beta_1 - 252) q^{75} + ( - 102 \beta_{2} - 12 \beta_1 - 302) q^{77} + (217 \beta_{2} + 14 \beta_1 + 831) q^{79} + ( - 336 \beta_{2} - 107) q^{81} + (350 \beta_{2} - 166 \beta_1 - 540) q^{83} + ( - 30 \beta_{2} - 26 \beta_1 - 16) q^{85} + ( - 87 \beta_{2} + 29) q^{87} + ( - 458 \beta_{2} + 74 \beta_1 + 398) q^{89} + (406 \beta_1 - 522) q^{91} + ( - 560 \beta_{2} - 275 \beta_1 - 43) q^{93} + ( - 172 \beta_{2} + 332 \beta_1 + 404) q^{95} + (162 \beta_{2} - 394 \beta_1 + 406) q^{97} + ( - 14 \beta_{2} - 38 \beta_1 - 178) q^{99}+O(q^{100})$$ q + (-3*b2 + 1) * q^3 + (4*b2 - 3*b1 - 1) * q^5 + (12*b2 - 2*b1 - 2) * q^7 + (-6*b2 - 9*b1 + 10) * q^9 + (-5*b2 + 4*b1 - 1) * q^11 + (-16*b2 - 13*b1 - 19) * q^13 + (-11*b2 - 49) * q^15 + (10*b2 + 8*b1 - 16) * q^17 + (-16*b2 - 26*b1 - 36) * q^19 + (6*b2 + 28*b1 - 146) * q^21 + (10*b2 + 10*b1 - 52) * q^23 + (22*b2 + 23*b1 + 12) * q^25 + (-9*b2 - 54*b1 + 55) * q^27 + 29 * q^29 + (7*b2 - 74*b1 + 41) * q^31 + (22*b2 + b1 + 59) * q^33 + (56*b2 + 10*b1 + 242) * q^35 + (-76*b2 - 34*b1 - 184) * q^37 + (-37*b2 - 100*b1 + 173) * q^39 + (42*b2 + 2*b1 + 402) * q^41 + (-77*b2 - 52*b1 - 33) * q^43 + (28*b2 + 48*b1 + 110) * q^45 + (59*b2 - 14*b1 + 261) * q^47 + (40*b2 - 84*b1 + 269) * q^49 + (106*b2 + 62*b1 - 136) * q^51 + (112*b2 + 53*b1 - 49) * q^53 + (-37*b2 - 24*b1 - 175) * q^55 + (-64*b2 - 152*b1 + 156) * q^57 + (62*b2 - 10*b1 + 32) * q^59 + (-126*b2 - 180*b1 - 120) * q^61 + (288*b2 + 184*b1 - 164) * q^63 + (-130*b2 + 177*b1 + 75) * q^65 + (12*b2 - 124*b1 - 76) * q^67 + (226*b2 + 70*b1 - 172) * q^69 + (-82*b2 + 16*b1 + 378) * q^71 + (-172*b2 + 102*b1 - 396) * q^73 + (124*b2 + 158*b1 - 252) * q^75 + (-102*b2 - 12*b1 - 302) * q^77 + (217*b2 + 14*b1 + 831) * q^79 + (-336*b2 - 107) * q^81 + (350*b2 - 166*b1 - 540) * q^83 + (-30*b2 - 26*b1 - 16) * q^85 + (-87*b2 + 29) * q^87 + (-458*b2 + 74*b1 + 398) * q^89 + (406*b1 - 522) * q^91 + (-560*b2 - 275*b1 - 43) * q^93 + (-172*b2 + 332*b1 + 404) * q^95 + (162*b2 - 394*b1 + 406) * q^97 + (-14*b2 - 38*b1 - 178) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 6 q^{3} - 4 q^{5} - 16 q^{7} + 45 q^{9}+O(q^{10})$$ 3 * q + 6 * q^3 - 4 * q^5 - 16 * q^7 + 45 * q^9 $$3 q + 6 q^{3} - 4 q^{5} - 16 q^{7} + 45 q^{9} - 2 q^{11} - 28 q^{13} - 136 q^{15} - 66 q^{17} - 66 q^{19} - 472 q^{21} - 176 q^{23} - 9 q^{25} + 228 q^{27} + 87 q^{29} + 190 q^{31} + 154 q^{33} + 660 q^{35} - 442 q^{37} + 656 q^{39} + 1162 q^{41} + 30 q^{43} + 254 q^{45} + 738 q^{47} + 851 q^{49} - 576 q^{51} - 312 q^{53} - 464 q^{55} + 684 q^{57} + 44 q^{59} - 54 q^{61} - 964 q^{63} + 178 q^{65} - 116 q^{67} - 812 q^{69} + 1200 q^{71} - 1118 q^{73} - 1038 q^{75} - 792 q^{77} + 2262 q^{79} + 15 q^{81} - 1804 q^{83} + 8 q^{85} + 174 q^{87} + 1578 q^{89} - 1972 q^{91} + 706 q^{93} + 1052 q^{95} + 1450 q^{97} - 482 q^{99}+O(q^{100})$$ 3 * q + 6 * q^3 - 4 * q^5 - 16 * q^7 + 45 * q^9 - 2 * q^11 - 28 * q^13 - 136 * q^15 - 66 * q^17 - 66 * q^19 - 472 * q^21 - 176 * q^23 - 9 * q^25 + 228 * q^27 + 87 * q^29 + 190 * q^31 + 154 * q^33 + 660 * q^35 - 442 * q^37 + 656 * q^39 + 1162 * q^41 + 30 * q^43 + 254 * q^45 + 738 * q^47 + 851 * q^49 - 576 * q^51 - 312 * q^53 - 464 * q^55 + 684 * q^57 + 44 * q^59 - 54 * q^61 - 964 * q^63 + 178 * q^65 - 116 * q^67 - 812 * q^69 + 1200 * q^71 - 1118 * q^73 - 1038 * q^75 - 792 * q^77 + 2262 * q^79 + 15 * q^81 - 1804 * q^83 + 8 * q^85 + 174 * q^87 + 1578 * q^89 - 1972 * q^91 + 706 * q^93 + 1052 * q^95 + 1450 * q^97 - 482 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 3$$ v^2 + v - 3 $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$ v^2 - v - 3
 $$\nu$$ $$=$$ $$( -\beta_{2} + \beta_1 ) / 2$$ (-b2 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{2} + \beta _1 + 6 ) / 2$$ (b2 + b1 + 6) / 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
 −1.86081 2.11491 −0.254102
0 −5.97021 0 12.4882 0 28.6773 0 8.64344 0
1.2 0 2.92622 0 −14.3315 0 −16.8804 0 −18.4372 0
1.3 0 9.04399 0 −2.15672 0 −27.7969 0 54.7938 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
$$2$$ $$-1$$
$$29$$ $$-1$$

## 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 1856.4.a.u 3
4.b odd 2 1 1856.4.a.p 3
8.b even 2 1 464.4.a.g 3
8.d odd 2 1 232.4.a.b 3
24.f even 2 1 2088.4.a.b 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
232.4.a.b 3 8.d odd 2 1
464.4.a.g 3 8.b even 2 1
1856.4.a.p 3 4.b odd 2 1
1856.4.a.u 3 1.a even 1 1 trivial
2088.4.a.b 3 24.f even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1856))$$:

 $$T_{3}^{3} - 6T_{3}^{2} - 45T_{3} + 158$$ T3^3 - 6*T3^2 - 45*T3 + 158 $$T_{5}^{3} + 4T_{5}^{2} - 175T_{5} - 386$$ T5^3 + 4*T5^2 - 175*T5 - 386 $$T_{7}^{3} + 16T_{7}^{2} - 812T_{7} - 13456$$ T7^3 + 16*T7^2 - 812*T7 - 13456

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3} - 6 T^{2} - 45 T + 158$$
$5$ $$T^{3} + 4 T^{2} - 175 T - 386$$
$7$ $$T^{3} + 16 T^{2} - 812 T - 13456$$
$11$ $$T^{3} + 2 T^{2} - 301 T - 106$$
$13$ $$T^{3} + 28 T^{2} - 3999 T - 137518$$
$17$ $$T^{3} + 66 T^{2} - 184 T - 1696$$
$19$ $$T^{3} + 66 T^{2} - 9616 T - 393632$$
$23$ $$T^{3} + 176 T^{2} + 8192 T + 106688$$
$29$ $$(T - 29)^{3}$$
$31$ $$T^{3} - 190 T^{2} - 54433 T + 9505226$$
$37$ $$T^{3} + 442 T^{2} + \cdots - 10323328$$
$41$ $$T^{3} - 1162 T^{2} + \cdots - 53735864$$
$43$ $$T^{3} - 30 T^{2} - 81277 T - 8035162$$
$47$ $$T^{3} - 738 T^{2} + \cdots - 10647458$$
$53$ $$T^{3} + 312 T^{2} - 97471 T + 4921082$$
$59$ $$T^{3} - 44 T^{2} - 23280 T - 849664$$
$61$ $$T^{3} + 54 T^{2} - 559656 T - 87366816$$
$67$ $$T^{3} + 116 T^{2} + \cdots + 19023808$$
$71$ $$T^{3} - 1200 T^{2} + \cdots - 44065376$$
$73$ $$T^{3} + 1118 T^{2} + \cdots - 19620512$$
$79$ $$T^{3} - 2262 T^{2} + \cdots - 199598902$$
$83$ $$T^{3} + 1804 T^{2} + \cdots - 652294144$$
$89$ $$T^{3} - 1578 T^{2} + \cdots + 1024346504$$
$97$ $$T^{3} - 1450 T^{2} + \cdots + 1820417096$$