# Properties

 Label 1856.4.a.s Level $1856$ Weight $4$ Character orbit 1856.a Self dual yes Analytic conductor $109.508$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.19816.1 Defining polynomial: $$x^{3} - x^{2} - 42 x - 54$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 58) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{3} + ( -6 - 2 \beta_{1} - \beta_{2} ) q^{5} + ( -8 + 4 \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} + 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{3} + ( -6 - 2 \beta_{1} - \beta_{2} ) q^{5} + ( -8 + 4 \beta_{2} ) q^{7} + ( 1 + 2 \beta_{1} + 3 \beta_{2} ) q^{9} + ( 3 + \beta_{1} - 2 \beta_{2} ) q^{11} + ( 4 - 8 \beta_{1} - 11 \beta_{2} ) q^{13} + ( 39 + 13 \beta_{1} + 2 \beta_{2} ) q^{15} + ( -18 - 12 \beta_{1} - 16 \beta_{2} ) q^{17} + ( -52 - 8 \beta_{1} + 10 \beta_{2} ) q^{19} + ( 28 + 4 \beta_{1} + 16 \beta_{2} ) q^{21} + ( 66 + 6 \beta_{1} - 12 \beta_{2} ) q^{23} + ( 13 + 40 \beta_{1} + 11 \beta_{2} ) q^{25} + ( -53 + 17 \beta_{1} + 6 \beta_{2} ) q^{27} -29 q^{29} + ( 25 + 11 \beta_{1} + 36 \beta_{2} ) q^{31} + ( -42 - 4 \beta_{1} - 11 \beta_{2} ) q^{33} + ( 24 \beta_{1} + 12 \beta_{2} ) q^{35} + ( 10 + 12 \beta_{1} - 38 \beta_{2} ) q^{37} + ( 121 + 31 \beta_{1} - 20 \beta_{2} ) q^{39} + ( 186 + 4 \beta_{1} - 6 \beta_{2} ) q^{41} + ( 11 - 15 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -132 - 26 \beta_{1} - 4 \beta_{2} ) q^{45} + ( -207 - 33 \beta_{1} - 2 \beta_{2} ) q^{47} + ( 201 - 64 \beta_{1} - 80 \beta_{2} ) q^{49} + ( 162 + 70 \beta_{1} - 28 \beta_{2} ) q^{51} + ( -276 + 116 \beta_{1} + 27 \beta_{2} ) q^{53} + ( -39 - 25 \beta_{1} - 8 \beta_{2} ) q^{55} + ( 254 + 66 \beta_{1} + 64 \beta_{2} ) q^{57} + ( 90 - 86 \beta_{1} + 4 \beta_{2} ) q^{59} + ( -134 + 80 \beta_{1} - 40 \beta_{2} ) q^{61} + ( 280 - 56 \beta_{1} - 56 \beta_{2} ) q^{63} + ( 468 + 90 \beta_{1} + 9 \beta_{2} ) q^{65} + ( -88 + 36 \beta_{1} - 72 \beta_{2} ) q^{67} + ( -204 - 72 \beta_{1} - 66 \beta_{2} ) q^{69} + ( 18 - 2 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -178 + 40 \beta_{1} + 30 \beta_{2} ) q^{73} + ( -968 - 144 \beta_{1} - 76 \beta_{2} ) q^{75} + ( -300 + 28 \beta_{1} + 24 \beta_{2} ) q^{77} + ( 691 + 37 \beta_{1} + 38 \beta_{2} ) q^{79} + ( -485 - 58 \beta_{1} - 108 \beta_{2} ) q^{81} + ( -150 + 162 \beta_{1} + 12 \beta_{2} ) q^{83} + ( 840 + 184 \beta_{1} + 38 \beta_{2} ) q^{85} + ( -29 + 29 \beta_{1} ) q^{87} + ( 282 + 68 \beta_{1} + 154 \beta_{2} ) q^{89} + ( -1064 + 208 \beta_{1} + 244 \beta_{2} ) q^{91} + ( 52 - 94 \beta_{1} + 111 \beta_{2} ) q^{93} + ( 552 + 244 \beta_{1} + 86 \beta_{2} ) q^{95} + ( 14 + 176 \beta_{1} - 34 \beta_{2} ) q^{97} + ( -114 + 38 \beta_{1} + 22 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} - 20 q^{5} - 24 q^{7} + 5 q^{9} + O(q^{10})$$ $$3 q + 2 q^{3} - 20 q^{5} - 24 q^{7} + 5 q^{9} + 10 q^{11} + 4 q^{13} + 130 q^{15} - 66 q^{17} - 164 q^{19} + 88 q^{21} + 204 q^{23} + 79 q^{25} - 142 q^{27} - 87 q^{29} + 86 q^{31} - 130 q^{33} + 24 q^{35} + 42 q^{37} + 394 q^{39} + 562 q^{41} + 18 q^{43} - 422 q^{45} - 654 q^{47} + 539 q^{49} + 556 q^{51} - 712 q^{53} - 142 q^{55} + 828 q^{57} + 184 q^{59} - 322 q^{61} + 784 q^{63} + 1494 q^{65} - 228 q^{67} - 684 q^{69} + 52 q^{71} - 494 q^{73} - 3048 q^{75} - 872 q^{77} + 2110 q^{79} - 1513 q^{81} - 288 q^{83} + 2704 q^{85} - 58 q^{87} + 914 q^{89} - 2984 q^{91} + 62 q^{93} + 1900 q^{95} + 218 q^{97} - 304 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 42 x - 54$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - 4 \nu - 27$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2} + 4 \beta_{1} + 27$$

## 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
 7.53003 −1.39712 −5.13291
0 −6.53003 0 −20.9205 0 −8.55839 0 15.6413 0
1.2 0 2.39712 0 3.28077 0 −33.9461 0 −21.2538 0
1.3 0 6.13291 0 −2.36031 0 18.5045 0 10.6126 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.s 3
4.b odd 2 1 1856.4.a.r 3
8.b even 2 1 464.4.a.i 3
8.d odd 2 1 58.4.a.d 3
24.f even 2 1 522.4.a.k 3
40.e odd 2 1 1450.4.a.h 3
232.b odd 2 1 1682.4.a.d 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.4.a.d 3 8.d odd 2 1
464.4.a.i 3 8.b even 2 1
522.4.a.k 3 24.f even 2 1
1450.4.a.h 3 40.e odd 2 1
1682.4.a.d 3 232.b odd 2 1
1856.4.a.r 3 4.b odd 2 1
1856.4.a.s 3 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1856))$$:

 $$T_{3}^{3} - 2 T_{3}^{2} - 41 T_{3} + 96$$ $$T_{5}^{3} + 20 T_{5}^{2} - 27 T_{5} - 162$$ $$T_{7}^{3} + 24 T_{7}^{2} - 496 T_{7} - 5376$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$96 - 41 T - 2 T^{2} + T^{3}$$
$5$ $$-162 - 27 T + 20 T^{2} + T^{3}$$
$7$ $$-5376 - 496 T + 24 T^{2} + T^{3}$$
$11$ $$2424 - 233 T - 10 T^{2} + T^{3}$$
$13$ $$-131706 - 5619 T - 4 T^{2} + T^{3}$$
$17$ $$-679368 - 10660 T + 66 T^{2} + T^{3}$$
$19$ $$-664448 - 124 T + 164 T^{2} + T^{3}$$
$23$ $$677376 + 4284 T - 204 T^{2} + T^{3}$$
$29$ $$( 29 + T )^{3}$$
$31$ $$4766172 - 48089 T - 86 T^{2} + T^{3}$$
$37$ $$7684896 - 79456 T - 42 T^{2} + T^{3}$$
$41$ $$-5982048 + 102432 T - 562 T^{2} + T^{3}$$
$43$ $$196488 - 10369 T - 18 T^{2} + T^{3}$$
$47$ $$3425124 + 98015 T + 654 T^{2} + T^{3}$$
$53$ $$-252120546 - 350571 T + 712 T^{2} + T^{3}$$
$59$ $$57362928 - 311444 T - 184 T^{2} + T^{3}$$
$61$ $$5254424 - 388372 T + 322 T^{2} + T^{3}$$
$67$ $$47608192 - 327840 T + 228 T^{2} + T^{3}$$
$71$ $$672 - 164 T - 52 T^{2} + T^{3}$$
$73$ $$-9410208 + 6112 T + 494 T^{2} + T^{3}$$
$79$ $$-285187172 + 1400543 T - 2110 T^{2} + T^{3}$$
$83$ $$-437606064 - 1038996 T + 288 T^{2} + T^{3}$$
$89$ $$598011552 - 664800 T - 914 T^{2} + T^{3}$$
$97$ $$-17006112 - 1500768 T - 218 T^{2} + T^{3}$$