# Properties

 Label 1856.4.a.p 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: 3.3.229.1 Defining polynomial: $$x^{3} - 4 x - 1$$ 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 3$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 6$$$$)/2$$

## 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
 −0.254102 2.11491 −1.86081
0 −9.04399 0 −2.15672 0 27.7969 0 54.7938 0
1.2 0 −2.92622 0 −14.3315 0 16.8804 0 −18.4372 0
1.3 0 5.97021 0 12.4882 0 −28.6773 0 8.64344 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.p 3
4.b odd 2 1 1856.4.a.u 3
8.b even 2 1 232.4.a.b 3
8.d odd 2 1 464.4.a.g 3
24.h odd 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.b even 2 1
464.4.a.g 3 8.d odd 2 1
1856.4.a.p 3 1.a even 1 1 trivial
1856.4.a.u 3 4.b odd 2 1
2088.4.a.b 3 24.h odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-158 - 45 T + 6 T^{2} + T^{3}$$
$5$ $$-386 - 175 T + 4 T^{2} + T^{3}$$
$7$ $$13456 - 812 T - 16 T^{2} + T^{3}$$
$11$ $$106 - 301 T - 2 T^{2} + T^{3}$$
$13$ $$-137518 - 3999 T + 28 T^{2} + T^{3}$$
$17$ $$-1696 - 184 T + 66 T^{2} + T^{3}$$
$19$ $$393632 - 9616 T - 66 T^{2} + T^{3}$$
$23$ $$-106688 + 8192 T - 176 T^{2} + T^{3}$$
$29$ $$( -29 + T )^{3}$$
$31$ $$-9505226 - 54433 T + 190 T^{2} + T^{3}$$
$37$ $$-10323328 + 7392 T + 442 T^{2} + T^{3}$$
$41$ $$-53735864 + 438636 T - 1162 T^{2} + T^{3}$$
$43$ $$8035162 - 81277 T + 30 T^{2} + T^{3}$$
$47$ $$10647458 + 159287 T + 738 T^{2} + T^{3}$$
$53$ $$4921082 - 97471 T + 312 T^{2} + T^{3}$$
$59$ $$849664 - 23280 T + 44 T^{2} + T^{3}$$
$61$ $$-87366816 - 559656 T + 54 T^{2} + T^{3}$$
$67$ $$-19023808 - 182096 T - 116 T^{2} + T^{3}$$
$71$ $$44065376 + 437756 T + 1200 T^{2} + T^{3}$$
$73$ $$-19620512 + 147744 T + 1118 T^{2} + T^{3}$$
$79$ $$199598902 + 1396799 T + 2262 T^{2} + T^{3}$$
$83$ $$652294144 + 124048 T - 1804 T^{2} + T^{3}$$
$89$ $$1024346504 - 475636 T - 1578 T^{2} + T^{3}$$
$97$ $$1820417096 - 1209748 T - 1450 T^{2} + T^{3}$$