# Properties

 Label 1856.4.a.y Level $1856$ Weight $4$ Character orbit 1856.a Self dual yes Analytic conductor $109.508$ Analytic rank $1$ Dimension $5$ 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: $$5$$ Coefficient field: 5.5.13458092.1 Defining polynomial: $$x^{5} - x^{4} - 14 x^{3} + 18 x^{2} + 20 x - 8$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 29) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta_{3} ) q^{3} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{5} + ( 8 + \beta_{2} - 2 \beta_{4} ) q^{7} + ( 8 + 5 \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -2 + \beta_{3} ) q^{3} + ( -2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{5} + ( 8 + \beta_{2} - 2 \beta_{4} ) q^{7} + ( 8 + 5 \beta_{1} + \beta_{2} - 6 \beta_{3} ) q^{9} + ( -1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - 7 \beta_{4} ) q^{11} + ( -1 - \beta_{1} + 4 \beta_{2} - 4 \beta_{3} + 10 \beta_{4} ) q^{13} + ( -17 - 13 \beta_{1} + 2 \beta_{2} + 12 \beta_{3} - \beta_{4} ) q^{15} + ( 13 + 7 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} ) q^{17} + ( -43 + 3 \beta_{1} - 5 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{19} + ( -5 + 7 \beta_{1} + 9 \beta_{3} + 7 \beta_{4} ) q^{21} + ( 26 - 6 \beta_{1} - 7 \beta_{2} + 20 \beta_{3} + 4 \beta_{4} ) q^{23} + ( 49 - 4 \beta_{1} + 2 \beta_{2} - 17 \beta_{3} + 15 \beta_{4} ) q^{25} + ( -75 - 35 \beta_{1} + 11 \beta_{2} + 24 \beta_{3} + 3 \beta_{4} ) q^{27} + 29 q^{29} + ( 80 - 6 \beta_{1} - 13 \beta_{2} + 13 \beta_{3} - 8 \beta_{4} ) q^{31} + ( -116 - 2 \beta_{1} - 20 \beta_{2} + 3 \beta_{3} + 11 \beta_{4} ) q^{33} + ( 8 + 12 \beta_{1} - 13 \beta_{2} + 30 \beta_{4} ) q^{35} + ( -88 - 18 \beta_{1} - 22 \beta_{2} + 40 \beta_{3} - 12 \beta_{4} ) q^{37} + ( -72 - 8 \beta_{1} + 11 \beta_{2} - 3 \beta_{3} - 8 \beta_{4} ) q^{39} + ( -241 - 3 \beta_{1} - 2 \beta_{2} + 25 \beta_{3} + 19 \beta_{4} ) q^{41} + ( 45 - 29 \beta_{1} + 27 \beta_{2} + 4 \beta_{3} + 7 \beta_{4} ) q^{43} + ( 329 + 43 \beta_{1} + 3 \beta_{2} - 91 \beta_{3} + 35 \beta_{4} ) q^{45} + ( 61 - 25 \beta_{1} + 58 \beta_{2} - 16 \beta_{3} + 9 \beta_{4} ) q^{47} + ( -53 - 30 \beta_{1} + 32 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} ) q^{49} + ( -58 - 42 \beta_{1} + 17 \beta_{2} + 52 \beta_{3} - 12 \beta_{4} ) q^{51} + ( 117 - 15 \beta_{1} - 26 \beta_{2} - 8 \beta_{3} - 2 \beta_{4} ) q^{53} + ( 98 + 26 \beta_{1} - 27 \beta_{2} + 15 \beta_{3} + 86 \beta_{4} ) q^{55} + ( 23 - 33 \beta_{1} - 5 \beta_{2} - 19 \beta_{3} - 9 \beta_{4} ) q^{57} + ( -80 - 20 \beta_{1} - 11 \beta_{2} - 10 \beta_{3} + 26 \beta_{4} ) q^{59} + ( -111 - 57 \beta_{1} + 40 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} ) q^{61} + ( 164 + 24 \beta_{1} + 10 \beta_{2} - 20 \beta_{3} + 40 \beta_{4} ) q^{63} + ( -317 + 41 \beta_{1} + 33 \beta_{2} - 56 \beta_{3} - 90 \beta_{4} ) q^{65} + ( -232 + 92 \beta_{1} + 22 \beta_{2} - 48 \beta_{3} + 48 \beta_{4} ) q^{67} + ( 409 + 73 \beta_{1} - 8 \beta_{2} - 75 \beta_{3} - 29 \beta_{4} ) q^{69} + ( -112 - 112 \beta_{1} + 32 \beta_{2} - 10 \beta_{3} + 16 \beta_{4} ) q^{71} + ( -382 - 36 \beta_{1} - 34 \beta_{2} - 2 \beta_{3} - 30 \beta_{4} ) q^{73} + ( -632 - 82 \beta_{1} - 10 \beta_{2} + 84 \beta_{3} - 24 \beta_{4} ) q^{75} + ( 365 - 143 \beta_{1} + 16 \beta_{2} - 33 \beta_{3} - 15 \beta_{4} ) q^{77} + ( 77 - 13 \beta_{1} - 38 \beta_{2} - 50 \beta_{3} - 127 \beta_{4} ) q^{79} + ( 404 + 107 \beta_{1} - 83 \beta_{2} - 163 \beta_{3} + 27 \beta_{4} ) q^{81} + ( -88 + 44 \beta_{1} + 55 \beta_{2} - 58 \beta_{3} - 2 \beta_{4} ) q^{83} + ( 357 + 25 \beta_{1} + 8 \beta_{2} - 91 \beta_{3} - 45 \beta_{4} ) q^{85} + ( -58 + 29 \beta_{3} ) q^{87} + ( 165 - 21 \beta_{1} + 28 \beta_{2} + 31 \beta_{3} + 121 \beta_{4} ) q^{89} + ( -516 + 72 \beta_{1} - 3 \beta_{2} - 36 \beta_{3} - 58 \beta_{4} ) q^{91} + ( -6 + 20 \beta_{1} - 39 \beta_{2} + 25 \beta_{3} - 23 \beta_{4} ) q^{93} + ( 459 - 85 \beta_{1} + 35 \beta_{2} + 17 \beta_{3} + 47 \beta_{4} ) q^{95} + ( 297 + 79 \beta_{1} - 22 \beta_{2} - \beta_{3} - 31 \beta_{4} ) q^{97} + ( 79 - 11 \beta_{1} - 5 \beta_{2} - 73 \beta_{3} + 107 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9} + O(q^{10})$$ $$5 q - 8 q^{3} - 10 q^{5} + 40 q^{7} + 33 q^{9} - 12 q^{11} - 14 q^{13} - 74 q^{15} + 66 q^{17} - 214 q^{19} + 164 q^{23} + 207 q^{25} - 362 q^{27} + 145 q^{29} + 420 q^{31} - 576 q^{33} + 52 q^{35} - 378 q^{37} - 374 q^{39} - 1158 q^{41} + 204 q^{43} + 1506 q^{45} + 248 q^{47} - 283 q^{49} - 228 q^{51} + 554 q^{53} + 546 q^{55} + 44 q^{57} - 440 q^{59} - 618 q^{61} + 804 q^{63} - 1656 q^{65} - 1164 q^{67} + 1968 q^{69} - 692 q^{71} - 1950 q^{73} - 3074 q^{75} + 1616 q^{77} + 272 q^{79} + 1801 q^{81} - 512 q^{83} + 1628 q^{85} - 232 q^{87} + 866 q^{89} - 2580 q^{91} + 40 q^{93} + 2244 q^{95} + 1562 q^{97} + 238 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 14 x^{3} + 18 x^{2} + 20 x - 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + 2 \nu - 6$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + \nu^{2} - 8 \nu - 2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{4} + \nu^{3} - 10 \nu^{2} - 2 \nu + 2$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} - \nu^{3} - 14 \nu^{2} + 18 \nu + 16$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - \beta_{3} + \beta_{2} + \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 11$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{4} - 5 \beta_{3} + 7 \beta_{2} + 3 \beta_{1} - 3$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-7 \beta_{4} + 9 \beta_{3} - 8 \beta_{2} + 4 \beta_{1} + 55$$

## 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
 2.27399 −0.957567 0.328194 −3.68360 3.03898
0 −9.87991 0 16.8209 0 5.21997 0 70.6126 0
1.2 0 −4.64574 0 −12.8729 0 26.0540 0 −5.41713 0
1.3 0 −1.84328 0 −18.3339 0 −16.8583 0 −23.6023 0
1.4 0 1.90549 0 6.52855 0 5.22706 0 −23.3691 0
1.5 0 6.46343 0 −2.14270 0 20.3573 0 14.7760 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
$$2$$ $$1$$
$$29$$ $$-1$$

## 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.y 5
4.b odd 2 1 1856.4.a.bb 5
8.b even 2 1 29.4.a.b 5
8.d odd 2 1 464.4.a.l 5
24.h odd 2 1 261.4.a.f 5
40.f even 2 1 725.4.a.c 5
56.h odd 2 1 1421.4.a.e 5
232.g even 2 1 841.4.a.b 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.4.a.b 5 8.b even 2 1
261.4.a.f 5 24.h odd 2 1
464.4.a.l 5 8.d odd 2 1
725.4.a.c 5 40.f even 2 1
841.4.a.b 5 232.g even 2 1
1421.4.a.e 5 56.h odd 2 1
1856.4.a.y 5 1.a even 1 1 trivial
1856.4.a.bb 5 4.b odd 2 1

## Hecke kernels

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

 $$T_{3}^{5} + 8 T_{3}^{4} - 52 T_{3}^{3} - 322 T_{3}^{2} + 187 T_{3} + 1042$$ $$T_{5}^{5} + 10 T_{5}^{4} - 366 T_{5}^{3} - 2904 T_{5}^{2} + 21453 T_{5} + 55534$$ $$T_{7}^{5} - 40 T_{7}^{4} + 84 T_{7}^{3} + 10768 T_{7}^{2} - 100288 T_{7} + 243968$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$1042 + 187 T - 322 T^{2} - 52 T^{3} + 8 T^{4} + T^{5}$$
$5$ $$55534 + 21453 T - 2904 T^{2} - 366 T^{3} + 10 T^{4} + T^{5}$$
$7$ $$243968 - 100288 T + 10768 T^{2} + 84 T^{3} - 40 T^{4} + T^{5}$$
$11$ $$-30997958 + 4398787 T - 50174 T^{2} - 4892 T^{3} + 12 T^{4} + T^{5}$$
$13$ $$13078418 + 1294565 T - 133312 T^{2} - 7558 T^{3} + 14 T^{4} + T^{5}$$
$17$ $$19935872 - 3694112 T + 205448 T^{2} - 2444 T^{3} - 66 T^{4} + T^{5}$$
$19$ $$-19441152 - 1091328 T + 342272 T^{2} + 15136 T^{3} + 214 T^{4} + T^{5}$$
$23$ $$-7938109184 + 24181632 T + 3316000 T^{2} - 18812 T^{3} - 164 T^{4} + T^{5}$$
$29$ $$( -29 + T )^{5}$$
$31$ $$-2094346 - 5574361 T - 1363354 T^{2} + 45552 T^{3} - 420 T^{4} + T^{5}$$
$37$ $$-23564115968 - 2452994048 T - 30918912 T^{2} - 69792 T^{3} + 378 T^{4} + T^{5}$$
$41$ $$59613728000 + 4409752000 T + 74423880 T^{2} + 462908 T^{3} + 1158 T^{4} + T^{5}$$
$43$ $$-198643410886 + 1855476667 T + 11715386 T^{2} - 94388 T^{3} - 204 T^{4} + T^{5}$$
$47$ $$-203435244846 + 8179300863 T + 76509294 T^{2} - 332696 T^{3} - 248 T^{4} + T^{5}$$
$53$ $$786854101018 - 14144920995 T + 72740336 T^{2} - 38334 T^{3} - 554 T^{4} + T^{5}$$
$59$ $$-109032704000 - 4107227200 T - 42988400 T^{2} - 95868 T^{3} + 440 T^{4} + T^{5}$$
$61$ $$-2140697762176 - 41066569056 T - 206442728 T^{2} - 204156 T^{3} + 618 T^{4} + T^{5}$$
$67$ $$39308070146048 + 25268520960 T - 457331392 T^{2} - 251984 T^{3} + 1164 T^{4} + T^{5}$$
$71$ $$98341318953856 + 153248941872 T - 572202480 T^{2} - 876848 T^{3} + 692 T^{4} + T^{5}$$
$73$ $$7201878016 + 19306358272 T + 267870432 T^{2} + 1168032 T^{3} + 1950 T^{4} + T^{5}$$
$79$ $$-240961986300538 + 543174633815 T + 545111474 T^{2} - 1497888 T^{3} - 272 T^{4} + T^{5}$$
$83$ $$-6057622580224 + 85821473600 T - 118532752 T^{2} - 520156 T^{3} + 512 T^{4} + T^{5}$$
$89$ $$21549994365568 + 138056266176 T + 69193352 T^{2} - 852420 T^{3} - 866 T^{4} + T^{5}$$
$97$ $$-20480102175488 - 81033100480 T + 290423688 T^{2} + 412988 T^{3} - 1562 T^{4} + T^{5}$$