# Properties

 Label 1856.4.a.bi Level $1856$ Weight $4$ Character orbit 1856.a Self dual yes Analytic conductor $109.508$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

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## 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: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 4 x^{9} - 135 x^{8} + 788 x^{7} + 3323 x^{6} - 26136 x^{5} + 2315 x^{4} + 188664 x^{3} - 265632 x^{2} - 18800 x + 128592$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: no (minimal twist has level 928) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + \beta_{2} q^{5} -\beta_{6} q^{7} + ( 5 - \beta_{5} + \beta_{9} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + \beta_{2} q^{5} -\beta_{6} q^{7} + ( 5 - \beta_{5} + \beta_{9} ) q^{9} + ( 4 \beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{11} + ( 22 - \beta_{2} - \beta_{5} + 2 \beta_{9} ) q^{13} + ( -\beta_{4} + 2 \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{15} + ( 4 + 5 \beta_{2} + 3 \beta_{5} + \beta_{9} ) q^{17} + ( -\beta_{1} + 3 \beta_{4} - \beta_{6} - 2 \beta_{7} - 3 \beta_{8} ) q^{19} + ( 12 - 6 \beta_{2} - \beta_{3} - 2 \beta_{5} + 4 \beta_{9} ) q^{21} + ( -8 \beta_{1} - \beta_{4} - 4 \beta_{7} - 2 \beta_{8} ) q^{23} + ( 43 + 11 \beta_{2} - 3 \beta_{3} - 7 \beta_{5} ) q^{25} + ( 4 \beta_{1} - \beta_{4} - 5 \beta_{6} - \beta_{8} ) q^{27} + 29 q^{29} + ( -7 \beta_{1} + \beta_{4} + 2 \beta_{6} + 8 \beta_{7} + 4 \beta_{8} ) q^{31} + ( -116 - \beta_{2} + 3 \beta_{3} + 9 \beta_{5} + 2 \beta_{9} ) q^{33} + ( 42 \beta_{1} - 7 \beta_{4} - 4 \beta_{6} + 8 \beta_{7} + 10 \beta_{8} ) q^{35} + ( 172 + 7 \beta_{2} + \beta_{3} - 9 \beta_{5} - \beta_{9} ) q^{37} + ( -53 \beta_{1} - 3 \beta_{4} - 12 \beta_{6} + \beta_{7} - \beta_{8} ) q^{39} + ( -98 + 11 \beta_{3} ) q^{41} + ( -12 \beta_{1} + \beta_{4} - 7 \beta_{6} - 7 \beta_{7} ) q^{43} + ( -40 + 2 \beta_{2} + 6 \beta_{3} + 15 \beta_{5} - 4 \beta_{9} ) q^{45} + ( -32 \beta_{1} + 7 \beta_{4} - 7 \beta_{7} ) q^{47} + ( 117 + 16 \beta_{2} - 8 \beta_{3} - 22 \beta_{5} ) q^{49} + ( -2 \beta_{1} - 14 \beta_{4} + 5 \beta_{6} + 22 \beta_{7} - 6 \beta_{8} ) q^{51} + ( 2 + 11 \beta_{2} + 6 \beta_{3} + 7 \beta_{5} - 12 \beta_{9} ) q^{53} + ( 3 \beta_{1} - 3 \beta_{4} - 18 \beta_{6} + \beta_{7} + 15 \beta_{8} ) q^{55} + ( 7 \beta_{2} - 17 \beta_{3} - 20 \beta_{5} + 3 \beta_{9} ) q^{57} + ( 15 \beta_{4} + 4 \beta_{6} - 16 \beta_{7} + 6 \beta_{8} ) q^{59} + ( 140 + 15 \beta_{2} - 18 \beta_{3} + 5 \beta_{5} + 3 \beta_{9} ) q^{61} + ( -72 \beta_{1} + 3 \beta_{4} - 7 \beta_{6} - 14 \beta_{7} - 2 \beta_{8} ) q^{63} + ( -212 + 16 \beta_{3} + 31 \beta_{5} - 5 \beta_{9} ) q^{65} + ( -8 \beta_{1} + 21 \beta_{4} + 15 \beta_{6} - 12 \beta_{7} + 12 \beta_{8} ) q^{67} + ( 238 + \beta_{2} - 10 \beta_{3} - 21 \beta_{5} + 7 \beta_{9} ) q^{69} + ( -18 \beta_{1} - 4 \beta_{4} - 6 \beta_{7} - 10 \beta_{8} ) q^{71} + ( -312 - \beta_{2} - \beta_{3} + 11 \beta_{5} - 5 \beta_{9} ) q^{73} + ( -72 \beta_{1} + 18 \beta_{4} + 16 \beta_{6} - 23 \beta_{7} - 23 \beta_{8} ) q^{75} + ( 290 + 53 \beta_{2} + 6 \beta_{3} - \beta_{5} - 23 \beta_{9} ) q^{77} + ( -108 \beta_{1} + 3 \beta_{4} + 2 \beta_{6} + 15 \beta_{7} - 16 \beta_{8} ) q^{79} + ( -215 - 21 \beta_{2} - 5 \beta_{3} + 24 \beta_{5} - 9 \beta_{9} ) q^{81} + ( -72 \beta_{1} - 28 \beta_{4} - 9 \beta_{6} - 4 \beta_{7} - 18 \beta_{8} ) q^{83} + ( 944 + 36 \beta_{2} - 5 \beta_{3} - 44 \beta_{5} + 8 \beta_{9} ) q^{85} -29 \beta_{1} q^{87} + ( 62 + 14 \beta_{2} - 19 \beta_{3} - 16 \beta_{5} + 6 \beta_{9} ) q^{89} + ( -178 \beta_{1} + 14 \beta_{4} - 13 \beta_{6} - 28 \beta_{7} - 10 \beta_{8} ) q^{91} + ( 238 + 11 \beta_{2} + 24 \beta_{3} + 26 \beta_{5} + 2 \beta_{9} ) q^{93} + ( -5 \beta_{1} + 8 \beta_{4} + 60 \beta_{6} - 11 \beta_{7} - 4 \beta_{8} ) q^{95} + ( 76 - \beta_{2} + 18 \beta_{3} - 15 \beta_{5} - 19 \beta_{9} ) q^{97} + ( 21 \beta_{1} - 11 \beta_{4} + 21 \beta_{6} + 28 \beta_{7} + 11 \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 4 q^{5} + 50 q^{9} + O(q^{10})$$ $$10 q + 4 q^{5} + 50 q^{9} + 220 q^{13} + 76 q^{17} + 104 q^{21} + 446 q^{25} + 290 q^{29} - 1120 q^{33} + 1708 q^{37} - 980 q^{41} - 348 q^{45} + 1146 q^{49} + 44 q^{53} - 40 q^{57} + 1492 q^{61} - 2016 q^{65} + 2328 q^{69} - 3100 q^{73} + 3016 q^{77} - 2174 q^{81} + 9440 q^{85} + 636 q^{89} + 2536 q^{93} + 620 q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 4 x^{9} - 135 x^{8} + 788 x^{7} + 3323 x^{6} - 26136 x^{5} + 2315 x^{4} + 188664 x^{3} - 265632 x^{2} - 18800 x + 128592$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-10854023209 \nu^{9} - 1714044196 \nu^{8} + 1354984318315 \nu^{7} - 3205965170646 \nu^{6} - 37184623963691 \nu^{5} + 132007765583264 \nu^{4} + 170250694648181 \nu^{3} - 1138500553462646 \nu^{2} + 644983415633540 \nu + 836484917813280$$$$)/ 52871845454856$$ $$\beta_{2}$$ $$=$$ $$($$$$-24688532 \nu^{9} + 182635072 \nu^{8} + 3815648153 \nu^{7} - 28379917644 \nu^{6} - 111970521934 \nu^{5} + 896651189380 \nu^{4} + 536042074105 \nu^{3} - 6466873013044 \nu^{2} + 3143136592192 \nu + 3934134509796$$$$)/ 71642067012$$ $$\beta_{3}$$ $$=$$ $$($$$$5620429 \nu^{9} + 6830280 \nu^{8} - 731642547 \nu^{7} + 691738306 \nu^{6} + 23855482859 \nu^{5} - 33518830920 \nu^{4} - 224253073397 \nu^{3} + 239979721678 \nu^{2} + 424655536480 \nu - 137905539504$$$$)/ 7960229668$$ $$\beta_{4}$$ $$=$$ $$($$$$-4877782454 \nu^{9} + 17799628726 \nu^{8} + 703860283880 \nu^{7} - 3453513745104 \nu^{6} - 21890281433683 \nu^{5} + 115234395846655 \nu^{4} + 156276691489090 \nu^{3} - 870527889420313 \nu^{2} + 167726291200639 \nu + 778464445245516$$$$)/ 6608980681857$$ $$\beta_{5}$$ $$=$$ $$($$$$-30036802 \nu^{9} + 95458676 \nu^{8} + 4237603342 \nu^{7} - 19853351823 \nu^{6} - 128192210690 \nu^{5} + 673071335138 \nu^{4} + 827115992750 \nu^{3} - 5130821138423 \nu^{2} + 1440220708808 \nu + 3779470536804$$$$)/ 35821033506$$ $$\beta_{6}$$ $$=$$ $$($$$$-6205797484 \nu^{9} + 17424622718 \nu^{8} + 859815349318 \nu^{7} - 3872373077583 \nu^{6} - 25412644310927 \nu^{5} + 133128926819267 \nu^{4} + 148929109835660 \nu^{3} - 1022745110657774 \nu^{2} + 416586995804369 \nu + 589724260107732$$$$)/ 6608980681857$$ $$\beta_{7}$$ $$=$$ $$($$$$9575261599 \nu^{9} - 32238786260 \nu^{8} - 1348953020665 \nu^{7} + 6542711940150 \nu^{6} + 40044976095365 \nu^{5} - 218449104064424 \nu^{4} - 233789192752655 \nu^{3} + 1597426781781734 \nu^{2} - 737668415783660 \nu - 870803080538256$$$$)/ 5874649494984$$ $$\beta_{8}$$ $$=$$ $$($$$$-42392072837 \nu^{9} + 105910314400 \nu^{8} + 5862590273195 \nu^{7} - 24690506425146 \nu^{6} - 176195958320839 \nu^{5} + 844661192384236 \nu^{4} + 1136324855611549 \nu^{3} - 6236281391484370 \nu^{2} + 1948020766737028 \nu + 3642904219917168$$$$)/ 17623948484952$$ $$\beta_{9}$$ $$=$$ $$($$$$-261879455 \nu^{9} + 525655384 \nu^{8} + 35733837842 \nu^{7} - 137637367368 \nu^{6} - 1067242411843 \nu^{5} + 4806636126976 \nu^{4} + 6710765534632 \nu^{3} - 36305714221612 \nu^{2} + 12976381677352 \nu + 21816689635044$$$$)/ 71642067012$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} - 2 \beta_{5} + \beta_{4} + 4$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{9} - 2 \beta_{7} + 5 \beta_{6} + 6 \beta_{5} - 7 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} - 6 \beta_{1} + 228$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$18 \beta_{9} + 16 \beta_{8} + 48 \beta_{7} + 35 \beta_{6} - 152 \beta_{5} + 75 \beta_{4} + 10 \beta_{3} + 54 \beta_{2} + 8 \beta_{1} - 512$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-292 \beta_{9} - 120 \beta_{8} - 466 \beta_{7} + 277 \beta_{6} + 1080 \beta_{5} - 723 \beta_{4} - 232 \beta_{3} - 456 \beta_{2} - 206 \beta_{1} + 15424$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$3002 \beta_{9} + 1704 \beta_{8} + 6120 \beta_{7} + 889 \beta_{6} - 15044 \beta_{5} + 7813 \beta_{4} + 1910 \beta_{3} + 6430 \beta_{2} + 240 \beta_{1} - 95352$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-35558 \beta_{9} - 17168 \beta_{8} - 62882 \beta_{7} + 16525 \beta_{6} + 139914 \beta_{5} - 81735 \beta_{4} - 26302 \beta_{3} - 60782 \beta_{2} - 6614 \beta_{1} + 1433244$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$379714 \beta_{9} + 192288 \beta_{8} + 720456 \beta_{7} - 18773 \beta_{6} - 1650272 \beta_{5} + 885691 \beta_{4} + 254034 \beta_{3} + 728054 \beta_{2} - 144 \beta_{1} - 12690928$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$-4194712 \beta_{9} - 2075960 \beta_{8} - 7633538 \beta_{7} + 1253921 \beta_{6} + 16909396 \beta_{5} - 9478879 \beta_{4} - 2999844 \beta_{3} - 7414180 \beta_{2} - 189534 \beta_{1} + 153648712$$$$)/8$$ $$\nu^{9}$$ $$=$$ $$($$$$45223890 \beta_{9} + 22357240 \beta_{8} + 84047344 \beta_{7} - 7015343 \beta_{6} - 188470276 \beta_{5} + 102616549 \beta_{4} + 30957030 \beta_{3} + 83494646 \beta_{2} - 646200 \beta_{1} - 1548739832$$$$)/8$$

## 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
 6.40120 1.25115 −3.21629 1.47788 −10.8075 −0.612154 5.97060 6.55010 −5.42259 2.40763
0 −8.07880 0 −14.4117 0 −28.4619 0 38.2670 0
1.2 0 −7.53335 0 7.78444 0 −5.94345 0 29.7513 0
1.3 0 −4.75556 0 22.0670 0 32.1228 0 −4.38468 0
1.4 0 −3.91971 0 −7.82384 0 17.9594 0 −11.6359 0
1.5 0 −0.0474689 0 −5.61587 0 −9.39051 0 −26.9977 0
1.6 0 0.0474689 0 −5.61587 0 9.39051 0 −26.9977 0
1.7 0 3.91971 0 −7.82384 0 −17.9594 0 −11.6359 0
1.8 0 4.75556 0 22.0670 0 −32.1228 0 −4.38468 0
1.9 0 7.53335 0 7.78444 0 5.94345 0 29.7513 0
1.10 0 8.07880 0 −14.4117 0 28.4619 0 38.2670 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$29$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1856.4.a.bi 10
4.b odd 2 1 inner 1856.4.a.bi 10
8.b even 2 1 928.4.a.f 10
8.d odd 2 1 928.4.a.f 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
928.4.a.f 10 8.b even 2 1
928.4.a.f 10 8.d odd 2 1
1856.4.a.bi 10 1.a even 1 1 trivial
1856.4.a.bi 10 4.b odd 2 1 inner

## Hecke kernels

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

 $$T_{3}^{10} - 160 T_{3}^{8} + 8686 T_{3}^{6} - 183092 T_{3}^{4} + 1287417 T_{3}^{2} - 2900$$ $$T_{5}^{5} - 2 T_{5}^{4} - 422 T_{5}^{3} - 1676 T_{5}^{2} + 21917 T_{5} + 108774$$ $$T_{7}^{10} - 2288 T_{7}^{8} + 1700448 T_{7}^{6} - 452968704 T_{7}^{4} + 37753198848 T_{7}^{2} - 839836258304$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$-2900 + 1287417 T^{2} - 183092 T^{4} + 8686 T^{6} - 160 T^{8} + T^{10}$$
$5$ $$( 108774 + 21917 T - 1676 T^{2} - 422 T^{3} - 2 T^{4} + T^{5} )^{2}$$
$7$ $$-839836258304 + 37753198848 T^{2} - 452968704 T^{4} + 1700448 T^{6} - 2288 T^{8} + T^{10}$$
$11$ $$-162994861358676 + 3903195573929 T^{2} - 13290129508 T^{4} + 14828158 T^{6} - 6576 T^{8} + T^{10}$$
$13$ $$( 28673602 + 4743389 T + 200580 T^{2} - 982 T^{3} - 110 T^{4} + T^{5} )^{2}$$
$17$ $$( -4092920640 + 97852992 T + 777728 T^{2} - 19952 T^{3} - 38 T^{4} + T^{5} )^{2}$$
$19$ $$-$$$$11\!\cdots\!44$$$$+ 59051104619872256 T^{2} - 11612885531136 T^{4} + 1126413056 T^{6} - 53684 T^{8} + T^{10}$$
$23$ $$-380625582252441600 + 2431079786796032 T^{2} - 4099837683456 T^{4} + 874773248 T^{6} - 56144 T^{8} + T^{10}$$
$29$ $$( -29 + T )^{10}$$
$31$ $$-22566053528844723284 + 1754820971781880849 T^{2} - 235495048039148 T^{4} + 10418827222 T^{6} - 179048 T^{8} + T^{10}$$
$37$ $$( 293417725952 - 3166162944 T - 9466624 T^{2} + 222064 T^{3} - 854 T^{4} + T^{5} )^{2}$$
$41$ $$( 5366289891808 + 12156041232 T - 104893392 T^{2} - 216624 T^{3} + 490 T^{4} + T^{5} )^{2}$$
$43$ $$-18446336888490645716 + 443836875230581513 T^{2} - 159021477695108 T^{4} + 12394430878 T^{6} - 260240 T^{8} + T^{10}$$
$47$ $$-$$$$19\!\cdots\!44$$$$+ 6175530906829580481 T^{2} - 599458293027132 T^{4} + 22739621670 T^{6} - 323384 T^{8} + T^{10}$$
$53$ $$( -2494384982 + 1420219021 T + 71220692 T^{2} - 338694 T^{3} - 22 T^{4} + T^{5} )^{2}$$
$59$ $$-$$$$40\!\cdots\!84$$$$+$$$$94\!\cdots\!08$$$$T^{2} - 36845080779079424 T^{4} + 326680293760 T^{6} - 1023472 T^{8} + T^{10}$$
$61$ $$( -53613741397184 + 35376452864 T + 560560096 T^{2} - 686592 T^{3} - 746 T^{4} + T^{5} )^{2}$$
$67$ $$-$$$$16\!\cdots\!04$$$$+$$$$11\!\cdots\!56$$$$T^{2} - 208052384778960896 T^{4} + 1068935873536 T^{6} - 1858240 T^{8} + T^{10}$$
$71$ $$-$$$$34\!\cdots\!44$$$$+ 84270205843877116160 T^{2} - 4415317003719424 T^{4} + 71079250144 T^{6} - 451824 T^{8} + T^{10}$$
$73$ $$( -801523248128 + 13974575872 T + 204249408 T^{2} + 870880 T^{3} + 1550 T^{4} + T^{5} )^{2}$$
$79$ $$-$$$$32\!\cdots\!56$$$$+$$$$45\!\cdots\!41$$$$T^{2} - 2252036127406682268 T^{4} + 4569001371654 T^{6} - 3685176 T^{8} + T^{10}$$
$83$ $$-$$$$57\!\cdots\!00$$$$+$$$$70\!\cdots\!92$$$$T^{2} - 3074576898391366400 T^{4} + 5638178231296 T^{6} - 4151792 T^{8} + T^{10}$$
$89$ $$( -33585949543200 + 184168432592 T + 225322768 T^{2} - 880832 T^{3} - 318 T^{4} + T^{5} )^{2}$$
$97$ $$( 2383191738336 + 67219679440 T - 459654672 T^{2} - 1521792 T^{3} - 310 T^{4} + T^{5} )^{2}$$
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