# Properties

 Label 2736.2.dc.c Level $2736$ Weight $2$ Character orbit 2736.dc Analytic conductor $21.847$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2736.dc (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$21.8470699930$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 16 x^{14} + 174 x^{12} + 1012 x^{10} + 4243 x^{8} + 9708 x^{6} + 15858 x^{4} + 12150 x^{2} + 6561$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 171) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{10} q^{5} -\beta_{14} q^{7} +O(q^{10})$$ $$q + \beta_{10} q^{5} -\beta_{14} q^{7} -\beta_{7} q^{11} + ( -2 - \beta_{4} - \beta_{5} ) q^{13} + ( \beta_{7} - \beta_{10} + \beta_{12} ) q^{17} + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{9} - \beta_{14} ) q^{19} + ( \beta_{6} + \beta_{15} ) q^{23} + ( 1 + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{8} - \beta_{9} - \beta_{14} ) q^{25} -\beta_{13} q^{29} + ( -\beta_{3} - \beta_{8} + 2 \beta_{9} ) q^{31} + ( -\beta_{1} - 2 \beta_{7} - 2 \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{35} + ( 1 + 2 \beta_{2} - \beta_{3} + 2 \beta_{9} - \beta_{14} ) q^{37} + ( \beta_{1} - \beta_{7} + \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{41} + ( \beta_{5} - \beta_{8} + \beta_{9} ) q^{43} + ( -2 \beta_{1} + \beta_{6} - 2 \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} + \beta_{15} ) q^{47} + ( 2 \beta_{5} + \beta_{8} ) q^{49} + ( -2 \beta_{7} - \beta_{12} ) q^{53} + ( 1 + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{55} + ( \beta_{1} - \beta_{7} + \beta_{12} ) q^{59} + ( 1 + \beta_{2} + \beta_{3} + 6 \beta_{4} - \beta_{5} - 2 \beta_{8} - \beta_{9} - \beta_{14} ) q^{61} + ( -\beta_{1} + \beta_{7} - 4 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{65} + ( 2 - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{9} - 2 \beta_{14} ) q^{67} + ( -2 \beta_{6} - \beta_{7} + \beta_{12} - \beta_{15} ) q^{71} + ( -4 - 3 \beta_{2} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{8} + \beta_{9} ) q^{73} + ( \beta_{6} - \beta_{7} - \beta_{11} ) q^{77} + ( 2 + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{8} - \beta_{9} ) q^{79} + ( -2 \beta_{6} + \beta_{7} + 3 \beta_{11} ) q^{83} + ( -2 - 2 \beta_{2} + 6 \beta_{4} - 2 \beta_{5} - 4 \beta_{8} + 2 \beta_{14} ) q^{85} + ( -\beta_{6} - 2 \beta_{7} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{15} ) q^{89} + ( 2 + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{9} + 2 \beta_{14} ) q^{91} + ( -\beta_{7} - 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{13} - \beta_{15} ) q^{95} + ( 2 + 4 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{8} - 2 \beta_{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 8q^{7} + O(q^{10})$$ $$16q - 8q^{7} - 24q^{13} + 12q^{19} + 20q^{25} + 4q^{55} - 44q^{61} + 24q^{67} - 20q^{73} + 48q^{79} - 56q^{85} + 24q^{91} + 48q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 16 x^{14} + 174 x^{12} + 1012 x^{10} + 4243 x^{8} + 9708 x^{6} + 15858 x^{4} + 12150 x^{2} + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-481967 \nu^{14} + 30766372 \nu^{12} + 473907666 \nu^{10} + 5119222054 \nu^{8} + 25255513579 \nu^{6} + 88135584891 \nu^{4} + 105715511190 \nu^{2} + 82286048988$$$$)/ 21904094763$$ $$\beta_{3}$$ $$=$$ $$($$$$326606 \nu^{14} + 4547720 \nu^{12} + 47592867 \nu^{10} + 231730940 \nu^{8} + 904757018 \nu^{6} + 962205174 \nu^{4} + 748162737 \nu^{2} - 7319943180$$$$)/ 2433788307$$ $$\beta_{4}$$ $$=$$ $$($$$$9939136 \nu^{14} + 150207814 \nu^{12} + 1606621224 \nu^{10} + 8773398223 \nu^{8} + 35915018668 \nu^{6} + 72060692802 \nu^{4} + 131635278990 \nu^{2} + 34847824212$$$$)/ 65712284289$$ $$\beta_{5}$$ $$=$$ $$($$$$3704024 \nu^{14} + 40373015 \nu^{12} + 381453396 \nu^{10} + 906590177 \nu^{8} + 887860574 \nu^{6} - 24565580241 \nu^{4} - 41971706802 \nu^{2} - 56770896384$$$$)/ 21904094763$$ $$\beta_{6}$$ $$=$$ $$($$$$26555897 \nu^{15} + 62210210 \nu^{13} - 770314944 \nu^{11} - 30249558199 \nu^{9} - 191131571350 \nu^{7} - 937524731802 \nu^{5} - 1670986351080 \nu^{3} - 2199042758754 \nu$$$$)/ 197136852867$$ $$\beta_{7}$$ $$=$$ $$($$$$-32058956 \nu^{15} - 505462190 \nu^{13} - 5463091302 \nu^{11} - 31353520355 \nu^{9} - 130718214368 \nu^{7} - 308344384614 \nu^{5} - 552063322863 \nu^{3} - 700940621358 \nu$$$$)/ 197136852867$$ $$\beta_{8}$$ $$=$$ $$($$$$10005800 \nu^{14} + 163560617 \nu^{12} + 1774630644 \nu^{10} + 10542157166 \nu^{8} + 43490411282 \nu^{6} + 100433630874 \nu^{4} + 123833655000 \nu^{2} + 71014305663$$$$)/ 21904094763$$ $$\beta_{9}$$ $$=$$ $$($$$$-30938182 \nu^{14} - 478042816 \nu^{12} - 5141477487 \nu^{10} - 28836857512 \nu^{8} - 119231635186 \nu^{6} - 262263231510 \nu^{4} - 440628437772 \nu^{2} - 337029762708$$$$)/ 65712284289$$ $$\beta_{10}$$ $$=$$ $$($$$$-17666630 \nu^{15} - 328891601 \nu^{13} - 3662652030 \nu^{11} - 23457056468 \nu^{9} - 96537911477 \nu^{7} - 231158174538 \nu^{5} - 254108418858 \nu^{3} - 151450330770 \nu$$$$)/ 65712284289$$ $$\beta_{11}$$ $$=$$ $$($$$$-19112531 \nu^{15} - 236592485 \nu^{13} - 2240929032 \nu^{11} - 8099390306 \nu^{9} - 20771370740 \nu^{7} + 33248580135 \nu^{5} + 63038114712 \nu^{3} + 95407816194 \nu$$$$)/ 65712284289$$ $$\beta_{12}$$ $$=$$ $$($$$$-58514042 \nu^{15} - 873827510 \nu^{13} - 9318113529 \nu^{11} - 50123726495 \nu^{9} - 204003532826 \nu^{7} - 386283003708 \nu^{5} - 612664504560 \nu^{3} + 89111629089 \nu$$$$)/ 197136852867$$ $$\beta_{13}$$ $$=$$ $$($$$$-19878272 \nu^{15} - 300415628 \nu^{13} - 3213242448 \nu^{11} - 17546796446 \nu^{9} - 71830037336 \nu^{7} - 144121385604 \nu^{5} - 263270557980 \nu^{3} - 201120217002 \nu$$$$)/ 65712284289$$ $$\beta_{14}$$ $$=$$ $$($$$$-6210188 \nu^{14} - 89119619 \nu^{12} - 904945566 \nu^{10} - 4406204120 \nu^{8} - 15342294773 \nu^{6} - 18295668252 \nu^{4} - 14225798826 \nu^{2} + 11675287323$$$$)/ 7301364921$$ $$\beta_{15}$$ $$=$$ $$($$$$324198086 \nu^{15} + 5217299756 \nu^{13} + 56522041548 \nu^{11} + 328205077598 \nu^{9} + 1350470930357 \nu^{7} + 2986631978124 \nu^{5} + 4206283806420 \nu^{3} + 2317137727815 \nu$$$$)/ 197136852867$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{9} + 4 \beta_{4} - \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{13} + 4 \beta_{12} + 2 \beta_{7} - 5 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-8 \beta_{9} - \beta_{8} + \beta_{5} - 23 \beta_{4} - 23$$ $$\nu^{5}$$ $$=$$ $$($$$$-2 \beta_{15} + 31 \beta_{13} - 20 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} - 40 \beta_{7} + 2 \beta_{6}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-3 \beta_{14} - 13 \beta_{8} - 26 \beta_{5} + 58 \beta_{3} + 154$$ $$\nu^{7}$$ $$=$$ $$($$$$-26 \beta_{15} - 168 \beta_{12} - 64 \beta_{11} - 32 \beta_{10} + 168 \beta_{7} - 52 \beta_{6} + 209 \beta_{1}$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$48 \beta_{14} + 422 \beta_{9} + 252 \beta_{8} + 126 \beta_{5} + 1049 \beta_{4} - 422 \beta_{3} - 48 \beta_{2} - 48$$ $$\nu^{9}$$ $$=$$ $$($$$$504 \beta_{15} - 1471 \beta_{13} + 2696 \beta_{12} + 348 \beta_{11} + 696 \beta_{10} + 1348 \beta_{7} + 252 \beta_{6} - 1471 \beta_{1}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$-3115 \beta_{9} - 1100 \beta_{8} + 1100 \beta_{5} - 7528 \beta_{4} + 522 \beta_{2} - 7528$$ $$\nu^{11}$$ $$=$$ $$($$$$-2200 \beta_{15} + 10643 \beta_{13} - 10630 \beta_{12} + 3244 \beta_{11} - 3244 \beta_{10} - 21260 \beta_{7} + 2200 \beta_{6}$$$$)/2$$ $$\nu^{12}$$ $$=$$ $$-4866 \beta_{14} - 9137 \beta_{8} - 18274 \beta_{5} + 23288 \beta_{3} + 60083$$ $$\nu^{13}$$ $$=$$ $$($$$$-18274 \beta_{15} - 83124 \beta_{12} - 56012 \beta_{11} - 28006 \beta_{10} + 83124 \beta_{7} - 36548 \beta_{6} + 78505 \beta_{1}$$$$)/2$$ $$\nu^{14}$$ $$=$$ $$42009 \beta_{14} + 175780 \beta_{9} + 147678 \beta_{8} + 73839 \beta_{5} + 411295 \beta_{4} - 175780 \beta_{3} - 42009 \beta_{2} - 42009$$ $$\nu^{15}$$ $$=$$ $$($$$$295356 \beta_{15} - 587075 \beta_{13} + 1293832 \beta_{12} + 231696 \beta_{11} + 463392 \beta_{10} + 646916 \beta_{7} + 147678 \beta_{6} - 587075 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$$.

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$-\beta_{4}$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 0.733987 + 1.27130i −1.13921 − 1.97317i 1.38883 + 2.40552i −0.484374 − 0.838961i 0.484374 + 0.838961i −1.38883 − 2.40552i 1.13921 + 1.97317i −0.733987 − 1.27130i 0.733987 − 1.27130i −1.13921 + 1.97317i 1.38883 − 2.40552i −0.484374 + 0.838961i 0.484374 − 0.838961i −1.38883 + 2.40552i 1.13921 − 1.97317i −0.733987 + 1.27130i
0 0 0 −3.12320 + 1.80318i 0 3.25512 0 0 0
449.2 0 0 0 −2.90154 + 1.67520i 0 −3.54697 0 0 0
449.3 0 0 0 −2.00395 + 1.15698i 0 0.442911 0 0 0
449.4 0 0 0 −0.557544 + 0.321898i 0 −2.15106 0 0 0
449.5 0 0 0 0.557544 0.321898i 0 −2.15106 0 0 0
449.6 0 0 0 2.00395 1.15698i 0 0.442911 0 0 0
449.7 0 0 0 2.90154 1.67520i 0 −3.54697 0 0 0
449.8 0 0 0 3.12320 1.80318i 0 3.25512 0 0 0
1889.1 0 0 0 −3.12320 1.80318i 0 3.25512 0 0 0
1889.2 0 0 0 −2.90154 1.67520i 0 −3.54697 0 0 0
1889.3 0 0 0 −2.00395 1.15698i 0 0.442911 0 0 0
1889.4 0 0 0 −0.557544 0.321898i 0 −2.15106 0 0 0
1889.5 0 0 0 0.557544 + 0.321898i 0 −2.15106 0 0 0
1889.6 0 0 0 2.00395 + 1.15698i 0 0.442911 0 0 0
1889.7 0 0 0 2.90154 + 1.67520i 0 −3.54697 0 0 0
1889.8 0 0 0 3.12320 + 1.80318i 0 3.25512 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1889.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.d odd 6 1 inner
57.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.2.dc.c 16
3.b odd 2 1 inner 2736.2.dc.c 16
4.b odd 2 1 171.2.m.a 16
12.b even 2 1 171.2.m.a 16
19.d odd 6 1 inner 2736.2.dc.c 16
57.f even 6 1 inner 2736.2.dc.c 16
76.f even 6 1 171.2.m.a 16
228.n odd 6 1 171.2.m.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.2.m.a 16 4.b odd 2 1
171.2.m.a 16 12.b even 2 1
171.2.m.a 16 76.f even 6 1
171.2.m.a 16 228.n odd 6 1
2736.2.dc.c 16 1.a even 1 1 trivial
2736.2.dc.c 16 3.b odd 2 1 inner
2736.2.dc.c 16 19.d odd 6 1 inner
2736.2.dc.c 16 57.f even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2736, [\chi])$$:

 $$T_{5}^{16} - \cdots$$ $$T_{17}^{16} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$104976 - 290304 T^{2} + 709504 T^{4} - 238608 T^{6} + 55740 T^{8} - 6848 T^{10} + 612 T^{12} - 30 T^{14} + T^{16}$$
$7$ $$( 11 - 20 T - 12 T^{2} + 2 T^{3} + T^{4} )^{4}$$
$11$ $$( 36 + 584 T^{2} + 240 T^{4} + 30 T^{6} + T^{8} )^{2}$$
$13$ $$( 225 + 1080 T + 1698 T^{2} - 144 T^{3} - 269 T^{4} + 24 T^{5} + 50 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$17$ $$26873856 - 19408896 T^{2} + 9787392 T^{4} - 2474496 T^{6} + 451008 T^{8} - 38208 T^{10} + 2320 T^{12} - 56 T^{14} + T^{16}$$
$19$ $$( 130321 - 41154 T + 8664 T^{2} + 2166 T^{3} - 662 T^{4} + 114 T^{5} + 24 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$23$ $$1296 - 103968 T^{2} + 8302096 T^{4} - 3079632 T^{6} + 949980 T^{8} - 64712 T^{10} + 3288 T^{12} - 66 T^{14} + T^{16}$$
$29$ $$429981696 + 199065600 T^{2} + 64954368 T^{4} + 9940992 T^{6} + 1086208 T^{8} + 64768 T^{10} + 2784 T^{12} + 64 T^{14} + T^{16}$$
$31$ $$( 9801 + 9948 T^{2} + 1798 T^{4} + 80 T^{6} + T^{8} )^{2}$$
$37$ $$( 793881 + 149796 T^{2} + 8910 T^{4} + 180 T^{6} + T^{8} )^{2}$$
$41$ $$16796160000 + 23377766400 T^{2} + 31211283456 T^{4} + 1799439360 T^{6} + 71537344 T^{8} + 1523392 T^{10} + 23616 T^{12} + 184 T^{14} + T^{16}$$
$43$ $$( 39601 - 8358 T + 10520 T^{2} + 1848 T^{3} + 1737 T^{4} + 84 T^{5} + 44 T^{6} + T^{8} )^{2}$$
$47$ $$857889103585536 - 50068122711552 T^{2} + 1886624680576 T^{4} - 42154191360 T^{6} + 687138864 T^{8} - 7611008 T^{10} + 61992 T^{12} - 312 T^{14} + T^{16}$$
$53$ $$8503056 + 45979488 T^{2} + 242331264 T^{4} + 33534000 T^{6} + 3243564 T^{8} + 162864 T^{10} + 5940 T^{12} + 90 T^{14} + T^{16}$$
$59$ $$13680577296 + 4937752224 T^{2} + 1362055968 T^{4} + 126843504 T^{6} + 8310604 T^{8} + 296320 T^{10} + 7644 T^{12} + 106 T^{14} + T^{16}$$
$61$ $$( 18879025 + 3328270 T + 951736 T^{2} + 126836 T^{3} + 28253 T^{4} + 3380 T^{5} + 400 T^{6} + 22 T^{7} + T^{8} )^{2}$$
$67$ $$( 45369 + 228762 T + 346578 T^{2} - 191172 T^{3} + 27175 T^{4} + 2136 T^{5} - 130 T^{6} - 12 T^{7} + T^{8} )^{2}$$
$71$ $$892616806656 + 916697461248 T^{2} + 909388238976 T^{4} + 32223619584 T^{6} + 799781040 T^{8} + 10267776 T^{10} + 95688 T^{12} + 360 T^{14} + T^{16}$$
$73$ $$( 2152089 - 2807838 T + 3974400 T^{2} + 376428 T^{3} + 62617 T^{4} + 1708 T^{5} + 312 T^{6} + 10 T^{7} + T^{8} )^{2}$$
$79$ $$( 154449 - 106110 T + 11724 T^{2} + 8640 T^{3} - 743 T^{4} - 768 T^{5} + 224 T^{6} - 24 T^{7} + T^{8} )^{2}$$
$83$ $$( 627264 + 448544 T^{2} + 50256 T^{4} + 456 T^{6} + T^{8} )^{2}$$
$89$ $$34867844010000 + 7622139398400 T^{2} + 1368386409456 T^{4} + 59930903376 T^{6} + 1972507788 T^{8} + 19509336 T^{10} + 141408 T^{12} + 438 T^{14} + T^{16}$$
$97$ $$( 944784 + 1831248 T + 1109280 T^{2} - 143184 T^{3} - 10268 T^{4} + 1824 T^{5} + 116 T^{6} - 24 T^{7} + T^{8} )^{2}$$