# Properties

 Label 2736.2.dc.d 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: $$x^{16} + 10 x^{14} + 46 x^{12} + 126 x^{10} + 315 x^{8} + 1134 x^{6} + 3726 x^{4} + 7290 x^{2} + 6561$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}\cdot 3^{7}$$ Twist minimal: no (minimal twist has level 684) 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{4} - \beta_{10} ) q^{5} + ( -\beta_{7} - \beta_{8} ) q^{7} +O(q^{10})$$ $$q + ( \beta_{4} - \beta_{10} ) q^{5} + ( -\beta_{7} - \beta_{8} ) q^{7} + ( \beta_{2} - \beta_{13} ) q^{11} + ( 1 + \beta_{3} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{13} + ( \beta_{4} - \beta_{10} + \beta_{11} ) q^{17} + ( 2 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{19} + ( -\beta_{11} + \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{23} + ( \beta_{6} + \beta_{8} ) q^{25} + ( -2 \beta_{2} + \beta_{15} ) q^{29} + ( 2 + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{31} + ( -2 \beta_{4} + 2 \beta_{10} + \beta_{11} - \beta_{12} ) q^{35} + ( \beta_{1} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{37} + ( -\beta_{2} + \beta_{4} + \beta_{10} + \beta_{11} + 2 \beta_{13} - \beta_{15} ) q^{41} + ( -1 - \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{43} + ( \beta_{12} - \beta_{14} - \beta_{15} ) q^{47} + ( 1 - \beta_{1} + \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{49} + ( 4 \beta_{4} - 2 \beta_{10} - \beta_{11} + \beta_{13} ) q^{53} + ( -3 - 3 \beta_{3} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{55} + ( 2 \beta_{4} + 2 \beta_{10} + \beta_{11} + 2 \beta_{13} ) q^{59} + ( 2 + 2 \beta_{5} - \beta_{6} + 3 \beta_{8} ) q^{61} + ( \beta_{2} + 2 \beta_{11} + \beta_{13} - 2 \beta_{15} ) q^{65} + ( -1 + \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{67} + ( -\beta_{12} + 2 \beta_{14} ) q^{71} + ( -\beta_{1} - 5 \beta_{5} - \beta_{9} ) q^{73} + ( \beta_{4} + \beta_{14} ) q^{77} + ( 3 - 2 \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{9} ) q^{79} + ( -2 \beta_{2} + 3 \beta_{4} + \beta_{13} ) q^{83} + ( 6 + 6 \beta_{5} + 2 \beta_{6} ) q^{85} + ( -2 \beta_{4} + \beta_{10} + 2 \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{14} ) q^{89} + ( 3 - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{91} + ( \beta_{2} + 3 \beta_{4} - 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{95} + ( 1 - \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + 12q^{19} + 4q^{25} + 4q^{43} - 20q^{55} + 12q^{61} - 36q^{67} + 44q^{73} + 12q^{79} + 56q^{85} + 60q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 10 x^{14} + 46 x^{12} + 126 x^{10} + 315 x^{8} + 1134 x^{6} + 3726 x^{4} + 7290 x^{2} + 6561$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{14} + 523 \nu^{12} + 2584 \nu^{10} + 5175 \nu^{8} + 8496 \nu^{6} + 57834 \nu^{4} + 214407 \nu^{2} + 282852$$$$)/12393$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{15} + 152 \nu^{13} + 1088 \nu^{11} + 2658 \nu^{9} + 3654 \nu^{7} + 18360 \nu^{5} + 84240 \nu^{3} + 165726 \nu$$$$)/12393$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{14} - 10 \nu^{12} - 46 \nu^{10} - 126 \nu^{8} - 315 \nu^{6} - 1134 \nu^{4} - 2997 \nu^{2} - 5832$$$$)/729$$ $$\beta_{4}$$ $$=$$ $$($$$$-23 \nu^{15} + 58 \nu^{13} + 850 \nu^{11} + 1845 \nu^{9} + 126 \nu^{7} + 8262 \nu^{5} + 63018 \nu^{3} + 87480 \nu$$$$)/37179$$ $$\beta_{5}$$ $$=$$ $$($$$$-50 \nu^{14} - 446 \nu^{12} - 1598 \nu^{10} - 3087 \nu^{8} - 8946 \nu^{6} - 41310 \nu^{4} - 111942 \nu^{2} - 126117$$$$)/12393$$ $$\beta_{6}$$ $$=$$ $$($$$$20 \nu^{14} + 158 \nu^{12} + 527 \nu^{10} + 939 \nu^{8} + 3303 \nu^{6} + 15606 \nu^{4} + 41472 \nu^{2} + 38880$$$$)/4131$$ $$\beta_{7}$$ $$=$$ $$($$$$56 \nu^{14} + 218 \nu^{12} - 34 \nu^{10} - 900 \nu^{8} + 3465 \nu^{6} + 13770 \nu^{4} - 26811 \nu^{2} - 122472$$$$)/12393$$ $$\beta_{8}$$ $$=$$ $$($$$$7 \nu^{14} + 40 \nu^{12} + 85 \nu^{10} + 168 \nu^{8} + 873 \nu^{6} + 3060 \nu^{4} + 5427 \nu^{2} + 2592$$$$)/1377$$ $$\beta_{9}$$ $$=$$ $$($$$$-22 \nu^{14} - 133 \nu^{12} - 493 \nu^{10} - 1038 \nu^{8} - 2853 \nu^{6} - 13311 \nu^{4} - 36531 \nu^{2} - 46899$$$$)/4131$$ $$\beta_{10}$$ $$=$$ $$($$$$133 \nu^{15} + 1168 \nu^{13} + 4012 \nu^{11} + 7119 \nu^{9} + 21483 \nu^{7} + 104652 \nu^{5} + 272484 \nu^{3} + 254421 \nu$$$$)/37179$$ $$\beta_{11}$$ $$=$$ $$($$$$-158 \nu^{15} - 2003 \nu^{13} - 8177 \nu^{11} - 16848 \nu^{9} - 39726 \nu^{7} - 195534 \nu^{5} - 625887 \nu^{3} - 794610 \nu$$$$)/37179$$ $$\beta_{12}$$ $$=$$ $$($$$$-19 \nu^{15} - 196 \nu^{13} - 646 \nu^{11} - 1221 \nu^{9} - 3834 \nu^{7} - 17901 \nu^{5} - 44631 \nu^{3} - 41067 \nu$$$$)/4131$$ $$\beta_{13}$$ $$=$$ $$($$$$-194 \nu^{15} - 635 \nu^{13} + 238 \nu^{11} + 1566 \nu^{9} - 12348 \nu^{7} - 34425 \nu^{5} + 82701 \nu^{3} + 325134 \nu$$$$)/37179$$ $$\beta_{14}$$ $$=$$ $$($$$$73 \nu^{15} + 541 \nu^{13} + 1819 \nu^{11} + 3690 \nu^{9} + 11115 \nu^{7} + 52326 \nu^{5} + 131544 \nu^{3} + 137781 \nu$$$$)/12393$$ $$\beta_{15}$$ $$=$$ $$($$$$-346 \nu^{15} - 2560 \nu^{13} - 7888 \nu^{11} - 14832 \nu^{9} - 53424 \nu^{7} - 231336 \nu^{5} - 545616 \nu^{3} - 433026 \nu$$$$)/37179$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} + 2 \beta_{10} - 4 \beta_{4} + \beta_{2}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - 3$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{15} - 2 \beta_{14} + 4 \beta_{12} - 6 \beta_{11} - 4 \beta_{10} + 2 \beta_{4} - 5 \beta_{2}$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{9} - 2 \beta_{8} + 5 \beta_{7} + \beta_{6} + 10 \beta_{5} - 5 \beta_{3} + 3 \beta_{1} + 3$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$7 \beta_{15} + 22 \beta_{14} + 12 \beta_{13} - 14 \beta_{12} + 18 \beta_{11} + 2 \beta_{10} - 4 \beta_{4} - 2 \beta_{2}$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$18 \beta_{9} - 5 \beta_{8} - 10 \beta_{7} + 19 \beta_{6} - 17 \beta_{5} - 5 \beta_{3} - 12 \beta_{1} - 3$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-59 \beta_{15} - 44 \beta_{14} + 24 \beta_{13} + 4 \beta_{12} + 24 \beta_{11} - 16 \beta_{10} - 34 \beta_{4} + 25 \beta_{2}$$$$)/6$$ $$\nu^{8}$$ $$=$$ $$($$$$-42 \beta_{9} + 52 \beta_{8} + 14 \beta_{7} - 56 \beta_{6} + 79 \beta_{5} - 26 \beta_{3} + 36 \beta_{1} - 135$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$70 \beta_{15} + 76 \beta_{14} - 144 \beta_{13} - 44 \beta_{12} - 96 \beta_{11} - 274 \beta_{10} + 290 \beta_{4} - 5 \beta_{2}$$$$)/6$$ $$\nu^{10}$$ $$=$$ $$($$$$78 \beta_{9} - 245 \beta_{8} - 22 \beta_{7} + 28 \beta_{6} - 422 \beta_{5} + 49 \beta_{3} - 132 \beta_{1} - 24$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-263 \beta_{15} - 122 \beta_{14} + 282 \beta_{13} + 58 \beta_{12} + 612 \beta_{11} + 758 \beta_{10} - 4 \beta_{4} + 322 \beta_{2}$$$$)/6$$ $$\nu^{12}$$ $$=$$ $$($$$$72 \beta_{9} + 589 \beta_{8} - 10 \beta_{7} - 413 \beta_{6} + 64 \beta_{5} + 238 \beta_{3} + 231 \beta_{1} + 780$$$$)/3$$ $$\nu^{13}$$ $$=$$ $$($$$$1237 \beta_{15} - 908 \beta_{14} - 1542 \beta_{13} - 158 \beta_{12} - 2136 \beta_{11} - 394 \beta_{10} - 466 \beta_{4} - 407 \beta_{2}$$$$)/6$$ $$\nu^{14}$$ $$=$$ $$($$$$-1284 \beta_{9} - 326 \beta_{8} - 175 \beta_{7} - 218 \beta_{6} - 164 \beta_{5} + 703 \beta_{3} - 396 \beta_{1} - 648$$$$)/3$$ $$\nu^{15}$$ $$=$$ $$($$$$-470 \beta_{15} + 1480 \beta_{14} - 576 \beta_{13} + 4168 \beta_{12} - 312 \beta_{11} + 2318 \beta_{10} + 884 \beta_{4} - 5 \beta_{2}$$$$)/6$$

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$1 + \beta_{5}$$ $$-1$$ $$1$$ $$1$$

## 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
 1.56478 − 0.742611i 0.750719 − 1.56090i −0.243856 − 1.71480i −0.654543 − 1.60361i 0.654543 + 1.60361i 0.243856 + 1.71480i −0.750719 + 1.56090i −1.56478 + 0.742611i 1.56478 + 0.742611i 0.750719 + 1.56090i −0.243856 + 1.71480i −0.654543 + 1.60361i 0.654543 − 1.60361i 0.243856 − 1.71480i −0.750719 − 1.56090i −1.56478 − 0.742611i
0 0 0 −2.99029 + 1.72644i 0 −0.128302 0 0 0
449.2 0 0 0 −2.47786 + 1.43059i 0 −3.93208 0 0 0
449.3 0 0 0 −1.11928 + 0.646214i 0 0.567493 0 0 0
449.4 0 0 0 −0.406956 + 0.234956i 0 3.49289 0 0 0
449.5 0 0 0 0.406956 0.234956i 0 3.49289 0 0 0
449.6 0 0 0 1.11928 0.646214i 0 0.567493 0 0 0
449.7 0 0 0 2.47786 1.43059i 0 −3.93208 0 0 0
449.8 0 0 0 2.99029 1.72644i 0 −0.128302 0 0 0
1889.1 0 0 0 −2.99029 1.72644i 0 −0.128302 0 0 0
1889.2 0 0 0 −2.47786 1.43059i 0 −3.93208 0 0 0
1889.3 0 0 0 −1.11928 0.646214i 0 0.567493 0 0 0
1889.4 0 0 0 −0.406956 0.234956i 0 3.49289 0 0 0
1889.5 0 0 0 0.406956 + 0.234956i 0 3.49289 0 0 0
1889.6 0 0 0 1.11928 + 0.646214i 0 0.567493 0 0 0
1889.7 0 0 0 2.47786 + 1.43059i 0 −3.93208 0 0 0
1889.8 0 0 0 2.99029 + 1.72644i 0 −0.128302 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.d 16
3.b odd 2 1 inner 2736.2.dc.d 16
4.b odd 2 1 684.2.bk.a 16
12.b even 2 1 684.2.bk.a 16
19.d odd 6 1 inner 2736.2.dc.d 16
57.f even 6 1 inner 2736.2.dc.d 16
76.f even 6 1 684.2.bk.a 16
228.n odd 6 1 684.2.bk.a 16

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

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16}$$
$5$ $$1296 - 6912 T^{2} + 31968 T^{4} - 24528 T^{6} + 14236 T^{8} - 2608 T^{10} + 348 T^{12} - 22 T^{14} + T^{16}$$
$7$ $$( 1 + 6 T - 14 T^{2} + T^{4} )^{4}$$
$11$ $$( 6084 + 5592 T^{2} + 1360 T^{4} + 70 T^{6} + T^{8} )^{2}$$
$13$ $$( 13689 + 12636 T - 1026 T^{2} - 4536 T^{3} + 1647 T^{4} - 42 T^{6} + T^{8} )^{2}$$
$17$ $$331776 - 2598912 T^{2} + 19243008 T^{4} - 8633856 T^{6} + 3350464 T^{8} - 161344 T^{10} + 5808 T^{12} - 88 T^{14} + T^{16}$$
$19$ $$( 130321 - 41154 T + 15884 T^{2} - 4674 T^{3} + 1242 T^{4} - 246 T^{5} + 44 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$23$ $$358892053776 - 82255531104 T^{2} + 14205954960 T^{4} - 880414416 T^{6} + 38411644 T^{8} - 919816 T^{10} + 15960 T^{12} - 154 T^{14} + T^{16}$$
$29$ $$34828517376 + 44503105536 T^{2} + 54661423104 T^{4} + 2744119296 T^{6} + 93457152 T^{8} + 1790208 T^{10} + 25056 T^{12} + 192 T^{14} + T^{16}$$
$31$ $$( 613089 + 126684 T^{2} + 7974 T^{4} + 168 T^{6} + T^{8} )^{2}$$
$37$ $$( 23409 + 71388 T^{2} + 6102 T^{4} + 156 T^{6} + T^{8} )^{2}$$
$41$ $$176319369216 + 88159684608 T^{2} + 38637886464 T^{4} + 2539579392 T^{6} + 122192064 T^{8} + 2379456 T^{10} + 33696 T^{12} + 216 T^{14} + T^{16}$$
$43$ $$( 2825761 + 47068 T + 152074 T^{2} + 4204 T^{3} + 6475 T^{4} + 124 T^{5} + 94 T^{6} - 2 T^{7} + T^{8} )^{2}$$
$47$ $$20736 - 235008 T^{2} + 2498688 T^{4} - 1841664 T^{6} + 1164976 T^{8} - 97408 T^{10} + 6600 T^{12} - 88 T^{14} + T^{16}$$
$53$ $$1386400841235216 + 81335129972832 T^{2} + 3149707672224 T^{4} + 68790854448 T^{6} + 1086958764 T^{8} + 11051424 T^{10} + 81756 T^{12} + 354 T^{14} + T^{16}$$
$59$ $$1386400841235216 + 81335129972832 T^{2} + 3149707672224 T^{4} + 68790854448 T^{6} + 1086958764 T^{8} + 11051424 T^{10} + 81756 T^{12} + 354 T^{14} + T^{16}$$
$61$ $$( 1990921 - 59262 T + 182372 T^{2} + 22308 T^{3} + 14721 T^{4} + 852 T^{5} + 164 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$67$ $$( 68121 + 112752 T + 68472 T^{2} + 10368 T^{3} - 1755 T^{4} - 432 T^{5} + 84 T^{6} + 18 T^{7} + T^{8} )^{2}$$
$71$ $$( 2916 + 54 T^{2} + T^{4} )^{4}$$
$73$ $$( 85849 + 2930 T + 31744 T^{2} - 13972 T^{3} + 11737 T^{4} - 2356 T^{5} + 376 T^{6} - 22 T^{7} + T^{8} )^{2}$$
$79$ $$( 700569 - 271188 T - 90558 T^{2} + 48600 T^{3} + 22311 T^{4} + 900 T^{5} - 138 T^{6} - 6 T^{7} + T^{8} )^{2}$$
$83$ $$( 46656 + 652320 T^{2} + 30352 T^{4} + 328 T^{6} + T^{8} )^{2}$$
$89$ $$403540761128976 + 118899414917568 T^{2} + 32779867590864 T^{4} + 639390771792 T^{6} + 8968479084 T^{8} + 56119176 T^{10} + 255096 T^{12} + 606 T^{14} + T^{16}$$
$97$ $$( 8503056 - 8503056 T + 3569184 T^{2} - 734832 T^{3} + 66420 T^{4} - 252 T^{6} + T^{8} )^{2}$$