# Properties

 Label 2736.3.m.f Level $2736$ Weight $3$ Character orbit 2736.m Analytic conductor $74.551$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2736 = 2^{4} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2736.m (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$74.5506003290$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} + 50 x^{10} - 136 x^{9} + 2215 x^{8} - 5020 x^{7} + 18282 x^{6} - 12094 x^{5} + 48457 x^{4} - 30372 x^{3} + 89392 x^{2} + 9344 x + 1024$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{16}$$ Twist minimal: no (minimal twist has level 912) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{2} ) q^{5} -\beta_{11} q^{7} +O(q^{10})$$ $$q + ( 1 - \beta_{2} ) q^{5} -\beta_{11} q^{7} + ( -\beta_{7} + \beta_{9} + \beta_{10} ) q^{11} + ( 3 + \beta_{1} ) q^{13} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{17} -\beta_{8} q^{19} + ( -2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{23} + ( 3 + \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{25} + ( -2 + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{29} + ( -2 \beta_{8} + \beta_{10} - \beta_{11} ) q^{31} + ( 2 \beta_{6} + \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + \beta_{10} - 3 \beta_{11} ) q^{35} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{37} + ( -11 + 3 \beta_{2} - 3 \beta_{3} + \beta_{5} ) q^{41} + ( 3 \beta_{6} - 2 \beta_{7} - 8 \beta_{8} + 2 \beta_{10} + 3 \beta_{11} ) q^{43} + ( \beta_{6} + 3 \beta_{7} + \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{47} + ( -3 - \beta_{1} + 5 \beta_{2} + 2 \beta_{5} ) q^{49} + ( 17 - \beta_{2} + \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{53} + ( -6 \beta_{6} - 6 \beta_{7} - 4 \beta_{8} + 6 \beta_{9} + 4 \beta_{10} + \beta_{11} ) q^{55} + ( -6 \beta_{6} - 3 \beta_{7} + 9 \beta_{8} + 4 \beta_{9} + 3 \beta_{10} + 4 \beta_{11} ) q^{59} + ( -16 - 3 \beta_{1} + \beta_{3} + 5 \beta_{4} + 4 \beta_{5} ) q^{61} + ( 5 - 7 \beta_{2} + 3 \beta_{5} ) q^{65} + ( -6 \beta_{6} - \beta_{7} - 7 \beta_{8} - 2 \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{67} + ( 2 \beta_{6} + 5 \beta_{7} + 5 \beta_{8} - 4 \beta_{9} - \beta_{10} ) q^{71} + ( -12 + 3 \beta_{1} + \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} ) q^{73} + ( 3 + 3 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} - 10 \beta_{4} - 3 \beta_{5} ) q^{77} + ( 5 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 9 \beta_{11} ) q^{79} + ( 2 \beta_{6} - 4 \beta_{7} + 9 \beta_{8} + 5 \beta_{9} - 7 \beta_{11} ) q^{83} + ( 16 + \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 7 \beta_{4} ) q^{85} + ( -37 - 8 \beta_{1} + 7 \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{89} + ( 6 \beta_{6} + \beta_{7} - 17 \beta_{8} - 6 \beta_{9} - \beta_{10} - 4 \beta_{11} ) q^{91} + ( -\beta_{6} - 2 \beta_{7} + \beta_{10} + 2 \beta_{11} ) q^{95} + ( 27 + 2 \beta_{1} - 13 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 3 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 10q^{5} + O(q^{10})$$ $$12q + 10q^{5} + 36q^{13} + 10q^{17} + 58q^{25} - 12q^{29} + 32q^{37} - 136q^{41} - 22q^{49} + 236q^{53} - 210q^{61} + 52q^{65} - 158q^{73} + 70q^{77} + 242q^{85} - 444q^{89} + 320q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} + 50 x^{10} - 136 x^{9} + 2215 x^{8} - 5020 x^{7} + 18282 x^{6} - 12094 x^{5} + 48457 x^{4} - 30372 x^{3} + 89392 x^{2} + 9344 x + 1024$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$1132830139545 \nu^{11} + 83463703932306 \nu^{10} + 82049035182874 \nu^{9} + 3546994404808800 \nu^{8} + 121097104652847 \nu^{7} + 148552108990073920 \nu^{6} + 8276942020880970 \nu^{5} + 214861540128950778 \nu^{4} + 1674390603566314313 \nu^{3} + 511350538329590112 \nu^{2} + 53532590571340032 \nu - 6360028955241199520$$$$)/ 524997139451464992$$ $$\beta_{2}$$ $$=$$ $$($$$$-38271754888553 \nu^{11} + 9342376003726 \nu^{10} - 1732279068255930 \nu^{9} + 1689016613416960 \nu^{8} - 72988651901379743 \nu^{7} + 34182444306290592 \nu^{6} - 243521736602383786 \nu^{5} - 1126465309466867194 \nu^{4} + 358570349069318247 \nu^{3} - 3052609166630207840 \nu^{2} - 319778864263561472 \nu - 2760047061250342272$$$$)/ 1049994278902929984$$ $$\beta_{3}$$ $$=$$ $$($$$$5763155937085 \nu^{11} + 13707431665502 \nu^{10} + 264163078401060 \nu^{9} + 394878866521450 \nu^{8} + 10622923163218199 \nu^{7} + 21967557357647790 \nu^{6} + 36863649425687030 \nu^{5} + 202391130690006256 \nu^{4} + 418238779655033811 \nu^{3} + 535662625147206784 \nu^{2} + 56107618738863104 \nu + 1155207366864617952$$$$)/ 131249284862866248$$ $$\beta_{4}$$ $$=$$ $$($$$$79024470131873 \nu^{11} + 62863161452290 \nu^{10} + 3600574397631482 \nu^{9} + 41309770325840 \nu^{8} + 148708082025817735 \nu^{7} + 76318485191843792 \nu^{6} + 503878265787402106 \nu^{5} + 2504032725025359002 \nu^{4} + 1334036523831820417 \nu^{3} + 6716120670536696672 \nu^{2} + 703519610675540224 \nu + 9455027681169833984$$$$)/ 1049994278902929984$$ $$\beta_{5}$$ $$=$$ $$($$$$-104157093719227 \nu^{11} - 31799241318102 \nu^{10} - 4701895587821294 \nu^{9} + 2138646768261440 \nu^{8} - 197245992312264669 \nu^{7} - 11488184600829152 \nu^{6} - 663478033868035454 \nu^{5} - 3189732782724601134 \nu^{4} - 2519212169482205707 \nu^{3} - 8595407063624990752 \nu^{2} - 900396216439284480 \nu - 19682932283508561344$$$$)/ 1049994278902929984$$ $$\beta_{6}$$ $$=$$ $$($$$$-2248559103131 \nu^{11} + 4524708578702 \nu^{10} - 112234893734550 \nu^{9} + 307066972550728 \nu^{8} - 4973193956999965 \nu^{7} + 11335519044013020 \nu^{6} - 40771871289610798 \nu^{5} + 27372180320175994 \nu^{4} - 107713265419379907 \nu^{3} + 76815419370702628 \nu^{2} - 197798094845688592 \nu - 9850181819940672$$$$)/ 5412341643829536$$ $$\beta_{7}$$ $$=$$ $$($$$$451147977297025 \nu^{11} - 972391254937206 \nu^{10} + 22576823616269066 \nu^{9} - 64660999224635840 \nu^{8} + 1002729201948563015 \nu^{7} - 2405490450613767224 \nu^{6} + 8328575736135637274 \nu^{5} - 6211993005620740054 \nu^{4} + 20321083445471531665 \nu^{3} - 15654047187007783992 \nu^{2} + 31606828010001537024 \nu + 1551585028059237120$$$$)/ 1049994278902929984$$ $$\beta_{8}$$ $$=$$ $$($$$$5084441 \nu^{11} - 10363066 \nu^{10} + 255384002 \nu^{9} - 701038728 \nu^{8} + 11325739791 \nu^{7} - 25986026436 \nu^{6} + 95500767898 \nu^{5} - 65799847374 \nu^{4} + 258687396929 \nu^{3} - 165859562764 \nu^{2} + 503176327088 \nu + 25239211968$$$$)/ 10877179584$$ $$\beta_{9}$$ $$=$$ $$($$$$563617568102453 \nu^{11} - 1247310895415622 \nu^{10} + 28517763625814898 \nu^{9} - 82512571560979552 \nu^{8} + 1268657916405993267 \nu^{7} - 3091614423655209552 \nu^{6} + 11061417044012668162 \nu^{5} - 8740333126476491310 \nu^{4} + 28277300774056917189 \nu^{3} - 17701743157976235184 \nu^{2} + 49984211369270773152 \nu + 2497291971551809920$$$$)/ 1049994278902929984$$ $$\beta_{10}$$ $$=$$ $$($$$$8386307706881 \nu^{11} - 16626138955879 \nu^{10} + 416130916219964 \nu^{9} - 1134473296241394 \nu^{8} + 18423362292467887 \nu^{7} - 41708784750609295 \nu^{6} + 147055547261325582 \nu^{5} - 98616276908108400 \nu^{4} + 382694351988536807 \nu^{3} - 290443011225962145 \nu^{2} + 673893276740335836 \nu + 33381840440796144$$$$)/ 11412981292423152$$ $$\beta_{11}$$ $$=$$ $$($$$$-288508475434596 \nu^{11} + 596180148308589 \nu^{10} - 14467096347365602 \nu^{9} + 40206134390967818 \nu^{8} - 641785219978602148 \nu^{7} + 1491221454877942883 \nu^{6} - 5377167426207954316 \nu^{5} + 3854624495713785258 \nu^{4} - 14215208134902360114 \nu^{3} + 9409745668273498125 \nu^{2} - 26382288658793309132 \nu - 1318602899492042416$$$$)/ 262498569725732496$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 2$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{11} + 4 \beta_{10} - 6 \beta_{9} + 16 \beta_{8} + 8 \beta_{7} + 31 \beta_{6} + 2 \beta_{5} + 5 \beta_{4} + 3 \beta_{3} + 7 \beta_{2} - 4 \beta_{1} - 64$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$12 \beta_{5} - 35 \beta_{4} + 35 \beta_{3} - 63 \beta_{2} - 4 \beta_{1} + 18$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$90 \beta_{11} - 96 \beta_{10} + 338 \beta_{9} - 672 \beta_{8} - 408 \beta_{7} - 1149 \beta_{6} + 106 \beta_{5} + 247 \beta_{4} + 113 \beta_{3} + 345 \beta_{2} - 168 \beta_{1} - 2360$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$-3850 \beta_{11} - 2554 \beta_{10} - 3762 \beta_{9} - 1858 \beta_{8} + 2338 \beta_{7} + 1135 \beta_{6} - 504 \beta_{5} + 1537 \beta_{4} - 1377 \beta_{3} + 2889 \beta_{2} + 72 \beta_{1} - 2702$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-4258 \beta_{5} - 11341 \beta_{4} - 3867 \beta_{3} - 16079 \beta_{2} + 6900 \beta_{1} + 97664$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$163514 \beta_{11} + 103394 \beta_{10} + 165862 \beta_{9} + 60614 \beta_{8} - 107558 \beta_{7} - 78445 \beta_{6} - 18652 \beta_{5} + 70235 \beta_{4} - 54587 \beta_{3} + 129927 \beta_{2} - 1388 \beta_{1} - 176930$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$-343514 \beta_{11} + 17776 \beta_{10} - 786242 \beta_{9} + 1080640 \beta_{8} + 840344 \beta_{7} + 2009317 \beta_{6} + 165610 \beta_{5} + 514511 \beta_{4} + 125817 \beta_{3} + 746449 \beta_{2} - 286040 \beta_{1} - 4139064$$$$)/8$$ $$\nu^{9}$$ $$=$$ $$($$$$670120 \beta_{5} - 3225385 \beta_{4} + 2178281 \beta_{3} - 5845345 \beta_{2} + 238472 \beta_{1} + 9957246$$$$)/4$$ $$\nu^{10}$$ $$=$$ $$($$$$18600482 \beta_{11} + 1973948 \beta_{10} + 37156070 \beta_{9} - 43542928 \beta_{8} - 37900232 \beta_{7} - 85899087 \beta_{6} + 6433250 \beta_{5} + 23319637 \beta_{4} + 3828083 \beta_{3} + 34550903 \beta_{2} - 11975428 \beta_{1} - 177340352$$$$)/8$$ $$\nu^{11}$$ $$=$$ $$($$$$-300468378 \beta_{11} - 177177090 \beta_{10} - 327495670 \beta_{9} - 48227286 \beta_{8} + 229470870 \beta_{7} + 241426709 \beta_{6} - 23665836 \beta_{5} + 148059987 \beta_{4} - 87645459 \beta_{3} + 263516047 \beta_{2} - 17431260 \beta_{1} - 523950514$$$$)/8$$

## Character values

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

 $$n$$ $$1009$$ $$1217$$ $$1711$$ $$2053$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1711.1
 −0.820808 + 1.42168i −0.820808 − 1.42168i 3.06079 + 5.30145i 3.06079 − 5.30145i 0.816029 − 1.41340i 0.816029 + 1.41340i −0.0525878 − 0.0910847i −0.0525878 + 0.0910847i 1.37340 − 2.37879i 1.37340 + 2.37879i −3.37682 + 5.84883i −3.37682 − 5.84883i
0 0 0 −8.90000 0 2.78940i 0 0 0
1711.2 0 0 0 −8.90000 0 2.78940i 0 0 0
1711.3 0 0 0 −1.38490 0 7.59081i 0 0 0
1711.4 0 0 0 −1.38490 0 7.59081i 0 0 0
1711.5 0 0 0 −0.606930 0 2.85501i 0 0 0
1711.6 0 0 0 −0.606930 0 2.85501i 0 0 0
1711.7 0 0 0 3.59607 0 9.49604i 0 0 0
1711.8 0 0 0 3.59607 0 9.49604i 0 0 0
1711.9 0 0 0 4.02901 0 7.12527i 0 0 0
1711.10 0 0 0 4.02901 0 7.12527i 0 0 0
1711.11 0 0 0 8.26675 0 9.51333i 0 0 0
1711.12 0 0 0 8.26675 0 9.51333i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1711.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 2736.3.m.f 12
3.b odd 2 1 912.3.m.a 12
4.b odd 2 1 inner 2736.3.m.f 12
12.b even 2 1 912.3.m.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.3.m.a 12 3.b odd 2 1
912.3.m.a 12 12.b even 2 1
2736.3.m.f 12 1.a even 1 1 trivial
2736.3.m.f 12 4.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 5 T_{5}^{5} - 77 T_{5}^{4} + 437 T_{5}^{3} + 16 T_{5}^{2} - 1644 T_{5} - 896$$ acting on $$S_{3}^{\mathrm{new}}(2736, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$( -896 - 1644 T + 16 T^{2} + 437 T^{3} - 77 T^{4} - 5 T^{5} + T^{6} )^{2}$$
$7$ $$1514143744 + 469984000 T^{2} + 48333216 T^{4} + 1920099 T^{6} + 35339 T^{8} + 305 T^{10} + T^{12}$$
$11$ $$187580416 + 2604011200 T^{2} + 547737172 T^{4} + 19804319 T^{6} + 240151 T^{8} + 885 T^{10} + T^{12}$$
$13$ $$( 2630656 - 631520 T - 3344 T^{2} + 9152 T^{3} - 416 T^{4} - 18 T^{5} + T^{6} )^{2}$$
$17$ $$( 8199352 + 2442812 T + 187502 T^{2} - 5163 T^{3} - 915 T^{4} - 5 T^{5} + T^{6} )^{2}$$
$19$ $$( 19 + T^{2} )^{6}$$
$23$ $$99230924406784 + 7515432550400 T^{2} + 108480069632 T^{4} + 629121792 T^{6} + 1711248 T^{8} + 2152 T^{10} + T^{12}$$
$29$ $$( 76842496 - 19631840 T + 1142320 T^{2} + 13856 T^{3} - 2096 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$31$ $$1396965376 + 7569993728 T^{2} + 898195712 T^{4} + 32421120 T^{6} + 417888 T^{8} + 1312 T^{10} + T^{12}$$
$37$ $$( 6873088 - 7755776 T + 1565568 T^{2} + 29088 T^{3} - 2740 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$41$ $$( 1137344512 + 128593408 T + 971264 T^{2} - 221328 T^{3} - 3012 T^{4} + 68 T^{5} + T^{6} )^{2}$$
$43$ $$14227016088027136 + 3925970460807936 T^{2} + 135793784837536 T^{4} + 197117772467 T^{6} + 87741547 T^{8} + 15745 T^{10} + T^{12}$$
$47$ $$1241462640530411776 + 14051430605107120 T^{2} + 52974525286600 T^{4} + 79986650915 T^{6} + 51615091 T^{8} + 12849 T^{10} + T^{12}$$
$53$ $$( 2520564736 - 198700896 T - 9459440 T^{2} + 516016 T^{3} - 1928 T^{4} - 118 T^{5} + T^{6} )^{2}$$
$59$ $$12\!\cdots\!84$$$$+ 670549120897515520 T^{2} + 953648696180736 T^{4} + 572168623104 T^{6} + 162332480 T^{8} + 21104 T^{10} + T^{12}$$
$61$ $$( 826249672 + 997736500 T + 1399834 T^{2} - 824113 T^{3} - 6647 T^{4} + 105 T^{5} + T^{6} )^{2}$$
$67$ $$23244689000026341376 + 227274567284686848 T^{2} + 380659045482496 T^{4} + 267388789760 T^{6} + 92423488 T^{8} + 15472 T^{10} + T^{12}$$
$71$ $$808242081226031104 + 29488294639173632 T^{2} + 143606242672640 T^{4} + 200228330496 T^{6} + 102523968 T^{8} + 19408 T^{10} + T^{12}$$
$73$ $$( 1441081768 + 46683116 T - 18273598 T^{2} - 1101111 T^{3} - 11631 T^{4} + 79 T^{5} + T^{6} )^{2}$$
$79$ $$49\!\cdots\!36$$$$+ 13408795741543137280 T^{2} + 10968066337497088 T^{4} + 3641501079296 T^{6} + 557919040 T^{8} + 38988 T^{10} + T^{12}$$
$83$ $$42831830207741034496 + 270269256104148992 T^{2} + 633481522282496 T^{4} + 663377369088 T^{6} + 285367104 T^{8} + 31516 T^{10} + T^{12}$$
$89$ $$( 307161915264 + 6872208480 T - 299866608 T^{2} - 6215184 T^{3} - 15936 T^{4} + 222 T^{5} + T^{6} )^{2}$$
$97$ $$( 100074188608 - 5675717376 T + 29639920 T^{2} + 1943872 T^{3} - 14132 T^{4} - 160 T^{5} + T^{6} )^{2}$$