Properties

 Label 27.9.b.d Level 27 Weight 9 Character orbit 27.b Analytic conductor 10.999 Analytic rank 0 Dimension 6 CM No Inner twists 2

Related objects

Newspace parameters

 Level: $$N$$ = $$27 = 3^{3}$$ Weight: $$k$$ = $$9$$ Character orbit: $$[\chi]$$ = 27.b (of order $$2$$ and degree $$1$$)

Newform invariants

 Self dual: No Analytic conductor: $$10.9992224717$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.6171673600.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}\cdot 3^{21}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( -131 + \beta_{2} ) q^{4}$$ $$+ ( 20 \beta_{1} - \beta_{4} ) q^{5}$$ $$+ ( -283 - 2 \beta_{2} - \beta_{3} ) q^{7}$$ $$+ ( -161 \beta_{1} + 7 \beta_{4} - \beta_{5} ) q^{8}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( -131 + \beta_{2} ) q^{4}$$ $$+ ( 20 \beta_{1} - \beta_{4} ) q^{5}$$ $$+ ( -283 - 2 \beta_{2} - \beta_{3} ) q^{7}$$ $$+ ( -161 \beta_{1} + 7 \beta_{4} - \beta_{5} ) q^{8}$$ $$+ ( -7713 + 53 \beta_{2} - \beta_{3} ) q^{10}$$ $$+ ( -280 \beta_{1} - 14 \beta_{4} - 5 \beta_{5} ) q^{11}$$ $$+ ( 6974 - 44 \beta_{2} - 2 \beta_{3} ) q^{13}$$ $$+ ( 165 \beta_{1} + 4 \beta_{4} - 20 \beta_{5} ) q^{14}$$ $$+ ( 28411 - 177 \beta_{2} + 16 \beta_{3} ) q^{16}$$ $$+ ( 552 \beta_{1} - 28 \beta_{4} - 42 \beta_{5} ) q^{17}$$ $$+ ( -6064 - 60 \beta_{2} + 14 \beta_{3} ) q^{19}$$ $$+ ( -17875 \beta_{1} + 133 \beta_{4} - 75 \beta_{5} ) q^{20}$$ $$+ ( 107883 - 23 \beta_{2} + 31 \beta_{3} ) q^{22}$$ $$+ ( 13552 \beta_{1} - 2 \beta_{4} - 46 \beta_{5} ) q^{23}$$ $$+ ( -274448 + 1108 \beta_{2} - 106 \beta_{3} ) q^{25}$$ $$+ ( 19310 \beta_{1} - 272 \beta_{4} ) q^{26}$$ $$+ ( -139831 - 1299 \beta_{2} - 72 \beta_{3} ) q^{28}$$ $$+ ( -32280 \beta_{1} - 336 \beta_{4} + 230 \beta_{5} ) q^{29}$$ $$+ ( 343079 + 2470 \beta_{2} - 193 \beta_{3} ) q^{31}$$ $$+ ( 39801 \beta_{1} + 265 \beta_{4} + 273 \beta_{5} ) q^{32}$$ $$+ ( -220050 - 246 \beta_{2} + 350 \beta_{3} ) q^{34}$$ $$+ ( -7480 \beta_{1} + 2124 \beta_{4} + 685 \beta_{5} ) q^{35}$$ $$+ ( -1565980 + 2100 \beta_{2} + 110 \beta_{3} ) q^{37}$$ $$+ ( 12832 \beta_{1} - 672 \beta_{4} + 368 \beta_{5} ) q^{38}$$ $$+ ( 4926681 - 11771 \beta_{2} + 552 \beta_{3} ) q^{40}$$ $$+ ( 60400 \beta_{1} - 2634 \beta_{4} + 1010 \beta_{5} ) q^{41}$$ $$+ ( -456214 - 10556 \beta_{2} - 438 \beta_{3} ) q^{43}$$ $$+ ( 46625 \beta_{1} - 4303 \beta_{4} - 575 \beta_{5} ) q^{44}$$ $$+ ( -5252436 + 11732 \beta_{2} + 412 \beta_{3} ) q^{46}$$ $$+ ( 215920 \beta_{1} + 4796 \beta_{4} + 126 \beta_{5} ) q^{47}$$ $$+ ( 4816776 + 21508 \beta_{2} - 306 \beta_{3} ) q^{49}$$ $$+ ( -604480 \beta_{1} + 9664 \beta_{4} - 3440 \beta_{5} ) q^{50}$$ $$+ ( -5680282 + 17022 \beta_{2} - 784 \beta_{3} ) q^{52}$$ $$+ ( -255644 \beta_{1} - 3471 \beta_{4} - 614 \beta_{5} ) q^{53}$$ $$+ ( -4445757 - 28678 \beta_{2} - 1979 \beta_{3} ) q^{55}$$ $$+ ( 264995 \beta_{1} - 6773 \beta_{4} - 5405 \beta_{5} ) q^{56}$$ $$+ ( 12540762 - 11762 \beta_{2} - 2406 \beta_{3} ) q^{58}$$ $$+ ( 88400 \beta_{1} - 12604 \beta_{4} + 630 \beta_{5} ) q^{59}$$ $$+ ( -6696796 + 23440 \beta_{2} + 3624 \beta_{3} ) q^{61}$$ $$+ ( -387273 \beta_{1} + 20764 \beta_{4} - 6716 \beta_{5} ) q^{62}$$ $$+ ( -8090243 - 3063 \beta_{2} + 1904 \beta_{3} ) q^{64}$$ $$+ ( 746040 \beta_{1} - 3372 \beta_{4} + 4370 \beta_{5} ) q^{65}$$ $$+ ( 18559214 - 10604 \beta_{2} + 5738 \beta_{3} ) q^{67}$$ $$+ ( 35018 \beta_{1} - 15190 \beta_{4} - 2806 \beta_{5} ) q^{68}$$ $$+ ( 2954547 - 49487 \beta_{2} - 4041 \beta_{3} ) q^{70}$$ $$+ ( -201280 \beta_{1} - 1054 \beta_{4} + 12320 \beta_{5} ) q^{71}$$ $$+ ( 2136953 - 22428 \beta_{2} + 4926 \beta_{3} ) q^{73}$$ $$+ ( -2152940 \beta_{1} + 12720 \beta_{4} + 320 \beta_{5} ) q^{74}$$ $$+ ( -6437296 + 34736 \beta_{2} - 400 \beta_{3} ) q^{76}$$ $$+ ( 883380 \beta_{1} + 68391 \beta_{4} + 15640 \beta_{5} ) q^{77}$$ $$+ ( 3636734 - 66848 \beta_{2} - 2468 \beta_{3} ) q^{79}$$ $$+ ( 3785635 \beta_{1} - 58285 \beta_{4} + 4715 \beta_{5} ) q^{80}$$ $$+ ( -23130972 + 188732 \beta_{2} - 11724 \beta_{3} ) q^{82}$$ $$+ ( -257704 \beta_{1} - 16014 \beta_{4} + 21301 \beta_{5} ) q^{83}$$ $$+ ( -13974066 - 91284 \beta_{2} - 11822 \beta_{3} ) q^{85}$$ $$+ ( 2508490 \beta_{1} - 66008 \beta_{4} + 920 \beta_{5} ) q^{86}$$ $$+ ( 9592029 + 159161 \beta_{2} + 8808 \beta_{3} ) q^{88}$$ $$+ ( -1376920 \beta_{1} + 4606 \beta_{4} + 7400 \beta_{5} ) q^{89}$$ $$+ ( 26105494 + 90376 \beta_{2} - 32 \beta_{3} ) q^{91}$$ $$+ ( -5087388 \beta_{1} + 74196 \beta_{4} - 14444 \beta_{5} ) q^{92}$$ $$+ ( -83668986 + 62818 \beta_{2} + 3662 \beta_{3} ) q^{94}$$ $$+ ( 1246640 \beta_{1} - 19886 \beta_{4} - 2990 \beta_{5} ) q^{95}$$ $$+ ( 8955851 - 36656 \beta_{2} - 4008 \beta_{3} ) q^{97}$$ $$+ ( -1372456 \beta_{1} + 156064 \beta_{4} - 28240 \beta_{5} ) q^{98}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q$$ $$\mathstrut -\mathstrut 786q^{4}$$ $$\mathstrut -\mathstrut 1698q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$6q$$ $$\mathstrut -\mathstrut 786q^{4}$$ $$\mathstrut -\mathstrut 1698q^{7}$$ $$\mathstrut -\mathstrut 46278q^{10}$$ $$\mathstrut +\mathstrut 41844q^{13}$$ $$\mathstrut +\mathstrut 170466q^{16}$$ $$\mathstrut -\mathstrut 36384q^{19}$$ $$\mathstrut +\mathstrut 647298q^{22}$$ $$\mathstrut -\mathstrut 1646688q^{25}$$ $$\mathstrut -\mathstrut 838986q^{28}$$ $$\mathstrut +\mathstrut 2058474q^{31}$$ $$\mathstrut -\mathstrut 1320300q^{34}$$ $$\mathstrut -\mathstrut 9395880q^{37}$$ $$\mathstrut +\mathstrut 29560086q^{40}$$ $$\mathstrut -\mathstrut 2737284q^{43}$$ $$\mathstrut -\mathstrut 31514616q^{46}$$ $$\mathstrut +\mathstrut 28900656q^{49}$$ $$\mathstrut -\mathstrut 34081692q^{52}$$ $$\mathstrut -\mathstrut 26674542q^{55}$$ $$\mathstrut +\mathstrut 75244572q^{58}$$ $$\mathstrut -\mathstrut 40180776q^{61}$$ $$\mathstrut -\mathstrut 48541458q^{64}$$ $$\mathstrut +\mathstrut 111355284q^{67}$$ $$\mathstrut +\mathstrut 17727282q^{70}$$ $$\mathstrut +\mathstrut 12821718q^{73}$$ $$\mathstrut -\mathstrut 38623776q^{76}$$ $$\mathstrut +\mathstrut 21820404q^{79}$$ $$\mathstrut -\mathstrut 138785832q^{82}$$ $$\mathstrut -\mathstrut 83844396q^{85}$$ $$\mathstrut +\mathstrut 57552174q^{88}$$ $$\mathstrut +\mathstrut 156632964q^{91}$$ $$\mathstrut -\mathstrut 502013916q^{94}$$ $$\mathstrut +\mathstrut 53735106q^{97}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6}\mathstrut -\mathstrut$$ $$2$$ $$x^{5}\mathstrut +\mathstrut$$ $$2$$ $$x^{4}\mathstrut +\mathstrut$$ $$40$$ $$x^{3}\mathstrut +\mathstrut$$ $$225$$ $$x^{2}\mathstrut +\mathstrut$$ $$150$$ $$x\mathstrut +\mathstrut$$ $$50$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$295 \nu^{5} - 669 \nu^{4} + 458 \nu^{3} + 13712 \nu^{2} + 58285 \nu + 20675$$$$)/1085$$ $$\beta_{2}$$ $$=$$ $$($$$$-792 \nu^{5} + 4239 \nu^{4} - 13464 \nu^{3} - 15840 \nu^{2} - 7920 \nu + 142110$$$$)/1085$$ $$\beta_{3}$$ $$=$$ $$($$$$5994 \nu^{5} - 50058 \nu^{4} + 101898 \nu^{3} + 119880 \nu^{2} + 59940 \nu - 4754700$$$$)/1085$$ $$\beta_{4}$$ $$=$$ $$($$$$1653 \nu^{5} - 3987 \nu^{4} + 6618 \nu^{3} + 62355 \nu^{2} + 377835 \nu + 133725$$$$)/217$$ $$\beta_{5}$$ $$=$$ $$($$$$-46063 \nu^{5} + 107679 \nu^{4} - 126212 \nu^{3} - 1708319 \nu^{2} - 9792715 \nu - 3469625$$$$)/1085$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5}\mathstrut +\mathstrut$$ $$9$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$18$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$68$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$972$$$$)/2916$$ $$\nu^{2}$$ $$=$$ $$($$$$10$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$27$$ $$\beta_{4}\mathstrut +\mathstrut$$ $$805$$ $$\beta_{1}$$$$)/2187$$ $$\nu^{3}$$ $$=$$ $$($$$$35$$ $$\beta_{5}\mathstrut +\mathstrut$$ $$171$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$23$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$306$$ $$\beta_{2}\mathstrut +\mathstrut$$ $$320$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$60264$$$$)/2916$$ $$\nu^{4}$$ $$=$$ $$($$$$-$$$$44$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$333$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$149202$$$$)/729$$ $$\nu^{5}$$ $$=$$ $$($$$$-$$$$2615$$ $$\beta_{5}\mathstrut -\mathstrut$$ $$11151$$ $$\beta_{4}\mathstrut -\mathstrut$$ $$1683$$ $$\beta_{3}\mathstrut -\mathstrut$$ $$18306$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$78680$$ $$\beta_{1}\mathstrut -\mathstrut$$ $$4968864$$$$)/8748$$

Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 −2.05620 + 2.05620i −0.356289 − 0.356289i 3.41249 − 3.41249i 3.41249 + 3.41249i −0.356289 + 0.356289i −2.05620 − 2.05620i
29.0422i 0 −587.450 1141.55i 0 −618.310 9626.03i 0 −33153.0
26.2 15.9629i 0 1.18662 230.735i 0 3842.93 4105.44i 0 3683.19
26.3 7.92067i 0 193.263 799.282i 0 −4073.62 3558.46i 0 6330.85
26.4 7.92067i 0 193.263 799.282i 0 −4073.62 3558.46i 0 6330.85
26.5 15.9629i 0 1.18662 230.735i 0 3842.93 4105.44i 0 3683.19
26.6 29.0422i 0 −587.450 1141.55i 0 −618.310 9626.03i 0 −33153.0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 26.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{2}^{6}$$ $$\mathstrut +\mathstrut 1161 T_{2}^{4}$$ $$\mathstrut +\mathstrut 283824 T_{2}^{2}$$ $$\mathstrut +\mathstrut 13483584$$ acting on $$S_{9}^{\mathrm{new}}(27, [\chi])$$.