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

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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:

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 \(S_{9}^{\mathrm{new}}(27, [\chi])\).