↑

Properties

Label 27.2.e.a.7.2
Level $27$
Weight $2$
Character 27.7
Analytic conductor $0.216$
Analytic rank $0$
Dimension $12$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,2,Mod(4,27)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("27.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 27.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.215596085457\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: 12.0.1952986685049.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 27 x^{10} - 80 x^{9} + 186 x^{8} - 330 x^{7} + 463 x^{6} - 504 x^{5} + 420 x^{4} + \cdots + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 7.2
Root \(0.500000 + 0.0126039i\) of defining polynomial
Character \(\chi\) \(=\) 27.7
Dual form 27.2.e.a.4.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.753189 - 0.274138i) q^{2} +(-1.68842 + 0.386327i) q^{3} +(-1.03995 + 0.872619i) q^{4} +(-0.477505 - 2.70806i) q^{5} +(-1.16579 + 0.753837i) q^{6} +(1.82076 + 1.52780i) q^{7} +(-1.34559 + 2.33062i) q^{8} +(2.70150 - 1.30456i) q^{9} +(-1.10204 - 1.90878i) q^{10} +(-0.0434396 + 0.246358i) q^{11} +(1.41875 - 1.87510i) q^{12} +(-2.45446 - 0.893351i) q^{13} +(1.79020 + 0.651581i) q^{14} +(1.85243 + 4.38787i) q^{15} +(0.0969067 - 0.549585i) q^{16} +(0.146688 + 0.254072i) q^{17} +(1.67711 - 1.72317i) q^{18} +(1.39237 - 2.41166i) q^{19} +(2.85969 + 2.39956i) q^{20} +(-3.66443 - 1.87615i) q^{21} +(0.0348180 + 0.197463i) q^{22} +(-5.12472 + 4.30015i) q^{23} +(1.37153 - 4.45490i) q^{24} +(-2.40714 + 0.876128i) q^{25} -2.09357 q^{26} +(-4.05728 + 3.24631i) q^{27} -3.22668 q^{28} +(0.333645 - 0.121437i) q^{29} +(2.59811 + 2.79707i) q^{30} +(2.11847 - 1.77761i) q^{31} +(-1.01231 - 5.74108i) q^{32} +(-0.0218307 - 0.432738i) q^{33} +(0.180135 + 0.151151i) q^{34} +(3.26796 - 5.66027i) q^{35} +(-1.67103 + 3.71406i) q^{36} +(3.49619 + 6.05558i) q^{37} +(0.387591 - 2.19814i) q^{38} +(4.48928 + 0.560124i) q^{39} +(6.95400 + 2.53105i) q^{40} +(9.13156 + 3.32362i) q^{41} +(-3.27434 - 0.408537i) q^{42} +(0.0452712 - 0.256746i) q^{43} +(-0.169802 - 0.294106i) q^{44} +(-4.82282 - 6.69291i) q^{45} +(-2.68104 + 4.64370i) q^{46} +(-8.75249 - 7.34421i) q^{47} +(0.0487006 + 0.965367i) q^{48} +(-0.234540 - 1.33014i) q^{49} +(-1.57285 + 1.31978i) q^{50} +(-0.345826 - 0.372309i) q^{51} +(3.33207 - 1.21277i) q^{52} +5.43137 q^{53} +(-2.16596 + 3.55734i) q^{54} +0.687897 q^{55} +(-6.01071 + 2.18772i) q^{56} +(-1.41922 + 4.60980i) q^{57} +(0.218007 - 0.182930i) q^{58} +(1.03788 + 5.88612i) q^{59} +(-5.75536 - 2.94669i) q^{60} +(-9.07515 - 7.61495i) q^{61} +(1.10830 - 1.91963i) q^{62} +(6.91190 + 1.75206i) q^{63} +(-1.77824 - 3.08001i) q^{64} +(-1.24723 + 7.07342i) q^{65} +(-0.135073 - 0.319949i) q^{66} +(-1.70113 - 0.619160i) q^{67} +(-0.374256 - 0.136218i) q^{68} +(6.99139 - 9.24026i) q^{69} +(0.909693 - 5.15912i) q^{70} +(0.185255 + 0.320871i) q^{71} +(-0.594663 + 8.05158i) q^{72} +(-2.51339 + 4.35333i) q^{73} +(4.29336 + 3.60255i) q^{74} +(3.72579 - 2.40921i) q^{75} +(0.656467 + 3.72301i) q^{76} +(-0.455479 + 0.382193i) q^{77} +(3.53483 - 0.808804i) q^{78} +(-0.754406 + 0.274581i) q^{79} -1.53459 q^{80} +(5.59624 - 7.04855i) q^{81} +7.78892 q^{82} +(2.58947 - 0.942488i) q^{83} +(5.44798 - 1.24655i) q^{84} +(0.617998 - 0.518562i) q^{85} +(-0.0362861 - 0.205789i) q^{86} +(-0.516417 + 0.333932i) q^{87} +(-0.515717 - 0.432738i) q^{88} +(-5.22533 + 9.05054i) q^{89} +(-5.46728 - 3.71891i) q^{90} +(-3.10412 - 5.37650i) q^{91} +(1.57704 - 8.94385i) q^{92} +(-2.89012 + 3.81976i) q^{93} +(-8.60560 - 3.13218i) q^{94} +(-7.19580 - 2.61906i) q^{95} +(3.92713 + 9.30225i) q^{96} +(-2.57600 + 14.6092i) q^{97} +(-0.541296 - 0.937552i) q^{98} +(0.204037 + 0.722208i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 3 q^{5} - 6 q^{7} + 6 q^{8} - 3 q^{10} + 3 q^{11} + 12 q^{12} - 6 q^{13} + 15 q^{14} + 9 q^{15} + 9 q^{17} + 9 q^{18} - 3 q^{19} - 3 q^{20} - 12 q^{21} + 3 q^{22} - 12 q^{23}+ \cdots - 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.753189 − 0.274138i 0.532585 − 0.193845i −0.0617072 − 0.998094i \(-0.519654\pi\)
0.594292 + 0.804249i \(0.297432\pi\)
\(3\) −1.68842 + 0.386327i −0.974808 + 0.223046i
\(4\) −1.03995 + 0.872619i −0.519974 + 0.436310i
\(5\) −0.477505 − 2.70806i −0.213547 − 1.21108i −0.883411 − 0.468600i \(-0.844759\pi\)
0.669864 − 0.742484i \(-0.266352\pi\)
\(6\) −1.16579 + 0.753837i −0.475932 + 0.307753i
\(7\) 1.82076 + 1.52780i 0.688183 + 0.577454i 0.918385 − 0.395689i \(-0.129494\pi\)
−0.230202 + 0.973143i \(0.573939\pi\)
\(8\) −1.34559 + 2.33062i −0.475736 + 0.823999i
\(9\) 2.70150 − 1.30456i 0.900501 − 0.434854i
\(10\) −1.10204 − 1.90878i −0.348494 − 0.603610i
\(11\) −0.0434396 + 0.246358i −0.0130975 + 0.0742798i −0.990656 − 0.136385i \(-0.956452\pi\)
0.977558 + 0.210665i \(0.0675628\pi\)
\(12\) 1.41875 − 1.87510i 0.409557 − 0.541296i
\(13\) −2.45446 − 0.893351i −0.680745 − 0.247771i −0.0215777 − 0.999767i \(-0.506869\pi\)
−0.659167 + 0.751996i \(0.729091\pi\)
\(14\) 1.79020 + 0.651581i 0.478452 + 0.174142i
\(15\) 1.85243 + 4.38787i 0.478294 + 1.13294i
\(16\) 0.0969067 − 0.549585i 0.0242267 − 0.137396i
\(17\) 0.146688 + 0.254072i 0.0355772 + 0.0616215i 0.883266 − 0.468873i \(-0.155340\pi\)
−0.847689 + 0.530494i \(0.822006\pi\)
\(18\) 1.67711 − 1.72317i 0.395299 − 0.406154i
\(19\) 1.39237 − 2.41166i 0.319432 − 0.553273i −0.660937 − 0.750441i \(-0.729841\pi\)
0.980370 + 0.197168i \(0.0631745\pi\)
\(20\) 2.85969 + 2.39956i 0.639446 + 0.536559i
\(21\) −3.66443 − 1.87615i −0.799645 − 0.409410i
\(22\) 0.0348180 + 0.197463i 0.00742323 + 0.0420992i
\(23\) −5.12472 + 4.30015i −1.06858 + 0.896643i −0.994923 − 0.100641i \(-0.967911\pi\)
−0.0736543 + 0.997284i \(0.523466\pi\)
\(24\) 1.37153 − 4.45490i 0.279962 − 0.909352i
\(25\) −2.40714 + 0.876128i −0.481428 + 0.175226i
\(26\) −2.09357 −0.410584
\(27\) −4.05728 + 3.24631i −0.780823 + 0.624752i
\(28\) −3.22668 −0.609786
\(29\) 0.333645 − 0.121437i 0.0619562 − 0.0225502i −0.310856 − 0.950457i \(-0.600616\pi\)
0.372812 + 0.927907i \(0.378394\pi\)
\(30\) 2.59811 + 2.79707i 0.474348 + 0.510673i
\(31\) 2.11847 − 1.77761i 0.380488 − 0.319268i −0.432406 − 0.901679i \(-0.642335\pi\)
0.812894 + 0.582411i \(0.197891\pi\)
\(32\) −1.01231 − 5.74108i −0.178952 − 1.01489i
\(33\) −0.0218307 − 0.432738i −0.00380023 − 0.0753299i
\(34\) 0.180135 + 0.151151i 0.0308929 + 0.0259222i
\(35\) 3.26796 − 5.66027i 0.552386 − 0.956760i
\(36\) −1.67103 + 3.71406i −0.278506 + 0.619010i
\(37\) 3.49619 + 6.05558i 0.574770 + 0.995531i 0.996067 + 0.0886080i \(0.0282418\pi\)
−0.421297 + 0.906923i \(0.638425\pi\)
\(38\) 0.387591 − 2.19814i 0.0628756 − 0.356585i
\(39\) 4.48928 + 0.560124i 0.718860 + 0.0896917i
\(40\) 6.95400 + 2.53105i 1.09952 + 0.400194i
\(41\) 9.13156 + 3.32362i 1.42611 + 0.519062i 0.935814 − 0.352494i \(-0.114666\pi\)
0.490296 + 0.871556i \(0.336888\pi\)
\(42\) −3.27434 − 0.408537i −0.505241 − 0.0630386i
\(43\) 0.0452712 − 0.256746i 0.00690379 − 0.0391534i −0.981161 − 0.193191i \(-0.938116\pi\)
0.988065 + 0.154037i \(0.0492276\pi\)
\(44\) −0.169802 − 0.294106i −0.0255986 − 0.0443381i
\(45\) −4.82282 − 6.69291i −0.718943 − 0.997720i
\(46\) −2.68104 + 4.64370i −0.395298 + 0.684677i
\(47\) −8.75249 − 7.34421i −1.27668 − 1.07126i −0.993692 − 0.112140i \(-0.964230\pi\)
−0.282989 − 0.959123i \(-0.591326\pi\)
\(48\) 0.0487006 + 0.965367i 0.00702933 + 0.139339i
\(49\) −0.234540 − 1.33014i −0.0335057 − 0.190020i
\(50\) −1.57285 + 1.31978i −0.222435 + 0.186645i
\(51\) −0.345826 − 0.372309i −0.0484253 − 0.0521337i
\(52\) 3.33207 − 1.21277i 0.462074 − 0.168181i
\(53\) 5.43137 0.746056 0.373028 − 0.927820i \(-0.378320\pi\)
0.373028 + 0.927820i \(0.378320\pi\)
\(54\) −2.16596 + 3.55734i −0.294750 + 0.484092i
\(55\) 0.687897 0.0927560
\(56\) −6.01071 + 2.18772i −0.803215 + 0.292346i
\(57\) −1.41922 + 4.60980i −0.187980 + 0.610583i
\(58\) 0.218007 − 0.182930i 0.0286257 − 0.0240198i
\(59\) 1.03788 + 5.88612i 0.135121 + 0.766308i 0.974776 + 0.223188i \(0.0716462\pi\)
−0.839655 + 0.543121i \(0.817243\pi\)
\(60\) −5.75536 − 2.94669i −0.743014 − 0.380416i
\(61\) −9.07515 − 7.61495i −1.16195 − 0.974995i −0.162023 − 0.986787i \(-0.551802\pi\)
−0.999930 + 0.0117924i \(0.996246\pi\)
\(62\) 1.10830 − 1.91963i 0.140754 − 0.243793i
\(63\) 6.91190 + 1.75206i 0.870817 + 0.220739i
\(64\) −1.77824 − 3.08001i −0.222281 − 0.385001i
\(65\) −1.24723 + 7.07342i −0.154700 + 0.877350i
\(66\) −0.135073 − 0.319949i −0.0166263 − 0.0393829i
\(67\) −1.70113 − 0.619160i −0.207826 − 0.0756424i 0.236010 − 0.971751i \(-0.424160\pi\)
−0.443835 + 0.896108i \(0.646383\pi\)
\(68\) −0.374256 − 0.136218i −0.0453852 − 0.0165189i
\(69\) 6.99139 − 9.24026i 0.841665 − 1.11240i
\(70\) 0.909693 − 5.15912i 0.108729 − 0.616633i
\(71\) 0.185255 + 0.320871i 0.0219857 + 0.0380804i 0.876809 − 0.480839i \(-0.159668\pi\)
−0.854823 + 0.518919i \(0.826334\pi\)
\(72\) −0.594663 + 8.05158i −0.0700817 + 0.948888i
\(73\) −2.51339 + 4.35333i −0.294171 + 0.509518i −0.974792 − 0.223117i \(-0.928377\pi\)
0.680621 + 0.732636i \(0.261710\pi\)
\(74\) 4.29336 + 3.60255i 0.499093 + 0.418789i
\(75\) 3.72579 − 2.40921i 0.430217 − 0.278192i
\(76\) 0.656467 + 3.72301i 0.0753019 + 0.427059i
\(77\) −0.455479 + 0.382193i −0.0519067 + 0.0435549i
\(78\) 3.53483 − 0.808804i 0.400240 − 0.0915790i
\(79\) −0.754406 + 0.274581i −0.0848773 + 0.0308928i −0.384110 − 0.923287i \(-0.625492\pi\)
0.299233 + 0.954180i \(0.403269\pi\)
\(80\) −1.53459 −0.171572
\(81\) 5.59624 − 7.04855i 0.621804 − 0.783173i
\(82\) 7.78892 0.860143
\(83\) 2.58947 − 0.942488i 0.284231 − 0.103452i −0.195971 − 0.980610i \(-0.562786\pi\)
0.480201 + 0.877158i \(0.340564\pi\)
\(84\) 5.44798 − 1.24655i 0.594424 − 0.136010i
\(85\) 0.617998 − 0.518562i 0.0670313 − 0.0562460i
\(86\) −0.0362861 − 0.205789i −0.00391283 − 0.0221908i
\(87\) −0.516417 + 0.333932i −0.0553657 + 0.0358012i
\(88\) −0.515717 − 0.432738i −0.0549756 − 0.0461300i
\(89\) −5.22533 + 9.05054i −0.553884 + 0.959356i 0.444105 + 0.895975i \(0.353522\pi\)
−0.997989 + 0.0633809i \(0.979812\pi\)
\(90\) −5.46728 − 3.71891i −0.576302 − 0.392007i
\(91\) −3.10412 − 5.37650i −0.325401 − 0.563611i
\(92\) 1.57704 − 8.94385i 0.164418 − 0.932461i
\(93\) −2.89012 + 3.81976i −0.299692 + 0.396091i
\(94\) −8.60560 − 3.13218i −0.887600 − 0.323060i
\(95\) −7.19580 − 2.61906i −0.738273 − 0.268709i
\(96\) 3.92713 + 9.30225i 0.400811 + 0.949407i
\(97\) −2.57600 + 14.6092i −0.261553 + 1.48334i 0.517120 + 0.855913i \(0.327004\pi\)
−0.778673 + 0.627430i \(0.784107\pi\)
\(98\) −0.541296 − 0.937552i −0.0546791 − 0.0947070i
\(99\) 0.204037 + 0.722208i 0.0205065 + 0.0725846i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 27.2.e.a.7.2 yes 12
3.2 odd 2 81.2.e.a.19.1 12
4.3 odd 2 432.2.u.c.385.2 12
5.2 odd 4 675.2.u.b.574.3 24
5.3 odd 4 675.2.u.b.574.2 24
5.4 even 2 675.2.l.c.601.1 12
9.2 odd 6 243.2.e.a.136.2 12
9.4 even 3 243.2.e.c.217.1 12
9.5 odd 6 243.2.e.b.217.2 12
9.7 even 3 243.2.e.d.136.1 12
27.2 odd 18 729.2.a.d.1.4 6
27.4 even 9 inner 27.2.e.a.4.2 ✓ 12
27.5 odd 18 243.2.e.b.28.2 12
27.7 even 9 729.2.c.e.487.4 12
27.11 odd 18 729.2.c.b.244.3 12
27.13 even 9 243.2.e.d.109.1 12
27.14 odd 18 243.2.e.a.109.2 12
27.16 even 9 729.2.c.e.244.4 12
27.20 odd 18 729.2.c.b.487.3 12
27.22 even 9 243.2.e.c.28.1 12
27.23 odd 18 81.2.e.a.64.1 12
27.25 even 9 729.2.a.a.1.3 6
108.31 odd 18 432.2.u.c.193.2 12
135.4 even 18 675.2.l.c.301.1 12
135.58 odd 36 675.2.u.b.274.3 24
135.112 odd 36 675.2.u.b.274.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.4.2 ✓ 12 27.4 even 9 inner
27.2.e.a.7.2 yes 12 1.1 even 1 trivial
81.2.e.a.19.1 12 3.2 odd 2
81.2.e.a.64.1 12 27.23 odd 18
243.2.e.a.109.2 12 27.14 odd 18
243.2.e.a.136.2 12 9.2 odd 6
243.2.e.b.28.2 12 27.5 odd 18
243.2.e.b.217.2 12 9.5 odd 6
243.2.e.c.28.1 12 27.22 even 9
243.2.e.c.217.1 12 9.4 even 3
243.2.e.d.109.1 12 27.13 even 9
243.2.e.d.136.1 12 9.7 even 3
432.2.u.c.193.2 12 108.31 odd 18
432.2.u.c.385.2 12 4.3 odd 2
675.2.l.c.301.1 12 135.4 even 18
675.2.l.c.601.1 12 5.4 even 2
675.2.u.b.274.2 24 135.112 odd 36
675.2.u.b.274.3 24 135.58 odd 36
675.2.u.b.574.2 24 5.3 odd 4
675.2.u.b.574.3 24 5.2 odd 4
729.2.a.a.1.3 6 27.25 even 9
729.2.a.d.1.4 6 27.2 odd 18
729.2.c.b.244.3 12 27.11 odd 18
729.2.c.b.487.3 12 27.20 odd 18
729.2.c.e.244.4 12 27.16 even 9
729.2.c.e.487.4 12 27.7 even 9