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 4.2
Root \(0.500000 - 0.0126039i\) of defining polynomial
Character \(\chi\) \(=\) 27.4
Dual form 27.2.e.a.7.2

$q$-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 \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{9}\right)\)

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 0.594292 − 0.804249i \(-0.297432\pi\)
−0.0617072 + 0.998094i \(0.519654\pi\)
\(3\) −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 0.669864 + 0.742484i \(0.266352\pi\)
−0.883411 + 0.468600i \(0.844759\pi\)
\(6\) −1.16579 − 0.753837i −0.475932 − 0.307753i
\(7\) 1.82076 − 1.52780i 0.688183 − 0.577454i −0.230202 − 0.973143i \(-0.573939\pi\)
0.918385 + 0.395689i \(0.129494\pi\)
\(8\) −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 0.977558 − 0.210665i \(-0.0675628\pi\)
−0.990656 + 0.136385i \(0.956452\pi\)
\(12\) 1.41875 + 1.87510i 0.409557 + 0.541296i
\(13\) −2.45446 + 0.893351i −0.680745 + 0.247771i −0.659167 − 0.751996i \(-0.729091\pi\)
−0.0215777 + 0.999767i \(0.506869\pi\)
\(14\) 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 −0.847689 − 0.530494i \(-0.822006\pi\)
0.883266 + 0.468873i \(0.155340\pi\)
\(18\) 1.67711 + 1.72317i 0.395299 + 0.406154i
\(19\) 1.39237 + 2.41166i 0.319432 + 0.553273i 0.980370 − 0.197168i \(-0.0631745\pi\)
−0.660937 + 0.750441i \(0.729841\pi\)
\(20\) 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 −0.0736543 − 0.997284i \(-0.523466\pi\)
−0.994923 + 0.100641i \(0.967911\pi\)
\(24\) 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 0.372812 − 0.927907i \(-0.378394\pi\)
−0.310856 + 0.950457i \(0.600616\pi\)
\(30\) 2.59811 − 2.79707i 0.474348 − 0.510673i
\(31\) 2.11847 + 1.77761i 0.380488 + 0.319268i 0.812894 − 0.582411i \(-0.197891\pi\)
−0.432406 + 0.901679i \(0.642335\pi\)
\(32\) −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 −0.421297 − 0.906923i \(-0.638425\pi\)
0.996067 − 0.0886080i \(-0.0282418\pi\)
\(38\) 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 0.490296 − 0.871556i \(-0.336888\pi\)
0.935814 + 0.352494i \(0.114666\pi\)
\(42\) −3.27434 + 0.408537i −0.505241 + 0.0630386i
\(43\) 0.0452712 + 0.256746i 0.00690379 + 0.0391534i 0.988065 − 0.154037i \(-0.0492276\pi\)
−0.981161 + 0.193191i \(0.938116\pi\)
\(44\) −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 −0.282989 + 0.959123i \(0.591326\pi\)
−0.993692 + 0.112140i \(0.964230\pi\)
\(48\) 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 −0.839655 − 0.543121i \(-0.817243\pi\)
0.974776 − 0.223188i \(-0.0716462\pi\)
\(60\) −5.75536 + 2.94669i −0.743014 + 0.380416i
\(61\) −9.07515 + 7.61495i −1.16195 + 0.974995i −0.999930 − 0.0117924i \(-0.996246\pi\)
−0.162023 + 0.986787i \(0.551802\pi\)
\(62\) 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 −0.443835 − 0.896108i \(-0.646383\pi\)
0.236010 + 0.971751i \(0.424160\pi\)
\(68\) −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 −0.854823 − 0.518919i \(-0.826334\pi\)
0.876809 + 0.480839i \(0.159668\pi\)
\(72\) −0.594663 − 8.05158i −0.0700817 − 0.948888i
\(73\) −2.51339 − 4.35333i −0.294171 − 0.509518i 0.680621 − 0.732636i \(-0.261710\pi\)
−0.974792 + 0.223117i \(0.928377\pi\)
\(74\) 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 0.299233 − 0.954180i \(-0.403269\pi\)
−0.384110 + 0.923287i \(0.625492\pi\)
\(80\) −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 0.480201 − 0.877158i \(-0.340564\pi\)
−0.195971 + 0.980610i \(0.562786\pi\)
\(84\) 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 −0.997989 − 0.0633809i \(-0.979812\pi\)
0.444105 − 0.895975i \(-0.353522\pi\)
\(90\) −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 −0.778673 − 0.627430i \(-0.784107\pi\)
0.517120 − 0.855913i \(-0.327004\pi\)
\(98\) −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.4.2 ✓ 12
3.2 odd 2 81.2.e.a.64.1 12
4.3 odd 2 432.2.u.c.193.2 12
5.2 odd 4 675.2.u.b.274.2 24
5.3 odd 4 675.2.u.b.274.3 24
5.4 even 2 675.2.l.c.301.1 12
9.2 odd 6 243.2.e.b.28.2 12
9.4 even 3 243.2.e.d.109.1 12
9.5 odd 6 243.2.e.a.109.2 12
9.7 even 3 243.2.e.c.28.1 12
27.2 odd 18 243.2.e.a.136.2 12
27.4 even 9 729.2.c.e.244.4 12
27.5 odd 18 729.2.c.b.487.3 12
27.7 even 9 inner 27.2.e.a.7.2 yes 12
27.11 odd 18 243.2.e.b.217.2 12
27.13 even 9 729.2.a.a.1.3 6
27.14 odd 18 729.2.a.d.1.4 6
27.16 even 9 243.2.e.c.217.1 12
27.20 odd 18 81.2.e.a.19.1 12
27.22 even 9 729.2.c.e.487.4 12
27.23 odd 18 729.2.c.b.244.3 12
27.25 even 9 243.2.e.d.136.1 12
108.7 odd 18 432.2.u.c.385.2 12
135.7 odd 36 675.2.u.b.574.3 24
135.34 even 18 675.2.l.c.601.1 12
135.88 odd 36 675.2.u.b.574.2 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.2.e.a.4.2 ✓ 12 1.1 even 1 trivial
27.2.e.a.7.2 yes 12 27.7 even 9 inner
81.2.e.a.19.1 12 27.20 odd 18
81.2.e.a.64.1 12 3.2 odd 2
243.2.e.a.109.2 12 9.5 odd 6
243.2.e.a.136.2 12 27.2 odd 18
243.2.e.b.28.2 12 9.2 odd 6
243.2.e.b.217.2 12 27.11 odd 18
243.2.e.c.28.1 12 9.7 even 3
243.2.e.c.217.1 12 27.16 even 9
243.2.e.d.109.1 12 9.4 even 3
243.2.e.d.136.1 12 27.25 even 9
432.2.u.c.193.2 12 4.3 odd 2
432.2.u.c.385.2 12 108.7 odd 18
675.2.l.c.301.1 12 5.4 even 2
675.2.l.c.601.1 12 135.34 even 18
675.2.u.b.274.2 24 5.2 odd 4
675.2.u.b.274.3 24 5.3 odd 4
675.2.u.b.574.2 24 135.88 odd 36
675.2.u.b.574.3 24 135.7 odd 36
729.2.a.a.1.3 6 27.13 even 9
729.2.a.d.1.4 6 27.14 odd 18
729.2.c.b.244.3 12 27.23 odd 18
729.2.c.b.487.3 12 27.5 odd 18
729.2.c.e.244.4 12 27.4 even 9
729.2.c.e.487.4 12 27.22 even 9