Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,9,Mod(2,27)] 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("27.2"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 27.f (of order \(18\), 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: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 2.19
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.19

$q$-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+(21.0020 + 3.70323i) q^{2} +(80.9924 + 1.11237i) q^{3} +(186.811 + 67.9935i) q^{4} +(674.968 + 804.395i) q^{5} +(1696.89 + 323.295i) q^{6} +(-2905.52 + 1057.52i) q^{7} +(-1056.43 - 609.927i) q^{8} +(6558.53 + 180.188i) q^{9} +(11196.8 + 19393.5i) q^{10} +(10222.7 - 12182.9i) q^{11} +(15054.6 + 5714.76i) q^{12} +(-2471.50 - 14016.6i) q^{13} +(-64938.1 + 11450.3i) q^{14} +(53772.5 + 65900.7i) q^{15} +(-58914.5 - 49435.1i) q^{16} +(51541.5 - 29757.5i) q^{17} +(137075. + 28072.0i) q^{18} +(-57614.1 + 99790.5i) q^{19} +(71397.5 + 196163. i) q^{20} +(-236501. + 82419.3i) q^{21} +(259814. - 218010. i) q^{22} +(134153. - 368583. i) q^{23} +(-84883.9 - 50574.6i) q^{24} +(-123639. + 701192. i) q^{25} -303529. i q^{26} +(530990. + 21889.3i) q^{27} -614687. q^{28} +(537040. + 94694.6i) q^{29} +(885286. + 1.58318e6i) q^{30} +(-1.32290e6 - 481496. i) q^{31} +(-853524. - 1.01719e6i) q^{32} +(841512. - 975353. i) q^{33} +(1.19268e6 - 434098. i) q^{34} +(-2.81180e6 - 1.62339e6i) q^{35} +(1.21295e6 + 479598. i) q^{36} +(-288762. - 500150. i) q^{37} +(-1.57956e6 + 1.88245e6i) q^{38} +(-184581. - 1.13799e6i) q^{39} +(-222430. - 1.26146e6i) q^{40} +(-1.84569e6 + 325445. i) q^{41} +(-5.27223e6 + 855155. i) q^{42} +(-2.02555e6 - 1.69964e6i) q^{43} +(2.73807e6 - 1.58082e6i) q^{44} +(4.28185e6 + 5.39727e6i) q^{45} +(4.18244e6 - 7.24419e6i) q^{46} +(93440.3 + 256725. i) q^{47} +(-4.71663e6 - 4.06940e6i) q^{48} +(2.90760e6 - 2.43977e6i) q^{49} +(-5.19335e6 + 1.42686e7i) q^{50} +(4.20757e6 - 2.35280e6i) q^{51} +(491334. - 2.78649e6i) q^{52} +1.42090e7i q^{53} +(1.10708e7 + 2.42610e6i) q^{54} +1.66999e7 q^{55} +(3.71448e6 + 654963. i) q^{56} +(-4.77730e6 + 8.01818e6i) q^{57} +(1.09283e7 + 3.97756e6i) q^{58} +(-1.70695e6 - 2.03426e6i) q^{59} +(5.56445e6 + 1.59671e7i) q^{60} +(17548.6 - 6387.15i) q^{61} +(-2.60005e7 - 1.50114e7i) q^{62} +(-1.92465e7 + 6.41225e6i) q^{63} +(-4.31471e6 - 7.47329e6i) q^{64} +(9.60669e6 - 1.14488e7i) q^{65} +(2.12854e7 - 1.73681e7i) q^{66} +(5.50666e6 + 3.12298e7i) q^{67} +(1.16518e7 - 2.05453e6i) q^{68} +(1.12754e7 - 2.97032e7i) q^{69} +(-5.30418e7 - 4.45073e7i) q^{70} +(-3.18584e7 + 1.83935e7i) q^{71} +(-6.81869e6 - 4.19058e6i) q^{72} +(1.44422e7 - 2.50147e7i) q^{73} +(-4.21241e6 - 1.15735e7i) q^{74} +(-1.07938e7 + 5.66536e7i) q^{75} +(-1.75480e7 + 1.47245e7i) q^{76} +(-1.68185e7 + 4.62085e7i) q^{77} +(337638. - 2.45836e7i) q^{78} +(1.46028e6 - 8.28165e6i) q^{79} -8.07577e7i q^{80} +(4.29818e7 + 2.36353e6i) q^{81} -3.99685e7 q^{82} +(6.33917e7 + 1.11777e7i) q^{83} +(-4.97850e7 - 683762. i) q^{84} +(5.87257e7 + 2.13744e7i) q^{85} +(-3.62466e7 - 4.31970e7i) q^{86} +(4.33908e7 + 8.26693e6i) q^{87} +(-1.82302e7 + 6.63526e6i) q^{88} +(7.86470e7 + 4.54069e7i) q^{89} +(6.99403e7 + 1.29210e8i) q^{90} +(2.20039e7 + 3.81118e7i) q^{91} +(5.01225e7 - 5.97336e7i) q^{92} +(-1.06609e8 - 4.04691e7i) q^{93} +(1.01173e6 + 5.73778e6i) q^{94} +(-1.19159e8 + 2.10109e7i) q^{95} +(-6.79974e7 - 8.33340e7i) q^{96} +(-8.25735e7 - 6.92874e7i) q^{97} +(7.01007e7 - 4.04726e7i) q^{98} +(6.92410e7 - 7.80601e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 447 q^{5} - 774 q^{6} - 6 q^{7} - 9 q^{8} - 12960 q^{9} - 3 q^{10} + 28668 q^{11} + 77421 q^{12} - 6 q^{13} - 120975 q^{14} + 105507 q^{15} - 774 q^{16} - 9 q^{17}+ \cdots + 228876057 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.0020 + 3.70323i 1.31263 + 0.231452i 0.785779 − 0.618507i \(-0.212262\pi\)
0.526849 + 0.849959i \(0.323373\pi\)
\(3\) 80.9924 + 1.11237i 0.999906 + 0.0137330i
\(4\) 186.811 + 67.9935i 0.729729 + 0.265600i
\(5\) 674.968 + 804.395i 1.07995 + 1.28703i 0.955559 + 0.294799i \(0.0952528\pi\)
0.124390 + 0.992233i \(0.460303\pi\)
\(6\) 1696.89 + 323.295i 1.30933 + 0.249456i
\(7\) −2905.52 + 1057.52i −1.21013 + 0.440451i −0.866749 − 0.498745i \(-0.833794\pi\)
−0.343381 + 0.939196i \(0.611572\pi\)
\(8\) −1056.43 − 609.927i −0.257916 − 0.148908i
\(9\) 6558.53 + 180.188i 0.999623 + 0.0274634i
\(10\) 11196.8 + 19393.5i 1.11968 + 1.93935i
\(11\) 10222.7 − 12182.9i 0.698224 − 0.832111i −0.294100 − 0.955775i \(-0.595020\pi\)
0.992324 + 0.123664i \(0.0394644\pi\)
\(12\) 15054.6 + 5714.76i 0.726013 + 0.275596i
\(13\) −2471.50 − 14016.6i −0.0865341 − 0.490759i −0.997015 − 0.0772082i \(-0.975399\pi\)
0.910481 − 0.413551i \(-0.135712\pi\)
\(14\) −64938.1 + 11450.3i −1.69039 + 0.298062i
\(15\) 53772.5 + 65900.7i 1.06217 + 1.30174i
\(16\) −58914.5 − 49435.1i −0.898964 − 0.754320i
\(17\) 51541.5 − 29757.5i 0.617108 − 0.356288i −0.158634 − 0.987337i \(-0.550709\pi\)
0.775742 + 0.631050i \(0.217376\pi\)
\(18\) 137075. + 28072.0i 1.30578 + 0.267414i
\(19\) −57614.1 + 99790.5i −0.442094 + 0.765728i −0.997845 − 0.0656202i \(-0.979097\pi\)
0.555751 + 0.831349i \(0.312431\pi\)
\(20\) 71397.5 + 196163.i 0.446235 + 1.22602i
\(21\) −236501. + 82419.3i −1.21606 + 0.423791i
\(22\) 259814. − 218010.i 1.10910 − 0.930647i
\(23\) 134153. − 368583.i 0.479391 − 1.31711i −0.430621 − 0.902533i \(-0.641706\pi\)
0.910012 − 0.414582i \(-0.136072\pi\)
\(24\) −84883.9 − 50574.6i −0.255847 − 0.152436i
\(25\) −123639. + 701192.i −0.316516 + 1.79505i
\(26\) − 303529.i − 0.664213i
\(27\) 530990. + 21889.3i 0.999151 + 0.0411887i
\(28\) −614687. −1.00005
\(29\) 537040. + 94694.6i 0.759302 + 0.133885i 0.539877 − 0.841744i \(-0.318471\pi\)
0.219425 + 0.975629i \(0.429582\pi\)
\(30\) 885286. + 1.58318e6i 1.09295 + 1.95454i
\(31\) −1.32290e6 − 481496.i −1.43245 − 0.521370i −0.494819 − 0.868996i \(-0.664766\pi\)
−0.937633 + 0.347626i \(0.886988\pi\)
\(32\) −853524. − 1.01719e6i −0.813984 − 0.970068i
\(33\) 841512. − 975353.i 0.709585 − 0.822444i
\(34\) 1.19268e6 − 434098.i 0.892497 − 0.324842i
\(35\) −2.81180e6 − 1.62339e6i −1.87375 − 1.08181i
\(36\) 1.21295e6 + 479598.i 0.722160 + 0.285540i
\(37\) −288762. − 500150.i −0.154075 − 0.266866i 0.778647 − 0.627463i \(-0.215906\pi\)
−0.932722 + 0.360597i \(0.882573\pi\)
\(38\) −1.57956e6 + 1.88245e6i −0.757533 + 0.902793i
\(39\) −184581. − 1.13799e6i −0.0797864 − 0.491901i
\(40\) −222430. − 1.26146e6i −0.0868868 − 0.492760i
\(41\) −1.84569e6 + 325445.i −0.653167 + 0.115171i −0.490405 − 0.871495i \(-0.663151\pi\)
−0.162761 + 0.986665i \(0.552040\pi\)
\(42\) −5.27223e6 + 855155.i −1.69433 + 0.274820i
\(43\) −2.02555e6 − 1.69964e6i −0.592475 − 0.497145i 0.296542 − 0.955020i \(-0.404166\pi\)
−0.889017 + 0.457874i \(0.848611\pi\)
\(44\) 2.73807e6 − 1.58082e6i 0.730523 − 0.421767i
\(45\) 4.28185e6 + 5.39727e6i 1.04420 + 1.31621i
\(46\) 4.18244e6 − 7.24419e6i 0.934110 − 1.61793i
\(47\) 93440.3 + 256725.i 0.0191488 + 0.0526110i 0.948899 − 0.315581i \(-0.102199\pi\)
−0.929750 + 0.368192i \(0.879977\pi\)
\(48\) −4.71663e6 − 4.06940e6i −0.888520 − 0.766594i
\(49\) 2.90760e6 − 2.43977e6i 0.504372 − 0.423218i
\(50\) −5.19335e6 + 1.42686e7i −0.830935 + 2.28298i
\(51\) 4.20757e6 − 2.35280e6i 0.621943 − 0.347779i
\(52\) 491334. − 2.78649e6i 0.0671991 − 0.381105i
\(53\) 1.42090e7i 1.80077i 0.435092 + 0.900386i \(0.356716\pi\)
−0.435092 + 0.900386i \(0.643284\pi\)
\(54\) 1.10708e7 + 2.42610e6i 1.30198 + 0.285321i
\(55\) 1.66999e7 1.82500
\(56\) 3.71448e6 + 654963.i 0.377699 + 0.0665985i
\(57\) −4.77730e6 + 8.01818e6i −0.452568 + 0.759585i
\(58\) 1.09283e7 + 3.97756e6i 0.965693 + 0.351484i
\(59\) −1.70695e6 − 2.03426e6i −0.140868 − 0.167880i 0.690998 − 0.722857i \(-0.257171\pi\)
−0.831866 + 0.554977i \(0.812727\pi\)
\(60\) 5.56445e6 + 1.59671e7i 0.429356 + 1.23203i
\(61\) 17548.6 − 6387.15i 0.00126742 − 0.000461305i −0.341386 − 0.939923i \(-0.610896\pi\)
0.342654 + 0.939462i \(0.388674\pi\)
\(62\) −2.60005e7 − 1.50114e7i −1.75960 − 1.01591i
\(63\) −1.92465e7 + 6.41225e6i −1.22177 + 0.407051i
\(64\) −4.31471e6 − 7.47329e6i −0.257177 − 0.445443i
\(65\) 9.60669e6 − 1.14488e7i 0.538171 − 0.641367i
\(66\) 2.12854e7 − 1.73681e7i 1.12178 − 0.915328i
\(67\) 5.50666e6 + 3.12298e7i 0.273268 + 1.54978i 0.744410 + 0.667722i \(0.232731\pi\)
−0.471143 + 0.882057i \(0.656158\pi\)
\(68\) 1.16518e7 − 2.05453e6i 0.544952 − 0.0960897i
\(69\) 1.12754e7 − 2.97032e7i 0.497433 − 1.31041i
\(70\) −5.30418e7 − 4.45073e7i −2.20915 − 1.85370i
\(71\) −3.18584e7 + 1.83935e7i −1.25369 + 0.723819i −0.971841 − 0.235639i \(-0.924282\pi\)
−0.281851 + 0.959458i \(0.590948\pi\)
\(72\) −6.81869e6 − 4.19058e6i −0.253729 − 0.155935i
\(73\) 1.44422e7 − 2.50147e7i 0.508561 − 0.880853i −0.491390 − 0.870939i \(-0.663511\pi\)
0.999951 − 0.00991319i \(-0.00315552\pi\)
\(74\) −4.21241e6 − 1.15735e7i −0.140477 − 0.385957i
\(75\) −1.07938e7 + 5.66536e7i −0.341138 + 1.79054i
\(76\) −1.75480e7 + 1.47245e7i −0.525986 + 0.441354i
\(77\) −1.68185e7 + 4.62085e7i −0.478437 + 1.31450i
\(78\) 337638. − 2.45836e7i 0.00912164 − 0.664150i
\(79\) 1.46028e6 − 8.28165e6i 0.0374910 − 0.212622i −0.960307 − 0.278944i \(-0.910015\pi\)
0.997798 + 0.0663224i \(0.0211266\pi\)
\(80\) − 8.07577e7i − 1.97162i
\(81\) 4.29818e7 + 2.36353e6i 0.998492 + 0.0549061i
\(82\) −3.99685e7 −0.884021
\(83\) 6.33917e7 + 1.11777e7i 1.33573 + 0.235526i 0.795482 − 0.605977i \(-0.207218\pi\)
0.540252 + 0.841503i \(0.318329\pi\)
\(84\) −4.97850e7 − 683762.i −0.999956 − 0.0137337i
\(85\) 5.87257e7 + 2.13744e7i 1.12500 + 0.409466i
\(86\) −3.62466e7 − 4.31970e7i −0.662634 − 0.789696i
\(87\) 4.33908e7 + 8.26693e6i 0.757392 + 0.144300i
\(88\) −1.82302e7 + 6.63526e6i −0.303991 + 0.110644i
\(89\) 7.86470e7 + 4.54069e7i 1.25349 + 0.723705i 0.971802 − 0.235799i \(-0.0757708\pi\)
0.281693 + 0.959505i \(0.409104\pi\)
\(90\) 6.99403e7 + 1.29210e8i 1.06600 + 1.96937i
\(91\) 2.20039e7 + 3.81118e7i 0.320873 + 0.555768i
\(92\) 5.01225e7 − 5.97336e7i 0.699650 − 0.833811i
\(93\) −1.06609e8 − 4.04691e7i −1.42516 − 0.540993i
\(94\) 1.01173e6 + 5.73778e6i 0.0129584 + 0.0734907i
\(95\) −1.19159e8 + 2.10109e7i −1.46296 + 0.257959i
\(96\) −6.79974e7 − 8.33340e7i −0.800585 − 0.981155i
\(97\) −8.25735e7 − 6.92874e7i −0.932725 − 0.782650i 0.0435792 − 0.999050i \(-0.486124\pi\)
−0.976305 + 0.216400i \(0.930568\pi\)
\(98\) 7.01007e7 − 4.04726e7i 0.760007 − 0.438791i
\(99\) 6.92410e7 − 7.80601e7i 0.720813 − 0.812621i
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 27.9.f.a.2.19 ✓ 138
3.2 odd 2 81.9.f.a.8.5 138
27.13 even 9 81.9.f.a.71.5 138
27.14 odd 18 inner 27.9.f.a.14.19 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.19 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.19 yes 138 27.14 odd 18 inner
81.9.f.a.8.5 138 3.2 odd 2
81.9.f.a.71.5 138 27.13 even 9