Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,8,Mod(10,27)] 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("27.10"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 27.c (of order \(3\), degree \(2\), not 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: \(8.43439568807\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{21} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 19.2
Root \(0.500000 - 9.08282i\) of defining polynomial
Character \(\chi\) \(=\) 27.19
Dual form 27.8.c.a.10.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-7.11595 + 12.3252i) q^{2} +(-37.2735 - 64.5595i) q^{4} +(145.304 + 251.673i) q^{5} +(-555.940 + 962.916i) q^{7} -760.739 q^{8} -4135.89 q^{10} +(2245.36 - 3889.07i) q^{11} +(-1218.29 - 2110.14i) q^{13} +(-7912.08 - 13704.1i) q^{14} +(10184.4 - 17639.9i) q^{16} -15905.4 q^{17} -49949.6 q^{19} +(10831.9 - 18761.5i) q^{20} +(31955.7 + 55348.9i) q^{22} +(34692.5 + 60089.2i) q^{23} +(-3163.73 + 5479.74i) q^{25} +34677.2 q^{26} +82887.2 q^{28} +(-47035.8 + 81468.4i) q^{29} +(9963.58 + 17257.4i) q^{31} +(96255.8 + 166720. i) q^{32} +(113182. - 196037. i) q^{34} -323120. q^{35} +331750. q^{37} +(355439. - 615638. i) q^{38} +(-110538. - 191457. i) q^{40} +(121133. + 209809. i) q^{41} +(-415713. + 720036. i) q^{43} -334769. q^{44} -987481. q^{46} +(-80005.3 + 138573. i) q^{47} +(-206366. - 357437. i) q^{49} +(-45025.8 - 77987.1i) q^{50} +(-90819.7 + 157304. i) q^{52} -311589. q^{53} +1.30503e6 q^{55} +(422925. - 732527. i) q^{56} +(-669408. - 1.15945e6i) q^{58} +(-156177. - 270506. i) q^{59} +(28723.9 - 49751.3i) q^{61} -283601. q^{62} -132603. q^{64} +(354044. - 613221. i) q^{65} +(2.05100e6 + 3.55243e6i) q^{67} +(592849. + 1.02684e6i) q^{68} +(2.29930e6 - 3.98251e6i) q^{70} +403110. q^{71} -823496. q^{73} +(-2.36072e6 + 4.08889e6i) q^{74} +(1.86180e6 + 3.22472e6i) q^{76} +(2.49656e6 + 4.32418e6i) q^{77} +(-489414. + 847689. i) q^{79} +5.91931e6 q^{80} -3.44791e6 q^{82} +(-1.85204e6 + 3.20782e6i) q^{83} +(-2.31111e6 - 4.00296e6i) q^{85} +(-5.91638e6 - 1.02475e7i) q^{86} +(-1.70813e6 + 2.95857e6i) q^{88} -2.09023e6 q^{89} +2.70918e6 q^{91} +(2.58622e6 - 4.47947e6i) q^{92} +(-1.13863e6 - 1.97216e6i) q^{94} +(-7.25786e6 - 1.25710e7i) q^{95} +(-1.75125e6 + 3.03325e6i) q^{97} +5.87396e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 9 q^{2} - 321 q^{4} + 180 q^{5} - 84 q^{7} - 5922 q^{8} + 252 q^{10} + 8460 q^{11} - 1848 q^{13} + 16272 q^{14} - 12417 q^{16} - 30564 q^{17} + 24432 q^{19} + 40788 q^{20} - 35001 q^{22} + 51588 q^{23}+ \cdots + 95833314 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.11595 + 12.3252i −0.628967 + 1.08940i 0.358792 + 0.933417i \(0.383189\pi\)
−0.987759 + 0.155985i \(0.950145\pi\)
\(3\) 0 0
\(4\) −37.2735 − 64.5595i −0.291199 − 0.504371i
\(5\) 145.304 + 251.673i 0.519854 + 0.900413i 0.999734 + 0.0230788i \(0.00734688\pi\)
−0.479880 + 0.877334i \(0.659320\pi\)
\(6\) 0 0
\(7\) −555.940 + 962.916i −0.612611 + 1.06107i 0.378188 + 0.925729i \(0.376547\pi\)
−0.990799 + 0.135344i \(0.956786\pi\)
\(8\) −760.739 −0.525316
\(9\) 0 0
\(10\) −4135.89 −1.30788
\(11\) 2245.36 − 3889.07i 0.508640 − 0.880991i −0.491310 − 0.870985i \(-0.663482\pi\)
0.999950 − 0.0100060i \(-0.00318506\pi\)
\(12\) 0 0
\(13\) −1218.29 − 2110.14i −0.153797 − 0.266385i 0.778823 − 0.627244i \(-0.215817\pi\)
−0.932620 + 0.360859i \(0.882484\pi\)
\(14\) −7912.08 − 13704.1i −0.770624 − 1.33476i
\(15\) 0 0
\(16\) 10184.4 − 17639.9i 0.621605 − 1.07665i
\(17\) −15905.4 −0.785187 −0.392593 − 0.919712i \(-0.628422\pi\)
−0.392593 + 0.919712i \(0.628422\pi\)
\(18\) 0 0
\(19\) −49949.6 −1.67069 −0.835343 − 0.549730i \(-0.814731\pi\)
−0.835343 + 0.549730i \(0.814731\pi\)
\(20\) 10831.9 − 18761.5i 0.302762 − 0.524399i
\(21\) 0 0
\(22\) 31955.7 + 55348.9i 0.639836 + 1.10823i
\(23\) 34692.5 + 60089.2i 0.594550 + 1.02979i 0.993610 + 0.112867i \(0.0360033\pi\)
−0.399060 + 0.916925i \(0.630663\pi\)
\(24\) 0 0
\(25\) −3163.73 + 5479.74i −0.0404957 + 0.0701406i
\(26\) 34677.2 0.386934
\(27\) 0 0
\(28\) 82887.2 0.713566
\(29\) −47035.8 + 81468.4i −0.358126 + 0.620292i −0.987648 − 0.156691i \(-0.949917\pi\)
0.629522 + 0.776983i \(0.283251\pi\)
\(30\) 0 0
\(31\) 9963.58 + 17257.4i 0.0600689 + 0.104042i 0.894496 − 0.447076i \(-0.147535\pi\)
−0.834427 + 0.551118i \(0.814201\pi\)
\(32\) 96255.8 + 166720.i 0.519280 + 0.899420i
\(33\) 0 0
\(34\) 113182. − 196037.i 0.493857 − 0.855385i
\(35\) −323120. −1.27387
\(36\) 0 0
\(37\) 331750. 1.07673 0.538363 − 0.842713i \(-0.319043\pi\)
0.538363 + 0.842713i \(0.319043\pi\)
\(38\) 355439. − 615638.i 1.05081 − 1.82005i
\(39\) 0 0
\(40\) −110538. − 191457.i −0.273087 − 0.473001i
\(41\) 121133. + 209809.i 0.274486 + 0.475423i 0.970005 − 0.243084i \(-0.0781591\pi\)
−0.695520 + 0.718507i \(0.744826\pi\)
\(42\) 0 0
\(43\) −415713. + 720036.i −0.797359 + 1.38107i 0.123971 + 0.992286i \(0.460437\pi\)
−0.921330 + 0.388781i \(0.872896\pi\)
\(44\) −334769. −0.592462
\(45\) 0 0
\(46\) −987481. −1.49581
\(47\) −80005.3 + 138573.i −0.112403 + 0.194687i −0.916738 − 0.399488i \(-0.869188\pi\)
0.804336 + 0.594175i \(0.202521\pi\)
\(48\) 0 0
\(49\) −206366. − 357437.i −0.250583 − 0.434023i
\(50\) −45025.8 − 77987.1i −0.0509409 − 0.0882323i
\(51\) 0 0
\(52\) −90819.7 + 157304.i −0.0895712 + 0.155142i
\(53\) −311589. −0.287486 −0.143743 − 0.989615i \(-0.545914\pi\)
−0.143743 + 0.989615i \(0.545914\pi\)
\(54\) 0 0
\(55\) 1.30503e6 1.05767
\(56\) 422925. − 732527.i 0.321814 − 0.557398i
\(57\) 0 0
\(58\) −669408. − 1.15945e6i −0.450498 − 0.780286i
\(59\) −156177. − 270506.i −0.0989997 − 0.171473i 0.812271 − 0.583280i \(-0.198231\pi\)
−0.911271 + 0.411807i \(0.864898\pi\)
\(60\) 0 0
\(61\) 28723.9 − 49751.3i 0.0162028 − 0.0280640i −0.857810 − 0.513966i \(-0.828176\pi\)
0.874013 + 0.485902i \(0.161509\pi\)
\(62\) −283601. −0.151125
\(63\) 0 0
\(64\) −132603. −0.0632302
\(65\) 354044. − 613221.i 0.159904 − 0.276962i
\(66\) 0 0
\(67\) 2.05100e6 + 3.55243e6i 0.833111 + 1.44299i 0.895559 + 0.444943i \(0.146776\pi\)
−0.0624478 + 0.998048i \(0.519891\pi\)
\(68\) 592849. + 1.02684e6i 0.228646 + 0.396026i
\(69\) 0 0
\(70\) 2.29930e6 − 3.98251e6i 0.801223 − 1.38776i
\(71\) 403110. 0.133666 0.0668328 − 0.997764i \(-0.478711\pi\)
0.0668328 + 0.997764i \(0.478711\pi\)
\(72\) 0 0
\(73\) −823496. −0.247760 −0.123880 − 0.992297i \(-0.539534\pi\)
−0.123880 + 0.992297i \(0.539534\pi\)
\(74\) −2.36072e6 + 4.08889e6i −0.677226 + 1.17299i
\(75\) 0 0
\(76\) 1.86180e6 + 3.22472e6i 0.486502 + 0.842646i
\(77\) 2.49656e6 + 4.32418e6i 0.623197 + 1.07941i
\(78\) 0 0
\(79\) −489414. + 847689.i −0.111682 + 0.193438i −0.916448 − 0.400153i \(-0.868957\pi\)
0.804767 + 0.593591i \(0.202290\pi\)
\(80\) 5.91931e6 1.29258
\(81\) 0 0
\(82\) −3.44791e6 −0.690569
\(83\) −1.85204e6 + 3.20782e6i −0.355530 + 0.615796i −0.987209 − 0.159434i \(-0.949033\pi\)
0.631678 + 0.775231i \(0.282366\pi\)
\(84\) 0 0
\(85\) −2.31111e6 − 4.00296e6i −0.408182 − 0.706993i
\(86\) −5.91638e6 − 1.02475e7i −1.00303 − 1.73729i
\(87\) 0 0
\(88\) −1.70813e6 + 2.95857e6i −0.267197 + 0.462799i
\(89\) −2.09023e6 −0.314289 −0.157145 − 0.987576i \(-0.550229\pi\)
−0.157145 + 0.987576i \(0.550229\pi\)
\(90\) 0 0
\(91\) 2.70918e6 0.376872
\(92\) 2.58622e6 − 4.47947e6i 0.346265 − 0.599748i
\(93\) 0 0
\(94\) −1.13863e6 − 1.97216e6i −0.141395 − 0.244903i
\(95\) −7.25786e6 − 1.25710e7i −0.868512 − 1.50431i
\(96\) 0 0
\(97\) −1.75125e6 + 3.03325e6i −0.194826 + 0.337448i −0.946843 − 0.321695i \(-0.895747\pi\)
0.752018 + 0.659143i \(0.229081\pi\)
\(98\) 5.87396e6 0.630435
\(99\) 0 0
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 27.8.c.a.19.2 12
3.2 odd 2 9.8.c.a.7.5 yes 12
4.3 odd 2 432.8.i.c.289.5 12
9.2 odd 6 81.8.a.e.1.2 6
9.4 even 3 inner 27.8.c.a.10.2 12
9.5 odd 6 9.8.c.a.4.5 ✓ 12
9.7 even 3 81.8.a.c.1.5 6
12.11 even 2 144.8.i.c.97.1 12
36.23 even 6 144.8.i.c.49.1 12
36.31 odd 6 432.8.i.c.145.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.5 ✓ 12 9.5 odd 6
9.8.c.a.7.5 yes 12 3.2 odd 2
27.8.c.a.10.2 12 9.4 even 3 inner
27.8.c.a.19.2 12 1.1 even 1 trivial
81.8.a.c.1.5 6 9.7 even 3
81.8.a.e.1.2 6 9.2 odd 6
144.8.i.c.49.1 12 36.23 even 6
144.8.i.c.97.1 12 12.11 even 2
432.8.i.c.145.5 12 36.31 odd 6
432.8.i.c.289.5 12 4.3 odd 2