Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [729,2,Mod(28,729)] 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("729.28"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(54)) chi = DirichletCharacter(H, H._module([44])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.g (of order \(27\), degree \(18\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [144,9,0,9,9,0,9,-18,0,-18,9,0,9,9,0,9,-18,0,-18,45] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(20)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(8\) over \(\Q(\zeta_{27})\)
Twist minimal: no (minimal twist has level 81)
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label 109.6
Character \(\chi\) \(=\) 729.109
Dual form 729.2.g.d.622.6

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.31250 + 1.39117i) q^{2} +(-0.0964037 + 1.65519i) q^{4} +(-2.46793 + 0.288460i) q^{5} +(-4.06877 - 2.04341i) q^{7} +(0.501084 - 0.420459i) q^{8} +(-3.64045 - 3.05470i) q^{10} +(0.125874 + 0.291810i) q^{11} +(-3.31064 - 0.784636i) q^{13} +(-2.49752 - 8.34231i) q^{14} +(4.53616 + 0.530202i) q^{16} +(0.885565 - 5.02229i) q^{17} +(-0.216164 - 1.22593i) q^{19} +(-0.239538 - 4.11270i) q^{20} +(-0.240746 + 0.558112i) q^{22} +(-1.70487 + 0.856218i) q^{23} +(1.14226 - 0.270720i) q^{25} +(-3.25365 - 5.63548i) q^{26} +(3.77448 - 6.53759i) q^{28} +(0.993943 - 3.32000i) q^{29} +(-5.67009 - 3.72928i) q^{31} +(4.43488 + 5.95708i) q^{32} +(8.14914 - 5.35978i) q^{34} +(10.6309 + 3.86933i) q^{35} +(-8.02231 + 2.91988i) q^{37} +(1.42176 - 1.90975i) q^{38} +(-1.11536 + 1.18221i) q^{40} +(4.35826 - 4.61948i) q^{41} +(-3.63719 + 4.88559i) q^{43} +(-0.495135 + 0.180214i) q^{44} +(-3.42878 - 1.24797i) q^{46} +(-3.40689 + 2.24075i) q^{47} +(8.19925 + 11.0135i) q^{49} +(1.87583 + 1.23375i) q^{50} +(1.61788 - 5.40409i) q^{52} +(-3.88136 + 6.72271i) q^{53} +(-0.394825 - 0.683857i) q^{55} +(-2.89797 + 0.686831i) q^{56} +(5.92322 - 2.97475i) q^{58} +(1.57205 - 3.64441i) q^{59} +(-0.142985 - 2.45496i) q^{61} +(-2.25394 - 12.7827i) q^{62} +(-0.880398 + 4.99299i) q^{64} +(8.39677 + 0.981441i) q^{65} +(0.988137 + 3.30061i) q^{67} +(8.22746 + 1.94994i) q^{68} +(8.57015 + 19.8678i) q^{70} +(-2.16099 - 1.81328i) q^{71} +(-3.24023 + 2.71887i) q^{73} +(-14.5913 - 7.32803i) q^{74} +(2.04998 - 0.239609i) q^{76} +(0.0841338 - 1.44452i) q^{77} +(4.97152 + 5.26950i) q^{79} -11.3479 q^{80} +12.1467 q^{82} +(8.29076 + 8.78769i) q^{83} +(-0.736786 + 12.6501i) q^{85} +(-11.5705 + 1.35239i) q^{86} +(0.185768 + 0.0932961i) q^{88} +(2.86182 - 2.40135i) q^{89} +(11.8669 + 9.95751i) q^{91} +(-1.25285 - 2.90442i) q^{92} +(-7.58878 - 1.79857i) q^{94} +(0.887111 + 2.96316i) q^{95} +(-14.5265 - 1.69791i) q^{97} +(-4.56012 + 25.8617i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 144 q + 9 q^{2} + 9 q^{4} + 9 q^{5} + 9 q^{7} - 18 q^{8} - 18 q^{10} + 9 q^{11} + 9 q^{13} + 9 q^{14} + 9 q^{16} - 18 q^{17} - 18 q^{19} + 45 q^{20} + 9 q^{22} - 45 q^{23} + 9 q^{25} + 45 q^{26} - 9 q^{28}+ \cdots + 81 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{7}{27}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.31250 + 1.39117i 0.928076 + 0.983703i 0.999906 − 0.0137139i \(-0.00436540\pi\)
−0.0718301 + 0.997417i \(0.522884\pi\)
\(3\) 0 0
\(4\) −0.0964037 + 1.65519i −0.0482019 + 0.827594i
\(5\) −2.46793 + 0.288460i −1.10369 + 0.129003i −0.648372 − 0.761324i \(-0.724550\pi\)
−0.455321 + 0.890327i \(0.650476\pi\)
\(6\) 0 0
\(7\) −4.06877 − 2.04341i −1.53785 − 0.772338i −0.540246 − 0.841507i \(-0.681669\pi\)
−0.997605 + 0.0691692i \(0.977965\pi\)
\(8\) 0.501084 − 0.420459i 0.177160 − 0.148655i
\(9\) 0 0
\(10\) −3.64045 − 3.05470i −1.15121 − 0.965981i
\(11\) 0.125874 + 0.291810i 0.0379526 + 0.0879840i 0.936138 − 0.351633i \(-0.114373\pi\)
−0.898185 + 0.439617i \(0.855114\pi\)
\(12\) 0 0
\(13\) −3.31064 − 0.784636i −0.918206 − 0.217619i −0.255770 − 0.966738i \(-0.582329\pi\)
−0.662436 + 0.749119i \(0.730477\pi\)
\(14\) −2.49752 − 8.34231i −0.667492 − 2.22958i
\(15\) 0 0
\(16\) 4.53616 + 0.530202i 1.13404 + 0.132550i
\(17\) 0.885565 − 5.02229i 0.214781 − 1.21808i −0.666505 − 0.745501i \(-0.732210\pi\)
0.881286 − 0.472584i \(-0.156679\pi\)
\(18\) 0 0
\(19\) −0.216164 − 1.22593i −0.0495915 − 0.281248i 0.949920 − 0.312493i \(-0.101164\pi\)
−0.999512 + 0.0312449i \(0.990053\pi\)
\(20\) −0.239538 − 4.11270i −0.0535622 − 0.919628i
\(21\) 0 0
\(22\) −0.240746 + 0.558112i −0.0513272 + 0.118990i
\(23\) −1.70487 + 0.856218i −0.355490 + 0.178534i −0.617577 − 0.786510i \(-0.711886\pi\)
0.262087 + 0.965044i \(0.415589\pi\)
\(24\) 0 0
\(25\) 1.14226 − 0.270720i 0.228452 − 0.0541440i
\(26\) −3.25365 − 5.63548i −0.638092 − 1.10521i
\(27\) 0 0
\(28\) 3.77448 − 6.53759i 0.713309 − 1.23549i
\(29\) 0.993943 − 3.32000i 0.184571 − 0.616509i −0.814756 − 0.579805i \(-0.803129\pi\)
0.999326 − 0.0367041i \(-0.0116859\pi\)
\(30\) 0 0
\(31\) −5.67009 − 3.72928i −1.01838 − 0.669798i −0.0735253 − 0.997293i \(-0.523425\pi\)
−0.944853 + 0.327496i \(0.893795\pi\)
\(32\) 4.43488 + 5.95708i 0.783984 + 1.05307i
\(33\) 0 0
\(34\) 8.14914 − 5.35978i 1.39757 − 0.919194i
\(35\) 10.6309 + 3.86933i 1.79695 + 0.654036i
\(36\) 0 0
\(37\) −8.02231 + 2.91988i −1.31886 + 0.480026i −0.903093 − 0.429445i \(-0.858709\pi\)
−0.415767 + 0.909471i \(0.636487\pi\)
\(38\) 1.42176 − 1.90975i 0.230639 − 0.309802i
\(39\) 0 0
\(40\) −1.11536 + 1.18221i −0.176353 + 0.186923i
\(41\) 4.35826 − 4.61948i 0.680646 − 0.721442i −0.291304 − 0.956630i \(-0.594089\pi\)
0.971950 + 0.235188i \(0.0755707\pi\)
\(42\) 0 0
\(43\) −3.63719 + 4.88559i −0.554666 + 0.745046i −0.987705 − 0.156329i \(-0.950034\pi\)
0.433039 + 0.901375i \(0.357441\pi\)
\(44\) −0.495135 + 0.180214i −0.0746444 + 0.0271683i
\(45\) 0 0
\(46\) −3.42878 − 1.24797i −0.505546 − 0.184004i
\(47\) −3.40689 + 2.24075i −0.496946 + 0.326846i −0.773119 − 0.634262i \(-0.781304\pi\)
0.276173 + 0.961108i \(0.410934\pi\)
\(48\) 0 0
\(49\) 8.19925 + 11.0135i 1.17132 + 1.57336i
\(50\) 1.87583 + 1.23375i 0.265282 + 0.174479i
\(51\) 0 0
\(52\) 1.61788 − 5.40409i 0.224359 − 0.749412i
\(53\) −3.88136 + 6.72271i −0.533146 + 0.923435i 0.466105 + 0.884729i \(0.345657\pi\)
−0.999251 + 0.0387058i \(0.987676\pi\)
\(54\) 0 0
\(55\) −0.394825 − 0.683857i −0.0532382 − 0.0922113i
\(56\) −2.89797 + 0.686831i −0.387257 + 0.0917817i
\(57\) 0 0
\(58\) 5.92322 − 2.97475i 0.777757 − 0.390604i
\(59\) 1.57205 − 3.64441i 0.204663 − 0.474462i −0.784710 − 0.619863i \(-0.787188\pi\)
0.989373 + 0.145401i \(0.0464473\pi\)
\(60\) 0 0
\(61\) −0.142985 − 2.45496i −0.0183074 − 0.314325i −0.995013 − 0.0997428i \(-0.968198\pi\)
0.976706 − 0.214582i \(-0.0688390\pi\)
\(62\) −2.25394 − 12.7827i −0.286250 − 1.62340i
\(63\) 0 0
\(64\) −0.880398 + 4.99299i −0.110050 + 0.624123i
\(65\) 8.39677 + 0.981441i 1.04149 + 0.121733i
\(66\) 0 0
\(67\) 0.988137 + 3.30061i 0.120720 + 0.403233i 0.996893 − 0.0787630i \(-0.0250970\pi\)
−0.876173 + 0.481996i \(0.839912\pi\)
\(68\) 8.22746 + 1.94994i 0.997726 + 0.236466i
\(69\) 0 0
\(70\) 8.57015 + 19.8678i 1.02433 + 2.37466i
\(71\) −2.16099 − 1.81328i −0.256462 − 0.215197i 0.505487 − 0.862834i \(-0.331313\pi\)
−0.761949 + 0.647637i \(0.775757\pi\)
\(72\) 0 0
\(73\) −3.24023 + 2.71887i −0.379240 + 0.318220i −0.812404 − 0.583095i \(-0.801842\pi\)
0.433164 + 0.901315i \(0.357397\pi\)
\(74\) −14.5913 − 7.32803i −1.69621 − 0.851866i
\(75\) 0 0
\(76\) 2.04998 − 0.239609i 0.235149 − 0.0274850i
\(77\) 0.0841338 − 1.44452i 0.00958793 − 0.164618i
\(78\) 0 0
\(79\) 4.97152 + 5.26950i 0.559339 + 0.592865i 0.943899 − 0.330235i \(-0.107128\pi\)
−0.384559 + 0.923100i \(0.625647\pi\)
\(80\) −11.3479 −1.26873
\(81\) 0 0
\(82\) 12.1467 1.34138
\(83\) 8.29076 + 8.78769i 0.910029 + 0.964575i 0.999506 − 0.0314144i \(-0.0100011\pi\)
−0.0894773 + 0.995989i \(0.528520\pi\)
\(84\) 0 0
\(85\) −0.736786 + 12.6501i −0.0799157 + 1.37210i
\(86\) −11.5705 + 1.35239i −1.24768 + 0.145832i
\(87\) 0 0
\(88\) 0.185768 + 0.0932961i 0.0198029 + 0.00994540i
\(89\) 2.86182 − 2.40135i 0.303352 − 0.254543i −0.478386 − 0.878150i \(-0.658778\pi\)
0.781738 + 0.623607i \(0.214334\pi\)
\(90\) 0 0
\(91\) 11.8669 + 9.95751i 1.24399 + 1.04383i
\(92\) −1.25285 − 2.90442i −0.130618 − 0.302807i
\(93\) 0 0
\(94\) −7.58878 − 1.79857i −0.782723 − 0.185509i
\(95\) 0.887111 + 2.96316i 0.0910157 + 0.304013i
\(96\) 0 0
\(97\) −14.5265 − 1.69791i −1.47495 − 0.172396i −0.659707 − 0.751523i \(-0.729320\pi\)
−0.815238 + 0.579126i \(0.803394\pi\)
\(98\) −4.56012 + 25.8617i −0.460642 + 2.61243i
\(99\) 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 729.2.g.d.109.6 144
3.2 odd 2 729.2.g.a.109.3 144
9.2 odd 6 729.2.g.b.352.3 144
9.4 even 3 81.2.g.a.76.3 yes 144
9.5 odd 6 243.2.g.a.118.6 144
9.7 even 3 729.2.g.c.352.6 144
81.11 odd 54 729.2.g.a.622.3 144
81.16 even 27 729.2.g.c.379.6 144
81.31 even 27 6561.2.a.c.1.15 72
81.38 odd 54 243.2.g.a.208.6 144
81.43 even 27 81.2.g.a.16.3 ✓ 144
81.50 odd 54 6561.2.a.d.1.58 72
81.65 odd 54 729.2.g.b.379.3 144
81.70 even 27 inner 729.2.g.d.622.6 144
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
81.2.g.a.16.3 ✓ 144 81.43 even 27
81.2.g.a.76.3 yes 144 9.4 even 3
243.2.g.a.118.6 144 9.5 odd 6
243.2.g.a.208.6 144 81.38 odd 54
729.2.g.a.109.3 144 3.2 odd 2
729.2.g.a.622.3 144 81.11 odd 54
729.2.g.b.352.3 144 9.2 odd 6
729.2.g.b.379.3 144 81.65 odd 54
729.2.g.c.352.6 144 9.7 even 3
729.2.g.c.379.6 144 81.16 even 27
729.2.g.d.109.6 144 1.1 even 1 trivial
729.2.g.d.622.6 144 81.70 even 27 inner
6561.2.a.c.1.15 72 81.31 even 27
6561.2.a.d.1.58 72 81.50 odd 54