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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [486,2,Mod(19,486)] 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("486.19"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(486, base_ring=CyclotomicField(54)) chi = DirichletCharacter(H, H._module([52])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 486 = 2 \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 486.g (of order \(27\), degree \(18\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [90] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.88072953823\)
Analytic rank: \(0\)
Dimension: \(90\)
Relative dimension: \(5\) over \(\Q(\zeta_{27})\)
Twist minimal: no (minimal twist has level 162)
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label 199.5
Character \(\chi\) \(=\) 486.199
Dual form 486.2.g.b.127.5

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.686242 + 0.727374i) q^{2} +(-0.0581448 + 0.998308i) q^{4} +(3.77668 - 0.441430i) q^{5} +(-3.13446 - 1.57418i) q^{7} +(-0.766044 + 0.642788i) q^{8} +(2.91280 + 2.44413i) q^{10} +(1.41437 + 3.27887i) q^{11} +(6.00411 + 1.42300i) q^{13} +(-1.00598 - 3.36019i) q^{14} +(-0.993238 - 0.116093i) q^{16} +(0.0237611 - 0.134756i) q^{17} +(-0.591493 - 3.35452i) q^{19} +(0.221089 + 3.79596i) q^{20} +(-1.41437 + 3.27887i) q^{22} +(-1.62119 + 0.814194i) q^{23} +(9.20321 - 2.18120i) q^{25} +(3.08522 + 5.34375i) q^{26} +(1.75377 - 3.03762i) q^{28} +(0.375745 - 1.25508i) q^{29} +(0.265030 + 0.174313i) q^{31} +(-0.597159 - 0.802123i) q^{32} +(0.114324 - 0.0751919i) q^{34} +(-12.5327 - 4.56154i) q^{35} +(-7.16975 + 2.60958i) q^{37} +(2.03408 - 2.73225i) q^{38} +(-2.60936 + 2.76576i) q^{40} +(-4.36125 + 4.62265i) q^{41} +(1.92691 - 2.58828i) q^{43} +(-3.35556 + 1.22132i) q^{44} +(-1.70475 - 0.620480i) q^{46} +(-3.99017 + 2.62437i) q^{47} +(3.16665 + 4.25355i) q^{49} +(7.90218 + 5.19734i) q^{50} +(-1.76970 + 5.91121i) q^{52} +(2.78294 - 4.82020i) q^{53} +(6.78900 + 11.7589i) q^{55} +(3.41300 - 0.808895i) q^{56} +(1.17076 - 0.587978i) q^{58} +(-1.85601 + 4.30271i) q^{59} +(-0.784513 - 13.4696i) q^{61} +(0.0550839 + 0.312396i) q^{62} +(0.173648 - 0.984808i) q^{64} +(23.3037 + 2.72382i) q^{65} +(-2.79770 - 9.34496i) q^{67} +(0.133146 + 0.0315562i) q^{68} +(-5.28253 - 12.2463i) q^{70} +(-3.05189 - 2.56084i) q^{71} +(-10.7055 + 8.98301i) q^{73} +(-6.81832 - 3.42429i) q^{74} +(3.38324 - 0.395444i) q^{76} +(0.728272 - 12.5039i) q^{77} +(2.36891 + 2.51090i) q^{79} -3.80239 q^{80} -6.35527 q^{82} +(-0.597161 - 0.632954i) q^{83} +(0.0302527 - 0.519418i) q^{85} +(3.20497 - 0.374608i) q^{86} +(-3.19108 - 1.60262i) q^{88} +(-6.42404 + 5.39041i) q^{89} +(-16.5795 - 13.9119i) q^{91} +(-0.718553 - 1.66579i) q^{92} +(-4.64712 - 1.10139i) q^{94} +(-3.71467 - 12.4078i) q^{95} +(-6.91065 - 0.807739i) q^{97} +(-0.920833 + 5.22230i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 90 q - 18 q^{13} + 9 q^{20} - 27 q^{23} - 18 q^{25} + 27 q^{26} - 18 q^{28} + 27 q^{29} + 54 q^{31} + 27 q^{35} + 18 q^{38} + 9 q^{41} - 36 q^{43} + 18 q^{46} + 27 q^{47} + 36 q^{52} + 27 q^{53} - 54 q^{55}+ \cdots - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/486\mathbb{Z}\right)^\times\).

\(n\) \(245\)
\(\chi(n)\) \(e\left(\frac{7}{27}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.686242 + 0.727374i 0.485246 + 0.514331i
\(3\) 0 0
\(4\) −0.0581448 + 0.998308i −0.0290724 + 0.499154i
\(5\) 3.77668 − 0.441430i 1.68898 − 0.197414i 0.783380 − 0.621543i \(-0.213494\pi\)
0.905602 + 0.424129i \(0.139420\pi\)
\(6\) 0 0
\(7\) −3.13446 − 1.57418i −1.18471 − 0.594985i −0.256323 − 0.966591i \(-0.582511\pi\)
−0.928391 + 0.371606i \(0.878807\pi\)
\(8\) −0.766044 + 0.642788i −0.270838 + 0.227260i
\(9\) 0 0
\(10\) 2.91280 + 2.44413i 0.921108 + 0.772901i
\(11\) 1.41437 + 3.27887i 0.426447 + 0.988616i 0.987033 + 0.160519i \(0.0513169\pi\)
−0.560585 + 0.828097i \(0.689424\pi\)
\(12\) 0 0
\(13\) 6.00411 + 1.42300i 1.66524 + 0.394669i 0.951963 − 0.306214i \(-0.0990621\pi\)
0.713277 + 0.700883i \(0.247210\pi\)
\(14\) −1.00598 − 3.36019i −0.268858 − 0.898049i
\(15\) 0 0
\(16\) −0.993238 − 0.116093i −0.248310 − 0.0290232i
\(17\) 0.0237611 − 0.134756i 0.00576291 − 0.0326831i −0.981791 − 0.189966i \(-0.939162\pi\)
0.987554 + 0.157283i \(0.0502734\pi\)
\(18\) 0 0
\(19\) −0.591493 − 3.35452i −0.135698 − 0.769580i −0.974371 − 0.224946i \(-0.927780\pi\)
0.838674 − 0.544634i \(-0.183332\pi\)
\(20\) 0.221089 + 3.79596i 0.0494371 + 0.848802i
\(21\) 0 0
\(22\) −1.41437 + 3.27887i −0.301544 + 0.699057i
\(23\) −1.62119 + 0.814194i −0.338042 + 0.169771i −0.609720 − 0.792617i \(-0.708718\pi\)
0.271677 + 0.962388i \(0.412422\pi\)
\(24\) 0 0
\(25\) 9.20321 − 2.18120i 1.84064 − 0.436240i
\(26\) 3.08522 + 5.34375i 0.605061 + 1.04800i
\(27\) 0 0
\(28\) 1.75377 − 3.03762i 0.331432 − 0.574057i
\(29\) 0.375745 − 1.25508i 0.0697741 − 0.233062i −0.915989 − 0.401204i \(-0.868592\pi\)
0.985763 + 0.168143i \(0.0537769\pi\)
\(30\) 0 0
\(31\) 0.265030 + 0.174313i 0.0476007 + 0.0313075i 0.573086 − 0.819495i \(-0.305746\pi\)
−0.525486 + 0.850802i \(0.676116\pi\)
\(32\) −0.597159 − 0.802123i −0.105564 − 0.141797i
\(33\) 0 0
\(34\) 0.114324 − 0.0751919i 0.0196064 − 0.0128953i
\(35\) −12.5327 − 4.56154i −2.11842 − 0.771041i
\(36\) 0 0
\(37\) −7.16975 + 2.60958i −1.17870 + 0.429012i −0.855743 − 0.517402i \(-0.826899\pi\)
−0.322958 + 0.946413i \(0.604677\pi\)
\(38\) 2.03408 − 2.73225i 0.329972 − 0.443229i
\(39\) 0 0
\(40\) −2.60936 + 2.76576i −0.412576 + 0.437305i
\(41\) −4.36125 + 4.62265i −0.681112 + 0.721937i −0.972042 − 0.234807i \(-0.924554\pi\)
0.290929 + 0.956745i \(0.406036\pi\)
\(42\) 0 0
\(43\) 1.92691 − 2.58828i 0.293850 − 0.394710i −0.630540 − 0.776157i \(-0.717167\pi\)
0.924391 + 0.381447i \(0.124574\pi\)
\(44\) −3.35556 + 1.22132i −0.505870 + 0.184121i
\(45\) 0 0
\(46\) −1.70475 − 0.620480i −0.251352 − 0.0914848i
\(47\) −3.99017 + 2.62437i −0.582026 + 0.382804i −0.806110 − 0.591765i \(-0.798431\pi\)
0.224085 + 0.974570i \(0.428061\pi\)
\(48\) 0 0
\(49\) 3.16665 + 4.25355i 0.452379 + 0.607650i
\(50\) 7.90218 + 5.19734i 1.11754 + 0.735015i
\(51\) 0 0
\(52\) −1.76970 + 5.91121i −0.245413 + 0.819737i
\(53\) 2.78294 − 4.82020i 0.382266 − 0.662105i −0.609119 − 0.793078i \(-0.708477\pi\)
0.991386 + 0.130974i \(0.0418103\pi\)
\(54\) 0 0
\(55\) 6.78900 + 11.7589i 0.915428 + 1.58557i
\(56\) 3.41300 − 0.808895i 0.456081 − 0.108093i
\(57\) 0 0
\(58\) 1.17076 − 0.587978i 0.153728 − 0.0772053i
\(59\) −1.85601 + 4.30271i −0.241631 + 0.560165i −0.995273 − 0.0971215i \(-0.969036\pi\)
0.753641 + 0.657286i \(0.228296\pi\)
\(60\) 0 0
\(61\) −0.784513 − 13.4696i −0.100447 − 1.72460i −0.553270 − 0.833002i \(-0.686620\pi\)
0.452824 − 0.891600i \(-0.350417\pi\)
\(62\) 0.0550839 + 0.312396i 0.00699566 + 0.0396744i
\(63\) 0 0
\(64\) 0.173648 − 0.984808i 0.0217060 − 0.123101i
\(65\) 23.3037 + 2.72382i 2.89047 + 0.337848i
\(66\) 0 0
\(67\) −2.79770 − 9.34496i −0.341793 − 1.14167i −0.940242 − 0.340507i \(-0.889401\pi\)
0.598449 − 0.801161i \(-0.295784\pi\)
\(68\) 0.133146 + 0.0315562i 0.0161464 + 0.00382676i
\(69\) 0 0
\(70\) −5.28253 − 12.2463i −0.631384 − 1.46371i
\(71\) −3.05189 − 2.56084i −0.362193 − 0.303916i 0.443471 − 0.896289i \(-0.353747\pi\)
−0.805664 + 0.592373i \(0.798191\pi\)
\(72\) 0 0
\(73\) −10.7055 + 8.98301i −1.25299 + 1.05138i −0.256595 + 0.966519i \(0.582601\pi\)
−0.996393 + 0.0848624i \(0.972955\pi\)
\(74\) −6.81832 − 3.42429i −0.792614 − 0.398066i
\(75\) 0 0
\(76\) 3.38324 − 0.395444i 0.388084 − 0.0453605i
\(77\) 0.728272 − 12.5039i 0.0829942 − 1.42496i
\(78\) 0 0
\(79\) 2.36891 + 2.51090i 0.266523 + 0.282498i 0.846789 − 0.531929i \(-0.178533\pi\)
−0.580266 + 0.814427i \(0.697051\pi\)
\(80\) −3.80239 −0.425120
\(81\) 0 0
\(82\) −6.35527 −0.701822
\(83\) −0.597161 − 0.632954i −0.0655470 − 0.0694757i 0.693770 − 0.720196i \(-0.255948\pi\)
−0.759317 + 0.650721i \(0.774467\pi\)
\(84\) 0 0
\(85\) 0.0302527 − 0.519418i 0.00328136 − 0.0563388i
\(86\) 3.20497 − 0.374608i 0.345601 − 0.0403950i
\(87\) 0 0
\(88\) −3.19108 − 1.60262i −0.340171 − 0.170840i
\(89\) −6.42404 + 5.39041i −0.680947 + 0.571382i −0.916283 − 0.400532i \(-0.868825\pi\)
0.235336 + 0.971914i \(0.424381\pi\)
\(90\) 0 0
\(91\) −16.5795 − 13.9119i −1.73801 − 1.45836i
\(92\) −0.718553 − 1.66579i −0.0749143 − 0.173671i
\(93\) 0 0
\(94\) −4.64712 − 1.10139i −0.479314 − 0.113599i
\(95\) −3.71467 − 12.4078i −0.381117 − 1.27302i
\(96\) 0 0
\(97\) −6.91065 − 0.807739i −0.701670 − 0.0820134i −0.242225 − 0.970220i \(-0.577877\pi\)
−0.459444 + 0.888207i \(0.651951\pi\)
\(98\) −0.920833 + 5.22230i −0.0930182 + 0.527532i
\(99\) 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 486.2.g.b.199.5 90
3.2 odd 2 162.2.g.b.103.3 ✓ 90
81.11 odd 54 162.2.g.b.151.3 yes 90
81.70 even 27 inner 486.2.g.b.127.5 90
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.b.103.3 ✓ 90 3.2 odd 2
162.2.g.b.151.3 yes 90 81.11 odd 54
486.2.g.b.127.5 90 81.70 even 27 inner
486.2.g.b.199.5 90 1.1 even 1 trivial