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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [162,2,Mod(7,162)] 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("162.7"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(54)) chi = DirichletCharacter(H, H._module([16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 162.g (of order \(27\), degree \(18\), 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: \(1.29357651274\)
Analytic rank: \(0\)
Dimension: \(90\)
Relative dimension: \(5\) over \(\Q(\zeta_{27})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{27}]$

Embedding invariants

Embedding label 151.3
Character \(\chi\) \(=\) 162.151
Dual form 162.2.g.b.103.3

$q$-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} +(0.851057 + 1.50854i) q^{3} +(-0.0581448 - 0.998308i) q^{4} +(-3.77668 - 0.441430i) q^{5} +(-1.68131 - 0.416189i) q^{6} +(-3.13446 + 1.57418i) q^{7} +(0.766044 + 0.642788i) q^{8} +(-1.55140 + 2.56771i) q^{9} +(2.91280 - 2.44413i) q^{10} +(-1.41437 + 3.27887i) q^{11} +(1.45651 - 0.937331i) q^{12} +(6.00411 - 1.42300i) q^{13} +(1.00598 - 3.36019i) q^{14} +(-2.54825 - 6.07296i) q^{15} +(-0.993238 + 0.116093i) q^{16} +(-0.0237611 - 0.134756i) q^{17} +(-0.803048 - 2.89052i) q^{18} +(-0.591493 + 3.35452i) q^{19} +(-0.221089 + 3.79596i) q^{20} +(-5.04232 - 3.38874i) q^{21} +(-1.41437 - 3.27887i) q^{22} +(1.62119 + 0.814194i) q^{23} +(-0.317725 + 1.70266i) q^{24} +(9.20321 + 2.18120i) q^{25} +(-3.08522 + 5.34375i) q^{26} +(-5.19384 - 0.155090i) q^{27} +(1.75377 + 3.03762i) q^{28} +(-0.375745 - 1.25508i) q^{29} +(6.16603 + 2.31399i) q^{30} +(0.265030 - 0.174313i) q^{31} +(0.597159 - 0.802123i) q^{32} +(-6.15002 + 0.656872i) q^{33} +(0.114324 + 0.0751919i) q^{34} +(12.5327 - 4.56154i) q^{35} +(2.65357 + 1.39948i) q^{36} +(-7.16975 - 2.60958i) q^{37} +(-2.03408 - 2.73225i) q^{38} +(7.25649 + 7.84640i) q^{39} +(-2.60936 - 2.76576i) q^{40} +(4.36125 + 4.62265i) q^{41} +(5.92513 - 1.34216i) q^{42} +(1.92691 + 2.58828i) q^{43} +(3.35556 + 1.22132i) q^{44} +(6.99262 - 9.01259i) q^{45} +(-1.70475 + 0.620480i) q^{46} +(3.99017 + 2.62437i) q^{47} +(-1.02043 - 1.39954i) q^{48} +(3.16665 - 4.25355i) q^{49} +(-7.90218 + 5.19734i) q^{50} +(0.183063 - 0.150530i) q^{51} +(-1.76970 - 5.91121i) q^{52} +(-2.78294 - 4.82020i) q^{53} +(3.67704 - 3.67143i) q^{54} +(6.78900 - 11.7589i) q^{55} +(-3.41300 - 0.808895i) q^{56} +(-5.56384 + 1.96260i) q^{57} +(1.17076 + 0.587978i) q^{58} +(1.85601 + 4.30271i) q^{59} +(-5.91452 + 2.89705i) q^{60} +(-0.784513 + 13.4696i) q^{61} +(-0.0550839 + 0.312396i) q^{62} +(0.820759 - 10.4906i) q^{63} +(0.173648 + 0.984808i) q^{64} +(-23.3037 + 2.72382i) q^{65} +(3.74261 - 4.92414i) q^{66} +(-2.79770 + 9.34496i) q^{67} +(-0.133146 + 0.0315562i) q^{68} +(0.151481 + 3.13857i) q^{69} +(-5.28253 + 12.2463i) q^{70} +(3.05189 - 2.56084i) q^{71} +(-2.83894 + 0.969758i) q^{72} +(-10.7055 - 8.98301i) q^{73} +(6.81832 - 3.42429i) q^{74} +(4.54202 + 15.7398i) q^{75} +(3.38324 + 0.395444i) q^{76} +(-0.728272 - 12.5039i) q^{77} +(-10.6870 - 0.106345i) q^{78} +(2.36891 - 2.51090i) q^{79} +3.80239 q^{80} +(-4.18629 - 7.96712i) q^{81} -6.35527 q^{82} +(0.597161 - 0.632954i) q^{83} +(-3.08982 + 5.23083i) q^{84} +(0.0302527 + 0.519418i) q^{85} +(-3.20497 - 0.374608i) q^{86} +(1.57355 - 1.63497i) q^{87} +(-3.19108 + 1.60262i) q^{88} +(6.42404 + 5.39041i) q^{89} +(1.75689 + 11.2711i) q^{90} +(-16.5795 + 13.9119i) q^{91} +(0.718553 - 1.66579i) q^{92} +(0.488514 + 0.251459i) q^{93} +(-4.64712 + 1.10139i) q^{94} +(3.71467 - 12.4078i) q^{95} +(1.71825 + 0.218187i) q^{96} +(-6.91065 + 0.807739i) q^{97} +(0.920833 + 5.22230i) q^{98} +(-6.22494 - 8.71853i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 90 q - 9 q^{6} - 18 q^{13} - 9 q^{18} - 9 q^{20} - 54 q^{21} + 27 q^{23} - 18 q^{25} - 27 q^{26} - 27 q^{27} - 18 q^{28} - 27 q^{29} + 9 q^{30} + 54 q^{31} - 63 q^{33} - 27 q^{35} - 9 q^{36} - 18 q^{38}+ \cdots - 81 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{20}{27}\right)\)

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
\(3\) 0.851057 + 1.50854i 0.491358 + 0.870958i
\(4\) −0.0581448 − 0.998308i −0.0290724 − 0.499154i
\(5\) −3.77668 − 0.441430i −1.68898 − 0.197414i −0.783380 − 0.621543i \(-0.786506\pi\)
−0.905602 + 0.424129i \(0.860580\pi\)
\(6\) −1.68131 − 0.416189i −0.686390 − 0.169908i
\(7\) −3.13446 + 1.57418i −1.18471 + 0.594985i −0.928391 − 0.371606i \(-0.878807\pi\)
−0.256323 + 0.966591i \(0.582511\pi\)
\(8\) 0.766044 + 0.642788i 0.270838 + 0.227260i
\(9\) −1.55140 + 2.56771i −0.517135 + 0.855904i
\(10\) 2.91280 − 2.44413i 0.921108 − 0.772901i
\(11\) −1.41437 + 3.27887i −0.426447 + 0.988616i 0.560585 + 0.828097i \(0.310576\pi\)
−0.987033 + 0.160519i \(0.948683\pi\)
\(12\) 1.45651 − 0.937331i 0.420457 − 0.270584i
\(13\) 6.00411 − 1.42300i 1.66524 − 0.394669i 0.713277 − 0.700883i \(-0.247210\pi\)
0.951963 + 0.306214i \(0.0990621\pi\)
\(14\) 1.00598 − 3.36019i 0.268858 − 0.898049i
\(15\) −2.54825 − 6.07296i −0.657956 − 1.56803i
\(16\) −0.993238 + 0.116093i −0.248310 + 0.0290232i
\(17\) −0.0237611 − 0.134756i −0.00576291 − 0.0326831i 0.981791 − 0.189966i \(-0.0608377\pi\)
−0.987554 + 0.157283i \(0.949727\pi\)
\(18\) −0.803048 − 2.89052i −0.189280 − 0.681302i
\(19\) −0.591493 + 3.35452i −0.135698 + 0.769580i 0.838674 + 0.544634i \(0.183332\pi\)
−0.974371 + 0.224946i \(0.927780\pi\)
\(20\) −0.221089 + 3.79596i −0.0494371 + 0.848802i
\(21\) −5.04232 − 3.38874i −1.10033 − 0.739484i
\(22\) −1.41437 − 3.27887i −0.301544 − 0.699057i
\(23\) 1.62119 + 0.814194i 0.338042 + 0.169771i 0.609720 − 0.792617i \(-0.291282\pi\)
−0.271677 + 0.962388i \(0.587578\pi\)
\(24\) −0.317725 + 1.70266i −0.0648554 + 0.347554i
\(25\) 9.20321 + 2.18120i 1.84064 + 0.436240i
\(26\) −3.08522 + 5.34375i −0.605061 + 1.04800i
\(27\) −5.19384 − 0.155090i −0.999554 − 0.0298472i
\(28\) 1.75377 + 3.03762i 0.331432 + 0.574057i
\(29\) −0.375745 − 1.25508i −0.0697741 − 0.233062i 0.915989 − 0.401204i \(-0.131408\pi\)
−0.985763 + 0.168143i \(0.946223\pi\)
\(30\) 6.16603 + 2.31399i 1.12576 + 0.422475i
\(31\) 0.265030 − 0.174313i 0.0476007 − 0.0313075i −0.525486 − 0.850802i \(-0.676116\pi\)
0.573086 + 0.819495i \(0.305746\pi\)
\(32\) 0.597159 − 0.802123i 0.105564 − 0.141797i
\(33\) −6.15002 + 0.656872i −1.07058 + 0.114347i
\(34\) 0.114324 + 0.0751919i 0.0196064 + 0.0128953i
\(35\) 12.5327 − 4.56154i 2.11842 − 0.771041i
\(36\) 2.65357 + 1.39948i 0.442262 + 0.233247i
\(37\) −7.16975 − 2.60958i −1.17870 − 0.429012i −0.322958 − 0.946413i \(-0.604677\pi\)
−0.855743 + 0.517402i \(0.826899\pi\)
\(38\) −2.03408 − 2.73225i −0.329972 − 0.443229i
\(39\) 7.25649 + 7.84640i 1.16197 + 1.25643i
\(40\) −2.60936 − 2.76576i −0.412576 − 0.437305i
\(41\) 4.36125 + 4.62265i 0.681112 + 0.721937i 0.972042 − 0.234807i \(-0.0754460\pi\)
−0.290929 + 0.956745i \(0.593964\pi\)
\(42\) 5.92513 − 1.34216i 0.914268 − 0.207099i
\(43\) 1.92691 + 2.58828i 0.293850 + 0.394710i 0.924391 − 0.381447i \(-0.124574\pi\)
−0.630540 + 0.776157i \(0.717167\pi\)
\(44\) 3.35556 + 1.22132i 0.505870 + 0.184121i
\(45\) 6.99262 − 9.01259i 1.04240 − 1.34352i
\(46\) −1.70475 + 0.620480i −0.251352 + 0.0914848i
\(47\) 3.99017 + 2.62437i 0.582026 + 0.382804i 0.806110 − 0.591765i \(-0.201569\pi\)
−0.224085 + 0.974570i \(0.571939\pi\)
\(48\) −1.02043 − 1.39954i −0.147287 − 0.202006i
\(49\) 3.16665 − 4.25355i 0.452379 − 0.607650i
\(50\) −7.90218 + 5.19734i −1.11754 + 0.735015i
\(51\) 0.183063 − 0.150530i 0.0256339 − 0.0210783i
\(52\) −1.76970 − 5.91121i −0.245413 − 0.819737i
\(53\) −2.78294 − 4.82020i −0.382266 − 0.662105i 0.609119 − 0.793078i \(-0.291523\pi\)
−0.991386 + 0.130974i \(0.958190\pi\)
\(54\) 3.67704 − 3.67143i 0.500381 − 0.499618i
\(55\) 6.78900 − 11.7589i 0.915428 − 1.58557i
\(56\) −3.41300 − 0.808895i −0.456081 − 0.108093i
\(57\) −5.56384 + 1.96260i −0.736948 + 0.259952i
\(58\) 1.17076 + 0.587978i 0.153728 + 0.0772053i
\(59\) 1.85601 + 4.30271i 0.241631 + 0.560165i 0.995273 − 0.0971215i \(-0.0309635\pi\)
−0.753641 + 0.657286i \(0.771704\pi\)
\(60\) −5.91452 + 2.89705i −0.763562 + 0.374008i
\(61\) −0.784513 + 13.4696i −0.100447 + 1.72460i 0.452824 + 0.891600i \(0.350417\pi\)
−0.553270 + 0.833002i \(0.686620\pi\)
\(62\) −0.0550839 + 0.312396i −0.00699566 + 0.0396744i
\(63\) 0.820759 − 10.4906i 0.103406 − 1.32169i
\(64\) 0.173648 + 0.984808i 0.0217060 + 0.123101i
\(65\) −23.3037 + 2.72382i −2.89047 + 0.337848i
\(66\) 3.74261 − 4.92414i 0.460683 − 0.606119i
\(67\) −2.79770 + 9.34496i −0.341793 + 1.14167i 0.598449 + 0.801161i \(0.295784\pi\)
−0.940242 + 0.340507i \(0.889401\pi\)
\(68\) −0.133146 + 0.0315562i −0.0161464 + 0.00382676i
\(69\) 0.151481 + 3.13857i 0.0182362 + 0.377839i
\(70\) −5.28253 + 12.2463i −0.631384 + 1.46371i
\(71\) 3.05189 − 2.56084i 0.362193 − 0.303916i −0.443471 − 0.896289i \(-0.646253\pi\)
0.805664 + 0.592373i \(0.201809\pi\)
\(72\) −2.83894 + 0.969758i −0.334572 + 0.114287i
\(73\) −10.7055 − 8.98301i −1.25299 − 1.05138i −0.996393 − 0.0848624i \(-0.972955\pi\)
−0.256595 − 0.966519i \(-0.582601\pi\)
\(74\) 6.81832 − 3.42429i 0.792614 − 0.398066i
\(75\) 4.54202 + 15.7398i 0.524468 + 1.81747i
\(76\) 3.38324 + 0.395444i 0.388084 + 0.0453605i
\(77\) −0.728272 − 12.5039i −0.0829942 − 1.42496i
\(78\) −10.6870 − 0.106345i −1.21006 − 0.0120412i
\(79\) 2.36891 − 2.51090i 0.266523 − 0.282498i −0.580266 − 0.814427i \(-0.697051\pi\)
0.846789 + 0.531929i \(0.178533\pi\)
\(80\) 3.80239 0.425120
\(81\) −4.18629 − 7.96712i −0.465143 − 0.885235i
\(82\) −6.35527 −0.701822
\(83\) 0.597161 − 0.632954i 0.0655470 − 0.0694757i −0.693770 − 0.720196i \(-0.744052\pi\)
0.759317 + 0.650721i \(0.225533\pi\)
\(84\) −3.08982 + 5.23083i −0.337127 + 0.570730i
\(85\) 0.0302527 + 0.519418i 0.00328136 + 0.0563388i
\(86\) −3.20497 − 0.374608i −0.345601 − 0.0403950i
\(87\) 1.57355 − 1.63497i 0.168703 − 0.175287i
\(88\) −3.19108 + 1.60262i −0.340171 + 0.170840i
\(89\) 6.42404 + 5.39041i 0.680947 + 0.571382i 0.916283 − 0.400532i \(-0.131175\pi\)
−0.235336 + 0.971914i \(0.575619\pi\)
\(90\) 1.75689 + 11.2711i 0.185192 + 1.18807i
\(91\) −16.5795 + 13.9119i −1.73801 + 1.45836i
\(92\) 0.718553 − 1.66579i 0.0749143 − 0.173671i
\(93\) 0.488514 + 0.251459i 0.0506565 + 0.0260750i
\(94\) −4.64712 + 1.10139i −0.479314 + 0.113599i
\(95\) 3.71467 − 12.4078i 0.381117 − 1.27302i
\(96\) 1.71825 + 0.218187i 0.175368 + 0.0222686i
\(97\) −6.91065 + 0.807739i −0.701670 + 0.0820134i −0.459444 − 0.888207i \(-0.651951\pi\)
−0.242225 + 0.970220i \(0.577877\pi\)
\(98\) 0.920833 + 5.22230i 0.0930182 + 0.527532i
\(99\) −6.22494 − 8.71853i −0.625630 − 0.876246i
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 162.2.g.b.151.3 yes 90
3.2 odd 2 486.2.g.b.127.5 90
81.22 even 27 inner 162.2.g.b.103.3 ✓ 90
81.59 odd 54 486.2.g.b.199.5 90
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.b.103.3 ✓ 90 81.22 even 27 inner
162.2.g.b.151.3 yes 90 1.1 even 1 trivial
486.2.g.b.127.5 90 3.2 odd 2
486.2.g.b.199.5 90 81.59 odd 54