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Properties

Label 162.2.g.b.103.3
Level $162$
Weight $2$
Character 162.103
Analytic conductor $1.294$
Analytic rank $0$
Dimension $90$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

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 103.3
Character \(\chi\) \(=\) 162.103
Dual form 162.2.g.b.151.3

$q$-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 \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{7}{27}\right)\)

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
\(3\) 0.851057 − 1.50854i 0.491358 − 0.870958i
\(4\) −0.0581448 + 0.998308i −0.0290724 + 0.499154i
\(5\) −3.77668 + 0.441430i −1.68898 + 0.197414i −0.905602 − 0.424129i \(-0.860580\pi\)
−0.783380 + 0.621543i \(0.786506\pi\)
\(6\) −1.68131 + 0.416189i −0.686390 + 0.169908i
\(7\) −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
\(9\) −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 −0.987033 − 0.160519i \(-0.948683\pi\)
0.560585 − 0.828097i \(-0.310576\pi\)
\(12\) 1.45651 + 0.937331i 0.420457 + 0.270584i
\(13\) 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
\(15\) −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 −0.987554 − 0.157283i \(-0.949727\pi\)
0.981791 + 0.189966i \(0.0608377\pi\)
\(18\) −0.803048 + 2.89052i −0.189280 + 0.681302i
\(19\) −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
\(21\) −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 −0.271677 − 0.962388i \(-0.587578\pi\)
0.609720 + 0.792617i \(0.291282\pi\)
\(24\) −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 −0.985763 − 0.168143i \(-0.946223\pi\)
0.915989 + 0.401204i \(0.131408\pi\)
\(30\) 6.16603 − 2.31399i 1.12576 − 0.422475i
\(31\) 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
\(33\) −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 −0.855743 − 0.517402i \(-0.826899\pi\)
−0.322958 + 0.946413i \(0.604677\pi\)
\(38\) −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 −0.290929 − 0.956745i \(-0.593964\pi\)
0.972042 + 0.234807i \(0.0754460\pi\)
\(42\) 5.92513 + 1.34216i 0.914268 + 0.207099i
\(43\) 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
\(45\) 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 −0.224085 − 0.974570i \(-0.571939\pi\)
0.806110 + 0.591765i \(0.201569\pi\)
\(48\) −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 −0.991386 − 0.130974i \(-0.958190\pi\)
0.609119 + 0.793078i \(0.291523\pi\)
\(54\) 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 −0.753641 − 0.657286i \(-0.771704\pi\)
0.995273 + 0.0971215i \(0.0309635\pi\)
\(60\) −5.91452 − 2.89705i −0.763562 − 0.374008i
\(61\) −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
\(63\) 0.820759 + 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 −0.940242 − 0.340507i \(-0.889401\pi\)
0.598449 − 0.801161i \(-0.295784\pi\)
\(68\) −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 0.805664 − 0.592373i \(-0.201809\pi\)
−0.443471 + 0.896289i \(0.646253\pi\)
\(72\) −2.83894 − 0.969758i −0.334572 − 0.114287i
\(73\) −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
\(75\) 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 0.846789 − 0.531929i \(-0.178533\pi\)
−0.580266 + 0.814427i \(0.697051\pi\)
\(80\) 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 0.759317 − 0.650721i \(-0.225533\pi\)
−0.693770 + 0.720196i \(0.744052\pi\)
\(84\) −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 −0.235336 − 0.971914i \(-0.575619\pi\)
0.916283 + 0.400532i \(0.131175\pi\)
\(90\) 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 −0.242225 − 0.970220i \(-0.577877\pi\)
−0.459444 + 0.888207i \(0.651951\pi\)
\(98\) 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.103.3 ✓ 90
3.2 odd 2 486.2.g.b.199.5 90
81.11 odd 54 486.2.g.b.127.5 90
81.70 even 27 inner 162.2.g.b.151.3 yes 90
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.b.103.3 ✓ 90 1.1 even 1 trivial
162.2.g.b.151.3 yes 90 81.70 even 27 inner
486.2.g.b.127.5 90 81.11 odd 54
486.2.g.b.199.5 90 3.2 odd 2