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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 13.5
Character \(\chi\) \(=\) 162.13
Dual form 162.2.g.b.25.5

$q$-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+(0.893633 + 0.448799i) q^{2} +(1.51179 + 0.845280i) q^{3} +(0.597159 + 0.802123i) q^{4} +(-0.379535 - 1.26773i) q^{5} +(0.971622 + 1.43386i) q^{6} +(-0.768169 - 1.78082i) q^{7} +(0.173648 + 0.984808i) q^{8} +(1.57100 + 2.55577i) q^{9} +(0.229793 - 1.30322i) q^{10} +(-2.10911 + 0.499867i) q^{11} +(0.224758 + 1.71741i) q^{12} +(-1.69934 - 1.11767i) q^{13} +(0.112768 - 1.93615i) q^{14} +(0.497815 - 2.23736i) q^{15} +(-0.286803 + 0.957990i) q^{16} +(-7.02180 + 2.55572i) q^{17} +(0.256873 + 2.98898i) q^{18} +(2.13356 + 0.776551i) q^{19} +(0.790236 - 1.06147i) q^{20} +(0.343980 - 3.34153i) q^{21} +(-2.10911 - 0.499867i) q^{22} +(1.34938 - 3.12822i) q^{23} +(-0.569919 + 1.63560i) q^{24} +(2.71434 - 1.78525i) q^{25} +(-1.01697 - 1.76145i) q^{26} +(0.214683 + 5.19172i) q^{27} +(0.969715 - 1.67960i) q^{28} +(-0.282496 - 4.85027i) q^{29} +(1.44899 - 1.77596i) q^{30} +(5.07563 - 0.593256i) q^{31} +(-0.686242 + 0.727374i) q^{32} +(-3.61105 - 1.02709i) q^{33} +(-7.42191 - 0.867497i) q^{34} +(-1.96605 + 1.64972i) q^{35} +(-1.11190 + 2.78634i) q^{36} +(4.70814 + 3.95060i) q^{37} +(1.55810 + 1.65149i) q^{38} +(-1.62429 - 3.12610i) q^{39} +(1.18257 - 0.593908i) q^{40} +(-6.98634 + 3.50867i) q^{41} +(1.80707 - 2.83173i) q^{42} +(-4.77609 - 5.06236i) q^{43} +(-1.66043 - 1.39326i) q^{44} +(2.64378 - 2.96162i) q^{45} +(2.60979 - 2.18988i) q^{46} +(4.18562 + 0.489229i) q^{47} +(-1.24336 + 1.20585i) q^{48} +(2.22247 - 2.35568i) q^{49} +(3.22684 - 0.377163i) q^{50} +(-12.7758 - 2.07167i) q^{51} +(-0.118264 - 2.03051i) q^{52} +(-5.39818 + 9.34993i) q^{53} +(-2.13819 + 4.73584i) q^{54} +(1.43418 + 2.48407i) q^{55} +(1.62037 - 1.06573i) q^{56} +(2.56908 + 2.97743i) q^{57} +(1.92435 - 4.46115i) q^{58} +(6.72062 + 1.59282i) q^{59} +(2.09191 - 0.936749i) q^{60} +(-5.37986 + 7.22641i) q^{61} +(4.80200 + 1.74779i) q^{62} +(3.34456 - 4.76093i) q^{63} +(-0.939693 + 0.342020i) q^{64} +(-0.771954 + 2.57851i) q^{65} +(-2.76599 - 2.53848i) q^{66} +(0.661912 - 11.3646i) q^{67} +(-6.24313 - 4.10617i) q^{68} +(4.68420 - 3.58860i) q^{69} +(-2.49732 + 0.591876i) q^{70} +(0.518333 - 2.93961i) q^{71} +(-2.24414 + 1.99094i) q^{72} +(1.80390 + 10.2304i) q^{73} +(2.43432 + 5.64339i) q^{74} +(5.61253 - 0.404540i) q^{75} +(0.651182 + 2.17510i) q^{76} +(2.51032 + 3.37195i) q^{77} +(-0.0485295 - 3.52257i) q^{78} +(9.16461 + 4.60264i) q^{79} +1.32333 q^{80} +(-4.06390 + 8.03024i) q^{81} -7.81791 q^{82} +(3.24547 + 1.62994i) q^{83} +(2.88573 - 1.71951i) q^{84} +(5.90499 + 7.93178i) q^{85} +(-1.99609 - 6.66740i) q^{86} +(3.67276 - 7.57137i) q^{87} +(-0.858516 - 1.99026i) q^{88} +(1.42927 + 8.10577i) q^{89} +(3.69174 - 1.46007i) q^{90} +(-0.684990 + 3.88477i) q^{91} +(3.31501 - 0.785672i) q^{92} +(8.17474 + 3.39345i) q^{93} +(3.52084 + 2.31569i) q^{94} +(0.174701 - 2.99951i) q^{95} +(-1.65229 + 0.519568i) q^{96} +(-2.30761 + 7.70796i) q^{97} +(3.04330 - 1.10767i) q^{98} +(-4.59096 - 4.60509i) 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 \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(e\left(\frac{4}{27}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.893633 + 0.448799i 0.631894 + 0.317349i
\(3\) 1.51179 + 0.845280i 0.872831 + 0.488023i
\(4\) 0.597159 + 0.802123i 0.298579 + 0.401062i
\(5\) −0.379535 − 1.26773i −0.169733 − 0.566948i −0.999955 − 0.00951452i \(-0.996971\pi\)
0.830222 − 0.557433i \(-0.188214\pi\)
\(6\) 0.971622 + 1.43386i 0.396663 + 0.585370i
\(7\) −0.768169 − 1.78082i −0.290341 − 0.673085i 0.709148 − 0.705060i \(-0.249080\pi\)
−0.999489 + 0.0319744i \(0.989821\pi\)
\(8\) 0.173648 + 0.984808i 0.0613939 + 0.348182i
\(9\) 1.57100 + 2.55577i 0.523668 + 0.851923i
\(10\) 0.229793 − 1.30322i 0.0726671 − 0.412115i
\(11\) −2.10911 + 0.499867i −0.635919 + 0.150716i −0.535916 − 0.844271i \(-0.680034\pi\)
−0.100004 + 0.994987i \(0.531885\pi\)
\(12\) 0.224758 + 1.71741i 0.0648821 + 0.495772i
\(13\) −1.69934 − 1.11767i −0.471312 − 0.309987i 0.291541 − 0.956558i \(-0.405832\pi\)
−0.762854 + 0.646571i \(0.776202\pi\)
\(14\) 0.112768 − 1.93615i 0.0301385 − 0.517458i
\(15\) 0.497815 − 2.23736i 0.128535 − 0.577683i
\(16\) −0.286803 + 0.957990i −0.0717008 + 0.239497i
\(17\) −7.02180 + 2.55572i −1.70304 + 0.619854i −0.996165 − 0.0874893i \(-0.972116\pi\)
−0.706870 + 0.707343i \(0.749893\pi\)
\(18\) 0.256873 + 2.98898i 0.0605456 + 0.704510i
\(19\) 2.13356 + 0.776551i 0.489471 + 0.178153i 0.574952 − 0.818187i \(-0.305021\pi\)
−0.0854813 + 0.996340i \(0.527243\pi\)
\(20\) 0.790236 − 1.06147i 0.176702 − 0.237352i
\(21\) 0.343980 − 3.34153i 0.0750626 − 0.729182i
\(22\) −2.10911 − 0.499867i −0.449663 − 0.106572i
\(23\) 1.34938 − 3.12822i 0.281366 − 0.652279i −0.717661 − 0.696393i \(-0.754787\pi\)
0.999027 + 0.0441139i \(0.0140465\pi\)
\(24\) −0.569919 + 1.63560i −0.116334 + 0.333866i
\(25\) 2.71434 − 1.78525i 0.542867 − 0.357049i
\(26\) −1.01697 − 1.76145i −0.199445 − 0.345449i
\(27\) 0.214683 + 5.19172i 0.0413159 + 0.999146i
\(28\) 0.969715 − 1.67960i 0.183259 − 0.317414i
\(29\) −0.282496 − 4.85027i −0.0524582 − 0.900673i −0.917193 − 0.398444i \(-0.869550\pi\)
0.864735 − 0.502229i \(-0.167487\pi\)
\(30\) 1.44899 − 1.77596i 0.264548 − 0.324244i
\(31\) 5.07563 − 0.593256i 0.911610 − 0.106552i 0.352658 − 0.935752i \(-0.385278\pi\)
0.558952 + 0.829200i \(0.311204\pi\)
\(32\) −0.686242 + 0.727374i −0.121312 + 0.128583i
\(33\) −3.61105 − 1.02709i −0.628603 − 0.178794i
\(34\) −7.42191 − 0.867497i −1.27285 − 0.148775i
\(35\) −1.96605 + 1.64972i −0.332324 + 0.278853i
\(36\) −1.11190 + 2.78634i −0.185317 + 0.464389i
\(37\) 4.70814 + 3.95060i 0.774013 + 0.649474i 0.941733 − 0.336360i \(-0.109196\pi\)
−0.167720 + 0.985835i \(0.553640\pi\)
\(38\) 1.55810 + 1.65149i 0.252757 + 0.267907i
\(39\) −1.62429 − 3.12610i −0.260095 − 0.500577i
\(40\) 1.18257 − 0.593908i 0.186981 − 0.0939052i
\(41\) −6.98634 + 3.50867i −1.09108 + 0.547962i −0.901006 − 0.433807i \(-0.857170\pi\)
−0.190077 + 0.981769i \(0.560874\pi\)
\(42\) 1.80707 − 2.83173i 0.278837 − 0.436945i
\(43\) −4.77609 − 5.06236i −0.728347 − 0.772003i 0.252329 − 0.967641i \(-0.418803\pi\)
−0.980676 + 0.195639i \(0.937322\pi\)
\(44\) −1.66043 − 1.39326i −0.250319 − 0.210042i
\(45\) 2.64378 − 2.96162i 0.394112 − 0.441492i
\(46\) 2.60979 − 2.18988i 0.384793 − 0.322880i
\(47\) 4.18562 + 0.489229i 0.610535 + 0.0713614i 0.415741 − 0.909483i \(-0.363522\pi\)
0.194794 + 0.980844i \(0.437596\pi\)
\(48\) −1.24336 + 1.20585i −0.179463 + 0.174049i
\(49\) 2.22247 − 2.35568i 0.317496 − 0.336526i
\(50\) 3.22684 − 0.377163i 0.456344 − 0.0533389i
\(51\) −12.7758 − 2.07167i −1.78896 − 0.290092i
\(52\) −0.118264 − 2.03051i −0.0164002 − 0.281581i
\(53\) −5.39818 + 9.34993i −0.741497 + 1.28431i 0.210316 + 0.977633i \(0.432551\pi\)
−0.951813 + 0.306678i \(0.900783\pi\)
\(54\) −2.13819 + 4.73584i −0.290971 + 0.644466i
\(55\) 1.43418 + 2.48407i 0.193384 + 0.334952i
\(56\) 1.62037 − 1.06573i 0.216531 − 0.142415i
\(57\) 2.56908 + 2.97743i 0.340283 + 0.394370i
\(58\) 1.92435 − 4.46115i 0.252680 − 0.585777i
\(59\) 6.72062 + 1.59282i 0.874950 + 0.207367i 0.643469 − 0.765472i \(-0.277495\pi\)
0.231481 + 0.972839i \(0.425643\pi\)
\(60\) 2.09191 − 0.936749i 0.270065 − 0.120934i
\(61\) −5.37986 + 7.22641i −0.688820 + 0.925246i −0.999682 − 0.0252087i \(-0.991975\pi\)
0.310862 + 0.950455i \(0.399382\pi\)
\(62\) 4.80200 + 1.74779i 0.609855 + 0.221969i
\(63\) 3.34456 − 4.76093i 0.421374 − 0.599821i
\(64\) −0.939693 + 0.342020i −0.117462 + 0.0427525i
\(65\) −0.771954 + 2.57851i −0.0957491 + 0.319824i
\(66\) −2.76599 − 2.53848i −0.340470 − 0.312465i
\(67\) 0.661912 − 11.3646i 0.0808654 − 1.38841i −0.676994 − 0.735988i \(-0.736718\pi\)
0.757860 − 0.652417i \(-0.226245\pi\)
\(68\) −6.24313 − 4.10617i −0.757091 − 0.497946i
\(69\) 4.68420 − 3.58860i 0.563911 − 0.432016i
\(70\) −2.49732 + 0.591876i −0.298487 + 0.0707427i
\(71\) 0.518333 − 2.93961i 0.0615148 − 0.348868i −0.938479 − 0.345336i \(-0.887765\pi\)
0.999994 − 0.00353124i \(-0.00112403\pi\)
\(72\) −2.24414 + 1.99094i −0.264474 + 0.234635i
\(73\) 1.80390 + 10.2304i 0.211131 + 1.19738i 0.887496 + 0.460816i \(0.152443\pi\)
−0.676365 + 0.736567i \(0.736446\pi\)
\(74\) 2.43432 + 5.64339i 0.282984 + 0.656031i
\(75\) 5.61253 − 0.404540i 0.648079 − 0.0467122i
\(76\) 0.651182 + 2.17510i 0.0746956 + 0.249501i
\(77\) 2.51032 + 3.37195i 0.286078 + 0.384269i
\(78\) −0.0485295 − 3.52257i −0.00549489 − 0.398852i
\(79\) 9.16461 + 4.60264i 1.03110 + 0.517837i 0.882131 − 0.471005i \(-0.156108\pi\)
0.148968 + 0.988842i \(0.452405\pi\)
\(80\) 1.32333 0.147953
\(81\) −4.06390 + 8.03024i −0.451544 + 0.892249i
\(82\) −7.81791 −0.863344
\(83\) 3.24547 + 1.62994i 0.356237 + 0.178909i 0.617912 − 0.786247i \(-0.287979\pi\)
−0.261675 + 0.965156i \(0.584275\pi\)
\(84\) 2.88573 − 1.71951i 0.314859 − 0.187614i
\(85\) 5.90499 + 7.93178i 0.640486 + 0.860323i
\(86\) −1.99609 − 6.66740i −0.215244 − 0.718964i
\(87\) 3.67276 − 7.57137i 0.393762 − 0.811736i
\(88\) −0.858516 − 1.99026i −0.0915181 − 0.212163i
\(89\) 1.42927 + 8.10577i 0.151502 + 0.859210i 0.961915 + 0.273350i \(0.0881319\pi\)
−0.810413 + 0.585860i \(0.800757\pi\)
\(90\) 3.69174 − 1.46007i 0.389144 − 0.153905i
\(91\) −0.684990 + 3.88477i −0.0718065 + 0.407235i
\(92\) 3.31501 − 0.785672i 0.345614 − 0.0819120i
\(93\) 8.17474 + 3.39345i 0.847681 + 0.351885i
\(94\) 3.52084 + 2.31569i 0.363147 + 0.238846i
\(95\) 0.174701 − 2.99951i 0.0179240 − 0.307743i
\(96\) −1.65229 + 0.519568i −0.168636 + 0.0530282i
\(97\) −2.30761 + 7.70796i −0.234302 + 0.782624i 0.757274 + 0.653098i \(0.226531\pi\)
−0.991576 + 0.129527i \(0.958654\pi\)
\(98\) 3.04330 − 1.10767i 0.307420 − 0.111892i
\(99\) −4.59096 − 4.60509i −0.461409 − 0.462829i
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.13.5 ✓ 90
3.2 odd 2 486.2.g.b.253.3 90
81.25 even 27 inner 162.2.g.b.25.5 yes 90
81.56 odd 54 486.2.g.b.73.3 90
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.2.g.b.13.5 ✓ 90 1.1 even 1 trivial
162.2.g.b.25.5 yes 90 81.25 even 27 inner
486.2.g.b.73.3 90 81.56 odd 54
486.2.g.b.253.3 90 3.2 odd 2