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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [270,2,Mod(49,270)] 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("270.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([14, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 270.p (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.15596085457\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 259.13
Character \(\chi\) \(=\) 270.259
Dual form 270.2.p.a.49.13

$q$-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+(0.642788 - 0.766044i) q^{2} +(-0.726309 - 1.57241i) q^{3} +(-0.173648 - 0.984808i) q^{4} +(1.96492 - 1.06728i) q^{5} +(-1.67140 - 0.454341i) q^{6} +(4.83804 + 0.853077i) q^{7} +(-0.866025 - 0.500000i) q^{8} +(-1.94495 + 2.28411i) q^{9} +(0.445446 - 2.19125i) q^{10} +(-1.60584 + 0.584478i) q^{11} +(-1.42240 + 0.988321i) q^{12} +(-1.95350 - 2.32809i) q^{13} +(3.76333 - 3.15781i) q^{14} +(-3.10534 - 2.31449i) q^{15} +(-0.939693 + 0.342020i) q^{16} +(-3.04109 + 1.75578i) q^{17} +(0.499540 + 2.95812i) q^{18} +(-0.936774 + 1.62254i) q^{19} +(-1.39227 - 1.74974i) q^{20} +(-2.17252 - 8.22699i) q^{21} +(-0.584478 + 1.60584i) q^{22} +(-2.79477 + 0.492792i) q^{23} +(-0.157204 + 1.72490i) q^{24} +(2.72184 - 4.19423i) q^{25} -3.03911 q^{26} +(5.00420 + 1.39929i) q^{27} -4.91268i q^{28} +(2.46183 + 2.06572i) q^{29} +(-3.76908 + 0.891101i) q^{30} +(0.994686 + 5.64114i) q^{31} +(-0.342020 + 0.939693i) q^{32} +(2.08538 + 2.10053i) q^{33} +(-0.609774 + 3.45820i) q^{34} +(10.4168 - 3.48730i) q^{35} +(2.58715 + 1.51877i) q^{36} +(1.92700 - 1.11256i) q^{37} +(0.640791 + 1.76056i) q^{38} +(-2.24187 + 4.76262i) q^{39} +(-2.23531 - 0.0581716i) q^{40} +(-0.451861 + 0.379157i) q^{41} +(-7.69871 - 3.62395i) q^{42} +(3.99619 + 10.9794i) q^{43} +(0.854450 + 1.47995i) q^{44} +(-1.38390 + 6.56390i) q^{45} +(-1.41894 + 2.45768i) q^{46} +(-13.0603 - 2.30288i) q^{47} +(1.22030 + 1.22917i) q^{48} +(16.1011 + 5.86031i) q^{49} +(-1.46341 - 4.78105i) q^{50} +(4.96957 + 3.50661i) q^{51} +(-1.95350 + 2.32809i) q^{52} -13.5050i q^{53} +(4.28856 - 2.93399i) q^{54} +(-2.53155 + 2.86233i) q^{55} +(-3.76333 - 3.15781i) q^{56} +(3.23169 + 0.294528i) q^{57} +(3.16487 - 0.558051i) q^{58} +(4.14855 + 1.50995i) q^{59} +(-1.74009 + 3.46007i) q^{60} +(1.58468 - 8.98717i) q^{61} +(4.96074 + 2.86408i) q^{62} +(-11.3583 + 9.39143i) q^{63} +(0.500000 + 0.866025i) q^{64} +(-6.32319 - 2.48959i) q^{65} +(2.94955 - 0.247296i) q^{66} +(4.47017 + 5.32734i) q^{67} +(2.25718 + 2.69000i) q^{68} +(2.80473 + 4.03660i) q^{69} +(4.02439 - 10.2214i) q^{70} +(2.54561 + 4.40913i) q^{71} +(2.82643 - 1.00562i) q^{72} +(-11.3098 - 6.52971i) q^{73} +(0.386387 - 2.19131i) q^{74} +(-8.57195 - 1.23354i) q^{75} +(1.76056 + 0.640791i) q^{76} +(-8.26773 + 1.45782i) q^{77} +(2.20733 + 4.77872i) q^{78} +(4.76174 + 3.99557i) q^{79} +(-1.48139 + 1.67496i) q^{80} +(-1.43433 - 8.88497i) q^{81} +0.589863i q^{82} +(2.47161 - 2.94555i) q^{83} +(-7.72475 + 3.56812i) q^{84} +(-4.10161 + 6.69565i) q^{85} +(10.9794 + 3.99619i) q^{86} +(1.46011 - 5.37136i) q^{87} +(1.68294 + 0.296747i) q^{88} +(-0.280096 + 0.485141i) q^{89} +(4.13869 + 5.27932i) q^{90} +(-7.46507 - 12.9299i) q^{91} +(0.970612 + 2.66673i) q^{92} +(8.14775 - 5.66127i) q^{93} +(-10.1591 + 8.52448i) q^{94} +(-0.108987 + 4.18796i) q^{95} +(1.72600 - 0.144711i) q^{96} +(2.45704 + 6.75067i) q^{97} +(14.8388 - 8.56719i) q^{98} +(1.78827 - 4.80470i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q + 6 q^{5} - 6 q^{6} + 12 q^{9} - 12 q^{11} + 6 q^{14} + 18 q^{15} + 12 q^{20} - 72 q^{21} - 18 q^{25} - 72 q^{26} + 6 q^{29} + 36 q^{30} + 36 q^{31} + 18 q^{35} - 30 q^{36} - 48 q^{39} - 12 q^{41}+ \cdots - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(217\)
\(\chi(n)\) \(e\left(\frac{2}{9}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.642788 − 0.766044i 0.454519 − 0.541675i
\(3\) −0.726309 − 1.57241i −0.419335 − 0.907832i
\(4\) −0.173648 − 0.984808i −0.0868241 − 0.492404i
\(5\) 1.96492 − 1.06728i 0.878740 − 0.477301i
\(6\) −1.67140 − 0.454341i −0.682346 − 0.185484i
\(7\) 4.83804 + 0.853077i 1.82861 + 0.322433i 0.978825 − 0.204700i \(-0.0656218\pi\)
0.849783 + 0.527133i \(0.176733\pi\)
\(8\) −0.866025 − 0.500000i −0.306186 − 0.176777i
\(9\) −1.94495 + 2.28411i −0.648317 + 0.761370i
\(10\) 0.445446 − 2.19125i 0.140862 − 0.692934i
\(11\) −1.60584 + 0.584478i −0.484179 + 0.176227i −0.572565 − 0.819860i \(-0.694051\pi\)
0.0883856 + 0.996086i \(0.471829\pi\)
\(12\) −1.42240 + 0.988321i −0.410612 + 0.285304i
\(13\) −1.95350 − 2.32809i −0.541803 − 0.645696i 0.423788 − 0.905762i \(-0.360700\pi\)
−0.965591 + 0.260066i \(0.916256\pi\)
\(14\) 3.76333 − 3.15781i 1.00579 − 0.843960i
\(15\) −3.10534 − 2.31449i −0.801795 − 0.597599i
\(16\) −0.939693 + 0.342020i −0.234923 + 0.0855050i
\(17\) −3.04109 + 1.75578i −0.737573 + 0.425838i −0.821186 − 0.570660i \(-0.806687\pi\)
0.0836130 + 0.996498i \(0.473354\pi\)
\(18\) 0.499540 + 2.95812i 0.117743 + 0.697235i
\(19\) −0.936774 + 1.62254i −0.214911 + 0.372236i −0.953245 − 0.302199i \(-0.902279\pi\)
0.738334 + 0.674435i \(0.235613\pi\)
\(20\) −1.39227 − 1.74974i −0.311321 − 0.391254i
\(21\) −2.17252 − 8.22699i −0.474084 − 1.79528i
\(22\) −0.584478 + 1.60584i −0.124611 + 0.342366i
\(23\) −2.79477 + 0.492792i −0.582749 + 0.102754i −0.457248 − 0.889339i \(-0.651165\pi\)
−0.125501 + 0.992094i \(0.540054\pi\)
\(24\) −0.157204 + 1.72490i −0.0320890 + 0.352094i
\(25\) 2.72184 − 4.19423i 0.544367 − 0.838847i
\(26\) −3.03911 −0.596018
\(27\) 5.00420 + 1.39929i 0.963058 + 0.269294i
\(28\) − 4.91268i − 0.928409i
\(29\) 2.46183 + 2.06572i 0.457150 + 0.383595i 0.842081 − 0.539351i \(-0.181330\pi\)
−0.384931 + 0.922945i \(0.625775\pi\)
\(30\) −3.76908 + 0.891101i −0.688136 + 0.162692i
\(31\) 0.994686 + 5.64114i 0.178651 + 1.01318i 0.933845 + 0.357678i \(0.116431\pi\)
−0.755194 + 0.655501i \(0.772457\pi\)
\(32\) −0.342020 + 0.939693i −0.0604612 + 0.166116i
\(33\) 2.08538 + 2.10053i 0.363017 + 0.365655i
\(34\) −0.609774 + 3.45820i −0.104575 + 0.593077i
\(35\) 10.4168 − 3.48730i 1.76077 − 0.589462i
\(36\) 2.58715 + 1.51877i 0.431191 + 0.253129i
\(37\) 1.92700 − 1.11256i 0.316797 − 0.182903i −0.333167 − 0.942868i \(-0.608117\pi\)
0.649964 + 0.759965i \(0.274784\pi\)
\(38\) 0.640791 + 1.76056i 0.103950 + 0.285601i
\(39\) −2.24187 + 4.76262i −0.358987 + 0.762629i
\(40\) −2.23531 − 0.0581716i −0.353434 − 0.00919774i
\(41\) −0.451861 + 0.379157i −0.0705689 + 0.0592143i −0.677390 − 0.735624i \(-0.736889\pi\)
0.606821 + 0.794839i \(0.292444\pi\)
\(42\) −7.69871 − 3.62395i −1.18794 − 0.559188i
\(43\) 3.99619 + 10.9794i 0.609413 + 1.67435i 0.731511 + 0.681830i \(0.238815\pi\)
−0.122098 + 0.992518i \(0.538962\pi\)
\(44\) 0.854450 + 1.47995i 0.128813 + 0.223111i
\(45\) −1.38390 + 6.56390i −0.206299 + 0.978489i
\(46\) −1.41894 + 2.45768i −0.209211 + 0.362364i
\(47\) −13.0603 − 2.30288i −1.90504 − 0.335909i −0.908416 − 0.418068i \(-0.862707\pi\)
−0.996619 + 0.0821587i \(0.973819\pi\)
\(48\) 1.22030 + 1.22917i 0.176136 + 0.177415i
\(49\) 16.1011 + 5.86031i 2.30015 + 0.837187i
\(50\) −1.46341 − 4.78105i −0.206957 − 0.676143i
\(51\) 4.96957 + 3.50661i 0.695879 + 0.491024i
\(52\) −1.95350 + 2.32809i −0.270902 + 0.322848i
\(53\) − 13.5050i − 1.85506i −0.373754 − 0.927528i \(-0.621930\pi\)
0.373754 − 0.927528i \(-0.378070\pi\)
\(54\) 4.28856 − 2.93399i 0.583598 − 0.399265i
\(55\) −2.53155 + 2.86233i −0.341354 + 0.385957i
\(56\) −3.76333 − 3.15781i −0.502896 − 0.421980i
\(57\) 3.23169 + 0.294528i 0.428048 + 0.0390113i
\(58\) 3.16487 − 0.558051i 0.415567 − 0.0732757i
\(59\) 4.14855 + 1.50995i 0.540095 + 0.196579i 0.597640 − 0.801764i \(-0.296105\pi\)
−0.0575451 + 0.998343i \(0.518327\pi\)
\(60\) −1.74009 + 3.46007i −0.224645 + 0.446693i
\(61\) 1.58468 − 8.98717i 0.202897 − 1.15069i −0.697817 − 0.716276i \(-0.745845\pi\)
0.900714 − 0.434412i \(-0.143044\pi\)
\(62\) 4.96074 + 2.86408i 0.630014 + 0.363739i
\(63\) −11.3583 + 9.39143i −1.43101 + 1.18321i
\(64\) 0.500000 + 0.866025i 0.0625000 + 0.108253i
\(65\) −6.32319 − 2.48959i −0.784295 − 0.308796i
\(66\) 2.94955 − 0.247296i 0.363065 − 0.0304401i
\(67\) 4.47017 + 5.32734i 0.546119 + 0.650839i 0.966548 − 0.256487i \(-0.0825649\pi\)
−0.420429 + 0.907325i \(0.638120\pi\)
\(68\) 2.25718 + 2.69000i 0.273723 + 0.326211i
\(69\) 2.80473 + 4.03660i 0.337650 + 0.485949i
\(70\) 4.02439 − 10.2214i 0.481007 − 1.22169i
\(71\) 2.54561 + 4.40913i 0.302109 + 0.523268i 0.976613 − 0.215003i \(-0.0689761\pi\)
−0.674505 + 0.738271i \(0.735643\pi\)
\(72\) 2.82643 − 1.00562i 0.333098 − 0.118514i
\(73\) −11.3098 − 6.52971i −1.32371 − 0.764245i −0.339392 − 0.940645i \(-0.610221\pi\)
−0.984319 + 0.176400i \(0.943555\pi\)
\(74\) 0.386387 − 2.19131i 0.0449165 − 0.254734i
\(75\) −8.57195 − 1.23354i −0.989804 − 0.142437i
\(76\) 1.76056 + 0.640791i 0.201950 + 0.0735038i
\(77\) −8.26773 + 1.45782i −0.942195 + 0.166134i
\(78\) 2.20733 + 4.77872i 0.249931 + 0.541084i
\(79\) 4.76174 + 3.99557i 0.535738 + 0.449537i 0.870077 − 0.492915i \(-0.164069\pi\)
−0.334340 + 0.942453i \(0.608513\pi\)
\(80\) −1.48139 + 1.67496i −0.165625 + 0.187266i
\(81\) −1.43433 − 8.88497i −0.159370 − 0.987219i
\(82\) 0.589863i 0.0651395i
\(83\) 2.47161 − 2.94555i 0.271294 − 0.323316i −0.613146 − 0.789970i \(-0.710096\pi\)
0.884440 + 0.466654i \(0.154541\pi\)
\(84\) −7.72475 + 3.56812i −0.842839 + 0.389314i
\(85\) −4.10161 + 6.69565i −0.444882 + 0.726245i
\(86\) 10.9794 + 3.99619i 1.18394 + 0.430920i
\(87\) 1.46011 − 5.37136i 0.156540 − 0.575870i
\(88\) 1.68294 + 0.296747i 0.179402 + 0.0316334i
\(89\) −0.280096 + 0.485141i −0.0296902 + 0.0514249i −0.880489 − 0.474067i \(-0.842785\pi\)
0.850799 + 0.525492i \(0.176119\pi\)
\(90\) 4.13869 + 5.27932i 0.436256 + 0.556489i
\(91\) −7.46507 − 12.9299i −0.782552 − 1.35542i
\(92\) 0.970612 + 2.66673i 0.101193 + 0.278026i
\(93\) 8.14775 − 5.66127i 0.844882 − 0.587046i
\(94\) −10.1591 + 8.52448i −1.04783 + 0.879233i
\(95\) −0.108987 + 4.18796i −0.0111819 + 0.429676i
\(96\) 1.72600 − 0.144711i 0.176159 − 0.0147695i
\(97\) 2.45704 + 6.75067i 0.249475 + 0.685427i 0.999706 + 0.0242509i \(0.00772006\pi\)
−0.750231 + 0.661176i \(0.770058\pi\)
\(98\) 14.8388 − 8.56719i 1.49895 − 0.865417i
\(99\) 1.78827 − 4.80470i 0.179728 − 0.482890i
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 270.2.p.a.259.13 yes 108
3.2 odd 2 810.2.p.a.289.2 108
5.4 even 2 inner 270.2.p.a.259.6 yes 108
15.14 odd 2 810.2.p.a.289.13 108
27.5 odd 18 810.2.p.a.199.13 108
27.22 even 9 inner 270.2.p.a.49.6 ✓ 108
135.49 even 18 inner 270.2.p.a.49.13 yes 108
135.59 odd 18 810.2.p.a.199.2 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
270.2.p.a.49.6 ✓ 108 27.22 even 9 inner
270.2.p.a.49.13 yes 108 135.49 even 18 inner
270.2.p.a.259.6 yes 108 5.4 even 2 inner
270.2.p.a.259.13 yes 108 1.1 even 1 trivial
810.2.p.a.199.2 108 135.59 odd 18
810.2.p.a.199.13 108 27.5 odd 18
810.2.p.a.289.2 108 3.2 odd 2
810.2.p.a.289.13 108 15.14 odd 2