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Properties

Label 279.2.o.a.212.9
Level $279$
Weight $2$
Character 279.212
Analytic conductor $2.228$
Analytic rank $0$
Dimension $60$
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] = [279,2,Mod(212,279)] 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("279.212"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(279, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 279 = 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 279.o (of order \(6\), degree \(2\), 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: \(2.22782621639\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 212.9
Character \(\chi\) \(=\) 279.212
Dual form 279.2.o.a.254.9

$q$-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+(-1.38533 + 0.799823i) q^{2} +(1.73203 - 0.00916385i) q^{3} +(0.279433 - 0.483993i) q^{4} -2.80402i q^{5} +(-2.39211 + 1.39801i) q^{6} -0.641206 q^{7} -2.30530i q^{8} +(2.99983 - 0.0317441i) q^{9} +(2.24272 + 3.88451i) q^{10} +(0.253632 - 0.439304i) q^{11} +(0.479551 - 0.840849i) q^{12} -4.81427i q^{13} +(0.888285 - 0.512851i) q^{14} +(-0.0256956 - 4.85664i) q^{15} +(2.40270 + 4.16160i) q^{16} +(3.17820 - 5.50481i) q^{17} +(-4.13038 + 2.44331i) q^{18} +(-3.47968 + 6.02698i) q^{19} +(-1.35713 - 0.783537i) q^{20} +(-1.11059 + 0.00587592i) q^{21} +0.811444i q^{22} +(-0.813677 + 1.40933i) q^{23} +(-0.0211255 - 3.99285i) q^{24} -2.86253 q^{25} +(3.85057 + 6.66938i) q^{26} +(5.19550 - 0.0824716i) q^{27} +(-0.179174 + 0.310339i) q^{28} +(-1.45418 - 2.51871i) q^{29} +(3.92005 + 6.70752i) q^{30} +(5.27752 + 1.77421i) q^{31} +(-2.66418 - 1.53817i) q^{32} +(0.435272 - 0.763211i) q^{33} +10.1680i q^{34} +1.79796i q^{35} +(0.822889 - 1.46077i) q^{36} +(6.66807 + 3.84981i) q^{37} -11.1325i q^{38} +(-0.0441173 - 8.33845i) q^{39} -6.46412 q^{40} +1.32592i q^{41} +(1.53383 - 0.896412i) q^{42} +0.998476i q^{43} +(-0.141747 - 0.245513i) q^{44} +(-0.0890110 - 8.41159i) q^{45} -2.60319i q^{46} +(0.634446 - 0.366298i) q^{47} +(4.19968 + 7.18598i) q^{48} -6.58885 q^{49} +(3.96557 - 2.28952i) q^{50} +(5.45429 - 9.56360i) q^{51} +(-2.33007 - 1.34527i) q^{52} +(3.07274 + 5.32214i) q^{53} +(-7.13154 + 4.26973i) q^{54} +(-1.23182 - 0.711191i) q^{55} +1.47817i q^{56} +(-5.97167 + 10.4708i) q^{57} +(4.02905 + 2.32617i) q^{58} +(-9.35927 + 5.40358i) q^{59} +(-2.35776 - 1.34467i) q^{60} +(10.1708 - 5.87209i) q^{61} +(-8.73018 + 1.76320i) q^{62} +(-1.92351 + 0.0203545i) q^{63} -4.68976 q^{64} -13.4993 q^{65} +(0.00743595 + 1.40544i) q^{66} -7.19582 q^{67} +(-1.77619 - 3.07646i) q^{68} +(-1.39640 + 2.44845i) q^{69} +(-1.43805 - 2.49077i) q^{70} +(-12.2001 + 7.04372i) q^{71} +(-0.0731797 - 6.91552i) q^{72} +(-3.46954 + 2.00314i) q^{73} -12.3167 q^{74} +(-4.95799 + 0.0262318i) q^{75} +(1.94468 + 3.36828i) q^{76} +(-0.162631 + 0.281685i) q^{77} +(6.73040 + 11.5163i) q^{78} -15.3265i q^{79} +(11.6692 - 6.73722i) q^{80} +(8.99798 - 0.190454i) q^{81} +(-1.06050 - 1.83684i) q^{82} +(-3.27638 + 5.67485i) q^{83} +(-0.307491 + 0.539158i) q^{84} +(-15.4356 - 8.91175i) q^{85} +(-0.798604 - 1.38322i) q^{86} +(-2.54176 - 4.34916i) q^{87} +(-1.01273 - 0.584700i) q^{88} -4.83358 q^{89} +(6.85109 + 11.5817i) q^{90} +3.08694i q^{91} +(0.454737 + 0.787628i) q^{92} +(9.15706 + 3.02462i) q^{93} +(-0.585947 + 1.01489i) q^{94} +(16.8998 + 9.75710i) q^{95} +(-4.62853 - 2.63973i) q^{96} +(5.02791 + 8.70860i) q^{97} +(9.12776 - 5.26992i) q^{98} +(0.746909 - 1.32589i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q - 6 q^{2} - 3 q^{3} + 26 q^{4} - 9 q^{6} + q^{9} - 4 q^{10} - 3 q^{11} - 9 q^{12} - 3 q^{14} - 3 q^{15} - 22 q^{16} + 5 q^{18} - 4 q^{19} - 21 q^{20} + 9 q^{21} - 9 q^{23} + 18 q^{24} - 52 q^{25}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/279\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(218\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38533 + 0.799823i −0.979579 + 0.565560i −0.902143 − 0.431437i \(-0.858007\pi\)
−0.0774360 + 0.996997i \(0.524673\pi\)
\(3\) 1.73203 − 0.00916385i 0.999986 − 0.00529075i
\(4\) 0.279433 − 0.483993i 0.139717 − 0.241996i
\(5\) − 2.80402i − 1.25400i −0.779021 − 0.626998i \(-0.784283\pi\)
0.779021 − 0.626998i \(-0.215717\pi\)
\(6\) −2.39211 + 1.39801i −0.976573 + 0.570735i
\(7\) −0.641206 −0.242353 −0.121177 − 0.992631i \(-0.538667\pi\)
−0.121177 + 0.992631i \(0.538667\pi\)
\(8\) − 2.30530i − 0.815048i
\(9\) 2.99983 − 0.0317441i 0.999944 − 0.0105814i
\(10\) 2.24272 + 3.88451i 0.709210 + 1.22839i
\(11\) 0.253632 − 0.439304i 0.0764731 − 0.132455i −0.825253 − 0.564763i \(-0.808967\pi\)
0.901726 + 0.432308i \(0.142301\pi\)
\(12\) 0.479551 − 0.840849i 0.138434 − 0.242732i
\(13\) − 4.81427i − 1.33524i −0.744502 − 0.667620i \(-0.767313\pi\)
0.744502 − 0.667620i \(-0.232687\pi\)
\(14\) 0.888285 − 0.512851i 0.237404 − 0.137065i
\(15\) −0.0256956 − 4.85664i −0.00663458 − 1.25398i
\(16\) 2.40270 + 4.16160i 0.600675 + 1.04040i
\(17\) 3.17820 − 5.50481i 0.770828 − 1.33511i −0.166282 − 0.986078i \(-0.553176\pi\)
0.937110 − 0.349035i \(-0.113490\pi\)
\(18\) −4.13038 + 2.44331i −0.973540 + 0.575894i
\(19\) −3.47968 + 6.02698i −0.798294 + 1.38268i 0.122433 + 0.992477i \(0.460930\pi\)
−0.920727 + 0.390208i \(0.872403\pi\)
\(20\) −1.35713 − 0.783537i −0.303463 − 0.175204i
\(21\) −1.11059 + 0.00587592i −0.242350 + 0.00128223i
\(22\) 0.811444i 0.173000i
\(23\) −0.813677 + 1.40933i −0.169663 + 0.293866i −0.938302 − 0.345818i \(-0.887601\pi\)
0.768638 + 0.639684i \(0.220935\pi\)
\(24\) −0.0211255 − 3.99285i −0.00431222 − 0.815036i
\(25\) −2.86253 −0.572507
\(26\) 3.85057 + 6.66938i 0.755158 + 1.30797i
\(27\) 5.19550 − 0.0824716i 0.999874 − 0.0158717i
\(28\) −0.179174 + 0.310339i −0.0338608 + 0.0586486i
\(29\) −1.45418 − 2.51871i −0.270035 − 0.467714i 0.698836 − 0.715282i \(-0.253702\pi\)
−0.968870 + 0.247569i \(0.920368\pi\)
\(30\) 3.92005 + 6.70752i 0.715700 + 1.22462i
\(31\) 5.27752 + 1.77421i 0.947870 + 0.318658i
\(32\) −2.66418 − 1.53817i −0.470966 − 0.271912i
\(33\) 0.435272 − 0.763211i 0.0757712 − 0.132858i
\(34\) 10.1680i 1.74380i
\(35\) 1.79796i 0.303910i
\(36\) 0.822889 − 1.46077i 0.137148 − 0.243461i
\(37\) 6.66807 + 3.84981i 1.09622 + 0.632905i 0.935227 − 0.354050i \(-0.115196\pi\)
0.160997 + 0.986955i \(0.448529\pi\)
\(38\) − 11.1325i − 1.80593i
\(39\) −0.0441173 − 8.33845i −0.00706442 − 1.33522i
\(40\) −6.46412 −1.02207
\(41\) 1.32592i 0.207073i 0.994626 + 0.103537i \(0.0330159\pi\)
−0.994626 + 0.103537i \(0.966984\pi\)
\(42\) 1.53383 − 0.896412i 0.236676 − 0.138319i
\(43\) 0.998476i 0.152266i 0.997098 + 0.0761331i \(0.0242574\pi\)
−0.997098 + 0.0761331i \(0.975743\pi\)
\(44\) −0.141747 − 0.245513i −0.0213691 − 0.0370124i
\(45\) −0.0890110 − 8.41159i −0.0132690 − 1.25393i
\(46\) − 2.60319i − 0.383820i
\(47\) 0.634446 − 0.366298i 0.0925435 − 0.0534300i −0.453014 − 0.891503i \(-0.649651\pi\)
0.545558 + 0.838073i \(0.316318\pi\)
\(48\) 4.19968 + 7.18598i 0.606171 + 1.03721i
\(49\) −6.58885 −0.941265
\(50\) 3.96557 − 2.28952i 0.560816 − 0.323787i
\(51\) 5.45429 − 9.56360i 0.763753 − 1.33917i
\(52\) −2.33007 − 1.34527i −0.323123 − 0.186555i
\(53\) 3.07274 + 5.32214i 0.422073 + 0.731052i 0.996142 − 0.0877547i \(-0.0279692\pi\)
−0.574069 + 0.818807i \(0.694636\pi\)
\(54\) −7.13154 + 4.26973i −0.970479 + 0.581037i
\(55\) −1.23182 − 0.711191i −0.166098 − 0.0958969i
\(56\) 1.47817i 0.197529i
\(57\) −5.97167 + 10.4708i −0.790967 + 1.38689i
\(58\) 4.02905 + 2.32617i 0.529040 + 0.305442i
\(59\) −9.35927 + 5.40358i −1.21847 + 0.703486i −0.964591 − 0.263749i \(-0.915041\pi\)
−0.253882 + 0.967235i \(0.581707\pi\)
\(60\) −2.35776 − 1.34467i −0.304385 − 0.173596i
\(61\) 10.1708 − 5.87209i 1.30223 − 0.751844i 0.321446 − 0.946928i \(-0.395831\pi\)
0.980787 + 0.195084i \(0.0624980\pi\)
\(62\) −8.73018 + 1.76320i −1.10873 + 0.223927i
\(63\) −1.92351 + 0.0203545i −0.242340 + 0.00256442i
\(64\) −4.68976 −0.586220
\(65\) −13.4993 −1.67439
\(66\) 0.00743595 + 1.40544i 0.000915303 + 0.172998i
\(67\) −7.19582 −0.879110 −0.439555 − 0.898216i \(-0.644864\pi\)
−0.439555 + 0.898216i \(0.644864\pi\)
\(68\) −1.77619 − 3.07646i −0.215395 − 0.373075i
\(69\) −1.39640 + 2.44845i −0.168106 + 0.294759i
\(70\) −1.43805 − 2.49077i −0.171879 − 0.297704i
\(71\) −12.2001 + 7.04372i −1.44788 + 0.835936i −0.998355 − 0.0573295i \(-0.981741\pi\)
−0.449529 + 0.893266i \(0.648408\pi\)
\(72\) −0.0731797 − 6.91552i −0.00862431 − 0.815002i
\(73\) −3.46954 + 2.00314i −0.406078 + 0.234449i −0.689103 − 0.724663i \(-0.741995\pi\)
0.283025 + 0.959113i \(0.408662\pi\)
\(74\) −12.3167 −1.43178
\(75\) −4.95799 + 0.0262318i −0.572499 + 0.00302899i
\(76\) 1.94468 + 3.36828i 0.223070 + 0.386368i
\(77\) −0.162631 + 0.281685i −0.0185335 + 0.0321009i
\(78\) 6.73040 + 11.5163i 0.762068 + 1.30396i
\(79\) − 15.3265i − 1.72437i −0.506595 − 0.862184i \(-0.669096\pi\)
0.506595 − 0.862184i \(-0.330904\pi\)
\(80\) 11.6692 − 6.73722i 1.30466 − 0.753245i
\(81\) 8.99798 − 0.190454i 0.999776 − 0.0211615i
\(82\) −1.06050 − 1.83684i −0.117113 − 0.202845i
\(83\) −3.27638 + 5.67485i −0.359629 + 0.622896i −0.987899 − 0.155100i \(-0.950430\pi\)
0.628270 + 0.777996i \(0.283763\pi\)
\(84\) −0.307491 + 0.539158i −0.0335500 + 0.0588269i
\(85\) −15.4356 − 8.91175i −1.67423 − 0.966615i
\(86\) −0.798604 − 1.38322i −0.0861157 − 0.149157i
\(87\) −2.54176 − 4.34916i −0.272505 − 0.466278i
\(88\) −1.01273 − 0.584700i −0.107957 − 0.0623292i
\(89\) −4.83358 −0.512359 −0.256179 − 0.966629i \(-0.582464\pi\)
−0.256179 + 0.966629i \(0.582464\pi\)
\(90\) 6.85109 + 11.5817i 0.722169 + 1.22082i
\(91\) 3.08694i 0.323599i
\(92\) 0.454737 + 0.787628i 0.0474096 + 0.0821159i
\(93\) 9.15706 + 3.02462i 0.949542 + 0.313639i
\(94\) −0.585947 + 1.01489i −0.0604358 + 0.104678i
\(95\) 16.8998 + 9.75710i 1.73388 + 1.00106i
\(96\) −4.62853 − 2.63973i −0.472398 − 0.269417i
\(97\) 5.02791 + 8.70860i 0.510507 + 0.884224i 0.999926 + 0.0121753i \(0.00387563\pi\)
−0.489419 + 0.872049i \(0.662791\pi\)
\(98\) 9.12776 − 5.26992i 0.922043 − 0.532342i
\(99\) 0.746909 − 1.32589i 0.0750672 − 0.133257i
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 279.2.o.a.212.9 ✓ 60
3.2 odd 2 837.2.o.a.584.22 60
9.2 odd 6 279.2.r.a.119.9 yes 60
9.7 even 3 837.2.r.a.305.22 60
31.6 odd 6 279.2.r.a.68.9 yes 60
93.68 even 6 837.2.r.a.719.22 60
279.223 odd 6 837.2.o.a.440.22 60
279.254 even 6 inner 279.2.o.a.254.9 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
279.2.o.a.212.9 ✓ 60 1.1 even 1 trivial
279.2.o.a.254.9 yes 60 279.254 even 6 inner
279.2.r.a.68.9 yes 60 31.6 odd 6
279.2.r.a.119.9 yes 60 9.2 odd 6
837.2.o.a.440.22 60 279.223 odd 6
837.2.o.a.584.22 60 3.2 odd 2
837.2.r.a.305.22 60 9.7 even 3
837.2.r.a.719.22 60 93.68 even 6