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Properties

Label 279.2.o.a.254.9
Level $279$
Weight $2$
Character 279.254
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 254.9
Character \(\chi\) \(=\) 279.254
Dual form 279.2.o.a.212.9

$q$-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 \(\left(\mathbb{Z}/279\mathbb{Z}\right)^\times\).

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

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 −0.0774360 − 0.996997i \(-0.524673\pi\)
−0.902143 + 0.431437i \(0.858007\pi\)
\(3\) 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 \(0.215717\pi\)
−0.779021 + 0.626998i \(0.784283\pi\)
\(6\) −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 0.901726 − 0.432308i \(-0.142301\pi\)
−0.825253 + 0.564763i \(0.808967\pi\)
\(12\) 0.479551 + 0.840849i 0.138434 + 0.242732i
\(13\) 4.81427i 1.33524i 0.744502 + 0.667620i \(0.232687\pi\)
−0.744502 + 0.667620i \(0.767313\pi\)
\(14\) 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 0.937110 + 0.349035i \(0.113490\pi\)
−0.166282 + 0.986078i \(0.553176\pi\)
\(18\) −4.13038 − 2.44331i −0.973540 − 0.575894i
\(19\) −3.47968 − 6.02698i −0.798294 − 1.38268i −0.920727 − 0.390208i \(-0.872403\pi\)
0.122433 − 0.992477i \(-0.460930\pi\)
\(20\) −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 0.768638 − 0.639684i \(-0.220935\pi\)
−0.938302 + 0.345818i \(0.887601\pi\)
\(24\) −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 −0.968870 − 0.247569i \(-0.920368\pi\)
0.698836 + 0.715282i \(0.253702\pi\)
\(30\) 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 0.160997 − 0.986955i \(-0.448529\pi\)
0.935227 + 0.354050i \(0.115196\pi\)
\(38\) 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 \(-0.966984\pi\)
0.994626 − 0.103537i \(-0.0330159\pi\)
\(42\) 1.53383 + 0.896412i 0.236676 + 0.138319i
\(43\) − 0.998476i − 0.152266i −0.997098 − 0.0761331i \(-0.975743\pi\)
0.997098 − 0.0761331i \(-0.0242574\pi\)
\(44\) −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 0.545558 − 0.838073i \(-0.316318\pi\)
−0.453014 + 0.891503i \(0.649651\pi\)
\(48\) 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 −0.574069 − 0.818807i \(-0.694636\pi\)
0.996142 + 0.0877547i \(0.0279692\pi\)
\(54\) −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 −0.253882 − 0.967235i \(-0.581707\pi\)
−0.964591 + 0.263749i \(0.915041\pi\)
\(60\) −2.35776 + 1.34467i −0.304385 + 0.173596i
\(61\) 10.1708 + 5.87209i 1.30223 + 0.751844i 0.980787 − 0.195084i \(-0.0624980\pi\)
0.321446 + 0.946928i \(0.395831\pi\)
\(62\) −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 −0.449529 − 0.893266i \(-0.648408\pi\)
−0.998355 + 0.0573295i \(0.981741\pi\)
\(72\) −0.0731797 + 6.91552i −0.00862431 + 0.815002i
\(73\) −3.46954 − 2.00314i −0.406078 − 0.234449i 0.283025 − 0.959113i \(-0.408662\pi\)
−0.689103 + 0.724663i \(0.741995\pi\)
\(74\) −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 \(0.330904\pi\)
−0.506595 + 0.862184i \(0.669096\pi\)
\(80\) 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 0.628270 − 0.777996i \(-0.283763\pi\)
−0.987899 + 0.155100i \(0.950430\pi\)
\(84\) −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 −0.489419 − 0.872049i \(-0.662791\pi\)
0.999926 − 0.0121753i \(-0.00387563\pi\)
\(98\) 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.254.9 yes 60
3.2 odd 2 837.2.o.a.440.22 60
9.4 even 3 837.2.r.a.719.22 60
9.5 odd 6 279.2.r.a.68.9 yes 60
31.26 odd 6 279.2.r.a.119.9 yes 60
93.26 even 6 837.2.r.a.305.22 60
279.212 even 6 inner 279.2.o.a.212.9 ✓ 60
279.274 odd 6 837.2.o.a.584.22 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
279.2.o.a.212.9 ✓ 60 279.212 even 6 inner
279.2.o.a.254.9 yes 60 1.1 even 1 trivial
279.2.r.a.68.9 yes 60 9.5 odd 6
279.2.r.a.119.9 yes 60 31.26 odd 6
837.2.o.a.440.22 60 3.2 odd 2
837.2.o.a.584.22 60 279.274 odd 6
837.2.r.a.305.22 60 93.26 even 6
837.2.r.a.719.22 60 9.4 even 3