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Properties

Label 279.2.r.a.119.5
Level $279$
Weight $2$
Character 279.119
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(68,279)] 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("279.68"); 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, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 279 = 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 279.r (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 119.5
Character \(\chi\) \(=\) 279.119
Dual form 279.2.r.a.68.5

$q$-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+(-1.98560 - 1.14638i) q^{2} +(1.70178 - 0.322417i) q^{3} +(1.62840 + 2.82047i) q^{4} +(-2.03476 - 1.17477i) q^{5} +(-3.74866 - 1.31070i) q^{6} +(-0.506971 - 0.878099i) q^{7} -2.88154i q^{8} +(2.79209 - 1.09736i) q^{9} +(2.69347 + 4.66523i) q^{10} -5.57529 q^{11} +(3.68054 + 4.27478i) q^{12} +(-3.01056 - 1.73815i) q^{13} +2.32473i q^{14} +(-3.84147 - 1.34315i) q^{15} +(-0.0465583 + 0.0806413i) q^{16} +(2.60265 - 4.50793i) q^{17} +(-6.80198 - 1.02189i) q^{18} +(-0.710547 + 1.23070i) q^{19} -7.65196i q^{20} +(-1.14587 - 1.33087i) q^{21} +(11.0703 + 6.39143i) q^{22} +(-1.43676 - 2.48854i) q^{23} +(-0.929057 - 4.90374i) q^{24} +(0.260163 + 0.450615i) q^{25} +(3.98517 + 6.90252i) q^{26} +(4.39771 - 2.76769i) q^{27} +(1.65110 - 2.85979i) q^{28} +(-1.63187 + 2.82649i) q^{29} +(6.08785 + 7.07077i) q^{30} +(-5.56667 + 0.110293i) q^{31} +(-4.80608 + 2.77479i) q^{32} +(-9.48790 + 1.79757i) q^{33} +(-10.3356 + 5.96728i) q^{34} +2.38229i q^{35} +(7.64172 + 6.08806i) q^{36} +(-0.529463 - 0.305685i) q^{37} +(2.82172 - 1.62912i) q^{38} +(-5.68371 - 1.98729i) q^{39} +(-3.38514 + 5.86324i) q^{40} +(6.32976 + 3.65449i) q^{41} +(0.749534 + 3.95618i) q^{42} +(-5.88649 + 3.39857i) q^{43} +(-9.07878 - 15.7249i) q^{44} +(-6.97039 - 1.04719i) q^{45} +6.58832i q^{46} +(8.02931 + 4.63572i) q^{47} +(-0.0532317 + 0.152245i) q^{48} +(2.98596 - 5.17184i) q^{49} -1.19299i q^{50} +(2.97570 - 8.51063i) q^{51} -11.3216i q^{52} +(-0.812196 - 1.40677i) q^{53} +(-11.9049 + 0.454044i) q^{54} +(11.3444 + 6.54968i) q^{55} +(-2.53028 + 1.46085i) q^{56} +(-0.812393 + 2.32348i) q^{57} +(6.48049 - 3.74151i) q^{58} -12.0193i q^{59} +(-2.46712 - 13.0219i) q^{60} +(2.77331 + 1.60117i) q^{61} +(11.1796 + 6.16255i) q^{62} +(-2.37910 - 1.89540i) q^{63} +12.9101 q^{64} +(4.08384 + 7.07343i) q^{65} +(20.8999 + 7.30754i) q^{66} +(6.96972 - 12.0719i) q^{67} +16.9526 q^{68} +(-3.24739 - 3.77171i) q^{69} +(2.73102 - 4.73027i) q^{70} +(-3.30857 + 1.91021i) q^{71} +(-3.16210 - 8.04553i) q^{72} +(2.92420 - 1.68829i) q^{73} +(0.700866 + 1.21394i) q^{74} +(0.588025 + 0.682965i) q^{75} -4.62821 q^{76} +(2.82651 + 4.89566i) q^{77} +(9.00737 + 10.4617i) q^{78} +(-0.0794528 + 0.0458721i) q^{79} +(0.189470 - 0.109390i) q^{80} +(6.59158 - 6.12789i) q^{81} +(-8.37890 - 14.5127i) q^{82} +15.8853 q^{83} +(1.88776 - 5.39906i) q^{84} +(-10.5915 + 6.11503i) q^{85} +15.5843 q^{86} +(-1.86578 + 5.33620i) q^{87} +16.0654i q^{88} -5.63299 q^{89} +(12.6399 + 10.0700i) q^{90} +3.52476i q^{91} +(4.67923 - 8.10466i) q^{92} +(-9.43768 + 1.98248i) q^{93} +(-10.6286 - 18.4094i) q^{94} +(2.89158 - 1.66946i) q^{95} +(-7.28424 + 6.27164i) q^{96} +(3.00571 - 5.20603i) q^{97} +(-11.8578 + 6.84612i) q^{98} +(-15.5667 + 6.11813i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q - 6 q^{2} - 3 q^{3} + 26 q^{4} - 6 q^{5} + 6 q^{6} - 5 q^{9} - 4 q^{10} - 6 q^{11} + 6 q^{12} - 3 q^{13} + 3 q^{15} - 22 q^{16} - 4 q^{18} - 4 q^{19} - 15 q^{22} + 9 q^{23} + 36 q^{24} + 26 q^{25}+ \cdots - 21 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.98560 − 1.14638i −1.40403 − 0.810617i −0.409226 − 0.912433i \(-0.634201\pi\)
−0.994803 + 0.101817i \(0.967535\pi\)
\(3\) 1.70178 − 0.322417i 0.982522 − 0.186148i
\(4\) 1.62840 + 2.82047i 0.814198 + 1.41023i
\(5\) −2.03476 − 1.17477i −0.909972 − 0.525372i −0.0295498 − 0.999563i \(-0.509407\pi\)
−0.880422 + 0.474191i \(0.842741\pi\)
\(6\) −3.74866 − 1.31070i −1.53038 − 0.535092i
\(7\) −0.506971 − 0.878099i −0.191617 − 0.331890i 0.754169 − 0.656680i \(-0.228040\pi\)
−0.945786 + 0.324790i \(0.894706\pi\)
\(8\) − 2.88154i − 1.01878i
\(9\) 2.79209 − 1.09736i 0.930698 − 0.365788i
\(10\) 2.69347 + 4.66523i 0.851751 + 1.47528i
\(11\) −5.57529 −1.68101 −0.840507 − 0.541801i \(-0.817743\pi\)
−0.840507 + 0.541801i \(0.817743\pi\)
\(12\) 3.68054 + 4.27478i 1.06248 + 1.23402i
\(13\) −3.01056 − 1.73815i −0.834979 − 0.482076i 0.0205751 − 0.999788i \(-0.493450\pi\)
−0.855555 + 0.517713i \(0.826784\pi\)
\(14\) 2.32473i 0.621311i
\(15\) −3.84147 − 1.34315i −0.991864 − 0.346801i
\(16\) −0.0465583 + 0.0806413i −0.0116396 + 0.0201603i
\(17\) 2.60265 − 4.50793i 0.631236 − 1.09333i −0.356063 − 0.934462i \(-0.615881\pi\)
0.987299 − 0.158871i \(-0.0507855\pi\)
\(18\) −6.80198 − 1.02189i −1.60324 − 0.240862i
\(19\) −0.710547 + 1.23070i −0.163011 + 0.282343i −0.935947 − 0.352141i \(-0.885454\pi\)
0.772936 + 0.634483i \(0.218787\pi\)
\(20\) − 7.65196i − 1.71103i
\(21\) −1.14587 − 1.33087i −0.250048 − 0.290420i
\(22\) 11.0703 + 6.39143i 2.36019 + 1.36266i
\(23\) −1.43676 − 2.48854i −0.299585 − 0.518897i 0.676456 − 0.736483i \(-0.263515\pi\)
−0.976041 + 0.217586i \(0.930182\pi\)
\(24\) −0.929057 − 4.90374i −0.189643 − 1.00097i
\(25\) 0.260163 + 0.450615i 0.0520325 + 0.0901230i
\(26\) 3.98517 + 6.90252i 0.781557 + 1.35370i
\(27\) 4.39771 − 2.76769i 0.846341 − 0.532642i
\(28\) 1.65110 − 2.85979i 0.312028 − 0.540449i
\(29\) −1.63187 + 2.82649i −0.303031 + 0.524866i −0.976821 − 0.214057i \(-0.931332\pi\)
0.673790 + 0.738923i \(0.264665\pi\)
\(30\) 6.08785 + 7.07077i 1.11148 + 1.29094i
\(31\) −5.56667 + 0.110293i −0.999804 + 0.0198092i
\(32\) −4.80608 + 2.77479i −0.849603 + 0.490518i
\(33\) −9.48790 + 1.79757i −1.65163 + 0.312917i
\(34\) −10.3356 + 5.96728i −1.77255 + 1.02338i
\(35\) 2.38229i 0.402681i
\(36\) 7.64172 + 6.08806i 1.27362 + 1.01468i
\(37\) −0.529463 − 0.305685i −0.0870431 − 0.0502544i 0.455847 − 0.890058i \(-0.349337\pi\)
−0.542890 + 0.839804i \(0.682670\pi\)
\(38\) 2.82172 − 1.62912i 0.457743 − 0.264278i
\(39\) −5.68371 − 1.98729i −0.910123 − 0.318220i
\(40\) −3.38514 + 5.86324i −0.535238 + 0.927059i
\(41\) 6.32976 + 3.65449i 0.988543 + 0.570735i 0.904838 − 0.425755i \(-0.139992\pi\)
0.0837045 + 0.996491i \(0.473325\pi\)
\(42\) 0.749534 + 3.95618i 0.115656 + 0.610452i
\(43\) −5.88649 + 3.39857i −0.897682 + 0.518277i −0.876447 − 0.481498i \(-0.840093\pi\)
−0.0212345 + 0.999775i \(0.506760\pi\)
\(44\) −9.07878 − 15.7249i −1.36868 − 2.37062i
\(45\) −6.97039 − 1.04719i −1.03908 − 0.156106i
\(46\) 6.58832i 0.971395i
\(47\) 8.02931 + 4.63572i 1.17119 + 0.676190i 0.953961 − 0.299930i \(-0.0969634\pi\)
0.217233 + 0.976120i \(0.430297\pi\)
\(48\) −0.0532317 + 0.152245i −0.00768333 + 0.0219746i
\(49\) 2.98596 − 5.17184i 0.426566 − 0.738834i
\(50\) − 1.19299i − 0.168714i
\(51\) 2.97570 − 8.51063i 0.416682 − 1.19173i
\(52\) − 11.3216i − 1.57002i
\(53\) −0.812196 − 1.40677i −0.111564 − 0.193234i 0.804837 − 0.593496i \(-0.202253\pi\)
−0.916401 + 0.400262i \(0.868919\pi\)
\(54\) −11.9049 + 0.454044i −1.62006 + 0.0617875i
\(55\) 11.3444 + 6.54968i 1.52967 + 0.883158i
\(56\) −2.53028 + 1.46085i −0.338122 + 0.195215i
\(57\) −0.812393 + 2.32348i −0.107604 + 0.307752i
\(58\) 6.48049 − 3.74151i 0.850930 − 0.491284i
\(59\) − 12.0193i − 1.56478i −0.622792 − 0.782388i \(-0.714002\pi\)
0.622792 − 0.782388i \(-0.285998\pi\)
\(60\) −2.46712 − 13.0219i −0.318504 − 1.68112i
\(61\) 2.77331 + 1.60117i 0.355086 + 0.205009i 0.666923 − 0.745126i \(-0.267611\pi\)
−0.311837 + 0.950136i \(0.600944\pi\)
\(62\) 11.1796 + 6.16255i 1.41981 + 0.782645i
\(63\) −2.37910 − 1.89540i −0.299739 − 0.238798i
\(64\) 12.9101 1.61377
\(65\) 4.08384 + 7.07343i 0.506539 + 0.877351i
\(66\) 20.8999 + 7.30754i 2.57259 + 0.899496i
\(67\) 6.96972 − 12.0719i 0.851487 − 1.47482i −0.0283798 − 0.999597i \(-0.509035\pi\)
0.879866 − 0.475221i \(-0.157632\pi\)
\(68\) 16.9526 2.05581
\(69\) −3.24739 − 3.77171i −0.390940 − 0.454060i
\(70\) 2.73102 − 4.73027i 0.326420 − 0.565376i
\(71\) −3.30857 + 1.91021i −0.392655 + 0.226700i −0.683310 − 0.730128i \(-0.739460\pi\)
0.290655 + 0.956828i \(0.406127\pi\)
\(72\) −3.16210 − 8.04553i −0.372657 − 0.948174i
\(73\) 2.92420 − 1.68829i 0.342252 − 0.197599i −0.319015 − 0.947750i \(-0.603352\pi\)
0.661268 + 0.750150i \(0.270019\pi\)
\(74\) 0.700866 + 1.21394i 0.0814741 + 0.141117i
\(75\) 0.588025 + 0.682965i 0.0678993 + 0.0788621i
\(76\) −4.62821 −0.530892
\(77\) 2.82651 + 4.89566i 0.322110 + 0.557912i
\(78\) 9.00737 + 10.4617i 1.01988 + 1.18455i
\(79\) −0.0794528 + 0.0458721i −0.00893914 + 0.00516102i −0.504463 − 0.863433i \(-0.668309\pi\)
0.495524 + 0.868594i \(0.334976\pi\)
\(80\) 0.189470 − 0.109390i 0.0211834 − 0.0122302i
\(81\) 6.59158 − 6.12789i 0.732398 − 0.680877i
\(82\) −8.37890 − 14.5127i −0.925295 − 1.60266i
\(83\) 15.8853 1.74364 0.871820 − 0.489827i \(-0.162940\pi\)
0.871820 + 0.489827i \(0.162940\pi\)
\(84\) 1.88776 − 5.39906i 0.205971 − 0.589086i
\(85\) −10.5915 + 6.11503i −1.14881 + 0.663268i
\(86\) 15.5843 1.68050
\(87\) −1.86578 + 5.33620i −0.200032 + 0.572101i
\(88\) 16.0654i 1.71258i
\(89\) −5.63299 −0.597096 −0.298548 − 0.954395i \(-0.596502\pi\)
−0.298548 + 0.954395i \(0.596502\pi\)
\(90\) 12.6399 + 10.0700i 1.33236 + 1.06148i
\(91\) 3.52476i 0.369495i
\(92\) 4.67923 − 8.10466i 0.487843 − 0.844970i
\(93\) −9.43768 + 1.98248i −0.978642 + 0.205574i
\(94\) −10.6286 − 18.4094i −1.09626 − 1.89878i
\(95\) 2.89158 − 1.66946i 0.296670 − 0.171283i
\(96\) −7.28424 + 6.27164i −0.743444 + 0.640096i
\(97\) 3.00571 − 5.20603i 0.305183 − 0.528593i −0.672119 − 0.740443i \(-0.734616\pi\)
0.977302 + 0.211850i \(0.0679490\pi\)
\(98\) −11.8578 + 6.84612i −1.19782 + 0.691563i
\(99\) −15.5667 + 6.11813i −1.56452 + 0.614895i
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.r.a.119.5 yes 60
3.2 odd 2 837.2.r.a.305.26 60
9.4 even 3 837.2.o.a.584.26 60
9.5 odd 6 279.2.o.a.212.5 ✓ 60
31.6 odd 6 279.2.o.a.254.5 yes 60
93.68 even 6 837.2.o.a.440.26 60
279.68 even 6 inner 279.2.r.a.68.5 yes 60
279.130 odd 6 837.2.r.a.719.26 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
279.2.o.a.212.5 ✓ 60 9.5 odd 6
279.2.o.a.254.5 yes 60 31.6 odd 6
279.2.r.a.68.5 yes 60 279.68 even 6 inner
279.2.r.a.119.5 yes 60 1.1 even 1 trivial
837.2.o.a.440.26 60 93.68 even 6
837.2.o.a.584.26 60 9.4 even 3
837.2.r.a.305.26 60 3.2 odd 2
837.2.r.a.719.26 60 279.130 odd 6