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Properties

Label 279.2.r.a.68.18
Level $279$
Weight $2$
Character 279.68
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 68.18
Character \(\chi\) \(=\) 279.68
Dual form 279.2.r.a.119.18

$q$-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+(0.529444 - 0.305675i) q^{2} +(-0.375794 - 1.69079i) q^{3} +(-0.813126 + 1.40838i) q^{4} +(-1.34378 + 0.775831i) q^{5} +(-0.715794 - 0.780310i) q^{6} +(-2.19077 + 3.79452i) q^{7} +2.21691i q^{8} +(-2.71756 + 1.27078i) q^{9} +(-0.474304 + 0.821519i) q^{10} -5.38476 q^{11} +(2.68684 + 0.845568i) q^{12} +(4.73363 - 2.73296i) q^{13} +2.67865i q^{14} +(1.81675 + 1.98050i) q^{15} +(-0.948599 - 1.64302i) q^{16} +(0.0887575 + 0.153733i) q^{17} +(-1.05035 + 1.50350i) q^{18} +(0.780073 + 1.35113i) q^{19} -2.52339i q^{20} +(7.23902 + 2.27817i) q^{21} +(-2.85093 + 1.64598i) q^{22} +(1.82782 - 3.16588i) q^{23} +(3.74833 - 0.833100i) q^{24} +(-1.29617 + 2.24503i) q^{25} +(1.67080 - 2.89390i) q^{26} +(3.16987 + 4.11727i) q^{27} +(-3.56274 - 6.17084i) q^{28} +(-0.377407 - 0.653689i) q^{29} +(1.56726 + 0.493228i) q^{30} +(-2.81234 + 4.80528i) q^{31} +(-4.84426 - 2.79683i) q^{32} +(2.02356 + 9.10451i) q^{33} +(0.0939843 + 0.0542619i) q^{34} -6.79866i q^{35} +(0.419983 - 4.86064i) q^{36} +(-6.83555 + 3.94651i) q^{37} +(0.826010 + 0.476897i) q^{38} +(-6.39975 - 6.97656i) q^{39} +(-1.71995 - 2.97903i) q^{40} +(7.78354 - 4.49383i) q^{41} +(4.52904 - 1.00662i) q^{42} +(8.96460 + 5.17571i) q^{43} +(4.37849 - 7.58376i) q^{44} +(2.66589 - 3.81601i) q^{45} -2.23488i q^{46} +(-9.41697 + 5.43689i) q^{47} +(-2.42153 + 2.22132i) q^{48} +(-6.09892 - 10.5636i) q^{49} +1.58483i q^{50} +(0.226575 - 0.207842i) q^{51} +8.88898i q^{52} +(2.69296 - 4.66434i) q^{53} +(2.93681 + 1.21092i) q^{54} +(7.23593 - 4.17767i) q^{55} +(-8.41210 - 4.85673i) q^{56} +(1.99133 - 1.82669i) q^{57} +(-0.399632 - 0.230728i) q^{58} +10.2172i q^{59} +(-4.26654 + 0.948276i) q^{60} +(-1.84270 + 1.06388i) q^{61} +(-0.0201261 + 3.40379i) q^{62} +(1.13154 - 13.0958i) q^{63} +0.374712 q^{64} +(-4.24064 + 7.34500i) q^{65} +(3.85438 + 4.20178i) q^{66} +(5.03541 + 8.72158i) q^{67} -0.288684 q^{68} +(-6.03973 - 1.90075i) q^{69} +(-2.07818 - 3.59951i) q^{70} +(-4.87193 - 2.81281i) q^{71} +(-2.81720 - 6.02457i) q^{72} +(-2.76988 - 1.59919i) q^{73} +(-2.41269 + 4.17891i) q^{74} +(4.28298 + 1.34789i) q^{75} -2.53719 q^{76} +(11.7967 - 20.4326i) q^{77} +(-5.52087 - 1.73746i) q^{78} +(-1.07418 - 0.620178i) q^{79} +(2.54942 + 1.47191i) q^{80} +(5.77024 - 6.90683i) q^{81} +(2.74730 - 4.75846i) q^{82} +2.88121 q^{83} +(-9.09476 + 8.34282i) q^{84} +(-0.238541 - 0.137722i) q^{85} +6.32834 q^{86} +(-0.963424 + 0.883770i) q^{87} -11.9375i q^{88} +4.66152 q^{89} +(0.244980 - 2.83526i) q^{90} +23.9491i q^{91} +(2.97250 + 5.14852i) q^{92} +(9.18159 + 2.94929i) q^{93} +(-3.32384 + 5.75706i) q^{94} +(-2.09649 - 1.21041i) q^{95} +(-2.90842 + 9.24167i) q^{96} +(2.43672 + 4.22052i) q^{97} +(-6.45807 - 3.72857i) q^{98} +(14.6334 - 6.84284i) 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 \(\left(\mathbb{Z}/279\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.529444 − 0.305675i 0.374374 − 0.216145i −0.300994 − 0.953626i \(-0.597318\pi\)
0.675367 + 0.737481i \(0.263985\pi\)
\(3\) −0.375794 − 1.69079i −0.216965 − 0.976179i
\(4\) −0.813126 + 1.40838i −0.406563 + 0.704188i
\(5\) −1.34378 + 0.775831i −0.600956 + 0.346962i −0.769418 − 0.638746i \(-0.779454\pi\)
0.168461 + 0.985708i \(0.446120\pi\)
\(6\) −0.715794 − 0.780310i −0.292222 − 0.318560i
\(7\) −2.19077 + 3.79452i −0.828032 + 1.43419i 0.0715480 + 0.997437i \(0.477206\pi\)
−0.899580 + 0.436756i \(0.856127\pi\)
\(8\) 2.21691i 0.783795i
\(9\) −2.71756 + 1.27078i −0.905853 + 0.423593i
\(10\) −0.474304 + 0.821519i −0.149988 + 0.259787i
\(11\) −5.38476 −1.62357 −0.811783 − 0.583959i \(-0.801503\pi\)
−0.811783 + 0.583959i \(0.801503\pi\)
\(12\) 2.68684 + 0.845568i 0.775623 + 0.244094i
\(13\) 4.73363 − 2.73296i 1.31287 − 0.757988i 0.330303 − 0.943875i \(-0.392849\pi\)
0.982571 + 0.185887i \(0.0595159\pi\)
\(14\) 2.67865i 0.715899i
\(15\) 1.81675 + 1.98050i 0.469084 + 0.511363i
\(16\) −0.948599 − 1.64302i −0.237150 − 0.410756i
\(17\) 0.0887575 + 0.153733i 0.0215269 + 0.0372856i 0.876588 − 0.481241i \(-0.159814\pi\)
−0.855061 + 0.518527i \(0.826481\pi\)
\(18\) −1.05035 + 1.50350i −0.247570 + 0.354377i
\(19\) 0.780073 + 1.35113i 0.178961 + 0.309970i 0.941525 − 0.336943i \(-0.109393\pi\)
−0.762564 + 0.646913i \(0.776060\pi\)
\(20\) − 2.52339i − 0.564248i
\(21\) 7.23902 + 2.27817i 1.57968 + 0.497138i
\(22\) −2.85093 + 1.64598i −0.607820 + 0.350925i
\(23\) 1.82782 − 3.16588i 0.381127 − 0.660131i −0.610097 − 0.792327i \(-0.708869\pi\)
0.991224 + 0.132196i \(0.0422027\pi\)
\(24\) 3.74833 − 0.833100i 0.765125 − 0.170056i
\(25\) −1.29617 + 2.24503i −0.259234 + 0.449007i
\(26\) 1.67080 − 2.89390i 0.327670 − 0.567541i
\(27\) 3.16987 + 4.11727i 0.610041 + 0.792370i
\(28\) −3.56274 − 6.17084i −0.673294 − 1.16618i
\(29\) −0.377407 − 0.653689i −0.0700828 − 0.121387i 0.828855 − 0.559464i \(-0.188993\pi\)
−0.898937 + 0.438077i \(0.855660\pi\)
\(30\) 1.56726 + 0.493228i 0.286141 + 0.0900507i
\(31\) −2.81234 + 4.80528i −0.505112 + 0.863054i
\(32\) −4.84426 − 2.79683i −0.856352 − 0.494415i
\(33\) 2.02356 + 9.10451i 0.352257 + 1.58489i
\(34\) 0.0939843 + 0.0542619i 0.0161182 + 0.00930584i
\(35\) − 6.79866i − 1.14918i
\(36\) 0.419983 − 4.86064i 0.0699971 − 0.810107i
\(37\) −6.83555 + 3.94651i −1.12376 + 0.648801i −0.942357 − 0.334608i \(-0.891396\pi\)
−0.181400 + 0.983409i \(0.558063\pi\)
\(38\) 0.826010 + 0.476897i 0.133997 + 0.0773629i
\(39\) −6.39975 − 6.97656i −1.02478 − 1.11714i
\(40\) −1.71995 − 2.97903i −0.271947 − 0.471027i
\(41\) 7.78354 − 4.49383i 1.21558 − 0.701818i 0.251614 − 0.967828i \(-0.419039\pi\)
0.963970 + 0.266009i \(0.0857052\pi\)
\(42\) 4.52904 − 1.00662i 0.698846 − 0.155325i
\(43\) 8.96460 + 5.17571i 1.36709 + 0.789289i 0.990555 − 0.137114i \(-0.0437827\pi\)
0.376533 + 0.926403i \(0.377116\pi\)
\(44\) 4.37849 − 7.58376i 0.660082 − 1.14330i
\(45\) 2.66589 − 3.81601i 0.397407 − 0.568858i
\(46\) − 2.23488i − 0.329514i
\(47\) −9.41697 + 5.43689i −1.37361 + 0.793052i −0.991380 − 0.131016i \(-0.958176\pi\)
−0.382227 + 0.924069i \(0.624843\pi\)
\(48\) −2.42153 + 2.22132i −0.349518 + 0.320620i
\(49\) −6.09892 − 10.5636i −0.871274 − 1.50909i
\(50\) 1.58483i 0.224128i
\(51\) 0.226575 − 0.207842i 0.0317269 − 0.0291038i
\(52\) 8.88898i 1.23268i
\(53\) 2.69296 − 4.66434i 0.369906 − 0.640696i −0.619645 − 0.784883i \(-0.712723\pi\)
0.989551 + 0.144187i \(0.0460566\pi\)
\(54\) 2.93681 + 1.21092i 0.399650 + 0.164785i
\(55\) 7.23593 − 4.17767i 0.975692 − 0.563316i
\(56\) −8.41210 − 4.85673i −1.12411 − 0.649007i
\(57\) 1.99133 − 1.82669i 0.263758 − 0.241950i
\(58\) −0.399632 − 0.230728i −0.0524743 − 0.0302960i
\(59\) 10.2172i 1.33017i 0.746768 + 0.665084i \(0.231604\pi\)
−0.746768 + 0.665084i \(0.768396\pi\)
\(60\) −4.26654 + 0.948276i −0.550807 + 0.122422i
\(61\) −1.84270 + 1.06388i −0.235934 + 0.136216i −0.613306 − 0.789845i \(-0.710161\pi\)
0.377373 + 0.926062i \(0.376828\pi\)
\(62\) −0.0201261 + 3.40379i −0.00255602 + 0.432282i
\(63\) 1.13154 − 13.0958i 0.142561 − 1.64992i
\(64\) 0.374712 0.0468390
\(65\) −4.24064 + 7.34500i −0.525987 + 0.911036i
\(66\) 3.85438 + 4.20178i 0.474441 + 0.517203i
\(67\) 5.03541 + 8.72158i 0.615173 + 1.06551i 0.990354 + 0.138559i \(0.0442471\pi\)
−0.375181 + 0.926951i \(0.622420\pi\)
\(68\) −0.288684 −0.0350081
\(69\) −6.03973 − 1.90075i −0.727098 − 0.228823i
\(70\) −2.07818 − 3.59951i −0.248390 − 0.430224i
\(71\) −4.87193 − 2.81281i −0.578191 − 0.333819i 0.182223 − 0.983257i \(-0.441671\pi\)
−0.760414 + 0.649438i \(0.775004\pi\)
\(72\) −2.81720 − 6.02457i −0.332010 − 0.710003i
\(73\) −2.76988 − 1.59919i −0.324190 − 0.187171i 0.329069 − 0.944306i \(-0.393265\pi\)
−0.653259 + 0.757135i \(0.726599\pi\)
\(74\) −2.41269 + 4.17891i −0.280470 + 0.485788i
\(75\) 4.28298 + 1.34789i 0.494556 + 0.155640i
\(76\) −2.53719 −0.291036
\(77\) 11.7967 − 20.4326i 1.34436 − 2.32851i
\(78\) −5.52087 − 1.73746i −0.625115 − 0.196728i
\(79\) −1.07418 − 0.620178i −0.120855 − 0.0697755i 0.438354 − 0.898802i \(-0.355562\pi\)
−0.559209 + 0.829027i \(0.688895\pi\)
\(80\) 2.54942 + 1.47191i 0.285033 + 0.164564i
\(81\) 5.77024 − 6.90683i 0.641138 − 0.767426i
\(82\) 2.74730 − 4.75846i 0.303389 − 0.525484i
\(83\) 2.88121 0.316254 0.158127 − 0.987419i \(-0.449454\pi\)
0.158127 + 0.987419i \(0.449454\pi\)
\(84\) −9.09476 + 8.34282i −0.992320 + 0.910276i
\(85\) −0.238541 − 0.137722i −0.0258734 − 0.0149380i
\(86\) 6.32834 0.682403
\(87\) −0.963424 + 0.883770i −0.103290 + 0.0947500i
\(88\) − 11.9375i − 1.27254i
\(89\) 4.66152 0.494120 0.247060 − 0.969000i \(-0.420536\pi\)
0.247060 + 0.969000i \(0.420536\pi\)
\(90\) 0.244980 − 2.83526i 0.0258231 − 0.298863i
\(91\) 23.9491i 2.51055i
\(92\) 2.97250 + 5.14852i 0.309904 + 0.536770i
\(93\) 9.18159 + 2.94929i 0.952087 + 0.305828i
\(94\) −3.32384 + 5.75706i −0.342828 + 0.593796i
\(95\) −2.09649 − 1.21041i −0.215096 − 0.124185i
\(96\) −2.90842 + 9.24167i −0.296840 + 0.943224i
\(97\) 2.43672 + 4.22052i 0.247411 + 0.428529i 0.962807 − 0.270191i \(-0.0870868\pi\)
−0.715396 + 0.698720i \(0.753753\pi\)
\(98\) −6.45807 − 3.72857i −0.652364 − 0.376642i
\(99\) 14.6334 − 6.84284i 1.47071 − 0.687731i
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.68.18 yes 60
3.2 odd 2 837.2.r.a.719.13 60
9.2 odd 6 279.2.o.a.254.18 yes 60
9.7 even 3 837.2.o.a.440.13 60
31.26 odd 6 279.2.o.a.212.18 ✓ 60
93.26 even 6 837.2.o.a.584.13 60
279.88 odd 6 837.2.r.a.305.13 60
279.119 even 6 inner 279.2.r.a.119.18 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
279.2.o.a.212.18 ✓ 60 31.26 odd 6
279.2.o.a.254.18 yes 60 9.2 odd 6
279.2.r.a.68.18 yes 60 1.1 even 1 trivial
279.2.r.a.119.18 yes 60 279.119 even 6 inner
837.2.o.a.440.13 60 9.7 even 3
837.2.o.a.584.13 60 93.26 even 6
837.2.r.a.305.13 60 279.88 odd 6
837.2.r.a.719.13 60 3.2 odd 2