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Properties

Label 279.2.o.a.212.5
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 212.5
Character \(\chi\) \(=\) 279.212
Dual form 279.2.o.a.254.5

$q$-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} +(0.571667 + 1.63499i) q^{3} +(1.62840 - 2.82047i) q^{4} -2.34954i q^{5} +(-3.00943 - 2.59108i) q^{6} +1.01394 q^{7} +2.88154i q^{8} +(-2.34639 + 1.86934i) q^{9} +(2.69347 + 4.66523i) q^{10} +(-2.78765 + 4.82834i) q^{11} +(5.54234 + 1.05005i) q^{12} +3.47630i q^{13} +(-2.01328 + 1.16237i) q^{14} +(3.84147 - 1.34315i) q^{15} +(-0.0465583 - 0.0806413i) q^{16} +(-2.60265 + 4.50793i) q^{17} +(2.51600 - 6.40163i) q^{18} +(-0.710547 + 1.23070i) q^{19} +(-6.62679 - 3.82598i) q^{20} +(0.579637 + 1.65778i) q^{21} -12.7829i q^{22} +(1.43676 - 2.48854i) q^{23} +(-4.71129 + 1.64728i) q^{24} -0.520325 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} +(2.87885 + 4.76573i) q^{31} +(-4.80608 - 2.77479i) q^{32} +(-9.48790 - 1.79757i) q^{33} -11.9346i q^{34} -2.38229i q^{35} +(1.45156 + 9.66195i) q^{36} +(-0.529463 - 0.305685i) q^{37} -3.25824i q^{38} +(-5.68371 + 1.98729i) q^{39} +6.77028 q^{40} +7.30898i q^{41} +(-3.05139 - 2.62721i) q^{42} -6.79714i q^{43} +(9.07878 + 15.7249i) q^{44} +(4.39209 + 5.51294i) q^{45} +6.58832i q^{46} +(8.02931 - 4.63572i) q^{47} +(0.105232 - 0.122222i) q^{48} -5.97192 q^{49} +(1.03316 - 0.596493i) q^{50} +(-8.85827 - 1.67828i) q^{51} +(9.80478 + 5.66079i) q^{52} +(0.812196 + 1.40677i) q^{53} +(11.9049 + 0.454044i) q^{54} +(11.3444 + 6.54968i) q^{55} +2.92171i q^{56} +(-2.41839 - 0.458185i) q^{57} +(-6.48049 - 3.74151i) q^{58} +(10.4090 - 6.00963i) 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} +8.16769 q^{65} +(20.8999 - 7.30754i) q^{66} -13.9394 q^{67} +(8.47630 + 14.6814i) q^{68} +(4.89009 + 0.926472i) q^{69} +(2.73102 + 4.73027i) q^{70} +(3.30857 - 1.91021i) q^{71} +(-5.38658 - 6.76122i) q^{72} +(2.92420 - 1.68829i) q^{73} +1.40173 q^{74} +(-0.297453 - 0.850727i) q^{75} +(2.31410 + 4.00815i) q^{76} +(-2.82651 + 4.89566i) q^{77} +(9.00737 - 10.4617i) q^{78} -0.0917442i q^{79} +(-0.189470 + 0.109390i) q^{80} +(2.01112 - 8.77242i) q^{81} +(-8.37890 - 14.5127i) q^{82} +(7.94265 - 13.7571i) q^{83} +(5.61960 + 1.06469i) q^{84} +(10.5915 + 6.11503i) q^{85} +(7.79214 + 13.4964i) q^{86} +(-3.68839 + 4.28391i) q^{87} +(-13.9131 - 8.03271i) q^{88} +5.63299 q^{89} +(-15.0409 - 5.91145i) q^{90} +3.52476i q^{91} +(-4.67923 - 8.10466i) q^{92} +(-6.14619 + 7.43131i) q^{93} +(-10.6286 + 18.4094i) q^{94} +(2.89158 + 1.66946i) q^{95} +(1.78928 - 9.44415i) q^{96} +(3.00571 + 5.20603i) q^{97} +(11.8578 - 6.84612i) q^{98} +(-2.48491 - 16.5403i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.98560 + 1.14638i −1.40403 + 0.810617i −0.994803 − 0.101817i \(-0.967535\pi\)
−0.409226 + 0.912433i \(0.634201\pi\)
\(3\) 0.571667 + 1.63499i 0.330052 + 0.943963i
\(4\) 1.62840 − 2.82047i 0.814198 − 1.41023i
\(5\) − 2.34954i − 1.05074i −0.850872 − 0.525372i \(-0.823926\pi\)
0.850872 − 0.525372i \(-0.176074\pi\)
\(6\) −3.00943 − 2.59108i −1.22859 − 1.05781i
\(7\) 1.01394 0.383234 0.191617 − 0.981470i \(-0.438627\pi\)
0.191617 + 0.981470i \(0.438627\pi\)
\(8\) 2.88154i 1.01878i
\(9\) −2.34639 + 1.86934i −0.782131 + 0.623114i
\(10\) 2.69347 + 4.66523i 0.851751 + 1.47528i
\(11\) −2.78765 + 4.82834i −0.840507 + 1.45580i 0.0489603 + 0.998801i \(0.484409\pi\)
−0.889467 + 0.456999i \(0.848924\pi\)
\(12\) 5.54234 + 1.05005i 1.59994 + 0.303122i
\(13\) 3.47630i 0.964151i 0.876130 + 0.482076i \(0.160117\pi\)
−0.876130 + 0.482076i \(0.839883\pi\)
\(14\) −2.01328 + 1.16237i −0.538071 + 0.310656i
\(15\) 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 \(0.384119\pi\)
−0.987299 + 0.158871i \(0.949215\pi\)
\(18\) 2.51600 − 6.40163i 0.593028 − 1.50888i
\(19\) −0.710547 + 1.23070i −0.163011 + 0.282343i −0.935947 − 0.352141i \(-0.885454\pi\)
0.772936 + 0.634483i \(0.218787\pi\)
\(20\) −6.62679 − 3.82598i −1.48180 − 0.855515i
\(21\) 0.579637 + 1.65778i 0.126487 + 0.361758i
\(22\) − 12.7829i − 2.72531i
\(23\) 1.43676 − 2.48854i 0.299585 − 0.518897i −0.676456 − 0.736483i \(-0.736485\pi\)
0.976041 + 0.217586i \(0.0698184\pi\)
\(24\) −4.71129 + 1.64728i −0.961688 + 0.336250i
\(25\) −0.520325 −0.104065
\(26\) −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 \(-0.0686680\pi\)
−0.673790 + 0.738923i \(0.735335\pi\)
\(30\) −6.08785 + 7.07077i −1.11148 + 1.29094i
\(31\) 2.87885 + 4.76573i 0.517057 + 0.855951i
\(32\) −4.80608 − 2.77479i −0.849603 − 0.490518i
\(33\) −9.48790 − 1.79757i −1.65163 − 0.312917i
\(34\) − 11.9346i − 2.04676i
\(35\) − 2.38229i − 0.402681i
\(36\) 1.45156 + 9.66195i 0.241926 + 1.61033i
\(37\) −0.529463 − 0.305685i −0.0870431 − 0.0502544i 0.455847 − 0.890058i \(-0.349337\pi\)
−0.542890 + 0.839804i \(0.682670\pi\)
\(38\) − 3.25824i − 0.528556i
\(39\) −5.68371 + 1.98729i −0.910123 + 0.318220i
\(40\) 6.77028 1.07048
\(41\) 7.30898i 1.14147i 0.821134 + 0.570735i \(0.193342\pi\)
−0.821134 + 0.570735i \(0.806658\pi\)
\(42\) −3.05139 − 2.62721i −0.470839 − 0.405387i
\(43\) − 6.79714i − 1.03655i −0.855213 − 0.518277i \(-0.826574\pi\)
0.855213 − 0.518277i \(-0.173426\pi\)
\(44\) 9.07878 + 15.7249i 1.36868 + 2.37062i
\(45\) 4.39209 + 5.51294i 0.654734 + 0.821820i
\(46\) 6.58832i 0.971395i
\(47\) 8.02931 − 4.63572i 1.17119 − 0.676190i 0.217233 − 0.976120i \(-0.430297\pi\)
0.953961 + 0.299930i \(0.0969634\pi\)
\(48\) 0.105232 − 0.122222i 0.0151889 − 0.0176413i
\(49\) −5.97192 −0.853132
\(50\) 1.03316 − 0.596493i 0.146110 − 0.0843568i
\(51\) −8.85827 − 1.67828i −1.24041 − 0.235006i
\(52\) 9.80478 + 5.66079i 1.35968 + 0.785010i
\(53\) 0.812196 + 1.40677i 0.111564 + 0.193234i 0.916401 − 0.400262i \(-0.131081\pi\)
−0.804837 + 0.593496i \(0.797747\pi\)
\(54\) 11.9049 + 0.454044i 1.62006 + 0.0617875i
\(55\) 11.3444 + 6.54968i 1.52967 + 0.883158i
\(56\) 2.92171i 0.390430i
\(57\) −2.41839 − 0.458185i −0.320323 − 0.0606881i
\(58\) −6.48049 − 3.74151i −0.850930 − 0.491284i
\(59\) 10.4090 − 6.00963i 1.35514 − 0.782388i 0.366172 − 0.930547i \(-0.380668\pi\)
0.988963 + 0.148159i \(0.0473349\pi\)
\(60\) 2.46712 − 13.0219i 0.318504 − 1.68112i
\(61\) −2.77331 + 1.60117i −0.355086 + 0.205009i −0.666923 − 0.745126i \(-0.732389\pi\)
0.311837 + 0.950136i \(0.399056\pi\)
\(62\) −11.1796 − 6.16255i −1.41981 − 0.782645i
\(63\) −2.37910 + 1.89540i −0.299739 + 0.238798i
\(64\) 12.9101 1.61377
\(65\) 8.16769 1.01308
\(66\) 20.8999 − 7.30754i 2.57259 − 0.899496i
\(67\) −13.9394 −1.70297 −0.851487 − 0.524376i \(-0.824299\pi\)
−0.851487 + 0.524376i \(0.824299\pi\)
\(68\) 8.47630 + 14.6814i 1.02790 + 1.78038i
\(69\) 4.89009 + 0.926472i 0.588698 + 0.111534i
\(70\) 2.73102 + 4.73027i 0.326420 + 0.565376i
\(71\) 3.30857 − 1.91021i 0.392655 − 0.226700i −0.290655 − 0.956828i \(-0.593873\pi\)
0.683310 + 0.730128i \(0.260540\pi\)
\(72\) −5.38658 − 6.76122i −0.634815 − 0.796817i
\(73\) 2.92420 − 1.68829i 0.342252 − 0.197599i −0.319015 − 0.947750i \(-0.603352\pi\)
0.661268 + 0.750150i \(0.270019\pi\)
\(74\) 1.40173 0.162948
\(75\) −0.297453 − 0.850727i −0.0343469 − 0.0982335i
\(76\) 2.31410 + 4.00815i 0.265446 + 0.459766i
\(77\) −2.82651 + 4.89566i −0.322110 + 0.557912i
\(78\) 9.00737 − 10.4617i 1.01988 − 1.18455i
\(79\) − 0.0917442i − 0.0103220i −0.999987 − 0.00516102i \(-0.998357\pi\)
0.999987 − 0.00516102i \(-0.00164281\pi\)
\(80\) −0.189470 + 0.109390i −0.0211834 + 0.0122302i
\(81\) 2.01112 − 8.77242i 0.223458 − 0.974714i
\(82\) −8.37890 − 14.5127i −0.925295 − 1.60266i
\(83\) 7.94265 − 13.7571i 0.871820 − 1.51004i 0.0117070 − 0.999931i \(-0.496273\pi\)
0.860113 − 0.510104i \(-0.170393\pi\)
\(84\) 5.61960 + 1.06469i 0.613149 + 0.116167i
\(85\) 10.5915 + 6.11503i 1.14881 + 0.663268i
\(86\) 7.79214 + 13.4964i 0.840248 + 1.45535i
\(87\) −3.68839 + 4.28391i −0.395437 + 0.459283i
\(88\) −13.9131 − 8.03271i −1.48314 − 0.856289i
\(89\) 5.63299 0.597096 0.298548 − 0.954395i \(-0.403498\pi\)
0.298548 + 0.954395i \(0.403498\pi\)
\(90\) −15.0409 − 5.91145i −1.58545 − 0.623121i
\(91\) 3.52476i 0.369495i
\(92\) −4.67923 − 8.10466i −0.487843 − 0.844970i
\(93\) −6.14619 + 7.43131i −0.637330 + 0.770591i
\(94\) −10.6286 + 18.4094i −1.09626 + 1.89878i
\(95\) 2.89158 + 1.66946i 0.296670 + 0.171283i
\(96\) 1.78928 − 9.44415i 0.182618 − 0.963890i
\(97\) 3.00571 + 5.20603i 0.305183 + 0.528593i 0.977302 − 0.211850i \(-0.0679490\pi\)
−0.672119 + 0.740443i \(0.734616\pi\)
\(98\) 11.8578 − 6.84612i 1.19782 − 0.691563i
\(99\) −2.48491 − 16.5403i −0.249743 − 1.66236i
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.212.5 ✓ 60
3.2 odd 2 837.2.o.a.584.26 60
9.2 odd 6 279.2.r.a.119.5 yes 60
9.7 even 3 837.2.r.a.305.26 60
31.6 odd 6 279.2.r.a.68.5 yes 60
93.68 even 6 837.2.r.a.719.26 60
279.223 odd 6 837.2.o.a.440.26 60
279.254 even 6 inner 279.2.o.a.254.5 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
279.2.o.a.212.5 ✓ 60 1.1 even 1 trivial
279.2.o.a.254.5 yes 60 279.254 even 6 inner
279.2.r.a.68.5 yes 60 31.6 odd 6
279.2.r.a.119.5 yes 60 9.2 odd 6
837.2.o.a.440.26 60 279.223 odd 6
837.2.o.a.584.26 60 3.2 odd 2
837.2.r.a.305.26 60 9.7 even 3
837.2.r.a.719.26 60 93.68 even 6