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Properties

Label 279.2.o.a.212.12
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 212.12
Character \(\chi\) \(=\) 279.212
Dual form 279.2.o.a.254.12

$q$-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+(-0.709222 + 0.409470i) q^{2} +(-1.73176 - 0.0319037i) q^{3} +(-0.664669 + 1.15124i) q^{4} +4.31120i q^{5} +(1.24126 - 0.686475i) q^{6} -3.40596 q^{7} -2.72653i q^{8} +(2.99796 + 0.110499i) q^{9} +(-1.76530 - 3.05760i) q^{10} +(1.40132 - 2.42715i) q^{11} +(1.18777 - 1.97246i) q^{12} -0.971533i q^{13} +(2.41558 - 1.39464i) q^{14} +(0.137543 - 7.46594i) q^{15} +(-0.212908 - 0.368768i) q^{16} +(-1.13384 + 1.96387i) q^{17} +(-2.17147 + 1.14921i) q^{18} +(-2.11869 + 3.66968i) q^{19} +(-4.96322 - 2.86552i) q^{20} +(5.89829 + 0.108663i) q^{21} +2.29519i q^{22} +(4.23384 - 7.33323i) q^{23} +(-0.0869862 + 4.72168i) q^{24} -13.5864 q^{25} +(0.397813 + 0.689033i) q^{26} +(-5.18822 - 0.287003i) q^{27} +(2.26383 - 3.92108i) q^{28} +(-0.335786 - 0.581598i) q^{29} +(2.95953 + 5.35133i) q^{30} +(-2.77764 + 4.82543i) q^{31} +(5.02448 + 2.90089i) q^{32} +(-2.50417 + 4.15853i) q^{33} -1.85709i q^{34} -14.6837i q^{35} +(-2.11987 + 3.37793i) q^{36} +(-2.20494 - 1.27302i) q^{37} -3.47016i q^{38} +(-0.0309955 + 1.68246i) q^{39} +11.7546 q^{40} +6.59167i q^{41} +(-4.22769 + 2.33811i) q^{42} -7.06710i q^{43} +(1.86282 + 3.22650i) q^{44} +(-0.476382 + 12.9248i) q^{45} +6.93452i q^{46} +(-2.71943 + 1.57007i) q^{47} +(0.356941 + 0.645410i) q^{48} +4.60055 q^{49} +(9.63578 - 5.56322i) q^{50} +(2.02619 - 3.36477i) q^{51} +(1.11847 + 0.645748i) q^{52} +(0.281383 + 0.487369i) q^{53} +(3.79712 - 1.92087i) q^{54} +(10.4639 + 6.04135i) q^{55} +9.28643i q^{56} +(3.78613 - 6.28740i) q^{57} +(0.476293 + 0.274988i) q^{58} +(5.29385 - 3.05640i) q^{59} +(8.50368 + 5.12073i) q^{60} +(3.65555 - 2.11054i) q^{61} +(-0.00590265 - 4.55966i) q^{62} +(-10.2109 - 0.376354i) q^{63} -3.89966 q^{64} +4.18847 q^{65} +(0.0732248 - 3.97470i) q^{66} -11.9830 q^{67} +(-1.50726 - 2.61065i) q^{68} +(-7.56594 + 12.5643i) q^{69} +(6.01255 + 10.4140i) q^{70} +(-10.5264 + 6.07741i) q^{71} +(0.301278 - 8.17403i) q^{72} +(-8.11830 + 4.68710i) q^{73} +2.08505 q^{74} +(23.5284 + 0.433456i) q^{75} +(-2.81646 - 4.87824i) q^{76} +(-4.77282 + 8.26677i) q^{77} +(-0.666933 - 1.20593i) q^{78} +0.510908i q^{79} +(1.58983 - 0.917890i) q^{80} +(8.97558 + 0.662543i) q^{81} +(-2.69909 - 4.67496i) q^{82} +(-4.01485 + 6.95392i) q^{83} +(-4.04551 + 6.71813i) q^{84} +(-8.46662 - 4.88821i) q^{85} +(2.89376 + 5.01215i) q^{86} +(0.562944 + 1.01790i) q^{87} +(-6.61769 - 3.82072i) q^{88} -11.6759 q^{89} +(-4.95446 - 9.36163i) q^{90} +3.30900i q^{91} +(5.62821 + 9.74834i) q^{92} +(4.96414 - 8.26785i) q^{93} +(1.28579 - 2.22705i) q^{94} +(-15.8207 - 9.13409i) q^{95} +(-8.60863 - 5.18393i) q^{96} +(-3.18229 - 5.51190i) q^{97} +(-3.26281 + 1.88378i) q^{98} +(4.46929 - 7.12167i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.709222 + 0.409470i −0.501496 + 0.289539i −0.729331 − 0.684161i \(-0.760169\pi\)
0.227835 + 0.973700i \(0.426835\pi\)
\(3\) −1.73176 − 0.0319037i −0.999830 − 0.0184196i
\(4\) −0.664669 + 1.15124i −0.332335 + 0.575620i
\(5\) 4.31120i 1.92803i 0.265856 + 0.964013i \(0.414345\pi\)
−0.265856 + 0.964013i \(0.585655\pi\)
\(6\) 1.24126 − 0.686475i 0.506744 − 0.280252i
\(7\) −3.40596 −1.28733 −0.643665 − 0.765307i \(-0.722587\pi\)
−0.643665 + 0.765307i \(0.722587\pi\)
\(8\) − 2.72653i − 0.963973i
\(9\) 2.99796 + 0.110499i 0.999321 + 0.0368329i
\(10\) −1.76530 − 3.05760i −0.558238 − 0.966897i
\(11\) 1.40132 − 2.42715i 0.422513 − 0.731813i −0.573672 − 0.819085i \(-0.694482\pi\)
0.996185 + 0.0872718i \(0.0278149\pi\)
\(12\) 1.18777 − 1.97246i 0.342881 − 0.569401i
\(13\) − 0.971533i − 0.269455i −0.990883 − 0.134727i \(-0.956984\pi\)
0.990883 − 0.134727i \(-0.0430159\pi\)
\(14\) 2.41558 − 1.39464i 0.645591 − 0.372732i
\(15\) 0.137543 − 7.46594i 0.0355134 − 1.92770i
\(16\) −0.212908 − 0.368768i −0.0532271 − 0.0921921i
\(17\) −1.13384 + 1.96387i −0.274997 + 0.476308i −0.970134 − 0.242568i \(-0.922010\pi\)
0.695138 + 0.718877i \(0.255343\pi\)
\(18\) −2.17147 + 1.14921i −0.511820 + 0.270871i
\(19\) −2.11869 + 3.66968i −0.486061 + 0.841882i −0.999872 − 0.0160215i \(-0.994900\pi\)
0.513811 + 0.857904i \(0.328233\pi\)
\(20\) −4.96322 − 2.86552i −1.10981 − 0.640750i
\(21\) 5.89829 + 0.108663i 1.28711 + 0.0237121i
\(22\) 2.29519i 0.489335i
\(23\) 4.23384 − 7.33323i 0.882817 − 1.52908i 0.0346213 − 0.999401i \(-0.488978\pi\)
0.848196 − 0.529683i \(-0.177689\pi\)
\(24\) −0.0869862 + 4.72168i −0.0177560 + 0.963809i
\(25\) −13.5864 −2.71728
\(26\) 0.397813 + 0.689033i 0.0780176 + 0.135130i
\(27\) −5.18822 − 0.287003i −0.998473 − 0.0552338i
\(28\) 2.26383 − 3.92108i 0.427825 − 0.741014i
\(29\) −0.335786 − 0.581598i −0.0623538 − 0.108000i 0.833163 − 0.553027i \(-0.186527\pi\)
−0.895517 + 0.445027i \(0.853194\pi\)
\(30\) 2.95953 + 5.35133i 0.540334 + 0.977015i
\(31\) −2.77764 + 4.82543i −0.498878 + 0.866672i
\(32\) 5.02448 + 2.90089i 0.888211 + 0.512809i
\(33\) −2.50417 + 4.15853i −0.435921 + 0.723907i
\(34\) − 1.85709i − 0.318489i
\(35\) − 14.6837i − 2.48201i
\(36\) −2.11987 + 3.37793i −0.353311 + 0.562989i
\(37\) −2.20494 − 1.27302i −0.362489 − 0.209283i 0.307683 − 0.951489i \(-0.400446\pi\)
−0.670172 + 0.742206i \(0.733780\pi\)
\(38\) − 3.47016i − 0.562934i
\(39\) −0.0309955 + 1.68246i −0.00496325 + 0.269409i
\(40\) 11.7546 1.85856
\(41\) 6.59167i 1.02945i 0.857357 + 0.514723i \(0.172105\pi\)
−0.857357 + 0.514723i \(0.827895\pi\)
\(42\) −4.22769 + 2.33811i −0.652347 + 0.360777i
\(43\) − 7.06710i − 1.07772i −0.842394 − 0.538861i \(-0.818855\pi\)
0.842394 − 0.538861i \(-0.181145\pi\)
\(44\) 1.86282 + 3.22650i 0.280831 + 0.486414i
\(45\) −0.476382 + 12.9248i −0.0710148 + 1.92672i
\(46\) 6.93452i 1.02244i
\(47\) −2.71943 + 1.57007i −0.396670 + 0.229018i −0.685046 − 0.728500i \(-0.740218\pi\)
0.288376 + 0.957517i \(0.406885\pi\)
\(48\) 0.356941 + 0.645410i 0.0515200 + 0.0931569i
\(49\) 4.60055 0.657221
\(50\) 9.63578 − 5.56322i 1.36271 − 0.786759i
\(51\) 2.02619 − 3.36477i 0.283723 − 0.471162i
\(52\) 1.11847 + 0.645748i 0.155104 + 0.0895492i
\(53\) 0.281383 + 0.487369i 0.0386509 + 0.0669453i 0.884704 − 0.466154i \(-0.154361\pi\)
−0.846053 + 0.533099i \(0.821027\pi\)
\(54\) 3.79712 − 1.92087i 0.516723 − 0.261397i
\(55\) 10.4639 + 6.04135i 1.41095 + 0.814615i
\(56\) 9.28643i 1.24095i
\(57\) 3.78613 − 6.28740i 0.501485 − 0.832786i
\(58\) 0.476293 + 0.274988i 0.0625404 + 0.0361077i
\(59\) 5.29385 − 3.05640i 0.689200 − 0.397910i −0.114112 − 0.993468i \(-0.536402\pi\)
0.803312 + 0.595558i \(0.203069\pi\)
\(60\) 8.50368 + 5.12073i 1.09782 + 0.661083i
\(61\) 3.65555 − 2.11054i 0.468046 − 0.270226i −0.247375 − 0.968920i \(-0.579568\pi\)
0.715421 + 0.698693i \(0.246235\pi\)
\(62\) −0.00590265 − 4.55966i −0.000749637 − 0.579077i
\(63\) −10.2109 − 0.376354i −1.28646 − 0.0474162i
\(64\) −3.89966 −0.487458
\(65\) 4.18847 0.519516
\(66\) 0.0732248 − 3.97470i 0.00901335 − 0.489252i
\(67\) −11.9830 −1.46396 −0.731980 − 0.681326i \(-0.761404\pi\)
−0.731980 + 0.681326i \(0.761404\pi\)
\(68\) −1.50726 − 2.61065i −0.182782 − 0.316587i
\(69\) −7.56594 + 12.5643i −0.910832 + 1.51256i
\(70\) 6.01255 + 10.4140i 0.718637 + 1.24472i
\(71\) −10.5264 + 6.07741i −1.24925 + 0.721256i −0.970960 − 0.239241i \(-0.923102\pi\)
−0.278292 + 0.960497i \(0.589768\pi\)
\(72\) 0.301278 − 8.17403i 0.0355059 − 0.963318i
\(73\) −8.11830 + 4.68710i −0.950175 + 0.548584i −0.893135 − 0.449788i \(-0.851500\pi\)
−0.0570399 + 0.998372i \(0.518166\pi\)
\(74\) 2.08505 0.242382
\(75\) 23.5284 + 0.433456i 2.71682 + 0.0500512i
\(76\) −2.81646 − 4.87824i −0.323070 − 0.559573i
\(77\) −4.77282 + 8.26677i −0.543914 + 0.942086i
\(78\) −0.666933 − 1.20593i −0.0755153 − 0.136545i
\(79\) 0.510908i 0.0574816i 0.999587 + 0.0287408i \(0.00914974\pi\)
−0.999587 + 0.0287408i \(0.990850\pi\)
\(80\) 1.58983 − 0.917890i 0.177749 − 0.102623i
\(81\) 8.97558 + 0.662543i 0.997287 + 0.0736159i
\(82\) −2.69909 − 4.67496i −0.298064 − 0.516263i
\(83\) −4.01485 + 6.95392i −0.440687 + 0.763291i −0.997741 − 0.0671848i \(-0.978598\pi\)
0.557054 + 0.830476i \(0.311932\pi\)
\(84\) −4.04551 + 6.71813i −0.441401 + 0.733008i
\(85\) −8.46662 − 4.88821i −0.918334 − 0.530201i
\(86\) 2.89376 + 5.01215i 0.312043 + 0.540474i
\(87\) 0.562944 + 1.01790i 0.0603539 + 0.109130i
\(88\) −6.61769 − 3.82072i −0.705448 − 0.407291i
\(89\) −11.6759 −1.23764 −0.618821 − 0.785532i \(-0.712389\pi\)
−0.618821 + 0.785532i \(0.712389\pi\)
\(90\) −4.95446 − 9.36163i −0.522246 − 0.986802i
\(91\) 3.30900i 0.346878i
\(92\) 5.62821 + 9.74834i 0.586781 + 1.01633i
\(93\) 4.96414 − 8.26785i 0.514758 − 0.857336i
\(94\) 1.28579 − 2.22705i 0.132619 − 0.229703i
\(95\) −15.8207 − 9.13409i −1.62317 − 0.937138i
\(96\) −8.60863 − 5.18393i −0.878615 − 0.529082i
\(97\) −3.18229 − 5.51190i −0.323113 − 0.559648i 0.658016 − 0.753004i \(-0.271396\pi\)
−0.981129 + 0.193356i \(0.938063\pi\)
\(98\) −3.26281 + 1.88378i −0.329594 + 0.190291i
\(99\) 4.46929 − 7.12167i 0.449181 − 0.715754i
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.212.12 ✓ 60
3.2 odd 2 837.2.o.a.584.19 60
9.2 odd 6 279.2.r.a.119.12 yes 60
9.7 even 3 837.2.r.a.305.19 60
31.6 odd 6 279.2.r.a.68.12 yes 60
93.68 even 6 837.2.r.a.719.19 60
279.223 odd 6 837.2.o.a.440.19 60
279.254 even 6 inner 279.2.o.a.254.12 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
279.2.o.a.212.12 ✓ 60 1.1 even 1 trivial
279.2.o.a.254.12 yes 60 279.254 even 6 inner
279.2.r.a.68.12 yes 60 31.6 odd 6
279.2.r.a.119.12 yes 60 9.2 odd 6
837.2.o.a.440.19 60 279.223 odd 6
837.2.o.a.584.19 60 3.2 odd 2
837.2.r.a.305.19 60 9.7 even 3
837.2.r.a.719.19 60 93.68 even 6