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Properties

Label 279.2.o.a.212.8
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 212.8
Character \(\chi\) \(=\) 279.212
Dual form 279.2.o.a.254.8

$q$-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+(-1.45499 + 0.840038i) q^{2} +(-1.65183 + 0.521020i) q^{3} +(0.411327 - 0.712440i) q^{4} +0.175209i q^{5} +(1.96572 - 2.14568i) q^{6} +4.05741 q^{7} -1.97803i q^{8} +(2.45708 - 1.72127i) q^{9} +(-0.147183 - 0.254928i) q^{10} +(0.246063 - 0.426194i) q^{11} +(-0.308247 + 1.39114i) q^{12} +2.38791i q^{13} +(-5.90349 + 3.40838i) q^{14} +(-0.0912876 - 0.289416i) q^{15} +(2.48427 + 4.30289i) q^{16} +(-0.0216897 + 0.0375676i) q^{17} +(-2.12908 + 4.56847i) q^{18} +(-1.32655 + 2.29766i) q^{19} +(0.124826 + 0.0720685i) q^{20} +(-6.70215 + 2.11399i) q^{21} +0.826809i q^{22} +(-1.81002 + 3.13505i) q^{23} +(1.03059 + 3.26737i) q^{24} +4.96930 q^{25} +(-2.00594 - 3.47438i) q^{26} +(-3.16185 + 4.12343i) q^{27} +(1.66893 - 2.89066i) q^{28} +(3.85815 + 6.68252i) q^{29} +(0.375943 + 0.344412i) q^{30} +(2.67208 - 4.88467i) q^{31} +(-3.80313 - 2.19574i) q^{32} +(-0.184399 + 0.832203i) q^{33} -0.0728806i q^{34} +0.710897i q^{35} +(-0.215639 - 2.45853i) q^{36} +(-3.97232 - 2.29342i) q^{37} -4.45742i q^{38} +(-1.24415 - 3.94442i) q^{39} +0.346569 q^{40} -4.87910i q^{41} +(7.97572 - 8.70590i) q^{42} +9.09531i q^{43} +(-0.202425 - 0.350610i) q^{44} +(0.301583 + 0.430503i) q^{45} -6.08196i q^{46} +(-5.61428 + 3.24141i) q^{47} +(-6.34549 - 5.81328i) q^{48} +9.46261 q^{49} +(-7.23028 + 4.17440i) q^{50} +(0.0162542 - 0.0733560i) q^{51} +(1.70124 + 0.982213i) q^{52} +(6.09442 + 10.5558i) q^{53} +(1.13662 - 8.65562i) q^{54} +(0.0746732 + 0.0431126i) q^{55} -8.02568i q^{56} +(0.994114 - 4.48650i) q^{57} +(-11.2271 - 6.48199i) q^{58} +(8.67969 - 5.01122i) q^{59} +(-0.243741 - 0.0540078i) q^{60} +(7.20110 - 4.15756i) q^{61} +(0.215453 + 9.35178i) q^{62} +(9.96938 - 6.98391i) q^{63} -2.55908 q^{64} -0.418385 q^{65} +(-0.430784 - 1.36575i) q^{66} -3.24025 q^{67} +(0.0178431 + 0.0309052i) q^{68} +(1.35643 - 6.12163i) q^{69} +(-0.597181 - 1.03435i) q^{70} +(-1.70536 + 0.984588i) q^{71} +(-3.40472 - 4.86017i) q^{72} +(3.33296 - 1.92429i) q^{73} +7.70623 q^{74} +(-8.20844 + 2.58910i) q^{75} +(1.09130 + 1.89018i) q^{76} +(0.998379 - 1.72924i) q^{77} +(5.12369 + 4.69395i) q^{78} -0.529369i q^{79} +(-0.753907 + 0.435268i) q^{80} +(3.07445 - 8.45859i) q^{81} +(4.09863 + 7.09904i) q^{82} +(-3.43452 + 5.94877i) q^{83} +(-1.25069 + 5.64442i) q^{84} +(-0.00658220 - 0.00380024i) q^{85} +(-7.64041 - 13.2336i) q^{86} +(-9.85473 - 9.02820i) q^{87} +(-0.843023 - 0.486720i) q^{88} +4.98238 q^{89} +(-0.800439 - 0.373036i) q^{90} +9.68875i q^{91} +(1.48903 + 2.57907i) q^{92} +(-1.86882 + 9.46084i) q^{93} +(5.44581 - 9.43242i) q^{94} +(-0.402571 - 0.232425i) q^{95} +(7.42615 + 1.64548i) q^{96} +(1.29180 + 2.23747i) q^{97} +(-13.7680 + 7.94895i) q^{98} +(-0.128999 - 1.47073i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.45499 + 0.840038i −1.02883 + 0.593997i −0.916650 − 0.399691i \(-0.869117\pi\)
−0.112182 + 0.993688i \(0.535784\pi\)
\(3\) −1.65183 + 0.521020i −0.953684 + 0.300811i
\(4\) 0.411327 − 0.712440i 0.205664 − 0.356220i
\(5\) 0.175209i 0.0783561i 0.999232 + 0.0391780i \(0.0124739\pi\)
−0.999232 + 0.0391780i \(0.987526\pi\)
\(6\) 1.96572 − 2.14568i 0.802500 − 0.875969i
\(7\) 4.05741 1.53356 0.766779 − 0.641911i \(-0.221858\pi\)
0.766779 + 0.641911i \(0.221858\pi\)
\(8\) − 1.97803i − 0.699339i
\(9\) 2.45708 − 1.72127i 0.819026 − 0.573757i
\(10\) −0.147183 − 0.254928i −0.0465432 − 0.0806152i
\(11\) 0.246063 − 0.426194i 0.0741908 − 0.128502i −0.826543 − 0.562873i \(-0.809696\pi\)
0.900734 + 0.434371i \(0.143029\pi\)
\(12\) −0.308247 + 1.39114i −0.0889833 + 0.401587i
\(13\) 2.38791i 0.662288i 0.943580 + 0.331144i \(0.107435\pi\)
−0.943580 + 0.331144i \(0.892565\pi\)
\(14\) −5.90349 + 3.40838i −1.57777 + 0.910928i
\(15\) −0.0912876 − 0.289416i −0.0235704 − 0.0747269i
\(16\) 2.48427 + 4.30289i 0.621069 + 1.07572i
\(17\) −0.0216897 + 0.0375676i −0.00526052 + 0.00911148i −0.868644 − 0.495437i \(-0.835008\pi\)
0.863383 + 0.504549i \(0.168341\pi\)
\(18\) −2.12908 + 4.56847i −0.501830 + 1.07680i
\(19\) −1.32655 + 2.29766i −0.304332 + 0.527119i −0.977112 − 0.212724i \(-0.931767\pi\)
0.672780 + 0.739842i \(0.265100\pi\)
\(20\) 0.124826 + 0.0720685i 0.0279120 + 0.0161150i
\(21\) −6.70215 + 2.11399i −1.46253 + 0.461311i
\(22\) 0.826809i 0.176276i
\(23\) −1.81002 + 3.13505i −0.377416 + 0.653704i −0.990686 − 0.136170i \(-0.956521\pi\)
0.613269 + 0.789874i \(0.289854\pi\)
\(24\) 1.03059 + 3.26737i 0.210369 + 0.666948i
\(25\) 4.96930 0.993860
\(26\) −2.00594 − 3.47438i −0.393396 − 0.681383i
\(27\) −3.16185 + 4.12343i −0.608499 + 0.793555i
\(28\) 1.66893 − 2.89066i 0.315397 − 0.546284i
\(29\) 3.85815 + 6.68252i 0.716441 + 1.24091i 0.962401 + 0.271632i \(0.0875635\pi\)
−0.245960 + 0.969280i \(0.579103\pi\)
\(30\) 0.375943 + 0.344412i 0.0686375 + 0.0628807i
\(31\) 2.67208 − 4.88467i 0.479921 − 0.877312i
\(32\) −3.80313 − 2.19574i −0.672305 − 0.388156i
\(33\) −0.184399 + 0.832203i −0.0320997 + 0.144868i
\(34\) − 0.0728806i − 0.0124989i
\(35\) 0.710897i 0.120164i
\(36\) −0.215639 − 2.45853i −0.0359399 − 0.409754i
\(37\) −3.97232 − 2.29342i −0.653045 − 0.377036i 0.136577 − 0.990629i \(-0.456390\pi\)
−0.789622 + 0.613594i \(0.789723\pi\)
\(38\) − 4.45742i − 0.723089i
\(39\) −1.24415 − 3.94442i −0.199223 − 0.631613i
\(40\) 0.346569 0.0547974
\(41\) − 4.87910i − 0.761988i −0.924578 − 0.380994i \(-0.875582\pi\)
0.924578 − 0.380994i \(-0.124418\pi\)
\(42\) 7.97572 − 8.70590i 1.23068 − 1.34335i
\(43\) 9.09531i 1.38702i 0.720446 + 0.693511i \(0.243937\pi\)
−0.720446 + 0.693511i \(0.756063\pi\)
\(44\) −0.202425 − 0.350610i −0.0305167 − 0.0528565i
\(45\) 0.301583 + 0.430503i 0.0449573 + 0.0641756i
\(46\) − 6.08196i − 0.896736i
\(47\) −5.61428 + 3.24141i −0.818928 + 0.472808i −0.850047 − 0.526708i \(-0.823426\pi\)
0.0311189 + 0.999516i \(0.490093\pi\)
\(48\) −6.34549 − 5.81328i −0.915892 − 0.839075i
\(49\) 9.46261 1.35180
\(50\) −7.23028 + 4.17440i −1.02252 + 0.590350i
\(51\) 0.0162542 − 0.0733560i 0.00227604 − 0.0102719i
\(52\) 1.70124 + 0.982213i 0.235920 + 0.136208i
\(53\) 6.09442 + 10.5558i 0.837133 + 1.44996i 0.892282 + 0.451478i \(0.149103\pi\)
−0.0551497 + 0.998478i \(0.517564\pi\)
\(54\) 1.13662 − 8.65562i 0.154675 − 1.17788i
\(55\) 0.0746732 + 0.0431126i 0.0100689 + 0.00581330i
\(56\) − 8.02568i − 1.07248i
\(57\) 0.994114 − 4.48650i 0.131674 − 0.594251i
\(58\) −11.2271 − 6.48199i −1.47420 − 0.851127i
\(59\) 8.67969 − 5.01122i 1.13000 − 0.652406i 0.186065 − 0.982537i \(-0.440426\pi\)
0.943935 + 0.330132i \(0.107093\pi\)
\(60\) −0.243741 − 0.0540078i −0.0314668 − 0.00697238i
\(61\) 7.20110 − 4.15756i 0.922007 − 0.532321i 0.0377320 − 0.999288i \(-0.487987\pi\)
0.884275 + 0.466967i \(0.154653\pi\)
\(62\) 0.215453 + 9.35178i 0.0273625 + 1.18768i
\(63\) 9.96938 − 6.98391i 1.25602 − 0.879890i
\(64\) −2.55908 −0.319885
\(65\) −0.418385 −0.0518942
\(66\) −0.430784 − 1.36575i −0.0530258 − 0.168112i
\(67\) −3.24025 −0.395860 −0.197930 − 0.980216i \(-0.563422\pi\)
−0.197930 + 0.980216i \(0.563422\pi\)
\(68\) 0.0178431 + 0.0309052i 0.00216380 + 0.00374780i
\(69\) 1.35643 − 6.12163i 0.163294 − 0.736958i
\(70\) −0.597181 − 1.03435i −0.0713768 − 0.123628i
\(71\) −1.70536 + 0.984588i −0.202388 + 0.116849i −0.597769 − 0.801668i \(-0.703946\pi\)
0.395381 + 0.918517i \(0.370613\pi\)
\(72\) −3.40472 − 4.86017i −0.401251 − 0.572776i
\(73\) 3.33296 − 1.92429i 0.390094 − 0.225221i −0.292107 − 0.956386i \(-0.594356\pi\)
0.682201 + 0.731165i \(0.261023\pi\)
\(74\) 7.70623 0.895831
\(75\) −8.20844 + 2.58910i −0.947828 + 0.298964i
\(76\) 1.09130 + 1.89018i 0.125180 + 0.216818i
\(77\) 0.998379 − 1.72924i 0.113776 − 0.197066i
\(78\) 5.12369 + 4.69395i 0.580143 + 0.531486i
\(79\) − 0.529369i − 0.0595586i −0.999556 − 0.0297793i \(-0.990520\pi\)
0.999556 − 0.0297793i \(-0.00948045\pi\)
\(80\) −0.753907 + 0.435268i −0.0842894 + 0.0486645i
\(81\) 3.07445 − 8.45859i 0.341606 − 0.939843i
\(82\) 4.09863 + 7.09904i 0.452618 + 0.783958i
\(83\) −3.43452 + 5.94877i −0.376988 + 0.652962i −0.990622 − 0.136628i \(-0.956373\pi\)
0.613635 + 0.789590i \(0.289707\pi\)
\(84\) −1.25069 + 5.64442i −0.136461 + 0.615857i
\(85\) −0.00658220 − 0.00380024i −0.000713940 − 0.000412193i
\(86\) −7.64041 − 13.2336i −0.823886 − 1.42701i
\(87\) −9.85473 − 9.02820i −1.05654 − 0.967925i
\(88\) −0.843023 − 0.486720i −0.0898666 − 0.0518845i
\(89\) 4.98238 0.528131 0.264066 − 0.964505i \(-0.414936\pi\)
0.264066 + 0.964505i \(0.414936\pi\)
\(90\) −0.800439 − 0.373036i −0.0843737 − 0.0393214i
\(91\) 9.68875i 1.01566i
\(92\) 1.48903 + 2.57907i 0.155242 + 0.268886i
\(93\) −1.86882 + 9.46084i −0.193787 + 0.981044i
\(94\) 5.44581 − 9.43242i 0.561693 − 0.972880i
\(95\) −0.402571 − 0.232425i −0.0413030 − 0.0238463i
\(96\) 7.42615 + 1.64548i 0.757928 + 0.167941i
\(97\) 1.29180 + 2.23747i 0.131163 + 0.227180i 0.924125 − 0.382090i \(-0.124796\pi\)
−0.792962 + 0.609271i \(0.791462\pi\)
\(98\) −13.7680 + 7.94895i −1.39078 + 0.802965i
\(99\) −0.128999 − 1.47073i −0.0129649 − 0.147814i
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.212.8 ✓ 60
3.2 odd 2 837.2.o.a.584.23 60
9.2 odd 6 279.2.r.a.119.8 yes 60
9.7 even 3 837.2.r.a.305.23 60
31.6 odd 6 279.2.r.a.68.8 yes 60
93.68 even 6 837.2.r.a.719.23 60
279.223 odd 6 837.2.o.a.440.23 60
279.254 even 6 inner 279.2.o.a.254.8 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
279.2.o.a.212.8 ✓ 60 1.1 even 1 trivial
279.2.o.a.254.8 yes 60 279.254 even 6 inner
279.2.r.a.68.8 yes 60 31.6 odd 6
279.2.r.a.119.8 yes 60 9.2 odd 6
837.2.o.a.440.23 60 279.223 odd 6
837.2.o.a.584.23 60 3.2 odd 2
837.2.r.a.305.23 60 9.7 even 3
837.2.r.a.719.23 60 93.68 even 6