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Properties

Label 279.2.o.a.254.6
Level $279$
Weight $2$
Character 279.254
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 254.6
Character \(\chi\) \(=\) 279.254
Dual form 279.2.o.a.212.6

$q$-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+(-1.80686 - 1.04319i) q^{2} +(-1.57828 + 0.713466i) q^{3} +(1.17649 + 2.03774i) q^{4} +3.18672i q^{5} +(3.59601 + 0.357313i) q^{6} -4.53398 q^{7} -0.736453i q^{8} +(1.98193 - 2.25210i) q^{9} +(3.32435 - 5.75794i) q^{10} +(1.14845 + 1.98917i) q^{11} +(-3.31069 - 2.37674i) q^{12} -4.87719i q^{13} +(8.19226 + 4.72980i) q^{14} +(-2.27362 - 5.02953i) q^{15} +(1.58472 - 2.74482i) q^{16} +(0.0460942 + 0.0798374i) q^{17} +(-5.93044 + 2.00169i) q^{18} +(0.475784 + 0.824082i) q^{19} +(-6.49370 + 3.74914i) q^{20} +(7.15589 - 3.23484i) q^{21} -4.79220i q^{22} +(-3.80275 - 6.58655i) q^{23} +(0.525434 + 1.16233i) q^{24} -5.15516 q^{25} +(-5.08784 + 8.81240i) q^{26} +(-1.52124 + 4.96848i) q^{27} +(-5.33419 - 9.23908i) q^{28} +(-0.484455 + 0.839100i) q^{29} +(-1.13865 + 11.4595i) q^{30} +(-5.56418 - 0.199836i) q^{31} +(-7.00231 + 4.04278i) q^{32} +(-3.23178 - 2.32009i) q^{33} -0.192340i q^{34} -14.4485i q^{35} +(6.92092 + 1.38909i) q^{36} +(7.68267 - 4.43559i) q^{37} -1.98533i q^{38} +(3.47971 + 7.69757i) q^{39} +2.34687 q^{40} -5.95396i q^{41} +(-16.3042 - 1.62005i) q^{42} -1.07365i q^{43} +(-2.70228 + 4.68048i) q^{44} +(7.17680 + 6.31585i) q^{45} +15.8680i q^{46} +(2.29154 + 1.32302i) q^{47} +(-0.542797 + 5.46273i) q^{48} +13.5570 q^{49} +(9.31465 + 5.37781i) q^{50} +(-0.129711 - 0.0931191i) q^{51} +(9.93846 - 5.73797i) q^{52} +(-0.328804 + 0.569506i) q^{53} +(7.93174 - 7.39040i) q^{54} +(-6.33893 + 3.65978i) q^{55} +3.33906i q^{56} +(-1.33888 - 0.961176i) q^{57} +(1.75068 - 1.01076i) q^{58} +(1.84976 + 1.06796i) q^{59} +(7.57399 - 10.5502i) q^{60} +(-13.1373 - 7.58482i) q^{61} +(9.84521 + 6.16557i) q^{62} +(-8.98604 + 10.2110i) q^{63} +10.5307 q^{64} +15.5422 q^{65} +(3.41908 + 7.56343i) q^{66} +12.6087 q^{67} +(-0.108459 + 0.187856i) q^{68} +(10.7011 + 7.68229i) q^{69} +(-15.0725 + 26.1064i) q^{70} +(-2.41191 - 1.39252i) q^{71} +(-1.65856 - 1.45960i) q^{72} +(-0.596742 - 0.344529i) q^{73} -18.5086 q^{74} +(8.13629 - 3.67804i) q^{75} +(-1.11951 + 1.93905i) q^{76} +(-5.20705 - 9.01887i) q^{77} +(1.74268 - 17.5384i) q^{78} -4.76336i q^{79} +(8.74696 + 5.05006i) q^{80} +(-1.14390 - 8.92701i) q^{81} +(-6.21111 + 10.7580i) q^{82} +(-4.12108 - 7.13792i) q^{83} +(15.0106 + 10.7761i) q^{84} +(-0.254419 + 0.146889i) q^{85} +(-1.12002 + 1.93993i) q^{86} +(0.165935 - 1.66998i) q^{87} +(1.46493 - 0.845778i) q^{88} -12.6005 q^{89} +(-6.37882 - 18.8986i) q^{90} +22.1131i q^{91} +(8.94779 - 15.4980i) q^{92} +(8.92440 - 3.65446i) q^{93} +(-2.76033 - 4.78103i) q^{94} +(-2.62612 + 1.51619i) q^{95} +(8.16720 - 11.3766i) q^{96} +(-2.92754 + 5.07064i) q^{97} +(-24.4955 - 14.1425i) q^{98} +(6.75596 + 1.35598i) 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 \(\left(\mathbb{Z}/279\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(218\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.80686 − 1.04319i −1.27764 − 0.737647i −0.301227 − 0.953552i \(-0.597396\pi\)
−0.976414 + 0.215906i \(0.930730\pi\)
\(3\) −1.57828 + 0.713466i −0.911220 + 0.411920i
\(4\) 1.17649 + 2.03774i 0.588245 + 1.01887i
\(5\) 3.18672i 1.42514i 0.701599 + 0.712572i \(0.252470\pi\)
−0.701599 + 0.712572i \(0.747530\pi\)
\(6\) 3.59601 + 0.357313i 1.46806 + 0.145872i
\(7\) −4.53398 −1.71368 −0.856842 − 0.515579i \(-0.827577\pi\)
−0.856842 + 0.515579i \(0.827577\pi\)
\(8\) − 0.736453i − 0.260375i
\(9\) 1.98193 − 2.25210i 0.660644 − 0.750700i
\(10\) 3.32435 − 5.75794i 1.05125 − 1.82082i
\(11\) 1.14845 + 1.98917i 0.346270 + 0.599758i 0.985584 − 0.169188i \(-0.0541146\pi\)
−0.639313 + 0.768946i \(0.720781\pi\)
\(12\) −3.31069 − 2.37674i −0.955714 − 0.686105i
\(13\) − 4.87719i − 1.35269i −0.736585 − 0.676345i \(-0.763563\pi\)
0.736585 − 0.676345i \(-0.236437\pi\)
\(14\) 8.19226 + 4.72980i 2.18947 + 1.26409i
\(15\) −2.27362 − 5.02953i −0.587045 − 1.29862i
\(16\) 1.58472 − 2.74482i 0.396180 − 0.686204i
\(17\) 0.0460942 + 0.0798374i 0.0111795 + 0.0193634i 0.871561 − 0.490287i \(-0.163108\pi\)
−0.860382 + 0.509650i \(0.829775\pi\)
\(18\) −5.93044 + 2.00169i −1.39782 + 0.471803i
\(19\) 0.475784 + 0.824082i 0.109152 + 0.189057i 0.915427 − 0.402484i \(-0.131853\pi\)
−0.806275 + 0.591541i \(0.798520\pi\)
\(20\) −6.49370 + 3.74914i −1.45204 + 0.838334i
\(21\) 7.15589 − 3.23484i 1.56154 − 0.705901i
\(22\) − 4.79220i − 1.02170i
\(23\) −3.80275 − 6.58655i −0.792928 − 1.37339i −0.924147 − 0.382038i \(-0.875222\pi\)
0.131219 − 0.991353i \(-0.458111\pi\)
\(24\) 0.525434 + 1.16233i 0.107254 + 0.237259i
\(25\) −5.15516 −1.03103
\(26\) −5.08784 + 8.81240i −0.997807 + 1.72825i
\(27\) −1.52124 + 4.96848i −0.292764 + 0.956185i
\(28\) −5.33419 − 9.23908i −1.00807 − 1.74602i
\(29\) −0.484455 + 0.839100i −0.0899610 + 0.155817i −0.907494 − 0.420064i \(-0.862008\pi\)
0.817533 + 0.575881i \(0.195341\pi\)
\(30\) −1.13865 + 11.4595i −0.207889 + 2.09220i
\(31\) −5.56418 − 0.199836i −0.999356 − 0.0358915i
\(32\) −7.00231 + 4.04278i −1.23784 + 0.714670i
\(33\) −3.23178 − 2.32009i −0.562581 − 0.403876i
\(34\) − 0.192340i − 0.0329860i
\(35\) − 14.4485i − 2.44224i
\(36\) 6.92092 + 1.38909i 1.15349 + 0.231515i
\(37\) 7.68267 − 4.43559i 1.26302 − 0.729206i 0.289364 − 0.957219i \(-0.406556\pi\)
0.973658 + 0.228013i \(0.0732228\pi\)
\(38\) − 1.98533i − 0.322064i
\(39\) 3.47971 + 7.69757i 0.557200 + 1.23260i
\(40\) 2.34687 0.371072
\(41\) − 5.95396i − 0.929852i −0.885350 − 0.464926i \(-0.846081\pi\)
0.885350 − 0.464926i \(-0.153919\pi\)
\(42\) −16.3042 − 1.62005i −2.51580 − 0.249979i
\(43\) − 1.07365i − 0.163730i −0.996643 − 0.0818649i \(-0.973912\pi\)
0.996643 − 0.0818649i \(-0.0260876\pi\)
\(44\) −2.70228 + 4.68048i −0.407384 + 0.705610i
\(45\) 7.17680 + 6.31585i 1.06985 + 0.941512i
\(46\) 15.8680i 2.33960i
\(47\) 2.29154 + 1.32302i 0.334256 + 0.192983i 0.657729 − 0.753255i \(-0.271517\pi\)
−0.323473 + 0.946237i \(0.604851\pi\)
\(48\) −0.542797 + 5.46273i −0.0783460 + 0.788478i
\(49\) 13.5570 1.93671
\(50\) 9.31465 + 5.37781i 1.31729 + 0.760538i
\(51\) −0.129711 − 0.0931191i −0.0181631 − 0.0130393i
\(52\) 9.93846 − 5.73797i 1.37822 − 0.795714i
\(53\) −0.328804 + 0.569506i −0.0451647 + 0.0782276i −0.887724 − 0.460376i \(-0.847715\pi\)
0.842559 + 0.538604i \(0.181048\pi\)
\(54\) 7.93174 − 7.39040i 1.07937 − 1.00571i
\(55\) −6.33893 + 3.65978i −0.854741 + 0.493485i
\(56\) 3.33906i 0.446201i
\(57\) −1.33888 − 0.961176i −0.177338 − 0.127311i
\(58\) 1.75068 − 1.01076i 0.229876 − 0.132719i
\(59\) 1.84976 + 1.06796i 0.240818 + 0.139036i 0.615552 − 0.788096i \(-0.288933\pi\)
−0.374735 + 0.927132i \(0.622266\pi\)
\(60\) 7.57399 − 10.5502i 0.977798 − 1.36203i
\(61\) −13.1373 − 7.58482i −1.68206 − 0.971137i −0.960290 − 0.279002i \(-0.909996\pi\)
−0.721768 − 0.692135i \(-0.756670\pi\)
\(62\) 9.84521 + 6.16557i 1.25034 + 0.783028i
\(63\) −8.98604 + 10.2110i −1.13213 + 1.28646i
\(64\) 10.5307 1.31633
\(65\) 15.5422 1.92778
\(66\) 3.41908 + 7.56343i 0.420859 + 0.930994i
\(67\) 12.6087 1.54039 0.770196 − 0.637807i \(-0.220158\pi\)
0.770196 + 0.637807i \(0.220158\pi\)
\(68\) −0.108459 + 0.187856i −0.0131526 + 0.0227809i
\(69\) 10.7011 + 7.68229i 1.28826 + 0.924839i
\(70\) −15.0725 + 26.1064i −1.80151 + 3.12031i
\(71\) −2.41191 − 1.39252i −0.286241 − 0.165261i 0.350004 − 0.936748i \(-0.386180\pi\)
−0.636245 + 0.771487i \(0.719513\pi\)
\(72\) −1.65856 − 1.45960i −0.195464 − 0.172015i
\(73\) −0.596742 − 0.344529i −0.0698433 − 0.0403241i 0.464672 − 0.885483i \(-0.346172\pi\)
−0.534515 + 0.845159i \(0.679506\pi\)
\(74\) −18.5086 −2.15159
\(75\) 8.13629 − 3.67804i 0.939497 − 0.424703i
\(76\) −1.11951 + 1.93905i −0.128417 + 0.222424i
\(77\) −5.20705 − 9.01887i −0.593398 − 1.02780i
\(78\) 1.74268 − 17.5384i 0.197320 − 1.98584i
\(79\) − 4.76336i − 0.535920i −0.963430 − 0.267960i \(-0.913651\pi\)
0.963430 − 0.267960i \(-0.0863495\pi\)
\(80\) 8.74696 + 5.05006i 0.977939 + 0.564614i
\(81\) −1.14390 − 8.92701i −0.127100 − 0.991890i
\(82\) −6.21111 + 10.7580i −0.685902 + 1.18802i
\(83\) −4.12108 − 7.13792i −0.452347 − 0.783488i 0.546184 − 0.837665i \(-0.316080\pi\)
−0.998531 + 0.0541770i \(0.982746\pi\)
\(84\) 15.0106 + 10.7761i 1.63779 + 1.17577i
\(85\) −0.254419 + 0.146889i −0.0275956 + 0.0159324i
\(86\) −1.12002 + 1.93993i −0.120775 + 0.209188i
\(87\) 0.165935 − 1.66998i 0.0177901 − 0.179040i
\(88\) 1.46493 − 0.845778i 0.156162 − 0.0901603i
\(89\) −12.6005 −1.33565 −0.667823 − 0.744320i \(-0.732774\pi\)
−0.667823 + 0.744320i \(0.732774\pi\)
\(90\) −6.37882 − 18.8986i −0.672387 − 1.99209i
\(91\) 22.1131i 2.31808i
\(92\) 8.94779 − 15.4980i 0.932872 − 1.61578i
\(93\) 8.92440 − 3.65446i 0.925417 − 0.378950i
\(94\) −2.76033 − 4.78103i −0.284706 − 0.493126i
\(95\) −2.62612 + 1.51619i −0.269434 + 0.155558i
\(96\) 8.16720 − 11.3766i 0.833562 − 1.16111i
\(97\) −2.92754 + 5.07064i −0.297246 + 0.514846i −0.975505 − 0.219978i \(-0.929402\pi\)
0.678259 + 0.734823i \(0.262735\pi\)
\(98\) −24.4955 − 14.1425i −2.47442 − 1.42861i
\(99\) 6.75596 + 1.35598i 0.678999 + 0.136281i
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.254.6 yes 60
3.2 odd 2 837.2.o.a.440.25 60
9.4 even 3 837.2.r.a.719.25 60
9.5 odd 6 279.2.r.a.68.6 yes 60
31.26 odd 6 279.2.r.a.119.6 yes 60
93.26 even 6 837.2.r.a.305.25 60
279.212 even 6 inner 279.2.o.a.212.6 ✓ 60
279.274 odd 6 837.2.o.a.584.25 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
279.2.o.a.212.6 ✓ 60 279.212 even 6 inner
279.2.o.a.254.6 yes 60 1.1 even 1 trivial
279.2.r.a.68.6 yes 60 9.5 odd 6
279.2.r.a.119.6 yes 60 31.26 odd 6
837.2.o.a.440.25 60 3.2 odd 2
837.2.o.a.584.25 60 279.274 odd 6
837.2.r.a.305.25 60 93.26 even 6
837.2.r.a.719.25 60 9.4 even 3