↑

Properties

Label 279.2.o.a.212.6
Level $279$
Weight $2$
Character 279.212
Analytic conductor $2.228$
Analytic rank $0$
Dimension $60$
Inner twists $2$

Related objects

Downloads

Learn more

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 212.6
Character \(\chi\) \(=\) 279.212
Dual form 279.2.o.a.254.6

$q$-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 \(\left(\mathbb{Z}/279\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.80686 + 1.04319i −1.27764 + 0.737647i −0.976414 − 0.215906i \(-0.930730\pi\)
−0.301227 + 0.953552i \(0.597396\pi\)
\(3\) −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 \(-0.747530\pi\)
0.701599 − 0.712572i \(-0.252470\pi\)
\(6\) 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 −0.639313 − 0.768946i \(-0.720781\pi\)
0.985584 + 0.169188i \(0.0541146\pi\)
\(12\) −3.31069 + 2.37674i −0.955714 + 0.686105i
\(13\) 4.87719i 1.35269i 0.736585 + 0.676345i \(0.236437\pi\)
−0.736585 + 0.676345i \(0.763563\pi\)
\(14\) 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 −0.860382 − 0.509650i \(-0.829775\pi\)
0.871561 + 0.490287i \(0.163108\pi\)
\(18\) −5.93044 − 2.00169i −1.39782 − 0.471803i
\(19\) 0.475784 − 0.824082i 0.109152 − 0.189057i −0.806275 − 0.591541i \(-0.798520\pi\)
0.915427 + 0.402484i \(0.131853\pi\)
\(20\) −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 0.131219 + 0.991353i \(0.458111\pi\)
−0.924147 + 0.382038i \(0.875222\pi\)
\(24\) 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 0.817533 − 0.575881i \(-0.195341\pi\)
−0.907494 + 0.420064i \(0.862008\pi\)
\(30\) −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 0.973658 − 0.228013i \(-0.0732228\pi\)
0.289364 + 0.957219i \(0.406556\pi\)
\(38\) 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 \(0.153919\pi\)
−0.885350 + 0.464926i \(0.846081\pi\)
\(42\) −16.3042 + 1.62005i −2.51580 + 0.249979i
\(43\) 1.07365i 0.163730i 0.996643 + 0.0818649i \(0.0260876\pi\)
−0.996643 + 0.0818649i \(0.973912\pi\)
\(44\) −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 −0.323473 − 0.946237i \(-0.604851\pi\)
0.657729 + 0.753255i \(0.271517\pi\)
\(48\) −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 0.842559 − 0.538604i \(-0.181048\pi\)
−0.887724 + 0.460376i \(0.847715\pi\)
\(54\) 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 −0.374735 − 0.927132i \(-0.622266\pi\)
0.615552 + 0.788096i \(0.288933\pi\)
\(60\) 7.57399 + 10.5502i 0.977798 + 1.36203i
\(61\) −13.1373 + 7.58482i −1.68206 + 0.971137i −0.721768 + 0.692135i \(0.756670\pi\)
−0.960290 + 0.279002i \(0.909996\pi\)
\(62\) 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 −0.636245 − 0.771487i \(-0.719513\pi\)
0.350004 + 0.936748i \(0.386180\pi\)
\(72\) −1.65856 + 1.45960i −0.195464 + 0.172015i
\(73\) −0.596742 + 0.344529i −0.0698433 + 0.0403241i −0.534515 − 0.845159i \(-0.679506\pi\)
0.464672 + 0.885483i \(0.346172\pi\)
\(74\) −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 \(0.0863495\pi\)
−0.963430 + 0.267960i \(0.913651\pi\)
\(80\) 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 −0.998531 − 0.0541770i \(-0.982746\pi\)
0.546184 + 0.837665i \(0.316080\pi\)
\(84\) 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 0.678259 − 0.734823i \(-0.262735\pi\)
−0.975505 + 0.219978i \(0.929402\pi\)
\(98\) −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.212.6 ✓ 60
3.2 odd 2 837.2.o.a.584.25 60
9.2 odd 6 279.2.r.a.119.6 yes 60
9.7 even 3 837.2.r.a.305.25 60
31.6 odd 6 279.2.r.a.68.6 yes 60
93.68 even 6 837.2.r.a.719.25 60
279.223 odd 6 837.2.o.a.440.25 60
279.254 even 6 inner 279.2.o.a.254.6 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
279.2.o.a.212.6 ✓ 60 1.1 even 1 trivial
279.2.o.a.254.6 yes 60 279.254 even 6 inner
279.2.r.a.68.6 yes 60 31.6 odd 6
279.2.r.a.119.6 yes 60 9.2 odd 6
837.2.o.a.440.25 60 279.223 odd 6
837.2.o.a.584.25 60 3.2 odd 2
837.2.r.a.305.25 60 9.7 even 3
837.2.r.a.719.25 60 93.68 even 6