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Properties

Label 266.2.bd.a.143.11
Level $266$
Weight $2$
Character 266.143
Analytic conductor $2.124$
Analytic rank $0$
Dimension $84$
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] = [266,2,Mod(33,266)] 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("266.33"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(266, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([15, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 266 = 2 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 266.bd (of order \(18\), degree \(6\), 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: \(2.12402069377\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(14\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 143.11
Character \(\chi\) \(=\) 266.143
Dual form 266.2.bd.a.173.11

$q$-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+(0.642788 + 0.766044i) q^{2} +(0.146596 + 0.831390i) q^{3} +(-0.173648 + 0.984808i) q^{4} +(-3.54275 + 0.624683i) q^{5} +(-0.542651 + 0.646706i) q^{6} +(0.729308 + 2.54325i) q^{7} +(-0.866025 + 0.500000i) q^{8} +(2.14936 - 0.782303i) q^{9} +(-2.75577 - 2.31237i) q^{10} -4.02182 q^{11} -0.844215 q^{12} +(0.694072 + 0.582396i) q^{13} +(-1.47945 + 2.19345i) q^{14} +(-1.03871 - 2.85383i) q^{15} +(-0.939693 - 0.342020i) q^{16} +(-2.36688 + 6.50295i) q^{17} +(1.98086 + 1.14365i) q^{18} +(0.0299394 - 4.35880i) q^{19} -3.59741i q^{20} +(-2.00752 + 0.979170i) q^{21} +(-2.58518 - 3.08089i) q^{22} +(3.93875 + 3.30500i) q^{23} +(-0.542651 - 0.646706i) q^{24} +(7.46242 - 2.71610i) q^{25} +0.906047i q^{26} +(2.23181 + 3.86561i) q^{27} +(-2.63125 + 0.276598i) q^{28} +(8.04746 + 1.41899i) q^{29} +(1.51849 - 2.63011i) q^{30} +(-0.875738 - 1.51682i) q^{31} +(-0.342020 - 0.939693i) q^{32} +(-0.589584 - 3.34370i) q^{33} +(-6.50295 + 2.36688i) q^{34} +(-4.17248 - 8.55452i) q^{35} +(0.397186 + 2.25255i) q^{36} +(8.42745 - 4.86559i) q^{37} +(3.35828 - 2.77885i) q^{38} +(-0.382449 + 0.662422i) q^{39} +(2.75577 - 2.31237i) q^{40} +(1.99784 - 1.67639i) q^{41} +(-2.04049 - 0.908448i) q^{42} +(-0.318122 - 0.115787i) q^{43} +(0.698382 - 3.96072i) q^{44} +(-7.12596 + 4.11418i) q^{45} +5.14167i q^{46} +(3.78296 + 10.3936i) q^{47} +(0.146596 - 0.831390i) q^{48} +(-5.93622 + 3.70962i) q^{49} +(6.87740 + 3.97067i) q^{50} +(-5.75346 - 1.01449i) q^{51} +(-0.694072 + 0.582396i) q^{52} +(-9.74419 - 1.71816i) q^{53} +(-1.52665 + 4.19443i) q^{54} +(14.2483 - 2.51236i) q^{55} +(-1.90322 - 1.83786i) q^{56} +(3.62825 - 0.614093i) q^{57} +(4.08580 + 7.07682i) q^{58} +(-3.41660 - 1.24354i) q^{59} +(2.99085 - 0.527367i) q^{60} +(4.89181 - 5.82983i) q^{61} +(0.599040 - 1.64585i) q^{62} +(3.55713 + 4.89581i) q^{63} +(0.500000 - 0.866025i) q^{64} +(-2.82274 - 1.62971i) q^{65} +(2.18244 - 2.60094i) q^{66} +(-1.82656 + 2.17680i) q^{67} +(-5.99315 - 3.46015i) q^{68} +(-2.17034 + 3.75914i) q^{69} +(3.87112 - 8.69504i) q^{70} +(3.37033 - 9.25992i) q^{71} +(-1.47025 + 1.75217i) q^{72} +(3.09988 - 0.546593i) q^{73} +(9.14432 + 3.32826i) q^{74} +(3.35210 + 5.80601i) q^{75} +(4.28738 + 0.786382i) q^{76} +(-2.93314 - 10.2285i) q^{77} +(-0.753278 + 0.132823i) q^{78} +(-4.23997 + 11.6492i) q^{79} +(3.54275 + 0.624683i) q^{80} +(2.36987 - 1.98856i) q^{81} +(2.56838 + 0.452874i) q^{82} +(-5.35520 - 3.09182i) q^{83} +(-0.615693 - 2.14705i) q^{84} +(4.32299 - 24.5169i) q^{85} +(-0.115787 - 0.318122i) q^{86} +6.89860i q^{87} +(3.48300 - 2.01091i) q^{88} +(1.12651 - 6.38877i) q^{89} +(-7.73212 - 2.81426i) q^{90} +(-0.974985 + 2.18994i) q^{91} +(-3.93875 + 3.30500i) q^{92} +(1.13269 - 0.950440i) q^{93} +(-5.53032 + 9.57880i) q^{94} +(2.61680 + 15.4608i) q^{95} +(0.731112 - 0.422108i) q^{96} +(-1.38142 - 7.83443i) q^{97} +(-6.65746 - 2.16291i) q^{98} +(-8.64434 + 3.14628i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84 q - 12 q^{7} + 12 q^{11} - 12 q^{12} + 6 q^{13} + 6 q^{14} - 30 q^{15} - 54 q^{17} - 36 q^{18} - 42 q^{19} + 12 q^{21} - 18 q^{22} + 12 q^{23} + 36 q^{25} - 24 q^{27} + 12 q^{28} + 36 q^{34} + 6 q^{35}+ \cdots - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/266\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.642788 + 0.766044i 0.454519 + 0.541675i
\(3\) 0.146596 + 0.831390i 0.0846375 + 0.480003i 0.997434 + 0.0715889i \(0.0228070\pi\)
−0.912797 + 0.408414i \(0.866082\pi\)
\(4\) −0.173648 + 0.984808i −0.0868241 + 0.492404i
\(5\) −3.54275 + 0.624683i −1.58437 + 0.279367i −0.895345 − 0.445373i \(-0.853071\pi\)
−0.689023 + 0.724740i \(0.741960\pi\)
\(6\) −0.542651 + 0.646706i −0.221536 + 0.264017i
\(7\) 0.729308 + 2.54325i 0.275652 + 0.961257i
\(8\) −0.866025 + 0.500000i −0.306186 + 0.176777i
\(9\) 2.14936 − 0.782303i 0.716453 − 0.260768i
\(10\) −2.75577 − 2.31237i −0.871452 − 0.731235i
\(11\) −4.02182 −1.21262 −0.606312 − 0.795227i \(-0.707352\pi\)
−0.606312 + 0.795227i \(0.707352\pi\)
\(12\) −0.844215 −0.243704
\(13\) 0.694072 + 0.582396i 0.192501 + 0.161528i 0.733944 − 0.679210i \(-0.237677\pi\)
−0.541443 + 0.840737i \(0.682122\pi\)
\(14\) −1.47945 + 2.19345i −0.395400 + 0.586224i
\(15\) −1.03871 − 2.85383i −0.268194 − 0.736857i
\(16\) −0.939693 − 0.342020i −0.234923 − 0.0855050i
\(17\) −2.36688 + 6.50295i −0.574052 + 1.57720i 0.223988 + 0.974592i \(0.428092\pi\)
−0.798040 + 0.602604i \(0.794130\pi\)
\(18\) 1.98086 + 1.14365i 0.466893 + 0.269561i
\(19\) 0.0299394 − 4.35880i 0.00686857 − 0.999976i
\(20\) − 3.59741i − 0.804405i
\(21\) −2.00752 + 0.979170i −0.438076 + 0.213672i
\(22\) −2.58518 − 3.08089i −0.551161 − 0.656849i
\(23\) 3.93875 + 3.30500i 0.821286 + 0.689141i 0.953273 − 0.302111i \(-0.0976912\pi\)
−0.131987 + 0.991251i \(0.542136\pi\)
\(24\) −0.542651 − 0.646706i −0.110768 − 0.132008i
\(25\) 7.46242 − 2.71610i 1.49248 − 0.543220i
\(26\) 0.906047i 0.177690i
\(27\) 2.23181 + 3.86561i 0.429512 + 0.743937i
\(28\) −2.63125 + 0.276598i −0.497260 + 0.0522720i
\(29\) 8.04746 + 1.41899i 1.49438 + 0.263499i 0.860307 − 0.509777i \(-0.170272\pi\)
0.634070 + 0.773276i \(0.281383\pi\)
\(30\) 1.51849 − 2.63011i 0.277238 − 0.480190i
\(31\) −0.875738 − 1.51682i −0.157287 − 0.272429i 0.776602 − 0.629991i \(-0.216942\pi\)
−0.933890 + 0.357562i \(0.883608\pi\)
\(32\) −0.342020 − 0.939693i −0.0604612 − 0.166116i
\(33\) −0.589584 − 3.34370i −0.102633 − 0.582063i
\(34\) −6.50295 + 2.36688i −1.11525 + 0.405916i
\(35\) −4.17248 − 8.55452i −0.705278 − 1.44598i
\(36\) 0.397186 + 2.25255i 0.0661976 + 0.375425i
\(37\) 8.42745 − 4.86559i 1.38546 − 0.799899i 0.392665 − 0.919682i \(-0.371553\pi\)
0.992800 + 0.119783i \(0.0382199\pi\)
\(38\) 3.35828 − 2.77885i 0.544784 − 0.450788i
\(39\) −0.382449 + 0.662422i −0.0612409 + 0.106072i
\(40\) 2.75577 − 2.31237i 0.435726 − 0.365618i
\(41\) 1.99784 − 1.67639i 0.312011 − 0.261808i −0.473312 − 0.880895i \(-0.656942\pi\)
0.785323 + 0.619087i \(0.212497\pi\)
\(42\) −2.04049 − 0.908448i −0.314855 − 0.140177i
\(43\) −0.318122 − 0.115787i −0.0485131 − 0.0176573i 0.317650 − 0.948208i \(-0.397106\pi\)
−0.366163 + 0.930551i \(0.619329\pi\)
\(44\) 0.698382 − 3.96072i 0.105285 − 0.597101i
\(45\) −7.12596 + 4.11418i −1.06228 + 0.613305i
\(46\) 5.14167i 0.758098i
\(47\) 3.78296 + 10.3936i 0.551802 + 1.51606i 0.831248 + 0.555901i \(0.187627\pi\)
−0.279447 + 0.960161i \(0.590151\pi\)
\(48\) 0.146596 − 0.831390i 0.0211594 − 0.120001i
\(49\) −5.93622 + 3.70962i −0.848031 + 0.529946i
\(50\) 6.87740 + 3.97067i 0.972611 + 0.561537i
\(51\) −5.75346 − 1.01449i −0.805645 − 0.142057i
\(52\) −0.694072 + 0.582396i −0.0962505 + 0.0807638i
\(53\) −9.74419 − 1.71816i −1.33847 − 0.236008i −0.541841 − 0.840481i \(-0.682273\pi\)
−0.796626 + 0.604473i \(0.793384\pi\)
\(54\) −1.52665 + 4.19443i −0.207750 + 0.570790i
\(55\) 14.2483 − 2.51236i 1.92124 − 0.338767i
\(56\) −1.90322 − 1.83786i −0.254329 − 0.245595i
\(57\) 3.62825 − 0.614093i 0.480573 − 0.0813386i
\(58\) 4.08580 + 7.07682i 0.536492 + 0.929232i
\(59\) −3.41660 − 1.24354i −0.444804 − 0.161895i 0.109901 − 0.993943i \(-0.464947\pi\)
−0.554705 + 0.832047i \(0.687169\pi\)
\(60\) 2.99085 − 0.527367i 0.386117 − 0.0680828i
\(61\) 4.89181 − 5.82983i 0.626332 − 0.746433i −0.355813 − 0.934557i \(-0.615796\pi\)
0.982145 + 0.188124i \(0.0602406\pi\)
\(62\) 0.599040 − 1.64585i 0.0760782 − 0.209023i
\(63\) 3.55713 + 4.89581i 0.448157 + 0.616815i
\(64\) 0.500000 − 0.866025i 0.0625000 − 0.108253i
\(65\) −2.82274 − 1.62971i −0.350118 − 0.202141i
\(66\) 2.18244 − 2.60094i 0.268640 − 0.320153i
\(67\) −1.82656 + 2.17680i −0.223149 + 0.265939i −0.865990 − 0.500061i \(-0.833311\pi\)
0.642841 + 0.766000i \(0.277756\pi\)
\(68\) −5.99315 − 3.46015i −0.726776 − 0.419604i
\(69\) −2.17034 + 3.75914i −0.261278 + 0.452547i
\(70\) 3.87112 − 8.69504i 0.462687 − 1.03926i
\(71\) 3.37033 − 9.25992i 0.399985 − 1.09895i −0.562306 − 0.826929i \(-0.690086\pi\)
0.962291 − 0.272021i \(-0.0876920\pi\)
\(72\) −1.47025 + 1.75217i −0.173270 + 0.206496i
\(73\) 3.09988 − 0.546593i 0.362814 − 0.0639739i 0.0107303 − 0.999942i \(-0.496584\pi\)
0.352084 + 0.935969i \(0.385473\pi\)
\(74\) 9.14432 + 3.32826i 1.06301 + 0.386903i
\(75\) 3.35210 + 5.80601i 0.387067 + 0.670420i
\(76\) 4.28738 + 0.786382i 0.491796 + 0.0902042i
\(77\) −2.93314 − 10.2285i −0.334263 − 1.16564i
\(78\) −0.753278 + 0.132823i −0.0852920 + 0.0150393i
\(79\) −4.23997 + 11.6492i −0.477034 + 1.31064i 0.434965 + 0.900447i \(0.356761\pi\)
−0.911999 + 0.410193i \(0.865461\pi\)
\(80\) 3.54275 + 0.624683i 0.396092 + 0.0698417i
\(81\) 2.36987 − 1.98856i 0.263319 − 0.220951i
\(82\) 2.56838 + 0.452874i 0.283630 + 0.0500116i
\(83\) −5.35520 − 3.09182i −0.587809 − 0.339372i 0.176422 − 0.984315i \(-0.443548\pi\)
−0.764231 + 0.644943i \(0.776881\pi\)
\(84\) −0.615693 − 2.14705i −0.0671776 − 0.234262i
\(85\) 4.32299 − 24.5169i 0.468894 − 2.65923i
\(86\) −0.115787 − 0.318122i −0.0124856 − 0.0343039i
\(87\) 6.89860i 0.739607i
\(88\) 3.48300 − 2.01091i 0.371289 − 0.214364i
\(89\) 1.12651 − 6.38877i 0.119410 − 0.677208i −0.865062 − 0.501666i \(-0.832721\pi\)
0.984472 − 0.175543i \(-0.0561681\pi\)
\(90\) −7.73212 − 2.81426i −0.815037 − 0.296649i
\(91\) −0.974985 + 2.18994i −0.102206 + 0.229568i
\(92\) −3.93875 + 3.30500i −0.410643 + 0.344570i
\(93\) 1.13269 − 0.950440i 0.117455 − 0.0985561i
\(94\) −5.53032 + 9.57880i −0.570409 + 0.987977i
\(95\) 2.61680 + 15.4608i 0.268478 + 1.58625i
\(96\) 0.731112 − 0.422108i 0.0746188 − 0.0430812i
\(97\) −1.38142 − 7.83443i −0.140262 − 0.795465i −0.971050 − 0.238876i \(-0.923221\pi\)
0.830788 − 0.556589i \(-0.187890\pi\)
\(98\) −6.65746 − 2.16291i −0.672505 − 0.218487i
\(99\) −8.64434 + 3.14628i −0.868789 + 0.316213i
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 266.2.bd.a.143.11 yes 84
7.5 odd 6 266.2.y.a.257.11 yes 84
19.2 odd 18 266.2.y.a.59.11 ✓ 84
133.40 even 18 inner 266.2.bd.a.173.11 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
266.2.y.a.59.11 ✓ 84 19.2 odd 18
266.2.y.a.257.11 yes 84 7.5 odd 6
266.2.bd.a.143.11 yes 84 1.1 even 1 trivial
266.2.bd.a.173.11 yes 84 133.40 even 18 inner