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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [483,2,Mod(10,483)] 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("483.10"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(483, base_ring=CyclotomicField(66)) chi = DirichletCharacter(H, H._module([0, 11, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.bf (of order \(66\), degree \(20\), 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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(640\)
Relative dimension: \(32\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

Embedding invariants

Embedding label 10.5
Character \(\chi\) \(=\) 483.10
Dual form 483.2.bf.a.145.5

$q$-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+(-0.647644 + 1.87125i) q^{2} +(0.458227 + 0.888835i) q^{3} +(-1.51001 - 1.18748i) q^{4} +(-3.91898 + 0.374217i) q^{5} +(-1.96000 + 0.281805i) q^{6} +(-1.01941 - 2.44148i) q^{7} +(-0.131596 + 0.0845715i) q^{8} +(-0.580057 + 0.814576i) q^{9} +(1.83785 - 7.57573i) q^{10} +(0.645660 - 0.223465i) q^{11} +(0.363552 - 1.88629i) q^{12} +(-0.0976592 - 0.332597i) q^{13} +(5.22881 - 0.326361i) q^{14} +(-2.12840 - 3.31185i) q^{15} +(-0.978813 - 4.03472i) q^{16} +(5.85483 + 2.34392i) q^{17} +(-1.14860 - 1.61298i) q^{18} +(1.61603 - 0.646960i) q^{19} +(6.36207 + 4.08865i) q^{20} +(1.70295 - 2.02484i) q^{21} +1.35291i q^{22} +(-4.15203 - 2.40013i) q^{23} +(-0.135471 - 0.0782141i) q^{24} +(10.3087 - 1.98684i) q^{25} +(0.685619 + 0.0326600i) q^{26} +(-0.989821 - 0.142315i) q^{27} +(-1.35989 + 4.89719i) q^{28} +(-0.951297 - 6.61641i) q^{29} +(7.57573 - 1.83785i) q^{30} +(-4.91106 + 0.233942i) q^{31} +(7.87244 + 0.751727i) q^{32} +(0.494482 + 0.471488i) q^{33} +(-8.17789 + 9.43779i) q^{34} +(4.90869 + 9.18661i) q^{35} +(1.84319 - 0.541209i) q^{36} +(-8.88359 - 6.32597i) q^{37} +(0.164009 + 3.44298i) q^{38} +(0.250874 - 0.239208i) q^{39} +(0.484073 - 0.380679i) q^{40} +(-10.0291 - 4.58013i) q^{41} +(2.68606 + 4.49801i) q^{42} +(2.14542 - 3.33833i) q^{43} +(-1.24031 - 0.429277i) q^{44} +(1.96840 - 3.40937i) q^{45} +(7.18027 - 6.21504i) q^{46} +(-6.49244 + 3.74841i) q^{47} +(3.13769 - 2.71882i) q^{48} +(-4.92160 + 4.97773i) q^{49} +(-2.95851 + 20.5769i) q^{50} +(0.599478 + 6.27802i) q^{51} +(-0.247487 + 0.618193i) q^{52} +(0.0565734 + 0.0593325i) q^{53} +(0.907358 - 1.76003i) q^{54} +(-2.44670 + 1.11737i) q^{55} +(0.340629 + 0.235075i) q^{56} +(1.31555 + 1.13993i) q^{57} +(12.9970 + 2.50497i) q^{58} +(-7.52852 - 1.82640i) q^{59} +(-0.718872 + 7.52836i) q^{60} +(6.51667 + 3.35958i) q^{61} +(2.74285 - 9.34130i) q^{62} +(2.58008 + 0.585807i) q^{63} +(-3.05580 + 6.69128i) q^{64} +(0.507188 + 1.26689i) q^{65} +(-1.20252 + 0.619941i) q^{66} +(-0.257243 - 1.33470i) q^{67} +(-6.05747 - 10.4919i) q^{68} +(0.230747 - 4.79028i) q^{69} +(-20.3695 + 3.23571i) q^{70} +(0.530127 + 0.611800i) q^{71} +(0.00744315 - 0.156251i) q^{72} +(-1.37288 + 1.74576i) q^{73} +(17.5909 - 12.5264i) q^{74} +(6.48970 + 8.25232i) q^{75} +(-3.20847 - 0.942092i) q^{76} +(-1.20378 - 1.34856i) q^{77} +(0.285139 + 0.624368i) q^{78} +(4.52434 - 4.74500i) q^{79} +(5.34581 + 15.4457i) q^{80} +(-0.327068 - 0.945001i) q^{81} +(15.0658 - 15.8006i) q^{82} +(5.09021 + 11.1460i) q^{83} +(-4.97593 + 1.03530i) q^{84} +(-23.8221 - 6.99479i) q^{85} +(4.85737 + 6.17665i) q^{86} +(5.44499 - 3.87736i) q^{87} +(-0.0660673 + 0.0840115i) q^{88} +(0.122645 - 2.57464i) q^{89} +(5.10495 + 5.89142i) q^{90} +(-0.712472 + 0.577485i) q^{91} +(3.41950 + 8.55469i) q^{92} +(-2.45831 - 4.25792i) q^{93} +(-2.80941 - 14.5766i) q^{94} +(-6.09107 + 3.14016i) q^{95} +(2.93920 + 7.34177i) q^{96} +(-6.54959 + 14.3416i) q^{97} +(-6.12712 - 12.4333i) q^{98} +(-0.192490 + 0.655561i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 640 q - 4 q^{2} + 36 q^{4} + 24 q^{8} - 32 q^{9} + 4 q^{18} - 28 q^{23} + 56 q^{25} - 84 q^{26} - 176 q^{28} - 24 q^{29} + 12 q^{31} + 36 q^{32} - 76 q^{35} + 28 q^{36} + 44 q^{37} - 110 q^{42} - 88 q^{43}+ \cdots - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{3}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.647644 + 1.87125i −0.457954 + 1.32317i 0.445806 + 0.895130i \(0.352917\pi\)
−0.903760 + 0.428041i \(0.859204\pi\)
\(3\) 0.458227 + 0.888835i 0.264557 + 0.513169i
\(4\) −1.51001 − 1.18748i −0.755005 − 0.593742i
\(5\) −3.91898 + 0.374217i −1.75262 + 0.167355i −0.921407 − 0.388599i \(-0.872959\pi\)
−0.831213 + 0.555954i \(0.812353\pi\)
\(6\) −1.96000 + 0.281805i −0.800165 + 0.115046i
\(7\) −1.01941 − 2.44148i −0.385301 − 0.922791i
\(8\) −0.131596 + 0.0845715i −0.0465261 + 0.0299005i
\(9\) −0.580057 + 0.814576i −0.193352 + 0.271525i
\(10\) 1.83785 − 7.57573i 0.581180 − 2.39566i
\(11\) 0.645660 − 0.223465i 0.194674 − 0.0673772i −0.227991 − 0.973663i \(-0.573216\pi\)
0.422665 + 0.906286i \(0.361095\pi\)
\(12\) 0.363552 − 1.88629i 0.104948 − 0.544524i
\(13\) −0.0976592 − 0.332597i −0.0270858 − 0.0922457i 0.944841 − 0.327530i \(-0.106216\pi\)
−0.971927 + 0.235284i \(0.924398\pi\)
\(14\) 5.22881 − 0.326361i 1.39746 − 0.0872237i
\(15\) −2.12840 − 3.31185i −0.549550 − 0.855116i
\(16\) −0.978813 − 4.03472i −0.244703 − 1.00868i
\(17\) 5.85483 + 2.34392i 1.42000 + 0.568484i 0.949459 − 0.313892i \(-0.101633\pi\)
0.470545 + 0.882376i \(0.344057\pi\)
\(18\) −1.14860 − 1.61298i −0.270728 − 0.380184i
\(19\) 1.61603 − 0.646960i 0.370742 − 0.148423i −0.178807 − 0.983884i \(-0.557224\pi\)
0.549549 + 0.835461i \(0.314800\pi\)
\(20\) 6.36207 + 4.08865i 1.42260 + 0.914251i
\(21\) 1.70295 − 2.02484i 0.371614 − 0.441856i
\(22\) 1.35291i 0.288442i
\(23\) −4.15203 − 2.40013i −0.865759 − 0.500461i
\(24\) −0.135471 − 0.0782141i −0.0276529 − 0.0159654i
\(25\) 10.3087 − 1.98684i 2.06174 − 0.397368i
\(26\) 0.685619 + 0.0326600i 0.134461 + 0.00640516i
\(27\) −0.989821 − 0.142315i −0.190491 − 0.0273885i
\(28\) −1.35989 + 4.89719i −0.256996 + 0.925481i
\(29\) −0.951297 − 6.61641i −0.176651 − 1.22864i −0.864444 − 0.502730i \(-0.832329\pi\)
0.687792 − 0.725908i \(-0.258580\pi\)
\(30\) 7.57573 − 1.83785i 1.38313 − 0.335544i
\(31\) −4.91106 + 0.233942i −0.882052 + 0.0420173i −0.483719 − 0.875224i \(-0.660714\pi\)
−0.398333 + 0.917241i \(0.630411\pi\)
\(32\) 7.87244 + 0.751727i 1.39166 + 0.132888i
\(33\) 0.494482 + 0.471488i 0.0860783 + 0.0820755i
\(34\) −8.17789 + 9.43779i −1.40250 + 1.61857i
\(35\) 4.90869 + 9.18661i 0.829720 + 1.55282i
\(36\) 1.84319 − 0.541209i 0.307198 − 0.0902015i
\(37\) −8.88359 − 6.32597i −1.46045 − 1.03998i −0.987276 − 0.159016i \(-0.949168\pi\)
−0.473176 − 0.880968i \(-0.656893\pi\)
\(38\) 0.164009 + 3.44298i 0.0266058 + 0.558525i
\(39\) 0.250874 − 0.239208i 0.0401720 − 0.0383039i
\(40\) 0.484073 − 0.380679i 0.0765386 − 0.0601906i
\(41\) −10.0291 − 4.58013i −1.56628 − 0.715295i −0.571815 − 0.820382i \(-0.693761\pi\)
−0.994463 + 0.105087i \(0.966488\pi\)
\(42\) 2.68606 + 4.49801i 0.414469 + 0.694058i
\(43\) 2.14542 − 3.33833i 0.327173 − 0.509091i −0.638233 − 0.769843i \(-0.720334\pi\)
0.965406 + 0.260753i \(0.0839707\pi\)
\(44\) −1.24031 − 0.429277i −0.186984 − 0.0647159i
\(45\) 1.96840 − 3.40937i 0.293432 − 0.508239i
\(46\) 7.18027 − 6.21504i 1.05867 − 0.916358i
\(47\) −6.49244 + 3.74841i −0.947020 + 0.546762i −0.892154 − 0.451732i \(-0.850806\pi\)
−0.0548659 + 0.998494i \(0.517473\pi\)
\(48\) 3.13769 − 2.71882i 0.452886 − 0.392428i
\(49\) −4.92160 + 4.97773i −0.703086 + 0.711105i
\(50\) −2.95851 + 20.5769i −0.418396 + 2.91001i
\(51\) 0.599478 + 6.27802i 0.0839438 + 0.879099i
\(52\) −0.247487 + 0.618193i −0.0343203 + 0.0857280i
\(53\) 0.0565734 + 0.0593325i 0.00777096 + 0.00814994i 0.727608 − 0.685993i \(-0.240632\pi\)
−0.719837 + 0.694143i \(0.755784\pi\)
\(54\) 0.907358 − 1.76003i 0.123476 − 0.239510i
\(55\) −2.44670 + 1.11737i −0.329913 + 0.150666i
\(56\) 0.340629 + 0.235075i 0.0455185 + 0.0314132i
\(57\) 1.31555 + 1.13993i 0.174248 + 0.150987i
\(58\) 12.9970 + 2.50497i 1.70659 + 0.328919i
\(59\) −7.52852 − 1.82640i −0.980129 − 0.237777i −0.286477 − 0.958087i \(-0.592484\pi\)
−0.693652 + 0.720310i \(0.743999\pi\)
\(60\) −0.718872 + 7.52836i −0.0928059 + 0.971908i
\(61\) 6.51667 + 3.35958i 0.834374 + 0.430150i 0.821803 − 0.569771i \(-0.192968\pi\)
0.0125705 + 0.999921i \(0.495999\pi\)
\(62\) 2.74285 − 9.34130i 0.348343 − 1.18635i
\(63\) 2.58008 + 0.585807i 0.325060 + 0.0738047i
\(64\) −3.05580 + 6.69128i −0.381976 + 0.836410i
\(65\) 0.507188 + 1.26689i 0.0629089 + 0.157139i
\(66\) −1.20252 + 0.619941i −0.148020 + 0.0763095i
\(67\) −0.257243 − 1.33470i −0.0314272 − 0.163060i 0.962824 − 0.270128i \(-0.0870659\pi\)
−0.994252 + 0.107068i \(0.965854\pi\)
\(68\) −6.05747 − 10.4919i −0.734577 − 1.27232i
\(69\) 0.230747 − 4.79028i 0.0277787 − 0.576682i
\(70\) −20.3695 + 3.23571i −2.43462 + 0.386742i
\(71\) 0.530127 + 0.611800i 0.0629145 + 0.0726073i 0.786334 − 0.617801i \(-0.211976\pi\)
−0.723420 + 0.690408i \(0.757431\pi\)
\(72\) 0.00744315 − 0.156251i 0.000877184 − 0.0184144i
\(73\) −1.37288 + 1.74576i −0.160683 + 0.204325i −0.859727 − 0.510753i \(-0.829367\pi\)
0.699044 + 0.715079i \(0.253609\pi\)
\(74\) 17.5909 − 12.5264i 2.04489 − 1.45616i
\(75\) 6.48970 + 8.25232i 0.749366 + 0.952896i
\(76\) −3.20847 − 0.942092i −0.368037 − 0.108065i
\(77\) −1.20378 − 1.34856i −0.137183 − 0.153683i
\(78\) 0.285139 + 0.624368i 0.0322857 + 0.0706957i
\(79\) 4.52434 − 4.74500i 0.509029 − 0.533854i −0.418197 − 0.908356i \(-0.637338\pi\)
0.927226 + 0.374502i \(0.122186\pi\)
\(80\) 5.34581 + 15.4457i 0.597680 + 1.72688i
\(81\) −0.327068 − 0.945001i −0.0363409 − 0.105000i
\(82\) 15.0658 − 15.8006i 1.66374 − 1.74488i
\(83\) 5.09021 + 11.1460i 0.558723 + 1.22343i 0.952588 + 0.304265i \(0.0984107\pi\)
−0.393864 + 0.919169i \(0.628862\pi\)
\(84\) −4.97593 + 1.03530i −0.542919 + 0.112960i
\(85\) −23.8221 − 6.99479i −2.58387 − 0.758691i
\(86\) 4.85737 + 6.17665i 0.523784 + 0.666045i
\(87\) 5.44499 − 3.87736i 0.583765 − 0.415697i
\(88\) −0.0660673 + 0.0840115i −0.00704280 + 0.00895565i
\(89\) 0.122645 − 2.57464i 0.0130004 − 0.272912i −0.983003 − 0.183592i \(-0.941228\pi\)
0.996003 − 0.0893200i \(-0.0284694\pi\)
\(90\) 5.10495 + 5.89142i 0.538109 + 0.621011i
\(91\) −0.712472 + 0.577485i −0.0746873 + 0.0605369i
\(92\) 3.41950 + 8.55469i 0.356507 + 0.891888i
\(93\) −2.45831 − 4.25792i −0.254915 − 0.441526i
\(94\) −2.80941 − 14.5766i −0.289768 − 1.50346i
\(95\) −6.09107 + 3.14016i −0.624931 + 0.322174i
\(96\) 2.93920 + 7.34177i 0.299981 + 0.749316i
\(97\) −6.54959 + 14.3416i −0.665010 + 1.45617i 0.212768 + 0.977103i \(0.431752\pi\)
−0.877778 + 0.479067i \(0.840975\pi\)
\(98\) −6.12712 − 12.4333i −0.618932 − 1.25596i
\(99\) −0.192490 + 0.655561i −0.0193460 + 0.0658864i
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 483.2.bf.a.10.5 ✓ 640
7.5 odd 6 inner 483.2.bf.a.355.28 yes 640
23.7 odd 22 inner 483.2.bf.a.283.28 yes 640
161.145 even 66 inner 483.2.bf.a.145.5 yes 640
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.bf.a.10.5 ✓ 640 1.1 even 1 trivial
483.2.bf.a.145.5 yes 640 161.145 even 66 inner
483.2.bf.a.283.28 yes 640 23.7 odd 22 inner
483.2.bf.a.355.28 yes 640 7.5 odd 6 inner