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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 283.28
Character \(\chi\) \(=\) 483.283
Dual form 483.2.bf.a.355.28

$q$-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+(1.94437 + 0.374746i) q^{2} +(0.998867 - 0.0475819i) q^{3} +(1.78340 + 0.713964i) q^{4} +(-2.28357 + 3.20683i) q^{5} +(1.96000 + 0.281805i) q^{6} +(2.17754 + 1.50276i) q^{7} +(-0.131596 - 0.0845715i) q^{8} +(0.995472 - 0.0950560i) q^{9} +(-5.64185 + 5.37949i) q^{10} +(-0.129304 - 0.670890i) q^{11} +(1.81535 + 0.628298i) q^{12} +(0.0976592 - 0.332597i) q^{13} +(3.67079 + 3.73795i) q^{14} +(-2.12840 + 3.31185i) q^{15} +(-3.00477 - 2.86504i) q^{16} +(4.95731 + 3.89847i) q^{17} +(1.97119 + 0.188225i) q^{18} +(1.36830 - 1.07604i) q^{19} +(-6.36207 + 4.08865i) q^{20} +(2.24658 + 1.39745i) q^{21} -1.35291i q^{22} +(-0.00255514 - 4.79583i) q^{23} +(-0.135471 - 0.0782141i) q^{24} +(-3.43370 - 9.92102i) q^{25} +(0.314525 - 0.610093i) q^{26} +(0.989821 - 0.142315i) q^{27} +(2.81051 + 4.23471i) q^{28} +(-0.951297 + 6.61641i) q^{29} +(-5.37949 + 5.64185i) q^{30} +(-2.25293 - 4.37007i) q^{31} +(-4.58724 - 6.44187i) q^{32} +(-0.161079 - 0.663978i) q^{33} +(8.17789 + 9.43779i) q^{34} +(-9.79168 + 3.55134i) q^{35} +(1.84319 + 0.541209i) q^{36} +(-1.03666 - 10.8564i) q^{37} +(3.06371 - 1.57945i) q^{38} +(0.0817230 - 0.336867i) q^{39} +(0.571714 - 0.228880i) q^{40} +(10.0291 - 4.58013i) q^{41} +(3.84449 + 3.55906i) q^{42} +(2.14542 + 3.33833i) q^{43} +(0.248392 - 1.28878i) q^{44} +(-1.96840 + 3.40937i) q^{45} +(1.79225 - 9.32582i) q^{46} +(-6.49244 + 3.74841i) q^{47} +(-3.13769 - 2.71882i) q^{48} +(2.48340 + 6.54467i) q^{49} +(-2.95851 - 20.5769i) q^{50} +(5.13719 + 3.65817i) q^{51} +(0.411627 - 0.523427i) q^{52} +(-0.0796701 - 0.0193278i) q^{53} +(1.97791 + 0.0942194i) q^{54} +(2.44670 + 1.11737i) q^{55} +(-0.159465 - 0.381916i) q^{56} +(1.31555 - 1.13993i) q^{57} +(-4.32915 + 12.5082i) q^{58} +(-5.34597 - 5.60669i) q^{59} +(-6.16032 + 4.38674i) q^{60} +(0.348856 - 7.32339i) q^{61} +(-2.74285 - 9.34130i) q^{62} +(2.31053 + 1.28897i) q^{63} +(-3.05580 - 6.69128i) q^{64} +(0.843568 + 1.07268i) q^{65} +(-0.0643742 - 1.35138i) q^{66} +(1.28451 - 0.444573i) 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} +(-0.139039 - 0.0716795i) q^{72} +(0.825431 - 2.06183i) q^{73} +(2.05274 - 21.4973i) q^{74} +(-3.90187 - 9.74640i) q^{75} +(3.20847 - 0.942092i) q^{76} +(0.726626 - 1.65521i) q^{77} +(0.285139 - 0.624368i) q^{78} +(-6.37146 + 1.54570i) q^{79} +(16.0493 - 3.09324i) q^{80} +(0.981929 - 0.189251i) q^{81} +(21.2166 - 5.14709i) q^{82} +(-5.09021 + 11.1460i) q^{83} +(3.00882 + 4.09619i) q^{84} +(-23.8221 + 6.99479i) q^{85} +(2.92045 + 7.29493i) q^{86} +(-0.635398 + 6.65418i) q^{87} +(-0.0397224 + 0.0992217i) q^{88} +(2.29103 + 1.18111i) 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} +(-14.0284 + 4.85528i) q^{94} +(0.326072 + 6.84510i) q^{95} +(-4.88856 - 6.21630i) q^{96} +(6.54959 + 14.3416i) q^{97} +(2.37605 + 13.6559i) 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 \(\left(\mathbb{Z}/483\mathbb{Z}\right)^\times\).

\(n\) \(323\) \(346\) \(442\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{19}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.94437 + 0.374746i 1.37488 + 0.264986i 0.822574 − 0.568658i \(-0.192537\pi\)
0.552302 + 0.833644i \(0.313749\pi\)
\(3\) 0.998867 − 0.0475819i 0.576696 − 0.0274714i
\(4\) 1.78340 + 0.713964i 0.891698 + 0.356982i
\(5\) −2.28357 + 3.20683i −1.02124 + 1.43414i −0.124167 + 0.992261i \(0.539626\pi\)
−0.897077 + 0.441875i \(0.854314\pi\)
\(6\) 1.96000 + 0.281805i 0.800165 + 0.115046i
\(7\) 2.17754 + 1.50276i 0.823034 + 0.567991i
\(8\) −0.131596 − 0.0845715i −0.0465261 − 0.0299005i
\(9\) 0.995472 − 0.0950560i 0.331824 − 0.0316853i
\(10\) −5.64185 + 5.37949i −1.78411 + 1.70114i
\(11\) −0.129304 − 0.670890i −0.0389865 − 0.202281i 0.957213 − 0.289385i \(-0.0934508\pi\)
−0.996199 + 0.0871041i \(0.972239\pi\)
\(12\) 1.81535 + 0.628298i 0.524046 + 0.181374i
\(13\) 0.0976592 − 0.332597i 0.0270858 − 0.0922457i −0.944841 − 0.327530i \(-0.893784\pi\)
0.971927 + 0.235284i \(0.0756020\pi\)
\(14\) 3.67079 + 3.73795i 0.981061 + 0.999010i
\(15\) −2.12840 + 3.31185i −0.549550 + 0.855116i
\(16\) −3.00477 − 2.86504i −0.751192 − 0.716260i
\(17\) 4.95731 + 3.89847i 1.20232 + 0.945517i 0.999433 − 0.0336840i \(-0.0107240\pi\)
0.202891 + 0.979201i \(0.434966\pi\)
\(18\) 1.97119 + 0.188225i 0.464613 + 0.0443652i
\(19\) 1.36830 − 1.07604i 0.313909 − 0.246861i −0.448756 − 0.893654i \(-0.648133\pi\)
0.762665 + 0.646794i \(0.223890\pi\)
\(20\) −6.36207 + 4.08865i −1.42260 + 0.914251i
\(21\) 2.24658 + 1.39745i 0.490244 + 0.304949i
\(22\) − 1.35291i − 0.288442i
\(23\) −0.00255514 − 4.79583i −0.000532784 − 1.00000i
\(24\) −0.135471 − 0.0782141i −0.0276529 − 0.0159654i
\(25\) −3.43370 − 9.92102i −0.686740 − 1.98420i
\(26\) 0.314525 − 0.610093i 0.0616834 − 0.119649i
\(27\) 0.989821 − 0.142315i 0.190491 − 0.0273885i
\(28\) 2.81051 + 4.23471i 0.531136 + 0.800286i
\(29\) −0.951297 + 6.61641i −0.176651 + 1.22864i 0.687792 + 0.725908i \(0.258580\pi\)
−0.864444 + 0.502730i \(0.832329\pi\)
\(30\) −5.37949 + 5.64185i −0.982156 + 1.03006i
\(31\) −2.25293 − 4.37007i −0.404638 − 0.784888i 0.595187 − 0.803587i \(-0.297078\pi\)
−0.999825 + 0.0186990i \(0.994048\pi\)
\(32\) −4.58724 − 6.44187i −0.810916 − 1.13877i
\(33\) −0.161079 − 0.663978i −0.0280403 − 0.115584i
\(34\) 8.17789 + 9.43779i 1.40250 + 1.61857i
\(35\) −9.79168 + 3.55134i −1.65510 + 0.600286i
\(36\) 1.84319 + 0.541209i 0.307198 + 0.0902015i
\(37\) −1.03666 − 10.8564i −0.170426 − 1.78478i −0.526352 − 0.850267i \(-0.676441\pi\)
0.355926 − 0.934514i \(-0.384165\pi\)
\(38\) 3.06371 − 1.57945i 0.497000 − 0.256221i
\(39\) 0.0817230 − 0.336867i 0.0130862 − 0.0539419i
\(40\) 0.571714 − 0.228880i 0.0903959 − 0.0361891i
\(41\) 10.0291 − 4.58013i 1.56628 − 0.715295i 0.571815 − 0.820382i \(-0.306239\pi\)
0.994463 + 0.105087i \(0.0335121\pi\)
\(42\) 3.84449 + 3.55906i 0.593218 + 0.549174i
\(43\) 2.14542 + 3.33833i 0.327173 + 0.509091i 0.965406 − 0.260753i \(-0.0839707\pi\)
−0.638233 + 0.769843i \(0.720334\pi\)
\(44\) 0.248392 − 1.28878i 0.0374465 − 0.194291i
\(45\) −1.96840 + 3.40937i −0.293432 + 0.508239i
\(46\) 1.79225 − 9.32582i 0.264253 − 1.37502i
\(47\) −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
\(49\) 2.48340 + 6.54467i 0.354771 + 0.934953i
\(50\) −2.95851 − 20.5769i −0.418396 − 2.91001i
\(51\) 5.13719 + 3.65817i 0.719350 + 0.512247i
\(52\) 0.411627 − 0.523427i 0.0570824 − 0.0725862i
\(53\) −0.0796701 − 0.0193278i −0.0109435 − 0.00265487i 0.230283 − 0.973124i \(-0.426035\pi\)
−0.241227 + 0.970469i \(0.577550\pi\)
\(54\) 1.97791 + 0.0942194i 0.269159 + 0.0128216i
\(55\) 2.44670 + 1.11737i 0.329913 + 0.150666i
\(56\) −0.159465 − 0.381916i −0.0213094 − 0.0510356i
\(57\) 1.31555 − 1.13993i 0.174248 − 0.150987i
\(58\) −4.32915 + 12.5082i −0.568445 + 1.64241i
\(59\) −5.34597 − 5.60669i −0.695985 − 0.729929i 0.276981 − 0.960875i \(-0.410666\pi\)
−0.972966 + 0.230947i \(0.925818\pi\)
\(60\) −6.16032 + 4.38674i −0.795294 + 0.566326i
\(61\) 0.348856 − 7.32339i 0.0446664 − 0.937664i −0.859672 − 0.510847i \(-0.829332\pi\)
0.904338 − 0.426817i \(-0.140365\pi\)
\(62\) −2.74285 − 9.34130i −0.348343 − 1.18635i
\(63\) 2.31053 + 1.28897i 0.291100 + 0.162395i
\(64\) −3.05580 − 6.69128i −0.381976 − 0.836410i
\(65\) 0.843568 + 1.07268i 0.104632 + 0.133050i
\(66\) −0.0643742 − 1.35138i −0.00792392 − 0.166344i
\(67\) 1.28451 − 0.444573i 0.156928 − 0.0543132i −0.247475 − 0.968894i \(-0.579601\pi\)
0.404402 + 0.914581i \(0.367480\pi\)
\(68\) 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 −0.723420 − 0.690408i \(-0.757431\pi\)
0.786334 + 0.617801i \(0.211976\pi\)
\(72\) −0.139039 − 0.0716795i −0.0163859 − 0.00844752i
\(73\) 0.825431 − 2.06183i 0.0966094 − 0.241319i −0.872189 − 0.489169i \(-0.837300\pi\)
0.968798 + 0.247851i \(0.0797243\pi\)
\(74\) 2.05274 − 21.4973i 0.238627 − 2.49901i
\(75\) −3.90187 − 9.74640i −0.450549 − 1.12542i
\(76\) 3.20847 − 0.942092i 0.368037 − 0.108065i
\(77\) 0.726626 − 1.65521i 0.0828067 − 0.188628i
\(78\) 0.285139 − 0.624368i 0.0322857 − 0.0706957i
\(79\) −6.37146 + 1.54570i −0.716845 + 0.173905i −0.577561 − 0.816348i \(-0.695995\pi\)
−0.139284 + 0.990252i \(0.544480\pi\)
\(80\) 16.0493 − 3.09324i 1.79436 − 0.345835i
\(81\) 0.981929 − 0.189251i 0.109103 − 0.0210279i
\(82\) 21.2166 − 5.14709i 2.34298 − 0.568401i
\(83\) −5.09021 + 11.1460i −0.558723 + 1.22343i 0.393864 + 0.919169i \(0.371138\pi\)
−0.952588 + 0.304265i \(0.901589\pi\)
\(84\) 3.00882 + 4.09619i 0.328289 + 0.446931i
\(85\) −23.8221 + 6.99479i −2.58387 + 0.758691i
\(86\) 2.92045 + 7.29493i 0.314920 + 0.786633i
\(87\) −0.635398 + 6.65418i −0.0681218 + 0.713403i
\(88\) −0.0397224 + 0.0992217i −0.00423442 + 0.0105771i
\(89\) 2.29103 + 1.18111i 0.242849 + 0.125197i 0.575355 − 0.817904i \(-0.304864\pi\)
−0.332506 + 0.943101i \(0.607894\pi\)
\(90\) −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
\(94\) −14.0284 + 4.85528i −1.44692 + 0.500783i
\(95\) 0.326072 + 6.84510i 0.0334543 + 0.702293i
\(96\) −4.88856 − 6.21630i −0.498936 − 0.634449i
\(97\) 6.54959 + 14.3416i 0.665010 + 1.45617i 0.877778 + 0.479067i \(0.159025\pi\)
−0.212768 + 0.977103i \(0.568248\pi\)
\(98\) 2.37605 + 13.6559i 0.240017 + 1.37945i
\(99\) −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.283.28 yes 640
7.5 odd 6 inner 483.2.bf.a.145.5 yes 640
23.10 odd 22 inner 483.2.bf.a.10.5 ✓ 640
161.33 even 66 inner 483.2.bf.a.355.28 yes 640
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.bf.a.10.5 ✓ 640 23.10 odd 22 inner
483.2.bf.a.145.5 yes 640 7.5 odd 6 inner
483.2.bf.a.283.28 yes 640 1.1 even 1 trivial
483.2.bf.a.355.28 yes 640 161.33 even 66 inner