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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(17,567)] 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("567.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([11, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.bd (of order \(18\), degree \(6\), not 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: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(22\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 189)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 467.17
Character \(\chi\) \(=\) 567.467
Dual form 567.2.bd.a.17.17

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.57155 + 0.277106i) q^{2} +(0.513586 + 0.186930i) q^{4} +(-1.78318 - 0.649025i) q^{5} +(-1.28547 - 2.31248i) q^{7} +(-2.00866 - 1.15970i) q^{8} +(-2.62250 - 1.51410i) q^{10} +(-0.432011 - 1.18694i) q^{11} +(0.326917 - 0.898197i) q^{13} +(-1.37938 - 3.99038i) q^{14} +(-3.67271 - 3.08177i) q^{16} +(2.00687 - 3.47600i) q^{17} +(6.64640 - 3.83730i) q^{19} +(-0.794495 - 0.666660i) q^{20} +(-0.350017 - 1.98504i) q^{22} +(-5.63807 + 0.994145i) q^{23} +(-1.07172 - 0.899282i) q^{25} +(0.762661 - 1.32097i) q^{26} +(-0.227929 - 1.42795i) q^{28} +(1.22870 + 3.37583i) q^{29} +(-3.04428 + 8.36408i) q^{31} +(-1.93608 - 2.30734i) q^{32} +(4.11711 - 4.90658i) q^{34} +(0.791374 + 4.95787i) q^{35} -7.99870 q^{37} +(11.5085 - 4.18874i) q^{38} +(2.82914 + 3.37163i) q^{40} +(3.01647 + 1.09790i) q^{41} +(0.111496 - 0.632326i) q^{43} -0.690352i q^{44} -9.13598 q^{46} +(8.03521 - 2.92458i) q^{47} +(-3.69512 + 5.94526i) q^{49} +(-1.43506 - 1.71024i) q^{50} +(0.335800 - 0.400191i) q^{52} +(11.3329 - 6.54305i) q^{53} +2.39691i q^{55} +(-0.0997058 + 6.13576i) q^{56} +(0.995499 + 5.64576i) q^{58} +(1.65418 - 1.38802i) q^{59} +(-1.05430 - 2.89668i) q^{61} +(-7.10196 + 12.3010i) q^{62} +(2.39111 + 4.14152i) q^{64} +(-1.16590 + 1.38947i) q^{65} +(0.371427 + 2.10647i) q^{67} +(1.68047 - 1.41008i) q^{68} +(-0.130175 + 8.01082i) q^{70} +(0.154696 - 0.0893139i) q^{71} -11.5040i q^{73} +(-12.5703 - 2.21649i) q^{74} +(4.13081 - 0.728373i) q^{76} +(-2.18944 + 2.52480i) q^{77} +(-0.211994 + 1.20228i) q^{79} +(4.54896 + 7.87903i) q^{80} +(4.43628 + 2.56129i) q^{82} +(5.11532 - 1.86183i) q^{83} +(-5.83462 + 4.89582i) q^{85} +(0.350443 - 0.962834i) q^{86} +(-0.508733 + 2.88517i) q^{88} +(-2.81958 - 4.88366i) q^{89} +(-2.49730 + 0.398619i) q^{91} +(-3.08147 - 0.543347i) q^{92} +(13.4381 - 2.36950i) q^{94} +(-14.3422 + 2.52892i) q^{95} +(5.97585 + 1.05370i) q^{97} +(-7.45452 + 8.31931i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} - 9 q^{10} - 9 q^{11} + 42 q^{14} - 15 q^{16} + 9 q^{17} - 9 q^{19} + 18 q^{20} - 12 q^{22} - 30 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31}+ \cdots + 180 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.57155 + 0.277106i 1.11125 + 0.195944i 0.698997 − 0.715125i \(-0.253630\pi\)
0.412255 + 0.911069i \(0.364741\pi\)
\(3\) 0 0
\(4\) 0.513586 + 0.186930i 0.256793 + 0.0934650i
\(5\) −1.78318 − 0.649025i −0.797463 − 0.290253i −0.0890280 − 0.996029i \(-0.528376\pi\)
−0.708435 + 0.705776i \(0.750598\pi\)
\(6\) 0 0
\(7\) −1.28547 − 2.31248i −0.485863 − 0.874035i
\(8\) −2.00866 − 1.15970i −0.710170 − 0.410017i
\(9\) 0 0
\(10\) −2.62250 − 1.51410i −0.829308 − 0.478801i
\(11\) −0.432011 − 1.18694i −0.130256 − 0.357876i 0.857370 − 0.514700i \(-0.172097\pi\)
−0.987627 + 0.156824i \(0.949874\pi\)
\(12\) 0 0
\(13\) 0.326917 − 0.898197i 0.0906704 − 0.249115i −0.886066 − 0.463560i \(-0.846572\pi\)
0.976736 + 0.214445i \(0.0687942\pi\)
\(14\) −1.37938 − 3.99038i −0.368654 − 1.06647i
\(15\) 0 0
\(16\) −3.67271 − 3.08177i −0.918177 − 0.770442i
\(17\) 2.00687 − 3.47600i 0.486737 − 0.843053i −0.513147 − 0.858301i \(-0.671520\pi\)
0.999884 + 0.0152477i \(0.00485368\pi\)
\(18\) 0 0
\(19\) 6.64640 − 3.83730i 1.52479 − 0.880338i 0.525221 − 0.850966i \(-0.323983\pi\)
0.999569 − 0.0293716i \(-0.00935062\pi\)
\(20\) −0.794495 − 0.666660i −0.177654 − 0.149070i
\(21\) 0 0
\(22\) −0.350017 − 1.98504i −0.0746239 − 0.423213i
\(23\) −5.63807 + 0.994145i −1.17562 + 0.207293i −0.727133 − 0.686496i \(-0.759148\pi\)
−0.448486 + 0.893790i \(0.648037\pi\)
\(24\) 0 0
\(25\) −1.07172 − 0.899282i −0.214344 − 0.179856i
\(26\) 0.762661 − 1.32097i 0.149570 − 0.259063i
\(27\) 0 0
\(28\) −0.227929 − 1.42795i −0.0430745 − 0.269857i
\(29\) 1.22870 + 3.37583i 0.228164 + 0.626876i 0.999960 − 0.00899503i \(-0.00286324\pi\)
−0.771795 + 0.635871i \(0.780641\pi\)
\(30\) 0 0
\(31\) −3.04428 + 8.36408i −0.546768 + 1.50223i 0.291280 + 0.956638i \(0.405919\pi\)
−0.838048 + 0.545596i \(0.816303\pi\)
\(32\) −1.93608 − 2.30734i −0.342255 − 0.407883i
\(33\) 0 0
\(34\) 4.11711 − 4.90658i 0.706078 − 0.841471i
\(35\) 0.791374 + 4.95787i 0.133767 + 0.838033i
\(36\) 0 0
\(37\) −7.99870 −1.31498 −0.657489 − 0.753464i \(-0.728381\pi\)
−0.657489 + 0.753464i \(0.728381\pi\)
\(38\) 11.5085 − 4.18874i 1.86692 − 0.679504i
\(39\) 0 0
\(40\) 2.82914 + 3.37163i 0.447326 + 0.533102i
\(41\) 3.01647 + 1.09790i 0.471093 + 0.171464i 0.566647 − 0.823960i \(-0.308240\pi\)
−0.0955544 + 0.995424i \(0.530462\pi\)
\(42\) 0 0
\(43\) 0.111496 − 0.632326i 0.0170030 − 0.0964289i −0.975125 − 0.221654i \(-0.928854\pi\)
0.992128 + 0.125226i \(0.0399654\pi\)
\(44\) − 0.690352i − 0.104074i
\(45\) 0 0
\(46\) −9.13598 −1.34703
\(47\) 8.03521 − 2.92458i 1.17206 − 0.426593i 0.318666 − 0.947867i \(-0.396765\pi\)
0.853389 + 0.521274i \(0.174543\pi\)
\(48\) 0 0
\(49\) −3.69512 + 5.94526i −0.527874 + 0.849323i
\(50\) −1.43506 − 1.71024i −0.202949 − 0.241865i
\(51\) 0 0
\(52\) 0.335800 − 0.400191i 0.0465671 − 0.0554965i
\(53\) 11.3329 − 6.54305i 1.55669 − 0.898757i 0.559123 − 0.829085i \(-0.311138\pi\)
0.997570 − 0.0696726i \(-0.0221955\pi\)
\(54\) 0 0
\(55\) 2.39691i 0.323200i
\(56\) −0.0997058 + 6.13576i −0.0133237 + 0.819926i
\(57\) 0 0
\(58\) 0.995499 + 5.64576i 0.130715 + 0.741324i
\(59\) 1.65418 − 1.38802i 0.215356 − 0.180705i −0.528728 − 0.848791i \(-0.677331\pi\)
0.744084 + 0.668086i \(0.232886\pi\)
\(60\) 0 0
\(61\) −1.05430 − 2.89668i −0.134990 − 0.370882i 0.853718 − 0.520735i \(-0.174342\pi\)
−0.988708 + 0.149853i \(0.952120\pi\)
\(62\) −7.10196 + 12.3010i −0.901950 + 1.56222i
\(63\) 0 0
\(64\) 2.39111 + 4.14152i 0.298889 + 0.517690i
\(65\) −1.16590 + 1.38947i −0.144613 + 0.172343i
\(66\) 0 0
\(67\) 0.371427 + 2.10647i 0.0453770 + 0.257346i 0.999054 − 0.0434866i \(-0.0138466\pi\)
−0.953677 + 0.300833i \(0.902735\pi\)
\(68\) 1.68047 − 1.41008i 0.203787 − 0.170997i
\(69\) 0 0
\(70\) −0.130175 + 8.01082i −0.0155589 + 0.957476i
\(71\) 0.154696 − 0.0893139i 0.0183591 − 0.0105996i −0.490792 − 0.871277i \(-0.663293\pi\)
0.509151 + 0.860677i \(0.329959\pi\)
\(72\) 0 0
\(73\) − 11.5040i − 1.34644i −0.739443 − 0.673220i \(-0.764911\pi\)
0.739443 − 0.673220i \(-0.235089\pi\)
\(74\) −12.5703 − 2.21649i −1.46127 − 0.257662i
\(75\) 0 0
\(76\) 4.13081 − 0.728373i 0.473836 − 0.0835501i
\(77\) −2.18944 + 2.52480i −0.249509 + 0.287727i
\(78\) 0 0
\(79\) −0.211994 + 1.20228i −0.0238512 + 0.135267i −0.994408 − 0.105605i \(-0.966322\pi\)
0.970557 + 0.240872i \(0.0774333\pi\)
\(80\) 4.54896 + 7.87903i 0.508589 + 0.880902i
\(81\) 0 0
\(82\) 4.43628 + 2.56129i 0.489906 + 0.282847i
\(83\) 5.11532 − 1.86183i 0.561480 − 0.204362i −0.0456599 − 0.998957i \(-0.514539\pi\)
0.607140 + 0.794595i \(0.292317\pi\)
\(84\) 0 0
\(85\) −5.83462 + 4.89582i −0.632853 + 0.531027i
\(86\) 0.350443 − 0.962834i 0.0377892 − 0.103825i
\(87\) 0 0
\(88\) −0.508733 + 2.88517i −0.0542311 + 0.307560i
\(89\) −2.81958 − 4.88366i −0.298875 − 0.517667i 0.677004 − 0.735980i \(-0.263278\pi\)
−0.975879 + 0.218313i \(0.929945\pi\)
\(90\) 0 0
\(91\) −2.49730 + 0.398619i −0.261789 + 0.0417866i
\(92\) −3.08147 − 0.543347i −0.321266 − 0.0566478i
\(93\) 0 0
\(94\) 13.4381 − 2.36950i 1.38604 − 0.244396i
\(95\) −14.3422 + 2.52892i −1.47148 + 0.259462i
\(96\) 0 0
\(97\) 5.97585 + 1.05370i 0.606755 + 0.106987i 0.468581 − 0.883421i \(-0.344765\pi\)
0.138174 + 0.990408i \(0.455877\pi\)
\(98\) −7.45452 + 8.31931i −0.753020 + 0.840377i
\(99\) 0 0
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 567.2.bd.a.467.17 132
3.2 odd 2 189.2.bd.a.47.6 yes 132
7.3 odd 6 567.2.ba.a.143.6 132
21.17 even 6 189.2.ba.a.101.17 ✓ 132
27.4 even 9 189.2.ba.a.131.17 yes 132
27.23 odd 18 567.2.ba.a.341.6 132
189.31 odd 18 189.2.bd.a.185.6 yes 132
189.185 even 18 inner 567.2.bd.a.17.17 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.17 ✓ 132 21.17 even 6
189.2.ba.a.131.17 yes 132 27.4 even 9
189.2.bd.a.47.6 yes 132 3.2 odd 2
189.2.bd.a.185.6 yes 132 189.31 odd 18
567.2.ba.a.143.6 132 7.3 odd 6
567.2.ba.a.341.6 132 27.23 odd 18
567.2.bd.a.17.17 132 189.185 even 18 inner
567.2.bd.a.467.17 132 1.1 even 1 trivial