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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 17.20
Character \(\chi\) \(=\) 567.17
Dual form 567.2.bd.a.467.20

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.26715 - 0.399760i) q^{2} +(3.10078 - 1.12859i) q^{4} +(1.73217 - 0.630457i) q^{5} +(0.678971 + 2.55715i) q^{7} +(2.59137 - 1.49613i) q^{8} +(3.67505 - 2.12179i) q^{10} +(0.0420027 - 0.115401i) q^{11} +(-0.0159063 - 0.0437022i) q^{13} +(2.56157 + 5.52601i) q^{14} +(0.221371 - 0.185752i) q^{16} +(-0.0786230 - 0.136179i) q^{17} +(-6.57855 - 3.79813i) q^{19} +(4.65954 - 3.90982i) q^{20} +(0.0490936 - 0.278423i) q^{22} +(0.957765 + 0.168880i) q^{23} +(-1.22730 + 1.02982i) q^{25} +(-0.0535324 - 0.0927208i) q^{26} +(4.99131 + 7.16286i) q^{28} +(3.33678 - 9.16772i) q^{29} +(-1.56144 - 4.29003i) q^{31} +(-3.41914 + 4.07477i) q^{32} +(-0.232689 - 0.277308i) q^{34} +(2.78826 + 4.00134i) q^{35} +9.37436 q^{37} +(-16.4329 - 5.98109i) q^{38} +(3.54543 - 4.22528i) q^{40} +(-5.76925 + 2.09983i) q^{41} +(1.67890 + 9.52153i) q^{43} -0.405238i q^{44} +2.23891 q^{46} +(-3.91757 - 1.42588i) q^{47} +(-6.07800 + 3.47246i) q^{49} +(-2.37078 + 2.82539i) q^{50} +(-0.0986438 - 0.117559i) q^{52} +(0.141686 + 0.0818025i) q^{53} -0.226376i q^{55} +(5.58527 + 5.61067i) q^{56} +(3.90009 - 22.1185i) q^{58} +(-9.06906 - 7.60984i) q^{59} +(1.03823 - 2.85251i) q^{61} +(-5.25501 - 9.10194i) q^{62} +(-6.41175 + 11.1055i) q^{64} +(-0.0551048 - 0.0656713i) q^{65} +(-1.70755 + 9.68399i) q^{67} +(-0.397483 - 0.333527i) q^{68} +(7.92099 + 7.95701i) q^{70} +(0.863276 + 0.498413i) q^{71} +9.18511i q^{73} +(21.2531 - 3.74749i) q^{74} +(-24.6852 - 4.35266i) q^{76} +(0.323617 + 0.0290528i) q^{77} +(0.451399 + 2.56001i) q^{79} +(0.266342 - 0.461318i) q^{80} +(-12.2403 + 7.06695i) q^{82} +(6.02764 + 2.19388i) q^{83} +(-0.222043 - 0.186316i) q^{85} +(7.61265 + 20.9156i) q^{86} +(-0.0638107 - 0.361889i) q^{88} +(5.30203 - 9.18338i) q^{89} +(0.100953 - 0.0703473i) q^{91} +(3.16041 - 0.557266i) q^{92} +(-9.45173 - 1.66659i) q^{94} +(-13.7897 - 2.43150i) q^{95} +(13.2580 - 2.33775i) q^{97} +(-12.3916 + 10.3023i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} - 9 q^{10} - 9 q^{11} + 42 q^{14} - 15 q^{16} + 9 q^{17} - 9 q^{19} + 18 q^{20} - 12 q^{22} - 30 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31}+ \cdots + 180 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.26715 − 0.399760i 1.60312 − 0.282673i 0.700674 − 0.713482i \(-0.252883\pi\)
0.902443 + 0.430809i \(0.141772\pi\)
\(3\) 0 0
\(4\) 3.10078 − 1.12859i 1.55039 − 0.564295i
\(5\) 1.73217 − 0.630457i 0.774649 − 0.281949i 0.0757093 − 0.997130i \(-0.475878\pi\)
0.698939 + 0.715181i \(0.253656\pi\)
\(6\) 0 0
\(7\) 0.678971 + 2.55715i 0.256627 + 0.966511i
\(8\) 2.59137 − 1.49613i 0.916186 − 0.528960i
\(9\) 0 0
\(10\) 3.67505 − 2.12179i 1.16215 − 0.670970i
\(11\) 0.0420027 − 0.115401i 0.0126643 − 0.0347949i −0.933199 − 0.359359i \(-0.882995\pi\)
0.945864 + 0.324564i \(0.105218\pi\)
\(12\) 0 0
\(13\) −0.0159063 − 0.0437022i −0.00441162 − 0.0121208i 0.937467 − 0.348073i \(-0.113164\pi\)
−0.941879 + 0.335953i \(0.890942\pi\)
\(14\) 2.56157 + 5.52601i 0.684609 + 1.47689i
\(15\) 0 0
\(16\) 0.221371 − 0.185752i 0.0553427 − 0.0464380i
\(17\) −0.0786230 − 0.136179i −0.0190689 − 0.0330283i 0.856334 − 0.516423i \(-0.172737\pi\)
−0.875402 + 0.483395i \(0.839404\pi\)
\(18\) 0 0
\(19\) −6.57855 − 3.79813i −1.50922 − 0.871350i −0.999942 − 0.0107502i \(-0.996578\pi\)
−0.509281 − 0.860600i \(-0.670089\pi\)
\(20\) 4.65954 − 3.90982i 1.04190 − 0.874261i
\(21\) 0 0
\(22\) 0.0490936 − 0.278423i 0.0104668 − 0.0593601i
\(23\) 0.957765 + 0.168880i 0.199708 + 0.0352139i 0.272607 − 0.962125i \(-0.412114\pi\)
−0.0728993 + 0.997339i \(0.523225\pi\)
\(24\) 0 0
\(25\) −1.22730 + 1.02982i −0.245459 + 0.205965i
\(26\) −0.0535324 − 0.0927208i −0.0104986 − 0.0181840i
\(27\) 0 0
\(28\) 4.99131 + 7.16286i 0.943269 + 1.35365i
\(29\) 3.33678 − 9.16772i 0.619624 − 1.70240i −0.0882816 − 0.996096i \(-0.528138\pi\)
0.707906 − 0.706307i \(-0.249640\pi\)
\(30\) 0 0
\(31\) −1.56144 − 4.29003i −0.280444 − 0.770512i −0.997310 − 0.0733016i \(-0.976646\pi\)
0.716866 − 0.697211i \(-0.245576\pi\)
\(32\) −3.41914 + 4.07477i −0.604424 + 0.720325i
\(33\) 0 0
\(34\) −0.232689 − 0.277308i −0.0399058 − 0.0475579i
\(35\) 2.78826 + 4.00134i 0.471303 + 0.676350i
\(36\) 0 0
\(37\) 9.37436 1.54113 0.770567 − 0.637358i \(-0.219973\pi\)
0.770567 + 0.637358i \(0.219973\pi\)
\(38\) −16.4329 − 5.98109i −2.66577 − 0.970261i
\(39\) 0 0
\(40\) 3.54543 − 4.22528i 0.560582 − 0.668076i
\(41\) −5.76925 + 2.09983i −0.901005 + 0.327939i −0.750656 − 0.660694i \(-0.770262\pi\)
−0.150350 + 0.988633i \(0.548040\pi\)
\(42\) 0 0
\(43\) 1.67890 + 9.52153i 0.256030 + 1.45202i 0.793414 + 0.608682i \(0.208301\pi\)
−0.537384 + 0.843338i \(0.680587\pi\)
\(44\) − 0.405238i − 0.0610919i
\(45\) 0 0
\(46\) 2.23891 0.330109
\(47\) −3.91757 − 1.42588i −0.571436 − 0.207986i 0.0401090 − 0.999195i \(-0.487229\pi\)
−0.611545 + 0.791210i \(0.709452\pi\)
\(48\) 0 0
\(49\) −6.07800 + 3.47246i −0.868285 + 0.496065i
\(50\) −2.37078 + 2.82539i −0.335279 + 0.399570i
\(51\) 0 0
\(52\) −0.0986438 − 0.117559i −0.0136794 − 0.0163025i
\(53\) 0.141686 + 0.0818025i 0.0194621 + 0.0112364i 0.509699 − 0.860353i \(-0.329757\pi\)
−0.490237 + 0.871589i \(0.663090\pi\)
\(54\) 0 0
\(55\) − 0.226376i − 0.0305245i
\(56\) 5.58527 + 5.61067i 0.746364 + 0.749758i
\(57\) 0 0
\(58\) 3.90009 − 22.1185i 0.512107 − 2.90430i
\(59\) −9.06906 − 7.60984i −1.18069 − 0.990717i −0.999974 − 0.00718212i \(-0.997714\pi\)
−0.180717 − 0.983535i \(-0.557842\pi\)
\(60\) 0 0
\(61\) 1.03823 − 2.85251i 0.132932 − 0.365227i −0.855312 − 0.518113i \(-0.826635\pi\)
0.988244 + 0.152886i \(0.0488568\pi\)
\(62\) −5.25501 − 9.10194i −0.667387 − 1.15595i
\(63\) 0 0
\(64\) −6.41175 + 11.1055i −0.801469 + 1.38818i
\(65\) −0.0551048 − 0.0656713i −0.00683491 − 0.00814552i
\(66\) 0 0
\(67\) −1.70755 + 9.68399i −0.208610 + 1.18309i 0.683046 + 0.730375i \(0.260655\pi\)
−0.891656 + 0.452713i \(0.850456\pi\)
\(68\) −0.397483 − 0.333527i −0.0482018 − 0.0404461i
\(69\) 0 0
\(70\) 7.92099 + 7.95701i 0.946739 + 0.951045i
\(71\) 0.863276 + 0.498413i 0.102452 + 0.0591507i 0.550351 − 0.834934i \(-0.314494\pi\)
−0.447899 + 0.894084i \(0.647827\pi\)
\(72\) 0 0
\(73\) 9.18511i 1.07504i 0.843252 + 0.537518i \(0.180638\pi\)
−0.843252 + 0.537518i \(0.819362\pi\)
\(74\) 21.2531 − 3.74749i 2.47062 − 0.435637i
\(75\) 0 0
\(76\) −24.6852 − 4.35266i −2.83158 − 0.499284i
\(77\) 0.323617 + 0.0290528i 0.0368796 + 0.00331087i
\(78\) 0 0
\(79\) 0.451399 + 2.56001i 0.0507864 + 0.288024i 0.999614 − 0.0277696i \(-0.00884048\pi\)
−0.948828 + 0.315793i \(0.897729\pi\)
\(80\) 0.266342 − 0.461318i 0.0297780 − 0.0515769i
\(81\) 0 0
\(82\) −12.2403 + 7.06695i −1.35172 + 0.780414i
\(83\) 6.02764 + 2.19388i 0.661619 + 0.240810i 0.650935 − 0.759133i \(-0.274377\pi\)
0.0106840 + 0.999943i \(0.496599\pi\)
\(84\) 0 0
\(85\) −0.222043 − 0.186316i −0.0240840 − 0.0202088i
\(86\) 7.61265 + 20.9156i 0.820893 + 2.25539i
\(87\) 0 0
\(88\) −0.0638107 − 0.361889i −0.00680225 − 0.0385775i
\(89\) 5.30203 − 9.18338i 0.562014 − 0.973436i −0.435307 − 0.900282i \(-0.643360\pi\)
0.997321 − 0.0731541i \(-0.0233065\pi\)
\(90\) 0 0
\(91\) 0.100953 − 0.0703473i 0.0105828 − 0.00737440i
\(92\) 3.16041 − 0.557266i 0.329496 − 0.0580990i
\(93\) 0 0
\(94\) −9.45173 − 1.66659i −0.974871 − 0.171896i
\(95\) −13.7897 − 2.43150i −1.41479 − 0.249466i
\(96\) 0 0
\(97\) 13.2580 − 2.33775i 1.34615 − 0.237363i 0.546313 − 0.837581i \(-0.316031\pi\)
0.799836 + 0.600218i \(0.204920\pi\)
\(98\) −12.3916 + 10.3023i −1.25174 + 1.04069i
\(99\) 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.17.20 132
3.2 odd 2 189.2.bd.a.185.3 yes 132
7.5 odd 6 567.2.ba.a.341.3 132
21.5 even 6 189.2.ba.a.131.20 yes 132
27.7 even 9 189.2.ba.a.101.20 ✓ 132
27.20 odd 18 567.2.ba.a.143.3 132
189.47 even 18 inner 567.2.bd.a.467.20 132
189.61 odd 18 189.2.bd.a.47.3 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.20 ✓ 132 27.7 even 9
189.2.ba.a.131.20 yes 132 21.5 even 6
189.2.bd.a.47.3 yes 132 189.61 odd 18
189.2.bd.a.185.3 yes 132 3.2 odd 2
567.2.ba.a.143.3 132 27.20 odd 18
567.2.ba.a.341.3 132 7.5 odd 6
567.2.bd.a.17.20 132 1.1 even 1 trivial
567.2.bd.a.467.20 132 189.47 even 18 inner