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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(143,567)] 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("567.143"); 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([7, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.ba (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 341.11
Character \(\chi\) \(=\) 567.341
Dual form 567.2.ba.a.143.11

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.00959490 - 0.0114348i) q^{2} +(0.347258 - 1.96940i) q^{4} +(-1.26073 - 1.05788i) q^{5} +(1.81613 - 1.92397i) q^{7} +(-0.0517058 + 0.0298524i) q^{8} +0.0245663i q^{10} +(-2.95219 - 3.51829i) q^{11} +(1.52751 + 4.19679i) q^{13} +(-0.0394257 - 0.00230675i) q^{14} +(-3.75751 - 1.36762i) q^{16} -3.77410 q^{17} -1.46832i q^{19} +(-2.52118 + 2.11552i) q^{20} +(-0.0119048 + 0.0675152i) q^{22} +(2.54129 + 6.98214i) q^{23} +(-0.397908 - 2.25665i) q^{25} +(0.0333330 - 0.0577345i) q^{26} +(-3.15839 - 4.24480i) q^{28} +(2.83246 - 7.78213i) q^{29} +(-8.27481 - 1.45907i) q^{31} +(0.0612550 + 0.168297i) q^{32} +(0.0362121 + 0.0431559i) q^{34} +(-4.32497 + 0.504358i) q^{35} +(0.397320 + 0.688178i) q^{37} +(-0.0167899 + 0.0140884i) q^{38} +(0.0967671 + 0.0170627i) q^{40} +(3.97658 - 1.44736i) q^{41} +(-0.303215 - 1.71962i) q^{43} +(-7.95407 + 4.59229i) q^{44} +(0.0554557 - 0.0960520i) q^{46} +(0.286275 + 1.62355i) q^{47} +(-0.403318 - 6.98837i) q^{49} +(-0.0219863 + 0.0262023i) q^{50} +(8.79559 - 1.55090i) q^{52} +(3.66888 - 2.11823i) q^{53} +7.55866i q^{55} +(-0.0364696 + 0.153696i) q^{56} +(-0.116164 + 0.0422802i) q^{58} +(10.8663 - 3.95501i) q^{59} +(2.96287 - 0.522435i) q^{61} +(0.0627118 + 0.108620i) q^{62} +(-3.99733 + 6.92357i) q^{64} +(2.51392 - 6.90693i) q^{65} +(-8.30643 - 6.96992i) q^{67} +(-1.31059 + 7.43270i) q^{68} +(0.0472649 + 0.0446158i) q^{70} +(-6.66808 - 3.84982i) q^{71} +(13.7076 + 7.91406i) q^{73} +(0.00405690 - 0.0111462i) q^{74} +(-2.89171 - 0.509886i) q^{76} +(-12.1307 - 0.709749i) q^{77} +(1.22109 - 1.02462i) q^{79} +(3.29043 + 5.69919i) q^{80} +(-0.0547050 - 0.0315840i) q^{82} +(4.61667 + 1.68033i) q^{83} +(4.75812 + 3.99254i) q^{85} +(-0.0167541 + 0.0199668i) q^{86} +(0.257675 + 0.0937859i) q^{88} +7.18505 q^{89} +(10.8487 + 4.68306i) q^{91} +(14.6331 - 2.58021i) q^{92} +(0.0158181 - 0.0188513i) q^{94} +(-1.55330 + 1.85115i) q^{95} +(1.56544 - 0.276029i) q^{97} +(-0.0760405 + 0.0716645i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} + 9 q^{11} - 3 q^{14} + 3 q^{16} + 18 q^{17} - 18 q^{20} - 12 q^{22} + 6 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31} - 3 q^{32} - 18 q^{34}+ \cdots - 27 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{11}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.00959490 − 0.0114348i −0.00678462 − 0.00808559i 0.762641 − 0.646821i \(-0.223902\pi\)
−0.769426 + 0.638736i \(0.779458\pi\)
\(3\) 0 0
\(4\) 0.347258 − 1.96940i 0.173629 − 0.984698i
\(5\) −1.26073 − 1.05788i −0.563815 − 0.473097i 0.315772 − 0.948835i \(-0.397737\pi\)
−0.879587 + 0.475738i \(0.842181\pi\)
\(6\) 0 0
\(7\) 1.81613 − 1.92397i 0.686434 − 0.727192i
\(8\) −0.0517058 + 0.0298524i −0.0182808 + 0.0105544i
\(9\) 0 0
\(10\) 0.0245663i 0.00776856i
\(11\) −2.95219 − 3.51829i −0.890120 − 1.06080i −0.997779 − 0.0666155i \(-0.978780\pi\)
0.107659 − 0.994188i \(-0.465665\pi\)
\(12\) 0 0
\(13\) 1.52751 + 4.19679i 0.423654 + 1.16398i 0.949600 + 0.313463i \(0.101489\pi\)
−0.525946 + 0.850518i \(0.676289\pi\)
\(14\) −0.0394257 − 0.00230675i −0.0105370 − 0.000616504i
\(15\) 0 0
\(16\) −3.75751 − 1.36762i −0.939379 − 0.341906i
\(17\) −3.77410 −0.915355 −0.457677 − 0.889118i \(-0.651318\pi\)
−0.457677 + 0.889118i \(0.651318\pi\)
\(18\) 0 0
\(19\) − 1.46832i − 0.336856i −0.985714 − 0.168428i \(-0.946131\pi\)
0.985714 − 0.168428i \(-0.0538690\pi\)
\(20\) −2.52118 + 2.11552i −0.563752 + 0.473044i
\(21\) 0 0
\(22\) −0.0119048 + 0.0675152i −0.00253810 + 0.0143943i
\(23\) 2.54129 + 6.98214i 0.529896 + 1.45588i 0.859193 + 0.511651i \(0.170966\pi\)
−0.329297 + 0.944226i \(0.606812\pi\)
\(24\) 0 0
\(25\) −0.397908 − 2.25665i −0.0795815 − 0.451329i
\(26\) 0.0333330 − 0.0577345i 0.00653714 − 0.0113227i
\(27\) 0 0
\(28\) −3.15839 − 4.24480i −0.596880 − 0.802192i
\(29\) 2.83246 − 7.78213i 0.525975 − 1.44511i −0.337795 − 0.941220i \(-0.609681\pi\)
0.863771 − 0.503885i \(-0.168097\pi\)
\(30\) 0 0
\(31\) −8.27481 − 1.45907i −1.48620 − 0.262057i −0.629147 − 0.777286i \(-0.716596\pi\)
−0.857052 + 0.515229i \(0.827707\pi\)
\(32\) 0.0612550 + 0.168297i 0.0108284 + 0.0297509i
\(33\) 0 0
\(34\) 0.0362121 + 0.0431559i 0.00621033 + 0.00740118i
\(35\) −4.32497 + 0.504358i −0.731054 + 0.0852520i
\(36\) 0 0
\(37\) 0.397320 + 0.688178i 0.0653190 + 0.113136i 0.896835 − 0.442364i \(-0.145860\pi\)
−0.831516 + 0.555500i \(0.812527\pi\)
\(38\) −0.0167899 + 0.0140884i −0.00272368 + 0.00228544i
\(39\) 0 0
\(40\) 0.0967671 + 0.0170627i 0.0153002 + 0.00269784i
\(41\) 3.97658 − 1.44736i 0.621037 − 0.226039i −0.0122884 − 0.999924i \(-0.503912\pi\)
0.633326 + 0.773885i \(0.281689\pi\)
\(42\) 0 0
\(43\) −0.303215 − 1.71962i −0.0462399 − 0.262239i 0.952920 − 0.303222i \(-0.0980623\pi\)
−0.999160 + 0.0409825i \(0.986951\pi\)
\(44\) −7.95407 + 4.59229i −1.19912 + 0.692313i
\(45\) 0 0
\(46\) 0.0554557 − 0.0960520i 0.00817649 − 0.0141621i
\(47\) 0.286275 + 1.62355i 0.0417575 + 0.236819i 0.998542 − 0.0539790i \(-0.0171904\pi\)
−0.956785 + 0.290798i \(0.906079\pi\)
\(48\) 0 0
\(49\) −0.403318 − 6.98837i −0.0576169 − 0.998339i
\(50\) −0.0219863 + 0.0262023i −0.00310933 + 0.00370556i
\(51\) 0 0
\(52\) 8.79559 − 1.55090i 1.21973 − 0.215071i
\(53\) 3.66888 − 2.11823i 0.503959 − 0.290961i −0.226388 − 0.974037i \(-0.572692\pi\)
0.730347 + 0.683076i \(0.239358\pi\)
\(54\) 0 0
\(55\) 7.55866i 1.01921i
\(56\) −0.0364696 + 0.153696i −0.00487346 + 0.0205385i
\(57\) 0 0
\(58\) −0.116164 + 0.0422802i −0.0152531 + 0.00555166i
\(59\) 10.8663 − 3.95501i 1.41467 − 0.514898i 0.482173 − 0.876076i \(-0.339848\pi\)
0.932497 + 0.361178i \(0.117625\pi\)
\(60\) 0 0
\(61\) 2.96287 − 0.522435i 0.379357 − 0.0668909i 0.0192817 − 0.999814i \(-0.493862\pi\)
0.360075 + 0.932923i \(0.382751\pi\)
\(62\) 0.0627118 + 0.108620i 0.00796441 + 0.0137948i
\(63\) 0 0
\(64\) −3.99733 + 6.92357i −0.499666 + 0.865447i
\(65\) 2.51392 − 6.90693i 0.311813 − 0.856700i
\(66\) 0 0
\(67\) −8.30643 − 6.96992i −1.01479 − 0.851512i −0.0258279 − 0.999666i \(-0.508222\pi\)
−0.988964 + 0.148155i \(0.952667\pi\)
\(68\) −1.31059 + 7.43270i −0.158932 + 0.901348i
\(69\) 0 0
\(70\) 0.0472649 + 0.0446158i 0.00564924 + 0.00533260i
\(71\) −6.66808 − 3.84982i −0.791355 − 0.456889i 0.0490844 − 0.998795i \(-0.484370\pi\)
−0.840439 + 0.541906i \(0.817703\pi\)
\(72\) 0 0
\(73\) 13.7076 + 7.91406i 1.60435 + 0.926271i 0.990603 + 0.136767i \(0.0436712\pi\)
0.613745 + 0.789504i \(0.289662\pi\)
\(74\) 0.00405690 − 0.0111462i 0.000471605 − 0.00129573i
\(75\) 0 0
\(76\) −2.89171 − 0.509886i −0.331701 − 0.0584879i
\(77\) −12.1307 − 0.709749i −1.38242 − 0.0808833i
\(78\) 0 0
\(79\) 1.22109 − 1.02462i 0.137384 − 0.115279i −0.571506 − 0.820598i \(-0.693641\pi\)
0.708890 + 0.705319i \(0.249196\pi\)
\(80\) 3.29043 + 5.69919i 0.367881 + 0.637189i
\(81\) 0 0
\(82\) −0.0547050 − 0.0315840i −0.00604116 − 0.00348787i
\(83\) 4.61667 + 1.68033i 0.506745 + 0.184440i 0.582725 − 0.812669i \(-0.301986\pi\)
−0.0759804 + 0.997109i \(0.524209\pi\)
\(84\) 0 0
\(85\) 4.75812 + 3.99254i 0.516091 + 0.433051i
\(86\) −0.0167541 + 0.0199668i −0.00180664 + 0.00215307i
\(87\) 0 0
\(88\) 0.257675 + 0.0937859i 0.0274682 + 0.00999761i
\(89\) 7.18505 0.761614 0.380807 − 0.924655i \(-0.375646\pi\)
0.380807 + 0.924655i \(0.375646\pi\)
\(90\) 0 0
\(91\) 10.8487 + 4.68306i 1.13725 + 0.490918i
\(92\) 14.6331 − 2.58021i 1.52561 − 0.269005i
\(93\) 0 0
\(94\) 0.0158181 − 0.0188513i 0.00163151 − 0.00194436i
\(95\) −1.55330 + 1.85115i −0.159365 + 0.189924i
\(96\) 0 0
\(97\) 1.56544 − 0.276029i 0.158946 − 0.0280265i −0.0936086 − 0.995609i \(-0.529840\pi\)
0.252555 + 0.967583i \(0.418729\pi\)
\(98\) −0.0760405 + 0.0716645i −0.00768125 + 0.00723921i
\(99\) 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.ba.a.341.11 132
3.2 odd 2 189.2.ba.a.131.12 yes 132
7.3 odd 6 567.2.bd.a.17.12 132
21.17 even 6 189.2.bd.a.185.11 yes 132
27.7 even 9 189.2.bd.a.47.11 yes 132
27.20 odd 18 567.2.bd.a.467.12 132
189.101 even 18 inner 567.2.ba.a.143.11 132
189.115 odd 18 189.2.ba.a.101.12 ✓ 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.12 ✓ 132 189.115 odd 18
189.2.ba.a.131.12 yes 132 3.2 odd 2
189.2.bd.a.47.11 yes 132 27.7 even 9
189.2.bd.a.185.11 yes 132 21.17 even 6
567.2.ba.a.143.11 132 189.101 even 18 inner
567.2.ba.a.341.11 132 1.1 even 1 trivial
567.2.bd.a.17.12 132 7.3 odd 6
567.2.bd.a.467.12 132 27.20 odd 18