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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 17.14
Character \(\chi\) \(=\) 567.17
Dual form 567.2.bd.a.467.14

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.877614 - 0.154747i) q^{2} +(-1.13313 + 0.412424i) q^{4} +(1.30288 - 0.474210i) q^{5} +(1.81190 + 1.92796i) q^{7} +(-2.47415 + 1.42845i) q^{8} +(1.07004 - 0.617790i) q^{10} +(0.197926 - 0.543796i) q^{11} +(1.85992 + 5.11010i) q^{13} +(1.88849 + 1.41162i) q^{14} +(-0.102835 + 0.0862885i) q^{16} +(-1.15616 - 2.00253i) q^{17} +(2.96522 + 1.71197i) q^{19} +(-1.28075 + 1.07468i) q^{20} +(0.0895514 - 0.507871i) q^{22} +(6.58928 + 1.16187i) q^{23} +(-2.35760 + 1.97826i) q^{25} +(2.42307 + 4.19688i) q^{26} +(-2.84824 - 1.43735i) q^{28} +(0.224543 - 0.616928i) q^{29} +(0.595291 + 1.63555i) q^{31} +(3.59586 - 4.28538i) q^{32} +(-1.32455 - 1.57853i) q^{34} +(3.27494 + 1.65268i) q^{35} -1.08507 q^{37} +(2.86724 + 1.04359i) q^{38} +(-2.54613 + 3.03436i) q^{40} +(1.37400 - 0.500093i) q^{41} +(-0.681623 - 3.86568i) q^{43} +0.697818i q^{44} +5.96264 q^{46} +(-5.79174 - 2.10802i) q^{47} +(-0.434042 + 6.98653i) q^{49} +(-1.76293 + 2.10098i) q^{50} +(-4.21505 - 5.02331i) q^{52} +(8.16395 + 4.71346i) q^{53} -0.802359i q^{55} +(-7.23690 - 2.18184i) q^{56} +(0.101595 - 0.576172i) q^{58} +(-3.88097 - 3.25652i) q^{59} +(4.62542 - 12.7082i) q^{61} +(0.775532 + 1.34326i) q^{62} +(2.62687 - 4.54987i) q^{64} +(4.84652 + 5.77585i) q^{65} +(-1.43047 + 8.11262i) q^{67} +(2.13596 + 1.79229i) q^{68} +(3.12988 + 0.943626i) q^{70} +(-13.6154 - 7.86086i) q^{71} +2.39174i q^{73} +(-0.952276 + 0.167912i) q^{74} +(-4.06603 - 0.716950i) q^{76} +(1.40704 - 0.603711i) q^{77} +(-0.465552 - 2.64028i) q^{79} +(-0.0930623 + 0.161189i) q^{80} +(1.12845 - 0.651511i) q^{82} +(-11.2771 - 4.10455i) q^{83} +(-2.45595 - 2.06079i) q^{85} +(-1.19640 - 3.28709i) q^{86} +(0.287088 + 1.62816i) q^{88} +(5.91604 - 10.2469i) q^{89} +(-6.48206 + 12.8448i) q^{91} +(-7.94566 + 1.40103i) q^{92} +(-5.40912 - 0.953774i) q^{94} +(4.67516 + 0.824357i) q^{95} +(-18.3532 + 3.23616i) q^{97} +(0.700224 + 6.19865i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} - 9 q^{10} - 9 q^{11} + 42 q^{14} - 15 q^{16} + 9 q^{17} - 9 q^{19} + 18 q^{20} - 12 q^{22} - 30 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31}+ \cdots + 180 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.877614 − 0.154747i 0.620567 − 0.109423i 0.145480 − 0.989361i \(-0.453528\pi\)
0.475087 + 0.879939i \(0.342416\pi\)
\(3\) 0 0
\(4\) −1.13313 + 0.412424i −0.566563 + 0.206212i
\(5\) 1.30288 − 0.474210i 0.582666 − 0.212073i −0.0338349 − 0.999427i \(-0.510772\pi\)
0.616501 + 0.787354i \(0.288550\pi\)
\(6\) 0 0
\(7\) 1.81190 + 1.92796i 0.684834 + 0.728700i
\(8\) −2.47415 + 1.42845i −0.874743 + 0.505033i
\(9\) 0 0
\(10\) 1.07004 − 0.617790i 0.338377 − 0.195362i
\(11\) 0.197926 − 0.543796i 0.0596768 − 0.163961i −0.906271 − 0.422696i \(-0.861084\pi\)
0.965948 + 0.258736i \(0.0833059\pi\)
\(12\) 0 0
\(13\) 1.85992 + 5.11010i 0.515850 + 1.41729i 0.875053 + 0.484027i \(0.160826\pi\)
−0.359203 + 0.933260i \(0.616951\pi\)
\(14\) 1.88849 + 1.41162i 0.504721 + 0.377271i
\(15\) 0 0
\(16\) −0.102835 + 0.0862885i −0.0257086 + 0.0215721i
\(17\) −1.15616 − 2.00253i −0.280410 − 0.485684i 0.691076 − 0.722782i \(-0.257137\pi\)
−0.971486 + 0.237098i \(0.923804\pi\)
\(18\) 0 0
\(19\) 2.96522 + 1.71197i 0.680269 + 0.392753i 0.799956 − 0.600058i \(-0.204856\pi\)
−0.119688 + 0.992812i \(0.538189\pi\)
\(20\) −1.28075 + 1.07468i −0.286385 + 0.240305i
\(21\) 0 0
\(22\) 0.0895514 − 0.507871i 0.0190924 − 0.108279i
\(23\) 6.58928 + 1.16187i 1.37396 + 0.242266i 0.811400 − 0.584491i \(-0.198706\pi\)
0.562559 + 0.826757i \(0.309817\pi\)
\(24\) 0 0
\(25\) −2.35760 + 1.97826i −0.471520 + 0.395652i
\(26\) 2.42307 + 4.19688i 0.475203 + 0.823076i
\(27\) 0 0
\(28\) −2.84824 − 1.43735i −0.538268 − 0.271633i
\(29\) 0.224543 − 0.616928i 0.0416967 − 0.114561i −0.917097 − 0.398664i \(-0.869474\pi\)
0.958794 + 0.284104i \(0.0916960\pi\)
\(30\) 0 0
\(31\) 0.595291 + 1.63555i 0.106917 + 0.293753i 0.981602 − 0.190940i \(-0.0611535\pi\)
−0.874684 + 0.484693i \(0.838931\pi\)
\(32\) 3.59586 − 4.28538i 0.635665 − 0.757556i
\(33\) 0 0
\(34\) −1.32455 − 1.57853i −0.227158 − 0.270716i
\(35\) 3.27494 + 1.65268i 0.553566 + 0.279354i
\(36\) 0 0
\(37\) −1.08507 −0.178385 −0.0891925 − 0.996014i \(-0.528429\pi\)
−0.0891925 + 0.996014i \(0.528429\pi\)
\(38\) 2.86724 + 1.04359i 0.465128 + 0.169293i
\(39\) 0 0
\(40\) −2.54613 + 3.03436i −0.402579 + 0.479775i
\(41\) 1.37400 − 0.500093i 0.214582 − 0.0781015i −0.232492 − 0.972598i \(-0.574688\pi\)
0.447074 + 0.894497i \(0.352466\pi\)
\(42\) 0 0
\(43\) −0.681623 − 3.86568i −0.103947 − 0.589510i −0.991636 − 0.129067i \(-0.958802\pi\)
0.887689 − 0.460443i \(-0.152309\pi\)
\(44\) 0.697818i 0.105200i
\(45\) 0 0
\(46\) 5.96264 0.879143
\(47\) −5.79174 − 2.10802i −0.844812 − 0.307486i −0.116889 − 0.993145i \(-0.537292\pi\)
−0.727923 + 0.685659i \(0.759514\pi\)
\(48\) 0 0
\(49\) −0.434042 + 6.98653i −0.0620060 + 0.998076i
\(50\) −1.76293 + 2.10098i −0.249316 + 0.297124i
\(51\) 0 0
\(52\) −4.21505 − 5.02331i −0.584523 − 0.696607i
\(53\) 8.16395 + 4.71346i 1.12141 + 0.647444i 0.941759 − 0.336288i \(-0.109171\pi\)
0.179646 + 0.983731i \(0.442505\pi\)
\(54\) 0 0
\(55\) − 0.802359i − 0.108190i
\(56\) −7.23690 − 2.18184i −0.967071 − 0.291561i
\(57\) 0 0
\(58\) 0.101595 − 0.576172i 0.0133400 − 0.0756551i
\(59\) −3.88097 − 3.25652i −0.505259 − 0.423963i 0.354198 − 0.935171i \(-0.384754\pi\)
−0.859457 + 0.511207i \(0.829198\pi\)
\(60\) 0 0
\(61\) 4.62542 − 12.7082i 0.592225 − 1.62712i −0.174141 − 0.984721i \(-0.555715\pi\)
0.766366 − 0.642404i \(-0.222063\pi\)
\(62\) 0.775532 + 1.34326i 0.0984927 + 0.170594i
\(63\) 0 0
\(64\) 2.62687 − 4.54987i 0.328359 − 0.568734i
\(65\) 4.84652 + 5.77585i 0.601136 + 0.716407i
\(66\) 0 0
\(67\) −1.43047 + 8.11262i −0.174760 + 0.991114i 0.763660 + 0.645619i \(0.223400\pi\)
−0.938420 + 0.345496i \(0.887711\pi\)
\(68\) 2.13596 + 1.79229i 0.259024 + 0.217347i
\(69\) 0 0
\(70\) 3.12988 + 0.943626i 0.374093 + 0.112785i
\(71\) −13.6154 − 7.86086i −1.61585 − 0.932913i −0.987977 − 0.154599i \(-0.950592\pi\)
−0.627875 − 0.778314i \(-0.716075\pi\)
\(72\) 0 0
\(73\) 2.39174i 0.279932i 0.990156 + 0.139966i \(0.0446994\pi\)
−0.990156 + 0.139966i \(0.955301\pi\)
\(74\) −0.952276 + 0.167912i −0.110700 + 0.0195194i
\(75\) 0 0
\(76\) −4.06603 − 0.716950i −0.466405 − 0.0822398i
\(77\) 1.40704 − 0.603711i 0.160347 − 0.0687993i
\(78\) 0 0
\(79\) −0.465552 − 2.64028i −0.0523787 − 0.297054i 0.947354 − 0.320189i \(-0.103746\pi\)
−0.999732 + 0.0231347i \(0.992635\pi\)
\(80\) −0.0930623 + 0.161189i −0.0104047 + 0.0180214i
\(81\) 0 0
\(82\) 1.12845 − 0.651511i 0.124616 − 0.0719473i
\(83\) −11.2771 − 4.10455i −1.23783 − 0.450532i −0.361556 − 0.932350i \(-0.617754\pi\)
−0.876271 + 0.481818i \(0.839977\pi\)
\(84\) 0 0
\(85\) −2.45595 − 2.06079i −0.266386 − 0.223524i
\(86\) −1.19640 − 3.28709i −0.129012 − 0.354456i
\(87\) 0 0
\(88\) 0.287088 + 1.62816i 0.0306037 + 0.173562i
\(89\) 5.91604 − 10.2469i 0.627099 − 1.08617i −0.361032 − 0.932553i \(-0.617576\pi\)
0.988131 − 0.153613i \(-0.0490911\pi\)
\(90\) 0 0
\(91\) −6.48206 + 12.8448i −0.679505 + 1.34651i
\(92\) −7.94566 + 1.40103i −0.828392 + 0.146068i
\(93\) 0 0
\(94\) −5.40912 − 0.953774i −0.557908 − 0.0983743i
\(95\) 4.67516 + 0.824357i 0.479661 + 0.0845773i
\(96\) 0 0
\(97\) −18.3532 + 3.23616i −1.86348 + 0.328582i −0.987974 − 0.154618i \(-0.950585\pi\)
−0.875510 + 0.483201i \(0.839474\pi\)
\(98\) 0.700224 + 6.19865i 0.0707333 + 0.626158i
\(99\) 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.17.14 132
3.2 odd 2 189.2.bd.a.185.9 yes 132
7.5 odd 6 567.2.ba.a.341.9 132
21.5 even 6 189.2.ba.a.131.14 yes 132
27.7 even 9 189.2.ba.a.101.14 ✓ 132
27.20 odd 18 567.2.ba.a.143.9 132
189.47 even 18 inner 567.2.bd.a.467.14 132
189.61 odd 18 189.2.bd.a.47.9 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.14 ✓ 132 27.7 even 9
189.2.ba.a.131.14 yes 132 21.5 even 6
189.2.bd.a.47.9 yes 132 189.61 odd 18
189.2.bd.a.185.9 yes 132 3.2 odd 2
567.2.ba.a.143.9 132 27.20 odd 18
567.2.ba.a.341.9 132 7.5 odd 6
567.2.bd.a.17.14 132 1.1 even 1 trivial
567.2.bd.a.467.14 132 189.47 even 18 inner