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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 17.9
Character \(\chi\) \(=\) 567.17
Dual form 567.2.bd.a.467.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.882178 + 0.155552i) q^{2} +(-1.12534 + 0.409592i) q^{4} +(-0.123522 + 0.0449584i) q^{5} +(1.69913 + 2.02805i) q^{7} +(2.48059 - 1.43217i) q^{8} +(0.101975 - 0.0588754i) q^{10} +(-1.37059 + 3.76567i) q^{11} +(-1.02604 - 2.81901i) q^{13} +(-1.81440 - 1.52479i) q^{14} +(-0.130766 + 0.109725i) q^{16} +(0.172311 + 0.298451i) q^{17} +(-4.24215 - 2.44921i) q^{19} +(0.120590 - 0.101187i) q^{20} +(0.623349 - 3.53519i) q^{22} +(6.27733 + 1.10686i) q^{23} +(-3.81699 + 3.20283i) q^{25} +(1.34365 + 2.32726i) q^{26} +(-2.74277 - 1.58630i) q^{28} +(-2.29851 + 6.31510i) q^{29} +(2.98538 + 8.20227i) q^{31} +(-3.58403 + 4.27128i) q^{32} +(-0.198433 - 0.236483i) q^{34} +(-0.301058 - 0.174119i) q^{35} -7.28599 q^{37} +(4.12331 + 1.50076i) q^{38} +(-0.242020 + 0.288428i) q^{40} +(-9.04416 + 3.29180i) q^{41} +(0.350643 + 1.98860i) q^{43} -4.79906i q^{44} -5.70989 q^{46} +(-4.17641 - 1.52009i) q^{47} +(-1.22594 + 6.89181i) q^{49} +(2.86905 - 3.41920i) q^{50} +(2.30929 + 2.75210i) q^{52} +(5.49931 + 3.17503i) q^{53} -0.526764i q^{55} +(7.11934 + 2.59731i) q^{56} +(1.04537 - 5.92857i) q^{58} +(-0.136314 - 0.114381i) q^{59} +(-1.59759 + 4.38933i) q^{61} +(-3.90952 - 6.77148i) q^{62} +(2.66805 - 4.62120i) q^{64} +(0.253476 + 0.302081i) q^{65} +(2.00174 - 11.3525i) q^{67} +(-0.316152 - 0.265283i) q^{68} +(0.292671 + 0.106774i) q^{70} +(-0.373587 - 0.215690i) q^{71} +1.52364i q^{73} +(6.42754 - 1.13335i) q^{74} +(5.77706 + 1.01865i) q^{76} +(-9.96576 + 3.61872i) q^{77} +(0.410757 + 2.32952i) q^{79} +(0.0112194 - 0.0194325i) q^{80} +(7.46651 - 4.31079i) q^{82} +(-6.81615 - 2.48087i) q^{83} +(-0.0347021 - 0.0291185i) q^{85} +(-0.618659 - 1.69975i) q^{86} +(1.99320 + 11.3040i) q^{88} +(7.70903 - 13.3524i) q^{89} +(3.97372 - 6.87070i) q^{91} +(-7.51752 + 1.32554i) q^{92} +(3.92079 + 0.691340i) q^{94} +(0.634113 + 0.111811i) q^{95} +(12.2061 - 2.15226i) q^{97} +(0.00946529 - 6.27050i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 132 q + 3 q^{2} - 3 q^{4} + 9 q^{5} - 6 q^{7} + 18 q^{8} - 9 q^{10} - 9 q^{11} + 42 q^{14} - 15 q^{16} + 9 q^{17} - 9 q^{19} + 18 q^{20} - 12 q^{22} - 30 q^{23} - 3 q^{25} - 12 q^{28} - 6 q^{29} - 9 q^{31}+ \cdots + 180 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.882178 + 0.155552i −0.623794 + 0.109992i −0.476606 − 0.879117i \(-0.658133\pi\)
−0.147188 + 0.989109i \(0.547022\pi\)
\(3\) 0 0
\(4\) −1.12534 + 0.409592i −0.562672 + 0.204796i
\(5\) −0.123522 + 0.0449584i −0.0552408 + 0.0201060i −0.369493 − 0.929234i \(-0.620469\pi\)
0.314252 + 0.949340i \(0.398246\pi\)
\(6\) 0 0
\(7\) 1.69913 + 2.02805i 0.642209 + 0.766529i
\(8\) 2.48059 − 1.43217i 0.877021 − 0.506348i
\(9\) 0 0
\(10\) 0.101975 − 0.0588754i 0.0322474 − 0.0186180i
\(11\) −1.37059 + 3.76567i −0.413249 + 1.13539i 0.542204 + 0.840247i \(0.317590\pi\)
−0.955453 + 0.295145i \(0.904632\pi\)
\(12\) 0 0
\(13\) −1.02604 − 2.81901i −0.284571 − 0.781852i −0.996802 − 0.0799075i \(-0.974538\pi\)
0.712231 − 0.701945i \(-0.247685\pi\)
\(14\) −1.81440 − 1.52479i −0.484918 − 0.407519i
\(15\) 0 0
\(16\) −0.130766 + 0.109725i −0.0326914 + 0.0274313i
\(17\) 0.172311 + 0.298451i 0.0417914 + 0.0723849i 0.886165 − 0.463371i \(-0.153360\pi\)
−0.844373 + 0.535756i \(0.820027\pi\)
\(18\) 0 0
\(19\) −4.24215 − 2.44921i −0.973216 − 0.561887i −0.0730010 − 0.997332i \(-0.523258\pi\)
−0.900215 + 0.435445i \(0.856591\pi\)
\(20\) 0.120590 − 0.101187i 0.0269649 − 0.0226262i
\(21\) 0 0
\(22\) 0.623349 − 3.53519i 0.132898 − 0.753704i
\(23\) 6.27733 + 1.10686i 1.30891 + 0.230797i 0.784214 − 0.620491i \(-0.213066\pi\)
0.524700 + 0.851287i \(0.324178\pi\)
\(24\) 0 0
\(25\) −3.81699 + 3.20283i −0.763397 + 0.640566i
\(26\) 1.34365 + 2.32726i 0.263511 + 0.456414i
\(27\) 0 0
\(28\) −2.74277 − 1.58630i −0.518335 − 0.299783i
\(29\) −2.29851 + 6.31510i −0.426822 + 1.17268i 0.520908 + 0.853613i \(0.325593\pi\)
−0.947731 + 0.319072i \(0.896629\pi\)
\(30\) 0 0
\(31\) 2.98538 + 8.20227i 0.536191 + 1.47317i 0.851588 + 0.524212i \(0.175640\pi\)
−0.315397 + 0.948960i \(0.602138\pi\)
\(32\) −3.58403 + 4.27128i −0.633573 + 0.755063i
\(33\) 0 0
\(34\) −0.198433 − 0.236483i −0.0340310 − 0.0405565i
\(35\) −0.301058 − 0.174119i −0.0508880 − 0.0294315i
\(36\) 0 0
\(37\) −7.28599 −1.19781 −0.598905 − 0.800820i \(-0.704397\pi\)
−0.598905 + 0.800820i \(0.704397\pi\)
\(38\) 4.12331 + 1.50076i 0.668889 + 0.243456i
\(39\) 0 0
\(40\) −0.242020 + 0.288428i −0.0382667 + 0.0456045i
\(41\) −9.04416 + 3.29180i −1.41246 + 0.514094i −0.931851 − 0.362841i \(-0.881807\pi\)
−0.480609 + 0.876935i \(0.659585\pi\)
\(42\) 0 0
\(43\) 0.350643 + 1.98860i 0.0534726 + 0.303258i 0.999801 − 0.0199486i \(-0.00635026\pi\)
−0.946328 + 0.323207i \(0.895239\pi\)
\(44\) − 4.79906i − 0.723485i
\(45\) 0 0
\(46\) −5.70989 −0.841878
\(47\) −4.17641 − 1.52009i −0.609192 − 0.221728i 0.0189578 − 0.999820i \(-0.493965\pi\)
−0.628150 + 0.778093i \(0.716187\pi\)
\(48\) 0 0
\(49\) −1.22594 + 6.89181i −0.175135 + 0.984545i
\(50\) 2.86905 − 3.41920i 0.405745 − 0.483549i
\(51\) 0 0
\(52\) 2.30929 + 2.75210i 0.320240 + 0.381648i
\(53\) 5.49931 + 3.17503i 0.755388 + 0.436123i 0.827637 − 0.561263i \(-0.189684\pi\)
−0.0722494 + 0.997387i \(0.523018\pi\)
\(54\) 0 0
\(55\) − 0.526764i − 0.0710288i
\(56\) 7.11934 + 2.59731i 0.951361 + 0.347081i
\(57\) 0 0
\(58\) 1.04537 − 5.92857i 0.137263 − 0.778460i
\(59\) −0.136314 − 0.114381i −0.0177466 − 0.0148912i 0.633871 − 0.773439i \(-0.281465\pi\)
−0.651618 + 0.758547i \(0.725909\pi\)
\(60\) 0 0
\(61\) −1.59759 + 4.38933i −0.204550 + 0.561996i −0.998970 − 0.0453721i \(-0.985553\pi\)
0.794420 + 0.607369i \(0.207775\pi\)
\(62\) −3.90952 − 6.77148i −0.496509 − 0.859979i
\(63\) 0 0
\(64\) 2.66805 − 4.62120i 0.333506 − 0.577649i
\(65\) 0.253476 + 0.302081i 0.0314399 + 0.0374686i
\(66\) 0 0
\(67\) 2.00174 − 11.3525i 0.244552 − 1.38692i −0.576979 − 0.816759i \(-0.695769\pi\)
0.821531 − 0.570164i \(-0.193120\pi\)
\(68\) −0.316152 − 0.265283i −0.0383390 − 0.0321702i
\(69\) 0 0
\(70\) 0.292671 + 0.106774i 0.0349809 + 0.0127619i
\(71\) −0.373587 − 0.215690i −0.0443366 − 0.0255977i 0.477668 − 0.878540i \(-0.341482\pi\)
−0.522004 + 0.852943i \(0.674816\pi\)
\(72\) 0 0
\(73\) 1.52364i 0.178329i 0.996017 + 0.0891645i \(0.0284197\pi\)
−0.996017 + 0.0891645i \(0.971580\pi\)
\(74\) 6.42754 − 1.13335i 0.747186 − 0.131749i
\(75\) 0 0
\(76\) 5.77706 + 1.01865i 0.662674 + 0.116847i
\(77\) −9.96576 + 3.61872i −1.13570 + 0.412392i
\(78\) 0 0
\(79\) 0.410757 + 2.32952i 0.0462138 + 0.262091i 0.999157 − 0.0410582i \(-0.0130729\pi\)
−0.952943 + 0.303150i \(0.901962\pi\)
\(80\) 0.0112194 − 0.0194325i 0.00125436 − 0.00217262i
\(81\) 0 0
\(82\) 7.46651 − 4.31079i 0.824538 − 0.476047i
\(83\) −6.81615 − 2.48087i −0.748169 − 0.272311i −0.0603341 − 0.998178i \(-0.519217\pi\)
−0.687835 + 0.725867i \(0.741439\pi\)
\(84\) 0 0
\(85\) −0.0347021 − 0.0291185i −0.00376397 − 0.00315834i
\(86\) −0.618659 − 1.69975i −0.0667117 − 0.183289i
\(87\) 0 0
\(88\) 1.99320 + 11.3040i 0.212476 + 1.20501i
\(89\) 7.70903 − 13.3524i 0.817156 − 1.41536i −0.0906137 − 0.995886i \(-0.528883\pi\)
0.907770 − 0.419469i \(-0.137784\pi\)
\(90\) 0 0
\(91\) 3.97372 − 6.87070i 0.416559 − 0.720245i
\(92\) −7.51752 + 1.32554i −0.783756 + 0.138197i
\(93\) 0 0
\(94\) 3.92079 + 0.691340i 0.404398 + 0.0713063i
\(95\) 0.634113 + 0.111811i 0.0650586 + 0.0114716i
\(96\) 0 0
\(97\) 12.2061 − 2.15226i 1.23934 − 0.218529i 0.484708 − 0.874676i \(-0.338926\pi\)
0.754634 + 0.656147i \(0.227815\pi\)
\(98\) 0.00946529 − 6.27050i 0.000956138 − 0.633416i
\(99\) 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.17.9 132
3.2 odd 2 189.2.bd.a.185.14 yes 132
7.5 odd 6 567.2.ba.a.341.14 132
21.5 even 6 189.2.ba.a.131.9 yes 132
27.7 even 9 189.2.ba.a.101.9 ✓ 132
27.20 odd 18 567.2.ba.a.143.14 132
189.47 even 18 inner 567.2.bd.a.467.9 132
189.61 odd 18 189.2.bd.a.47.14 yes 132
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.ba.a.101.9 ✓ 132 27.7 even 9
189.2.ba.a.131.9 yes 132 21.5 even 6
189.2.bd.a.47.14 yes 132 189.61 odd 18
189.2.bd.a.185.14 yes 132 3.2 odd 2
567.2.ba.a.143.14 132 27.20 odd 18
567.2.ba.a.341.14 132 7.5 odd 6
567.2.bd.a.17.9 132 1.1 even 1 trivial
567.2.bd.a.467.9 132 189.47 even 18 inner