Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [882,2,Mod(37,882)] 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("882.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(882, base_ring=CyclotomicField(42)) chi = DirichletCharacter(H, H._module([0, 32])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.z (of order \(21\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,-3,0,3,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.04280545828\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(3\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 294)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label 289.3
Character \(\chi\) \(=\) 882.289
Dual form 882.2.z.e.235.3

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.988831 - 0.149042i) q^{2} +(0.955573 - 0.294755i) q^{4} +(0.292828 - 3.90752i) q^{5} +(-2.59908 - 0.494735i) q^{7} +(0.900969 - 0.433884i) q^{8} +(-0.292828 - 3.90752i) q^{10} +(-1.49328 + 3.80482i) q^{11} +(-4.20929 - 5.27829i) q^{13} +(-2.64379 - 0.101836i) q^{14} +(0.826239 - 0.563320i) q^{16} +(4.10859 - 3.81221i) q^{17} +(-2.25402 + 3.90407i) q^{19} +(-0.871942 - 3.82023i) q^{20} +(-0.909524 + 3.98488i) q^{22} +(2.59177 + 2.40481i) q^{23} +(-10.2388 - 1.54325i) q^{25} +(-4.94897 - 4.59197i) q^{26} +(-2.62944 + 0.293338i) q^{28} +(-0.826235 - 3.61997i) q^{29} +(-2.37449 - 4.11273i) q^{31} +(0.733052 - 0.680173i) q^{32} +(3.49452 - 4.38199i) q^{34} +(-2.69427 + 10.0111i) q^{35} +(0.386751 + 0.119297i) q^{37} +(-1.64697 + 4.19641i) q^{38} +(-1.43158 - 3.64760i) q^{40} +(2.99795 - 1.44374i) q^{41} +(2.08120 + 1.00225i) q^{43} +(-0.305449 + 4.07593i) q^{44} +(2.92124 + 1.99167i) q^{46} +(-2.59340 + 0.390893i) q^{47} +(6.51047 + 2.57172i) q^{49} -10.3544 q^{50} +(-5.57809 - 3.80308i) q^{52} +(6.76643 - 2.08717i) q^{53} +(14.4301 + 6.94918i) q^{55} +(-2.55635 + 0.681959i) q^{56} +(-1.35654 - 3.45639i) q^{58} +(-0.0803400 - 1.07206i) q^{59} +(-6.65779 - 2.05366i) q^{61} +(-2.96093 - 3.71289i) q^{62} +(0.623490 - 0.781831i) q^{64} +(-21.8576 + 14.9023i) q^{65} +(-0.550599 - 0.953665i) q^{67} +(2.80239 - 4.85388i) q^{68} +(-1.17210 + 10.3008i) q^{70} +(2.17212 - 9.51666i) q^{71} +(9.93622 + 1.49764i) q^{73} +(0.400212 + 0.0603222i) q^{74} +(-1.00313 + 4.39501i) q^{76} +(5.76354 - 9.15026i) q^{77} +(4.87249 - 8.43939i) q^{79} +(-1.95924 - 3.39350i) q^{80} +(2.74929 - 1.87443i) q^{82} +(8.58328 - 10.7631i) q^{83} +(-13.6932 - 17.1707i) q^{85} +(2.20733 + 0.680872i) q^{86} +(0.305449 + 4.07593i) q^{88} +(-1.54978 - 3.94878i) q^{89} +(8.32896 + 15.8012i) q^{91} +(3.18546 + 1.53404i) q^{92} +(-2.50618 + 0.773054i) q^{94} +(14.5952 + 9.95083i) q^{95} +14.3895 q^{97} +(6.82105 + 1.57266i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 3 q^{2} + 3 q^{4} - q^{7} + 6 q^{8} - 13 q^{11} + 2 q^{13} + 4 q^{14} + 3 q^{16} - q^{17} - 2 q^{19} - 7 q^{20} - 12 q^{22} - 42 q^{23} + 33 q^{25} - 34 q^{26} - 9 q^{28} + 4 q^{29} - q^{31} - 3 q^{32}+ \cdots - 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(e\left(\frac{4}{21}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.988831 − 0.149042i 0.699209 − 0.105389i
\(3\) 0 0
\(4\) 0.955573 − 0.294755i 0.477786 − 0.147378i
\(5\) 0.292828 − 3.90752i 0.130957 − 1.74749i −0.415127 − 0.909764i \(-0.636263\pi\)
0.546083 − 0.837731i \(-0.316118\pi\)
\(6\) 0 0
\(7\) −2.59908 − 0.494735i −0.982361 − 0.186992i
\(8\) 0.900969 − 0.433884i 0.318541 − 0.153401i
\(9\) 0 0
\(10\) −0.292828 − 3.90752i −0.0926003 − 1.23567i
\(11\) −1.49328 + 3.80482i −0.450241 + 1.14720i 0.508031 + 0.861339i \(0.330374\pi\)
−0.958272 + 0.285857i \(0.907722\pi\)
\(12\) 0 0
\(13\) −4.20929 − 5.27829i −1.16745 − 1.46393i −0.858469 − 0.512865i \(-0.828584\pi\)
−0.308979 − 0.951069i \(-0.599987\pi\)
\(14\) −2.64379 − 0.101836i −0.706583 − 0.0272168i
\(15\) 0 0
\(16\) 0.826239 − 0.563320i 0.206560 − 0.140830i
\(17\) 4.10859 − 3.81221i 0.996479 − 0.924598i −0.000673998 − 1.00000i \(-0.500215\pi\)
0.997153 + 0.0754022i \(0.0240241\pi\)
\(18\) 0 0
\(19\) −2.25402 + 3.90407i −0.517107 + 0.895656i 0.482695 + 0.875788i \(0.339658\pi\)
−0.999803 + 0.0198677i \(0.993676\pi\)
\(20\) −0.871942 − 3.82023i −0.194972 − 0.854229i
\(21\) 0 0
\(22\) −0.909524 + 3.98488i −0.193911 + 0.849580i
\(23\) 2.59177 + 2.40481i 0.540422 + 0.501438i 0.902489 − 0.430714i \(-0.141738\pi\)
−0.362066 + 0.932152i \(0.617929\pi\)
\(24\) 0 0
\(25\) −10.2388 − 1.54325i −2.04776 − 0.308650i
\(26\) −4.94897 − 4.59197i −0.970573 − 0.900560i
\(27\) 0 0
\(28\) −2.62944 + 0.293338i −0.496917 + 0.0554357i
\(29\) −0.826235 − 3.61997i −0.153428 − 0.672212i −0.991874 − 0.127227i \(-0.959392\pi\)
0.838446 − 0.544985i \(-0.183465\pi\)
\(30\) 0 0
\(31\) −2.37449 − 4.11273i −0.426470 − 0.738668i 0.570086 − 0.821585i \(-0.306910\pi\)
−0.996556 + 0.0829168i \(0.973576\pi\)
\(32\) 0.733052 − 0.680173i 0.129586 − 0.120239i
\(33\) 0 0
\(34\) 3.49452 − 4.38199i 0.599305 − 0.751505i
\(35\) −2.69427 + 10.0111i −0.455415 + 1.69218i
\(36\) 0 0
\(37\) 0.386751 + 0.119297i 0.0635815 + 0.0196123i 0.326383 − 0.945238i \(-0.394170\pi\)
−0.262801 + 0.964850i \(0.584646\pi\)
\(38\) −1.64697 + 4.19641i −0.267174 + 0.680748i
\(39\) 0 0
\(40\) −1.43158 − 3.64760i −0.226353 − 0.576737i
\(41\) 2.99795 − 1.44374i 0.468201 − 0.225474i −0.184881 − 0.982761i \(-0.559190\pi\)
0.653082 + 0.757287i \(0.273476\pi\)
\(42\) 0 0
\(43\) 2.08120 + 1.00225i 0.317380 + 0.152842i 0.585791 − 0.810462i \(-0.300784\pi\)
−0.268411 + 0.963305i \(0.586498\pi\)
\(44\) −0.305449 + 4.07593i −0.0460482 + 0.614470i
\(45\) 0 0
\(46\) 2.92124 + 1.99167i 0.430714 + 0.293656i
\(47\) −2.59340 + 0.390893i −0.378287 + 0.0570176i −0.335434 − 0.942064i \(-0.608883\pi\)
−0.0428531 + 0.999081i \(0.513645\pi\)
\(48\) 0 0
\(49\) 6.51047 + 2.57172i 0.930068 + 0.367388i
\(50\) −10.3544 −1.46434
\(51\) 0 0
\(52\) −5.57809 − 3.80308i −0.773542 − 0.527392i
\(53\) 6.76643 − 2.08717i 0.929440 − 0.286694i 0.207170 − 0.978305i \(-0.433575\pi\)
0.722271 + 0.691611i \(0.243099\pi\)
\(54\) 0 0
\(55\) 14.4301 + 6.94918i 1.94576 + 0.937027i
\(56\) −2.55635 + 0.681959i −0.341607 + 0.0911307i
\(57\) 0 0
\(58\) −1.35654 − 3.45639i −0.178122 − 0.453847i
\(59\) −0.0803400 − 1.07206i −0.0104594 − 0.139571i 0.989529 − 0.144332i \(-0.0461034\pi\)
−0.999989 + 0.00476154i \(0.998484\pi\)
\(60\) 0 0
\(61\) −6.65779 − 2.05366i −0.852442 − 0.262944i −0.162411 − 0.986723i \(-0.551927\pi\)
−0.690031 + 0.723780i \(0.742403\pi\)
\(62\) −2.96093 − 3.71289i −0.376039 − 0.471538i
\(63\) 0 0
\(64\) 0.623490 − 0.781831i 0.0779362 − 0.0977289i
\(65\) −21.8576 + 14.9023i −2.71110 + 1.84840i
\(66\) 0 0
\(67\) −0.550599 − 0.953665i −0.0672663 − 0.116509i 0.830431 − 0.557122i \(-0.188094\pi\)
−0.897697 + 0.440613i \(0.854761\pi\)
\(68\) 2.80239 − 4.85388i 0.339839 − 0.588619i
\(69\) 0 0
\(70\) −1.17210 + 10.3008i −0.140093 + 1.23119i
\(71\) 2.17212 − 9.51666i 0.257783 − 1.12942i −0.665833 − 0.746101i \(-0.731924\pi\)
0.923616 − 0.383319i \(-0.125219\pi\)
\(72\) 0 0
\(73\) 9.93622 + 1.49764i 1.16295 + 0.175286i 0.702023 − 0.712154i \(-0.252280\pi\)
0.460923 + 0.887440i \(0.347518\pi\)
\(74\) 0.400212 + 0.0603222i 0.0465237 + 0.00701232i
\(75\) 0 0
\(76\) −1.00313 + 4.39501i −0.115067 + 0.504142i
\(77\) 5.76354 − 9.15026i 0.656816 − 1.04277i
\(78\) 0 0
\(79\) 4.87249 − 8.43939i 0.548198 − 0.949506i −0.450201 − 0.892927i \(-0.648648\pi\)
0.998398 − 0.0565786i \(-0.0180191\pi\)
\(80\) −1.95924 − 3.39350i −0.219049 − 0.379405i
\(81\) 0 0
\(82\) 2.74929 − 1.87443i 0.303608 − 0.206996i
\(83\) 8.58328 − 10.7631i 0.942138 − 1.18140i −0.0411146 − 0.999154i \(-0.513091\pi\)
0.983252 − 0.182249i \(-0.0583377\pi\)
\(84\) 0 0
\(85\) −13.6932 − 17.1707i −1.48523 − 1.86242i
\(86\) 2.20733 + 0.680872i 0.238023 + 0.0734203i
\(87\) 0 0
\(88\) 0.305449 + 4.07593i 0.0325610 + 0.434496i
\(89\) −1.54978 − 3.94878i −0.164276 − 0.418569i 0.824904 − 0.565273i \(-0.191229\pi\)
−0.989181 + 0.146703i \(0.953134\pi\)
\(90\) 0 0
\(91\) 8.32896 + 15.8012i 0.873112 + 1.65642i
\(92\) 3.18546 + 1.53404i 0.332107 + 0.159934i
\(93\) 0 0
\(94\) −2.50618 + 0.773054i −0.258493 + 0.0797344i
\(95\) 14.5952 + 9.95083i 1.49744 + 1.02093i
\(96\) 0 0
\(97\) 14.3895 1.46103 0.730516 − 0.682895i \(-0.239280\pi\)
0.730516 + 0.682895i \(0.239280\pi\)
\(98\) 6.82105 + 1.57266i 0.689030 + 0.158862i
\(99\) 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 882.2.z.e.289.3 36
3.2 odd 2 294.2.m.d.289.1 yes 36
49.39 even 21 inner 882.2.z.e.235.3 36
147.137 odd 42 294.2.m.d.235.1 ✓ 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.2.m.d.235.1 ✓ 36 147.137 odd 42
294.2.m.d.289.1 yes 36 3.2 odd 2
882.2.z.e.235.3 36 49.39 even 21 inner
882.2.z.e.289.3 36 1.1 even 1 trivial