Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [546,2,Mod(211,546)] 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("546.211"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.l (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-2,-2,-2,2,-2,-2,4,-2,-1,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 211.1
Root \(-1.63746 + 1.52274i\) of defining polynomial
Character \(\chi\) \(=\) 546.211
Dual form 546.2.l.j.295.1

$q$-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+(-0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} -3.27492 q^{5} +(-0.500000 + 0.866025i) q^{6} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(1.63746 + 2.83616i) q^{10} +(2.00000 + 3.46410i) q^{11} +1.00000 q^{12} +(3.50000 - 0.866025i) q^{13} +1.00000 q^{14} +(1.63746 + 2.83616i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(-1.50000 + 2.59808i) q^{17} +1.00000 q^{18} +(4.27492 - 7.40437i) q^{19} +(1.63746 - 2.83616i) q^{20} +1.00000 q^{21} +(2.00000 - 3.46410i) q^{22} +(-1.13746 - 1.97014i) q^{23} +(-0.500000 - 0.866025i) q^{24} +5.72508 q^{25} +(-2.50000 - 2.59808i) q^{26} +1.00000 q^{27} +(-0.500000 - 0.866025i) q^{28} +(-0.362541 - 0.627940i) q^{29} +(1.63746 - 2.83616i) q^{30} +6.27492 q^{31} +(-0.500000 + 0.866025i) q^{32} +(2.00000 - 3.46410i) q^{33} +3.00000 q^{34} +(1.63746 - 2.83616i) q^{35} +(-0.500000 - 0.866025i) q^{36} +(3.63746 + 6.30026i) q^{37} -8.54983 q^{38} +(-2.50000 - 2.59808i) q^{39} -3.27492 q^{40} +(-0.362541 - 0.627940i) q^{41} +(-0.500000 - 0.866025i) q^{42} +(-5.41238 + 9.37451i) q^{43} -4.00000 q^{44} +(1.63746 - 2.83616i) q^{45} +(-1.13746 + 1.97014i) q^{46} +8.54983 q^{47} +(-0.500000 + 0.866025i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(-2.86254 - 4.95807i) q^{50} +3.00000 q^{51} +(-1.00000 + 3.46410i) q^{52} +11.5498 q^{53} +(-0.500000 - 0.866025i) q^{54} +(-6.54983 - 11.3446i) q^{55} +(-0.500000 + 0.866025i) q^{56} -8.54983 q^{57} +(-0.362541 + 0.627940i) q^{58} +(-5.41238 + 9.37451i) q^{59} -3.27492 q^{60} +(4.50000 - 7.79423i) q^{61} +(-3.13746 - 5.43424i) q^{62} +(-0.500000 - 0.866025i) q^{63} +1.00000 q^{64} +(-11.4622 + 2.83616i) q^{65} -4.00000 q^{66} +(5.13746 + 8.89834i) q^{67} +(-1.50000 - 2.59808i) q^{68} +(-1.13746 + 1.97014i) q^{69} -3.27492 q^{70} +(1.13746 - 1.97014i) q^{71} +(-0.500000 + 0.866025i) q^{72} +13.2749 q^{73} +(3.63746 - 6.30026i) q^{74} +(-2.86254 - 4.95807i) q^{75} +(4.27492 + 7.40437i) q^{76} -4.00000 q^{77} +(-1.00000 + 3.46410i) q^{78} -8.00000 q^{79} +(1.63746 + 2.83616i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.362541 + 0.627940i) q^{82} +10.8248 q^{83} +(-0.500000 + 0.866025i) q^{84} +(4.91238 - 8.50848i) q^{85} +10.8248 q^{86} +(-0.362541 + 0.627940i) q^{87} +(2.00000 + 3.46410i) q^{88} +(-4.13746 - 7.16629i) q^{89} -3.27492 q^{90} +(-1.00000 + 3.46410i) q^{91} +2.27492 q^{92} +(-3.13746 - 5.43424i) q^{93} +(-4.27492 - 7.40437i) q^{94} +(-14.0000 + 24.2487i) q^{95} +1.00000 q^{96} +(-7.27492 + 12.6005i) q^{97} +(-0.500000 + 0.866025i) q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{5} - 2 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{9} - q^{10} + 8 q^{11} + 4 q^{12} + 14 q^{13} + 4 q^{14} - q^{15} - 2 q^{16} - 6 q^{17} + 4 q^{18} + 2 q^{19} - q^{20}+ \cdots - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) −0.500000 − 0.866025i −0.288675 − 0.500000i
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −3.27492 −1.46459 −0.732294 − 0.680989i \(-0.761550\pi\)
−0.732294 + 0.680989i \(0.761550\pi\)
\(6\) −0.500000 + 0.866025i −0.204124 + 0.353553i
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i
\(8\) 1.00000 0.353553
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 1.63746 + 2.83616i 0.517810 + 0.896873i
\(11\) 2.00000 + 3.46410i 0.603023 + 1.04447i 0.992361 + 0.123371i \(0.0393705\pi\)
−0.389338 + 0.921095i \(0.627296\pi\)
\(12\) 1.00000 0.288675
\(13\) 3.50000 − 0.866025i 0.970725 − 0.240192i
\(14\) 1.00000 0.267261
\(15\) 1.63746 + 2.83616i 0.422790 + 0.732294i
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −1.50000 + 2.59808i −0.363803 + 0.630126i −0.988583 − 0.150675i \(-0.951855\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(18\) 1.00000 0.235702
\(19\) 4.27492 − 7.40437i 0.980733 − 1.69868i 0.321187 − 0.947016i \(-0.395918\pi\)
0.659546 − 0.751664i \(-0.270749\pi\)
\(20\) 1.63746 − 2.83616i 0.366147 − 0.634185i
\(21\) 1.00000 0.218218
\(22\) 2.00000 − 3.46410i 0.426401 − 0.738549i
\(23\) −1.13746 − 1.97014i −0.237177 − 0.410802i 0.722726 − 0.691134i \(-0.242889\pi\)
−0.959903 + 0.280332i \(0.909555\pi\)
\(24\) −0.500000 − 0.866025i −0.102062 − 0.176777i
\(25\) 5.72508 1.14502
\(26\) −2.50000 − 2.59808i −0.490290 − 0.509525i
\(27\) 1.00000 0.192450
\(28\) −0.500000 − 0.866025i −0.0944911 − 0.163663i
\(29\) −0.362541 − 0.627940i −0.0673222 − 0.116606i 0.830400 − 0.557168i \(-0.188112\pi\)
−0.897722 + 0.440563i \(0.854779\pi\)
\(30\) 1.63746 − 2.83616i 0.298958 − 0.517810i
\(31\) 6.27492 1.12701 0.563504 − 0.826113i \(-0.309453\pi\)
0.563504 + 0.826113i \(0.309453\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 2.00000 − 3.46410i 0.348155 − 0.603023i
\(34\) 3.00000 0.514496
\(35\) 1.63746 − 2.83616i 0.276781 − 0.479399i
\(36\) −0.500000 − 0.866025i −0.0833333 − 0.144338i
\(37\) 3.63746 + 6.30026i 0.597995 + 1.03576i 0.993117 + 0.117128i \(0.0373689\pi\)
−0.395122 + 0.918629i \(0.629298\pi\)
\(38\) −8.54983 −1.38697
\(39\) −2.50000 − 2.59808i −0.400320 − 0.416025i
\(40\) −3.27492 −0.517810
\(41\) −0.362541 − 0.627940i −0.0566195 − 0.0980678i 0.836326 − 0.548232i \(-0.184699\pi\)
−0.892946 + 0.450164i \(0.851366\pi\)
\(42\) −0.500000 − 0.866025i −0.0771517 − 0.133631i
\(43\) −5.41238 + 9.37451i −0.825380 + 1.42960i 0.0762493 + 0.997089i \(0.475706\pi\)
−0.901629 + 0.432511i \(0.857628\pi\)
\(44\) −4.00000 −0.603023
\(45\) 1.63746 − 2.83616i 0.244098 − 0.422790i
\(46\) −1.13746 + 1.97014i −0.167709 + 0.290481i
\(47\) 8.54983 1.24712 0.623561 − 0.781775i \(-0.285685\pi\)
0.623561 + 0.781775i \(0.285685\pi\)
\(48\) −0.500000 + 0.866025i −0.0721688 + 0.125000i
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) −2.86254 − 4.95807i −0.404824 − 0.701177i
\(51\) 3.00000 0.420084
\(52\) −1.00000 + 3.46410i −0.138675 + 0.480384i
\(53\) 11.5498 1.58649 0.793246 − 0.608901i \(-0.208390\pi\)
0.793246 + 0.608901i \(0.208390\pi\)
\(54\) −0.500000 − 0.866025i −0.0680414 − 0.117851i
\(55\) −6.54983 − 11.3446i −0.883179 − 1.52971i
\(56\) −0.500000 + 0.866025i −0.0668153 + 0.115728i
\(57\) −8.54983 −1.13245
\(58\) −0.362541 + 0.627940i −0.0476040 + 0.0824526i
\(59\) −5.41238 + 9.37451i −0.704631 + 1.22046i 0.262193 + 0.965015i \(0.415554\pi\)
−0.966824 + 0.255442i \(0.917779\pi\)
\(60\) −3.27492 −0.422790
\(61\) 4.50000 − 7.79423i 0.576166 − 0.997949i −0.419748 − 0.907641i \(-0.637882\pi\)
0.995914 − 0.0903080i \(-0.0287851\pi\)
\(62\) −3.13746 − 5.43424i −0.398458 − 0.690149i
\(63\) −0.500000 − 0.866025i −0.0629941 − 0.109109i
\(64\) 1.00000 0.125000
\(65\) −11.4622 + 2.83616i −1.42171 + 0.351783i
\(66\) −4.00000 −0.492366
\(67\) 5.13746 + 8.89834i 0.627640 + 1.08711i 0.988024 + 0.154301i \(0.0493125\pi\)
−0.360383 + 0.932804i \(0.617354\pi\)
\(68\) −1.50000 − 2.59808i −0.181902 − 0.315063i
\(69\) −1.13746 + 1.97014i −0.136934 + 0.237177i
\(70\) −3.27492 −0.391427
\(71\) 1.13746 − 1.97014i 0.134992 − 0.233812i −0.790603 − 0.612329i \(-0.790233\pi\)
0.925594 + 0.378517i \(0.123566\pi\)
\(72\) −0.500000 + 0.866025i −0.0589256 + 0.102062i
\(73\) 13.2749 1.55371 0.776856 − 0.629679i \(-0.216813\pi\)
0.776856 + 0.629679i \(0.216813\pi\)
\(74\) 3.63746 − 6.30026i 0.422846 − 0.732391i
\(75\) −2.86254 − 4.95807i −0.330538 − 0.572508i
\(76\) 4.27492 + 7.40437i 0.490367 + 0.849340i
\(77\) −4.00000 −0.455842
\(78\) −1.00000 + 3.46410i −0.113228 + 0.392232i
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.63746 + 2.83616i 0.183073 + 0.317092i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −0.362541 + 0.627940i −0.0400360 + 0.0693444i
\(83\) 10.8248 1.18817 0.594085 − 0.804402i \(-0.297514\pi\)
0.594085 + 0.804402i \(0.297514\pi\)
\(84\) −0.500000 + 0.866025i −0.0545545 + 0.0944911i
\(85\) 4.91238 − 8.50848i 0.532822 − 0.922875i
\(86\) 10.8248 1.16726
\(87\) −0.362541 + 0.627940i −0.0388685 + 0.0673222i
\(88\) 2.00000 + 3.46410i 0.213201 + 0.369274i
\(89\) −4.13746 − 7.16629i −0.438570 − 0.759625i 0.559010 − 0.829161i \(-0.311181\pi\)
−0.997579 + 0.0695360i \(0.977848\pi\)
\(90\) −3.27492 −0.345207
\(91\) −1.00000 + 3.46410i −0.104828 + 0.363137i
\(92\) 2.27492 0.237177
\(93\) −3.13746 − 5.43424i −0.325339 − 0.563504i
\(94\) −4.27492 − 7.40437i −0.440924 − 0.763703i
\(95\) −14.0000 + 24.2487i −1.43637 + 2.48787i
\(96\) 1.00000 0.102062
\(97\) −7.27492 + 12.6005i −0.738656 + 1.27939i 0.214445 + 0.976736i \(0.431206\pi\)
−0.953101 + 0.302653i \(0.902128\pi\)
\(98\) −0.500000 + 0.866025i −0.0505076 + 0.0874818i
\(99\) −4.00000 −0.402015
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 546.2.l.j.211.1 ✓ 4
3.2 odd 2 1638.2.r.x.757.2 4
13.3 even 3 7098.2.a.ca.1.1 2
13.9 even 3 inner 546.2.l.j.295.1 yes 4
13.10 even 6 7098.2.a.bm.1.2 2
39.35 odd 6 1638.2.r.x.1387.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.j.211.1 ✓ 4 1.1 even 1 trivial
546.2.l.j.295.1 yes 4 13.9 even 3 inner
1638.2.r.x.757.2 4 3.2 odd 2
1638.2.r.x.1387.2 4 39.35 odd 6
7098.2.a.bm.1.2 2 13.10 even 6
7098.2.a.ca.1.1 2 13.3 even 3