Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1638,2,Mod(757,1638)] 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("1638.757"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.r (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,0,-2,-2,0,-2,-4,0,-1,-8,0,14,-4,0,-2,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.0794958511\)
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: no (minimal twist has level 546)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1387.2
Root \(2.13746 + 0.656712i\) of defining polynomial
Character \(\chi\) \(=\) 1638.1387
Dual form 1638.2.r.x.757.2

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1638\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(703\) \(911\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 − 0.866025i 0.353553 − 0.612372i
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.250000 − 0.433013i
\(5\) 3.27492 1.46459 0.732294 − 0.680989i \(-0.238450\pi\)
0.732294 + 0.680989i \(0.238450\pi\)
\(6\) 0 0
\(7\) −0.500000 − 0.866025i −0.188982 − 0.327327i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.63746 − 2.83616i 0.517810 − 0.896873i
\(11\) −2.00000 + 3.46410i −0.603023 + 1.04447i 0.389338 + 0.921095i \(0.372704\pi\)
−0.992361 + 0.123371i \(0.960630\pi\)
\(12\) 0 0
\(13\) 3.50000 + 0.866025i 0.970725 + 0.240192i
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) 1.50000 + 2.59808i 0.363803 + 0.630126i 0.988583 − 0.150675i \(-0.0481447\pi\)
−0.624780 + 0.780801i \(0.714811\pi\)
\(18\) 0 0
\(19\) 4.27492 + 7.40437i 0.980733 + 1.69868i 0.659546 + 0.751664i \(0.270749\pi\)
0.321187 + 0.947016i \(0.395918\pi\)
\(20\) −1.63746 − 2.83616i −0.366147 − 0.634185i
\(21\) 0 0
\(22\) 2.00000 + 3.46410i 0.426401 + 0.738549i
\(23\) 1.13746 − 1.97014i 0.237177 − 0.410802i −0.722726 − 0.691134i \(-0.757111\pi\)
0.959903 + 0.280332i \(0.0904447\pi\)
\(24\) 0 0
\(25\) 5.72508 1.14502
\(26\) 2.50000 − 2.59808i 0.490290 − 0.509525i
\(27\) 0 0
\(28\) −0.500000 + 0.866025i −0.0944911 + 0.163663i
\(29\) 0.362541 − 0.627940i 0.0673222 − 0.116606i −0.830400 − 0.557168i \(-0.811888\pi\)
0.897722 + 0.440563i \(0.145221\pi\)
\(30\) 0 0
\(31\) 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
\(33\) 0 0
\(34\) 3.00000 0.514496
\(35\) −1.63746 − 2.83616i −0.276781 − 0.479399i
\(36\) 0 0
\(37\) 3.63746 − 6.30026i 0.597995 − 1.03576i −0.395122 − 0.918629i \(-0.629298\pi\)
0.993117 − 0.117128i \(-0.0373689\pi\)
\(38\) 8.54983 1.38697
\(39\) 0 0
\(40\) −3.27492 −0.517810
\(41\) 0.362541 − 0.627940i 0.0566195 − 0.0980678i −0.836326 − 0.548232i \(-0.815301\pi\)
0.892946 + 0.450164i \(0.148634\pi\)
\(42\) 0 0
\(43\) −5.41238 − 9.37451i −0.825380 − 1.42960i −0.901629 − 0.432511i \(-0.857628\pi\)
0.0762493 − 0.997089i \(-0.475706\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) −1.13746 − 1.97014i −0.167709 − 0.290481i
\(47\) −8.54983 −1.24712 −0.623561 − 0.781775i \(-0.714315\pi\)
−0.623561 + 0.781775i \(0.714315\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 2.86254 − 4.95807i 0.404824 − 0.701177i
\(51\) 0 0
\(52\) −1.00000 − 3.46410i −0.138675 − 0.480384i
\(53\) −11.5498 −1.58649 −0.793246 − 0.608901i \(-0.791610\pi\)
−0.793246 + 0.608901i \(0.791610\pi\)
\(54\) 0 0
\(55\) −6.54983 + 11.3446i −0.883179 + 1.52971i
\(56\) 0.500000 + 0.866025i 0.0668153 + 0.115728i
\(57\) 0 0
\(58\) −0.362541 − 0.627940i −0.0476040 − 0.0824526i
\(59\) 5.41238 + 9.37451i 0.704631 + 1.22046i 0.966824 + 0.255442i \(0.0822209\pi\)
−0.262193 + 0.965015i \(0.584446\pi\)
\(60\) 0 0
\(61\) 4.50000 + 7.79423i 0.576166 + 0.997949i 0.995914 + 0.0903080i \(0.0287851\pi\)
−0.419748 + 0.907641i \(0.637882\pi\)
\(62\) 3.13746 − 5.43424i 0.398458 − 0.690149i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 11.4622 + 2.83616i 1.42171 + 0.351783i
\(66\) 0 0
\(67\) 5.13746 − 8.89834i 0.627640 − 1.08711i −0.360383 − 0.932804i \(-0.617354\pi\)
0.988024 − 0.154301i \(-0.0493125\pi\)
\(68\) 1.50000 − 2.59808i 0.181902 − 0.315063i
\(69\) 0 0
\(70\) −3.27492 −0.391427
\(71\) −1.13746 − 1.97014i −0.134992 − 0.233812i 0.790603 − 0.612329i \(-0.209767\pi\)
−0.925594 + 0.378517i \(0.876434\pi\)
\(72\) 0 0
\(73\) 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
\(75\) 0 0
\(76\) 4.27492 − 7.40437i 0.490367 − 0.849340i
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −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 0
\(82\) −0.362541 − 0.627940i −0.0400360 − 0.0693444i
\(83\) −10.8248 −1.18817 −0.594085 − 0.804402i \(-0.702486\pi\)
−0.594085 + 0.804402i \(0.702486\pi\)
\(84\) 0 0
\(85\) 4.91238 + 8.50848i 0.532822 + 0.922875i
\(86\) −10.8248 −1.16726
\(87\) 0 0
\(88\) 2.00000 − 3.46410i 0.213201 − 0.369274i
\(89\) 4.13746 − 7.16629i 0.438570 − 0.759625i −0.559010 − 0.829161i \(-0.688819\pi\)
0.997579 + 0.0695360i \(0.0221519\pi\)
\(90\) 0 0
\(91\) −1.00000 − 3.46410i −0.104828 − 0.363137i
\(92\) −2.27492 −0.237177
\(93\) 0 0
\(94\) −4.27492 + 7.40437i −0.440924 + 0.763703i
\(95\) 14.0000 + 24.2487i 1.43637 + 2.48787i
\(96\) 0 0
\(97\) −7.27492 − 12.6005i −0.738656 − 1.27939i −0.953101 − 0.302653i \(-0.902128\pi\)
0.214445 − 0.976736i \(-0.431206\pi\)
\(98\) 0.500000 + 0.866025i 0.0505076 + 0.0874818i
\(99\) 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 1638.2.r.x.1387.2 4
3.2 odd 2 546.2.l.j.295.1 yes 4
13.3 even 3 inner 1638.2.r.x.757.2 4
39.17 odd 6 7098.2.a.bm.1.2 2
39.29 odd 6 546.2.l.j.211.1 ✓ 4
39.35 odd 6 7098.2.a.ca.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.l.j.211.1 ✓ 4 39.29 odd 6
546.2.l.j.295.1 yes 4 3.2 odd 2
1638.2.r.x.757.2 4 13.3 even 3 inner
1638.2.r.x.1387.2 4 1.1 even 1 trivial
7098.2.a.bm.1.2 2 39.17 odd 6
7098.2.a.ca.1.1 2 39.35 odd 6