Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1386,2,Mod(793,1386)] 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("1386.793"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1386, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,0,-2,-4,0,-2,-4,0,4,2] 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: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 154)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 991.1
Root \(-0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1386.991
Dual form 1386.2.k.t.793.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-1.70711 - 2.95680i) q^{5} +(-2.62132 + 0.358719i) q^{7} -1.00000 q^{8} +(1.70711 - 2.95680i) q^{10} +(0.500000 - 0.866025i) q^{11} +1.82843 q^{13} +(-1.62132 - 2.09077i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-3.82843 + 6.63103i) q^{17} +(1.70711 + 2.95680i) q^{19} +3.41421 q^{20} +1.00000 q^{22} +(1.12132 + 1.94218i) q^{23} +(-3.32843 + 5.76500i) q^{25} +(0.914214 + 1.58346i) q^{26} +(1.00000 - 2.44949i) q^{28} +8.65685 q^{29} +(2.00000 - 3.46410i) q^{31} +(0.500000 - 0.866025i) q^{32} -7.65685 q^{34} +(5.53553 + 7.13834i) q^{35} +(3.29289 + 5.70346i) q^{37} +(-1.70711 + 2.95680i) q^{38} +(1.70711 + 2.95680i) q^{40} +2.58579 q^{41} +5.65685 q^{43} +(0.500000 + 0.866025i) q^{44} +(-1.12132 + 1.94218i) q^{46} +(3.24264 + 5.61642i) q^{47} +(6.74264 - 1.88064i) q^{49} -6.65685 q^{50} +(-0.914214 + 1.58346i) q^{52} +(-5.94975 + 10.3053i) q^{53} -3.41421 q^{55} +(2.62132 - 0.358719i) q^{56} +(4.32843 + 7.49706i) q^{58} +(-4.20711 + 7.28692i) q^{59} +(-3.08579 - 5.34474i) q^{61} +4.00000 q^{62} +1.00000 q^{64} +(-3.12132 - 5.40629i) q^{65} +(-5.62132 + 9.73641i) q^{67} +(-3.82843 - 6.63103i) q^{68} +(-3.41421 + 8.36308i) q^{70} -3.07107 q^{71} +(3.29289 - 5.70346i) q^{73} +(-3.29289 + 5.70346i) q^{74} -3.41421 q^{76} +(-1.00000 + 2.44949i) q^{77} +(2.37868 + 4.11999i) q^{79} +(-1.70711 + 2.95680i) q^{80} +(1.29289 + 2.23936i) q^{82} -16.1421 q^{83} +26.1421 q^{85} +(2.82843 + 4.89898i) q^{86} +(-0.500000 + 0.866025i) q^{88} +(-2.24264 - 3.88437i) q^{89} +(-4.79289 + 0.655892i) q^{91} -2.24264 q^{92} +(-3.24264 + 5.61642i) q^{94} +(5.82843 - 10.0951i) q^{95} +1.82843 q^{97} +(5.00000 + 4.89898i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{5} - 2 q^{7} - 4 q^{8} + 4 q^{10} + 2 q^{11} - 4 q^{13} + 2 q^{14} - 2 q^{16} - 4 q^{17} + 4 q^{19} + 8 q^{20} + 4 q^{22} - 4 q^{23} - 2 q^{25} - 2 q^{26} + 4 q^{28} + 12 q^{29}+ \cdots + 20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.500000 + 0.866025i 0.353553 + 0.612372i
\(3\) 0 0
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) −1.70711 − 2.95680i −0.763441 − 1.32232i −0.941067 − 0.338221i \(-0.890175\pi\)
0.177625 − 0.984098i \(-0.443158\pi\)
\(6\) 0 0
\(7\) −2.62132 + 0.358719i −0.990766 + 0.135583i
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.70711 − 2.95680i 0.539835 − 0.935021i
\(11\) 0.500000 − 0.866025i 0.150756 − 0.261116i
\(12\) 0 0
\(13\) 1.82843 0.507114 0.253557 − 0.967320i \(-0.418399\pi\)
0.253557 + 0.967320i \(0.418399\pi\)
\(14\) −1.62132 − 2.09077i −0.433316 − 0.558782i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) −3.82843 + 6.63103i −0.928530 + 1.60826i −0.142747 + 0.989759i \(0.545593\pi\)
−0.785783 + 0.618502i \(0.787740\pi\)
\(18\) 0 0
\(19\) 1.70711 + 2.95680i 0.391637 + 0.678335i 0.992666 − 0.120892i \(-0.0385755\pi\)
−0.601028 + 0.799228i \(0.705242\pi\)
\(20\) 3.41421 0.763441
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 1.12132 + 1.94218i 0.233811 + 0.404973i 0.958927 − 0.283654i \(-0.0915468\pi\)
−0.725115 + 0.688628i \(0.758213\pi\)
\(24\) 0 0
\(25\) −3.32843 + 5.76500i −0.665685 + 1.15300i
\(26\) 0.914214 + 1.58346i 0.179292 + 0.310543i
\(27\) 0 0
\(28\) 1.00000 − 2.44949i 0.188982 − 0.462910i
\(29\) 8.65685 1.60754 0.803769 − 0.594942i \(-0.202825\pi\)
0.803769 + 0.594942i \(0.202825\pi\)
\(30\) 0 0
\(31\) 2.00000 − 3.46410i 0.359211 − 0.622171i −0.628619 − 0.777714i \(-0.716379\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 0.500000 − 0.866025i 0.0883883 − 0.153093i
\(33\) 0 0
\(34\) −7.65685 −1.31314
\(35\) 5.53553 + 7.13834i 0.935676 + 1.20660i
\(36\) 0 0
\(37\) 3.29289 + 5.70346i 0.541348 + 0.937643i 0.998827 + 0.0484222i \(0.0154193\pi\)
−0.457479 + 0.889221i \(0.651247\pi\)
\(38\) −1.70711 + 2.95680i −0.276929 + 0.479656i
\(39\) 0 0
\(40\) 1.70711 + 2.95680i 0.269917 + 0.467510i
\(41\) 2.58579 0.403832 0.201916 − 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(42\) 0 0
\(43\) 5.65685 0.862662 0.431331 − 0.902194i \(-0.358044\pi\)
0.431331 + 0.902194i \(0.358044\pi\)
\(44\) 0.500000 + 0.866025i 0.0753778 + 0.130558i
\(45\) 0 0
\(46\) −1.12132 + 1.94218i −0.165330 + 0.286359i
\(47\) 3.24264 + 5.61642i 0.472988 + 0.819239i 0.999522 − 0.0309151i \(-0.00984215\pi\)
−0.526534 + 0.850154i \(0.676509\pi\)
\(48\) 0 0
\(49\) 6.74264 − 1.88064i 0.963234 − 0.268662i
\(50\) −6.65685 −0.941421
\(51\) 0 0
\(52\) −0.914214 + 1.58346i −0.126779 + 0.219587i
\(53\) −5.94975 + 10.3053i −0.817261 + 1.41554i 0.0904325 + 0.995903i \(0.471175\pi\)
−0.907693 + 0.419634i \(0.862158\pi\)
\(54\) 0 0
\(55\) −3.41421 −0.460372
\(56\) 2.62132 − 0.358719i 0.350289 − 0.0479359i
\(57\) 0 0
\(58\) 4.32843 + 7.49706i 0.568350 + 0.984412i
\(59\) −4.20711 + 7.28692i −0.547719 + 0.948677i 0.450712 + 0.892670i \(0.351170\pi\)
−0.998430 + 0.0560070i \(0.982163\pi\)
\(60\) 0 0
\(61\) −3.08579 − 5.34474i −0.395094 − 0.684324i 0.598019 − 0.801482i \(-0.295955\pi\)
−0.993113 + 0.117158i \(0.962621\pi\)
\(62\) 4.00000 0.508001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.12132 − 5.40629i −0.387152 − 0.670567i
\(66\) 0 0
\(67\) −5.62132 + 9.73641i −0.686754 + 1.18949i 0.286129 + 0.958191i \(0.407632\pi\)
−0.972882 + 0.231301i \(0.925702\pi\)
\(68\) −3.82843 − 6.63103i −0.464265 − 0.804131i
\(69\) 0 0
\(70\) −3.41421 + 8.36308i −0.408077 + 0.999579i
\(71\) −3.07107 −0.364469 −0.182234 − 0.983255i \(-0.558333\pi\)
−0.182234 + 0.983255i \(0.558333\pi\)
\(72\) 0 0
\(73\) 3.29289 − 5.70346i 0.385404 − 0.667539i −0.606421 − 0.795144i \(-0.707395\pi\)
0.991825 + 0.127604i \(0.0407288\pi\)
\(74\) −3.29289 + 5.70346i −0.382791 + 0.663014i
\(75\) 0 0
\(76\) −3.41421 −0.391637
\(77\) −1.00000 + 2.44949i −0.113961 + 0.279145i
\(78\) 0 0
\(79\) 2.37868 + 4.11999i 0.267622 + 0.463536i 0.968247 − 0.249994i \(-0.0804287\pi\)
−0.700625 + 0.713530i \(0.747095\pi\)
\(80\) −1.70711 + 2.95680i −0.190860 + 0.330580i
\(81\) 0 0
\(82\) 1.29289 + 2.23936i 0.142776 + 0.247296i
\(83\) −16.1421 −1.77183 −0.885915 − 0.463848i \(-0.846468\pi\)
−0.885915 + 0.463848i \(0.846468\pi\)
\(84\) 0 0
\(85\) 26.1421 2.83551
\(86\) 2.82843 + 4.89898i 0.304997 + 0.528271i
\(87\) 0 0
\(88\) −0.500000 + 0.866025i −0.0533002 + 0.0923186i
\(89\) −2.24264 − 3.88437i −0.237719 − 0.411742i 0.722340 − 0.691538i \(-0.243067\pi\)
−0.960060 + 0.279796i \(0.909733\pi\)
\(90\) 0 0
\(91\) −4.79289 + 0.655892i −0.502432 + 0.0687562i
\(92\) −2.24264 −0.233811
\(93\) 0 0
\(94\) −3.24264 + 5.61642i −0.334453 + 0.579289i
\(95\) 5.82843 − 10.0951i 0.597984 − 1.03574i
\(96\) 0 0
\(97\) 1.82843 0.185649 0.0928243 − 0.995683i \(-0.470411\pi\)
0.0928243 + 0.995683i \(0.470411\pi\)
\(98\) 5.00000 + 4.89898i 0.505076 + 0.494872i
\(99\) 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 1386.2.k.t.991.1 4
3.2 odd 2 154.2.e.e.67.1 yes 4
7.2 even 3 inner 1386.2.k.t.793.1 4
7.3 odd 6 9702.2.a.ch.1.1 2
7.4 even 3 9702.2.a.cx.1.2 2
12.11 even 2 1232.2.q.f.529.2 4
21.2 odd 6 154.2.e.e.23.1 ✓ 4
21.5 even 6 1078.2.e.m.177.2 4
21.11 odd 6 1078.2.a.t.1.2 2
21.17 even 6 1078.2.a.x.1.1 2
21.20 even 2 1078.2.e.m.67.2 4
84.11 even 6 8624.2.a.cc.1.1 2
84.23 even 6 1232.2.q.f.177.2 4
84.59 odd 6 8624.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
154.2.e.e.23.1 ✓ 4 21.2 odd 6
154.2.e.e.67.1 yes 4 3.2 odd 2
1078.2.a.t.1.2 2 21.11 odd 6
1078.2.a.x.1.1 2 21.17 even 6
1078.2.e.m.67.2 4 21.20 even 2
1078.2.e.m.177.2 4 21.5 even 6
1232.2.q.f.177.2 4 84.23 even 6
1232.2.q.f.529.2 4 12.11 even 2
1386.2.k.t.793.1 4 7.2 even 3 inner
1386.2.k.t.991.1 4 1.1 even 1 trivial
8624.2.a.bh.1.2 2 84.59 odd 6
8624.2.a.cc.1.1 2 84.11 even 6
9702.2.a.ch.1.1 2 7.3 odd 6
9702.2.a.cx.1.2 2 7.4 even 3