Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6840,2,Mod(1,6840)] 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("6840.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6840.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,3,0,-1,0,0,0,0,0,-11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(54.6176749826\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 6840.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +1.19656 q^{7} -4.97858 q^{11} -6.63565 q^{13} -1.48929 q^{17} +1.00000 q^{19} +0.510711 q^{23} +1.00000 q^{25} +7.88240 q^{29} -2.97858 q^{31} +1.19656 q^{35} -7.14637 q^{37} -1.66442 q^{41} -6.39312 q^{43} +9.95715 q^{47} -5.56825 q^{49} +11.4219 q^{53} -4.97858 q^{55} +11.8396 q^{59} +3.66442 q^{61} -6.63565 q^{65} +7.61423 q^{67} +13.8396 q^{73} -5.95715 q^{77} +12.6858 q^{79} +8.68585 q^{83} -1.48929 q^{85} +4.87819 q^{89} -7.93994 q^{91} +1.00000 q^{95} -6.81079 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - q^{7} - 11 q^{13} + 3 q^{17} + 3 q^{19} + 9 q^{23} + 3 q^{25} + 7 q^{29} + 6 q^{31} - q^{35} - 20 q^{37} + 22 q^{41} - 10 q^{43} + 12 q^{49} + 7 q^{53} - 11 q^{59} - 16 q^{61} - 11 q^{65}+ \cdots + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.19656 0.452256 0.226128 − 0.974098i \(-0.427393\pi\)
0.226128 + 0.974098i \(0.427393\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.97858 −1.50110 −0.750549 − 0.660815i \(-0.770211\pi\)
−0.750549 + 0.660815i \(0.770211\pi\)
\(12\) 0 0
\(13\) −6.63565 −1.84040 −0.920200 − 0.391449i \(-0.871974\pi\)
−0.920200 + 0.391449i \(0.871974\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.48929 −0.361206 −0.180603 − 0.983556i \(-0.557805\pi\)
−0.180603 + 0.983556i \(0.557805\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.510711 0.106491 0.0532453 − 0.998581i \(-0.483043\pi\)
0.0532453 + 0.998581i \(0.483043\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.88240 1.46373 0.731863 − 0.681452i \(-0.238651\pi\)
0.731863 + 0.681452i \(0.238651\pi\)
\(30\) 0 0
\(31\) −2.97858 −0.534968 −0.267484 − 0.963562i \(-0.586192\pi\)
−0.267484 + 0.963562i \(0.586192\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.19656 0.202255
\(36\) 0 0
\(37\) −7.14637 −1.17486 −0.587428 − 0.809277i \(-0.699859\pi\)
−0.587428 + 0.809277i \(0.699859\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.66442 −0.259939 −0.129970 − 0.991518i \(-0.541488\pi\)
−0.129970 + 0.991518i \(0.541488\pi\)
\(42\) 0 0
\(43\) −6.39312 −0.974941 −0.487470 − 0.873139i \(-0.662080\pi\)
−0.487470 + 0.873139i \(0.662080\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.95715 1.45240 0.726200 − 0.687483i \(-0.241285\pi\)
0.726200 + 0.687483i \(0.241285\pi\)
\(48\) 0 0
\(49\) −5.56825 −0.795464
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.4219 1.56892 0.784458 − 0.620182i \(-0.212941\pi\)
0.784458 + 0.620182i \(0.212941\pi\)
\(54\) 0 0
\(55\) −4.97858 −0.671311
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.8396 1.54138 0.770690 − 0.637211i \(-0.219912\pi\)
0.770690 + 0.637211i \(0.219912\pi\)
\(60\) 0 0
\(61\) 3.66442 0.469181 0.234591 − 0.972094i \(-0.424625\pi\)
0.234591 + 0.972094i \(0.424625\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.63565 −0.823052
\(66\) 0 0
\(67\) 7.61423 0.930226 0.465113 − 0.885251i \(-0.346014\pi\)
0.465113 + 0.885251i \(0.346014\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 13.8396 1.61980 0.809899 − 0.586570i \(-0.199522\pi\)
0.809899 + 0.586570i \(0.199522\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.95715 −0.678881
\(78\) 0 0
\(79\) 12.6858 1.42727 0.713635 − 0.700518i \(-0.247048\pi\)
0.713635 + 0.700518i \(0.247048\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.68585 0.953395 0.476698 − 0.879067i \(-0.341834\pi\)
0.476698 + 0.879067i \(0.341834\pi\)
\(84\) 0 0
\(85\) −1.48929 −0.161536
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.87819 0.517087 0.258544 − 0.966000i \(-0.416757\pi\)
0.258544 + 0.966000i \(0.416757\pi\)
\(90\) 0 0
\(91\) −7.93994 −0.832332
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −6.81079 −0.691531 −0.345765 − 0.938321i \(-0.612381\pi\)
−0.345765 + 0.938321i \(0.612381\pi\)
\(98\) 0 0
\(99\) 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 6840.2.a.bm.1.2 3
3.2 odd 2 760.2.a.i.1.1 ✓ 3
12.11 even 2 1520.2.a.q.1.3 3
15.2 even 4 3800.2.d.n.3649.6 6
15.8 even 4 3800.2.d.n.3649.1 6
15.14 odd 2 3800.2.a.w.1.3 3
24.5 odd 2 6080.2.a.bx.1.3 3
24.11 even 2 6080.2.a.br.1.1 3
60.59 even 2 7600.2.a.bp.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.1 ✓ 3 3.2 odd 2
1520.2.a.q.1.3 3 12.11 even 2
3800.2.a.w.1.3 3 15.14 odd 2
3800.2.d.n.3649.1 6 15.8 even 4
3800.2.d.n.3649.6 6 15.2 even 4
6080.2.a.br.1.1 3 24.11 even 2
6080.2.a.bx.1.3 3 24.5 odd 2
6840.2.a.bm.1.2 3 1.1 even 1 trivial
7600.2.a.bp.1.1 3 60.59 even 2