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 1.1
Root \(-1.81361\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} -4.91638 q^{7} -0.578337 q^{11} -6.39194 q^{13} +0.710831 q^{17} +1.00000 q^{19} +2.71083 q^{23} +1.00000 q^{25} -6.54359 q^{29} +1.42166 q^{31} -4.91638 q^{35} -9.10278 q^{37} +11.0489 q^{41} +5.83276 q^{43} +1.15667 q^{47} +17.1708 q^{49} -13.2736 q^{53} -0.578337 q^{55} -11.3869 q^{59} -9.04888 q^{61} -6.39194 q^{65} +2.97028 q^{67} -9.38692 q^{73} +2.84333 q^{77} +4.37279 q^{79} +0.372787 q^{83} +0.710831 q^{85} +16.6167 q^{89} +31.4252 q^{91} +1.00000 q^{95} +3.94610 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - q^{7} - 11 q^{13} + 3 q^{17} + 3 q^{19} + 9 q^{23} + 3 q^{25} + 7 q^{29} + 6 q^{31} - q^{35} - 20 q^{37} + 22 q^{41} - 10 q^{43} + 12 q^{49} + 7 q^{53} - 11 q^{59} - 16 q^{61} - 11 q^{65}+ \cdots + 8 q^{97}+O(q^{100}) \) 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
\(7\) −4.91638 −1.85822 −0.929109 − 0.369807i \(-0.879424\pi\)
−0.929109 + 0.369807i \(0.879424\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.578337 −0.174375 −0.0871876 − 0.996192i \(-0.527788\pi\)
−0.0871876 + 0.996192i \(0.527788\pi\)
\(12\) 0 0
\(13\) −6.39194 −1.77281 −0.886403 − 0.462914i \(-0.846804\pi\)
−0.886403 + 0.462914i \(0.846804\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.710831 0.172402 0.0862010 − 0.996278i \(-0.472527\pi\)
0.0862010 + 0.996278i \(0.472527\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.71083 0.565247 0.282624 − 0.959231i \(-0.408795\pi\)
0.282624 + 0.959231i \(0.408795\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.54359 −1.21512 −0.607558 − 0.794276i \(-0.707851\pi\)
−0.607558 + 0.794276i \(0.707851\pi\)
\(30\) 0 0
\(31\) 1.42166 0.255338 0.127669 − 0.991817i \(-0.459250\pi\)
0.127669 + 0.991817i \(0.459250\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.91638 −0.831020
\(36\) 0 0
\(37\) −9.10278 −1.49649 −0.748243 − 0.663424i \(-0.769103\pi\)
−0.748243 + 0.663424i \(0.769103\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.0489 1.72554 0.862772 − 0.505593i \(-0.168726\pi\)
0.862772 + 0.505593i \(0.168726\pi\)
\(42\) 0 0
\(43\) 5.83276 0.889488 0.444744 − 0.895658i \(-0.353295\pi\)
0.444744 + 0.895658i \(0.353295\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.15667 0.168718 0.0843591 − 0.996435i \(-0.473116\pi\)
0.0843591 + 0.996435i \(0.473116\pi\)
\(48\) 0 0
\(49\) 17.1708 2.45297
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −13.2736 −1.82327 −0.911633 − 0.411004i \(-0.865178\pi\)
−0.911633 + 0.411004i \(0.865178\pi\)
\(54\) 0 0
\(55\) −0.578337 −0.0779830
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.3869 −1.48245 −0.741225 − 0.671256i \(-0.765755\pi\)
−0.741225 + 0.671256i \(0.765755\pi\)
\(60\) 0 0
\(61\) −9.04888 −1.15859 −0.579295 − 0.815118i \(-0.696672\pi\)
−0.579295 + 0.815118i \(0.696672\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.39194 −0.792823
\(66\) 0 0
\(67\) 2.97028 0.362878 0.181439 − 0.983402i \(-0.441925\pi\)
0.181439 + 0.983402i \(0.441925\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −9.38692 −1.09866 −0.549328 − 0.835607i \(-0.685116\pi\)
−0.549328 + 0.835607i \(0.685116\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.84333 0.324027
\(78\) 0 0
\(79\) 4.37279 0.491977 0.245988 − 0.969273i \(-0.420887\pi\)
0.245988 + 0.969273i \(0.420887\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.372787 0.0409187 0.0204593 − 0.999791i \(-0.493487\pi\)
0.0204593 + 0.999791i \(0.493487\pi\)
\(84\) 0 0
\(85\) 0.710831 0.0771005
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.6167 1.76136 0.880681 − 0.473710i \(-0.157086\pi\)
0.880681 + 0.473710i \(0.157086\pi\)
\(90\) 0 0
\(91\) 31.4252 3.29426
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 3.94610 0.400666 0.200333 − 0.979728i \(-0.435798\pi\)
0.200333 + 0.979728i \(0.435798\pi\)
\(98\) 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.1.1 3
3.2 odd 2 760.2.a.i.1.3 ✓ 3
12.11 even 2 1520.2.a.q.1.1 3
15.2 even 4 3800.2.d.n.3649.2 6
15.8 even 4 3800.2.d.n.3649.5 6
15.14 odd 2 3800.2.a.w.1.1 3
24.5 odd 2 6080.2.a.bx.1.1 3
24.11 even 2 6080.2.a.br.1.3 3
60.59 even 2 7600.2.a.bp.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.3 ✓ 3 3.2 odd 2
1520.2.a.q.1.1 3 12.11 even 2
3800.2.a.w.1.1 3 15.14 odd 2
3800.2.d.n.3649.2 6 15.2 even 4
3800.2.d.n.3649.5 6 15.8 even 4
6080.2.a.br.1.3 3 24.11 even 2
6080.2.a.bx.1.1 3 24.5 odd 2
6840.2.a.bm.1.1 3 1.1 even 1 trivial
7600.2.a.bp.1.3 3 60.59 even 2