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(3761,6840)] 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("6840.3761"); 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, 1, 0, 1])) 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.r (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [40,0,0,0,0,0,-8,0,0,0,0,0,0,0,0,0,0,0,-4,0,0,0,0,0,-40,0,0,0, -24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(54.6176749826\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3761.16
Character \(\chi\) \(=\) 6840.3761
Dual form 6840.2.r.a.3761.15

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{5} +2.46536 q^{7} -1.44253i q^{11} +1.57852i q^{13} +6.94693i q^{17} +(-0.533717 + 4.32610i) q^{19} +3.74828i q^{23} -1.00000 q^{25} +4.38868 q^{29} -9.23576i q^{31} +2.46536i q^{35} +3.43570i q^{37} +5.54404 q^{41} -11.7551 q^{43} +4.85456i q^{47} -0.921997 q^{49} +6.64314 q^{53} +1.44253 q^{55} -8.07310 q^{59} +0.246147 q^{61} -1.57852 q^{65} +3.28800i q^{67} -0.181507 q^{71} -9.19331 q^{73} -3.55635i q^{77} +1.05686i q^{79} +13.1132i q^{83} -6.94693 q^{85} -10.2511 q^{89} +3.89163i q^{91} +(-4.32610 - 0.533717i) q^{95} -1.14988i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 8 q^{7} - 4 q^{19} - 40 q^{25} - 24 q^{29} + 8 q^{41} + 8 q^{43} + 64 q^{49} - 32 q^{53} - 16 q^{59} - 16 q^{61} - 16 q^{65} + 32 q^{71} + 24 q^{89} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6840\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(2737\) \(3421\) \(5321\) \(6481\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\) \(-1\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) 2.46536 0.931819 0.465909 − 0.884832i \(-0.345727\pi\)
0.465909 + 0.884832i \(0.345727\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.44253i − 0.434938i −0.976067 − 0.217469i \(-0.930220\pi\)
0.976067 − 0.217469i \(-0.0697801\pi\)
\(12\) 0 0
\(13\) 1.57852i 0.437804i 0.975747 + 0.218902i \(0.0702475\pi\)
−0.975747 + 0.218902i \(0.929752\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.94693i 1.68488i 0.538792 + 0.842439i \(0.318881\pi\)
−0.538792 + 0.842439i \(0.681119\pi\)
\(18\) 0 0
\(19\) −0.533717 + 4.32610i −0.122443 + 0.992476i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.74828i 0.781570i 0.920482 + 0.390785i \(0.127796\pi\)
−0.920482 + 0.390785i \(0.872204\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.38868 0.814958 0.407479 − 0.913215i \(-0.366408\pi\)
0.407479 + 0.913215i \(0.366408\pi\)
\(30\) 0 0
\(31\) − 9.23576i − 1.65879i −0.558661 − 0.829396i \(-0.688685\pi\)
0.558661 − 0.829396i \(-0.311315\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.46536i 0.416722i
\(36\) 0 0
\(37\) 3.43570i 0.564826i 0.959293 + 0.282413i \(0.0911348\pi\)
−0.959293 + 0.282413i \(0.908865\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.54404 0.865834 0.432917 − 0.901434i \(-0.357484\pi\)
0.432917 + 0.901434i \(0.357484\pi\)
\(42\) 0 0
\(43\) −11.7551 −1.79263 −0.896317 − 0.443414i \(-0.853767\pi\)
−0.896317 + 0.443414i \(0.853767\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.85456i 0.708110i 0.935225 + 0.354055i \(0.115197\pi\)
−0.935225 + 0.354055i \(0.884803\pi\)
\(48\) 0 0
\(49\) −0.921997 −0.131714
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.64314 0.912505 0.456252 − 0.889850i \(-0.349191\pi\)
0.456252 + 0.889850i \(0.349191\pi\)
\(54\) 0 0
\(55\) 1.44253 0.194510
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.07310 −1.05103 −0.525514 − 0.850785i \(-0.676127\pi\)
−0.525514 + 0.850785i \(0.676127\pi\)
\(60\) 0 0
\(61\) 0.246147 0.0315158 0.0157579 − 0.999876i \(-0.494984\pi\)
0.0157579 + 0.999876i \(0.494984\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.57852 −0.195792
\(66\) 0 0
\(67\) 3.28800i 0.401694i 0.979623 + 0.200847i \(0.0643694\pi\)
−0.979623 + 0.200847i \(0.935631\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.181507 −0.0215410 −0.0107705 − 0.999942i \(-0.503428\pi\)
−0.0107705 + 0.999942i \(0.503428\pi\)
\(72\) 0 0
\(73\) −9.19331 −1.07600 −0.537998 − 0.842946i \(-0.680819\pi\)
−0.537998 + 0.842946i \(0.680819\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 3.55635i − 0.405283i
\(78\) 0 0
\(79\) 1.05686i 0.118906i 0.998231 + 0.0594530i \(0.0189357\pi\)
−0.998231 + 0.0594530i \(0.981064\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.1132i 1.43936i 0.694306 + 0.719680i \(0.255712\pi\)
−0.694306 + 0.719680i \(0.744288\pi\)
\(84\) 0 0
\(85\) −6.94693 −0.753500
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.2511 −1.08662 −0.543308 − 0.839534i \(-0.682828\pi\)
−0.543308 + 0.839534i \(0.682828\pi\)
\(90\) 0 0
\(91\) 3.89163i 0.407954i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.32610 − 0.533717i −0.443849 − 0.0547582i
\(96\) 0 0
\(97\) − 1.14988i − 0.116753i −0.998295 − 0.0583766i \(-0.981408\pi\)
0.998295 − 0.0583766i \(-0.0185924\pi\)
\(98\) 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.r.a.3761.16 yes 40
3.2 odd 2 6840.2.r.b.3761.25 yes 40
19.18 odd 2 6840.2.r.b.3761.26 yes 40
57.56 even 2 inner 6840.2.r.a.3761.15 ✓ 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6840.2.r.a.3761.15 ✓ 40 57.56 even 2 inner
6840.2.r.a.3761.16 yes 40 1.1 even 1 trivial
6840.2.r.b.3761.25 yes 40 3.2 odd 2
6840.2.r.b.3761.26 yes 40 19.18 odd 2