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 3761.17
Character \(\chi\) \(=\) 6840.3761
Dual form 6840.2.r.a.3761.18

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{5} -0.0196792 q^{7} -1.52125i q^{11} +1.31603i q^{13} +3.56187i q^{17} +(-3.58217 - 2.48355i) q^{19} -2.94390i q^{23} -1.00000 q^{25} +3.61087 q^{29} -5.57722i q^{31} +0.0196792i q^{35} +1.59074i q^{37} -9.70402 q^{41} +8.05211 q^{43} +6.62374i q^{47} -6.99961 q^{49} +2.61659 q^{53} -1.52125 q^{55} +10.0038 q^{59} -10.0837 q^{61} +1.31603 q^{65} -3.94872i q^{67} -8.97732 q^{71} -3.08046 q^{73} +0.0299369i q^{77} -1.89948i q^{79} +5.48778i q^{83} +3.56187 q^{85} -13.7225 q^{89} -0.0258983i q^{91} +(-2.48355 + 3.58217i) q^{95} -14.3328i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q - 8 q^{7} - 4 q^{19} - 40 q^{25} - 24 q^{29} + 8 q^{41} + 8 q^{43} + 64 q^{49} - 32 q^{53} - 16 q^{59} - 16 q^{61} - 16 q^{65} + 32 q^{71} + 24 q^{89} - 4 q^{95}+O(q^{100}) \) 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
\(7\) −0.0196792 −0.00743802 −0.00371901 − 0.999993i \(-0.501184\pi\)
−0.00371901 + 0.999993i \(0.501184\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.52125i − 0.458674i −0.973347 − 0.229337i \(-0.926344\pi\)
0.973347 − 0.229337i \(-0.0736559\pi\)
\(12\) 0 0
\(13\) 1.31603i 0.365000i 0.983206 + 0.182500i \(0.0584190\pi\)
−0.983206 + 0.182500i \(0.941581\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.56187i 0.863881i 0.901902 + 0.431940i \(0.142171\pi\)
−0.901902 + 0.431940i \(0.857829\pi\)
\(18\) 0 0
\(19\) −3.58217 − 2.48355i −0.821807 − 0.569766i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 2.94390i − 0.613846i −0.951734 − 0.306923i \(-0.900701\pi\)
0.951734 − 0.306923i \(-0.0992994\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.61087 0.670522 0.335261 − 0.942125i \(-0.391176\pi\)
0.335261 + 0.942125i \(0.391176\pi\)
\(30\) 0 0
\(31\) − 5.57722i − 1.00170i −0.865535 − 0.500849i \(-0.833021\pi\)
0.865535 − 0.500849i \(-0.166979\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.0196792i 0.00332638i
\(36\) 0 0
\(37\) 1.59074i 0.261516i 0.991414 + 0.130758i \(0.0417412\pi\)
−0.991414 + 0.130758i \(0.958259\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.70402 −1.51551 −0.757756 − 0.652538i \(-0.773704\pi\)
−0.757756 + 0.652538i \(0.773704\pi\)
\(42\) 0 0
\(43\) 8.05211 1.22794 0.613968 − 0.789331i \(-0.289573\pi\)
0.613968 + 0.789331i \(0.289573\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.62374i 0.966171i 0.875573 + 0.483086i \(0.160484\pi\)
−0.875573 + 0.483086i \(0.839516\pi\)
\(48\) 0 0
\(49\) −6.99961 −0.999945
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.61659 0.359416 0.179708 − 0.983720i \(-0.442485\pi\)
0.179708 + 0.983720i \(0.442485\pi\)
\(54\) 0 0
\(55\) −1.52125 −0.205125
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.0038 1.30238 0.651192 − 0.758913i \(-0.274269\pi\)
0.651192 + 0.758913i \(0.274269\pi\)
\(60\) 0 0
\(61\) −10.0837 −1.29109 −0.645546 − 0.763721i \(-0.723370\pi\)
−0.645546 + 0.763721i \(0.723370\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.31603 0.163233
\(66\) 0 0
\(67\) − 3.94872i − 0.482413i −0.970474 − 0.241207i \(-0.922457\pi\)
0.970474 − 0.241207i \(-0.0775432\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.97732 −1.06541 −0.532706 − 0.846301i \(-0.678825\pi\)
−0.532706 + 0.846301i \(0.678825\pi\)
\(72\) 0 0
\(73\) −3.08046 −0.360541 −0.180270 − 0.983617i \(-0.557697\pi\)
−0.180270 + 0.983617i \(0.557697\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.0299369i 0.00341163i
\(78\) 0 0
\(79\) − 1.89948i − 0.213708i −0.994275 − 0.106854i \(-0.965922\pi\)
0.994275 − 0.106854i \(-0.0340778\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.48778i 0.602362i 0.953567 + 0.301181i \(0.0973808\pi\)
−0.953567 + 0.301181i \(0.902619\pi\)
\(84\) 0 0
\(85\) 3.56187 0.386339
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −13.7225 −1.45458 −0.727291 − 0.686329i \(-0.759221\pi\)
−0.727291 + 0.686329i \(0.759221\pi\)
\(90\) 0 0
\(91\) − 0.0258983i − 0.00271488i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.48355 + 3.58217i −0.254807 + 0.367523i
\(96\) 0 0
\(97\) − 14.3328i − 1.45528i −0.685961 − 0.727639i \(-0.740618\pi\)
0.685961 − 0.727639i \(-0.259382\pi\)
\(98\) 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.3761.17 ✓ 40
3.2 odd 2 6840.2.r.b.3761.24 yes 40
19.18 odd 2 6840.2.r.b.3761.23 yes 40
57.56 even 2 inner 6840.2.r.a.3761.18 yes 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6840.2.r.a.3761.17 ✓ 40 1.1 even 1 trivial
6840.2.r.a.3761.18 yes 40 57.56 even 2 inner
6840.2.r.b.3761.23 yes 40 19.18 odd 2
6840.2.r.b.3761.24 yes 40 3.2 odd 2