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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1860,2,Mod(929,1860)] 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("1860.929"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1860.i (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.8521747760\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 929.12
Character \(\chi\) \(=\) 1860.929
Dual form 1860.2.i.b.929.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.37690 + 1.05079i) q^{3} +(2.02481 + 0.948759i) q^{5} +3.69636i q^{7} +(0.791696 - 2.89365i) q^{9} +2.12856 q^{11} -5.24567 q^{13} +(-3.78490 + 0.821298i) q^{15} +0.528347i q^{17} -6.33048 q^{19} +(-3.88408 - 5.08950i) q^{21} +1.52572i q^{23} +(3.19971 + 3.84211i) q^{25} +(1.95052 + 4.81617i) q^{27} -9.72778 q^{29} +(5.56700 + 0.0925425i) q^{31} +(-2.93081 + 2.23666i) q^{33} +(-3.50695 + 7.48442i) q^{35} -2.30913 q^{37} +(7.22275 - 5.51207i) q^{39} +2.60377i q^{41} -0.258635 q^{43} +(4.34841 - 5.10796i) q^{45} +1.54591 q^{47} -6.66305 q^{49} +(-0.555180 - 0.727480i) q^{51} -2.24918i q^{53} +(4.30993 + 2.01949i) q^{55} +(8.71642 - 6.65198i) q^{57} -7.92457i q^{59} -14.0648i q^{61} +(10.6960 + 2.92639i) q^{63} +(-10.6215 - 4.97687i) q^{65} +11.0285i q^{67} +(-1.60321 - 2.10076i) q^{69} +9.46231i q^{71} +5.21421 q^{73} +(-8.44292 - 1.92799i) q^{75} +7.86791i q^{77} -4.55416i q^{79} +(-7.74643 - 4.58179i) q^{81} +14.6947i q^{83} +(-0.501274 + 1.06980i) q^{85} +(13.3942 - 10.2218i) q^{87} -10.0978 q^{89} -19.3898i q^{91} +(-7.76243 + 5.72230i) q^{93} +(-12.8180 - 6.00610i) q^{95} -10.5063i q^{97} +(1.68517 - 6.15931i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{9} - 8 q^{19} + 64 q^{25} + 24 q^{31} + 8 q^{39} - 16 q^{45} - 160 q^{49} + 68 q^{51} - 24 q^{69} + 52 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1860\mathbb{Z}\right)^\times\).

\(n\) \(931\) \(1117\) \(1241\) \(1801\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

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
\(3\) −1.37690 + 1.05079i −0.794952 + 0.606672i
\(4\) 0 0
\(5\) 2.02481 + 0.948759i 0.905523 + 0.424298i
\(6\) 0 0
\(7\) 3.69636i 1.39709i 0.715566 + 0.698546i \(0.246169\pi\)
−0.715566 + 0.698546i \(0.753831\pi\)
\(8\) 0 0
\(9\) 0.791696 − 2.89365i 0.263899 − 0.964550i
\(10\) 0 0
\(11\) 2.12856 0.641785 0.320892 − 0.947116i \(-0.396017\pi\)
0.320892 + 0.947116i \(0.396017\pi\)
\(12\) 0 0
\(13\) −5.24567 −1.45489 −0.727443 − 0.686168i \(-0.759291\pi\)
−0.727443 + 0.686168i \(0.759291\pi\)
\(14\) 0 0
\(15\) −3.78490 + 0.821298i −0.977257 + 0.212058i
\(16\) 0 0
\(17\) 0.528347i 0.128143i 0.997945 + 0.0640715i \(0.0204086\pi\)
−0.997945 + 0.0640715i \(0.979591\pi\)
\(18\) 0 0
\(19\) −6.33048 −1.45231 −0.726156 − 0.687530i \(-0.758695\pi\)
−0.726156 + 0.687530i \(0.758695\pi\)
\(20\) 0 0
\(21\) −3.88408 − 5.08950i −0.847576 − 1.11062i
\(22\) 0 0
\(23\) 1.52572i 0.318134i 0.987268 + 0.159067i \(0.0508487\pi\)
−0.987268 + 0.159067i \(0.949151\pi\)
\(24\) 0 0
\(25\) 3.19971 + 3.84211i 0.639942 + 0.768423i
\(26\) 0 0
\(27\) 1.95052 + 4.81617i 0.375379 + 0.926872i
\(28\) 0 0
\(29\) −9.72778 −1.80640 −0.903202 − 0.429216i \(-0.858790\pi\)
−0.903202 + 0.429216i \(0.858790\pi\)
\(30\) 0 0
\(31\) 5.56700 + 0.0925425i 0.999862 + 0.0166211i
\(32\) 0 0
\(33\) −2.93081 + 2.23666i −0.510188 + 0.389353i
\(34\) 0 0
\(35\) −3.50695 + 7.48442i −0.592783 + 1.26510i
\(36\) 0 0
\(37\) −2.30913 −0.379619 −0.189809 − 0.981821i \(-0.560787\pi\)
−0.189809 + 0.981821i \(0.560787\pi\)
\(38\) 0 0
\(39\) 7.22275 − 5.51207i 1.15657 − 0.882638i
\(40\) 0 0
\(41\) 2.60377i 0.406640i 0.979112 + 0.203320i \(0.0651732\pi\)
−0.979112 + 0.203320i \(0.934827\pi\)
\(42\) 0 0
\(43\) −0.258635 −0.0394415 −0.0197208 − 0.999806i \(-0.506278\pi\)
−0.0197208 + 0.999806i \(0.506278\pi\)
\(44\) 0 0
\(45\) 4.34841 − 5.10796i 0.648223 − 0.761450i
\(46\) 0 0
\(47\) 1.54591 0.225494 0.112747 − 0.993624i \(-0.464035\pi\)
0.112747 + 0.993624i \(0.464035\pi\)
\(48\) 0 0
\(49\) −6.66305 −0.951864
\(50\) 0 0
\(51\) −0.555180 − 0.727480i −0.0777407 − 0.101868i
\(52\) 0 0
\(53\) − 2.24918i − 0.308949i −0.987997 − 0.154474i \(-0.950632\pi\)
0.987997 − 0.154474i \(-0.0493684\pi\)
\(54\) 0 0
\(55\) 4.30993 + 2.01949i 0.581150 + 0.272308i
\(56\) 0 0
\(57\) 8.71642 − 6.65198i 1.15452 − 0.881076i
\(58\) 0 0
\(59\) − 7.92457i − 1.03169i −0.856682 − 0.515846i \(-0.827478\pi\)
0.856682 − 0.515846i \(-0.172522\pi\)
\(60\) 0 0
\(61\) − 14.0648i − 1.80081i −0.435054 − 0.900404i \(-0.643271\pi\)
0.435054 − 0.900404i \(-0.356729\pi\)
\(62\) 0 0
\(63\) 10.6960 + 2.92639i 1.34756 + 0.368691i
\(64\) 0 0
\(65\) −10.6215 − 4.97687i −1.31743 − 0.617305i
\(66\) 0 0
\(67\) 11.0285i 1.34734i 0.739032 + 0.673670i \(0.235283\pi\)
−0.739032 + 0.673670i \(0.764717\pi\)
\(68\) 0 0
\(69\) −1.60321 − 2.10076i −0.193003 − 0.252902i
\(70\) 0 0
\(71\) 9.46231i 1.12297i 0.827487 + 0.561485i \(0.189770\pi\)
−0.827487 + 0.561485i \(0.810230\pi\)
\(72\) 0 0
\(73\) 5.21421 0.610277 0.305139 − 0.952308i \(-0.401297\pi\)
0.305139 + 0.952308i \(0.401297\pi\)
\(74\) 0 0
\(75\) −8.44292 − 1.92799i −0.974904 − 0.222625i
\(76\) 0 0
\(77\) 7.86791i 0.896632i
\(78\) 0 0
\(79\) − 4.55416i − 0.512383i −0.966626 − 0.256192i \(-0.917532\pi\)
0.966626 − 0.256192i \(-0.0824678\pi\)
\(80\) 0 0
\(81\) −7.74643 − 4.58179i −0.860715 − 0.509087i
\(82\) 0 0
\(83\) 14.6947i 1.61295i 0.591267 + 0.806476i \(0.298628\pi\)
−0.591267 + 0.806476i \(0.701372\pi\)
\(84\) 0 0
\(85\) −0.501274 + 1.06980i −0.0543708 + 0.116036i
\(86\) 0 0
\(87\) 13.3942 − 10.2218i 1.43600 − 1.09589i
\(88\) 0 0
\(89\) −10.0978 −1.07036 −0.535182 − 0.844737i \(-0.679757\pi\)
−0.535182 + 0.844737i \(0.679757\pi\)
\(90\) 0 0
\(91\) − 19.3898i − 2.03261i
\(92\) 0 0
\(93\) −7.76243 + 5.72230i −0.804926 + 0.593375i
\(94\) 0 0
\(95\) −12.8180 − 6.00610i −1.31510 − 0.616213i
\(96\) 0 0
\(97\) − 10.5063i − 1.06676i −0.845877 − 0.533378i \(-0.820922\pi\)
0.845877 − 0.533378i \(-0.179078\pi\)
\(98\) 0 0
\(99\) 1.68517 − 6.15931i 0.169366 − 0.619034i
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 1860.2.i.b.929.12 yes 48
3.2 odd 2 inner 1860.2.i.b.929.9 ✓ 48
5.4 even 2 inner 1860.2.i.b.929.37 yes 48
15.14 odd 2 inner 1860.2.i.b.929.40 yes 48
31.30 odd 2 inner 1860.2.i.b.929.38 yes 48
93.92 even 2 inner 1860.2.i.b.929.39 yes 48
155.154 odd 2 inner 1860.2.i.b.929.11 yes 48
465.464 even 2 inner 1860.2.i.b.929.10 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1860.2.i.b.929.9 ✓ 48 3.2 odd 2 inner
1860.2.i.b.929.10 yes 48 465.464 even 2 inner
1860.2.i.b.929.11 yes 48 155.154 odd 2 inner
1860.2.i.b.929.12 yes 48 1.1 even 1 trivial
1860.2.i.b.929.37 yes 48 5.4 even 2 inner
1860.2.i.b.929.38 yes 48 31.30 odd 2 inner
1860.2.i.b.929.39 yes 48 93.92 even 2 inner
1860.2.i.b.929.40 yes 48 15.14 odd 2 inner