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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [150,16,Mod(49,150)] 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("150.49"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 150.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-32768,0,559872,0,0,-9565938,0,220510104] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(214.040257650\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 150.49
Dual form 150.16.c.i.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q-128.000i q^{2} +2187.00i q^{3} -16384.0 q^{4} +279936. q^{6} +2.02506e6i q^{7} +2.09715e6i q^{8} -4.78297e6 q^{9} +1.10255e8 q^{11} -3.58318e7i q^{12} -5.60479e7i q^{13} +2.59207e8 q^{14} +2.68435e8 q^{16} -1.93010e9i q^{17} +6.12220e8i q^{18} -2.16319e9 q^{19} -4.42880e9 q^{21} -1.41126e10i q^{22} -6.22897e9i q^{23} -4.58647e9 q^{24} -7.17413e9 q^{26} -1.04604e10i q^{27} -3.31785e10i q^{28} -6.47437e10 q^{29} -2.02376e10 q^{31} -3.43597e10i q^{32} +2.41128e11i q^{33} -2.47053e11 q^{34} +7.83642e10 q^{36} +4.88968e11i q^{37} +2.76888e11i q^{38} +1.22577e11 q^{39} -7.72359e11 q^{41} +5.66886e11i q^{42} -1.30677e12i q^{43} -1.80642e12 q^{44} -7.97309e11 q^{46} +3.35182e12i q^{47} +5.87068e11i q^{48} +6.46710e11 q^{49} +4.22114e12 q^{51} +9.18288e11i q^{52} -9.38781e12i q^{53} -1.33893e12 q^{54} -4.24685e12 q^{56} -4.73089e12i q^{57} +8.28720e12i q^{58} -2.89304e13 q^{59} +4.23931e13 q^{61} +2.59041e12i q^{62} -9.68578e12i q^{63} -4.39805e12 q^{64} +3.08644e13 q^{66} -5.22472e13i q^{67} +3.16228e13i q^{68} +1.36228e13 q^{69} -2.71945e13 q^{71} -1.00306e13i q^{72} +9.16042e13i q^{73} +6.25879e13 q^{74} +3.54417e13 q^{76} +2.23273e14i q^{77} -1.56898e13i q^{78} -6.28821e13 q^{79} +2.28768e13 q^{81} +9.88620e13i q^{82} +2.23567e14i q^{83} +7.25614e13 q^{84} -1.67266e14 q^{86} -1.41595e14i q^{87} +2.31222e14i q^{88} -5.54199e14 q^{89} +1.13500e14 q^{91} +1.02056e14i q^{92} -4.42597e13i q^{93} +4.29033e14 q^{94} +7.51447e13 q^{96} -1.38887e15i q^{97} -8.27788e13i q^{98} -5.27346e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32768 q^{4} + 559872 q^{6} - 9565938 q^{9} + 220510104 q^{11} + 518414336 q^{14} + 536870912 q^{16} - 4326376360 q^{19} - 8857594944 q^{21} - 9172942848 q^{24} - 14348252672 q^{26} - 129487438140 q^{29}+ \cdots - 10\!\cdots\!76 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 128.000i − 0.707107i
\(3\) 2187.00i 0.577350i
\(4\) −16384.0 −0.500000
\(5\) 0 0
\(6\) 279936. 0.408248
\(7\) 2.02506e6i 0.929398i 0.885469 + 0.464699i \(0.153837\pi\)
−0.885469 + 0.464699i \(0.846163\pi\)
\(8\) 2.09715e6i 0.353553i
\(9\) −4.78297e6 −0.333333
\(10\) 0 0
\(11\) 1.10255e8 1.70590 0.852950 − 0.521993i \(-0.174811\pi\)
0.852950 + 0.521993i \(0.174811\pi\)
\(12\) − 3.58318e7i − 0.288675i
\(13\) − 5.60479e7i − 0.247733i −0.992299 − 0.123867i \(-0.960471\pi\)
0.992299 − 0.123867i \(-0.0395295\pi\)
\(14\) 2.59207e8 0.657184
\(15\) 0 0
\(16\) 2.68435e8 0.250000
\(17\) − 1.93010e9i − 1.14081i −0.821363 − 0.570406i \(-0.806786\pi\)
0.821363 − 0.570406i \(-0.193214\pi\)
\(18\) 6.12220e8i 0.235702i
\(19\) −2.16319e9 −0.555191 −0.277595 − 0.960698i \(-0.589537\pi\)
−0.277595 + 0.960698i \(0.589537\pi\)
\(20\) 0 0
\(21\) −4.42880e9 −0.536588
\(22\) − 1.41126e10i − 1.20625i
\(23\) − 6.22897e9i − 0.381468i −0.981642 − 0.190734i \(-0.938913\pi\)
0.981642 − 0.190734i \(-0.0610867\pi\)
\(24\) −4.58647e9 −0.204124
\(25\) 0 0
\(26\) −7.17413e9 −0.175174
\(27\) − 1.04604e10i − 0.192450i
\(28\) − 3.31785e10i − 0.464699i
\(29\) −6.47437e10 −0.696968 −0.348484 − 0.937315i \(-0.613303\pi\)
−0.348484 + 0.937315i \(0.613303\pi\)
\(30\) 0 0
\(31\) −2.02376e10 −0.132113 −0.0660567 − 0.997816i \(-0.521042\pi\)
−0.0660567 + 0.997816i \(0.521042\pi\)
\(32\) − 3.43597e10i − 0.176777i
\(33\) 2.41128e11i 0.984901i
\(34\) −2.47053e11 −0.806676
\(35\) 0 0
\(36\) 7.83642e10 0.166667
\(37\) 4.88968e11i 0.846773i 0.905949 + 0.423387i \(0.139159\pi\)
−0.905949 + 0.423387i \(0.860841\pi\)
\(38\) 2.76888e11i 0.392579i
\(39\) 1.22577e11 0.143029
\(40\) 0 0
\(41\) −7.72359e11 −0.619356 −0.309678 − 0.950841i \(-0.600221\pi\)
−0.309678 + 0.950841i \(0.600221\pi\)
\(42\) 5.66886e11i 0.379425i
\(43\) − 1.30677e12i − 0.733136i −0.930391 − 0.366568i \(-0.880533\pi\)
0.930391 − 0.366568i \(-0.119467\pi\)
\(44\) −1.80642e12 −0.852950
\(45\) 0 0
\(46\) −7.97309e11 −0.269738
\(47\) 3.35182e12i 0.965044i 0.875884 + 0.482522i \(0.160279\pi\)
−0.875884 + 0.482522i \(0.839721\pi\)
\(48\) 5.87068e11i 0.144338i
\(49\) 6.46710e11 0.136219
\(50\) 0 0
\(51\) 4.22114e12 0.658648
\(52\) 9.18288e11i 0.123867i
\(53\) − 9.38781e12i − 1.09773i −0.835911 − 0.548865i \(-0.815060\pi\)
0.835911 − 0.548865i \(-0.184940\pi\)
\(54\) −1.33893e12 −0.136083
\(55\) 0 0
\(56\) −4.24685e12 −0.328592
\(57\) − 4.73089e12i − 0.320540i
\(58\) 8.28720e12i 0.492831i
\(59\) −2.89304e13 −1.51343 −0.756717 − 0.653742i \(-0.773198\pi\)
−0.756717 + 0.653742i \(0.773198\pi\)
\(60\) 0 0
\(61\) 4.23931e13 1.72711 0.863557 − 0.504251i \(-0.168231\pi\)
0.863557 + 0.504251i \(0.168231\pi\)
\(62\) 2.59041e12i 0.0934182i
\(63\) − 9.68578e12i − 0.309799i
\(64\) −4.39805e12 −0.125000
\(65\) 0 0
\(66\) 3.08644e13 0.696430
\(67\) − 5.22472e13i − 1.05318i −0.850120 − 0.526590i \(-0.823470\pi\)
0.850120 − 0.526590i \(-0.176530\pi\)
\(68\) 3.16228e13i 0.570406i
\(69\) 1.36228e13 0.220240
\(70\) 0 0
\(71\) −2.71945e13 −0.354849 −0.177425 − 0.984134i \(-0.556777\pi\)
−0.177425 + 0.984134i \(0.556777\pi\)
\(72\) − 1.00306e13i − 0.117851i
\(73\) 9.16042e13i 0.970496i 0.874376 + 0.485248i \(0.161271\pi\)
−0.874376 + 0.485248i \(0.838729\pi\)
\(74\) 6.25879e13 0.598759
\(75\) 0 0
\(76\) 3.54417e13 0.277595
\(77\) 2.23273e14i 1.58546i
\(78\) − 1.56898e13i − 0.101137i
\(79\) −6.28821e13 −0.368404 −0.184202 − 0.982888i \(-0.558970\pi\)
−0.184202 + 0.982888i \(0.558970\pi\)
\(80\) 0 0
\(81\) 2.28768e13 0.111111
\(82\) 9.88620e13i 0.437951i
\(83\) 2.23567e14i 0.904321i 0.891937 + 0.452161i \(0.149347\pi\)
−0.891937 + 0.452161i \(0.850653\pi\)
\(84\) 7.25614e13 0.268294
\(85\) 0 0
\(86\) −1.67266e14 −0.518405
\(87\) − 1.41595e14i − 0.402394i
\(88\) 2.31222e14i 0.603126i
\(89\) −5.54199e14 −1.32813 −0.664065 − 0.747675i \(-0.731170\pi\)
−0.664065 + 0.747675i \(0.731170\pi\)
\(90\) 0 0
\(91\) 1.13500e14 0.230243
\(92\) 1.02056e14i 0.190734i
\(93\) − 4.42597e13i − 0.0762756i
\(94\) 4.29033e14 0.682389
\(95\) 0 0
\(96\) 7.51447e13 0.102062
\(97\) − 1.38887e15i − 1.74531i −0.488333 − 0.872657i \(-0.662395\pi\)
0.488333 − 0.872657i \(-0.337605\pi\)
\(98\) − 8.27788e13i − 0.0963216i
\(99\) −5.27346e14 −0.568633
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 150.16.c.i.49.1 2
5.2 odd 4 150.16.a.h.1.1 1
5.3 odd 4 6.16.a.a.1.1 ✓ 1
5.4 even 2 inner 150.16.c.i.49.2 2
15.8 even 4 18.16.a.f.1.1 1
20.3 even 4 48.16.a.c.1.1 1
60.23 odd 4 144.16.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.16.a.a.1.1 ✓ 1 5.3 odd 4
18.16.a.f.1.1 1 15.8 even 4
48.16.a.c.1.1 1 20.3 even 4
144.16.a.o.1.1 1 60.23 odd 4
150.16.a.h.1.1 1 5.2 odd 4
150.16.c.i.49.1 2 1.1 even 1 trivial
150.16.c.i.49.2 2 5.4 even 2 inner