Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,8,Mod(49,50)] 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("50.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-128,0,-912] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(15.6192512742\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.8.b.b.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.00000i q^{2} +57.0000i q^{3} -64.0000 q^{4} -456.000 q^{6} +1174.00i q^{7} -512.000i q^{8} -1062.00 q^{9} -7563.00 q^{11} -3648.00i q^{12} +5372.00i q^{13} -9392.00 q^{14} +4096.00 q^{16} -24021.0i q^{17} -8496.00i q^{18} +51235.0 q^{19} -66918.0 q^{21} -60504.0i q^{22} -57618.0i q^{23} +29184.0 q^{24} -42976.0 q^{26} +64125.0i q^{27} -75136.0i q^{28} -47040.0 q^{29} -192358. q^{31} +32768.0i q^{32} -431091. i q^{33} +192168. q^{34} +67968.0 q^{36} -197066. i q^{37} +409880. i q^{38} -306204. q^{39} -237723. q^{41} -535344. i q^{42} +653012. i q^{43} +484032. q^{44} +460944. q^{46} +826884. i q^{47} +233472. i q^{48} -554733. q^{49} +1.36920e6 q^{51} -343808. i q^{52} +569022. i q^{53} -513000. q^{54} +601088. q^{56} +2.92040e6i q^{57} -376320. i q^{58} -1.50108e6 q^{59} -2.06892e6 q^{61} -1.53886e6i q^{62} -1.24679e6i q^{63} -262144. q^{64} +3.44873e6 q^{66} +3.44435e6i q^{67} +1.53734e6i q^{68} +3.28423e6 q^{69} +4.12105e6 q^{71} +543744. i q^{72} -83653.0i q^{73} +1.57653e6 q^{74} -3.27904e6 q^{76} -8.87896e6i q^{77} -2.44963e6i q^{78} -1.45403e6 q^{79} -5.97772e6 q^{81} -1.90178e6i q^{82} +1.62657e6i q^{83} +4.28275e6 q^{84} -5.22410e6 q^{86} -2.68128e6i q^{87} +3.87226e6i q^{88} -6.00434e6 q^{89} -6.30673e6 q^{91} +3.68755e6i q^{92} -1.09644e7i q^{93} -6.61507e6 q^{94} -1.86778e6 q^{96} -3.41175e6i q^{97} -4.43786e6i q^{98} +8.03191e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 128 q^{4} - 912 q^{6} - 2124 q^{9} - 15126 q^{11} - 18784 q^{14} + 8192 q^{16} + 102470 q^{19} - 133836 q^{21} + 58368 q^{24} - 85952 q^{26} - 94080 q^{29} - 384716 q^{31} + 384336 q^{34} + 135936 q^{36}+ \cdots + 16063812 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000i 0.707107i
\(3\) 57.0000i 1.21885i 0.792844 + 0.609425i \(0.208600\pi\)
−0.792844 + 0.609425i \(0.791400\pi\)
\(4\) −64.0000 −0.500000
\(5\) 0 0
\(6\) −456.000 −0.861858
\(7\) 1174.00i 1.29367i 0.762628 + 0.646837i \(0.223909\pi\)
−0.762628 + 0.646837i \(0.776091\pi\)
\(8\) − 512.000i − 0.353553i
\(9\) −1062.00 −0.485597
\(10\) 0 0
\(11\) −7563.00 −1.71325 −0.856623 − 0.515943i \(-0.827442\pi\)
−0.856623 + 0.515943i \(0.827442\pi\)
\(12\) − 3648.00i − 0.609425i
\(13\) 5372.00i 0.678163i 0.940757 + 0.339082i \(0.110116\pi\)
−0.940757 + 0.339082i \(0.889884\pi\)
\(14\) −9392.00 −0.914766
\(15\) 0 0
\(16\) 4096.00 0.250000
\(17\) − 24021.0i − 1.18582i −0.805268 − 0.592911i \(-0.797978\pi\)
0.805268 − 0.592911i \(-0.202022\pi\)
\(18\) − 8496.00i − 0.343369i
\(19\) 51235.0 1.71368 0.856839 − 0.515584i \(-0.172425\pi\)
0.856839 + 0.515584i \(0.172425\pi\)
\(20\) 0 0
\(21\) −66918.0 −1.57680
\(22\) − 60504.0i − 1.21145i
\(23\) − 57618.0i − 0.987440i −0.869621 − 0.493720i \(-0.835637\pi\)
0.869621 − 0.493720i \(-0.164363\pi\)
\(24\) 29184.0 0.430929
\(25\) 0 0
\(26\) −42976.0 −0.479534
\(27\) 64125.0i 0.626981i
\(28\) − 75136.0i − 0.646837i
\(29\) −47040.0 −0.358158 −0.179079 − 0.983835i \(-0.557312\pi\)
−0.179079 + 0.983835i \(0.557312\pi\)
\(30\) 0 0
\(31\) −192358. −1.15970 −0.579848 − 0.814725i \(-0.696888\pi\)
−0.579848 + 0.814725i \(0.696888\pi\)
\(32\) 32768.0i 0.176777i
\(33\) − 431091.i − 2.08819i
\(34\) 192168. 0.838503
\(35\) 0 0
\(36\) 67968.0 0.242798
\(37\) − 197066.i − 0.639596i −0.947486 − 0.319798i \(-0.896385\pi\)
0.947486 − 0.319798i \(-0.103615\pi\)
\(38\) 409880.i 1.21175i
\(39\) −306204. −0.826580
\(40\) 0 0
\(41\) −237723. −0.538676 −0.269338 − 0.963046i \(-0.586805\pi\)
−0.269338 + 0.963046i \(0.586805\pi\)
\(42\) − 535344.i − 1.11496i
\(43\) 653012.i 1.25251i 0.779618 + 0.626256i \(0.215413\pi\)
−0.779618 + 0.626256i \(0.784587\pi\)
\(44\) 484032. 0.856623
\(45\) 0 0
\(46\) 460944. 0.698226
\(47\) 826884.i 1.16172i 0.814003 + 0.580861i \(0.197284\pi\)
−0.814003 + 0.580861i \(0.802716\pi\)
\(48\) 233472.i 0.304713i
\(49\) −554733. −0.673593
\(50\) 0 0
\(51\) 1.36920e6 1.44534
\(52\) − 343808.i − 0.339082i
\(53\) 569022.i 0.525005i 0.964931 + 0.262503i \(0.0845478\pi\)
−0.964931 + 0.262503i \(0.915452\pi\)
\(54\) −513000. −0.443342
\(55\) 0 0
\(56\) 601088. 0.457383
\(57\) 2.92040e6i 2.08872i
\(58\) − 376320.i − 0.253256i
\(59\) −1.50108e6 −0.951528 −0.475764 − 0.879573i \(-0.657828\pi\)
−0.475764 + 0.879573i \(0.657828\pi\)
\(60\) 0 0
\(61\) −2.06892e6 −1.16705 −0.583524 − 0.812096i \(-0.698327\pi\)
−0.583524 + 0.812096i \(0.698327\pi\)
\(62\) − 1.53886e6i − 0.820029i
\(63\) − 1.24679e6i − 0.628204i
\(64\) −262144. −0.125000
\(65\) 0 0
\(66\) 3.44873e6 1.47657
\(67\) 3.44435e6i 1.39909i 0.714589 + 0.699544i \(0.246614\pi\)
−0.714589 + 0.699544i \(0.753386\pi\)
\(68\) 1.53734e6i 0.592911i
\(69\) 3.28423e6 1.20354
\(70\) 0 0
\(71\) 4.12105e6 1.36648 0.683241 − 0.730193i \(-0.260570\pi\)
0.683241 + 0.730193i \(0.260570\pi\)
\(72\) 543744.i 0.171684i
\(73\) − 83653.0i − 0.0251682i −0.999921 − 0.0125841i \(-0.995994\pi\)
0.999921 − 0.0125841i \(-0.00400574\pi\)
\(74\) 1.57653e6 0.452263
\(75\) 0 0
\(76\) −3.27904e6 −0.856839
\(77\) − 8.87896e6i − 2.21638i
\(78\) − 2.44963e6i − 0.584480i
\(79\) −1.45403e6 −0.331802 −0.165901 − 0.986142i \(-0.553053\pi\)
−0.165901 + 0.986142i \(0.553053\pi\)
\(80\) 0 0
\(81\) −5.97772e6 −1.24979
\(82\) − 1.90178e6i − 0.380902i
\(83\) 1.62657e6i 0.312247i 0.987738 + 0.156124i \(0.0498998\pi\)
−0.987738 + 0.156124i \(0.950100\pi\)
\(84\) 4.28275e6 0.788398
\(85\) 0 0
\(86\) −5.22410e6 −0.885659
\(87\) − 2.68128e6i − 0.436541i
\(88\) 3.87226e6i 0.605724i
\(89\) −6.00434e6 −0.902817 −0.451409 − 0.892317i \(-0.649078\pi\)
−0.451409 + 0.892317i \(0.649078\pi\)
\(90\) 0 0
\(91\) −6.30673e6 −0.877322
\(92\) 3.68755e6i 0.493720i
\(93\) − 1.09644e7i − 1.41350i
\(94\) −6.61507e6 −0.821461
\(95\) 0 0
\(96\) −1.86778e6 −0.215464
\(97\) − 3.41175e6i − 0.379556i −0.981827 − 0.189778i \(-0.939223\pi\)
0.981827 − 0.189778i \(-0.0607768\pi\)
\(98\) − 4.43786e6i − 0.476302i
\(99\) 8.03191e6 0.831947
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 50.8.b.b.49.2 2
3.2 odd 2 450.8.c.q.199.1 2
4.3 odd 2 400.8.c.d.49.1 2
5.2 odd 4 50.8.a.c.1.1 ✓ 1
5.3 odd 4 50.8.a.f.1.1 yes 1
5.4 even 2 inner 50.8.b.b.49.1 2
15.2 even 4 450.8.a.p.1.1 1
15.8 even 4 450.8.a.l.1.1 1
15.14 odd 2 450.8.c.q.199.2 2
20.3 even 4 400.8.a.q.1.1 1
20.7 even 4 400.8.a.d.1.1 1
20.19 odd 2 400.8.c.d.49.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.8.a.c.1.1 ✓ 1 5.2 odd 4
50.8.a.f.1.1 yes 1 5.3 odd 4
50.8.b.b.49.1 2 5.4 even 2 inner
50.8.b.b.49.2 2 1.1 even 1 trivial
400.8.a.d.1.1 1 20.7 even 4
400.8.a.q.1.1 1 20.3 even 4
400.8.c.d.49.1 2 4.3 odd 2
400.8.c.d.49.2 2 20.19 odd 2
450.8.a.l.1.1 1 15.8 even 4
450.8.a.p.1.1 1 15.2 even 4
450.8.c.q.199.1 2 3.2 odd 2
450.8.c.q.199.2 2 15.14 odd 2