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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [450,8,Mod(199,450)] 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("450.199"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 450.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-128,0,0,0,0,0,0,15126] 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: \(140.573261468\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 450.199
Dual form 450.8.c.q.199.1

$q$-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)
 
\(f(q)\) \(=\) \(q+8.00000i q^{2} -64.0000 q^{4} -1174.00i q^{7} -512.000i q^{8} +7563.00 q^{11} -5372.00i q^{13} +9392.00 q^{14} +4096.00 q^{16} -24021.0i q^{17} +51235.0 q^{19} +60504.0i q^{22} -57618.0i q^{23} +42976.0 q^{26} +75136.0i q^{28} +47040.0 q^{29} -192358. q^{31} +32768.0i q^{32} +192168. q^{34} +197066. i q^{37} +409880. i q^{38} +237723. q^{41} -653012. i q^{43} -484032. q^{44} +460944. q^{46} +826884. i q^{47} -554733. q^{49} +343808. i q^{52} +569022. i q^{53} -601088. q^{56} +376320. i q^{58} +1.50108e6 q^{59} -2.06892e6 q^{61} -1.53886e6i q^{62} -262144. q^{64} -3.44435e6i q^{67} +1.53734e6i q^{68} -4.12105e6 q^{71} +83653.0i q^{73} -1.57653e6 q^{74} -3.27904e6 q^{76} -8.87896e6i q^{77} -1.45403e6 q^{79} +1.90178e6i q^{82} +1.62657e6i q^{83} +5.22410e6 q^{86} -3.87226e6i q^{88} +6.00434e6 q^{89} -6.30673e6 q^{91} +3.68755e6i q^{92} -6.61507e6 q^{94} +3.41175e6i q^{97} -4.43786e6i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 128 q^{4} + 15126 q^{11} + 18784 q^{14} + 8192 q^{16} + 102470 q^{19} + 85952 q^{26} + 94080 q^{29} - 384716 q^{31} + 384336 q^{34} + 475446 q^{41} - 968064 q^{44} + 921888 q^{46} - 1109466 q^{49}+ \cdots - 13230144 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000i 0.707107i
\(3\) 0 0
\(4\) −64.0000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 1174.00i − 1.29367i −0.762628 − 0.646837i \(-0.776091\pi\)
0.762628 − 0.646837i \(-0.223909\pi\)
\(8\) − 512.000i − 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 7563.00 1.71325 0.856623 − 0.515943i \(-0.172558\pi\)
0.856623 + 0.515943i \(0.172558\pi\)
\(12\) 0 0
\(13\) − 5372.00i − 0.678163i −0.940757 − 0.339082i \(-0.889884\pi\)
0.940757 − 0.339082i \(-0.110116\pi\)
\(14\) 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 \(-0.202022\pi\)
\(18\) 0 0
\(19\) 51235.0 1.71368 0.856839 − 0.515584i \(-0.172425\pi\)
0.856839 + 0.515584i \(0.172425\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 60504.0i 1.21145i
\(23\) − 57618.0i − 0.987440i −0.869621 − 0.493720i \(-0.835637\pi\)
0.869621 − 0.493720i \(-0.164363\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 42976.0 0.479534
\(27\) 0 0
\(28\) 75136.0i 0.646837i
\(29\) 47040.0 0.358158 0.179079 − 0.983835i \(-0.442688\pi\)
0.179079 + 0.983835i \(0.442688\pi\)
\(30\) 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
\(33\) 0 0
\(34\) 192168. 0.838503
\(35\) 0 0
\(36\) 0 0
\(37\) 197066.i 0.639596i 0.947486 + 0.319798i \(0.103615\pi\)
−0.947486 + 0.319798i \(0.896385\pi\)
\(38\) 409880.i 1.21175i
\(39\) 0 0
\(40\) 0 0
\(41\) 237723. 0.538676 0.269338 − 0.963046i \(-0.413195\pi\)
0.269338 + 0.963046i \(0.413195\pi\)
\(42\) 0 0
\(43\) − 653012.i − 1.25251i −0.779618 − 0.626256i \(-0.784587\pi\)
0.779618 − 0.626256i \(-0.215413\pi\)
\(44\) −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 \(0.802716\pi\)
\(48\) 0 0
\(49\) −554733. −0.673593
\(50\) 0 0
\(51\) 0 0
\(52\) 343808.i 0.339082i
\(53\) 569022.i 0.525005i 0.964931 + 0.262503i \(0.0845478\pi\)
−0.964931 + 0.262503i \(0.915452\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −601088. −0.457383
\(57\) 0 0
\(58\) 376320.i 0.253256i
\(59\) 1.50108e6 0.951528 0.475764 − 0.879573i \(-0.342172\pi\)
0.475764 + 0.879573i \(0.342172\pi\)
\(60\) 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
\(63\) 0 0
\(64\) −262144. −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 3.44435e6i − 1.39909i −0.714589 − 0.699544i \(-0.753386\pi\)
0.714589 − 0.699544i \(-0.246614\pi\)
\(68\) 1.53734e6i 0.592911i
\(69\) 0 0
\(70\) 0 0
\(71\) −4.12105e6 −1.36648 −0.683241 − 0.730193i \(-0.739430\pi\)
−0.683241 + 0.730193i \(0.739430\pi\)
\(72\) 0 0
\(73\) 83653.0i 0.0251682i 0.999921 + 0.0125841i \(0.00400574\pi\)
−0.999921 + 0.0125841i \(0.995994\pi\)
\(74\) −1.57653e6 −0.452263
\(75\) 0 0
\(76\) −3.27904e6 −0.856839
\(77\) − 8.87896e6i − 2.21638i
\(78\) 0 0
\(79\) −1.45403e6 −0.331802 −0.165901 − 0.986142i \(-0.553053\pi\)
−0.165901 + 0.986142i \(0.553053\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.90178e6i 0.380902i
\(83\) 1.62657e6i 0.312247i 0.987738 + 0.156124i \(0.0498998\pi\)
−0.987738 + 0.156124i \(0.950100\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.22410e6 0.885659
\(87\) 0 0
\(88\) − 3.87226e6i − 0.605724i
\(89\) 6.00434e6 0.902817 0.451409 − 0.892317i \(-0.350922\pi\)
0.451409 + 0.892317i \(0.350922\pi\)
\(90\) 0 0
\(91\) −6.30673e6 −0.877322
\(92\) 3.68755e6i 0.493720i
\(93\) 0 0
\(94\) −6.61507e6 −0.821461
\(95\) 0 0
\(96\) 0 0
\(97\) 3.41175e6i 0.379556i 0.981827 + 0.189778i \(0.0607768\pi\)
−0.981827 + 0.189778i \(0.939223\pi\)
\(98\) − 4.43786e6i − 0.476302i
\(99\) 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 450.8.c.q.199.2 2
3.2 odd 2 50.8.b.b.49.1 2
5.2 odd 4 450.8.a.l.1.1 1
5.3 odd 4 450.8.a.p.1.1 1
5.4 even 2 inner 450.8.c.q.199.1 2
12.11 even 2 400.8.c.d.49.2 2
15.2 even 4 50.8.a.f.1.1 yes 1
15.8 even 4 50.8.a.c.1.1 ✓ 1
15.14 odd 2 50.8.b.b.49.2 2
60.23 odd 4 400.8.a.d.1.1 1
60.47 odd 4 400.8.a.q.1.1 1
60.59 even 2 400.8.c.d.49.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.8.a.c.1.1 ✓ 1 15.8 even 4
50.8.a.f.1.1 yes 1 15.2 even 4
50.8.b.b.49.1 2 3.2 odd 2
50.8.b.b.49.2 2 15.14 odd 2
400.8.a.d.1.1 1 60.23 odd 4
400.8.a.q.1.1 1 60.47 odd 4
400.8.c.d.49.1 2 60.59 even 2
400.8.c.d.49.2 2 12.11 even 2
450.8.a.l.1.1 1 5.2 odd 4
450.8.a.p.1.1 1 5.3 odd 4
450.8.c.q.199.1 2 5.4 even 2 inner
450.8.c.q.199.2 2 1.1 even 1 trivial