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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [400,8,Mod(49,400)] 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("400.49"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-2124] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(124.954010194\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 400.49
Dual form 400.8.c.d.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)
 
\(f(q)\) \(=\) \(q+57.0000i q^{3} +1174.00i q^{7} -1062.00 q^{9} +7563.00 q^{11} -5372.00i q^{13} +24021.0i q^{17} -51235.0 q^{19} -66918.0 q^{21} -57618.0i q^{23} +64125.0i q^{27} -47040.0 q^{29} +192358. q^{31} +431091. i q^{33} +197066. i q^{37} +306204. q^{39} -237723. q^{41} +653012. i q^{43} +826884. i q^{47} -554733. q^{49} -1.36920e6 q^{51} -569022. i q^{53} -2.92040e6i q^{57} +1.50108e6 q^{59} -2.06892e6 q^{61} -1.24679e6i q^{63} +3.44435e6i q^{67} +3.28423e6 q^{69} -4.12105e6 q^{71} +83653.0i q^{73} +8.87896e6i q^{77} +1.45403e6 q^{79} -5.97772e6 q^{81} +1.62657e6i q^{83} -2.68128e6i q^{87} -6.00434e6 q^{89} +6.30673e6 q^{91} +1.09644e7i q^{93} +3.41175e6i q^{97} -8.03191e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2124 q^{9} + 15126 q^{11} - 102470 q^{19} - 133836 q^{21} - 94080 q^{29} + 384716 q^{31} + 612408 q^{39} - 475446 q^{41} - 1109466 q^{49} - 2738394 q^{51} + 3002160 q^{59} - 4137836 q^{61} + 6568452 q^{69}+ \cdots - 16063812 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 57.0000i 1.21885i 0.792844 + 0.609425i \(0.208600\pi\)
−0.792844 + 0.609425i \(0.791400\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1174.00i 1.29367i 0.762628 + 0.646837i \(0.223909\pi\)
−0.762628 + 0.646837i \(0.776091\pi\)
\(8\) 0 0
\(9\) −1062.00 −0.485597
\(10\) 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 \(-0.110116\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 24021.0i 1.18582i 0.805268 + 0.592911i \(0.202022\pi\)
−0.805268 + 0.592911i \(0.797978\pi\)
\(18\) 0 0
\(19\) −51235.0 −1.71368 −0.856839 − 0.515584i \(-0.827575\pi\)
−0.856839 + 0.515584i \(0.827575\pi\)
\(20\) 0 0
\(21\) −66918.0 −1.57680
\(22\) 0 0
\(23\) − 57618.0i − 0.987440i −0.869621 − 0.493720i \(-0.835637\pi\)
0.869621 − 0.493720i \(-0.164363\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 64125.0i 0.626981i
\(28\) 0 0
\(29\) −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 \(-0.303112\pi\)
0.579848 + 0.814725i \(0.303112\pi\)
\(32\) 0 0
\(33\) 431091.i 2.08819i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 197066.i 0.639596i 0.947486 + 0.319798i \(0.103615\pi\)
−0.947486 + 0.319798i \(0.896385\pi\)
\(38\) 0 0
\(39\) 306204. 0.826580
\(40\) 0 0
\(41\) −237723. −0.538676 −0.269338 − 0.963046i \(-0.586805\pi\)
−0.269338 + 0.963046i \(0.586805\pi\)
\(42\) 0 0
\(43\) 653012.i 1.25251i 0.779618 + 0.626256i \(0.215413\pi\)
−0.779618 + 0.626256i \(0.784587\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 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
\(51\) −1.36920e6 −1.44534
\(52\) 0 0
\(53\) − 569022.i − 0.525005i −0.964931 − 0.262503i \(-0.915452\pi\)
0.964931 − 0.262503i \(-0.0845478\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.92040e6i − 2.08872i
\(58\) 0 0
\(59\) 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 \(0.698327\pi\)
\(62\) 0 0
\(63\) − 1.24679e6i − 0.628204i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.44435e6i 1.39909i 0.714589 + 0.699544i \(0.246614\pi\)
−0.714589 + 0.699544i \(0.753386\pi\)
\(68\) 0 0
\(69\) 3.28423e6 1.20354
\(70\) 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 \(0.995994\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.87896e6i 2.21638i
\(78\) 0 0
\(79\) 1.45403e6 0.331802 0.165901 − 0.986142i \(-0.446947\pi\)
0.165901 + 0.986142i \(0.446947\pi\)
\(80\) 0 0
\(81\) −5.97772e6 −1.24979
\(82\) 0 0
\(83\) 1.62657e6i 0.312247i 0.987738 + 0.156124i \(0.0498998\pi\)
−0.987738 + 0.156124i \(0.950100\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 2.68128e6i − 0.436541i
\(88\) 0 0
\(89\) −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
\(92\) 0 0
\(93\) 1.09644e7i 1.41350i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.41175e6i 0.379556i 0.981827 + 0.189778i \(0.0607768\pi\)
−0.981827 + 0.189778i \(0.939223\pi\)
\(98\) 0 0
\(99\) −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 400.8.c.d.49.2 2
4.3 odd 2 50.8.b.b.49.1 2
5.2 odd 4 400.8.a.q.1.1 1
5.3 odd 4 400.8.a.d.1.1 1
5.4 even 2 inner 400.8.c.d.49.1 2
12.11 even 2 450.8.c.q.199.2 2
20.3 even 4 50.8.a.c.1.1 ✓ 1
20.7 even 4 50.8.a.f.1.1 yes 1
20.19 odd 2 50.8.b.b.49.2 2
60.23 odd 4 450.8.a.p.1.1 1
60.47 odd 4 450.8.a.l.1.1 1
60.59 even 2 450.8.c.q.199.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.8.a.c.1.1 ✓ 1 20.3 even 4
50.8.a.f.1.1 yes 1 20.7 even 4
50.8.b.b.49.1 2 4.3 odd 2
50.8.b.b.49.2 2 20.19 odd 2
400.8.a.d.1.1 1 5.3 odd 4
400.8.a.q.1.1 1 5.2 odd 4
400.8.c.d.49.1 2 5.4 even 2 inner
400.8.c.d.49.2 2 1.1 even 1 trivial
450.8.a.l.1.1 1 60.47 odd 4
450.8.a.p.1.1 1 60.23 odd 4
450.8.c.q.199.1 2 60.59 even 2
450.8.c.q.199.2 2 12.11 even 2