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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1900,2,Mod(1749,1900)] 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("1900.1749"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-4,0,0] 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: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1749.3
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1900.1749
Dual form 1900.2.c.e.1749.2

$q$-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)
 
\(f(q)\) \(=\) \(q+0.732051i q^{3} +2.00000i q^{7} +2.46410 q^{9} +3.46410 q^{11} -0.732051i q^{13} +3.46410i q^{17} -1.00000 q^{19} -1.46410 q^{21} -3.46410i q^{23} +4.00000i q^{27} +3.46410 q^{29} +5.46410 q^{31} +2.53590i q^{33} +3.26795i q^{37} +0.535898 q^{39} -6.00000 q^{41} -8.92820i q^{43} -0.928203i q^{47} +3.00000 q^{49} -2.53590 q^{51} +7.26795i q^{53} -0.732051i q^{57} +6.92820 q^{59} -8.39230 q^{61} +4.92820i q^{63} +3.26795i q^{67} +2.53590 q^{69} -9.46410 q^{71} +7.46410i q^{73} +6.92820i q^{77} +10.9282 q^{79} +4.46410 q^{81} +3.46410i q^{83} +2.53590i q^{87} +8.53590 q^{89} +1.46410 q^{91} +4.00000i q^{93} +14.5885i q^{97} +8.53590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} - 4 q^{19} + 8 q^{21} + 8 q^{31} + 16 q^{39} - 24 q^{41} + 12 q^{49} - 24 q^{51} + 8 q^{61} + 24 q^{69} - 24 q^{71} + 16 q^{79} + 4 q^{81} + 48 q^{89} - 8 q^{91} + 48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0.732051i 0.422650i 0.977416 + 0.211325i \(0.0677778\pi\)
−0.977416 + 0.211325i \(0.932222\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) 2.46410 0.821367
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 − 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) − 0.732051i − 0.203034i −0.994834 − 0.101517i \(-0.967630\pi\)
0.994834 − 0.101517i \(-0.0323697\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410i 0.840168i 0.907485 + 0.420084i \(0.137999\pi\)
−0.907485 + 0.420084i \(0.862001\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.46410 −0.319493
\(22\) 0 0
\(23\) − 3.46410i − 0.722315i −0.932505 − 0.361158i \(-0.882382\pi\)
0.932505 − 0.361158i \(-0.117618\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.00000i 0.769800i
\(28\) 0 0
\(29\) 3.46410 0.643268 0.321634 − 0.946864i \(-0.395768\pi\)
0.321634 + 0.946864i \(0.395768\pi\)
\(30\) 0 0
\(31\) 5.46410 0.981382 0.490691 − 0.871334i \(-0.336744\pi\)
0.490691 + 0.871334i \(0.336744\pi\)
\(32\) 0 0
\(33\) 2.53590i 0.441443i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.26795i 0.537248i 0.963245 + 0.268624i \(0.0865688\pi\)
−0.963245 + 0.268624i \(0.913431\pi\)
\(38\) 0 0
\(39\) 0.535898 0.0858124
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 8.92820i − 1.36154i −0.732498 − 0.680769i \(-0.761646\pi\)
0.732498 − 0.680769i \(-0.238354\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 0.928203i − 0.135392i −0.997706 − 0.0676962i \(-0.978435\pi\)
0.997706 − 0.0676962i \(-0.0215649\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −2.53590 −0.355097
\(52\) 0 0
\(53\) 7.26795i 0.998330i 0.866507 + 0.499165i \(0.166360\pi\)
−0.866507 + 0.499165i \(0.833640\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 0.732051i − 0.0969625i
\(58\) 0 0
\(59\) 6.92820 0.901975 0.450988 − 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) −8.39230 −1.07452 −0.537262 − 0.843415i \(-0.680541\pi\)
−0.537262 + 0.843415i \(0.680541\pi\)
\(62\) 0 0
\(63\) 4.92820i 0.620895i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.26795i 0.399244i 0.979873 + 0.199622i \(0.0639713\pi\)
−0.979873 + 0.199622i \(0.936029\pi\)
\(68\) 0 0
\(69\) 2.53590 0.305286
\(70\) 0 0
\(71\) −9.46410 −1.12318 −0.561591 − 0.827415i \(-0.689811\pi\)
−0.561591 + 0.827415i \(0.689811\pi\)
\(72\) 0 0
\(73\) 7.46410i 0.873607i 0.899557 + 0.436804i \(0.143889\pi\)
−0.899557 + 0.436804i \(0.856111\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.92820i 0.789542i
\(78\) 0 0
\(79\) 10.9282 1.22952 0.614759 − 0.788715i \(-0.289253\pi\)
0.614759 + 0.788715i \(0.289253\pi\)
\(80\) 0 0
\(81\) 4.46410 0.496011
\(82\) 0 0
\(83\) 3.46410i 0.380235i 0.981761 + 0.190117i \(0.0608868\pi\)
−0.981761 + 0.190117i \(0.939113\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.53590i 0.271877i
\(88\) 0 0
\(89\) 8.53590 0.904803 0.452402 − 0.891814i \(-0.350567\pi\)
0.452402 + 0.891814i \(0.350567\pi\)
\(90\) 0 0
\(91\) 1.46410 0.153480
\(92\) 0 0
\(93\) 4.00000i 0.414781i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.5885i 1.48123i 0.671928 + 0.740617i \(0.265467\pi\)
−0.671928 + 0.740617i \(0.734533\pi\)
\(98\) 0 0
\(99\) 8.53590 0.857890
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 1900.2.c.e.1749.3 4
5.2 odd 4 1900.2.a.d.1.2 2
5.3 odd 4 380.2.a.d.1.1 ✓ 2
5.4 even 2 inner 1900.2.c.e.1749.2 4
15.8 even 4 3420.2.a.h.1.1 2
20.3 even 4 1520.2.a.l.1.2 2
20.7 even 4 7600.2.a.bf.1.1 2
40.3 even 4 6080.2.a.bj.1.1 2
40.13 odd 4 6080.2.a.z.1.2 2
95.18 even 4 7220.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.d.1.1 ✓ 2 5.3 odd 4
1520.2.a.l.1.2 2 20.3 even 4
1900.2.a.d.1.2 2 5.2 odd 4
1900.2.c.e.1749.2 4 5.4 even 2 inner
1900.2.c.e.1749.3 4 1.1 even 1 trivial
3420.2.a.h.1.1 2 15.8 even 4
6080.2.a.z.1.2 2 40.13 odd 4
6080.2.a.bj.1.1 2 40.3 even 4
7220.2.a.h.1.2 2 95.18 even 4
7600.2.a.bf.1.1 2 20.7 even 4