Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1425,2,Mod(1,1425)] 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("1425.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1425.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1,1,-1,0,-1,-4,3,1,0,4] 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: yes
Analytic conductor: \(11.3786822880\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 285)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1425.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} -4.00000 q^{7} +3.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +4.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{19} -4.00000 q^{21} -4.00000 q^{22} +4.00000 q^{23} +3.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +4.00000 q^{28} -2.00000 q^{29} -5.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} -1.00000 q^{36} +6.00000 q^{37} +1.00000 q^{38} -2.00000 q^{39} -6.00000 q^{41} +4.00000 q^{42} -8.00000 q^{43} -4.00000 q^{44} -4.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -2.00000 q^{51} +2.00000 q^{52} +14.0000 q^{53} -1.00000 q^{54} -12.0000 q^{56} -1.00000 q^{57} +2.00000 q^{58} +4.00000 q^{59} +14.0000 q^{61} -4.00000 q^{63} +7.00000 q^{64} -4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} +4.00000 q^{69} +3.00000 q^{72} +14.0000 q^{73} -6.00000 q^{74} +1.00000 q^{76} -16.0000 q^{77} +2.00000 q^{78} +16.0000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{84} +8.00000 q^{86} -2.00000 q^{87} +12.0000 q^{88} -6.00000 q^{89} +8.00000 q^{91} -4.00000 q^{92} -12.0000 q^{94} -5.00000 q^{96} +10.0000 q^{97} -9.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −4.00000 −1.51186 −0.755929 − 0.654654i \(-0.772814\pi\)
−0.755929 + 0.654654i \(0.772814\pi\)
\(8\) 3.00000 1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) −4.00000 −0.852803
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 4.00000 0.755929
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −5.00000 −0.883883
\(33\) 4.00000 0.696311
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 1.00000 0.162221
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000 0.617213
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 12.0000 1.75038 0.875190 − 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) −1.00000 −0.144338
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 2.00000 0.277350
\(53\) 14.0000 1.92305 0.961524 − 0.274721i \(-0.0885855\pi\)
0.961524 + 0.274721i \(0.0885855\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −12.0000 −1.60357
\(57\) −1.00000 −0.132453
\(58\) 2.00000 0.262613
\(59\) 4.00000 0.520756 0.260378 − 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 − 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) −4.00000 −0.503953
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 2.00000 0.242536
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 3.00000 0.353553
\(73\) 14.0000 1.63858 0.819288 − 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) −6.00000 −0.697486
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −16.0000 −1.82337
\(78\) 2.00000 0.226455
\(79\) 16.0000 1.80014 0.900070 − 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000 0.662589
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) −2.00000 −0.214423
\(88\) 12.0000 1.27920
\(89\) −6.00000 −0.635999 −0.317999 − 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) −4.00000 −0.417029
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −9.00000 −0.909137
\(99\) 4.00000 0.402015
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 1425.2.a.c.1.1 1
3.2 odd 2 4275.2.a.j.1.1 1
5.2 odd 4 1425.2.c.f.799.1 2
5.3 odd 4 1425.2.c.f.799.2 2
5.4 even 2 285.2.a.c.1.1 ✓ 1
15.14 odd 2 855.2.a.a.1.1 1
20.19 odd 2 4560.2.a.w.1.1 1
95.94 odd 2 5415.2.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.c.1.1 ✓ 1 5.4 even 2
855.2.a.a.1.1 1 15.14 odd 2
1425.2.a.c.1.1 1 1.1 even 1 trivial
1425.2.c.f.799.1 2 5.2 odd 4
1425.2.c.f.799.2 2 5.3 odd 4
4275.2.a.j.1.1 1 3.2 odd 2
4560.2.a.w.1.1 1 20.19 odd 2
5415.2.a.e.1.1 1 95.94 odd 2