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,-1,0,1,0,-3,1,0,5] 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: yes
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} +1.00000 q^{6} -3.00000 q^{8} +1.00000 q^{9} +5.00000 q^{11} -1.00000 q^{12} -4.00000 q^{13} -1.00000 q^{16} +4.00000 q^{17} +1.00000 q^{18} -1.00000 q^{19} +5.00000 q^{22} +9.00000 q^{23} -3.00000 q^{24} -4.00000 q^{26} +1.00000 q^{27} +7.00000 q^{29} +3.00000 q^{31} +5.00000 q^{32} +5.00000 q^{33} +4.00000 q^{34} -1.00000 q^{36} +10.0000 q^{37} -1.00000 q^{38} -4.00000 q^{39} -2.00000 q^{41} -4.00000 q^{43} -5.00000 q^{44} +9.00000 q^{46} -8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +4.00000 q^{51} +4.00000 q^{52} -11.0000 q^{53} +1.00000 q^{54} -1.00000 q^{57} +7.00000 q^{58} +8.00000 q^{59} +13.0000 q^{61} +3.00000 q^{62} +7.00000 q^{64} +5.00000 q^{66} -9.00000 q^{67} -4.00000 q^{68} +9.00000 q^{69} +10.0000 q^{71} -3.00000 q^{72} +5.00000 q^{73} +10.0000 q^{74} +1.00000 q^{76} -4.00000 q^{78} -15.0000 q^{79} +1.00000 q^{81} -2.00000 q^{82} -9.00000 q^{83} -4.00000 q^{86} +7.00000 q^{87} -15.0000 q^{88} +3.00000 q^{89} -9.00000 q^{92} +3.00000 q^{93} -8.00000 q^{94} +5.00000 q^{96} +10.0000 q^{97} -7.00000 q^{98} +5.00000 q^{99} +O(q^{100})\)

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 \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) −3.00000 −1.06066
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 − 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) −1.00000 −0.288675
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 4.00000 0.970143 0.485071 − 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 5.00000 1.06600
\(23\) 9.00000 1.87663 0.938315 − 0.345782i \(-0.112386\pi\)
0.938315 + 0.345782i \(0.112386\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 7.00000 1.29987 0.649934 − 0.759991i \(-0.274797\pi\)
0.649934 + 0.759991i \(0.274797\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 − 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 5.00000 0.883883
\(33\) 5.00000 0.870388
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 10.0000 1.64399 0.821995 − 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(38\) −1.00000 −0.162221
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −5.00000 −0.753778
\(45\) 0 0
\(46\) 9.00000 1.32698
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) −1.00000 −0.144338
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 4.00000 0.560112
\(52\) 4.00000 0.554700
\(53\) −11.0000 −1.51097 −0.755483 − 0.655168i \(-0.772598\pi\)
−0.755483 + 0.655168i \(0.772598\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 7.00000 0.919145
\(59\) 8.00000 1.04151 0.520756 − 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) 13.0000 1.66448 0.832240 − 0.554416i \(-0.187058\pi\)
0.832240 + 0.554416i \(0.187058\pi\)
\(62\) 3.00000 0.381000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 5.00000 0.615457
\(67\) −9.00000 −1.09952 −0.549762 − 0.835321i \(-0.685282\pi\)
−0.549762 + 0.835321i \(0.685282\pi\)
\(68\) −4.00000 −0.485071
\(69\) 9.00000 1.08347
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 − 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) −3.00000 −0.353553
\(73\) 5.00000 0.585206 0.292603 − 0.956234i \(-0.405479\pi\)
0.292603 + 0.956234i \(0.405479\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 0 0
\(78\) −4.00000 −0.452911
\(79\) −15.0000 −1.68763 −0.843816 − 0.536633i \(-0.819696\pi\)
−0.843816 + 0.536633i \(0.819696\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.00000 −0.220863
\(83\) −9.00000 −0.987878 −0.493939 − 0.869496i \(-0.664443\pi\)
−0.493939 + 0.869496i \(0.664443\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 7.00000 0.750479
\(88\) −15.0000 −1.59901
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −9.00000 −0.938315
\(93\) 3.00000 0.311086
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) 10.0000 1.01535 0.507673 − 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) −7.00000 −0.707107
\(99\) 5.00000 0.502519
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.h.1.1 yes 1
3.2 odd 2 4275.2.a.f.1.1 1
5.2 odd 4 1425.2.c.h.799.2 2
5.3 odd 4 1425.2.c.h.799.1 2
5.4 even 2 1425.2.a.b.1.1 ✓ 1
15.14 odd 2 4275.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1425.2.a.b.1.1 ✓ 1 5.4 even 2
1425.2.a.h.1.1 yes 1 1.1 even 1 trivial
1425.2.c.h.799.1 2 5.3 odd 4
1425.2.c.h.799.2 2 5.2 odd 4
4275.2.a.f.1.1 1 3.2 odd 2
4275.2.a.l.1.1 1 15.14 odd 2