Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5415,2,Mod(1,5415)] 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("5415.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5415, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5415 = 3 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5415.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,1,2,-1,2,-2,0,1,-2,1] 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: \(43.2389926945\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 5415.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -1.00000 q^{5} +2.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} -2.00000 q^{10} +1.00000 q^{11} +2.00000 q^{12} -2.00000 q^{13} -4.00000 q^{14} -1.00000 q^{15} -4.00000 q^{16} +2.00000 q^{17} +2.00000 q^{18} -2.00000 q^{20} -2.00000 q^{21} +2.00000 q^{22} -4.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +5.00000 q^{29} -2.00000 q^{30} -9.00000 q^{31} -8.00000 q^{32} +1.00000 q^{33} +4.00000 q^{34} +2.00000 q^{35} +2.00000 q^{36} +6.00000 q^{37} -2.00000 q^{39} -6.00000 q^{41} -4.00000 q^{42} -10.0000 q^{43} +2.00000 q^{44} -1.00000 q^{45} -8.00000 q^{46} -4.00000 q^{48} -3.00000 q^{49} +2.00000 q^{50} +2.00000 q^{51} -4.00000 q^{52} +2.00000 q^{53} +2.00000 q^{54} -1.00000 q^{55} +10.0000 q^{58} -7.00000 q^{59} -2.00000 q^{60} -7.00000 q^{61} -18.0000 q^{62} -2.00000 q^{63} -8.00000 q^{64} +2.00000 q^{65} +2.00000 q^{66} -8.00000 q^{67} +4.00000 q^{68} -4.00000 q^{69} +4.00000 q^{70} -3.00000 q^{71} -2.00000 q^{73} +12.0000 q^{74} +1.00000 q^{75} -2.00000 q^{77} -4.00000 q^{78} +11.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} -12.0000 q^{82} +6.00000 q^{83} -4.00000 q^{84} -2.00000 q^{85} -20.0000 q^{86} +5.00000 q^{87} -15.0000 q^{89} -2.00000 q^{90} +4.00000 q^{91} -8.00000 q^{92} -9.00000 q^{93} -8.00000 q^{96} -8.00000 q^{97} -6.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.41421 0.707107 − 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) −1.00000 −0.447214
\(6\) 2.00000 0.816497
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) 1.00000 0.301511 0.150756 − 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 2.00000 0.577350
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −4.00000 −1.06904
\(15\) −1.00000 −0.258199
\(16\) −4.00000 −1.00000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 2.00000 0.471405
\(19\) 0 0
\(20\) −2.00000 −0.447214
\(21\) −2.00000 −0.436436
\(22\) 2.00000 0.426401
\(23\) −4.00000 −0.834058 −0.417029 − 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −4.00000 −0.784465
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) 5.00000 0.928477 0.464238 − 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) −2.00000 −0.365148
\(31\) −9.00000 −1.61645 −0.808224 − 0.588875i \(-0.799571\pi\)
−0.808224 + 0.588875i \(0.799571\pi\)
\(32\) −8.00000 −1.41421
\(33\) 1.00000 0.174078
\(34\) 4.00000 0.685994
\(35\) 2.00000 0.338062
\(36\) 2.00000 0.333333
\(37\) 6.00000 0.986394 0.493197 − 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) −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
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 2.00000 0.301511
\(45\) −1.00000 −0.149071
\(46\) −8.00000 −1.17954
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) −4.00000 −0.577350
\(49\) −3.00000 −0.428571
\(50\) 2.00000 0.282843
\(51\) 2.00000 0.280056
\(52\) −4.00000 −0.554700
\(53\) 2.00000 0.274721 0.137361 − 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 2.00000 0.272166
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) −7.00000 −0.911322 −0.455661 − 0.890153i \(-0.650597\pi\)
−0.455661 + 0.890153i \(0.650597\pi\)
\(60\) −2.00000 −0.258199
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) −18.0000 −2.28600
\(63\) −2.00000 −0.251976
\(64\) −8.00000 −1.00000
\(65\) 2.00000 0.248069
\(66\) 2.00000 0.246183
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 4.00000 0.485071
\(69\) −4.00000 −0.481543
\(70\) 4.00000 0.478091
\(71\) −3.00000 −0.356034 −0.178017 − 0.984027i \(-0.556968\pi\)
−0.178017 + 0.984027i \(0.556968\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 12.0000 1.39497
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) −4.00000 −0.452911
\(79\) 11.0000 1.23760 0.618798 − 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) −12.0000 −1.32518
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −4.00000 −0.436436
\(85\) −2.00000 −0.216930
\(86\) −20.0000 −2.15666
\(87\) 5.00000 0.536056
\(88\) 0 0
\(89\) −15.0000 −1.59000 −0.794998 − 0.606612i \(-0.792528\pi\)
−0.794998 + 0.606612i \(0.792528\pi\)
\(90\) −2.00000 −0.210819
\(91\) 4.00000 0.419314
\(92\) −8.00000 −0.834058
\(93\) −9.00000 −0.933257
\(94\) 0 0
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) −6.00000 −0.606092
\(99\) 1.00000 0.100504
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 5415.2.a.l.1.1 1
19.8 odd 6 285.2.i.c.121.1 yes 2
19.12 odd 6 285.2.i.c.106.1 ✓ 2
19.18 odd 2 5415.2.a.b.1.1 1
57.8 even 6 855.2.k.a.406.1 2
57.50 even 6 855.2.k.a.676.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.i.c.106.1 ✓ 2 19.12 odd 6
285.2.i.c.121.1 yes 2 19.8 odd 6
855.2.k.a.406.1 2 57.8 even 6
855.2.k.a.676.1 2 57.50 even 6
5415.2.a.b.1.1 1 19.18 odd 2
5415.2.a.l.1.1 1 1.1 even 1 trivial