Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2175,2,Mod(1,2175)] 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("2175.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-2,1,2,0,-2,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2175.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +1.00000 q^{3} +2.00000 q^{4} -2.00000 q^{6} -2.00000 q^{7} +1.00000 q^{9} -3.00000 q^{11} +2.00000 q^{12} +4.00000 q^{13} +4.00000 q^{14} -4.00000 q^{16} +8.00000 q^{17} -2.00000 q^{18} -2.00000 q^{21} +6.00000 q^{22} -1.00000 q^{23} -8.00000 q^{26} +1.00000 q^{27} -4.00000 q^{28} +1.00000 q^{29} -8.00000 q^{31} +8.00000 q^{32} -3.00000 q^{33} -16.0000 q^{34} +2.00000 q^{36} -7.00000 q^{37} +4.00000 q^{39} +7.00000 q^{41} +4.00000 q^{42} +9.00000 q^{43} -6.00000 q^{44} +2.00000 q^{46} -12.0000 q^{47} -4.00000 q^{48} -3.00000 q^{49} +8.00000 q^{51} +8.00000 q^{52} +9.00000 q^{53} -2.00000 q^{54} -2.00000 q^{58} +10.0000 q^{59} +2.00000 q^{61} +16.0000 q^{62} -2.00000 q^{63} -8.00000 q^{64} +6.00000 q^{66} +8.00000 q^{67} +16.0000 q^{68} -1.00000 q^{69} -8.00000 q^{71} -1.00000 q^{73} +14.0000 q^{74} +6.00000 q^{77} -8.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} -14.0000 q^{82} +9.00000 q^{83} -4.00000 q^{84} -18.0000 q^{86} +1.00000 q^{87} +10.0000 q^{89} -8.00000 q^{91} -2.00000 q^{92} -8.00000 q^{93} +24.0000 q^{94} +8.00000 q^{96} +13.0000 q^{97} +6.00000 q^{98} -3.00000 q^{99} +O(q^{100})\)

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 \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) −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
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 − 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 2.00000 0.577350
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 8.00000 1.94029 0.970143 − 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) −2.00000 −0.471405
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 6.00000 1.27920
\(23\) −1.00000 −0.208514 −0.104257 − 0.994550i \(-0.533247\pi\)
−0.104257 + 0.994550i \(0.533247\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −8.00000 −1.56893
\(27\) 1.00000 0.192450
\(28\) −4.00000 −0.755929
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 8.00000 1.41421
\(33\) −3.00000 −0.522233
\(34\) −16.0000 −2.74398
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) −7.00000 −1.15079 −0.575396 − 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 − 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 4.00000 0.617213
\(43\) 9.00000 1.37249 0.686244 − 0.727372i \(-0.259258\pi\)
0.686244 + 0.727372i \(0.259258\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) 2.00000 0.294884
\(47\) −12.0000 −1.75038 −0.875190 − 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −4.00000 −0.577350
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 8.00000 1.10940
\(53\) 9.00000 1.23625 0.618123 − 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) −2.00000 −0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −2.00000 −0.262613
\(59\) 10.0000 1.30189 0.650945 − 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 16.0000 2.03200
\(63\) −2.00000 −0.251976
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 6.00000 0.738549
\(67\) 8.00000 0.977356 0.488678 − 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 16.0000 1.94029
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 − 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 − 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 14.0000 1.62747
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000 0.683763
\(78\) −8.00000 −0.905822
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −14.0000 −1.54604
\(83\) 9.00000 0.987878 0.493939 − 0.869496i \(-0.335557\pi\)
0.493939 + 0.869496i \(0.335557\pi\)
\(84\) −4.00000 −0.436436
\(85\) 0 0
\(86\) −18.0000 −1.94099
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) −2.00000 −0.208514
\(93\) −8.00000 −0.829561
\(94\) 24.0000 2.47541
\(95\) 0 0
\(96\) 8.00000 0.816497
\(97\) 13.0000 1.31995 0.659975 − 0.751288i \(-0.270567\pi\)
0.659975 + 0.751288i \(0.270567\pi\)
\(98\) 6.00000 0.606092
\(99\) −3.00000 −0.301511
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 2175.2.a.a.1.1 1
3.2 odd 2 6525.2.a.l.1.1 1
5.2 odd 4 435.2.c.a.349.1 ✓ 2
5.3 odd 4 435.2.c.a.349.2 yes 2
5.4 even 2 2175.2.a.j.1.1 1
15.2 even 4 1305.2.c.a.784.2 2
15.8 even 4 1305.2.c.a.784.1 2
15.14 odd 2 6525.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.c.a.349.1 ✓ 2 5.2 odd 4
435.2.c.a.349.2 yes 2 5.3 odd 4
1305.2.c.a.784.1 2 15.8 even 4
1305.2.c.a.784.2 2 15.2 even 4
2175.2.a.a.1.1 1 1.1 even 1 trivial
2175.2.a.j.1.1 1 5.4 even 2
6525.2.a.b.1.1 1 15.14 odd 2
6525.2.a.l.1.1 1 3.2 odd 2