Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,22,Mod(49,50)] 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("50.49"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(50, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 50.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4194304,0,259465216] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(139.738672144\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 78961x^{2} + 1558670400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 5^{2}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(199.196i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.f.49.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00i q^{2} +174616. i q^{3} -1.04858e6 q^{4} +1.78807e8 q^{6} -9.60462e8i q^{7} +1.07374e9i q^{8} -2.00304e10 q^{9} +8.54661e10 q^{11} -1.83098e11i q^{12} -9.74302e11i q^{13} -9.83513e11 q^{14} +1.09951e12 q^{16} -1.17045e13i q^{17} +2.05111e13i q^{18} -1.41128e13 q^{19} +1.67712e14 q^{21} -8.75173e13i q^{22} +2.65862e13i q^{23} -1.87492e14 q^{24} -9.97685e14 q^{26} -1.67108e15i q^{27} +1.00712e15i q^{28} -1.45075e15 q^{29} -7.63253e15 q^{31} -1.12590e15i q^{32} +1.49237e16i q^{33} -1.19854e16 q^{34} +2.10034e16 q^{36} +1.09912e16i q^{37} +1.44515e16i q^{38} +1.70129e17 q^{39} +7.93639e16 q^{41} -1.71737e17i q^{42} +8.36751e16i q^{43} -8.96177e16 q^{44} +2.72243e16 q^{46} +3.57127e17i q^{47} +1.91992e17i q^{48} -3.63941e17 q^{49} +2.04379e18 q^{51} +1.02163e18i q^{52} +8.17016e17i q^{53} -1.71119e18 q^{54} +1.03129e18 q^{56} -2.46432e18i q^{57} +1.48557e18i q^{58} -8.21011e17 q^{59} +4.53683e18 q^{61} +7.81571e18i q^{62} +1.92384e19i q^{63} -1.15292e18 q^{64} +1.52819e19 q^{66} +8.02826e18i q^{67} +1.22730e19i q^{68} -4.64238e18 q^{69} -5.25728e19 q^{71} -2.15075e19i q^{72} -9.28698e18i q^{73} +1.12549e19 q^{74} +1.47984e19 q^{76} -8.20869e19i q^{77} -1.74212e20i q^{78} -1.08861e19 q^{79} +8.22724e19 q^{81} -8.12687e19i q^{82} +4.79994e19i q^{83} -1.75859e20 q^{84} +8.56833e19 q^{86} -2.53325e20i q^{87} +9.17685e19i q^{88} -2.35270e20 q^{89} -9.35779e20 q^{91} -2.78777e19i q^{92} -1.33276e21i q^{93} +3.65698e20 q^{94} +1.96600e20 q^{96} +7.25967e20i q^{97} +3.72675e20i q^{98} -1.71192e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4194304 q^{4} + 259465216 q^{6} - 23733475652 q^{9} - 82653663552 q^{11} - 599242104832 q^{14} + 4398046511104 q^{16} - 14351589304880 q^{19} + 399437207788528 q^{21} - 272068998332416 q^{24} - 38\!\cdots\!44 q^{26}+ \cdots - 54\!\cdots\!24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1024.00i − 0.707107i
\(3\) 174616.i 1.70730i 0.520844 + 0.853652i \(0.325617\pi\)
−0.520844 + 0.853652i \(0.674383\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) 1.78807e8 1.20725
\(7\) − 9.60462e8i − 1.28514i −0.766227 − 0.642570i \(-0.777868\pi\)
0.766227 − 0.642570i \(-0.222132\pi\)
\(8\) 1.07374e9i 0.353553i
\(9\) −2.00304e10 −1.91489
\(10\) 0 0
\(11\) 8.54661e10 0.993507 0.496753 − 0.867892i \(-0.334525\pi\)
0.496753 + 0.867892i \(0.334525\pi\)
\(12\) − 1.83098e11i − 0.853652i
\(13\) − 9.74302e11i − 1.96015i −0.198640 − 0.980073i \(-0.563652\pi\)
0.198640 − 0.980073i \(-0.436348\pi\)
\(14\) −9.83513e11 −0.908732
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) − 1.17045e13i − 1.40812i −0.710143 − 0.704058i \(-0.751370\pi\)
0.710143 − 0.704058i \(-0.248630\pi\)
\(18\) 2.05111e13i 1.35403i
\(19\) −1.41128e13 −0.528081 −0.264041 − 0.964512i \(-0.585055\pi\)
−0.264041 + 0.964512i \(0.585055\pi\)
\(20\) 0 0
\(21\) 1.67712e14 2.19413
\(22\) − 8.75173e13i − 0.702515i
\(23\) 2.65862e13i 0.133818i 0.997759 + 0.0669090i \(0.0213137\pi\)
−0.997759 + 0.0669090i \(0.978686\pi\)
\(24\) −1.87492e14 −0.603623
\(25\) 0 0
\(26\) −9.97685e14 −1.38603
\(27\) − 1.67108e15i − 1.56199i
\(28\) 1.00712e15i 0.642570i
\(29\) −1.45075e15 −0.640346 −0.320173 − 0.947359i \(-0.603741\pi\)
−0.320173 + 0.947359i \(0.603741\pi\)
\(30\) 0 0
\(31\) −7.63253e15 −1.67252 −0.836259 − 0.548335i \(-0.815262\pi\)
−0.836259 + 0.548335i \(0.815262\pi\)
\(32\) − 1.12590e15i − 0.176777i
\(33\) 1.49237e16i 1.69622i
\(34\) −1.19854e16 −0.995688
\(35\) 0 0
\(36\) 2.10034e16 0.957443
\(37\) 1.09912e16i 0.375773i 0.982191 + 0.187886i \(0.0601637\pi\)
−0.982191 + 0.187886i \(0.939836\pi\)
\(38\) 1.44515e16i 0.373410i
\(39\) 1.70129e17 3.34656
\(40\) 0 0
\(41\) 7.93639e16 0.923406 0.461703 − 0.887035i \(-0.347239\pi\)
0.461703 + 0.887035i \(0.347239\pi\)
\(42\) − 1.71737e17i − 1.55148i
\(43\) 8.36751e16i 0.590442i 0.955429 + 0.295221i \(0.0953933\pi\)
−0.955429 + 0.295221i \(0.904607\pi\)
\(44\) −8.96177e16 −0.496753
\(45\) 0 0
\(46\) 2.72243e16 0.0946236
\(47\) 3.57127e17i 0.990365i 0.868789 + 0.495183i \(0.164899\pi\)
−0.868789 + 0.495183i \(0.835101\pi\)
\(48\) 1.91992e17i 0.426826i
\(49\) −3.63941e17 −0.651586
\(50\) 0 0
\(51\) 2.04379e18 2.40408
\(52\) 1.02163e18i 0.980073i
\(53\) 8.17016e17i 0.641702i 0.947130 + 0.320851i \(0.103969\pi\)
−0.947130 + 0.320851i \(0.896031\pi\)
\(54\) −1.71119e18 −1.10449
\(55\) 0 0
\(56\) 1.03129e18 0.454366
\(57\) − 2.46432e18i − 0.901595i
\(58\) 1.48557e18i 0.452793i
\(59\) −8.21011e17 −0.209124 −0.104562 − 0.994518i \(-0.533344\pi\)
−0.104562 + 0.994518i \(0.533344\pi\)
\(60\) 0 0
\(61\) 4.53683e18 0.814310 0.407155 − 0.913359i \(-0.366521\pi\)
0.407155 + 0.913359i \(0.366521\pi\)
\(62\) 7.81571e18i 1.18265i
\(63\) 1.92384e19i 2.46090i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) 1.52819e19 1.19941
\(67\) 8.02826e18i 0.538066i 0.963131 + 0.269033i \(0.0867041\pi\)
−0.963131 + 0.269033i \(0.913296\pi\)
\(68\) 1.22730e19i 0.704058i
\(69\) −4.64238e18 −0.228468
\(70\) 0 0
\(71\) −5.25728e19 −1.91668 −0.958338 − 0.285638i \(-0.907794\pi\)
−0.958338 + 0.285638i \(0.907794\pi\)
\(72\) − 2.15075e19i − 0.677014i
\(73\) − 9.28698e18i − 0.252921i −0.991972 − 0.126460i \(-0.959638\pi\)
0.991972 − 0.126460i \(-0.0403616\pi\)
\(74\) 1.12549e19 0.265711
\(75\) 0 0
\(76\) 1.47984e19 0.264041
\(77\) − 8.20869e19i − 1.27680i
\(78\) − 1.74212e20i − 2.36638i
\(79\) −1.08861e19 −0.129356 −0.0646781 − 0.997906i \(-0.520602\pi\)
−0.0646781 + 0.997906i \(0.520602\pi\)
\(80\) 0 0
\(81\) 8.22724e19 0.751902
\(82\) − 8.12687e19i − 0.652947i
\(83\) 4.79994e19i 0.339560i 0.985482 + 0.169780i \(0.0543057\pi\)
−0.985482 + 0.169780i \(0.945694\pi\)
\(84\) −1.75859e20 −1.09706
\(85\) 0 0
\(86\) 8.56833e19 0.417506
\(87\) − 2.53325e20i − 1.09327i
\(88\) 9.17685e19i 0.351258i
\(89\) −2.35270e20 −0.799783 −0.399891 − 0.916563i \(-0.630952\pi\)
−0.399891 + 0.916563i \(0.630952\pi\)
\(90\) 0 0
\(91\) −9.35779e20 −2.51906
\(92\) − 2.78777e19i − 0.0669090i
\(93\) − 1.33276e21i − 2.85549i
\(94\) 3.65698e20 0.700294
\(95\) 0 0
\(96\) 1.96600e20 0.301812
\(97\) 7.25967e20i 0.999571i 0.866149 + 0.499786i \(0.166588\pi\)
−0.866149 + 0.499786i \(0.833412\pi\)
\(98\) 3.72675e20i 0.460741i
\(99\) −1.71192e21 −1.90245
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 50.22.b.f.49.2 4
5.2 odd 4 50.22.a.f.1.2 2
5.3 odd 4 10.22.a.b.1.1 ✓ 2
5.4 even 2 inner 50.22.b.f.49.3 4
20.3 even 4 80.22.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.b.1.1 ✓ 2 5.3 odd 4
50.22.a.f.1.2 2 5.2 odd 4
50.22.b.f.49.2 4 1.1 even 1 trivial
50.22.b.f.49.3 4 5.4 even 2 inner
80.22.a.d.1.2 2 20.3 even 4