Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [50,26,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, 26); 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, 26, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
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,-67108864,0,3111714816] 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: \(197.998389976\)
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: \( x^{4} + 53353x^{2} + 711608976 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 2)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(-162.829i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.26.b.e.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)
 
\(f(q)\) \(=\) \(q-4096.00i q^{2} +1.75788e6i q^{3} -1.67772e7 q^{4} +7.20027e9 q^{6} +3.04153e10i q^{7} +6.87195e10i q^{8} -2.24285e12 q^{9} +2.58704e12 q^{11} -2.94923e13i q^{12} -9.57327e13i q^{13} +1.24581e14 q^{14} +2.81475e14 q^{16} +1.64685e15i q^{17} +9.18671e15i q^{18} -4.95030e15 q^{19} -5.34664e16 q^{21} -1.05965e16i q^{22} -1.07650e16i q^{23} -1.20801e17 q^{24} -3.92121e17 q^{26} -2.45323e18i q^{27} -5.10284e17i q^{28} +1.36741e18 q^{29} -4.42000e18 q^{31} -1.15292e18i q^{32} +4.54771e18i q^{33} +6.74549e18 q^{34} +3.76288e19 q^{36} -1.01944e19i q^{37} +2.02764e19i q^{38} +1.68287e20 q^{39} +1.58687e20 q^{41} +2.18999e20i q^{42} +1.83575e20i q^{43} -4.34034e19 q^{44} -4.40934e19 q^{46} -1.40203e21i q^{47} +4.94799e20i q^{48} +4.15978e20 q^{49} -2.89496e21 q^{51} +1.60613e21i q^{52} -1.99903e21i q^{53} -1.00484e22 q^{54} -2.09012e21 q^{56} -8.70202e21i q^{57} -5.60091e21i q^{58} +4.16691e21 q^{59} +3.42128e22 q^{61} +1.81043e22i q^{62} -6.82170e22i q^{63} -4.72237e21 q^{64} +1.86274e22 q^{66} -8.67051e22i q^{67} -2.76295e22i q^{68} +1.89236e22 q^{69} -5.13159e22 q^{71} -1.54128e23i q^{72} +3.49147e22i q^{73} -4.17563e22 q^{74} +8.30522e22 q^{76} +7.86857e22i q^{77} -6.89302e23i q^{78} -2.91588e23 q^{79} +2.41214e24 q^{81} -6.49981e23i q^{82} -1.64916e24i q^{83} +8.97018e23 q^{84} +7.51925e23 q^{86} +2.40374e24i q^{87} +1.77780e23i q^{88} -8.74435e23 q^{89} +2.91174e24 q^{91} +1.80607e23i q^{92} -7.76982e24i q^{93} -5.74273e24 q^{94} +2.02670e24 q^{96} -1.00608e25i q^{97} -1.70384e24i q^{98} -5.80235e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 67108864 q^{4} + 3111714816 q^{6} - 6589062865332 q^{9} + 16646069220528 q^{11} + 3084590645248 q^{14} + 11\!\cdots\!24 q^{16} + 954158485898800 q^{19} - 18\!\cdots\!12 q^{21} - 52\!\cdots\!56 q^{24}+ \cdots - 23\!\cdots\!24 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4096.00i − 0.707107i
\(3\) 1.75788e6i 1.90974i 0.297031 + 0.954868i \(0.404004\pi\)
−0.297031 + 0.954868i \(0.595996\pi\)
\(4\) −1.67772e7 −0.500000
\(5\) 0 0
\(6\) 7.20027e9 1.35039
\(7\) 3.04153e10i 0.830552i 0.909696 + 0.415276i \(0.136315\pi\)
−0.909696 + 0.415276i \(0.863685\pi\)
\(8\) 6.87195e10i 0.353553i
\(9\) −2.24285e12 −2.64709
\(10\) 0 0
\(11\) 2.58704e12 0.248539 0.124270 − 0.992248i \(-0.460341\pi\)
0.124270 + 0.992248i \(0.460341\pi\)
\(12\) − 2.94923e13i − 0.954868i
\(13\) − 9.57327e13i − 1.13964i −0.821769 − 0.569821i \(-0.807012\pi\)
0.821769 − 0.569821i \(-0.192988\pi\)
\(14\) 1.24581e14 0.587289
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) 1.64685e15i 0.685555i 0.939417 + 0.342777i \(0.111368\pi\)
−0.939417 + 0.342777i \(0.888632\pi\)
\(18\) 9.18671e15i 1.87178i
\(19\) −4.95030e15 −0.513111 −0.256555 − 0.966530i \(-0.582588\pi\)
−0.256555 + 0.966530i \(0.582588\pi\)
\(20\) 0 0
\(21\) −5.34664e16 −1.58613
\(22\) − 1.05965e16i − 0.175744i
\(23\) − 1.07650e16i − 0.102427i −0.998688 − 0.0512136i \(-0.983691\pi\)
0.998688 − 0.0512136i \(-0.0163089\pi\)
\(24\) −1.20801e17 −0.675194
\(25\) 0 0
\(26\) −3.92121e17 −0.805849
\(27\) − 2.45323e18i − 3.14551i
\(28\) − 5.10284e17i − 0.415276i
\(29\) 1.36741e18 0.717668 0.358834 − 0.933401i \(-0.383174\pi\)
0.358834 + 0.933401i \(0.383174\pi\)
\(30\) 0 0
\(31\) −4.42000e18 −1.00786 −0.503931 − 0.863744i \(-0.668113\pi\)
−0.503931 + 0.863744i \(0.668113\pi\)
\(32\) − 1.15292e18i − 0.176777i
\(33\) 4.54771e18i 0.474645i
\(34\) 6.74549e18 0.484760
\(35\) 0 0
\(36\) 3.76288e19 1.32355
\(37\) − 1.01944e19i − 0.254590i −0.991865 − 0.127295i \(-0.959371\pi\)
0.991865 − 0.127295i \(-0.0406294\pi\)
\(38\) 2.02764e19i 0.362824i
\(39\) 1.68287e20 2.17642
\(40\) 0 0
\(41\) 1.58687e20 1.09835 0.549177 − 0.835706i \(-0.314941\pi\)
0.549177 + 0.835706i \(0.314941\pi\)
\(42\) 2.18999e20i 1.12157i
\(43\) 1.83575e20i 0.700582i 0.936641 + 0.350291i \(0.113917\pi\)
−0.936641 + 0.350291i \(0.886083\pi\)
\(44\) −4.34034e19 −0.124270
\(45\) 0 0
\(46\) −4.40934e19 −0.0724270
\(47\) − 1.40203e21i − 1.76009i −0.474890 − 0.880045i \(-0.657512\pi\)
0.474890 − 0.880045i \(-0.342488\pi\)
\(48\) 4.94799e20i 0.477434i
\(49\) 4.15978e20 0.310184
\(50\) 0 0
\(51\) −2.89496e21 −1.30923
\(52\) 1.60613e21i 0.569821i
\(53\) − 1.99903e21i − 0.558946i −0.960154 − 0.279473i \(-0.909840\pi\)
0.960154 − 0.279473i \(-0.0901598\pi\)
\(54\) −1.00484e22 −2.22421
\(55\) 0 0
\(56\) −2.09012e21 −0.293644
\(57\) − 8.70202e21i − 0.979906i
\(58\) − 5.60091e21i − 0.507468i
\(59\) 4.16691e21 0.304905 0.152452 − 0.988311i \(-0.451283\pi\)
0.152452 + 0.988311i \(0.451283\pi\)
\(60\) 0 0
\(61\) 3.42128e22 1.65031 0.825156 − 0.564904i \(-0.191087\pi\)
0.825156 + 0.564904i \(0.191087\pi\)
\(62\) 1.81043e22i 0.712666i
\(63\) − 6.82170e22i − 2.19855i
\(64\) −4.72237e21 −0.125000
\(65\) 0 0
\(66\) 1.86274e22 0.335624
\(67\) − 8.67051e22i − 1.29452i −0.762268 − 0.647261i \(-0.775914\pi\)
0.762268 − 0.647261i \(-0.224086\pi\)
\(68\) − 2.76295e22i − 0.342777i
\(69\) 1.89236e22 0.195609
\(70\) 0 0
\(71\) −5.13159e22 −0.371127 −0.185564 − 0.982632i \(-0.559411\pi\)
−0.185564 + 0.982632i \(0.559411\pi\)
\(72\) − 1.54128e23i − 0.935888i
\(73\) 3.49147e22i 0.178432i 0.996012 + 0.0892159i \(0.0284361\pi\)
−0.996012 + 0.0892159i \(0.971564\pi\)
\(74\) −4.17563e22 −0.180022
\(75\) 0 0
\(76\) 8.30522e22 0.256555
\(77\) 7.86857e22i 0.206425i
\(78\) − 6.89302e23i − 1.53896i
\(79\) −2.91588e23 −0.555176 −0.277588 − 0.960700i \(-0.589535\pi\)
−0.277588 + 0.960700i \(0.589535\pi\)
\(80\) 0 0
\(81\) 2.41214e24 3.36000
\(82\) − 6.49981e23i − 0.776654i
\(83\) − 1.64916e24i − 1.69351i −0.531986 − 0.846753i \(-0.678554\pi\)
0.531986 − 0.846753i \(-0.321446\pi\)
\(84\) 8.97018e23 0.793067
\(85\) 0 0
\(86\) 7.51925e23 0.495386
\(87\) 2.40374e24i 1.37056i
\(88\) 1.77780e23i 0.0878720i
\(89\) −8.74435e23 −0.375277 −0.187639 − 0.982238i \(-0.560083\pi\)
−0.187639 + 0.982238i \(0.560083\pi\)
\(90\) 0 0
\(91\) 2.91174e24 0.946532
\(92\) 1.80607e23i 0.0512136i
\(93\) − 7.76982e24i − 1.92475i
\(94\) −5.74273e24 −1.24457
\(95\) 0 0
\(96\) 2.02670e24 0.337597
\(97\) − 1.00608e25i − 1.47227i −0.676835 − 0.736134i \(-0.736649\pi\)
0.676835 − 0.736134i \(-0.263351\pi\)
\(98\) − 1.70384e24i − 0.219333i
\(99\) −5.80235e24 −0.657907
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 50.26.b.e.49.2 4
5.2 odd 4 2.26.a.b.1.2 ✓ 2
5.3 odd 4 50.26.a.c.1.1 2
5.4 even 2 inner 50.26.b.e.49.3 4
15.2 even 4 18.26.a.e.1.2 2
20.7 even 4 16.26.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.26.a.b.1.2 ✓ 2 5.2 odd 4
16.26.a.c.1.1 2 20.7 even 4
18.26.a.e.1.2 2 15.2 even 4
50.26.a.c.1.1 2 5.3 odd 4
50.26.b.e.49.2 4 1.1 even 1 trivial
50.26.b.e.49.3 4 5.4 even 2 inner