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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 49.1
Root \(163.829i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.26.b.e.49.4

$q$-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)
 
\(f(q)\) \(=\) \(q-4096.00i q^{2} -1.37803e6i q^{3} -1.67772e7 q^{4} -5.64442e9 q^{6} -3.00388e10i q^{7} +6.87195e10i q^{8} -1.05168e12 q^{9} +5.73599e12 q^{11} +2.31195e13i q^{12} -1.07343e13i q^{13} -1.23039e14 q^{14} +2.81475e14 q^{16} -2.97473e15i q^{17} +4.30769e15i q^{18} +5.42738e15 q^{19} -4.13944e16 q^{21} -2.34946e16i q^{22} -1.04540e17i q^{23} +9.46976e16 q^{24} -4.39677e16 q^{26} +2.81659e17i q^{27} +5.03967e17i q^{28} -3.09182e18 q^{29} -4.26809e18 q^{31} -1.15292e18i q^{32} -7.90437e18i q^{33} -1.21845e19 q^{34} +1.76443e19 q^{36} +4.51028e19i q^{37} -2.22305e19i q^{38} -1.47922e19 q^{39} -7.56724e19 q^{41} +1.69551e20i q^{42} -1.39036e20i q^{43} -9.62339e19 q^{44} -4.28196e20 q^{46} -4.67328e20i q^{47} -3.87881e20i q^{48} +4.38741e20 q^{49} -4.09927e21 q^{51} +1.80092e20i q^{52} -1.24315e21i q^{53} +1.15368e21 q^{54} +2.06425e21 q^{56} -7.47909e21i q^{57} +1.26641e22i q^{58} +1.32468e22 q^{59} -2.36984e20 q^{61} +1.74821e22i q^{62} +3.15912e22i q^{63} -4.72237e21 q^{64} -3.23763e22 q^{66} +5.36922e22i q^{67} +4.99076e22i q^{68} -1.44059e23 q^{69} -2.27032e23 q^{71} -7.22710e22i q^{72} +2.78528e23i q^{73} +1.84741e23 q^{74} -9.10563e22 q^{76} -1.72302e23i q^{77} +6.05889e22i q^{78} -6.36958e23 q^{79} -5.02942e23 q^{81} +3.09954e23i q^{82} +1.19662e24i q^{83} +6.94482e23 q^{84} -5.69491e23 q^{86} +4.26063e24i q^{87} +3.94174e23i q^{88} +3.22134e24 q^{89} -3.22445e23 q^{91} +1.75389e24i q^{92} +5.88156e24i q^{93} -1.91418e24 q^{94} -1.58876e24 q^{96} -3.58465e24i q^{97} -1.79708e24i q^{98} -6.03243e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 67108864 q^{4} + 3111714816 q^{6} - 6589062865332 q^{9} + 16646069220528 q^{11} + 3084590645248 q^{14} + 11\!\cdots\!24 q^{16} + 954158485898800 q^{19} - 18\!\cdots\!12 q^{21} - 52\!\cdots\!56 q^{24}+ \cdots - 23\!\cdots\!24 q^{99}+O(q^{100}) \) 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
\(3\) − 1.37803e6i − 1.49707i −0.663093 − 0.748537i \(-0.730757\pi\)
0.663093 − 0.748537i \(-0.269243\pi\)
\(4\) −1.67772e7 −0.500000
\(5\) 0 0
\(6\) −5.64442e9 −1.05859
\(7\) − 3.00388e10i − 0.820270i −0.912025 − 0.410135i \(-0.865482\pi\)
0.912025 − 0.410135i \(-0.134518\pi\)
\(8\) 6.87195e10i 0.353553i
\(9\) −1.05168e12 −1.24123
\(10\) 0 0
\(11\) 5.73599e12 0.551061 0.275531 − 0.961292i \(-0.411146\pi\)
0.275531 + 0.961292i \(0.411146\pi\)
\(12\) 2.31195e13i 0.748537i
\(13\) − 1.07343e13i − 0.127786i −0.997957 − 0.0638928i \(-0.979648\pi\)
0.997957 − 0.0638928i \(-0.0203516\pi\)
\(14\) −1.23039e14 −0.580018
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) − 2.97473e15i − 1.23833i −0.785262 − 0.619164i \(-0.787472\pi\)
0.785262 − 0.619164i \(-0.212528\pi\)
\(18\) 4.30769e15i 0.877683i
\(19\) 5.42738e15 0.562561 0.281281 − 0.959626i \(-0.409241\pi\)
0.281281 + 0.959626i \(0.409241\pi\)
\(20\) 0 0
\(21\) −4.13944e16 −1.22800
\(22\) − 2.34946e16i − 0.389659i
\(23\) − 1.04540e17i − 0.994683i −0.867555 − 0.497341i \(-0.834310\pi\)
0.867555 − 0.497341i \(-0.165690\pi\)
\(24\) 9.46976e16 0.529296
\(25\) 0 0
\(26\) −4.39677e16 −0.0903581
\(27\) 2.81659e17i 0.361141i
\(28\) 5.03967e17i 0.410135i
\(29\) −3.09182e18 −1.62270 −0.811352 − 0.584558i \(-0.801268\pi\)
−0.811352 + 0.584558i \(0.801268\pi\)
\(30\) 0 0
\(31\) −4.26809e18 −0.973222 −0.486611 − 0.873619i \(-0.661767\pi\)
−0.486611 + 0.873619i \(0.661767\pi\)
\(32\) − 1.15292e18i − 0.176777i
\(33\) − 7.90437e18i − 0.824980i
\(34\) −1.21845e19 −0.875630
\(35\) 0 0
\(36\) 1.76443e19 0.620616
\(37\) 4.51028e19i 1.12637i 0.826330 + 0.563186i \(0.190424\pi\)
−0.826330 + 0.563186i \(0.809576\pi\)
\(38\) − 2.22305e19i − 0.397791i
\(39\) −1.47922e19 −0.191305
\(40\) 0 0
\(41\) −7.56724e19 −0.523769 −0.261884 − 0.965099i \(-0.584344\pi\)
−0.261884 + 0.965099i \(0.584344\pi\)
\(42\) 1.69551e20i 0.868330i
\(43\) − 1.39036e20i − 0.530605i −0.964165 − 0.265302i \(-0.914528\pi\)
0.964165 − 0.265302i \(-0.0854718\pi\)
\(44\) −9.62339e19 −0.275531
\(45\) 0 0
\(46\) −4.28196e20 −0.703347
\(47\) − 4.67328e20i − 0.586676i −0.956009 − 0.293338i \(-0.905234\pi\)
0.956009 − 0.293338i \(-0.0947662\pi\)
\(48\) − 3.87881e20i − 0.374269i
\(49\) 4.38741e20 0.327158
\(50\) 0 0
\(51\) −4.09927e21 −1.85387
\(52\) 1.80092e20i 0.0638928i
\(53\) − 1.24315e21i − 0.347595i −0.984781 − 0.173797i \(-0.944396\pi\)
0.984781 − 0.173797i \(-0.0556038\pi\)
\(54\) 1.15368e21 0.255365
\(55\) 0 0
\(56\) 2.06425e21 0.290009
\(57\) − 7.47909e21i − 0.842196i
\(58\) 1.26641e22i 1.14743i
\(59\) 1.32468e22 0.969303 0.484652 − 0.874707i \(-0.338946\pi\)
0.484652 + 0.874707i \(0.338946\pi\)
\(60\) 0 0
\(61\) −2.36984e20 −0.0114313 −0.00571565 − 0.999984i \(-0.501819\pi\)
−0.00571565 + 0.999984i \(0.501819\pi\)
\(62\) 1.74821e22i 0.688172i
\(63\) 3.15912e22i 1.01814i
\(64\) −4.72237e21 −0.125000
\(65\) 0 0
\(66\) −3.23763e22 −0.583349
\(67\) 5.36922e22i 0.801634i 0.916158 + 0.400817i \(0.131274\pi\)
−0.916158 + 0.400817i \(0.868726\pi\)
\(68\) 4.99076e22i 0.619164i
\(69\) −1.44059e23 −1.48911
\(70\) 0 0
\(71\) −2.27032e23 −1.64194 −0.820971 − 0.570970i \(-0.806567\pi\)
−0.820971 + 0.570970i \(0.806567\pi\)
\(72\) − 7.22710e22i − 0.438842i
\(73\) 2.78528e23i 1.42342i 0.702473 + 0.711710i \(0.252079\pi\)
−0.702473 + 0.711710i \(0.747921\pi\)
\(74\) 1.84741e23 0.796465
\(75\) 0 0
\(76\) −9.10563e22 −0.281281
\(77\) − 1.72302e23i − 0.452019i
\(78\) 6.05889e22i 0.135273i
\(79\) −6.36958e23 −1.21275 −0.606376 − 0.795178i \(-0.707378\pi\)
−0.606376 + 0.795178i \(0.707378\pi\)
\(80\) 0 0
\(81\) −5.02942e23 −0.700576
\(82\) 3.09954e23i 0.370360i
\(83\) 1.19662e24i 1.22880i 0.788996 + 0.614398i \(0.210601\pi\)
−0.788996 + 0.614398i \(0.789399\pi\)
\(84\) 6.94482e23 0.614002
\(85\) 0 0
\(86\) −5.69491e23 −0.375194
\(87\) 4.26063e24i 2.42931i
\(88\) 3.94174e23i 0.194830i
\(89\) 3.22134e24 1.38249 0.691244 − 0.722621i \(-0.257063\pi\)
0.691244 + 0.722621i \(0.257063\pi\)
\(90\) 0 0
\(91\) −3.22445e23 −0.104819
\(92\) 1.75389e24i 0.497341i
\(93\) 5.88156e24i 1.45699i
\(94\) −1.91418e24 −0.414843
\(95\) 0 0
\(96\) −1.58876e24 −0.264648
\(97\) − 3.58465e24i − 0.524566i −0.964991 − 0.262283i \(-0.915525\pi\)
0.964991 − 0.262283i \(-0.0844754\pi\)
\(98\) − 1.79708e24i − 0.231335i
\(99\) −6.03243e24 −0.683994
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.49.1 4
5.2 odd 4 2.26.a.b.1.1 ✓ 2
5.3 odd 4 50.26.a.c.1.2 2
5.4 even 2 inner 50.26.b.e.49.4 4
15.2 even 4 18.26.a.e.1.1 2
20.7 even 4 16.26.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.26.a.b.1.1 ✓ 2 5.2 odd 4
16.26.a.c.1.2 2 20.7 even 4
18.26.a.e.1.1 2 15.2 even 4
50.26.a.c.1.2 2 5.3 odd 4
50.26.b.e.49.1 4 1.1 even 1 trivial
50.26.b.e.49.4 4 5.4 even 2 inner