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Properties

Label 50.22.b.e.49.2
Level $50$
Weight $22$
Character 50.49
Analytic conductor $139.739$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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,-4194304,0,63430656] 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: \( x^{4} + 589825x^{2} + 86973087744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(542.558i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.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)
 
\(f(q)\) \(=\) \(q-1024.00i q^{2} +145820. i q^{3} -1.04858e6 q^{4} +1.49320e8 q^{6} +2.36011e8i q^{7} +1.07374e9i q^{8} -1.08031e10 q^{9} +1.49202e11 q^{11} -1.52903e11i q^{12} +4.89350e11i q^{13} +2.41675e11 q^{14} +1.09951e12 q^{16} -5.46288e12i q^{17} +1.10624e13i q^{18} +1.12049e13 q^{19} -3.44151e13 q^{21} -1.52783e14i q^{22} +1.78791e13i q^{23} -1.56573e14 q^{24} +5.01094e14 q^{26} -4.99812e13i q^{27} -2.47475e14i q^{28} +2.82120e15 q^{29} +4.24797e15 q^{31} -1.12590e15i q^{32} +2.17567e16i q^{33} -5.59399e15 q^{34} +1.13279e16 q^{36} -5.74457e16i q^{37} -1.14738e16i q^{38} -7.13570e16 q^{39} -3.51283e16 q^{41} +3.52410e16i q^{42} -2.62144e17i q^{43} -1.56450e17 q^{44} +1.83081e16 q^{46} -1.08406e17i q^{47} +1.60331e17i q^{48} +5.02845e17 q^{49} +7.96597e17 q^{51} -5.13120e17i q^{52} -2.47953e18i q^{53} -5.11807e16 q^{54} -2.53415e17 q^{56} +1.63389e18i q^{57} -2.88891e18i q^{58} -2.14259e17 q^{59} +3.64272e18 q^{61} -4.34992e18i q^{62} -2.54965e18i q^{63} -1.15292e18 q^{64} +2.22788e19 q^{66} +1.13372e19i q^{67} +5.72825e18i q^{68} -2.60712e18 q^{69} +2.44087e19 q^{71} -1.15998e19i q^{72} -4.12244e19i q^{73} -5.88244e19 q^{74} -1.17492e19 q^{76} +3.52133e19i q^{77} +7.30695e19i q^{78} -6.83832e19 q^{79} -1.05716e20 q^{81} +3.59714e19i q^{82} +1.39311e20i q^{83} +3.60868e19 q^{84} -2.68435e20 q^{86} +4.11387e20i q^{87} +1.60205e20i q^{88} +4.04817e20 q^{89} -1.15492e20 q^{91} -1.87475e19i q^{92} +6.19438e20i q^{93} -1.11008e20 q^{94} +1.64179e20 q^{96} +9.18963e20i q^{97} -5.14913e20i q^{98} -1.61185e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4194304 q^{4} + 63430656 q^{6} - 27065634372 q^{9} + 210383554368 q^{11} + 901036761088 q^{14} + 4398046511104 q^{16} - 47642433431600 q^{19} - 21983998409232 q^{21} - 66511863545856 q^{24} + 631719217012736 q^{26}+ \cdots - 29\!\cdots\!24 q^{99}+O(q^{100}) \) 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
\(3\) 145820.i 1.42575i 0.701290 + 0.712876i \(0.252608\pi\)
−0.701290 + 0.712876i \(0.747392\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) 1.49320e8 1.00816
\(7\) 2.36011e8i 0.315793i 0.987456 + 0.157896i \(0.0504712\pi\)
−0.987456 + 0.157896i \(0.949529\pi\)
\(8\) 1.07374e9i 0.353553i
\(9\) −1.08031e10 −1.03277
\(10\) 0 0
\(11\) 1.49202e11 1.73441 0.867206 − 0.497949i \(-0.165913\pi\)
0.867206 + 0.497949i \(0.165913\pi\)
\(12\) − 1.52903e11i − 0.712876i
\(13\) 4.89350e11i 0.984497i 0.870455 + 0.492248i \(0.163825\pi\)
−0.870455 + 0.492248i \(0.836175\pi\)
\(14\) 2.41675e11 0.223299
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) − 5.46288e12i − 0.657216i −0.944466 − 0.328608i \(-0.893421\pi\)
0.944466 − 0.328608i \(-0.106579\pi\)
\(18\) 1.10624e13i 0.730277i
\(19\) 1.12049e13 0.419270 0.209635 − 0.977780i \(-0.432772\pi\)
0.209635 + 0.977780i \(0.432772\pi\)
\(20\) 0 0
\(21\) −3.44151e13 −0.450242
\(22\) − 1.52783e14i − 1.22641i
\(23\) 1.78791e13i 0.0899916i 0.998987 + 0.0449958i \(0.0143275\pi\)
−0.998987 + 0.0449958i \(0.985673\pi\)
\(24\) −1.56573e14 −0.504079
\(25\) 0 0
\(26\) 5.01094e14 0.696144
\(27\) − 4.99812e13i − 0.0467183i
\(28\) − 2.47475e14i − 0.157896i
\(29\) 2.82120e15 1.24525 0.622623 − 0.782522i \(-0.286067\pi\)
0.622623 + 0.782522i \(0.286067\pi\)
\(30\) 0 0
\(31\) 4.24797e15 0.930858 0.465429 − 0.885085i \(-0.345900\pi\)
0.465429 + 0.885085i \(0.345900\pi\)
\(32\) − 1.12590e15i − 0.176777i
\(33\) 2.17567e16i 2.47284i
\(34\) −5.59399e15 −0.464722
\(35\) 0 0
\(36\) 1.13279e16 0.516384
\(37\) − 5.74457e16i − 1.96399i −0.188909 − 0.981995i \(-0.560495\pi\)
0.188909 − 0.981995i \(-0.439505\pi\)
\(38\) − 1.14738e16i − 0.296469i
\(39\) −7.13570e16 −1.40365
\(40\) 0 0
\(41\) −3.51283e16 −0.408721 −0.204361 − 0.978896i \(-0.565512\pi\)
−0.204361 + 0.978896i \(0.565512\pi\)
\(42\) 3.52410e16i 0.318369i
\(43\) − 2.62144e17i − 1.84978i −0.380233 − 0.924891i \(-0.624156\pi\)
0.380233 − 0.924891i \(-0.375844\pi\)
\(44\) −1.56450e17 −0.867206
\(45\) 0 0
\(46\) 1.83081e16 0.0636337
\(47\) − 1.08406e17i − 0.300626i −0.988638 − 0.150313i \(-0.951972\pi\)
0.988638 − 0.150313i \(-0.0480282\pi\)
\(48\) 1.60331e17i 0.356438i
\(49\) 5.02845e17 0.900275
\(50\) 0 0
\(51\) 7.96597e17 0.937027
\(52\) − 5.13120e17i − 0.492248i
\(53\) − 2.47953e18i − 1.94748i −0.227662 − 0.973740i \(-0.573108\pi\)
0.227662 − 0.973740i \(-0.426892\pi\)
\(54\) −5.11807e16 −0.0330348
\(55\) 0 0
\(56\) −2.53415e17 −0.111650
\(57\) 1.63389e18i 0.597775i
\(58\) − 2.88891e18i − 0.880521i
\(59\) −2.14259e17 −0.0545748 −0.0272874 − 0.999628i \(-0.508687\pi\)
−0.0272874 + 0.999628i \(0.508687\pi\)
\(60\) 0 0
\(61\) 3.64272e18 0.653826 0.326913 − 0.945054i \(-0.393992\pi\)
0.326913 + 0.945054i \(0.393992\pi\)
\(62\) − 4.34992e18i − 0.658216i
\(63\) − 2.54965e18i − 0.326141i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) 2.22788e19 1.74856
\(67\) 1.13372e19i 0.759835i 0.925020 + 0.379918i \(0.124048\pi\)
−0.925020 + 0.379918i \(0.875952\pi\)
\(68\) 5.72825e18i 0.328608i
\(69\) −2.60712e18 −0.128306
\(70\) 0 0
\(71\) 2.44087e19 0.889882 0.444941 − 0.895560i \(-0.353225\pi\)
0.444941 + 0.895560i \(0.353225\pi\)
\(72\) − 1.15998e19i − 0.365138i
\(73\) − 4.12244e19i − 1.12270i −0.827578 − 0.561350i \(-0.810282\pi\)
0.827578 − 0.561350i \(-0.189718\pi\)
\(74\) −5.88244e19 −1.38875
\(75\) 0 0
\(76\) −1.17492e19 −0.209635
\(77\) 3.52133e19i 0.547715i
\(78\) 7.30695e19i 0.992529i
\(79\) −6.83832e19 −0.812578 −0.406289 − 0.913745i \(-0.633177\pi\)
−0.406289 + 0.913745i \(0.633177\pi\)
\(80\) 0 0
\(81\) −1.05716e20 −0.966159
\(82\) 3.59714e19i 0.289009i
\(83\) 1.39311e20i 0.985522i 0.870165 + 0.492761i \(0.164012\pi\)
−0.870165 + 0.492761i \(0.835988\pi\)
\(84\) 3.60868e19 0.225121
\(85\) 0 0
\(86\) −2.68435e20 −1.30799
\(87\) 4.11387e20i 1.77541i
\(88\) 1.60205e20i 0.613207i
\(89\) 4.04817e20 1.37614 0.688072 − 0.725643i \(-0.258458\pi\)
0.688072 + 0.725643i \(0.258458\pi\)
\(90\) 0 0
\(91\) −1.15492e20 −0.310897
\(92\) − 1.87475e19i − 0.0449958i
\(93\) 6.19438e20i 1.32717i
\(94\) −1.11008e20 −0.212575
\(95\) 0 0
\(96\) 1.64179e20 0.252040
\(97\) 9.18963e20i 1.26530i 0.774436 + 0.632652i \(0.218034\pi\)
−0.774436 + 0.632652i \(0.781966\pi\)
\(98\) − 5.14913e20i − 0.636590i
\(99\) −1.61185e21 −1.79124
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.e.49.2 4
5.2 odd 4 10.22.a.d.1.2 ✓ 2
5.3 odd 4 50.22.a.d.1.1 2
5.4 even 2 inner 50.22.b.e.49.3 4
20.7 even 4 80.22.a.c.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.d.1.2 ✓ 2 5.2 odd 4
50.22.a.d.1.1 2 5.3 odd 4
50.22.b.e.49.2 4 1.1 even 1 trivial
50.22.b.e.49.3 4 5.4 even 2 inner
80.22.a.c.1.1 2 20.7 even 4