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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 = [6,0,0,-6291456,0,94992384] 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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 61918633x^{4} + 958456664523288x^{2} + 127494184747700656656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 5^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.3
Root \(5371.39i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.g.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)
 
\(f(q)\) \(=\) \(q-1024.00i q^{2} +176613. i q^{3} -1.04858e6 q^{4} +1.80851e8 q^{6} +1.28981e9i q^{7} +1.07374e9i q^{8} -2.07317e10 q^{9} -8.21479e9 q^{11} -1.85192e11i q^{12} -2.70307e11i q^{13} +1.32077e12 q^{14} +1.09951e12 q^{16} +1.05954e13i q^{17} +2.12292e13i q^{18} +2.60103e13 q^{19} -2.27798e14 q^{21} +8.41194e12i q^{22} +3.27037e14i q^{23} -1.89636e14 q^{24} -2.76795e14 q^{26} -1.81405e15i q^{27} -1.35247e15i q^{28} -2.01684e15 q^{29} -7.82354e15 q^{31} -1.12590e15i q^{32} -1.45084e15i q^{33} +1.08496e16 q^{34} +2.17387e16 q^{36} +2.12834e16i q^{37} -2.66345e16i q^{38} +4.77397e16 q^{39} -9.00182e16 q^{41} +2.33265e17i q^{42} +6.03505e16i q^{43} +8.61383e15 q^{44} +3.34886e17 q^{46} +3.50369e17i q^{47} +1.94188e17i q^{48} -1.10508e18 q^{49} -1.87127e18 q^{51} +2.83438e17i q^{52} -1.24046e18i q^{53} -1.85758e18 q^{54} -1.38493e18 q^{56} +4.59374e18i q^{57} +2.06524e18i q^{58} +5.56450e18 q^{59} -5.81826e18 q^{61} +8.01130e18i q^{62} -2.67400e19i q^{63} -1.15292e18 q^{64} -1.48566e18 q^{66} +2.21752e19i q^{67} -1.11100e19i q^{68} -5.77588e19 q^{69} +3.61226e19 q^{71} -2.22605e19i q^{72} +4.54704e19i q^{73} +2.17942e19 q^{74} -2.72738e19 q^{76} -1.05956e19i q^{77} -4.88854e19i q^{78} +9.81840e19 q^{79} +1.03523e20 q^{81} +9.21786e19i q^{82} +4.19887e19i q^{83} +2.38863e20 q^{84} +6.17989e19 q^{86} -3.56199e20i q^{87} -8.82056e18i q^{88} +3.94595e20 q^{89} +3.48646e20 q^{91} -3.42923e20i q^{92} -1.38174e21i q^{93} +3.58778e20 q^{94} +1.98848e20 q^{96} -6.34667e20i q^{97} +1.13160e21i q^{98} +1.70306e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6291456 q^{4} + 94992384 q^{6} - 50125674708 q^{9} - 155131852698 q^{11} + 1867315679232 q^{14} + 6597069766656 q^{16} + 47919252129450 q^{19} - 452827014956148 q^{21} - 99606734045184 q^{24} + 60018091106304 q^{26}+ \cdots + 23\!\cdots\!64 q^{99}+O(q^{100}) \) 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
\(3\) 176613.i 1.72683i 0.504497 + 0.863413i \(0.331678\pi\)
−0.504497 + 0.863413i \(0.668322\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) 1.80851e8 1.22105
\(7\) 1.28981e9i 1.72583i 0.505349 + 0.862915i \(0.331364\pi\)
−0.505349 + 0.862915i \(0.668636\pi\)
\(8\) 1.07374e9i 0.353553i
\(9\) −2.07317e10 −1.98193
\(10\) 0 0
\(11\) −8.21479e9 −0.0954934 −0.0477467 − 0.998859i \(-0.515204\pi\)
−0.0477467 + 0.998859i \(0.515204\pi\)
\(12\) − 1.85192e11i − 0.863413i
\(13\) − 2.70307e11i − 0.543817i −0.962323 − 0.271908i \(-0.912345\pi\)
0.962323 − 0.271908i \(-0.0876548\pi\)
\(14\) 1.32077e12 1.22035
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 1.05954e13i 1.27468i 0.770582 + 0.637341i \(0.219966\pi\)
−0.770582 + 0.637341i \(0.780034\pi\)
\(18\) 2.12292e13i 1.40144i
\(19\) 2.60103e13 0.973267 0.486634 − 0.873606i \(-0.338225\pi\)
0.486634 + 0.873606i \(0.338225\pi\)
\(20\) 0 0
\(21\) −2.27798e14 −2.98021
\(22\) 8.41194e12i 0.0675240i
\(23\) 3.27037e14i 1.64609i 0.567974 + 0.823046i \(0.307727\pi\)
−0.567974 + 0.823046i \(0.692273\pi\)
\(24\) −1.89636e14 −0.610525
\(25\) 0 0
\(26\) −2.76795e14 −0.384536
\(27\) − 1.81405e15i − 1.69562i
\(28\) − 1.35247e15i − 0.862915i
\(29\) −2.01684e15 −0.890208 −0.445104 − 0.895479i \(-0.646833\pi\)
−0.445104 + 0.895479i \(0.646833\pi\)
\(30\) 0 0
\(31\) −7.82354e15 −1.71437 −0.857187 − 0.515006i \(-0.827790\pi\)
−0.857187 + 0.515006i \(0.827790\pi\)
\(32\) − 1.12590e15i − 0.176777i
\(33\) − 1.45084e15i − 0.164900i
\(34\) 1.08496e16 0.901336
\(35\) 0 0
\(36\) 2.17387e16 0.990965
\(37\) 2.12834e16i 0.727649i 0.931467 + 0.363825i \(0.118529\pi\)
−0.931467 + 0.363825i \(0.881471\pi\)
\(38\) − 2.66345e16i − 0.688204i
\(39\) 4.77397e16 0.939077
\(40\) 0 0
\(41\) −9.00182e16 −1.04737 −0.523684 − 0.851912i \(-0.675443\pi\)
−0.523684 + 0.851912i \(0.675443\pi\)
\(42\) 2.33265e17i 2.10733i
\(43\) 6.03505e16i 0.425855i 0.977068 + 0.212928i \(0.0682999\pi\)
−0.977068 + 0.212928i \(0.931700\pi\)
\(44\) 8.61383e15 0.0477467
\(45\) 0 0
\(46\) 3.34886e17 1.16396
\(47\) 3.50369e17i 0.971624i 0.874063 + 0.485812i \(0.161476\pi\)
−0.874063 + 0.485812i \(0.838524\pi\)
\(48\) 1.94188e17i 0.431707i
\(49\) −1.10508e18 −1.97849
\(50\) 0 0
\(51\) −1.87127e18 −2.20115
\(52\) 2.83438e17i 0.271908i
\(53\) − 1.24046e18i − 0.974284i −0.873323 − 0.487142i \(-0.838039\pi\)
0.873323 − 0.487142i \(-0.161961\pi\)
\(54\) −1.85758e18 −1.19899
\(55\) 0 0
\(56\) −1.38493e18 −0.610173
\(57\) 4.59374e18i 1.68066i
\(58\) 2.06524e18i 0.629472i
\(59\) 5.56450e18 1.41736 0.708680 − 0.705530i \(-0.249291\pi\)
0.708680 + 0.705530i \(0.249291\pi\)
\(60\) 0 0
\(61\) −5.81826e18 −1.04431 −0.522155 − 0.852850i \(-0.674872\pi\)
−0.522155 + 0.852850i \(0.674872\pi\)
\(62\) 8.01130e18i 1.21224i
\(63\) − 2.67400e19i − 3.42047i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) −1.48566e18 −0.116602
\(67\) 2.21752e19i 1.48621i 0.669172 + 0.743107i \(0.266649\pi\)
−0.669172 + 0.743107i \(0.733351\pi\)
\(68\) − 1.11100e19i − 0.637341i
\(69\) −5.77588e19 −2.84252
\(70\) 0 0
\(71\) 3.61226e19 1.31694 0.658471 − 0.752606i \(-0.271204\pi\)
0.658471 + 0.752606i \(0.271204\pi\)
\(72\) − 2.22605e19i − 0.700718i
\(73\) 4.54704e19i 1.23834i 0.785258 + 0.619168i \(0.212530\pi\)
−0.785258 + 0.619168i \(0.787470\pi\)
\(74\) 2.17942e19 0.514526
\(75\) 0 0
\(76\) −2.72738e19 −0.486634
\(77\) − 1.05956e19i − 0.164805i
\(78\) − 4.88854e19i − 0.664028i
\(79\) 9.81840e19 1.16669 0.583346 − 0.812224i \(-0.301743\pi\)
0.583346 + 0.812224i \(0.301743\pi\)
\(80\) 0 0
\(81\) 1.03523e20 0.946115
\(82\) 9.21786e19i 0.740602i
\(83\) 4.19887e19i 0.297038i 0.988910 + 0.148519i \(0.0474507\pi\)
−0.988910 + 0.148519i \(0.952549\pi\)
\(84\) 2.38863e20 1.49010
\(85\) 0 0
\(86\) 6.17989e19 0.301125
\(87\) − 3.56199e20i − 1.53723i
\(88\) − 8.82056e18i − 0.0337620i
\(89\) 3.94595e20 1.34139 0.670697 − 0.741732i \(-0.265995\pi\)
0.670697 + 0.741732i \(0.265995\pi\)
\(90\) 0 0
\(91\) 3.48646e20 0.938535
\(92\) − 3.42923e20i − 0.823046i
\(93\) − 1.38174e21i − 2.96042i
\(94\) 3.58778e20 0.687042
\(95\) 0 0
\(96\) 1.98848e20 0.305263
\(97\) − 6.34667e20i − 0.873862i −0.899495 − 0.436931i \(-0.856065\pi\)
0.899495 − 0.436931i \(-0.143935\pi\)
\(98\) 1.13160e21i 1.39900i
\(99\) 1.70306e20 0.189261
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.g.49.3 6
5.2 odd 4 50.22.a.h.1.3 yes 3
5.3 odd 4 50.22.a.g.1.1 ✓ 3
5.4 even 2 inner 50.22.b.g.49.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.g.1.1 ✓ 3 5.3 odd 4
50.22.a.h.1.3 yes 3 5.2 odd 4
50.22.b.g.49.3 6 1.1 even 1 trivial
50.22.b.g.49.4 6 5.4 even 2 inner