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 = [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 49.5
Root \(-366.307i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.g.49.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1024.00i q^{2} -26460.2i q^{3} -1.04858e6 q^{4} +2.70953e7 q^{6} +3.30980e8i q^{7} -1.07374e9i q^{8} +9.76021e9 q^{9} +1.32998e9 q^{11} +2.77455e10i q^{12} +1.22401e11i q^{13} -3.38924e11 q^{14} +1.09951e12 q^{16} +1.18162e12i q^{17} +9.99446e12i q^{18} -1.96281e13 q^{19} +8.75781e12 q^{21} +1.36190e12i q^{22} +1.45964e14i q^{23} -2.84114e13 q^{24} -1.25339e14 q^{26} -5.35040e14i q^{27} -3.47058e14i q^{28} +2.54346e15 q^{29} -8.14052e13 q^{31} +1.12590e15i q^{32} -3.51915e13i q^{33} -1.20998e15 q^{34} -1.02343e16 q^{36} -5.96053e15i q^{37} -2.00992e16i q^{38} +3.23876e15 q^{39} +2.16034e16 q^{41} +8.96800e15i q^{42} +5.16519e15i q^{43} -1.39458e15 q^{44} -1.49467e17 q^{46} -4.47352e17i q^{47} -2.90933e16i q^{48} +4.48998e17 q^{49} +3.12660e16 q^{51} -1.28347e17i q^{52} -2.16134e17i q^{53} +5.47881e17 q^{54} +3.55387e17 q^{56} +5.19363e17i q^{57} +2.60450e18i q^{58} -3.65866e18 q^{59} -6.71330e17 q^{61} -8.33589e16i q^{62} +3.23044e18i q^{63} -1.15292e18 q^{64} +3.60361e16 q^{66} -5.53900e18i q^{67} -1.23902e18i q^{68} +3.86223e18 q^{69} +9.93356e18 q^{71} -1.04799e19i q^{72} +4.06075e19i q^{73} +6.10358e18 q^{74} +2.05815e19 q^{76} +4.40197e17i q^{77} +3.31649e18i q^{78} +6.51458e19 q^{79} +8.79380e19 q^{81} +2.21218e19i q^{82} +1.48741e20i q^{83} -9.18323e18 q^{84} -5.28916e18 q^{86} -6.73005e19i q^{87} -1.42805e18i q^{88} +2.72017e20 q^{89} -4.05124e19 q^{91} -1.53054e20i q^{92} +2.15400e18i q^{93} +4.58089e20 q^{94} +2.97916e19 q^{96} -1.44113e19i q^{97} +4.59774e20i q^{98} +1.29809e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6291456 q^{4} + 94992384 q^{6} - 50125674708 q^{9} - 155131852698 q^{11} + 1867315679232 q^{14} + 6597069766656 q^{16} + 47919252129450 q^{19} - 452827014956148 q^{21} - 99606734045184 q^{24} + 60018091106304 q^{26}+ \cdots + 23\!\cdots\!64 q^{99}+O(q^{100}) \) 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
\(3\) − 26460.2i − 0.258714i −0.991598 − 0.129357i \(-0.958709\pi\)
0.991598 − 0.129357i \(-0.0412913\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) 2.70953e7 0.182939
\(7\) 3.30980e8i 0.442866i 0.975176 + 0.221433i \(0.0710735\pi\)
−0.975176 + 0.221433i \(0.928927\pi\)
\(8\) − 1.07374e9i − 0.353553i
\(9\) 9.76021e9 0.933067
\(10\) 0 0
\(11\) 1.32998e9 0.0154604 0.00773021 − 0.999970i \(-0.497539\pi\)
0.00773021 + 0.999970i \(0.497539\pi\)
\(12\) 2.77455e10i 0.129357i
\(13\) 1.22401e11i 0.246252i 0.992391 + 0.123126i \(0.0392920\pi\)
−0.992391 + 0.123126i \(0.960708\pi\)
\(14\) −3.38924e11 −0.313154
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 1.18162e12i 0.142156i 0.997471 + 0.0710780i \(0.0226439\pi\)
−0.997471 + 0.0710780i \(0.977356\pi\)
\(18\) 9.99446e12i 0.659778i
\(19\) −1.96281e13 −0.734455 −0.367227 − 0.930131i \(-0.619693\pi\)
−0.367227 + 0.930131i \(0.619693\pi\)
\(20\) 0 0
\(21\) 8.75781e12 0.114576
\(22\) 1.36190e12i 0.0109322i
\(23\) 1.45964e14i 0.734688i 0.930085 + 0.367344i \(0.119733\pi\)
−0.930085 + 0.367344i \(0.880267\pi\)
\(24\) −2.84114e13 −0.0914693
\(25\) 0 0
\(26\) −1.25339e14 −0.174127
\(27\) − 5.35040e14i − 0.500112i
\(28\) − 3.47058e14i − 0.221433i
\(29\) 2.54346e15 1.12265 0.561327 − 0.827594i \(-0.310291\pi\)
0.561327 + 0.827594i \(0.310291\pi\)
\(30\) 0 0
\(31\) −8.14052e13 −0.0178383 −0.00891917 − 0.999960i \(-0.502839\pi\)
−0.00891917 + 0.999960i \(0.502839\pi\)
\(32\) 1.12590e15i 0.176777i
\(33\) − 3.51915e13i − 0.00399983i
\(34\) −1.20998e15 −0.100519
\(35\) 0 0
\(36\) −1.02343e16 −0.466533
\(37\) − 5.96053e15i − 0.203782i −0.994796 − 0.101891i \(-0.967511\pi\)
0.994796 − 0.101891i \(-0.0324894\pi\)
\(38\) − 2.00992e16i − 0.519338i
\(39\) 3.23876e15 0.0637090
\(40\) 0 0
\(41\) 2.16034e16 0.251357 0.125678 − 0.992071i \(-0.459889\pi\)
0.125678 + 0.992071i \(0.459889\pi\)
\(42\) 8.96800e15i 0.0810173i
\(43\) 5.16519e15i 0.0364475i 0.999834 + 0.0182237i \(0.00580112\pi\)
−0.999834 + 0.0182237i \(0.994199\pi\)
\(44\) −1.39458e15 −0.00773021
\(45\) 0 0
\(46\) −1.49467e17 −0.519503
\(47\) − 4.47352e17i − 1.24057i −0.784376 − 0.620286i \(-0.787016\pi\)
0.784376 − 0.620286i \(-0.212984\pi\)
\(48\) − 2.90933e16i − 0.0646785i
\(49\) 4.48998e17 0.803869
\(50\) 0 0
\(51\) 3.12660e16 0.0367778
\(52\) − 1.28347e17i − 0.123126i
\(53\) − 2.16134e17i − 0.169756i −0.996391 − 0.0848782i \(-0.972950\pi\)
0.996391 − 0.0848782i \(-0.0270501\pi\)
\(54\) 5.47881e17 0.353632
\(55\) 0 0
\(56\) 3.55387e17 0.156577
\(57\) 5.19363e17i 0.190014i
\(58\) 2.60450e18i 0.793836i
\(59\) −3.65866e18 −0.931913 −0.465957 − 0.884808i \(-0.654290\pi\)
−0.465957 + 0.884808i \(0.654290\pi\)
\(60\) 0 0
\(61\) −6.71330e17 −0.120496 −0.0602480 − 0.998183i \(-0.519189\pi\)
−0.0602480 + 0.998183i \(0.519189\pi\)
\(62\) − 8.33589e16i − 0.0126136i
\(63\) 3.23044e18i 0.413224i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) 3.60361e16 0.00282831
\(67\) − 5.53900e18i − 0.371232i −0.982622 − 0.185616i \(-0.940572\pi\)
0.982622 − 0.185616i \(-0.0594281\pi\)
\(68\) − 1.23902e18i − 0.0710780i
\(69\) 3.86223e18 0.190074
\(70\) 0 0
\(71\) 9.93356e18 0.362153 0.181077 − 0.983469i \(-0.442042\pi\)
0.181077 + 0.983469i \(0.442042\pi\)
\(72\) − 1.04799e19i − 0.329889i
\(73\) 4.06075e19i 1.10590i 0.833214 + 0.552951i \(0.186498\pi\)
−0.833214 + 0.552951i \(0.813502\pi\)
\(74\) 6.10358e18 0.144096
\(75\) 0 0
\(76\) 2.05815e19 0.367227
\(77\) 4.40197e17i 0.00684690i
\(78\) 3.31649e18i 0.0450491i
\(79\) 6.51458e19 0.774108 0.387054 − 0.922057i \(-0.373493\pi\)
0.387054 + 0.922057i \(0.373493\pi\)
\(80\) 0 0
\(81\) 8.79380e19 0.803681
\(82\) 2.21218e19i 0.177736i
\(83\) 1.48741e20i 1.05223i 0.850414 + 0.526114i \(0.176351\pi\)
−0.850414 + 0.526114i \(0.823649\pi\)
\(84\) −9.18323e18 −0.0572879
\(85\) 0 0
\(86\) −5.28916e18 −0.0257723
\(87\) − 6.73005e19i − 0.290447i
\(88\) − 1.42805e18i − 0.00546609i
\(89\) 2.72017e20 0.924700 0.462350 − 0.886697i \(-0.347006\pi\)
0.462350 + 0.886697i \(0.347006\pi\)
\(90\) 0 0
\(91\) −4.05124e19 −0.109057
\(92\) − 1.53054e20i − 0.367344i
\(93\) 2.15400e18i 0.00461503i
\(94\) 4.58089e20 0.877217
\(95\) 0 0
\(96\) 2.97916e19 0.0457346
\(97\) − 1.44113e19i − 0.0198427i −0.999951 − 0.00992134i \(-0.996842\pi\)
0.999951 − 0.00992134i \(-0.00315811\pi\)
\(98\) 4.59774e20i 0.568421i
\(99\) 1.29809e19 0.0144256
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.49.5 6
5.2 odd 4 50.22.a.g.1.2 ✓ 3
5.3 odd 4 50.22.a.h.1.2 yes 3
5.4 even 2 inner 50.22.b.g.49.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.g.1.2 ✓ 3 5.2 odd 4
50.22.a.h.1.2 yes 3 5.3 odd 4
50.22.b.g.49.2 6 5.4 even 2 inner
50.22.b.g.49.5 6 1.1 even 1 trivial