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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 = [2,0,0,-2097152,0,-146644992] 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: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.a.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1024.00i q^{2} +71604.0i q^{3} -1.04858e6 q^{4} -7.33225e7 q^{6} +8.53202e8i q^{7} -1.07374e9i q^{8} +5.33322e9 q^{9} +8.67312e10 q^{11} -7.50822e10i q^{12} -8.95323e11i q^{13} -8.73679e11 q^{14} +1.09951e12 q^{16} -3.25757e12i q^{17} +5.46122e12i q^{18} -2.30325e13 q^{19} -6.10927e13 q^{21} +8.88127e13i q^{22} +1.46496e14i q^{23} +7.68842e13 q^{24} +9.16811e14 q^{26} +1.13088e15i q^{27} -8.94648e14i q^{28} +7.34052e14 q^{29} -3.14666e15 q^{31} +1.12590e15i q^{32} +6.21030e15i q^{33} +3.33575e15 q^{34} -5.59229e15 q^{36} +1.29638e16i q^{37} -2.35852e16i q^{38} +6.41087e16 q^{39} +4.57146e16 q^{41} -6.25589e16i q^{42} -2.40736e16i q^{43} -9.09442e16 q^{44} -1.50012e17 q^{46} +4.49992e17i q^{47} +7.87294e16i q^{48} -1.69408e17 q^{49} +2.33255e17 q^{51} +9.38815e17i q^{52} +2.06484e18i q^{53} -1.15802e18 q^{54} +9.16119e17 q^{56} -1.64922e18i q^{57} +7.51669e17i q^{58} +3.78050e18 q^{59} -7.61981e18 q^{61} -3.22218e18i q^{62} +4.55032e18i q^{63} -1.15292e18 q^{64} -6.35935e18 q^{66} +1.87912e19i q^{67} +3.41581e18i q^{68} -1.04897e19 q^{69} -4.52649e18 q^{71} -5.72650e18i q^{72} -2.55715e19i q^{73} -1.32749e19 q^{74} +2.41513e19 q^{76} +7.39992e19i q^{77} +6.56473e19i q^{78} -9.93364e19 q^{79} -2.51884e19 q^{81} +4.68118e19i q^{82} +2.95818e18i q^{83} +6.40603e19 q^{84} +2.46514e19 q^{86} +5.25610e19i q^{87} -9.31269e19i q^{88} -1.18803e20 q^{89} +7.63892e20 q^{91} -1.53612e20i q^{92} -2.25314e20i q^{93} -4.60792e20 q^{94} -8.06189e19 q^{96} +5.69053e20i q^{97} -1.73474e20i q^{98} +4.62556e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2097152 q^{4} - 146644992 q^{6} + 10666440774 q^{9} + 173462359224 q^{11} - 1747358498816 q^{14} + 2199023255552 q^{16} - 46064935288840 q^{19} - 122185408153536 q^{21} + 153768419131392 q^{24} + 18\!\cdots\!28 q^{26}+ \cdots + 92\!\cdots\!88 q^{99}+O(q^{100}) \) 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
\(3\) 71604.0i 0.700106i 0.936730 + 0.350053i \(0.113837\pi\)
−0.936730 + 0.350053i \(0.886163\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) −7.33225e7 −0.495050
\(7\) 8.53202e8i 1.14162i 0.821081 + 0.570811i \(0.193371\pi\)
−0.821081 + 0.570811i \(0.806629\pi\)
\(8\) − 1.07374e9i − 0.353553i
\(9\) 5.33322e9 0.509851
\(10\) 0 0
\(11\) 8.67312e10 1.00821 0.504106 − 0.863642i \(-0.331822\pi\)
0.504106 + 0.863642i \(0.331822\pi\)
\(12\) − 7.50822e10i − 0.350053i
\(13\) − 8.95323e11i − 1.80125i −0.434594 − 0.900627i \(-0.643108\pi\)
0.434594 − 0.900627i \(-0.356892\pi\)
\(14\) −8.73679e11 −0.807249
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) − 3.25757e12i − 0.391904i −0.980613 − 0.195952i \(-0.937220\pi\)
0.980613 − 0.195952i \(-0.0627797\pi\)
\(18\) 5.46122e12i 0.360519i
\(19\) −2.30325e13 −0.861842 −0.430921 − 0.902390i \(-0.641811\pi\)
−0.430921 + 0.902390i \(0.641811\pi\)
\(20\) 0 0
\(21\) −6.10927e13 −0.799258
\(22\) 8.88127e13i 0.712914i
\(23\) 1.46496e14i 0.737365i 0.929555 + 0.368683i \(0.120191\pi\)
−0.929555 + 0.368683i \(0.879809\pi\)
\(24\) 7.68842e13 0.247525
\(25\) 0 0
\(26\) 9.16811e14 1.27368
\(27\) 1.13088e15i 1.05706i
\(28\) − 8.94648e14i − 0.570811i
\(29\) 7.34052e14 0.324002 0.162001 − 0.986791i \(-0.448205\pi\)
0.162001 + 0.986791i \(0.448205\pi\)
\(30\) 0 0
\(31\) −3.14666e15 −0.689529 −0.344765 − 0.938689i \(-0.612041\pi\)
−0.344765 + 0.938689i \(0.612041\pi\)
\(32\) 1.12590e15i 0.176777i
\(33\) 6.21030e15i 0.705856i
\(34\) 3.33575e15 0.277118
\(35\) 0 0
\(36\) −5.59229e15 −0.254925
\(37\) 1.29638e16i 0.443215i 0.975136 + 0.221608i \(0.0711304\pi\)
−0.975136 + 0.221608i \(0.928870\pi\)
\(38\) − 2.35852e16i − 0.609414i
\(39\) 6.41087e16 1.26107
\(40\) 0 0
\(41\) 4.57146e16 0.531894 0.265947 − 0.963988i \(-0.414315\pi\)
0.265947 + 0.963988i \(0.414315\pi\)
\(42\) − 6.25589e16i − 0.565160i
\(43\) − 2.40736e16i − 0.169872i −0.996386 − 0.0849361i \(-0.972931\pi\)
0.996386 − 0.0849361i \(-0.0270686\pi\)
\(44\) −9.09442e16 −0.504106
\(45\) 0 0
\(46\) −1.50012e17 −0.521396
\(47\) 4.49992e17i 1.24789i 0.781467 + 0.623946i \(0.214472\pi\)
−0.781467 + 0.623946i \(0.785528\pi\)
\(48\) 7.87294e16i 0.175027i
\(49\) −1.69408e17 −0.303303
\(50\) 0 0
\(51\) 2.33255e17 0.274375
\(52\) 9.38815e17i 0.900627i
\(53\) 2.06484e18i 1.62177i 0.585206 + 0.810885i \(0.301014\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(54\) −1.15802e18 −0.747452
\(55\) 0 0
\(56\) 9.16119e17 0.403625
\(57\) − 1.64922e18i − 0.603381i
\(58\) 7.51669e17i 0.229104i
\(59\) 3.78050e18 0.962948 0.481474 − 0.876460i \(-0.340102\pi\)
0.481474 + 0.876460i \(0.340102\pi\)
\(60\) 0 0
\(61\) −7.61981e18 −1.36767 −0.683835 − 0.729637i \(-0.739689\pi\)
−0.683835 + 0.729637i \(0.739689\pi\)
\(62\) − 3.22218e18i − 0.487571i
\(63\) 4.55032e18i 0.582057i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) −6.35935e18 −0.499116
\(67\) 1.87912e19i 1.25941i 0.776833 + 0.629706i \(0.216825\pi\)
−0.776833 + 0.629706i \(0.783175\pi\)
\(68\) 3.41581e18i 0.195952i
\(69\) −1.04897e19 −0.516234
\(70\) 0 0
\(71\) −4.52649e18 −0.165025 −0.0825123 − 0.996590i \(-0.526294\pi\)
−0.0825123 + 0.996590i \(0.526294\pi\)
\(72\) − 5.72650e18i − 0.180260i
\(73\) − 2.55715e19i − 0.696411i −0.937418 − 0.348205i \(-0.886791\pi\)
0.937418 − 0.348205i \(-0.113209\pi\)
\(74\) −1.32749e19 −0.313400
\(75\) 0 0
\(76\) 2.41513e19 0.430921
\(77\) 7.39992e19i 1.15100i
\(78\) 6.56473e19i 0.891710i
\(79\) −9.93364e19 −1.18039 −0.590193 − 0.807262i \(-0.700948\pi\)
−0.590193 + 0.807262i \(0.700948\pi\)
\(80\) 0 0
\(81\) −2.51884e19 −0.230201
\(82\) 4.68118e19i 0.376106i
\(83\) 2.95818e18i 0.0209269i 0.999945 + 0.0104634i \(0.00333068\pi\)
−0.999945 + 0.0104634i \(0.996669\pi\)
\(84\) 6.40603e19 0.399629
\(85\) 0 0
\(86\) 2.46514e19 0.120118
\(87\) 5.25610e19i 0.226836i
\(88\) − 9.31269e19i − 0.356457i
\(89\) −1.18803e20 −0.403861 −0.201931 − 0.979400i \(-0.564722\pi\)
−0.201931 + 0.979400i \(0.564722\pi\)
\(90\) 0 0
\(91\) 7.63892e20 2.05635
\(92\) − 1.53612e20i − 0.368683i
\(93\) − 2.25314e20i − 0.482744i
\(94\) −4.60792e20 −0.882393
\(95\) 0 0
\(96\) −8.06189e19 −0.123763
\(97\) 5.69053e20i 0.783519i 0.920068 + 0.391759i \(0.128133\pi\)
−0.920068 + 0.391759i \(0.871867\pi\)
\(98\) − 1.73474e20i − 0.214467i
\(99\) 4.62556e20 0.514038
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.a.49.2 2
5.2 odd 4 2.22.a.a.1.1 ✓ 1
5.3 odd 4 50.22.a.c.1.1 1
5.4 even 2 inner 50.22.b.a.49.1 2
15.2 even 4 18.22.a.e.1.1 1
20.7 even 4 16.22.a.a.1.1 1
40.27 even 4 64.22.a.f.1.1 1
40.37 odd 4 64.22.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.22.a.a.1.1 ✓ 1 5.2 odd 4
16.22.a.a.1.1 1 20.7 even 4
18.22.a.e.1.1 1 15.2 even 4
50.22.a.c.1.1 1 5.3 odd 4
50.22.b.a.49.1 2 5.4 even 2 inner
50.22.b.a.49.2 2 1.1 even 1 trivial
64.22.a.b.1.1 1 40.37 odd 4
64.22.a.f.1.1 1 40.27 even 4