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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 = [4,0,0,-4194304,0,259465216] 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: \( x^{4} + 78961x^{2} + 1558670400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 5^{2}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(198.196i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.f.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)
 
\(f(q)\) \(=\) \(q+1024.00i q^{2} +47924.0i q^{3} -1.04858e6 q^{4} -4.90741e7 q^{6} -6.67863e8i q^{7} -1.07374e9i q^{8} +8.16365e9 q^{9} -1.26793e11 q^{11} -5.02519e10i q^{12} +9.12750e11i q^{13} +6.83892e11 q^{14} +1.09951e12 q^{16} -8.60406e12i q^{17} +8.35957e12i q^{18} +6.93702e12 q^{19} +3.20066e13 q^{21} -1.29836e14i q^{22} -3.30974e14i q^{23} +5.14580e13 q^{24} -9.34656e14 q^{26} +8.92536e14i q^{27} +7.00305e14i q^{28} +3.90534e15 q^{29} +3.32185e15 q^{31} +1.12590e15i q^{32} -6.07642e15i q^{33} +8.81056e15 q^{34} -8.56020e15 q^{36} +4.33804e16i q^{37} +7.10351e15i q^{38} -4.37426e16 q^{39} -9.56620e16 q^{41} +3.27748e16i q^{42} -7.81706e16i q^{43} +1.32952e17 q^{44} +3.38917e17 q^{46} +2.03634e17i q^{47} +5.26930e16i q^{48} +1.12505e17 q^{49} +4.12341e17 q^{51} -9.57088e17i q^{52} +1.42121e18i q^{53} -9.13957e17 q^{54} -7.17112e17 q^{56} +3.32449e17i q^{57} +3.99907e18i q^{58} -2.28919e18 q^{59} -5.32798e18 q^{61} +3.40158e18i q^{62} -5.45220e18i q^{63} -1.15292e18 q^{64} +6.22226e18 q^{66} +1.31401e19i q^{67} +9.02201e18i q^{68} +1.58616e19 q^{69} -2.98159e19 q^{71} -8.76565e18i q^{72} +1.37094e18i q^{73} -4.44215e19 q^{74} -7.27399e18 q^{76} +8.46803e19i q^{77} -4.47924e19i q^{78} -1.74281e19 q^{79} +4.26208e19 q^{81} -9.79579e19i q^{82} +3.75717e19i q^{83} -3.35614e19 q^{84} +8.00467e19 q^{86} +1.87159e20i q^{87} +1.36143e20i q^{88} -2.43231e20 q^{89} +6.09592e20 q^{91} +3.47051e20i q^{92} +1.59196e20i q^{93} -2.08521e20 q^{94} -5.39576e19 q^{96} +1.80740e20i q^{97} +1.15205e20i q^{98} -1.03509e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4194304 q^{4} + 259465216 q^{6} - 23733475652 q^{9} - 82653663552 q^{11} - 599242104832 q^{14} + 4398046511104 q^{16} - 14351589304880 q^{19} + 399437207788528 q^{21} - 272068998332416 q^{24} - 38\!\cdots\!44 q^{26}+ \cdots - 54\!\cdots\!24 q^{99}+O(q^{100}) \) 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
\(3\) 47924.0i 0.468576i 0.972167 + 0.234288i \(0.0752758\pi\)
−0.972167 + 0.234288i \(0.924724\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) −4.90741e7 −0.331333
\(7\) − 6.67863e8i − 0.893631i −0.894626 − 0.446815i \(-0.852558\pi\)
0.894626 − 0.446815i \(-0.147442\pi\)
\(8\) − 1.07374e9i − 0.353553i
\(9\) 8.16365e9 0.780437
\(10\) 0 0
\(11\) −1.26793e11 −1.47391 −0.736957 − 0.675940i \(-0.763738\pi\)
−0.736957 + 0.675940i \(0.763738\pi\)
\(12\) − 5.02519e10i − 0.234288i
\(13\) 9.12750e11i 1.83631i 0.396219 + 0.918156i \(0.370322\pi\)
−0.396219 + 0.918156i \(0.629678\pi\)
\(14\) 6.83892e11 0.631892
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) − 8.60406e12i − 1.03512i −0.855648 − 0.517559i \(-0.826841\pi\)
0.855648 − 0.517559i \(-0.173159\pi\)
\(18\) 8.35957e12i 0.551852i
\(19\) 6.93702e12 0.259573 0.129787 − 0.991542i \(-0.458571\pi\)
0.129787 + 0.991542i \(0.458571\pi\)
\(20\) 0 0
\(21\) 3.20066e13 0.418733
\(22\) − 1.29836e14i − 1.04221i
\(23\) − 3.30974e14i − 1.66591i −0.553341 − 0.832955i \(-0.686648\pi\)
0.553341 − 0.832955i \(-0.313352\pi\)
\(24\) 5.14580e13 0.165666
\(25\) 0 0
\(26\) −9.34656e14 −1.29847
\(27\) 8.92536e14i 0.834269i
\(28\) 7.00305e14i 0.446815i
\(29\) 3.90534e15 1.72377 0.861886 − 0.507103i \(-0.169284\pi\)
0.861886 + 0.507103i \(0.169284\pi\)
\(30\) 0 0
\(31\) 3.32185e15 0.727918 0.363959 − 0.931415i \(-0.381425\pi\)
0.363959 + 0.931415i \(0.381425\pi\)
\(32\) 1.12590e15i 0.176777i
\(33\) − 6.07642e15i − 0.690640i
\(34\) 8.81056e15 0.731939
\(35\) 0 0
\(36\) −8.56020e15 −0.390218
\(37\) 4.33804e16i 1.48312i 0.670889 + 0.741558i \(0.265913\pi\)
−0.670889 + 0.741558i \(0.734087\pi\)
\(38\) 7.10351e15i 0.183546i
\(39\) −4.37426e16 −0.860451
\(40\) 0 0
\(41\) −9.56620e16 −1.11304 −0.556518 − 0.830836i \(-0.687863\pi\)
−0.556518 + 0.830836i \(0.687863\pi\)
\(42\) 3.27748e16i 0.296089i
\(43\) − 7.81706e16i − 0.551601i −0.961215 − 0.275800i \(-0.911057\pi\)
0.961215 − 0.275800i \(-0.0889428\pi\)
\(44\) 1.32952e17 0.736957
\(45\) 0 0
\(46\) 3.38917e17 1.17798
\(47\) 2.03634e17i 0.564707i 0.959310 + 0.282354i \(0.0911151\pi\)
−0.959310 + 0.282354i \(0.908885\pi\)
\(48\) 5.26930e16i 0.117144i
\(49\) 1.12505e17 0.201425
\(50\) 0 0
\(51\) 4.12341e17 0.485031
\(52\) − 9.57088e17i − 0.918156i
\(53\) 1.42121e18i 1.11625i 0.829756 + 0.558127i \(0.188480\pi\)
−0.829756 + 0.558127i \(0.811520\pi\)
\(54\) −9.13957e17 −0.589917
\(55\) 0 0
\(56\) −7.17112e17 −0.315946
\(57\) 3.32449e17i 0.121630i
\(58\) 3.99907e18i 1.21889i
\(59\) −2.28919e18 −0.583089 −0.291545 − 0.956557i \(-0.594169\pi\)
−0.291545 + 0.956557i \(0.594169\pi\)
\(60\) 0 0
\(61\) −5.32798e18 −0.956311 −0.478155 − 0.878275i \(-0.658694\pi\)
−0.478155 + 0.878275i \(0.658694\pi\)
\(62\) 3.40158e18i 0.514716i
\(63\) − 5.45220e18i − 0.697422i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) 6.22226e18 0.488356
\(67\) 1.31401e19i 0.880668i 0.897834 + 0.440334i \(0.145140\pi\)
−0.897834 + 0.440334i \(0.854860\pi\)
\(68\) 9.02201e18i 0.517559i
\(69\) 1.58616e19 0.780604
\(70\) 0 0
\(71\) −2.98159e19 −1.08702 −0.543508 − 0.839404i \(-0.682904\pi\)
−0.543508 + 0.839404i \(0.682904\pi\)
\(72\) − 8.76565e18i − 0.275926i
\(73\) 1.37094e18i 0.0373361i 0.999826 + 0.0186681i \(0.00594257\pi\)
−0.999826 + 0.0186681i \(0.994057\pi\)
\(74\) −4.44215e19 −1.04872
\(75\) 0 0
\(76\) −7.27399e18 −0.129787
\(77\) 8.46803e19i 1.31713i
\(78\) − 4.47924e19i − 0.608431i
\(79\) −1.74281e19 −0.207093 −0.103546 − 0.994625i \(-0.533019\pi\)
−0.103546 + 0.994625i \(0.533019\pi\)
\(80\) 0 0
\(81\) 4.26208e19 0.389519
\(82\) − 9.79579e19i − 0.787035i
\(83\) 3.75717e19i 0.265792i 0.991130 + 0.132896i \(0.0424276\pi\)
−0.991130 + 0.132896i \(0.957572\pi\)
\(84\) −3.35614e19 −0.209367
\(85\) 0 0
\(86\) 8.00467e19 0.390040
\(87\) 1.87159e20i 0.807717i
\(88\) 1.36143e20i 0.521107i
\(89\) −2.43231e20 −0.826845 −0.413423 − 0.910539i \(-0.635667\pi\)
−0.413423 + 0.910539i \(0.635667\pi\)
\(90\) 0 0
\(91\) 6.09592e20 1.64098
\(92\) 3.47051e20i 0.832955i
\(93\) 1.59196e20i 0.341084i
\(94\) −2.08521e20 −0.399308
\(95\) 0 0
\(96\) −5.39576e19 −0.0828332
\(97\) 1.80740e20i 0.248858i 0.992229 + 0.124429i \(0.0397099\pi\)
−0.992229 + 0.124429i \(0.960290\pi\)
\(98\) 1.15205e20i 0.142429i
\(99\) −1.03509e21 −1.15030
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.f.49.4 4
5.2 odd 4 10.22.a.b.1.2 ✓ 2
5.3 odd 4 50.22.a.f.1.1 2
5.4 even 2 inner 50.22.b.f.49.1 4
20.7 even 4 80.22.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.b.1.2 ✓ 2 5.2 odd 4
50.22.a.f.1.1 2 5.3 odd 4
50.22.b.f.49.1 4 5.4 even 2 inner
50.22.b.f.49.4 4 1.1 even 1 trivial
80.22.a.d.1.1 2 20.7 even 4