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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 49.1
Root \(-5738.70i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.22.b.g.49.6

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00i q^{2} -156690. i q^{3} -1.04858e6 q^{4} -1.60450e8 q^{6} -4.70592e7i q^{7} +1.07374e9i q^{8} -1.40914e10 q^{9} -7.06811e10 q^{11} +1.64301e11i q^{12} +4.22014e11i q^{13} -4.81886e10 q^{14} +1.09951e12 q^{16} +9.39783e12i q^{17} +1.44296e13i q^{18} +1.75774e13 q^{19} -7.37370e12 q^{21} +7.23775e13i q^{22} +2.12289e13i q^{23} +1.68244e14 q^{24} +4.32143e14 q^{26} +5.68943e14i q^{27} +4.93451e13i q^{28} -3.71024e15 q^{29} +4.95433e15 q^{31} -1.12590e15i q^{32} +1.10750e16i q^{33} +9.62338e15 q^{34} +1.47759e16 q^{36} +2.52608e15i q^{37} -1.79993e16i q^{38} +6.61254e16 q^{39} -1.20948e17 q^{41} +7.55067e15i q^{42} +1.31006e17i q^{43} +7.41145e16 q^{44} +2.17384e16 q^{46} -6.43094e17i q^{47} -1.72282e17i q^{48} +5.56331e17 q^{49} +1.47254e18 q^{51} -4.42514e17i q^{52} +3.87349e17i q^{53} +5.82598e17 q^{54} +5.05294e16 q^{56} -2.75420e18i q^{57} +3.79929e18i q^{58} +6.13339e18 q^{59} +9.11348e18 q^{61} -5.07323e18i q^{62} +6.63128e17i q^{63} -1.15292e18 q^{64} +1.13408e19 q^{66} -1.33011e19i q^{67} -9.85434e18i q^{68} +3.32636e18 q^{69} +4.55939e18 q^{71} -1.51305e19i q^{72} -3.93714e19i q^{73} +2.58671e18 q^{74} -1.84313e19 q^{76} +3.32620e18i q^{77} -6.77124e19i q^{78} +7.55381e19 q^{79} -5.82531e19 q^{81} +1.23850e20i q^{82} +3.70451e19i q^{83} +7.73188e18 q^{84} +1.34151e20 q^{86} +5.81358e20i q^{87} -7.58933e19i q^{88} -1.83128e19 q^{89} +1.98596e19 q^{91} -2.22602e19i q^{92} -7.76293e20i q^{93} -6.58528e20 q^{94} -1.76417e20 q^{96} +1.19926e21i q^{97} -5.69683e20i q^{98} +9.95993e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6291456 q^{4} + 94992384 q^{6} - 50125674708 q^{9} - 155131852698 q^{11} + 1867315679232 q^{14} + 6597069766656 q^{16} + 47919252129450 q^{19} - 452827014956148 q^{21} - 99606734045184 q^{24} + 60018091106304 q^{26}+ \cdots + 23\!\cdots\!64 q^{99}+O(q^{100}) \) 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
\(3\) − 156690.i − 1.53203i −0.642822 − 0.766016i \(-0.722236\pi\)
0.642822 − 0.766016i \(-0.277764\pi\)
\(4\) −1.04858e6 −0.500000
\(5\) 0 0
\(6\) −1.60450e8 −1.08331
\(7\) − 4.70592e7i − 0.0629673i −0.999504 − 0.0314836i \(-0.989977\pi\)
0.999504 − 0.0314836i \(-0.0100232\pi\)
\(8\) 1.07374e9i 0.353553i
\(9\) −1.40914e10 −1.34712
\(10\) 0 0
\(11\) −7.06811e10 −0.821637 −0.410819 − 0.911717i \(-0.634757\pi\)
−0.410819 + 0.911717i \(0.634757\pi\)
\(12\) 1.64301e11i 0.766016i
\(13\) 4.22014e11i 0.849028i 0.905421 + 0.424514i \(0.139555\pi\)
−0.905421 + 0.424514i \(0.860445\pi\)
\(14\) −4.81886e10 −0.0445246
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 9.39783e12i 1.13061i 0.824881 + 0.565306i \(0.191242\pi\)
−0.824881 + 0.565306i \(0.808758\pi\)
\(18\) 1.44296e13i 0.952559i
\(19\) 1.75774e13 0.657722 0.328861 − 0.944378i \(-0.393335\pi\)
0.328861 + 0.944378i \(0.393335\pi\)
\(20\) 0 0
\(21\) −7.37370e12 −0.0964679
\(22\) 7.23775e13i 0.580985i
\(23\) 2.12289e13i 0.106853i 0.998572 + 0.0534264i \(0.0170143\pi\)
−0.998572 + 0.0534264i \(0.982986\pi\)
\(24\) 1.68244e14 0.541655
\(25\) 0 0
\(26\) 4.32143e14 0.600353
\(27\) 5.68943e14i 0.531801i
\(28\) 4.93451e13i 0.0314836i
\(29\) −3.71024e15 −1.63766 −0.818830 − 0.574037i \(-0.805377\pi\)
−0.818830 + 0.574037i \(0.805377\pi\)
\(30\) 0 0
\(31\) 4.95433e15 1.08564 0.542821 − 0.839848i \(-0.317356\pi\)
0.542821 + 0.839848i \(0.317356\pi\)
\(32\) − 1.12590e15i − 0.176777i
\(33\) 1.10750e16i 1.25877i
\(34\) 9.62338e15 0.799464
\(35\) 0 0
\(36\) 1.47759e16 0.673561
\(37\) 2.52608e15i 0.0863633i 0.999067 + 0.0431816i \(0.0137494\pi\)
−0.999067 + 0.0431816i \(0.986251\pi\)
\(38\) − 1.79993e16i − 0.465080i
\(39\) 6.61254e16 1.30074
\(40\) 0 0
\(41\) −1.20948e17 −1.40724 −0.703618 − 0.710578i \(-0.748433\pi\)
−0.703618 + 0.710578i \(0.748433\pi\)
\(42\) 7.55067e15i 0.0682131i
\(43\) 1.31006e17i 0.924430i 0.886768 + 0.462215i \(0.152945\pi\)
−0.886768 + 0.462215i \(0.847055\pi\)
\(44\) 7.41145e16 0.410819
\(45\) 0 0
\(46\) 2.17384e16 0.0755564
\(47\) − 6.43094e17i − 1.78339i −0.452635 − 0.891696i \(-0.649516\pi\)
0.452635 − 0.891696i \(-0.350484\pi\)
\(48\) − 1.72282e17i − 0.383008i
\(49\) 5.56331e17 0.996035
\(50\) 0 0
\(51\) 1.47254e18 1.73213
\(52\) − 4.42514e17i − 0.424514i
\(53\) 3.87349e17i 0.304232i 0.988363 + 0.152116i \(0.0486088\pi\)
−0.988363 + 0.152116i \(0.951391\pi\)
\(54\) 5.82598e17 0.376040
\(55\) 0 0
\(56\) 5.05294e16 0.0222623
\(57\) − 2.75420e18i − 1.00765i
\(58\) 3.79929e18i 1.15800i
\(59\) 6.13339e18 1.56226 0.781132 − 0.624366i \(-0.214643\pi\)
0.781132 + 0.624366i \(0.214643\pi\)
\(60\) 0 0
\(61\) 9.11348e18 1.63576 0.817882 − 0.575386i \(-0.195148\pi\)
0.817882 + 0.575386i \(0.195148\pi\)
\(62\) − 5.07323e18i − 0.767665i
\(63\) 6.63128e17i 0.0848246i
\(64\) −1.15292e18 −0.125000
\(65\) 0 0
\(66\) 1.13408e19 0.890088
\(67\) − 1.33011e19i − 0.891461i −0.895167 − 0.445730i \(-0.852944\pi\)
0.895167 − 0.445730i \(-0.147056\pi\)
\(68\) − 9.85434e18i − 0.565306i
\(69\) 3.32636e18 0.163702
\(70\) 0 0
\(71\) 4.55939e18 0.166224 0.0831121 − 0.996540i \(-0.473514\pi\)
0.0831121 + 0.996540i \(0.473514\pi\)
\(72\) − 1.51305e19i − 0.476279i
\(73\) − 3.93714e19i − 1.07224i −0.844143 − 0.536118i \(-0.819890\pi\)
0.844143 − 0.536118i \(-0.180110\pi\)
\(74\) 2.58671e18 0.0610681
\(75\) 0 0
\(76\) −1.84313e19 −0.328861
\(77\) 3.32620e18i 0.0517363i
\(78\) − 6.77124e19i − 0.919760i
\(79\) 7.55381e19 0.897598 0.448799 − 0.893633i \(-0.351852\pi\)
0.448799 + 0.893633i \(0.351852\pi\)
\(80\) 0 0
\(81\) −5.82531e19 −0.532385
\(82\) 1.23850e20i 0.995066i
\(83\) 3.70451e19i 0.262066i 0.991378 + 0.131033i \(0.0418294\pi\)
−0.991378 + 0.131033i \(0.958171\pi\)
\(84\) 7.73188e18 0.0482339
\(85\) 0 0
\(86\) 1.34151e20 0.653670
\(87\) 5.81358e20i 2.50895i
\(88\) − 7.58933e19i − 0.290493i
\(89\) −1.83128e19 −0.0622528 −0.0311264 − 0.999515i \(-0.509909\pi\)
−0.0311264 + 0.999515i \(0.509909\pi\)
\(90\) 0 0
\(91\) 1.98596e19 0.0534610
\(92\) − 2.22602e19i − 0.0534264i
\(93\) − 7.76293e20i − 1.66324i
\(94\) −6.58528e20 −1.26105
\(95\) 0 0
\(96\) −1.76417e20 −0.270828
\(97\) 1.19926e21i 1.65124i 0.564229 + 0.825618i \(0.309173\pi\)
−0.564229 + 0.825618i \(0.690827\pi\)
\(98\) − 5.69683e20i − 0.704303i
\(99\) 9.95993e20 1.10685
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.49.1 6
5.2 odd 4 50.22.a.h.1.1 yes 3
5.3 odd 4 50.22.a.g.1.3 ✓ 3
5.4 even 2 inner 50.22.b.g.49.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.g.1.3 ✓ 3 5.3 odd 4
50.22.a.h.1.1 yes 3 5.2 odd 4
50.22.b.g.49.1 6 1.1 even 1 trivial
50.22.b.g.49.6 6 5.4 even 2 inner