Properties

Label 50.22.a.g.1.3
Level $50$
Weight $22$
Character 50.1
Self dual yes
Analytic conductor $139.739$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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(1,50)] 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("50.1"); 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([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3072,-46383] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(139.738672144\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 30959316x - 11291332284 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(5738.70\) of defining polynomial
Character \(\chi\) \(=\) 50.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} +156690. q^{3} +1.04858e6 q^{4} -1.60450e8 q^{6} -4.70592e7 q^{7} -1.07374e9 q^{8} +1.40914e10 q^{9} -7.06811e10 q^{11} +1.64301e11 q^{12} -4.22014e11 q^{13} +4.81886e10 q^{14} +1.09951e12 q^{16} +9.39783e12 q^{17} -1.44296e13 q^{18} -1.75774e13 q^{19} -7.37370e12 q^{21} +7.23775e13 q^{22} -2.12289e13 q^{23} -1.68244e14 q^{24} +4.32143e14 q^{26} +5.68943e14 q^{27} -4.93451e13 q^{28} +3.71024e15 q^{29} +4.95433e15 q^{31} -1.12590e15 q^{32} -1.10750e16 q^{33} -9.62338e15 q^{34} +1.47759e16 q^{36} +2.52608e15 q^{37} +1.79993e16 q^{38} -6.61254e16 q^{39} -1.20948e17 q^{41} +7.55067e15 q^{42} -1.31006e17 q^{43} -7.41145e16 q^{44} +2.17384e16 q^{46} -6.43094e17 q^{47} +1.72282e17 q^{48} -5.56331e17 q^{49} +1.47254e18 q^{51} -4.42514e17 q^{52} -3.87349e17 q^{53} -5.82598e17 q^{54} +5.05294e16 q^{56} -2.75420e18 q^{57} -3.79929e18 q^{58} -6.13339e18 q^{59} +9.11348e18 q^{61} -5.07323e18 q^{62} -6.63128e17 q^{63} +1.15292e18 q^{64} +1.13408e19 q^{66} -1.33011e19 q^{67} +9.85434e18 q^{68} -3.32636e18 q^{69} +4.55939e18 q^{71} -1.51305e19 q^{72} +3.93714e19 q^{73} -2.58671e18 q^{74} -1.84313e19 q^{76} +3.32620e18 q^{77} +6.77124e19 q^{78} -7.55381e19 q^{79} -5.82531e19 q^{81} +1.23850e20 q^{82} -3.70451e19 q^{83} -7.73188e18 q^{84} +1.34151e20 q^{86} +5.81358e20 q^{87} +7.58933e19 q^{88} +1.83128e19 q^{89} +1.98596e19 q^{91} -2.22602e19 q^{92} +7.76293e20 q^{93} +6.58528e20 q^{94} -1.76417e20 q^{96} +1.19926e21 q^{97} +5.69683e20 q^{98} -9.95993e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3072 q^{2} - 46383 q^{3} + 3145728 q^{4} + 47496192 q^{6} + 911775234 q^{7} - 3221225472 q^{8} + 25062837354 q^{9} - 77565926349 q^{11} - 48636100608 q^{12} - 29305708548 q^{13} - 933657839616 q^{14}+ \cdots - 11\!\cdots\!82 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1024.00 −0.707107
\(3\) 156690. 1.53203 0.766016 − 0.642822i \(-0.222236\pi\)
0.766016 + 0.642822i \(0.222236\pi\)
\(4\) 1.04858e6 0.500000
\(5\) 0 0
\(6\) −1.60450e8 −1.08331
\(7\) −4.70592e7 −0.0629673 −0.0314836 − 0.999504i \(-0.510023\pi\)
−0.0314836 + 0.999504i \(0.510023\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) 1.40914e10 1.34712
\(10\) 0 0
\(11\) −7.06811e10 −0.821637 −0.410819 − 0.911717i \(-0.634757\pi\)
−0.410819 + 0.911717i \(0.634757\pi\)
\(12\) 1.64301e11 0.766016
\(13\) −4.22014e11 −0.849028 −0.424514 − 0.905421i \(-0.639555\pi\)
−0.424514 + 0.905421i \(0.639555\pi\)
\(14\) 4.81886e10 0.0445246
\(15\) 0 0
\(16\) 1.09951e12 0.250000
\(17\) 9.39783e12 1.13061 0.565306 − 0.824881i \(-0.308758\pi\)
0.565306 + 0.824881i \(0.308758\pi\)
\(18\) −1.44296e13 −0.952559
\(19\) −1.75774e13 −0.657722 −0.328861 − 0.944378i \(-0.606665\pi\)
−0.328861 + 0.944378i \(0.606665\pi\)
\(20\) 0 0
\(21\) −7.37370e12 −0.0964679
\(22\) 7.23775e13 0.580985
\(23\) −2.12289e13 −0.106853 −0.0534264 − 0.998572i \(-0.517014\pi\)
−0.0534264 + 0.998572i \(0.517014\pi\)
\(24\) −1.68244e14 −0.541655
\(25\) 0 0
\(26\) 4.32143e14 0.600353
\(27\) 5.68943e14 0.531801
\(28\) −4.93451e13 −0.0314836
\(29\) 3.71024e15 1.63766 0.818830 − 0.574037i \(-0.194623\pi\)
0.818830 + 0.574037i \(0.194623\pi\)
\(30\) 0 0
\(31\) 4.95433e15 1.08564 0.542821 − 0.839848i \(-0.317356\pi\)
0.542821 + 0.839848i \(0.317356\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) −1.10750e16 −1.25877
\(34\) −9.62338e15 −0.799464
\(35\) 0 0
\(36\) 1.47759e16 0.673561
\(37\) 2.52608e15 0.0863633 0.0431816 − 0.999067i \(-0.486251\pi\)
0.0431816 + 0.999067i \(0.486251\pi\)
\(38\) 1.79993e16 0.465080
\(39\) −6.61254e16 −1.30074
\(40\) 0 0
\(41\) −1.20948e17 −1.40724 −0.703618 − 0.710578i \(-0.748433\pi\)
−0.703618 + 0.710578i \(0.748433\pi\)
\(42\) 7.55067e15 0.0682131
\(43\) −1.31006e17 −0.924430 −0.462215 − 0.886768i \(-0.652945\pi\)
−0.462215 + 0.886768i \(0.652945\pi\)
\(44\) −7.41145e16 −0.410819
\(45\) 0 0
\(46\) 2.17384e16 0.0755564
\(47\) −6.43094e17 −1.78339 −0.891696 − 0.452635i \(-0.850484\pi\)
−0.891696 + 0.452635i \(0.850484\pi\)
\(48\) 1.72282e17 0.383008
\(49\) −5.56331e17 −0.996035
\(50\) 0 0
\(51\) 1.47254e18 1.73213
\(52\) −4.42514e17 −0.424514
\(53\) −3.87349e17 −0.304232 −0.152116 − 0.988363i \(-0.548609\pi\)
−0.152116 + 0.988363i \(0.548609\pi\)
\(54\) −5.82598e17 −0.376040
\(55\) 0 0
\(56\) 5.05294e16 0.0222623
\(57\) −2.75420e18 −1.00765
\(58\) −3.79929e18 −1.15800
\(59\) −6.13339e18 −1.56226 −0.781132 − 0.624366i \(-0.785357\pi\)
−0.781132 + 0.624366i \(0.785357\pi\)
\(60\) 0 0
\(61\) 9.11348e18 1.63576 0.817882 − 0.575386i \(-0.195148\pi\)
0.817882 + 0.575386i \(0.195148\pi\)
\(62\) −5.07323e18 −0.767665
\(63\) −6.63128e17 −0.0848246
\(64\) 1.15292e18 0.125000
\(65\) 0 0
\(66\) 1.13408e19 0.890088
\(67\) −1.33011e19 −0.891461 −0.445730 − 0.895167i \(-0.647056\pi\)
−0.445730 + 0.895167i \(0.647056\pi\)
\(68\) 9.85434e18 0.565306
\(69\) −3.32636e18 −0.163702
\(70\) 0 0
\(71\) 4.55939e18 0.166224 0.0831121 − 0.996540i \(-0.473514\pi\)
0.0831121 + 0.996540i \(0.473514\pi\)
\(72\) −1.51305e19 −0.476279
\(73\) 3.93714e19 1.07224 0.536118 − 0.844143i \(-0.319890\pi\)
0.536118 + 0.844143i \(0.319890\pi\)
\(74\) −2.58671e18 −0.0610681
\(75\) 0 0
\(76\) −1.84313e19 −0.328861
\(77\) 3.32620e18 0.0517363
\(78\) 6.77124e19 0.919760
\(79\) −7.55381e19 −0.897598 −0.448799 − 0.893633i \(-0.648148\pi\)
−0.448799 + 0.893633i \(0.648148\pi\)
\(80\) 0 0
\(81\) −5.82531e19 −0.532385
\(82\) 1.23850e20 0.995066
\(83\) −3.70451e19 −0.262066 −0.131033 − 0.991378i \(-0.541829\pi\)
−0.131033 + 0.991378i \(0.541829\pi\)
\(84\) −7.73188e18 −0.0482339
\(85\) 0 0
\(86\) 1.34151e20 0.653670
\(87\) 5.81358e20 2.50895
\(88\) 7.58933e19 0.290493
\(89\) 1.83128e19 0.0622528 0.0311264 − 0.999515i \(-0.490091\pi\)
0.0311264 + 0.999515i \(0.490091\pi\)
\(90\) 0 0
\(91\) 1.98596e19 0.0534610
\(92\) −2.22602e19 −0.0534264
\(93\) 7.76293e20 1.66324
\(94\) 6.58528e20 1.26105
\(95\) 0 0
\(96\) −1.76417e20 −0.270828
\(97\) 1.19926e21 1.65124 0.825618 − 0.564229i \(-0.190827\pi\)
0.825618 + 0.564229i \(0.190827\pi\)
\(98\) 5.69683e20 0.704303
\(99\) −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.a.g.1.3 ✓ 3
5.2 odd 4 50.22.b.g.49.1 6
5.3 odd 4 50.22.b.g.49.6 6
5.4 even 2 50.22.a.h.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.22.a.g.1.3 ✓ 3 1.1 even 1 trivial
50.22.a.h.1.1 yes 3 5.4 even 2
50.22.b.g.49.1 6 5.2 odd 4
50.22.b.g.49.6 6 5.3 odd 4