Properties

Label 2.22.a.a.1.1
Level $2$
Weight $22$
Character 2.1
Self dual yes
Analytic conductor $5.590$
Analytic rank $1$
Dimension $1$
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] = [2,22,Mod(1,2)] 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("2.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-1024] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.58954688574\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{2} +71604.0 q^{3} +1.04858e6 q^{4} -2.86938e7 q^{5} -7.33225e7 q^{6} -8.53202e8 q^{7} -1.07374e9 q^{8} -5.33322e9 q^{9} +2.93824e10 q^{10} +8.67312e10 q^{11} +7.50822e10 q^{12} -8.95323e11 q^{13} +8.73679e11 q^{14} -2.05459e12 q^{15} +1.09951e12 q^{16} +3.25757e12 q^{17} +5.46122e12 q^{18} +2.30325e13 q^{19} -3.00876e13 q^{20} -6.10927e13 q^{21} -8.88127e13 q^{22} +1.46496e14 q^{23} -7.68842e13 q^{24} +3.46495e14 q^{25} +9.16811e14 q^{26} -1.13088e15 q^{27} -8.94648e14 q^{28} -7.34052e14 q^{29} +2.10390e15 q^{30} -3.14666e15 q^{31} -1.12590e15 q^{32} +6.21030e15 q^{33} -3.33575e15 q^{34} +2.44816e16 q^{35} -5.59229e15 q^{36} -1.29638e16 q^{37} -2.35852e16 q^{38} -6.41087e16 q^{39} +3.08097e16 q^{40} +4.57146e16 q^{41} +6.25589e16 q^{42} -2.40736e16 q^{43} +9.09442e16 q^{44} +1.53030e17 q^{45} -1.50012e17 q^{46} -4.49992e17 q^{47} +7.87294e16 q^{48} +1.69408e17 q^{49} -3.54811e17 q^{50} +2.33255e17 q^{51} -9.38815e17 q^{52} +2.06484e18 q^{53} +1.15802e18 q^{54} -2.48864e18 q^{55} +9.16119e17 q^{56} +1.64922e18 q^{57} +7.51669e17 q^{58} -3.78050e18 q^{59} -2.15439e18 q^{60} -7.61981e18 q^{61} +3.22218e18 q^{62} +4.55032e18 q^{63} +1.15292e18 q^{64} +2.56902e19 q^{65} -6.35935e18 q^{66} -1.87912e19 q^{67} +3.41581e18 q^{68} +1.04897e19 q^{69} -2.50692e19 q^{70} -4.52649e18 q^{71} +5.72650e18 q^{72} -2.55715e19 q^{73} +1.32749e19 q^{74} +2.48104e19 q^{75} +2.41513e19 q^{76} -7.39992e19 q^{77} +6.56473e19 q^{78} +9.93364e19 q^{79} -3.15491e19 q^{80} -2.51884e19 q^{81} -4.68118e19 q^{82} +2.95818e18 q^{83} -6.40603e19 q^{84} -9.34719e19 q^{85} +2.46514e19 q^{86} -5.25610e19 q^{87} -9.31269e19 q^{88} +1.18803e20 q^{89} -1.56703e20 q^{90} +7.63892e20 q^{91} +1.53612e20 q^{92} -2.25314e20 q^{93} +4.60792e20 q^{94} -6.60888e20 q^{95} -8.06189e19 q^{96} -5.69053e20 q^{97} -1.73474e20 q^{98} -4.62556e20 q^{99} +O(q^{100})\)

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
\(3\) 71604.0 0.700106 0.350053 − 0.936730i \(-0.386163\pi\)
0.350053 + 0.936730i \(0.386163\pi\)
\(4\) 1.04858e6 0.500000
\(5\) −2.86938e7 −1.31402 −0.657011 − 0.753881i \(-0.728179\pi\)
−0.657011 + 0.753881i \(0.728179\pi\)
\(6\) −7.33225e7 −0.495050
\(7\) −8.53202e8 −1.14162 −0.570811 − 0.821081i \(-0.693371\pi\)
−0.570811 + 0.821081i \(0.693371\pi\)
\(8\) −1.07374e9 −0.353553
\(9\) −5.33322e9 −0.509851
\(10\) 2.93824e10 0.929154
\(11\) 8.67312e10 1.00821 0.504106 − 0.863642i \(-0.331822\pi\)
0.504106 + 0.863642i \(0.331822\pi\)
\(12\) 7.50822e10 0.350053
\(13\) −8.95323e11 −1.80125 −0.900627 − 0.434594i \(-0.856892\pi\)
−0.900627 + 0.434594i \(0.856892\pi\)
\(14\) 8.73679e11 0.807249
\(15\) −2.05459e12 −0.919955
\(16\) 1.09951e12 0.250000
\(17\) 3.25757e12 0.391904 0.195952 − 0.980613i \(-0.437220\pi\)
0.195952 + 0.980613i \(0.437220\pi\)
\(18\) 5.46122e12 0.360519
\(19\) 2.30325e13 0.861842 0.430921 − 0.902390i \(-0.358189\pi\)
0.430921 + 0.902390i \(0.358189\pi\)
\(20\) −3.00876e13 −0.657011
\(21\) −6.10927e13 −0.799258
\(22\) −8.88127e13 −0.712914
\(23\) 1.46496e14 0.737365 0.368683 − 0.929555i \(-0.379809\pi\)
0.368683 + 0.929555i \(0.379809\pi\)
\(24\) −7.68842e13 −0.247525
\(25\) 3.46495e14 0.726653
\(26\) 9.16811e14 1.27368
\(27\) −1.13088e15 −1.05706
\(28\) −8.94648e14 −0.570811
\(29\) −7.34052e14 −0.324002 −0.162001 − 0.986791i \(-0.551795\pi\)
−0.162001 + 0.986791i \(0.551795\pi\)
\(30\) 2.10390e15 0.650507
\(31\) −3.14666e15 −0.689529 −0.344765 − 0.938689i \(-0.612041\pi\)
−0.344765 + 0.938689i \(0.612041\pi\)
\(32\) −1.12590e15 −0.176777
\(33\) 6.21030e15 0.705856
\(34\) −3.33575e15 −0.277118
\(35\) 2.44816e16 1.50012
\(36\) −5.59229e15 −0.254925
\(37\) −1.29638e16 −0.443215 −0.221608 − 0.975136i \(-0.571130\pi\)
−0.221608 + 0.975136i \(0.571130\pi\)
\(38\) −2.35852e16 −0.609414
\(39\) −6.41087e16 −1.26107
\(40\) 3.08097e16 0.464577
\(41\) 4.57146e16 0.531894 0.265947 − 0.963988i \(-0.414315\pi\)
0.265947 + 0.963988i \(0.414315\pi\)
\(42\) 6.25589e16 0.565160
\(43\) −2.40736e16 −0.169872 −0.0849361 − 0.996386i \(-0.527069\pi\)
−0.0849361 + 0.996386i \(0.527069\pi\)
\(44\) 9.09442e16 0.504106
\(45\) 1.53030e17 0.669955
\(46\) −1.50012e17 −0.521396
\(47\) −4.49992e17 −1.24789 −0.623946 − 0.781467i \(-0.714472\pi\)
−0.623946 + 0.781467i \(0.714472\pi\)
\(48\) 7.87294e16 0.175027
\(49\) 1.69408e17 0.303303
\(50\) −3.54811e17 −0.513821
\(51\) 2.33255e17 0.274375
\(52\) −9.38815e17 −0.900627
\(53\) 2.06484e18 1.62177 0.810885 − 0.585206i \(-0.198986\pi\)
0.810885 + 0.585206i \(0.198986\pi\)
\(54\) 1.15802e18 0.747452
\(55\) −2.48864e18 −1.32481
\(56\) 9.16119e17 0.403625
\(57\) 1.64922e18 0.603381
\(58\) 7.51669e17 0.229104
\(59\) −3.78050e18 −0.962948 −0.481474 − 0.876460i \(-0.659898\pi\)
−0.481474 + 0.876460i \(0.659898\pi\)
\(60\) −2.15439e18 −0.459978
\(61\) −7.61981e18 −1.36767 −0.683835 − 0.729637i \(-0.739689\pi\)
−0.683835 + 0.729637i \(0.739689\pi\)
\(62\) 3.22218e18 0.487571
\(63\) 4.55032e18 0.582057
\(64\) 1.15292e18 0.125000
\(65\) 2.56902e19 2.36689
\(66\) −6.35935e18 −0.499116
\(67\) −1.87912e19 −1.25941 −0.629706 − 0.776833i \(-0.716825\pi\)
−0.629706 + 0.776833i \(0.716825\pi\)
\(68\) 3.41581e18 0.195952
\(69\) 1.04897e19 0.516234
\(70\) −2.50692e19 −1.06074
\(71\) −4.52649e18 −0.165025 −0.0825123 − 0.996590i \(-0.526294\pi\)
−0.0825123 + 0.996590i \(0.526294\pi\)
\(72\) 5.72650e18 0.180260
\(73\) −2.55715e19 −0.696411 −0.348205 − 0.937418i \(-0.613209\pi\)
−0.348205 + 0.937418i \(0.613209\pi\)
\(74\) 1.32749e19 0.313400
\(75\) 2.48104e19 0.508735
\(76\) 2.41513e19 0.430921
\(77\) −7.39992e19 −1.15100
\(78\) 6.56473e19 0.891710
\(79\) 9.93364e19 1.18039 0.590193 − 0.807262i \(-0.299052\pi\)
0.590193 + 0.807262i \(0.299052\pi\)
\(80\) −3.15491e19 −0.328505
\(81\) −2.51884e19 −0.230201
\(82\) −4.68118e19 −0.376106
\(83\) 2.95818e18 0.0209269 0.0104634 − 0.999945i \(-0.496669\pi\)
0.0104634 + 0.999945i \(0.496669\pi\)
\(84\) −6.40603e19 −0.399629
\(85\) −9.34719e19 −0.514970
\(86\) 2.46514e19 0.120118
\(87\) −5.25610e19 −0.226836
\(88\) −9.31269e19 −0.356457
\(89\) 1.18803e20 0.403861 0.201931 − 0.979400i \(-0.435278\pi\)
0.201931 + 0.979400i \(0.435278\pi\)
\(90\) −1.56703e20 −0.473730
\(91\) 7.63892e20 2.05635
\(92\) 1.53612e20 0.368683
\(93\) −2.25314e20 −0.482744
\(94\) 4.60792e20 0.882393
\(95\) −6.60888e20 −1.13248
\(96\) −8.06189e19 −0.123763
\(97\) −5.69053e20 −0.783519 −0.391759 − 0.920068i \(-0.628133\pi\)
−0.391759 + 0.920068i \(0.628133\pi\)
\(98\) −1.73474e20 −0.214467
\(99\) −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 2.22.a.a.1.1 ✓ 1
3.2 odd 2 18.22.a.e.1.1 1
4.3 odd 2 16.22.a.a.1.1 1
5.2 odd 4 50.22.b.a.49.1 2
5.3 odd 4 50.22.b.a.49.2 2
5.4 even 2 50.22.a.c.1.1 1
8.3 odd 2 64.22.a.f.1.1 1
8.5 even 2 64.22.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.22.a.a.1.1 ✓ 1 1.1 even 1 trivial
16.22.a.a.1.1 1 4.3 odd 2
18.22.a.e.1.1 1 3.2 odd 2
50.22.a.c.1.1 1 5.4 even 2
50.22.b.a.49.1 2 5.2 odd 4
50.22.b.a.49.2 2 5.3 odd 4
64.22.a.b.1.1 1 8.5 even 2
64.22.a.f.1.1 1 8.3 odd 2