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Properties

Label 10.22.a.d.1.2
Level $10$
Weight $22$
Character 10.1
Self dual yes
Analytic conductor $27.948$
Analytic rank $0$
Dimension $2$
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] = [10,22,Mod(1,10)] 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("10.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(10, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,2048,30972] 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: \(27.9477344287\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1179649}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 294912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-542.558\) of defining polynomial
Character \(\chi\) \(=\) 10.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1024.00 q^{2} +145820. q^{3} +1.04858e6 q^{4} -9.76562e6 q^{5} +1.49320e8 q^{6} -2.36011e8 q^{7} +1.07374e9 q^{8} +1.08031e10 q^{9} -1.00000e10 q^{10} +1.49202e11 q^{11} +1.52903e11 q^{12} +4.89350e11 q^{13} -2.41675e11 q^{14} -1.42402e12 q^{15} +1.09951e12 q^{16} +5.46288e12 q^{17} +1.10624e13 q^{18} -1.12049e13 q^{19} -1.02400e13 q^{20} -3.44151e13 q^{21} +1.52783e14 q^{22} +1.78791e13 q^{23} +1.56573e14 q^{24} +9.53674e13 q^{25} +5.01094e14 q^{26} +4.99812e13 q^{27} -2.47475e14 q^{28} -2.82120e15 q^{29} -1.45820e15 q^{30} +4.24797e15 q^{31} +1.12590e15 q^{32} +2.17567e16 q^{33} +5.59399e15 q^{34} +2.30479e15 q^{35} +1.13279e16 q^{36} +5.74457e16 q^{37} -1.14738e16 q^{38} +7.13570e16 q^{39} -1.04858e16 q^{40} -3.51283e16 q^{41} -3.52410e16 q^{42} -2.62144e17 q^{43} +1.56450e17 q^{44} -1.05499e17 q^{45} +1.83081e16 q^{46} +1.08406e17 q^{47} +1.60331e17 q^{48} -5.02845e17 q^{49} +9.76562e16 q^{50} +7.96597e17 q^{51} +5.13120e17 q^{52} -2.47953e18 q^{53} +5.11807e16 q^{54} -1.45705e18 q^{55} -2.53415e17 q^{56} -1.63389e18 q^{57} -2.88891e18 q^{58} +2.14259e17 q^{59} -1.49320e18 q^{60} +3.64272e18 q^{61} +4.34992e18 q^{62} -2.54965e18 q^{63} +1.15292e18 q^{64} -4.77881e18 q^{65} +2.22788e19 q^{66} -1.13372e19 q^{67} +5.72825e18 q^{68} +2.60712e18 q^{69} +2.36011e18 q^{70} +2.44087e19 q^{71} +1.15998e19 q^{72} -4.12244e19 q^{73} +5.88244e19 q^{74} +1.39065e19 q^{75} -1.17492e19 q^{76} -3.52133e19 q^{77} +7.30695e19 q^{78} +6.83832e19 q^{79} -1.07374e19 q^{80} -1.05716e20 q^{81} -3.59714e19 q^{82} +1.39311e20 q^{83} -3.60868e19 q^{84} -5.33484e19 q^{85} -2.68435e20 q^{86} -4.11387e20 q^{87} +1.60205e20 q^{88} -4.04817e20 q^{89} -1.08031e20 q^{90} -1.15492e20 q^{91} +1.87475e19 q^{92} +6.19438e20 q^{93} +1.11008e20 q^{94} +1.09423e20 q^{95} +1.64179e20 q^{96} -9.18963e20 q^{97} -5.14913e20 q^{98} +1.61185e21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2048 q^{2} + 30972 q^{3} + 2097152 q^{4} - 19531250 q^{5} + 31715328 q^{6} - 439959356 q^{7} + 2147483648 q^{8} + 13532817186 q^{9} - 20000000000 q^{10} + 105191777184 q^{11} + 32476495872 q^{12}+ \cdots + 14\!\cdots\!12 q^{99}+O(q^{100}) \) 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
\(3\) 145820. 1.42575 0.712876 − 0.701290i \(-0.247392\pi\)
0.712876 + 0.701290i \(0.247392\pi\)
\(4\) 1.04858e6 0.500000
\(5\) −9.76562e6 −0.447214
\(6\) 1.49320e8 1.00816
\(7\) −2.36011e8 −0.315793 −0.157896 − 0.987456i \(-0.550471\pi\)
−0.157896 + 0.987456i \(0.550471\pi\)
\(8\) 1.07374e9 0.353553
\(9\) 1.08031e10 1.03277
\(10\) −1.00000e10 −0.316228
\(11\) 1.49202e11 1.73441 0.867206 − 0.497949i \(-0.165913\pi\)
0.867206 + 0.497949i \(0.165913\pi\)
\(12\) 1.52903e11 0.712876
\(13\) 4.89350e11 0.984497 0.492248 − 0.870455i \(-0.336175\pi\)
0.492248 + 0.870455i \(0.336175\pi\)
\(14\) −2.41675e11 −0.223299
\(15\) −1.42402e12 −0.637615
\(16\) 1.09951e12 0.250000
\(17\) 5.46288e12 0.657216 0.328608 − 0.944466i \(-0.393421\pi\)
0.328608 + 0.944466i \(0.393421\pi\)
\(18\) 1.10624e13 0.730277
\(19\) −1.12049e13 −0.419270 −0.209635 − 0.977780i \(-0.567228\pi\)
−0.209635 + 0.977780i \(0.567228\pi\)
\(20\) −1.02400e13 −0.223607
\(21\) −3.44151e13 −0.450242
\(22\) 1.52783e14 1.22641
\(23\) 1.78791e13 0.0899916 0.0449958 − 0.998987i \(-0.485673\pi\)
0.0449958 + 0.998987i \(0.485673\pi\)
\(24\) 1.56573e14 0.504079
\(25\) 9.53674e13 0.200000
\(26\) 5.01094e14 0.696144
\(27\) 4.99812e13 0.0467183
\(28\) −2.47475e14 −0.157896
\(29\) −2.82120e15 −1.24525 −0.622623 − 0.782522i \(-0.713933\pi\)
−0.622623 + 0.782522i \(0.713933\pi\)
\(30\) −1.45820e15 −0.450862
\(31\) 4.24797e15 0.930858 0.465429 − 0.885085i \(-0.345900\pi\)
0.465429 + 0.885085i \(0.345900\pi\)
\(32\) 1.12590e15 0.176777
\(33\) 2.17567e16 2.47284
\(34\) 5.59399e15 0.464722
\(35\) 2.30479e15 0.141227
\(36\) 1.13279e16 0.516384
\(37\) 5.74457e16 1.96399 0.981995 − 0.188909i \(-0.0604951\pi\)
0.981995 + 0.188909i \(0.0604951\pi\)
\(38\) −1.14738e16 −0.296469
\(39\) 7.13570e16 1.40365
\(40\) −1.04858e16 −0.158114
\(41\) −3.51283e16 −0.408721 −0.204361 − 0.978896i \(-0.565512\pi\)
−0.204361 + 0.978896i \(0.565512\pi\)
\(42\) −3.52410e16 −0.318369
\(43\) −2.62144e17 −1.84978 −0.924891 − 0.380233i \(-0.875844\pi\)
−0.924891 + 0.380233i \(0.875844\pi\)
\(44\) 1.56450e17 0.867206
\(45\) −1.05499e17 −0.461868
\(46\) 1.83081e16 0.0636337
\(47\) 1.08406e17 0.300626 0.150313 − 0.988638i \(-0.451972\pi\)
0.150313 + 0.988638i \(0.451972\pi\)
\(48\) 1.60331e17 0.356438
\(49\) −5.02845e17 −0.900275
\(50\) 9.76562e16 0.141421
\(51\) 7.96597e17 0.937027
\(52\) 5.13120e17 0.492248
\(53\) −2.47953e18 −1.94748 −0.973740 − 0.227662i \(-0.926892\pi\)
−0.973740 + 0.227662i \(0.926892\pi\)
\(54\) 5.11807e16 0.0330348
\(55\) −1.45705e18 −0.775653
\(56\) −2.53415e17 −0.111650
\(57\) −1.63389e18 −0.597775
\(58\) −2.88891e18 −0.880521
\(59\) 2.14259e17 0.0545748 0.0272874 − 0.999628i \(-0.491313\pi\)
0.0272874 + 0.999628i \(0.491313\pi\)
\(60\) −1.49320e18 −0.318808
\(61\) 3.64272e18 0.653826 0.326913 − 0.945054i \(-0.393992\pi\)
0.326913 + 0.945054i \(0.393992\pi\)
\(62\) 4.34992e18 0.658216
\(63\) −2.54965e18 −0.326141
\(64\) 1.15292e18 0.125000
\(65\) −4.77881e18 −0.440280
\(66\) 2.22788e19 1.74856
\(67\) −1.13372e19 −0.759835 −0.379918 − 0.925020i \(-0.624048\pi\)
−0.379918 + 0.925020i \(0.624048\pi\)
\(68\) 5.72825e18 0.328608
\(69\) 2.60712e18 0.128306
\(70\) 2.36011e18 0.0998625
\(71\) 2.44087e19 0.889882 0.444941 − 0.895560i \(-0.353225\pi\)
0.444941 + 0.895560i \(0.353225\pi\)
\(72\) 1.15998e19 0.365138
\(73\) −4.12244e19 −1.12270 −0.561350 − 0.827578i \(-0.689718\pi\)
−0.561350 + 0.827578i \(0.689718\pi\)
\(74\) 5.88244e19 1.38875
\(75\) 1.39065e19 0.285150
\(76\) −1.17492e19 −0.209635
\(77\) −3.52133e19 −0.547715
\(78\) 7.30695e19 0.992529
\(79\) 6.83832e19 0.812578 0.406289 − 0.913745i \(-0.366823\pi\)
0.406289 + 0.913745i \(0.366823\pi\)
\(80\) −1.07374e19 −0.111803
\(81\) −1.05716e20 −0.966159
\(82\) −3.59714e19 −0.289009
\(83\) 1.39311e20 0.985522 0.492761 − 0.870165i \(-0.335988\pi\)
0.492761 + 0.870165i \(0.335988\pi\)
\(84\) −3.60868e19 −0.225121
\(85\) −5.33484e19 −0.293916
\(86\) −2.68435e20 −1.30799
\(87\) −4.11387e20 −1.77541
\(88\) 1.60205e20 0.613207
\(89\) −4.04817e20 −1.37614 −0.688072 − 0.725643i \(-0.741542\pi\)
−0.688072 + 0.725643i \(0.741542\pi\)
\(90\) −1.08031e20 −0.326590
\(91\) −1.15492e20 −0.310897
\(92\) 1.87475e19 0.0449958
\(93\) 6.19438e20 1.32717
\(94\) 1.11008e20 0.212575
\(95\) 1.09423e20 0.187503
\(96\) 1.64179e20 0.252040
\(97\) −9.18963e20 −1.26530 −0.632652 − 0.774436i \(-0.718034\pi\)
−0.632652 + 0.774436i \(0.718034\pi\)
\(98\) −5.14913e20 −0.636590
\(99\) 1.61185e21 1.79124
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 10.22.a.d.1.2 ✓ 2
4.3 odd 2 80.22.a.c.1.1 2
5.2 odd 4 50.22.b.e.49.3 4
5.3 odd 4 50.22.b.e.49.2 4
5.4 even 2 50.22.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.22.a.d.1.2 ✓ 2 1.1 even 1 trivial
50.22.a.d.1.1 2 5.4 even 2
50.22.b.e.49.2 4 5.3 odd 4
50.22.b.e.49.3 4 5.2 odd 4
80.22.a.c.1.1 2 4.3 odd 2