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Properties

Label 36.22.a.a.1.1
Level $36$
Weight $22$
Character 36.1
Self dual yes
Analytic conductor $100.612$
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] = [36,22,Mod(1,36)] 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("36.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(36, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 36.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,11268090] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(100.611843943\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 12)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 36.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.12681e7 q^{5} +2.81914e8 q^{7} +3.61721e10 q^{11} -4.49099e11 q^{13} -2.12186e12 q^{17} -4.60941e12 q^{19} -9.50953e13 q^{23} -3.49867e14 q^{25} +2.24574e15 q^{29} -3.15569e15 q^{31} +3.17663e15 q^{35} -1.81785e16 q^{37} +1.69650e17 q^{41} -1.58969e17 q^{43} +1.34697e17 q^{47} -4.79070e17 q^{49} +1.56374e16 q^{53} +4.07590e17 q^{55} -2.97724e18 q^{59} +3.60386e18 q^{61} -5.06048e18 q^{65} +2.10662e19 q^{67} -2.19801e19 q^{71} -1.70544e19 q^{73} +1.01974e19 q^{77} -1.15020e20 q^{79} +9.66285e19 q^{83} -2.39093e19 q^{85} -6.04276e19 q^{89} -1.26607e20 q^{91} -5.19392e19 q^{95} -4.07820e20 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.12681e7 0.516018 0.258009 − 0.966142i \(-0.416933\pi\)
0.258009 + 0.966142i \(0.416933\pi\)
\(6\) 0 0
\(7\) 2.81914e8 0.377214 0.188607 − 0.982053i \(-0.439603\pi\)
0.188607 + 0.982053i \(0.439603\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.61721e10 0.420485 0.210242 − 0.977649i \(-0.432575\pi\)
0.210242 + 0.977649i \(0.432575\pi\)
\(12\) 0 0
\(13\) −4.49099e11 −0.903517 −0.451759 − 0.892140i \(-0.649203\pi\)
−0.451759 + 0.892140i \(0.649203\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.12186e12 −0.255272 −0.127636 − 0.991821i \(-0.540739\pi\)
−0.127636 + 0.991821i \(0.540739\pi\)
\(18\) 0 0
\(19\) −4.60941e12 −0.172477 −0.0862387 − 0.996275i \(-0.527485\pi\)
−0.0862387 + 0.996275i \(0.527485\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −9.50953e13 −0.478648 −0.239324 − 0.970940i \(-0.576926\pi\)
−0.239324 + 0.970940i \(0.576926\pi\)
\(24\) 0 0
\(25\) −3.49867e14 −0.733725
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.24574e15 0.991245 0.495622 − 0.868538i \(-0.334940\pi\)
0.495622 + 0.868538i \(0.334940\pi\)
\(30\) 0 0
\(31\) −3.15569e15 −0.691508 −0.345754 − 0.938325i \(-0.612377\pi\)
−0.345754 + 0.938325i \(0.612377\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.17663e15 0.194649
\(36\) 0 0
\(37\) −1.81785e16 −0.621498 −0.310749 − 0.950492i \(-0.600580\pi\)
−0.310749 + 0.950492i \(0.600580\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.69650e17 1.97389 0.986944 − 0.161061i \(-0.0514916\pi\)
0.986944 + 0.161061i \(0.0514916\pi\)
\(42\) 0 0
\(43\) −1.58969e17 −1.12174 −0.560870 − 0.827904i \(-0.689533\pi\)
−0.560870 + 0.827904i \(0.689533\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.34697e17 0.373536 0.186768 − 0.982404i \(-0.440199\pi\)
0.186768 + 0.982404i \(0.440199\pi\)
\(48\) 0 0
\(49\) −4.79070e17 −0.857710
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.56374e16 0.0122819 0.00614097 − 0.999981i \(-0.498045\pi\)
0.00614097 + 0.999981i \(0.498045\pi\)
\(54\) 0 0
\(55\) 4.07590e17 0.216978
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.97724e18 −0.758347 −0.379173 − 0.925326i \(-0.623792\pi\)
−0.379173 + 0.925326i \(0.623792\pi\)
\(60\) 0 0
\(61\) 3.60386e18 0.646851 0.323425 − 0.946254i \(-0.395166\pi\)
0.323425 + 0.946254i \(0.395166\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.06048e18 −0.466232
\(66\) 0 0
\(67\) 2.10662e19 1.41189 0.705945 − 0.708267i \(-0.250523\pi\)
0.705945 + 0.708267i \(0.250523\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.19801e19 −0.801340 −0.400670 − 0.916222i \(-0.631223\pi\)
−0.400670 + 0.916222i \(0.631223\pi\)
\(72\) 0 0
\(73\) −1.70544e19 −0.464458 −0.232229 − 0.972661i \(-0.574602\pi\)
−0.232229 + 0.972661i \(0.574602\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.01974e19 0.158613
\(78\) 0 0
\(79\) −1.15020e20 −1.36675 −0.683375 − 0.730067i \(-0.739489\pi\)
−0.683375 + 0.730067i \(0.739489\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.66285e19 0.683574 0.341787 − 0.939777i \(-0.388968\pi\)
0.341787 + 0.939777i \(0.388968\pi\)
\(84\) 0 0
\(85\) −2.39093e19 −0.131725
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.04276e19 −0.205419 −0.102709 − 0.994711i \(-0.532751\pi\)
−0.102709 + 0.994711i \(0.532751\pi\)
\(90\) 0 0
\(91\) −1.26607e20 −0.340819
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.19392e19 −0.0890015
\(96\) 0 0
\(97\) −4.07820e20 −0.561520 −0.280760 − 0.959778i \(-0.590587\pi\)
−0.280760 + 0.959778i \(0.590587\pi\)
\(98\) 0 0
\(99\) 0 0
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 36.22.a.a.1.1 1
3.2 odd 2 12.22.a.a.1.1 ✓ 1
12.11 even 2 48.22.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.22.a.a.1.1 ✓ 1 3.2 odd 2
36.22.a.a.1.1 1 1.1 even 1 trivial
48.22.a.b.1.1 1 12.11 even 2