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Properties

Label 9.100.a.e.1.4
Level $9$
Weight $100$
Character 9.1
Self dual yes
Analytic conductor $558.609$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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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] = [9,100,Mod(1,9)] 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("9.1"); S:= CuspForms(chi, 100); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 100, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 100 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0] 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: \(558.609014683\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: multiple of \( 2^{224}\cdot 3^{268}\cdot 5^{33}\cdot 7^{18}\cdot 11^{6}\cdot 13^{2}\cdot 17^{4} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-5.00999e13\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-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)
 
\(f(q)\) \(=\) \(q-9.01798e14 q^{2} +1.79413e29 q^{4} -1.38624e34 q^{5} +4.17482e41 q^{7} +4.09787e44 q^{8} +1.25011e49 q^{10} +7.26502e50 q^{11} +1.32702e55 q^{13} -3.76484e56 q^{14} -4.83262e59 q^{16} +3.50079e60 q^{17} -9.78599e62 q^{19} -2.48710e63 q^{20} -6.55158e65 q^{22} -3.97461e67 q^{23} -1.38556e69 q^{25} -1.19670e70 q^{26} +7.49019e70 q^{28} +2.93386e71 q^{29} -5.40043e73 q^{31} +1.76071e74 q^{32} -3.15701e75 q^{34} -5.78730e75 q^{35} -3.83467e77 q^{37} +8.82498e77 q^{38} -5.68064e78 q^{40} -9.99042e79 q^{41} +6.44544e80 q^{43} +1.30344e80 q^{44} +3.58430e82 q^{46} -2.50777e82 q^{47} -2.87777e83 q^{49} +1.24949e84 q^{50} +2.38085e84 q^{52} -6.43286e84 q^{53} -1.00711e85 q^{55} +1.71079e86 q^{56} -2.64575e86 q^{58} -2.92393e87 q^{59} +2.48164e88 q^{61} +4.87009e88 q^{62} +1.47523e89 q^{64} -1.83957e89 q^{65} -2.95084e90 q^{67} +6.28090e89 q^{68} +5.21898e90 q^{70} +4.11037e91 q^{71} +9.19776e91 q^{73} +3.45809e92 q^{74} -1.75574e92 q^{76} +3.03301e92 q^{77} -4.32925e92 q^{79} +6.69918e93 q^{80} +9.00934e94 q^{82} +5.19373e94 q^{83} -4.85294e94 q^{85} -5.81249e95 q^{86} +2.97711e95 q^{88} -6.89330e95 q^{89} +5.54006e96 q^{91} -7.13099e96 q^{92} +2.26150e97 q^{94} +1.35657e97 q^{95} -1.14776e98 q^{97} +2.59517e98 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 50\!\cdots\!12 q^{4} + 19\!\cdots\!20 q^{7} + 68\!\cdots\!20 q^{10} + 29\!\cdots\!80 q^{13} + 16\!\cdots\!24 q^{16} + 10\!\cdots\!56 q^{19} + 95\!\cdots\!00 q^{22} + 95\!\cdots\!80 q^{25} + 62\!\cdots\!40 q^{28}+ \cdots + 54\!\cdots\!20 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.01798e14 −1.13272 −0.566362 − 0.824156i \(-0.691650\pi\)
−0.566362 + 0.824156i \(0.691650\pi\)
\(3\) 0 0
\(4\) 1.79413e29 0.283065
\(5\) −1.38624e34 −0.348998 −0.174499 − 0.984657i \(-0.555831\pi\)
−0.174499 + 0.984657i \(0.555831\pi\)
\(6\) 0 0
\(7\) 4.17482e41 0.614164 0.307082 − 0.951683i \(-0.400647\pi\)
0.307082 + 0.951683i \(0.400647\pi\)
\(8\) 4.09787e44 0.812090
\(9\) 0 0
\(10\) 1.25011e49 0.395319
\(11\) 7.26502e50 0.205257 0.102629 − 0.994720i \(-0.467275\pi\)
0.102629 + 0.994720i \(0.467275\pi\)
\(12\) 0 0
\(13\) 1.32702e55 0.960905 0.480453 − 0.877021i \(-0.340472\pi\)
0.480453 + 0.877021i \(0.340472\pi\)
\(14\) −3.76484e56 −0.695679
\(15\) 0 0
\(16\) −4.83262e59 −1.20294
\(17\) 3.50079e60 0.433456 0.216728 − 0.976232i \(-0.430462\pi\)
0.216728 + 0.976232i \(0.430462\pi\)
\(18\) 0 0
\(19\) −9.78599e62 −0.492381 −0.246191 − 0.969221i \(-0.579179\pi\)
−0.246191 + 0.969221i \(0.579179\pi\)
\(20\) −2.48710e63 −0.0987890
\(21\) 0 0
\(22\) −6.55158e65 −0.232500
\(23\) −3.97461e67 −1.56231 −0.781155 − 0.624337i \(-0.785369\pi\)
−0.781155 + 0.624337i \(0.785369\pi\)
\(24\) 0 0
\(25\) −1.38556e69 −0.878200
\(26\) −1.19670e70 −1.08844
\(27\) 0 0
\(28\) 7.49019e70 0.173848
\(29\) 2.93386e71 0.119878 0.0599388 − 0.998202i \(-0.480909\pi\)
0.0599388 + 0.998202i \(0.480909\pi\)
\(30\) 0 0
\(31\) −5.40043e73 −0.812876 −0.406438 − 0.913678i \(-0.633229\pi\)
−0.406438 + 0.913678i \(0.633229\pi\)
\(32\) 1.76071e74 0.550508
\(33\) 0 0
\(34\) −3.15701e75 −0.490986
\(35\) −5.78730e75 −0.214342
\(36\) 0 0
\(37\) −3.83467e77 −0.907282 −0.453641 − 0.891185i \(-0.649875\pi\)
−0.453641 + 0.891185i \(0.649875\pi\)
\(38\) 8.82498e77 0.557732
\(39\) 0 0
\(40\) −5.68064e78 −0.283418
\(41\) −9.99042e79 −1.46819 −0.734096 − 0.679045i \(-0.762394\pi\)
−0.734096 + 0.679045i \(0.762394\pi\)
\(42\) 0 0
\(43\) 6.44544e80 0.896529 0.448264 − 0.893901i \(-0.352042\pi\)
0.448264 + 0.893901i \(0.352042\pi\)
\(44\) 1.30344e80 0.0581011
\(45\) 0 0
\(46\) 3.58430e82 1.76967
\(47\) −2.50777e82 −0.427016 −0.213508 − 0.976941i \(-0.568489\pi\)
−0.213508 + 0.976941i \(0.568489\pi\)
\(48\) 0 0
\(49\) −2.87777e83 −0.622802
\(50\) 1.24949e84 0.994759
\(51\) 0 0
\(52\) 2.38085e84 0.271998
\(53\) −6.43286e84 −0.286252 −0.143126 − 0.989704i \(-0.545715\pi\)
−0.143126 + 0.989704i \(0.545715\pi\)
\(54\) 0 0
\(55\) −1.00711e85 −0.0716345
\(56\) 1.71079e86 0.498757
\(57\) 0 0
\(58\) −2.64575e86 −0.135788
\(59\) −2.92393e87 −0.643861 −0.321931 − 0.946763i \(-0.604332\pi\)
−0.321931 + 0.946763i \(0.604332\pi\)
\(60\) 0 0
\(61\) 2.48164e88 1.04933 0.524664 − 0.851309i \(-0.324191\pi\)
0.524664 + 0.851309i \(0.324191\pi\)
\(62\) 4.87009e88 0.920765
\(63\) 0 0
\(64\) 1.47523e89 0.579365
\(65\) −1.83957e89 −0.335354
\(66\) 0 0
\(67\) −2.95084e90 −1.20017 −0.600084 − 0.799937i \(-0.704866\pi\)
−0.600084 + 0.799937i \(0.704866\pi\)
\(68\) 6.28090e89 0.122696
\(69\) 0 0
\(70\) 5.21898e90 0.242791
\(71\) 4.11037e91 0.947529 0.473765 − 0.880651i \(-0.342895\pi\)
0.473765 + 0.880651i \(0.342895\pi\)
\(72\) 0 0
\(73\) 9.19776e91 0.536044 0.268022 − 0.963413i \(-0.413630\pi\)
0.268022 + 0.963413i \(0.413630\pi\)
\(74\) 3.45809e92 1.02770
\(75\) 0 0
\(76\) −1.75574e92 −0.139376
\(77\) 3.03301e92 0.126062
\(78\) 0 0
\(79\) −4.32925e92 −0.0505676 −0.0252838 − 0.999680i \(-0.508049\pi\)
−0.0252838 + 0.999680i \(0.508049\pi\)
\(80\) 6.69918e93 0.419824
\(81\) 0 0
\(82\) 9.00934e94 1.66306
\(83\) 5.19373e94 0.526155 0.263077 − 0.964775i \(-0.415263\pi\)
0.263077 + 0.964775i \(0.415263\pi\)
\(84\) 0 0
\(85\) −4.85294e94 −0.151275
\(86\) −5.81249e95 −1.01552
\(87\) 0 0
\(88\) 2.97711e95 0.166688
\(89\) −6.89330e95 −0.220608 −0.110304 − 0.993898i \(-0.535182\pi\)
−0.110304 + 0.993898i \(0.535182\pi\)
\(90\) 0 0
\(91\) 5.54006e96 0.590154
\(92\) −7.13099e96 −0.442234
\(93\) 0 0
\(94\) 2.26150e97 0.483691
\(95\) 1.35657e97 0.171840
\(96\) 0 0
\(97\) −1.14776e98 −0.518383 −0.259192 − 0.965826i \(-0.583456\pi\)
−0.259192 + 0.965826i \(0.583456\pi\)
\(98\) 2.59517e98 0.705463
\(99\) 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 9.100.a.e.1.4 ✓ 16
3.2 odd 2 inner 9.100.a.e.1.13 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.100.a.e.1.4 ✓ 16 1.1 even 1 trivial
9.100.a.e.1.13 yes 16 3.2 odd 2 inner