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Properties

Label 9.100.a.d.1.4
Level $9$
Weight $100$
Character 9.1
Self dual yes
Analytic conductor $558.609$
Analytic rank $1$
Dimension $8$
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] = [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 = [8,208040616902520] 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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{109}\cdot 3^{62}\cdot 5^{13}\cdot 7^{9}\cdot 11^{3}\cdot 13\cdot 17 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(4.33987e12\) 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)
 
\(f(q)\) \(=\) \(q-2.86466e14 q^{2} -5.51763e29 q^{4} +4.79915e34 q^{5} -6.92478e41 q^{7} +3.39630e44 q^{8} -1.37479e49 q^{10} -1.55674e51 q^{11} -2.14548e55 q^{13} +1.98371e56 q^{14} +2.52429e59 q^{16} +1.39488e61 q^{17} +1.51628e63 q^{19} -2.64799e64 q^{20} +4.45952e65 q^{22} -2.55387e67 q^{23} +7.25462e68 q^{25} +6.14606e69 q^{26} +3.82084e71 q^{28} -1.39180e72 q^{29} -1.44293e73 q^{31} -2.87578e74 q^{32} -3.99585e75 q^{34} -3.32331e76 q^{35} -5.19313e77 q^{37} -4.34363e77 q^{38} +1.62994e79 q^{40} +3.58403e79 q^{41} +2.84099e80 q^{43} +8.58951e80 q^{44} +7.31597e81 q^{46} +7.73779e82 q^{47} +1.74580e82 q^{49} -2.07820e83 q^{50} +1.18380e85 q^{52} -1.89817e84 q^{53} -7.47103e85 q^{55} -2.35187e86 q^{56} +3.98704e86 q^{58} +2.13540e87 q^{59} +4.67707e88 q^{61} +4.13350e87 q^{62} -7.76144e88 q^{64} -1.02965e90 q^{65} +3.33536e90 q^{67} -7.69642e90 q^{68} +9.52013e90 q^{70} -2.25284e91 q^{71} +2.47879e90 q^{73} +1.48765e92 q^{74} -8.36629e92 q^{76} +1.07801e93 q^{77} -1.29965e94 q^{79} +1.21144e94 q^{80} -1.02670e94 q^{82} +1.54205e95 q^{83} +6.69423e95 q^{85} -8.13847e94 q^{86} -5.28716e95 q^{88} -2.29359e96 q^{89} +1.48570e97 q^{91} +1.40913e97 q^{92} -2.21661e97 q^{94} +7.27688e97 q^{95} -3.70442e98 q^{97} -5.00113e96 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 208040616902520 q^{2} + 28\!\cdots\!24 q^{4} + 48\!\cdots\!60 q^{5} - 56\!\cdots\!00 q^{7} - 59\!\cdots\!60 q^{8} - 20\!\cdots\!60 q^{10} - 66\!\cdots\!36 q^{11} - 53\!\cdots\!40 q^{13} + 27\!\cdots\!92 q^{14}+ \cdots + 12\!\cdots\!60 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.86466e14 −0.359822 −0.179911 − 0.983683i \(-0.557581\pi\)
−0.179911 + 0.983683i \(0.557581\pi\)
\(3\) 0 0
\(4\) −5.51763e29 −0.870528
\(5\) 4.79915e34 1.20823 0.604114 − 0.796898i \(-0.293527\pi\)
0.604114 + 0.796898i \(0.293527\pi\)
\(6\) 0 0
\(7\) −6.92478e41 −1.01872 −0.509358 − 0.860555i \(-0.670117\pi\)
−0.509358 + 0.860555i \(0.670117\pi\)
\(8\) 3.39630e44 0.673057
\(9\) 0 0
\(10\) −1.37479e49 −0.434747
\(11\) −1.55674e51 −0.439823 −0.219912 − 0.975520i \(-0.570577\pi\)
−0.219912 + 0.975520i \(0.570577\pi\)
\(12\) 0 0
\(13\) −2.14548e55 −1.55356 −0.776781 − 0.629771i \(-0.783149\pi\)
−0.776781 + 0.629771i \(0.783149\pi\)
\(14\) 1.98371e56 0.366557
\(15\) 0 0
\(16\) 2.52429e59 0.628347
\(17\) 1.39488e61 1.72709 0.863545 − 0.504272i \(-0.168239\pi\)
0.863545 + 0.504272i \(0.168239\pi\)
\(18\) 0 0
\(19\) 1.51628e63 0.762917 0.381459 − 0.924386i \(-0.375422\pi\)
0.381459 + 0.924386i \(0.375422\pi\)
\(20\) −2.64799e64 −1.05180
\(21\) 0 0
\(22\) 4.45952e65 0.158258
\(23\) −2.55387e67 −1.00386 −0.501928 − 0.864910i \(-0.667376\pi\)
−0.501928 + 0.864910i \(0.667376\pi\)
\(24\) 0 0
\(25\) 7.25462e68 0.459816
\(26\) 6.14606e69 0.559006
\(27\) 0 0
\(28\) 3.82084e71 0.886821
\(29\) −1.39180e72 −0.568692 −0.284346 − 0.958722i \(-0.591776\pi\)
−0.284346 + 0.958722i \(0.591776\pi\)
\(30\) 0 0
\(31\) −1.44293e73 −0.217191 −0.108595 − 0.994086i \(-0.534635\pi\)
−0.108595 + 0.994086i \(0.534635\pi\)
\(32\) −2.87578e74 −0.899150
\(33\) 0 0
\(34\) −3.99585e75 −0.621445
\(35\) −3.32331e76 −1.23084
\(36\) 0 0
\(37\) −5.19313e77 −1.22869 −0.614347 − 0.789036i \(-0.710580\pi\)
−0.614347 + 0.789036i \(0.710580\pi\)
\(38\) −4.34363e77 −0.274514
\(39\) 0 0
\(40\) 1.62994e79 0.813207
\(41\) 3.58403e79 0.526709 0.263354 − 0.964699i \(-0.415171\pi\)
0.263354 + 0.964699i \(0.415171\pi\)
\(42\) 0 0
\(43\) 2.84099e80 0.395168 0.197584 − 0.980286i \(-0.436690\pi\)
0.197584 + 0.980286i \(0.436690\pi\)
\(44\) 8.58951e80 0.382878
\(45\) 0 0
\(46\) 7.31597e81 0.361209
\(47\) 7.73779e82 1.31757 0.658784 − 0.752332i \(-0.271071\pi\)
0.658784 + 0.752332i \(0.271071\pi\)
\(48\) 0 0
\(49\) 1.74580e82 0.0377824
\(50\) −2.07820e83 −0.165452
\(51\) 0 0
\(52\) 1.18380e85 1.35242
\(53\) −1.89817e84 −0.0844655 −0.0422328 − 0.999108i \(-0.513447\pi\)
−0.0422328 + 0.999108i \(0.513447\pi\)
\(54\) 0 0
\(55\) −7.47103e85 −0.531407
\(56\) −2.35187e86 −0.685654
\(57\) 0 0
\(58\) 3.98704e86 0.204628
\(59\) 2.13540e87 0.470224 0.235112 − 0.971968i \(-0.424454\pi\)
0.235112 + 0.971968i \(0.424454\pi\)
\(60\) 0 0
\(61\) 4.67707e88 1.97764 0.988819 − 0.149121i \(-0.0476444\pi\)
0.988819 + 0.149121i \(0.0476444\pi\)
\(62\) 4.13350e87 0.0781501
\(63\) 0 0
\(64\) −7.76144e88 −0.304813
\(65\) −1.02965e90 −1.87706
\(66\) 0 0
\(67\) 3.33536e90 1.35656 0.678282 − 0.734802i \(-0.262725\pi\)
0.678282 + 0.734802i \(0.262725\pi\)
\(68\) −7.69642e90 −1.50348
\(69\) 0 0
\(70\) 9.52013e90 0.442884
\(71\) −2.25284e91 −0.519328 −0.259664 − 0.965699i \(-0.583612\pi\)
−0.259664 + 0.965699i \(0.583612\pi\)
\(72\) 0 0
\(73\) 2.47879e90 0.0144464 0.00722319 − 0.999974i \(-0.497701\pi\)
0.00722319 + 0.999974i \(0.497701\pi\)
\(74\) 1.48765e92 0.442111
\(75\) 0 0
\(76\) −8.36629e92 −0.664141
\(77\) 1.07801e93 0.448055
\(78\) 0 0
\(79\) −1.29965e94 −1.51805 −0.759025 − 0.651062i \(-0.774324\pi\)
−0.759025 + 0.651062i \(0.774324\pi\)
\(80\) 1.21144e94 0.759187
\(81\) 0 0
\(82\) −1.02670e94 −0.189521
\(83\) 1.54205e95 1.56219 0.781094 − 0.624414i \(-0.214662\pi\)
0.781094 + 0.624414i \(0.214662\pi\)
\(84\) 0 0
\(85\) 6.69423e95 2.08672
\(86\) −8.13847e94 −0.142190
\(87\) 0 0
\(88\) −5.28716e95 −0.296026
\(89\) −2.29359e96 −0.734025 −0.367013 − 0.930216i \(-0.619619\pi\)
−0.367013 + 0.930216i \(0.619619\pi\)
\(90\) 0 0
\(91\) 1.48570e97 1.58264
\(92\) 1.40913e97 0.873884
\(93\) 0 0
\(94\) −2.21661e97 −0.474090
\(95\) 7.27688e97 0.921778
\(96\) 0 0
\(97\) −3.70442e98 −1.67309 −0.836545 − 0.547898i \(-0.815428\pi\)
−0.836545 + 0.547898i \(0.815428\pi\)
\(98\) −5.00113e96 −0.0135949
\(99\) 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.d.1.4 8
3.2 odd 2 1.100.a.a.1.5 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.100.a.a.1.5 ✓ 8 3.2 odd 2
9.100.a.d.1.4 8 1.1 even 1 trivial