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Properties

Label 9.100.a.d.1.6
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 1.6
Root \(-9.79932e12\) 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)
 
\(f(q)\) \(=\) \(q+7.31556e14 q^{2} -9.86512e28 q^{4} -5.46163e34 q^{5} -9.42405e41 q^{7} -5.35848e44 q^{8} -3.99548e49 q^{10} -6.80844e51 q^{11} +1.27244e55 q^{13} -6.89422e56 q^{14} -3.29475e59 q^{16} -1.40992e60 q^{17} -1.83028e63 q^{19} +5.38796e63 q^{20} -4.98075e66 q^{22} -7.80909e66 q^{23} +1.40521e69 q^{25} +9.30860e69 q^{26} +9.29694e70 q^{28} +4.49461e72 q^{29} -6.78211e73 q^{31} +9.86044e73 q^{32} -1.03144e75 q^{34} +5.14706e76 q^{35} +1.72780e77 q^{37} -1.33895e78 q^{38} +2.92660e79 q^{40} +3.13802e78 q^{41} +1.67273e80 q^{43} +6.71660e80 q^{44} -5.71278e81 q^{46} -5.53499e82 q^{47} +4.26060e83 q^{49} +1.02799e84 q^{50} -1.25527e84 q^{52} +9.83283e84 q^{53} +3.71851e86 q^{55} +5.04986e86 q^{56} +3.28806e87 q^{58} +2.85418e87 q^{59} -7.36889e87 q^{61} -4.96150e88 q^{62} +2.80964e89 q^{64} -6.94958e89 q^{65} +2.22058e90 q^{67} +1.39091e89 q^{68} +3.76537e91 q^{70} -7.03212e91 q^{71} +1.65280e92 q^{73} +1.26398e92 q^{74} +1.80560e92 q^{76} +6.41631e93 q^{77} -2.18798e93 q^{79} +1.79947e94 q^{80} +2.29564e93 q^{82} +6.95611e94 q^{83} +7.70048e94 q^{85} +1.22369e95 q^{86} +3.64828e96 q^{88} -8.71787e95 q^{89} -1.19915e97 q^{91} +7.70375e95 q^{92} -4.04915e97 q^{94} +9.99632e97 q^{95} +1.85375e98 q^{97} +3.11686e98 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 208040616902520 q^{2} + 28\!\cdots\!24 q^{4} + 48\!\cdots\!60 q^{5} - 56\!\cdots\!00 q^{7} - 59\!\cdots\!60 q^{8} - 20\!\cdots\!60 q^{10} - 66\!\cdots\!36 q^{11} - 53\!\cdots\!40 q^{13} + 27\!\cdots\!92 q^{14}+ \cdots + 12\!\cdots\!60 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.31556e14 0.918888 0.459444 − 0.888207i \(-0.348049\pi\)
0.459444 + 0.888207i \(0.348049\pi\)
\(3\) 0 0
\(4\) −9.86512e28 −0.155644
\(5\) −5.46163e34 −1.37501 −0.687506 − 0.726178i \(-0.741295\pi\)
−0.687506 + 0.726178i \(0.741295\pi\)
\(6\) 0 0
\(7\) −9.42405e41 −1.38639 −0.693194 − 0.720751i \(-0.743797\pi\)
−0.693194 + 0.720751i \(0.743797\pi\)
\(8\) −5.35848e44 −1.06191
\(9\) 0 0
\(10\) −3.99548e49 −1.26348
\(11\) −6.80844e51 −1.92358 −0.961788 − 0.273794i \(-0.911721\pi\)
−0.961788 + 0.273794i \(0.911721\pi\)
\(12\) 0 0
\(13\) 1.27244e55 0.921384 0.460692 − 0.887560i \(-0.347601\pi\)
0.460692 + 0.887560i \(0.347601\pi\)
\(14\) −6.89422e56 −1.27394
\(15\) 0 0
\(16\) −3.29475e59 −0.820131
\(17\) −1.40992e60 −0.174572 −0.0872860 − 0.996183i \(-0.527819\pi\)
−0.0872860 + 0.996183i \(0.527819\pi\)
\(18\) 0 0
\(19\) −1.83028e63 −0.920906 −0.460453 − 0.887684i \(-0.652313\pi\)
−0.460453 + 0.887684i \(0.652313\pi\)
\(20\) 5.38796e63 0.214013
\(21\) 0 0
\(22\) −4.98075e66 −1.76755
\(23\) −7.80909e66 −0.306953 −0.153477 − 0.988152i \(-0.549047\pi\)
−0.153477 + 0.988152i \(0.549047\pi\)
\(24\) 0 0
\(25\) 1.40521e69 0.890660
\(26\) 9.30860e69 0.846649
\(27\) 0 0
\(28\) 9.29694e70 0.215783
\(29\) 4.49461e72 1.83650 0.918250 − 0.396001i \(-0.129602\pi\)
0.918250 + 0.396001i \(0.129602\pi\)
\(30\) 0 0
\(31\) −6.78211e73 −1.02085 −0.510424 − 0.859923i \(-0.670512\pi\)
−0.510424 + 0.859923i \(0.670512\pi\)
\(32\) 9.86044e73 0.308299
\(33\) 0 0
\(34\) −1.03144e75 −0.160412
\(35\) 5.14706e76 1.90630
\(36\) 0 0
\(37\) 1.72780e77 0.408797 0.204399 − 0.978888i \(-0.434476\pi\)
0.204399 + 0.978888i \(0.434476\pi\)
\(38\) −1.33895e78 −0.846209
\(39\) 0 0
\(40\) 2.92660e79 1.46014
\(41\) 3.13802e78 0.0461164 0.0230582 − 0.999734i \(-0.492660\pi\)
0.0230582 + 0.999734i \(0.492660\pi\)
\(42\) 0 0
\(43\) 1.67273e80 0.232668 0.116334 − 0.993210i \(-0.462886\pi\)
0.116334 + 0.993210i \(0.462886\pi\)
\(44\) 6.71660e80 0.299393
\(45\) 0 0
\(46\) −5.71278e81 −0.282056
\(47\) −5.53499e82 −0.942481 −0.471241 − 0.882005i \(-0.656194\pi\)
−0.471241 + 0.882005i \(0.656194\pi\)
\(48\) 0 0
\(49\) 4.26060e83 0.922071
\(50\) 1.02799e84 0.818417
\(51\) 0 0
\(52\) −1.25527e84 −0.143408
\(53\) 9.83283e84 0.437545 0.218773 − 0.975776i \(-0.429795\pi\)
0.218773 + 0.975776i \(0.429795\pi\)
\(54\) 0 0
\(55\) 3.71851e86 2.64494
\(56\) 5.04986e86 1.47222
\(57\) 0 0
\(58\) 3.28806e87 1.68754
\(59\) 2.85418e87 0.628503 0.314251 − 0.949340i \(-0.398247\pi\)
0.314251 + 0.949340i \(0.398247\pi\)
\(60\) 0 0
\(61\) −7.36889e87 −0.311584 −0.155792 − 0.987790i \(-0.549793\pi\)
−0.155792 + 0.987790i \(0.549793\pi\)
\(62\) −4.96150e88 −0.938046
\(63\) 0 0
\(64\) 2.80964e89 1.10342
\(65\) −6.94958e89 −1.26691
\(66\) 0 0
\(67\) 2.22058e90 0.903159 0.451580 − 0.892231i \(-0.350861\pi\)
0.451580 + 0.892231i \(0.350861\pi\)
\(68\) 1.39091e89 0.0271711
\(69\) 0 0
\(70\) 3.76537e91 1.75168
\(71\) −7.03212e91 −1.62106 −0.810528 − 0.585699i \(-0.800820\pi\)
−0.810528 + 0.585699i \(0.800820\pi\)
\(72\) 0 0
\(73\) 1.65280e92 0.963250 0.481625 − 0.876377i \(-0.340047\pi\)
0.481625 + 0.876377i \(0.340047\pi\)
\(74\) 1.26398e92 0.375639
\(75\) 0 0
\(76\) 1.80560e92 0.143334
\(77\) 6.41631e93 2.66682
\(78\) 0 0
\(79\) −2.18798e93 −0.255566 −0.127783 − 0.991802i \(-0.540786\pi\)
−0.127783 + 0.991802i \(0.540786\pi\)
\(80\) 1.79947e94 1.12769
\(81\) 0 0
\(82\) 2.29564e93 0.0423758
\(83\) 6.95611e94 0.704695 0.352347 − 0.935869i \(-0.385384\pi\)
0.352347 + 0.935869i \(0.385384\pi\)
\(84\) 0 0
\(85\) 7.70048e94 0.240039
\(86\) 1.22369e95 0.213796
\(87\) 0 0
\(88\) 3.64828e96 2.04266
\(89\) −8.71787e95 −0.279000 −0.139500 − 0.990222i \(-0.544550\pi\)
−0.139500 + 0.990222i \(0.544550\pi\)
\(90\) 0 0
\(91\) −1.19915e97 −1.27740
\(92\) 7.70375e95 0.0477755
\(93\) 0 0
\(94\) −4.04915e97 −0.866035
\(95\) 9.99632e97 1.26626
\(96\) 0 0
\(97\) 1.85375e98 0.837239 0.418620 − 0.908162i \(-0.362514\pi\)
0.418620 + 0.908162i \(0.362514\pi\)
\(98\) 3.11686e98 0.847280
\(99\) 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.1.6 8
3.2 odd 2 1.100.a.a.1.3 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.100.a.a.1.3 ✓ 8 3.2 odd 2
9.100.a.d.1.6 8 1.1 even 1 trivial