↑

Properties

Label 9.16.a.a.1.1
Level $9$
Weight $16$
Character 9.1
Self dual yes
Analytic conductor $12.842$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,16,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, 16); 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, 16, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-216] 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: \(12.8424154590\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q-216.000 q^{2} +13888.0 q^{4} -52110.0 q^{5} +2.82246e6 q^{7} +4.07808e6 q^{8} +1.12558e7 q^{10} -2.05869e7 q^{11} -1.90073e8 q^{13} -6.09650e8 q^{14} -1.33595e9 q^{16} -1.64653e9 q^{17} +1.56326e9 q^{19} -7.23704e8 q^{20} +4.44676e9 q^{22} -9.45112e9 q^{23} -2.78021e10 q^{25} +4.10558e10 q^{26} +3.91983e10 q^{28} +3.69026e10 q^{29} +7.15885e10 q^{31} +1.54934e11 q^{32} +3.55650e11 q^{34} -1.47078e11 q^{35} -1.03365e12 q^{37} -3.37664e11 q^{38} -2.12509e11 q^{40} -1.64197e12 q^{41} -4.92403e11 q^{43} -2.85910e11 q^{44} +2.04144e12 q^{46} +3.41068e12 q^{47} +3.21870e12 q^{49} +6.00526e12 q^{50} -2.63974e12 q^{52} -6.79715e12 q^{53} +1.07278e12 q^{55} +1.15102e13 q^{56} -7.97095e12 q^{58} -9.85886e12 q^{59} +4.93184e12 q^{61} -1.54631e13 q^{62} +1.03106e13 q^{64} +9.90472e12 q^{65} -2.88378e13 q^{67} -2.28670e13 q^{68} +3.17689e13 q^{70} -1.25050e14 q^{71} -8.21715e13 q^{73} +2.23269e14 q^{74} +2.17105e13 q^{76} -5.81055e13 q^{77} -2.54131e13 q^{79} +6.96162e13 q^{80} +3.54666e14 q^{82} +2.81737e14 q^{83} +8.58006e13 q^{85} +1.06359e14 q^{86} -8.39548e13 q^{88} -7.15619e14 q^{89} -5.36474e14 q^{91} -1.31257e14 q^{92} -7.36708e14 q^{94} -8.14613e13 q^{95} +6.12786e14 q^{97} -6.95238e14 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −216.000 −1.19324 −0.596621 − 0.802523i \(-0.703491\pi\)
−0.596621 + 0.802523i \(0.703491\pi\)
\(3\) 0 0
\(4\) 13888.0 0.423828
\(5\) −52110.0 −0.298295 −0.149148 − 0.988815i \(-0.547653\pi\)
−0.149148 + 0.988815i \(0.547653\pi\)
\(6\) 0 0
\(7\) 2.82246e6 1.29536 0.647682 − 0.761911i \(-0.275739\pi\)
0.647682 + 0.761911i \(0.275739\pi\)
\(8\) 4.07808e6 0.687513
\(9\) 0 0
\(10\) 1.12558e7 0.355938
\(11\) −2.05869e7 −0.318526 −0.159263 − 0.987236i \(-0.550912\pi\)
−0.159263 + 0.987236i \(0.550912\pi\)
\(12\) 0 0
\(13\) −1.90073e8 −0.840129 −0.420065 − 0.907494i \(-0.637993\pi\)
−0.420065 + 0.907494i \(0.637993\pi\)
\(14\) −6.09650e8 −1.54568
\(15\) 0 0
\(16\) −1.33595e9 −1.24420
\(17\) −1.64653e9 −0.973200 −0.486600 − 0.873625i \(-0.661763\pi\)
−0.486600 + 0.873625i \(0.661763\pi\)
\(18\) 0 0
\(19\) 1.56326e9 0.401216 0.200608 − 0.979672i \(-0.435708\pi\)
0.200608 + 0.979672i \(0.435708\pi\)
\(20\) −7.23704e8 −0.126426
\(21\) 0 0
\(22\) 4.44676e9 0.380079
\(23\) −9.45112e9 −0.578794 −0.289397 − 0.957209i \(-0.593455\pi\)
−0.289397 + 0.957209i \(0.593455\pi\)
\(24\) 0 0
\(25\) −2.78021e10 −0.911020
\(26\) 4.10558e10 1.00248
\(27\) 0 0
\(28\) 3.91983e10 0.549012
\(29\) 3.69026e10 0.397257 0.198629 − 0.980075i \(-0.436351\pi\)
0.198629 + 0.980075i \(0.436351\pi\)
\(30\) 0 0
\(31\) 7.15885e10 0.467337 0.233669 − 0.972316i \(-0.424927\pi\)
0.233669 + 0.972316i \(0.424927\pi\)
\(32\) 1.54934e11 0.797117
\(33\) 0 0
\(34\) 3.55650e11 1.16126
\(35\) −1.47078e11 −0.386401
\(36\) 0 0
\(37\) −1.03365e12 −1.79003 −0.895017 − 0.446031i \(-0.852837\pi\)
−0.895017 + 0.446031i \(0.852837\pi\)
\(38\) −3.37664e11 −0.478748
\(39\) 0 0
\(40\) −2.12509e11 −0.205082
\(41\) −1.64197e12 −1.31670 −0.658351 − 0.752711i \(-0.728746\pi\)
−0.658351 + 0.752711i \(0.728746\pi\)
\(42\) 0 0
\(43\) −4.92403e11 −0.276253 −0.138127 − 0.990415i \(-0.544108\pi\)
−0.138127 + 0.990415i \(0.544108\pi\)
\(44\) −2.85910e11 −0.135000
\(45\) 0 0
\(46\) 2.04144e12 0.690642
\(47\) 3.41068e12 0.981991 0.490996 − 0.871162i \(-0.336633\pi\)
0.490996 + 0.871162i \(0.336633\pi\)
\(48\) 0 0
\(49\) 3.21870e12 0.677968
\(50\) 6.00526e12 1.08707
\(51\) 0 0
\(52\) −2.63974e12 −0.356070
\(53\) −6.79715e12 −0.794800 −0.397400 − 0.917645i \(-0.630087\pi\)
−0.397400 + 0.917645i \(0.630087\pi\)
\(54\) 0 0
\(55\) 1.07278e12 0.0950147
\(56\) 1.15102e13 0.890580
\(57\) 0 0
\(58\) −7.97095e12 −0.474024
\(59\) −9.85886e12 −0.515747 −0.257873 − 0.966179i \(-0.583022\pi\)
−0.257873 + 0.966179i \(0.583022\pi\)
\(60\) 0 0
\(61\) 4.93184e12 0.200926 0.100463 − 0.994941i \(-0.467968\pi\)
0.100463 + 0.994941i \(0.467968\pi\)
\(62\) −1.54631e13 −0.557647
\(63\) 0 0
\(64\) 1.03106e13 0.293044
\(65\) 9.90472e12 0.250606
\(66\) 0 0
\(67\) −2.88378e13 −0.581302 −0.290651 − 0.956829i \(-0.593872\pi\)
−0.290651 + 0.956829i \(0.593872\pi\)
\(68\) −2.28670e13 −0.412470
\(69\) 0 0
\(70\) 3.17689e13 0.461070
\(71\) −1.25050e14 −1.63172 −0.815862 − 0.578247i \(-0.803737\pi\)
−0.815862 + 0.578247i \(0.803737\pi\)
\(72\) 0 0
\(73\) −8.21715e13 −0.870562 −0.435281 − 0.900295i \(-0.643351\pi\)
−0.435281 + 0.900295i \(0.643351\pi\)
\(74\) 2.23269e14 2.13595
\(75\) 0 0
\(76\) 2.17105e13 0.170047
\(77\) −5.81055e13 −0.412607
\(78\) 0 0
\(79\) −2.54131e13 −0.148886 −0.0744430 − 0.997225i \(-0.523718\pi\)
−0.0744430 + 0.997225i \(0.523718\pi\)
\(80\) 6.96162e13 0.371138
\(81\) 0 0
\(82\) 3.54666e14 1.57114
\(83\) 2.81737e14 1.13961 0.569807 − 0.821779i \(-0.307018\pi\)
0.569807 + 0.821779i \(0.307018\pi\)
\(84\) 0 0
\(85\) 8.58006e13 0.290301
\(86\) 1.06359e14 0.329637
\(87\) 0 0
\(88\) −8.39548e13 −0.218991
\(89\) −7.15619e14 −1.71497 −0.857485 − 0.514509i \(-0.827974\pi\)
−0.857485 + 0.514509i \(0.827974\pi\)
\(90\) 0 0
\(91\) −5.36474e14 −1.08827
\(92\) −1.31257e14 −0.245309
\(93\) 0 0
\(94\) −7.36708e14 −1.17175
\(95\) −8.14613e13 −0.119681
\(96\) 0 0
\(97\) 6.12786e14 0.770054 0.385027 − 0.922905i \(-0.374192\pi\)
0.385027 + 0.922905i \(0.374192\pi\)
\(98\) −6.95238e14 −0.808981
\(99\) 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 9.16.a.a.1.1 1
3.2 odd 2 1.16.a.a.1.1 ✓ 1
4.3 odd 2 144.16.a.f.1.1 1
12.11 even 2 16.16.a.d.1.1 1
15.2 even 4 25.16.b.a.24.2 2
15.8 even 4 25.16.b.a.24.1 2
15.14 odd 2 25.16.a.a.1.1 1
21.2 odd 6 49.16.c.c.18.1 2
21.5 even 6 49.16.c.b.18.1 2
21.11 odd 6 49.16.c.c.30.1 2
21.17 even 6 49.16.c.b.30.1 2
21.20 even 2 49.16.a.a.1.1 1
24.5 odd 2 64.16.a.i.1.1 1
24.11 even 2 64.16.a.c.1.1 1
33.32 even 2 121.16.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.16.a.a.1.1 ✓ 1 3.2 odd 2
9.16.a.a.1.1 1 1.1 even 1 trivial
16.16.a.d.1.1 1 12.11 even 2
25.16.a.a.1.1 1 15.14 odd 2
25.16.b.a.24.1 2 15.8 even 4
25.16.b.a.24.2 2 15.2 even 4
49.16.a.a.1.1 1 21.20 even 2
49.16.c.b.18.1 2 21.5 even 6
49.16.c.b.30.1 2 21.17 even 6
49.16.c.c.18.1 2 21.2 odd 6
49.16.c.c.30.1 2 21.11 odd 6
64.16.a.c.1.1 1 24.11 even 2
64.16.a.i.1.1 1 24.5 odd 2
121.16.a.a.1.1 1 33.32 even 2
144.16.a.f.1.1 1 4.3 odd 2