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Properties

Label 9.18.a.a.1.1
Level $9$
Weight $18$
Character 9.1
Self dual yes
Analytic conductor $16.490$
Analytic rank $0$
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] = [9,18,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, 18); 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, 18, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-204] 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: \(16.4899878610\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
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)
 
\(f(q)\) \(=\) \(q-204.000 q^{2} -89456.0 q^{4} +163554. q^{5} -2.08466e7 q^{7} +4.49877e7 q^{8} -3.33650e7 q^{10} -8.17372e8 q^{11} +2.99590e8 q^{13} +4.25270e9 q^{14} +2.54768e9 q^{16} +4.47756e10 q^{17} +7.87487e10 q^{19} -1.46309e10 q^{20} +1.66744e11 q^{22} +7.04672e11 q^{23} -7.36190e11 q^{25} -6.11163e10 q^{26} +1.86485e12 q^{28} +1.63794e11 q^{29} +1.04986e12 q^{31} -6.41636e12 q^{32} -9.13422e12 q^{34} -3.40954e12 q^{35} -1.98057e13 q^{37} -1.60647e13 q^{38} +7.35792e12 q^{40} -1.46600e13 q^{41} +1.16039e14 q^{43} +7.31189e13 q^{44} -1.43753e14 q^{46} +1.76607e14 q^{47} +2.01949e14 q^{49} +1.50183e14 q^{50} -2.68001e13 q^{52} -1.52863e14 q^{53} -1.33685e14 q^{55} -9.37839e14 q^{56} -3.34139e13 q^{58} +2.62797e14 q^{59} -1.35855e15 q^{61} -2.14172e14 q^{62} +9.75007e14 q^{64} +4.89991e13 q^{65} +4.44864e14 q^{67} -4.00545e15 q^{68} +6.95546e14 q^{70} +4.00327e15 q^{71} +9.24833e14 q^{73} +4.04037e15 q^{74} -7.04454e15 q^{76} +1.70394e16 q^{77} +1.47473e16 q^{79} +4.16684e14 q^{80} +2.99065e15 q^{82} -2.64230e16 q^{83} +7.32323e15 q^{85} -2.36719e16 q^{86} -3.67717e16 q^{88} +3.88837e16 q^{89} -6.24542e15 q^{91} -6.30371e16 q^{92} -3.60277e16 q^{94} +1.28797e16 q^{95} -2.53744e16 q^{97} -4.11975e16 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −204.000 −0.563476 −0.281738 − 0.959491i \(-0.590911\pi\)
−0.281738 + 0.959491i \(0.590911\pi\)
\(3\) 0 0
\(4\) −89456.0 −0.682495
\(5\) 163554. 0.187248 0.0936238 − 0.995608i \(-0.470155\pi\)
0.0936238 + 0.995608i \(0.470155\pi\)
\(6\) 0 0
\(7\) −2.08466e7 −1.36679 −0.683394 − 0.730050i \(-0.739497\pi\)
−0.683394 + 0.730050i \(0.739497\pi\)
\(8\) 4.49877e7 0.948045
\(9\) 0 0
\(10\) −3.33650e7 −0.105509
\(11\) −8.17372e8 −1.14969 −0.574847 − 0.818261i \(-0.694938\pi\)
−0.574847 + 0.818261i \(0.694938\pi\)
\(12\) 0 0
\(13\) 2.99590e8 0.101861 0.0509306 − 0.998702i \(-0.483781\pi\)
0.0509306 + 0.998702i \(0.483781\pi\)
\(14\) 4.25270e9 0.770152
\(15\) 0 0
\(16\) 2.54768e9 0.148295
\(17\) 4.47756e10 1.55677 0.778387 − 0.627785i \(-0.216038\pi\)
0.778387 + 0.627785i \(0.216038\pi\)
\(18\) 0 0
\(19\) 7.87487e10 1.06374 0.531872 − 0.846824i \(-0.321489\pi\)
0.531872 + 0.846824i \(0.321489\pi\)
\(20\) −1.46309e10 −0.127796
\(21\) 0 0
\(22\) 1.66744e11 0.647824
\(23\) 7.04672e11 1.87629 0.938146 − 0.346240i \(-0.112542\pi\)
0.938146 + 0.346240i \(0.112542\pi\)
\(24\) 0 0
\(25\) −7.36190e11 −0.964938
\(26\) −6.11163e10 −0.0573963
\(27\) 0 0
\(28\) 1.86485e12 0.932826
\(29\) 1.63794e11 0.0608015 0.0304008 − 0.999538i \(-0.490322\pi\)
0.0304008 + 0.999538i \(0.490322\pi\)
\(30\) 0 0
\(31\) 1.04986e12 0.221084 0.110542 − 0.993871i \(-0.464741\pi\)
0.110542 + 0.993871i \(0.464741\pi\)
\(32\) −6.41636e12 −1.03161
\(33\) 0 0
\(34\) −9.13422e12 −0.877204
\(35\) −3.40954e12 −0.255928
\(36\) 0 0
\(37\) −1.98057e13 −0.926993 −0.463496 − 0.886099i \(-0.653405\pi\)
−0.463496 + 0.886099i \(0.653405\pi\)
\(38\) −1.60647e13 −0.599394
\(39\) 0 0
\(40\) 7.35792e12 0.177519
\(41\) −1.46600e13 −0.286729 −0.143365 − 0.989670i \(-0.545792\pi\)
−0.143365 + 0.989670i \(0.545792\pi\)
\(42\) 0 0
\(43\) 1.16039e14 1.51399 0.756993 − 0.653424i \(-0.226668\pi\)
0.756993 + 0.653424i \(0.226668\pi\)
\(44\) 7.31189e13 0.784660
\(45\) 0 0
\(46\) −1.43753e14 −1.05724
\(47\) 1.76607e14 1.08187 0.540935 − 0.841064i \(-0.318070\pi\)
0.540935 + 0.841064i \(0.318070\pi\)
\(48\) 0 0
\(49\) 2.01949e14 0.868109
\(50\) 1.50183e14 0.543719
\(51\) 0 0
\(52\) −2.68001e13 −0.0695197
\(53\) −1.52863e14 −0.337255 −0.168628 − 0.985680i \(-0.553934\pi\)
−0.168628 + 0.985680i \(0.553934\pi\)
\(54\) 0 0
\(55\) −1.33685e14 −0.215277
\(56\) −9.37839e14 −1.29578
\(57\) 0 0
\(58\) −3.34139e13 −0.0342602
\(59\) 2.62797e14 0.233012 0.116506 − 0.993190i \(-0.462831\pi\)
0.116506 + 0.993190i \(0.462831\pi\)
\(60\) 0 0
\(61\) −1.35855e15 −0.907346 −0.453673 − 0.891168i \(-0.649887\pi\)
−0.453673 + 0.891168i \(0.649887\pi\)
\(62\) −2.14172e14 −0.124575
\(63\) 0 0
\(64\) 9.75007e14 0.432990
\(65\) 4.89991e13 0.0190732
\(66\) 0 0
\(67\) 4.44864e14 0.133842 0.0669208 − 0.997758i \(-0.478683\pi\)
0.0669208 + 0.997758i \(0.478683\pi\)
\(68\) −4.00545e15 −1.06249
\(69\) 0 0
\(70\) 6.95546e14 0.144209
\(71\) 4.00327e15 0.735731 0.367865 − 0.929879i \(-0.380089\pi\)
0.367865 + 0.929879i \(0.380089\pi\)
\(72\) 0 0
\(73\) 9.24833e14 0.134220 0.0671102 − 0.997746i \(-0.478622\pi\)
0.0671102 + 0.997746i \(0.478622\pi\)
\(74\) 4.04037e15 0.522338
\(75\) 0 0
\(76\) −7.04454e15 −0.726001
\(77\) 1.70394e16 1.57139
\(78\) 0 0
\(79\) 1.47473e16 1.09366 0.546830 − 0.837244i \(-0.315834\pi\)
0.546830 + 0.837244i \(0.315834\pi\)
\(80\) 4.16684e14 0.0277678
\(81\) 0 0
\(82\) 2.99065e15 0.161565
\(83\) −2.64230e16 −1.28771 −0.643855 − 0.765148i \(-0.722666\pi\)
−0.643855 + 0.765148i \(0.722666\pi\)
\(84\) 0 0
\(85\) 7.32323e15 0.291502
\(86\) −2.36719e16 −0.853094
\(87\) 0 0
\(88\) −3.67717e16 −1.08996
\(89\) 3.88837e16 1.04702 0.523508 − 0.852021i \(-0.324623\pi\)
0.523508 + 0.852021i \(0.324623\pi\)
\(90\) 0 0
\(91\) −6.24542e15 −0.139223
\(92\) −6.30371e16 −1.28056
\(93\) 0 0
\(94\) −3.60277e16 −0.609608
\(95\) 1.28797e16 0.199184
\(96\) 0 0
\(97\) −2.53744e16 −0.328727 −0.164364 − 0.986400i \(-0.552557\pi\)
−0.164364 + 0.986400i \(0.552557\pi\)
\(98\) −4.11975e16 −0.489158
\(99\) 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 9.18.a.a.1.1 1
3.2 odd 2 3.18.a.a.1.1 ✓ 1
12.11 even 2 48.18.a.e.1.1 1
15.2 even 4 75.18.b.a.49.2 2
15.8 even 4 75.18.b.a.49.1 2
15.14 odd 2 75.18.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.18.a.a.1.1 ✓ 1 3.2 odd 2
9.18.a.a.1.1 1 1.1 even 1 trivial
48.18.a.e.1.1 1 12.11 even 2
75.18.a.a.1.1 1 15.14 odd 2
75.18.b.a.49.1 2 15.8 even 4
75.18.b.a.49.2 2 15.2 even 4