Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,22,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, 22); 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, 22, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2844] 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: \(25.1529609858\)
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)
 
\(f(q)\) \(=\) \(q+2844.00 q^{2} +5.99118e6 q^{4} -3.10995e6 q^{5} +3.63304e8 q^{7} +1.10746e10 q^{8} -8.84470e9 q^{10} -1.45818e10 q^{11} +1.13351e11 q^{13} +1.03324e12 q^{14} +1.89318e13 q^{16} +8.58939e12 q^{17} -2.92029e13 q^{19} -1.86323e13 q^{20} -4.14707e13 q^{22} +1.55899e14 q^{23} -4.67165e14 q^{25} +3.22370e14 q^{26} +2.17662e15 q^{28} -2.40079e15 q^{29} +2.23982e15 q^{31} +3.06169e16 q^{32} +2.44282e16 q^{34} -1.12986e15 q^{35} -3.07851e16 q^{37} -8.30532e16 q^{38} -3.44415e16 q^{40} +1.03208e17 q^{41} -1.65557e17 q^{43} -8.73624e16 q^{44} +4.43377e17 q^{46} +6.65872e16 q^{47} -4.26556e17 q^{49} -1.32862e18 q^{50} +6.79105e17 q^{52} -4.35423e17 q^{53} +4.53488e16 q^{55} +4.02346e18 q^{56} -6.82784e18 q^{58} -5.53437e18 q^{59} -7.17621e18 q^{61} +6.37005e18 q^{62} +4.73716e19 q^{64} -3.52515e17 q^{65} -1.57554e19 q^{67} +5.14606e19 q^{68} -3.21331e18 q^{70} -2.64579e19 q^{71} +1.34712e19 q^{73} -8.75527e19 q^{74} -1.74960e20 q^{76} -5.29764e18 q^{77} -1.68861e19 q^{79} -5.88770e19 q^{80} +2.93522e20 q^{82} +1.70688e20 q^{83} -2.67126e19 q^{85} -4.70845e20 q^{86} -1.61488e20 q^{88} +3.12592e20 q^{89} +4.11808e19 q^{91} +9.34021e20 q^{92} +1.89374e20 q^{94} +9.08197e19 q^{95} +9.49015e20 q^{97} -1.21313e21 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2844.00 1.96388 0.981939 − 0.189196i \(-0.0605882\pi\)
0.981939 + 0.189196i \(0.0605882\pi\)
\(3\) 0 0
\(4\) 5.99118e6 2.85682
\(5\) −3.10995e6 −0.142419 −0.0712096 − 0.997461i \(-0.522686\pi\)
−0.0712096 + 0.997461i \(0.522686\pi\)
\(6\) 0 0
\(7\) 3.63304e8 0.486117 0.243058 − 0.970012i \(-0.421849\pi\)
0.243058 + 0.970012i \(0.421849\pi\)
\(8\) 1.10746e10 3.64657
\(9\) 0 0
\(10\) −8.84470e9 −0.279694
\(11\) −1.45818e10 −0.169508 −0.0847538 − 0.996402i \(-0.527010\pi\)
−0.0847538 + 0.996402i \(0.527010\pi\)
\(12\) 0 0
\(13\) 1.13351e11 0.228044 0.114022 − 0.993478i \(-0.463627\pi\)
0.114022 + 0.993478i \(0.463627\pi\)
\(14\) 1.03324e12 0.954674
\(15\) 0 0
\(16\) 1.89318e13 4.30460
\(17\) 8.58939e12 1.03335 0.516676 − 0.856181i \(-0.327169\pi\)
0.516676 + 0.856181i \(0.327169\pi\)
\(18\) 0 0
\(19\) −2.92029e13 −1.09273 −0.546366 − 0.837546i \(-0.683989\pi\)
−0.546366 + 0.837546i \(0.683989\pi\)
\(20\) −1.86323e13 −0.406866
\(21\) 0 0
\(22\) −4.14707e13 −0.332892
\(23\) 1.55899e14 0.784696 0.392348 − 0.919817i \(-0.371663\pi\)
0.392348 + 0.919817i \(0.371663\pi\)
\(24\) 0 0
\(25\) −4.67165e14 −0.979717
\(26\) 3.22370e14 0.447851
\(27\) 0 0
\(28\) 2.17662e15 1.38875
\(29\) −2.40079e15 −1.05968 −0.529840 − 0.848097i \(-0.677748\pi\)
−0.529840 + 0.848097i \(0.677748\pi\)
\(30\) 0 0
\(31\) 2.23982e15 0.490812 0.245406 − 0.969420i \(-0.421079\pi\)
0.245406 + 0.969420i \(0.421079\pi\)
\(32\) 3.06169e16 4.80714
\(33\) 0 0
\(34\) 2.44282e16 2.02938
\(35\) −1.12986e15 −0.0692323
\(36\) 0 0
\(37\) −3.07851e16 −1.05250 −0.526250 − 0.850330i \(-0.676402\pi\)
−0.526250 + 0.850330i \(0.676402\pi\)
\(38\) −8.30532e16 −2.14599
\(39\) 0 0
\(40\) −3.44415e16 −0.519341
\(41\) 1.03208e17 1.20083 0.600414 − 0.799689i \(-0.295002\pi\)
0.600414 + 0.799689i \(0.295002\pi\)
\(42\) 0 0
\(43\) −1.65557e17 −1.16823 −0.584117 − 0.811670i \(-0.698559\pi\)
−0.584117 + 0.811670i \(0.698559\pi\)
\(44\) −8.73624e16 −0.484252
\(45\) 0 0
\(46\) 4.43377e17 1.54105
\(47\) 6.65872e16 0.184656 0.0923280 − 0.995729i \(-0.470569\pi\)
0.0923280 + 0.995729i \(0.470569\pi\)
\(48\) 0 0
\(49\) −4.26556e17 −0.763690
\(50\) −1.32862e18 −1.92404
\(51\) 0 0
\(52\) 6.79105e17 0.651482
\(53\) −4.35423e17 −0.341991 −0.170995 − 0.985272i \(-0.554698\pi\)
−0.170995 + 0.985272i \(0.554698\pi\)
\(54\) 0 0
\(55\) 4.53488e16 0.0241411
\(56\) 4.02346e18 1.77266
\(57\) 0 0
\(58\) −6.82784e18 −2.08108
\(59\) −5.53437e18 −1.40968 −0.704842 − 0.709364i \(-0.748982\pi\)
−0.704842 + 0.709364i \(0.748982\pi\)
\(60\) 0 0
\(61\) −7.17621e18 −1.28805 −0.644023 − 0.765006i \(-0.722736\pi\)
−0.644023 + 0.765006i \(0.722736\pi\)
\(62\) 6.37005e18 0.963896
\(63\) 0 0
\(64\) 4.73716e19 5.13604
\(65\) −3.52515e17 −0.0324779
\(66\) 0 0
\(67\) −1.57554e19 −1.05595 −0.527977 − 0.849258i \(-0.677049\pi\)
−0.527977 + 0.849258i \(0.677049\pi\)
\(68\) 5.14606e19 2.95210
\(69\) 0 0
\(70\) −3.21331e18 −0.135964
\(71\) −2.64579e19 −0.964588 −0.482294 − 0.876009i \(-0.660196\pi\)
−0.482294 + 0.876009i \(0.660196\pi\)
\(72\) 0 0
\(73\) 1.34712e19 0.366875 0.183437 − 0.983031i \(-0.441278\pi\)
0.183437 + 0.983031i \(0.441278\pi\)
\(74\) −8.75527e19 −2.06698
\(75\) 0 0
\(76\) −1.74960e20 −3.12174
\(77\) −5.29764e18 −0.0824005
\(78\) 0 0
\(79\) −1.68861e19 −0.200653 −0.100326 − 0.994955i \(-0.531989\pi\)
−0.100326 + 0.994955i \(0.531989\pi\)
\(80\) −5.88770e19 −0.613057
\(81\) 0 0
\(82\) 2.93522e20 2.35828
\(83\) 1.70688e20 1.20749 0.603744 − 0.797178i \(-0.293675\pi\)
0.603744 + 0.797178i \(0.293675\pi\)
\(84\) 0 0
\(85\) −2.67126e19 −0.147169
\(86\) −4.70845e20 −2.29427
\(87\) 0 0
\(88\) −1.61488e20 −0.618121
\(89\) 3.12592e20 1.06263 0.531317 − 0.847173i \(-0.321697\pi\)
0.531317 + 0.847173i \(0.321697\pi\)
\(90\) 0 0
\(91\) 4.11808e19 0.110856
\(92\) 9.34021e20 2.24174
\(93\) 0 0
\(94\) 1.89374e20 0.362642
\(95\) 9.08197e19 0.155626
\(96\) 0 0
\(97\) 9.49015e20 1.30668 0.653341 − 0.757064i \(-0.273367\pi\)
0.653341 + 0.757064i \(0.273367\pi\)
\(98\) −1.21313e21 −1.49980
\(99\) 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 9.22.a.d.1.1 1
3.2 odd 2 3.22.a.a.1.1 ✓ 1
12.11 even 2 48.22.a.e.1.1 1
15.2 even 4 75.22.b.a.49.1 2
15.8 even 4 75.22.b.a.49.2 2
15.14 odd 2 75.22.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.a.1.1 ✓ 1 3.2 odd 2
9.22.a.d.1.1 1 1.1 even 1 trivial
48.22.a.e.1.1 1 12.11 even 2
75.22.a.c.1.1 1 15.14 odd 2
75.22.b.a.49.1 2 15.2 even 4
75.22.b.a.49.2 2 15.8 even 4