Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,26,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, 26); 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, 26, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.6397101957\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1287001}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 321750 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(567.730\) 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)
 
\(f(q)\) \(=\) \(q-6644.76 q^{2} +1.05985e7 q^{4} -1.29474e8 q^{5} -4.99182e10 q^{7} +1.52537e11 q^{8} +8.60326e11 q^{10} +2.40245e12 q^{11} -1.10132e14 q^{13} +3.31695e14 q^{14} -1.36920e15 q^{16} -1.88823e15 q^{17} -5.16785e15 q^{19} -1.37223e15 q^{20} -1.59637e16 q^{22} -2.07157e17 q^{23} -2.81260e17 q^{25} +7.31801e17 q^{26} -5.29056e17 q^{28} +3.91886e17 q^{29} +7.70306e18 q^{31} +3.97970e18 q^{32} +1.25469e19 q^{34} +6.46313e18 q^{35} -2.99716e18 q^{37} +3.43391e19 q^{38} -1.97496e19 q^{40} +2.45123e20 q^{41} -2.88038e20 q^{43} +2.54623e19 q^{44} +1.37651e21 q^{46} +3.01578e20 q^{47} +1.15076e21 q^{49} +1.86890e21 q^{50} -1.16723e21 q^{52} +4.35340e21 q^{53} -3.11056e20 q^{55} -7.61438e21 q^{56} -2.60399e21 q^{58} -6.92162e21 q^{59} -2.89150e22 q^{61} -5.11850e22 q^{62} +1.94985e22 q^{64} +1.42593e22 q^{65} -1.41680e22 q^{67} -2.00123e22 q^{68} -4.29460e22 q^{70} +6.66992e22 q^{71} +1.33600e23 q^{73} +1.99154e22 q^{74} -5.47712e22 q^{76} -1.19926e23 q^{77} +8.16786e23 q^{79} +1.77276e23 q^{80} -1.62878e24 q^{82} +7.24814e23 q^{83} +2.44478e23 q^{85} +1.91394e24 q^{86} +3.66463e23 q^{88} -1.55091e23 q^{89} +5.49760e24 q^{91} -2.19554e24 q^{92} -2.00391e24 q^{94} +6.69104e23 q^{95} -2.30974e24 q^{97} -7.64654e24 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 324 q^{2} + 25607696 q^{4} - 570861756 q^{5} - 29687385728 q^{7} + 23299970112 q^{8} - 2215598822136 q^{10} + 18187785618552 q^{11} - 105143636679236 q^{13} + 472678969464576 q^{14} - 27\!\cdots\!40 q^{16}+ \cdots - 14\!\cdots\!76 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6644.76 −1.14711 −0.573554 − 0.819167i \(-0.694436\pi\)
−0.573554 + 0.819167i \(0.694436\pi\)
\(3\) 0 0
\(4\) 1.05985e7 0.315859
\(5\) −1.29474e8 −0.237169 −0.118585 − 0.992944i \(-0.537836\pi\)
−0.118585 + 0.992944i \(0.537836\pi\)
\(6\) 0 0
\(7\) −4.99182e10 −1.36312 −0.681559 − 0.731763i \(-0.738698\pi\)
−0.681559 + 0.731763i \(0.738698\pi\)
\(8\) 1.52537e11 0.784785
\(9\) 0 0
\(10\) 8.60326e11 0.272059
\(11\) 2.40245e12 0.230806 0.115403 − 0.993319i \(-0.463184\pi\)
0.115403 + 0.993319i \(0.463184\pi\)
\(12\) 0 0
\(13\) −1.10132e14 −1.31106 −0.655529 − 0.755170i \(-0.727554\pi\)
−0.655529 + 0.755170i \(0.727554\pi\)
\(14\) 3.31695e14 1.56365
\(15\) 0 0
\(16\) −1.36920e15 −1.21609
\(17\) −1.88823e15 −0.786039 −0.393019 − 0.919530i \(-0.628569\pi\)
−0.393019 + 0.919530i \(0.628569\pi\)
\(18\) 0 0
\(19\) −5.16785e15 −0.535661 −0.267830 − 0.963466i \(-0.586307\pi\)
−0.267830 + 0.963466i \(0.586307\pi\)
\(20\) −1.37223e15 −0.0749120
\(21\) 0 0
\(22\) −1.59637e16 −0.264759
\(23\) −2.07157e17 −1.97107 −0.985533 − 0.169484i \(-0.945790\pi\)
−0.985533 + 0.169484i \(0.945790\pi\)
\(24\) 0 0
\(25\) −2.81260e17 −0.943751
\(26\) 7.31801e17 1.50393
\(27\) 0 0
\(28\) −5.29056e17 −0.430553
\(29\) 3.91886e17 0.205677 0.102838 − 0.994698i \(-0.467208\pi\)
0.102838 + 0.994698i \(0.467208\pi\)
\(30\) 0 0
\(31\) 7.70306e18 1.75648 0.878238 − 0.478225i \(-0.158720\pi\)
0.878238 + 0.478225i \(0.158720\pi\)
\(32\) 3.97970e18 0.610205
\(33\) 0 0
\(34\) 1.25469e19 0.901672
\(35\) 6.46313e18 0.323290
\(36\) 0 0
\(37\) −2.99716e18 −0.0748495 −0.0374248 − 0.999299i \(-0.511915\pi\)
−0.0374248 + 0.999299i \(0.511915\pi\)
\(38\) 3.43391e19 0.614461
\(39\) 0 0
\(40\) −1.97496e19 −0.186127
\(41\) 2.45123e20 1.69662 0.848311 − 0.529498i \(-0.177620\pi\)
0.848311 + 0.529498i \(0.177620\pi\)
\(42\) 0 0
\(43\) −2.88038e20 −1.09924 −0.549621 − 0.835414i \(-0.685228\pi\)
−0.549621 + 0.835414i \(0.685228\pi\)
\(44\) 2.54623e19 0.0729020
\(45\) 0 0
\(46\) 1.37651e21 2.26103
\(47\) 3.01578e20 0.378596 0.189298 − 0.981920i \(-0.439379\pi\)
0.189298 + 0.981920i \(0.439379\pi\)
\(48\) 0 0
\(49\) 1.15076e21 0.858093
\(50\) 1.86890e21 1.08258
\(51\) 0 0
\(52\) −1.16723e21 −0.414109
\(53\) 4.35340e21 1.21725 0.608626 − 0.793457i \(-0.291721\pi\)
0.608626 + 0.793457i \(0.291721\pi\)
\(54\) 0 0
\(55\) −3.11056e20 −0.0547400
\(56\) −7.61438e21 −1.06975
\(57\) 0 0
\(58\) −2.60399e21 −0.235934
\(59\) −6.92162e21 −0.506475 −0.253237 − 0.967404i \(-0.581495\pi\)
−0.253237 + 0.967404i \(0.581495\pi\)
\(60\) 0 0
\(61\) −2.89150e22 −1.39476 −0.697381 − 0.716701i \(-0.745651\pi\)
−0.697381 + 0.716701i \(0.745651\pi\)
\(62\) −5.11850e22 −2.01487
\(63\) 0 0
\(64\) 1.94985e22 0.516120
\(65\) 1.42593e22 0.310943
\(66\) 0 0
\(67\) −1.41680e22 −0.211530 −0.105765 − 0.994391i \(-0.533729\pi\)
−0.105765 + 0.994391i \(0.533729\pi\)
\(68\) −2.00123e22 −0.248277
\(69\) 0 0
\(70\) −4.29460e22 −0.370849
\(71\) 6.66992e22 0.482382 0.241191 − 0.970478i \(-0.422462\pi\)
0.241191 + 0.970478i \(0.422462\pi\)
\(72\) 0 0
\(73\) 1.33600e23 0.682764 0.341382 − 0.939925i \(-0.389105\pi\)
0.341382 + 0.939925i \(0.389105\pi\)
\(74\) 1.99154e22 0.0858605
\(75\) 0 0
\(76\) −5.47712e22 −0.169193
\(77\) −1.19926e23 −0.314616
\(78\) 0 0
\(79\) 8.16786e23 1.55514 0.777570 − 0.628796i \(-0.216452\pi\)
0.777570 + 0.628796i \(0.216452\pi\)
\(80\) 1.77276e23 0.288420
\(81\) 0 0
\(82\) −1.62878e24 −1.94621
\(83\) 7.24814e23 0.744304 0.372152 − 0.928172i \(-0.378620\pi\)
0.372152 + 0.928172i \(0.378620\pi\)
\(84\) 0 0
\(85\) 2.44478e23 0.186424
\(86\) 1.91394e24 1.26095
\(87\) 0 0
\(88\) 3.66463e23 0.181133
\(89\) −1.55091e23 −0.0665596 −0.0332798 − 0.999446i \(-0.510595\pi\)
−0.0332798 + 0.999446i \(0.510595\pi\)
\(90\) 0 0
\(91\) 5.49760e24 1.78713
\(92\) −2.19554e24 −0.622578
\(93\) 0 0
\(94\) −2.00391e24 −0.434291
\(95\) 6.69104e23 0.127042
\(96\) 0 0
\(97\) −2.30974e24 −0.337999 −0.169000 − 0.985616i \(-0.554054\pi\)
−0.169000 + 0.985616i \(0.554054\pi\)
\(98\) −7.64654e24 −0.984326
\(99\) 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 9.26.a.b.1.1 2
3.2 odd 2 3.26.a.a.1.2 ✓ 2
12.11 even 2 48.26.a.f.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.26.a.a.1.2 ✓ 2 3.2 odd 2
9.26.a.b.1.1 2 1.1 even 1 trivial
48.26.a.f.1.1 2 12.11 even 2