Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,26,Mod(1,18)] 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("18.1"); S:= CuspForms(chi, 26); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 26, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,8192,0,33554432,697960704] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(71.2794203914\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 8439938686 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(91869.1\) of defining polynomial
Character \(\chi\) \(=\) 18.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4096.00 q^{2} +1.67772e7 q^{4} +5.87105e8 q^{5} -6.29827e10 q^{7} +6.87195e10 q^{8} +2.40478e12 q^{10} -1.65508e13 q^{11} +6.25240e13 q^{13} -2.57977e14 q^{14} +2.81475e14 q^{16} +4.12137e15 q^{17} +1.78860e16 q^{19} +9.84999e15 q^{20} -6.77922e16 q^{22} -5.26580e16 q^{23} +4.66692e16 q^{25} +2.56098e17 q^{26} -1.05667e18 q^{28} +4.67370e15 q^{29} +5.16669e18 q^{31} +1.15292e18 q^{32} +1.68811e19 q^{34} -3.69774e19 q^{35} -1.72094e19 q^{37} +7.32610e19 q^{38} +4.03456e19 q^{40} +1.59590e20 q^{41} +3.44114e20 q^{43} -2.77677e20 q^{44} -2.15687e20 q^{46} -4.59775e20 q^{47} +2.62575e21 q^{49} +1.91157e20 q^{50} +1.04898e21 q^{52} +4.24862e21 q^{53} -9.71708e21 q^{55} -4.32814e21 q^{56} +1.91435e19 q^{58} +6.73928e19 q^{59} -8.44753e21 q^{61} +2.11628e22 q^{62} +4.72237e21 q^{64} +3.67082e22 q^{65} +5.11670e21 q^{67} +6.91451e22 q^{68} -1.51460e23 q^{70} -4.17949e22 q^{71} -1.19268e22 q^{73} -7.04896e22 q^{74} +3.00077e23 q^{76} +1.04242e24 q^{77} -1.70505e23 q^{79} +1.65255e23 q^{80} +6.53679e23 q^{82} +1.85865e24 q^{83} +2.41968e24 q^{85} +1.40949e24 q^{86} -1.13736e24 q^{88} +3.40122e23 q^{89} -3.93793e24 q^{91} -8.83455e23 q^{92} -1.88324e24 q^{94} +1.05010e25 q^{95} +4.10439e24 q^{97} +1.07551e25 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8192 q^{2} + 33554432 q^{4} + 697960704 q^{5} - 19285401800 q^{7} + 137438953472 q^{8} + 2858847043584 q^{10} - 5751609553920 q^{11} + 70123116734020 q^{13} - 78993005772800 q^{14} + 562949953421312 q^{16}+ \cdots + 13\!\cdots\!24 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4096.00 0.707107
\(3\) 0 0
\(4\) 1.67772e7 0.500000
\(5\) 5.87105e8 1.07545 0.537726 − 0.843120i \(-0.319284\pi\)
0.537726 + 0.843120i \(0.319284\pi\)
\(6\) 0 0
\(7\) −6.29827e10 −1.71987 −0.859935 − 0.510404i \(-0.829496\pi\)
−0.859935 + 0.510404i \(0.829496\pi\)
\(8\) 6.87195e10 0.353553
\(9\) 0 0
\(10\) 2.40478e12 0.760459
\(11\) −1.65508e13 −1.59005 −0.795026 − 0.606575i \(-0.792543\pi\)
−0.795026 + 0.606575i \(0.792543\pi\)
\(12\) 0 0
\(13\) 6.25240e13 0.744312 0.372156 − 0.928170i \(-0.378619\pi\)
0.372156 + 0.928170i \(0.378619\pi\)
\(14\) −2.57977e14 −1.21613
\(15\) 0 0
\(16\) 2.81475e14 0.250000
\(17\) 4.12137e15 1.71565 0.857827 − 0.513938i \(-0.171814\pi\)
0.857827 + 0.513938i \(0.171814\pi\)
\(18\) 0 0
\(19\) 1.78860e16 1.85393 0.926965 − 0.375149i \(-0.122408\pi\)
0.926965 + 0.375149i \(0.122408\pi\)
\(20\) 9.84999e15 0.537726
\(21\) 0 0
\(22\) −6.77922e16 −1.12434
\(23\) −5.26580e16 −0.501033 −0.250517 − 0.968112i \(-0.580600\pi\)
−0.250517 + 0.968112i \(0.580600\pi\)
\(24\) 0 0
\(25\) 4.66692e16 0.156596
\(26\) 2.56098e17 0.526308
\(27\) 0 0
\(28\) −1.05667e18 −0.859935
\(29\) 4.67370e15 0.00245293 0.00122647 − 0.999999i \(-0.499610\pi\)
0.00122647 + 0.999999i \(0.499610\pi\)
\(30\) 0 0
\(31\) 5.16669e18 1.17812 0.589062 − 0.808087i \(-0.299497\pi\)
0.589062 + 0.808087i \(0.299497\pi\)
\(32\) 1.15292e18 0.176777
\(33\) 0 0
\(34\) 1.68811e19 1.21315
\(35\) −3.69774e19 −1.84964
\(36\) 0 0
\(37\) −1.72094e19 −0.429777 −0.214889 − 0.976639i \(-0.568939\pi\)
−0.214889 + 0.976639i \(0.568939\pi\)
\(38\) 7.32610e19 1.31093
\(39\) 0 0
\(40\) 4.03456e19 0.380230
\(41\) 1.59590e20 1.10460 0.552301 − 0.833644i \(-0.313750\pi\)
0.552301 + 0.833644i \(0.313750\pi\)
\(42\) 0 0
\(43\) 3.44114e20 1.31325 0.656625 − 0.754217i \(-0.271984\pi\)
0.656625 + 0.754217i \(0.271984\pi\)
\(44\) −2.77677e20 −0.795026
\(45\) 0 0
\(46\) −2.15687e20 −0.354284
\(47\) −4.59775e20 −0.577194 −0.288597 − 0.957451i \(-0.593189\pi\)
−0.288597 + 0.957451i \(0.593189\pi\)
\(48\) 0 0
\(49\) 2.62575e21 1.95795
\(50\) 1.91157e20 0.110730
\(51\) 0 0
\(52\) 1.04898e21 0.372156
\(53\) 4.24862e21 1.18795 0.593976 − 0.804482i \(-0.297557\pi\)
0.593976 + 0.804482i \(0.297557\pi\)
\(54\) 0 0
\(55\) −9.71708e21 −1.71002
\(56\) −4.32814e21 −0.608066
\(57\) 0 0
\(58\) 1.91435e19 0.00173448
\(59\) 6.73928e19 0.00493133 0.00246566 − 0.999997i \(-0.499215\pi\)
0.00246566 + 0.999997i \(0.499215\pi\)
\(60\) 0 0
\(61\) −8.44753e21 −0.407480 −0.203740 − 0.979025i \(-0.565310\pi\)
−0.203740 + 0.979025i \(0.565310\pi\)
\(62\) 2.11628e22 0.833060
\(63\) 0 0
\(64\) 4.72237e21 0.125000
\(65\) 3.67082e22 0.800471
\(66\) 0 0
\(67\) 5.11670e21 0.0763932 0.0381966 − 0.999270i \(-0.487839\pi\)
0.0381966 + 0.999270i \(0.487839\pi\)
\(68\) 6.91451e22 0.857827
\(69\) 0 0
\(70\) −1.51460e23 −1.30789
\(71\) −4.17949e22 −0.302270 −0.151135 − 0.988513i \(-0.548293\pi\)
−0.151135 + 0.988513i \(0.548293\pi\)
\(72\) 0 0
\(73\) −1.19268e22 −0.0609518 −0.0304759 − 0.999536i \(-0.509702\pi\)
−0.0304759 + 0.999536i \(0.509702\pi\)
\(74\) −7.04896e22 −0.303898
\(75\) 0 0
\(76\) 3.00077e23 0.926965
\(77\) 1.04242e24 2.73468
\(78\) 0 0
\(79\) −1.70505e23 −0.324637 −0.162319 − 0.986738i \(-0.551897\pi\)
−0.162319 + 0.986738i \(0.551897\pi\)
\(80\) 1.65255e23 0.268863
\(81\) 0 0
\(82\) 6.53679e23 0.781072
\(83\) 1.85865e24 1.90863 0.954316 − 0.298798i \(-0.0965857\pi\)
0.954316 + 0.298798i \(0.0965857\pi\)
\(84\) 0 0
\(85\) 2.41968e24 1.84510
\(86\) 1.40949e24 0.928608
\(87\) 0 0
\(88\) −1.13736e24 −0.562168
\(89\) 3.40122e23 0.145969 0.0729843 − 0.997333i \(-0.476748\pi\)
0.0729843 + 0.997333i \(0.476748\pi\)
\(90\) 0 0
\(91\) −3.93793e24 −1.28012
\(92\) −8.83455e23 −0.250517
\(93\) 0 0
\(94\) −1.88324e24 −0.408138
\(95\) 1.05010e25 1.99381
\(96\) 0 0
\(97\) 4.10439e24 0.600623 0.300312 − 0.953841i \(-0.402909\pi\)
0.300312 + 0.953841i \(0.402909\pi\)
\(98\) 1.07551e25 1.38448
\(99\) 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 18.26.a.g.1.2 yes 2
3.2 odd 2 18.26.a.f.1.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.26.a.f.1.1 ✓ 2 3.2 odd 2
18.26.a.g.1.2 yes 2 1.1 even 1 trivial