Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [729,2,Mod(1,729)] 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("729.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(5.82109430735\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{36})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 6x^{4} + 9x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.28558\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.28558 q^{2} -0.347296 q^{4} +0.446476 q^{5} -3.53209 q^{7} -3.01763 q^{8} +0.573978 q^{10} -2.78006 q^{11} +3.29086 q^{13} -4.54077 q^{14} -3.18479 q^{16} -7.03936 q^{17} -5.18479 q^{19} -0.155059 q^{20} -3.57398 q^{22} +7.27693 q^{23} -4.80066 q^{25} +4.23065 q^{26} +1.22668 q^{28} +3.61916 q^{29} -1.93582 q^{31} +1.94096 q^{32} -9.04963 q^{34} -1.57699 q^{35} -3.22668 q^{37} -6.66544 q^{38} -1.34730 q^{40} -4.86084 q^{41} -5.75877 q^{43} +0.965505 q^{44} +9.35504 q^{46} +3.01763 q^{47} +5.47565 q^{49} -6.17161 q^{50} -1.14290 q^{52} +8.77141 q^{53} -1.24123 q^{55} +10.6585 q^{56} +4.65270 q^{58} +2.96377 q^{59} +7.88713 q^{61} -2.48865 q^{62} +8.86484 q^{64} +1.46929 q^{65} -9.43882 q^{67} +2.44474 q^{68} -2.02734 q^{70} +5.30731 q^{71} -1.55438 q^{73} -4.14814 q^{74} +1.80066 q^{76} +9.81942 q^{77} -11.9017 q^{79} -1.42193 q^{80} -6.24897 q^{82} -16.2573 q^{83} -3.14290 q^{85} -7.40333 q^{86} +8.38919 q^{88} +18.4258 q^{89} -11.6236 q^{91} -2.52725 q^{92} +3.87939 q^{94} -2.31488 q^{95} +10.1138 q^{97} +7.03936 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64}+ \cdots - 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.28558 0.909039 0.454519 − 0.890737i \(-0.349811\pi\)
0.454519 + 0.890737i \(0.349811\pi\)
\(3\) 0 0
\(4\) −0.347296 −0.173648
\(5\) 0.446476 0.199670 0.0998350 − 0.995004i \(-0.468169\pi\)
0.0998350 + 0.995004i \(0.468169\pi\)
\(6\) 0 0
\(7\) −3.53209 −1.33500 −0.667502 − 0.744608i \(-0.732636\pi\)
−0.667502 + 0.744608i \(0.732636\pi\)
\(8\) −3.01763 −1.06689
\(9\) 0 0
\(10\) 0.573978 0.181508
\(11\) −2.78006 −0.838220 −0.419110 − 0.907935i \(-0.637658\pi\)
−0.419110 + 0.907935i \(0.637658\pi\)
\(12\) 0 0
\(13\) 3.29086 0.912720 0.456360 − 0.889795i \(-0.349153\pi\)
0.456360 + 0.889795i \(0.349153\pi\)
\(14\) −4.54077 −1.21357
\(15\) 0 0
\(16\) −3.18479 −0.796198
\(17\) −7.03936 −1.70730 −0.853648 − 0.520850i \(-0.825615\pi\)
−0.853648 + 0.520850i \(0.825615\pi\)
\(18\) 0 0
\(19\) −5.18479 −1.18947 −0.594736 − 0.803921i \(-0.702744\pi\)
−0.594736 + 0.803921i \(0.702744\pi\)
\(20\) −0.155059 −0.0346723
\(21\) 0 0
\(22\) −3.57398 −0.761975
\(23\) 7.27693 1.51734 0.758672 − 0.651473i \(-0.225848\pi\)
0.758672 + 0.651473i \(0.225848\pi\)
\(24\) 0 0
\(25\) −4.80066 −0.960132
\(26\) 4.23065 0.829698
\(27\) 0 0
\(28\) 1.22668 0.231821
\(29\) 3.61916 0.672061 0.336031 − 0.941851i \(-0.390915\pi\)
0.336031 + 0.941851i \(0.390915\pi\)
\(30\) 0 0
\(31\) −1.93582 −0.347684 −0.173842 − 0.984774i \(-0.555618\pi\)
−0.173842 + 0.984774i \(0.555618\pi\)
\(32\) 1.94096 0.343117
\(33\) 0 0
\(34\) −9.04963 −1.55200
\(35\) −1.57699 −0.266560
\(36\) 0 0
\(37\) −3.22668 −0.530463 −0.265232 − 0.964185i \(-0.585448\pi\)
−0.265232 + 0.964185i \(0.585448\pi\)
\(38\) −6.66544 −1.08128
\(39\) 0 0
\(40\) −1.34730 −0.213026
\(41\) −4.86084 −0.759135 −0.379568 − 0.925164i \(-0.623927\pi\)
−0.379568 + 0.925164i \(0.623927\pi\)
\(42\) 0 0
\(43\) −5.75877 −0.878204 −0.439102 − 0.898437i \(-0.644703\pi\)
−0.439102 + 0.898437i \(0.644703\pi\)
\(44\) 0.965505 0.145555
\(45\) 0 0
\(46\) 9.35504 1.37932
\(47\) 3.01763 0.440166 0.220083 − 0.975481i \(-0.429367\pi\)
0.220083 + 0.975481i \(0.429367\pi\)
\(48\) 0 0
\(49\) 5.47565 0.782236
\(50\) −6.17161 −0.872797
\(51\) 0 0
\(52\) −1.14290 −0.158492
\(53\) 8.77141 1.20485 0.602423 − 0.798177i \(-0.294202\pi\)
0.602423 + 0.798177i \(0.294202\pi\)
\(54\) 0 0
\(55\) −1.24123 −0.167367
\(56\) 10.6585 1.42431
\(57\) 0 0
\(58\) 4.65270 0.610930
\(59\) 2.96377 0.385851 0.192925 − 0.981213i \(-0.438203\pi\)
0.192925 + 0.981213i \(0.438203\pi\)
\(60\) 0 0
\(61\) 7.88713 1.00984 0.504922 − 0.863165i \(-0.331521\pi\)
0.504922 + 0.863165i \(0.331521\pi\)
\(62\) −2.48865 −0.316058
\(63\) 0 0
\(64\) 8.86484 1.10810
\(65\) 1.46929 0.182243
\(66\) 0 0
\(67\) −9.43882 −1.15313 −0.576567 − 0.817050i \(-0.695608\pi\)
−0.576567 + 0.817050i \(0.695608\pi\)
\(68\) 2.44474 0.296469
\(69\) 0 0
\(70\) −2.02734 −0.242314
\(71\) 5.30731 0.629862 0.314931 − 0.949115i \(-0.398019\pi\)
0.314931 + 0.949115i \(0.398019\pi\)
\(72\) 0 0
\(73\) −1.55438 −0.181926 −0.0909631 − 0.995854i \(-0.528995\pi\)
−0.0909631 + 0.995854i \(0.528995\pi\)
\(74\) −4.14814 −0.482212
\(75\) 0 0
\(76\) 1.80066 0.206550
\(77\) 9.81942 1.11903
\(78\) 0 0
\(79\) −11.9017 −1.33904 −0.669521 − 0.742793i \(-0.733501\pi\)
−0.669521 + 0.742793i \(0.733501\pi\)
\(80\) −1.42193 −0.158977
\(81\) 0 0
\(82\) −6.24897 −0.690083
\(83\) −16.2573 −1.78447 −0.892233 − 0.451576i \(-0.850862\pi\)
−0.892233 + 0.451576i \(0.850862\pi\)
\(84\) 0 0
\(85\) −3.14290 −0.340896
\(86\) −7.40333 −0.798322
\(87\) 0 0
\(88\) 8.38919 0.894290
\(89\) 18.4258 1.95313 0.976567 − 0.215214i \(-0.0690450\pi\)
0.976567 + 0.215214i \(0.0690450\pi\)
\(90\) 0 0
\(91\) −11.6236 −1.21849
\(92\) −2.52725 −0.263484
\(93\) 0 0
\(94\) 3.87939 0.400128
\(95\) −2.31488 −0.237502
\(96\) 0 0
\(97\) 10.1138 1.02690 0.513451 − 0.858119i \(-0.328367\pi\)
0.513451 + 0.858119i \(0.328367\pi\)
\(98\) 7.03936 0.711083
\(99\) 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 729.2.a.c.1.5 yes 6
3.2 odd 2 inner 729.2.a.c.1.2 ✓ 6
9.2 odd 6 729.2.c.c.244.5 12
9.4 even 3 729.2.c.c.487.2 12
9.5 odd 6 729.2.c.c.487.5 12
9.7 even 3 729.2.c.c.244.2 12
27.2 odd 18 729.2.e.q.568.2 12
27.4 even 9 729.2.e.m.406.2 12
27.5 odd 18 729.2.e.r.649.1 12
27.7 even 9 729.2.e.m.325.2 12
27.11 odd 18 729.2.e.r.82.1 12
27.13 even 9 729.2.e.q.163.1 12
27.14 odd 18 729.2.e.q.163.2 12
27.16 even 9 729.2.e.r.82.2 12
27.20 odd 18 729.2.e.m.325.1 12
27.22 even 9 729.2.e.r.649.2 12
27.23 odd 18 729.2.e.m.406.1 12
27.25 even 9 729.2.e.q.568.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.c.1.2 ✓ 6 3.2 odd 2 inner
729.2.a.c.1.5 yes 6 1.1 even 1 trivial
729.2.c.c.244.2 12 9.7 even 3
729.2.c.c.244.5 12 9.2 odd 6
729.2.c.c.487.2 12 9.4 even 3
729.2.c.c.487.5 12 9.5 odd 6
729.2.e.m.325.1 12 27.20 odd 18
729.2.e.m.325.2 12 27.7 even 9
729.2.e.m.406.1 12 27.23 odd 18
729.2.e.m.406.2 12 27.4 even 9
729.2.e.q.163.1 12 27.13 even 9
729.2.e.q.163.2 12 27.14 odd 18
729.2.e.q.568.1 12 27.25 even 9
729.2.e.q.568.2 12 27.2 odd 18
729.2.e.r.82.1 12 27.11 odd 18
729.2.e.r.82.2 12 27.16 even 9
729.2.e.r.649.1 12 27.5 odd 18
729.2.e.r.649.2 12 27.22 even 9