Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,8,Mod(1,27)] 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("27.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 27.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,9,0,77] 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: \(8.43439568807\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.53113\) of defining polynomial
Character \(\chi\) \(=\) 27.1

$q$-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)
 
\(f(q)\) \(=\) \(q-7.59339 q^{2} -70.3405 q^{4} +65.8132 q^{5} -738.405 q^{7} +1506.08 q^{8} -499.745 q^{10} +4961.26 q^{11} +5967.24 q^{13} +5606.99 q^{14} -2432.64 q^{16} +36651.6 q^{17} +22378.9 q^{19} -4629.33 q^{20} -37672.8 q^{22} -51473.6 q^{23} -73793.6 q^{25} -45311.5 q^{26} +51939.7 q^{28} -68495.7 q^{29} +150655. q^{31} -174306. q^{32} -278310. q^{34} -48596.8 q^{35} +489027. q^{37} -169932. q^{38} +99119.7 q^{40} +590635. q^{41} -842643. q^{43} -348978. q^{44} +390859. q^{46} +1.22637e6 q^{47} -278301. q^{49} +560343. q^{50} -419738. q^{52} +958904. q^{53} +326517. q^{55} -1.11209e6 q^{56} +520114. q^{58} -316269. q^{59} -29722.9 q^{61} -1.14398e6 q^{62} +1.63495e6 q^{64} +392723. q^{65} +293025. q^{67} -2.57809e6 q^{68} +369014. q^{70} +714537. q^{71} -3.96273e6 q^{73} -3.71337e6 q^{74} -1.57415e6 q^{76} -3.66342e6 q^{77} +2.53805e6 q^{79} -160100. q^{80} -4.48492e6 q^{82} +1.66311e6 q^{83} +2.41216e6 q^{85} +6.39851e6 q^{86} +7.47204e6 q^{88} +4.64819e6 q^{89} -4.40624e6 q^{91} +3.62068e6 q^{92} -9.31228e6 q^{94} +1.47283e6 q^{95} +1.47010e7 q^{97} +2.11325e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{2} + 77 q^{4} + 180 q^{5} + 700 q^{7} + 1827 q^{8} + 1395 q^{10} + 10890 q^{11} - 5480 q^{13} + 29475 q^{14} - 15967 q^{16} + 16416 q^{17} + 16024 q^{19} + 12195 q^{20} + 60705 q^{22} + 24372 q^{23}+ \cdots + 22779738 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.59339 −0.671167 −0.335583 − 0.942010i \(-0.608933\pi\)
−0.335583 + 0.942010i \(0.608933\pi\)
\(3\) 0 0
\(4\) −70.3405 −0.549535
\(5\) 65.8132 0.235461 0.117730 − 0.993046i \(-0.462438\pi\)
0.117730 + 0.993046i \(0.462438\pi\)
\(6\) 0 0
\(7\) −738.405 −0.813676 −0.406838 − 0.913500i \(-0.633369\pi\)
−0.406838 + 0.913500i \(0.633369\pi\)
\(8\) 1506.08 1.04000
\(9\) 0 0
\(10\) −499.745 −0.158033
\(11\) 4961.26 1.12387 0.561937 − 0.827180i \(-0.310056\pi\)
0.561937 + 0.827180i \(0.310056\pi\)
\(12\) 0 0
\(13\) 5967.24 0.753306 0.376653 − 0.926354i \(-0.377075\pi\)
0.376653 + 0.926354i \(0.377075\pi\)
\(14\) 5606.99 0.546112
\(15\) 0 0
\(16\) −2432.64 −0.148476
\(17\) 36651.6 1.80935 0.904674 − 0.426104i \(-0.140114\pi\)
0.904674 + 0.426104i \(0.140114\pi\)
\(18\) 0 0
\(19\) 22378.9 0.748518 0.374259 − 0.927324i \(-0.377897\pi\)
0.374259 + 0.927324i \(0.377897\pi\)
\(20\) −4629.33 −0.129394
\(21\) 0 0
\(22\) −37672.8 −0.754308
\(23\) −51473.6 −0.882139 −0.441069 − 0.897473i \(-0.645401\pi\)
−0.441069 + 0.897473i \(0.645401\pi\)
\(24\) 0 0
\(25\) −73793.6 −0.944558
\(26\) −45311.5 −0.505594
\(27\) 0 0
\(28\) 51939.7 0.447143
\(29\) −68495.7 −0.521519 −0.260760 − 0.965404i \(-0.583973\pi\)
−0.260760 + 0.965404i \(0.583973\pi\)
\(30\) 0 0
\(31\) 150655. 0.908275 0.454137 − 0.890932i \(-0.349948\pi\)
0.454137 + 0.890932i \(0.349948\pi\)
\(32\) −174306. −0.940344
\(33\) 0 0
\(34\) −278310. −1.21437
\(35\) −48596.8 −0.191589
\(36\) 0 0
\(37\) 489027. 1.58718 0.793591 − 0.608451i \(-0.208209\pi\)
0.793591 + 0.608451i \(0.208209\pi\)
\(38\) −169932. −0.502380
\(39\) 0 0
\(40\) 99119.7 0.244878
\(41\) 590635. 1.33837 0.669184 − 0.743096i \(-0.266644\pi\)
0.669184 + 0.743096i \(0.266644\pi\)
\(42\) 0 0
\(43\) −842643. −1.61623 −0.808117 − 0.589023i \(-0.799513\pi\)
−0.808117 + 0.589023i \(0.799513\pi\)
\(44\) −348978. −0.617609
\(45\) 0 0
\(46\) 390859. 0.592062
\(47\) 1.22637e6 1.72297 0.861486 − 0.507782i \(-0.169534\pi\)
0.861486 + 0.507782i \(0.169534\pi\)
\(48\) 0 0
\(49\) −278301. −0.337932
\(50\) 560343. 0.633956
\(51\) 0 0
\(52\) −419738. −0.413968
\(53\) 958904. 0.884727 0.442364 − 0.896836i \(-0.354140\pi\)
0.442364 + 0.896836i \(0.354140\pi\)
\(54\) 0 0
\(55\) 326517. 0.264628
\(56\) −1.11209e6 −0.846220
\(57\) 0 0
\(58\) 520114. 0.350027
\(59\) −316269. −0.200482 −0.100241 − 0.994963i \(-0.531961\pi\)
−0.100241 + 0.994963i \(0.531961\pi\)
\(60\) 0 0
\(61\) −29722.9 −0.0167663 −0.00838315 − 0.999965i \(-0.502668\pi\)
−0.00838315 + 0.999965i \(0.502668\pi\)
\(62\) −1.14398e6 −0.609604
\(63\) 0 0
\(64\) 1.63495e6 0.779604
\(65\) 392723. 0.177374
\(66\) 0 0
\(67\) 293025. 0.119026 0.0595130 − 0.998228i \(-0.481045\pi\)
0.0595130 + 0.998228i \(0.481045\pi\)
\(68\) −2.57809e6 −0.994300
\(69\) 0 0
\(70\) 369014. 0.128588
\(71\) 714537. 0.236930 0.118465 − 0.992958i \(-0.462203\pi\)
0.118465 + 0.992958i \(0.462203\pi\)
\(72\) 0 0
\(73\) −3.96273e6 −1.19224 −0.596121 − 0.802894i \(-0.703292\pi\)
−0.596121 + 0.802894i \(0.703292\pi\)
\(74\) −3.71337e6 −1.06526
\(75\) 0 0
\(76\) −1.57415e6 −0.411337
\(77\) −3.66342e6 −0.914470
\(78\) 0 0
\(79\) 2.53805e6 0.579168 0.289584 − 0.957153i \(-0.406483\pi\)
0.289584 + 0.957153i \(0.406483\pi\)
\(80\) −160100. −0.0349603
\(81\) 0 0
\(82\) −4.48492e6 −0.898269
\(83\) 1.66311e6 0.319263 0.159631 − 0.987177i \(-0.448969\pi\)
0.159631 + 0.987177i \(0.448969\pi\)
\(84\) 0 0
\(85\) 2.41216e6 0.426030
\(86\) 6.39851e6 1.08476
\(87\) 0 0
\(88\) 7.47204e6 1.16883
\(89\) 4.64819e6 0.698906 0.349453 − 0.936954i \(-0.386368\pi\)
0.349453 + 0.936954i \(0.386368\pi\)
\(90\) 0 0
\(91\) −4.40624e6 −0.612947
\(92\) 3.62068e6 0.484766
\(93\) 0 0
\(94\) −9.31228e6 −1.15640
\(95\) 1.47283e6 0.176246
\(96\) 0 0
\(97\) 1.47010e7 1.63548 0.817738 − 0.575590i \(-0.195228\pi\)
0.817738 + 0.575590i \(0.195228\pi\)
\(98\) 2.11325e6 0.226809
\(99\) 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 27.8.a.e.1.1 yes 2
3.2 odd 2 27.8.a.b.1.2 ✓ 2
4.3 odd 2 432.8.a.q.1.1 2
9.2 odd 6 81.8.c.h.28.1 4
9.4 even 3 81.8.c.d.55.2 4
9.5 odd 6 81.8.c.h.55.1 4
9.7 even 3 81.8.c.d.28.2 4
12.11 even 2 432.8.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.8.a.b.1.2 ✓ 2 3.2 odd 2
27.8.a.e.1.1 yes 2 1.1 even 1 trivial
81.8.c.d.28.2 4 9.7 even 3
81.8.c.d.55.2 4 9.4 even 3
81.8.c.h.28.1 4 9.2 odd 6
81.8.c.h.55.1 4 9.5 odd 6
432.8.a.j.1.2 2 12.11 even 2
432.8.a.q.1.1 2 4.3 odd 2