Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [432,8,Mod(1,432)] 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("432.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 432.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-180,0,-700] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(134.950331009\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 27)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.53113\) of defining polynomial
Character \(\chi\) \(=\) 432.1

$q$-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)
 
\(f(q)\) \(=\) \(q-65.8132 q^{5} +738.405 q^{7} +4961.26 q^{11} +5967.24 q^{13} -36651.6 q^{17} -22378.9 q^{19} -51473.6 q^{23} -73793.6 q^{25} +68495.7 q^{29} -150655. q^{31} -48596.8 q^{35} +489027. q^{37} -590635. q^{41} +842643. q^{43} +1.22637e6 q^{47} -278301. q^{49} -958904. q^{53} -326517. q^{55} -316269. q^{59} -29722.9 q^{61} -392723. q^{65} -293025. q^{67} +714537. q^{71} -3.96273e6 q^{73} +3.66342e6 q^{77} -2.53805e6 q^{79} +1.66311e6 q^{83} +2.41216e6 q^{85} -4.64819e6 q^{89} +4.40624e6 q^{91} +1.47283e6 q^{95} +1.47010e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 180 q^{5} - 700 q^{7} + 10890 q^{11} - 5480 q^{13} - 16416 q^{17} - 16024 q^{19} + 24372 q^{23} - 138880 q^{25} + 143280 q^{29} + 38708 q^{31} + 115650 q^{35} + 455620 q^{37} - 731880 q^{41} + 1088840 q^{43}+ \cdots + 4098670 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −65.8132 −0.235461 −0.117730 − 0.993046i \(-0.537562\pi\)
−0.117730 + 0.993046i \(0.537562\pi\)
\(6\) 0 0
\(7\) 738.405 0.813676 0.406838 − 0.913500i \(-0.366631\pi\)
0.406838 + 0.913500i \(0.366631\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 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 \(0.377075\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −36651.6 −1.80935 −0.904674 − 0.426104i \(-0.859886\pi\)
−0.904674 + 0.426104i \(0.859886\pi\)
\(18\) 0 0
\(19\) −22378.9 −0.748518 −0.374259 − 0.927324i \(-0.622103\pi\)
−0.374259 + 0.927324i \(0.622103\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −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
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 68495.7 0.521519 0.260760 − 0.965404i \(-0.416027\pi\)
0.260760 + 0.965404i \(0.416027\pi\)
\(30\) 0 0
\(31\) −150655. −0.908275 −0.454137 − 0.890932i \(-0.650052\pi\)
−0.454137 + 0.890932i \(0.650052\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −48596.8 −0.191589
\(36\) 0 0
\(37\) 489027. 1.58718 0.793591 − 0.608451i \(-0.208209\pi\)
0.793591 + 0.608451i \(0.208209\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −590635. −1.33837 −0.669184 − 0.743096i \(-0.733356\pi\)
−0.669184 + 0.743096i \(0.733356\pi\)
\(42\) 0 0
\(43\) 842643. 1.61623 0.808117 − 0.589023i \(-0.200487\pi\)
0.808117 + 0.589023i \(0.200487\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 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
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −958904. −0.884727 −0.442364 − 0.896836i \(-0.645860\pi\)
−0.442364 + 0.896836i \(0.645860\pi\)
\(54\) 0 0
\(55\) −326517. −0.264628
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −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 \(0.502668\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −392723. −0.177374
\(66\) 0 0
\(67\) −293025. −0.119026 −0.0595130 − 0.998228i \(-0.518955\pi\)
−0.0595130 + 0.998228i \(0.518955\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 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 \(0.703292\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.66342e6 0.914470
\(78\) 0 0
\(79\) −2.53805e6 −0.579168 −0.289584 − 0.957153i \(-0.593517\pi\)
−0.289584 + 0.957153i \(0.593517\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 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
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.64819e6 −0.698906 −0.349453 − 0.936954i \(-0.613632\pi\)
−0.349453 + 0.936954i \(0.613632\pi\)
\(90\) 0 0
\(91\) 4.40624e6 0.612947
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.47283e6 0.176246
\(96\) 0 0
\(97\) 1.47010e7 1.63548 0.817738 − 0.575590i \(-0.195228\pi\)
0.817738 + 0.575590i \(0.195228\pi\)
\(98\) 0 0
\(99\) 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 432.8.a.j.1.2 2
3.2 odd 2 432.8.a.q.1.1 2
4.3 odd 2 27.8.a.b.1.2 ✓ 2
12.11 even 2 27.8.a.e.1.1 yes 2
36.7 odd 6 81.8.c.h.28.1 4
36.11 even 6 81.8.c.d.28.2 4
36.23 even 6 81.8.c.d.55.2 4
36.31 odd 6 81.8.c.h.55.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.8.a.b.1.2 ✓ 2 4.3 odd 2
27.8.a.e.1.1 yes 2 12.11 even 2
81.8.c.d.28.2 4 36.11 even 6
81.8.c.d.55.2 4 36.23 even 6
81.8.c.h.28.1 4 36.7 odd 6
81.8.c.h.55.1 4 36.31 odd 6
432.8.a.j.1.2 2 1.1 even 1 trivial
432.8.a.q.1.1 2 3.2 odd 2