Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,3,Mod(26,81)] 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("81.26"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 81.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] 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: no
Analytic conductor: \(2.20709014132\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 53.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 81.53
Dual form 81.3.d.b.26.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.59808 - 1.50000i) q^{2} +(2.50000 - 4.33013i) q^{4} +(2.59808 + 1.50000i) q^{5} +(-2.50000 - 4.33013i) q^{7} -3.00000i q^{8} +9.00000 q^{10} +(-12.9904 + 7.50000i) q^{11} +(5.00000 - 8.66025i) q^{13} +(-12.9904 - 7.50000i) q^{14} +(5.50000 + 9.52628i) q^{16} +18.0000i q^{17} -16.0000 q^{19} +(12.9904 - 7.50000i) q^{20} +(-22.5000 + 38.9711i) q^{22} +(10.3923 + 6.00000i) q^{23} +(-8.00000 - 13.8564i) q^{25} -30.0000i q^{26} -25.0000 q^{28} +(25.9808 - 15.0000i) q^{29} +(0.500000 - 0.866025i) q^{31} +(38.9711 + 22.5000i) q^{32} +(27.0000 + 46.7654i) q^{34} -15.0000i q^{35} +20.0000 q^{37} +(-41.5692 + 24.0000i) q^{38} +(4.50000 - 7.79423i) q^{40} +(-51.9615 - 30.0000i) q^{41} +(-25.0000 - 43.3013i) q^{43} +75.0000i q^{44} +36.0000 q^{46} +(-5.19615 + 3.00000i) q^{47} +(12.0000 - 20.7846i) q^{49} +(-41.5692 - 24.0000i) q^{50} +(-25.0000 - 43.3013i) q^{52} -27.0000i q^{53} -45.0000 q^{55} +(-12.9904 + 7.50000i) q^{56} +(45.0000 - 77.9423i) q^{58} +(25.9808 + 15.0000i) q^{59} +(38.0000 + 65.8179i) q^{61} -3.00000i q^{62} +91.0000 q^{64} +(25.9808 - 15.0000i) q^{65} +(5.00000 - 8.66025i) q^{67} +(77.9423 + 45.0000i) q^{68} +(-22.5000 - 38.9711i) q^{70} -90.0000i q^{71} +65.0000 q^{73} +(51.9615 - 30.0000i) q^{74} +(-40.0000 + 69.2820i) q^{76} +(64.9519 + 37.5000i) q^{77} +(-7.00000 - 12.1244i) q^{79} +33.0000i q^{80} -180.000 q^{82} +(2.59808 - 1.50000i) q^{83} +(-27.0000 + 46.7654i) q^{85} +(-129.904 - 75.0000i) q^{86} +(22.5000 + 38.9711i) q^{88} +90.0000i q^{89} -50.0000 q^{91} +(51.9615 - 30.0000i) q^{92} +(-9.00000 + 15.5885i) q^{94} +(-41.5692 - 24.0000i) q^{95} +(42.5000 + 73.6122i) q^{97} -72.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{4} - 10 q^{7} + 36 q^{10} + 20 q^{13} + 22 q^{16} - 64 q^{19} - 90 q^{22} - 32 q^{25} - 100 q^{28} + 2 q^{31} + 108 q^{34} + 80 q^{37} + 18 q^{40} - 100 q^{43} + 144 q^{46} + 48 q^{49} - 100 q^{52}+ \cdots + 170 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.59808 − 1.50000i 1.29904 − 0.750000i 0.318800 − 0.947822i \(-0.396720\pi\)
0.980238 + 0.197822i \(0.0633868\pi\)
\(3\) 0 0
\(4\) 2.50000 − 4.33013i 0.625000 − 1.08253i
\(5\) 2.59808 + 1.50000i 0.519615 + 0.300000i 0.736777 − 0.676136i \(-0.236347\pi\)
−0.217162 + 0.976136i \(0.569680\pi\)
\(6\) 0 0
\(7\) −2.50000 − 4.33013i −0.357143 − 0.618590i 0.630339 − 0.776320i \(-0.282916\pi\)
−0.987482 + 0.157730i \(0.949582\pi\)
\(8\) − 3.00000i − 0.375000i
\(9\) 0 0
\(10\) 9.00000 0.900000
\(11\) −12.9904 + 7.50000i −1.18094 + 0.681818i −0.956233 − 0.292607i \(-0.905477\pi\)
−0.224711 + 0.974425i \(0.572144\pi\)
\(12\) 0 0
\(13\) 5.00000 − 8.66025i 0.384615 − 0.666173i −0.607100 − 0.794625i \(-0.707667\pi\)
0.991716 + 0.128452i \(0.0410008\pi\)
\(14\) −12.9904 − 7.50000i −0.927884 − 0.535714i
\(15\) 0 0
\(16\) 5.50000 + 9.52628i 0.343750 + 0.595392i
\(17\) 18.0000i 1.05882i 0.848365 + 0.529412i \(0.177587\pi\)
−0.848365 + 0.529412i \(0.822413\pi\)
\(18\) 0 0
\(19\) −16.0000 −0.842105 −0.421053 − 0.907036i \(-0.638339\pi\)
−0.421053 + 0.907036i \(0.638339\pi\)
\(20\) 12.9904 − 7.50000i 0.649519 − 0.375000i
\(21\) 0 0
\(22\) −22.5000 + 38.9711i −1.02273 + 1.77142i
\(23\) 10.3923 + 6.00000i 0.451839 + 0.260870i 0.708607 − 0.705604i \(-0.249324\pi\)
−0.256767 + 0.966473i \(0.582657\pi\)
\(24\) 0 0
\(25\) −8.00000 − 13.8564i −0.320000 − 0.554256i
\(26\) − 30.0000i − 1.15385i
\(27\) 0 0
\(28\) −25.0000 −0.892857
\(29\) 25.9808 − 15.0000i 0.895888 − 0.517241i 0.0200244 − 0.999799i \(-0.493626\pi\)
0.875864 + 0.482558i \(0.160292\pi\)
\(30\) 0 0
\(31\) 0.500000 − 0.866025i 0.0161290 − 0.0279363i −0.857848 − 0.513903i \(-0.828199\pi\)
0.873977 + 0.485967i \(0.161532\pi\)
\(32\) 38.9711 + 22.5000i 1.21785 + 0.703125i
\(33\) 0 0
\(34\) 27.0000 + 46.7654i 0.794118 + 1.37545i
\(35\) − 15.0000i − 0.428571i
\(36\) 0 0
\(37\) 20.0000 0.540541 0.270270 − 0.962784i \(-0.412887\pi\)
0.270270 + 0.962784i \(0.412887\pi\)
\(38\) −41.5692 + 24.0000i −1.09393 + 0.631579i
\(39\) 0 0
\(40\) 4.50000 − 7.79423i 0.112500 − 0.194856i
\(41\) −51.9615 − 30.0000i −1.26735 − 0.731707i −0.292868 − 0.956153i \(-0.594610\pi\)
−0.974487 + 0.224446i \(0.927943\pi\)
\(42\) 0 0
\(43\) −25.0000 − 43.3013i −0.581395 − 1.00701i −0.995314 − 0.0966925i \(-0.969174\pi\)
0.413919 − 0.910314i \(-0.364160\pi\)
\(44\) 75.0000i 1.70455i
\(45\) 0 0
\(46\) 36.0000 0.782609
\(47\) −5.19615 + 3.00000i −0.110556 + 0.0638298i −0.554259 − 0.832345i \(-0.686998\pi\)
0.443702 + 0.896174i \(0.353665\pi\)
\(48\) 0 0
\(49\) 12.0000 − 20.7846i 0.244898 − 0.424176i
\(50\) −41.5692 − 24.0000i −0.831384 − 0.480000i
\(51\) 0 0
\(52\) −25.0000 − 43.3013i −0.480769 − 0.832717i
\(53\) − 27.0000i − 0.509434i −0.967016 − 0.254717i \(-0.918018\pi\)
0.967016 − 0.254717i \(-0.0819823\pi\)
\(54\) 0 0
\(55\) −45.0000 −0.818182
\(56\) −12.9904 + 7.50000i −0.231971 + 0.133929i
\(57\) 0 0
\(58\) 45.0000 − 77.9423i 0.775862 − 1.34383i
\(59\) 25.9808 + 15.0000i 0.440352 + 0.254237i 0.703747 − 0.710451i \(-0.251509\pi\)
−0.263395 + 0.964688i \(0.584842\pi\)
\(60\) 0 0
\(61\) 38.0000 + 65.8179i 0.622951 + 1.07898i 0.988933 + 0.148361i \(0.0473997\pi\)
−0.365982 + 0.930622i \(0.619267\pi\)
\(62\) − 3.00000i − 0.0483871i
\(63\) 0 0
\(64\) 91.0000 1.42188
\(65\) 25.9808 − 15.0000i 0.399704 − 0.230769i
\(66\) 0 0
\(67\) 5.00000 − 8.66025i 0.0746269 − 0.129258i −0.826297 − 0.563235i \(-0.809557\pi\)
0.900924 + 0.433977i \(0.142890\pi\)
\(68\) 77.9423 + 45.0000i 1.14621 + 0.661765i
\(69\) 0 0
\(70\) −22.5000 − 38.9711i −0.321429 − 0.556731i
\(71\) − 90.0000i − 1.26761i −0.773495 − 0.633803i \(-0.781493\pi\)
0.773495 − 0.633803i \(-0.218507\pi\)
\(72\) 0 0
\(73\) 65.0000 0.890411 0.445205 − 0.895428i \(-0.353131\pi\)
0.445205 + 0.895428i \(0.353131\pi\)
\(74\) 51.9615 − 30.0000i 0.702183 − 0.405405i
\(75\) 0 0
\(76\) −40.0000 + 69.2820i −0.526316 + 0.911606i
\(77\) 64.9519 + 37.5000i 0.843531 + 0.487013i
\(78\) 0 0
\(79\) −7.00000 − 12.1244i −0.0886076 − 0.153473i 0.818315 − 0.574770i \(-0.194908\pi\)
−0.906923 + 0.421297i \(0.861575\pi\)
\(80\) 33.0000i 0.412500i
\(81\) 0 0
\(82\) −180.000 −2.19512
\(83\) 2.59808 − 1.50000i 0.0313021 − 0.0180723i −0.484267 − 0.874920i \(-0.660914\pi\)
0.515569 + 0.856848i \(0.327580\pi\)
\(84\) 0 0
\(85\) −27.0000 + 46.7654i −0.317647 + 0.550181i
\(86\) −129.904 − 75.0000i −1.51051 − 0.872093i
\(87\) 0 0
\(88\) 22.5000 + 38.9711i 0.255682 + 0.442854i
\(89\) 90.0000i 1.01124i 0.862757 + 0.505618i \(0.168735\pi\)
−0.862757 + 0.505618i \(0.831265\pi\)
\(90\) 0 0
\(91\) −50.0000 −0.549451
\(92\) 51.9615 − 30.0000i 0.564799 − 0.326087i
\(93\) 0 0
\(94\) −9.00000 + 15.5885i −0.0957447 + 0.165835i
\(95\) −41.5692 − 24.0000i −0.437571 − 0.252632i
\(96\) 0 0
\(97\) 42.5000 + 73.6122i 0.438144 + 0.758888i 0.997546 − 0.0700082i \(-0.0223025\pi\)
−0.559402 + 0.828896i \(0.688969\pi\)
\(98\) − 72.0000i − 0.734694i
\(99\) 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 81.3.d.b.53.2 4
3.2 odd 2 inner 81.3.d.b.53.1 4
4.3 odd 2 1296.3.q.j.1025.2 4
9.2 odd 6 inner 81.3.d.b.26.2 4
9.4 even 3 27.3.b.b.26.2 yes 2
9.5 odd 6 27.3.b.b.26.1 ✓ 2
9.7 even 3 inner 81.3.d.b.26.1 4
12.11 even 2 1296.3.q.j.1025.1 4
36.7 odd 6 1296.3.q.j.593.1 4
36.11 even 6 1296.3.q.j.593.2 4
36.23 even 6 432.3.e.c.161.2 2
36.31 odd 6 432.3.e.c.161.1 2
45.4 even 6 675.3.c.h.26.1 2
45.13 odd 12 675.3.d.d.674.1 2
45.14 odd 6 675.3.c.h.26.2 2
45.22 odd 12 675.3.d.a.674.2 2
45.23 even 12 675.3.d.a.674.1 2
45.32 even 12 675.3.d.d.674.2 2
72.5 odd 6 1728.3.e.m.1025.1 2
72.13 even 6 1728.3.e.m.1025.2 2
72.59 even 6 1728.3.e.g.1025.1 2
72.67 odd 6 1728.3.e.g.1025.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.3.b.b.26.1 ✓ 2 9.5 odd 6
27.3.b.b.26.2 yes 2 9.4 even 3
81.3.d.b.26.1 4 9.7 even 3 inner
81.3.d.b.26.2 4 9.2 odd 6 inner
81.3.d.b.53.1 4 3.2 odd 2 inner
81.3.d.b.53.2 4 1.1 even 1 trivial
432.3.e.c.161.1 2 36.31 odd 6
432.3.e.c.161.2 2 36.23 even 6
675.3.c.h.26.1 2 45.4 even 6
675.3.c.h.26.2 2 45.14 odd 6
675.3.d.a.674.1 2 45.23 even 12
675.3.d.a.674.2 2 45.22 odd 12
675.3.d.d.674.1 2 45.13 odd 12
675.3.d.d.674.2 2 45.32 even 12
1296.3.q.j.593.1 4 36.7 odd 6
1296.3.q.j.593.2 4 36.11 even 6
1296.3.q.j.1025.1 4 12.11 even 2
1296.3.q.j.1025.2 4 4.3 odd 2
1728.3.e.g.1025.1 2 72.59 even 6
1728.3.e.g.1025.2 2 72.67 odd 6
1728.3.e.m.1025.1 2 72.5 odd 6
1728.3.e.m.1025.2 2 72.13 even 6