↑

Properties

Label 81.5.d.b.26.1
Level $81$
Weight $5$
Character 81.26
Analytic conductor $8.373$
Analytic rank $0$
Dimension $4$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,5,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, 5); 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, 5, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 81.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-14] 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: no
Analytic conductor: \(8.37296700979\)
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 26.1
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 81.26
Dual form 81.5.d.b.53.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.59808 - 1.50000i) q^{2} +(-3.50000 - 6.06218i) q^{4} +(28.5788 - 16.5000i) q^{5} +(9.50000 - 16.4545i) q^{7} +69.0000i q^{8} -99.0000 q^{10} +(106.521 + 61.5000i) q^{11} +(-151.000 - 261.540i) q^{13} +(-49.3634 + 28.5000i) q^{14} +(47.5000 - 82.2724i) q^{16} -414.000i q^{17} -304.000 q^{19} +(-200.052 - 115.500i) q^{20} +(-184.500 - 319.563i) q^{22} +(-259.808 + 150.000i) q^{23} +(232.000 - 401.836i) q^{25} +906.000i q^{26} -133.000 q^{28} +(-587.165 - 339.000i) q^{29} +(-119.500 - 206.980i) q^{31} +(709.275 - 409.500i) q^{32} +(-621.000 + 1075.60i) q^{34} -627.000i q^{35} +740.000 q^{37} +(789.815 + 456.000i) q^{38} +(1138.50 + 1971.94i) q^{40} +(-197.454 + 114.000i) q^{41} +(491.000 - 850.437i) q^{43} -861.000i q^{44} +900.000 q^{46} +(1875.81 + 1083.00i) q^{47} +(1020.00 + 1766.69i) q^{49} +(-1205.51 + 696.000i) q^{50} +(-1057.00 + 1830.78i) q^{52} +1593.00i q^{53} +4059.00 q^{55} +(1135.36 + 655.500i) q^{56} +(1017.00 + 1761.50i) q^{58} +(2530.53 - 1461.00i) q^{59} +(158.000 - 273.664i) q^{61} +717.000i q^{62} -3977.00 q^{64} +(-8630.81 - 4983.00i) q^{65} +(-2311.00 - 4002.77i) q^{67} +(-2509.74 + 1449.00i) q^{68} +(-940.500 + 1628.99i) q^{70} -1818.00i q^{71} -3031.00 q^{73} +(-1922.58 - 1110.00i) q^{74} +(1064.00 + 1842.90i) q^{76} +(2023.90 - 1168.50i) q^{77} +(5225.00 - 9049.97i) q^{79} -3135.00i q^{80} +684.000 q^{82} +(10940.5 + 6316.50i) q^{83} +(-6831.00 - 11831.6i) q^{85} +(-2551.31 + 1473.00i) q^{86} +(-4243.50 + 7349.96i) q^{88} +7002.00i q^{89} -5738.00 q^{91} +(1818.65 + 1050.00i) q^{92} +(-3249.00 - 5627.43i) q^{94} +(-8687.97 + 5016.00i) q^{95} +(3258.50 - 5643.89i) q^{97} -6120.00i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{4} + 38 q^{7} - 396 q^{10} - 604 q^{13} + 190 q^{16} - 1216 q^{19} - 738 q^{22} + 928 q^{25} - 532 q^{28} - 478 q^{31} - 2484 q^{34} + 2960 q^{37} + 4554 q^{40} + 1964 q^{43} + 3600 q^{46} + 4080 q^{49}+ \cdots + 13034 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\)

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 −0.649519 − 0.375000i 0.138753 − 0.990327i \(-0.455691\pi\)
−0.788272 + 0.615327i \(0.789024\pi\)
\(3\) 0 0
\(4\) −3.50000 − 6.06218i −0.218750 − 0.378886i
\(5\) 28.5788 − 16.5000i 1.14315 − 0.660000i 0.195944 − 0.980615i \(-0.437223\pi\)
0.947210 + 0.320615i \(0.103890\pi\)
\(6\) 0 0
\(7\) 9.50000 − 16.4545i 0.193878 − 0.335806i −0.752654 − 0.658416i \(-0.771227\pi\)
0.946532 + 0.322610i \(0.104560\pi\)
\(8\) 69.0000i 1.07812i
\(9\) 0 0
\(10\) −99.0000 −0.990000
\(11\) 106.521 + 61.5000i 0.880340 + 0.508264i 0.870770 − 0.491690i \(-0.163621\pi\)
0.00956942 + 0.999954i \(0.496954\pi\)
\(12\) 0 0
\(13\) −151.000 − 261.540i −0.893491 − 1.54757i −0.835661 − 0.549246i \(-0.814915\pi\)
−0.0578301 − 0.998326i \(-0.518418\pi\)
\(14\) −49.3634 + 28.5000i −0.251854 + 0.145408i
\(15\) 0 0
\(16\) 47.5000 − 82.2724i 0.185547 − 0.321377i
\(17\) − 414.000i − 1.43253i −0.697830 − 0.716263i \(-0.745851\pi\)
0.697830 − 0.716263i \(-0.254149\pi\)
\(18\) 0 0
\(19\) −304.000 −0.842105 −0.421053 − 0.907036i \(-0.638339\pi\)
−0.421053 + 0.907036i \(0.638339\pi\)
\(20\) −200.052 − 115.500i −0.500130 − 0.288750i
\(21\) 0 0
\(22\) −184.500 − 319.563i −0.381198 − 0.660255i
\(23\) −259.808 + 150.000i −0.491130 + 0.283554i −0.725043 − 0.688704i \(-0.758180\pi\)
0.233913 + 0.972257i \(0.424847\pi\)
\(24\) 0 0
\(25\) 232.000 − 401.836i 0.371200 − 0.642937i
\(26\) 906.000i 1.34024i
\(27\) 0 0
\(28\) −133.000 −0.169643
\(29\) −587.165 − 339.000i −0.698175 − 0.403092i 0.108492 − 0.994097i \(-0.465398\pi\)
−0.806667 + 0.591006i \(0.798731\pi\)
\(30\) 0 0
\(31\) −119.500 − 206.980i −0.124350 − 0.215380i 0.797129 − 0.603809i \(-0.206351\pi\)
−0.921479 + 0.388429i \(0.873018\pi\)
\(32\) 709.275 − 409.500i 0.692651 − 0.399902i
\(33\) 0 0
\(34\) −621.000 + 1075.60i −0.537197 + 0.930453i
\(35\) − 627.000i − 0.511837i
\(36\) 0 0
\(37\) 740.000 0.540541 0.270270 − 0.962784i \(-0.412887\pi\)
0.270270 + 0.962784i \(0.412887\pi\)
\(38\) 789.815 + 456.000i 0.546963 + 0.315789i
\(39\) 0 0
\(40\) 1138.50 + 1971.94i 0.711562 + 1.23246i
\(41\) −197.454 + 114.000i −0.117462 + 0.0678168i −0.557580 − 0.830123i \(-0.688270\pi\)
0.440118 + 0.897940i \(0.354937\pi\)
\(42\) 0 0
\(43\) 491.000 − 850.437i 0.265549 − 0.459944i −0.702158 − 0.712021i \(-0.747780\pi\)
0.967707 + 0.252077i \(0.0811135\pi\)
\(44\) − 861.000i − 0.444731i
\(45\) 0 0
\(46\) 900.000 0.425331
\(47\) 1875.81 + 1083.00i 0.849168 + 0.490267i 0.860370 − 0.509670i \(-0.170233\pi\)
−0.0112024 + 0.999937i \(0.503566\pi\)
\(48\) 0 0
\(49\) 1020.00 + 1766.69i 0.424823 + 0.735815i
\(50\) −1205.51 + 696.000i −0.482203 + 0.278400i
\(51\) 0 0
\(52\) −1057.00 + 1830.78i −0.390902 + 0.677063i
\(53\) 1593.00i 0.567106i 0.958957 + 0.283553i \(0.0915131\pi\)
−0.958957 + 0.283553i \(0.908487\pi\)
\(54\) 0 0
\(55\) 4059.00 1.34182
\(56\) 1135.36 + 655.500i 0.362041 + 0.209024i
\(57\) 0 0
\(58\) 1017.00 + 1761.50i 0.302319 + 0.523631i
\(59\) 2530.53 − 1461.00i 0.726954 − 0.419707i −0.0903529 − 0.995910i \(-0.528800\pi\)
0.817307 + 0.576203i \(0.195466\pi\)
\(60\) 0 0
\(61\) 158.000 − 273.664i 0.0424617 − 0.0735458i −0.844013 − 0.536322i \(-0.819813\pi\)
0.886475 + 0.462776i \(0.153147\pi\)
\(62\) 717.000i 0.186524i
\(63\) 0 0
\(64\) −3977.00 −0.970947
\(65\) −8630.81 − 4983.00i −2.04280 − 1.17941i
\(66\) 0 0
\(67\) −2311.00 − 4002.77i −0.514814 − 0.891684i −0.999852 − 0.0171910i \(-0.994528\pi\)
0.485038 − 0.874493i \(-0.338806\pi\)
\(68\) −2509.74 + 1449.00i −0.542764 + 0.313365i
\(69\) 0 0
\(70\) −940.500 + 1628.99i −0.191939 + 0.332448i
\(71\) − 1818.00i − 0.360643i −0.983608 − 0.180321i \(-0.942286\pi\)
0.983608 − 0.180321i \(-0.0577138\pi\)
\(72\) 0 0
\(73\) −3031.00 −0.568775 −0.284387 − 0.958709i \(-0.591790\pi\)
−0.284387 + 0.958709i \(0.591790\pi\)
\(74\) −1922.58 − 1110.00i −0.351091 − 0.202703i
\(75\) 0 0
\(76\) 1064.00 + 1842.90i 0.184211 + 0.319062i
\(77\) 2023.90 − 1168.50i 0.341356 − 0.197082i
\(78\) 0 0
\(79\) 5225.00 − 9049.97i 0.837206 − 1.45008i −0.0550164 − 0.998485i \(-0.517521\pi\)
0.892222 − 0.451597i \(-0.149146\pi\)
\(80\) − 3135.00i − 0.489844i
\(81\) 0 0
\(82\) 684.000 0.101725
\(83\) 10940.5 + 6316.50i 1.58811 + 0.916897i 0.993618 + 0.112796i \(0.0359806\pi\)
0.594493 + 0.804101i \(0.297353\pi\)
\(84\) 0 0
\(85\) −6831.00 − 11831.6i −0.945467 − 1.63760i
\(86\) −2551.31 + 1473.00i −0.344958 + 0.199162i
\(87\) 0 0
\(88\) −4243.50 + 7349.96i −0.547973 + 0.949116i
\(89\) 7002.00i 0.883979i 0.897020 + 0.441990i \(0.145727\pi\)
−0.897020 + 0.441990i \(0.854273\pi\)
\(90\) 0 0
\(91\) −5738.00 −0.692911
\(92\) 1818.65 + 1050.00i 0.214869 + 0.124055i
\(93\) 0 0
\(94\) −3249.00 − 5627.43i −0.367700 − 0.636876i
\(95\) −8687.97 + 5016.00i −0.962656 + 0.555789i
\(96\) 0 0
\(97\) 3258.50 − 5643.89i 0.346317 − 0.599839i −0.639275 − 0.768978i \(-0.720765\pi\)
0.985592 + 0.169139i \(0.0540987\pi\)
\(98\) − 6120.00i − 0.637234i
\(99\) 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 81.5.d.b.26.1 4
3.2 odd 2 inner 81.5.d.b.26.2 4
9.2 odd 6 27.5.b.c.26.1 ✓ 2
9.4 even 3 inner 81.5.d.b.53.2 4
9.5 odd 6 inner 81.5.d.b.53.1 4
9.7 even 3 27.5.b.c.26.2 yes 2
36.7 odd 6 432.5.e.e.161.2 2
36.11 even 6 432.5.e.e.161.1 2
45.2 even 12 675.5.d.d.674.1 2
45.7 odd 12 675.5.d.a.674.1 2
45.29 odd 6 675.5.c.h.26.2 2
45.34 even 6 675.5.c.h.26.1 2
45.38 even 12 675.5.d.a.674.2 2
45.43 odd 12 675.5.d.d.674.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.b.c.26.1 ✓ 2 9.2 odd 6
27.5.b.c.26.2 yes 2 9.7 even 3
81.5.d.b.26.1 4 1.1 even 1 trivial
81.5.d.b.26.2 4 3.2 odd 2 inner
81.5.d.b.53.1 4 9.5 odd 6 inner
81.5.d.b.53.2 4 9.4 even 3 inner
432.5.e.e.161.1 2 36.11 even 6
432.5.e.e.161.2 2 36.7 odd 6
675.5.c.h.26.1 2 45.34 even 6
675.5.c.h.26.2 2 45.29 odd 6
675.5.d.a.674.1 2 45.7 odd 12
675.5.d.a.674.2 2 45.38 even 12
675.5.d.d.674.1 2 45.2 even 12
675.5.d.d.674.2 2 45.43 odd 12