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Properties

Label 81.5.d.b.53.1
Level $81$
Weight $5$
Character 81.53
Analytic conductor $8.373$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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 53.1
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 81.53
Dual form 81.5.d.b.26.1

$q$-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 \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

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

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 −0.649519 + 0.375000i −0.788272 − 0.615327i \(-0.789024\pi\)
0.138753 + 0.990327i \(0.455691\pi\)
\(3\) 0 0
\(4\) −3.50000 + 6.06218i −0.218750 + 0.378886i
\(5\) 28.5788 + 16.5000i 1.14315 + 0.660000i 0.947210 − 0.320615i \(-0.103890\pi\)
0.195944 + 0.980615i \(0.437223\pi\)
\(6\) 0 0
\(7\) 9.50000 + 16.4545i 0.193878 + 0.335806i 0.946532 − 0.322610i \(-0.104560\pi\)
−0.752654 + 0.658416i \(0.771227\pi\)
\(8\) − 69.0000i − 1.07812i
\(9\) 0 0
\(10\) −99.0000 −0.990000
\(11\) 106.521 − 61.5000i 0.880340 − 0.508264i 0.00956942 − 0.999954i \(-0.496954\pi\)
0.870770 + 0.491690i \(0.163621\pi\)
\(12\) 0 0
\(13\) −151.000 + 261.540i −0.893491 + 1.54757i −0.0578301 + 0.998326i \(0.518418\pi\)
−0.835661 + 0.549246i \(0.814915\pi\)
\(14\) −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 \(0.254149\pi\)
−0.697830 + 0.716263i \(0.745851\pi\)
\(18\) 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 0.233913 − 0.972257i \(-0.424847\pi\)
−0.725043 + 0.688704i \(0.758180\pi\)
\(24\) 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 −0.806667 − 0.591006i \(-0.798731\pi\)
0.108492 + 0.994097i \(0.465398\pi\)
\(30\) 0 0
\(31\) −119.500 + 206.980i −0.124350 + 0.215380i −0.921479 − 0.388429i \(-0.873018\pi\)
0.797129 + 0.603809i \(0.206351\pi\)
\(32\) 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 0.440118 − 0.897940i \(-0.354937\pi\)
−0.557580 + 0.830123i \(0.688270\pi\)
\(42\) 0 0
\(43\) 491.000 + 850.437i 0.265549 + 0.459944i 0.967707 − 0.252077i \(-0.0811135\pi\)
−0.702158 + 0.712021i \(0.747780\pi\)
\(44\) 861.000i 0.444731i
\(45\) 0 0
\(46\) 900.000 0.425331
\(47\) 1875.81 − 1083.00i 0.849168 − 0.490267i −0.0112024 − 0.999937i \(-0.503566\pi\)
0.860370 + 0.509670i \(0.170233\pi\)
\(48\) 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 \(-0.908487\pi\)
0.958957 − 0.283553i \(-0.0915131\pi\)
\(54\) 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 0.817307 − 0.576203i \(-0.195466\pi\)
−0.0903529 + 0.995910i \(0.528800\pi\)
\(60\) 0 0
\(61\) 158.000 + 273.664i 0.0424617 + 0.0735458i 0.886475 − 0.462776i \(-0.153147\pi\)
−0.844013 + 0.536322i \(0.819813\pi\)
\(62\) − 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 0.485038 + 0.874493i \(0.338806\pi\)
−0.999852 + 0.0171910i \(0.994528\pi\)
\(68\) −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 \(0.0577138\pi\)
−0.983608 + 0.180321i \(0.942286\pi\)
\(72\) 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 0.892222 + 0.451597i \(0.149146\pi\)
−0.0550164 + 0.998485i \(0.517521\pi\)
\(80\) 3135.00i 0.489844i
\(81\) 0 0
\(82\) 684.000 0.101725
\(83\) 10940.5 − 6316.50i 1.58811 − 0.916897i 0.594493 − 0.804101i \(-0.297353\pi\)
0.993618 − 0.112796i \(-0.0359806\pi\)
\(84\) 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 \(-0.854273\pi\)
0.897020 − 0.441990i \(-0.145727\pi\)
\(90\) 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 0.985592 − 0.169139i \(-0.0540987\pi\)
−0.639275 + 0.768978i \(0.720765\pi\)
\(98\) 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.53.1 4
3.2 odd 2 inner 81.5.d.b.53.2 4
9.2 odd 6 inner 81.5.d.b.26.1 4
9.4 even 3 27.5.b.c.26.1 ✓ 2
9.5 odd 6 27.5.b.c.26.2 yes 2
9.7 even 3 inner 81.5.d.b.26.2 4
36.23 even 6 432.5.e.e.161.2 2
36.31 odd 6 432.5.e.e.161.1 2
45.4 even 6 675.5.c.h.26.2 2
45.13 odd 12 675.5.d.a.674.2 2
45.14 odd 6 675.5.c.h.26.1 2
45.22 odd 12 675.5.d.d.674.1 2
45.23 even 12 675.5.d.d.674.2 2
45.32 even 12 675.5.d.a.674.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.5.b.c.26.1 ✓ 2 9.4 even 3
27.5.b.c.26.2 yes 2 9.5 odd 6
81.5.d.b.26.1 4 9.2 odd 6 inner
81.5.d.b.26.2 4 9.7 even 3 inner
81.5.d.b.53.1 4 1.1 even 1 trivial
81.5.d.b.53.2 4 3.2 odd 2 inner
432.5.e.e.161.1 2 36.31 odd 6
432.5.e.e.161.2 2 36.23 even 6
675.5.c.h.26.1 2 45.14 odd 6
675.5.c.h.26.2 2 45.4 even 6
675.5.d.a.674.1 2 45.32 even 12
675.5.d.a.674.2 2 45.13 odd 12
675.5.d.d.674.1 2 45.22 odd 12
675.5.d.d.674.2 2 45.23 even 12