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Properties

Label 243.5.d.a.161.1
Level $243$
Weight $5$
Character 243.161
Analytic conductor $25.119$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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] = [243,5,Mod(80,243)] 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("243.80"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(243, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 243.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-9,0,11,45,0,-50] 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: no
Analytic conductor: \(25.1189010294\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 161.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 243.161
Dual form 243.5.d.a.80.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.50000 + 2.59808i) q^{2} +(5.50000 - 9.52628i) q^{4} +(22.5000 + 12.9904i) q^{5} +(-25.0000 - 43.3013i) q^{7} -25.9808i q^{8} -135.000 q^{10} +(-18.0000 + 10.3923i) q^{11} +(-127.000 + 219.970i) q^{13} +(225.000 + 129.904i) q^{14} +(155.500 + 269.334i) q^{16} -342.946i q^{17} +251.000 q^{19} +(247.500 - 142.894i) q^{20} +(54.0000 - 93.5307i) q^{22} +(292.500 + 168.875i) q^{23} +(25.0000 + 43.3013i) q^{25} -1319.82i q^{26} -550.000 q^{28} +(940.500 - 542.998i) q^{29} +(-190.000 + 329.090i) q^{31} +(-1039.50 - 600.156i) q^{32} +(891.000 + 1543.26i) q^{34} -1299.04i q^{35} -1324.00 q^{37} +(-1129.50 + 652.117i) q^{38} +(337.500 - 584.567i) q^{40} +(2682.00 + 1548.45i) q^{41} +(1793.00 + 3105.57i) q^{43} +228.631i q^{44} -1755.00 q^{46} +(1210.50 - 698.883i) q^{47} +(-49.5000 + 85.7365i) q^{49} +(-225.000 - 129.904i) q^{50} +(1397.00 + 2419.67i) q^{52} +5035.07i q^{53} -540.000 q^{55} +(-1125.00 + 649.519i) q^{56} +(-2821.50 + 4886.98i) q^{58} +(-1557.00 - 898.934i) q^{59} +(-913.000 - 1581.36i) q^{61} -1974.54i q^{62} +1261.00 q^{64} +(-5715.00 + 3299.56i) q^{65} +(-2612.50 + 4524.98i) q^{67} +(-3267.00 - 1886.20i) q^{68} +(3375.00 + 5845.67i) q^{70} +9399.84i q^{71} -3031.00 q^{73} +(5958.00 - 3439.85i) q^{74} +(1380.50 - 2391.10i) q^{76} +(900.000 + 519.615i) q^{77} +(2180.00 + 3775.87i) q^{79} +8080.02i q^{80} -16092.0 q^{82} +(-5850.00 + 3377.50i) q^{83} +(4455.00 - 7716.29i) q^{85} +(-16137.0 - 9316.70i) q^{86} +(270.000 + 467.654i) q^{88} -10444.3i q^{89} +12700.0 q^{91} +(3217.50 - 1857.62i) q^{92} +(-3631.50 + 6289.94i) q^{94} +(5647.50 + 3260.59i) q^{95} +(159.500 + 276.262i) q^{97} -514.419i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{2} + 11 q^{4} + 45 q^{5} - 50 q^{7} - 270 q^{10} - 36 q^{11} - 254 q^{13} + 450 q^{14} + 311 q^{16} + 502 q^{19} + 495 q^{20} + 108 q^{22} + 585 q^{23} + 50 q^{25} - 1100 q^{28} + 1881 q^{29}+ \cdots + 319 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/243\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.50000 + 2.59808i −1.12500 + 0.649519i −0.942673 − 0.333719i \(-0.891696\pi\)
−0.182327 + 0.983238i \(0.558363\pi\)
\(3\) 0 0
\(4\) 5.50000 − 9.52628i 0.343750 − 0.595392i
\(5\) 22.5000 + 12.9904i 0.900000 + 0.519615i 0.877200 − 0.480125i \(-0.159409\pi\)
0.0227998 + 0.999740i \(0.492742\pi\)
\(6\) 0 0
\(7\) −25.0000 − 43.3013i −0.510204 − 0.883699i −0.999930 − 0.0118230i \(-0.996237\pi\)
0.489726 − 0.871876i \(-0.337097\pi\)
\(8\) − 25.9808i − 0.405949i
\(9\) 0 0
\(10\) −135.000 −1.35000
\(11\) −18.0000 + 10.3923i −0.148760 + 0.0858868i −0.572533 − 0.819882i \(-0.694039\pi\)
0.423772 + 0.905769i \(0.360706\pi\)
\(12\) 0 0
\(13\) −127.000 + 219.970i −0.751479 + 1.30160i 0.195626 + 0.980678i \(0.437326\pi\)
−0.947106 + 0.320922i \(0.896007\pi\)
\(14\) 225.000 + 129.904i 1.14796 + 0.662775i
\(15\) 0 0
\(16\) 155.500 + 269.334i 0.607422 + 1.05209i
\(17\) − 342.946i − 1.18666i −0.804958 − 0.593332i \(-0.797812\pi\)
0.804958 − 0.593332i \(-0.202188\pi\)
\(18\) 0 0
\(19\) 251.000 0.695291 0.347645 − 0.937626i \(-0.386981\pi\)
0.347645 + 0.937626i \(0.386981\pi\)
\(20\) 247.500 − 142.894i 0.618750 − 0.357235i
\(21\) 0 0
\(22\) 54.0000 − 93.5307i 0.111570 − 0.193245i
\(23\) 292.500 + 168.875i 0.552930 + 0.319234i 0.750303 − 0.661094i \(-0.229908\pi\)
−0.197373 + 0.980328i \(0.563241\pi\)
\(24\) 0 0
\(25\) 25.0000 + 43.3013i 0.0400000 + 0.0692820i
\(26\) − 1319.82i − 1.95240i
\(27\) 0 0
\(28\) −550.000 −0.701531
\(29\) 940.500 − 542.998i 1.11831 − 0.645657i 0.177342 − 0.984149i \(-0.443250\pi\)
0.940969 + 0.338492i \(0.109917\pi\)
\(30\) 0 0
\(31\) −190.000 + 329.090i −0.197711 + 0.342445i −0.947786 − 0.318908i \(-0.896684\pi\)
0.750075 + 0.661353i \(0.230017\pi\)
\(32\) −1039.50 − 600.156i −1.01514 − 0.586089i
\(33\) 0 0
\(34\) 891.000 + 1543.26i 0.770761 + 1.33500i
\(35\) − 1299.04i − 1.06044i
\(36\) 0 0
\(37\) −1324.00 −0.967129 −0.483565 − 0.875309i \(-0.660658\pi\)
−0.483565 + 0.875309i \(0.660658\pi\)
\(38\) −1129.50 + 652.117i −0.782202 + 0.451605i
\(39\) 0 0
\(40\) 337.500 − 584.567i 0.210938 − 0.365354i
\(41\) 2682.00 + 1548.45i 1.59548 + 0.921150i 0.992343 + 0.123512i \(0.0394156\pi\)
0.603136 + 0.797639i \(0.293918\pi\)
\(42\) 0 0
\(43\) 1793.00 + 3105.57i 0.969713 + 1.67959i 0.696380 + 0.717674i \(0.254793\pi\)
0.273334 + 0.961919i \(0.411874\pi\)
\(44\) 228.631i 0.118094i
\(45\) 0 0
\(46\) −1755.00 −0.829395
\(47\) 1210.50 − 698.883i 0.547986 − 0.316380i −0.200324 − 0.979730i \(-0.564199\pi\)
0.748309 + 0.663350i \(0.230866\pi\)
\(48\) 0 0
\(49\) −49.5000 + 85.7365i −0.0206164 + 0.0357087i
\(50\) −225.000 − 129.904i −0.0900000 − 0.0519615i
\(51\) 0 0
\(52\) 1397.00 + 2419.67i 0.516642 + 0.894850i
\(53\) 5035.07i 1.79248i 0.443571 + 0.896239i \(0.353711\pi\)
−0.443571 + 0.896239i \(0.646289\pi\)
\(54\) 0 0
\(55\) −540.000 −0.178512
\(56\) −1125.00 + 649.519i −0.358737 + 0.207117i
\(57\) 0 0
\(58\) −2821.50 + 4886.98i −0.838734 + 1.45273i
\(59\) −1557.00 − 898.934i −0.447285 − 0.258240i 0.259398 − 0.965771i \(-0.416476\pi\)
−0.706683 + 0.707530i \(0.749809\pi\)
\(60\) 0 0
\(61\) −913.000 − 1581.36i −0.245364 − 0.424983i 0.716870 − 0.697207i \(-0.245574\pi\)
−0.962234 + 0.272224i \(0.912241\pi\)
\(62\) − 1974.54i − 0.513668i
\(63\) 0 0
\(64\) 1261.00 0.307861
\(65\) −5715.00 + 3299.56i −1.35266 + 0.780960i
\(66\) 0 0
\(67\) −2612.50 + 4524.98i −0.581978 + 1.00802i 0.413267 + 0.910610i \(0.364388\pi\)
−0.995245 + 0.0974057i \(0.968946\pi\)
\(68\) −3267.00 − 1886.20i −0.706531 − 0.407916i
\(69\) 0 0
\(70\) 3375.00 + 5845.67i 0.688776 + 1.19299i
\(71\) 9399.84i 1.86468i 0.361586 + 0.932339i \(0.382235\pi\)
−0.361586 + 0.932339i \(0.617765\pi\)
\(72\) 0 0
\(73\) −3031.00 −0.568775 −0.284387 − 0.958709i \(-0.591790\pi\)
−0.284387 + 0.958709i \(0.591790\pi\)
\(74\) 5958.00 − 3439.85i 1.08802 − 0.628169i
\(75\) 0 0
\(76\) 1380.50 − 2391.10i 0.239006 − 0.413971i
\(77\) 900.000 + 519.615i 0.151796 + 0.0876396i
\(78\) 0 0
\(79\) 2180.00 + 3775.87i 0.349303 + 0.605011i 0.986126 − 0.166000i \(-0.0530851\pi\)
−0.636823 + 0.771010i \(0.719752\pi\)
\(80\) 8080.02i 1.26250i
\(81\) 0 0
\(82\) −16092.0 −2.39322
\(83\) −5850.00 + 3377.50i −0.849180 + 0.490274i −0.860374 − 0.509663i \(-0.829770\pi\)
0.0111942 + 0.999937i \(0.496437\pi\)
\(84\) 0 0
\(85\) 4455.00 − 7716.29i 0.616609 − 1.06800i
\(86\) −16137.0 − 9316.70i −2.18186 − 1.25969i
\(87\) 0 0
\(88\) 270.000 + 467.654i 0.0348657 + 0.0603892i
\(89\) − 10444.3i − 1.31855i −0.751900 − 0.659277i \(-0.770862\pi\)
0.751900 − 0.659277i \(-0.229138\pi\)
\(90\) 0 0
\(91\) 12700.0 1.53363
\(92\) 3217.50 − 1857.62i 0.380139 − 0.219474i
\(93\) 0 0
\(94\) −3631.50 + 6289.94i −0.410989 + 0.711854i
\(95\) 5647.50 + 3260.59i 0.625762 + 0.361284i
\(96\) 0 0
\(97\) 159.500 + 276.262i 0.0169519 + 0.0293615i 0.874377 − 0.485247i \(-0.161270\pi\)
−0.857425 + 0.514609i \(0.827937\pi\)
\(98\) − 514.419i − 0.0535630i
\(99\) 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 243.5.d.a.161.1 2
3.2 odd 2 243.5.d.d.161.1 2
9.2 odd 6 inner 243.5.d.a.80.1 2
9.4 even 3 243.5.b.d.242.1 ✓ 2
9.5 odd 6 243.5.b.d.242.2 yes 2
9.7 even 3 243.5.d.d.80.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
243.5.b.d.242.1 ✓ 2 9.4 even 3
243.5.b.d.242.2 yes 2 9.5 odd 6
243.5.d.a.80.1 2 9.2 odd 6 inner
243.5.d.a.161.1 2 1.1 even 1 trivial
243.5.d.d.80.1 2 9.7 even 3
243.5.d.d.161.1 2 3.2 odd 2