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Properties

Label 243.4.c.d.163.1
Level $243$
Weight $4$
Character 243.163
Analytic conductor $14.337$
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] = [243,4,Mod(82,243)] 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("243.82"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(243, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 243 = 3^{5} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 243.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,3,0,-1,-3,0,10] 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: \(14.3374641314\)
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: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 163.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 243.163
Dual form 243.4.c.d.82.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.50000 - 2.59808i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-1.50000 - 2.59808i) q^{5} +(5.00000 - 8.66025i) q^{7} +21.0000 q^{8} -9.00000 q^{10} +(6.00000 - 10.3923i) q^{11} +(-4.00000 - 6.92820i) q^{13} +(-15.0000 - 25.9808i) q^{14} +(35.5000 - 61.4878i) q^{16} +126.000 q^{17} -91.0000 q^{19} +(-1.50000 + 2.59808i) q^{20} +(-18.0000 - 31.1769i) q^{22} +(-46.5000 - 80.5404i) q^{23} +(58.0000 - 100.459i) q^{25} -24.0000 q^{26} -10.0000 q^{28} +(91.5000 - 158.483i) q^{29} +(-85.0000 - 147.224i) q^{31} +(-22.5000 - 38.9711i) q^{32} +(189.000 - 327.358i) q^{34} -30.0000 q^{35} -172.000 q^{37} +(-136.500 + 236.425i) q^{38} +(-31.5000 - 54.5596i) q^{40} +(237.000 + 410.496i) q^{41} +(86.0000 - 148.956i) q^{43} -12.0000 q^{44} -279.000 q^{46} +(-79.5000 + 137.698i) q^{47} +(121.500 + 210.444i) q^{49} +(-174.000 - 301.377i) q^{50} +(-4.00000 + 6.92820i) q^{52} -603.000 q^{53} -36.0000 q^{55} +(105.000 - 181.865i) q^{56} +(-274.500 - 475.448i) q^{58} +(282.000 + 488.438i) q^{59} +(215.000 - 372.391i) q^{61} -510.000 q^{62} +433.000 q^{64} +(-12.0000 + 20.7846i) q^{65} +(219.500 + 380.185i) q^{67} +(-63.0000 - 109.119i) q^{68} +(-45.0000 + 77.9423i) q^{70} +351.000 q^{71} -727.000 q^{73} +(-258.000 + 446.869i) q^{74} +(45.5000 + 78.8083i) q^{76} +(-60.0000 - 103.923i) q^{77} +(-616.000 + 1066.94i) q^{79} -213.000 q^{80} +1422.00 q^{82} +(249.000 - 431.281i) q^{83} +(-189.000 - 327.358i) q^{85} +(-258.000 - 446.869i) q^{86} +(126.000 - 218.238i) q^{88} +1044.00 q^{89} -80.0000 q^{91} +(-46.5000 + 80.5404i) q^{92} +(238.500 + 413.094i) q^{94} +(136.500 + 236.425i) q^{95} +(-575.500 + 996.795i) q^{97} +729.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - q^{4} - 3 q^{5} + 10 q^{7} + 42 q^{8} - 18 q^{10} + 12 q^{11} - 8 q^{13} - 30 q^{14} + 71 q^{16} + 252 q^{17} - 182 q^{19} - 3 q^{20} - 36 q^{22} - 93 q^{23} + 116 q^{25} - 48 q^{26} - 20 q^{28}+ \cdots + 1458 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.50000 − 2.59808i 0.530330 − 0.918559i −0.469044 − 0.883175i \(-0.655401\pi\)
0.999374 − 0.0353837i \(-0.0112653\pi\)
\(3\) 0 0
\(4\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(5\) −1.50000 − 2.59808i −0.134164 − 0.232379i 0.791114 − 0.611669i \(-0.209502\pi\)
−0.925278 + 0.379290i \(0.876168\pi\)
\(6\) 0 0
\(7\) 5.00000 − 8.66025i 0.269975 − 0.467610i −0.698880 − 0.715239i \(-0.746318\pi\)
0.968855 + 0.247629i \(0.0796514\pi\)
\(8\) 21.0000 0.928078
\(9\) 0 0
\(10\) −9.00000 −0.284605
\(11\) 6.00000 − 10.3923i 0.164461 − 0.284854i −0.772003 − 0.635619i \(-0.780745\pi\)
0.936464 + 0.350765i \(0.114078\pi\)
\(12\) 0 0
\(13\) −4.00000 − 6.92820i −0.0853385 − 0.147811i 0.820197 − 0.572081i \(-0.193864\pi\)
−0.905535 + 0.424271i \(0.860530\pi\)
\(14\) −15.0000 − 25.9808i −0.286351 − 0.495975i
\(15\) 0 0
\(16\) 35.5000 − 61.4878i 0.554688 − 0.960747i
\(17\) 126.000 1.79762 0.898808 − 0.438342i \(-0.144434\pi\)
0.898808 + 0.438342i \(0.144434\pi\)
\(18\) 0 0
\(19\) −91.0000 −1.09878 −0.549390 − 0.835566i \(-0.685140\pi\)
−0.549390 + 0.835566i \(0.685140\pi\)
\(20\) −1.50000 + 2.59808i −0.0167705 + 0.0290474i
\(21\) 0 0
\(22\) −18.0000 − 31.1769i −0.174437 − 0.302134i
\(23\) −46.5000 − 80.5404i −0.421562 − 0.730166i 0.574531 − 0.818483i \(-0.305185\pi\)
−0.996092 + 0.0883167i \(0.971851\pi\)
\(24\) 0 0
\(25\) 58.0000 − 100.459i 0.464000 − 0.803672i
\(26\) −24.0000 −0.181030
\(27\) 0 0
\(28\) −10.0000 −0.0674937
\(29\) 91.5000 − 158.483i 0.585901 − 1.01481i −0.408862 − 0.912596i \(-0.634074\pi\)
0.994762 − 0.102214i \(-0.0325925\pi\)
\(30\) 0 0
\(31\) −85.0000 − 147.224i −0.492466 − 0.852976i 0.507496 − 0.861654i \(-0.330571\pi\)
−0.999962 + 0.00867760i \(0.997238\pi\)
\(32\) −22.5000 − 38.9711i −0.124296 − 0.215287i
\(33\) 0 0
\(34\) 189.000 − 327.358i 0.953330 − 1.65122i
\(35\) −30.0000 −0.144884
\(36\) 0 0
\(37\) −172.000 −0.764233 −0.382117 − 0.924114i \(-0.624805\pi\)
−0.382117 + 0.924114i \(0.624805\pi\)
\(38\) −136.500 + 236.425i −0.582716 + 1.00929i
\(39\) 0 0
\(40\) −31.5000 − 54.5596i −0.124515 − 0.215666i
\(41\) 237.000 + 410.496i 0.902761 + 1.56363i 0.823895 + 0.566742i \(0.191796\pi\)
0.0788651 + 0.996885i \(0.474870\pi\)
\(42\) 0 0
\(43\) 86.0000 − 148.956i 0.304997 − 0.528271i −0.672264 − 0.740312i \(-0.734678\pi\)
0.977261 + 0.212041i \(0.0680112\pi\)
\(44\) −12.0000 −0.0411152
\(45\) 0 0
\(46\) −279.000 −0.894268
\(47\) −79.5000 + 137.698i −0.246729 + 0.427347i −0.962616 − 0.270869i \(-0.912689\pi\)
0.715887 + 0.698216i \(0.246022\pi\)
\(48\) 0 0
\(49\) 121.500 + 210.444i 0.354227 + 0.613540i
\(50\) −174.000 − 301.377i −0.492146 − 0.852422i
\(51\) 0 0
\(52\) −4.00000 + 6.92820i −0.0106673 + 0.0184763i
\(53\) −603.000 −1.56280 −0.781400 − 0.624030i \(-0.785494\pi\)
−0.781400 + 0.624030i \(0.785494\pi\)
\(54\) 0 0
\(55\) −36.0000 −0.0882589
\(56\) 105.000 − 181.865i 0.250557 − 0.433978i
\(57\) 0 0
\(58\) −274.500 − 475.448i −0.621442 − 1.07637i
\(59\) 282.000 + 488.438i 0.622259 + 1.07778i 0.989064 + 0.147486i \(0.0471182\pi\)
−0.366805 + 0.930298i \(0.619548\pi\)
\(60\) 0 0
\(61\) 215.000 − 372.391i 0.451278 − 0.781636i −0.547188 − 0.837010i \(-0.684302\pi\)
0.998466 + 0.0553740i \(0.0176351\pi\)
\(62\) −510.000 −1.04468
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) −12.0000 + 20.7846i −0.0228987 + 0.0396617i
\(66\) 0 0
\(67\) 219.500 + 380.185i 0.400242 + 0.693239i 0.993755 − 0.111585i \(-0.0355928\pi\)
−0.593513 + 0.804824i \(0.702260\pi\)
\(68\) −63.0000 − 109.119i −0.112351 − 0.194598i
\(69\) 0 0
\(70\) −45.0000 + 77.9423i −0.0768361 + 0.133084i
\(71\) 351.000 0.586705 0.293352 − 0.956004i \(-0.405229\pi\)
0.293352 + 0.956004i \(0.405229\pi\)
\(72\) 0 0
\(73\) −727.000 −1.16560 −0.582801 − 0.812615i \(-0.698043\pi\)
−0.582801 + 0.812615i \(0.698043\pi\)
\(74\) −258.000 + 446.869i −0.405296 + 0.701993i
\(75\) 0 0
\(76\) 45.5000 + 78.8083i 0.0686738 + 0.118946i
\(77\) −60.0000 − 103.923i −0.0888004 − 0.153807i
\(78\) 0 0
\(79\) −616.000 + 1066.94i −0.877284 + 1.51950i −0.0229738 + 0.999736i \(0.507313\pi\)
−0.854310 + 0.519764i \(0.826020\pi\)
\(80\) −213.000 −0.297677
\(81\) 0 0
\(82\) 1422.00 1.91504
\(83\) 249.000 − 431.281i 0.329293 − 0.570352i −0.653079 − 0.757290i \(-0.726523\pi\)
0.982372 + 0.186938i \(0.0598564\pi\)
\(84\) 0 0
\(85\) −189.000 − 327.358i −0.241176 − 0.417728i
\(86\) −258.000 − 446.869i −0.323498 − 0.560316i
\(87\) 0 0
\(88\) 126.000 − 218.238i 0.152632 − 0.264367i
\(89\) 1044.00 1.24341 0.621707 − 0.783250i \(-0.286440\pi\)
0.621707 + 0.783250i \(0.286440\pi\)
\(90\) 0 0
\(91\) −80.0000 −0.0921569
\(92\) −46.5000 + 80.5404i −0.0526952 + 0.0912708i
\(93\) 0 0
\(94\) 238.500 + 413.094i 0.261696 + 0.453270i
\(95\) 136.500 + 236.425i 0.147417 + 0.255334i
\(96\) 0 0
\(97\) −575.500 + 996.795i −0.602404 + 1.04339i 0.390052 + 0.920793i \(0.372457\pi\)
−0.992456 + 0.122601i \(0.960876\pi\)
\(98\) 729.000 0.751430
\(99\) 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 243.4.c.d.163.1 2
3.2 odd 2 243.4.c.a.163.1 2
9.2 odd 6 243.4.a.d.1.1 yes 1
9.4 even 3 inner 243.4.c.d.82.1 2
9.5 odd 6 243.4.c.a.82.1 2
9.7 even 3 243.4.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
243.4.a.a.1.1 ✓ 1 9.7 even 3
243.4.a.d.1.1 yes 1 9.2 odd 6
243.4.c.a.82.1 2 9.5 odd 6
243.4.c.a.163.1 2 3.2 odd 2
243.4.c.d.82.1 2 9.4 even 3 inner
243.4.c.d.163.1 2 1.1 even 1 trivial