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Properties

Label 273.4.i.b.235.3
Level $273$
Weight $4$
Character 273.235
Analytic conductor $16.108$
Analytic rank $0$
Dimension $6$
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] = [273,4,Mod(79,273)] 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("273.79"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(273, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 273.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.1075214316\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.432216027.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 11x^{4} - 14x^{3} + 121x^{2} - 77x + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 235.3
Root \(1.46758 + 2.54192i\) of defining polynomial
Character \(\chi\) \(=\) 273.235
Dual form 273.4.i.b.79.3

$q$-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+(1.46758 + 2.54192i) q^{2} +(-1.50000 + 2.59808i) q^{3} +(-0.307557 + 0.532705i) q^{4} +(8.90273 + 15.4200i) q^{5} -8.80545 q^{6} +(-15.3087 + 10.4232i) q^{7} +21.6758 q^{8} +(-4.50000 - 7.79423i) q^{9} +(-26.1309 + 45.2600i) q^{10} +(-7.37567 + 12.7750i) q^{11} +(-0.922672 - 1.59811i) q^{12} -13.0000 q^{13} +(-48.9616 - 23.6166i) q^{14} -53.4164 q^{15} +(34.2713 + 59.3596i) q^{16} +(12.8034 - 22.1761i) q^{17} +(13.2082 - 22.8772i) q^{18} +(46.3162 + 80.2220i) q^{19} -10.9524 q^{20} +(-4.11722 - 55.4080i) q^{21} -43.2974 q^{22} +(-40.5294 - 70.1989i) q^{23} +(-32.5136 + 56.3153i) q^{24} +(-96.0171 + 166.307i) q^{25} +(-19.0785 - 33.0449i) q^{26} +27.0000 q^{27} +(-0.844187 - 11.3608i) q^{28} -75.8952 q^{29} +(-78.3926 - 135.780i) q^{30} +(-31.7976 + 55.0751i) q^{31} +(-13.8884 + 24.0553i) q^{32} +(-22.1270 - 38.3251i) q^{33} +75.1596 q^{34} +(-297.015 - 143.265i) q^{35} +5.53603 q^{36} +(-140.278 - 242.969i) q^{37} +(-135.945 + 235.464i) q^{38} +(19.5000 - 33.7750i) q^{39} +(192.973 + 334.240i) q^{40} -96.3809 q^{41} +(134.800 - 91.7811i) q^{42} +10.5359 q^{43} +(-4.53688 - 7.85811i) q^{44} +(80.1245 - 138.780i) q^{45} +(118.960 - 206.045i) q^{46} +(120.280 + 208.331i) q^{47} -205.628 q^{48} +(125.713 - 319.132i) q^{49} -563.650 q^{50} +(38.4101 + 66.5282i) q^{51} +(3.99825 - 6.92516i) q^{52} +(-50.4866 + 87.4454i) q^{53} +(39.6245 + 68.6317i) q^{54} -262.654 q^{55} +(-331.828 + 225.931i) q^{56} -277.897 q^{57} +(-111.382 - 192.919i) q^{58} +(65.3496 - 113.189i) q^{59} +(16.4286 - 28.4552i) q^{60} +(89.7306 + 155.418i) q^{61} -186.662 q^{62} +(150.130 + 72.4152i) q^{63} +466.812 q^{64} +(-115.735 - 200.460i) q^{65} +(64.9461 - 112.490i) q^{66} +(-462.913 + 801.789i) q^{67} +(7.87553 + 13.6408i) q^{68} +243.176 q^{69} +(-71.7243 - 965.239i) q^{70} +870.348 q^{71} +(-97.5409 - 168.946i) q^{72} +(351.488 - 608.795i) q^{73} +(411.739 - 713.152i) q^{74} +(-288.051 - 498.920i) q^{75} -56.9795 q^{76} +(-20.2448 - 272.447i) q^{77} +114.471 q^{78} +(101.251 + 175.372i) q^{79} +(-610.216 + 1056.92i) q^{80} +(-40.5000 + 70.1481i) q^{81} +(-141.446 - 244.992i) q^{82} +1286.15 q^{83} +(30.7824 + 14.8479i) q^{84} +455.939 q^{85} +(15.4622 + 26.7813i) q^{86} +(113.843 - 197.181i) q^{87} +(-159.873 + 276.908i) q^{88} +(-266.380 - 461.383i) q^{89} +470.355 q^{90} +(199.013 - 135.502i) q^{91} +49.8604 q^{92} +(-95.3929 - 165.225i) q^{93} +(-353.040 + 611.484i) q^{94} +(-824.681 + 1428.39i) q^{95} +(-41.6651 - 72.1660i) q^{96} +986.092 q^{97} +(995.700 - 148.797i) q^{98} +132.762 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} + 2 q^{4} + 27 q^{5} + 38 q^{7} + 42 q^{8} - 27 q^{9} - 66 q^{10} + 19 q^{11} + 6 q^{12} - 78 q^{13} - 137 q^{14} - 162 q^{15} + 94 q^{16} - 25 q^{17} + 169 q^{19} + 162 q^{20} - 66 q^{21}+ \cdots - 342 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.46758 + 2.54192i 0.518866 + 0.898703i 0.999760 + 0.0219238i \(0.00697912\pi\)
−0.480893 + 0.876779i \(0.659688\pi\)
\(3\) −1.50000 + 2.59808i −0.288675 + 0.500000i
\(4\) −0.307557 + 0.532705i −0.0384447 + 0.0665881i
\(5\) 8.90273 + 15.4200i 0.796284 + 1.37920i 0.922021 + 0.387141i \(0.126537\pi\)
−0.125736 + 0.992064i \(0.540129\pi\)
\(6\) −8.80545 −0.599135
\(7\) −15.3087 + 10.4232i −0.826593 + 0.562800i
\(8\) 21.6758 0.957942
\(9\) −4.50000 − 7.79423i −0.166667 − 0.288675i
\(10\) −26.1309 + 45.2600i −0.826330 + 1.43125i
\(11\) −7.37567 + 12.7750i −0.202168 + 0.350165i −0.949227 − 0.314593i \(-0.898132\pi\)
0.747059 + 0.664758i \(0.231465\pi\)
\(12\) −0.922672 − 1.59811i −0.0221960 − 0.0384447i
\(13\) −13.0000 −0.277350
\(14\) −48.9616 − 23.6166i −0.934682 − 0.450843i
\(15\) −53.4164 −0.919470
\(16\) 34.2713 + 59.3596i 0.535489 + 0.927494i
\(17\) 12.8034 − 22.1761i 0.182663 − 0.316381i −0.760124 − 0.649778i \(-0.774862\pi\)
0.942786 + 0.333397i \(0.108195\pi\)
\(18\) 13.2082 − 22.8772i 0.172955 − 0.299568i
\(19\) 46.3162 + 80.2220i 0.559245 + 0.968641i 0.997560 + 0.0698195i \(0.0222423\pi\)
−0.438314 + 0.898822i \(0.644424\pi\)
\(20\) −10.9524 −0.122452
\(21\) −4.11722 − 55.4080i −0.0427834 − 0.575763i
\(22\) −43.2974 −0.419592
\(23\) −40.5294 − 70.1989i −0.367433 − 0.636413i 0.621730 − 0.783231i \(-0.286430\pi\)
−0.989163 + 0.146819i \(0.953097\pi\)
\(24\) −32.5136 + 56.3153i −0.276534 + 0.478971i
\(25\) −96.0171 + 166.307i −0.768137 + 1.33045i
\(26\) −19.0785 − 33.0449i −0.143908 − 0.249255i
\(27\) 27.0000 0.192450
\(28\) −0.844187 − 11.3608i −0.00569773 − 0.0766779i
\(29\) −75.8952 −0.485979 −0.242989 − 0.970029i \(-0.578128\pi\)
−0.242989 + 0.970029i \(0.578128\pi\)
\(30\) −78.3926 − 135.780i −0.477082 − 0.826330i
\(31\) −31.7976 + 55.0751i −0.184227 + 0.319090i −0.943316 − 0.331897i \(-0.892311\pi\)
0.759089 + 0.650987i \(0.225645\pi\)
\(32\) −13.8884 + 24.0553i −0.0767231 + 0.132888i
\(33\) −22.1270 − 38.3251i −0.116722 − 0.202168i
\(34\) 75.1596 0.379111
\(35\) −297.015 − 143.265i −1.43442 − 0.691892i
\(36\) 5.53603 0.0256298
\(37\) −140.278 − 242.969i −0.623288 − 1.07957i −0.988869 − 0.148786i \(-0.952463\pi\)
0.365582 − 0.930779i \(-0.380870\pi\)
\(38\) −135.945 + 235.464i −0.580347 + 1.00519i
\(39\) 19.5000 − 33.7750i 0.0800641 − 0.138675i
\(40\) 192.973 + 334.240i 0.762794 + 1.32120i
\(41\) −96.3809 −0.367126 −0.183563 − 0.983008i \(-0.558763\pi\)
−0.183563 + 0.983008i \(0.558763\pi\)
\(42\) 134.800 − 91.7811i 0.495241 − 0.337194i
\(43\) 10.5359 0.0373653 0.0186826 − 0.999825i \(-0.494053\pi\)
0.0186826 + 0.999825i \(0.494053\pi\)
\(44\) −4.53688 − 7.85811i −0.0155446 − 0.0269240i
\(45\) 80.1245 − 138.780i 0.265428 − 0.459735i
\(46\) 118.960 − 206.045i 0.381297 − 0.660426i
\(47\) 120.280 + 208.331i 0.373291 + 0.646558i 0.990070 − 0.140578i \(-0.0448961\pi\)
−0.616779 + 0.787136i \(0.711563\pi\)
\(48\) −205.628 −0.618329
\(49\) 125.713 − 319.132i 0.366512 − 0.930413i
\(50\) −563.650 −1.59424
\(51\) 38.4101 + 66.5282i 0.105460 + 0.182663i
\(52\) 3.99825 − 6.92516i 0.0106626 − 0.0184682i
\(53\) −50.4866 + 87.4454i −0.130847 + 0.226633i −0.924003 − 0.382385i \(-0.875103\pi\)
0.793157 + 0.609018i \(0.208436\pi\)
\(54\) 39.6245 + 68.6317i 0.0998559 + 0.172955i
\(55\) −262.654 −0.643932
\(56\) −331.828 + 225.931i −0.791828 + 0.539130i
\(57\) −277.897 −0.645761
\(58\) −111.382 − 192.919i −0.252158 − 0.436750i
\(59\) 65.3496 − 113.189i 0.144200 − 0.249762i −0.784874 − 0.619655i \(-0.787272\pi\)
0.929074 + 0.369893i \(0.120606\pi\)
\(60\) 16.4286 − 28.4552i 0.0353487 − 0.0612258i
\(61\) 89.7306 + 155.418i 0.188341 + 0.326217i 0.944697 − 0.327943i \(-0.106356\pi\)
−0.756356 + 0.654160i \(0.773022\pi\)
\(62\) −186.662 −0.382356
\(63\) 150.130 + 72.4152i 0.300232 + 0.144817i
\(64\) 466.812 0.911741
\(65\) −115.735 − 200.460i −0.220849 − 0.382523i
\(66\) 64.9461 − 112.490i 0.121126 − 0.209796i
\(67\) −462.913 + 801.789i −0.844087 + 1.46200i 0.0423250 + 0.999104i \(0.486524\pi\)
−0.886412 + 0.462897i \(0.846810\pi\)
\(68\) 7.87553 + 13.6408i 0.0140448 + 0.0243264i
\(69\) 243.176 0.424275
\(70\) −71.7243 − 965.239i −0.122467 − 1.64812i
\(71\) 870.348 1.45481 0.727403 − 0.686210i \(-0.240727\pi\)
0.727403 + 0.686210i \(0.240727\pi\)
\(72\) −97.5409 − 168.946i −0.159657 − 0.276534i
\(73\) 351.488 − 608.795i 0.563542 − 0.976083i −0.433642 − 0.901085i \(-0.642772\pi\)
0.997184 − 0.0749978i \(-0.0238950\pi\)
\(74\) 411.739 − 713.152i 0.646806 − 1.12030i
\(75\) −288.051 − 498.920i −0.443484 − 0.768137i
\(76\) −56.9795 −0.0860000
\(77\) −20.2448 − 272.447i −0.0299625 − 0.403224i
\(78\) 114.471 0.166170
\(79\) 101.251 + 175.372i 0.144198 + 0.249758i 0.929073 − 0.369895i \(-0.120606\pi\)
−0.784875 + 0.619654i \(0.787273\pi\)
\(80\) −610.216 + 1056.92i −0.852802 + 1.47710i
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) −141.446 − 244.992i −0.190489 − 0.329937i
\(83\) 1286.15 1.70088 0.850439 − 0.526074i \(-0.176336\pi\)
0.850439 + 0.526074i \(0.176336\pi\)
\(84\) 30.7824 + 14.8479i 0.0399838 + 0.0192861i
\(85\) 455.939 0.581806
\(86\) 15.4622 + 26.7813i 0.0193876 + 0.0335803i
\(87\) 113.843 − 197.181i 0.140290 − 0.242989i
\(88\) −159.873 + 276.908i −0.193665 + 0.335438i
\(89\) −266.380 − 461.383i −0.317261 − 0.549512i 0.662655 − 0.748925i \(-0.269430\pi\)
−0.979915 + 0.199413i \(0.936096\pi\)
\(90\) 470.355 0.550887
\(91\) 199.013 − 135.502i 0.229256 − 0.156093i
\(92\) 49.8604 0.0565034
\(93\) −95.3929 − 165.225i −0.106363 − 0.184227i
\(94\) −353.040 + 611.484i −0.387376 + 0.670955i
\(95\) −824.681 + 1428.39i −0.890636 + 1.54263i
\(96\) −41.6651 − 72.1660i −0.0442961 − 0.0767231i
\(97\) 986.092 1.03219 0.516095 − 0.856531i \(-0.327385\pi\)
0.516095 + 0.856531i \(0.327385\pi\)
\(98\) 995.700 − 148.797i 1.02634 − 0.153375i
\(99\) 132.762 0.134779
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 273.4.i.b.235.3 yes 6
7.2 even 3 inner 273.4.i.b.79.3 ✓ 6
7.3 odd 6 1911.4.a.i.1.1 3
7.4 even 3 1911.4.a.j.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.4.i.b.79.3 ✓ 6 7.2 even 3 inner
273.4.i.b.235.3 yes 6 1.1 even 1 trivial
1911.4.a.i.1.1 3 7.3 odd 6
1911.4.a.j.1.1 3 7.4 even 3