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Properties

Label 273.4.i.e.79.3
Level $273$
Weight $4$
Character 273.79
Analytic conductor $16.108$
Analytic rank $0$
Dimension $26$
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 = [26,-3] 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: \(26\)
Relative dimension: \(13\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.3
Character \(\chi\) \(=\) 273.79
Dual form 273.4.i.e.235.3

$q$-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+(-2.18545 + 3.78531i) q^{2} +(-1.50000 - 2.59808i) q^{3} +(-5.55236 - 9.61696i) q^{4} +(8.51830 - 14.7541i) q^{5} +13.1127 q^{6} +(6.69768 - 17.2668i) q^{7} +13.5704 q^{8} +(-4.50000 + 7.79423i) q^{9} +(37.2326 + 64.4887i) q^{10} +(21.2911 + 36.8772i) q^{11} +(-16.6571 + 28.8509i) q^{12} +13.0000 q^{13} +(50.7226 + 63.0883i) q^{14} -51.1098 q^{15} +(14.7615 - 25.5677i) q^{16} +(-59.2726 - 102.663i) q^{17} +(-19.6690 - 34.0677i) q^{18} +(7.52988 - 13.0421i) q^{19} -189.187 q^{20} +(-54.9069 + 8.49907i) q^{21} -186.122 q^{22} +(-47.9772 + 83.0989i) q^{23} +(-20.3556 - 35.2568i) q^{24} +(-82.6230 - 143.107i) q^{25} +(-28.4108 + 49.2090i) q^{26} +27.0000 q^{27} +(-203.242 + 31.4599i) q^{28} -19.8473 q^{29} +(111.698 - 193.466i) q^{30} +(-114.889 - 198.993i) q^{31} +(118.803 + 205.772i) q^{32} +(63.8732 - 110.632i) q^{33} +518.148 q^{34} +(-197.703 - 245.902i) q^{35} +99.9424 q^{36} +(-25.3808 + 43.9608i) q^{37} +(32.9123 + 57.0058i) q^{38} +(-19.5000 - 33.7750i) q^{39} +(115.596 - 200.219i) q^{40} +383.836 q^{41} +(87.8245 - 226.414i) q^{42} -457.523 q^{43} +(236.431 - 409.511i) q^{44} +(76.6647 + 132.787i) q^{45} +(-209.703 - 363.216i) q^{46} +(-99.8115 + 172.879i) q^{47} -88.5692 q^{48} +(-253.282 - 231.294i) q^{49} +722.272 q^{50} +(-177.818 + 307.989i) q^{51} +(-72.1806 - 125.021i) q^{52} +(15.4973 + 26.8421i) q^{53} +(-59.0071 + 102.203i) q^{54} +725.455 q^{55} +(90.8899 - 234.316i) q^{56} -45.1793 q^{57} +(43.3752 - 75.1280i) q^{58} +(-132.894 - 230.179i) q^{59} +(283.780 + 491.521i) q^{60} +(168.268 - 291.449i) q^{61} +1004.33 q^{62} +(104.442 + 129.904i) q^{63} -802.362 q^{64} +(110.738 - 191.804i) q^{65} +(279.183 + 483.559i) q^{66} +(-468.914 - 812.183i) q^{67} +(-658.205 + 1140.04i) q^{68} +287.863 q^{69} +(1362.88 - 210.962i) q^{70} +533.347 q^{71} +(-61.0667 + 105.771i) q^{72} +(-151.591 - 262.563i) q^{73} +(-110.937 - 192.148i) q^{74} +(-247.869 + 429.321i) q^{75} -167.234 q^{76} +(779.351 - 120.636i) q^{77} +170.465 q^{78} +(-314.245 + 544.288i) q^{79} +(-251.486 - 435.587i) q^{80} +(-40.5000 - 70.1481i) q^{81} +(-838.854 + 1452.94i) q^{82} +585.226 q^{83} +(386.598 + 480.848i) q^{84} -2019.61 q^{85} +(999.892 - 1731.86i) q^{86} +(29.7709 + 51.5647i) q^{87} +(288.928 + 500.437i) q^{88} +(-318.886 + 552.326i) q^{89} -670.187 q^{90} +(87.0698 - 224.468i) q^{91} +1065.55 q^{92} +(-344.666 + 596.979i) q^{93} +(-436.266 - 755.634i) q^{94} +(-128.284 - 222.194i) q^{95} +(356.408 - 617.316i) q^{96} -598.031 q^{97} +(1429.05 - 453.269i) q^{98} -383.239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26 q - 3 q^{2} - 39 q^{3} - 55 q^{4} - 15 q^{5} + 18 q^{6} - 13 q^{7} - 12 q^{8} - 117 q^{9} - 11 q^{10} - 57 q^{11} - 165 q^{12} + 338 q^{13} + 105 q^{14} + 90 q^{15} - 311 q^{16} - 162 q^{17} - 27 q^{18}+ \cdots + 1026 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.18545 + 3.78531i −0.772672 + 1.33831i 0.163421 + 0.986556i \(0.447747\pi\)
−0.936094 + 0.351751i \(0.885586\pi\)
\(3\) −1.50000 − 2.59808i −0.288675 − 0.500000i
\(4\) −5.55236 − 9.61696i −0.694045 − 1.20212i
\(5\) 8.51830 − 14.7541i 0.761900 − 1.31965i −0.179970 − 0.983672i \(-0.557600\pi\)
0.941870 − 0.335978i \(-0.109067\pi\)
\(6\) 13.1127 0.892205
\(7\) 6.69768 − 17.2668i 0.361641 − 0.932318i
\(8\) 13.5704 0.599731
\(9\) −4.50000 + 7.79423i −0.166667 + 0.288675i
\(10\) 37.2326 + 64.4887i 1.17740 + 2.03931i
\(11\) 21.2911 + 36.8772i 0.583591 + 1.01081i 0.995050 + 0.0993803i \(0.0316860\pi\)
−0.411459 + 0.911428i \(0.634981\pi\)
\(12\) −16.6571 + 28.8509i −0.400707 + 0.694045i
\(13\) 13.0000 0.277350
\(14\) 50.7226 + 63.0883i 0.968298 + 1.20436i
\(15\) −51.1098 −0.879767
\(16\) 14.7615 − 25.5677i 0.230649 − 0.399496i
\(17\) −59.2726 − 102.663i −0.845630 − 1.46467i −0.885073 − 0.465452i \(-0.845892\pi\)
0.0394432 − 0.999222i \(-0.487442\pi\)
\(18\) −19.6690 − 34.0677i −0.257557 − 0.446102i
\(19\) 7.52988 − 13.0421i 0.0909196 − 0.157477i −0.816979 − 0.576668i \(-0.804353\pi\)
0.907898 + 0.419190i \(0.137686\pi\)
\(20\) −189.187 −2.11517
\(21\) −54.9069 + 8.49907i −0.570555 + 0.0883166i
\(22\) −186.122 −1.80370
\(23\) −47.9772 + 83.0989i −0.434954 + 0.753362i −0.997292 − 0.0735456i \(-0.976569\pi\)
0.562338 + 0.826907i \(0.309902\pi\)
\(24\) −20.3556 − 35.2568i −0.173127 − 0.299866i
\(25\) −82.6230 − 143.107i −0.660984 − 1.14486i
\(26\) −28.4108 + 49.2090i −0.214301 + 0.371180i
\(27\) 27.0000 0.192450
\(28\) −203.242 + 31.4599i −1.37175 + 0.212334i
\(29\) −19.8473 −0.127088 −0.0635439 − 0.997979i \(-0.520240\pi\)
−0.0635439 + 0.997979i \(0.520240\pi\)
\(30\) 111.698 − 193.466i 0.679771 − 1.17740i
\(31\) −114.889 − 198.993i −0.665633 − 1.15291i −0.979113 − 0.203315i \(-0.934828\pi\)
0.313480 − 0.949595i \(-0.398505\pi\)
\(32\) 118.803 + 205.772i 0.656298 + 1.13674i
\(33\) 63.8732 − 110.632i 0.336936 − 0.583591i
\(34\) 518.148 2.61358
\(35\) −197.703 − 245.902i −0.954799 − 1.18757i
\(36\) 99.9424 0.462696
\(37\) −25.3808 + 43.9608i −0.112772 + 0.195327i −0.916887 − 0.399147i \(-0.869306\pi\)
0.804115 + 0.594474i \(0.202640\pi\)
\(38\) 32.9123 + 57.0058i 0.140502 + 0.243357i
\(39\) −19.5000 − 33.7750i −0.0800641 − 0.138675i
\(40\) 115.596 − 200.219i 0.456935 − 0.791435i
\(41\) 383.836 1.46208 0.731038 − 0.682336i \(-0.239036\pi\)
0.731038 + 0.682336i \(0.239036\pi\)
\(42\) 87.8245 − 226.414i 0.322657 − 0.831818i
\(43\) −457.523 −1.62260 −0.811298 − 0.584633i \(-0.801238\pi\)
−0.811298 + 0.584633i \(0.801238\pi\)
\(44\) 236.431 − 409.511i 0.810076 − 1.40309i
\(45\) 76.6647 + 132.787i 0.253967 + 0.439883i
\(46\) −209.703 − 363.216i −0.672153 − 1.16420i
\(47\) −99.8115 + 172.879i −0.309766 + 0.536531i −0.978311 − 0.207141i \(-0.933584\pi\)
0.668545 + 0.743672i \(0.266917\pi\)
\(48\) −88.5692 −0.266330
\(49\) −253.282 − 231.294i −0.738432 − 0.674328i
\(50\) 722.272 2.04289
\(51\) −177.818 + 307.989i −0.488225 + 0.845630i
\(52\) −72.1806 − 125.021i −0.192493 − 0.333408i
\(53\) 15.4973 + 26.8421i 0.0401644 + 0.0695669i 0.885409 − 0.464813i \(-0.153878\pi\)
−0.845244 + 0.534380i \(0.820545\pi\)
\(54\) −59.0071 + 102.203i −0.148701 + 0.257557i
\(55\) 725.455 1.77855
\(56\) 90.8899 − 234.316i 0.216887 − 0.559140i
\(57\) −45.1793 −0.104985
\(58\) 43.3752 − 75.1280i 0.0981972 − 0.170083i
\(59\) −132.894 − 230.179i −0.293242 − 0.507910i 0.681332 − 0.731974i \(-0.261401\pi\)
−0.974574 + 0.224064i \(0.928068\pi\)
\(60\) 283.780 + 491.521i 0.610597 + 1.05759i
\(61\) 168.268 − 291.449i 0.353190 − 0.611742i −0.633617 − 0.773647i \(-0.718430\pi\)
0.986806 + 0.161905i \(0.0517637\pi\)
\(62\) 1004.33 2.05726
\(63\) 104.442 + 129.904i 0.208863 + 0.259783i
\(64\) −802.362 −1.56711
\(65\) 110.738 − 191.804i 0.211313 − 0.366005i
\(66\) 279.183 + 483.559i 0.520682 + 0.901848i
\(67\) −468.914 − 812.183i −0.855029 − 1.48095i −0.876618 − 0.481188i \(-0.840206\pi\)
0.0215882 − 0.999767i \(-0.493128\pi\)
\(68\) −658.205 + 1140.04i −1.17381 + 2.03310i
\(69\) 287.863 0.502241
\(70\) 1362.88 − 210.962i 2.32708 − 0.360211i
\(71\) 533.347 0.891503 0.445751 − 0.895157i \(-0.352937\pi\)
0.445751 + 0.895157i \(0.352937\pi\)
\(72\) −61.0667 + 105.771i −0.0999552 + 0.173127i
\(73\) −151.591 − 262.563i −0.243046 − 0.420969i 0.718534 − 0.695492i \(-0.244813\pi\)
−0.961581 + 0.274523i \(0.911480\pi\)
\(74\) −110.937 − 192.148i −0.174272 − 0.301848i
\(75\) −247.869 + 429.321i −0.381619 + 0.660984i
\(76\) −167.234 −0.252409
\(77\) 779.351 − 120.636i 1.15344 − 0.178542i
\(78\) 170.465 0.247453
\(79\) −314.245 + 544.288i −0.447536 + 0.775154i −0.998225 − 0.0595558i \(-0.981032\pi\)
0.550689 + 0.834710i \(0.314365\pi\)
\(80\) −251.486 − 435.587i −0.351463 − 0.608752i
\(81\) −40.5000 − 70.1481i −0.0555556 − 0.0962250i
\(82\) −838.854 + 1452.94i −1.12971 + 1.95671i
\(83\) 585.226 0.773938 0.386969 − 0.922093i \(-0.373522\pi\)
0.386969 + 0.922093i \(0.373522\pi\)
\(84\) 386.598 + 480.848i 0.502158 + 0.624581i
\(85\) −2019.61 −2.57714
\(86\) 999.892 − 1731.86i 1.25373 − 2.17153i
\(87\) 29.7709 + 51.5647i 0.0366871 + 0.0635439i
\(88\) 288.928 + 500.437i 0.349997 + 0.606213i
\(89\) −318.886 + 552.326i −0.379796 + 0.657826i −0.991032 − 0.133622i \(-0.957339\pi\)
0.611236 + 0.791448i \(0.290672\pi\)
\(90\) −670.187 −0.784932
\(91\) 87.0698 − 224.468i 0.100301 − 0.258578i
\(92\) 1065.55 1.20751
\(93\) −344.666 + 596.979i −0.384303 + 0.665633i
\(94\) −436.266 − 755.634i −0.478695 − 0.829125i
\(95\) −128.284 − 222.194i −0.138543 − 0.239964i
\(96\) 356.408 − 617.316i 0.378914 − 0.656298i
\(97\) −598.031 −0.625988 −0.312994 − 0.949755i \(-0.601332\pi\)
−0.312994 + 0.949755i \(0.601332\pi\)
\(98\) 1429.05 − 453.269i 1.47302 − 0.467215i
\(99\) −383.239 −0.389060
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.e.79.3 ✓ 26
7.2 even 3 1911.4.a.bc.1.11 13
7.4 even 3 inner 273.4.i.e.235.3 yes 26
7.5 odd 6 1911.4.a.bb.1.11 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.4.i.e.79.3 ✓ 26 1.1 even 1 trivial
273.4.i.e.235.3 yes 26 7.4 even 3 inner
1911.4.a.bb.1.11 13 7.5 odd 6
1911.4.a.bc.1.11 13 7.2 even 3