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Properties

Label 273.4.k.d.22.8
Level $273$
Weight $4$
Character 273.22
Analytic conductor $16.108$
Analytic rank $0$
Dimension $20$
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(22,273)] 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("273.22"); 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, 0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 273.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,5] 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: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 5 x^{19} + 81 x^{18} - 194 x^{17} + 3136 x^{16} - 5035 x^{15} + 81800 x^{14} - 49122 x^{13} + \cdots + 43877376 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 22.8
Root \(-1.25980 - 2.18205i\) of defining polynomial
Character \(\chi\) \(=\) 273.22
Dual form 273.4.k.d.211.8

$q$-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+(1.75980 - 3.04807i) q^{2} +(-1.50000 + 2.59808i) q^{3} +(-2.19383 - 3.79982i) q^{4} -10.3397 q^{5} +(5.27941 + 9.14421i) q^{6} +(3.50000 + 6.06218i) q^{7} +12.7141 q^{8} +(-4.50000 - 7.79423i) q^{9} +(-18.1959 + 31.5162i) q^{10} +(9.94415 - 17.2238i) q^{11} +13.1630 q^{12} +(-14.0592 - 44.7140i) q^{13} +24.6373 q^{14} +(15.5096 - 26.8634i) q^{15} +(39.9249 - 69.1519i) q^{16} +(-24.2906 - 42.0726i) q^{17} -31.6765 q^{18} +(-52.1007 - 90.2411i) q^{19} +(22.6835 + 39.2890i) q^{20} -21.0000 q^{21} +(-34.9995 - 60.6209i) q^{22} +(-18.2766 + 31.6560i) q^{23} +(-19.0711 + 33.0321i) q^{24} -18.0904 q^{25} +(-161.033 - 35.8343i) q^{26} +27.0000 q^{27} +(15.3568 - 26.5987i) q^{28} +(20.2020 - 34.9908i) q^{29} +(-54.5876 - 94.5485i) q^{30} -66.3288 q^{31} +(-89.6637 - 155.302i) q^{32} +(29.8324 + 51.6713i) q^{33} -170.987 q^{34} +(-36.1890 - 62.6812i) q^{35} +(-19.7444 + 34.1984i) q^{36} +(145.390 - 251.823i) q^{37} -366.748 q^{38} +(137.259 + 30.5440i) q^{39} -131.460 q^{40} +(79.3858 - 137.500i) q^{41} +(-36.9559 + 64.0095i) q^{42} +(-60.2130 - 104.292i) q^{43} -87.2629 q^{44} +(46.5287 + 80.5901i) q^{45} +(64.3265 + 111.417i) q^{46} -291.585 q^{47} +(119.775 + 207.456i) q^{48} +(-24.5000 + 42.4352i) q^{49} +(-31.8356 + 55.1408i) q^{50} +145.744 q^{51} +(-139.061 + 151.517i) q^{52} +467.908 q^{53} +(47.5147 - 82.2979i) q^{54} +(-102.820 + 178.089i) q^{55} +(44.4992 + 77.0749i) q^{56} +312.604 q^{57} +(-71.1030 - 123.154i) q^{58} +(238.888 + 413.766i) q^{59} -136.101 q^{60} +(-387.778 - 671.650i) q^{61} +(-116.726 + 202.175i) q^{62} +(31.5000 - 54.5596i) q^{63} +7.63542 q^{64} +(145.368 + 462.329i) q^{65} +209.997 q^{66} +(-20.7293 + 35.9042i) q^{67} +(-106.579 + 184.600i) q^{68} +(-54.8298 - 94.9680i) q^{69} -254.742 q^{70} +(355.220 + 615.260i) q^{71} +(-57.2133 - 99.0963i) q^{72} -84.8236 q^{73} +(-511.716 - 886.317i) q^{74} +(27.1356 - 47.0002i) q^{75} +(-228.600 + 395.947i) q^{76} +139.218 q^{77} +(334.649 - 364.624i) q^{78} -511.447 q^{79} +(-412.812 + 715.011i) q^{80} +(-40.5000 + 70.1481i) q^{81} +(-279.407 - 483.947i) q^{82} +256.721 q^{83} +(46.0703 + 79.7962i) q^{84} +(251.158 + 435.018i) q^{85} -423.852 q^{86} +(60.6059 + 104.972i) q^{87} +(126.430 - 218.984i) q^{88} +(-132.448 + 229.407i) q^{89} +327.526 q^{90} +(221.857 - 241.728i) q^{91} +160.383 q^{92} +(99.4932 - 172.327i) q^{93} +(-513.132 + 888.771i) q^{94} +(538.707 + 933.067i) q^{95} +537.982 q^{96} +(-165.754 - 287.095i) q^{97} +(86.2304 + 149.355i) q^{98} -178.995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 5 q^{2} - 30 q^{3} - 57 q^{4} + 30 q^{5} + 15 q^{6} + 70 q^{7} - 144 q^{8} - 90 q^{9} + 27 q^{10} + 40 q^{11} + 342 q^{12} + 166 q^{13} + 70 q^{14} - 45 q^{15} - 165 q^{16} - 46 q^{17} - 90 q^{18}+ \cdots - 720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.75980 − 3.04807i 0.622185 − 1.07766i −0.366893 − 0.930263i \(-0.619579\pi\)
0.989078 − 0.147393i \(-0.0470881\pi\)
\(3\) −1.50000 + 2.59808i −0.288675 + 0.500000i
\(4\) −2.19383 − 3.79982i −0.274228 − 0.474977i
\(5\) −10.3397 −0.924812 −0.462406 − 0.886668i \(-0.653014\pi\)
−0.462406 + 0.886668i \(0.653014\pi\)
\(6\) 5.27941 + 9.14421i 0.359219 + 0.622185i
\(7\) 3.50000 + 6.06218i 0.188982 + 0.327327i
\(8\) 12.7141 0.561887
\(9\) −4.50000 − 7.79423i −0.166667 − 0.288675i
\(10\) −18.1959 + 31.5162i −0.575404 + 0.996629i
\(11\) 9.94415 − 17.2238i 0.272570 − 0.472106i −0.696949 − 0.717121i \(-0.745460\pi\)
0.969519 + 0.245015i \(0.0787930\pi\)
\(12\) 13.1630 0.316651
\(13\) −14.0592 − 44.7140i −0.299948 − 0.953955i
\(14\) 24.6373 0.470328
\(15\) 15.5096 − 26.8634i 0.266970 − 0.462406i
\(16\) 39.9249 − 69.1519i 0.623826 − 1.08050i
\(17\) −24.2906 − 42.0726i −0.346549 − 0.600241i 0.639085 − 0.769136i \(-0.279313\pi\)
−0.985634 + 0.168896i \(0.945980\pi\)
\(18\) −31.6765 −0.414790
\(19\) −52.1007 − 90.2411i −0.629091 − 1.08962i −0.987734 − 0.156143i \(-0.950094\pi\)
0.358643 − 0.933475i \(-0.383239\pi\)
\(20\) 22.6835 + 39.2890i 0.253609 + 0.439264i
\(21\) −21.0000 −0.218218
\(22\) −34.9995 − 60.6209i −0.339178 − 0.587474i
\(23\) −18.2766 + 31.6560i −0.165693 + 0.286988i −0.936901 − 0.349595i \(-0.886319\pi\)
0.771208 + 0.636583i \(0.219653\pi\)
\(24\) −19.0711 + 33.0321i −0.162203 + 0.280944i
\(25\) −18.0904 −0.144723
\(26\) −161.033 − 35.8343i −1.21466 − 0.270295i
\(27\) 27.0000 0.192450
\(28\) 15.3568 − 26.5987i 0.103649 − 0.179524i
\(29\) 20.2020 − 34.9908i 0.129359 − 0.224056i −0.794069 − 0.607827i \(-0.792041\pi\)
0.923428 + 0.383771i \(0.125375\pi\)
\(30\) −54.5876 − 94.5485i −0.332210 − 0.575404i
\(31\) −66.3288 −0.384290 −0.192145 − 0.981367i \(-0.561544\pi\)
−0.192145 + 0.981367i \(0.561544\pi\)
\(32\) −89.6637 − 155.302i −0.495327 − 0.857931i
\(33\) 29.8324 + 51.6713i 0.157369 + 0.272570i
\(34\) −170.987 −0.862471
\(35\) −36.1890 − 62.6812i −0.174773 − 0.302716i
\(36\) −19.7444 + 34.1984i −0.0914094 + 0.158326i
\(37\) 145.390 − 251.823i 0.645999 − 1.11890i −0.338071 − 0.941121i \(-0.609774\pi\)
0.984070 − 0.177782i \(-0.0568922\pi\)
\(38\) −366.748 −1.56564
\(39\) 137.259 + 30.5440i 0.563565 + 0.125409i
\(40\) −131.460 −0.519640
\(41\) 79.3858 − 137.500i 0.302390 − 0.523754i −0.674287 − 0.738469i \(-0.735549\pi\)
0.976677 + 0.214715i \(0.0688823\pi\)
\(42\) −36.9559 + 64.0095i −0.135772 + 0.235164i
\(43\) −60.2130 − 104.292i −0.213544 − 0.369869i 0.739277 − 0.673401i \(-0.235167\pi\)
−0.952821 + 0.303532i \(0.901834\pi\)
\(44\) −87.2629 −0.298986
\(45\) 46.5287 + 80.5901i 0.154135 + 0.266970i
\(46\) 64.3265 + 111.417i 0.206183 + 0.357120i
\(47\) −291.585 −0.904936 −0.452468 − 0.891781i \(-0.649456\pi\)
−0.452468 + 0.891781i \(0.649456\pi\)
\(48\) 119.775 + 207.456i 0.360166 + 0.623826i
\(49\) −24.5000 + 42.4352i −0.0714286 + 0.123718i
\(50\) −31.8356 + 55.1408i −0.0900445 + 0.155962i
\(51\) 145.744 0.400160
\(52\) −139.061 + 151.517i −0.370853 + 0.404070i
\(53\) 467.908 1.21268 0.606340 − 0.795205i \(-0.292637\pi\)
0.606340 + 0.795205i \(0.292637\pi\)
\(54\) 47.5147 − 82.2979i 0.119740 − 0.207395i
\(55\) −102.820 + 178.089i −0.252076 + 0.436609i
\(56\) 44.4992 + 77.0749i 0.106187 + 0.183921i
\(57\) 312.604 0.726412
\(58\) −71.1030 − 123.154i −0.160970 − 0.278809i
\(59\) 238.888 + 413.766i 0.527128 + 0.913012i 0.999500 + 0.0316132i \(0.0100645\pi\)
−0.472372 + 0.881399i \(0.656602\pi\)
\(60\) −136.101 −0.292843
\(61\) −387.778 − 671.650i −0.813932 − 1.40977i −0.910093 − 0.414405i \(-0.863990\pi\)
0.0961609 − 0.995366i \(-0.469344\pi\)
\(62\) −116.726 + 202.175i −0.239100 + 0.414133i
\(63\) 31.5000 − 54.5596i 0.0629941 − 0.109109i
\(64\) 7.63542 0.0149129
\(65\) 145.368 + 462.329i 0.277396 + 0.882229i
\(66\) 209.997 0.391649
\(67\) −20.7293 + 35.9042i −0.0377983 + 0.0654686i −0.884306 − 0.466908i \(-0.845368\pi\)
0.846507 + 0.532377i \(0.178701\pi\)
\(68\) −106.579 + 184.600i −0.190067 + 0.329206i
\(69\) −54.8298 − 94.9680i −0.0956628 − 0.165693i
\(70\) −254.742 −0.434965
\(71\) 355.220 + 615.260i 0.593759 + 1.02842i 0.993721 + 0.111889i \(0.0356901\pi\)
−0.399962 + 0.916532i \(0.630977\pi\)
\(72\) −57.2133 − 99.0963i −0.0936479 − 0.162203i
\(73\) −84.8236 −0.135998 −0.0679990 − 0.997685i \(-0.521661\pi\)
−0.0679990 + 0.997685i \(0.521661\pi\)
\(74\) −511.716 − 886.317i −0.803861 − 1.39233i
\(75\) 27.1356 − 47.0002i 0.0417780 − 0.0723616i
\(76\) −228.600 + 395.947i −0.345029 + 0.597608i
\(77\) 139.218 0.206044
\(78\) 334.649 − 364.624i 0.485790 − 0.529302i
\(79\) −511.447 −0.728384 −0.364192 − 0.931324i \(-0.618655\pi\)
−0.364192 + 0.931324i \(0.618655\pi\)
\(80\) −412.812 + 715.011i −0.576922 + 0.999258i
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) −279.407 − 483.947i −0.376284 − 0.651744i
\(83\) 256.721 0.339503 0.169751 − 0.985487i \(-0.445704\pi\)
0.169751 + 0.985487i \(0.445704\pi\)
\(84\) 46.0703 + 79.7962i 0.0598415 + 0.103649i
\(85\) 251.158 + 435.018i 0.320493 + 0.555110i
\(86\) −423.852 −0.531455
\(87\) 60.6059 + 104.972i 0.0746854 + 0.129359i
\(88\) 126.430 − 218.984i 0.153154 − 0.265270i
\(89\) −132.448 + 229.407i −0.157747 + 0.273226i −0.934056 − 0.357127i \(-0.883756\pi\)
0.776309 + 0.630353i \(0.217090\pi\)
\(90\) 327.526 0.383603
\(91\) 221.857 − 241.728i 0.255570 − 0.278462i
\(92\) 160.383 0.181751
\(93\) 99.4932 − 172.327i 0.110935 − 0.192145i
\(94\) −513.132 + 888.771i −0.563038 + 0.975210i
\(95\) 538.707 + 933.067i 0.581791 + 1.00769i
\(96\) 537.982 0.571954
\(97\) −165.754 − 287.095i −0.173503 − 0.300516i 0.766139 − 0.642675i \(-0.222175\pi\)
−0.939642 + 0.342158i \(0.888842\pi\)
\(98\) 86.2304 + 149.355i 0.0888836 + 0.153951i
\(99\) −178.995 −0.181713
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.k.d.22.8 ✓ 20
13.3 even 3 inner 273.4.k.d.211.8 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.4.k.d.22.8 ✓ 20 1.1 even 1 trivial
273.4.k.d.211.8 yes 20 13.3 even 3 inner