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Properties

Label 273.4.i.b.235.2
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 235.2
Root \(0.331419 + 0.574035i\) of defining polynomial
Character \(\chi\) \(=\) 273.235
Dual form 273.4.i.b.79.2

$q$-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+(0.331419 + 0.574035i) q^{2} +(-1.50000 + 2.59808i) q^{3} +(3.78032 - 6.54771i) q^{4} +(5.49426 + 9.51633i) q^{5} -1.98851 q^{6} +(15.8128 - 9.64127i) q^{7} +10.3142 q^{8} +(-4.50000 - 7.79423i) q^{9} +(-3.64180 + 6.30779i) q^{10} +(18.7445 - 32.4665i) q^{11} +(11.3410 + 19.6431i) q^{12} -13.0000 q^{13} +(10.7751 + 5.88182i) q^{14} -32.9655 q^{15} +(-26.8243 - 46.4610i) q^{16} +(3.25735 - 5.64190i) q^{17} +(2.98277 - 5.16631i) q^{18} +(-3.65187 - 6.32523i) q^{19} +83.0803 q^{20} +(1.32948 + 55.5449i) q^{21} +24.8492 q^{22} +(-9.17374 - 15.8894i) q^{23} +(-15.4713 + 26.7971i) q^{24} +(2.12627 - 3.68281i) q^{25} +(-4.30845 - 7.46245i) q^{26} +27.0000 q^{27} +(-3.35058 - 139.985i) q^{28} +283.439 q^{29} +(-10.9254 - 18.9234i) q^{30} +(-94.2406 + 163.229i) q^{31} +(59.0369 - 102.255i) q^{32} +(56.2336 + 97.3994i) q^{33} +4.31820 q^{34} +(178.629 + 97.5087i) q^{35} -68.0458 q^{36} +(-47.5704 - 82.3943i) q^{37} +(2.42060 - 4.19261i) q^{38} +(19.5000 - 33.7750i) q^{39} +(56.6688 + 98.1533i) q^{40} +262.907 q^{41} +(-31.4441 + 19.1718i) q^{42} +63.3129 q^{43} +(-141.721 - 245.467i) q^{44} +(49.4483 - 85.6470i) q^{45} +(6.08070 - 10.5321i) q^{46} +(143.894 + 249.232i) q^{47} +160.946 q^{48} +(157.092 - 304.912i) q^{49} +2.81875 q^{50} +(9.77206 + 16.9257i) q^{51} +(-49.1442 + 85.1202i) q^{52} +(52.4469 - 90.8406i) q^{53} +(8.94832 + 15.4989i) q^{54} +411.949 q^{55} +(163.097 - 99.4419i) q^{56} +21.9112 q^{57} +(93.9370 + 162.704i) q^{58} +(-134.722 + 233.346i) q^{59} +(-124.620 + 215.849i) q^{60} +(328.175 + 568.416i) q^{61} -124.933 q^{62} +(-146.304 - 79.8632i) q^{63} -350.924 q^{64} +(-71.4253 - 123.712i) q^{65} +(-37.2738 + 64.5600i) q^{66} +(-22.4325 + 38.8542i) q^{67} +(-24.6277 - 42.6564i) q^{68} +55.0424 q^{69} +(3.22781 + 134.856i) q^{70} -382.976 q^{71} +(-46.4139 - 80.3912i) q^{72} +(-22.6615 + 39.2509i) q^{73} +(31.5315 - 54.6141i) q^{74} +(6.37881 + 11.0484i) q^{75} -55.2210 q^{76} +(-16.6137 - 694.108i) q^{77} +25.8507 q^{78} +(293.193 + 507.825i) q^{79} +(294.759 - 510.537i) q^{80} +(-40.5000 + 70.1481i) q^{81} +(87.1325 + 150.918i) q^{82} -90.0422 q^{83} +(368.718 + 201.272i) q^{84} +71.5869 q^{85} +(20.9831 + 36.3438i) q^{86} +(-425.158 + 736.395i) q^{87} +(193.335 - 334.865i) q^{88} +(-481.365 - 833.748i) q^{89} +65.5525 q^{90} +(-205.567 + 125.336i) q^{91} -138.719 q^{92} +(-282.722 - 489.688i) q^{93} +(-95.3784 + 165.200i) q^{94} +(40.1287 - 69.5049i) q^{95} +(177.111 + 306.765i) q^{96} -1281.59 q^{97} +(227.093 - 10.8773i) q^{98} -337.401 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.331419 + 0.574035i 0.117174 + 0.202952i 0.918647 − 0.395080i \(-0.129283\pi\)
−0.801472 + 0.598032i \(0.795950\pi\)
\(3\) −1.50000 + 2.59808i −0.288675 + 0.500000i
\(4\) 3.78032 − 6.54771i 0.472540 − 0.818464i
\(5\) 5.49426 + 9.51633i 0.491421 + 0.851167i 0.999951 − 0.00987764i \(-0.00314420\pi\)
−0.508530 + 0.861044i \(0.669811\pi\)
\(6\) −1.98851 −0.135301
\(7\) 15.8128 − 9.64127i 0.853813 − 0.520579i
\(8\) 10.3142 0.455827
\(9\) −4.50000 − 7.79423i −0.166667 − 0.288675i
\(10\) −3.64180 + 6.30779i −0.115164 + 0.199470i
\(11\) 18.7445 − 32.4665i 0.513790 − 0.889910i −0.486083 − 0.873913i \(-0.661575\pi\)
0.999872 − 0.0159966i \(-0.00509211\pi\)
\(12\) 11.3410 + 19.6431i 0.272821 + 0.472540i
\(13\) −13.0000 −0.277350
\(14\) 10.7751 + 5.88182i 0.205698 + 0.112284i
\(15\) −32.9655 −0.567444
\(16\) −26.8243 − 46.4610i −0.419129 − 0.725953i
\(17\) 3.25735 − 5.64190i 0.0464720 − 0.0804919i −0.841854 − 0.539706i \(-0.818536\pi\)
0.888326 + 0.459214i \(0.151869\pi\)
\(18\) 2.98277 − 5.16631i 0.0390581 − 0.0676506i
\(19\) −3.65187 − 6.32523i −0.0440946 − 0.0763741i 0.843136 − 0.537701i \(-0.180707\pi\)
−0.887230 + 0.461327i \(0.847374\pi\)
\(20\) 83.0803 0.928866
\(21\) 1.32948 + 55.5449i 0.0138151 + 0.577185i
\(22\) 24.8492 0.240812
\(23\) −9.17374 − 15.8894i −0.0831677 − 0.144051i 0.821441 − 0.570293i \(-0.193170\pi\)
−0.904609 + 0.426242i \(0.859837\pi\)
\(24\) −15.4713 + 26.7971i −0.131586 + 0.227914i
\(25\) 2.12627 − 3.68281i 0.0170102 − 0.0294625i
\(26\) −4.30845 − 7.46245i −0.0324983 − 0.0562887i
\(27\) 27.0000 0.192450
\(28\) −3.35058 − 139.985i −0.0226143 − 0.944810i
\(29\) 283.439 1.81494 0.907469 − 0.420118i \(-0.138011\pi\)
0.907469 + 0.420118i \(0.138011\pi\)
\(30\) −10.9254 − 18.9234i −0.0664899 − 0.115164i
\(31\) −94.2406 + 163.229i −0.546003 + 0.945706i 0.452540 + 0.891744i \(0.350518\pi\)
−0.998543 + 0.0539614i \(0.982815\pi\)
\(32\) 59.0369 − 102.255i 0.326136 − 0.564884i
\(33\) 56.2336 + 97.3994i 0.296637 + 0.513790i
\(34\) 4.31820 0.0217813
\(35\) 178.629 + 97.5087i 0.862682 + 0.470914i
\(36\) −68.0458 −0.315027
\(37\) −47.5704 − 82.3943i −0.211365 − 0.366096i 0.740777 − 0.671751i \(-0.234458\pi\)
−0.952142 + 0.305656i \(0.901124\pi\)
\(38\) 2.42060 − 4.19261i 0.0103335 − 0.0178982i
\(39\) 19.5000 − 33.7750i 0.0800641 − 0.138675i
\(40\) 56.6688 + 98.1533i 0.224003 + 0.387985i
\(41\) 262.907 1.00144 0.500722 − 0.865608i \(-0.333068\pi\)
0.500722 + 0.865608i \(0.333068\pi\)
\(42\) −31.4441 + 19.1718i −0.115522 + 0.0704351i
\(43\) 63.3129 0.224538 0.112269 − 0.993678i \(-0.464188\pi\)
0.112269 + 0.993678i \(0.464188\pi\)
\(44\) −141.721 − 245.467i −0.485573 − 0.841036i
\(45\) 49.4483 − 85.6470i 0.163807 − 0.283722i
\(46\) 6.08070 − 10.5321i 0.0194902 − 0.0337581i
\(47\) 143.894 + 249.232i 0.446576 + 0.773493i 0.998161 − 0.0606263i \(-0.0193098\pi\)
−0.551584 + 0.834119i \(0.685976\pi\)
\(48\) 160.946 0.483969
\(49\) 157.092 − 304.912i 0.457994 − 0.888955i
\(50\) 2.81875 0.00797262
\(51\) 9.77206 + 16.9257i 0.0268306 + 0.0464720i
\(52\) −49.1442 + 85.1202i −0.131059 + 0.227001i
\(53\) 52.4469 − 90.8406i 0.135927 − 0.235432i −0.790024 − 0.613076i \(-0.789932\pi\)
0.925951 + 0.377643i \(0.123265\pi\)
\(54\) 8.94832 + 15.4989i 0.0225502 + 0.0390581i
\(55\) 411.949 1.00995
\(56\) 163.097 − 99.4419i 0.389191 − 0.237294i
\(57\) 21.9112 0.0509161
\(58\) 93.9370 + 162.704i 0.212664 + 0.368345i
\(59\) −134.722 + 233.346i −0.297277 + 0.514899i −0.975512 − 0.219946i \(-0.929412\pi\)
0.678235 + 0.734845i \(0.262745\pi\)
\(60\) −124.620 + 215.849i −0.268140 + 0.464433i
\(61\) 328.175 + 568.416i 0.688828 + 1.19309i 0.972217 + 0.234080i \(0.0752078\pi\)
−0.283389 + 0.959005i \(0.591459\pi\)
\(62\) −124.933 −0.255910
\(63\) −146.304 − 79.8632i −0.292581 − 0.159711i
\(64\) −350.924 −0.685399
\(65\) −71.4253 − 123.712i −0.136296 − 0.236071i
\(66\) −37.2738 + 64.5600i −0.0695164 + 0.120406i
\(67\) −22.4325 + 38.8542i −0.0409039 + 0.0708477i −0.885753 − 0.464158i \(-0.846357\pi\)
0.844849 + 0.535005i \(0.179690\pi\)
\(68\) −24.6277 − 42.6564i −0.0439198 − 0.0760713i
\(69\) 55.0424 0.0960337
\(70\) 3.22781 + 134.856i 0.00551139 + 0.230262i
\(71\) −382.976 −0.640153 −0.320076 − 0.947392i \(-0.603709\pi\)
−0.320076 + 0.947392i \(0.603709\pi\)
\(72\) −46.4139 − 80.3912i −0.0759712 − 0.131586i
\(73\) −22.6615 + 39.2509i −0.0363333 + 0.0629311i −0.883620 − 0.468204i \(-0.844901\pi\)
0.847287 + 0.531135i \(0.178234\pi\)
\(74\) 31.5315 − 54.6141i 0.0495332 − 0.0857940i
\(75\) 6.37881 + 11.0484i 0.00982082 + 0.0170102i
\(76\) −55.2210 −0.0833459
\(77\) −16.6137 − 694.108i −0.0245884 − 1.02728i
\(78\) 25.8507 0.0375258
\(79\) 293.193 + 507.825i 0.417554 + 0.723226i 0.995693 − 0.0927132i \(-0.0295540\pi\)
−0.578138 + 0.815939i \(0.696221\pi\)
\(80\) 294.759 − 510.537i 0.411938 − 0.713497i
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) 87.1325 + 150.918i 0.117344 + 0.203245i
\(83\) −90.0422 −0.119077 −0.0595387 − 0.998226i \(-0.518963\pi\)
−0.0595387 + 0.998226i \(0.518963\pi\)
\(84\) 368.718 + 201.272i 0.478933 + 0.261436i
\(85\) 71.5869 0.0913493
\(86\) 20.9831 + 36.3438i 0.0263101 + 0.0455704i
\(87\) −425.158 + 736.395i −0.523928 + 0.907469i
\(88\) 193.335 − 334.865i 0.234199 − 0.405645i
\(89\) −481.365 − 833.748i −0.573310 − 0.993001i −0.996223 − 0.0868312i \(-0.972326\pi\)
0.422914 − 0.906170i \(-0.361007\pi\)
\(90\) 65.5525 0.0767760
\(91\) −205.567 + 125.336i −0.236805 + 0.144383i
\(92\) −138.719 −0.157200
\(93\) −282.722 − 489.688i −0.315235 − 0.546003i
\(94\) −95.3784 + 165.200i −0.104655 + 0.181267i
\(95\) 40.1287 − 69.5049i 0.0433381 − 0.0750637i
\(96\) 177.111 + 306.765i 0.188295 + 0.326136i
\(97\) −1281.59 −1.34150 −0.670749 − 0.741685i \(-0.734027\pi\)
−0.670749 + 0.741685i \(0.734027\pi\)
\(98\) 227.093 − 10.8773i 0.234080 − 0.0112120i
\(99\) −337.401 −0.342526
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.235.2 yes 6
7.2 even 3 inner 273.4.i.b.79.2 ✓ 6
7.3 odd 6 1911.4.a.i.1.2 3
7.4 even 3 1911.4.a.j.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.4.i.b.79.2 ✓ 6 7.2 even 3 inner
273.4.i.b.235.2 yes 6 1.1 even 1 trivial
1911.4.a.i.1.2 3 7.3 odd 6
1911.4.a.j.1.2 3 7.4 even 3