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Properties

Label 273.2.bd.b.127.8
Level $273$
Weight $2$
Character 273.127
Analytic conductor $2.180$
Analytic rank $0$
Dimension $16$
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] = [273,2,Mod(43,273)] 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("273.43"); S:= CuspForms(chi, 2); 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, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bd (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 10 x^{14} - 8 x^{13} - 3 x^{12} + 32 x^{11} - 5 x^{10} - 44 x^{9} + 214 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 127.8
Root \(1.31463 - 1.12772i\) of defining polynomial
Character \(\chi\) \(=\) 273.127
Dual form 273.2.bd.b.43.8

$q$-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+(2.35660 - 1.36058i) q^{2} +(0.500000 + 0.866025i) q^{3} +(2.70236 - 4.68063i) q^{4} +1.58278i q^{5} +(2.35660 + 1.36058i) q^{6} +(-0.866025 - 0.500000i) q^{7} -9.26480i q^{8} +(-0.500000 + 0.866025i) q^{9} +(2.15350 + 3.72996i) q^{10} +(-4.79969 + 2.77110i) q^{11} +5.40472 q^{12} +(-1.34211 + 3.34645i) q^{13} -2.72116 q^{14} +(-1.37073 + 0.791388i) q^{15} +(-7.20078 - 12.4721i) q^{16} +(2.40288 - 4.16192i) q^{17} +2.72116i q^{18} +(0.772957 + 0.446267i) q^{19} +(7.40839 + 4.27723i) q^{20} -1.00000i q^{21} +(-7.54062 + 13.0607i) q^{22} +(-2.31507 - 4.00983i) q^{23} +(8.02355 - 4.63240i) q^{24} +2.49482 q^{25} +(1.39030 + 9.71228i) q^{26} -1.00000 q^{27} +(-4.68063 + 2.70236i) q^{28} +(0.941087 + 1.63001i) q^{29} +(-2.15350 + 3.72996i) q^{30} -1.47545i q^{31} +(-17.8916 - 10.3297i) q^{32} +(-4.79969 - 2.77110i) q^{33} -13.0773i q^{34} +(0.791388 - 1.37073i) q^{35} +(2.70236 + 4.68063i) q^{36} +(4.32400 - 2.49646i) q^{37} +2.42873 q^{38} +(-3.56917 + 0.510922i) q^{39} +14.6641 q^{40} +(-8.45175 + 4.87962i) q^{41} +(-1.36058 - 2.35660i) q^{42} +(0.506349 - 0.877022i) q^{43} +29.9541i q^{44} +(-1.37073 - 0.791388i) q^{45} +(-10.9114 - 6.29969i) q^{46} +12.5703i q^{47} +(7.20078 - 12.4721i) q^{48} +(0.500000 + 0.866025i) q^{49} +(5.87927 - 3.39440i) q^{50} +4.80577 q^{51} +(12.0366 + 15.3252i) q^{52} +6.25784 q^{53} +(-2.35660 + 1.36058i) q^{54} +(-4.38604 - 7.59684i) q^{55} +(-4.63240 + 8.02355i) q^{56} +0.892533i q^{57} +(4.43552 + 2.56085i) q^{58} +(2.98511 + 1.72345i) q^{59} +8.55447i q^{60} +(1.79275 - 3.10513i) q^{61} +(-2.00746 - 3.47703i) q^{62} +(0.866025 - 0.500000i) q^{63} -27.4144 q^{64} +(-5.29669 - 2.12426i) q^{65} -15.0812 q^{66} +(12.7653 - 7.37003i) q^{67} +(-12.9869 - 22.4940i) q^{68} +(2.31507 - 4.00983i) q^{69} -4.30699i q^{70} +(-11.3794 - 6.56988i) q^{71} +(8.02355 + 4.63240i) q^{72} -5.33654i q^{73} +(6.79328 - 11.7663i) q^{74} +(1.24741 + 2.16058i) q^{75} +(4.17762 - 2.41195i) q^{76} +5.54221 q^{77} +(-7.71593 + 6.06018i) q^{78} +0.779028 q^{79} +(19.7406 - 11.3972i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-13.2782 + 22.9986i) q^{82} -1.47136i q^{83} +(-4.68063 - 2.70236i) q^{84} +(6.58738 + 3.80323i) q^{85} -2.75571i q^{86} +(-0.941087 + 1.63001i) q^{87} +(25.6737 + 44.4682i) q^{88} +(6.63332 - 3.82975i) q^{89} -4.30699 q^{90} +(2.83553 - 2.22706i) q^{91} -25.0247 q^{92} +(1.27777 - 0.737723i) q^{93} +(17.1029 + 29.6231i) q^{94} +(-0.706341 + 1.22342i) q^{95} -20.6594i q^{96} +(-0.657035 - 0.379340i) q^{97} +(2.35660 + 1.36058i) q^{98} -5.54221i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{3} + 14 q^{4} - 8 q^{9} - 4 q^{10} + 28 q^{12} - 12 q^{13} - 4 q^{14} - 12 q^{15} - 10 q^{16} - 2 q^{17} + 18 q^{20} - 18 q^{22} - 6 q^{23} - 20 q^{25} + 20 q^{26} - 16 q^{27} - 12 q^{29} + 4 q^{30}+ \cdots - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.35660 − 1.36058i 1.66636 − 0.962076i 0.696791 − 0.717274i \(-0.254610\pi\)
0.969573 − 0.244802i \(-0.0787230\pi\)
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 2.70236 − 4.68063i 1.35118 − 2.34031i
\(5\) 1.58278i 0.707839i 0.935276 + 0.353920i \(0.115151\pi\)
−0.935276 + 0.353920i \(0.884849\pi\)
\(6\) 2.35660 + 1.36058i 0.962076 + 0.555455i
\(7\) −0.866025 − 0.500000i −0.327327 − 0.188982i
\(8\) − 9.26480i − 3.27560i
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 2.15350 + 3.72996i 0.680995 + 1.17952i
\(11\) −4.79969 + 2.77110i −1.44716 + 0.835519i −0.998311 − 0.0580877i \(-0.981500\pi\)
−0.448850 + 0.893607i \(0.648166\pi\)
\(12\) 5.40472 1.56021
\(13\) −1.34211 + 3.34645i −0.372235 + 0.928139i
\(14\) −2.72116 −0.727261
\(15\) −1.37073 + 0.791388i −0.353920 + 0.204336i
\(16\) −7.20078 − 12.4721i −1.80020 − 3.11803i
\(17\) 2.40288 − 4.16192i 0.582785 − 1.00941i −0.412363 − 0.911020i \(-0.635296\pi\)
0.995148 − 0.0983932i \(-0.0313703\pi\)
\(18\) 2.72116i 0.641384i
\(19\) 0.772957 + 0.446267i 0.177328 + 0.102381i 0.586037 − 0.810284i \(-0.300687\pi\)
−0.408708 + 0.912665i \(0.634021\pi\)
\(20\) 7.40839 + 4.27723i 1.65657 + 0.956419i
\(21\) − 1.00000i − 0.218218i
\(22\) −7.54062 + 13.0607i −1.60767 + 2.78456i
\(23\) −2.31507 − 4.00983i −0.482726 − 0.836107i 0.517077 − 0.855939i \(-0.327020\pi\)
−0.999803 + 0.0198323i \(0.993687\pi\)
\(24\) 8.02355 − 4.63240i 1.63780 − 0.945584i
\(25\) 2.49482 0.498963
\(26\) 1.39030 + 9.71228i 0.272661 + 1.90474i
\(27\) −1.00000 −0.192450
\(28\) −4.68063 + 2.70236i −0.884555 + 0.510698i
\(29\) 0.941087 + 1.63001i 0.174755 + 0.302685i 0.940077 − 0.340963i \(-0.110753\pi\)
−0.765321 + 0.643649i \(0.777420\pi\)
\(30\) −2.15350 + 3.72996i −0.393173 + 0.680995i
\(31\) − 1.47545i − 0.264998i −0.991183 − 0.132499i \(-0.957700\pi\)
0.991183 − 0.132499i \(-0.0423001\pi\)
\(32\) −17.8916 − 10.3297i −3.16281 − 1.82605i
\(33\) −4.79969 − 2.77110i −0.835519 − 0.482387i
\(34\) − 13.0773i − 2.24273i
\(35\) 0.791388 − 1.37073i 0.133769 − 0.231695i
\(36\) 2.70236 + 4.68063i 0.450393 + 0.780104i
\(37\) 4.32400 − 2.49646i 0.710862 − 0.410416i −0.100518 − 0.994935i \(-0.532050\pi\)
0.811380 + 0.584519i \(0.198717\pi\)
\(38\) 2.42873 0.393992
\(39\) −3.56917 + 0.510922i −0.571524 + 0.0818130i
\(40\) 14.6641 2.31860
\(41\) −8.45175 + 4.87962i −1.31994 + 0.762069i −0.983719 − 0.179713i \(-0.942483\pi\)
−0.336223 + 0.941782i \(0.609150\pi\)
\(42\) −1.36058 − 2.35660i −0.209942 − 0.363631i
\(43\) 0.506349 − 0.877022i 0.0772174 − 0.133745i −0.824831 − 0.565379i \(-0.808730\pi\)
0.902048 + 0.431635i \(0.142063\pi\)
\(44\) 29.9541i 4.51575i
\(45\) −1.37073 − 0.791388i −0.204336 − 0.117973i
\(46\) −10.9114 − 6.29969i −1.60880 − 0.928839i
\(47\) 12.5703i 1.83357i 0.399385 + 0.916783i \(0.369224\pi\)
−0.399385 + 0.916783i \(0.630776\pi\)
\(48\) 7.20078 − 12.4721i 1.03934 − 1.80020i
\(49\) 0.500000 + 0.866025i 0.0714286 + 0.123718i
\(50\) 5.87927 − 3.39440i 0.831455 − 0.480041i
\(51\) 4.80577 0.672942
\(52\) 12.0366 + 15.3252i 1.66918 + 2.12523i
\(53\) 6.25784 0.859581 0.429790 − 0.902929i \(-0.358587\pi\)
0.429790 + 0.902929i \(0.358587\pi\)
\(54\) −2.35660 + 1.36058i −0.320692 + 0.185152i
\(55\) −4.38604 − 7.59684i −0.591413 − 1.02436i
\(56\) −4.63240 + 8.02355i −0.619030 + 1.07219i
\(57\) 0.892533i 0.118219i
\(58\) 4.43552 + 2.56085i 0.582413 + 0.336256i
\(59\) 2.98511 + 1.72345i 0.388628 + 0.224374i 0.681565 − 0.731757i \(-0.261300\pi\)
−0.292938 + 0.956132i \(0.594633\pi\)
\(60\) 8.55447i 1.10438i
\(61\) 1.79275 − 3.10513i 0.229538 − 0.397571i −0.728134 − 0.685435i \(-0.759612\pi\)
0.957671 + 0.287864i \(0.0929453\pi\)
\(62\) −2.00746 − 3.47703i −0.254948 − 0.441583i
\(63\) 0.866025 − 0.500000i 0.109109 − 0.0629941i
\(64\) −27.4144 −3.42681
\(65\) −5.29669 − 2.12426i −0.656973 − 0.263483i
\(66\) −15.0812 −1.85637
\(67\) 12.7653 − 7.37003i 1.55953 − 0.900392i 0.562223 − 0.826986i \(-0.309946\pi\)
0.997302 − 0.0734069i \(-0.0233872\pi\)
\(68\) −12.9869 − 22.4940i −1.57489 − 2.72780i
\(69\) 2.31507 − 4.00983i 0.278702 − 0.482726i
\(70\) − 4.30699i − 0.514784i
\(71\) −11.3794 − 6.56988i −1.35048 − 0.779702i −0.362166 − 0.932114i \(-0.617963\pi\)
−0.988317 + 0.152412i \(0.951296\pi\)
\(72\) 8.02355 + 4.63240i 0.945584 + 0.545933i
\(73\) − 5.33654i − 0.624595i −0.949984 − 0.312298i \(-0.898901\pi\)
0.949984 − 0.312298i \(-0.101099\pi\)
\(74\) 6.79328 − 11.7663i 0.789703 − 1.36781i
\(75\) 1.24741 + 2.16058i 0.144038 + 0.249482i
\(76\) 4.17762 − 2.41195i 0.479205 − 0.276669i
\(77\) 5.54221 0.631593
\(78\) −7.71593 + 6.06018i −0.873657 + 0.686180i
\(79\) 0.779028 0.0876475 0.0438237 − 0.999039i \(-0.486046\pi\)
0.0438237 + 0.999039i \(0.486046\pi\)
\(80\) 19.7406 − 11.3972i 2.20707 − 1.27425i
\(81\) −0.500000 − 0.866025i −0.0555556 − 0.0962250i
\(82\) −13.2782 + 22.9986i −1.46634 + 2.53977i
\(83\) − 1.47136i − 0.161503i −0.996734 − 0.0807513i \(-0.974268\pi\)
0.996734 − 0.0807513i \(-0.0257320\pi\)
\(84\) −4.68063 − 2.70236i −0.510698 − 0.294852i
\(85\) 6.58738 + 3.80323i 0.714502 + 0.412518i
\(86\) − 2.75571i − 0.297156i
\(87\) −0.941087 + 1.63001i −0.100895 + 0.174755i
\(88\) 25.6737 + 44.4682i 2.73683 + 4.74032i
\(89\) 6.63332 − 3.82975i 0.703130 − 0.405953i −0.105382 − 0.994432i \(-0.533606\pi\)
0.808512 + 0.588479i \(0.200273\pi\)
\(90\) −4.30699 −0.453997
\(91\) 2.83553 − 2.22706i 0.297244 − 0.233459i
\(92\) −25.0247 −2.60900
\(93\) 1.27777 − 0.737723i 0.132499 − 0.0764983i
\(94\) 17.1029 + 29.6231i 1.76403 + 3.05539i
\(95\) −0.706341 + 1.22342i −0.0724690 + 0.125520i
\(96\) − 20.6594i − 2.10854i
\(97\) −0.657035 − 0.379340i −0.0667118 − 0.0385161i 0.466273 − 0.884641i \(-0.345596\pi\)
−0.532985 + 0.846125i \(0.678930\pi\)
\(98\) 2.35660 + 1.36058i 0.238052 + 0.137439i
\(99\) − 5.54221i − 0.557013i
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 273.2.bd.b.127.8 yes 16
3.2 odd 2 819.2.ct.c.127.1 16
13.2 odd 12 3549.2.a.bc.1.8 8
13.4 even 6 inner 273.2.bd.b.43.8 ✓ 16
13.11 odd 12 3549.2.a.ba.1.1 8
39.17 odd 6 819.2.ct.c.316.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.bd.b.43.8 ✓ 16 13.4 even 6 inner
273.2.bd.b.127.8 yes 16 1.1 even 1 trivial
819.2.ct.c.127.1 16 3.2 odd 2
819.2.ct.c.316.1 16 39.17 odd 6
3549.2.a.ba.1.1 8 13.11 odd 12
3549.2.a.bc.1.8 8 13.2 odd 12