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Properties

Label 273.4.i.e.79.2
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 79.2
Character \(\chi\) \(=\) 273.79
Dual form 273.4.i.e.235.2

$q$-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+(-2.42742 + 4.20442i) q^{2} +(-1.50000 - 2.59808i) q^{3} +(-7.78477 - 13.4836i) q^{4} +(-2.12186 + 3.67518i) q^{5} +14.5645 q^{6} +(-14.6768 + 11.2957i) q^{7} +36.7489 q^{8} +(-4.50000 + 7.79423i) q^{9} +(-10.3013 - 17.8424i) q^{10} +(-18.2620 - 31.6307i) q^{11} +(-23.3543 + 40.4508i) q^{12} +13.0000 q^{13} +(-11.8652 - 89.1267i) q^{14} +12.7312 q^{15} +(-26.9271 + 46.6391i) q^{16} +(-53.6488 - 92.9224i) q^{17} +(-21.8468 - 37.8398i) q^{18} +(-13.3306 + 23.0893i) q^{19} +66.0729 q^{20} +(51.3622 + 21.1878i) q^{21} +177.319 q^{22} +(18.0405 - 31.2470i) q^{23} +(-55.1234 - 95.4765i) q^{24} +(53.4954 + 92.6567i) q^{25} +(-31.5565 + 54.6575i) q^{26} +27.0000 q^{27} +(266.562 + 109.961i) q^{28} +18.8748 q^{29} +(-30.9040 + 53.5272i) q^{30} +(86.5487 + 149.907i) q^{31} +(16.2689 + 28.1786i) q^{32} +(-54.7860 + 94.8922i) q^{33} +520.913 q^{34} +(-10.3716 - 77.9076i) q^{35} +140.126 q^{36} +(-35.1008 + 60.7964i) q^{37} +(-64.7182 - 112.095i) q^{38} +(-19.5000 - 33.7750i) q^{39} +(-77.9762 + 135.059i) q^{40} +307.899 q^{41} +(-213.760 + 164.517i) q^{42} +180.460 q^{43} +(-284.331 + 492.476i) q^{44} +(-19.0968 - 33.0766i) q^{45} +(87.5837 + 151.699i) q^{46} +(-119.244 + 206.536i) q^{47} +161.562 q^{48} +(87.8144 - 331.568i) q^{49} -519.424 q^{50} +(-160.946 + 278.767i) q^{51} +(-101.202 - 175.287i) q^{52} +(81.0710 + 140.419i) q^{53} +(-65.5404 + 113.519i) q^{54} +154.998 q^{55} +(-539.355 + 415.105i) q^{56} +79.9838 q^{57} +(-45.8171 + 79.3576i) q^{58} +(-118.923 - 205.981i) q^{59} +(-99.1093 - 171.662i) q^{60} +(79.9054 - 138.400i) q^{61} -840.361 q^{62} +(-21.9958 - 165.225i) q^{63} -588.799 q^{64} +(-27.5842 + 47.7773i) q^{65} +(-265.978 - 460.687i) q^{66} +(155.525 + 269.377i) q^{67} +(-835.287 + 1446.76i) q^{68} -108.243 q^{69} +(352.733 + 145.508i) q^{70} -381.477 q^{71} +(-165.370 + 286.430i) q^{72} +(436.758 + 756.487i) q^{73} +(-170.409 - 295.157i) q^{74} +(160.486 - 277.970i) q^{75} +415.104 q^{76} +(625.318 + 257.954i) q^{77} +189.339 q^{78} +(254.897 - 441.495i) q^{79} +(-114.271 - 197.923i) q^{80} +(-40.5000 - 70.1481i) q^{81} +(-747.401 + 1294.54i) q^{82} +1141.74 q^{83} +(-114.155 - 857.490i) q^{84} +455.342 q^{85} +(-438.052 + 758.728i) q^{86} +(-28.3122 - 49.0381i) q^{87} +(-671.110 - 1162.40i) q^{88} +(440.809 - 763.503i) q^{89} +185.424 q^{90} +(-190.798 + 146.844i) q^{91} -561.763 q^{92} +(259.646 - 449.720i) q^{93} +(-578.910 - 1002.70i) q^{94} +(-56.5716 - 97.9849i) q^{95} +(48.8068 - 84.5358i) q^{96} +1752.27 q^{97} +(1180.89 + 1174.07i) q^{98} +328.716 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.42742 + 4.20442i −0.858224 + 1.48649i 0.0153980 + 0.999881i \(0.495098\pi\)
−0.873622 + 0.486606i \(0.838235\pi\)
\(3\) −1.50000 − 2.59808i −0.288675 − 0.500000i
\(4\) −7.78477 − 13.4836i −0.973096 − 1.68545i
\(5\) −2.12186 + 3.67518i −0.189785 + 0.328718i −0.945179 − 0.326554i \(-0.894113\pi\)
0.755393 + 0.655272i \(0.227446\pi\)
\(6\) 14.5645 0.990991
\(7\) −14.6768 + 11.2957i −0.792470 + 0.609910i
\(8\) 36.7489 1.62409
\(9\) −4.50000 + 7.79423i −0.166667 + 0.288675i
\(10\) −10.3013 − 17.8424i −0.325756 − 0.564227i
\(11\) −18.2620 − 31.6307i −0.500564 − 0.867002i −1.00000 0.000651462i \(-0.999793\pi\)
0.499436 − 0.866351i \(-0.333541\pi\)
\(12\) −23.3543 + 40.4508i −0.561817 + 0.973096i
\(13\) 13.0000 0.277350
\(14\) −11.8652 − 89.1267i −0.226507 − 1.70144i
\(15\) 12.7312 0.219145
\(16\) −26.9271 + 46.6391i −0.420735 + 0.728735i
\(17\) −53.6488 − 92.9224i −0.765396 − 1.32571i −0.940037 − 0.341073i \(-0.889210\pi\)
0.174640 − 0.984632i \(-0.444124\pi\)
\(18\) −21.8468 − 37.8398i −0.286075 − 0.495496i
\(19\) −13.3306 + 23.0893i −0.160961 + 0.278793i −0.935214 − 0.354084i \(-0.884793\pi\)
0.774253 + 0.632877i \(0.218126\pi\)
\(20\) 66.0729 0.738717
\(21\) 51.3622 + 21.1878i 0.533722 + 0.220169i
\(22\) 177.319 1.71838
\(23\) 18.0405 − 31.2470i 0.163552 − 0.283280i −0.772588 − 0.634908i \(-0.781038\pi\)
0.936140 + 0.351627i \(0.114372\pi\)
\(24\) −55.1234 − 95.4765i −0.468834 − 0.812044i
\(25\) 53.4954 + 92.6567i 0.427963 + 0.741254i
\(26\) −31.5565 + 54.6575i −0.238028 + 0.412277i
\(27\) 27.0000 0.192450
\(28\) 266.562 + 109.961i 1.79912 + 0.742169i
\(29\) 18.8748 0.120861 0.0604304 − 0.998172i \(-0.480753\pi\)
0.0604304 + 0.998172i \(0.480753\pi\)
\(30\) −30.9040 + 53.5272i −0.188076 + 0.325756i
\(31\) 86.5487 + 149.907i 0.501439 + 0.868518i 0.999999 + 0.00166238i \(0.000529153\pi\)
−0.498560 + 0.866855i \(0.666138\pi\)
\(32\) 16.2689 + 28.1786i 0.0898740 + 0.155666i
\(33\) −54.7860 + 94.8922i −0.289001 + 0.500564i
\(34\) 520.913 2.62753
\(35\) −10.3716 − 77.9076i −0.0500891 − 0.376251i
\(36\) 140.126 0.648731
\(37\) −35.1008 + 60.7964i −0.155961 + 0.270132i −0.933408 − 0.358816i \(-0.883181\pi\)
0.777448 + 0.628947i \(0.216514\pi\)
\(38\) −64.7182 − 112.095i −0.276281 − 0.478533i
\(39\) −19.5000 − 33.7750i −0.0800641 − 0.138675i
\(40\) −77.9762 + 135.059i −0.308228 + 0.533867i
\(41\) 307.899 1.17282 0.586411 − 0.810013i \(-0.300540\pi\)
0.586411 + 0.810013i \(0.300540\pi\)
\(42\) −213.760 + 164.517i −0.785331 + 0.604416i
\(43\) 180.460 0.639996 0.319998 − 0.947418i \(-0.396318\pi\)
0.319998 + 0.947418i \(0.396318\pi\)
\(44\) −284.331 + 492.476i −0.974194 + 1.68735i
\(45\) −19.0968 − 33.0766i −0.0632618 − 0.109573i
\(46\) 87.5837 + 151.699i 0.280728 + 0.486236i
\(47\) −119.244 + 206.536i −0.370074 + 0.640988i −0.989577 − 0.144007i \(-0.954001\pi\)
0.619502 + 0.784995i \(0.287334\pi\)
\(48\) 161.562 0.485823
\(49\) 87.8144 − 331.568i 0.256019 − 0.966672i
\(50\) −519.424 −1.46915
\(51\) −160.946 + 278.767i −0.441902 + 0.765396i
\(52\) −101.202 − 175.287i −0.269888 − 0.467460i
\(53\) 81.0710 + 140.419i 0.210112 + 0.363925i 0.951750 − 0.306876i \(-0.0992837\pi\)
−0.741637 + 0.670801i \(0.765950\pi\)
\(54\) −65.5404 + 113.519i −0.165165 + 0.286075i
\(55\) 154.998 0.379999
\(56\) −539.355 + 415.105i −1.28704 + 0.990548i
\(57\) 79.9838 0.185862
\(58\) −45.8171 + 79.3576i −0.103726 + 0.179658i
\(59\) −118.923 − 205.981i −0.262415 − 0.454517i 0.704468 − 0.709736i \(-0.251186\pi\)
−0.966883 + 0.255219i \(0.917852\pi\)
\(60\) −99.1093 − 171.662i −0.213249 − 0.369359i
\(61\) 79.9054 − 138.400i 0.167719 − 0.290497i −0.769899 − 0.638166i \(-0.779693\pi\)
0.937617 + 0.347669i \(0.113027\pi\)
\(62\) −840.361 −1.72139
\(63\) −21.9958 − 165.225i −0.0439875 − 0.330418i
\(64\) −588.799 −1.15000
\(65\) −27.5842 + 47.7773i −0.0526370 + 0.0911699i
\(66\) −265.978 − 460.687i −0.496055 − 0.859192i
\(67\) 155.525 + 269.377i 0.283588 + 0.491188i 0.972266 − 0.233879i \(-0.0751420\pi\)
−0.688678 + 0.725067i \(0.741809\pi\)
\(68\) −835.287 + 1446.76i −1.48961 + 2.58008i
\(69\) −108.243 −0.188854
\(70\) 352.733 + 145.508i 0.602280 + 0.248451i
\(71\) −381.477 −0.637648 −0.318824 − 0.947814i \(-0.603288\pi\)
−0.318824 + 0.947814i \(0.603288\pi\)
\(72\) −165.370 + 286.430i −0.270681 + 0.468834i
\(73\) 436.758 + 756.487i 0.700256 + 1.21288i 0.968377 + 0.249493i \(0.0802640\pi\)
−0.268121 + 0.963385i \(0.586403\pi\)
\(74\) −170.409 − 295.157i −0.267698 − 0.463667i
\(75\) 160.486 − 277.970i 0.247085 − 0.427963i
\(76\) 415.104 0.626522
\(77\) 625.318 + 257.954i 0.925476 + 0.381775i
\(78\) 189.339 0.274852
\(79\) 254.897 − 441.495i 0.363015 − 0.628760i −0.625441 − 0.780272i \(-0.715081\pi\)
0.988456 + 0.151511i \(0.0484140\pi\)
\(80\) −114.271 − 197.923i −0.159699 − 0.276606i
\(81\) −40.5000 − 70.1481i −0.0555556 − 0.0962250i
\(82\) −747.401 + 1294.54i −1.00654 + 1.74339i
\(83\) 1141.74 1.50991 0.754953 − 0.655779i \(-0.227660\pi\)
0.754953 + 0.655779i \(0.227660\pi\)
\(84\) −114.155 − 857.490i −0.148278 − 1.11381i
\(85\) 455.342 0.581044
\(86\) −438.052 + 758.728i −0.549260 + 0.951346i
\(87\) −28.3122 − 49.0381i −0.0348895 − 0.0604304i
\(88\) −671.110 − 1162.40i −0.812960 − 1.40809i
\(89\) 440.809 − 763.503i 0.525007 − 0.909339i −0.474569 − 0.880219i \(-0.657396\pi\)
0.999576 − 0.0291208i \(-0.00927076\pi\)
\(90\) 185.424 0.217171
\(91\) −190.798 + 146.844i −0.219792 + 0.169159i
\(92\) −561.763 −0.636607
\(93\) 259.646 − 449.720i 0.289506 − 0.501439i
\(94\) −578.910 − 1002.70i −0.635213 − 1.10022i
\(95\) −56.5716 − 97.9849i −0.0610960 − 0.105821i
\(96\) 48.8068 − 84.5358i 0.0518888 − 0.0898740i
\(97\) 1752.27 1.83419 0.917093 − 0.398672i \(-0.130529\pi\)
0.917093 + 0.398672i \(0.130529\pi\)
\(98\) 1180.89 + 1174.07i 1.21722 + 1.21019i
\(99\) 328.716 0.333709
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.79.2 ✓ 26
7.2 even 3 1911.4.a.bc.1.12 13
7.4 even 3 inner 273.4.i.e.235.2 yes 26
7.5 odd 6 1911.4.a.bb.1.12 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.4.i.e.79.2 ✓ 26 1.1 even 1 trivial
273.4.i.e.235.2 yes 26 7.4 even 3 inner
1911.4.a.bb.1.12 13 7.5 odd 6
1911.4.a.bc.1.12 13 7.2 even 3