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Properties

Label 273.4.i.b.235.1
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 235.1
Root \(-1.79899 - 3.11595i\) of defining polynomial
Character \(\chi\) \(=\) 273.235
Dual form 273.4.i.b.79.1

$q$-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+(-1.79899 - 3.11595i) q^{2} +(-1.50000 + 2.59808i) q^{3} +(-2.47277 + 4.28296i) q^{4} +(-0.896985 - 1.55362i) q^{5} +10.7940 q^{6} +(18.4959 + 0.950110i) q^{7} -10.9899 q^{8} +(-4.50000 - 7.79423i) q^{9} +(-3.22734 + 5.58992i) q^{10} +(-1.86885 + 3.23695i) q^{11} +(-7.41830 - 12.8489i) q^{12} -13.0000 q^{13} +(-30.3135 - 59.3415i) q^{14} +5.38191 q^{15} +(39.5530 + 68.5078i) q^{16} +(-28.5607 + 49.4686i) q^{17} +(-16.1910 + 28.0436i) q^{18} +(41.8357 + 72.4615i) q^{19} +8.87213 q^{20} +(-30.2123 + 46.6285i) q^{21} +13.4482 q^{22} +(35.7031 + 61.8396i) q^{23} +(16.4849 - 28.5527i) q^{24} +(60.8908 - 105.466i) q^{25} +(23.3869 + 40.5074i) q^{26} +27.0000 q^{27} +(-49.8052 + 76.8676i) q^{28} +66.4566 q^{29} +(-9.68203 - 16.7698i) q^{30} +(67.0382 - 116.114i) q^{31} +(98.3514 - 170.350i) q^{32} +(-5.60656 - 9.71084i) q^{33} +205.522 q^{34} +(-15.1144 - 29.5878i) q^{35} +44.5098 q^{36} +(112.349 + 194.594i) q^{37} +(150.524 - 260.716i) q^{38} +(19.5000 - 33.7750i) q^{39} +(9.85782 + 17.0742i) q^{40} -232.526 q^{41} +(199.644 + 10.2555i) q^{42} +524.151 q^{43} +(-9.24247 - 16.0084i) q^{44} +(-8.07286 + 13.9826i) q^{45} +(128.459 - 222.498i) q^{46} +(76.8259 + 133.066i) q^{47} -237.318 q^{48} +(341.195 + 35.1462i) q^{49} -438.169 q^{50} +(-85.6821 - 148.406i) q^{51} +(32.1460 - 55.6784i) q^{52} +(231.540 - 401.039i) q^{53} +(-48.5729 - 84.1307i) q^{54} +6.70533 q^{55} +(-203.269 - 10.4417i) q^{56} -251.014 q^{57} +(-119.555 - 207.075i) q^{58} +(213.873 - 370.438i) q^{59} +(-13.3082 + 23.0505i) q^{60} +(-129.406 - 224.137i) q^{61} -482.406 q^{62} +(-75.8261 - 148.437i) q^{63} -74.8872 q^{64} +(11.6608 + 20.1971i) q^{65} +(-20.1723 + 34.9395i) q^{66} +(65.8454 - 114.048i) q^{67} +(-141.248 - 244.648i) q^{68} -214.219 q^{69} +(-65.0035 + 100.324i) q^{70} +551.628 q^{71} +(49.4548 + 85.6582i) q^{72} +(-111.826 + 193.689i) q^{73} +(404.230 - 700.147i) q^{74} +(182.673 + 316.398i) q^{75} -413.799 q^{76} +(-37.6415 + 58.0945i) q^{77} -140.322 q^{78} +(235.556 + 407.994i) q^{79} +(70.9568 - 122.901i) q^{80} +(-40.5000 + 70.1481i) q^{81} +(418.314 + 724.541i) q^{82} -360.104 q^{83} +(-125.000 - 244.699i) q^{84} +102.474 q^{85} +(-942.945 - 1633.23i) q^{86} +(-99.6849 + 172.659i) q^{87} +(20.5386 - 35.5739i) q^{88} +(354.244 + 613.569i) q^{89} +58.0922 q^{90} +(-240.446 - 12.3514i) q^{91} -353.142 q^{92} +(201.115 + 348.341i) q^{93} +(276.419 - 478.772i) q^{94} +(75.0520 - 129.994i) q^{95} +(295.054 + 511.049i) q^{96} +562.493 q^{97} +(-504.293 - 1126.37i) q^{98} +33.6393 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.79899 − 3.11595i −0.636041 − 1.10165i −0.986294 − 0.165000i \(-0.947238\pi\)
0.350253 − 0.936655i \(-0.386096\pi\)
\(3\) −1.50000 + 2.59808i −0.288675 + 0.500000i
\(4\) −2.47277 + 4.28296i −0.309096 + 0.535369i
\(5\) −0.896985 − 1.55362i −0.0802288 − 0.138960i 0.823119 − 0.567868i \(-0.192232\pi\)
−0.903348 + 0.428908i \(0.858898\pi\)
\(6\) 10.7940 0.734437
\(7\) 18.4959 + 0.950110i 0.998683 + 0.0513011i
\(8\) −10.9899 −0.485692
\(9\) −4.50000 − 7.79423i −0.166667 − 0.288675i
\(10\) −3.22734 + 5.58992i −0.102058 + 0.176769i
\(11\) −1.86885 + 3.23695i −0.0512255 + 0.0887251i −0.890501 − 0.454981i \(-0.849646\pi\)
0.839276 + 0.543706i \(0.182979\pi\)
\(12\) −7.41830 − 12.8489i −0.178456 − 0.309096i
\(13\) −13.0000 −0.277350
\(14\) −30.3135 − 59.3415i −0.578687 − 1.13283i
\(15\) 5.38191 0.0926402
\(16\) 39.5530 + 68.5078i 0.618015 + 1.07043i
\(17\) −28.5607 + 49.4686i −0.407470 + 0.705759i −0.994605 − 0.103730i \(-0.966922\pi\)
0.587136 + 0.809489i \(0.300256\pi\)
\(18\) −16.1910 + 28.0436i −0.212014 + 0.367218i
\(19\) 41.8357 + 72.4615i 0.505146 + 0.874938i 0.999982 + 0.00595187i \(0.00189455\pi\)
−0.494837 + 0.868986i \(0.664772\pi\)
\(20\) 8.87213 0.0991934
\(21\) −30.2123 + 46.6285i −0.313946 + 0.484532i
\(22\) 13.4482 0.130326
\(23\) 35.7031 + 61.8396i 0.323679 + 0.560628i 0.981244 − 0.192769i \(-0.0617469\pi\)
−0.657565 + 0.753398i \(0.728414\pi\)
\(24\) 16.4849 − 28.5527i 0.140207 − 0.242846i
\(25\) 60.8908 − 105.466i 0.487127 − 0.843728i
\(26\) 23.3869 + 40.5074i 0.176406 + 0.305544i
\(27\) 27.0000 0.192450
\(28\) −49.8052 + 76.8676i −0.336154 + 0.518808i
\(29\) 66.4566 0.425541 0.212770 − 0.977102i \(-0.431751\pi\)
0.212770 + 0.977102i \(0.431751\pi\)
\(30\) −9.68203 − 16.7698i −0.0589229 − 0.102058i
\(31\) 67.0382 − 116.114i 0.388401 − 0.672730i −0.603834 − 0.797110i \(-0.706361\pi\)
0.992235 + 0.124380i \(0.0396943\pi\)
\(32\) 98.3514 − 170.350i 0.543320 − 0.941058i
\(33\) −5.60656 − 9.71084i −0.0295750 − 0.0512255i
\(34\) 205.522 1.03667
\(35\) −15.1144 − 29.5878i −0.0729943 − 0.142893i
\(36\) 44.5098 0.206064
\(37\) 112.349 + 194.594i 0.499190 + 0.864623i 1.00000 0.000934894i \(-0.000297586\pi\)
−0.500809 + 0.865558i \(0.666964\pi\)
\(38\) 150.524 − 260.716i 0.642586 − 1.11299i
\(39\) 19.5000 − 33.7750i 0.0800641 − 0.138675i
\(40\) 9.85782 + 17.0742i 0.0389664 + 0.0674919i
\(41\) −232.526 −0.885720 −0.442860 − 0.896591i \(-0.646036\pi\)
−0.442860 + 0.896591i \(0.646036\pi\)
\(42\) 199.644 + 10.2555i 0.733470 + 0.0376774i
\(43\) 524.151 1.85889 0.929445 − 0.368960i \(-0.120286\pi\)
0.929445 + 0.368960i \(0.120286\pi\)
\(44\) −9.24247 − 16.0084i −0.0316671 − 0.0548491i
\(45\) −8.07286 + 13.9826i −0.0267429 + 0.0463201i
\(46\) 128.459 − 222.498i 0.411746 − 0.713165i
\(47\) 76.8259 + 133.066i 0.238430 + 0.412973i 0.960264 − 0.279093i \(-0.0900339\pi\)
−0.721834 + 0.692066i \(0.756701\pi\)
\(48\) −237.318 −0.713623
\(49\) 341.195 + 35.1462i 0.994736 + 0.102467i
\(50\) −438.169 −1.23933
\(51\) −85.6821 − 148.406i −0.235253 − 0.407470i
\(52\) 32.1460 − 55.6784i 0.0857277 − 0.148485i
\(53\) 231.540 − 401.039i 0.600084 − 1.03938i −0.392724 − 0.919656i \(-0.628467\pi\)
0.992808 − 0.119719i \(-0.0381994\pi\)
\(54\) −48.5729 − 84.1307i −0.122406 − 0.212014i
\(55\) 6.70533 0.0164390
\(56\) −203.269 − 10.4417i −0.485052 − 0.0249165i
\(57\) −251.014 −0.583292
\(58\) −119.555 − 207.075i −0.270661 − 0.468799i
\(59\) 213.873 − 370.438i 0.471930 − 0.817406i −0.527554 − 0.849521i \(-0.676891\pi\)
0.999484 + 0.0321148i \(0.0102242\pi\)
\(60\) −13.3082 + 23.0505i −0.0286347 + 0.0495967i
\(61\) −129.406 − 224.137i −0.271618 − 0.470456i 0.697658 − 0.716431i \(-0.254225\pi\)
−0.969276 + 0.245975i \(0.920892\pi\)
\(62\) −482.406 −0.988155
\(63\) −75.8261 − 148.437i −0.151638 − 0.296845i
\(64\) −74.8872 −0.146264
\(65\) 11.6608 + 20.1971i 0.0222515 + 0.0385406i
\(66\) −20.1723 + 34.9395i −0.0376219 + 0.0651630i
\(67\) 65.8454 − 114.048i 0.120064 − 0.207957i −0.799729 − 0.600362i \(-0.795023\pi\)
0.919793 + 0.392404i \(0.128357\pi\)
\(68\) −141.248 − 244.648i −0.251894 − 0.436294i
\(69\) −214.219 −0.373752
\(70\) −65.0035 + 100.324i −0.110992 + 0.171300i
\(71\) 551.628 0.922059 0.461030 − 0.887385i \(-0.347480\pi\)
0.461030 + 0.887385i \(0.347480\pi\)
\(72\) 49.4548 + 85.6582i 0.0809486 + 0.140207i
\(73\) −111.826 + 193.689i −0.179292 + 0.310542i −0.941638 − 0.336627i \(-0.890714\pi\)
0.762346 + 0.647169i \(0.224047\pi\)
\(74\) 404.230 − 700.147i 0.635011 − 1.09987i
\(75\) 182.673 + 316.398i 0.281243 + 0.487127i
\(76\) −413.799 −0.624553
\(77\) −37.6415 + 58.0945i −0.0557097 + 0.0859804i
\(78\) −140.322 −0.203696
\(79\) 235.556 + 407.994i 0.335469 + 0.581050i 0.983575 − 0.180500i \(-0.0577717\pi\)
−0.648105 + 0.761551i \(0.724438\pi\)
\(80\) 70.9568 − 122.901i 0.0991652 − 0.171759i
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) 418.314 + 724.541i 0.563354 + 0.975758i
\(83\) −360.104 −0.476223 −0.238112 − 0.971238i \(-0.576528\pi\)
−0.238112 + 0.971238i \(0.576528\pi\)
\(84\) −125.000 − 244.699i −0.162365 − 0.317844i
\(85\) 102.474 0.130763
\(86\) −942.945 − 1633.23i −1.18233 − 2.04786i
\(87\) −99.6849 + 172.659i −0.122843 + 0.212770i
\(88\) 20.5386 − 35.5739i 0.0248798 − 0.0430931i
\(89\) 354.244 + 613.569i 0.421908 + 0.730767i 0.996126 − 0.0879360i \(-0.0280271\pi\)
−0.574218 + 0.818703i \(0.694694\pi\)
\(90\) 58.0922 0.0680383
\(91\) −240.446 − 12.3514i −0.276985 − 0.0142284i
\(92\) −353.142 −0.400191
\(93\) 201.115 + 348.341i 0.224243 + 0.388401i
\(94\) 276.419 − 478.772i 0.303302 − 0.525335i
\(95\) 75.0520 − 129.994i 0.0810544 − 0.140390i
\(96\) 295.054 + 511.049i 0.313686 + 0.543320i
\(97\) 562.493 0.588789 0.294395 − 0.955684i \(-0.404882\pi\)
0.294395 + 0.955684i \(0.404882\pi\)
\(98\) −504.293 − 1126.37i −0.519810 − 1.16103i
\(99\) 33.6393 0.0341503
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.235.1 yes 6
7.2 even 3 inner 273.4.i.b.79.1 ✓ 6
7.3 odd 6 1911.4.a.i.1.3 3
7.4 even 3 1911.4.a.j.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.4.i.b.79.1 ✓ 6 7.2 even 3 inner
273.4.i.b.235.1 yes 6 1.1 even 1 trivial
1911.4.a.i.1.3 3 7.3 odd 6
1911.4.a.j.1.3 3 7.4 even 3