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Properties

Label 147.4.e.m.79.1
Level $147$
Weight $4$
Character 147.79
Analytic conductor $8.673$
Analytic rank $0$
Dimension $4$
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] = [147,4,Mod(67,147)] 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("147.67"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,3,6,-17,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.1
Root \(-1.63746 + 1.52274i\) of defining polynomial
Character \(\chi\) \(=\) 147.79
Dual form 147.4.e.m.67.1

$q$-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+(-1.13746 + 1.97014i) q^{2} +(1.50000 + 2.59808i) q^{3} +(1.41238 + 2.44631i) q^{4} +(-2.27492 + 3.94027i) q^{5} -6.82475 q^{6} -24.6254 q^{8} +(-4.50000 + 7.79423i) q^{9} +(-5.17525 - 8.96379i) q^{10} +(20.3746 + 35.2898i) q^{11} +(-4.23713 + 7.33892i) q^{12} -53.2990 q^{13} -13.6495 q^{15} +(16.7114 - 28.9450i) q^{16} +(2.27492 + 3.94027i) q^{17} +(-10.2371 - 17.7312i) q^{18} +(61.2990 - 106.173i) q^{19} -12.8522 q^{20} -92.7010 q^{22} +(-65.6736 + 113.750i) q^{23} +(-36.9381 - 63.9787i) q^{24} +(52.1495 + 90.3256i) q^{25} +(60.6254 - 105.006i) q^{26} -27.0000 q^{27} -216.598 q^{29} +(15.5257 - 26.8914i) q^{30} +(-125.897 - 218.060i) q^{31} +(-60.4846 - 104.762i) q^{32} +(-61.1238 + 105.869i) q^{33} -10.3505 q^{34} -25.4228 q^{36} +(-5.94851 + 10.3031i) q^{37} +(139.450 + 241.535i) q^{38} +(-79.9485 - 138.475i) q^{39} +(56.0208 - 97.0308i) q^{40} +111.752 q^{41} +369.196 q^{43} +(-57.5531 + 99.6850i) q^{44} +(-20.4743 - 35.4624i) q^{45} +(-149.402 - 258.772i) q^{46} +(-131.347 + 227.500i) q^{47} +100.268 q^{48} -237.272 q^{50} +(-6.82475 + 11.8208i) q^{51} +(-75.2782 - 130.386i) q^{52} +(283.550 + 491.123i) q^{53} +(30.7114 - 53.1937i) q^{54} -185.402 q^{55} +367.794 q^{57} +(246.371 - 426.728i) q^{58} +(419.945 + 727.366i) q^{59} +(-19.2782 - 33.3909i) q^{60} +(-242.897 + 420.710i) q^{61} +572.811 q^{62} +542.577 q^{64} +(121.251 - 210.013i) q^{65} +(-139.051 - 240.844i) q^{66} +(166.846 + 288.985i) q^{67} +(-6.42608 + 11.1303i) q^{68} -394.042 q^{69} +590.248 q^{71} +(110.814 - 191.936i) q^{72} +(245.350 + 424.960i) q^{73} +(-13.5324 - 23.4387i) q^{74} +(-156.449 + 270.977i) q^{75} +346.309 q^{76} +363.752 q^{78} +(-60.8455 + 105.388i) q^{79} +(76.0340 + 131.695i) q^{80} +(-40.5000 - 70.1481i) q^{81} +(-127.114 + 220.168i) q^{82} -609.608 q^{83} -20.7010 q^{85} +(-419.945 + 727.366i) q^{86} +(-324.897 - 562.738i) q^{87} +(-501.733 - 869.026i) q^{88} +(359.519 - 622.705i) q^{89} +93.1545 q^{90} -371.023 q^{92} +(377.691 - 654.180i) q^{93} +(-298.804 - 517.544i) q^{94} +(278.900 + 483.070i) q^{95} +(181.454 - 314.287i) q^{96} +637.877 q^{97} -366.743 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 6 q^{3} - 17 q^{4} + 6 q^{5} + 18 q^{6} - 174 q^{8} - 18 q^{9} - 66 q^{10} + 6 q^{11} + 51 q^{12} - 32 q^{13} + 36 q^{15} - 137 q^{16} - 6 q^{17} + 27 q^{18} + 64 q^{19} - 444 q^{20} - 552 q^{22}+ \cdots - 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.13746 + 1.97014i −0.402152 + 0.696548i −0.993985 − 0.109512i \(-0.965071\pi\)
0.591833 + 0.806061i \(0.298404\pi\)
\(3\) 1.50000 + 2.59808i 0.288675 + 0.500000i
\(4\) 1.41238 + 2.44631i 0.176547 + 0.305788i
\(5\) −2.27492 + 3.94027i −0.203475 + 0.352429i −0.949646 − 0.313326i \(-0.898557\pi\)
0.746171 + 0.665754i \(0.231890\pi\)
\(6\) −6.82475 −0.464366
\(7\) 0 0
\(8\) −24.6254 −1.08830
\(9\) −4.50000 + 7.79423i −0.166667 + 0.288675i
\(10\) −5.17525 − 8.96379i −0.163656 − 0.283460i
\(11\) 20.3746 + 35.2898i 0.558470 + 0.967298i 0.997624 + 0.0688867i \(0.0219447\pi\)
−0.439155 + 0.898412i \(0.644722\pi\)
\(12\) −4.23713 + 7.33892i −0.101929 + 0.176547i
\(13\) −53.2990 −1.13711 −0.568557 − 0.822644i \(-0.692498\pi\)
−0.568557 + 0.822644i \(0.692498\pi\)
\(14\) 0 0
\(15\) −13.6495 −0.234952
\(16\) 16.7114 − 28.9450i 0.261115 − 0.452265i
\(17\) 2.27492 + 3.94027i 0.0324558 + 0.0562151i 0.881797 − 0.471629i \(-0.156334\pi\)
−0.849341 + 0.527844i \(0.823001\pi\)
\(18\) −10.2371 − 17.7312i −0.134051 − 0.232183i
\(19\) 61.2990 − 106.173i 0.740156 − 1.28199i −0.212269 − 0.977211i \(-0.568085\pi\)
0.952424 − 0.304776i \(-0.0985815\pi\)
\(20\) −12.8522 −0.143691
\(21\) 0 0
\(22\) −92.7010 −0.898360
\(23\) −65.6736 + 113.750i −0.595387 + 1.03124i 0.398106 + 0.917340i \(0.369668\pi\)
−0.993492 + 0.113900i \(0.963666\pi\)
\(24\) −36.9381 − 63.9787i −0.314165 − 0.544150i
\(25\) 52.1495 + 90.3256i 0.417196 + 0.722605i
\(26\) 60.6254 − 105.006i 0.457293 − 0.792055i
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −216.598 −1.38694 −0.693470 − 0.720486i \(-0.743919\pi\)
−0.693470 + 0.720486i \(0.743919\pi\)
\(30\) 15.5257 − 26.8914i 0.0944867 − 0.163656i
\(31\) −125.897 − 218.060i −0.729412 − 1.26338i −0.957132 − 0.289652i \(-0.906460\pi\)
0.227720 − 0.973727i \(-0.426873\pi\)
\(32\) −60.4846 − 104.762i −0.334134 − 0.578736i
\(33\) −61.1238 + 105.869i −0.322433 + 0.558470i
\(34\) −10.3505 −0.0522087
\(35\) 0 0
\(36\) −25.4228 −0.117698
\(37\) −5.94851 + 10.3031i −0.0264305 + 0.0457790i −0.878938 − 0.476936i \(-0.841747\pi\)
0.852508 + 0.522715i \(0.175081\pi\)
\(38\) 139.450 + 241.535i 0.595311 + 1.03111i
\(39\) −79.9485 − 138.475i −0.328257 − 0.568557i
\(40\) 56.0208 − 97.0308i 0.221442 − 0.383548i
\(41\) 111.752 0.425678 0.212839 − 0.977087i \(-0.431729\pi\)
0.212839 + 0.977087i \(0.431729\pi\)
\(42\) 0 0
\(43\) 369.196 1.30935 0.654673 − 0.755912i \(-0.272806\pi\)
0.654673 + 0.755912i \(0.272806\pi\)
\(44\) −57.5531 + 99.6850i −0.197192 + 0.341547i
\(45\) −20.4743 − 35.4624i −0.0678249 − 0.117476i
\(46\) −149.402 − 258.772i −0.478872 − 0.829431i
\(47\) −131.347 + 227.500i −0.407637 + 0.706049i −0.994624 − 0.103548i \(-0.966981\pi\)
0.586987 + 0.809596i \(0.300314\pi\)
\(48\) 100.268 0.301510
\(49\) 0 0
\(50\) −237.272 −0.671105
\(51\) −6.82475 + 11.8208i −0.0187384 + 0.0324558i
\(52\) −75.2782 − 130.386i −0.200754 − 0.347716i
\(53\) 283.550 + 491.123i 0.734879 + 1.27285i 0.954777 + 0.297324i \(0.0960942\pi\)
−0.219898 + 0.975523i \(0.570572\pi\)
\(54\) 30.7114 − 53.1937i 0.0773943 − 0.134051i
\(55\) −185.402 −0.454538
\(56\) 0 0
\(57\) 367.794 0.854658
\(58\) 246.371 − 426.728i 0.557761 − 0.966070i
\(59\) 419.945 + 727.366i 0.926648 + 1.60500i 0.788890 + 0.614535i \(0.210656\pi\)
0.137758 + 0.990466i \(0.456010\pi\)
\(60\) −19.2782 − 33.3909i −0.0414801 − 0.0718457i
\(61\) −242.897 + 420.710i −0.509832 + 0.883056i 0.490103 + 0.871665i \(0.336959\pi\)
−0.999935 + 0.0113909i \(0.996374\pi\)
\(62\) 572.811 1.17334
\(63\) 0 0
\(64\) 542.577 1.05972
\(65\) 121.251 − 210.013i 0.231374 − 0.400752i
\(66\) −139.051 − 240.844i −0.259334 − 0.449180i
\(67\) 166.846 + 288.985i 0.304230 + 0.526942i 0.977090 − 0.212828i \(-0.0682675\pi\)
−0.672859 + 0.739770i \(0.734934\pi\)
\(68\) −6.42608 + 11.1303i −0.0114599 + 0.0198492i
\(69\) −394.042 −0.687493
\(70\) 0 0
\(71\) 590.248 0.986613 0.493306 − 0.869856i \(-0.335788\pi\)
0.493306 + 0.869856i \(0.335788\pi\)
\(72\) 110.814 − 191.936i 0.181383 − 0.314165i
\(73\) 245.350 + 424.960i 0.393371 + 0.681339i 0.992892 − 0.119020i \(-0.0379754\pi\)
−0.599521 + 0.800359i \(0.704642\pi\)
\(74\) −13.5324 − 23.4387i −0.0212582 − 0.0368203i
\(75\) −156.449 + 270.977i −0.240868 + 0.417196i
\(76\) 346.309 0.522689
\(77\) 0 0
\(78\) 363.752 0.528037
\(79\) −60.8455 + 105.388i −0.0866539 + 0.150089i −0.906095 − 0.423075i \(-0.860951\pi\)
0.819441 + 0.573164i \(0.194284\pi\)
\(80\) 76.0340 + 131.695i 0.106261 + 0.184049i
\(81\) −40.5000 − 70.1481i −0.0555556 − 0.0962250i
\(82\) −127.114 + 220.168i −0.171187 + 0.296505i
\(83\) −609.608 −0.806183 −0.403091 − 0.915160i \(-0.632064\pi\)
−0.403091 + 0.915160i \(0.632064\pi\)
\(84\) 0 0
\(85\) −20.7010 −0.0264157
\(86\) −419.945 + 727.366i −0.526556 + 0.912023i
\(87\) −324.897 − 562.738i −0.400375 − 0.693470i
\(88\) −501.733 − 869.026i −0.607783 − 1.05271i
\(89\) 359.519 − 622.705i 0.428190 − 0.741648i −0.568522 − 0.822668i \(-0.692485\pi\)
0.996712 + 0.0810204i \(0.0258179\pi\)
\(90\) 93.1545 0.109104
\(91\) 0 0
\(92\) −371.023 −0.420455
\(93\) 377.691 − 654.180i 0.421126 − 0.729412i
\(94\) −298.804 − 517.544i −0.327865 − 0.567878i
\(95\) 278.900 + 483.070i 0.301206 + 0.521704i
\(96\) 181.454 − 314.287i 0.192912 − 0.334134i
\(97\) 637.877 0.667697 0.333849 − 0.942627i \(-0.391653\pi\)
0.333849 + 0.942627i \(0.391653\pi\)
\(98\) 0 0
\(99\) −366.743 −0.372313
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 147.4.e.m.79.1 4
3.2 odd 2 441.4.e.p.226.2 4
7.2 even 3 147.4.a.i.1.2 2
7.3 odd 6 147.4.e.l.67.1 4
7.4 even 3 inner 147.4.e.m.67.1 4
7.5 odd 6 21.4.a.c.1.2 ✓ 2
7.6 odd 2 147.4.e.l.79.1 4
21.2 odd 6 441.4.a.r.1.1 2
21.5 even 6 63.4.a.e.1.1 2
21.11 odd 6 441.4.e.p.361.2 4
21.17 even 6 441.4.e.q.361.2 4
21.20 even 2 441.4.e.q.226.2 4
28.19 even 6 336.4.a.m.1.1 2
28.23 odd 6 2352.4.a.bz.1.2 2
35.12 even 12 525.4.d.g.274.3 4
35.19 odd 6 525.4.a.n.1.1 2
35.33 even 12 525.4.d.g.274.2 4
56.5 odd 6 1344.4.a.bg.1.2 2
56.19 even 6 1344.4.a.bo.1.2 2
84.47 odd 6 1008.4.a.ba.1.2 2
105.89 even 6 1575.4.a.p.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.c.1.2 ✓ 2 7.5 odd 6
63.4.a.e.1.1 2 21.5 even 6
147.4.a.i.1.2 2 7.2 even 3
147.4.e.l.67.1 4 7.3 odd 6
147.4.e.l.79.1 4 7.6 odd 2
147.4.e.m.67.1 4 7.4 even 3 inner
147.4.e.m.79.1 4 1.1 even 1 trivial
336.4.a.m.1.1 2 28.19 even 6
441.4.a.r.1.1 2 21.2 odd 6
441.4.e.p.226.2 4 3.2 odd 2
441.4.e.p.361.2 4 21.11 odd 6
441.4.e.q.226.2 4 21.20 even 2
441.4.e.q.361.2 4 21.17 even 6
525.4.a.n.1.1 2 35.19 odd 6
525.4.d.g.274.2 4 35.33 even 12
525.4.d.g.274.3 4 35.12 even 12
1008.4.a.ba.1.2 2 84.47 odd 6
1344.4.a.bg.1.2 2 56.5 odd 6
1344.4.a.bo.1.2 2 56.19 even 6
1575.4.a.p.1.2 2 105.89 even 6
2352.4.a.bz.1.2 2 28.23 odd 6