↑

Properties

Label 147.4.e.b.67.1
Level $147$
Weight $4$
Character 147.67
Analytic conductor $8.673$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

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 = [2,-4,-3,-8,-4] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 67.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 147.67
Dual form 147.4.e.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)
 
\(f(q)\) \(=\) \(q+(-2.00000 - 3.46410i) q^{2} +(-1.50000 + 2.59808i) q^{3} +(-4.00000 + 6.92820i) q^{4} +(-2.00000 - 3.46410i) q^{5} +12.0000 q^{6} +(-4.50000 - 7.79423i) q^{9} +(-8.00000 + 13.8564i) q^{10} +(-31.0000 + 53.6936i) q^{11} +(-12.0000 - 20.7846i) q^{12} +62.0000 q^{13} +12.0000 q^{15} +(32.0000 + 55.4256i) q^{16} +(42.0000 - 72.7461i) q^{17} +(-18.0000 + 31.1769i) q^{18} +(50.0000 + 86.6025i) q^{19} +32.0000 q^{20} +248.000 q^{22} +(21.0000 + 36.3731i) q^{23} +(54.5000 - 94.3968i) q^{25} +(-124.000 - 214.774i) q^{26} +27.0000 q^{27} -10.0000 q^{29} +(-24.0000 - 41.5692i) q^{30} +(-24.0000 + 41.5692i) q^{31} +(128.000 - 221.703i) q^{32} +(-93.0000 - 161.081i) q^{33} -336.000 q^{34} +72.0000 q^{36} +(123.000 + 213.042i) q^{37} +(200.000 - 346.410i) q^{38} +(-93.0000 + 161.081i) q^{39} +248.000 q^{41} +68.0000 q^{43} +(-248.000 - 429.549i) q^{44} +(-18.0000 + 31.1769i) q^{45} +(84.0000 - 145.492i) q^{46} +(162.000 + 280.592i) q^{47} -192.000 q^{48} -436.000 q^{50} +(126.000 + 218.238i) q^{51} +(-248.000 + 429.549i) q^{52} +(-129.000 + 223.435i) q^{53} +(-54.0000 - 93.5307i) q^{54} +248.000 q^{55} -300.000 q^{57} +(20.0000 + 34.6410i) q^{58} +(60.0000 - 103.923i) q^{59} +(-48.0000 + 83.1384i) q^{60} +(311.000 + 538.668i) q^{61} +192.000 q^{62} -512.000 q^{64} +(-124.000 - 214.774i) q^{65} +(-372.000 + 644.323i) q^{66} +(-452.000 + 782.887i) q^{67} +(336.000 + 581.969i) q^{68} -126.000 q^{69} -678.000 q^{71} +(-321.000 + 555.988i) q^{73} +(492.000 - 852.169i) q^{74} +(163.500 + 283.190i) q^{75} -800.000 q^{76} +744.000 q^{78} +(-370.000 - 640.859i) q^{79} +(128.000 - 221.703i) q^{80} +(-40.5000 + 70.1481i) q^{81} +(-496.000 - 859.097i) q^{82} -468.000 q^{83} -336.000 q^{85} +(-136.000 - 235.559i) q^{86} +(15.0000 - 25.9808i) q^{87} +(100.000 + 173.205i) q^{89} +144.000 q^{90} -336.000 q^{92} +(-72.0000 - 124.708i) q^{93} +(648.000 - 1122.37i) q^{94} +(200.000 - 346.410i) q^{95} +(384.000 + 665.108i) q^{96} +1266.00 q^{97} +558.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 3 q^{3} - 8 q^{4} - 4 q^{5} + 24 q^{6} - 9 q^{9} - 16 q^{10} - 62 q^{11} - 24 q^{12} + 124 q^{13} + 24 q^{15} + 64 q^{16} + 84 q^{17} - 36 q^{18} + 100 q^{19} + 64 q^{20} + 496 q^{22} + 42 q^{23}+ \cdots + 1116 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 − 3.46410i −0.707107 − 1.22474i −0.965926 − 0.258819i \(-0.916667\pi\)
0.258819 − 0.965926i \(-0.416667\pi\)
\(3\) −1.50000 + 2.59808i −0.288675 + 0.500000i
\(4\) −4.00000 + 6.92820i −0.500000 + 0.866025i
\(5\) −2.00000 − 3.46410i −0.178885 − 0.309839i 0.762614 − 0.646854i \(-0.223916\pi\)
−0.941499 + 0.337016i \(0.890582\pi\)
\(6\) 12.0000 0.816497
\(7\) 0 0
\(8\) 0 0
\(9\) −4.50000 − 7.79423i −0.166667 − 0.288675i
\(10\) −8.00000 + 13.8564i −0.252982 + 0.438178i
\(11\) −31.0000 + 53.6936i −0.849714 + 1.47175i 0.0317500 + 0.999496i \(0.489892\pi\)
−0.881464 + 0.472252i \(0.843441\pi\)
\(12\) −12.0000 − 20.7846i −0.288675 − 0.500000i
\(13\) 62.0000 1.32275 0.661373 − 0.750057i \(-0.269974\pi\)
0.661373 + 0.750057i \(0.269974\pi\)
\(14\) 0 0
\(15\) 12.0000 0.206559
\(16\) 32.0000 + 55.4256i 0.500000 + 0.866025i
\(17\) 42.0000 − 72.7461i 0.599206 − 1.03785i −0.393733 − 0.919225i \(-0.628817\pi\)
0.992939 − 0.118630i \(-0.0378502\pi\)
\(18\) −18.0000 + 31.1769i −0.235702 + 0.408248i
\(19\) 50.0000 + 86.6025i 0.603726 + 1.04568i 0.992251 + 0.124246i \(0.0396511\pi\)
−0.388526 + 0.921438i \(0.627016\pi\)
\(20\) 32.0000 0.357771
\(21\) 0 0
\(22\) 248.000 2.40335
\(23\) 21.0000 + 36.3731i 0.190383 + 0.329753i 0.945377 − 0.325979i \(-0.105694\pi\)
−0.754994 + 0.655731i \(0.772360\pi\)
\(24\) 0 0
\(25\) 54.5000 − 94.3968i 0.436000 − 0.755174i
\(26\) −124.000 − 214.774i −0.935323 − 1.62003i
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −10.0000 −0.0640329 −0.0320164 − 0.999487i \(-0.510193\pi\)
−0.0320164 + 0.999487i \(0.510193\pi\)
\(30\) −24.0000 − 41.5692i −0.146059 − 0.252982i
\(31\) −24.0000 + 41.5692i −0.139049 + 0.240840i −0.927137 − 0.374723i \(-0.877738\pi\)
0.788088 + 0.615563i \(0.211071\pi\)
\(32\) 128.000 − 221.703i 0.707107 − 1.22474i
\(33\) −93.0000 − 161.081i −0.490582 − 0.849714i
\(34\) −336.000 −1.69481
\(35\) 0 0
\(36\) 72.0000 0.333333
\(37\) 123.000 + 213.042i 0.546516 + 0.946593i 0.998510 + 0.0545719i \(0.0173794\pi\)
−0.451994 + 0.892021i \(0.649287\pi\)
\(38\) 200.000 − 346.410i 0.853797 − 1.47882i
\(39\) −93.0000 + 161.081i −0.381844 + 0.661373i
\(40\) 0 0
\(41\) 248.000 0.944661 0.472330 − 0.881422i \(-0.343413\pi\)
0.472330 + 0.881422i \(0.343413\pi\)
\(42\) 0 0
\(43\) 68.0000 0.241161 0.120580 − 0.992704i \(-0.461524\pi\)
0.120580 + 0.992704i \(0.461524\pi\)
\(44\) −248.000 − 429.549i −0.849714 − 1.47175i
\(45\) −18.0000 + 31.1769i −0.0596285 + 0.103280i
\(46\) 84.0000 − 145.492i 0.269242 − 0.466341i
\(47\) 162.000 + 280.592i 0.502769 + 0.870821i 0.999995 + 0.00319997i \(0.00101858\pi\)
−0.497226 + 0.867621i \(0.665648\pi\)
\(48\) −192.000 −0.577350
\(49\) 0 0
\(50\) −436.000 −1.23319
\(51\) 126.000 + 218.238i 0.345952 + 0.599206i
\(52\) −248.000 + 429.549i −0.661373 + 1.14553i
\(53\) −129.000 + 223.435i −0.334330 + 0.579077i −0.983356 − 0.181689i \(-0.941843\pi\)
0.649026 + 0.760767i \(0.275177\pi\)
\(54\) −54.0000 − 93.5307i −0.136083 − 0.235702i
\(55\) 248.000 0.608006
\(56\) 0 0
\(57\) −300.000 −0.697122
\(58\) 20.0000 + 34.6410i 0.0452781 + 0.0784239i
\(59\) 60.0000 − 103.923i 0.132396 − 0.229316i −0.792204 − 0.610256i \(-0.791066\pi\)
0.924600 + 0.380941i \(0.124400\pi\)
\(60\) −48.0000 + 83.1384i −0.103280 + 0.178885i
\(61\) 311.000 + 538.668i 0.652778 + 1.13064i 0.982446 + 0.186548i \(0.0597300\pi\)
−0.329668 + 0.944097i \(0.606937\pi\)
\(62\) 192.000 0.393291
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) −124.000 − 214.774i −0.236620 − 0.409838i
\(66\) −372.000 + 644.323i −0.693788 + 1.20168i
\(67\) −452.000 + 782.887i −0.824188 + 1.42754i 0.0783505 + 0.996926i \(0.475035\pi\)
−0.902538 + 0.430609i \(0.858299\pi\)
\(68\) 336.000 + 581.969i 0.599206 + 1.03785i
\(69\) −126.000 −0.219835
\(70\) 0 0
\(71\) −678.000 −1.13329 −0.566646 − 0.823961i \(-0.691759\pi\)
−0.566646 + 0.823961i \(0.691759\pi\)
\(72\) 0 0
\(73\) −321.000 + 555.988i −0.514660 + 0.891418i 0.485195 + 0.874406i \(0.338749\pi\)
−0.999855 + 0.0170119i \(0.994585\pi\)
\(74\) 492.000 − 852.169i 0.772890 − 1.33868i
\(75\) 163.500 + 283.190i 0.251725 + 0.436000i
\(76\) −800.000 −1.20745
\(77\) 0 0
\(78\) 744.000 1.08002
\(79\) −370.000 − 640.859i −0.526940 − 0.912687i −0.999507 − 0.0313921i \(-0.990006\pi\)
0.472567 − 0.881295i \(-0.343327\pi\)
\(80\) 128.000 − 221.703i 0.178885 − 0.309839i
\(81\) −40.5000 + 70.1481i −0.0555556 + 0.0962250i
\(82\) −496.000 − 859.097i −0.667976 − 1.15697i
\(83\) −468.000 −0.618912 −0.309456 − 0.950914i \(-0.600147\pi\)
−0.309456 + 0.950914i \(0.600147\pi\)
\(84\) 0 0
\(85\) −336.000 −0.428757
\(86\) −136.000 − 235.559i −0.170526 − 0.295360i
\(87\) 15.0000 − 25.9808i 0.0184847 − 0.0320164i
\(88\) 0 0
\(89\) 100.000 + 173.205i 0.119101 + 0.206289i 0.919412 − 0.393297i \(-0.128665\pi\)
−0.800311 + 0.599585i \(0.795332\pi\)
\(90\) 144.000 0.168655
\(91\) 0 0
\(92\) −336.000 −0.380765
\(93\) −72.0000 − 124.708i −0.0802801 − 0.139049i
\(94\) 648.000 − 1122.37i 0.711022 − 1.23153i
\(95\) 200.000 − 346.410i 0.215995 − 0.374115i
\(96\) 384.000 + 665.108i 0.408248 + 0.707107i
\(97\) 1266.00 1.32518 0.662592 − 0.748981i \(-0.269456\pi\)
0.662592 + 0.748981i \(0.269456\pi\)
\(98\) 0 0
\(99\) 558.000 0.566476
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.b.67.1 2
3.2 odd 2 441.4.e.n.361.1 2
7.2 even 3 inner 147.4.e.b.79.1 2
7.3 odd 6 21.4.a.b.1.1 ✓ 1
7.4 even 3 147.4.a.g.1.1 1
7.5 odd 6 147.4.e.c.79.1 2
7.6 odd 2 147.4.e.c.67.1 2
21.2 odd 6 441.4.e.n.226.1 2
21.5 even 6 441.4.e.m.226.1 2
21.11 odd 6 441.4.a.b.1.1 1
21.17 even 6 63.4.a.a.1.1 1
21.20 even 2 441.4.e.m.361.1 2
28.3 even 6 336.4.a.h.1.1 1
28.11 odd 6 2352.4.a.l.1.1 1
35.3 even 12 525.4.d.b.274.1 2
35.17 even 12 525.4.d.b.274.2 2
35.24 odd 6 525.4.a.b.1.1 1
56.3 even 6 1344.4.a.i.1.1 1
56.45 odd 6 1344.4.a.w.1.1 1
84.59 odd 6 1008.4.a.m.1.1 1
105.59 even 6 1575.4.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.b.1.1 ✓ 1 7.3 odd 6
63.4.a.a.1.1 1 21.17 even 6
147.4.a.g.1.1 1 7.4 even 3
147.4.e.b.67.1 2 1.1 even 1 trivial
147.4.e.b.79.1 2 7.2 even 3 inner
147.4.e.c.67.1 2 7.6 odd 2
147.4.e.c.79.1 2 7.5 odd 6
336.4.a.h.1.1 1 28.3 even 6
441.4.a.b.1.1 1 21.11 odd 6
441.4.e.m.226.1 2 21.5 even 6
441.4.e.m.361.1 2 21.20 even 2
441.4.e.n.226.1 2 21.2 odd 6
441.4.e.n.361.1 2 3.2 odd 2
525.4.a.b.1.1 1 35.24 odd 6
525.4.d.b.274.1 2 35.3 even 12
525.4.d.b.274.2 2 35.17 even 12
1008.4.a.m.1.1 1 84.59 odd 6
1344.4.a.i.1.1 1 56.3 even 6
1344.4.a.w.1.1 1 56.45 odd 6
1575.4.a.k.1.1 1 105.59 even 6
2352.4.a.l.1.1 1 28.11 odd 6