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Properties

Label 147.4.e.c.67.1
Level $147$
Weight $4$
Character 147.67
Analytic conductor $8.673$
Analytic rank $0$
Dimension $2$
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 = [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.c.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 0.941499 − 0.337016i \(-0.109418\pi\)
−0.762614 + 0.646854i \(0.776084\pi\)
\(6\) −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 \(-0.730026\pi\)
−0.661373 + 0.750057i \(0.730026\pi\)
\(14\) 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 \(0.371183\pi\)
−0.992939 + 0.118630i \(0.962150\pi\)
\(18\) −18.0000 + 31.1769i −0.235702 + 0.408248i
\(19\) −50.0000 − 86.6025i −0.603726 − 1.04568i −0.992251 − 0.124246i \(-0.960349\pi\)
0.388526 − 0.921438i \(-0.372984\pi\)
\(20\) −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 −0.788088 − 0.615563i \(-0.788929\pi\)
0.927137 + 0.374723i \(0.122262\pi\)
\(32\) 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 \(-0.656587\pi\)
−0.472330 + 0.881422i \(0.656587\pi\)
\(42\) 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 \(-0.998981\pi\)
0.497226 − 0.867621i \(-0.334352\pi\)
\(48\) 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 −0.924600 − 0.380941i \(-0.875600\pi\)
0.792204 + 0.610256i \(0.208934\pi\)
\(60\) −48.0000 + 83.1384i −0.103280 + 0.178885i
\(61\) −311.000 − 538.668i −0.652778 − 1.13064i −0.982446 − 0.186548i \(-0.940270\pi\)
0.329668 − 0.944097i \(-0.393063\pi\)
\(62\) −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 \(-0.661251\pi\)
0.999855 − 0.0170119i \(-0.00541532\pi\)
\(74\) 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 \(-0.399853\pi\)
0.309456 + 0.950914i \(0.399853\pi\)
\(84\) 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 0.800311 − 0.599585i \(-0.204668\pi\)
−0.919412 + 0.393297i \(0.871335\pi\)
\(90\) −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 \(-0.730544\pi\)
−0.662592 + 0.748981i \(0.730544\pi\)
\(98\) 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.c.67.1 2
3.2 odd 2 441.4.e.m.361.1 2
7.2 even 3 inner 147.4.e.c.79.1 2
7.3 odd 6 147.4.a.g.1.1 1
7.4 even 3 21.4.a.b.1.1 ✓ 1
7.5 odd 6 147.4.e.b.79.1 2
7.6 odd 2 147.4.e.b.67.1 2
21.2 odd 6 441.4.e.m.226.1 2
21.5 even 6 441.4.e.n.226.1 2
21.11 odd 6 63.4.a.a.1.1 1
21.17 even 6 441.4.a.b.1.1 1
21.20 even 2 441.4.e.n.361.1 2
28.3 even 6 2352.4.a.l.1.1 1
28.11 odd 6 336.4.a.h.1.1 1
35.4 even 6 525.4.a.b.1.1 1
35.18 odd 12 525.4.d.b.274.1 2
35.32 odd 12 525.4.d.b.274.2 2
56.11 odd 6 1344.4.a.i.1.1 1
56.53 even 6 1344.4.a.w.1.1 1
84.11 even 6 1008.4.a.m.1.1 1
105.74 odd 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 7.4 even 3
63.4.a.a.1.1 1 21.11 odd 6
147.4.a.g.1.1 1 7.3 odd 6
147.4.e.b.67.1 2 7.6 odd 2
147.4.e.b.79.1 2 7.5 odd 6
147.4.e.c.67.1 2 1.1 even 1 trivial
147.4.e.c.79.1 2 7.2 even 3 inner
336.4.a.h.1.1 1 28.11 odd 6
441.4.a.b.1.1 1 21.17 even 6
441.4.e.m.226.1 2 21.2 odd 6
441.4.e.m.361.1 2 3.2 odd 2
441.4.e.n.226.1 2 21.5 even 6
441.4.e.n.361.1 2 21.20 even 2
525.4.a.b.1.1 1 35.4 even 6
525.4.d.b.274.1 2 35.18 odd 12
525.4.d.b.274.2 2 35.32 odd 12
1008.4.a.m.1.1 1 84.11 even 6
1344.4.a.i.1.1 1 56.11 odd 6
1344.4.a.w.1.1 1 56.53 even 6
1575.4.a.k.1.1 1 105.74 odd 6
2352.4.a.l.1.1 1 28.3 even 6