Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [147,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-10,9,-68,-106] 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: \(23.5764215125\)
Analytic rank: \(1\)
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 79.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 147.79
Dual form 147.6.e.b.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+(-5.00000 + 8.66025i) q^{2} +(4.50000 + 7.79423i) q^{3} +(-34.0000 - 58.8897i) q^{4} +(-53.0000 + 91.7987i) q^{5} -90.0000 q^{6} +360.000 q^{8} +(-40.5000 + 70.1481i) q^{9} +(-530.000 - 917.987i) q^{10} +(-46.0000 - 79.6743i) q^{11} +(306.000 - 530.008i) q^{12} -670.000 q^{13} -954.000 q^{15} +(-712.000 + 1233.22i) q^{16} +(-111.000 - 192.258i) q^{17} +(-405.000 - 701.481i) q^{18} +(-454.000 + 786.351i) q^{19} +7208.00 q^{20} +920.000 q^{22} +(588.000 - 1018.45i) q^{23} +(1620.00 + 2805.92i) q^{24} +(-4055.50 - 7024.33i) q^{25} +(3350.00 - 5802.37i) q^{26} -729.000 q^{27} +1118.00 q^{29} +(4770.00 - 8261.88i) q^{30} +(1848.00 + 3200.83i) q^{31} +(-1360.00 - 2355.59i) q^{32} +(414.000 - 717.069i) q^{33} +2220.00 q^{34} +5508.00 q^{36} +(-2091.00 + 3621.72i) q^{37} +(-4540.00 - 7863.51i) q^{38} +(-3015.00 - 5222.13i) q^{39} +(-19080.0 + 33047.5i) q^{40} +6662.00 q^{41} -3700.00 q^{43} +(-3128.00 + 5417.85i) q^{44} +(-4293.00 - 7435.69i) q^{45} +(5880.00 + 10184.5i) q^{46} +(-3528.00 + 6110.68i) q^{47} -12816.0 q^{48} +81110.0 q^{50} +(999.000 - 1730.32i) q^{51} +(22780.0 + 39456.1i) q^{52} +(18789.0 + 32543.5i) q^{53} +(3645.00 - 6313.33i) q^{54} +9752.00 q^{55} -8172.00 q^{57} +(-5590.00 + 9682.16i) q^{58} +(16350.0 + 28319.0i) q^{59} +(32436.0 + 56180.8i) q^{60} +(-5401.00 + 9354.81i) q^{61} -36960.0 q^{62} -18368.0 q^{64} +(35510.0 - 61505.1i) q^{65} +(4140.00 + 7170.69i) q^{66} +(-32498.0 - 56288.2i) q^{67} +(-7548.00 + 13073.5i) q^{68} +10584.0 q^{69} -61320.0 q^{71} +(-14580.0 + 25253.3i) q^{72} +(19461.0 + 33707.4i) q^{73} +(-20910.0 - 36217.2i) q^{74} +(36499.5 - 63219.0i) q^{75} +61744.0 q^{76} +60300.0 q^{78} +(44048.0 - 76293.4i) q^{79} +(-75472.0 - 130721. i) q^{80} +(-3280.50 - 5681.99i) q^{81} +(-33310.0 + 57694.6i) q^{82} -71892.0 q^{83} +23532.0 q^{85} +(18500.0 - 32042.9i) q^{86} +(5031.00 + 8713.95i) q^{87} +(-16560.0 - 28682.8i) q^{88} +(55909.0 - 96837.2i) q^{89} +85860.0 q^{90} -79968.0 q^{92} +(-16632.0 + 28807.5i) q^{93} +(-35280.0 - 61106.8i) q^{94} +(-48124.0 - 83353.2i) q^{95} +(12240.0 - 21200.3i) q^{96} +150846. q^{97} +7452.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{2} + 9 q^{3} - 68 q^{4} - 106 q^{5} - 180 q^{6} + 720 q^{8} - 81 q^{9} - 1060 q^{10} - 92 q^{11} + 612 q^{12} - 1340 q^{13} - 1908 q^{15} - 1424 q^{16} - 222 q^{17} - 810 q^{18} - 908 q^{19}+ \cdots + 14904 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.00000 + 8.66025i −0.883883 + 1.53093i −0.0368954 + 0.999319i \(0.511747\pi\)
−0.846988 + 0.531612i \(0.821586\pi\)
\(3\) 4.50000 + 7.79423i 0.288675 + 0.500000i
\(4\) −34.0000 − 58.8897i −1.06250 − 1.84030i
\(5\) −53.0000 + 91.7987i −0.948093 + 1.64214i −0.198656 + 0.980069i \(0.563658\pi\)
−0.749437 + 0.662076i \(0.769676\pi\)
\(6\) −90.0000 −1.02062
\(7\) 0 0
\(8\) 360.000 1.98874
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) −530.000 − 917.987i −1.67601 − 2.90293i
\(11\) −46.0000 − 79.6743i −0.114624 − 0.198535i 0.803005 − 0.595972i \(-0.203233\pi\)
−0.917629 + 0.397437i \(0.869900\pi\)
\(12\) 306.000 − 530.008i 0.613435 − 1.06250i
\(13\) −670.000 −1.09955 −0.549777 − 0.835312i \(-0.685287\pi\)
−0.549777 + 0.835312i \(0.685287\pi\)
\(14\) 0 0
\(15\) −954.000 −1.09476
\(16\) −712.000 + 1233.22i −0.695312 + 1.20432i
\(17\) −111.000 − 192.258i −0.0931538 − 0.161347i 0.815683 − 0.578499i \(-0.196362\pi\)
−0.908837 + 0.417152i \(0.863028\pi\)
\(18\) −405.000 − 701.481i −0.294628 − 0.510310i
\(19\) −454.000 + 786.351i −0.288517 + 0.499727i −0.973456 − 0.228874i \(-0.926496\pi\)
0.684939 + 0.728601i \(0.259829\pi\)
\(20\) 7208.00 4.02939
\(21\) 0 0
\(22\) 920.000 0.405258
\(23\) 588.000 − 1018.45i 0.231770 − 0.401438i −0.726559 − 0.687104i \(-0.758882\pi\)
0.958329 + 0.285666i \(0.0922149\pi\)
\(24\) 1620.00 + 2805.92i 0.574099 + 0.994369i
\(25\) −4055.50 − 7024.33i −1.29776 − 2.24779i
\(26\) 3350.00 − 5802.37i 0.971877 − 1.68334i
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 1118.00 0.246858 0.123429 − 0.992353i \(-0.460611\pi\)
0.123429 + 0.992353i \(0.460611\pi\)
\(30\) 4770.00 − 8261.88i 0.967643 − 1.67601i
\(31\) 1848.00 + 3200.83i 0.345380 + 0.598216i 0.985423 − 0.170123i \(-0.0544166\pi\)
−0.640042 + 0.768340i \(0.721083\pi\)
\(32\) −1360.00 − 2355.59i −0.234782 − 0.406654i
\(33\) 414.000 − 717.069i 0.0661783 − 0.114624i
\(34\) 2220.00 0.329348
\(35\) 0 0
\(36\) 5508.00 0.708333
\(37\) −2091.00 + 3621.72i −0.251102 + 0.434921i −0.963829 − 0.266520i \(-0.914126\pi\)
0.712728 + 0.701441i \(0.247460\pi\)
\(38\) −4540.00 − 7863.51i −0.510031 − 0.883400i
\(39\) −3015.00 − 5222.13i −0.317414 − 0.549777i
\(40\) −19080.0 + 33047.5i −1.88551 + 3.26580i
\(41\) 6662.00 0.618935 0.309467 − 0.950910i \(-0.399849\pi\)
0.309467 + 0.950910i \(0.399849\pi\)
\(42\) 0 0
\(43\) −3700.00 −0.305162 −0.152581 − 0.988291i \(-0.548759\pi\)
−0.152581 + 0.988291i \(0.548759\pi\)
\(44\) −3128.00 + 5417.85i −0.243576 + 0.421887i
\(45\) −4293.00 − 7435.69i −0.316031 − 0.547382i
\(46\) 5880.00 + 10184.5i 0.409716 + 0.709649i
\(47\) −3528.00 + 6110.68i −0.232961 + 0.403501i −0.958678 − 0.284493i \(-0.908175\pi\)
0.725717 + 0.687993i \(0.241508\pi\)
\(48\) −12816.0 −0.802878
\(49\) 0 0
\(50\) 81110.0 4.58827
\(51\) 999.000 − 1730.32i 0.0537824 − 0.0931538i
\(52\) 22780.0 + 39456.1i 1.16828 + 2.02351i
\(53\) 18789.0 + 32543.5i 0.918785 + 1.59138i 0.801263 + 0.598312i \(0.204162\pi\)
0.117522 + 0.993070i \(0.462505\pi\)
\(54\) 3645.00 − 6313.33i 0.170103 − 0.294628i
\(55\) 9752.00 0.434697
\(56\) 0 0
\(57\) −8172.00 −0.333151
\(58\) −5590.00 + 9682.16i −0.218194 + 0.377922i
\(59\) 16350.0 + 28319.0i 0.611488 + 1.05913i 0.990990 + 0.133937i \(0.0427620\pi\)
−0.379502 + 0.925191i \(0.623905\pi\)
\(60\) 32436.0 + 56180.8i 1.16319 + 2.01470i
\(61\) −5401.00 + 9354.81i −0.185844 + 0.321892i −0.943861 − 0.330344i \(-0.892835\pi\)
0.758016 + 0.652236i \(0.226169\pi\)
\(62\) −36960.0 −1.22110
\(63\) 0 0
\(64\) −18368.0 −0.560547
\(65\) 35510.0 − 61505.1i 1.04248 − 1.80563i
\(66\) 4140.00 + 7170.69i 0.116988 + 0.202629i
\(67\) −32498.0 − 56288.2i −0.884443 − 1.53190i −0.846351 − 0.532625i \(-0.821206\pi\)
−0.0380913 − 0.999274i \(-0.512128\pi\)
\(68\) −7548.00 + 13073.5i −0.197952 + 0.342863i
\(69\) 10584.0 0.267625
\(70\) 0 0
\(71\) −61320.0 −1.44363 −0.721816 − 0.692085i \(-0.756692\pi\)
−0.721816 + 0.692085i \(0.756692\pi\)
\(72\) −14580.0 + 25253.3i −0.331456 + 0.574099i
\(73\) 19461.0 + 33707.4i 0.427423 + 0.740319i 0.996643 − 0.0818666i \(-0.0260881\pi\)
−0.569220 + 0.822185i \(0.692755\pi\)
\(74\) −20910.0 − 36217.2i −0.443889 − 0.768839i
\(75\) 36499.5 − 63219.0i 0.749262 − 1.29776i
\(76\) 61744.0 1.22620
\(77\) 0 0
\(78\) 60300.0 1.12223
\(79\) 44048.0 − 76293.4i 0.794069 − 1.37537i −0.129359 − 0.991598i \(-0.541292\pi\)
0.923428 − 0.383771i \(-0.125375\pi\)
\(80\) −75472.0 − 130721.i −1.31844 − 2.28361i
\(81\) −3280.50 − 5681.99i −0.0555556 − 0.0962250i
\(82\) −33310.0 + 57694.6i −0.547066 + 0.947547i
\(83\) −71892.0 −1.14547 −0.572737 − 0.819739i \(-0.694118\pi\)
−0.572737 + 0.819739i \(0.694118\pi\)
\(84\) 0 0
\(85\) 23532.0 0.353274
\(86\) 18500.0 − 32042.9i 0.269728 − 0.467182i
\(87\) 5031.00 + 8713.95i 0.0712617 + 0.123429i
\(88\) −16560.0 − 28682.8i −0.227957 − 0.394834i
\(89\) 55909.0 − 96837.2i 0.748181 − 1.29589i −0.200513 − 0.979691i \(-0.564261\pi\)
0.948694 − 0.316196i \(-0.102406\pi\)
\(90\) 85860.0 1.11734
\(91\) 0 0
\(92\) −79968.0 −0.985024
\(93\) −16632.0 + 28807.5i −0.199405 + 0.345380i
\(94\) −35280.0 − 61106.8i −0.411821 − 0.713296i
\(95\) −48124.0 − 83353.2i −0.547082 − 0.947574i
\(96\) 12240.0 − 21200.3i 0.135551 − 0.234782i
\(97\) 150846. 1.62781 0.813906 − 0.580996i \(-0.197337\pi\)
0.813906 + 0.580996i \(0.197337\pi\)
\(98\) 0 0
\(99\) 7452.00 0.0764161
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 147.6.e.b.79.1 2
7.2 even 3 147.6.a.g.1.1 1
7.3 odd 6 147.6.e.a.67.1 2
7.4 even 3 inner 147.6.e.b.67.1 2
7.5 odd 6 21.6.a.d.1.1 ✓ 1
7.6 odd 2 147.6.e.a.79.1 2
21.2 odd 6 441.6.a.b.1.1 1
21.5 even 6 63.6.a.a.1.1 1
28.19 even 6 336.6.a.a.1.1 1
35.12 even 12 525.6.d.a.274.2 2
35.19 odd 6 525.6.a.a.1.1 1
35.33 even 12 525.6.d.a.274.1 2
84.47 odd 6 1008.6.a.bc.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.d.1.1 ✓ 1 7.5 odd 6
63.6.a.a.1.1 1 21.5 even 6
147.6.a.g.1.1 1 7.2 even 3
147.6.e.a.67.1 2 7.3 odd 6
147.6.e.a.79.1 2 7.6 odd 2
147.6.e.b.67.1 2 7.4 even 3 inner
147.6.e.b.79.1 2 1.1 even 1 trivial
336.6.a.a.1.1 1 28.19 even 6
441.6.a.b.1.1 1 21.2 odd 6
525.6.a.a.1.1 1 35.19 odd 6
525.6.d.a.274.1 2 35.33 even 12
525.6.d.a.274.2 2 35.12 even 12
1008.6.a.bc.1.1 1 84.47 odd 6