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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [147,6,Mod(1,147)] 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("147.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-3,-18,65] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(23.5764215125\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{249}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 62 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(8.38987\) of defining polynomial
Character \(\chi\) \(=\) 147.1

$q$-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)
 
\(f(q)\) \(=\) \(q-9.38987 q^{2} -9.00000 q^{3} +56.1696 q^{4} -71.7291 q^{5} +84.5088 q^{6} -226.949 q^{8} +81.0000 q^{9} +673.526 q^{10} -560.610 q^{11} -505.526 q^{12} +533.509 q^{13} +645.562 q^{15} +333.597 q^{16} +1005.70 q^{17} -760.579 q^{18} +1368.53 q^{19} -4028.99 q^{20} +5264.05 q^{22} +3228.08 q^{23} +2042.54 q^{24} +2020.06 q^{25} -5009.58 q^{26} -729.000 q^{27} -753.456 q^{29} -6061.74 q^{30} +8206.42 q^{31} +4129.95 q^{32} +5045.49 q^{33} -9443.39 q^{34} +4549.74 q^{36} -2808.66 q^{37} -12850.3 q^{38} -4801.58 q^{39} +16278.9 q^{40} +245.827 q^{41} -17504.5 q^{43} -31489.2 q^{44} -5810.05 q^{45} -30311.2 q^{46} -16345.5 q^{47} -3002.37 q^{48} -18968.1 q^{50} -9051.30 q^{51} +29967.0 q^{52} -29641.7 q^{53} +6845.21 q^{54} +40212.0 q^{55} -12316.7 q^{57} +7074.85 q^{58} -10356.1 q^{59} +36260.9 q^{60} +954.179 q^{61} -77057.2 q^{62} -49454.8 q^{64} -38268.1 q^{65} -47376.5 q^{66} -19815.2 q^{67} +56489.8 q^{68} -29052.7 q^{69} +62125.4 q^{71} -18382.9 q^{72} +27109.6 q^{73} +26373.0 q^{74} -18180.5 q^{75} +76869.6 q^{76} +45086.2 q^{78} +44687.4 q^{79} -23928.6 q^{80} +6561.00 q^{81} -2308.29 q^{82} +15606.6 q^{83} -72137.9 q^{85} +164365. q^{86} +6781.10 q^{87} +127230. q^{88} +13635.3 q^{89} +54555.6 q^{90} +181320. q^{92} -73857.8 q^{93} +153482. q^{94} -98163.1 q^{95} -37169.5 q^{96} -12919.5 q^{97} -45409.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 18 q^{3} + 65 q^{4} - 33 q^{5} + 27 q^{6} - 375 q^{8} + 162 q^{9} + 921 q^{10} - 1137 q^{11} - 585 q^{12} + 925 q^{13} + 297 q^{15} - 895 q^{16} - 324 q^{17} - 243 q^{18} + 2311 q^{19} - 3687 q^{20}+ \cdots - 92097 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.38987 −1.65991 −0.829955 − 0.557831i \(-0.811634\pi\)
−0.829955 + 0.557831i \(0.811634\pi\)
\(3\) −9.00000 −0.577350
\(4\) 56.1696 1.75530
\(5\) −71.7291 −1.28313 −0.641564 − 0.767069i \(-0.721714\pi\)
−0.641564 + 0.767069i \(0.721714\pi\)
\(6\) 84.5088 0.958349
\(7\) 0 0
\(8\) −226.949 −1.25373
\(9\) 81.0000 0.333333
\(10\) 673.526 2.12988
\(11\) −560.610 −1.39694 −0.698472 − 0.715637i \(-0.746137\pi\)
−0.698472 + 0.715637i \(0.746137\pi\)
\(12\) −505.526 −1.01342
\(13\) 533.509 0.875555 0.437777 − 0.899083i \(-0.355766\pi\)
0.437777 + 0.899083i \(0.355766\pi\)
\(14\) 0 0
\(15\) 645.562 0.740815
\(16\) 333.597 0.325778
\(17\) 1005.70 0.844007 0.422004 − 0.906594i \(-0.361327\pi\)
0.422004 + 0.906594i \(0.361327\pi\)
\(18\) −760.579 −0.553303
\(19\) 1368.53 0.869699 0.434850 − 0.900503i \(-0.356801\pi\)
0.434850 + 0.900503i \(0.356801\pi\)
\(20\) −4028.99 −2.25228
\(21\) 0 0
\(22\) 5264.05 2.31880
\(23\) 3228.08 1.27240 0.636201 − 0.771523i \(-0.280505\pi\)
0.636201 + 0.771523i \(0.280505\pi\)
\(24\) 2042.54 0.723841
\(25\) 2020.06 0.646419
\(26\) −5009.58 −1.45334
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −753.456 −0.166365 −0.0831827 − 0.996534i \(-0.526508\pi\)
−0.0831827 + 0.996534i \(0.526508\pi\)
\(30\) −6061.74 −1.22969
\(31\) 8206.42 1.53373 0.766866 − 0.641807i \(-0.221815\pi\)
0.766866 + 0.641807i \(0.221815\pi\)
\(32\) 4129.95 0.712968
\(33\) 5045.49 0.806526
\(34\) −9443.39 −1.40098
\(35\) 0 0
\(36\) 4549.74 0.585100
\(37\) −2808.66 −0.337284 −0.168642 − 0.985677i \(-0.553938\pi\)
−0.168642 + 0.985677i \(0.553938\pi\)
\(38\) −12850.3 −1.44362
\(39\) −4801.58 −0.505502
\(40\) 16278.9 1.60870
\(41\) 245.827 0.0228387 0.0114193 − 0.999935i \(-0.496365\pi\)
0.0114193 + 0.999935i \(0.496365\pi\)
\(42\) 0 0
\(43\) −17504.5 −1.44371 −0.721853 − 0.692047i \(-0.756709\pi\)
−0.721853 + 0.692047i \(0.756709\pi\)
\(44\) −31489.2 −2.45206
\(45\) −5810.05 −0.427710
\(46\) −30311.2 −2.11207
\(47\) −16345.5 −1.07933 −0.539663 − 0.841881i \(-0.681449\pi\)
−0.539663 + 0.841881i \(0.681449\pi\)
\(48\) −3002.37 −0.188088
\(49\) 0 0
\(50\) −18968.1 −1.07300
\(51\) −9051.30 −0.487288
\(52\) 29967.0 1.53686
\(53\) −29641.7 −1.44948 −0.724741 − 0.689021i \(-0.758041\pi\)
−0.724741 + 0.689021i \(0.758041\pi\)
\(54\) 6845.21 0.319450
\(55\) 40212.0 1.79246
\(56\) 0 0
\(57\) −12316.7 −0.502121
\(58\) 7074.85 0.276151
\(59\) −10356.1 −0.387317 −0.193659 − 0.981069i \(-0.562035\pi\)
−0.193659 + 0.981069i \(0.562035\pi\)
\(60\) 36260.9 1.30035
\(61\) 954.179 0.0328326 0.0164163 − 0.999865i \(-0.494774\pi\)
0.0164163 + 0.999865i \(0.494774\pi\)
\(62\) −77057.2 −2.54586
\(63\) 0 0
\(64\) −49454.8 −1.50924
\(65\) −38268.1 −1.12345
\(66\) −47376.5 −1.33876
\(67\) −19815.2 −0.539276 −0.269638 − 0.962962i \(-0.586904\pi\)
−0.269638 + 0.962962i \(0.586904\pi\)
\(68\) 56489.8 1.48149
\(69\) −29052.7 −0.734622
\(70\) 0 0
\(71\) 62125.4 1.46259 0.731296 − 0.682060i \(-0.238916\pi\)
0.731296 + 0.682060i \(0.238916\pi\)
\(72\) −18382.9 −0.417910
\(73\) 27109.6 0.595410 0.297705 − 0.954658i \(-0.403779\pi\)
0.297705 + 0.954658i \(0.403779\pi\)
\(74\) 26373.0 0.559861
\(75\) −18180.5 −0.373210
\(76\) 76869.6 1.52658
\(77\) 0 0
\(78\) 45086.2 0.839087
\(79\) 44687.4 0.805595 0.402798 − 0.915289i \(-0.368038\pi\)
0.402798 + 0.915289i \(0.368038\pi\)
\(80\) −23928.6 −0.418015
\(81\) 6561.00 0.111111
\(82\) −2308.29 −0.0379101
\(83\) 15606.6 0.248665 0.124332 − 0.992241i \(-0.460321\pi\)
0.124332 + 0.992241i \(0.460321\pi\)
\(84\) 0 0
\(85\) −72137.9 −1.08297
\(86\) 164365. 2.39642
\(87\) 6781.10 0.0960511
\(88\) 127230. 1.75139
\(89\) 13635.3 0.182469 0.0912347 − 0.995829i \(-0.470919\pi\)
0.0912347 + 0.995829i \(0.470919\pi\)
\(90\) 54555.6 0.709959
\(91\) 0 0
\(92\) 181320. 2.23345
\(93\) −73857.8 −0.885500
\(94\) 153482. 1.79159
\(95\) −98163.1 −1.11594
\(96\) −37169.5 −0.411632
\(97\) −12919.5 −0.139417 −0.0697086 − 0.997567i \(-0.522207\pi\)
−0.0697086 + 0.997567i \(0.522207\pi\)
\(98\) 0 0
\(99\) −45409.4 −0.465648
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.a.i.1.1 2
3.2 odd 2 441.6.a.t.1.2 2
7.2 even 3 21.6.e.b.4.2 ✓ 4
7.3 odd 6 147.6.e.l.79.2 4
7.4 even 3 21.6.e.b.16.2 yes 4
7.5 odd 6 147.6.e.l.67.2 4
7.6 odd 2 147.6.a.k.1.1 2
21.2 odd 6 63.6.e.c.46.1 4
21.11 odd 6 63.6.e.c.37.1 4
21.20 even 2 441.6.a.s.1.2 2
28.11 odd 6 336.6.q.e.289.2 4
28.23 odd 6 336.6.q.e.193.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.2 ✓ 4 7.2 even 3
21.6.e.b.16.2 yes 4 7.4 even 3
63.6.e.c.37.1 4 21.11 odd 6
63.6.e.c.46.1 4 21.2 odd 6
147.6.a.i.1.1 2 1.1 even 1 trivial
147.6.a.k.1.1 2 7.6 odd 2
147.6.e.l.67.2 4 7.5 odd 6
147.6.e.l.79.2 4 7.3 odd 6
336.6.q.e.193.2 4 28.23 odd 6
336.6.q.e.289.2 4 28.11 odd 6
441.6.a.s.1.2 2 21.20 even 2
441.6.a.t.1.2 2 3.2 odd 2