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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 \(-0.278286\pi\)
0.641564 + 0.767069i \(0.278286\pi\)
\(6\) −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 \(-0.644234\pi\)
−0.437777 + 0.899083i \(0.644234\pi\)
\(14\) 0 0
\(15\) 645.562 0.740815
\(16\) 333.597 0.325778
\(17\) −1005.70 −0.844007 −0.422004 − 0.906594i \(-0.638673\pi\)
−0.422004 + 0.906594i \(0.638673\pi\)
\(18\) −760.579 −0.553303
\(19\) −1368.53 −0.869699 −0.434850 − 0.900503i \(-0.643199\pi\)
−0.434850 + 0.900503i \(0.643199\pi\)
\(20\) 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 \(-0.778185\pi\)
−0.766866 + 0.641807i \(0.778185\pi\)
\(32\) 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 \(-0.503635\pi\)
−0.0114193 + 0.999935i \(0.503635\pi\)
\(42\) 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 \(-0.318551\pi\)
0.539663 + 0.841881i \(0.318551\pi\)
\(48\) 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 \(-0.437965\pi\)
0.193659 + 0.981069i \(0.437965\pi\)
\(60\) 36260.9 1.30035
\(61\) −954.179 −0.0328326 −0.0164163 − 0.999865i \(-0.505226\pi\)
−0.0164163 + 0.999865i \(0.505226\pi\)
\(62\) 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 \(-0.596221\pi\)
−0.297705 + 0.954658i \(0.596221\pi\)
\(74\) 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 \(-0.539679\pi\)
−0.124332 + 0.992241i \(0.539679\pi\)
\(84\) 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 \(-0.529081\pi\)
−0.0912347 + 0.995829i \(0.529081\pi\)
\(90\) −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 \(-0.477793\pi\)
0.0697086 + 0.997567i \(0.477793\pi\)
\(98\) 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.k.1.1 2
3.2 odd 2 441.6.a.s.1.2 2
7.2 even 3 147.6.e.l.67.2 4
7.3 odd 6 21.6.e.b.16.2 yes 4
7.4 even 3 147.6.e.l.79.2 4
7.5 odd 6 21.6.e.b.4.2 ✓ 4
7.6 odd 2 147.6.a.i.1.1 2
21.5 even 6 63.6.e.c.46.1 4
21.17 even 6 63.6.e.c.37.1 4
21.20 even 2 441.6.a.t.1.2 2
28.3 even 6 336.6.q.e.289.2 4
28.19 even 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 7.5 odd 6
21.6.e.b.16.2 yes 4 7.3 odd 6
63.6.e.c.37.1 4 21.17 even 6
63.6.e.c.46.1 4 21.5 even 6
147.6.a.i.1.1 2 7.6 odd 2
147.6.a.k.1.1 2 1.1 even 1 trivial
147.6.e.l.67.2 4 7.2 even 3
147.6.e.l.79.2 4 7.4 even 3
336.6.q.e.193.2 4 28.19 even 6
336.6.q.e.289.2 4 28.3 even 6
441.6.a.s.1.2 2 3.2 odd 2
441.6.a.t.1.2 2 21.20 even 2