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Properties

Label 147.8.e.g.79.2
Level $147$
Weight $8$
Character 147.79
Analytic conductor $45.921$
Analytic rank $0$
Dimension $4$
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] = [147,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 147.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,9,-54] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(45.9205987462\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-355})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 88x^{2} - 89x + 7921 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 79.2
Root \(-7.90858 + 5.14337i\) of defining polynomial
Character \(\chi\) \(=\) 147.79
Dual form 147.8.e.g.67.2

$q$-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+(10.4086 - 18.0282i) q^{2} +(-13.5000 - 23.3827i) q^{3} +(-152.677 - 264.445i) q^{4} +(73.1717 - 126.737i) q^{5} -562.064 q^{6} -3692.02 q^{8} +(-364.500 + 631.333i) q^{9} +(-1523.23 - 2638.31i) q^{10} +(3745.84 + 6487.99i) q^{11} +(-4122.29 + 7140.01i) q^{12} -8553.34 q^{13} -3951.27 q^{15} +(-18886.0 + 32711.5i) q^{16} +(-2609.83 - 4520.35i) q^{17} +(7587.86 + 13142.6i) q^{18} +(-23568.4 + 40821.7i) q^{19} -44686.6 q^{20} +155956. q^{22} +(-11916.9 + 20640.7i) q^{23} +(49842.3 + 86329.3i) q^{24} +(28354.3 + 49111.1i) q^{25} +(-89028.2 + 154201. i) q^{26} +19683.0 q^{27} -183928. q^{29} +(-41127.1 + 71234.3i) q^{30} +(-90135.9 - 156120. i) q^{31} +(156864. + 271696. i) q^{32} +(101138. - 175176. i) q^{33} -108658. q^{34} +222603. q^{36} +(202061. - 349979. i) q^{37} +(490628. + 849793. i) q^{38} +(115470. + 200000. i) q^{39} +(-270151. + 467916. i) q^{40} -632653. q^{41} +443131. q^{43} +(1.14381e6 - 1.98114e6i) q^{44} +(53342.2 + 92391.3i) q^{45} +(248076. + 429680. i) q^{46} +(-262495. + 454654. i) q^{47} +1.01984e6 q^{48} +1.18051e6 q^{50} +(-70465.3 + 122050. i) q^{51} +(1.30590e6 + 2.26189e6i) q^{52} +(-260333. - 450911. i) q^{53} +(204872. - 354849. i) q^{54} +1.09636e6 q^{55} +1.27270e6 q^{57} +(-1.91443e6 + 3.31589e6i) q^{58} +(-710794. - 1.23113e6i) q^{59} +(603269. + 1.04489e6i) q^{60} +(-308195. + 533809. i) q^{61} -3.75275e6 q^{62} +1.69611e6 q^{64} +(-625863. + 1.08403e6i) q^{65} +(-2.10540e6 - 3.64666e6i) q^{66} +(818851. + 1.41829e6i) q^{67} +(-796922. + 1.38031e6i) q^{68} +643513. q^{69} -595589. q^{71} +(1.34574e6 - 2.33089e6i) q^{72} +(-2.20593e6 - 3.82078e6i) q^{73} +(-4.20633e6 - 7.28558e6i) q^{74} +(765566. - 1.32600e6i) q^{75} +1.43935e7 q^{76} +4.80752e6 q^{78} +(-1.01954e6 + 1.76589e6i) q^{79} +(2.76384e6 + 4.78711e6i) q^{80} +(-265720. - 460241. i) q^{81} +(-6.58502e6 + 1.14056e7i) q^{82} +1.08171e6 q^{83} -763862. q^{85} +(4.61236e6 - 7.98885e6i) q^{86} +(2.48302e6 + 4.30072e6i) q^{87} +(-1.38297e7 - 2.39538e7i) q^{88} +(4.40312e6 - 7.62643e6i) q^{89} +2.22087e6 q^{90} +7.27776e6 q^{92} +(-2.43367e6 + 4.21524e6i) q^{93} +(5.46440e6 + 9.46461e6i) q^{94} +(3.44908e6 + 5.97399e6i) q^{95} +(4.23532e6 - 7.33580e6i) q^{96} -6.13951e6 q^{97} -5.46144e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 9 q^{2} - 54 q^{3} - 317 q^{4} - 360 q^{5} - 486 q^{6} - 10134 q^{8} - 1458 q^{9} - 9030 q^{10} + 4932 q^{11} - 8559 q^{12} - 15416 q^{13} + 19440 q^{15} - 20033 q^{16} - 28584 q^{17} + 6561 q^{18}+ \cdots - 7190856 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.4086 − 18.0282i 0.919998 − 1.59348i 0.120583 − 0.992703i \(-0.461524\pi\)
0.799415 − 0.600779i \(-0.205143\pi\)
\(3\) −13.5000 − 23.3827i −0.288675 − 0.500000i
\(4\) −152.677 − 264.445i −1.19279 − 2.06597i
\(5\) 73.1717 − 126.737i 0.261787 − 0.453428i −0.704930 − 0.709277i \(-0.749022\pi\)
0.966717 + 0.255849i \(0.0823549\pi\)
\(6\) −562.064 −1.06232
\(7\) 0 0
\(8\) −3692.02 −2.54946
\(9\) −364.500 + 631.333i −0.166667 + 0.288675i
\(10\) −1523.23 − 2638.31i −0.481687 − 0.834306i
\(11\) 3745.84 + 6487.99i 0.848546 + 1.46972i 0.882506 + 0.470301i \(0.155855\pi\)
−0.0339601 + 0.999423i \(0.510812\pi\)
\(12\) −4122.29 + 7140.01i −0.688658 + 1.19279i
\(13\) −8553.34 −1.07978 −0.539889 − 0.841736i \(-0.681534\pi\)
−0.539889 + 0.841736i \(0.681534\pi\)
\(14\) 0 0
\(15\) −3951.27 −0.302286
\(16\) −18886.0 + 32711.5i −1.15271 + 1.99655i
\(17\) −2609.83 − 4520.35i −0.128837 − 0.223152i 0.794389 − 0.607409i \(-0.207791\pi\)
−0.923226 + 0.384257i \(0.874458\pi\)
\(18\) 7587.86 + 13142.6i 0.306666 + 0.531161i
\(19\) −23568.4 + 40821.7i −0.788303 + 1.36538i 0.138703 + 0.990334i \(0.455707\pi\)
−0.927006 + 0.375047i \(0.877627\pi\)
\(20\) −44686.6 −1.24903
\(21\) 0 0
\(22\) 155956. 3.12264
\(23\) −11916.9 + 20640.7i −0.204228 + 0.353734i −0.949887 − 0.312595i \(-0.898802\pi\)
0.745658 + 0.666329i \(0.232135\pi\)
\(24\) 49842.3 + 86329.3i 0.735967 + 1.27473i
\(25\) 28354.3 + 49111.1i 0.362935 + 0.628622i
\(26\) −89028.2 + 154201.i −0.993393 + 1.72061i
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) −183928. −1.40041 −0.700203 − 0.713943i \(-0.746907\pi\)
−0.700203 + 0.713943i \(0.746907\pi\)
\(30\) −41127.1 + 71234.3i −0.278102 + 0.481687i
\(31\) −90135.9 − 156120.i −0.543416 − 0.941223i −0.998705 − 0.0508798i \(-0.983797\pi\)
0.455289 − 0.890344i \(-0.349536\pi\)
\(32\) 156864. + 271696.i 0.846249 + 1.46575i
\(33\) 101138. − 175176.i 0.489908 − 0.848546i
\(34\) −108658. −0.474119
\(35\) 0 0
\(36\) 222603. 0.795194
\(37\) 202061. − 349979.i 0.655806 − 1.13589i −0.325885 − 0.945409i \(-0.605662\pi\)
0.981691 − 0.190480i \(-0.0610045\pi\)
\(38\) 490628. + 849793.i 1.45047 + 2.51229i
\(39\) 115470. + 200000.i 0.311705 + 0.539889i
\(40\) −270151. + 467916.i −0.667417 + 1.15600i
\(41\) −632653. −1.43358 −0.716790 − 0.697289i \(-0.754389\pi\)
−0.716790 + 0.697289i \(0.754389\pi\)
\(42\) 0 0
\(43\) 443131. 0.849948 0.424974 − 0.905206i \(-0.360283\pi\)
0.424974 + 0.905206i \(0.360283\pi\)
\(44\) 1.14381e6 − 1.98114e6i 2.02428 − 3.50615i
\(45\) 53342.2 + 92391.3i 0.0872623 + 0.151143i
\(46\) 248076. + 429680.i 0.375779 + 0.650868i
\(47\) −262495. + 454654.i −0.368789 + 0.638761i −0.989376 − 0.145376i \(-0.953561\pi\)
0.620587 + 0.784137i \(0.286894\pi\)
\(48\) 1.01984e6 1.33103
\(49\) 0 0
\(50\) 1.18051e6 1.33560
\(51\) −70465.3 + 122050.i −0.0743841 + 0.128837i
\(52\) 1.30590e6 + 2.26189e6i 1.28795 + 2.23079i
\(53\) −260333. − 450911.i −0.240195 − 0.416030i 0.720575 − 0.693377i \(-0.243878\pi\)
−0.960770 + 0.277347i \(0.910545\pi\)
\(54\) 204872. − 354849.i 0.177054 − 0.306666i
\(55\) 1.09636e6 0.888553
\(56\) 0 0
\(57\) 1.27270e6 0.910254
\(58\) −1.91443e6 + 3.31589e6i −1.28837 + 2.23152i
\(59\) −710794. − 1.23113e6i −0.450569 − 0.780409i 0.547852 − 0.836575i \(-0.315446\pi\)
−0.998421 + 0.0561662i \(0.982112\pi\)
\(60\) 603269. + 1.04489e6i 0.360564 + 0.624514i
\(61\) −308195. + 533809.i −0.173848 + 0.301114i −0.939762 − 0.341829i \(-0.888954\pi\)
0.765914 + 0.642943i \(0.222287\pi\)
\(62\) −3.75275e6 −1.99976
\(63\) 0 0
\(64\) 1.69611e6 0.808767
\(65\) −625863. + 1.08403e6i −0.282672 + 0.489602i
\(66\) −2.10540e6 − 3.64666e6i −0.901429 − 1.56132i
\(67\) 818851. + 1.41829e6i 0.332616 + 0.576107i 0.983024 − 0.183478i \(-0.0587355\pi\)
−0.650408 + 0.759585i \(0.725402\pi\)
\(68\) −796922. + 1.38031e6i −0.307351 + 0.532348i
\(69\) 643513. 0.235823
\(70\) 0 0
\(71\) −595589. −0.197489 −0.0987444 − 0.995113i \(-0.531483\pi\)
−0.0987444 + 0.995113i \(0.531483\pi\)
\(72\) 1.34574e6 − 2.33089e6i 0.424911 − 0.735967i
\(73\) −2.20593e6 − 3.82078e6i −0.663685 − 1.14954i −0.979640 − 0.200762i \(-0.935658\pi\)
0.315955 − 0.948774i \(-0.397675\pi\)
\(74\) −4.20633e6 − 7.28558e6i −1.20668 − 2.09003i
\(75\) 765566. − 1.32600e6i 0.209541 − 0.362935i
\(76\) 1.43935e7 3.76112
\(77\) 0 0
\(78\) 4.80752e6 1.14707
\(79\) −1.01954e6 + 1.76589e6i −0.232653 + 0.402967i −0.958588 − 0.284796i \(-0.908074\pi\)
0.725935 + 0.687764i \(0.241407\pi\)
\(80\) 2.76384e6 + 4.78711e6i 0.603529 + 1.04534i
\(81\) −265720. − 460241.i −0.0555556 − 0.0962250i
\(82\) −6.58502e6 + 1.14056e7i −1.31889 + 2.28438i
\(83\) 1.08171e6 0.207653 0.103826 − 0.994595i \(-0.466891\pi\)
0.103826 + 0.994595i \(0.466891\pi\)
\(84\) 0 0
\(85\) −763862. −0.134911
\(86\) 4.61236e6 − 7.98885e6i 0.781950 − 1.35438i
\(87\) 2.48302e6 + 4.30072e6i 0.404263 + 0.700203i
\(88\) −1.38297e7 − 2.39538e7i −2.16334 − 3.74701i
\(89\) 4.40312e6 − 7.62643e6i 0.662058 − 1.14672i −0.318016 − 0.948085i \(-0.603017\pi\)
0.980074 − 0.198632i \(-0.0636500\pi\)
\(90\) 2.22087e6 0.321125
\(91\) 0 0
\(92\) 7.27776e6 0.974407
\(93\) −2.43367e6 + 4.21524e6i −0.313741 + 0.543416i
\(94\) 5.46440e6 + 9.46461e6i 0.678570 + 1.17532i
\(95\) 3.44908e6 + 5.97399e6i 0.412735 + 0.714878i
\(96\) 4.23532e6 − 7.33580e6i 0.488582 − 0.846249i
\(97\) −6.13951e6 −0.683019 −0.341509 − 0.939878i \(-0.610938\pi\)
−0.341509 + 0.939878i \(0.610938\pi\)
\(98\) 0 0
\(99\) −5.46144e6 −0.565697
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 147.8.e.g.79.2 4
7.2 even 3 147.8.a.c.1.1 2
7.3 odd 6 147.8.e.h.67.2 4
7.4 even 3 inner 147.8.e.g.67.2 4
7.5 odd 6 21.8.a.b.1.1 ✓ 2
7.6 odd 2 147.8.e.h.79.2 4
21.2 odd 6 441.8.a.m.1.2 2
21.5 even 6 63.8.a.f.1.2 2
28.19 even 6 336.8.a.n.1.2 2
35.19 odd 6 525.8.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.8.a.b.1.1 ✓ 2 7.5 odd 6
63.8.a.f.1.2 2 21.5 even 6
147.8.a.c.1.1 2 7.2 even 3
147.8.e.g.67.2 4 7.4 even 3 inner
147.8.e.g.79.2 4 1.1 even 1 trivial
147.8.e.h.67.2 4 7.3 odd 6
147.8.e.h.79.2 4 7.6 odd 2
336.8.a.n.1.2 2 28.19 even 6
441.8.a.m.1.2 2 21.2 odd 6
525.8.a.e.1.2 2 35.19 odd 6