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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,8,Mod(361,588)] 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("588.361"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 588.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-54,0,264] 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: \(183.682394985\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{3649})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 913x^{2} + 912x + 831744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.1
Root \(-14.8517 - 25.7240i\) of defining polynomial
Character \(\chi\) \(=\) 588.373
Dual form 588.8.i.j.361.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-13.5000 - 23.3827i) q^{3} +(-115.221 + 199.568i) q^{5} +(-364.500 + 631.333i) q^{9} +(2513.55 + 4353.59i) q^{11} +13772.6 q^{13} +6221.93 q^{15} +(16237.4 + 28123.9i) q^{17} +(4825.98 - 8358.83i) q^{19} +(-16017.2 + 27742.7i) q^{23} +(12510.8 + 21669.4i) q^{25} +19683.0 q^{27} +103672. q^{29} +(120968. + 209523. i) q^{31} +(67865.7 - 117547. i) q^{33} +(63573.3 - 110112. i) q^{37} +(-185930. - 322040. i) q^{39} -607138. q^{41} +443862. q^{43} +(-83996.0 - 145485. i) q^{45} +(345889. - 599097. i) q^{47} +(438409. - 759346. i) q^{51} +(-336412. - 582683. i) q^{53} -1.15845e6 q^{55} -260603. q^{57} +(-1.29323e6 - 2.23994e6i) q^{59} +(767965. - 1.33015e6i) q^{61} +(-1.58689e6 + 2.74858e6i) q^{65} +(2.10411e6 + 3.64443e6i) q^{67} +864931. q^{69} +1.51772e6 q^{71} +(2.73889e6 + 4.74391e6i) q^{73} +(337792. - 585073. i) q^{75} +(3.37928e6 - 5.85308e6i) q^{79} +(-265720. - 460241. i) q^{81} -8.36612e6 q^{83} -7.48353e6 q^{85} +(-1.39957e6 - 2.42413e6i) q^{87} +(-2.58504e6 + 4.47741e6i) q^{89} +(3.26614e6 - 5.65711e6i) q^{93} +(1.11211e6 + 1.92622e6i) q^{95} +1.06286e7 q^{97} -3.66475e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 54 q^{3} + 264 q^{5} - 1458 q^{9} + 4980 q^{11} + 20296 q^{13} - 14256 q^{15} + 17832 q^{17} + 6256 q^{19} - 14052 q^{23} - 141326 q^{25} + 78732 q^{27} + 487176 q^{29} + 470824 q^{31} + 134460 q^{33}+ \cdots - 7260840 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −13.5000 − 23.3827i −0.288675 − 0.500000i
\(4\) 0 0
\(5\) −115.221 + 199.568i −0.412227 + 0.713998i −0.995133 − 0.0985419i \(-0.968582\pi\)
0.582906 + 0.812539i \(0.301915\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −364.500 + 631.333i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 2513.55 + 4353.59i 0.569393 + 0.986218i 0.996626 + 0.0820766i \(0.0261552\pi\)
−0.427233 + 0.904142i \(0.640511\pi\)
\(12\) 0 0
\(13\) 13772.6 1.73866 0.869329 − 0.494234i \(-0.164551\pi\)
0.869329 + 0.494234i \(0.164551\pi\)
\(14\) 0 0
\(15\) 6221.93 0.475998
\(16\) 0 0
\(17\) 16237.4 + 28123.9i 0.801575 + 1.38837i 0.918579 + 0.395237i \(0.129338\pi\)
−0.117004 + 0.993131i \(0.537329\pi\)
\(18\) 0 0
\(19\) 4825.98 − 8358.83i 0.161416 − 0.279581i −0.773961 − 0.633234i \(-0.781727\pi\)
0.935377 + 0.353653i \(0.115060\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −16017.2 + 27742.7i −0.274499 + 0.475446i −0.970009 − 0.243071i \(-0.921845\pi\)
0.695510 + 0.718517i \(0.255179\pi\)
\(24\) 0 0
\(25\) 12510.8 + 21669.4i 0.160138 + 0.277368i
\(26\) 0 0
\(27\) 19683.0 0.192450
\(28\) 0 0
\(29\) 103672. 0.789347 0.394674 − 0.918821i \(-0.370858\pi\)
0.394674 + 0.918821i \(0.370858\pi\)
\(30\) 0 0
\(31\) 120968. + 209523.i 0.729297 + 1.26318i 0.957181 + 0.289491i \(0.0934861\pi\)
−0.227884 + 0.973688i \(0.573181\pi\)
\(32\) 0 0
\(33\) 67865.7 − 117547.i 0.328739 − 0.569393i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 63573.3 − 110112.i 0.206333 − 0.357379i −0.744224 − 0.667930i \(-0.767180\pi\)
0.950557 + 0.310551i \(0.100514\pi\)
\(38\) 0 0
\(39\) −185930. − 322040.i −0.501907 − 0.869329i
\(40\) 0 0
\(41\) −607138. −1.37576 −0.687882 − 0.725823i \(-0.741459\pi\)
−0.687882 + 0.725823i \(0.741459\pi\)
\(42\) 0 0
\(43\) 443862. 0.851350 0.425675 − 0.904876i \(-0.360037\pi\)
0.425675 + 0.904876i \(0.360037\pi\)
\(44\) 0 0
\(45\) −83996.0 − 145485.i −0.137409 − 0.237999i
\(46\) 0 0
\(47\) 345889. − 599097.i 0.485953 − 0.841695i −0.513917 − 0.857840i \(-0.671806\pi\)
0.999870 + 0.0161452i \(0.00513941\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 438409. − 759346.i 0.462789 − 0.801575i
\(52\) 0 0
\(53\) −336412. − 582683.i −0.310389 − 0.537609i 0.668058 − 0.744110i \(-0.267126\pi\)
−0.978447 + 0.206500i \(0.933793\pi\)
\(54\) 0 0
\(55\) −1.15845e6 −0.938877
\(56\) 0 0
\(57\) −260603. −0.186388
\(58\) 0 0
\(59\) −1.29323e6 − 2.23994e6i −0.819775 − 1.41989i −0.905848 − 0.423603i \(-0.860765\pi\)
0.0860734 − 0.996289i \(-0.472568\pi\)
\(60\) 0 0
\(61\) 767965. − 1.33015e6i 0.433198 − 0.750322i −0.563948 − 0.825810i \(-0.690718\pi\)
0.997147 + 0.0754884i \(0.0240516\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.58689e6 + 2.74858e6i −0.716721 + 1.24140i
\(66\) 0 0
\(67\) 2.10411e6 + 3.64443e6i 0.854687 + 1.48036i 0.876935 + 0.480609i \(0.159584\pi\)
−0.0222477 + 0.999752i \(0.507082\pi\)
\(68\) 0 0
\(69\) 864931. 0.316964
\(70\) 0 0
\(71\) 1.51772e6 0.503254 0.251627 − 0.967824i \(-0.419034\pi\)
0.251627 + 0.967824i \(0.419034\pi\)
\(72\) 0 0
\(73\) 2.73889e6 + 4.74391e6i 0.824034 + 1.42727i 0.902655 + 0.430365i \(0.141615\pi\)
−0.0786205 + 0.996905i \(0.525052\pi\)
\(74\) 0 0
\(75\) 337792. − 585073.i 0.0924559 − 0.160138i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 3.37928e6 − 5.85308e6i 0.771132 − 1.33564i −0.165811 − 0.986158i \(-0.553024\pi\)
0.936943 − 0.349482i \(-0.113643\pi\)
\(80\) 0 0
\(81\) −265720. − 460241.i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −8.36612e6 −1.60602 −0.803009 − 0.595966i \(-0.796769\pi\)
−0.803009 + 0.595966i \(0.796769\pi\)
\(84\) 0 0
\(85\) −7.48353e6 −1.32172
\(86\) 0 0
\(87\) −1.39957e6 − 2.42413e6i −0.227865 − 0.394674i
\(88\) 0 0
\(89\) −2.58504e6 + 4.47741e6i −0.388688 + 0.673228i −0.992273 − 0.124071i \(-0.960405\pi\)
0.603585 + 0.797299i \(0.293738\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.26614e6 − 5.65711e6i 0.421060 − 0.729297i
\(94\) 0 0
\(95\) 1.11211e6 + 1.92622e6i 0.133080 + 0.230502i
\(96\) 0 0
\(97\) 1.06286e7 1.18242 0.591212 − 0.806516i \(-0.298650\pi\)
0.591212 + 0.806516i \(0.298650\pi\)
\(98\) 0 0
\(99\) −3.66475e6 −0.379596
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 588.8.i.j.373.1 4
7.2 even 3 588.8.a.f.1.2 2
7.3 odd 6 588.8.i.k.361.2 4
7.4 even 3 inner 588.8.i.j.361.1 4
7.5 odd 6 84.8.a.c.1.1 ✓ 2
7.6 odd 2 588.8.i.k.373.2 4
21.5 even 6 252.8.a.c.1.2 2
28.19 even 6 336.8.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.a.c.1.1 ✓ 2 7.5 odd 6
252.8.a.c.1.2 2 21.5 even 6
336.8.a.q.1.1 2 28.19 even 6
588.8.a.f.1.2 2 7.2 even 3
588.8.i.j.361.1 4 7.4 even 3 inner
588.8.i.j.373.1 4 1.1 even 1 trivial
588.8.i.k.361.2 4 7.3 odd 6
588.8.i.k.373.2 4 7.6 odd 2