Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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,-18,0,90,0,0,0,-162,0,-566] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(94.3056860500\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{2641})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 661x^{2} + 660x + 435600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.2
Root \(13.0977 - 22.6858i\) of defining polynomial
Character \(\chi\) \(=\) 588.373
Dual form 588.6.i.j.361.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.50000 - 7.79423i) q^{3} +(48.1953 - 83.4768i) q^{5} +(-40.5000 + 70.1481i) q^{9} +(89.7580 + 155.465i) q^{11} +56.8747 q^{13} -867.516 q^{15} +(708.446 + 1227.06i) q^{17} +(-646.297 + 1119.42i) q^{19} +(2334.31 - 4043.14i) q^{23} +(-3083.08 - 5340.05i) q^{25} +729.000 q^{27} -5003.10 q^{29} +(-3722.17 - 6446.99i) q^{31} +(807.822 - 1399.19i) q^{33} +(4540.45 - 7864.29i) q^{37} +(-255.936 - 443.295i) q^{39} -6628.36 q^{41} -12924.5 q^{43} +(3903.82 + 6761.62i) q^{45} +(-2749.00 + 4761.41i) q^{47} +(6376.01 - 11043.6i) q^{51} +(-14769.5 - 25581.6i) q^{53} +17303.7 q^{55} +11633.4 q^{57} +(529.253 + 916.693i) q^{59} +(4743.93 - 8216.72i) q^{61} +(2741.10 - 4747.72i) q^{65} +(-665.516 - 1152.71i) q^{67} -42017.5 q^{69} +49811.2 q^{71} +(-37879.1 - 65608.6i) q^{73} +(-27747.7 + 48060.5i) q^{75} +(-31185.4 + 54014.7i) q^{79} +(-3280.50 - 5681.99i) q^{81} -53221.5 q^{83} +136575. q^{85} +(22513.9 + 38995.3i) q^{87} +(-59340.9 + 102782. i) q^{89} +(-33499.6 + 58023.0i) q^{93} +(62297.0 + 107902. i) q^{95} +136608. q^{97} -14540.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{3} + 90 q^{5} - 162 q^{9} - 566 q^{11} + 1872 q^{13} - 1620 q^{15} - 558 q^{17} - 324 q^{19} + 2862 q^{23} - 3082 q^{25} + 2916 q^{27} - 8912 q^{29} - 1116 q^{31} - 5094 q^{33} + 23712 q^{37}+ \cdots + 91692 q^{99}+O(q^{100}) \) 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
\(3\) −4.50000 − 7.79423i −0.288675 − 0.500000i
\(4\) 0 0
\(5\) 48.1953 − 83.4768i 0.862144 − 1.49328i −0.00771076 − 0.999970i \(-0.502454\pi\)
0.869855 − 0.493307i \(-0.164212\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 89.7580 + 155.465i 0.223662 + 0.387393i 0.955917 − 0.293637i \(-0.0948656\pi\)
−0.732255 + 0.681030i \(0.761532\pi\)
\(12\) 0 0
\(13\) 56.8747 0.0933385 0.0466693 − 0.998910i \(-0.485139\pi\)
0.0466693 + 0.998910i \(0.485139\pi\)
\(14\) 0 0
\(15\) −867.516 −0.995518
\(16\) 0 0
\(17\) 708.446 + 1227.06i 0.594545 + 1.02978i 0.993611 + 0.112859i \(0.0360009\pi\)
−0.399066 + 0.916922i \(0.630666\pi\)
\(18\) 0 0
\(19\) −646.297 + 1119.42i −0.410722 + 0.711392i −0.994969 − 0.100184i \(-0.968057\pi\)
0.584247 + 0.811576i \(0.301390\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2334.31 − 4043.14i 0.920107 − 1.59367i 0.120859 − 0.992670i \(-0.461435\pi\)
0.799248 − 0.601002i \(-0.205232\pi\)
\(24\) 0 0
\(25\) −3083.08 − 5340.05i −0.986586 − 1.70882i
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −5003.10 −1.10470 −0.552349 − 0.833613i \(-0.686269\pi\)
−0.552349 + 0.833613i \(0.686269\pi\)
\(30\) 0 0
\(31\) −3722.17 − 6446.99i −0.695653 − 1.20491i −0.969960 − 0.243264i \(-0.921782\pi\)
0.274307 − 0.961642i \(-0.411551\pi\)
\(32\) 0 0
\(33\) 807.822 − 1399.19i 0.129131 − 0.223662i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4540.45 − 7864.29i 0.545249 − 0.944399i −0.453342 − 0.891336i \(-0.649768\pi\)
0.998591 − 0.0530623i \(-0.0168982\pi\)
\(38\) 0 0
\(39\) −255.936 − 443.295i −0.0269445 − 0.0466693i
\(40\) 0 0
\(41\) −6628.36 −0.615809 −0.307905 − 0.951417i \(-0.599628\pi\)
−0.307905 + 0.951417i \(0.599628\pi\)
\(42\) 0 0
\(43\) −12924.5 −1.06597 −0.532983 − 0.846126i \(-0.678929\pi\)
−0.532983 + 0.846126i \(0.678929\pi\)
\(44\) 0 0
\(45\) 3903.82 + 6761.62i 0.287381 + 0.497759i
\(46\) 0 0
\(47\) −2749.00 + 4761.41i −0.181522 + 0.314406i −0.942399 − 0.334490i \(-0.891436\pi\)
0.760877 + 0.648896i \(0.224769\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 6376.01 − 11043.6i 0.343260 − 0.594545i
\(52\) 0 0
\(53\) −14769.5 − 25581.6i −0.722233 − 1.25094i −0.960103 − 0.279648i \(-0.909782\pi\)
0.237869 − 0.971297i \(-0.423551\pi\)
\(54\) 0 0
\(55\) 17303.7 0.771314
\(56\) 0 0
\(57\) 11633.4 0.474261
\(58\) 0 0
\(59\) 529.253 + 916.693i 0.0197940 + 0.0342842i 0.875753 − 0.482760i \(-0.160366\pi\)
−0.855959 + 0.517044i \(0.827032\pi\)
\(60\) 0 0
\(61\) 4743.93 − 8216.72i 0.163235 − 0.282731i −0.772792 − 0.634659i \(-0.781140\pi\)
0.936027 + 0.351928i \(0.114474\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2741.10 − 4747.72i 0.0804713 − 0.139380i
\(66\) 0 0
\(67\) −665.516 − 1152.71i −0.0181122 − 0.0313713i 0.856827 − 0.515604i \(-0.172432\pi\)
−0.874939 + 0.484232i \(0.839099\pi\)
\(68\) 0 0
\(69\) −42017.5 −1.06245
\(70\) 0 0
\(71\) 49811.2 1.17268 0.586342 − 0.810064i \(-0.300567\pi\)
0.586342 + 0.810064i \(0.300567\pi\)
\(72\) 0 0
\(73\) −37879.1 − 65608.6i −0.831942 − 1.44097i −0.896496 − 0.443051i \(-0.853896\pi\)
0.0645543 − 0.997914i \(-0.479437\pi\)
\(74\) 0 0
\(75\) −27747.7 + 48060.5i −0.569605 + 0.986586i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −31185.4 + 54014.7i −0.562190 + 0.973742i 0.435115 + 0.900375i \(0.356708\pi\)
−0.997305 + 0.0733670i \(0.976626\pi\)
\(80\) 0 0
\(81\) −3280.50 − 5681.99i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −53221.5 −0.847993 −0.423996 − 0.905664i \(-0.639373\pi\)
−0.423996 + 0.905664i \(0.639373\pi\)
\(84\) 0 0
\(85\) 136575. 2.05033
\(86\) 0 0
\(87\) 22513.9 + 38995.3i 0.318899 + 0.552349i
\(88\) 0 0
\(89\) −59340.9 + 102782.i −0.794108 + 1.37544i 0.129296 + 0.991606i \(0.458728\pi\)
−0.923404 + 0.383829i \(0.874605\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −33499.6 + 58023.0i −0.401635 + 0.695653i
\(94\) 0 0
\(95\) 62297.0 + 107902.i 0.708204 + 1.22664i
\(96\) 0 0
\(97\) 136608. 1.47416 0.737081 − 0.675804i \(-0.236203\pi\)
0.737081 + 0.675804i \(0.236203\pi\)
\(98\) 0 0
\(99\) −14540.8 −0.149108
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 588.6.i.j.373.2 4
7.2 even 3 588.6.a.j.1.1 yes 2
7.3 odd 6 588.6.i.k.361.1 4
7.4 even 3 inner 588.6.i.j.361.2 4
7.5 odd 6 588.6.a.i.1.2 ✓ 2
7.6 odd 2 588.6.i.k.373.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.6.a.i.1.2 ✓ 2 7.5 odd 6
588.6.a.j.1.1 yes 2 7.2 even 3
588.6.i.j.361.2 4 7.4 even 3 inner
588.6.i.j.373.2 4 1.1 even 1 trivial
588.6.i.k.361.1 4 7.3 odd 6
588.6.i.k.373.1 4 7.6 odd 2