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,-6,0,0,0,-162,0,90] 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: \(\Q(\sqrt{-3}, \sqrt{5569})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 1393x^{2} + 1392x + 1937664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(18.9064 - 32.7469i\) of defining polynomial
Character \(\chi\) \(=\) 588.361
Dual form 588.6.i.i.373.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-4.50000 + 7.79423i) q^{3} +(-38.8129 - 67.2259i) q^{5} +(-40.5000 - 70.1481i) q^{9} +(-238.690 + 413.423i) q^{11} +63.7544 q^{13} +698.632 q^{15} +(518.813 - 898.610i) q^{17} +(-333.509 - 577.654i) q^{19} +(-1625.81 - 2815.99i) q^{23} +(-1450.38 + 2512.13i) q^{25} +729.000 q^{27} +2300.97 q^{29} +(1858.53 - 3219.06i) q^{31} +(-2148.21 - 3720.81i) q^{33} +(-6122.93 - 10605.2i) q^{37} +(-286.895 + 496.916i) q^{39} +1829.65 q^{41} -20794.2 q^{43} +(-3143.84 + 5445.29i) q^{45} +(-2141.68 - 3709.51i) q^{47} +(4669.32 + 8087.49i) q^{51} +(-12859.2 + 22272.8i) q^{53} +37057.0 q^{55} +6003.16 q^{57} +(-1419.36 + 2458.40i) q^{59} +(8401.60 + 14552.0i) q^{61} +(-2474.49 - 4285.94i) q^{65} +(31267.5 - 54157.0i) q^{67} +29264.6 q^{69} +72301.0 q^{71} +(-27838.4 + 48217.6i) q^{73} +(-13053.4 - 22609.1i) q^{75} +(1994.60 + 3454.74i) q^{79} +(-3280.50 + 5681.99i) q^{81} +46092.2 q^{83} -80546.5 q^{85} +(-10354.4 + 17934.3i) q^{87} +(67692.3 + 117247. i) q^{89} +(16726.7 + 28971.6i) q^{93} +(-25888.9 + 44840.8i) q^{95} -142878. q^{97} +38667.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 18 q^{3} - 6 q^{5} - 162 q^{9} + 90 q^{11} - 1536 q^{13} + 108 q^{15} + 1926 q^{17} + 2248 q^{19} - 6354 q^{23} - 4906 q^{25} + 2916 q^{27} + 21144 q^{29} - 3312 q^{31} + 810 q^{33} - 2104 q^{37}+ \cdots - 14580 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −38.8129 − 67.2259i −0.694306 − 1.20257i −0.970414 − 0.241446i \(-0.922378\pi\)
0.276109 − 0.961126i \(-0.410955\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −40.5000 − 70.1481i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) −238.690 + 413.423i −0.594775 + 1.03018i 0.398804 + 0.917036i \(0.369425\pi\)
−0.993579 + 0.113144i \(0.963908\pi\)
\(12\) 0 0
\(13\) 63.7544 0.104629 0.0523145 − 0.998631i \(-0.483340\pi\)
0.0523145 + 0.998631i \(0.483340\pi\)
\(14\) 0 0
\(15\) 698.632 0.801715
\(16\) 0 0
\(17\) 518.813 − 898.610i 0.435400 − 0.754135i −0.561928 − 0.827186i \(-0.689940\pi\)
0.997328 + 0.0730511i \(0.0232736\pi\)
\(18\) 0 0
\(19\) −333.509 − 577.654i −0.211945 − 0.367100i 0.740378 − 0.672191i \(-0.234646\pi\)
−0.952323 + 0.305091i \(0.901313\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1625.81 − 2815.99i −0.640842 − 1.10997i −0.985245 − 0.171149i \(-0.945252\pi\)
0.344403 − 0.938822i \(-0.388081\pi\)
\(24\) 0 0
\(25\) −1450.38 + 2512.13i −0.464121 + 0.803881i
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) 2300.97 0.508061 0.254031 − 0.967196i \(-0.418244\pi\)
0.254031 + 0.967196i \(0.418244\pi\)
\(30\) 0 0
\(31\) 1858.53 − 3219.06i 0.347348 − 0.601624i −0.638430 − 0.769680i \(-0.720416\pi\)
0.985777 + 0.168056i \(0.0537490\pi\)
\(32\) 0 0
\(33\) −2148.21 − 3720.81i −0.343393 − 0.594775i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6122.93 − 10605.2i −0.735284 − 1.27355i −0.954599 − 0.297895i \(-0.903716\pi\)
0.219315 − 0.975654i \(-0.429618\pi\)
\(38\) 0 0
\(39\) −286.895 + 496.916i −0.0302038 + 0.0523145i
\(40\) 0 0
\(41\) 1829.65 0.169984 0.0849920 − 0.996382i \(-0.472914\pi\)
0.0849920 + 0.996382i \(0.472914\pi\)
\(42\) 0 0
\(43\) −20794.2 −1.71503 −0.857513 − 0.514463i \(-0.827991\pi\)
−0.857513 + 0.514463i \(0.827991\pi\)
\(44\) 0 0
\(45\) −3143.84 + 5445.29i −0.231435 + 0.400858i
\(46\) 0 0
\(47\) −2141.68 − 3709.51i −0.141420 − 0.244947i 0.786612 − 0.617448i \(-0.211833\pi\)
−0.928032 + 0.372502i \(0.878500\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 4669.32 + 8087.49i 0.251378 + 0.435400i
\(52\) 0 0
\(53\) −12859.2 + 22272.8i −0.628818 + 1.08914i 0.358972 + 0.933348i \(0.383127\pi\)
−0.987789 + 0.155796i \(0.950206\pi\)
\(54\) 0 0
\(55\) 37057.0 1.65182
\(56\) 0 0
\(57\) 6003.16 0.244733
\(58\) 0 0
\(59\) −1419.36 + 2458.40i −0.0530837 + 0.0919437i −0.891346 − 0.453323i \(-0.850238\pi\)
0.838262 + 0.545267i \(0.183572\pi\)
\(60\) 0 0
\(61\) 8401.60 + 14552.0i 0.289093 + 0.500723i 0.973593 − 0.228289i \(-0.0733131\pi\)
−0.684501 + 0.729012i \(0.739980\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2474.49 − 4285.94i −0.0726445 − 0.125824i
\(66\) 0 0
\(67\) 31267.5 − 54157.0i 0.850955 − 1.47390i −0.0293918 − 0.999568i \(-0.509357\pi\)
0.880347 − 0.474330i \(-0.157310\pi\)
\(68\) 0 0
\(69\) 29264.6 0.739981
\(70\) 0 0
\(71\) 72301.0 1.70215 0.851077 − 0.525042i \(-0.175950\pi\)
0.851077 + 0.525042i \(0.175950\pi\)
\(72\) 0 0
\(73\) −27838.4 + 48217.6i −0.611417 + 1.05901i 0.379584 + 0.925157i \(0.376067\pi\)
−0.991002 + 0.133849i \(0.957266\pi\)
\(74\) 0 0
\(75\) −13053.4 − 22609.1i −0.267960 − 0.464121i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1994.60 + 3454.74i 0.0359573 + 0.0622799i 0.883444 − 0.468537i \(-0.155219\pi\)
−0.847487 + 0.530817i \(0.821885\pi\)
\(80\) 0 0
\(81\) −3280.50 + 5681.99i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 46092.2 0.734400 0.367200 − 0.930142i \(-0.380316\pi\)
0.367200 + 0.930142i \(0.380316\pi\)
\(84\) 0 0
\(85\) −80546.5 −1.20920
\(86\) 0 0
\(87\) −10354.4 + 17934.3i −0.146665 + 0.254031i
\(88\) 0 0
\(89\) 67692.3 + 117247.i 0.905867 + 1.56901i 0.819749 + 0.572723i \(0.194113\pi\)
0.0861180 + 0.996285i \(0.472554\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 16726.7 + 28971.6i 0.200541 + 0.347348i
\(94\) 0 0
\(95\) −25888.9 + 44840.8i −0.294309 + 0.509759i
\(96\) 0 0
\(97\) −142878. −1.54183 −0.770914 − 0.636939i \(-0.780200\pi\)
−0.770914 + 0.636939i \(0.780200\pi\)
\(98\) 0 0
\(99\) 38667.8 0.396517
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.i.361.1 4
7.2 even 3 inner 588.6.i.i.373.1 4
7.3 odd 6 84.6.a.c.1.1 ✓ 2
7.4 even 3 588.6.a.k.1.2 2
7.5 odd 6 588.6.i.l.373.2 4
7.6 odd 2 588.6.i.l.361.2 4
21.17 even 6 252.6.a.h.1.2 2
28.3 even 6 336.6.a.x.1.1 2
84.59 odd 6 1008.6.a.bo.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.a.c.1.1 ✓ 2 7.3 odd 6
252.6.a.h.1.2 2 21.17 even 6
336.6.a.x.1.1 2 28.3 even 6
588.6.a.k.1.2 2 7.4 even 3
588.6.i.i.361.1 4 1.1 even 1 trivial
588.6.i.i.373.1 4 7.2 even 3 inner
588.6.i.l.361.2 4 7.6 odd 2
588.6.i.l.373.2 4 7.5 odd 6
1008.6.a.bo.1.2 2 84.59 odd 6