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 = [2,0,9,0,-34,0,0,0,-81,0,332] 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: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 373.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 588.373
Dual form 588.6.i.e.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)
 
\(f(q)\) \(=\) \(q+(4.50000 + 7.79423i) q^{3} +(-17.0000 + 29.4449i) q^{5} +(-40.5000 + 70.1481i) q^{9} +(166.000 + 287.520i) q^{11} +1026.00 q^{13} -306.000 q^{15} +(461.000 + 798.475i) q^{17} +(226.000 - 391.443i) q^{19} +(1888.00 - 3270.11i) q^{23} +(984.500 + 1705.20i) q^{25} -729.000 q^{27} +1166.00 q^{29} +(-4896.00 - 8480.12i) q^{31} +(-1494.00 + 2587.68i) q^{33} +(-1195.00 + 2069.80i) q^{37} +(4617.00 + 7996.88i) q^{39} +7230.00 q^{41} +4652.00 q^{43} +(-1377.00 - 2385.03i) q^{45} +(12336.0 - 21366.6i) q^{47} +(-4149.00 + 7186.28i) q^{51} +(-555.000 - 961.288i) q^{53} -11288.0 q^{55} +4068.00 q^{57} +(23446.0 + 40609.7i) q^{59} +(-4881.00 + 8454.14i) q^{61} +(-17442.0 + 30210.4i) q^{65} +(13126.0 + 22734.9i) q^{67} +33984.0 q^{69} +65440.0 q^{71} +(-2803.00 - 4854.94i) q^{73} +(-8860.50 + 15346.8i) q^{75} +(4920.00 - 8521.69i) q^{79} +(-3280.50 - 5681.99i) q^{81} -61108.0 q^{83} -31348.0 q^{85} +(5247.00 + 9088.07i) q^{87} +(-31479.0 + 54523.2i) q^{89} +(44064.0 - 76321.1i) q^{93} +(7684.00 + 13309.1i) q^{95} +37838.0 q^{97} -26892.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{3} - 34 q^{5} - 81 q^{9} + 332 q^{11} + 2052 q^{13} - 612 q^{15} + 922 q^{17} + 452 q^{19} + 3776 q^{23} + 1969 q^{25} - 1458 q^{27} + 2332 q^{29} - 9792 q^{31} - 2988 q^{33} - 2390 q^{37} + 9234 q^{39}+ \cdots - 53784 q^{99}+O(q^{100}) \) 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
\(5\) −17.0000 + 29.4449i −0.304105 + 0.526726i −0.977062 − 0.212956i \(-0.931691\pi\)
0.672956 + 0.739682i \(0.265024\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 166.000 + 287.520i 0.413644 + 0.716452i 0.995285 − 0.0969935i \(-0.0309226\pi\)
−0.581641 + 0.813445i \(0.697589\pi\)
\(12\) 0 0
\(13\) 1026.00 1.68379 0.841897 − 0.539638i \(-0.181439\pi\)
0.841897 + 0.539638i \(0.181439\pi\)
\(14\) 0 0
\(15\) −306.000 −0.351150
\(16\) 0 0
\(17\) 461.000 + 798.475i 0.386882 + 0.670099i 0.992028 − 0.126015i \(-0.0402187\pi\)
−0.605146 + 0.796114i \(0.706885\pi\)
\(18\) 0 0
\(19\) 226.000 − 391.443i 0.143623 − 0.248763i −0.785235 − 0.619198i \(-0.787458\pi\)
0.928858 + 0.370435i \(0.120791\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1888.00 − 3270.11i 0.744188 − 1.28897i −0.206386 − 0.978471i \(-0.566170\pi\)
0.950573 − 0.310500i \(-0.100496\pi\)
\(24\) 0 0
\(25\) 984.500 + 1705.20i 0.315040 + 0.545665i
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 1166.00 0.257456 0.128728 − 0.991680i \(-0.458911\pi\)
0.128728 + 0.991680i \(0.458911\pi\)
\(30\) 0 0
\(31\) −4896.00 − 8480.12i −0.915034 − 1.58489i −0.806852 − 0.590754i \(-0.798830\pi\)
−0.108182 − 0.994131i \(-0.534503\pi\)
\(32\) 0 0
\(33\) −1494.00 + 2587.68i −0.238817 + 0.413644i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1195.00 + 2069.80i −0.143504 + 0.248556i −0.928814 − 0.370547i \(-0.879170\pi\)
0.785310 + 0.619103i \(0.212504\pi\)
\(38\) 0 0
\(39\) 4617.00 + 7996.88i 0.486069 + 0.841897i
\(40\) 0 0
\(41\) 7230.00 0.671705 0.335853 − 0.941915i \(-0.390976\pi\)
0.335853 + 0.941915i \(0.390976\pi\)
\(42\) 0 0
\(43\) 4652.00 0.383679 0.191840 − 0.981426i \(-0.438555\pi\)
0.191840 + 0.981426i \(0.438555\pi\)
\(44\) 0 0
\(45\) −1377.00 − 2385.03i −0.101368 − 0.175575i
\(46\) 0 0
\(47\) 12336.0 − 21366.6i 0.814572 − 1.41088i −0.0950621 − 0.995471i \(-0.530305\pi\)
0.909635 − 0.415410i \(-0.136362\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4149.00 + 7186.28i −0.223366 + 0.386882i
\(52\) 0 0
\(53\) −555.000 − 961.288i −0.0271396 − 0.0470072i 0.852137 − 0.523319i \(-0.175307\pi\)
−0.879276 + 0.476312i \(0.841973\pi\)
\(54\) 0 0
\(55\) −11288.0 −0.503165
\(56\) 0 0
\(57\) 4068.00 0.165842
\(58\) 0 0
\(59\) 23446.0 + 40609.7i 0.876877 + 1.51880i 0.854750 + 0.519041i \(0.173711\pi\)
0.0221275 + 0.999755i \(0.492956\pi\)
\(60\) 0 0
\(61\) −4881.00 + 8454.14i −0.167952 + 0.290901i −0.937700 − 0.347447i \(-0.887049\pi\)
0.769748 + 0.638348i \(0.220382\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −17442.0 + 30210.4i −0.512051 + 0.886898i
\(66\) 0 0
\(67\) 13126.0 + 22734.9i 0.357228 + 0.618737i 0.987497 − 0.157640i \(-0.0503886\pi\)
−0.630269 + 0.776377i \(0.717055\pi\)
\(68\) 0 0
\(69\) 33984.0 0.859314
\(70\) 0 0
\(71\) 65440.0 1.54063 0.770313 − 0.637666i \(-0.220100\pi\)
0.770313 + 0.637666i \(0.220100\pi\)
\(72\) 0 0
\(73\) −2803.00 − 4854.94i −0.0615625 − 0.106629i 0.833602 − 0.552366i \(-0.186275\pi\)
−0.895164 + 0.445737i \(0.852942\pi\)
\(74\) 0 0
\(75\) −8860.50 + 15346.8i −0.181888 + 0.315040i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 4920.00 − 8521.69i 0.0886946 − 0.153624i −0.818265 − 0.574841i \(-0.805064\pi\)
0.906960 + 0.421218i \(0.138397\pi\)
\(80\) 0 0
\(81\) −3280.50 − 5681.99i −0.0555556 − 0.0962250i
\(82\) 0 0
\(83\) −61108.0 −0.973650 −0.486825 − 0.873500i \(-0.661845\pi\)
−0.486825 + 0.873500i \(0.661845\pi\)
\(84\) 0 0
\(85\) −31348.0 −0.470611
\(86\) 0 0
\(87\) 5247.00 + 9088.07i 0.0743212 + 0.128728i
\(88\) 0 0
\(89\) −31479.0 + 54523.2i −0.421256 + 0.729636i −0.996063 − 0.0886527i \(-0.971744\pi\)
0.574807 + 0.818289i \(0.305077\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 44064.0 − 76321.1i 0.528295 − 0.915034i
\(94\) 0 0
\(95\) 7684.00 + 13309.1i 0.0873531 + 0.151300i
\(96\) 0 0
\(97\) 37838.0 0.408318 0.204159 − 0.978938i \(-0.434554\pi\)
0.204159 + 0.978938i \(0.434554\pi\)
\(98\) 0 0
\(99\) −26892.0 −0.275762
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.e.373.1 2
7.2 even 3 588.6.a.b.1.1 1
7.3 odd 6 588.6.i.c.361.1 2
7.4 even 3 inner 588.6.i.e.361.1 2
7.5 odd 6 84.6.a.b.1.1 ✓ 1
7.6 odd 2 588.6.i.c.373.1 2
21.5 even 6 252.6.a.c.1.1 1
28.19 even 6 336.6.a.e.1.1 1
84.47 odd 6 1008.6.a.u.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.6.a.b.1.1 ✓ 1 7.5 odd 6
252.6.a.c.1.1 1 21.5 even 6
336.6.a.e.1.1 1 28.19 even 6
588.6.a.b.1.1 1 7.2 even 3
588.6.i.c.361.1 2 7.3 odd 6
588.6.i.c.373.1 2 7.6 odd 2
588.6.i.e.361.1 2 7.4 even 3 inner
588.6.i.e.373.1 2 1.1 even 1 trivial
1008.6.a.u.1.1 1 84.47 odd 6