Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [592,2,Mod(33,592)] 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("592.33"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(592, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 0, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.bc (of order \(9\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,3] 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: \(4.72714379966\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 497.1
Root \(-0.173648 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 592.497
Dual form 592.2.bc.b.81.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.326352 - 0.118782i) q^{3} +(-0.0209445 + 0.118782i) q^{5} +(-0.233956 + 1.32683i) q^{7} +(-2.20574 + 1.85083i) q^{9} +(2.26604 + 3.92490i) q^{11} +(-0.592396 - 0.497079i) q^{13} +(0.00727396 + 0.0412527i) q^{15} +(-2.29813 + 1.92836i) q^{17} +(-1.91875 + 0.698367i) q^{19} +(0.0812519 + 0.460802i) q^{21} +(0.0282185 - 0.0488759i) q^{23} +(4.68479 + 1.70513i) q^{25} +(-1.02094 + 1.76833i) q^{27} +(2.89053 + 5.00654i) q^{29} +3.34730 q^{31} +(1.20574 + 1.01173i) q^{33} +(-0.152704 - 0.0555796i) q^{35} +(5.60607 + 2.36051i) q^{37} +(-0.252374 - 0.0918566i) q^{39} +(-5.47565 - 4.59462i) q^{41} -9.31315 q^{43} +(-0.173648 - 0.300767i) q^{45} +(4.25877 - 7.37641i) q^{47} +(4.87211 + 1.77330i) q^{49} +(-0.520945 + 0.902302i) q^{51} +(0.482926 + 2.73881i) q^{53} +(-0.513671 + 0.186961i) q^{55} +(-0.543233 + 0.455827i) q^{57} +(2.25624 + 12.7958i) q^{59} +(-8.82295 - 7.40333i) q^{61} +(-1.93969 - 3.35965i) q^{63} +(0.0714517 - 0.0599551i) q^{65} +(-0.889185 + 5.04282i) q^{67} +(0.00340357 - 0.0193026i) q^{69} +(12.6236 - 4.59462i) q^{71} -8.71688 q^{73} +1.73143 q^{75} +(-5.73783 + 2.08840i) q^{77} +(-0.720285 + 4.08494i) q^{79} +(1.37686 - 7.80856i) q^{81} +(4.53596 - 3.80612i) q^{83} +(-0.180922 - 0.313366i) q^{85} +(1.53802 + 1.29055i) q^{87} +(1.32295 + 7.50281i) q^{89} +(0.798133 - 0.669713i) q^{91} +(1.09240 - 0.397600i) q^{93} +(-0.0427664 - 0.242540i) q^{95} +(5.12061 - 8.86916i) q^{97} +(-12.2626 - 4.46324i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 3 q^{5} - 6 q^{7} - 3 q^{9} + 9 q^{11} + 18 q^{15} - 9 q^{19} + 3 q^{21} + 15 q^{23} + 21 q^{25} - 3 q^{27} + 18 q^{31} - 3 q^{33} - 3 q^{35} + 9 q^{37} - 18 q^{39} + 6 q^{41} - 12 q^{43}+ \cdots - 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{1}{9}\right)\) \(1\) \(1\)

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
\(3\) 0.326352 − 0.118782i 0.188419 − 0.0685790i −0.246087 − 0.969248i \(-0.579145\pi\)
0.434507 + 0.900669i \(0.356923\pi\)
\(4\) 0 0
\(5\) −0.0209445 + 0.118782i −0.00936668 + 0.0531211i −0.989133 − 0.147024i \(-0.953030\pi\)
0.979766 + 0.200146i \(0.0641415\pi\)
\(6\) 0 0
\(7\) −0.233956 + 1.32683i −0.0884269 + 0.501494i 0.908137 + 0.418672i \(0.137504\pi\)
−0.996564 + 0.0828217i \(0.973607\pi\)
\(8\) 0 0
\(9\) −2.20574 + 1.85083i −0.735246 + 0.616944i
\(10\) 0 0
\(11\) 2.26604 + 3.92490i 0.683238 + 1.18340i 0.973987 + 0.226604i \(0.0727622\pi\)
−0.290749 + 0.956799i \(0.593904\pi\)
\(12\) 0 0
\(13\) −0.592396 − 0.497079i −0.164301 − 0.137865i 0.556930 − 0.830559i \(-0.311979\pi\)
−0.721231 + 0.692694i \(0.756424\pi\)
\(14\) 0 0
\(15\) 0.00727396 + 0.0412527i 0.00187813 + 0.0106514i
\(16\) 0 0
\(17\) −2.29813 + 1.92836i −0.557379 + 0.467697i −0.877431 − 0.479703i \(-0.840744\pi\)
0.320051 + 0.947400i \(0.396300\pi\)
\(18\) 0 0
\(19\) −1.91875 + 0.698367i −0.440191 + 0.160216i −0.552602 − 0.833445i \(-0.686365\pi\)
0.112411 + 0.993662i \(0.464143\pi\)
\(20\) 0 0
\(21\) 0.0812519 + 0.460802i 0.0177306 + 0.100555i
\(22\) 0 0
\(23\) 0.0282185 − 0.0488759i 0.00588396 − 0.0101913i −0.863068 − 0.505087i \(-0.831460\pi\)
0.868952 + 0.494896i \(0.164794\pi\)
\(24\) 0 0
\(25\) 4.68479 + 1.70513i 0.936959 + 0.341025i
\(26\) 0 0
\(27\) −1.02094 + 1.76833i −0.196481 + 0.340315i
\(28\) 0 0
\(29\) 2.89053 + 5.00654i 0.536758 + 0.929692i 0.999076 + 0.0429778i \(0.0136845\pi\)
−0.462318 + 0.886714i \(0.652982\pi\)
\(30\) 0 0
\(31\) 3.34730 0.601192 0.300596 − 0.953752i \(-0.402814\pi\)
0.300596 + 0.953752i \(0.402814\pi\)
\(32\) 0 0
\(33\) 1.20574 + 1.01173i 0.209892 + 0.176120i
\(34\) 0 0
\(35\) −0.152704 − 0.0555796i −0.0258116 − 0.00939466i
\(36\) 0 0
\(37\) 5.60607 + 2.36051i 0.921632 + 0.388066i
\(38\) 0 0
\(39\) −0.252374 − 0.0918566i −0.0404122 − 0.0147088i
\(40\) 0 0
\(41\) −5.47565 − 4.59462i −0.855153 − 0.717559i 0.105765 − 0.994391i \(-0.466271\pi\)
−0.960918 + 0.276832i \(0.910715\pi\)
\(42\) 0 0
\(43\) −9.31315 −1.42024 −0.710121 − 0.704080i \(-0.751360\pi\)
−0.710121 + 0.704080i \(0.751360\pi\)
\(44\) 0 0
\(45\) −0.173648 − 0.300767i −0.0258859 − 0.0448358i
\(46\) 0 0
\(47\) 4.25877 − 7.37641i 0.621206 − 1.07596i −0.368056 − 0.929804i \(-0.619977\pi\)
0.989262 − 0.146156i \(-0.0466901\pi\)
\(48\) 0 0
\(49\) 4.87211 + 1.77330i 0.696016 + 0.253329i
\(50\) 0 0
\(51\) −0.520945 + 0.902302i −0.0729468 + 0.126348i
\(52\) 0 0
\(53\) 0.482926 + 2.73881i 0.0663350 + 0.376204i 0.999844 + 0.0176510i \(0.00561877\pi\)
−0.933509 + 0.358553i \(0.883270\pi\)
\(54\) 0 0
\(55\) −0.513671 + 0.186961i −0.0692633 + 0.0252098i
\(56\) 0 0
\(57\) −0.543233 + 0.455827i −0.0719530 + 0.0603757i
\(58\) 0 0
\(59\) 2.25624 + 12.7958i 0.293738 + 1.66587i 0.672288 + 0.740290i \(0.265312\pi\)
−0.378550 + 0.925581i \(0.623577\pi\)
\(60\) 0 0
\(61\) −8.82295 − 7.40333i −1.12966 − 0.947900i −0.130611 − 0.991434i \(-0.541694\pi\)
−0.999052 + 0.0435341i \(0.986138\pi\)
\(62\) 0 0
\(63\) −1.93969 − 3.35965i −0.244378 − 0.423276i
\(64\) 0 0
\(65\) 0.0714517 − 0.0599551i 0.00886250 − 0.00743652i
\(66\) 0 0
\(67\) −0.889185 + 5.04282i −0.108631 + 0.616079i 0.881076 + 0.472974i \(0.156820\pi\)
−0.989708 + 0.143104i \(0.954292\pi\)
\(68\) 0 0
\(69\) 0.00340357 − 0.0193026i 0.000409741 − 0.00232376i
\(70\) 0 0
\(71\) 12.6236 − 4.59462i 1.49815 − 0.545281i 0.542567 − 0.840013i \(-0.317453\pi\)
0.955580 + 0.294732i \(0.0952304\pi\)
\(72\) 0 0
\(73\) −8.71688 −1.02023 −0.510117 − 0.860105i \(-0.670398\pi\)
−0.510117 + 0.860105i \(0.670398\pi\)
\(74\) 0 0
\(75\) 1.73143 0.199928
\(76\) 0 0
\(77\) −5.73783 + 2.08840i −0.653886 + 0.237995i
\(78\) 0 0
\(79\) −0.720285 + 4.08494i −0.0810384 + 0.459592i 0.917103 + 0.398650i \(0.130521\pi\)
−0.998141 + 0.0609412i \(0.980590\pi\)
\(80\) 0 0
\(81\) 1.37686 − 7.80856i 0.152984 − 0.867617i
\(82\) 0 0
\(83\) 4.53596 − 3.80612i 0.497886 − 0.417776i −0.358956 − 0.933354i \(-0.616867\pi\)
0.856843 + 0.515578i \(0.172423\pi\)
\(84\) 0 0
\(85\) −0.180922 − 0.313366i −0.0196238 − 0.0339894i
\(86\) 0 0
\(87\) 1.53802 + 1.29055i 0.164893 + 0.138362i
\(88\) 0 0
\(89\) 1.32295 + 7.50281i 0.140232 + 0.795297i 0.971072 + 0.238785i \(0.0767492\pi\)
−0.830840 + 0.556511i \(0.812140\pi\)
\(90\) 0 0
\(91\) 0.798133 − 0.669713i 0.0836671 − 0.0702050i
\(92\) 0 0
\(93\) 1.09240 − 0.397600i 0.113276 − 0.0412292i
\(94\) 0 0
\(95\) −0.0427664 − 0.242540i −0.00438774 − 0.0248841i
\(96\) 0 0
\(97\) 5.12061 − 8.86916i 0.519920 − 0.900527i −0.479812 − 0.877371i \(-0.659295\pi\)
0.999732 − 0.0231560i \(-0.00737146\pi\)
\(98\) 0 0
\(99\) −12.2626 − 4.46324i −1.23244 − 0.448572i
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 592.2.bc.b.497.1 6
4.3 odd 2 74.2.f.a.53.1 yes 6
12.11 even 2 666.2.x.c.127.1 6
37.7 even 9 inner 592.2.bc.b.81.1 6
148.7 odd 18 74.2.f.a.7.1 ✓ 6
148.83 odd 18 2738.2.a.m.1.2 3
148.139 odd 18 2738.2.a.p.1.2 3
444.155 even 18 666.2.x.c.451.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.a.7.1 ✓ 6 148.7 odd 18
74.2.f.a.53.1 yes 6 4.3 odd 2
592.2.bc.b.81.1 6 37.7 even 9 inner
592.2.bc.b.497.1 6 1.1 even 1 trivial
666.2.x.c.127.1 6 12.11 even 2
666.2.x.c.451.1 6 444.155 even 18
2738.2.a.m.1.2 3 148.83 odd 18
2738.2.a.p.1.2 3 148.139 odd 18