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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 81.1
Root \(-0.173648 + 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 592.81
Dual form 592.2.bc.b.497.1

$q$-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 \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\) \(1\) \(1\)

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 0.434507 − 0.900669i \(-0.356923\pi\)
−0.246087 + 0.969248i \(0.579145\pi\)
\(4\) 0 0
\(5\) −0.0209445 − 0.118782i −0.00936668 − 0.0531211i 0.979766 − 0.200146i \(-0.0641415\pi\)
−0.989133 + 0.147024i \(0.953030\pi\)
\(6\) 0 0
\(7\) −0.233956 − 1.32683i −0.0884269 − 0.501494i −0.996564 − 0.0828217i \(-0.973607\pi\)
0.908137 − 0.418672i \(-0.137504\pi\)
\(8\) 0 0
\(9\) −2.20574 − 1.85083i −0.735246 − 0.616944i
\(10\) 0 0
\(11\) 2.26604 − 3.92490i 0.683238 − 1.18340i −0.290749 − 0.956799i \(-0.593904\pi\)
0.973987 − 0.226604i \(-0.0727622\pi\)
\(12\) 0 0
\(13\) −0.592396 + 0.497079i −0.164301 + 0.137865i −0.721231 − 0.692694i \(-0.756424\pi\)
0.556930 + 0.830559i \(0.311979\pi\)
\(14\) 0 0
\(15\) 0.00727396 − 0.0412527i 0.00187813 − 0.0106514i
\(16\) 0 0
\(17\) −2.29813 − 1.92836i −0.557379 − 0.467697i 0.320051 − 0.947400i \(-0.396300\pi\)
−0.877431 + 0.479703i \(0.840744\pi\)
\(18\) 0 0
\(19\) −1.91875 − 0.698367i −0.440191 − 0.160216i 0.112411 − 0.993662i \(-0.464143\pi\)
−0.552602 + 0.833445i \(0.686365\pi\)
\(20\) 0 0
\(21\) 0.0812519 − 0.460802i 0.0177306 − 0.100555i
\(22\) 0 0
\(23\) 0.0282185 + 0.0488759i 0.00588396 + 0.0101913i 0.868952 − 0.494896i \(-0.164794\pi\)
−0.863068 + 0.505087i \(0.831460\pi\)
\(24\) 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 −0.462318 − 0.886714i \(-0.652982\pi\)
0.999076 − 0.0429778i \(-0.0136845\pi\)
\(30\) 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 −0.960918 − 0.276832i \(-0.910715\pi\)
0.105765 + 0.994391i \(0.466271\pi\)
\(42\) 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 0.989262 + 0.146156i \(0.0466901\pi\)
−0.368056 + 0.929804i \(0.619977\pi\)
\(48\) 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 −0.933509 − 0.358553i \(-0.883270\pi\)
0.999844 − 0.0176510i \(-0.00561877\pi\)
\(54\) 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 −0.378550 − 0.925581i \(-0.623577\pi\)
0.672288 − 0.740290i \(-0.265312\pi\)
\(60\) 0 0
\(61\) −8.82295 + 7.40333i −1.12966 + 0.947900i −0.999052 − 0.0435341i \(-0.986138\pi\)
−0.130611 + 0.991434i \(0.541694\pi\)
\(62\) 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 −0.989708 − 0.143104i \(-0.954292\pi\)
0.881076 − 0.472974i \(-0.156820\pi\)
\(68\) 0 0
\(69\) 0.00340357 + 0.0193026i 0.000409741 + 0.00232376i
\(70\) 0 0
\(71\) 12.6236 + 4.59462i 1.49815 + 0.545281i 0.955580 − 0.294732i \(-0.0952304\pi\)
0.542567 + 0.840013i \(0.317453\pi\)
\(72\) 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 −0.998141 − 0.0609412i \(-0.980590\pi\)
0.917103 − 0.398650i \(-0.130521\pi\)
\(80\) 0 0
\(81\) 1.37686 + 7.80856i 0.152984 + 0.867617i
\(82\) 0 0
\(83\) 4.53596 + 3.80612i 0.497886 + 0.417776i 0.856843 − 0.515578i \(-0.172423\pi\)
−0.358956 + 0.933354i \(0.616867\pi\)
\(84\) 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 −0.830840 − 0.556511i \(-0.812140\pi\)
0.971072 − 0.238785i \(-0.0767492\pi\)
\(90\) 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 0.999732 + 0.0231560i \(0.00737146\pi\)
−0.479812 + 0.877371i \(0.659295\pi\)
\(98\) 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.81.1 6
4.3 odd 2 74.2.f.a.7.1 ✓ 6
12.11 even 2 666.2.x.c.451.1 6
37.16 even 9 inner 592.2.bc.b.497.1 6
148.107 odd 18 2738.2.a.m.1.2 3
148.115 odd 18 2738.2.a.p.1.2 3
148.127 odd 18 74.2.f.a.53.1 yes 6
444.275 even 18 666.2.x.c.127.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.a.7.1 ✓ 6 4.3 odd 2
74.2.f.a.53.1 yes 6 148.127 odd 18
592.2.bc.b.81.1 6 1.1 even 1 trivial
592.2.bc.b.497.1 6 37.16 even 9 inner
666.2.x.c.127.1 6 444.275 even 18
666.2.x.c.451.1 6 12.11 even 2
2738.2.a.m.1.2 3 148.107 odd 18
2738.2.a.p.1.2 3 148.115 odd 18