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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [592,2,Mod(417,592)] 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("592.417"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 592.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2] 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: \(2\)
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 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 433.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 592.433
Dual form 592.2.i.a.417.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 + 1.73205i) q^{3} +(-1.50000 + 2.59808i) q^{5} +(-2.00000 + 3.46410i) q^{7} +(-0.500000 - 0.866025i) q^{9} +6.00000 q^{11} +(-1.00000 + 1.73205i) q^{13} +(-3.00000 - 5.19615i) q^{15} +(-1.50000 - 2.59808i) q^{17} +(1.00000 - 1.73205i) q^{19} +(-4.00000 - 6.92820i) q^{21} -6.00000 q^{23} +(-2.00000 - 3.46410i) q^{25} -4.00000 q^{27} +3.00000 q^{29} -2.00000 q^{31} +(-6.00000 + 10.3923i) q^{33} +(-6.00000 - 10.3923i) q^{35} +(5.50000 - 2.59808i) q^{37} +(-2.00000 - 3.46410i) q^{39} +(-1.50000 + 2.59808i) q^{41} +4.00000 q^{43} +3.00000 q^{45} +6.00000 q^{47} +(-4.50000 - 7.79423i) q^{49} +6.00000 q^{51} +(3.00000 + 5.19615i) q^{53} +(-9.00000 + 15.5885i) q^{55} +(2.00000 + 3.46410i) q^{57} +(0.500000 - 0.866025i) q^{61} +4.00000 q^{63} +(-3.00000 - 5.19615i) q^{65} +(1.00000 - 1.73205i) q^{67} +(6.00000 - 10.3923i) q^{69} +(-6.00000 + 10.3923i) q^{71} -10.0000 q^{73} +8.00000 q^{75} +(-12.0000 + 20.7846i) q^{77} +(7.00000 - 12.1244i) q^{79} +(5.50000 - 9.52628i) q^{81} +(3.00000 + 5.19615i) q^{83} +9.00000 q^{85} +(-3.00000 + 5.19615i) q^{87} +(-1.50000 - 2.59808i) q^{89} +(-4.00000 - 6.92820i) q^{91} +(2.00000 - 3.46410i) q^{93} +(3.00000 + 5.19615i) q^{95} -13.0000 q^{97} +(-3.00000 - 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 3 q^{5} - 4 q^{7} - q^{9} + 12 q^{11} - 2 q^{13} - 6 q^{15} - 3 q^{17} + 2 q^{19} - 8 q^{21} - 12 q^{23} - 4 q^{25} - 8 q^{27} + 6 q^{29} - 4 q^{31} - 12 q^{33} - 12 q^{35} + 11 q^{37}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) −1.00000 + 1.73205i −0.577350 + 1.00000i 0.418432 + 0.908248i \(0.362580\pi\)
−0.995782 + 0.0917517i \(0.970753\pi\)
\(4\) 0 0
\(5\) −1.50000 + 2.59808i −0.670820 + 1.16190i 0.306851 + 0.951757i \(0.400725\pi\)
−0.977672 + 0.210138i \(0.932609\pi\)
\(6\) 0 0
\(7\) −2.00000 + 3.46410i −0.755929 + 1.30931i 0.188982 + 0.981981i \(0.439481\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) −0.500000 − 0.866025i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) −1.00000 + 1.73205i −0.277350 + 0.480384i −0.970725 − 0.240192i \(-0.922790\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 0 0
\(15\) −3.00000 − 5.19615i −0.774597 − 1.34164i
\(16\) 0 0
\(17\) −1.50000 − 2.59808i −0.363803 − 0.630126i 0.624780 − 0.780801i \(-0.285189\pi\)
−0.988583 + 0.150675i \(0.951855\pi\)
\(18\) 0 0
\(19\) 1.00000 − 1.73205i 0.229416 − 0.397360i −0.728219 − 0.685344i \(-0.759652\pi\)
0.957635 + 0.287984i \(0.0929851\pi\)
\(20\) 0 0
\(21\) −4.00000 − 6.92820i −0.872872 − 1.51186i
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −2.00000 − 3.46410i −0.400000 − 0.692820i
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 − 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) −6.00000 + 10.3923i −1.04447 + 1.80907i
\(34\) 0 0
\(35\) −6.00000 − 10.3923i −1.01419 − 1.75662i
\(36\) 0 0
\(37\) 5.50000 − 2.59808i 0.904194 − 0.427121i
\(38\) 0 0
\(39\) −2.00000 − 3.46410i −0.320256 − 0.554700i
\(40\) 0 0
\(41\) −1.50000 + 2.59808i −0.234261 + 0.405751i −0.959058 − 0.283211i \(-0.908600\pi\)
0.724797 + 0.688963i \(0.241934\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 3.00000 0.447214
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −4.50000 − 7.79423i −0.642857 − 1.11346i
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 3.00000 + 5.19615i 0.412082 + 0.713746i 0.995117 − 0.0987002i \(-0.0314685\pi\)
−0.583036 + 0.812447i \(0.698135\pi\)
\(54\) 0 0
\(55\) −9.00000 + 15.5885i −1.21356 + 2.10195i
\(56\) 0 0
\(57\) 2.00000 + 3.46410i 0.264906 + 0.458831i
\(58\) 0 0
\(59\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 0.500000 − 0.866025i 0.0640184 − 0.110883i −0.832240 − 0.554416i \(-0.812942\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 0 0
\(63\) 4.00000 0.503953
\(64\) 0 0
\(65\) −3.00000 − 5.19615i −0.372104 − 0.644503i
\(66\) 0 0
\(67\) 1.00000 − 1.73205i 0.122169 − 0.211604i −0.798454 − 0.602056i \(-0.794348\pi\)
0.920623 + 0.390453i \(0.127682\pi\)
\(68\) 0 0
\(69\) 6.00000 − 10.3923i 0.722315 − 1.25109i
\(70\) 0 0
\(71\) −6.00000 + 10.3923i −0.712069 + 1.23334i 0.252010 + 0.967725i \(0.418908\pi\)
−0.964079 + 0.265615i \(0.914425\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 8.00000 0.923760
\(76\) 0 0
\(77\) −12.0000 + 20.7846i −1.36753 + 2.36863i
\(78\) 0 0
\(79\) 7.00000 − 12.1244i 0.787562 − 1.36410i −0.139895 − 0.990166i \(-0.544677\pi\)
0.927457 − 0.373930i \(-0.121990\pi\)
\(80\) 0 0
\(81\) 5.50000 − 9.52628i 0.611111 − 1.05848i
\(82\) 0 0
\(83\) 3.00000 + 5.19615i 0.329293 + 0.570352i 0.982372 − 0.186938i \(-0.0598564\pi\)
−0.653079 + 0.757290i \(0.726523\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 0 0
\(87\) −3.00000 + 5.19615i −0.321634 + 0.557086i
\(88\) 0 0
\(89\) −1.50000 − 2.59808i −0.159000 − 0.275396i 0.775509 − 0.631337i \(-0.217494\pi\)
−0.934508 + 0.355942i \(0.884160\pi\)
\(90\) 0 0
\(91\) −4.00000 − 6.92820i −0.419314 − 0.726273i
\(92\) 0 0
\(93\) 2.00000 − 3.46410i 0.207390 − 0.359211i
\(94\) 0 0
\(95\) 3.00000 + 5.19615i 0.307794 + 0.533114i
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 − 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 0 0
\(99\) −3.00000 − 5.19615i −0.301511 − 0.522233i
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.i.a.433.1 2
4.3 odd 2 74.2.c.b.63.1 yes 2
12.11 even 2 666.2.f.d.433.1 2
37.10 even 3 inner 592.2.i.a.417.1 2
148.11 odd 6 2738.2.a.c.1.1 1
148.47 odd 6 74.2.c.b.47.1 ✓ 2
148.63 odd 6 2738.2.a.a.1.1 1
444.47 even 6 666.2.f.d.343.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.c.b.47.1 ✓ 2 148.47 odd 6
74.2.c.b.63.1 yes 2 4.3 odd 2
592.2.i.a.417.1 2 37.10 even 3 inner
592.2.i.a.433.1 2 1.1 even 1 trivial
666.2.f.d.343.1 2 444.47 even 6
666.2.f.d.433.1 2 12.11 even 2
2738.2.a.a.1.1 1 148.63 odd 6
2738.2.a.c.1.1 1 148.11 odd 6