Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [567,2,Mod(190,567)] 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("567.190"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(567, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.f (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-6,3,0,-3,18,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1156923.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 190.1
Root \(0.500000 - 1.51496i\) of defining polynomial
Character \(\chi\) \(=\) 567.190
Dual form 567.2.f.m.379.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.33454 - 2.31149i) q^{2} +(-2.56199 + 4.43750i) q^{4} +(0.727452 - 1.25998i) q^{5} +(-0.500000 - 0.866025i) q^{7} +8.33816 q^{8} -3.88325 q^{10} +(0.772548 + 1.33809i) q^{11} +(-2.94163 + 5.09505i) q^{13} +(-1.33454 + 2.31149i) q^{14} +(-6.00362 - 10.3986i) q^{16} -6.79306 q^{17} -6.24797 q^{19} +(3.72745 + 6.45614i) q^{20} +(2.06199 - 3.57147i) q^{22} +(-1.45490 + 2.51997i) q^{23} +(1.44163 + 2.49697i) q^{25} +15.7029 q^{26} +5.12398 q^{28} +(-1.94163 - 3.36300i) q^{29} +(-1.00000 + 1.73205i) q^{31} +(-7.68597 + 13.3125i) q^{32} +(9.06561 + 15.7021i) q^{34} -1.45490 q^{35} +5.00000 q^{37} +(8.33816 + 14.4421i) q^{38} +(6.06561 - 10.5059i) q^{40} +(1.12398 - 1.94680i) q^{41} +(3.56561 + 6.17582i) q^{43} -7.91705 q^{44} +7.76651 q^{46} +(2.66908 + 4.62298i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(3.84782 - 6.66461i) q^{50} +(-15.0728 - 26.1069i) q^{52} -9.79306 q^{53} +2.24797 q^{55} +(-4.16908 - 7.22106i) q^{56} +(-5.18236 + 8.97610i) q^{58} +(2.33816 - 4.04981i) q^{59} +(1.18236 + 2.04790i) q^{61} +5.33816 q^{62} +17.0145 q^{64} +(4.27979 + 7.41281i) q^{65} +(-1.68236 + 2.91393i) q^{67} +(17.4038 - 30.1442i) q^{68} +(1.94163 + 3.36300i) q^{70} +1.36471 q^{71} -1.88325 q^{73} +(-6.67270 - 11.5575i) q^{74} +(16.0072 - 27.7253i) q^{76} +(0.772548 - 1.33809i) q^{77} +(-1.68236 - 2.91393i) q^{79} -17.4694 q^{80} -6.00000 q^{82} +(-1.12398 - 1.94680i) q^{83} +(-4.94163 + 8.55915i) q^{85} +(9.51690 - 16.4837i) q^{86} +(6.44163 + 11.1572i) q^{88} +0.793062 q^{89} +5.88325 q^{91} +(-7.45490 - 12.9123i) q^{92} +(7.12398 - 12.3391i) q^{94} +(-4.54510 + 7.87234i) q^{95} +(-5.12398 - 8.87500i) q^{97} +2.66908 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 3 q^{5} - 3 q^{7} + 18 q^{8} + 6 q^{10} + 6 q^{11} - 3 q^{13} - 12 q^{16} - 6 q^{17} + 21 q^{20} + 3 q^{22} - 6 q^{23} - 6 q^{25} + 54 q^{26} + 12 q^{28} + 3 q^{29} - 6 q^{31} - 18 q^{32}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.33454 − 2.31149i −0.943662 − 1.63447i −0.758408 − 0.651780i \(-0.774023\pi\)
−0.185254 − 0.982691i \(-0.559311\pi\)
\(3\) 0 0
\(4\) −2.56199 + 4.43750i −1.28100 + 2.21875i
\(5\) 0.727452 − 1.25998i 0.325326 − 0.563482i −0.656252 − 0.754542i \(-0.727859\pi\)
0.981578 + 0.191060i \(0.0611925\pi\)
\(6\) 0 0
\(7\) −0.500000 − 0.866025i −0.188982 − 0.327327i
\(8\) 8.33816 2.94798
\(9\) 0 0
\(10\) −3.88325 −1.22799
\(11\) 0.772548 + 1.33809i 0.232932 + 0.403450i 0.958670 − 0.284522i \(-0.0918347\pi\)
−0.725738 + 0.687972i \(0.758501\pi\)
\(12\) 0 0
\(13\) −2.94163 + 5.09505i −0.815861 + 1.41311i 0.0928478 + 0.995680i \(0.470403\pi\)
−0.908708 + 0.417432i \(0.862930\pi\)
\(14\) −1.33454 + 2.31149i −0.356671 + 0.617772i
\(15\) 0 0
\(16\) −6.00362 − 10.3986i −1.50090 − 2.59964i
\(17\) −6.79306 −1.64756 −0.823780 − 0.566910i \(-0.808139\pi\)
−0.823780 + 0.566910i \(0.808139\pi\)
\(18\) 0 0
\(19\) −6.24797 −1.43338 −0.716691 − 0.697391i \(-0.754344\pi\)
−0.716691 + 0.697391i \(0.754344\pi\)
\(20\) 3.72745 + 6.45614i 0.833484 + 1.44364i
\(21\) 0 0
\(22\) 2.06199 − 3.57147i 0.439618 − 0.761441i
\(23\) −1.45490 + 2.51997i −0.303368 + 0.525450i −0.976897 − 0.213712i \(-0.931445\pi\)
0.673528 + 0.739161i \(0.264778\pi\)
\(24\) 0 0
\(25\) 1.44163 + 2.49697i 0.288325 + 0.499394i
\(26\) 15.7029 3.07959
\(27\) 0 0
\(28\) 5.12398 0.968342
\(29\) −1.94163 − 3.36300i −0.360551 − 0.624493i 0.627501 − 0.778616i \(-0.284078\pi\)
−0.988052 + 0.154123i \(0.950745\pi\)
\(30\) 0 0
\(31\) −1.00000 + 1.73205i −0.179605 + 0.311086i −0.941745 − 0.336327i \(-0.890815\pi\)
0.762140 + 0.647412i \(0.224149\pi\)
\(32\) −7.68597 + 13.3125i −1.35870 + 2.35334i
\(33\) 0 0
\(34\) 9.06561 + 15.7021i 1.55474 + 2.69289i
\(35\) −1.45490 −0.245924
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 − 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 8.33816 + 14.4421i 1.35263 + 2.34282i
\(39\) 0 0
\(40\) 6.06561 − 10.5059i 0.959057 − 1.66114i
\(41\) 1.12398 − 1.94680i 0.175537 − 0.304038i −0.764810 − 0.644256i \(-0.777167\pi\)
0.940347 + 0.340217i \(0.110501\pi\)
\(42\) 0 0
\(43\) 3.56561 + 6.17582i 0.543750 + 0.941803i 0.998684 + 0.0512782i \(0.0163295\pi\)
−0.454934 + 0.890525i \(0.650337\pi\)
\(44\) −7.91705 −1.19354
\(45\) 0 0
\(46\) 7.76651 1.14511
\(47\) 2.66908 + 4.62298i 0.389325 + 0.674331i 0.992359 − 0.123385i \(-0.0393749\pi\)
−0.603034 + 0.797716i \(0.706042\pi\)
\(48\) 0 0
\(49\) −0.500000 + 0.866025i −0.0714286 + 0.123718i
\(50\) 3.84782 − 6.66461i 0.544163 − 0.942519i
\(51\) 0 0
\(52\) −15.0728 − 26.1069i −2.09023 − 3.62038i
\(53\) −9.79306 −1.34518 −0.672590 − 0.740015i \(-0.734818\pi\)
−0.672590 + 0.740015i \(0.734818\pi\)
\(54\) 0 0
\(55\) 2.24797 0.303116
\(56\) −4.16908 − 7.22106i −0.557117 − 0.964954i
\(57\) 0 0
\(58\) −5.18236 + 8.97610i −0.680477 + 1.17862i
\(59\) 2.33816 − 4.04981i 0.304402 − 0.527240i −0.672726 − 0.739892i \(-0.734877\pi\)
0.977128 + 0.212652i \(0.0682100\pi\)
\(60\) 0 0
\(61\) 1.18236 + 2.04790i 0.151385 + 0.262207i 0.931737 − 0.363134i \(-0.118293\pi\)
−0.780352 + 0.625341i \(0.784960\pi\)
\(62\) 5.33816 0.677947
\(63\) 0 0
\(64\) 17.0145 2.12681
\(65\) 4.27979 + 7.41281i 0.530842 + 0.919445i
\(66\) 0 0
\(67\) −1.68236 + 2.91393i −0.205533 + 0.355993i −0.950302 − 0.311329i \(-0.899226\pi\)
0.744770 + 0.667322i \(0.232559\pi\)
\(68\) 17.4038 − 30.1442i 2.11052 − 3.65552i
\(69\) 0 0
\(70\) 1.94163 + 3.36300i 0.232069 + 0.401955i
\(71\) 1.36471 0.161962 0.0809808 − 0.996716i \(-0.474195\pi\)
0.0809808 + 0.996716i \(0.474195\pi\)
\(72\) 0 0
\(73\) −1.88325 −0.220418 −0.110209 − 0.993908i \(-0.535152\pi\)
−0.110209 + 0.993908i \(0.535152\pi\)
\(74\) −6.67270 − 11.5575i −0.775685 − 1.34353i
\(75\) 0 0
\(76\) 16.0072 − 27.7253i 1.83616 − 3.18032i
\(77\) 0.772548 − 1.33809i 0.0880400 − 0.152490i
\(78\) 0 0
\(79\) −1.68236 − 2.91393i −0.189280 − 0.327842i 0.755730 − 0.654883i \(-0.227282\pi\)
−0.945010 + 0.327040i \(0.893949\pi\)
\(80\) −17.4694 −1.95314
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −1.12398 − 1.94680i −0.123373 − 0.213689i 0.797723 − 0.603024i \(-0.206038\pi\)
−0.921096 + 0.389336i \(0.872705\pi\)
\(84\) 0 0
\(85\) −4.94163 + 8.55915i −0.535995 + 0.928370i
\(86\) 9.51690 − 16.4837i 1.02623 − 1.77749i
\(87\) 0 0
\(88\) 6.44163 + 11.1572i 0.686680 + 1.18936i
\(89\) 0.793062 0.0840644 0.0420322 − 0.999116i \(-0.486617\pi\)
0.0420322 + 0.999116i \(0.486617\pi\)
\(90\) 0 0
\(91\) 5.88325 0.616733
\(92\) −7.45490 − 12.9123i −0.777227 − 1.34620i
\(93\) 0 0
\(94\) 7.12398 − 12.3391i 0.734783 − 1.27268i
\(95\) −4.54510 + 7.87234i −0.466317 + 0.807685i
\(96\) 0 0
\(97\) −5.12398 − 8.87500i −0.520262 − 0.901120i −0.999723 − 0.0235564i \(-0.992501\pi\)
0.479461 − 0.877563i \(-0.340832\pi\)
\(98\) 2.66908 0.269618
\(99\) 0 0
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 567.2.f.m.190.1 6
3.2 odd 2 567.2.f.l.190.3 6
9.2 odd 6 567.2.f.l.379.3 6
9.4 even 3 567.2.a.e.1.3 ✓ 3
9.5 odd 6 567.2.a.f.1.1 yes 3
9.7 even 3 inner 567.2.f.m.379.1 6
36.23 even 6 9072.2.a.cb.1.2 3
36.31 odd 6 9072.2.a.bu.1.2 3
63.13 odd 6 3969.2.a.o.1.3 3
63.41 even 6 3969.2.a.n.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.e.1.3 ✓ 3 9.4 even 3
567.2.a.f.1.1 yes 3 9.5 odd 6
567.2.f.l.190.3 6 3.2 odd 2
567.2.f.l.379.3 6 9.2 odd 6
567.2.f.m.190.1 6 1.1 even 1 trivial
567.2.f.m.379.1 6 9.7 even 3 inner
3969.2.a.n.1.1 3 63.41 even 6
3969.2.a.o.1.3 3 63.13 odd 6
9072.2.a.bu.1.2 3 36.31 odd 6
9072.2.a.cb.1.2 3 36.23 even 6