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 379.3
Root \(0.500000 + 1.51496i\) of defining polynomial
Character \(\chi\) \(=\) 567.379
Dual form 567.2.f.l.190.3

$q$-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 \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\)

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 0.185254 − 0.982691i \(-0.440689\pi\)
0.758408 − 0.651780i \(-0.225977\pi\)
\(3\) 0 0
\(4\) −2.56199 − 4.43750i −1.28100 − 2.21875i
\(5\) −0.727452 − 1.25998i −0.325326 − 0.563482i 0.656252 − 0.754542i \(-0.272141\pi\)
−0.981578 + 0.191060i \(0.938808\pi\)
\(6\) 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 \(-0.908165\pi\)
0.725738 + 0.687972i \(0.241499\pi\)
\(12\) 0 0
\(13\) −2.94163 − 5.09505i −0.815861 − 1.41311i −0.908708 − 0.417432i \(-0.862930\pi\)
0.0928478 − 0.995680i \(-0.470403\pi\)
\(14\) 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 \(-0.191861\pi\)
0.823780 + 0.566910i \(0.191861\pi\)
\(18\) 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 \(-0.0685554\pi\)
−0.673528 + 0.739161i \(0.735222\pi\)
\(24\) 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 \(-0.715922\pi\)
0.988052 + 0.154123i \(0.0492553\pi\)
\(30\) 0 0
\(31\) −1.00000 − 1.73205i −0.179605 − 0.311086i 0.762140 − 0.647412i \(-0.224149\pi\)
−0.941745 + 0.336327i \(0.890815\pi\)
\(32\) 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 \(-0.222833\pi\)
−0.940347 + 0.340217i \(0.889499\pi\)
\(42\) 0 0
\(43\) 3.56561 − 6.17582i 0.543750 − 0.941803i −0.454934 − 0.890525i \(-0.650337\pi\)
0.998684 − 0.0512782i \(-0.0163295\pi\)
\(44\) 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 \(-0.960625\pi\)
0.603034 + 0.797716i \(0.293958\pi\)
\(48\) 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 \(-0.265182\pi\)
0.672590 + 0.740015i \(0.265182\pi\)
\(54\) 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 \(-0.265123\pi\)
−0.977128 + 0.212652i \(0.931790\pi\)
\(60\) 0 0
\(61\) 1.18236 − 2.04790i 0.151385 − 0.262207i −0.780352 − 0.625341i \(-0.784960\pi\)
0.931737 + 0.363134i \(0.118293\pi\)
\(62\) −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 0.744770 − 0.667322i \(-0.232559\pi\)
−0.950302 + 0.311329i \(0.899226\pi\)
\(68\) −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 \(-0.525805\pi\)
−0.0809808 + 0.996716i \(0.525805\pi\)
\(72\) 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 −0.945010 − 0.327040i \(-0.893949\pi\)
0.755730 + 0.654883i \(0.227282\pi\)
\(80\) 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 \(-0.793962\pi\)
0.921096 + 0.389336i \(0.127295\pi\)
\(84\) 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 \(-0.513383\pi\)
−0.0420322 + 0.999116i \(0.513383\pi\)
\(90\) 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 0.479461 + 0.877563i \(0.340832\pi\)
−0.999723 + 0.0235564i \(0.992501\pi\)
\(98\) −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.l.379.3 6
3.2 odd 2 567.2.f.m.379.1 6
9.2 odd 6 567.2.a.e.1.3 ✓ 3
9.4 even 3 inner 567.2.f.l.190.3 6
9.5 odd 6 567.2.f.m.190.1 6
9.7 even 3 567.2.a.f.1.1 yes 3
36.7 odd 6 9072.2.a.cb.1.2 3
36.11 even 6 9072.2.a.bu.1.2 3
63.20 even 6 3969.2.a.o.1.3 3
63.34 odd 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 9.2 odd 6
567.2.a.f.1.1 yes 3 9.7 even 3
567.2.f.l.190.3 6 9.4 even 3 inner
567.2.f.l.379.3 6 1.1 even 1 trivial
567.2.f.m.190.1 6 9.5 odd 6
567.2.f.m.379.1 6 3.2 odd 2
3969.2.a.n.1.1 3 63.34 odd 6
3969.2.a.o.1.3 3 63.20 even 6
9072.2.a.bu.1.2 3 36.11 even 6
9072.2.a.cb.1.2 3 36.7 odd 6