Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [666,2,Mod(127,666)] 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("666.127"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(666, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.x (of order \(9\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,6,0,6,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 24x^{10} + 264x^{8} - 1687x^{6} + 6600x^{4} - 15000x^{2} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 145.2
Root \(-2.20976 - 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 666.145
Dual form 666.2.x.g.271.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.766044 - 0.642788i) q^{2} +(0.173648 + 0.984808i) q^{4} +(1.43969 + 0.524005i) q^{5} +(4.53424 + 1.65033i) q^{7} +(0.500000 - 0.866025i) q^{8} +(-0.766044 - 1.32683i) q^{10} +(-0.546896 + 0.947252i) q^{11} +(0.307284 + 1.74269i) q^{13} +(-2.41262 - 4.17878i) q^{14} +(-0.939693 + 0.342020i) q^{16} +(-0.511542 + 2.90110i) q^{17} +(-0.632167 + 0.530451i) q^{19} +(-0.266044 + 1.50881i) q^{20} +(1.02783 - 0.374099i) q^{22} +(-0.121620 - 0.210653i) q^{23} +(-2.03209 - 1.70513i) q^{25} +(0.884789 - 1.53250i) q^{26} +(-0.837893 + 4.75193i) q^{28} +(-2.78587 + 4.82526i) q^{29} -5.73495 q^{31} +(0.939693 + 0.342020i) q^{32} +(2.25665 - 1.89356i) q^{34} +(5.66313 + 4.75193i) q^{35} +(5.84038 - 1.69999i) q^{37} +0.825235 q^{38} +(1.17365 - 0.984808i) q^{40} +(-0.505653 - 2.86770i) q^{41} +4.37987 q^{43} +(-1.02783 - 0.374099i) q^{44} +(-0.0422383 + 0.239545i) q^{46} +(1.13249 + 1.96153i) q^{47} +(12.4734 + 10.4664i) q^{49} +(0.460637 + 2.61240i) q^{50} +(-1.66286 + 0.605232i) q^{52} +(5.77859 - 2.10324i) q^{53} +(-1.28373 + 1.07717i) q^{55} +(3.69634 - 3.10160i) q^{56} +(5.23571 - 1.90564i) q^{58} +(8.29301 - 3.01841i) q^{59} +(-2.04075 - 11.5737i) q^{61} +(4.39322 + 3.68635i) q^{62} +(-0.500000 - 0.866025i) q^{64} +(-0.470787 + 2.66996i) q^{65} +(-8.44220 - 3.07271i) q^{67} -2.94585 q^{68} +(-1.28373 - 7.28038i) q^{70} +(0.933535 - 0.783329i) q^{71} +8.79418 q^{73} +(-5.56672 - 2.45185i) q^{74} +(-0.632167 - 0.530451i) q^{76} +(-4.04303 + 3.39251i) q^{77} +(-2.02379 - 0.736599i) q^{79} -1.53209 q^{80} +(-1.45597 + 2.52181i) q^{82} +(-0.884528 + 5.01641i) q^{83} +(-2.25665 + 3.90864i) q^{85} +(-3.35518 - 2.81533i) q^{86} +(0.546896 + 0.947252i) q^{88} +(8.43486 - 3.07004i) q^{89} +(-1.48272 + 8.40891i) q^{91} +(0.186333 - 0.156352i) q^{92} +(0.393310 - 2.23057i) q^{94} +(-1.18809 + 0.432428i) q^{95} +(-2.17954 - 3.77507i) q^{97} +(-2.82750 - 16.0355i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{5} + 6 q^{7} + 6 q^{8} - 3 q^{11} - 6 q^{13} - 3 q^{14} + 3 q^{17} - 3 q^{19} + 6 q^{20} - 3 q^{22} + 21 q^{23} - 6 q^{25} - 3 q^{28} - 6 q^{29} + 42 q^{31} - 3 q^{34} + 9 q^{35} - 3 q^{37}+ \cdots - 51 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.766044 − 0.642788i −0.541675 − 0.454519i
\(3\) 0 0
\(4\) 0.173648 + 0.984808i 0.0868241 + 0.492404i
\(5\) 1.43969 + 0.524005i 0.643850 + 0.234342i 0.643248 − 0.765658i \(-0.277586\pi\)
0.000601695 1.00000i \(0.499808\pi\)
\(6\) 0 0
\(7\) 4.53424 + 1.65033i 1.71378 + 0.623765i 0.997272 − 0.0738128i \(-0.0235167\pi\)
0.716509 + 0.697578i \(0.245739\pi\)
\(8\) 0.500000 − 0.866025i 0.176777 − 0.306186i
\(9\) 0 0
\(10\) −0.766044 − 1.32683i −0.242245 − 0.419580i
\(11\) −0.546896 + 0.947252i −0.164895 + 0.285607i −0.936618 − 0.350352i \(-0.886062\pi\)
0.771723 + 0.635959i \(0.219395\pi\)
\(12\) 0 0
\(13\) 0.307284 + 1.74269i 0.0852253 + 0.483337i 0.997308 + 0.0733298i \(0.0233626\pi\)
−0.912082 + 0.410007i \(0.865526\pi\)
\(14\) −2.41262 − 4.17878i −0.644799 − 1.11682i
\(15\) 0 0
\(16\) −0.939693 + 0.342020i −0.234923 + 0.0855050i
\(17\) −0.511542 + 2.90110i −0.124067 + 0.703619i 0.857791 + 0.513999i \(0.171837\pi\)
−0.981858 + 0.189620i \(0.939274\pi\)
\(18\) 0 0
\(19\) −0.632167 + 0.530451i −0.145029 + 0.121694i −0.712416 − 0.701758i \(-0.752399\pi\)
0.567387 + 0.823451i \(0.307954\pi\)
\(20\) −0.266044 + 1.50881i −0.0594893 + 0.337381i
\(21\) 0 0
\(22\) 1.02783 − 0.374099i 0.219134 − 0.0797582i
\(23\) −0.121620 − 0.210653i −0.0253596 − 0.0439241i 0.853067 − 0.521801i \(-0.174740\pi\)
−0.878427 + 0.477877i \(0.841406\pi\)
\(24\) 0 0
\(25\) −2.03209 − 1.70513i −0.406418 − 0.341025i
\(26\) 0.884789 − 1.53250i 0.173521 − 0.300548i
\(27\) 0 0
\(28\) −0.837893 + 4.75193i −0.158347 + 0.898030i
\(29\) −2.78587 + 4.82526i −0.517322 + 0.896028i 0.482475 + 0.875910i \(0.339738\pi\)
−0.999798 + 0.0201189i \(0.993596\pi\)
\(30\) 0 0
\(31\) −5.73495 −1.03003 −0.515013 − 0.857182i \(-0.672213\pi\)
−0.515013 + 0.857182i \(0.672213\pi\)
\(32\) 0.939693 + 0.342020i 0.166116 + 0.0604612i
\(33\) 0 0
\(34\) 2.25665 − 1.89356i 0.387013 − 0.324742i
\(35\) 5.66313 + 4.75193i 0.957243 + 0.803223i
\(36\) 0 0
\(37\) 5.84038 − 1.69999i 0.960152 − 0.279477i
\(38\) 0.825235 0.133871
\(39\) 0 0
\(40\) 1.17365 − 0.984808i 0.185570 − 0.155712i
\(41\) −0.505653 − 2.86770i −0.0789697 − 0.447859i −0.998496 − 0.0548301i \(-0.982538\pi\)
0.919526 − 0.393029i \(-0.128573\pi\)
\(42\) 0 0
\(43\) 4.37987 0.667924 0.333962 − 0.942587i \(-0.391614\pi\)
0.333962 + 0.942587i \(0.391614\pi\)
\(44\) −1.02783 − 0.374099i −0.154951 − 0.0563975i
\(45\) 0 0
\(46\) −0.0422383 + 0.239545i −0.00622770 + 0.0353191i
\(47\) 1.13249 + 1.96153i 0.165191 + 0.286119i 0.936723 − 0.350071i \(-0.113843\pi\)
−0.771532 + 0.636190i \(0.780509\pi\)
\(48\) 0 0
\(49\) 12.4734 + 10.4664i 1.78192 + 1.49521i
\(50\) 0.460637 + 2.61240i 0.0651439 + 0.369450i
\(51\) 0 0
\(52\) −1.66286 + 0.605232i −0.230597 + 0.0839305i
\(53\) 5.77859 − 2.10324i 0.793751 − 0.288902i 0.0868564 − 0.996221i \(-0.472318\pi\)
0.706894 + 0.707319i \(0.250096\pi\)
\(54\) 0 0
\(55\) −1.28373 + 1.07717i −0.173098 + 0.145246i
\(56\) 3.69634 − 3.10160i 0.493945 − 0.414469i
\(57\) 0 0
\(58\) 5.23571 − 1.90564i 0.687483 − 0.250223i
\(59\) 8.29301 − 3.01841i 1.07966 − 0.392963i 0.259878 − 0.965642i \(-0.416318\pi\)
0.819780 + 0.572678i \(0.194096\pi\)
\(60\) 0 0
\(61\) −2.04075 − 11.5737i −0.261292 − 1.48186i −0.779390 − 0.626539i \(-0.784471\pi\)
0.518098 − 0.855321i \(-0.326640\pi\)
\(62\) 4.39322 + 3.68635i 0.557940 + 0.468167i
\(63\) 0 0
\(64\) −0.500000 − 0.866025i −0.0625000 − 0.108253i
\(65\) −0.470787 + 2.66996i −0.0583939 + 0.331168i
\(66\) 0 0
\(67\) −8.44220 − 3.07271i −1.03138 − 0.375391i −0.229773 − 0.973244i \(-0.573798\pi\)
−0.801606 + 0.597853i \(0.796021\pi\)
\(68\) −2.94585 −0.357237
\(69\) 0 0
\(70\) −1.28373 − 7.28038i −0.153435 − 0.870172i
\(71\) 0.933535 − 0.783329i 0.110790 − 0.0929640i −0.585710 − 0.810521i \(-0.699184\pi\)
0.696500 + 0.717557i \(0.254740\pi\)
\(72\) 0 0
\(73\) 8.79418 1.02928 0.514640 − 0.857406i \(-0.327925\pi\)
0.514640 + 0.857406i \(0.327925\pi\)
\(74\) −5.56672 − 2.45185i −0.647119 − 0.285022i
\(75\) 0 0
\(76\) −0.632167 − 0.530451i −0.0725145 − 0.0608469i
\(77\) −4.04303 + 3.39251i −0.460746 + 0.386612i
\(78\) 0 0
\(79\) −2.02379 − 0.736599i −0.227694 − 0.0828739i 0.225654 − 0.974208i \(-0.427548\pi\)
−0.453348 + 0.891334i \(0.649770\pi\)
\(80\) −1.53209 −0.171293
\(81\) 0 0
\(82\) −1.45597 + 2.52181i −0.160785 + 0.278488i
\(83\) −0.884528 + 5.01641i −0.0970895 + 0.550622i 0.896997 + 0.442036i \(0.145744\pi\)
−0.994087 + 0.108586i \(0.965368\pi\)
\(84\) 0 0
\(85\) −2.25665 + 3.90864i −0.244768 + 0.423951i
\(86\) −3.35518 − 2.81533i −0.361798 − 0.303584i
\(87\) 0 0
\(88\) 0.546896 + 0.947252i 0.0582993 + 0.100977i
\(89\) 8.43486 − 3.07004i 0.894093 − 0.325423i 0.146210 − 0.989254i \(-0.453292\pi\)
0.747883 + 0.663830i \(0.231070\pi\)
\(90\) 0 0
\(91\) −1.48272 + 8.40891i −0.155431 + 0.881494i
\(92\) 0.186333 − 0.156352i 0.0194266 − 0.0163008i
\(93\) 0 0
\(94\) 0.393310 − 2.23057i 0.0405669 − 0.230066i
\(95\) −1.18809 + 0.432428i −0.121895 + 0.0443661i
\(96\) 0 0
\(97\) −2.17954 − 3.77507i −0.221298 − 0.383300i 0.733904 − 0.679253i \(-0.237696\pi\)
−0.955203 + 0.295953i \(0.904363\pi\)
\(98\) −2.82750 − 16.0355i −0.285620 − 1.61983i
\(99\) 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 666.2.x.g.145.2 12
3.2 odd 2 74.2.f.b.71.1 yes 12
12.11 even 2 592.2.bc.d.145.2 12
37.12 even 9 inner 666.2.x.g.271.2 12
111.44 odd 18 2738.2.a.t.1.1 6
111.86 odd 18 74.2.f.b.49.1 ✓ 12
111.104 odd 18 2738.2.a.q.1.1 6
444.419 even 18 592.2.bc.d.49.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.49.1 ✓ 12 111.86 odd 18
74.2.f.b.71.1 yes 12 3.2 odd 2
592.2.bc.d.49.2 12 444.419 even 18
592.2.bc.d.145.2 12 12.11 even 2
666.2.x.g.145.2 12 1.1 even 1 trivial
666.2.x.g.271.2 12 37.12 even 9 inner
2738.2.a.q.1.1 6 111.104 odd 18
2738.2.a.t.1.1 6 111.44 odd 18