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 127.1
Root \(2.00752 - 0.984808i\) of defining polynomial
Character \(\chi\) \(=\) 666.127
Dual form 666.2.x.g.451.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.939693 + 0.342020i) q^{2} +(0.766044 + 0.642788i) q^{4} +(0.326352 - 1.85083i) q^{5} +(-0.711830 + 4.03699i) q^{7} +(0.500000 + 0.866025i) q^{8} +(0.939693 - 1.62760i) q^{10} +(1.67088 + 2.89404i) q^{11} +(1.48512 + 1.24616i) q^{13} +(-2.04963 + 3.55007i) q^{14} +(0.173648 + 0.984808i) q^{16} +(-3.40626 + 2.85819i) q^{17} +(0.0932770 - 0.0339500i) q^{19} +(1.43969 - 1.20805i) q^{20} +(0.580289 + 3.29098i) q^{22} +(4.76146 - 8.24709i) q^{23} +(1.37939 + 0.502055i) q^{25} +(0.969343 + 1.67895i) q^{26} +(-3.14022 + 2.63496i) q^{28} +(0.387288 + 0.670802i) q^{29} +10.3914 q^{31} +(-0.173648 + 0.984808i) q^{32} +(-4.17840 + 1.52081i) q^{34} +(7.23948 + 2.63496i) q^{35} +(-3.50302 + 4.97281i) q^{37} +0.0992633 q^{38} +(1.76604 - 0.642788i) q^{40} +(-4.36218 - 3.66031i) q^{41} -1.11986 q^{43} +(-0.580289 + 3.29098i) q^{44} +(7.29498 - 6.12122i) q^{46} +(-3.55724 + 6.16132i) q^{47} +(-9.21271 - 3.35315i) q^{49} +(1.12449 + 0.943555i) q^{50} +(0.336649 + 1.90923i) q^{52} +(-0.142188 - 0.806388i) q^{53} +(5.90168 - 2.14804i) q^{55} +(-3.85205 + 1.40203i) q^{56} +(0.134504 + 0.762808i) q^{58} +(-1.40642 - 7.97622i) q^{59} +(-8.19737 - 6.87841i) q^{61} +(9.76476 + 3.55408i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(2.79111 - 2.34202i) q^{65} +(2.28630 - 12.9663i) q^{67} -4.44656 q^{68} +(5.90168 + 4.95210i) q^{70} +(-2.79386 + 1.01688i) q^{71} -2.55740 q^{73} +(-4.99257 + 3.47481i) q^{74} +(0.0932770 + 0.0339500i) q^{76} +(-12.8726 + 4.68524i) q^{77} +(-0.943888 + 5.35305i) q^{79} +1.87939 q^{80} +(-2.84721 - 4.93152i) q^{82} +(-0.504249 + 0.423115i) q^{83} +(4.17840 + 7.23720i) q^{85} +(-1.05232 - 0.383014i) q^{86} +(-1.67088 + 2.89404i) q^{88} +(-0.612149 - 3.47167i) q^{89} +(-6.08790 + 5.10835i) q^{91} +(8.94862 - 3.25703i) q^{92} +(-5.45001 + 4.57310i) q^{94} +(-0.0323948 - 0.183720i) q^{95} +(-1.81012 + 3.13522i) q^{97} +(-7.51027 - 6.30186i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{5} + 6 q^{7} + 6 q^{8} - 3 q^{11} - 6 q^{13} - 3 q^{14} + 3 q^{17} - 3 q^{19} + 6 q^{20} - 3 q^{22} + 21 q^{23} - 6 q^{25} - 3 q^{28} - 6 q^{29} + 42 q^{31} - 3 q^{34} + 9 q^{35} - 3 q^{37}+ \cdots - 51 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.939693 + 0.342020i 0.664463 + 0.241845i
\(3\) 0 0
\(4\) 0.766044 + 0.642788i 0.383022 + 0.321394i
\(5\) 0.326352 − 1.85083i 0.145949 − 0.827718i −0.820651 − 0.571429i \(-0.806389\pi\)
0.966600 − 0.256289i \(-0.0824997\pi\)
\(6\) 0 0
\(7\) −0.711830 + 4.03699i −0.269046 + 1.52584i 0.488214 + 0.872724i \(0.337649\pi\)
−0.757260 + 0.653113i \(0.773463\pi\)
\(8\) 0.500000 + 0.866025i 0.176777 + 0.306186i
\(9\) 0 0
\(10\) 0.939693 − 1.62760i 0.297157 − 0.514691i
\(11\) 1.67088 + 2.89404i 0.503788 + 0.872586i 0.999990 + 0.00437929i \(0.00139397\pi\)
−0.496203 + 0.868207i \(0.665273\pi\)
\(12\) 0 0
\(13\) 1.48512 + 1.24616i 0.411898 + 0.345623i 0.825071 − 0.565029i \(-0.191135\pi\)
−0.413173 + 0.910653i \(0.635580\pi\)
\(14\) −2.04963 + 3.55007i −0.547787 + 0.948795i
\(15\) 0 0
\(16\) 0.173648 + 0.984808i 0.0434120 + 0.246202i
\(17\) −3.40626 + 2.85819i −0.826140 + 0.693214i −0.954401 − 0.298527i \(-0.903505\pi\)
0.128261 + 0.991740i \(0.459060\pi\)
\(18\) 0 0
\(19\) 0.0932770 − 0.0339500i 0.0213992 − 0.00778868i −0.331298 − 0.943526i \(-0.607487\pi\)
0.352698 + 0.935737i \(0.385264\pi\)
\(20\) 1.43969 − 1.20805i 0.321925 − 0.270127i
\(21\) 0 0
\(22\) 0.580289 + 3.29098i 0.123718 + 0.701640i
\(23\) 4.76146 − 8.24709i 0.992833 − 1.71964i 0.392920 − 0.919573i \(-0.371465\pi\)
0.599913 − 0.800065i \(-0.295202\pi\)
\(24\) 0 0
\(25\) 1.37939 + 0.502055i 0.275877 + 0.100411i
\(26\) 0.969343 + 1.67895i 0.190104 + 0.329269i
\(27\) 0 0
\(28\) −3.14022 + 2.63496i −0.593445 + 0.497960i
\(29\) 0.387288 + 0.670802i 0.0719175 + 0.124565i 0.899742 − 0.436423i \(-0.143755\pi\)
−0.827824 + 0.560988i \(0.810421\pi\)
\(30\) 0 0
\(31\) 10.3914 1.86636 0.933179 − 0.359413i \(-0.117023\pi\)
0.933179 + 0.359413i \(0.117023\pi\)
\(32\) −0.173648 + 0.984808i −0.0306970 + 0.174091i
\(33\) 0 0
\(34\) −4.17840 + 1.52081i −0.716590 + 0.260817i
\(35\) 7.23948 + 2.63496i 1.22370 + 0.445389i
\(36\) 0 0
\(37\) −3.50302 + 4.97281i −0.575894 + 0.817525i
\(38\) 0.0992633 0.0161026
\(39\) 0 0
\(40\) 1.76604 − 0.642788i 0.279236 − 0.101634i
\(41\) −4.36218 − 3.66031i −0.681259 − 0.571644i 0.235115 − 0.971968i \(-0.424453\pi\)
−0.916374 + 0.400324i \(0.868898\pi\)
\(42\) 0 0
\(43\) −1.11986 −0.170777 −0.0853884 − 0.996348i \(-0.527213\pi\)
−0.0853884 + 0.996348i \(0.527213\pi\)
\(44\) −0.580289 + 3.29098i −0.0874818 + 0.496134i
\(45\) 0 0
\(46\) 7.29498 − 6.12122i 1.07559 − 0.902524i
\(47\) −3.55724 + 6.16132i −0.518877 + 0.898721i 0.480883 + 0.876785i \(0.340316\pi\)
−0.999759 + 0.0219357i \(0.993017\pi\)
\(48\) 0 0
\(49\) −9.21271 − 3.35315i −1.31610 − 0.479022i
\(50\) 1.12449 + 0.943555i 0.159026 + 0.133439i
\(51\) 0 0
\(52\) 0.336649 + 1.90923i 0.0466848 + 0.264763i
\(53\) −0.142188 − 0.806388i −0.0195310 − 0.110766i 0.973484 − 0.228756i \(-0.0734659\pi\)
−0.993015 + 0.117990i \(0.962355\pi\)
\(54\) 0 0
\(55\) 5.90168 − 2.14804i 0.795782 − 0.289641i
\(56\) −3.85205 + 1.40203i −0.514751 + 0.187354i
\(57\) 0 0
\(58\) 0.134504 + 0.762808i 0.0176612 + 0.100162i
\(59\) −1.40642 − 7.97622i −0.183101 − 1.03842i −0.928372 − 0.371653i \(-0.878791\pi\)
0.745271 − 0.666762i \(-0.232320\pi\)
\(60\) 0 0
\(61\) −8.19737 − 6.87841i −1.04957 − 0.880691i −0.0565183 − 0.998402i \(-0.518000\pi\)
−0.993048 + 0.117711i \(0.962444\pi\)
\(62\) 9.76476 + 3.55408i 1.24013 + 0.451369i
\(63\) 0 0
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) 2.79111 − 2.34202i 0.346195 − 0.290492i
\(66\) 0 0
\(67\) 2.28630 − 12.9663i 0.279317 − 1.58408i −0.445591 − 0.895237i \(-0.647006\pi\)
0.724907 − 0.688846i \(-0.241883\pi\)
\(68\) −4.44656 −0.539225
\(69\) 0 0
\(70\) 5.90168 + 4.95210i 0.705386 + 0.591889i
\(71\) −2.79386 + 1.01688i −0.331570 + 0.120682i −0.502440 − 0.864612i \(-0.667564\pi\)
0.170870 + 0.985294i \(0.445342\pi\)
\(72\) 0 0
\(73\) −2.55740 −0.299321 −0.149660 − 0.988737i \(-0.547818\pi\)
−0.149660 + 0.988737i \(0.547818\pi\)
\(74\) −4.99257 + 3.47481i −0.580374 + 0.403938i
\(75\) 0 0
\(76\) 0.0932770 + 0.0339500i 0.0106996 + 0.00389434i
\(77\) −12.8726 + 4.68524i −1.46697 + 0.533932i
\(78\) 0 0
\(79\) −0.943888 + 5.35305i −0.106196 + 0.602265i 0.884540 + 0.466464i \(0.154472\pi\)
−0.990736 + 0.135802i \(0.956639\pi\)
\(80\) 1.87939 0.210122
\(81\) 0 0
\(82\) −2.84721 − 4.93152i −0.314422 − 0.544595i
\(83\) −0.504249 + 0.423115i −0.0553485 + 0.0464429i −0.670042 − 0.742323i \(-0.733724\pi\)
0.614694 + 0.788766i \(0.289280\pi\)
\(84\) 0 0
\(85\) 4.17840 + 7.23720i 0.453211 + 0.784985i
\(86\) −1.05232 − 0.383014i −0.113475 − 0.0413015i
\(87\) 0 0
\(88\) −1.67088 + 2.89404i −0.178116 + 0.308506i
\(89\) −0.612149 − 3.47167i −0.0648877 − 0.367996i −0.999910 − 0.0134078i \(-0.995732\pi\)
0.935022 − 0.354589i \(-0.115379\pi\)
\(90\) 0 0
\(91\) −6.08790 + 5.10835i −0.638185 + 0.535501i
\(92\) 8.94862 − 3.25703i 0.932958 − 0.339569i
\(93\) 0 0
\(94\) −5.45001 + 4.57310i −0.562125 + 0.471679i
\(95\) −0.0323948 − 0.183720i −0.00332363 − 0.0188493i
\(96\) 0 0
\(97\) −1.81012 + 3.13522i −0.183790 + 0.318334i −0.943168 − 0.332316i \(-0.892170\pi\)
0.759378 + 0.650650i \(0.225503\pi\)
\(98\) −7.51027 − 6.30186i −0.758652 − 0.636584i
\(99\) 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.127.1 12
3.2 odd 2 74.2.f.b.53.1 yes 12
12.11 even 2 592.2.bc.d.497.2 12
37.7 even 9 inner 666.2.x.g.451.1 12
111.44 odd 18 74.2.f.b.7.1 ✓ 12
111.65 odd 18 2738.2.a.q.1.5 6
111.83 odd 18 2738.2.a.t.1.5 6
444.155 even 18 592.2.bc.d.81.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.7.1 ✓ 12 111.44 odd 18
74.2.f.b.53.1 yes 12 3.2 odd 2
592.2.bc.d.81.2 12 444.155 even 18
592.2.bc.d.497.2 12 12.11 even 2
666.2.x.g.127.1 12 1.1 even 1 trivial
666.2.x.g.451.1 12 37.7 even 9 inner
2738.2.a.q.1.5 6 111.65 odd 18
2738.2.a.t.1.5 6 111.83 odd 18