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(289,666)] 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("666.289"); 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, 7])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.bj (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,-12,0,0,6,6,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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 \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 469.1
Root \(-0.642788 - 0.766044i\) of defining polynomial
Character \(\chi\) \(=\) 666.469
Dual form 666.2.bj.c.595.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.642788 - 0.766044i) q^{2} +(-0.173648 + 0.984808i) q^{4} +(1.45842 + 4.00698i) q^{5} +(-3.39364 + 1.23518i) q^{7} +(0.866025 - 0.500000i) q^{8} +(2.13207 - 3.69285i) q^{10} +(-1.05840 - 1.83321i) q^{11} +(2.84019 + 0.500802i) q^{13} +(3.12760 + 1.80572i) q^{14} +(-0.939693 - 0.342020i) q^{16} +(-0.0263137 + 0.00463982i) q^{17} +(-2.07522 + 2.47315i) q^{19} +(-4.19936 + 0.740460i) q^{20} +(-0.723990 + 1.98915i) q^{22} +(-2.57421 - 1.48622i) q^{23} +(-10.0987 + 8.47380i) q^{25} +(-1.44200 - 2.49762i) q^{26} +(-0.627119 - 3.55657i) q^{28} +(-4.96493 + 2.86650i) q^{29} -6.76932i q^{31} +(0.342020 + 0.939693i) q^{32} +(0.0204685 + 0.0171751i) q^{34} +(-9.89872 - 11.7968i) q^{35} +(-4.49375 + 4.09954i) q^{37} +3.22847 q^{38} +(3.26652 + 2.74094i) q^{40} +(-0.259000 + 1.46886i) q^{41} +5.53737i q^{43} +(1.98915 - 0.723990i) q^{44} +(0.516159 + 2.92728i) q^{46} +(1.30654 - 2.26300i) q^{47} +(4.62880 - 3.88403i) q^{49} +(12.9826 + 2.28918i) q^{50} +(-0.986387 + 2.71008i) q^{52} +(1.79389 + 0.652924i) q^{53} +(5.80203 - 6.91459i) q^{55} +(-2.32139 + 2.76652i) q^{56} +(5.38726 + 1.96080i) q^{58} +(-1.92380 + 5.28560i) q^{59} +(-5.65366 - 0.996892i) q^{61} +(-5.18560 + 4.35124i) q^{62} +(0.500000 - 0.866025i) q^{64} +(2.13549 + 12.1110i) q^{65} +(-6.50406 + 2.36728i) q^{67} -0.0267197i q^{68} +(-2.67412 + 15.1657i) q^{70} +(7.10830 + 5.96457i) q^{71} +16.2707 q^{73} +(6.02896 + 0.807274i) q^{74} +(-2.07522 - 2.47315i) q^{76} +(5.85619 + 4.91392i) q^{77} +(-0.484006 - 1.32980i) q^{79} -4.26414i q^{80} +(1.29170 - 0.745761i) q^{82} +(0.294580 + 1.67065i) q^{83} +(-0.0569682 - 0.0986718i) q^{85} +(4.24187 - 3.55935i) q^{86} +(-1.83321 - 1.05840i) q^{88} +(-2.82882 + 7.77213i) q^{89} +(-10.2572 + 1.80861i) q^{91} +(1.91065 - 2.27702i) q^{92} +(-2.57339 + 0.453757i) q^{94} +(-12.9364 - 4.70847i) q^{95} +(5.65105 + 3.26264i) q^{97} +(-5.95067 - 1.04926i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7} + 6 q^{10} + 6 q^{11} + 6 q^{13} + 18 q^{14} - 18 q^{19} - 18 q^{25} - 12 q^{26} - 6 q^{28} - 18 q^{29} + 12 q^{34} - 18 q^{35} + 30 q^{37} + 24 q^{38} + 12 q^{40} - 24 q^{41} - 6 q^{44}+ \cdots - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.642788 − 0.766044i −0.454519 − 0.541675i
\(3\) 0 0
\(4\) −0.173648 + 0.984808i −0.0868241 + 0.492404i
\(5\) 1.45842 + 4.00698i 0.652226 + 1.79198i 0.609358 + 0.792895i \(0.291427\pi\)
0.0428683 + 0.999081i \(0.486350\pi\)
\(6\) 0 0
\(7\) −3.39364 + 1.23518i −1.28268 + 0.466856i −0.891316 − 0.453382i \(-0.850217\pi\)
−0.391359 + 0.920238i \(0.627995\pi\)
\(8\) 0.866025 − 0.500000i 0.306186 − 0.176777i
\(9\) 0 0
\(10\) 2.13207 − 3.69285i 0.674220 − 1.16778i
\(11\) −1.05840 − 1.83321i −0.319120 − 0.552733i 0.661184 − 0.750223i \(-0.270054\pi\)
−0.980305 + 0.197491i \(0.936721\pi\)
\(12\) 0 0
\(13\) 2.84019 + 0.500802i 0.787726 + 0.138897i 0.553019 − 0.833169i \(-0.313476\pi\)
0.234707 + 0.972066i \(0.424587\pi\)
\(14\) 3.12760 + 1.80572i 0.835885 + 0.482598i
\(15\) 0 0
\(16\) −0.939693 − 0.342020i −0.234923 − 0.0855050i
\(17\) −0.0263137 + 0.00463982i −0.00638202 + 0.00112532i −0.176838 − 0.984240i \(-0.556587\pi\)
0.170456 + 0.985365i \(0.445476\pi\)
\(18\) 0 0
\(19\) −2.07522 + 2.47315i −0.476088 + 0.567380i −0.949623 − 0.313395i \(-0.898534\pi\)
0.473534 + 0.880775i \(0.342978\pi\)
\(20\) −4.19936 + 0.740460i −0.939005 + 0.165572i
\(21\) 0 0
\(22\) −0.723990 + 1.98915i −0.154355 + 0.424087i
\(23\) −2.57421 − 1.48622i −0.536760 − 0.309899i 0.207005 − 0.978340i \(-0.433628\pi\)
−0.743765 + 0.668441i \(0.766962\pi\)
\(24\) 0 0
\(25\) −10.0987 + 8.47380i −2.01974 + 1.69476i
\(26\) −1.44200 − 2.49762i −0.282800 − 0.489823i
\(27\) 0 0
\(28\) −0.627119 − 3.55657i −0.118514 − 0.672129i
\(29\) −4.96493 + 2.86650i −0.921964 + 0.532296i −0.884261 − 0.466993i \(-0.845337\pi\)
−0.0377027 + 0.999289i \(0.512004\pi\)
\(30\) 0 0
\(31\) − 6.76932i − 1.21581i −0.794011 − 0.607903i \(-0.792011\pi\)
0.794011 − 0.607903i \(-0.207989\pi\)
\(32\) 0.342020 + 0.939693i 0.0604612 + 0.166116i
\(33\) 0 0
\(34\) 0.0204685 + 0.0171751i 0.00351031 + 0.00294550i
\(35\) −9.89872 − 11.7968i −1.67319 − 1.99403i
\(36\) 0 0
\(37\) −4.49375 + 4.09954i −0.738767 + 0.673961i
\(38\) 3.22847 0.523727
\(39\) 0 0
\(40\) 3.26652 + 2.74094i 0.516482 + 0.433380i
\(41\) −0.259000 + 1.46886i −0.0404490 + 0.229398i −0.998330 − 0.0577674i \(-0.981602\pi\)
0.957881 + 0.287165i \(0.0927129\pi\)
\(42\) 0 0
\(43\) 5.53737i 0.844441i 0.906493 + 0.422221i \(0.138749\pi\)
−0.906493 + 0.422221i \(0.861251\pi\)
\(44\) 1.98915 − 0.723990i 0.299875 − 0.109146i
\(45\) 0 0
\(46\) 0.516159 + 2.92728i 0.0761035 + 0.431605i
\(47\) 1.30654 − 2.26300i 0.190579 − 0.330092i −0.754863 − 0.655882i \(-0.772297\pi\)
0.945442 + 0.325790i \(0.105630\pi\)
\(48\) 0 0
\(49\) 4.62880 − 3.88403i 0.661257 − 0.554861i
\(50\) 12.9826 + 2.28918i 1.83602 + 0.323740i
\(51\) 0 0
\(52\) −0.986387 + 2.71008i −0.136787 + 0.375820i
\(53\) 1.79389 + 0.652924i 0.246410 + 0.0896860i 0.462273 − 0.886738i \(-0.347034\pi\)
−0.215862 + 0.976424i \(0.569256\pi\)
\(54\) 0 0
\(55\) 5.80203 − 6.91459i 0.782345 − 0.932363i
\(56\) −2.32139 + 2.76652i −0.310208 + 0.369692i
\(57\) 0 0
\(58\) 5.38726 + 1.96080i 0.707382 + 0.257466i
\(59\) −1.92380 + 5.28560i −0.250457 + 0.688126i 0.749210 + 0.662333i \(0.230433\pi\)
−0.999667 + 0.0257935i \(0.991789\pi\)
\(60\) 0 0
\(61\) −5.65366 − 0.996892i −0.723876 − 0.127639i −0.200443 − 0.979705i \(-0.564238\pi\)
−0.523434 + 0.852066i \(0.675349\pi\)
\(62\) −5.18560 + 4.35124i −0.658572 + 0.552608i
\(63\) 0 0
\(64\) 0.500000 − 0.866025i 0.0625000 − 0.108253i
\(65\) 2.13549 + 12.1110i 0.264875 + 1.50218i
\(66\) 0 0
\(67\) −6.50406 + 2.36728i −0.794597 + 0.289210i −0.707246 − 0.706968i \(-0.750063\pi\)
−0.0873516 + 0.996178i \(0.527840\pi\)
\(68\) − 0.0267197i − 0.00324024i
\(69\) 0 0
\(70\) −2.67412 + 15.1657i −0.319619 + 1.81265i
\(71\) 7.10830 + 5.96457i 0.843600 + 0.707865i 0.958371 − 0.285527i \(-0.0921688\pi\)
−0.114770 + 0.993392i \(0.536613\pi\)
\(72\) 0 0
\(73\) 16.2707 1.90434 0.952169 − 0.305571i \(-0.0988473\pi\)
0.952169 + 0.305571i \(0.0988473\pi\)
\(74\) 6.02896 + 0.807274i 0.700852 + 0.0938437i
\(75\) 0 0
\(76\) −2.07522 − 2.47315i −0.238044 − 0.283690i
\(77\) 5.85619 + 4.91392i 0.667374 + 0.559993i
\(78\) 0 0
\(79\) −0.484006 − 1.32980i −0.0544549 − 0.149614i 0.909483 − 0.415742i \(-0.136478\pi\)
−0.963938 + 0.266128i \(0.914256\pi\)
\(80\) − 4.26414i − 0.476745i
\(81\) 0 0
\(82\) 1.29170 − 0.745761i 0.142644 − 0.0823556i
\(83\) 0.294580 + 1.67065i 0.0323343 + 0.183377i 0.996697 − 0.0812054i \(-0.0258770\pi\)
−0.964363 + 0.264583i \(0.914766\pi\)
\(84\) 0 0
\(85\) −0.0569682 − 0.0986718i −0.00617907 − 0.0107025i
\(86\) 4.24187 − 3.55935i 0.457413 − 0.383815i
\(87\) 0 0
\(88\) −1.83321 − 1.05840i −0.195421 − 0.112826i
\(89\) −2.82882 + 7.77213i −0.299855 + 0.823844i 0.694669 + 0.719330i \(0.255551\pi\)
−0.994523 + 0.104514i \(0.966671\pi\)
\(90\) 0 0
\(91\) −10.2572 + 1.80861i −1.07524 + 0.189594i
\(92\) 1.91065 − 2.27702i 0.199199 − 0.237396i
\(93\) 0 0
\(94\) −2.57339 + 0.453757i −0.265425 + 0.0468015i
\(95\) −12.9364 − 4.70847i −1.32725 − 0.483079i
\(96\) 0 0
\(97\) 5.65105 + 3.26264i 0.573777 + 0.331270i 0.758657 − 0.651491i \(-0.225856\pi\)
−0.184879 + 0.982761i \(0.559189\pi\)
\(98\) −5.95067 − 1.04926i −0.601109 − 0.105992i
\(99\) 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.bj.c.469.1 12
3.2 odd 2 74.2.h.a.25.2 yes 12
12.11 even 2 592.2.bq.b.321.1 12
37.3 even 18 inner 666.2.bj.c.595.1 12
111.59 even 36 2738.2.a.s.1.1 6
111.77 odd 18 74.2.h.a.3.2 ✓ 12
111.89 even 36 2738.2.a.r.1.2 6
444.299 even 18 592.2.bq.b.225.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.3.2 ✓ 12 111.77 odd 18
74.2.h.a.25.2 yes 12 3.2 odd 2
592.2.bq.b.225.1 12 444.299 even 18
592.2.bq.b.321.1 12 12.11 even 2
666.2.bj.c.469.1 12 1.1 even 1 trivial
666.2.bj.c.595.1 12 37.3 even 18 inner
2738.2.a.r.1.2 6 111.89 even 36
2738.2.a.s.1.1 6 111.59 even 36