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 289.2
Root \(0.342020 + 0.939693i\) of defining polynomial
Character \(\chi\) \(=\) 666.289
Dual form 666.2.bj.c.613.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.342020 + 0.939693i) q^{2} +(-0.766044 + 0.642788i) q^{4} +(-2.57176 + 0.453471i) q^{5} +(-0.361075 - 2.04776i) q^{7} +(-0.866025 - 0.500000i) q^{8} +(-1.30572 - 2.26157i) q^{10} +(2.99810 - 5.19285i) q^{11} +(2.64632 + 3.15377i) q^{13} +(1.80077 - 1.03967i) q^{14} +(0.173648 - 0.984808i) q^{16} +(0.618710 - 0.737350i) q^{17} +(0.534946 - 1.46975i) q^{19} +(1.67860 - 2.00048i) q^{20} +(5.90509 + 1.04123i) q^{22} +(5.51705 - 3.18527i) q^{23} +(1.70986 - 0.622339i) q^{25} +(-2.05847 + 3.56538i) q^{26} +(1.59287 + 1.33658i) q^{28} +(3.51193 + 2.02761i) q^{29} -3.39997i q^{31} +(0.984808 - 0.173648i) q^{32} +(0.904494 + 0.329209i) q^{34} +(1.85720 + 5.10261i) q^{35} +(6.07068 - 0.383130i) q^{37} +1.56408 q^{38} +(2.45395 + 0.893164i) q^{40} +(-7.94502 + 6.66666i) q^{41} -3.76932i q^{43} +(1.04123 + 5.90509i) q^{44} +(4.88011 + 4.09490i) q^{46} +(-3.08750 - 5.34771i) q^{47} +(2.51491 - 0.915354i) q^{49} +(1.16962 + 1.39389i) q^{50} +(-4.05440 - 0.714901i) q^{52} +(1.39401 - 7.90585i) q^{53} +(-5.35558 + 14.7143i) q^{55} +(-0.711179 + 1.95395i) q^{56} +(-0.704183 + 3.99362i) q^{58} +(-5.02269 - 0.885636i) q^{59} +(-6.25519 - 7.45465i) q^{61} +(3.19493 - 1.16286i) q^{62} +(0.500000 + 0.866025i) q^{64} +(-8.23586 - 6.91071i) q^{65} +(-1.83263 - 10.3934i) q^{67} +0.962542i q^{68} +(-4.15968 + 3.49039i) q^{70} +(10.1503 + 3.69442i) q^{71} +3.55293 q^{73} +(2.43632 + 5.57354i) q^{74} +(0.534946 + 1.46975i) q^{76} +(-11.7162 - 4.26436i) q^{77} +(2.51098 - 0.442753i) q^{79} +2.61144i q^{80} +(-8.98197 - 5.18574i) q^{82} +(-5.29798 - 4.44553i) q^{83} +(-1.25681 + 2.17686i) q^{85} +(3.54200 - 1.28918i) q^{86} +(-5.19285 + 2.99810i) q^{88} +(16.0165 + 2.82414i) q^{89} +(5.50263 - 6.55778i) q^{91} +(-2.17885 + 5.98635i) q^{92} +(3.96922 - 4.73033i) q^{94} +(-0.709264 + 4.02243i) q^{95} +(-14.1175 + 8.15074i) q^{97} +(1.72030 + 2.05018i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7} + 6 q^{10} + 6 q^{11} + 6 q^{13} + 18 q^{14} - 18 q^{19} - 18 q^{25} - 12 q^{26} - 6 q^{28} - 18 q^{29} + 12 q^{34} - 18 q^{35} + 30 q^{37} + 24 q^{38} + 12 q^{40} - 24 q^{41} - 6 q^{44}+ \cdots - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(e\left(\frac{7}{18}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.342020 + 0.939693i 0.241845 + 0.664463i
\(3\) 0 0
\(4\) −0.766044 + 0.642788i −0.383022 + 0.321394i
\(5\) −2.57176 + 0.453471i −1.15013 + 0.202798i −0.716031 − 0.698068i \(-0.754043\pi\)
−0.434096 + 0.900867i \(0.642932\pi\)
\(6\) 0 0
\(7\) −0.361075 − 2.04776i −0.136473 − 0.773979i −0.973822 − 0.227311i \(-0.927007\pi\)
0.837349 − 0.546669i \(-0.184104\pi\)
\(8\) −0.866025 − 0.500000i −0.306186 − 0.176777i
\(9\) 0 0
\(10\) −1.30572 − 2.26157i −0.412904 − 0.715171i
\(11\) 2.99810 − 5.19285i 0.903960 − 1.56570i 0.0816522 − 0.996661i \(-0.473980\pi\)
0.822308 − 0.569043i \(-0.192686\pi\)
\(12\) 0 0
\(13\) 2.64632 + 3.15377i 0.733958 + 0.874697i 0.995907 − 0.0903843i \(-0.0288095\pi\)
−0.261949 + 0.965082i \(0.584365\pi\)
\(14\) 1.80077 − 1.03967i 0.481275 − 0.277864i
\(15\) 0 0
\(16\) 0.173648 − 0.984808i 0.0434120 − 0.246202i
\(17\) 0.618710 − 0.737350i 0.150059 − 0.178834i −0.685778 − 0.727810i \(-0.740538\pi\)
0.835838 + 0.548977i \(0.184982\pi\)
\(18\) 0 0
\(19\) 0.534946 − 1.46975i 0.122725 − 0.337184i −0.863083 − 0.505063i \(-0.831469\pi\)
0.985808 + 0.167878i \(0.0536916\pi\)
\(20\) 1.67860 − 2.00048i 0.375346 − 0.447320i
\(21\) 0 0
\(22\) 5.90509 + 1.04123i 1.25897 + 0.221990i
\(23\) 5.51705 − 3.18527i 1.15038 − 0.664174i 0.201403 − 0.979508i \(-0.435450\pi\)
0.948981 + 0.315334i \(0.102117\pi\)
\(24\) 0 0
\(25\) 1.70986 − 0.622339i 0.341973 − 0.124468i
\(26\) −2.05847 + 3.56538i −0.403700 + 0.699229i
\(27\) 0 0
\(28\) 1.59287 + 1.33658i 0.301025 + 0.252590i
\(29\) 3.51193 + 2.02761i 0.652149 + 0.376519i 0.789279 − 0.614035i \(-0.210454\pi\)
−0.137130 + 0.990553i \(0.543788\pi\)
\(30\) 0 0
\(31\) − 3.39997i − 0.610653i −0.952248 − 0.305326i \(-0.901234\pi\)
0.952248 − 0.305326i \(-0.0987655\pi\)
\(32\) 0.984808 − 0.173648i 0.174091 − 0.0306970i
\(33\) 0 0
\(34\) 0.904494 + 0.329209i 0.155119 + 0.0564588i
\(35\) 1.85720 + 5.10261i 0.313924 + 0.862498i
\(36\) 0 0
\(37\) 6.07068 − 0.383130i 0.998014 − 0.0629862i
\(38\) 1.56408 0.253727
\(39\) 0 0
\(40\) 2.45395 + 0.893164i 0.388003 + 0.141222i
\(41\) −7.94502 + 6.66666i −1.24080 + 1.04116i −0.243343 + 0.969940i \(0.578244\pi\)
−0.997461 + 0.0712179i \(0.977311\pi\)
\(42\) 0 0
\(43\) − 3.76932i − 0.574816i −0.957808 − 0.287408i \(-0.907206\pi\)
0.957808 − 0.287408i \(-0.0927936\pi\)
\(44\) 1.04123 + 5.90509i 0.156971 + 0.890227i
\(45\) 0 0
\(46\) 4.88011 + 4.09490i 0.719534 + 0.603760i
\(47\) −3.08750 − 5.34771i −0.450359 − 0.780044i 0.548050 − 0.836446i \(-0.315371\pi\)
−0.998408 + 0.0564019i \(0.982037\pi\)
\(48\) 0 0
\(49\) 2.51491 − 0.915354i 0.359273 − 0.130765i
\(50\) 1.16962 + 1.39389i 0.165409 + 0.197126i
\(51\) 0 0
\(52\) −4.05440 − 0.714901i −0.562245 − 0.0991389i
\(53\) 1.39401 − 7.90585i 0.191483 − 1.08595i −0.725857 − 0.687846i \(-0.758556\pi\)
0.917339 − 0.398106i \(-0.130332\pi\)
\(54\) 0 0
\(55\) −5.35558 + 14.7143i −0.722146 + 1.98408i
\(56\) −0.711179 + 1.95395i −0.0950352 + 0.261107i
\(57\) 0 0
\(58\) −0.704183 + 3.99362i −0.0924638 + 0.524388i
\(59\) −5.02269 − 0.885636i −0.653899 − 0.115300i −0.163150 − 0.986601i \(-0.552166\pi\)
−0.490749 + 0.871301i \(0.663277\pi\)
\(60\) 0 0
\(61\) −6.25519 − 7.45465i −0.800895 − 0.954470i 0.198778 − 0.980045i \(-0.436303\pi\)
−0.999673 + 0.0255750i \(0.991858\pi\)
\(62\) 3.19493 − 1.16286i 0.405756 − 0.147683i
\(63\) 0 0
\(64\) 0.500000 + 0.866025i 0.0625000 + 0.108253i
\(65\) −8.23586 − 6.91071i −1.02153 − 0.857168i
\(66\) 0 0
\(67\) −1.83263 − 10.3934i −0.223892 − 1.26975i −0.864792 − 0.502130i \(-0.832550\pi\)
0.640901 − 0.767624i \(-0.278561\pi\)
\(68\) 0.962542i 0.116725i
\(69\) 0 0
\(70\) −4.15968 + 3.49039i −0.497177 + 0.417181i
\(71\) 10.1503 + 3.69442i 1.20462 + 0.438447i 0.864835 − 0.502056i \(-0.167423\pi\)
0.339787 + 0.940502i \(0.389645\pi\)
\(72\) 0 0
\(73\) 3.55293 0.415839 0.207920 − 0.978146i \(-0.433331\pi\)
0.207920 + 0.978146i \(0.433331\pi\)
\(74\) 2.43632 + 5.57354i 0.283217 + 0.647911i
\(75\) 0 0
\(76\) 0.534946 + 1.46975i 0.0613625 + 0.168592i
\(77\) −11.7162 − 4.26436i −1.33519 − 0.485969i
\(78\) 0 0
\(79\) 2.51098 − 0.442753i 0.282507 − 0.0498136i −0.0305991 − 0.999532i \(-0.509742\pi\)
0.313106 + 0.949718i \(0.398630\pi\)
\(80\) 2.61144i 0.291967i
\(81\) 0 0
\(82\) −8.98197 − 5.18574i −0.991893 − 0.572670i
\(83\) −5.29798 − 4.44553i −0.581529 − 0.487960i 0.303920 − 0.952698i \(-0.401704\pi\)
−0.885449 + 0.464737i \(0.846149\pi\)
\(84\) 0 0
\(85\) −1.25681 + 2.17686i −0.136320 + 0.236113i
\(86\) 3.54200 − 1.28918i 0.381944 − 0.139016i
\(87\) 0 0
\(88\) −5.19285 + 2.99810i −0.553560 + 0.319598i
\(89\) 16.0165 + 2.82414i 1.69774 + 0.299358i 0.936905 − 0.349584i \(-0.113677\pi\)
0.760838 + 0.648942i \(0.224788\pi\)
\(90\) 0 0
\(91\) 5.50263 − 6.55778i 0.576832 − 0.687442i
\(92\) −2.17885 + 5.98635i −0.227161 + 0.624120i
\(93\) 0 0
\(94\) 3.96922 − 4.73033i 0.409394 − 0.487896i
\(95\) −0.709264 + 4.02243i −0.0727689 + 0.412693i
\(96\) 0 0
\(97\) −14.1175 + 8.15074i −1.43342 + 0.827583i −0.997380 − 0.0723469i \(-0.976951\pi\)
−0.436036 + 0.899929i \(0.643618\pi\)
\(98\) 1.72030 + 2.05018i 0.173777 + 0.207099i
\(99\) 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.289.2 12
3.2 odd 2 74.2.h.a.67.1 yes 12
12.11 even 2 592.2.bq.b.289.2 12
37.21 even 18 inner 666.2.bj.c.613.2 12
111.50 even 36 2738.2.a.s.1.4 6
111.95 odd 18 74.2.h.a.21.1 ✓ 12
111.98 even 36 2738.2.a.r.1.3 6
444.95 even 18 592.2.bq.b.465.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.21.1 ✓ 12 111.95 odd 18
74.2.h.a.67.1 yes 12 3.2 odd 2
592.2.bq.b.289.2 12 12.11 even 2
592.2.bq.b.465.2 12 444.95 even 18
666.2.bj.c.289.2 12 1.1 even 1 trivial
666.2.bj.c.613.2 12 37.21 even 18 inner
2738.2.a.r.1.3 6 111.98 even 36
2738.2.a.s.1.4 6 111.50 even 36