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 613.2
Root \(0.342020 - 0.939693i\) of defining polynomial
Character \(\chi\) \(=\) 666.613
Dual form 666.2.bj.c.289.2

$q$-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 \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(e\left(\frac{11}{18}\right)\)

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 −0.434096 − 0.900867i \(-0.642932\pi\)
−0.716031 + 0.698068i \(0.754043\pi\)
\(6\) 0 0
\(7\) −0.361075 + 2.04776i −0.136473 + 0.773979i 0.837349 + 0.546669i \(0.184104\pi\)
−0.973822 + 0.227311i \(0.927007\pi\)
\(8\) −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 0.822308 + 0.569043i \(0.192686\pi\)
0.0816522 + 0.996661i \(0.473980\pi\)
\(12\) 0 0
\(13\) 2.64632 − 3.15377i 0.733958 − 0.874697i −0.261949 − 0.965082i \(-0.584365\pi\)
0.995907 + 0.0903843i \(0.0288095\pi\)
\(14\) 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 0.835838 − 0.548977i \(-0.184982\pi\)
−0.685778 + 0.727810i \(0.740538\pi\)
\(18\) 0 0
\(19\) 0.534946 + 1.46975i 0.122725 + 0.337184i 0.985808 − 0.167878i \(-0.0536916\pi\)
−0.863083 + 0.505063i \(0.831469\pi\)
\(20\) 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 0.948981 − 0.315334i \(-0.102117\pi\)
0.201403 + 0.979508i \(0.435450\pi\)
\(24\) 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 −0.137130 − 0.990553i \(-0.543788\pi\)
0.789279 + 0.614035i \(0.210454\pi\)
\(30\) 0 0
\(31\) 3.39997i 0.610653i 0.952248 + 0.305326i \(0.0987655\pi\)
−0.952248 + 0.305326i \(0.901234\pi\)
\(32\) 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 −0.997461 − 0.0712179i \(-0.977311\pi\)
−0.243343 − 0.969940i \(-0.578244\pi\)
\(42\) 0 0
\(43\) 3.76932i 0.574816i 0.957808 + 0.287408i \(0.0927936\pi\)
−0.957808 + 0.287408i \(0.907206\pi\)
\(44\) 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 −0.998408 − 0.0564019i \(-0.982037\pi\)
0.548050 + 0.836446i \(0.315371\pi\)
\(48\) 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 0.917339 + 0.398106i \(0.130332\pi\)
−0.725857 + 0.687846i \(0.758556\pi\)
\(54\) 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 −0.490749 − 0.871301i \(-0.663277\pi\)
−0.163150 + 0.986601i \(0.552166\pi\)
\(60\) 0 0
\(61\) −6.25519 + 7.45465i −0.800895 + 0.954470i −0.999673 − 0.0255750i \(-0.991858\pi\)
0.198778 + 0.980045i \(0.436303\pi\)
\(62\) 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 0.640901 + 0.767624i \(0.278561\pi\)
−0.864792 + 0.502130i \(0.832550\pi\)
\(68\) − 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 0.339787 − 0.940502i \(-0.389645\pi\)
0.864835 + 0.502056i \(0.167423\pi\)
\(72\) 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 0.313106 − 0.949718i \(-0.398630\pi\)
−0.0305991 + 0.999532i \(0.509742\pi\)
\(80\) − 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 −0.885449 − 0.464737i \(-0.846149\pi\)
0.303920 + 0.952698i \(0.401704\pi\)
\(84\) 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 0.760838 − 0.648942i \(-0.224788\pi\)
0.936905 + 0.349584i \(0.113677\pi\)
\(90\) 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 −0.436036 − 0.899929i \(-0.643618\pi\)
−0.997380 + 0.0723469i \(0.976951\pi\)
\(98\) 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.613.2 12
3.2 odd 2 74.2.h.a.21.1 ✓ 12
12.11 even 2 592.2.bq.b.465.2 12
37.30 even 18 inner 666.2.bj.c.289.2 12
111.17 even 36 2738.2.a.r.1.3 6
111.20 even 36 2738.2.a.s.1.4 6
111.104 odd 18 74.2.h.a.67.1 yes 12
444.215 even 18 592.2.bq.b.289.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.h.a.21.1 ✓ 12 3.2 odd 2
74.2.h.a.67.1 yes 12 111.104 odd 18
592.2.bq.b.289.2 12 444.215 even 18
592.2.bq.b.465.2 12 12.11 even 2
666.2.bj.c.289.2 12 37.30 even 18 inner
666.2.bj.c.613.2 12 1.1 even 1 trivial
2738.2.a.r.1.3 6 111.17 even 36
2738.2.a.s.1.4 6 111.20 even 36