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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [666,2,Mod(343,666)] 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("666.343"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(666, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,0,-1,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 433.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 666.433
Dual form 666.2.f.d.343.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(1.50000 - 2.59808i) q^{5} +(2.00000 - 3.46410i) q^{7} +1.00000 q^{8} -3.00000 q^{10} +6.00000 q^{11} +(-1.00000 + 1.73205i) q^{13} -4.00000 q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.50000 + 2.59808i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(1.50000 + 2.59808i) q^{20} +(-3.00000 - 5.19615i) q^{22} -6.00000 q^{23} +(-2.00000 - 3.46410i) q^{25} +2.00000 q^{26} +(2.00000 + 3.46410i) q^{28} -3.00000 q^{29} +2.00000 q^{31} +(-0.500000 + 0.866025i) q^{32} +(1.50000 - 2.59808i) q^{34} +(-6.00000 - 10.3923i) q^{35} +(5.50000 - 2.59808i) q^{37} +2.00000 q^{38} +(1.50000 - 2.59808i) q^{40} +(1.50000 - 2.59808i) q^{41} -4.00000 q^{43} +(-3.00000 + 5.19615i) q^{44} +(3.00000 + 5.19615i) q^{46} +6.00000 q^{47} +(-4.50000 - 7.79423i) q^{49} +(-2.00000 + 3.46410i) q^{50} +(-1.00000 - 1.73205i) q^{52} +(-3.00000 - 5.19615i) q^{53} +(9.00000 - 15.5885i) q^{55} +(2.00000 - 3.46410i) q^{56} +(1.50000 + 2.59808i) q^{58} +(0.500000 - 0.866025i) q^{61} +(-1.00000 - 1.73205i) q^{62} +1.00000 q^{64} +(3.00000 + 5.19615i) q^{65} +(-1.00000 + 1.73205i) q^{67} -3.00000 q^{68} +(-6.00000 + 10.3923i) q^{70} +(-6.00000 + 10.3923i) q^{71} -10.0000 q^{73} +(-5.00000 - 3.46410i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(12.0000 - 20.7846i) q^{77} +(-7.00000 + 12.1244i) q^{79} -3.00000 q^{80} -3.00000 q^{82} +(3.00000 + 5.19615i) q^{83} +9.00000 q^{85} +(2.00000 + 3.46410i) q^{86} +6.00000 q^{88} +(1.50000 + 2.59808i) q^{89} +(4.00000 + 6.92820i) q^{91} +(3.00000 - 5.19615i) q^{92} +(-3.00000 - 5.19615i) q^{94} +(3.00000 + 5.19615i) q^{95} -13.0000 q^{97} +(-4.50000 + 7.79423i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + 3 q^{5} + 4 q^{7} + 2 q^{8} - 6 q^{10} + 12 q^{11} - 2 q^{13} - 8 q^{14} - q^{16} + 3 q^{17} - 2 q^{19} + 3 q^{20} - 6 q^{22} - 12 q^{23} - 4 q^{25} + 4 q^{26} + 4 q^{28} - 6 q^{29}+ \cdots - 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.500000 − 0.866025i −0.353553 − 0.612372i
\(3\) 0 0
\(4\) −0.500000 + 0.866025i −0.250000 + 0.433013i
\(5\) 1.50000 − 2.59808i 0.670820 − 1.16190i −0.306851 − 0.951757i \(-0.599275\pi\)
0.977672 − 0.210138i \(-0.0673912\pi\)
\(6\) 0 0
\(7\) 2.00000 − 3.46410i 0.755929 − 1.30931i −0.188982 − 0.981981i \(-0.560519\pi\)
0.944911 − 0.327327i \(-0.106148\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −3.00000 −0.948683
\(11\) 6.00000 1.80907 0.904534 − 0.426401i \(-0.140219\pi\)
0.904534 + 0.426401i \(0.140219\pi\)
\(12\) 0 0
\(13\) −1.00000 + 1.73205i −0.277350 + 0.480384i −0.970725 − 0.240192i \(-0.922790\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 1.50000 + 2.59808i 0.363803 + 0.630126i 0.988583 − 0.150675i \(-0.0481447\pi\)
−0.624780 + 0.780801i \(0.714811\pi\)
\(18\) 0 0
\(19\) −1.00000 + 1.73205i −0.229416 + 0.397360i −0.957635 − 0.287984i \(-0.907015\pi\)
0.728219 + 0.685344i \(0.240348\pi\)
\(20\) 1.50000 + 2.59808i 0.335410 + 0.580948i
\(21\) 0 0
\(22\) −3.00000 − 5.19615i −0.639602 − 1.10782i
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −2.00000 − 3.46410i −0.400000 − 0.692820i
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 2.00000 + 3.46410i 0.377964 + 0.654654i
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) −0.500000 + 0.866025i −0.0883883 + 0.153093i
\(33\) 0 0
\(34\) 1.50000 − 2.59808i 0.257248 − 0.445566i
\(35\) −6.00000 − 10.3923i −1.01419 − 1.75662i
\(36\) 0 0
\(37\) 5.50000 − 2.59808i 0.904194 − 0.427121i
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) 1.50000 − 2.59808i 0.237171 − 0.410792i
\(41\) 1.50000 − 2.59808i 0.234261 − 0.405751i −0.724797 − 0.688963i \(-0.758066\pi\)
0.959058 + 0.283211i \(0.0913998\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 − 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −3.00000 + 5.19615i −0.452267 + 0.783349i
\(45\) 0 0
\(46\) 3.00000 + 5.19615i 0.442326 + 0.766131i
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −4.50000 − 7.79423i −0.642857 − 1.11346i
\(50\) −2.00000 + 3.46410i −0.282843 + 0.489898i
\(51\) 0 0
\(52\) −1.00000 − 1.73205i −0.138675 − 0.240192i
\(53\) −3.00000 − 5.19615i −0.412082 − 0.713746i 0.583036 − 0.812447i \(-0.301865\pi\)
−0.995117 + 0.0987002i \(0.968532\pi\)
\(54\) 0 0
\(55\) 9.00000 − 15.5885i 1.21356 − 2.10195i
\(56\) 2.00000 − 3.46410i 0.267261 − 0.462910i
\(57\) 0 0
\(58\) 1.50000 + 2.59808i 0.196960 + 0.341144i
\(59\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) 0.500000 − 0.866025i 0.0640184 − 0.110883i −0.832240 − 0.554416i \(-0.812942\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) −1.00000 − 1.73205i −0.127000 − 0.219971i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) −1.00000 + 1.73205i −0.122169 + 0.211604i −0.920623 − 0.390453i \(-0.872318\pi\)
0.798454 + 0.602056i \(0.205652\pi\)
\(68\) −3.00000 −0.363803
\(69\) 0 0
\(70\) −6.00000 + 10.3923i −0.717137 + 1.24212i
\(71\) −6.00000 + 10.3923i −0.712069 + 1.23334i 0.252010 + 0.967725i \(0.418908\pi\)
−0.964079 + 0.265615i \(0.914425\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) −5.00000 − 3.46410i −0.581238 − 0.402694i
\(75\) 0 0
\(76\) −1.00000 − 1.73205i −0.114708 − 0.198680i
\(77\) 12.0000 − 20.7846i 1.36753 − 2.36863i
\(78\) 0 0
\(79\) −7.00000 + 12.1244i −0.787562 + 1.36410i 0.139895 + 0.990166i \(0.455323\pi\)
−0.927457 + 0.373930i \(0.878010\pi\)
\(80\) −3.00000 −0.335410
\(81\) 0 0
\(82\) −3.00000 −0.331295
\(83\) 3.00000 + 5.19615i 0.329293 + 0.570352i 0.982372 − 0.186938i \(-0.0598564\pi\)
−0.653079 + 0.757290i \(0.726523\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 2.00000 + 3.46410i 0.215666 + 0.373544i
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) 1.50000 + 2.59808i 0.159000 + 0.275396i 0.934508 − 0.355942i \(-0.115840\pi\)
−0.775509 + 0.631337i \(0.782506\pi\)
\(90\) 0 0
\(91\) 4.00000 + 6.92820i 0.419314 + 0.726273i
\(92\) 3.00000 − 5.19615i 0.312772 − 0.541736i
\(93\) 0 0
\(94\) −3.00000 − 5.19615i −0.309426 − 0.535942i
\(95\) 3.00000 + 5.19615i 0.307794 + 0.533114i
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 − 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) −4.50000 + 7.79423i −0.454569 + 0.787336i
\(99\) 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.f.d.433.1 2
3.2 odd 2 74.2.c.b.63.1 yes 2
12.11 even 2 592.2.i.a.433.1 2
37.10 even 3 inner 666.2.f.d.343.1 2
111.11 odd 6 2738.2.a.c.1.1 1
111.26 odd 6 2738.2.a.a.1.1 1
111.47 odd 6 74.2.c.b.47.1 ✓ 2
444.47 even 6 592.2.i.a.417.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.c.b.47.1 ✓ 2 111.47 odd 6
74.2.c.b.63.1 yes 2 3.2 odd 2
592.2.i.a.417.1 2 444.47 even 6
592.2.i.a.433.1 2 12.11 even 2
666.2.f.d.343.1 2 37.10 even 3 inner
666.2.f.d.433.1 2 1.1 even 1 trivial
2738.2.a.a.1.1 1 111.26 odd 6
2738.2.a.c.1.1 1 111.11 odd 6