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 379.1
Root \(-2.14169 + 0.642788i\) of defining polynomial
Character \(\chi\) \(=\) 666.379
Dual form 666.2.x.g.181.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.173648 - 0.984808i) q^{2} +(-0.939693 + 0.342020i) q^{4} +(-0.266044 - 0.223238i) q^{5} +(-0.365982 - 0.307095i) q^{7} +(0.500000 + 0.866025i) q^{8} +(-0.173648 + 0.300767i) q^{10} +(1.29268 + 2.23899i) q^{11} +(4.21288 - 1.53336i) q^{13} +(-0.238878 + 0.413749i) q^{14} +(0.766044 - 0.642788i) q^{16} +(1.88864 + 0.687407i) q^{17} +(0.611631 - 3.46873i) q^{19} +(0.326352 + 0.118782i) q^{20} +(1.98050 - 1.66184i) q^{22} +(2.60486 - 4.51175i) q^{23} +(-0.847296 - 4.80526i) q^{25} +(-2.24162 - 3.88261i) q^{26} +(0.448944 + 0.163402i) q^{28} +(0.114111 + 0.197646i) q^{29} -6.74658 q^{31} +(-0.766044 - 0.642788i) q^{32} +(0.349006 - 1.97931i) q^{34} +(0.0288122 + 0.163402i) q^{35} +(2.42322 + 5.57925i) q^{37} -3.52224 q^{38} +(0.0603074 - 0.342020i) q^{40} +(10.0701 - 3.66521i) q^{41} +8.85889 q^{43} +(-1.98050 - 1.66184i) q^{44} +(-4.89554 - 1.78183i) q^{46} +(3.72890 - 6.45864i) q^{47} +(-1.17590 - 6.66887i) q^{49} +(-4.58512 + 1.66885i) q^{50} +(-3.43437 + 2.88178i) q^{52} +(-3.35194 + 2.81261i) q^{53} +(0.155916 - 0.884246i) q^{55} +(0.0829614 - 0.470498i) q^{56} +(0.174828 - 0.146698i) q^{58} +(-3.43016 + 2.87825i) q^{59} +(1.35175 - 0.491998i) q^{61} +(1.17153 + 6.64409i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(-1.46312 - 0.532531i) q^{65} +(-8.06111 - 6.76408i) q^{67} -2.00984 q^{68} +(0.155916 - 0.0567489i) q^{70} +(-1.54193 + 8.74474i) q^{71} +7.18667 q^{73} +(5.07370 - 3.35524i) q^{74} +(0.611631 + 3.46873i) q^{76} +(0.214485 - 1.21641i) q^{77} +(2.70857 + 2.27276i) q^{79} -0.347296 q^{80} +(-5.35818 - 9.28064i) q^{82} +(1.32932 + 0.483834i) q^{83} +(-0.349006 - 0.604496i) q^{85} +(-1.53833 - 8.72431i) q^{86} +(-1.29268 + 2.23899i) q^{88} +(-2.92954 + 2.45818i) q^{89} +(-2.01273 - 0.732572i) q^{91} +(-0.904658 + 5.13057i) q^{92} +(-7.00804 - 2.55072i) q^{94} +(-0.937074 + 0.786298i) q^{95} +(-9.24175 + 16.0072i) q^{97} +(-6.36336 + 2.31607i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{5} + 6 q^{7} + 6 q^{8} - 3 q^{11} - 6 q^{13} - 3 q^{14} + 3 q^{17} - 3 q^{19} + 6 q^{20} - 3 q^{22} + 21 q^{23} - 6 q^{25} - 3 q^{28} - 6 q^{29} + 42 q^{31} - 3 q^{34} + 9 q^{35} - 3 q^{37}+ \cdots - 51 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(e\left(\frac{4}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.173648 − 0.984808i −0.122788 − 0.696364i
\(3\) 0 0
\(4\) −0.939693 + 0.342020i −0.469846 + 0.171010i
\(5\) −0.266044 − 0.223238i −0.118979 − 0.0998350i 0.581357 − 0.813649i \(-0.302522\pi\)
−0.700336 + 0.713814i \(0.746966\pi\)
\(6\) 0 0
\(7\) −0.365982 − 0.307095i −0.138328 − 0.116071i 0.570998 − 0.820951i \(-0.306556\pi\)
−0.709326 + 0.704880i \(0.751001\pi\)
\(8\) 0.500000 + 0.866025i 0.176777 + 0.306186i
\(9\) 0 0
\(10\) −0.173648 + 0.300767i −0.0549124 + 0.0951110i
\(11\) 1.29268 + 2.23899i 0.389758 + 0.675081i 0.992417 − 0.122918i \(-0.0392253\pi\)
−0.602659 + 0.797999i \(0.705892\pi\)
\(12\) 0 0
\(13\) 4.21288 − 1.53336i 1.16844 − 0.425278i 0.316336 − 0.948647i \(-0.397547\pi\)
0.852106 + 0.523369i \(0.175325\pi\)
\(14\) −0.238878 + 0.413749i −0.0638428 + 0.110579i
\(15\) 0 0
\(16\) 0.766044 − 0.642788i 0.191511 − 0.160697i
\(17\) 1.88864 + 0.687407i 0.458062 + 0.166721i 0.560737 − 0.827994i \(-0.310518\pi\)
−0.102675 + 0.994715i \(0.532740\pi\)
\(18\) 0 0
\(19\) 0.611631 − 3.46873i 0.140318 − 0.795782i −0.830690 − 0.556735i \(-0.812054\pi\)
0.971008 − 0.239047i \(-0.0768350\pi\)
\(20\) 0.326352 + 0.118782i 0.0729745 + 0.0265605i
\(21\) 0 0
\(22\) 1.98050 − 1.66184i 0.422245 − 0.354305i
\(23\) 2.60486 − 4.51175i 0.543151 − 0.940765i −0.455570 − 0.890200i \(-0.650565\pi\)
0.998721 − 0.0505649i \(-0.0161022\pi\)
\(24\) 0 0
\(25\) −0.847296 − 4.80526i −0.169459 − 0.961051i
\(26\) −2.24162 − 3.88261i −0.439619 − 0.761442i
\(27\) 0 0
\(28\) 0.448944 + 0.163402i 0.0848424 + 0.0308801i
\(29\) 0.114111 + 0.197646i 0.0211899 + 0.0367019i 0.876426 − 0.481537i \(-0.159921\pi\)
−0.855236 + 0.518239i \(0.826588\pi\)
\(30\) 0 0
\(31\) −6.74658 −1.21172 −0.605861 − 0.795571i \(-0.707171\pi\)
−0.605861 + 0.795571i \(0.707171\pi\)
\(32\) −0.766044 − 0.642788i −0.135419 − 0.113630i
\(33\) 0 0
\(34\) 0.349006 − 1.97931i 0.0598540 − 0.339449i
\(35\) 0.0288122 + 0.163402i 0.00487015 + 0.0276200i
\(36\) 0 0
\(37\) 2.42322 + 5.57925i 0.398376 + 0.917222i
\(38\) −3.52224 −0.571383
\(39\) 0 0
\(40\) 0.0603074 − 0.342020i 0.00953543 − 0.0540781i
\(41\) 10.0701 − 3.66521i 1.57268 − 0.572410i 0.599086 − 0.800685i \(-0.295531\pi\)
0.973597 + 0.228275i \(0.0733086\pi\)
\(42\) 0 0
\(43\) 8.85889 1.35097 0.675484 − 0.737374i \(-0.263935\pi\)
0.675484 + 0.737374i \(0.263935\pi\)
\(44\) −1.98050 − 1.66184i −0.298572 − 0.250532i
\(45\) 0 0
\(46\) −4.89554 − 1.78183i −0.721807 − 0.262716i
\(47\) 3.72890 − 6.45864i 0.543916 − 0.942090i −0.454758 − 0.890615i \(-0.650274\pi\)
0.998674 − 0.0514750i \(-0.0163923\pi\)
\(48\) 0 0
\(49\) −1.17590 − 6.66887i −0.167986 − 0.952696i
\(50\) −4.58512 + 1.66885i −0.648434 + 0.236011i
\(51\) 0 0
\(52\) −3.43437 + 2.88178i −0.476261 + 0.399631i
\(53\) −3.35194 + 2.81261i −0.460424 + 0.386342i −0.843287 − 0.537464i \(-0.819383\pi\)
0.382863 + 0.923805i \(0.374938\pi\)
\(54\) 0 0
\(55\) 0.155916 − 0.884246i 0.0210238 − 0.119232i
\(56\) 0.0829614 − 0.470498i 0.0110862 − 0.0628729i
\(57\) 0 0
\(58\) 0.174828 − 0.146698i 0.0229560 − 0.0192624i
\(59\) −3.43016 + 2.87825i −0.446569 + 0.374716i −0.838161 − 0.545423i \(-0.816369\pi\)
0.391592 + 0.920139i \(0.371924\pi\)
\(60\) 0 0
\(61\) 1.35175 − 0.491998i 0.173074 − 0.0629939i −0.254030 − 0.967196i \(-0.581756\pi\)
0.427104 + 0.904203i \(0.359534\pi\)
\(62\) 1.17153 + 6.64409i 0.148785 + 0.843800i
\(63\) 0 0
\(64\) −0.500000 + 0.866025i −0.0625000 + 0.108253i
\(65\) −1.46312 − 0.532531i −0.181477 − 0.0660523i
\(66\) 0 0
\(67\) −8.06111 − 6.76408i −0.984822 − 0.826363i −1.18267e−5 − 1.00000i \(-0.500004\pi\)
−0.984810 + 0.173637i \(0.944448\pi\)
\(68\) −2.00984 −0.243729
\(69\) 0 0
\(70\) 0.155916 − 0.0567489i 0.0186356 − 0.00678280i
\(71\) −1.54193 + 8.74474i −0.182994 + 1.03781i 0.745512 + 0.666492i \(0.232205\pi\)
−0.928506 + 0.371318i \(0.878906\pi\)
\(72\) 0 0
\(73\) 7.18667 0.841136 0.420568 − 0.907261i \(-0.361831\pi\)
0.420568 + 0.907261i \(0.361831\pi\)
\(74\) 5.07370 − 3.35524i 0.589805 − 0.390038i
\(75\) 0 0
\(76\) 0.611631 + 3.46873i 0.0701589 + 0.397891i
\(77\) 0.214485 − 1.21641i 0.0244429 − 0.138622i
\(78\) 0 0
\(79\) 2.70857 + 2.27276i 0.304738 + 0.255705i 0.782313 − 0.622885i \(-0.214040\pi\)
−0.477575 + 0.878591i \(0.658484\pi\)
\(80\) −0.347296 −0.0388289
\(81\) 0 0
\(82\) −5.35818 − 9.28064i −0.591712 − 1.02487i
\(83\) 1.32932 + 0.483834i 0.145912 + 0.0531077i 0.413944 − 0.910302i \(-0.364151\pi\)
−0.268032 + 0.963410i \(0.586373\pi\)
\(84\) 0 0
\(85\) −0.349006 − 0.604496i −0.0378550 − 0.0655668i
\(86\) −1.53833 − 8.72431i −0.165882 − 0.940766i
\(87\) 0 0
\(88\) −1.29268 + 2.23899i −0.137800 + 0.238677i
\(89\) −2.92954 + 2.45818i −0.310531 + 0.260567i −0.784711 − 0.619861i \(-0.787189\pi\)
0.474180 + 0.880428i \(0.342744\pi\)
\(90\) 0 0
\(91\) −2.01273 − 0.732572i −0.210991 − 0.0767944i
\(92\) −0.904658 + 5.13057i −0.0943172 + 0.534899i
\(93\) 0 0
\(94\) −7.00804 − 2.55072i −0.722824 − 0.263086i
\(95\) −0.937074 + 0.786298i −0.0961417 + 0.0806725i
\(96\) 0 0
\(97\) −9.24175 + 16.0072i −0.938358 + 1.62528i −0.169824 + 0.985474i \(0.554320\pi\)
−0.768534 + 0.639809i \(0.779013\pi\)
\(98\) −6.36336 + 2.31607i −0.642797 + 0.233959i
\(99\) 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.379.1 12
3.2 odd 2 74.2.f.b.9.1 ✓ 12
12.11 even 2 592.2.bc.d.305.2 12
37.33 even 9 inner 666.2.x.g.181.1 12
111.62 odd 18 2738.2.a.q.1.3 6
111.86 odd 18 2738.2.a.t.1.3 6
111.107 odd 18 74.2.f.b.33.1 yes 12
444.107 even 18 592.2.bc.d.33.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.f.b.9.1 ✓ 12 3.2 odd 2
74.2.f.b.33.1 yes 12 111.107 odd 18
592.2.bc.d.33.2 12 444.107 even 18
592.2.bc.d.305.2 12 12.11 even 2
666.2.x.g.181.1 12 37.33 even 9 inner
666.2.x.g.379.1 12 1.1 even 1 trivial
2738.2.a.q.1.3 6 111.62 odd 18
2738.2.a.t.1.3 6 111.86 odd 18