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(307,666)] 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("666.307"); 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, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,2,6,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 307.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 666.307
Dual form 666.2.s.e.397.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.50000 + 0.866025i) q^{5} +(-2.00000 + 3.46410i) q^{7} -1.00000i q^{8} +1.73205 q^{10} -1.26795 q^{11} +(5.19615 + 3.00000i) q^{13} +4.00000i q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.50000 - 0.866025i) q^{17} +(4.09808 + 2.36603i) q^{19} +(1.50000 - 0.866025i) q^{20} +(-1.09808 + 0.633975i) q^{22} +4.73205i q^{23} +(-1.00000 - 1.73205i) q^{25} +6.00000 q^{26} +(2.00000 + 3.46410i) q^{28} -7.73205i q^{29} +4.73205i q^{31} +(-0.866025 - 0.500000i) q^{32} +(0.866025 - 1.50000i) q^{34} +(-6.00000 + 3.46410i) q^{35} +(-0.500000 - 6.06218i) q^{37} +4.73205 q^{38} +(0.866025 - 1.50000i) q^{40} +(-3.23205 + 5.59808i) q^{41} -2.53590i q^{43} +(-0.633975 + 1.09808i) q^{44} +(2.36603 + 4.09808i) q^{46} +5.66025 q^{47} +(-4.50000 - 7.79423i) q^{49} +(-1.73205 - 1.00000i) q^{50} +(5.19615 - 3.00000i) q^{52} +(-4.73205 - 8.19615i) q^{53} +(-1.90192 - 1.09808i) q^{55} +(3.46410 + 2.00000i) q^{56} +(-3.86603 - 6.69615i) q^{58} +(-8.19615 + 4.73205i) q^{59} +(2.30385 + 1.33013i) q^{61} +(2.36603 + 4.09808i) q^{62} -1.00000 q^{64} +(5.19615 + 9.00000i) q^{65} +(2.09808 - 3.63397i) q^{67} -1.73205i q^{68} +(-3.46410 + 6.00000i) q^{70} +(4.73205 - 8.19615i) q^{71} +8.39230 q^{73} +(-3.46410 - 5.00000i) q^{74} +(4.09808 - 2.36603i) q^{76} +(2.53590 - 4.39230i) q^{77} +(-1.90192 - 1.09808i) q^{79} -1.73205i q^{80} +6.46410i q^{82} +(-2.83013 - 4.90192i) q^{83} +3.00000 q^{85} +(-1.26795 - 2.19615i) q^{86} +1.26795i q^{88} +(-4.50000 + 2.59808i) q^{89} +(-20.7846 + 12.0000i) q^{91} +(4.09808 + 2.36603i) q^{92} +(4.90192 - 2.83013i) q^{94} +(4.09808 + 7.09808i) q^{95} +5.19615i q^{97} +(-7.79423 - 4.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 6 q^{5} - 8 q^{7} - 12 q^{11} - 2 q^{16} + 6 q^{17} + 6 q^{19} + 6 q^{20} + 6 q^{22} - 4 q^{25} + 24 q^{26} + 8 q^{28} - 24 q^{35} - 2 q^{37} + 12 q^{38} - 6 q^{41} - 6 q^{44} + 6 q^{46}+ \cdots + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 − 0.500000i 0.612372 − 0.353553i
\(3\) 0 0
\(4\) 0.500000 − 0.866025i 0.250000 − 0.433013i
\(5\) 1.50000 + 0.866025i 0.670820 + 0.387298i 0.796387 − 0.604787i \(-0.206742\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0 0
\(7\) −2.00000 + 3.46410i −0.755929 + 1.30931i 0.188982 + 0.981981i \(0.439481\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) 1.73205 0.547723
\(11\) −1.26795 −0.382301 −0.191151 − 0.981561i \(-0.561222\pi\)
−0.191151 + 0.981561i \(0.561222\pi\)
\(12\) 0 0
\(13\) 5.19615 + 3.00000i 1.44115 + 0.832050i 0.997927 − 0.0643593i \(-0.0205004\pi\)
0.443227 + 0.896410i \(0.353834\pi\)
\(14\) 4.00000i 1.06904i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 1.50000 − 0.866025i 0.363803 − 0.210042i −0.306944 − 0.951727i \(-0.599307\pi\)
0.670748 + 0.741685i \(0.265973\pi\)
\(18\) 0 0
\(19\) 4.09808 + 2.36603i 0.940163 + 0.542803i 0.890011 − 0.455938i \(-0.150696\pi\)
0.0501517 + 0.998742i \(0.484030\pi\)
\(20\) 1.50000 − 0.866025i 0.335410 − 0.193649i
\(21\) 0 0
\(22\) −1.09808 + 0.633975i −0.234111 + 0.135164i
\(23\) 4.73205i 0.986701i 0.869831 + 0.493350i \(0.164228\pi\)
−0.869831 + 0.493350i \(0.835772\pi\)
\(24\) 0 0
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) 6.00000 1.17670
\(27\) 0 0
\(28\) 2.00000 + 3.46410i 0.377964 + 0.654654i
\(29\) − 7.73205i − 1.43581i −0.696143 − 0.717903i \(-0.745102\pi\)
0.696143 − 0.717903i \(-0.254898\pi\)
\(30\) 0 0
\(31\) 4.73205i 0.849901i 0.905216 + 0.424951i \(0.139709\pi\)
−0.905216 + 0.424951i \(0.860291\pi\)
\(32\) −0.866025 − 0.500000i −0.153093 − 0.0883883i
\(33\) 0 0
\(34\) 0.866025 − 1.50000i 0.148522 − 0.257248i
\(35\) −6.00000 + 3.46410i −1.01419 + 0.585540i
\(36\) 0 0
\(37\) −0.500000 − 6.06218i −0.0821995 − 0.996616i
\(38\) 4.73205 0.767640
\(39\) 0 0
\(40\) 0.866025 − 1.50000i 0.136931 − 0.237171i
\(41\) −3.23205 + 5.59808i −0.504762 + 0.874273i 0.495223 + 0.868766i \(0.335086\pi\)
−0.999985 + 0.00550690i \(0.998247\pi\)
\(42\) 0 0
\(43\) − 2.53590i − 0.386721i −0.981128 − 0.193360i \(-0.938061\pi\)
0.981128 − 0.193360i \(-0.0619387\pi\)
\(44\) −0.633975 + 1.09808i −0.0955753 + 0.165541i
\(45\) 0 0
\(46\) 2.36603 + 4.09808i 0.348851 + 0.604228i
\(47\) 5.66025 0.825633 0.412816 − 0.910814i \(-0.364545\pi\)
0.412816 + 0.910814i \(0.364545\pi\)
\(48\) 0 0
\(49\) −4.50000 − 7.79423i −0.642857 − 1.11346i
\(50\) −1.73205 − 1.00000i −0.244949 − 0.141421i
\(51\) 0 0
\(52\) 5.19615 − 3.00000i 0.720577 − 0.416025i
\(53\) −4.73205 − 8.19615i −0.649997 − 1.12583i −0.983123 − 0.182946i \(-0.941437\pi\)
0.333126 − 0.942882i \(-0.391897\pi\)
\(54\) 0 0
\(55\) −1.90192 − 1.09808i −0.256455 − 0.148065i
\(56\) 3.46410 + 2.00000i 0.462910 + 0.267261i
\(57\) 0 0
\(58\) −3.86603 − 6.69615i −0.507634 − 0.879248i
\(59\) −8.19615 + 4.73205i −1.06705 + 0.616061i −0.927373 − 0.374137i \(-0.877939\pi\)
−0.139675 + 0.990197i \(0.544606\pi\)
\(60\) 0 0
\(61\) 2.30385 + 1.33013i 0.294977 + 0.170305i 0.640184 − 0.768221i \(-0.278858\pi\)
−0.345207 + 0.938527i \(0.612191\pi\)
\(62\) 2.36603 + 4.09808i 0.300486 + 0.520456i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 5.19615 + 9.00000i 0.644503 + 1.11631i
\(66\) 0 0
\(67\) 2.09808 − 3.63397i 0.256321 − 0.443961i −0.708933 − 0.705276i \(-0.750823\pi\)
0.965253 + 0.261316i \(0.0841563\pi\)
\(68\) − 1.73205i − 0.210042i
\(69\) 0 0
\(70\) −3.46410 + 6.00000i −0.414039 + 0.717137i
\(71\) 4.73205 − 8.19615i 0.561591 − 0.972704i −0.435767 − 0.900060i \(-0.643523\pi\)
0.997358 − 0.0726447i \(-0.0231439\pi\)
\(72\) 0 0
\(73\) 8.39230 0.982245 0.491122 − 0.871091i \(-0.336587\pi\)
0.491122 + 0.871091i \(0.336587\pi\)
\(74\) −3.46410 − 5.00000i −0.402694 − 0.581238i
\(75\) 0 0
\(76\) 4.09808 − 2.36603i 0.470082 − 0.271402i
\(77\) 2.53590 − 4.39230i 0.288992 − 0.500550i
\(78\) 0 0
\(79\) −1.90192 − 1.09808i −0.213983 − 0.123543i 0.389178 − 0.921163i \(-0.372759\pi\)
−0.603161 + 0.797619i \(0.706092\pi\)
\(80\) − 1.73205i − 0.193649i
\(81\) 0 0
\(82\) 6.46410i 0.713841i
\(83\) −2.83013 − 4.90192i −0.310647 − 0.538056i 0.667856 − 0.744291i \(-0.267212\pi\)
−0.978503 + 0.206235i \(0.933879\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) −1.26795 − 2.19615i −0.136726 − 0.236817i
\(87\) 0 0
\(88\) 1.26795i 0.135164i
\(89\) −4.50000 + 2.59808i −0.476999 + 0.275396i −0.719165 − 0.694839i \(-0.755475\pi\)
0.242166 + 0.970235i \(0.422142\pi\)
\(90\) 0 0
\(91\) −20.7846 + 12.0000i −2.17882 + 1.25794i
\(92\) 4.09808 + 2.36603i 0.427254 + 0.246675i
\(93\) 0 0
\(94\) 4.90192 − 2.83013i 0.505595 − 0.291905i
\(95\) 4.09808 + 7.09808i 0.420454 + 0.728247i
\(96\) 0 0
\(97\) 5.19615i 0.527589i 0.964579 + 0.263795i \(0.0849741\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) −7.79423 − 4.50000i −0.787336 − 0.454569i
\(99\) 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.s.e.307.2 4
3.2 odd 2 74.2.e.a.11.1 ✓ 4
12.11 even 2 592.2.w.d.529.2 4
37.27 even 6 inner 666.2.s.e.397.2 4
111.8 even 12 2738.2.a.f.1.1 2
111.29 even 12 2738.2.a.j.1.1 2
111.101 odd 6 74.2.e.a.27.1 yes 4
444.323 even 6 592.2.w.d.545.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.a.11.1 ✓ 4 3.2 odd 2
74.2.e.a.27.1 yes 4 111.101 odd 6
592.2.w.d.529.2 4 12.11 even 2
592.2.w.d.545.2 4 444.323 even 6
666.2.s.e.307.2 4 1.1 even 1 trivial
666.2.s.e.397.2 4 37.27 even 6 inner
2738.2.a.f.1.1 2 111.8 even 12
2738.2.a.j.1.1 2 111.29 even 12