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,-6,0,4] 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.a.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)
 
\(f(q)\) \(=\) \(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.50000 - 0.866025i) q^{5} +(1.00000 - 1.73205i) q^{7} -1.00000i q^{8} -1.73205 q^{10} +1.26795 q^{11} +(-3.00000 - 1.73205i) q^{13} -2.00000i q^{14} +(-0.500000 - 0.866025i) q^{16} +(3.69615 - 2.13397i) q^{17} +(-4.09808 - 2.36603i) q^{19} +(-1.50000 + 0.866025i) q^{20} +(1.09808 - 0.633975i) q^{22} +1.26795i q^{23} +(-1.00000 - 1.73205i) q^{25} -3.46410 q^{26} +(-1.00000 - 1.73205i) q^{28} -8.66025i q^{29} +4.73205i q^{31} +(-0.866025 - 0.500000i) q^{32} +(2.13397 - 3.69615i) q^{34} +(-3.00000 + 1.73205i) q^{35} +(4.69615 + 3.86603i) q^{37} -4.73205 q^{38} +(-0.866025 + 1.50000i) q^{40} +(-1.96410 + 3.40192i) q^{41} -12.9282i q^{43} +(0.633975 - 1.09808i) q^{44} +(0.633975 + 1.09808i) q^{46} -1.26795 q^{47} +(1.50000 + 2.59808i) q^{49} +(-1.73205 - 1.00000i) q^{50} +(-3.00000 + 1.73205i) q^{52} +(4.73205 + 8.19615i) q^{53} +(-1.90192 - 1.09808i) q^{55} +(-1.73205 - 1.00000i) q^{56} +(-4.33013 - 7.50000i) q^{58} +(8.19615 - 4.73205i) q^{59} +(1.50000 + 0.866025i) q^{61} +(2.36603 + 4.09808i) q^{62} -1.00000 q^{64} +(3.00000 + 5.19615i) q^{65} +(-0.0980762 + 0.169873i) q^{67} -4.26795i q^{68} +(-1.73205 + 3.00000i) q^{70} +(-1.73205 + 3.00000i) q^{71} +4.00000 q^{73} +(6.00000 + 1.00000i) q^{74} +(-4.09808 + 2.36603i) q^{76} +(1.26795 - 2.19615i) q^{77} +(14.4904 + 8.36603i) q^{79} +1.73205i q^{80} +3.92820i q^{82} +(5.83013 + 10.0981i) q^{83} -7.39230 q^{85} +(-6.46410 - 11.1962i) q^{86} -1.26795i q^{88} +(-14.8923 + 8.59808i) q^{89} +(-6.00000 + 3.46410i) q^{91} +(1.09808 + 0.633975i) q^{92} +(-1.09808 + 0.633975i) q^{94} +(4.09808 + 7.09808i) q^{95} -4.26795i q^{97} +(2.59808 + 1.50000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 6 q^{5} + 4 q^{7} + 12 q^{11} - 12 q^{13} - 2 q^{16} - 6 q^{17} - 6 q^{19} - 6 q^{20} - 6 q^{22} - 4 q^{25} - 4 q^{28} + 12 q^{34} - 12 q^{35} - 2 q^{37} - 12 q^{38} + 6 q^{41} + 6 q^{44}+ \cdots + 6 q^{95}+O(q^{100}) \) 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 0.125567 − 0.992085i \(-0.459925\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(6\) 0 0
\(7\) 1.00000 − 1.73205i 0.377964 − 0.654654i −0.612801 − 0.790237i \(-0.709957\pi\)
0.990766 + 0.135583i \(0.0432908\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) −1.73205 −0.547723
\(11\) 1.26795 0.382301 0.191151 − 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 0 0
\(13\) −3.00000 − 1.73205i −0.832050 − 0.480384i 0.0225039 − 0.999747i \(-0.492836\pi\)
−0.854554 + 0.519362i \(0.826170\pi\)
\(14\) − 2.00000i − 0.534522i
\(15\) 0 0
\(16\) −0.500000 − 0.866025i −0.125000 − 0.216506i
\(17\) 3.69615 − 2.13397i 0.896449 − 0.517565i 0.0204023 − 0.999792i \(-0.493505\pi\)
0.876046 + 0.482227i \(0.160172\pi\)
\(18\) 0 0
\(19\) −4.09808 − 2.36603i −0.940163 − 0.542803i −0.0501517 − 0.998742i \(-0.515970\pi\)
−0.890011 + 0.455938i \(0.849304\pi\)
\(20\) −1.50000 + 0.866025i −0.335410 + 0.193649i
\(21\) 0 0
\(22\) 1.09808 − 0.633975i 0.234111 − 0.135164i
\(23\) 1.26795i 0.264386i 0.991224 + 0.132193i \(0.0422018\pi\)
−0.991224 + 0.132193i \(0.957798\pi\)
\(24\) 0 0
\(25\) −1.00000 − 1.73205i −0.200000 − 0.346410i
\(26\) −3.46410 −0.679366
\(27\) 0 0
\(28\) −1.00000 − 1.73205i −0.188982 − 0.327327i
\(29\) − 8.66025i − 1.60817i −0.594515 − 0.804084i \(-0.702656\pi\)
0.594515 − 0.804084i \(-0.297344\pi\)
\(30\) 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
\(34\) 2.13397 − 3.69615i 0.365974 − 0.633885i
\(35\) −3.00000 + 1.73205i −0.507093 + 0.292770i
\(36\) 0 0
\(37\) 4.69615 + 3.86603i 0.772043 + 0.635571i
\(38\) −4.73205 −0.767640
\(39\) 0 0
\(40\) −0.866025 + 1.50000i −0.136931 + 0.237171i
\(41\) −1.96410 + 3.40192i −0.306741 + 0.531291i −0.977647 − 0.210251i \(-0.932572\pi\)
0.670906 + 0.741542i \(0.265905\pi\)
\(42\) 0 0
\(43\) − 12.9282i − 1.97153i −0.168122 − 0.985766i \(-0.553770\pi\)
0.168122 − 0.985766i \(-0.446230\pi\)
\(44\) 0.633975 − 1.09808i 0.0955753 − 0.165541i
\(45\) 0 0
\(46\) 0.633975 + 1.09808i 0.0934745 + 0.161903i
\(47\) −1.26795 −0.184949 −0.0924747 − 0.995715i \(-0.529478\pi\)
−0.0924747 + 0.995715i \(0.529478\pi\)
\(48\) 0 0
\(49\) 1.50000 + 2.59808i 0.214286 + 0.371154i
\(50\) −1.73205 − 1.00000i −0.244949 − 0.141421i
\(51\) 0 0
\(52\) −3.00000 + 1.73205i −0.416025 + 0.240192i
\(53\) 4.73205 + 8.19615i 0.649997 + 1.12583i 0.983123 + 0.182946i \(0.0585633\pi\)
−0.333126 + 0.942882i \(0.608103\pi\)
\(54\) 0 0
\(55\) −1.90192 − 1.09808i −0.256455 − 0.148065i
\(56\) −1.73205 − 1.00000i −0.231455 − 0.133631i
\(57\) 0 0
\(58\) −4.33013 − 7.50000i −0.568574 − 0.984798i
\(59\) 8.19615 − 4.73205i 1.06705 − 0.616061i 0.139675 − 0.990197i \(-0.455394\pi\)
0.927373 + 0.374137i \(0.122061\pi\)
\(60\) 0 0
\(61\) 1.50000 + 0.866025i 0.192055 + 0.110883i 0.592944 − 0.805243i \(-0.297965\pi\)
−0.400889 + 0.916127i \(0.631299\pi\)
\(62\) 2.36603 + 4.09808i 0.300486 + 0.520456i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 3.00000 + 5.19615i 0.372104 + 0.644503i
\(66\) 0 0
\(67\) −0.0980762 + 0.169873i −0.0119819 + 0.0207533i −0.871954 − 0.489587i \(-0.837147\pi\)
0.859972 + 0.510341i \(0.170481\pi\)
\(68\) − 4.26795i − 0.517565i
\(69\) 0 0
\(70\) −1.73205 + 3.00000i −0.207020 + 0.358569i
\(71\) −1.73205 + 3.00000i −0.205557 + 0.356034i −0.950310 − 0.311305i \(-0.899234\pi\)
0.744753 + 0.667340i \(0.232567\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 6.00000 + 1.00000i 0.697486 + 0.116248i
\(75\) 0 0
\(76\) −4.09808 + 2.36603i −0.470082 + 0.271402i
\(77\) 1.26795 − 2.19615i 0.144496 − 0.250275i
\(78\) 0 0
\(79\) 14.4904 + 8.36603i 1.63030 + 0.941251i 0.984001 + 0.178163i \(0.0570155\pi\)
0.646294 + 0.763088i \(0.276318\pi\)
\(80\) 1.73205i 0.193649i
\(81\) 0 0
\(82\) 3.92820i 0.433797i
\(83\) 5.83013 + 10.0981i 0.639940 + 1.10841i 0.985446 + 0.169991i \(0.0543740\pi\)
−0.345506 + 0.938417i \(0.612293\pi\)
\(84\) 0 0
\(85\) −7.39230 −0.801808
\(86\) −6.46410 − 11.1962i −0.697042 − 1.20731i
\(87\) 0 0
\(88\) − 1.26795i − 0.135164i
\(89\) −14.8923 + 8.59808i −1.57858 + 0.911394i −0.583523 + 0.812096i \(0.698326\pi\)
−0.995058 + 0.0992979i \(0.968340\pi\)
\(90\) 0 0
\(91\) −6.00000 + 3.46410i −0.628971 + 0.363137i
\(92\) 1.09808 + 0.633975i 0.114482 + 0.0660964i
\(93\) 0 0
\(94\) −1.09808 + 0.633975i −0.113258 + 0.0653895i
\(95\) 4.09808 + 7.09808i 0.420454 + 0.728247i
\(96\) 0 0
\(97\) − 4.26795i − 0.433345i −0.976244 − 0.216672i \(-0.930480\pi\)
0.976244 − 0.216672i \(-0.0695203\pi\)
\(98\) 2.59808 + 1.50000i 0.262445 + 0.151523i
\(99\) 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.a.307.2 4
3.2 odd 2 74.2.e.b.11.1 ✓ 4
12.11 even 2 592.2.w.e.529.1 4
37.27 even 6 inner 666.2.s.a.397.2 4
111.8 even 12 2738.2.a.e.1.2 2
111.29 even 12 2738.2.a.i.1.2 2
111.101 odd 6 74.2.e.b.27.1 yes 4
444.323 even 6 592.2.w.e.545.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.e.b.11.1 ✓ 4 3.2 odd 2
74.2.e.b.27.1 yes 4 111.101 odd 6
592.2.w.e.529.1 4 12.11 even 2
592.2.w.e.545.1 4 444.323 even 6
666.2.s.a.307.2 4 1.1 even 1 trivial
666.2.s.a.397.2 4 37.27 even 6 inner
2738.2.a.e.1.2 2 111.8 even 12
2738.2.a.i.1.2 2 111.29 even 12